Advances in
ATOMIC, MOLECULAR, AND OPTICAL PHYSICS V O L U M E 54
Editors PAUL R. B ERMAN
University of Michigan Ann Arbor, Michigan C HUN C. L IN
University of Wisconsin Madison, Wisconsin E NNIO A RIMONDO
University of Pisa Pisa, Italy
Editorial Board C. J OACHAIN
Université Libre de Bruxelles Brussels, Belgium M. G AVRILA
F.O.M. Insituut voor Atoom- en Molecuulfysica Amsterdam, The Netherlands M. I NOKUTI
Argonne National Laboratory Argonne, Illinois
ADVANCES IN
ATOMIC, MOLECULAR, AND OPTICAL PHYSICS Edited by
P.R. Berman PHYSICS DEPARTMENT UNIVERSITY OF MICHIGAN ANN ARBOR , MI , USA
C.C. Lin DEPARTMENT OF PHYSICS UNIVERSITY OF WISCONSIN MADISON , WI , USA
and
E. Arimondo PHYSICS DEPARTMENT UNIVERSITY OF PISA PISA , ITALY
Volume 54
AMSTERDAM • BOSTON • HEIDELBERG • LONDON • NEW YORK OXFORD • PARIS • SAN DIEGO • SAN FRANCISCO • SINGAPORE SYDNEY • TOKYO Academic Press is an imprint of Elsevier
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ISBN-13: 978-0-12-003854-1 ISBN-10: 0-12-003854-4 ISSN: 1049-250X
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07 08 09 10 11
10 9 8 7 6 5 4 3 2 1
IN MEMORIAM HERBERT WALTHER (1935–2006) It is with great sadness that the Editors report the passing of Professor Herbert Walther on July 22, 2006. Professor Walther expanded the horizons of our understanding of the interaction of radiation with matter. The physics community will sorely miss his presence. Professor Walther assumed co-editorship of the Advances series starting with Volume 33 and continued in that role through Volume 51, which was a special issue, edited by Henry Stroke, to honor his co-editor Benjamin Bederson. Volume 53, edited by Gerhard Rempe and Marlan Scully, was published as a tribute to Professor Walther.
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Contents C ONTRIBUTORS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P REFACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Experimental Realization of the BCS-BEC Crossover with a Fermi Gas of Atoms C.A. Regal and D.S. Jin 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . BCS-BEC Crossover Physics . . . . . . . . . . . . . . . . . . . . . . . Feshbach Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . Cooling a Fermi Gas and Measuring its Temperature . . . . . . . . . . Elastic Scattering near Feshbach Resonances between Fermionic Atoms Creating Molecules from a Fermi Gas of Atoms . . . . . . . . . . . . Inelastic Collisions near a Fermionic Feshbach Resonance . . . . . . . Creating Condensates from a Fermi Gas of Atoms . . . . . . . . . . . The Momentum Distribution of a Fermi Gas in the Crossover . . . . . Conclusions and Future Directions . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 10 18 24 36 42 51 56 64 71 72 72
Deterministic Atom–Light Quantum Interface Jacob Sherson, Brian Julsgaard and Eugene S. Polzik 1. 2. 3. 4. 5. 6. 7. 8. A. B. 9.
Introduction . . . . . . . . . . . Atom–Light Interaction . . . . . Quantum Information Protocols Experimental Methods . . . . . Experimental Results . . . . . . Conclusions . . . . . . . . . . . Acknowledgements . . . . . . . Appendices . . . . . . . . . . . Effect of Atomic Motion . . . . Technical Details . . . . . . . . References . . . . . . . . . . . .
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Cold Rydberg Atoms J.-H. Choi, B. Knuffman, T. Cubel Liebisch, A. Reinhard and G. Raithel 1. 2. 3. 4. 5. 6. 7.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Preparation and Analysis of Cold Rydberg-Atom Clouds . . . . . . . Collision-Induced Rydberg-Atom Gas Dynamics . . . . . . . . . . . . Towards Coherent Control of Rydberg-Atom Interactions . . . . . . . Rydberg-Atom Trapping . . . . . . . . . . . . . . . . . . . . . . . . . Experimental Realization of Rydberg-Atom Trapping . . . . . . . . . Landau Quantization and State Mixing in Cold, Strongly Magnetized Rydberg Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
132 135 149 159 176 186 192 196 197 198
Non-Perturbative Quantal Methods for Electron–Atom Scattering Processes D.C. Griffin and M.S. Pindzola 1. 2. 3. 4. 5. 6. 7. 8.
Introduction . . . . . . . . . . . . . . . . . . . . . . The Configuration-Average Distorted-Wave Method The R-Matrix with Pseudo-States Method . . . . . . The Time-Dependent Close-Coupling Method . . . Results . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . .
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R-Matrix Theory of Atomic, Molecular and Optical Processes P.G. Burke, C.J. Noble and V.M. Burke 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
Introduction . . . . . . . . . . . . . . . . . . . . . . Electron Atom Scattering at Low Energies . . . . . Electron Scattering at Intermediate Energies . . . . Atomic Photoionization and Photorecombination . . Electron Molecule Scattering . . . . . . . . . . . . . Positron Atom Scattering . . . . . . . . . . . . . . . Atomic and Molecular Multiphoton Processes . . . Electron Energy Loss from Transition Metal Oxides Conclusions . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . .
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Electron-Impact Excitation of Rare-Gas Atoms from the Ground Level and Metastable Levels John B. Boffard, R.O. Jung, L.W. Anderson and C.C. Lin 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electronic Structure . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . Background: Excitation of Helium and the Multipole Field Picture Argon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Neon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Krypton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Xenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison to Theoretical Calculations . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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320 321 325 342 348 372 384 397 406 410 418 418
Internal Rotation in Symmetric Tops I. Ozier and N. Moazzen-Ahmadi 1. 2. 3. 4. 5. 6.
Introduction . . . . . . . . . . . . . . . . Theory . . . . . . . . . . . . . . . . . . . Spectroscopy from 50 kHz to 1000 cm−1 Discussion . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . References . . . . . . . . . . . . . . . . .
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424 436 449 498 505 506
Introduction . . . . . . . . . . . . . . . . . . . . . . . Tunnel Ionization and Electron Re-collision . . . . . . Producing and Measuring Attosecond Optical Pulses . Measuring an Attosecond Electron Pulse . . . . . . . Attosecond Imaging . . . . . . . . . . . . . . . . . . . Imaging Electrons and their Dynamics . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . .
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1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Attosecond and Angstrom Science Hiromichi Niikura and P.B. Corkum 1. 2. 3. 4. 5. 6. 7. 8.
Atomic Processing of Optically Carried RF Signals Jean-Louis Le Gouët, Fabien Bretenaker and Ivan Lorgeré
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2. 3. 4. 5. 6. 7. 8. 9. 10.
Radio Frequency Spectral Analyzers . . . . . . . . . . . Spectrum Photography Architecture . . . . . . . . . . . Frequency Selective Materials as Programmable Filters Rainbow Analyzer . . . . . . . . . . . . . . . . . . . . . Photon Echo Chirp Transform Spectrum Analyzer . . . Frequency Agile Laser Technology . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . .
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I NDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C ONTENTS OF VOLUMES IN T HIS S ERIAL . . . . . . . . . . . . . . .
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Controlling Optical Chaos, Spatio-Temporal Dynamics, and Patterns Lucas Illing, Daniel J. Gauthier and Rajarshi Roy 1. 2. 3. 4. 5. 6. 7. 8. 9.
Introduction . . . . . . . . . . . . . Recent Examples . . . . . . . . . . Control . . . . . . . . . . . . . . . . Synchronization . . . . . . . . . . . Communication . . . . . . . . . . . Spatio-Temporal Chaos and Patterns Outlook . . . . . . . . . . . . . . . Acknowledgement . . . . . . . . . . References . . . . . . . . . . . . . .
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CONTRIBUTORS
Numbers in parentheses indicate the pages on which the author’s contributions begin.
C.A. R EGAL (1), JILA, National Institute of Standards and Technology Quantum Physics Division and University of Colorado, and Department of Physics, University of Colorado, Boulder, CO 80309-0440, USA D.S. J IN (1), JILA, National Institute of Standards and Technology Quantum Physics Division and University of Colorado and Department of Physics, University of Colorado, Boulder, CO 80309-0440, USA JACOB S HERSON (81), QUANTOP, Danish Research Foundation Center for Quantum Optics, Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark; Department of Physics and Astronomy, University of Aarhus, Ny Munkegade, bygning 1520, DK-8000 Aarhus C, Denmark B RIAN J ULSGAARD (81), QUANTOP, Danish Research Foundation Center for Quantum Optics, Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark E UGENE S. P OLZIK (81), Department of Physics and Astronomy, University of Aarhus, Ny Munkegade, bygning 520, DK-8000 Aarhus C, Denmark J.-H. C HOI (131), FOCUS Center, Department of Physics, University of Michigan, Ann Arbor, MI 48109-1040, USA B. K NUFFMAN (131), FOCUS Center, Department of Physics, University of Michigan, Ann Arbor, MI 48109-1040, USA T. C UBEL L IEBISCH (131), FOCUS Center, Department of Physics, University of Michigan, Ann Arbor, MI 48109-1040, USA A. R EINHARD (131), FOCUS Center, Department of Physics, University of Michigan, Ann Arbor, MI 48109-1040, USA G. R AITHEL (131), FOCUS Center, Department of Physics, University of Michigan, Ann Arbor, MI 48109-1040, USA D.C. G RIFFIN (203), Department of Physics, Rollins College, Winter Park, FL 32789, USA M.S. P INDZOLA (203), Department of Physics, Auburn University, Auburn, AL 36849, USA xi
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P.G. B URKE (237), Department of Applied Mathematics and Theoretical Physics, The Queen’s University of Belfast, Belfast BT7 1NN, UK C.J. N OBLE (237), CSE Department, CCLRC Daresbury Laboratory, Warrington WA4 4AD, UK V.M. B URKE (237), CSE Department, CCLRC Daresbury Laboratory, Warrington WA4 4AD, UK J OHN B. B OFFARD (319), Department of Physics, University of Wisconsin, Madison, WI 53706, USA R.O. J UNG (319), Department of Physics, University of Wisconsin, Madison, WI 53706, USA L.W. A NDERSON (319), Department of Physics, University of Wisconsin, Madison, WI 53706, USA C.C. L IN (319), Department of Physics, University of Wisconsin, Madison, WI 53706, USA I. O ZIER (423), Department of Physics and Astronomy, University of British Columbia, 6224 Agricultural Road, Vancouver, British Columbia, Canada V6T 1Z1 N. M OAZZEN -A HMADI (423), Department of Physics and Astronomy, University of Calgary, 2500 University Drive NW, Calgary, Alberta, Canada T2N 1N4 H IROMICHI N IIKURA (511), National Research Council of Canada, 100 Sussex Drive, Ottawa, Ontario, Canada K1A0R6; PRESTO, Japan Science and Technology Agency (JST), 4-1-8 Honcho, Kawaguchi-city, Saitama, Japan P.B. C ORKUM (511), National Research Council of Canada, 100 Sussex Drive, Ottawa, Ontario, Canada K1A0R6 J EAN -L OUIS L E G OUËT (549), Laboratoire Aimé Cotton, CNRS UPR3321, bâtiment 505, campus universitaire, 91405 Orsay Cedex, France FABIEN B RETENAKER (549), Laboratoire Aimé Cotton, CNRS UPR3321, bâtiment 505, campus universitaire, 91405 Orsay Cedex, France I VAN L ORGERÉ (549), Laboratoire Aimé Cotton, CNRS UPR3321, bâtiment 505, campus universitaire, 91405 Orsay Cedex, France L UCAS I LLING (615), Department of Physics and Center for Nonlinear and Complex Systems, Duke University, Durham, NC 27708, USA DANIEL J. G AUTHIER (615), Department of Physics and Center for Nonlinear and Complex Systems, Duke University, Durham, NC 27708, USA R AJARSHI ROY (615), Department of Physics and Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742, USA
PREFACE Volume 54 of the Advances Series contains ten contributions, covering a diversity of subject areas in atomic, molecular and optical physics. The chapter by Regal and Jin reviews the properties of a Fermi degenerate gas of cold potassium atoms in the crossover regime between the Bose–Einstein condensation of molecules and the condensation of fermionic atom pairs. The transition between the two regions can be probed by varying an external magnetic field. Sherson, Julsgaard, and Polzik explore the manner in which light and atoms can be entangled, with applications to quantum information processing and communication. They report on the result of recent experiments involving the state entanglement of distant ensembles and a method for storing the quantum state of an optical field in an atomic ensemble. Recent developments in cold Rydberg atom physics are reviewed in the chapter by Choi, Knuffman, Cubel Liebisch, Reinhard, and Raithel. Fascinating experiments are described in which cold, highly excited atoms (“Rydberg” atoms) and cold plasmas are generated. Evidence for a collective excitation of Rydberg matter is also presented. Griffin and Pindzola offer an account of non-perturbative quantal methods for electron–atom scattering processes. Included in the discussion are the R-matrix with pseudo-states method and the time-dependent close-coupling method. An extensive review of the R-matrix theory of atomic, molecular, and optical processes is given by Burke, Noble, and Burke. They present a systematic development of the R-matrix method and its applications to various processes such as electron–atom scattering, atomic photoionization, electron–molecule scattering, positron–atom scattering, and atomic/molecular multiphoton processes. Electron impact excitation of raregas atoms from both their ground and metastable states is discussed in the chapter by Boffard, Jung, Anderson, and Lin. Excitation cross sections measured by the optical method are reviewed with emphasis on the physical interpretation in terms of electronic structure of the target atoms. Ozier and Moazzen-Ahmadi explore internal rotation of symmetric top molecules. Developments of new experimental methods based on high-resolution torsional, vibrational, and molecular beam spectroscopy allow accurate determination of internal barriers for these symmetric molecules. The subject of attosecond and angstrom science is reviewed by Niikura and Corkum. The underlying physical mechanisms allowing one to generate attosecond radiation pulses are described and the technology needed for the preparation of such pulses is discussed. Le Gouët, Bretenaker, and Lorgeré describe how rare earth ions embedded in crystals can be used for processing optically carried broadband radio-frequency signals. Methods for reaching tens of gigahertz xiii
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instantaneous bandwidth with submegahertz resolution using such devices are analyzed in detail and demonstrated experimentally. Finally, in the chapter by Illing, Gauthier, and Roy, it is shown that small perturbations applied to optical systems can be used to suppress or control optical chaos, spatio-temporal dynamics, and patterns. Applications of these techniques to communications, laser stabilization, and improving the sensitivity of low-light optical switches are explored. The Editors would like to thank all the contributing authors for their contributions and for their cooperation in assembling this volume. They would also like to express their appreciation to Dr. Anita Koch at Elsevier for her invaluable assistance. Ennio Arimondo
Paul Berman
Chun Lin
ADVANCES IN ATOMIC, MOLECULAR AND OPTICAL PHYSICS, VOL. 54
EXPERIMENTAL REALIZATION OF THE BCS-BEC CROSSOVER WITH A FERMI GAS OF ATOMS C.A. REGAL and D.S. JIN JILA, National Institute of Standards and Technology Quantum Physics Division and University of Colorado, and Department of Physics, University of Colorado, Boulder, CO 80309-0440, USA 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Historical Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. BCS-BEC Crossover Physics . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Pairing in a Fermi Gas of Atoms . . . . . . . . . . . . . . . . . . . . 2.2. Varying Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Simple Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Beyond T = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5. Modern Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Feshbach Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. A Specific Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Cooling a Fermi Gas and Measuring its Temperature . . . . . . . . . . . 4.1. Cooling 40 K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Measuring the Temperature of a Fermi Gas . . . . . . . . . . . . . . 5. Elastic Scattering near Feshbach Resonances between Fermionic Atoms 5.1. Measuring the Elastic Collision Cross Section . . . . . . . . . . . . 5.2. Anisotropic Expansion . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Measuring the Mean-Field Interaction Energy . . . . . . . . . . . . 5.4. 40 K Feshbach Resonance Summary . . . . . . . . . . . . . . . . . . 6. Creating Molecules from a Fermi Gas of Atoms . . . . . . . . . . . . . . 6.1. Magnetic-Field Association . . . . . . . . . . . . . . . . . . . . . . 6.2. Rf Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3. Understanding Molecule Conversion Efficiency . . . . . . . . . . . 6.4. A Precise Measurement of B0 . . . . . . . . . . . . . . . . . . . . . 7. Inelastic Collisions near a Fermionic Feshbach Resonance . . . . . . . . 7.1. Expected Inelastic Decay Processes . . . . . . . . . . . . . . . . . . 7.2. Lifetime of Feshbach Molecules . . . . . . . . . . . . . . . . . . . . 7.3. Three-Body Recombination . . . . . . . . . . . . . . . . . . . . . . 7.4. Comparison of 40 K and 6 Li . . . . . . . . . . . . . . . . . . . . . . 8. Creating Condensates from a Fermi Gas of Atoms . . . . . . . . . . . . . 8.1. Emergence of a Molecular Condensate from a Fermi Gas of Atoms 8.2. Observing Condensates in the Crossover . . . . . . . . . . . . . . . 1
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8.3. Measurement of a Phase Diagram . . . . . . . . . . . . . 9. The Momentum Distribution of a Fermi Gas in the Crossover 9.1. Measuring the Momentum Distribution . . . . . . . . . . 9.2. Extracting the Kinetic Energy . . . . . . . . . . . . . . . 9.3. Comparing the Kinetic Energy to Theory . . . . . . . . . 9.4. Temperature Dependence . . . . . . . . . . . . . . . . . . 10. Conclusions and Future Directions . . . . . . . . . . . . . . . 11. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . 12. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Abstract Ultracold atomic gases have proven to be remarkable model systems for exploring quantum mechanical phenomena. Experimental work on gases of fermionic atoms in particular has seen astounding recent progress. In the short span of time between 2001 and 2004 accessible Fermi gas systems evolved from normal Fermi liquids at moderate temperatures to superfluids in the BCS-BEC crossover. This was made possible by unique control over interparticle interactions using Feshbach resonances in 6 Li and 40 K gases. In this chapter we present the story of the experimental realization of BCS-BEC crossover physics from the point of view of studies using 40 K at JILA. We start with some historical context and an introduction to the theory of the BCS-BEC crossover and Feshbach resonances. We then present studies of a normal 40 K Fermi gas at a Feshbach resonance and the work required to cool the gas to temperatures where superfluidity in the crossover is predicted. These studies culminated in the first observation of a phase transition in the BCS-BEC crossover regime, a task accomplished through detection of condensation of fermionic atom pairs. We also discuss subsequent work that confirmed the crossover nature of the pairs in these condensates.
1. Introduction 1.1. H ISTORICAL P ERSPECTIVE The phenomenon of superconductivity/superfluidity has fascinated and occupied physicists since the beginning of the 20th century. In 1911 superconductivity was discovered when the resistance of mercury was observed to go to zero below a critical temperature (Onnes, 1911). Although liquid 4 He was actually used in this discovery, the superfluid phase of liquid 4 He was not revealed until the 1930s when the viscosity of the liquid below the λ point (2.17 K) was measured (Allen and Misener, 1938; Kapitza, 1938). Much later, 3 He, the fermionic helium isotope, was also found to be superfluid at yet a much colder temperature than 4 He
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F IG . 1. Classic experimental realizations of superfluidity/superconductivity arranged according to the binding energy (twice the excitation gap, Egap ) of the constituent fermions. The vertical axis shows the corresponding transition temperature, Tc , to a superfluid/superconducting state compared to the Fermi temperature, TF . (Figure reproduced with permission from Ref. (Holland et al., 2001).)
(Osheroff et al., 1972). Relatively recently in 1986, high-temperature superconductors in Copper-oxide materials further enlarged the list of superconducting materials (Bednorz and Mueller, 1986). These “super” systems, which we will refer to in general as superfluids, are listed in Fig. 1, but they are only classic examples. There are many other physical systems that display superfluid properties from astrophysical phenomena such as neutron stars, to excitons in semiconductors, to atomic nuclei (Snoke and Baym, 1995). Although the physical properties of these systems vary widely, they are all linked by their counterintuitive behaviors such as frictionless flow and quantized vorticity. The manifestation of these effects depends upon, for example, whether the system in question is electrically charged (superconductors) or neutral (superfluids). Besides these intriguing properties, there are many practical reasons for the intense research in this field; arguably the most useful super-systems are superconductors, and if a robust room-temperature superconductor were created it would be an amazing discovery. Some of the first attempts to understand the phenomenon of superfluidity were in the context of Bose–Einstein condensation (BEC) of an ensemble of bosonic particles (Randeria, 1995). BEC is a consequence of the quantum statistics of bosons, which are particles with integer spin, and it results in a macroscopic occupation of a single quantum state (Fig. 2) (Bose, 1924; Einstein, 1925). Fritz London proposed in 1938 that superfluid 4 He was a consequence of Bose–Einstein condensation of bosonic 4 He (London, 1938). (4 He behaves as a boson because it is made up of an even number of 1/2 integer spin fermions—electrons, protons, and neutrons.) Physicists such as Blatt et al. pushed a similar idea in the context of superconductors in proposing that “at low temper-
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F IG . 2. Quantum statistics: Bosons versus fermions with weak interactions at T = 0. Bosons form a BEC in which all of the bosons macroscopically occupy a single quantum state. Due to the Pauli exclusion principle, fermions form a Fermi sea in which each energy state up to the Fermi energy is occupied.
ature, charge carrying bosons occur, e.g., because of the interaction of electrons with lattice vibrations” (Blatt and Butler, 1954). For the case of tightly bound bosons, such as 4 He, London’s hypothesis turned out to be correct. However, the very strong interactions in 4 He made it difficult to verify the existence of condensation (Sokol, 1995), and for many years 4 He studies did not mention BEC. In the case of superconductors, discussion of BEC was overshadowed by the amazing success in 1957 of the Bardeen–Cooper–Schrieffer (BCS) theory of superconductivity (Bardeen et al., 1957a, 1957b). In 1956 Cooper found that a pair of fermions in the presence of a filled Fermi sea (Fig. 2) will form a bound pair with an arbitrarily small attractive interaction (Cooper, 1956). The BCS theory solved this problem in the case where many pairs can form in the Fermi sea. The result predicted (among other things) the formation of a minimum excitation energy, or energy gap, in the conductor below a critical temperature Tc . A great many properties of conventional superconductors can be understood as consequences of this energy gap. Figure 1 sorts the classic superfluid systems according to the strength of the interaction between the fermions. A key aspect of the classic BCS theory is that it applies to the perturbative limit of weak attractive interactions and hence is only an exact theory for the far right side of Fig. 1. The theory perfectly described conventional superconductors for which the attraction between fermions is ∼10,000 times less than the Fermi energy, EF . The BCS ground state was also able to accurately describe superfluid 3 He and many (although arguably not all (Chen et al., 2005b)) aspects of high-Tc superconductors. The theory in its original form, however, did not at all apply to the case of the tightly bound boson, 4 He. In 1995 a completely new system joined 4 He on the left side of Fig. 1. Here the composite bosons were alkali atoms, such as 87 Rb, that had been cooled as a gas down to nanoKelvin temperatures via laser cooling and evaporative cooling (Phillips, 1998; Ketterle et al., 1999; Cornell et al., 1999). At
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F IG . 3. Bose–Einstein condensation in a dilute gas of 87 Rb atoms. (a) Phase space density criterion for condensation. (b) Momentum distributions of 87 Rb atoms at three values of the temperature compared to the critical temperature. (Figure adapted by M.R. Matthews from data in Ref. (Anderson et al., 1995).)
2π h¯ 2 these temperatures the thermal deBroglie wavelength λdeBroglie = mkb T of the particles becomes on the order of the interparticle spacing in the gas and a Bose–Einstein condensate is formed (Fig. 3(a)) (Anderson et al., 1995; Davis et al., 1995). In contrast to 4 He the alkali BEC that was created was weakly interacting, making the condensation stunningly visible, as shown in Fig. 3(b). Experiments also verified the superfluid nature of these gases (Matthews et al., 1999; Raman et al., 1999). In this way both the BEC and superfluid properties could be clearly seen in one system and understood extremely well theoretically (see, e.g., Ref. (Dalfovo et al., 1999)). However, the fact that the dilute gas BEC was found to be a superfluid was not at all a surprise. Although a long and complicated history was required, physicists now understand the basic connection between BEC and superfluidity (see, e.g., Ref. (Tilley, 1986)). It is commonly accepted that superfluidity is always intimately connected to the macroscopic occupation of a quantum state. Besides providing the first clear evidence for BEC, ultracold alkali gases opened a world of possibilities for studying superfluid systems. Many of the initial experiments with alkali BEC could be perfectly described by existing theories. However, recent work in the field of BEC has developed techniques to reach a regime that is more relevant for the outstanding theoretical questions in condensed matter physics, which are most commonly in strongly correlated systems. For example, experiments achieved BEC with much stronger interatomic interactions than typical alkali gases; furthermore, these interactions could even be controllably tuned (Inouye et al., 1998; Roberts et al., 1998). A phase transition to the highly-correlated Mott insulator state was observed through studies of quantum gases in optical lattice potentials (Greiner et al., 2002). These bosonic systems require theory that goes beyond mean-field interactions; yet they have a controllability rarely found in solid state materials.
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At the same time as the creation of the first strongly interacting Bose gases, the techniques used to create an alkali BEC were applied to the other class of quantum particles, fermions. To create a Fermi gas of atoms experimenters applied the same cooling techniques as those used to achieve BEC, simply replacing a bosonic atom, such as 87 Rb or 23 Na, with an alkali atom with an odd number of electrons, protons, and neutrons. (The two such stable alkali atoms are 40 K and 6 Li.) Still, evaporatively cooling fermions required ingenuity. Due to the quantum statistics of fermions, the s-wave collisions required for evaporative cooling are not present at ultracold temperatures in a gas of spin-polarized, identical fermions. The solution to this problem was to introduce a second particle for the evaporative cooling, either another state of the fermionic atom or another species entirely. The first gas of fermionic atoms to enter the quantum degenerate regime was created at JILA in 1999 using 40 K (DeMarco and Jin, 1999). The observation in these experiments was not a phase transition, as in the Bose gas, but rather the presence of more and more energy than would be expected classically as the Fermi gas was cooled below the Fermi temperature. Many more Fermi gas experiments, using a variety of cooling techniques, followed (Truscott et al., 2001; Schreck et al., 2001; Roati et al., 2002; Granade et al., 2002; Hadzibabic et al., 2002; Goldwin et al., 2004; Bartenstein et al., 2004b; Köhl et al., 2005; Ospelkaus et al., 2006). The next goal after the creation of a normal Fermi gas of atoms was to form a superfluid in a paired Fermi gas. In conventional superconductors swave pairing occurs between spin up and spin down electrons. The hope was that s-wave pairing could similarly occur with the creation of a two-component atomic gas with an equal Fermi energy for each component. Indeed, such a two-component gas can be realized using an equal mixture of alkali atoms in two different hyperfine spin states. The simplistic view was that a BCS state would appear if the temperature of this two-component gas were cold enough and the interaction between fermions attractive and large enough. However, for typical interactions the temperatures required to reach a true BCS state were far too low compared to achievable temperatures (at that point) to imagine creating Cooper pairs. Stoof et al. noted that the interaction between 6 Li atoms was large compared to typical values (|a| ≈ 2000a0 ), as well as attractive, bringing the BCS transition temperature closer to realistic temperatures (Stoof et al., 1996; Combescot, 1999). It was then recognized that a type of scattering resonance, known as a Feshbach resonance, could allow arbitrary changes in the interaction strength. Theories were developed that explicitly treated the case where the interactions were enhanced by a Feshbach resonance (Timmermans et al., 2001; Holland et al., 2001; Ohashi and Griffin, 2002). In these proposals, however, increasing the interactions beyond the perturbative limit of BCS theory meant that the physical system would not be simply a BCS state, but rather something much more interesting. It was predicted that a
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Feshbach resonance would allow the realization of a system with an excitation gap on the order of the Fermi energy and provide the ability to continually tune within this region (shaded box in Fig. 1) (Holland et al., 2001). This system, if achievable, would be the experimental realization of a theoretical topic that dated back to the late 1960s, the BCS-BEC crossover. In a theory originally put forth by Eagles and later by Leggett, it was proposed that the BCS wave-function was more generally applicable than just to the weakly interacting limit (Eagles, 1969; Leggett, 1980). As long as the chemical potential is found self-consistently as the interaction is increased, the BCS ground state can (at least qualitatively) describe everything from Cooper pairing to a BEC of composite bosons made up of two fermions, i.e., the fundamental physics behind all of the systems in Fig. 1 (Eagles, 1969; Leggett, 1980; Nozieres and Schmitt-Rink, 1985; Randeria et al., 1990; Dreschler and Zwerger, 1992; Haussman, 1994). After nearly a century of 4 He and superconductors being considered separate entities, an experimental realization of a superfluid in the BCS-BEC crossover regime would provide a physical link between the two. More recent interest in crossover theories has come in response to the possibility that they could apply to high-Tc superconductors. These superconductors differ from normal superconductors both in their high transition temperature and the apparent presence of a pseudogap, which are both characteristics expected to be found in a Fermi gas in the crossover (Randeria, 1995; Chen et al., 2005b). Thus, starting in about 2001 a major goal in dilute Fermi gas experiments was to achieve a superfluid Fermi gas at a Feshbach resonance, often referred to as a “resonance superfluid” (Holland et al., 2001). However, achievement of this experimental goal was a number of years and many steps away. The existence of Feshbach resonances had been predicted by atomic physicists both in the 6 Li and in the 40 K systems (Houbiers et al., 1998; Bohn, 2000), and the first step was to locate these resonances (Loftus et al., 2002; O’Hara et al., 2002b; Dieckmann et al., 2002; Regal et al., 2003b). Subsequent experimental studies appeared at an amazingly fast rate with contributions from a large number of groups, in particular those of R. Grimm (Innsbruck), R. Hulet (Rice), D. Jin (JILA), W. Ketterle (MIT), C. Salomon (ENS), and J. Thomas (Duke). Experimenters discovered interesting properties of the normal state of a strongly interacting Fermi gas (O’Hara et al., 2002a; Regal and Jin, 2003; Gupta et al., 2003; Gehm et al., 2003; Bourdel et al., 2003). Then Fermi gases were reversibly converted to gases of diatomic molecules using Feshbach resonances (Regal et al., 2003a; Strecker et al., 2003; Cubizolles et al., 2003; Jochim et al., 2003a). The observation that these molecules were surprisingly long-lived created many opportunities for further study (Strecker et al., 2003; Jochim et al., 2003a; Cubizolles et al., 2003; Regal et al., 2004a). Condensates of diatomic molecules in the BEC limit were achieved (Greiner et al., 2003; Jochim et al., 2003b; Zwierlein et al., 2003; Bourdel et al., 2004; Partridge et al., 2005); then these
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condensates were found to exist in the crossover regime (Regal et al., 2004b; Zwierlein et al., 2004), signalling the existence of a phase transition in the BCSBEC crossover regime. Collective excitations (Bartenstein et al., 2004a; Kinast et al., 2004a, 2004b) and thermodynamic properties (Bartenstein et al., 2004b; Bourdel et al., 2004; Regal et al., 2005; Kinast et al., 2005b) were also measured, and the nature of the pairs was probed in a variety of ways (Chin et al., 2004; Greiner et al., 2005a; Partridge et al., 2005). Most recently a vortex lattice was even created in the crossover (Zwierlein et al., 2005a). Developing these techniques to access and probe the BCS-BEC crossover was a challenging adventure for the field. Experiments in the crossover are inherently difficult because the strong interactions make probing difficult. Some of the techniques used in the end were borrowed from those developed for alkali BEC, while others were taken from condensed matter physics. Some were new inventions that relied on the unique ability to tune the interactions in the system at arbitrary rates using the Feshbach resonance. So far the experiments that have been carried out with dilute Fermi gases near Feshbach resonances have been qualitatively consistent with classic BCS-BEC crossover theory. The excitation gap is on the order of the Fermi energy; the system crosses a phase transition to a superfluid state. However, quantitatively there is much work to be done. In tandem with these experiments, sophisticated theories of the crossover have been developed that are too numerous to list here, but are actively being pursued in groups such as those of A. Bulgac, K. Burnett, J. Carlson, S. Giorgini, A. Griffin, H. Heiselberg, T.L. Ho, M. Holland, K. Levin, E.J. Mueller, M. Randeria, C.A.R. Sa de Melo, G. Shlyapnikov, S. Strinati, S. Stringari, B. Svistunov, E. Timmermans, and P. Torma. In time it is expected that the BCS-BEC crossover system provided by dilute Fermi gases should be able to rigorously test these theories. The power to test these many-body theories comes from the fact that by using ultracold, dilute gases one can create a very clean strongly interacting Fermi system. In principle the density and two-fermion interaction in the sample can be known precisely and the s-wave pairing fully characterized as a function of the interaction strength. The end result could be a fully understood physical system that connects the spectrum of pairing from BCS to BEC, uniting the basic physics surrounding “super” systems. On the other hand, the complicated materials physics involved in, for example, high-Tc superconductors, cannot be elucidated in these clean crossover experiments. Still the hope is that an understanding of the basic physics will provide a solid foundation for studies of real materials. 1.2. C ONTENTS In this chapter we will present some of the first experimental work studying fermions in the BCS-BEC crossover in a dilute atomic system. We and our co-workers
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performed these experiments at JILA between 2001 and 2005 using 40 K atoms. As outlined above this was an exciting time in the field of Fermi gases. In 2001 in our lab at JILA the technology existed to create two-component Fermi gases at temperatures around 0.25TF (DeMarco, 2001). Predictions had been made for the existence of a superfluid state near a Feshbach resonance for a Fermi gas on the order of this temperature (Timmermans et al., 2001; Holland et al., 2001; Ohashi and Griffin, 2002). Yet in the early days of this work many physicists were skeptical about the feasibility of experimentally realizing such a state. The Feshbach resonances that had been observed in Bose gases were associated with extremely fast inelastic decay of the trapped gas (Inouye et al., 1998; Stenger et al., 1999; Roberts et al., 2000). These decay processes, which most often stem from three-body collisions, can quickly turn a hard-earned quantum gas into a classical gas of atoms (Fedichev et al., 1996; Nielsen and Macek, 1999; Esry et al., 1999; Braaten et al., 2001). Carl Wieman’s group at JILA produced the only experiments studying BEC near a Feshbach resonance over long time scales (Roberts et al., 1998). In this group 85 Rb BECs were studied at very low densities, where two-body elastic collisions dominate over threebody collisions. For two-component Fermi gases three-body decay processes were expected to be suppressed compared to the Bose case (Esry et al., 2001; Petrov, 2003), due to the Pauli exclusion principle. Still there was a fair amount of contention about the degree of this suppression. Further difficulties stemmed from the fact that not all researchers in the atomic physics community were familiar with theories from condensed matter physics such as BCS-BEC crossover theory. Even when these theories were understood, there was significant confusion about the relation between classic crossover theory and the two-body physics of the Feshbach resonances in 6 Li and 40 K. An additional challenge was creating a sufficiently cold Fermi gas near a Feshbach resonance. To be certain of achieving the predicted phase transition, temperatures well below the predicted Tc would have to be achieved. A technical challenge was that the states required to access Feshbach resonances in the 40 K and 6 Li gases could not be confined in a magnetic trap, which was the most proven trap in studies of ultracold gases up to that point. Instead the experiments would have to achieve an ultracold Fermi gas in an alternative trap, such as an optical dipole trap. Much of the work to create the results presented in this chapter involved overcoming these difficulties and sources of confusion, and as we shall see there are now clear answers to many of these questions. These answers were found through careful studies of the normal state of a Fermi gas near the Feshbach resonance and significant work to cool a 40 K gas to the coldest temperatures possible. In late 2003 we observed a phase transition in the crossover and since then have been able to study BCS-BEC crossover physics with a Fermi gas of atoms.
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In the first few sections we discuss theory that is necessary to understand experiments presented in later sections. The goal is not to rigorously derive modern theories, but rather to convey the mindset that many experimentalists currently use to think about the crossover problem with atomic Fermi gases. Section 2 presents the ideas of BCS-BEC crossover physics through simplified theory. Section 3 introduces Feshbach resonances and discusses how well the Feshbach resonance systems reproduce conditions for the classic “universal” BCS-BEC crossover problem. The main body of our chapter presents experiments using an ultracold gas of 40 K atoms. Section 4 describes cooling methods and temperature measurement techniques. Section 5 contains experiments that probe the presence of Feshbach resonances in the 40 K system and study their ability to tune atomic scattering properties. Section 6 introduces the creation of molecules from a Fermi gas of atoms, which is the analog of the BCS-BEC crossover in the normal state. Section 7 concentrates on the stability of fermionic atoms and pairs against inelastic processes in the presence of a Feshbach resonance; this is a subject of practical importance to the ability to study the BCS-BEC crossover with a Fermi gas of atoms. Section 8 describes the first experiments to observe a phase transition in a Fermi gas of atoms in the BCS-BEC crossover regime. Section 9 focuses on a measurement of the momentum distribution of the pairs in the crossover; this measurement is an important probe of the nature of the pairs in the crossover. Some of the experiments we discuss in this chapter were also performed in 6 Li gases in the groups of R. Grimm, R. Hulet, W. Ketterle, C. Salomon, or J. Thomas. In addition the experiments presented in this chapter represent only a fraction of the experimental work in atomic physics on the BCS-BEC crossover. There are numerous experiments using 6 Li gases not mentioned here that have studied fascinating aspects of BCS-BEC crossover physics.
2. BCS-BEC Crossover Physics To understand the experimental work presented in this chapter it is helpful to be familiar with BCS-BEC crossover physics. In this section we present the theory of the BCS-BEC crossover, first from a purely qualitative point of view and then from a slightly more quantitative perspective. In this quantitative perspective it is not our goal to present the most sophisticated theory, but rather a theory that introduces important parameters and illustrates the key differences between the crossover problem and the BCS limit. Lastly we discuss some of the unanswered questions related to the crossover problem; this conveys a sense of the importance of experimental studies of crossover physics.
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F IG . 4. Cartoon illustration of the continuum of pairing in the BCS-BEC crossover.
2.1. PAIRING IN A F ERMI G AS OF ATOMS As we have seen, superfluids are fundamentally associated with the quantum properties of bosons. Since all visible matter is made up of fermions, creating a superfluid most often requires pairing of fermions. The simplest (although historically not the most famous) way to imagine pairing fermions is to create a two-body bound state with two fermions. Two half-integer spin fermions when paired will produce an integer spin particle, which is a composite boson. In the case of experiments discussed in this chapter the fermionic particles are atoms; this makes such a two-body bound state a diatomic molecule. Below a critical temperature an ensemble of these diatomic molecules will form a BEC. The left side of Fig. 4 represents a superfluid containing these type of pairs. The two shades of gray represent fermions in two different spin states; two states are required if the fermions are to pair via s-wave (l = 0) interactions. In other pairing mechanisms, such as Cooper pairing, the underlying fermionic nature of the system is much more apparent. Cooper considered the problem of two fermions with equal and opposite momentum outside a perfect Fermi sea (Cooper, 1956). The energy of the two fermions turns out to be less than the expected value of 2EF for arbitrarily weak attractive interactions. This result is in surprising contrast to the result of the problem of two fermions in vacuum; in this case there will not be a bound state until the interaction reaches a certain threshold. The key difference between the two situations arises from Pauli blocking, which in the Cooper pair case prevents the two fermions under consideration from occupying momentum states k < kF , where kF is the Fermi wave-vector (de Gennes, 1966). Considering only one pair of electrons as free to pair on top of a static Fermi sea is not a sufficient solution to the pairing problem in a Fermi sea. All fermions should be allowed to participate in the pairing, and we expect that pairs should form until an equilibrium point is reached. At this equilibrium point the remain-
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ing ensemble of fermions is disturbed enough from a Fermi sea configuration to no longer lead to a bound state at the given interaction strength (Tinkham, 1980). The BCS state was an approximate solution to this many-body problem. A description of the full BCS theory is beyond the scope of this current discussion, but is presented in the original papers (Bardeen et al., 1957a, 1957b) and discussed in numerous textbooks, for example, Refs. (de Gennes, 1966; Tinkham, 1980). Qualitatively the BCS state consists of loose correlations between fermions across the Fermi surface in momentum space (Fig. 4 right side). Spatially the pairs are highly overlapping and cannot simply be considered to be composite bosons. In the BCS limit the momentum distribution only deviates from that of a normal Fermi sea in an exponentially small region near the Fermi surface. It is interesting to consider what happens if diatomic molecules become more and more weakly bound, to the point where the binding energy of the molecules, Eb , becomes less than the Fermi energy, EF . One could also consider increasing the interaction energy of a Cooper paired state until it is close to EF . The essence of the BCS-BEC crossover is that these two sentences describe the same physical state. As the interaction between fermions is increased there will be a continual change, or crossover, between a BCS state and a BEC of diatomic molecules. The point where two fermions in vacuum would have zero binding energy is considered the cusp of the crossover problem, and pairing in such a state is represented in the middle of Fig. 4. These pairs have some properties of diatomic molecules and some properties of Cooper pairs. Many-body effects are required for the pairing, as with the BCS state, but there is some amount of spatial correlation, as with diatomic molecules. The pair size is on the order of the spacing between fermions, and the system is strongly interacting.
2.2. VARYING I NTERACTIONS It is instructive to consider a physical situation that will allow the realization pairing throughout the crossover (Fig. 4) (Leggett, 1980). Suppose we start with an attractive potential between two atomic fermions in vacuum, such as the square well potential with characteristic range r0 shown in Fig. 5. If this potential is very shallow there is a weak attractive interaction between the fermions. If we make this potential deeper the interaction between fermions becomes stronger, and for a strong enough attraction a bound molecular state will appear. This molecule will become more and more deeply bound as the potential becomes deeper. The interaction in this system can be characterized by the s-wave scattering length a. The quantity a comes out of studying two-body, low-energy scattering and is related to the s-wave collision cross section through σ = 4πa 2 . The top of Fig. 5 shows a pictorial representation of a. Just before the bound state appears
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F IG . 5. Scattering wave-functions in the presence of an attractive potential (right) and a more deeply attractive potential (left), in a regime where a weakly bound state of the potential (dashed line) is near threshold. r here describes the relative position of two distinguishable fermions. As in the bottom plot the scattering length diverges as the bound state moves through threshold.
a is large and negative, corresponding to a strong attractive interaction. As the bound state comes through threshold a diverges and then becomes large and positive, corresponding to a strong repulsive interaction. When a is much larger than r0 , the interaction is independent of the exact form of the potential, and a > 0 is universally related to the binding energy of the two-body bound state through h¯ 2 Eb = ma 2 , where m is the mass of a fermion (see, e.g., Ref. (Sakurai, 1994)). Now if we consider an ensemble of many fermions under the situation in Fig. 5, we have a system that can be tuned from BCS to BEC simply by tweaking the attractive potential. To the far right we have a small negative a and thus the BCS limit. In the opposite limit we have an ensemble of diatomic molecules; hence the BEC limit. It is important to note that although the interaction between fermions in pairs is strongest in the BEC limit, from the point of view of collisions in the gas, the BEC limit is actually weakly interacting because the molecule–molecule
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interaction is weak.1 The most strongly interacting gas from the point of view of collisions occurs near the divergence of a. Here many-body calculations are difficult because there is no small expansion parameter. The precise point at which a diverges is known as unitarity. Here the only length scale in the problem is 1/kF , giving this point many unique properties (Baker, 1999; Heiselberg, 2001; Kinast et al., 2005a; Thomas et al., 2005).
2.3. S IMPLE T HEORY BCS theory was originally applied in the limit where the interaction energy is extremely small compared to the Fermi energy. In this case the chemical potential, μ, can be fixed at EF , and many calculations become reasonably simple. Leggett pointed out that if the BCS gap equation is examined allowing μ to vary, the gap equation actually becomes precisely the Schrödinger equation for a diatomic molecule in the limit where μ dominates (Leggett, 1980). Below we briefly illustrate the key steps in an application of BCS theory to the entire crossover. This gives qualitatively correct results for the entire spectrum of pairing. The structure of the theory below originates in the work of Nozieres and Schmitt-Rink (NSR) in Ref. (Nozieres and Schmitt-Rink, 1985) and of Randeria et al. in Ref. (Randeria et al., 1990). We will consider a homogeneous Fermi system in three dimensions in an equal mixture of two different states at T = 0. Application of usual BCS theory results in the gap equation Δk = −
k
Ukk
k , 2Ek
(1)
2 2 where Ek = ξk2 + Δ2 , ξk = k − μ, and k = h¯2mk . Ukk < 0 is the attractive interaction for scattering of fermions with momenta k and −k to k and −k. We also obtain the number equation ξk Ntot = (2) 1− , Ek k
where Ntot is the total number of fermions in both states. To solve Eq. (1) in the BCS limit the standard approach is to assume that the potential is constant at a value U < 0, which implies that the gap is constant as
1 In the limit a r the molecule–molecule scattering length a mm is predicted to be 0.6a, where a 0 is the scattering length for atoms scattering above threshold (Petrov et al., 2004).
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well, i.e., Δk = Δ. In this case the gap equation (1) becomes 1 1 . − = U 2Ek
15
(3)
k
One will find, however, that this equation diverges. For BCS superconductors this problem is resolved because the interaction is limited to within the Debye energy, hω ¯ D , of EF . This is due to the nature of the phonon-mediated interaction between the electrons that gives rise to the attractive interaction. Further simplifications in the BCS limit are that μ = EF and that, since hω ¯ D EF , the density of states is constant at the value N (ξ = 0). The gap equation then becomes 1 = − N (0)U
h ¯ ωD
−h¯ ωD
dξ
2 Δ2 + ξ 2
.
(4)
Solving Eq. (4) produces the BCS result ≈ 2h¯ ωD e−1/N(0)|U | . To extend this calculation to the crossover in atomic systems we can no longer apply the h¯ ωD cutoff. The solution to the divergence problem in this case is nontrivial and requires a renormalization procedure, the full description of which we will not present here (but see, for example, Randeria (1995) and references therein). The result of such a procedure is a “renormalized” gap equation 1 1 1 m − = , − (5) V 2Ek 2k 4π h¯ 2 a k where the interaction is now described by the s-wave scattering length a instead of U and V is the volume of the system. Furthermore, in the crossover we cannot assume μ = EF ; instead we must solve the gap equation (5) and number equation (2) simultaneously for μ and the gap parameter Δ. We will solve for these √ parameters as a function of the dimensionless parameter kF a, where kF = 2mEF /h¯ . As pointed out in a useful paper by M. Marini et al. this can actually be done analytically using elliptic integrals (Marini et al., 1998). The solid lines in Fig. 6 show the result of this calculation of Δ and μ. We also plot the values of Δ and μ as they would be calculated in the BCS and BEC limits to find that the crossover occurs in a relatively small region of the parameter 1/kF a, namely from approximately −1 < 1/kF a < 1. In typical crossover experiments with 40 K or 6 Li, this regime corresponds to varying a from −2000a0 through ∞ and to 2000a0 , where a0 is the Bohr radius (0.0529 nm). It is useful to explicitly understand the value and meaning of both μ and Δ in the two limits. μ is EF in the BCS limit and −Eb /2 = −( kF1 a )2 EF in the BEC limit. /EF −π 16 1 is e kF |a| in the BCS limit and 3π kF a in the BEC limit (Marini et al., 1998). Although referred to as the gap parameter, Δ only has meaning as the excitation gap, i.e., the smallest possible energy that can create a hole (remove a fermion) in
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F IG . 6. Gap parameter, Δ, and chemical potential, μ, of a homogeneous Fermi gas at T = 0 as determined through NSR theory. The dashed lines show the BCS and BEC limits of the theory. Note that the limiting theories only deviate significantly from the full theory in approximately the range −1 < 1/kF a < 1.
the superfluid, in the BCS limit. In general the excitation energy is
2 h¯ 2 k 2 Egap = min Ek = min − μ + Δ2 2m (Randeria, 1995). This is Δ when μ is positive (as in the BCS limit), but becomes μ2 + Δ2 when μ is negative. 2.4. B EYOND T = 0 The phase transition temperature, Tc , is an important parameter for any superfluid system. In the BCS-BEC crossover the transition temperature increases as the interaction is increased, i.e., it is lowest in the perturbative BCS regime and highest in the BEC limit (Fig. 1). In a homogeneous system, in the BCS limit −π
Tc /TF = π8 eγ −2 e 2kF |a| where γ = 0.58 (Haussman, 1994), and in the BEC limit Tc /TF = 0.22 (Pethick and Smith, 2002). Note that BCS transition temperatures can be extremely small due to the exponential dependence upon 1/kF a. For example, at a typical interaction strength for alkali gases (a = −100a0 ) and a typical kF (1/kF = 2000a0 ) the BCS transition temperature would be ∼10−14 TF , which is a completely inaccessible temperature in contemporary atomic systems. Still, at 1/kF a = −1 where BCS theory nearly applies, the transition temperature is on the order of 0.1TF , which is accessible in current atomic systems. In the BCS limit pairing and the phase transition to a superfluid state occur at the same temperature. However, in the BEC limit this is not the case; because
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EXPERIMENTAL REALIZATION OF BCS-BEC CROSSOVER
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the constituent fermions are very tightly bound, pairs can form far above Tc . It is natural to expect that there would be a crossover between these two behaviors in the BCS-BEC crossover, i.e., at the cusp of the crossover the pairing temperature, T ∗ , would be distinct from Tc , yet not far from it. In a simple picture T ∗ should be related to a pair dissociation temperature, which in the case of molecules is ∼Eb /kb (Randeria, 1995). It is important to differentiate between the superconducting order parameter, which exists below Tc , and the pairing gap, which exists below T ∗ (Chen et al., 2005b). The pairing gap is associated with so-called pre-formed pairs, which are pairs that exist at temperatures above the critical temperature for condensation, Tc . Aspects of the pseudogap observed in high-Tc superconductors may be a manifestation of pre-formed pairs (Chen et al., 2005b).
2.5. M ODERN C HALLENGES The discussions and calculations above, while providing an introduction to basic crossover theory, are far from the state-of-the-art for theory in the field. There are noticeable problems with the NSR theory presented in Section 2.3. For example, the result for the chemical potential at unitarity (1/kF a = 0) is significantly different from the result of more accurate calculations using full Monte Carlo simulations (Carlson et al., 2003; Astrakharchik et al., 2004). The chemical potential is often written as μ = (1 + β)EF , and the NSR theory produced β = −0.41 at unitarity, while the Monte Carlo simulation of Ref. (Astrakharchik et al., 2004) finds β = −0.58 ± 0.01 at unitarity. As another example, extension of the NSR theory predicts that amm = 2aaa , while a full 4-body calculation in the BEC limit yields amm = 0.6aaa (Petrov et al., 2004). (Experiments at ENS have shown that amm = 0.6+0.3 −0.2 aaa (Bourdel et al., 2004), in agreement with the full calculation but not the NSR theory.) Both of these problems point to the fact that the NSR ground state, which only includes two-particle correlations, is not sufficient to accurately describe the system. Thus, it is clear that adding higher-order correlations to BCS-BEC theory is necessary, yet not a simple task (Holland et al., 2005). An even greater challenge is to extend accurate theories to nonzero temperatures where predictions can be made about the critical temperatures and the role of pre-formed pairs. Furthermore, all of the calculations considered thus far are carried out for a homogeneous Fermi system. However, the experiments with ultracold gases take place in traps (most often harmonic traps). The use of an inhomogeneous density gas can lead to qualitative changes in the system that must be accounted for in theory: Strong interactions can modify the density of the trapped gas (Vichi and Stringari, 1999), and signals can become blurred as the density, and hence the gap, varies across the sample (Chin et al., 2004).
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3. Feshbach Resonances In the previous section we determined that varying the fermion–fermion interaction is the key to accessing BCS-BEC crossover physics. We also observed that to arbitrarily vary the interactions we could imagine using a variable attractive potential with a bound state near threshold. Amazingly, atomic systems can achieve exactly such a physical situation. The attractive potential is provided by the van der Waals interaction between two atoms, and the knob to tune this potential is a homogeneous magnetic field. The sensitive magneticfield dependence of the potential can be provided through a scattering resonance known as a Feshbach resonance (Fano, 1961; Feshbach, 1962; Stwalley, 1976; Tiesinga et al., 1993). The goal of this section will be to understand exactly how a Feshbach resonance allows us to arbitrarily tune the interaction using a magnetic field. In this section we will be discussing only two-body physics, namely the problem of two fermions scattering in vacuum. We will find that the Feshbach resonance used for the experiments in this chapter approximates well the classic two-body system required for study of the BCS-BEC crossover problem of Section 2.
3.1. D ESCRIPTION Calculating the interaction between two ground state alkali atoms is a nontrivial problem that has been studied extensively in atomic physics (Burke, 1999). The result of studies of this problem show that for S-state atoms the interatomic potential is repulsive for very small r and has a weak attractive tail that goes as −C6 /r 6 as r → ∞. This weak attractive tail is a result of the interaction between mutually induced dipole moments of the atoms, which is known as the van der Waals interaction. The interatomic potentials are deep enough to contain a large number of bound vibrational states. A Feshbach resonance occurs when one of these bound states (often called the bare molecule state) coincides with the collision energy of two free atoms in a different scattering channel. Such a situation is depicted in Fig. 7(a). The interatomic potential of the two free atoms is often referred to as the open channel, while the potential containing the bare molecule state is referred to as the closed channel. When the closed and open channels describe atoms in different magnetic sublevels, they can be shifted with respect to each other through the Zeeman effect using an external magnetic field (Fig. 7(b)). Typically the effect of the coupling between the closed and open channels is small, but at a Feshbach resonance when the open-channel dissociation threshold is nearly degenerate with the bare molecule state, the effect of the coupling can be significantly enhanced. This coupling changes the effective interatomic potential, which we will refer to as the multichannel potential (Szyma´nska et al., 2005).
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F IG . 7. (a) Feshbach resonances are the result of coupling between a molecular state in one interatomic potential with the threshold of another. (b) The bare molecule state of the closed channel tunes differently with magnetic field than the open channel threshold. This can lead to a crossing of the two levels.
A bound state will be added to this multichannel potential at a magnetic field value near (but not exactly at) the magnetic field position of the crossing of the bare molecule and open channel threshold. When we use the term “molecule” in this chapter we are always referring to this additional bound state of the multichannel potential, a so-called dressed molecule. The wave-function of these molecules is generally a linear superposition of open and closed channel wave-functions. As we will see in the next section, the open-channel component dominates for the molecules we study. As the magnetic field is tuned this multichannel bound state moves through threshold, and the scattering length between atoms in the open channel diverges. The scattering length near a Feshbach resonance varies with the magnetic field, B, according to the following equation (Burke, 1999): w . a(B) = abg 1 − (6) B − B0 Here abg is the triplet background (nonresonant) scattering length for atoms scattering in the open channel, B0 is the magnetic field position at which the molecular bound state of the coupled system goes through threshold, and w is the width of the Feshbach resonance, defined as the distance in magnetic field between B0 and the magnetic field at which a = 0. Figure 8 shows how a diverges according to Eq. (6) for the 40 K resonance described in the next section. 3.2. A S PECIFIC E XAMPLE To further illustrate Feshbach resonance physics it is useful to consider a specific atomic example. Here we will discuss the case of the 40 K resonance that is used for most of our BCS-BEC crossover studies. The open channel in this case describes the scattering of |f, mf = |9/2, −9/2 and |9/2, −7/2 atoms, which are
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F IG . 8. Behavior of the scattering length at a Feshbach resonance in 40 K between the mf = −7/2, −9/2 spin states.
F IG . 9. 40 K ground state level diagram, with exaggerated Zeeman splittings. The two levels represent the hyperfine structure, which originates from the coupling of the nuclear spin (I = 4) with the electron spin (S = 1/2). Note the hyperfine structure of 40 K is inverted.
the two lowest energy states in the 40 K system (see Fig. 9). f describes the total atomic angular momentum and mf is the magnetic quantum number. For this particular problem the open channel couples to only one closed channel, |f, mf = |9/2, −9/2 and |7/2, −7/2 atoms. We will compare the results of this example to the requirements for studying the classic BCS-BEC crossover problem of Section 2. In the classic problem instead of two coupled channels a single variable depth potential with r0 a is considered (where r0 is the range of the potential). To calculate exactly the properties of our 40 K resonance we would need to carry out a full coupled channels calculation using realistic potassium potentials. The description of such a calculation is beyond the scope of this chapter but is de-
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EXPERIMENTAL REALIZATION OF BCS-BEC CROSSOVER
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scribed nicely in, for example, Ref. (Burke, 1999).2 Instead, for the demonstrative purposes of this section, we will examine the results of a simpler technique that is derived from K. Góral et al. in Ref. (Góral et al., 2004) and applied to the case of 40 K by M. Szyma´nska et al. in Ref. (Szyma´nska et al., 2005). We will solve the coupled Schrödinger equations that describe our two-channel system with a few simplifying assumptions. This technique reproduces the important physics of our 40 K resonance using a small set of experimentally measurable parameters. The main approximation is the so-called “pole approximation” described in Ref. (Góral et al., 2004) that holds when the open channel is strongly coupled to only one bare molecule state. Our main goal will be to use the simplified two-channel calculation to determine the binding energy of the molecular state in the multichannel potential as a function of the magnetic field. The Hamiltonian for our two-channel system is
2 − h¯m ∇ 2 + Voc W H = (7) , 2 W − h¯m ∇ 2 + Vcc where Voc is the uncoupled open channel potential, Vcc is the uncoupled closed channel potential, the potential W describes the coupling between the open and closed channel, and m is the mass of 40 K. It can be shown that the solution to this problem depends on only a few accessible parameters of the 40 K system (Góral et al., 2004). For the 40 K resonance we are considering, these parameters include the background scattering length abg = 174a0 (Loftus et al., 2002), the van der Waals coefficient C6 = 3897 a.u. (Derevianko et al., 1999), the resonance width w = 7.8 G (Regal et al., 2003b), and the resonance position B0 = 202.1 G (Regal et al., 2004b). Also useful is the binding energy of the first bound state in Voc , E−1 = 8.75 MHz (which can be attained from abg and C6 ) (Góral et al., 2004; Szyma´nska et al., 2005). Finally, we need the difference in magnetic moment (change in energy with magnetic field) of the open channel threshold with respect to the closed channel bare molecule. In our simplified calculation we will assume this to be the linear value that best approximates the result of a full coupled channels calculation, μco = 1.679μB = h × 2.35 MHz/G (Szyma´nska et al., 2005; Szyma´nska, 2005).3 These parameters, which in the end come from experimentally measured values, of course all have uncertainties, which we will ignore for now. The solution to the coupled Schrödinger equations based on the Hamiltonian above using the pole approximation is outlined in Ref. (Góral et al., 2004). The 2 We have compared some of the data presented in this chapter to full coupled channels calculations carried out by Chris Ticknor in John Bohn’s group at JILA. See Sections 5 and 6. 3 For 40 K µ from the result of the full calculation comes quite close to the difference in magnetic co moments of the closed and open channel thresholds, which is μco = h × 2.49 MHz/G at 200 G according to the Breit–Rabi formula (Corney, 1977).
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F IG . 10. Multichannel bound state of a Feshbach resonance in 40 K determined through the simplified calculation of Refs. (Góral et al., 2004; Szyma´nska et al., 2005) described in the text.
calculation is not computationally intensive and, after inserting the parameters above, provides the multichannel binding energy Eb , which we plot as the solid line in Fig. 10. Here Eb is plotted with respect to the energy of the open channel threshold, i.e., the open channel threshold is zero for all values of B. The dotted line shows the movement of the bare molecule energy with magnetic field, and the flat dashed line is the value of E−1 . The multichannel bound state is the dressed state of the avoided crossing of these two levels. The bare molecule state crosses threshold about 9 G higher than the position where the multichannel bound state comes through threshold, B0 . The difference between these two crossings is proportional to the resonance width parameter w and related to the interchannel coupling parameter in the Hamiltonian, W (Góral et al., 2004). Note that for 40 K the multichannel bound state at low field adiabatically connects to the highest-lying vibrational state of the open channel, rather than the bare molecule of the closed channel. The physics we study in this chapter actually occurs over a very small region near threshold in Fig. 10. The solid line in Fig. 11 shows a closeup of the multichannel bound state near threshold. Within a few gauss range of the Feshbach h¯ 2 resonance position the result obeys the expectation for a r0 , Eb = ma 2 (where a is determined through Eq. (6)), which is shown by the dashed line in Fig. 11. Farther away there is a clear deviation from quadratic behavior in 1/a. This behavior can be estimated by subtracting the range of the van der Waals potential r0 h¯ 2 from a in the calculation of the binding energy, Eb = m(a−r 2 (Góral et al., 2004; 0) Szyma´nska et al., 2005; Chin, 2005). The range of the van der Waals potential is given by (Gribakin and Flambaum, 1993) 1 (3/4) mC6 1/4 . r0 = √ (8) 8 (5/4) h¯ 2
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F IG . 11. Binding energy near the Feshbach resonance peak using three calculations of varying degrees of approximation. These results agree with those in Ref. (Szyma´nska et al., 2005).
F IG . 12. Binding energy in the magnetic-field regime used for the BCS-BEC crossover studies of this chapter.
The value of r0 is ∼60a0 for 40 K (and ∼30a0 for 6 Li). In Fig. 12 we examine the binding energy even closer to threshold. Here the h¯ 2 result of the simplified two-channel calculation and Eb = m(a−r (solid line) )2 2
0
h¯ are indistinguishable, while the Eb = ma 2 prediction (dashed line) only deviates slightly. We also plot the closed channel contribution to the molecule b wave-function, which is given by dE dB /μco . The Feshbach molecules that we will be interested in for the BCS-BEC crossover studies will have binding energies on order of or smaller than EF . A typical value of EF in our experiments (h × 15 kHz) is shown by the horizontal dotted line in Fig. 12. At this point the closed channel fraction according to our two-channel calculation is only 2%; this result is in agreement with full coupled channels calculations (Ticknor, 2005;
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Szyma´nska, 2005). This open-channel dominance means that, for our crossover studies, the two coupled channels problem of the Feshbach resonance can be approximated by small effective changes to the open channel potential. A resonance for which this is true is often referred to as a “broad resonance”, while the limit in which the closed channel dominates for Eb ≈ EF is referred to as a “narrow resonance” (Diener and Ho, 2004; Bruun and Pethick, 2004; h¯ 2 De Palo et al., 2004). In the case in which Eb ≈ m(a−r 2 we can derive a simple 0) equation that describes this criterion for a broad resonance in a Fermi system, w 2 μ2co 4EF
h¯ 2 2 mabg
1.
(9)
For the 40 K resonance we have been considering the left side of this expression has the numerical value of 43, and for the 6 Li resonance at B0 = 834 G it is over 10,000 (Chin, 2005; Bartenstein et al., 2005), indicating both Feshbach resonances used for BCS-BEC crossover studies thus far can be considered broad. In conclusion, we have seen how in atomic systems a Feshbach resonance can be used to add an additional bound state to an interatomic potential, leading to a divergence in the zero-energy scattering cross section for atoms colliding through the open channel. The Feshbach resonances that have so far been used for BCS-BEC crossover studies are broad resonances. These resonances can be approximated by a single channel problem and display universal properties, i.e., they have no dependence on the details of the atomic structure, but rather only on the parameters a and kF . For the 40 K resonance there are slight deviations from the a r0 limit that must be taken into account for precise measurements (Fig. 12). Still the physics of a 40 K gas at this resonance should basically reproduce the classic BCS-BEC crossover scenario envisioned by Leggett (Leggett, 1980) and described in Section 2.2.
4. Cooling a Fermi Gas and Measuring its Temperature To access the superfluid state of a Fermi gas in the crossover, the gas must be cooled below the critical temperature, Tc . To achieve such a temperature with a trapped gas of fermions was one of our largest challenges. In this section we describe how we cool 40 K and assess the success of the cooling through temperature measurements. 4.1. C OOLING 40 K The apparatus used to cool 40 K for these BCS-BEC crossover experiments employs the strategy used for some of the first experiments with 87 Rb BEC
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(Anderson et al., 1995; Myatt, 1997; Ensher, 1998). We perform the “usual” combination of trapping and cooling in a magneto-optical trap (MOT) followed by evaporative cooling (Phillips, 1998; Ketterle et al., 1999; Cornell et al., 1999). The laser cooling uses light from semiconductor diode lasers on the 40 K D2 line (4S1/2 to 4P3/2 transition at 766.7 nm), and a two chamber apparatus allows for a ultra high vacuum region for evaporative cooling (DeMarco, 2001). The major difference compared to the 87 Rb experiments stems from the fact that elastic collisions between identical fermions are suppressed at ultracold temperatures. This is because quantum statistics require antisymmetry of the total wave-function for two colliding fermions, which forbids s-wave collisions for identical fermions. While odd partial wave collisions, such as p-wave, are allowed, these collisions are suppressed below T ≈ 100 µK because of the angular momentum barrier (DeMarco et al., 1999). Since evaporative cooling requires collisions to rethermalize the gas, a mixture of two distinguishable particles is required to cool fermions. 40 K provides an elegant solution to this problem. Figure 9 shows the ground state energy levels of 40 K. The large angular momentum of the lowest ground state hyperfine level (f = 9/2) provides 10 distinct spin states. The two highest energy states, mf = +9/2 and mf = +7/2, can both be held with reasonable spatial overlap in a magnetic trap, which is the type of trap most proven for evaporative cooling when starting from a MOT. In this way an apparatus designed for only one atomic species could provide two distinguishable states for cooling. To remove the highest energy atoms for evaporative cooling, microwaves at ∼1.3 GHz were used to transfer atoms to untrapped spin states in the upper hyperfine state (DeMarco and Jin, 1999; DeMarco, 2001). With this technique, quantum degeneracy was reached in 1999, and by 2001 two-component 40 K Fermi gases at temperatures of 0.25TF could be created. One of the first steps in accessing the BCS-BEC crossover was to create a degenerate Fermi gas in an equal mixture of the f = 9/2 Zeeman states between which a Feshbach resonance was predicted, mf = −9/2 and −7/2 (Fig. 9). To accomplish this we needed to trap the high field seeking spin states of the f = 9/2 manifold; these states cannot be trapped in a magnetic trap. Thus, we utilized a far-off-resonance optical dipole trap (FORT) to confine the required spin states for the crossover experiments. Such a trap can be formed at the focus of a Gaussian laser beam whose optical frequency is far detuned from the 40 K transitions. We found that the best way to realize a cold gas in an optical dipole trap was to load a relatively hot gas of fermions after some evaporation in the magnetic trap. Typically we load the optical trap when the sample in the magnetic trap reaches T /TF ≈ 3 (T ≈ 5 µK with a few 107 atoms). After transfer to the mf = −9/2, −7/2 spin states the gas is evaporated in the optical trap simply by lowering the depth of the trap and allowing the hottest atoms to spill out. This evaporation and the subsequent temperature measurements were typically performed at a magnetic field of ∼235 G. At this field the scattering length is
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near the background (nonresonant) value of 174a0 , and from this field Feshbach resonances can easily be accessed.4 Evaporative cooling in the optical trap actually allowed us to achieve colder temperatures than previous records. We could cool to T /TF ≈ 0.1 (and possibly colder) with up to 106 atoms in each spin state. At these temperatures another impediment to cooling fermions appears—Pauli blocking of collisions (DeMarco et al., 2001). In a degenerate Fermi sea the Pauli exclusion principle forbids collisions for which the final state would place fermions in already occupied levels. This results in a suppression of the collision rate compared to the classical expectation and may make evaporative cooling more difficult in an all-fermion system. Many recent experiments have used a bosonic atom as their “second particle” as a possible approach to avoiding this problem (Truscott et al., 2001; Schreck et al., 2001; Roati et al., 2002; Hadzibabic et al., 2002; Goldwin et al., 2004).
4.2. M EASURING THE T EMPERATURE OF A F ERMI G AS In atomic gas experiments the standard technique for measuring the temperature of a gas is to suddenly turn off the trapping potential confining the atoms and then wait for an “expansion time” t. A two-dimensional image of the atomic distribution then reveals the velocity, or momentum, distribution of the gas (Ketterle et al., 1999). This technique is often referred to as time-of-flight imaging. For a classical gas the velocity spread is directly related to the temperature of the gas through the Maxwell–Boltzmann distribution. In the quantum regime measuring temperature from such distributions becomes trickier. In this section we will describe techniques for measuring the temperature of an ideal Fermi gas using images of gas distributions. 4.2.1. An Optically Trapped Fermi Gas Before presenting temperature measurements, it is useful to describe the expected distribution of an optically trapped Fermi gas. First, we analyze the optical dipole potential created from a single Gaussian laser beam using a harmonic approximation; this is the potential used for most of the BCS-BEC crossover experiments presented in this chapter. Then we present the most relevant formulas for describing the spatial and momentum distributions of a Fermi gas in such a harmonic trap. 4 The background scattering length is reasonably large compared to many other alkali atoms, resulting in nice evaporative cooling. Yet the interaction it provides still allows us to approximate the gas as an ideal Fermi gas for the purpose of temperature measurement.
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F IG . 13. Potential for potassium atoms created by a single 1064 nm Gaussian laser beam with w = 15 µm and a power of 10 mW. The y-direction includes the effect of gravity. Note that the horizontal scale in the third graph (z-direction) is a factor of 100 larger than the scale in the first two graphs. The dashed line in the left-most graph shows the parabola that approximates the potential for small x.
An optical dipole trap results from the interaction between a light field and the oscillating atomic electric dipole moment that is induced by the light field. This effect, known as the ac Stark shift, forms the conservative part of the interaction of atoms with light. The shift is proportional to the light intensity, and the sign depends upon the sign of the detuning of the light compared to the atomic resonance. When the light is detuned red of the atomic transition and the light intensity varies in space, an attractive potential well can be formed. While the photon-scattering rate decreases quadratically with the frequency detuning between the light and the atomic transition, the ac Stark shift varies linearly with this detuning. Thus, a trap with a large detuning results in a significant dipole potential at a low photonscattering rate. This allows use of the dipole potential for atom traps with long storage times, as was first realized in Ref. (Chu et al., 1986). The simplest optical dipole trap is formed by the focus of a single red-detuned Gaussian laser beam. The trap potential from the beam will be proportional to the laser intensity, which is given by 2
I (r, z) =
2
−2r /w Ipk 1+(z/zr )2 e 1 + (z/zR )2
(10)
for a beam pointing along zˆ . Here w is the beam waist (1/e2 radius), zR = πw2 /λ 2p is the Rayleigh range, Ipk = πw 2 is the peak intensity, and p is the total optical power. The strength of the potential created can easily be calculated from atomic properties (Grimm et al., 2000). Figure 13 shows cross sections of the potential for a typical trap that would hold a degenerate Fermi gas in our system, where we have included the effect of gravity in the y-direction. Gravity effectively
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lowers the trap depth in the y-direction, making evaporation in shallow traps onedimensional. At the bottom of the trap we can approximate the Gaussian potential as a parabolic potential (dashed line in Fig. 13). In this case we have a harmonic trap with oscillator frequencies given by 4U0 4U0 ωr = 2πνr = (11) , ωz = 2πνz = , 2 mw2 mzR where U0 is the depth of the potential. In shallow traps the harmonic approximation breaks down since atoms sample the potential near the top of the well where the Gaussian rises less steeply than the parabola. We account for this effect in experiments by modelling the real trap as a harmonic trap with a lower effective frequency. We determine the effective frequency experimentally through observation of the Fermi gas excitation frequencies in the weakly interacting regime. We now consider a Fermi gas in a harmonic potential described by the Hamiltonian for a particle in a harmonic well H =
1 p2 + mωr2 ρ 2 , 2m 2
(12)
where p 2 = px2 + py2 + pz2 and ρ 2 = x 2 + y 2 + λ2 z2 (λ = ωz /ωr ). The density 2
, where ω¯ = (ωr2 ωz )1/3 (see, e.g., Ref. (Pethick and of states is g() = 2(hω) ¯¯ 3 Smith, 2002)). The statistics of the Fermi gas are described by the Fermi–Dirac distribution function 1 , f () = (13) e kb T /ζ + 1 where ζ = eμ/kb T is the fugacity. We can now calculate the Fermi energy, which is defined as the energy of the highest occupied level of the potential at T = 0. We simply equate the integral over all states up to EF to the number of particles in one fermion spin state, N , to find EF h¯ ω¯ = (6N )1/3 . TF = (14) kb kb The temperature compared to TF describes the degeneracy of the Fermi gas and in the classical limit is related to peak phase space density through PSDpk = (T /TF )−3 /6. Also using the distribution function and the density of states we can obtain thermodynamic quantities such as the energy per particle, which will become important in Section 9. ∞ g()f () d Li4 (−ζ ) U = 3kb T = 0 ∞ . E= (15) N Li3 (−ζ ) 0 g()f () d
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F IG . 14. Energy of an ideal, trapped Fermi gas.
∞ k n The function Lin (x) = k=1 x /k appears often in the analysis of a harmonically trapped Fermi gas. Lin is the Poly-Logarithmic function of order n, sometimes written gn . Figure 14 plots the result of Eq. (15). In the classical regime the energy is proportional to the temperature, while in the Fermi gas limit the energy asymptotes to 34 EF ( 38 EF kinetic energy and 38 EF potential energy). Distribution functions in position and momentum can be determined through standard statistical mechanics techniques and the Thomas–Fermi approximation, which holds when many oscillator states are occupied (see Refs. (Butts and Rokhsar, 1997; DeMarco, 2001)). Table I shows the resulting Fermi–Dirac distribution functions in position and momentum space. The full expression is shown first, but it is also useful to understand the classical and T = 0 limits. In the classical limit the distribution is Gaussian, and at T = 0 the distributions are only nonzero for values less than the Fermi radius rF or Fermi momentum pF = h¯ kF . h2 k 2
Note that kF is defined through EF = ¯2mF and is related to the inverse distance between fermions at the center of the trap; at T = 0 the number density of one fermion spin state at the center of the trap is written in terms of kF as npk =
kF3 . 6π 2
4.2.2. Measuring Temperature from the Momentum Distribution In the experiment we access the distribution of the Fermi gas through absorption images of an expanded gas. Absorption images are taken by illuminating the atoms with a resonant laser beam pulse and imaging the shadow cast by the atoms onto a CCD camera. These images effectively integrate through one dimension to give a two-dimensional image (for example, Fig. 15(a)). The appropriate function for this distribution is n(ρ) or Π(p) integrated over one dimension (here x) (DeMarco, 2001). Writing the result in terms of the experimentally observed optical depth (OD), we find for the Fermi–Dirac distribution
ODFD (y, z) = ODpk Li2 −ζ e
−
y2 2σy2
e
−
z2 2σz2
/Li2 (−ζ ).
(16)
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Table I Distribution functions for a harmonically trapped Fermi gas Validity
Spatial distribution where σr2 =
all T /TF
n(ρ) =
kb T 2E and rF2 = F2 mωr2 mωr
2 2 λN Li3/2 (−ζ e−ρ /2σr )/Li3 (−ζ ) (2π )3/2 σr3 2 2 λN nc (ρ) = e−ρ /2σr (2π )3/2 σr3 8 1 − ρ 2 3/2 for ρ < r , 0 otherwise n0 (ρ) = λN F rF2 rF3 π 2
T /TF 1 T =0 Validity
2 = 2mE Momentum distribution where σp2 = mkb T and pF F
all T /TF
Π(p) =
2 2 N Li3/2 (−ζ e−p /2σp )/Li3 (−ζ ) (2π )3/2 σp3 2 2 N Πc (p) = e−p /2σp (2π )3/2 σp3 2 3/2 Π0 (p) = N3 82 1 − p2 for p < pF , 0 otherwise p p π
T /TF 1 T =0
F
F
F IG . 15. Nonclassical momentum distribution of Fermi gas. (a) Sample absorption image of the momentum distribution of a degenerate Fermi gas. White corresponds to many atoms and black to zero atoms. Here the integration is along the z-direction. (b) Azimuthally averaged profile of the absorption image.
In the classical limit this equation becomes a two-dimensional Gaussian function: ODgauss (y, z) = ODpk e
−
y2 2σy2
e
−
z2 2σz2
(17)
.
These forms are applicable for both the spatial and momentum profiles, and for kb T 2 2 arbitrary expansion times through the relations σy2 = mω 2 [1 + (ωr t) ] and σz = kb T [1 + (ωz t)2 ], mωz2
r
where t is the expansion time.
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F IG . 16. Theoretical cross section of a harmonically trapped Fermi gas momentum distribution integrated along one dimension. TF is held fixed as T /TF varies.
Examples of cross sections of ODFD for various degeneracies, and constant TF , are shown in Fig. 16. As the gas is cooled the overall distribution becomes narrower, while the shape of the distribution becomes flatter. Note that as the temperature is lowered far below TF the changes in the distribution become small compared to TF . Still, down to T /TF ≈ 0.1 the temperature can be determined from least-squared fits to such distributions. Figure 15(a) is a sample absorption image of an expanded Fermi gas. The points in Fig. 15(b) are the result of an azimuthal average of the image. The solid line shows the result of a surface fit of the two-dimensional image to the Fermi–Dirac distribution (Eq. (16)), which reveals that the gas is at a temperature of 0.1TF . For comparison we plot as a dashed line the result of a fit appropriate for a classical distribution (Eq. (17)). Clearly the experimental distribution is flatter than a classical distribution. To evaluate this thermometry, we can examine the results of least-squared surface fits for gases at a variety of expected temperatures. In the fits ODpk , σy , σz , and ζ are independent fit parameters. σy and σz tell us the temperature; ζ can be viewed as a shape parameter that is directly related to T /TF through Li3 (−ζ ) = −(T /TF )−3 /6. As a check on the fits we compare the result for ζ to T /TF calculated through σy2 mωr2 T . = (18) TF h¯ ω(6N ¯ )1/3 (1 + (ωr t)2 ) We use the measured trap frequencies for ωr and ωz and the number of atoms in each spin state N calculated from the total absorption in the image. Figure 17(a) shows a comparison of ζ with T /TF from Eq. (18). The line shows the expected relationship for an ideal Fermi gas. In Fig. 17(b) we convert ζ to T /TF for a more direct comparison. In general we find the two values agree, indicating the fits are extracting the correct information. Note that the noise in the points becomes large
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F IG . 17. Analysis of fits to Eq. (16) for expansion images of an optically trapped gas with an equal mixture of mf = −9/2, −7/2 atoms (Regal and Jin, 2003). For these data the integration was in the x-direction, and T is extracted from σy .
at temperatures >0.5TF . This is expected because the changes in the shape of the distribution become small in this limit. A similar effect occurs in the low temperature limit where the distribution changes little as the T = 0 Fermi gas limit is approached. However, the success of this thermometer in the 0.1 < T /TF < 0.5 range has made this method the workhorse of temperature measurements in our experiments. 4.2.3. Measuring Temperature Using an Impurity Spin State A second technique for measuring temperature that we have explored is impurity spin-state thermometry. Eric Cornell proposed this idea as a method to check the Fermi–Dirac surface-fit technique outlined in the previous section. A check is especially necessary for the coldest temperature gases at 0.1TF and below, because of the decrease in the sensitivity of the Fermi–Dirac fits at these temperatures. We have not done extensive work using the impurity thermometer; however, as we will see here this thermometer works quite well and has unexplored potential, in particular as a technique that could measure temperatures less than 0.1TF . The idea of the impurity spin-state technique is to embed a small number of atoms in a third state within the usual two-component gas (Fig. 18). In the limit where the number of atoms in the impurity spin, Nim , is small compared to the particle number in the original states, the Fermi energy of the impurity state will be low enough that the impurity gas will be nondegenerate. Provided all of the spin states in the system are in thermal equilibrium, the temperature of the sysmσ 2
tem will be k tim2 , where σim is simply determined from a fit of the impurity gas b momentum distribution to a Gaussian distribution (Eq. (17)).
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F IG . 18. Measurement of T through an embedded impurity spin state. All three components are overlapped in the optical trap.
A difficulty with this method is that EF scales weakly with particle number. Suppose our original gas has a particle number of 105 at T /TF = 0.1. For T /TF of the impurity to be 1, Nim would need to be 100, and detecting the distribution of 100 atoms with good signal to noise is not trivial. However, the fully classical limit does not need to be reached to gain information about the temperature from the impurity state. It is only required that the T /TF be large enough that the energy of the impurity gas changes significantly with temperature. Figure 14 illustrates the range of T /TF for which this is the case. To see if an impurity spin-state thermometer was feasible we designed an experiment to test this thermometer against the surface-fit technique described previously. We started with a (not necessarily equal) mixture of atoms in the mf = +7/2, +9/2 spin states. Part way through the evaporative cooling process a small fraction of the mf = +7/2 atoms were transferred to the mf = +5/2 state, which serves as our impurity (Fig. 18). For this experiment the gas was prepared at a low magnetic field of a few gauss where the three-state mixture is fully stable. Here the scattering length between any pair of the three spin states is 174a0 . The spin states were selectively imaged by applying a large magnetic field gradient of ∼80 G/cm during the expansion to spatially separate the spin states (Stern–Gerlach imaging) (DeMarco, 2001). For analysis of the impurity spin-state data in the most general case where the impurity is not fully classical, it is useful to introduce the variable Tgauss . For mσ 2
a momentum space distribution Tgauss is defined as k gauss 2 , where σgauss is the bt result from a least-squared fit to the Gaussian distribution of Eq. (17). As we saw in Fig. 15 the Fermi distributions will not be well fit by a Gaussian, but the result is a well-defined quantity. Figure 19 shows the result of Tgauss /TF as a function of degeneracy, as determined through least-squared fits to theoretical distributions. Tgauss provides much the same information as the Fermi gas energy E (Fig. 14). However, it is more useful for our current purposes because it can be extracted
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F IG . 19. Dimensionless plot of the variable Tgauss (defined in the text) versus temperature.
F IG . 20. Impurity spin-state thermometry. Here we plot Tgauss for all of the states in the gas for multiple experiment iterations in which T was held fixed. The mf = 9/2 points have an average degeneracy of T /TF = 0.13.
from real images with better signal to noise than E, and it is convenient because in the classical limit it becomes precisely the real temperature of the system. In Fig. 20 we plot Tgauss as a function of N for a cold gas of atoms distributed among the three states mf = 9/2, 7/2, and 5/2. N is the measured number of atoms in the spin state from which Tgauss is extracted. The seven sets of points come from multiple iterations of the experiment for which the temperature was expected to be constant. Since the trap strength was held constant, N uniquely defines EF , and given that the temperature T is constant for all points on the plot, we can translate the dimensionless theoretical result of Fig. 19 to the situation of Fig. 20. The line in Fig. 20 shows the best-fit curve to the data, in which the only free parameter is the real temperature T . Note T is the value of Tgauss as N goes to zero. If the gas were fully classical Tgauss would be constant as a function of N ;
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F IG . 21. Comparison of thermometers. The y-axis shows the result of the Fermi–Dirac fits to the mf = 9/2 distribution, and the x-axis shows the result of the impurity spin-state technique.
it is a manifestation of Pauli blocking that different components of an equilibrium gas can have the same temperature but different energy (DeMarco et al., 2001). Figure 21 shows the results of four experiments like the one shown in Fig. 20. The temperature result from the impurity measurement is compared to the result from the surface fits described in Section 4.2.2 (applied to the mf = 9/2 cloud). We see that both methods agree to within the uncertainty for clouds in the T /TF = 0.1–0.2 range. 4.2.4. Temperature in the BCS-BEC Crossover Our focus thus far has been on measuring the temperature of an ideal, normal Fermi gas. For the BCS-BEC crossover experiments we would like to know the temperature of our gas at any interaction strength. To this point there have been no experiments that have directly measured the temperature of a gas throughout the entire crossover. The main technique that has been used thus far is to instead measure the entropy of the system, which, through theory, can be translated to a temperature at any point in the crossover. The entropy, S, can be determined through temperature measurements in the weakly interacting regime. Then if experiments are performed using adiabatic magnetic-field ramps to the crossover regime, the entropy will be held constant at the weakly interacting value. As we have seen in 40 K it is convenient to measure the temperature of the weakly interacting gas above the Feshbach resonance with a Fermi gas, while in most of the 6 Li experiments the temperature is measured through the condensate fraction close to the BEC limit. In either case, the limitation of this technique is that it relies upon theory to convert between entropy and temperature in the crossover.
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Some initial work on this theoretical problem can be found in Ref. (Chen et al., 2005a). An alternative temperature measurement in the crossover was applied in the group of J. Thomas where they measured the temperature at unitarity by using fits to the momentum distribution in the hydrodynamic limit (Kinast et al., 2005b); here the distribution is simply a rescaled Fermi distribution and thus amenable to the same fitting procedure as the weakly interacting regime (Jackson et al., 2004). The impurity spin state thermometer described in Section 4.2.3 is another possible, but as of yet unexplored, direct thermometer in the crossover regime.
5. Elastic Scattering near Feshbach Resonances between Fermionic Atoms Some of the first signatures of the presence of fermionic Feshbach resonances were the observation of magnetic-field-dependent changes in the elastic-scattering properties of a normal Fermi gas (Loftus et al., 2002; O’Hara et al., 2002a; 2002b; Regal et al., 2003b; Regal and Jin, 2003; Gupta et al., 2003; Gehm et al., 2003; Bourdel et al., 2003). Here we present three different techniques that we used to probe changes in scattering properties at a fermionic Feshbach resonance. Two techniques probe the collision cross section, which tells us the magnitude of a. In the third measurement we observe the sign of a change and see evidence for unitarity-limited interactions.
5.1. M EASURING THE E LASTIC C OLLISION C ROSS S ECTION The first Feshbach resonance we searched for experimentally is the resonance described in Section 3 that occurs between the mf = −9/2 and mf = −7/2 spin states. The original theoretical prediction for the location of this resonance was B0 = 196+9 −21 G, based on available potassium potentials (Bohn, 2000). We first experimentally measured the position of this resonance using the technique of cross-dimensional rethermalization, which measures the collision cross section (Monroe et al., 1993). This was a technique that had provided much information about a Feshbach resonance in the bosonic 85 Rb gas (Roberts et al., 1998, 2001a). For this measurement we started with a gas of fermions in the mf = −7/2, −9/2 spin states at T ≈ 2TF . The gas was taken out of thermal equilibrium by modulating the optical trap intensity at 2νy , which caused selective heating in the y-direction. (We could selectively modulate one radial direction because for this measurement our optical trap was not cylindrically symmetric (νx = 1.7νy ).) The exponential time constant for energy transfer between the two radial directions, τ , was measured as a function of magnetic field. τ is related to the s-wave
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F IG . 22. Collision cross section measured near an s-wave Feshbach resonance between 40 K atoms in the mf = −7/2, −9/2 spin states at T = 4.4 µK (Regal et al., 2003b). In between the peak and dip in σ the interaction is attractive; everywhere else it is repulsive.
√ collision cross section through 1/τ = 2nσ v/α. v = 4 kb T /πm is the mean relative speed between colliding fermions and n = N1tot n7 (r)n9 (r) d 3 r is the density weighted density. α is the calculated number of binary s-wave collisions required for rethermalization (DeMarco et al., 1999). Figure 22 plots the result of this measurement as a function of the magnetic field B. The magnetic field was calibrated through radio-frequency (rf) transitions between mf levels in the 40 K system. An advantage of the cross-dimensional rethermalization technique is that it allows measurements of σ over a large range. Through cross section measurements that extend over four orders of magnitude, both the position of the divergence of the scattering length, B0 , and the position of the zero crossing could be measured (Fig. 22). This allows a measurement of the magnetic-field width of the resonance w, which as we saw in Section 3 describes the coupling strength. The line in Fig. 22 is the result of a full coupled channels calculation of σ carried out by C. Ticknor and J. Bohn, in which the parameters of the potassium potential were adjusted to achieve a best fit to our data from two different 40 K resonance (Regal et al., 2003b; Burke et al., 1999). This calculation takes into account the distribution of collision energies in the gas by thermally averaging over a Gaussian distribution defined by a temperature of 4.4 µK. The fit result places the Feshbach resonance at B0 = 201.6 ± 0.6 G and the zero crossing at 209.9 ± 0.6 G.
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5.2. A NISOTROPIC E XPANSION A disadvantage of the cross-dimensional rethermalization method is that it only provides a valid measurement of σ in the so-called collisionless regime. A trapped gas is considered collisionless if the trap oscillator period 1/ν is much shorter than the mean time between collisions in the gas, 1/Γ . In the opposite limit where Γ ν the gas is collisionally hydrodynamic and the cross-dimensional rethermalization time will be determined by the oscillator period 1/ν instead of the mean time between collisions. At the peak of the Feshbach resonance the fermion–fermion interactions can easily become strong enough to make the gas collisionally hydrodynamic. In this regime a technique more suited to measuring changes in the elastic cross section is anisotropic expansion. In the hydrodynamic limit collisions during the expansion transfer kinetic energy from the elongated axial (z) cloud dimension into the radial (r) direction. This changes the aspect ratio of the expanded cloud (σz /σr ) compared to the collisionless expectation. This effect was first observed in a 6 Li Fermi gas in Ref. (O’Hara et al., 2002a). Figure 23 presents a measurement of anisotropic expansion in a 40 K gas at T /TF = 0.34. Here we enhance interactions between the mf = −9/2, −5/2 spin states using a Feshbach resonance between these states at ∼224 G. At the position of the Feshbach resonance where σ is large, the aspect ratio σz /σy decreases. As the gas becomes collisionless away from the resonance the aspect ratio smoothly evolves to the collisionless value. The key to observing anisotropic expansion in these systems is that during the beginning of the expansion the magnetic field must remain at the value near the Feshbach resonance. In these experiments the magnetic field remained high for 5 ms of expansion and an absorption image was taken after a total expansion time of 20 ms. In this measurement the mf =
F IG . 23. Anisotropic expansion of a strongly interacting Fermi gas near a Feshbach resonance between the mf = −5/2, −9/2 spin states (Regal and Jin, 2003).
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−9/2, −5/2 gas is created at the field B by starting with a mf = −9/2, −7/2 gas and applying a π pulse between the −5/2 and −7/2 states 0.3 ms before expansion. This technique avoids complications due to atom loss but creates a nonequilibrium gas. In general the expected aspect ratio in the regime between collisionless and hydrodynamic behavior is difficult to calculate. We can however check to see if some degree of hydrodynamic expansion of the normal gas is expected. We can calculate the elastic collision rate Γ = 2nσ v in the gas, using an elastic 2 and |a | = 2000a (as was measured collision cross section given by σ = 4πa59 59 0 near the resonance peak (Regal and Jin, 2003)) to find Γ = 46 kHz. Comparing this rate to the trapping frequencies we find Γ /νr = 37 and Γ /νz = 2400. Hence, it is not surprising that we observe anisotropic expansion. For a gas that was fully hydrodynamic, with Γ νr , νz , we would expect our measured aspect ratio to reach 0.4 (Menotti et al., 2002; Kagan et al., 1997). In ultracold gas experiments anisotropic expansion is often associated with superfluidity of a BEC. This is because a typical ultracold Bose gas is collisionless, while below the superfluid transition the gas obeys superfluid hydrodynamic equations. Near a Feshbach resonance, however, where the gas can be collisionally hydrodynamic, anisotropic expansion can be observed both above and below the superfluid transition temperature. Still anisotropic expansion has been put forth as a possible signature of superfluidity in the BCS-BEC crossover regime (Menotti et al., 2002; Kagan et al., 1997). In pursuing this signature it is important to carefully distinguish between collisional and superfluid hydrodynamics and take into account that changes during expansion will affect the many-body state (Gupta et al., 2004). Such analysis is possible and has been considered in Refs. (O’Hara et al., 2002a; Bourdel et al., 2004).
5.3. M EASURING THE M EAN -F IELD I NTERACTION E NERGY Measurements of the collision cross section are useful for detecting the strength of the interaction but are not sensitive to whether the interactions are attractive or repulsive. The mean-field energy, on the other hand, is a quantum mechanical, many-body effect that is proportional to na. For Bose–Einstein condensates with repulsive interactions the mean-field energy (and therefore a) can be determined from the size of the trapped condensate (Inouye et al., 1998; Cornish et al., 2000), while attractive interactions cause condensates with large atom number to become mechanically unstable (Gerton et al., 2000; Roberts et al., 2001b). For an atomic Fermi gas the mean-field interaction energy has a smaller impact on the thermodynamics. Here we discuss a novel spectroscopic technique that measures the mean-field energy of a two-component Fermi gas directly (Regal and Jin, 2003; Gupta et al., 2003).
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F IG . 24. Radio-frequency spectra. (a) Transition of interest. (b) rf line-shapes with (solid line) and without (dotted line) interactions (Regal and Jin, 2003).
In this measurement we again used the Feshbach resonance between the mf = −5/2, −9/2 spin states. At magnetic fields near the resonance peak, the meanfield energy in the Fermi gas was measured using rf spectroscopy (Fig. 24(a)). First, optically trapped atoms were evaporatively cooled in a 72/28 mixture of the mf = −9/2 and mf = −7/2 spin states. After the evaporation the optical trap was recompressed to achieve a larger density, and the magnetic field was ramped to the desired value near the resonance. We then quickly turned on the resonant interaction by transferring atoms from the mf = −7/2 state to the mf = −5/2 state with a 73 µs rf π-pulse. The fraction of mf = −7/2 atoms remaining after the pulse was measured as a function of the rf frequency. The relative number of mf = −7/2 atoms was obtained from an absorption image of the gas taken after 1 ms of expansion from the optical trap. Atoms in the mf = −7/2 state were probed selectively by leaving the magnetic field high and taking advantage of nonlinear Zeeman shifts. Sample rf absorption spectra are shown in Fig. 24(b). At magnetic fields well away from the Feshbach resonance we were able to transfer all of the mf = −7/2 atoms to the mf = −5/2 state and the rf line-shape had a Fourier width defined by the rf pulse duration. At the Feshbach resonance we observed two changes to the rf spectra. First, the frequency for maximum transfer was shifted relative to the expected value from a magnetic field calibration. Second, the maximum transfer was reduced and the measured line-shape is wider. Both of these effects arise from the mean-field energy caused by strong interactions between mf = −9/2, −5/2 atoms at the Feshbach resonance. The mean-field energy produces a density-dependent frequency shift given by ν =
2h¯ n9 (a59 − a79 ), m
(19)
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F IG . 25. Scattering length as measured through the mean-field interaction (Regal and Jin, 2003). These data were taken for a normal Fermi gas at T /TF = 0.4 and at two different densities of the mf = −9/2 gas: npk = 1.8 × 1014 cm−3 (circles) and npk = 0.58 × 1014 cm−3 (squares).
where n9 is the number density of atoms in the mf = −9/2 state, and a59 (a79 ) is the scattering length for collisions between atoms in the mf = −9/2 and mf = −5/2 (mf = −7/2) states (Harber et al., 2002). Here we have ignored a nonresonant interaction term proportional to the population difference between the mf = −7/2 and mf = −5/2 states; this term equals 0 for a perfect π-pulse. For our spatially inhomogeneous trapped gas, the density dependence broadens the line-shape and lowers the maximum transfer. This effect occurs on both sides of the Feshbach resonance peak. In contrast, the frequency shift for maximum transfer reflects the scattering length and changes sign across the resonance. We measured the mean-field shift ν as a function of B near the Feshbach resonance peak. The rf frequency for maximum transfer was obtained from Lorentzian fits to spectra like those shown in Fig. 24(b). The expected resonance frequency was then subtracted to yield ν. The scattering length a59 was obtained using Eq. (19) with n9 = 0.5npk and a79 = 174a0 , where npk is the peak density of the mf = −9/2 gas (Loftus et al., 2002). The numerical factor 0.5 multiplying npk was determined by modelling the transfer with a pulse-width limited Lorentzian integrated over a Gaussian density profile. The measured scattering length as a function of B is shown in Fig. 25. The solid line shows a fit to the expected form for a Feshbach resonance, Eq. (6). Data within ±0.5 G of the peak were excluded from the fit. We find that the Feshbach resonance peak occurs at 224.21 ± 0.05 G and the resonance has a width w of 9.7 ± 0.6 G. When B is tuned very close to the Feshbach resonance peak, we expect the measured a59 to have a maximum value of on the order of 1/kF due to the unitarity limit. This saturation can be seen in the data shown in Fig. 25. Two points
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Table II Observed Feshbach resonances in 40 K Open channel |f, mf
l B0 (G)
|9/2, −9/2 + |9/2, −7/2 |9/2, −9/2 + |9/2, −5/2 |9/2, −7/2 + |9/2, −5/2 |9/2, −7/2
s s s p
w (G)
202.10 ± 0.07 7.8 ± 0.6 224.21 ± 0.05 9.7 ± 0.6 ∼174 ∼7 ∼198.8
Reference (Regal et al., 2003b, 2004b) (Regal and Jin, 2003) unpublished (Regal et al., 2003b; Ticknor et al., 2004)
that were taken within ±0.5 G of the Feshbach resonance peak, one on either side of the resonance, clearly lie below the fit curve. We find that the unitaritylimited point on the attractive interaction side of the resonance (higher B) has an effective scattering length of ∼ 2/kF . (Here h¯ kF is the Fermi momentum for the mf = −9/2 gas.)
5.4.
40 K
F ESHBACH R ESONANCE S UMMARY
Table II lists the Feshbach resonances we have studied experimentally in 40 K. All of these resonances were originally located by measuring scattering properties using the techniques described above. We include the states between which the resonance occurs, the partial wave of the resonant collision l, our most precise measurement of the resonance position B0 , and the resonance width w.
6. Creating Molecules from a Fermi Gas of Atoms After locating Feshbach resonances in our 40 K system, we wanted to observe evidence of a molecular bound state near threshold on the low-field side of the Feshbach resonances. Creating molecules in this bound state, referred to as “Feshbach molecules”, would be the first step towards achieving the BEC limit of the crossover problem. We were motivated to believe that it would be possible to create Feshbach molecules by experiments carried out in the Wieman group at JILA (Donley et al., 2002; Claussen et al., 2002). By pulsing the magnetic field quickly towards the Feshbach resonance they were able to observe coherent oscillations between atoms and Feshbach molecules in a 85 Rb BEC. We hoped to employ a slightly different approach to creating molecules in which we would ramp the magnetic field fully across the Feshbach resonance. In this section we will present how, using this technique, we were able to efficiently and reversibly create Feshbach molecules from a Fermi gas of atoms. Another of our contributions to the study of Feshbach molecules was a spectroscopic detection technique that firmly
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F IG . 26. Creating molecules using magnetic-field ramps across a Feshbach resonance.
established that Feshbach molecules had been created. We also present our current understanding of the physics of the conversion of atoms to molecules using adiabatic magnetic-field ramps; this understanding was gained through a study of the conversion dependences led by the Wieman group (Hodby et al., 2005). Lastly, we describe how the dissociation of molecules in a sample at low density provided our most precise measurement of the Feshbach resonance position. 6.1. M AGNETIC -F IELD A SSOCIATION Figure 26 shows the behavior of the bound molecular state at a Feshbach resonance presented in Section 3. Given this picture, one would expect that atoms could be converted to molecules simply by ramping the magnetic field in time across the Feshbach resonance position B0 (Timmermans et al., 1999; Abeelen and Verhaar, 1999; Mies et al., 2000). The only requirement to creating molecules in this way is that the magnetic-field ramp must be slow enough to be adiabatic with respect to the two-body physics of the Feshbach resonance (twobody adiabatic). To a very good approximation the Feshbach molecules would have twice the polarizability of the atoms (Ratcliff et al., 1987) and therefore would be confined in the optical dipole trap along with the atoms. In fact we would expect that the atoms and molecules have the same trapping frequency, but the molecule trap depth would be twice as large as the atom trap depth. We performed such an experiment using a magnetic-field ramp across the mf = −5/2, −9/2 resonance introduced in the previous section. We started with a nearly equal mixture of the two spin states mf = −5/2, −9/2 at a magnetic field of 227.8 G. The field was ramped at a rate of (40 µs/G)−1 across the resonance to various final values. The number of atoms remaining following the ramp was determined from an absorption image of the cloud at ∼4 G after expansion from the optical trap. Since the light used for these images was resonant with the atomic transition, but not with any molecular transitions, we selectively detected only the atoms. In Fig. 27 we present the observed total atom number in
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F IG . 27. Creation of molecules as seen through atom loss (Regal et al., 2003a). A fit to an error function provides a guide to the eye. Atom loss occurs at precisely the expected position of the Feshbach resonance given a previous measurement of the scattering length divergence (Regal and Jin, 2003).
the mf = −5/2, −9/2 states as a function of the final magnetic-field value of the ramp. We found that the atoms disappear abruptly at the Feshbach resonance peak (dashed line). We also found in similar experiments that we could recover the lost atoms with an immediate magnetic-field ramp back above the Feshbach resonance. This ruled out many atom loss processes and strongly suggested that all of the lost atoms were converted to Feshbach molecules. We were surprised at the efficiency of the conversion of our Fermi gas of atoms to a Bose gas of molecules; we could easily create hundreds of thousands of Feshbach molecules. 6.2. R F S PECTROSCOPY While suggestive of molecule creation, the measured atom loss was not conclusive proof for the existence of Feshbach molecules. We thus employed a spectroscopic technique to probe the molecules. First, we created the molecules with a magneticfield ramp across the Feshbach resonance and stopped at a magnetic field Bhold . At Bhold , a 13 µs radio frequency (rf) pulse was applied to the cloud; the rf frequency was chosen so that the photon energy was near the energy splitting between the mf = −5/2 and mf = −7/2 atom states (see Fig. 24(a)). The resulting population in the mf = −7/2 state, which was initially unoccupied, was then probed selectively either by separating the spin states spatially using a strong magneticfield gradient during free expansion (Stern–Gerlach imaging) or by leaving the magnetic field high (215 G) and taking advantage of nonlinear Zeeman shifts. Figure 28(a) shows a sample rf spectrum at Bhold = 222.49 G; the resulting number of atoms in the mf = −7/2 state is plotted as a function of the fre-
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F IG . 28. Rf spectrum for an atom/Feshbach molecule mixture (Regal et al., 2003a). (a) Transfer to the mf = −7/2 states as a function of rf frequency. The left feature is the molecule dissociation spectrum and the right feature represents the transfer of atoms between mf = −5/2 and mf = −7/2. (b) Corresponding kinetic energy of the mf = −7/2 state.
quency of the rf pulse. We observe two distinct features in the spectrum. The sharp, symmetric peak is very near the expected mf = −5/2 to mf = −7/2 transition frequency for free atoms. With the Stern–Gerlach imaging we see that the total number of mf = −5/2 and mf = −7/2 atoms is constant, consistent with transfer between these two atom states. The width of this peak is defined by the Fourier width of the applied rf pulse. Nearby is a broader, asymmetric peak shifted lower in frequency. Here we find that after the rf pulse the total number of observed atoms (mf = −5/2 + −7/2) actually increases. Also, the resulting mf = −7/2 gas in this region has a significantly increased kinetic energy, which grows linearly for larger frequency shifts from the atom peak (Fig. 28(b)). The asymmetric peak corresponds to the dissociation of molecules into free mf = 7/2 and mf = −9/2 atoms. Since the applied rf pulse stimulates a transition to a lower energy Zeeman state, we expect hνrf = hν0 − Eb − E, where Eb is the binding energy of the molecule, ν0 is the atom–atom transition frequency for noninteracting atoms, and we have ignored mean-field interaction energy shifts. The remaining energy, E, must by imparted to the dissociating atom pair as kinetic energy. Two separate linear fits are applied to the kinetic energy data in
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F IG . 29. The frequency shift (ν) from the expected mf = −5/2 → −7/2 transition plotted versus magnetic field for the mf = −7/2 atoms (squares) and the molecules (circles). The solid line corresponds to a calculation of the binding energy of the molecules as a function of detuning from the Feshbach resonance (Regal et al., 2003a).
Fig. 28(b) to determine the threshold position. The slope beyond threshold for the data is 0.49 ± 0.03; this indicates that the atom pair (mf = −7/2 + mf = −9/2) does indeed receive the additional energy, E, beyond the binding energy when the molecule is dissociated. The observed line-shape of the asymmetric peak in Fig. 28(a) should depend upon the Franck–Condon factor, which gives the overlap of the molecular wavefunction with the atomic wave-function. C. Ticknor and J. Bohn calculated this multichannel Franck–Condon overlap as a function of energy. The resulting transition rate, convolved with the frequency width of the applied rf and scaled vertically, is shown as the solid line in Fig. 28(a). The agreement between theory and experiment for the dissociation spectrum is quite good. This well-resolved spectrum provides much information about the molecular wave-function. A useful discussion of the theoretical aspects of these dissociation spectra and their relation to the wave-function of the initial and final states can be found in Ref. (Chin and Julienne, 2005). In Fig. 29 we plot the magnetic-field dependence of the frequency shift ν = νrf − ν0 , which to first order should correspond to the molecular binding energy. While ν could in principle be obtained directly from the transfer spectrum (Fig. 28(a)), we use the appearance of the threshold in the energy of the mf = −7/2 cloud, as it is more clear (Fig. 28(b)). We compare the position of this energy threshold to the expected atom–atom transition frequency ν0 based upon a calibration of the magnetic-field strength. The data are consistent with a theoretical calculation of the binding energy (solid line) based upon a full coupled channels calculation with no free parameters carried out by C. Ticknor and
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F IG . 30. Dissociation of molecules with radio frequencies (Greiner et al., 2005b). (a) The atoms resulting from the dissociation have equal and opposite momenta. (b) Absorption image of a dissociated molecular cloud.
J. Bohn. This measurement of Eb accentuates the fact that these Feshbach molecules are not typical molecules. With binding energies on the order of h×100 kHz (4 × 10−10 eV) they are extremely weakly bound compared to the molecules chemists are accustomed to studying. The excellent agreement with theory in Fig. 29 left no doubt that efficient creation of Feshbach molecules is possible. In addition, our rf spectroscopy technique was extended for a variety of other measurements in paired systems. It was proposed that rf spectroscopy could be used to measure the excitation gap in a superfluid Fermi gas (Petrosyan, 1999; Törmä and Zoller, 2000); such a measurement is published in Ref. (Chin et al., 2004). Our rf spectroscopy technique was also extended to detect confinement induced molecules in a one-dimensional Fermi gas (Moritz et al., 2005). Molecule dissociation via rf spectroscopy has proven useful for giving atoms in a molecule a large relative momentum, as in Ref. (Greiner et al., 2005b). Dissociating the molecules far above threshold produces a fun absorption image. The dissociated atoms fly out in a spherical shell, and the resulting absorption image is a ring structure (Fig. 30).
6.3. U NDERSTANDING M OLECULE C ONVERSION E FFICIENCY While using magnetic-field ramps to create molecules was very successful, there were many outstanding questions about the physics of the process. For example, what parameters define the conversion efficiency from atoms to molecules? The first important parameter turns out to be the rate of the magnetic-field ramp across the resonance. If the ramp is too fast no molecules will be created because the ramp will be diabatic with respect to the atom–molecule coupling. As the ramp is made slower, however, atoms will start to be coupled into molecules. This effect is shown in Fig. 31 where we observe molecule creation through atom loss as a function of the inverse magnetic field ramp rate across the resonance, (dB/dt)−1 .
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F IG . 31. Time scale for two-body adiabaticity at a Feshbach resonance in 40 K (Regal et al., 2003a).
Theoretical predictions find that this effect can be well modelled by the Landau– Zener formula for the transition probability at a two-level crossing f = fm 1 − e−δLZ , (20) where f is the fraction of atoms converted to molecules, fm is the maximum fraction of atoms that can be converted to molecules, and δLZ = β(dB/dt)−1 is the Landau–Zener parameter (Mies et al., 2000; Góral et al., 2004). We can fit Fig. 31 with β as the fitting parameter to find in this case β ≈ 20 µs/G (Regal et al., 2003a; Hodby et al., 2005). Reference (Góral et al., 2004) predicts that β = αnwabg , where n is the atomic gas density, w is the width of the Feshbach resonance, abg is the background scattering length, and α is a proportionality constant. A study by Hodby et al. verified the linear dependence of β upon the density n, but the proportionality constant α for this expression is still under investigation (Hodby et al., 2005). Notice in Fig. 31 that even at rates a few times slower than β not all the atoms are converted to molecules. One would expect that if the atom–molecule system were in chemical equilibrium and the temperature of the molecular sample were much less than Eb /kb then 100% of the atoms should be converted to molecules. An important point to recognize for all of the experiments in this section is that we are not operating in chemical equilibrium. At the final magnetic-field value in these experiments the time scale for chemical equilibrium is significantly longer than time scales in Fig. 31, and for many of the experiments described in this chapter we work on time scales intermediate between the time scale of β and the chemical equilibrium time scale. Thus, the observed saturation in molecule conversion is important to understand. Reference (Hodby et al., 2005) studies this phenomenon both for a bosonic gas of 85 Rb and our fermionic gas.
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F IG . 32. Dependence of molecule conversion on initial T /TF of a two-component Fermi gas (Hodby et al., 2005).
Eric Cornell suggested that the saturation in molecule conversion was likely related to phase space density (or T /TF in our case) based on intuitive arguments. An adiabatic process smoothly alters the wave function describing atom pairs but does not change the occupation of states in phase space. Thus to form a molecule a pair of atoms must initially be sufficiently close in phase space that their combined wave-function can evolve smoothly into the Feshbach molecule state as the resonance is crossed. In other words one would expect a molecule to form if a pair of atoms has a relative position rrel and relative velocity vrel such that |δrrel mδvrel | < γ h,
(21)
where γ will be an experimentally determined constant. Figure 32 shows the result of a measurement of the saturation in molecule conversion (fm ) for a 40 K gas as a function of T /TF , which is monotonically related to the phase space density of the gas. We see that indeed the conversion fraction does increase as T /TF decreases (phase space density increases) with a maximum conversion at our lowest temperatures of ∼90%. The work in 85 Rb found a similar dependence. To quantify the conversion expected for the many-body problem given the two-body criterion above (Eq. (21)) an algorithm described in Ref. (Hodby et al., 2005) was developed. The line in Fig. 32 is the result of this algorithm for the best fit value of γ . We found that for the fermion data γ = 0.38 ± 0.04 and for the boson data γ = 0.44 ± 0.03, indicating that a similar process is at work in both the Fermi and Bose cases.
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F IG . 33. Determination of the position of the Feshbach resonance via molecule dissociation in a low-density Fermi gas (Regal et al., 2004b). A fit of the data to an error function reveals B0 = 202.10 ± 0.07 G, where the uncertainty is given by the 10–90% width.
6.4. A P RECISE M EASUREMENT OF B0 A precise determination of the magnetic-field location of the Feshbach resonance B0 is an essential ingredient for exploring the BCS-BEC crossover regime. Knowledge of the position and the width of the resonance allows a precise calculation of the interaction strength at a particular magnetic field (see Section 3). As we saw in Section 5, B0 can be measured via scattering properties of the resonance, but our most precise measurement of B0 actually comes from the study of Feshbach molecules. In particular we looked for dissociation of Feshbach molecules in a low density gas as a function of magnetic field. To determine if the molecules had been dissociated or not we probed the gas at low magnetic field; here atoms not bound in molecules can be selectively detected. Figure 33 shows the result of such a measurement. Molecules created by a slow magnetic-field ramp across the resonance were dissociated by raising the magnetic field to a value Bprobe near the Feshbach resonance (inset to Fig. 33). Note that to avoid many-body effects, we dissociated the molecules after allowing the gas to expand from the trap to much lower density. This plan also allowed us to be certain we would not create molecules in the ramp of the magnetic field to near zero field for imaging. The measured number of atoms increases sharply at B0 = 202.10 ± 0.07 G. This measurement of the resonance position agreed well with previous less precise results (Regal et al., 2003b, 2004a).5
5 Note that this measurement of the resonance position is quite similar to the molecule association result of Fig. 27. However, the disadvantages of the association method are: (1) The molecule creation must take place in a high density sample; hence many-body effects may play a role. (2) The details of the magnetic-field ramp to go to low field for imaging, in particular its initial rate, are crucial.
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7. Inelastic Collisions near a Fermionic Feshbach Resonance The elastic collisions between atoms discussed in Section 5 are often referred to as good collisions. These collisions allow rethermalization in the gas but do not change the internal state of the atoms or molecules. In atomic gas experiments a constant concern is inelastic collisions between atoms, often referred to as bad collisions. In these collisions the products often are particles in lowerenergy internal states. The difference in energy between the incoming particles and the products of the collision must be carried away in the form of kinetic energy. When this energy difference is large compared to other energy scales such as the trap depth particles can be lost from the trap and significant heating of the sample can occur. Near a Feshbach resonance inelastic collisions can be enhanced along with the elastic collisions. To accomplish the work in this chapter we spent a large fraction of our time understanding the inelastic processes near 40 K Feshbach resonances and designing experiments that minimize the effect of inelastic collisions. In this section we will discuss the inelastic collisions near a 40 K Feshbach resonance and present measurements of relevant inelastic collision rates. We observe clear evidence of inelastic processes near the fermionic Feshbach resonance but find that despite these inelastic processes the lifetime of the sample is long enough to study BCS-BEC crossover physics.
7.1. E XPECTED I NELASTIC D ECAY P ROCESSES Let us first consider the stability of free fermionic atoms on the BCS side of a fermionic Feshbach resonance. In particular consider the Feshbach resonance between the |f, mf = |9/2, −9/2 and |9/2, −7/2 states that is used for many of the experiments in 40 K. Since these are the two lowest energy states of 40 K the only inelastic collision involving two of these fermions that is energetically favorable is |9/2, −9/2 + |9/2, −7/2 → |9/2, −9/2 + |9/2, −9/2. This process however is forbidden due to the fermionic nature of the particles (Bohn, 2000). Thus, any inelastic collision with these states must involve at least three fermions. A three-body inelastic collision in a two-component Fermi gas with components X and Y would take the form X + X + Y → X + (XY )− .
(22)
Here the subscript − represents a lower-energy molecular state. Such lowerenergy molecular states are always present in these atomic gas systems as there are many vibrational levels of the interatomic potential. To conserve energy
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F IG . 34. Particles involved in inelastic collisions in a Feshbach molecule/atom mixture. (a) Eq. (23); (b) Eq. (24).
and momentum in the collision the products X and (XY )− carry away the binding energy of the (XY )− molecule in the form of relative kinetic energy. Theory predicts that this three-body collision process will be suppressed for swave interactions between fermions because it requires that two identical fermions approach each other (Esry et al., 2001; Petrov, 2003; Petrov et al., 2004; D’Incao and Esry, 2005). However, while the rate of this inelastic collision process is suppressed, it is not forbidden, making it an important process near a Feshbach resonance (Regal et al., 2003b; Suno et al., 2003). As we cross the Feshbach resonance to the BEC side with a cold 40 K Fermi gas we want to consider the stability of a mixture of fermionic atoms and Feshbach molecules. An isolated Feshbach molecule for the mf = −7/2, −9/2 40 K resonance will be stable for the same reason as the two fermion mixture is stable. Note that the case in which the two atoms in the molecule are not in the lowest energy internal states, such as molecules created using 85 Rb, is quite different (Donley et al., 2002; Hodby et al., 2005). These 85 Rb Feshbach molecules will spontaneously dissociate as observed in Ref. (Thompson et al., 2005). For our 40 K molecules however we again expect that any decay processes will require more than two fermions, for example, (Petrov et al., 2004; D’Incao and Esry, 2006) X + XY → X + (XY )− ,
(23)
XY + XY → XY + (XY )− .
(24)
The first process is reminiscent of Eq. (22) above. These processes are often referred to as collisional quenching of vibrations (Balakrishnan et al., 1998; Forrey et al., 1999; Soldán et al., 2002). Again, we expect some suppression of these decay channels due to Fermi statistics since two identical fermions must approach each other, as shown schematically in Fig. 34.
7.2. L IFETIME OF F ESHBACH M OLECULES In this section we present experimental data on the stability of a mixture of atoms and Feshbach molecules. To obtain these data we created a molecule sample at
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F IG . 35. Feshbach molecule loss rate as a function of the atom–atom scattering length near a Feshbach resonance in 40 K (Regal et al., 2004a). N here is the number of molecules. The line is a fit of the closed circles (•) to a power law. The open circles (◦) are data for which the pair size expected from two-body theory is larger than the interparticle spacing.
the mf = −7/2, −9/2 Feshbach resonance in which typically 50% of original atom gas was converted to molecules. We then measured the molecule number as a function of time while holding the molecule/atom mixture in a relatively shallow optical dipole trap (Regal et al., 2004a). Figure 35 shows the result of this measurement at a variety of magnetic fields on the BEC side of the Feshbach resonance. The plot shows N˙ /N versus the atom–atom scattering length a. Here N is the number of molecules and N˙ is the initial linear decay rate. We find that far from resonance the molecules decay quickly, but the decay rate changes by orders of magnitude as the Feshbach resonance is approached. Physically this effect is partially related to the overlap between the wave-functions of the XY molecule and the (XY )− molecule. As the Feshbach resonance is approached the XY molecules become extremely weakly bound and quite large; hence, they have less overlap with the small (XY )− molecules. A scaling law for the dependence of the molecule decay rate upon the atom– atom scattering length a was found in (Petrov et al., 2004) and later in (D’Incao and Esry, 2005) for both the processes of Eqs. (23) and (24) schematized in Fig. 34. The scaling law is found by solving the full few-body problem in the limit where the molecules are smaller than the interparticle spacing, yet a r0 . Physical effects important to the result are the Fermi statistics and the wave-function overlap. The prediction for Eq. (23) (atom–molecule collisions) is that the decay rate should scale with a −3.33 and for Eq. (24) (molecule–molecule collisions) with a −2.55 . Since our measurement was carried out with thermal molecules the density of the molecule gas remains approximately constant over the a = 1000a0 to 3000a0 range. (The peak atom density in one spin state in the weakly interacting regime was n0pk = 7.5 × 1012 cm−3 .) Thus, we can measure the power law by fitting the data in Fig. 35 to the functional form Ca −p , where C and p are constants. We fit
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only points for which the interatomic spacing at the peak of the cloud is larger than the expected size of a two-body molecule, a/2. We find p = 2.3 ± 0.4, consistent with the predicted power law for molecule–molecule collisions. A similar power law was observed in a gas of Li2 molecules at the 834 G Feshbach resonance (Zhang et al., 2005). In general we find that the lifetime of the molecules is surprisingly long near the Feshbach resonance. The molecule lifetime for magnetic fields at which a > 3000a0 is greater than 100 ms. This is much longer than lifetimes observed in bosonic systems for similar densities and internal states (Xu et al., 2003; Dürr et al., 2003). 100 ms is actually a long time compared to many other time scales in our Fermi gas—for example, the time scale for two-body adiabaticity, the mean time between elastic collisions, and the radial trap period. This comparison suggested that it would indeed be possible to study BCS-BEC crossover physics using atomic 40 K gases. 7.3. T HREE -B ODY R ECOMBINATION We also observe inelastic decay of fermionic atoms on the BCS side of the Feshbach resonance, where the decay is due to Eq. (22) (Regal et al., 2003b). These collisions cause both particle loss and heating. The heating can result from a combination of two processes: First, the density dependence of the three-body process results in preferential loss in high density regions of the cloud (Weber et al., 2003). Second, the products of the inelastic collision, which have kinetic energies on order the binding energy of the XY− molecules, can collide with other particles on the way out of the sample in a hydrodynamic gas causing transfer of energy to the gas. We have found that one relevant measure of the effect of inelastic decay processes on our ability to study BCS-BEC crossover physics is the heating of our Fermi gas during magnetic-field ramps that are sufficiently slow to be adiabatic compared to many-body time scales (see Section 8). We performed an experiment in which we approached the Feshbach resonance at rate of (6 ms/G)−1 , waited 1 ms, and then ramped back at the same slow rate to the weakly interacting regime. The result of this experiment is shown in Fig. 36. If we start with a cloud initially at T /TF = 0.10, T /TF upon return increases by less than 10% for a ramp to 1/kF0 a = 0 (yet by 80% for a ramp to 1/kF0 a = 0.5). (kF0 is the Fermi wave-vector measured in the weakly interacting regime.) For this measurement the peak density in one spin state was npk = 1.2 × 1013 cm−3 . 7.4. C OMPARISON OF 40 K AND 6 L I In both 40 K and 6 Li gases the scaling law of the decay rate as a function of molecule size is the same (Regal et al., 2004a; Zhang et al., 2005). However,
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F IG . 36. T /TF measured in the weakly interacting regime as a function of the final magnetic field in an adiabatic ramp towards the Feshbach resonance and back (Regal et al., 2005). The magnetic field is represented through the dimensionless parameter 1/kF0 a.
comparison of the absolute decay rates for similar densities and a ≈ 5000a0 shows that the inelastic decay of 40 K occurs more than two orders of magnitude faster than at the 6 Li Feshbach resonance at 834 G (Bartenstein et al., 2004b; Zhang et al., 2005; Bartenstein et al., 2005). This difference has so far not been fully explained theoretically. However, it must be related in some way to the difference in the full three-body potentials for 40 K versus 6 Li atoms. This difference in lifetime between the two atomic species affects how experiments in the BCSBEC crossover are approached in 40 K versus 6 Li. For example, in our experiments we obtain the coldest gases in 40 K by evaporating a Fermi gas in the weakly interacting regime where inelastic decay is negligible. Experimenters using 6 Li, however, obtain their coldest gases through evaporation near the Feshbach resonance or slightly to the molecular side of the resonance (Chin and Grimm, 2004; Jochim et al., 2003b). In our discussions thus far we have only considered inelastic collisions in which the final state is a more deeply bound molecule. In the 6 Li systems experimenters make use of inelastic collisions that interconvert atoms on the repulsive side of the Feshbach resonance and Feshbach molecules (Chin and Grimm, 2004; Jochim et al., 2003b). The rate of these inelastic collisions in the 6 Li system is faster than inelastic decay to more deeply bound molecules, and the binding energy h¯ 2 of the Feshbach molecules Eb ≈ ma 2 is small enough to not cause significant problems in the cooling process. Hence, they typically start with a hot gas of atoms on the BEC side of the resonance and after evaporative cooling observe a pure sample of Feshbach molecules (Jochim et al., 2003a, 2003b). Thus, in contrast to the 40 K experiments in this chapter, 6 Li experiments can operate with chemical equilibrium between atoms and Feshbach molecules.
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8. Creating Condensates from a Fermi Gas of Atoms So far we have considered the normal state of the Feshbach resonance/Fermi gas system and found that it has all the elements necessary to study BCS-BEC crossover physics. However, the true test of whether we could access BCS-BEC crossover physics with our atomic gas would be to observe a phase transition. The phase transition could be distinguished through observation of the onset of either superfluid behavior or condensation. Due to the linked nature of these phenomena in massive systems one would necessarily imply the other. Just as with alkali BEC in 87 Rb and 23 Na (Fig. 3), the observable of choice for the first experiments to observe this phase transition in the BCS-BEC crossover was condensation. In this section we will discuss how we were able to show condensation of fermionic atom pairs in the BCS-BEC crossover regime. This demonstration relied heavily upon our previous knowledge of the normal state of a Fermi gas at a Feshbach resonance. First we present our observation of condensation of Feshbach molecules to create one of the first molecular BECs (Greiner et al., 2003). This work led the way to observation of condensation of fermionic atom pairs in the crossover regime. Then we present a phase diagram of the BCS-BEC crossover regime attained through measurements of condensate fraction. 8.1. E MERGENCE OF A M OLECULAR C ONDENSATE FROM A F ERMI G AS OF ATOMS We have seen that a Feshbach resonance can be used to create a large number of ultracold molecules starting with a Fermi gas of atoms. After observing that these molecules can be long lived, the creation of a BEC from these bosonic Feshbach molecules was an obvious goal. Previously we had created molecules by applying a magnetic-field ramp just slow enough to be two-body adiabatic; in the experiments here the idea is to apply a magnetic-field ramp that is not only two-body adiabatic, but also slow with respect to the many-body physics timescale (manybody adiabatic). With such a magnetic-field ramp across the Feshbach resonance the entropy of the original quantum Fermi gas should be conserved (Cubizolles et al., 2003; Carr et al., 2004). For an initial atom gas with a sufficiently low T /TF the result should be a low entropy sample of bosonic molecules, which for a low enough entropy is a BEC. To pursue this idea experimentally we again used the Feshbach resonance between the mf = −9/2 and mf = −7/2 spin states starting with a Fermi gas at temperatures below quantum degeneracy. We applied a time-dependent ramp of the magnetic field starting above the Feshbach resonance and ending below the resonance. The magnetic field was typically ramped in 7 ms from B = 202.78 G to either B = 201.54 G or B = 201.67 G, where a sample of 78% to 88% Feshbach molecules was observed. A critical element of this experiment is that the
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lifetime of the Feshbach molecules can be much longer than the typical collision time in the gas and longer than the radial trapping period (see Section 7). The relatively long molecule lifetime near the Feshbach resonance allows the atom/molecule mixture to achieve thermal equilibrium during the magnetic-field ramp. Note however that since the optical trap is strongly anisotropic (νr /νz ≈ 80) we may attain only local equilibrium in the axial direction. To study the resulting atom–molecule gas mixture after the magnetic-field ramp, we measured the momentum distribution of both the molecules and the residual atoms using time-of-flight absorption imaging. After typically 10 to 20 ms of expansion we applied a radio frequency (rf) pulse that dissociated the molecules into free atoms in the mf = −5/2 and mf = −9/2 spin states (Regal et al., 2003a). Immediately after this rf dissociation pulse we took a spin-selective absorption image. The rf pulse had a duration of 140 µs and was detuned 50 kHz beyond the molecule dissociation threshold where it did not affect the residual unpaired atoms in the mf = −7/2 state. We selectively detected the expanded molecule cloud by imaging atoms transferred into the previously unoccupied mf = −5/2 state by the rf dissociation pulse. Alternatively we could image only the expanded atom cloud by detecting atoms in the mf = −7/2 spin state. Close to the Feshbach resonance, the atoms and molecules are strongly interacting with effectively repulsive interactions. The scattering length for atom– molecule and molecule–molecule collisions close to the Feshbach resonance was calculated by Petrov et al. to be 1.2a and 0.6a, respectively, where a is the atom– atom scattering length (Petrov et al., 2004). During the initial stage of expansion the positive interaction energy is converted into additional kinetic energy of the expanding cloud. Therefore the measured momentum distribution is very different from the original momentum distribution of the trapped gas. In order to reduce the effect of these interactions on the molecule time-of-flight images we used the magnetic-field Feshbach resonance to control the interparticle interaction strength during expansion. We could significantly reduce the momentum kick due to the interaction energy by rapidly changing the magnetic field before we switched off the optical trap for expansion. The field was typically lowered by 4 G in 10 µs. At this magnetic field farther away from the resonance the atom–atom scattering length a is reduced to ∼500a0 . We found that this magnetic-field jump resulted in a loss of typically 50% of the molecules, which we attribute to the reduced molecule lifetime away from the Feshbach resonance. To attempt to observe condensation of molecules we monitored the molecule momentum distribution while varying the temperature of the initial weakly interacting Fermi gas, (T /TF )0 . Below (T /TF )0 of 0.17 we observed the sudden onset of a pronounced bimodal momentum distribution. Figure 37 shows such a bimodal distribution for an experiment starting with an initial temperature of 0.1TF ; for comparison we also show the resulting molecule momentum distribution for an experiment starting at 0.19TF . The bimodal mo-
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F IG . 37. Momentum distribution of a molecule sample created by applying a magnetic-field ramp to an atomic Fermi gas with an initial temperature of 0.19TF (0.1TF ) for the left (right) picture (Greiner et al., 2003). In the right sample the molecules form a Bose–Einstein condensate. The lines illustrate the result of bimodal surface fits.
F IG . 38. Molecular condensate fraction N0 /N versus the scaled temperature T /Tc (Greiner et al., 2003). The temperature of the molecules is varied by changing the initial temperature of the fermionic atoms prior to the formation of the molecules, yet measured through the momentum distribution of the molecular thermal gas.
mentum distribution is a striking indication that the cloud of weakly bound molecules has undergone a phase transition to a BEC (Anderson et al., 1995; Davis et al., 1995). To obtain thermodynamic information about the molecule cloud we fit the momentum distribution with a two-component fit. The fit function is the sum of an inverted parabola describing the Thomas–Fermi momentum distribution of a bosonic condensate and a Gaussian momentum distribution describing the noncondensed component of the molecule cloud. In Fig. 38 the measured condensate fraction is plotted as a function of the fitted temperature of the molecular thermal component in units of the critical temperature for an ideal
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F IG . 39. Time scale for many-body adiabaticity (Greiner et al., 2003). We plot the fraction of condensed molecules versus the time in which the magnetic field is ramped across the Feshbach resonance from 202.78 to 201.54 G.
Bose gas Tc = 0.94(N νr2 νz )1/3 h/kB . In this calculated Tc , N is the total number of molecules measured without changing the magnetic field for the expansion. From Fig. 38 we determine an actual critical temperature for the strongly interacting molecules and for our trap geometry of 0.8 ± 0.1Tc . Such a decrease of the critical temperature relative to the ideal gas prediction is expected for a strongly interacting gas (Giorgini et al., 1996). As expected we found that the creation of a BEC of molecules requires that the Feshbach resonance be traversed sufficiently slowly to be many-body adiabatic. We expect that this many-body time scale to be determined by the time it takes atoms to collide and move in the trap. In Fig. 39 the measured condensate fraction is plotted versus the ramp time across the Feshbach resonance starting with a Fermi gas at a temperature ∼0.1TF . Our fastest ramps resulted in a much smaller condensate fraction while the largest condensate fraction appeared for a B-field ramps slower than ∼3 ms/G. In conclusion, we have discussed the creation of a BEC of weakly bound molecules starting with a gas of ultracold fermionic atoms. With a relatively slow ramp of an applied magnetic field that converts most of the fermionic atoms into bosonic molecules and with an initial atomic gas below T /TF = 0.17, we observe a molecular condensate in time-of-flight absorption images. Our experiment approaches the BEC limit, in which superfluidity occurs due to BEC of essentially local pairs whose binding energy is larger than the Fermi energy. Strikingly, our 40 K molecular condensate is not formed by any active cooling of the molecules, but rather merely by traversing the BCS-BEC crossover regime. At the same time as these experiments in 40 K, experiments using 6 Li created a similar BEC of Feshbach molecules. Their approach however was direct evaporative cooling of the Feshbach molecules (Jochim et al., 2003b; Zwierlein et al., 2003).
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8.2. O BSERVING C ONDENSATES IN THE C ROSSOVER To create a molecular condensate we started with a quantum Fermi gas, slowly traversed the BCS-BEC crossover regime, and ended up with a BEC of molecules. An obvious question was whether condensation also had occurred in the crossover regime that we had passed through. To answer this question we needed to overcome a number of challenges. First, we required a probe of the momentum distribution of pairs in the crossover. In the BEC limit the momentum distribution of the molecules could be measured using standard time-of-flight absorption imaging. However, this method is problematic in the crossover because the pairs depend on many-body effects and are not bound throughout expansion of the gas. Second, to prove observation of condensation in the BCS-BEC crossover regime we had to show that we were not simply seeing condensation of pairs in the two-body bound state (two-body pairs), but rather condensation of pairs requiring many-body effects to form (many-body pairs). A clear example of condensation of many-body pairs would be condensation on the BCS side of the Feshbach resonance. Here the two-body physics of the resonance no longer supports the weakly bound molecular state; hence, only many-body effects can give rise to a condensation of fermion pairs. To solve the problem of measuring the momentum distribution of pairs in the crossover we introduced a technique that takes advantage of the Feshbach resonance to pairwise project the fermionic atoms onto Feshbach molecules. We were able to probe the system by rapidly ramping the magnetic field to the BEC side of the resonance, where time-of-flight imaging could be used to measure the momentum distribution of the weakly bound molecules. The projecting magneticfield ramp was completed on a timescale that allowed molecule formation but was still too brief for particles to collide or move significantly in the trap. This is possible due to the clear separation of the two-body and many-body time scales. The timescale for many-body adiabaticity in Fig. 39 is two orders of magnitude longer than the timescale for two-body adiabaticity shown in Fig. 31. The key to the second problem, verifying condensation of many-body pairs, came from careful understanding of the two-body physics. As discussed in Section 6 we were able to precisely measure the magnetic-field position above which a two-body bound state no longer exists, B0 (Fig. 33). If we observed condensation of fermionic atom pairs at B > B0 we could be assured that these were pairs that were the result of many-body effects. To perform experiments making use of these ideas, we continued with the same experimental setup as the last section where we discussed the creation of molecular condensates. We initially prepared the ultracold two-component atom gas at a magnetic field far above the Feshbach resonance. Here the gas is not strongly interacting, and we measured (T /TF )0 through surface fits to time-of-flight images of the Fermi gas (Section 4). The field was then slowly lowered at the many-body adiabatic rate of 10 ms/G to a value Bhold near the resonance. Whereas before we
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F IG . 40. A typical magnetic-field ramp used to measure the fraction of condensed fermionic atom pairs (Regal et al., 2004b). The system is first prepared by a slow magnetic-field ramp towards the resonance to a variable position Bhold , indicated by the two-sided arrow. After a time thold the optical trap is turned off at t = 0 and the magnetic field is quickly lowered by ∼10 G. After free expansion, the molecules are imaged on the BEC side of the resonance (◦).
had considered only values of Bhold below B0 , on the BEC side, now we explore the behavior of the sample when ramping slowly to values of Bhold on either side of the Feshbach resonance. To probe the system we projected the fermionic atoms pairwise onto molecules and measured the momentum distribution of the resulting molecular gas. This projection was accomplished by rapidly lowering the magnetic field by ∼10 G at a rate of (40 µs/G)−1 while simultaneously releasing the gas from the trap (Fig. 40). This puts the gas far on the BEC side of the resonance, where it is weakly interacting. After a total of typically 17 ms of expansion the molecules were selectively detected using rf photodissociation immediately followed by spin-selective absorption imaging. To look for condensation, these absorption images were again surface-fit to a two-component function that is the sum of a Thomas–Fermi profile for a condensate and a Gaussian function for noncondensed molecules. Figures 41 and 42 present the main result of this section. In Fig. 41(a) we plot the measured condensate fraction N0 /N as a function of the magnetic-field detuning from the resonance, B = Bhold − B0 . The data in Fig. 41(a) was taken for a Fermi gas initially at T /TF = 0.08 and for two different wait times at Bhold . Condensation is observed on both the BCS (B > 0) and BEC (B < 0) sides of the resonance. We further found that the condensation that occurs on the BCS side of the Feshbach resonance is distinguished by its longer lifetime (Fig. 41(a)). An essential aspect of these measurements is the fast magnetic-field ramp that projects the fermionic atoms pairwise onto molecules. It is a potential concern that the condensation might occur during this ramp rather than at Bhold . To verify that condensation did not occur during the ramp we studied the measured condensate fraction for different magnetic-field ramp rates. Figure 41 compares the condensate fraction measured using the 40 µs/G (circles) rate to that using a ramp that was ∼7 times faster (open squares). We found that the measured con-
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F IG . 41. Measured condensate fraction as a function of detuning from the Feshbach resonance B = Bhold − B0 (Regal et al., 2004b). (a) Data for thold = 2 ms (") and thold = 30 ms (P) with an initial cloud at T /TF = 0.08. (b) Data for two different projection magnetic-field ramp rates: 40 µs/G (circles) and ∼6 µs/G (squares). The dashed lines reflect the uncertainty in the Feshbach resonance position.
densate fraction is identical for these two very different rates, indicating that this measurement constitutes a projection with respect to the many-body physics. The validity of the magnetic-field projection technique was also explored in studies of a 6 Li gas at MIT. Researchers there first reproduced the observation of condensation using the pairwise projection technique with a 6 Li gas (Zwierlein et al., 2004). They also monitored the delayed response of the many-body system after modulating the interaction strength (Zwierlein et al., 2005b). They found that the response time of the many-body system was slow compared to the rate of the rapid projection magnetic-field ramp. There have been a number of theoretical papers on the subject of the pairwise projection technique for measuring condensate fraction in the crossover (Avdeenkov and Bohn, 2005; Altman and Vishwanath, 2005; Perali et al., 2005). Work thus far has established that observation of condensation of molecules following a rapid projection ramp indicates a pre-existence of condensation of fermionic atom pairs before the projection ramp. To summarize, in this section we have introduced a method for probing the momentum distribution of fermionic atom pairs and employed this technique to observe condensation near a Feshbach resonance. By projecting the system onto a molecule gas, we observe condensation of fermionic pairs as a function of the magnetic-field detuning from the resonance as shown in Fig. 42. While Fig. 42 is
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F IG . 42. Time-of-flight images showing condensation of fermionic atom pairs. The images, taken after the projection of the fermionic system onto a molecule gas, are shown for B = 0.12, 0.25, and 0.55 G (right to left) on the BCS side of the resonance. The original atom cloud starts at (T /TF )0 = 0.07. 3D artistry is courtesy of Markus Greiner.
reminiscent of Fig. 3 where condensation was observed a function of T /Tc , note that the condensate here actually appears as a function of interaction strength, not temperature.
8.3. M EASUREMENT OF A P HASE D IAGRAM In addition to varying B and measuring the condensate fraction, we can also vary the initial temperature of the Fermi gas. Figure 43 is a phase diagram created from data varying both B and (T /TF )0 . B is converted to the dimensionless parameter 1/kF0 a, where a is calculated directly from B through Eq. (6) and kF0 is extracted from the weakly interacting Fermi gas. The shades of gray represent the measured condensate fraction using the projection technique. The boundary between the black and light gray regions shows where the phase transition occurs in the BCS-BEC crossover. On the BCS side of the resonance we find the condensate forms for higher initial T /TF as B decreases, as expected based upon BCS-BEC crossover theories (Section 2). These data lie precisely in the regime that is neither described by BCS nor by BEC physics, −1 < 1/kF a < 1 (see Fig. 6). The condensed pairs in these experiments are expected to be pairs with some qualities of diatomic molecules and some qualities of Cooper pairs. Thus, these experiments realize a phase transition in the BCS-BEC crossover regime and initiate experimental study of this physics. Finally we note that, as in our previous measurements performed in the BEC limit, the measured condensate fraction in Fig. 43 always remains well below one (Greiner et al., 2003). This is not observed in the case of 6 Li experiments
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F IG . 43. Transition to condensation as a function of both interaction and T /TF (Regal et al., 2004b). The data for this phase diagram are collected with the same procedure as shown in Fig. 40 with thold ≈ 2 ms. The contour plot is obtained using a Renka–Cline interpolation of approximately 200 distinct data points.
(Zwierlein et al., 2004), suggesting that technical issues particular to 40 K may play a role. As part of our probing procedure the magnetic field is set well below the Feshbach resonance where the molecule lifetime is only on the order of milliseconds (Fig. 35) (Regal et al., 2004a; Petrov et al., 2004). This results in a measured loss of 50% of the molecules and may also reduce the measured condensate fraction. Further, temperature measurements using the surface-fitting procedure become less accurate below T /TF = 0.1 (see Section 4). Further, it is expected that the isentropic ramp from a weakly interacting Fermi gas to the crossover results in a larger value of T /TF than (T /TF )0 . The extent of this adiabatic heating is currently under theoretical investigation (Chen et al., 2005a). Density-dependent loss processes could also play a role in heating of the sample, especially as the BEC limit is approached (see Section 7). Whether any or all of these processes play a role in the small condensate fraction is a subject of current study in our group.
9. The Momentum Distribution of a Fermi Gas in the Crossover The measurements of condensation in the crossover described in the last section probed the phase coherence between fermionic atom pairs. As discussed in Sec-
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tion 2, it is only in the BCS limit that the pairs are always coherent, and when the interaction becomes large so-called preformed pairs are predicted to exist above the phase transition temperature Tc (Randeria, 1995; Chen et al., 2005b). To verify such theories it is important that techniques for detecting pairing be developed alongside studies probing the phase transition. There have been a number of probing techniques used to detect pairing including rf spectroscopy (Chin et al., 2004), magnetic-field modulation (Greiner et al., 2005a), spectroscopic probes (Partridge et al., 2005), and measurements of the atomic momentum distribution (Bourdel et al., 2003; Regal et al., 2005). This section focuses on measurements of the atomic momentum distribution in the BCS-BEC crossover using 40 K (Regal et al., 2005). The classic characteristic change of the momentum distribution of a superfluid Fermi system is a broadening of the Fermi surface compared to the ideal Fermi sea (see, for example, de Gennes (1966)). Figure 44 (inset) shows the expected momentum distribution of a homogeneous, zero-temperature Fermi system. In the BCS limit (1/kF a → −∞) the amount of broadening is small and associated with . As the interaction increases this effect grows until at unitarity (1/kF a = 0) the effect is on order of EF , and in the BEC limit (1/kF a → ∞) the momentum distribution becomes the square of the Fourier transform of the molecule wave-function (Nozieres and Schmitt-Rink, 1985). This kinetic energy increase can be interpreted as a cost of pairing. Amazingly, the total energy of the system is lowered despite this large kinetic energy increase. Here we will discuss the atom momentum distribution of a trapped Fermi gas. The case of an inhomogeneous trapped gas is more complicated than the homogeneous case. Nonetheless, in the strongly interacting regime the momentum distribution probes the pair wave-function and consequently the nature of the pairs: Small, tightly bound pairs will broaden the momentum distribution more than large, weakly bound pairs. Similarly a fully paired gas will broaden the distribution more than a gas with a small fraction of paired atoms. It is expected that this measurement will probe the pairs independent of whether they have formed a condensate. Throughout this section we will compare our data to theory of the momentum distribution of a trapped gas. This theoretical work is the result of a collaboration between our group, Murray Holland of JILA, and Stefano Giorgini, a JILA visiting fellow from Trento, Italy.
9.1. M EASURING THE M OMENTUM D ISTRIBUTION We found that the momentum distribution of a Fermi gas in the crossover could actually be measured using the standard technique of time-of-flight expansion followed by absorption imaging (Anderson et al., 1995). The key to measuring the atom momentum distribution is that the gas must expand freely without any interatomic interactions; to achieve this we quickly changed the scattering length
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F IG . 44. Theoretical column integrated momentum distributions of a trapped Fermi gas n(k) calculated from a mean-field theory at T = 0 (Viverit et al., 2004; Regal et al., 2005). The normalization is given by 2π n(k)k dk = N. The lines, in order of decreasing peak amplitude, correspond to a = 0, 1/kF0 a = −0.66, 1/kF0 a = 0, and 1/kF0 a = 0.59. (inset) Corresponding distributions for a homogeneous system.
to zero for the expansion. This was particularly convenient using 40 K because the zero crossing of the scattering length occurs only 7.8 G above the resonance. Bourdel et al. pioneered this type of measurement using a gas of 6 Li atoms at T /TF ≈ 0.6, where TF is the Fermi temperature (Bourdel et al., 2003). In this work we carried out measurements down to T /TF ≈ 0.1, where pairing becomes a significant effect and condensates have been observed (Regal et al., 2004b; Zwierlein et al., 2004). To understand what we expect for our trapped atomic system, we can predict the atomic momentum distribution using a local density approximation and the results for the homogeneous case. In the trapped gas case, in addition to the local broadening of the momentum distribution due to pairing, attractive interactions compress the density profile and thereby enlarge the overall momentum distribution. Figure 44 shows a calculation of an integrated column density from the result of a mean-field calculation at T = 0 as described in Ref. (Viverit et al., 2004). First we will discuss the atomic momentum distribution measured with a low temperature Fermi gas. We started with a weakly interacting mf = −7/2, −9/2 gas at T = 0.12TF in a trap with a radial frequency of νr = 280 Hz and νr /νz = 14.6 We then adiabatically increased the interaction strength by ramping the magnetic field at a rate of (6.5 ms/G)−1 to near the mf = −7/2, −9/2 Feshbach resonance. After a delay of 1 ms, the optical trap was switched off 6 For this measurement we introduced the use of a crossed dipole trap configuration. One beam is oriented parallel to the force of gravity (y) ˆ with a waist of wy = 200 µm, and the second beam is perpendicular to the first (ˆz) and has wz = 15 µm.
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F IG . 45. Experimental, azimuthally averaged, momentum distributions of a trapped Fermi gas at (T /TF )0 = 0.12 normalized such that the area under the curves is the same as in Fig. 44 (Regal et al., 2005). The curves, in order of decreasing peak OD, correspond to 1/kF0 a = −71, −0.66, 0, and 0.59. Error bars represent the standard deviation of the mean of averaged pixels. (inset) Curves for 1/kF0 a = −71 (top) and 0 (bottom) weighted by k 3 . The lines are the results of a fit to Eq. (16).
and simultaneously a magnetic-field ramp to a ≈ 0 (B = 209.6 G) at a rate of (2 µs/G)−1 was initiated. The rate of this magnetic-field ramp was designed to be fast compared to typical many-body timescales as determined by EhF = 90 µs. The cloud was allowed to freely expand for 12.2 ms, and then an absorption image is taken. The imaging beam propagated along zˆ and selectively probed the mf = −9/2 state. Samples of these absorption images, azimuthally averaged, are shown in Fig. 45 for various values of 1/kF0 a. We observe a dramatic change in the distribution that is qualitatively very similar to the prediction in Fig. 44. Some precautions need to be taken in the quantitative comparison of Figs. 44 and 45. First, the magneticfield ramp to the Feshbach resonance, while adiabatic with respect to most time scales, is not fully adiabatic with respect to the axial trap period. Second, in the experiment an adiabatic field ramp keeps the entropy of the gas, not T /TF , constant. However, we expect the resulting change in T /TF to have a minimal effect on the distribution for 1/kF0 a < 0 (Chen et al., 2005a). Third, the theory assumes T = 0 and does not include the Hartree term, thus underestimating the broadening on the BCS side compared to a full theory (Astrakharchik et al., 2004).
9.2. E XTRACTING THE K INETIC E NERGY It is natural now to consider extracting the kinetic energy from the momentum distribution. While the momentum distribution should be universal for small momenta, for large momenta it is influenced by details of the interatomic scattering
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potential. In the extreme case of a delta potential, which we used for the calculation in Fig. 44, the momentum distribution has a tail with a 1/k 4 dependence, giving rise to a divergence of the kinetic energy. In the experiment we avoid a dependence of the measured kinetic energy on details of the interatomic potential because the magnetic-field ramp is never fast enough to access features on the order of the interaction length of the Van der Waals potential, r0 ≈ 60a0 for 40 K (Gribakin and Flambaum, 1993). Thus, the results presented here represent a universal quantity, independent of the details of the interatomic potential. Although universal in this sense, the measured kinetic energy is intrinsically dependent on the dynamics of the magnetic-field ramp, with faster ramps corresponding to higher measured energies. To exactly obtain the kinetic energy from the experimental data we would need 3 to take the second moment of the distribution, which is proportional to k OD/ kOD. As illustrated in Fig. 45 (inset) this is difficult because of the decreased signal-to-noise ratio for large k. Thus, our approach will be to apply a 2D surface fit to the image and extract an energy from the fitted function. In the limit of weak interactions the appropriate function is that for an ideal, harmonically trapped Fermi gas (Eq. (16)). Using the results of a fit to this function the kinetic energy per particle is given by Ekin =
3 mσx σy Li4 (−ζ ) , 2 t 2 Li3 (−ζ )
(25)
where t is the expansion time. Although Eq. (16) is not an accurate theoretical description of the Fermi gas in the crossover, empirically we find it to be a fitting function that describes the data reasonably well throughout the crossover, as illustrated in Fig. 45 (inset). Figure 46 shows the result of extracting Ekin as a function of 1/kF0 a; we see that Ekin more than doubles between the noninteracting regime and unitarity.
9.3. C OMPARING THE K INETIC E NERGY TO T HEORY As mentioned earlier Ekin of a trapped gas is affected both by the broadening due to pairing (Fig. 44 (inset)) and by changes in the trapped gas density profile. In 0 the BCS limit, the broadening due to pairing scales with e−π/2kF |a| and is thus exponentially small compared to density profile changes, which scale linearly with 0 using a mean-field calculation in kF0 |a|. In this limit we can calculate Ekin /Ekin the normal state (Vichi and Stringari, 1999). S. Giorgini found that to lowest order 0 = 2048 k 0 |a| + 1. We plot this result in Fig. 46 (inset) and in kF0 |a|, Ekin /Ekin 945π 2 F find good agreement for the weakly interacting regime (1/kF0 a < −1). However, caution must be taken with the apparent agreement for −0.5 < 1/kF0 a < −1. The agreement could be explained by the lack of pairing in this theory being
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F IG . 46. The measured energy Ekin of a Fermi gas at (T /TF )0 = 0.12 in the crossover nor0 = 0.25k µK (Regal et al., 2005). The dash–dot line is the expected energy ratio malized to Ekin b from a calculation only valid in the weakly interacting regime (1/kF0 a < −1). In the molecule limit (1/kF0 a > 1) we calculate the expected energy for an isolated molecule (dashed line). (inset) A focus on the weakly interacting regime.
compensated by the theory’s overestimation of the density profile change when |a| becomes larger than 1/kF . In the crossover regime where the pairs are more tightly bound, pairing provides a significant contribution to the change in the momentum distribution. At unitarity a full Monte Carlo calculation predicts the radius of the Fermi gas density profile will become (1 + β)1/4 R0 = 0.81R0 , where R0 is the Thomas–Fermi radius of a noninteracting Fermi gas (Astrakharchik et al., 2004). Just this rescal0 = 1.54. Thus, at unitarity, pairing effects on the ing would result in Ekin /Ekin momentum distribution must account for a large fraction of the measured value 0 = 2.3 ± 0.3 (Fig. 46). of Ekin /Ekin In the BEC limit and at T = 0 we expect the measured energy would be that of an isolated diatomic molecule after dissociation by the magnetic-field ramp. Provided the scattering length associated with the initial molecular state, a(t = 0), is much larger than r0 ≈ 60a0 , the wave function for the molecule is given by ψ = Ae−r/a(t=0) /r where r is the internuclear separation and A is a normalization constant. M. Holland and S. Giorgini calculated the measured energy from the solution of the Schrödinger equation with a time-dependent boundary condi1 tion on the two-particle wave-function d log(rψ) = − a(t) , where a(t) is the dr r=0 scattering length fixed by the magnetic field at time t. In Fig. 46 we show the result of this calculation for a pure gas of molecules with zero center of mass momentum and a (2 µs/G)−1 ramp rate. We find reasonable agreement considering that there is a large systematic uncertainty in the theory prediction resulting from the experimental uncertainty in the magnetic-field ramp rate and also that this two-body theory should match the data only in the BEC limit (1/kF0 a 1).
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0 normalized to the Fermi energy at F IG . 47. Temperature dependence of Ekin = Ekin − Ekin 0 a = 0. (T /TF ) is the temperature of the noninteracting gas.
A greater theoretical challenge is to calculate the expected kinetic energy for all values of 1/kF0 a in the crossover. This is a difficult problem because it requires an accurate many-body wave-function at all points in the crossover and the ability to time-evolve this wave-function. Recent work in Ref. (Chiofalo et al., 2005) has addressed this calculation using the NSR ground state (Nozieres and Schmitt-Rink, 1985). In the strongly interacting regime the result does not accurately reproduce the measured kinetic energy; this suggests that more sophisticated crossover theories are necessary.
9.4. T EMPERATURE D EPENDENCE We have also studied the dependence of the momentum distribution on (T /TF )0 . To vary the temperature of our gas, we typically recompress the optical dipole trap after evaporation and heat the cloud through resonant modulation of the optical trap strength. The experimental sequence for measuring the momentum distribution was the same as before except the ramp rate to a = 0 for expansion was ∼(8 µs/G)−1 . Figure 47 shows the measured kinetic energy change 0 . Ideally for this measurement we would have liked to Ekin = Ekin − Ekin have varied the temperature while maintaining a fixed density. However, for these data the density does vary some. For example, for the coldest dataset (circles) the peak density, for atoms in one spin state, in the weakly interacting regime is n0pk = 1.4 × 1013 cm−3 and EF0 = 0.56 µK. For the hottest dataset (stars) n0pk decreases to 6 × 1012 cm−3 and EF0 = 0.79 µK. On the BEC side of the resonance (1/kF a > 0), Ekin decreases dramatically with (T /TF )0 . Because Ekin should be proportional to the molecule fraction, this result is closely related to our recent observation that the molecule conversion efficiency scales with T /TF (Hodby et al., 2005). In the strongly interacting
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regime we also observe a decrease in Ekin with increasing (T /TF )0 . However, the decrease is not as steep as would be expected if the phase transition, which occurs at (T /TF )0 ≈ 0.15 as measured through N0 /N (Fig. 43), were responsible for the change in the momentum distribution. Instead the temperature dependence of Ekin is consistent with the expectation that the changes in the kinetic energy are caused by pairing and not condensation (Regal et al., 2004b; Chin et al., 2004; Chen et al., 2005b).
10. Conclusions and Future Directions In this chapter we have presented the story of the realization of BCS-BEC crossover physics with a gas of 40 K atoms. Experiments with 40 K and 6 Li have shown that a Fermi gas at a broad Feshbach resonance crosses a phase transition to a superfluid state and displays properties of the classic BCS-BEC crossover problem. With this system we can study the evolution from fermionic superfluidity of pairs nearly described by BCS theory to BEC of diatomic molecules. This new system provides a physical link between two descriptions of superfluid systems, BCS and BEC, that were historically thought to be distinct. The fermionic superfluids created in these gases have extremely high transition temperatures, Tc ≈ 0.1TF . In these model systems the absolute value of Tc ≈ 100 nK is very cold. However, for typical values of TF in metals the corresponding transition temperature would be above room temperature. Fully understanding this model system will perhaps contribute to the efforts to create higher transition temperature superconductors in real materials. These atomic gas systems also provide the opportunity to study aspects of quantum systems that are not typically accessible in solid state materials. For example, experiments such as those in Section 8 utilize dynamics for measurements. The ability to study the physics of the BCS-BEC crossover in real time may provide new insight into this many-body quantum system. In the immediate future there are a plethora of experiments that could be done to study BCS-BEC crossover physics with this new system. This list includes both more precise versions of previous measurements as well as experiments designed to study entirely new phenomena. More precise experiments will be vital to rigorous comparison with sophisticated crossover theories. New phenomena that could be studied include, for example, higher partial wave pairing (Regal et al., 2003b; Zhang et al., 2005; Schunck et al., 2005). This would be an important contribution due to the relevance to cuprate superconductors, which have been shown to have d-wave symmetry. Predictions have been made for the behavior of an atomic Fermi gas in the BCS-BEC crossover with unequal particle number in each spin state of the two-component gas (Mizushima et al., 2005; Sheehy and
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Radzihovsky, 2006). Such a “magnetized” system has been the topic of a number of recent experimental pursuits (Zwierlein et al., 2006). Further, quantum Fermi gases have now been studied in optical lattice potentials (Ott et al., 2004; Köhl et al., 2005); studying crossover physics in the presence of such a lattice could more closely mimic conditions in real solids.
11. Acknowledgements We are grateful to many different people for their contributions to this work. We thank our experimental co-workers Markus Greiner and Jayson Stewart, and we are indebted to Eric Cornell, Carl Wieman, and the rest of the BEC collaboration at JILA for stimulating discussions. We have benefited from collaboration with many theorists, including Murray Holland, Stefano Giorgini, Marilu Chiofalo, John Bohn, Chris Ticknor, Alexander Avdeenkov, Marzena Szyma´nska, Paul Julienne, Chris Greene, Brett Esry, Leo Radzihovsky, Kathy Levin, and Qijin Chen. Funding for this work was provided by NSF, NIST, ONR, NASA, and the Hertz foundation.
12. References Abeelen, F.A., Verhaar, B.J. (1999). Time-dependent Feshbach resonance scattering and anomalous decay of a Na Bose–Einstein condensate. Phys. Rev. Lett. 83, 1550. Allen, J.F., Misener, A.D. (1938). Flow of liquid helium II. Nature 141, 75. Altman, E., Vishwanath, A. (2005). Dynamic projection on Feshbach molecules: A probe of pairing and phase fluctuations. Phys. Rev. Lett. 95 (11), 110404. Anderson, M.H., Ensher, J.R., Matthews, M.R., Wieman, C.E., Cornell, E.A. (1995). Observation of Bose–Einstein condensation in a dilute atomic vapor. Science 269, 198. Astrakharchik, G.E., Boronat, J., Casulleras, J., Giorgini, S. (2004). Equation of state of a Fermi gas in the BEC-BCS crossover: A quantum Monte Carlo study. Phys. Rev. Lett. 93 (20), 2004040. Avdeenkov, A.V., Bohn, J.L. (2005). Pair wave functions in atomic Fermi condensates. Phys. Rev. A 71, 023609. Baker, G.A. (1999). Neutron matter model. Phys. Rev. C 60, 054311. Balakrishnan, N., Forrey, R.C., Dalgarno, A. (1998). Quenching of H2 vibrations in ultracold 3 He and 4 He collisions. Phys. Rev. Lett. 80 (15), 3224. Bardeen, J., Cooper, L.N., Schrieffer, J.R. (1957a). Microscopic theory of superconductivity. Phys. Rev. 106, 162. Bardeen, J., Cooper, L.N., Schrieffer, J.R. (1957b). Theory of superconductivity. Phys. Rev. 108, 1175. Bartenstein, M., Altmeyer, A., Riedl, S., Jochim, S., Chin, C., Hecker Denschlag, J., Grimm, R. (2004a). Collective excitations of a degenerate gas at the BEC-BCS crossover. Phys. Rev. Lett. 92, 203201. Bartenstein, M., Altmeyer, A., Riedl, S., Jochim, S., Chin, C., Hecker Denschlag, J., Grimm, R. (2004b). Crossover from a molecular Bose–Einstein condensate to a degenerate Fermi gas. Phys. Rev. Lett. 92, 120401.
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EXPERIMENTAL REALIZATION OF BCS-BEC CROSSOVER
73
Bartenstein, M., Altmeyer, A., Riedl, S., Geursen, R., Jochim, S., Chin, C., Denschlag, J.H., Grimm, R., Simoni, A., Tiesinga, E., Williams, C.J., Julienne, P.S. (2005). Precise determination of 6 Li cold collision parameters by radio-frequency spectroscopy on weakly bound molecules. Phys. Rev. Lett. 94, 103201. Bednorz, J.G., Mueller, K. (1986). Possible high-Tc superconductivity in the Ba–La–Cu–O system. Z. Physik B 64, 189. Blatt, J.M., Butler, S.T. (1954). Superfluidity of a boson gas. Phys. Rev. 96, 1149. Bohn, J.L. (2000). Cooper pairing in ultracold 40 K using Feshbach resonances. Phys. Rev. A 61, 053409. Bose, S. (1924). Plancks Gesetz und Lichtquantenhypothese. Z. Phys. 26 (3), 178. Bourdel, T., Cubizolles, J., Khaykovich, L., Magalhães, K.M.F., Kokkelmans, S.J.J.M.F., Shlyapnikov, G.V., Salomon, C. (2003). Measurement of the interaction energy near a Feshbach resonance in a 6 Li Fermi gas. Phys. Rev. Lett. 91 (2), 020402. Bourdel, T., Khaykovich, L., Cubizolles, J., Zhang, J., Chevy, F., Teichmann, M., Tarruell, L., Kokkelmans, S., Salomon, C. (2004). Experimental study of the BEC-BCS crossover region in 6 Li. Phys. Rev. Lett. 93, 050401. Braaten, E., Hammer, H.-W., Hermans, S. (2001). Nonuniversal effects in the homogeneous Bose gas. Phys. Rev. A 63, 063609. Bruun, G.M., Pethick, C.J. (2004). Effective theory of Feshbach resonances and many-body properties of Fermi gases. Phys. Rev. Lett. 92, 140404. Burke, J.P. (1999). Theoretical investigations of cold alkali atom collisions. Ph.D. thesis, University of Colorado, and references therein. Burke Jr., J.P., Greene, C.H., Bohn, J.L., Wang, H., Gould, P.L., Stwalley, W.C. (1999). Determination of 39 K scattering lengths using photoassociation spectroscopy of the Og− state. Phys. Rev. A 60, 4417. Butts, D.A., Rokhsar, D.S. (1997). Trapped Fermi gases. Phys. Rev. A 55 (6), 4346. Carlson, J., Chang, S.-Y., Pandharipande, V.R., Schmidt, K.E. (2003). Superfluid Fermi gases with large scattering length. Phys. Rev. Lett. 91, 050401. Carr, L.D., Shlyapnikov, G.V., Castin, Y. (2004). Achieving a BCS transition in an atomic Fermi gas. Phys. Rev. Lett. 92, 150404. Chen, Q., Stajic, J., Levin, K. (2005a). Thermodynamics of interacting fermions in atomic traps. Phys. Rev. Lett. 95, 260405. Chen, Q., Stajic, J., Tan, S., Levin, K. (2005b). BCS-BEC crossover: From high temperature superconductors to ultracold superfluids. Phys. Rep. 412, 1. Chin, C. (2005). A simple model of Feshbach molecules. cond-mat/0506313. Chin, C., Grimm, R. (2004). Thermal equilibrium and efficient evaporation of an ultracold atom– molecule mixture. Phys. Rev. A 69, 033612. Chin, C., Julienne, P.S. (2005). Radio-frequency transitions on weakly-bound ultracold molecules. Phys. Rev. A 71, 012713. Chin, C., Bartenstein, M., Altmeyer, A., Riedl, S., Jochim, S., Denschlag, J.H., Grimm, R. (2004). Observation of the pairing gap in a strongly interacting Fermi gas. Science 305, 1128. Chiofalo, M.L., Giorgini, S., Holland, M. (2005). Released momentum distribution of a Fermi gas in the BCS-BEC crossover. cond-mat/0512460. Chu, S., Bjorkholm, J.E., Ashkin, A., Cable, A. (1986). Experimental observation of optically trapped atoms. Phys. Rev. Lett. 57 (3), 314. Claussen, N.R., Donley, E.A., Thompson, S.T., Wieman, C.E. (2002). Microscopic dynamics in a strongly interacting Bose–Einstein condensate. Phys. Rev. Lett. 89, 010401. Combescot, R. (1999). Trapped 6 Li: A high Tc superfluid? Phys. Rev. Lett. 83, 3766. Cooper, L.N. (1956). Bound electron pairs in a degenerate Fermi gas. Phys. Rev. 104, 1189.
74
C.A. Regal and D.S. Jin
[12
Cornell, E.A., Ensher, J.R., Wieman, C.E. (1999). Experiments in dilute atomic Bose–Einstein condensation. In: Inguscio, M., Stringari, S., Wieman, C.E. (Eds.), “Proceedings of the International School of Physics—Enrico Fermi”, IOS Press, Amsterdam, p. 15. Corney, A. (1977). “Atomic and Laser Spectroscopy”. Clarendon Press, Oxford, UK. Cornish, S.L., Claussen, N.R., Roberts, J.L., Cornell, E.A., Wieman, C.E. (2000). Stable 85 Rb Bose– Einstein condensates with widely tunable interactions. Phys. Rev. Lett. 85 (9), 1795. Cubizolles, J., Bourdel, T., Kokkelmans, S.J.J.M.F., Shlyapnikov, G.V., Salomon, C. (2003). Production of long-lived ultracold Li2 molecules from a Fermi gas of atoms. Phys. Rev. Lett. 91, 240401. Dalfovo, F., Giorgini, S., Pitaevskii, L.P., Stringari, S. (1999). Theory of Bose–Einstein condensation in trapped gases. Rev. Mod. Phys. 71 (3), 463. Davis, K.B., Mewes, M.-O., Andrews, M.R., van Druten, N.J., Durfee, D.S., Kurn, D.M., Ketterle, W. (1995). Bose–Einstein condensation in a gas of sodium atoms. Phys. Rev. Lett. 75 (22), 3969. de Gennes, P.G. (1966). “Superconductivity of Metals and Alloys”. Addison–Wesley, California. DeMarco, B. (2001). Quantum behavior of an atomic Fermi gas. Ph.D. thesis, University of Colorado. DeMarco, B., Jin, D.S. (1999). Onset of Fermi degeneracy in a trapped atomic gas. Science 285 (5434), 1703. DeMarco, B., Bohn, J.L., Burke Jr., J.P., Holland, M., Jin, D.S. (1999). Measurement of p-wave threshold law using evaporatively cooled fermionic atoms. Phys. Rev. Lett. 82 (21), 4208. DeMarco, B., Papp, S.B., Jin, D.S. (2001). Pauli blocking of collisions in a quantum degenerate atomic Fermi gas. Phys. Rev. Lett. 86 (24), 5409. De Palo, S., Chiofalo, M.L., Holland, M.J., Kokkelmans, S.J.J.M.F. (2004). Resonance effects on the crossover of bosonic to fermionic superfluidity. Phys. Lett. A 327, 490. Derevianko, A., Johnson, W.R., Safronova, M.S., Babb, J.F. (1999). High-precision calculations of dispersion coefficients, static dipole polarizabilities, and atom–wall interaction constants for alkalimetal atoms. Phys. Rev. Lett. 82, 3589. Dieckmann, K., Stan, C.A., Gupta, S., Hadzibabic, Z., Schunck, C.H., Ketterle, W. (2002). Decay of an ultracold fermionic lithium gas near a Feshbach resonance. Phys. Rev. Lett. 89, 203201. Diener, R.B., Ho, T.-L. (2004). The condition for universality at resonance and direct measurement of pair wave-functions using rf spectroscopy. cond-mat/0405174. D’Incao, J.P., Esry, B.D. (2005). Scattering length scaling laws for ultracold three-body collisions. Phys. Rev. Lett. 94, 213201. D’Incao, J.P., Esry, B.D. (2006). Enhancing observability of the Efimov effect in ultracold atomic gas mixtures. Phys. Rev. A 73, 030703. Donley, E.A., Claussen, N.R., Thompson, S.T., Wieman, C.E. (2002). Atom–molecule coherence in a Bose–Einstein condensate. Nature 417, 529. Dreschler, M., Zwerger, W. (1992). Crossover from BCS-superconductivity to Bose-condensation. Ann. Physik 1, 15. Dürr, S., Volz, T., Marte, A., Rempe, G. (2003). Observation of molecules produced from a Bose– Einstein condensate. Phys. Rev. Lett. 92, 020406. Eagles, D.M. (1969). Possible pairing without superconductivity at low carrier concentrations in bulk and thin-film superconducting semiconductors. Phys. Rev. 186, 456. Einstein, A. (1925). Quantentheorie des einatomigen idealen Gases. Zweite Abhandlung. Sitzungsber. Preuss. Akad. Wiss. 1925, 3. Ensher, J.R. (1998). The first experiments with Bose–Einstein condensation of 87 Rb. Ph.D. thesis, University of Colorado. Esry, B.D., Greene, C.H., Burke, J.P. (1999). Recombination of three atoms in the ultracold limit. Phys. Rev. Lett. 83, 1751. Esry, B.D., Greene, C.H., Suno, H. (2001). Threshold laws for three-body rescombination. Phys. Rev. A 65, 010705(R). Fano, U. (1961). Effects of configuration interaction on intensities and phase shifts. Phys. Rev. 124, 1866.
12]
EXPERIMENTAL REALIZATION OF BCS-BEC CROSSOVER
75
Fedichev, P.O., Reynolds, M.W., Shlyapnikov, G.V. (1996). Three-body recombination of ultracold atoms to a weakly bound s-level. Phys. Rev. Lett. 77 (14), 2921. Feshbach, H. (1962). A unified theory of nuclear reactions II. Ann. Phys. 19, 287. Forrey, R.C., Balakrishnan, N., Dalgarno, A., Haggerty, M.R., Heller, E.J. (1999). Quasiresonant energy transfer in ultracold atom–diatom collisions. Phys. Rev. Lett. 82 (13), 2657. Gehm, M.E., Hemmer, S.L., O’Hara, K.M., Thomas, J.E. (2003). Unitarity-limited elastic collision rate in a harmonically trapped Fermi gas. Phys. Rev. A 68, 011603(R). Gerton, J.M., Strekalov, D., Prodan, I., Hulet, R.G. (2000). Direct observation of growth and collapse of a Bose–Einstein condensate with attractive interactions. Nature 408, 692. Giorgini, S., Pitaevskii, L.P., Stringari, S. (1996). Condensate fraction and critical temperature of a trapped interacting Bose gas. Phys. Rev. A 54 (6), R4633. Goldwin, J., Inouye, S., Olsen, M., Newman, B., DePaola, B.D., Jin, D.S. (2004). Measurement of the interaction strength in a Bose–Fermi mixture with 87 Rb and 40 K. Phys. Rev. A 70, 021601. Góral, K., Köhler, T., Gardiner, S.A., Tiesinga, E., Julienne, P.S. (2004). Adiabatic association of ultracold molecules via magnetic-field tunable interactions. J. Phys. B 37, 3457. Granade, S.R., Gehm, M.E., O’Hara, K.M., Thomas, J.E. (2002). All-optical production of a degenerate Fermi gas. Phys. Rev. Lett. 88, 120405. Greiner, M., Mandel, O., Esslinger, T., Hänsch, T.W., Bloch, I. (2002). Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms. Nature 415, 39. Greiner, M., Regal, C.A., Jin, D.S. (2003). Emergence of a molecular Bose–Einstein condensate from a Fermi gas. Nature 426, 537. Greiner, M., Regal, C.A., Jin, D.S. (2005a). Probing the excitation spectrum of a Fermi gas in the BCS-BEC crossover regime. Phys. Rev. Lett. 94 (7), 070403. Greiner, M., Regal, C.A., Stewart, J.T., Jin, D.S. (2005b). Probing pair-correlated fermionic atoms through correlations in atom shot noise. Phys. Rev. Lett. 94 (11), 110401. Gribakin, G.F., Flambaum, V.V. (1993). Calculation of the scattering length in atomic collisions using the semiclassical approximation. Phys. Rev. A 48 (1), 546. Grimm, R., Weidemüller, M., Ovchinnikov, Y.B. (2000). Optical dipole traps for neutral atoms. Adv. At. Mol. Opt. Phys. 42, 95–170. Gupta, S., Hadzibabic, Z., Zwierlein, M.W., Stan, C.A., Dieckmann, K., Schunck, C.H., van Kempen, E.G.M., Verhaar, B.J., Ketterle, W. (2003). Radio-frequency spectroscopy of ultracold fermions. Science 300, 1723. Gupta, S., Hadzibabic, Z., Anglin, J.R., Ketterle, W. (2004). Collisions in zero temperature Fermi gases. Phys. Rev. Lett. 92 (10), 100401. Hadzibabic, Z., Stan, C.A., Dieckmann, K., Gupta, S., Zwierlein, M.W., Görlitz, A., Ketterle, W. (2002). Two species mixture of quantum degenerate Bose and Fermi gases. Phys. Rev. Lett. 88, 160401. Harber, D.M., Lewandowski, H.J., McGuirck, J.M., Cornell, E.A. (2002). Effect of cold collisions in spin coherence and resonance shifts in a magnetically trapped ultracold gas. Phys. Rev. A 66, 053616. Haussman, R. (1994). Properties of a Fermi liquid at the superfluid transition in the crossover region between BCS superconductivity and Bose–Einstein condensation. Phys. Rev. B 49, 12975. Heiselberg, H. (2001). Fermi systems with long scattering lengths. Phys. Rev. A 63, 043606. Hodby, E., Thompson, S.T., Regal, C.A., Greiner, M., Wilson, A.C., Jin, D.S., Cornell, E.A., Wieman, C.E. (2005). Production efficiency of ultracold Feshbach molecules in bosonic and fermionic systems. Phys. Rev. Lett. 94 (12), 120402. Holland, M., Kokkelmans, S.J.J.M.F., Chiofalo, M.L., Walser, R. (2001). Resonance superfluidity in a quantum degenerate Fermi gas. Phys. Rev. Lett. 87 (12), 120406. Holland, M.J., Menotti, C., Viverit, L. (2005). The role of boson–fermion correlations in the resonance theory of superfluids. cond-mat/0404234.
76
C.A. Regal and D.S. Jin
[12
Houbiers, M., Stoof, H.T.C., McAlexander, W.I., Hulet, R.G. (1998). Elastic and inelastic collisions of Li-6 atoms in magnetic and optical traps. Phys. Rev. A 57 (3), R1497. Inouye, S., Andrews, M.R., Stenger, J., Miesner, H.-J., Stamper-Kurn, D.M., Ketterle, W. (1998). Observation of Feshbach resonances in a Bose–Einstein condensate. Nature 392 (0), 151. Jackson, B., Pedri, P., Stringari, S. (2004). Collisions and expansion of an ultracold dilute Fermi gas. Europhys. Lett. 67 (4), 524. Jochim, S., Bartenstein, M., Altmeyer, A., Hendl, G., Chin, C., Hecker Denschlag, J., Grimm, R. (2003a). Pure gas of optically trapped molecules created from fermionic atoms. Phys. Rev. Lett. 91, 240402. Jochim, S., Bartenstein, M., Altmeyer, A., Hendl, G., Riedl, S., Chin, C., Hecker Denschlag, J., Grimm, R. (2003b). Bose–Einstein condensation of molecules. Science 302, 2101. Kagan, Y., Surkov, E.L., Shlyapnikov, G.V. (1997). Evolution of a Bose gas in anisotropic timedependent traps. Phys. Rev. A 55 (1), R18. Kapitza, P. (1938). Viscosity of liquid helium at temperatures below the lambda point. Nature 141, 74. Ketterle, W., Durfee, D.S., Stamper-Kurn, D.M. (1999). Making, probing and understanding Bose– Einstein condensates. In: Inguscio, M., Stringari, S., Wieman, C.E. (Eds.), “Proceedings of the International School of Physics—Enrico Fermi”, IOS Press, p. 67. Kinast, J., Hemmer, S.L., Gehm, M.E., Turlapov, A., Thomas, J.E. (2004a). Evidence for superfluidity in a resonantly interacting Fermi gas. Phys. Rev. Lett. 92, 150402. Kinast, J., Turlapov, A., Thomas, J.E. (2004b). Breakdown of hydrodynamics in the radial breathing mode of a strongly interacting Fermi gas. Phys. Rev. A 70 (5), 051401. Kinast, J., Turlapov, A., Thomas, J.E. (2005a). Damping of a unitary Fermi gas. Phys. Rev. Lett. 94, 170404. Kinast, J., Turlapov, A., Thomas, J.E., Chen, Q., Stajic, J., Levin, K. (2005b). Heat capacity of a strongly interacting Fermi gas. Science 307, 1296. Köhl, M., Moritz, H., Stöferle, T., Günter, K., Esslinger, T. (2005). Fermionic atoms in a 3D optical lattice: Observing Fermi surfaces, dynamics, and interactions. Phys. Rev. Lett. 94, 080403. Leggett, A.J. (1980). Cooper pairing in spin-polarized Fermi systems. J. Phys. C (Paris) 41, 7. Loftus, T., Regal, C.A., Ticknor, C., Bohn, J.L., Jin, D.S. (2002). Resonant control of elastic collisions in an optically trapped Fermi gas of atoms. Phys. Rev. Lett. 88 (17), 173201. London, F. (1938). On the Bose–Einstein condensation. Phys. Rev. 54 (0), 947. Marini, M., Pistolesi, F., Strinati, G.C. (1998). Evolution from BCS superconductivity to Bose condensation: Analytic results for the crossover in three dimensions. Eur. Phys. J. B 1, 151. Matthews, M.R., Anderson, B.P., Haljan, P.C., Hall, D.S., Wieman, C.E., Cornell, E.A. (1999). Vortices in a Bose–Einstein condensate. Phys. Rev. Lett. 83 (13), 2498. Menotti, C., Pedri, P., Stringari, S. (2002). Expansion of an interacting Fermi gas. Phys. Rev. Lett. 89 (25), 250402. Mies, F.H., Tiesinga, E., Julienne, P.S. (2000). Manipulation of Feshbach resonances in ultracold atomic collisions using time-dependent magnetic fields. Phys. Rev. A 61, 022721. Mizushima, T., Machida, K., Ichioka, M. (2005). Direct imaging of spatially modulated superfluid phases in atomic fermion systems. Phys. Rev. Lett. 94, 060404. Monroe, C.R., Cornell, E.A., Sackett, C.A., Myatt, C.J., Wieman, C.E. (1993). Measurement of Cs–Cs elastic scattering at T = 30 µK. Phys. Rev. Lett. 70 (4), 414. Moritz, H., Stöferle, T., Günter, K., Köhl, M., Esslinger, T. (2005). Confinement induced molecules in a 1D Fermi gas. Phys. Rev. Lett. 94, 210401. Myatt, C.J. (1997). Bose–Einstein condensation experiments in a dilute vapor of rubidium. Ph.D. thesis, University of Colorado. Nielsen, E., Macek, J.H. (1999). Low-energy recombination of identical bosons by three-body collisions. Phys. Rev. Lett. 83, 1566. Nozieres, P., Schmitt-Rink, S. (1985). Bose condensation in an attractive fermion gas: From weak to strong coupling superconductivity. J. Low-Temp. Phys. 59, 195.
12]
EXPERIMENTAL REALIZATION OF BCS-BEC CROSSOVER
77
O’Hara, K.M., Hemmer, S.L., Gehm, M.E., Granade, S.R., Thomas, J.E. (2002a). Observation of a strongly interacting degenerate Fermi gas of atoms. Science 298, 2179. O’Hara, K.M., Hemmer, S.L., Granade, S.R., Gehm, M.E., Thomas, J.E., Venturi, V., Tiesinga, E., Williams, C.J. (2002b). Measurement of the zero crossing in a Feshbach resonance of fermionic 6 Li. Phys. Rev. A 66, 041401(R). Ohashi, Y., Griffin, A. (2002). BCS-BEC crossover in a gas of Fermi atoms with a Feshbach resonance. Phys. Rev. Lett. 89, 130402. Onnes, H.K. (1911). On the sudden rate at which the resistance of mercury disappears. Akad. van Wetenschappen 14 (113), 818. Osheroff, D.D., Richardson, R.C., Lee, D.M. (1972). Evidence for a new phase of solid 3 He. Phys. Rev. Lett. 28, 885. Ospelkaus, C., Ospelkaus, S., Sengstock, K., Bongs, K. (2006). Interaction-driven dynamics of 40 K/87 Rb Fermi–Bose gas mixtures in the large particle number limit. Phys. Rev. Lett. 96, 020401. Ott, H., de Mirandes, E., Ferlaino, F., Roati, G., Modugno, G., Inguscio, M. (2004). Collisionally induced transport in periodic potentials. Phys. Rev. Lett. 92, 140405. Partridge, G.B., Strecker, K.E., Kamar, R.I., Jack, M.W., Hulet, R.G. (2005). Molecular probe of pairing in the BEC-BCS crossover. Phys. Rev. Lett. 95, 020404. Perali, A., Pieri, P., Strinati, G.C. (2005). Extracting the condensate density from projection experiments with Fermi gases. Phys. Rev. Lett. 95, 010407. Pethick, C.J., Smith, H. (2002). “Bose–Einstein Condensation in Dilute Gases”. Cambridge Univ. Press, Cambridge, UK. Petrosyan, K.G. (1999). Fermionic atom laser. JETP Lett. 70, 11. Petrov, D.S. (2003). Three-body problem in Fermi gases with short-range interparticle interaction. Phys. Rev. A 67, 010703(R). Petrov, D.S., Salomon, C., Shlyapnikov, G.V. (2004). Weakly bound dimers of fermionic atoms. Phys. Rev. Lett. 93, 090404. Phillips, W.D. (1998). Laser cooling and trapping of neutral atoms. Rev. Mod. Phys. 70 (3), 721. Raman, C., Köhl, M., Onofrio, R., Durfee, D.S., Kuklewicz, C.E., Hadzibabic, Z., Ketterle, W. (1999). Evidence for a critical velocity in a Bose–Einstein condensed gas. Phys. Rev. Lett. 83 (13), 2502. Randeria, M. (1995). Crossover from BCS theory to Bose–Einstein condensation. In: Griffin, A., Snoke, D.W., Stringari, S. (Eds.), “Bose–Einstein Condensation”, Cambridge Univ. Press, Cambridge, UK, p. 355. Randeria, M., Duan, J.M., Shieh, L.Y. (1990). Superconductivity in two-dimensional Fermi gas: Evolution from Cooper pairing to Bose condensation. Phys. Rev. B 41, 327. Ratcliff, L.B., Fish, J.L., Konowalow, D.D. (1987). Electronic transition dipole moment functions for a transitions among the twenty-six lowest-lying states of Li2 . J. Mol. Spectrosc. 122, 293–312. Regal, C.A., Jin, D.S. (2003). Measurement of positive and negative scattering lengths in a Fermi gas of atoms. Phys. Rev. Lett. 90 (23), 230404. Regal, C.A., Ticknor, C., Bohn, J.L., Jin, D.S. (2003a). Creation of ultracold molecules from a Fermi gas of atoms. Nature 424, 47. Regal, C.A., Ticknor, C., Bohn, J.L., Jin, D.S. (2003b). Tuning p-wave interactions in an ultracold Fermi gas of atoms. Phys. Rev. Lett. 90 (5), 053201. Regal, C.A., Greiner, M., Jin, D.S. (2004a). Lifetime of atom–molecule mixtures near a Feshbach resonance in 40 K. Phys. Rev. Lett. 92, 083201. Regal, C.A., Greiner, M., Jin, D.S. (2004b). Observation of resonance condensation of fermionic atom pairs. Phys. Rev. Lett. 92, 040403. Regal, C.A., Greiner, M., Giorgini, S., Holland, M., Jin, D.S. (2005). Momentum distribution of a Fermi gas of atoms in the BCS-BEC crossover. Phys. Rev. Lett. 95, 250404. Roati, G., Riboli, F., Modugno, G., Inguscio, M. (2002). Fermi–Bose quantum degenerate 40 K–87 Rb mixture with attractive interaction. Phys. Rev. Lett. 89, 1804.
78
C.A. Regal and D.S. Jin
[12
Roberts, J.L., Claussen, N.R., Burke Jr., J.P., Greene, C.H., Cornell, E.A., Wieman, C.E. (1998). Resonant magnetic field control of elastic scattering of cold 85 Rb. Phys. Rev. Lett. 81 (23), 5109. Roberts, J.L., Claussen, N.R., Cornish, S.L., Wieman, C.E. (2000). Magnetic field dependence of ultracold inelastic collisions near a Feshbach resonance. Phys. Rev. Lett. 85 (4), 728. Roberts, J.L., Burke Jr., J.P., Claussen, N.R., Cornish, S.L., Donley, E.A., Wieman, C.E. (2001a). Improved characterization of elastic scattering near a Feshbach resonance in 85 Rb. Phys. Rev. A 64, 024702. Roberts, J.L., Claussen, N.R., Cornish, S.L., Donley, E.A., Cornell, E.A., Wieman, C.E. (2001b). Controlled collapse of a Bose–Einstein condensate. Phys. Rev. Lett. 86 (19), 4211. Sakurai, J.J. (1994). “Modern Quantum Mechanics”. Addison–Wesley, Reading, MA. Schreck, F., Khaykovich, L., Corwin, K.L., Ferrari, G., Bourdel, T., Cubizolles, J., Salomon, C. (2001). Quasipure Bose–Einstein condensate immersed in a Fermi sea. Phys. Rev. Lett. 87 (8), 080403. Schunck, C.H., Zwierlein, M.W., Stan, C.A., Raupach, S.M.F., Ketterle, W., Simoni, A., Tiesinga, E., Williams, C.J., Julienne, P.S. (2005). Feshbach resonances in 6 Li. Phys. Rev. A 71, 045601. Sheehy, D.E., Radzihovsky, L. (2006). BEC-BCS crossover in magnetized Feshbach-resonantly paired superfluids. Phys. Rev. Lett. 96, 060401. Snoke, D.W., Baym, G. (1995). Introduction: Unifying themes of Bose–Einstein condensation. In: Griffin, A., Snoke, D.W., Stringari, S. (Eds.), “Bose–Einstein Condensation”, Cambridge Univ. Press, Cambridge, UK, p. 1. Sokol, P.E. (1995). Bose–Einstein condensation in liquid helium. In: Griffin, A., Snoke, D.W., Stringari, S. (Eds.), “Bose–Einstein Condensation”, Cambridge Univ. Press, Cambridge, UK, p. 51. Soldán, P., Cvita˘s, M.T., Hutson, J.M., Honvault, P., Launay, J.-M. (2002). Quantum dynamics of ultracold Na and Na2 collisions. Phys. Rev. Lett. 89 (15), 153201. Stenger, J., Inouye, S., Andrews, M.R., Miesner, H.-J., Stamper-Kurn, D.M., Ketterle, W. (1999). Strongly enhanced inelastic collisions in a Bose–Einstein condensate near Feshbach resonances. Phys. Rev. Lett. 82 (12), 2422. Stoof, H.T.C., Houbiers, M., Sackett, C.A., Hulet, R.G. (1996). Superfluidity of spin-polarized 6 Li. Phys. Rev. Lett. 76 (1), 10. Strecker, K.E., Partridge, G.B., Hulet, R.G. (2003). Conversion of an atomic Fermi gas to a long-lived molecular Bose gas. Phys. Rev. Lett. 91, 080406. Stwalley, W.C. (1976). Stability of spin-aligned hydrogen at low temperatures and high magnetic fields: New field-dependent scattering resonance and predissociations. Phys. Rev. Lett. 37, 1628– 1631. Suno, H., Esry, B.D., Greene, C.H. (2003). Recombination of three ultracold Fermionic atoms. Phys. Rev. Lett. 90 (5), 053202. Szyma´nska, M.H. (2005). Private communication. The full coupled channels calculation was done by P. Julienne. Szyma´nska, M.H., Góral, K., Köhler, T., Burnett, K. (2005). Conventional character of the BCS-BEC crossover in ultracold gases of 40 K. Phys. Rev. A 72, 013610. Thomas, J.E., Kinast, J., Turpalov, A. (2005). Virial theorem and universality in a unitary Fermi gas. Phys. Rev. Lett. 95, 120402. Thompson, S.T., Hodby, E., Wieman, C.E. (2005). Spontaneous dissociation of 85 Rb Feshbach molecules. Phys. Rev. Lett. 94, 020401. Ticknor, C. (2005). Private communication. Ticknor, C., Regal, C.A., Jin, D.S., Bohn, J.L. (2004). Multiplet structure of Feshbach resonances in nonzero partial waves. Phys. Rev. A 69 (4), 042712. Tiesinga, E., Verhaar, B.J., Stoof, H.T.C. (1993). Threshold and resonance phenomena in ultracold ground-state collisions. Phys. Rev. A 47 (5), 4114. Tilley, D.R. (1986). “Superfluidity and Superconductivity”. Adam Hilger, Ltd., Accord, MA.
12]
EXPERIMENTAL REALIZATION OF BCS-BEC CROSSOVER
79
Timmermans, E., Tommasini, P., Hussein, M., Kerman, A. (1999). Feshbach resonances in atomic Bose–Einstein condensates. Phys. Rep. 315, 199. Timmermans, E., Furuya, K., Milonni, P.W., Kerman, A.K. (2001). Prospect of creating a composite Fermi–Bose superfluid. Phys. Lett. A 285, 228. Tinkham, M. (1980). “Introduction to Superconductivity”. Krieger, Malabar, FL. Törmä, P., Zoller, P. (2000). Laser probing of atomic Cooper pairs. Phys. Rev. Lett. 85 (3), 487. Truscott, A.G., Strecker, K.E., McAlexander, W.I., Partridge, G.B., Hulet, R.G. (2001). Observation of Fermi pressure in a gas of trapped atoms. Science 291 (5513), 2570. Vichi, L., Stringari, S. (1999). Collective oscillations of an interacting trapped Fermi gas. Phys. Rev. A 60 (6), 4734. Viverit, L., Giorgini, S., Pitaevskii, L., Stringari, S. (2004). Momentum distribution of a trapped Fermi gas with large scattering length. Phys. Rev. A 69 (1), 013607. Weber, T., Herbig, J., Mark, M., Nagerl, H.-C., Grimm, R. (2003). Three-body recombination at large scattering lengths in an ultracold atomic gas. Phys. Rev. Lett. 91, 123201. Xu, K., Mukaiyama, T., Abo-Shaeer, J.R., Chin, J.K., Miller, D.E., Ketterle, W. (2003). Formation of quantum-degenerate sodium molecules. Phys. Rev. Lett. 91, 210402. Zhang, J., van Kempen, E.G.M., Bourdel, T., Khaykovich, L., Cubizolles, J., Chevy, F., Teichmann, M., Tuarruell, L., Kokkelmans, S.J.J.M.F., Salomon, C. (2005). Expansion of a lithium gas in the BEC-BCS crossover. In: Marcassa, L., Bagnato, V., Helmerson, K. (Eds.), In: “Atomic Physics”, vol. 19. American Institute of Physics, New York, p. 228. Zwierlein, M.W., Stan, C.A., Schunck, C.H., Raupach, S.M.F., Gupta, S., Hadzibabic, Z., Ketterle, W. (2003). Observation of Bose–Einstein condensation of molecules. Phys. Rev. Lett. 91, 250401. Zwierlein, M.W., Stan, C.A., Schunck, C.H., Raupach, S.M.F., Kerman, A.J., Ketterle, W. (2004). Condensation of pairs of fermionic atoms near a Feshbach resonance. Phys. Rev. Lett. 92, 120403. Zwierlein, M., Abo-Shaeer, J., Schirotzek, A., Schunck, C., Ketterle, W. (2005a). Observation of highTc superfluidity. Nature 435, 1047–1051. Zwierlein, M.W., Schunck, C.H., Stan, C.A., Raupach, S.M.F., Ketterle, W. (2005b). Formation dynamics of a fermion pair condensate. Phys. Rev. Lett. 94, 180401. Zwierlein, M.W., Schirotzek, A., Schunck, C.H., Ketterle, W. (2006). Fermionic superfluidity with imbalanced spin populations and quantum phase transition to the normal state. Sience 311, 492.
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ADVANCES IN ATOMIC, MOLECULAR AND OPTICAL PHYSICS, VOL. 54
DETERMINISTIC ATOM–LIGHT QUANTUM INTERFACE JACOB SHERSON1,2 , BRIAN JULSGAARD1,* and EUGENE S. POLZIK1 1 QUANTOP, Danish Research Foundation Center for Quantum Optics, Niels Bohr Institute,
Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark 2 Department of Physics and Astronomy, University of Aarhus, Ny Munkegade, bygning 1520,
DK-8000 Aarhus C, Denmark 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . 2. Atom–Light Interaction . . . . . . . . . . . . . . . . 2.1. Atomic Spin Operators . . . . . . . . . . . . . 2.2. Polarization States of Light . . . . . . . . . . . 2.3. Off-Resonant Coupling . . . . . . . . . . . . . 2.4. Propagation Equations . . . . . . . . . . . . . . 2.5. The Rotating Frame . . . . . . . . . . . . . . . 2.6. Two Oppositely Oriented Spin Samples . . . . 3. Quantum Information Protocols . . . . . . . . . . . . 3.1. Entanglement—Two Mode Squeezing Protocol 3.2. Quantum Memory . . . . . . . . . . . . . . . . 4. Experimental Methods . . . . . . . . . . . . . . . . . 4.1. Paraffin Coated Vapor Cells . . . . . . . . . . . 4.2. Detection of Polarization States . . . . . . . . . 4.3. Magnetic Fields . . . . . . . . . . . . . . . . . 5. Experimental Results . . . . . . . . . . . . . . . . . 5.1. Projection Noise Level . . . . . . . . . . . . . . 5.2. Decoherence . . . . . . . . . . . . . . . . . . . 5.3. Entanglement Results . . . . . . . . . . . . . . 5.4. Quantum Memory Results . . . . . . . . . . . . 6. Conclusions . . . . . . . . . . . . . . . . . . . . . . . 7. Acknowledgements . . . . . . . . . . . . . . . . . . 8. Appendices . . . . . . . . . . . . . . . . . . . . . . . A. Effect of Atomic Motion . . . . . . . . . . . . . . . . A.1. Modeling Atomic Motion . . . . . . . . . . . . A.2. Atomic Motion as a Source of Decoherence . . B. Technical Details . . . . . . . . . . . . . . . . . . . . B.1. Light Polarization and Stark Shifts . . . . . . . B.2. Influence of Laser Noise . . . . . . . . . . . . . 9. References . . . . . . . . . . . . . . . . . . . . . . .
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* Present address: Lund Institute of Technology, Box 118, S-221 00 Lund, Sweden.
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© 2007 Elsevier Inc. All rights reserved ISSN 1049-250X DOI: 10.1016/S1049-250X(06)54002-9
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Abstract The notion of an atom–light quantum interface has been developed in the past decade, to a large extent due to demands within the new field of quantum information processing and communication. A promising type of such interface using large atomic ensembles has emerged in the past several years. In this chapter we review this area of research with a special emphasis on deterministic high fidelity quantum information protocols. Two recent experiments, entanglement of distant atomic objects and quantum memory for light are described in detail.
1. Introduction Interaction of light with atoms has always been one of the most exciting areas in AMO physics. A complete quantum-mechanical approach to atom–light interactions allows one to consider aspects such as quantum state transfer between atoms and light and generation of entangled states of light and atoms. Entangled or non-separable states form the basis for Quantum Information Processing and Communication where information is encoded, processed and transferred in quantum states of light and matter (for a collection of review articles on QIPC see, for example, Zoller et al., 2005). Processing information encoded in quantum states provides parallelism and unconditional security, the two most promising properties within this field. State transfer between matter, where quantum information is processed and stored, and light, the prime long distance carrier of information, is crucial for quantum networks and other applications. A well-designed interaction between two or more quantum systems is an efficient tool for creation of a desired joint entangled state of the systems. Such interaction alone, or combined with a projective measurement on one of the systems, can lead to the creation of a non-trivial quantum state of the remaining system(s). In this review we describe recent progress in the development of a tool box for deterministic quantum state generation of light and atomic ensembles via the well known dipole interaction. This tool box forms the basis for a quantum atom–light interface capable of generating entangled states of atoms and light on demand, performing efficient quantum state exchange between light and atoms— quantum memory for light, and long distance transfer of a quantum state of atoms via teleportation and other quantum transfer protocols. A key to successful quantum state engineering is a favorable balance between efficient unitary interaction between the systems in question (light and atoms within the context of this chapter) and decoherence caused by interaction with the environment. Historically the first setting considered most promising for the light– atoms interaction was Cavity Quantum Electrodynamics (Cavity QED) where the Jaynes–Cummings type of interaction could be realized. Driven by the interest in single photon–single atom interactions (Cirac et al., 1997), this field of research has achieved spectacular successes in both optical (McKeever et al., 2004;
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Kuhn et al., 2002) and microwave (Raimond et al., 2001) domains. Through the dramatic improvements in cavity quality factors and with the use of cold atoms, coupling of single atoms to single photon cavity modes has been demonstrated. However, formidable technical challenges of the strong coupling cavity QED regime promoted searches for alternative routes to atom–light quantum interfaces. One of the most successful of these emerged in the second half of the 90s with the recognition of the fact that the use of collective quantum states of large atomic ensembles instead of single atoms can provide efficient “strong” coupling to light outside of the cavity QED regime. An intuitive explanation goes as follows. In order to be efficiently coupled to a single atom, light has to be focused and shaped to match the atomic dipole pattern. Since this is very difficult in free space, a cavity enhancing interaction of light with a specific mode is necessary. On the other hand, a collection of atoms with transverse dimensions large compared to the wavelength of light collectively couples very efficiently to any spatial light mode which matches its shape. In classical electrodynamics this is manifested by large absorption and dispersion that characterizes an atomic ensemble interacting with nearly resonant light. Luckily, the same scaling persists for some collective quantum properties of light and atoms. The first approach to coupling quantum features of light to an atomic ensemble was based on a straightforward assumption that if the input light is completely absorbed by atoms, its quantum features should be to a certain extent transferred to the atoms. The proposal of Kuzmich et al. (1997) considered a V-type atomic level scheme where excitation in the two arms of the V-transition is carried out by quantum correlated—entangled—light modes. Absorption of these modes leads to the creation of entanglement in the excited state of atoms. If excitation, for example, is performed with squeezed light, the excited state of atoms becomes spin squeezed. In case of cw excitation spontaneous emission was shown to limit the degree of spin squeezing of atoms to 50%. The experimental implementation of this proposal has been carried out for cold cesium atoms excited by squeezed light resulting in the first demonstration of a macroscopic (106 atoms) ensemble of entangled atoms (Hald et al., 1999). Efficient coupling at the quantum level also requires that no other modes of light but the desired input mode couple to the atoms. In classical physics other modes are of no interest provided they do not contain any light. In quantum physics any empty mode is in a vacuum state. Coupling to such modes leads to spontaneous emission of atoms, the major source of decoherence which a quantum state engineer has to fight. In the Cavity QED setting interaction with vacuum modes of the field is made small compared to coupling to the relevant input mode by the strong coupling provided by the cavity. How can this type of decoherence be suppressed for a large atomic ensemble in the absence of any cavity? The solution is to use for coupling with light two substates of the atomic ground state rather than a ground and an excited state as in the standard Jaynes–Cummings
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approach. The advantages of using ground state levels are multiple. Their spontaneous decay is negligible leading to long coherence times. Their energy spacing is in the radio-frequency or microwave domain which means that with the size of the ensemble smaller than the radio—or microwave—wavelength the position of each atom is irrelevant and, hence, collective coupling insensitive to atomic motion can be achieved. In order to circumvent limitations due to spontaneous emission, the idea of complete absorption has been taken further by Kozhekin et al. (2000) where a driven Raman transition involving a weak quantum mode in one arm has been shown to be capable of faithful transfer of the light quantum state onto ground state collective atomic coherence. In a parallel development, a celebrated electro-magnetically induced transparency process utilizing carefully timed Raman pulses has been proposed for quantum storage of light (Fleischhauer and Lukin, 2002). The first experiment testing these ideas has been recently carried out (Eisaman et al., 2005). In all of the above approaches to light–atom quantum interfaces the tool box has included light–atoms interactions only. A significant next step allowing implementation of atomic entanglement and quantum memory has been made by adding a quantum measurement and feedback to the picture. The two experiments which are central to this chapter, the deterministic entanglement of distant atomic objects (Julsgaard et al., 2001) and the deterministic quantum memory for light (Julsgaard et al., 2004a), have been carried out following essentially the same scenario. A pulse of light interacts with atoms via a quantum non-demolition (QND) Hamiltonian, a projective homodyning measurement is performed on the transmitted light, and the feedback conditioned on the result of the measurement is applied to atoms (Figs. 1 and 2). Early proposals for using the QND interaction for atomic quantum state engineering have been published by Sanders and Milburn (1989), Corney and Milburn (1998), and Wiseman (1998). In particular, spin squeezing in an atomic ensemble generated by a QND measurement has been studied theoretically in Kuzmich et al. (1998), Takahashi et al. (1999), Madsen and Mølmer (2004), and experimentally demonstrated in Kuzmich et al. (2000) and Geremia et al. (2004). Quantum feedback has been explored in a series of papers by Wiseman and co-authors. In particular, quantum feedback in relation to spin squeezing has been analyzed in Thomsen et al. (2002). To complete this brief introduction into atom–light quantum interface research, we wish to draw a line between the deterministic interface reviewed in this chapter and probabilistic schemes. The latter are based on generation of entanglement between light and atoms conditioned on a random event of detection of a photon in a certain mode. Spectacular experimental progress along these lines has been achieved for schemes involving a single ion (Blinov et al., 2004; Polzik, 2004), single atom (Volz et al., 2006), and atomic ensembles (Kuzmich
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et al., 2003; van der Wal et al., 2003; Chaneliere et al., 2005; Chou et al., 2005; Eisaman et al., 2005). The chapter is organized as follows. In Section 2 we introduce quantum variables for light—Stokes operators, and for atoms—collective spin operators. An effective Hamiltonian and equations of motion for the atomic and light variables are then derived. We then expand the theory to two atomic ensembles with an addition of a bias magnetic field. In Section 3 we formulate the equations of motion in the language of canonical variables for light and atoms. Two main quantum information protocols are then described theoretically: entanglement of two atomic ensembles and quantum memory for light. In both cases the treatment includes a simple decoherence model. The quantum memory protocol which we dubbed “the direct mapping protocol” represents an alternative to the teleportation method for a high-fidelity quantum state transfer from one system to another. Unlike teleportation it involves only two objects: one carrying the input state and another the target on which the input state is transferred. Section 4 describes experimental methods and the key elements of the setup including atomic cells, magnetic fields and detection of light. Application of the magneto-optical resonance method for characterization of the collective atomic spin states is outlined. Magnetic feedback used for the spin manipulation is discussed. Section 5 contains the main experimental results on atomic entanglement and quantum memory for light. The determination of the benchmark atomic quantum noise level—the projection noise—is described in detail. The effects leading to decoherence of generated atomic entangled states are summarized. In the appendices we discuss various technical aspects, such as the effect of atomic motion on quantum state generation and decoherence, the influence of polarization of light, and of technical noise of the probe laser. We conclude with a brief summary and outlook.
2. Atom–Light Interaction In this section we introduce the quantum variables for light and atoms and describe the off-resonant dipole interaction between the ground state 6S1/2 and the excited state 6P3/2 in cesium. We use spin operators for atoms and Stokes operators for light as a convenient way to describe the interaction, and present an effective Hamiltonian which describes the dynamics of the ground state spin and the light. For simplicity the Hamiltonian is specific to the cesium ground state, although, the same procedure can be applied for any other atom with a magnetically non-degenerate ground state. With this as a starting point we derive equations of motion for the light and atomic operators of a single atomic ensemble.
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F IG . 1. Schematic layout illustrating main components of the experiments. Two clouds of Cs atoms contained in paraffin coated glass cells are placed inside the magnetic shields. A light pulse consisting of orthogonally polarized strong classical and weak quantum fields passes through the cells and is detected by the detector on the right. Electronic feedback is applied to the rf magnetic coils surrounding cells to rotate the collective atomic spin. The bias magnetic field is applied to atoms in both cells. Atoms in the two cells are optically pumped as shown in the figure.
A significant next step for quantum information protocols discussed in this chapter is the transition from a single atomic ensemble to two oppositely oriented ensembles accompanied by the addition of a constant magnetic field (Fig. 1). If this field is oriented along the mean atomic spin direction it allows for achieving entanglement between two ensembles with a single pulse of light instead of two pulses required in the absence of the field. An even more important advantage is brought about by the possibility to make measurements at a rather high Zeeman frequency, thus achieving quantum limits of sensitivity with macroscopic numbers of atoms via spectral filtering of classical noise. 2.1. ATOMIC S PIN O PERATORS The ground states of cesium are characterized by its outermost electron which is in the 6S1/2 state, i.e. the orbital angular momentum L is zero. The electron spin S and thus the total electronic angular momentum J has quantum number S = J = 1/2. The nuclear spin I of cesium-133 has I = 7/2, and the coupling between the nucleus and the electron gives rise to the total angular momentum F = I + J with quantum numbers F = 3 and F = 4. It is indeed the total angular momentum F which interests us in this work since F and the magnetic quantum numbers mF define the energy levels of the ground states in the limit of low magnetic field discussed here. Furthermore, we restrict ourselves to one hyperfine level, F = 4, which is possible experimentally since the hyperfine splitting νhfs = 9.1926 GHz is large compared to typical resolutions
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of our laser systems. We choose to denote the total angular momentum of a single atom by j and for a collection of atoms (in the F = 4 state) we denote the collective total angular momentum by J, i.e. J=
N
j(i) ,
(1)
i=1
where N is the number of atoms in the F = 4 state and j(i) is the total angular momentum of the ith atom. The reason for using J and not F is conventional, and indeed, we wish to think about our spins more abstractly than just the properties of some atoms. Many of the results should be applicable in a broader sense than to a collection of cesium atoms. In our experiments the number of atoms N is of order 1012 and we will almost always aim at having all atoms polarized along one direction which we denote as the x-axis. With the x-axis as quantization axis we have mF = 4 for all atoms to a high degree of accuracy, and the collective spin Jˆx is really a macroscopic entity. With this experimental choice, we may treat the x-component of the collective spin as a classical c-number, i.e. we replace the operator Jˆx by the number Jx . The transverse spin components Jˆy and Jˆz maintain their quantum nature. They typically have zero or a small mean value. The quantum fluctuations are governed by the commutation relation and the Heisenberg uncertainty relation (with h¯ = 1) Jˆy , Jˆz = iJx (2) J2 ⇒ Var Jˆy · Var Jˆz x . 4
(3)
With 1012 atoms the quantum uncertainty of the angle of the collective spin direction is of order 10−6 .
2.2. P OLARIZATION S TATES OF L IGHT All our experiments involve narrow-band light interacting with atomic spin states, and it turns out that the polarization states of the light form a convenient language to describe the light degrees of freedom. Consider a pulse of light, or a collection of photons, propagating in the z-direction. The polarization state is well described by the Stokes operators 1 1 Sˆx = nˆ ph (x) − nˆ ph (y) = aˆ x† aˆ x − aˆ y† aˆ y , 2 2 1 1 Sˆy = nˆ ph (+45◦ ) − nˆ ph (−45◦ ) = aˆ x† aˆ y + aˆ y† aˆ x , 2 2 1 1 † aˆ x aˆ y − aˆ y† aˆ x , Sˆz = nˆ ph (σ+ ) − nˆ ph (σ− ) = 2 2i
(4)
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where nˆ ph (x) is the number of photons in the pulse with x-polarization, and so on. The Stokes operators are dimensionless as written here, they count photons. At our convenience we later break these up into temporal or spatial slices. We make the following assumption central to the experiments described below. We assume that light consists of a strong component linearly polarized along the x-direction and a much weaker component polarized in the y-plane (Fig. 1). This means that we can treat the x-mode operators aˆ x , aˆ x† , and hence Sˆx → Sx as a cnumber. Note, this is very similar to the approximation of a well polarized sample of spins in the previous section. Specifically we find (with aˆ x† = aˆ x = Ax being real) that Sˆy = Ax · (aˆ y + aˆ y† )/2 and Sˆz = Ax · (aˆ y − aˆ y† )/2i. We see that in our approximation the quantum properties of Sˆy and Sˆz are solely encoded in the y-polarized part of the light. It can be shown that the Stokes vector satisfies angular commutation relations Sˆy , Sˆz = iSx (5) S2 ⇒ Var Sˆy · Var Sˆz x 4
(pulse of light).
(6)
2.3. O FF -R ESONANT C OUPLING We consider the special case with a propagating beam of light coupled offresonantly to the 6S1/2,F =4 → 6P3/2,F =3,4,5 transitions in cesium. Then absorption effects can be neglected and all the dynamics are of a dispersive nature. Furthermore, the optically excited states can be adiabatically eliminated and an effective Hamiltonian describing the light interacting with only ground state degrees of freedom is obtained (Julsgaard, 2003; Hammerer et al., 2005a)1 . h¯ cγ λ2 eff =− Hˆ int 8AΔ 2π
L
ˆ t) + a1 · Sˆz (z, t)jˆz (z, t) a0 · φ(z,
0
ˆ t)jˆz2 (z, t) − Sˆ− (z, t)jˆ+2 (z, t) − Sˆ+ (z, t)jˆ−2 (z, t) + a2 φ(z, × ρA dz.
(7)
Here we have assumed a one-dimensional theory for the light which is sufficient for a beam cross section A that is much larger than the squared wavelength λ2 . The small letter spin operators jˆ(z, t) are dimensionless and refer to single atoms ˆ t) are taken to count the number at position z at time t. The Stokes operators S(z, of photons per unit length at position z and time t. The integration then runs over 1 Julsgaard (2003) contains a factor of two error which has been corrected in Eq. (7).
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the entire sample of length L with atomic density ρ. The factor γ in front of Eq. (7) is the natural FWHM line width of the optical transition 6S1/2 → 6P3/2 and Δ is the detuning from the F = 4 to F = 5 transition with red being positive. ˆ t) is the photon flux per unit length, Sˆ+ = Sˆx + As for the operators, φ(z, † † aˆ − and Sˆ− = Sˆx − i Sˆy = −aˆ − aˆ + are raising and lowering operators i Sˆy = −aˆ + converting σ+ -photons into σ− -photons or vice versa, and jˆ± = jˆx ± i jˆy are the usual raising and lowering operators for the spin. The parameters a0 , a1 , and a2 for the F = 4 ground state in cesium are given by 1 7 1 + + 8 → 4 (F = 4), a0 = 4 1 − 35 /Δ 1 − Δ45 /Δ 1 35 21 − − + 176 → 1, a1 = (8) 120 1 − Δ35 /Δ 1 − Δ45 /Δ 1 21 5 − + 16 → 0, a2 = 240 1 − Δ35 /Δ 1 − Δ45 /Δ where the limit is calculated for very large values of the detuning. The detunings Δ35 /2π = 452.2 MHz and Δ45 /2π = 251.0 MHz are given by the splitting in the excited state. Let us comment on the different terms in the Hamiltonian (7). The first term containing a0 just gives a Stark shift to all atoms independent of ˆ t). The second term the internal state but proportional to the photon density φ(z, containing a1 rotates the Stokes vector S and the spin vector J around the z-axis, known as Faraday rotation—we look more closely into this below. The last terms proportional to a2 are higher order couplings between the light and the atoms and since a2 is small for a sufficiently large detuning these can normally be neglected. All these terms conserve individually the z-projection of the total angular momentum of light and atoms, e.g., the Sˆ− jˆ+2 term can change a σ+ photon into a σ− photon (changing the light angular momentum along z by −2h¯ while the atoms receive 2h¯ mediated by the atomic raising operator jˆ+2 . The total angular momentum must have its z-projection invariant since the physical system is axially symmetric around the direction of light propagation (the z-axis). For us the term proportional to a1 is useful and relevant. This term represents the QND interaction. The higher order terms proportional to a2 create some minor problems which can be minimized with large detuning (this will be discussed further in Appendices B.1 and B.2). For a detailed treatment of the higher order—atomic alignment—effects we refer to Kupriyanov et al. (2005). The zeroth order term proportional to a0 produces an overall phase shift and can be omitted.
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2.4. P ROPAGATION E QUATIONS The Hamiltonian (7) is a very convenient starting point for many calculations and we now show the procedure to derive equations of motion. For the spin operators we need the Heisenberg evolution ∂ jˆi /∂t = i1h¯ [jˆi , Hˆ ] and for the Stokes operators the Maxwell–Bloch equations (∂/∂t + c · ∂/∂z)Sˆi (z, t) = i1h¯ [Sˆi (z, t), Hˆ int ], see Julsgaard (2003). For the latter we neglect retardation effects, i.e. we do not calculate dynamics on the time scale L/c of propagation across the sample. This is equivalent to setting the speed of light to infinity and hence leaving out the ∂/∂t term. If we consider only the term proportional to a1 and neglect the other we find ∂ jˆx (z, t) cγ λ2 ˆ =+ a1 Sz (z, t)jy (z, t), ∂t 8AΔ 2π ∂ jˆy (z, t) cγ λ2 ˆ =− a1 Sz (z, t)jx (z, t), ∂t 8AΔ 2π ∂ jˆz (z, t) = 0. ∂t
(9)
∂ ˆ γρ λ2 ˆ Sx (z, t) = + a1 Sy (z, t)jˆz (z, t), ∂z 8Δ 2π γρ λ2 ˆ ∂ ˆ Sy (z, t) = − a1 Sx (z, t)jˆz (z, t), ∂z 8Δ 2π ∂ ˆ Sz (z, t) = 0. ∂z
(10)
and
We observe that jˆz (z, t) and Sˆz (z, t) are individually conserved during the interaction which is also apparent from the Hamiltonian (7) since these operators commute with the a1 -term. The effect of the interaction is that the spin rotates around the z-axis with an amount proportional to Sˆz , and the Stokes vector rotates around the z-axis proportionally with jˆz . Below we assume that these rotations are small and that the dominant classical (mean) polarization vector of light and the orientation vector of the collective atomic spin stay oriented along the x-direction after the interaction. This turns out to be a very good approximation for experimentally attainable values of the interaction strength. Under this assumption the first line of the systems (9) and (10), respectively, can be omitted. Furthermore, given the QND structure of the remaining equations, we can sum over the individual atomic spins and obtain the same equation for the collective spin variables (1). In our continuous notation L we have Jˆi (t) = 0 jˆi (z, t)ρA dz. As for the light operators we concentrate on the in- and out-going parts only. Hence we define Sˆiin = cSˆi (z = 0, t) and
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Sˆiout = cSˆi (z = L, t). The multiplication by the speed of light c turns the normalization into photons per unit time instead of per unit length. With the assumption of small rotation angles, integrating Eqs. (9) and (10) over space from z = 0 to z = L leads to the following very important equations: Sˆyout (t) = Sˆyin (t) + aSx Jˆz (t),
(11)
Sˆzout (t)
(12)
=
Sˆzin (t),
∂ ˆ Jy (t) = aJx Sˆzin (t), ∂t ∂ ˆ Jz (t) = 0, ∂t
(13) (14)
2
γ λ where a = − 8AΔ 2π a1 . In and out refer to light before and after passing the atomic sample, respectively. We note from Eqs. (11) and (14) that in the case of a large interaction strength (i.e. if aSx Jˆz dominates Sˆyin ) a measurement on Sˆyout amounts to a QND measurement of Jˆz . Using off-resonant light for QND measurements of spins has also been discussed in Kuzmich et al. (1998) and Takahashi et al. (1999). Equation (13) implies that a part of the state of light is also mapped onto the atoms—we denote this as back action. This opens up the possibility of using this sort of system for quantum memory which will be the topic of Sections 3.2 and 5.4.
2.5. T HE ROTATING F RAME In the experiment a constant and homogeneous magnetic field is added in the x-direction. We discuss the experimental reason for this below. For our modeling, the magnetic field adds a term Hˆ B = Ω Jˆx to the Hamiltonian. This makes the transverse spin components precess at the Larmor frequency Ω depending on the strength of the field. Introducing the rotating frame coordinates: ˆ ˆ Jy Jy cos(Ωt) sin(Ωt) (15) = − sin(Ωt) cos(Ωt) Jˆz Jˆ z
we can easily show that Eqs. (11)–(14) transform into: Sˆyout (t) = Sˆyin (t) + aSx Jˆy (t) sin(Ωt) + Jˆz (t) cos(Ωt) , Sˆzout (t)
=
Sˆzin (t),
∂ ˆ J (t) = aJx Sˆzin (t) cos(Ωt), ∂t y ∂ ˆ J (t) = aJx Sˆzin (t) sin(Ωt). ∂t z
(16) (17) (18) (19)
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Thus, the atomic imprint on the light is encoded at the Ω-sideband instead of at the carrier frequency. The primary motivation for adding the magnetic field is the fact that lasers are generally much more quiet at high sideband frequencies than at the carrier frequency. A measurement without a magnetic field is a DC measurement for which the technical noise dominates the subtle quantum signal. Also, as the measurement time is longer than 1/Ω, Eq. (16) enables us to access both Jy and Jz at the same time. We are of course not allowed to perform nondestructive measurements on these two operators simultaneously since they are non-commuting. This is also reflected by the fact that neither Jˆy nor Jˆz are constant in Eqs. (18) and (19). Below we consider two atomic samples and we see that a QND-type interaction can be regained in this setting. 2.6. T WO O PPOSITELY O RIENTED S PIN S AMPLES Inspired by the above we now assume that we have two atomic samples with oriented spins such that Jx1 = −Jx2 ≡ Jx . We re-express the equations of motion (16)–(19) for two samples in a way which is much more convenient for the understanding of our entanglement creation and verification procedure. For two atomic samples we write equations of motion: (t) + Jˆy2 (t) sin(Ωt) Sˆyout (t) = Sˆyin (t) + aSx Jˆy1 (t) + Jˆz2 (t) cos(Ωt) , + Jˆz1 (20) ∂ ˆ (21) (t) = a(Jx1 + Jx2 )Sˆzin (t) cos(Ωt) = 0, J (t) + Jˆy2 ∂t y1 ∂ ˆ (t) = a(Jx1 + Jx2 )Sˆzin (t) sin(Ωt) = 0. J (t) + Jˆz2 (22) ∂t z1 (t)+ Jˆ (t) and Jˆ (t)+ Jˆ (t) have zero time derivative The fact that the sums Jˆy1 y2 z1 z2 relies on the assumption of opposite spins of equal magnitude. The constancy of these terms together with Eq. (20) allows us to perform QND measurements on the two sums. We note that each of the sums can be accessed by considering the two operators T
Sˆyout cos(Ωt) dt =
T
0
0
T
T
0
Sˆyout sin(Ωt) dt =
0
aSx T ˆ (t) , Jz1 (t) + Jˆz2 Sˆyin cos(Ωt) dt + 2
(23)
aSx T ˆ (t) . Jy1 (t) + Jˆy2 Sˆyin sin(Ωt) dt + 2
(24)
T T We have used the fact that 0 cos2 (Ωt) dt ≈ 0 sin2 (Ωt) dt ≈ T /2 and that T 0 cos(Ωt) sin(Ωt) dt ≈ 0. Each of the operators on the left-hand side can
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be measured simultaneously by making an Sˆy -measurement and multiplying the photo-current by cos(Ωt) or sin(Ωt) followed by integration over the duration T . (t) + Jˆ (t) and Jˆ (t) + Jˆ (t) enThe possibility to gain information about Jˆy1 y2 z1 z2 ables us to generate entangled states, the topic of Sections 3.1 and 5.3. At the same time we must lose information about some other physical variable. This is indeed (t) − Jˆ (t) and Jˆ (t) − Jˆ (t), true, the conjugate variables to these sums are Jˆz2 z1 y1 y2 respectively. These have the time evolution ∂ ˆ (25) (t) = 2aJx Sˆzin (t) cos(Ωt), Jy1 (t) − Jˆy2 ∂t ∂ ˆ (t) = 2aJx Sˆzin (t) sin(Ωt). (26) Jz1 (t) − Jˆz2 ∂t We see how noise from the input Sˆz -variable is piling up in the difference components while we are allowed to learn about the sum components via Sˆy measurements. The equations above clearly describe the physical ingredients for light and atoms. However, when transfer of a quantum state between two very different systems—light and atoms—is concerned, an isomorphic formulation, such as rendered by canonical variables, is desirable. In Section 3 we will re-write the theory in this more convenient language.
3. Quantum Information Protocols In this section we introduce in detail the two main protocols of this chapter: deterministic generation of entanglement of two macroscopic objects and deterministic quantum state mapping from one system onto another—the so-called direct mapping protocol. The latter is then applied as the quantum memory for light protocol. In order to simplify the description of the protocols we re-write equations of motion of the previous section in the language of canonical variables. This also allows for unified treatment of a single ensemble without- and two ensembles, with a constant magnetic field. For a single sample we define: Jˆy Xˆ As = √ , Jx 1 Xˆ Ls = √ Sx T
Jˆz PˆAs = √ , Jx T
Sˆy (t) dt,
0
1 PˆLs = √ Sx T
T
(27) Sˆz (t) dt.
0
For two samples we get two sets of canonical variables by defining the atomic operators: Xˆ A1 =
− Jˆ Jˆy1 y2 , √ 2Jx
(28a)
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+ Jˆ Jˆz1 √ z2 , 2Jx
Jˆ − Jˆz2 , Xˆ A2 = − z1 √ 2Jx
PˆA2 =
+ Jˆ Jˆy1 y2 √ 2Jx
[3 (28b) (28c) (28d)
and the light operators: 2 Sx T
Xˆ L1 =
2 Sx T 2 Sx T
T
Sˆz (t) cos(Ωt) dt,
(29b)
T
Sˆy (t) sin(Ωt) dt,
(29c)
Sˆz (t) sin(Ωt) dt.
(29d)
0
PˆL2 =
(29a)
0
Xˆ L2 =
Sˆy (t) cos(Ωt) dt,
0
PˆL1 =
T
2 Sx T
T 0
ˆ Pˆ operators satisfy the usual commutation relation, e.g., Each pair of X, [Xˆ L1 , PˆL1 ] = i. Equations (11)–(14) and (20)–(22) now translate into out in in = Xˆ Li + κ PˆAi , Xˆ Li
(30a)
out in = PˆLi , PˆLi
(30b)
out Xˆ Ai
=
in Xˆ Ai
out in PˆAi = PˆAi ,
in + κ PˆLi ,
(30c) (30d)
where we recall that i = 1, 2, s refer to the definitions above and not the two samples. Note, that in the case of two samples the two sets of interacting light and atomic operators are decoupled. The √parameter describing the strength of light/matter-interactions is given by κ = a Jx Sx T . The limit to strong coupling is reached around κ ≈ 1. This set of equations represents the starting point for numerous applications, which in the context of this paper are implemented in the double sample setup.
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3.1. E NTANGLEMENT—T WO M ODE S QUEEZING P ROTOCOL It follows from Eqs. (30a) and (30d) that a sufficiently precise measurement of Xˆ L of a light pulse transmitted through two atomic ensembles renders knowledge about PˆA . The measurement projects PˆA into a two-mode squeezed or Einstein– Podolsky–Rosen entangled state (Julsgaard et al., 2001). The necessary and sufficient condition for such an entangled state has been derived by Duan et al. (2000). Demonstration of entanglement is thus reduced to the fulfillment of this criterion. The criterion, which is introduced shortly, utilizes the vacuum state noise of a canonical variable, which for atomic ensembles reduces to the coherent spin state fluctuations. Hence, these fluctuations, which are also called projection noise, form an extremely important benchmark in experiments with atomic ensembles. 3.1.1. The Coherent Spin State To create entangled or squeezed states one has to generate states which exhibit less fluctuations than all equivalent classical states. The boundary occurs at the coherent spin state (CSS) in which all spins are independent realizations of a single spin oriented along a specific direction. The characteristics of this state are discussed in more detail in Section 5.1 so for now we need only to quantify the role of the coherent spin state as a boundary between classical and purely quantum mechanical states. From the canonical atomic operators a Heisenberg uncertainty relation can be formed: |[Xˆ Ai , PˆAi ]|2 1 = . Var Xˆ Ai Var PˆAi (31) 4 4 For the coherent spin state the two variances are both equal to one half, thus confirming that it is the classical state with the least possible noise. For a state to be squeezed, it has to have less noise in one of the quadratures. Since the Heisenberg uncertainty relation still has to be fulfilled it follows that the other quadrature has to exhibit excessive fluctuations, it is anti-squeezed. Entanglement is the non-local interconnection of two systems, such that it is impossible to write the total density matrix as a product of density matrices for each system. It has been shown that for effectively continuous variable systems such as the ones described so far in this paper the necessary and sufficient criterion for entanglement is (Duan et al., 2000) Var PˆA1 + Var PˆA2 < 1. (32) 3.1.2. Entanglement Generation and Verification We now turn to the actual understanding of entanglement generation and verification. We concentrate here on generation of an entangled state conditioned on the
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result of a measurement. Generation of an unconditional entangled state with the help of feedback is described in the experimental section below. For generation of conditional entanglement we perform the following steps: First the atoms are prepared in the oppositely oriented coherent states corresponding to creating the vacuum states of the two modes (Xˆ A1 , PˆA1 ) and (Xˆ A2 , PˆA2 ). Next a pulse of light called the entangling pulse is sent through the atoms and we measure the two opout and X ˆ out with outcomes A1 and B1 , respectively. These results bear erators Xˆ L1 L2 information about the atomic operators PˆA1 and PˆA2 and hence we reduce variances Var(PˆA1 ) and Var(PˆA2 ). To prove that we have an entangled state we must confirm that the variances of PˆA1 and PˆA2 fulfill the criterion (32). That is we need to know the mean values of PˆA1 and PˆA2 with a total precision better than unity. To demonstrate that, we send a second verifying pulse through the atomic out and X ˆ out with outcomes A2 and B2 . Now it is a samples again measuring Xˆ L1 L2 matter of comparing A1 with A2 and B1 with B2 . If the results are sufficiently close the state created by the first pulse was entangled. Let us be more quantitative. The interaction (30a) mapping the atomic operators PˆAi to field operators Xˆ Li is very useful for large κ, and useless if κ 1. We will describe in detail the role of κ for all values. To this end we first describe the natural way to determine κ experimentally. If we repeatedly perform the first two steps of the measurement cycle, i.e. prepare coherent states of the atomic spins, send in the first measurement pulse, and record outcomes A1 and B1 , we may deduce the statistical properties of the measurements. Theoretically we expect from (30a) 1 κ2 + . (33) 2 2 The first term in the variances is the shot noise (SN) of light. This can be measured in the absence of the interaction where κ = 0. The quantum nature of the shot noise level is confirmed by checking the linear scaling with the photon number of the pulse. The second term arises from the projection noise (PN) of atoms. Hence, we may calibrate κ 2 to be the ratio κ 2 = PN/SN of atomic projection noise to shot noise of light. Theoretically κ 2 has a linear scaling κ 2 = aJx Sx T with a macroscopic spin Jx that must be confirmed in the experiment (see Section 5.1). Next we describe how to deduce the statistical properties of the state created by the entangling pulse. Based on the measurement results A1 and B1 of this pulse we must predict the mean value of the second measurement outcome. If κ → ∞ in we ought to trust the first measurement completely since the initial noise of Xˆ Li is negligible, i.e. A2 = A1 and B2 = B1 . On the other hand, if κ = 0 we know that atoms must still be in the vacuum state such that A2 = B2 = 0. It is natural to take in general A2 = αA1 and B2 = αB1 . We need not know a theoretical value for α. The actual value can be deduced from the data. If we (i) (i) (i) (i) repeat the measurement cycle N times with outcomes A1 , B1 , A2 , and B2 , A1 = B1 = 0 and
Var(A1 ) = Var(B1 ) =
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the correct α is found by minimizing the conditional variance Var(A2 |A1 ) + Var(B2 |B1 ) (i) 1 (i) (i) 2 (i) 2 A2 − αA1 . + B2 − αB1 N −1 N
= min α
(34)
i
In order to deduce whether we fulfill the entanglement criterion (32) we compare the above to our expectation from (30a). For the verifying pulse we get 2 out out 2 in,2nd = Xˆ + κ Pˆ ent − Pˆ ent Xˆ − Xˆ Li
Li
Ai
Li
ent 1 , = + κ 2 Var PˆAi 2
Ai
(35)
in,2nd refers to the incoming light of the verifying pulse which has zero where Xˆ Li ent ˆ mean. PAi refers to the atoms after being entangled. We see that the practical entanglement criterion becomes ent ent Var(A2 |A1 ) + Var(B2 |B1 ) = 1 + κ 2 Var PˆA1 + Var PˆA2
< 1 + κ 2 = Var(A1 ) + Var(B1 ).
(36)
Simply stated, we must predict the outcomes A2 and B2 with a precision better than the statistical spreading of the outcomes A1 and B1 with the additional constraint that A1 and B1 are outcomes of quantum noise limited measurements. 3.1.3. Theoretical Entanglement Modeling We described above the experimental procedure for generating and verifying the entangled states. Here we present a simple way to derive what we expect for the ent ). mean values (i.e. the α-parameter) and for the variances Var(PˆAi We calculate directly the expected conditional variance of A2 based on A1 : out,2nd out,1st 2 − α Xˆ L1 Xˆ L1 in in,2nd in,1st ent 2 − α Xˆ L1 + κ PˆA1 − α PˆA1 = Xˆ L1 1 = 1 + α 2 + κ 2 (1 − α)2 . (37) 2 In the second step we assume that a perfect QND measurement without any dein = Pˆ ent . By taking the derivative with respect to coherence is performed, i.e. PˆA1 A1 α we obtain the theoretical minimum κ2 1 + κ2 ent ent 1 + Var PˆA2 = ⇒ Var PˆA1 1 + κ2
Var(A2 |A1 ) + Var(B2 |B1 ) = 1 +
(38)
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obtained with the α-parameter α=
κ2 . 1 + κ2
(39)
It is interesting that, in principle, any value of κ leads to the creation of entanglement. The reason for this is our prior knowledge of the input state. The atoms are in a coherent state which is as well defined in terms of variances as is possible for separable states. We need only an “infinitesimal” extra knowledge about the spin state to go into the entangled regime. It is also interesting to see what happens to the conjugate variables Xˆ Ai in the entangling process. This is governed by Eq. (30c). We do not perform measurein so all we know is that both X ˆ in and Pˆ in are in ments of the light operator PˆLi Li Ai ent ) = (1 + κ 2 )/2 and we preserve the minimum the vacuum state. Hence Var(Xˆ Ai ent ) Var(Pˆ ent ) = 1/4. uncertainty relation Var(Xˆ Ai Ai 3.1.4. Entanglement Model with Decoherence Practically our spin states decohere between the light pulses and also in the presence of the light. We model this decoherence naively by attributing the entire effect to the time interval between the two pulses, i.e. we assume that there is no decoherence in the presence of the light but instead a larger decoherence between the pulses. We may then perform an analysis in complete analogy with the above ent = β Pˆ in + 1 − β 2 Vˆ where Vˆ is a vacuum with the only difference that PˆA1 p p A1 operator admixed such that β = 0 corresponds to a complete decay to the vacuum state and β = 1 corresponds to no decoherence. Completing the analysis we find the theoretical conditional variances 1 + (1 − β 2 )κ 2 1 + κ2 ent 1 + (1 − β 2 )κ 2 ent + Var PˆA2 = ⇒ Var PˆA1 1 + κ2
Var(A2 |A1 ) + Var(B2 |B1 ) = 1 + κ 2
(40)
obtained with α-parameter α=
βκ 2 . 1 + κ2
(41)
In the limit β → 1 these results agree with (38) and (39). For β → 0 we have α → 0 (outcomes A1 and B1 are useless) and the variance approaches that of the vacuum state which is a separable state. Note that in the analysis above, the decoherence is assumed to be towards the coherent spin state. To model decoherence towards, e.g., a thermal state, the more sophisticated methods discussed in Section 3.1.5 have to be employed.
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3.1.5. Gaussian State Modeling Following the extensive study of the evolution of Gaussian states during arbitrary interactions and measurements (Giedke and Cirac, 2002; Eisert and Plenio, 2003) the development of spin squeezing in a single atomic sample and entanglement in two samples were treated in detail theoretically in Madsen and Mølmer (2004) and Sherson and Mølmer (2005). Arbitrary Gaussian states of n canonical modes are fully characterized by a 2n×1 vector, v, describing the mean values and a 2n×2n matrix, γ , describing the correlations within the atomic and light systems and the cross-correlations between these. In this formalism an interaction between light and atoms is governed by: v → Sv and γ → Sγ S T . For the Faraday interaction, the coefficients of the matrix S are easily determined from Eqs. (30). Decoherence is easily included but most importantly, there is an explicit expression for the state of the remaining modes after an arbitrary homodyne measurement on a number of the modes is performed. If the incoming pulse of light is split into a large number of segments and the interaction with atoms and subsequent measurement of each segment is treated sequentially differential equations for v(t) and γ (t) can be obtained in the limit of infinitesimal segment durations. The former is a stochastic differential equation determined by the outcome of the measurement, whereas the latter is deterministic, although non-linear because of the measurement dynamics. In Madsen and Mølmer (2004) and Sherson and Mølmer (2005) such differential equations are solved giving the time resolved dynamics in the presence of, e.g., atomic decoherence and light losses. In this way, e.g., analytic expressions for the optimal degree of spin squeezing and the degree of entanglement at arbitrary rotation frequencies (the verification and generalization of Eq. (38)) can be obtained easily.
3.2. Q UANTUM M EMORY For complete quantum memory we require (1) that the light state to be stored is supplied by a third party in an unknown state, (2) that this state is mapped onto an atomic state with a fidelity higher than the best classical fidelity, and finally (3) that the stored state can be retrieved from memory. As is described in Section 5.4 the first two criteria have been met experimentally in Julsgaard et al. (2004a) whereas the last one still remains an unsolved experimental challenge. A recently developed experimentally feasible protocol for retrieval now exists and is discussed in Section 3.2.3. 3.2.1. Direct Mapping Protocol In Eqs. (30a)–(30c) one of the light variables is mapped onto one of the atomic variables. This represents a natural starting point for a quantum memory protocol
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in which the entire light mode described by the two non-commuting variables Xˆ in and Pˆ in is faithfully stored. In the so-called “direct mapping protocol” of Julsgaard et al. (2004a) the mapping is completed by measuring the remaining light quadrature Xˆ Lout = Xˆ Lin + κ PˆAin and feeding the result back into the atomic Xˆ A with a gain of g: out in Xˆ A = Xˆ A + κ PˆLin ,
(42a)
PˆAout = PˆAout − g Xˆ Lout = PˆAin (1 − κg) − g Xˆ Lin .
(42b)
If κ = g = 1 and the initial atomic state is assumed to be a coherent state with zero mean value the mean values of both light variables are stored faithfully in the atoms. Although the initial atomic state has zero mean, it is a quantum mechanically fluctuating state, and any uncanceled atomic part increases the noise of the final state and thus degrades the mapping performance. Although this protocol works for any state, in the following we discuss storage of coherent states of light, i.e. vacuum states which are displaced by an unknown amount in phase space. For in contribution limthe storage of an arbitrary coherent light state the remaining Xˆ A its the storage fidelity to 82%. This can be remedied by initially squeezing the atomic state, in which case 100% fidelity can be reached in the limit of infinite squeezing. 3.2.2. Mapping with Decoherence Just as in the case of entanglement generation the spin states decohere. Again we can model this by a beam splitter type admixture of vacuum components right after the passage of the first light pulse. We can furthermore model light damping (e.g., reflection losses) in a similar way to obtain: in out → β Xˆ A + κ PˆLin + 1 − β 2 VXA , Xˆ A (43a) PˆAout → β − gκ ζ PˆAin − g ζ Xˆ Lin + 1 − β 2 VP A − g 1 − ζ VXL . (43b) Here 1 − ζ is the fraction of the light lost after the interaction with the atoms. √ = βκ and gF = g ζ , We see that PˆLin and Xˆ Lin are mapped with gains gBA respectively. The variances can easily be calculated to be: 2 , Var X out = 1 + gBA Var P out = 1 +
gF2 ζ
+
(44a) 2 gF2 gBA β2
− 2gF gBA .
(44b)
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3.2.3. Quantum Memory Retrieval As mentioned in Julsgaard et al. (2004a) the stored state can in principle be retrieved by inverting the roles of light and atoms in the direct mapping protocol. This would involve first an interaction between a read-out light beam with the atomic sample acting as a storage medium. According to Eq. (30a) this would map PˆA onto the light. Next Xˆ A has to be measured and feedback applied to the read-out beam according to the result of the measurement. However, since the atomic measurement requires a certain time during which the read-out pulse propagates at the speed of light, the feedback is only practical for pulse durations shorter than a microsecond. In the experiments of Julsgaard et al. (2004a) pulses of millisecond duration (∼ 300 km) are required in order to obtain a sufficiently high interaction strength, and the inverse direct mapping protocol is thus infeasible for this experimental realization. Several years ago a retrieval scheme, which did not involve measurements, but instead used two orthogonal passages of the read-out pulse was proposed (Kuzmich and Polzik, 2003). In each passage one of the atomic quadratures is mapped onto the light pulse, and in this way retrieval fidelities of up to 82% can be achieved with coherent readout light and up to 100% with perfectly squeezed readout light. Unfortunately in order to preserve the QND nature of each of the two interactions the light has to pass entirely through the atomic medium before proceeding to the second passage. This again renders the protocol inapplicable to all setups requiring “long” pulses. Recently this problem was eliminated by solving the complex dynamics arising when a light beam passes through the atomic medium along two orthogonal directions simultaneously (Sherson et al., 2005b). This protocol works with arbitrary pulse durations and the fidelity of this two-pass protocol has been calculated both for a coherent input state and for a light qubit. It was also shown that if the light is reflected back after the second passage, thus completing four passages, and a time dependent interaction strength is applied perfect retrieval can be achieved without requiring squeezed initial states. In a related proposal (Fiurášek et al., 2005) retrieval is achieved by sending two different beams through the atomic samples simultaneously, each in a direction orthogonal to the other. The fidelity of retrieval for coherent states including realistic light losses is calculated and shown to exceed the classical bound. 3.2.4. Alternative Quantum Memory Proposals In Kuzmich and Polzik (2003) various protocols which can be implemented using the QND-Faraday interaction are described in detail, including atom-atom teleportation. Instead of reviewing these, we would like to mention some recent theoretical proposals for improving the quantum memory performance. All of these break with the simple interaction scheme either by introducing a non-QND
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interaction (as was also done in Sherson et al., 2005b; Fiurášek et al., 2005) or by exploiting the fact that the atoms involved are not simple spin-1/2 atoms and thus contain more than two magnetic sublevels. In Opatrny and Fiurasek (2005) it is proposed to exploit additional atomic coherences to enhance the capacity of an atomic quantum memory. Remember that during the usual QND Faraday interaction m = 1 atomic coherences are coupled to the sideband of the light at the Larmor frequency. If, in addition, circularly polarized classical fields are present during the interaction, coupling between m = 2 coherences and a light sideband at twice the Larmor frequency is also created. In this way an additional quantum channel is added to the quantum memory, thus enhancing its capacity. By appropriately tuning the detuning of the additional coupling field two-mode squeezer and beam splitter Hamiltonians can also be realized. With the latter, a quantum memory is realized in a single passage without a requirement of measurements or prior squeezing of atoms or light. In Opatrny (2005) the author proposes to create a single cell atomic memory by optically pumping the sample into an incoherent mixture of the two extreme magnetic sublevels (m = −4 and m = 4 for the F = 4 hyperfine level of cesium). This is done with suitable linearly polarized light. A light pulse sent through the atomic sample interacts with both extreme coherences simultaneously, creating the usual QND–Faraday interaction. Apart from the prospect of making the memory more compact, this proposal would also substantially decrease the reflection losses associated with the many air-glass transitions in a two-cell setup. A main problem in this approach is that in the presence of the probe light the two coherences experience different AC-Stark shifts and thus different phase evolutions. For relevant pulse durations this effect is in fact significant. The author proposes to solve this by either decreasing the bias magnetic field or by introducing an additional AC-Stark shift by adding a light field with appropriate polarization and detuning. An atomic memory could of course also be implemented using light–atom teleportation. This could be achieved by first sending an auxiliary pulse through the two oppositely oriented atomic samples. This would entangle the light and atomic systems via the QND–Faraday interaction. After this, the auxiliary light beam should be mixed on a 50/50 beam splitter with the quantum light signal to be teleported. Measurements of Sy and Sz in the two output ports respectively and subsequent feedback to the atomic system would complete the teleportation. Unfortunately the achievable fidelity is limited to 67% with the usual QNDinteraction. It was therefore proposed in Hammerer et al. (2005a) to replace the two atomic samples in the protocol just sketched by a single sample, still with a constant bias magnetic field. This changes the interaction dramatically. Both atomic quadratures are transferred to the light, albeit at the cost of additional new modes of light. The resulting multi-mode light–atom entanglement enables teleportation with up to 77% fidelity.
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F IG . 2. (a) A photographic view of a typical experimental setup. Atomic vapor cells are placed inside the cylindrical magnetic shields. The pumping beams are indicated with dashed arrows and the path of the quantum probe field is marked with the solid arrows. (b) A schematic view of the setup. The probe pulses reach a detection system measuring Sˆy (t). The photo current is sent to a lock-in amplifier which singles out the sin(Ωt) and cos(Ωt) parts as we discuss in Section 4.2. These are integrated and stored in a PC. Typically, the pulse sequence consists of (1) a pumping pulse, (2) and (3) two laser pulses for quantum manipulation and detection of the atomic states. A very small portion of the probe field is sent through each sample in the x-direction to measure the magnitude Jx of the macroscopic spins by Faraday rotation measurements.
4. Experimental Methods In this section we describe the typical setup for our experiments. It is centered around two glass cells filled with Cs vapor at room temperature placed in two separate magnetically insulating shields with a bias magnetic field inside. Additional coils are used to apply an rf magnetic field with the frequency equal to the Larmor frequency of the bias field. Atoms in the cells are optically pumped. The setup, therefore, is similar to that of a classical magneto-optical resonance experiment. The typical experimental setup is shown in Fig. 2. A Verdi V8 pumped Ti:sapphire laser delivers what we call the probe field. This is used for quantum manipulation of the atomic samples and also for detecting the macroscopic spin Jx . Diode lasers provide optical pumping fields for creating highly polarized spin states. The cw lasers are modulated by AOMs or EOMs to achieve the desired pulse shapes. Atoms are contained in vapor cells and placed in stable magnetic fields. The magnetic field homogeneity must be of order 10−3 across the vapor cell volume.
4.1. PARAFFIN C OATED VAPOR C ELLS In our experiments the atomic samples are contained in a paraffin coated vapor cell. The coating prevents depolarization of the spin state when atoms hit the
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F IG . 3. The setup for measuring Sˆy . A strong classical pulse is mixed on a polarizing beam splitter with the quantum field which can be in a vacuum state, as in the entanglement experiment. Measurement is performed in the 45◦ - and −45◦ -basis by two detectors. Half of the difference of the two photo currents is Sˆy . An optional λ/4-retardation plate turns this measurement into the measurement of Sˆz component.
walls. We have measured spin coherence times exceeding 40 ms and spin polarizations exceeding 99% by the methods described in Section 4.3.1. The atomic density and thereby the macroscopic spin Jx can be controlled by heating or cooling the vapor cell. In order to achieve a stable vapor density, temperature gradients across the cell should be avoided. Temperature control by air flow is a convenient solution. Metal heating/cooling elements cause severe problems since the atoms are disturbed by random magnetic fields created by thermal currents even if aluminum is used, see Julsgaard (2003). Further information on paraffin coated cells can be found in Bouchiat and Brossel (1966) and Alexandrov et al. (1996, 2002).
4.2. D ETECTION OF P OLARIZATION S TATES The Stokes parameters are measured with low noise photo detectors. We use high quantum efficiency photo diodes and home made amplifiers characterized by negligible electronic noise compared to the shot noise of light at optical power higher than 1 mW. In Fig. 3 we depict how the Sˆy -component of light is measured. The differential photo current i(t) from the two detectors corresponds to a realization of the measurement of Sˆy . By passing i(t) through a lock-in amplifier we can detect the sine and cosine components at the Larmor precession frequency Ω. Practically, the current i(t) is multiplied by cos(Ωt) and sin(Ωt) and integrated over time for this purpose. According to (29a)–(29d), with appropriate scaling, this exactly corresponds to measuring the Xˆ L1 and Xˆ L2 components of light. For Sˆz -detection we would measure PˆL1 and PˆL2 .
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4.3. M AGNETIC F IELDS As was discussed in Section 2.5, external magnetic fields are added to control the dynamics of the atomic samples. The interaction of atoms with a magnetic field is governed by the Hamiltonian Hˆ mag = gF μB J · B + O B 2 . (45) We stress here that J is the total angular momentum of the atom including the nuclear spin. For the F = 4 ground state of cesium, gF ≈ 1/4. The second term O(B 2 ) reminds us that the above linear equation is only approximately true. When the magnetic energy becomes comparable to the hyperfine splitting of the ground state the response is non-linear. We comment on this below. Adding a constant bias magnetic field Bx in the x-direction leads to the equations of motion (15) with Larmor precession at frequency ΩL = gF μB Bx /h¯ . If we furthermore add an RF magnetic field at frequency Ω along the y-direction such that Bext = Bx ex + Bc cos(Ωt + φ) + Bs sin(Ωt + φ) ey (46) with constants Bc and Bs we may derive for the rotating frame coordinates Jˆy and Jˆz of (15) that ∂ Jˆy (t) ∂t
= −ωs sin(ΩL t) sin(Ωt + φ)Jx ,
(47) ˆ ∂ Jz (t) = −ωc cos(ΩL t) cos(Ωt + φ)Jx , ∂t with ωc,s = gF μB Bc,s /h¯ . Choosing the phase and the frequency of the RF-drive such that φ = 0 and Ω = ΩL we obtain: ∂ Jˆy (t)
∂ Jˆz (t) ω s Jx ω c Jx (48) , =− . ∂t 2 ∂t 2 These equations are valid for interaction times T such that ωc T , ωs T 1 ΩT . We see that with pulses of RF-magnetic fields we are able to change the spin components Jˆy and Jˆz by an amount controlled by the sine and cosine components Bs and Bc . This has several experimental applications, which are discussed below. Jx (t) = Jx (0),
=−
4.3.1. Characterizing the Spin State with the Magneto-Optical Resonance Method Equations (20) and (47) describe the Magneto-Optical Resonance method (MORS) which we use extensively for the spin state characterization. Application
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of MORS to our experiments is described in detail in Julsgaard et al. (2004b). Within this method the RF frequency is scanned around ΩL and the oscillating transverse spin components are probed via their introduction of oscillating polarization rotation of the optical probe through the Faraday interaction. In order to quantitatively explain the MORS signal as the RF-frequency is scanned across ΩL , we need to return to the second order term mentioned in Eq. (45). The single atom transverse spin components jˆy and jˆz can be expressed in terms of coherences σˆ m,m±1 in the following way: 1 jˆy = F (F + 1) − m(m + 1)(σˆ m+1,m + σˆ m,m+1 ), 2 m (49) 1 jˆz = F (F + 1) − m(m + 1)(σˆ m+1,m − σˆ m,m+1 ). 2i m In the absence of the second order term in (45) the energy separation h¯ ΩL between states |m and |m + 1 is the same for all m and all terms σˆ m+1,m have the same resonant frequency. The second order term in (45), however, makes the frequency of the coherences σˆ m+1,m slightly different for different values of m. It can be shown, that the quadratic Zeeman frequency difference ωQZ between σˆ m,m+1 and σˆ m−1,m is ωQZ = 2ΩL /ωhfs where ωhfs = 2π · 9.1926 GHz is the hyperfine splitting of the Cesium ground state. We typically have ΩL = 2π · 322 kHz and the effect is small but detectable. In the special case that the amplitude and frequency of the driving RF-field vary slowly compared to the spin coherence time, the off-diagonal coherences follow the diagonal populations adiabatically and we may write e.g. Jˆy as:
F −1 iΩt [F (F + 1) − m(m + 1)] · e [σˆ m+1,m+1 − σˆ m,m ] Jˆy = Re const i(Ωm+1,m − Ω) − Γm+1,m /2 m=−F (50) where Γm+1,m are the FWHM line-widths giving an exponential e−Γ t/2 decay of each coherence. The Larmor frequency ΩL has been replaced by the individual coherence evolution frequencies Ωm+1,m . For Jˆz we have to take the imaginary part instead of the real part in Eq. (50). We see that two adjacent magnetic sublevels act as a single two level system with the usual Lorentzian response to a driving RF field. Scanning the RF frequency we get eight Lorentzian peaks, the magnitudes of which will depend on the populations of the magnetic sublevels. The MORS signal is proportional to the square of the term in the square brackets in Eq. (50). An example of such a signal can be seen in Fig. 4. The application of this method is twofold. First, we gain information on the distribution of population among the different ground state magnetic sublevels. From this we infer that we are able to optically pump the atoms to such an extent that only the outermost coherence (mF = 4 ↔ mF = 3) becomes significant. Second, measuring
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F IG . 4. MORS signal, in arbitrary units (a.u.) of a poorly oriented sample. Each peak corresponds to a mF ↔ mF + 1 Zeeman resonance. The height of each resonance is proportional to the population difference of the two relevant sublevels.
the widths of the resonances under different experimental conditions allows us to quantify the effect of different decoherence mechanisms as discussed in Section 5.2. A second implication of the second order term is that the equations of motion ∂ jˆy /∂t = −Ω jˆz and ∂ jˆz /∂t = Ω jˆy describing Larmor precession are too simple and must, in principle, be generalized. However, this is not necessary for our case with highly polarized samples. We have almost all the population in the mF = 4 and mF = −4 states for the two samples, respectively. Then there is effectively only one non-vanishing frequency component in the sums (49). For the jx = 4 sample the single term is σˆ 3,4 and for the jx = −4 sample the only term is σˆ −3,−4 .
4.3.2. Manipulating the Spin State Of course, as suggested by Eq. (48), an appropriate choice of phase, strength and envelope function of the RF-field allows us to create an arbitrary mean value of either of the spins or any combination of these. This has two applications. First, it enables us to create a large classical mean value and observe the evolution of this state, thus constituting another mechanism for studying decoherence and calibrating the system as is discussed briefly at the end of Appendix B.1. Secondly, we can actively feed back the result of a quantum probing of the spin state, thus creating a particular desired state. This is the keystone element for creation of deterministic entanglement, discussed in Section 5.3.2, and for the quantum mapping experiment discussed in Section 5.4.
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5. Experimental Results 5.1. P ROJECTION N OISE L EVEL Since the Heisenberg uncertainty relation sets the starting point of all our calculations, one of the most important tasks in our experiments is the achievement of quantum noise limited performance. Practically, it is also one of the most difficult tasks. When we detect polarization states of light we observe noise in the signals. After the light has passed the atomic samples, there is a contribution to this noise from the light itself and from the atomic spins. The noise contribution from atoms in the minimum uncertainty state (the coherent spin state) is called projection noise. We discussed the ratio of the projection noise to the quantum noise of light (shot noise) already in Section 3.1. We found that theoretically this ratio should be κ 2 = a 2 Jx Sx T .
(51)
κ2
came from the more general interaction equations (30a)–(30d). In The ratio the present section we discuss how to determine this projection noise level experimentally and how to predict the noise level from independent measurements. 5.1.1. Measuring the Macroscopic Spin The ratio of projection noise to shot noise is proportional to the macroscopic spin Jx . This linearity is the fingerprint of quantum noise and is essential to establish experimentally. To this end we need a good measure of Jx . We measure Jx by detecting polarization rotation of a linearly polarized laser field propagating through the atomic samples along the x-direction. To see what happens in this setting we consider Eqs. (9) and (10). These equations assume propagation along the z-direction so we assume the spin to be polarized along the z-direction in the following. For linearly polarized light we have Sˆz = 0 and the effect on the transverse spin components jˆx , jˆy is negligible. It can also be shown easily that the a0 and a2 terms of the Hamiltonian (7) play no role in this calculation. We are left with the evolution of Sˆx and Sˆy according to (10) and after integration over the sample we find Sxout = Sxin cos(2θF ) − Syin sin(2θF ),
(52)
Syout = Sxin sin(2θF ) + Syin cos(2θF ), where “in” refers to the polarization state before the sample at z = 0 and “out” refers to the state after the sample at z = L. The angle θF is given by (in radians) θF = −
a1 γ λ2 ρL · jˆz . 32πΔ
(53)
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If a linearly polarized beam of light is rotated by the angle θ , the Stokes vector is rotated by 2θ. Thus, in the above, θF is the polarization rotation caused by the spin orientation along the direction of light propagation. We note that the angle θF depends on the density ρ of atoms times the length L that the light traverses. We wish to re-express this in terms of the macroscopic spin size Jz = Nat jˆz of the entire sample (remember we have the spins polarized along z in this discussion). To this end we observe that Nat = ρV ≡ ρAcell L where V is the vapor cell volume and Acell is the area of the vapor cell transverse to the beam direction. This will conveniently be an effective area for cells that are not exactly box like. Returning to the usual convention of spin polarization along the x-axis we then rewrite (53) as θF = −
a 1 γ λ2 Jx . 32πAcell Δ
(54)
5.1.2. Predicting the Projection Noise Level Now, let us return to the predicted ratio of projection to shot noise (51). This prediction relies on Eqs. (11)–(14) which are derived under the assumption that all atoms in the sample interact with the laser beam which has a cross sectional area A. In experiments the laser beam does not intersect all the atoms. In Appendix A we show that the random motion of atoms in and out of the beam modifies the expected variance of the transverse spin components Jˆy and Jˆz by statistical effects from the usual Jx /2 to p 2 (1 + σ 2 )Jx /2 where p = A/Acell is the mean time of an atom inside the laser beam and σ 2 is the relative variance of p. We furthermore present a simple model for p and σ and show that the atomic motion acts as an effective source of decoherence between two probe pulses. We incorporate atomic motion into Eq. (51) by replacing A with Acell in the factor a and multiplying the whole expression by 1 + σ 2 . We then find 2 γ λ2 a 1 Jx Sx T 1 + σ 2 κ 2 = a 2 Jx Sx T · p 2 1 + σ 2 = 8Acell Δ 2π =
(1 + σ 2 )γ λ3 a1 P · T · θF . 32π 2 Acell Δh¯ c
(55)
In the last step we replaced Sx = φ/2 = P /2h¯ ω = P λ/4π hc ¯ where P is the optical power. We also inserted Eq. (54) to express κ 2 as a function of θF . However, we must remember that the area Acell in (54) refers to the transverse area for a beam propagating in the x-direction while the Acell from the relation p = A/Acell refers to the transverse area for a beam along z. Hence, the last step above is valid for a vapor cell of cubic symmetry only, but it can still be an irregularly shaped cell. In other cases the generalization is straightforward. We have reached an expression for κ 2 in terms of convenient parameters from an
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F IG . 5. Measured atomic noise relative to shot noise of light. The linearity is a clear signature of 2 = 0.104(2) · θ should be compared to the theoretical the projection noise limitation. The slope κexp F 2 = 0.140 · θ from Eq. (56) with σ 2 = 0. value of κth F
F IG . 6. Slope of measured κ 2 vs. θ normalized to the experimentally predicted level (without the 2 /κ 2 = 1 + 0.47(13)/T [ms]. factor 1 + σ 2 ) vs. Tprobe . The fit gives κexp th
experimental point of view. With γ /2π = 5.21 MHz and λ = 852.3 nm together with hc ¯ we reach our final theoretical estimate for the projection to shot noise ratio expressed in convenient units: 2 = κth
56.4 · P [mW] · T [ms] · θF [deg] · a1 (Δ) · (1 + σ 2 ) , Acell [cm2 ] · Δ[MHz]
(56)
where a1 (Δ) was defined in Eq. (8). 5.1.3. Experimental Investigation Turning to experiment, in Fig. 5 we see an example where the measured noise relative to the shot noise of light is plotted. The data are clearly linear. With
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Δ/2π = 700 MHz, T = 2.0 ms, P = 4.5 mW, and σ 2 = 0 for the moment, we predict a linear slope of 0.140 which is somewhat higher than the measured value of 0.104. Including the σ 2 from atomic motion makes the discrepancy slightly worse but given the simplicity of the theoretical model we consider the agreement satisfactory. To test the scaling properties predicted in the atomic motion calculations, we fix the power P , detuning Δ and macroscopic spin size J but vary the probe duration T . The measured noise is plotted in Fig. 6 relative to the prediction (56) with σ 2 = 0. We see that as T is increased we do see a lower and lower noise level which corresponds to decreasing σ 2 . The solid line in the figure represents a fit where σ 2 = (0.47±0.13)/T [ms]. To compare this to the simple model described in (A.4) we estimate our beam diameter to be 1.6 cm which gives A ≈ 2.0 cm; moreover, L = 3.0 cm, v0 = 13.7 cm/ms (cesium at room temperature). For T = 1 ms we get the prediction σ 2 = 0.44. This is in very good agreement with the measured data, but this agreement must be viewed as fortuitous. As mentioned in Appendix A.1, numerical simulations of atomic motion have shown that the variance estimate (A.4) is almost four times too high. The high experimental value must be attributed to the additional Doppler broadening effect. We also note the relatively high uncertainty of 0.13. But all together, however, we have a qualitative understanding of the physics and a quantitative agreement within a few tens of percent. 5.1.4. Thermal Spin Noise Another issue concerning the projection noise level is the question of thermal spin noise. For the establishment of the correct noise level we must be in the CSS with high precision. For the CSS the spin is completely polarized along the x-direction and Var(jy ) = Var(jz ) = F /2 = 2 for the F = 4 ground state. As a very different example we may consider a completely unpolarized sample. We then have by symmetry Var(jx ) = Var(jy ) = Var(jz ) = (jx2 + jy2 + jz2 )/3 = F (F + 1)/3 = 20/3. This is a factor of 10/3 higher than the CSS noise and, even for fairly good polarization, the thermal noise may be significant. In our experiments with quantum information protocols we exceed a spin polarization of 99% which means that the thermal noise must be very small compared to the true projection noise. The degree of spin polarization has been measured independently as described in Section 4.3.1 with methods similar to those in Julsgaard et al. (2004b). A nice illustration of the fact that we get lower noise for the CSS than in the unpolarized case is given in Fig. 7. Experimentally, we perform measurements on very poor vapor cells where the macroscopic spin life time is small. We optically pump the sample and wait for some variable delay time before probing the spin noise. For long times the spins reach thermal equilibrium, where the noise of each atom in F = 4 contributes 20/3. The fraction of atoms in F = 4 is 9/16,
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F IG . 7. Coherent state noise compared to the completely unpolarized spin noise. The data is taken with a vapor cell in which the spin life time is very short. The noise level increases on a time scale of roughly 8 ms to the thermal equilibrium level. The increase in noise is consistent with predictions for the coherent and unpolarized spin states.
whereas the remaining 7/16 are in the F = 3 state and do not contribute because of the large detuning. Initially, all atoms are in F = 4 in the CSS and they each contribute the value 2 to the noise. Hence the measured noise must be on the form 20 9 Measured noise ∝ 2 · exp(−Γ t) + (57) · 1 − exp(−Γ t) . 3 16 The predicted ratio of final to initial noise is thus 15/8 ≈ 1.88. Experimentally we find the ratio 2.05 ± 0.09 which is consistent. To sum up, there is strong evidence that we really do create the CSS with the correct minimum uncertainty noise. 5.1.5. Concluding Remarks on the Projection Noise Level Let us sum up the discussion of the projection noise level. To reach the quantum noise limited performance one should first observe the atomic noise grow linearly with the macroscopic spin size Jx . An experimental example of this was shown in Fig. 5. The linearity of the noise basically arises from the fact that different atoms yield independent measurements when their spin state is detected. Technical noise sources from e.g. external electromagnetic fields couple to all atoms and the effect on the noise variance would be quadratic. However, linearity alone is not enough. An ensemble of independent and unpolarized atoms would also show a linear increase in the spin noise with an increase in the number of atoms. Since unpolarized atoms have larger noise variance than the 100% polarized atomic sample, we must know independently that the spin orientation is high. In our experiments the spin samples are polarized better than 99%.
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One may argue that the small fraction of atoms that are not in the completely polarized state could, in principle, form exotic-multi particle states with a very high variance of the detected spin noise. The results discussed in Section 5.1.4 prove that this is not the case. Finally, as derived in Appendix A the atomic motion leads to an increased ratio of atomic to shot noise. Generally a large ratio of atomic projection noise to shot noise is good for the quantum information protocols. However, as discussed in Appendix A.2, we do not gain anything by the increase of the atomic noise caused by atomic motion since there is an accompanying increase in the decoherence rate.
5.2. D ECOHERENCE As mentioned earlier, all atoms are optically pumped into an extreme Zeeman sublevel with the x-axis as quantization axis. A conventional way of categorizing sources of decoherence is according to whether they affect the magnitude of the spin along this axis or merely along transverse directions. The appropriate life times of these are called T1 and T2 and defined as: Jx (t) = e−t/T1 Jx (0)
and Jtrans (t) = e−t/T2 Jtrans (0)
(58)
in the absence of additional interactions. In the absence of the probe light the dominant processes are collisions with the walls and other atoms and, typically T1 ≈ 300 ms. Even though the probe detunings are quite high in our experiments (700–1200 MHz) making the desired refractive Faraday interaction dominant by far, the small probability of absorption still reduces T1 by about a factor of 2 depending on probe power and detuning. It is, however, still very large compared to typical probe durations (0.5–2 ms). The lifetime of the transverse spin components, however, turns out to be much more critical to our experiments. As can be seen from Eq. (49) the transverse spin components are determined by coherences between magnetic sublevels in the x-basis. Therefore anything that affects the lifetime of Jx will also affect T2 . In addition, however, the total transverse spin components are also sensitive to random phase changes in each atom. As discussed in Section 4.3 we can use the widths obtained in MORS signals in different experimental settings to quantify and separate the effect of different decoherence mechanisms. The FWHM obtained from such signals are related to T2 by: Γtrans [Hz] =
1 . πT2 [s]
(59)
We can separate the mechanisms into two main categories: some are mediated by the probe light and the rest are independent of the presence of the probe. Starting with the latter, these combine to a decay rate, Γdark , and consist mainly of phase
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F IG . 8. Line-width of the σ34 coherence in the MORS signal as a function of optical power. In a single trace it is impossible to separate the expected power broadening from absorption from the light induced collisions.
changing and spin-flip collisions with the walls and other atoms and random phase changes because of atomic motion through inhomogeneous magnetic fields. The effect of these are reduced by the paraffin coating on the inside of the glass cells, the diluteness of the atomic sample, and the application of additional dc-magnetic fields to cancel field gradients. To determine Γdark we measure the width of the mF = 3 ↔ 4 coherence for different probe powers and find the residual width in the absence of light. An example of this is shown in Fig. 8. As can be seen we obtain a width of the order of 12 Hz corresponding to a lifetime of the transverse spin of T2dark ≈ 27 ms. We see that this will limit but not destroy all correlations between two subsequent pulses in e.g. an entanglement experiment as described in Section 3.1. Turning to the probe induced decoherence mechanism, we have already mentioned absorption and subsequent spontaneous emission in the discussion of T1 . Adding this effect to the other decoherence mechanism, we would expect a total decoherence rate of the general form: Γideal = a + b · n + c · P ,
(60)
n is the atomic density, P is the light power, and a, b, and c are coefficients, which can be determined experimentally. If Γ is plotted vs. n we would expect a line with constant slope b and offset determined by the optical power. In Fig. 9 we show measurements of the decoherence rate vs. atomic density for different optical powers. The results clearly contradict the simple model of Eq. (60), since the slope grows with increasing power. It turns out that the experiments fit a model: Γexp = a + b · n + c · P + d · n · P ,
(61)
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F IG . 9. Line-width of the σ34 coherence in the MORS signal as a function of atomic density for different optical powers. The fact that the slopes are not equal reveals light induced collisions.
where the size of the expected pure power broadening term, c, agrees with solutions of the Maxwell–Bloch equations for the full multi-level atomic system in the presence of Doppler broadening. The last term could represent light induced collisions, but a clear theoretical understanding of the nature of these is still missing. For the experimentally relevant densities and powers this term contributes around 30 Hz of broadening and is thus the main source of decoherence. We stress that this is a pure T2 process since we do not observe similar features when investigating the decay of the longitudinal spin. Hence, the atoms practically decay towards the fully polarized state, i.e. the coherent spin state. This was also assumed implicitly in the inclusion of decoherence for the entanglement and quantum mapping protocols in Section 3, where decoherence was modeled by an admixture of a vacuum state with the same variance as the coherent spin state. 5.3. E NTANGLEMENT R ESULTS 5.3.1. Conditional Entanglement We now turn to the experimental demonstration of entanglement generation. First the boundary between the classical and the quantum fluctuations has to be established. As discussed in Section 5.1 this projection noise level is found by performing several measurements of Xˆ L as a function of the macroscopic spin size and verifying a linear increase of the atomic noise of each measurement. This linearity combined with nearly perfect orientation of the sample ensures the correct projection noise level. Once this is established we implement a probing sequence (see Fig. 2) in which the initial probing pulse is followed by a second one after a short delay. To verify entanglement we need to fulfill the criterion (36), in which case our ability to predict the outcome of the second probing of the atomic state conditioned on the result of the first measurement exceeds the classical limit.
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F IG . 10. Atomic noise in units of shot noise as a function of the macroscopic spin size (measured by DC Faraday rotation). The dotted curve is a quadratic fit to the first pulse variances (dots) and the dashed curve is the linear part of this. Dash-dotted and full-drawn lines are fits to the optimal weight factors, α (x’s) and the conditional variances (+’s), respectively.
For the experimental data we calculate the atomic part of the noise by subtracting the shot and electronic noise of a single light pulse and then normalize to the light shot noise level. In Fig. 10 the resulting atomic noise is shown for the first and second pulses (with filled and empty circles respectively). Since the measurement is of a QND-type and the variance is calculated based on 10,000 independent repetitions of the pulse sequence, the variance of the first and the second pulses should be identical. We make a quadratic + linear fit to the first pulses’ variances (dotted curve) and from this extract the linear part, which represent the coherent spin state level. The slope of this is 0.176(12), which can be compared to the theoretical values 0.187 and 0.165 obtained from Eq. (56) with and without the effect of atomic motion included. In this fit we used: T = 2.0 ms, P = 5.0 mW, and Δ/2π = 825 MHz. As discussed in Appendix A the results including atomic motion are very preliminary, since this is an ongoing research topic. The overall agreement with experimental results are, however, very encouraging. Note that the small quadratic component is caused by various classical noise sources. Next we calculate the correlations between the first and the second pulse measurements as discussed in Section 3.1. In Fig. 10 the pulses show the conditional variance Var(A2 |A1 ) + Var(B2 |B1 ) normalized to shot noise and with shot and electronics noise subtracted. The points below the straight line (36) indicate that we have in fact created an entangled state between the two atomic samples. For the highest densities the noise reduction is up to 30%. The corresponding α-parameters from the minimization procedure (34) are plotted in Fig. 10 with crosses. Ideally we would expect these to follow Eq. (39) but since the atoms decohere the appropriate expression is instead Eq. (41). Fitting, we ob-
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tain β = 0.619(11), which is inserted into Eq. (40) (solid curve). Considering the fact that the solid and dash-dotted curves are obtained from a single free parameter, β, the agreement between experimental results and the simple model of decoherence must be considered very satisfactory. In addition, we would like to stress the very important point that the two atomic samples, which are entangled, are in completely separate environments about 0.5 meters apart (Sherson et al., 2005a). This represents a major breakthrough towards the creation of truly distant entanglement, which combined with quantum teleportation will enable quantum communication over long distances. 5.3.2. Unconditional Entanglement As we have just seen, the results of two probes of the spin state yield correlated results. The actual results, however, vary from shot to shot, representing random realizations of the probability distribution of the spins. That is, we create a nonlocal state with reduced variance but with a non-deterministic mean value. Thus, the entanglement only appears when the knowledge gained in the first pulse is applied. To create a deterministically entangled state in which no knowledge of measurement results is necessary would of course constitute a very important advance. We have realized this experimentally by simply feeding the result of the first measurement pulse back to the atoms using an RF-magnetic pulse as discussed in Section 4.3.2. This procedure is very closely related to the way in which unconditional spin squeezing is generated in Geremia et al. (2004) except that there the feedback is applied continuously in time, which is more robust against errors in the feedback strength. In the experiment we first choose a certain atomic density. Using the linear fit of Fig. 10 we find the projection noise level relative to which entanglement is estimated. Next, we vary the strength or the phase of the RF feedback pulse, and observe the variance of the second pulse fluctuations as a function of these feedback parameters. The feedback giving the lowest variance corresponds to the feedback with the optimal gain. In Fig. 11 we alternate between having feedback and not having it. As can be seen clearly, with feedback we obtain the same variance for the second pulse as we do without feedback for the conditional variance. This means that we have, in fact, created a state which is deterministically entangled. That is, the state after the first probe and subsequent feedback of the measurement result has zero mean and a variance reduced by 19% compared to the projection noise level. When feedback is turned on the variance of the second pulse and the conditional variance coincide, which means that knowledge of the measurement result cannot improve the degree of entanglement beyond the entanglement established by the second pulse measurement alone.
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F IG . 11. Deterministic entanglement generation. For a fixed atomic density negative feedback of the first pulse measurement is alternately turned on and off, thus switching between conditional and unconditional entanglement generation.
5.4. Q UANTUM M EMORY R ESULTS We now present the experimental demonstration of quantum memory, which has been published in (Julsgaard et al., 2004a). The used protocol was discussed in Section 3.2 and, briefly, involves (1) preparation of the initial atomic state in a coherent state via optical pumping, (2) mapping of one of the light quadratures through the off-resonant Faraday interaction, Eqs. (30a)–(30d), and (3) mapping of the second light quadrature by a direct measurement and subsequent feedback (an illustration of the timing sequence is shown in the inset of Fig. 12). For the reasons discussed in Section 3.2.3 we have not been able to retrieve the mapped state. Instead we have performed a destructive reconstruction of the mapped state. This is done by waiting for a time τ and then sending a readout light pulse through ˆ the atomic sample. Measuring the X-component of the outgoing light then gives information about the stored atomic Pˆ -component according to Eq. (30a). If a π/2 rotation in the atomic XP space is performed prior to the readout pulse we obtain ˆ information about the stored atomic X-component. Repeating this 10,000 times, we can reconstruct the statistics for the atomic variables after the storage procedure. The first thing to check is that the mean value of the stored state depends linearly on the mean value of the input light state. This is shown in Fig. 12. First we note that the linear dependence is clear for both quadratures. This completes the proof of classical memory performance. The next thing to note is that the slope is not unity, which means that the stored state has a different mean value than the input state. The reason for this is discussed further below. For the quadrature mapped straight from the back action of the light onto the atoms we have the gain of gBA = 0.836 and for the quadrature mapped via the feedback we
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F IG . 12. Mean value of the read out pulse as a function of the mean values of the input light variables, Xˆ Lin and PˆLin , to be stored. Inset: the strong classical and weak quantum pulses in opposite polarizations. Between the input and the output pulses are the feedback pulse and the optional π/2 pulse.
out (o’s) and Pˆ out (x’s), as a function of the mean F IG . 13. Variance of the atomic quadratures, Xˆ A A values of the input light variables. Full drawn curve: the variance for perfect quantum storage. Dotted curve: classical limit on the variances for n0 = 4.
have gF = 0.797. In order to verify quantum storage we also need to consider the shot-to-shot fluctuations in the stored state, which for a Gaussian state are fully characterized by the variance of the state. The experimentally reconstructed variances of the atomic quadratures, σx2 , σp2 normalized to shot noise, that is the variance of the readout pulse with the one unit of shot noise intrinsic to the readout pulse subtracted, followed by a rescaling by 1/κ 2 , are shown in Fig. 13. Also shown is the ideal quantum limit on the variance for a perfect mapping and the
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classical limit for a Gaussian distribution of input states with mean photon number n0 = 4. As can be seen the variance is more or less independent of the mean value of the input light quadratures. The fidelity of the stored state for a given n0 can be calculated given the measured gains and variances, σx2 and σp2 , according to: F =
2 (2n0 (1 − gBA )2 + 1 + σx2 )(2n0 (1 − gF )2 + 1 + σp2 )
.
(62)
With the experimentally measured values we get F = (66.7 ± 1.7)% for n0 = 4 and F = (70.0 ± 2.0)% for n0 = 2. The best classical fidelity was recently derived by Hammerer et al. (2005b) Fclass =
1 + n0 1 → , 1 + 2n0 2
n0 → ∞
(63)
for coherent states drawn out of a Gaussian distribution with mean photon number n0 . This means that Fclass decreases monotonically from unity for the vacuum state to 1/2 for an arbitrary coherent state. For n0 = 4, 2 we obtain the classical boundaries of 55.6% and 60.0%, respectively. This verifies that the storage of the light state in fact constitutes a quantum mapping. The results shown were obtained for a pulse duration of 1 ms and the memory has been shown to work for up to 4 ms delay between the two probe pulses. Note that we have chosen to calculate the fidelity as the average of the squared overlap between the stored state and the ideally stored state. For non-unity gain this decreases very rapidly with coherent states having large amplitudes. However, one could argue that a storage with an arbitrary but known gain constitutes just as useful a memory as unity gain memory. If analyzed solely in terms of the added noise, our memory would therefore perform much better than the previously stated results, which can therefore be viewed as a lower bound on the memory capability. Note also that the choice of the optimal gain which maximizes the fidelity depends on the class of available states. For classes of coherent states limited in their amplitudes discussed in this section, the experimental gains quoted above are, in fact, close to the optimal ones. 5.4.1. Decoherence In Fig. 14 we show the final atomic variances as a function of the feedback gain. These are compared to the variances expected from Eqs. (44a) with independently determined decoherence values of β = 0.61 and ζ = 0.75. We stress that this is not a fit to the data, which means that we understand the level of experimentally determined atomic noise quite well. The figure clearly shows that, because of the decoherence and light loss, if the feedback gain is increased towards unity the noise grows dramatically. With this, the fidelity quantified by Eq. (62) can be optimized with respect to the feedback gain. As can be seen from the values of β
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out (o’s) and Pˆ out (x’s) as a function of feedback gain. The F IG . 14. Reconstructed variances of Xˆ A A curves are theoretical curves based on independently determined noise parameters.
and ζ the light loss and atomic decoherence are significant. The high light loss is due to the fact that the glass cells containing the atomic vapor were not antireflection coated. Therefore each glass-air interface contributes about 4% loss. As discussed in Section 5.2 the main source of atomic decoherence is light assisted collisions, which change the phase of the atomic coherence without affecting the macroscopic spin size Jx . The atoms are driven towards a coherent state, which justifies the use of the simple model of beam splitter admixture of vacuum.
6. Conclusions We have described the progress in the development of the deterministic quantum interface between light and macroscopic atomic ensembles. Theory of the interface is based on canonical variables which provide a convenient language when different physical systems, such as light and atoms, are considered. Surprisingly enough, simple dipole off-resonant interaction of light with spin polarized atomic samples with high optical density provides a powerful tool for quantum state engineering and transfer. To perform sophisticated quantum information protocols, the interaction can be combined with a quantum measurement on light and subsequent feedback to the atoms. Two central experiments of this review, entanglement of two macroscopic objects and quantum memory for light, are performed following these general steps. Future perspectives include multipass protocols towards more efficient quantum memory including the retrieval process, as well as various types of quantum teleportation involving atomic ensembles. Another exciting future direction
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is demonstration of deterministic qubit state manipulation with the tool box described in this paper. It is appropriate to stress that the language of canonical variables is fully applicable not only to Gaussian states but to all single mode states, including qubit and Fock states. This language is associated with homodyne measurements in quantum optics. Addition of single photon counting to the experiments described in this chapter may pave the road towards more efficient and robust quantum information processing and communication.
7. Acknowledgements This research has been supported by the Danish National Research Foundation and by EU grants QUICOV and COVAQIAL. B. Julsgaard is supported by the Carlsberg Foundation. We would like to thank J.I. Cirac, L.M. Duan, K. Hammerer, J. Fiurasek, A. Kozhekin, D. Kupriyanov, A. Kuzmich, and K. Mølmer for fruitful collaborations. The contributions of J. Hald and J.L. Sørensen to the early experiments are gratefully acknowledged.
8. Appendices Appendix A. Effect of Atomic Motion Since in our experiments the atoms are at room temperature and, for experimental reasons, the light beam does not cover the entire cross section of the atomic sample, the atoms move across the beam several times (∼ 10) during the time of a pulse. This averaging effect insures that all atoms spend roughly the same amount of time inside the beam but, as we shall see, it still has important implications for the noise properties. In brief, the atomic motion modifies the projection noise level and acts as an additional source of decoherence since two subsequent probe pulses interact with the atoms differently. The results are related to the work of Kuzmich and Kennedy (2004). A PPENDIX A.1. M ODELING ATOMIC M OTION To be more quantitative, we introduce new pseudo-angular momentum operators (i) Jq → i pi Fq , where pi is the fraction of time the ith atom spends interacting with the laser beam and q = x, y, z. These have the commutator:
N Nat Nat Nat at (i) (i) (A.1) pi F y , pi F z pi2 Fy(i) , Fz(i) = i pi2 Fx(i) . = i=1
i=1
i=1
i=1
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This leads to the Heisenberg uncertainty relation (for a highly polarized sample with Fx ≈ F )
N
N at at 2 J 2 (i) (i) Var (A.2) pi F y pi F z Var p 1 + σ2 2 i=1
i=1
where we have introduced the mean p = pi and variance Var(pi ) = σ 2 · p 2 of pi . With this definition, σ is the relative standard deviation of p. Since, for the coherent spin state,
N at J (i) Var (A.3) pi F z = p 2 1 + σ 2 = Var(CSS), 2 i=1
this highly polarized state corresponds to a minimum uncertainty state. The measured noise is then limited by the Heisenberg uncertainty principle and we confidently call this projection noise. To maintain the correct commutation relation [X, P ] = i we experimentally normalize the atomic√ operators to the measured projection noise, i.e. instead of defining X = Jy / J we effectively (i) have X = i pi Fy / Jp 2 (1 + σ 2 ). The average fraction of time p each atom spends inside the beam is clearly p = Abeam /Acell . Let us now discuss the scaling of σ 2 with simple physical parameters. The fact that the variance may be non-zero arises from the finite time available for the averaging process carried out by the atomic motion. A typical traversing time across the vapor cell is τ = L/v0 where L is the cell dimension and v0 is, e.g., the one-dimensional rms speed of the atoms. We may think of this atomic motion as n independent journeys across the vapor cell volume, where n ≈ T /τ = T v0 /L. We then model the motion through the beam with mean occupancy p by assuming in each walk across the cell volume that either (1) the atom spends all the time τ inside the beam, which should happen with probability p, or (2) the atom spends all the time τ outside the beam which should happen with probability 1 − p. We then count the number of times ninside that an atom was inside the beam out of the possible n journeys. In this simple model ninside is a stochastic variable which is binomially distributed with mean np and variance np(1 − p). We are interested in the fraction of time (≈ ninside /n) spent inside the beam. It follows ninside /n = p and σ 2 = Var([ninside /n]/p) = (1 − p)/np. Hence the simple model leads to p=
Abeam Acell
and σ 2 =
(Acell − Abeam )L . Abeam T v0
(A.4)
Note the characteristic scaling with T −1 and with the area (Acell − Abeam ) not covered by the light beam (when Abeam is close to its maximum value Acell ). We
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note that due to the simplicity of the above model the absolute numbers should only hold as an order of magnitude estimate. Numerical simulations performed for a cubic cell have shown that the relative variance σ 2 is roughly four times smaller than the estimate above. Also, due to the Doppler broadening, the effective detuning differs from atom to atom and causes an increase in σ 2 .
A PPENDIX A.2. ATOMIC M OTION AS A S OURCE OF D ECOHERENCE To see how atomic motion acts as an effective source of decoherence, imagine that we perform some manipulations of atoms using one laser pulse and subsequently probe these manipulations with another laser pulse. Since atoms move during interactions the probed quantum operator changes in time. Comparing the operator at the 1st and 2nd times we get
N Nat at (i) (i) pi,2nd Fz,2nd − pi,1st Fz,1st Var i=1
=
Nat
i=1
(i) Var Fz,1st (pi,2nd − pi,1st )2
i=1
J 1 (A.5) . · 2p 2 σ 2 = 2Var(CSS)(1 − β) with β = 2 1 + σ2 We assumed pi,1st and pi,2nd to be uncorrelated, which is reasonable since a collision with the cell wall randomizes the velocity of the atoms. Also, we took (i) (i) Fz,1st = Fz,2nd . This corresponds to having no decoherence at all apart from the effect of atomic motion which is the only effect studied in this calculation. To interpret the above calculations we consider a standard decoherence calculation. Consider a true spin operator Jz subject to decoherence parametrized by the number β such that Jz,1st → Jz,2nd = βJz,1st + 1 − β 2 Jvac =
J = Var(CSS). 2 Then the operator changes by an amount characterized by the variance Var Jz2nd − Jz1st = Var Jz1st (1 − β) − 1 − β 2 Jvac with Var(Jvac ) =
= J (1 − β) = 2 Var(CSS)(1 − β)
(A.6)
(A.7)
which is exactly the same as in (A.5). We are led to the conclusion that atomic motion inevitably gives rise to an effective decoherence. We thus see that, whereas the increased coherent spin state noise with increased σ 2 might seem to suggest
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that non-classical states are more easily created (by producing states with noise lower than the CSS), this is compensated for by an increased decoherence of the state. Therefore, higher σ 2 does not lead to higher fidelity protocols.
Appendix B. Technical Details A PPENDIX B.1. L IGHT P OLARIZATION AND S TARK S HIFTS Let us calculate the Stark effect from the probe laser on the magnetic sublevels |F, m. We let the light be strong and linearly polarized along the vector e1 = ex cos α + ey sin α,
(B.1)
i.e. α is the angle between the macroscopic spin direction (the x-axis) and the probe polarization direction. Light is propagating in the z-direction. The Stark effect on magnetic sub-levels is much weaker than the splitting caused by the constant bias magnetic field and can be calculated in non-degenerate perturbation theory from the interaction Hamiltonian (7). The a0 term is common to all levels, the a1 term is zero on average since Sˆz = 0, and we are left with the higher order components proportional to a2 . With aˆ 1 the creation operator along the strong direction we have aˆ x = aˆ 1 cos α and aˆ y = aˆ 1 sin α (neglecting the direction orthogonal to aˆ 1 which is in the vacuum state for linear polarization). With Sˆ± = Sˆx ± i Sˆy we derive from (4) that φ(t) 2iα φ(t) −2iα and Sˆ− (t) = , e e Sˆ+ (t) = (B.2) 2 2 where φ(t) is the photon flux and Stokes operators are normalized to photons per second. In order to calculate the higher order terms of the interaction Hamiltonian for a single atom we leave out the integral . . . ρA dz in (7) and renormalize light operators (by absorbing the speed of light c) to photons per second and find
h¯ γ λ2 eff Hˆ int =− a2 · φ(t) · jˆz2 − jˆx2 − jˆy2 cos(2α) 8AΔ 2π − [jˆx jˆy + jˆy jˆx ] sin(2α) .
(B.3)
We also replaced jˆ± by jˆx ± i jˆy . We need to calculate the expectation value of this Hamiltonian for the different energy eigenstates |m quantized along the x-direction. We have m|jˆx2 |m = m2 , m|jˆy2 |m =
F (F + 1) − m2 , 2
(B.4)
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F (F + 1) − m2 , 2 m|jˆx jˆy + jˆy jˆx |m = 0. m|jˆz2 |m =
The first of these is obvious since |m is quantized along the x-axis. We have jˆy2 = jˆz2 by symmetry and the value is found from the fact that jˆx2 + jˆy2 + jˆz2 = F (F + 1). Also, by symmetry we have in an eigenstate of jˆx that m|jˆy jˆx |m = m · m|jˆy |m = 0 which leads to the final line. Calculating the expectation value of (B.3) we get Stark Em =
h¯ γ λ2 a2 · φ(t) 8AΔ 2π 1 + 3 cos(2α) 1 + cos(2α) 2 × ·m − F (F + 1) . 2 2
(B.5)
Stark − E Stark )/ h of a magnetic What is important for us is the shift δνm+1,m = (Em+1 m resonance line which then becomes Stark [Hz] = δνm+1,m
γ λ 2 a2 · φ(t) · 1 + 3 cos(2α) · (2m + 1). 2 64π AΔ
(B.6)
This Stark shift is problematic for several protocols, especially the setup with two oppositely oriented samples. Note, that for atoms polarized in the mF = 4 state the relevant transition mF = 4 ↔ mF = 3 has a Stark shift proportional to 2 · 3 + 1 = 7. An oppositely oriented sample with mF = −4 has for the transition mF = −3 ↔ mF = −4 a Stark shift proportional to 2 · (−4) + 1 = −7. Hence, these two transition frequencies cannot be overlapped both in the presence and absence of light (see Fig. 15 for an illustration). One remedy for this is to choose the light to be linearly polarized at an angle α = 54.7◦ corresponding to 3 cos(2α) = −1, and the Stark term disappears. As we shall see in Appendix B.2 this gives rise to other problems. Another remedy is to add an extra bias magnetic field along the x-direction when the laser light is on. In this way the frequency of the desired transitions can be kept stable. This is the approach we have taken and it works well. One should note though, that with our laser pulse timing in the vicinity of one millisecond, it is not completely trivial to make a magnetic pulse following the laser intensity since eddy currents in metallic parts and induced electric fields in other current loops for magnetic fields should be taken into account. A convenient diagnostic method is to apply a classical shift to the spin states along the lines of Eq. (48) prior to the application of two laser pulses. An Sˆy detection of the light will show in real time the mean value of the spin state components Jˆy and Jˆz . These should be constant through each laser pulse and conserved in the dark time in between (apart from a possible decay) if all frequencies are well adjusted.
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F IG . 15. An illustration of the problem with the light induced Stark shift of the Zeeman sublevels. Without applying additional fields the two important Zeeman resonances cannot be overlapped both without light (a) and with light (b).
A PPENDIX B.2. I NFLUENCE OF L ASER N OISE In this section we make a brief comment on laser noise entering the atomic samples. This is by no means a complete analysis but it describes some important issues in connection with the choice of laser polarization. We start out with the interaction Hamiltonian (7) and also assume that the atoms are polarized along the x-axis and placed in a bias magnetic field, leading to the magnetic Hamiltonian Hˆ mag = Ω jˆx . This leads to the following equations of motion for the transverse spin components jˆy and jˆz : ∂ cγ λ2 jˆy (z, t) = −Ω jˆz + −a1 Sˆz jˆx ∂t 8AΔ 2π + a2 − 2Sˆx + φˆ [jˆx jˆz + jˆz jˆx ] − 2Sˆy [jˆz jˆy + jˆy jˆz ] , (B.7) cγ λ2 ˆ ∂ jˆz (z, t) = Ω jˆy + a2 4Sx [jˆx jˆy + jˆy jˆx ] − 4Sˆy jˆx2 − jˆy2 . ∂t 8AΔ 2π (B.8) The Larmor precession terms take all interesting dynamics to the frequency component around Ω. Let us see which terms above couple Ω-components of light into the spin state. First, the a1 Sˆz jˆx term is our favorite interaction term (13) used in all quantum information protocols. It consists of an atomic operator jˆx which is constant equal to ±F for all practical purposes. This is multiplied by Sˆz , whose Ω-components drive ∂ jˆy /∂t. In Eq. (B.8) the final term is proportional to jˆx2 − jˆy2 . Practically, this is also constant, equal to F 2 − F /2. It is then multiplied by Sˆy , whose Ω-components
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drive ∂ jˆz /∂t. This is an unwanted effect. The ratio of the unwanted to wanted noise is found by squaring these contributions. We get 2 Noise(Sˆy ) Unwanted noise 2 a2 (B.9) . = 4(2F − 1) Wanted noise a1 Noise(Sˆz ) For our typical values of detuning we have a2 /a1 ≈ 10−2 and the above ratio becomes ≈ 0.02 · Noise(Sˆy )/Noise(Sˆz ) for F = 4. If our laser beam is polarized along the x- or y-axis with a clean linear polarization, the noise of Sˆy and Sˆz at frequency Ω can be shot noise limited, i.e. by quantum noise (amplitude noise of the laser does not couple to Sˆy and Sˆz in the case of clean linear polarization). In this case the unwanted noise only contributes ≈ 2% of the total noise pile up. But if we choose arbitrary polarization directions in the xy-plane, the Sˆy -component will have non-zero mean value, and the fluctuations at Ω will essentially be the amplitude noise of the laser at Ω. In this case, to keep the last term of Eq. (B.8) from accumulating extra noise, one requires the laser intensity to be shot noise limited at Ω (which is a more difficult condition to meet than clean linear polarization). We thus have one motivation for choosing the laser to be polarized along the x- or y-direction and not in between. We typically do this in our experiments and for this reason we have to compensate the Stark splitting discussed in Appendix B.1. For a further discussion of the different higher order terms in the interaction, see Julsgaard (2003). A more thorough discussion of quantum noise with the higher order terms included is given by Kupriyanov et al. (2005).
9. References Alexandrov, E.B., Balabas, M.V., Pasgalev, A.S., Vershovskii, A.K., Yakobson, N.N. (1996). Doubleresonance atomic magnetometers: From gas discharge to laser pumping. Laser Phys 6, 244. Alexandrov, E.B., Balabas, M.V., Budker, D., English, D.F., Kimball, C.-H.L., Yashchuk, V.V. (2002). Light-induced desorption of alkali-metal atoms from paraffin coating. Phys. Rev. A 66, 042903. Blinov, B.B., Moehring, D.L., Duan, L.M., Monroe, C. (2004). Observation of entanglement between a single trapped atom and a single photon. Nature 428, 153. Bouchiat, M.A., Brossel, J. (1966). Relaxation of optically pumped rb atoms on paraffin-coated walls. Phys. Rev. 147, 41. Chaneliere, T., Matsukevich, D.N., Jenkins, S.D., Lan, S.-Y., Kennedy, T.A.B., Kuzmich, A. (2005). Storage and retrieval of single photons transmitted between remote quantum memories. Nature 438, 833–836. Chou, C.W., de Riedmatten, H., Felinto, D., Polyakov, S.V., van Enk, S.J., Kimble, H.J. (2005). Measurement-induced entanglement for excitation stored in remote atomic ensembles. Nature 438, 828–832. Cirac, J.I., Zoller, P., Kimble, H.J., Mabuchi, H. (1997). Quantum state transfer and entanglement distribution among distant nodes in a quantum network. Phys. Rev. Lett. 78, 3221. Corney, J.F., Milburn, G.J. (1998). Homodyne measurements on a Bose–Einstein condensate. Phys. Rev. A 58, 2399–2406.
9]
DETERMINISTIC ATOM–LIGHT INTERFACE
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Duan, L.M., Giedke, G., Cirac, J.I., Zoller, P. (2000). Inseparability criterion for continuous variable systems. Phys. Rev. Lett. 84, 2722. Eisaman, M.D., Andre, A., Massou, F., Fleischhauer, M., Zibrov, A.S., Lukin, M.D. (2005). Electromagnetically induced transparency with tunable single-photon pulses. Nature 438, 837–841. Eisert, J., Plenio, M.B. (2003). Introduction to the basics of entanglement theory in continuousvariable systems. Internat. J. Quant. Inf. 1, 479. Fiurášek, J., Sherson, J., Opatrny, T., Polzik, E.S. (2005). Single-passage read-out of atomic quantum memory. quant-ph/0510099. Fleischhauer, M., Lukin, M.D. (2002). Quantum memory for photons: Dark-state polaritons. Phys. Rev. A 65, 022314. Geremia, J., Stockton, J.K., Mabuchi, H. (2004). Real-time quantum feedback control of atomic spinsqueezing. Science 304, 270. Giedke, G., Cirac, J.I. (2002). Characterization of Gaussian operations and distillation of Gaussian states. Phys. Rev. A 66, 032316. Hald, J., Sørensen, J.L., Schori, C., Polzik, E.S. (1999). Spin squeezed atoms: A macroscopic entangled ensemble created by light. Phys. Rev. Lett. 83, 1319. Hammerer, K., Polzik, E.S., Cirac, J.I. (2005a). Teleportation and spin squeezing utilizing multimode entanglement of light with atoms. Phys. Rev. A 72, 052313. Hammerer, K., Wolf, M.M., Polzik, E.S., Cirac, J.I. (2005b). Quantum benchmark for storage and transmission of coherent states. Phys. Rev. Lett. 94, 150503. Julsgaard, B. (2003). Entanglement and quantum interactions with macroscopic gas samples. Ph.D. thesis, Department of Physics and Astronomy, University of Aarhus, Denmark, available at http://www.phys.au.dk/main/publications/PhD/. Julsgaard, B., Kozhekin, A., Polzik, E.S. (2001). Experimental long-lived entanglement of two macroscopic objects. Nature 413, 400. Julsgaard, B., Sherson, J., Cirac, J., Fiurášek, J., Polzik, E. (2004a). Experimental demonstration of quantum memory for light. Nature 432, 482. Julsgaard, B., Sherson, J., Sørensen, J.L., Polzik, E.S. (2004b). Characterizing the spin state of an atomic ensemble using the magneto-optical resonance method. J. Opt. B: Quantum Semiclass. Opt. 6, 5. Kozhekin, A.E., Mølmer, K., Polzik, E.S. (2000). Quantum memory for light. Phys. Rev. A 62, 033809. Kuhn, A., Hennrich, M., Rempe, G. (2002). Deterministic single-photon source for distributed quantum networking. Phys. Rev. Lett. 89, 067901. Kupriyanov, D., Mishina, O., Sokolov, I., Julsgaard, B., Polzik, E. (2005). Multimode entanglement of light and atomic ensembles via off-resonant coherent forward scattering. Phys. Rev. A 71, 032348. Kuzmich, A., Polzik, E.S. (2003). In: Braunstein, S.L., Pati, A.K. (Eds.), “Quantum Information with Continuous Variables”, Kluwer, Dordrecht, pp. 231–265. Kuzmich, A., Mølmer, K., Polzik, E.S. (1997). Spin squeezing in an ensemble of atoms illuminated with squeezed light. Phys. Rev. Lett. 79, 4782. Kuzmich, A., Bigelow, N.P., Mandel, L. (1998). Atomic quantum non-demolition measurements and squeezing. Europhys. Lett. 42, 481. Kuzmich, A., Mandel, L., Bigelow, N.P. (2000). Generation of spin squeezing via continuous quantum nondemolition measurement. Phys. Rev. Lett. 85, 1594. Kuzmich, A., Bowen, W.P., Boozer, A.D., Boca, A., Chou, C.W., Duan, L.M., Kimble, H.J. (2003). Generation of nonclassical photon pairs for scalable quantum communication with atomic ensembles. Nature 423, 731. Kuzmich, A., Kennedy, T.A.B. (2004). Nonsymmetric entanglement of atomic ensembles. Phys. Rev. Lett. 92, 030407. Madsen, L.B., Mølmer, K. (2004). Spin squeezing and precision probing with light and samples of atoms in the Gaussian description. Phys. Rev. A 70, 052324.
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J. Sherson et al.
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McKeever, J., Boca, A., Boozer, A.D., Miller, R., Buck, J.R., Kuzmich, A., Kimble, H.J. (2004). Deterministic generation of single photons from one atom trapped in a cavity. Science 303, 1992. Opatrny, T. (2005). Single-cell atomic quantum memory for light. quant-ph/0509094. Opatrny, T., Fiurasek, J. (2005). Enhancing the capacity and performance of collective atomic quantum memory. Phys. Rev. Lett. 95, 053602. Polzik, E.S. (2004). Quantum information: Flight of the qubit. Nature 428, 129. Raimond, J.M., Brune, M., Haroche, S. (2001). Manipulating quantum entanglement with atoms and photons in a cavity. Rev. Mod. Phys. 73, 565. Sanders, B.C., Milburn, G.J. (1989). Quantum nondemolition measurement of quantum beats and the enforcement of complementarity. Phys. Rev. A 40, 7087–7092. Sherson, J., Mølmer, K. (2005). Entanglement of large atomic samples: A Gaussian state analysis. Phys. Rev. A 71, 033813. Sherson, J., Julsgaard, B., Polzik, E.S. (2005a). Distant entanglement of macroscopic gas samples. In: Akulin, W.M., Sarfati, A., Kurizki, G., Pellegrin, S. (Eds.), “Decoherence, Entanglement and Information Protection in Complex Quantum Systems”, In: NATO Sci. Ser. II. Math. Phys. Chem., vol. 189. Springer, Dordrecht, pp. 353–372. Sherson, J., Sørensen, A.S., Fiurášek, J., Mølmer, K., Polzik, E. (2005b). Light qubit storage and retrieval using macroscopic atomic ensembles. quant-ph/0505170. Takahashi, Y., Honda, K., Tanaka, N., Toyoda, K., Ishikawa, K., Yabuzaki, T. (1999). Quantum nondemolition measurement of spin via the paramagnetic Faraday rotation. Phys. Rev. A 60, 4974. Thomsen, L.K., Mancini, S., Wiseman, H.M. (2002). Continuous quantum non-demolition feedback and unconditional atomic spin squeezing. J. Phys. B 35, 4937–4952. van der Wal, C.H., Eisaman, M.D., Andre, A., Walsworth, R.L., Phillips, D.F., Zibrov, A.S., Lukin, M.D. (2003). Atomic memory for correlated photon states. Science 301, 196. Volz, K., Weber, M., Schlenk, D., Rosenfeld, W., Vrana, J., Saucke, K., Kurtsiefer, C., Weinfurter, H. (2006). Observation of entaglement of a single photon with a trapped atom. Phys. Rev. Lett. 96, 030404. Wiseman, H.M. (1998). In-loop squeezing is like real squeezing to an in-loop atom. Phys. Rev. Lett. 81, 3840–3843. Zoller, P., Beth, T., Binosi, D., Blatt, R., Briegel, H., Bruss, D., Calarco, T., Cirac, J., Deutsch, D., Eisert, J., Ekert, A., Fabre, C., Gisin, N., Grangier, P., Grassl, M., Haroche, S., Imamoglu, A., Karlson, A., Kempe, J., Kouwenhoven, L., Kröll, S., Leuchs, G., Lewenstein, M., Loss, D., Lütkenhaus, N., Massar, S., Mooij, J., Plenio, M., Polzik, E., Popescu, S., Rempe, G., Sergienko, A., Adn, J., Twamley, D.S., Wendin, G., Werner, R., Winter, A., Wrachtrup, J., Zeilinger, A. (2005). Quantum information processing and communication: Strategic report on current status, visions and goals for research in Europe. European J. Phys. D 36 (2), 203–228.
ADVANCES IN ATOMIC, MOLECULAR AND OPTICAL PHYSICS, VOL. 54
COLD RYDBERG ATOMS J.-H. CHOI, B. KNUFFMAN, T. CUBEL LIEBISCH, A. REINHARD and G. RAITHEL FOCUS Center, Department of Physics, University of Michigan, Ann Arbor, MI 48109-1040 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Preparation and Analysis of Cold Rydberg-Atom Clouds . . . . . . . . . . . . . . . . 2.1. Atom Trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Rydberg-Atom Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. STIRAP Excitation into Rydberg States . . . . . . . . . . . . . . . . . . . . . . . 2.4. Rydberg-Atom Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5. Laser Cooling and Magnetic Trapping in Strong Magnetic Fields . . . . . . . . . 3. Collision-Induced Rydberg-Atom Gas Dynamics . . . . . . . . . . . . . . . . . . . . 3.1. State-Mixing Collisions in Cold Rydberg-Atom Gases . . . . . . . . . . . . . . . 3.2. Relation between Cold Rydberg-Atom Gases and Cold Plasmas . . . . . . . . . 3.3. Collision-Induced Production of Fast Rydberg Atoms . . . . . . . . . . . . . . . 4. Towards Coherent Control of Rydberg-Atom Interactions . . . . . . . . . . . . . . . . 4.1. Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Coherent Rydberg Excitations in Many-Body Systems . . . . . . . . . . . . . . . 4.3. The Yin and Yang of Rydberg–Rydberg Interactions . . . . . . . . . . . . . . . . 4.4. Control Options for Rydberg–Rydberg Interactions . . . . . . . . . . . . . . . . 4.5. Methods to Measure the Blockade Effect . . . . . . . . . . . . . . . . . . . . . . 4.6. Experimental Implementation of an Excitation-Statistics Measurement . . . . . . 4.7. Results of Excitation-Statistics Measurements . . . . . . . . . . . . . . . . . . . 4.8. Effect of the Excitation Blockade on the Transition Linewidth . . . . . . . . . . 4.9. Experiments in Progress and Planned Research . . . . . . . . . . . . . . . . . . . 5. Rydberg-Atom Trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Electrostatic Rydberg-Atom Trapping . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Rydberg-Atom Trapping in Weak Magnetic Fields . . . . . . . . . . . . . . . . . 5.3. Ponderomotive Optical Lattices for Rydberg Atoms . . . . . . . . . . . . . . . . 5.4. Trapping of Rydberg Atoms in Strong Magnetic Fields . . . . . . . . . . . . . . 6. Experimental Realization of Rydberg-Atom Trapping . . . . . . . . . . . . . . . . . . 6.1. Production and Decay of Long-Lived Rydberg Atoms . . . . . . . . . . . . . . . 6.2. Oscillations in Trapped Rydberg-Atom Clouds . . . . . . . . . . . . . . . . . . . 6.3. State Analysis of Trapped Rydberg Atoms . . . . . . . . . . . . . . . . . . . . . 7. Landau Quantization and State Mixing in Cold, Strongly Magnetized Rydberg Atoms 8. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Abstract We employ ensembles of cold Rydberg-atom gases produced via photo-excitation of laser-cooled atom clouds in order to investigate collision-induced dynamics, exci131
© 2007 Elsevier Inc. All rights reserved ISSN 1049-250X DOI: 10.1016/S1049-250X(06)54003-0
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tation blockades, and Rydberg-atom trapping. After a general description of typical experimental methods used in this research, we review state-mixing and ionizing collision processes that have been observed in dense, cold Rydberg atom ensembles excited with broad-band (bandwidth 15 GHz) laser pulses. We then discuss how inter-atomic forces lead to Penning-ionizing collisions between Rydberg atoms and to the partial conversion of internal energy to center-of-mass energy. By exciting Rydberg-atom samples using narrow-band laser pulses (bandwidth 5 MHz), we create many-body states in which Rydberg excitations are coherently shared among several atoms. In this domain, van der Waals and electric–dipole interactions between Rydberg atoms give rise to an excitation blockade. We describe the implications of the blockade on the probability distribution of the number of Rydberg excitations that can be created in small atom ensembles. As a signature of the blockade, sub-Poissonian distributions of the Rydberg-atom number are presented. We further review possible methods of Rydberg-atom trapping in static electric and magnetic fields and in optical fields. A recent experimental demonstration of Rydberg-atom trapping in a Ioffe–Pritchard trap with a central magnetic field of 2.9 Tesla is discussed. The strongly magnetized Rydberg atoms we have trapped using this apparatus are atoms in long-lived, high-angular momentum states which are populated via Rydberg-atom collisions. We also present results on the photo-, electric-field-, and auto-ionization of cold, strongly magnetized Rydberg atoms.
1. Introduction Recently, dense clouds of laser-cooled atoms excited into high-lying Rydberg states (Stebbings and Dunning, 1983; Gallagher, 1994) have been studied by several research groups. A variety of dynamics can be observed, ranging from complex collision-induced dynamics that is closely linked to the physics of cold plasmas to coherent many-body interactions that have potential applications in quantum-information processing. With the availability of cold Rydberg atoms, it has also become possible to trap these atoms, opening new avenues for future research. In this chapter, we provide an overview of some recent work in the field, with an emphasis on contributions at the University of Michigan. One avenue in this research is to excite laser-cooled atom clouds with broadband laser pulses or with laser pulses derived from narrow-band continuous-wave laser sources into Rydberg levels, and to study the subsequent evolution of the generated Rydberg-atom gas over tens to millions of microseconds. In Section 2 we outline the typical laser cooling, Rydberg-atom excitation, and detection methods that are used to accomplish this. Specialized tools such as excitation of Rydberg states using Stimulated Raman Adiabatic Passage (STIRAP; see Sec-
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tion 2.3) and laser cooling in strong magnetic fields (Section 2.5) are presented in more detail. The variety of processes that have been identified in dense, cold clouds of Rydberg atoms include state-changing collisions of Rydberg atoms with other Rydberg atoms and with free electrons. These include l and n-mixing (Dutta et al., 2001; Walz-Flannigan et al., 2004), Penning ionizing collisions (Robinson et al., 2000) and electron-Rydberg collisions (Vanhaecke et al., 2005). Over a wide range of the experimental parameter space (atom density, atom number, principal quantum number) a large fraction of the Rydberg-atom gas ionizes and forms a cold plasma. It has also been determined experimentally (Walz-Flannigan et al., 2004; Li et al., 2004, 2005) and in calculations (Robicheaux, 2005) that Penning-ionizing collisions between Rydberg atoms can be triggered by interatomic forces. The forces cause atoms to “snap” together, leading to an exchange of energy between the internal and external degrees of freedom of the Rydberg atoms. Due to the presence of ionizing collisions, the study of cold Rydberg-atom gases is closely linked to cold-plasma research (Killian et al., 1999), which has become a main interest of some research groups. Conversely, three-body recombination leads to the formation of Rydberg atoms in cold, laser-excited plasmas (Killian et al., 2001). Experimental details such as atom and/or ion density, atomic states, evolution time, etc., determine which process is dominant. The field has also attracted considerable interest from theory groups (Mazevet et al., 2002; Robicheaux and Hanson, 2002; Kuzmin and O’Neil, 2002). In Section 3 of this chapter we review recent work on the evolution of cold Rydberg-atom gases. Recent results on collision-induced energy exchange between the internal energy and the external degrees of freedom of the Rydberg atoms are discussed in detail in Section 3.3. At lower densities and quantum numbers, and at shorter interaction times, a few resonant electric–dipole energy-exchanging Rydberg–Rydberg collisions have been observed to dominate the dynamics (Anderson et al., 1998; Mourachko et al., 1998; Anderson et al., 2002). Typically, narrow-band lasers, carefully controlled bias electric fields, and narrow-band microwave radiation are employed in order to spectrally resolve these binary Rydberg–Rydberg interactions. In these studies, the utilized interaction times are typically kept short, in order to avoid the ionizing and state-mixing collisions discussed in Section 3 of this chapter. Rydberg excitations may travel from atom to atom while the atoms largely remain frozen (up to a certain time). This interesting physical situation has been studied theoretically (Akulin et al., 1999; Frasier et al., 1999), and has recently undergone further investigation in new sophisticated experiments (Anderson et al., 2002; Afrousheh et al., 2004) and advanced modeling (Robicheaux et al., 2004). While the complex collision-induced dynamics described in Section 3 is incoherent, processes driven by a few well-defined, binary Rydberg–Rydberg interactions can, to a certain extent, exhibit experimentally relevant coherence. Among those
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processes, energy shifts caused by van der Waals and electric–dipole interactions between Rydberg atoms have attracted particular interest, because they lead to what is known as a “blockade” of Rydberg excitation. This blockade is important in proposed quantum information processing schemes (Jaksch et al., 2000; Lukin et al., 2001; Saffman and Walker, 2005). The proper physical picture that needs to be stressed in this context is that Rydberg atoms excited in a system of stationary atoms must be viewed as excitations that are phase-coherently distributed among all atoms. This situation is described by a many-body wave-function with only a few (mostly two) internal-state components per atom. A “Rydberg blockade” occurs if many-body states with more than one shared excitation cannot be excited because they are shifted out of resonance due to an interatomic interaction. There are analogies of “blockaded” Rydberg systems with quantum dots, in which a Coulomb blockade can occur. Recently, significant progress has been made in experimental realizations of Rydberg-blockade effects of different kinds (Tong et al., 2004; Singer et al., 2004; Liebisch et al., 2005), numerical modeling by Robicheaux and Hernández (2005), and by Ates et al. (2006). In Section 4 of this chapter we review some aspects of many-body systems with coherent Rydberg excitations. Recent results on a novel measurement of the Rydberg blockade using statistics techniques is presented in detail. An interesting area of new research with cold Rydberg atoms could emerge if one were to demonstrate the trapping of Rydberg atoms. Possible avenues involving the control of center-of-mass motion to deterministically load Rydberg atoms and to coherently control their interactions in cavity-QED environments are already on the horizon (Zheng and Guo, 2000; Osnaghi et al., 2001; Hyafil et al., 2004). Elaborate forms of Rydberg-atom trapping such as periodic ponderomotive optical lattices (Dutta et al., 2000) may open new possibilities in the precision measurement of Rydberg-atom transitions. Also, trapping of Rydberg atoms may become useful in cold antihydrogen research (Amoretti et al., 2002; Gabrielse et al., 2002a) where antihydrogen Rydberg atoms are formed by threebody recombination in strongly magnetized plasmas (Gabrielse et al., 2002b). There, it would be beneficial to trap these antihydrogen Rydberg atoms as they radiatively decay into the ground state, thereby maximizing the yield of groundstate antihydrogen atoms. The latter are highly sought after in order to test fundamental symmetries. In Section 5 we discuss a few potential options of how Rydberg atom traps could be realized. We focus on trapping in strong magnetic fields (Section 5.4) and on a recent first experimental demonstration of a highmagnetic-field Rydberg atom trap (Section 6). The chapter is concluded with a special topic dealing with cold Rydberg atoms in strong magnetic fields, namely the photo-, electric-field, and auto-ionization behavior of such atoms (Section 7). This topic also is related to ongoing studies of antihydrogen (Vrinceanu et al., 2004).
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2. Preparation and Analysis of Cold Rydberg-Atom Clouds Rydberg atoms are produced when an atom absorbs energy from a collision partner or its surroundings and one of its valence shell electrons is promoted into a state of high principal quantum number n. The transfer of energy can occur via impact with a free electron, photo-excitation, or a combination of collisional and optical excitation (Gallagher, 1994). Collisional charge exchange between ions and excited atoms can also be used to prepare Rydberg states of the initially ionic species (Gallagher, 1994). In this chapter, we are concerned with ensembles of cold Rydberg atoms, i.e., Rydberg atoms with low center-of-mass velocities. Since there are, at present, no techniques available for the active cooling of Rydberg atoms, the only appropriate method for the preparation of cold Rydberg atoms is photo-excitation of laser-cooled ground-state atoms. In this section we describe the methods we use to obtain cold samples of ground-state atoms, photoexcite these atoms into Rydberg states, and detect the internal-state distribution and spatial distribution of the Rydberg-atom ensembles. A novel method to prepare cold atoms in magnetic fields of several Tesla is also presented at the end of the section. 2.1. ATOM T RAPPING In all experiments presented, a vapor-cell magneto-optical trap (MOT) is used to initially laser-cool and trap 85 Rb ground-state atoms. The MOT is the most widely used, versatile, and simplest method of obtaining ensembles of cold ground-state atoms. The MOT cools and traps atoms as a result of position-dependent radiation pressure forces. The vapor-cell MOTs we employ consist of a pair of antiHelmholtz coils wrapped around rubidium vapor cells and three pairs of counterpropagating and suitably σ -polarized laser beams (Raab et al., 1987). The coils produce a quadrupole magnetic field of the form B(r) = (αx/2, αy/2, −αz), with α ∼ 20 Gauss/cm, and thus induce a linear Zeeman shift of the atomic states with non-zero magnetic quantum number mF . The laser beams propagate in the ±ˆx-, ±ˆy- and ±ˆz-directions, and are slightly red-detuned from the transition 5S1/2 , F = 3 → 5P3/2 , F = 4. The beams are σ -polarized in a way that an atom displaced from the trap center in any given direction predominantly scatters light from the laser beam that provides a restoring radiation-pressure force pointing towards the trap center. This property is achieved by a combination of optical pumping into the lowest mF sublevel and the red-shift of the dominant transition due to the Zeeman effect. Details on the implementation of magneto-optical traps are provided by Raab et al. (1987) and Metcalf and van der Straten (1999). Our vapor-cell MOTs yield densities of 85 Rb ground-state atoms up to 1010 cm−3 at temperatures near the Doppler limit (140 µK or an average atomic velocity of 15 cm/s) (Metcalf and van der Straten, 1999). A typical vapor-cell
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MOT has a diameter of ∼1 mm and contains ∼107 atoms. For most of the experiments described in this chapter, these densities and atom numbers are sufficient. In the research described in Section 2.3 and Section 7 it was necessary to limit the interactions between the excited Rydberg atoms by an artificial reduction of the MOT atom density or by using an attenuated Rydberg excitation beam. The density reduction was achieved by reducing the intensity and the diameter of the trapping beams. In the experiments on collision-induced Rydberg-gas dynamics discussed in Section 3, however, we seek to enhance atom–atom interactions by providing atom numbers and densities larger than those achievable in plain vapor-cell MOTs. The atom number can be increased using a double-trap design, in which a secondary, ultra-high-vacuum MOT is loaded by a cold atomic beam extracted from a primary, vapor-cell trap. In this design, most atoms entering the range of the secondary MOT have a velocity less than the MOT capture velocity (a few tens of m/s), while the background pressure and the collision-induced atom loss rate are suppressed by a few orders of magnitude. Due to these factors, the atom numbers achievable in double-MOT systems are larger than those in plain vapor-cell MOTs. In the experiments presented in this chapter, the primary traps used in doubleMOT systems are of the so-called LVIS type [Low Velocity Intense Source, first demonstrated by Lu et al. (1996)]. An LVIS can be implemented in several ways. In the experiments discussed in Section 3 we use the original method of Lu et al. (1996). There, one of the retro-reflecting mirror-waveplate assemblies used in standard MOTs is placed in the vacuum tube connecting the primary and the secondary trap volume. The assembly has a central hole with a diameter of order 1 mm. The trap magnetic field is adjusted such that a trappedatom cloud forms about 5 to 10 mm vertically above the hole. Due to the radiation pressure imbalance caused by the hole, cold atoms are expelled from the trap through the hole, thereby forming a cold atomic beam. The atomic beam has velocities on the order of 10–15 m/s and can be used to load secondary MOTs (as done in the experiments described in Section 3). Alternatively, one may employ a separate, small-diameter pusher beam that is focused through a standard 6-beam vapor-cell MOT. The pusher beam expels trapped atoms and thereby forms a directed, cold atomic beam (Swanson et al., 1998; Myatt et al., 1996). We have used pusher lasers in the experiments described in Sections 4, 6 and 7. In a few experiments, we utilize a pyramidal LVIS in which a large diameter, circularly polarized laser beam is incident on a pyramidally shaped mirror assembly with a hole drilled at the apex. The pyramidal mirror is situated inside an appropriate quadrupole magnetic field. The pyramidal MOT, proposed by Lee et al. (1996), was first implemented by Williamson et al. (1998) as an LVIS. We have found that among the three methods to prepare cold atomic beams the pyramidal LVIS is the most stable and the easiest to align.
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In addition to double-trap setups which allow one to prepare background-vaporfree cold-atom clouds larger than those in plain vapor-cell MOTs, higher densities can be temporarily achieved using a magneto-optic compression phase (Petrich et al., 1994) or by compressing the atoms in an electric–dipole laser trap (Chu et al., 1986).
2.2. RYDBERG -ATOM E XCITATION In all experiments discussed here, gases of cold Rydberg atoms are created by photo-excitation of cold ground-state atoms in MOTs and high-magnetic-field atom traps. In the following discussion we consider the case of excitation in zero magnetic field. The described experimental techniques translate directly to the case of strong magnetic fields. The main difference between the two cases lies in the nature of the Rydberg atoms and cold plasmas that are generated. Strongly magnetized, cold Rydberg atoms are discussed in depth in Sections 5.4, 6 and 7. The excitation of Rb Rydberg atoms can, in principle, be achieved in several ways. One could turn off the atom trapping light at the time of excitation and excite directly from the Rb 5S1/2 ground state to nP Rydberg states via a one-photon UV transition (transition wavelength ≈ 297 nm). Alternatively, one could employ a two-photon transition to the Rydberg state. A particularly convenient twophoton transition of Rb into nS or nD Rydberg states is 5S1/2 → 5P3/2 → nS or nD, because both the lower and upper transitions have optical wavelengths (780.23 nm and ≈ 480 nm, respectively) and the lower transition can be excited by the already-available laser light that drives the atom-trap cycling transition. In all experiments discussed in this chapter this excitation method is used. An added benefit of the two-photon scheme is that the oscillator strengths of the 5P3/2 → nD transitions are considerably higher than those of the 5S1/2 → nP transitions. The respective photo-ionization cross-sections, which represent a measure for the excitation probabilities one can obtain for equal photon fluences and laser linewidths, are σPI = 1.7 × 10−20 cm−2 for Rb 5S1/2 (Lowell et al., 2002) and 1.5 × 10−17 cm−2 for Rb 5P3/2 (Gabbanini et al., 1998). In the experiments on collision-induced Rydberg-atom gas dynamics (Section 3) and on Rydberg atoms in high magnetic fields (Sections 6 and 7), atoms are excited to the 5P3/2 state by the atom-trap laser and are then excited to Rydberg states by a Nd:YAG-laser-pumped tunable pulsed dye laser (λ ≈ 480 nm, pulse width ≈ 10 ns, bandwidth ≈ 15 GHz, pulse energy 3 mJ). In Section 4 we describe an experiment which quantifies a “blockade” of Rydberg atom excitation. To realize a blockade of Rydberg excitation, it is necessary to achieve narrowband excitation into Rydberg states (laser bandwidth 10 MHz). The broadband pulsed dye lasers discussed above are not suitable to measure excitation-blockade effects. In the experiments in Section 4, the upper transition laser pulse is created
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by passing light from a narrow-band continuous-wave frequency-doubled external cavity diode laser (∼5 MHz linewidth at 480 nm) through amplitude-modulated acousto-optic modulators (AOMs). In addition, we perform the first step in the two-step excitation to the Rydberg state by turning off the MOT trapping light during the time of excitation and applying a resonant laser pulse coupling the 5S1/2 ground state to the 5P3/2 state. Typically, the lower- and upper-transition laser pulses are chosen to be 100 ns in duration. In all discussed experiments, during the excitation, the atoms move less than a few tens of nanometers, which is a few orders of magnitude less than the average inter-atomic separation. In Fig. 1 we show two examples of Rydberg excitation spectra obtained at low atom density. In the spectra, the frequency of the lower-transition laser is held constant and the frequency of the laser that excites the transition from 5P3/2 to Rydberg states is varied. Details on how the Rydberg atom signal is detected can be found in Section 2.4. Figure 1(a) shows a spectrum obtained by broadband excitation with a pulsed dye laser, as in Section 3, and Fig. 1(b) shows a spectrum of the 44D fine structure doublet obtained by narrowband diode laser excitation, as in Section 4. The frequency of the dye laser is scanned in steps of 6.75 GHz while the frequency of the diode laser is scanned in steps of 1.6 MHz. The fine frequency scan of the diode laser was accomplished by locking the laser to a scanning pressure-tuned Fabry–Perot interferometer (Hansis et al., 2005). The scan range in Fig. 1(a) approximately equals 104 times the range in Fig. 1(b). In Fig. 1(a), the width of the Rydberg resonances is about 15 GHz. Since the natural linewidth of the displayed Rydberg resonances is only a few kHz, and because line broadening due to environmental fields amounts to less than about 10 MHz, the linewidth observed in Fig. 1(a) is entirely due to the bandwidth of the laser that excites atoms from 5P3/2 into Rydberg states. Another feature which can be seen in Fig. 1(a) is that nD and (n + 2)S levels appear in pairs with energy spacings that are considerably smaller than the spacing between adjacent nD levels or nS levels. This pattern reflects the quantum defects of the Rb nD and nS levels, which are δD = 1.35 and δS = 3.13, respectively. Due to electric–dipole selection rules and due to the absence of substantial electric fields in the excitation region, the spectrum does not exhibit any nP -lines. It is further apparent in Fig. 1(a) that adjacent nD and (n + 2)S states become unresolvable above n ≈ 44. All states become unresolvable above n ≈ 70 (as seen in Fig. 5 below). This agrees with the fact that for n 70 the linewidth of the laser used in Fig. 1(a) approaches or exceeds the hydrogenic energy spacing, 27.2 eV/n3 (it is 27.2 eV/n3 = h×19 GHz for n = 70). In the spectrum in Fig. 1(b), which was obtained with a narrow-band diode laser, the width of the Rydberg resonance lines is about 7 MHz, which slightly exceeds the estimated laser linewidth of 5 MHz (presumably due to some Rydberg level broadening caused by weak stray electric fields). The line pair in Fig. 1(b) is a fine-structure doublet of 44D lines, which are separated by 140 MHz [i.e., about one-hundredth of the resolution limit of the laser used in Fig. 1(a)].
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(a)
(b) F IG . 1. Rydberg atom signal as a function of the wavenumber of the upper-transition (5P3/2 → nD) excitation laser. The frequency of the lower-transition laser (5S1/2 → 5P3/2 ) remains fixed. The spectrum in (a) is obtained by broadband excitation with a pulsed dye laser (linewidth about 15 GHz), while the spectrum in (b) has been taken with narrowband diode laser excitation (linewidth about 5 MHz).
2.3. STIRAP E XCITATION INTO RYDBERG S TATES The excitation schemes described in Section 2.2 are sufficient for most studies of cold Rydberg atoms. However, they are limited fundamentally in their achievable excitation efficiency. In applications of a Rydberg blockade to quantum information processing, it is necessary to maximize and stabilize the Rydberg excitation efficiency. If one were to saturate the transition between the ground state and the Rydberg state with an intense UV laser beam, the maximum achievable Rydberg excitation efficiency would only be about 50%. A sequential, highly saturated two-step excitation process would yield about 25% excitation efficiency. Even if
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25% or 50% excitation efficiency were acceptable for a given application, saturating the transition from 5S or 5P into a Rydberg state could still be impractical due to the large laser intensity that would be required (because of the low oscillator strengths of optical transitions into Rydberg states). Excitation efficiencies larger than 50% require some form of coherent control of the excitation process, such as the application of π-pulses or STIRAP (Stimulated Raman Adiabatic Passage), discussed here. Efficient Rydberg-atom excitation using π-pulses has been demonstrated by Mossberg et al. (1977), but was achieved with a laser with a large spectral width. Such broadband Rydberg excitation is impractical for applications of a Rydberg blockade to quantum information science. One could, in principle, derive narrowband π-pulses from a narrow-band continuous-wave laser using acousto-optic modulators driven with suitable radio-frequency pulses. However, even if sufficient laser power were available to accomplish a narrow-band π-pulse Rydberg excitation, the transfer efficiency would suffer from a high sensitivity to laserpulse inhomogeneities and other instabilities, such as variations in laser power and frequency which would be hard to eradicate. For these reasons, we have demonstrated the use of STIRAP as an alternative excitation scheme. Using STIRAP one can, in principle, achieve 100% excitation efficiency while avoiding the high susceptibility of π-pulses to laser frequency and intensity variations. A detailed discussion of the benefits of STIRAP as opposed to π-pulses is provided by Shore et al. (1992). STIRAP is a modern technique employed widely to coherently transfer population in a three-level system (Bergmann et al., 1998), but is equally appropriate to a ladder excitation scheme, such as that discussed here. STIRAP is achieved using a “counterintuitive” pulse sequence in which the “Stokes pulse” (uppertransition pulse) precedes in time but overlaps the pump pulse (lower-transition pulse). The pair of pulses is tuned to the two-photon resonance, and both transitions ideally have Rabi frequencies much greater than the intermediate-state decay rate. In a three level system such as that shown in Fig. 2(a), the upper-transition pulse is applied first and couples the initially unpopulated states |2 and |3. In a dressed-state picture, the resultant superposition states are tuned out of resonance from the |1-level by the upper-transition Rabi frequency. Thus, while the uppertransition pulse is turned on, atoms in level |1 remain in level |1. Once the lower transition pulse turns on, the state |1 adiabatically evolves through a coherent superposition of states |1 and |3 into state |3, during the temporal overlap region of the pulses. The atoms are in the pure target state |3 once the upper-transition pulse has turned off. Finally, the lower-transition pulse, which couples states |1 and |2, is turned off at some arbitrary later time, without affecting the atoms any more. During the whole evolution, the radiatively decaying intermediate state |2 is not significantly populated. Thus, in the whole process the state vector of the system is adiabatically rotated from |1 into |3 in a plane perpendicular to |2,
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F IG . 2. (a) Level diagram for STIRAP excitation. (b) Experimental timing diagram showing one experimental cycle, which includes two identical STIRAP and Rydberg-atom detection sequences. Two sequences per experimental cycle are employed in order to obtain a reliable measure of the absolute STIRAP excitation efficiency. Prior to the STIRAP excitations, optical-pumping pulses on the lower transition are applied in order to ensure that the STIRAP dynamics is restricted to a three-level system.
and 100% of the population is transferred to state |3 with no radiative loss from the intermediate state. A detailed theoretical description of STIRAP can be found in Bergmann et al. (1998). In our experiment (Cubel et al., 2005), population is adiabatically transferred from the 85 Rb ground state 5S1/2 |F = 3, mF = 3 to the 44D5/2 |F = 5, mF = 5 Rydberg state, with negligible population in the rapidly decaying intermediate state 5P3/2 |F = 4, mF = 4. We laser cool and trap 85 Rb ground-state atoms, turn off the MOT trapping light for 280 µs, and apply the Rydberg excitation pulses. The lower- and upper-transition pulses are derived from diode lasers and are shaped using acousto-optic modulators. The main challenge in this experiment has been the development of an experimental technique that allows one to measure the absolute Rydberg-atom excitation efficiency without having to rely on uncertain parameters such as the atom detection efficiency and the number of atoms in the excitation region. In the utilized method, two identical sequences of excitation pulses (i.e., two identical STIRAP sequences) are applied to the same sample of atoms during each experimental cycle. The two STIRAP sequences have the same time delay of the upper-transition pulse relative to the lower-transition pulse. A negative time delay corresponds to a counter-intuitive sequence, in which the upper-transition pulse precedes the lower-transition pulse.
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F IG . 3. Number of Rydberg atoms detected after the first (triangles up) and second (triangles down) STIRAP sequences as a function of time delay T between the upper- and lower-transition laser pulses. T < 0 corresponds to the upper-transition pulse preceding the lower-transition pulse. The experimental results are compared with theory for a lower-transition Rabi frequency of 10 MHz and upper-transition Rabi frequencies of Ω/(2π ) = 7.5 MHz (dashed lines) and 5 MHz (solid lines). (The upper-transition Rabi frequency is not precisely known in the experiment.) Maximum excitation efficiency is observed for negative T , as expected for STIRAP. [Reprinted figure with permission from Cubel et al. (2005).] © 2005 American Physical Society
The two STIRAP sequences have durations of less than 1 µs and are separated in time by 60 µs [see Fig. 2(b)]. This separation is sufficiently short that atoms do not migrate into or out of the excitation region between the two STIRAP sequences. After each of the two individual STIRAP sequences, we count the number of Rydberg excitations via electric-field ionization and thereby remove all Rydberg atoms from the sample (details on Rydberg-atom detection can be found in Section 2.4). Therefore, the second STIRAP sequence of the pair applied in each experimental cycle starts with the ground-state population left over after the detection pulse of the first sequence. The Rydberg atom signal detected after the second sequence is thus reduced relative to that after the first sequence. This signal reduction is a robust measure for the absolute efficiency of the Rydberg atom excitation. In the experiment described next, we have measured the absolute excitation efficiency and how it depends on the time delay between the upper- and lower-transition pulses in the STIRAP sequences. Figure 3 shows an example of typical data we obtained for upper- and lowertransition pulses of 500 ns duration and Rabi frequencies in the range from 5 MHz to 10 MHz. The Rabi frequencies were limited by the laser power available for
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the upper transition and were roughly the same as the radiative level width of the intermediate state. Under these constraints, the STIRAP excitation efficiency is highest when both laser pulses are tuned into resonance with the respective singlephoton transitions. As seen in Fig. 3, maximum excitation efficiency occurs for small, negative time delay, corresponding to counter-intuitive pulse ordering. At the maxima of the displayed data, the Rydberg-atom yield measured after the second STIRAP sequence is reduced by roughly one-half relative to that measured after the first sequence. Under the assumption of a spatially constant excitation efficiency, this result indicates that we achieve roughly 50% excitation efficiency into the Rydberg state 44D5/2 (Cubel et al., 2005). However, due to the fact that our excitation pulses have Gaussian beam profiles, the Rabi frequencies on the lower and upper transitions are inhomogeneous (mostly in the radial direction). Therefore, a careful modeling of the experiment is required in which the density matrix equations for the system are solved accounting for the inhomogeneous Rabi frequencies and using experimentally measured temporal pulse profiles. A detailed analysis of the experiment along these lines can be found in Cubel et al. (2005), showing a maximum excitation efficiency of roughly 70% at the center of the beams where the Rabi frequencies are highest. While this excitation efficiency is significantly higher than what could be expected for a standard, saturated two-photon transition, it still is significantly below the theoretically possible 100%. In simulations we have seen that the excitation efficiency could be increased from our present best value of 70% by three main improvements. First, by using higher laser powers one would be able to increase the Rabi frequencies of the involved transitions and therefore minimize radiative decay from the intermediate level. Second, using arbitrary wave-form generators the pulses could be carefully shaped in a manner that optimizes the adiabaticity of the transition process. Further, in the experiment one could use smaller atom clouds in order to eliminate the effects of the spatial variation of the Rabi frequencies across the extent of the atom cloud.
2.4. RYDBERG -ATOM D ETECTION Since the photo-absorption cross-sections of atoms for transitions into Rydberg states are very small, in most spectroscopic investigations of Rydberg atoms detection methods are employed in which the Rydberg atoms themselves are detected. Rydberg atoms may be detected directly in one of two ways. First, one may use photomultiplier tubes to observe the visible fluorescence emitted by Rydberg atoms as they radiatively decay to low-energy states. Second, the valence electron in a Rydberg atom is in a highly excited state characterized (classically) by a large, weakly bound orbit. As a result, Rydberg atoms ionize easily due to either
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collisional ionization or due to the application of a modest electric field. The free electrons and/or ions can then be detected (Gallagher, 1994). In the STIRAP experiment described in Section 2.3 and in all subsequently described experiments, we employ electric-field ionization to detect Rydberg atoms. State-selective field ionization (SSFI) is a technique that allows one to not only detect and count all atoms excited to a Rydberg state but also to obtain limited information on the distribution of Rydberg states. The latter aspect is important in experiments in which the state distribution of Rydberg atoms changes due to collisions, such as in Sections 3, 6, and 7. SSFI involves the application of a timevarying electric field pulse to a sample of Rydberg atoms and identification of Rydberg states by the electric field at which they ionize. One typically applies an electric-field ramp with rise-times up to tens of μs and detects the electrons liberated from the loosely-bound Rydberg atoms as a function of time. Using the known shape of the electric-field pulse, the time-dependent Rydberg-atom signal is then transformed into a population distribution as a function of ionization electric field. Peaks in this signal typically correspond to Rydberg states of a certain energy. This simple interpretation of SSFI signal neglects ambiguities in the assignment of Rydberg states that result from the time dependence of the ionization process and from the mapping of the multi-dimensional Rydberg atom state space onto the single dimension space of the electric field. In pulsed field ionization, the ionization field for a given initial state will, in general, depend on the rise time of the electric field pulse. This is important in the SSFI of non-hydrogenic atoms such as rubidium. Due to the non-zero quantum defects of non-hydrogenic atoms, the energy levels belonging to adjacent n-manifolds exhibit avoided crossings when plotted versus the electric field (these plots are referred to as Stark maps). During the field ionization pulse, atoms starting in a given initial state traverse the Stark map, ionize and produce a signal when they reach the ionization threshold of the Stark energy level on which they are traveling. Typically, while traversing the Stark map the atoms encounter anti-crossings with other Stark levels. Depending on the separation of the anti-crossings in the Stark map and on the rise time (or slew rate, S = dE/dt) of the electric-field ramp, the atoms may pass through these anti-crossings adiabatically (remaining in the same energy level) or diabatically (jumping energy levels), resulting in different ionization electric fields. Diabatic passage typically results in higher ionization electric fields for a given initial state than adiabatic passage. If the slew rate is sufficiently low and the separation of the anti-crossings in the Stark map sufficiently large for the passage to be adiabatic, the ionization electric field is given by the classical ionization threshold for m = 0 states. In this case, the ionization threshold for low- states is 2 1 E = W4 → 16n 4 (in atomic units), where W is the level energy and ns denotes an s effective quantum number in the presence of an electric field. Typically, this SSFI behavior is found for atoms that are initially in a low-angular-momentum (low- )
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state with a large quantum defect. If atoms follow this classical SSFI behavior, the relationship between E and n is well-defined. Rydberg states with large |m| have Stark maps with very narrow anti-crossings, leading to diabatic field ionization behavior. For given n, the ionization electric fields of high-|m| states range from E = 9n1 4 to E = 4n1 4 , depending on the s s exact parabolic quantum numbers of the state. Blue-shifted levels tend to ionize at higher and red-shifted levels at lower electric fields. As a result, for high-|m| states SSFI does not allow for an unambiguous determination of n. Much more detail on electric field ionization of Rydberg atoms can be found in Gallagher (1994) and references therein. In the experiments described in this chapter we use mostly SSFI electric-field ramps that are slow enough that atoms in low- states ionize adiabatically, allowing for a well-defined assignment of n-values to peaks in SSFI measurements. We do, however, see evidence of diabatic passage in the SSFI of very high-lying low- states (n 70) and of high-|m| states produced in collisions. In the following, we discuss some of the experimental details of the SSFI method. The SSFI pulse is typically applied via solid or wire-mesh electrodes situated around the atom trapping region. The SSFI electric-field pulse accelerates the liberated Rydberg electrons toward an electron detector. In the region between the SSFI zone and the electron detector, we typically employ guiding electrodes with a circular or rectangular cross section, held at potentials of order 10 to 100 V relative to the SSFI location, in order to control the electron trajectories and to increase the electron collection efficiency. The electrons then impinge on microchannel plate detectors (MCP). An MCP is a periodic array of fused thin lead glass capillaries (or channels) sliced into a thin plate. A single incident electron enters a channel, hits a channel wall, and releases secondary electrons. The latter are accelerated by voltages 1000 V applied across the MCP, and produce more secondary electrons. We use two-stage MCPs with gains in the range of 108 . Pulses are capacitively coupled out of the MCP rear face. In this way, each electron detected by the MCP creates a voltage pulse of the order of 10 mV in height and several nanosecond duration. These pulses are typically pre-amplified and counted via a pulse-counting apparatus such as a single-photon counter, as in the experiments described in Sections 3, 4 and 6. Dense sequences of MCP pulses can also be averaged and displayed as a function of time (or as a function of applied electric field) on an oscilloscope in order to infer the distribution of states in a Rydberg-atom gas. This method is used widely in our experiments. MCPs are often used in conjunction with a phosphor screen mounted behind the MCP output face. The phosphor screen emits a localized fluorescent light burst when impacted by an electron shower. The screen allows one not only to determine the temporal distribution of electron counts, but also the spatial distribution of the counts. If the electron transfer function between the SSFI region and the MCP is sufficiently well defined, one can use the spatial image acquired
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on the phosphor screen to infer the spatial distribution of Rydberg atoms at the time of ionization. This method is used in Section 6, where a strong magnetic field is employed in order to image SSFI electrons onto an MCP-phosphor-screen assembly. 2.5. L ASER C OOLING AND M AGNETIC T RAPPING IN S TRONG M AGNETIC F IELDS In order to conduct research on cold Rydberg atoms and plasmas in strong magnetic fields, we have developed a special trap for Rb ground-state atoms (high-B trap) operating in a bias field of several Tesla (Guest et al., 2005). This work represents a nearly factor-of-20 increase in the magnitude of the bias B-field compared to previous atom traps. The high-B atom trap is a prerequisite to the study of cold Rydberg-atom gases and plasmas in the strong-magnetization regime, discussed in Sections 6 and 7. The high-B trap provides a spin-polarized cold sample of 85 Rb atoms in a single hyperfine state and therefore a well-defined launch state for the Rydberg-atom excitation. Also, the Rydberg atoms are essentially stationary during the experiments, enabling a careful time-resolved study of the evolution of the initially excited states over hundreds of milliseconds. Perturbations by motional electric fields, which can be significant in atom-beam experiments, are very small because the velocity |v| of the atoms is only of order 10 cm/s. In the following, we describe laser cooling in variable magnetic fields up to 6 T and magnetic trapping at fields up to 2.9 T. Our high-B trapping apparatus is a superconducting Ioffe–Pritchard trap. Magnetic trapping conditions are realized with simultaneous operation of superconducting dipole and quadrupole coils [Fig. 4(a)]. The typical operating trap depth is a few hundred Gauss (equivalent to a few tens of mK), and the curvature of the field is a few mT/cm2 . The trap is loaded with a continuous, accelerated cold atomic beam of 85 Rb from a pyramidal Low Velocity Intense Source (LVIS) (Lu et al., 1996) located 68 cm from the center of the superconducting magnetic trap along the axis of the superconducting dipole coils. Atoms leaving the LVIS region are accelerated by a narrowly focused (500 µm beam waist) blue-detuned (≈10–25 MHz) high intensity pusher laser beam aligned parallel to the atomic beam generated by the LVIS. The strong magnetic field of the high-B trap produces a magnetic-dipole potential hill that prohibits loading of the high-B trap unless the acceleration beam √ is applied. At 6 T, a velocity of the incident atoms larger than 28 m/s (= 2μB B/M; M is the atomic mass of 85 Rb) is required for the loading. In the LVIS and during the acceleration, atoms scattered into the level 5S1/2 , F = 2 are optically re-pumped into the state 5S1/2 , F = 3 via excitation into the level 5P3/2 , F = 3 by optical sidebands on the trapping and accelerating beams (Myatt et al., 1996). The atoms emitted by the LVIS-pusher-beam combination remain in the low-field-seeking state
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F IG . 4. (a) Experimental setup of the high-B atom trap. An LVIS, located 68 cm from the high-B trap center, produces a beam of slow atoms. The curve drawn on the coils represents the magnitude of the magnetic field generated by the coils. (b) Two-step photo-ionization process used for atom counting. (c) Photo-electrons escaping the trapped-atom cloud follow the magnetic field lines and strike the charged-particle detector (MCP). A circular distribution at the center of the trap is mapped onto an ellipse with an aspect ratio of 23 : 1. The electron trajectories are not drawn to the scale. A typical phosphor image is also shown. (d) Electron counts (proportional to the number of trapped atoms) as a function of the molasses laser frequency. The laser-cooled and trapped atoms are collected near the minimum-field location of the trap (B = 2.9 T). The inset is an optical image of the atom cloud.
(5S1/2 |F = 3, mF = +3 = 5S1/2 |mJ = +1/2, mI = +5/2) as they enter the region of strong magnetic fields. Because of the large separation of the LVIS and the atom trap (68 cm), the capture efficiency is only a few percent. The atoms reaching the center of the high-B trap are slowed and cooled by a six-beam optical molasses. Laser cooling in the superconducting atom trap is performed along all three axes by counter-propagating optical molasses beams,
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which are tuned slightly to the red of the 5S1/2 |mJ = +1/2, mI = +5/2 → 5P3/2 |mJ = +3/2, mI = +5/2 transition. Since both the ground- and the excited-state Zeeman manifolds are in the Paschen–Back regime of the hyperfine structure, no re-pumping laser is necessary. In a 6 T field, the Zeeman shifts of the involved states change the frequency of the transition by ≈80 GHz. Therefore, the molasses laser cannot be stabilized using conventional saturation spectroscopy in rubidium vapor-cells. Instead, we use a home-built temperature-stabilized and pressure-tuned Fabry–Perot interferometer to lock the high-B trap laser (drift rate a few MHz per hour) (Hansis et al., 2005). The cooled and trapped atoms are counted at a repetition rate of 10 Hz cycle through two-step photo-ionization and electron detection, as indicated in Fig. 4(b). The optical molasses operates most efficiently when the molasses beam is approximately a linewidth (6 MHz) red-detuned from the atomic transition, while the population in the 5P state is maximized by introducing on-resonance laser light. For this reason, we switch the frequency of the 5S–5P molasses laser using an acousto-optical modulator. During a brief (5 µs) probe phase, the molasses light is upshifted into resonance and increased in intensity (I Isat ), causing a brief, temporary enhancement of the population in the 5P -state. During this phase, a blue laser pulse (10 ns pulse width) tuned above the photo-ionization threshold of the 5P -level (λ < 479.0 nm) is introduced to photo-ionize a fraction of the trapped atoms. The photo-electrons are detected on a micro-channel plate detector (MCP) located 47 cm from the trap center. Alternatively, the blue laser can be tuned below the photo-ionization threshold in order to excite 5P -atoms into Rydberg states, and the Rydberg atoms can be counted using electric-field ionization. Since in magnetic fields of several Tesla the liberated electrons are pinned to the magnetic field lines passing through the parent atoms, the electrons maintain spatial information about the atom cloud as they stream down the magnetic field lines and strike the MCP. The imaging geometry of an electron distribution located on a circle centered at the location of the trap is shown in Fig. 4(c). Due to the presence of the quadrupole field component in the magnetic field, the circular electron distribution is mapped onto an ellipse with an aspect ratio of 23 : 1. The inset on the lower right of Fig. 4(c) shows a measured phosphor image of an electron distribution with an initially circular crosssection. When both the dipole and quadrupole coils are operating, the magnetic field has the geometry of a Ioffe–Pritchard trap with a local field minimum at the trap center. Laser-cooled atoms become trapped near the local field minimum. In our experiment, this Ioffe–Pritchard mode of operation can be employed for central magnetic fields up to 2.9 T. The steady-state spectral signature of atoms captured and cooled from the atomic beam into the high-B atom trap can be studied by measuring the electron signal versus the trap-laser frequency. In Fig. 4(d), the measured population of atoms is shown by the filled dots. The result shows a
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slightly asymmetric peak that has a linewidth smaller than the natural linewidth, indicating that most of the atoms are tightly localized in space. The highest atom numbers and densities in the high-B trap are achieved with intensities of the molasses beams close to the saturation intensity of the cooling transition (1.6 mW/cm2 ). In the presented case, we count 2.4×107 trapped atoms on the peak, equivalent to a central density of 2 × 109 cm−3 . In the figure, a comparison is made with a spectrum from the atomic beam source without laser cooling (open dots, scale magnified 500 times). For the reference spectrum, the molasses laser is switched off during the laser-cooling phase and is turned on only during the probe phase. The atomic-beam spectrum is characterized by a broad and weak peak at and above the transition frequency at the trap center. The shape of the atomic-beam spectrum reflects the spatial variation of trap-field-induced shift of the probe transition over the atomic-beam cross-section. The spatial distribution and temperature of the laser-cooled atomic cloud can be obtained by examining the image of the electrons generated through photoionization. Although the acquired phosphor images are distorted due to the asymmetry in the magnetic field, the shape of the original atom cloud can be inferred from the phosphor images using the calculated magnetic-field geometry. From the deduced size of the original atom cloud, the temperature of the atoms confined in the nearly harmonic trap can be determined using the virial theorem. Under the above-mentioned operating conditions, we find that the atoms are laser-cooled to the Doppler limit (140 µK). Leaving the quadrupole coils off, we have slowed and laser-cooled atoms in bias fields up to B = 6 T. In this mode of operation, the magnetic field exhibits a saddle point at the trap center (and not a minimum, as in the case of the Ioffe–Pritchard trap geometry). Thus, atoms are magnetically confined only in the axial direction, while in the transverse directions the negative curvature of the magnetic field leads to anti-trapping. Without the transverse confinement, the optical-molasses lifetime reduces to tens of milliseconds, which compares to tens of seconds in the Ioffe–Pritchard mode. Nevertheless, using a total molasses laser power of 30 mW we can still continuously maintain about 2 × 105 atoms at the magnetic-field saddle point with B = 6 T. Operating the LVIS atomic beam in a pulsed rather than a continuous fashion and synchronizing the molasses cycle with that of the atomic beam, the atom number can be temporarily pushed up to 3 × 106 , with central atom densities of 4 × 107 cm−3 .
3. Collision-Induced Rydberg-Atom Gas Dynamics Because Rydberg atoms are large, with an atomic radius that scales as n2 , the cross-sections for a variety of interactions are enormous. Therefore, ensembles of Rydberg atoms may exhibit collision-dominated evolution at densities as low
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as 107 cm−3 to 108 cm−3 . In Rydberg-atom gases excited in a room-temperature atomic gas or in a dense atomic beam, collisions between Rydberg atoms simply occur due to the thermal center-of-mass motion of the atoms. If the Rydberg-atom gas is “cold”, center-of-mass velocities (∼ 0.1 m/s) are too slow to cause collisions on experimentally relevant time scales ( 100 µs) since the Rydberg atoms are essentially fixed for these short times. However, collisions between Rydberg atoms may be triggered by attractive van der Waals and dipole–dipole interactions, which can cause atom pairs to “snap” together. Typically, some collisions will result in ionization and the production of free electrons. Thermal ionization of Rydberg atoms immersed in the blackbody radiation field is also an important source of electrons. If a significant density of free electrons builds up, collisions between electrons and Rydberg atoms become dominant because of the high sensitivity of Rydberg atoms to the electric fields produced by these electrons. In this section, we discuss the collision-induced dynamics of Rydberg-atom gases in a magnetic-field-free environment.
3.1. S TATE -M IXING C OLLISIONS IN C OLD RYDBERG -ATOM G ASES Interactions in cold Rydberg-atom gases can be divided into three classes; Rydberg atoms can collide with electrons, ground-state atoms, and other Rydberg atoms. Collisions between Rydberg atoms and ground-state atoms do not significantly affect the dynamics of the system since these collisions have a small cross-section compared to those associated with Rydberg–Rydberg and Rydbergelectron collisions. Furthermore, the experimental data presented throughout this section was obtained from measurements in a MOT loaded from a cold atomic beam in the absence of room-temperature Rb background pressure. Collisions in a cold Rydberg-atom gas alter the distribution of the constituent atoms over the available quantum states, which can then be probed using state-selective field ionization (SSFI) techniques (see Section 2.4). Figure 5 contains a 2-dimensional plot showing SSFI spectra taken over a range of wavelengths. At each wavelength, a cold Rydberg-atom gas of 85 Rb is created via a two-step excitation of atoms in a magneto-optic trap at t = 0, as discussed in Section 2.2. This excitation scheme utilizes two optical transitions from the 5S ground state to the 5P state (λ ≈ 780 nm) and from the 5P to nS or nD Rydberg states (λ ≈ 480 nm). The spectral linewidth of the pulsed Rydberg excitation laser is ≈ 15 GHz. At n lower than about 45, the closely spaced (n+2)S and nD lines can be resolved, as seen in Figs. 1(a) and 5. After the excitation, the Rydberg-atom gas is allowed to evolve for 40 µs before a SSFI ramp is applied to the Rydberg-atom cloud. Figure 5 demonstrates many of the effects of collisions in a cold Rydberg-atom gas. The peak labeled A, found below the photo-ionization threshold, shows the initial state into which the Rydberg atoms were excited. These initially excited,
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F IG . 5. Two-dimensional data set showing state-selective field ionization spectra over a range of excitation wavelength. The initial Rydberg-atom density ranges from ρ0 ≈ 2 × 108 cm−3 at low n to ρ0 ≈ 2 × 107 cm−3 at high n. The letters imprinted on the data identify the following: (A) Initial state peak. (B) Free electron/plasma signal. (C) High-angular momentum states resulting from -mixing collisions between electrons and Rydberg atoms. (D) High-n states resulting from electron-Rydberg collisions and thermal redistribution. (E) Low-n states resulting from electron-Rydberg collisions and thermal redistribution. (F) Three-body recombination signal. (G) Low-n states resulting from Penning ionization. [Reprinted figure with permission from Walz-Flannigan (2004).] © 2004 Walz-Flannigan
low-angular momentum states ionize at electric fields of 1/16n4s (in atomic units), where ns denotes an effective quantum number in the presence of an electric field. As a result of collisions in the gas, the initial state population (signal A) is redistributed into other bound states (signals C–G) and unbound states (signal B) in Fig. 5. The peak labeled C corresponds to high-angular momentum states (WalzFlannigan et al., 2004), which ionize at electric fields between 1/9n4s and 1/4n4s atomic units. The promotion of low-angular-momentum Rydberg atoms to highangular momentum states occurs predominately as a result of -mixing collisions between an electron and a Rydberg atom (Dutta et al., 2001). During an electronRydberg collision, the atom experiences an electric field due to the electron. This field increases as the electron moves closer to the atom until the initially excited atomic state is significantly mixed with high-angular momentum states of
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a hydrogenic manifold nearby in energy. (Typical hydrogenic manifolds are the triangular regions that are densely filled with levels in Fig. 11.) As the electron leaves the Rydberg atom, the probability that the atom remains in a high-angular momentum state is appreciable. The cross-section of this nearly elastic process is of order n5 , in atomic units, as calculated in detail for rubidium Rydberg levels by Walz-Flannigan et al. (2002), and therefore larger than that of inelastic processes, which have cross-sections that scale as n4 . The lifetime of high-angular momentum states scales as n5 , as opposed to n3 for low-angular-momentum states (Gallagher, 1994). Furthermore, atoms in high-angular momentum states can decay radiatively only into other high-angular momentum states, as given by the dipole selection rules, which means that the principal quantum number does not change greatly in a single decay process. Therefore, high-angular momentum Rydberg atoms in cold gases can remain in Rydberg states for tens of milliseconds (Dutta et al., 2001). The signal labeled D in Fig. 5 reveals a significant shift in the population from the initial Rydberg state to states that ionize at lower electric fields. In addition to some thermal redistribution of the initial state, most of this signal can be attributed to inelastic, n-changing collisions between electrons and Rydberg atoms (Walz-Flannigan et al., 2004). These collisions can result in the excitation, ionization, and de-excitation of Rydberg atoms. In the lower range of initial principal quantum numbers in Fig. 5 and for electric fields less than the ionization field of the initial state, the SSFI signal clearly exhibits a grid-like structure; the SSFI peaks observed in the signal labeled D line up with SSFI peaks of initially excited Rydberg levels (signal labeled A). Therefore, we attribute the D-signal to atoms that are scattered into higher principal quantum numbers by n-changing collisions after their initial excitation. Electron–Rydberg collisions can also result in the ionization of Rydberg atoms in the cold gas (Mansbach and Keck, 1969; Vanhaecke et al., 2005). The resulting electrons provide the most dominant contribution to the broad free-electron signal below the photo-ionization threshold occurring for times in the range 2 µs < t < 40 µs (B in Fig. 5). The free-electron signal is closely related to the D signal except that these electron–Rydberg collisions result in excitations to unbound states instead of scattering into different bound Rydberg states. Following Robicheaux and Hanson (2003), the total excitation rate due to electron–Rydberg scattering, which includes scattering into both bound and unbound states, scales as n4.66 , while the de-excitation rate due to electron–Rydberg scattering scales as n2.66 . Initially populated high-n states near the photo-ionization threshold become ionized with a high probability. Whereas, for states with lower initial principal quantum numbers, the excitation rate due to electron–Rydberg collisions is not as dominant, and the redistribution of population exhibits greater balance between scattering into more deeply bound (signal E) and less deeply bound states (signal D).
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For initial excitation above the photo-ionization threshold, Fig. 5 also exhibits a signal from bound Rydberg atoms (labeled F ). These Rydberg atoms are the product of three-body recombination, which is described in Section 3.2. Finally, the signal labeled G in Fig. 5 is the result of Penning-ionizing collisions between Rydberg atoms. These collisions are discussed in more detail in Section 3.3. In this section, attention was given to how the internal states of the Rydberg atoms evolve through interactions in the gas. In the next section, we discuss briefly the case in which a very large fraction of the Rydberg-atom gas evolves into a cold plasma through ionizing collisions. Then, in Section 3.3, we examine what effects Penning-ionizing Rydberg–Rydberg collisions can have on the center-ofmass motion of the constituent Rydberg atoms.
3.2. R ELATION BETWEEN C OLD RYDBERG -ATOM G ASES AND C OLD P LASMAS Cold plasmas were first created by photo-ionizing laser cooled atoms (Killian et al., 1999). The low binding energy of Rydberg atoms establishes an important link between cold Rydberg-atom gases and cold plasmas. Rydberg atoms are formed in expanding cold plasmas as a result of a process called threebody recombination (Killian et al., 2001). Furthermore, cold Rydberg-atom gases are known to ionize and evolve into plasmas at high densities (Robinson et al., 2000). Several theoretical investigations have been done to characterize the plasma dynamics of these systems (Murillo, 2001; Robicheaux and Hanson, 2002; Kuzmin and O’Neil, 2002; Mazevet et al., 2002). Since the kinetic energy of these systems is very low compared to conventional plasma systems, reaching a strongly-coupled, two-component plasma regime (Ichimaru, 1982) in these systems appeared promising but has proven difficult, as the internal energy of Rydberg atoms seems to provide a source of heat for plasma electrons. In this section, we briefly describe the evolution of cold plasmas created by photo-exciting cold atoms above the ionization threshold. In such systems, the initial electron temperature can be controlled by the frequency of the photo-ionizing laser. Using a higher photon energy to ionize a sample of cold atoms translates into a higher initial electron temperature. Immediately after excitation, the plasma is a neutral collection of ions and electrons. The kinetic energy of the electrons is much greater than that of the ions. A fraction of the electrons leave the plasma, thereby creating a positive space charge in the region of the ions. This positive space charge creates a Coulomb well for the remaining electrons, which effectively traps the remaining electrons. The space-charge well becomes an effective trap for an electron when the potential depth, U , provided by the ions, is equal to thekinetic energy of the electron, WKE (Killian et al., 1999). If we take
U=
2 N e2 π σ ,
where N is the number of ions in a spatial Gaussian distribution of
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F IG . 6. State-selective field ionization spectra of a cold cloud of Rb atoms prepared in a MOT and excited above the photo-ionization threshold (a) and into a Rydberg state (b). (a) At t = 0, the gas of atoms is excited above the photo-ionization threshold. A fraction of the photo-electrons escape during the first microsecond, creating a positive space-charge trap for the remaining electrons. As the plasma expands, electrons “boil off” the cloud. At t = 40 µs, the field ionization ramp is applied, and the remaining plasma electrons are immediately removed from the cloud. The late arriving bound-state signal reflects the presence of Rydberg atoms formed via three-body recombination. (b) At early times, the Rydberg-atom gas produces a plasma signature similar to that seen in (a). However, the bound-state signal is more complex, containing signals attributed to the initially excited Rydberg state, -mixing collisions, n-changing collisions, and Penning ionization. These bound state features have been discussed in greater detail in Section 3.1.
rms radius, σ , then the trapping condition can be expressed as 2 N e2 WKE . (1) π σ Electrons confined within the positive space charge region exert a thermodynamic pressure on the ions, causing an expansion of the ion cloud. Another contribution to the expansion is due to Coulombic repulsion of the ions resulting from the positive space charge. However, experimental investigations have shown the electron-pressure effect to be dominant (Kulin et al., 2000; Simien et al., 2004), unless the initial kinetic energy of the electrons, WKE , is so large that most electrons escape within a few nanoseconds after the photoionization (Feldbaum et al., 2002). From Eq. (1) we see that, as the plasma expands, the trapping potential decreases. Furthermore, for cold plasmas with very low initial electron temperatures, intrinsic heating can increase the electron temperature (Kuzmin and O’Neil, 2002). Therefore, as time passes, electrons “boil off” the expanding plasma. The signal B in Fig. 5 reflects the continuous electron “boil-off” from expanding plasmas. Collisions between electrons and ions can result in the production of Rydberg atoms through a process called three-body recombination. In this process,
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two electrons collide with an ion resulting in the production of a Rydberg atom and an accelerated electron, whose kinetic energy has increased corresponding to the binding energy of the newly created Rydberg atom. Since three-body recombination increases the kinetic energy of the product electron, it causes heating of the electron gas in plasmas with initially low temperatures and leads to larger-than-expected expansion velocities of plasmas with low initial electron temperatures (Kulin et al., 2000; Robicheaux and Hanson, 2002). As the elec−9/2 tron temperature, Te , increases, the three-body recombination rate, Nr ∝ Te , becomes less dominant, and the characteristic expansion velocity becomes sim√ ply v ∝ kB Te /Mion (Mion : ion mass). Using the time-of-flight measurement scheme discussed in Section 3.3, detecting and characterizing the velocity distributions of fast Rydberg atoms created in the expanding plasma should become possible in the future. Cold plasmas can also be formed when a large fraction of a cold Rydbergatom gas ionizes (Robinson et al., 2000). The dynamics of these plasmas proceed in much the same way as described in the previous section. One important difference is that the ions and electrons that constitute the plasma are products of gradual ionization of Rydberg atoms instead of rapid, direct photo-ionization of atoms in a low-lying state. Electrons resulting from Rydberg-atom ionization first escape the Rydberg cloud, gradually creating a space-charge potential well that may become deep enough to trap subsequently produced electrons. Whether the well becomes deep enough to trap many electrons depends on the initial density, the Rydberg-atom number, and the principal quantum number. Once many electrons are trapped in the space charge well, they collide very efficiently with the remaining Rydberg atoms as they move about in the cold gas. These electron–Rydberg collisions cause state mixing and further ionize Rydberg atoms. Under sufficiently favorable initial conditions, an ionization avalanche develops and a cold plasma is formed (Vitrant et al., 1982; Robinson et al., 2000). The underlying mechanism that leads to the production of the first “seed” electrons in a cold Rydberg-atom gas has not been entirely determined. The two processes that most likely produce the initial ionization events in a cold Rydberg-atom gas are thermal ionization of Rydberg atoms by blackbody radiation (Walz-Flannigan, 2004; Pohl et al., 2003) and Penning-ionizing collisions of Rydberg atoms that are excited in close proximity to one another (Li et al., 2005). Available data suggests that both mechanisms influence plasma formation, but the extent to which each one contributes is not yet known precisely. 3.3. C OLLISION -I NDUCED P RODUCTION OF FAST RYDBERG ATOMS As a cold Rydberg-atom gas evolves in time, not only does the internal-state energy distribution of the Rydberg atoms change, but the center-of-mass en-
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ergy distribution also changes due to inter-atomic forces between the Rydberg atoms. These forces are expected to enhance Penning ionization in cold Rydberg-atom gases. One of the products of a Penning-ionizing collision is a fast Rydberg atom having a velocity on the order of 10 m/s (Robicheaux, 2005), which is roughly 50 times faster than Rb atoms laser-cooled to the Doppler limit (velocity 0.15 m/s). The increase in velocity results from the conversion of internal energy to center-of-mass kinetic energy. Another process that results in fast Rydberg atoms is three-body recombination of ions that have been accelerated in an expanding cold plasma. In this process, the Rydberg-atom velocity approximately equals that of the ion, or about 40 m/s. In the remainder of this section, we discuss recent experiments in which the production of fast Rydberg atoms via Penning ionization has been demonstrated (Walz-Flannigan et al., 2004; Knuffman and Raithel, 2006). Penning ionization can occur when two Rydberg atoms collide. This process can be represented in the following way: Ryd(n∗0 ) + Ryd(n∗0 ) → Ryd(n) + ion + e− ,
(2)
with n∗0 and n denoting the initial and final principal quantum numbers, respectively. Two closely spaced Rydberg atoms can be accelerated towards each other by attractive forces, thereby increasing their kinetic energy. For atoms in highangular momentum states, the attraction is due to electric–dipole forces, which depend on the inter-atomic separation as R −4 , and on the mutual orientation of the permanent dipole moments of the atoms. When R becomes small enough that the magnitude of the Rydberg–Rydberg interaction energy is on the same order as the binding energy of the Rydberg state, one of the atoms is likely to become ionized and the other atom to undergo a transition to a more deeply bound state. The freed electron leaves the area in less than a nanosecond, leaving the ion and the Rydberg atom in close proximity to each other. The ion and atom then separate due to a repulsive force created by the positive net charge that results from the absence of the escaped electron, further increasing the kinetic energy of the collision products (Robicheaux, 2005). The change in internal energy of the bound atom provides at least the energy needed to ionize its collision √ partner, constraining the final state of the bound atom such that n n0 / 2. Assuming that the initial kinetic energy of the two atoms is negligible, the internal energy difference in the bound atom equals the sum of the initial binding energy of the ionized atom and the kinetic energy of the three collision products. Most of the energy is transferred to the electron, but enough kinetic energy is transferred to the product ion and Rydberg atom that their velocities should increase by a couple of orders of magnitude. In the experiment discussed next, laser-cooled rubidium atoms in a MOT are excited into a Rydberg state, creating a cold gas with Rydberg atom densities
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(a)
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F IG . 7. (a) Experimental apparatus used to perform time-of-flight measurements on atoms emitted from cold Rydberg-atom gases. Fast Rydberg atoms which travel the 2 cm from the Rydberg gas to the remote field-ionization region are ionized and the resulting electrons are counted by the micro-channel plate (MCP) detector. (b) Typical measured time-of-flight distribution. This data was taken for an initial principal quantum number of n0 = 90 and an initial central Rydberg-atom density of 1.4 × 108 cm−3 . (c) The velocity distribution calculated from time-of-flight distribution in (b). (d) Total production of fast Rydberg atoms () vs initial Rydberg atom density ρ at n0 = 90, where ρscale = 9.1 × 108 cm−3 . The production of fast Rydberg atoms is compared with integrated Penning ionization (◦) and plasma (•) signals obtained from state-selective field ionization spectra. [Reprinted figure with permission from Knuffman and Raithel (2006).] © 2006 American Physical Society
on the order of 108 cm−3 . Figure 7(a) shows the apparatus used to make timeof-flight measurements of Rydberg atoms emitted from the cold gas. Fast Rydberg atoms created in the gas travel 2 cm from the MOT location to a remote field ionization region located between the right semi-circular electrode and the first extraction electrode. Electrons resulting from Rydberg atoms ionized in this region are then counted by an MCP located 12.5 cm away. A photon counter then records the MCP pulses over a 0.5–1 ms window with variable delay relative to the Rydberg-atom excitation. To avoid mistaking electrons produced in the Rydberg-atom gas for fast Rydberg atoms, a small electric field is applied to the Rydberg-atom gas to direct electrons away from the remote field ionization region. Figure 7(b) shows typical time-of-flight data for n0 = 90 Rydberg atoms traveling a distance L = 2 cm from the Rydberg-gas location to the Rydberg-atom detection region. The velocity distribution, PV (v), of the fast Rydberg atoms [Fig. 7(c)], can be calculated from the time-of-flight data, PT (t) [Fig. 7(a)], using
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the relation PV (v) = PT ( Lv ) vL2 . The most probable velocity of the detected fast Rydberg atoms is found to be slightly less than 5 m/s. The corresponding kineticenergy range of this distribution exceeds that of Rb atoms in a MOT (velocity ∼ 0.15 m/s) by a factor of about 1000. The most probable velocity from Fig. 7(c) agrees to within 20% with classical simulations of Penning-ionizing collisions between Rydberg atoms (Robicheaux, 2005). Many of the processes in cold Rydberg-atom gases display a strong dependence on the initial Rydberg-atom density. With this in mind, we have made time-offlight measurements as in Fig. 7(b) at various densities and compared them to field ionization spectra of the cold Rydberg-atom gas at those densities [see Fig. 7(d)]. In this way, we hope to establish experimentally what mechanism is responsible for the production of fast Rydberg atoms. The net Penning ionization signal [◦ in Fig. 7(d)], which is obtained by integrating over the Penning ionization signal in the SSFI spectra, can be compared to the total number of fast Rydberg atoms [ in Fig. 7(d)]. Both quantities increase rapidly with density at low densities, exhibit broad maxima at intermediate density values, and decrease with density at high densities. The similarity in the behavior of these quantities indicates that Penningionizing collisions also result in the production of fast Rydberg atoms. The steady increase in the Penning-ionization and fast-Rydberg signals at low densities is due to the fact that with increasing density two Rydberg atoms are more likely to be sufficiently close to “snap together” and undergo a Penning-ionizing collision. At densities higher than the optimal density for the production of fast Rydberg atoms, more and more of the Rydberg-atom gas evolves into a plasma, as evident from the plasma signal [• in Fig. 7(d)]. Electron–Rydberg-atom collisions in the plasma cause many fast Rydberg atoms to ionize (Vanhaecke et al., 2005) or to undergo transitions into lower Rydberg states (Walz-Flannigan et al., 2004) that are so low in energy that they cannot be detected in the remote field ionization region. At high Rydberg-gas densities, presumably both of these effects become important, explaining why in the high-density domain both the Penning-ionization signature and the number of fast Rydberg atoms decrease with increasing density. Many questions remain to be answered and warrant further exploration. First, the most probable velocity of emitted Rydberg atoms is expected to scale as √ √ BE = ( 2n)−1 (Robicheaux, 2005), where BE is the binding energy of the initial state of the Rydberg atoms participating in the collision. Qualitatively, this can be understood to result from the fact that Penning-ionizing collisions involve more energy exchange as n decreases, so more internal energy can be converted to the kinetic energy of the product Rydberg atom. This trend has yet to be demonstrated experimentally. Also, by measuring the state distribution of atoms emitted from the Rydberg-atom gas, the final distribution of states resulting from Penning ionization could be determined experimentally. Furthermore, if the velocities of the emitted atoms were correlated to their final state, this would shed light on how internal-state energy is distributed as a function of the kinetic energy of
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the products of Penning-ionizing collisions. Finally, we may study how a static electric field applied to a Rydberg atom sample may affect the Penning-ionizing collisions. After creating a sample of Rydberg atoms in a state with low angular momentum, permanent electric dipole moments can by generated by application of uniform, static electric fields of order n−5 , in atomic units. We expect that applying such fields would enhance inter-atomic interactions between neighboring Rydberg atoms and lead to the production of more fast Rydberg atoms.
4. Towards Coherent Control of Rydberg-Atom Interactions 4.1. M OTIVATION In this section, we highlight how the rich dynamics of Rydberg-atom gases, discussed in Section 3, can be controlled in small systems in order to create an entangled Rydberg-atom sample. As discussed by Lukin (2003), the ability to coherently control a collective quantum state is pursued by many research groups in diverse areas such as condensed matter, AMO physics and NMR because of a range of important applications. Controlling an entangled sample allows one to take advantage of many-body quantum superpositions for applications ranging from atomic clocks (Bouchoule and Mølmer, 2002) to secure quantum cryptography (Saffman and Walker, 2002) to quantum information processing (Jaksch et al., 2000; Lukin et al., 2001; Saffman and Walker, 2005). Manipulating quantum states generally requires strong interactions, but on the other hand, a system with long quantum coherence times is desired (hence weak interaction with the environment). In this respect, a promising scheme employs neutral atoms, localized at regular sites of an optical lattice, to store quantum information in ground state hyperfine levels, as in ion quantum information systems, which have long coherence times. Quantum gate operations are implemented by transferring selected atoms into a Rydberg state for short times. During the short gate time, atoms mutually interact via van der Waals or dipole–dipole interactions, which scale as n11 /R 6 and n4 /R 3 , respectively. Rydberg atoms have long radiative lifetimes given by τ = τ0 (n − δl )α , where for a Rubidium D-state α = 2.85, δl = 1.34, and τ0 = 2.09 ns, giving τ = 70 µs for n = 40 at 0 K (Gallagher, 1994). Decay is thus not a significant cause of decoherence in Rydberg systems, especially for the time scales needed for quantum gate operations ( 100 ns). Two sources of decoherence are mechanical forces between interacting Rydberg atoms, discussed in Section 3.3, and - or n-changing collisions inherent to dense Rydberg samples (Mudrich et al., 2005), discussed in Section 3.1. Mechanical forces are a source of decoherence because they would heat the laser-cooled atoms. In an architecture in which atoms are kept at well-defined sites of a optical lattice, heating would cause excitations
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F IG . 8. Lowest collective states of an ensemble of N atoms containing k Rydberg excitations. The energy of a single, isolated excitation is WRyd . Rydberg–Rydberg interaction causes an energy shift Vint of the second excited many-body state, |N, k = 2. [Reprinted figure with permission from Liebisch et al. (2005).] © 2005 American Physical Society
of center-of-mass oscillations and, in severe cases, displacement of atoms from their original lattice sites. State-changing collisions are a source of decoherence because they have the effect of irreversibly spreading the internal-state wavefunction over a vast, quasi-continuous array of background states. To alleviate both of these sources of decoherence, quantum information schemes are desired in which at most one Rydberg excitation is present at any given time. Quantum gates of that type can, in principle, be designed using a blockade effect between Rydberg excitations, as proposed by Jaksch et al. (2000) and Lukin et al. (2001). At the heart of these schemes is the ability to coherently share a single Rydberg excitation among a sample of atoms.
4.2. C OHERENT RYDBERG E XCITATIONS IN M ANY-B ODY S YSTEMS In a small sample of non-moving but interacting atoms, the stationary quantum states involving Rydberg excitations are collective many-body states in which the Rydberg excitations are coherently shared among all atoms of the ensemble. In an ensemble of N atoms, the ground state is |N, 0 = |g1 , g2 , . . . , gN , where the subscripts are the atom labels and |g denotes an atom in the ground state. The lowest state, which carries one excitation, can be written as √ excited |N, 1 = (1/ N ) N i=1 |g1 , g2 , . . . , ri , . . . , gN , where |r denotes an atom in a Rydberg state. In the first excited state, the Rydberg excitation evidently is shared among all atoms, and the many-body quantum state is an entangled state. Without Rydberg–Rydberg interactions, the energies of the many-body states |N, k would fall on an equidistant ladder, Wk = kWRyd , where WRyd denotes the energy of a single, isolated Rydberg excitation. However, as discussed in Section 3, Rydberg atoms are highly sensitive to external fields due to their large polarizabilities, and in turn have large van der Waals or dipole–dipole interaction
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potentials. Due to the resultant significant interaction between multiple Rydberg excitations in a many-body system, the energy levels of the system deviate from the equidistant ladder Wk = kWRyd (see Fig. 8). While the separation of states |N, 0 and |N, 1 equals WRyd , the energy of the second excited state, |N, 2, is modified by a binary Rydberg–Rydberg interaction term Vint and is not equal to 2WRyd . States with k > 2 experience even larger interaction-induced shifts. Consequently, a laser pulse tuned to the transition |N, 0 → |N, 1 is detuned from all transitions leading into states with more than one Rydberg excitation, provided that the laser bandwidth is less than the Rydberg–Rydberg interaction strength |Vint |/ h. In this way there is negligible probability of any excitation other than the first collective excitation |N, 1. The inability to access any states with k 2 is commonly referred to as dipole blockade, or Rydberg excitation blockade. In experimental schemes that utilize narrow-band lasers and access only the states |N, 0 and |N, 1, any decoherence effects that require the presence of at least two Rydberg excitations are eliminated. This includes decoherence due to mechanical forces between interacting atomic dipoles and decoherence due to state-changing Rydberg–Rydberg collisions. However, any experimental strategy to realize a Rydberg-excitation blockade has to proceed in a manner that complex collision-induced dynamics, as discussed in Section 3, are avoided.
4.3. T HE Y IN AND YANG OF RYDBERG –RYDBERG I NTERACTIONS While it is understood that van der Waals and dipole–dipole interactions between Rydberg atoms can be used to achieve a blockade effect, in actual experimental implementations of the blockade one has to ensure that these same interactions do not introduce state-changing and/or ionizing collisions. There is a trade-off between achieving a high single-atom Rydberg excitation probability, which is required for blockade effects to become noticeable, and avoiding collisions and plasma formation. Further, when choosing a Rydberg–Rydberg interaction mechanism (dipole–dipole versus van der Waals interaction), there is a trade-off between having external blockade control parameters (electric fields and/or mutual dipole orientation angle) and having an interaction potential that is fairly insensitive to these parameters and thus more reliable. There have been a number of papers recently addressing these issues of cold Rydberg-atom gas dynamics in the context of using Rydberg atoms as elements in a quantum phase gate (Li et al., 2005; Farooqi et al., 2003; de Oliveira et al., 2003; Afrousheh et al., 2004; Carroll et al., 2004). To illustrate the importance of these considerations, in Fig. 9 we first show two Rydberg-excitation spectra in the vicinity of the 86S Rydberg state with durations of the Rydberg excitation pulse of 200 ns (left) and 420 µs (right). In both cases, the lower-transition excitation laser 5S → 5P is locked on-resonance and
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F IG . 9. Rydberg counts per pulse versus level energy in wave-numbers relative to the 5P3/2 level. The upper-transition (from 5P into Rydberg states) excitation laser has a linewidth of about 5 MHz and is scanned in steps of 10 MHz, while the lower-transition laser has a linewidth of a few MHz and is tuned on-resonance with the 5S → 5P transition. The two plots highlight the differences between (a) short-pulse (200 ns) and (b) long-pulse (20 µs) laser excitation.
the upper-transition laser 5P → nD is scanned in 10 MHz steps. In the case of short-pulse excitation, the Rydberg resonances are narrow and there are no lines that violate the electric–dipole selection rules (only S- and D-lines appear). This result is consistent with the absence of ionizing collisions and plasma-electricfield-induced effects. In contrast, in the case of long-pulse excitation we observe line broadening and severe violations of selection rules. In Fig. 9(b), we observe P - and F -lines; even high-angular momentum hydrogenic states are excited [broad feature in Fig. 9(b)]. These signatures are characteristic of the presence of free charges in the excitation region (Feldbaum et al., 2002) and hence ionizing Rydberg-atom interactions. From Fig. 9 it is concluded that the study of coherent Rydberg excitations in many-body systems requires short excitation pulses, fast experimental processing and rapid state analysis in order to avoid interactions that cause ionization or decoherence. To evaluate the importance of state-mixing collisions (n- and -mixing) in our studies of Rydberg-blockade phenomena, we routinely perform SSFI scans. A typical example is shown in Fig. 10, where we have excited the 44D5/2 -state using STIRAP excitation with 500 ns laser pulses at moderate atom densities of order 108 cm−3 . The SSFI scan does not exhibit any signatures of -mixing and ionizing collisions, showing that for 44D5/2 -Rydberg states an excitation duration of 500 ns is sufficiently short to avoid decoherence effects caused by these collision types. The SSFI scan in Fig. 10 does, however, show two side-peaks of approximately equal area next to the main signal. These peaks indicate a singular interaction that yields equal numbers of product atoms with effective n-quantum
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F IG . 10. Field-ionization spectrum obtained after exciting 44D5/2 Rydberg atoms using excitation pulses of 500 ns duration applied at time 0. The field ionization pulse is also shown. [Reprinted figure with permission from Cubel et al. (2005).] © 2005 American Physical Society
numbers changed by about ±2 from the initial effective n-quantum number. The side-peaks are due to a near-elastic Rydberg–Rydberg interaction that yields product atoms in states 42F7/2 and 46P3/2 . This near-elastic interaction is of the type |n, l, j ⊗ |n, l, j → |n , l , j ⊗ |n , l , j with large radial matrix elements n, l|r|n , l and n, l|r|n , l and a small energy mismatch W . In the present case W ∼ h × 100 MHz for |n, l, j = |44, 2, 2.5. For this particular reso3 nance, we calculate a coupling strength of about 0.4×n∗4 0 /R atomic units, where n∗0 is the effective initial-state principal quantum number and R is the interatomic separation. The numerical pre-factor 0.4 has been found in detailed calculations of the energy shifts. It depends somewhat on the angular orientation of the interacting atoms and on the detailed mj -quantum numbers used in the calculation. Depending on R, the coupling strength is either smaller or larger than W and the interaction may accordingly be classified as a van der Waals or a dipole– dipole interaction (Li et al., 2005). In the present case, the transition between the van der Waals and dipole–dipole regimes occurs at an interatomic separation of R ∼ 2.5 µm. We believe that under the conditions of Fig. 10 a small fraction of excited atom pairs are close enough that the product states 42F7/2 and 46P3/2 become significantly populated, causing the side-peaks in the SSFI spectrum and thereby providing evidence for the importance of the Rydberg–Rydberg interaction 2 × 44D5/2 → 42F7/2 + 46P3/2 .
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The product of electric–dipole matrix elements that enters into the coupling strength of Rydberg–Rydberg interactions of the type discussed in the previous paragraph typically scales as n∗4 0 , while the energy mismatch W typically exhibits an n∗0 -dependence that is very specific to the individual Rydberg–Rydberg interaction channels. In a few cases, the mismatch W of a channel with a reasonably large product of electric–dipole matrix elements becomes very small over a narrow range of n∗0 -values. This occurs, for instance, for the interaction channel 2 × nD5/2 → (n − 2)F7/2 + (n + 2)P3/2 for n ≈ 43; note that this channel causes the side-peaks in Fig. 10. More typically, however, there exist a few Rydberg–Rydberg interaction channels with a very large product of electric–dipole matrix elements and large W . For Rb D5/2 states the channel 2 × nD5/2 → (n − 1)F7/2 + (n + 1)P3/2 has the largest coupling strength, 3 namely about 1 × n∗4 0 /R , and a W -value in the range of about 1 GHz. (The numerical pre-factor 1 in the coupling strength depends somewhat on the angular orientation of the interacting atoms and on the involved mj -quantum numbers.) For n-quantum numbers differing from 43 by more than about 7, this rather offresonant channel dominates the level shifts of Rb D5/2 states. Due to the large 6 detuning W , the shifts then scale as n∗11 0 /R , i.e., they follow the scaling law expected for van der Waals type shifts. Numerically, for Rb D5/2 -states with n 50 ∗11 6 6 we calculate shifts ranging from about −30 × n∗11 0 /R to −80 × n0 /R atomic units, depending on orientation angles and involved mj -quantum numbers. Note that, while the shift moderately varies as a function of these parameters, it does not switch sign or approach zero under any condition. For n∗0 = 60 and R = 5 µm the shifts range from −10 MHz to −27 MHz. Since these shifts are larger than the linewidth of the laser that excites the atoms from 5P into Rydberg states, and because R = 5 µm corresponds to fairly easily realizable atom densities of a few 109 cm−3 , they are well suited to produce blockade effects. In calculations we have found similar characteristics for the Rydberg–Rydberg energy shifts of Rb D3/2 -states and Rb S1/2 -states. For S-states, the shifts are isotropic and positive, but magnitude-wise of the same order as for D-states. Due to these properties, Rydberg-atom van der Waals shifts are a very robust mechanism that should allow one to implement Rydberg excitation blockades over a wide range of principal quantum numbers, as seen in recent experiments by Tong et al. (2004) and Liebisch et al. (2005). Further, the van der Waals interactions, which scale 6 as n∗11 0 /R , fall off more rapidly as a function of R than the dipole–dipole in3 teractions, which scale as n∗4 0 /R . This property may become important in the realization of lattice architectures of Rydberg atoms with only nearest-neighbor interactions. Summarizing this section, we generally observe that for high atom densities and long laser-excitation pulses Rydberg-atom ionization and plasma formation, as well as state-mixing collisions are inevitable. Therefore, in research dealing
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with Rydberg excitations that are coherently shared among many atoms, the duration of the laser excitation pulses plus the duration allowed for Rydberg–Rydberg interactions must be kept short (of order a few hundred 100 ns). This will, in most cases, ensure conditions free of ionization and unwanted state-changing collisions. Then, van der Waals-type Rydberg–Rydberg interactions are a robust mechanism suitable to implement Rydberg-excitation blockades and to prepare coherent Rydberg excitations in many-body systems.
4.4. C ONTROL O PTIONS FOR RYDBERG –RYDBERG I NTERACTIONS In this section, we discuss two options for controlling the strength of Rydberg– Rydberg interactions. A blockade that could be controlled using an easily accessible control parameter, such as an applied electric field, would be advantageous. Having the ability to externally turn on and off the blockade effect in a realtime, reproducible way will facilitate testing and tuning the blockade efficiency. Furthermore, having accurate control of the blockade mechanism will become a necessity in future research on quantum information schemes based on the blockade. In one method, a fairly strong electric field is introduced so that the optically accessible Rydberg states have permanent electric dipole moments. In the second method, a somewhat weaker electric field is employed to tune a welldefined Rydberg–Rydberg molecular resonance |n, l, j ⊗|n, l, j → |n , l , j ⊗ |n , l , j , discussed in Section 4.3, exactly into resonance. In both cases, the resonant Rydberg-atom interaction potential dominates any van der Waals shift that may also be present but has a strong angular dependence. We do not discuss the complications that may arise from the strong angular dependence of these resonant interactions. We also do not discuss experimental issues associated with the homogeneity requirements for the utilized electric fields. An induced dipole moment is achieved by application of an external electric field large enough that the desired state acquires a significant permanent dipole moment, which is given by the derivative of the level energy W with respect to the applied field E, −dW/dE. Therefore, desirable states with large interatomic dipole–dipole interactions have large slopes in the Stark map (for an example, see Fig. 11). Due to their quantum defects, states with low angular momentum have quadratic Stark shifts at low electric fields. At higher electric fields, these levels blend into nearby hydrogenic manifolds, thereby acquiring large permanent dipole moments. For effective quantum number n∗0 , the permanent dipole moment p is limited to |p| = 1.5n∗2 0 . For an atomic separation vector R, the corresponding ˆ 2 )/R 3 , has an extremum limit of the dipole–dipole interaction energy, (p2 −3(p·R) ∗4 ∗ 3 of −4.5n0 /R (for p E). Thus, for n0 = 40 and R = 5 µm, the dipole–dipole interaction energy reaches about 90 MHz. In this way, one may be able to turn on
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F IG . 11. Stark map of Rb in the vicinity of the 44D level [energy (W ) relative to the ionization potential]. The manifolds mj = 1/2, 3/2 and 5/2 are shown. Levels with both large permanent electric dipole moment and high oscillator strength with the 5P3/2 -state are found in the intersection region of the 45D levels with the manifold of n = 43 hydrogenic states (levels that exhibit linear Stark effect).
and off a strong blockade simply by changing the external electric field applied at the time of Rydberg-atom excitation. Experimental challenges and additional control possibilities may arise from the fact that the interaction energy depends on the angle θ between R and E as 1 − 3 cos2 (θ ). Note that the interaction vanishes and switches sign at θ = 55 degrees. In practice, it is not possible to excite the Stark states with maximal electric– dipole moment, because the oscillator strength between these states and low-lying atomic levels is too small. One has to settle for states with an intermediate electric dipole moment and with a reasonably high oscillator strength. As an example, in Fig. 12 we show experimental Stark spectra for excitation from 5P3/2 into Rydberg states near the 54D-levels with a scan range of about 1.5 GHz, taken in steps of about 0.6 V/cm. With increasing electric field, the five levels originating in the D3/2 and D5/2 fine-structure components shift towards lower energies at increasing rates, in close analogy with the behavior of the D-levels shown in the Stark map in Fig. 11. For electric fields between about 4.7 V/cm and 6.5 V/cm, which are well below the electric field at which the displayed levels start to mix with linear Stark levels (which happens for fields 11 V/cm), the pair of states that red-shift the fastest have both large oscillator strengths and electric–dipole moments of about 2.6 × 10−27 Cm, equivalent to 300 (= 0.11n∗2 0 ) atomic units. The spatial range over which the mutual interaction of electric dipoles of that strength could be resolved with a 5 MHz linewidth laser would be about 3 µm,
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F IG . 12. Stark spectra shown for 54D3/2 and 54D5/2 Rydberg states excited with mixed π - and σ -polarized light and a homogeneous electric field scanned from 0-6.5 V/cm. Energy (W ) is relative to the ionization potential.
which is within the range of values for which experimental investigations appear promising. Another possibility for achieving large dipole–dipole interactions is to tune a comparatively weak background electric field such that a Rydberg–Rydberg molecular interaction, such as the ones discussed in Section 4.3, comes into resonance (Anderson et al., 1998; Lukin et al., 2001). We have calculated that in rubidium the process 2 × 46D5/2 → 44F7/2 + 48P3/2 , which has an energy detuning of about 200 MHz under electric-field-free conditions, becomes resonant at an external electric field of about E = 0.3 V/cm. The resonant coupling strength, 3 estimated to be 0.4 × n∗4 0 /R for this interaction channel, is quite strong, reaching about h × 12 MHz at an atomic separation of 5 µm, while the van der Waals shift of the same levels at E = 0 would only be about h×1.5 MHz. In detailed calculations we have seen that the interaction strength in this scheme is only moderately sensitive to the relative atomic orientation if the electric field is kept somewhat below the value at which the Rydberg–Rydberg interaction is resonant. This may be viewed as an advantage over the previously discussed method, in which permanent electric–dipole moments are induced by strong electric fields. In that scheme, the interaction strength is strongly dependent on the atomic orientation.
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F IG . 13. Model of an extended atomic ensemble with a Rydberg-excitation blockade of a range Rb . Each “bubble” represents a region in which, ideally, there exists exactly one Rydberg excitation. Since the number of bubbles that fit into the excitation volume is well-defined, the number of Rydberg excitations does not fluctuate significantly. [Reprinted figure with permission from Liebisch et al. (2005).] © 2005 American Physical Society
4.5. M ETHODS TO M EASURE THE B LOCKADE E FFECT Regardless of which of the methods described in the previous sections is used to achieve large interaction potentials, an important element in implementing a blockade between Rydberg atoms is how to quantify the effectiveness of the blockade. In particular, a method is needed to determine how the blockade effectiveness varies as a function of the atom density, the excitation-laser detunings, and the chosen Rydberg states. One straightforward approach is to measure the suppression of Rydberg excitations in a bulk sample of atoms (Tong et al., 2004; Singer et al., 2004). However, one would eventually like to use the blockade effect for small atomic samples loaded into an optical lattice, where only one or no Rydberg excitations are created in each sample. In the following we discuss a method that allows one to measure the blockade effect using an atom counting approach, which is particularly well-suited for small atomic samples. Assuming that an excitation blockade is effective, an extended atomic ensemble, as sketched in Fig. 13, is expected to break up into a closely packed set of regions with uniform volumes within which only one collective Rydberg excitation occurs. The typical radius Rb of these regions, represented as bubbles in Fig. 13, is determined by the strength of the Rydberg–Rydberg interaction and the laser bandwidth, which is 5 MHz in our experiment (Teo et al., 2003). The number of Rydberg excitations in the whole ensemble then approximately equals the number of bubbles that fit into the whole excitation volume. In this way, the number of Rydberg excitations does not fluctuate greatly from one realization of the excitation process to another. Therefore, the Rydberg-excitation blockade is expected to result in a sub-Poissonian distribution of the number of Rydberg excitations. This effect is observed by repeated generation and measurement of Rydberg excitations in identically prepared atomic ensembles. This establishes a close connection between the Rydberg atom counting statistics and the effec-
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tiveness of a blockade in the Rydberg-atom excitation. A recent simulation of the excitation process under presence of a blockade by Robicheaux and Hernández (2005) and Ates et al. (2006) appears to be consistent with our findings. 4.6. E XPERIMENTAL I MPLEMENTATION OF AN E XCITATION -S TATISTICS M EASUREMENT In the experiment described next, small samples of 85 Rb ground-state atoms are prepared by collecting atoms in a vapor-cell magneto-optical trap (MOT). The MOT is run at a higher-than-usual magnetic field gradient of 50 G/cm and small-diameter laser beams in order to limit the MOT diameter to 500 µm while maintaining a central atom density of about 5 × 109 atoms/cm3 . Rydberg atoms are excited via a two-step excitation, as discussed in Section 2.2, with coincident, counter-propagating, on-resonant laser pulses of 100 ns duration. The lower-transition laser 5S → 5P (780 nm) is collimated to 3 mm through the atomic sample. The upper-transition laser, 5P → nD (≈ 480 nm), is focused into the MOT with a full width at half maximum diameter of the intensity distribution of 16 µm ± 1 µm. The upper-transition laser is focused in order to achieve high Rabi frequencies on the upper transition. In this experimental setup, the lower-transition laser has a uniform intensity profile across the whole atomic sample, whereas the intensity profile of the upper-transition beam varies radially from the beam axis. In the longitudinal direction, the intensity of the upper-transition beam does not vary significantly because the Rayleigh length of the beam (1.1 mm ± 0.1 mm) is larger than the diameter of the atomic sample. In this way, a cylindrical excitation volume such as in Fig. 13 is achieved. As discussed in Section 2.4, the Rydberg atoms are counted by field-ionizing them with moderate external electric fields and by subsequent detection of electrons released by field ionization using a micro-channel plate detector. Some care needs to be exercised to ensure that at the time instant of excitation the field ionization electrodes do not generate a substantial stray electric field in the excitation region. Due to the high susceptibility of Rydberg atoms to electric fields, such a field would result in quadratic Stark shifts and broadening of the optical Rydberg resonance line, thereby spoiling the spectral resolution needed for the blockade effect to emerge (see Fig. 8). Therefore, in the experiment we employ two large, flat field plates placed symmetrically about the MOT location that shield stray electric fields and allow us to apply ionization electric fields with a high degree of homogeneity. The plates have a spacing of 2.2 cm, a diameter of about 6 cm, central holes 1.6 cm in diameter for electron extraction, as well as shielding structures behind the holes. Further, clamp circuits are installed that suppress stray potentials originating in the high-voltage electronics that produce the SSFI pulse. The process of exciting, ionizing, and counting the Rydberg atoms is typically repeated 5000 times to obtain one Rydberg-atom counting statistics.
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F IG . 14. (a): The histogram shows the probability distribution of the number of detected 84D5/2 Rydberg atoms, obtained from 5000 realizations of a photo-excitation experiment. A highly sub-Poissonian Q-value of −0.51 ± 0.04 is found. To make the sub-Poissonian character of the measured distribution more apparent, the histogram is compared with a Poissonian with the same average (line). (b) Reference experiment at a Rydberg-atom density that is ≈ 100 times lower than in (a). In the reference experiment, a close-to-Poissonian Q-value of −0.08 ± 0.04 is found. [Reprinted figure with permission from Liebisch et al. (2005).] © 2005 American Physical Society
4.7. R ESULTS OF E XCITATION -S TATISTICS M EASUREMENTS Figure 14(a) shows typical counting statistics for the 84D5/2 Rydberg state. By comparing the profile of a Poissonian distribution [solid line in Fig. 14(a)] with the atom counting statistics [histogram in Fig. 14(a)], it is clear that the measured statistics is sub-Poissonian. One can, however, quantify the width of the atom-counting statistics, and thereby measure the blockade effectiveness, by calculating the commonly used Mandel Q parameter of the distribution. The Mandel Q parameter is defined as the variance in the count distribution divided by the 2 −N2 mean number of counts minus one, Q = N N − 1 (Mandel, 1979). A value Q = 0 corresponds to a Poissonian distribution, Q > 0 to a super-Poissonian distribution, and Q < 0 to a sub-Poissonian distribution. If the number of detected Rydberg atoms would be the same in all realizations of the experiment, i.e., for a number state, the Q-value would take its minimum possible value, which is −1. The distribution in Fig. 14(a) has a Q-value of −0.51, which qualifies as highly sub-Poissonian. Furthermore the Q-value −0.51 of detected Rydberg atoms, henceforth referred to as QD , is not the same as the actual Q value of the Rydberg atoms present in the atomic sample before detection, referred to as QA . QA differs from QD due to detection inefficiencies of the MCP detector. Denoting the probability of detecting a Rydberg excitation by η, then QD = ηQA . If QA had its minimum possible value, −1, then η would have the value −0.51. MCPs have a quoted detection efficiencies for low-energy electrons in the range 0.5 η 0.85 (Wiza, 1979). In order to be consistent with our measurement result, QD = −0.51, this range of detection efficiencies must correspond to a range −1 QA −0.64 for the Q-value of the distribution of the actual Rydberg atom
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F IG . 15. Measured QD -values as a function of principal quantum number n. The average number of detected Rydberg atoms is kept constant at about 30. [Reprinted figure with permission from Liebisch et al. (2005).] © 2005 American Physical Society
number present in the sample. We therefore conclude from our measurement that QA −0.64. Using atom-counting statistics, it is possible to measure the blockade effectiveness as a function of Rydberg atom density. In Fig. 14(b) the focal diameter of the upper-transition laser beam was increased by a factor of 10, from its initial FWHM of 16 µm ± 1 µm. The excitation volume was thereby increased 100 times. Since the atom number was held fixed using the laser intensity as a control parameter, the Rydberg-atom density was reduced by a factor of 100. As shown in Fig. 14(b) the statistics now yields a QD -value of −0.08, corresponding to a nearly Poissonian distribution. We attribute this change of the statistics to a reduction of the level shifts caused by van der Waals interactions, which scale as n11 /R 6 . In the low-density measurement, the counting statistics mostly depends on the ground-state atom-number fluctuations and on the value of the single-atom excitation probability, which in the low-density measurement is very small, leading to near-Poissonian measurement results. In the low-density measurement, the “bubble” concept advocated in Fig. 13 becomes useless, because on average much fewer than one atom per bubble is excited, and the blockade effect has no consequences. Viewed in context with Fig. 8, the result in Fig. 14(a) corresponds to a saturated excitation on the lowest step of the ladder, while the higher-lying steps are out of resonance due to the blockade effect. The result in Fig. 14(b) corresponds to a weak excitation of the lowest step of the ladder, in which case it doesn’t make any difference whether the second and higher steps on the ladder are blockaded or not. Based on detailed calculations, we expect that in the range n 50 the level shifts of rubidium nD5/2 Rydberg states due to Rydberg–Rydberg interactions ∗ 6 should follow the scaling of van der Waals interactions, n∗11 0 /R , where n0 is the effective quantum number (see Section 4.3). Therefore, in the range n 50 we
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expect to observe a smooth onset of blockade effectiveness as a function of n. Statistics measurements were taken for a range of Rydberg states from 54D5/2 to 88D5/2 . At each tested n, the upper-transition laser intensity was adjusted such that approximately 50 Rydberg excitations were generated in the atomic sample. An intensity adjustment is necessary since the single-atom oscillator strength of optical transitions into Rydberg levels scales as n−3 . The values QD of the Rydberg-atom counting statistics plotted versus n are shown in Fig. 15. For n 60 we find QD ≈ 0, indicative of a random excitation and Poissonian excitation statistics. In the range 60 n 77, we observe the expected smooth transition from a Poissonian to highly sub-Poissonian statistics. For n 77, the QD -value saturates at −0.5. The interpretation of Fig. 15 needs to account for the fact that the effective bubble radius Rb in Fig. 13 increases rapidly with n. Using our calculated re∗ 6 sult that the level shift δW ≈ 100 n∗11 0 /R atomic units (n0 is the effective quantum number; see Section 4.3), noting that in the fully blockaded domain n 77 the Rydberg-atom separation R ≈ 2Rb , and assuming a laser linewidth ∗11/6 , or, in laboratory units, Rb (n∗0 ) ≈ of 5 MHz, we estimate Rb (n∗0 ) ≈ 36 a0 n0 ∗ 11/6 . Thus, over the blockaded range 77 n 88 in Fig. 15 5.8 µm × (n0 /80) the estimated Rydberg atom density NR ≈ 1.2 × 109 cm−3 × (80/n∗0 )11/2 drops from about 1.6 × 109 cm−3 to about 8 × 108 cm−3 . At an estimated groundstate atom density of 5 × 109 cm−3 , there are about three to six ground-state atoms in each “bubble”. Assuming a fairly efficient Rydberg-excitation mechanism, which we believe we have, it appears reasonable that the experimentally observed Rydberg-excitation blockade is fully developed under these conditions. An estimate of the overall excitation volume is given by the count number of about 30 divided by the Rydberg-atom detection efficiency and multiplied by the aforementioned estimate of Rydberg-atom densities. The excitation volume estimated in this manner is consistent with the geometry described in Section 4.6. In the blockaded regime, n 77, the experimental boundary condition of maintaining a fixed Rydberg-atom number entails that the overall excitation volume must increase with n. Since the upper-transition excitation laser has a Gaussian beam waist much smaller than the MOT diameter, such an increase of the effective excitation volume can arise if with increasing n the laser intensity is scaled up faster than n3 . As n is reduced from about 77, the bubble radius shrinks and, due to limited ground-state atom density, the system enters a regime in which there are too few ground-state atoms per bubble to ensure the generation of a Rydberg excitation in each bubble, and the significance of the blockade effect gradually diminishes. In the case of Fig. 15, the estimated number of ground-state atoms per bubble reaches one at n ∼ 63, which is quite close to where the Rydberg excitation statistics becomes Poissonian. We close this section by pointing out that, while much of the observed behavior could be reasonably well explained using calculations of
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F IG . 16. (a) Laser scans of the fine-structure doublet 76D5/2 and 76D3/2 vs frequency offset for powers of the 5P → 76D excitation laser ranging from 4.6 µW to 1.02 mW. The beam has a full width at half maximum (FWHM) diameter of 16 µm. The slight shifts of the resonance line one might perceive are attributed to slow drifts of the frequency reference used for the Rydberg-atom excitation laser. (b) The FWHM of the D5/2 component of the Rydberg-excitation lines shown in (a) is plotted as a function of the laser power. The FWHM of the line increases by only 1–2 MHz.
the van der Waals interaction strength, a few experimental parameters such as the laser linewidth and the ground-state atom density have uncertainties. Also, all estimates were based on the assumption of binary Rydberg–Rydberg interactions, while the physical system might exhibit interactions among three or more atoms. Therefore, in the future we will perform more measurements in order to solidify our interpretations.
4.8. E FFECT OF THE E XCITATION B LOCKADE ON THE T RANSITION L INEWIDTH Varying the frequency of the upper-transition (5P3/2 → nD5/2 ) laser, we have studied the width of the Rydberg-excitation line, the peak excitation number, and the area of the line for different values of the intensity of the upper-transition laser. As shown in Fig. 16, the FWHM line width increases slightly from ≈ 12 MHz to ≈ 14 MHz. This result is in strong contrast to a result reported by Singer et al. (2004), where substantial line-broadening is found. This difference may be attributable to an absence of plasma electric fields in our experiment, as discussed in Section 4.3. Simulations by Robicheaux and Hernández (2005) show a broadening of the Rydberg resonance lines that is similar to what we have found experimentally. Under the valid assumption that power broadening of the Rydberg-atom excitation is much less than the laser linewidth, the absence of significant intensityinduced broadening of the Rydberg-atom excitation line in a blockaded system
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F IG . 17. Peak counts (a) and area (b) of the D5/2 -component of the Rydberg-excitation lines in Fig. 16(a) versus laser power. At low laser powers both curves increase linearly, as shown by the solid reference lines, while at higher powers a saturation trend is observed.
can be qualitatively understood as follows. Assuming that the excitation laser is tuned into resonance with the lowest excitation step of the uneven energy ladder shown in Fig. 8, all excitations further up the ladder into states higher than |N, 1 are far-detuned relative to the laser line width. Therefore, in blockaded systems only the lowest step of the ladder is actually excited, and the observed width of the Rydberg-excitation line should be just about the same as in the case of isolated, non-interacting atoms, as observed in our experiments. In Fig. 17, we plot the peak counts and areas of the D5/2 -component of the Rydberg-excitation lines shown in Fig. 16(a) as a function of laser power. We observe a saturation trend, as has been observed previously by Tong et al. (2004). The saturation is an expected feature of blockaded Rydberg systems; it immediately follows from the “bubble” picture presented in Fig. 13. In our experiment, the saturation effect is partially compensated by the fact that with increasing intensity an increasing number of Rydberg atoms are excited in the wings of the upper-transition laser, which has a Gaussian profile with a FWHM width of much less than the atom-cloud diameter. With increasing laser intensity, more Rydberg atoms are added in the un-blockaded peripheral regions of the uppertransition laser beam, while the Rydberg-atom density in a central region of the beam saturates at a value equivalent to the bubble density (which depends only on the laser line width and the Rydberg-state quantum numbers). As a result, the Rydberg-atom number still increases with intensity even if the Rydberg-excitation is already blockaded throughout most of the excitation volume. Nevertheless, both numerical models and the experiment in Fig. 17 show that an observable saturation effect remains. However, the effect is somewhat less pronounced than in experiments with a fixed excitation volume (Tong et al., 2004).
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4.9. E XPERIMENTS IN P ROGRESS AND P LANNED R ESEARCH Applications of the Rydberg-excitation blockade in fields such as quantum information processing and research on single-photon sources requires implementations of the blockade in smaller atom samples. Current experiments are therefore pushing towards the demonstration of the blockade effect in atomic samples with small enough spatial extent that only one or no Rydberg excitation is created (i.e., the excitation volume is sufficiently small to fit into a single bubble of Fig. 13). Possible experimental schemes include the use of optical dipole traps formed by discrete laser beams in order to prepare atom clouds with diameters 10 µm. Many advances have been made in creating small, dense dipole traps (SebbyStrabley et al., 2005) and dipole traps with a single loaded atom (Schlosser et al., 2001). These clouds are small enough to achieve a situation in which the spatial range of the blockade encompasses the whole cloud. The atom-counting statistics technique discussed in Section 4.5 is ideal for measuring the blockade effectiveness in such small dipole traps. Small arrays of two or a few of such interacting clouds could then be used to proceed towards the realizations of rudimentary quantum gates, as required in quantum information processing. The simplified bubble model advocated in Fig. 13 qualitatively explains many phenomena associated with the Rydberg excitation blockade. However, there are no physical bubbles present in the system; they are meant merely to represent a typical minimal correlation length between neighboring Rydberg excitations. Anisotropy in the Rydberg–Rydberg interaction will have an effect on the threedimensional shape of the spatial correlation function between Rydberg excitations. This effect is expected to be particularly strong in the case of dipolar interactions. In this context, we are currently interested in a new method of characterizing the properties of the Rydberg excitation blockade in a spatially sensitive manner. In the method, correlations of spatially detected Rydberg counts are measured by position-resolved measurement of individual Rydberg excitations using position-sensitive MCP electron detectors. Measurements of this type allow one to verify directly calculations of spatial correlation functions of Rydberg excitations (Robicheaux and Hernández, 2005). An interesting aspect of this measurement is that before the Rydberg excitations are extracted from the many-body system via SSFI they are coherently shared among all atoms (see Section 4.2). A localization of excitations onto individual atoms is only achieved via a spatially sensitive (destructive) quantum measurement process in which the excitations, and their mutual correlations, are projected onto well-defined atoms. The spectra of the higher-lying transitions in the excitation ladder of Fig. 8 deserve further experimental consideration, because the magnitude of the interactions causing the blockade effect and the nature of these interactions could be directly revealed. One interesting experimental approach to this problem is to laser-excite the blockaded system with two independent lasers. A first excitation pulse has a fixed center frequency and is tuned to saturate the lowest stage of the
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excitation ladder in Fig. 8. A second laser pulse is scanned in frequency in order to probe the second (and possibly higher) excitation steps of the ladder. In the case of electric–dipole interactions, the second transition is blue- or red-shifted relative to the first excitation step, dependent on the detailed excitation geometry and the vector of the inter-atomic separation. In the case of a disordered atom sample and a large excitation volume one would therefore expect a spectrum of higher excitations featuring both a red- and a blue-shifted peak, similar to a plot of spectral densities in Fig. 8 of Lukin (2003). If van der Waals interactions were to dominate the blockaded system, the spectrum of the second excitation step should exhibit only one blue- or red-shifted peak. In the case of Rb nD Rydberg states, which typically have negative van der Waals interactions, that peak should be red-shifted relative to the frequency of the first excitation step. In the described manner, the higher excitations of the many-body quantum system could be probed explicitly and the interaction potential causing the blockade could be measured directly.
5. Rydberg-Atom Trapping Some proposed avenues of quantum information processing (Raimond et al., 2001; Jaksch et al., 2000; Lukin et al., 2001; Sørensen et al., 2004) and coherent control of Rydberg interactions in cavity-QED experiments (Hyafil et al., 2004; Zheng and Guo, 2000; Osnaghi et al., 2001) benefit from an ability to trap Rydberg atoms. Rydberg-atom trapping techniques could also become useful in other areas of physics. In antihydrogen research, Rydberg atoms are produced via recombination in positron and anti-proton plasmas. The trapping of slowly decaying anti-atoms in Rydberg states could assist in collecting a large enough number of ground-state anti-atoms for proposed measurements on trapped anti-hydrogen (Amoretti et al., 2002; Gabrielse et al., 2002a). Rydberg-atom traps may also be useful to study collective “excitation-hopping” effects in cold Rydberg-atom gases (Côté, 2000; Robicheaux et al., 2004) and to perform precision microwave spectroscopy on small samples of trapped Rydberg atoms (Dutta et al., 2000). The trapping of cold atoms in highly excited Rydberg states has been considered theoretically in a number of papers (Dutta et al., 2000; Hyafil et al., 2004; Lesanovsky and Schmelcher, 2005). In this section, we provide an overview of several possible Rydberg-atom trapping methods, and we discuss very recent experimental progress in the field of Rydberg-atom trapping. The trapping of long-lived, highly excited atoms is distinct from other atom-trapping methodologies in several respects. First, these near-macroscopic atoms may not be treated as a point-like particle because the length scale over which the trapping fields vary can be of the same order as the size of the atom to be trapped. The high density of electronic states and the couplings among them induced by the trapping field can make the situation quite complicated. The combined quantum states of the
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center-of-mass and electronic degrees of freedom can differ from the trappingfield-free states (Dutta et al., 2000; Lesanovsky and Schmelcher, 2005). Second, in Rydberg-atom trapping it is important to consider how stable the trapped atoms are against transitions into un-trapped states due to radiative interactions and couplings induced by stray electric or magnetic fields. Also, the trapped atoms may collide with each other, causing transitions into un-trapped states. These challenges must be addressed in experimental implementations of Rydberg-atom trapping.
5.1. E LECTROSTATIC RYDBERG -ATOM T RAPPING Neglecting core effects and spin, the Schrödinger equation of a Rydberg electron is separable in parabolic coordinates (Bethe and Salpeter, 1957; Gallagher, 1994), leading to a complete set of solutions, {|n, n1 , n2 , m}. The parabolic states are related to the spherical basis {|n, , m} via a well-known unitary transformation. Both sets share the usual principal and the magnetic quantum number, n and m. The parabolic quantum numbers n1 and n2 in the set of parabolic solutions satisfy the relation n = n1 + n2 + |m| + 1, where n1 and n2 are non-negative integers. Notice that as |m| increases, the maximum values allowed for n1 and n2 decrease. The potential of a Rydberg electron in a uniform external electric field is given by V = − 1r − Ez (core effects and spin neglected). In lowest order in E, the parabolic states {|n, n1 , n2 , m} are eigenstates of the hydrogen atom in an electric field. To first order, the energy of a Stark state can be written as 1 3 (3) + E(n1 − n2 )n, 2 2n2 indicating that the parabolic states have a permanent electric dipole moment of 3 2 n(n2 − n1 ). In electric fields that vary slowly in position, the internal and external motion can be, to some extent, adiabatically separated. The linear Stark shift of the internal motion, evident from Eq. (3), then generates Born–Oppenheimer potential surfaces Wn,n1 ,n2 ,m (R) = 32 n(n1 −n2 )|E(R)| for the center-of-mass degree of freedom R. Since states of equal (n1 − n2 ) are degenerate and share the same surface, internal and external dynamics cannot be de-coupled within sub-spaces of fixed (n1 − n2 ). This is not very relevant in the weak-field limit and under the assumption of slowly varying fields, because atoms in states of equal (n1 − n2 ), which mix as the atoms move through the field, experience equal forces (so the mixing does not have any consequences on the center-of-mass motion). In many cases, a classical treatment of the center-of-mass motion is sufficient. An atom moving adiabatically through a position-dependent electric field experiences a force, in the sense that (n1 − n2 ) is conserved, 3 F(R) = −∇R Wn,n1 ,n2 ,m (R) = − (n1 − n2 )n∇R E(R). (4) 2 Wn,n1 ,n2 ,m = −
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Depending on the sign of (n1 −n2 ) Rydberg-atoms are low- or high-electric-fieldseeking; states with positive (n1 − n2 ) are low-field-seeking. Trapping fields can be designed in close analogy with standard magnetic traps for atoms with magnetic dipole moments. A quadrupole electric field E(R) = β(X, Y, −2Z) is fairly well-suited to trap Rydberg atoms. The center-of-mass potential experienced by the atoms as they move in a trap formed by this electric field is 3 Wn,n1 ,n2 ,m = (n1 − n2 )nβ X 2 + Y 2 + 4Z 2 . (5) 2 This potential has a extremum at R = 0, towards which atoms with positive (n1 − n2 ) are attracted. Assuming a field gradient β = 100 Vcm−2 , (n1 − n2 ) ∼ 10 and n ∼ 40, which are all fairly easily achieved in the laboratory, trap accelerations in the range 100g to 1000g (g = 9.8 m/s2 ) are obtained. Our discussion of atom trapping using electric fields is based on the Stark effect of an atom in a weak, uniform electric field. On the other hand, following Eq. (4) the generation of trapping forces requires a certain degree of non-uniformity in the field. For the analysis presented above to be valid, the spatial variation of the field must be smooth enough so that it can be considered uniform over the volume of the Rydberg-electron wave-function. If macroscopic electrodes are used to generate the trapping field, this spatial condition is easily satisfied, because even for small electrodes (size ∼ 1 mm), the Rydberg-atom radius is at least a factor 103 smaller than the typical field variation length. Furthermore, Eq. (4) applies only if the quantum number n and the difference n1 − n2 are adiabatically conserved. Therefore, in the frame of reference of the moving atoms, the temporal variation of the field must be sufficiently smooth. To evaluate the validity of this temporal adiabaticity condition, we first note that the direction and the magnitude of the field in the atomic rest frame change on a time scale given by the center-of-mass oscillation period of the atoms in the trap, TCM . The evolution of the Rydbergelectron wave-function is adiabatic in time if TCM is large compared to the time scale of the internal atomic dynamics, which is given by the Stark time, TS . In laboratory units, TS = h/WS , where WS = 3ea0 nE/2 is the energy splitting due to the Stark effect. For most trap accelerations and sizes that can be experimentally realized, TCM exceeds TS by many orders of magnitude. For instance, rubidium atoms in the trap potential given by Eq. (5) with the listed parameters oscillate with a period of order 30 ms, while the Stark time ranges in fractions of μs. Therefore, in electrostatic Rydberg-atom traps we generally expect the internal state of the atoms to adiabatically follow the rotation of the electric field. Quadrupole trapping potentials such as Eq. (5) have zero electric field at the trap center. While the Stark time becomes infinite at that point, in initial realizations of electric Rydberg-atom traps it is highly unlikely that atoms ever come sufficiently close to the zero of the electric field that non-adiabatic transitions into un-trapped
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states begin to matter. To avoid this issue, one could, in close analogy with Ioffe– Pritchard magnetic traps, also build electric traps that have a trapping-potential minimum with a non-zero electric field. The perturbing effect of possible stray magnetic fields in the trapping region scales with the ratio between Larmor and Stark frequencies. Since in typical experimental settings (electric fields of order tens of V/cm and magnetic fields below 1 Gauss) this ratio is very small, magnetic perturbations of electric Rydbergatom traps are assumed to be minimal. The highest uncertainty regarding the realizability of electric Rydberg atom traps may come from electromagnetic transitions and collisions among Rydberg atoms, causing transitions into un-trapped states [states with negative (n1 − n2 )]. Since there are as many un-trapped states as there are trapped ones, the likelihood of such transitions could be high. In future experiments the stability of electrostatic Rydberg-atom traps will be determined.
5.2. RYDBERG -ATOM T RAPPING IN W EAK M AGNETIC F IELDS The usual treatment of magnetic trapping of ground-state atoms translates directly to Rydberg atoms provided that the magnetic fields are weak enough that diamagnetic effects as discussed in Sections 5.4 and 6 can be neglected and that the fields are varying slowly in space. The trapping of Rydberg atoms in quadrupole magnetic fields, Ioffe–Pritchard-type fields, etc., may therefore be possible. However, a fundamental concern is that moderate magnetic fields in the range below about 1000 Gauss may not generate trapping forces F(R) = −mμB ∇R |B(R)| large enough to clearly dominate electric–dipole forces due to environmental electric fields, given by Eq. (4). A possible solution to this problem could lie in the use of extreme B-field gradients produced by micro-fabricated current-carrying wires or superconducting magnets. The quantum dynamics of Rydberg atoms trapped in moderate magnetic fields (a few hundred Gauss) with extreme inhomogeneities has recently been discussed theoretically by Lesanovsky and Schmelcher (2005). This work accurately accounts for the mixing of internal atomic states caused by the large field gradients, due to which the magnetic field significantly varies within the volume of the internal Rydberg-electron wave-function. The authors have found that low-field-seeking, trappable Rydberg-atom states in high-gradient quadrupole magnetic fields exist. The realizability of magnetic Rydberg-atom traps in the field range below about 1000 Gauss is linked to the question of whether one is able to control electromagnetic transitions and collision-induced transitions into magnetically unbound states. Future experiments are required to answer this question.
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5.3. P ONDEROMOTIVE O PTICAL L ATTICES FOR RYDBERG ATOMS A quite different trapping method for Rydberg atoms involves the use of ponderomotive forces caused by inhomogeneous optical fields, which oscillate faster than any internal frequency of Rydberg atoms. The ponderomotive potential is the time-averaged kinetic energy of free electrons in an oscillating electric field. The form of this potential for a standing wave with optical frequency, ω, and positiondependent amplitude, |E(R)|, is given by e2 |E(R)|2 . (6) 4me ω2 Since the potential is proportional to the intensity of the light, in standing light waves, it has spatially periodic structure from which electrons can be diffracted. This leads to the well-known Kapitza–Dirac effect (Bucksbaum et al., 1988). A Rydberg electron is essentially a quasi-free electron that is weakly bound to an ionic core. Consequently, when placed in an optical field, the Rydberg electron is subject to the ponderomotive force on free electrons. Since the ionic core is still weakly bound to the electron, the ponderomotive force can be used to act on the whole Rydberg atom via the interaction between ionic core and Rydberg electron. The dynamics of the system can be formulated using the Born–Oppenheimer approximation (Dutta et al., 2000). Given the periodic nature of standing-wave ponderomotive potentials, optical lattices for Rydberg atoms can be formed. Ponderomotive optical lattices can be created in one, two or three dimensions, and it should be possible to trap Rydberg atoms in them. In proposed experiments, a standing-wave light field is created using a 1064 nm Nd:YAG laser. Fig. 18(a) shows a laser beam configuration which yields a onedimensional ponderomotive potential for trapping Rydberg atoms. For the anticipated YAG laser intensities, the depth of the ponderomotive optical lattice is of order h × 50 kHz. To load atoms into a trap of that depth, they must be pre-cooled using a sub-Doppler cooling method (Metcalf and van der Straten, 1999) before excitation to the Rydberg state. One method for experimental demonstration of a ponderomotive optical lattice for Rydberg-atom trapping involves the observation of trap oscillations of Rydberg atoms confined in the wells of the optical potential. The ponderomotive lattice has an oscillation period, TCM , associated with it. At a delay of TCM /4 after the application of the ponderomotive potential, the spatial distribution of the confined Rydberg atoms should collapse near the minima of the intensity profile. Using field-ionization imaging of Rydberg electrons onto a position-sensitive MCP detector at variable delay times, it should be possible to observe such breathing-mode trap oscillations. Despite the anharmonicity of the potential, several periods of this oscillation should still be observable. By modulating the intensity or the spatial phase of the applied optical standing waves at a frequency Ω, the Rydberg electron can be subjected to a timeperiodic perturbation (namely a time-periodic modulation of the ponderomotive W Q (R) =
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(b) F IG . 18. (a) Rydberg atoms confined in one dimension by a light field created by two laser beams propagating with a mutual angle, φ. The period of the lattice is given by λ/ sin(φ/2), where λ is the wavelength of the laser beams. (b) Rydberg atom located at the minimum of a periodic ponderomotive potential that is amplitude-modulated at a frequency Ω.
potential). Generally, the perturbation potential can be developed in spherical multipole components Al,m r l Yl,m (θ, φ) cos(Ωt), where (r, θ, φ) are relative coordinates of the Rydberg electron. The amplitude coefficients Al,m depend on the lattice geometry, the modulation type and amplitude, and the center-of-mass location of the Rydberg atom in the ponderomotive potential. According to firstorder time-dependent perturbation theory, the modulation allows us to couple Rydberg states that differ in energy by hΩ. The coupling strengths are given ¯ by ψf |Al,m r l Yl,m |ψi , with initial- and final-state Rydberg wave-functions |ψi and |ψf . Since the ponderomotive lattice varies on length scales that are of the same order as the size of the Rydberg-electron wave-function, it is fairly simple to construct situations in which higher-order multipole transitions become dominant or at least significant. For instance, in the one-dimensional case depicted in Fig. 18(b) the leading non-trivial term is A2,0 r 2 Y2,0 (θ, φ) cos(Ωt). In this way, a wide variety of multipole transitions between Rydberg states can be excited. By directing crossed, modulated lattice beams at well-defined locations within a cloud of Rydberg atoms, these transitions can, in principle, be applied in a spatially selective way. Also, because these transitions between Rydberg states are driven optically, there is no need to introduce microwave radiation into the enclosure of the Rydberg-atom sample. Since the ponderomotive lattice itself causes only small, known trap-induced internal-state level shifts in the trapped
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atoms, this technique offers the possibility of performing precision measurements of Rydberg-atom transitions. Using this method, high-precision Rydberg-atom spectroscopy could be performed, for instance, on electric–dipole and on electric– quadrupole transitions. By selecting transitions with vanishing DC-Stark or Zeeman shifts, errors caused by stray electric and magnetic fields could be minimized. Spectroscopic measurements in modulated ponderomotive lattices could be used to determine atomic properties (core polarizabilities, quantum defects, hyperfine splittings, etc.) and presumably the Rydberg constant. As an example, we consider the case of driving quadrupole transitions between Rydberg states by application of an amplitude-modulated standing-wave ponderomotive potential to a low-density, cold Rydberg-atom gas. Figure 18(b) shows a Rydberg atom located at the minimum of a one-dimensional ponderomotive potential with an intensity modulated at a frequency Ω. The potential experienced by the Rydberg electron is symmetric, so that it couples Rydberg states of the same parity. The Rydberg atom experiences a time-dependent potential V (z, t) = −1/r + Az2 (1 + cos(Ωt)), where the constant A depends on lattice depth and beam angles and is the amplitude modulation index. The value of A anticipated in the intended experiments is of order h × 1 MHz/(1000 nm)2 in laboratory units. Atomic transitions are excited if the modulation frequency Ω is tuned into resonance with the frequency of an allowed quadrupole transition frequency. For experiments around n = 40, using ψf |z2 |ψi ∼ a02 n4 and = 1, the estimated Rabi frequencies for quadrupole transitions are on the order ten kHz. It should therefore be possible to observe these transitions. Most Rydberg–Rydberg transitions have frequencies in the microwave range. For example, the quadrupole transition 47D to 48S occurs at ∼ 25 GHz. Electrooptic fiber modulators can provide amplitude modulation of standing waves up to about 30 GHz and can be used over a wide range of modulation frequencies Ω. For the estimated Rabi frequencies, the modulated ponderomotive potential would need be applied to the Rydberg atoms for times of order 10 µs in order to drive transitions significantly. State-selective field ionization techniques can be used to determine the final-state atom distribution in the Rydberg-atom gas to measure the extent to which the initial population is transferred to the selected Rydberg state.
5.4. T RAPPING OF RYDBERG ATOMS IN S TRONG M AGNETIC F IELDS In this section, we consider a fourth Rydberg-atom trapping method, namely the magnetic trapping of strongly magnetized, diamagnetic Rydberg atoms. In magnetic fields of a few Tesla and more, the interaction of the Rydberg electron with the magnetic field is stronger than the internal atomic Coulomb interaction. Atoms in strongly magnetized, high-angular momentum states have Penning-trap-like
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F IG . 19. (a) Classical trajectories of Rydberg electrons with m = 0 and −300 (energy = −2 × 10−5 in atomic units). The displayed orbit for m = −300 is characteristic of guiding-center drift atoms, because it features clearly distinct cyclotron, z-bounce and magnetron motions. (b) Guiding-center drift Rydberg atom in an inhomogeneous magnetic field. The magnetic-bottle force on the electron and internal Coulomb forces are shown. The net force on the atom is equal to the magnetic force on the Rydberg electron.
electronic orbits, and are called “guiding-center” drift atoms (Fig. 19). Classically, the motion of the Rydberg electron in guiding-center atoms is characterized by three decoupled motions: a fast, small-diameter cyclotron oscillation, a bounce motion parallel to the magnetic-field lines (“z-bounce” or axial motion), and a slow E × B magnetron drift of the electron in the plane transverse to B. The cyclotron component of the motion is the dominant electron-magnetic field interaction; the fact that its frequency is higher than all other frequencies of the motion reflects the dominance of magnetic forces in guiding-center atoms. The z-bounce motion is driven by the component of the atomic Coulomb force parallel to B. The magnetron drift is an E × B-drift of the Rydberg electron due to the external magnetic field B and the component of the atomic Coulomb field transverse to B. Classically, the motion can be well described using the guiding-center approximation, hence the name guiding-center drift atoms. Due to the clearly distinct time scales of the three components of the Rydberg-electron motion, the drift Rydbergelectron motion is stable (Glinsky and O’Neil, 1991) and can be quantized using semiclassical quantization rules (Raithel and Fauth, 1995). Quantum mechanically, drift Rydberg-atom states are solutions with large negative values of the magnetic quantum number m. The cyclotron component of the motion generates an energy-level structure similar to that of free-electron Landau levels (Landau and Lifshitz, 1977), while the z-bounce motion generates a Rydberg-like level series below each Landau level (Raithel and Fauth, 1995;
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Wang and Greene, 1989; Guest and Raithel, 2003). This quantization structure entails multiple ionization potentials and a complex electric-field ionization behavior of Rydberg atoms laser-excited in strong magnetic fields (Choi et al., 2005a), discussed in Section 7. Due to the fairly simple dynamics of guiding-center drift atoms, it is straightforward to estimate their magnetic dipole moments and the suitability for Rydbergatom trapping experiments. Since the motion of a drift-state Rydberg electron is dominated by its interaction with the magnetic field, the quasi-free Rydberg electron experiences forces similar to those acting on free electrons gyrating in inhomogeneous magnetic fields. Therefore, a conceivable strategy to trap drift Rydberg atoms is to apply an inhomogeneous magnetic field that would also be suitable to trap a free electron. Because of the presence of the weak residual atomic binding between the atomic core and the trapped electron, the entire atom may become trapped [as suggested in Fig. 19(b)]. In this simplified model, other intra-atomic interactions such as the effect of the magnetron drift are not accounted for. Since the atomic Coulomb forces (FCoul ) are internal and therefore cancel, in this model the total trapping force is equivalent to the magnetic force on the quasi-free Rydberg electron. Consequently, in inhomogeneous, strong magnetic fields with variation lengths much larger than the atomic size, the quantummechanical trapping potentials, VR (R), for drift Rydberg atoms are given by the position-dependent free-electron Landau energy levels, which are, in atomic units (Landau and Lifshitz, 1977), VR (R) = (nc + ms + 1/2)B(r) = −μeff B(R). (7) Here, nc and ms are the cyclotron and electron spin quantum numbers, respectively. Note that for the present case, m < 0, the magnetic quantum number m does not enter into this expression. The trapping potential can be phrased in terms of an effective dipole moment μeff . In classical electrodynamics, the directional derivative of −VR (R) parallel to the magnetic-field lines is equivalent to the magnetic-bottle force acting on charged particles (Jackson, 1999). Note that in Eq. (7) μeff is always negative, implying that all atoms would be low-field-seeking and could therefore be magnetically trapped. Considering the Rydberg electron’s magnetron motion, however, we find a mitigating positive (or high-field-seeking) contribution to μeff due to the anti-parallel nature of the magnetron and cyclotron magnetic moments. The magnetic moment of the magnetron motion is given by the product of the area of the circular magnetron orbit and the magnetron revolution frequency, both of which follow from the E × B-drift velocity of the Rydberg electron and the radius of the magnetron orbit, ρe . The magnetron magnetic moment is Ftrans ρe /B, where ρe is related to the negative magnetic quantum number m via m = −1/(Bρe3 ) − (1/2)Bρe2 and Ftrans denotes the transverse component of the atomic Coulomb field averaged over one period of the z-bounce motion. The actual calculation of Ftrans requires
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an integration of a Rydberg-electron trajectory over a half-period of the z-bounce motion. The magnetic moments of drift atoms after correcting for the effect of the magnetron motion are then obtained as Ftrans ρe (8) . B The positive correction term caused by the magnetron motion reduces the magnetic-trapping force and, for sufficiently small ρe , reverses the sign of μeff . Therefore, due to the magnetron motion not all drift Rydberg states are magnetically trapped. For light ionic cores such as in hydrogen, the nuclear motion becomes significant. While in principle the solution of a genuine two-body problem is required to deal with this situation (Johnson et al., 1983), it can be seen that for stable drift Rydberg atoms with significant ionic motion the latter simply consists of a planar, circular motion in the plane transverse to B (Choi et al., 2006). The frequency and the rotation sense of the circular ionic motion equal those of the electron magnetron motion. Notably, the frequency of the circular ionic motion in guidingcenter drift atoms does not equal the cyclotron frequency of an isolated ion. When the magnetic moment due to the circular ionic motion is taken into consideration, the expressions for μeff become slightly more complicated. The corrections due to the ion motion lead to more negative values of μeff , i.e., the ion motion tends to enhance the low-field-seeking nature of the atoms. Although this effect is noticeable when the ionic mass is relatively small (for example, in hydrogen Rydberg atoms), the ionic motion effect can be safely neglected in heavy-mass cases, including rubidium atoms. Therefore, we do not discuss the effect of the ion motion in further detail. Henceforth, Eq. (8) is used as a good approximation for the effective magnetic moment of rubidium drift Rydberg atoms. Before we move on to a discussion of a recent experimental realization of Rydberg-atom trapping, we comment briefly on the significance of positive-m Rydberg states in strong magnetic fields. Both in quantum and classical dynamics, positive- and negative-m solutions are connected by well-defined, simple transformations (Guest et al., 2003). They differ in magnetic moment, energy, radiative decay rates, and most likely also in collisional properties. The magnetic moments of positive-m states are more negative than those of the equivalent negative-m states by an amount m (in atomic units). Thus, the magnetic trapping forces acting on atoms in positive-m states can exceed those acting on atoms in the corresponding negative-m states by large factors. However, strongly magnetized Rydberg atoms with large, positive m-values are highly energetic, as they have cyclotron radii as large as their magnetron motion radii. Therefore, they are less likely to be produced in collisions than negative-m atoms. Even when produced, they will decay very rapidly due to cyclotron decay. For these reasons, we believe that the Rydberg-atom population in cold, collision-rich environments such μeff = −(nc + ms + 1/2) +
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F IG . 20. During the collision between laser-excited Rydberg atoms and a nearby free electron, the transverse electric field from the bypassing electron can cause an E × B drift motion of the bound Rydberg electron. The numbers in the parentheses represent the order in which the collision proceeds. A drift state can be produced after multiple collision events.
as cold, magnetized plasmas and Rydberg-atom gases consist mostly of negativem guiding-center atoms. This assessment is strengthened by the results of the Rydberg-atom trapping experiments discussed in the next section.
6. Experimental Realization of Rydberg-Atom Trapping Recently, the trapping of Rydberg atoms has been demonstrated for the first time (Choi et al., 2005b). In the experiment long-lived, guiding-center Rydberg atoms are produced via Rydberg atom collisions and trapped in a Ioffe-Pritchard magnetic trap with a bias field of 2.9 T. Here, we present two independent evidences for trapping. First, the decay behavior of Rydberg atoms with and without the presence of transverse confinement in the trap is compared. Second, trapped Rydberg atoms are observed to undergo induced oscillations due to a sudden change in the trap depth.
6.1. P RODUCTION AND D ECAY OF L ONG -L IVED RYDBERG ATOMS To produce the guiding-center drift Rydberg atoms, we first laser-excite Rb atoms collected in the high-magnetic-field laser-cooling apparatus described in Section 2.5 into strongly magnetized low-|m| Rydberg states using a pulsed dye laser (λ ≈ 480 nm, pulse width ≈ 10 ns, bandwidth ≈ 15 GHz, pulse energy 3 mJ; see Section 2.2). Within tens of microseconds, these atoms collide with free electrons and presumably with other Rydberg atoms, similar to what is observed in numerous experiments on cold Rydberg atoms in environments with vanishing magnetic field (Dutta et al., 2001; Walz-Flannigan et al., 2004; Li et al., 2004). During electron–Rydberg-atom collisions, transient transverse electric fields produced by free electrons can cause m-mixing and thereby produce
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high-|m| drift Rydberg states (Fig. 20). Free electrons necessary for the production of the long-lived drift states may be generated through Penning-ionizing collisions between Rydberg atoms (Li et al., 2005), which have been observed to generate free electrons with energies in the range of 10 meV at low magnetic field. These collisions may be favored in strong magnetic fields due to the large permanent electric quadrupole moments which characterize strongly magnetized Rydberg atoms. After the initial laser excitation, the decay of the population of the trapped Rydberg atoms is investigated by field ionization, explained in detail in Section 2.4. By applying an electric field ramp to the Rydberg atoms, the weakly bound electrons are stripped from their parent atoms. These electrons are magnetically imaged onto a MCP particle detector located outside the superconducting magnet used for atom trapping (see Fig. 4). To measure the number of Rydberg atoms, the electron pulses are pre-amplified and counted using a photon counter. Background electron counts are less than 0.002 per shot, which is significantly lower than the Rydberg-atom signal level. This implies that the measurement is virtually background-free. The strong magnetic field provides high collection efficiency and maintains the information on the spatial distribution of the Rydberg-atom cloud. We have detected Rydberg atoms in the trap region at times up to 200 ms after the excitation, as shown in Fig. 21(a). One out of 105 laser-excited Rydberg atoms survives for 200 ms. This observation indicates both the transfer of the initially laser-excited low-|m| Rydberg atoms, which are short-lived, into long-lived, high-|m|, guiding-center drift Rydberg atoms as well as the magnetic trapping of the latter. The lifetime extracted from the decay curve increases from ≈ 10 ms for detection times less than 80 ms to ≈ 80 ms thereafter. We believe that most of the decay is due to radiative transitions of the trapped Rydberg atoms into lower-lying states that cannot be detected. Decay into high-field-seeking states that are expelled from the trap also is possible. Because of the low radiation temperature (4K) and the low densities of trapped drift Rydberg atoms, decay of the Rydberg atoms due to thermal ionization or ionizing Rydberg–Rydberg collisions is deemed unlikely. To verify that a magnetic trapping force is confining the atoms, we have performed a reference measurement with the transverse confinement coils turned off. In this case, the magnetic-field magnitude generated by the trap magnet has a saddle point at the location of the atoms, as opposed to a minimum in the trapon-case [compare the insets in Fig. 21(a)]. In the trap-off case, low-field-seeking atoms should be expelled from the trap region in the directions transverse to the magnetic field. If a magnetic force was confining the Rydberg atoms, the measured Rydberg-atom population in the trapping region should reveal a much faster decay in the trap-off case than in the trap-on case. Indeed, comparing the opencircle data with the filled-circle data in Fig. 21(a), one sees that such a difference
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F IG . 21. (a) Rydberg atom counts vs detection time. Trap ON: Measurement results for a typical trapping configuration. Filled circles: initial Rydberg-atom number 105 and density 107 cm−3 . Half-filled circles: initial Rydberg-atom number 104 and density 106 cm−3 . Trap OFF, open circles: Measurement result with the transverse confinement coils turned off and with an initial Rydberg-atom number 105 and density 106 cm−3 . Insets: B-field magnitude has a minimum at the center location in the trap-on case, while it has a saddle point in the trap-off case. (b) Depending on the location of the atom cloud at the time of Rydberg excitation, sloshing and breathing motion can be observed. (c) A vertical profile of the Rydberg atom cloud is obtained by integrating the rows of the phosphor image. (d), (e) Observed oscillatory motion of the Rydberg atom cloud. (f) Vertical width of the cloud versus detection time for the breathing-mode case displayed in (d). (g) Center-of-mass position of the cloud versus detection time for the sloshing-mode case in part (e). [Reprinted figure with permission from Choi et al. (2005a).] © 2005 American Physical Society
is clearly observed. Without the transverse confinement, the number of Rydberg atoms detected at 40 ms delay is a factor of 400 less than the number detected with the trap enabled. Although the significantly different decay behavior observed in Fig. 21(a) strongly suggests the presence of the magnetic confinement, there could, in principle, be a possibility that the difference between the trap-on case of Fig. 21(a) (filled circles) and the trap-off case (open circles) originates mostly from the lack of collisions that promote Rydberg atoms from their initial state into the long-lived drift states. This suspicion might arise because without transverse confinement (trap-off case) the initial density of Rydberg atoms that can be achieved at the trap center is reduced by about a factor of 10 with respect to the trap-on case. Thus, in the trap-off case there tend to be fewer collisions that produce drift-state Rydberg atoms. To diffuse this doubt, we have performed a trap-on measurement in which the initial density of Rydberg atoms has been reduced artificially to a value
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equivalent to that of the trap-off case. The result [half-filled circles in Fig. 21(a)] affirms that it is indeed the magnetic confinement, not the density, which is responsible for the long dwell times of the Rydberg atoms in the detection region observed in the trap-on case.
6.2. O SCILLATIONS IN T RAPPED RYDBERG -ATOM C LOUDS The magnetic moment of the ground-state Rb atoms prepared in the trap before excitation into Rydberg states is μG = −1/2 (in atomic units). Comparing this with the μeff given in Eq. (8), we note that the average trapping force suddenly increases by a factor 2μeff as ground-state atoms are laser-excited and evolve into magnetically trapped drift Rydberg states. As a result of the sudden trap depth change, we expect to observe induced breathing and sloshing oscillations of Rydberg clouds [Fig. 21(b)]. Due to the spatial imaging capability of our high-B trapping setup, this oscillation appears as modulation in the vertical width of the detected electron images, or as a change in the vertical center-of-mass position of the electron images. Further, measuring the oscillation period of the Rydberg cloud and comparing the result with the known oscillation period of the groundstate atom trap (52 ms), we can experimentally determine the magnetic moments μeff of the trapped Rydberg atoms. This information is valuable because it allows us to obtain further insight into the nature of the trapped Rydberg atoms. In Fig. 21(d), the initial cloud position is close to its equilibrium position, leading to a Rydberg-atom cloud evolution that is strongly dominated by a breathingmode oscillation. A modulation in the size of the atom cloud is observed as the Rydberg-atom detection time is varied. In Fig. 21(f), the standard deviation of the distributions from Fig. 21(d) is plotted versus detection time (circles). The breathing-mode oscillation period is determined to be ≈ 9 ms. Considering that the breathing frequency equals twice the fundamental oscillation frequency, we conclude that μeff ≈ −4.0 (represents an ensemble average). Thus, the trap depth approximately increases by an order of magnitude as the atoms become excited into long-lived, trappable Rydberg states. We are further able to learn about the width of the μeff -distribution using the measured oscillation curve. The breathing mode oscillation in Fig. 21(f) displays a pronounced decay of the oscillation amplitude, which is attributed to the fact that different atoms, with different values of μeff , oscillate at different frequencies. The resultant inhomogeneous dephasing leads to a decay of the net oscillation signal. To obtain an estimate for the width of the μeff -distribution, we have fit the data assuming a Gaussian probability distribution for μeff . We have found best agreement for a probability distribution with a standard deviation μeff = 1.4 [dotted line in Fig. 21(f), with an average μeff = −4.0]. Best agreement between experimental data and fit result is obtained by an additional shift of the fit result
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by −2 ms in time, indicating a surprisingly fast initial collapse of the Rydberg atom cloud [solid line in Fig. 21(f)]. This shift could be due to a transient phase in the early stage of the Rydberg-atom evolution in which the atoms in low-angular momentum states have above-average magnetic moments, causing a rapid initial collapse. The vertical position of the ground-state atom cloud prior to Rydberg-atom excitation can be varied by modification of the detailed radiation-pressure conditions in the ground-state atom trap. Therefore, a sloshing motion of the Rydberg-atom cloud can be initiated by preparing the cloud such that it is initially displaced from its vertical equilibrium position [Figs. 21(e) and (g)]. As expected, the oscillation period of the sloshing motion of the average cloud position is ≈ 18 ms, or twice the period of the breathing oscillation.
6.3. S TATE A NALYSIS OF T RAPPED RYDBERG ATOMS The presented measurements of magnetic moments of trapped Rydberg atoms have yielded an average value of −8μB (equivalent to −4 atomic units). Based on this finding, the nature of the drift Rydberg states that have been trapped can be further investigated. In the following, we use classical trajectory calculations to estimate the quantum-number range of guiding-center drift states that have magnetic moments of the experimentally observed order. For this characterization of the trapped Rydberg atoms, we assume that the Rydberg-atom energy does not change very much during the evolution from the laser-excited low-|m| states into drift-state atoms. This assumption is plausible because m-mixing typically is a quasi-elastic process, and because the energies of the free electrons present in the trapping region are very low. In the following classical computation, we therefore focus on drift-atom trajectories with a fixed total energy of Wtot = −3.0 × 10−5 (in atomic units), equivalent to the laser excitation energy used in the described Rydberg-atom trapping experiment. For a given drift trajectory, the number of cyclotron quanta, nc , follows from the cyclotron energy, Wc , via nc = Wc /B − 1/2. In a classical computation, nc can take a continuous range of real numbers. The drift trajectory is further characterized by the z-component of the canonical angular momentum, |m| (m < 0), which follows from the value of the magnetron radius, ρe . The z-bounce motion is fully characterized by this value of ρe and the longitudinal energy, W , which is specified by known parameters (W = Wtot − Wc ). Consequently, the drift trajectories for given total energy Wtot can be represented on the (|m|, nc )plane. We obtain the magnetic moments of these trajectories using Eq. (8). Since the Rydberg atoms are prepared in Rydberg states with low-field-seeking orientation of the electronic spin in the experiment, we further account for the spin contribution by adding −μB to the magnetic moments. The result in Fig. 22
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F IG . 22. Magnetic moments of Rb drift Rydberg atoms in units of the Bohr magneton vs magnetic quantum number |m| (m < 0) and cyclotron excitation number nc , obtained from classical calculations. Wtot = −3 × 10−5 is assumed.
shows that for a total energy of Wtot = −3.0 × 10−5 all drift atoms on the (|m|, nc )-plane would be magnetically trapped (i.e., μeff < 0). Inspecting Fig. 22, we further learn that the experimentally observed range of magnetic moments, −11μB μ −5μB , corresponds to an m range of |m| 400 and an nc -range of 3 nc 12. These are the states that likely were populated in our high-B Rydberg-atom trapping experiment. In the future, the measured decay lifetimes could be used for a further characterization of the trapped Rydberg atoms (Guest et al., 2003). Finally, we would like to discuss the experimental realization of high-B Rydberg atom trapping in the context of problems that generally arise in Rydbergatom traps, mentioned several times throughout Section 5. Rydberg atoms in strong magnetic fields have several properties which we believe have been beneficial in realizing a robust trap. As seen in Fig. 22, we have found that the majority of drift Rydberg atoms produced have negative magnetic moments and are therefore attracted to the minimum of the B-field, regardless of their internal quantum numbers. Thus, in strong magnetic fields the density of states of lowfield-seeking, trappable states by far exceeds that of non-trappable states. As a result, state-changing collisions and radiative interactions are quite unlikely to turn a magnetically trapped Rydberg atom into an un-trapped one. Further, Rydberg states in strong magnetic fields are non-degenerate and have no permanent electric dipole moments, and consequently their susceptibilities to stray electric fields are lower than those of most Rydberg states in weak magnetic fields. Therefore, strong-magnetic-field Rydberg-atom traps are less likely to be compromised by stray electric fields than low-magnetic-field traps.
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7. Landau Quantization and State Mixing in Cold, Strongly Magnetized Rydberg Atoms We conclude this chapter with a discussion of photo-ionization of atoms in strong magnetic fields and field ionization of Rydberg atoms in strong parallel electric and magnetic fields. Auto-ionization of strongly magnetized Rydberg atoms is also discussed. The presentation of this topic builds on some of the concepts explained in Sections 5.4 and 6. The experiments discussed in the following have also been performed in the high-B atom trap described in Section 2.5. In contrast to the Rydberg-atom trapping experiments shown in the previous sections, here the atom density is kept low enough that collisions are suppressed. Thus, here we are dealing with laser-excited, strongly magnetized low-|m| Rydberg atoms, not with high-|m| guiding-center drift atoms. We first consider the Hamiltonian of a Rydberg atom in a strong magnetic field B = B zˆ in atomic units and in cylindrical coordinates, H =
p2 m2 − 1/4 B 2 ρ 2 B 1 + + − + (m + 2ms ) , 2 2 2 2 8 2 2ρ ρ +z
(9)
where the spin g-factor is taken to be 2. For simplicity, the spin–orbit interaction is neglected. The solutions from the Hamiltonian fall into uncoupled subspaces of constant m and ms , which we label as [m, ms ]. For the range of Rydbergexcitation energies shown in Fig. 23, the diamagnetic term in the Hamiltonian is, on average, larger than the Coulomb energy. As suggested by the discussion in Section 5.4, for large values of |m| a suitable approach to find the energies and eigenstates of the Hamiltonian in Eq. (9) is to adiabatically separate the dynamics into a fast component transverse and a slow component parallel to B. This Born– Oppenheimer separation leads to drift-atom solutions (Wang and Greene, 1989; Glinsky and O’Neil, 1991; Friedrich, 1998; Guest and Raithel, 2003). The transverse cyclotron and spin energies of these states are 1 1 Wnc ,m,ms = nc + m + |m| + + ms h¯ ωc . (10) 2 2 This expression applies to both positive and negative m. The z-bounce motion parallel toB is described by a one-dimensional potential Vnc ,m,ms (z) ≈ √ Wnc ,m,ms − 1/ z2 + r02 , where r0 ≈ 2|m|/B is the ρ-component of the separation between the gyration center of the Rydberg electron and the ionic core. The z-motion generates an additional quantum number, k. The adiabatic Rydberg states are fully characterized by quantum numbers |nc , m, ms , k and energies Wnc ,m,ms − ξ(nc , m, k). For large k, the binding energy of the z-motion ξ(nc , m, k) → 2/k 2 . Therefore, each set of cyclotron and spin quantum numbers,
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F IG . 23. Density plots of the electron signal versus time and level energy relative to the field-free ionization potential for B-values of (a) 6 T and (b) 2.9 T and mixed laser polarization. A linear SSFI pulse, indicated on top of panels (a) and (b), is applied at t ≈ 11 µs. A linear gray scale is used to render the data. Details of the experiment are explained in the text. (c) Electron signal for σ − -excitation at the indicated energy. Arrays of 200 traces of this type, taken over a range of excitation energies, are used to generate the density plots in (a) and (b). The trace displayed in (c) clearly exhibits a slow, auto-ionizing decay of Rydberg atoms during the electric-field-free time interval 0 < t < 10 µs. As seen in the inset, the decay signal is fitted fairly well by a decaying exponential with a time constant of τ = 10.4 µs. (d) Sketches of z-bounce potentials of the [m, ms ]-subspaces and relevant couplings between subspaces for an initial laser-excitation into the [0, 1/2]-manifold. Each z-bounce potential contains densely spaced energy levels. [Reprinted figure with permission from Choi et al. (2005a).] © 2005 American Physical Society
nc , m, ms , is associated with a Rydberg series that converges to an ionization potential given by Wnc ,m,ms in Eq. (10). For states with low values of |m| that we study in this section, the adiabatic approximation fails (Wang and Greene, 1989; Friedrich, 1998; Guest and Raithel, 2003) and the Landau channels of same m, same ms , and different nc become mixed. This mixing has a significant implication on the ionization potentials of the [m, ms ]-subspaces. Since all low-|m| Rydberg states contain non-vanishing nc = 0 character, the ionization potentials of the [m, ms ]-subspaces follow from
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Eq. (10) by setting nc = 0. Each low-|m| subspace [m, ms ] is, therefore, expected to have an ionization potential Wm,ms = [ 12 (m + |m|) + 12 + ms ]h¯ ωc . In our experiment, we observe the shifts Wm,ms of the ionization potentials relative to the field-free ionization potential of the 5P3/2 state of Rb. Relative to that reference ionization potential, the ionization-potential change observed in a strong magnetic field equals Wm,ms = Wm,ms − W5P , where W5P denotes the Zeeman shift of the utilized 5P3/2 Zeeman component from which the ionization potential is measured. In the high-B trap, we prepare atoms in a spin-polarized 5P3/2 -state with magnetic and spin quantum numbers m0 = 1 and ms0 = +1/2, which has a Zeeman shift of W5P = (m0 +2ms0 )h¯ ωc /2 (see Section 2.5). The observed magnetic-field-induced shifts Wm,ms of the ionization potentials should thus follow 1 1 Wm,ms = Wm,ms − W5P = (11) m + |m| + ms − h¯ ωc . 2 2 Since we use photo-ionization in order to measure the ionization potentials, the spin quantum number should not change during excitation, i.e., one may expect to observe ionization potentials associated only with ms = 1/2-manifolds. Based on optical selection rules we further expect to observe three relative ionization potentials only, namely Wm,ms =1/2 = mh¯ ωc with m = 0, 1 or 2. The lowest observed ionization potential—the one for m = 0—should be independent of the magnetic field, while the separation between adjacent ionization potentials should be h¯ ωc . The results of photo-excitation and field ionization experiments are shown in Fig. 23. Rb atoms in the state 5P3/2 are excited into Rydberg states using a pulsed dye laser (λ ≈ 480 nm, pulse width ≈ 10 ns, bandwidth ≈ 15 GHz; see Section 2.2). The polarization of the excitation laser is linear and perpendicular to the magnetic field. The excitation energy, plotted on the y-axes of Figs. 23(a) and (b), is scanned over a range from −hc × 18.7 cm−1 to hc × 7.4 cm−1 relative to the field-free photo-ionization threshold. The excitation occurs near time t = 0; time is plotted on the x-axes of Figs. 23(a) and (b). In the range 0 < t 11 µs, the excited atoms evolve freely, i.e., without applied electric field (and practically under absence of collisions). After t ≈ 11 µs a linear state-selective electric-field ionization ramp (SSFI) is applied in order to measure the distribution of Rydberg atoms over bound states. The electric field of the SSFI pulse is parallel to B; the time dependence of the pulse is indicated on top of Figs. 23(a) and (b). The photo- and field-electrons emerging from the trap area in response to the photoexcitation and SSFI are detected using a MCP detector, seen in Fig. 4. For each probed photo-excitation energy, 40 MCP traces are averaged on a digital scope and recorded. In Figs. 23(a) and (b), the records taken at ≈ 200 different excitation energy values and at respective magnetic fields of 6 T and 2.9 are assembled to produce two-dimensional plots.
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The results in Figs. 23(a) and (b) show Rydberg-state distributions over bound energy levels characterized by a prominent band pattern during the SSFI ramp. In both Figs. 23(a) and (b), multiple SSFI bands with associated ionization potentials (solid and dotted horizontal lines) are observed. The ionization potentials, defined by the high-energy ends of the SSFI bands, largely follow the optical selection rules stated above. In both Figs. 23(a) and (b), there are three dominant SSFI bands with ionization thresholds given by 0, h¯ ωc and 2h¯ ωc , equivalent to the mquantum numbers of 0, 1 and 2 that are allowed by electric–dipole excitation from the intermediate m0 = 1 level. We also observe multiple weaker SSFI bands that obviously violate the optical selection rules. Those bands are indicative of mmixing present in the experiment. Most notably, there also is one SSFI band that appears below the B-independent band. According to Eq. (11), this band must correspond to Rydberg atoms in subspaces [m 0, ms = −1/2], which have a relative ionization potential of Wm0,ms =−1/2 = −h¯ ωc . We believe that the spin-flip interaction that is needed in order to populate ms = −1/2-states is the spin–orbit interaction (see estimate below). During the electric-field-free parts of Figs. 23(a), (b), and (c) (t 11µs), we also find slowly decaying electron signals for excitation energies Wexc relative to the field-free ionization potential in the range −h¯ ωc < Wexc < 2h¯ ωc . These decay signals are caused by gradual auto-ionization of metastable Rydberg levels. The selected SSFI trace displayed in Fig. 23(c) shows an auto-ionization time scale of order 10 µs. Furthermore, Fig. 23(c) demonstrates that for laser excitation into the range −h¯ ωc < Wexc < 0 there is a short time window 0 t 2 µs in which the auto-ionization current increases in time before it transitions into a decaying regime. For laser excitation into the range 0 < Wexc < 2h¯ ωc we find that the auto-ionization current always continuously decreases with time. As explained below, the slow auto-ionization behavior is due to a combination of spin-flips and m-mixing, as is the presence of the SSFI bands that violate the photo-excitation selection rules. The observations in Figs. 23(a), (b), and (c) can be explained qualitatively with a model illustrated in Fig. 23(d). For simplicity we assume that atoms are initially excited into the manifold [m = 0, ms = 1/2] using a σ − -polarized excitation beam [as used in Fig. 23(c)]. The spin-flip and m-mixing effects are predominantly energy-conserving, as can be concluded from the fact that in Fig. 23 all SSFI bands are related via translations by integer multiples of h¯ ωc in energy (i.e., the mixing effects do not involve radiative interaction). Thus, all couplings between levels in the various z-bounce potentials sketched in Fig. 23(d) can be assumed to be energy-conserving (i.e., the arrows indicating mixing-induced transitions are horizontal). Spin–orbit coupling conserves m+ms even in the presence of strong magnetic fields. In the case considered in Fig. 23(d), the spin–orbit interaction couples states in the [m = 0, ms = 1/2]-subspace to states in the [m = 1, ms = −1/2]-subspace [thick horizontal arrow in Fig. 23(d)]. At our ex-
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citation energy, the spin–orbit interaction strength can be estimated roughly to be of order h × 1 MHz [in this estimate, we use spin–orbit coupling constants from (Zimmerman et al., 1979)]. Further, based on the time it takes for the higher-lying SSFI bands to develop (Choi et al., 2005a), the strength of the m-mixing is estimated to be of order h × 0.1 MHz. This estimate is consistent with stray electric fields and residual motional electric fields of order a few mV/cm, and average diameters of the Rydberg states transverse to B of order 1000 a0 . An upper-limit estimate for the m-mixing strength is obtained by taking the product of these numbers. The observable consequences of the outlined coupling scheme are as follows. Upward m-mixing starting from m = 0, corresponding to transitions leading to the right in Fig. 23(d), explains the experimentally observed population of higher SSFI bands. Downward m-mixing starting from m = 0 does not change the ionization potential and therefore does not have any immediate effects, according to Eq. (11). Similarly, spin–orbit interaction couples states in the initially populated [m = 0, ms = 1/2]-subspace to states in the [m = 1, ms = −1/2]-subspace, which according to Eq. (11) does not change the ionization potential and therefore has no immediate effects. However, a combination of downward m-mixing and spin–orbit coupling, starting from the manifold [m = 0, ms = 1/2], populates the manifold [m = 0, ms = −1/2], which, according to Eq. (11), has an ionization potential of −h¯ ωc . Thus, spin–orbit coupling and m-mixing between the center column and the leftmost column in Fig. 23(d) explains the appearance of the lowest SSFI band in Fig. 23(a) and (b). Finally, by inspecting Fig. 23(d) further, it is seen that the presence of both downward m-mixing and spin–orbit coupling explains the slow auto-ionization of metastable [m = 0, ms = 1/2]-Rydberg levels with energies in the range −h¯ ωc < Wexc < 0. The fact that this is a second-order process leading through an intermediate bound manifold, e.g., [m = 0, ms = 1/2] → [m = 1, ms = −1/2] → [m = 0, ms = −1/2], might explain why the auto-ionization current in Fig. 23(c) first progresses through an increasing phase, before it turns into an exponential decay. In contrast, for initial excitation into the manifold [m = 1, ms = 1/2] in an energy range 0 < Wexc < hω ¯ c , and for initial excitation into the manifold [m = 2, ms = 1/2] in an energy range h¯ ωc < Wexc < 2h¯ ωc , a single m-mixing step is sufficient to cause auto-ionization. This might explain why in the energy range 0 < Wexc < 2h¯ ωc the auto-ionization signals observed in Figs. 23(a) and (b) are continuously decreasing at all times.
8. Conclusion In this chapter, we have provided an overview of various types of dynamics that can occur in ensembles of cold Rydberg atoms prepared in close-to-zero and in
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strong magnetic fields. If the samples are moderately dense, a large variety of state-changing collision processes involving Rydberg atoms is observed (m-, -, and n-mixing collisions with electrons, ionization by Rydberg-atom-electron collisions, Penning-ionizing collisions between pairs of Rydberg atoms). We have discussed in detail the effect of Penning-ionizing collisions on the distribution of the center-of-mass velocities of the Rydberg atoms contained in the Rydberg-atom gas. In the case of very high density and long interaction time, large fractions of cold Rydberg-atom gases ionize and form cold plasmas. At low densities and atom numbers, a window of parameters exists in which these drastic, state-changing collision processes take a sufficiently long time to develop so that a more fragile, coherent type of dynamics becomes observable. In small ensembles of cold atoms, Rydberg excitations can be coherently shared among atoms in a manner equivalent to delocalized excitonic excitations. Manybody states of this kind that carry more than one Rydberg excitation can be energy-shifted due to coherent Rydberg–Rydberg interaction. We have discussed in detail an experiment in which manifestations of this shift have been observed in the Rydberg excitation number statistics of small atom samples. In this context, we have also described an efficient STIRAP excitation technique for Rydberg atoms. We have outlined several strategies of how cold Rydberg atoms could, in principle, be trapped using electromagnetic fields, and we have discussed the merits which this techniques could have in future experiments. A recent experimental demonstration of Rydberg-atom trapping in a superconducting highmagnetic-field trap has been presented in detail. A closely related special topic, namely the effect of Landau quantization on the ionization behavior, was discussed.
9. Acknowledgements We would like to thank former students and postdoctoral research associates for valuable contributions, making this review chapter possible. These are, in alphabetical order, S. Dutta, D. Feldbaum, J.R. Guest, E. Hansis, A.P. Povilus, B.K. Teo, and A. Walz-Flannigan. We also thank Professor P.R. Berman for his important contributions to the work and Professors P.H. Bucksbaum and D. Steel for the generous loaning of equipment. This work was supported by the National Science Foundation (Grants No. PHY-0245532 and PHY-0114336) and by the Chemical Sciences, Geosciences and Biosciences Division of the Office of Basic Energy Sciences, Office of Science, U.S. Department of Energy.
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10. References Afrousheh, K., Bohlouli-Zanjani, P., Vagale, D., Mugford, A., Federov, M., Martin, J.D.D. (2004). Spectroscopic observation of resonant electric dipole–dipole interactions between cold Rydberg atoms. Phys. Rev. Lett. 93, 233001. Akulin, V.M., de Tomasi, F., Mourachko, I., Pillet, P. (1999). Level-band problem and many-body effects in cold Rydberg atoms. Phys. D 131, 125. Amoretti, M., Amsler, C., Bonomi, G., Bouchta, A., Bowe, P., Carraro, C., Cesar, C.L., Charlton, M., Collier, M.J.T., Doser, M., Filippini, V., Fine, K.S., Fontana, A., Fujiwara, M.C., Funakoshi, R., Genova, P., Hangst, J.S., Hayano, R.S., Holzscheiter, M.H., Jørgensen, L.V., Lagomarsino, V., Landua, R., Lindelöf, D., Lodi Rizzini, E., Macrì, M., Madsen, N., Manuzio, G., Marchesotti, M., Montagna, P., Pruys, H., Regenfus, C., Riedler, P., Rochet, J., Rotondi, A., Rouleau, G., Testera, G., Variola, A., Watson, T.L., van der Werf, D.P. (2002). Production and detection of cold antihydrogen atoms. Nature 419, 456. Anderson, W.R., Robinson, M.P., Martin, J.D.D., Gallagher, T.F. (2002). Dephasing of resonant energy transfer in a cold Rydberg gas. Phys. Rev. A 65, 063404. Anderson, W.R., Veale, J.R., Gallagher, T.F. (1998). Resonant dipole–dipole energy transfer in a nearly frozen Rydberg gas. Phys. Rev. Lett. 80, 249. Ates, C., Pohl, T., Pattard, T., Rost, J.M. (2006). Strong interaction effects on the atom counting statistics of ultracold Rydberg gases. J. Phys. B 39, L233. Bergmann, K., Theuer, H., Shore, B.W. (1998). Coherent population transfer among quantum states of atoms and molecules. Rev. Mod. Phys. 70, 1003. Bethe, H.A., Salpeter, E.E. (1957). “Quantum Mechanics of One- and Two-Electron Atoms”. Academic Press, New York. Bouchoule, I., Mølmer, K. (2002). Spin squeezing of atoms by the dipole interaction in virtually excited Rydberg states. Phys. Rev. A 65, 041803(R). Bucksbaum, P.H., Schumacher, D.W., Bashkansky, M. (1988). High-intensity Kapitza–Dirac effect. Phys. Rev. Lett. 61, 1182. Carroll, T.J., Claringbould, K., Goodsell, A., Lim, M.J., Noel, M.W. (2004). Angular dependence of the dipole–dipole interaction in a nearly one-dimensional sample of Rydberg atoms. Phys. Rev. Lett. 93, 153001. Côté, R. (2000). From classical mobility to hopping conductivity: Charge hopping in an ultracold gas. Phys. Rev. Lett. 85, 5316. Choi, J.H., Guest, J.R., Hansis, E., Povilus, A.P., Raithel, G. (2005a). Landau quantization and time dependence in the ionization of cold, strongly-magnetized Rydberg atoms. Phys. Rev. Lett. 95, 253005. Choi, J.H., Guest, J.R., Povilus, A.P., Hansis, E., Raithel, G. (2005b). Magnetic trapping of long-lived cold Rydberg atoms. Phys. Rev. Lett. 95, 243001. Choi, J.-H., Guest, J.R., Raithel, G., (2006). Magnetic trapping of strongly-magnetized Rydberg atoms. European J. Phys. D, in press. Chu, S., Bjorkholm, J., Ashkin, A., Cable, A. (1986). Experimental observation of optically trapped atoms. Phys. Rev. Lett. 57, 314. Cubel, T., Teo, B.K., Malinovsky, V.S., Guest, J.R., Reinhard, A., Knuffman, B., Berman, P.R., Raithel, G. (2005). Coherent population transfer of ground-state atoms into Rydberg states. Phys. Rev. A 72, 023405. de Oliveira, A.L., Mancini, M.W., Bagnato, V.S., Marcassa, L.G. (2003). Rydberg cold collisions dominated by ultralong range potential. Phys. Rev. Lett. 90, 143002. Dutta, S.K., Feldbaum, D., Walz-Flannigan, A., Guest, J.R., Raithel, G. (2001). High-angularmomentum states in cold Rydberg gases. Phys. Rev. Lett. 86, 3993.
10]
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199
Dutta, S.K., Guest, J.R., Feldbaum, D., Walz-Flannigan, A., Raithel, G. (2000). Ponderomotive optical lattice for Rydberg atoms. Phys. Rev. Lett. 85, 5551. Farooqi, S.M., Tong, D., Krishnan, S., Stanojevic, J., Zhang, Y.P., Ensher, J.R., Estrin, A.S., Boisseau, C., Côte, R., Eyler, E.E., Gould, P.L. (2003). Long-range molecular resonances in a cold Rydberg gas. Phys. Rev. Lett. 91, 183002. Feldbaum, D., Morrow, N.V., Dutta, S.K., Raithel, G. (2002). Coulomb expansion of laser-excited ion plasmas. Phys. Rev. Lett. 89, 173004. Frasier, J.S., Celli, V., Blum, T. (1999). Resonant processes in a frozen gas. Phys. Rev. A 59, 4358. Friedrich, H. (1998). “Theoretical Atomic Physics”. Springer, Berlin. Gabbanini, C., Ceccherini, F., Gozzini, S., Lucchesini, A. (1998). Partial photoionization cross section measurement in a Rb magneto-optical trap. J. Phys. B 31, 4143. Gabrielse, G., Bowden, N.S., Oxley, P., Speck, A., Storry, C.H., Tan, J.N., Wessels, M., Grzonka, D., Oelert, W., Schepers, G., Sefzick, T., Walz, J., Pittner, H., Hänsch, T.W., Hessels, E.A. (2002a). Background-free observation of cold antihydrogen with field-ionization analysis of its states. Phys. Rev. Lett. 89, 213401. Gabrielse, G., Bowden, N.S., Oxley, P., Speck, A., Storry, C.H., Tan, J.N., Wessels, M., Grzonka, D., Oelert, W., Schepers, G., Sefzick, T., Walz, J., Pittner, H., Hänsch, T.W., Hessels, E.A. (2002b). Driven production of cold antihydrogen and the first measured distribution of antihydrogen states. Phys. Rev. Lett. 89, 233401. Gallagher, T.F. (1994). “Rydberg Atoms”. Cambridge University Press, New York. Glinsky, M.E., O’Neil, T.M. (1991). Guiding center atoms: Three-body recombination in a strongly magnetized plasma. Phys. Fluids B 3, 1279. Guest, J.R., Choi, J.H., Hansis, E., Povilus, A.P., Raithel, G. (2005). Laser cooling and magnetic trapping at several tesla. Phys. Rev. Lett. 94, 073003. Guest, J.R., Choi, J.H., Raithel, G. (2003). Decay rates of high-|m| Rydberg states in strong magnetic fields. Phys. Rev. A 68, 022509. Guest, J.R., Raithel, G. (2003). High-|m| Rydberg states in strong magnetic fields. Phys. Rev. A 68, 052502. Hansis, E., Cubel, T., Choi, J.H., Guest, J.R., Raithel, G. (2005). Simple pressure-tuned Fabry–Pérot interferometer. Rev. Sci. Instrum. 76, 033105. Hyafil, P., Mozley, J., Perrin, A., Tailleur, J., Nogues, G., Brune, M., Raimond, J.M., Haroche, S. (2004). Coherence-preserving trap architecture for long-term control of giant Rydberg atoms. Phys. Rev. Lett. 93, 103001. Ichimaru, S. (1982). Strongly coupled plasmas. Rev. Mod. Phys. 54, 1017. Jackson, J.D. (1999). “Classical Electrodynamics”. Wiley, New York. Jaksch, D., Cirac, J.I., Zoller, P., Rolston, S.L., Côte, R., Lukin, M.D. (2000). Fast quantum gates for neutral atoms. Phys. Rev. Lett. 85, 2208. Johnson, B.R., Hirschfelder, J.O., Yang, K.-H. (1983). Interaction of atoms, molecules, and ions with constant electric and magnetic fields. Rev. Mod. Phys. 55, 109. Killian, T.C., Kulin, S., Bergeson, S.D., Orozco, L.A., Orzel, C., Rolston, S.L. (1999). Creation of an ultracold neutral plasma. Phys. Rev. Lett. 83, 4776. Killian, T.C., Lim, M.J., Kulin, S., Dumke, R., Bergeson, S.D., Rolston, S.L. (2001). Formation of Rydberg atoms in an expanding ultracold neutral plasma. Phys. Rev. Lett. 86, 3759. Knuffman, B., Raithel, G. (2006). Emission of fast atoms from a cold Rydberg gas. Phys. Rev. A 73, 020704(R). Kulin, S., Killian, T.C., Bergeson, S.D., Rolston, S.L. (2000). Plasma oscillations and expansion of an ultracold neutral plasma. Phys. Rev. Lett. 85, 318. Kuzmin, S.G., O’Neil, T.M. (2002). Numerical simulation of ultracold plasmas: How rapid intrinsic heating limits the development of correlation. Phys. Rev. Lett. 88, 065003. Landau, L.D., Lifshitz, E.M. (1977). “Quantum Mechanics: Non-Relativistic Theory”. Pergamon Press.
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J.-H. Choi et al.
[10
Lee, K.I., Kim, J.A., Noh, H.R., Jhe, W. (1996). Single-beam atom trap in a pyramidal and conical hollow mirror. Opt. Lett. 21, 1177. Lesanovsky, I., Schmelcher, P. (2005). Magnetic trapping of ultracold Rydberg atoms. Phys. Rev. Lett. 95, 053001. Li, L., Noel, M.W., Robinson, M.P., Tanner, P.J., Gallagher, T.F., Comparat, D., Tolra, B.L., Vanhaecke, N., Vogt, T., Zahzam, N., Pillet, P., Tate, D.A. (2004). Evolution dynamics of a dense frozen Rydberg gas to plasma. Phys. Rev. A 70, 042713. Li, W., Tanner, P.J., Gallagher, T.F. (2005). Dipole–dipole excitation and ionization in an ultracold gas of Rydberg atoms. Phys. Rev. Lett. 94, 173001. Liebisch, T.C., Reinhard, A., Berman, P.R., Raithel, G. (2005). Atom counting statistics in ensembles of interacting Rydberg atoms. Phys. Rev. Lett. 95, 253002. Lowell, J., Northup, T., Patterson, B., Takekoshi, T., Knize, R. (2002). Measurement of the photoionization cross section of the 5S1/2 state of Rubidium. Phys. Rev. A 66, 62704. Lu, Z.T., Corwin, K.L., Renn, M.J., Anderson, M.H., Cornell, E.A., Weimann, C.E. (1996). Lowvelocity intense source of atoms from a magneto-optical trap. Phys. Rev. Lett. 77, 3331. Lukin, M.D. (2003). Colloquium: trapping and manipulating photon states in atomic ensembles. Rev. Mod. Phys. 75, 457. Lukin, M.D., Fleischhauer, M., Côte, R., Duan, L.M., Jaksch, D., Cirac, J.I., Zoller, P. (2001). Dipole blockade and quantum information processing in mesoscopic atomic ensembles. Phys. Rev. Lett. 87, 037901. Mandel, L. (1979). Sub-Poissonian photon statistics in resonance fluorescence. Opt. Lett. 4, 205. Mansbach, P., Keck, J. (1969). Monte Carlo trajectory calculations of atomic excitation and ionization by thermal electrons. Phys. Rev. A 181, 275. Mazevet, S., Collins, L.A., Kress, J.D. (2002). Evolution of ultracold neutral plasmas. Phys. Rev. Lett. 88, 055001. Metcalf, H.J., van der Straten, P. (1999). “Laser Cooling and Trapping”. Springer, New York. Mossberg, T., Flusberg, A., Kachru, R., Hartmann, S.R. (1977). Tri-level echoes. Phys. Rev. Lett. 39, 1523. Mourachko, I., Comparat, D., de Tomasi, F., Fioretti, A., Nosbaum, P., Akulin, V.M., Pillet, P. (1998). Many-body effects in a frozen Rydberg gas. Phys. Rev. Lett. 80, 253. Mudrich, M., Zahzam, N., Vogt, T., Comparat, D., Pillet, P. (2005). Back and forth transfer and coherent coupling in a cold Rydberg dipole gas. Phys. Rev. Lett. 95, 233002. Murillo, M.S. (2001). Using Fermi statistics to create strongly coupled ion plasmas in atom traps. Phys. Rev. Lett. 87, 115003. Myatt, C.J., Newbury, N.R., Ghrist, R.W., Loutzenhiser, S., Wieman, C.E. (1996). Multiply loaded magneto-optical trap. Opt. Lett. 21, 290. Osnaghi, S., Bertet, P., Auffeves, A., Maioli, P., Brune, M., Raimond, J.M., Haroche, S. (2001). Coherent control of an atomic collision in a cavity. Phys. Rev. Lett. 87, 037902. Petrich, W., Anderson, M.H., Ensher, J.R., Cornell, E.A. (1994). Behavior of atoms in a compressed magneto-optical trap. J. Opt. Soc. Amer. B 11, 1332. Pohl, T., Pattard, T., Rost, J.M. (2003). Plasma formation from ultracold Rydberg gases. Phys. Rev. A 68, 010703(R). Raab, E., Prentiss, M., Cable, A., Chu, S., Pritchard, D. (1987). High-angular-momentum states in cold Rydberg gases. Phys. Rev. Lett. 59, 2631. Raimond, J.M., Brune, M., Haroche, S. (2001). Manipulating quantum entanglement with atoms and photons in a cavity. Rev. Mod. Phys. 73, 565. Raithel, G., Fauth, M. (1995). Observation of high angular momentum states of Rydberg atoms in strong magnetic and weak crossed electric fields. J. Phys. B 28, 1687. Robicheaux, F. (2005). Ionization due to the interaction between two Rydberg atoms. J. Phys. B 38, S333.
10]
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201
Robicheaux, F., Hanson, J.D. (2002). Simulation of the expansion of an ultracold neutral plasma. Phys. Rev. Lett. 88, 055002. Robicheaux, F., Hanson, J.D. (2003). Simulated expansion of an ultra-cold, neutral plasma. Phys. Plasmas 10, 2217. Robicheaux, F., Hernández, J.V. (2005). Many-body wave function in a dipole blockade configuration. Phys. Rev. A 72, 063403. Robicheaux, F., Hernández, J.V., Topçu, T., Noordam, L.D. (2004). Simulation of coherent interactions between Rydberg atoms. Phys. Rev. A 70, 042703. Robinson, M.P., Tolra, B.L., Noel, M.W., Gallagher, T.F., Pillet, P. (2000). Spontaneous evolution of Rydberg atoms into an ultracold plasma. Phys. Rev. Lett. 85, 4466. Saffman, M., Walker, T.G. (2002). Creating single-atom and single-photon sources from entangled atomic ensembles. Phys. Rev. A 66, 065403. Saffman, M., Walker, T.G. (2005). Analysis of a quantum logic device based on dipole–dipole interactions of optically trapped Rydberg atoms. Phys. Rev. A 72, 022347. Schlosser, N., Reymond, G., Protsenko, I., Grangier, P. (2001). Sub-Poissonian loading of single atoms in a microscopic dipole trap. Nature 411, 1024. Sebby-Strabley, J., Newell, R., Day, J., Brekke, E., Walker, T. (2005). High-density mesoscopic atom clouds in a holographic atom trap. Phys. Rev. A 71, 21401. Shore, B.W., Bergmann, K., Kuhn, A., Schiemann, S., Oreg, J. (1992). Laser-induced population transfer in multistate systems: A comparative study. Phys. Rev. A 45, 5297. Simien, C.E., Chen, Y.C., Gupta, P., Laha, S., Mickelson, Y.N., Mickelson, P.G., Nagel, S.B., Killian, T.C. (2004). Using absorption imaging to study ion dynamics in an ultracold neutral plasma. Phys. Rev. Lett. 92, 143001. Singer, K., Reetz-Lamour, M., Amthor, T., Marcasse, L.G., Weidemueller, M. (2004). Suppression of excitation and spectral broadening induced by interactions in a cold gas of Rydberg atoms. Phys. Rev. Lett. 93, 163001. Sørensen, A.S., van der Wal, C.H., Childress, L.I., Lukin, M.D. (2004). Capacitive coupling of atomic systems to mesoscopic conductors. Phys. Rev. Lett. 92, 063601. Stebbings, R.F., Dunning, F.B. (1983). “Rydberg States of Atoms and Molecules”. Cambridge University Press, New York. Swanson, T.B., Asgeirsson, D., Behr, J.A., Gorelov, A., Melconian, D. (1998). Efficient transfer in a double magneto-optical trap system. J. Opt. Soc. Amer. B 15, 2641. Teo, B.K., Feldbaum, D., Cubel, T., Guest, J.R., Berman, P.R., Raithel, G. (2003). Autler–Townes spectroscopy of the 5s1/2 –5p3/2 –44d cascade of cold 85 Rb atoms. Phys. Rev. A 68, 053407. Tong, D., Farooqi, S.M., Stanojevic, J., Krishnan, S., Zhang, Y.P., Côte, R., Eyler, E.E., Gould, P.L. (2004). Local blockade of Rydberg excitation in an ultracold gas. Phys. Rev. Lett. 93, 063001. Vanhaecke, N., Comparat, D., Tate, D.A., Pillet, P. (2005). Ionization of Rydberg atoms embedded in an ultracold plasma. Phys. Rev. A 71, 013416. Vitrant, G., Raimond, J.M., Gross, M., Haroche, S. (1982). Rydberg to plasma evolution in a dense gas of very excited atoms. J. Phys. B 15, L49. Vrinceanu, D., Granger, B., Parrott, R., Sadeghpour, H., Cederbaum, L., Mody, A., Tan, J., Gabrielse, G. (2004). Strongly magnetized antihydrogen and its field ionization. Phys. Rev. Lett. 92, 133402. Walz-Flannigan, A., Feldbaum, D., Dutta, S.K., Guest, J.R., Raithel, G. (2002). l-changing collisions in cold Rydberg gases. In: Burgdoerfer, J., Cohen, J.S., Datz, S., Vane, C.R. (Eds.), “Photonic, Electronic and Atomic Collisions (XXII ICPEAC Proceedings)”, Rinton Press, Princeton. Walz-Flannigan, A.I. (2004). Cold Rydberg gas dynamics. Ph.D. in Applied Physics, University of Michigan, Ann Arbor. Walz-Flannigan, A.I., Guest, J.R., Choi, J.-H., Raithel, G. (2004). Cold-Rydberg-gas dynamics. Phys. Rev. A 69, 063405. Wang, Q., Greene, C.H. (1989). Quantal description of diamagnetic quasi-Landau-resonances. Phys. Rev. A 40, 742.
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J.-H. Choi et al.
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Williamson, R.S.I., Voytas, P.A., Newell, R.T., Walker, T. (1998). A magneto-optical trap loaded from a pyramidal funnel. Optics Express 3, 111. Wiza, J.L. (1979). Microchannel plate detectors. Nuclear Instr. Methods, 162. Zheng, S.B., Guo, G.C. (2000). Efficient scheme for two-atom entanglement and quantum information processing in cavity QED. Phys. Rev. Lett. 85, 2392. Zimmerman, M.L., Littman, M.G., Kash, M.M., Kleppner, D. (1979). Stark structure of the Rydberg states of alkali-metal atoms. Phys. Rev. A 20, 2251.
ADVANCES IN ATOMIC, MOLECULAR AND OPTICAL PHYSICS, VOL. 54
NON-PERTURBATIVE QUANTAL METHODS FOR ELECTRON–ATOM SCATTERING PROCESSES D.C. GRIFFIN1 and M.S. PINDZOLA2 1 Department of Physics, Rollins College, Winter Park, FL 32789 2 Department of Physics, Auburn University, Auburn, AL 36849
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. The Configuration-Average Distorted-Wave Method . . . . . . 3. The R-Matrix with Pseudo-States Method . . . . . . . . . . . . 3.1. The R-Matrix Method . . . . . . . . . . . . . . . . . . . . 3.2. The Addition of Pseudo States . . . . . . . . . . . . . . . . 4. The Time-Dependent Close-Coupling Method . . . . . . . . . . 4.1. Exact Solutions to One-Electron Atomic Systems . . . . . 4.2. Approximate Solutions to Multi-Electron Atomic Systems 4.3. Exact Solutions to Two-Electron Atomic Systems . . . . . 5. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Excitation and Ionization of All Ionization Stages of Be . 5.2. Electron-Impact Excitation and Ionization of Ne . . . . . . 5.3. Ionization out of Excited States of H-Like Ions . . . . . . 5.4. Electron-Impact Single and Double Ionization of He . . . 6. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . 8. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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204 204 206 206 209 211 211 214 215 218 218 221 227 230 232 233 234
Abstract In just over a decade, a number of non-perturbative close-coupling methods have been developed that are capable of very accurate calculations of electron–atom collisions. For electron collisions with hydrogen-like targets, one of the most important three-body Coulomb problems, they are limited only by the accuracy of the two-electron continuum and the completeness of the partial-wave expansion; for multielectron systems, the accuracy of the target description is normally the limiting factor. Here we consider two non-perturbative methods: the R-matrix with pseudo states (RMPS) method and the time-dependent close-coupling (TDCC) method. We present descriptions of both of these methods and consider their application to electron-impact excitation and ionization. 203
© 2007 Elsevier Inc. All rights reserved ISSN 1049-250X DOI: 10.1016/S1049-250X(06)54004-2
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1. Introduction In the last twelve years, enormous progress has been made on the development of non-perturbative close-coupling methods for treating the collision of electrons with atoms and ions. These include the intermediate-energy R-matrix [1], the converged close-coupling (CCC) [2], the hyperspherical close-coupling [3], the exterior complex scaling (ECS) [4–6], the R-matrix with pseudo states (RMPS) [7,8], and the time-dependent close-coupling (TDCC) [9,10] methods. All of these have been used to determine accurate total cross sections for ionization from the ground state of hydrogen; furthermore, the ECS [6], the CCC [11], and the TDCC [12] methods have produced triply differential cross sections for the ionization of ground-state hydrogen, and the ECS and TDCC differential cross sections [12] have been shown to be in excellent agreement with each other. The TDCC method is the only non-perturbative method that has been applied successfully to the electron-impact double ionization of He [13]. In addition, only the CCC, TDCC, and RMPS methods have been employed to determine ionization from excited states of H-like species, as well as ionization from the ground and excited states of more complex targets. Finally, the CCC and RMPS methods have been used extensively to calculate excitation cross sections between a large number of levels in many light non-hydrogenic species, and TDCC calculations of excitation for selected transitions have been made to provide an independent check of the RMPS excitation cross sections. Here we will focus on the TDCC method and our version of the RMPS method and their applications to electron-impact ionization and excitation. The remainder of this chapter is organized as follows. In the next section, we describe the configuration-average distorted-wave method that we employ to provide high partial-wave contributions for TDCC calculations and to compare with TDCC and RMPS ionization cross sections. In Sections 3 and 4, we describe the RMPS and TDCC methods, respectively. In Section 5, we present results for electron-impact ionization and excitation using these two advanced closecoupling methods. Finally, in Section 6, we provide a brief summary and discuss future applications of these methods.
2. The Configuration-Average Distorted-Wave Method Our emphasis in this chapter is on non-perturbative advanced close-coupling methods. However, for electron-impact ionization, we often compare ionization cross sections obtained from these methods with results from configurationaverage distorted-wave (CADW) theory [14]. In addition, we employ the CADW method to provide high partial-wave contributions to TDCC calculations of electron-impact ionization. Therefore, we give a brief description of the CADW
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approximation in which one averages over the states of the initial target configuration and sums over the states of the final configuration of the ion. For ionization of an electron from the (nt lt ) subshell with occupation number wt e− + (nt lt )wt → (nt lt )wt −1 + e− + e− ,
(1)
the configuration-average cross section is given by σ =
32wt ki3 ×
E/2
k2
d( 2e ) ke kf
0
(2li + 1)(2le + 1)(2lf + 1)P(li , le , lf , ki , ke , kf ),
(2)
li ,le ,lf
where the linear momenta (ki , ke , kf ) and the angular momentum quantum numbers (li , le , lf ) correspond to the incoming, ejected, and outgoing electrons, rek2
k2
k2
spectively. The total energy E = 2i − I = 2e + 2f , where I is the subshell ionization energy. The first-order scattering probability is given by 2 P(li , le , lf , ki , ke , kf ) = Aλli le lf R λ (ke le , kf lf , nt lt , ki li ) +
λ
+
λ
Blλi le lf
λ
λ
λ 2 R (kf lf , ke le , nt lt , ki li )
Clλλ R λ (ke le , kf lf , nt lt , ki li )R λ (kf lf , ke le , nt lt , ki li ), (3) i le lf
where the angular coefficients A, B, and C may be expressed in terms of 3j and 6j symbols as: 1 le λ lt 2 lf λ li 2 Aλli le lf = (4) , 0 0 0 2λ + 1 0 0 0 1 lf λ lt 2 le λ li 2 λ Bli le lf = (5) , 0 0 0 2λ + 1 0 0 0 and 1/2 Clλλ = (2λ + 1)(2λ + 1)Aλli le lf Blλi le lf i le lf
le lf
lt li
λ λ
.
The R λ are radial Slater integrals ∞ ∞ R λ (ke le , kf lf , nt lt , ki li ) = 0
0
(r, r )λ< (r, r )λ+1 >
Pke le (r)Pkf lf (r )
(6)
206
D.C. Griffin and M.S. Pindzola × Pnt lt (r)Pki li (r ) dr dr ,
[3 (7)
where Pnt lt (r) is the bound radial orbital of the active electron and the Pkl (r) are the distorted waves. They are solutions to a radial Schrödinger equation given by: 1 d2 l(l + 1) Z k2 − (8) + − + VD (r) + VX (r) − Pkl (r) = 0, 2 dr 2 r 2 2r 2 where Z is the atomic number. The direct VD potential is given by: VD (r) =
occ u
∞ wu
Pn2u lu (r ) (r , r)>
dr ,
(9)
0
where the sum is over the occupied bound orbitals Pnu lu (r). The exchange potential VX is calculated in a local density approximation [15]. We often employ two different forms for the scattering potentials. In the prior form, the incident and scattered electron continuum orbitals are evaluated in a V N potential, while the ejected continuum orbital is calculated in a V N−1 potential, where N = u wu is the total number of target electrons [16]. The prior form has proven especially accurate for high angular momentum scattering. In a second form that we refer to as the post form, the incident, scattered, and ejected electrons are all calculated in a V N−1 potential [17]. The continuum normalization for all distorted-waves is one times a sine function.
3. The R-Matrix with Pseudo-States Method The R-matrix with pseudo-states (RMPS) method is an extension of the standard R-matrix close-coupling method in which the high Rydberg states and the target continuum are represented with an L2 pseudo-state basis. Thus, we begin with a brief introduction to the R-matrix method.
3.1. T HE R-M ATRIX M ETHOD The R-matrix method was first developed for applications to nuclear physics by Wigner [18] and Wigner and Eisenbud [19], and a review of R-matrix theory for nuclear reactions is given by Lane and Thomas [20]. It was first applied to atomic scattering by Burke and colleagues [21,22], and since that time, has been used widely to study electron–atom, as well as photon–atom collisions. The central idea of this method is that the scattering problem can be divided into two regions. In the inner region, the many-body interactions are strong and must be treated in great detail. In the asymptotic region, the many-body interactions are much weaker and one can solve close-coupling equations which include
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only the electron–nuclear interaction plus the direct part of the electron–electron interactions. In the inner region, one expands the wave-function ΨE for the (N + 1)-electron atom at energy E in terms of R-matrix basis functions Ψk ΨE = AEk Ψk . (10) k
The basis functions are given by aij k Φ¯ i uj + bj k φj , Ψk = A ij
(11)
j
where uj are a finite set of orbitals used to represent the continuum wave-function for a given value of the angular momentum l, Φ¯ i are a finite set of wave-functions for the N-electron target combined with the angular and spin parts of the continuum orbital, the operator A antisymmetrizes the scattered electron coordinate with the N atomic electron coordinates, and the φj are (N + 1)-electron wavefunctions formed from the target orbitals, which must be included to ensure that the total wave-function is complete. The coefficients aij k and bj k are determined by diagonalizing the (N + 1)-electron Hamiltonian H N+1 in the inner region Ψk |H N+1 |Ψk = EkN+1 δkk . (12) The radius of the inner region a is chosen so that the bound orbitals Pnl (r) are all contained inside; i.e., Pnl (r) 0 for r > a. For each value of the angular momentum l, the continuum basis orbitals uj are determined from the solutions to the differential equation kj2 l(l + 1) 1 d2 + + V (r) − λj i Pni l (r), uj (r) = − − (13) 2 2 2 dr 2 2r i
subject to the R-matrix boundary conditions a duj uj (0) = 0, = b, uj (a) dr r=a
(14)
and the orthonormality condition a ui uj dr = δij .
(15)
0
V (r) is some suitable potential chosen to represent an approximate charge distribution of the target atom or ion and the λj i are Lagrange multipliers used to force orthogonality of the continuum basis with all bound orbitals Pni l (r) with the same value of the angular momentum l. As pointed out by Burke and Robb [22], the
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right-hand side of Eq. (13) also plays the role of an exchange potential. The value of the logarithmic derivative b is arbitrary as long as a sufficient number of terms is retained in the R-matrix basis, and it is normally set equal to zero. It is convenient to define aij k uj . wik = (16) j
Then the scattered electron radial wave-function in channel i at energy E is given by AEk wik . yi = (17) k
It can be shown [22] that the coefficients AEk are given by dyj 1 wj k (a) a . − byj AEk = dr 2a(EkN+1 − E) r=a
(18)
j
Thus the scattered radial wave-function at the R-matrix boundary is given by dyj Rij a , − byj yi (a) = (19) dr r=a j
where Rij is an element of the R-matrix defined by the equation 1 wik (a)wj k (a) Rij = . 2a EkN+1 − E k
(20)
The surface amplitudes wik (a) and poles EkN+1 of the R-matrix are then determined from the eigenvectors and eigenvalues of the (N + 1)-electron Hamiltonian (Eq. (12)). The largest source of error in this method arises from the finite number of terms in the R-matrix expansion in Eq. (10), and therefore, the truncation of the sum in the right-hand side of Eq. (20). The high-lying eigenvalues of the Hamiltonian matrix, not included in this expansion, contribute primarily to the diagonal elements of the R-matrix, where they add coherently. An approximate method for completing the sum was developed by Buttle [23] and is described by Burke et al. [21]. In order to make use of the R-matrix from the inner-region calculation to determine the scattering matrices, and thereby the collision strengths and cross sections, one must solve the close-coupling equations in the outer region (r > a) at each electron energy. However, since a is chosen so that Pnl (r) 0, electronexchange is negligible and the close-coupling equations simplify to λ n λ max cij l(l + 1) (Z − N ) ki2 1 d2 + − (r) = − yj (r), − y − i 2 dr 2 r 2 2r 2 r λ+1 λ=1 j =1
3]
NON-PERTURBATIVE QUANTAL METHODS i = 1, . . . , n,
209 (21)
where n is the total number of scattering channels, open and closed, and λmax is the maximum value in the multipole expansion as allowed by the triangular λ are given by relations that result from the angular integrals. The coefficients cij N λ λ ¯ rk Pλ (cos θk,N +1 )Φ¯ j , cij = Φi (22) k
where θk,N+1 is the angle between the vectors rˆ k and rˆ N+1 . In many early R-matrix calculations on ions, the multipole expansion of the direct electron– electron interaction potential was dropped and Coulomb functions were used to describe the continuum orbitals in the asymptotic region. However, more recently, it has been found that the retention of these terms is important for the determination of accurate cross sections. As r → ∞, the solution to these coupled equations becomes yij ∼ √1k (sin θi δij + cos θi Kij ) for ki2 > 0 i for ki2 < 0, yij ∼ O r −2
(23)
where i = 1, . . . , no for the open channels, i = no + 1, . . . , n for the closed channels, Kij is an element of the K-matrix, and 1 (Z − N ) (Z − N ) ln 2ki r + arg Γ li + 1 − i . (24) θi = ki r − li π + 2 ki ki The second index j is introduced to label the no linearly independent solutions. Thus, by solving the approximate close-coupling equations in the asymptotic region, it is possible to relate the K-matrix to the R-matrix and thereby determine the collision strengths and cross sections.
3.2. T HE A DDITION OF P SEUDO S TATES In the above description of the R-matrix method, we have assumed that the target orbitals are all spectroscopic. However, even in many of the earliest R-matrix calculations, non-spectroscopic pseudo orbitals were used in order to improve the target structure. However, in such cases, the pseudo states arising from these orbitals were included only in the configuration-interaction expansion of the target, and not the close-coupling expansion. The importance of coupling of the bound states to the target continuum was recognized quite early. For example, it was shown by Castillejo et al. [24] that inclusion of the target continuum in the close-coupling expansion contributed substantially to the polarizability of the ground state of hydrogen. In a series of papers, Burke and collaborators added a small set of pseudo states with appreciable
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continuum character to the close-coupling expansion for hydrogen to demonstrate the importance of coupling to the continuum on the elastic scattering phase shifts for hydrogen [25] and excitation to the 2s and 2p states of hydrogen [26,27]. Whenever pseudo orbitals, designated by the notation nl, ¯ are included in the target description, one must generate continuum basis orbitals that are orthogonal to not only the spectroscopic orbitals, but also the pseudo orbitals. However, it is important to avoid Lagrange orthogonalization of the continuum basis to the pseudo orbitals since the right-hand side of Eq. (13) mimics electron exchange and this would introduce unphysical character into the continuum basis. If only a small set of pseudo orbitals is added to the target description, the generation of a new continuum basis that is orthogonal to the pseudo orbitals can be accomplished using the standard Gram–Schmidt orthogonalization procedure [22]. However, when more than a few pseudo orbitals of a given angular momentum l are included, this method becomes unstable. In order to provide an accurate representation of the high Rydberg states and the target continuum, one must include a large number of pseudo states in the close-coupling expansion. However, this requires a much more stable method for generating new continuum basis orbitals that are orthogonal to the pseudo orbitals. This was accomplished by Bartschat et al. [7] in what they referred to as the R-matrix with pseudo-state (RMPS) method. Their pseudo-state expansion consists of a set of Sturmian orbitals, and they employ a recursive Gram–Schmidt orthogonalization procedure to generate a new continuum basis, which together with the spectroscopic and pseudo orbitals, forms an orthonormal set. In our implementation of the RMPS method, we employ a set of nonorthogonal Laguerre pseudo orbitals l+1 −λnl Pnl e ¯ zr/2 L2l+1 (λnl ¯ (r) = Nnl ¯ (λnl ¯ zr) ¯ zr). n+l ¯
(25)
Here z = Z − N + 1, where Z is the nuclear charge and N is the number of target electrons; L2l+1 is the associated Laguerre polynomial; and Nnl ¯ is a normalization n+l ¯ constant. We then orthogonalize these orbitals to each other and the spectroscopic orbitals. The scaling parameters λnl ¯ can be used to adjust the energy and the radial extent of the pseudo states. In general, we adjust the scaling parameters so that for each value of l there are one or two layers of pseudo states below the ionization limit and the rest above, and so that the layers immediately above and below the ionization limit have nearly equal energy spacing with respect to the limit. By spacing the energies of the pseudo states in this way, we are able to reduce the continuum character of the pseudo states below the ionization limit and the bound character of those pseudo states above the ionization limit [28]. We employ a method developed by Gorczyca and Badnell [29] to generate new continuum basis orbitals that are orthogonal to the pseudo orbitals. It requires a modification of the Buttle correction, since the standard Buttle is based on the use of the original continuum basis orbitals uj (r) determined from Eq. (13), rather
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than the new continuum basis. However, it has proven to be exceedingly stable, even when a large number of pseudo orbitals is included in the close-coupling expansion. In order to provide accurate cross sections for excitation or ionization of neutral or low-charge-state species using the RMPS method, a large number of target pseudo states must be included in the close-coupling expansion. Therefore, even for relatively simple targets, RMPS calculations can grow quite large. For that reason, we have been involved in a project to develop a suite of efficient parallel R-matrix programs that will run equally well on small computer clusters and massively parallel computers [30,31]. These parallel programs were developed from serial codes that are themselves significantly modified versions of the Belfast RMATRIX I programs [32]. The R-matrix codes consist of a set of stages that are run in series. PSTG1 generates the orbital basis for the (N + 1)-electron continuum and calculates all radial integrals needed for the generation of the (N + 1)-electron Hamiltonian. The calculations of the radial integrals are distributed over the processors in this parallel code. PSTG2 carries out the angular algebra calculations and generates the (N + 1)-electron matrix elements in LS coupling. These calculations are distributed over the processors by LSΠ partial wave. For calculations that are to be done in intermediate coupling using the Breit–Pauli approximation, the program PRECUPD calculates term-coupling coefficients for the N -electron target states and transforms the (N + 1)-electron Hamiltonian from LS to j K coupling; these calculations are distributed over the processors by J Π partial wave. PSTG3 reads the inner-region information from PSTG2 for an LS calculation, or PRECUPD for an intermediate-coupling calculation, and then forms the (N + 1)-electron Hamiltonian and diagonalizes it in parallel to determine the R-matrix poles and surface amplitudes. PSTGF solves the coupled equations in the asymptotic region and matches to the R-matrix on the inner-region boundary to determine the collision strengths; this portion of the calculation is distributed over the electron energy. These parallel programs have enabled us to perform RMPS calculations on a variety of neutral and near-neutral species that would have been completely impractical using the serial versions of these programs.
4. The Time-Dependent Close-Coupling Method 4.1. E XACT S OLUTIONS TO O NE -E LECTRON ATOMIC S YSTEMS The time-dependent close-coupling (TDCC) method was first applied to calculate the total cross section for electron-impact ionization of hydrogen [9,10]; i.e., the simplest quantal three-body Coulomb breakup problem. As pointed out by Bottcher [33], the time evolution of a wave-packet localized in space obviates the
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need for answers to questions about the asymptotic form of the wave-function in position space or its singularities in momentum space. The TDCC method is a wave-packet solution of the same set of close-coupled partial differential equations used in the time-independent electron–atom scattering method of Wang and Callaway [34,35]. For electron scattering from a one-electron H-like atom, the full electrostatic Hamiltonian is given by: 2 1 2 Z 1 , − ∇i − + H = (26) 2 ri |r1 − r2 | i=1
where r1 and r2 are the coordinates of the two electrons and Z is the nuclear charge. The total electronic wave-function is expanded in coupled spherical harmonics for each total angular momentum, L: Ψ L (r1 , r2 , t) =
PlLl (r1 , r2 , t) 1 2 l1 l2
×
r1 r2
l1 l2 L Cm Yl1 m1 (! r1 )Yl2 m2 (! r2 ), 1 m2 0
(27)
m1 m2 l l l
3 r ) is a spherical harwhere Cm11 m2 2 m 3 is a Clebsch–Gordan coefficient and Ylm (ˆ monic. From projection onto the time-dependent Schrödinger equation, we obtain the following set of time-dependent close-coupled partial differential equations for each L symmetry:
i
∂PlL1 l2 (r1 , r2 , t) ∂t
= Tl1 l2 (r1 , r2 )PlL1 l2 (r1 , r2 , t) + VlL1 l2 ,l l (r1 , r2 )PlL l (r1 , r2 , t), l1 l2
1 2
(28)
1 2
where Tl1 l2 (r1 , r2 ) =
2 li (li + 1) Z 1 ∂2 + − − , 2 ∂ri2 ri 2ri2 i
(29)
and the coupling operator is given in terms of 3j and 6j symbols by VlL1 l2 ,l l (r1 , r2 )
= (−1)l1 +l1 +L (2l1 + 1)(2l1 + 1)(2l2 + 1)(2l2 + 1) (r1 , r2 )λ l1 λ l l2 λ l l1 l2 < 1 2 × l2 l1 0 0 0 0 0 0 (r1 , r2 )λ+1 > 1 2
λ
L . (30) λ
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We solve the time-dependent close-coupled equations using a discrete representation of the radial wave-functions and all operators on a two-dimensional lattice. Our specific implementation on massively parallel computers is to partition both the r1 and r2 coordinates over the many processors, a so-called domain decomposition. The initial condition for the solution of the time-dependent close-coupling equations is given by PlL1 l2 (r1 , r2 , t = 0) = P1s (r1 )Gk0 L (r2 )δl1 ,0 δl2 ,L ,
(31)
where P1s (r) is the ground state radial wave-function for the H-like atom and k2
the Gaussian wave-packet, Gk0 L (r), has a propagation energy of 20 . Both explicit and implicit methods have been employed to time propagate the close-coupled partial differential equations. Probabilities for all the many collision processes possible are obtained by t → ∞ projection onto fully antisymmetric spatial and spin wave-functions. As an example, for electron ionization of the ground state of the hydrogen atom, the partial collision probability is given by: R(12, t) + (−1)S R(21, t)2 , Pl1 l2 L,s1 s2 S (t) = (32) k1
k2
where ∞ R(ij, t) =
∞ dr2 Pk1 l1 (ri )Pk2 l2 (rj )PlL1 l2 (r1 , r2 , t).
dr1 0
(33)
0
The Pkl (r) are continuum radial wave-functions for the hydrogen atom, s1 = s2 = 12 , and S = 0 or S = 1. Alternatively, the time-dependent wave-function, PlL1 l2 (r1 , r2 , t), may be radially symmetrized for S = 0 or radially antisymmetrized for S = 1 at t = 0 or as t → ∞, and then projected onto product wave-functions to obtain collision probabilities. Partial ionization probabilities may be obtained by subtracting the projection onto purely bound wave-functions from unity. Partial ionization probabilities may even be determined approximately by summing the squares of PlL1 l2 (r1 , r2 , t) as t → ∞ over a < r1 < ∞ and a < r2 < ∞, where a is the approximate radial extent of the ground and first few excited states. The total cross section for the electron ionization of the ground state of hydrogen is given by: π LS σion = 2 (34) (2L + 1)(2S + 1)Pion , 4k0 L,S LS is found by summing the partial collision probabilities over all diswhere Pion tinct determinantal states of a given LS symmetry. Differential cross sections in
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the emitted energies and angles of the two outgoing electrons following ionization of hydrogen have also been derived in terms of the asymptotic values of the time-dependent radial wave-functions [12].
4.2. A PPROXIMATE S OLUTIONS TO M ULTI -E LECTRON ATOMIC S YSTEMS The time-dependent close-coupling method has been applied to calculate the total cross section for electron-impact ionization of a target electron in the outer subshells of a multi-electron atom or ion [36,37]. For one active electron outside closed subshells, e.g., the 2s orbital in Li (1s 2 2s), the TDCC method yields cross sections from the initial doublet term for the various electron scattering processes. For other cases, such as the 2s or 2p orbital in Be (1s 2 2s2p), the TDCC method yields configuration-averaged cross sections for the various electron scattering processes. To handle multi-electron atomic systems, the single particle operator in the time-dependent close-coupling equations is now given by: Tl1 l2 (r1 , r2 ) =
2 1 ∂2 li + V (r ) − PP i , 2 ∂ri2
(35)
i
VPl P (r)
is an l-dependent core pseudo potential, and the subshell occupawhere tion number of the active electron now multiplies the expression for the scattering cross sections. Our method for determining the pseudo potentials, VPl P (r), is best illustrated through the example of Li (1s 2 2s). The 1s orbital is first obtained by solving the Hartree–Fock equations for Li+ (1s 2 ). The core orbital is then used to construct the radial Hamiltonian: h(r) = −
1 ∂2 + VHl X (r), 2 ∂r 2
(36)
where VHl X (r) =
l(l + 1) Z αl − + VH (r) − r 2 2r 2
24ρ π
1/3 .
(37)
VH (r) is the direct Hartree potential and ρ is the probability density in the local exchange potential. The excited-state spectrum is then obtained for each l by diagonalizing h(r) on the lattice. The parameter αl is varied to obtain experimental energy splittings for the first few excited states. For l = 0, the inner node of the 2s orbital is smoothly removed and VP0 P (r) is obtained by inverting the radial Schrödinger equation with the newly constructed 2s pseudo orbital. For l > 0, VPl P (r) = VHl X (r). The introduction of pseudo potentials removes the problem of unphysical de-excitation from the active orbital to closed subshells with the
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same angular momentum, i.e. 2s → 1s for Li. The new radial Hamiltonian: 1 ∂2 (38) + VP0 P (r), 2 ∂r 2 is then diagonalized on the lattice to obtain an l = 0 excited pseudo-state spectrum. h(r) = −
4.3. E XACT S OLUTIONS TO T WO -E LECTRON ATOMIC S YSTEMS The time-dependent close-coupling method has also been applied to calculate the total cross section for electron-impact double ionization of helium [13]; i.e., the simplest quantal four-body Coulomb breakup problem. The extension of the three-body TDCC method to address the four-body problem is straightforward. The TDCC method currently stands alone as the only quantal non-perturbative method providing predictions for the total electron-impact double ionization of helium. For electron scattering from a two-electron helium-like atom, the full electrostatic Hamiltonian is given by: 3 1 Z 1 H = (39) . − ∇i2 − + 2 ri |ri − rj | i<j
i=1
The total electronic wave-function is expanded in coupled spherical harmonics for each total angular momentum, L, symmetry: Ψ L (r1 , r2 , r3 , t) = ×
PlLl
1 2 Ll3
r1 r2 r3
l1 ,l2 L,l3 L l3 L m3 0
CM
(r1 , r2 , r3 , t)
l1 l2 L Cm Yl1 m1 (! r1 )Yl2 m2 (! r2 )Yl3 m3 (! r3 ). 1 m2 M
(40)
m1 ,m2
M,m3
From projection onto the time-dependent Schrödinger equation, we obtain the following set of time-dependent close-coupled partial differential equations for each L symmetry: i
∂PlL (r1 , r2 , r3 , t) 1 l2 Ll3 +
∂t 3 l1 ,l2 ,L ,l3
(r , r2 , r3 , t) = Tl1 l2 l3 (r1 , r2 , r3 )PlL 1 l2 Ll3 1
L VlL (ri , rj )Pl l L l (r1 , r2 , r3 , t), 1 l2 Ll3 ,l l L l
i<j
where Tl1 l2 l3 (r1 , r2 , r3 ) =
1 2
3
1 2
(41)
3
3 li (li + 1) Z 1 ∂2 + − − , 2 ∂ri2 ri 2ri2 i
(42)
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and the coupling operators are given in terms of 3j and 6j symbols by:
l1 +l1 +L VlL δl3 ,l3 δL,L (r1 , r2 ) = (−1) 1 l2 Ll3 ,l1 l2 L l3 × (2l1 + 1)(2l1 + 1)(2l2 + 1)(2l2 + 1) (r1 , r2 )λ l1 λ l l2 λ l < 1 2 × λ+1 0 0 0 0 0 0 (r , r ) 1 2 > λ l1 l2 L , × (43) l2 l1 λ l2 +L VlL δl2 ,l2 (r1 , r3 ) = (−1) 1 l2 Ll3 ,l1 l2 L l3 × (2l1 + 1)(2l1 + 1)(2l3 + 1)(2l3 + 1)(2L + 1)(2L + 1) (r1 , r3 )λ< l3 λ l3 l1 λ l1 × (−1)λ 0 0 0 0 0 0 (r1 , r3 )λ+1 > λ L l3 L l1 l2 L × , l3 L λ L λ l1
(44)
and
l +l2 +l2 +L+L +L VlL δl1 ,l1 (r2 , r3 ) = (−1) 1 1 l2 Ll3 ,l1 l2 L l3 × (2l2 + 1)(2l2 + 1)(2l3 + 1)(2l3 + 1)(2L + 1)(2L + 1) (r2 , r3 )λ< l3 λ l3 l2 λ l2 × (−1)λ 0 0 0 0 0 0 (r2 , r3 )λ+1 > λ l1 l2 L L l3 L . × l3 L λ λ L l2
(45)
We solve the time-dependent close-coupled equations using a discrete representation of the radial wave-functions and all operators on a three-dimensional lattice; partitioning the r1 , r2 , and r3 coordinates over the many processors on a massively parallel computer. The initial condition for the solution of the timedependent close-coupling equations is given by L (r , r2 , r3 , t = 0) = P¯l l0 (r1 , r2 )Gk0 l3 (r3 )δl1 ,l1 δl2 ,l2 δl3 ,l3 δL,L0 , PlL 1 l2 Ll3 1 l1 ,l2
1 2
(46)
where P¯l l0 (r1 , r2 ) with L0 = 0 and l1 = l2 = l are the ground-state radial wave1 2 functions for the helium atom, obtained by relaxation of the two-electron TDCC equations in imaginary time. Probabilities for all the many collision processes are L
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obtained by t → ∞ projection onto fully antisymmetric spatial and spin wavefunctions. As an example, for electron double ionization of the ground state of the helium atom, the partial collision probability is given by: Pl1 l2 Ll3 L,s1 s2 Ss3 S (t) = δL,L Qa R(123, t) k1
k2
k3
L
l2 l2 +l3 +L+L − (−1) (2L + 1)(2L + 1) l3 L (−1)l1 +l2 −L δL,L Qc R(213, t) −
l1 L
L Qb R(132, t) L
L
l2 l1 L (−1)l1 +l2 +L (2L + 1)(2L + 1) + Qc R(312, t) l 3 L L L l1 l2 L l2 +l3 +L + (−1) (2L + 1)(2L + 1) Qb R(231, t) l 3 L L L 2 l1 l2 L , (2L + 1)(2L + 1) R(321, t) − (47) Q a l3 L L L
where ∞ R(ij k, t) =
∞ dr1
∞ dr2
dr3 Pk1 l1 (ri )Pk2 l2 (rj )Pk3 l3 (rk )
0
0 0 L × Pl1 l2 L l3 (r1 , r2 , r3 , t).
(48)
+ atomic ion, s = radial The Pkl (r) are continuum wave-functions for the He 1 1 1 1 2 1 s2 = s3 = 2 , Qa = 2 δS,0 − 6 δS,1 , Qb = 3 δS,1 , Qc = − 2 δS,0 − 16 δS,1 ,
and S = 12 . To guard against the unwanted contribution to the partial collision probability coming from the continuum correlation part of the two-electron bound wavefunctions, one may project out the two-electron bound states from the threeelectron time-propagated radial wave-function and then project onto all electron momenta ki . Alternatively, we found that a simple restriction of the sums over the electron momenta ki , so that the energy, k02 k2 k2 k2 (49) = 1 + 2 + 3, 2 2 2 2 was approximately conserved, greatly reduced contamination from the continuum piece of the two-electron bound wave-functions. In addition, this method Eatom +
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of restricted momenta sums should become more accurate as the lattice size increases. We note that the collision probability for electron single ionization of the 1 S ground state of helium leaving the He+ ion in an nl bound state is almost identical to Eq. (47). Simply eliminate one of the sums over electron momenta, change one of the Pkl (r) radial wave-functions to Pnl (r), calculate the remaining two continuum radial wave-functions in a V (r) potential that screens the Coulomb field, and apply the relevant equation for the conservation of energy. Finally, the electron-impact single or double ionization cross section is given by: π (2L + 1)(2S + 1)P LS , σion = 2 (50) 2k0 L,S where P LS is found by summing the partial collision probabilities over all distinct determinantal states of a given LS symmetry. Work is in progress to calculate differential cross sections in the emitted energies and angles of the three outgoing electrons following double ionization of helium.
5. Results 5.1. E XCITATION AND I ONIZATION OF A LL I ONIZATION S TAGES OF B E Accurate data for excitation and ionization of beryllium and its ions are very important to controlled fusion research. Beryllium has been used as a surface material at the Joint European Torus and is being proposed for the plasma facing walls for ITER. For this reason, we have carried out advanced close-coupling calculations of ionization [38] and excitation [39] for beryllium and all its ions. We completed RMPS calculations of ionization for all charge states of Be. For Be+ , Be2+ , and Be3+ , we employed spectroscopic orbitals for the subshells with n = 1 and n = 2 and pseudo orbitals for n = 3 up to n = 14 and orbital angular momentum from l = 0 up to l = 4. However, for neutral beryllium, we developed a target basis that could be employed for both excitation and ionization. Therefore, we used spectroscopic orbitals up to n = 4 and pseudo orbitals for n = 5 to 11 for the 2snl configurations and n = 5 to 10 for the 2pnl configurations, with the orbital angular momentum ranging from l = 0 to l = 4. TDCC calculations of ionization for Be, Be+ , and Be2+ were performed, in which we used the pseudo-potential approach discussed in Section 4.2. For all calculations, we carried out TDCC calculations for the lower partial waves and used the prior form of the CADW method to provide the contributions from the higher partial waves, beginning at that partial wave where the TDCC and CADW results were in good agreement. A Fourier-transform method [40] was used to
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extract the ionization cross section for many incident energies from only one time propagation of the close-coupling equations. For all the beryllium targets, we determined the ionization cross sections from both the ground configuration and the first excited configuration. There was reasonably good agreement between the RMPS and TDCC cross sections for Be, Be+ , and Be2+ and with earlier CCC results for the ground-state ionization of Be [41] and Be+ [42]. However, CADW calculations yielded cross sections that were significantly higher than the non-perturbative results, although this difference decreased with ionization stage. As an example, we show the cross sections for ionization from the 2s 2 and 2s2p configurations of neutral Be in Fig. 1. As can be seen, the RMPS cross sections are restricted to relatively low electron energies. Because there were 280 terms in the RMPS close-coupling expansion for neutral Be, we had to restrict the size of the basis set for the (N + 1)-electron continuum to 45; to move to higher energies would have required a larger (N +1)-electron continuum basis set, and most likely, a larger pseudo-state expansion of the N -electron continuum. Near the peak in the ground-state cross section, the TDCC results are about 9% above the RMPS cross section. This is at least partially due to the fact that the two-electron correlation (2s 2 + 2p 2 + · · ·) that is included in the RMPS calculation, could not be included in the TDCC calculation, because of the use of a 2s pseudo orbital [38]. However, this does not explain the smaller difference between the RMPS and CCC results. The relatively good agreement between the 2s2p configuration-average TDCC results and the 2s2p 3 P RMPS results indicate that 2s and 2p orbitals used in the TDCC calculation are very similar to the 2s and 2p orbitals that make up the 3 P term. We completed RMPS calculations of electron-impact excitation through the n = 4 terms in Be, Be+ , Be2+ , and Be3+ . For these calculations, we employed spectroscopic orbitals through n = 4 and pseudo states with n 5 to represent the high-Rydberg states and the continuum. We also performed standard R-matrix calculations without pseudo states for these targets. In the non pseudo-state calculations, we employed the same configuration-interaction expansion of the target as used in the RMPS calculations; however the close-coupling expansions included only the spectroscopic states plus the n = 5 pseudo states. In order to remove the pseudo resonances attached to the pseudo states included in the configurationinteraction expansions, but not the close-coupling expansions, we employed the pseudo-resonance removal method described in Gorczycal et al. [43]. By comparing the standard R-matrix cross sections with those from the RMPS calculation, we determined the effects of coupling to the high-Rydberg states and the target continuum on electron-impact excitation. In order to provide an independent test on the accuracy of the RMPS calculations, we also performed TDCC calculations on electron-impact excitation from the ground term of Be+ and the metastable term of Be3+ . We employed a pseudo
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F IG . 1. Electron-impact ionization of neutral Be from the ground and first excited configurations. The dashed lines are from the TDCC calculations; the solid lines are from the RMPS calculations; the dot-dashed lines are from a CCC calculation [41]; the long dashed lines with crosses are CADW prior-form results; and the dotted lines with diamonds are CADW post-form results. In (b), the calculations include ionization of both the 2s and 2p subshells, and the RMPS results are for the 2s2p 3 P term only, while the TDCC are configuration-average results (1.0 Mb = 10−18 cm2 ).
potential to generate the 2s orbital for Be+ and again employed distorted-wave calculations to generate the cross sections for the high partial waves. An example from this work that clearly demonstrates the importance of coupling to the high-Rydberg states and the target continuum is shown in Fig. 2 for excitation to the ns and nd terms from the 2s ground term of Be+ . These coupling effects are included in all the advanced close-coupling calculations, but not in the standard R-matrix calculation. The agreement between the results from the RMPS, the CCC, and the TDCC calculations is in general quite good. However, the differences between the standard R-matrix cross sections and advanced close-
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F IG . 2. Electron-impact excitation of Be+ from the ground term to the ns and nd excited terms. The short dashed lines are from the standard R-matrix calculation; the solid lines are from the RMPS calculation; the dashed-dot lines are from a CCC calculation [44]; the filled squares are from the TDCC calculations (1.0 Mb = 10−18 cm2 ).
coupling cross sections are large and increase with principal quantum number; for example, near the peak of the cross sections, the standard R-matrix results are a factor of 1.8 above the RMPS cross section for excitation to the 4s term and 2.3 above the RMPS cross section for excitation to the 4d term. These coupling effects decrease slowly with ionization stage, but are still significant in Be3+ .
5.2. E LECTRON -I MPACT E XCITATION AND I ONIZATION OF N E Electron collisions with neutral neon are of both fundamental and applied interest. There have been a number of experimental measurements and theoretical calculations of total and differential excitation cross sections [45–49] that have
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been used to test the accuracy of various theoretical approaches. In addition, there have been measurements of ionization from ground [50,51] and metastable [52] neon. Electron collisions of electrons with neon are also important to controlled fusion research. For example, neon gas-puff experiments have been conducted in Tokamaks in the U.S. and the U.K. to study disruption mitigation and particle confinement. Although there have been a large number of R-matrix calculations for the excitation of neon [45,48,49], none of these included the effects of coupling to the target continuum. Because of the importance of these effects in other neutral and low-charge state species, we applied the RMPS method to the electron-impact excitation of Ne [31]. However, this presented a significant computational challenge. The large spin–orbit interaction within the 2p subshell causes a breakdown in LS coupling for the majority of excited states of Ne. Thus, excitation calculations for this atom must be performed in intermediate coupling (IC); this adds significantly to the number of scattering channels, making the RMPS calculation particularly large. However, we were able to complete such an calculation that employed spectroscopic orbitals for the 2p 6 ground configuration and all excited configurations from 2p 5 3s to 2p 5 4f as well as 2p 5 5s and 2p 5 5p. We then used Laguerre pseudo orbitals for the configurations 2p 5 5d; 2p 5 5f ; 2p 5 nl, with n = 6 through n = 8 and l = 0 to l = 3; 2p 5 9s; 2p 5 9p; and 2p 5 10s. This led to a total of 235 levels, 79 of which were spectroscopic. In order to determine the effects of coupling to the target continuum, we compared the results of this RMPS calculation with a standard R-matrix calculation that included the 2p 6 ground configuration and all excited configurations from 2p 5 3s to 2p 5 5g, for a total of 115 levels. Such a comparison for excitation from the ground state to two 2p 5 3s and two 2p 5 3d levels is shown in Fig. 3, where the excited states are designated in j K coupling. The first things we notice in this figure are the oscillations in the RMPS cross sections, which are especially pronounced for the 2p 6 1 S0 → 2p 5 3d 3/2[3/2]2 transition. Clearly our pseudo-state expansion was not nearly large enough to provide reliable excitation cross sections. Nevertheless, it is clear by comparing the RMPS cross sections with those from the standard R-matrix calculation that the effects of coupling to the target continuum are quite large, especially for the strong dipole 2p 6 1 S0 → 2p 5 3d 3/2[3/2]1 transition. Even with the use of available massively parallel computers, a much larger intermediate-coupling RMPS calculation was not feasible. Instead, we performed a series of model LS-coupling RMPS calculations on Ne in order to complete a pseudo-state convergence study. In each successive calculation, we added additional Laguerre pseudo states until convergence for excitation to the terms of the 2p 5 3l configurations was achieved. We found that this required a 243-term RMPS calculation, which included all the configurations that comprise the 125 terms and 235 levels from our IC RMPS calculation plus the configurations 2p 5 5g, 2p 5 6g,
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F IG . 3. Intermediate-coupling cross sections for electron-impact excitation of Ne from the 2p6 1 S0 ground level to (a) the 2p5 3s 3/2[3/2]2 level; (b) the 2p5 3s 1/2[1/2]1 level (c) the 2p 5 3d 3/2[3/2]2 level; and (d) the 2p5 3d 3/2[3/2]1 level, where all excited levels are given in j K coupling notation. The dashed curves are from a 115-level R-matrix calculation and the solid lines are from a 235-level RMPS calculation. The solid circles are the experimental results of Khakoo et al. [48] (1.0 Mb = 10−18 cm2 ).
2p 5 7g, 2p 5 8g, 2p 5 9d, 2p 5 9f , 2p 5 9g, and 2p 5 nl, with n = 10 through n = 12 and l = 0 through l = 4. The results from this 243-term RMPS calculation are compared to those from a 61-term standard R-matrix calculation for excitation from the ground term to two 2p 5 3s terms and two 2p 5 3d terms in Fig. 4. As can be seen, the effects of continuum coupling are indeed very large; in fact, the difference between the RMPS and standard R-matrix cross section for the 2p 6 1 S → 2p5 3d 1 P excitation is the largest we have yet seen in any target species. Although we were able to achieve
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F IG . 4. LS-coupling cross sections for electron-impact excitation of Ne from the 2p6 1 S ground term to (a) the 2p5 3s 3 P term; (b) the 2p5 3s 1 P term (c) the 2p5 3d 3 P term; and (d) the 2p5 3d 1 P term. The dashed curves are from a 61-term R-matrix calculation and the solid lines are from a 243-term RMPS calculation (1.0 Mb = 10−18 cm2 ).
convergence for excitation to the 2p 5 3l terms using an LS-coupling RMPS approach, such a calculation would still be impractical in intermediate coupling as it would require 465-levels in the close-coupling expansion. Furthermore, even more pseudo states might be required to achieve convergence for excitation to the n = 4 levels. Although intermediate coupling is needed for the majority of excited states in Ne, this is not true for the ground and 2p 5 3s excited states. Thus a full IC approach is not required to study ionization from the ground and metastable states of Ne. A number of years ago, we applied the configuration-average TDCC method to ionization from the ground-state of Ne [37]. The TDCC ionization
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F IG . 5. Electron-impact ionization of ground-state Ne. Solid squares from a configuration-average TDCC calculation for the 2s 2 2p 6 → 2s 2 2p 5 ionization transition combined with a CADW calculation for the 2s 2 2p 6 → 2s2p 6 ionization transition. Solid line from CADW calculations for both the 2s 2 2p 6 → 2s 2 2p 5 and 2s 2 2p 6 → 2s2p6 ionization transitions. Short dashed line from a LS term-dependent distorted-wave calculation for 2s 2 2p 6 → 2s 2 2p 5 ionization transition and a CADW calculation for 2s 2 2p 6 → 2s2p6 ionization transition. Filled circles are experimental points from Krishnakumar and Srivastava [51] (1.0 Mb = 10−18 cm2 ).
cross section at three energies (solid squares) is compared to the CADW cross section (solid line) in Fig. 5. It can be seen, from the close-agreement between the TDCC and the CADW results that interchannel coupling for ground-state ionization is quite small. However, both configuration-average cross sections are well above the experimental measurements of Krishnakumar and Srivastava [51]. Previous DW calculations of ionization from ground-state argon [53,54] have shown that certain ejected electron continua are not properly represented by a configuration-average potential. In particular, these studies indicated that the 2p 6 1 S → 2p 5 kd 1 P continuum excitation contribution should be large and should show strong LS term dependence in the kd continuum wave. Thus we repeated the DW ionization calculation using the 2p5 kd 1 P term-dependent Hartree-Fock potential, while using the CA potential for all other continuum
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states. The result is shown by the dashed line in Fig. 5, which is in relatively good agreement with the experimental measurements. Thus, for ground-state neon, interchannel coupling is not large, but continuum term-dependence is. Both effects are included in an RMPS description of ionization, but our 243term RMPS LS calculation did not have a sufficiently large (N + 1)-electron continuum basis set to allow us to extend it to energies above about 40 eV, well below the peak in the ground-state cross section. However, this RMPS calculation is accurate to sufficiently high energies to easily determine ionization out of the 2p 5 3s 3 P metastable term in Ne. In addition to our RMPS ionization cross section, we present the results of configuration-average TDCC and CADW calculations for ionization from the 2p5 3s excited configuration [55]. The RMPS, TDCC, and CADW results are compared with the semiclassical cross section of Deutsch et al. [56] and the experimental measurements of Johnston et al. [52] in Fig. 6. The difference between the CADW cross section and the TDCC and
F IG . 6. Electron-impact ionization of metastable Ne. The dashed curve is from the CADW calculation, the dotted curve is from the semiclassical calculation of Deutsch et al. [56], and the filled squares are from the configuration-average TDCC calculation, all for ionization from the 2p5 3s configuration; the solid line is the RMPS cross section for ionization from the 2p5 3s 3 P term; and the filled circles are the experimental measurements of Johnston et al. [52], where the relatively large error bars represent the systematic error and the smaller ones the statistical error (1.0 Mb = 10−18 cm2 ).
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RMPS cross sections would indicate that interchannel coupling effects are large for ionization from 2p 5 3s. We also note the RMPS and TDCC cross sections are in good agreement with each other and both are in reasonable agreement with experiment; however, the error bars are quite large. Although the TDCC results are for ionization from the 2p 5 3s configuration, while the RMPS results are for the 2p 5 3s 3 P metastable term, we have also calculated a configuration average RMPS cross section by statistically averaging over the 3 P and 1 P cross sections and the result is only 3% above the 3 P cross section shown in Fig. 6.
5.3. I ONIZATION OUT OF E XCITED S TATES OF H-L IKE I ONS Ionization out of excited states is of importance from both a fundamental and applied perspective. Excited-state ionization is a more sensitive probe of collision dynamics because of the stronger coupling between such states and the target continuum. Furthermore, in a plasma environment, excited states are populated by charge exchange reactions, excitation from ground and metastable states, and for multi-electron species, by dielectronic recombination. Ionization cross sections grow rapidly with principal quantum number, and at sufficient electron densities, excited state ionization can dominate the total ionization rate. However, advanced close-coupling calculations of ionization out of excited states also presents a significant computational challenge. The excited state target orbitals extend out to large radii, which makes an accurate representation of the continuum orbitals more difficult. In addition, the small ionization energies of these excited states leads to strong coupling between a large number of scattering channels. Because of these difficulties, plasma modeling studies have used excited-state ionization rate coefficients determined from classical and perturbative quantal methods. Recently, we undertook a study of excited-state ionization in several hydrogenlike species in which we compared cross sections determined from RMPS and TDCC calculations with those calculated using both distorted-wave (DW) and classical trajectory Monte Carlo (CTMC) calculations [57]. In particular, we studied ionization out of excited states through n = 4 from H, Li2+ , and B4+ . Since both the TDCC and RMPS methods would be expected to give quite accurate cross sections, they were employed as “benchmarks” to test the accuracy of the DW and CTMC methods as a function of principal quantum number and ionization stage. The RMPS calculations were performed for the three hydrogenic targets using physical orbitals through n = 4 and Laguerre pseudo orbitals from 5s to 12h. The n = 5 and n = 6 states were bound and all others were in the continuum. TDCC calculations were performed for ionization of H and Li2+ out of selected ns subshells to check on the accuracy of the RMPS cross sections. Distortedwave calculations were performed for the three targets using both the prior and
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post approximations described in Section 2. Finally, we calculated CTMC cross sections as a function of n for all three hydrogenic species. With this method, one solves Hamilton’s equations to compute a classical cross section from the probability that an incoming electron with a given impact parameter ionizes the atom. The probability of ionization is determined from the fraction of trajectories for which both electrons have positive energy after the collision. The two-electron simulation begins with the incident electron at a large distance from the atom and ends when one of the electrons reaches a comparably large distance. Although it is possible to generate CTMC cross sections as a function of both n and l, normally these calculations include a statistical distribution over the angular momentum of the target electron so as to produce cross sections that depend only on n. We found excellent agreement between the RMPS and TDCC results for ionization from the 1s and 2s subshells of H and Li2+ as well as from the 3s subshell of H and the 4s subshell of Li2+ . There was also excellent agreement with the CCC cross sections of Bartschat and Bray [8] for ionization from the 1s and 2s subshells of H, the experimental cross sections of Shah et al. [58] for the ionization from the 1s subshell of H, and the measurements of DeFrance et al. [59] for ionization from the 2s subshell of H. These other advanced close-coupling calculations and measurements all support the general accuracy of the RMPS calculations for ionization of hydrogenic targets. For plasma-modeling studies of H-like systems, collisions with protons tend to distribute the nl states statistically. Therefore comparisons of n-bundled cross sections that are averaged over the angular momentum of the target are appropriate. In Fig. 7, we compare such bundled cross sections from our RMPS, DW, and CTMC calculations for ionization from the n = 1 to n = 4 shells of hydrogen. For ionization from the excited states, the CTMC cross sections are in much better agreement with the RMPS results than the prior form DW cross sections. This indicates that the inclusion of the full three-body interactions included in the CTMC calculation is more important for hydrogen than the quantal effects included in the DW approximation. In addition, we see from the comparisons of the DW and RMPS cross sections that the effects of interchannel coupling that are included in the RMPS cross sections, but not the DW results, increase with principal quantum number. We tested the accuracy of the CTMC and DW methods as a function of ionization stage by making similar comparisons of n-bundled cross sections for Li2+ and B4+ . The results for Li2+ are shown in Fig. 8. As one would expect, the effects of interchannel coupling decrease with ionization stage and the differences between the DW and RMPS results are smaller for Li2+ than they are for H; however, these differences still increase with principal quantum number and are far from insignificant for n = 3 and n = 4. However, the most striking feature of this figure is the large difference between the CTMC and RMPS cross sections. Clearly the accuracy of the CTMC method decreases with ionization stage and
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F IG . 7. Electron-impact ionization of H as a function of n. Dashed lines are DW (prior form); dot-dashed lines are CTMC; and solid lines are RMPS (1.0 Mb = 10−18 cm2 ).
does not improve with n. The fact that the DW cross sections are in better agreement with the RMPS results than the CTMC cross sections seems to indicate that quantal effects are becoming more important than the three-body interactions in ions. For B4+ the effects of interchannel coupling decrease further and the differences between the DW and RMPS cross sections are somewhat smaller than they are for Li2+ . However, the CTMC results are about as far from the RMPS cross sections for B4+ as they are for Li2+ . Further work is now in progress to carry out RMPS and TDCC calculations to higher values of the principal quantum number. Although this is a significant computational challenge, it is needed to determine at what value of n, the CTMC ionization cross sections will converge to the advanced close-coupling results for neutral hydrogen and whether the DW method will continue to grow worse with principal quantum numbers for ions.
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F IG . 8. Electron-impact ionization of Li2+ as a function of n. Dashed lines are DW (prior form); dot-dashed lines are CTMC; and solid lines are RMPS (1.0 Mb = 10−18 cm2 ).
5.4. E LECTRON -I MPACT S INGLE AND D OUBLE I ONIZATION OF H E Recently, we developed a non-perturbative theoretical method to treat three continuum electrons moving in the field of a charged core, that is, Coulomb four-body breakup. The formalism found in Section 4.3 was first applied to the electronimpact single and double ionization of helium [13]. For single ionization, leaving He+ in the 1s ground state, we can compare to previous non-perturbative methods, which freeze one of the K shell electrons [60–62], and to absolute experimental measurements [50,63,64]. For double ionization, we can compare to absolute experimental measurements [64]. The electron-impact single and double ionization cross sections for the 1 S ground state of helium were calculated at incident electron energies of 100 eV, 150 eV, and 200 eV, all above the double ionization threshold. We employed
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a three-dimensional lattice with 192 points in each radial direction from 0.0 to 38.4 a.u. and a uniform mesh with spacing r = 0.20 a.u. Initially the wavepacket of Eq. (46) was centered at r3 = 20.0 a.u. with a spread in coordinate space of 6.0 a.u. After relaxation to obtain a fully correlated ground state of helium on the lattice, the time-dependent close-coupled equations of Eq. (41) were propagated for up to 6200 time steps to obtain total cross sections from Eq. (50). Explicit time-dependent calculations were made for the partial waves L = 0 to L = 5, including up to 87 coupled channels for L = 5, with extrapolation to higher L guided by scaled first and second order distorted-wave calculations. Total single ionization cross sections, leaving He+ in the 1s ground state, are shown in Fig. 9. The 3D time-dependent calculations are represented as filled squares, frozen-core 2D time-dependent calculations [62] are shown as open squares, and absolute experimental measurements [64] are filled circles with error
F IG . 9. Single ionization of helium, leaving He+ in the 1s ground state. The filled squares are from the 3D time-dependent close-coupling calculations, the open squares are from the frozen-core 2D time-dependent close-coupling calculations [62], and the filled circles are the absolute experimental measurements [64] (1 Mb = 10−18 cm2 ).
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F IG . 10. Double ionization of helium. The filled squares are the 3D time-dependent close-coupling calculations and the filled circles are the absolute experimental measurements [64] (1 Kb = 10−21 cm2 ).
bars. The 3D results are slightly lower than the frozen-core 2D results; however, the 2D cross sections are just above the error bars for three energies and within the error bars for the highest energy, while the 3D results are within the error bars of the absolute experimental measurements for all three energies. Total double ionization cross sections are shown in Fig. 10. The time-dependent close-coupling results are shown as filled squares and absolute experimental measurements [64] are filled circles with error bars. The time-dependent close-coupling results are found to lie within the error bars of the absolute experimental measurements.
6. Summary We have described the R-matrix with pseudo states (RMPS) and the timedependent close-coupling (TDCC) methods and have presented examples of applications of these methods to calculations of electron-impact excitation and ion-
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ization. The results clearly indicate that these methods are capable of generating exceptionally accurate results for neutral and low-charge species, where coupling of the bound states to the target continuum is very important in excitation and interchannel coupling is critical to ionization. The RMPS method is especially effective for electron-impact excitation, where with a single calculation, one can generate accurate cross sections for a large number of transitions that include both the contributions of resonant excitation, which are especially important at low energies, and the effects of coupling to the target continuum, which are most important in the intermediate-energy region. It may also be employed to determine accurate ionization cross sections from both ground and excited states. The accuracy of this method is determined primarily by the quality of the bound target states and the size of the pseudo-state expansion. In addition, it is limited in energy by the size of the (N +1)-electron basis set, and this grows rapidly with the radius of the R-matrix “box” required to contain the target states. Although they have proven very effective for collisions of electrons with light atoms and ions, RMPS calculations can grow very large for more complex targets, especially when intermediate coupling must be employed, as in the case of neutral Ne. Additional work is needed to develop methods for more accurately representing the target continuum with smaller pseudo-state expansions. The TDCC method is especially effective for the determination of total and differential ionization cross sections of one-electron species, and more recently, it is the only method thus far that has been applied successfully to determine the total double ionization cross section of a two-electron target. It can also be applied to determine highly accurate excitation cross sections for selected transitions in one- and two-electron atoms and ions at high enough energies that the contributions from resonant excitation are not important. In addition, one can consider electron-impact ionization of more complex targets by including the exact interaction between the ejected and scattered electrons in an approximate average core pseudo potential to generate configuration-average cross sections. In the future, with further improvements in theoretical and computational methods and access to larger and faster massively parallel computers, it should be possible to apply these methods to more complex atomic species.
7. Acknowledgements This work was carried out in collaboration with C.P. Ballance of Rollins College; F. Robicheaux, S.D. Loch, and J.D. Ludlow of Auburn University; J. Colgan of Los Alamos National Laboratory; N.R. Badnell and M.C. Witthoeft of the University of Strathclyde; D.M. Mitnik of the University of Argentina, T.W. Gorczyca of Western Michigan University; and D.R. Schultz of Oak Ridge National Laboratory. It was supported in part by a U.S. DOE grant (Grant No.
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DE-FG02-99ER54367) to Rollins College, a U.S. DOE grant (Grant No. DEFG02-96ER54348) to Auburn University, and a U.S. DOE SciDAC grant (Grant No. DE-FG02-01ER54644) through Auburn University. A large portion of the computational work was carried out at the National Energy Research Scientific Computing Center in Oakland, California, and at the National Center for Computational Sciences in Oak Ridge, Tennessee.
8. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
[15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34]
M.P. Scott, P.G. Burke, J. Phys. B 226 (1993) L191. I. Bray, A.T. Stelbovics, Phys. Rev. Lett. 70 (1993) 746. D. Kato, S. Watanabe, Phys. Rev. Lett. 74 (1995) 2443. C.W. McCurdy, T.N. Rescigno, D. Byrum, Phys. Rev. A 56 (1997) 1958. C.W. McCurdy, T.N. Rescigno, Phys. Rev. A 56 (1997) R4369. T.N. Rescigno, M. Baertschy, W.A. Issacs, C.W. McCurdy, Science 286 (1999) 2474. K. Bartschat, E.T. Hudson, M.P. Scott, P.G. Burke, V.M. Burke, J. Phys. B 29 (1996) 115. K. Bartschat, I. Bray, J. Phys. B 29 (1996) L577. M.S. Pindzola, D.R. Schultz, Phys. Rev. A 53 (1996) 1525. M.S. Pindzola, F. Robicheaux, Phys. Rev. A 54 (1996) 2142. I. Bray, J. Phys. B 33 (2000) 581. J. Colgan, M.S. Pindzola, F.J. Robicheaux, D.C. Griffin, M. Baertschy, Phys. Rev. A 65 (2002) 042721. M.S. Pindzola, F. Robicheaux, J.P. Colgan, M.C. Witthoeft, J.A. Ludlow, Phys. Rev. A 70 (2004) 032705. M.S. Pindzola, D.C. Griffin, C. Bottcher, in: F. Brouillard (Ed.), “Atomic Processes in Electron– Ion and Ion–Ion Collisions”, in: NATO Adv. Sci. Inst. Ser. B Phys., vol. 145, Plenum, New York, 1986, p. 75. M.E. Riley, D.G. Truhlar, J. Chem. Phys. 63 (1975) 2182. S.M. Younger, Phys. Rev. A 24 (1981) 1278. J. Botero, J.H. Macek, J. Phys. B 24 (1991) L405. E.P. Wigner, Phys. Rev. 70 (1946) 15. E.P. Wigner, L. Eisenbud, Phys. Rev. 72 (1947) 86. A.M. Lane, R.G. Thomas, Rev. Mod. Phys. 30 (1958) 257. P.G. Burke, A. Hibbert, W.D. Robb, J. Phys. B 4 (1971) 153. P.G. Burke, W.D. Robb, Adv. At. Mol. Phys. 11 (1975) 143. P.J.A. Buttle, Phys. Rev. 160 (1967) 719. L. Castillejo, I.C. Percival, M.J. Seaton, Proc. Roy. Soc. London Ser. A 254 (1960) 259. P.G. Burke, D.F. Gallaher, S. Geltman, J. Phys. B 2 (1969) 1142. S. Geltman, P.G. Burke, J. Phys. B 3 (1970) 1062. P.G. Burke, T.G. Webb, J. Phys. B 3 (1970) L131. D.M. Mitnik, M.S. Pindzola, D.C. Griffin, N.R. Badnell, J. Phys. B 32 (1999) L479. T.W. Gorczyca, N.R. Badnell, J. Phys. B 30 (1997) 3897. D.M. Mitnik, D.C. Griffin, C.P. Ballance, N.R. Badnell, J. Phys. B 36 (2003) 717. C.P. Ballance, D.C. Griffin, J. Phys. B 37 (2004) 2943. K.A. Berrington, W.B. Eissner, P.H. Norrington, Comput. Phys. Commun. 92 (1995) 290. C. Bottcher, Adv. At. Mol. Phys. 20 (1985) 241. Y.D. Wang, J. Callaway, Phys. Rev. A 48 (1993) 2058.
8] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48]
[49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64]
NON-PERTURBATIVE QUANTAL METHODS
235
Y.D. Wang, J. Callaway, Phys. Rev. A 50 (1994) 2327. M.S. Pindzola, F. Robicheaux, N.R. Badnell, T.W. Gorczyca, Phys. Rev. A 56 (1997) 1994. M.S. Pindzola, J. Colgan, F. Robicheaux, D.C. Griffin, Phys. Rev. A 62 (2000) 042705. J. Colgan, S.D. Loch, M.S. Pindzola, C.P. Ballance, D.C. Griffin, Phys. Rev. A 68 (2003) 032712. C.P. Ballance, D.C. Griffin, J. Colgan, S.D. Loch, M.S. Pindzola, Phys. Rev. A 68 (2003) 062705. J. Colgan, M.S. Pindzola, F. Robicheaux, Phys. Rev. A 66 (2002) 012718. D.V. Fursa, I. Bray, J. Phys. B 30 (1997) 754. I. Bray, J. Phys. B 28 (1995) L247. T.W. Gorczyca, F. Robicheaux, M.S. Pindzola, D.C. Griffin, N.R. Badnell, Phys. Rev. A 52 (1995) 3877. K. Bartschat, I. Bray, J. Phys. B 30 (1997) L109. V. Zeman, K. Bartschat, J. Phys. B 30 (1997) 4609. J.E. Chilton, M.D. Stewart, C.C. Lin, Phys. Rev. A 61 (2000) 052708. J.B. Boffard, M.L. Keeler, G.A. Piech, L.W. Anderson, C.C. Lin, Phys. Rev. A 64 (2001) 032708. M.A. Khakoo, J. Wrkich, M. Larsen, G. Kleiban, I. Kanik, S. Trajmar, M.J. Brunger, P.J.O. Teubner, A. Crowe, C.J. Fontes, R.E.H. Clark, V. Zeman, K. Bartschat, D.H. Madison, R. Srivastava, A.D. Stauffer, Phys. Rev. A 65 (2002) 062711. J. Bömmels, K. Franz, T.H. Hoffmann, A. Gopalan, O. Zatsarinny, K. Bartschat, M.W. Ruf, H. Hotop, Phys. Rev. A 71 (2005) 012704. R.C. Wetzel, F.A. Baiocchi, T.R. Hayes, R.S. Freund, Phys. Rev. A 35 (1987) 559. E. Krishnakumar, S.K. Srivastava, J. Phys. B 21 (1988) 1055. M. Johnston, K. Fujii, J. Nickel, S. Trajmar, J. Phys. B 29 (1996) 531. S.M. Younger, Phys. Rev. A 26 (1982) 3177. D.C. Griffin, M.S. Pindzola, T.W. Gorczyca, N.R. Badnell, Phys. Rev. A 51 (1995) 2265. C.P. Ballance, D.C. Griffin, J.A. Ludlow, M.S. Pindzola, J. Phys. B 37 (2004) 4779. H. Deutsch, K. Becker, S. Matt, T.D. Märk, J. Phys. B 32 (1999) 4249. D.C. Griffin, C.P. Ballance, M.S. Pindzola, F. Robicheaux, S.D. Loch, J.A. Ludlow, M.C. Witthoeft, J. Colgan, C.J. Fontes, D.R. Schultz, J. Phys. B 38 (2005) L199. M.B. Shah, D.S. Elliott, H.B. Gilbody, J. Phys. B 20 (1987) 3501. P. Defrance, W. Claeys, A. Cornet, G. Poulaert, J. Phys. B 14 (1981) 111. D.V. Fursa, I. Bray, Phys. Rev. A 52 (1995) 1279. E.T. Hudson, K. Bartschat, M.P. Scott, P.G. Burke, V.M. Burke, J. Phys. B 29 (1996) 5513. M.S. Pindzola, F. Robicheaux, Phys. Rev. A 61 (2000) 052707. R.K. Montague, M.F.A. Harrison, A.C.H. Smith, J. Phys. B 17 (1984) 3295. M.B. Shah, D.S. Elliott, P. McCallion, H.B. Gilbody, J. Phys. B 21 (1988) 2751.
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ADVANCES IN ATOMIC, MOLECULAR AND OPTICAL PHYSICS, VOL. 54
R-MATRIX THEORY OF ATOMIC, MOLECULAR AND OPTICAL PROCESSES P.G. BURKE1 , C.J. NOBLE2 and V.M. BURKE2 1 Department of Applied Mathematics and Theoretical Physics, The Queen’s University of Belfast,
Belfast BT7 1NN, UK 2 CSE Department, CCLRC Daresbury Laboratory, Warrington WA4 4AD, UK
1. Introduction . . . . . . . . . . . . . . . . . . . . . . 2. Electron Atom Scattering at Low Energies . . . . . 2.1. R-Matrix Theory . . . . . . . . . . . . . . . . 2.2. Computer Programs . . . . . . . . . . . . . . . 2.3. Illustrative Results . . . . . . . . . . . . . . . 3. Electron Scattering at Intermediate Energies . . . . 3.1. Pseudostate Methods . . . . . . . . . . . . . . 3.2. Distorted Wave and Born-Series Methods . . . 4. Atomic Photoionization and Photorecombination . 4.1. Photoionization . . . . . . . . . . . . . . . . . 4.2. Photorecombination and Radiation Damping . 5. Electron Molecule Scattering . . . . . . . . . . . . 6. Positron Atom Scattering . . . . . . . . . . . . . . . 7. Atomic and Molecular Multiphoton Processes . . . 7.1. Atomic R-Matrix-Floquet Theory . . . . . . . 7.2. Molecular R-Matrix–Floquet Theory . . . . . 7.3. Time-Dependent R-Matrix Theory . . . . . . . 8. Electron Energy Loss from Transition Metal Oxides 9. Conclusions . . . . . . . . . . . . . . . . . . . . . . 10. Acknowledgements . . . . . . . . . . . . . . . . . . 11. References . . . . . . . . . . . . . . . . . . . . . . .
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1. Introduction R-matrix theory was introduced by Wigner (1946a, 1946b) and Wigner and Eisenbud (1947) in fundamental papers describing nuclear resonance reactions. This and other early work applying R-matrix theory in nuclear physics were comprehensively reviewed by Lane and Thomas (1958), Breit (1959) and Mahaux and 237
© 2007 Elsevier Inc. All rights reserved ISSN 1049-250X DOI: 10.1016/S1049-250X(06)54005-4
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F IG . 1. Partitioning of configuration space in R-matrix theory of electron atom scattering.
Weidenmüller (1969). In the early 1970s R-matrix theory was introduced and developed as an ab initio procedure for calculating electron atom scattering cross sections, where the Coulomb interactions between the electrons and the nuclei, unlike the nuclear interactions, are known exactly. R-matrix theory has since been developed and applied as an ab initio procedure for calculating a wide range of atomic, molecular and optical processes, where this work has been largely stimulated by the urgent need for a detailed understanding of these processes in many applications. For example, atomic, molecular and optical processes are of crucial importance in the astrophysics of gaseous nebulae, in atmospheric physics and chemistry including global warming, in plasma physics including controlled thermonuclear fusion, in the interaction of super intense lasers with atoms, molecules and plasmas, in isotope separation, in electrical discharges in gases and in electron surface interaction processes. The essential idea in R-matrix theory is to partition configuration space describing the scattering process into two or more regions where the process in each of these regions has distinctly different physical properties. A different representation of the wave function describing the process is then adopted in each region, which are connected by the R-matrix defined on their common boundaries. As an example we consider electron atom scattering, illustrated in Fig. 1, where a sphere of radius r = a0 separates an internal region from an external region, r being the radial distance of the scattered electron from the target nucleus. The radius a0 of the sphere is chosen so that the target eigenstates of interest are completely contained in the internal region. In the internal region, where electron exchange and correlation effects between the scattered electron and the target electrons are important, a configuration interaction expansion of the total wave function is adopted. The R-matrix on the boundary r = a0 of the internal region is then obtained by diagonalizing the total Hamiltonian in this configuration interaction basis. The R-matrix, which is a meromorphic function of the energy E with the poles on the real energy axis, has the general
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form Rij (E) =
1 wik wj k , 2a0 Ek − E
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(1)
k
where wik are surface amplitudes. Having determined the R-matrix, the logarithmic derivative of the reduced radial wave function Fi (r), describing the radial motion of the scattered electron in the ith channel on the boundary of the internal region is then given by dFj Rij (E)a0 . Fi (a0 ) = (2) dr j
r=a0
In Wigner and Eisenbud’s original theory, each term in the summation in Eq. (1) was identified with a separate resonance at energy Ek , where the wik were related to the partial decay widths of the kth resonance in the ith channel. In the applications of R-matrix theory discussed in this review, the terms in the summation in Eq. (1) represent both non-resonant and resonant scattering and their mutual interference effects. The R-matrix thus represents the full complexity of the multi-electron scattering process in the internal region. In the external region the scattered electron moves in the long-range local multipole potential of the target atom in the eigenstates of interest. The R-matrix can then be propagated outwards in this potential from r = a0 to r = ap where it can be fitted in an asymptotic region to an expansion yielding the K-matrix, S-matrix, scattering amplitudes and cross sections. Reviews of earlier work on R-matrix theory of atomic, molecular and optical processes have been written by Burke and Robb (1975) and by Burke and Berrington (1993). In this review, we introduce R-matrix theory in Section 2.1 by considering in some detail its application to low-energy electron scattering by atoms and atomic ions, where only elastic scattering and excitation processes are energetically allowed. We first describe the partitioning of configuration space into three regions depending on the properties of the electron target atom or ion interaction in each region. We then discuss the form of the solution in each of these regions, resulting in the calculation of the K-matrix, S-matrix, scattering amplitudes and cross sections. The atomic structure and scattering computer programs, written over the last thirty years to implement this theory, are then briefly reviewed. Finally in this section we present the results of some recent R-matrix calculations illustrating the application of this theory to low-energy electron scattering. This shows the accuracy now obtainable for electron scattering by light atoms and ions and the difficulties still experienced for more complex targets. Next, in Section 3, we consider the extension of the theory to treat electron scattering at intermediate energies. In this energy region an infinite number of channels, including ionizing channels are energetically allowed and an expansion in terms
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of pseudostates, which makes approximate allowance for these open channels, is developed. Some recent applications to excitation and ionization at intermediate energies are presented. We also consider in this section an approach which can be used when the energies of the incident and scattered electrons are well above the ionization threshold so that the corresponding wave functions can be accurately represented by distorted waves or Born-series. Then in Section 4 we extend the theory to treat atomic photoionization and photorecombination. We show how both the initial bound state and the final continuum state in photoionization can be represented by R-matrix expansions and we present results showing the application of this theory in the analysis of experiments using synchrotron radiation sources and in the calculation of opacities of importance in the analysis of stellar structure and understanding solar element abundances. We then discuss the developments of R-matrix theory to describe photorecombination where the incident electron is captured by an ion with the emission of a photon. In addition we consider the situation where the photon is reabsorbed by the ion giving rise to a modification of the electron ion scattering amplitude, called radiation damping. Applications of both of these processes are presented. The extension of the theory to treat electron molecule scattering is considered next in Section 5. The processes which can then occur are more varied than those that arise in electron scattering by atoms and atomic ions because of the possibility of exciting degrees of freedom associated with the nuclear motion. We show in this section how these degrees of freedom can be included in R-matrix theory and we consider recent applications of this theory to electron scattering by both diatomic and polyatomic molecules. Then in Section 6 we consider a further extension of the theory to treat the scattering of positrons by atoms and atomic ions. In this case, as well as the processes that can occur in electron atom scattering, the possibility of positronium formation must also be represented in the theory. Recent results for both positron and positronium scattering are then presented. Next in Section 7 we consider the interaction of intense laser fields with atoms and atomic ions. A wide range of multiphoton processes are then possible, including multiphoton ionization, harmonic generation and laser-assisted electron scattering. We consider first atomic R-matrix–Floquet theory which is fully non-perturbative and is applicable to atoms and atomic ions exposed to laser pulses consisting of many field cycles. We then extend R-matrix–Floquet theory to treat the interaction of intense laser fields with diatomic molecules. Finally in this section we describe a new time-dependent R-matrix theory, which can be adopted when atoms and ions are exposed to few femtosecond (10−15 sec) and attosecond (10−18 sec) laser pulses consisting of only a few cycles of the laser field. The last application that we briefly review in Section 8 is a generalization of R-matrix theory to describe lowenergy electron energy-loss spectroscopy from transition metal oxides. We show that 3d–3d energy loss spectra observed in recent experiments can be explained in a straightforward way by this theory. Finally, in Section 9 we conclude our review
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of R-matrix theory and applications by mentioning some emerging directions of research in this very active field.
2. Electron Atom Scattering at Low Energies 2.1. R-M ATRIX T HEORY R-matrix theory was introduced and developed by Burke et al. (1971) as an ab initio procedure for the accurate calculation of S-matrix elements and cross sections for electron scattering by complex atoms and atomic ions. Following this early work, we introduce R-matrix theory in this review by considering its application to low-energy electron atom and electron ion elastic scattering and excitation. That is, we consider the following process: e− + Ai → Aj + e− ,
(3)
where Ai and Aj are the initial and final bound states of the target atom which we assume contains N electrons and has nuclear charge number Z. We will consider the extension to excitation and ionization at intermediate energies in Section 3. For light atoms and ions, where relativistic effects are negligible, this process is described by the time-independent Schrödinger equation HN+1 Ψ = EΨ,
(4)
where HN+1 is the non-relativistic Hamiltonian defined in atomic units by N+1 N+1 1 1 Z . HN+1 = − ∇i2 − + 2 ri rij i=1
(5)
i>j =1
In this equation we have taken the origin of coordinates to be the target nucleus, which we assume has infinite mass, and we have written rij = |ri − rj | where ri and rj are the vector coordinates of the ith and j th electrons. As mentioned in the introduction, in order to describe the scattering of electrons by atoms and ions, we partition configuration space into three regions, an internal region, an external region and an asymptotic region, as illustrated in Fig. 2. The Schrödinger equation is then solved independently in each region. We consider below the solution in each of these regions in turn. Internal Region Solution The radius of the internal region r = a0 in Fig. 2 is chosen so that the wave functions of the target states of importance in the scattering process, given by Eq. (3), are negligible for r a0 . In this internal region electron exchange and correlation effects between the scattered electron and the target electrons are important and
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F IG . 2. Partitioning of configuration space in R-matrix theory of electron scattering by atoms and atomic ions.
the (N + 1)-electron complex behaves in a similar way to a bound state. Consequently, the total wave function is expanded in a configuration interaction basis which takes the following general form: −1 Φ LSπ (XN ; rˆ N+1 σN+1 )rN+1 uij (rN+1 )aijLSπ ψkLSπ = A i k ij
+
χjLSπ (XN+1 )bjLSπ k ,
(6)
j
for each set of quantum numbers L, S and π which are conserved in nonrelativistic scattering, where L is the total orbital angular momentum, S is the total spin angular momentum and π is the parity. In later equations we will omit these superscripts unless needed for clarity. Also in Eq. (6) A is the usual antisymmetrization operator, which ensures that the first term is completely antisymmetric with respect to exchange of any pair of electrons. The Φ i are channel functions formed by coupling the target eigenstates, and possibly pseudostates Φi , with the spin-angle functions of the scattered electron, uij are radial continuum basis orbitals representing the scattered electron which are non-zero on the boundary r = a0 and χj are additional quadratically integrable functions which are zero for r a0 and which allow for electron-electron correlation effects not included in the first expansion. Also in this equation xi ≡ (ri σi ) represents the space and spin coordinates of the ith electron and we have written XN = x1 , . . . , xN and XN+1 = x1 , . . . , xN+1 . We determine the coefficients aij k and bj k in Eq. (6) by diagonalizing the operator HN+1 + LN+1 in this basis as follows: ψk |HN+1 + LN+1 |ψk int = Ek δkk ,
(7)
where the integrations over the space and spin coordinates of the N + 1 electrons in this equation are confined to the internal region. Also LN+1 in Eq. (7) is the
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Bloch operator (Bloch, 1957; Lane and Robson, 1966) defined by LN+1 =
N+1 i=1
1 d b0 − 1 − δ(ri − a0 ) , 2 dri ri
(8)
where b0 is an arbitrary constant, which ensures that HN+1 +LN+1 is Hermitian in the space of the internal region configuration interaction basis. The Schrödinger equation (4) describing the collision process in the internal region can then be written as (HN+1 + LN+1 − E)Ψ = LN+1 Ψ,
(9)
which has the formal solution Ψ = (HN+1 + LN+1 − E)−1 LN+1 Ψ.
(10)
We then project this equation onto the channel functions Φ i , which we assume are n in number and evaluate it on the boundary r = a0 of the internal region yielding the following expression for the reduced radial wave functions, describing the scattered electron on the boundary of the internal region dFj Rij (E) a0 , − b 0 Fj Fi (a0 ) = (11) dr r=a0 j
which reduces to Eq. (2) when we set the arbitrary constant b0 = 0 in Eq. (8). We obtain in this way the following explicit expressions for the n × n dimensional R-matrix 1 wik wj k Rij (E) = (12) , 2a0 Ek − E k
and for the reduced radial wave functions −1 Ψ . Fi (rN+1 ) = Φ i rN+1
(13)
Also the surface amplitudes wik in Eq. (12) are defined in terms of the radial continuum basis orbitals uij (r) on the boundary r = a0 by −1 ψk r =a = uij (a0 )aij k , wik = Φ i rN+1 (14) N+1
0
j
where the primes on the Dirac brackets in Eqs. (13) and (14) mean that the integrations are carried out over the space and spin coordinates of all N + 1 electrons except the radial coordinate rN+1 of the scattered electron. It follows from the above equations that this approach results in a very efficient procedure where a single diagonalization of the Hamiltonian matrix defined by Eq. (7) yields the R-matrix defined by Eq. (12) at all energies.
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Radial Continuum Basis Orbitals We now consider the form adopted for the radial continuum basis orbitals uij in expansion (6). In applications of R-matrix theory to electron atom and electron ion scattering, they are usually chosen to be solutions of a zero-order differential equation for each orbital angular momentum i as follows: 2 d i ( i + 1) 2 − − U0 (r) + kij uij (r) = λij n Pn i (r), 2 2 dr r n j = 1, . . . , nc ,
(15)
satisfying the homogeneous boundary conditions uij (0) = 0, a0 duij = b0 , uij (a0 ) dr r=a0
(16)
where b0 is an arbitrary constant which is usually taken to be zero. The summation n in Eq. (15) goes over the reduced radial physical orbitals Pn i (r) used to construct the target states included in expansion (6) for the given i . Also, λij n in Eq. (15) are Lagrange multipliers which are chosen so that the orthogonality constraints a0 uij (r)Pn i (r) dr = 0,
(17)
0
are satisfied for all j and n. It follows that the radial continuum basis orbitals uij generated in this way are mutually orthogonal and can be normalized so that a0 uij (r)uij (r) dr = δjj .
(18)
0
In order to obtain rapid convergence in the R-matrix expansion over these continuum basis orbitals, the zero-order potential U0 (r) in Eq. (15) is usually taken to provide a good representation of the charge distribution of the target atom or ion in its ground state. For example, a Thomas–Fermi statistical model potential is often adopted. The terms on the right-hand side of Eq. (15), first introduced in a model e− –H scattering calculation by Lippmann and Schey (1961), act as an effective exchange potential which also improves the convergence. For most applications, the number of continuum basis orbitals nc is chosen in the range 15 to 20, which gives cross sections at arbitrary low-energies which are converged, from the point of view of the radial continuum basis orbital expansion, to several decimal places.
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In addition to the physical orbitals included on the right-hand side of Eq. (15), additional pseudo-orbitals are often required, both to allow for electron-electron correlation effects in the target eigenstates, and also in the construction of target pseudostates, discussed in Section 3.1. However, these pseudo-orbitals are not included on the right-hand side of Eq. (15) since this would spoil the rapid convergence of the expansion over the radial continuum basis orbitals discussed above. Instead, the radial continuum basis orbitals are Schmidt orthogonalized to these additional pseudo-orbitals and additional quadratically integrable functions containing these pseudo-orbitals are included in the second expansion in Eq. (6) for completeness. Since the radial continuum basis orbitals uij (r) satisfy homogeneous boundary conditions (16), it is necessary to add a “Buttle correction” to the R-matrix expansion (12) in order to obtain accurate results. This correction, first introduced by Buttle (1967), corrects for the omission of high-lying pole terms in expansion (12). In practice, it is only necessary to correct the diagonal elements of the R-matrix since the off-diagonal elements oscillate in sign and converge rapidly. If nc radial continuum basis orbitals are included in expansion (6), for a given orbital angular momentum i , then the Buttle correction to the diagonal elements of the R-matrix is given by RiiBC (E) =
∞ 1 [uij (a0 )]2 , 2a0 Eij0 − E j =n +1
i = 1, . . . , n.
(19)
c
In this equation, the zero-order energy Eij0 = ei + 12 kij2 , where ei is the energy of the target state in the ith channel and kij2 is the corresponding zero-order eigenvalue in Eq. (15). The Buttle correction given by Eq. (19) can be rewritten as −1 nc [uij (a0 )]2 a0 dui 1 BC − , − b0 Rii (E) = ui dr 2a0 Eij0 − E r=a0 j =1 i = 1, . . . , n,
(20)
where ui is the solution of the zero-order differential Eq. (15) at the total energy E of interest and the expansion on the right-hand side of this equation goes over the zero-order radial continuum basis orbitals included in the original R-matrix expansion. It follows that the Buttle correction is a smooth function of energy, since it does not contain poles in the low-energy range of interest. It can therefore be rapidly evaluated at a few energy values and interpolated at the energies of interest. In most applications of R-matrix theory to electron scattering by atoms and atomic ions and to atomic photoionization, use of radial continuum basis orbitals satisfying homogeneous boundary conditions incorporating a Buttle correction to the R-matrix has proved to be rapidly convergent and convenient to use. However,
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it was pointed out by Yu and Seaton (1985) that in order to calculate highly accurate photoionization cross sections, which involves an integral over the initial bound state and final continuum state wave functions (see Section 4.1), a correction to the wave function on the boundary is also sometimes desirable. In recent work there has been increasing interest in methods which use radial continuum basis orbitals satisfying arbitrary boundary conditions on the surface of the internal region r = a0 . These methods do not require a correction to the R-matrix nor to the wave function and, as a result they are derivable from a variational principle (e.g., Kohn, 1948; Jackson, 1951; Nesbet, 1980; Aymar et al., 1996). However, this does not necessarily mean that the R-matrix expansion obtained using arbitrary boundary condition methods converges faster than that obtained using homogeneous boundary condition methods including a Buttle correction. Recent arbitrary boundary condition methods that have been considered include the use of Lagrange mesh bases by Malegat (1994) and by Plummer and Noble (1999), and the use of B-spline bases by van der Hart (1997), Zatsarinny and Froese Fischer (2000), Zatsarinny and Bartschat (2004a, 2004b) and Zatsarinny (2006). We consider some applications of these methods later in this review. External Region Solution In the external region in Fig. 2, electron exchange and correlation effects between the colliding electron and the target electrons are negligible. The scattered electron then moves in the long-range local multipole potential of the target atom or ion. Hence the expansion corresponding to Eq. (6) for the total wave function reduces to Ψ =
n
−1 Φ i (XN ; rˆ N+1 σN+1 )rN+1 Fi (rN+1 ),
r a0 ,
(21)
i=1
where n is the number of channel functions retained in Eq. (6). Substituting this expansion into the Schrödinger equation (4) and projecting onto the channel functions Φ i then yields the following set of coupled second-order differential equations, satisfied by the reduced radial functions Fi (r), 2 n d i ( i + 1) 2(Z − N ) 2 − + Vij (r)Fj (r), + ki Fi (r) = 2 r dr 2 r2 j =1
r a0 ,
i = 1, . . . , n.
(22)
In this equation i is the orbital angular momentum and ki is the wave number of the scattered electron in the ith channel, defined in terms of the total energy E by ki2 = 2(E − ei ),
(23)
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where ei is the energy of the target state in the ith channel. Also the potential matrix elements Vij (r) are defined by N 1 −1 Vij (rN+1 ) = Φ i (XN ; rˆ N+1 σN+1 )rN+1 rkN+1 k=1 N −1 ˆ − Φ (X ; r σ )r (24) j N N+1 N+1 N+1 . rN+1 Since the target states included in Φ i vanish in the external region then Eq. (24) reduces to the following simple form Vij (r) =
λ max
aij λ r −λ−1 ,
r a0 ,
(25)
λ=1
where the coefficients aij λ are related to the multipole moments of the target states retained in the original R-matrix expansion. The coupled second-order differential equations (22) must be solved over the range a0 r ap subject to the R-matrix boundary condition at r = a0 , defined by Eqs. (11) and (12). This can be accomplished using a number of standard methods (e.g., Burke and Seaton, 1971; Press et al., 1992). Here we briefly discuss R-matrix propagator methods used in many recent applications. We will also mention an important approach by Seaton (1985) applicable to electron ion scattering, when we discuss computer programs in Section 2.2. In R-matrix propagator methods, the R-matrix calculated on the boundary r = a0 can be propagated from r = a0 to r = ap by subdividing the external region into p subregions, as illustrated in Fig. 2. We then use the R-matrix propagator equation −1 as R(as ) = Gs (as , as ) − Gs (as , as−1 ) Gs (as−1 , as−1 ) + as−1 R(as−1 ) × Gs (as−1 , as ),
s = 1, . . . , p.
(26)
Also, it is sometimes necessary to propagate the R-matrix inwards across these subregions using the R-matrix propagator equation −1 as−1 R(as−1 ) = −Gs (as−1 , as−1 ) + Gs (as−1 , as ) Gs (as , as ) − as R(as ) × Gs (as , as−1 ),
s = 1, . . . , p.
(27)
The Green’s functions Gs in each subregion are determined either by assuming that the potential matrix elements Vij (r) in Eq. (22) are slowly varying in each subregion so that they can be approximated by constants (Light and Walker, 1976; Stechel et al., 1978; Schneider and Walker, 1979) or that the reduced radial functions Fi can be expanded in each subregion in a basis of shifted orthogonal
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polynomials (Baluja et al., 1982). The choice of method depends on the required number of energy values and the rapidity of variation of the potential in the external region. In this way the R-matrix can be determined on the outer boundary r = ap of the external region. Having determined the R-matrix on the boundaries r = as of the p subregions, we can then propagate the reduced radial wave function F(as ), if required, across these subregions. We obtain the following equations for outward and inward propagation: −1 −1 F(as ) = as R(as ) Gs (as , as ) − as R(as ) Gs (as , as−1 )as−1 ×R(as−1 )−1 F(as−1 )
(28)
and −1 F(as−1 ) = as−1 R(as−1 ) Gs (as−1 , as−1 ) + as−1 R(as−1 ) ×Gs (as−1 , as )as−1 R(as )−1 F(as ).
(29)
The wave function is required in the calculation of matrix elements such as occur in the distorted wave and Born-series methods, discussed in Section 3.2 and in the study of photoionization processes, discussed in Section 4.1. Asymptotic Region Solution The boundary r = ap between the external and asymptotic regions in Fig. 2, is chosen large enough that the solution of the coupled second-order differential equations (22) can be accurately represented by an asymptotic expansion for r ap (Burke and Schey, 1962; Burke et al., 1964; Gailitis, 1976; Noble and Nesbet, 1984). The leading term in this asymptotic expansion has the following form in the open channels where ki2 0 −1/2
Fij (r) ∼ ki r→∞
(sin θi δij + cos θi Kij ),
(30)
where the second index j on Fij denotes the linear independent solutions of Eq. (22). Also θi = ki r − 12 i π for a neutral target and includes logarithmic terms for an ionic target, and Kij are the elements of the real symmetric K-matrix. The S-matrix and T-matrix are then defined in terms of the K-matrix by 2iK I + iK , T= , (31) I − iK I − iK and the total cross section for a transition from an initial state i to final state j is given in atomic units a02 by 2 π . (32) σ (i, j ) = 2 (2L + 1)(2S + 1)TjLSπ i 2ki (2Li + 1)(2Si + 1) LSπ S=
i j
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In this equation, Li and Si are the target orbital and spin angular momenta and i and j are the incident and scattered electron orbital angular momenta. The whole calculation has to be repeated in the internal, external and asymptotic regions for all significant L, S and π values which are conserved in the collision. A further quantity Ω(i, j ), first introduced by Hebb and Menzel (1940), and subsequently called the collision strength by Seaton (1975), is often used in electron scattering by ions. It is defined by (2Li + 1)(2Si + 1)ki2 ωi ki2 (33) σ (i, j ) = σ (i, j ), π π where ωi = (2Li + 1)(2Si + 1) is the statistical weight of the initial state when fine-structure is not considered and is equal to (2Ji + 1) when it is. Since ki has the dimensions of a reciprocal length, Ω(i, j ) is dimensionless. It is also symmetric so that Ω(i, j ) = Ω(j, i). In an ionized plasma, we also need to determine the electron ion scattering cross section averaged over a Maxwell distribution of electron velocities. The rate coefficient for de-excitation (ei > ej ) is then given by Ω(i, j ) =
qij (Te ) =
8.63 × 10−6 Υij (Te ) , ωi (Te )1/2
(34)
where we have introduced the effective collision strength ∞ Υij (Te ) =
i i Ω(i, j ) exp − d . k B Te k B Te
(35)
0
In this equation i is the kinetic energy in Rydbergs of the electron in the ith state, Te is the electron temperature in Kelvin K and kB = 6.339 × 10−6 Rydbergs/K is Boltzmann’s constant. The rate coefficient can be calculated for each temperature once the cross section has been calculated over a sufficient range of energies, at a mesh fine enough to accurately resolve any resonant structure.
2.2. C OMPUTER P ROGRAMS A number of general R-matrix computer programs which enable electron atom and electron ion scattering amplitudes and cross sections to be calculated for arbitrary multi-electron targets have been written over the last thirty years. In addition, atomic structure programs, which provide target state input data have also been written. We will see later in this review that some of these programs have been extended to enable electron scattering by atoms and ions at intermediate energies, atomic photoionization and photorecombination and atomic multiphoton processes to be calculated. In this section we give a brief summary of these
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programs, where we note that in many cases detailed descriptions have been published in Computer Physics Communications (CPC) with copies obtainable from the associated CPC Program Library. General atomic structure programs, based on the non-relativistic Hamiltonian, where relativistic effects are included by retaining one- and two-body terms of the Breit–Pauli Hamiltonian (e.g., Bethe and Salpeter, 1957 and Akhiezer and Berestetsky, 1965) have been written by many authors. The programs which have been particularly important in the calculation of atomic target states required in R-matrix calculations are MCHF, written by Froese Fischer (1969, 1972, 1978, 1991), SUPERSTRUCTURE, written by Eissner et al. (1974) and Nussbaumer and Storey (1978), AUTOSTRUCTURE (which incorporates SUPERSTRUCTURE) written by Badnell (1986, 1997) and CIV3, written by Hibbert (1975). Atomic structure programs, which include relativistic effects using the Dirac Hamiltonian, have been used to determine atomic target states for Dirac R-matrix calculations. These are MCDF, written by Grant et al. (1980) and major updates of this program are GRASP, written by Dyall et al. (1989) and GRASP92, written by Parpia et al. (1996). R-matrix programs applicable for electron scattering by arbitrary light atoms and ions based on the non-relativistic Hamiltonian defined by Eq. (5) have been written by Berrington et al. (1974, 1978, 1987), Burke et al. (1994) and Zatsarinny and Froese Fischer (2000). As the nuclear charge number Z increases, relativistic effects become important and for intermediate Z are included, as in the atomic structure programs, by retaining terms of the Breit–Pauli Hamiltonian. An R-matrix program using this approximation, based on the non-relativistic program by Berrington et al. (1974, 1978) has been written by Scott and Burke (1980), Scott and Taylor (1982) and Berrington et al. (1995). In addition, an independent R-matrix program including relativistic effects via the Breit–Pauli Hamiltonian and representing the continuum basis orbitals by B-splines has been written by Zatsarinny (2006). For electron scattering by the heaviest atoms and ions, R-matrix programs using the Dirac Hamiltonian have been written by Chang (1975) and more recently by Norrington and Grant (1981, 1987). In addition to these general programs, a number of useful more specific programs have been written. These include a program JAJOM, written by Saraph (1972, 1978) which transforms the K-matrix calculated in LSπ coupling, neglecting relativistic terms in the Hamiltonian, to a pair coupling scheme, enabling scattering amplitudes and cross sections between fine-structure levels of the target to be calculated. This approach operates on the open channel K-matrix and hence is most appropriate for scattering energies when all channels are open. A more general approach, valid for electron–ion scattering, has been developed by Badnell et al. (1998) and Griffin et al. (1998). In this approach the R-matrix program STGF, written by Seaton (1985), which solves the coupled second-order differential equations (22) in the external region, representing the non-Coulomb potentials
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as a perturbation, was modified to calculate unphysical K-matrices in LSπ coupling using multichannel quantum defect theory MQDT (Seaton 1966a, 1966b, 1983). The unphysical K-matrix, which includes both open and closed channels, is transformed to pair coupling, enabling scattering amplitudes and cross sections between fine-structure levels to be calculated. This method has yielded results in very good agreement with calculations which include Breit–Pauli terms explicitly in the Hamiltonian. Other useful programs include a no-exchange program, valid for high incident electron angular momenta, written by V.M. Burke et al. (1992) and a program FARM, written by V.M. Burke and Noble (1995), which solves the coupled second-order differential equations (22) in the external region, using the R-matrix propagator equation (26). Finally, we note that recently considerable effort is being devoted to the development of parallel R-matrix programs, which can take full advantage of massively parallel computer architectures. These programs include the development of a parallel version of the scalar program written by Burke et al. (1994) together with the parallel external region program developed by Sunderland et al. (2002). In addition a parallel program based on the scalar program by Berrington et al. (1995) is being developed by Mitnik et al. (2003) and Ballance and Griffin (2004). These parallel programs will be required to obtain converged excitation and ionization cross sections for complex atomic targets such as those with open d- and f-shells, including relativistic effects, which can involve thousands of coupled channels.
2.3. I LLUSTRATIVE R ESULTS Over the last thirty years a vast number of low-energy electron atom and electron ion scattering cross sections have been calculated using the R-matrix computer programs described in Section 2.2, both in support of experiment and to provide data required in the analysis of applications, for example in plasma physics, laser physics and astrophysics. We present here three examples of this work which illustrate the accuracy now attainable for light atoms and ions and the difficulties still experienced for more complex targets. In our first example, we present in Fig. 3 a comparison of a recent R-matrix calculation by Zatsarinny and Bartschat (2004a) for the following electron impact excitation process in Neon compared with experiment (Buckman et al., 1983; Buckman and Clark, 1994) e− + Ne 2p6 1 S0 → e− + Ne 2P5 3s 3 P2 + 3 P0 . (36) The R-matrix calculations were carried out including relativistic effects using the Breit–Pauli Hamiltonian and retaining 31 fine-structure target states in expansions (6) and (21). In addition, the scattered electron radial continuum basis orbitals
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F IG . 3. Angle-integrated excitation cross section for production of neon in the metastable 2p5 3s 3 P2 and 2p5 3s 3 P0 states. Solid line and thin dotted line theory, thick dotted line experiment (see text).
were represented by B-splines and non-orthogonal orbitals were used to represent the outer target orbitals, which were state-dependent. These state-dependent orbitals avoided the need to include pseudo-orbitals in the calculation, which we will see later in this review give rise to unphysical pseudo-resonances at intermediate energies. The experimental data of Buckman et al. (1983) were normalized to provide a good visual fit to the theory at energies above the excitation threshold. The solid line includes the cascade contribution from all states included in the model, while the dotted line starting about 18.4 eV represents the theoretical results without cascade. Zatsarinny and Bartschat (2004b) have recently extended this work to argon with equally impressive results. As our second example we consider electron scattering by Si III (Si2+ ). There is considerable demand for excitation rates for this ion, particularly in the analysis of solar spectra and in the study of laboratory plasmas. Of particular importance is the following excitation process e− + Si III(3s2 1 S)−→e− + Si III(3s3p 3 Po ) , Si II∗∗ (3s3p 1 Po n )
(37)
which is dominated by Feshbach resonances close to threshold. In these resonances, the incident electron is captured into a bound state in the field of Si III in an excited 3s3p 1 Po state, which then decays leaving the target in its 3s3p 3 Po first excited state. R-matrix calculations for this process have been carried out by Baluja et al. (1980) and Griffin et al. (1993), where the later authors also studied transitions in Ar VII (Ar6+ ). In these calculations the 12 lowest target states
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F IG . 4. Collision strengths for the transition 3s2 1 S → 3s3p 3 Po in Si III. Full line: R-matrix calculation (Baluja et al., 1980); dashed line: distorted wave calculation (Nicolas, 1977).
of Si III, each represented by configuration interaction wave functions, were included in expansions (6) and (21). We present in Fig. 4 the collision strength for the transition corresponding to Eq. (37), compared with distorted wave results calculated by Blaha (quoted by Nicolas, 1977). We see at these low energies the collision strength is dominated by the resonance contribution while the distorted wave results represent the much smaller non-resonant background. An indication of this domination is given by the effective collision strengths at electron temperatures below about 7000 K where the R-matrix result is six times larger than the distorted wave result. More recently, the absolute excitation cross sections for the 3s2 1 S → 3s3p 3 Po and 1 Po transitions in Si III were measured close to threshold by Wallbank et al. (1997) using a merged electron–ion beams energy-loss technique. We show in Fig. 5 the measured cross section for the 3s2 1 S → 3s3p 3 Po transition compared with the R-matrix calculations of Baluja et al. (1980) and Griffin et al. (1993). The strong resonance effect close to threshold seen in the R-matrix calculations is confirmed by experiment which is also in good agreement with the predicted height of the resonance peak. The agreement between theory and experiment is less good at energies more than about 1 eV above threshold, which might be partly due to omission of back scatter electrons in the experiment. However, overall this important experiment confirms the crucial role resonances play in the scattering
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F IG . 5. Cross section for 3s2 1 S → 3s3p 3 Po electron impact excitation in Si III. Points represent average experimental values and bars display relative uncertainties at 90% confidence level (Wallbank et al., 1997). Dashed curve: R-matrix theory (Baluja et al., 1980); solid curve: R-matrix theory (Griffin et al., 1993), both theories convoluted with a Gaussian of width 0.24 eV.
of low-energy electrons by ions and provides a vital check on the R-matrix theory predictions. As our third example, we consider electron scattering by the iron peak element Fe II. Electron impact excitation cross sections and related effective collision strengths for low ionization stages of open d-shell iron peak elements are of crucial importance in the quantitative analysis of many astrophysical spectra (e.g., Bautista and Pradhan, 1998). The calculations reported here were carried out as part of the international “IRON Project” collaboration to study these elements using the R-matrix program package (Hummer et al., 1993). To illustrate the complexity of this problem, we show in Fig. 6 the energy level diagram of Fe II below 30,000 cm−1 (∼3.72 eV) with some of the forbidden infra-red and optical lines observed in gaseous nebulae indicated. Of particular importance are the transitions between the fine-structure levels belonging to the four lowest LS-coupled terms of the target corresponding to the 3d6 4s a6 D ground state and the 3d7 a4 F, 3d6 4s a4 D and 3d7 a4 P excited states. However, in order to obtain accurate effective collision strengths for transitions between these levels it is not sufficient to just include these four target states in expansions (6) and (21), since when the incident electron penetrates the internal R-matrix region many higher states will be virtually excited and must be included in the expansion to obtain converged results. We show in Fig. 7 the total collision strength for the 3d6 4s a6 D → 3d7 a4 F transition calculated in LS coupling by Ramsbottom et al. (2005), including all target states corresponding to the 3d6 4s, 3d7 , 3d6 4p, 3d5 4s2 and 3d5 4s4p target
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F IG . 6. The 16 LS terms of Fe II below 30,000 cm−1 showing the corresponding 46 fine-structure levels and some observed infra-red and optical lines with their wavelengths given in Å.
configurations. As a result 113 target states were included in expansions (6) and (21), where each target state was represented by a configuration interaction expansion. This calculation led to 301 coupled channels for the higher L values. We see that the low-energy collision strength is dominated by resonances, which required evaluation at 15,000 distinct incident electron energies to accurately delineate. As pointed out earlier, this was achieved by a single diagonalization of the internal region Hamiltonian corresponding to Eq. (7) for each LSπ involved. However, in the external and asymptotic regions the coupled second-order differential equations (22) had to be solved for each required energy. It was also found that the positions of some of the most significant resonances were very sensitive to the configuration interaction effects included in expansion (6) and in the representation of the target states. Since there are no experimental measurements to compare with, accurate results can only be obtained by systematically increasing the number of configuration interaction terms retained in expansion (6) and in
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F IG . 7. Total collision strengths for 3d6 4s a6 D → 3d7 a4 F transition in e− –Fe II scattering at low energies (Ramsbottom et al., 2005).
the target state expansions until convergence is obtained. In addition, it should be noted that the observed infra-red and optical lines shown in Fig. 6 involve transitions between fine-structure levels of the target. An accurate calculation of these transitions will require the inclusion of relativistic terms in the Hamiltonian which will increase the maximum number of coupled target states, corresponding to the five configurations 3d6 4s, 3d7 , 3d6 4p, 3d5 4s2 and 3d5 4s4p, from 113 to 716 and the number of coupled channels from 301 to 5076. To carry out these calculations will require the development of the new parallel R-matrix programs mentioned in the previous section.
3. Electron Scattering at Intermediate Energies 3.1. P SEUDOSTATE M ETHODS In the examples that we have considered so far, the energy of the incident electron is insufficient to ionize the target. We now consider scattering of electrons by atoms or atomic ions at intermediate energies, which are defined as incident electron energies ranging from close to the ionization threshold to several times this threshold. In this energy range, the target can be ionized as well as excited and since this energy is not high, strong coupling effects will exist between the channels leading to excitation and to ionization. Hence theoretical methods which give reliable results for excitation and ionization cross sections at intermediate energies must accurately represent this coupling.
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F IG . 8. Exact spectrum of an atom or ion and its approximate representation by a bound state plus pseudostate spectrum.
A successful approach for representing the continuum in electron atom scattering is the R-matrix with pseudostates (RMPS) method (Bartschat et al., 1996a, 1996b; Bartschat and Bray, 1996; Badnell and Gorczyca (1997); Gorczyca and Badnell, 1997a). In this method expansion (6) over target eigenstates is augmented by including additional quadratically integrable target pseudostates, which are constructed by including additional contracted pseudo-orbitals in the orbital basis. These pseudostates represent, in an average way, the high-lying Rydberg states and continuum states of the target and are chosen to diagonalize the target Hamiltonian as follows: Φi |HN |Φj = wi δij ,
(38)
where the Φi represent both target eigenstates and pseudostates retained in expansions (6) and (21). We compare the exact spectrum of a typical atom or ion with its approximate representation by a bound state plus pseudostate spectrum in Fig. 8. We see that the lowest bound eigenstates are accurately represented by the approximate spectrum while the infinite number of high-lying Rydberg states and the continuum spectrum are represented by the discrete pseudostate spectrum. As a result electron impact ionization can be approximately represented by an appropriate summation over the cross sections for excitation of the pseudostates lying in
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the continuum. It is clear from this figure that in order to obtain an accurate representation of excitation and ionization at intermediate energies, the continuum must be spanned by a sufficiently dense pseudostate spectrum. In practice, it has proved necessary to retain several times more pseudostates than eigenstates in expansions (6) and (21) in order to obtain accurate cross sections at intermediate energies. This is usually achieved by expanding the pseudostate basis in terms of Sturmian type pseudo-orbitals which give a satisfactory representation of the continuum for multi-electron targets over a wide energy range. In order to calculate accurate cross sections close to the ionization threshold it is necessary that there is a high density of pseudostates in the neighbourhood of this threshold. This can be achieved by taking the radius r = a0 of the internal region large and subdividing this region into a number of subregions. This is the basis of the Intermediate Energy R-matrix Method (IERM) introduced by Burke et al. (1987) and further developed by LeDourneuf et al. (1990) and Dunseath et al. (1996). In order to illustrate this method, we consider electron scattering by atoms or atomic ions with one active electron, such as atomic hydrogen or alkalilike metal atoms and ions. We introduce a two-dimensional plane, illustrated in Fig. 9, where the radial coordinates r1 and r2 of the scattered and active target electrons label the axes. We then introduce an expansion in each subregion for each set of conserved quantum numbers LSπ with the following general form: ψkLSπ (r1 , r2 ) = 1 + (−1)S P12 r1−1 un1 1 (r1 )r2−1 un2 2 (r2 )Y 1 2 LML (ˆr1 , rˆ 2 )anLSπ , × (39) 1 1 n2 2 k n1 1 n2 2
where P12 interchanges the coordinates of the two electrons so that (1 + (−1)S P12 ), which is included in the diagonal subregions, symmetrizes or antisymmetrizes the wave function. Also un (r) are radial continuum basis orbitals, defined in each subregion, and Y 1 2 LML (ˆr1 , rˆ 2 ) are angular functions which are eigenstates of the total orbital angular momentum and its z component. The twoelectron Hamiltonian is then diagonalized in this basis in each subregion to yield and the R-matrix on the boundaries of the subregions the coefficients anLSπ 1 1 n2 2 k propagated outwards across the subregions, where from symmetry we need only consider subregions where r1 r2 . For example, the Hamiltonian is diagonalized in the (s, t) subregion in Fig. 9 and the R-matrix is then propagated across the subregion from edges labelled 5, 1, 2, 6, 7 to edges labelled 5, 4, 3, 6, 7. A parallel strategy has been developed for propagating the R-matrix outwards, across the subregions in Fig. 9, on distributed memory parallel computers by Heggarty et al. (1998). The R-matrix on the outer boundary r1 = aq of the internal region (corresponding to r = a0 in Figs. 1 and 2) is then propagated outwards, as in standard R-matrix theory discussed in Section 2.1 and fitted to an asymptotic expansion yielding the K-matrix, S-matrix and cross sections.
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F IG . 9. Division of the (r1 , r2 ) plane into elementary subregions in the IERM method. The shaded subregion with edges labelled 1, 2, 3 and 4 is referred to as the (s, t) subregion.
As our first example of electron atom scattering at intermediate energies, we consider RMPS and convergent close-coupling (CCC) calculations for electron impact ionization of the 1s state of atomic hydrogen obtained by Bartschat and Bray (1996). The CCC method, which like the RMPS method represents the continuum spectrum by a quadratically integrable basis of pseudostates, has been very successful in calculating electron impact excitation and ionization cross sections at intermediate energies for a wide range of atomic targets with one active electron (e.g., Bray et al., 2002). We present in Fig. 10 results obtained by RMPS and CCC calculations for the total ionization cross section σI and for the ionization spin asymmetry AI of atomic hydrogen compared with experiment over the energy range from 13.6 eV to 100 eV. The total ionization cross section measurements were made by Shah et al. (1987). The ionization spin asymmetry AI measurements were made by Fletcher et al. (1985) and by Crowe et al. (1990)
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(a)
(b) F IG . 10. Total ionization cross section σI (a) and ionization spin asymmetry AI (b) for electron impact ionization of atomic hydrogen compared with experiment.
where AI is defined by AI =
σ S − σIT 1 NA − NP = IS . Pe PA NA + NP σI + σIT
(40)
In this equation NA and NP are the count rates for ionization with antiparallel and parallel scattered electron (Pe ) and target (PA ) spin polarizations, respectively,
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while σIS and σIT are the singlet and triplet contributions to the total ionization cross section, respectively. The agreement between the results from the two independent calculations is very good for both σI and AI . Also the agreement between theory and experiment for σI is excellent and, while there are some minor discrepancies between these calculations and the experiments for AI , we note that there is also some scattering in the experimental data. This comparison shows that the RMPS and CCC pseudostate methods can yield reliable ionization results over a wide energy range. More recent work by Mouret et al. (2003) has shown that the IERM approach also gives accurate total ionization cross sections and spin asymmetries for electron hydrogen atom scattering. In the last few years, there has been increasing interest in the study of electron impact ionization of atomic hydrogen at energies close to threshold. In addition to the RMPS and CCC methods, other approaches adopted have included hyperspherical close coupling (Kato and Watanabe, 1997), time-dependent closecoupling (TDCC) (Pindzola and Robicheaux, 2000) and exterior complex scaling (ECS) (Rescigno et al., 1999; McCurdy et al., 2004). At these energies, the IERM approach discussed above has enabled accurate cross sections to be obtained. For example, Scott et al. (2002) have propagated the R-matrix outwards to aq = 600 atomic units, shown in Fig. 9, yielding a dense pseudostate spectrum in the neighbourhood of the ionization threshold and giving the 1 Se single differential cross section at 17.6 eV in excellent agreement with recent TDCC and ECS calculations. As a second example of electron scattering at intermediate energies we present in Fig. 11 independent RMPS calculations by Mitnik et al. (1999) and Scott et al. (2000) for electron impact ionization of C3+ given by e− + C3+ 1s2 2s → C4+ 1s2 + 2e− , (41) compared with a CCC calculation by Bray (1995), a distorted wave calculation by Younger (1981) and with experiment by Crandall et al. (1979). In the RMPS calculation by Scott et al. (2000), the five lowest target eigenstates of C3+ , 1s2 2s 2 Se , 1s2 2p 2 Po , 1s2 3s 2 Se , 1s2 3p 2 Po and 1s2 3d 2 De , together with 21 pseudostates of the form 1s2 n up to 9s, 9p, 7d, 6f and 6g were included in expansions (6) and (21). The small oscillations in the cross section before smoothing resulted from the representation of the continuum by a discrete pseudostate basis and are damped as this basis is increased. We see from Fig. 11 that there is good overall agreement between the RMPS and CCC calculations and experiment. Recently, RMPS calculations have been carried out yielding reliable cross sections at intermediate energies for electron scattering by many atoms and atomic ions. We will see in Section 5 that the RMPS method has also been used to obtain cross sections at intermediate energies for electron molecule scattering.
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F IG . 11. Electron impact ionization of C3+ . Full curve: RMPS smoothed (Scott et al., 2000); short-broken curve: RMPS unsmoothed (Scott et al., 2000); long-broken curve: RMPS unsmoothed (Mitnik et al., 1999); dotted curve with diamonds: CCC (Bray, 1995); chain curve: distorted wave (Younger, 1981); full circles: experiment (Crandall et al., 1979).
In addition to the direct ionization process considered above, several important indirect ionization processes can occur which involve the active participation of more than one target electron. We illustrate these processes by considering again electron impact ionization of C3+ . These additional indirect processes are excitation-autoionization (EA), where an intermediate resonance state is excited which then decays with the ejection of an electron as follows: e− + C3+ 1s2 2s → C3+∗∗ 1sn n + e− ↓ 4+
C
2
1s
(42) −
+ 2e ,
resonant-excitation-double-autoionization (REDA) where first the incident electron is captured and a 1s electron is excited, and then two electrons are ejected sequentially as follows:
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F IG . 12. Electron impact ionization of C3+ . Full curve: convoluted RMPS results including direct and indirect processes, dashed curve: experimental data of Müller et al. (1988).
e− + C3+ 1s2 2s → C2+∗∗∗ 1sn n n ↓
C3+∗∗ 1sn n + e−
(43)
↓
C4+ 1s2 + 2e− , and resonant-excitation-auto-double-ionization (READI) where again first the incident electron is captured and a 1s electron is excited, and then two electrons are ejected simultaneously as follows e− + C3+ 1s2 2s → C2+∗∗ 1sn n n ↓ 4+
C
2
1s
(44) −
+ 2e ,
where intermediate resonance states, where two or three electrons are excited, are indicated by asterisks. In addition, the scattered electron can be captured into an intermediate resonance state which can decay by emission of a photon giving rise to dielectronic recombination, discussed in Section 4.2. An RMPS calculation of the indirect processes in electron impact ionization of C3+ was made by Scott et al. (2000) by augmenting the five eigenstates and 21
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pseudostates retained in expansions (6) and (21) in their earlier calculation, discussed above, with an additional 16 autoionizing states, where only one electron was retained in the 1s orbital. In this way the intermediate states in Eqs. (42), (43) and (44) are represented in the calculation. We compare their calculated ionization cross section in the energy range from 284 eV to 356 eV with the experimental data of Müller et al. (1988) in Fig. 12, where the calculated cross section was convoluted with a 2.0 eV full-width-at-half-maximum Gaussian to simulate the experimental energy resolution. The RMPS calculation shows a number of resonance features observed by Müller et al. However the calculated non-resonant background is about 10% higher than experiment and there is an energy shift of the calculated autoionizing peaks by about 2.0 eV relative to their observed positions. The shift in position of the autoionizing peaks indicates that more pseudostates should be included in the calculation to give convergence, while the discrepancy in normalization with experiment is larger than that shown for direct ionization, shown in Fig. 11, indicating that part of this discrepancy could be attributed to experiment. However, in view of the complexity of these indirect processes and the difficulty of the experiment, the overall agreement is very encouraging. In concluding our discussion of indirect ionization processes, we observe that the RMPS method has enabled the main features of these processes to be obtained using a completely ab initio theory. In view of the importance of these indirect processes in many applications, a systematic study of the convergence of the RMPS method with respect to the number of target states, pseudostates and autoionizing states included in expansions (6) and (21) for electron scattering by C3+ and more complex ions would be of considerable interest. 3.2. D ISTORTED WAVE AND B ORN -S ERIES M ETHODS The pseudostate methods discussed in Section 3.1 have been successfully applied to a wide range of electron-atom and electron–ion scattering processes yielding accurate excitation and ionization cross sections at intermediate energies. These methods have also been extended enabling accurate atomic photoionization, electron molecule scattering, positron atom scattering and multiphoton processes to be calculated at intermediate energies. However, as the incident electron energy increases the number of pseudostates that need to be included in expansions (6) and (21) to yield reliable results also increases. Since the resultant computing requirements increase as the cube of the number of target eigenstates and pseudostates included in expansions (6) and (21) then these requirements can quickly become excessive. On the other hand, if the energies of the incident and scattered electron are well above the ionization threshold, then, in the absence of resonance capture, the interaction time of this electron with the target is short. In this case, the interaction of
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this electron with the target can often be reliably treated using distorted waves or Born-series expansions rather than by the non-perturbative approach of expanding in pseudostates. As an example, we consider the excitation-autoionization process illustrated by Eq. (42). If the energies of the incident and scattered electrons are well above the energy necessary to excite the autoionizing state C3+∗∗ (1sn n ), then their wave functions can often be accurately represented by distorted wave or Born series approximations. However, the slower electron emitted in the decay of this autoionizing state will usually be strongly coupled to the residual ion and this interaction will need to be treated non-perturbatively using a close coupling or R-matrix expansion, possibly including pseudostates. This is the basis of the Distorted-Wave R-matrix (DWRM) method introduced by Bartschat and Burke (1987) which extended the Coulomb–Born exchange and distortedwave exchange approximations using close-coupling wave functions considered by Jacubowicz and Moores (1981). In the following discussion we will describe the approach developed by Bartschat and Burke and we will then mention further developments which have been made to extend its range of validity. We consider the following electron atom or electron ion ionization process (q+1)+ q+ e− (k0 ) + Ai → e− (k1 ) + Af (45) + e− (k2 ) , where we make the following approximations: (i) the fast incident and scattered electrons e− (k0 ) and e− (k1 ) with momenta k0 and k1 are described by distorted waves; q+ (ii) both the initial bound state of the target Ai and the final continuum state of (q+1)+ + e− (k2 )] containing N + 1 electrons are described by the target [Af R-matrix expansions analogous to Eqs. (6) and (21); (iii) only the Coulomb interaction between the fast incident and scattered electrons and the N + 1 target electrons are retained in the calculation; (iv) exchange interactions between the fast incident and scattered electrons and the N + 1 target electrons are neglected; (v) relativistic effects are neglected, although their inclusion using the Breit– Pauli Hamiltonian would be straightforward. The scattering amplitude describing the ionization process (45) is then given in the distorted wave approximation (see, for example, Madison et al., 1977) f (k0 , k1 , k2 ) = −(2π)−5/2 Ψf−E (k2 ; XN+1 )φ1− (k1 ; xN+2 ) × V (XN+1 , xN+2 )Ψi (XN+1 )φ0+ (k0 ; xN+2 ) , (46) where, as in Eq. (6), the space and spin coordinates of the ith electron are denoted by xi ≡ (ri σi ) and where XN+1 = x1 , . . . , xN+1 . Also V (XN+1 , xN+2 ) is the Coulomb interaction between the incident and scattered electrons and the target
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electrons defined by V (XN+1 , xN+2 ) =
N+1 i=1
1 , |ri − rN+2 |
(47)
and the integration in the matrix element (46) goes over the space and spin coordinates of all N + 2 electrons. We now consider the approximations made for the initial and final state wave functions in Eq. (46). The distorted wave describing the incident and scattered electrons in Eq. (46) are expanded in partial waves as 4π ∗ i exp(iδ )Y m (θk0 , φk0 )Y m (θ, φ) φ0+ (k0 ; x) = √ k0 r m
× Pk0 (r)χ 1 m0 (σ ), 2
(48)
and 4π ∗ i exp(−iδ )Y m (θk1 , φk1 )Y m (θ, φ) φ1− (k1 ; x) = √ k1 r m
× Pk1 (r)χ 1 m1 (σ ), 2
(49)
where χ 1 m0 (σ ) and χ 1 m1 (σ ) are the spin functions of the incident and scattered 2 2 electrons and Y m (θ, φ) are spherical harmonics. The reduced radial functions Pk (r) are usually taken to satisfy a differential equation of the form
( + 1) d2 2 − + 2U (r) + k Pk (r) = 0, dr 2 r2
(50)
subject to the boundary conditions Pk (0) = 0,
Pk (r) ∼ k −1/2 sin kr − 12 π + δk , r→∞
(51)
where we have assumed that the target is neutral. If it is charged then the usual Coulomb phase and logarithmic terms would have to be included in Eqs. (48), (49) and (51). Finally, U (r) in Eq. (50) is a model potential which represents the charge distribution of the target. We consider next the calculation of the initial bound state Ψi and the final continuum state Ψf−E in Eq. (46). Both of these states are represented in the internal region by expansions over the basis ψkLSπ defined by Eq. (6). We consider first
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the expansion of the initial bound state wave function Ψi . We rewrite Eq. (10) in this case as 1 |ψk |ψk Aki , ψk |LN+1 |Ψi = |Ψi = (52) Ek − E k
k
where we have expanded the Green’s function in Eq. (10) in terms of the basis ψk defined by Eqs. (6) and (7). Substituting for the Bloch operator LN+1 defined by Eq. (8) and using Eqs. (14) and (21), then yields the following expression for the expansion coefficients Aki in Eq. (52) dFj i 1 wj k a0 . − b 0 Fj i Aki = (53) 2a0 (Ek − E) dr r=a0 j
In this equation Fj i is the j th component of the reduced radial wave function describing the motion of the (N + 1)th electron in the external and asymptotic regions where r a0 . We can determine Fj i and dFj i / dr at r = a0 by integrating the coupled second-order differential equations (22) from r = a0 to r = ap using a propagator method for the wave function, as discussed in Section 2.1, and fitting on the boundary r = ap to a decaying asymptotic wave solution Fj i (r) ∼ Nj i exp(−κj i r), r→∞
all j.
(54)
Since all the channels are closed, the wave number kj i of the (N + 1)th electron in the j th channel satisfies kj2i = −κj2i or kj i = iκj i . Also Nj i in Eq. (54) are normalization factors. The boundary condition given by Eq. (54) is achieved by iteratively varying the total energy E and resolving the coupled second-order differential equations (22) in the external and asymptotic region equations until a continuous solution, decaying asymptotically is obtained. This yields the energy eigenvalue of the initial bound state. The corresponding wave function is obtained by normalizing the total wave function in the internal, external and asymptotic regions to unity so that Ψi , Ψi = δii ,
(55)
which yields the normalization factor Nj i in Eq. (54). We then consider the expansion of the final continuum state wave function Ψf−E in Eq. (46). We also expand this wave function in the basis ψkLSπ defined by Eq. (6). We now write |Ψf−E = (56) |ψk A− kf , k
Lπ
for each continuum state energy E, where the summation over the total orbital angular momentum L and parity π goes over all values which give a significant
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contribution to the scattering amplitude defined by Eq. (46). We note that there is no summation over the total spin angular momentum S since this quantity is conserved in the non-relativistic approximation considered here. The expansion coefficients A− kf can be determined for each L, S and π in the same way as those in Eq. (52) giving − dFjf 1 − = w F . − b a A− (57) jk 0 0 jf kf 2a0 (Ek − E) dr r=a0 j
− − Again we determine Fjf and dFjf /dr at r = a0 by integrating Eq. (22) from r = a0 to r = ap and fitting to an asymptotic boundary condition analogous to Eq. (30) which can be written in matrix notation as 1/2 2 (sin θ + cos θK)(I + iK)−1 , F− (r) ∼ (58) r→∞ πk
where K is the usual K-matrix. The total wave function then satisfies the required normalization condition Ψf−E |Ψf− E = δff δ E − E , (59) and ingoing wave boundary condition Ψf−E
∼
rN+1 →∞
ing
Ψfinc E + Ψf E ,
(60)
corresponding to a Coulomb modified incident plane wave plus ingoing waves in all channels. An R-matrix computer program package RMATRX-ION, which implements the above theory for a general atom or ion, was developed by Bartschat (1993) based on the non-relativistic R-matrix program of Berrington et al. (1974, 1978) and the Breit–Pauli R-matrix program of Scott and Taylor (1982), discussed in Section 2.2, although the relativistic options in the latter program were not included. This program has been used to calculate electron impact ionization cross sections for a number of atoms and ions, including He, Ar, Cr, Ne6+ and Ar9+ (Bartschat and Burke, 1988; Bartschat et al., 1990; Reid et al., 1992; Raeker et al., 1994; Laghdas et al., 1995, 1999). The computer program RMATRX-ION was extended by Schweinhorst et al. (1995) to enable double-differential cross sections (DDCS) and triple-differential cross sections (TDCS) to be calculated. They then used this program to calculate DDCS and TDCS for helium, which they compared with experiment. The experimental data for DDCS were well reproduced by the theory at incident electron energies of 200 eV and 300 eV. However, a more detailed comparison with TDCS data showed that higher-order effects should be included to further improve the theoretical model. This conclusion is consistent with studies of TDCS for ionization of helium by Byron et al. (1986) and other workers that has demonstrated
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the necessity of including second Born approximation terms to account for details of the observations. In order to explore this possibility further, Reid et al. (1998, 2000) included second-order effects in the DWRM method by replacing V in the expression (46) for the scattering amplitude by V + V lim (E − H0 + iη)−1 V , η→0+
(61)
where H0 + V is the Hamiltonian for the entire N + 2 electron system. In order to calculate the contribution from the second-order term in Eq. (61) several approximations were made, including restricting the evaluation of the additional matrix elements to the internal region. The inclusion of second-order effects in this way improved the overall agreement between the experimental data for helium and theoretical predictions. However, the results for the TDCS were found to be sensitive to the representation of the initial bound state. A further important extension of the theory was made by Fang and Bartschat (2001a, 2001b, 2001c) who replaced the distorted waves used by Reid et al. (1998, 2000) by plane waves in a second-Born R-matrix with pseudostates (RMPS) theory. This enabled analytical simplifications to be made in the theory and, as a result, it was not only possible to perform calculations at higher incident electron energies, where a partial wave expansion converges very slowly, but it also enabled a larger number of coupled target eigenstates and pseudostates to be included in the R-matrix expansion. As an example, we consider calculations by Fang and Bartschat (2001a) for the following ionization–excitation process in He: e− + He 1s2 1 S → e− + He+ (2s, 2p) + e− , (62) where the He+ ion is left in a 2s or 2p state. In this calculation the incident and scattered electron were described using the second-Born approximation, while the He bound state and the final He+ excited state plus ejected electron were described by a 23-state RMPS expansion which included the first 6 1s, 2s, 2p, 3s, 3p and 3d eigenstates, together with up to 17 pseudostates of He+ . We compare these calculations with experiments of Avaldi et al. (1998) in Fig. 13. We see that the 23-state and 12-state second-Born RMPS results are in good agreement with the experiment. However the 6-state second-Born R-matrix results, which only include the lowest 6 eigenstates in the R-matrix expansion, show a deviation at small ejected angles indicating lack of convergence of the bound and continuum states in this case. We conclude that the 23-state RMPS results are converged in the description of the He bound and continuum states and the second-Born approximation gives an accurate representation of the incident and scattered electrons at this energy. Finally we mention recent work by Bartschat and Grum-Grzhiamailo (2002) in which simultaneous electron impact ionization-excitation is reformulated in
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(a)
(b) F IG . 13. In-plane angular distribution of the ejected electron for ionization-excitation to the He+ n = 2 states. The experimental data of Avaldi et al. (1998) for an incident electron energy of 1585 eV, final electron energies of 1500 eV and 20 eV and a detection angle of 4◦ for the fast electron are compared with results from several models. (a) Second-Born results with: 23-state RMPS model, full curve; 12-state RMPS model, broken curve; 6-state eigenstate model, chain curve. (b) Second-Born RMPS 23-state model for intermediate-state energies of the fast electron: full curve, 1543 eV; chain curve, 1521 eV; broken curve, 1565 eV. These energies correspond to the geometric mean of the initial and final energies as well as one value closer to each of these energies.
terms of irreducible tensors and benchmark calculations carried out for e− –He scattering where the decay photon from the He+ (2p 2 Po ) state is observed in triple coincidence with the two outgoing electrons. Also Andersen and Bartschat (2004) have considered the dipole polarization of a coherently excited Stark manifold for the simplest case of ionization-excitation of the He+ (2s, 2p) manifold in e− –He scattering. Clearly the distorted wave R-matrix method and the associated secondBorn RMPS method have opened up an important new area of study in electron atom and electron ion scattering at intermediate energies.
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4. Atomic Photoionization and Photorecombination 4.1. P HOTOIONIZATION In this section we consider the extension of R-matrix theory to atomic photoionization given by − hν + Ai → A+ j +e ,
(63)
where Ai is an atom in state i and A+ j is the residual positive ion in state j . The differential cross section for photoionization can be written in atomic units, either in the dipole length form dσijL
2 = 4π 2 αa02 ωΨf−E |ˆ .DL |Ψi ,
(64)
dΩ or in the dipole velocity form
4π 2 αa02 − Ψ |ˆ .DV |Ψi 2 , (65) fE dΩ ω where the dipole length and dipole velocity operators DL and DV are defined by dσijV
=
DL =
N+1 i=1
ri ,
DV =
N+1
∇i .
(66)
i=1
In these equations, ω is the incident photon energy in atomic units, α is the fine structure constant, a0 is the Bohr radius of the hydrogen atom and the initial bound state Ψi and the final continuum state Ψf−E are assumed to contain N +1 electrons. Also in equations (64) and (65), Ψi and Ψf−E satisfy the normalization conditions given by Eqs. (55) and (59), respectively. Chandrasekhar (1945) introduced a further form for the photoionization cross section using the dipole acceleration operator which emphasizes the wave function close to the nucleus. However, the dipole acceleration form usually gives poorer results when approximate wave functions are used and we will therefore not discuss it further. The extension of R-matrix theory of electron atom scattering, and the corresponding computer program, to treat atomic photoionization was first made by Burke and Taylor (1975), and developed further by Berrington et al. (1978, 1987). In current applications of this theory, both the initial bound state and the final continuum state wave functions are represented in the internal region by expansion (6). Following our discussion of the distorted wave R-matrix method in Section 3.2 (see Eqs. (52) and (56)), we rewrite Eq. (10) defining the initial and final state wave functions in the internal region as follows: |Ψi = (67) |ψk Aki , |Ψf−E = |ψk A− kf , k
k
Lπ
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where the basis functions ψk are defined by Eqs. (6) and (7). In non-relativistic R-matrix theory considered here, the initial bound state Ψi is an eigenstate of the total orbital angular momentum L, the total spin angular momentum S and the parity π. Also the final continuum state Ψf−E involves a summation over all L and π values which give a significant contribution to the photoionization cross section defined by Eqs. (64) and (65), where the total spin S is conserved by the dipole operators in these equations. The expansion coefficients Aki and A− kf in Eqs. (67) are defined, as described in Section 3.2, in terms of the surface amplitudes wik and the reduced radial wave − functions Fj i (a0 ) and Fjf (a0 ) and their derivatives on the boundary of the internal region, by Eqs. (53) and (57). The reduced radial wave functions are then determined by integrating the corresponding coupled second-order differential equations (22) in the external region, as discussed in Section 2.1, and fitting on the outer boundary r = ap to a decaying asymptotic wave solution (54) for Aki , and to a Coulomb modified plane wave plus ingoing wave solution (58) for A− kf . In this way both the initial bound state Ψi and the final continuum state Ψf−E can be calculated in the internal, external and asymptotic regions, and hence the photoionization cross section defined by Eqs. (64) and (65) determined. In recent years many R-matrix calculations have been carried out in support of experiments at synchrotron radiation facilities. As an example, we mention photoelectron measurements of hollow-atom–hollow-ion decay paths of triply excited lithium atoms, carried out at the Advanced Light Source at Berkeley by Diehl et al. (1997). In this experiment the following resonant two-step decay process was observed hν + Li 1s2 2s 2 S → Li∗∗∗ n n , n ↓
Li+∗∗ n n + e−
(68)
↓ Li2+ (1s) + 2e− . Observing the secondary Li+ Auger decays as a function of the incident photon energy was shown to be a valuable technique for the detection of new triply excited resonances of the parent lithium ion. In the R-matrix calculation, 29 Li+ states were included in expansions (6) and (21) consisting of 19 Li+ ground and singly excited states and 10 Li+ doubly excited hollow-ion states. We compare in Fig. 14 the experimental partial cross sections to the Li+ 2s2 , 2s2p 3 P, 2s2p 1 P states and their sum with the corresponding ab initio R-matrix calculation predictions for incident photons in the energy range 160.1–163.6 eV. The R-matrix resonance predictions show a small systematic energy shift from the experimental values, but the number and relative magnitudes of the resonances and even
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F IG . 14. Comparison of the experimental partial photoionization cross section for lithium atoms to (a) 2s2 , (b) 2s2p 3 P, (c) 2s2p 1 P and (d) their sum, with R-matrix calculations for photons in the energy range 160.1–163.6 eV. The theoretical curves are convoluted with the experimental bandpass of 0.11 eV. The vertical scale on the right-hand side gives the theoretical cross section in MBarn (×10−3 ) whereas the scales on the left-hand side provide relative experimental values only.
small spectral features are very well reproduced by the theory. Overall the good agreement between theory and experiment has enabled a clear understanding of the excitation and decay mechanisms summarized in Eq. (68) to be obtained. We conclude our discussion of atomic photoionization by mentioning the international “Opacity Project” led by M.J. Seaton (Seaton, 1987). This has involved research workers from about a dozen groups in Europe and North and South America which have collaborated for over 20 years in the calculation of boundbound, bound-free and free-free transitions, using atomic structure and R-matrix photoionization programs (Berrington et al., 1987). Calculations have been carried out for all transitions and ionizations stages of the 17 elements H, He, C, N, O, Ne, Na, Mg, Al, Si, S, Ar, Ca, Cr, Mn, Fe, Ni which provide a significant contribution to stellar opacities. This work required further developments of the
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F IG . 15. Calculated photoionization cross sections (in Mb) from (a) 3p6 3d4 p 1 Po and (b) 3p5 (2 Po )3d3 (2 Db ) 1 Po states of Fe6+ .
R-matrix program and the establishment of a data base “Topbase” to store the enormous amount of calculated data (Cunto et al., 1993). As one example of these calculations, we briefly discuss the work of Saraph et al. (1992) who carried out detailed calculations for radiative transitions in Fe6+ and Fe7+ . We show in Fig. 15 their results for the following two transitions in Fe6+ : hν + Fe6+ 3p6 3d4p 1 Po → Fe7+ + e− , 2 2 hν + Fe6+ 3p5 Po 3d3 Db 1 Po → Fe7+ + e− ,
(69)
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where 31 states of Fe7+ with configurations 3p6 3d, 3p5 3d2 , 3s3p6 3d2 , 3p6 4s, 3p6 4p, 3p6 4d and 3p6 4f, were retained in expansions (6) and (21), representing the initial bound and final continuum states of Fe6+ . The cross section in Fig. 15(a) is strongly enhanced near 5.8 Rydbergs by PEC (photoexcitation of the core) resonances with configurations 3p5 3d2 4p and 3p6 4p2 . In general, the photoionization cross sections of these and similar ions are dominated by resonances at low energies which must be accurately represented in order to determine reliable opacities. Recent developments of the “Opacity Project” work have been reviewed by Seaton (2005) and Badnell et al. (2005), and its importance in our understanding of elemental abundances in the sun has been discussed by Bahcall (2005).
4.2. P HOTORECOMBINATION AND R ADIATION DAMPING In this section we summarize recent developments in R-matrix theory to treat photorecombination and radiation damping. In photorecombination the electron is accelerated by a charged atomic ion and, as a result, emits a photon of sufficient energy that the electron is captured into a bound state. This is an important cooling mechanism in hot plasmas and is also used as a plasma diagnostic (e.g., Summers, 1994). In the absence of radiation damping, which we discuss below, the cross section for photorecombination σP R is related to the cross section for photoionization σP I through detailed balance given by the Milne relation (e.g., Osterbrock, 1989) σ P R = σP I
2 2 gi α Eph , gj 4(Eph − I )
(70)
in atomic units. In this equation gi and gj are the statistical weights of the initial and residual ions respectively, Eph and I are the photon energy and ionization potential of the final ionic state respectively and α is the fine structure constant. Photorecombination can proceed non-resonantly as follows: e− + An+ i ↔ Ak
(n−1)+
+ hν,
(71)
which is called radiative recombination (RR). It can also proceed resonantly through an intermediate doubly excited state as follows: (n−1)+ ∗∗ (n−1)+ ↔ Ak + hν, e− + An+ (72) i ↔ Al which is called dielectronic recombination (DR). In both cases, photorecombination occurs in competition with electron ion scattering. In addition, the emitted photon can be reabsorbed by the ion, giving rise to modification of the electron ion scattering amplitude, called radiation damping. We also mention that a photon
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can be emitted leaving behind an electron–ion scattering state as follows: n+ − e− + An+ i ↔ Aj + e + hν,
(73)
which is known as free–free scattering or bremsstrahlung. Dielectronic recombination was first considered theoretically by Massey and Bates (1942) and Bates and Massey (1943), who were concerned with recombination processes that occur in the earth’s ionosphere. This process was investigated further by Bates (1962), Bates and Dalgarno (1962) and Seaton (1962). Also, Burgess (1964, 1965) showed that DR can give an increase of one to two orders of magnitude over the RR process at temperatures found in the solar corona, where complete Rydberg series of resonances contribute to recombination. An interesting review of this early work has been written by Seaton and Storey (1976). The role of radiation damping in electron ion scattering, as well as in photoionization and photorecombination was analyzed by Davies and Seaton (1969). Later, an R-matrix theory of free-free scattering was developed by Bell et al. (1977) and a general theory of dielectronic recombination using MQDT, enabling complete Rydberg series of resonances to be accurately treated, was developed by Bell and Seaton (1985) and first applied by Pradhan and Seaton (1985). In recent years, the R-matrix program developed for photoionization calculations, mentioned in Section 4.1, has been extended and applied to a wide range of photorecombination and radiation damping studies. In an early application, Sakimoto et al. (1990) used this program to show the importance of radiation damping in electron scattering by highly charged ions. Further work using the R-matrix approach to photorecombination and radiation damping has been carried out by Terao and Burke (1990), Terao et al. (1991), Nahar and Pradhan (1994, 1995), Zhang and Pradhan (1995, 1997), Pradhan and Zhang (1997) and Zhang et al. (1999, 2001). In addition, Robicheaux et al. (1995) developed an important new approach for including radiation damping in the close coupling equations describing electron ion scattering. In this approach they defined an additional non-local imaginary potential which gives rise to absorption from the electron ion channels into photorecombination. These workers implemented this approach in the R-matrix computer program, where they used MQDT to treat Rydberg series of resonances, and radiation damping calculations have been carried out, for example, by Gorczyca et al. (1995, 1996, 2002), Gorczyca and Badnell (1996, 1997b) and Badnell et al. (1998). Photorecombination and radiation damping in low energy electron scattering by an N electron atom or ion with nuclear charge Z is described by the timedependent Schrödinger equation
∂ HN+1 + Hrad (t) Ψ (XN+1 , t) = i Ψ (XN+1 , t). ∂t
(74)
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In this equation HN+1 is the N +1 electron Hamiltonian defined by Eq. (5), where we may retain terms from the relativistic Breit–Pauli Hamiltonian for higher Z targets. Also Hrad (t) is the interaction with the radiation field, which can be written in the length gauge as ∞ 2ω3 α 3 1/2 Hrad (t) = Rμ e−iωt + Rμ∗ eiωt dω, 3π μ
(75)
0
in atomic units (Bell and Seaton, 1985). The first term in Eq. (75) corresponds to absorption (annihilation) and the second term to emission (creation) of a photon with angular frequency ω and polarization μ. Also in Eq. (75) Rμ =
N+1
(76)
riμ ,
i=1
where riμ are the spherical tensor components of the position coordinates of the ith electron defined by 1 ri−1 = √ (xi − iyi ), 2
ri0 = z,
1 ri+1 = − √ (xi + iyi ). 2
(77)
We now consider the solution of Eq. (74) corresponding to an element Vμ (t) of the interaction with the radiation field Hrad (t), defined by Vμ (t) = Dμ e−iωt + Dμ∗ eiωt ,
(78)
with Dμ =
2ω3 α 3 3π
1/2 (79)
Rμ .
We expand the time-dependent wave function Ψ (XN+1 , t) in Eq. (74) as follows: Ψ (XN+1 , t) = e−iEt
0
e−inωt ψn (XN+1 ),
(80)
n=−1
which we will see in Section 7.1 corresponds to two terms in the Floquet–Fourier expansion (109). In the present context ψ0 corresponds to electron ion scattering in the initial state in the absence of a photon in Eqs. (71)–(73), and ψ−1 corresponds to the final state in these equations with one emitted photon. Substituting Eq. (80) into Eq. (74) and equating coefficients of exp(−i(E+nω)t) to zero yields the following coupled equations (HN+1 − E + ω)ψ−1 + Dμ∗ (ω)ψ0 = 0,
(81)
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and Dμ (ω)ψ−1 + (HN+1 − E)ψ0 = 0.
(82)
If we consider processes represented by Eqs. (71) and (72), where ψ−1 corresponds to a bound state of the electron ion system, then we can substitute the solution of Eq. (81) for ψ−1 into Eq. (82) yielding the equation
∞ Dμ (ω)|ψb ψb |Dμ∗ (ω) dω − E ψ0 = 0, HN+1 + lim (83) δ→0 E − Eb − ω ± iδ μb 0
where Eb = ψ−1 |HN+1 |ψ−1 ,
(84)
and where we have summed over the photon polarization direction μ and over the bound states and integrated over the photon energy ω in Eq. (75). The real part of the integral over ω in the second term in Eq. (83) diverges. However, after mass renormalization using quantum electrodynamics (e.g., Jauch and Rohrlich, 1955) this yields an energy shift, analogous to the Lamb shift, which is small for most systems of interest. We will therefore not consider this shift further. The remaining imaginary part of the integral corresponds to radiation damping in the electron ion scattering process. After choosing the appropriate sign in the denominator we obtain immediately an additional potential contribution arising from this term given by Vrad = −iπ (85) Dμ (ωb )|ψb ψb |Dμ∗ (ωb ), μb
where ωb = E − E b ,
(86)
is the energy difference between the electron ion scattering state and the bound state. Equation (85) defines a non-local, imaginary radiation damping potential in the length gauge. This potential, which is antihermitian and thus gives rise to absorption, was first derived in the velocity gauge by Robicheaux et al. (1995). As pointed out by Robicheaux (1998), it corresponds to the additional complex term that Hickman (1984) added to the electron ion interaction potential in order to describe dielectronic recombination. Equation (83), which now becomes (HN+1 + Vrad − E)ψ0 = 0,
(87)
can be solved using a straightforward modification of standard R-matrix theory to treat electron atom and electron ion scattering described in Section 2.1. The main
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modification arises since the radiation damping potential is complex symmetric and energy dependent. This energy dependence means that the usual efficient procedure for determining the R-matrix, where a single diagonalization of the Hamiltonian matrix, defined by Eq. (7), yields the R-matrix, defined by Eq. (12) at all energies, must be modified. As a result, a further matrix, corresponding to the radiation damping potential, must be diagonalized at each energy. However, the radiation damping potential matrix is of low rank compared with the Hamiltonian matrix. Hence the dimension of the additional matrix which must be diagonalized at each energy is much less than the dimension of the Hamiltonian matrix, so that the additional computational effort is not excessive. Also, the extension of MQDT to treat this problem, mentioned earlier, greatly reduces the number of energy values where detailed R-matrix calculations need to be carried out. As a result of the antihermitian properties of the radiation damping potential, the K-matrix is no longer real and the S-matrix is no longer unitary. Flux will therefore be lost from the electron ion scattering channels into photorecombination. The photorecombination cross section for each J π combination then becomes
nt na π 2J + 1 ∗ Sj i S j i , 1− σP R = 2 (88) ki 2(2Ji + 1 i=1
j =1
where na is the number of open channels at the energy under consideration and nt is the number of channels coupled to the target state. We also mention here a similar optical potential R-matrix approach, introduced by Gorczyca and Robicheaux (1999) to treat photoionization in the vicinity of inner shell excited resonance series. In this work they considered the 2p−1 ns(nd) inner shell photoexcited resonance states in argon. Following inner shell photoexcitation of the argon ground state as follows: hν + Ar 2p6 3s2 3p6 1 S → Ar∗ 2p5 3s2 3p6 ns, nd 1 Po , (89) there are two competing Auger decay routes. First there is participator Auger decay Ar 2p5 3s2 3p6 ns, nd 1 Po → Ar+ 2p6 3s2 3p5 2 Po + e− → Ar+ 2p6 3s3p6 2 S + e− , (90) where the valence electron ns or nd takes part in the autoionization process. Then there is the more important spectator Auger decay Ar 2p5 3s2 3p6 ns, nd 1 Po → Ar+ 2p6 3s2 3p4 ns, nd + e− → Ar+ 2p6 3s3p5 ns, nd + e− → Ar+ 2p6 3p6 ns, nd + e− . (91)
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We see that the spectator Auger decay route gives rise to an infinite number of channels, which clearly cannot be explicitly included in the R-matrix expansion. The standard R-matrix method has therefore been extended by the inclusion of an optical potential which allows implicitly for these channels, based on similar ideas to the radiation damping potential introduced by Robichaux et al. (1995). As well as argon other applications include inner shell photoexcited resonances in neon by Gorczyca (2000) and in atomic oxygen by Gorczyca and McLaughlin (2000). This optical potential approach has also been applied to K-shell photodetachment of Li− by Gorczyca et al. (2003). As an example of the above theory we consider the effect of radiation damping on electron impact excitation of the n = 2 levels of the hydrogen-like ion Fe XXVI. Calculations carried out by Gorczyca and Badnell (1996) used an R-matrix program, which included the one-body terms from the Breit–Pauli Hamiltonian (Scott and Taylor, 1982; Berrington et al., 1995), which had been modified to include the radiation damping potential Vrad defined by Eq. (85). In these calculations all nine target states up to n = 3 were included in the R-matrix expansion and excitation cross sections from the ground state to the 2s1/2 2p1/2 and 2p2/3 states were calculated, both including and omitting the radiation damping potential. The resultant cross sections are given in Fig. 16. We see the important role that damping plays in these excitation cross sections. Thus the KLn resonances, primarily the 2pn (n 20) ones, just above the 2s1/2 and 2p1/2 thresholds are completely damped, as are the KMn resonances for n 6 (E 590 Ryd). Also we see that even the KMM resonances at E ∼ 535 Ryd show approximately 10% damping. Overall these results show the importance of radiation damping in highly ionized ions even for low-lying resonances. As a second example, we consider electron–ion photorecombination calculations carried out by Zhang et al. (1999). In this work results were obtained for C V, C VI, O VIII and Fe XXV, using both non-relativistic and relativistic R-matrix programs, where in the latter calculations one-body terms from the Breit–Pauli Hamiltonian were retained (Scott and Taylor, 1982; Berrington et al., 1995). This work was based on the theory developed by Bell and Seaton (1985) where recombination to high n states used MQDT (Nahar and Pradhan, 1994). In these calculations all target states up to and including n = 3 were included in the R-matrix expansion. We illustrate this work by showing in Fig. 17 recombination cross sections for the H-like ion O VIII, where experimental results have been reported by Kilgus et al. (1990), using the heavy-ion Test Storage Ring in Heidelberg. We see by comparing Figs. 17(a) and (b) that inclusion of relativistic effects introduces additional fine-structure resonances and modifies the cross sections. Also the effect of including radiation damping can be seen, by comparing Figs. 17(b) and (c), to significantly reduce the photorecombination cross sections. The resultant photorecombination cross section is found to be in good agreement with experiment.
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F IG . 16. R-matrix calculation for electron impact excitation to the (a) 2s1/2 (b) 2p1/2 and (c) 2p3/2 levels in Fe XXVI, convoluted with 2.5 Ryd FWHM Gaussian: dotted line, undamped; full line, damped.
Finally we mention a more recent study of electron ion photorecombination by Gorczyca et al. (2002). In this work, they performed radiation-damped R-matrix scattering calculations for photorecombination of electrons incident on the fluorine-like ion Fe17+ , of importance in the analysis of observations made by the X-ray satellites Chandra and XMM-Newton. While these calculations significantly improve earlier R-matrix calculations for this ion, a difficulty arises in the calculation of the following inner shell dielectronic recombination
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F IG . 17. Photorecombination cross sections for e− + O VIII → O VII + hν: (a) non-relativistic LS coupling calculations, (b) relativistic Breit–Pauli calculations without radiation damping, and (c) relativistic Breit–Pauli calculations with radiation damping.
process
e− + Fe17+ 2s2 2p5 2 Po3/2
Fe17+ 2s2 2p5 2 Po3/2 1/2 + e− ↔ Fe16+∗∗ 2s2p6 2 S1/2 n . Fe16+ 2s2 2p5 2 Po3/2 1/2 n + hν
(92)
We see that the outer n electron is a spectator in the inner shell process leading to the emitted photon. Hence, while the autoionizing width scales as n−3 , the core radiation width is independent of n. It follows that, for sufficiently high n, radiative decay is followed by autoionization which is not accurately predicted using this theory.
5. Electron Molecule Scattering The processes that occur in electron molecule scattering are considerably more varied than those that arise in electron scattering by atoms and atomic ions because of the possibility of exciting degrees of freedom associated with the motion
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F IG . 18. Molecular frame of reference for electron diatomic molecule scattering.
of the nuclei. Thus, as well as electronic excitation and ionization, vibrational and rotational excitation, dissociative attachment and dissociation can also occur, which in the case of diatomic molecules can be written as follows: ABi v j + e−
e− + ABivj →A + B − A + B + e−
electronic and vibronic excitation dissociative attachment
(93)
dissociation
A further distinctive feature of electron molecule scattering arises in the role resonances play in vibrational excitation, dissociative attachment and dissociation. This is because when the scattered electron energy is near a resonance the electron has a high probability of being captured into the resonance state where it has an enhanced probability of transferring energy to the nuclear motion. Also, from a computational point of view, the multicentre nature of the electron molecule interaction raises additional problems not found in electron atom scattering. R-matrix theory was first extended to treat electron molecule scattering in the fixed-nuclei approximation by Schneider (1975a, 1975b), Schneider and Hay (1976) and Burke et al. (1977). In this approximation, the nuclei are held fixed and the calculations are carried out in a molecular frame of reference which is rigidly attached to the molecule. In the case of diatomic molecules we introduce a frame of reference where the z-axis is chosen to lie along the internuclear axis as illustrated in Fig. 18. Also in this figure G is the centre of gravity of the two nuclei labelled A and B, which are fixed in space, R = RA + RB is the distance between the nuclei and the vector distances between A, B and G and the ith electron are rAi , rBi and ri , respectively. We also assume the target has N electrons and the nuclear charge numbers corresponding to A and B are ZA and ZB . For light molecules, where relativistic effects are negligible, we then have to solve the
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time-independent Schrödinger equation HN+1 Ψ = EΨ,
(94)
where HN+1 is the non-relativistic fixed-nuclei Hamiltonian, defined in atomic units by N+1 N+1 1 1 ZA ZB ZA ZB HN+1 = (95) − + , − ∇i2 − + 2 rAi rBi rij R i=1
i>j =1
where rij = |ri − rj |. Following our discussion of R-matrix theory of electron atom scattering in Section 2.1, we partition configuration space into three regions, as illustrated in Fig. 2, where the internal region is usually defined to be a sphere of radius r = a0 , centred on G which completely contains the target eigenstates and pseudostates of interest. In the internal region an expansion analogous to expansion (6) is adopted, where the channel functions Φ i are now formed from multicentre target eigenstates and pseudostates. In addition, the radial continuum basis orbitals uij , representing the scattered electron, are centred on the centre of gravity G and are non-vanishing on the boundary r = a0 . Multicentre bound state codes have been modified to calculate the matrix elements, analogous to those in equations (7), and the R-matrix at r = a0 , constructed from the radial continuum basis orbitals uij (a0 ) using Eqs. (12) and (14). A single-centre expansion centred on G, analogous to expansion (21), is adopted in the external region and the resultant coupled second-order differential equations (22) solved, as in electron atom scattering, to yield the R-matrix at r = ap . Finally, in the asymptotic region, the solution is fitted to an asymptotic expansion from which the K-matrix, S-matrix and cross sections for electronic transitions at fixed internuclear separation are calculated. The nuclear motion is included using non-adiabatic R-matrix theory, developed by Schneider et al. (1979). It is necessary to solve the following time-independent Schrödinger equation (HN+1 + TR − E)Ψ (XN+1 , R) = 0,
(96)
where HN+1 is the electronic Hamiltonian and TR is the nuclear kinetic energy operator. The partitioning of configuration space is extended to include the internuclear coordinates, as illustrated in Fig. 19 for a diatomic molecule with internuclear coordinate R, where we assume that the molecule does not rotate appreciably during the collision. The internal region is taken to be a rectangle defined by 0 r a0 and Ai R A0 , where a0 is defined as in the fixed-nuclei approximation, Ai is chosen to exclude the nuclear singularity at R = 0 and A0 is chosen to contain the target vibrational states of interest. For r > a0 the molecule separates into an electron plus a molecule, which may be vibrationally and electronically excited. For R > A0 the molecule separates into an atom plus a
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F IG . 19. Partitioning of configuration space in non-adiabatic R-matrix theory.
negative ion or into two atoms corresponding to dissociative attachment or dissociation. In the internal region, the total wave function in Eq. (96) is expanded in the following basis ψi (XN+1 , R)ηj (R)cij k , Ψk (XN+1 , R) = (97) ij
where ψi (XN+1 , R) are the fixed-nuclei internal region basis functions, calculated as described above for a range of R values satisfying Ai R A0 , and ηj (R) are basis functions representing the nuclear motion which are orthonormal over this range of R. The operator HN+1 + TR is diagonalized in the internal region in Fig. 19 in this basis including appropriate Bloch operators on the boundaries r = a0 and R = A0 . In this way the R-matrix on these boundaries is calculated, providing the boundary conditions for integrating solutions outwards from r = a0 and R = A0 yielding the K-matrix, S-matrix cross sections for vibrational and electronic excitation and for dissociative attachment and dissociation. Over the last twenty years general R-matrix computer program packages have been developed, both for electron scattering by diatomic molecules (e.g., Gillan et al., 1987) and for electron scattering by polyatomic molecules (e.g., Pfingst et al., 1994; Morgan et al., 1997). We conclude this section by presenting results from two of many recent electron molecule R-matrix scattering calculations. We consider first low-energy electron scattering by oxygen molecules which is of importance in understanding the physics of a wide range of processes occurring in the upper atmosphere, gaseous discharges and laboratory plasmas. We give in Fig. 20 the low-energy potential energy curves of O2 and the corresponding resonance curves of O− 2 which control these low-energy processes. The nine O2 potential energy curves shown in this figure have the assignments 1πu4 1πg2
X 3 Σg− ,
a 1 Δg ,
b 1 Σg+ ,
1πu3 1πg3
c 1 Σu− ,
C 3 Δu ,
A3 Σu+ ,
B 3 Σu− ,
1
Δu ,
1
Σu+ ,
(98)
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F IG . 20. Calculated potential energy curves for O2 target states (solid curves) and O− 2 resonances and bound state (dotted curves), in Rydbergs.
and the four O− 2 resonance curves have the assignments 1πu4 1πg3 2 $g ,
1πu3 1πg4 2 $u ,
1πu4 1πg2 3 Σg− 3σu 2 Σu− .
1πu4 1πg2 3 Σg− 3σu 4 Σu− ,
(99)
R-matrix calculations, carried out by Noble and Burke (1992) and Higgins et al. (1994), including the nine target states defined by Eq. (98) in the R-matrix expansion, found that the X 3 Σg− → a 1 Δg and X 3 Σg− → b 1 Σg+ electronic excitation cross sections were dominated by the 2 $g and 2 $u resonances from threshold to 15 eV. The calculated cross sections were in excellent agreement with experiments by Middleton et al. (1994). In addition, Noble et al. (1996) found that the 4 Σu− resonance, and to a lesser extent the 2 Σu− resonance, dominated the
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F IG . 21. Total vibrational excitation cross sections in 10−18 cm2 for e− −O2 scattering in the X 3 Σg− ground state. Solid line: R-matrix calculations including only the 4 Σu− symmetry; dotted line: including in addition the 2 Σu− symmetry; vertical lines including errors: experiments reported by Noble et al. (1996); crosses: experiments by Shyn and Sweeney (1993).
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F IG . 22. Total cross section for electron impact excitation of the first excited electronic state in H+3 at intermediate energies. Dark full line: 6-state calculation; light full line: 64-state calculation; dashed line: fit to 64-state calculation above the ionization threshold, denoted by I.T.
vibrational excitation cross sections in the X 3 Σg− ground state in the energy range 0–15 eV. We compare their calculations with experiments reported by Noble et al. (1996) and by Shyn and Sweeney (1993) in Fig. 21, where we see that there is good overall agreement between R-matrix calculations and experiment. We consider next a recent R-matrix calculation for electron scattering from the simplest polyatomic molecule H+3 by Gorfinkiel and Tennyson (2004). H+3 is particularly important as it is the dominant ion in low-temperature hydrogen plasmas. It also plays a fundamental role in interstellar chemistry and has been seen in planetary aurora. We show in Fig. 22 the total cross section for excitation of H+3 from the X 1 A1 ground state to the 3 Σ first excited state at intermediate energies. In this calculation two approximations were considered. The first included the six lowest target eigenstates in the R-matrix expansion and the second included a further 58 pseudostates giving a 64-state RMPS calculation. We see from Fig. 22 that the 6-state and the 64-state results are in reasonable agreement at energies below about 20 eV. However, at higher energies, above the highest eigenstate energy 20.78 eV included in the expansion, the RMPS calculation results in much reduced cross sections due to loss of flux into the pseudostate channels. We also observe that the 6-state calculation exhibits more pronounced pseudoresonance structure at intermediate energies than the 64-state calculation, which is closer to a fully converged solution. This calculation shows that cross sections for electron scattering from simple polyatomic molecules can now be reliably calculated both at low and at intermediate energies.
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6. Positron Atom Scattering In this section we review extensions made to R-matrix theory of electron atom scattering to enable positron atom scattering cross sections to be calculated. The following processes can occur in positron atom scattering
e+ + Ai
→ Ai + e+ → Aj + e+
elastic scattering excitation
− + → A+ j +e +e
ionization
→
Ps formation
→ → → →
A+ j + Ps − A2+ j + Ps − A2+ j + Ps + e − − A3+ j + Ps + e A+ j + γ rays
Ps− formation
(100)
transfer ionization transfer ionization with Ps− formation annihilation
where Ps (positronium) is a bound state of a positron and an electron. Only the first three of these processes have their analogue in electron scattering. Also we observe that while the positron annihilation process is sufficiently weak that it can be ignored in calculating the scattering cross sections, the annihilation rate, which is proportional to the probability of finding the positron and an electron at the same position, provides a critical test of the approximations made in the R-matrix calculations. Since the positron is distinguishable from the target electrons, we no longer have to antisymmetrize the total wave function with respect to interchange of the positron coordinates with those of the electrons. However, this simplification is balanced by the additional positronium atom formation channels that have to be included in the expansion of the total wave function in the internal and external regions. For example, in the case of positron scattering by hydrogen atoms, we introduce the Jacobi coordinates defined in Fig. 23. The R-matrix expansion in the internal region that replaces Eq. (6) then becomes ˆ R −1 vij (R)bij k , ψk = (101) Φ i (t; rˆ )r −1 uij (r)aij k + Θ i s; R ij
ij
where Φ i are channel functions formed from the hydrogen atom eigenstates and pseudostates, Θ i are channel functions formed from the positronium eigenstates and pseudostates and uij and vij are radial continuum basis functions representing the scattered positron and the scattered positronium atom, which are nonvanishing on the boundaries r = a0 and R = A0 of the internal region, illustrated in Fig. 24. We see that the internal region is two-dimensional, as in electron molecule scattering illustrated in Fig. 19. After diagonalizing the positron hydrogen atom Hamiltonian in the internal region, the R-matrix on the boundaries r = a0
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F IG . 23. Jacobi coordinate system for positron hydrogen atom scattering.
F IG . 24. Partitioning of configuration space in positron atom scattering.
and R = A0 can be calculated and the R-matrix in the external region propagated outwards yielding the K-matrix, S-matrix and cross sections for positron scattering and positronium formation. R-matrix computer programs have been developed which have enabled detailed calculations to be carried out for positron scattering by atomic hydrogen, by the “one-electron” alkali metal atoms Li, Na, K, Rb and Cs and by the “two-electron” atoms He, Mg, Ca and Zn (Walters, 1999; Walters et al., 2005). As an example, we show in Fig. 25 the results of R-matrix calculations for positron scattering by atomic hydrogen at low and intermediate energies obtained by Kernoghan et al. (1996) compared with experimental measurements by Jones et al. (1993) and
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F IG . 25. Positron scattering by atomic hydrogen: (a) total positronium formation cross section, (b) ionization cross section, (c) total cross section. Solid curve: 33-state calculation (Kernoghan et al., 1996); Points: experiment (Jones et al., 1993; Zhou et al., 1997).
Zhou et al. (1997). The curves in this figure were obtained from a 33-state RMPS calculation which included Ps(1s, 2s, 2p) and H(1s, 2s, 2p + 27 pseudostates). Excellent agreement with experiment is obtained for total positronium formation, for ionization and for the total cross sections for all energies up to 100 eV, showing this RMPS calculation provides an accurate representation of positron scattering by hydrogen atoms over a wide energy range. The emergence in recent years of the experimental capability to produce monoenergetic beams of positronium, for example by Garner et al. (1996, 2000) and Laricchia et al. (2004) has stimulated considerable interest in positronium scattering by atoms and molecules. Positronium exists in two states “ortho” and “para” where the electron and positron spin states are in triplet and singlet spin states respectively. The ortho (para) states decay into three (two) photons with life-times of 142 ns (0.125 ns). As a result, positronium beams consist of the longer lifetime ortho positronium. Recent R-matrix calculations have been carried out for positronium scattering by H (Campbell et al., 1998; Blackwood et al., 2002a, 2002b), and by He, Ne, Ar, Kr and Xe (Blackwood et al., 1999, 2002c). In these calculations pseudostates were used to represent the positronium continuum, but the inert gas targets were frozen in their ground state.
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F IG . 26. Fragmentation cross section for Ps–He scattering. Solid curve: R-matrix calculation by Blackwood et al. (1999); dashed curve: Born approximation calculation by Biswas and Adhikari (1999); solid circles: experiment by Armitage et al. (2002).
We conclude this section by considering the fragmentation (break-up) of positronium in collision with He atoms given by Ps + He → He + e− + e+ .
(102)
We compare in Fig. 26 the measurement of this process by Armitage et al. (2002) with R-matrix calculations by Blackwood et al. (1999) and Born approximation calculations by Biswas and Adhikari (1999). In the R-matrix calculation, the expansion over the target atom eigenstates included only the ground state of He, which was represented by a Hartree–Fock wave function, while the positronium expansion included the 1s, 2s and 2p eigenstates and up to 19 pseudostates, representing the highly excited and continuum states of positronium. In this way, we see from the comparison with experiment, shown in Fig. 26, that a good representation of the continuum, and hence of the fragmentation process, was obtained. However, it is also clear from this figure that the Born approximation, as expected, is not accurate at these relatively low energies. Finally, we mention that the R-matrix calculation also gives predictions for the total cross section, the elastic scattering cross section and the Ps (n = 2) excitation cross section (Walters et al., 2005). It was found that the calculated total cross section is slightly smaller than the measured total cross section of Garner et al. (1996, 2000) over the energy range from 15 to 40 eV and does not show the down-turn seen in the measurements at 10 eV. However, in spite of these discrepancies, the overall representation of the positronium helium scattering process by R-matrix theory is very encouraging.
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7. Atomic and Molecular Multiphoton Processes The study of the interaction of intense laser fields with atoms, ions and molecules has attracted considerable attention in recent years. In particular, the availability of increasingly intense lasers has made possible the observation of a wide variety of multiphoton processes, including multiphoton ionization, harmonic generation and laser-assisted electron atom scattering (see for example Gavrila, 1992; Protopapas et al., 1997; Joachain et al., 2000). In discussing these processes we observe that the atomic unit of electric field strength experienced by an electron in the ground state of atomic hydrogen a ≈ 5.1×109 V cm−1 corresponds to a laser intensity Ia ≈ 3.6 × 1016 W cm−2 . Lasers delivering pulses with intensities very much larger than this are now available using the “chirped-pulse-amplification” (CPA) scheme, in which the laser pulse is stretched, amplified and then compressed (Strickland and Mourou, 1985). As a result, atoms in intense laser fields exhibit many new properties which require a fully non-perturbative approach going beyond the first-order perturbation theory treatment of atomic photoionization considered in Section 4.1.
7.1. ATOMIC R-M ATRIX -F LOQUET T HEORY We commence our discussion in this section by considering atomic R-matrix– Floquet (RMF) theory which, since its introduction by Burke et al. (1990, 1991) and Dörr et al. (1992), has been used to describe a wide range of atomic multiphoton processes. In particular, we are interested in the following atomic multiphoton processes: multiphoton ionization − nhν + Ai → A+ j +e ,
(103)
harmonic generation nhν + Ai → Ai + hν ,
(104)
where ν = nν, and laser-assisted electron atom scattering nhν + e− + Ai → Aj + e− .
(105)
We will consider processes where there is at most one ejected or scattered electron. However, the possibility of ejecting more than one electron can be treated by a straightforward extension of the RMPS method discussed in Section 3.1. In RMF theory the laser field is treated classically and is assumed to be monochromatic, monomode and spatially homogeneous. In applications of RMF theory considered so far, the laser field is also assumed to be linearly polarized. The corresponding vector potential describing the laser field is then given by A(t) = ˆ A0 sin ωt,
(106)
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where ˆ is a unit vector along the laser polarization direction, ω is its angular frequency and the electric field strength E0 = −ωA0 /c. Neglecting relativistic effects, the atom or atomic ion, which we assume has (N +1) electrons, is described in the presence of the laser field by the time-dependent Schrödinger equation 1 N +1 2 HN+1 + A(t).PN+1 + A (t) Ψ (XN+1 , t) c 2c2 ∂ = i Ψ (XN+1 , t), (107) ∂t where HN+1 is the non-relativistic Hamiltonian in the absence of the field, defined by Eq. (5) and where PN+1 =
N+1
pi
(108)
i=1
is the total electron momentum operator. Again, following our discussion of R-matrix theory of electron atom scattering in Section 2.1, we partition configuration space into three regions as illustrated in Fig. 2, where we use the same criteria for defining the boundaries a0 and ap between these regions. Since the vector potential A(t) defined by Eq. (106) has constant amplitude A0 and frequency ω we can expand the wave function in each of these regions by a Floquet–Fourier expansion (Floquet, 1883; Shirley, 1965) as follows: ∞ Ψ (XN+1 , t) = e−iEt (109) e−inωt ψn (XN+1 ). n=−∞
Substituting this expansion into Eq. (107) yields an infinite set of coupled timeindependent equations for the functions ψn . The R-matrices obtained by solving these equations are then matched on the boundaries a0 and ap between the regions. We now briefly consider the determination of the solution of these equations in each region. In the internal region it is convenient to use the length gauge where the laseratom interaction tends to zero at the origin. This is obtained by transforming Eq. (107) using the relation i Ψ (XN+1 , t) = exp − A(t).RN+1 ΨL (XN+1 , t), (110) c where RN+1 =
N+1
ri .
(111)
i=1
Substituting Eq. (110) into Eq. (107) and using Eq. (109) yields an infinite set of coupled time-independent equations given by
7]
R-MATRIX THEORY L L (HN+1 − E − nω)ψnL + DN+1 ψn−1 + ψn+1 = 0,
295 (112)
where 1 (113) E 0 ˆ .RN+1 . 2 We truncate Eqs. (112) to a finite number of Floquet components ψnL and expand each of these components in a configuration interaction basis analogous to Eq. (6). Following the analysis given in Section 2.1, we then determine the R-matrix in the length gauge which relates the reduced radial wave functions corresponding to the Floquet components to their derivatives on the boundary r = a0 . In the external region, we transform Eq. (107) to a mixed gauge where the ejected or scattered electron is described in the velocity gauge and the remaining N electrons are described in the length gauge, which avoids the divergence of the length gauge at large distances. The required transformation is given by DN+1 =
t i i Ψ (XN+1 , t) = exp − A(t).RN − 2 A2 (t ) dt ΨV (XN+1 , t), (114) c 2c where ri a0 ,
i = 1, . . . , N,
rN+1 a0 .
(115)
Substituting Eq. (114) into Eq. (107) and using Eq. (109) yields an infinite set of coupled time-independent equations given by V V + ψn+1 (HN+1 − EV − nω)ψnV + DN ψn−1 V V − ψn+1 = 0, + PN+1 ψn−1 (116) where A0 (117) ˆ .pN+1 . 2c Again, we truncate Eqs. (116) to a finite number of Floquet components ψnV and expand each of these components, as in Eq. (21). We then substitute these expansions into Eq. (116) and project onto the channel functions to yield a set of coupled second-order differential equations satisfied by the corresponding reduced radial wave functions. The R-matrix in the length gauge, obtained from the internal region solution, is transformed to the velocity gauge and propagated outwards across the p subregions from r = a0 to r = ap , where it is fitted to an asymptotic expansion. In the case of multiphoton ionization and harmonic generation (e.g., Gebarowski et al., 1997a, 1997b) the radial wave functions are fitted to Siegert (1939) outgoing wave boundary conditions, defined by PN+1 = i
Fi (r) = Ni exp(iki r),
all i,
(118)
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where ki are complex wave numbers of the ejected electron in the ith channel and where Ni are normalization factors. This boundary condition is achieved by iteratively varying the total energy E and resolving the external and asymptotic region equations until a convergent solution is obtained. In general the resultant total energy E is complex and is given by 1 E = Er − iΓ, (119) 2 where Er is the real part of the energy and Γ corresponds to the multiphoton ionization rate. Programs to calculate multiphoton ionization rates and harmonic generation rates have been developed for general multi-electron atoms and ions, by modifying the general R-matrix photoionization programs, used in the “Opacity Project” discussed in Section 4.1 (Berrington et al., 1987), and later by modifying the electron atom and ion scattering program developed by Burke et al. (1994). We now consider results of some recent RMF calculations of atomic multiphoton processes. We commence by discussing a calculation by van der Hart et al. (2005) who studied the following multiphoton ionization process in helium nhν + He(1s2 1 S)−→He+ (1s2 S) + e− , hν + He∗ (1s n )
(120)
where the laser wave length was 390 nm, corresponding to the frequency doubled Ti:sapphire laser, and the laser intensity was varied from 1 × 1014 W cm−2 to 2.5 × 1014 W cm−2 . We show the corresponding calculated multiphoton ionization rate in Fig. 27, where it is compared with completely independent timedependent numerical integration calculations and with an ADK model electron tunnelling calculation (Perelomov et al., 1966; Ammosov et al., 1986). At the lowest intensity shown in this figure, at least 8 photons are required to ionize the atom. However, as the intensity of the laser field increases, the excited states and the ionization threshold of He are shifted by the ponderomotive energy U0 = E02 /4ω2 relative to the ground state and, as a result, for intensities between 0.58 × 1014 W cm−2 and 2.75 × 1014 W cm−2 at least 9 photons are required for ionization, and above 2.75 × 1014 W cm−2 at least 10 photons are required to ionize the atom. At energies just above the threshold at 0.58 × 1014 W cm−2 , corresponding to n = 9 in Eq. (120), the first 8 photons excite one of the 1sn Rydberg states of the He atom and the last photon ionizes this state. This gives rise to the resonance-enhanced-multiphoton-ionization (REMPI) peaks seen in Fig. 27. In a similar way REMPI peaks are obtained just above the threshold at 2.75 × 1014 W cm−2 when n = 10 in Eq. (120). In the RMF calculation, 31 terms were retained in the Floquet–Fourier expansion (109), where 22 of these terms correspond to absorption and 8 correspond
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F IG . 27. Multiphoton ionization of He at 390 nm as a function of laser intensity using the RMF approach (dashed line) and the time-dependent numerical integration approach (solid line). Also shown are ADK model calculations.
to emission. In the configuration interaction expansion (6) of the corresponding Floquet states ψn , only the He+ 1s ground state was included and the maximum angular momentum retained was = 13. This gave a total of 217 coupled channels of which 98 were open. The complex energy in Eq. (119) was obtained through an iterative root-finding procedure. Including higher He+ states in the configuration interaction expansion of the Floquet states only changed the results by about 2%. An important feature of the results presented in Fig. 27 is the excellent agreement between the completely independent RMF and time-dependent calculations, providing firm evidence for the correctness of the calculations. Also we see that the tunnelling ADK model calculation is more than an order of magnitude smaller showing that considerable care must be taken in drawing quantitative conclusions from this approximation. As our second example of atomic multiphoton processes we consider resonant enhancement of third harmonic generation in argon at the fundamental KrF laser wavelength of 248 nm given by 3hν + Ar 3p6 1 Se → Ar∗ 3p5 4d 1 Po → Ar 3p6 1 Se + hν , (121) where ν = 3ν. In RMF calculations by Plummer and Noble (2000, 2002), between 8 and 10 terms corresponding to absorption and between three and five terms corresponding to emission were retained in the Floquet–Fourier expansion (109). Also, both the 3s2 3p5 2 Po ground and 3s3p6 2 Se excited states of Ar+ were included in the configuration interaction expansions (6) and (21) of the corre-
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F IG . 28. Third harmonic generation rate in Argon as a function of laser intensity at the KrF laser wavelength of 248 nm.
sponding Floquet states ψn . We show in Fig. 28 the third harmonic generation rate as a function of laser intensity. We see that there is a strong resonant enhancement of the rate at a laser intensity of ∼ 7.5 × 1012 W cm−2 caused by the relative ponderomotive shifts of energies of the argon 1 Se ground state and the 1 Po excited state. Resonance peaks were also found in the fifth and seventh harmonic generation rates although the maximum height of these peaks were several orders of magnitude smaller than the third harmonic generation peak. These calculations have demonstrated the importance of resonant enhancement of harmonic generation in multi-electron atoms, suggesting that this process could be a possible low-cost source of sustainable harmonic radiation. As an introduction to our next example of applications of RMF theory, we observe that a bound state and an autoionizing state of a multi-electron atom or ion, which are coupled by a strong laser field, can become degenerate at certain laser frequencies and intensities. These degeneracies, referred to as “laser induced degenerate states” or LIDS, were first demonstrated in RMF calculations by Latinne et al. (1995) and Cyr et al. (1997). To illustrate the LIDS mechanism we consider the following processes e− + A+ →A∗∗ 1
A+ + e− .
nhν
+ A→A∗∗ 2
(122)
The upper process in Eq. (122) corresponds to electron ion scattering which proceeds through an intermediate autoionizing state A∗∗ 1 lying in the continuum with
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F IG . 29. Trajectories of the complex Floquet energies for the 3s3p6 4p 1 Po autoionizing state and the 3s2 3p6 1 Se ground state of Ar dressed by one photon (each shifted down by the laser angular frequency), for intensities varying from 0 to 5×1013 W cm−2 . The corresponding value of the angular frequency is indicated on the trajectories and the dots on the trajectories give the intensity increase in steps of 9 × 1012 W cm−2 .
a complex energy E1 . The lower process in Eq. (122) corresponds to multiphoton ionization which proceeds through a “laser-induced-continuum-structure” LICS state A∗∗ 2 (Knight, 1984; Knight et al., 1990) which also lies in the continuum with a complex energy E2 . By varying both the laser frequency and intensity, the real and imaginary parts of the energies E1 and E2 can be made degenerate giving rise to a double pole in the corresponding laser-assisted electron–ion scattering Smatrix. Latinne et al. (1995) first observed LIDS in an RMF study of multiphoton ionization of Ar at the KrF laser wavelength of 248 nm. In this work, they included the 3s2 3p5 2 Po ground state and the 3s3p6 2 Se first excited state of Ar+ in the R-matrix expansions corresponding to Eqs. (6) and (21). In this way the interaction between the 3s3p6 4p 1 Po autoionizing state and the 3s2 3p6 1 Se ground state of Ar dressed by one photon was studied. We show in Fig. 29 the resultant trajectories of the complex Floquet energies E1 and E2 (shifted down in each case by the laser angular frequency ω) as the laser intensity is varied from zero to 5 × 1013 W cm−2 , where several fixed values of ω in the neighbourhood of 0.99 a.u. are considered. The zero-field position of the Ar ground state lies on the real axis at Eg = −0.57816 a.u., while the shifted zero-field position of the autoionizing state (denoted by circles in Fig. 29) lies at an energy of 0.40936 − 0.00119i − ω a.u., where the zero-field width Γa = 2 × 0.00119 a.u.. For each frequency there are two trajectories, one connected adiabatically with the zero-field position of the ground state and the other with the shifted zero-field position of the autoionizing state. At large positive or negative detunings from
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the double pole (e.g., ω = 0.984 or ω = 0.991) the autoionizing state does not move far from its zero-field position while the width of the ground state increases rapidly from zero with increasing intensity. At small detunings (ω ≈ 0.987) just the opposite occurs; the trajectories connected to the shifted autoionizing state increases with intensity while the ground state trajectory is “trapped” close to the real energy axis. For intermediate detunings, both on the positive and negative side, there are two complex energies about which the trajectories of the ground state and the shifted autoionizing state exchange their roles. At each of these complex energies there is a critical laser frequency and intensity such that the two complex quasi-energies are degenerate, resulting in double poles in the S-matrix. It has been known for many years that multiple poles in the S-matrix give rise to new phenomena including a modification of the exponential decay law and the Breit–Wigner resonance profile (e.g., Goldberger and Watson, 1964; Newton, 1966). In addition, the physical implications of laser induced degenerate states has been discussed by Kylstra and Joachain (1998). However, while the LIDS process is general and has been demonstrated for a number of targets including Ar, He and H− , more work needs to be carried out both theoretically and experimentally to reveal the full implication of this interesting phenomenon in atomic multiphoton processes. Finally, we briefly mention recent applications of R-matrix–Floquet theory to laser-assisted electron atom scattering. Several experimental groups have investigated free-free scattering of electrons on rare gas atoms, where the atom in Eq. (105) remains in its ground state. Also, experimental investigations of simultaneous electron-photon excitation (SEPE) of the target atom have been carried out. A review of earlier work in this area has been written by Mason (1993). Recently, R-matrix–Floquet theory has been extended and applied to treat both free-free scattering and SEPE processes by Charlo et al. (1998), Terao-Dunseath et al. (2001), Terao-Dunseath and Dunseath (2002) and Dunseath and Terao-Dunseath (2004). In order to describe these processes the radial wave function must be fitted to an asymptotic form described by Eq. (30), where the total energy E is real, yielding the K-matrix, S-matrix and cross sections for transitions between the electronic states corresponding to each Floquet component. To achieve this, the scattered electron wave function in the asymptotic region is transformed from the velocity gauge to the acceleration frame using a Kramers–Henneberger transformation (Kramers, 1956; Henneberger, 1968). In this recent work, the angular distribution of electron hydrogen atom scattering in a CO2 laser field, involving the absorption or emission of up to three photons was studied by Charlo et al. (1998), yielding good agreement with the low frequency, soft photon formula of Kroll and Watson (1973). The SEPE process in electron helium scattering in a Nd-YAG laser field was studied by Terao-Dunseath et al. (2001), where the first five target states of He were included in the R-matrix expansion. This showed the importance of the process where the incident electron is first captured into the
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well-known He− 1s2s2 2 S resonance at ∼19.2 eV, which then absorbs one or two photons, leaving the atom in a 1s2s 3 S or 1s2p 3 Po excited state. We also mention work on low energy electron helium scattering in a CO2 laser field by Dunseath and Terao-Dunseath (2004). Differential cross sections for elastic scattering with net absorption or emission of up to two photons were calculated over a range of laser intensities. The resultant differential cross sections were found to be considerably smaller than the experiments of Wallbank and Holmes (2001), suggesting that the experiments may include double scattering effects not included in the theory. In conclusion, while considerable work has been carried out applying R-matrix– Floquet theory to multi-electron atoms in intense laser fields, there is a need to extend this work in several new directions. These include the application to heavier atoms where relativistic effects become important and allowing for laser fields with arbitrary polarization.
7.2. M OLECULAR R-M ATRIX –F LOQUET T HEORY In this section we extend our discussion of R-matrix–Floquet theory to briefly consider its recent application to molecules. Compared with atomic multiphoton processes, the study of molecular multiphoton processes is much more complicated. Firstly, as in electron molecule scattering, the loss of spherical symmetry due to the presence of more than one nucleus means that the electronic structure of the target is more difficult to treat computationally. Secondly, the additional degrees of freedom associated with nuclear motion give rise to new effects. For example, modification of the nuclear potential energy curves by the laser field resulting in bond softening or hardening and the alignment of the molecular axis by the laser field both play an important role in the dynamics of multiphoton dissociation. Recent reviews of molecules in intense laser fields have been written, for example, by Bandrauk (1994), Giusti-Suzor et al. (1995) and Posthumus (2001). R-matrix–Floquet theory has recently been extended to treat the interaction of intense laser field with diatomic molecules by Colgan et al. (1998, 2000, 2001), Burke et al. (2000) and McKenna (2004) in the fixed-nuclei approximation. In the absence of relativistic effects, the molecule in the presence of the laser field is then described by the following time-dependent Schrödinger equation: 1 N +1 2 HN+1 + A(t).PN+1 + A (t) Ψ (XN+1 , t) c 2c2 ∂ = i Ψ (XN+1 , t), (123) ∂t where HN+1 is the fixed-nuclei Hamiltonian defined by Eq. (95) and where the total momentum operator PN+1 is defined by Eq. (108). As in atomic R-matrix–
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F IG . 30. Molecular frame of reference for a diatomic molecule in a laser field with polarization direction ˆ .
Floquet theory, the laser field is treated classically and is assumed to be monochromatic, monomode, spatially homogeneous and linearly polarized. The vector potential describing the laser field is then given by A(t) = ˆ A0 sin ωt.
(124)
The corresponding molecular frame of reference, in which the nuclei are held fixed, is illustrated in Fig. 30 where the notation in this figure is the same as that adopted in Fig. 18 for electron molecule scattering. The solution of Eq. (123) then proceeds as in our treatment of atomic R-matrix– Floquet theory, discussed in Section 7.1. We partition configuration space into three regions as illustrated in Fig. 2, where we use the same criteria for defining the boundaries a0 and ap between these regions. We then expand the wave function in each of these regions by the Floquet–Fourier expansion (109). Substituting this expansion into Eq. (123) yields an infinite set of coupled time-independent equations for the functions ψn (XN+1 ), where the length gauge is used in the internal region and the velocity gauge is used for the ejected or scattered electron in the external and asymptotic regions. After truncating these equations to a finite number of Floquet components, these coupled equations are solved in the internal region, using a modified version of the R-matrix computer program package developed by Gillan et al. (1987) for electron scattering by diatomic molecules. In the external region we obtain a finite number of coupled second-order differential equations which are solved using an R-matrix propagator method, as in atomic R-matrix–Floquet theory. Finally, in the asymptotic region we fit the solution to Siegert (1939) outgoing wave boundary conditions defined by Eq. (118), yielding the multiphoton ionization rate.
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F IG . 31. Potential energy curves of H2 and H+2 . The broken curves represent Rydberg bound states converging to the H+2 (X 2 Σg+ ) ground state and doubly excited states converging to the H+2 (2 Σu+ ) excited state. Two- and four-photon ionization processes are illustrated by the vertical lines with arrows.
We illustrate this theory by discussing calculations of multiphoton ionization of H2 molecules, carried out in the fixed-nuclei approximation by Colgan et al. (2001). The potential energy curves of the X 1 Σg+ ground state of H2 and the X 2 Σg+ ground state and 2 Σu+ first excited state of H+2 are shown in Fig. 31, together with potential energy curves of Rydberg bound states converging to the X 2 Σg+ ground state of H+2 and potential energy curves of doubly excited resonance states converging to the 2 Σu+ first excited state of H+2 . These calculations were carried out both in a one-state approximation, where only the X 2 Σg+ ground state of H+2 was included in the R-matrix expansions, and in a two-state approximation, where both the X 2 Σg+ and 2 Σu+ states of H+2 were included in these expansions. Four-photon ionization rates in H2 , calculated using both the onestate and two-state approximations at the equilibrium internuclear separation of 1.4 a.u. are presented in Fig. 32. These calculations were carried out at three laser intensities 1013 W cm−2 , 3 × 1013 W cm−2 and 1014 W cm−2 for a range
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F IG . 32. Four-photon ionization rates in H2 , calculated at the equilibrium internuclear separation of 1.4 a.u. for three laser intensities and for a range of frequencies. full curves: one-state approximation; broken curves: two-state approximation.
of frequencies, which includes the fundamental KrF laser frequency and the third harmonic of the Ti:sapphire laser frequency, which are marked on this figure. We see that the multiphoton ionization rates are dominated by resonance-enhancedmultiphoton-ionization (REMPI) peaks, where the first three photons excite a Rydberg bound state of the molecule and the fourth photon ionizes this bound state, as shown in Fig. 31. Due to the ponderomotive shift of the ionization threshold and the associated Rydberg states, these peaks go in and out of resonance as the intensity of the laser is increased, as already seen in Fig. 27, where we discussed a similar situation in He. The good agreement between the results obtained using the one-state and two-state approximations indicates that the two-state calculation is close to convergence at the laser intensities and the low ejected electron energies considered in these calculations. Calculations were also carried out by Colgan et al. (2001) which showed that the positions of the REMPI peaks were strongly dependent on the internuclear separation. This suggests that the population in different final H+2 X 2 Σg+ vibrational states can be controlled by varying the laser intensity and frequency. In
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addition, the probability of dissociation via the H+2 2 Σu+ repulsive state, illustrated in Fig. 31, will also depend strongly on the laser parameters. However, a full understanding of these processes will require the inclusion of the nuclear motion using non-adiabatic R-matrix theory of multiphoton ionization, analogous to that used in electron molecule scattering, discussed in Section 5. In conclusion, we have seen that recent work using molecular R-matrix– Floquet theory can yield interesting and potentially important results in the fixednuclei approximation. The extension of this work to include nuclear motion using non-adiabatic R-matrix theory and to consider more complex multi-electron molecules is an important objective for future work in this field.
7.3. T IME -D EPENDENT R-M ATRIX T HEORY We conclude our discussion of atomic and molecular multiphoton processes by briefly considering the interaction of ultrashort laser pulses with atoms and atomic ions. In recent years experiments with few femtosecond (10−15 sec) and attosecond (10−18 sec) pulse lasers have enabled the electronic motion of atoms and ions to be resolved for the first time, opening up an exciting new area of research in atomic physics (e.g., Hentschel et al., 2001; Rudenko et al., 2004). One approach to the treatment of atoms exposed to short laser pulses was proposed by Day et al. (2000). In this paper they expanded the wave-function describing an atom in a laser field as a time-dependent non-Hermitian superposition of Floquet states. The coefficients in this expansion satisfy a set of coupled time-dependent equations which can be solved to yield the probabilities of population transfer between the states. This approach was used by Plummer and Noble (2003) to study the dynamics of argon atoms in a KrF laser field for intensities up to 1014 W cm−2 , where the Floquet states were calculated using the R-matrix– Floquet method, discussed in Section 7.1. This non-Hermitian Floquet dynamics approach is of considerable interest in the study of atoms in short laser pulses. However, for ultrashort laser pulses, which may involve only a few cycles of the field, the full time-dependent Schrödinger equation (107) must be solved directly. In recent years considerable progress has been made in the direct solution of the time-dependent Eq. (107) for He and He-like two-electron systems (e.g., Parker et al., 1996, 1998; Scrinzi and Piraux, 1998 and Lagmago Kamta and Starace, 2001). There is now a need to develop an approach which is able to describe the full dynamics of a general multi-electron atom or ion in an ultrashort pulse laser field. In this section we briefly describe an ab initio non-perturbative timedependent R-matrix TDRM theory which is capable of achieving this objective.
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In oder to solve the time-dependent Schrödinger equation (107) we first transform the wave function using the gauge transformation
N+1 t i 2 Ψ (XN+1 , t) = exp − 2 (125) A t dt ΨV (XN+1 , t), 2c i=1
to eliminate the
A2 (t)
term which yields the equation N+1 1 ∂ HN+1 + A(t).pi ΨV (XN+1 , t) = i ΨV (XN+1 , t). c ∂t
(126)
i=1
This is the velocity form of the time-dependent Schrödinger equation which is adopted in both the internal and external R-matrix regions. In order to solve this equation we write it in the form ∂ (127) Ψ (XN+1 , t), ∂t where for notational convenience we omit the subscript V on Ψ . We then introduce a discrete mesh in time defined by H (t)Ψ (XN+1 , t) = i
tm = mΔt,
m = 0, 1, 2, . . . ,
(128)
where Δt is the time interval. Equation (127) can then be propagated forward in time using the unitary Cayley form of the time evolution operator exp(−itH (t)) giving 1 − (1/2)iΔtH (tm+1/2 ) Ψ (XN+1 , tm ) + O Δt 3 , 1 + (1/2)iΔtH (tm+1/2 ) m = 0, 1, 2, . . . , (129)
Ψ (XN+1 , tm+1 ) =
where tm+1/2 = tm + 12 Δt. Equation (129) can be rewritten as H (tm+1/2 ) − E Ψ (XN+1 , tm+1 ) = Θ(XN+1 , tm ), m = 0, 1, 2, . . . ,
(130)
where
Θ(XN+1 , tm ) = − H (tm+1/2 ) + E Ψ (XN+1 , tm ),
(131)
and where the energy E = 2i(Δt)−1 is pure imaginary. Equation (130) is an inhomogeneous equation which enables the wave function to be propagated forward in time since the right-hand side can be calculated in terms of the wave function from the previous time-step. Equation (130) can be solved for each time-step tm by expanding the wave function in both the internal and external regions as in standard R-matrix theory described in Section 2.1, modified to include the inhomogeneous term on the
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right-hand side. In the external region this involves using an inhomogeneous form of the R-matrix propagator equations, introduced by Schneider and Taylor (1982). Near the outer boundary of the external region an absorbing mask function (e.g., Krause et al., 1992) can be imposed to ensure that the wave function in multiphoton ionization vanishes on this boundary. For ultrashort laser pulses the outer boundary r = ap can be kept reasonably small without introducing significant error. The TDRM theory has been applied by Burke and Burke (1997) to analyze the multiphoton ionization of a charged particle initially bound in a one-dimensional well. The method was found to be stable for integration over a large time range and to give results for long laser pulses in excellent agreement with completely independent RMF calculations. A TDRM computer program is now being developed for a general multi-electron atom, taking advantage wherever possible of the multi-electron RMF program. It is expected that this new program will enable recent ultrashort laser pulse experiments on multi-electron atoms and ions to be analyzed in detail for the first time.
8. Electron Energy Loss from Transition Metal Oxides In this section we summarize recent work which generalizes R-matrix theory to describe low-energy electron energy-loss spectroscopy from transition metal oxides. Electron energy-loss spectroscopy (EELS) experiments provide an important probe of the electronic structure of solids, yielding information on the momentum and energy transfer associated with excitations (e.g., Fuggle and Inglesfield, 1992). At high incident electron energies this process can be described in the Born approximation by a dielectric loss function. However, recently there has been increasing interest in low-energy EELS (LE-EELS) in which incident electrons, with energy typically in the range 20–100 eV excite non-dipole allowed transitions, including electron exchange effects which can give rise to multiplicity-changing transitions. In this way, the experiments show a wealth of angle, spin-polarization and energy-dependent structure (e.g., Gorschlüter and Merz, 1994; Fromme et al., 1994, 1996). These LE-EELS experiments have been used to study the localized 3d–3d excitations in transitionmetal compounds, such as NiO and CoO and also the even more localized 4f–4f excitations in rare earth metals, such as Gd (e.g., Matthew et al., 1991; Porter et al., 1994). However, while the energy-loss spectra, measured in this way, can be described by parametrized crystal field models, the R-matrix approach described in this section is one of the first ab initio attempts to explain the energyloss spectra and their dependence on incident energy, angle of scattering and spin polarization.
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The generalization of R-matrix theory to describe electron energy loss from transition metal oxides was made by Michaels et al. (1997a, 1997b) and Jones et al. (2000). In this work, the localized 3d–3d excitations in the transition metal compound NiO were studied. We now show how R-matrix theory can be extended to describe the electronic transitions of the Ni2+ ion which is situated in a crystal field. In contrast to the case of a free Ni2+ ion, we will see that the crystal field potential has a strong effect on the interaction between the scattered electron and the target ion. We consider first the target states of the Ni2+ ion in the octahedral crystal field. We limit our discussion to states associated with 3d–3d excitation although other transitions can be treated by a straightforward extension of the theory considered here. The 3d8 configuration gives rise to the following five terms in the spherical environment of the free ion: 1 e 3 e 1
S , P , D e , 3 Fe , 1 G e ,
(132)
where 3 Fe is the ground term. The crystal field potential has the form (Sugano et al., 1970) 1/2 7 VC (r, θ, φ) = βr 4 12 1/2 5 Y44 (θ, φ) + Y4−4 (θ, φ) × Y40 (θ, φ) + 14 + VM , (133) where VM is a constant energy shift due to the Madelung potential, fitted to Hartree–Fock band-structure calculations (Towler et al., 1994). This potential splits the five spherical terms into eleven target states, as follows: S → 1 A1g
1 e
P → 3 T1g
3 e 1
De → 1 Σg + 1 T2g
(134)
F → A2g + T1g + T2g
3 e 1
3
3
3
Ge → 1 A1g + 1 Σg + 1 T1g + 1 T2g ,
where the labelling of the states on the right correspond to the irreducible representation of the octahedral Oh symmetry group. These states are determined as linear combinations of the five spherical states, determined by a Hartree–Fock calculation of Ni2+ . Having determined the target states, we are now in a position to construct the configuration interaction basis in the internal region, corresponding to Eq. (6) in electron ion scattering. The channel functions are formed by coupling the target
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states, defined by Eq. (134) with the spin-angle function of the scattered electron. The angular functions which are appropriate to cubic symmetry are constructed from spherical harmonics, as follows: pμ pμ Xh (θN+1 , φN+1 ) = (135) Y m (θN+1 , φN+1 )bh m , m
where p denotes the irreducible representation (IR) and μ its component. Also h labels the different possible linear combinations of the spherical harmonics with angular momentum that transform according to the pth IR. The radial continuum basis functions uij (r) representing the scattered electron are chosen to satisfy a zero-order differential equation corresponding to Eq. (15). The additional quadratically integrable functions which are included in Eq. (6) come from the 3d9 configuration which gives rise to 2 Eg and 2 T2g symmetries in the octahedral crystal field. Finally, the coefficients corresponding to aij k and bj k in Eq. (6) CF are obtained by diagonalizing the (N +1)-electron crystal field Hamiltonian HN+1 in the internal region where C HNCF +1 = HN+1 + VN+1 .
(136)
In this equation HN+1 is defined by Eq. (5) and C VN+1
=
N+1
VC (ri , θi , φi ),
(137)
i=1
where VC is the crystal field potential defined by Eq. (133). The radius a0 of the internal region should, in principle, extend further out than the distance between neighbouring atoms in the crystal. Typically, for Ni2+ ions in free space a0 ∼ 7 a.u. However, in a solid state environment the scattered electron “feels” the full Coulomb potential of the ion over a much shorter distance, typically the atomic sphere radius, and beyond this radius it interacts predominantly with the neighbouring atoms. This corresponds to the muffin-tin or atomic sphere approximation that is frequently made in band structure calculations (e.g., Gonis, 1992). The atomic sphere radius of Ni2+ in NiO is taken to be 2.58 a.u. from conventional band structure calculations and the scattering amplitude is determined at this radius. In a full multiple-scattering calculation, scattering by all the atomic spheres would be included. However, the calculations carried out so far use a single-scattering approximation taking a constant potential outside the atomic sphere radius. In order to determine the scattering amplitude at the smaller radius, the internal region calculation is carried out using a radius a0 = 7 a.u. The R-matrix on this boundary is then propagated backwards from r = 7 a.u. to r = 2.58 a.u., using Eq. (27), where the Green’s function in this equation is calculated using the same one-electron Hamiltonian used to calculate the radial continuum basis
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(a)
(b) F IG . 33. Comparison of calculated spin-flip spectra (a) with experimental data (b) for electrons scattered from NiO as a function of scattering angle α and energy loss. The scattering angles are in the range 73◦ α 123◦ in intervals of 3.125◦ with 73◦ being the lowest line.
functions uij (r). Given the R-matrix on the boundary r = 2.58 a.u., the corresponding K-matrix and S-matrix can be determined. Hence the differential and total cross sections for transitions between the target states defined by Eq. (134) can be calculated. We illustrate this theory by comparing in Fig. 33 calculations carried out by Jones et al. (2000) for the spin-flip spectra of electrons scattered from NiO with experimental data obtained by Müller et al. (1999). In this experiment, the energy
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loss of polarized electrons incident at an angle of 45◦ to the normal to the surface with an incident energy of 33 eV and undergoing a spin-flip was measured for sixteen scattering angles α in the range 73◦ α 123◦ at an interval of 3.125◦ . (The scattering angle α is related to the angle to the normal θf by θf = 135◦ − α.) In the theoretical model the elastic peak is ignored as is the 0.6 eV loss peak, which is due to a surface excitation. The small 1.05 eV loss peak is due to 3 T2g excitation, the big peak at 1.7 eV is due to two overlapping transitions 1 Eg (1.70 eV) and 3 T (1.75 eV). Also the 3.2 eV loss peak is due to two overlapping transitions 1g 3 T (3.13.eV) and 1 T (3.28 eV). The peak at 3.2 eV, albeit too narrow, drops 1g 1g off dramatically at large α, apparently in agreement with experiment. More significantly, at 2.7 eV we see a small shoulder at α = 73◦ that slowly increases and becomes dominant at 123◦ . This peak is of particular interest as it is a combination of two triplet-singlet excitation to the states 1 A1g (2.80 eV) and 1 T2g (2.70 eV) for which spin-flip dominates. Overall the model shows reasonable agreement with experiment bearing in mind the approximations made. Although this theory provides a clear understanding of the main experimental features using a single scattering approximation, a full theory of LE-EELS must also treat multiple-elastic scattering events that occur before and after inelastic scattering. The inclusion of multiple scattering in this theory is a challenge for future work on LE-EELS from NiO and other transition metal oxides.
9. Conclusions We have shown in this review that R-matrix theory is applicable to a wide range of atomic, molecular and optical processes. We have seen that general R-matrix computer programs have been developed, and are being used, to analyze experiments and to provide data required in order to understand the physics and chemistry of processes in many fields. These include the astrophysics of gaseous nebulae, stellar opacities, atmospheric physics and chemistry, plasma physics and controlled thermonuclear fusion, positron interactions, the interaction of intense lasers with atoms and molecules and the interaction of electrons with surfaces. However, it is also clear that there are many major challenges ahead which R-matrix theory can and should address. These fields include the study of electron and photon interactions with open d-shell and f-shell atoms and ions, involving many thousands of coupled channels; electron scattering by polyatomic molecules, including the rapidly increasing interest in bio-molecules; the interaction of positrons with molecules; the interaction of ultra-short laser pulses with multielectron atoms and molecules; and the scattering of electrons with surfaces, taking full account of inelastic and multiple scattering. Many of these processes are currently being addressed by groups world-wide. This involves both the development of new theoretical and computational approaches and the development of new
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computer programs that can take full advantage of the rapid advance in the power of massively parallel computers. We can therefore expect a continuing interest over the forthcoming years in the solution of many new and exciting problems using R-matrix theory of atomic, molecular and optical processes.
10. Acknowledgements The authors wish to acknowledge the continued support, received over many years, for R-matrix theory, program development and applications, from Queen’s University Belfast and from the UK Collaborative Computational Project on the Continuum States of Atoms and Molecules (CCP2), coordinated through the Daresbury Laboratory.
11. References Akhiezer, A.I., Berestetsky, V.B. (1965). “Quantum Electrodynamics”. Interscience, New York. Ammosov, M.V., Delone, N.B., Krainov, V.P. (1986). Sov. Phys. JETP 64, 1191. Andersen, N., Bartschat, K. (2004). J. Phys. B 37, 3809. Armitage, S., Leslie, D.E., Garner, A.J., Laricchia, G. (2002). Phys. Rev. Lett. 89, 173402. Avaldi, L., Camilloni, R., Multari, R., Stefani, G., Robaux, O., Tweed, R.J., Vien, G.N. (1998). J. Phys. B 31, 2981. Aymar, M., Greene, C.H., Luc-Koenig, E. (1996). Rev. Mod. Phys. 68, 1015. Badnell, N.R. (1986). J. Phys. B 19, 3827. Badnell, N.R. (1997). J. Phys. B 30, 1. Badnell, N.R., Gorczyca, T.W. (1997). J. Phys. B 30, 2011. Badnell, N.R., Gorczyca, T.W., Price, A.D. (1998). J. Phys. B 31, L239. Badnell, N.R., Bautista, M.A., Butler, K., Delahaye, F., Mendoza, C., Palmeri, P., Zeippen, C.J., Seaton, M.J. (2005). Monthly Notices Roy. Astronom. Soc. 360, 458. Bahcall, J.N. (2005). Physics World 18 (2), 26. Ballance, C.P., Griffin, D.C. (2004). J. Phys. B 37, 2943. Baluja, K.L., Burke, P.G., Kingston, A.E. (1980). J. Phys. B 13, L543. Baluja, K.L., Burke, P.G., Morgan, L.A. (1982). Comput. Phys. Commun. 27, 299. Bandrauk, A.D. (Ed.) (1994). “Molecules in Laser Fields”, Dekker, New York. Bartschat, K. (1993). Comput. Phys. Commun. 75, 219. Bartschat, K., Bray, I. (1996). J. Phys. B 29, L577. Bartschat, K., Burke, P.G. (1987). J. Phys. B 20, 3191. Bartschat, K., Burke, P.G. (1988). J. Phys. B 21, 2969. Bartschat, K., Grum-Grzhiamailo, A.N. (2002). J. Phys. B 35, 5035. Bartschat, K., Reid, R.H.G., Burke, P.G., Summers, H.P. (1990). J. Phys. B 23, L721. Bartschat, K., Hudson, E.T., Scott, M.P., Burke, P.G., Burke, V.M. (1996a). J. Phys. B 29, 115. Bartschat, K., Hudson, E.T., Scott, M.P., Burke, P.G., Burke, V.M. (1996b). Phys. Rev. A 54, R998. Bates, D.R. (1962). Planet. Space Sci. 9, 77. Bates, D.R., Dalgarno, A. (1962). In: Bates, D.R. (Ed.), “Atomic and Molecular Processes”, Academic Press, New York, p. 245. Bates, D.R., Massey, H.S.W. (1943). Philos. Trans. Roy. Soc. London Ser. A 239, 269.
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Bautista, M.A., Pradhan, A.H. (1998). Astrophys. J. 492, 650. Bell, K.L., Burke, P.G., Kingston, A.E. (1977). J. Phys. B 10, 3117. Bell, R.H., Seaton, M.J. (1985). J. Phys. B 18, 1589. Berrington, K.A., Burke, P.G., Chang, J.J., Chivers, A.T., Robb, W.D., Taylor, K.T. (1974). Comput. Phys. Commun. 8, 149. Berrington, K.A., Burke, P.G., Le Dourneuf, M., Robb, W.D., Taylor, K.T., Lan, Vo Ky (1978). Comput. Phys. Commun. 14, 367. Berrington, K.A., Burke, P.G., Butler, K., Seaton, M.J., Storey, P.J., Taylor, K.T., Yan, Yu (1987). J. Phys. B 20, 6379. Berrington, K.A., Eissner, W.B., Norrington, P.H. (1995). Comput. Phys. Commun. 92, 290. Bethe, H.A., Salpeter, E.E. (1957). “Quantum Mechanics of One- and Two-Electron Atoms”. Springer-Verlag, Berlin. Biswas, P.K., Adhikari, S.K. (1999). Phys. Rev. A 59, 363. Blackwood, J.E., McAlinden, M.T., Walters, H.R.J. (1999). Phys. Rev. A 60, 4454. Blackwood, J.E., McAlinden, M.T., Walters, H.R.J. (2002a). Phys. Rev. A 65, 030502. Blackwood, J.E., McAlinden, M.T., Walters, H.R.J. (2002b). Phys. Rev. A 65, 032517. Blackwood, J.E., McAlinden, M.T., Walters, H.R.J. (2002c). J. Phys. B 35, 2661. Bloch, C. (1957). Nucl. Phys. 4, 503. Bray, I. (1995). J. Phys. B 28, L247. Bray, I., Fursa, D.V., Kheifets, A.S., Stelbovics, A.T. (2002). J. Phys. B 35, R117. Breit, G. (1959). In: Flügge, S. (Ed.), Handbuch der Physik, vol. 41/1. Springer-Verlag, Berlin. Buckman, S.J., Clark, C.W. (1994). Rev. Mod. Phys. 66, 539. Buckman, S.J., Hammond, P., Read, F.H., King, G.C. (1983). J. Phys. B 16, 4039. Burgess, A. (1964). Astrophys. J. 139, 776. Burgess, A. (1965). Ann. d’Astrophysique 28, 774. Burke, P.G., Berrington, K.A. (1993). “Atomic and Molecular Processes: An R-matrix Approach”. Institute of Physics Publishing, Bristol and Philadelphia. Burke, P.G., Burke, V.M. (1997). J. Phys. B 30, L383. Burke, P.G., Robb, W.D. (1975). Adv. At. Mol. Phys. 11, 143. Burke, P.G., Schey, H.M. (1962). Phys. Rev. 126, 147. Burke, P.G., Seaton, M.J. (1971). In: Alder, B., Fernbach, S., Rotenberg, M. (Eds.), Methods in Computational Physics, vol. 10. Academic Press, New York, p. 1. Burke, P.G., Taylor, K.T. (1975). J. Phys. B 8, 2620. Burke, P.G., McVicar, D.D., Smith, K. (1964). Proc. Phys. Soc. London 83, 397. Burke, P.G., Hibbert, A., Robb, W.D. (1971). J. Phys. B 4, 153. Burke, P.G., Mackey, I., Shimamura, I. (1977). J. Phys. B 10, 2497. Burke, P.G., Noble, C.J., Scott, M.P. (1987). Proc. Roy. Soc. London Ser. A 410, 289. Burke, P.G., Francken, P., Joachain, C.J. (1990). Europhys. Lett. 13, 617. Burke, P.G., Francken, P., Joachain, C.J. (1991). J. Phys. B 24, 761. Burke, P.G., Burke, V.M., Dunseath, K.M. (1994). J. Phys. B 27, 5341. Burke, P.G., Colgan, J., Glass, D.H., Higgins, K. (2000). J. Phys. B 33, 143. Burke, V.M., Noble, C.J. (1995). Comput. Phys. Commun. 85, 471. Burke, V.M., Burke, P.G., Scott, N.S. (1992). Comput. Phys. Commun. 69, 76. Buttle, P.J.A. (1967). Phys. Rev. 160, 719. Byron, F.W., Joachain, C.J., Piraux, B. (1986). J. Phys. B 19, 1201. Campbell, C.P., McAlinden, M.T., MacDonald, F.G.R.S., Walters, H.R.J. (1998). Phys. Rev. Lett. 80, 5097. Chandrasekhar, S. (1945). Astrophys. J. 102, 223. Chang, J.J. (1975). J. Phys. B 8, 2327. Charlo, D., Terao-Dunseath, M., Dunseath, K.M., Launay, J.-M. (1998). J. Phys. B 31, L539.
314
P.G. Burke et al.
[11
Colgan, J., Glass, D.H., Higgins, K., Burke, P.G. (1998). Comput. Phys. Commun. 114, 27. Colgan, J., Glass, D.H., Higgins, K., Burke, P.G. (2000). In: DiMauro, L.F., Freeman, R.R., Kulander, K.C. (Eds.), “Multiphoton Processes”, AIP Conf. Proc., vol. 525, p. 427. Colgan, J., Glass, D.H., Higgins, K., Burke, P.G. (2001). J. Phys. B 34, 2089. Crandall, D.H., Phaneuf, R.A., Hasselquist, B.E., Gregory, D.C. (1979). J. Phys. B 12, L249. Crowe, D.M., Gou, X.Q., Lubell, M.S., Slevin, J., Eminyan, M. (1990). J. Phys. B 23, L325. Cunto, W., Mendoza, C., Ochsenbein, F., Zeippen, C.J. (1993). Astron. and Astrophys. 275, L5; http://hizier.u-strasbg.fr/topbase/. Cyr, A., Latinne, O., Burke, P.G. (1997). J. Phys. B 30, 659. Davies, P.C.W., Seaton, M.J. (1969). J. Phys. B 2, 757. Day, H.C., Piraux, B., Potvliege, R.M. (2000). Phys. Rev. A 61, 031402. Diehl, S., Cubaynes, C., Kennedy, E.T., Wuilleumier, F.J., Bizau, J.-M., Journe, L., VoKy, L., Faucher, P., Hibbert, A., Blancard, C., Berrah, N., Morgan, T.J., Bozek, J., Schlachter, A.S. (1997). J. Phys. B 30, L595. Dörr, M., Terao-Dunseath, M., Purvis, J., Noble, C.J., Burke, P.G., Joachain, C.J. (1992). J. Phys. B 25, 2809. Dunseath, K.M., Terao-Dunseath, M. (2004). J. Phys. B 37, 1305. Dunseath, K.M., LeDourneuf, M., Terao-Dunseath, M., Launay, J.-M. (1996). Phys. Rev. A 54, 561. Dyall, K.G., Grant, I.P., Johnson, C.T., Parpia, F.A., Plummer, E.P. (1989). Comput. Phys. Commun. 55, 425. Eissner, W., Jones, M., Nussbaumer, H. (1974). Comput. Phys. Commun. 8, 270. Fang, Y., Bartschat, K. (2001a). J. Phys. B 34, L19. Fang, Y., Bartschat, K. (2001b). J. Phys. B 34, 2745. Fang, Y., Bartschat, K. (2001c). Phys. Rev. A 64, 020701(R). Fletcher, G.D., Alguard, M.J., Gay, T.J., Wainwright, P.F., Lubell, M.S., Raith, W., Hughes, V.W. (1985). Phys. Rev. A 31, 2854. Floquet, G. (1883). Ann. Ec. Norm. (2) 13, 47. Froese Fischer, C. (1969). Comput. Phys. Commun. 1, 151. Froese Fischer, C. (1972). Comput. Phys. Commun. 4, 107. Froese Fischer, C. (1978). Comput. Phys. Commun. 14, 145. Froese Fischer, C. (1991). Comput. Phys. Commun. 64, 369. Fromme, B., Schmitt, M., Kister, E., Gorschlüter, A., Merz, H. (1994). Phys. Rev. B 50, 1874. Fromme, B., Möller, M., Anschütz, Th., Bethe, C., Kister, E. (1996). Phys. Rev. Lett. 77, 1548. Fuggle, J.C., Inglesfield, J.E. (Eds.) (1992). “Unoccupied Electronic States”, Springer-Verlag, Berlin. Gailitis, M. (1976). J. Phys. B 9, 843. Garner, A.J., Laricchia, G., Özen, A. (1996). J. Phys. B 29, 5961. Garner, A.J., Özen, A., Laricchia, G. (2000). J. Phys. B 33, 1149. Gavrila, M. (Ed.) (1992). “Atoms in Intense Laser Fields”, Adv. At. Mol. Opt. Phys. Suppl., vol. 1. Academic Press, New York. Gebarowski, R., Burke, P.G., Taylor, K.T., Dörr, M., Bensaid, M., Joachain, C.J. (1997a). J. Phys. B 30, 1837. Gebarowski, R., Taylor, K.T., Burke, P.G. (1997b). J. Phys. B 30, 2505. Gillan, C.J., Nagy, O., Burke, P.G., Morgan, L.A., Noble, C.J. (1987). J. Phys. B 20, 4585. Giusti-Suzor, A., Mies, F.H., DiMauro, L.F., Charron, E., Yang, B. (1995). J. Phys. B 28, 309. Goldberger, M.L., Watson, K.M. (1964). “Collision Theory”. Wiley, New York. Gonis, A. (1992). Green Functions for Ordered and Disordered Systems. North-Holland, Amsterdam. Gorczyca, T.W. (2000). Phys. Rev. A 61, 024702. Gorczyca, T.W., Badnell, N.R. (1996). J. Phys. B 29, L283. Gorczyca, T.W., Badnell, N.R. (1997a). J. Phys. B 30, 3897. Gorczyca, T.W., Badnell, N.R. (1997b). Phys. Rev. Lett. 79, 2783.
11]
R-MATRIX THEORY
315
Gorczyca, T.W., McLaughlin, B.M. (2000). J. Phys. B 33, L859. Gorczyca, T.W., Robicheaux, F. (1999). Phys. Rev. A 60, 1216. Gorczyca, T.W., Robicheaux, F., Pindzola, M.S., Badnell, N.R. (1995). Phys. Rev. A 52, 3852. Gorczyca, T.W., Robicheaux, F., Pindzola, M.S., Badnell, N.R. (1996). Phys. Rev. A 54, 2107. Gorczyca, T.W., Badnell, N.R., Savin, D.W. (2002). Phys. Rev. A 65, 062707. Gorczyca, T.W., Zatsarinny, O., Zhou, H.-L., Manson, S.T., Felfli, Z. (2003). Phys. Rev. A 68, 050703. Gorfinkiel, J.D., Tennyson, J. (2004). J. Phys. B 37, L343. Gorschlüter, A., Merz, H. (1994). Phys. Rev. B 49, 17293. Grant, I.P., McKenzie, B.J., Norrington, P.H., Mayers, D.F., Pyper, N.C. (1980). Comput. Phys. Commun. 21, 207. Griffin, D.C., Pindzola, M.S., Badnell, N.R. (1993). Phys. Rev. A 47, 2871. Griffin, D.C., Badnell, N.R., Pindzola, M.S. (1998). J. Phys. B 31, 3713. Hebb, M.H., Menzel, D.H. (1940). Astrophys. J. 92, 408. Heggarty, J.W., Scott, M.P., Scott, N.S., Burke, P.G. (1998). Comput. Phys. Commun. 114, 195. Henneberger, W.C. (1968). Phys. Rev. Lett. 21, 838. Hentschel, M., Kienberger, R., Spielman, Ch., Reider, G.A., Milosovic, N., Brabec, T., Corkum, P., Heinzmann, U., Drescher, M., Krausz, F. (2001). Nature 414, 509. Hibbert, A. (1975). Comput. Phys. Commun. 9, 141. Hickman, A.P. (1984). J. Phys. B 17, L101. Higgins, K., Noble, C.J., Burke, P.G. (1994). J. Phys. B 27, 3203. Hummer, D.G., Berrington, K.A., Eissner, W., Pradhan, A.K., Saraph, H.E., Tully, J.A. (1993). Astron. Astrophys. 279, 298. Jackson, J.L. (1951). Phys. Rev. 83, 301. Jacubowicz, H., Moores, D.L. (1981). J. Phys. B 14, 3733. Jauch, J.M., Rohrlich (1955). “The Theory of Photons and Electrons”. Addison-Wesley, Cambridge, MA. Joachain, C.J., Dörr, M., Kylstra, N.J. (2000). Adv. At. Mol. Opt. Phys. 42, 226. Jones, G.O., Charlton, M., Slevin, J., Larichia, G., Kövér, Á., Poulsen, M.R., Nic Chormaic, S. (1993). J. Phys. B 26, L483. Jones, P., Inglesfield, J.E., Michiels, J.J.M., Noble, C.J., Burke, V.M., Burke, P.G. (2000). Phys. Rev. B 62, 13508. Kato, D., Watanabe, S. (1997). Phys. Rev. A 56, 3687. Kernoghan, A.A., Robinson, D.J.R., McAlinden, M.T., Walters, H.R.J. (1996). J. Phys. B 29, 2089. Kilgus, G., Berger, J., Blatt, P., Grieser, M., Habs, D., Hochadel, B., Jaeschke, E., Krämer, D., Neumann, R., Neureither, G., Ott, W., Schwalm, D., Steck, M., Stokstad, R., Szmola, E., Wolf, A., Schuch, R., Müller, A., Wagner, M. (1990). Phys. Rev. Lett. 64, 737. Knight, P.L. (1984). Comm. At. Mol. Phys. 15, 193. Knight, P.L., Lauder, M.A., Dalton, B.J. (1990). Phys. Rep. 190, 1. Kohn, W. (1948). Phys. Rev. 74, 1763. Kramers, H.A. (1956). “Collected Scientific Papers”. North-Holland, Amsterdam. Krause, J.L., Schafer, K.J., Kulander, K.C. (1992). Phys. Rev. A 45, 4998. Kroll, N.M., Watson, K.M. (1973). Phys. Rev. A 8, 804. Kylstra, N.J., Joachain, C.J. (1998). Phys. Rev. A 57, 412. Laghdas, K., Reid, R.H.G., Joachain, C.J., Burke, P.G. (1995). J. Phys. B 28, 4811. Laghdas, K., Reid, R.H.G., Joachain, C.J., Burke, P.G. (1999). J. Phys. B 32, 1439. Lagmago Kamta, G., Starace, A.F. (2001). Phys. Rev. Lett. 86, 5687. Lane, A.M., Thomas, R.G. (1958). Rev. Mod. Phys. 30, 257. Lane, A.M., Robson, D. (1966). Phys. Rev. 151, 774. Laricchia, G., Armitage, S., Leslie, D.E. (2004). Nucl. Instrum. Methods B 221, 60. Latinne, O., Kylstra, N.J., Dörr, M., Purvis, J., Terao-Dunseath, M., Joachain, C.J., Burke, P.G., Noble, C.J. (1995). Phys. Rev. Lett. 74, 46.
316
P.G. Burke et al.
[11
LeDourneuf, M., Launay, J.-M., Burke, P.G. (1990). J. Phys. B 23, L559. Light, J.C., Walker, R.B. (1976). J. Chem. Phys. 65, 4272. Lippmann, B., Schey, H.M. (1961). Phys. Rev. 121, 1112. McCurdy, C.W., Baertschy, M., Rescigno, T.N. (2004). J. Phys. B 37, R137. McKenna, C. (2004). Ph.D. thesis, Queen’s University Belfast. Madison, D.H., Calhoun, R.V., Shelton, W.N. (1977). Phys. Rev. A 16, 552. Mahaux, C., Weidenmüller, H.A. (1969). “Shell-Model Approach to Nuclear Reactions”. NorthHolland, Amsterdam. Malegat, L. (1994). J. Phys. B 27, L691. Mason, N.J. (1993). Rep. Prog. Phys. 56, 1275. Massey, H.S.W., Bates, D.R. (1942). Rep. Prog. Phys. 9, 62. Matthew, J.A.D., Henle, W.A., Ramsey, M.G., Netzer, F.P. (1991). Phys. Rev. B 43, 4897. Michiels, J.J.M., Inglesfield, J.E., Noble, C.J., Burke, V.M., Burke, P.G. (1997a). Phys. Rev. Lett. 78, 2851. Michiels, J.J.M., Inglesfield, J.E., Noble, C.J., Burke, V.M., Burke, P.G. (1997b). J. Phys. Condens. Matter 9, L543. Middleton, A.G., Brunger, M.J., Teubner, P.J.O., Anderson, M.W.B., Noble, C.J., Wöste, G., Blum, K., Burke, P.G., Fullerton, C. (1994). J. Phys. B 27, 4057. Mitnik, D., Pindzola, M.S., Griffin, D.C., Badnell, N.R. (1999). J. Phys. B 32, L479. Mitnik, D., Griffin, D.C., Ballance, C.P., Badnell, N.R. (2003). J. Phys. B 36, 717. Morgan, L.A., Gillan, C.J., Tennyson, J., Chen, X. (1997). J. Phys. B 30, 4087. Mouret, L., Dunseath, K.M., Terao-Dunseath, M., Launay, J.-M. (2003). J. Phys. B 36, L39. Müller, A., Hofmann, G., Tinschert, K., Satzborn, E. (1988). Phys. Rev. Lett. 61, 1352. Müller, F., Steiner, P., Straub, Th., Reinicke, D., Palm, S., de Masi, R., Hüfner, S. (1999). Surf. Sci. 442, 485. Nahar, S.N., Pradhan, A.K. (1994). Phys. Rev. A 49, 1816. Nahar, S.N., Pradhan, A.K. (1995). Astrophys. J. 447, 966. Nesbet, R.K. (1980). “Variational Methods in Electron–Atom Scattering Theory”. Plenum Press, New York. Newton, R.G. (1966). “Scattering Theory of Waves and Particles”. McGraw-Hill, New York. Nicolas, K.R. (1977). Ph.D. thesis, University of Maryland. Noble, C.J., Burke, P.G. (1992). Phys. Rev. Lett. 68, 2011. Noble, C.J., Nesbet, R.K. (1984). Comput. Phys. Commun. 33, 399. Noble, C.J., Higgins, K., Wöste, G., Duddy, P., Burke, P.G., Teubner, P.J.O., Middleton, A.G., Brunger, M.J. (1996). Phys. Rev. Lett. 76, 3534. Norrington, P.H., Grant, I.P. (1981). J. Phys. B 14, L261. Norrington, P.H., Grant, I.P. (1987). J. Phys. B 20, 4869. See also http://www.am.qub.ac.uk/DARC/. Nussbaumer, H., Storey, P.J. (1978). Astron. Astrophys. 70, 37. Osterbrock, D.E. (1989). “Astrophysics of Gaseous Nebulae and Active Galactic Nuclei”. University Science Books, Sausalito, CA. Parker, J., Taylor, K.T., Clark, C.W., Blodgett-Ford, S. (1996). J. Phys. B 29, L33. Parker, J.S., Smyth, E.S., Taylor, K.T. (1998). J. Phys. B 31, L571. Parpia, F.A., Froese Fischer, C., Grant, I.P. (1996). Comput. Phys. Commun. 94, 249. Perelomov, A.M., Popov, V.S., Terent’ev, M.V. (1966). Sov. Phys. JETP 23, 924. Pfingst, K., Nestmann, B.M., Peyerimhoff, S.D. (1994). J. Phys. B 27, 2283. Pindzola, M.S., Robicheaux, F.J. (2000). Phys. Rev. A 61, 052707. Plummer, M., Noble, C.J. (1999). J. Phys. B 32, L345. Plummer, M., Noble, C.J. (2000). J. Phys. B 33, L807. Plummer, M., Noble, C.J. (2002). J. Phys. B 35, L51. Plummer, M., Noble, C.J. (2003). J. Phys. B 36, L219.
11]
R-MATRIX THEORY
317
Porter, S.J., Matthew, J.A.D., Leggott, R.J. (1994). Phys. Rev. B 50, 2638. Posthumus, J.H. (2001). “Molecules and Clusters in Intense Laser Fields”. Cambridge University Press. Pradhan, A.K., Seaton, M.J. (1985). J. Phys. B 18, 1631. Pradhan, A.K., Zhang, H.L. (1997). J. Phys. B 30, L571. Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P. (1992). “Numerical Recipes in Fortran. The Art of Scientific Computing”, second ed. Cambridge University Press. Protopapas, M., Keitel, C.H., Knight, P.L. (1997). Rep. Prog. Phys. 60, 389. Raeker, A., Bartschat, K., Reid, R.H.G. (1994). J. Phys. B 27, 3129. Ramsbottom, C.A., Noble, C.J., Burke, V.M., Scott, M.P., Kisielius, R., Burke, P.G. (2005). J. Phys. B 38, 2999. Reid, R.H.G., Bartschat, K., Burke, P.G. (1992). J. Phys. B 25, 3175. Reid, R.H.G., Bartschat, K., Raeker, A. (1998). J. Phys. B 31, 563. Reid, R.H.G., Bartschat, K., Raeker, A. (2000). J. Phys. B 33, 5261. Rescigno, T.N., Baertschy, M., Isaacs, W.A., McCurdy, C.W. (1999). Science 286, 2474. Robicheaux, F. (1998). J. Phys. B 31, L109. Robicheaux, F., Gorczyca, T.W., Pindzola, M.S., Badnell, N.R. (1995). Phys. Rev. A 52, 1319. Rudenko, A., Zrost, K., Feuarstein, B., de Jesus, V.L.B., Schröter, C.D., Moshammer, R., Ullrich, J. (2004). Phys. Rev. Lett. 93, 253001. Sakimoto, K., Terao, M., Berrington, K.A. (1990). Phys. Rev. A 42, 291. Saraph, H.E. (1972). Comput. Phys. Commun. 3, 25. Saraph, H.E. (1978). Comput. Phys. Commun. 15, 247. Saraph, H.E., Storey, P.J., Taylor, K.T. (1992). J. Phys. B 25, 4409. Schneider, B.I. (1975a). Chem. Phys. Lett. 31, 237. Schneider, B.I. (1975b). Phys. Rev. A 11, 1957. Schneider, B.I., Hay, P.J. (1976). Phys. Rev. A 13, 2049. Schneider, B.I., Taylor, H.S. (1982). J. Chem. Phys. 77, 379. Schneider, B.I., Walker, R.B. (1979). J. Chem. Phys. 70, 2466. Schneider, B.I., LeDourneuf, M., Burke, P.G. (1979). J. Phys. B 12, L365. Schwienhorst, R., Raeker, A., Reid, R.H.G., Bartschat, K. (1995). J. Phys. B 28, 4651. Scott, M.P., Teng, H., Burke, P.G. (2000). J. Phys. B 33, L63. Scott, M.P., Stitt, T., Scott, N.S., Burke, P.G. (2002). J. Phys. B 35, L323. Scott, N.S., Burke, P.G. (1980). J. Phys. B 13, 4299. Scott, N.S., Taylor, K.T. (1982). Comput. Phys. Commun. 25, 347. Scrinzi, A., Piraux, B. (1998). Phys. Rev. A 58, 1310. Seaton, M.J. (1962). The Observatory 82, 111. Seaton, M.J. (1966a). Proc. Phys. Soc. 88, 801. Seaton, M.J. (1966b). Proc. Phys. Soc. 88, 815. Seaton, M.J. (1975). Adv. At. Mol. Phys. 11, 83. Seaton, M.J. (1983). Rep. Prog. Phys. 46, 167. Seaton, M.J. (1985). J. Phys. B 18, 2111. Seaton, M.J. (1987). J. Phys. B 20, 6363. Seaton, M.J. (2005). Monthly Notices Roy. Astronom. Soc., in press. Seaton, M.J., Storey, P.J. (1976). In: Burke, P.G., Moiseiwitsch, B.L. (Eds.), “Atomic Processes and Applications”, North-Holland, Amsterdam, p. 133. Shah, M.B., Elliott, D.S., Gilbody, H.B. (1987). J. Phys. B 20, 3501. Shirley, J.H. (1965). Phys. Rev. B 138, 979. Shyn, T.W., Sweeney, C.J. (1993). Phys. Rev. A 48, 1214. Siegert, A.J.F. (1939). Phys. Rev. 56, 750. Stechel, E.B., Walker, R.B., Light, J.C. (1978). J. Chem. Phys. 69, 3518.
318
P.G. Burke et al.
[11
Strickland, D., Mourou, G. (1985). Opt. Commun. 56, 219. Sugano, S., Tanabe, Y., Kamimura, H. (1970). “Multiplets of Transition-Metal Ions in Crystals”. Academic Press, New York. Summers, H.P. (1994). Adv. At. Mol. Opt. Phys. 33, 275. Sunderland, A.G., Noble, C.J., Burke, V.M., Burke, P.G. (2002). Comput. Phys. Commun. 145, 311. Terao, M., Burke, P.G. (1990). J. Phys. B 23, 1815. Terao, M., Bell, K.L., Burke, P.G., Hibbert, A. (1991). J. Phys. B 24, L321. Terao-Dunseath, M., Dunseath, K.M. (2002). J. Phys. B 35, 125. Terao-Dunseath, M., Dunseath, K.M., Charlo, D., Hibbert, A., Allan, R.J. (2001). J. Phys. B 34, L263. Towler, M.D., Allan, N.L., Harrison, N.M., Saunders, V.R., Mackrodt, W.C., Aprà, E. (1994). Phys. Rev. B 50, 5041. van der Hart, H.W. (1997). J. Phys. B 30, 453. van der Hart, H.W., Doherty, B.J.S., Parker, J.S., Taylor, K.T. (2005). J. Phys. B 38, L207. Wallbank, B., Holmes, J.K. (2001). Can. J. Phys. 79, 1237. Wallbank, B., Djuri´c, N., Woitke, O., Zhou, S., Dunn, G.H., Smith, A.C.H., Bannister, M.E. (1997). Phys. Rev. A 56, 3714. Walters, H.R.J. (1999). In: Whelan, C.T., Dreizler, R.M., Macek, H.R.J., Walters, J.H. (Eds.), “New Directions in Atomic Physics”, Kluwer Academic/Plenum, New York/Boston, p. 105. Walters, H.R.J., Sahoo, S., Gilmore, S. (2005). Nucl. Instrum. Methods B 233, 78. Wigner, E.P. (1946a). Phys. Rev. 70, 15. Wigner, E.P. (1946b). Phys. Rev. 70, 606. Wigner, E.P., Eisenbud, L. (1947). Phys. Rev. 72, 29. Younger, S.M. (1981). J. Quant. Spectroc. Radiat. Transfer 26, 329. Yu, Y., Seaton, M.J. (1985). J. Phys. B 18, 2577. Zatsarinny, O. (2006). Comput. Phys. Commun. 174, 273. Zatsarinny, O., Bartschat, K. (2004a). J. Phys. B 37, 2173. Zatsarinny, O., Bartschat, K. (2004b). J. Phys. B 37, 4693. Zatsarinny, O., Froese Fischer, C. (2000). J. Phys. B 33, 313. Zhang, H.L., Pradhan, A.K. (1995). J. Phys. B 28, L285. Zhang, H.L., Pradhan, A.K. (1997). Phys. Rev. Lett. 78, 195. Zhang, H.L., Nahar, S.N., Pradhan, A.K. (1999). J. Phys. B 32, 1459. Zhang, H.L., Nahar, S.N., Pradhan, A.K. (2001). Phys. Rev. A 64, 032719. Zhou, S., Li, H., Kauppila, W.E., Kwan, C.K., Stein, T.S. (1997). Phys. Rev. A 55, 361.
ADVANCES IN ATOMIC, MOLECULAR AND OPTICAL PHYSICS, VOL. 54
ELECTRON-IMPACT EXCITATION OF RARE-GAS ATOMS FROM THE GROUND LEVEL AND METASTABLE LEVELS JOHN B. BOFFARD, R.O. JUNG, L.W. ANDERSON and C.C. LIN Department of Physics, University of Wisconsin, Madison, WI 53706
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Electronic Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Neon and Argon . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Krypton and Xenon . . . . . . . . . . . . . . . . . . . . . . . 2.3. Helium as a Misfit . . . . . . . . . . . . . . . . . . . . . . . . 3. Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Optical Method . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Resonance Radiation Trapping . . . . . . . . . . . . . . . . . 3.3. Measurements with Ground State Targets . . . . . . . . . . . 3.4. Measurements with Metastable Targets . . . . . . . . . . . . 3.5. Special Optical Techniques . . . . . . . . . . . . . . . . . . . 4. Background: Excitation of Helium and the Multipole Field Picture . 4.1. Excitation out of the Ground Level . . . . . . . . . . . . . . . 4.2. Excitation out of He Metastable Levels . . . . . . . . . . . . 4.3. Multipole Field Model Applied to the Heavy Rare Gases . . . 5. Argon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Excitation out of the Ground Level . . . . . . . . . . . . . . . 5.2. Excitation out of Metastable Levels . . . . . . . . . . . . . . 5.3. Experimental Uncertainty of Excitation Cross Sections . . . . 6. Neon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. Excitation out of the Ground Level . . . . . . . . . . . . . . . 6.2. Excitation out of the Metastable Levels . . . . . . . . . . . . 7. Krypton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1. Excitation out of the Ground Level . . . . . . . . . . . . . . . 7.2. Excitation out of the Metastable Levels . . . . . . . . . . . . 8. Xenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1. Excitation out of the Ground Level . . . . . . . . . . . . . . . 8.2. Excitation out of the Metastable Levels . . . . . . . . . . . . 9. Comparison to Theoretical Calculations . . . . . . . . . . . . . . . 10. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1. Systematic Trends in Rare-Gas Cross Sections . . . . . . . . 10.2. Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3. Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . 319
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1. Introduction Electron-impact excitation of atoms has been a subject of research for over seven decades. It is one of the most basic processes in atomic physics and is of fundamental importance for many areas of science and technology such as atmospheric research, astrophysics, plasma science, gas discharges, and controlled nuclear fusion. Continual advances in experimental technology have made it possible to constantly explore new frontiers of research. Novel applications of the results to various related areas in turn stimulate further studies. Measurements of electron-impact excitation cross sections can be made by monitoring either the colliding electron or the target atom. In the first case the angular distribution of the scattered electrons with a given energy loss is measured (or inferred by detecting the target atom’s recoil) whereas the second approach utilizes the optical emission from a target atom that was excited by the projectile. Both techniques have been well documented in the literature. Our discussion in this chapter is limited to the second method (the “optical method”) which was used in the majority of excitation experiments on rare gases. Of all the rare gases, the largest bulk of research on electron excitation has been directed towards helium. The past decade, however, has seen a much higher level of activity devoted to the heavier rare gases which are interesting because of the vast variety of excited levels and the relevance of the cross sections to other areas of research and technology. In this chapter we focus on Ne, Ar, Kr, and Xe; studies of He are discussed only so far as they form the foundation for understanding the heavier rare gases. For orientation, the electronic structure of the excited levels is reviewed in the next section. Description of the experimental methods in Section 3 includes excitation from both the ground state and the metastable levels. Section 4 summarizes the main features of excitation of He that provide the background for studying the heavier rare gases. Results of excitation cross sections for Ar, Ne, Kr, and Xe are presented in Sections 5, 6, 7, and 8, respectively. The basic theoretical framework is that the electron-impact excitation processes discussed in this chapter are due to the Coulomb and electron-exchange interactions of the projectile with the target atoms. To understand the observed excitation behaviors of the various atoms we apply to this theoretical premise fundamental physical principles and simple, well-known analytic procedures to arrive at generalizations and rules that provide insights into the collision dynamics (Section 4). A very important objective of studying electron-impact excitation is to understand the major features and trends of the cross section data from the standpoint of basic principles in order to develop physical insight and an intuitive view that
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can be extended to other systems for a unified picture. Thus the semi-quantitative theoretical analyses indicated above is presented along with the description of experimental results of cross sections in Sections 5–8. Of special interest is to ascertain from the Ne, Ar, Kr, Xe series which aspects of the cross section behaviors are universal to the rare gases and which are specific to individual atoms. A full ab initio calculation of the electron excitation cross sections of rare gas atoms by rigorously solving the Schrodinger equations for the projectile-target system is well beyond the current computing capabilities. Strenuous approximations particularly for the heavier atoms must be made to make computation tractable. Even for electron excitation where the theoretical framework consists of only Coulomb and electron-exchange interactions the solution of the scattering equations is nevertheless exceedingly complicated. Comparison of experiments with theoretical calculations of cross sections serves to test the approximation steps employed in the latter. This is addressed in Section 9.
2. Electronic Structure The interpretation of electron-impact excitation cross sections is greatly aided by an understanding of the electronic structure of the levels involved. For example, in a strongly LS-coupled atom such as helium, the cross sections to various excited levels differ systematically depending on whether the final state is a singlet or triplet level (cf. Section 4.1). Hence, while the heavy rare-gases are generally not well characterized by LS-coupling, the LS-decomposition of the levels is still of considerable value. Furthermore, while the energy levels of the heavy rare-gases are generally similar, there are significant differences between the lighter (Ne and Ar) and heavier (Kr and Xe) ones. To represent a typical excited configuration we use the notation np 5 nl where nl pertains to the excited electron and n = 2, 3, 4, 5 for Ne, Ar, Kr, and Xe, respectively. 2.1. N EON AND A RGON The ground state of Ne is 2p 6 1 S0 . The electronic structure of an excited configuration 2p 5 nl is complicated by the coupling of the orbital and spin angular momenta of the 2p 5 core (lc and sc ) and of the outer nl-electron (lo and so ) to form the total angular momentum J through: (a) Coulomb and exchange of the 2p 5 core with the outer electron, (b) spin–orbit coupling of the 2p 5 core, and (c) spin– orbit coupling of the outer electron (which is much smaller than that of the 2p 5 core). If interaction (a) dominates, we have the standard LS-coupling which is not valid here. Nonetheless, it is convenient to start with the LS-representation and transform into the intermediate coupling (IC) by mixing LS-eigenfunctions of the same J . Thus the four levels of the 2p 5 3s configuration, designated in Paschen’s
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notation as 1s2 , . . . , 1s5 , are composed of 1 P1 , 3 P0 , 3 P1 , 3 P2 LS-terms, |1s2 = α|2p 5 3s,1 P1 + β|2p 5 3s,3 P1 , |1s4 = β|2p 5 3s,1 P1 − α|2p 5 3s,3 P1 , |1s3 = |2p 5 3s,3 P0 ,
(1)
|1s5 = |2p 5 3s,3 P2 . Of these four levels, the 1s2 and 1s4 levels both contain a 1 P1 component and therefore are optically connected to the 1 S0 ground state. The remaining 1s3 and 1s5 levels are dipole forbidden from decaying to any lower levels and are therefore metastable. Higher levels in the 2p5 ns (n > 3) configurations can be similarly decomposed into LS-eigenfunctions, but with different α and β coefficients. For example, the four levels of the 2p 5 4s configuration are designated as 2s2 . . . 2s5 in Paschen’s notation. Note that in these higher levels the J = 0 and J = 2 levels are no longer metastable since they can decay to 2p 5 3p levels. The 2p 5 3p configuration yields ten LS-terms: 1 S0 , 1 P1 , 1 D2 , 3 S1 , 3 P0 , 3 P1 , 3 P , 3 D , 3 D , 3 D , and barring configuration interactions the wave functions for 2 1 2 3 the ten levels (2p1 through 2p10 in Paschen’s notation) are obtained by mixing these LS-terms of the same J in the IC scheme. Of the ten levels only the J = 3 level (2p9 ) is a pure LS-entity (3 D3 ). The other nine levels all have both singlet and triplet characters. For qualitative applications a semi-empirical method [1,2] pioneered in the early days of quantum mechanics [3] is widely used to estimate the mixing coefficients in the IC wave functions. The twelve levels in the 2p5 3d configuration include two pure triplet LS-levels, 3 P and 3 F . While the IC wave functions of all the others are singlet-triplet 0 4 mixtures, it is not uncommon to find one with an exceptionally small coefficient of the singlet member so that it acts like a triplet level. The electronic structure of Ar (including Paschen’s notation used for labeling the resulting energy levels) is essentially the same as that described for Ne. Above the argon 3p 6 1 S0 ground state are the 1s3 and 1s5 metastable levels and the two 3 P1 /1 P1 mixed levels (1s2 and 1s4 ) from the 3p 5 4s configuration. The next configuration 3p 5 4p consists of ten levels (2p1 through 2p10 ); only one of which (2p9 ) is a pure triplet level. The Paschen designation of a few of the levels, however, differ from Ne to Ar. For example, in Ne the two levels with J = 0 are designated as 2p1 and 2p3 , whereas these levels are labeled as 2p1 and 2p5 in Ar. To minimize confusion, we will indicate a particular level’s J -value when appropriate. 2.2. K RYPTON AND X ENON Both Ne and Ar are not well approximated by any of the standard vector coupling schemes such as LS, jj , and j K; within a configuration only J is a good
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F IG . 1. Energy levels of the two lowest configurations of Krypton.
quantum number. In contrast, the 4p 5 core in Kr(4p 5 nl) has a strong spin–orbit coupling which is much larger than the interaction (Coulomb and exchange) of the core with the outer electron. A useful description is to introduce the core angular momentum jc ≡ lc + sc (with jc equal to 3/2 or 1/2) and then couple jc with lo and so to form J. This allows us to treat jc as a quantum number in the first approximation. Since we are not concerned whether jc and lo are first combined to form K as an intermediate step (j K coupling) or jo ≡ lo + so is chosen as the intermediary (jj coupling), we write the coupling scheme as [(lc sc )jc , lo , so ]J . Thus for the 4p 5 5s configuration we start with the two 4p 5 core levels (jc = 3/2 and 1/2) separated by 0.6 eV. The jc = 3/2 core level couples to the 5s electron to form a doublet with J = 2 and J = 1, and likewise from the jc = 1/2 core level we generate a pair of levels with J = 1 and J = 0. The four levels segregate into two tiers as depicted in Fig. 1. It is convenient to label the levels not only by J but also by jc which is, however, only approximately a good quantum number. For a more accurate description, admixture with the other value of jc possibly needs to be considered. The next configuration 4p 5 5p consists of a lower sextet (J = 1, 3, 2, 1, 2, 0) made from coupling jc = 3/2 to the outer 5p electron (lo = 1, so = 1/2) and an upper quartet (J = 1, 1, 2, 0) with a parentage of jc = 1/2. As evident in Fig. 1, the levels also segregate into two tiers; in Paschen’s notation the four jc = 1/2 upper tier levels are designated as 2p1 through 2p4 , and the six lower tier levels are labeled as 2p5 through 2p10 .
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F IG . 2. Energy levels of the np5 (n +1)s and np5 (n +1)p configurations of neon, argon, krypton and xenon. The zero of the energy scale for each atom has been set equal to the ionization energy of the atom.
In Xe we have an extreme case of very large splitting of 1.4 eV between the jc = 3/2 and jc = 1/2 levels of the 5p 5 core of the Xe(5p 5 6s) and Xe(5p 5 6p) configurations. As illustrated in Fig. 2, the two tier structure in each configuration is so accentuated that the upper tier of Xe(5p 5 6s) almost overlaps the lower tier of Xe(5p 5 6p). For comparison we include in the same figure the energy levels of the Ne(2p 5 3s), Ne(2p 5 3p), Ar(3p 5 4s), Ar(3p 5 4p), Kr(4p 5 5s), and Kr(4p 5 5p) configurations. The Ne and Ar levels do not exhibit the twotier structure of Kr and Xe. This is because the small spin–orbit splittings of the Ne(2p 5 ) and Ar(3p 5 ) cores, 0.097 eV and 0.17 eV, respectively, are overwhelmed by the coupling of the core with the outer electron. In addition to Paschen’s notation, Racah notation is also widely used for labeling excited levels of the heavy rare gases. In this scheme, based upon j K coupling, a np 5 nl level is labeled as nl[K]J , where K is the intermediate vector sum of jc and lo which may or may not be a good quantum number even as an approximation. A prime on nl indicates the level has jc = 1/2 whereas levels with jc = 3/2 are left unprimed. The total angular momentum J is the vector sum of K and so = 1/2. While Racah notation can be used to label all of the heavy rare-gases (Ne to Xe), the j K coupling scheme upon which it is based is not helpful for understanding the lowest excited configurations of Ne and Ar. Particularly for these atoms, the values of jc and K should be viewed as labels rather than as good quantum numbers.
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2.3. H ELIUM AS A M ISFIT For a unified view of electron-impact excitation processes we also discuss the case of helium in the context of the heavier rare-gas atoms. An excited configuration in helium consists of a 1s-core (lc = 0, sc = 1/2) plus an outer nl-electron (lo = l, so = 1/2). Unlike the np 5 -core for Ne, . . . , Xe, the 1s-core of He has no internal fine structure. Furthermore, LS-coupling is valid for the S, P , and D states of He, though not applicable to the F states [4]. Although the energy levels of (1s)(nl) are labeled by L, S, and J as 1 LJ and 3 LJ , the three members of the 3 L triplet corresponding to J = L and J = L ± 1 are so close together that they J are not resolved in collision experiments and only the sum of the cross sections for excitation into, for example, the J = 0, 1, 2 levels of n3 P is measured. If one is interested only in excitation into the entire n3 P manifold, one can equally well use the representation where L and S are decoupled (LSML MS representation) and visualize the process as excitation into all the ML and MS members. Thus the quantum number J is not needed to specify the final state. Each (1s)(nl) configuration turns into effectively two components, 1 L and 3 L, as opposed to ten levels for the heavy rare-gas (np)5 (np) configuration and twelve levels for (np)5 (nd). It is this large variety of energy levels for each (nl) excited configuration along with their very different excitation properties that makes the heavier rare gases intellectually interesting and technologically important.
3. Experimental Methods 3.1. O PTICAL M ETHOD The method for measuring excitation cross sections via optical measurements has been well documented in the literature [5–7], hence we only present the basic principles and definitions here. A mono-energetic electron beam with variable energy is passed through an atomic target which can be either a static gas target in the case of ground state excitation, or some form of an atomic beam in the case of excitation from metastable levels. In steady state, the rate of decay out of a particular level is equal to the rate at which atoms are excited into the level. The former can be measured by monitoring the fluorescence whereas the latter is proportional to the desired excitation cross section. Fluorescence from atoms excited to level-i decaying into level-j (i → j ) is monitored at wavelength λij , and the number of such photons emitted per unit time per electron beam length, Φij , is measured to give the optical emission cross section, Qij . Assuming the only mechanism for populating excited levels is electron-impact excitation, and the only mechanism for depopulating any excited level is radiative decay, we write opt
Qij =
Φij , n0 (I /e)
(2)
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where I is the electron beam current, e is the electron charge, and n0 is the number density of target atoms. Note that in writing Eq. (2) we have made the further simplifying assumption that the target density is uniform (as would be the case for a static gas target); a more complicated integration over the electron current density, atomic beam density and spatial detection efficiency is needed for beam targets. Summation of the emission cross sections for all transitions out of the level-i gives the apparent excitation cross section (also sometimes referred to as the effective cross section) which includes the contribution from both direct excitation and cascades from higher-lying levels excited by the same electron beam that decay into level-i. The direct excitation cross section (also sometimes referred to as the level cross section) for level-i can be obtained from measurements of the optical emission cross sections for all transitions out of and into level-i, opt opt app casc Qdir (3) = Qij − Qki . i = Qi − Qi j
k>i
The i → j and k → i emission wavelengths for the rare gases span a wide spectral range from the vacuum ultraviolet (for the resonance transitions to the ground state) to the far-infrared (IR). Detection schemes employed for measurements covering this large spectral range are discussed in Section 3.3. Rather than measure all of the optical emission cross sections out of a level to determine the apparent cross section, if the branching fraction, Γij , for the i → j transition is known, one need only measure the i → j optical emission cross section (which may have a favorable wavelength) and obtain the apparent cross section from app
Qi
=
1 opt Q . Γij ij
(4)
In addition to the basic principles described above, there are a number of techniques that extend the optical method to special cases. For example, the optical method can not be employed to measure cross sections into metastable levels, since these levels do not decay via radiative transitions; but this limitation can be overcome by the technique of laser-induced-fluorescence (LIF). To create an optical signal, a laser is used to transfer the population of atoms from the metastable level to an excited level which decays to a lower level producing a detectable fluorescence signal (cf. Section 3.5.1). Another extension to the optical method is to differentiate the direct and cascade contributions to an optical emission cross section by making time-resolved measurements as described in Section 3.5.2.
3.2. R ESONANCE R ADIATION T RAPPING One potential limitation of the optical method involves the assumption that only electron-impact excitation is responsible for populating excited levels. If the target
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density is high enough, a photon emitted by a particular transition (usually terminating in the ground state) can be reabsorbed by another atom before it can leave the collision chamber thus providing an additional excitation mechanism. When this second atom decays, the photon is reradiated in a random direction, which is unlikely to be in the direction of the detector. As an example, Tsurubuchi et al. [8] have measured the apparent cross section into the Kr(1s4 ) resonance level by monitoring the 123.6 nm emission line from this level as it decays to the ground state using a vacuum ultraviolet (VUV) monochromator and channel electron multiplier (cf. Section 3.3). For an optical path length between the electron excitation region and the entrance slits of the VUV monochromator of 48 cm, the transmission of the 123.6 nm resonance radiation decreased to one half at a pressure of 6 × 10−6 Torr [8]. Thus, measurements of resonance lines must either be made at very low pressures, or corrected for reabsorption. In addition to the two principal resonance levels of the np 5 (n + 1)s configuration of each rare gas, there are numerous other J = 1 resonance levels in np 5 ns (n > n + 1) and np 5 n d configurations that are also optically connected to the J = 0 np 6 ground state. However, these higher resonance levels can also decay to other excited states, without the decay of a photon that can be reabsorbed by a ground state atom. To differentiate these higher resonance levels from the principal resonance levels that can only decay to the ground state, we refer to the former as resonant levels. For example, whereas the 1s2 and 1s4 levels of the 3p 5 4s configuration of Ar are resonance levels, we refer to the 2s2 and 2s4 levels of the 3p 5 5s configuration as resonant levels. In addition to their decay channels to the ground state (at wavelengths of 88.0 and 87.0 nm, respectively), these 3p 5 5s levels also decay into the levels of 4p 5 5p configuration with transition wavelengths in the range of 0.92 to 2.03 µm. As with the principal resonance levels, emissions to the ground state from these resonant levels can be reabsorbed by a nearby Ar atom repopulating the 2s2 or 2s4 level. This second atom can then either decay to the ground state, or it can alternatively decay to a level of the 3p 5 4p configuration. In case of the latter, the photon escapes from the collision chamber, while in the former the photon is still trapped. Each reabsorption/emission cycle generates additional emissions for transitions terminating in 3p 5 4p levels at the expense of emissions to the ground level. Since the probability of a photon being reabsorbed increases with the target density, this recycling process results in the 3p 5 5s → 3p 5 4p emission cross sections having a dependence on target gas pressure. In Fig. 3 the increase in the Ar(2s2 → 2p3 ) optical emission cross section with increasing target gas pressure is displayed. Gabriel and Heddle have presented a quantitative analysis of resonance radiation trapping and its effect on the measurement of electron-impact excitation cross sections [9,12]. The key parameter in their model is g, the fraction of resonant photons that escape the collision region, which in turn is a function of the target gas pressure. This function can be expressed as a universal function of the
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F IG . 3. Pressure dependence of the 1300.8 nm Ar(2s2 → 2p3 ) optical emission cross section at an incident electron energy of 100 eV. The solid line is a fit to the data using the radiation trapping model of [9] along with the transition probabilities of [10] and [11].
dimensionless quantity kρ where ρ is the characteristic collision radius which depends on the experimental geometry (i.e. the radius of the Faraday cup) and k is the absorption coefficient of the resonant line which is proportional to the gas pressure, P . At very low pressures essentially all the resonant photons escape, so g(kρ) → 1; whereas at very high pressures virtually all of the resonant photons are reabsorbed, i.e. g(kρ) → 0. Between these two limits, the i → j optical emission cross section can be written as opt
Qij (P ) =
Aij app Q , g(k(P )ρ)Ai0 + n
(5)
where the subscript 0 represents the ground level. When the transition probabilities are known with some accuracy, Eq. (5) does an excellent job of reproducing the measured pressure dependence of resonant levels as displayed in Fig. 3 for the Ar(2s2 → 2p3 ) optical emission cross section. Due to the dependence on the characteristic collision length ρ, which is a function of the experimental geometry, the exact pressure dependence of cross sections (as in Fig. 3) measured in other laboratories may not be identical to the curves shown here. The value of ρ = 1.4 cm that describes our experimental apparatus (Section 3.3), however, is reasonably characteristic of most electron-beam experiments. Resonance radiation trapping affects cross section measurements of the noble gases in three major ways. First, as previously stated, optical emission cross sections for the VUV resonance transitions need to be corrected for the signal decrease due to radiation trapping. Secondly this radiation trapping, however, can also be used to the experimentalist’s advantage. By taking measurements at
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high pressures, in the limit where the resonance transition is completely reabsorbed, it is possible to measure cross sections for resonant levels without any VUV measurements of the resonance transition. The use of Eq. (5) to measure cross sections for resonant levels is illustrated in Section 5.1.3. Third, radiation trapping of resonant levels can affect in a very profound way the measurements of emission cross sections for non-resonance levels via pressure-dependent cascades from resonant levels. For example, the observed pressure dependence of the non-resonance Ar(3p5 4p) levels introduced by cascades will be illustrated in Section 5.1.1. 3.3. M EASUREMENTS WITH G ROUND S TATE TARGETS The primary challenge in measuring absolute optical emission cross sections for ground state atoms is the detection of transitions over a wide spectral range. For example, to measure all of the optical emission cross sections for the four lowest excited configurations of Ar would require measurements for wavelengths ranging from 87 nm to 2.4 µm. Before turning to the measurements in the ultraviolet and infrared regions, let us first describe the techniques necessary to measure cross sections in the ‘visible’ region. Figure 4 illustrates the experimental apparatus used by Chilton et al. [13] to measure the Ar 3p 5 4s − 3p 5 4p transition array. A detailed description of how this apparatus is used to measure cross sections is found in [5]. A stainless-steel vacuum chamber is evacuated to a base pressure of 10−8 Torr before being filled with ultra-high purity gas. Since the heavy rare gases are inert, a getter pump can be left open to the chamber while data is being collected. This pump maintains the purity of the target by removing molecular gases that are given off by the electron-gun’s heated cathode. The pressure of the gas in the chamber is recorded to high accuracy by either a capacitive manometer or a spinning rotor gauge. A variable-energy, mono-energetic electron beam is formed by a pentodegeometry electron gun using an indirectly heated BaO cathode. The beam current is measured by a deep, shielded Faraday cup. Fluorescence from excited atoms exits the Faraday cup through a narrow slit and is imaged using a spherical mirror onto the entrance slits of a monochromator and ultimately detected by a photomultiplier tube (PMT). The spectral sensitivity of the optical system is determined by measuring the signal from a calibrated spectral-irradiance lamp. Photon counting along with a modulation of the electron beam is used to extract the electron excitation signal from background sources. In terms of experimentally measurable quantities, Eq. (2) for the optical emisopt sion cross section, Qij , translates into exc
opt
Qij =
4π e kT H λij F (λij ) Sij , lamp hc IP Ω Sij
(6)
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F IG . 4. Apparatus for measuring electron-impact excitation cross sections for a static ground state target using a monochromator [5]. Fluorescence signal from electron-impact excitation within the collision chamber passes though the fused-silica window W , is reflected off the rotatable mirror M1 and directed into the 1.26 m monochromator. Radiometric calibration is performed by rotating mirror M1 towards a standard lamp and measuring the signal. The static target gas pressure is measured with either a capacitive manometer or a spinning rotor gauge (SRG).
where λij is the wavelength of the observed transition, Sijexc is the observed electron-excitation signal recorded by the PMT integrated over the width of the lamp bandpass of the monochromator, Sij is the signal from the standard lamp which has spectral irradiance F (λij ), H is the height of an auxiliary slit used in the lamp calibration portion of the experiment, Ω is the solid angle of the electron beam as observed by the optical system, I is the electron beam current, P is the target gas pressure, T is the gas temperature, and e, k, h and c are the standard atomic constants. Of these quantities, only a few contribute significantly to the uncertainty of optical emission cross sections; others, such as geometric quantities Ω and H , can be measured quite accurately. For example, the lamp calibration procedure introduces approximately 8% to the total uncertainty via the 2% uncertainty in the lamp calibration, the statistical uncertainty in the lamp signal (5%) and an additional systematic uncertainty of 5% introduced by the calibration of neutral
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density filters used to reduce the intensity of the lamp. While the pressure of gas can be measured to high accuracy (1%) with the spinning-rotor gauge, the uncertainty in the target number density is dominated by the systematic uncertainty (5%) introduced by the correction to the gas temperature T due to the localized heating of the gas in the collision region by the hot cathode [14]. In total, the systematic uncertainty of the optical emission cross section measurements is estimated to be 12–15%. An excellent test of the radiometric calibration procedure and associated uncertainties is obtained by comparing the experimentally measured values for the He (n1 S → 21 P ) optical emission cross sections with the benchmark values measured by Van Zyl et al. [15]. For example, Chilton et al. and Weber et al. [16] have used the apparatus in Fig. 4 and the technique described in the preceding paragraph to measure the He emission lines at 443.8, 504.8, and 728.1 nm and found them to agree with the values of Van Zyl et al. to within 4 to 10%, well within the ±15% estimated uncertainty. Comparisons of cross section measurements among different experimental groups is also greatly aided by this common technique (i.e. [16,17]). Since the path of the projectile electron defines an externally imposed axis on the atomic target, the excited atomic state may be oriented along this axis. This leads to an anisotropy in the emitted fluorescence and polarization of this fluorescence (see for example [6]). Much can be learned about the excitation process by studying the polarization of fluorescence. Indeed, a rich body of work on this subject exists in this context of the study of heavy rare-gas atoms [18–20]. However, for the purposes of measuring cross sections, the important point is that due to the anisotropy in the emitted fluorescence, the light emitted at 90◦ to the electron beam is not representative of the average fluorescence, but must be corrected by a factor of (1 − P /3) where P is the polarization of the emitted light [5,6]. Typical measured polarizations of the np 5 ns–np 5 np emission arrays in the heavy rare gases are less than 10% [13,16,21], which leads to a very small 3% correction to the measured cross sections. In the ultraviolet region of the spectrum, the difficulties that exist in the visible/PMT range (radiometric calibration, polarization correction, etc.) are magnified by the limited choice of UV transmitting materials. In the near-UV (200– 400 nm), there are still suitable windows (MgF2 , CaF2 ), and deuterium lamps (which have little visible/IR output and thus limit the problem of scattered light) that can be used as calibration lamps. But at the wavelengths of the resonance lines of the heavy rare-gases, no suitable window materials are generally available, and the radiometric calibration is particularly difficult as detailed in the review article of van der Burgt et al. [22]. In place of photomultiplier tubes, the VUV emissions are typically detected with a channel electron multiplier. To eliminate the problems associated with the polarization correction, VUVmeasurements are often conducted in a polarization-free geometry [17,23]. At the
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F IG . 5. The effects of polarization on the measurement of electron-impact excitation cross sections can be eliminated by taking data in the double-magic angle configuration as in [17]. Fluorescence is detected along an axis oriented at 54.7◦ to the electron beam axis. Since the light projected along this axis can still be polarized, detection with a polarization sensitive detector (such as a grating monochromator) must also be compensated for by orienting the detector at 45◦ .
‘magic angle’ of 54.7◦ to the electron beam axis, the intensity of the light is equal to the isotropic average intensity, independent of the polarization of the light [6]. The light emitted along this direction, however, is still polarized. For detection systems that have a strong polarization sensitivity (such as grating monochromators), polarization can introduce serious errors. This polarization effect can also be compensated for by orienting the detection system at 45◦ to the electron beam as illustrated in Fig. 5. At the opposite end of the spectrum, the main problem associated with measuring cross sections in the infrared is the poor detector sensitivity in this spectral range in comparison to those available in the visible/UV (although there are now available photomultiplier tubes with InGaAs photocathodes, sensitive to 1.7 µm). The low detector sensitivity combined with the large number of emission lines that need to be measured, pose a difficult challenge to measuring rare-gas infrared transitions one-at-a-time. For example, there are 75 emission lines (almost all of which are infrared) in the Ar(3p5 4p–3p 5 3d) transition array. These transitions form the bulk of the cascades into the 3p 5 4p levels whose emissions are in a far more favorable (PMT) spectral range. Until these infrared cascades could be measured, however, experiments were generally limited to only reporting apparent cross sections for the 3p 5 4p levels. The problem of measuring the infrared cascade radiation was resolved by the pioneering efforts of DeJoseph and Clark [24] whereupon they used a Fourier transform spectrometer (FTS) to detect infrared radiation from an electron-beam excitation experiment. The FTS introduces two main advantages over sequential measurements made with a monochromator. First, for a given resolution, an FTS allows a much larger solid angle of the emitted fluorescence to be collected. Second, the FTS system possesses a multiplex advantage in that all emission lines within the range of the detector are collected
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F IG . 6. Apparatus for measuring electron-impact excitation cross sections in the near-IR portion of the spectrum using Fourier transform spectroscopy [13,21]. Fluorescence signal from electron-impact excitation within the collision chamber passes though the CaF2 window W1 , and is collimated by the rotatable, off-axis parabolic mirror M1 . The relative spectral response of the FTS is determined by measuring the signal from a black-body source along an identical optical path.
in parallel, rather than sequentially. Combined, these two advantages offset the losses due to the poor sensitivity of the IR detectors available. The apparatus used by Chilton et al. [13,25,26] and Fons and Lin [21] to measure near-IR optical emission cross sections for the rare-gases is displayed in Fig. 6. The vacuum chamber and electron beam are essentially the same as the visible-wavelength apparatus described earlier. The main difference is the use of an off-axis parabolic mirror to collect a larger solid-angle of emitted light from the collision chamber, and the use of a commercial FTS system (based on a Michelson-interferometer geometry). Three different IR detectors can be used depending on the wavelength range being studied: Ge and InGaAs detectors both have good sensitivity in the 0.85–1.7 µm range, whereas InSb detectors (which has much lower sensitivity) were used in the 1.3–6.3 µm range.
3.4. M EASUREMENTS WITH M ETASTABLE TARGETS Measuring excitation cross sections from the ground state of the rare gases is straight forward. Most experiments use a static gas target, although a gas jet target has also been used in some experiments to reduce the reabsorption of resonance radiation. Measurements of excitation cross sections out of metastable atoms are
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much more complicated since a source of the metastable atoms is required. At least three main varieties of sources have been employed to produce heavy rare gas metastable atoms: (i) discharges, (ii) near-resonance charge exchange, and (iii) electron-excitation. 3.4.1. Hollow Cathode Discharge Source The hollow cathode discharge displayed in Fig. 7 has been used to measure electron-impact excitation cross sections out of the metastable levels of all the rare-gas atoms [27,29–32]. As described in greater detail in [28], metastable atoms are created in a ∼3 Torr hollow cathode DC discharge. For studies with Xe, a 85% He-15% Xe gas mixture was required for a stable discharge [30]. Atoms flowing out of a small 1 mm hole located at the base of the hollow cathode form a thermal atomic beam containing approximately 107 –108 metastable atoms/cm3
F IG . 7. Hollow cathode discharge experiment for studying electron-impact excitation of rare-gas atoms in metastable levels [27,28]. Atoms in the 1s3 and 1s5 metastable levels are created in the hollow cathode discharge and flow out a 1 mm hole at the base of the discharge. The effusive atomic beam is crossed at right angles by a mono-energetic electron beam. The fluorescence from excited atoms is collected by a lens, passed though a narrow-band interference filter (for spectral isolation), and imaged onto a PMT. A single-mode Ti:sapphire laser is used to either (a) measure the 1s5 : 1s3 number density in the target, or (b) remove the 1s5 atoms in the target via laser quenching.
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along with approximately 1013 –1014 ground state atoms/cm3 . A mono-energetic electron beam located 1 cm below the base of the cathode intersects the atomic beam at right angles and excites the metastable atoms into higher levels. Fluorescence from the decay of excited atoms is collected by a lens, passed through a narrow-band (0.3–1.0 nm FWHM) interference filter for spectral isolation, and detected by a photomultiplier. In comparison to the monochromator arrangement used for ground state measurements, the use of interference filters greatly increases the solid angle from which data is collected—a necessity with the small signal sizes. The optical axis is oriented at 60◦ relative to the electron beam. No polarization corrections are necessary since this is close to the “magic angle” of 54.7◦ and the optical system itself is polarization insensitive. The atomic beam contains atoms in the ground state and the two metastable levels. To ensure that the detected radiation is solely due to excitation from metastable levels, the energy of the electron beam is kept below the threshold for excitation out of the ground state. To separate out the signal contributions from the two metastable levels, one must have a pure target with atoms in only one of the metastable levels. For Ar this was achieved by Piech et al. [27,33] using a single-mode Ti:sapphire laser pumped by an 8 W argon-ion laser to quench atoms in the 1s5 metastable level. The 801.5 nm laser beam, positioned immediately below the base of the hollow cathode (which ensures that each atom interacts with the laser prior to entering the collision region), pumps atoms from the 1s5 (J = 2) metastable level into the 2p8 (J = 1) level. Atoms in the 2p8 level subsequently decay to either the 1s2 or 1s4 resonance levels or back to the 1s5 metastable level. Those atoms that decay into the 1s2 and 1s4 resonance levels quickly decay to the ground state, effectively removing them from the target. Atoms that leak back to the 1s5 metastable level interact with the quenching laser beam repeatedly so as to be pumped back to the 2p8 level, allowing additional opportunities to decay to the resonance levels. With a laser power greater than 500 mW, more than 95% of all atoms in the 1s5 metastable level are eliminated from the target, so that any remaining electron-impact excitation signal can be unambiguously assigned to excitation from the 1s3 level [27]. In a similar fashion, the Kr(1s5 ) or Kr(1s3 ) metastable levels can also be quenched [31]. From their statistical weights, the ratio of atoms in the 1s5 (J = 2) and 1s3 (J = 0) metastable levels is expected to be 5 : 1. This ratio can be measured directly using laser-induced fluorescence (cf. Section 3.5.1). We use the same single-mode Ti:sapphire laser as in the quenching procedure except that the beam is now positioned so that it interacts with atoms in the collision region and the power is reduced to several µW using neutral density filters. The fluorescence from the excited atoms is collected using the same optics used to measure the electron-excitation signal. Atoms in one of the metastable levels are laser pumped into an upper 2px level and fluorescence from that upper level is observed. This is then repeated for the other metastable level and the ratio of the two fluores-
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cence signals is analyzed to yield the number density ratio 1s5 : 1s3 . For argon, atoms in the metastable levels were pumped into the 2p4 level using the 714.7 nm (1s5 –2p4 ) and 794.8 nm (1s3 –2p4 ) transitions while the fluorescence from the 794.8 nm emission line was monitored. These measurements yielded a ratio of (5.6 ± 1.6) : 1 which is close to the statistical weighting of the levels [33]. In the cases of krypton and xenon, a common upper level cannot readily be used; this requires the use of two separate upper levels and additional calibration steps to compensate for measuring fluorescence from different transitions [30]. In Kr the 1s5 : 1s3 number density ratio was measured to be (12 ± 2) : 1 [34], and a ratio of >200 : 1 was found for the Xe target [30]. These far-from-statistical ratios are due in part to the large energy difference between 1s5 and 1s3 energy levels in Kr and Xe. 3.4.2. Charge Exchange Fast Beam Source The limitation on electron energies less than the ground state excitation threshold can be overcome by using near-resonant charge exchange between a fast ion beam and an alkali vapor target to produce a target with much higher fraction of metastable atoms. For example, argon metastable atoms can be created via charge exchange between an Ar ion beam and a Cs vapor target via the reaction Ar+ (3p 5 ) + Cs → Ar(3p 5 4s) + Cs+ + E,
(7)
where the energy defect E is difference between the ionization potential of the Ar energy level of interest (i.e. 4.04 eV to 4.21 eV) and the ionization potential of Cs (3.89 eV). A considerable amount of research on near-resonant charge exchange has found that this process produces a neutral beam with an appreciable fraction (∼0.5) of the atoms in metastable levels [35–37]. The remainder of atoms in the fast beam target are ground state atoms that arise from the decay of atoms created in the 3p5 4s J = 1 resonance levels, with a negligible fraction formed directly in the ground state due to the very large energy defect between the Cs ground state and Ar atom ground state (ionization potential of 15.76 eV). Similarly, near-resonant charge-exchange cross sections for the production of the other heavy rare-gas atoms in metastable levels are also relatively large [36–39]. Since the cross sections for electron-impact excitation from metastable levels are two to three orders of magnitude larger than the corresponding cross sections from the ground state, the approximately 50% of ground-state atoms in the fast beam target contribute a negligible fraction to the electron-excitation signal at all electron energies. Charge-exchange fast beam sources have been used in the past to measure electron-impact ionization cross sections for atoms in metastable levels [41,42]. The apparatus illustrated in Fig. 8 is the first used to study electron-impact excitation cross sections [40,43]. An rf-ion source is used to create a 1.6–3 keV ion
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F IG . 8. Apparatus for studying electron-impact excitation of metastable atoms using a fast beam target produced by charge-exchange [40].
beam of the desired heavy rare-gas. The ion beam is passed though a recirculating Cs vapor target, creating atoms in all of the np 5 (n +1)s levels. A set of deflection plates in the next chamber removes the remaining ions from the beam. Atoms created in the two J = 1 resonance levels decay to the ground state before reaching the collision region, yielding an atomic target of approximately 50% metastable atoms and 50% ground state atoms (with the exact ratios depending on the particular rare gas [27,29,30,34]). The fast neutral beam is crossed at right angles with a variable-energy, mono-energetic electron beam and the resulting fluorescence is detected by an optical system orientated at right angles to both beams. This optical system is essentially identical to that used with the hollow-cathode discharge source, with the substitution of a high-sensitivity GaAs PMT optimized for detecting the very weak excitation signal. The number density of metastable atoms in the target is determined by measuring the spatial distribution of atoms in the fast beam target and the ‘particle current’ of the neutral beam [40,43]. Since the number density of metastable atoms in the target is determined with a reasonable degree of accuracy, the fast beam target can be used for absolute cross section measurements [43]. The optical sensitivity is determined by measuring the optical signal for a ground state target formed by filling the collision chamber with gas. The metastable excitation cross section is then determined relative to the known cross section for excitation from the ground state (i.e. [13]). The fast beam target thus has two primary advantages over the previously described hollow cathode discharge source: the ability to take measurements at high electron energies, and the ability to make absolute measurements. The rather substantial limitation of the fast beam target is the very low target density (106 cm−3 ) which limits measurements to only the largest electron-impact excitation cross sections. 3.4.3. Electron-Beam Production of Metastable Atoms A third route used to create atoms in metastable levels is electron-beam excitation. Mityureva, Smirnov and Penkin have reported results using this technique for the
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heavy rare gases [44–47]. In their experiments, a cell is filled to a pressure of 40–100 mTorr of gas, and a high energy (25 eV in the case of Ar) electron pulse is used to excite atoms into the metastable levels (and higher levels that cascade into the metastable levels). After the population of atoms in metastable levels saturates (20 µs), the energy of the electron beam is shifted to a value below the threshold for ground state excitation. After a short delay (10 µs) to allow the decay of atoms in non-metastable levels, any excitation signal detected during this low energy pulse is attributed to excitation from metastable levels. The authors claim to achieve metastable densities on the order of 1010 cm−3 with electron currents on the order of 1 mA and pressures on the order of 10−1 Torr [44]. Absolute calibration is performed by comparing the metastable excitation signal to higherenergy signals due to ground state excitation for which the cross sections are known. The 1s5 metastable target density is determined from a separate absorption measurement. The results obtained by Mityureva et al. differ substantially from the measurements made in our laboratory using the hollow cathode and charge-exchange sources described earlier. In general, the energy dependence of measured cross sections are in reasonable accord, although there are exceptions for each of the heavy rare-gases studied. The more important difference lies in the magnitude of the cross sections measured in the two experiments. For Ar, the results of Mityureva et al. [47] are on average a factor of six larger than those of Boffard et al. [33]; for Kr, the values of Mityureva et al. [45] are approximately a factor of 20 larger than those of Jung et al. [34]; and for Xe, the results of Mityureva and Smirnov [46] are larger than the results of Jung et al. [30] by a factor of five. These differences far exceed the combined uncertainties in the two sets of experiments. 3.4.4. Other Sources of Metastable Atoms Excitation of metastable atoms can also be studied in the afterglows of pulsed plasmas. While the plasma is ‘on’, the electron temperature is high, and a large population of metastable atoms builds up. Immediately after the plasma is turned ‘off’ the electron temperature drops to a very low value, and all excited levels except the metastable levels decay away. Due to the low electron temperature, any excited levels populated during this time will generally be due to excitation from metastable levels. For example, Kolokolov and Terekhova [48] have derived metastable excitation rates into the Kr(4p5 5p) levels as a function of the electron temperature from afterglow measurements in a He–Kr plasma. Further analysis allows one to extract cross sections from these emission rates. However, the cross sections derived in this manner differ by about a factor of five in magnitude from measurements made with an electron-beam [31]. DeJoseph and Demidov [49] have also recently studied the role of excitation from metastable levels of Ar in
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populating Ar(3p 5 5p) levels in the afterglow of a pulsed inductively coupled plasma.
3.5. S PECIAL O PTICAL T ECHNIQUES 3.5.1. Laser-Induced Fluorescence The measurement of cross sections into metastable levels is complicated by the fact that the metastable levels do not radiatively decay. Hence, alternative methods other than direct optical emission must be adopted to determine the number density of the metastable atoms produced by electron impact. This can be accomplished by the technique of laser-induced fluorescence illustrated in Fig. 9 [50]. An electron beam excites atoms from the ground level g into a metastable level m. The metastable atoms are pumped to a higher level a by means of a laser tuned to the m → a absorption wavelength and the subsequent a → b fluorescence is measured. However, the electron beam also excites atoms into level a. The difference of the a → b fluorescence with the laser on and with the laser off is due to the laser-induced-fluorescence. If one uses a high-power cw laser so that all the metastable atoms are pumped to level a, the laser-induced a → b emission rate, corrected for the branching fraction, gives the production rate of the metastable atoms by electron collision (apparent excitation cross section). If the laser is not intense enough to pump all the metastable atoms into level a, the laser-induced a → b fluorescence gives only the relative apparent excitation cross section as a function of the electron energy. For absolute calibration we recognize that the
F IG . 9. Scheme of laser-induced fluorescence.
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direct excitation cross section for the metastable levels (3 P0 and 3 P2 ), which are purely triplet levels, diminishes steeply with increasing electron energy (cf. Section 4.1). At 100 eV, for instance, the metastable level’s population is essentially entirely due to cascade, so that the uncalibrated relative apparent excitation cross section of the metastable level can be set equal to the total cascade cross section from the higher levels which can be measured absolutely. Once the absolute scale at 100 eV is established, absolute apparent cross sections at other energies are obtained from the relative measurements. This calibration step can be performed at any energy above 75 eV making no difference in the final results as confirmed experimentally [50]. Alternatively a pulsed laser also works for this experiment [51]. With a steady electron beam the metastable atoms are pumped to level a by a pulsed laser over a short duration tL (∼10 ns). Immediately after the laser pulse the a → b fluorescence is collected over a period τ which is large compared to the lifetime of the signal decay so that the a → b fluorescence has reached its steady-state value which corresponds to electron excitation from the ground level into level a. Again subtracting the laser-off signal from the laser-on signal allows us to extract the laser-induced part which is proportional to the apparent excitation cross section for the metastable level so that the calibration procedure for the cw laser experiment is also applicable here. 3.5.2. Time-Resolved Measurements While optical emission cross sections are the fundamental quantity in experimental measurements based on the optical method, direct excitation cross sections are the more fundamental quantity from the perspective of collision dynamics. The extraction of the direct cross section for level-i from a set of optical emission measurements requires one to make measurements of all the cascade transitions into level-i. Alternatively, one can make time-resolved measurements of a transition out of level-i. To illustrate how this is possible, assume a target is excited by an electron beam until the decay rate out of each level is equal to the production rate via direct excitation and cascades. If the electron beam is suddenly turned off, the fluorescence from level-i will decay away with a time dependence governed by the lifetime of level-i and the lifetimes of the levels cascading into level-i. The magnitude of the components at various decay rates can be converted into individual cascade cross sections. While this technique works well on a gas like He (see for example [52]) where only a limited number of excited levels generally contribute to the cascades, in the heavy rare gases many levels in each higher configuration cascade into the level of interest. The separation of the components at each decay rate poses a difficult challenge. Direct cross sections can also be measured via time-resolved techniques by observing the fluorescence signal at the end of a very short electron beam pulse. If
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the length of the pulse is much shorter than the lifetimes of cascading levels, the prompt fluorescence is dominated by the contribution from direct excitation. Bogdanova and coworkers, in particular, have used this technique to measure direct cross sections for excitation of np 5 (n + 1)p levels of the rare gases [54–56]. It is interesting to note that the use of a fast beam target, as used in the creation of metastable atom targets (Section 3.4), also permits a form of time resolved measurements by studying the spatial dependence of fluorescence from the decay of excited atoms after passing though the electron beam [57]. The velocity of keV energy atoms is on the order of 10−2 cm/ns. Hence, an electron beam of width ∼0.2 cm corresponds to a short electron beam pulse of ∼20 ns. Fluorescence from atoms decaying at the location of the electron beam thus are dominated by direct excitation (as in the preceding paragraph), whereas the fluorescence signal far downstream from the location of the electron beam is dominated by cascades from long-lived levels. Studies of the spatial dependence of the downstream fluorescence signal can thus be used to place limits on the cascade contribution to the measured signal. For example, the cascade analysis of the Ar(1s5 → 2p9 ) metastable excitation cross section is presented in Fig. 10.
F IG . 10. Cascade analysis of the Ar(1s5 → 2p9 ) excitation cross section at an incident electron energy of 50 eV made with a fast beam target [53]. The dotted line is the spatial signal pattern for a static target (including the widths of the electron beam and optical system). The solid line is a model calculation that includes the motion of atoms in the 2.1 keV beam, and the 30 ns lifetime of the 2p9 level. The dashed line also includes a 20% cascade contribution from the cascading level with the shortest possible lifetime.
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4. Background: Excitation of Helium and the Multipole Field Picture There is a large amount of research on electron-impact excitation of He in the literature (see review article [7] and references therein). Much of the findings have become the foundation of this field. In this section we will highlight only the results that are most relevant in understanding the excitation behaviors of the heavier rare-gas atoms and then present a multipole field model to analyze the trends observed in the cross sections for He. This will be followed by an application of the multipole field picture to the heavy rare-gas atoms in Section 4.3.
4.1. E XCITATION OUT OF THE G ROUND L EVEL Early experiments on electron excitation out of the He 11 S ground level into a final level revealed a fundamental correlation of the excitation cross section and its energy dependence to the character of the final level. In Fig. 11 we show the excitation functions of the 31 S, 31 P , 31 D, 33 S, 33 P and 33 D levels; each curve is separately normalized to the same peak height [58]. Levels of the same spin, same L, and different n exhibit a very similar excitation function except for a decreasing peak magnitude with increasing n and a slight shift in the energy scale due to differences in threshold energy [58]. Three kinds of excitation functions are evident. For final states that are optically connected to the initial level (11 S →
F IG . 11. Normalized direct excitation cross sections for He(11 S → n1 L, n3 L) excitation processes [58].
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n1 P ) we find a broad maximum at about four times the threshold energy followed by a slow decline. On the other extreme is the very narrow peak with a steep decline with energy exhibited by the 33 S, 33 P and 33 D excitation functions. It may at first appear that such a 11 S → n3 L spin-changing excitation would require a magnetic spin-dependent interaction between the collision partners which is much smaller than the Coulomb interaction responsible for the spin-preserving excitations. In reality, however, the mechanism for changing spin involves not a direct spin flip, but instead can be pictured as an electron exchange process in which the incident electron is captured in the excited-state orbital and the atomic electron with opposite spin to the incident electron is ejected. A major difference between the Coulomb interaction and electron exchange mechanisms is that the former is felt so long as the incident electron is in the proximity of the target atom whereas the latter requires a significant direct overlap in charge cloud between the colliding and atomic electrons. At higher electron energies the duration of overlapping encounter is short causing a drastic decline in the cross section. Intermediate between the very broad and very narrow varieties are the excitation functions of the n1 S and n1 D levels which correspond to spin-allowed but dipole-forbidden transitions from the initial level. The magnitude of the cross sections also depend on the nature of the transition. For example the dipole-allowed excitation into the 31 P level has a much larger peak cross section (3.2 × 10−18 cm2 ) than the spin-preserving, dipole-forbidden excitation into the 31 S and 31 D levels (0.47 × 10−18 and 0.35 × 10−18 cm2 , respectively) [58]. In the spin-changing series the peak cross sections for 33 S and 33 P levels (0.89 × 10−18 and 0.67 × 10−18 cm2 , respectively) are larger than the 33 D cross section (0.25 × 10−18 cm2 ) but are still much smaller than the 31 P cross section. Because of the steep decline of the spin-changing excitation functions in Fig. 11, at high energies the magnitude of the excitation cross sections, like the excitation functions, fall into three categories: largest for the dipole-allowed case and smallest for spin-changing excitation with the spin-allowed, dipole-forbidden group in the middle. The qualitative features of these categories of excitation behaviors outlined above are in agreement with theoretical calculations based on the Born-type approximation. The Born–Bethe theory shows that at high energies the cross sections for excitation to the n1 P levels are proportional to E −1 ln E where E is the incident electron energy and the cross sections for excitation to n1 S and n1 D levels are proportional to E −1 . Calculations of spin-changing excitation often employ the Born–Ochkur and Born–Rudge approximations which add an ad hoc step of approximation to the Born theory and predict an E −3 dependence for the n3 L cross sections at high energies. A simple, intuitive way to understand the behaviors of the cross sections is to think of the excitation process as an absorption-like transition caused by the electromagnetic field associated with the colliding electron. A multipole-type analysis
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of the electromagnetic field attributes the excitation from the 11 S ground level to the n1 P levels to the dipole component (k = 1) which is the major constituent. The quadrupole component (k = 2) is responsible for excitation into the n1 D levels resulting in smaller cross sections. Because of the net charge of the colliding electron, one must introduce a monopole component (k = 0) which causes excitation into the n1 S levels. The genesis of this simple picture traces back to Bohr’s work on the stopping power of matter for charged particles [59,60] and to Purcell’s analysis of the lifetime of the 22 S1/2 level of hydrogen in an ionized atmosphere [61]. A more rigorous visualization of this intuitive idea is to make use of the correlation between excitation cross section to the integral containing the initial and final wave functions of the projectile-target system and the Coulomb interaction of the incident electron at r (r, θ, φ) with each atomic electron at ri (ri , θi , φi ). Expansion of the Coulomb denominator in terms of spherical harmonics and the greater and lesser of r and ri (r> and r< ), k 1 4π r< 1 ∗ Ykq (θ, φ)Ykq (θ , φ ), = (8) |r − ri | r> 2k + 1 r> kq
shows that the k = 0, 1, and 2 terms are responsible for excitation from the ground state into the 1 S, 1 P and 1 D levels. This picture is akin to a high-energy collision approximation where the impact duration is much shorter than the classical period of the target electron being excited. For slow collisions allowances may have to be made for distortion of the target by the projectile. This simple picture is found to be very useful for understanding many aspects of excitation of rare gases as is seen in the subsequent sections. We must emphasize that this multipole field picture does not take into account the electron-exchange interaction between the collision partners and hence is not applicable without modification to such processes as He(11 S → n3 L) excitation where the cross section arises predominately from electron exchange. We have seen that the effect of exchange is felt primarily at energies just above threshold, thus inclusion of electron exchange in the collision model may significantly alter the cross section near the excitation threshold but with much less change at higher energies.
4.2. E XCITATION OUT OF H E M ETASTABLE L EVELS Measurements of electron excitation cross sections out of the He 23 S metastable level into numerous higher triplet levels have been reported [32,62]. The 23 S → 23 P excitation has an exceptionally large cross section, the peak value being 22 × 10−15 cm2 [32]. This is not surprising since the excitation corresponds to an optically allowed transition with a large oscillator strength of 0.539 and a very small excitation threshold of 1.15 eV. The peak of the excitation function occurs
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at 4.5 eV, about four times the threshold energy just like the case of excitation from 11 S to n1 P . A very interesting finding comes to light upon comparing the cross sections for excitation from the 23 S metastable level into the 33 S, 33 P and 33 D levels. Based on what we learned from excitation experiments out of the ground level, we expect the 23 S → 33 P excitation which corresponds to a dipole-allowed transition, to have a larger cross section than the 23 S → 33 S and 23 S → 33 D, but experiments show otherwise. For instance, at 10 eV the 23 S → 33 P excitation cross section is 0.97 × 10−16 cm2 which is much smaller than the cross sections of 1.9 × 10−16 cm2 and 4.0 × 10−16 cm2 for excitation into the 33 S and 33 D, respectively [62]. Although this reversal is consistent with earlier theoretical calculations, it was unsettling to see such a deviation from an established rule of great intuitive appeal without an explanation based upon physical arguments. To understand this anomaly we note the analogy of electron excitation with optical excitation and associate the 23 S → n3 P and 23 S → n3 D electron excitations with their optical dipole and quadrupole absorption counterparts. For forward scattering this analogy is quantitatively fulfilled but we apply it to integrated cross sections to provide physical insight. The dipole matrix element for the 23 S → n3 P , m = 0 series satisfy the sum rule 23 S zn3 P , m = 0 2 = 23 S z2 23 S .
(9)
n
The 23 S and 23 P levels are close together with strongly overlapping radial wave functions resulting in an exceptionally large dipole matrix element between them. Consequently the other matrix elements in the n3 P series are diminished on account of the sum rule. Carrying this analogy to electron excitation we expect a reduction of the 23 S → 33 P electron excitation cross section because of the abnormally large 23 S → 23 P cross section referred to in the preceding paragraph. Hence the 23 S → 23 P acts like a robber, victimizing the cross sections of the higher members of 23 S → n3 P series. The same kind of cross section reduction does not happen to 23 S → 33 D because even though there exists a quadrupole sum rule, 23 S 2z2 − x 2 − y 2 n3 D, m = 0 2 n
2 = 23 S 2z2 − x 2 − y 2 23 S
(10)
for the 23 S → n3 D excitations as a counterpart to the dipole sum rule for 23 S → n3 P , there is no extraordinarily large cross section in the S → D series to assume the role of the robber for reducing the 23 S → 33 D cross section.
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4.3. M ULTIPOLE F IELD M ODEL A PPLIED TO THE H EAVY R ARE G ASES 4.3.1. Excitation from the Ground Level In the comparison of the electronic structure of He and the heavy rare-gases (Section 2.3) a major difference between He and the other rare gases emerges in the way we characterize an excitation process. In the case of, say Ar, we speak of an excitation from a given initial state (ground or metastable) into a level in the 3p 5 4p configuration with total angular momentum J so that the final state is associated with both quantum numbers l and J . For He, although an excited configuration such as (1s)(2p) consists of four levels 1 P1 , 3 P2 , 3 P1 , 3 P0 , the last three levels are not resolved in an electron excitation experiment and the measured cross section includes all three J -values of the 3 P manifold. Here the individual J values are of no consequence; l is the only relevant angular momentum quantum number for the triplet series. In the case of the singlet levels, J is always equal to l and again is not needed. Accordingly, excitation of He in Section 4.1 and Section 4.2 is described in terms of the quantum number l of the final level without reference to J . The multipole field model and Eq. (8) are used to select the favored values of l for large cross sections. When applied to the heavier rare gases, the multipole field model has a double task of addressing both l and J . To see how this is done, let us consider the particular case of argon. Excitation from the 3p6 1 S0 ground level into the 3p 5 4p configuration involves an l = 1 → l = 1 transition for the active electron. This can be accomplished by the k = 0 and k = 2 components of the multipole interaction. The levels are further characterized by J with J = 0 for the initial level and J ranging from 0 to 3 for the final level. The k = 0 component is effective for J = 0 → J = 0 excitations and the k = 2 component for J = 0 → J = 2, but neither one can supply the coupling potential necessary for J = 0 → 1 and J = 0 → 3 excitations. Thus, the multipole field analysis favors excitation into 3p 5 4p levels with even-J over levels with odd-J values. Although excitation into the odd-J levels can proceed through “higher order” effects such as target distortion and electron exchange, the resulting cross sections are small compared to the cross sections of the even-J levels. For a more quantum-mechanical grounding of this argument let us consider an incident electron of coordinate r(rθ φ) exciting a target atom with electron coordinates r (r θ φ ) from an initial state of angular momentum Ji and wave function Ψi (Ji |r1 . . . rn ) into a final state of Jf and Ψf (Jf |r1 . . . rn ). If we consider only the Coulomb interaction of the projectile with the target and neglect exchange excitation, the collisional coupling potential connecting the initial and final states Cf i (r ) is
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e2 1 . . . dr n . Ψi (Ji |r1 . . . rn )dr |r − rj | (11)
For excitation from the ground state into the 3p 5 4p levels we have Ji = 0 and Jf = 0, 1, 2, 3. If we adopt the one-configuration approximation for both the initial (3p 6 ) and final (3p 5 4p) wave functions and expand the |r − rj | term according to Eq. (8), the above integral survives only for the k = 0 and k = 2 terms because the active electron undergoes a change from 3p to 4p. The coupling potential in Eq. (11) decomposes into integrals of triple products Ψf∗ Ykm (θj φj )Ψi with k = 0 and 2. Since Ψf , Ykm , and Ψi are eigenfunctions of J corresponding to eigenvalues J = Jf , J = k, and J = Ji = 0, respectively, the integrals of the triple products vanish unless Jf = 0 (when k = 0) or Jf = 2 (when k = 2). To the first approximation the i → f excitation cross section is governed by the Cf i coupling potential. Thus excitation from the ground level into the J = 0 and J = 2 levels are “allowed” even under the first approximation with the Coulomb interaction only, as opposed to excitation into the J = 1 and J = 3 levels which is possible only with a more refined treatment. As a further illustration of this multipole field picture, consider excitation from the ground level into the twelve levels of the Ar(3p 5 nd) configuration. Here the active electron makes an l = 1 → l = 2 transition which selects out the k = 1 and k = 3 components of the multipole field. The k = 1 component allows an excitation from the ground level (J = 0) into the J = 1 levels of the 3p 5 nd configuration and the k = 3 component is responsible for excitation into the J = 3 levels, but levels with J = 0, 2, and 4 cannot be reached by this route making excitation less favorable. Thus for excitation into the 3p5 nd configuration the multipole field picture predicts larger cross sections for the odd-J levels, as opposed to larger cross sections for even-J in the case of 3p 5 np. The preference for odd-J levels over the even-J levels can also be demonstrated by the same quantum-mechanical approach sketched in the preceding paragraph. For the 3p 6 → 3p 5 nd excitation it is the k = 1 and k = 3 terms in the spherical harmonics expansion that survive the integration in Eq. (11) and permit excitation from the ground level (J = 0) into the J = 1 and J = 3 final levels with “large” cross sections. This prediction is confirmed by experimental measurements to be discussed later (Section 5.1.5). One can, of course, apply the same line of argument to excitation from the ground level to the four levels of 3p 5 ns configurations with J ranging over 0, 1, 2. In this case one finds again that the odd-J levels have larger cross sections than the even-J levels, a result also confirmed by experiment (Section 5.1.3). Since electron exchange is not included in the multipole field model, comments on its effects on the cross sections are in order. It is responsible for a singlet-totriplet transition such as excitation from the ground level of Ar (1 S0 ) to the 2p9 level which is purely 3 D3 . All the other 3p 5 4p levels have both singlet and triplet
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components. Electron exchange provides a linkage from the ground level to those triplet components supplementing the cross sections at low energies. 4.3.2. Excitation from the Metastable Levels In addition to excitation from the ground state, the multipole field model can also be applied to excitation from metastable levels. For illustrative purposes we once again consider the particular case of argon. The 3p 5 4s configuration contains two metastable levels: the 1s3 level with J = 0 and the 1s5 level with J = 2. Here excitation from 3p 5 4s to 3p 5 4p has l = 1 for the active electron, and thus entails the k = 1 component of the multipole field which gives preference to excitations with J = 0, ±1 but excluding J = 0 → J = 0. Hence, excitation cross sections from the J = 2 1s5 metastable level are expected to be largest into the 3p 5 4p levels with J = 1, 2, and 3; whereas cross sections from the J = 0 1s3 level are only expected to be large into levels with J = 1. This is indeed observed experimentally in the case of excitation from the metastable levels of Ar (Section 5.2.2). Note that in contrast to excitation from the ground state, the multipole field model yields no even-J /odd-J dichotomy in the 3p 5 4p cross sections for excitation from the J = 2 1s5 metastable level. In the case of excitation from the Ar(3p 6 ) ground state into the 3p 5 3d levels, the same J = 0, ±1 (J = 0 J = 0) dipole selection rule applies, but the Ji = 0 ground level only permits excitation into levels with Jf = 1. When combined with the excitations permitted by the k = 3 term, the Ji = 0 value of the initial level limits the preferred 3p 5 3d final levels to only those with odd-Jf . Similarly, excitation from the Ji = 0 1s3 metastable level is only preferred for 3p 5 4p levels with Jf = 1, but the non-zero Ji = 2 value of the 1s5 metastable level permits excitation into 3p 5 4p levels with both even and odd Jf . Interestingly, a sort of even/odd preference for excitation from the Ar metastable levels reemerges when one considers excitation into 3p 5 ns and 3p 5 nd levels, but these cross sections have not been experimentally measured.
5. Argon Of the four p-shell rare gases the largest bulk of electron excitation experiments have been performed with argon. We will first discuss excitation out of the ground level and then move on to the more recent experiments on excitation out of the metastable levels.
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5.1. E XCITATION OUT OF THE G ROUND L EVEL 5.1.1. Historical Perspective: Pressure Effects Of all the excited levels of argon, the excitation cross sections into the ten levels of the 3p 5 4p configuration have been those studied most extensively dating back to the work of Fischer [63] in 1933 through papers by Tsurubuchi et al. [17] in 1996 and by Chilton et al. in 1998 [13]. The 3p 5 4p levels (2p1 through 2p10 in Paschen’s notation) have been favorite candidates for measurements by the optical method since they radiatively decay only to the 3p 5 4s levels (1s2 . . . 1s5 ) with wavelengths (697–1149 nm) in a favorable spectral region (see also Fig. 12). Optical cross sections for 2px → 1sy emissions have been measured by numerous groups, but significant discrepancies exist among the early results. Part of the confusion is due to the fact that the measured emission cross sections depend significantly on the target gas pressure, a point not fully appreciated in the early works. Pressure dependence of the measured cross section in this type of experiment normally arises from two sources: radiation trapping and excitation transfer. Since the 3p 5 4p levels are not optically connected to the ground state, radiation from the former are not expected to be reabsorbed. Excitation transfer through a collision of an excited atom during its short lifetime with another argon atom is not expected to significantly influence the cross section measurements at pressures below 10 mTorr. Thus the pressure effects escaped attention for a long time.
F IG . 12. Energy level diagram of argon.
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F IG . 13. (a) Variation in shape of the excitation function for the Ar(2p7 ) level with target gas pressure. (b) The Ar(2p7 ) apparent cross section at 100 eV as a function of target gas pressure. The heavy solid line is a model calculation that includes a pressure independent direct cross section contribution, a weakly pressure dependent contribution from non-resonant levels (i.e. 3d1 and 2s5 ) and a strongly pressure dependent contribution from resonant levels (i.e. 3d2 and 2s4 ). The pressure dependence of the resonant cascade levels was calculated using Eq. (5) and the appropriate transition probabilities [10,66,67]. In both plots, only statistical error bars are shown.
To offer a concrete example, let us examine excitation into the 2p7 level as a function of target gas pressure as illustrated in Fig. 13. Note that the 2p7 has some of the most extreme pressure dependence of the 3p 5 4p levels. For values taken at an intermediate target gas pressure (approximately 1 mTorr), the shape of the 2p7 excitation function is that of a sharp peak at low energies along with a broad shoulder extending from 50 eV up [13,64,65]. At even higher pressures, the shoulder transforms into a second peak near 75 eV as illustrated in the 5 mTorr data displayed in Fig. 13(a). At much lower pressures ( 0.2 mTorr) the shoulder virtually disappears leaving only the sharp low-energy peak [17]. Similar secondary peaks at 75 eV show up in the excitation functions of the 2p1 and 2p5 levels at moderate to high pressures [64]. If one were to compare the magnitude of the cross section results at a fixed energy (such as 100 eV in Fig. 13(b)), one finds that the value is highly sensitive to the target pressure used in the experiment. Notwithstanding the puzzling pressure effects which will be revisited later in this section, let us turn to the measurements of cross sections. The apparent excitation cross sections for all ten 3p 5 4p levels are obtained readily by summing all the relevant 3p 5 4p → 3p 5 4s emission cross sections measured by the standard optical technique (Section 3). Determination of the direct excitation cross section, however, is an entirely different matter. Numerous energy levels, espe-
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cially those of the 3p 5 5s and 3p 5 3d configurations, cascade into the 3p 5 4p levels through predominately infrared radiation not detectable by PMT’s. Without a reliable assessment of cascades the direct excitation cross sections remained elusive. Attempts have been made to estimate the cascade contribution by theoretical calculations of the excitation cross sections of the cascading levels and the relevant branching fractions. Another stop-gap approach is to detect the fluorescence in the excitation experiment using time-resolved measurements with a very short pulsed electron beam in order to reduce the amount of cascade into a particular 3p 5 4p level from levels that have a much longer lifetime [54]. Reliability of the direct excitation cross sections by these methods is difficult to establish. Nonetheless, some of the early time-resolved measurements pointed to cascades as a contributor to the pressure effects observed in apparent cross section measurements [68]. Incorporation of a commercially available weak-emission FTS into an electronbeam excitation apparatus has made it possible to obtain by direct measurement the cascade contributions into the 3p 5 4p levels from 3p 5 5s, 3p 5 3d, 3p 5 6s, 3p 5 4d levels [13] and thus obtain direct excitation cross sections via Eq. (3). These measurements reveal that in most cases the cascade component is as large as the direct excitation underscoring the importance of accurate cascade correction. More importantly, however, the measurement of the infrared cascade cross sections provided a definitive answer to the origin of the pressure effects. The total cascade cross sections for all ten 3p 5 4p levels were found to vary with pressure in the same manner as do the apparent excitation cross sections so that when the cascade cross section is subtracted from the apparent excitation cross section the resulting direct excitation cross sections are independent of gas pressure. This is illustrated in Fig. 14. The cascading levels include those with J = 1 from the 3p 5 5s, 3p 5 3d, . . . configurations which decay partly into the 3p 5 4p levels and partly to the ground level (via resonance transitions). Reabsorption of the resonance radiation by a nearby Ar atom repopulates the cascading level which in turn partly cascade into the 3p 5 4p levels as was illustrated by the Ar(2s2 → 2p3 ) cross section results in Fig. 3 (Section 3.2). Each reabsorption/emission cycle generates additional cascade radiation into 3p 5 4p levels at the expense of cascade into the ground level. It is this recycling process, which occurs more frequently at elevated gas pressure, that makes the cascade cross sections increase with pressure. The apparent cross sections of the 3p 5 4p levels then acquire the observed pressure dependence through cascades from higher resonance levels (J = 1), whereas the direct excitation cross sections, as shown in Fig. 14, do not vary with pressure as one expects. For the case of the Ar(2p7 ) level discussed earlier, the extreme pressure dependence observed at 100 eV (Fig. 3(b)) can be traced to the pressure dependence of the large cascade contributions from the 3d2 and 2s2 resonant levels.
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F IG . 14. Pressure curves of direct (•), cascade ( ) and apparent ( ) cross sections at 100 eV for select Ar(3p5 4p) levels [13]. Error bars combine the statistical and systematic uncertainties. Slight horizontal offsets have been added to make the three data sets more readable.
5.1.2. Excitation of 3p5 4s Levels The first excited configuration 3p 5 4s consists of two metastable levels (1s3 and 1s5 ) and two resonance levels (1s2 and 1s4 ). The former do not radiate, thus their cross sections cannot be obtained directly by the optical method. However, by incorporating the technique of laser-induced fluorescence to the apparatus, the 1s3 and 1s5 cross sections have been reported by Schappe et al. [51]. The apparent and direct excitation cross sections for excitation into the Ar metastable levels are displayed in Fig. 15. Both metastable levels are pure triplet levels (3 P0 and 3 P2 ) in the one-configuration approximation, thus excitation into these levels from the 1 S0 ground state proceeds primarily through electron exchange. While the apparent cross sections for these two levels have moderately wide peaks, upon subtracting off the very substantial cascade contribution from higher levels, the direct cross sections have the very sharp peaks at low energies characteristic of spin-changing cross sections.
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F IG . 15. Apparent (solid line) and direct (dashed line) excitation cross sections into the Ar(3p5 4s) resonance [17] and metastable [51] levels. Note that the apparent cross sections for the resonance levels were measured at very low target pressures, whereas the apparent cross sections for the metastable levels were made at 3 mTorr. Estimated uncertainties in the direct cross sections are on the order of ±15% for the 1s2 level, and ±30% for the remaining levels.
Excitation into the 1s2 and 1s4 levels can be accessed by optical means, via the resonance transitions at 104.8 and 106.6 nm, but this entails spectroscopy in the vacuum ultraviolet region where absolute radiometry is difficult (see Section 3.3). To overcome this challenge, in the measurements of Tsurubuchi et al. [17] two independent calibration methods were employed. In the first method, the authors first measured the relative optical emission cross section as a function of incident electron energy (from threshold to 1000 eV) using a vacuum ultraviolet monochromator with an electron multiplier. A second visible/near-IR monochromator was used to measure on an absolute scale almost all of the 3p 5 4p → 3p 5 4s and 3p 5 5p → 3p 4 4s optical emission cross sections that form the majority of the cascades into the 1s2 and 1s4 levels. The absolute value of the apparent cross section at 500 eV was obtained by combining the direct cross section values of Li et
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al. [69] obtained by electron-energy loss spectroscopy with the measured cascade contribution from the 3p5 np levels. This value at 500 eV was then used to place the relative results at other energies on an absolute scale. The direct cross section at other energies was obtained by subtracting off the measured cascade contribution. The second calibration method used by Tsurubuchi et al. was to measure the relative wavelength sensitivity of their apparatus between the argon resonance lines and the hydrogen Lyman-α emission line. The hydrogen Lyman-α optical emission cross section for electron-impact excitation of H2 at 100 eV was then used as a radiometric standard. For the 1s4 level the two methods gave essentially the same result, but for the 1s2 level the results of the two calibration methods differed by approximately 30% [69]. A number of previous measurements of the optical emission cross sections for the Ar resonance line have used variations on these two calibration techniques. For example, McConkey and Donaldson [23] placed their relative VUV results on an absolute scale by normalizing to the Born– Bethe approximation at energies over 200 eV, where the cascade contribution to the apparent cross section should be small. Direct cross sections were obtained by iteratively subtracting off the 3p 5 4p → 3p 5 4s cascades. Other measurements of the Ar resonance lines prior to 1989 are reviewed in the article of van der Burgt et al. [22]; subsequent measurements include [17] and [70]. The 1s2 and 1s4 cross section results of Tsurubuchi et al. [17] are included in Fig. 15. The direct cross sections for both resonance levels have the very broad peaks characteristic of dipole-allowed excitation processes. The cascade contribution to the 1s4 level, which is larger than that of the 1s2 level, distorts the shape of the 1s4 level’s apparent cross section at low energies giving it more of a sharp peak. Note that the 1s2 and 1s4 levels are mixtures of 1 P1 and 3 P1 LS terms (Section 2.1). At high incident electron energies the magnitude of the direct cross section into a given resonance level will be determined by the singlet component in the level’s wave function. In the case of the Ar(1s2 ) and Ar(1s4 ) levels the 1s2 level has a larger singlet weighting factor (80% |1 P1 ) than the 1s4 level (20% |1 P1 ) which is also evident in the factor of four larger direct cross section. 5.1.3. Excitation of 3p 5 ns (n 5) Levels The cross sections for the higher 3p5 ns configurations follow many of the same trends as those of the 3p 5 4s configuration described in the last section. The higher 3p 5 ns levels, however, have other decay channels open to them besides decays to the ground state. For example, the 3p 5 5s configuration contains two purely triplet levels, 2s3 (3 P0 ) and 2s5 (3 P2 ), and two J = 1 resonance levels, 2s2 and 2s4 . All four levels can decay to levels of the 3p5 4p configuration with wavelengths in the 0.92 to 2.03 µm range. Using a near-IR FTS system, the optical cross sections have been measured for these 3p 5 5s → 3p 5 4p emission lines from which the
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F IG . 16. Excitation functions for Ar(3p5 5s) levels [71,72]. Error bars on direct cross sections include both the statistical uncertainty and the uncertainty from the radiometric calibration.
apparent excitation cross sections for the 2s3 and 2s5 levels are derived [71]. Further measurements of the 3p5 5p → 3p 5 5s, 3p 5 6p → 3p 5 5s, 3p 5 7p → 3p 5 5s optical emission cross sections (all in the infrared) provide the cascade corrections. The resulting apparent and direct cross sections for the 2s3 and 2s5 levels are displayed in Fig. 16. Similar to the 1s3 and 1s5 levels, the direct cross sections for the 2s3 and 2s5 levels are sharply peaked due to the spin-changing nature of excitation into these purely triplet levels. In addition to the near-IR decay channels to the 3p 5 4p levels, the 2s2 and 2s4 resonant levels are also optically connected to the ground level. No measurements of the VUV resonance transitions were made in the near-IR work of Chilton and Lin [71]. Nevertheless, it is still possible to extract apparent cross sections for the 2s2 and 2s4 levels strictly from the near-IR measurements [72]. At very high target gas pressures the resonance transitions to the ground state are completely reabsorbed, leaving only the 3p 5 5s → 3p 5 4p decay channels open (Section 3.2). By measuring the size of 3p 5 5s → 3p 5 4p emission cross sections as a function
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of target gas pressure, it is possible to extract this high pressure limit from data taken at moderate pressures, before secondary processes complicate the experimental measurements [72]. For example, the 2s2 → 2p3 optical emission cross section displayed in Fig. 3 indicates that the 2s2 resonance transition is about 87% suppressed at 5 mTorr. Thus, the 2s2 apparent cross section can be obtained by multiplying the sum of the 2s2 → 2px optical emission cross sections measured at 5 mTorr by a factor of 1.15. The values of the apparent and direct cross sections derived in this fashion by Stewart et al. [72] for the 2s2 and 2s4 levels are included in Fig. 16. The apparent excitation cross sections for 2s2 and 2s4 are found to contain a significant amount of cascade from the 3p 5 np levels and therefore do not correspond very well to the energy dependence of the direct excitation cross sections. Upon subtracting off the cascades, however, both direct excitation cross section curves are seen to have broad peaks of the dipole-allowed variety which arise from the 1 P1 component of the wave function. Note that the 2s2 and 2s4 direct cross sections only differ by a factor of two at high energies, as compared to the factor of four difference for the 1s2 and 1s4 cross sections, reflecting a more equal mixing of the singlet and triplet components of the wave functions. The wavelengths of the 3p 5 6s → 3p 5 4p (3s → 2p in Paschen’s notation) transitions lie in the 586 to 906 nm range that can be easily measured by a photomultiplier tube [64,71]. The 3p 5 6s → 3p 5 5p decay channels, however, have much longer wavelengths (3.2–4.6 µm) which are difficult to measure accurately due to the small size of the cross sections and poor sensitivity of InSb detectors required in this spectral range. Apparent cross sections, nonetheless, can be obtained via Eq. (4) if one employs theoretical branching fractions [73] to estimate the missing 3p 5 6s → 3p 5 5p optical emission cross sections. The results so obtained are displayed in Fig. 17. The excitation functions for the 3s3 and 3s5 levels are both sharply peaked, as expected for pure triplet levels. The excitation functions for the 3s2 and 3s4 levels both have very broad peaks, which is somewhat surprising considering that the 1s2 , 1s4 , 2s2 , and 2s4 apparent excitation functions all have a low energy peak due to cascades. The limited distortion of the 3s2 and 3s4 levels (which both display a slight peak near threshold) indicates a reduced role of cascades in populating these levels. Also surprising is the size of the 3s2 apparent cross section. The apparent cross sections for the other three 3sx levels are all substantially smaller than the corresponding 2sx cross sections as expected. In contrast, the apparent cross section at 100 eV for the 3s2 level, (9±4)×10−19 cm2 , is larger than the 2s2 direct cross section, (7.6 ± 1.4) × 10−19 cm2 . While this may be due to a large cascade contribution to the 3s2 apparent cross section, this seems very unlikely due to the limited distortion of the excitation function shape. For dipole-allowed transitions, the electron-impact excitation cross section is expected to be proportional to the oscillator strength of the optical transition (cf. Section 5.2.3). In this context, it is interesting to note that the spectroscopic value
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F IG . 17. Apparent cross sections for Ar(3p5 6s) levels measured at 5 mTorr. Estimated uncertainties (including the large contribution by the use of branching fractions) are on the order of ±35%. The 3s2 and 3s4 cross sections have been extrapolated to the limit of complete reabsorption of the VUV resonance transition.
of the 1 S0 –3s2 oscillator strength (0.0224) is also larger than the 1 S0 –2s2 oscillator strength (0.0124) [67]. 5.1.4. Excitation of 3p 5 np levels The direct excitation cross sections for the ten 3p 5 4p levels and their energy dependence obtained by Chilton et al. [13] are given in Fig. 18. The most conspicuous observation within this set of cross section results is the very sharp peak found for the 2p9 (J = 3) level. The 2p9 is the only level with J = 3 in the 3p 5 4p configuration and therefore is a pure triplet LS-level (3 D3 ). Excitation from the 1 S0 ground state into the 2p9 level involves a change in the spin, hence a sharply peaked energy dependence. The IC wave functions of all nine other levels contain a singlet component which is dipole-forbidden with respect to the ground level on account of parity. Analogous to He, the excitation functions for these
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F IG . 18. Apparent (solid line), cascade ( ) and direct (•) cross sections for the ten Ar(3p5 4p) levels [13]. Note that the apparent and cascade cross sections were measured at a pressure of 3 mTorr. Error bars included on the direct cross sections combine the statistical and systematic uncertainties.
nine levels exhibit peaks of “intermediate” width. The 2p10 level shows a sharper peak than the others. Using the semi-empirical method referred to in Section 2.1 to calculate the IC wave functions of these nine levels, we find the 2p10 level to
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Table I Direct excitation cross sections for the ten levels of the Ar(3p5 4p) configuration at select incident electron energies [13]. Error bars represent the combined statistical and systematic uncertainties. Given in the last two rows are the average cross sections over all even-J levels and over all odd-J levels. Cross section (10−19 cm2 ) Level
J
20 eV
40 eV
50 eV
75 eV
100 eV
150 eV
200 eV
2p1 2p2 2p3 2p4 2p5 2p6 2p7 2p8 2p9 2p10
0 1 2 1 0 2 1 2 3 1
50 ± 7 14 ± 6 29 ± 9 22 ± 7 16 ± 3 33 ± 11 26 ± 8 53 ± 14 52 ± 22 47 ± 20
33 ± 5 8.9 ± 4.5 17 ± 6 9.8 ± 3.7 17 ± 3 19 ± 6 12 ± 7 29 ± 8 6.9 ± 5.0 25 ± 10
32 ± 5 7.5 ± 3.9 14 ± 4 9.3 ± 3.4 15 ± 4 18 ± 5 8.2 ± 4.0 27 ± 7 5.7 ± 3.2 12 ± 5
29 ± 5 7.2 ± 3.9 12 ± 3 9.1 ± 3.4 14 ± 4 16 ± 4 7.2 ± 5.0 23 ± 6 4.4 ± 2.2 7.4 ± 3.0
25 ± 5 6.7 ± 3.9 8.7 ± 3.3 8.3 ± 3.5 11 ± 3 11 ± 3 4.7 ± 3.6 19 ± 5 3.0 ± 1.9 6.7 ± 2.4
20 ± 4 5.6 ± 2.9 7.4 ± 2.4 7.4 ± 3.2 10 ± 3 11 ± 3 3.7 ± 3.3 16 ± 4 2.0 ± 1.1 3.6 ± 1.7
18 ± 4 5.1 ± 2.5 6.5 ± 1.9 7.0 ± 3.2 9.3 ± 2.7 9.7 ± 2.3 3.0 ± 3.4 14 ± 4 1.7 ± 0.8 2.5 ± 1.4
36 32
23 12
21 8.5
19 7.1
15 5.9
13 4.5
11 3.9
even-J odd-J
have the smallest singlet admixture (1% |1 P1 ), consistent with the very narrow excitation function. From Table I we see an interesting correlation of the direct excitation cross section with J , i.e. the levels with even values of J on a whole have larger cross sections than the odd-J levels. This parity relation is a consequence of the multipole field picture discussed in Section 4. While there have been numerous measurements of the apparent cross sections for the levels of 3p 5 4p configuration [13,17,64,65], the difficulty of measuring near-IR cascades has limited optical measurements of the direct cross sections to only those of Chilton et al. [13] and those of Bogdanova and Yurgenson [54]. The latter measurements used a current pulse of 10–15 ns to resolve the direct excitation component for the 2px levels (with lifetimes between 20 and 30 ns) from the cascade component due to excitation into higher levels which have much longer lifetimes. For most levels there is generally good agreement between the two experimental methods, although both experiments have relatively large uncertainties (on the order of 30%). In the measurements of Chilton et al. the uncertainties in the apparent cross sections are generally much less, on the order of 12%. The larger uncertainties in the direct cross sections are a result of the error propagation in subtracting the large cascade correction from the apparent cross sections. Comparison of apparent cross sections from different laboratories is difficult due to the dependence on target gas pressure, but when extrapolated to zero pressure
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the recent measurements of Chilton et al. [13] and Tsurubuchi et al. [17] are in good agreement. In the absence of configuration mixing, the 2p9 level is a pure triplet level (3 D3 ). Direct excitation into this level from the 1 S0 ground state should proceed though the exchange interaction. As a result, the energy dependence of the 2p9 direct cross section at high electron energies is expected to have an E −3 dependence according to Born–Ochkur theory. The large error bars on the direct cross section measurements of Chilton et al. [13] (particularly at high energies) make it difficult to verify this prediction with their results. Tsurubuchi et al. [17] did observe an E −3.0±0.1 energy dependence to the apparent cross section in the energy range from 22 to 50 eV. In this energy range the apparent cross section should be dominated by direct excitation (and cascades from triplet levels such as the 3d4 level). At higher energies (>300 eV) where the cascade contribution is larger, they found only an E −0.86±0.04 dependence. Subsequent time-resolved measurements which attempted to separate the direct and cascade components of the Ar(2p9 ) cross section by Tsurubuchi and Kobayashi [74] confirmed a weaker than expected decrease in the direct cross section with increasing electron energy. Apparent cross sections for higher 3p5 np (n = 5, 6) levels have also been measured by Weber et al. [16]. In the case of the 3p 5 5p levels, the 3p 5 5p → 3p 5 4s emissions lie in the 395 to 470 nm wavelength range, while the 3p 5 5p → 3p 5 5s, 3p 5 3d emissions lie in the 1.4 to 7.7 µm range. Due to the difficulty in measuring these IR channels (particularly those greater than 3 µm), Weber et al. used Eq. (4) to obtain apparent cross sections from only the 3p 5 5p → 3p 5 4s optical emission cross sections and branching fractions derived from various experimental measurements. As with the 2px levels, the 3px levels display significant pressure effects, particularly at high energies [16]. To minimize the contributions due to cascades, the values at 0.5 mTorr are displayed in Fig. 19. The cross section results for the levels of the 3p 5 5p and 3p 5 6p configurations conform to the same patterns observed in the Ar(3p 5 4p) results: (i) levels with even J -values have larger cross sections on average than levels with odd J -values; (ii) the pure triplet 3p9 and 4p9 levels both have very sharp excitation functions; (iii) the excitation functions of the J = 0 levels (3p1 , 3p5 , 4p1 , and 4p5 ) are generally wider than those of the other levels; and (iv) the shape of the apparent excitation functions of 3p7 and 4p7 levels are highly sensitive to the target gas pressure. A number of other measurements of the 3p 5 5p → 3p 5 4s optical emission cross sections have also been reported in the literature [17,75] including direct cross section measurements using time resolved techniques [76]. Little can be said about the agreement, or lack thereof, between the different experiments due to the complications of pressure effects. In the zero pressure limit, however, the 100 eV values of Weber et al. [16] are found to agree within error bars with the values of Tsurubuchi et al. [17].
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F IG . 19. Apparent cross sections for the ten Ar(3p5 5p) levels [16] measured at 0.5 mTorr (to minimize the pressure-dependent cascade contribution from resonant levels). Total uncertainties, including those introduced by the branching fractions used in Eq. (4), are on the order of ±30%.
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5.1.5. Excitation of 3p5 nd Levels The levels of the 3p 5 3d configuration lie approximately 0.8 eV above the 3p 5 4p configuration. There are twelve LS-terms possible from the combination of a p and d orbital: 3 P0 , 3 P1 , 3 P2 , 1 P1 , 3 D1 , 3 D2 , 3 D3 , 1 D2 , 3 F2 , 3 F3 , 3 F4 , and 1 F3 . With spin–orbit mixing of terms of the same J , we obtain: (a) two unmixed triplet levels, 3 P0 and 3 F4 (3d6 and 3d4 respectively in Paschen’s notation); (b) three resonance levels with J = 1 (3d5 , 3d2 , 3s1 ) with LS-constituents 1 P1 , 3 P1 , 3 D1 ; (c) four levels of J = 2 (3d3 , 3d1 , 3s1 , 3s1 ) with LS-constituents 3 P2 , 3 D2 , 1 D , 3 F ; and (d) three levels of J = 3 (3d , 3d , 3s ) with LS-constituents 2 2 4 1 1 3 D , 3 F , 1 F . The 3d and 3d levels, being pure triplet levels, should have 3 3 3 6 4 the most narrow excitation functions. Assuming that the LS-singlet components in none of the ten remaining levels is unusually small, we expect the J = 1 excitation functions to exhibit a very broad maximum characteristic of dipoleallowed levels and expect the J = 2 and J = 3 to have excitation functions of intermediate width. As for the magnitude of the cross sections, the multipole field predicts larger cross sections for the odd-J levels than the even-J levels at high energies. With the exception of the J = 1 resonance levels, the only radiative decay channels for the 3p 5 3d levels are the 3p 5 3d → 3p 5 4p transitions with emissions in the 0.89–2.4 µm wavelength range. While some of these are detectable with PMTs, the longer wavelengths require an InSb detector. The J = 1 levels emit, in addition, VUV resonance radiation at 89.9, 87.6, and 86.7 nm. Cascades into the 3p 5 3d manifold are predominately from the 3p 5 5p and 3p 5 6p groups (1.0–3.2 µm) as well as the 3p 5 4f and 3p 5 5f levels (0.9–1.6 µm). Chilton and Lin [71] measured the optical cross sections for all the relevant infrared emission lines into and out of the 3p 5 3d configuration using a FTS but made no measurements of the VUV lines. Their experiment yielded absolute cross sections, both direct and apparent, for the J = 1 levels, but only relative apparent excitation cross sections for the J = 1 levels. Using the technique of measuring the 3p 5 3d → 3p 5 4p emissions at a high enough pressure to trap most of the resonance radiation, however, made it possible to obtain the cross sections for the J = 1 levels as well [72]. Apparent, direct and cascade cross sections for the twelve levels of the 3p 5 3d configuration are shown in Fig. 20. Of the three J = 1 levels, the 3d2 and 3s1 curves bear the signature of a dipole-allowed level through the very broad maximum, but the narrow spike in the 3d5 curve is contrary to one’s expectations. This anomaly is resolved by examining the composition of the IC wave functions as a superposition of 1 P1 , 3 P1 , 3 D1 eigenfunctions; a semi-empirical calculation (cf. Section 2.1) gives the weighting of the 1 P1 components in the 3s1 , 3d2 and 3d5 wave functions as 57%, 42%, and 0.09%, respectively. Because of the exceptionally small 1 P1 component, the 3d5 level is practically a triplet level hence the sharp peak. The two purely triplet levels 3d6 (3 P0 ) and 3d4 (3 F4 ), also have
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F IG . 20. Apparent (solid line), cascade ( ) and direct (•) cross sections for the twelve Ar(3p5 3d) levels [71,72]. Apparent and cascade measurements were performed at 3 mTorr. The cross sections for the J = 1 resonant levels have been extrapolated from the value at 3 mTorr to the high pressure limit of complete trapping of the VUV resonance transition. No cascade measurements are available for the 3s1 level. The error bars on the direct cross sections include both the statistical and systematic uncertainties.
very narrow excitation functions as expected. The remaining levels with J = 2 and J = 3 all have narrow to intermediate width excitation functions. As to the magnitude of the cross sections listed in Table II, the odd-J levels have larger values than the even-J levels at high energies (above 75 eV) as predicted by the multipole field analysis. This relationship also holds if we exclude the J = 1 resonance levels and only compare the J = 3 levels to the even-J levels. However at
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Table II Direct excitation cross sections for the levels of the Ar(3p5 3d) configuration at select incident electron energies [71,72]. Error bars represent the combined statistical and systematic uncertainties. For the 3s1 resonance level, no cascade correction has been made to the apparent cross section since all significant cascades into this level occur at wavelengths outside the range measured in [71]. Cross section (10−19 cm2 ) Level
J
3s1 a 3s1 3s1 3s1 3d1 3d1 3d2 3d3 3d4 3d4 3d5 3d6
1
3
15
20
27
31
31
2 3 2 3 2 1 2 3 4 1 0
6.9 ± 1.3 7.4 ± 1.6 7.3 ± 1.6 0.9 ± 0.9 12 ± 3 12 ± 2 14 ± 7 9.7 ± 2.8 18 ± 5 2±5 5.7±0.9
11 ± 2 8.7 ± 2.7 14 ± 3 2.2 ± 1.7 23 ± 6 19 ± 3 25 ± 6 16 ± 4 34 ± 8 24 ± 5 8.6 ± 1.6
7.3 ± 1.4 8.0 ± 1.9 8.4 ± 1.6 5.3 ± 2.3 14 ± 3 25 ± 4 18 ± 4 11 ± 3 19 ± 4 17 ± 3 6.0 ± 1.0
2.3 ± 0.7 5.5 ± 1.3 2.2 ± 0.8 4.6 ± 1.9 3.2 ± 1.7 34 ± 5 3.3 ± 1.8 3.3 ± 1.8 4.3 ± 1.8 4.7 ± 1.5 1.6 ± 0.3
0.44 ± 0.30 3.7 ± 0.9 0.53 ± 0.41 3.7 ± 1.4 0.75 ± 0.64 39 ± 5 0.6 ± 0.2 1.8 ± 1.3 1.0 ± 0.6 1.8 ± 1.4 0.56 ± 0.17
0.13 ± 0.07 3.0 ± 0.8 0.18 ± 0.18 3.3 ± 1.4 0.21 ± 0.19 40 ± 6 0.3 ± 0.4 1.3 ± 0.9 0.31 ± 0.20 0.6 ± 0.5 0.26 ± 0.09
11 6 6
19 14 9
12 15 8
2.8 13 4.5
0.65 14 3.1
0.23 13 2.5
even-J odd-J J = 3
15 eV
20 eV
30 eV
50 eV
75 eV
100 eV
a Values for the 3s level are apparent excitation cross sections. 1
lower energies the cross sections of the even-J level, especially 3d4 , rise steeply presumably due to electron exchange, which is neglected in the multipole field analysis, overshadowing the J = 3 levels. A limited number of measurements have been reported for some of the higher members of the 3p 5 nd series, namely some of the levels of the 3p 5 4d [64] and 3p 5 5d [77] configurations. In the measurements of Ballou et al. [64] relative excitation functions for nine of the twelve 3p 5 4d levels were obtained by measuring the 3p 5 4d → 3p 5 4p emissions in the 590 to 910 nm spectral range. No measurements of the 3p 5 4d → 3p 5 5p infrared emissions lines were made. The shapes of the measured excitation functions follow the same trends as those observed by Chilton and Lin [71] for the 3p 5 3d levels: the pure triplet levels have the narrowest excitation functions, the J = 1 resonance levels have broad excitation functions with the notable exception of the 4d5 level which also has a small 1 P1 component in its wave function, and the remaining levels with J = 2 and J = 3 have peaks of intermediate width. Blanco et al. [77] measured cross sections for the three J = 2 levels of the 3p 5 5d configuration. Optical emission cross sections were measured for 3p 5 5d → 3p 5 4p emission lines and combined with experi-
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mental branching fractions to obtain apparent cross section results. The authors also used time-resolved techniques to separate out the direct and cascade portions of the apparent cross sections. They interpret the larger size of the measured cross sections versus theoretical values based on the Born approximation as being due to indirect coupling involving the 1 D2 component in each level’s wave function.
5.2. E XCITATION OUT OF M ETASTABLE L EVELS 5.2.1. Measurements of 3p 5 4p Cross Sections The electron-impact excitation cross sections into the levels of the 3p 5 4p configuration from the metastable levels of argon have been measured by Boffard et al. [27] using the metastable atom targets described in Section 3.4. The fast beam target was used for measuring excitation cross sections from the metastable levels into the 2p4 , 2p6 , and 2p9 levels on an absolute scale for electron energies up to 400 eV, whereas relative cross sections for the other levels of the 2p group (sans the 2p7 and 2p10 levels) were obtained with the hollow cathode discharge source for electron energies from threshold to 11 eV. The optimal emission lines for studying the 2p7 and 2p10 levels occur at near-infrared wavelengths where the PMT sensitivity is greatly reduced and thus no cross sections for these two levels were reported in [27]. Since the absolute cross section measurements for the 1s5 → 2p9 excitation made with the fast beam experiment extend down to an energy of 5 eV, the relative cross sections from the hollow cathode source could be placed on an absolute scale by normalizing the results in the 5 to 11 eV overlap range. The observed fluorescence signal from a 2p level is due to excitation from both metastable levels. To sort out the contribution from each metastable level the technique of laser quenching is employed to remove one of the two metastable species. To quench the Ar(1s5 ) level a Ti:sapphire laser is tuned to a wavelength of 801.5 nm which pumps the 1s5 metastable atoms into the 2p8 (J = 2) level. The 2p8 atoms can either decay back to the 1s5 level (27% branching fraction) or they can decay to the two J = 1 levels, 1s2 and 1s4 , which subsequently decay to the ground state. Repeated excitation followed by spontaneous decay during the optical pumping removes all the 1s5 atoms. Although it is possible to pump the 1s3 (J = 0) atoms into one of the 2p levels with J = 1, the subsequent J = 1 → J = 2 radiative decay would cause an increase in the 1s5 population. This complicates the analysis of the results since in addition to removing the 1s3 contribution to the signal rate, the 1s5 signal rate is increased. For this reason it is more practical to quench only the 1s5 atoms and perform two data runs, one with the unquenched 1s3 /1s5 mixed target and one with a quenched target with all the 1s5 atoms removed but no change in the 1s3 population. The two sets of data, for excitation into each 2p level, are combined to apportion the mix-target signal
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F IG . 21. Metastable excitation signal for a mixed 1s3 /1s5 target (open circles) and a 1s3 -only target (filled squares) obtained by laser quenching [33]. Error bars are statistical only.
into contributions from excitation out of the 1s3 and 1s5 metastable levels [33]. Since the signal rate is proportional to the target density, the absolute 1s3 → 2px cross sections must be corrected for the 1s5 : 1s3 ratio of the mixed target which is measured in a separate laser-induced-fluorescence experiment [27]. For Ar we find the 1s5 : 1s3 ratio to be (5.6 ± 1.6) : 1 which is nearly the statistical weight of J = 2 versus J = 0. To measure the 1s5 → 2p9 excitation cross section, for example, one detects the 2p9 → 1s5 emission signal (811.5 nm) emerging from the collision chamber with and without quenching. The data in Fig. 21 show that the signal due to excitation out of the 1s3 atoms alone (with laser quenching) is virtually zero, thus the excitation from the mixed target (unquenched) arises entirely from the 1s5 metastables. This observation makes it possible to use the 1s5 → 2p9 absolute cross sections from the fast-beam-target experiment to put the relative signals from the hollow cathode experiment on an absolute basis, as listed in Table III. As with experiments performed with ground state targets, the observed fluorescence signal contains contributions from both direct excitation and cascades from excitation of higher levels. For excitation from the ground state into the levels of the Ar(3p 5 4p) configuration, the cascade contribution was often greater than the direct excitation component [13]. For excitation from the metastable levels, however, we present two lines of reasoning why the cascade component
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Table III Apparent cross sections for excitation from the 1s3 and 1s5 metastable levels of Ar to the levels of the Ar(3p5 4p) configuration at select incident electron energies [27]. Total uncertainties are on the order of ±35%. Cross section (10−16 cm2 ) Level
12 eV
50 eV
100 eV
200 eV
Excitation from the 1s3 (J = 0) metastable level 2p2 1 5.7 9.1 9.8 9.8 1 7.4 17 18 18 2p4
18
10
5.7
3.8
Excitation from the 1s5 (J = 2) metastable level 1 0.20 0.52 0.61 0.65 2p2 2p3 2 0.29 1.4 1.1 0.96 1 0.26 0.62 0.60 0.50 2p4 2p5 0 0.44 0.29 0.17 0.11 2 2.2 8.3 8.8 9.0 2p6 2p8 2 5.7 6.3 5.2 4.9 3 12 24 25 24 2p9
0.82 0.44 0.08 8.5 4.6 23
4.9
3.1
1.8
12
6.8
4.0
J
2 eV
4 eV
6 eV
8 eV
makes up much less of the observed apparent cross section. First, cross sections for processes corresponding to dipole-allowed excitation processes are generally larger than cross sections for dipole-forbidden processes. For excitation from the 3p 6 ground state, direct excitation into all the levels of the 3p 5 4p configuration are dipole-forbidden, whereas excitation is dipole-allowed into cascading levels of the 3p 5 5s and 3p 5 3d configurations that have J = 1. Hence, at high energies, the cascades from these levels constitute a significant portion of the 3p 5 4p apparent cross sections. In contrast, excitation from the 3p 5 4s metastable levels into the levels of the 3p 5 4p configuration are generally dipole-allowed, whereas excitation into the cascading 3p 5 5s and 3p 5 3d configurations are dipole-forbidden. Thus, the same factor that enhances the cascades for excitation from the ground state attenuates the cascade contribution for the metastable excitation signal. Second, due to the motion of atoms in the fast beam target, atoms populated from long lived cascade levels decay far downstream from the location of the data collection optics, leaving the signal at small separations dominated by direct excitation contribution [57]. This suppression of the cascade population is significant even if the cascading levels have lifetimes on the order of the 3p 5 4p level of interest. This causes the ‘apparent’ cross section measurements for the 2p4 , 2p6 , and 2p9 levels made with the fast beam experiment to be essentially direct cross section measurements, without any need for a cascade correction. Furthermore, the pseudo-time-resolved nature of the fast beam experiment can be used to estimate directly the cascade contribution by comparing the signal rates close to the electron beam location and far downstream from the collision region [57]. Mea-
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surements for the 2p9 level (illustrated in Fig. 10) and the 2p6 level reveal the cascade contribution (as would be measured in a non-time-resolved experiment) to be less than 10% of the direct excitation cross section for these two levels. 5.2.2. Results of Cross Sections: Relation to J Comparison of the measured fluorescence signals for excitation into the J = 2 levels of the 2p group (2p3 , 2p6 , 2p8 ) using the mixed 1s3 /1s5 target (no laser quenching) and the pure 1s3 target (with laser quenching) indicates that for the mixed target the signal rates for excitation from the 1s3 level are completely overwhelmed by the signal contribution from the 1s5 metastable atoms (see Fig. 21). In contrast, the 3p 5 4p levels with J = 1 can be excited from both the 1s5 and 1s3 metastable levels as is evident in Fig. 21 where removal of the 1s5 atoms in the target still leaves behind a substantial part of the excitation signal. Thus the 1s5 level (J = 2) is readily excited to the J = 1, 2, 3 levels of the 3p 5 4p configuration, whereas the 1s3 level (J = 0) only to the J = 1 levels, reminiscent of the optical dipole selection rules. This is also the pattern one expects from the multipole field analysis discussed in Section 4.3.2. Since excitation into the J = 0 levels of 3p 5 4p (2p1 and 2p5 ) does not satisfy the dipole selection rules from either metastable level, the cross sections into these levels are expected to be small. Experimentally the 1s5 → 2p5 (J = 2) excitation data in Fig. 22 has both the smallest cross section and the sharpest peak in its excitation function which distinguishes it from the broad maximum in the other cases which all conform to the dipole selection rules. Due to the small direct excitation cross section into these dipole-forbidden levels, the cascade contribution to these levels may be more significant than for the dipole-allowed excitation cross sections. 5.2.3. Relation of Cross Sections to Optical Oscillator Strengths For electron-impact excitation from level-i to a level-j corresponding to an allowed optical dipole transition, the cross section at high energies can be expressed as a series with a leading term proportional to the optical oscillator strength, fij , as Qij (E) = 4πa02
fij 1 ln(E/R) + · · · , Eij /R E/R
(12)
where a0 is the Bohr radius, Eij is the threshold energy of the i → j excitation, and R is the Rydberg constant. If we retain only the first term, usually referred to as the Born–Bethe approximation, the above equation offers a simple relation between the cross section and oscillator strength. A plot of Q × E against ln E (a Bethe plot) should yield at high energies a straight line with a slope proportional to the oscillator strength. Values of the 1s5 → 2p9 , 1s5 → 2p6 , and 1s3 → 2p4 excitation cross sections were measured up to an energy of 400 eV
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F IG . 22. Apparent cross sections for excitation from the J = 0 1s3 () and J = 2 1s5 (•) metastable levels of Ar into levels of the 3p5 4p configuration [27]. Error bars are statistical only. There is an additional ±35% systematic uncertainty (common to all levels) from the absolute calibration of the 1s5 → 2p9 cross section.
in [27]. A comparison of the oscillator strengths extracted from fitting the high energy slopes of the data in Bethe plots provide a check on the absolute calibration procedure. Including only the uncertainties in the fitting procedure (and not the estimated ±35% absolute calibration uncertainty) the Bethe plots yield oscillator strengths f (1s5 → 2p9 ) = 0.39 ± 0.10, f (1s5 → 2p6 ) = 0.21 ± 0.05 and f (1s3 → 2p4 ) = 0.37 ± 0.05. These values are in good agreement with the spectroscopic values of 0.46, 0.21, and 0.53, respectively [78]. One may inquire to what extent the proportionality relation between electron excitation cross section and optical oscillator strength holds at low electron energies. In Table IV are listed the cross sections at 10 eV for excitations of the type 1sy → 2px along with the scaled oscillator strengths C × f (1sy → 2px )/Eij where the scaling factor C was chosen to match the 1s5 → 2p9 cross section. Note that the direct cross section Qij in Eq. (12) depends also on the excitation
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Table IV Comparison of the measured Ar(1sy → 2px ) cross sections [27] at 10 eV with the corresponding optical oscillator strengths fij obtained from [78]. To aid in the comparison, in the fifth column we have multiplied fij /Eij by a normalization factor, C, chosen to match the 1s5 → 2p9 cross section. Eij is the energy threshold of the i → j excitation process. Error bars include both the statistical and systematic uncertainties. Excitation i→j
Eij (eV)
fij
Q (10 eV) (10−16 cm2 )
fij /Eij × C (scaled)
1s3 (J = 0) → 2p2 (J = 1) → 2p4 (J = 1)
1.61 1.56
0.314 0.531
9.1 ± 3.6 18 ± 7
15 26
1s5 (J = 2) → 2p2 → 2p3 → 2p4 → 2p5 → 2p6 → 2p8 → 2p9
1.78 1.75 1.73 1.72 1.62 1.55 1.53
0.0278 0.0285 0.003 — 0.214 0.089 0.461
0.65 ± 0.25 0.94 ± 0.5 0.44 ± 0.20 0.11 ± 0.05 8.9 ± 3.2 4.7 ± 1.7 23 ± 8
1.2 1.2 0.13 — 10. 4.4 (23)
(J (J (J (J (J (J (J
= 1) = 2) = 1) = 0) = 2) = 2) = 3)
threshold energy Eij which varies somewhat for the various 1sy → 2px excitations. Indeed the cross sections at 10 eV track somewhat better with fij /Eij than with fij . One finds generally good agreement between the experimental cross sections and the scaled oscillator strengths with the only exception of 1s5 → 2p4 in which the cross section at 10 eV is more than three times the value expected from fij /Eij scaling. This transition has an unusually small oscillator strength (0.003) so it is possible that the first term in Eq. (12) is no longer the major contributor. Alternatively, since the direct cross section into this level is expected to be small based on the fij /Eij scaling, the observed apparent cross section may be dominated by the cascade contribution. Interestingly the excitation function for the 1s5 → 2p4 process has a somewhat narrower peak compared to the other dipole-allowed excitation functions in Fig. 22 which is consistent with the presence of higher order effects contributing to the observed shape via either cascades (which would have a sharper energy dependence) or through the contributions to the direct cross section from higher order terms in Eq. (12). This point will be discussed more fully in the cases of Kr and Xe where the 1sy → 2px excitation cross sections cover a wide range of optical oscillator strengths. While the scaling of the cross section by fij /Eij is motivated by Eq. (12), an important distinction between them must be made. Eq. (12) is intended for collisions at high energies and direct quantitative applications. The true energy dependence of the excitation cross sections at low electron energies differs substantially from the E −1 ln E form of Eq. (12). Nevertheless, to the extent that the excitation functions for all the dipole-allowed processes have the same shape/energy depen-
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dence, if the magnitudes of the cross sections scale with fij /Eij at high energies, they will also scale with fij /Eij at low energies. Hence we adopt at low electron energies the empirical form Qij = C × fij /Eij with an adjustable parameter C. The same scaling relationship may also be used as a means of estimating the 1s2 → 2px and 1s4 → 2px cross sections in the absence of experimental measurements. 5.2.4. Excitation of 3p 5 5p Levels Excitation from the 1s5 metastable level into the levels of the 3p 5 5p configuration are also dipole-allowed (with the exception of the two J = 0 levels). The 3p 5 5p → 3p 5 4s emissions from these levels is easily detected by a PMT. If the magnitude of the excitation cross section into these levels is given by the same Born–Bethe scaling relationship as discussed in the preceding section, the electron-impact excitation cross sections for the 3px levels should be rather small due to the fact that the oscillator strengths of the 1s5 –3px transitions are generally two orders of magnitude smaller than the corresponding 1s5 –2px transitions [78]. The limited data available, however, does not support this generalization [79]. While the 1s5 → 3p9 cross section is indeed small, only 4% the size of the 1s5 → 2p9 cross section at 8 eV, the fij /Eij scaling would predict that it should be even smaller, closer to 0.4%. The observed 1s5 → 3px cross sections, however, are in reasonable accord with the magnitudes from a more complete Born calculation [80].
5.3. E XPERIMENTAL U NCERTAINTY OF E XCITATION C ROSS S ECTIONS Having presented cross section results for excitation from both the ground state and metastable levels, it is possible to better explain the estimated experimental uncertainties of the cross section data measured in our laboratory. For excitation out of the ground level the individual optical emission cross sections generally have an experimental uncertainty of 10–15% (Section 3.3). Since the apparent cross section for level-i is the sum of all the optical emission cross sections out of level-i, the apparent cross sections also generally have a similar 10–15% uncertainty. When the cascade contribution is subtracted to obtain the direct excitation cross sections, however, the propagation of the uncertainties of the individual optical cross sections may result in a substantially larger percentage error especially if the cascades contribute a significant fraction of the apparent cross section. Since the size of the cascade contribution is a function of electron energy, the uncertainty of the direct excitation cross section is listed separately for each value as in Tables I and II. For excitation out of the metastable levels the situation is quite different. The cascade correction is typically no more than 10% so that no distinction needs to
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be made between apparent and direct excitation cross sections. Consider, for instance excitation out of the Ar(1s5 ) metastable level. The ratio of the 1s5 → 2px and 1s5 → 2py excitation cross sections or the ratio of the 1s5 → 2px at two different electron energies are generally accurate to better than 15%. This network of relative cross section results is subject to one overall scaling factor, determined by measuring the absolute magnitude of the 1s5 → 2p9 cross section at 20 eV. This absolute calibration introduces an additional 30% uncertainty that applies uniformly to all of the reported cross sections. An individual excitation cross section (i.e. for a particular level at a particular electron energy) therefore has a total uncertainty of about 35–40%, whereas the ratio of all other cross sections (i.e. a different electron energy) relative to this one is subject to only a 10–15% uncertainty. Thus the error analysis varies depending on how the cross section data are used. For instance, in a plot of the cross section versus energy (Fig. 22) we generally only include the statistical error bars which correspond to the uncertainty of the relative values to exhibit the shape of the excitation function. The systematic uncertainty from the absolute calibration does not affect the shape of the excitation functions, only the absolute magnitude of the all the 1s5 → 2px cross sections as a group. The same consideration applies to excitation from the 1s3 metastable level albeit with a slightly larger uncertainty due to the additional 1s3 : 1s5 number density measurement needed in the absolute calibration of these cross sections. These same uncertainties apply to the cross section results for the other rare-gas atoms presented in the following three sections.
6. Neon 6.1. E XCITATION OUT OF THE G ROUND L EVEL Experiments on electron-impact excitation of neon date back to the work of Hanle published in 1930 [81]. It was not until 40 years later, however, that a systematic study of excitation cross sections in relation to the electronic structure of the excited levels was reported [82]. To the extent that these earlier measurements were mainly limited to either apparent or optical emission cross sections, it was difficult to use these measurements as a comprehensive test of collision theory due to the large role of cascades in populating excited levels. Recent optical experiments, however, that combine measurements in radically different spectral ranges (i.e. the visible and VUV [83] or visible and IR [25]) determine direct cross sections that allow straightforward comparisons with theoretical calculations. 6.1.1. Pressure Effects As discussed in Section 5.1.1 at pressures below the onset of excitation transfer, the primary source of pressure effects in connection with measurement of
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excitation cross sections is radiation trapping. As with the pressure effects observed in Ar, radiation trapping affects not only levels directly connected to the Ne 2p 6 ground state, i.e., J = 1 levels in the 2p 5 ns and 2p 5 nd configurations, but also non-resonance levels such as 2p5 np levels via cascades from the higher resonance levels. Interestingly, since the absorption coefficients of Ne resonance lines are smaller than those of Ar, experimental measurements of Ne excitation cross sections for non-resonance levels have historically been less confounded by pressure effects. Where the pressure effects are most evident in Ne are higher resonance levels. Take for instance the 3d2 (2p 5 3d, J = 1) → 2p8 (2p 5 3p, J = 2) optical emission cross section. The 3d2 level decays partially to the ground level emitting a resonant photon and partially to 2p 5 3p levels (such as the 2p8 level). Reabsorption of the resonant photon by a Ne atom regenerates an atom in the 3d2 level allowing the possibility of additional 3d2 → 2p8 emission. As discussed in Section 3.2 this idea can be expressed quantitatively by Eq. (5) in terms of transition probabilities and the universal photon escape function g. The transition probabilities needed for application of Eq. (5) are generally not known to adequate accuracy. Using the values of [10,66] predicts a pressure dependence which differs noticeably from the observed data, but this discrepancy all but disappears upon a moderate adjustment of the relevant transition probabilities as shown in Fig. 23. The upper limit of the S-shaped curve corresponds to the case of complete reabsorption where the decay channel to the ground level is completely blocked off.
F IG . 23. Optical emission cross section as a function of target gas pressure for the 3d2 → 2p8 transition [72]. Dashed line is model calculation using the transition probabilities of [10] and [66]. Solid line is model calculation with transition probabilities adjusted to fit the cross section data.
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F IG . 24. Dependence of apparent ( ), cascade ( ) and direct (•) excitation cross sections on target gas pressure for select Ne(2p5 3p) levels at 100 eV [25].
In addition to resonance levels, weaker pressure dependence is observed in nonresonance levels like the 2p8 that receive radiative cascades from resonant levels. For instance, part of the 2p8 population results from the 3d2 → 2p8 emission which was previously shown to be pressure dependent. Even though the direct excitation cross section of the 2p8 level is not pressure-dependent, the apparent excitation cross section is. The photon escape factor g(kρ) depends on the reabsorption coefficient k which scales as λ3 . Among the heavy rare-gases, Ne has the shortest resonance wavelengths, and the smallest reabsorption coefficients. Thus it is not very surprising that the pressure effects for the 2p levels illustrated in Fig. 24 are weak compared to the heavier rare gases. 6.1.2. Excitation into the 2p 5 3s Levels The electronic structure of the 2p 5 3s configuration was reviewed in Section 2.1. Excitation cross sections into the two metastable levels, 1s3 and 1s5 , have been measured by means of the cw-LIF technique described in Section 3.5.1 using a multimode tunable dye-laser pumped by an Ar-ion laser [50]. The results are shown in Fig. 25 and Table V. The direct cross sections into the purely triplet metastable levels are sharply peaked at low-energies. The apparent cross sections
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F IG . 25. Apparent (solid line) and direct (dashed line) cross sections for excitation into the four levels of the Ne(2p5 3s) configuration [50].
for these two levels, however, are relatively broad due to the substantial cascade contribution from the decay of 2p 5 3p levels. The same LIF technique was also used by Phillips et al. [50] to obtain the cross sections for excitation into the two resonance levels, 1s2 and 1s4 . At sufficiently high atomic densities (above 3×1014 atoms/cm3 ) the lifetimes of these two levels are greatly lengthened through reabsorption of the resonance radiation so that a substantial population of these two levels is maintained in an electron-beam excitation experiment in spite of their short radiative lifetimes. For example, to measure the cross section from the ground level into the 1s2 , atoms that have been excited to the 1s2 level by electron impact are pumped to the 2p1 level by a cw-dye laser and the laser-induced 2p1 → 1s4 fluorescence provides the relative apparent excitation cross section. Such measurements have been reported for electron energies from threshold to 300 eV. A Bethe plot of the 1s2 relative apparent cross sections indeed yields a straight line (as in Eq. (12)) at energies above 120 eV. For absolute calibration one may at
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Table V Direct excitation cross sections of the four levels of the Ne(2p5 3s) configuration [50]. Uncertainties are ±25% for the metastable levels, ±28% for the resonance levels. Energy (eV) 25 30 40 60 80 100 200 300
Cross section (10−19 cm2 ) 1s2 (J = 1)
1s3 (J = 0)
1s4 (J = 1)
1s5 (J = 2)
62 76 91 100 100 97 67 51
1.8 1.4 0.93 0.42
6.5 7.5 7.9 8.8 8.6 8.0 5.7 4.4
10. 7.3 4.1 1.3
first think of normalizing the slope of the Bethe plot to the known optical oscillator strength of the 1 S0 –1s2 transition. This procedure, however, is not correct. The direct excitation cross section of the 1s2 level at high energies can be written as (b/E) log E + (a/E) where b is proportional to the oscillator strength of the 1 S0 –1s2 transition. In addition to direct excitation, the 1s2 level also is populated to a large degree by radiative cascades from higher levels such as those of the 2p 5 3p configuration (Section 6.1.3). The 2p 5 3p levels are non-resonant, and typically have direct excitation cross sections with an E −1 (or E −3 ) energy dependence at high energies. The 2p 5 3p levels, however, in turn are also additionally populated by cascades from higher levels some of which are resonant levels. Indeed, in the 100 to 300 eV range, the cascade contribution makes up about half of the total apparent cross section for six of the ten 2p 5 3p levels. Thus the apparent excitation cross sections for 2p 5 3p levels have a (log E)/E component and the cross section of the 2p → 1s2 cascade is of the form (b /E) log E + (a /E). The slope of the Bethe plot of the 1s2 apparent cross section is b + b , but only b is proportional to the oscillator strength. For a level with significant amounts of cascade (like the 1s2 ), normalization of the slope of the Bethe plot to the oscillator strength may lead to serious error. The normalization procedure adopted in [50] and [84] is as follows. Based on Fajen’s Born-approximation calculation, a set of theoretical direct excitation cross sections of the 1s2 level is obtained which exhibits the expected E −1 log E dependence at energies above 100 eV and yields an optical oscillator strength in good agreement with experiment. Setting the apparent excitation cross section at 300 eV equal to the Born (direct) excitation cross section plus the experimentally measured 2px → 1s2 cascade cross sections provides the normalization and yields absolute apparent excitation cross sections at all other energies. Subtraction of the measured cascade cross sections from this set of absolute apparent
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cross sections gives the experimental absolute direct excitation cross sections. These values are found to agree with the Born cross sections for energies down to 200 eV supporting the normalization procedure. Absolute and direct excitation cross sections for the 1s2 and 1s4 levels are shown in Fig. 25. Excitation cross sections for the 1s2 and 1s4 resonance levels can also be measured by detecting the 73.7 nm and 74.4 nm emission lines, respectively. At 100 eV Kanik et al. [85] have measured values of 8.42 and 2.08 × 10−18 cm2 , respectively, for the 1s2 and 1s4 levels, with an estimated total uncertainty of ±41%. A second set of VUV measurements made by Tsurubuchi et al. [83] found values of 9.00 and 1.80 × 10−18 cm2 , respectively, with an uncertainty of approximately ±14%. In comparison, the LIF values of Phillips et al. [50] at 100 eV are 12.4 and 2.7 × 10−18 cm2 with an estimated uncertainty of ±20%. While the apparent cross section values measured in the two VUV experiments of Tsurubuchi et al. and Kanik et al. are in excellent agreement, the LIF measurements of Phillips et al. are 30–40% higher. Note, however, that the VUV measurements are generally carried out at very low target pressures (i.e. 5 × 10−6 Torr in [85]) to minimize resonance reabsorption, whereas the LIF measurements of Phillips et al. were carried out at much higher target pressures (1–36 mTorr). To the extent that the 2p → 1s emission lines that provide most of the cascades into the 1s levels exhibit a 10–40% pressure dependence between 0.1 and 20 mTorr, and that cascades from these levels make a substantial contribution to the 1s apparent cross sections, most of these differences can thus be easily understood. Additional references for earlier VUV measurements of the Ne resonance lines can be found in [83,85] and review articles [7,22]. A more rigorous comparison between the VUV and LIF experiments is to examine direct cross sections which should be independent of target pressure. In addition to measuring the VUV resonance lines, Tsurubuchi et al. [83] have also measured the optical emission cross sections for the 2p 5 3p → 2p 5 3s cascading lines. Upon subtracting off this cascade contribution, Tsurubuchi et al. found direct cross section values of 7.02 and 0.76 × 10−18 cm2 , respectively, for the 1s2 and 1s4 levels at 100 eV, with an estimated total uncertainty of ±20% [83]. For the 1s4 level this is in excellent agreement with the value of 0.80 × 10−18 cm2 measured by Phillips et al. [50]. The Phillips et al. value for the 1s2 level of 9.7 × 10−18 cm2 is approximately 30% larger than that of Tsurubuchi et al., but still well within the range of the two experiment’s quoted uncertainties. From the radiative lifetimes of 1.6 ns for the 1s2 level and 21 ns for the 1s4 level [10], one can estimate the singlet-triplet mixing coefficients in Eq. (1) as α = 0.96 and β = 0.27, i.e., Ne(1s2 ) is largely singlet and Ne(1s4 ) is largely triplet. Thus the 1s4 level has much smaller direct excitation cross section from the 1 S0 ground state and consequently a much larger part of the 1s4 population is due to cascades from 2px levels which have both singlet and triplet components.
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6.1.3. Cross Sections of the 2p 5 3p Levels Since the 2p 5 3p → 2p 5 3s (2py → 1sx ) transitions are the only radiative decay channels of the 2p levels and are all in a favorable spectral region for photomultiplier detection, the apparent excitation cross sections are readily obtained by summing the appropriate optical emission cross sections from the 2py → 1sx array. The next step in determining the direct excitation cross sections for the 2p levels is to measure the cascades into the 2p 5 3p levels. These cascades are primarily from the decay of levels in the 2p 5 4s and 2p 5 3d configurations with transition wavelengths in the infrared region of the spectrum and have eluded direct measurement in electron excitation studies until the incorporation of weak-emission FTS techniques in recent years [25]. Additional cascades from the next two higher configurations 2p5 5s and 2p 5 4d have been measured but are quite insignificant for making cascade corrections to the apparent cross section. From these complete cascade measurements the resulting direct excitation cross sections are summarized in Table VI. As explained by the multi-pole field model (Section 4.3), except at the lowest energy the even-J levels on the average have larger cross sections than the odd-J levels, and this trend is more apparent at higher energies. Of the two J = 0 levels, note that the 2p1 level has a much larger cross section than the 2p3 level. The relative size of the two J = 0 levels in the np 5 np levels of all the rare-gases vary in a systematic way. The 2p1 state can be written in the LS-basis as a combination of |1 S0 and |3 P0 which are mixed through spin–orbit coupling. In the absence of spin–orbit mixing, the 1 S term has the highest energy of the 2p 5 3p LS-terms. Looking at the Ne energy level diagram of the 2p group, Table VI Direct excitation cross sections for the ten levels of the Ne(2p5 3p) configuration at select incident electron energies [25]. Error bars represent the combined statistical and systematic uncertainties. Cross section (10−19 cm2 ) Level
J
25 eV
30 eV
50 eV
75 eV
100 eV
150 eV
200 eV
2p1 2p2 2p3 2p4 2p5 2p6 2p7 2p8 2p9 2p10
0 1 0 2 1 2 1 2 3 1
7.3 ± 0.9 3.2 ± 0.5 1.3 ± 0.2 5.7 ± 1.3 2.4 ± 0.5 6.4 ± 1.0 2.8 ± 0.5 4.6 ± 1.0 5.2 ± 1.0 2.7 ± 0.7
11.5 ± 1.5 3.4 ± 0.5 1.8 ± 0.4 7.1 ± 1.4 2.3 ± 0.7 7.7 ± 1.3 2.9 ± 0.7 5.0 ± 1.3 5.5 ± 2.0 2.7 ± 0.8
19 ± 2 3.7 ± 0.6 2.7 ± 0.6 10.6 ± 1.7 2.8 ± 0.7 10.9 ± 1.7 3.6 ± 0.8 7.0 ± 1.6 5.9 ± 1.2 3.1 ± 0.9
16 ± 2 3.2 ± 0.5 2.4 ± 0.5 9.9 ± 1.5 2.4 ± 0.6 11 ± 2 3.5 ± 0.8 6.7 ± 1.4 3.9 ± 0.7 2.8 ± 0.7
14 ± 2 2.9 ± 0.5 2.2 ± 0.5 8.8 ± 1.3 2.0 ± 0.6 10 ± 2 3.2 ± 0.8 5.8 ± 1.5 3.0 ± 0.6 2.5 ± 0.7
10.4 ± 1.3 2.4 ± 0.4 1.9 ± 0.4 6.7 ± 1.1 1.3 ± 0.4 8.2 ± 1.2 2.6 ± 0.7 4.6 ± 1.0 1.8 ± 0.4 1.8 ± 0.6
8.6 ± 1.1 2.0 ± 0.4 1.4 ± 0.4 5.4 ± 0.9 1.0 ± 0.4 6.8 ± 1.0 2.2 ± 0.6 3.5 ± 0.9 1.2 ± 0.3 1.3 ± 0.5
5.1 3.3
6.6 3.4
10. 3.8
9.2 3.2
8.2 2.7
6.4 2.0
5.1 1.5
even-J odd-J
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one finds the 2p1 level is much higher than the other nine members. Hence it is not that surprising that an intermediate coupling calculation based on the method of [1] gives the Ne(2p1 ) level as predominately 1 S0 (91%) and the Ne(2p3 ) level as mostly 3 P0 -like. Since the ground state is 2p 6 1 S0 , excitation into the 2p1 is enhanced on account of its larger singlet component at the expense of the 2p3 level. In the heavier rare-gases, as well as the higher members of the Ne(2p 5 np, n 4), the two J = 0 levels have less disparity in the weighting of the two LS basis terms and thus more equally sized cross sections. For example, at 100 eV the 2p1 and 2p3 apparent cross sections are 140 and 36 × 10−20 cm2 , whereas the cross sections for 4p1 and 4p3 levels are 12 and 11 × 10−20 cm2 [86]. The energy dependence of the direct cross sections for the ten levels of the Ne(2p 5 3p) configuration displayed in Fig. 26 are generally very similar to one another, when compared to the wider variations observed in the levels of the Ar(3p 5 4p) configuration. Only the pure triplet Ne 2p9 (3 D3 ) level stands out as being slightly narrower than the other nine levels. A resonance structure at 18.6 eV in the excitation function of the 2p10 was first reported by Sharpton et al. [82]. The same resonance feature has also been seen in the study of electron scattering and metastable excitation cross sections. Due to the reduced pressure dependence of the Ne apparent cross sections compared to those observed in Ar, it is far easier to compare cross sections measured in different experiments. For example the values of Chilton and Lin [25] differ by only 2% on average from the high pressure values of Sharpton et al. [82], and also agree well with the low pressure values of Tsurubuchi et al. [83]. A number of other apparent cross section measurements are listed in the review of Heddle and Gallagher [7]. While the experiment of Sharpton et al. [82] also reported direct cross sections, the 2s, 3d → 2p IR-cascade transitions were generally not measured directly, but obtained by poorly known branching fractions. Bogdanova and Yurgenson [55] have also reported direct cross sections for the 2p levels using time-resolved measurements to eliminate the need for a cascade correction. With the exception of the 2p4 , 2p5 and 2p10 levels the direct cross sections of Chilton and Lin [25] agree within error bars with the values of Bogdanova and Yurgenson [55]. In the case of the 2p5 and 2p10 levels the two experimental value’s uncertainties just fail to overlap, whereas for the 2p4 level the difference is closer to 50%. 6.1.4. Excitation into the 2p 5 4s, 2p 5 3d, and Higher Levels The levels in the 2p 5 4s and 2p 5 3d configurations with J = 1 decay only to the 2p 5 3p levels with emission wavelengths in the infrared range which can be measured with FTS techniques. The J = 1 levels can also decay to the ground state and measurements of such emission cross sections would normally entail VUV spectrometry; however, at high pressures the VUV radiation is reabsorbed
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F IG . 26. Apparent (solid line), direct (•) and cascade ( ) cross sections for levels of the Ne(2p5 3p) group [25]. Error bars on the direct and cascade cross sections are the combined statistical and systematic uncertainties. The apparent and cascade cross sections were measured at 20 mTorr.
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and converted into IR emissions (Section 6.1.1). For the particular case of the 3d2 level, Fig. 23 demonstrates that at 20 mTorr reabsorption is almost complete. By fitting cross section measurements at different pressures to the theoretical model of radiation trapping (i.e. Eq. (5)) one can extrapolate the data to the limiting case of complete reabsorption. In this manner one can obtain the apparent excitation cross sections for resonant levels of 2p 5 4s and 2p 5 3d by detecting only the IR transitions into the 2p 5 3p manifold at high pressure. To find the direct excitation cross sections for levels of these two configurations one must consider cascades from levels in the 2p 5 4p configuration. Such emissions are in the range of 2–5 µm where detector sensitivity (even using the FTS technique) is relatively poor. Thus, for the 2p 5 4s and 2p 5 3d configurations only the apparent excitation cross sections have been reported [25]. Since the pattern of results for the Ne(2p5 4s) and Ne(2p 5 3d) levels generally parallel those of the corresponding processes in Ar, only the main features are summarized here and the readers are referred to the original papers for further details [25,82]. In the 2p 5 4s configuration, the 2s3 (J = 0) and 2s5 (J = 2) levels are purely triplet states and exhibit sharp excitation functions as displayed in Fig. 27. The apparent excitation functions of the 2s2 and 2s4 levels, both optically
F IG . 27. Apparent cross sections for the excitation into the four levels of the Ne(2p5 4s) configuration [25]. Cross sections for the two J = 1 resonant levels were obtained in the limit of complete reabsorption of the VUV resonance transitions. Total uncertainties (statistical and systematic) are approximately ±15%.
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connected to the ground level, however, do not have a broad maximum, typical of dipole-allowed excitations, due to distortion by cascades from higher levels. This distortion generally disappears for the next higher pair, 3s2 and 3s4 , indicating a much smaller cascade contribution relative to the direct excitation cross sections [25]. This is the same situation as exists for the 2s2 /2s4 and 3s2 /3s4 in Ar (Section 5.1.3). The 2p 5 3d configuration contains two purely triplet levels, 3d6 (J = 0) and 3d4 (J = 4); both of which have sharply peaked excitation functions. In contrast, a broad peak is clearly evident in the excitation functions for the three resonant levels (3d2 , 3d5 , 3s1 ). Measurements of optical emission cross sections for excitation into selected levels of the 2p 5 5s, 2p 5 4p and 2p 5 4d configurations have also been reported [25, 82]. While these results allow one to study the energy dependence of the apparent excitation cross sections, the lack of reliable branching fractions prevent one from determining the absolute magnitude of the cross sections. Nevertheless, the shapes of the excitation functions conform to the patterns one expects based upon the multipole field model. Both resonant levels of the 2p 5 5s and all three resonant levels of the 2p 5 4d configurations have the usual broad maximum. On the other extreme are the two purely triplet levels of 2p 5 5s (3s2 and 3s4 ) with very spikey excitation functions; data for the purely triplet levels of the 2p5 4d configuration are unavailable due to spectral contamination.
6.2. E XCITATION OUT OF THE M ETASTABLE L EVELS The energy dependence of the excitation cross sections measured with a mixed target of atoms in the 1s3 and 1s5 metastable levels into the ten levels of the 2p 5 3p configuration is shown in Fig. 28 [29]. Excitation into the two 2p 5 3p J = 0 levels
F IG . 28. Energy dependence of apparent cross sections for excitation into Ne(2p5 3p) levels from a mixed target of atoms in the 1s3 and 1s5 metastable levels at low electron energies [29].
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Table VII Apparent cross sections for electron excitation into J = 2 and J = 3 levels of the Ne(2p5 3p) configuration from the J = 2 1s5 metastable level [29]. Total uncertainties are on the order of ±30%. For excitation cross sections into 2p5 3p levels with J = 1, see Table VIII. Energy (eV) 3 6 8 10 20 50 100 200 400
Measured cross section (10−16 cm2 ) 2p4 (J = 2)
2p6 (J = 2)
2p8 (J = 2)
2p9 (J = 3)
1.9 2.6 2.6 2.5 2.4 1.6 0.98 0.66 0.35
6.7 9.9 10. 9.6 9.4 6.5 4.2 2.7 1.4
3.9 5.7 5.1 4.8 3.8 2.8 1.8 1.2 0.63
13 19 19 19 18 12 8.4 5.0 2.7
(2p1 and 2p3 ) is dipole-forbidden from both the J = 2 1s5 and the J = 0 1s3 metastable levels. The excitation functions for these two levels have a sharp peak at low energies, as compared to the broad peaks characteristic of dipole-allowed excitations for the other eight levels. In the work of Boffard et al. [29] no laser quenching experiments were performed to separate out the 1s3 and 1s5 sources of the excitation signal. As a result, cross section values were only reported into those levels where the vast majority of the signal could be assumed to arise from 1s5 excitation. Thus, results were limited to the 2p 5 3p levels with J = 2 and 3 since these levels are only dipole-allowed from the J = 2 1s5 metastable level. Cross section values for excitation out of the 1s5 metastable level into the 2p4 , 2p6 , 2p8 , and 2p9 levels at energies from 2 to 400 eV are listed in Table VII. Measurements made with the fast beam target reveal that the cascade correction to the observed signal is very small for the reported levels [29]. Hence, the apparent cross sections for these excitation processes are very good approximations of the direct cross sections. As was the case for excitation from the metastable levels of Ar (Section 5.2.3), the Ne(1s5 → 2px ) cross sections are found to scale well with oscillator strengths [87] even at low energies as illustrated by the values in Table VIII. In addition to the four cross sections reported in [29], this scaling relationship can also be tested on excitation cross sections into the four J = 1 levels for which only mixed target results were obtained in [29]. If one assumes (i) 5 : 1 statistical weighing for the 1s5 : 1s3 number density ratio in the hollow cathode discharge source target, and (ii) the excitation cross sections are equal to scaled values listed in Table VIII, one obtains weighted sum (i.e. Q(1s5 → 2px ) + (1/5)Q(1s3 → 2px )) cross sections (at 10 eV) of 3.4, 4.3, 2.7 and
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Table VIII Comparison of the measured Ne(1sy → 2px ) cross sections [29] at 10 eV with the corresponding optical oscillator strengths fij obtained from [87]. Eij is the energy threshold of the i → j excitation process. The scaled cross section values are equal to fij /Eij × C where C is a normalization factor chosen to match the experimentally measured 1s5 → 2p9 cross section. Error bars include both the statistical and systematic uncertainties. Excitation i→j
Eij (eV)
fij
Cross section (10−16 cm2 ) QMeas (10 eV)
Qscaled (10 eV)
1s3 (J = 0) → 2p2 → 2p5 → 2p7 → 2p10
(J (J (J (J
= 1) = 1) = 1) = 1)
2.01 1.98 1.90 1.67
0.242 0.427 0.191 0.057
— — — —
10. 19 8.7 2.9
1s5 (J = 2) → 2p2 → 2p3 → 2p4 → 2p5 → 2p6 → 2p7 → 2p8 → 2p9 → 2p10
(J (J (J (J (J (J (J (J (J
= 1) = 0) = 2) = 1) = 2) = 1) = 2) = 3) = 1)
2.11 2.09 2.08 2.07 2.02 1.99 1.96 1.93 1.76
0.034 — 0.058 0.011 0.158 0.021 0.099 0.428 0.111
— 0.2 2.5 ± 0.8 — 9.0 ± 2.8 — 4.8 ± 1.4 19 ± 6 —
1.4 — 2.4 0.46 6.7 0.91 4.4 (19) 5.4
5.9 × 10−16 cm2 for the 2p2 , 2p5 , 2p7 and 2p10 levels, respectively. These scaled values agree very well with the measured weighted sum cross sections of 3.1, 4.2, 1.7 and 5.3 × 10−16 cm2 . Thus the individual scaled cross sections for excitation into 2p2 , 2p5 , 2p7 and 2p10 levels from the 1s3 and 1s5 levels given in Table VIII can be used as estimates of the respective excitation cross sections in the absence of direct experimental measurements. Excitation into the 2p3 (J = 0) level is dipole-forbidden from both metastable levels, so oscillator strengths can not be used to apportion signals. Nonetheless, due to the 5 : 1 weighting of 1s5 : 1s3 atoms in the target, it is safe to assign almost all of the observed 2p3 excitation signal to excitation from the 1s5 level and thus obtain a value at 10 eV of 0.2 × 10−16 cm2 .
7. Krypton As discussed in Section 2.2 the electronic structure of Kr differs from Ne and Ar in that the core angular momentum jc is approximately a good quantum number. In a given electron configuration an energy level is labeled by jc and J . As it happens, the cross sections, especially those for excitation out of the metastable levels, are strongly influenced by the additional quantum number jc .
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7.1. E XCITATION OUT OF THE G ROUND L EVEL 7.1.1. Pressure Effects The same pressure dependence of optical emission cross sections observed in Ne and Ar is also evident in Kr for both resonance and non-resonant levels. Due to the longer wavelengths of the Kr resonance lines, however, the pressure at which radiation trapping effects become significant shifts to lower pressures than in Ne and Ar because of the resulting larger reabsorption coefficients. For example, the Ne(3d2 ) cross section data displayed in Fig. 23 approach the completereabsorption asymptotic value near 10 mTorr, the Ar(2s2 ) data in Fig. 3 near 5 mTorr, but the Kr(3s4 ) data in Fig. 29 near the asymptotic value at only 2 mTorr [26]. Note that in Fig. 29 the Kr(3s4 ) experimental data agree well with the radiation trapping model developed by Gabriel and Heddle (i.e. Eq. (5)) using the theoretical transition probabilities of [88], but the same scheme does a poor job for the Kr(3s2 ) level. This disagreement, however, disappears upon using the experimentally derived transition probabilities from [67]. It is also possible to derive transition probabilities from the cross section measurements by including them as free parameters in the fit to Eq. (5) [72]. While this does not give a unique choice of transition probabilities, measurements of the pressure dependence of optical emission cross sections can, in some cases, provide a useful check on transition probability values. The resonance radiation trapping of the 4p 5 ns and 4p 5 nd levels also influence via cascades the observed pressure dependence of the 4p 5 5p → 4p 5 5s optical emission cross sections. As in the case of argon, the apparent excitation cross
F IG . 29. Pressure dependence of the Kr(3s2 → 2p2 ) and Kr(3s4 → 2p8 ) optical emission cross sections at 100 eV [72]. The dashed lines are a fit to the data using the theoretical transition probabilities of [88], the solid lines are a fit to data using Eq. (5) with the transition probabilities as free parameters.
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F IG . 30. Dependence of apparent ( ), cascade ( ) and direct (•) cross sections for select levels of the Kr(4p 5 5p) configuration at 100 eV [26]. The representative error bars on the 2 mTorr direct cross sections include both the statistical and systematic uncertainties.
section and cascade cross section for a 2px level both have the same pressure dependence so that the difference, which gives the direct excitation cross section, is independent of pressure. This is illustrated for a few select levels of the 4p 5 5p configuration in Fig. 30. 7.1.2. Excitation of 4p 5 5s Levels In comparison to Ar and Ne, there are far fewer measurements of excitation cross sections into the 1sx resonance and metastable levels of Kr (see reviews [7,22]). One of the few attempts to measure the direct cross sections into the 4p 5 5s (J = 1) resonance levels is the recent experiment of Tsurubuchi et al. [8]. Figure 31 includes both the apparent cross sections they obtained by measuring the 116.5 nm and 123.6 nm resonance emission lines, and their cascade corrected direct cross sections. Note that in contrast to Ne and Ar, for Kr the direct cross section into the 1s4 level is larger than the cross section into the 1s2 level. Particularly at low energies, the apparent cross section is dominated by the cascade contribution from the 4p 5 5p levels. Mityureva and Smirnov likewise find that the apparent cross sections into the J = 0, 2 metastable levels are also dominated by cascades at low energies [89]. 7.1.3. Excitation of 4p 5 5p Levels Recall that the energy levels of each Kr 4p 5 nl configuration are split into two tiers based on the value of the core angular momentum jc (Section 2.2). One consequence of this structure is that the energy levels of the upper tier of 4p 5 5p overlap with the energy levels of the lower tier of 4p 5 4d. Emissions from 4p 5 5p → 4p 5 4d decays, however, have low frequency and very small dipole matrix elements on account of the change in jc ; their branching fractions are estimated to be no more than 0.3%. Thus, the apparent cross sections for the 4p 5 5p
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F IG . 31. Apparent (solid line) and direct (dashed line) cross sections for excitation into the J = 1 resonance levels of the Kr(4p5 5s) configuration [8]. Total uncertainties in the apparent (and cascade) cross sections are on the order of ±28%.
levels are still determined from the 4p 5 5p → 4p 5 5s emission array. Cascades from the 4p 5 ns (n = 6, 7, 8) and 4p 5 nd (n = 4, 5, 6) levels were measured by Chilton et al. [26] in order to obtain the direct excitation cross sections which are presented in Fig. 32 and Table IX. Table IX reveals a new feature not seen in the lighter rare gases. Here the uppertier levels (2p1 through 2p4 ) have smaller cross sections than the lower-tier levels, differing on the average by a factor of two. A possible explanation is that the upper group has an average ionization energy of 1.84 eV which is significantly lower than that of the lower group, 2.51 eV. For a one-electron atomic model the orbit radius is inversely proportional to the ionization energy. Excitation from the ground level into an upper-tier level requires a larger expansion of the electron cloud, and hence is generally less likely to take place compared with excitation into the lower-tier. Within each tier the parity relation holds well; the even-J levels have larger cross sections than the levels with odd-J at energies above 30 eV. Even at the peak energy there are only a few exceptions. It is also interesting to note that the cross section for the J = 0 2p5 level is larger than the cross section for the J = 0 2p1 level, which is opposite the pattern observed in Ne and Ar. This change is related to two factors: first, as described in the last paragraph the upper tier jc = 1/2 levels (including the 2p1 ) have smaller cross sections on average than the lower tier jc = 3/2 levels (including the 2p5 ). Second, in Kr the spin–orbit splitting of the 3 P multiplet pushes the 3 P0 level above the 1 S term, so that in Kr the upper J = 0 level has a higher triplet weighting than singlet weighting in contrast to Ne where the upper level has a very high singlet weighting.
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F IG . 32. Apparent (solid line), direct (•) and cascade ( ) cross sections for levels of the Kr 4p5 5p configuration [26]. Error bars on the direct cross sections and the combined statistical and systematic uncertainties. Apparent and cascade cross sections were obtained at a target pressure of 2 mTorr.
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Table IX Direct excitation cross sections for the ten levels of the Kr(4p5 5p) configuration at select incident electron energies [26]. Error bars represent the combined statistical and systematic uncertainties. Included in the table are also the average cross sections over all even-J levels, all odd-J levels, all levels with jc = 1/2, and all jc = 3/2 levels. Cross section (10−19 cm2 ) Level
J
jc
20 eV
30 eV
50 eV
75 eV
100 eV
200 eV
2p1 2p2 2p3 2p4 2p5 2p6 2p7 2p8 2p9 2p10
0 2 1 1 0 2 1 2 3 1
1/2 1/2 1/2 1/2 3/2 3/2 3/2 3/2 3/2 3/2
28 ± 4 59 ± 15 34 ± 6 26 ± 5 62 ± 9 65 ± 9 77 ± 15 136 ± 20 75 ± 13 36 ± 9
26 ± 3 44 ± 8 17 ± 4 16 ± 3 55 ± 8 48 ± 7 64 ± 15 107 ± 17 43 ± 7 28 ± 7
24 ± 3 30 ± 6 9.3 ± 2.2 8.5 ± 2.6 50 ± 8 35 ± 6 44 ± 14 71 ± 15 24 ± 5 16 ± 6
20 ± 3 23 ± 5 7.2 ± 1.5 7.0 ± 2.2 44 ± 8 28 ± 5 33 ± 13 47 ± 13 16 ± 3 9.4 ± 4.7
17 ± 2 19 ± 4 5.6 ± 1.3 5.6 ± 1.3 37 ± 6 22 ± 4 23 ± 11 38 ± 11 12 ± 3 8.2 ± 4.6
10 ± 1 11 ± 3 3.8 ± 0.8 4.2 ± 1.3 21 ± 4 13 ± 3 7.7 ± 5.2 17 ± 7 7.9 ± 1.8 2.4 ± 2.4
70 50 37 75
56 34 26 57
42 20 18 40
32 15 14 30
27 11 12 23
14 5.2 7.2 11
even-J odd-J jc = 1/2 jc = 3/2
A number of other experiments have measured apparent cross sections into the 4p 5 5p levels of Kr [8,65,74,90]. Due to the observed pressure dependent cascade process, however, it is difficult to make comparisons among the experiments except in the limit of zero pressure. Comparing the low-pressure limit apparent cross sections of Chilton et al. [26] to those of Tsurubuchi et al. [8] there is good agreement at 100 eV for the 2p1 , 2p3 , 2p4 , 2p5 , 2p6 , 2p7 , and 2p10 levels with an average difference of only 5%. For the remaining three levels (2p2 , 2p8 , 2p9 ) the values of Tsurubuchi et al. are on average over 70% larger than values of Chilton et al. It is difficult to explain this difference on pressure effects since the measured pressure dependence for these three levels is not as severe as many of the other seven levels. Likewise, the wavelengths of optical emission cross sections used to observe these three levels are intermixed with those of the wavelengths observed for the other seven levels. Bogdanova et al. [56,91] have measured the direct cross sections into nine of the 4p 5 5p levels using time-resolved techniques. The direct excitation cross section values of Chilton et al. and Bogdanova et al. generally overlap within error bars at both 20 and 100 eV except for the 2p3 , 2p5 , 2p6 , 2p7 , and 2p9 levels. For the 2p3 , 2p5 and 2p7 levels, however, the values at 100 eV are within error bars. For the remaining 100 eV values and the values at 20 eV the results of Chilton et al. are approximately a factor of 2.5 times larger than those of Bogdanova et al.
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7.1.4. Excitation of 4p5 6s, 4p 5 4d, and Higher Levels For the 4p 5 6s and 4p 5 7s levels (2sx and 3sx in Paschen’s notation), absolute apparent cross sections for the 2s3 and 2s5 , have been reported by Chilton et al. [26]. While the 2s5 excitation function has the expected sharp peak, the 2s3 level has a much broader excitation function in addition to the narrow structure as seen in Fig. 33. It is possible that the broad maximum is due to cascades from 4p5 np levels since the 2s3 cross sections are so small compared to 2s5 , but no definite explanation was given to this anomaly. It was, however, pointed out in [26] that the 2s3 and 2s2 levels are much closer together (45 cm−1 ) than the 2s4 /2s5 pair (267 cm−1 ) so that excitation transfer may also play a larger role for this pair of levels. The excitation function for the 3s5 level also has a narrow peak, but only relative results are available for this level. All four resonant levels (2s2 , 2s4 , 3s2 and 3s4 ) display the broad-peak energy dependence as expected; in particular the 2s2 and 2s4 excitation functions are not distorted, unlike their counterparts in Ar and Ne. Absolute apparent cross sections for the 2s2 and 2s4 levels were obtained by extrapolating the 2 mTorr values reported in [26] to the limit of com-
F IG . 33. Apparent cross sections for the Kr 4p5 6s levels [26] measured at 2 mTorr. The cross sections for the 2s2 and 2s4 resonant levels have been extrapolated from their 2 mTorr values to the high pressure limit of complete reabsorption of the resonance transitions [72]. Uncertainties are on the order of ±15%.
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plete trapping of the resonance transitions. This amounted to a 12% correction for the 2s4 level and a 18% correction for the 2s2 level. Excitation data for the 4p 5 4d levels are rather fragmentary because many of the 4p 5 4d → 4p 5 5p transitions have wavelengths greater than 4 µm. The InSb detectors used in this spectral region have low sensitivity which limits measurements for these wavelengths to only the largest optical emission cross sections. Absolute apparent excitation cross sections were only obtained for the 3s1 , 3d1 , and 3s1 levels. For the J = 1 3s1 resonant level the cross section was obtained by fitting the pressure dependence of the 3s1 → 2p10 optical emission cross section to obtain the asymptotic limit for the 3s1 → 2px cross sections when the decay channel to the ground state is blocked by radiation trapping. Additional relative excitation functions were reported for the other eight levels of the 4p 5 4d configuration and eleven levels of the 4p 5 5d configuration. All of the resonant levels (3s1 , 3d2 , 3d5 , 4s1 , 4d2 , 4d5 ) have the expected broad-peak excitation functions, although some distortion is seen in the 4s1 and 4d5 curves. The two observed levels that are pure triplet levels in the one-configuration approximation, namely the 4d1 (3 F4 ) and 4d6 (3 P0 ) levels have sharply peaked excitation functions. The J = 2 levels of the 4p 5 nd configurations generally (with two exceptions) have much narrower excitation functions than the observed J = 3 levels. The two exceptions, the 3s1 and 4s1 excitation functions are much broader than those of their J = 2 fellow members; the reason is not understood. The readers are referred to [26] for more details.
7.2. E XCITATION OUT OF THE M ETASTABLE L EVELS 7.2.1. Characterization in Terms of jc and Oscillator Strength The two-tier structure of the energy levels and inclusion of the core angular momentum quantum number jc to characterize the 4p 5 nl states have profound ramifications on the excitation cross sections. As described in Section 5.2, experiments on Ar show a preference for electron-impact excitation processes that conform to the optical dipole selection rules; for these cases cross sections track well with the oscillator strengths and have a broad maximum in the energy dependence [27]. With the new quantum number jc , we further characterize an excitation as being either core-preserving (jc = 0) or core-changing (jc = ±1). To the extent that jc is approximately a good quantum number, a core-preserving transition is expected to be favored over a core-changing one. As a matter of notation we denote the upper tier levels (jc = 1/2) of 4p 5 5p as 2p (upper) and the lower tier levels as 2p (lower). The metastable excitation processes of interest are thus divided into four classes: 1s5 → 2p (lower), 1s3 → 2p (upper), 1s5 → 2p (upper), and 1s3 → 2p (lower). Within each class we have both dipole-like excitation and
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non-dipole-like excitation. The first two classes are core-preserving; the cross sections are most likely governed by J according to the dipole selection rules as we have seen in Ar. The last two involve a change in jc which we have not considered previously. In relation to core-changing excitation let us consider the 1s5 → 2p (upper) cross sections. While the 1s5 → 2p2 (J = 2), 1s5 → 2p3 (J = 1), and 1s5 → 2p4 (J = 1) excitations all satisfy the dipole selection rules, their oscillator strengths are all very small, about two orders of magnitude weaker than a typical dipole-allowed transition with no change in jc . A general question then arises as to whether the signature of a broad peak for dipole-allowed excitations would appear in the excitation functions of weakly dipole-allowed processes. The answer is not dictated by the magnitude of the oscillator strength per se but rather depends on whether the dipole-type interaction dominates all other “higher order” effects in the context of the Born theory. Take as example excitation from the ground state of helium into the various n1 P levels, i.e., 1s 2 1 S → 1snp 1 P . The oscillator strengths for these transitions range from 0.276 to 0.0015 for n = 2 through n = 11 [92], yet the excitation functions for all these n values retain the same broad-peak shape [34]. We find no shape change because at higher n the final-state charge cloud moves further away from the domain of the ground state so that the dipole term and the higher order terms, which all depend on the overlap of the initial and final states, decrease in unison leaving the dipole dominance unchallenged. In the present case of Kr we have a drastic reduction in the dipole term from 1s5 → 2p (lower) to 1s5 → 2p (upper) but the extent to which the higher order terms change is not apparent. In general, if the dipole interaction term should lose its controlling influence, the excitation function would have a narrower peak and the magnitude of the cross section would be greater than what is expected from a direct scaling to the oscillator strength. This applies to electron excitation processes regardless of any core consideration, and in fact has been observed in excitation of Ar out of the metastable levels. As discussed in Section 5.2.3, the Ar(1s5 → 2p4 ) excitation, which has the smallest non-zero oscillator strength (0.003) in Table IV, deviates appreciably from the norm in both the shape of the excitation function and scaling of cross section with optical oscillator strength. 7.2.2. Results of Cross Sections Cross sections for excitation from the metastable levels of Kr measured by Jung et al. [31,34] for core-preserving excitations are presented in Table X and Fig. 34. All of the excitations in Fig. 34 correspond to dipole-allowed transitions with three exceptions: 1s5 (J = 2) → 2p5 (J = 0), 1s3 (J = 0) → 2p1 (J = 0), and 1s3 (J = 0) → 2p2 (J = 2). The excitation functions for these three are indeed narrower than the others, whereas the other excitations in Fig. 34 show a
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Table X Apparent cross sections for core-preserving excitations from the 1s3 and 1s5 metastable levels of Kr to the levels of the Kr(4p5 5p) configuration at select incident electron energies [34]. Total uncertainties are ±35% for cross sections out of the 1s5 metastable level, and ±40% for cross sections out of the 1s3 metastable level. Cross section (10−16 cm2 ) Level
J
jc
2 eV
3 eV
6 eV
8 eV
15 eV
50 eV
100 eV
200 eV
8.8
4.8
3.1
23
16
8.9
Excitation from the 1s3 (J = 0, jc = 1/2) metastable level 0 1/2 0.90 1.3 0.63 0.23 2p1 2p2 2 1/2 7.0 18 5.5 3.6 1 1/2 17 43 59 60 2p3 2p4 1 1/2 18 48 62 63 Excitation from the 1s5 (J = 2, jc = 3/2) metastable level 2p5 0 3/2 0.32 0.76 0.11 0.04 2 3/2 8.6 18 25 24 17 2p6 2p7 1 3/2 4.7 3.8 2.6 2.2 2 3/2 4.8 7.6 9.6 8.6 2p8 2p9 3 3/2 27 47 52 51 41 1 3/2 9.8 15 17 17 2p10
broad maximum near 6 eV. Note, however, that a prominent feature near threshold is also apparent in the 1s5 → 2p7 excitation which has the smallest oscillator strength (0.023) [93] of the core-preserving transitions. To examine this issue we list the oscillator strengths for the seven core-preserving transitions along with the excitation cross sections at two different energies in Table XI. At 6 eV the cross sections are proportional to the oscillator strength. This linear relation continues down to 3.5 eV aside from the sole case of 1s5 → 2p7 where the cross section is larger than the prediction. Thus the 1s5 → 2p7 excitation fits into the pattern of a small optical oscillator strength with a concomitant deviation from the broad-maximum in the excitation function and the proportionality relation of cross section versus oscillator strength that can be attributed to the manifestation of higher order terms as discussed in Section 7.2.1. We should mention, however, that these deviations are also consistent with cascades into the 2p7 level from higher levels which assume a more prominent role when the direct excitation is reduced due to the small oscillator strength. For core-changing processes, only the 1s5 (J = 2) → 2p2 (J = 2), 1s5 (J = 2) → 2p3 (J = 1) and 1s5 (J = 2) → 2p4 (J = 1) excitation cross sections have been reported [34] and these values are summarized in Table XII and Fig. 35. Because of the very low concentration of atoms in the 1s3 metastable level in the collision target, no cross sections for core-changing excitation out of the 1s3 levels were reported. All three of the reported excitation
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F IG . 34. Apparent cross sections out of the metastable levels of Kr for core-preserving processes [34]. Error bars are statistical only. Total uncertainties are ±35% for cross sections out of the 1s5 metastable level, and ±40% for cross sections out of the 1s3 metastable level.
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Table XI Comparison of the measured Kr(1sy → 2px ) cross sections [34] at 3.5 eV and 6 eV with the corresponding optical oscillator strengths fij obtained from [93]. To aid in the comparison, in the fourth column we have multiplied fij by a normalization factor, C, chosen to match the 1s5 → 2p9 cross section. Error bars include both the statistical and systematic uncertainties. fij
Q (6 eV) (10−16 cm2 )
fij × C (scaled)
Q (3.5 eV) (10−16 cm2 )
0.57 0.46 0.24 0.023 0.088 0.50 0.16
59 ± 24 62 ± 25 25 ± 9 2.6 ± 0.9 9.6 ± 3.4 52 ± 18 17 ± 6
59 48 25 2.4 9.2 (52) 17
55 ± 21 64 ± 25 21 ± 7 3.2 ± 1.1 8.9 ± 3.1 52 ± 18 15 ± 7
1s5 → 2p2 1s5 → 2p3 1s5 → 2p4
5 × 10−4 0.003 4 × 10−5
0.3 ± 0.15 0.2 ± 0.1 0.1 ± 0.05
0.05 0.28 0.004
0.9 ± 0.4 0.5 ± 0.3 0.4 ± 0.2
1s3 → 2p1 1s3 → 2p2 1s5 → 2p5
0 0 0
0.63 ± 0.25 5.5 ± 2.2 0.11 ± 0.05
0 0 0
1.3 ± 0.5 16 ± 6 0.4 ± 0.2
Excitation i→j 1s3 1s3 1s5 1s5 1s5 1s5 1s5
→ 2p3 → 2p4 → 2p6 → 2p7 → 2p8 → 2p9 → 2p10
Table XII Apparent cross sections for core-changing excitations from the 1s5 (J = 2, jc = 3/2) metastable level of Kr to the levels of the Kr(4p5 5p) configuration at select incident electron energies [34]. Total uncertainties are ±50%. Cross section (10−16 cm2 ) Level
J
jc
3 eV
4 eV
6 eV
8 eV
2p2 2p3 2p4
2 1 1
1/2 1/2 1/2
0.47 0.5 0.5
0.66 0.4 0.3
0.30 0.2 0.1
0.10 0.2 0.05
processes correspond to optically allowed transitions albeit with very small oscillator strengths on account of the core-changing nature. The excitation functions all have a narrow peak bearing no resemblance to the broad maximum normally associated with dipole-allowed excitation. Likewise the excitation cross sections listed in Table XI are significantly larger than expected from consideration of oscillator strengths alone. Here we find a complete departure from the commonly expected behavior of dipole-allowed excitation when it comes to core-changing processes. The results of Figs. 34 and 35 together establish a continuous shift of excitation behavior in accordance with the oscillator strength (i.e. Table XI). An
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F IG . 35. Core-changing apparent cross sections for excitation out of the 1s5 (J = 2, jc = 3/2) metastable level of Kr into the 2p (upper) (jc = 1/2) levels [34]. Error bars are statistical only, total uncertainties are ±50%.
excitation corresponding to an oscillator strength of “normal” size conforms to the conventional rules whereas a complete departure is seen when the oscillator strength is down in magnitude to 0.003 and below. Intermediate between these two limits is the 1s5 → 2p7 case where we begin to recognize that modifications to the conventional rules are needed to analyze the data. In the preceding paragraphs we have examined the magnitudes and energy dependence of the excitation cross sections at low energies, i.e. less than 10 eV. At much higher electron energies the fij E −1 ln E dipole term should dominate over any higher order components. For the core-preserving, dipole-allowed transitions 1s5 → 2p6 and 1s5 → 2p9 for which measurements extend to electron energies of 400 eV, the cross sections are indeed observed to have the expected Born–Bethe behavior [34]. The extracted oscillator strengths of 0.28 ± 0.02 and 0.55 ± 0.05 (statistical uncertainties only) agree well with the spectroscopically derived values of 0.24 and 0.50 [93] for the 1s5 → 2p6 and 1s5 → 2p9 transitions, respectively. No high energy data for core-changing, dipole-allowed processes have been reported, however, in comparing the cross section values in Table XI at 3.5 eV to those at 6 eV, we generally observe a convergence towards the scaling relation based on oscillator strengths. It is clear that core-preserving, dipole-allowed excitations have the largest cross sections whereas the other excitation classes are in varying degrees less favorable. No core-changing, dipole-forbidden excitations, such as 1s5 (J = 2) → 2p1 (J = 0), have been measured, but should be very small due to its doubly unfavorable nature. One instructive exercise is to compare the core-preserving, dipole-forbidden with the core-changing, dipole-allowed cases. Inspection of Tables X and XII suggest that the former have marginally larger cross sections than the latter. A more concrete illustration is furnished by excitation into the 2p2 (J = 2) level where the core-preserving but dipole-forbidden excitation out of the
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1s3 (J = 0) metastable level overpowers the core-changing but dipole-allowed excitation out of the 1s5 (J = 2) level. Mityureva et al. [45] have also measured excitation cross sections into the 4p5 5p levels using the technique described in Section 3.4. While the magnitudes of their cross sections are a factor of 10 to 20 larger than the cross sections reported by Jung et al. [31,34], they still share many of the same general behaviors. In particular, Mityureva et al. found a close correlation between the magnitude of the cross section and the size of the corresponding optical oscillator strength, especially for the strongest transitions (i.e. 2p6 , 2p8 , 2p9 ). They also reported that the excitation function for the 1s5 → 2p7 cross section was narrower than that of other dipole-allowed excitation processes. Mityureva et al. also measured cross sections into the 2p (upper) group, which they attributed to excitation from the 1s5 metastable level since the concentration of 1s5 atoms in the target exceeded the population of other 1s levels by an order of magnitude. However, the experiment of Jung et al. [31] demonstrates that the primary excitation mechanism for the upper-tier levels is the core-preserving excitation from the 1s3 metastable atoms even when the concentration of this metastable level is an order of magnitude less than that of the 1s5 level.
8. Xenon 8.1. E XCITATION OUT OF THE G ROUND L EVEL Xenon stands out from the lighter rare gases in its notoriously strong pressure effects on the 5p 5 6p → 5p 5 6s emission cross sections. An initial report by Walker [94] has triggered additional experiments which confirmed such effects to persist in some cases at pressures down to 0.1 mTorr. Although the increase of cross section with pressure is usually attributed to radiation trapping and/or atomatom collisional excitation transfer, the origin of this exceptionally large effect remained an enigma for a long time. It was not until the direct measurements of the infrared cascade radiation by Fourier transform spectroscopy that the cascades were unambiguously identified as being entirely responsible for the variation of cross section by as much as a factor of three over the pressure range from 0.1 to 2.0 mTorr [21]. As an illustration, Fig. 36 shows the apparent, direct, and cascade cross section of the 2p7 level (5p 5 6p, J = 1) at various pressures at 50 eV electron energy. The two-tier structure of the energy levels in each configuration is more pronounced in Xe than in Kr. Referring to the energy level diagram in Fig. 37, radiation decay channels of the 2p5 , . . . , 2p10 (i.e. jc = 3/2, lower-tier of 5p 5 6p) consist entirely of the 2p → 1s transition array, but the upper-tier members of the 5p 5 6p group (2p1 . . . 2p4 ) have the additional decay channels of transitions into
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F IG . 36. Pressure dependence of the Xe(2p7 ) apparent (), cascade (×) and direct (•) cross sections at 50 eV [21]. Upon subtracting the pressure-dependent cascade contribution from the pressure-dependent apparent cross section, the direct cross section is pressure independent as expected. For clarity, only the error bars (combined statistical and systematic uncertainties) of the direct cross section are included.
F IG . 37. Energy level diagram for xenon with both the Paschen and Racah labeling schemes. In Paschen’s notation, for example, the four levels of the 5p5 6s configuration are labeled as 1s2 , 1s3 , 1s4 and 1s5 (highest to lowest energy). In Racah notation, these four levels are labeled 6s [1/2]o1 , 6s [1/2]o0 , 6s[3/2]o1 , and 6s[3/2]o2 .
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Table XIII Direct excitation cross sections for the ten levels of the Xe(5p5 6p) configuration at select incident electron energies [21]. Error bars represent the combined statistical and systematic uncertainties. Cross section (10−19 cm2 ) Level
J
jc
15 eV
20 eV
30 eV
50 eV
100 eV
150 eV
2p1 2p2 2p3 2p4 2p5 2p6 2p7 2p8 2p9 2p10
0 1 2 1 0 2 1 3 2 1
1/2 1/2 1/2 1/2 3/2 3/2 3/2 3/2 3/2 3/2
6.0 ± 1.2 8.5 ± 1.5 36 ± 6 15 ± 2 38 ± 9 60 ± 16 115 ± 20 150 ± 25 175 ± 28 110 ± 17
21 ± 3 8.6 ± 1.8 30 ± 7 5.4 ± 2.4 72 ± 16 72 ± 20 92 ± 20 74 ± 32 135 ± 29 44 ± 12
44 ± 6 3.9 ± 0.9 22 ± 4 2.3 ± 1.4 145 ± 24 54 ± 10 50 ± 15 20 ± 9 85 ± 22 22 ± 8
26 ± 4 2.7 ± 0.9 20 ± 4 2.0 ± 1.3 100 ± 20 42 ± 8 37 ± 15 5.1 ± 5.4 67 ± 20 8.4 ± 6.8
19 ± 3 2.4 ± 0.8 17 ± 3 1.7 ± 1.2 87 ± 18 33 ± 6 27 ± 13 4.1 ± 3.6 40 ± 16 2.1 ± 5.9
15 ± 2 2.3 ± 0.8 14 ± 3 1.3 ± 1.1 75 ± 15 25 ± 5 24 ± 12 3.7 ± 2.9 36 ± 14 1.9 ± 5.1
63 80 16 110
66 45 16 81
70 20 18 63
51 11 13 43
39 7.5 10 32
33 6.6 8 28
even-J odd-J jc = 1/2 jc = 3/2
the lower tier of the 5p 5 7s and 5p 5 5d configurations which, however, involve a change of the core angular momentum jc . Such channels indeed are found to make an insignificant contribution to the apparent excitation cross section. To obtain the direct excitation cross sections, cascades into all 2p levels from the relevant levels of the 5p5 7s, 5p 5 8s, 5p 5 5d, 5p 5 6d, 5p 5 7d, 5p 5 8d configurations have been examined and subtracted from the apparent cross sections. Values of direct excitation cross sections are given in Table XIII with plots of apparent, direct, and cascade cross sections versus energy in Fig. 38. A number of features are apparent from the direct excitation cross sections. First, the excitation cross sections for Xe are significantly larger than the cross sections of Kr and the lighter rare gases. Second, the cross sections for excitation into the upper tier levels (2p1 through 2p4 ) are distinctly smaller than those into the lower tier levels. The disparity in cross sections between the two tiers is even more pronounced here than in Kr and this is in line with the relation to ionization energies. The upper levels have on the average an ionization energy of 1.1 eV which is less than one-half of that of the lower-tier. This makes excitation into the upper tier levels much less favorable because of their diffusive electron charge cloud. In comparison the ionization energy ratio for the corresponding case in Kr is 1.8 eV : 2.5 eV, hence a less drastic disparity in cross section. Third, within each tier the average cross sections of the even-J levels are larger than those of the odd-J levels as expected based on the parity rule. Finally the excitation
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F IG . 38. Apparent, direct and cascade cross sections for levels of the Xe(5p5 6p) levels [21]. Error bars on the direct cross sections include the combined statistical and systematic uncertainties. Apparent and cascade cross sections measurements were performed at a target pressure of 1.0 mTorr.
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functions of all ten 2p levels have quite narrow peaks in contrast to the lighter rare gases. Closer examinations, however, do reveal different degrees of decline with increasing energy that are correlated with J : least steep for J = 0, and most steep for J = 3. The later, of course, is not too surprising since the J = 3 level is a pure triplet level in the one configuration approximation. The excitation functions of all the odd-J levels are in general steeper than the levels with even-J . The 5p5 6p levels are the only ones into which extensive measurements of the direct excitation cross section have been made. Bogdanova and Yurgenson have also reported direct cross sections for four 5p 5 6p levels using time-resolved techniques to alleviate the need for IR-cascade measurements [95]. Their results are on average about 25% the size of the values of Fons and Lin listed in Table XIII. While Fons and Lin found that radiation trapping of cascading resonant levels was responsible for most of the observed pressure dependence in the 2p cross sections in the 0.1–2.0 mTorr range (i.e. Fig. 36), their measurements do not rule out the importance of additional mechanisms at higher pressures (i.e. collision transfer). Since the time-resolved measurements of Bogdanova and Yurgenson were made at higher pressures than those explored by Fons and Lin, it is unclear if these additional mechanisms may explain the differences between these two sets of measurements. Earlier measurements of the 5p5 6p levels (i.e. [65], and those reviewed in [7]) have been limited to either optical emission or apparent cross sections and are thus highly sensitive to the pressure used in the experiment. Kanik et al. [96] have conducted VUV measurements of the emission lines arising from the J = 1 resonance levels of the 5p 5 6s, 5p 5 7s, and 5p 5 5d configurations. As expected of dipole-allowed excitation processes, their observed excitation functions are relatively broad, with a small peak near threshold presumably due to cascades from higher levels. The measured cross section into the 1s4 level was approximately 2.7× larger than the cross section into the 1s2 level. Earlier measurements of cross sections into the 5p 5 6s resonance and metastable levels are reviewed in [7,22,89]. DeJoseph and Clark [24] have measured apparent cross sections for four 5p 5 5d levels by using a FTS with an InSb detector to measure IR 5p 5 5d → 5p 5 6p emission lines. Optical emission cross sections from an additional three levels were also reported. The shape of the excitation function for the J = 1 3d2 resonance level was a broad maximum, whereas the excitation function for the pure triplet 3d6 (3 P0 ) level consisted of a sharp peak at threshold that decreases to a near constant value for energies between 40 and 150 eV. Since the measurements of DeJoseph and Clark were made at 4 mTorr, pressure dependent radiation trapping of higher resonant levels and possibly collision transfer effects may contribute substantially to the observed high energy tail. Fons and Lin [21] also measured excitation cross sections into numerous 5p 5 ns (n = 7–11) and 5p 5 nd (n = 5–9) for the purposes of determining the cascade correction to the 5p 5 6p direct cross sections. The excitation function of the J = 1 resonance levels gen-
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erally have the broad maximum expected of dipole-allowed transitions, with the notable exception of the 2s2 level which has a sharper peak, possibly due to a large cascade contribution. The pure triplet levels of the 5p 5 ns and 5p 5 nd configurations generally had very narrow excitation functions, with the exception of the 2s3 level which had a more intermediate width.
8.2. E XCITATION OUT OF THE M ETASTABLE L EVELS The large spin–orbit splitting (1.3 eV) of the Xe+ (5p 5 ) core makes the core angular momentum jc even closer to an exact quantum number than in Kr. The energy levels of the Xe(5p 5 6s) and Xe(5p 5 6p) configurations, which are reproduced in Fig. 37, group themselves into two tiers according to the two values of jc . Again we distinguish the core-preserving excitations, 1s5 → 2p (lower) and 1s3 → 2p (upper), from the core-changing ones, 1s5 → 2p (upper) and 1s3 → 2p (lower), which are expected to have smaller cross sections. Note, however, that the 1s3 level is very close in energy to the lower members of the 2p manifold (i.e. only 0.13 eV below the 2p10 level). Such a near-resonance situation may boost the 1s3 → 2p (lower) cross sections over what one may expect from core considerations alone. In a typical plasma or discharge, the large energy separation between the 1s5 and 1s3 levels leads to vastly different number densities of atoms in the two metastable levels; well beyond the 5 : 1 ratio between the levels based upon their statistical weights. This is partially due to the Boltzmann factor of exp[−E/kTe ], where E is the energy difference between the 1s3 and 1s5 metastable levels (1.1 eV), and kTe is the electron temperature of the plasma discharge. Additionally, it has been recognized that the 1s3 population in a Xe discharge is greatly reduced by the excitation transfer from 1s3 into the nearby 2p levels through collisions with ground-state Xe atoms [97]. Indeed, the decay rate of the 1s3 population in a discharge was found to increase with the Xe pressure [97]. The 1s5 -to-1s3 density measured in the hollow cathode discharge source target of Jung et al. [30] was in excess of 200 : 1. With such a lopsided mixed target, the signal for electron-impact excitation into the 2p5 , . . . , 2p10 levels at energies below the ground state excitation threshold (9.5 eV) are effectively all due to core-preserving excitation from the 1s5 level with a negligible contribution from excitation out of 1s3 . The 1s5 → 2p (lower) excitation cross sections resulting from the measurements of Jung et al. [30] are included in Fig. 39 and Table XIV. Since the 1s3 level is 1.1 eV above the 1s5 level, the energy threshold for the 1s3 → 2p (lower) excitation processes is 1.1 eV lower than the corresponding 1s5 → 2p (lower) threshold. Thus, one can place a limit on the contribution to the reported 1s5 → 2p (lower) cross sections from excitation of the 1s3 level by
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F IG . 39. Apparent cross sections for excitation from the 1s5 metastable level of Xe into the jc = 3/2 (lower tier) levels of the Xe(5p5 6p) configuration [30]. The shape and magnitude of the i → j excitation cross sections at low electron energies are correlated with the value of the corresponding optical oscillator strength, fij [88].
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Table XIV Apparent cross sections for excitation from the 1s5 metastable level (J = 2, jc = 3/2) into the lower-tier levels (jc = 3/2) of the Xe(5p5 6p) configuration [30]. The total (statistical and systematic) uncertainties in the values are ±35%. Energy (eV) 2 3 4 6 8 12 25 50 100 150
Cross section (10−16 cm2 ) 2p5 (J = 0)
2p6 (J = 2)
2p7 (J = 1)
2p8 (J = 3)
2p9 (J = 2)
2p10 (J = 1)
0.70 0.81 0.47 0.32 0.28
9.1 13 15 15 14 14 10. 7.1 4.3 3.6
5.7 4.7 3.2 2.2 1.4
15 25 32 34 33 32 25 15 10. 7.6
5.8 9.9 10. 8.4 6.4
6.7 11 13 15 14
examining the signal rate in the 0.2-to-1.3 eV energy range between the thresholds for excitation from the two metastable levels. Examination of Fig. 39 does reveal some hints of non-zero signals in this energy range, but if present at all constitute a very small correction to the reported 1s5 excitation cross sections. If the core-changing 1s3 → 2p (lower) excitation cross sections were two orders of magnitude smaller than the core-preserving 1s5 → 2p (lower) cross sections as has been measured for Kr (Section 7.2), upon folding in the reduced 1s3 : 1s5 number density ratio one would not expect any discernable excitation signal. If, in fact, the signals in the 1s5 sub-threshold range are non-zero, this suggests the possibility that the 1s3 → 2p (lower) cross sections are considerably larger than expected for a core-changing excitation because of the near resonance of the initial and final states. At the present time there is no adequate metastable target with sufficient 1s3 atoms available for accurate measurements of core-changing excitations into the 2p (lower) levels. Such experiments are, however, of significant interest as they may provide quantitative insights into how the cross sections are influenced by various dynamic factors. Figure 39 provides a full view to the core-preserving excitations from the 1s5 metastable level into the 2p (lower) levels. Of the six excitation processes, only the 1s5 (J = 2) → 2p5 (J = 0) fails to meet the dipole criteria and is expected to have a sharp peak in its excitation function. While this is indeed what is observed for the 2p5 level, the excitation function for the 2p7 level is almost as sharp, despite being a dipole-allowed process. The shapes of the excitation functions, however, do correlate with the values of corresponding optical oscillator strengths. The three curves with a broad maximum (1s5 → 2p8 , 1s5 → 2p6 ,
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Table XV Comparison of excitation cross sections out of the Xe(1s5 ) metastable level with optical oscillator strengths. The experimental values were obtained at an electron energy of 8 eV [30]. In the fourth column the optical oscillator strengths (from [88]) have been normalized to match the experimental 1s5 → 2p8 value. Cross section (10−16 cm2 ) Level
Eij (eV)
fij
C × fij /Eij
Exp. (8 eV)
2p1 2p2 2p3 2p4 2p5 2p6 2p7 2p8 2p9 2p10
2.82 2.76 2.74 2.64 1.62 1.51 1.48 1.41 1.37 1.27
— 0.0014 0.0024 0.0006 — 0.24 0.013 0.56 0.12 0.24
0 0.040 0.068 0.018 0 14 0.77 (33) 7.1 14
small small small small 0.28 14 1.7 33 6.4 14
1s5 → 2p10 ) that is generally regarded as a signature of dipole-allowed excitation, are indeed associated with the three largest optical oscillator strengths (0.56, 0.24, 0.24, respectively). After this trio is the 1s5 → 2p9 transition with an oscillator strength of 0.12. The reduction in oscillator strength is accompanied by an appreciable narrowing of the excitation function. The ultimate case is the aforementioned 1s5 → 2p7 excitation where the narrow peak echoes an unusually small oscillator strength 0.013. This correlation also is apparent in the magnitude of the excitation cross sections. Table XV compares the cross sections at 8 eV with the relative oscillator strengths scaled to match the value of the observed 1s5 → 2p8 cross section. The proportionality holds very well for the three levels with the largest oscillator strengths (2p6 , 2p8 , 2p10 ), whereas at the other extreme of the smallest oscillator strength (1s5 → 2p7 ) the experimentally observed value is more than twice the scaled value based upon fij . The intermediate case of 2p9 conforms to the linear relation to within 10%. These results closely parallel the 1s5 → 2px cross section results observed for excitation from the metastable levels of Kr (Section 7.2). As with excitation from the metastable levels of Kr, it is also unclear if the change in excitation function shape is solely due to the influence of higher order processes, or if an increased role for cascades when the direct cross section is very small is also important. In their experiments on electron-impact excitation from the Xe metastable levels, Jung et al. [30] detected no discernable fluorescence from the 2p1 , . . . , 2p4 levels which may arise from core-changing excitation out of the 1s5 level and/or core-preserving excitation out of the minute fraction of 1s3 atoms in the target.
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Such unsuccessful attempts nevertheless provide an estimate of the upper limit for the 1s5 → 2p (upper) cross sections. For instance, Ref. [30] gives the value of 0.2 × 10−16 cm2 as the upper limit of the peak 1s5 → 2p4 cross section, confirming the core propensity in the 1s → 2p excitation of Xe. Mityureva and Smirnov have reported experimental measurements for excitation from the 1s5 metastable level of Xe into the 2p levels [46] and higher levels [98] using a dual-energy electron-beam technique described in Section 3.4. Qualitatively their results for the 2p levels are very similar to those of Jung et al. [30]. For example, they find that cross sections into the 2p (lower) levels are an order of magnitude larger than those into the 2p (upper) group, and the dipoleforbidden excitation into the 2p5 level has a sharply peaked excitation function. The magnitudes of the cross sections measured by Mityureva and Smirnov, however, differ dramatically from those of Jung et al. and do not have the same close correlation with oscillator strengths.
9. Comparison to Theoretical Calculations As indicated in Section 1, the underlying theoretical framework for the electronimpact processes studied in this chapter is that cross sections are completely accounted for by the Coulomb and exchange interactions of the incident electron with the target atom. The challenge of the theoretical calculation of cross sections lies in finding approximate procedures of sufficient accuracy to solve the (N + 1)-electron problem (an N -electron target plus a projectile electron) and implementing the computational work. In view of the complexity of the electronic structure of the heavy rare-gases (in comparison to the alkalis or helium) it is not surprising that theoretical calculations of excitation cross sections for rare-gases had not appeared to be a popular and rewarding endeavor. Stimulated by the new experiments and the prospects of technological applications (Section 10.2), the situation has greatly improved. Papers on the calculation of cross sections from both the ground state and metastable levels using the latest theoretical methods are now abundant for all the heavy rare gases: Ne [99–102], Ar [103–108], Kr [108– 112], and Xe [108,113,114]. The experimental data provides a reference point to assess the approximation procedures in the calculations. Improvements of the computational methods in the last few years have led to a better understanding of the factors that influence the accuracy of the calculations. To appreciate the vast degree of computational complexity we should first note that prior to calculating excitation cross sections, one must have a set of accurate wave functions for the atomic target states. The electronic structure of excited states of rare-gas atoms have been of considerable interest since the early days of quantum mechanics. The energy spanning of the group of levels associated with a typical configuration np 5 nl depends intricately on the coupling of the
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orbital and spin angular momenta of the np 5 core and the outer nl electron (Section 2). A satisfactory ab initio calculation requires not only an extensive analysis of configuration interactions but also an accurate account of the coupling among the various angular momentum vectors. For instance, a recent multiconfiguration calculation with a moderate basis set failed to reproduce the proper ordering of the energy levels in the Ar(3p 5 4p) manifold [104]. Even with satisfactory results for energy levels, one may still question the accuracy of the cross sections derived from that set of target wave functions. A practical way to check the target wave functions of the heavy rare-gases is to calculate the radiative transition probabilities of the 1sx –2py array and compare them to experimental spectroscopic values. This, however, has generally not been done in theoretical works cited in the preceding paragraph although there are some noticeable exceptions including [99] and [105]. Several researchers reported that the calculated cross sections are sensitive to the choice of target wave functions. This often is a symptom of inadequate target wave functions. Computer codes for determining multiconfiguration Hartree–Fock wave functions for target states are available but the calculated cross sections based on these wave functions may not be adequate without a careful testing. The next task is to allow the projectile electron to interact with the target atom and solve the (N + 1)-electron problem. A method based on the principle of close coupling and formulated using R-matrix theory (RM) has been used extensively. It entails expanding the (N + 1)-electron in terms of the N -electron target-state functions, thus it is desirable to use a basis function set which covers not only the initial and final states of the excitation but also numerous intermediate states including the continuum. Therefore, the choice of basis set is a very important issue. In addition to RM, the method of distorted waves has proved effective for calculating excitation cross sections largely through the efforts of Madison et al. and Stauffer et al. Again options have been offered to modify the original version of Madison and Shelton [115] in order to include relativistic effects and to ensure unitarity of the S matrix. Perusal of the theoretical calculations cited in the first paragraph of this section reveals significant progress in the two areas in the past ten years. In the earlier papers, calculations of cross sections using certain methods were reported but issues like the adequacy of the target functions and basis sets were not well addressed. In some cases the cross section results show problematic features such as sensitivity to the target functions or large variations depending on whether orthogonality is imposed. Refinements of the computational scheme were then made to overcome these deficiencies leading to a higher level of sophistication. With the on-going evolvement it is not surprising to find varying degrees of agreement between theoretical calculations and experimental measurements. To illustrate the current status take the case of excitation of the 4p 5 5p levels of Kr from the two metastable levels, 1s3 (J = 0) and 1s5 (J = 2), for
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which experimental cross sections were reported in 2005 [31] and 2006 [34] and four sets of theoretical calculations (from 2002 [112] and 2005 [111]) are available. The calculations include results using (a) the Breit–Pauli R-matrix method with 15 basis functions (BP15) and with 51 basis functions including pseudostates (BP51) [112], (b) two versions of the method of distorted waves (DW-1 and DW-2) [112], and (c) a relativistic RM calculation with a much more extended basis set (RMEB) [111]. In calculations (a) and (b) the target wave functions are composed entirely of the 4p 6 , 4p 5 5s, and 4p 5 5p configurations for the DW-1 and BP15, but supplemented by 4p 5 4d and 4p 5 6s configurations along with pseudostate configurations 4p 5 6p and 4p 5 7p for BP51. Calculation (c) published three years after (a) and (b), employed a much more extensive configuration-interaction treatment and demonstrated good agreement with the energies of the lowest 23 levels, although no calculations of the 1sx –2py transition probabilities were included [111]. As to the issue of the basis set for R-matrix calculations, the cross section values of BP15 and BP51 sometimes differ by a factor of two, which raises the question of the adequacy of the basis function sets. A much larger basis set was used in the RMEB calculation which yields cross sections that are different from either BP15 or BP51, but are generally closer to BP15 than BP51. In connection with the concern of the size of the basis set, a recent detailed examination by Ballance and Griffin [102] led to the conclusion that in their 235-level Breit–Pauli RMPS calculation of Ne excitation cross sections, the pseudo-state expansion is not sufficiently complete to represent the target continuum. It is clear that important progress is being made concerning the issue of basis sets, but no definite conclusion has been obtained. How do the theoretical cross sections compare with experiment? On one hand, comparing experiment with theory for excitation of metastable krypton is not an ideal choice. Compared to excitation from the ground state, the experimental uncertainties for the absolute excitation cross sections from metastable levels are much larger, typically 35–40% [34]. Furthermore, in contrast to excitation from the ground state [26], no explicit cascade correction has been applied to the measured apparent cross sections. On the other hand, the opportunity to study the systematic dependence of the excitation cross sections on changes in the angular momentum values J and jc is compelling. This is most evident for excitation from the metastable levels since the ground state does not have a well defined core angular momentum. Furthermore, while the experimental uncertainties in the absolute cross sections results are large, much of this uncertainty is limited to one overall scaling parameter for the entire 1s5 → 2px manifold. Hence, the uncertainties in comparisons between levels, or in comparing the energy dependence of the cross section is much less (<10% for the larger cross sections). A detailed assessment would require a case-by-case comparison of theoretical and experimental cross sections at various energies. Instead, we will give only an overall picture with cross sections for only a couple of representative levels dis-
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F IG . 40. Sample comparison of experimental and theoretical calculations for excitation from the 1s5 metastable level of Kr. Theoretical calculations include the Breit–Pauli R-matrix calculations with 15 and 51 basis states [112], an R-matrix calculation with an extended basis [111], and two distorted wave approximation calculations from [112]. The error bars on the experimental results of Jung et al. [34] include only statistical errors which represent the uncertainty of the relative values (i.e. the uncertainty in the energy dependence). An additional ±35% uncertainty due to the calibration of the 1s5 → 2p9 cross section affects identically the absolute magnitude of all cross section values. Looking at one energy only, all of the theoretical calculations in (a) are consistent with the experimental values when one includes the full experimental uncertainty, but considerable differences are seen in the energy dependence.
played in Fig. 40. Among the R-matrix calculation, for core-preserving excitation from both 1s3 and 1s5 , the experimental data agrees the best with RMEB, but still exhibits reasonable accord with BP15 (see [34]). One distinct feature of the experimental data is the unexpected sharp peak near threshold seen in the dipoleallowed but core-changing excitation (i.e. 1s5 → 2p2 , 2p3 , 2p4 ). However, only a mild shoulder riding on a broad peak is seen in RMEB and BP15. Interestingly the BP51 calculation which generally doesn’t agree well in magnitude with the experimental values, does offer a sharp peak with good qualitative resemblance to experiment (see Fig. 40(b)). As to the method of distorted waves, DW-1 compares more favorably to experiment than does DW-2, although the later does an adequate job at higher energies. In fact, the DW-1 calculation reproduces the experimental data remarkably well for core-preserving excitation out of the 1s5 metastable level with large cross sections (i.e. from 1s5 to 2p6 , 2p8 , 2p9 , 2p10 ). This is encouraging in view of the relatively simple computational procedure involved in the DW-1 method. The bulk of research efforts in the past decade has made us aware of the capabilities and limitations with regard to the current status in the development of computational techniques for excitation cross sections of rare gases. Clearly there are major challenges ahead. With continual growth of computer technology one can look forward to progress that will enrich our understanding of what is surely one of the most computationally demanding electron–atom processes.
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10. Conclusions 10.1. S YSTEMATIC T RENDS IN R ARE -G AS C ROSS S ECTIONS Having examined each rare-gas in turn, it is worthwhile to now examine the global view of the heavy rare-gases as a whole. In Table XVI we summarize the direct cross section results at 75 eV incident electron energy for excitation from the ground state to the np 5 (n + 1)p levels for Ne (n = 2), Ar (n = 3), Kr (n = 4), and Xe (n = 5). Also included in Table XVI are apparent cross sections (measured at low pressures to reduce the cascades) for the Ar(3p 5 5p) and Ar(3p 5 6p) configurations. Since Paschen’s notation is not consistent across all of the rare gases (i.e. the two J = 0 levels are p1 and p3 in Ne, but p1 and p5 in all the other rare-gases), we use Racah notation to denote the levels. The sum of the direct cross sections summed over the entire np 5 (n + 1)p manifold are plotted in Fig. 41. A number of interesting patterns arise from this comparison. First, the sum of the excitation cross section increases steadily with n , this pattern is also evident at electron energies of 50, 100, and 150 eV. At the lower energies (50 and 75 eV), however, the cross section total for Xe is not much larger than the Kr sum, but this difference increases at the higher energies [26]. While the total cross sections have a clear n dependence, the patterns of the individual excitation cross sections are much more varied. For example, it has already been pointed out that the cross sections into the jc = 1/2 upper-tier levels of Kr and Xe are smaller on Table XVI Excitation cross sections from the ground state into np5 np configurations of Ne, Ar, Kr, and Xe at 75 eV. Values for np5 (n + 1)p configurations of Ne, Ar, Kr, and Xe are direct cross sections, values for Ar(3p5 5p) and Ar(3p5 6p) configurations are apparent cross sections at 0.5 mTorr (which include some cascade contribution). Levels are designated with Racah notation to maintain the same set of labels for all the rare-gases. Cross section (10−19 cm2 ) Ne(2p 5 3p)
Ar(3p5 4p)
Level
jc
J
Ref. [25]
Ref. [13]
p [1/2]0 p [1/2]1 p [3/2]1 p [3/2]2 p[1/2]0 p[1/2]1 p[3/2]1 p[3/2]2 p[5/2]2 p[5/2]3
1/2 1/2 1/2 1/2 3/2 3/2 3/2 3/2 3/2 3/2
0 1 1 2 0 1 1 2 2 3
16 3.2 2.4 9.9 2.4 2.8 3.5 11 6.7 3.9
30 6.9 7.2 12 13 7.3 7.2 16 23 4.3
Ar(3p5 5p) Ar(3p5 6p) Kr(4p5 5p) Xe(5p 5 6p) Ref. [16] Ref. [16] Ref. [26] Ref. [21] 5.3 1.5 1.7 2.9 6.3 1.3 3.2 3.9 6.9 2.5
1.0 0.59 0.37 1.8 1.9 0.62 0.93 0.87 2.3 0.29
20 7.2 7.0 23 44 9.4 29 28 47 16
22 2.4 1.5 19 94 5.3 32 38 50 3.1
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F IG . 41. Direct excitation cross section summed over the entire np5 (n + 1)p manifold at an incident electron energy of 75 eV, n = 2 Ne, n = 3 Ar, n = 4 Kr, n = 5 Xe [26]. Error bars include statistical and systematic uncertainties.
average than those of the jc = 3/2 lower-tier levels. As a result, the cross sections for J = 1 upper-tier levels of Kr and Xe decrease with increasing n , with the Xe values smaller than the corresponding cross sections in Ne– in sharp contrast to the general trend of increasing cross sections. Another interesting case is that of the np[5/2]3 (3 D3 ) levels. The direct excitation cross sections for these levels are expected to be very small at high energies since this is a singlet-to-triplet spinchanging excitation processes. While the cross section for the Kr(4p[5/2]3 ) level is one of the smallest of the lower-tier Kr cross sections, it appears to be unusually large in comparison to the trend for this level observed in the other rare gases. Tsurubuchi and Kobayashi [74] have used time-resolved techniques to measure direct cross sections for the 3 D3 level in Ne, Ar and Kr. For all three atoms, none had the expected E −3 energy dependence. For Kr, in particular, their measured direct excitation cross section had a much more gradual decline with electron energy above 50 eV. The increase in total cross section size with n is not surprising since the nporbital’s expectation value for r (i.e. the ‘size’ of an atom in its ground state) also increases with n , although this alone is insufficient to explain the factor of 4.4 increase in total np 5 (n + 1)p cross sections from Ne to Xe for which the corresponding increase in atomic radius is only about 1.4 [116]. The Ar(3p 6 → 3p 5 np, n = 4, 5, 6) data allows a similar comparison of cross section size with the ‘size’ of the final state atomic cloud (since the initial state is the same for the three cases). The total excitation cross sections summed over the entire 3p 5 np manifold
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at 75 eV are 127, 36 and 11 × 10−19 cm2 for the n = 4, 5, and 6 configurations, respectively. It is also enlightening to examine the difference in the average cross sections for the two possible jc -values for the various Ar(3p 5 np, n = 4, 5, 6) configurations. In Kr and Xe the jc = 1/2 upper-tier levels on average have smaller cross sections than the lower-tier levels, whereas this difference is non-existent in the levels of the Ne(2p 5 3p) and Ar(3p 5 4p) configurations. The spin–orbit splitting of Ar is small enough that energy levels of the Ar(3p 5 4p) configuration do not conform to the two tier pattern observed in Kr and Xe where the spin–orbit splitting is much larger (i.e. Fig. 2). However, if one looks at the energy levels of the higher Ar(3p 5 5p) and Ar(3p 5 6p) configurations, as illustrated in Fig. 42, there is a noticeable separation into two tiers based on the value of jc . The average jc = 1/2 cross sections for these two configurations given in Table XVI are also somewhat smaller than the corresponding jc = 3/2 average cross sections. Thus, in addition to moving down the periodic table, the dominance of the spin–orbit interaction over the Coulomb and exchange interactions can also be achieved within a given atomic system by looking at higher configurations for which the spin–orbit splitting remains fixed but the Coulomb and exchange terms are diminished. This particularly interesting point can be better illustrated by restricting ourselves to the two J = 0 levels of the np 5 np configurations [86]. Similar to the mixing of the two J = 1 levels of the Ne(2p 5 3s) configuration discussed in Section 2.1, the two J = 0 levels of a np 5 np configuration can be expressed as linear
F IG . 42. Energy levels of the np5 np configurations of neon, argon, krypton and xenon. The zero of the energy scale for each atom has been set equal to the ionization energy of the atom.
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combinations of the two LS-basis terms with J = 0, namely 1 3 |a = α S0 + β P0 , 3 1 |b = β S0 − α P0 .
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(13)
In the absence of any spin–orbit interaction, the energy levels of the “unperturbed system” can be expressed in terms of Slater–Condon parameters (i.e. F2 and G0 ) involving the np and np electrons. Of the six LS-basis terms, the 1 S term has the highest energy [3]. Introduction of a small or moderate spin–orbit coupling as a perturbation splits the 3 P term into an inverted triplet (J = 0, 1, 2) and allows the top member 3 P0 to interact with 1 S0 creating the scrambled wave functions given in Eq. (13). For the case of weak spin–orbit interactions, the 1 S0 term is well above the 3 P0 term, so that with only a small admixture in Eq. (13) the upper level is mostly singlet and the lower level is mostly triplet. At high incident electron energies, excitation from the singlet np 6 ground state is favored into the np 5 np J = 0 level with the higher singlet weighting (i.e. the upper level). If one were to increase the size of the spin–orbit interaction, there is a greater admixture of the two LS-basis terms, leading to a lessening of the disparity in singlet composition between the two levels and resulting differences in cross sections. Further increase of the spin–orbit splitting ζ eventually pushes the |3 P0 energy above the |1 S0 . When their mixing is taken into account as in Eq. (13), the lower J = 0 level now has a larger singlet component, and therefore larger cross section than the upper one. This general pattern has already been pointed out in the discussion of the individual atom’s cross sections. In Ne, where there is little spin–orbit splitting (ζ = 521 cm−1 ), the 2p1 (upper level) cross section is 6.7 times the size of the 2p3 (lower level) cross section (Section 6); whereas in Xe, where the spin– orbit splitting is much larger (ζ = 7025 cm−1 ), the 2p1 (upper level) cross section is only 0.23 times the size of the 2p5 (lower level) cross section (Section 8). For the 3p5 np excited configurations of Ar, the spin–orbit term of the 3p 5 core is fixed (ζ = 954 cm−1 ), but the ‘unperturbed’ energy spacing between the 1 S and 3 P LS-basis terms decrease with increasing n. The enhanced role of spin–orbit mixing is evident in the progression of cross section ratios for the np5 /np1 pairs, with the cross section into the 2p1 level being much larger than the cross section into the 2p5 level, the cross section into the 3p1 being nearly equal to the 3p5 , and eventually the 4p5 cross section being much larger than the 4p1 cross section. When combined with cross section results into the higher configurations of the other rare-gases in Fig. 43, the clear universality of cross section behavior for all the rare-gases is evident [86]. A corresponding analysis of the global patterns observed in the excitation cross sections out of the metastable levels of the heavy rare-gases is complicated by the fact that many of the cross sections have not been measured out of both metastable
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F IG . 43. Ratio of np5 np cross sections for the np [1/2]0 and np[1/2]0 levels for Ne, Ar, Kr and Xe [86]. Note that ζ is the spin–orbit splitting of the ion-core levels, whereas 6G0 − 15F2 is the energy difference of the unperturbed 1 S0 and 3 P0 LS-basis levels. Table XVII Excitation cross sections from the 1s5 (J = 2) metastable level into np5 (n + 1)p configurations of Ne, Ar, Kr, and Xe at 8 eV. In Racah notation, the 1s5 level is designated ns[3/2]o2 . Entries in parentheses are cross sections where the 1s5 excitation cross sections have been apportioned from the observed sum of the 1s3 and 1s5 excitation cross sections based upon oscillator strengths. Cross section (10−16 cm2 ) Ne(2p 5 3p) Level
jc
J
Ref. [29]
Ar(3p5 4p) Ref. [33]
Kr(4p 5 5p) Ref. [34]
Xe(5p5 6p) Ref. [30]
p[1/2]0 p[1/2]1 p[3/2]1 p[3/2]2 p[5/2]2 p[5/2]3
3/2 3/2 3/2 3/2 3/2 3/2
0 1 1 2 2 3
(0.2) (5.4) (0.9) 10 4.8 19
0.12 (6.9) (1.6) 19 4.9 24
0.04 17 2.2 24 8.6 51
0.28 14 1.4 14 6.4 33
levels. Hence, we will restrict our analysis to excitation cross sections out of the 1s5 metastable level into the 2p (lower) levels. Even for this limited set, however, some of the Ne and Ar cross sections in Table XVII have been extrapolated due to difficulties in separating the 1s3 and 1s5 components of the observed excitation signal [29]. In comparison to the clear variation with n observed in ground state excitation, there is no clear pattern in the sum of the 1s5 → 2p (lower) cross sections. For example, the ‘total’ 1s5 → 2p (lower) cross section for metastable
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Ar is only marginally (∼35%) larger than that of the ‘total’ 1s5 → 2p (lower) excitation cross section for metastable Ne, in contrast to the factor of two increase observed between the two gases in the size of their ground state excitation cross sections. Both of these observations, however, are consistent with the effective ‘size’ of an atom in an excited level being inversely related to the ionization energy of that level. The ionization energies from the ground states of the heavy rare gases are 21.6, 15.8, 14.0, and 12.1 eV for Ne to Xe, respectively. In contrast, the ionization energies of the 1s5 metastable level of the rare gases show much less variation, with values of 4.9, 4.2, 4.1, and 3.8 eV and have a concomitant reduction in cross section variation.
10.2. A PPLICATIONS The optical emissions from electron-impact excitation of atoms and molecules have historically been used in two distinct ways, as a light source and as a remote probe of the environment giving rise to the emissions whether it is a distant astrophysical object or a laboratory plasma. Knowledge of electron–atom collision cross sections have permitted many of these devices such as neon lights [117], gas-discharge lasers, and laboratory plasmas [118–121] to be modeled in great detail. Electron collisions with rare-gas atoms continue to play an important role in interesting new applications in the fields of lighting and remote diagnostics [79, 122]. In the former category, for example, xenon rare gas mixtures are being considered as a mercury-free fluorescent lamp replacement [123,124]. While the UV resonance transitions are the primary optical output of these devices, other excitation channels, including excitation of the metastable levels, play an important role in the energy efficiency of the devices [124–126]. Optical emissions from rare-gas discharges are also finding new uses in plasma display panels [127]. It is now recognized that in these low-temperature plasmas excitation of metastable levels, due to their low excitation thresholds, play an important role in spite of their low number densities [125,128]. Low-temperature plasmas are also ubiquitous in semiconductor chip manufacturing and other plasma processing applications. Here, the optical emissions are not the primary output of the plasma, but a byproduct of the plasma chemistry. Nonetheless, plasma emissions provide a non-invasive diagnostic of many important plasma properties such as the electron temperature, electron energy distribution function (EEDF), and reactive species concentrations that are crucial for optimizing the desired plasma chemistry [79,122]. While Ar containing plasmas are used in some plasma etching and sputtering applications, most industrial etching and deposition plasmas use reactive molecules such as O2 , Cl2 and various fluorocarbons. By adding small amounts of rare-gas atoms to these plasmas, however, the considerable knowledge of electron–atom collision cross sections with
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rare-gas atoms can be exploited to learn much about these plasma discharges [129, 130]. Donnelly and co-workers have refined this method of adding trace amounts of rare-gases as an optical plasma diagnostic to a high art in the form of a technique called trace rare gases optical emission spectroscopy (TRG-OES) [130, 131]. A small admixture of rare-gases (1% each of Ne, Ar, Kr, Xe) is added to the plasma of interest, and emissions from the 1s–2p emission lines are used in conjunction with electron–atom collision cross sections to determine the electron temperature and other relevant plasma properties. Since the energy thresholds for excitation from the ground state to the 2p levels (Paschen’s notation) range from 18.5 eV for Ne to 9.5 eV for Xe, the different gases each probe a different region of the electron energy distribution. By including in the analysis 2p emission lines originating in levels populated mainly through excitation of metastable levels (i.e. Ar(2p9 ), Kr(2p9 ), . . .), even lower energy ranges (∼1.5 eV) in the EEDF can be probed. Knowledge of electron–atom cross sections, from the ground and metastable levels, are used both in a fitting procedure used to extract the electrontemperature and as a guide in selecting which levels are populated primarily from the ground state and which are populated primarily via step-wise excitation from metastable levels [132,133].
10.3. C ONCLUDING R EMARKS Because of their structural simplicity, the helium and alkali atoms are two cases for which extensive measurements have been made of electron-impact excitation of an active electron from its lowest level (l = 0) into an excited nl shell. The cross sections were characterized according to the quantum numbers n and l. In the case of He each nl shell is divided into a singlet and a triplet member with very different cross sections. A very well known and all-important finding is the preference of excitation corresponding to optical dipole-allowed transitions. The complexity of the Ne, . . . , Xe rare-gas series arises from the unique (np)5 ion core in conjunction with each excited nl shell which gives rise to 4, 10, and 12 levels, respectively, for the (np)5 nl configuration with l = 0, 1, and 2. The variation of cross sections from one level to another within a shell represents a new horizon over the simple helium/alkali case. For excitation out of the ground level, much of the variation in cross sections for the levels within a shell can be traced to the total angular momentum value of the final state. This general J -dependence is augmented by a dependence on the core angular momentum upon moving towards the heavier rare-gas atoms as well as moving to higher n for the (np)5 nl series of Ne and Ar. As one turns to the study of excitation out of the metastable levels, the situation becomes even more complex. In the case of excitation out of the 1s5 metastable level of Kr into the 4p 5 5p levels that correspond to optical dipole
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transitions, for instance, the disparity in cross sections between different levels is as large as a factor of 600. Understanding the observed excitation behaviors in terms of fundamental physical principles is a major challenge. The ensuing outcome, which includes the multipole field model, the J rules, the jc propensity rule, and a revised and more general view of the dominance of dipole-type excitation, has proved to be rewarding not only from the intellectual standpoint but also in the broader perspective of technological applications. As an example to the latter point, consider the 2p9 (J = 3) and 2p1 (J = 0) levels of Ar atoms in a plasma with an electron energy distribution corresponding to a temperature of a few eV and a metastable atom concentration of ∼10−4 of the ground state density. The 2p levels can be populated by (a) excitation of a ground-state Ar atom by an electron of ∼14 eV energy (fast electrons) and (b) excitation of a metastable Ar atom by an electron of ∼1.5 eV energy (slow electrons). The 2p9 level has a relatively large cross section for excitation from the 1s5 (J = 2) metastable level since the 1s5 → 2p9 excitation process satisfies the dipole selection rule; whereas the 1s5 → 2p1 excitation cross section is relatively small since this is a dipole-forbidden excitation process. For excitation from the ground state, the peak (low energy) cross sections into the 2p1 and 2p9 levels are about equal (at higher energies the cross section into the 2p1 level is larger since it has a even J value whereas the 2p9 level has odd J , but at low energies exchange greatly enhances the 2p9 cross section). Under the specified plasma conditions, the 2p1 level is populated primarily by mechanism (a), whereas the 2p9 population is significantly enhanced by the metastable step-wise mechanism (b). Moshkalyov et al. [134], for example, have taken advantage of this sharp distinction in production mechanisms in an experiment where the intensity ratio of the 2p9 → 1s5 versus the 2p1 → 1s2 emission lines was utilized as a quantitative index in monitoring the spatial and temporal variations of metastable atoms and slow electrons in a plasma. In a typical electron excitation experiment, optical measurements are made to yield apparent excitation cross sections from which the direct excitation cross sections are obtained after the often-difficult task of assessing the cascade contributions. The latter is necessary since direct cross sections are the fundamental quantity pertaining to an inelastic process whereas the apparent cross sections represents a convolution of numerous excitation processes. However, in many applications where, for instance, argon emissions are used to probe the environment, the total intensity is of major concern; hence the apparent excitation cross sections are the parameters of prime importance. It has proved useful for experimenters to publish both apparent and direct excitation cross sections. In the case of excitation of rare gases out of the ground level, there is the complication that the apparent cross sections are pressure dependent due to the radiation trapping of cascading levels. To address the needs of prospective users it is suggested that results of measurements of pressure dependence of apparent excitation cross sections be
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included in the publications. Theoretical calculations of the apparent excitation cross sections are impractical because one must compute the direct excitation cross section for the level of interest as well as the cross sections for numerous high-lying levels along with the associated branching fraction for cascade into the level of interest, not to mention the additional complexity of taking into account the effects of radiation trapping. Availability of apparent excitation cross sections at various pressures could be of considerable value [131]. The very rich energy level spectra of the target atoms make the study of electron excitation of the rare gases an especially challenging task. It is precisely the complexity and widely varied nature of the results that provide new grounds of technological applications. As in many areas of physics, a sound fundamental understanding of properties of atoms eventually lead to significant applications which in turn stimulate further exploration.
11. Acknowledgements The authors wish to thank B. Chiaro for obtaining a few additional measurements used in some of the plots. This work was supported by the National Science Foundation and the U.S. Air Force Office of Scientific Research.
12. References [1] R.D. Cowan, K.L. Andrew, J. Opt. Soc. Amer. 55 (1965) 502. [2] R.D. Cowan, “The Theory of Atomic Structure and Spectra”, University of California Press, Berkeley, 1981. [3] E.U. Condon, G.H. Shortley, “The Theory of Atomic Spectra”, Cambridge University Press, London, 1963. [4] J.E. Chilton, C.C. Lin, Phys. Rev. A 58 (1998) 4572. [5] A.R. Filippelli, C.C. Lin, L.W. Anderson, J.W. McConkey, Adv. At. Mol. Opt. Phys. 33 (1994) 1. [6] B.L. Moiseiwitsch, S.J. Smith, Rev. Mod. Phys. 40 (1968) 238. [7] D.W.O. Heddle, J.W. Gallagher, Rev. Mod. Phys. 61 (1989) 221. [8] S. Tsurubuchi, H. Kobayashi, M. Hyodo, J. Phys. B 36 (2003) 2629. [9] A.H. Gabriel, D.W.O. Heddle, Proc. Roy. Soc. London Ser. A 258 (1960) 124. [10] NIST Atomic Spectra Database, http://physics.nist.gov/asd. [11] I. Tanarro, J. Campos, Can. J. Phys. 63 (1985) 1389. [12] D.W.O. Heddle, M.J. Samuel, J. Phys. B 3 (1970) 1593. [13] J.E. Chilton, J.B. Boffard, R.S. Schappe, C.C. Lin, Phys. Rev. A 57 (1998) 267. [14] A.R. Filippelli, S. Chung, C.C. Lin, Phys. Rev. A 29 (1984) 1709. [15] B. Van Zyl, G.H. Dunn, G. Chamberlain, D.W.O. Heddle, Phys. Rev. A 22 (1980) 1916. [16] T. Weber, J.B. Boffard, C.C. Lin, Phys. Rev. A 68 (2003) 032719. [17] S. Tsurubuchi, T. Miyazaki, K. Motohashi, J. Phys. B 29 (1996) 1785. [18] J.E. Furst, W.M.K.P. Wijayaratna, D.H. Madison, T.J. Gay, Phys. Rev. A 47 (1993) 3775.
12] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58]
ELECTRON-IMPACT EXCITATION OF RARE-GAS ATOMS
419
T.J. Gay, J.E. Furst, K.W. Trantham, W.M.K.P. Wijayaratna, Phys. Rev. A 53 (1996) 1623. D.H. Yu, P.A. Hayes, J.F. Williams, V. Zeman, K. Bartschat, J. Phys. B 33 (2000) 1881. J.T. Fons, C.C. Lin, Phys. Rev. A 58 (1998) 4603. P.J.M. van der Burgt, W.B. Westerveld, J.S. Risley, J. Phys. Chem. Ref. Data 18 (1989) 1757. J.W. McConkey, F.G. Donaldson, Can. J. Phys. 51 (1973) 914. C.A. DeJoseph Jr., J.D. Clark, J. Phys. B 23 (1990) 1879. J.E. Chilton, M.D. Stewart Jr., C.C. Lin, Phys. Rev. A 61 (2000) 052708. J.E. Chilton, M.D. Stewart Jr., C.C. Lin, Phys. Rev. A 62 (2000) 032714. J.B. Boffard, G.A. Piech, M.F. Gehrke, L.W. Anderson, C.C. Lin, Phys. Rev. A 59 (1999) 2749. R.B. Lockwood, L.W. Anderson, C.C. Lin, Z. Phys. D 24 (1992) 155. J.B. Boffard, M.L. Keeler, G.A. Piech, L.W. Anderson, C.C. Lin, Phys. Rev. A 64 (2001) 032708. R.O. Jung, J.B. Boffard, L.W. Anderson, C.C. Lin, Phys. Rev. A 72 (2005) 022723. R.O. Jung, T.E. Stone, J.B. Boffard, L.W. Anderson, C.C. Lin, Phys. Rev. Lett. 94 (2005) 163202. G.A. Piech, J.E. Chilton, L.W. Anderson, C.C. Lin, J. Phys. B 31 (1998) 859. G.A. Piech, J.B. Boffard, M.F. Gehrke, L.W. Anderson, C.C. Lin, Phys. Rev. Lett. 81 (1998) 309. R.O. Jung, T.E. Stone, J.B. Boffard, L.W. Anderson, C.C. Lin, Phys. Rev. A 73 (2006) 022722. F.W. Meyer, L.W. Anderson, Phys. Rev. A 9 (1974) 1909. G.E. Ice, R.E. Olson, Phys. Rev. A 11 (1975) 111. J.R. Peterson, D.C. Lorents, Phys. Rev. 182 (1969) 152. M.J. Coggiola, T.D. Gaily, K.T. Gillen, J.R. Peterson, J. Chem. Phys. 70 (1979) 2576. F.W. Meyer, C.J. Anderson, L.W. Anderson, Phys. Rev. A 15 (1977) 455. J.B. Boffard, M.E. Lagus, L.W. Anderson, C.C. Lin, Rev. Sci. Instrum. 67 (1996) 2738. A.J. Dixon, M.F.A. Harrison, A.C.H. Smith, J. Phys. B 9 (1976) 2617. M. Johnston, K. Fujii, J. Nickel, S. Trajmar, J. Phys. B 29 (1996) 531. M.E. Lagus, J.B. Boffard, L.W. Anderson, C.C. Lin, Phys. Rev. A 53 (1996) 1505. A.A. Mityureva, V.V. Smirnov, Opt. Spektrosk. 97 (2004) 544; Opt. Spectrosc. (USSR) 97 (2004) 508. A.A. Mityureva, N.P. Penkin, V.V. Smirnov, Opt. Spektrosk. 67 (1989) 785; Opt. Spectrosc. (USSR) 67 (1989) 461. A.A. Mityureva, V.V. Smirnov, Opt. Spektrosk. 74 (1993) 12; Opt. Spectrosc. 74 (1993) 6. A.A. Mityureva, N.P. Penkin, V.V. Smirnov, Opt. Spektrosk. 66 (1989) 790; Opt. Spectrosc. (USSR) 66 (1989) 463. N.B. Kolokolov, O.V. Terekhova, Opt. Spektrosk. 86 (1999) 547; Opt. Spectrosc. 86 (1999) 481. C.A. DeJoseph Jr., V.I. Demidov, J. Phys. B 38 (2005) 3805. M.H. Phillips, L.W. Anderson, C.C. Lin, Phys. Rev. A 32 (1985) 2117. R.S. Schappe, M.B. Schulman, L.W. Anderson, C.C. Lin, Phys. Rev. A 50 (1994) 444. R.J. Anderson, R.H. Hughes, J.H. Tung, S.T. Chen, Phys. Rev. A 8 (1973) 810. J.B. Boffard, G.A. Piech, M.F. Gehrke, M.E. Lagus, L.W. Anderson, C.C. Lin, J. Phys. B 29 (1996) L795. I.P. Bogdanova, S.V. Yurgenson, Opt. Spektrosk. 62 (1987) 471; Opt. Spectrosc. (USSR) 62 (1987) 281. I.P. Bogdanova, S.V. Yurgenson, Opt. Spektrosk. 63 (1987) 1373; Opt. Spectrosc. (USSR) 63 (1987) 815. I.P. Bogdanova, S.V. Yurgenson, Opt. Spektrosk. 62 (1987) 713; Opt. Spectrosc. (USSR) 62 (1987) 424. J.B. Boffard, M.F. Gehrke, M.E. Lagus, L.W. Anderson, C.C. Lin, Eur. Phys. J. D 8 (2000) 193. R.M.St. John, F.L. Miller, C.C. Lin, Phys. Rev. 134 (1964) A888.
420 [59] [60] [61] [62] [63] [64] [65] [66] [67] [68] [69] [70] [71] [72] [73] [74]
[75] [76] [77] [78] [79] [80] [81] [82] [83] [84] [85] [86] [87] [88] [89] [90] [91] [92]
[93] [94] [95] [96] [97]
J.B. Boffard et al.
[12
N. Bohr, Phil. Mag. 25 (1913) 10. N. Bohr, Phil. Mag. 30 (1915) 581. E.M. Purcell, Astrophys. J. 116 (1952) 457. J.B. Boffard, M.E. Lagus, L.W. Anderson, C.C. Lin, Phys. Rev. A 59 (1999) 4079. O. Fischer, Z. Phys. 86 (1933) 646. J.K. Ballou, C.C. Lin, F.E. Fajen, Phys. Rev. A 8 (1973) 1797. I.P. Zapesochnyi, P.V. Feltsan, Opt. Spektrosk. 20 (1966) 521; Opt. Spectrosc. (USSR) 20 (1966) 291. R.A. Lilly, J. Opt. Soc. Am. 66 (1976) 245. J. Berkowitz, “Photoabsorption, Photoionization, and Photoelectron Spectroscopy”, Academic Press, New York, 1979. J.W. Keto, C.-Y. Kuo, J. Chem. Phys. 74 (1981) 6188. G.P. Li, T. Takayanagi, K. Wakiya, H. Suzuki, T. Ajiro, S. Yagi, S.S. Kano, H. Takuma, Phys. Rev. A 38 (1988) 1240. J.M. Ajello, G.K. James, B. Franklin, S. Howell, J. Phys. B 23 (1990) 4355. J.E. Chilton, C.C. Lin, Phys. Rev. A 60 (1999) 3712. M.D. Stewart Jr., J.E. Chilton, J.B. Boffard, C.C. Lin, Phys. Rev. A 65 (2002) 032704. K. Katsonis, H.W. Drawin, J. Quant. Spectrosc. Radiat. Transfer 23 (1980) 1. S. Tsurubuchi, H. Kobayashi, in: D.H. Madison, M. Schulz (Eds.), “Correlations, Polarization, and Ionization in Atomic Systems”, AIP Conf. Proc., vol. 604, American Institute of Physics, Melville, NY, 2002, p. 166. P.V. Feltsan, I.P. Zapesochny, Ukr. Fiz. Zh. 12 (1967) 633. I.P. Bogdanova, S.V. Yurgenson, Opt. Spektrosk. 68 (1990) 1246; Opt. Spectrosc. (USSR) 68 (1990) 730. F. Blanco, J.A. Sanchez, J. Campos, J. Phys. B 25 (1992) 3531. W.L. Wiese, J.W. Brault, K. Danzmann, V. Helbig, M. Kock, Phys. Rev. A 39 (1989) 2461. J.B. Boffard, C.C. Lin, C.A. DeJoseph Jr., J. Phys. D 37 (2004) R143. H.A. Hyman, Phys. Rev. A 18 (1978) 441. W. Hanle, Z. Physik 65 (1930) 512. F.A. Sharpton, R.M.St. John, C.C. Lin, F.E. Fajen, Phys. Rev. A 2 (1970) 1305. S. Tsurubuchi, K. Arakawa, S. Kinokuni, K. Motohashi, J. Phys. B 33 (2000) 3713. R.E. Miers, J.E. Gastineau, M.H. Phillips, L.W. Anderson, C.C. Lin, Phys. Rev. A 25 (1982) 1185. I. Kanik, J.M. Ajello, G.K. James, J. Phys. B 29 (1996) 2355. T. Weber, J.B. Boffard, C.C. Lin, Int. J. Mass Spectrom. 233 (2004) 75. M.J. Seaton, J. Phys. B. 31 (1998) 5315. M. Aymar, M. Coulombe, At. Data Nucl. Data Tables 21 (1978) 537. A.A. Mityureva, V.V. Smirnov, J. Phys. B 27 (1994) 1869. P.V. Fel’tsan, Ukr. Fiz. Zh. 12 (1967) 1425. I.P. Bogdanova, O.V. Veide, V.I. Yakovleva, Opt. Spektrosk. 81 (1996) 366; Opt. Spectrosc. (USSR) 81 (1996) 330. W.L. Wiese, M.W. Smith, B.M. Glennon (Eds.), “Atomic Transition Probabilities”, Natl. Bur. Stand. (U.S.) Natl. Stand. Ref. Data. Series Circ. No. NSRDS-NBS 4, vol. I, U.S. GPO, Washington, DC, 1966. K. Dzierzega, U. Volz, G. Nave, U. Griesmann, Phys. Rev. A 62 (2000) 022505. K.G. Walker, Bull. Amer. Phys. Soc. 29 (1984) 158. I.P. Bogdanova, S.V. Yurgenson, Opt. Spektrosk. 70 (1991) 486; Opt. Spectrosc. (USSR) 70 (1991) 285. I. Kanik, P.V. Johnson, G.K. James, J. Phys. B 34 (2001) 1685. N. Sadeghi, J. Sabbagh, Phys. Rev. A 16 (1977) 2336.
12]
ELECTRON-IMPACT EXCITATION OF RARE-GAS ATOMS
421
[98] A.A. Mityureva, V.V. Smirnov, Opt. Spektrosk. 75 (1993) 705; Opt. Spektrosc. (USSR) 75 (1993) 417. [99] O. Zatsarinny, K. Bartschat, J. Phys. B 37 (2004) 2173. [100] V. Zeman, K. Bartschat, J. Phys. B 30 (1997) 4609. [101] G.D. Meneses, R.E.H. Clark, J. Abdallah Jr., G. Csanak, J. Phys. B 35 (2002) 3119. [102] C.P. Ballance, D.C. Griffin, J. Phys. B 37 (2004) 2943. [103] A. Dasgupta, M. Blaha, J.L. Giuliani, Phys. Rev. A 61 (1999) 012703; note addenda: Phys. Rev. A 65 (2002) 039905. [104] K. Bartschat, V. Zeman, Phys. Rev. A 59 (1999) R2552. [105] C.M. Maloney, J.L. Peacher, K. Bartschat, D.H. Madison, Phys. Rev. A 61 (2000) 022701. [106] D.H. Madison, C.M. Maloney, J.B. Wang, J. Phys. B 31 (1998) 873. [107] D.H. Madison, A. Dasgupta, K. Bartschat, D. Vaid, J. Phys. B 37 (2004) 1073. [108] S. Kaur, R. Srivastava, R.P. McEachran, A.D. Stauffer, J. Phys. B 31 (1998) 4833. [109] A. Dasgupta, K. Bartschat, D. Vaid, A.N. Grum-Grzhimailo, D.H. Madison, M. Blaha, J.L. Giuliani, Phys. Rev. A 64 (2001) 052710. [110] K. Bartschat, A.N. Grum-Grzhimailo, J. Phys. B 33 (2000) 4603. [111] J. Zeng, J. Wu, F. Jin, G. Zhao, J. Yuan, Phys. Rev. A 72 (2005) 042707. [112] A. Dasgupta, K. Bartschat, D. Vaid, A.N. Grum-Grzhimailo, D.H. Madison, M. Blaha, J.L. Giuliani, Phys. Rev. A 65 (2002) 042724. [113] K. Bartschat, A. Dasgupta, D.H. Madison, Phys. Rev. A 69 (2004) 062706. [114] S. Nakazaki, K.A. Berrington, W.B. Eissner, Y. Itikawa, J. Phys. B 30 (1997) 5805. [115] D.H. Madison, W.N. Shelton, Phys. Rev. A 7 (1973) 499. [116] H. Haken, H.C. Wolf, “Atomic and Quantum Physics”, Springer-Verlag, New York, 1987. [117] W.L. Nighan, IEEE Trans. Electron. Devices 28 (1981) 625. [118] A. Bultel, B. von Ootegem, A. Bourdon, P. Vervisch, Phys. Rev. E 65 (2002) 046406. [119] B. Pokrzywka, Phys. Scripta 66 (2002) 437. [120] R.S. Stewart, D.J. Smith, J. Phys. D 35 (2002) 1777. [121] D.J. Smith, C.J. Whitehead, R. Stewart, Plasma Sources Sci. Technol. 11 (2002) 115. [122] K. Behringer, U. Fantz, Contrib. Plasma Phys. 39 (1999) 411. [123] T.J. Sommerer, D.A. Doughty, J. Phys. D 31 (1998) 2803. [124] S. Gortchakov, H. Lange, D. Uhrlandt, J. Appl. Phys. 93 (2003) 9508. [125] T.J. Sommerer, J. Phys. D 29 (1996) 769. [126] T. Shiga, L.C. Pitchford, J.P. Boeuf, S. Mikoshiba, J. Phys. D 36 (2003) 512. [127] M.J. Kushner, J. Appl. Phys. 95 (2004) 846. [128] L.C. Pitchford, J. Kang, C. Punset, J.P. Boeuf, J. Appl. Phys. 92 (2002) 6990. [129] T. Czerwiec, D.B. Graves, J. Phys. D 37 (2004) 2827. [130] V.M. Donnelly, J. Phys. D 37 (2004) R217. [131] M.V. Malyshev, V.M. Donnelly, J. Vac. Sci. Technol. A 15 (1997) 550. [132] N.C.M. Fuller, M.V. Malyshev, V.M. Donnelly, I.P. Herman, Plasma Sources Sci. Technol. 9 (2000) 116. [133] V.M. Donnelly, M.J. Schabel, J. Appl. Phys. 91 (2002) 6288. [134] S.A. Moshkalyov, P.G. Steen, S. Gomez, W.G. Graham, Appl. Phys. Lett. 75 (1999) 328.
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ADVANCES IN ATOMIC, MOLECULAR AND OPTICAL PHYSICS, VOL. 54
INTERNAL ROTATION IN SYMMETRIC TOPS I. OZIER1 and N. MOAZZEN-AHMADI2 1 Department of Physics and Astronomy, University of British Columbia, 6224 Agricultural Road,
Vancouver, British Columbia V6T 1Z1, Canada 2 Department of Physics and Astronomy, University of Calgary, 2500 University Drive NW, Calgary, Alberta T2N 1N4, Canada 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. The Zeroth-Order Hamiltonian and the Hybrid Approach 2.2. The Vibration–Torsion–Rotation Hamiltonian . . . . . . 2.3. Basis Functions, Eigenfunctions, and Eigenvalues . . . . 3. Spectroscopy from 50 kHz to 1000 cm−1 . . . . . . . . . . . 3.1. Rotational Spectroscopy . . . . . . . . . . . . . . . . . . 3.2. The Avoided Crossing Molecular Beam Method . . . . . 3.3. Torsion–Rotation Spectroscopy . . . . . . . . . . . . . . 3.4. Vibration–Torsion–Rotation Spectroscopy . . . . . . . . 4. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . 6. References . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Abstract Internal rotation in C2 H6 and CH3 SiH3 and similar symmetric tops has been investigated from 50 kHz to 1000 cm−1 using high resolution rotational, torsional, vibrational, and molecular beam spectroscopy. Electric dipole transitions that are forbidden in first order have been measured at high sensitivity with a Fourier transform microwave cavity spectrometer and infrared interferometers. The molecular beam avoided crossing technique has been developed to detect transitions that break the standard rotation–torsion–nuclear spin selection rules. The torsional mode has been studied from the harmonic limit to the case of nearly free internal rotation. Intensity and Stark shift measurements have been carried out in an investigation of the torsional dependence of the electric dipole operator. Fragmentation of the lowest lying parallel and perpendicular vibrational bands has been observed due to resonant and non-resonant interactions with dark states that differ from the bright states by as many as five torsional quanta. Giant torsional splittings are observed reflecting the fact that the frequency of tunneling through the hindering potential has been increased by more than a factor of ten. The results have serious implications for methyl enhancement of intramolecular vibrational relaxation. An effective 423
© 2007 Elsevier Inc. All rights reserved ISSN 1049-250X DOI: 10.1016/S1049-250X(06)54007-8
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Hamiltonian has been developed showing that the fragmentation is caused by Coriolis and anharmonic interactions mediated by the large amplitude motion. For each molecule studied, a multi-band fit to a large global data set has been made to within experimental error. The isotopic dependence of the torsional parameters for C2 H6 , CH3 CD3 , and C2 D6 is discussed. This work is reviewed and possible future directions suggested.
1. Introduction The nature of internal rotation about single bonds is still a matter of some controversy even after seven decades of theoretical and experimental study. The interest in resolving basic physical questions relating to internal rotation stems in large part from the fact that torsional motion by methyl groups can dramatically accelerate intramolecular vibrational relaxation in those molecules in which they are present. Hindered internal rotation occurs in a large number of molecular systems ranging in size from four atoms to hundreds or thousands, including large molecules that are active in biological environments. In many cases, the internal rotation of the CH3 group determines crucial characteristics of the structure, and consequently greatly influences the dynamic properties and functions of these important molecules. Simple symmetric tops such ethane and methyl silane offer two significant advantages in the study of internal rotation over other types of molecules. First, symmetric rotors possess a high degree of symmetry about the torsional axis, a property which dramatically simplifies analysis both of high resolution spectra and of computational results. Second, the simplicity of electronic structure of these molecular systems make them ideal candidates for ab initio studies. Historically, studies of ethane-like molecules have succeeded, where those of lesssymmetrical molecules have had great difficulty. Internal rotation in simple symmetric tops offers an excellent opportunity to investigate a single internal degree of freedom over a wide range in the amplitude of motion. The variation in the torsional angle about the symmetry axis of one coaxial rotor relative to the other is hindered by a periodic potential V , whose height in lowest order is indicated by V3 [1]. In the zero point state, for a large but finite V3 , the internal rotation is highly hindered and acts as a harmonic torsional oscillator. As the number of quanta in this angular degree of freedom is increased, the relative motion approaches free internal rotation. In the initial theoretical development of the problem, the coupling between the internal and overall rotation is fully taken into account even for highly excited internal rotor states, but the coupling to the small amplitude harmonic vibrations in the molecular skeleton
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is assumed to be accounted for implicitly in an effective Hamiltonian. However, when resonant interactions are present, the coupling to the normal vibrations must be treated in zeroth order, a problem that is particularly challenging in cases where the torsional state involved is undergoing large amplitude motion. For X2 Y6 and XY3 AB3 symmetric tops, the analysis of this problem is greatly simplified by the high degree of symmetry and the associated selection rules which severely limit the terms that can appear in the Hamiltonian. However, the corresponding electric dipole selection rules similarly limit the information available from microwave spectroscopy, a standard spectroscopic method that works well for molecules of lower symmetry in the determination of the height and the shape of the barrier [1]. Experimental methods that break the first order dipole selection rules are therefore needed to determine the necessary energy differences to high precision, and such methods have been developed. The lessons learned in the analysis of symmetric rotors should be very helpful in investigating the more complicated, but much more common case of asymmetric tops, such as methyl alcohol [2]. In the four decades since the review of Lin and Swalen [1], there has been considerable progress on both the theoretical and experimental aspects of the problem of internal rotation in symmetric tops, and it is this progress which will be the primary subject of the present review. Although the focus of this work has been on the vibration–torsion–rotation interactions, the results have implications for the electronic structure. An understanding of the origin of the torsional potential V and the preference in X2 Y6 and XY3 AB3 molecules for the staggered configuration as the equilibrium structure can provide valuable insight into electronic structure of the molecules and how the electronic degrees of freedom are affected by the torsional motion. The staggered conformation was established as the equilibrium structure for ethane in 1949 [3], but agreement on the explanation for the height V3 of the hindering potential was difficult to achieve [4]. For some time, the structural preference for the staggered conformation in ethane was attributed to steric effects; i.e. to a combination of Coulomb and Pauli exchange repulsion [5]. Very recently [6], it has been argued that the dominant effect is hyperconjugation, in which the mixing between a bonding and an antibonding orbital stabilizes the staggered configuration. The stretching of the C–C bond in ethane during internal rotation plays a critical role in this explanation. When a similar analysis is carried out for disilane [7], the lengthening of the central bond as compared to ethane greatly reduces the hyperconjugation contribution so that it is no longer dominant. However, the stretching of the central bond (Si–Si in this case) again plays a central role in the barrier energetics. Presumably, the situation in CH3 SiH3 falls between that in C2 H6 and that in Si2 H6 . In any case, it is clear that the physical process conventionally called “internal rotation” involves mode mixing in an essential manner, in particular the mixing of the torsional motion with the stretching of the central bond. Thus the vibration–torsion–rotation spectroscopy discussed here reflects directly
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on the mechanisms responsible for the barrier height V3 and the preference for either the staggered or eclipsed conformation. Intramolecular vibrational redistribution (IVR) is critical to the understanding of the rates of chemical reactions [8]. It has been shown that this redistribution can be greatly accelerated by methyl substitution at a site where the CH3 group undergoes internal rotation [9,10]. This pioneering work was done on asymmetric rotors with nearly free internal rotation using electronic spectroscopy at vibrational resolution. There has been one high resolution IVR study of CH3 SiH3 [11]. The first overtone of the C–H stretch near 6000 cm−1 was shown to be considerably more fragmented than the nearly isoenergetic second overtone of the Si–H stretch. The individual transitions were resolved, but the torsional quantum numbers could not be assigned, and a statistical analysis was carried out. In the work discussed in the present review, mechanisms for the IVR enhancement have been identified in a series of symmetric rotors by fitting resonantly perturbed, high resolution spectra to within experimental error. In each case, gateway states through which the energy can be rapidly transferred have been identified and the specific Coriolis-like or Fermi-like coupling constants mediating the transfer have been measured. Interactions off-diagonal by as much as 5 units in the principal torsion quantum number have been shown to produce fragmentation of the spectrum. Because methyl groups are common in the much more complex molecules that occur in biological environments, the insight obtained into the mechanisms responsible for IVR in relatively simple molecules considered here will be useful in understanding the corresponding dynamic processes occurring in biological systems. Considerations similar to those relevant to IVR also apply to collisional relaxation of vibrationally excited molecules. The spectroscopy of two of the molecules studied here is of atmospheric/astrophysical interest. From studies in the 3μ region, the abundance of C2 H6 has been measured in the troposphere of the Earth [12] and in comets [13]. Using studies of the ν9 band, the abundance of C2 H6 has been determined in the stratospheres of Jupiter [14] and Saturn [15]. Recently, the Cassini mission has measured the ν9 band in the atmospheres of Saturn [16] and of Titan [17]. Laboratory work is in progress [18] on the absolute intensities of these ν9 transitions and the corresponding line widths in order to refine the HITRAN data base [19] used in these investigations. As has been noted in [19], the ethane component of the present HITRAN data base is “quite dated”. The studies of CH3 CF3 reviewed here have application in designing measures to reduce global warming. Because chlorofluorocarbon compounds have been shown to contribute significantly to the depletion of the ozone layer, work is underway to evaluate potential commercial alternatives, including hydrofluorocarbons such as 1,1,1-trifluorethane (HFC-143a) [20, 21]. The spectroscopy of CH3 CF3 and the understanding of its thermodynamic implications will be helpful in this process.
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F IG . 1. Energy level scheme probed for CH3 CD3 . Each level shown has (J = 1, G = 0) and the value of v6 indicated. For v6 = 4, the σ -splitting is large enough to see; the (σ = 0) sublevel is below its (σ = ±1) counterpart. For v6 = 5, the σ -ordering is reversed. For each vibrational state, only one of the three equivalent wells in the hindering potential V (α) is shown, with α plotted from −60◦ to +60◦ .
The energy level scheme probed for a typical XY3 AB3 symmetric top is shown in Fig. 1, which is drawn for CH3 CD3 . This has been adapted from Fig. 4 of [22]. The three-fold hindering potential V is drawn for the ground vibrational state (gs), the lowest lying non-degenerate vibration ν5 , and the lowest lying degenerate vibration ν12 . The ν5 mode is predominantly the C–C stretch, while the ν12 mode is primarily the CD3 rocking motion [23]. For each vibrational state, only one of the three equivalent wells forming V (α) is shown. The energy is a minimum in the staggered configuration, where the torsional angle α = 0. For ν5 and ν12 , the wells have been shifted to the right for clarity. In general, the height of each of the three potentials will be different. Within each well, the torsion–vibration energies are shown labelled with the principal torsional quantum number v6 . Because of the three-fold symmetry, each of these levels consists of three torsional sublevels labelled by σ = 0, +1, −1. For each vibrational state, the levels are drawn for the states where the torsional angular momentum Jα does not couple to the other angular momenta in the molecule. In this case, the sublevels with σ = +1 and −1 are degenerate, but are split from σ = 0. It is only for v6 4 that this splitting is large enough to be seen in Fig. 1.
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In the gs and v5 , Jα will couple strongly to Jz , the component along the symmetry axis of the total angular momentum J (exclusive of the nuclear spin). This coupling lifts the degeneracy between σ = +1 and −1 when the eigenvalue k of Jz is non-zero. In the degenerate mode ν12 , Jα will also couple to the vibrational angular momentum 12 with eigenvalue l12 = ±1. For this state, the quantum number G ≡ (k − l12 ) plays the same role with regard to the symmetry as k does for the gs and v5 . Here it is for G = 0 that the degeneracy between σ = +1 and −1 is lifted. As will be seen in more detail in Section 3.4.2, the energy level scheme for C2 H6 looks qualitatively much like that for its less symmetrical isotopomer CH3 CD3 . Aside from increasing the energy spacings by 10 to 20% depending on the mode involved, the change in symmetry has to be taken into account. The permutation-inversion group for C2 H6 is G†36 , while that for CH3 CD3 is G18 [24]. Consequently, the quantum numbers v5 , v12 , and v6 appropriate for CH3 CD3 are changed to v3 , v9 , and v4 , respectively, for ethane. These will be written in turn as vs , vt , and vi when a common labelling scheme is required here for both symmetries. Furthermore, because the two ends of the molecule are equivalent, the torsional potential V in C2 H6 is six-fold and there are four distinct torsional sublevels labelled by σ = 0, 1, 2 and 3. Symmetry considerations require that (G+σ ) be even [25,26], so that only two of the σ -states are populated for each value of G. In the C2 H6 version of Fig. 1, each well shown for V is plotted as a function of the torsional angle γ from 0◦ to +60◦ , with (γ = 0) corresponding to the eclipsed configuration. The experimental data being reviewed here are summarized in Table I. Seven symmetric rotors have been investigated: two methyl silane isotopomers (CH3 SiH3 and CH3 SiD3 ); three ethane isotopomers (C2 H6 , CH3 CD3 , and C2 D6 ); and two near spherical tops (CH3 CF3 and CH3 SiF3 ). All seven are prolate. The two X2 Y6 molecules are non-polar, while the XY3 AB3 molecules have a permanent dipole moment μ0 . For each molecule, several different spectra have been reported, each with an accompanying analysis incorporating the data then available. The reference for the most recent analysis covering all the data listed (referred to as “Primary”) is noted on the first line for the molecule in question. The different components to the global data set are numbered in the first column. For each component, the selection rules are given, along with an indication of the range of quantum numbers included and the experimental error typical of the measurements involved. The attribution is given in the last column, with the experimental method being specified in the accompanying footnotes. For CH3 CD3 , components 3 to 6 are indicated schematically in Fig. 1 by solid arrows [22]. The (v6 = 3 ← 0) transitions become observable through resonant Coriolis mixing between the states (v12 = 0, v6 = 3) and (v12 = 1, v6 = 0). The dotted arrow in Fig. 1 indicates torsion–vibration transitions (v12 = 1, v6 = 1) ← (v12 = 0, v6 = 0) that become allowed only through
1]
Table I Measurements Analyzed for AB3 XY3 and X2 Y6 Symmetric Rotors Selection rules/Transitionsa
1 2 3 4 5 1 2 3 4 5
CH3 SiH3 (Primary Ref. [27]) gs v6 = 0; (1, ±1, σα ) ↔ (2, 0, σβ ) σ =0, ±1 (2, 0) gs v6 = 0; (J, ±1, ∓1) ↔ (J, ∓1, σβ ) σβ =0, ∓1 (6, 1)e gs v6 = 0, 1, 2, 3, 4; J = +1 (45, 13) ν12 l12 = ±1 v6 = 0; J = +1 (8, 7) Microwave difference bandm (19, 6) Microwave direct l-doublingn σ = 0, ±1 (26, 1) gs v6 = 2 ← 0; J = 0, ±1 (25, 12) gs v6 = 3 ← 1; J = 0, ±1 (25, 9) gs v6 = 3 ← 0; J = 0; (k, σ ) = (6, −1) (16, 6) gs v6 = 5 ← 0; J = ±1; (k, σ ) = (1, −1) (25, 1) v12 = 1 ← 0; v6 = 0; J = 0, ±1; k = l12 = ±1 (25, 9) v5 = 1 ← 0; v6 = 0; J = 0, ±1 (25, 9) CH3 SiD3 (Primary Ref. [38]) gs v6 = 0; (4, ∓1, σ ) ↔ (3, ±2, σ ) σ = 0, ±1, ∓1 (3, 2)e gs v6 = 0; (J, ±1, ∓1) ↔ (J, ∓1, σβ ) σβ = 0, ∓1 (2, 1) gs v6 = 0, 1, 2; ν12 l12 = ±1 v6 = 0; J = ±1 (1, 1) gs v6 = 2 ← 0; v6 = 3 ← 1; J = 0, ±1 (30, 13) v12 = 1 ← 0; v6 = 0; J = 0, ±1; k = l12 = ±1 (40, 22) CH3 CH3 (Primary Ref. [40]) gs v4 = 1 ← 0; J = 0, ±1 (36, 15) gs v4 = 2 ← 1; J = 0, ±1 (35, 11) gs v4 = 3 ← 0; J = −1; (K, σ ) = (18, 0) (30, 18) v9 = 1 ← 0; v4 = 0; J = 0, ±1; k = l9 = ±1 (30, 20) v3 = 1 ← 0; v4 = 0; J = 0 (24, 5)
c,d
Reference
30 13 10 to 1000 10, 50 10 8 20, 100 10 15 6 4 6
[28]f [28]f , [29]f [28]g , [30]h , [31]h , [32]h , [27]i,j [32]h , [27]j [27]j [27]j [33]k , [34]l [35]k [35]k [36]k [37]k , [35]k [36]k
175 14 100 35, 70 2 20 to 100 50 to 150 5 to 20 2 to 20 10 to 20
[29]f [29]f [39]g [38]k [38]k [41]k , [42]k [42]k [41]k [41]k [43]o
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1 2 3 4 5 6 7 8 9 10 11 12
(Jm , Km )b
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Selection rules/Transitionsa
1 2 3 4 5 6 1 2 3 4 5 6 7 8 9 10 11
(Jm , Km )b
CH3 CD3 (Primary Ref. [22]) gs v6 = 0; (J, ±1, ∓1) ↔ (J, ∓1, σβ ) σβ = 0, ∓1 (2, 1) gs v6 = 0, 1, 2; J = +1 (10, 8) gs v6 = 1 ← 0; J = ±1 (19, 9) gs v6 = 3 ← 0; J = 0, ±1; (k, σ ) = (1, 1); (3, 0); (4, 0) (28, 4) v12 = 1 ← 0; v6 = 0; J = 0, ±1; k = l12 = ±1 (30, 9) v5 = 1 ← 0; v6 = 0; J = ±1 (29, 9) CD3 CD3 (Primary Ref. [52]) gs v4 = 1 ← 0; J = ±1; (k, σ ) = (8, 2) (24, 2) gs v4 = 2 ← 1; Band origin (0, 0) gs v4 = 2 ← 0; v4 = 3 ← 1; Band origins (0, 0) gs v4 = 3 ← 0; J = −1; (k, σ ) = (15, 1); (16, 2); (17, 3) (34, 17) gs v4 = 1; J = +1 (14, 6) v9 = 1 ← 0; v4 = 0, 1; J = 0, ±1; k = l9 = ±1 (40, 18) CH3 CF3 (Primary Refs. [56] and [57]) v6 = 0; (J, ±2, σ ) ↔ (J, ∓1, σ ) σ = 0, ±1, ∓1 (6, 1)e gs v6 = 0, 1, 2, 3; J = +1 (86, 69) gs v6 = 0, 1; J = 0; k = ±3 (75, 8)q ν5 v6 = 0; J = +1 (34, 30) ν12 l12 = ±1 v6 = 0, 1; J = +1 (20, 20) ν11 l12 = ±1 v6 = 0, 1; J = +1 (12, 12) gs v6 = 2 ← 0; J = ±1 (38, 21) v5 = 1 ← 0; v6 = 0, 1; J = ±1 (70, 46) v12 = 1 ← 0; v6 = 0, 1; J = ±1; k = l12 = ±1 (65, 52) v11 = 1 ← 0; v6 = 0, 1; J = 0, ±1; k = l11 = ±1 (60, 60) v11 = 1 ← 0; v6 = 0; J = 0, −1; G = ±3; l11 = −1 (49, 15)
c,d
Reference
6 10 to 200 200 4 2 2
[44]f , [29]f [45]g , [46]g , [47]g , [48]h , [49]h [50]k [51]k [51]k [22]k
500, 1000 1000 10000 2 to 10 2 to 5 2 to 10
[53]k [53]k [54]o [55]k [53]k,p [55]k , [52]k
18 3 to 500 30, 50 5 to 50 24, 36 30 to 65 15 3, 4 3, 5 3, 5 3
[58]f [58]g,h , [59]h,i , [60]g,j , [61]h [60]j [56]h [57]h , [62]h [57]h , [62]h , [63]j [61]k [56]k [57]k,q [57]k [57]k
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Table I Continued.
[1
Selection rules/Transitionsa
(Jm , Km )b
CH3 SiF3 (Primary Ref. [57]) gs v6 = 0; (J, k, σ ) ↔ (J, k ∓ 3, σ ) σ = 0, ±1, ∓1 (5, 1)e gs v6 = 0; (J, ±1, ∓1) ↔ (J, ∓1, σβ ) σβ = 0, ∓1 (5, 1)e gs v6 = 0, 1; J = +1 (60, 48) gs v6 = 0; J = 0; k = ±3 (65, 7) ν5 v6 = 0; ν11 l11 = ±1 v6 = 0; J = +1 (1, 1) v5 = 1 ← 0; v6 = 0; J = 0, ±1 (60, 27) v11 = 1 ← 0; v6 = 0; J = ±1; k = l11 = ±1 (66, 51)
a Unless otherwise specified, v , v , l , v , k, and σ are conserved. s t t i b Maximum values included in the data set for the quantum numbers of the lower state. c A typical experimental uncertainty in a transition frequency. d Unless otherwise noted, the units are 10−4 cm−1 for infrared transitions and kHz for other measurements. e Relative difference measurements were also included; see the original reference. f Avoided crossing electric resonance spectroscopy. g Microwave spectroscopy. h Mmwave spectroscopy. i Sub-mmwave spectroscopy. j Fourier transform microwave spectroscopy (FTMW). k Fourier transform infrared spectroscopy (FTIR).
Reference
20, 62 2 to 8 5 to 200 5, 10 100 3 4
[64]f , [29]f [64]f , [29]f [64]g,h , [65]g , [66]j , [67]i [66]j [68]j [57]k [57]k
431
l Diode laser spectroscopy. m (v = 0, l = 0; J, v = 3, k = 6, σ = −1) ↔ (v = 1, l = −1; J, v = 0, k = 5, σ = −1). 12 12 6 12 12 6 n (v = 0; v = 1, A ; J, v = 0, σ ) ↔ (v = 0; v = 1, A ; J, v = 0, σ ). + − 5 12 6 5 12 6 o Raman spectroscopy: stimulated in [43] and conventional in [54]. p Lower-state combination differences from the ν + ν − ν band. The value of is given in MHz. 9 4 4 q (A − A ) splittings were measured for v = 0. 1 2 6
c,d
INTERNAL ROTATION IN SYMMETRIC TOPS
1 2 3 4 5 6 7
1]
Table I Continued.
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a higher order interaction between the states (v12 = 0, v5 = 1, v6 = 0) and (v12 = 1, v5 = 0, v6 = 1). These frequencies are well predicted by the model used, but they have not been put into the data set, pending a thorough investigation of the state (v12 = 1, v5 = 0, v6 = 1). The different types of spectra listed in Table I can be divided into four broad categories. The first consists of pure rotational transitions including the normal (k = 0) R-branch and the “forbidden” (k = ±3) Q-branch. The former is driven by the parallel dipole moment μ0 [69], while the latter is driven by the small distortion dipole moment μD [70], that is generated perpendicular to the symmetry axis by centrifugal distortion [71,72]. The second type of spectrum was obtained by the electric-resonance molecular-beam avoided-crossing technique [73]. In this method, energy differences are measured to high precision between levels for which k = 0 and, in some cases, σ = 0. In the “hyperfine anticrossings” [28], many of the normal symmetric top dipole selection rules are violated. In components 1 and 2 given in Table I for CH3 SiH3 , for example, the vibration–torsion–rotation symmetry is in effect broken by the nuclear hyperfine interactions [28]. The third type of spectrum listed in Table I consists of infrared torsional bands with vi = 1, 2, 3, 5 and vibrational bands with vs = 1 or vt = 1. The torsional bands are effectively dipole forbidden in lowest order, arising primarily (or exclusively) through Coriolis-like or Fermi-like mixing with infrared active vibrational states. On the other hand, the vibrational bands observed are dipole allowed. This same mixing can be observed, often producing effects which dramatically increase the frequencies with which the methyl group tunnels through the barrier. The fourth type of spectrum was obtained by Raman spectroscopy in X2 Y6 molecules [43,54]. The torsional bands observed with v4 = 2 are forbidden in lowest order, while the vibrational band with v3 = 1 is Raman active. The three different types of dipole spectra will be reviewed, going into more detail for the avoided crossing method because it is generally less widely known. For a recent discussion of Raman studies of C2 H6 , see [74]. Because of the internal degree of freedom, the molecule-fixed co-ordinate system to be used in analyzing the spectroscopic data must be defined with some care. In an XY3 AB3 symmetric top, the XY3 group will be assumed to have a smaller moment of inertia about the symmetry axis than the AB3 group. Then XY3 and AB3 , respectively, are designated as the top and frame. The torsional Hamiltonian can be easily diagonalized with either the Principal Axis Method (PAM) or the Internal Axis Method (IAM) [1]. In the PAM, the molecule-fixed co-ordinate system is taken to be fixed with respect to the AB3 frame. In the IAM, the molecule-fixed co-ordinate system is not fixed with respect to either the frame or the top, but is chosen, in lowest order, so that the angular momentum of the internal motion vanishes. Both the PAM and the IAM have major advantages and disadvantages. Recently, a third alternative, termed the hybrid approach, has been developed [57,61] which combines the important advantages of both of the older
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methods. This new approach will be summarized briefly below. In X2 Y6 molecules, the top and frame are identical, and the IAM treats the two XY3 rotors in a symmetrical manner. The difficulties inherent in the IAM treatment of XY3 AB3 molecules do not arise, and analysis for molecules with identical rotors is carried out in the IAM. The early work on internal rotation was analyzed with what today would be called a one-band model, i.e. a model designed to deal with a single stack of gs torsional levels that do not interact (except possibly weakly) with any of the skeletal vibrations [1]. However, the data summarized in Table I involve a wide range of levels in the gs, in νt , and in νs , many of which are very sensitive to intervibrational interactions. In order to analyze this large body of diverse data, it is necessary to set up the effective vibration–torsion–rotation Hamiltonian HVTR in a three-band model [36] which includes in zeroth order the intervibrational interactions between the gs stack of torsional levels and the corresponding stacks for νt and νs . The leading (gs, νt ) interactions are Coriolis-like [37], while the leading (gs, νs ) interactions are Fermi-like [36]. Such a model has been used in the Primary References for C2 H6 [40], CH3 CD3 [22], CH3 SiH3 [27], and CH3 SiF3 [57]. In the remaining molecules listed in Table I, a two-band model was used. For C2 D6 [55] and CH3 SiD3 [38], only the gs and νt have been investigated to date. For CH3 CF3 , gs, νt , and νs have been studied, but the (gs, νt ) coupling was negligible [56,57]. For most of the molecules, the Hamiltonian matrix was diagonal in G. However, for CH3 CD3 [22] and CH3 CF3 [56,57], the matrix had to be expanded to account for matrix elements off-diagonal in G. The Hamiltonian HVTR will be discussed, along with the basis set and the calculational procedures used. In the progression from the two-band (gs, νt ) analysis to the three-band (gs, νt , νs ) model with the introduction of the νs data, some of the molecular parameters characterizing the gs energy change substantially in value. The most striking change occurs in the parameter V6 which characterizes the second term in the Fourier expansion of the gs potential. In C2 H6 , for example, V6 equals (8.782 ± 0.019) cm−1 in the two-band (gs, νt ) analysis [41], but vanishes in the three-band (gs, νt , νs ) analysis [40]. It has been shown that the two-band value for V6 can be interpreted as arising entirely from the Fermi-like (gs, νs ) coupling [40]. Similar results have been obtained for CH3 SiH3 [36], CH3 CF3 [56], and CH3 CD3 [22]. The changes in the molecular parameters in going from the two-band to three-band models will be discussed in the context of the contact transformations that are implicitly used to remove non-resonant intervibrational terms in the Hamiltonian. The torsional dependence of the effective dipole moment operator μ in the gs has been investigated by measuring the intensities of the torsional bands in CH3 SiH3 [75], C2 H6 [76], CH3 CD3 [50], and C2 D6 [53]. In CH3 SiH3 , Stark effect measurements were made as well [77,75]. The results have been interpreted in terms of four torsional dipole constants resulting from non-resonant Coriolis-
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F IG . 2. Resonance-enhanced tunneling in ν5 of CH3 SiH3 . The two of the three equivalent wells of the potential V (α) are shown with α = 0 corresponding to the staggered configuration. For (v6 , k = 1, σ = −1) with v6 = 0, 1 and 2, the energy E is plotted after subtraction of the vibrational quantum ν 5 . The zeroth-order rotational energy has also been removed for illustration purposes. For each of the torsional states, the probability density P (α) is plotted with the zero for the probability distribution being coincident with the associated energy. For v6 = 0, note the local maxima inside the classically forbidden region.
like (gs, νt ) interactions and Fermi-like (gs, νs ) coupling [72]. This work will be discussed in the context again of the contact transformations that are used to remove non-resonant intervibrational terms in the Hamiltonian [72,78]. One of the major surprises encountered in the current program is illustrated in Fig. 2 using the best molecular parameters from [36]. The potential V (α) is plotted with α running from 0◦ to 240◦ for (k, σ ) = (1, −1) in ν5 . (The third well has been omitted for clarity.) For v6 = 0, the total energy is shown as a horizontal straight line, with the vibrational quantum ν 5 and zeroth-order rotational energy having been removed for illustration purposes. The probability density P (α) is plotted with the zero for the probability distribution being coincident with the associated energy. The corresponding curves are also shown for v6 = 1 and 2. In the simplest case, each P (α) is just the square of the modulus of the corresponding torsional wave function. Curves such as this are well known in introductory quantum mechanical treatments of the harmonic oscillator; see, for example, Fig. 2.7
1]
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of [79]. However, in Fig. 2 here, P (α) for v6 = 0 displays some striking differences from what is expected for a small amplitude torsional oscillator. In the classically forbidden region inside the potential, P (α) does not fall rapidly to zero. In fact, the minimum value for P (α) in this region is more than 10% of the probability maximum at α = 120◦ . Furthermore, P (α) has a local maximum inside the torsional barrier. On the other hand, for v6 = 1 and 2, P (α) behaves qualitatively as expected, at least as far as is known at present. The experimental and theoretical considerations leading to Fig. 2 and the resulting implications form a significant part of the work reviewed here. The current review will be largely limited in scope to the data presented in Table I and the associated analyses. Although several related investigations will be discussed, there are many other interesting and important works on internal rotation in XY3 AB3 and X2 Y6 molecules that will not be reviewed. For example, there are the studies of the ν9 + ν4 − ν4 band in C2 H6 by Susskind et al. [80]; the ν5 band in 12 CH3 CD3 and 13 CH3 CD3 by Lattanzi et al. [81]; the torsional Raman Q-branch transitions in CH3 CD3 by Van Helvoort et al. [82]; the molecular-beam optothermal spectrum of the Fermi triad in CH3 CF3 near 970 cm−1 by Fraser et al. [83]; and the microwave spectrum of PH3 BH3 by Durig et al. [84]. These various papers fall outside the scope of the present work. They deal with bands for which the upper state falls above the current upper limit on the energy; or present data which have been superseded by later, more accurate measurements; or investigate molecules which are not considered here. Some of these works discuss intervibrational perturbations. However, to our knowledge, the only global multiband fits that have been made for XY3 AB3 or X2 Y6 molecules undergoing internal rotation are those referred to in Table I. The rest of the present chapter consists of five parts. Section 2 outlines the formalism for obtaining the vibration–torsion–rotation energy. In Section 2.1, the hybrid method is introduced, and compared to the PAM and IAM. In Section 2.2, the Hamiltonian is presented, and the High Barrier and Free Rotor Models are compared. In Section 2.3, the symmetry properties of the eigenstates are discussed. The eigenfunctions are expressed in terms of free rotor basis functions and the cases where a Wang transformation is required are considered. The various methods developed for diagonalizing the Hamiltonian matrix are described. Section 3 reviews the three different types of dipole spectra which have been principally used to investigate internal rotation in symmetric tops, and discusses briefly some of the Raman studies. Section 3.1 treats primarily rotational spectroscopy, but includes direct l-doubling transitions and a very unusual torsion–rotation difference band that falls in the microwave region. Section 3.2 describes the avoided crossing molecular beam method, and introduces the different types of anticrossings possible. In Section 3.3, the torsional dependence of the dipole moment is considered, and torsional studies of isotopomers of CH3 SiH3 and C2 H6 are reviewed. Section 3.4 discusses the lowest lying parallel and perpendicular vibrational bands of
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CH3 SiH3 and C2 H6 , as well as the lowest frequency degenerate band of C2 D6 . In each case, the emphasis is on (near) resonant intervibrational interactions. Finally, Section 4 presents a discussion of some of the implications of this body of work and several of the puzzles that still remain. A series of different experimental methods are outlined which might be used to investigate internal rotation in symmetric tops. The more casual reader may prefer to omit the more mathematical parts of the review. The text is structured in such a way that the reader primarily interested in the different types of spectroscopy and the results can follow the later sections of the current work without Section 2 on the theoretical development. In part to assist with such an approach, a simplified expression for the vibration–torsion–rotation energy is given in Section 3.1.
2. Theory 2.1. T HE Z EROTH -O RDER H AMILTONIAN AND THE H YBRID A PPROACH (0)
The zeroth-order torsion–rotation Hamiltonian HTR,gs for the gs of an XY3 AB3 symmetric top will be considered first. In the PAM, this operator has the form [1]: (0)
HTR,gs = (A0,F − B0 )J2z + B0 J2 + F0 J2α − 2A0,F Jα Jz 1 + V0,3 (1 − cos 3α). (1) 2 Here Jα represents the total angular momentum of the top about the symmetry axis, including the contributions from both the external and internal rotation. It commutes with the three components of J in the molecule-fixed co-ordinate system. The reduced rotational constant F0 = (A0,F + A0,T ), where A0,F and A0,T are the rotational constants about the z-axis of the frame and top, respectively. The subscript 0 on the molecular parameters indicates explicitly that these refer to the gs. The large zeroth-order term in Jα Jz couples the top and total angular momenta, thus preventing the separation of the internal and external motion. The traditional way of overcoming this difficulty has been to change from the PAM to the IAM [1]. This involves two steps. First, Jα in Eq. (1) is expressed in terms of a new torsional angular momentum operator Jα ≡ Jα − ρ0 Jz , where the prime indicates that the operator is defined in the IAM. With the structural parameter ρ0 taken to be A0,F /F0 , the IAM counterpart of Eq. (1) has no cross term in Jα and Jz . As a result, the torsion and overall rotation are now independent of one another. Of course, Jα commutes with Jz , but not Jx or Jy . Second, a rotation is made about the symmetry axis through angle ρ0 α; see, for example, Section 2.A and Fig. 1 of [85]. Because of this change in reference frame, the meaning of “overall rotation” has been changed. In the IAM co-ordinate system, Jα commutes with all three components Jx , Jy and Jz of J.
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Both the PAM and the IAM have major advantages and disadvantages. On the one hand, the PAM has the advantage that the derivation of the classical Hamiltonian is more direct [1]. Furthermore, conventional group theory can be applied in the PAM, but not in the IAM [85]. It is only in the PAM that the vibrational, torsional, and rotational wave-functions individually transform properly according to the irreducible representations of the G18 permutation-inversion group. The same is true for some of the operators that enter in higher order. For example, the PAM operators (J+ , J− ) that ladder k (down, up) form a basis for the irreducible representation Γ = E1 , but their IAM counterparts (J+ , J− ) do not transform properly according to the irreducible representations of G18 . As a result, the derivation of selection rules can be carried out by conventional means in the PAM, but not the IAM. On the other hand, as argued above, the IAM has the advantage that the problem is separable. In the limit that the barrier height goes to infinity, the zeroth-order Hamiltonian in the IAM (but not in the PAM) separates into two parts, one for a harmonic oscillator and one for a rigid rotator. It follows then that the IAM molecular parameters go over to their C3v counterparts in this limit. This can be very helpful, particularly in interpreting the parameters and in making comparisons with simpler molecules without internal rotation. Recently, the hybrid model was presented by Wang et al. [61] as an alternative method of dealing with the large cross term in Eq. (1). In this approach, the first step in the change from the PAM to the IAM given above is carried out, but not the second. In the hybrid method, the new torsional angular momentum operator is written: " Jα ≡ (Jα − ρ0 Jz ). Since Jα and ρ0 are defined as before, " Jα = Jα . However, the tilde rather than the prime is used to indicate that the hybrid method is used, rather than the IAM. No co-ordinate transformation is performed; all that occurs is a re-grouping of the angular momentum operators. The PAM co-ordinate system, wave functions and operators are used throughout. Nevertheless, the cross term is formally removed; one can write: (0) (0) H(0) TR,gs = HR,gs + HT ,gs ;
(2)
2 2 H(0) R,gs = (A0 − B0 )Jz + B0 J ;
(3)
1 (0) HT ,gs = F0" J2α + V0,3 (1 − cos 3α). 2
(4)
(0)
(0)
Here HR,gs is the rigid rotor Hamiltonian, while HT ,gs is the zeroth-order internal rotor Hamiltonian. The separation of variables that occurs in the transformation to (0) the IAM has not been carried out. While HR,gs involves only differential operators (0)
in Euler angles θ, φ and χ [1], HT ,gs involves differential operators in both χ and α. In spite of this, it has been shown [61] that the Hamiltonian matrix for (0) HTR,gs in the PAM basis has exactly the same form as that for the corresponding IAM Hamiltonian in the IAM basis. Furthermore, this equivalence of form has
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been shown to occur for the full effective torsion–rotation-vibration Hamiltonian HVTR . This striking result arises because rotation of the top about the symmetry axis in a symmetric rotor with a single torsional degree of freedom does not alter the moment-of-inertia tensor. Of course, " Jα does not commute with J+ or J− . As a result, terms in HVTR involving products of " Jα with J+ and/or J− , such as those in the Coriolis-like interaction, must be symmetrized. However, this complication has proven to be relatively minor. Aside from this symmetrization requirement, the results derived in the IAM in our earlier papers can be taken over directly by substituting " Jα for Jα (or p, the symbol sometimes used for the IAM torsional angular momentum operator [37]). The hybrid approach has the major advantages of both the PAM and the IAM. First, the rotational, torsional and vibrational wave functions as well as all the relevant operators retain their well defined symmetries in the permutation-inversion group G18 . Second, a re-grouping of the PAM molecular parameters occurs so that the new linear combinations go over to their C3v counterparts in the limit of infinite barrier height. Third, the treatment of intervibrational interactions is simplified, and the interpretation of the associated molecular parameters is clarified. For example, in the first study of Coriolis-like interactions in CH3 SiH3 [37], the vibrational operators are treated in the PAM, while the torsional and rotational operators are treated in the IAM. This introduces some difficulties in the physical interpretation of the parameters which characterize these intervibrational interactions involving products of torsional, rotational and vibrational operators. In the hybrid approach, these difficulties are avoided, since all the operators are in the PAM. Furthermore, the application of contact transformations to eliminate nonresonant intervibrational interactions is simplified. For an X2 Y6 symmetric top, the situation is much simpler. Because the two co-axial tops are identical, ρ0 = 1/2 and the IAM treats the two tops in an identical manner [26]. Furthermore, conventional group theory works in the IAM [24]. The torsional angular momentum operator in the IAM is defined as Jγ = (1/i)(∂/∂γ ). Terms such as Jγ Jz which involve Jγ raised to an odd power are forbidden by symmetry [86]. Thus there is no need to use the PAM or the hybrid method, and no primes or tildes are required. Eqs. (2) and (3) apply to C2 H6 , while Eq. (4) is replaced by: 1 2 H(0) T ,gs = A0,R Jγ + V0,3 (1 + cos 6γ ). 2
(5)
Here A0,R is the gs reduced rotational constant. In lowest order, A0,R = A0 , but in higher order the two parameters can differ [87]; the subscript R is introduced to allow for these higher order effects. The plus sign in the potential energy term in Eq. (5) arises because (γ = 0) is the eclipsed configuration [86].
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2.2. T HE V IBRATION –T ORSION –ROTATION H AMILTONIAN The effective vibration–torsion–rotation Hamiltonian HG VTR used to fit the data for CH3 SiH3 and C2 H6 is given in Table II, where the index i labels the operators Oi and the associated molecular parameters Xi (or Xi ). As is indicated by the superscript, HG VTR is diagonal in G. For each molecule, the High Barrier Model [27,38] (to be described below) is used, and the fit is to well within the experimental error. In Table II, (qt+ , qt− ) are ladder operators for the degenerate normal coordinates of νt , and (pt+ , pt− ) are the corresponding operators for the conjugate momenta [78]. Similarly, qs is the non-degenerate normal coordinate for νs , and ps will be used as its conjugate momentum. In addition, [Oa , Ob ]+ is the anticommutator of Oa and Ob . The overall Hamiltonian is divided into five groups of terms according to the selection rules and the states involved. Three of these groups have only matrix elements diagonal in vt and vs : Hgs with i = 1 to 25; Hνt with i = 26 to 40; and Hνs with i = 41 to 46. The last two groups in Table II involve matrix elements off-diagonal in the vibrational quantum numbers: Hgs,νt with i = 47 to 52 for (gs, νt ) coupling; and Hgs,νs with i = 53 to 57 for (gs, νs ) interaction. For discussion purposes, Hν ≡ [Hgs + Hνt + Hνs ], while Hkν is defined to be Hν without the l-doubling terms. Since Hkν does not involve O39 , it is diagonal in k as well as vt and vs . With respect to its order, the operator 12 (1 − cos 3α) is considered to be equivalent to terms like " J2α and J2 (i.e. quadratic); while 12 (1 − cos 6α) is considered to be equivalent to terms J4α (i.e. quartic). In Table II and this discussion, the operators are like " J2α J2z and " written in the form appropriate for XY3 AB3 molecules. When these operators are to be applied to X2 Y6 molecules, α is replaced by 2(γ + π/6); " Jα by Jγ ; and Fv by Av,R . In addition, in the first term of O47 and in O49 , the factor of 1/2 and the anticommutator, while formally correct, are not required. To form HG VTR using Table II, all the terms Xi Oi are added, allowing for the fact that different rows apply to different matrix elements as noted above. In carrying out the sum, three further modifications are required. First, the parameter ρ does not multiply any individual operator. For CH3 SiH3 , ρ appears implicitly through operators Oi that contain " Jα . F or C2 H6 , ρ does not enter at all. Note that the subscript 0 has been dropped from ρ because its vibrational dependence was found to gs be negligible. Second, for some of the parameters in Hνt , Xi ≡ [Xiνt − Xi ] is listed rather than Xi itself. In these cases, the parameter multiplying Oi in Hνt is gs [Xi +Xi ]. The corresponding statement for Hνs also applies. Finally, the quartic and sextic distortion terms for νt and νs require special treatment. Consider as an example the allowed νt quartic operator J2z" J2α which does not appear in Table II. 2 2 In spite of its absence, Jz" Jα is entered into Hνt , but it is multiplied by the gs parameter associated with this operator. The corresponding statements are true for all allowed νt and νs quartic operators for CH3 SiH3 and C2 H6 . In CH3 SiH3 , the ν12 sextic operators are also treated in this manner, but the ν5 sextic operators were
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Table II Operatorsa and Spectroscopic Parametersb in the (gs/νt /νs ) Analysis of CH3 SiH3 and C2 H6 i
Xi or Xi
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
ρ F0 V0,3 A0 B0 D0,J D0,J K D0,K D0,m D0,J m D0,Km D0,sJ D0,sK D0,sm F0,3J F0,3K F0,6J F0,6K F0,3J J F0,3J K F0,3KK H0,J K H0,KJ H0,J J m H0,J Km
26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
" νt Ft Vt,3 At Bt Dt,J Dt,J K Dt,K Ft,3J Aζtz ηt,J ηt,K Aζtm qt qt,J
Units
Operator Oi
CH3 SiH3
C2 H6
Hgs terms for diagonal matrix elements with (vs = 0, vt = 0) Unitless Implicit 0.3519209(34) 0.5c " MHz J2α 246 311.1d 80 380.4d 1 (1 − cos 3α) cm−1 603.3878(37) 1 047.736(28) 2 MHz J2z 56 176.811(56) 80 380.4(4.0) MHz J2 − J2z 10 984.4451(34) 19 913.487(42) kHz −J4 10.7087(13) 30.973(31) kHz −J2 J2z 45.732(15) 79.28(34) kHz −J4z 189.65e 283e MHz −" J4α 3.3368(96) 0.395(19) MHz −J2" 0.61649(65) 0.3033(11) J2α MHz −J2z" 4.198(24) 2.0823(79) J2α 2 MHz −J Jz" −0.125460(59) 0.0f Jα MHz −J3z" −0.268(28) 0.0f Jα MHz −Jz" 2.262(43) 0.0f J3α 1 2 MHz J 2 (1 − cos 3α) −132.275(37) −395.2(2.1) MHz J2z 21 (1 − cos 3α) 395.3(1.4) 1 010.7(1.4) MHz J2 21 (1 − cos 6α) 0.0g 8.56(63) MHz J2z 21 (1 − cos 6α) 0.0g 9.00(54) kHz J4 21 (1 − cos 3α) 0.3107(95) 1.669(80) kHz J2 J2z 12 (1 − cos 3α) −2.96(18) 0.0g kHz J4z 21 (1 − cos 3α) 0.0g 21.4(1.6) 4 2 Hz J Jz 0.184(11) 0.0g Hz J2 J4z 0.0g 2.94(86) J2α Hz J4" 2.80(20) 0.0g 2 2 2 " Hz J Jz Jα 29.2(4.0) −7.4(1.8) Hνt terms for diagonal matrix elements with (vs = 0, vt = 1) cm−1 unity 624.55258(28)h 973.97125(71)h " MHz J2α 247 215.8d 80 622.5d 1 (1 − cos 3α) cm−1 659.04(36) 1 116.4(1.4) 2 MHz J2z 206.341(35) 242.100(89) MHz J2 − J2z 0.0g −41.619(36) kHz −J4 0.0278(23) −0.456(95) kHz −J2 J2z 0.770(56) 2.750(53) kHz −J4z 0.0g 3.70(15) MHz J2 21 (1 − cos 3α) −30.842(16) 0.0g MHz −2Jz t 13 681.26(19) 21 121.4(4.0) MHz J2 Jz t 0.01688(21) 0.0237(13) MHz J3z t 0.5507(109) 0.6171(27) MHz 2" Jα t 87 000i 0.0f 1 [J2 q2 + J2 q2 ] MHz 22.0035(40) 49.233(69) − t− 4 + t+ Hz J2 O39 −72.29(14) 0.0g
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Table II Continued. i Xi or Xi Units
Operator Oi
CH3 SiH3
C2 H6
50 Qi,ta
Hνs terms for diagonal matrix elements with (vs = 1, vt = 0) cm−1 unity 993.428(85)h 993.959(94)h " MHz J2α 246 167.8d 80 292.6d 1 (1 − cos 3α) cm−1 577.79(21) 1 040.3(1.3) 2 MHz J2z −32.69(20) −87.84(30) MHz J2 − J2z −76.27(19) −183.895(94) J2α MHz −J2" −0.412(33) 0.0g Hgs,νt terms for (gs, νt ) coupling 1 [" MHz 2 Jα , J+ qt− + J− qt+ ]+ −2λζ sin 3α[J+ pt− + J− pt+ ] 428.56(33)j 343.7(1.7)j 2 j kHz J O47 −3.0836(69) −3.98(36)j i " MHz − 2 [Jα , {(Jz J+ + J+ Jz )qt− −(Jz J− + J− Jz )qt+ }]+ 0.0g −9.27(17)j MHz −2i[Jz , J+ qt− − J− qt+ ]+ sin 3α 13.392(27)j 0.0g
51 Mi,ta
MHz
41 ν˜ s 42 Fs 43 Vs,3 44 As 45 Bs 46 Ds,J m x ) 47 (Bζi,ta x,J 48 (Bζi,ta ) yz 49 Pi,ta yz
yz,3
yz,3,J
52 Mi,ta
kHz
αα 53 Ms,0
cm−1
αα,J 54 Ms,0 3 55 Cs,0 3,J 56 Cs,0 6 57 Cs,0
MHz cm−1 MHz cm−1
1 [J , J q + J q ] − t+ + 2 z + t− 1 2 (1 − cos 3α) J2 O51
30.8(1.6)j
7.3(1.7)j Hgs,νs terms for (gs, νs ) coupling qs" −0.22530(41)k J2α J2 O
53 qs 21 (1 − cos 3α) J2 O55 1 qs 2 (1 − cos 6α)
0.1569(93)k
0.0g 0.0g 0.0g
128.246(13)k
0.0g 260.50(24)k
−1.25(19)k 1.066(17)k
−166.3(7.9)k 2.40(11)k
a The operators are given in the form appropriate to the High Barrier Model used for CH SiH . For 3 3 Jα by Jγ , and Fv by Av,R . Furthermore, in the first term of O47 C2 H6 , α is replaced by 2(γ + π/6), " and in O49 , the factor of 1/2 and the anti-commutator are not required. For terms diagonal in both vt and vs , the vibrational operators can be taken to be unity. The treatment of the distortion constants in the excited vibrational states is discussed in Section 2.2. The (ht = hgs , hs = 0) sextic option was used for CH3 SiH3 , while the (h12 = hgs , h5 = hgs ) sextic option was used for C2 H6 . b The parameter values are taken from [27] (CH SiH ) and [40] (C H ). 3 3 2 6 c For C H , ρ ≡ 1/2. 2 6 d This was held fixed at A /ρ(1 − ρ) (CH SiH ) and A (C H ). v v 3 3 2 6 e This is fixed to the force field value given in [88] (CH SiH ) and [89] (C H ). 3 3 2 6 f This is fixed at zero because the associated operator is forbidden by symmetry. g This is fixed at zero because the parameter (although allowed by symmetry) could not be determined. h The relationship of the fitting parameter value to the vibrational quantum is discussed in Section 3.4.1. i This was held fixed at A 12,F = [A12 /(1 − ρ)]. See Section 3.4.1 here and [37]. j The relative signs of the (gs, ν ) coupling constants are determined, but not the absolute signs. t k The relative signs of the (gs, ν ) coupling constants are determined, but not the absolute signs. s
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not taken into account. This will here be referred to as the (h12 = hgs , h5 = 0) sextic option. On the other hand, in C2 H6 , the (h12 = hgs , h5 = hgs ) sextic option was used, where all the νt and νs sextic parameters were fixed at the gs values. In forming HG νt and " νs can be interpreted at this VTR , the fitting parameters " stage as the respective vibrational quanta ν t and ν s . However, the numerical values given in Table II correspond to specialized definitions selected for reducing correlations in the fitting process. These definitions are discussed in Section 3.4.1. The effective Hamiltonian HVTR used to date in the XY3 AB3 analyses listed in Table I is the sum of HG VTR and two terms off-diagonal in G. These terms are: HS = + J J2 Jz , J3+ + J3− + ; (6) J 2 " 2 H12,5 = 12,5 + 12,5 J Jα , J+ + p12+ q5 + q12+ p5 . (7) The term HS is diagonal in the vibrational quantum numbers. For CH3 CF3 , HS has been used to treat (A1 − A2 ) splittings in the gs [60] and in ν12 [57], as well as (k = ±3) resonant perturbations in ν11 [57]. The term H12,5 has matrix elements off-diagonal in v6 , v12 , and v5 . For CH3 CD3 , H12,5 has been used to treat a strong local (k = ∓2, l12 = ±1) resonance in the ν5 band [22]. A similar term exists with all the (up/down) ladder operators replaced by their (down/up) counterparts, but was not needed to fit the data.1 To obtain the form of the effective Hamiltonian HVTR described above, a series of contact transformations have been applied implicitly. A vibrational contact transformation has been used to eliminate the leading terms off-diagonal in the vibrational quantum numbers which are not considered explicitly [78,90]. A torsion–rotation contact transformation has been employed to eliminate terms such as [" J2α , 12 (1 − cos 3α)]+ [72,91]. Finally, a rotational contact transformation has led to a simplified form for HS [60,72]. For an XY3 AB3 symmetric top in lowest order, Fν = Aν /[ρν (1 − ρν )] and ρν = Iν,α /Iν,a , where Iν,α is the moment of inertia of the top about the symmetry axis in vibrational state ν, while Iν,a is the corresponding moment of inertia for the entire molecule. In the least squares analysis, ρν has been treated as a fitting parameter and Fν has been treated as a derived quantity [27,28,37]. While HG VTR as given in Table II fits all the data now available for CH3 SiH3 and C2 H6 to within experimental error, three important terms are absent from the ground state Hamiltonian Hgs . First, " Jα Jz does not appear for CH3 SiH3 because ρ is allowed to vary freely. This step eliminates not only the A0,F term in the zerothorder torsional Hamiltonian, but also the quartic and higher-order contributions in " HG VTR that can generate terms in Jα Jz . For C2 H6 , the corresponding operators are 1 In Eq. (1) of [22], q 12+ and q12− should be interchanged, as should p12+ and p12− . The corresponding typographical error occurs in the last two lines of Table 1 of [22]. This correction has been made here in Eq. (7).
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forbidden. Second, in CH3 SiH3 , F0 can differ in higher order from A0 /[ρ(1−ρ)]. In earlier work, the term F0,m" J2α was introduced to accommodate this difference and the magnitude of F0,m was the order of a quartic distortion constant [32]. However, in the more recent analyses [27,36], F0,m was found to be negligible. Similarly, in C2 H6 , the deviation of A0,R from A0 can be neglected [40]. Third, and perhaps most striking, the hindering potential V is represented as a pure sinusoid. Correction terms such as V0,6 12 (1 − cos 6α) in CH3 SiH3 [32] and V0,6 12 (1 − cos 12γ ) in C2 H6 [41] were used when two-band analyses were carried out, but became superfluous when the coupling to the parallel band νs was introduced explicitly. In Hgs , aside from the exceptions just noted, all the allowed quartic operators are included in Table II, but several of the sextic parameters have been fixed at zero. The particular selection of sextic parameters included in the analyses is somewhat arbitrary, as there are other selections that lead to fits of comparable quality. For a molecule with no torsional degrees of freedom, the order of each term Hl,m in the Hamiltonian is determined according to its degrees l and m in the vibrational and rotational operators, respectively [78]. When a molecule with large amplitude internal rotation is considered, the question arises as to whether the torsional motion should be classified as a vibration or a rotation. In the original work [37] on the resonant interactions between the gs and ν12 in CH3 SiH3 , the torsional normal co-ordinate q6 was treated as a vibration, and the leading (gs, νt ) xy,ν operator O47 in Table II was derived as the Coriolis interaction H2,1 12 of degree two in vibration and one in rotation (e.g., " Jα qt− and J+ , respectively). In O47 , the terms in " Jα and sin 3α are of the same order. The derivation followed a limiting procedure introduced in [26] and discussed with respect to CH3 SiH3 in Section III of [72]. In this procedure, the Coriolis coupling in the PAM was first derived in terms of q6 and its conjugate momentum p6 for the limiting case of small amplitude torsional motion. Then p6 was replaced by a suitable multiple of Jα and q6 was replaced by a suitable multiple of sin 3α; see Eqs. (9) of [37]. These multi(0) ples were selected so that the HT ,gs goes to its C3v limit as V3 → ∞. Moreover, xy,ν H2,1 12 has all the required symmetry properties. This approach has come to be known as the High Barrier (HB) Model [27,38], and has been used in all of the fits listed in Table I involving (gs, νt ) coupling. On the other hand, if the torsional motion is grouped with the degrees of freedom for overall rotation, the leading (gs, νt ) operator is of degree one in vibration and two in rotation, an (l, m) combination normally associated with centrifugal distortion [38,78]. From Eq. (2) of [38], it is seen that this interaction Hamiltonian αy,ν αy,ν H1,2 12 is identical in form to the " Jα term in O47 . For a derivation of H1,2 12 from the classical Hamiltonian, see [57]. This approach has come to be known as the Free Rotor (FR) Model [27,38]. In this model, an operator identical in form to the sin 3α term in O47 does arise, but only in higher order [57]. For CH3 SiH3 [27]
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and CH3 SiD3 [38], the data have been fit with both the FR and HB models. For each molecule, the higher order (gs, νt ) coupling terms were different in the two models, but the qualities of the fits were comparable. Of course, internal rotation has some of the characteristics of both vibration and overall rotation. The question remains open as to which model is better. There are theoretical arguments on both sides, and it is not clear whether the HB and FR models are distinguishable on experimental grounds alone.
2.3. BASIS F UNCTIONS , E IGENFUNCTIONS , AND E IGENVALUES The symmetry properties of the eigenstates of Hkν for an XY3 AB3 molecule are summarized in Table II of [37]. Recall that, for this Hamiltonian, k is a good quantum number. In the gs, for σ = 0 and/or k = 0, ±3n (n = 1, 2, 3, . . .), the torsion–rotation symmetry ΓTR = E1 , E2 , E3 , or E4 . For σ = 0 and k = 0, ΓTR = A1 if (J + vi ) is even, or A2 if (J + vi ) is odd. For σ = 0 and k = ±3n, ΓTR = A1 ⊕ A2 . For the gs and νs , the vibration–torsion–rotation symmetry ΓVTR = ΓTR since the vibrational symmetry ΓV in these cases is A1 . For νt , ΓV = E1 so that ΓVTR = E1 ⊗ ΓTR . The ΓVTR in νt for specific J , G, lt , k, and σ equals the ΓTR in the gs for k = G and the same J and σ . There is one exception: when σ = 0 and G = 0 in νt , ΓVTR = A1 ⊕ A2 , rather than A1 or A2 . In order to treat the effects of HS in the gs and of the l-doubling term qt O39 in νt , the symmetry labels A+ and A− are introduced following [92] and [60]. When this convention is adapted to the current problem, A+ = A1 /A2 and A− = A2 /A1 as (J + vi + vt ) is even/odd [57,60]. Here the gs levels with k = 0, σ = 0 are always of symmetry A+ . The eigenfunctions of Hkν can be written [36,57]: Ψ (vs ; vt , lt ; J, vi , k, σ ) = |vs |vt , lt |J, k|vi , k, σ ; 1 lt ;vi ,k,σ iα(3kf +σ ) |vi , k, σ = √ Akf e , 2π k
(8) (9)
f
where |vs and |vt , lt are the harmonic oscillator functions for the onedimensional νs and two-dimensional isotropic νt cases, respectively [78]. In Eq. (8), |J, k is the symmetric top rotational function [69]; the dependence on the magnetic quantum number mJ has been suppressed. In Eq. (9), the torsional wave function |vi , k, σ is expanded in terms of free rotor functions |σ ; kf = √1 eiα(3kf +σ ) where kf = 0, ±1, ±2, . . . is the free rotor quan2π tum number. The sum on kf runs from −kfmax to +kfmax , with the truncation limit kfmax being determined numerically. The Alktf;vi ,k,σ are real superposition constants l ;v ,k,σ
are functions of J , vs , and vt , but obtained by diagonalizing Hkν . The Aktf i this dependence is suppressed to simplify the notation. In each column vector
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Alktf;vi ,k,σ , the relative signs of the (2kfmax + 1) components are completely constrained, but the vector as a whole is determined only to within a common phase v factor lt i;k,σ which equals +1 or −1. Because all terms off-diagonal in k have been excluded, there are cases for Hkν where A+ and A− eigenstates occur in degenerate pairs. In the degenerate mode νt , such pairs occur for (σ = 0, G = 0) and for (σ = 0, G = ±3n). In these cases, a Wang transformation is applied to the eigenfunctions so that the analysis can accommodate the splittings produced when terms off-diagonal in k are subsequently taken into account [57]. Φ(vs ; vt ; J, vi , G, σ = 0; ΓVTR = A± ) 1 = √ |vs |vt , lt = +1|J, k|vi , k, 0 2 ± (−1)k−1 |vt , lt = −1|J, −k|vi , −k, 0 .
(10)
Since the values of (k − lt ) in the two wave-functions on the right-hand side are of opposite sign, G is no longer a good quantum number in these cases in general. To resolve this ambiguity, the convention is adopted that G on the left-hand side is taken to be the value for the (lt = +1) case and k on the right-hand side is G + 1. Equation (10) reduces to Eq. (25) of [37] for G = 0 i.e. for the l-doubling levels; A± here corresponds to τ = ±1 in [37]. In the non-degenerate modes (gs and νs ), only the (σ = 0, G = ±3n) case arises. The Wang transformation can be read from Eq. (10) by setting vt = lt = 0 and G = k = 3n. This result is equivalent to that given in Eq. (9) of [60]. In both the gs/νs and the νt versions, the transformation in Eq. (10) is an extension of its counterpart for a C3v molecule. In the present case, however, Φ(vs ; vt ; J, vi , G, 0; A± ) transforms by vi irreducible representation A± only if the phases lvt i;k,σ =0 and −l are propt ;−k,0 erly related. For the gs and νs , this can be arranged, for example, by applying the phase-locking procedure described in [61]; see also the Appendix of [60]. For νt , see the discussion in [57]. Note that Eq. (21a) of [37] and the associated phase implications are not valid, and have been corrected in these later works. Because of this inconsistency, the symmetries given for τ = ±1 in [37] do not match those given here for A± if vi is odd. The eigenvalues of Hkν (and of HVTR ) do not change if the signs of lt , k, and σ are changed simultaneously. Thus the full range of eigenvalues can be spanned by using only the positive values of k. This leads to the following convention. In the Ψ basis where ΓVTR = A± , states occur in degenerate pairs, e.g., (lt = ∓1, k = ±4, G = ±5, σ = ∓1) with k positive in one state and negative in the other. In such cases, the pair of states will be labelled with the set of quantum numbers for k positive, e.g., (lt = −1, k = +4, G = +5, σ = −1). The symbol k is retained, even though it is positive definite in this application. In the Φ basis where ΓVTR = A± , states can occur in degenerate pairs, but k is both positive
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and negative in each state. In cases of this type, the quantum number K ≡ |k| will be used. The degeneracy can be lifted and the symmetry label A± is used to distinguish the two states. While K is never negative, there are special cases where negative values of k are required, as for example in labelling a transition k = ±2 ← ∓1. The ambiguities that arise will be clarified by the context. This convention and similar ones have been widely used, although often the use has been implicit. For an X2 Y6 molecule, the symmetry properties of the eigenstates of Hkν are summarized in Fig. 1 of [86] and in Fig. 1 of [26]. In each figure, leaving the case k = 0 aside, the results apply to k modulo 6. The torsional symmetry ΓT is independent of k, while the rotational symmetry ΓR is independent of σ . In Fig. 1 of [86], gs values of ΓR , ΓT and ΓTR are given for k from 1 to 6, and for even and odd values of vi . The results are independent of J . In Fig. 1 of [26], ΓTR and ΓVTR are given for four different vibrational symmetries ΓV , including A1s (ν3 ) and E1d (ν9 ). The value of k runs from 0 to 6, but only even vi are considered. For k = 0, the results apply to J even. For J odd, all the symmetry labels for k = 0 must be multiplied by A2s . For the cases not explicitly treated in one or other of these two figures (e.g., gs with k = 0 and vi odd), the symmetries can be deduced from the figures and simple group theory. If the origin for γ is chosen at the staggered configuration, then the (σ = 3) symmetry labels for even and odd vi must be interchanged [86]. The eigenfunctions of Hkν can be written [40,41] as in Eqs. (8) and (9) provided that two changes are made. First, α is replaced by 2(γ + π/6). Second, the torsional function |vi , σ is independent of k, and the expansion coefficients are l ;v ,σ written Aktf i . There are again cases where the eigenstates occur in degenerate pairs. For ν9 , such pairs arise for G = 0 and for G = ±3n. For the gs and ν3 , they occur only for G = ±3n. For the pairs involved, ΓVTR is (A1s , A2s ), (A3s , A4s ), or (E3s , E4s ). In these cases, a Wang transformation similar to Eq. (10) is applied to |vt , lt = +1|J, k and |vt , lt = −1|J, −k to deal with the splittings that will occur in general when matrix elements off-diagonal in k are considered. The resulting eigenstates of Hkν are written Φ(vs ; vt ; J, vi , G, σ ; ΓVTR ). Since the torsional function |vi , σ are not affected by the transformation, the phases of the l ;v ,σ Aktf i do not enter the symmetry considerations. See [86] for further discussion of these transformations in the gs. To determine the eigenvalues and eigenfunctions of HG VTR in a (gs/νt ) problem for an XY3 AB3 molecule, the original two-step procedure [37] now called J 2 –vi first order can be used. In the first step, the torsional problem is solved. For the gs, a pseudo zeroth-order torsional Hamiltonian HT ,gs is defined which includes all the terms of Hgs in Table II that involve torsional operators except the torsional distortion terms such as −D0,J m J2" J2α that involve both torsional operators 2 and powers of J . This Hamiltonian is then diagonalized using a free-rotor basis
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|σ ; kf with a truncation limit kfmax that typically is 10. The torsional eigenfuncgs,v ,k,σ
tions obtained are of the form in Eq. (9), but the superposition constants Akf i do not depend on J . For νt , the corresponding calculation is done, but now the pseudo zeroth-order torsional Hamiltonian HT ,νt also excludes the terms in Hνt ν ,vi ,k,σ
involving t . In this case, the superposition constants Akft
do not depend on
J or lt . In the second step, the matrix for the (gs/νt ) part of HG VTR is set up in the Ψ (vt , lt ; J, vi , k, σ ) basis using these approximate torsional eigenfunctions for the gs and νt . The general features of this 17 × 17 matrix are illustrated in Fig. 2 of [37]. (In the first line of the figure, (l = −1) and (l = +1) should be interchanged.) The terms in Hν previously excluded are introduced at this stage, but the torsional distortion terms are inserted only on the diagonal; their matrix elements off-diagonal in vi are neglected. The resulting matrix for HG VTR is then diagonalized. If ΓVTR = A± , the eigenstates are labelled by (vt , lt ; J, vi , k, σ ) according to the quantum numbers of the dominant Ψ basis function. Similarly, if ΓVTR = A± , the eigenstates are labelled by (vt ; J, vi , G, σ = 0; ΓVTR ). Two other methods of diagonalizing HG VTR in such cases have been introduced. The original J 2 –vi first order procedure has the advantage in computer time that the torsional eigenfunctions are independent of J . The torsional Hamiltonian has to be diagonalized only once for each (vt , k, σ ) combination of interest [37]. However, the method has the disadvantage in accuracy that the torsional distortion terms are treated with first order perturbation theory. The J 2 –vi exact two-step procedure overcomes this difficulty [40,42]. In the second step, when the torsional distortion terms are introduced, the matrix elements off-diagonal in vi are taken into account. An equivalent direct diagonalization method has been used [36]. In this one-step procedure, the matrix for HG VTR is set up in the free rotor basis i.e. in the basis defined by Eq. (8), but with |vi , k, σ replaced by |σ ; kf . The eigenvalues obtained by the one-step and the J 2 –vi exact two-step methods are the same. However, the eigenfunctions are generated as linear combinations of |vt , lt |J, k|σ ; kf for the one-step, and as linear combinations of Ψ/Φ functions in the J 2 –vi exact two-step. Compare Eq. (26) of [36] with Eqs. (1) and (2) of [27]. The computer programming is simpler for the one-step, but the J 2 –vi exact two-step often offers more insight into the problem. The discussion here has been cast in terms appropriate to an XY3 AB3 molecule, but it applies equally well to the X2 Y6 case once allowance is made for the fact that the torsional eigenfunctions are independent of k. The improvement in accuracy over the J 2 –vi first order procedure was essential when the analysis was extended to (gs/νt /νs ) problems in CH3 SiH3 [36], C2 H6 [40], and CH3 CD3 [22]. For CH3 CF3 , splittings have been observed in the gs between A+ and A− levels for K = 3 that can be accounted for by introducing the matrix elements off-diagonal in k due HS [60]. Since the gs could be treated in isolation for the torsional levels involved, a two-step one-band method can be applied [61]. For
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ΓTR = A± , in the first of the two steps, only Hgs is considered and k is a good quantum number. For given J , k, σ , the matrix for Hgs is diagonalized using the free-rotor basis |J, k|σ ; kf . The eigenvectors Ψ (J, vi , k, σ ) are found in the form of Eqs. (8) and (9). Since all the torsional distortion terms are included, J,v ,k,σ this is a J 2 –vi exact procedure and the superposition constants Akf i are a function of J . In the second step, the matrix elements off-diagonal in k are taken into account. The matrix for (Hgs + HS ) is set up in the Ψ (J, vi , k, σ ) basis with the rows/columns labelled by k /k . The k-value of interest is placed at the center of the matrix, with k and k running from (k − 15) to (k + 15) in increments of 3. The A-coefficients obtained in the first step are used to calculate the required torsional matrix elements and overlap sums i.e. sums on kf over J,v ,k,σ
J,v ,k±3,σ
products Akf i Akf i as defined in Eq. (14) of [61]. After the matrix is diagonalized, the number of increments included in the k /k range is checked to see if sufficient accuracy has been attained. For ΓTR = A± , a similar two-step process can be used. The modifications required to take the change in symmetry into account are discussed in [60]. A similar J 2 –vi first order computer program has been developed [60] that runs the order of 10 times faster, but can lead to difficulties in the interpretation of the sextic parameters. The J 2 –vi exact twostep procedure has been extended to (gs/νs ) problems in CH3 CF3 ; see [56]. For CH3 CD3 , a strong local resonance has been observed in the ν5 band for the (k = 7) lines for all three values of σ [22]. The resonant shifts can be accounted for by introducing the matrix elements of H12,5 that couple the upper states in the transitions (v5 = 1; v12 = 0, l12 = 0; J, v6 = 0, k, σ ) to their interaction = 1, l = +1; J, v = 1, k = k − 2, σ ). The eigenvalpartners (v5 = 0; v12 12 6 ues and eigenvectors of HVTR can be calculated by extending the two-step J 2 –vi exact procedure used to diagonalize HG VTR for XY3 AB3 molecules [22]. In the second step of the original version of this procedure [40], each matrix HG VTR is labelled by its values for (J, k, σ ) in the gs, which equals (J, k, σ ) in ν5 , and equals (J, G, σ ) in ν12 . Each such matrix has 4 blocks across, one each for the gs, ν5 , ν12 with l12 = +1, and ν12 with l12 = −1. In the extended version [22], = +1; in this block, k = k − 2 and a fifth block is added labelled ν12 with l12 G = G − 3. The matrix elements for H12,5 are then entered in the appropriate positions, and the expanded matrix is diagonalized to give the final energies E(v5 ; v12 , l12 ; J, v6 , k, σ ). This procedure neglects several higher order effects such as the coupling to neighboring states within the (G = ±3) chain. However, these approximations are justified because the resonance is highly localized and the shifts for the resonant cases are relatively small (the order of 0.100 cm−1 ). Furthermore, no transitions to or from the perturbing partners enter the data set. The matrix elements for most of the terms in HVTR can be found in the original literature. For an XY3 AB3 molecule, Moazzen-Ahmadi et al. [37] deal with the leading terms in the (gs, νt ) part of HG VTR using the IAM form of the Ψ basis.
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Using the same basis, Ozier et al. [60] treat the various terms that can couple levels with k = ±3 and k = ±6. In the free rotor basis using the hybrid method, Schroderus et al. [36] deal with the torsional operators and the l-doubling Hamiltonian qt O39 as well as the leading (gs, νt ) and (gs, νs ) interactions. Wang et al. [56] give the matrix elements for a different selection of (gs, νs ) coupling terms. The phases lvt i;k,σ of the vectors formed by the superposition constants Alktf;vi ,k,σ are discussed in [60,57,61]. For an X2 Y6 molecule, Moazzen-Ahmadi et al. [41] give the matrix elements for the l-doubling Hamiltonian and the leading (gs, νt ) terms using the ethane version of the Ψ basis.
3. Spectroscopy from 50 kHz to 1000 cm−1 3.1. ROTATIONAL S PECTROSCOPY In order to discuss rotational spectroscopy in XY3 AB3 molecules, the eigenvalue problem is treated with a simplified model that explains many of the features observed. Only Hamiltonian terms diagonal in G, vt , and vs are taken into account. The Hamiltonian is treated in first order, and distortion terms no higher than quartic are included. The vibration–torsion–rotation energy for vibrational state ν can be written in the hybrid method as: E (1) (ν, J, vi , k, σ ) = Bν J (J + 1) + (Aν − Bν )k 2 − Dν,J J 2 (J + 1)2 − Dν,J K J (J + 1)k 2 − Dν,K k 4 + ν − 2Aζtz − ηt,J J (J + 1) − ηt,K k 2 klt (0) Jα ν,v ,k,σ + Eld + ET (ν, vi , k, σ ) + 2Aζtm lt " i + Fν,3J J (J + 1) + Fν,3K k 2 12 (1 − cos 3α) ν,v ,k,σ i 4 2 "2 Jα ν,v ,k,σ − Dν,J m J (J + 1) + Dν,Km k Jα ν,v ,k,σ − Dν,m " i i 3 2 " − Dν,sJ J (J + 1) + Dν,sK k k Jα ν,v ,k,σ − Dν,sm k " Jα ν,v ,k,σ . (11) i
E (1) (ν, J, v
i
The dependence of i , k, σ ) on ΓVTR as well as on G and lt is suppressed to simplify the notation. The third line and the first two terms of the fourth line apply in full only for νt ; only the vibrational quantum ν is included for νs and all these terms are omitted for the gs. In lowest order, the l-doubling term (0) Eld enters only for G = 0. The zeroth-order torsional energy ET (ν, vi , k, σ ) is (0) the eigenvalue of the Hamiltonian HT ,ν defined in analogy with Eq. (4), and the corresponding zeroth-order eigenfunctions are used to calculate the expectation values Oi ν,vi ,k,σ of the torsional operators. These matrix elements either are
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invariant or change sign under the operation (k, σ ) → (−k, −σ ); the operators 1 "2n "2n−1 is odd [30]. E (1) (ν, J, vi , k, σ ) can 2 (1 − cos 3nα) and Jα are even, while Jα be written in terms of the effective rotational constants: !ν (vi , k, σ ) = Aν + Fν,3K 1 (1 − cos 3α) A 2 ν,vi ,k,σ 2 Jα ν,v ,k,σ ; − Dν,Km " (12) i 1 !ν (vi , k, σ ) = Bν + Fν,3J (1 − cos 3α) B 2 ν,vi ,k,σ 2 " − Dν,J m Jα ν,v ,k,σ . (13) i
For the molecules of interest here, the reduced barrier height s0 ≡ 4V0,3 /9F0 is moderate (e.g., 32.6 in CH3 SiH3 ) or large (e.g., 91.0 in CH3 CF3 ). In such cases, for low vi , the magnitude of Oodd ν,vi ,k,σ is very small and Oeven ν,vi ,k,σ has a weak dependence on (k, σ ). For given ν, vi , and k, the unweighted average of Oodd ν,vi ,k,σ over σ vanishes, while the corresponding average of Oeven ν,vi ,k,σ !ν (vi , k, σ ) is independent of k [30]. Consequently, the unweighted averages of A !ν (vi , k, σ ) over σ are also independent of k; these are here indicated Aν (vi ) and B and B ν (vi ), respectively. The first method for using frequency measurements to determine the barrier height in a symmetric top was introduced in 1954 by Kivelson, who considered the gs problem [93]. The expression for the effective B-value obtained is equiv!0 (vi , k, σ ). The electric dipole selection rules for alent to that in Eq. (13) for B rotational transitions in the gs are J = 0, ±1; vi = 0; k = 0; σ = 0. (Only the Ψ basis of Eq. (8) is considered here.) Because ET(0) (ν, vi , k, σ ) does not depend on J , the normal microwave spectrum is not affected by the zeroth (1) order torsional splittings. The transition frequencies νgs (J, vi , k, σ ) for the microwave R-branch are then given by the standard expression for a C3v molecule !0 (vi , k, σ ) is to be used instead of B0 . The Kivelson torsional [69] except that B satellite method of determining V0,3 is based on the torsional splittings produced !0 as well as by the contribution from D0,sJ . by the (vi , k, σ )-dependence of B Because the microwave R-branch is sensitive to V0,3 only through its effect on the expectation values, the results are far less accurate than those obtained for asymmetric rotors, where the spectrum is directly sensitive to the zeroth order splittings because of the less stringent electric dipole selection rules that apply in that case [1,69]. Furthermore, when this method is used alone, F0 and ρ must be calculated from the structure. Nonetheless, the torsional satellite method was a major advance and can provide a great deal of valuable information about internal rotation in symmetric tops, particularly when it is combined with other techniques. The torsional splittings in the R-branch increase rapidly with vi . In the gs of CH3 SiH3 , the torsional splittings for vi = 0 are too small to be resolved with conventional absorption microwave spectroscopy, but they can be measured with spe-
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cial techniques. For k = 1, the (J = 4 ← 3) transition consists of three lines. The (σ = 0) component is 43 kHz below its (σ = +1) partner, which in turn is 75 kHz below the (σ = −1) component, with the respective spin statistical weights g [39] being in the ratio 2 : 2 : 1. Using sub-Doppler saturation spectroscopy at the Universität zu Köln, in Köln, Germany, Pelz et al. [31] have resolved this triplet and the corresponding triplet for k = 2. The (J = 1 ← 0) gs transition consists of a doublet because the (σ = +1) and (σ = −1) levels are degenerate for k = 0. Using the best current model, the splitting for vi = 0 is calculated to be 25 kHz [27]. This (vi = 0) gs transition has been investigated in CH3 SiH3 with molecular beam methods at the University of Nijmegen (U.N.) in Nijmegen, The Netherlands, and a partially resolved doublet was observed with a splitting of about 28 kHz [28]. The same spectrum was subsequently re-measured with a Fourier transform microwave (FTMW) cavity spectrometer at the Eidgenössische Technische Hochschule (E.T.H.) in Zürich, Switzerland [27]. In this case, the fitting of the time domain signal yielded a doublet with a splitting of about 35 kHz. The (σ = 0) component was clearly broader than its (σ = ±1) partner and the ratio of the peak intensities (former:latter) was clearly less than unity, whereas a ratio of 2 is expected from the spin statistical weights. The observations were explained qualitatively in terms of the nuclear hyperfine interactions [27]. For vi > 0, the torsional splittings have been resolved in CH3 SiH3 using a variety of experimental techniques, including FTMW waveguide spectroscopy at 22 GHz (J = 1 ← 0) (E.T.H.) [27]; conventional mm-wave Stark modulation near 100 GHz (J = 5 ← 4) at the University of British Columbia (U.B.C.) [30]; saturation modulation mm-wave methods at 285 GHz (J = 13 ← 12) (Freie Universität, Berlin, Germany) [30]; and tunable sideband far-infrared spectroscopy near 1 THz (J = 45 ← 44) (U.N.) [27]. These splittings have also been measured for vi > 0 in CH3 SiD3 , CH3 CD3 , CH3 CF3 , and CH3 SiF3 . See Table I for the citations to the original works in which the data involved in the corresponding global analyses have been obtained. The requirements on the spectrometer sensitivity were particularly stringent in the case of CH3 CD3 [48] because the permanent dipole moment μ0 is only 0.010 861 7(5) D [44]. In the rotational spectrum for νt , the l-doubling can introduce some interesting new effects [37,39,94]. The l-doubling operator qt O39 in Table II has non-zero matrix elements between wave-functions Ψ (vs ; vt = 1, lt = 1; J, vi , k = 1, σ ) and Ψ (vs ; vt = 1, lt = −1; J, vi , k = −1, σ ). With qt O39 turned off, the corresponding energy levels are degenerate for σ = 0 (ΓVTR = A± ) because the energy does not change when lt , k and σ change sign simultaneously. Then for moderate (or high) values of sνt , Eld in Eq. (11) equals ± 12 qt J (J + 1) for ΓVTR = A± . The situation is more complicated for the two cases σ = +1 and σ = −1 (ΓVTR = E4 ). In each such case, with qt O39 turned off, the energies of the states coupled by qt O39 are not equal. As can be seen from Eqs. (31) of [37], the energy difference (v6 , σ ) between the two coupled levels with qt O39 turned off is given
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in the simplified model by the sum of two terms. When these are re-written using the symmetry properties of the matrix elements, the first term is seen to arise from the σ -splitting in the zeroth-order torsional energies between (k = +1, σ = +1) and (+1, −1). The second term is generated by the difference in the expectation values of 2Aζtm lt" Jα between (+1, +1) and (+1, −1). (Note that the sign between the two terms in the second bracket of Eq. (31b) of [37] has been corrected here from + to −.) The energies for l-doublet levels with ΓVTR = E4 can be calculated by diagonalizing a 2 × 2 matrix whose diagonal elements differ by (v6 , σ ) and whose off-diagonal element is 12 qt J (J + 1). However, the first order treatment of the term 2Aζtm lt" Jα can lead to substantial Jα errors in the value of (v6 , σ ) [37]. The magnitude of the matrix elements of " with vi = ±1 is large (∼1), and Aζtm in CH3 SiH3 , for example, is ∼3 cm−1 . Jα must be moved into the zeroth-order torsional Hamiltonian The term 2Aζtm lt" or treated by an equivalent procedure (such as those discussed in Section 2.3) in order to obtain an accurate value of (v6 , σ ) [37]. This corrected value will here be indicated by C (v6 , σ ). When qt O39 is turned on, the energy separation between the E4 l-doubling levels will be significantly different from that of their A± counterparts if |C (v6 , σ )| | 12 qt J (J + 1)|. However, for the cases reported to date in CH3 SiH3 , CH3 SiD3 , CH3 CF3 and CH3 SiF3 , |C (v6 , σ )| is negligible. A Wang transformation analogous to that in Eq. (10) can then be carried out, and the resulting functions Φ can be assigned pseudo-symmetry labels E4+ or E4− according to whether their associated energies match more closely their A+ or A− counterparts [57]. In the R-branch, the selection rules on ΓVTR are A± ← A± for σ = 0 and E4± ← E4± for σ = ±1. For positive qt , the high frequency l-doublet in the R-branch consists of two lines, one for σ = 0 (A+ ) and one for σ = ±1 (E4+ ); the same is true for the low frequency l-doublet, except the symmetries are A− and E4− , respectively. Within the simplified model, the σ -splittings on the two sides are equal to each other and are small for low vi , being primarily due to !ν (vi , k, σ ). Such torsional splittings that are explained by the simplified model B are termed intrinsic because they arise from Hν , the part of the Hamiltonian diagonal in the vibrational quantum numbers. (0) (0) As vi increases, ET (gs, vi , k, σ ) becomes the order of [ν t +ET (νt , 0, k, σ )], and near-resonant/resonant interactions due to Coriolis-like (gs, νt ) coupling can become important. When this occurs, the σ -splittings can become much larger than the corresponding intrinsic values and can be a very sensitive probe of the coupling mechanism. The direct l-doubling spectrum in νt for a given J consists of a two lines obeying the selection rules J = 0; vi = 0; G = 0 ← 0; σ = 0; ΓVTR = A+ ← A− for σ = 0 and ΓVTR = E4+ ← E4− for σ = ±1. In the simplified model, the intrinsic σ -splitting is very small. This spectrum has been investigated in CH3 SiH3 using the FTMW waveguide spectrometer at E.T.H. Lines for J from 17 to 26 were measured between 8 and
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18 GHz to an experimental uncertainty ∼ 10 kHz [27]. The σ -splitting at J = 22 was just over 1 GHz, largely due to the Coriolis coupling between (vt = 1; vi = 0) and (vt = 0; vi = 3). For the coupling between these levels, because of the Wang transformation, the Coriolis-like matrix element is increased √ by a factor of 2 for ΓVTR = A+ and E4+ , and vanishes for ΓVTR = A− and E4− [37]. Thus the σ -splitting between the A and E4 frequencies is dominated by the energy difference between the A+ and E4+ states. In the R-branch spectrum of CH3 SiH3 , near-resonant and resonant effects have been observed for both (vt = 0; vi = 3) and (vt = 1; vi = 0) [32]. For example, the frequency νgs (J = 6, vi = 3, k = 6, σ = −1) is shifted by over 600 MHz by the resonant coupling between (vt = 0, lt = 0; J, vi = 3, k = 6, σ = −1) and (vt = 1, lt = −1; J, vi = 0, k = 5, σ = −1). Similarly large shifts have been observed for the frequency ννt ,lt =−1 (J = 6, vi = 0, k = 5, σ = −1). These lines have been measured for J = 6, 7 and 8 to an accuracy 50 kHz using tone burst modulation at the Jet Propulsion Laboratory in Pasadena, California [32]. Q-branch transitions between the interacting partners form a torsion–vibration difference band. With the FTMW waveguide spectrometer at E.T.H., this band has been measured for J from 6 to 19. The frequencies fall between 17.8 and 26.6 GHz and were determined with ∼10 kHz [27]. Because of the resonant mixing, these lines are weaker than the l-doubling lines for the same value of J by only a factor of ∼4. In C2 H6 , the permanent dipole moment μ0 vanishes. Furthermore, C2 H6 has a center of symmetry in its equilibrium configuration and so does not have a distortion dipole moment perpendicular to the symmetry axis [71]. However, because of the torsional motion, the molecule does not have an instantaneous center of symmetry, and pure rotational transitions are not strictly forbidden [95]. As can be seen from the discussion in Section 3.3.3, the R-branch is allowed for σ = 1 and 2 through intensity borrowing from the E1d and A4s infrared active vibrational bands. The transition moment matrix element is proportional to |v4 , σ |Jγ |v4 , σ |, which is very small for low values of v4 . At room temperature, the strongest lines with frequencies below 360 GHz occur for v4 = 3 or 4 and J 8 [76]. For such lines with integrated intensities above 10−11 cm−2 amagat−1 , a table of predicted frequencies and intensities is given in Table VII of [76]. The intensity calculations used (μ − μ⊥ ) = −31.7(2.0) μD, as determined from the intensity of the torsional fundamental [76]; see Section 3.3.3. The estimated accuracy of the frequency predictions for v4 = 3 was 30 MHz based on the data available in 1992. Today, these predictions can be made to considerably higher accuracy. For an XY3 AB3 molecule, the gs rotational spectrum driven by the small distortion dipole moment μD perpendicular to the z-axis offers the opportunity !0 (vi , k, σ ) to make direct measurements of the effective A-rotational constant A (0) and the zeroth-order torsional energy ET (gs, vi , k, σ ), as well as to study the
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(A+ − A− ) splittings [60]. The value of μD is determined primarily by centrifugal distortion effects, but can receive a torsional distortion contribution as well; see Eq. (17) of [72]. In the distortion moment Q-branch, the selection rules are (k = ±3; σ = 0) for ΓTR = E1 , E2 , E3 or E4 ; and (k = ±3; σ = 0; A± ↔ A∓ ) for ΓTR = A± . For given J and k = ±3n with σ = 0, the two A± levels are degenerate in the simplified model of Eq. (11). When the splitting Hamiltonian HS in Eq. (6) is introduced to account for the (A+ − A− ) splittings, the degeneracy is lifted. For given J and k = ±3n with σ = +1, the two levels coupled by HS with ΓTR = E4 are not degenerate. However, if the energy difference is negligible, one can make a Wang transformation from the Ψ basis of Eq. (8) to the Φ basis of the gs counterpart of Eq. (10). One Φ function is given a pseudo-symmetry labels E4+ and the other E4− , according to whether the associated energies follow more closely A+ or A− . The same can be done for the corresponding states with σ = −1. For these cases, the ΓTR selection rule is E4± ↔ E4∓ . In the R- and P -branch, the selection rule on ΓTR is A± ↔ A± for σ = 0 and E4± ↔ E4± for σ = ±1. The energy level pattern and selection rules are analogous to those for the l-doubling states in νt . For CH3 CF3 in the gs, the Q-branch of this distortion moment spectrum has been studied from 8 to 18 GHz with the FTMW waveguide spectrometer at E.T.H. [60]. For vi = 0, six (k + 3 ← k) series were measured with 3 k 8 and 27 J 75. Because s0 is so large, the σ -splitting was only ∼100 kHz and could only be resolved for a few cases under optimal spectrometer performance. For vi = 1, three (k + 3 ← k) series were measured with 5 k 7 and 50 J 67. The J -value in the original (vi = 1) assignment has been increased by one in accordance with the correction made in [61]. In this higher torsional state, the σ -triplet spanned ∼6.8 MHz and all three components were measured for k = 5 and 7. Effective values for V0,3 and ρ were obtained, and determinations were made of Aν (0) and Aν (1). In the (vi = 0) series (K = 6 ← 3), the (A+ − A− ) splitting was measured for 37 J 74. Two different operators can account for these splittings, one involving J3+ + J3− and the other involving J6+ + J6− . In general, there are three options available for treating these operators. One or the other can be eliminated by a contact transformation, or both can be retained [92,96]. The different options are investigated for CH3 CF3 in [60]. In the current review (see Eq. (6)), only the operator involving J3+ + J3− is retained, corresponding to Model F 1 of Ozier et al. [60] and to Reduction B in the notation of Sarka [96]. In a similar study of CH3 SiF3 , the distortion moment Qbranch for vi = 0 was measured from 11 to 17 GHz [66]. Because s0 is much lower, the σ -splitting was ∼10 MHz and was easily resolved. No series involving K = 3 was measured and so no (A+ − A− ) splitting was observed. However, the (A+ −A− ) splitting has been observed in the (vi = 0) R-branch for CH3 SiF3 [67] as well as CH3 CF3 [59] at the Université de Lille, in Lille, France; no torsional effects were reported.
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3.2. T HE AVOIDED C ROSSING M OLECULAR B EAM M ETHOD In the avoided-crossing molecular-beam method for polar symmetric tops [70,97], the normal electric dipole selection rules k = 0 and σ = 0 are broken and precision measurements can be made of the terms in the energy which depend only on (vi , k, σ ) [28,73]. In favourable cases for G18 molecules, the method yields effective values for A0 , V0,3 and ρ; none of these constants can be determined directly by conventional microwave spectroscopy. The energy difference measurements are confined to the lower rotational states of (vi = 0) in the gs because the effective torsion–rotation temperature in the molecular beam is typically ∼6 K. Although the full theory described in Section 2 has been used to analyze the data, the simplified model is sufficient when the anticrossing data is treated in isolation and Eq. (11) will be used for discussion purposes. The major features of the anticrossing method can be described in terms of a simple two-level problem. A full discussion is given in [97]; see Fig. 1, in particular. Consider two states |α and |β in field free space with kα = kβ . The respective energies are Eα and Eβ , where Eα > Eβ . States |α and |β have negative and positive Stark effects, respectively. A static homogeneous external electric field E is applied along the space fixed Z axis. In the presence of the electric field, the eigenstates are |a and |b with eigenvalues Ea > Eb . As E is increased from zero, the two levels initially approach one another. The crossing field EC is eventually reached at which the difference ST in the Stark energies due to the dipole moment along the symmetry axis and the anisotropy in the polarizability cancels the difference 0 ≡ (Eα − Eβ ) in the zero-field energies. At EC , the two levels would cross if they did not interact. However, if appropriate selection rules are satisfied, |α and |β will have a non-zero matrix element η and will undergo an avoided crossing. At EC , the two levels will then have a minimum separation νm = 2|η|. As E is increased further, the two levels separate and, as E → ∞, the upper eigenstate |a → |β, while the lower eigenstate |b → |α. The physical properties of the two states have been interchanged; for example, the upper state now has the positive Stark effect. Each anticrossing is traditionally identified by the quantum numbers for |α and |β appropriate for values of E far enough below EC that the (α, β) mixing is negligible, but still large enough that the normal Stark effect decouples J and the nuclear spins from one another (for all but a few exceptional cases) [97]. With the static field E set near EC , an oscillating electric field of amplitude ERF is applied along E; the frequency in these experiments was usually in the range from 50 kHz to 5 MHz. The transition operator V acts through the normal dipole operator μ and the mixing between |α and |β. An expression for the transition matrix element |a|V|b| is given in terms of η and the relevant quantum numbers by Eq. (12) of [97]. Provided that |η| is large enough [58,85], the transition probability can easily be made unity by increasing ERF . For the transitions |a ↔ |b to be detected, the states |α and |β must meet the focussing requirements [98] of
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the electric-resonance molecular beam apparatus employed [99,100]. In practice, these requirements are met if |α and/or |β has a linear Stark effect [97], provided always that |α’s Stark effect is negative and |β’s is positive. Typically, EC is measured by observing the spectrum with E set first below and then above EC . Most often, the minimum separation νm was too small to be measured directly. For each molecule studied, a careful investigation of the normal Stark effect is required so that the measurements of crossing fields can be converted to zero-field splittings i.e. to the frequencies of the corresponding forbidden transitions. The effective dipole moment function to be used in calculating ST is discussed in Section 3.3.1 below, and applied to the case of CH3 SiD3 in Section II.B of [29]. The procedure used in calibrating the electric field in given in [101], and the determination of the experimental error in 0 is discussed in [97]. In the ideal case, /0 ∼20 ppm, as determined by the long term stability and resetability of the primary voltage source. However, /0 ∼60 ppm is more usual. For a given value of μ0 , the largest zero field splitting that can be measured is ultimately set by the difficulty in generating large electric fields of the required homogeneity [97]. Fields ∼11 kV/cm have been generated with inhomogeneities ∼10 ppm [101]. The current limit on 0 /μ0 is ∼5 GHz/D [101]. There are several different types of avoided crossings. For rotational Stark anticrossings, η arises from the interaction of the distortion dipole moment μD with E. The selection rules are J = 0, ±1; k = ±3; σ = 0; mJ = 0 [58,64]. As a result of the focussing requirements, these anticrossings can be observed only for (kα = ±2) ↔ (kβ = ∓1) and (kα = ±3) ↔ (kβ = 0). An expression for η in terms of J , k and mJ is given in Eq. (13) of [97]. For rotational hyperfine anticrossings, η arises from the hyperfine Hamiltonian Hhyp produced, for example, by the interaction of the nuclear spins with the internal molecular magnetic fields. The selection rules are J = 0, ±1, ±2; k = ±1, ±2; σ = 0, ±1, ±2; mJ = 0, ±1, ±2 [28,58,64]. For this set of selection rules, the usual convention of upper signs going with upper and lower with lower does not apply. In [85], a discussion of Hhyp is presented with emphasis on the different dipolar interactions possible. For both types of rotational anticrossings, |k| = 0 and the energy of rigid rotation contributes to 0 . For barrier hyperfine anticrossings, η again arises from the hyperfine Hamiltonian Hhyp , but |k| = 0 and the energy of rigid rotation does not contribute to 0 . The selection rules are (J, k, σ, ΓTR ) = (J, ±1, ∓1, E3 ) ↔ (J, ∓1, ∓1, E2 ); (J, ±1, ∓1, E3 ) ↔ (J, ∓1, 0, E1 ); (J, ±1, ±1, E2 ) ↔ (J, ∓1, 0, E1 ), where upper signs go with upper and lower with lower [28,85]. For all three cases, mJ = 0, ±1, ∓1, ±2, ∓2. This type of anticrossing only occurs for k = ±1 ↔ ∓1. For all avoided crossings, of course, mF = 0, where mF is the magnetic quantum number for the total angular momentum F (including the nuclear spin). The major features of a typical energy level system probed by the avoided crossing method are illustrated in Fig. 3, which is drawn for CH3 SiF3 . This is
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F IG . 3. Schematic plot of the energy against electric field for CH3 SiF3 in the gs with v6 = 0. The values of (J, k) = (2, ±2) and (2, ∓1), while mJ = ±2 for all the levels except the two with the smallest intercepts on the horizontal axis, in which cases mJ = ∓2. In the labelling of the levels, upper signs go with upper and lower with lower. The solid dots and open circles, respectively, indicate the rotational and barrier anticrossings. The anticrossing selection rules are upper/lower ↔ upper/lower at the solid dots, and upper/lower ↔ lower/upper at the open circles. For clarity, the σ -splitting has been enlarged relative to the k-splitting.
adapted from Fig. 1 in [64]. Each level shown has mJ = ±2 (or ∓2). The zerofield energies shown form two σ -triplets, the upper one with (J = 2, k = ±2, σ )
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and the lower one with (J = 2, k = ∓1, σ ). The separation between one triplet and the other is ∼ 3(A0 − B0 ), while the structure within each triplet is primarily (0) due to the zeroth order torsional Hamiltonian HT ,gs . Each energy level is doubly degenerate due to the invariance of the Stark-torsion–rotation Hamiltonian under the operation (k, σ, mJ ) → (−k, −σ, −mJ ). This sign-doubling is characteristic of the linear Stark effect. The solid dots indicate rotational Stark anticrossings, while the open circles indicated barrier hyperfine anticrossings. In each avoided crossing, the matching of upper/lower signs for |α with those for |β is dictated by the fact that mJ = 0; the alternative of mJ = ±4 is not allowed. All six avoided crossings have been observed except that involving ΓTR = E2 ↔ E1 . In spite of several attempts, this particular anticrossing could not be observed in initial studies of several molecules, and was thought for some time to be forbidden by symmetry. In due course, it was shown that |η| is very small but not zero, and an (E2 ↔ E1 ) avoided crossing was subsequently observed in CH3 SiH3 [85]. In Fig. 3, for the (E3 ↔ E1 ) barrier anticrossing, the frequency of the corresponding forbidden zero-field transition (i.e. 0 ) is written νEA , where the subscript indicates the change in ΓT from E (σ = ±1) to A (σ = 0) [28,64]. For the (E3 ↔ E2 ) barrier anticrossing, the frequency is written νEE because ΓT = E for both |α and |β [28,64]. The rotational anticrossings yield direct in(0) formation on both (A0 − B0 ) and ET (gs, vi = 0, k, σ ). The barrier anticrossings (0) yield direct information only on ET (gs, vi = 0, k = ±1, σ ), but can be detected for molecules where the rotational constants are so large that EC for the rotational anticrossings is too high to be reached. CH3 CD3 is just such a case [44]. The information content in νEA and νEE is summarized in Eqs. (13) of [28]. To excellent approximation, ρ is a function only of the ratio νEA /νEE . Thus the value obtained for ρ is very insensitive to the dipole parameters that characterize ST and the correlations with the other torsion–rotation parameters are minimal. Each level shown in Fig. 3 is actually a family of levels with different values of the nuclear spin quantum numbers. For each XY3 AB3 molecule studied, nuclei A and B have zero spin. The total nuclear spin for the three Y nuclei is IY and for the three B nuclei is IB . The corresponding magnetic quantum numbers are mY and mB . Of course, mF = mY + mB + mJ . In almost all cases, the anticrossings can be treated as a series of two-level problems [28,97]. Typically, the observed spectrum for a transition |a ↔ |b near the crossing field EC consists of a single envelope with all the hyperfine structure blended together. The centre of this line can be shifted from the hyperfine-free value and a hyperfine shift δνhyp can be produced in the zero field splittings 0 obtained. In the original works on the barrier anticrossings [28,44,64], it was argued that δνhyp = 0 in these cases. However, this argument was subsequently shown to be incorrect and the situation was re-assessed in 1998. In the Appendix of [29], |δνhyp | was estimated for each case studied to that time; the largest values were 7 kHz. The overall effect on the molecular parameters was small, but the error limits on the various measurements
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of 0 were adjusted in subsequent reports. For the rotational Stark anticrossings, the centre of gravity will not be shifted in first order [29,97]. In CH3 SiD3 , it was estimated for this case that |δνhyp | 8 kHz [29]. For the rotational hyperfine avoided crossings, the centre of gravity will be shifted, but the shifts tend to be smaller when mJ = 0. In CH3 SiH3 , it was estimated for this case that |δνhyp | ∼ a few kHz [28]. Avoided crossing studies have been carried out at U.N. for all five polar molecules listed in Table I. For CH3 28 SiH3 [28], the twelve possible rotational hyperfine anticrossings (J = 1; k = 1; σ = 0, +1, −1; ΓTR = E1 , E2 , E3 ; mJ = 1) ↔ (2; 0; 0, ±1; A1 , E4 ; 0, 1) have been observed; see Fig. 1 of [28]. The EA and EE barrier anticrossings have been measured for J 6. The isotope shifts to CH3 30 SiH3 has been determined for one of the rotational avoided crossings, and three of the barrier anticrossings. For CH3 SiD3 [29], the three possible rotational Stark avoided crossings (J = 4; k = −1; σ ; ΓTR ; mJ = −3) ↔ (3; 2; σ ; ΓTR ; −3) have been measured; see Fig. 1 of [29]. At E = F W of 42 kHz was observed 13900.5 V/cm, a full width at half maximum νH M with ST = 3280.1 MHz, for a fractional homogeneity of ∼13 ppm. The corresponding time-of-flight line width was 12.5 kHz. For J = 1 and 2, EA and EE barrier anticrossings have been measured. For CH3 CD3 [44], these same four barrier anticrossings were observed. In each case, the transition |a ↔ |b could be followed right though the crossing and νm could be determined. It appears that the hydrogen-hydrogen dipolar interaction provides the mixing for the EA cases, while the deuterium quadrupole interaction drives the EE cases. For CH3 CF3 [58], a series of rotational Stark and hyperfine rotational anticrossings were studied. For (J = 6; k = 2; σ ; ΓTR ; mJ = 6) ↔ (6; −1; σ ; ΓTR ; 6), νm was measured for each value of σ . To within the experimental uncertainty, all three values were equal. From the mean of 114.31(38) kHz, it was determined that μD = 3.220(11) μD. For CH3 SiF3 [64], all three types of anticrossings were investigated: rotational Stark, rotational hyperfine, and barrier. It was found that μD = 2.13(57) μD in this case. The hyperfine anticrossings, both rotational and barrier, have some features that appear very strange in the context of vibration–torsion–rotation spectroscopy. When α|Hhyp |β provides the mixing, ΓVTR,α = ΓVTR,β ; see, for example, the open circles in Fig. 3. In traditional microwave and infrared spectroscopy, matrix elements of this type are strictly forbidden. After all, any term in HVTR has symmetry A1 and so ΓVTR,α = ΓVTR,β [24]. However, once the nuclear spin degrees of freedom are introduced, the situation changes. If ΓN is the symmetry of the nuclear spin part of the overall eigenfunction, ΓVTRN = ΓVTR ⊗ ΓN . In this case, α|Hhyp |β can be non-zero for ΓVTR,α = ΓVTR,β provided that ΓN,α and ΓN,β are chosen so that ΓVTRN,α = ΓVTRN,β . As a result, one can have mixing terms which are off-diagonal in IY and/or IB . For CH3 SiF3 , the E3 ↔ E1 barrier anticrossing illustrated in Fig. 3 follows the selection rule (IY = 12 , IB = 12 ) ↔ ( 12 , 32 ),
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while the E3 ↔ E2 barrier anticrossing follows (IY = 12 , IB = 12 ) ↔ ( 32 , 12 ); see Table XIII of [85]. In such cases, the avoided crossing spectrum involves ortho-para transitions. The possibility of such mixing was first pointed with application to CH4 in the molecular beam literature by Anderson and Ramsey [102], and first observed experimentally in the ortho-para spectrum of CH4 [103]. Of course, the mixing due to α|Hhyp |β is present in zero-field as well. However, when states |α and |β are over 100 MHz apart, off-diagonal matrix elements |α|Hhyp |β| 100 kHz have little effect for the isolated molecule. On the other hand, when the levels are only 500 kHz apart, as is the case in a typical anticrossing experiment, these matrix elements can lead to strong transitions between the mixed levels.
3.3. T ORSION –ROTATION S PECTROSCOPY 3.3.1. Torsional Dependence of the Electric Dipole Moment for XY3 AB3 Unlike the case for asymmetric rotors such as methyl alcohol, the torsional bands in XY3 AB3 symmetric tops are very weak, being driven by higher dipole constants that arise from intervibrational mixing. In spite of the “forbidden” nature of these spectra, a great deal of information on the hindering potential and the mixing of the torsional mode with the skeletal vibrations has been obtained by measuring the frequencies and intensities of the torsional transitions. The torsional operators in the dipole moment not only determine the strength of these spectra, but also produce important changes in the Stark effect. By investigating both the intensities and the Stark effect, an unexpected link between the two was discovered, which will be discussed below. For the gs, the dipole operator μ along the symmetry axis can be expanded as a Fourier series in 3α and a power series in " Jα and the components of J. The leading terms that involve torsional operators have molecule-fixed components μTz and μT±1 in the spherical basis. When Eqs. (3a) and (3b) of [75] are converted to the hybrid method, 1 Jα J z J2α + μT + 2ρμT2 " μTz = μT0 (1 − cos 3α) + μT2 " 2 + ρμT + ρ 2 μT2 J2z ; μT±1
1 1 = μT⊥ " Jα J±1 + J±1" Jα + ρμT⊥ [Jz J±1 + J±1 Jz ]. 2 2
(14) (15)
In the PAM, the leading even torsional operators that appear in the expansion are characterized by μT0 and μT2 , while the corresponding odd operators are characterized by μT⊥ and μT [72]. The even parameters μT0 and μT2 are given in Eqs. (27a)
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and (27b), respectively, of [72].2 The dipole constant μT0 consists of two terms, one produced by Fermi coupling to the A1 vibrations and a second proportional to ∂ 2 μz /∂α 2 that does not arise from intervibrational mixing. However, in CH3 SiH3 , the Fermi mixing contribution is dominant because μz is only very weakly dependent on α [75]. The dipole constant μT2 is produced entirely by Fermi coupling to the A1 vibrations. The intervibrational mixing mechanisms giving rise to the odd parameters μT⊥ and μT are discussed in Section 3.3.3. The leading rotational terms in the expansion of μ can be expressed in terms jk of the centrifugal distortion coefficients Θi introduced by Watson and cojk workers [71,104] for C3v molecules. By using Eqs. (3) of [77], the Θi can be R related to the three parameters μQ J , μJ , and μK that characterize the dependence of the dipole matrix elements on rotational state in the limit that the torsional Q contributions to μ vanish. Note that μJ here is denoted μJ in [77]. The matrix elements of μ can be obtained from the corresponding expressions for a C3v molecule [69] by replacing the usual moment μ0 with an effective dipole moment function [75]. The selection rules are J = 0, ±1; k = 0; σ = 0; vi arbitrary. For J = +1 and 0, the dipole functions are denoted μR and μQ , respectively, where [75]: 2 μR (J, vi , k, σ, vi ) = " μ 0 + μR μK k 2 δ(vi , 0) J (J + 1) + " + μT0 12 (1 − cos 3α) v ,v i i T T "2 Jα v ,v ; + μ2 Jα v ,v + " μ − μT⊥ k " (16) i i i i Q μ0 + " μJ J (J + 1) + " μK k 2 δ(vi , 0) μQ (J, vi , k, σ, vi ) = " 2 Jα v ,v + μT0 12 (1 − cos 3α) v ,v + μT2 " i i i i T T T " + μ⊥ J (J + 1)/k + " μ − μ⊥ k Jα v ,v . (17) i
i
These expressions apply to transitions from initial state |J, vi , k, σ to final state |J , vi , k, σ in the Ψ basis, with vi ≡ vi − vi . Since only gs transitions are considered, the dependence on vs and vt is not indicated. In each off-diagonal matrix element Oi vi ,vi between these two states, the dependence on k and σ is suppressed. In Column I of Table II of [75], the four effective dipole parameters Q " μ0 , " μK , " μJ , and " μT in Eqs. (16) and (17) are defined in terms of μ0 and the seven torsional and rotational dipole parameters introduced above. In Eq. (17), 2 In Section III of [72], several corrections should be made. The labelling of Eqs. (19a) and (19b) should be interchanged. The right-hand side of Eq. (20c) should be multiplied by a factor of 1/2. The right-hand side of Eqs. (21), (23a), and (23b) should be multiplied by a factor of 2. In the sum in Eq. (27a), the cubic potential constant K66s should be inserted as a multiplicative factor. These corrections were made in subsequent publications and are reflected in the current work.
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the k in the denominator of the term in μT⊥ J (J + 1) causes no difficulty because it is cancelled by the factor k which appears in the numerator of the standard (J = 0) rotational matrix element; see Eq. (A-1) of [77]. If σ = ±1, this term produces a linear Stark shift for k = 0, a rather unusual effect that does not occur for a C3v molecule. Furthermore, for given mJ , this shift is independent of J [77]. For the gs, Eqs. (16) and (17) are the electric dipole counterparts of Eq. (11) for the energy in the simplified model, where resonant intervibrational terms are not taken into account. Of course, μR and μQ , are given only up to quadratic expansion terms, while E (1) (ν, J, vi , k, σ ) is expanded to quartic terms. In order to characterize completely the state dependence of the Stark effect in the gs, it requires nine parameters: the eight dipole constants that appear in Eqs. (16) and (17), and the anisotropy (α − α⊥ ) in the polarizability. However, for an individual low vi in a molecule like CH3 SiH3 with an intermediate value of the reduced barrier height s, only four independent constants are required. First, the diagonal matrix elements of the odd operators are small and the odd torsional dipole parameters can be fixed at zero. Second, the (k, σ )-dependence of the even matrix elements is very weak. For the particular vi of interest, a mean dipole constant μgs (vi ) can be defined in complete analogy with the mean rotational constants Aν (vi ) and B ν (vi ) introduced in Section 3.1. This step absorbs " μ0 , μT0 and μT2 into a single constant. Finally, for a typical symmetric top, the (J, k)-dependence of the contribution of the second order Stark effect due to μR J is identical to that for (α − α⊥ ) [77]. The two parameters can then be combined in an effective polarizability anisotropy (α − α⊥ )eff = (α − α⊥ ) − μR μ0 / hB0 ; J" see Section II.2 of [77]. This leaves only four constants: (α − α⊥ )eff , μgs (vi ), Q μK , with the last two characterizing the J - and k-dependence of μQ . " μJ , and " In representing the intensities of the torsional transitions within the simplified model, the effects of μT0 and μT2 are completely correlated with one another [75]. The off-diagonal matrix elements appearing in Eqs. (16) and (17) are evaluated (0) with the eigenfunctions of the zeroth order torsional Hamiltonian HT ,ν . As a result, " J2α vi ,vi = −(V0,3 /F0 ) 12 (1 − cos 3α)vi ,vi ; see Eq. (4). So long as the simplified model is valid, μT0 and μT2 cannot be separated from one another using intensity measurements alone, regardless of the number of torsional bands investigated. Jagannath et al. [75] introduced (μT0 )eff ≡ μT0 −(V0,3 /F0 )μT2 to represent the contribution of the even dipole constants to the intensities. The difficulties in evaluating the individual dipole parameters from Stark effect and intensity measurements are typical of the redundancies [72,91] that occur for internal rotor molecules. 3.3.2. CH3 SiH3 and CH3 SiD3 In the investigation of the torsional dependence of the dipole moment functions μQ and μR in the gs of CH3 SiH3 , the first step was the detailed study of the
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Stark effect. For vi = 0, the molecular beam electric resonance method was applied at U.N. to attain accuracies in dipole moments of ∼5 ppm and ∼42 ppm for relative and absolute measurements, respectively [77]. In the transitions studied, only the Stark energy changed; the selection rules obeyed were mJ = ±1; J = 0; k = 0; σ = 0. Nine different spectra with linear Stark effect were measured, each denoted by J±K , ∓|mJ | → (∓|mJ | − 1). In order to separate the four parameters characterizing the Stark effect, six different values of J±K from 2±1 to 5±5 were selected for study. Furthermore, in each of three cases, transitions for two different initial values of |mJ | were investigated. The electric fields used were 8 and 12.3 kV/cm. Each spectrum consisted of a single structureless enveF W ranged from 9 kHz lope of the hyperfine and σ components. The value of νH M for transitions frequencies ν ∼ 100 MHz to 24 kHz for ν ∼ 760 MHz, with inhomogeneity broadening dominating at the higher frequencies. It was found that Q μgs (0) = 0.7345600(33) D, " μJ = 8.83(35) μD, " μK = −32.82(37) μD, and (α − α⊥ )eff = 1.99(16) × 10−24 cm−3 , where the numbers in parentheses are the relative errors. Only the relative signs of the dipole parameters were determined. To study the dependence of μR on vi and σ in CH3 SiH3 , a conventional Stark modulation spectrometer at U.B.C. was used to measure the quadratic Stark effect in the gs transitions (J = 1, vi = 2, k = 0, σ ) ← (J = 0, vi = 2, k = 0, σ ) for σ = 0 and ±1 [75]. Since the spectrometer was set up for mJ = 0 selection rules, mJ vanishes in both the upper and lower states. At each field, the spectrum consisted of two well resolved lines with the intensity ratio (σ = 0) to (σ = ±1) F W varied from 340 kHz at being 2 due to the spin statistics. For each line, νH M zero field up to 810 kHz for a field of 3520 V/cm, corresponding to Stark shift of ∼43 MHz. All the Stark shifts were measured relative to the corresponding shift for the (J = 1 ← 0) line for vi = 0 at the same field. The analysis showed that μgs (vi ) increases by 0.01094(33) D when vi goes from 0 to 2. Furthermore, for (vi = 2, k = 0), μR decreases by 0.00166(46) D when σ goes from 0 to ±1. In the first analysis, the error in μT2 exceeded its magnitude. Subsequently, μT2 was fixed at zero and μT0 was found to be 0.0297(10) D; see Model D in Table I μ0 . of [75]. The sign of μT0 was determined relative to that of " The next step in the investigation of μQ and μR in CH3 SiH3 was a study of the intensities of the torsional spectrum from 150 to 380 cm−1 [75]. The spectra were taken at U.B.C. with a resolution of 0.15 cm−1 using a Michelson interferometer made by Beckman-RIIC and equipped with a 4.2 K germanium bolometer detector supplied by Infrared Laboratories. The absorption cell was a simple polished brass light pipe with an internal diameter of 1.3 cm and a length of 12.4 m. Four different average sample interferograms were obtained with sample temperatures between 295.2(1.0) and 298.1(1.0) K, and sample densities between 0.61(1) and 0.96(1) amagat. The use of relatively high densities was dictated by the low intensity at the absorption path length available. The spectrum was dominated by five bands: vi = 1 ← 0, 2 ← 1, 2 ← 0, 3 ← 1, and 4 ← 2. In the Q-branch of the
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torsional fundamental, the torsional fine-structure together with the J -dependence of the energy produced a broad, asymmetric envelope; the peak was at 189 cm−1 . For the other bands, the (J, k, σ ) fine-structure was more pronounced, particularly for the higher values of the upper state torsional quantum number vi . For the bands 1 ← 0 and 2 ← 1, the odd moments dominate the intensity, while for vi = 2 ← 0, the even moments dominate. Because the even transition moments are much larger, the torsional overtone is the strongest band in the spectrum. For transitions where vi > 2, the even–odd separation breaks down because the top of the barrier is close to E (1) (gs, J, vi = 3, k, σ ). For example, both even and odd moments contribute significantly to the highly blended high J transitions of the (4 ← 2) P -branch and the (4 ← 3) R-branch shown in Fig. 1 of [75]. To obtain the CH3 SiH3 torsional dipole parameters, a theoretical spectrum was synthesized and compared to the experimental data. A one-band model was used that is equivalent to the first step in the two-step J 2 –vi first order procedure discussed above in Section 3.3. The parameters appearing in the Hamiltonian HG VTR were left fixed at the best values available at the time [33]. For each frequency calculated, a line strength was determined using the eigenfunctions of HG VTR to evaluate the effective moments μQ and μR . Since the individual lines were pressure broadened, each transition was assigned a Lorentzian line shape. The instrumental line shape appropriate to the apodization applied was taken into account and baseline matching was carried out to deal with the broad underlying continuum. Because of the 100% correlation between μT0 and μT2 , one of the two had to be fixed. Since the latter was indeterminate in the Stark analysis, μT2 was fixed at zero, and the fit was carried out by varying μT0 , μT⊥ , (" μT − μT⊥ ) and the pressure broadening parameter γ0 . Two frequency regions were used in the analysis: 180 to 200 cm−1 and 345 to 360 cm−1 . The former was dominated by the (1 ← 0) Q-branch and used to determine primarily the odd dipole constants. The latter was dominated by the (2 ← 0) Q-branches and used to determine primarily μT0 . It was found that μT − μT⊥ ) = −85(10) μD, and μT0 = 0.0370(40) D, μT⊥ = 24.8(1.0) μD, (" −1 γ0 = 0.16(3) cm /amagat; see Model B in Table I of [75]. As was the case with the Stark effect, only the relative signs were determined. The intensity value for μT0 is in reasonable agreement with its Stark counterpart. In general, a good fit was obtained in the intensity analysis, as can be seen, for example, from Fig. 3 of [75]. This compares the experimental and optimum synthetic spectrum from 260.5 to 270.5 cm−1 , a region which was not used in the fit, but is sensitive to both the even and odd dipole parameters. Unfortunately, there were frequency intervals where the fit was less satisfactory. These were found to be the regions where the upper levels in the spectrum were closest to their counterparts in νt to which they are coupled by the intervibrational interactions. It would be very instructive to repeat this intensity measurement with longer absorption path length, lower
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density, and higher resolution using with the instrumentation now available [42]; see Section 3.3.3. T With μR J having been absorbed into (α − α⊥ )eff and μ2 having been fixed at zero, it appears that there are five independent parameters required to characterize the variation with torsion–rotation state of the dipole moment functions μQ and Q μR : two arising from centrifugal distortion (μJ and μK ) and three generated by torsion–distortion effects (μT0 , μT⊥ , and μT − μT⊥ ). However, there is strong evidence that, at least for CH3 SiH3 to the accuracy attained in [75], only the three Q torsional constants are needed. If μJ and μK are set to zero, it follows that the Q tilded parameters " μJ = ρμT⊥ and " μK = ρ(" μT − μT⊥ ); see Eqs. (9a) and (9b), reT T spectively, of [75] and note that " μ = μ if μT2 = 0. If the intensity values given μT − μT⊥ ) are used along with the value of ρ in Table II above, above of μT⊥ and (" Q
Q
" μJ = 8.73(35) μD and " μK = −29.9(3.5) μD. These intensity results for " μJ and " μK agree with the corresponding Stark determinations to within the errors in the Q Q μJ | and |μK | |" μK |; the centrifugal distortion conformer. Clearly, |μJ | |" tributions are much smaller than torsion distortion effects. The (J, k)-dependence of the dipole moment function μQ that is normally ascribed to centrifugal disQ tortion (i.e. to μJ and μK ) in a C3v molecule [71,104] arises primarily from a torsion-distortion effects (i.e. from μT⊥ and μT ) in CH3 SiH3 [75]. Conversely, the dipole parameters that determine the intensity of the perturbation-allowed torsional fundamental can be calculated from the (J, k)-dependence of the dipole moment function μQ that determines the linear Stark effect for vi = 0 [75]. This analysis underlines the importance of proper use of the IAM and PAM (as well as now the hybrid method). In [77], the IAM was not applied in a consistent manner. The correct form for the (J, k)-dependence of the dipole moment function μQ was obtained, but the dipole expansion constants were incorrectly ascribed to the centrifugal distortion effects that arise in C3v molecules. Three different investigations of the CH3 SiH3 torsional bands aimed at frequency determinations have been carried out. In the first, the (vi = 2 ← 0) torsional overtone was measured from 330 to 380 cm−1 at a resolution of 0.015 cm−1 [33]. The spectra were recorded with a BOMEM DA3.002 interferometer at the National Research Council (N.R.C.) in Ottawa, Canada; the detector was a 4.2 K copper-doped germanium photoconductor supplied by Santa Barbara Research. A Wilks multiple reflection absorption cell was used to provide an absorption path length of 22 m. The sample temperature was 295 K and pressure was 18 Torr (2400 Pa). The transitions are so weak that the final mean sample interferogram required an accumulation time of 18 hr to yield a signal-to-noise ratio of 10 for the strongest identified line. Although far better resolved that the Beckman measurements used in the intensity investigation, the spectra were still congested with a considerable amount of blending. Only extended series of lines
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were considered for addition to the global data set; 8 different (k, σ ) series made up from 339 lines passed this test. In the second torsional frequency study, the (vi = 2 ← 0) band was measured in the same frequency range using a Laser-Analytics LS-3 diode-laser spectrometer at U.B.C. equipped with a similar detector [33]. The absorption path length was 64 m. The data were taken at room temperature with a sample pressure between 2 F W was and 6 Torr, depending on the spectral region involved. The observed νH M −1 ∼0.002 cm , but the absolute frequency error was limited to the order of the line width by the frequency calibration. With a typical scan width of 0.25 cm−1 , a signal-to-noise ratio of 20 was obtained for the strong lines for an accumulation time ∼450 s. Blending was not a serious problem. Assignments of individual transitions could be made with confidence and 82 frequency determinations from 22 different (k, σ ) series were added to the data set. An indication of the sensitivity is provided in Fig. 1 of [33]. A line due to H2 18 O arising from trace amount of water in the sample is observed to be comparable in strength to the stronger CH3 SiH3 lines. This H2 18 O line is weaker by a factor of over 1500 than the strongest water line between 330 and 380 cm−1 . In both the BOMEM and diode laser studies, a one-band J 2 –vi first order model was used to analyze the data. In the third torsional frequency study of CH3 SiH3 , the (vi = 3 ← 1) hot band was measured from 250 to 342 cm−1 at a resolution of 0.01 cm−1 [35]. The spectra taken at U.B.C. with a slightly modified version of the Fourier transform spectrometer described by Buijs and Gush [105]. The absorption path length was 124 m. The gas was at room temperature; the pressure was 20 Torr. The total accumulation time for the final mean sample interferogram was ∼6 hr. The strongest identified lines in this band were ∼10% absorbing. A total of 15 different (k, σ ) sub-bands were added to the global data set, with the shortest series accepted having ∼10 members. The upper state (vt = 0, lt = 0; J, vi = 3, k, σ ) in this band is strongly coupled to its partner level (vt = 1, lt = −1; J, vi = 0, k − 1, σ ) with the same value of G, primarily by Coriolis-like interactions. Consequently, the data set involving the (vi = 3 ← 1) band was analyzed with a two-band J 2 –vi first order model described in Section 3.3. At the stage that this was done, the νt spectrum had already been measured [37], and so was included in the global data set by Moazzen-Ahmadi et al. [35]. The νt spectrum is discussed in Section 3.4.1 below, along with the torsional transitions (vi = 3 ← 0) and (vi = 5 ← 0) recorded while the νt and νs bands, respectively, were investigated. These two torsional bands with vi > 2 arise from resonant coupling, in contrast to the torsional bands with vi 2, which derive their intensity from non-resonant interactions. In CH3 SiD3 , the gs torsional bands (vi = 2 ← 0) and (vi = 3 ← 1) were recorded between 230 and 350 cm−1 at a resolution of 0.002 cm−1 using a Bruker IFS120 HR interferometer [38]. In a conventional experiment in this frequency region, the interferometer uses a thermal radiation source commonly referred to as
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a “globar”. However, in this case, the interferometer was coupled to the infrared beam line of the MAX-I electron storage ring in Lund, Sweden [106,107]. As compared to a globar, this synchrotron light source should provide a gain in intensity by over a factor of 3 at 250 cm−1 under ideal conditions. For such weak spectra, the resulting gain of an order of magnitude in integration time for a given signal-to-noise ratio is very important. While the gain was not realized in this case, primarily because of instabilities in the electron beam orbit, the study provided a valuable introduction to the application of synchrotrons; see [38] for a detailed discussion. The sample was cooled to 203 K at a pressure of 2.1 Torr. The absorption path length was 114 m. With an accumulation time of 5.8 hr, a signal-to noise-ratio of ∼10 was obtained for the stronger lines. In this same work, Schroderus et al. [38] reported a study of the νt band. Crossings were observed between the states (vt = 0, lt = 0; J, vi = 3, k, σ ) and their interaction partners (vt = 1, lt = +1; J, vi = 0, k + 1, σ ) with the same value of G for (k, σ ) = (15, −1), (18, −1) and (19, 0). The global data set involving these resonantly coupled states was analyzed with the one-step procedure described in Section 2.3. Both the high barrier and free rotor models were used and the results compared. 3.3.3. CH3 CH3 The most direct determination of the hindering potential for ethane is through the observation of its torsional spectrum in the gs. Unfortunately, as for the XY3 AB3 case, these bands in C2 H6 are forbidden in the lowest order, and are weakly allowed due to intensity borrowing from the infrared active bands [95,108]. As a result, the progress on the characterization of the torsional potential has been incremental. The first observation of the torsional bands was made by Weiss and Leroi [109] at high pressures (∼6 atm) and low resolution (∼2 cm−1 ) using an absorption path length of 10 m. From the Q-branch origin of the ν4 fundamental and the two Q-branch origins for the 2ν4 − ν4 hot band (one for each torsional symmetry), they computed the barrier height in the gs of ethane to be 1024 cm−1 . Subsequently, three medium resolution spectra (0.015 cm−1 and 0.010 cm−1 ) at lower pressures and longer absorption path lengths were reported by MoazzenAhmadi and co-workers [41,76,86]. The relatively high spectral resolution made it possible to achieve partial resolution of the torsion–rotation structure. Frequency analyses of the torsional bands together with the lower state combination differences from frequencies belonging to the vibrational bands ν9 and ν9 +ν4 −ν4 were made to characterize the torsion–rotation Hamiltonian in the gs. More recently, the two torsional bands were studied at a very high resolution of 0.0016 cm−1 using a Bruker IFS120 HR Fourier transform spectrometer at the University of Oulu (U.O.) in Oulu, Finland [42]. The sample temperature and pressure were 294 K and 10 Torr, respectively. The absorption path length was 172 m with a
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Table III Molecular Parameters for the gs of CH3 CH3 Parameter
Units
Operatora
V0,3 V0,6 A0 B0 D0,J D0,J K D0,K D0,m D0,J m D0,Km F0,3J F0,3K F0,6J F0,6K H0,J J m H0,J Km H0,KKm
cm−1 cm−1 MHz MHz kHz kHz kHz MHz MHz MHz MHz MHz MHz MHz Hz Hz Hz
1 (1 + cos 6γ ) 2 1 (1 − cos 12γ ) 2 J2z 2 J − J2z −J4
−J2 J2z −J4z J4γ −J2 J2γ −J2z J2γ J2 21 (1 + cos 6γ ) J2z 21 (1 + cos 6γ ) J2 21 (1 − cos 12γ ) J2z 21 (1 − cos 12γ ) J4 J2γ J2 J2z J2γ J4z J2γ
Valueb 1013.069(27) 10.119(18) 80257.5(4.6) 19916.530(17) 31.073(19) 77.89(20) 283.c 1.037(21) 0.3110(11) 2.0531(92) −356.12(17) 1016.8(1.5) −16.095(81) 18.58(64) 8.12(18) −60.3(1.3) 125.1(5.4)
a The operator form is appropriate to the origin for γ being at the eclipsed configuration. b From [42] calculated with a one-band analysis. c From the empirical force field of [89].
total recording time of 60 hr. The earlier studies of ethane led to the identification of 168 lines in the torsional fundamental and 36 in the hot band. In the most recent investigation [42], 770 transitions were assigned in the fundamental and 302 in the hot band. The substantial increase in the number of available lines and the significant reduction in experimental error allowed the determination of all the torsion–rotation parameters in the gs Hamiltonian up to the first-order. Included in this set of parameters were V0,3 , V0,6 , and A0,R . The analysis was carried out with a one-band two-step procedure using the J 2 –vi exact approach as described in Section 2.3. Molecular parameters from this analysis are given in Table III [42]. A portion of the spectrum recorded in [42] is shown in the lower trace of Fig. 4, which is adapted from Fig. 2 of that report. The transitions in this trace are primarily from the Q-branch of the torsional fundamental v4 . Here three of the most prominent rotational series are marked: (k, σ ) = (1, 1) lines are labelled by a, (2, 2) lines by b, and (3, 3) lines by c. The spectrum is almost fully resolved and blending is not a serious concern. The upper trace in Fig. 4 is the simulated spectrum whose transition frequencies were calculated using the molecular parameters
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F IG . 4. A portion of the torsional spectrum of ethane in the region of the Q-branch of the torsional fundamental ν4 . The spectral resolution was 0.0016 cm−1 . The pressure was 10 Torr and the temperature was 294 K. Three of the most prominent rotational series are marked. Lines with (k, σ ) = (1, 1), (2, 2), and (3, 3) are labelled by a, b, and c, respectively. The lower trace is the experimental spectrum, while the upper trace is its calculated counterpart, which includes the transitions from the torsional fundamental and the first torsional hot band.
Table IV Pressure Broadening Parameter and Torsional Dipole Moments of Ethane Isotopomersa CH3 CH3 b
Parameter γ0 (μT − μT⊥ ) μT⊥ (μT − μT⊥ )/μT⊥
cm−1 amagat−1 μD μD
CD3 CD3 c
CH3 CD3 d
0.14(1)
0.16(3)
0.12(2)
−31.7(2.0)
−19.9(2.0)
−39.6(4.0)
9.37(70) −3.39(6)
7.41(74)
14.0(1.5)
−2.68(13)
−2.84(14)
a In each case, the sign of μT is arbitrarily taken to be positive, but the sign of the ratio (μT −μT )/μT ⊥ ⊥ ⊥
is determined experimentally. See Sections 3.3.3 and 3.3.4. b From [76]. c From [53]. d From [32].
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of Table III. The line intensities were calculated using procedures described below and torsional dipole moments given in Table IV. It is satisfying to see that the theoretical model is quite successful with respect to both frequency and intensity. The lines which appear in the observed spectrum but are absent in the simulated spectrum are due to impurities in the sample and/or difference bands of ethane, such as ν12 − ν9 , which occurs in this region [109]. The work in [42] represents an essential step in addressing many questions relating to internal rotation in ethane. The effective torsion–rotation Hamiltonian characterizing the three lowest torsional states in this one-band model can be used to predict energies for higher torsional states in investigations of the excited vibrational states. These higher torsional levels act as bath states for ν9 and ν3 , for example. The electric dipole moment which drives the torsional bands of ethane arises from the Coriolis interactions of the E1d and A4s vibrational modes with the gs [34,95,108,109]. In the molecular fixed frame, μz = μ Jγ Jz and μ±1 = μ⊥ Jγ J±1 , where: −1 x μ⊥ = 4(A0,R /9V0,3 )1/4 Bx ζ4,t (ω4 ωt )1/2 ω42 − ωt2 (∂μx /∂qta ); a t⊃E1d
μ = 4(A0,R /9V0,3 )
1/4
z Bz ζ4,s (ω4 ωs )1/2 ω42
−1 − ωs2 (∂μz /∂qs ).
(18)
s⊃A4s
(19) Here ωi is the frequency of the ıth fundamental, is the Coriolis coupling constant between the torsion and the ith vibration, and (∂μα /∂qi ) is the derivative of the α-component of the dipole moment with respect to ith dimensionless normal coordinate. The selection rules are J = 0, ±1; k = 0; σ = 0. Furthermore, v4 = odd for σ = 0 and 3, while v4 = arbitrary for σ = 1 and 2 [86]. The integrated intensity for a transition (J , k, v4 , σ ) ← (J, k, v4 , σ ) is given by [69]: α ζ4,i
A(J, k, v4 , σ ; J , k, v4 , σ ) = (β/ZVTR )S(J, k, v4 , σ ; J , k, v4 , σ )ν × e−E(J,k,v4 ,σ )hc/kT − e−E(J ,k,v4 ,σ )hc/kT . (20) For J = 0 and +1, the line strength S is given by SQ and SR , respectively, where [86]: 2 SQ (J, k, v4 , σ J, k, v4 , σ ) = g (μ − μ⊥ )k 2 + μ⊥ J (J + 1) 2 × v4 , σ |Jγ |v4 , σ (2J + 1)/J (J + 1); (21) 2 2 2 SR (J, k, v4 , σ ; J , k, v4 , σ ) = g(μ − μ⊥ ) k v4 , σ |Jγ |v4 , σ × (J + 1)2 − k 2 /(J + 1). (22)
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For J = −1, the factor (J + 1) in Eq. (22) must be replaced by J . In Eq. (20), E(J, k, v4 , σ ) is the gs torsion–rotation energy in cm−1 , ν is the frequency of the transition, β = 8π 3 N/3hc, ZVTR is the vibration–torsion–rotation partition function, N is the number density, and the other symbols have their usual spectroscopic meaning. The magnitude of the matrix element v4 , σ |Jγ |v4 , σ is large for v4 = odd and has its largest value for v4 = 1 ← 0. Primarily because of the difference in symmetry, the values of S for the strongest lines in the ethane spectrum are substantially smaller than their counterparts for methyl silane. For the pure rotational spectrum discussed in Section 3.1, Eq. (22) gives the line strength for σ = 1 and 2. The partition function ZVTR can be written as the product ZTR ZV . The torsion– rotation partition function can be evaluated using v4,max Jmax kmax
ZTR =
3
(2J + 1)ge−ETR (J,k,v4 ,σ )hc/kT ,
(23)
v4 =0 J =0 k=0 σ =0
with k + σ = even. The vibrational partition function can be obtained from [110] # −di 1 − e−hcωi /kT , ZV = (24) i
where di represents the degeneracy of ith vibrational fundamental. The vibrational frequencies necessary for computing ZV are given in [89]. The torsional mode ν4 must be excluded in evaluating ZV . The nuclear spin statistical weights g are given in Table III of [26]. The vibration–torsion–rotation partition function ZVTR is used in the modeling of planetary atmospheres and in analyzing ground-, air-, and space-based measurements of ethane concentrations in the Earth’s atmosphere [111]. For temperatures between 100 and 310 K in steps of 10 K, ZVTR and ZTR have been calculated with a one-band J 2 –vi exact procedure (see Section 2.3) using the gs parameters listed in Table III [42]. The results are given in Table V. The torsion–rotation partition sum ZTR was evaluated using v4,max = 10, Jmax = 60, and kmax = 40; these limits are high enough for convergence at the highest temperature. Earlier values for ZVTR are given in column 5 of Table 2 in [111], as calculated by Pine and Rinsland using the parameters in [76]. The present values for ZVTR are larger than their counterparts in [111] by ∼4% at 100 K, but the difference decreases to ∼1.4% at 300 K. The changes are caused primarily by the use of the more accurate gs energies now available. Both here and in [111], the reference energy for determining ZVTR was chosen to be the zero-point energy, whereas the reference in [76] was the bottom of the torsional potential. The ground state energies which are also needed to obtain the absolute integrated transition intensities are available from Journal of Molecular Spectroscopy Supplementary Material Archives as calculated in [42].
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Table V Partition Functions for Ethane T /K
ZTR
ZVTR
T /K
ZTR
ZVTR
100.0 110.0 120.0 130.0 140.0 150.0 160.0 170.0 180.0 190.0 200.0
10723.8 12439.1 14274.9 16236.5 18329.3 20558.3 22928.9 25446.1 28114.8 30940.0 33926.1
10723.9 12439.6 14276.3 16239.9 18336.6 20573.3 22957.0 25495.7 28197.7 31072.5 34130.0
210.0 220.0 230.0 240.0 250.0 260.0 270.0 280.0 290.0 300.0 310.0
37077.8 40399.5 43895.2 47569.0 51424.8 55466.2 59696.6 64119.3 68737.5 73554.0 78571.5
37381.4 40838.6 44514.8 48424.3 52582.4 57006.0 61713.1 66723.2 72057.5 77738.6 83790.8
Determinations of the torsional dipole moments μ and μ⊥ were made from the torsional spectra observed in [76]. Measurement were carried out under various experimental conditions at resolutions of 0.08, 0.02, and 0.016 cm−1 . The spectrum at the highest resolution was recorded at a temperature of 208 K. A section of this spectrum in the Q-branch of the torsional fundamental is shown (dotted curve) in Fig. 5 (taken from Fig. 3 of [76]). Synthesized spectra were obtained from a stick spectrum with the line intensities calculated from Eq. (20) in terms of the two torsional dipole moments. Each transition was assigned a line profile consisting of a pressure broadened Lorentzian convolved with the instrumental function. The pressure broadening parameter γ0 and the dipole components were varied along with the background level, its slope and curvature to obtain the best fit between the calculated and the experimental intensities. The parameters relevant for frequency determination were fixed during the line-shape analysis. The quality of the best fit is illustrated in Fig. 5. Here the calculated spectrum is represented by the solid curve. As can be seen, the calculated spectrum is quite successful. The sharp feature at 289.445 cm−1 in the experimental spectrum is due to water impurity in the sample. The experimentally determined values of γ0 and the torsional dipole moments are given in Table IV. The molecular constants in Eqs. (18) and (19) required for calculation of μ and μ⊥ are available in the literature. These are collected in Table V of [76]. Dipole derivatives have been determined by Nyquist et al. [112] from absolute intensity measurements on the fundamental vibrations of CH3 CH3 . Coriolis coupling constants have been calculated by Duncan et al. [89]. The experimental measurement of intensities yields only the absolute magnitude of the derivatives. Furthermore, there is ambiguity in the sign of the calculated Coriolis coupling constants. Thus, a comparison of the calculated and observed torsional dipole moments can yield in-
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F IG . 5. A portion of the torsional spectrum of CH3 CH3 illustrating the comparison between the calculated spectrum (solid curve) and the experimental spectrum (dotted curve) in the region of the Q-branch of the torsional fundamental ν4 . The spectral resolution was 0.016 cm−1 . The pressure was 72 Torr and the temperature was 208 K. The sharp feature at 289.445 cm−1 in the experimental trace is due to water impurity in the sample.
formation on the relative signs of the various contributions to the torsional dipole α ∂μ /∂q for the different moments i.e. on the relative signs of the products ζ4,i α i vibrational modes i. The calculated contributions to the torsional dipole moments and the resulting total dipole moments for various combinations of signs are given in Table VI of [76]. The best match between the theory and experiment was obα ∂μ /∂q is +, +, +, −, and − for i = 5, 6, 7, 8, and 9, tained when the sign of ζ4,i α i respectively. 3.3.4. CD3 CD3 and CH3 CD3 For CD3 CD3 , the intensity analysis of the torsional bands was made in much the same way as for CH3 CH3 . The torsional spectrum was recorded at a resolution of 0.016 cm−1 ; the sample pressure and temperature were 72 Torr and −105 ◦ C, respectively [53]. A portion of the torsional band illustrating the comparison between the calculated spectrum (solid curve) and the experimental spectrum (dotted curve) in the region of the P -branch of the torsional fundamental (v4 = 1 ← 0) is shown in Fig. 6 (taken from Fig. 2 of [53]). Unlike CH3 CH3 , the observation of σ -splittings in the torsional fundamental of CD3 CD3 at a medium resolution of 0.016 cm−1 is not possible. For small values of v4 , the intrinsic σ dependence of the energy of state v4 is given by Nσ Xv4 , where Nσ = −2, −1,
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F IG . 6. A portion of the torsional spectrum of CD3 CD3 illustrating the comparison between the calculated spectrum (solid curve) and the experimental spectrum (dotted curve) in the region of the P -branch of the torsional fundamental ν4 . The calculation also includes lines from the first torsional hot band 2ν4 − ν4 . The weak features in the calculated spectrum to the right of P(12), P(11), and P(10) are the beginning of the R-branch of the hot band. The spectral resolution was 0.016 cm−1 .
+2, +1 for σ = 0, 1, 2, 3, respectively [25,86]. For CD3 CD3 , X0 < 1 MHz and X1 = 30 MHz. These result in an overall torsional splitting in the spectrum of 3(X0 + X1 ) = 0.003 cm−1 , well below the spectral resolution of 0.016 cm−1 . For ethane, however, X0 = 0.002 cm−1 and X1 = 0.079 cm−1 , so that 3(X0 + X1 ) = 0.243 cm−1 , which is over 15 times the spectral resolution. In the P - and R-branch, the torsion–rotation transitions for a given J cluster to form a broad envelope with a width ∼0.2 cm−1 . This width can be explained by the contribution from the term in F0,3K , which has a k 2 dependence. Consequently, it is possible to determine the value of F0,3K by fitting the shapes of the J -envelopes, even though individual lines within the envelope are not resolved. The molecular parameters required for the calculations of μ and μ⊥ are also available in the literature. These are given in Table III of [53]. A comparison of the calculated and the observed torsional dipole moments showed that the relative α ∂μ /∂q for the various values of i are the same as those signs of the product ζ4,i α i α ∂μ /∂q for ethane. This leads to the appealing conclusion that the sign of ζ4,i α i is unlikely to change by the simple substitution of the hydrogen atoms with deuterium atoms. The experimentally determined pressure broadening parameter γ0 and torsional dipole moments for CD3 CD3 are give in Table IV.
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For CH3 CD3 , the torsional dipole operators μTz and μT±1 are discussed in Section 3.3.1 and given in terms of the dipole constants μT0 , μT2 , μT⊥ , and μT in Eqs. (14) and (15). For the purpose of analyzing the intensity of the (v6 = 1 ← 0) band, these two expressions can be simplified a great deal. First, |μT0 | and |μT2 | are expected to be small since they vanish by symmetry for CH3 CH3 and CD3 CD3 . Second, for two levels well below the top of the barrier and connected by v6 odd, the matrix elements of the even operators 12 (1−cos 3α) and " J2α are small, whereas those for the odd operator " Jα are large. Finally, only terms involving torsional operators need be considered. It then follows that the transition dipole operators for the torsional fundamental of CH3 CD3 can be written to a good approximation as μTz = μT " Jα J±1 + J±1" Jα Jz and μT±1 = μT⊥ 12 [" Jα ]. T T Expressions for μ⊥ and μ are given by [72]:3 −1 x μT⊥ = 4(4F0 /9V0,3 )1/4 Bx ζ6,t (ω6 ωt )1/2 ω62 − ωt2 (∂μx /∂qta ) a t⊃E1
− 2(4F0 /9V0,3 )
(25) B6 ω6−1 μ0 ; −1 z Bz ζ6,s (ω6 ωs )1/2 ω62 − ωs2 (∂μz /∂qs ). (26)
1/4
μT = 4(4F0 /9V0,3 )1/4
xy
s⊃A1
As has been pointed out in [32], all the doubly degenerate vibrations of CH3 CD3 have E1 symmetry. Therefore, the torsional fundamental borrows intensity from ν7 to ν12 . However, as is also the case for CH3 CH3 and CD3 CD3 , the lowest energy degenerate vibration (ν12 ) makes the dominant contribution from the modes of this type. In addition, of the five A1 vibrations which contribute to μT , two (ν1 and ν2 ) make by far the largest contribution. See Table IV of [32]. The torsional spectrum of CH3 CD3 was observed [32] in the same way as for CH3 CH3 and CD3 CD3 . The spectral resolution was 0.016 cm−1 . The gas pressure and temperature were 82 Torr and 190 K, respectively. The torsional dipole components that were determined from the intensity analysis are given in Table IV. The values for the various molecular parameters in Eqs. (25) and (26) required for the calculation of the torsional dipole moments are accumulated in Table IV α and the harmonic frequencies ω of [32]. The Coriolis coupling parameters ζ6,i i were taken from empirical force field calculations [89]. The dipole derivatives xy (∂μα /∂qi ) and the inertial coefficient B6 were calculated by numerical differenxy ∗∗ tiation at the MP2/6-311G level [113,114]. Since |B6 | was determined to be less than 10−5 cm−1 , the last term in Eq. (25) is negligible. α ∂μ /∂q which can be obtained from The sign information on the product ζ4,i α i comparison of the calculated and the observed torsional dipole moments is complicated significantly by the large number of modes which contribute. However, 3 See footnote 2 on page 461.
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it is possible to find a satisfactory match between the experiment and the theory [32]. In Table IV, the results are given and compared with those for CH3 CH3 and CD3 CD3 . The values of the broadening parameter γ0 for the three isotopic forms average to 0.14 cm−1 amagat−1 , which equals that reported for CH3 CH3 in [115]. Both μT⊥ and (μT − μT⊥ ) are larger in magnitude for CH3 CD3 than for either of the other two isotopomers. This arises very likely from the lower symmetry of CH3 CD3 . However, the ratio (μT − μT⊥ )/μT⊥ for CH3 CD3 does fall between the values for CH3 CH3 and CD3 CD3 . 3.4. V IBRATION –T ORSION –ROTATION S PECTROSCOPY 3.4.1. CH3 SiH3 The two lowest vibrational modes in CH3 SiH3 have been studied at sufficiently high resolution to measure the torsional splittings: the perpendicular band ν12 near 525 cm−1 [37] and the parallel band ν5 near 700 cm−1 [36]. In each case, resonant interactions have been observed with excited torsional levels in the gs. For ν12 , the resonance is caused by Coriolis-like interactions with the (v6 = 3) state; the x O47 as defined in Taleading term in the coupling Hamiltonian Hgs,ν12 is Bζ6,12a ble II. For ν5 , the resonance is caused by Fermi-like interactions with the (v6 = 5) αα O and state; here the leading terms in the coupling Hamiltonian Hgs,ν5 are M5,0 53 3 C5,0 O55 . In each case, the resonant frequency shifts have a (k, σ )-dependence that is ∼1 cm−1 . In each case, the mixing is strong enough that normally forbidden torsional transitions have been measured: v6 = 3 ← 0 and v6 = 5 ← 0 for resonances involving ν12 and ν5 , respectively. These results provide the first direct experimental evidence that interactions involving large changes in the torsional quantum number can lead to complete mixing of bright and dark states. In turn, this demonstrates that the sea of dark states capable of fragmenting a spectrum includes levels with a wide range of torsional quantum number. These interactions change dramatically the tunneling motion of the internal rotor in the bright state, and provide gateway levels through which vibration energy can flow easily from an excited vibrational state to the gs. The silyl rock ν12 was measured in CH3 SiH3 from 475 to 575 cm−1 with the BOMEM DA3.002 interferometer at N.R.C. [37] The sample temperature was ∼170 K, and its pressure was ∼ 3/4 Torr. The spectrum is relatively strong. The absorption path length was only 15 cm. An accumulation time for the final sample interferogram of ∼100 minutes was sufficient to yield a signal-to-noise ratio 10 F W of the apodized instrumental line for a large number of weaker lines. The νH M shape was 0.0045 cm−1 . The spectrum was subjected to a numerical deconvoluF W to 0.0022 cm−1 . tion which reduced νH M Since ν12 is a degenerate vibrational level, the selection rules [37,57] appropriate to the Ψ -basis defined in Eq. (8) are: v12 = 1; J = 0, ±1; v6 = 0;
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k = l12 = ±1; σ = 0. For cases where (A+ − A− ) splittings are observed, the Φ-basis of Eq. (10) must be used. In this basis, k and l12 are not good quantum numbers when ΓVTR = A± or E4± . In these cases, when J = 0, the selection rules on k and l12 are replaced by ΓVTR = A± ← A± and E4± ← E4± . When |J | = 1, ΓVTR = A± ← A∓ and E4± ← E4∓ . In both cases, G = 0. When the transitions are up to an l-doubling state in ν12 , the gs level involved has k = 0 and ΓVTR = A+ or E4+ . See Section III.2 of [37] and Section 2.3 above. Then in the Q-branch, the upper state must have ΓVTR = A+ or E4+ , while in the P /Rbranches the upper state must have ΓVTR = A− or E4− . In each √ of these cases, the electric dipole matrix element for the allowed transition is 2 times its counterpart before the Wang transformation [37]. Symbols such as p Rkσ (J ) are used to label individual transitions, where J and k are lower state quantum numbers, the lower case left superscript specifies k, and the capital letter specifies J . For the l-doubling transitions, artificial labels are used: p P0σ (J ), r Qσ0 (J ), and p R σ (J ); see [37] for the rationalization. 0 One of the striking features of the spectrum is the splitting of the multiplets r R σ (J ) and p P σ (J ) into their σ -components with their characteristic spin k k statistical weights g [37]. The characteristic pattern in the r R-branch for all the possible relative σ -weights is illustrated in Fig. 7, adapted from Fig. 4 of [37]. (The values of g apply for k = 1 modulo 3, with (k = 0) being an exception [39].) In interpreting the σ -patterns, the original spectrum in Fig. 7a is better than its deconvolved counterpart in Fig. 7b because the deconvolution can alter the relative intensities to some extent. In r R2σ (16), for example, the σ -components have intensities in the ratio 2 : 2 : 1 for σ = −1, 0, +1, as would be expected from the corresponding values of g. For the σ -doublet r R0σ (22) (the lowest frequency multiplet in Fig. 7), g = 2 and 1, respectively, for σ = 0 and ±1. The corresponding upper levels have ΓVTR = A− or E4− . The two σ -components have intensities in the ratio 2 : 1, as is the case when the splitting C (v6 , σ ) as defined in Section 3.1 is negligible and the E4± states behave as their A± counterparts. In the r R-branch, the σ -splitting pattern shown in Fig. 7 is dominated by intrinsic effects arising primarily from the changes between ν12 and the gs in the barrier height and the reduced rotational constant. For k = +1, the σ -pattern changes very slowly with J . The situation in the p P -branch is very different. The σ splitting in p P8σ (J ) is illustrated in Fig. 8, adapted from Fig. 6 of [37]. In the multiplet p P8σ (8) shown in Fig. 8a, the (σ = 0) transition (g = 2) is the highest frequency member of the triplet. However, in p P8σ (11) shown in Fig. 8b, the (σ = 0) component is the lowest frequency member. In contrast to this, in both cases, the (σ = +1) line (g = 1) is a low frequency shoulder on the line for its (σ = −1) partner (g = 2). For many of the p Pkσ (J ) multiplets, the position in the pattern for one member of the σ -triplet changes considerably as a function of J . The cause for the anomalous variation with J in the p P -branch is the near
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F IG . 7. Part of the r R-branch of the ν12 band of CH3 SiH3 : (a) original and (b) after deconvolution. The values k and σ are indicated in (a) and (b), respectively. From left to right, the values of J for the six multiplets are 22, 7, 10, 13, 16, and 19. Note the characteristic spin statistical weights for the components within each multiplet.
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F IG . 8. Two sections of the p P -branch of the ν12 band of CH3 SiH3 after deconvolution. Each line is labelled with the lower state values of J , k , and σ . In particular, note the triplet p P8σ (8) in (a) and p P8σ (11) in (b). The change in the σ -ordering is primarily due to near-resonant Coriolis-like perturbation of the (σ = 0) component. The features labelled A are high J -lines in the p P0 sub-branch. The frequency scales in (a) and (b) are the same.
resonant Coriolis-like coupling between levels with (v12 = 1, l12 = −1, v6 = 0) and (v12 = 0, l12 = 0, v6 = 3). The explanation [37] for the behaviour of the σ -multiplets in the r R- and p P branch can be seen in Fig. 9, even though this figure (taken from Fig. 1 of [37]) illustrates r Q- and p Q-transitions. One has only to change the gs value of J from 11 to 10 for r R8σ (10), and from 11 to 12 for p P8σ (12). The corresponding changes to the gs energies are not important here. On each level, the quantum numbers shown are J , v6 , k, and σ . On the primary energy scale in cm−1 , the torsional splittings for v6 = 0 are too small to be seen for both gs and ν12 , and so a secondary energy scale in MHz is used to illustrate these splittings. Since the Coriolis-like matrix elements coupling ν12 to the gs do not depend strongly
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F IG . 9. Energy level diagram to illustrate near-resonant Coriolis-like shifts for (k = 8) in the ν12 band of CH3 SiH3 . Note the break in the vertical axis. For each level, J , k, v6 , and σ are given in that order. For v6 = 0, the σ -splittings are too small to see on the cm−1 energy scale, and so an expanded splitting scale in MHz is used. The dashed line in the left-hand column shows the unperturbed energy for the (σ = 0) level. The arrows show p Qσ8 (11) to the left and r Qσ8 (11) to the right. The patterns for p P σ (12) and r R σ (10) are similar to their Q-branch counterparts on the left and right, respectively. 8 8
on σ , the perturbation pattern can be understood by examining the unperturbed energy differences between interacting levels. Consider the σ -triplet with (v12 = 1, l12 = +1, v6 = 0, k = 9, σ ) in the right-hand column. Each of these levels
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is above its interaction partner with (v12 = 0, l12 = 0, v6 = 3, k = 8, σ ) by 17 cm−1 . Each level is shifted upward by the Coriolis-like interaction, but the shift is too small to show clearly even on the MHz splitting scale. The same is true for the (σ = +1) and (σ = −1) members of the σ -triplet with (v12 = 1, l12 = −1, v6 = 0, k = 7, σ ) in the left-hand column. However, for the (σ = 0) member, the corresponding unperturbed energy difference has the opposite sign and is smaller in magnitude by a factor ∼25. This level is shifted downward by over 700 MHz from the unperturbed position (the dashed line in Fig. 9). For this level, the Coriolis shift is larger than the intrinsic σ -dependence of the energy by a factor ∼3. On the other hand, for the other five σ -levels shown for ν12 in Fig. 9, the intrinsic σ -dependence dominates. In the multiplet p P8σ (8) shown in Fig. 8a, (J = 7, k = 7) in the upper state. From the Coriolis selection rules (J = 0, k = l12 = ±1), the interaction gs partner with v6 = 3 in this case would have (J = 7, k = 8); since J < k, the state does not exist. Consequently, p P σ (8) is unperturbed and the σ -splitting is entirely intrinsic, causing the (σ = 8 0) member to be the highest frequency member of the σ -triplet. In p P8σ (11), it is primarily the large downward Coriolis-like shift for σ = 0 that causes p P80 (11) in Fig. 8b to be the lowest frequency member. The effect of this near-resonant interaction is also seen the p Q-branch; see Fig. 5 of [37]. For the near-resonance illustrated in Fig. 9 for σ = 0, the gs dark state |d = |v12 = 0, l12 = 0; J = 11, v6 = 3, k = 8, σ = 0 is within 20 cm−1 of the two ν12 bright states |b± = |v12 = 1, l12 = ±1; J = 11, v6 = 0, k = 8 ± 1, σ = 0 with which it interacts. However, the corresponding three coupled torsional stacks of states must be taken into account in diagonalizing the Hamiltonian, even though the additional states are much further away [37]. The energies for the different torsional states that can interact with |d, |b− or |b+ are shown in Fig. 10, which is taken from Fig. 3 of [37]. All the levels that can interact with |d are linked to |d with solid lines, while all the levels that can interact with |b− are linked to |b− with dashed lines. Of course, the dashed line directly between |b− and |d is superimposed on the corresponding solid line. The levels that can interact with |b+ are indicated by a similar fan of dashed lines. All pairwise interactions possible in Fig. 10 (not just those illustrated) were taken into account in the diagonalization in [37]. For the levels included in Fig. 10, the maximum values of v6 are 4 and 5 for the gs and ν12 , respectively. These truncation limits correspond to the 17 × 17 matrix diagonalized in the second step of the twostep J 2 –vi first order analysis used at the time; see Section 2.3. Both limits were raised to 6 in [35] as data set was expanded and greater accuracy was required. The energy level pattern in Fig. 9 has another very unusual feature [37]. If the interaction coupling the three stacks of levels is turned off, then the splitting pattern in the left-hand column (that with the dashed line) is the same as the pattern in the right-hand column to good approximation. In the former case, k = 7
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F IG . 10. Energy level diagram to illustrate the interactions in CH3 SiH3 that contribute to the Coriolis-like shifts in the bright states |b± ≡ |v12 = 1, l12 = ±1; J = 11, v6 = 0, k = 8 ± 1, σ = 0. Each state is labelled by the quantum numbers at the bottom of its column and the value of v6 on the level itself. The dashed diagonal lines indicate the pairwise coupling to |b± , while the solid diagonal lines indicate pairwise coupling to the dark state |d in near resonance with |b− . The energy gap between |d and |b− is 1.726 cm−1 , as measured in [37]. A similar fans of lines can be drawn for the other levels in the diagram. The top and bottom of the gs potential are indicated, with the top being ∼19 cm−1 below the dark state |d.
and G = 8, while in the latter, k = 9 and G = 8. Thus it seems that the σ -ordering and the σ -spacing are determined by G rather than by k. Furthermore, except for a common scale factor, each of these ν12 patterns is the same as that in the gs, where k = G = 8. These conclusions are based on the fit to the global data set,
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but they are substantiated by direct examination of the spectra. For cases where the Coriolis-like shifts are small, the J -dependence of the patterns is very weak and the multiplets p Pkσ (J ) and r Rkσ (J ) look very similar. As an example, compare r R σ (16) shown in Fig. 7 here with r P σ (17) shown in Fig. 5 of [37]. From a 2 2 casual visual inspection of these figures (rather than of the original spectra), it is difficult to make a detailed comparison of the spacings, but certainly it is clear that the component with g = 1 is the high frequency member of the σ -triplet in both cases. These observations seem to be in striking disagreement with the predictions of the simplified model discussed in Section 3.1. This model should apply here since the observations involved concern the case where the interstack interactions are negligible. From Eq. (11), the leading σ -dependent terms in the transition frequencies for k = ±1 can be written [37]: νk=±1 (k, σ ) = ET(0) (ν12 , 0, k ± 1, σ ) − ET(0) (gs, 0, k, σ ) m" Jα ν ,0,k±1,σ . ± 2Aζ12 (27) 12
(0)
As can be seen from Eq. (11) of [28], the σ -ordering of ET is different in each of the three cases: k, k + 1, and k − 1. Then the σ -ordering of the square bracket in Eq. (27) is different for k = +1 and k = −1. Furthermore, the σ -spacing will be modified by the last term in Eq. (27). This argument leads to the conclusion that the σ -pattern (including the ordering) will be different p Pkσ (J ) and r Rkσ (J ) in a way that depends on the molecular parameters. This apparent contradiction is m" resolved by taking into account the matrix elements of 2Aζ12 Jα 12 off-diagonal in v6 . Then this is done numerically, the σ -ordering for each k is changed in just such a way that the σ -pattern is a function of G rather than k, and the ν12 pattern differs from that for the gs only by a scale factor. The reason [37] for this unexpected result lies in the form of the zeroth-order (0) torsion–rotation PAM Hamiltonian HTR,ν12 for ν12 . This Hamiltonian is obtained mJ by adding the term 2Aζ12 α 12 to the ν12 counterpart of the right-hand side of Eq. (1), where, as indicated in Section 2.1, Jα is the PAM torsional angular m = A m momentum. If Aζ12 12,F , the terms −2A12,F Jα Jz and 2Aζ12 Jα 12 can be combined to form −2A12,F Jα (Jz − 12 ). Then the torsional angular momentum in the hybrid method can be re-defined: " Jα ≡ [Jα − ρ(Jz − 12 )]. This will elim(0) inate the term −2A12,F Jα (Jz − 12 ). In addition, the zeroth-order torsional ET for ν12 will depend on G in just the same manner as the zeroth-order torsional (0) ET for the gs depends on k. The σ -patterns in the energy levels and the spectra will then have the properties noted above. m should There is considerable support for this empirical argument that Aζ12 equal A12,F . In the two-band (gs, ν12 ) analysis of the global data set for CH3 SiH3 m = 86,709(351) MHz, in good agreement with in [37], the fitting parameter Aζ12
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the value for A12,F of 86,985(1) MHz calculated from the zeroth-order relationship A12,F = A12 /(1 − ρ). In the three-band analyses (gs, ν12 , ν5 ) applied m could not be varied independently, but once the ν5 band was investigated, Aζ12 m a good fit was obtained with Aζ12 fixed at the derived value of A12,F [36,27]. m /A However, similar analyses show Aζ12 12,F ∼ 0.70 in CH3 CD3 [51,22] and m Aζ12 /A12,F = 0.86(1) in CH3 SiD3 [38]. On one hand, it is possible that the results in CH3 CD3 and CH3 SiD3 arise from correlations with other parameters and limitations of the data sets. In particular, for these molecules, the intrinsic splitting is too small to be seen directly in the spectra. On the other hand, it is possible that m /A the relationship Aζ12 12,F = 1 is specific to CH3 SiH3 . After all, no theoretical explanation for this (near) equality has been given. Clearly, further investigation m and A is required into the relationship between Aζ12 12,F . The strongest (gs, ν12 ) coupling occurs between the bright state |b− = |v12 = 1, l12 = −1; J, v6 = 0, k = 5, σ = −1 and its dark state partner |d = |v12 = 0, l12 = 0; J, v6 = 3, k = 6, σ = −1. In the two-level approximation, the eigenstates can be written |β− = c(β− , b− )|b− + c(β− , d)|d and |δ = c(δ, b− )|b− + c(δ, d)|d, where the expansion coefficients c are such that |β− is predominantly |b− and |δ is predominantly |d. Nearly complete mixing occurs for values of J near the crossing, which falls between 16 and 17. In the 1989 study [37], 14 lines in the ν12 Q-branch series p Q−1 6 (J ) with 8 J 25 were recorded in which the upper eigenstate was |β− , along with two gs (v6 = 3 ← 0) lines q Q−1 6 (J ) with J = 15 and 16 in which the upper eigenstate was |δ. Only the six ν12 p Q−1 6 (J ) lines with J 15 were reported in 1989 [37]; the remainder were identified near 512 cm−1 in the 1989 spectrum once the (v6 = 3 ← 1) band discussed in Section 3.3.2 was analyzed in 1995. See Table IV of [35]. The vibration–torsion difference band between |β− and |δ was subsequently measured in the microwave region [27]; see Section 3.1. At the crossing, |β− is shifted in energy by ∼10 GHz. In this state, traditionally labelled (v12 = 1, v6 = 0), the frequency for tunneling through the torsional barrier is increased from ∼600 MHz to ∼10 GHz by the mixing. The lowest lying parallel band ν5 was measured in CH3 SiH3 from 670 to 730 cm−1 with the BOMEM DA3.002 interferometer at U.B.C. [36]. The experimental conditions were very similar to those used for the ν12 study. The F W for the instrumental line shape in the deconvolved spectrum was again νH M about 0.0022 cm−1 . Since ν5 is a non-degenerate vibrational level, the selection rules are the same as those given in Section 3.1 for gs pure rotational transitions except that v5 = 1. Within the simplified model, the spectrum should look like the parallel band of a C3v molecule with some torsional fine structure [36]. Consider the q Rkσ (8) manifold of lines as an example. From Eq. (11) and the values of A5 and B5 in Table II, there should be a strong, blended feature for the low values of k, with the higher k-lines shaded to the high frequency side. These should be separated
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F IG . 11. The (J = 9 ← 8) region of the ν5 band of CH3 SiH3 : (A) observed spectrum after deconvolution; (B) corresponding synthesized spectrum. In (A), for the q Rkσ (8) transitions, the Latin letters indicate the values of (k, σ ): a → (2, −1); b → (5, 0); c → (0, ±1); d → (3, 0); e → [(0, 0), (1, 0), (1, +1), (2, 0), (3, −1)]; f → (2, +1); g → (3, +1); h → [(4, +1), (4, −1), (6, +1)]; i → (5, +1); j → (5, −1); k → [(6, 0), (8, +1)]; l → (6, −1); m → [(7, 0), (7, −1)]; n → (8, −1); o → (8, 0); p → (7, +1); and q → (4, 0). The label β indicates the highly perturbed ν5 line q R1−1 (7), while δ indicates the perturbation-allowed gs (v6 = 5 ← 0) transition q R1−1 (11).
by increasing amounts and have decreasing intensity as k goes up. Since ν5 is predominantly the C–Si stretch, the central bond length is expected to be slightly longer in ν5 than in the gs. Consequently, the ν5 barrier height should be slightly (0) lower, and the σ -splitting due to ET (ν5 , 0, k, σ ) should be a little larger than (0) that due to ET (gs, 0, k, σ ). From these simple arguments, the resulting intrinsic σ -splitting in the spectrum is ∼0.004 cm−1 , smaller than the k-splitting except for low values of k [36]. However, this description bears little resemblance to the observed spectrum shown in Fig. 11, which is taken from Fig. 1 of [36]. Clearly, the q Rkσ (8) spectrum is severely perturbed. Similar effects are seen throughout the P - and R-branches in ν5 . The serious fragmentation of the ν5 band can be understood in terms of the Fermi-like (gs, ν5 ) coupling with reference to Fig. 12, which is taken from Fig. 2 of [36]. The final J = 12 energies are plotted for the ν5 levels with v6 = 0, as well as for the gs levels with v6 = 4 and 5. As was the case for the perturbation of ν12 , the fragmentation of ν5 can be assessed in terms of the energy differences between interacting (gs, ν5 ) partners because the coupling matrix elements have only a
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F IG . 12. Energy level diagram to illustrate the interactions in CH3 SiH3 that make the largest contributions to the Fermi-like shifts in the ν5 bright states |b ≡ |v5 = 1; J = 12, v6 = 0, k, σ . Each state is labelled by the quantum numbers at the bottom of its column and the value of k on the level itself. The diagonal lines indicate the (near) resonance between |b and its dark interaction gs partner |d ≡ |v5 = 0; J = 12, v6 = 5, k, σ for the four most severely perturbed (k, σ ) components. For ν5 , the σ -splitting is too small to be seen clearly on the energy scale used, and all three σ -levels are shown as coincident.
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weak (k, σ ) dependence. For each bright state |b = |v5 = 1; J, v6 = 0, k, σ , its closest gs interaction partner is the dark state |d = |v5 = 0; J, v6 = 5, k, σ . However, the more distant gs and ν5 levels with the same (J, k, σ ) also have an important effect. These gs levels with v6 < 5 are all lower than |b. Similarly, the corresponding ν5 levels with v6 > 0 are all higher than |d. The resulting non-resonant (gs, ν5 ) coupling pushes the energy for |b up and the energy for |d down, raising the former relative to the latter by ∼10 cm−1 . (Note the large magnitudes of the Fermi-like coupling constants in Table II.) These non-resonant effects have a weak dependence on (J, k, σ ). As can be seen from Fig. 12, the shift in the energy of |b produced by the direct coupling between |b and |d can have either sign, depending on the values of k and σ . Furthermore, the magnitude of this shift can vary considerably; the (k, σ )-dependence can be ∼1 cm−1 . The J -dependence of the Fermi-like shift vanishes in first order, as can be seen from the expression given for Hgs,ν5 in Table II and from the fact that the associated unperturbed energy difference depends only weakly on J . As a result, the fragmentation pattern is much the same for all the J -values investigated. The non-resonant shifts are critical to understanding the final energy pattern. In cases such as (k = 1, σ = −1), the sign of the (near) resonant shift is changed by the non-resonant contribution. As is indicated by the dot-dashed diagonal lines in Fig. 12, three of the rotational series showing the strongest near-resonance effects have (k, σ ) = (2, −1), (5, 0), and (4, 0). In Fig. 11, the corresponding transitions in the q Rkσ (8) spectral region are labelled a, b, and q. As expected from Fig. 12, the (2, −1) and (4, 0) components in q Rkσ (8) are shifted to the low and high frequency sides of the pattern, respectively. The σ -splitting between the (2, −1) transition and its (2, +1) counterpart (labelled f in Fig. 11) is 0.227 cm−1 , a factor of ∼50 larger than the predicted intrinsic value. For each value of J , an accidental degeneracy occurs between the states |b and |d with (k = 1, σ = −1); see the dashed diagonal line in Fig. 12. In this resonance case, it is convenient to label the eigenstates |β and |δ in analogy with the corresponding discussion of the resonances in ν12 , again using the two-level approximation. While the lines labelled with Latin letters in Fig. 11 are part of q Rkσ (8), the (1, −1) component labelled β is a (J = 8 ← 7) transition shifted up in frequency from the q Rkσ (7) region by more than 2B5 . As a result of the nearly 50/50 mixing, transitions to upper level |δ that is predominantly |d become allowed through the admixture that is |b. In Fig. 11, the line labelled δ is just such a (v6 = 5 ← 0) transition with J = 12 ← 11. A total of 29 members of the gs (k = 1, σ = −1) series with v6 = 5 ← 0 have been identified and added to the data set; see Table III of [36]. The analysis was carried out with the one-step procedure [36] in which the matrix for HG VTR is set up in the free rotor basis; see Section 2.3. In Fig. 11, the observed spectrum (upper panel) and its synthesized counterpart (lower panel) agree very well with respect to both
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frequency and relative intensity. The agreement is particularly satisfying for the two (k = 1, σ = −1) lines. The anomalous form of the probability distribution P (α) plotted for the lowest energy level in Fig. 2 can be easily explained in the light of the preceding discussion. The level in question is typically thought of as the ν5 state for v6 = 0 with given J , k = 1, and σ = −1 i.e. as the bright state |b. However, the lowest level in Fig. 2 is |β, which is here a linear combination of |b and |d. Because the mixing is nearly 50/50, |β has a good deal of |d-character. As can be seen from Fig. 3 of [36], the probability distribution for a pure |d-state shows a maximum at the peak of the potential. This is expected classically. After all, |d is well above the top of the potential (see Fig. 10, for example), and classically the velocity is a minimum when the potential energy is a maximum. The surprising maximum of P (α) in the classically forbidden region in Fig. 2 arises for |β through its |d-character. Furthermore, because of the |d-character of |β, there is a “giant” σ -splitting between q R1−1 (8) and q R10 (8)/q R1+1 (8) (blended line e of Fig. 11) of ∼0.9 cm−1 . The frequency for tunneling through the torsional barrier for these mixed ν5 states is increased from ∼720 MHz to ∼27 GHz. The anharmonic interactions can clearly produce very strong (gs, ν5 ) coupling between levels which differ in v6 by 5 units. This (near) resonant mixing has serious implications for energy transfer between the different vibrational states i.e. for IVR [36]. Consider the case where the gs and ν5 stacks of levels are isolated from one another. If a specific ν5 state with v6 = 0 is excited with a laser pulse, the rotational and torsional states in ν5 would typically equilibrate well before the excess energy appears in the gs. However, the gs and ν5 stacks are coupled by the Fermi-like interactions that lead to the fragmentation of the ν5 spectrum, as illustrated in Fig. 11. As a result, the energy deposited in ν5 would find its way into the gs on a time scale comparable to that for the rotational and torsional equilibration. Similar effects occur as a result of the Coriolis-like coupling between the gs and ν12 stacks of levels. Because the magnitudes of the Fermi-like and Coriolis-like matrix elements fall off relatively slowly with |v6 |, the pool of candidate gateway states for such energy transfer is much larger than would be expected if all the torsional states involved harmonic motion. A three-band analysis of ν5 , ν12 , and the gs has been carried out for CH3 SiH3 using a one-step procedure to diagonalize the effective Hamiltonian HG VTR ; see Section 2.3 and [36]. A good fit was obtained by varying 41 molecular parameters; the values obtained are given in Table I of [36]. The model was subsequently refined using the results of the microwave and mm-wave study reported in [27] and discussed here in Section 3.1. The data set used involving a total of 3423 frequencies is summarized in Table I. Again, a fit to well within experimental error was obtained. The determinations made for the molecular constants are listed in Table II. The characteristics of the fit for this study are given in
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Table 1 of [27], together with a more detailed specification of the global data set. Each of the vibrational quanta ν 5 and ν 12 is defined as the energy difference between the bottom of the potential in the excited state and the bottom of the potential in the gs [32,36,37]. In the data available for ν5 , only levels with v6 = 0 have been probed. As a result, in the least squares analysis, ν 5 is very highly correlated with the average zero-point energy E T ,ZP (ν5 ) associated with the torsional motion in ν5 . Since E T ,ZP (ν5 ) is determined by V5,3 , there is a very strong correlation between ν 5 and V5,3 . In order to deal with this difficulty, ν 5 is replaced as a fitting parameter by ν˜ 5 ≡ [ν 5 + E T ,ZP (ν5 )]. Once V5,3 is determined by the torsional splittings, ν 5 can be calculated from ν˜ 5 . For computational convenience in the case of ν5 , E T ,ZP is replaced by 12 V5,3 . In ν12 , a similar situation arises. Two different measures of E T ,ZP (ν12 ) have been used. In the two-step procedure [37,32], E T ,ZP (ν12 ) was set equal to the unweighted average over (0) the three values of σ of ET (ν12 , v6 = 0, k = 0, σ ). (The simplified model is used here for illustration purposes.) As can be seen from the discussion of Oeven ν,vi ,k,σ in Section 3.1, this average is independent of k. In the one-step procedure, the two-step definition is awkward to implement. Instead, E T ,ZP (ν12 ) was set equal to the unweighted average over the three values of σ of the eigenvalues of Hν (as defined in Section 2.2) for J = v6 = k = 0 [36], an average which is also independent of k (and l12 ). The one-step measure of E T ,ZP (ν12 ) ml " includes the effect of the matrix elements of 2Aζ12 12 Jα off-diagonal in v6 , but the two-step definition does not. This operator acts like an anharmonic term and depresses the (v12 = 1, v6 = 0) energies by ∼1 cm−1 for low J and k [32]. The one-step value of ν˜ 12 is consequently ∼1 cm−1 lower than the two-step value. However, after correction for this effect, the former and latter yield the same result for ν 12 . In traditional normal mode analyses, a vibrational quantum with a different physical meaning is adopted. For the ν12 mode, the traditional definition can be expressed as the difference ν12,∞ between the eigenvalue of Hν for (v12 = 1, l12 ; J, v6 = 0, k, σ ) and that for (v12 = 0, l12 = 0; J, v6 = 0, k, σ ) in the limit that J → 0 and k → 0 under the assumption that V12,3 = V0,3 → ∞ [32]. This definition is often implicitly adopted when torsional effects are not considered, as in [116]. To convert ν 12 to ν12,∞ when the two-step procedure is used, ν 12 must increased by the difference [E T ,ZP (ν12 ) − E T ,ZP (gs)] and decreased m" by the anharmonic shift due to 2Aζ12 Jα l12 . When the one-step procedure is used, the correction due to the anharmonic shift is not required. For a non-degenerate mode, only the zero point correction enters. In terms of the traditional definition, ν12,∞ = 519.92(18) cm−1 and ν5,∞ = 704.55(13) cm−1 for CH3 SiH3 [36]. The relatively large error arises from the high correlation between ν 12 /ν 5 and the corresponding excited state barrier heights V12,3 /V5,3 .
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3.4.2. CH3 CH3 Several studies of the lowest lying perpendicular band of ethane at moderate or high resolution have been reported. One of the first such investigations of ν9 was carried out by Daunt et al. [117]. Unfortunately, the resolution of 0.04 cm−1 did not allow the observation of any torsional splittings. Patterson et al. [118] indirectly observed the effect of torsional splittings from an intensity perturbation in the r Q0 branch using a diode laser spectrometer. A more complete investigation of the methyl rock in ethane was made by Susskind et al. [87] who observed diode laser spectra of the Q-branches. A deconvolution procedure was used to obtain an effective resolution below the Doppler limit. The measured torsional splittings were fitted to a model which treated the Coriolis interaction between the excited vibrational level (v9 = 1, v4 = 0) and the excited torsional level (v9 = 0, v4 = 3) in the gs with second order perturbation theory. Three parameters were used in the fit: the barrier to internal rotation in v9 , found to be 1123(10) cm−1 ; the energy difference between the torsional sublevel (v9 = 1, v4 = 0, σ = 0) and its interacting partner (v9 = 0, v4 = 3, σ = 0); and an effective Coriolis coupling constant. Torsional splittings in the gs were calculated using the torsional parameters V0,3 = 1024 cm−1 and V0,6 = 0 cm−1 [109]; the reduced rotational constant A0,R was taken as A0 = 2.671 cm−1 [119]. In a subsequent study, Daunt et al. [120] used both high resolution Fourier transform and tunable diode laser spectra to carry out a whole band analysis. The essential features of the theoretical treatment were the same as those in [87]. Henry et al. [121] carried out a whole band analysis of ν9 also using high resolution Fourier transform and diode laser spectra. The model used to analyze the data included the standard unperturbed Hamiltonian for the lower and upper states as well as an unperturbed Hamiltonian for (v9 = 0, v4 = 3) with rotational constants that are slightly different from those for (v9 = 0, v4 = 0). In the analysis, it was assumed that all four torsional components in (v9 = 0, v4 = 3) had the same rotational parameters. In addition, l-type doubling and a σ -independent Coriolis interaction between (v9 = 0, v4 = 3) and (v9 = 1, v4 = 0) were considered. Other torsional levels in the gs and in ν9 were assumed to have negligible effects. Henry et al. [121] found that rotational levels of the (v9 = 0; v4 = 3, k = 18, σ = 0) cross interacting rotational levels for (v9 = 1, l9 = −1; v4 = 0, k = 17, σ = 0). The spectrum was followed to k = 19 on the p P side of the band to permit the inclusion of ν9 levels beyond this crossing. No transitions belonging to 3ν4 arising from resonant mixing were identified. The most recent study of the ν9 band is the two-band analysis reported in [41]. Three spectra were recorded in this study. Two of these, one at a temperature of 133 K and the other at room temperature, were measured at a resolution of 0.0014 cm−1 using the modified BOMEM spectrometer at N.R.C. [76]. The low temperature spectrum was recorded for the ν9 band, while the room temperature spectrum was recorded for the ν9 +ν4 −ν4 band. The sample pressure was 0.2 Torr
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F IG . 13. Torsional energy level diagram for the gs, ν9 , and ν3 in CH3 CH3 and CD3 CD3 . For each vibrational state, only one of the six equivalent wells in the hindering potential V (γ ) is shown, with γ plotted from 0◦ to +60◦ .
in the low temperature case and 0.5 Torr for the room temperature measurement. The absorption pathlength was 2 m. A third spectrum was recorded at a resolution of 0.00125 cm−1 using the Bruker IFS120 HR Fourier transform spectrometer at U.O. The sample temperature and pressure were 293 K and 3.0 Torr, respectively. Over 1400 transitions from the ν9 band, 24 transitions from 3ν4 , 212 lower state combination differences from the ν9 + ν4 − ν4 hot band, and the frequencies from the far-infrared torsional bands available at that time were used in the analysis. This two-step J 2 –vi first order procedure used 21 torsional states: 7 for the gs, 7 for ν9 with l9 = +1, and 7 for ν9 with l9 = −1. Furthermore, it was shown that the mechanism responsible for the fragmentation of the ν9 band is the Coriolis coupling given by O47 . This far more rigorous model allowed an excellent fit of both the highly perturbed ν9 transitions and a perturbation-allowed rotational series identified in the 3ν4 band. The energy level pattern for ethane is relatively simple. The torsional energy levels and the torsional splittings for the gs, ν9 , and ν3 are shown in Fig. 13, which is adapted from Fig. 3 of [52]. Because (k + σ ) must be even in the gs, each transition in the ν9 band consists of a doublet with one component having the label σ and the other having the label σ + 2. In vibrational state ν, the intrinsic
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σ -splitting in the energy for the lowest torsional level is ET (ν, v4 = 0, σ = 2) (0) (0) (0) − ET (ν, 0, 0) ET (ν, 0, 3) − ET (ν, 0, 1), where the notation of Section 3.1 (0) has been used, adjusted for the fact that ET is independent of k. In the gs, this −1 σ -splitting is 0.006 cm and decreases to ∼0.004 cm−1 in ν9 , due to a higher effective barrier height for this state. Thus in the absence of any other effects, the σ -component of the doublet would be higher in frequency by 0.002 cm−1 relative to its (σ + 2) partner. This prediction is confirmed for medium or high values of k in the branch with k > 0. This is illustrated in Fig. 14a, which contains part of the r Q8 sub-branch. In each doublet, the weaker member at higher frequency is the σ = 0 component. The lower trace is the experimental spectrum and the upper trace is the simulated spectrum. For these transitions, kk = +8. However, as is shown in Figs. 14b–14d, the σ -splitting gradually increases as the algebraic value of kk decreases, and grows to as large as several tenths of a cm−1 for kk = −18. (Figs. 14a–14d are adapted from Figs. 1–4 in [41].) This behaviour is most easily explained by considering the torsional levels nearest to the upper state (v9 = 1, v4 = 0) in the transition (see Fig. 13) and the expression for the Coriolis coupling operator O47 in Table II. To higher energy, the closest neighbour is the gs torsional level v4 = 4, but this state can be discounted because the magnitude of the torsional factor v4 = 0, σ |Jγ |v4 = 4, σ in the leading coupling matrix element is either zero (for σ = 0 and 3) or nearly zero (for σ = 1 and 2). The same can be said about v4 = 0, σ | sin 6γ |v4 = 4, σ . (The notation here suppresses the vibrational dependence of the torsional functions.) To lower energy, the nearest neighbour is the gs torsional level v4 = 3. Here, the magnitudes of the matrix elements of Jγ and sin 6γ coupling the two states are relatively large with a small σ -dependence. In addition, the energy of the bright state (v9 = 1, v4 = 0, σ ) is virtually independent of σ and is higher than that of its dark counterpart (v9 = 0, v4 = 3, σ ) by 49, 63, 82, and 90 cm−1 for σ = 0, 1, 2, and 3, respectively. See Table I of [41]. It is therefore apparent that there is a shift to higher energy introduced by O47 which is largest for the σ = 0 sublevel of the bright state and smallest for σ = 3. As a result, the σ and (σ + 2) components of each doublet experience different frequency shifts. To explain the fragmentation of the spectrum as a function of kk, we find the zeroth order J -independent energy gap E(k, k, σ ) between the levels in (0) (0) question. If EV T (σ ) ≡ ν 9 + ET (ν9 , ν4 = 0, σ ) − ET (gs, ν4 = 3, σ ), then [41]: E(k, k, σ ) = EV T (σ ) + (A − B) − 2(Aζ9z ) k 2 + 2 (A − B) − (Aζ9z ) kk = EV T (σ ) + (0.6147 + 2.61893kk) cm−1 .
(28)
For each σ , EV T (σ ) > 0 and the p P branch transitions are perturbed more than their r R counterparts because the term 2.61893 kk reduces the energy gap
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F IG . 14. Four small portions of the ν9 band of CH3 CH3 illustrating the progression of torsional splitting as a function of kk as it approaches the energy σ =0 (28) labelled with level crossing: (a) kk = +8; (b) kk = +2; (c) kk = −8; (d) kk = −18. In (d), note the perturbation-allowed 3ν4 transition q P18 the asterisk. In each panel, the lower trace is the experimental spectrum, while the upper trace is its synthesized counterpart.
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for negative k. Equation (28) can be used to calculate the values of k for which level crossing will occur. These are k 18, 25, 32, and 35 for σ = 0, 1, 2, and 3, respectively. Except for k 18, these crossings occur at unreasonably large k. Note that the energy level crossing for σ = 0 occurs at 25 < Jc < 26. The resonance for (k = 18, σ = 0) has been observed and the associated perturbationσ =0 (J ) in the 3ν band has been identified. In the allowed rotational series q Pk=18 4 lower trace of Fig. 14d, the doublet p P18 (29) is labelled. The (σ = 0) component is resonantly shifted to higher frequency and the resulting σ -splitting is very large 0 (28) in the 3ν band is also (∼0.4 cm−1 ). The perturbation-allowed transition q P18 4 p identified along with other P doublets which fall in this frequency region. The characteristics of the fit and molecular parameters deduced in this work are given in Tables II and III, respectively, of [41]. The lowest lying parallel band in CH3 CH3 is the C–C stretch ν3 , which is infrared inactive and has been studied by a variety of Raman techniques. Al-Kahtani et al. [122] examined ν3 using high resolution coherent anti-Stokes Raman spectroscopy. At room temperature, a complex Q-branch pattern was observed. To simplify the spectrum, the authors obtained jet spectra for ethane itself and for 50% mixtures in helium. The spectra of these cold samples showed that the torsional component with torsional symmetry ΓT = A3d (σ = 3) is red shifted by over 1 cm−1 ; see Fig. 2 in [122]. A more comprehensive study of the ν3 band was made by Bermejo et al. [43], who recorded the stimulated Raman spectrum with a resolution of 0.0055 cm−1 . The analysis revealed a strong dependence of the torsional barrier on the normal co-ordinate q3 . This dependence was attributed to Fermi coupling between the gs, the fundamental level v3 = 1, and the overtone level v3 = 2. With first order perturbation theory, a value of −275.9(1.4) cm−1 was determined for the derivative of the potential barrier with respect q3 . A three-band analysis of ν3 , ν9 , and the gs has subsequently been made for ethane using the data set summarized in Table I involving a total of 2889 frequencies [40]. The data set is illustrated graphically by the solid arrows in Fig. 13. 2 To diagonalize the effective Hamiltonian HG VTR , a two-step J –vi exact procedure was used which included 9 torsional states for each of the gs and ν3 , as well as 7 torsional states for each of the two l-components of the ν9 state. A good fit to well within the experimental error was obtained. The resulting values for the molecular parameters are listed in Table II. The characteristics of the fit for this study are given in Table 4 of [40], along with a more detailed specification of the global data set. From the values of ν˜ 3 and ν˜ 9 , respectively, in Table II, it has been found that ν3,∞ = 993.310 cm−1 and ν9,∞ = 821.290 cm−1 [40]. These results agree well with the deharmonized values of ν3,∞ = 994 cm−1 and ν9,∞ = 820 cm−1 obtained by Duncan et al. [89]. (For the definitions of νν,∞ , see Section 3.4.1 and [41].) The severe fragmentation of the ν3 band by the Fermi interaction can be understood by considering Fig. 13 here and Table 2 in [40], as well as the ex-
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pression for the leading Fermi coupling operator O55 given in Table II of the present work. First, the dark state providing the resonant interaction must be v4 = 4 of the gs. It is clear from Fig. 13 that this is the nearest neighbour to the upper state (v3 = 1, v4 = 0) in the transition. Furthermore, the torsional factor in the matrix element connecting the bright and dark states in this case is v4 = 0, σ | 12 (1 + cos 6γ )|v4 = 4, σ , which has a sizable magnitude 0.019 with only a 15% variation depending on σ . The next nearest neighbours are v4 = 3 and 5 of the gs. However, these levels can be discounted since |v4 = 0, σ | 12 (1+cos 6γ )|v4 , σ | either vanishes or is very small if v4 is odd. Second, the resulting large energy shifts (∼1 cm−1 ) have different signs for different values of σ . Relative to the bright sublevel (v3 = 1, v4 = 0, σ ), the perturbing dark sublevels (v3 = 0, v4 = 4, σ ) are located below for σ = 0, 1, and 2, but above for σ = 3. Thus the sublevels with σ < 3 are shifted up by the resonant component of the interaction, while that with σ = 3 is shifted down. See Table 2 of [40]. Finally, these effects are very insensitive to J and k. With the resonance terms set to zero, the energy gaps between the interacting states are relatively slow functions of J and k. Furthermore, the matrix elements of O55 are independent of J and k. Thus, as shown in [40], the (gs, ν3 ) Fermi interaction explains the observed serious fragmentation of the ν3 band for all J and k, with the (σ = 3) sub-branch red shifted by more than 1 cm−1 . The situation is similar to that found in CH3 SiH3 [36] and CH3 CD3 [22]. 3.4.3. CD3 CD3 The first frequency analysis of the ν9 band of CD3 CD3 was reported in [55]. Three spectra were recorded for this study at N.R.C. The first spectrum was recorded at room temperature with a resolution of 0.0024 cm−1 . A low temperature spectrum (128 K) at the same resolution was also measured. After a preliminary analysis of the ν9 band, it was realized that the high k sub-branches with k = −1 showed large torsional splitting. To follow these rotational series with large negative kk, a third spectrum with a longer absorption path (20 m) was recorded. In this case, the sample was at room temperature and the resolution was 0.0041 cm−1 . In the hot band ν9 + ν4 − ν4 , twelve (k, σ ) rotational series were identified and used to generate lower state combination differences. As for CH3 CH3 , each rotational line in the ν9 band is expected to consist of a doublet. However, the reduced barrier height for CD3 CD3 is much larger than that for CH3 CH3 . This results in vanishingly small intrinsic σ -splittings for v4 = 0 in both the gs and ν9 . Therefore, no torsional doubling is expected in the spectrum. This is, in fact, what we observe in the entire r R-branch (kk > 0) and for lines in the p P -branch (kk < 0) with low |kk|. The picture is, however, different for lines with k = −1 and large |kk|. For kk = −9 and −10, there is a gradual broadening of the lines indicating the onset of torsional splitting. At kk = −11,
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the separation between the doublet components becomes clearly visible and it grows to about 0.25 cm−1 for kk = −15. The fragmentation of the spectrum as a function of kk can again be explained with reference to the energy level diagram in Fig. 13. The nearest state which can interact strongly with the upper level (v9 = 1, v4 = 0) in the transition is v4 = 3 of the gs. When Eq. (28) is modified for CD3 CD3 , the zeroth order J -independent energy gap is given by [55]: E(k, k, σ ) = EV T (σ ) + (0.3114 + 1.1926 kk) cm−1 . cm−1
(29)
with a variation of about 1 cm−1 The average of EV T (σ ) over σ is 16.9 due to the σ -splitting for v4 = 3 of the gs. See the discussion of Table 2 in [55]. As with CH3 CH3 , (near) resonant interactions are possible only in the branch with kk < 0 because EV T (σ ) is positive. Equation (29) can be used to predict the k sub-branches where level crossing will take place. By setting k = −1 and E(k, k, σ ) = 0, we obtain k ≈ 15 for the crossing value. Resonances for p P sub-branches with (k, σ ) = (15, 1), (16, 2), and (17, 3) have, in fact, been observed. In addition, perturbation-allowed 3ν4 transitions have been identified in each case and successfully included in the final data set; see Table I. The largest effects were observed in the case of p P15 . Here the bright state σ =1 (J ) are comes between the two σ -levels of the dark state. The lines for p P15 σ =3 p pushed to lower frequency, while those for P15 (J ) are pushed upward. Because the unperturbed levels with σ = 1 are much closer to each other than those 1 is much more severe. In the region where the with σ = 3, the perturbation of p P15 mixing for σ = 1 is close to 50/50, three rotational series are observed, the extra 1 (J ) in the “forbidden” band 3ν . In the resonance region, the series being q P15 4 dark 3ν4 sublevel assumes a good deal of the character of the bright sublevel with which it is mixed and becomes bright as well. For σ = 1, the ν9 and gs sublevels involved act much like a two-level system. The level crossing for (k, σ ) = (15, 1) is illustrated in Fig. 15, which is adapted from Fig. 5 of [55]. (The original article can be found on-line through the web site http://www.tandf.co.uk.) The frequency 1 (J ) and shift from the unperturbed value is plotted as a function of J for p P15 p P 3 (J ) (shown by • and solid triangles, respectively). The corresponding shift 15 1 (J ) (shown by ◦). As J is also plotted for the perturbation-allowed 3ν4 series q P15 1 1 p q increases, the frequencies for P15 (J ) and P15 (J ) move away from each other because the Coriolis interaction gets larger. Finally, the two series reach their maximum separation at J = 22. The level crossing is marked by the dramatic change in sign of the large frequency shift experienced by each (σ = 1) series. For most of the available data on the gs of CD3 CD3 , the experimental uncertainty is relatively large (see Table I), and so it was not possible to determine all the quartic parameters at the same time. To overcome this difficulty, the quartic parameters F0,3J and F0,3K were calculated by ab intio methods [55]. The reliability of this procedure was tested for ethane, for which the data set allows for the
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F IG . 15. The deviation from the unperturbed frequency of CD3 CD3 for the two rotational series p P σ =1 (J ) and p P σ =3 (J ) in the ν band, and for the series q P σ =1 (J ) in the perturbation-allowed 9 15 15 15 1 (J ) and q P 1 (J ) are resonantly mixed and cross between J = 21 3ν4 band. The upper levels for p P15 15
and 22.
experimental determination of all quartic parameters. These tests indicate that the ab initio results are accurate to 25% for CH3 CH3 . Consequently, F0,3J and F0,3K were held fixed at their ab initio values in the analysis of the CD3 CD3 data set. A two-band J 2 –vi exact procedure was used. The molecular parameters obtained are listed in Table 4 of [55]. The details of the global data set and the characteristics of the fit are given in Table 3 of the same reference. Note that only the lower state-combination differences generated for ν9 + ν4 − ν4 were included in the analysis, rather than the infrared frequencies themselves. The constants F0,3J and F0,3K are a measure of the change of the rotational constants B and A, respectively, between the staggered and eclipsed configurations. The ab initio methods for calculating these parameters are of broader interest, as they can be used for molecular systems which are beyond the reach of current experimental methods. The band ν9 + ν4 − ν4 has very recently been investigated in detail [52] using the room temperature spectrum recorded by Szott et al. at a resolution of 0.0024 cm−1 . The band center for the hot band is blue-shifted by a few cm−1 relative to that for the fundamental because the effective barrier height in ν9 is larger than in the gs. The intensity of the hot band is down by a factor ∼3 at a temperature of 300 K, mainly due to the lower thermal population of the (v4 = 1) level. However, the structure of the two bands is the same, and so the lines in the hot band are easily assignable by the usual spectroscopic means. The intrinsic σ -splitting is about 0.001 cm−1 for v4 = 1 in both the gs and ν9 . Consequently,
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in the absence of intervibrational interactions, no σ -doubling is expected in the spectrum. No resonances are expected as can be seen from Fig. 13, and none was detected. However, line broadening and doubling were observed at J 25 due to non-resonant perturbations by bath levels in the gs. In general, these doubled lines were very weak and not well resolved, and were therefore excluded from the fit. Nevertheless, their frequencies and intensities were successfully modelled using the best fit parameters obtained. In Fig. 1 of [52], a segment of the experimental spectrum is compared to its simulated counterpart. In Fig. 2 of [52], the corresponding comparison is given for the 128 K spectrum recorded at the same resolution. Most of the lines not reproduced in the room temperature simulation are absent in the low temperature experimental spectrum, and are therefore likely to be from the next hot band ν9 + 2ν4 − 2ν4 . A global two-band analysis including the ν9 +ν4 −ν4 band was carried out and a good fit obtained; the determinations made for the molecular parameters are given in Table 1 of [52]. The data set is illustrated graphically by the arrows in Fig. 13. The analysis required the addition of several barrier-dependent terms to the effec1 tive Hamiltonian HG VTR . These include η9,3 2 (1 + cos 6γ )Jz 9 , which is associated z with Coriolis coupling constant Aζt , and q9,3 14 (J2+ q29− + J2− q29+ ) 12 (1 + cos 6γ ), which is associated with the l-doubling constant q9 . (See Table II.) Several higher order terms were also necessary; the full Hamiltonian is given in [52]. This work by Cooper and Moazzen-Ahmadi [52] represents the first report on an X2 Y6 symmetric top in which the hot band of a degenerate fundamental has been included in the global analysis. The first such study of an XY3 AB3 symmetric rotor for a non-degenerate fundamental (and its associated hot band) has been carried out recently by Wang et al. [56] for the ν5 band of CH3 CF3 ; see Table I. It has been demonstrated that non-resonant interactions between a bright state and a distant dark state differing by a large number of torsional quanta can produce significant fragmentation of the bright state. In addition, these studies have important implications for the parameters assigned to the vibrationally excited states involved; the barrier dependence of these parameters is necessarily undetermined in studies which consider data only in the lowest torsional state of the excited vibrational mode.
4. Discussion To assess the importance of hindered internal rotation in dynamic processes which involve vibrationally excited molecules, it is necessary to understand in detail the mechanisms that couple the torsional coordinate and to the skeletal vibrations. This requires the formulation of the vibration–torsion–rotation Hamiltonian and subsequently the determination of the leading coupling parameters such as the
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Coriolis and cubic potential constants. Many of the difficulties in treating problems of this type arise from the essential features of the torsional mode. It has a frequency which is much lower than that of the normal vibrations and so tends to have a high density of states near the vibrational state in question. Furthermore, internal rotation is highly anharmonic, and so the mixing matrix elements between torsional oscillator states which differ by a large number of quanta νi are often orders of magnitude larger than their counterparts for coupled harmonic states. As a result, the torsional mode is much more likely to mix strongly with the skeletal vibrations. The analysis of vibration–torsion–rotation interactions is most easily done at low energy where the degree of vibrational excitation is low. The normal mode description is then valid and the number of the background states is small enough that the coupling of the skeletal mode in question with other vibrational states, including those involving the torsional degree of freedom, is characterized by the lowest order terms. Once the coupling mechanisms are understood and the associated molecular parameters are determined, the results can be used to analyze the multitude of torsion–vibration interactions which are required for the interpretation of spectra of the vibrational fundamentals at high frequency, and of spectra involving high degree of vibrational excitation. It is in this spirit that much of the work described here has been carried out. The values of the leading torsional molecular parameters are significantly different in some cases depending on whether the different skeletal modes are treated explicitly or implicitly in the analysis. For CH3 CH3 , the parameters values obtained are given in Table II for the most recent (gs, ν9 , ν3 ) three-band analysis [40] and in Table VI for the earlier (gs, ν9 ) two-band treatment [41]. These values will "i , respectively. In the two-band case, the (gs, ν3 ) here be referred to as Xi and X Fermi interaction is eliminated implicitly by the a priori application of a contact transformation, and the effects of the elimination are absorbed into the effective "i − Xi Hamiltonian. This elimination process generates a difference Xi = X between the two- and three-band values. For several of the molecular parameters, theoretical expressions for Xi in terms of the Fermi-type parameters have been derived for XY3 AB3 symmetric tops [36,57]. These expressions are given here in Table VII in the form appropriate for ethane for the four cases where Xi is most significant [40]. For each of these cases, Table VII presents a comparison between the theoretical value Xicalc and its observed counterpart Xiobs . Here Xicalc is determined from the associated theoretical expression using the molecular constants given in Table II, while Xiobs is calculated by subtracting the value in Table II from its counterpart in Table VI. For the large, lower-order constants V0,3 and V0,6 , the agreement is very good. For the small higher-order parameters V0,9 and F0,3J , additional effects must be taken into account. When the Fermi interaction is eliminated, both V0,3 and V0,6 are observed to decrease by a significant amounts: 34.4 cm−1 and 8.8 cm−1 , respectively. The "0,6 ; that is, the entire decrease for V0,6 is particularly striking, because V0,6 = V
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Table VI Comparison of the Molecular Parameters for CH3 CH3 , CH3 CD3 , and CD3 CD3 a Parameter
Units
V0,3 V0,6 A0 B0 D0,J M D0,KM F0,3J F0,3K Vt,3 At Bt ζtz x ζi,t
cm−1 cm−1 MHz MHz MHz MHz MHz MHz cm−1 MHz MHz
CH3 CH3 b 1013.28(10) 8.798(41) 80184.8(7.7) 19913.371(44) 0.3393(43) 1.867(32) −348.0(1.3) 963.9(9.7) 1117.0(1.2) 240.910(86) −42.029(26) 0.2610(12) 0.2267(20)
CH3 CD3 c
CD3 CD3 d
1001.876(23) 9.328(18) 53651.5(1.1) 16500.653(40) 0.7585(45) 3.32(43) −288.86(24) 720.(29.) 1079.8(4.7) 152.55(22) −31.981(18) 0.248657(50) 0.2245(45)
989.946(90) 9.51(10) 40158.7(7.3) 13801.412(33) 0.1356(24) 0.298(13) −239.2(1.4) 448.1(9.3) 1066.95(24) 99.055(19) −14.1986(35) 0.20959(18) 0.2319(13)
a A (gs, ν ) two-band model was used in each case. See Section 4. t b Ref. [41]. Here V −1 and D 0,9 = 0.879(24) cm 0,m is fixed at zero. c Ref. [51]. Here V 0,9 is fixed at zero and D0,m = 7.976(60) MHz. d Ref. [52]. Here V 0,9 and D0,m are fixed at zero.
Table VII Corrections Xi to the ground state molecular parameters Xi in Hgs for CH3 CH3 from the elimination of the Fermi-type terms by a contact transformationa,b "i X
Theoretical expression for Xi
"0,3 V " V0,6
3 )2 + − 2ν1 [(C3,0 3 1 6 )2 − − 2ν [(C3,0
"0,9 V "0,3J F
3
1 C3 C6 ] 2 3,0 3,0 1 (C 3 )2 + C 3 C 6 ] 3,0 3,0 4 3,0
3 C6 + 4ν1 C3,0 3,0 3 3,J 3 + 1 C 6 )] − 2ν1 [C3,0 (2C3,0 2 3,0 3
Xicalc
Xiobs
−34.293
−34.404
8.217
8.782
0.157 44.
0.891 67.
a See Section 4. bV "0,6 , and V "0,9 in cm−1 ; F "0,3J is in MHz. "0,3 , V
two-band value of V0,6 arises from the (gs, ν3 ) Fermi interaction [40]. Corresponding conclusions have been reached for CH3 SiH3 [36], CH3 CF3 [56], and CH3 CD3 [22]. It is certainly not clear the electronic contribution to V0,6 vanishes. However, it is clear that the vibrational contributions such as that from the Fermi interaction are large, perhaps dominant. It is likely that similar effects occur in a wide variety of internal rotor molecules, including asymmetric tops. Since V0,6
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is the leading shape parameter determining the deviation of the potential from a pure cosine, this result has significance in the investigation of the electronic mechanism generating the barrier [123]. Equilibrium values for the first and second derivatives of the gs barrier height V0,3 with respect to the normal coordinate qs for the lowest lying non-degenerate mode can be determined by using the parameter values given in Table II. If V0,3 is expanded as a Taylor series in the normal coordinates and only the dependence on qs is considered, then [36,40]: ∂V3 1 ∂ 2 V3 V0,3 = V3,e + (30) qs + q2 + · · · . ∂qs e 2 ∂qs2 e s 3 = (∂V /∂q ) . FurIt can be shown that the leading Fermi coupling constant Cs,0 3 s e thermore, the difference (Vs,3 − V0,3 ) between the barrier heights in νs and in the gs equals (1/2)(∂ 2 V3 /∂qs2 )e . From these relationships and Table II, it has been determined for CH3 SiH3 that (∂V3 /∂q5 )e = 128.25 cm−1 and (∂ 2 V3 /∂q52 )e = −51.50 cm−1 [36]. Similarly, for CH3 CH3 , (∂V3 /∂q3 )e = 260.50 cm−1 and (∂ 2 V3 /∂q32 )e = −14.9 cm−1 [40]. The corresponding results for CH3 CD3 are (∂V3 /∂q5 )e = 257.06 cm−1 and (∂ 2 V3 /∂q52 )e = −101.92 cm−1 [22]. While no definitive conclusions can be drawn concerning the convergence of these series, the results for the different molecules compare favorably with one another. In CH3 SiH3 , it has also been determined that the derivative (∂F0 /∂q5 )e at equilibrium of the reduced rotational constant equals −0.225 cm−1 [36]. A selection of the molecular parameters for CH3 CH3 , CH3 CD3 , and CD3 CD3 are compared in Table VI. Ideally, the parameters shifts Xi due to the implicit contact transformations would be treated the same for all three molecules. In Table VI, this requirement is met to good approximation by using values from a (gs, νt ) two-band analysis in each case. A (gs, νt , νs ) three-band study has been reported for CH3 CH3 and CH3 CD3 , but not for CD3 CD3 . For each isotopomer, the gs barrier shape parameter V0,6 is approximately the same, and the barrier height in the excited vibrational state is about 10% higher than that in the ground state. As can be seen from the determinations of the gs barrier height V0,3 given in Table VI, the average of the values for CH3 CH3 and CD3 CD3 is 1001.61(14) cm−1 , which differs from the CH3 CD3 value by less than twice the statistical uncertainty. Based on information obtained from rotational spectra of three asymmetrically substituted isotopomers of ethane, Hirota et al. [124] suggested that the barrier height is a linear function of the number nD of deuterium atoms in the isotopomer with a negative slope mD . For the case that V0,6 is assumed to vanish, the determinations for V0,3 by these authors are summarized in cm−1 in Table 6 of [44]. From the values of V0,3 for CH3 CH2 D and CH3 CHD2 (the best determined cases), the change per deuteron is mD = −3.5(1.0) cm−1 , which agrees
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with the value of mD = −3.89(9) cm−1 obtained from Table VI. The agreement may well be fortuitous. The microwave study necessarily treated the data from the different isotopomers differently. Furthermore, the present global analyses include different higher order torsional parameters for the different isotopomers; see Table VI. Moreover, the microwave and current analyses are very different from each other. Nonetheless, these various effects are likely to be small and the agreement is very persuasive. The effect of the vibrational motion on the barrier height in the ethane isotopomers has been investigated by Kirtman et al. using Hartree–Fock methods [125]. For CH3 CH3 , distortion of the molecular skeleton during internal rotation was found to decrease the gs barrier height V0,3 by 56(7) cm−1 . Vibrational averaging over the zero-point motion associated with the skeletal vibrations was found to increase V0,3 by 131(31) cm−1 . The distortion effect is isotopically invariant, while the zero-point effect is not. From the isotopic change in the zero-point contribution, it was predicted that the CD3 CD3 barrier height would be lower than that for CH3 CH3 by 36 cm−1 . This is of the correct order, but somewhat larger than the decrease of 23.33(14) cm−1 measured here; see Table VI. A simple physical model can be developed that explains qualitatively this zero-point effect and its isotopic dependence. Consider the ν12 mode √ as an example. In the harmonic limit, the zero-point energy E12,ZP (γ ) ∝ k/μ, where k and μ are effective values of the force constant and reduced mass, respectively. As with any structural quantity, E12,ZP can be expanded in a Fourier (0) (3) 1 + E12,ZP series in γ . To first order, E12,ZP = E12,ZP 2 (1 + cos 6γ ). The term (3)
in E12,ZP will be absorbed into the gs barrier height. As γ moves from the staggered to the eclipsed configuration, the ν12 oscillator becomes strained, the (3) force constant k increases, and E12,ZP (γ ) gets larger. Thus E12,ZP is positive; moreover, the effect will be smaller for larger values of μ. Thus as the number of deuterons goes up, this positive contribution to the barrier height will get smaller. z From the calculations of Duncan et al. [89], the Coriolis constants ζ9z (ζ12 ) and x x ζ4,9 (ζ6,12 ) are expected to be largely mass invariant and, as can be seen from Table VI, this is indeed found to be the case. The experimental and calculated z x (ζ x ); values of ζ9z (ζ12 ) agree very well, but there is clear disagreement for ζ4,9 6,12 x x see Table X in [89]. In fact, the experimental values of ζ4,9 and ζ6,12 are 2 to 3 times smaller than their calculated counterparts, which are larger in magnitude than any other off-diagonal x, y Coriolis coupling constant. However, in [89], it is assumed that the ethane isotopomers are semirigid, and this assumption may be the cause of the disagreement. There is preliminary evidence that the torsional motion enhances the mixing of the skeletal vibrations. This mixing may serve to average the different Coriolis effects, thereby reducing the values observed for x and ζ x . ζ4,9 6,12
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Considerable progress has been made in the understanding of internal rotation in molecules like CH3 SiH3 and C2 H6 . However, several puzzles remain. The differences between the High Barrier and Free Rotor models require additional investigation; see Section 2.2. Perhaps a thorough study of the rotational spectrum of CH3 SiH3 and CH3 SiD3 in the excited torsional states of the gs and in the v6 = 0 levels of ν12 and ν5 will clarify the issues involved. The link between m and the rotational constant A the Coriolis-like coupling constant Aζ12 12,F of the frame about the symmetry axis in ν12 is still unresolved; see Section 3.4.1. A full investigation of the hyperfine Hamiltonian Hhyp in the gs remains to be carried out. Several difficulties have been pointed out in the rotational hyperfine anticrossing spectra of CH3 CF3 and two possible explanations have been suggested [58]. There may be a spin-torsion interaction needed in Hhyp . Alternatively, the fact that the nuclear hyperfine matrix elements are almost the same order as the torsional splittings may lead to additional lines in the spectrum. The torsional splittings determined for v6 = 0 in the gs of CH3 CD3 by the anticrossing method [29,44] seem to be somewhat inconsistent with measurements of the pure rotational spectrum [48] and infrared bands [22,51]. In the study of the ν5 band [22], the (J = 2) molecular beam values for νEE and νEA disagree by two to three times the experimental error, but the disagreement for J = 1 is larger. It has been suggested that this is due to the deuterium quadrupole interaction and/or higher order terms in the Stark Hamiltonian that become important when the leading dipole constant μ0 is very small [48]. The data sets included in the global analyses discussed here have been largely confined to the gs and to the (vi = 0) torsional levels of the lowest lying perpendicular and parallel bands of molecules with intermediate (or large) barrier heights. The next step may well be to extend the investigation in such molecules to the higher torsional levels in νt and νs . Several initial studies have been carried out for vi = 1. As indicated in Table I and Section 3.4.3, the first excited torsional level has been investigated for ν9 of CD3 CD3 and for ν5 , ν11 , and ν12 of CH3 CF3 . However, no intervibrational resonances were observed, and the reduced barrier height for each of these molecules is rather large. Work is currently underway on the ν9 + ν4 − ν4 hot band in CH3 CH3 ; see the dot-dashed arrow in Fig. 13. For this molecule, the intrinsic σ -splitting is much larger. In general, the analysis of the higher torsional levels in νt and νs might require the development of a four-band computer program, as these levels might have (near) resonant interactions with the next higher small amplitude vibrational mode. Even if this is not the case, the development of the full Hamiltonian will be an interesting challenge as the torsional effects in νt and νs increase rapidly with vi . Another direction research in this area can go is to molecules with much lower barrier heights. A great deal of work has already been done; for example, see the infrared study of the methyl rocking mode in CH3 CCCH3 [126], the microwave investigation of CH3 CCCD3 [127], and the mm-wave study of CH3 CCSiH3 [128].
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In each case, it would be a major challenge to obtain data sets comparable to those for CH3 CH3 and CH3 SiH3 , and carry out a similar global multi-band analysis. For XY3 AB3 symmetric tops, methods in addition to those described in Section 3 can be used in favorable cases for investigating the torsional splittings. Each technique will be discussed in terms of the rotational R-branch, although in principle other types of spectroscopy can be employed. For discussion purposes, the simplified model will be assumed. The first method involves a study of the l-doublets. As an example, consider the (v12 = 1, v6 = 0) state in a molecule similar to CH3 SiH3 , but where the reduced barrier height is somewhat lower. As discussed in Section 3.1, the E4 states with (k = +1, l12 = +1) and (k = −1, l12 = −1) coupled by the l-doubling operator q12 O39 have a non-zero intrinsic splitting C when q12 O39 is turned off. If q12 is positive, the energy of the l-doubling level E4+ /E4− will be above/below that of its A+ /A− counterpart by | 12 {[q12 J (J + 1)]2 + 2C }1/2 − 12 q12 J (J + 1)|. In the (J + 1 ← J ) spectrum, the E4± component will be between its A± partner and the “central” cluster of lines due to other values of G. If 2C is large enough, the σ -components of the low frequency l-doublet can be resolved. This σ -splitting δνL (J ) between (σ = 0) and (σ = ±1) can then be measured if |C | is small enough that the (σ = ±1) component is isolated from the central lines. Similarly, a measurement of the corresponding splitting δνH (J ) can be made for the two σ -components of the high frequency l-doublet. In each case, |C | can be determined. A sugml " gestion along these lines was first made by Laurie [94], but the term 2Aζ12 12 Jα was omitted from the Hamiltonian. This operator was introduced by Hirota [39], but the large effects due to the matrix elements off-diagonal in v6 were not taken into account. If a homogeneous static external electric field E is applied, there is a value of E for which the σ -splitting in the spectrum is a maximum, as can be shown from Eqs. (29a) and (29b) in [39]. If the σ -splittings are too small to be resolved for E = 0, then they can be measured under favorable circumstances when E is the order of this optimum value. If (near) resonant Coriolis-like effects are present, the difference between δνH and δνL will be very useful in separating out the part due to C . If |C | is too small to produce significant effects for (v12 = 1, v6 = 0), it may still be possible for |C | to modify the spectrum seriously for (v12 = 1, v6 = 1), where the torsional effects are much larger. An analogous method of investigating the torsional splittings is available involving a study of the sextic splittings for K = 3 in the gs. As an example, consider a molecule with v6 = 0 where these sextic splittings in the (J + 1 ← J ) spectrum are large enough to shift the (K = 3) lines for ΓVTR = A± well away from the central cluster of components with other values of K. The E4 states with k = +3 and k = −3 coupled by the splitting Hamiltonian HS in Eq. (6) have a non-vanishing intrinsic splitting T when HS is set to zero. If |T | is
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large enough compared to the quartic centrifugal distortion constant characterizing HS , then each E4± transition can be resolved from its A± counterpart in the spectrum. If the value of |T | is small enough that these E4± components are well separated from the central cluster, then |T | can be determined from the splittings. A third technique of this type involves the study of the (K = 1) doublets that can arise due to the hyperfine Hamiltonian Hhyp . Consider as an example the (J + 1 ← J ) spectrum in field-free space for the gs with v6 = 0. There are terms in Hhyp that have selection rules k = ±2 and σ = 0. See, for example, Row 11 in Table XIV of [85], where the (k = ±1 ↔ ∓1) matrix element for the fluorine-fluorine dipolar interaction in CH3 SiF3 is given for J = 2 and σ = ∓1. For σ = 0, matrix elements of this type couple levels that are degenerate with respect to HVTR and produce a K-doubling in the gs analogous to the l-doubling in ν12 . On the other hand, for σ = ∓1, the two vibration–torsion–rotation levels coupled by the terms are separated by νEE ; see Fig. 3 and Section 3.2. The various (k = ±2) terms in Hhyp for CH3 SiF3 are very small ( a few kHz) compared to νEE , which is 17.879(2) MHz for J = 3 [29,64]. As a result, these terms produce only second order energy shift for σ = ∓1 that is typically negligible in the (J + 1 ← J ) spectrum. However, in an XY3 AB3 molecule where the spin of nucleus B is 1, there will be a quadrupole term in Hhyp with selection rules k = ±2 and σ = 0 [44]. If the associated matrix elements are comparable to νEE , a sizeable shift will be produced for σ = ∓1. If this frequency shift can be measured with a Fourier transform cavity spectrometer, for example, and compared to its (σ = 0) counterpart, then in favorable cases the splitting νEE can be determined.
5. Acknowledgements The authors wish to thank their many collaborators without whom much of the work reviewed here could not have been done. The authors would like particularly to thank Dr. W.L. Meerts for his partnership in the earlier phases of the research, especially in the development of the avoided crossing molecular beam method. I.O. wishes to express his appreciation to Dr. A. Bauder for his hospitality and stimulating participation in the Fourier transform microwave experiments. Both of us wish to express their gratitude to Dr. A.R.W. McKellar, Dr. J. Schroderus, and Dr. S.-X. Wang for their substantial contributions to the infrared projects. N.M.-A. gratefully acknowledges the financial support of a Killam Resident Fellowship. Both N.M.-A. and I.O. would like to thank the Natural Sciences and Engineering Research Council of Canada for partial financial support.
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6. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37]
C.C. Lin, J.D. Swalen, Rev. Mod. Phys. 31 (1959) 841–892. J.T. Hougen, J. Mol. Spectrosc. 207 (2001) 60–65. L.G. Smith, J. Chem. Phys. 17 (1949) 139–167. J.P. Lowe, Science 179 (1973) 527–532. F.A. Carey, R.J. Sunberg, “Advanced Organic Chemistry”, Kluwer Academic/Plenum, New York, 2000. V. Pophristic, L. Goodman, Nature 411 (2001) 565–568. V. Pophristic, L. Goodman, C.T. Wu, J. Phys. Chem. 105 (2001) 7454–7459. D.J. Nesbitt, R.W. Field, J. Phys. Chem. 100 (1996) 12735–12756. C.S. Parmenter, B.M. Stone, J. Chem. Phys. 88 (1986) 4710–4711. R.J. Longfellow, C.S. Parmenter, J. Chem. Soc. Faraday Trans. II 84 (1988) 1499–1510. J.W. Dolce, A. Callegari, B. Meyer, K.K. Lehmann, G. Scoles, J. Chem. Phys. 107 (1997) 6549–6561. C.P. Rinsland, E. Mahieu, R. Zander, P. Demoulin, J. Forrer, B. Buchmann, J. Geophys. Res. 105 (2000) 24235–24249. M.J. Mumma, N. Dello Russo, M.A. DiSanti, K. Magee-Sauer, R.E. Novak, S. Brittain, T. Rettig, I.S. McLean, D.C. Reuter, L.-H. Xu, Science 292 (2001) 1334–1338. P.V. Sada, G.L. Bjoraker, D.E. Jennings, G.H. McCabe, P.N. Romani, Icarus 136 (1998) 192– 201. P.V. Sada, G.L. Bjoraker, D.E. Jennings, P.N. Romani, G.H. McCabe, Icarus 173 (2005) 499– 507. F.M. Flasar, et al., Science 307 (2005) 1247–1251. F.M. Flasar, et al., Science 308 (2005) 975–978. N. Moazzen-Ahmadi, J. Vander Auwera, private communication (2005). L.S. Rothman, et al., J. Quant. Spectrosc. Radiat. Transfer 82 (2003) 5–44. D.A. Fisher, et al., Nature 344 (1990) 508–512. G. Di Lonardo, G. Masciarelli, J. Quant. Spectrosc. Radiat. Transfer 66 (2000) 129–142. J.R. Cooper, A.R.W. McKellar, I. Ozier, N. Moazzen-Ahmadi, Chem. Phys. Lett. 382 (2003) 338–348. I. Nakagawa, T. Shimanouchi, J. Mol. Spectrosc. 39 (1971) 255–274. P.R. Bunker, P. Jensen, “Molecular Symmetry and Spectroscopy”, third ed., NRC Research, Ottawa, 1998. J. Susskind, J. Mol. Spectrosc. 49 (1974) 1–17. J.T. Hougen, J. Mol. Spectrosc. 82 (1980) 92–116. I. Ozier, A. Bauder, W.L. Meerts, N. Moazzen-Ahmadi, J. Schroderus, J. Mol. Spectrosc. 228 (2004) 279–292. W.L. Meerts, I. Ozier, J. Mol. Spectrosc. 94 (1982) 38–54. I. Ozier, W.L. Meerts, J. Chem. Phys. 109 (1998) 4823–4832. M. Wong, I. Ozier, W.L. Meerts, J. Mol. Spectrosc. 102 (1983) 89–111. G. Pelz, P. Mittler, K.M.T. Yamada, G. Winnewisser, J. Mol. Spectrosc. 156 (1992) 390–402. N. Moazzen-Ahmadi, I. Ozier, G.A. McRae, E.A. Cohen, J. Mol. Spectrosc. 175 (1996) 54–61. N. Moazzen-Ahmadi, I. Ozier, H. Jagganath, J. Mol. Spectrosc. 119 (1986) 299–312. N. Moazzen-Ahmadi, I. Ozier, J. Mol. Spectrosc. 123 (1987) 26–36. N. Moazzen-Ahmadi, I. Ozier, E.H. Wishnow, H.P. Gush, J. Mol. Spectrosc. 170 (1995) 516– 533. J. Schroderus, N. Moazzen-Ahmadi, I. Ozier, J. Chem. Phys. 115 (2001) 1392–1404. N. Moazzen-Ahmadi, I. Ozier, W.L. Meerts, J. Mol. Spectrosc. 137 (1989) 166–203.
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[38] J. Schroderus, V.-M. Horneman, M.S. Johnson, N. Moazzen-Ahmadi, I. Ozier, J. Mol. Spectrosc. 215 (2002) 134–143. [39] E. Hirota, J. Mol. Spectrosc. 43 (1972) 36–64. [40] N. Moazzen-Ahmadi, J. Mol. Spectrosc. 214 (2002) 144–151. [41] N. Moazzen-Ahmadi, J. Schroderus, A.R.W. McKellar, J. Chem. Phys. 111 (1999) 9609–9617. [42] N. Moazzen-Ahmadi, E. Kelly, J. Schroderus, V.-M. Horneman, J. Mol. Spectrosc. 209 (2001) 228–232. [43] D. Bermejo, J. Santos, P. Cancio, J.M. Fernández-Sánchez, S. Montero, J. Chem. Phys. 97 (1992) 7055–7063. [44] I. Ozier, W.L. Meerts, Can. J. Phys. 62 (1984) 1844–1854; Erratum, Can. J. Phys. 63 (1985) 1375. [45] E. Hirota, C. Matsumura, J. Chem. Phys. 55 (1971) 981–982. [46] E. Hirota, K. Matsumura, M. Imachi, M. Fujio, Y. Tsuno, C. Matsumura, J. Chem. Phys. 66 (1977) 2660–2663. [47] W. Hüttner, H. Häussler, W. Majer, Chem. Phys. Lett. 109 (1984) 359–361. [48] N. Moazzen-Ahmadi, I. Ozier, I. Mukhopadhyay, T. Amano, J. Chem. Phys. 99 (1993) 2429– 2438. [49] G. Cazzoli, C. Degli Esposti, G.D. Nivellini, F. Tullini, J. Mol. Spectrosc. 59 (1993) 279–281. [50] N. Moazzen-Ahmadi, I. Ozier, A.R.W. McKellar, F. Zerbetto, J. Chem. Phys. 105 (1996) 8536– 8542. [51] N. Moazzen-Ahmadi, I. Ozier, I. Mukhopadhyay, A.R.W. McKellar, J. Chem. Phys. 108 (1998) 838–848. [52] J.R. Cooper, N. Moazzen-Ahmadi, Chem. Phys. Lett. 413 (2005) 100–104. [53] N. Moazzen-Ahmadi, A.R.W. McKellar, J.W.C. Johns, I. Ozier, J. Chem. Phys. 94 (1991) 2387– 2394. [54] J.M. Fernández-Sánchez, A.G. Valdenebro, S. Montero, J. Chem. Phys. 91 (1989) 3327–3334. [55] A.C. Szott, J.R. Cooper, R.I. Thompson, A.R.W. McKellar, N. Moazzen-Ahmadi, Mol. Phys. 101 (2003) 2267–2277. [56] S.-X. Wang, J. Schroderus, I. Ozier, N. Moazzen-Ahmadi, V.-M. Horneman, V.V. Ilyushyn, E.A. Alekseev, A.A. Katrich, S.F. Dyubko, J. Mol. Spectrosc. 214 (2002) 69–79. [57] S.-X. Wang, Ph.D. thesis, University of British Columbia, Vancouver, Canada (2000). [58] W.L. Meerts, I. Ozier, Chem. Phys. 152 (1991) 241–259. [59] R. Bocquet, J. Burie, W.D. Chen, G. Wlodarczak, J. Demaison, J. Mol. Spectrosc. 165 (1994) 494–499. [60] I. Ozier, J. Schroderus, S.-X. Wang, G.A. McRae, M.C.L. Gerry, B. Vogelsanger, A. Bauder, J. Mol. Spectrosc. 190 (1998) 324–340. [61] S.-X. Wang, J. Schroderus, I. Ozier, N. Moazzen-Ahmadi, A.R.W. McKellar, V.V. Ilyushyn, E.A. Alekseev, A.A. Katrich, S.F. Dyubko, J. Mol. Spectrosc. 205 (2001) 146–163. [62] V.V. Ilyushyn, E.A. Alekseev, A.A. Katrich, S.F. Dyubko, private communication (1999). [63] R.M. Spycher, I. Ozier, A. Bauder, private communication (1998). [64] W.L. Meerts, I. Ozier, Chem. Phys. 71 (1982) 401–415. [65] J.R. Durig, Y.S. Li, C.C. Tong, J. Mol. Struct. 14 (1972) 255–260. [66] C. Styger, Ph.D. thesis, Eidgenössische Technische Hochschule, Zürich, Switzerland (1991). [67] P. Dréan, J.-M. Colmont, J. Demaison, L. Dore, C. Degli Esposti, J. Mol. Spectrosc. 176 (1996) 23–27. [68] I. Ozier, A. Bauder, private communication (1998). [69] W. Gordy, R.L. Cook, “Microwave Molecular Spectra”, third ed., Wiley, New York, 1984. [70] I. Ozier, W.L. Meerts, Phys. Rev. Lett. 40 (1978) 226–229. [71] J.K.G. Watson, J. Mol. Spectrosc. 40 (1971) 536–544. [72] N. Moazzen-Ahmadi, I. Ozier, J. Mol. Spectrosc. 126 (1987) 99–112.
508 [73] [74] [75] [76] [77] [78] [79] [80] [81] [82] [83] [84] [85] [86] [87] [88] [89] [90] [91] [92] [93] [94] [95] [96] [97] [98] [99] [100] [101] [102] [103] [104] [105] [106] [107] [108] [109] [110] [111] [112] [113]
I. Ozier and N. Moazzen-Ahmadi
[6
W.L. Meerts, I. Ozier, Phys. Rev. Lett. 41 (1978) 1109–1112. J.M. Fernández-Sánchez, S. Montero, J. Chem. Phys. 118 (2003) 2657–2672. H. Jagannath, I. Ozier, N. Moazzen-Ahmadi, J. Mol. Spectrosc. 119 (1986) 313–330. N. Moazzen-Ahmadi, A.R.W. McKellar, J.W.C. Johns, I. Ozier, J. Chem. Phys. 97 (1992) 3981– 3988. I. Ozier, W.L. Meerts, J. Mol. Spectrosc. 93 (1982) 164–178. D. Papousek, M.R. Aliev, “Molecular Vibrational-Rotational Spectra”, Elsevier, Amsterdam, 1982. D.J. Griffiths, “Introduction to Quantum Mechanics”, second edition, Pearson Prentice-Hall, Upper Saddle River, NJ, 2005. J. Susskind, L. Iredell, W.E. Blass, D.C. Reuter, G.W. Halsey, D.E. Jennings, J. Mol. Spectrosc. 192 (1998) 119–138. F. Lattanzi, C. di Lauro, C. Claveau, A. Valentin, G.D. Nivellini, J. Mol. Spectrosc. 207 (2001) 193–200. K. Van Helvoort, R. Fantoni, W.L. Meerts, J. Reuss, Chem. Phys. Lett. 128 (1986) 494–500. G.T. Fraser, A.S. Pine, J.L. Domenech, B.H. Pate, J. Chem. Phys. 99 (1993) 2396–2404. J.R. Durig, Y.S. Li, L.A. Carreira, J.D. Odom, J. Amer. Chem. Soc. 95 (1973) 2491–2496. J.T. Hougen, W.L. Meerts, I. Ozier, J. Mol. Spectrosc. 146 (1991) 8–48. N. Moazzen-Ahmadi, H.P. Gush, M. Halpern, H. Jagannath, A. Leung, I. Ozier, J. Chem. Phys. 88 (1988) 563–577. J. Susskind, D.C. Reuter, D.E. Jennings, S.J. Daunt, W.E. Blass, G.W. Halsey, J. Chem. Phys. 77 (1982) 2728–2744. E.A. Clark, A.J. Weber, J. Chem. Phys. 45 (1966) 1759–1766. J.L. Duncan, R.A. Kelly, G.D. Nivellini, F. Tullini, J. Mol. Spectrosc. 98 (1983) 87–110. M.R. Aliev, J.K.G. Watson, J. Mol. Spectrosc. 61 (1976) 29–52. R.M. Lees, J.G. Baker, J. Chem. Phys. 48 (1968) 5299–5318. G. Tarrago, M. Dang Nhu, J. Mol. Spectrosc. 111 (1985) 425–439. D. Kivelson, J. Chem. Phys. 22 (1954) 1733–1739; Erratum, J. Chem. Phys. 27 (1957) 980. V.W. Laurie, J. Mol. Spectrosc. 13 (1964) 283–287. A. Rosenberg, J. Susskind, Phys. Rev. Lett. 42 (1979) 1613–1616. K.J. Sarka, J. Mol. Spectrosc. 134 (1989) 354–361. I. Ozier, W.L. Meerts, Can. J. Phys. 59 (1981) 150–171. T.C. English, J.C. Zorn, in: D. Williams (Ed.), “Methods of Experimental Physics, vol. 3, Part B”, Academic Press, New York, 1974. F.H. De Leeuw, Ph.D. thesis, Katholieke Universiteit, Nijmegen, The Netherlands (1971). F.H. De Leeuw, A. Dymanus, J. Mol. Spectrosc. 48 (1973) 427–445. W.L. Meerts, I. Ozier, J. Chem. Phys. 75 (1981) 596–603. C.H. Anderson, N.F. Ramsey, Phys. Rev. 149 (1966) 14–24. I. Ozier, P.N. Yi, A. Khosla, N.F. Ramsey, Phys. Rev. Lett. 24 (1970) 642–646. J.K.G. Watson, M. Takami, T. Oka, J. Chem. Phys. 70 (1979) 5376–5380. H.L. Buijs, H.P. Gush, Can. J. Phys. 49 (1971) 2366–2375. B. Nelander, Vib. Spectrosc. 9 (1995) 29–41. M.S. Johnson, B. Nelander, Nuovo Cimento D 20 (1998) 449–462. D.F. Eggers Jr., J. Chem. Phys. 48 (1967) 1393–1394. S. Weiss, G.E. Leroi, J. Chem. Phys. 48 (1968) 962–967. G. Herzberg, “Molecular Spectra and Molecular Structure II. Infrared and Raman Spectra of Polyatomic Molecules”, Van Nostrand, New York, 1972. A.S. Pine, C.P. Rinsland, J. Quant. Spectrosc. Radiat. Transfer 62 (1999) 445–458. I.M. Nyquist, I.M. Mills, W.B. Person, Jr. B. Crawford, J. Chem. Phys. 26 (1957) 552–558. R. Krishnan, J.S. Binkley, R. Seeger, J.A. Pople, J. Chem. Phys. 72 (1980) 650–654.
6]
INTERNAL ROTATION IN SYMMETRIC TOPS
509
[114] M.J. Frisch, G.W. Trucks, H.B. Schlegel, P.M.W. Gill, B.G. Johnson, M.W. Wong, J.B. Foresman, M.A. Robb, M. Head-Gordon, E.S. Replogle, R. Gomperts, J.L. Andres, K. Raghavachari, J.S. Binkley, C. Gonzalez, R.L. Martin, D.J. Fox, D.J. Defrees, J. Baker, J.J.P. Stewart, J.A. Pople, GAUSSIAN 92/DFT, Revision G.1, Gaussian, Inc., Pittsburgh, PA (1993). [115] W.E. Blass, G.W. Halsey, D.E. Jennings, J. Quant. Spectrosc. Radiat. Transfer 38 (1987) 183– 184. [116] J.L. Duncan, A.M. Ferguson, D.C. McKean, J. Mol. Spectrosc. 168 (1994) 522–539. [117] S.J. Daunt, W.E. Blass, G.W. Halsey, K. Fox, R.J. Lovell, H. Flicker, J.D. King, J. Mol. Spectrosc. 86 (1981) 327–343. [118] C.W. Patterson, H. Flicker, R.S. McDowell, N.G. Nereson, Mol. Phys. 43 (1981) 517–526. [119] D.W. Lepard, D.E. Shaw, H.L. Welsh, Can. J. Phys. 44 (1966) 2353–2362. [120] S.J. Daunt, A.K. Atakan, W.E. Blass, G.W. Halsey, D.E. Jennings, D.C. Reuter, J. Susskind, J.W. Brault, Astrophys. J. 280 (1984) 921–936. [121] L. Henry, A. Valentin, W.J. Lafferty, J.T. Hougen, V. Malathy Devi, P.P. Das, K. Narahari Rao, J. Mol. Spectrosc. 100 (1983) 260–289. [122] A. Al-Kahtani, S. Montero, J.W. Nibler, J. Chem. Phys. 98 (1993) 101–109. [123] J.P. Lowe, R.G. Parr, J. Chem. Phys. 44 (1966) 3001–3009. [124] E. Hirota, Y. Endo, S. Saito, J.L. Duncan, J. Mol. Spectrosc. 89 (1981) 285–295. [125] B. Kirtman, W.E. Palke, C.S. Ewig, J. Chem. Phys. 64 (1976) 1883–1890. [126] P.R. Bunker, J.W.C. Johns, A.R.W. McKellar, C. di Lauro, J. Mol. Spectrosc. 162 (1993) 142– 151. [127] J. Nakagawa, M. Hayashi, Y. Endo, S. Saito, E. Hirota, J. Chem. Phys. 80 (1984) 5922–5925. [128] J. Nakagawa, K. Yamada, M. Bester, G. Winnewisser, J. Mol. Spectrosc. 110 (1985) 74–85.
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ADVANCES IN ATOMIC, MOLECULAR AND OPTICAL PHYSICS, VOL. 54
ATTOSECOND AND ANGSTROM SCIENCE HIROMICHI NIIKURA1,2 and P.B. CORKUM1 1 National Research Council of Canada, 100 Sussex Drive, Ottawa, Ontario, Canada K1A0R6 2 PRESTO, Japan Science and Technology Agency (JST), 4-1-8 Honcho, Kawaguchi-city, Saitama,
Japan 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Tunnel Ionization and Electron Re-collision . . . . . . . . . . . . . . . . . . . . . . . 2.1. Tunnel Ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Classical Electron Motion in an Intense Laser Field . . . . . . . . . . . . . . . . 2.3. Re-collision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Quantum Perspective of the Re-collision Process . . . . . . . . . . . . . . . . . 3. Producing and Measuring Attosecond Optical Pulses . . . . . . . . . . . . . . . . . . 3.1. Producing Single Attosecond Pulses . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Attosecond Streak Camera . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Measuring an Attosecond Electron Pulse . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Forming an Electron Wave Packet/Launching a Vibrational Wave Packet in H+ 2 4.2. Spatial Distribution of the Re-collision Electron Wave Packet . . . . . . . . . . 4.3. Time-Structure of the Re-collision Electron . . . . . . . . . . . . . . . . . . . . 4.4. Reading the Molecular Clock–the Vibrational Wave Packet . . . . . . . . . . . . 4.5. Confirming the Time-Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6. The Importance of Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7. Single, Attosecond Electron Pulse . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Attosecond Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Observing Vibrational Wave Packet Motion of D+ 2 . . . . . . . . . . . . . . . . 5.2. Laser Induced Electron Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Controlling and Imaging a Vibrational Wave Packet . . . . . . . . . . . . . . . . 6. Imaging Electrons and Their Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. Tomographic Imaging of the Electron Orbital . . . . . . . . . . . . . . . . . . . 6.2. Attosecond Electron Wave Packet Motion . . . . . . . . . . . . . . . . . . . . . 7. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Abstract When a strong laser field ionizes atoms (or molecules), the electron wave packet that tunnels from the molecule moves under the influence of the strong field and can re-collide with its parent ion. The maximum re-collision electron kinetic energy depends on the laser wavelength. Timed by the laser field oscillations, the re-colliding electron interferes with the bound state wave function from which it 511
© 2007 Elsevier Inc. All rights reserved ISSN 1049-250X DOI: 10.1016/S1049-250X(06)54008-X
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tunneled. The oscillating dipole caused by the quantum interference produces attosecond optical pulses. Interference can characterize both interfering beams—their wavelength, phase and spatial structure. Thus, written on the attosecond pulse is an image of the bound state orbital and the wave function at the re-collision electron. In addition to interfering, the re-collision electron can elastically or inelastically scatter from its parent ion, diffracting from the ion, and exciting or even exploding it. We review attosecond technology while emphasizing the underlying electron–ion re-collision physics.
1. Introduction Observing the internal motion of matter on an ever-faster time scale is one of the major aims of science. During the past few decades, optical science has dominated this quest. As shown in Fig. 1, during the 25 years following the invention of the laser, the pulse duration of optical pulses decreased from nanoseconds to a few femtoseconds. However, once the pulse duration reached 6 fs in 1986 [1], the record stood for the next 10 years. A 6 fs laser pulse at 600 nm is so short that the electric field oscillates only a few times in the pulse. Therefore, in order to reach the attosecond time scale (as, 10−18 s), a new approach based on a new physical mechanism was required. Although the minimum laser pulse duration remained fixed for the next 15 years, other aspects of laser technology improved, especially the technology of generating intense pulses [2]. Producing intense, well-controlled femtosecond pulses has proven to be a critical technology for attosecond science [3].
F IG . 1. Achieved laser pulse duration as a function of year.
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If we apply the intense laser fields to gaseous atoms or molecules, then an attosecond photon or electron (when the electron is viewed from the perspective of its parent ion) pulses can be produced. The basic physics is tunnel ionization and electron re-collision [4]. The intense laser pulse (∼1014 W/cm2 ) transfers part of the bound electron wave function to the ionization continuum via tunnel ionization. In many ways tunneling is like a beam splitter for light, splitting the wave function in two. In the continuum, the newly formed electron wave packet is pulled away from the parent ion by the strong laser field but, when the laser field changes its sign, it can return to the parent ion with the high kinetic energy obtained from the laser field where it can “recollide”. In quantum mechanics, what we can know about an object depends upon how it is measured. The coherently re-colliding electron wave packet interferes with the remaining bound state electron wave-function and the dipole oscillation (or transition of the continuum electron back to the ground state where the “which way” information is lost) caused by this interference produces the coherent light in a short burst of radiation extending into the XUV. If we observe the radiation, we observe the interference. Since an electron wave packet that is born near any field maximum re-collides about 2/3 of a period later, the short burst of radiation is well-timed with respect to the laser field oscillation. Repeated over many 1/2 laser periods, a train of attosecond pulses, with correspondingly high harmonics of the fundamental, is generated. The spectrum of high harmonics is characterized by a long plateau region and cut-off [5]. Producing a single attosecond pulse instead of a train of pulses requires controlling a laser pulse, which in turn controls the electron recollision, so that it can only occur over a small fraction of one period of laser field oscillation. Attosecond optical pulse trains [6–8] and single attosecond optical pulses (250 as) [9–11] were first measured in 2001. Since that time, single attosecond pulses have been used to measure Auger decay dynamics of krypton [12,13] and to trace out the time-dependent electric field of a light pulse [11,14,15]. Spatial coherence of the high harmonics has been also measured [16,17]. The photon energy reaches to the water window [18,19]. From the spectrum of XUV radiation, we can obtain information of the highest occupied molecular orbital [20,21], internal attosecond electron wave packet motion [22], or the molecular vibrational motion of its parent ion [23]. Those are alternative approaches to attosecond measurement. If our observable is electrons instead of photons, then we know that the electron tunneled. In that case, interference between the bound and continuum parts of the wave function is not possible—we know that the electron is not in the bound state. However, the continuum wave packet is still coherent and the electron can
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elastically scatter [24,25] (and diffract) from its parent ion or can inelastically scatter from the ion. Attosecond electron pulses were first measured in 2002 [26]. Since then, electron pulses have been used to monitor the D+ 2 vibrational dynamics with 200 attosecond and 0.05 Å precision [27,28], as well as the attosecond dynamics of double ionization in neon [29] and orientation dependence of the branching ratio of double ionization in N2 between attosecond and slower dynamics [30]. Thus, attosecond science combines both optics and collision physics and opens new opportunities for both. Looking from the optics perspective the new technology produces the shortest duration optical pulses and the shortest wavelength coherent light that can be currently produced. In addition, anyone with an optics background will immediately recognize that interferometry can fully characterize an optical beam—its spatial, frequency and phase characteristics. By analogy, measuring the photons produced by the electron interferometer, can fully characterize the electron—both the bound state wave function and re-collision electron wave packet. From a collision physics perspective, attosecond science allows one to transfer optics concepts and methods to electrons. The field of a laser pulse can be used to time an electron–ion collision to attosecond precision with respect to the laser field. This allows collision experiments to borrow pump-probe technology from optics—the collision being either the probe to a photon pump or vice versa. In addition, if a collision leads to the rapid emission of one or more charged particles, then the strong laser field maps the time of release of the products onto the direction and energy of the electron. (Mapping is often called streaking—referring to the attosecond streak camera [10,31,32] which we will briefly describe below.) Through collisions, ultrafast science may even extend its reach into measurements of the dynamics of atomic nuclei [27,33,34]. Thus attosecond science is truly a synthesis of optical and collision physics, each enhanced by the interplay with the other and the coherence of the process. Imaging the highest occupied molecular orbital of N2 [20] is an example of the new opportunities that arise from this synthesis. This review will cover both attosecond electrons and photons. However we will place greater emphasis on the electrons since they are used in their own right and they are needed to produce attosecond photon pulses. In addition, attosecond electrons can be very efficiently used if the target atom or molecule of interest is consistent with re-collision, since we avoid the steps of generating an attosecond optical pulse, shining it to the target molecule and then observing the consequence. In a re-collision experiment electrons are delivered to their target with combined attosecond and angstrom precision. The probability of recollision is extremely high. An external source would need a current density of ∼1011 Amperes/cm2 to match it.
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The chapter is organized as follows. Section 2 discusses tunnel ionization and electron re-collision using the semi-classical, three-step model. Sections 3 and 4 describe how attosecond optical and electron pulses are produced and characterized. Section 5 concentrates on how the attosecond measurement of the vibrational wave packet motion of D+ 2 can be combined with measurement of the position of the wave packet. In Section 6 we discuss how electron “interferometry” can be used to measure the highest occupied molecular orbital of a small molecule and how the motion of the bound state electron wave packet can be observed.
2. Tunnel Ionization and Electron Re-collision The process of tunnel ionization and electron re-collision of a one-electron system in an intense laser field is fully described by the time-dependent Schrödinger equation [35]. However, in order to present an intuitive understanding of the process we use the semi-classical, three-step approach [4]. In this model, the tunnel ionization probability of an atom is calculated as a function of the laser intensity, the motion of the electron under influence of the field is treated as a classical particle ionized at a particular phase of the laser field, and the electron– ion interaction is considered if the newly ionized electron returns to the ion.
2.1. T UNNEL I ONIZATION The potential energy of the bound, single electron is the addition of the Coulomb potential from the ion core with the potential from the laser field: V = −e/4πε0 r + eE(t)x,
E(t) = f (t) cos(ωt).
Here e is the charge on the electron, ε0 is permittivity of free space, the f (t) is the envelope function of the laser field, ω is the angular frequency of the laser field, and x is the coordinate. Figure 2(a) is a sketch of a 1-dimensional cut along the electric field direction through the center of a singly charged ion, evaluated for a constant electric field equivalent to the peak of the laser pulse at an intensity of 1 × 1014 W/cm2 . If the potential barrier is lower than the vertical ionization energy (IP ) of the electron, then the electron is released in the ionization continuum according to classical physics (Barrier Suppression Ionization, BSI). The laser intensity where BSI occurs is given by EBSI = IP4 /4 in atomic units [36]. However, before the laser intensity reaches that value, the bound state electron can tunnel through the potential barrier to the ionization continuum. Figure 2(b) shows the tunnel ionization rate calculated using an atomic ionization model that is tested widely against strong field experiments. It is often referred to as the ADK
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F IG . 2. (a) The potential energy that a bound state electron feels under the presence of a laser field. The electron can tunnel through the barrier of the combined Coulomb and laser interaction (tunnel ionization). (b) The ionization rate as a function of laser intensity calculated using the ADK model [37]. The abbreviation a.u. stands for arbitrary units throughout the manuscript.
tunneling model named after the initials of the three authors of the paper [37]. The tunneling probability is highly non-linear as a function of the field intensity in the range <1015 W/cm2 . Tunnel ionization is a valid approximation to multiphoton ionization when the electric field oscillates slowly compared to the time the electron spends below the barrier (tunneling time). The ratio between the electron’s tunneling √ time and the laser period is defined as a Keldysh parameter, given by γ = IP /2UP using atomic units [38]. Here UP is referred to as the ponderomotive energy, given by UP = e2 |E|2 /4mω2 , where m is the electron mass, E is the strength of the electric field, such that UP [eV] = 9.34 × I [1014 W/cm2 ] × λ2 [µm2 ]. If γ 1, tunnel ionization dominates while, for γ 1, perturbation theory dominates. The terms multiphoton ionization and perturbation theory are often used interchangeably in strong field science, but of course tunnel ionization also involves many photons. For λ = 800 nm and I = 1014 W/cm2 , UP is ∼6 eV. In general, for small to medium size molecules, the tunnel ionization probability of a molecule is suppressed compared to an atom with the same ionization potential [39–41]. If the molecular size approaches the dimensions of the electron oscillation, multielectron effects [42,43] must be considered.
2.2. C LASSICAL E LECTRON M OTION IN AN I NTENSE L ASER F IELD Since the tunnel ionization probability is non-linear with respect to the laser intensity, tunneling occurs near the peak of each laser cycle. After ionization, the electron wave packet propagates in the field, E. In a semi-classical approach, instead of treating the electron wave packet motion quantum mechanically, the
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F IG . 3. The relation between time of the tunnel ionization (t0 ) and the re-collision time (tc ). The re-collision time and energy relates to the laser phase. Two electron trajectories yields the same re-collision energy. The earlier is referred as a short trajectory, and the other is as a long trajectory. The re-collision time which provides the maximum re-collision energy is ∼2/3 of the optical period after the tunnel ionization.
classical motion of the electron ionized at the phase of the laser field is calculated by solving Newton’s equation. For simplicity, we assume that the Coulomb field can be neglected. This would be the case if the electric field of the laser is larger than the Coulomb field (strong field approximation) over most of the electron trajectory. In case of the linear polarization and with E(t) = |E| cos ωt, the position of the electron along the laser polarization as a function of time is give by x(t) = e|E|/mω2 (cos ωt0 − cos ωt) + ω(t0 − t) sin ωt0 + x(t0 ) where t0 is the time of ionization. The time of the re-collision (tc ) can be calculated as a function of t0 by x(tc ) = 0. Figure 3 shows the relation between ionization time and the re-collision time, schematically. If tunnel ionization occurs before the laser intensity reaches its peak (t0 < 0), then the ionized electron never returns to the parent ion. Recollision occurs only for 0 ωt0 π/2 and π ωt0 3/2, modulo 2π. If tunneling occurs just at the peak of the laser field (t0 = 0), the electron returns to the parent ion after the one period. If t0 > 0, the electron re-collides
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at tc with the net kinetic energy that it gained from the laser field (and IP when the Coulomb potential is included in the calculation) as it traverses its trajectory. In the case of 0 < t0 < (17/180)π/ω, the kinetic energy at tc (re-collision energy) increases as t0 increases. When ionization occurs at ∼17 degrees of the laser phase following the peak field, the re-collision energy reaches at its maximum value, ∼3.17 UP in the absence of the Coulomb potential. As we show later, the re-collision probability has a maximum value at this time also. For t0 > (17/180)π/ω, the re-collision energy and probability decrease as t0 increases. Thus, two classes of trajectories contribute the same re-collision energy each 1/2 period. The one that collides earlier is referred as a short trajectory, and the other as a long trajectory. In real atoms or molecules, the electron moves in the Coulomb field of the ion as well as the laser field. It attracts (Coulomb focusing) [43] the electron, modifying these statements a bit. Coulomb focusing increases the re-collision probability and modifies the time of re-collision. If we increase the ellipticity of the laser fields, then the electron is displaced along the direction of the minor axis of the ellipse. As the ellipticity increases, the electron can miss its parent ion. The re-collision probability drops rapidly with ellipticity [44].
2.3. R E - COLLISION When the electron re-collides with its parent ion, a number of physical processes are induced, as is shown in Figure 4. The electron can scatter elastically [24,25]. In that case, if the parent ion is a molecule, the momentum distribution of the scattered electron (and the re-coil momentum of the ion) carries diffractive information of the molecular structure at the time of scattering. The electron can scatter inelastically. In that case, the ion is excited or further ionized. Inelastic scattering gives rise to the non-sequential double ionization or two-electron excitation [26– 30]. The electron can interfere with its parent orbital (i.e. re-combine) [20–22]. In that case, the re-collision energy is converted to XUV radiation, producing attosecond pulses containing high harmonics of the fundamental. Because re-collision occurs within one optical cycle, molecular and electron dynamics can be probed with sub-laser-cycle time resolution using electron recollision. This is illustrated in Section 5 where we show how vibrational wave packet motion of D+ 2 can be observed using the inelastic process [27,28]. In Section 6, we show how the electron wave packet motion can be observed using radiative re-combination (high harmonic generation) [22]. Since UP is proportional to the square of the wavelength, the maximum recollision energy (∼3.17UP ) increases with laser wavelength for the same laser intensity. The maximum photon energy of the high harmonics is given by 3.17UP + 1.32IP [45]. For reference, at the laser intensity of I = 1.5 × 1014 W/cm2 ,
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F IG . 4. Processes caused by re-collision: (1) inelastic scattering, (2) excitation or double ionization, (3) double excitation, and (4) radiative re-combination (high harmonic generation). Since the electron returns within one optical laser cycle, dynamics of molecules or electrons can be probed using these processes with attosecond time precision.
the maximum re-collision energy at 800 nm is ∼31 eV while it is ∼190 eV at 2000 nm.
2.4. Q UANTUM P ERSPECTIVE OF THE R E - COLLISION P ROCESS The semi-classical three-step approach that we have just introduced is evident in the quantum mechanical approach of Lewenstein et al. [45]. In the semi-classical approach, we have regarded the electron wave packet as the sum of the electron trajectories ionized at different laser phases. Of course, there is nothing in this process that destroys the coherence of the re-collision electron with respect to its parent orbital. Thus, we refer to the analogy with optical interferometry (upper panel in Figure 5). From a quantum perspective, tunnel ionization splits a bound state electron wave packet into two, one (ψb ) remains in the bound potential and the other (ψc ) propagates in the ionization continuum (lower panel in Fig. 5). Re-collision recombines them. At the time of the re-combination, coherent interaction between two wave packets induces the electron’s dipole moment which generates the radiation (high harmonics). The spectrum of the high harmonics is given by ¨ a Fourier transformation of the dipole acceleration, d(t) ≡ ψ|∂V /∂r|ψ ∼ ¨ exp(−iωt) dt. From the spectrum, we can reψb |∂V /∂r|ψc and d(ω) = d(t) construct ψb (Section 6.1) and its time-evolution with attosecond time-resolution (Section 6.2).
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F IG . 5. Quantum perspective of the tunnel ionization and the re-collision process. Lower panel: Tunnel ionization splits the bound electron wave function into two, one remains in the bound state and the other propagates in the continuum. At the time of re-collision, two parts of the wave-function coherently interact and the dipole induced by their interaction produces high harmonics. The high harmonic spectrum contains information of both bound and continuum electron wave-function. This process is analogous to an optical interferometer (upper panel).
3. Producing and Measuring Attosecond Optical Pulses Attosecond optical pulses are produced during the electron ion re-collision occurring in an intermediate density gas. Essential to the process is the coherence of the electron wave packet with the wave function from which it has tunneled. At the single atom level, coherence ensures that, when the electron re-collides, it can interfere with the bound portion of the wave function. At the multi-atom level coherence plays another role. It ensures that each atom in a gas interferes in an identical fashion, synchronized by the fundamental pulse. That is, high harmonic generation is phase matched just like other nonlinear optics processes are also phase matched. Synchronized re-collisions produce attosecond optical pulses. The characteristics of attosecond optical pulses are largely imposed by the electrons. The optical pulses are chirped (except at the cut-off) because the electron pulses are chirped. The electrons are perfectly phased with the laser field and therefore so are the photons that they produce. Comparing attosecond optical and electron pulses, the conversion efficiency from laser light to high harmonic photons is ∼10−6 for mid-plateau photons in argon (considerably lower for helium and neon). As we shall see below, in many
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ways the electron pulses are more intense. They also have a much shorter wavelength. It is the short wavelength that offers the potential for imaging the structure of matter. However, the electrons are only seen as attosecond bursts by their parent atom while the photons can be transported out of the system. Ideas for how to generate attosecond pulses are more than a decade old. The main hold-up that kept attosecond pulses out of the lab was finding a method of measuring them. We begin by briefly reviewing the two methods that are used to produce isolated attosecond pulses. Then we move to the attosecond streak camera [31,32], one of the approaches to attosecond pulse duration measurements. We choose the streak camera measurement because it provides attosecond time resolved measurement in collision physics as well. We refer the reader to other approaches to characterize the attosecond optical pulses [46–49].
3.1. P RODUCING S INGLE ATTOSECOND P ULSES In a multi-cycle laser pulse, attosecond optical pulses are generated at every half laser cycle. If we select an electron trajectory so that the electron re-collision occurs one time during the laser pulse, then single and isolated attosecond optical pulses can be generated. Two approaches have been proposed so far. One uses a laser pulse whose polarization changes rapidly during the pulse so that the polarization is circular at the rising and falling part of the pulse while it is a linear in the middle range of the pulse [50]. Since the electron re-collision probability decreases rapidly with ellipticity, only in the middle range of the pulse can the attosecond optical burst be generated effectively. Another approach uses few-cycle, carrier-envelope phase stabilized laser pulses where only the middle part of the laser pulse has a sufficient intensity to ionize a gas. Adjusting the carrier-envelop phase of the 5 fs, 800 nm laser pulses, the electron trajectories that contributes to the re-collision can be restricted to only one path near the cut-off region. Using this approach, isolated attosecond optical pulses have been produced for the first time [10,11]. If one combines the carrier-envelope phase stabilized, few-cycle laser pulse with time-dependent polarization techniques, reduction of the attosecond pulse duration to about one atomic unit seems possible [51,52].
3.2. ATTOSECOND S TREAK C AMERA The key to measuring the duration of attosecond optical pulses has been to produce a photo-electron replica of the attosecond pulse and then to measure it. There are two ways to produce a replica pulse. It can be accomplished by using atomic photoionization—the atom being a photocathode appropriate for attosecond technology—or by using the re-collision electron—an already existing
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attosecond replica pulse. Here we concentrate on photoionization. It is used in most attosecond metrology experiments so far. In general, a process that was able to produce attosecond pulses is a good place to look for measurement. The attosecond streak camera [10,11,31,32] exploits the phase dependent drift energy transferred to a photoelectron by a strong laser field. This energy depends on the phase at the birth of the electron and it remains after the optical pulse is terminated. We can characterize the photoelectron by its velocity Vi (1/2mVi2 = hω ¯ − IP where h¯ ω is the photon energy and IP is the ionization potential of the atom being ionized). Auger decay, or an inelastic scattering could equally produce an electron with velocity Vi . If photoionization occurs in the presence of a strong laser field, the electron gains an additional velocity from a strong laser field E(t) = E0 (t) cos(ωt): V = Vi + eE0 (t)/mω sin(ωt) − eE0 (t0 )/mω sin(ωt0 ) where E0 (t) is the envelope of the laser field and t0 is the moment that the photoelectron is released into the laser field. Here we have assumed no re-collision has occurred. This is ensured if |Vi | > |(eE(t)/mω)|. The term (eE0 (t)/mω) sin(ωt) goes to zero after the optical pulse has gone, but the term (eE0 (t0 )/mω) sin(ωt0 ) remains, labeling the time of birth of the photoelectron into the laser field. Since, in re-collision physics, an attosecond optical pulse is perfectly phased with the laser field, the photoelectron velocity distribution depends on the range of times over which the electron can be released into the laser field. A long pulse releases electrons over a long time interval, while a short pulse has a very short range of release times. Therefore, the photoelectron spectrum is smeared more by the field for a long pulse than for a short one. At the optimum phase (the attosecond pulse placed at a field maximum) the attosecond streak camera is capable of resolving ∼70 attosecond transformed limited pulses [53]. A non-transform limited pulse is easier to resolve than a transform limited pulse. Scanning the phase, any attosecond optical pulse can be fully characterized [53,54]. During the past few years it has become apparent that all of the measurement technology developed for visible laser pulses can be transferred to attosecond optical pulses. Thus, the measurement problem is fully solved for attosecond optical pulses. It is interesting, however, that the solution has been to transfer visible technology to the XUV with only one small change. The measurement is performed on a photoelectron replica rather than on the pulse itself. This contrasts with the underlying technology of attosecond pulse generation which is a major departure from the ultrafast technology that preceded it. An alternate approach would be to measure the re-collision electron—a preexisting replica of the optical pulse. This requires developing radically different technology for metrology. We now turn our attention to this seemingly more complex task.
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4. Measuring an Attosecond Electron Pulse In this section, we borrow collision physics techniques to characterize the recollision electron wave packet seen from the parent ion. We determine the recollision probability (current density) as a function of time using inelastic scattering in H2 . Since the electron pulse duration is mapped onto the optical pulses, this is a first step towards a new, uniquely attosecond, measurement technology. However it does not allow the precision of the streak camera yet. We include it for two important reasons. First, it shows how attosecond metrology can make use of collision physics techniques (and vice versa). We use inelastic scattering for the measurement. Second, it shows the important role that correlation can play in attosecond science. For our measurement, tunnel ionization of H2 produces two correlated wave packets, the electron and the vibrational wave packet. We use the vibrational wave packet to clock the time and intensity (current density) of the re-colliding electron wave packet. Correlated measurements extend the range of technology of ultrafast science and will allow ultrafast methods to be used in completely new areas of science, such as nuclear dynamics [27,33,34]. Full characterization of the re-collision electron—as complete as any optical measurement—has just been achieved [55]. It is too early to be included it in a review. However, it is clear that the key to full characterization of the re-collision electron is interferometry. From general principles we know that interferometry allows all aspects of the interfering waves to be measured. The only uncertainty is the details of how the measurement can be performed. 4.1. F ORMING AN E LECTRON WAVE PACKET /L AUNCHING A V IBRATIONAL WAVE PACKET IN H+ 2 Figure 6 is a plot of the important potential energy surfaces of H2 and its ions. Tunnel ionization launches an electron wave packet in the continuum. Using H2 as the parent molecule, it simultaneously launches a vibrational wave function + on H+ 2 (Σg ). The transition from H2 to H2 is essentially (but not quite) vertical since the tunnel ionization probability is only slightly dependent on the internuclear co-ordinate (through the co-ordinate dependence of the ionization potential). It is confined to a single potential surface because tunnel ionization transfers very ++ [56]. Until the electron little population to the other excited state of H+ 2 or H2 returns to the parent ion, the vibrational wave packet moves on the H+ 2 X potential. Inelastic scattering caused by re-collision promotes the vibrational wave packet + to the H+ 2 (AΣu ) state or other excited states, leading to H fragments. The kinetic energy of the fragments indicates the internuclear separation at the time of the electron re-collision. Using the vibrational wave packet motion as a molecular clock, we can evaluate when the electron re-collides with the parent ion.
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F IG . 6. The potential energy surfaces of H2 and H+ 2 . Tunnel ionization launches a vibrational 2 wave function to the H+ 2 (X Σg ) state and produces an electron wave packet simultaneously. Until 2 the electron returns to the parent ion, the vibrational wave packet propagates on the H+ 2 (X Σg ) state 2 Σ ) vibrational wave (X with a vibrational period of ∼25 fs. Re-collision further promotes the H+ g 2 2 packet to the H+ 2 (A Σu ) state or the other states, leading to dissociation. The kinetic energy of the H+ fragment indicates the time of the re-collision.
Since the laser field is present throughout the measurement, to simplify the interpretation of the vibrational wave packet motion, we require that the potential 2 energy surface H+ 2 (X Σg ) is not affected by the laser fields. If the molecules are aligned parallel to the laser polarization direction, the potential energy surface + 2 2 of H+ 2 (X Σg ) is modified by the laser-induced coupling with H2 (A Σu ). Therefore, we select the kinetic energy distribution of H+ dissociating from the parent molecule aligning perpendicular to the laser polarization. With the molecule perpendicular to the laser field the vibrational motion is a clock that can time the electron re-collision.
4.2. S PATIAL D ISTRIBUTION OF THE R E - COLLISION E LECTRON WAVE PACKET After the tunnel ionization, the electron wave packet spreads in all three directions. In the direction perpendicular to the laser field, spreading occurs because of the initial lateral velocity dv⊥ that the electron acquires as it exits the tunnel. In the direction of the laser polarization the shear imposed by the laser field is
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responsible mainly for spreading of the electron wave packet. The lateral velocity determines the re-collision probability for linearly polarized light. We estimate the value of the dv⊥ by measuring the ellipticity dependence of the re-collision yield of H+ 2 . Figure 7(a) shows schematically how the electron wave packet is influenced by the ellipticity. For linear polarization, the electron wave packet moves along the x axis, but if the light is elliptical with its minor axis of the laser fields along the y axis, then the wave packet is pushed laterally. The electron offset of the classical trajectory from the ion core (dy) at the time of re-collision with the maximum re-collision energy caused by the laser ellipticity is proportional to the ellipticity (ε = Ey /Ex ) and given by dy = 5.14εE/mω2 where Ey and Ex are the components of the laser fields in each direction, respectively [44]. If we observe double ionization (or high harmonic generation) then the lateral initial velocity compensates for this offset. By measuring the strength of the double ionization signal as a function of ellipticity, we measure dv⊥ . It is by dv⊥ = dy/dt, where dt is the time between the tunnel ionization and the re-collision. Figure 7(b) is a plot of the signal counts of H+ produced by the electron recollision as a function of the laser ellipticity. At each ellipticity, we measure the kinetic energy spectrum of H+ and integrate the signal counts at >4 eV (see Section 4.4). The figure includes the data points when the main laser polarization axis is parallel to the molecular axis (circles) and perpendicular to the molecular axis (triangles). To keep the tunnel ionization probability the same, we maintain the laser intensity of the main polarization axis and increase the intensity of the minor axis. In either cases, the re-collision probability has its maximum value at ε = 0 and decreases as the ellipticity increases. The measured data points are well-fitted by the Gaussian curve (solid line). Taking the 1/e width of the curve, we estimate the average spatial distribution of dx = 9 Å and the average lateral initial velocity of dv⊥ = 5.0 Å/fs for parallel to the molecular axis, dy = 7.7 Å and dv⊥ = 4.2 Å/fs for the perpendicular case. Therefore, the re-collision electron wave packet is a “nano-beam” with the diameter of ∼15 Å at the maximum re-collision time. For comparison, we show the ellipticity dependence curve of the non-sequential double ionization probability of argon measured by the same laser conditions (squares). The data points are also well-fitted by Gaussian curve and the 1/e average initial lateral velocity is dv⊥ = 5.4 Å/fs. The lateral initial velocity agrees √ with the prediction of the atomic tunneling theory that gives dv⊥ = (|E|/ 2IP )1/2 = 5.6 Å/fs. Although argon has the same ionization energy as H2 , the observed value of the lateral initial velocity of H2 is smaller than the value for argon. Molecular tunneling theory [40,41] or recent study of high harmonic generation [21,57,58] may find the origin of the differences.
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F IG . 7. (a) Until the re-collision, the electron wave packet spreads spatially. In the case of a linearly polarized laser pulse, the electron wave packet moves along its polarization axis by the laser field and spreads vertically by the lateral initial velocity at the time of tunnel ionization. An elliptically polarized laser pulse pushes the electron wave packet away from the parent ion, leading to the smaller re-collision probability. (b) The ellipticity dependence of the number of H+ ions produced by re-collision when the main axis of the laser polarization is parallel (circles) and perpendicular (triangles) to the molecular axis for a 40 fs, 800 nm pulse having I = 1.5 × 1014 W/cm2 . For comparison, the ellipticity dependence of Ar+ ionization yield due to the re-collision is also plotted (squares). The data points in each case are well-fitted by the Gaussian curve (solid or dotted lines). The upper axis plots the distance of the electron from the parent ion at the time of re-collision with maximum re-collision energy.
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4.3. T IME -S TRUCTURE OF THE R E - COLLISION E LECTRON We have obtained the initial velocity that the electron acquires on tunneling. With it we can calculate the time-structure of the re-collision electron wave packet seen from the parent ion using the semi-classical three-step model. We regard the electron wave packet as a sum of the electron trajectories ionized at different laser phases and positions and calculate those motions under the Coulomb potential combined with the laser fields by solving the Newton’s equation. We calculate the equivalent current density [Amperes/cm2 ]—that is, the ratio of the number of the electron trajectories returning to the parent ion per unit time and unit area with the total number of the electron trajectories. The bond distance of H+ 2 is assumed 2
to 0.9 Å and the calculation used an area of 1 Å . However, the current density is 2 insensitive to the area used as long as the area 15 Å . We include the electron trajectories only with kinetic energy larger than the energy difference between + H+ 2 (X) and H2 (A) at a bond distance of 0.9 Å, as the trajectories that contributes the re-collision. Figure 8(a) is a plot of the calculated electron equivalent current densities as a function of time at a laser intensity of 1.5 × 1014 W/cm2 and the pulse du-
F IG . 8. The calculated equivalent current densities as a function of time for a laser pulse duration of (a) 40 fs and (b) 8 fs (I = 1.5 × 1014 W/cm2 , 800 nm). Panel (c) is a schematic plot of the relation between the return time and the laser phase. After ionization, the electron wave packet returns several times during the 40 fs laser pulse while multiple re-collision probability is suppressed for the 8 fs laser pulse.
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ration of 40 fs (800 nm). The electron returns at ∼2/3 of the laser period (T ) with a maximum re-collision probability and returns several times at ∼5/2T , ∼7/4T and so on. The upper panel of the Fig. 8(c) shows the relation between the re-collision time and the laser phase schematically. The re-collision probability decreases drastically after the first peak, but the probability remains relatively high for a few-femtoseconds. That is because Coulomb focusing, that keeps the electron wave packet near the ion core [59]. The first peak of the equivalent current density contains 50% of all re-collision probability and its duration is <1 fs. We can use the first peak as a probe for observing the dynamics in the parent ion.
4.4. R EADING THE M OLECULAR C LOCK – THE V IBRATIONAL WAVE PACKET Figure 9 is a schematic of the laser set up. The 800 nm, 40 fs, 0.8 mJ laser pulse is generated by the chirp pulse amplified, Ti:Sapphire laser system. Depending on the aim of the experiments, we guide the laser pulse into different optical systems. We put the laser pulse directly into the vacuum chamber with attenuating the pulse intensity for a 800 nm, 40 fs laser pulse. For longer laser wavelengths (used in Section 5), we convert the wavelength by the optical parametric amplifier (OPA, TOPAS). The tuning wavelength range is 1200–1550 nm for the signal output, and 1700–2100 nm for the idler output. To generate a few-cycle, 8 fs (800 nm) pulse, we couple ∼400 µJ of the laser output into a hollow core fiber filled with argon gas [60,61] where self-phase modulation broadens the bandwidth to 650–900 nm. The output pulse from the fiber is compressed by six reflections on two pairs of the chirped mirrors. We further compensate the chirp by inserting thin quartz plates in the optical path before the vacuum chamber. Each pulse is focused by a parabolic mirror (5 cm focal length) in the vacuum chamber. The duration of the few-cycle laser pulse is measured by SPIDER [62]. We measure the kinetic energy distribution of the fragment (H+ ) by timeof-flight (TOF) mass spectrometry. The ions are accelerated to the direction of multi-channel plates by a DC electric field (1600 V) applied between two electrodes separated by 3 cm (Fig. 10). A 1-mm hole in the middle of the electrode selects the fragments dissociating from the parent ion aligning parallel to the TOF axis. The detection angle is ∼8 degrees for 8 eV of H+ . Therefore, if the laser polarization is parallel (perpendicular) to the TOF axis, then we observe the kinetic energy distribution of the fragments dissociating from the parent ion parallel (perpendicular) to the laser polarization. Dissociation of H+ 2 can be identified by measuring the kinetic energy distribution of fragment H+ . Figure 11 shows the measured kinetic energy distribution when the laser polarization is parallel (a) and perpendicular (b) to the molecular
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F IG . 9. Schematic diagram for the laser setup. The chirp pulse amplified Ti:Sapphire laser system generates 800 nm, ∼0.8 mJ, and 40 fs laser pulses. The wavelength is shifted to longer values by an optical parametric amplifier (OPA, TOPAS). The 8 fs laser pulse is generated by optical fiber compression techniques.
F IG . 10. Schematic diagram for the time-of-flight (TOF) mass spectrometer. The ions are accelerated to the multi-channel plate (MCP) by a DC electric field applied between two electrodes. A 1 mm hole on the electrodes selects alignment of the parent ion which produces the fragments detected by the MCP. At the laser polarization vertical to the TOF axis, we select the kinetic energy distribution of the fragments whose parent ion is aligned perpendicular to the laser polarization axis.
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F IG . 11. The kinetic energy distribution of H+ dissociating from H+ 2 for a 40 fs, 800 nm pulse having I = 1.5 × 1014 W/cm2 . The polarization of the laser pulse is (a) parallel and (b) perpendicular to the molecular axis. The square data points in both panels are the signal counts measured using a linearly polarized laser pulse. The peaks of 0.5 eV and 2.5 eV are produced by bond softening dissociation and enhanced ionization, respectively. At the ellipticity of 0.3, the signal with energies >4 eV in (b) disappears (open circles). The difference of the signal between the square and the open circles is responsible for the electron re-collision.
axis at the laser intensity of 1.5 × 1014 W/cm2 , with wavelength of 800 nm and pulse duration of 40 fs (FWHM). The peak in (a) ∼0.5 eV and ∼3 eV is caused by the bond softening dissociation and the enhanced ionization, respectively. Contributions of the re-collision are found at >4 eV, but cannot be seen in this vertical scale. If the laser polarization is vertical to the molecular axis, then the signal due to enhanced ionization disappears since the potential energy surfaces of H+ 2 (X) and H+ (A) are closed. Only the signal responsible for the re-collision is observed 2 in the higher kinetic energy region (squares). If the laser pulse duration is <10 fs and I > 5 × 1014 W/cm2 , then the other dissociation channel opens, referred as the double sequential ionization [61]. First, tunnel ionization of H2 produces the H+ 2 vibrational wave packet at the leading edge of the pulse. Next, further ionization of H+ 2 occurs in the vicinity of the peak ++ of the laser pulse that leads to H2 before the vibrational wave packet reaches the classical outer turning point. The kinetic energy distribution of the correlated H+ fragments indicates the time between first and the second ionizations. Here again, the vibrational wave packet motion on H+ 2 is used as a molecular clock. Figure 12 is a plot of the kinetic energy spectrum of D+ with a pulse duration of 8 fs 800 nm and I = 6 × 1014 W/cm2 . The peak is ∼6.5 eV for 8 fs. This method allows us to check the pulse duration without optical techniques. Re-collision is the fastest pathway of all. To identify the kinetic energy distribution of H+ caused by re-collision, we measure the kinetic energy spectrum at both linear (square data points in Fig. 11(b)) and elliptical polarization (circle data points in Fig. 11(b), ellipticity ε = 0.3). If the ellipticity of the laser pulse
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14 2 F IG . 12. The kinetic energy distribution of D+ dissociation from D+ 2 at I = 6 × 10 W/cm and a pulse duration of 8 fs.
increases, then the ionized electron wave packet is pushed away from the parent ion and the re-collision probability decreases. As is mentioned earlier, at the ellipticity of 0.3, re-collision between H+ 2 and the electron becomes impossible. From the Fig. 11(b) the energetic fragments >4 eV are caused by electron re-collision.
4.5. C ONFIRMING THE T IME -S TRUCTURE Using the molecular clock based on H+ 2 vibration, we experimentally confirm the time structure obtained by the semi-classical calculation. Figure 13(a) also contains a plot of the observed kinetic energy distribution of H+ (squares). We measured the distribution for the case of linear and elliptical polarization (ε = 0.3), and subtract the signal counts measured by the linearly polarized pulse from those measured by the elliptically polarized pulse. To compare the experimental results with calculations, we predict the kinetic energy distribution of H+ using the current density shown in Fig. 8. Specifically, we calculate the vibrational wave packet motion on H+ 2 (XΣg ) by solving the time-dependent Schrödinger equation under field-free conditions. The initial wave packet is obtained from the H2 ground state vibrational wave function weighted by the tunnel ionization probability that depends on the internuclear separation. Assuming that the vibrational wave packet is excited to the H+ 2 (AΣu ) state with the excitation probability according to the current density, we calculate the kinetic energy distribution of H+ . The dotted line in Fig. 13(a) is the calculated kinetic energy distribution when only the first peak of the current density is included, and the dashed line is the distribution when only the third peak of the current density is included. The solid line includes all five peaks. The first peak is separated from the third peak by
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+ + F IG . 13. (a) The kinetic energy distribution of H+ created from H+ 2 (Σu ) → H , H. H2 is produced by electron re-collision. The laser intensity is I = 1.5 × 1014 W/cm2 , the pulse duration is 40 fs (800 nm) and the laser polarization is perpendicular to the molecular axis. The three curves are calculated results using the current density shown in Fig. 8(a). The dotted curve is produced by the first re-collision, the dashed curve is produced by the third re-collision, and the solid line includes all five + + peaks. (b) The kinetic energy distribution of D+ created from D++ 2 → D , D. D2 is produced by the electron re-collision. The laser intensity is I = 1.5 × 1014 W/cm2 , the pulse duration is 8 fs (800 nm) and the laser polarization is perpendicular to the molecular axis. The two curves are calculated results using the current density shown in Fig. 8. The dotted curve is produced by the first re-collision in Fig. 8(b) and the solid line includes all five peaks in Fig. 8(a). The experimental results are consistent with a single re-collision with a small satellite pulse.
2.7 fs, and then the differences can be resolved in the kinetic energy spectrum. In the upper axis of Fig. 13(a), we plot the time scale converted from the kinetic energy distribution of H+ with a molecular clock. The observed spectrum agrees well the calculated spectrum. The dotted vertical line in the figure (8.2 eV) is the kinetic energy if the dissociation of H+ 2 occurs just after the tunnel ionization (t = 0). These results indicate that the re-collision electron wave packet contributing to the excitation is well localized spatially and temporally. Recent quantum mechanical calculations agree with the results of our calculation [63,64]. 4.6. T HE I MPORTANCE OF C ORRELATION We have just described a measurement of the electron packet in which we achieve a time resolution of ∼1 fs. We achieved this in spite of using a 40 fs laser pulse. How is this ultrafast measurement without ultrashort pulses possible? In our case it is possible because the vibrational wave packet and the electron wave packet were correlated (strictly speaking they were entangled). Because of the entanglement, vibrational wave packets that are launched at different peaks of the laser
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field (separated by the half optical period in time) are not coherent with each other. Therefore, the shape of the vibrational wave packet does not depend on the laser bandwidth or the laser pulse duration. In the next chapter we extend the use of correlated wave packets. We control the electron wave packet and use it to probe the vibrational wave packet motion. We achieve a combined resolution of 200 attoseconds and 0.05 Å. As you reflect on the past sections and read the next section, it is interesting to keep nuclear physics in mind. Can real-time measurements be extended to nuclear physics? However, first we show that we can achieve a single attosecond electron pulse.
4.7. S INGLE , ATTOSECOND E LECTRON P ULSE As we have seen in Section 3.1, one method of producing single attosecond optical pulses is to use a few-cycle laser pulse. In the same way, a few-cycle laser pulse can reduce multiple electron re-collisions shown in Fig. 8, favoring a single recollision. Figure 8(b) is a plot of the calculated current density for a 800 nm pulse with duration of 8 fs and I = 1.5 × 1014 W/cm2 . Compared to the pulse duration of 40 fs, the magnitudes of the current density after the first re-collision are suppressed. Since only three cycles are included in the laser pulse, the tunnel ionization and the re-collision probability depends on both the carrier-envelope phase and the crest of the laser field in the envelope. We calculate the motion of the electron trajectories ionized at different peaks of the laser pulse with the different carrier envelope phases, and average over them to obtain the curve in Fig. 8(b). We confirm the result of the calculation experimentally. Figure 13(b) shows the kinetic energy distribution of D+ measured by a 8 fs, 800 nm, I = 1.5 × 1014 W/cm2 laser pulse (linear polarization). We measure only the correlated fragments that have the same magnitude of momentum, but opposite direction. This specifies the dissociation potential only for the D++ 2 potential. We confirm it by comparing the spectrum measured with linearly and the elliptically polarized laser pulses. In the case of elliptical polarization, the signal counts of H+ disappear at energies >2 eV. Therefore, the spectrum is caused by electron re-collision. The dotted line in the spectrum is the calculated kinetic energy distribution using the first peak of the current density in Fig. 13(b) (8 fs). The solid line is the calculated spectrum using the current density in Fig. 13(a) (40 fs). The measured spectrum (squares) is in between them. The contribution of the second and third re-collision is reduced for this ∼8 fs laser pulse. In summary, we have characterized the re-collision electron wave packet using a molecular clock in H+ 2 . Since its pulse duration is <1 fs, the current density is large. As we noted at the beginning of the section, it is possible to dramatically improve this measurement by concentrating on the optical emission of the nonlinear
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medium. This emission gives us access to electron interferometry. However, from this example, it is clear that collision technology can be exploited for attosecond measurements. In the next section we extend this idea, showing that molecular structure can be measured simultaneously.
5. Attosecond Imaging There are at least four ways that strong field and attosecond science can be used to image molecular structure. These will be the subject of this and the following chapter. First, we show how inelastic scattering of the re-colliding electron allows us to trace wave packet dynamics in D+ 2 . Then we discuss how a re-collision electron that elastically scatters from its parent molecule is, in fact, diffracting from it. Thus, molecular structure is encoded in the above threshold ionization (often called ATI) spectrum of the scattered electron. We complete the chapter by exploiting a third method of measuring molecular structure. We use Coulomb Explosion Imaging to measure vibrational wave packets in D2 , and monitor the influence of a control pulse that guides dissociation. Section 5 focuses on measuring the position of the atoms in a molecule. Section 6 turns to imaging electrons—both their wave function (static structure) and their wave packet motion (dynamical structure). By the end of these two sections it will be clear that strong field and attosecond science allows structural determination in partnership with dynamics. This combination goes to the heart of what we might wish to know about quantum systems. The determination of structure and dynamics is a unique contribution that attosecond science has to offer science as a whole. 5.1. O BSERVING V IBRATIONAL WAVE PACKET M OTION OF D+ 2 In Section 4 we demonstrated that ionization launched a vibrational wave packet. Although we do not know when it was launched in laboratory time, we did not need this information in order to measure the electron wave packet. All that we needed was the knowledge that the electron and vibrational wave packets were launched together. If we modify our perspective on the measurement, we can think of the electron wave packet as a probe of the vibrational dynamics. From this point of view, we have an additional tool to bring to bear on the measurement. Once launched, the electron wave packet is controlled by the laser field. If we change the field, we modify the motion of the electron wave packet. We show that this control can be used to measure the position of the vibrational wave packet as a function of time with combined 200 attosecond, 0.05 Å resolution. This section, therefore, introduces the first experiment that uses attosecond technology for molecular imaging.
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F IG . 14. The observed kinetic energy distribution of D+ produced from D+ 2 by the electron re-collision at four different laser wavelengths (square data points). By shifting the laser wavelength from 800 nm to 1850 nm, the re-collision time increases from 1.7 fs to 4.2 fs.
As is seen in the next two sections, imaging and dynamics develop hand-in-hand in attosecond technology. In some ways our approach is analogous to conventional pump-probe methods. The pump is the tunnel ionization of D2 . It produces both the vibrational wave packet on D+ 2 (X) and the electron wave packet in a correlated fashion. Recollision between the D+ 2 vibrational wave packet and the electron wave packet is the probe. Since the electron wave packet returns to the parent ion for the first time at ∼2/3 of the optical period, the pump-probe delay can be changed by changing the laser wavelength. We use the laser wavelength of 800 nm, 1200 nm, 1530 nm, and 1850 nm, where the corresponding re-collision time is 1.7 fs, 2.7 fs, 3.4 fs, and 4.2 fs, respectively. The position of the vibrational wave packet at the time of the re-collision is determined by the kinetic energy distribution of D+ .
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F IG . 15. The measured peak position of the vibrational wave packet on the D+ 2 (XΣg ) state as a function of time (squares). The solid line is the average position of the wave packet calculated by solving time-dependent Schrödinger equation. A 200 attoseconds and 0.05 Å resolution is achieved.
Figure 14 is a plot of the observed kinetic energy distributions at four different wavelengths. In each case, we subtracted the signal counts measured by the elliptically polarized laser pulse from those measured by the linear polarized laser pulse. As the laser wavelength increases, the spectrum shifts to lower kinetic energy. This agrees with the qualitative prediction that the vibrational wave packet propagates to larger internuclear separation as the re-collision time becomes longer. We evaluate the average motion of the vibrational wave packet on D+ 2 (X) from the measured spectrum. First, we reflect the measured kinetic energy spectrum + to the D+ 2 (A) state and back to the D2 (X) state including the dependence of + the cross section between D2 (X) and D+ 2 (A) on the internuclear separation. In Fig. 15, we plot the peak position of the reflected spectrum as a function of the re-collision time (squares). Figure 15 includes the calculated average motion of the vibrational wave packet (solid line). In the measured time range (1.7–2.4 fs), the vibrational wave packet motion is almost linear because it is very short range compared to one vibrational period of ∼25 fs. So the vibrational wave packet motion is measured with 0.05 Å and 200 attosecond resolution. In Fig. 14, we plot the contribution of D+ 2 (A) to the dissociation as triangles. In the lower energy side, the contribution of the second and third re-collision or other excited states should be included. Recent results of the quantum mechanical calculation agree with our calculation and the observed spectra [63]. 5.2. L ASER I NDUCED E LECTRON D IFFRACTION Section 5.1 emphasized opportunities that arise because the re-collision electron inelastically scatters from its parent ion. It also scatters elastically. It was recog-
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F IG . 16. Ratios of momentum distributions for an electron wave packet diffracted by a diatomic core. (a) Diffraction of a Gaussian wave packet with its initial momentum matching the effective cutoff momentum at I = 1 × 1014 W/cm2 . (b) Laser induced diffraction at the same intensity.
nized very quickly after the idea of the re-collision electron was introduced that elastic scattering of the short wavelength (∼1 Å wavelength) electron requires that the electron diffracts and therefore yields structural information on its parent ion at the moment of re-collision [24]. Thus, controlled by the laser field, the re-collision electron becomes a probe of molecular structure that is almost as convenient to use as an optical probe. Since the re-collision electron is delivered to an atom or molecule with very high timing precision and high current density, re-collision allows diffraction from molecules undergoing chemical dynamics on any time scale relevant to atomic motion in molecules. Here we show one theoretical result in Fig. 16 [25]. There are other theoretical studies [24,65]. Experiments exploiting laser induced electron diffraction have not yet been demonstrated. Figure 16(b) shows a calculated diffraction pattern found by solving the 3-D time dependent Schrödinger equation for a two atom, one electron molecule [25] with its internuclear axis aligned perpendicular to the laser field. Plotted is the ratio of the electron probability scattered parallel to the internuclear axis to that perpendicular to the axis. The diffraction pattern is clearly evident in the image. Figure 16(a) is for an external, monoenergetic electron source. It is included for reference. Analyzed in this manner, the diffraction pattern is as clearly identified with laser induced diffraction as with conventional diffraction. By concentrating on atoms and molecules we may have given the reader the mistaken impression that re-collision is important only for moderate intensity optical pulses. The impression is incorrect. While we will not review it here, by intensities of 1022 W/cm2 (just now becoming experimentally feasible) the re-
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collision electron has sufficient energy to interact elastically or inelastically with the nucleus. It may be used to stimulate and probe nuclear dynamics or to probe nuclear structure by re-collision [33].
5.3. C ONTROLLING AND I MAGING A V IBRATIONAL WAVE PACKET The intense laser pulse can influence the bonding electron of the molecule to change the bond strength of the molecule. Therefore, controlling the laser field can control vibrational motion. In this section we show how the vibrational wave packet of D+ 2 is controlled and offer another approach to its measurement during propagation. Figure 17 is a plot of the potential energy surfaces in the presence (dotted lines) ++ and the absence (solid lines) of the laser fields of D+ 2 together with D2 and D2 . When an intense laser field is applied parallel to the molecular axis, the ground
F IG . 17. The potential energy surface of D+ 2 under the presence (dotted lines), I = 2 × 1014 W/cm2 and the absence (solid lines) of the laser field together with D2 and D++ 2 . The laser pulse is parallel to the molecular axis. The laser field couples two potential energy surface of D+ 2 (XΣg ) and (AΣu ) states, leading the bound (upper, dotted line) and dissociative (lower, dotted line) potentials. Using three 8 fs laser pulses, control and imaging of molecular bond dissociation is possible [68]. The first laser pulse launches the vibrational wave packet to the D+ 2 state by tunnel ion++ ization. The delayed, control pulse modifies the potentials. The third pulse further ionizes D+ 2 to D2 , leading the correlated D+ . From the kinetic energy distribution of D+ , we obtain the time-evolution of the square of the wave packet on D+ 2 at the time of probing.
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+ (D+ 2 XΣg ) and excited states (D2 AΣu ) are coupled by the laser field. This coupling shifts the energy levels (Stark shifts). Applying the laser field at an appropriate time during one vibrational period, one can control the D+ 2 vibrational wave packet motion, dissociation and further ionization [66–68]. If the vibrational wave packet moves to the shorter bond distance when the laser pulse is applied, then the vibrational wave packet motion is slowed. On the other hand, if the vibrational wave packet moves towards the longer bond distance when the laser pulse is applied, then the vibrational wave packet is accelerated. If the vibrational wave packet reaches at outer classical turning point (∼2.5 Å) when the laser pulse is present, then the vibrational wave packet of D+ 2 can propagate towards larger internuclear separation beyond the field-free classical outer turning point (bond softening dissociation [69]). If the vibrational wave packet reaches critical distances where the binding electron is localized to one of the protons when the laser field is still present, then further ++ ionization of D+ and a pair of the fragments (en2 is enhanced leading to D2 hanced ionization [70]). We have demonstrated experimental control over the dissociation of D+ 2 by using a series of intense, few-cycle (8 fs) laser pulses [68]. The first laser pulse launches the vibrational wave packet to D+ 2 state by tunnel ionization. The delayed, control pulse modifies the potentials. The third pulse further ionizes D+ 2 + + to D++ 2 , leading to correlated D . From the kinetic energy distribution of D , we obtain the time-evolution of the square of the wave packet on D+ 2 at the time of probing. Using this approach, we have imaged the shape of the vibrational wave packet as a function of time when it undergoes field-induced dissociation. We have observed that the vibrational wave packet separates into two parts, one remains in a bound state and the other propagates to the ionization continuum with a few-femtosecond time-resolution. This is an imaging of molecular bond breaking as it occurs.
6. Imaging Electrons and Their Dynamics In Section 5 we discussed how elastic or inelastic scattering can yield images of the positions of the atoms in a molecule. These images can be obtained with attosecond precision. Once this technology developed, it will be possible to image atomic positions and their changes on any time-scales using the molecular equivalent of time-lapse photography. Now we show that it is also possible to image electrons and their dynamics.
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6.1. T OMOGRAPHIC I MAGING OF THE E LECTRON O RBITAL We have described how tunneling splits the ground state wave function, ψg , launching a coherent wave packet in the continuum, ψc . It is possible to think of tunneling as a beam splitter for the wave function. We have then described how the electron wave packet moves coherently in the continuum first away from the molecule and then back. It is possible to think of this motion as a delay line in an electron interferometer. During the re-collision the two parts of the electron wave function interfere just as light might interfere in an interferometer. We fully characterize optical pulses with shearing interferometry (SPIDER [62]) by observing the harmonic emission. Equivalently, we can fully characterize the wave function through the high harmonics that are emitted during the electron ion re-collision. Regarding the continuum wave packet as a sum of the plane waves ψc = a(k) exp(ikx − iwt), the potential can be readily seen by considering that the Fourier transform of the radiating dipole is given by: d(ω) ∼ a(k) ψg r exp(ikx) where a(k) is the current density of the re-collision electron with momentum k, and x is the co-ordinate along the direction or the laser polarization. Aside from a(k), d(ω) is a one-dimensional spatial Fourier transform times the initial wave function. Measuring d(ω) for different projections of the wave function measures the full wave function. The transform d(ω) can be obtained from the spectrum of high harmonics (|d(ω)|2 ) by assuming an appropriate spectral phase. Experimentally it is achieved by measuring the spectrum of the high harmonics from different molecular alignments with respect to the laser polarization. Intense infrared pulses with pulse durations shorter than the rotational period makes coherent superposition of the rotational levels [71,72]. It allows field-free control of the molecular alignment. We refer the reader to a recent paper for a detailed description of how “orbital tomography” is accomplished [20], as well as for a discussion of recent experiments involving high harmonic generation with aligned molecules [21,73–75].
6.2. ATTOSECOND E LECTRON WAVE PACKET M OTION The same approach can be used to measure the bound state electron wave packet motion. Consider the case where the electron wave packet is produced in the bound state by coherent superposition of the ground and excited electronic wave functions. In the presence of laser fields, part of the excited state wave function (with a lower ionization potential) tunnels to the ionization continuum. When the continuum electron wave packet returns to the parent ion, it can coherently interacts with both the ground and excited state wave function (Fig. 18). It generates
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F IG . 18. Schematic diagram showing how the continuum electron wave packet (ψc ) interacts with both the ground (ψg ) and excited state (ψe ) electron wave-function. Tunnel ionization produces the continuum electron wave packet from the excited state. When the continuum wave packet returns, it interacts coherently with both wave-function. The spectrum of the high harmonics contains information on the bound state wave packet motion.
the high harmonics whose spectrum contains the information of the wave packet motion, i.e., the phase relation between two bound state electron wave functions. We demonstrate this process by solving the one-dimensional Schrödinger equation. We assume the 1600 nm, 8 fs laser pulse whose carrier-envelope phase is adjusted so that only one re-collision dominates over the entire laser 2 2 pulse, E(t) √ = sin(ω(t − t0 )) exp(−(t − t0 ) /2σ ), where t0 = 8 fs and σ = 6/ 8 ln 2 fs. (Carrier-envelope phase stabilization has been achieved using 800 nm laser pulses [14,76–79] and carrier-envelope phase stabilized laser pulses will be achieved in the infrared using recent advances in laser technology, i.e., OPCPA [80].) In Figure 19(a), we plot the electric field as a function of time. We use a one-dimensional, one electron and two-center potential to adjust the level spacing and ionization potential: V = −e/(4πε0 (x − R/2)2 + a 2 ) − e/(4πε0 (x + R/2)2 + a 2 ), where a is a smoothing parameter, and R is the internuclear distance between two nuclei. The initial wave function is defined as a coherent sum of the ground and excited electronic wave functions: ψ(t) = ψ0 + ψ1 exp(−i(Et/h¯ + φ)), where E is the energy difference between the ground (ψ0 ) and excited (ψ1 ) electronic states, and φ is the initial phase difference. Solving the time-dependent Schrödinger equation, we calculate time¨ evolution of the electron wave packet as well as the dipole acceleration, d(t). The ¨ Fourier transform of d(t) yields the spectrum of high harmonics. Figure 19(b) is the calculated spectrum when only the first excited state is populated, and (c) is the spectrum when the ground and the excited states are equally initially populated. The vertical ionization energy is 32.8 eV for the ground and 18.6 eV for the first excited state (E = 14.2 eV, R = 0.4 Å and a = 0.65 Å), respectively. The period of the electron wave packet is 290 attoseconds. With a laser intensity of 1 × 1014 W/cm2 , the electron wave function tunnels from the excited state only to the ionization continuum. The spectrum (b) is continuous since only one re-collision occurs during the laser pulse. On the other hand, the
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F IG . 19. (a) The laser field profile that we use in the calculation (1600 nm, 8 fs). The carrier-envelop phase is adjusted so that only one re-collision dominates. (b) The calculated high harmonic spectrum when only the excited state is populated initially. Because of the single electron re-collision, the spectrum is continuous. (c) The calculated high harmonic spectrum when the ground and the excited states are equally populated (E = 14.2 eV). The periodic dips in the spectrum reflect the bound state electron wave packet motion.
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F IG . 20. The relation between the photon energy of the high harmonics and time. The solid line in the middle panel is a plot of frequency chirp of the high harmonics that connects the photon energy and time. Left panel is a plot of the laser field and the bottom is the same high harmonics spectrum as Fig. 19(c).
spectrum in Fig. 19(c) shows periodic intensity dips that reflect the bound state wave packet motion. The photon energy of the high harmonics can be converted to time from ionization to re-collision. The re-collision energy sweeps (short trajectory) until it reaches its maximum energy at ∼2/3 of the optical period, (∼3.4 fs for 1600 nm) and then decreases (long trajectory). In that case the photon energy is uniquely related to the time. The solid line in the middle panel of Fig. 20 is a plot of the frequency chirp of the high harmonics together with the spectrum (lower panel) and the laser field (left panel). This analysis shows the intensity dips in the spectrum (therefore destructive interference) appear when the bound state wave packet counter-propagates against the incoming continuum electron wave packet. Thus, attosecond pulse generation maps bound state electron wave packet motion onto the spectrum. It is a single laser shot measurement. A pump-probe approach could be used. A pump pulse produces the bound state electron wave
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packet, and the delayed carrier-envelope phase stabilized, short laser pulse probes the wave packet motion. In the mapping approach, it is not necessary to change the delay between pump and probe since the dynamics is mapped in the spectrum. However, if we change the delay, then the intensity dips slide in photon energy. Figure 21(a) is a plot of the calculated spectrum as a function of the delay at the laser field of 3 fs, 800 nm, 1 × 1014 W/cm2 . Increasing the delay, the spectrum
F IG . 21. (a) The intensity map of the calculated high harmonic spectrum as a function of the pump-probe delay (upper panel). The lower panel shows the spectrum zero delay. We assume a laser pulse characterized by 1×1014 W/cm2 , 800 nm, and 3 fs. (b) The radiation intensity at photon energy of 33 eV as a function of the pump-probe delay.
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shifts periodically. Taking a particular photon energy and changing the delay, the bound state wave packet motion is also observed. Figure 21(b) shows the radiation intensity at ∼33 eV as a function of the pump-probe delay. The period of the curve agrees with the period of the electron wave packet motion. The pump-probe approach is useful when the bound state electron wave packet motion is slower than the mapping time window (photon energy) of the high harmonic spectrum. Finally, if the electron wave packet loses its coherence, then the depth of the dips decreases. The decay of the electron wave packet motion can be also observed. We have shown that the attosecond bound electron wave packet motion is mapped onto the spectrum of attosecond pulses. An alternative approach is to use attosecond optical pulses as a probe of the bound state electron wave packet [81].
7. Conclusion Attosecond technology is a radical departure from the ultra-fast technology that preceded it. Although it relies heavily on the advances of femtosecond technology, attosecond science is a mixture of electron and photon physics where their mutual coherence can play an important role. This mixture offers optical science the potential to spatially resolve electron and vibrational wave functions of atoms, molecules and solids. Even measurements within the atomic nucleus may be possible [27,33,34]. Spatial resolution is available on any time scale up to and including attoseconds. In this review we have presented concepts and experimental techniques for attosecond measurement. There are two key features. One is that intense laser fields can split the molecular (or atomic) system coherently into two or more parts and control them. First, we have demonstrated that the tunnel ionization of H2 produces the correlated vibrational and electron wave packets (Sections 4 and 5). Controlling one of the wave packets is a measure of the other wave packet’s motion with attosecond precision. We use the inelastic scattering between two wave packets to measure one of the wave packet’s motion. We have also shown that tunnel ionization splits the bound state electron wave function to the two wave packets, one remaining in the bound and other propagating in the continuum (Section 6). One (or both) of the wave packets can be characterized by measuring their dipole interaction which appears as the high harmonic generation. We have shown how this process is essentially electron interferometry. The other key feature is that the laser phase provides a well-defined attosecond clock. The streak method for measuring an attosecond optical pulse relies on the attosecond clock. (Section 3). Measuring the vibrational wave packet motion also uses the fact that the electron re-collides with its maximum re-collision energy
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at a well-defined time (Section 5). Finally, measurement of the attosecond bound state wave packet motion makes use of the relation between the photon energy and the time of re-collision (Section 6). It has always been clear that attosecond science would allow dynamics measurements to be extended from the femtosecond to the attosecond time scale. However, as you have seen, this single statement does not capture the essence of the attosecond science. Attosecond science, at its core, borrows much from collision science. That is why we chose the title “Attosecond and Angstrom Science”. “Attoseconds and Angstrom Science” represents an inextricable mixture of both optical and collision science.
8. References [1] R.L. Fork, C.H. Brito Cruz, P.C. Becker, C.V. Shank, Opt. Lett. 12 (1987) 483. [2] P. Maine, D. Strickland, P. Bado, M. Pessot, G. Mourou, IEEE J. Quantum Electron. 24 (1988) 398. [3] T. Brabec, F. Krausz, Rev. Mod. Phys. 72 (2000) 545. [4] P.B. Corkum, Phys. Rev. Lett. 71 (1993) 1994. [5] M. Ferray, A. l’Huillier, X.F. Li, L.A. Lompré, G. Mainfray, C. Manus, J. Phys. B 21 (1988) L31. [6] P.M. Paul, E.S. Toma, P. Breger, G. Mullot, F. Augé, Ph. Balcou, H.G. Muller, P. Agostini, Science 292 (2001) 1689. [7] Y. Mairesse, A. de Bohan, L.J. Frasinski, H. Merdji, L.C. Dinu, P. Monchicourt, P. Breger, M. Kovacev, R. Taïeb, B. Carré, H.G. Muller, P. Agostini, P. Salières, Science 302 (2003) 1540. [8] P. Tzallas, D. Charalambidis, N.A. Papadogiannis, K. Witte, G.D. Tsakiris, Nature 426 (2003) 267. [9] M. Drescher, M. Hentshel, R. Kienberger, G. Tempea, C. Spielmann, G. Reider, P. Corkum, F. Krausz, Science 291 (2001) 1923. [10] M. Hentschel, R. Kienberger, Ch. Spielmann, G.A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, F. Krausz, Nature 414 (2001) 509. [11] R. Kienberger, E. Goulielmakis, M. Uiberacker, A. Baltuška, V. Yakovlev, F. Bammer, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, F. Krausz, Nature 427 (2004) 817. [12] M. Drescher, M. Hentschel, R. Kienberger, M. Uiberacker, V. Yakolev, A. Scrinzi, Th. Westerwalbesloh, U. Kleineberg, U. Heinzmann, F. Krausz, Nature 419 (2002) 803. [13] R. Kienberger, M. Hentschel, M. Uiberacker, Ch. Spielmann, M. Kitzler, A. Scrinzi, M. Wieland, Th. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, F. Krausz, Science 297 (2002) 1144. [14] A. Baltuška, Th. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, Ch. Gohle, R. Holzwarth, V.S. Yakovlev, A. Scrinzi, T.W. Hänsch, F. Krausz, Nature 421 (2003) 611. [15] E. Goulielmakis, M. Uiberacker, R. Kienberger, A. Baltuška, V. Yakovlev, A. Scrinzi, Th. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, F. Krausz, Science 305 (2004) 1267. [16] R.A. Bartels, A. Paul, H. Green, H.C. Kapteyn, M.M. Murnane, S. Backus, I.P. Christov, Y. Liu, D. Attwood, C. Jacobsen, Science 297 (2002) 376. [17] A. Paul, R.A. Bartels, R. Tobey, H. Green, S. Weiman, I.P. Christov, M.M. Murnane, H.C. Kapteyn, S. Backus, Nature 421 (2003) 51.
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[18] E.A. Gibson, A. Paul, N. Wagner, R. Tobey, D. Gaudiosi, S. Backus, I.P. Christov, A. Aquila, E.M. Gullikson, D.T. Attwood, M.M. Murnane, H.C. Kapteyn, Science 302 (2003) 95. [19] J. Seres, E. Seres, A.J. Verhoef, G. Tempea, C. Streli, P. Wobrauschek, V. Yakovlev, A. Scrinzi, C. Spielmann, F. Krausz, Nature 433 (2005) 596. [20] J. Itatani, J. Levesque, D. Zeidler, H. Niikura, H. Pépin, J.C. Kieffer, P.B. Corkum, D.M. Villeneuve, Nature 432 (2004) 867. [21] J. Itatani, D. Zeidler, J. Levesque, M. Spanner, D.M. Villeneuve, P.B. Corkum, Phys. Rev. Lett. 94 (2005) 123902. [22] H. Niikura, D.M. Villeneuve, P.B. Corkum, Phys. Rev. Lett. 94 (2005) 083003. [23] M. Lein, Phys. Rev. Lett. 94 (2005) 053004. [24] T. Zuo, A.D. Bandrauk, P.B. Corkum, Chem. Phys. Lett. 259 (1996) 313. [25] S.N. Yurchenko, S. Patchkovskii, I.V. Litvinyuk, P.B. Corkum, G.L. Yudin, Phys. Rev. Lett. 93 (2004) 223003. [26] H. Niikura, F. Légaré, R. Hasbani, M. Ivanov, A.D. Bandrauk, D.M. Villeneuve, P.B. Corkum, Nature 417 (2002) 917. [27] H. Niikura, F. Légaré, R. Hasbani, M. Ivanov, D.M. Villeneuve, P.B. Corkum, Nature 421 (2003) 826. [28] H. Niikura, F. Légaré, D.M. Villeneuve, P.B. Corkum, J. Mod. Optics 52 (2005) 453. [29] M. Weckenbrock, D. Zeidler, A. Staudte, Th. Weber, M. Schoffler, M. Meckel, S. Kammer, M. Smolarski, O. Jagutzki, V.R. Bhardwaj, D.M. Rayner, D.M. Villeneuve, P.B. Corkum, R. Doerner, Phys. Rev. Lett. 92 (2004) 213002. [30] D. Zeidler, A. Staudte, A.B. Bardon, D.M. Villeneuve, R. Dörner, P.B. Corkum, Phys. Rev. Lett. 95 (2005) 203003. [31] E. Constant, V. Taranukhin, A. Stolow, P.B. Corkum, Phys. Rev. A 56 (1997) 3870. [32] J. Itatani, F. Quéré, G.L. Yudin, M.Y. Ivanov, F. Krausz, P.B. Corkum, Phys. Rev. Lett. 88 (2002) 173903. [33] N. Milosevic, P.B. Corkum, T. Brabec, Phys. Rev. Lett. 92 (2004) 013002. [34] S. Chelkowski, A.D. Bandrauk, P.B. Corkum, Phys. Rev. Lett. 93 (2004) 083602. [35] K. Burnett, V.C. Reed, J. Cooper, P.L. Knight, Phys. Rev. A 45 (1992) 3347. [36] S. Augst, D.D. Meyerhofer, D. Strickland, S.L. Chin, J. Opt. Soc. Am. B 8 (1991) 858. [37] A.V. Ammosov, N.D. Delone, V.P. Krainov, Sov. Phys. JETP 64 (1986) 1191. [38] L.V. Keldysh, Sov. Phys. JETP 20 (1965) 1307. [39] S.M. Hankin, D.M. Villeneuve, P.B. Corkum, D.M. Rayner, Phys. Rev. A 64 (2001) 013405. [40] X.M. Tong, Z.X. Zhao, C.D. Lin, Phys. Rev. A 66 (2002) 033402. [41] T. Brabec, M. Côte, P. Boulanger, L. Ramunno, Phys. Rev. Lett. 95 (2005) 073001. [42] M. Lezius, V. Blanchet, M.Y. Ivanov, A. Stolow, J. Chem. Phys. 117 (2002) 1575; M.J. Dewitt, R.J. Levis, J. Chem. Phys. 108 (1998) 7739. [43] V.R. Bhardwaj, S.A. Aseyev, M. Mehendale, G.L. Yudin, D.M. Villeneuve, D.M. Rayner, M.Y. Ivanov, P.B. Corkum, Phys. Rev. Lett. 86 (2001) 3522. [44] P. Dietrich, N.H. Burnett, M.Y. Ivanov, P.B. Corkum, Phys. Rev. A 50 (1994) R3585. [45] M. Lewenstein, Ph. Balcou, M.Y. Ivanov, A. L’Huillier, P.B. Corkum, Phys. Rev. A 49 (1994) 2117. [46] P. Agostini, L.F. DiMauro, Rep. Prog. Phys. 67 (2004) 813. [47] Y. Mairesse, A. de Bohan, L.J. Frasinski, H. Merdji, L.C. Dinu, P. Monchicourt, P. Breger, M. Kovacev, T. Auguste, B. Carré, H.G. Muller, P. Agostini, P. Salières, Phys. Rev. Lett. 93 (2004) 163901. [48] K. Varjú, Y. Mairesse, P. Agostini, P. Breger, B. Carré, L.J. Frasinski, E. Gustafsson, P. Johnsson, J. Mauritsson, H. Merdji, P. Monchicourt, A. L’Huillier, P. Salières, Phys. Rev. Lett. 95 (2005) 243901. [49] T. Sekikawa, A. Kosuge, T. Kanai, S. Watanabe, Nature 432 (2004) 605.
548
H. Niikura and P.B. Corkum
[8
[50] P.B. Corkum, N.H. Burnett, M.Y. Ivanov, Opt. Lett. 19 (1994) 1870. [51] R. López-Martens, J. Mauritsson, P. Johnsson, A. L’Huillier, O. Tcherbakoff, A. Zaïr, E. Mével, E. Constant, Phys. Rev. A 69 (2004) 053811. [52] E. Constant, private communications. [53] Y. Mairesse, F. Quéré, Phys. Rev. A 71 (2005) R011401. [54] F. Quéré, Y. Mairesse, J. Itatani, J. Mod. Opt. 52 (2005) 339. [55] N. Doudovich, O. Smirnova, J. Levesque, M.Y. Ivanov, D.M. Villeneuve, P.B. Corkum, to be submitted. [56] I.V. Litvinyuk, F. Légaré, P.W. Dooley, D.M. Villeneuve, P.B. Corkum, J. Zanghellini, A. Pegarkov, C. Fabian, T. Brabec, Phys. Rev. Lett. 94 (2005) 033003. [57] R. de Nalda, E. Heesel, M. Lein, N. Hay, R. Velotta, E. Springate, M. Castillejo, J.P. Marangos, Phys. Rev. A 69 (2004) R031804. [58] C. Vozzi, F. Calegari, E. Benedetti, J.P. Caumes, G. Sansone, S. Stagira, M. Nisoli, R. Torres, E. Heesel, N. Kajumba, J. Marangos, C. Altucci, R. Velotta, Phys. Rev. Lett. 95 (2005) 153902. [59] G.L. Yudin, M.Y. Ivanov, Phys. Rev. A 64 (2001) 013409. [60] G. Steinmeyer, D.H. Sutter, L. Gallmann, N. Matuschek, U. Keller, Science 286 (1999) 1507. [61] F. Légaré, I.V. Litvinyuk, P.W. Dooley, F. Quéré, A.D. Bandrauk, D.M. Villeneuve, P.B. Corkum, Phys. Rev. Lett. 91 (2003) 093002. [62] C. Iaconis, I.A. Walmsley, Opt. Lett. 23 (1998) 792. [63] J. Hu, K.L. Han, G. He, Phys. Rev. Lett. 95 (2005) 123001. [64] M. Kitzler, M. Lezius, Phys. Rev. Lett. 95 (2005) 253001. [65] M. Lein, J.P. Marangos, P.L. Knight, Phys. Rev. A 66 (2002) 051404. [66] H. Niikura, D.M. Villeneuve, P.B. Corkum, Phys. Rev. Lett. 92 (2004) 133002. [67] H. Niikura, P.B. Corkum, D.M. Villeneuve, Phys. Rev. Lett. 90 (2003) 203601. [68] H. Niikura, D.M. Villeneuve, P.B. Corkum, Phys. Rev. A 73 (2006) R021402. [69] A. Zavriyev, P.H. Bucksbaum, J. Squier, F. Salin, Phys. Rev. Lett 70 (1993) 1077. [70] T. Seideman, M.Yu. Ivanov, P.B. Corkum, Phys. Rev. Lett. 75 (1995) 2819. [71] H. Stapelfeldt, T. Seideman, Rev. Mod. Phys. 75 (2003) 543. [72] P.W. Dooley, I.V. Litvinyuk, K.F. Lee, D.M. Rayner, D.M. Villeneuve, P.B. Corkum, Phys. Rev. A 68 (2003) 023406. [73] T. Kanai, S. Minemoto, H. Sakai, Nature 435 (2005) 470. [74] K. Miyazaki, M. Kaku, G. Miyaji, A. Adburrouf, F.H.M. Faisal, Phys. Rev. Lett. 95 (2005) 243903. [75] F. Lindner, G.G. Paulus, H. Walther, A. Baltuška, E. Goulielmakis, M. Lezius, F. Krausz, Phys. Rev. 92 (2004) 113001. [76] M. Kakehata, Y. Fujihara, H. Takada, Y. Kobayashi, K. Torizuka, T. Homma, H. Takahashi, Appl. Phys. B 74 (2002) S43. [77] A. Baltuška, M. Uiberacker, E. Goulielmakis, R. Kienberger, V. Yakovlev, Th. Udem, T. Hänsch, F. Krausz, IEEE J. Selected Topics Quantum Electron. 9 (2003) 972. [78] I. Thomann, E. Gagnon, R.J. Jones, A.S. Sandhu, A. Lytle, R. Anderson, J. Ye, M.M. Murnane, H. Kapteyn, Opt. Exp. 12 (2004) 3493. [79] T. Fuji, J. Rauschenberger, A. Apolonski, V.S. Yakovlev, G. Tempea, T. Udem, C. Gohle, T.W. Hänsch, W. Lehnert, M. Scherer, F. Krausz, Opt. Lett. 29 (2004) 332. [80] X. Fang, T. Kobayashi, Opt. Lett. 29 (2004) 1282. [81] G.L. Yudin, S. Chelkowski, J. Itatani, A.D. Bandrauk, P.B. Corkum, Phys. Rev. A 72 (2005) 051401 (R).
ADVANCES IN ATOMIC, MOLECULAR AND OPTICAL PHYSICS, VOL. 54
ATOMIC PROCESSING OF OPTICALLY CARRIED RF SIGNALS JEAN-LOUIS LE GOUËT* , FABIEN BRETENAKER and IVAN LORGERÉ Laboratoire Aimé Cotton, CNRS UPR3321, bâtiment 505, campus universitaire, 91405 Orsay Cedex, France
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Radio Frequency Spectral Analyzers . . . . . . . . . . . . . . . . . . . . . 3. Spectrum Photography Architecture . . . . . . . . . . . . . . . . . . . . . . 3.1. Principle of Operation . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Basic Spectroscopic Properties of Tm3+ :YAG for SHB Experiments . 3.3. Experimental Demonstration . . . . . . . . . . . . . . . . . . . . . . . 4. Frequency Selective Materials as Programmable Filters . . . . . . . . . . . 4.1. Reconfigurable Filtering . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Linear Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Time-Delayed Four-Wave Mixing . . . . . . . . . . . . . . . . . . . . 4.4. Coherent Combination of Local Response: Photon Echo . . . . . . . . 5. Rainbow Analyzer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Principle of Operation . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Programming Stage . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Spectral Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4. Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5. Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6. Future Improvements . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Photon Echo Chirp Transform Spectrum Analyzer . . . . . . . . . . . . . . 6.1. Photon Echo Chirp Transform . . . . . . . . . . . . . . . . . . . . . . 6.2. Experimental Demonstration in Er3+ :YSO . . . . . . . . . . . . . . . 6.3. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Frequency Agile Laser Technology . . . . . . . . . . . . . . . . . . . . . . 7.1. Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. Electro-Optic Tuning of Diode Laser Extended Cavity . . . . . . . . . 7.3. Laser Chirp Spectral Purity Characterization . . . . . . . . . . . . . . 7.4. Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Abstract Rare earth ions embedded in crystals are natural high-quality-factor resonators that can be used for processing optically-carried broadband radio-frequency signals. This chapter focuses on the radio-frequency spectrum analysis function and describes different architectures that are designed to reach tens of gigahertz instantaneous bandwidth with sub-megahertz resolution. Various approaches are considered. The active material may act as a spectral buffer memory. Instead one may store a processing function inside the crystal. The latter may then operate either as a frequencyto-angle converter or a frequency-to-time transformer. All those architectures have been explored experimentally. To meet the specific requirements of these processors, frequency agile lasers have been developed. These keynote devices are described in detail.
1. Introduction When absorbing centers are embedded in a host matrix and the medium is cooled to a few degrees Kelvin, homogeneous absorption line widths are dramatically reduced as a result of strongly diminished environmental fluctuations. While the cooling decreases the homogeneous widths, it has little effect on the inhomogeneous line width that results from the static interaction of the absorbing centers with the host. When exposed to monochromatic light, the material is excited only within the homogeneous-width of the incoming light and a narrow spectral hole is burned at this position within the absorption profile [1]. This work is part of a long quest for practical applications of spectral hole burning (SHB) in solid materials, an effort that has been pursued since the beginning of the 70’s. The first application considered was optical memory, with the goal of overcoming the storage density diffraction limit by adding a fourth non-spatial dimension, namely frequency, to the memory volume. The initial two patents in this area consider page-oriented storage, where a spatial page of information is stored at a spectral address [2,3]. To retrieve information one tunes the readout laser to the engraving wavelength. After discovery of permanent SHB in organic materials [4,5], storage of 2000 [6], 6000 [7] and ultimately 10,000 images [8] was demonstrated in chlorine, a porphyrine derivative. A storage density of up to 10 Gb/cm2 is reached at a temperature of 1.7 K by the ETH group [9]. Transfer rate is then limited by the inverse spectral channel width. Time domain encoding of data address can be substituted for spectral addressing [10]. In this approach the Fourier transform of the data sequence is engraved over the inhomogeneously broadened spectral memory absorption band. Then, one has simultaneous access to all the spectral channels. More precisely the data transfer rate equals the inhomogeneous
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line-width, a figure that reaches tens of GHz in rare earth ion doped materials such as Tm3+ :YAG. This excessively high speed proves difficult to manage and a hybrid chirped carrier technique is proposed. Combining temporal encoding with spectral scan, one has an approach that conveys flexibility to data transfer at both engraving and readout [11–13]. Despite excellent performance in terms of storage density and capacity, access time and transfer rate, SHB memories suffer from operating at liquid helium temperature, a severe limitation for mass market application. In the quest of niches for SHB technology, it was soon realized that the specific time/frequency dimension should not be considered on an equal footing with the spatial ones, as it was in large memory application. Instead, the inhomogeneous line-width can be regarded as the spectral bandwidth of an optically carried signal analog processor. In this respect SHB materials can outdo any existing electronic processor and appear to be potentially attractive microwave photonics components, offering an instantaneous bandwidth in excess of 10 GHz with a channel capacity larger than 104 . Application to range and Doppler processing of RADAR return signals has been proposed recently. The cross-correlation function of the encoded RADAR pulse and the return signal is stored as a spectral grating within the SHB material. Even if the pulse encoding is renewed at each transmission, the correlation grating remains unchanged and can be accumulated for milliseconds. Scanning a monochromatic laser through the engraved spectral structure and using a narrow band detector one may retrieve information on target distance and speed with large dynamic range [14]. Another application has been considered for array antennas. For narrow band operation, one directs the gain to the target by adjusting the relative phase shift of the array elements. The phase shift procedure fails to provide an achromatic delay between the array elements in broad band operation, giving rise to the so called “squint effect”. Then “true time delays” can be stored in a SHB material and be applied to arbitrary waveform optically carried signals without latency, either at transmission or reception [15–17]. Broadband devices are also needed for spectral analysis at the back-end of the heterodyne receivers used in (sub)millimeter astrophysics. The molecular lines observed in astrophysics are generally broadened by either the Doppler effect in the interstellar medium or by pressure in planet atmospheres. In the former case, a 1000 km/s-wide spectral window is generally enough to analyze a 700 km/s Doppler broadening. If at 100 GHz this interval is covered by a 330 MHz bandwidth spectrometer, the required bandwidth raises to 10 GHz at 3 THz, the position of the HD line, and 20 GHz at 6 THz, the position of the atomic oxygen narrow line. As for the lines observed in planetary atmospheres, their pressure broadening may reach several GHz and they often exhibit a narrow feature at their center, corresponding to high altitude emission. Both the broad global profile and the narrow central structure usually need be explored. In an heterodyne
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receiver, the THz range signal, mixed with a local oscillator, is shifted down to the GHz range where its spectral analysis is usually performed either by acoustooptic spectrometers (AOS) or by digital auto-correlators (DAC). The AOS cover a bandwidth of up to 2 GHz in a single unit. The DAC offer a smaller bandwidth but their flexible resolution can accommodate fine details such as narrow lines in the dark clouds of our galaxy, with Doppler broadening as small as 0.1–1 km/s. Combining more than 10 GHz instantaneous bandwidth with less than 1 MHz resolution, SHB materials have the potential to accomplish a breakthrough in RF spectral analysis. In addition they would offer improved power consumption with respect to DAC’s together with resolution flexibility and better stability than AOS’s. In this chapter we review recent experimental achievements that demonstrate SHB capabilities for broadband spectral analysis with a unity probability of intercept. Different approaches have been considered. Some use the SHB material as a storage medium where the optically carried RF signal spectrum is accumulated before readout. Unlike this “spectral photography” scheme, other approaches avoid the storage of the signal under investigation in the SHB medium. Instead the material is programmed as an analog processor that continuously processes the signal-carrying beam. Various programming functions have been explored. Indeed the SHB material is opened to a wealth of wide-band spectrum analysis architectures through diverse combinations of space and frequency variables. After a brief review of conventional RF spectrum analysis techniques in Section 2, we present the spectrum photography architecture in Section 3. Although this is the most recently considered approach it is also the conceptually simplest one. Then we introduce the programmable filtering approach in Section 4, reestablishing the mathematical expression of the SHB material optical response within the framework of a two-level atom ensemble picture. Two demonstrations of analog RF signal processing are then presented in Sections 5 and 6. One processor— the rainbow analyzer—achieves the angular separation of the spectral components while the other one—the photon echo chirp transform analyzer—displays the angular components in the time domain. Keystones of these architectures, specific lasers had to be developed for these experiments. Section 7 is devoted to these devices.
2. Radio Frequency Spectral Analyzers In this section we review the RF spectrum analyzers that have been considered so far for spatial missions. They have been used at the back end of heterodyne receivers for atmospheric and astronomic observation in the submillimeter and millimeter wavelength range. The THz signal collected by the antenna is mixed
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with a local oscillator, which shifts its frequency to the GHz range. After amplification the signal is then directed to various RF spectrum analyzers that may include filter banks, acousto-optic analyzers, digital and analog autocorrelators, and surface acoustic wave devices. In a filter bank analyzer the signal is divided into different spectral channels, each being equipped with its own bandpass filter and detector. Weight and power consumption limit the channel number to a few tens. Bandwidth and resolution are fixed. Missions devoted to solar and atmospheric observation carry those devices [18,19], each filter bank covering a specific molecular line, with optimum resolution at the line center [20,21]. Acousto-optic spectral analyzers (AOS) may offer up to 2 GHz instantaneous bandwidth in a single compact and low consumption unit. First demonstrated in the 70’s [22,23], these analyzers were used for the first time in a civilian space mission in 1998 [24–26]. The ODIN satellite [27], launched in 2001, carries an AOS with 1000 channel resolution over 1 GHz bandwidth [28]. The forthcoming HERSCHEL mission [29] will also be equipped with AOS [30–32]. In an AOS the RF signal to be analyzed is fed to the transducer of an AO modulator. The acoustic wave generates an index grating that diffracts a monochromatic laser beam. The different spectral components of the RF signal give rise to different grating components that diffract the beam in different directions. Diffracted intensity distribution along a photodetector array then reflects the RF signal power spectrum. Since the Wiener–Khinchin theorem relates the power spectrum to the autocorrelation function, one may record the latter quantity and then retrieve the power spectrum by Fourier transform computation. The quantity to be measured is the time averaged product of the signal and its time delayed replica, at various settings of the time delay. The inverse time delay half size and the number of delay settings respectively determine the analyzer bandwidth and channel number. In a digital autocorrelator the signal is first processed by an analog to digital converter (ADC). Because of time averaging a poor dynamic range converter can result in excellent dynamic range autocorrelation measurement [33,34]. Digital processing offers numerous benefits such as stability and absence of additional noise. Delays, supplied by cascaded shift registers, are easily controlled through the clock rate, which gives resolution flexibility. Finally the device is easy to assemble and reproduce [35]. Limitation comes from the sampling rate of the ADC. With a 2 GS/s rate (GS/s = Gigasamples/second), available units can offer up to 1 GHz bandwidth [36]. Another important limitation is set by power consumption that grows with the bandwidth. ODIN and HERSCHEL missions are equipped with such devices [27,29,37,38]. Increased bandwidth is made available in analog auto-correlators. In these devices two signal replica counter-propagate along two parallel micro-strip lines.
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Table I Features of current RF spectrum analyzers for sub-millimeter astronomy applications
Bandwidth Channel number Dynamic range Spectral flexibility Best resolution Interception probability Access time
Filter bank
Digital autocorrelator
Analog autocorrelator
SAW
AOS
1.2 GHz 25
700 MHz 100
3.6 GHz 128
205 MHz 4096
1 GHz 2000
60 dB
48 dB
38 dB
40 dB
30 dB
no
yes
no
no
no
NA
140 kHz
NA
NA
NA
100%
100%
100%
100%
100%
FT limited
Integration time
Integration time (>10 ms)
FT limited
Readout time
The two traveling waves are sampled at taps, evenly spaced along the lines. Different pairs of opposite taps supply different delays between the sampled signals that are subsequently multiplied and digitally converted [39,40]. The most advanced system offers an instantaneous bandwidth of 4 GHz and 128 channels. Signal attenuation along the microstrips limits the available delay and thus the spectral resolution. High resolution broadband spectral filtering and large delay generation are tasks that are difficult to achieve with pure electronic means. Hybridizing may help to address these issues as already illustrated by the AOS where optics is combined with electronic processing. Another example of hybridizing is offered by surface acoustic wave (SAW) dispersion lines. By converting the electric signal into a surface acoustic wave one reduces both the wavelength and the delay line dimension in the ratio of sound and light speeds. With the small size acoustic resonant circuits and delay lines that are readily feasible one can build dispersion lines with large group delay dispersion rate. Multiplication by a chirped reference followed by dispersion in the appropriate SAW line performs the signal “chirp transform” [41] and results in the desired power spectrum profile [42,43]. The device to be placed in The Stratospheric Observatory For Infrared Astronomy (SOFIA) will cover 205 MHz with 50 kHz resolution and 40 dB dynamic range [44–47]. The features of these RF spectrum analyzers are summarized in Table I.
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F IG . 1. Basic principle of the spectrum photography analyzer.
3. Spectrum Photography Architecture 3.1. P RINCIPLE OF O PERATION Although it has been considered only recently [48–52], the spectrum photography architecture is actually the simplest of the three architectures we consider here. It consequently constitutes a good introduction to more complicated schemes which will be discussed later. Its principle of operation can be understood as a two-step process from the scheme of Fig. 1. The RF signal to be analyzed is transposed on a monochromatic fixed frequency laser beam with the help of a Mach–Zehnder modulator. The laser frequency is adjusted in such a way that one of the RF-signal-carrying side-bands lies inside the SHB material absorption band [see Fig. 2(a)]. Upon crossing the SHB material, the beam excites the ions that are resonant with the different optically carried spectral components of the RF signal. Consequently, the optically carried RF spectrum is recorded in the material absorption spectrum [see Fig. 2(b)]. By probing the material transmission with a frequency chirped laser [see Fig. 2(c)] one readily obtains a temporal image of the engraved spectrum [see Fig. 2(d)]. Although the read out scanning procedure is reminiscent of usual electronic spectrum analyzers, the SHB analyzer exhibits a very distinctive feature. Indeed, although spectral addresses are probed sequentially the RF signal is continuously recorded in the SHB material that operates as a spectral buffer memory. By simply using two separate lasers for engraving and probing one may reach 100% probability of interception. Storage lifetime in the buffer memory is limited by the excited level population lifetime that can exceed 10 ms in the materials we consider. The spectral resolution is limited by the laser noise and the square root of the chirp rate of the reading laser. Given the broad bandwidth to be scanned in less than 10 ms, the chirp rate limited resolution will dominate the homogeneous line-width of the optical transition. The geometry of the experiment can be rather
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F IG . 2. Basic steps of the spectrum photography analyzer. The RF spectrum (a) is engraved in the absorption spectrum (b) which is then read by a frequency-swept laser (c) leading to signal (d).
simple, allowing us to consider this experiment as an elementary pixel of a future broadband spectrally resolved RF imaging system. The principle schematized in Fig. 1 has been implemented experimentally. This experiment makes use of a particular type of SHB material, namely a Tm3+ :YAG crystal. Before describing this experiment we turn to a short summary of the basic properties of rare earth doped crystals for SHB, with Tm3+ :YAG as an illustration.
3.2. BASIC S PECTROSCOPIC P ROPERTIES OF T M 3+ :YAG FOR SHB E XPERIMENTS Rare earth ion-doped crystals (REIDC) represent a satisfactory candidate for high resolution applications. The common structure of rare earth atoms is 5s2 5p6 4fn+1 6s2 , where 0 n 13. In triply charged ions, one 4f electron and both 6s electrons are removed. Optical transitions within the 4fn configuration are forbidden for parity reason. Nevertheless some weak lines, with oscillator strength of the order of 10−8 , result from mixing with the 4f(n−1) 5d configuration in sites without inversion symmetry. In addition the sharpness of the lines is preserved, in a solid state matrix, by the shielding of 4f electrons by the 5s and 5p electrons. At 5 K in a crystal host matrix the homogeneous line-width is usually much smaller than 1 MHz. If profitable as far as spectral resolution is concerned,
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F IG . 3. Level scheme in Tm3+ : YAG. The laser is coupled to the 3 H4 ↔ 3 H6 transition at 793 nm. The level decay rates are denoted by γi .
a weak oscillator strength unfortunately means that a larger amount of energy is needed for engraving. On some transitions the ions behave as two-level atoms, with an excited state that directly decays to the ground state. Then the spectral hole lifetime coincides with that of the excited state. The 4 I13/2 ↔ 4 I15/2 transition of Er3+ near 1.5 µm belongs to this class. It offers a 10 ms storage time in the upper level [53]. This material gives access to the wealth of tools that has been developed for fiber-optics communications, including versatile lasers and amplifiers [54]. Different processes may slow down the return to initial state and the spontaneous erasure of the engraved structure. In Eu3+ and Pr3+ doped crystals, resonant excitation results in an optical pumping alteration of the atom distribution over the ground state hyperfine structure [55]. This alteration may survive for hours before return to thermal equilibrium. The long memory life is obtained at the expense of the bandwidth. Indeed the lines connected to the different hyperfine sublevels overlap, which limits the available bandwidth to a few MHz. A hyperfine-structure-free-ion, such as Thulium Tm3+ is preferred for our broadband processing application. On the 3 H4 ↔ 3 H6 transition in Tm3+ doped crystals, a bottleneck metastable state, with a lifetime of about 10 ms, delays the excited state decay to the ground state [56] (see Fig. 3). The 3 H4 ↔ 3 H6 transition wavelength at 793 nm is compatible with integrated electro-optic LiNbO3 ultrafast modulators. This wavelength also falls within the range of common semiconductor lasers and of the Titanium–Sapphire laser. Optical pumping from the ground state to the bottleneck state 3 F4 via the upper state 3 H6 offers a convenient way to accumulate engraving [57]. Owing to the 500 µs lifetime of the upper level, an optical pumping cycle lasts about 1 ms. Several cycles can take place
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F IG . 4. Collinear spectrum photography analysis experiment.
during the 10 ms lifetime of 3 F4 . We experimentally verified that a large fraction of the ground state population can actually be transferred to 3 F4 . This procedure entails two benefits. First optimal engraving can be reached without saturating the 3 H ← 3 H transition, which means lower laser intensity requirements. Second, 4 6 with a 1 kHz refresh rate and a 10 ms storage lifetime, the diffraction efficiency is nearly stationary.
3.3. E XPERIMENTAL D EMONSTRATION 3.3.1. Collinear Experiment The experimental demonstration of the spectrum photography architecture has been first performed using the experiment schematized in Fig. 4 [51]. A very similar experiment has been performed at the University of Colorado [52]. The experiment reported in [51] is based on a 2.5-mm long 0.5-at.% doped Tm3+ :YAG crystal cooled to 4.5 K. Under these conditions, the peak absorption at 793 nm is 85%. As can be seen in Fig. 4, this proof of principle experiment has been performed with only one laser. This laser is a frequency agile external cavity diode laser (ECDL) [58]. It is used alternatively as the engraving laser and as the probe laser of Fig. 1. During the engraving stage, the frequency of this laser is tuned by steps
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F IG . 5. (a) Example of spectrum obtained using the setup of Fig. 4. The 10-GHz bandwidth readout is performed in 2 ms. 15 equally spaced tones have been engraved in the sample, except for a 5 MHz doublet, which is perfectly resolved as can be seen in (b).
to different spectral positions, which mimics an optically carried RF signal. The power of the beam is controlled by the acousto-optic modulator AO1 and is focused to a waist radius w0 = 90 µm inside the crystal. The zeroth-order beam at the output of AO1 can be frequency shifted using AO1 before being mixed with the beam emerging from the cryostat. This allows performing a heterodyne detection at 5 MHz of the light emitted by the ions, as will be illustrated later. Light is detected using a PIN photodiode (PD) followed by a 10-dB gain amplifier. An example of a 10-GHz bandwidth spectral analysis is reproduced in Fig. 5. The engraved spectrum consists of a series of 16 spikes each lasting 150 µs (pulse energy 450 nJ) with the laser tuned to 16 different frequencies. The reading is performed 1.6 ms later, with a 10-GHz bandwidth scanned in 2 ms. During the reading phase, one reduces the optical power incident on the crystal to 750 µW, in order not to erase the engraved spectrum. The resulting signal is demodulated and its amplitude is normalized to the unsaturated transmission of the crystal. Among the 16 engraved peaks, 15 are equally spaced by 620 MHz all over the 10-GHz bandwidth [see Fig. 5(a)]. In this experiment, the 16th engraved peak is located 5 MHz apart from one of the 15 equally separated peaks. This doublet is perfectly resolved by our analyzer, as can be seen in Fig. 5(b). The line-width of each peak corresponds to 2 MHz in this experiment. Of course, this resolution depends on the chirp rate r, as can be seen from the experimental results reproduced in Fig. 6. The open circles in Fig. 6 represent
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F IG . 6. (a) Experimental (circles) and theoretical (thick line) evolutions of the line-width (full width at half maximum) of a single-frequency readout signal versus chirp rate r. The thin line is r 1/2 .
the measured evolution of the width of a single peak engraved in 200 µs (pulse energy 600 nJ) versus the reading chirp rate. Indeed, it is well known [59] that as soon as the width Γ /2π of any spectral feature of interest is not much larger than r 1/2 , the readout gets distorted and, in particular, broadened. If we consider a Lorentzian lineshape of width Γ /2π (in Hz) probed by a light beam of varying detuning δ(t) = rt (exact resonance occurs at t = 0), the resulting time evolution s(t) of the transmitted field amplitude normalized to the undistorted one is given by [60]: Γ 1 + i − i (Γ −2iπrt)2 1−i 1 4πr s(t) = Im √ √ e erfc √ √ (Γ − 2iπrt) , (1) r 2 2 2 2 πr where erfc stands for the complementary error function. This equation leads to the thick line of Fig. 6 obtained with Γ /2π = 700 kHz, which is in very good agreement with the measurements. This value of Γ /2π is larger than the absolute limit given by twice the homogeneous line-width of the ions at 4.5 K (2 × 150 kHz = 300 kHz) because of the contribution of the laser frequency jitter, which will be shown in Section 7.3. This limit line-width Γ /2π becomes relevant when it is larger than r 1/2 whose value is reproduced as a thin line in Fig. 6. This shows that a sub-MHz resolution can be reached if the 10 GHz bandwidth is probed in 10 ms, leading to a number of independent frequency bins equal to 10,000. Of course, due to the finite lifetime of the engraved spectrum that is limited by the lifetime of the metastable level population (10 ms), the use of a slower chirp in order to improve the frequency resolution and to reach a number of frequency channels equal to 10,000 will lead to an increase of the delay before the engraved spectrum is read and hence to a reduction of the detected signal.
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F IG . 7. Experimental (circles and squares) and theoretical (full line) evolutions of the detected amplitude of a peak versus the engraving optical energy. The delay between the engraving and the reading of the peak is 1.4 ms. The engraving times are 400 µs (open circles) or 600 µs (filled squares). The theoretical curve is computed with saturation energy equal to 75 nJ. The inset shows the distribution of the signal value in the background absorption region (thin line) together with a Gaussian fit (thick line) leading to a standard deviation of 50 µV.
Another important property of a spectrum analyzer is its dynamic range. It can be extracted from the experimental results reproduced in Fig. 7. This figure represents the evolution of single peak readout amplitude versus engraving optical energy. The experimental points have been obtained for two values of the engraving pulse duration (400 µs and 600 µs) and by varying the engraving power. The engraved peak is read after a time delay equal to 1.3 ms. These measurements exhibit a typical saturation behavior, which can be reproduced from a simple three-level rate equation description of the system represented as a full line in Fig. 7. We estimate the linear part (with a maximum non-linearity of 20%) of this response to correspond to a signal voltage between 0 and 2 mV. To compare this signal with the typical signal noise, we record 10,333 successive samples separated by 8 ns while the laser is chirped in a region where no peak has been engraved. The inset in Fig. 7 then reproduces the distribution of the signal value together with a Gaussian fit. The average value of the signal (5.15 mV) corresponds to the transmitted signal when the absorption is not saturated. The standard deviation of this Gaussian noise is found to be equal to 50 µV. The main components of this noise are (i) the thermal noise of the detector and amplifier, due to the low value of the detected optical power (a 750 µW incident power is incident on the crystal during the reading stage), and (ii) the low-frequency components of the laser intensity noise. This leads to a value of 16 dB for the linear dynamic range in
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terms of optical field amplitude of our “photographic scheme” spectral analyzer. Since in this architecture the field emitted by the atoms is directly proportional to the RF signal power, this corresponds to a 16 dB dynamic range for the RF signal also. This relatively poor value is limited by two factors. First, the signal amplitude is rather low, due to the small number of atoms probed by the optical beams. Second, the noise is rather important, due to the fact that our measurements are performed on a non-dark background. Indeed, in this direct transmission experiment, the peaks of Fig. 5 are sitting on a background due to the transmitted laser power. The shot noise associated with this background intensity is actually the limiting factor here. This is why we now turn to the realization of a dark background experiment based on a non-collinear geometry. 3.3.2. Non-Collinear Experiment The principle of this non-collinear spectrum photography analyzer is summarized in Fig. 8, in comparison with the collinear one which has just been described. Fig. 8(a) represents the writing field EW impinging on the SHB sample in the collinear experiment. During the read-out phase of this experiment [see Fig. 8(b)], the field with real amplitude EP probes the susceptibility of the material which has been spectrally shaped during the engraving phase. This susceptibility gives rise to a macroscopic polarization that radiates the field ERF , with a real part ER and an imaginary part EI . Since the two fields are collinear, the detected intensity is given by I = EP2 + 2EP ER + ER2 + EI2 ≈ EP2 + 2EP ER .
(2)
In Eq. (2), the intensity of the RF field emitted by the ions has been neglected because it is usually much smaller than the two other terms. This equation perfectly
F IG . 8. Principle of the (a), (b) collinear and (c), (d) non-collinear geometries.
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illustrates the pros and cons of the collinear architecture. Indeed, it shows that the term EI , i.e., the dispersive response of the atoms, can be neglected thanks to the homodyning or the real part ER by the probe field amplitude EP . This leads to the very good frequency resolution of Fig. 6. The drawback of this approach is that the term EP2 is very large and leads to a strong noise which severely limits the signal to noise ratio of the analyzer. To circumvent this problem, we choose to illuminate the sample with two rather than one engraving beams, as shown in Fig. 8(c). The two engraving beams both carry the same RF signal and are separated by a small angle in the SHB crystal. Consequently, they will engrave a spatial grating only in atoms which are resonant with their spectral components. To read this family of gratings, the chirped probe beam is then incident along the direction of one of the engraving beams, as seen in Fig. 8(d). This beam will then be diffracted by the gratings engraved in the spectrum only when it corresponds to an engraved frequency. This is why the RF field ERF emitted by the ions can now be detected on a dark background. The detected intensity is given by I = ER2 + EI2 .
(3)
As expected, the probe field intensity and its associated shot noise have disappeared. However, we now have two new problems. First, the detected intensity becomes proportional to the square of the RF power, leading to a decrease of the sensitivity of the analyzer. Second, the system is now sensitive to the imaginary part of the atomic response also. This leads to a strong degradation of spectral resolution, as illustrated by the experimental result shown in Fig. 9(a). In this figure, one can clearly see that the detection of both quadratures of the emitted field leads to a dispersive-like response of the system. This strongly degrades the spectral resolution of the system. To get rid of the dispersive part of the atomic response, we perform a heterodyne detection of the diffracted beam. To this aim, a fraction of the probe beam is frequency shifted by an acousto-optic modulator operating at frequency f and mixed with the diffracted field before it reaches the detector. The detected intensity is given by 2 I = ELO + 2ELO ER cos(2πf t) − 2ELO EI sin(2πf t) + ER2 + EI2 2 ≈ ELO + 2ELO ER cos(2πf t) − 2ELO EI sin(2πf t),
(4)
where ELO is the local oscillator field amplitude. The detected beat note can be demodulated with the correct phase reference to isolate the term 2ELO ER , which contains only the real part ER of the emitted field. This leads to the experimental result of Fig. 9(b). One can see that the dispersive part of the atomic response has been almost perfectly eliminated, in agreement with Eq. (4). According to this equation, one can see that (i) the signal intensity is again proportional to the RF signal intensity, and (ii) the detection no longer occurs on a dark background,
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F IG . 9. Detected spectrum for the non-collinear experiment. In (a) and (b), the engraved spectrum is represented by the rectangular curve. (a) The total diffracted field is detected. (b) A heterodyne detection permits to isolate the real part of the emitted field and to discard the dispersive part of the atomic response.
due to the presence of the local oscillator intensity. However, the system is now sensitive to the spectral component of the local oscillator intensity noise at frequency f , and no longer to the low-frequency part of the intensity noise as in the case of the collinear geometry. A shot-noise limited signal-to-noise ratio is then easier to reach here. Optimization of the signal to noise ratio of the analyzer, i.e. of its linear dynamic range, hence consists in increasing the local oscillator power to increase the signal of Eq. (4), until the total detected power reaches the saturation limit of the detector for the maximum value of the power diffracted by the ions. Recent results [61], obtained with a 500-µm value for the beam waist inside the crystal, a 10 mW probe beam power, an optimized local oscillator power and a PIN photodiode led to a linear dynamic range larger than 30 dB, together with MHz resolution and 10-GHz bandwidth.
4. Frequency Selective Materials as Programmable Filters 4.1. R ECONFIGURABLE F ILTERING In Section 3 the spectrum of the RF signal to be analyzed is stored within the SHB material and is then retrieved through the transmission spectrum of a probe beam. In other words the probe beam is filtered by the SHB medium that has been previously programmed by the optically carried RF signal. With such a representation of the SHB material function one insists on its processing capability. This picture is substantiated by the mathematical expression of the optical response, as established in the pioneering works [10] and [62]. It is shown that the optical signal radiated by a SHB material can be made proportional to a three field
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product: E˜1∗ (r , ν)E˜2 (r , ν)E˜3 (r , ν) in the spectral domain, the first two fields being involved in the engraving stage while E˜3 (r , ν) represents the probe field spectral shape. At first sight this is nothing but a trivial third order non-linear optical process. Atomic spectral selectivity makes a difference. In non-linear optics spectral selectivity generally results from phase matching and transparency conditions. In the processes we consider, spectral selectivity is imposed by atomic coherence lifetime. The probe field component E˜3 (r , ν) at frequency ν does not interact with the engraved structure E˜1∗ (r , ν )E˜2 (r , ν ) at separation ν − ν larger than the inverse atomic coherence lifetime. In this section we briefly re-establish this specific behavior within the elementary framework of a two-level atom ensemble.
4.2. L INEAR R ESPONSE Whatever the medium, response to a probe field E3 (r , t) is described by the macroscopic polarization density P (r , t). In the weak field regime the response reduces to the linear component and can be expressed in the frequency domain as P˜ (r , ν) = ε0 χ( (5) ˜ r , ν)E˜ 3 (r , ν), where f˜(ν) = f (t)e−2iπνt dt and χ(r , t) represents the electric susceptibility. In the present work the optical fields interact with an absorbing medium made of resonantly excited two-level atoms. Lower and upper states |a and |b are connected by an electric dipole transition whose dipole moment a|μ|b is denoted by μab . The polarization density of such an atomic ensemble is P (r , t) = N μab g(νab )ρab (νab , r, t) dνab + c.c., (6) where N and ρab (νab , r, t) respectively stand for the atom spatial density and for the off-diagonal density matrix element and g(νab ) represents the atom distribution as a function of the transition frequency. The width of g(νab ) corresponds to the transition inhomogeneous line-width. To relate the general expression (5) with the microscopic description given by Eq. (6) we have to solve the atomic density matrix equation ⎧ dρ ⎨ i h¯ ρ˙ = [H, ρ] + i h¯ (7) dt relaxation ⎩ H = H0 − μ · E(r , t), where H0 represents the atom Hamiltonian. The relaxation term accounts for the upper state decay to the ground state at rate γb and for the off-diagonal matrix element ρab decay at rate γab . The level population difference ρaa − ρbb is denoted by nab . The level populations also satisfy the closure relation, ρaa + ρbb = Tr(ρ) = 1.
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The incident traveling waves are decomposed in their complex components according to 1 (8) E (r , t) + c.c. , 2 where the complex field E (r , t) is related to the complex amplitude A(r , t) by E(r , t) =
E (r , t) = A(r , t) exp(2iπν0 t).
(9)
Resonant interaction means that the reference frequency ν0 is contained within the inhomogeneously broadened atomic absorption line and A(r , t) is a slowly varying function of time. Atoms are coupled to the field by electric dipole interaction. Because the transition frequency νab is close to ν0 , rapidly oscillating terms can be eliminated from the density matrix equation, which is known as the rotating wave approximation. One is left with the optical Bloch equation ⎧ ⎨ n˙ ab = i Ω ρ˜ba − Ω ∗ ρ˜ab − γb (nab − 1), (10) i ⎩ ρ˙˜ ab = − Ωnab + (iΔ − γab )ρ˜ab , 2 where Δ = 2π(νab − ν0 ), ρ˜ab = ρab e−2iπν0 t and the Rabi frequency Ω is defined as Ω = μab A(r , t)/h¯ . The Bloch equation can be written in an equivalent integral form as ⎧ nab (t) = 1 + nab (t0 ) − 1 e−γb (t−t0 ) ⎪ ⎪ ⎪ ⎪ t ⎪ ⎪ ⎪ ⎪ ⎪ + i dt Ω t ρ˜ba t − c.c. e−γb (t−t ) , ⎨ (11) t0 ⎪ ⎪ t ⎪ ⎪ ⎪ i ⎪ ⎪ ρ˜ab (t) = ρ˜ab (t0 )e(iΔ−γab )(t−t0 ) − dt Ω t nab t e(iΔ−γab )(t−t ) . ⎪ ⎪ 2 ⎩ t0
We assume the optical density is sufficiently small so that we can neglect the incoming field attenuation. In the linear response limit nab is replaced by its initial (0) value nab = 1 in the expression of ρ˜ab (t) and one obtains (1) ρab (νab , r, t)
iμab =− 2h¯
∞
E3 (r , t − τ )e(2iπνab −γab )τ dτ.
0
Substitution of Eq. (12) into Eq. (6) leads to P˜ (r , ν) =
∞ −∞
P (r , t)e−2iπνt dt
(12)
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1 (+) ε0 χ˜ (r , ν)E˜3 (r , ν) + χ˜ (−) (r , ν)E˜3 (r , −ν)∗ , 2
(13)
where χ˜ (±) (r , ν) = ∓i
N μ2ab (1 − iH ) g(±ν) ⊗ L(±ν) . 2h¯ ε0
(14)
In this expression ⊗ stands for the convolution product, L(ν) represents the homogeneous line profile, L(ν) =
1 γab /2π , 2 π ν + (γab /2π)2
and the Hilbert transform is defined as 1 f (y) H f (x) = P dy, π x−y
(15)
(16)
where P denotes a principal value. The atomic response, as given by Eq. (14), reveals two components. One is proportional to the inhomogeneously broadened absorption profile, g(±ν) ⊗ L(±ν). The other component, given by the Hilbert transform of the absorption profile, is the dispersion contribution that expresses causality in the spectral domain. These two components correspond to the response terms ER and EI we already met in Section 3.3.2. Since the transition frequency distribution function g(ν) only differs from zero in the vicinity of the positive frequency ν0 , χ˜ (+) (r , ν) and χ˜ (−) (r , ν) respectively appear to be positive and negative frequency components of susceptibility, the same as E˜3 (r , ν) and E˜3 (r , −ν)∗ are positive and negative frequency components of the field. Therefore the polarization amplitude can be written as 1 P˜ (r , ν) = ε0 χ˜ (−) (r , ν) + χ˜ (+) (r , ν) E˜3 (r , ν) + E˜3 (r , −ν)∗ , 2
(17)
such that χ( ˜ r , ν) = χ˜ (−) (r , ν) + χ˜ (+) (r , ν).
(18)
4.3. T IME -D ELAYED F OUR -WAVE M IXING The considered architectures all involve a preparation stage where an engraving field Ee (r , t) spectrum is stored in the atom population difference. Although Ee (r , t) may result from combination of different beams (typically two beams) with controlled relative phase, the decomposition given in Eq. (8) remains valid. From Eq. (11) one readily obtains the lowest order perturbation to the population
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difference (2) nab (νab , r, t)
1 =− 2
t
dt
−∞
×e
t
∗ dt Ωe r, t Ωe r, t eiΔ(t −t ) + c.c.
−∞
−γab |t −t | −γb (t−t )
e
(19)
.
Since the integrand is unchanged under t and t permutation, nab (t) can be written as (2)
(2) nab (νab , r, t)
1 =− 4
t dt −∞
×e =−
t
∗ dt Ωe r, t Ωe r, t eiΔ(t −t ) + c.c.
−∞
−γab |t −t | −γb (t−t )
μ2ab 4h2 ¯
t −∞
dt
e t
∗ dt Ee r, t Ee r, t e2iπνab (t −t )
−∞
+ c.c. e−γab |t −t | e−γb (t−t ) .
(20)
Let us assume the experiment time scale is much shorter than the upper level (2) lifetime. After the field is switched off nab (νab , r) reduces to (2)
nab (νab , r) = −
μ2ab 2h¯ 2
S(r , νab ) ⊗ L(νab ).
(21)
The engraved structure appears to be proportional to the light field power spectrum, ∞ 2 2 −2iπνab t ˜ S(r , νab ) = Ee (r , νab ) = (22) Ee (r , t)e dt , −∞
(2)
convolved with the homogeneous line profile. Inserting nab (νab , r) in the expression for the coherence, one readily obtains the engraving field-dependent contribution to the susceptibility χ˜ (±) (r , ν) = ±i
N μ4ab h3
4ε0 ¯
(1 − iH ) g(±ν)S(r , ±ν) ⊗ L(±ν) ⊗ L(±ν) .
(23) If, as is the case in the following, the engraving field Ee (r , t) is made up of two components E1 (r , t) and E2 (r , t), the power spectrum is 2 2 S(r , ν) = E˜1 (r , ν) + E˜2 (r , ν) + E˜1 (r , ν)∗ E˜2 (r , ν) + c.c., (24)
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such that the polarization density contains the term 1 N μ4ab (+) (1 − iH ) g(ν)E˜1∗ (r , ν)E˜2 (r , ν) ⊗ L(ν) ⊗ L(ν) P˜1∗ 23 (r , ν) = i 3 4 h¯ × E˜3 (r , ν). (25) ∗ ˜ ˜ ˜ This expression still differs from the expected product E1 (r , ν)E2 (r , ν)E3 (r , ν). The population distribution, as modified by the engraving fields, reflects spectral profiles of those fields provided two conditions are satisfied: (i) the initial population distribution g(ν) should be much broader than E˜1∗ (r , ν) × E˜2 (r , ν); (ii) the spectral resolution, as limited by the homogeneous line profile L(ν), should accommodate the fine details of E˜1∗ (r , ν)E˜2 (r , ν). If these conditions are sufficient to make the absorption component in Eq. (25) coincide with E˜1∗ (r , ν)E˜2 (r , ν)E˜3 (r , ν), one is still left with the Hilbert transform distortion that affects the dispersion component spectral selectivity, each atom contributing in inverse proportion to its spectral distance from the probing frequency. Being in quadrature with the incident field, the dispersion term is washed out in a simple transmission experiment where the probe field co-propagates with the signal and automatically plays the part of a homodyning reference, as seen in Eq. (2) above. In holographic configuration, when the signal does not copropagate with the probe field, one must resort to other means in order to restore spectral selectivity. One may heterodyne the signal with a reference that is phaselocked to the probe and demodulate the signal with appropriate phase to reject the dispersion term, as we have seen in Section 3.3.2. Engraving submodulation using a sine wave offers another means to eliminate the dispersive behavior. Let E˜1∗ (r , ν)E˜2 (r , ν) be replaced by E˜1∗ (r , ν)E˜2 (r , ν)e−2iπνT , where T is adjusted so that T −1 is smaller than the characteristic variation scale of E˜1∗ (r , ν)E˜2 (r , ν), i.e. smaller than the desired resolution. Then H E˜1∗ (r , ν)E˜2 (r , ν)e−2iπνT i E˜1∗ (r , ν)E˜2 (r , ν)e−2iπνT , (26) which eliminates the undesired dispersive behavior. One easily generates the spectral submodulation by time delaying the field E2 (r , t). Indeed the spectral amplitude of E2 (r , t − T ) reads as E˜2 (r , ν)e−2iπνT . Cross-engraving by those time-delayed fields corresponds to the so-called photon echo configuration that is now going to be discussed. 4.4. C OHERENT C OMBINATION OF L OCAL R ESPONSE : P HOTON E CHO By definition, polarization density expresses a local response to excitation by the light fields. Actually Eq. (25) appears to be local both spatially and spectrally.
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It should be stressed that the sample emission coherently combines the local responses. Response to pulsed excitation represents an important example of such a coherent combination. Let the system be excited by a sequence of two pulses E1 (r , t) and E2 (r , t − T ). According to Eq. (26), provided the characteristic variation scale of E˜1∗ (r , ν)E˜2 (r , ν) is much larger than T −1 and the homogeneous line-width, and provided E˜1∗ (r , ν)E˜2 (r , ν) is much narrower than g(ν), the local atomic response is (+) P˜1∗ 23 (r , ν) = i
N μ4ab
g(ν0 )E˜1∗ (r , ν)E˜2 (r , ν)E˜3 (r , ν)e−2iπνT −2γab T . (27) 2h¯ 3 Contributions from all frequency classes combine into the time domain response (+) (+) P1∗ 23 (r , t) = P˜1∗ 23 (r , ν)e2iπνt dν N μ4ab
g(ν0 )E1∗ (r , −t + T ) ⊗ E2 (r , t − T ) ⊗ E3 (r , t − T ) 2h¯ 3 × e−2γab T .
=i
This quantity gives rise to the stimulated photon echo signal. The decay factor reflects the atomic coherence relaxation along the signal formation.
5. Rainbow Analyzer 5.1. P RINCIPLE OF O PERATION The rainbow analyzer concept relies on the engraving of monochromatic gratings in a frequency selective material. Each grating is able to diffract a single spectral component, with a resolution ultimately determined by the homogeneous line-width of the medium, which is usually less than 1 MHz at the temperature of 5 K. A large number of gratings can coexist within the inhomogeneous width of the absorption line, which may reach tens of GHz. By varying the laser frequency synchronously with the angle of incidence during the engraving procedure, one associates a specific diffraction angle with each specific spectral component. Therefore, the different spectral components of an incident polychromatic probe beam are diffracted and simultaneously retrieved in different directions. The stack of monochromatic gratings works as a spectrometer which is expected to exhibit a resolution of less than 1 MHz and a bandwidth of several tens of GHz. As depicted in Fig. 10, radiofrequency spectral analysis can be performed after transfer of the test microwave signal to an optical carrier with the help of a Mach–Zehnder electro-optic modulator (MZM). The bandwidth of the frequency selective medium indeed matches that of integrated MZMs developed for high flow telecommunication. To be more specific, frequency selective
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F IG . 10. Rainbow spectrometer concept. The insert shows the architecture of the analyzer. A Mach–Zehnder modulator MZM transfers the RF signal S(t) to the optical carrier from laser L. The crystal C diffracts the RF signal frequency components in different directions. The angular separation of the side-band spectral components is effected by the gratings engraved in the SHB crystal. The 3D picture shows the counter propagating box geometry giving independent access to engraving (#1, #2), reading (#3) and diffracted beams.
media can be devised to cover a bandwidth in excess of 50 GHz offered by fast integrated MZMs [63]. Since the spectrometer relies on the angular separation of the frequency components, the number of frequency channels equals the number of different angular directions that can be addressed by the device. For a given setup this number is fixed. However, the engraving laser frequency scanning range, which determines the spectrometer bandwidth, can be varied easily. By reducing this range while the channel number is kept fixed, one is able to zoom in on a specific spectral region with improved spectral resolution. Since energy is detected in each spectral channel, this processor is independent of the spectral coherence features of the channeled signal, quite in the same way as the spectrum photography architecture presented in Section 3. The potential of a similar arrangement to process the spectral phase and operate as a time-integrating correlator has been addressed in [64]. This rainbow analyzer is reminiscent of the well known acousto-optic spectrometer. In the latter device a Bragg cell achieves two functions. It both transfers the RF signal on the optical carrier and accomplishes the angular separation of the optically carried spectral components. Acoustic wave absorption limits the bandwidth to ∼1 GHz. The integrated MZM offer a much larger bandwidth. However a MZM transfers only the RF signal on the optical carrier but does not achieve the angular separation of the spectral components, since the carrier and the side bands propagate along the same direction. In the SHB spectrometer, the SHB
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crystal is intended to complement the MZM component by providing the angular separation. The beam configuration is strongly constrained by the material limitations. Owing to the short lifetime of the engraving, the gratings must be refreshed continuously and simultaneously diffract the impinging signal beam. This condition is satisfied in the “box configuration” presented in Fig. 10. The probe beam that carries the RF signal to be processed propagates along k3 . It lies out of the plane defined by the engraving beam wave vectors k1 and k2 . This non-coplanar arrangement is consistent with simultaneous writing and readout since the diffracted beam is directed along k3 + k1 − k2 which differs from all the incident wave vectors. The wave vectors k2 and k3 are headed in fixed directions while k1 rotates synchronously with the frequency scan of the engraving beams. Therefore the different spectral components that are carried by the probe beam are diffracted in different directions. The “box configuration” satisfies the phase matching condition, k1 − k2 being orthogonal to k3 + k2 .
5.2. P ROGRAMMING S TAGE In the active crystal one engraves the interference pattern of two beams that simultaneously undergo angular and frequency scan. The angular scan is achieved by a frequency-shift compensated pair of acousto-optic deflectors that are crossed successively by beam #1. The deflectors respectively diffract the light beam in order −1 and +1 so that the respective frequency downshift and upshift are subtracted from each other. The two deflectors are oriented in perpendicular directions and are slanted at 45◦ from the horizontal. When the two deflectors are driven synchronously with a fixed detuning, the emerging beam is scanned horizontally (see Fig. 11). An optical relay images the deflectors on the active crystal. This way, the illuminated spot on the crystal does not move as the beam direction is varied. More specifically, the phase pattern generated at the deflectors is transported to the active medium without distortion. Let k1.0 denote beam #1 wave vector at time t = 0. During engraving by fields E1 and E2 the wave vector k2 is kept fixed while k1 is scanned around the Oy axis, perpendicular to the (k1.0 , k2 ) plane, by the AO deflector assembly represented in the insert of Fig. 11. The phase shift imparted at position ρ in deflector Dj by the acoustic wave obeys the following relation, κˆ j · ρ Φj (ρ, t) = Φj 0, t − , v
(28)
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F IG . 11. 100-channel set-up. After the beam splitter BS the beam #1 is directed to the acousto-optic deflector AOD. Through the lenses L1 and L2, the fixed beam #2 and the scanned beam #1 are made to overlap in the SHB crystal. The RF signal carrier beam #3 is diffracted on the engraved gratings. The diffracted beams counter propagate beneath the light sheet formed by beam #1. The non-coplanar arrangement is not visible in this top view of the set up. Through the lens L2, the mirror M and the lens L3 the diffracted beams converge to the pin-hole PH, placed in imaging position with respect to the crystal. Finally they reach the photodiode array PDA that is positioned in the Fourier plane of PH through the lens L4. The focal distance of lens Li is denoted by fi . f2 /f1 = 0.25, f3 /f2 = 0.5, f4 = 150 mm. The AO deflector assembly is shown in the insert. The acoustic wave vectors are denoted by κ1 and κ2 .
where κˆ j and v respectively represent the sound wave unit vector and the sound velocity. The input phase shift is t Φj (0, t) = 2π
fj t dt ,
(29)
0
where the linearly chirped acoustic frequency fj (t) is expressed as fj (t) = fj0 + rA t,
(30)
where fj0 and rA stand for the AOM central frequency and chirp rate, respectively. The sound wave unit vectors are oriented at 45◦ from the deflection axis Ox, directed along K = k2 − k1 , and they are orthogonal to each other. The combined phase shift imparted to E1 is Φ(ρ, t) = Φ1 (ρ, t) − Φ2 (ρ, t)
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x y = 2π − √ f10 + f20 − √ f10 − f20 + f10 − f20 t v 2 v 2 √ rA rA − 2 xt + 2 xy . v v
(31)
The first two terms on the right-hand side respectively represent the horizontal and vertical deflection at time t = 0. Despite of the chirp they undergo, the acoustic waves convey a fixed optical frequency shift f10 − f20 to the light beam. This is reflected in the third term. The fourth term in Eq. (31) corresponds to the time dependent angular deflection. The last term is a time-independent, purely spatial quadratic phase. In other words a chirped acousto-optic deflector behaves as a lens, the focal length of which depends on the chirp rate. An AO deflector would mimic a cylindrical lens since it operates along a single direction. In our setup this quadratic phase factor modifies imaging conditions and degrades the spectral resolution. The first two terms in the phase expression (Eq. (31)) are incorporated in the wave vector k1.0 at t = 0. The laser frequency is swept at rate r synchronously with the angular scan. Then the complex fields are E1 (ρ, t) = E0 (ρ)e E2 (ρ, t) = E0 (ρ)e
√ r r 2iπ(ν1 t+ 12 rt 2 )−2iπ 2 vA xt+2iπ A2 xy v
ρ 2iπ(ν2 t+ 12 rt 2 )−i K·
,
,
(32)
where the Gaussian beam Rabi frequency spatial distribution is expressed in terms of the beam waist at the AOD, wAO , as ρ2 E0 (ρ) (33) = E0 (0) exp − 2 wAO and where ν1 = ν2 +f10 −f20 . The AOD is imaged on the active crystal with magnification factors mx and my along directions x and y, respectively. As recalled in Section 4, low intensity engraving can be described in terms of the incident ν) + E˜2 (ρ, ν)|2 . The cross-term E˜1∗ (ρ, ν)E˜2 (ρ, ν) = light power spectrum |E˜1 (ρ, rA ρ iΨ (ρ,ν)−2iπ xy−i K· 1 2 v2 E ( ρ)e , r 0
where
x √ 2 π π ν − ν1 + rA 2 − (ν − ν2 )2 , Ψ (ρ, ν) = r v r
(34)
as derived from the Fourier transform of Eq. (32), gives rise to the diffraction gratings dedicated to the spectral analysis of the optically carried RF signal. The phase Ψ (ρ, ν) can be rearranged as √ rA2 2 rA 2 (ν − ν1 )x − πrT 2 , Ψ (ρ, ν) = −2π(ν − ν1 )T + 2π 2 x + 2π rv rv (35)
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where T = (f10 − f20 )/r. The acoustic wave detuning that appears in Eq. (32) leads to a fixed frequency shift of the deflected beam with respect to the fixed direction engraving beam. Since the laser is frequency chirped, each ion interacts with the two beams at successive moments separated by delay T . As discussed in [65] and recalled in Sections 4.3 and 4.4, the interaction delay makes the setup operate in a photon echo configuration and results in submodulation of the level population difference spectral distribution, which should optimize the spectral resolution and the signal diffraction efficiency. However, in contrast to conventional photon echo experiments in which all atoms at a specific transition frequency undergo the same excitation independent of their location, in the present case, atoms at different spatial positions are excited in a different way. As a result, the photon echo behavior is not visible at the local level, i.e. before summing over both the spatial and frequency distribution of the atoms. The spectral behavior of the crystal response results from the coherent combination of local emissions all over the illuminated spot.
5.3. S PECTRAL R ESOLUTION Let a monochromatic Gaussian probe beam at frequency ν be directed to the sample along k3 . The probe beam is assumed to exhibit the same spatial distribution E0 (ρ) as the engraving beams. The linear response of the engraved diffraction grating at the image of position ρ is given by Eq. (25). The local radiated field, proportional to this response, can be expressed as ˜ ρ, E˜ (ρ, ν) ∝ A( ν)e−i(k3 +K)·ρ ,
(36)
where rA −iΨ (ρ,ν)−2iπ xy ˜ ρ, v2 ⊗ L(ν) ⊗ L(ν) . A( ν) ∝ (1 − iH ) E03 (ρ)e
(37)
In the frame of the Huygens–Fresnel principle Fraunhofer approximation, the signal emitted at angular distance (ϕ, θ ) from direction k3 + K combines the local ˜ ρ, contributions through the following Fourier transform of A( ν), 2iπ ˜ ˜ ρ, A(ϕ, (38) θ, ν) = dX dY A( ν)e λ (ϕX+θY ) , where space coordinates X and Y are now expressed at the crystal. Performing the space integration first, one finds 2 ˜ S(ν, ϕ, θ ) = A(ϕ, θ, ν) = (1 − iH ) e−2iπ(ν−ν1 )T g(ν − ν1 , ϕ, θ) ⊗ L(ν − ν1 ) 2 ⊗ L(ν − ν1 ) (39)
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where ϕ¯
g(ν, ϕ, θ ) = e−2 ln 2{[(ν ν¯ +ϕ)/δϕ]
2 +[θ/δθ]2 }
√ 6 ln 2 λ δϕ = 1 + α2, π mx wAO
√ δθ =
(40)
, 6 ln 2 λ 1 + α2, π my wAO
(41)
2 π rA wAO (42) , 2 v2 and ν¯ represents the frequency interval over which the laser is scanned while ϕ is varied over the angular range ϕ. ¯ If the submodulation period 1/T is much smaller than the width νδϕ/ ¯ ϕ¯ of g(ν, ϕ, θ ), the signal intensity reduces to 2 S(ν, ϕ, θ ) = e−4t12 /T2 g(ν − ν1 , ϕ, θ) . (43)
α=
This is the well-known photon echo situation that leads to optimal spectral resolution by eliminating the dispersive part of the diffracted signal [66]. According to Eq. (43) the spectral component ν1 is diffracted in direction (ϕ = 0, θ = 0). The angular full width at half maximum of the diffracted beam is given by (δϕ, δθ ). ¯ ν , θ = 0), The spectral component ν is diffracted in direction (ϕν = (ν − ν1 )ϕ/¯ with the same angular spreading as component ν1 . The angular spreading of each frequency channel limits the spectral resolution of the spectrum analyzer to √ δϕ 6 ln 2 v δν = ν¯ 1 + α2, = ν¯ ϕ¯ wAO f¯ π where f¯ stands for the acoustic frequency scanning range of the AOD. The frequency resolution is independent of the optics magnification factor. It depends only on the engraving laser scanning range and on the number of directions that can be addressed by the AOD. The photon echo configuration also limits the effects of bleaching by the probe, as discussed in [67]. The parameter α characterizes the AOD lens effect. On account of the AOD driver frequency chirp, the two sides of the laser spot undergo deflection at different angles in the AOD. The angle difference can be expressed in terms of the chirp rate and of the acoustic wave transit time across the light beam, wAO /v, as λrA wAO /v 2 . This angle has to be compared with the divergence of the laser beam, λ/wAO . The parameter α is nothing but the ratio of these two quantities. The lensing effect is negligible when λrA wAO /v 2 λ/wAO . The channel broadening caused by the acousto-optic deflector pair lens effect is not a fundamental limitation. On account of the holographic storage properties, the corresponding wave front distortion does not entail irreversible signal alteration. The quadratic cross phase factor can be alleviated by an appropriate non-axially symmetric wave front correction to any of the four fields involved in the signal formation.
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The last piece of information we derive from Eq. (39) is the signal intensity variation as a function of the two engraving beam detuning. Integrating Eq. (39) one obtains 2 S(ν, ϕν , 0) = 1 + erf(u) e−4t12 /T2 , (44) where u=
√ π ν¯ t12 δϕ ϕ¯ 2 ln 2 . − √ π νT ¯ 2 δϕ ϕ¯ 2 ln 2
5.4. E XPERIMENTAL S ETUP We have investigated various aspects of the rainbow analyzer, such as bandwidth flexibility [65,68] and dynamic range [65]. The largest channel number was obtained with the setup we describe here. The programming beam #1 goes through a 2D deflector device (A&A DTS XY 250) that offers 5 mm clear aperture, and 69 mrad scanning range as the common driving frequency is varied from 85 MHz to 125 MHz. An optical relay, with a magnification factor mx = 0.25, images the deflector on the SHB crystal so that the illuminated spot does not move as the beam direction is varied. The two output channels of a 1 Gigasample/s arbitrary wave form generator (Sony/Tektronix AWG520) separately drive the two AOs that are combined in the deflector. This enables us to set a fixed frequency detuning between the two chirped synchronized driving waves. The synchronized laser frequency chirp and beam angular scan must be repeated over the spectrometer bandwidth with precision better than the desired resolution, since the grating storage is accumulated in the active crystal for several engraving cycles. To satisfy the repeatability requirements of our application it was necessary to develop a novel electro-optically-tuned extended-cavity diodelaser [69,58]. The laser we use in this experiment, equipped with an improved AR coated chip, is easily scanned without mode hopping over a 3.5 GHz interval. In more recent experiments demonstrating the spectrum photography architecture we have been able to increase the scanning range to 10 GHz (see Section 3). In previously studied photon echo applications, accumulation is regarded as a demanding procedure [70,71]. In these applications the storage coordinate is the time delay of the engraving fields. The delay dependent phase shift that is stored in the material is increasingly affected by laser frequency fluctuations as one increases the delay range. A 1 µs delay storage range requires at least 100 kHz laser stability during a 10 ms long accumulation process [71]. The present situation is somewhat different. All the gratings associated with the different spectral channels are engraved with the same delay that only has to be larger than the inverse spectral resolution. For 33 MHz resolution, this delay need not exceed ∼10 ns,
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which is much smaller than the inverse jitter range of a free running extended cavity diode laser over a 10 ms time interval. Since we split the three needed beams from a single laser, we have to operate in a sequenced mode, where grating engraving and signal diffraction alternate. Engraving takes place during the first half repetition period, while the laser frequency is kept fixed for readout during the second half period. With a repetition rate of 2 kHz, the 250 µs-long writing step is followed by a same duration step devoted to RF signal spectral analysis. Since the shelving state lifetime is ∼10 ms, engraving is accumulated over ∼20 writing steps. The modulated probe beam to be spectrally analyzed is directed to the Tm3+ : YAG crystal and is diffracted on the engraved gratings into a frequency-dependent direction (see Fig. 11). In order to filter out the stray light scattered off the cryostat windows, we make the diffracted beams converge to a pin hole with the help of the optical relay formed by the lenses L2 and L3 . The pin hole is sized to the dimensions of the active area image formed through lenses L2 and L3 . Emerging from this precisely defined emitting area, the diffracted beams are collected on a photodiode array (PDA) through the lens L4 operating as an angleto-position Fourier transformer. The PDA includes 1024 pixels. The pixels are 2.5 mm high and 25 µm wide. They are sequentially read out every 10 ms. Since the detector is operating continuously, this interval also represents the integration time. The illuminated position coordinate on the PDA is proportional to the diffraction angle and so to the RF signal frequency. Specifically, a beam diffracted in direction ϕ strikes the PDA at position x − x0 = (ϕ − ϕ0 )f4 f2 /f3 , where the focal length of lens Li is denoted by fi . The direction ϕ is connected to the RF ¯ − ν0 )/¯ν where ϕ¯ and ν¯ respectively represent the frequency by ϕ − ϕ0 = ϕ(ν angular scanning range of beam #2 and the frequency chirp range of the laser during the engraving step. The spectral channel number of the device is given by the number of different directions that can be addressed by the AO deflector.
5.5. E XPERIMENTAL R ESULTS Using this setup we demonstrate a 3.3 GHz instantaneous bandwidth with 100channel resolution. Experiments have been performed with a 2.5 mm-thick, Tm3+ :YAG (0.5 at.%) crystal. We simulate a multiple-line RF signal by making the laser perform various discrete frequency jumps during the readout step of the writing/readout sequence. The staircase frequency scan of the probe beam is illustrated in Fig. 12. The laser stays for 10 µs at each frequency step. In a previous publication [68] we demonstrated the spectral analysis of a RF signal that was put on the probe beam with the help of a Mach–Zehnder modulator.
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F IG . 12. Time diagram of the laser frequency scan. During the first 250 µs the laser is continuously scanned over the selected bandwidth as it engraves the spectrally selective holograms. During the second half of the sequence, a multiple-line spectrum is simulated by a stair case frequency scan. The laser stays for 10 µs at each step.
In order to increase the intensity, we have reduced the vertical spot size to 150 µm. With a beam waist at the deflector wAO = 1.5 mm, and a magnification factor mx = 0.25, the horizontal spot size is 750 µm. The fixed engraving, deflected engraving, and probe beam powers are amount to 4.7 mW, 3.2 mW and 0.7 mW, respectively. An experimental spectrum is displayed in Fig. 13. Given the acoustic wave velocity v = 650 m/s, the acoustic wave chirp rate rA = 1.2 · 1011 s−1 , Eq. (41) predicts an angular resolution δϕ = 1.64 mrad, which corresponds to ∼5 pixels of the PDA. We actually measure an angular resolution of ∼6 pixels. Combined with a 205 mrad scanning range, this resolution offers a 100-channel capacity. The angular scanning range is accidentally limited to 205 mrad because of a beam folding mirror size. The XY AO deflector (AOD) actually offers more than 270 mrad scanning range, which corresponds to 135 channels. Compensation of the acousto-optic lens effect mentioned in Section 5.3 would probably increase this number by reducing the channel width.
5.6. F UTURE I MPROVEMENTS In order to continuously detect the optically carried RF signal, without dead time, one has to refresh the engraved gratings and to analyze the signal simultaneously. The single laser we used in the reported experiment was not able to accomplish these two tasks. Two different lasers are needed since the engraving beam frequency has to be continuously scanned at fixed rate, while the RF signal to be
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F IG . 13. RF spectral analyzer demonstration. The simulated multi-frequency RF signal undergoes 19 successive 175 MHz-wide jumps, and dwells only 10 µs at each spectral position. This demonstrates 3.3 GHz instantaneous bandwidth and 100-channel capacity.
investigated must be transferred onto a fixed frequency carrier. The relative stability of the two lasers should be better than the expected resolution of the spectrum analyzer. In order to proceed toward 100% duty cycle operation, we demonstrated an active stabilization scheme where the laser frequencies are periodically compared after each frequency scan of the engraving laser [72]. The number of resolved spectral channels is closely related to the timebandwidth product of the AOD. Indeed this product represents the number of different directions that can be addressed by the AOD. Used with a Gaussian beam expanded to the maximum held diameter of 4.2 mm at 1/e2 , and operated over a 40 MHz acoustic wave bandwidth, the A&A DTS XY 250 deflector offers a time-bandwidth product of ∼350. A larger device with a clear aperture of ∼14 mm would be needed to reach a 1000 channel resolution. Such a wide aperture probably excludes the use of an XY AO device. The frequency shift caused by a single dimension deflector could be compensated easily with an AO shifter placed on the other engraving beam. As for the dynamic range, our best 35 dB result [65] could be improved by reduction of stray light. The cryostat windows probably contribute significantly to this undesired background. Their number should be reduced. This demands that most optics be enclosed within an evacuated box, the active crystal itself being attached to a cold finger inside this box.
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6. Photon Echo Chirp Transform Spectrum Analyzer As the rainbow analyzer, the photon echo chirp transform analyzer uses the crystal as a processor. Whereas the rainbow analyzer makes the crystal a frequency-toangle converter, the chirp transform analyzer uses the crystal as a frequency-totime converter. No spatial dimension is used in this architecture. Instead one uses the photon echo process to produce the Fourier transform of the RF carrying beam, thus producing a temporal image of the RF spectrum. This transformation relies on the coherent combination of the emissions from the ions with resonant frequencies distributed over a wide domain, typically equal to the crystal inhomogeneous width. This must be contrasted with the rainbow or the photographic plate analyzers that require no phase relation between the atomic coherences carried by ions with different transition frequencies.
6.1. P HOTON E CHO C HIRP T RANSFORM 6.1.1. Chirp Transform and Photon Echoes The chirp transform algorithm [41] computes the Fourier transform s˜ (ν) of a signal s(t) according to 2 2 2 s˜ (ν = rt) = s(t)e−iπrt ⊗ eiπrt eiπrt . (45) This process is also called the MCM scheme since it involves a multiplication followed by a convolution and a last multiplication. The last multiplication simply corrects for a phase factor and can be dropped when only the modulus of the Fourier transform is of interest. We then have the MC scheme. In paraxial diffraction theory, the MC process is the one that yields, in the focal plane of a lens, the space frequency spectrum of an object situated just before the lens. The object is first multiplied by the transmission factor of the lens. Diffraction to the focal plane makes the convolution. In the time domain (Eq. (45)), the phase quadratic factor represents a linear frequency chirp with chirp rate r. The multiplication means modulating a chirp by the signal s(t), or equivalently performing quadratic phase modulation of the signal. This multiplication is the operation of a so-called time lens [73,74] whose focal time is ν0 /r with ν0 the optical frequency. The convolution assigns different delays to the different input frequencies, with a linear relation of coefficient 1/r between delay and frequency. It is a dispersion process of dispersion rate 1/r. The aperture of the time lens—the maximum duration of the signal that can be quadratically phase modulated—sets the spectral resolution of the chirp transform. The maximum signal bandwidth that can pass through the lens sets both the lens and the process bandwidth. However, the dispersive delay line features must be taken into account. Not only must its dispersion rate match the lens focal time,
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F IG . 14. Architecture of the photon echo chirp transform RF spectrum analyzer. FALO: frequency agile laser oscillator, MZM: Mach–Zehnder modulator, PD: photodiode.
but its bandwidth must also exceed that of the lens, and its maximum delay—the delay between the two extreme frequencies of the bandwidth—must be greater than the lens aperture. The chirp transform algorithm has already been implemented for RF spectral analysis purposes with acousto-optic and surface acoustic wave (SAW) devices [75]. The latter are the sole commercial devices. They perform a maximum bandwidth of 500 MHz, an ultimate resolution down to 10 kHz, and a time bandwidth product up to 2500 [76]. Although impressive, these devices cannot cope with multi GHz bandwidth. Optical solutions may overcome this limitation. Photon echoes in rare earth doped crystals are potentially a good solution. The optoelectronic architecture we propose is sketched in Fig. 14. A cw laser with chirping capability, the so-called frequency agile laser source, produces a microsecond long optical chirp spanning a multi GHz bandwidth [69]. This optical carrier is then modulated by the RF signal in an integrated Mach Zehnder electro optic modulator (MZM). This makes the multiplication, i.e., our time lens effect. The beam is then directed toward a dispersive delay line engraved in a rare earth doped crystal by another frequency agile laser. The output of the crystal is detected on a fast photodiode which directly yields the rf spectrum. Thanks to the matching features of frequency agile lasers, rare earth doped crystals, fast MZM and photodiodes, development of a multi GHz bandwidth spectral analyzer with resolution below 1 MHz, and great flexibility, should be possible. As demonstrated in Section 4.4, in the perturbation regime the stimulated photon echo signal Ee (t) is described by Ee (t) = E1∗ (−t) ⊗ E2 (t) ⊗ E3 (t), E˜e (ν) = E˜1∗ (ν)E˜2 (ν)E˜3 (ν),
(46)
where Ei (t) and E˜i (ν) are the complex amplitudes of the three excitation pulses and their Fourier transforms. The first two pulses engrave the frequency domain
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interference pattern E˜1∗ (ν)E˜2 (ν) in the absorption band of the material. This engraving is a coherent filter which lasts for the lifetime T1 of the upper level of the transition. The third pulse, usually called the reading or probing pulse, is spectrally filtered by this coherent filter to yield the echo signal. As explained in Section 4.4, Eqs. (46) implicitly suppose that the engraved filter contains a sinusoidal submodulation of period smaller than that of the fine details of the field spectral amplitudes, so as to remove the dispersive component of the filter response. The bandwidth of the process is limited by that of the material, which is given by the inhomogeneous width Γinh of the optical transition. The spectral resolution is limited by the homogeneous line-width 1/πT2 of the transition, where T2 is the optical transition coherence time. Expression (46) can be arranged such that the echo signal gives the spectrum of the RF signal s(t) according to the MC scheme. Let the first two pulses be such that their correlation is the quadratic phase factor exp(iπrt 2 ). This is the case, for instance, if the first pulse is a short pulse represented by the Dirac distribution δ(t) and the second pulse the optical chirp exp(iπrt 2 ). Now let the third pulse be an optical chirp exp(−iπrt 2 ) modulated by an RF signal s(t). The echo signal becomes Ee (t) = s˜ (rt) exp iπrt 2 . (47) The echo signal time evolution therefore gives the Fourier transform of the RF signal s(t), carried by an optical chirp. Photo-detection of the echo power directly yields the power spectrum |˜s (rt)2 |. 6.1.2. A Physical Picture The process is conveniently represented in the time and frequency plane. Let us first consider the configuration of Fig. 15. It represents the three exciting pulses and the echo signal in the time and frequency plane. The first pulse is a short pulse whose bandwidth spans an interval Δ. The second pulse is an optical chirp of duration τ2 − τ1 which spans the same Δ bandwidth. For the excited ions, the delay between the first and second excitations is a linear function of the ion resonant frequency. More precisely, as one easily deduces from Fig. 15, the delay is τ (ν) = τ1 + (ν − ν1 )/r, r = Δ/(τ2 − τ1 ),
(48)
where ν1 (resp. ν2 ) is the frequency corresponding to delay τ1 (resp. τ2 ) and the parameter r is the chirp rate of the second pulse. The ions excited by the two pulses record the delay between the two excitations. Since this delay is a linear function of frequency, one actually records a dispersive filter. Recording is
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F IG . 15. Representation of the photon echo chirp transform process in the time-frequency plane. The first excitation is provided by a short pulse of bandwidth Δ drawn as a line parallel to the frequency axis. The next slanted line of slope r represents the chirp which makes the second excitation. The third slanted continuous line is the laser reading chirp with rate −r, and duration Ts . It generates the echo peak represented as a continuous vertical line. The dotted (resp. dashed) line parallel to the laser reading chirp line is the lower (resp. upper) side chirp due to modulation by the RF signal of frequency F . It generates the echo peak in dotted line (resp. dashed). (a) represents the reading chirp configuration for double sideband RF spectral analysis, while (b) represents the reading chirp for single sideband analysis. In (b) the continuous line is the laser reading chirp while the dashed line is the upper RF sideband chirp.
physically represented by a chirped sinusoidal modulation of the crystal inhomogeneous absorption band. The dispersion rate dτ/dν is the inverse 1/r of the chirp rate. The third pulse is an optical chirp of rate −r and duration Ts modulated by an RF signal. Let us first consider the chirped optical carrier. It excites the ions of frequency ν at time t3 = t3 (ν2 ) − (ν − ν2 )/r. The different ions then emit an echo signal which is delayed with respect to excitation time by the very delay τ (ν) the ions have recorded between the first two excitations. Therefore the echo generated by ions of frequency ν appears at time te (ν) = t3 (ν) + τ (ν) = t3 (ν1 ) + τ1 . Consequently, all the elementary echoes generated by the ions of different frequencies actually appear at the same time. Their coherent addition gives an echo whose duration is naturally given by the inverse of the bandwidth rTs which participates to signal formation. Now let us consider RF modulation. Modulating an optical chirp by an RF signal of frequency F generates two symmetric side chirps shifted by ±F from the optical carrier. This RF modulated chirp will therefore generate three echo peaks. One for the carrier as discussed above and one for each side chirp, time delayed by ±F /r with respect to the optical-carrier echo peak. Hence the echo signal displays the RF signal spectrum with the time-to-frequency correspondence F = (t − t0 )/r where t0 corresponds to the optical-carrier echo peak. One notices on Fig. 15 that the three echo peaks appear centered at different frequencies. This
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results from direct graphical construction of these echo peaks. It means that the echo peaks are on a chirped carrier. The chirp rate is easily computed to be r from Fig. 15, which is in agreement with Eq. (47). 6.1.3. Expected Features The features of this photon echo MC scheme, regarding its application to spectral analysis of RF signals, can easily be deduced from the graphical representation given above. The bandwidth Δ and the maximum delay τ2 of the dispersive filter engraved in the crystal are limited, respectively, by the inhomogeneous width Γinh and the coherence time T2 of the transition. Some conditions must be imposed to the RF signal. First, one requires that the total bandwidth occupied by the reading chirp, with its RF sidebands, be less than Δ. This implies that 2Δs + rTs Δ,
(49)
with rTs the reading chirp bandwidth and Δs the signal bandwidth. This necessitates that the spectral domain swept by the reading chirp be in the center of the spectral domain occupied by the dispersive filter. One also requires that the whole echo signal appears after the end of the reading chirp. This is a strict condition for the echo to give the Fourier transform of the RF signal. This time separation condition is τ1 0,
(50)
as deduced from Fig. 15. We also want to be able to perform continuous spectral analysis of an incoming RF signal. For that purpose, one must generate a continuous succession of identical reading chirps, with ideally zero delay between two successive chirps. Then each reading chirp generates an echo which gives the spectrum of the RF signal carried by the chirp. Real time spectral analysis is performed with 100% probability of interception. The echo from one chirp arises during the next reading chirp. Angular separation of the echo from the reading chirp allows such timing. However one must ensure that the successive echoes do not overlap in time. In other words, the duration of the echo generated by one reading chirp must be less than the chirp duration 2Δs /r Ts .
(51)
In order to fully exploit the capabilities of the dispersive filter, one considers Eqs. (49) and (50) as equalities. We therefore end up with the conditions Δs = 12 Δ(1 − Ts /τ2 ), τ1 = 0, Ts τ2 /2.
(52)
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The maximum time bandwidth Δs Ts product of the analyzer is obtained for Δs = Δ/4 and Ts = τ2 /2. For greater bandwidth, that is shorter Ts , the condition for continuous analysis cannot be respected and the probability of interception is less than 100%. 6.1.4. Single Side Band Analysis For spectrum analysis purpose, one need only see a single RF sideband. The reading chirp configuration of Fig. 15(b) achieves spectrum analysis of the upper RF sideband. For that purpose the lower frequency of the spectral domain swept by the reading chirp is set to coincide with the lower frequency of the spectral filter. Consequently, the lower RF sideband chirp falls partly out of the dispersive filter domain. But we are interested only in the upper RF sideband, and condition (49) now becomes rTs + Δs Δ. We also set τ1 = 0. This ensures that the echo signal corresponding to the upper sideband appears only after the end of the reading chirp. The signal generated by the lower sideband appears during the chirp. The condition for continuous analysis is unchanged. We therefore end up with the following conditions: Δs = Δ(1 − Ts /τ2 ), τ1 = 0,
(53)
Ts 23 τ2 . The maximum time bandwidth product is obtained for Δs = Δ/2 and Ts = τ2 /2 and is 1/4 of the available product ΔT2 in the crystal. However these values do not satisfy the condition for continuous analysis. At the limit of this condition, one has Ts = 2τ2 /3 and Δs = Δ/3. Let us note that the continuous analysis condition can be somewhat relaxed depending on the application. 6.1.5. Influence of the Material Finite Coherence Time As noted above the finite coherence time T2 of the optical transition limits the maximum delay of the dispersive line. Hence it limits the maximum duration Ts of the reading chirp, and therefore the spectral resolution 1/Ts of the spectrum analyzer. The coherence time also affects the signal amplitude. As can be observed from Fig. 15, the spectral domains contributing to the formation of the different echo peaks are different. Different spectral domains also mean different delay ranges in the dispersive filter. Thus, higher RF frequencies probe longer delays in the dispersive filter. From simple graphical construction one expects to see an exp(−4t/T2 ) time dependence of the echo signal power. In order to investigate this feature, we compute the expected signal accounting for finite coherence time. Assuming a low intensity reading chirp with constant
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F IG . 16. Response peak to a monochromatic RF signal, for T2 = ∞ (continuous line), Ts (dotted line), Ts /2 (dashed line).
amplitude over its duration Ts , one gets the expression sinh(Ts /T2 − iπrTs (t − F /r)) 2 Ee (t)2 = e−4t/T2 s ˜ (F ) , 2/T2 − 2iπr(t − F /r)
(54)
F
where the sum runs over the RF frequencies F and s˜ (F ) is the amplitude of the RF component of frequency F . This expression confirms the expected decay of the signal. It also indicates that the peaks corresponding to the different RF frequencies all have the same width which depends on both the reading chirp duration and the crystal coherence time. Figure 16 displays the expected peak shape for different values of Ts /T2 . One observes that for Ts significantly longer than T2 , the peak assumes a Lorentzian-like shape, strongly asymmetric, and with shifted maximum. On the other hand, the peak is little affected by the coherence time, as long as Ts < T2 . Let us point out that the exponential decay can be neglected in most cases. According to Eq. (51), the RF spectrum is displayed in a time equal to half the reading pulse duration Ts . The exponential decay can therefore be neglected as long as the target resolution 1/Ts is more than the crystal limit 1/T2 . Since T2 usually lies in the 10 kHz range, one can easily operate with resolutions down to 100 kHz and negligible echo exponential decay.
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F IG . 17. Practical chirps for engraving the dispersive filter in the crystal: (a) two symmetric chirps, (b) two co-temporal chirps of duration Tw .
6.1.6. Engraving Chirps Chirps are now widely used for optical processing of RF signals with optical coherent transients in rare earth doped crystals. The reason is twofold. First a chirp yields a time-to-frequency correspondence that offers considerable versatility in the shaping of a spectrum. Second, the bandwidth and duration required for the chirps are compatible with the capabilities of either a cw laser with chirp capabilities [69], or external modulation of a cw laser [77]. Considering such cw laser sources, one cannot generate short intense pulses. Therefore, the configuration of Fig. 15 for engraving the dispersive filter is not practical. Instead one can use the configuration of Fig. 17(a). This is the one used in the very first demonstration of the chirp transform algorithm with photon echoes [78]. In this configuration two symmetric chirps with opposite chirp rate ±2r engrave the dispersive filter with dispersion rate 1/r. There is no more short pulse required. However, with this configuration the engraving duration τ2 must still be shorter than the coherence time T2 of the excited transition in the crystal. Therefore one needs high output power to engrave a dispersive filter with good efficiency. The configuration of Fig. 17(b) allows more efficient engraving. It stems from the configuration used in [79] for memory application of photon echoes. In this configuration, one uses two co-temporal chirps with the same duration Tw but different chirp rates. With such an excitation the delay τ (ν) recorded by the ions of the crystal still depends on the minimum and maximum delays τ1 and τ2 , as given by Eq. (48), but is independent of the chirp duration. Consequently, the engraving duration is no longer limited by the coherence time of the crystal but only by the much longer lifetime of the engraving. Therefore, for a given cw laser power, one can put more energy into the filter, and control this energy via Tw , independently of the filter dispersion rate. It is this configuration we have used in the experiments reported below. Let us point out that one still needs a fast chirp at readout. This is not a problem from the power point of view since reading must be carried out at lower power than engraving, but still requires a fast chirp source.
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F IG . 18. Energy levels of the Er3+ ion, showing the first two levels of the ground and excited Stark multiplets in the YSO matrix.
6.2. E XPERIMENTAL D EMONSTRATION IN E R 3+ :YSO After a first proof of principle demonstration performed using a Tm3+ : YAG crystal [78] at a wavelength of 793 nm, the algorithm was tested with a new setup based on a Er3+ :Y2 SiO5 crystal. Figure 18 shows the relevant energy levels of the erbium ions. One uses the 4 I15/2 → 4 I13/2 optical transition. The choice of the crystal parameters followed from T. Böttger’s Ph.D. [80,81]. We ordered from Scientific Materials a 0.005 %at. doped crystal cut perpendicular to the b axis. The crystal is placed in a magnetic field created between permanent NdFeB magnets. The field was measured to be about 1.1 Tesla. With our crystal cut, the magnetic field is oriented at 135◦ from the D1 axis which maximizes the coherence time of the transition [80]. The magnetic field is necessary for erbium doped crystals because of Kramers degeneracy. Indeed for ions with an odd number of electrons, the degeneracy of the multiplets cannot be completely lifted by the crystal field Hamiltonian [82]. In the Y2 SiO5 matrix low symmetry sites, a two-fold spin degeneracy remains. Phonon assisted electronic spin flips of neighbouring Er3+ ions is then an important cause of dephasing, limiting the transition coherence time to a few microseconds [53]. An external magnetic field can lift the Kramers degeneracy, eventually freezing the electronic spins in the lower component of the Kramers doublet for high fields. Then one can reach very long coherence times, up to several milliseconds, which is close to the 11 ms lifetime of the 4 I13/2 level [81]. The wavelength of the transition in site 1 was found to be 1536.2612 nm in vacuum, with a Burleigh 1650 wave-meter, whose calibration was checked against CH2 spectral lines. The apparent discrepancy with the published value of 1536.48 nm in vacuum is actually due to displacement of the transition frequency with applied magnetic field. With the displacement of
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F IG . 19. Experimental setup sketch. PD: photodiode, AO: acousto-optic cell, PMF: polarization maintaining fiber, λ/2: half wave plate.
0.719 cm−1 /Tesla measured by T. Böttger and Y.C. Sun [83] for a field oriented at 135◦ from D1 , one computes a field strength of 1.3 Tesla, which is consistent with our direct measurement. We also measured an inhomogeneous full width at half maximum of 1.2 GHz. This larger value as compared to the one found by Böttger is due to magnetic field inhomogeneities. A new wavelength gives the opportunity to test new laser sources for photon echoes. The following experiments were performed with a fiber laser from Koheras. Based on a distributed feedback fiber laser followed by an amplification stage, it yields 6 mW of polarization maintained output. With this laser, Koheras measured the line-width of the output beat from an asymmetric Mach–Zehnder with 120 µs delay, and the result was a few kHz. Such a measurement does not directly yield the laser line-width, and what is more, does not give information on frequency jitter occurring on a time scale longer than the Mach–Zehnder asymmetry. However, fiber lasers are expected to, and actually do, as experiments below demonstrate, have much better frequency stability than the diode lasers used mainly in photon echo experiments. The price to pay is the lack of chirping capabilities of our fiber laser. But the research for integrated chirped lasers has already begun. The setup for the first echo experiments is a simple collinear setup sketched in Fig. 19. The laser beam is collimated and focused in an acousto-optic AO1 (custom model by A&A, with > 50% diffraction efficiency over a 40 MHz RF-frequency range). Controlled by an arbitrary waveform generator (Tektronics AWG520), it is used to produce the chirps, as well as amplitude modulation of the laser beam. The beam is focused to a spot of 25 µm waist in the crystal which is maintained at about 1.5 K. The available power is typically 1–2 mW in front of the cryostat. A half wave plate before the crystal controls the polarization of the beam. After the cryostat the beam is then gated by AO2, which protects the photo detector (New Focus Mod 1811, 125 MHz bandwidth). The photodetected signal is digitized and displayed by an oscilloscope with 2 GS/s sampling rate and 10 kpoints of memory depth. In the setup, the fiber laser output, AO1, the crystal, AO2, and the photodiode, are all in image planes with respect to each others.
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F IG . 20. Power spectrum of an input 3 µs square signal as given by the photon echo chirp transform, in log scale. The signal was averaged over 16 shots. The optical power was 2 mW. Successive engraving and reading cycles were repeated at 100 Hz. (a) Experimental signal. (b) Fit (line) of the same experimental record (dots) by the expected sinc2 function.
The Fourier transform of a square function is a cardinal sine. We checked this basic property with our photon echo Fourier transformer. We used the cotemporal chirp configuration of Fig. 17(b) for engraving, with parameter values Tw = 30 µs, Δ = 40 MHz, τ1 = 0, τ2 = 12 µs. This double chirp was programmed on a single output of the AWG, which was fed to AO1. The power at engraving was 1.8 mW in front of the cryostat. The reading pulse is a simple chirp of duration Ts = 3 µs, spectrally centered in the dispersive filter spectral band (cf. Fig. 15). The engraving plus reading sequence was cycled at 100 Hz, which was found to optimize the signal strength, giving about twice the signal strength at 10 Hz cycling rate. For higher repetition rates the signal strength was about stationary up to 300 Hz and then slowly decreased down to zero. Such a signal decrease at high cycling rate is expected since the atomic system is a two level system. A precise study of the accumulated engraving regime with our experimental system remains to be done. Still, the gain on signal strength observed under accumulation indicates that laser frequency stability is sufficient for this accumulation regime to work, which is necessary from the application point of view. Figure 20(a) shows the detected echo signal in logarithmic scale while Fig. 20(b) shows the fit by the sinc2 function. A very good fit is obtained down to the last detectable side-lobe with sinc width of 100 ns, as expected. This demonstrates faithful Fourier transform with a power optical dynamic range of 27 dB. We then tested the response of the Fourier transformer to a multi-frequency RF signal. The engraving sequence was the same as above with τ1 = 0.2 µs
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F IG . 21. Experimental spectrum of an input RF signal containing 10 frequencies. Single shot record. Time translates into frequency with the coefficient 2.67 MHz/µs. The first peak on the left corresponds to the optical carrier (RF frequency = 0).
and τ2 = 15.2 µs. The small τ1 value was chosen so that the zero-RF-frequency peak is visible in the echo to serve as a frequency reference. We used the single sideband chirp configuration of Fig. 15(b) for reading, with a power of 0.3 mW in front of the cryostat. Figure 21 presents the spectrum of an input RF signal s(t) = 0.3 + i sin(2πFi t) with 10 frequencies Fi spanning a 13.4 MHz interval, and lasting 7.5 µs. The observed peak width of 39 ± 2 ns, where the error is due to the low time resolution of the recording, compares well with the 1/rTs = 38 ns value. The peaks are also situated at the expected positions. Apart from peakto-peak height fluctuations the spectrum analyzer response is rather flat over its bandwidth. This was obtained via tuning of the Bragg angle of the gate cell AO2 which was found to modify considerably the overall shape of the response. This stems from the fact that the echo is on a chirped optical carrier which is also angularly rotating and the AO2 cell diffraction efficiency is not a flat function of incident angle. AO1 Bragg angle tuning was also found to alter the peak-to-peak amplitude fluctuations. This seems to indicate that the peaks unevenness is due to the modulation process in AO1. We checked with dedicated experiments that this was indeed the case [84].
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F IG . 22. High resolution single side band spectra given by the photon echo spectrum analyzer. The input RF signal contains 12 RF frequencies spanning a 20 MHz wide interval. (c) is a zoom on part of (a) and corresponds to 50 kHz resolution. (d) is a zoom on part of (b) and corresponds to 40 kHz resolution.
Higher resolution was then tested. Figures 22(a) and (c) (respectively Figs. 22(b) and (d)) show records from single sideband experiment with Tw = 100 µs, Δ = 40 MHz, τ1 = 0.2 µs, τ2 = 40.2 µs (respectively 50.2 µs), 10 Hz cycling rate, and a power of 1.8 mW at engraving. The reading signal, with 0.3 mW of power, lasted 20 µs (respectively 25 µs) and contained 12 frequencies spanning a 20 MHz bandwidth, set so that no second order harmonic combination of any two frequencies coincides with another frequency. The peak width is 48 ns, again in agreement with the computed 1/rTs value, and the peaks are all at expected positions. This corresponds to a 1/Ts resolution of 50 kHz for the records of Figs. 22(a) and (c), and 40 kHz for the records of Figs. 22(b) and (d). It demonstrates a time bandwidth product of 500. One notices a decay of the signal as a function of time. These recordings were taken after checking the correct Bragg tuning of the gate AO2. Therefore the observed decay should be attributed to the coherence time of the crystal. Actually, the observed decay along the 20 µs (respectively 25 µs) duration of the signal
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would correspond to a coherence time of 50 µs. This is compatible with previous measurements we made.
6.3. D ISCUSSION The above results fully demonstrate the high resolution capability of the photon echo chirp transform spectrum analyzer. However this architecture has yet to be demonstrated over multi-GHz bandwidth. The difficulty is then to produce the necessary chirps. If one resorts to electronic modulation to produce these chirps, the photon echo approach adds little as compared to surface acoustic wave devices. Indeed, in order to reach the 10 GHz bandwidth capability, one must be able to produce the corresponding chirps using frequency agile lasers. Because these chirps must span the 10 GHz in a few microseconds only, this puts severe constraints on laser technology. All the more since the chirp linearity must be very good, the echo signal resulting from coherent superposition of the whole excited bandwidth. Laser technology is advancing in this direction. Demonstration of the photon echo chirp transform over a 1 GHz bandwidth using a frequency agile laser is a challenging goal, whose success would open the way to many other applications. As compared to chirp transform spectrum analyzer developed with SAW dispersive delay lines, the photon echo chirp transform approach has two main advantages. First, the bandwidth capability is much greater since rare earth doped crystals inhomogeneous bands are several tens of GHz wide. Let us point out that the limited bandwidth or the Erbium doped crystal used in our first demonstration is not representative. YSO crystals co-doped with Er and Eu have bandwidth greater than 10 GHz with very similar coherence properties. The Er:LiNbO3 crystals has an inhomogeneous width of 270 GHz [81]. The other advantage is flexibility. Whereas the bandwidth and resolution of a SAW chirp transform analyzer is fixed at construction, the bandwidth and resolution of our analyzer can be modified in a short time, thanks to the limited lifetime of engraving in the crystal. One may also point out the features of this spectrum analyzer as compared to the other architectures discussed here. The chirp transform analyzer gives fastest access to the RF spectrum, since that spectrum is displayed in a time limited only by the spectral resolution. The immediate drawback is that detection of the spectrum requires a bandwidth equal to the RF signal bandwidth, which is not favorable to dynamic range. From an engineering point of view, the optical setup is rather straightforward because signal processing is in the time-frequency domain only. The main technological challenge lies in laser technology.
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7. Frequency Agile Laser Technology 7.1. R EQUIREMENTS Development of the different hole-burning spectrum analyzers presented above, and more generally, of hole-burning based optical processors, relies strongly on laser technology. More specifically one needs high spectral purity, rapidly tunable, monochromatic lasers, so-called frequency agile lasers. In the following, we discuss the laser requirements for the three spectrum analyzers presented above, considering the aim of 10 GHz bandwidth and 1 MHz resolution. Then we describe our present laser architectures. Section 7.3 is devoted to the techniques we developed to measure the chirp spectral purity. Finally, Section 7.4 mentions present laser technology developments for applications to RF signal processing. In the photographic architecture described in Section 3, the laser must scan the absorption band of the crystal so as to probe the engraved RF spectrum. The sweeps must cover a bandwidth of typically 10 GHz, in a time of typically 1 ms, since the engraving lifetime is 10 ms. Obviously, the spectral resolution is limited by the laser line-width during the scan. The absolute precision of the retrieved RF spectrum depends on the linearity of the sweep. An absolute precision of 1 MHz over a 10 GHz wide sweep is a severe constraint. For the rainbow analyzer (see Section 5), the constraints are a bit more severe. In this architecture, the laser frequency must be linearly scanned over 10 GHz, synchronously with the angular scanning of the engraving beams. This engraves in the crystal the set of monochromatic Bragg gratings which make the frequency-toangle processor used for angular separation of the RF spectral components. Here again the precision of the frequency-to-angle law engraved in the crystal critically depends on the linearity of the laser sweep. We therefore have the same linearity constraint as for the photographic analyzer. In addition, since engraving must be performed within the grating lifetime, the same chirp rate as for the photographic analyzer is required. However, one also has to consider the necessity to refresh the gratings. This forces one to repeat the engraving cycles at a rate higher than the gratings decay rate. Then the engraved processor is the result of accumulation of successive cycles. With a 2 kHz cycling rate one typically accumulates 20 cycles in the 10 ms grating lifetime. Supposing the chirps are perfectly linear, should the laser frequency shift between two successive sweeps, this shifts the frequencyto-angle law. As a result, not only is the spectral resolution degraded, but also the diffraction efficiency of the accumulated gratings. Hence a central frequency stability constraint is added to the linearity constraint. The photon echo chirp transform spectral analyzer is by far the most demanding, mainly because in this architecture the signal results from coherent addition of the atomic response from the whole analyzer bandwidth. For this architecture, one need laser chirps both at engraving and at reading. Let us first consider
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the programming stage. Using the co-temporal chirp approach discussed in Section 6.1.6, the laser frequency must sweep the whole analyzer bandwidth in a time limited by the grating lifetime. We therefore need the same chirp rate, say 10 GHz in typically 1 ms, as required for the rainbow and the photographic analyzers. Linearity is also required. One must also consider the necessary refreshing, as for the rainbow analyzer, which again adds the constraint of central frequency stability. At reading, one needs a laser chirp that covers the whole bandwidth in a time given by the inverse of the resolution. This means covering 10 GHz in a few microseconds, only, a challenging task indeed. Obviously, linearity is required. Let us point out that the linearity requirement actually does not mean the same for the chirp transform as for the other two analyzers. In the latter ones, phase defaults in the laser chirps have limited consequence since the different spectral channels are treated independently. Such defaults have drastic consequences in the chirp transform analyzers since the signal coherently combines atomic response from the whole analyzer spectral bandwidth. Indeed a phase continuous chirp is required. 7.2. E LECTRO -O PTIC T UNING OF D IODE L ASER E XTENDED C AVITY Diode lasers have many interesting features such as compactness, high gain and high wall plug efficiency. However, their spectral line-width is usually several tens of MHz. Using the diode as a mere gain medium in an external cavity closed by a diffraction grating is the usual way to lower the laser line-width to a few tens of kHz or below. In addition, the diffraction grating is a convenient way to control the laser frequency. Mechanically moving the grating in a well controlled way produces laser frequency mode-hop-free tuning over several tens of nm. Piezo-electrically controlled grating movements can produce frequency sweeps a few tens of GHz wide at a few hundred hertz repetition rate. Our linearity constraint forbids the use of such sweeping solutions. Instead, because of its fast response and high linearity, we chose to rely on the electro-optic effect for frequency agility. Figure 23 gives the principle of the Littrow type cavities we have developed. In Fig. 23(a), the intracavity electro-optic crystal is cut as a prism. Applying a voltage between the crystal faces modifies the refractive index, which has two consequences. It shifts the laser cavity mode frequency ν according to L n ν = =e , (55) ν L L where e is the beam path length in the crystal and L is the total optical length of the cavity. It also changes the angle of incidence of the beam on the grating, shifting the Littrow wavelength according to (dθ/dn)n ν = , ν tan θ
(56)
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F IG . 23. Principle of extended cavity diode laser with intracavity electro-optic crystal for fast mode hop free frequency tuning. (a) Prism type cavity. (b) Wedge type cavity.
where θ is the angle of incidence on the grating. In order to perform mode-hopfree frequency sweeps, the two variations must be the same, which determines the crystal geometrical parameters versus the cavity optical length. With the prism set in the minimum deviation orientation, the mode hop free condition imposes e = L
2 sin(α/2) . tan θ 1 − n2 sin2 (α/2)
(57)
First demonstration of this principle was performed in 2000 [69]. A similar idea was later developed [58], in which the prism is replaced by a wedge (see Fig. 23(b)). Applying a voltage to the crystal then results in an index gradient. This again shifts the laser cavity mode according to Eq. (55), and also deviates the beam incidence on the grating via the mirage effect according to Eq. (56) where dθ/dn now depends on the wedge angle ε and thickness h. Mode hop free tuning is achieved if ε = h tan θ/L.
(58)
Whereas the first laser built to demonstrate the prism approach principle was a crude one, better engineering work was performed, in collaboration with Lund University, to develop a laser based on the wedge principle. Based on a LiTaO3 crystal wedge and antireflection coated diodes from JDS-SDL and operating around 795 nm these lasers can sweep over up to 50 GHz wide intervals, with an electro-optic sensitivity of 12 MHz/V. The laser sweep rate is limited by the high voltage source performance. One easily performs 10 GHz wide chirps in 1 ms. Several copies of the same model were built, which have been used to demonstrate the rainbow spectral analyzer, and later the spectrum photography analyzer.
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Experiments using the rainbow analyzer in accumulation mode showed that the laser sweeps can be repeated with a precision of about 500 kHz over a 10 ms time. The laser coherence time and technical noise were characterized with dedicated experiments described in the next section. One copy of this laser also proves very useful in coherent control experiments of thulium ions in a YAG host [85]. In these experiments, the laser frequency is locked to a high finesse cavity using the Pound–Drever–Hall technique [86]. The intracavity crystal was found to be a very convenient element on which to close the feedback loop. Thanks to the DC-2 MHz bandwidth of the crystal’s electro-optic response function, a single loop was implemented, instead of the usual combination of a low-frequency loop (acting for example on a piezoelectric transducer) and a high-frequency loop (acting for example on the diode injection current).
7.3. L ASER C HIRP S PECTRAL P URITY C HARACTERIZATION As we have seen above, the spectral purity of the laser during the chirp determines the resolution and the frequency precision of the RF spectral analysis. Ideally, we need to maintain the laser frequency deviation from a perfectly linear chirp lower than 1 MHz. In order to fulfill this condition, a complete measurement of laser frequency noise during the chirp is necessary. Two different families of phase errors occurring during the laser chirp can be distinguished. First, deterministic errors, such as, e.g., the non-linearities of the chirp or some extra frequency modulations have to be isolated. Second, stochastic errors, such as the phase random walk induced by the spontaneous emission or the technical 1/f frequency noise, have to be characterized. This noise exists of course in the case of fixed frequency lasers and can be measured with well-known techniques [86–89]. The problem here is to measure this noise while the average frequency is swept rapidly. Some experiments dealing with the phase noise characterization of frequencychirped CW lasers have already been reported [90–94]. However, these techniques are either not adapted to the values of the chirp rates and amplitudes used here or are not sensitive enough to be able to deal with the different kinds of noise investigated here. Consequently, we needed develop a specific experiment for the complete characterization of single linear frequency chirps in the range necessary for RF signal processing (10 GHz in 1 ms or 1 µs) with the required resolution (1 MHz) [95]. For a chirped laser operating above threshold with constant amplitude, the phase fluctuations are the major causes of spectral impurity. A well-known process to detect the phase fluctuations of usual, i.e., fixed frequency lasers is provided by path-difference interferometers. This is why we choose here also a self-heterodyne interferometry technique for characterizing the deterministic and
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F IG . 24. Principle of the unbalanced interferometer. PD: photodetector; AOM: acousto-optic modulator creating a frequency shift ν.
stochastic phase errors of a rapidly chirped laser. This consists of an unbalanced Mach–Zehnder interferometer with a time delay τd between the two paths, as schematized in Fig. 24. At the laser output and in the absence of noise, the electrical field can be expressed as 1
E(t) = E (t) + c.c. = E0 e2iπ(ν0 t+ 2 rt
2 )+iφ
0
+ c.c.,
(59)
where E0 is the light field amplitude which we suppose constant, ν0 is the average optical frequency, r is the chirp rate value and φ0 is a constant phase. In the unbalanced Mach–Zehnder interferometer schematized in Fig. 24, the laser beam is split in two arms. One of them experiences a delay τd while the other is frequency shifted by an acousto-optic modulator operating at frequency ν. If the contrast is supposed to be equal to 1, the recombination of these two beams at the output of the interferometer leads to the following field: 1 ET (t) = √ E (t)e−2iπνt + E (t − τd ) + c.c. . 2 Consequently, the detected intensity is I (t) = 2E02 1 + cos 2πfb t + 2πν0 τd − πrτd2 ,
(60)
(61)
where fb = rτd − ν is the beat signal frequency. We notice that fb is a linear function of the chirp rate r. The laser chirp rate r can consequently be determined by a simple Fourier analysis of the beat note signal. Of course, the typical resolution of the resulting spectrum will be of the order of 1/T , where T is the duration of the considered chirp. However, the precision on the measurement of r can be adjusted by modifying the delay τd . Moreover, even for low values of τd , the signal can be shifted out of the low frequency noise region to higher frequencies thanks to the acousto-optic modulator frequency ν.
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F IG . 25. Experimental result for a 20-ms-long chirp of amplitude 10 GHz presenting a variation of r. The delay is τd = 150 ns. (a) Spectrum of the whole signal. (b) Ten spectra corresponding to the ten successive 2-ms-long parts of the whole signal. (c) Corresponding evolution of the averaged beat note frequency versus time. Each point corresponds to a 2-ms duration sample.
As we have just seen, a perfect chirp will lead to a beat-note at frequency fb = rτd − ν at the output of the interferometer. Any discrepancy with respect to a perfect beat note will reveal the existence of an error. In order to illustrate the sensitivity of this technique, we choose two examples: a deterministic error and a stochastic error. A first example of result is reproduced in Fig. 25. This has been obtained with the external cavity diode laser at 793 nm that we used for the spectral photographic analyzer (see Section 3) and the rainbow analyzer (see Section 5). Figure 25(a) is the FFT of the beat signal obtained at the output of the unbalanced interferometer for a chirp amplitude νlaser = 10 GHz in a duration T = 20 ms. The delay of the interferometer is τd = 150 ns. The width of this spectrum is much broader (of the order of a few kHz) than what could be expected from the resolution of the FFT (1/T = 50 Hz), indicating an important deviation of the laser instantaneous frequency with respect to a perfectly linear chirp. To analyze this deviation, one can divide the 20 ms-long signal in ten successive records lasting 2 ms each. The FFT spectra of these ten successive records are shown in Fig. 25(b). As can be seen in Fig. 25(c), the average beat frequency fb increases quasi-linearly as a function of time during the 20 ms-long chirp. Consequently, the laser instantaneous frequency can be modeled by the expression ν(t) = ν0 + rt + αt 2 , with r = 0.47 THz/s and α = 1.5 THz/s2 . In this case, this error was due to a bad alignment of the grating closing the laser cavity. The result of Fig. 25 illustrates the capability of our unbalanced interferometer to detect deterministic deviations of the laser chirp with respect to perfectly linear ones. We have also put into evidence the ability of this system to detect other types of deterministic errors, such as an, extra sinusoidal frequency mod-
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ulation superimposed on the linear frequency chirp. This type of error leads to frequency-modulation side bands on the FFT of the beat signal, allowing a precise measurement of the amplitude and frequency of this spurious modulation [95]. One can show that the precision on the measurement of such deterministic errors is better than 1 MHz, as required. Another illustration of the usefulness of this technique is illustrated by its ability to detect stochastic laser phase fluctuations during the chirp. As with most lasers, we can expect the lasers described above (see Section 7.2) to be affected by two main types of frequency noise: (i) white frequency noise due to spontaneous emission which leads to the Schawlow–Townes line-width in the case of usual stable frequency lasers, and (ii) low frequency (typically 1/f ) technical frequency noise induced by mechanical or acoustic variations of the laser cavity length. In both cases, we can write the instantaneous phase of the linearly chirped laser as r 2 Φ(t) = 2π ν0 t + t + φ(t). (62) 2 Here, φ(t) can represent both the white quantum and the low-frequency technical noise. At the output of the interferometer the optical intensity incident on the detector can be written as I (t) = 2E02 1 + cos 2πfb t + ψ0 + φ(t) − φ(t − τd ) . (63) A sine wave is observed at the output of the photo detector with an average frequency fb . To obtain the spectrum of this signal, we need to calculate the auto-correlation function of I (t) and apply the Wiener–Khinchin theorem. The auto-correlation function is given by RI (t, τ ) = I (t)I (t + τ ) , (64) where the brackets denote ensemble averaging. This leads to RI (t, τ ) = 4E04 + 2E04 cos H (t, τ ) cos(2πfb τ ),
(65)
where we have defined H (t, τ ) = φ(t + τ ) − φ(t) − φ(t + τ − τd ) + φ(t − τd ).
(66)
H (t, τ ) depends on the optical phase at four different times. We suppose that the phase jitter defined by φ(t, τ ) = φ(t + τ ) − φ(t)
(67)
is a zero-mean stationary Gaussian process. Then the dependence in t disappears in the moments of H (t, τ ), and the auto-correlation function of the optical intensity can be written as 1
RI (τ ) = 4E04 + 2E04 e− 2 H
2 (t,τ )
cos(2πfb τ ).
(68)
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The Fourier transform of RI (t) leads to the spectrum of the intensity detected at the output of the interferometer. Let us first suppose that the only noise occurring in the frequency chirped laser is white quantum noise. In this case, the auto-correlation of the frequency error is given by
1 ˙ + τ )φ(t) ˙ φ(t = δ(τ ), τc
(69)
where τc is the laser coherence time. From Eqs. (66) and (67), one can express RI (τ ) as a function of the phase jitter as H (t, τ ) = φ(t, τ ) − φ(t − τd , τ ). Using Eq. (69), this leads to 2 2|τ |/τc H (t, τ ) = 2τd /τc
for |τ | < τd , for |τ | > τd .
This can be used together with Eq. (68) to obtain 4 2E0 cos(2πfb τ )e−|τ |/τc for |τ | < τd , 4 RI (τ ) = 4E0 + 2E04 cos(2πfb τ )e−τd /τc for |τ | > τd .
(70)
(71)
(72)
Consequently, the power spectral density of the detected signal is given by the Fourier transform of Eq. (72) SI (ω) = 4E04 δ(ω) + E04 e−τd /τc δ(ω − 2πfb ) τc /π + E04 e−τd /τc 1 + (ω − 2πfb )2 τc2 sin(ω − 2πfb )τd × eτd /τc − cos(ω − 2πfb )τd − (ω − 2πfb )τc + {fb → −fb }.
(73)
This result is similar to the self-heterodyne spectrum obtained for laser sources operating at fixed frequency [87–89], except for the frequency shift fb which varies with the chirp rate. Consequently, the quantum frequency noise of our chirped laser can be obtained as usual by taking τd as long as possible in order to favor the Lorentzian part of Eq. (73), which is just the image of the Lorentzian Schawlow–Townes broadening, with respect to the Dirac term. We have used this technique to isolate the white part of the stochastic frequency noise of our frequency agile lasers while it is chirped. As expected, we have checked that the Schawlow–Townes line-width of the laser is not modified by the fact that the laser average frequency is chirped [94]. Let us now turn to the case of a low-frequency noise affecting the laser frequency, induced for example by mechanical or acoustic noise. We suppose in the following calculation that this noise is the predominant process, with respect to
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which all other noise can be neglected. If we choose the interferometer delay τd short enough with respect to the characteristic time of variation of the laser fre˙ quency noise φ(t), i.e., with respect to the laser frequency noise coherence time, the following simplification can be used: ˙ φ(t) − φ(t − τd ) ≈ τd φ(t).
(74)
This can be used to simplify Eq. (66), leading to ˙ + τ ) − φ(t) ˙ H (t, τ ) = τd φ(t .
(75)
To obtain the auto-correlation function of the detected intensity, one needs to calculate the variance of H , 2 H (t, τ ) = 2τd2 σφ2˙ − Rφ˙ (τ ) , (76) ˙ where Rφ˙ (τ ) is the auto-correlation function of the frequency noise φ(t) and 2 ˙ where we have supposed that the variance σφ˙ of φ(t) is finite. In the case of 1/f noise, this latter condition can be easily fulfilled by bounding the frequency noise spectrum to a finite interval [ωmin , ωmax ] with ωmin > 0. Equation (68) then leads to RI (τ ) = 4E04 + 2E04 e
−τd2 σ 2˙ τ 2 R ˙ (τ ) φ d φ
e
cos(2πfb τ ).
(77)
An analytical expression of the spectrum can be obtained if τd is short enough to fulfill the following condition 2 τ R ˙ (τ ) τ 2 σ 2 1. (78) d φ d φ˙ Then the exponential term exp[τd2 Rφ˙ (τ )] in Eq. (77) can be expanded to first order. The Fourier transform of Eq. (76) can be expressed as SI (ω) = 4E04 δ(ω) + E04 e
−τd2 σ 2˙
+ {fb → −fb }.
φ
δ(ω − 2πfb ) + τd2 Sφ˙ (ω − 2πfb ) (79)
This equation shows that the interferometer has transferred the low-frequency noise to the base of the Dirac peak created by the beat note at fb . This is illustrated by the experimental result of Fig. 26. We have just seen [see Eq. (74)] that to measure the spectrum of such a frequency noise, we must choose the delay τd of the interferometer much shorter than the typical timescale of the variations of the instantaneous laser frequency. This is why we choose here a rather small interferometer path difference using a 20 m-long fibre, leading to τd = 100 ns. Contrary to the results of Fig. 25 which led to the measurement of deterministic frequency errors, we keep the acousto-optic frequency shifter at 80 MHz, in order to shift the spectrum far from the zero frequency. In order to maintain a sufficient
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F IG . 26. (a) Experimental self-heterodyne spectrum for τd = 100 ns, ν = 80 MHz, and r = 0 recorded with a 1 GHz bandwidth photodiode. (b) Zoom on the central part of the self-heterodyne spectrum with a resolution bandwidth RBW = 1 kHz and for r = 0. (c) Instantaneous frequency spectrum extracted from (b). (d), (e) Same as (b), (c) for r = 1.5 GHz/s. The photodiode is followed by a 9 dB gain amplifier.
dynamical range for the measurements, we have to analyze the signal using the spectrum analyzer. The resolution we want (1 kHz) obliges us to work with very slow chirps only (T = 2.5 s). Figure 26(a) shows the typical spectra (here with r = 0) that we obtain with a resolution bandwidth of 30 kHz, when the laser is not chirped. Three different parts can be distinguished in this spectrum: (i) the Dirac term at fb , (ii) the oscillations due to the white frequency noise and given by Eq. (73), that are consistent with the value τC = 20 µs determined from other measurements, and (iii) a broadening, together with extra lines at the base of the Dirac peak, over a bandwidth of
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the order of 500 kHz. This last low-frequency component is the one we wish to investigate here. As required [see the discussion of Eq. (74)], its bandwidth is much smaller than 1/τd = 107 s−1 . We hence zoom on a 2 MHz bandwidth around 80 MHz with a resolution bandwidth of 1 kHz, leading to the spectrum of Fig. 26(b). We can clearly see the extra noise component lying at the base of the Dirac peak and on top of the background noise given by the white noise component of the self-heterodyne spectrum. We use Eq. (79) to extract the low-frequency component of the power spectral density Sδν (ω) of the instantaneous frequency error ˙ δν = φ/2π. This leads to the spectrum of Fig. 26(c). This noise is quite well fit by a 1/f law between 5 kHz and 400 kHz, suggesting a technical origin. The total power of this noise, which gives the variance σδν of the frequency due to this noise component, is given by the area below the spectrum of Fig. 26(c) and is of the order of 18 kHz. This shows that τd σφ˙ = 2πτd σδν ≈ 10−2 1, as required to derive Eq. (79). The peak at 864 kHz should not be attributed to the laser frequency: it is due to the ‘Radio Bleue’ broadcasting transmitter of Villebonsur-Yvette, close to our laboratory. Now, at very slow chirp r = 1.5 MHz/ms (νlaser = 3.75 GHz in T = 2.5 s), the central part of the self-heterodyne spectrum becomes the one of Fig. 26(d). After transformation using Eq. (79), we obtain the power spectral density of the instantaneous frequency noise of Fig. 26(e). The noise is larger than in the case of Fig. 26(c), with a variance σδν of the order of 60 kHz. This is still consistent with the hypothesis τd σφ˙ 1. The increase of the low-frequency noise has been observed to be independent of the value of r, in the range of small values of r achievable with the present experiment. Actually, the extra noise observed in Fig. 26(e) with respect to Fig. 26(c) is due to the highvoltage amplifier which amplifies the ramp applied to the electro-optic crystal. This illustrates the high sensitivity of the present setup. In the two preceding examples, the Fourier analysis of the signals delivered by the interferometer has allowed us to measure the different noise affecting the frequency of our chirped laser, but only a posteriori. The orders of magnitude of the different errors we have observed permit us to conclude that if we want to reach a precision better than, say, 1 MHz over the laser frequency during the chirp, we have to take care of the possible deterministic frequency errors, either periodic or not, and of the low-frequency component of the stochastic frequency noise. The white frequency noise has been shown to lead to a Lorentzian broadening of the order of 10 kHz and can be ignored [95]. Consequently, all the frequency noise we have to deal with for our RF spectral analysis applications using rare earth ion doped crystals exhibit bandwidths lying below 1 MHz. This is within the range of conventional electronic servo control loops and one would be tempted to servolock the laser chirp to a perfect linear chirp within an error smaller than 1 MHz. However, to realize this, we need to measure the laser frequency instantaneously. This can be performed using our interferometer, provided that we measure the phase difference Ψ between the two arms at the output of the interferometer.
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F IG . 27. (a) Quadrature signals measured at the output of the interferometer for τd = 100 ns and r = 1.6 GHz/ms. This figure reproduces only a small portion of the record. (b) Corresponding phase error. The sampling rate of the digital oscilloscope is 6.25 MHz.
This phase can be obtained provided we detect two signals in quadrature at the output of the interferometer [96] using two photodiodes. In general, the phase difference between these two signals and the equality of their amplitudes can be adjusted by tuning the orientations of a quarter-wave plate and a half-wave plate located at the output of the fiber, together with one polarizer in front of each detector. Here we simply use one half-wave plate at the output of the fiber. The rotation of this half-wave plate allows us to perfectly adjust the π/2 dephasing between the two signals, even if their amplitudes cannot be equalized. Some resulting typical signals are reproduced in Fig. 27(a). They were obtained with a 20-m-long fiber, corresponding to τd = 100 ns, with a chirp rate r = 1.6 GHz/ms (νlaser = 6.4 GHz in T = 4 ms). Since a 2π dephasing of the interferometer corresponds to a frequency variation of 1/τc = 10 MHz, reaching a precision better than 1 MHz on the laser frequency requires one to measure Ψ with a resolution better than π/5. Figure 27(a) shows that this is easily achieved with the sampling rate of 6.25 MHz that we use. The phase error δΨ (t) = Ψ (t) − 2πrτd t induced by the laser frequency error can easily be reconstructed from the two quadrature signals, as evidenced in Fig. 27(b). This signal contains the different types of errors we have observed. It is the superposition of (i) a quasi-linear drift, (ii) a sinusoidal modulation, and (iii) a stochastic component. Such a signal could be filtered and used as an error signal to lock the laser frequency in a digital servo loop with a correction voltage applied on the electrooptic crystal, as already used for a fixed frequency laser [85]. As we have seen above, all the errors we have to correct for, i.e., the deterministic frequency errors and the low-frequency part of the stochastic noise, lie in a bandwidth smaller than a few hundreds of kHz. A loop bandwidth of 1 MHz would consequently be sufficient. Besides, each interference fringe of Fig. 27(a) corresponds to a variation of the laser frequency of 1/τd = 10 MHz. Consequently, a resolution better than
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1 MHz, as required for spectral analysis applications, could easily be achieved by digitalizing the quadrature signals with 8-bit converters.
7.4. P ERSPECTIVES While the laser model described in Section 7.2 proved efficient in the demonstration of the rainbow and the photographic analyzers, frequency agile laser technology must be improved. From an application point of view, the model is too big, heavy, and sensitive to external perturbations. In addition, the high voltage source requirement limits the chirp speed, is a source of noise, and consumes electrical power. Also, the chirp linearity must still be improved, especially for high bandwidth operation of the chirp transform analyzer. We are pursuing two paths for achieving these goals. One consists in developing more compact and lightweight laser structures, the other in working out efficient schemes for opto-electronic control of the laser chirp purity. In collaboration with the team of Professor W. Sohler from Paderborn University, we have recently designed an extended cavity diode laser whose light is coupled into a monomode waveguide inscribed in a LiNbO3 wafer by titanium indiffusion [97]. The cavity is closed by a Bragg grating, holographically written via the photorefractive effect in a Fe doped section of the waveguide. This approach leads to a very compact design which could be made monolithic. Also, the waveguide approach considerably increases the electro-optic frequency tuning sensitivity since the electrode spacing is reduced to about 10 µm instead of 1 mm for the unguided laser cavities described in Section 7.2. A sensitivity of 55 MHz/V was obtained, which could be increased by a factor of 5 with a different crystal cut. Active stabilization of the laser chirps is also under study. Analog and digital high bandwidth phase locked loops are under development to lock the frequency chirp to a Mach–Zehnder interferometer, along the lines discussed in Section 7.3. These are derived from the low bandwidth frequency locking loop developed by Juncar and Pinard [96]. We believe these should yield laser frequency chirps whose beat note line-width is limited by the inverse chirp duration.
8. Conclusion As illustrated by these few examples, hybridization of fast electronic with atomic and laser physics appears to be a powerful tool for broadband signal processing. The proposed RF spectral analysis architectures offer unrivaled capabilities in terms of instantaneous bandwidth, number of frequency channels, and probability of interception. The spectrum photography configuration has already
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demonstrated 10 GHz instantaneous bandwidth with 10,000 channel capacity. The rainbow analyzer, by offering simultaneous access to the different spectral channels, is well suited for long time spectral integration. As for the photon echo chirp transform architecture, its coherent coverage of the bandwidth guarantees Fourier transform limited access time to the spectral addresses. To meet the requirements in terms of stability, scanning speed, range and repeatability, original tunable lasers have been developed together with specific diagnostic, monitoring and control techniques. Integrated cryogenics now represents the next important milestone on the roadmap to a demonstrator assembly. However mechanical cryocoolers have accomplished dramatic progress in recent years, stimulated by development initiatives such as the NASA’s Advanced Cryocooler Technology Development Program. Multistage refrigerators, able to offer a few milliwatts cooling power at 5 K, starting from room temperature, should not exceed 20 kg payload and 200 Watts input power [98].
9. Acknowledgements The authors are much indebted to the many collaborators who took part in the development of atomic RF spectral analyzers. Among them, the Ph.D. students L. Ménager, V. Lavielle, V. Crozatier, and G. Gorju deserve special acknowledgments of their contributions. Discussions with J.-P. Huignard, D. Dolfi, I. Zayer, K. Wagner, S. Tonda, L. Morvan, F. Schlottau, L. Levin were very motivating and enriching. The help of T. Böttger, R. Equall and Y.C. Sun was invaluable in the understanding of the erbium crystal parameters. We gratefully acknowledge support from European Space Agency under contracts ESTEC/12876/98/NL/MV and ESTEC/14174/00/NL/SB, from Délégation Générale de l’Armement under contract DGA STTC/34047 and from the NICOP Program of the Office of Naval Research under contract N00014-03-10770.
10. References [1] A. Szabo, Observation of the optical analog of the Mössbauer effect in ruby, Phys. Rev. Lett. 27 (1971) 323–326. [2] A. Szabo, Frequency selective optical memory, US patent 3 896 420, 22 July 1975. [3] G. Castro, D. Haarer, R.M. Macfarlane, H.P. Trommsdorff, Frequency selective optical data storage system, US patent 4 101 976, 18 July 1978. [4] A.A. Gorokhovskii, R.K. Kaarli, L.A. Rebane, Hole burning in the contour of a pure electronic line in a Shpol’skii system, JETP Lett. 20 (1974) 216–218. [5] B.M. Kharlamov, R.I. Personov, L.A. Bykovskaya, Stable “gap” in absorption spectra of solid solutions of organic molecules by laser irradiation, Opt. Commun. 12 (1974) 191–193. [6] B. Kohler, S. Bernet, A. Renn, U.P. Wild, Storage of 2000 holograms in a photochemical holeburning system, Opt. Lett. 18 (1993) 2144–2146.
10]
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[7] E.S. Maniloff, S.B. Altner, S. Bernet, F.R. Graf, A. Renn, U.P. Wild, Recording of 6000 holograms by use of spectral hole burning, Appl. Opt. 34 (1995) 4140–4148. [8] B. Plagemann, F.R. Graf, S.B. Altner, A. Renn, U.P. Wild, Exploring the limits of optical storage using persistent spectral hole-burning: holographic recording of 12000 images, Appl. Phys. B 66 (1998) 67–74. [9] A. Renn, U.P. Wild, A. Rebane, Multidimensional holography by persistent spectral hole burning, J. Phys. Chem. A 106 (2002) 3045–3060. [10] T.W. Mossberg, Time-domain frequency-selective optical data storage, Opt. Lett. 7 (1982) 77–79. [11] T.W. Mossberg, Swept-carrier time-domain optical memory, Opt. Lett. 17 (1992) 535–537. [12] H. Lin, T. Wang, G.A. Wilson, T.W. Mossberg, Experimental demonstration of swept-carrier time-domain optical memory, Opt. Lett. 20 (1995) 91–93. [13] H. Lin, T. Wang, T.W. Mossberg, Demonstration of 8-Gbit/in.2 areal storage density based on swept-carrier frequency selective optical memory, Opt. Lett. 20 (1995) 1658–1660. [14] Z. Cole, T. Böttger, R. Krishna Mohan, R. Reibel, W.R. Babbitt, R.L. Cone, K.D. Merkel, Coherent integration of 0.5 GHz spectral holograms at 1536 nm using dynamic biphase codes, Appl. Phys. Lett. 81 (2002) 3525–3527. [15] K.D. Merkel, W.R. Babbitt, Optical coherent-transient true-time-delay regenerator, Opt. Lett. 21 (1996) 1102–1104. [16] K.D. Merkel, Z. Cole, W.R. Babbitt, Signal correlator with programmable variable time delay based on optical coherent transients, J. Lumin. 86 (2000) 375–382. [17] R. Reibel, Z. Barber, M. Tian, W.R. Babbitt, Temporally overlapped linear frequency-chirped pulse programming for true-time-delay applications, Opt. Lett. 27 (2002) 494–496. [18] M.R. Schoeberl, A.R. Douglass, E. Hilsenrath, P.K. Bhartia, J. Barnett, J. Gille, R. Beer, M. Gunson, J. Waters, P.F. Levelt, P. DeCola, Earth observing system missions benefit atmospheric research, EOS Trans. Amer. Geophys. Union 85 (2004) 177–184. [19] J.W. Waters, W.G. Read, L. Froidevaux, R.F. Jarnot, R.E. Cofield, D.A. Flower, G.K. Lau, H.M. Pickett, M.L. Santee, D.L. Wu, M.A. Boyles, J.R. Burke, R.R. Lay, M.S. Loo, N.J. Livesey, T.A. Lungu, G.L. Manney, L.L. Nakamura, V.S. Perun, B.P. Ridenoure, Z. Shippony, P.H. Siegel, R.P. Thurstans, The UARS and EOS Microwave Limb Sounder (MLS) experiments, J. Atmospheric Sci. 56 (1998) 194–218. [20] J.W. Waters, An overview of the EOS MLS experiment, JPL D-15745, Version 1.0, 1999. [21] R.F. Jarnot, EOS MLS level 1 data processing algorithm theoretical basis, JPL D-15210, Version 1.1, 1999. [22] N. Kaifu, N. Ukita, Y. Chikada, T. Miyaji, A high-resolution acousto-optical radiospectrometer for millimeter-wave astronomy, Publ. Astronom. Japan 29 (1977) 429–435. [23] T.W. Cole, D.K. Milne, An acousto-optical radio spectrograph for spectral integration, Proc. Astronom. Soc. Australia 3 (1977) 108–111. [24] M. Klumb, J. Frerick, R. Schieder, G. Winnewisser, V. Tolls, The SWAS acousto optical spectrometer, in: M.S. Scholl (Ed.), “Proc. of Infrared Spaceborne Remote Sensing II, 27–29 July 1994, San Diego, CA”, in: Proc. SPIE, vol. 2268, 1994, pp. 305–315. [25] V. Tolls, G.J. Melnick, N. Erickson, P. Goldsmith, M. Harwit, R. Schieler, R. Snell, J.R. Stauffer, The submillimeter wave astronomy satellite—the instrument, in: G. Winnewisser, G.C. Pelz (Eds.), “Proc. of The Physics and Chemistry of Interstellar Molecular Clouds, 21–24 September 1993, Zermatt, Switzerland”, in: Lecture Notes in Phys., vol. 459, Springer-Verlag, 1995. [26] G.J. Melnick, J.R. Stauffer, M.L.N. Ashby, E.A. Bergin, G. Chin, N.R. Erickson, P.F. Goldsmith, M. Harwit, J.E. Howe, S.C. Kleiner, D.G. Koch, D.A. Neufeld, B.M. Patten, R. Plume, R. Schieder, R.L. Snell, V. Tolls, Z. Wang, G. Winnewisser, Y.F. Zhang, The submillimeter wave astronomy satellite: Science objectives and instrument description, Astrophys. J. 539 (2000) 77– 85.
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J.-L. Le Gouët et al.
[10
[27] U. Frisk, M. Hagström, J. Ala-Laurinaho, S. Andersson, J.-C. Berges, J.-P. Chabaud, M. Dahlgren, A. Emrich, H.-G. Florén, G. Florin, M. Fredrixon, T. Gaier, R. Haas, T. Hirvonen, Å. Hjalmarsson, B. Jakobsson, P. Jukkala, P.S. Kildal, E. Kollberg, J. Lassing, A. Lecacheux, P. Lehikoinen, A. Lehto, J. Mallat, C. Marty, D. Michet, J. Narbonne, M. Nexon, M. Olberg, A.O.H. Olofsson, G. Olofsson, A. Origné, M. Petersson, P. Piironen, R. Pons, D. Pouliquen, I. Ristorcelli, C. Rosolen, G. Rouaix, A.V. Räisänen, G. Serra, F. Sjöberg, L. Stenmark, S. Torchinsky, J. Tuovinen, C. Ullberg, E. Vinterhav, N. Wadefalk, H. Zirath, P. Zimmermann, R. Zimmermann, The Odin satellite, Astronom. Astrophys. 402 (2003) L27–L34. [28] A. Lecacheux, C. Rosolen, D. Michet, V. Clerc, Space qualified, wide band and ultra wide band acousto-optical spectrometers for millimeter and sub-millimeter radio astronomy, in: T.G. Phillips (Ed.), “Proc. of Advanced Technology MMW, Radio, and Terahertz Telescopes Conference, 20–28 March 1998, Kona, HI”, in: Proc. SPIE, vol. 3357, 1998, pp. 519–532. [29] G.L. Pilbratt, The Herschel mission, scientific objectives, and this Meeting, in: G.L. Pilbratt, J. Cernicharo, A.M. Heras, T. Prusti, R. Harris (Eds.), “Proc. Symposium “The Promise of the Herschel Space Observatory”, 12–15 December 2000, Toledo, Spain”, in: ESA, vol. SP-460, 2001. [30] J. Horn, The development of an array acousto-optical spectrometer, Ph.D. thesis, Köln University (1997). [31] J. Horn, O. Siebertz, F. Schmülling, C. Kunz, R. Schieder, G. Winnewisser, A 4 × 1 GHz array acousto-optical spectrometer, Exp. Astron. 9 (1999) 17–38. [32] R. Schieder, J. Horn, O. Siebertz, C. Möckel, F. Schlöder, C. Macke, F. Schmülling, Design of large bandwidth acousto-optical spectrometers, in: T.G. Phillips (Ed.), “Proc. of Advanced Technology MMW, Radio, and Terahertz Telescopes Conference, 20–28 March 1998, Kona, HI”, in: Proc. SPIE, vol. 3357, 1998, pp. 359–367. [33] A.R. Thompson, J.M. Moran, G.W. Swenson, “Interferometry and Synthesis in Radio Astronomy”, Wiley, 1986, Chapter 8. [34] T. Cole, Finite sample correlations of quantized Gaussians, Austral. J. Phys. 21 (1968) 273–282. [35] A. Emrich, Autocorrelation spectrometers for space borne (sub)millimeter spectroscopy, in: “Proc. of the Far InfraRed and Submillimeter Universe Symposium, 15–17 April 1997, Grenoble France”, in: ESA, vol. SP-401, 1997, pp. 361–364. [36] A.I. Harris, Spectrometers for heterodyne detection, in: J. Wolf, J. Farhoomand, C.R. McCreight (Eds.), “Proc. of Far-IR, Sub-mm & mm Detector Technology Workshop, 01–03 April 2002, Monterey, CA”, 2002. [37] L. Ravera, M. Giard, D. Lagrange, E. Caux, A. Cros, G. Serra, J.-L. Noullet, A. Ferreira, P. Cais, A. Baudry, J.M. Desbat, A. Escobar, G. Montignac, M. Torres, J.Y. Mayvial, Wideband digital autocorrelator for FIRST, in: T.G. Phillips (Ed.), “Proc. of Advanced Technology MMW, Radio, and Terahertz Telescopes Conference, 20–28 March 1998, Kona, HI”, in: Proc. SPIE, vol. 3357, 1998, pp. 368–375. [38] T. de Graauw, F.P. Helmich, Herschel-HIFI: The heterodyne instrument for the far-infrared, in: G.L. Pilbratt, J. Cernicharo, A.M. Heras, T. Prusti, R. Harris (Eds.), “Proc. Symposium “The Promise of the Herschel Space Observatory”, 12–15 December 2000, Toledo, Spain”, in: ESA, vol. SP-460, 2001, pp. 45–51. [39] A.I. Harris, K.G. Isaak, J. Zmuidzinas, WASP: A wide-band spectrometer for heterodyne spectroscopy, in: T.G. Phillips (Ed.), “Proc. of Advanced Technology MMW, Radio, and Terahertz Telescopes Conference, 20–28 March 1998, Kona, HI”, in: Proc. SPIE, vol. 3357, 1998, pp. 384– 395. [40] A.I. Harris, J. Zmuidzinas, A wide-band lag correlator for heterodyne spectroscopy of broad astronomical and atmospheric spectral lines, Rev. Sci. Inst. 72 (2001) 1531–1538. [41] L.I. Bluestein, A linear filtering approach to the computation of the discrete Fourier transform, IEEE Trans. Electroacoustics 18 (1970) 451–455.
10]
ATOMIC PROCESSING OF RF SIGNALS
611
[42] H.R. Fetterman, P.E. Tannenwald, C.D. Parker, J. Melngailis, R.C. Williamson, P. Woskoboinikow, H.C. Praddaude, W.J. Mulligan, Real-time spectral analysis of far-infrared laser pulses using SAW dispersive delay line, Appl. Phys. Lett. 34 (1979) 123–125. [43] P. Hartogh, G.K. Hartmann, A high-resolution chirp transform spectrometer for microwave measurements, Meas. Sci. Technol. 1 (1990) 592–595. [44] R. Güsten, I. Camara, P. Hartogh, H.-W. Hübers, U. Graf, K. Jacobs, C. Kasemann, H.-P. Röser, R. Schieder, G. Schnieder, O. Siebertz, J. Stutzki, G. Villanueva, A. Wagner, P. van der Wal, A. Wunsch, GREAT: The German receiver for astronomy at Terahertz frequencies, in: R.K. Melugin, H.-P. Röser (Eds.), “Proc. of Airborne Telescope System II Conference, 27–28 August 2002, Waikoloa, HI”, in: Proc. SPIE, vol. 4857, 2003, pp. 56–61. [45] R. Güsten, F. Schäfer, R. Stark, P. van der Wal, U. Graf, R. Schieder, J. Stutzki, H. Hübers, A. Krabbe, H.-P. Röser, P. Hartogh, The next step: FIR heterodyne spectroscopy on board SOFIA, in: D. Danesy (Ed.), “ISO beyond the Peaks: The 2nd ISO Workshop on Analytical Spectroscopy, 2–4 February 2000, Satellite Tracking Station of Villafranca del Castillo, Spain”, in: ESA, vol. SP-456, 2000. [46] R. Güsten, P. Hartogh, H.-W. Hübers, U. Graf, K. Jacobs, H.-P. Röser, F. Schäfer, R. Schieder, R. Stark, J. Stutzki, P. van der Wal, A. Wunsch, GREAT—The first-generation German heterodyne receiver for SOFIA, in: R.K. Melugin, H.-P. Röser (Eds.), “Proc. of Airborne Telescope Systems, 27–28 March 2000, Munich, Germany”, in: Proc. SPIE, vol. 4014, 2000, pp. 23–30. [47] G. Villanueva, P. Hartogh, L. Reindl, Microwave technologies for SOFIA’s high resolution spectrometer, in: “Proc. of The 33rd European Microwave Conference, 6–10 October 2003, Munich, Germany, Vol. 3”, 2003. [48] M. Colice, F. Schlottau, K. Wagner, K. Mohan, W.R. Babbitt, I. Lorgeré, J.-L. Le Gouët, RF spectrum analysis in spectral hole burning media, in: “Optical Information Systems II”, in: Proc. SPIE, vol. 5557, 2004, pp. 132–139. [49] R.K. Mohan, Z. Cole, R.R. Reibel, T. Chang, K.D. Merkel, W.R. Babbitt, M. Colice, F. Schlottau, K.H. Wagner, Microwave spectral analysis using optical spectral holeburning, in: “Proc. Microwave Photonics (MWP), Ogunquit, MA”, October 2004. [50] J.-L. Le Gouët, I. Lorgeré, F. Bretenaker, V. Lavielle, V. Crozatier, Atomic spectral analyzers for radio frequency signals processing, in: “Proc. IEEE/LEOS Annual Meeting, Puerto-Rico”, October 2004. [51] G. Gorju, V. Crozatier, I. Lorgeré, J.-L. Le Gouët, F. Bretenaker, 10-GHz bandwidth RF spectral analyzer with MHz resolution based on spectral hole burning in Tm3+ :YAG, IEEE Photonics Technol. Lett. 17 (2005) 2385–2387. [52] F. Schlottau, M. Colice, K.H. Wagner, W.R. Babbitt, Spectral hole burning for wide-band, highresolution radio-frequency spectrum analysis, Opt. Lett. 30 (2005) 3003–3005. [53] R.M. Macfarlane, T.L. Harris, Y. Sun, R.L. Cone, R.W. Equall, Measurement of photon echoes in Er:Y2 SiO5 at 1.5 µm with a diode laser and an amplifier, Opt. Lett. 22 (1997) 871–873. [54] T.L. Harris, Y. Sun, R.L. Cone, R.M. Macfarlane, R.W. Equall, Demonstration of real-time address header decoding for optical data routing at 1536 nm, Opt. Lett. 23 (1998) 636–638. [55] R.W. Equall, Y. Sun, R.L. Cone, R.M. Macfarlane, Ultraslow optical dephasing in Eu3+ :Y2 SiO5 , Phys. Rev. Lett. 72 (1994) 2179–2182. [56] R.M. Macfarlane, Spectral hole burning in the trivalent thulium ion, Opt. Lett. 18 (1993) 829– 831. [57] W.H. Hesselink, D.A. Wiersma, Photon echoes stimulated from an accumulated grating: Theory of generation and detection, J. Chem. Phys. 75 (1981) 4192–4197. [58] L. Levin, Mode-hop-free electro-optically tuned diode laser, Opt. Lett. 27 (2002) 237–239. [59] T. Chang, R.K. Mohan, M. Tian, T.L. Harris, W.R. Babbitt, K.D. Merkel, Frequency-chirped readout of spatial-spectral absorption features, Phys. Rev. A 70 (2004) 063803.
612
J.-L. Le Gouët et al.
[10
[60] J. Poirson, F. Bretenaker, M. Vallet, A. Le Floch, Analytical and experimentally study of ringing effects in a Fabry–Perot cavity. Application to the measurement of high finesses, J. Opt. Soc. Amer. B 14 (1997) 2811–2817. [61] G. Gorju, A. Chauve, I. Lorgeré, J.-L. Le Gouët, F. Bretenaker, 10-GHz bandwidth RF spectral analyzer with MHz resolution based on spectral-spatial holography in Tm3+ :YAG: Experimental and theoretical study, J. Opt. Soc. Amer. B, submitted for publication. [62] A. Rebane, R. Kaarli, P. Saari, A. Anijalg, K. Timpmann, Photochemical time-domain holography of weak picosecond pulses, Opt. Commun. 47 (1983) 173–176. [63] Y. Sun, C.W. Thiel, R.L. Cone, R.W. Equall, R.L. Hutcheson, Recent progress in developing new rare earth materials for hole burning and coherent transient applications, J. Lumin. 98 (2002) 281–287. [64] F. Schlottau, K. Wagner, Demonstration of a continuous scanner and time-integrating correlator using spatial-spectral holography, J. Lumin. 107 (2004) 90–102. [65] I. Lorgeré, L. Ménager, V. Lavielle, J.-L. Le Gouët , D. Dolfi, S. Tonda, J.-P. Huignard, Demonstration of a radio-frequency spectrum analyser based on spectral hole burning, J. Mod. Opt. 49 (2002) 2459–2475. [66] S. Bernet, S.B. Altner, F.R. Graf, E.S. Maniloff, A. Renn, U.P. Wild, Frequency and phase swept holograms in spectral hole-burning materials, Appl. Opt. 34 (1995) 4674. [67] V. Lavielle, I. Lorgeré, J.-L. Le Gouët, Dispersion only stimulated photon echo, Opt. Commun. 232 (2004) 263–271. [68] V. Lavielle, I. Lorgeré, J.-L. Le Gouët, S. Tonda, D. Dolfi, Wideband versatile radio-frequency spectrum analyzer, Opt. Lett. 28 (2003) 384–386. [69] L. Ménager, L. Cabaret, I. Lorgeré, J.-L. Le Gouët, Diode laser extended cavity for broadrange fast ramping, Opt. Lett. 25 (2000) 1246–1248. [70] U. Elman, B. Luo, S. Kröll, Influence of laser phase and frequency fluctuations on photon-echo data erasure, J. Opt. Soc. Amer. B 13 (1996) 1905–1915. [71] K.D. Merkel, W.R. Babbitt, Optical coherent transient continuously programmed continuous processor, Opt. Lett. 24 (1999) 172–174. [72] V. Lavielle, F. De Seze, I. Lorgeré, J.-L. Le Gouët, Wideband radio frequency spectrum analyzer: improved design and experimental results, J. Lumin. 107 (2004) 75–89. [73] T. Jannson, Real time Fourier transformation in dispersive optical fibers, Opt. Lett. 8 (1983) 232. [74] B.H. Kolner, Space time duality and the theory of temporal imaging, IEEE J. Quantum Electron. 30 (1994) 1951–1963. [75] M.A. Jack, P.M. Grant, J.H. Collins, The theory design and applications of surface acoustic wave Fourier transform processors, Proc. IEEE 68 (1980) 450–468. [76] See product line of, e.g., Phonon Corporation, Temex Microsonics. [77] R.R. Reibel, Z.W. Barber, J.A. Fisher, M. Tian, W.R. Babbitt, Broadband demonstration of true time delay using linear sideband chirped programming and optical coherent transients, J. Lumin. 107 (2004) 103–113. [78] L. Ménager, J.-L. Le Gouët, I. Lorgeré, Time to frequency Fourier transform with photon echoes, Opt. Lett. 26 (2001) 1397–1399. [79] H. Lin, T. Wang, G.A. Wilson, T.W. Mossberg, Experimental demonstration of swept carrier time-domain optical memory, Opt. Lett. 20 (1995) 91–93. [80] T. Böttger, Laser frequency stabilization to spectral hole burning references in Er doped crystals: materials and optimization, Ph.D. thesis, Montana State University, 2002. [81] T. Böttger, Y.C. Sun, C.W. Thiel, Material optimization of Er3+ :Y2 SiO5 at 1.5 µm for optical processing, memory, and laser frequency stabilization applications, in: “SPIE Proceedings of Advanced Optical Data Storage (OE05)”, 2003. [82] A. Messiah, “Mécanique Quantique 2”, Dunod, Paris, 1964. [83] Y.C. Sun, Private communication.
10]
ATOMIC PROCESSING OF RF SIGNALS
613
[84] V. Crozatier, G. Gorju, F. Bretenaker, J.-L. Le Gouët, I. Lorgeré, High resolution radio frequency spectral analysis with photon echo chirp transform in an Er:YSO crystal, IEEE J. Quantum Electron. 40 (2004) 1450–1457. [85] V. Crozatier, F. de Seze, L. Haals, F. Bretenaker, I. Lorgeré, J.-L. Le Gouët, Laser diode stabilisation for coherent driving of rare earth ions, Opt. Commun. 241 (2004) 203–213. [86] R.W.P. Drever, J.L. Hall, F.V. Kowalski, J. Hough, G.M. Ford, A.J. Munley, H. Ward, Laser phase and frequency stabilization using an optical resonator, Appl. Phys. B 31 (1983) 97–105. [87] M.P. van Exter, S.J. Kuppens, J.P. Woerdman, Excess phase noise in self-heterodyne detection, IEEE J. Quantum Electron. 28 (1992) 580–584. [88] L.E. Richter, H.I. Mandelberg, M.S. Kruger, P.A. McGrath, Linewidth determination from selfheterodyne measurements with subcoherence delay times, IEEE J. Quantum Electron. 22 (1986) 2070–2074. [89] P.B. Gallion, G. Debarge, Quantum phase noise and field correlation in single frequency semiconductor laser systems, IEEE J. Quantum Electron. 20 (1984) 343–349. [90] C.J. Karlsson, F.A. Olsson, Linearization of the frequency sweep of a frequency-modulated continuous-wave semiconductor laser radar and the resulting ranging performance, Appl. Opt. 38 (1999) 3376–3386. [91] K. Repasky, J.L. Carlsten, Simple method for measuring frequency chirps with a Fabry–Perot interferometer, Appl. Opt. 39 (2000) 5500–5504. [92] C. Greiner, B. Boggs, T. Wang, T.W. Mossberg, Laser frequency stabilization by means of optical self-heterodyne beat-frequency control, Opt. Lett. 23 (1998) 1280–1282. [93] B. Boggs, C. Greiner, T. Wang, H. Lin, T.W. Mossberg, Simple high-coherence rapidly tunable external-cavity diode laser, Opt. Lett. 23 (1998) 1906–1908. [94] S. Kakuma, K. Ohmura, R. Ohba, Improved uncertainty of optical frequency domain reflectometry based length measurement by linearizing the frequency chirping of a laser diode, Opt. Rev. 10 (2003) 182–184. [95] G. Gorju, V. Crozatier, V. Lavielle, I. Lorgeré, J.-L. Le Gouët, F. Bretenaker, Experimental investigation of deterministic and stochastic frequency noises of a rapidly frequency chirped laser, European Phys. J. Appl. Phys. 30 (2005) 175–183. [96] P. Juncar, J. Pinard, Instrument to measure wave numbers of cw and pulsed laser lines: the sigmameter, Rev. Sci. Instrum. 53 (1982) 939–948. [97] V. Crozatier, B.K. Das, G. Baïli, G. Gorju, F. Bretenaker, J.-L. Le Gouët, I. Lorgeré, W. Sohler, Highly coherent electronically tunable waveguide extended cavity diode laser, IEEE Photon. Tech. Lett. 18 (2006) 1527–1529. [98] R.G. Ross Jr., R.F. Boyle, R.W. Key, D.R. Coulter, NASA advanced cryocooler technology development program, in: Proc. SPIE, vol. 4850, 2003, pp. 1020–1028.
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ADVANCES IN ATOMIC, MOLECULAR AND OPTICAL PHYSICS, VOL. 54
CONTROLLING OPTICAL CHAOS, SPATIO-TEMPORAL DYNAMICS, AND PATTERNS LUCAS ILLING1 , DANIEL J. GAUTHIER1 and RAJARSHI ROY2 1 Department of Physics and Center for Nonlinear and Complex Systems, Duke University, Durham,
NC 27708, USA 2 Department of Physics and Institute for Physical Science and Technology, University of Maryland,
College Park, MD 20742, USA 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Recent Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. The ‘Green Problem’ . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Synchronizing Laser Chaos . . . . . . . . . . . . . . . . . . . . . 2.3. Optical Chaos Communication . . . . . . . . . . . . . . . . . . . 2.4. Spatio-Temporal Chaos . . . . . . . . . . . . . . . . . . . . . . . 3. Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Controlling Unstable Periodic Orbits . . . . . . . . . . . . . . . . 3.3. Controlling Unstable Steady States . . . . . . . . . . . . . . . . . 3.4. Summary of Controlling Chaos Research . . . . . . . . . . . . . . 4. Synchronization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Introduction and Connection to Control . . . . . . . . . . . . . . . 4.2. Identical Synchronization . . . . . . . . . . . . . . . . . . . . . . 4.3. Generalized Synchronization . . . . . . . . . . . . . . . . . . . . 4.4. Phase Synchronization . . . . . . . . . . . . . . . . . . . . . . . . 4.5. Synchronization Errors . . . . . . . . . . . . . . . . . . . . . . . . 4.6. Summary of Synchronizing Chaos Research . . . . . . . . . . . . 5. Communication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Chaos Communication Using Fiber Lasers . . . . . . . . . . . . . 5.3. Adverse Effects of Realistic Communication Channels . . . . . . 5.4. Minimizing the Effect of Channel Distortions on Synchronization 5.5. Summary of Chaos Communication Research . . . . . . . . . . . 6. Spatio-Temporal Chaos and Patterns . . . . . . . . . . . . . . . . . . . 6.1. Spatio-Temporal Chaos Communication . . . . . . . . . . . . . . 6.2. All-Optical Switching . . . . . . . . . . . . . . . . . . . . . . . . 7. Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615
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© 2007 Elsevier Inc. All rights reserved ISSN 1049-250X DOI: 10.1016/S1049-250X(06)54010-8
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Abstract We describe how small perturbations applied to optical systems can be used to suppress or control optical chaos, spatio-temporal dynamics, and patterns. This research highlights the fact that complex behavior, such as chaos, has a beautiful and orderly underlying structure. We demonstrate that this orderly structure can be exploited for a variety of applications, such as stabilizing laser behavior in a regime where the device would normally produce erratic behavior, communicating information masked in a seemingly noise-like chaotic carrier, and improving the sensitivity of ultra-low-light level optical switches. Keywords: chaos, synchronization, spatio-temporal chaos, transverse patterns, controlling chaos
1. Introduction Why do some lasers produce noise-like intensity spikes? What is the optimum implementation of an optical switch? Such questions have motivated scientists over the last four decades to uncover the sources of instability in nonlinear optical systems. Recently, there has been a surge in the number of researchers investigating the dynamics of optical systems because it now seems possible to tackle problems that seemed intractable two decades ago, such as the origin of optical ‘turbulence’ and controlling optical chaos. This renaissance has been spurred by our understanding of relatively simple optical systems and by advances in nonlinear dynamics, mathematics, computational physics and experimental techniques. Many laser users have been confronted by the appearance of ‘noise-like’ intensity fluctuations or ‘turbulent’ spatial patterns in the beam emitted from the laser. This type of behavior was clearly evident even during the earliest investigations of lasers in the 1960’s (Collins et al., 1960; Nelson and Boyle, 1962) where it was found that the intensity of the light generated by the ruby laser displayed irregular spiking, as shown in Fig. 1. Was the spiking due to inadequate shielding of the laser from the environment or was is it due to some intrinsic property of the laser? After many years of research on nonlinear optical systems, we now understand that the instabilities can arise from deterministic and stochastic effects that govern the interplay between the radiation field and matter. The path to this realization was rather long because it took considerable effort to uncover and model with precision the origin of the instabilities. Then, it took several years to place the research in the framework of universal behavior of dynamical systems. From the general perspective of the field of nonlinear dynamics, optical devices are fascinating because seemingly simple textbook devices, such as a single mode laser, can show exceedingly erratic, noise-like behavior that is a manifestation of
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F IG . 1. Temporal evolution of the intensity of the beam emitted by a ruby laser. From Nelson and Boyle (1962).
deterministic chaos. Deterministic refers to the idea that the future behavior of the system can be predicted using a mathematical model that does not include random or stochastic influences. Chaos refers to the idea that the system displays extreme sensitivity to initial conditions so that arbitrarily small errors in measuring the initial state of the system grow large exponentially and hence practical, long-term predictability of the future state of the system is lost. Although mathematicians have studied chaotic solutions of dynamical systems since the works of Henri Poincaré at the turn of the 20th century, the possibility of chaos in real physical systems was not widely appreciated until relatively recently. A major turning point was the seminal paper by the meteorologist Edward Lorenz (Lorenz, 1963), in which he studied the problem of convection of a fluid heated from below (a highly simplified model of the Earth’s atmosphere). Lorenz’s numerical computations revealed a totally new aspect of the behavior of this dynamical system: large, irregular fluctuations appeared to originate from an innocuous looking set of three-coupled, nonlinear ordinary differential equations without any sources of noise or fluctuations included in them. Even more surprising was the incredible sensitivity of solutions of these equations to a small difference in initial conditions. Lorenz engraved this aspect of chaotic dynamics in our minds through the title of his talk to the American Association for the Advancement of Science in 1972, “Does the Flap of a Butterfly’s Wings in Brazil Set off a Tornado in Texas?” (Lorenz, 1993). The connection between chaotic dynamics and laser instabilities was made by Hermann Haken, who, in a short paper (Haken, 1975), remarked on a beautiful similarity that he had discovered between the chaotic equations for a fluid studied by Lorenz a dozen years earlier (Lorenz, 1963), and the semi-classical equations that describe the operation of a single-mode laser.
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In the decades following the works of Lorenz and Haken, most scientists and engineers have begun to recognize that chaotic behavior is commonplace in physical, chemical, and biological systems. Early nonlinear dynamics research in the 1980s focused on identifying systems that display chaos, developing mathematical models to describe them, developing new nonlinear statistical methods for characterizing chaos, and identifying the way in which a nonlinear system goes from simple to chaotic behavior as a parameter is varied (the so-called route to chaos). One outcome of this research was the understanding that the behavior of nonlinear systems falls into just a few universal categories. For example, the route to chaos for a pendulum, a nonlinear electronic circuit, and a piece of paced heart tissue are all identical under appropriate conditions. This observation is very exciting because experiments conducted with an optical device can be used to understand some aspects of the behavior of a fibrillating heart, for example. Such universality has fueled a large increase in research on nonlinear systems that transcends disciplinary boundaries and often involves interdisciplinary or multidisciplinary research teams. In the 1990s, the focus of research shifted toward the possibility of controlling chaotic systems. The main idea behind the research on controlling chaos (Ott et al., 1990; Shinbrot et al., 1993; Gauthier, 2003) is to stabilize the unstable periodic orbits that are contained within the chaotic dynamics of a nonlinear system, or to stabilize the unstable steady state of the system. The most amazing aspect of these novel schemes to control chaotic systems is that the perturbations necessary are vanishingly small as the system approaches the dynamical state that is desired, whether it is periodic or steady. It is the very sensitivity of chaotic systems to perturbations that allows one to control them with tiny perturbations. A second topic that evolved through the 1990s is the problem of synchronization of chaotic systems. Though the synchronization of clocks (periodic systems) has been studied with great care over centuries, it was the surprising discovery of temporal synchronization between two chaotic systems that initiated the field of chaotic communications. At first, even the notion of synchronization of chaotic systems appears self-contradictory. How can two chaotic systems that inevitably start from slightly different initial conditions ever be synchronized? After all, the exponentially fast divergence of state-space trajectories from ever so slightly different initial conditions is one of the hallmarks of chaos. The crucial realization was that, when two chaotic systems are coupled to each other in a suitable way, they exert a form of control on each other, and it is possible for both systems to synchronize their dynamics, even when they start from very different initial conditions (Fujisaka and Yamada, 1983; Afraimovich et al., 1986; Pecora and Carroll, 1990). The first experimental demonstrations of synchronization of chaotic systems were done on electronic circuits (Pecora and Carroll,
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1990), and soon lead to the question: Can the chaotic dynamics of these systems be used for something practical? The possibility that chaotic lasers could be synchronized was examined very shortly after the appearance of Pecora and Carroll’s work, in an important numerical study by Winful and Rahman (1990) on a linear array of coupled semiconductor lasers. They showed that synchronization can be achieved by mutually coupling nearest-neighbor lasers through injection of their optical fields. Their work stimulated experiments a few years later on mutually coupled solid state lasers (Roy and Thornburg, 1994), as described in Section 4. Independently, Sugawara et al. (1994) studied synchronization of one-way coupled CO2 lasers. These early studies demonstrated that synchronization of chaotic optical systems could provide a means for communications through free space or optical fibers, and motivated much of the research on chaos synchronization and chaos communication. Instabilities of physical systems are not confined to the temporal domain but also arise in the spatial domain leading to such phenomena as pattern formation, spatio-temporal chaos, and turbulence. Optics addressed the issue of spatiotemporal effects much later than the more traditional fields of nonlinear chemical reactions and, especially, hydrodynamics. This is despite the fact that it was well known that optical systems exhibit spatio-temporal effects such as spontaneous pattern formation in the structure of the electromagnetic field in the planes orthogonal to the propagation direction. However, these phenomena were mostly considered undesirable and difficult to understand and control. This view changed in the 1980’s when pattern formation in laser beams became a major topic of the newly born field of transverse nonlinear optics. During this period, many different spatio-temporal phenomena were revealed and explained (Lugiato, 1994). It is the greater understanding of these intriguing phenomena that allowed recent progress in manipulating the transverse optical patterns for practical applications, such as all-optical switches. It also lead, in combination with the experience gained from temporal chaos, to recent efforts to synchronize and control spatio-temporal chaos, as discussed in Section 6. The purpose of this chapter is to review important aspects of controlling optical chaos, spatio-temporal dynamics and patterns. In this review we discuss mainly facets of our own research, but start off by briefly highlighting a few fascinating results by other groups on chaos control and synchronization that have been published recently. Subsequently, chaos control is discussed in Section 3, whereas a review of chaos synchronization is given in Section 4, and its uses for chaos communication are described in Section 5. Spatio-temporal optical systems are the focus of Section 6 and we conclude with an outlook in Section 7.
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2. Recent Examples In this section, we summarize a few recent advances on controlling optical chaos, turbulence, and patterns to demonstrate the vibrancy of the field. A quick search on a publication database using the keywords ‘optical’ and ‘chaos’ reveals that the field began around 1981 and has grown steadily; there have been consistently greater than 200 papers published per year since 1994. Unfortunately, it is not possible to describe each of these studies in this review; we only highlight a few examples to whet the reader’s appetite.
2.1. T HE ‘G REEN P ROBLEM ’ Green and blue lasers are very important light sources for CD players and information processing applications, as well as pump sources for other lasers. Since 1986, it has been know that a 532-nm (green color) Nd:YAG laser with an intercavity frequency-doubling crystal displays complex dynamics when operated at moderate to high pump rates, known familiarly as the ‘green problem’. Given the technical importance of these lasers and that many users require a steady beam, it is important to identify methods of suppressing the instability. In Section 3.2.1, we describe some of the early research that used closed-loop feedback methods for controlling the laser behavior at moderate pump rates. At higher pump rates, more complex ‘type-II’ chaos occurs where modes in two perpendicular polarization directions are active; this type of behavior has been difficult to suppress using standard chaos-control methods. Recently, Ahlborn and Parlitz (2004) have devised a new method, known as multiple delayed feedback control, that is effective in suppressing type-II chaos. In their setup (see Fig. 2(a)), they measure the intensity of the light emitted by a frequency-doubled Nd:YAG laser in two orthogonal linear polarization components. They generate a feedback error signal by comparing the intensity in each state of polarization with the intensities measured at a previous (or delayed) time. This error signal is applied to the laser through the injection current of the diode laser that serves as the pump of the Nd:YAG crystal. By appropriate adjustment of the feedback parameters (six total), they are able to obtain robust suppression of the chaotic intensity and polarization fluctuations, as shown in Fig. 2(b), where control is switched on at t = 0. Using this method, they were able to run the laser up to three times above the lasing threshold with stable behavior. The controller can be made compact so that it should be possible to integrate it in existing commercial products at low cost.
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F IG . 2. Controlling the “green problem”. (a) Experimental setup using modified delayed feedback control to suppress chaotic intensity fluctuations in a frequency-doubled Nd:YAG laser. (b) Suppression of chaos using chaos control. Temporal evolution of both linear polarization components of the beam emitted by the laser. Reprinted with permission from A. Ahlborn and U. Parlitz, Phys. Rev. Lett. 93, 264101 (2004). © 2004 American Physical Society
2.2. S YNCHRONIZING L ASER C HAOS While many laser users desire stable behavior, there are some potential applications of optical chaos. One is communicating information that is ‘hidden’ in a chaotic carrier, which requires synchronization of the chaos produced by two lasers (one in the transmitter and one in the receiver). There have been several demonstrations of synchronizing optical chaos (see Section 4); the current push is to obtain synchronization in devices that match with current telecommunication technologies or to investigate the case when there is a large distance between transmitter and receiver. Hong et al. (2004) have investigated synchronized chaos in vertical-cavity surface-emitting semiconductor lasers (VCSELs) using the setup shown in Fig. 3(a). VCSELs have many technologically important characteristics, such as low threshold current, single-longitudinal-mode operation, circular output beam profiles, and wafer-scale integration. On the other hand, the lasers tend to exhibit changes in the emission polarization and hence polarization effects need to be considered when attempting to synchronize their behavior. In their setup, chaos is induced in one laser (VCSEL1) by retro-reflecting about 20% of the beam emitted by the laser using mirror M1. By unidirectional coupling of some of the light emitted by VCSEL1 in one linear polarization component
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into the other laser (VCSEL2) and properly adjusting the state of polarization of the light, they obtain high-quality chaos synchronization. Figure 3(b) shows the temporal evolution of the light injected into VCSEL2 in the y-polarization component and the resulting light emitted by the laser in this component. It is seen that the two traces are highly correlated, indicating synchronization. Surprisingly, they find that the light emitted in the x-polarization component is anti-correlated (so-called anti-synchronization), as shown in Fig. 3(c). Note that the time scale of the chaotic fluctuations is nanoseconds and hence these devices could be used for transmitting information at data rates that are compatible with existing commercial optical telecommunication systems. Using a similar experimental system, but with bidirectional rather than unidirectional coupling, Mulet et al. (2004) have investigated what happens when the distance between the two lasers is so large that the transit time of light propagating from one laser to the other is comparable to or longer than the time scale of the chaotic fluctuations. They observe achronal synchronization (e.g., high correlation between the signals at a non-zero time lag) between the lasers once the coupling strength is greater than a critical value. The role of ‘leader’ and ‘follower’ can be switched depending on the initial conditions, even under symmetric coupling conditions.
F IG . 3. Chaos synchronization in coupled vertical-cavity surface-emitting semiconductor lasers. (a) Experimental setup. Temporal evolution of (b) the injected beam (top trace) and the receiver output (bottom trace) in one linear polarization component, and (c) the injected beam (top trace) and the receiver output (bottom trace) in the opposite linear polarization component. From Hong et al. (2004).
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These results demonstrate that it is possible to achieve synchronization of sub-nanosecond optical chaos, paving the way for development of high-bit-rate communication systems based on chaotic waveforms. 2.3. O PTICAL C HAOS C OMMUNICATION The primary schemes being considered for practical chaos communication systems are based on chaotic waveforms generated by lasers with external cavity feedback (the systems described in the previous subsection), opto-electronic feedback (Goedgebuer et al., 1998; Blakely et al., 2004b), and electro-optic feedback (Goedgebuer et al., 2002). To date, the electro-optic feedback methods have progressed most rapidly toward deployment and we briefly mention some of the most recent achievements. The electro-optic feedback devices described by Gastaud et al. (2004) consists of a 1.55-µm telecommunications laser whose output beam passes through a Mach–Zehnder electro-optic modulator, a long length of standard telecommunication fiber, and is converted to a voltage via a photoreceiver, as shown in Fig. 4(a). The voltage is amplified and is used to drive the Mach–Zehnder modulator. The source of nonlinearity is the modulator, whose intensity transmission is proportional to the sine squared of the drive voltage; the laser is merely a passive source of high-power optical radiation. The system is known as a delay dynamical system because the delay in the optical fiber is long in comparison to the response time of the modulator. What is impressive about this system is that it uses off-theshelf optical telecommunication components, can operate at data rates exceeding 1 Gbit/s, and can be readily integrated into existing underground systems. To mask information within the chaotic waveform produced by the electrooptic feedback loop, a binary message is encoded on the beam produced by an auxiliary laser using a standard telecommunication modulation protocol, where an eye-diagram for the message is shown in the top trace of Fig. 4(b). The message beam is ‘folded’ into the electro-optic chaos device using a 2 × 2 fiber coupler. Specifically, half of the message beam is coupled directly into the electro-optic feedback loop, while half of it is sent to the receiver side of the communication system. Simultaneously, half of the beam circulating in the electro-optic feedback loop is coupled out of the loop and sent to the receiver side of the system, while the other half is combined with the message and circulated around the feedback loop. The resulting masked signal that propagates over the communication channel is shown in the middle trace of Fig. 4(b), where it is seen that signal has little resemblance to the data waveform and the eye-diagram is completely closed. The receiver side of the communication system consists of an identical electrooptic feedback loop that has been split apart, as shown in the right-hand side of Fig. 4(a). A fraction of the incoming signal is sent to a photoreceiver and converted to a voltage. The rest of the signal propagates through an optical fiber,
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F IG . 4. Chaos communication using synchronized electro-optic feedback devices. (a) Experimental system setup. (b) Eye diagrams for a 3 Gbit/s pseudo-random bit sequence (upper: original; middle: encoded; lower: decoded). (c) Bit error rate versus the masking coefficient α. Reprinted with permission from N. Gastaud, S. Poinsot, L. Larger, J.-M. Merolla, M. Hana, J.-P. Goedgebuer, and F. Malassenet, Electron. Lett. 40, 898 (2004). © 2004 Institute of Engineering & Technology
which delays the signal by an amount that is identical to the delay produced by the long fiber in the transmitter, and the resulting signal is used to drive an identical Mach–Zehnder modulator. An auxiliary laser beam passes through this modulator and is converted to a voltage via a photoreceiver. This voltage is subtracted from the voltage proportional to the incoming signal. The resulting difference signal contains the original message; the chaos part of the signal is removed from the waveform with high rejection, as shown in the bottom trace of Fig. 4(b). The re-opened eye-diagram is clearly evident, indicating high quality message recovery. This type of receiver is known as an open-loop device and does not display chaos in the absence of an input signal. Gastaud et al. (2004) have also demonstrated that the bit error rate (BER) is orders-of-magnitude smaller using the chaos-based receiver versus direct detection of the masked signal, as shown in Fig. 4(c), demonstrating that a high level of signal privacy is possible using this approach.
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F IG . 5. Practical implementation of chaos-based optical communication. (a) Network topology in Athens, Greece. (b) Temporal sequence of 1 Gbit/s data (upper trace), message masked in the chaotic waveform (middle trace) and decoded message (lower trace). (c) The bit-error-rate of the encoded signal (squares), back-to-back decoded messages (circles) and decoded message after transmission through the network for two different code lengths (triangles). Reprinted with permission from Macmillan Publishers Ltd: Nature (A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C.R. Mirasso, L. Pesquera, and K.A. Shore, Nature 437, 343 (2005)). © 2004
Very recently, Argyris et al. (2005) have used two different time-delay chaotic devices to communicate high-speed digital data over 120 km of optical fiber in the metropolitan area network of Athens, Greece (a schematic of the network topology is shown in Fig. 5(a)). They independently investigated the use of an electro-optic feedback device similar to the system described by Gastaud et al. (2004) and a device based on laser external cavity feedback similar to the systems described by Hong et al. (2004) and Mulet et al. (2004). The optical fiber used for the experiments in Athens was temporarily free of network traffic, but was still installed and connected to the switches of the network nodes. The authors measured the characteristics of the fiber, such as its attenuation and chromatic dispersion, before the experiment. This allowed them, for example, to exactly counter the effects of dispersion by inserting an appropriate length of dispersion-compensating fiber at the beginning of the link. Three amplifiers were
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used, one at the transmitter, one 50 kilometers from the transmitter and one at the receiver, followed by optical filters with bandwidths of around 1 nanometer— this compensated for optical losses and removed spontaneous noise, respectively. Figures 5(b) and (c) show the message, masked message, and decoded message, and the corresponding bit-error-rate as a function of transmission data rate, respectively. These results demonstrate that chaos-based communication methods can be integrated into a real-world system and operate at high data rates with reasonably low errors. Whether this technology will be adopted depends, in part, on whether commercial vendors will be willing to incorporate ‘non-standard’ transmitters and receivers in their devices.
2.4. S PATIO -T EMPORAL C HAOS In all the previous examples described above, complexity appeared in the temporal behavior of the device. Here, we discuss recent investigations of systems that display complexity in both space and time. In optical devices, the spatial dimensions displaying complexity are usually the coordinates transverse to the propagation vector of a beam of light and hence they are sometimes referred to as arising from transverse-beam instabilities. Pastur et al. (2004) have recently investigated spatio-temporal instabilities occurring in a nonlinear optical system containing a liquid crystal light valve (LCLV). The LCLV consists of an absorptive photoconductor on one side and a mirror coated with a thin layer of liquid crystal on the other side. Light falling on the photoconductor at one spatial location changes an electric field on the opposite side of the device at the same spatial (transverse) location, thereby changing the orientation of the liquid crystal. The device, when illuminated from either side, behaves approximately as a material described by an extremely large third-order nonlinear optical susceptibility—a sensitive Kerr medium. Figure 6(a) shows the experimental setup of their oscillator, where a laser beam is split, part reflects off the reflective side of the LCLV and is redirected to the photoconductive side. Under appropriate conditions, the oscillator can display complex spatio-temporal patterns such as those shown in Figs. 6(b) and (c). The time scale of the changes in the observed pattern dynamics is of the order of one second. Using small perturbations via optical feedback, they are able to suppress spatiotemporal complexity and control the system to various regular patterns (Figs. 6(d) and (e)) or they can target a specific complex spatio-temporal pattern (Fig. 6(f)). To control the device, they measure in real time the pattern emanating from the oscillator using a video camera, compare this image to a known target pattern that is stored in a computer, and use this difference signal to drive a liquid crystal display. A portion of the laser beam passes through the liquid crystal display and is injected into the optical resonator. The observed controlled or targeted pattern
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F IG . 6. Targeting and control of spatio-temporal chaos. (a) Experimental setup. (b) Snapshot of the uncontrolled spatio-temporally complex state. (c) Space- (vertical) time (horizontal) dynamical evolution of the central vertical line of pixels from (b). (d)–(f) Examples of target patterns (top row), controlled pattern (center row), and corresponding far-field images (bottom row) for the control trials of (d) hexagonal pattern, (e) square pattern, and (f) a snapshot of the uncontrolled dynamics. Reprinted with permission from L. Pastur, L. Gostiaux, U. Bortolozzo, S. Boccaletti, and P.L. Ramazza, Phys. Rev. Lett. 93, 063902 (2004). © 2004 American Physical Society
is highly correlated with the desired pattern, with a normalized correlation coefficient exceeding 0.6 for all conditions shown in the middle panels of Figs. 6(d)–(f). Another type of spatio-temporal behavior displayed by a cavity containing a nonlinear optical material driven by a coherent field (holding beam) is known as a cavity soliton. Cavity solitons are localized intensity peaks that can form in a homogeneous background of light arising from the holding beam and are formed when focused laser pulses are injected into the cavity. The ability to turn them on and off and to control their location suggests that they can be used as ‘pixels’ for reconfigurable arrays or all-optical processing units. Recently, Barland et al. (2002) used a VCSEL that was pumped electrically to just below the laser threshold and illuminated by a holding beam. The ability to create a cavity soliton is very sensitive to the precise resonance frequency of the VCSEL. Slight manufacturing imperfections allow only the creation of cavity solitons in a narrow vertical stripe shown in Fig. 7(a) bounded by the complex intensity pattern to the left and the uniform field to the right. When a focused beam is injected into this region, a high-intensity spot is created (the dark spot in Fig. 7(b)). After the injected beam is removed, the spot reshapes slightly but remains in the same spatial location (Fig. 7(c)). They also find that additional cavity solitons can be written in the device and that the cavity solitons can be
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F IG . 7. Experimental demonstration of independent writing of a cavity soliton. The intensity distribution of the output field is shown over a 60 µm × 60 µm region in the sample center. (a) The underlying pattern when the writing beam is blocked. (b) The 15-µm-diameter writing beam (single dark high-intensity spot) is directed to the edge of the homogeneous region. The writing beam is blocked and a 10-µm-diameter spot remains stable (the cavity soliton). Reprinted with permission from Macmillan Publishers Ltd: Nature (S. Barland, J. Tredicce, M. Brambilla, L. Lugiato, S. Balle, M. Giudicl, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knodl, M. Miller, and R. Jager, Nature 419, 699 (2002)). © 2002
erased using a writing beam that has its phase shifted by π with respect to the beam that originally wrote the cavity soliton. Future work will require a method for writing and manipulating a large number of cavity solitons in a single device. This completes our brief overview of recent research. We now turn to the more detailed discussion of controlling optical chaos, spatio-temporal dynamics, and patterns.
3. Control 3.1. I NTRODUCTION As mentioned at the beginning of Section 1, chaos arising in a dynamical system can be controlled by applying appropriately designed minute perturbations to an accessible system parameter (a ‘knob’ that affects the state of the system) that forces it to follow a desired behavior rather than the erratic, noise-like behavior indicative of chaos. In greater detail, the key idea underlying most controlling-chaos schemes is to take advantage of the unstable steady states (USSs) and unstable periodic orbits (UPOs) of the system (infinite in number) that are embedded in the chaotic attractor characterizing the dynamics in state space. Figure 8(a) shows an example of chaotic oscillations in which the presence of UPOs is clearly evident with the appearance of nearly periodic oscillations during short intervals. (This figure illustrates the dynamical evolution of current flowing through an electronic diode resonator circuit described in Sukow et al., 1997.) Many of the control protocols attempt to stabilize one such UPO by making small adjustments to an accessible parameter when the system is close to the targeted UPO. Techniques for stabilizing unstable states in nonlinear dynamical systems using small perturbations fall into three general categories: feedback, non-feedback schemes, and a combination of feedback and non-feedback. In non-feedback
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F IG . 8. (a) Chaotic behavior observed in a nonlinear electronic circuit. The system naturally visits the unstable period orbits embedded in the strange attractor, three of which are indicated. (b) Closed-loop feedback scheme for controlling a chaotic system. From Sukow et al. (1997).
(open-loop) schemes, an orbit similar to the desired unstable state is entrained by adjusting an accessible system parameter about its nominal value by a weak periodic signal, usually in the form of a continuous sinusoidal modulation. This is somewhat simpler than feedback schemes because it does not require real-time measurement of the state of the system and processing of a feedback signal. Unfortunately, periodic modulation fails in many cases to entrain the UPO (its success or failure is highly dependent on the specific form of the dynamical system). The possibility that chaos and instabilities can be controlled efficiently using feedback (closed-loop) schemes to stabilize UPOs was described by Ott et al. (1990) (OGY). The basic building blocks of a generic feedback scheme consist of the chaotic system that is to be controlled, a device to sense the dynamical state of the system, a processor to generate the feedback signal, and an actuator that adjusts the accessible system parameter, as shown schematically in Fig. 8(b). In concept, closed-loop feedback works because, by observing the dynamics of the system in the neighborhood of a fixed point or periodic state, one can find the direction and amount of instability. One can then use that information to keep the system near the fixed point or periodic orbit. A (non-chaotic) example may serve to make this point clear. Imagine trying to balance a ball at the center of a saddle. The saddle surface is unstable in the direction of convexity; the ball will fall off along the sides of the saddle. The amount of instability, or how fast the ball falls off, is determined by the curvature of the saddle. In the other direction, the saddle
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F IG . 9. A segment of a trajectory in a three-dimensional state space and a possible surface-of-section through which the trajectory passes. Some control algorithms require only knowledge of the coordinates where the trajectory pierces the surface, indicated by the dots. From Gauthier (2003).
is stable; the ball returns towards the center if displaced along the ridge of the saddle. Feedback control schemes, such as the OGY algorithm, tell us essentially how to move a saddle under the ball so as to keep it balanced at the center. Once the curvature of the saddle in the unstable direction is known, one can balance the ball at the center by making observations of the position of the ball from time to time. If control is initiated when the ball is sufficiently close to the center, control can be maintained in a small neighborhood of the center with only small corrective motions. In their original conceptualization of the control scheme, OGY suggested the use of discrete proportional feedback because of its simplicity and because the control parameters can be determined straightforwardly from experimental observations. In this particular form of feedback control, the state of the system is sensed and adjustments are made to the accessible system parameter as the system passes through a surface-of-section. Figure 9 illustrates a portion of a trajectory in a three-dimensional state space and one possible surface-of-section that is oriented so that all trajectories pass through it. The dots on the plane indicate the locations where the trajectory pierces the surface. In the OGY control algorithm, the size of the adjustments is proportional to the difference between the current and desired states of the system. Specifically, consider a system whose dynamics on a surface-of-section is governed by the m-dimensional map zi+1 = F(zi , pi ),
(1)
where zi is its location on the ith piercing of the surface and pi is the value of an externally accessible control parameter that can be adjusted about a nominal value p. ¯ The map F is a nonlinear vector function that transforms a point
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F IG . 10. Chaotic evolution of the logistic map for r = 53.9. The circles denote the value of xn on each iterate of the map. The solid line connecting the circles is a guide to the eye. The horizontal line indicates the location of the period-1 fixed point. From Gauthier (2003).
on the plane with position vector zi to a new point with position vector zi+1 . Feedback control of the desired UPO (characterized by the location z∗ (p) ¯ of its piercing through the section) is achieved by adjusting the accessible parameter by an amount i = pi − p¯ = −γ n · zi − z∗ (p) (2) ¯ ¯ where on each piercing of the section when zi is in a small neighborhood of z∗ (p), γ is the feedback gain and n is an m-dimensional unit vector that is directed along the measurement direction. The location of the unstable fixed-point z∗ (p) ¯ must be determined before control is initiated; fortunately, it can be determined from experimental observations of zi in the absence of control (a learning phase). The feedback gain γ and the measurement direction n necessary to obtain control are determined from the local linear dynamics of the system about z∗ (p) ¯ using standard techniques of modern control engineering (see, for example, Ogata, 1990 and Romeiras et al., 1992), and they are chosen so that the adjustments i force the system onto the local stable manifold of the fixed point on the next piercing of the section. Successive iterations of the map in the presence of control direct the system to z∗ (po ). It is important to note that i vanishes when the system is stabilized; the control only has to counteract the destabilizing effects of noise. As a simple example, consider control of the one-dimensional logistic map defined as xn+1 = f (xn , r) = rxn (1 − xn ).
(3)
This map can display chaotic behavior when the ‘bifurcation parameter’ r is greater than ∼3.57. Figure 10 shows xn (closed circles) as a function of the it-
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erate number n for r = 3.9. The non-trivial period-1 fixed point of the map, denoted by x∗ , satisfies the condition xn+1 = xn = x∗ and hence can be determined through the relation x∗ = f (x∗ , r).
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Using the function given in Eq. (3), it can be shown that x∗ = 1 − 1/r.
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A linear stability analysis reveals that the fixed point is unstable when r > 3. For r¯ = 3.9, x∗ = 0.744, which is indicated by the thin horizontal line in Fig. 10. It is seen that the trajectory naturally visits a neighborhood of this point when n ∼ 32, n ∼ 64, and again when n ∼ 98 as it explores state space in a chaotic fashion. Surprisingly, it is possible to stabilize this unstable fixed point by making only slight adjustments to the bifurcation parameter of the form rn = r¯ + n ,
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where n = −γ (xn − x∗ ).
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When the system is in a neighborhood of the fixed point (i.e., when xn is close to x∗ ), the dynamics can be approximated by a locally linear map given by xn+1 = x∗ + α(xn − x∗ ) + βn . The Floquet multiplier of the uncontrolled map is given by ∂f (x, r) α= = r¯ (1 − 2x∗ ), ∂x x=x∗ ,r=¯r and the perturbation sensitivity by ∂f (x, r) β= = x∗ (1 − x∗ ), ∂r x=x∗ ,r=¯r
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where that fact that n = 0 when x = x∗ and r = r¯ has been used to obtain these results. For future reference, α = −1.9 and β = 0.191 when r¯ = 3.9 (the value used to generate Fig. 10). Defining the deviation from the fixed point as yn = xn − x∗ ,
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one finds that the behavior of the controlled system in a neighborhood of the fixed point is governed by yn+1 = (α + βγ )yn ,
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where the size of the perturbations is given by n = βγ yn .
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In the absence of control (γ = 0), yn+1 = αyn so that a perturbation to the system grows (i.e., the fixed point is unstable) when |α| 1. With control, it is seen from Eq. (12) that an initial perturbation shrinks when |α + βγ | < 0,
(14)
and hence control stabilizes successfully the fixed point when condition (14) is satisfied. Any value of γ satisfying condition (14) will control chaos, but the time to achieve control and the sensitivity of the system to noise is affected by the specific choice. For the proportional feedback scheme (7) considered in this simple example, the optimum choice for the control gain occurs when γ = −α/β. In this situation, a single control perturbation is sufficient to direct the trajectory to the fixed point and no other control perturbations are required if control is applied when the trajectory is close to the fixed point and there is no noise in the system. If control is applied when the yn is large, nonlinear effects become important and additional control perturbations are required to stabilize the fixed point. Figure 11 shows the behavior of the controlled logistic map for r¯ = 3.9 and the same initial condition used to generate Fig. 10. Control is turned on suddenly as soon as the trajectory is somewhat close to the fixed point near n ∼ 32 with the control gain set to γ = −α/β = 9.965. It is seen in Figs. 11(a) and (b) that only a few control perturbations are required to drive the system to the fixed point. Also, the size of the perturbations vanish as n becomes large since they are proportional to yn (see Eq. (13)). When random noise is added to the map on each iterate, the control perturbations remain finite to counteract the effects of noise, as shown in Figs. 11(c) and (d). This simple example illustrates the basic features of control of an unstable fixed point in a nonlinear discrete-time dynamical system using small perturbations. It also demonstrates the control of unstable periodic orbits since periodic orbits correspond to fixed points of the map describing the dynamics on the surfaceof-section. Over the past decade, since the early work on controlling chaos, researchers have devised many techniques for controlling chaos that go beyond the closed-loop proportional method described above. Below, we describe some of our own work on advancing the control of chaos in optical systems. 3.2. C ONTROLLING U NSTABLE P ERIODIC O RBITS In this section, we focus on closed-loop feedback methods we have investigated for controlling UPOs in optical systems. This problem is very rich because there are a very large (probably infinite) number of UPOs embedded in a chaotic attractor, as mentioned in Section 3 and seen in Fig. 8(a). Thus, it should be possible
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F IG . 11. Controlling chaos in the logistic map. (a) Proportional control is turned on when the trajectory approaches the fixed point. (b) The perturbations n vanish once the system is controlled in this example where there is no noise in the system. (c) Uniformly distributed random numbers between ±0.1 are added to the logistic map on each iterate. (d) The perturbations n remain finite to counteract the effects of the noise. From Gauthier (2003).
to switch between the various orbits just by changing the settings of the feedback loop, while always using tiny perturbations. Such control might have applications as an agile waveform generator or for transmitting information by symbolic dynamics, where each orbit corresponds to a different letter in the communication alphabet. For convenience, we define here a notation that is used in the remaining sections on chaos control. We denote the continuous-time state of the system by the m-dimensional state-vector z(t), the measured dynamical state of the system by ξ(t) = n · z(t), where n is an m-dimensional unit vector along the measurement direction, the nominal value of the accessible system parameter by p¯ and the closed-loop feedback perturbations by (t). The key to the problem of controlling chaos is to devise an algorithm that relates the perturbations (t) to the measured state of the system ξ(t) so that the system is directed to the desired state. 3.2.1. Applying Occasional Proportional Feedback Control to the ‘Green Problem’ Baer (1986) studied the generation of green light from a diode laser pumped Nd:YAG laser with an intracavity KTP crystal. He found that though the Nd:YAG laser operated in a stable steady state without the intracavity crystal, large irreg-
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ular intensity fluctuations were sometimes observed when the intracavity KTP crystal was used to generate green light from the system. Baer noted that this behavior occurred when the laser operated in three or more longitudinal modes and hypothesized that sum-frequency generation in the KTP crystal could provide mode-mode coupling that would destabilize the laser. This was not a desirable situation for proposed practical applications of the system, in optical disk readers, for example. The unstable behavior of this system soon came to be known as the “green problem.” The chaotic nature of the green laser was investigated in some detail and connected to the destabilization of relaxation oscillations (Bracikowski and Roy, 1991). Relaxation oscillations are always present in a laser; they are the result of power exchange between the atoms of the active medium and the electric field in the cavity and are normally very small in amplitude. It was found that the nonlinear coupling of the modes through sum-frequency generation resulted in the destabilization of relaxation oscillations in the green laser system. A reasonably accurate model was developed for the system, predicting many aspects of system behavior, both chaotic and non-chaotic. As may be expected, several methods were proposed and implemented to get rid of the fluctuations. These methods consisted of system modifications such as restricting the laser to operate in two orthogonally polarized modes by adding wave plates to the laser cavity (Oka and Kubota, 1988; Anthon et al., 1992) or proper orientation of the YAG and KTP crystals (James et al., 1990). These are typical examples of what has been the traditional reaction of scientists and engineers when faced with irregular fluctuations in a laser system—redesign the system so that it is inherently stable or try to find a parameter regime where chaos does not exist. A departure from this traditional mindset required a new perspective and approach towards working with chaotic systems. It was clear, soon after publication of the OGY paper (Ott et al., 1990), that it would be of great interest to try and apply these new chaos control techniques to the chaotic green laser. There was the purely scientific motivation—could one demonstrate control of a chaotic laser in an experiment and stabilize several different periodic waveforms for the same laser parameters? There was also the practical motivation—could such control techniques be used to stabilize chaotic lasers without having to redesign the system? It was at this point that one of the authors (R.R.) fortuitously happened to learn that Earle Hunt (of Ohio University) had developed an analog circuit to stabilize periodic waveforms generated by a chaotic diode resonator circuit through implementation of a variant of the OGY approach which he called Occasional Proportional Feedback (OPF) (Hunt, 1991; Hunt and Johnson, 1993). The name arose from the fact that the feedback consisted of a series of perturbations of limited duration δt (“kicks”) delivered to the input drive signal at periodic intervals (T ) in proportion to the difference of the chaotic output signal from a reference value. The OPF algorithm is shown schematically in Fig. 12.
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F IG . 12. The Occasional Proportional Feedback (OPF) algorithm used for control of the chaotic laser. The four parameters of the control circuit, T , δt, p and Iref are shown. From Roy et al. (1994).
The OPF technique appeared to be perfectly suited for an attempt to stabilize periodic orbits of the green laser, because the operation of the circuit could easily be speeded up to the microsecond time scale required for the laser. The laser intensity was detected with a fast photodiode and this signal provided the input for the control circuit. The output of the control circuit modified the injection current-of the diode laser used to pump the Nd:YAG crystal. This seemed to be the most natural and convenient choice of control parameter. To adapt Hunt’s circuit for control of the autonomously chaotic laser, Roy et al. (1992) had to supply an external timing signal from a function generator. This determined the interval T between “kicks” applied to the pump laser injection current. Even though there was no external periodic modulation responsible for the chaotic dynamics, the relaxation oscillations of the laser intensity provided a natural time scale for perturbative corrections. The interval between kicks was thus adjusted to be roughly at the relaxation oscillation period (approximately 100 kHz), or a fractional multiple of it. The period T , duration of the kicks δt, reference level Iref with respect to
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F IG . 13. Control signal and time traces of the Nd:YAG laser intensity. (a) The chaotic intensity fluctuations of the laser output without any applied control signal. (b) A period-1 orbit obtained by adjusting the interval T between “kicks” to approximately the relaxation oscillation frequency. (c) A period-4 orbit obtained by adjusting T to approximately 1/4 the relaxation oscillation frequency. (d) A period-9 orbit obtained by adjusting the interval T to approximately 6/9 of the relaxation oscillation frequency. The complex nature of the control signal is clearly visible. From Roy et al. (1992).
which the deviation of the signal is measured and the proportionality factor p to determine the amplitude of the kicks are the four parameters of the control circuit. The results of application of OPF to the laser were quite remarkable. Roy et al. (1992) were able to demonstrate stabilization of a large variety of periodic waveforms with perturbations of a few percent applied to the pump laser injection current. A typical chaotic waveform, together with several periodic waveforms stabilized in this way are shown in Fig. 13. The control signal fluctuations are shown above the intensity waveforms. The particular waveforms stabilized can be selected by changes of control circuit parameters, mainly the time period T and the reference level Iref . We note here that the laser had to be operated so as to generate very little green light for the control circuit to work successfully. The laser is “weakly” chaotic in this regime; the rate of separation of initially close trajectories in phase space is small, and there is only one direction of instability. If a significant amount of green light was generated, and the laser was highly chaotic (particularly if the laser has more than one direction of instability in phase space), the circuit was unable to stabilize the laser. 3.2.2. Extended Time-Delay Autosynchronization Occasional proportional feedback and related approaches have been very successful in controlling chaos in slow systems (characteristic time scale <1 µs), but
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scaling these schemes to significantly higher frequencies, such as those encountered in high-speed electronic or optical systems, for example, is challenging for several reasons. One important issue in high-speed feedback control of chaotic systems is the latency through the control loop, that is, the time τ between the sensing of the state of the system and the application of the control signal. The latency of the control loop is affected by the propagation speed of the signals through the components of the loop and the processing time of the feedback signal. An additional important issue is that it is difficult to accurately sample the state of the system at discrete times in order to compare it with the reference value and to rapidly adjust the control parameter on a time-scale comparable to the response time of the system. Continuous feedback schemes avoid or reduce many of these problems and hence may be useful for controlling high-speed chaos. An obvious extension of the original OGY suggestion for controlling UPOs is to use continuous adjustment of the accessible system parameter by an amount (t) = −γ n · z(t) − z∗ (t) , (15) where γ is a constant feedback gain (Pyragas, 1992; Qu et al., 1993). This scheme is not amenable for controlling the dynamics of high-speed systems, however, because it is difficult to accurately determine, store, and regenerate z∗ (t). As first suggested by Pyragas (1992), the UPOs of a dynamical system can be controlled using continuous feedback that does not require knowledge of z∗ (t). In this scheme, which we refer to as ‘time-delay autosynchronization’ (TDAS), the control perturbations are designed to synchronize the current state of the system and a time-delayed version of itself, with the time delay equal to one period of the desired orbit. Specifically, UPOs of period τ can be stabilized by continuous adjustment of the accessible parameter by an amount (t) = −γ ξ(t) − ξ(t − τ ) , (16) where γ is the feedback gain. Note that (t) vanishes when the system is on the UPO since ξ(t) = ξ(t − τ ) for all t. This control scheme has been demonstrated experimentally in electronic circuits (Pyragas and Tamasevicius, 1993; Gauthier et al., 1994) operating at frequencies of 10 MHz, corresponding to τ ∼ 100 ns. The main drawback to TDAS is that it is not effective at controlling highly unstable orbits. Socolar et al. (1994) introduced a generalization of TDAS that is capable of extending the domain of effective control significantly (Pyragas, 1995; Sukow et al., 1997) and is easy to implement in high-speed systems. Stabilizing UPOs is achieved by feedback of an error signal that is proportional to the difference between the value of a state variable and an infinite series of values of the state variable delayed in time by integral multiples of τ . Specifically, ETDAS
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∞ k−1 R ξ(t − kτ ) , (t) = −γ ξ(t) − (1 − R)
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where −1 R < 1 regulates the weight of information from the past. As we discuss below, highly unstable orbits can be stabilized as R → 1. The case R = 0 corresponds to TDAS, the scheme introduced by Pyragas (1992). We emphasize that, for any R, (t) vanishes when the UPO is stabilized since ξ(t − kτ ) = ξ(t) for all t and k, so there is no power dissipated in the feedback loop whenever ETDAS is successful. Note that no property of the UPO must be known in advance except its period. In periodically driven systems, where the period of the orbit is determined from the driving, no features of the UPO need ever be determined explicitly. The control parameters γ and R can be determined empirically in an experiment or by performing a linear stability analysis of the system in the presence of ETDAS feedback control for fixed n and p¯ (Bleich and Socolar, 1996; Just et al., 1997). Some insight into the underlying reasons for effectiveness of ETDAS in stabilizing highly unstable orbits as R → 1 can be gained using a frequency-domain analysis. To begin, note that the ETDAS feedback signal given by Eq. (17) linearly relates the input signal ξ(t) with the output signal (t); hence (ω) = −γ T (ω)ξ(ω), where ξ(ω) and (ω) are the Fourier amplitudes of the input and output signals, respectively, and T (ω) =
1 − exp(iωτ ) 1 − R exp(iωτ )
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is the transfer function for ETDAS feedback. The transfer function ‘filters’ the observed state of the dynamical system, characterized by ξ(f ), to produce the necessary feedback signal. Figure 14 shows the frequency dependence of |T (ω)| for R = 0 (TDAS) and R = 0.65 (ETDAS) where it is seen that there is a series of notches that drop to zero at multiples of the characteristic frequency of the orbit ω∗ = 2π/τ and that the notches become narrower for larger R. The existence of the notches in the transfer function can be understood easily by considering that (t), and hence (ω) must vanish when the UPO is stabilized. Recall that the spectrum ξ(ω) of the system when it is on the UPO consists, in general, of a series of δ-functions at multiples of the characteristic frequency of the orbit ω∗ = 2π/τ ; therefore, the filter must remove these frequencies (via the notches) so that (ω) = 0. The ETDAS feedback is more effective in stabilizing UPOs for larger R partly because it is more sensitive to frequencies that could potentially destabilize the UPO. The narrower notches imply that more feedback is generated for signals with frequency components slightly different from the desired set. In addition, |T (ω)| is flatter
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F IG . 14. Transfer function |T (ω)| of ETDAS feedback. The dashed line is for R = 0 and the solid line is for R = 0.65. Note that notches at ω/ω∗ = q (q = 0, 1, 2, . . .) are narrower for R = 0.65. From Sukow et al. (1997).
and remains near one between the notches for larger R, so the system is less likely to be destabilized by a large feedback response at these intermediate frequencies. We note that other transfer functions that possess notches at multiples of ω∗ could stabilize the dynamics of the UPO; however, Eq. (18) is easy to implement experimentally. For example, Sukow et al. (1997) constructed an analog circuit, consisting of high-speed video amplifiers and coaxial-cable delay lines, that implements ETDAS feedback. They were able to stabilize highly unstable UPOs in a chaotic electronic circuit, known as a diode resonator, which consists of a sinusoidally-driven series-connected switching diode, inductor, and resistor. The measured system variable was a voltage drop V (t) across the resistor and the system was driven into the chaotic regime by increasing the amplitude of the sinusoidal drive voltage beyond a critical value. Figures 15(a) and (b) show the temporal evolution of V (t) when the ETDAS feedback circuitry is adjusted to stabilize the period-1 and period-4 UPOs, respectively. It is seen that the feedback signal V (t) is a small fraction of the drive amplitude (<2×10−3 for both cases). The slight increase in V (t) for the period-4 orbit was due mainly to imperfect reproduction of the form of the ETDAS feedback signal by the analog circuitry. This effect is more prevalent for the period-4 setup because the delay line is longer and hence causes more distortion of the signals due to the dispersion and frequency-dependent loss of the coaxial cable. Figures 15(c) and (d) presents further evidence that the ETDAS feedback indeed stabilizes the UPOs embedded within the strange attractor of the diode resonator. Shown are the return maps for the uncontrolled system (light dots) and the controlled system (dark dots indicated by arrows) for the stabilized period-1
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F IG . 15. Time series data and first return maps illustrating successful ETDAS control. Temporal evolution (solid lines, scale on left) of the stabilized (a) period-1 UPO (γ = 6.2, R = 0.28) and (b) period-4 UPO (γ = 4.2, R = 0.26) along with their associated ETDAS error signals V (t) (dashed lines, scale on right), expressed as a fraction of the drive amplitude. Similar data in the form of a first return map for the controlled (c) period-1 (γ = 4.4, R = 0) and (d) period-4 (γ = 3.1, R = 0.26) trajectories are highlighted by the dark concentrations of points indicated by arrows. These maps are superimposed on maps of the uncontrolled chaotic resonator. From Sukow et al. (1997).
orbit (Fig. 15(c)) and the period-4 orbit (Fig. 15(d)). It is clear that the stabilized orbits lie on the unperturbed map, indicating that they are periodic orbits internal to the dynamics of the uncontrolled system. Finally, we note that a useful feature of ETDAS feedback is the ability to generate the error signal using an all-optical technique. Specifically, the form of the ETDAS error signal given by Eq. (17) is identical to an equation that describes the reflection of light from a Fabry–Perot interferometer consisting of two equal-reflectivity mirrors, where R = r 2 corresponds to the square of amplitudereflection-coefficient r of the mirrors, and τ corresponds to the round-trip transittime of light in the cavity. In one possible scenario, a transducer converts the measured dynamical state of the system ξ(t) into a laser beam of field strength Einc (t) that is directed toward an optical attenuator or amplifier (adjusting γ ) and
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F IG . 16. All-optical implementation of ETDAS feedback using a Fabry–Perot interferometer consisting of two mirrors with amplitude reflection coefficient r separated by a distance such that the round-trip-time of light in the cavity is equal to the period τ of the desired UPO. An optical attenuator or amplifier adjusted the feedback gain γ . From Sukow et al. (1997).
a Fabry–Perot interferometer, as shown in Fig. 16. The field Eref (t) reflected by the interferometer passes through the attenuator/amplifier and is converted to the ETDAS error signal (t) by an output transducer. It may be possible to control fast dynamics of optical systems that generate a laser beam directly, such as semiconductor lasers, using the field generated by the laser as the measured system parameter ξ(t) and as the accessible system parameter (t); no transducers are required. 3.2.3. Controlling UPOs with Large Control-Loop Latency In some situations it may be impossible to produce a feedback signal that faithfully reproduces the form of Eq. (17) due to the latency of the control loop. Latency is the time τ that it takes to measure the state of the system, determine the feedback signal, and apply the perturbation to the system. As a rough guideline, the latency becomes an issue when τ is comparable to or larger than the correlation time of the chaotic system, which is approximately equal to λ−1 , where λ is the largest (local) positive Lyapunov exponent of the desired UPO in the absence of control. (A positive Lyapunov exponent is a measure of the exponential divergence of nearby initial points in state space.) Another issue that arises in fast dynamical systems is that the time it takes for signals to propagate through the device components is comparable to the time scale of the fluctuations and hence many fast systems are most accurately described by time-delay differential equations (Ikeda et al., 1982; VanWiggeren and
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F IG . 17. (a) Schematic of a typical topology of a time-delay system consisting of a nonlinear element and a long loop connecting the output to the input of the element, introducing a total time-delay TD . (b) Schematic of feedback control that measures the state at point p1 and perturbs the system at p2 . The propagation time through the controller (control loop latency) is denoted by τ , and τ21 is the propagation time from p2 to p1 (along the path through the nonlinear element). Note that the signal takes a time TD − τ21 to get from p1 to p2 . From Blakely et al. (2004a).
Roy, 1998a). An advantageous feature of these time-delay devices is that the complexity of the dynamics can be tuned by adjusting the delay. The control of very fast chaotic systems with time scales <10 ns is difficult because of these two challenges: control-loop latencies are unavoidable, and complex high-dimensional behavior of systems is common due to inherent timedelays. Recently, Blakely et al. (2004b) described a novel approach for controlling time-delay systems even in the presence of substantial control-loop latency. This is a general approach that can be applied to any fast systems describable with time-delay differential equations. As a specific example, they demonstrated this general approach by using TDAS to stabilize fast periodic oscillations (τ ∼ 12 ns) in a photonic device, the fastest controlled chaotic system at that time. In principle, much faster oscillations can be controlled using, for example, high speed electronic or all-optical control components, paving the way to using controlled chaotic devices in high-bandwidth applications. Blakely et al. (2004a) discovered that the effects of control-loop latency can be mitigated when controlling chaotic systems consisting of a nonlinear element and an inherent time-delay TD , as shown schematically in Fig. 17(a). Chaos can
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be controlled in time-delay systems by taking advantage of the fact that it is often possible to measure the state of the system at one point in the time-delay loop (p1 ) and to apply perturbations at a different point (p2 ), as shown in Fig. 17(b). Such distributed feedback is effective because the state of the system at p2 is just equal to its state p1 delayed by the propagation time TD − τ21 through the loop between the points. The arrival of the control perturbations at p2 is timed correctly if τ + τ21 = TD .
(19)
Hence, it is possible to compensate for a reasonable amount of control loop latency τ by appropriate choice of p1 and p2 . The advantage of this approach is that the propagation time through the controller does not have to be faster than the controlled dynamics. In contrast, the conventional approach for controlling chaos is to perform the measurement and apply the perturbations instantaneously (τ → 0), which requires controller components that are much faster than the components of the chaotic device to approximate instantaneous feedback. Note that the method of computing the control perturbations has not been specified. In principle, any existing method may be used as long as they can be implemented with latency satisfying Eq. (19). To demonstrate the feasibility of controlling fast chaos using this general concept, Blakely et al. (2004a) applied it to a chaotic photonic device shown schematically in Fig. 18(a). The device consists of commercially-available components including a semiconductor laser, a Mach–Zehnder interferometer, and an electronic time-delay feedback, and can display nanosecond-scale chaotic fluctuations (Blakely et al., 2004b). The semiconductor laser acts as a simple currentcontrolled source that converts current oscillations into oscillations of the optical frequency and, to a lesser extent, into amplitude oscillations. Light generated by the laser traverses an unequal-path Mach–Zehnder interferometer whose output is a nonlinear function of the optical frequency. The light exiting the interferometer is converted to a voltage using a fast silicon photodiode and a resistor. This voltage propagates through a delay line (a short piece of coaxial cable), is amplified and bandpass filtered, and is then used to modulate the laser injection current by combining it with the dc injection current via a bias-T. The system is subject to an external driving force provided by adding an RF voltage to the feedback signal. (The driven system has more prominent bifurcations than the undriven device.) The length of the coaxial cable can be adjusted to obtain values of the time-delay TD in the range 11–20 ns. This photonic device displays a range of periodic and chaotic behavior that is set by the amplifier gain and the ratio of the time-delay to the characteristic response time of the system (typically set to a large value). Figure 19(a) shows the chaotic temporal evolution of the voltage measured immediately after the photodiode when the device is operated in the absence of control. The corresponding broadband power spectrum is shown in Fig. 19(b). The behavior of the system is well described by a delay-differential equation,
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F IG . 18. (a) Experimental setup of a chaotic time-delay device of the type shown in Fig. 17(a). (b) Experimental system with controller. The measurement point p1 is the second beam splitter of the interferometer. Perturbations are applied at p2 , an RF-power combiner. The time τ21 for a signal to propagate from point p2 to p1 is ∼3 ns. The controller contains two delay lines, the first sets τk , the period of the orbit to be controlled, the second is used to adjust the latency τ to properly time the arrival of perturbations at p2 . The state of the system is monitored through a directional coupler positioned directly after the photodiode in the delay loop of the photonic device. The control signal is measured through a directional coupler at the output of the controller. From Blakely et al. (2004a).
which they used to investigate numerically the observed dynamics (Blakely et al., 2004b). Blakely et al. (2004a) applied this control method to the photonic device using the setup shown in Fig. 18(b) by measuring the state of the system at point p1 and injecting continuously a control signal (t + τ ) at point p2 . For a given device time-delay TD , coaxial cable was added or removed from the control loop to obtain a value of τ + τ21 satisfying Eq. (19). The control perturbations were computed using the TDAS scheme described in Section 3.2.2. Specifically, (t) = γ [ξ(t) − ξ(t − τk )] where τk is a control-loop delay that is nominally set equal to the period τ of the desired orbit. Controlling the fast photonic device was initiated by setting the various controlloop time delays (τ + τ21 and τk ) and applying the output of the TDAS controller to point p2 with γ set to a low value (γ = 0.1 mV/mW). Upon increasing γ to 10.3 mV/mW, Blakely et al. (2004a) observe that (t) decreased, which they
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F IG . 19. Experimental data showing control of fast chaos. The state of the system was monitored by measuring the voltage in the delay loop before the amplifier (see Fig. 18(b)). (a) The chaotic time series of the monitored voltage in the absence of control, (b) the corresponding broad power spectrum, (c) the periodic time series of the stabilized orbit with control on, and (d) the corresponding power spectrum. From Blakely et al. (2004a).
further minimized by making fine adjustments to τk and τ + τ21 . Successful control was indicated when (t) drops to the noise level of the device. Figure 19(c) shows the periodic temporal evolution of the controlled orbit with a period of τ ∼ 12 ns. The corresponding power spectrum, shown in Fig. 19(d), is dominated by a single fundamental frequency of 81 MHz and its harmonics. The data shown in Fig. 19 demonstrates the feasibility of controlling chaos in high-bandwidth systems even when the latency is comparable to the characteristic time scales of the chaotic device (compare τ ∼ 12 ns and τ ∼ 8 ns). Blakely et al. (2004a) also inferred what would happen if p1 = p2 (the conventional method of implementing chaos control with nearly instantaneous feedback). In this case, τ21 = TD and they found that control would be effective only when τ < 0.5 ns, which was not possible using their implementation of TDAS. Note that the shortest reported control-loop latency of a chaos controller is 4.4 ns (Corron et al., 2003), much too large to control their device. In principle, faster time-delay chaotic systems can be controlled using this approach as long as the controller uses technology (e.g., integrated circuits, alloptical) that is as fast as the system to be controlled so that τ is comparable to TD . Traditional chaos control schemes require that τ be much shorter than TD , increasing substantially the cost and complexity of the controller. Note that their control approach is equally useful for non-chaotic fast and ultrafast time-delay devices, where the fast time scale makes the suppression of
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F IG . 20. Stabilization of the steady state of a frequency-doubled Nd:YAG laser. The transition towards the steady state is shown when the control signal is applied to the chaotic laser. From Roy et al. (1994).
undesired instabilities challenging (e.g., the double pulsing instability in femtosecond fiber lasers such as that described by Ilday et al. (2003).) 3.3. C ONTROLLING U NSTABLE S TEADY S TATES Most research on controlling chaos has focused on stabilizing UPOs. Of course, these experiments beg the question—can the unstable steady state (USS) of a chaotic devices, such as a laser, be stabilized by such control techniques? That is, if the laser is in a chaotic state, can one apply small corrective perturbations to obtain a stable output? This is of course the case of obvious engineering interest and practical application. 3.3.1. Stabilizing USSs Using Occasional Proportional Feedback Gills et al. (1992) found that one could indeed achieve this result. They used occasional proportional feedback control (described in Section 3.2.1) to stabilize the output of a chaotic multimode Nd:YAG laser with a nonlinear intracavity KTP crystal. Without control the diode laser pumped solid state laser system displays periodic and chaotic fluctuations of the output intensity for certain operating parameter regimes, a feature known as the ‘green problem’. Stable laser output was obtained by adjustment of two parameters of the controller: the reference level Iref was set to the mean of the chaotic fluctuations and the period T was chosen to match the relaxation oscillation period. The control voltage fluctuations became extremely small once the steady state was controlled.
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F IG . 21. OPF combined with a tracking algorithm stabilizes the USS of a chaotic Nd:YAG laser. (a) No control: The symbols designate stable steady-state (•), periodic (◦), and chaotic (∗) behavior of the laser. Stable operation is obtained only very near to threshold; for higher pump powers, a complex sequence of periodic and chaotic behavior is found. (b) With control: the regime of stable operation was extended from about 20% above threshold to more than 300% above threshold. From Gills et al. (1992).
In Fig. 20 is shown the transient behavior of the laser intensity fluctuations as they are reduced to small fluctuations about the steady state as well as the control signal fluctuations during the stabilization process. If the control parameters are fixed and the pump power of the laser is increased or decreased after the steady state is stabilized, the control signal fluctuations increase rapidly, and control is lost as the laser goes into periodic or chaotic oscillations. Clearly, one needs to change the control circuit parameters as the laser pump power is changed. A procedure called “tracking” accomplishes this change of control parameters in a systematic fashion. Gills et al. (1992) applied such a tracking procedure to the frequency-doubled Nd:YAG laser; the control circuit parameters are varied so as to minimize the control signal fluctuations at each value of the pump power, which is increased in small increments. Their experiment was an illustration of the general algorithms for tracking periodic orbits developed by Schwarz and Triandaf (1992) and Carroll et al. (1992). Figure 21(a) shows that without the control circuit the laser intensity lost stability at a pump power of about 25 mW, and either periodic or chaotic fluctuations were observed for higher pump powers. Figure 21(b) shows that a combination of stabilization and tracking could maintain a stable steady state as the laser pump power is increased from threshold (21 mW) to more than three times above threshold (about 80 mW). As we noted in Section 3.2.1, the laser is highly chaotic and stabilization of the USS becomes very challenging for even larger pump powers. Stabilization of this highly chaotic regime (‘type-II’ chaos) was achieve recently by Ahlborn and Parlitz (2004) using a new method, known as multiple delayed feedback control, as discussed in Section 2.1.
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3.3.2. Controlling USSs Using Extended Time-Delay Autosynchronization In a subsequent investigation of the OPF feedback scheme for stabilizing USSs, Johnson and Hunt (1993) determined that the only part of the feedback signal responsible for control was proportional to the derivative of ξ(t). In a separate set of experiments, Bielawski et al. (1993) stabilized the dynamics of a fiber laser, and Parmananda et al. (1994) controlled an electrochemical cell using a feedback signal of the form dξ(t) (20) , dt in which ξ(t) was taken to be the laser intensity. We note that modern control engineers use derivative control to stabilize non-chaotic systems. Derivative control has several key advantages over previous schemes for controlling USSs: it eliminates the need for a reference state with which to compare the system’s current behavior, and it offers the possibility of stabilizing fast systems because it can be generated easily using analog electronics. Despite these advantages, derivative control is not perfect in an experimental setting because it can accentuate fluctuations that are often unrelated to the unstable behavior inherent in the nonlinear system. To understand the origin of the noise sensitivity, it is useful to consider the effects of the controller in the frequency domain. For feedback of the form of Eq. (20), T (ω) = iω. It is apparent that derivative control will be very sensitive to high-frequency information contained in ξ(ω) because |T (ω)| → ∞ as ω → ∞. Because the spectrum of the chaotic fluctuations is often band limited, the controller may feedback a high-frequency signal arising from random fluctuations (noise) in the system rather than the chaotic fluctuations. This effect can lead to high-frequency instabilities and large power dissipation in the feedback loop. Chang et al. (1998) proposed a control scheme that efficiently stabilizes USSs while avoiding the disadvantages of derivative control. It is based on a limiting form of the ETDAS scheme given by Eq. (17), which can be motivated most easily in the frequency domain. The ETDAS transfer function is given by Eq. (18) and shown in Fig. 14 for two different values of the parameter R. Recall, from Section 3.2.2 that the notches in the transfer function correspond to the location of the δ-functions of the UPO spectrum. The filtering action of the ETDAS feedback must remove these frequencies (via the notches) so that (ω) = 0. Based on this reasoning, an appropriate transfer function for stabilizing USSs consists of a notch at ω = 0 because the spectrum of a USS is a single δ-function at ω = 0. All other frequencies are passed by the filter with equal weight, thereby generating a negative feedback signal from this information. We can obtain this transfer function by decreasing the value of τ , thereby increasing the distance between notches, while simultaneously increasing the value of R in order to keep the controller sensitive to frequencies near ω = 0. Placing this discussion on a (t) = −γ
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rigorous foundation, we rewrite Eq. (17) in an equivalent, recursive form as (t) = −γ ξ(t) − ξ(t − τ ) + R(t − τ ), (21) and perform the limits τ → 0 and R → 1 with (1 − R)/τ ≡ ω0 held finite, thereby obtaining a new feedback relation given by dξ(t) d(t) = −ω0 (t) − γ , (22) dt dt which is effective for stabilizing USSs. The feedback signal prescribed by Eq. (22) is identical to a single-pole high-pass filter familiar from elementary electronics. We note that ETDAS feedback in the form of Eq. (17) with τ small but finite and R near to but less than 1 does not necessarily stabilize USSs (Ahlborn and Parlitz, 2004). Similar to derivative feedback, the feedback scheme given by Eq. (22) does not require a reference state. This is advantageous in situations where it is difficult to obtain an accurate reference state such as in high-speed dynamical systems, but it may lead to complications when the system possesses more than one USS with similar properties because the controller cannot explicitly select one stable state from the others. Selection of specific states may be possible if the basin of attraction for the states in the presence of control are distinct, or if the domain of feedback parameters that successfully stabilize the USSs are distinct. A linear stability analysis a dynamical system in the presence of EDTAS control given by Eq. (22) further reveals that it can stabilize only a certain class of USSs. Consider a system whose evolution is governed by dz = F(z, p), (23) dt possessing an USS z∗ . Introduce a new variable x ≡ z − z∗ and linearize the dynamics about the USS so that dx = Ax + B, (24) dt where A = ∂F/∂x|p=p¯ is the Jacobian of the system in the absence of control, and B = ∂F/∂p|p=p¯ quantifies the manner by which the control signal affects the state of the system. The combined dynamics of the system and the controller can be stated succinctly as dy = Gy, dt where the (N + 1)-dimensional expanded state vector is given by x y= ,
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where T denotes the transpose of the vector. The system (25) is stable if and only if G possesses eigenvalues whose real parts are negative. The values of the control parameters γ , ω0 , and B rendering the system stable can be determined by calculating explicitly the eigenvalues of the matrix or by using the Routh–Hurwitz stability criterion (Ogata, 1990) that does not require an explicit calculation of the eigenvalues of G. Briefly, the Routh–Hurwitz criterion imposes constraints on the values of the characteristic polynomial coefficients of G. Chang et al. (1998) successfully stabilized the USSs of an electronic circuit and found very good agreement between the experimental observations and these theoretical predictions. We also use this approach below when studying the stabilization of USSs in a laser model. While predictions regarding the stability of a USS under feedback control requires a detailed analysis of the eigenvalues or characteristic polynomial, it is possible to make some general statements regarding the inability of the controller to stabilize certain classes of USSs. First proposed by Ushio (1996), the main idea of this type of argument is to show, by an ingenious application of the intermediate value theorem, that if the Jacobian of the uncontrolled system linearized around the target dynamics (matrix A) has an odd number of positive real eigenvalues, then control is not possible. Ushio (1996) showed this limitation of controllability for the case of stabilizing unstable periodic orbits (UPOs) in discrete-time systems using delayed-feedback controllers (TDAS). Nakajima (1997) extended this type of reasoning to show that the same limitations apply to TDAS-control of UPOs in continuous-time systems (ODEs). Chang et al. (1998) showed that the argument also holds for USS of ODEs, where control using feedback of the form given by Eq. (22) fails for the case when the associated matrix A has an odd number of positive real eigenvalues. Finally, Chang et al. (1998) noted that Eq. (25) can be written in a slightly dif"T has terms proportional "T , where the matrix K ferent form in which G = " A+" BK only to the feedback parameters γ and ω0 . This decomposition facilitates the use of modern control engineering techniques such as pole-placement and tests of controllability and observability (Ogata, 1990). We do not pursue this approach since, in general, these techniques require access to all dynamical variables or that they be reconstructed using mathematical models, an approach that is not in the spirit of most controlling chaos research. Moving on to an example from optics, Gauthier (1998) demonstrated that feedback of the form prescribed by Eq. (22) may be ideally suited for stabilizing the USSs of lasers. For an idealized laser model, it was found theoretically that there exists a wide range of feedback parameters giving rise to stable behavior (known
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as the domain of control) and the controller can automatically track slow variation or drift of the laser parameters. While it is well known that this idealized laser model does not describe quantitatively the behavior of typical lasers, these observations highlight the potential of the feedback schemes. Detailed studies of specific laser systems must be undertaken to ascertain whether the controlling chaos techniques will be useful in real-world applications. For simplicity, consider a single mode, homogeneously broadened, resonant, two level laser whose dynamics is governed by the following set of dimensionless equations dE = −κ(E − υ + Einj ), dt dυ = −(υ − Ew), dt dw = −γ (w + Eυ − wp ), dt
(28)
where E is the laser field strength inside the cavity, υ (w) is the atomic polarization (population inversion), κ (γ ) is the cavity (atomic inversion) decay rate, wp is the inversion due to the pumping process in the absence of a field, and Einj accounts for the possibility of injecting a field into the cavity. In Eqs. (28), the field strength is normalized to its saturation value, the inversion to its value at the first laser threshold, the polarization to its value for a field at the saturation strength, and time to the inverse of the polarization decay rate. The dynamical state vector in state space is denoted by z = [E, υ, w]T . For low pump rates (wp < 1) the laser is below threshold and resides on the stable steady state z0∗ = [0, 0, wp ]T . At wp = 1, the state z0∗ becomes unstable via a pitchfork bifurcation (the first laser threshold) and, for slightly higher pump rates, the system resides on either of two symmetric steady-states T , where E = [±E , ±υ , 1] = υ = wp − 1. For a bad cavity z± ss ss ss ss ∗ (κ > γ + 1) and high pump rates these steady states are destabilized via a Hopf bifurcation (the second laser threshold) and the system fluctuates in a stable periodic or chaotic manner. For example, Fig. 22 shows a projection of the chaotic attractor for wp = 17, κ = 4, and γ = 0.5. The locations of z± ∗ are indicated by the solid dots. The practical goal of the controller is to stabilize either of these two states, corresponding to the case when the laser is generating a powerful, continuous-wave beam devoid of erratic intensity or frequency fluctuations. Fortuitously, feedback of the form given by Eq. (22) stabilizes always one of the 0 lasing USS (z± ∗ ) because it is incapable of stabilizing the non-lasing USS (z∗ ), since this state is destabilized via a pitchfork bifurcation in the absence of control. In the first feedback scheme investigated by Gauthier (1998), called ‘incoherent control’, the pump rate is adjusted about its nominal value wp by a feedback
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F IG . 22. Chaotic attractor of the single mode laser. From Gauthier (1998).
F IG . 23. (a) Possible realization of the ‘incoherent’ control scheme with its (b) associated domain of control. The USS is unstable in the absence of control at wp = 12. From Gauthier (1998).
signal proportional to the high-pass-filtered intensity I = Tm E 2 of the beam generated by the laser and sensed by a square-law detector, where Tm is the intensity transmission coefficient of the laser output coupler, and I is normalized to the saturation intensity. A schematic of one possible realization of this feedback technique is shown in Fig. 23(a). The pump rate in the presence of feedback is given by wp = wp + (t) where dI d = −ω0 + γi . (29) dt dt The control law Eq. (29) is of the form of Eq. (22). For simplicity, Eq. (29) does not account for the control loop latency discussed in Section 3.2.3.
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The stability of the USSs in the presence of feedback can be determined using the linear stability analysis outlined above. The laser dynamics (28) is linearized about the desired state and the range of feedback parameters ω0 and γi rendering the state stable is determined using the Routh–Hurwitz criterion. Note that the local linear dynamics of the laser can be determined directly from experimental measurements. It is found that stabilization of the USSs is not overly sensitive to the parameter ω0 so long as it is much smaller than the characteristic frequency of the chaotic fluctuations. For a narrow filter (ω0 κ, γ , 1), the approximate − domain of control for either state z+ ∗ or z∗ is given by 1 > γi > 1 − (1 + γ + κ)(κ + wp )/2κ(w p − 1).
(30)
The precise domain (between the solid lines) is determined for κ = 4, γ = 0.5, ω0 = 0.1, and Einj = 0 and compared to the approximate domain (between the dashed lines) in Fig. 23(b). It is seen that control is possible for all wp and the two results are in close agreement. Note that selecting between the states z+ ∗ and z− ∗ is accomplished by the choice of initial conditions. This analysis is applicable only for the case when the trajectory of the chaotic system is in a neighborhood of the USS, whereas the chaotic trajectory never visits this neighborhood as seen in Fig. 22. Direct numerical integration of Eqs. (28) and (29) indicates that the states are globally stable in the presence of feedback. However, the size of the transient control perturbations can often attain unphysical values. A method for circumventing this problem is suggested by Fig. 23(b). A careful inspection of the figure reveals that the domain of control encompasses a horizontal band in the approximate range 0.9 > γi Tm > 0.3, implying that the states can be controlled for all wp without adjustment of γi . Thus, the control loop can automatically track slow changes or drift in the pump rate so long as the loop can adiabatically follow these changes (i.e., slow in comparison to the response time of the loop ∼ω0−1 ). The procedure for stabilizing the USS using only small perturbations is to turn on control when the pump rate is low so that the USSs z± ∗ are stable in the absence of control, set γi in the range between 0.3 and 0.9, then slowly adjust wp to the desired value. In the second feedback scheme, called ‘coherent control’, the control perturbation is an optical field Einj injected into the laser whose nominal value is zero. It is generated all-optically by filtering a fraction of the optical field emitted by the laser with a short, high-finesse Fabry–Perot interferometer as shown schematically in Fig. 24(a). When the interferometer is adjusted so that one of its longitudinal modes coincides with the optical carrier frequency of the laser and its free-spectral-range is much larger than the spectral content of the chaotic fluctuations, the injected field is governed approximately by dE dEinj = −ω0 Einj + γc Tm , dt dt
(31)
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F IG . 24. (a) Possible realization of the ‘coherent’ control scheme with its (b) associated domain of control. From Gauthier (1998).
where an optical attenuator or amplifier in the beam path adjusts the feedback parameter γc and the cavity length and finesse adjusts the cut-off frequency ω0 . Note that Einj vanishes when control is successful; this distinguishes the controlling chaos technique from other methods of frequency locking and narrowing of lasers using a field reflected from a Fabry–Perot interferometer. The stability of the USSs in the presence of coherent feedback can be determined using the methods outlined in the previous discussion on incoherent control. The exact domain of control is shown in Fig. 24(b) for κ = 4, γ = 0.5, − ω0 = 0.1, and ε = 0. For a narrow filter, the domain for either state z+ ∗ or z∗ is given approximately by γc >
1 κ(2γ + 1)/(γ + 1) + γ + 1 − 4γ wp κ/(γ + 1) − 1 2κ ( 1/2 + (γ + 1)2 − κ 6γ2 + 4γ − κ − 2 (γ + 1)2 . (32)
The approximate result is indistinguishable from the exact result shown in Fig. 24. As with incoherent control, it is seen that stabilization of the USS is effective for arbitrarily large pump rates. Also, the procedure for stabilization of a USS using only small perturbations is to turn on control when the pump rate is low so that the USSs z± ∗ are stable in the absence of control, set γc to a large enough value (determined from Fig. 24(b)), then slowly adjust wp to the desired value. 3.4. S UMMARY OF C ONTROLLING C HAOS R ESEARCH In this section, we described methods for controlling complex periodic orbits (UPOs) and steady state behaviors (USSs) in optical systems. This research has taught us a lot about the underlying structure of chaotic systems and that there is, in fact, considerable order in chaos—the skeleton of UPOs and USSs. This order can be taken advantage of for potentially practical benefit. It serves as the structure on which chaotic dynamics can be controlled using small perturbations.
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4. Synchronization 4.1. I NTRODUCTION AND C ONNECTION TO C ONTROL Huygens made the acute observation that two slightly out of step pendulum clocks synchronized when placed close to each other, but not when kept far apart (Sargent III et al., 1974). This discovery of the entrainment of coupled clocks has been seminal in the development of synchronized oscillators of all types. Synchronization of chaotic systems may be viewed as a generalization of the entrainment or synchronization of Huygens’ coupled clocks. Now, the synchronized motion considered is not periodic, but chaotic. Just as is the case for Huygens’ clocks, two uncoupled chaotic systems will, after some transient time, evolve on their individual attractors in state space. The location of one has no relation to the other, which is particularly true for chaotic systems because of the extreme sensitivity to initial conditions. What happens if we couple them? Since coupling can be viewed as a form of control, it is not entirely surprising that coupling synchronizes the systems in some cases. That is, coupling can “lock” one system to the other, so that knowledge of the state of one system allows the prediction of the state of the other system. In a loose sense, we talk about chaos synchronization when some dynamic property that is uncorrelated when comparing two uncoupled chaotic oscillators becomes correlated when we couple the systems. Each of the systems remains chaotic even in the coupled case, but knowledge about one oscillator gives us predictive power about the second one. Whereas it is the goal of chaos-control schemes to stabilize UPOs or USSs using small perturbations, synchronization can be viewed as an attempt to use small perturbations to control a dynamical system to a particular chaotic trajectory. For example, it is possible, in some cases, to achieve a replication of chaotic oscillations through the transmission of a single scalar signal from one high-dimensional dynamical system (the transmitter) to another high-dimensional dynamical system (the receiver). Furthermore, the coupling can be chosen such that the perturbations of the receiver-dynamics are zero when the receiver replicates the chaotic oscillations of the transmitter. In this sense, chaos synchronization may be viewed as a generalization of chaos control. Chaos synchronization is also intimately linked with the observer problem in the control literature (Nijmeijer, 2001). This connection arises from asking: How does one construct a synchronizing system (or observer) for a given chaotic system? The purpose of this construction is to allow the dynamic state of the given chaotic system, which typically has ‘inaccessible’ degrees of freedom, to be determined using the synchronized system (the observer). Several types of chaos synchronization can be distinguished depending on the dynamic property that is investigated. In the following, we use our own work to
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illustrate the main types of synchronization, namely identical synchronization, generalized synchronization, and phase synchronization. (The interested reader may find further information in the book by Pikovsky et al. (2001) and the articles of Pecora et al. (1997b) and Boccaletti et al. (2002). Also, Uchida et al. (2005) provide a thorough review of chaos synchronization in optical systems.)
4.2. I DENTICAL S YNCHRONIZATION Consider, as an example, a pair of identical and unidirectionally-coupled chaotic oscillators described by ordinary differential equations dxT (33) = F(xT ), dt dxR = F(xR ) + cK(xR − xT ), (34) dt where xT (t) denotes the n-dimensional state-space variable of the transmitter (or drive) system and xR (t) the n-dimensional state-space variable of the receiver (or response) system, F is the nonlinear function governing the flow of a single oscillator, K is an n × n coupling matrix, and c is the scalar coupling strength. Energy is dissipated in realistic devices. Therefore, the transmitter (and receiver) is modeled as a dissipative chaotic system. In this context, dissipative means that the volume of a ball of initial conditions in n-dimensional state-space will contract as the system evolves in time according to Eq. (33). As a consequence, the ball of initial conditions will be mapped, asymptotically, onto an attracting limit set, i.e. the chaotic attractor. In contrast, being chaotic, which is synonymous with extreme sensitivity to initial conditions, means that the distance between any two points in the ball of initial conditions will grow exponentially with time. This implies a stretching of the ball of initial conditions in at least one direction as the system evolves in time according to Eq. (33). Thus, the evolution of a dissipative chaotic system in its state-space is characterized both by contraction in some directions and by stretching in other directions. The average rate of contraction and stretching is quantified by the Lyapunov exponents of the system (for a technical definition of Lyapunov exponents see Eckmann and Ruelle, 1985). For a chaotic system at least one Lyapunov exponent is positive, while the sum over all exponents is negative because of dissipation. Lyapunov exponents are useful quantities for characterizing chaotic systems because they are topological invariants, i.e. their values are independent of the chosen coordinate system. Furthermore, numerical methods to determine the exponents exist (Benettin et al., 1980; Eckmann and Ruelle, 1985). In Eqs. (33) and (34), if transmitter and receiver are uncoupled (c = 0), then the chaotic attractor in the combined 2n-dimensional state space is the product of the attractors of transmitter and receiver in their respective n-dimensional state
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F IG . 25. Projection of the six-dimensional state space of two coupled Lorenz oscillators. The motion of the synchronized system takes place on a chaotic attractor, which is confined to a hyperplane (synchronization manifold) defined by xR = xT .
spaces. The 2n-Lyapunov exponents of the combined system are those of, for example, the chaotic transmitter system with each value appearing twice. If transmitter and receiver are coupled (c = 0), then the attractor in the 2n-dimensional state space will not be a simple product anymore and chaos synchronization becomes possible. Identical synchronization (sometimes also referred to as complete synchronization) occurs when xT (t) = xR (t)
(35)
is a stable solution of Eqs. (33) and (34). Geometrically, condition (35) implies that the attractor of the combined transmitter-response system in its 2ndimensional state space is confined to an n-dimensional hyperplane (or synchronization manifold), as depicted in Fig. 25 for the case of two coupled Lorenz systems. The stability of the synchronous solution (or, equivalently, the synchronization manifold) can sometimes be proven analytically, e.g., by using Lyapunov functions. In general, however, the local stability has to be determined numerically. One method is to compute the Lyapunov exponents; synchronization occurs if and only if all transverse Lyapunov exponents are negative (Pecora and Carroll, 1990). The transverse Lyapunov exponents (also known as condi-
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tional Lyapunov exponents) are the average rate of exponential expansion or contraction in directions transverse to the synchronization manifold. In the case of unidirectionally-coupled identical transmitter and receiver systems, there are precisely n-transverse Lyapunov exponents. The remaining n-Lyapunov exponents characterize the chaotic dynamics within the synchronization manifold and have values identical to those of the chaotic transmitter. The first experimental demonstration of identical synchronization in optical systems used Nd:YAG lasers (Roy and Thornburg, 1994). Since then, identical synchronization has been demonstrated in many optical and opto-electronic systems, including fiber ring lasers, which we discuss next. In fiber lasers, the optical gain is provided by rare-earth elements (such as erbium, neodymium and ytterbium) embedded in silica fiber. Under optical pumping, those atoms provide light amplification at a characteristic wavelength; in the particular case of erbium that wavelength is of the order of 1.55 µm, which lies within the spectral region of minimal loss of silica fibers. For that reason, erbiumdoped fiber amplifiers have been used widely since the mid 1990s in fiber-optics communication systems. Laser emission is obtained in the system by providing feedback through closing the fiber on itself using a piece undoped silica fiber. A defining characteristic of such fiber ring lasers is that, due to the waveguiding properties of optical fiber, their cavities can be very long (on the order of tens of meters), orders of magnitude longer than most other lasers. For that reason, the frequency separation between consecutive longitudinal modes is very small (on the order of ∼1 MHz). Additionally, the amorphous character of the host medium leads to a very broad gain profile (on the order of hundreds of GHz). As a consequence, a large number of longitudinal cavity modes can experience gain and coexist inside the cavity, coupled through gain sharing. Hence, fiber lasers usually operate in a strongly multimode regime, and, consequently, their dynamics cannot be described in general by single-mode models. For fiber lengths of tens of meters, the round-trip time taken by the light to travel once along the laser cavity is of the order of hundreds of nanoseconds. It is therefore natural to use a modeling approach based on delay-differential equations, which was done in a series of papers with increasingly detailed description of the laser physics (Williams and Roy, 1996; Williams et al., 1997; Abarbanel and Kennel, 1998; Abarbanel et al., 1999; Lewis et al., 2000). For example, in Abarbanel and Kennel (1998) and in Lewis et al. (2000), the behavior of two coupled erbium-doped fiber ring lasers operating in a high-dimensional chaotic regime was investigated, showing that the two lasers become identically synchronized if sufficient light from the first laser is injected into the second. The transmitter laser is modeled schematically as ET (t + τR ) = M wT (t), ET (t) , (36)
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2 dwT (t) = Q − 2γ wT (t) + 1 + ET (t) eGwT (t) − 1 /G , (37) dt where ET (t) is the complex envelope of the electric field, measured at a given reference point inside the cavity, and wT (t) is the averaged population inversion of the nonlinear medium. The propagation round-trip time around the cavity is τR , and M denotes the mapping of the electric field from time t to time t + τR . The mapping M(w, E) takes into account birefringence, group velocity dispersion, gain dispersion, nonlinear polarization effects associated with the Kerr term, and the contribution of the pump light that is injected into the cavity (Lewis et al., 2000). In the equation for the population inversion (Eq. (37)), Q is the pumping strength, G the overall gain and γ is the decay time of the atomic transition. The receiver is an identical copy of the transmitter and is described by ER (t + τR ) = M wR (t), cET (t) + (1 − c)ER (t) , (38) dwR (t) = Q − 2γ wR (t) + 1 dt 2 (39) + cET (t) + (1 − c)ER (t) eGwR (t) − 1 /G . When c = 0, the lasers are uncoupled and run independently. If all of the physical parameters in the two laser subsystems are identical, the electric field in each laser visits the same chaotic attractor. However, ET (t) and ER (t), as well as wT (t) and wR (t), are uncorrelated due to the instabilities in the state-space of the system. As c increases away from zero, Lewis et al. (2000) find numerically that the lasers asymptotically achieve identical synchronization, ET (t) = ER (t), and wT (t) = wR (t), for a certain minimum coupling (as low as ccrit ∼ 1.3 × 10−3 ) even though the population inversions are not physically coupled. Identical synchronization is most easily demonstrated for the case c = 1, which is commonly referred to as open loop configuration. Inspection of Eqs. (36)–(39) shows that ET (t) = ER (t) and wT (t) = wR (t) is always a solution. Identical synchronization happens when this solution is asymptotically stable such that starting the transmitter and receiver from different initial conditions their dynamics will converge. To demonstrate asymptotic stability, consider Eqs. (37) and (39) with c = 1. It can be shown that any initial difference between the population inversion of the transmitter and the receiver evolves according to the equation 2 d[wT (t) − wR (t)] = −2γ wT (t) − wR (t) + ET (t) eGwR (t) dt × eG[wT (t)−wR (t)] − 1 /G .
(40)
Since ex − 1 x for any real x, 2 d[wT (t) − wR (t)] −2γ wT (t) − wR (t) 1 + ET (t) eGwR (t) , dt
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so that |wT (t) − wR (t)| tends to 0 faster than e−2γ t . Using the exponential convergence of the receiver’s population inversion to that of the transmitter, it is straightforward to see that the maps M for receiver and transmitter become identical, and therefore ET (t) − ER (t) tends to 0 also. Thus, in the open loop configuration two erbium-doped fiber ring lasers synchronize identically. Experimental demonstrations of the synchronization between chaotic fiber lasers exist. The first experimental observation of this phenomenon was made by VanWiggeren and Roy (1998a) using the experimental setup shown in Fig. 26(a). The transmitter is an erbium-doped fiber ring laser, whose dynamical regime can be manipulated by means of an intracavity polarization controller. The laser is set to operate in a chaotic regime, from which 10% of the intracavity radiation is extracted via a 90/10 output coupler and transmitted to the receiver. Half of this transmitted signal is injected into an erbium-doped fiber amplifier (EDFA) whose physical characteristics (fiber length, dopant concentration, and pump diode laser) are matched as closely as possible to those of the transmitter EDFA. It should be noted that the receiver output is not reinjected back into itself, i.e. the system operates in an open loop. A second 90/10 coupler in the transmitter cavity allows for the introduction of an external signal, whose role is discussed in Section 5, in the context of chaotic communications. Plot A in Fig. 26(b) shows a sample time trace of the transmitted field in the absence of an external signal. The state-space reconstruction of the dynamics displayed in plot A is shown in plot B and exhibits no low-dimensional structure. The signal detected by photodiode B (see experimental setup in Fig. 26(a)) after passing through the receiver s EDFA is shown in plot C. This signal has been shifted in time an amount τ = 51 ns, equal to the time mismatch corresponding to the fiber length difference between the paths leading to photodiodes A and B. Visual inspection already indicates that the time series of the transmitter and receiver are very similar. The existence of synchronization is confirmed by plotting the output of the receiver (again shifted 51 ns) versus that of the transmitter; the resulting straight line with slope unity in this synchronization plot is a clear indicator of the occurrence of synchronization. 4.3. G ENERALIZED S YNCHRONIZATION Identical synchronization is easy to recognize in experiment. However, it is a very restrictive definition of synchronization, which, for instance, doesn’t allow the discussion of synchronization of non-identical systems. A first logical step toward relaxing the definition of synchronization is to require a (static) functional relationship of the two coupled chaotic systems instead of identity: xR (t) = F xT (t) . (42) This type of chaos synchronization was discussed for two mutually coupled systems in Afraimovich et al. (1986), and has been introduced for unidirectionally
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F IG . 26. (a) schematic experimental setup to observe synchronization of fiber-laser chaos. (b) experimental results showing the transmitter (A) and receiver (C) outputs, the state-space reconstruction plot of the transmitted signal (B) and the synchronization plot (D). From VanWiggeren and Roy (1998a).
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coupled systems under the name of generalized synchronization by Rulkov et al. (1995). We restrict ourselves here to the discussion of unidirectionally coupled systems, where we have an autonomous transmitter (or drive system) and a receiver (or response system), which are characterized by their respective statespace variables xT (t) and xR (t). Generalized synchronization has been shown to exist through predictability (Rulkov et al., 1995) or the existence of a functional relationship (Pecora et al., 1997a; Brown, 1998) between the transmitter and receiver. These approaches are often difficult to implement in experimental measurements due to the presence of noise and lack of precision in measurements. When replicas or duplicates of the receiver are available, the auxiliary-system method introduced by Abarbanel et al. (1996) can be used for detecting generalized synchronization. In this method, two or more receivers are coupled with the transmitter. If the receivers, starting from different initial conditions, display identical synchronization between them (after transients have disappeared), one can conclude that the receiver signal is synchronized with the transmitter in a generalized way (Kocarev and Parlitz, 1996a; Rulkov et al., 2001). Most examples of generalized synchronization discussed in the literature consist of situations where the transmitter and receivers are different from each other, or those involving the same system operated at different parameter values (Lewis et al., 2000). One may expect that strongly coupled identical systems with similar parameter values display identical synchronization, if they synchronize at all. However, generalized synchronization of chaos can occur with identical transmitter and receivers with similar parameter values. This was shown by Uchida et al. (2003) (also McAllister et al., 2004), who used a two-longitudinal-mode Nd:YAG microchip laser as a laser source, as shown in Fig. 27(a). The total intensity of the laser output is detected by a photodiode and the voltage signal is fed back into an intracavity acousto-optic modulator (AOM) in the laser cavity through an electronic low pass filter with an amplifier. The loss of the laser cavity was modulated by the self-feedback signal through the AOM, which induced chaotic oscillations. Temporal waveforms of the laser output were measured by a digital oscilloscope and stored in a computer for later use as a drive signal. In order to test for synchronization, the same laser was used as a receiver, which ensured identical parameter settings between transmitter and receiver. The drive signal was sent to the AOM in the same laser cavity using an arbitrary waveform generator connected to the computer. The original feedback loop was disconnected (dashed line in Fig. 27(a)), i.e., the open loop configuration was used for the receiver (dotted line in Fig. 27(a)). The total intensity of the laser output was detected with a digital oscilloscope. Typical temporal waveforms of the transmitter and the receiver are shown in Fig. 27(b). There is no obvious correlation between the two outputs and the correlation plot between the transmitter and receiver waveforms (Fig. 27(c)) shows no evidence of identical synchronization. Next, the drive sig-
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F IG . 27. (a) Experimental setup of a diode-pumped Nd:YAG microchip laser with optoelectronic feedback. The dashed line corresponds to the closed-loop transmitter, and the dotted line corresponds to the open-loop receiver. AFG, arbitrary function generator; AOM, acousto-optic modulator; BS, beam splitter; COM, computer; F-P, Fabry–Perot interferometer; L, lens; LD, laser diode for pumping; LPF-A, low pass filter and amplifier; M, mirror; Nd:YAG, Nd:YAG laser crystal; OC, output coupler; OSC, digital oscilloscope; PD, photodetector. (b) Temporal waveforms of experimentally measured total intensity of the transmitter and two receivers. (c) Correlation plots between the transmitter and receiver outputs. (d) Correlation plots between the two receiver outputs; (c) and (d) are obtained from (b). (e) Temporal waveforms of the total intensity as obtained from numerical calculations. From Uchida et al. (2003).
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nal was fed back into the receiver laser repeatedly, in order to apply the auxiliary system method. Figure 27(b) shows two receiver outputs driven by the same drive signal at different times. The correlation plot between the two receiver outputs shows linear correlation, as shown in Fig. 27(d). This implies that the receiver laser driven by the same drive signal always generates identical outputs, independent of initial conditions. Because the dynamics of the receiver laser are repeatable and reproducible, generalized synchronization can be achieved stably in this system. Numerical results obtained from Tang–Statz–deMars equations agree well with the experimental observations (Fig. 27(e)). Analysis of these equations reveals that modal dynamics is responsible for the occurrence of generalized synchronization in identical systems.
4.4. P HASE S YNCHRONIZATION Both identical synchronization and generalized synchronization are defined as deterministic relationships between all variables of the coupled systems. Phase synchronization is different. The definition of phase synchronization was inspired by the concept of frequency locking of periodic oscillators, where very minute coupling is sufficient to lock the phases of drive and response. That is, their oscillation period becomes identical, yet the amplitude of the oscillations of drive and response might be very different. The concept of phase synchronization of chaotic systems such as lasers is based on the fact that, in some cases, the chaotic oscillations can be decomposed in terms of an amplitude and a phase. Under certain conditions, it can happen that the amplitudes of the chaotic oscillations of two coupled lasers are desynchronized, while a clear synchronization exists between the phases. This is called phase synchronization (Rosenblum et al., 1996). We note that “phase” in this context indicates a suitably defined phase of chaotic oscillations of laser intensity, not the optical phase of the electric field. Let us illustrate the concept of phase synchronization with the example of a periodically driven chaotic Rössler oscillator, whose dynamics is described by x˙ = −(y + z), y˙ = −x + 0.15y + k sin(ωt),
(43)
z˙ = 0.2 + z(x − 10), where k denotes the coupling strength, and the drive frequency is close to the average frequency of the chaotic Rössler system, namely ω = 1.0335. The projection of the Rössler attractor onto the x–y plane (see Fig. 28) reveals that we can introduce phase through an angle variable by simply going to cylindrical coordinates, yielding a phase φ = arcsin(y/r), where r = x 2 + y 2 . A good
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F IG . 28. Phase synchronization of a periodically driven Rössler oscillator. In (A) and (B) projections of the chaotic attractor onto the x–y plane are shown and the black squares are a stroboscopic plot in which the coordinates of the driven Rössler system are sampled after every full period of the drive. (A) In the uncoupled case, k = 0, the points spread relatively evenly over the whole attractor, which corresponds to a drift of the phase of the Rössler oscillator with respect to the drive. (B) Phase synchronization occurs for a coupling k = 0.05, as seen by the concentration of squares in one region of the attractor. In (C) the evolution of the phase difference (φ(t) − ωt)/2π is plotted for different k.
measure for phase synchronization between the chaotic Rössler system and the periodic drive is the phase difference φ(t) − ωt. For weak or no coupling the phase difference varies either without bound in a random-walk like manner if the drive frequency equals the mean rate of phase increase of the chaotic oscillator or steadily increases (or decreases) if the drive frequency and mean oscillator frequency differ. This is shown in Fig. 28(C), where the evolution of the phase difference is depicted for the case of no coupling, weak coupling, and coupling above the synchronization threshold. For no coupling, the phase difference grows steadily. An increase in k to k = 0.015 leads to partial synchronization, where long periods of phase locking are interspersed by phase slips. The rate of slipping decreases continually as k increases, until the synchronization threshold is
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reached. Above the synchronization threshold, there are no more phase slips and the phase difference is bounded by a fraction of 2π for all times. Although phase synchronization has been observed in many experimental systems, the phenomena is still not completely understood. A central problem is that no general definition of “phase” exists that works for all chaotic systems. That is, only for very special chaotic oscillators are we able to introduce a phase variable in the straightforward manner discussed above and many alternative definitions of phase have been proposed and applied. One example is the use of the Gabor analytic signal (Gabor, 1946) defined through the Hilbert transform (Born and Wolf, 1999) of the intensity time series, as used in the experimental investigation of phase synchronization in a chaotic laser array by Deshazer et al. (2001). The chaotic system of Deshazer et al. (2001) consists of three parallel, laterally coupled single-mode Nd:YAG laser arrays. Coupling through the electric fields of the individual beams exists only for adjacent pairs. Experimental intensity measurements are displayed in Figs. 29(a)–(c). The two outer lasers in the array (lasers 1 and 3) have nearly identical intensity fluctuations. However, no synchronization relationship is obvious between the center laser (laser 2) and the outer lasers (lasers 1 and 3), even though the center laser mediates the identical synchronization of the outer lasers. To test for interdependence between the time series of the outer and center lasers, an analytic phase and a Gaussian filtered phase are introduced (see details in the article by Deshazer et al. (2001)). Phase synchronization between the outer and the center laser (lasers 1 and 2) is shown by plotting their relative phase versus time (Figs. 29(d)–(e)). Figure 29(d) shows the difference of the analytic signal phases for these lasers, which has a large range of variation (∼130 rotations). Phase synchronization is not discernible. Next, in Fig. 29(e), the difference of the Gaussian filtered phase for these two lasers is plotted at different frequencies of the Gaussian filter. Synchronization of the side and central lasers in the frequency regime of 140 kHz is apparent immediately, since the flat portion of this plot extends across essentially the entire time of observation (solid line). Periods of phase synchronization and phase slipping are found in the less correlated frequency regime of 80 kHz (dotted line). No indication of synchronization is found when one of the component phases is replaced with a surrogate phase extracted from another experimental data set taken from this array under identical conditions (dashed line). These results illustrate that the detection of phase synchronization may require careful consideration of the nature of the time series measured. The time series considered in this experiment are of a distinctly nonstationary nature, and it is clearly advantageous to introduce a Gaussian filtered phase variable. One is then able to quantitatively assess phase synchronization for different frequency components of the dynamics.
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F IG . 29. (a)–(c) Experimental intensity time series showing chaotic bursts of three Nd:YAG lasers evanescently coupled in a linear array. (d), (e) Time series for the differences between the (d) analytic and (e) Gaussian filtered phases of the center and an outer laser. The solid line in (e) corresponds to a Gaussian filter centered at 140 kHz; for the dotted line it is centered at 80 kHz; and the dashed line is the phase difference with respect to a surrogate time series. From Deshazer et al. (2001).
4.5. S YNCHRONIZATION E RRORS In realistic devices there is always noise and no two physical devices are exactly identical. One therefore has to consider the effects of noise and slight mismatches of transmitter and receiver on the quality of chaos synchronization. For simplicity, we focus on the case of identical synchronization, but similar considerations apply to generalized synchronization and phase synchronization. As a concrete example consider the two coupled opto-electronic devices shown in Fig. 30(A) that were investigated theoretically by Abarbanel et al. (2001) and experimentally by Tang and Liu (2001b). The transmitter and receiver each consist of a single-mode semiconductor laser with time-delayed feedback formed by an optoelectronic loop from the laser optical output back to the bias current across the laser itself. This particular feedback is insensitive to the optical phase of the laser output, which simplifies experiments as compared to optical
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F IG . 30. Synchronization of two hyperchaotic opto-electronic devices with delayed feedback. (A) Setup. LD: laser diode; PD: photodetector; A: amplifier; τ : feedback delay time; T : transmission time. (B) Synchronization error from mismatches in the photon decay rate (γc ) and the carrier decay rate (γs ). From Abarbanel et al. (2001).
feedback schemes, such as those described in Section 2.2. The transmitter and receiver generate chaotic oscillations at gigahertz frequencies due to the interplay of laser nonlinearities and time-delayed feedback, which makes the state-space of the system formally infinite-dimensional. The two chaotic devices are coupled by injecting light from the chaotic transmitter system into the receiver with coupling strength c. The equations of motion describing the time-delay dynamics of the transmitter-receiver system read ⎧ dsT √ sp ⎪ ⎪ = γcT g(nT , sT ) − 1 sT + 2 sT FT , ⎨ dt (44) ⎪ dn ⎪ ⎩ T = γsT J − nT − J g(nT , sT )sT + γsT ξ(J + 1)sT (t − τ ), dt ⎧ ds √ R sp ⎪ ⎪ = γcR g(nR , sR ) − 1 sR + 2 sR FR , ⎪ ⎪ dt ⎪ ⎨ dnR (45) = γsR J − nR − J g(nR , sR )sR ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎩ + γsR ξ(J + 1) csT (t − τ ) + (1 − c)sR (t − τ ) ,
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where s is the intracavity photon density of the laser, n is the carrier density, g(n, s) is the gain, J is the dc-bias current, γc is the photon decay rate, γs is the spontaneous carrier decay rate, τ is the feedback delay, and F sp is the noise due to spontaneous emission. For the case without noise and equal parameters, i.e. γcT = γcR , γsT = γsR and τ = τ , the systems synchronize identically for coupling strengths c 0.1, as shown by numerical simulations of Eqs. (44) and (45) as well as computations of the transverse Lyapunov exponents (Abarbanel et al., 2001). In an experiment, the transmitter and receiver systems are inevitably different from each other. In this situation, identical synchronization is not a mathematical solution of the model equations. Nonetheless, the deviation from the identity solution can be small over some range of parameters. An evaluation of the effect of parameter mismatches between the transmitter and the receiver on the quality of synchronization, quantified by the synchronization error |sT (t) − sR (t)| E= (46) , sT (t) is shown in Fig. 30(B) for the two coupling strengths c = 0.4 and c = 1. In these computations, a realistic level of spontaneous emission noise is present, so E > 0 even for perfectly matched parameters. While the error is tolerable for a 5% or so mismatch at c = 1, for c = 0.4 the error is large for almost all mismatches. Thus, as one would expect, the open loop configuration (c = 1) is most robust with respect to parameter mismatch. This conclusion was confirmed experimentally by Tang and Liu (2001b), where approximate identical synchronization was found for a range of coupling strengths c with a minimum synchronization error for the open loop configuration. The case of two coupled opto-electronic devices is an example of a system where the noise-free model with identical transmitter and receiver predicts identical synchronization (all transverse Lyapunov exponents are negative) and where the inclusion of noise and parameter mismatches does not destroy synchronized behavior in any essential way. In other words, in this example, the asymptotic stability of the synchronization manifold is a good predictor of synchronization in the physical system despite the presence of noise and parameter mismatch. However, this is not always true. Illing et al. (2002) show that slight parameter mismatches can lead to large synchronization errors in some systems even when no noise is present. In this case, synchronization errors arise due to the fact that even when the slightly mismatched systems exhibit generalized synchronization, the deviation of the manifold associated with generalized synchronization (defined by xR = F (xT )) from the hyperplane associated with identical synchronization can be large. Aside from parameter mismatches, the presence of noise can prevent high quality synchronization. In some synchronization experiments where all transverse
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Lyapunov exponents are negative, long intervals of synchronization are interrupted irregularly by large (comparable to the size of the chaotic attractor), brief desynchronization events. This behavior has been called attractor bubbling (Ashwin et al., 1994, 1996). Ashwin et al. (1994, 1996) pointed out that the synchronization manifold may contain unstable sets that are embedded in or close to the chaotic attractor. These unstable sets, such as unstable periodic orbits, are characterized by positive transverse Lyapunov exponents, even though the transverse exponents characterizing the entire chaotic attractor are all negative. Near such sets, the manifold is locally repelling so that a small perturbation arising from experimental noise results in a brief excursion away from the synchronization manifold. These desynchronization events recur because the chaotic evolution of the system brings it into the neighborhood of the repelling set an infinite number of times. This explanation implies that the transition from bubble-free synchronization to bubbling as the coupling strength is varied occurs when a set first becomes transversely unstable. Thus, a criterion for high-quality, bubble-free synchronization is that the largest transverse exponents characterizing each of the unstable sets embedded in the attractor must be negative. Since Lyapunov exponents are topological invariants, this criterion is independent of the choice of metric. Unfortunately, it is impossible to apply this criterion since there are typically an infinite number of unstable sets embedded in a chaotic attractor whose stability must be determined. There is some indication that the criterion might be applied in an approximate sense because it has been suggested that a low-period UPO typically yields the largest exponent (Hunt and Ott, 1996). However, it is known that this conjecture does not hold for an attractor near a crisis (Zoldi and Greenside, 1998; Hunt and Ott, 1998; Yang et al., 2000). To circumvent the problems associated with applying the criterion proposed by Ashwin et al. (1994), synchronization criteria that are much simpler to apply were suggested, e.g., by Gauthier and Bienfang (1996) and Brown and Rulkov (1997). Unfortunately, although simple to apply, these criteria are often too conservative: experimental systems synchronize even when the condition that guarantees synchronization is not satisfied. This was demonstrated by Blakely and Gauthier (2000) who compared regimes of high-quality synchronization observed in an experimental system of coupled hyperchaotic electronic oscillators to the predictions of the different criteria for high-quality synchronization. In summary, both parameter mismatches and experimental noise can destroy high quality identical synchronization in certain chaotic systems. Furthermore, there is no simple and generally reliable way to determine how to couple two oscillators to achieve high-quality synchronization. Indeed, in many cases, the computationally most efficient approach is to include realistic levels of noise and parameter mismatches in the model and check synchronization quality by direct simulation.
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4.6. S UMMARY OF S YNCHRONIZING C HAOS R ESEARCH In this section, we described methods for synchronizing the behavior of two or more systems displaying chaotic behavior. This research drives home the fact that chaotic systems are acutely sensitive to tiny perturbations and that any behavior that is consistent with the system dynamics, including even chaotic behavior, can serve as a target state.
5. Communication 5.1. I NTRODUCTION Synchronization leads to communication—even when the signals used are chaotic. The main idea of chaos communication schemes is to mix the message with a chaotic carrier in the transmitter and to send the mixed signal to the receiver. At the receiver, the message can be recovered by comparing the transmitted mixed signal to the message-free chaotic carrier. The crux of synchronization based chaos communication is that it solves the problem of how to obtain the message-free chaotic carrier at the receiver. That is, although it is not possible to store the non-repeating chaotic carrier that is needed for message retrieval, the chaotic carrier can be regenerated at the receiver using chaos synchronization. Communication based on chaotic synchronization was already proposed by Pecora and Carroll (1990) in their seminal paper on chaos synchronization and developed further by Frey (1993); Volkovskii and Rulkov (1993); Kocarev and Parlitz (1995); and Feldmann et al. (1996). The feasibility of chaos communication using synchronization was confirmed experimentally soon thereafter in electronic circuits (Kocarev et al., 1992; Cuomo and Oppenheim, 1993) and laser systems (VanWiggeren and Roy, 1998a, 1998b; Goedgebuer et al., 1998). Synchronization based communication is not the only known approach to chaos communication. For example, the unmodulated chaotic waveform can be transmitted along with the modulated signal (known as transmitted reference scheme) using either a separate channel or time division (Kolumban et al., 1998). Thus, reliable detection can be achieved at the expense of efficiency, since at least 3 dB of the signal-to-noise ratio is lost. Another method encodes the message into the symbolic dynamics of the chaotic transmitter (Hayes et al., 1993, 1994; Corron et al., 2002), with the result that the information can be read off simply by observing the transmitted signal. However, this method requires detailed and precise control of the transmitter dynamics, which is an extremely challenging task in high-speed optical systems. For this reason essentially all optical chaos communication schemes are based on chaos synchronization. In the context of synchronization-based chaos communication, one key issue is how to mix the message with the chaotic carrier. One approach is chaos masking,
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F IG . 31. Schematic of chaos communication based on synchronization where the message is encoded using (a) chaos masking, i.e. the message m(t) is added to the transmitter signal without influencing the transmitter dynamics, and (b) chaos modulation, i.e. the message m(t) is incorporated into the transmitter dynamics.
where the information signal m(t) is added (or multiplied) to the chaotic signal x(t) generated by a chaotic transmitter whose oscillations do not depend on the information m(t), as shown in Fig. 31(a). The mixture u(t) = x(t) + m(t) is transmitted to the receiver where it is used as a driving signal for the matched receiver. Various implementations of the matched receivers were proposed; see, for example, Carroll and Pecora (1993); Cuomo and Oppenheim (1993); Kocarev et al. (1992); Dedieu et al. (1993); Murali and Lakshmanan (1993); Yu et al. (1995), and references therein. The message is then recovered by subtracting (or dividing) the synchronized chaotic signal in the receiver (xr ) from the transmitted signal. The common shortcoming of such methods of communication is that the driving signal at the receiver (u = x + m), which is “distorted” by the message m(t), does not match the corresponding “driving” signal in the transmitter (x) and perfect synchronization of transmitter and receiver dynamics is not possible. As a result, the recovered message mr (t) will always contain some traces of chaotic waveforms no matter how perfectly the parameters of the receiver match those of the transmitter. In contrast, perfect synchronization and message recovery is possible, in principle, in chaos communication schemes that utilize chaos modulation to encode and decode information (Volkovskii and Rulkov, 1993; Halle et al., 1993; Frey, 1993; Kocarev and Parlitz, 1995; Feldmann et al., 1996; Parlitz et al., 1996). The key
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idea of this approach is that the information signal m(t) is injected into one of the feedback loops of the chaotic transmitter system, as depicted in Fig. 31(b). The feedback is selected in such a way that the remaining subsystem (SS) is conditionally stable. In this case, the distorted chaotic signal u(t) = x(t) + m(t) returns back to the generator and drives the dynamics of the transmitter. The signal u(t) is also transmitted to the receiver, where it drives the dynamics and generates oscillations of the receiver system that are identical to the chaotic oscillations in the transmitter. The message can be recovered using the open feedback loop of the receiver, as shown in Fig. 31(b). In this case (in the absence of noise) the information can be restored exactly after initial transients. Since the first numerical proposal of communications with synchronized chaotic lasers was carried out in a solid-state Nd:YAG laser model by Colet and Roy (1994), many communication systems based on chaotic lasers have been investigated. We now turn to describe communication using fiber lasers as a representative example of optical chaos communication schemes (for more comprehensive reviews on optical chaos communication see Donati and Mirasso, 2002; Larger and Goedgebuer, 2004; Gavrielides et al., 2004; Ohtsubo, 2005, and Uchida et al., 2005).
5.2. C HAOS C OMMUNICATION U SING F IBER L ASERS In their theoretical analysis of chaos synchronization of fiber ring laser systems (see Section 4.2), Abarbanel and Kennel (1998) examined the possibility of using that property for communication purposes. They showed that the states of two fiber lasers can become synchronized even when a message m(t) is added to the transmitter field ET (t). Part of the resulting field ET (t) + m(t) is used for transmission to the receiver and part for internal time-delayed feedback, which in turn determines the transmitter dynamics. Thus, they suggest a chaos modulation scheme (see Fig. 31) and demonstrate that subtraction of the synchronized receiver signal ER (t) from the total transmitted signal allows the recovery of the message. As an example of the feasibility of their proposal, Abarbanel and Kennel (1998) used their communication scheme to model the transmission of a segment of speech, both analog and digital, at frequencies on the order of hundreds of MHz. Several experimental demonstrations of chaotic communications in fiber lasers have been reported. VanWiggeren and Roy (1998a) demonstrated one of the first experiments for chaos communications using an erbium doped fiber ring laser (EDFRL), as shown in Fig. 26(a). The ability to produce chaos synchronization was discussed in Section 4.2. A small-amplitude, 10-MHz square-wave message is introduced into the transmitter’s cavity via an output coupler (chaos modulation). Under these conditions synchronization between the receiver’s output (ER )
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F IG . 32. Message recovery in the experimental setup of Fig. 26(a). (A): Difference between the signals of photodiodes A and B before filtering. (B): The same signal after low-pass filtering (solid line); the dashed line represents the original message (as detected by photodiode A when the transmitter erbium doped fiber amplifier (EDFA) is turned off). (C): Low-pass-filtered transmitter signal. From VanWiggeren and Roy (1998a).
and the transmitter’s unperturbed electric field (ET , i.e. without the message) is preserved. To recover the message they subtract the signals from photodiodes A and B (Fig. 26). The result of such a subtraction is shown in Fig. 32. In the absence of low-pass filtering (Fig. 32(A)), not much trace of the original squarewave message is to be found, because photodiodes record only intensities, so that the resulting signal corresponds to ET (t) + m(t)2 − ER (t)2 = 2 Re ET (t)m(t) + m(t)2 , (47) given that ER (t) = ET (t). Low-pass filtering that signal difference reproduces the message intensity |m(t)|2 , since the typical frequency of the chaotic carrier fluctuations (hundreds of MHz) is much faster than the message frequency (10 MHz). Figure 32(B) shows the result of such low-pass filtering (solid line), and compares it with the original message (dashed line). The good quality of the message recovery is evident. When a similar filtering is applied to the raw transmitted signal (as detected by photodiode A), the result shows no trace of the original message (Fig. 32(C)). As we have seen, the transmission bit-rate in the experimental implementation described above is limited by the requirement that the message frequency be sufficiently smaller than the characteristic frequency of the chaotic carrier so that message recovery can be performed by low-pass filtering of the subtracted signals. In order to overcome this limitation, VanWiggeren and Roy (1998b) modified their original experimental setup in such a way that the message was injected in the transmitter by acting upon an intracavity intensity modulator, as shown in the left panel of Fig. 33. This encoding results in a signal m(t)(ET 1 + ET 2 ), ten percent of which is transmitted and ninety percent of which returns via a feedback loop to the transmitter fiber amplifier EDFA 1. This type of parametric encoding via chaos modulation requires decoding by division, instead of subtraction. Photodiode A at the receiver measures the intensity m(ET 1 + ET 2 )2 = |m|2 |ET 1 + ET 2 |2 , (48)
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F IG . 33. A second experimental implementation of chaotic communications in coupled fiber lasers. Left: Experimental setup. Right: transmitted signal (top), signal after filtering by the receiver (middle), and result of dividing the two (bottom). From VanWiggeren and Roy (1998b).
whereas photodiode B measures |ER1 + ER2 |2 and E + E 2 = |ET 1 + ET 2 |2 R1 R2
(49)
under optimum conditions (VanWiggeren and Roy, 1998b). Dividing Eq. (48) by Eq. (49) gives the message |m|2 . Therefore, no low-pass filtering of the decoded signal is required, and the frequency limitation discussed above disappears. Successful transmission of a 126 Mbits/sec pseudorandom digital message is observed, as shown in the right plot of Fig. 33. In this new experimental setup, the structures of the transmitter and receiver were modified by the addition of a second fiber loop, which introduced the re-
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F IG . 34. (A) shows both the transmitted signal measured by photodiode A (thin line) and the signal measured by photodiode B (thick-line). (B) shows the results of subtracting the thick-line from the thin-line. (C) gives an optical spectrum showing the lasing wavelengths of the EDFRL. The message injection is occurring at a wavelength of 1.533 µm. (D) The transmitted signal after passing through a 1 nm bandpass filter at 1.533 µm. The chaotic light at this wavelength still masks the message. From VanWiggeren and Roy (1999).
quirement of multiple parameter matching for extraction of the message. In this case, accurate recovery of the message requires multiple matched parameters in the receiver. The geometrical configuration of the receiver must be the same as in the transmitter. The lengths of the fiber in the outer loop and the time-delay between photodiode A and B must be matched fairly precisely. Finally, the relative power levels in the receiver must be properly matched to the power levels in the transmitter. This method suggests that more complicated geometries and systems requiring additional parameters for message recovery may also be possible to construct using an EDFRL as a basic element. VanWiggeren and Roy (1999) also performed experiments with message injection at wavelengths which were not resonant with the lasing wavelength of the ring laser. Figure 34(A) shows signals measured by photodiodes A and B when the wavelength of the injected message is 1533.01 nm. In this case the fiber amplifiers
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were pumped at about 85 mW, many times threshold. This resulted in an optical power in the ring of 9.1 dBm without any message injection. The injected message power was −3.1 dBm. The subtraction of the traces in Fig. 34(A) is seen in Fig. 34(B). Once again, the same pattern of bits is obtained. The optical spectrum (Fig. 34(C)) shows two distinct peaks. The first of these peaks (1533 nm) corresponds to the message injection, whereas the second peak (1558 nm) corresponds to the natural lasing wavelength of the ring laser. The very broad line-width is characteristic of the EDFRL. The message light at 1533 nm stimulates the laser to emit at the same wavelength and the fraction of the light that remains in the ring continues to circulate stimulating additional emission. Consequently, the light detected at 1533 nm consists of a combination of the message itself plus chaotic light produced by the laser. If a bandpass filter was used to isolate the message wavelength (1533.01 nm), it was observed that the message was well obscured by the chaotic laser light. Figure 34(D) shows one of these measurements. The sequence of bits is not visible even after isolating just the message wavelength. This experiment indicates that wavelength division multiplexing may be possible, while still using a chaotic waveform as carrier. In summary, the message wavelength can be varied around the natural lasing wavelength of the EDFRL and chaotic communication can still occur. Bit-rates of 125 Mbits/s and 250 Mbits/s (for a non-return to zero waveform) were demonstrated by VanWiggeren and Roy (1999). Taking full advantage of the large bandwidth available in the optical system would permit even faster rates, but these experiments were limited by the bandwidth of their photodiodes (125 MHz 3-dB roll-off) and oscilloscope (1 GS/s).
5.3. A DVERSE E FFECTS OF R EALISTIC C OMMUNICATION C HANNELS All practical communication channels introduce signal distortions that alter the chaotic waveform shape, resulting in received chaotic oscillations that do not precisely represent the transmitted oscillations. Channel noise, filtering, attenuation variability and other distortions in the channel corrupt the chaotic carrier and information signal. The presence of these channel distortions significantly hamper the onset of identical synchronization of the chaotic systems and imperfect synchronization results in information loss. As an example, Abarbanel et al. (2001) considered information loss due to signal distortions in an open air channel for a communication system based on the synchronized opto-electronic devices described in Section 4.5 (see Fig. 35(A)). Figure 35(B) summarizes the information loss of the proposed communication scheme due to channel noise by showing the bit error rate (BER) versus the signal to noise ratio (SNR), where the latter is defined as the energy per bit divided by the
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F IG . 35. (A) Chaos communication setup based on coupled optoelectronic devices in an open loop configuration (c = 1). LD: laser diode, PD: photo-diode, M: electro-optic modulator which multiplicatively modulates information onto the light intensity. (B) Numerical estimated bit error rate (BER) versus effective signal to noise ratio (SNR). Adapted from Abarbanel et al. (2001).
spectral density of the noise. Results are shown of calculations for the open loop configuration (c = 1), and, as a reference, the case of “direct signaling”, where it is assumed that the receiver has a perfect copy of the transmitter’s chaotic carrier signal. In general, the BER curves for synchronization-based chaos communication lie to the right of the curve for direct signaling for any non-zero coupling because some channel noise enters the feedback loop of the receiver. This noise causes desynchronization and thus information loss in addition to the information loss that results from channel-noise-induced bit flips. It is seen that one of the chaos-communication BER curves (diamond-symbol) is close to the direct-signaling curve, indicating good synchronization of transmitter and receiver despite the noisy channel. This case corresponds to a situation where the device dynamics is in a regime of low-dimensional chaos, i.e. the Lyapunov dimension DL of the chaotic attractor is in the range of three to four. In contrast, the chaos-communication BER curve shown with circle symbols in Fig. 35(B) is far to the right of the direct-signaling curve, implying substantial information losses due to noise-induced desynchronization events. This curve corresponds to the case of high-dimensional chaos (DL > 5). Nevertheless, even for the case of high-dimensional chaos, the noise entering the feedback loop does not cause global desynchronization and thus it is still possible to communicate. This result demonstrates the structural stability of the synchronized state. In the example above, information is encoded in the signal amplitude and no error correction codes are applied. In this context, reliable communication (low BER) can be achieved only for low noise levels in the communication channel
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(SNR > 1) even for the direct signaling case, which requires no chaos synchronization. The potentially significant additional information losses for chaos communication schemes that result because of imperfect synchronization can be largely avoided through a careful choice of the chaotic carrier signal. Furthermore, the levels of channel-noise that are required for reliable communication can be straightforwardly achieved in communication applications that use optical fiber links (Argyris et al., 2005) and, of course, in bench-top free-space experiments. The latter was conclusively demonstrated by Tang and Liu (2001a) who achieved 2.5 Gbit/s chaos communication. 5.4. M INIMIZING THE E FFECT OF C HANNEL D ISTORTIONS ON S YNCHRONIZATION The enhanced sensitivity to chaotic signal waveform shape distortions and the resulting problems with chaos synchronization are major challenges for practical implementation of chaos-based communications systems. In some cases, this challenge is overcome by correcting at the receiver the signal waveform distortions that arise in the communication channel. This was done, for example, in the successful demonstration of chaos communication by Argyris et al. (2005) mentioned in Section 2.3. Another way to minimize the effect of channel distortions was suggested by Rulkov and Volkovskii (1993) who substituted continuous chaotic waveforms with chaotically-timed pulse sequences. Each pulse in the sequence has identical shape, but the time delay between them varies chaotically. Since the information about the state of the chaotic system is contained entirely in the timing between pulses, the distortions that affect the pulse shape do not significantly influence the ability of the chaotic pulse generators to synchronize. Therefore, synchronizing chaotic impulse generators can be utilized in communications schemes for realistic channels and, at the same time, allow the use of filters for noise reduction. The information can be encoded in the pulse train by alteration of time position of pulses with respect to chaotic carrier. This is the essence of the Chaotic Pulse Position Modulation (CPPM) system described by Sushchik et al. (2000). The ability of the self-synchronizing CPPM method to communicate in the presence of significant non-stationary signal distortions in the channel has been studied experimentally using a free-space laser communication link by Rulkov et al. (2002). A schematic representation of the chaotic free-space laser communication system is shown in Fig. 36(a). Here, the communication carrier signal consists of a sequence of optical pulses that travel through air. The CPPM communication method encodes information in the interpulse intervals and can therefore tolerate strong signal distortions and amplitude variations like the ones caused by atmospheric turbulence as long as fluctuations of the propagation time in the turbulent channel remain small.
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F IG . 36. Robust chaos communication over a free-space laser link. (a) Experimental setup. (b) Fluctuations of the CPPM pulses after traveling through atmospheric turbulence. Pulse amplitude Ap measured in volts at the output of the amplifier (top). Propagation times τm (middle) and τt (bottom). From Rulkov et al. (2002).
Figure 36(b) illustrates the severe pulse amplitude fluctuations of the signal entering the CPPM demodulator. It also shows that the pulse propagation time τm , which is measured between the leading front of the TTL pulse applied to the laser and the maximal point of the received pulse, varies only within a 0.2 µs time interval. However, in order to trigger the CPPM demodulator circuit, the received pulse amplitude has to exceed a certain threshold level. Therefore, the actual delay time τt , as seen by the transceiver, depends on the amplitude of the received pulses and fluctuates (see Fig. 36, bottom). Nevertheless, the variations of the pulse propagation time are small enough for the CPPM controller to selfsynchronize and to maintain the stability of the communication link. The gaps visible in Fig. 36(b) are caused by pulse amplitude fading when the pulse amplitude falls to the photo-receiver noise level, which is below the threshold level. The gaps are audible as occasional clicks when using the free-space laser communication system for real-time voice communication. They are the main contribution to the total BER of 1.92 × 10−2 measured in this experiment. Indeed, errors that are not related to the complete failure of the channel because of fading instances contributed to the BER only ∼5.5×10−5 . Thus, chaos communication using pulses and encoding information in the timing of the pulses appears to be a particularly robust communication scheme.
5.5. S UMMARY OF C HAOS C OMMUNICATION R ESEARCH Chaos communication research has shown that gigabit-per-second transmission rates and very low bit-error rates, comparable to those of traditional communication systems, can be achieved. Thus, within little more than a decade, optical
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chaos communication developed from an idea to a communication technology that can now be compared in a meaningful way to traditional communication approaches. There are clearly practical applications of such non-traditional communication systems that one can envision such as cost-effective private optical communication. However, in our opinion, the main merit of the research on chaos communication is that it challenges us to think more deeply about fundamental issues of information transmission. Whereas most traditional communication techniques utilize regular and repeating sinusoidal signals, chaos communication shows that parties can communicate equally well using irregular signals that never repeat exactly. This research generalizes standard communication approaches and provides insights about information transmission in biological systems, where non-sinusoidal, pulse-like waveforms are used to convey information.
6. Spatio-Temporal Chaos and Patterns Spatio-temporal chaos and pattern formation can often take place in a nonlinear optical device when the Fresnel number is large; that is, when the device can support a large number of transverse modes. The behavior can often be visualized by measuring the intensity profile of the light generated by the device in a plane that is perpendicular to the propagation direction. For some time, it was thought that transverse behavior was just too complex (too high-dimensional) to be responsive to the control and synchronization methods described in the previous sections. But we see below that this is not the case.
6.1. S PATIO -T EMPORAL C HAOS C OMMUNICATION All communication systems discussed in Section 5 involve serial transmission of data through a single communication channel, using temporal chaotic signals as information carriers. A generalization of this approach to systems with spatial degrees of freedom would enable the use of spatio-temporal chaos for the parallel transfer of information, which would yield a substantial increase in channel capacity. Implementing a parallel chaotic communication scheme in electronic systems would be a complex task (due mainly to the need of a comprehensive extended coupling between transmitter and receiver). Optical systems, on the other hand, provide a natural arena for the parallel transfer of information, as we review in what follows. Multichannel chaotic communications have been recently proposed in models of multimode semiconductor lasers with optical feedback by White and Moloney (1999) (see also Buldú et al., 2004). In that case, only variations of
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F IG . 37. Scheme for communicating spatio-temporal information using optical chaos. CM is a coupling mirror. Adapted from García-Ojalvo and Roy (2001b).
the electric field along its propagation direction are considered. Information is encoded in the different longitudinal cavity modes, giving rise to a technique for multiplexing. Spatio-temporal communication, on the other hand, utilizes the inherent large scale parallelism of information transfer that is possible with broad-area optical wavefronts. As in the previous systems considered, spatiotemporal chaotic communications require the existence of synchronization between transmitter and receiver. Synchronization of spatio-temporal chaos has been investigated extensively in arrays of nonlinear oscillators (Kocarev and Parlitz, 1996b) and in model partial differential equations (Amengual et al., 1997; Kocarev et al., 1997). García-Ojalvo and Roy (2001b) proposed a communication system based on the synchronization of the spatio-temporal chaos generated by a broad-area nonlinear optical cavity. The setup is shown schematically in Fig. 37. Two optical ring cavities are unidirectionally coupled by a light beam extracted from the left ring (the transmitter) and partially injected into the right one (the receiver). Each cavity contains a broad-area nonlinear absorbing medium, and is subject to a continuously injected plane wave Ai . Light diffraction is taken into account during propagation through the medium, in such a way that a nonuniform distribution of light in the plane transverse to the propagation direction appears. In fact, an infinite number of transverse modes can oscillate within the cavity. In the absence of a message, the transmitter is a standard nonlinear ring cavity, well known to exhibit temporal optical chaos (Ikeda, 1979). When transverse effects due to light diffraction are taken into account, a rich variety of spatiotemporal instabilities appear, including solitary waves (McLaughlin et al., 1983) and spatio-temporal chaos (Sauer and Kaiser, 1996; Le Berre et al., 1997). This latter behavior is the one in which we are interested here, since such chaotic waveforms can be used as information carriers. The propagation of light through the nonlinear medium can be described by the following equation for the slowlyvarying complex envelope En (x, z) of the electric field (assumed to be linearly
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F IG . 38. (a) Space–time representation of a spatio-temporal chaotic state. (b) Intensity difference between the transmitter and the receiver; coupling is switched on at n = 100. (c) Temporal evolution of the synchronization error (see text) for the case shown in plot (b). Parameters common to the two cavities are α = 100.0, Δ −10.0, R = 0.9, T = 0.1, k = 100.0, = 0.01, L = 0.015, A = 7.0. In (b), c = 0.4. From García-Ojalvo and Roy (2001a).
polarized) in the nth passage through the resonator (García-Ojalvo and Roy, 2001b): ∂En (x, z) α(1 + iΔ) i 2 En (x, z). = ∇ En (x, z) − ∂z 2k 1 + 4|En |2
(50)
The first term on the right-hand side of Eq. (50) describes diffraction, and the second saturable absorption. The propagation direction is denoted by z, whereas x is a vector in the plane orthogonal to the propagation direction. Equation (50) obeys the boundary condition √ En (x, 0) = T A + R exp(ikL)En−1 (x, ), (51) which corresponds to an infinite-dimensional map. The value z = 0 in Eq. (51) denotes the input of the nonlinear medium, which has length . The total length of the cavity is L. Other parameters of the model are the absorption coefficient α of the medium, the detuning Δ between the atomic transition and cavity resonance frequencies, the transmittivity T of the input mirror, and the total return coefficient R of the cavity (fraction of light intensity remaining in the cavity after one round trip). The injected signal, with amplitude A and wavenumber k, is taken to be in resonance with a longitudinal cavity mode. Previous studies by Sauer and Kaiser (1996) have shown that, for Δ < 0, model Eqs. (50) and (51) exhibits irregular dynamics in both space and time for A large enough. An example of this regime is shown in Fig. 38(a). This spatiotemporally chaotic behavior can become synchronized to that of a second cavity,
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also operating in a chaotic regime, coupled to the first one as shown in Fig. 37. The coupling mechanism can be modeled by: (1) En(1) (x, 0) = F (1) En−1 (x, ) , (52) (2) (1) En(2) (x, 0) = F (2) (1 − c)En−1 (x, ) + cEn−1 (x, ) where the application F (i) represents the action of the map (51) in every round trip. The coupling coefficient c is given by the transmittivity of the coupling mirror CM (Fig. 37). The super-indices 1 and 2 represent the transmitter and receiver, respectively. Junge and Parlitz (2000) have shown that local sensor coupling is enough to achieve synchronization of spatio-temporal chaos in model continuous equations. In the optical model shown in Eqs. (50)–(51), however, the whole spatial domain can be coupled to the receiver in a natural way. Figure 38(b) shows a space-time representation of the intensity difference between the transmitter and the receiver before and after coupling between the two systems is activated (at the 100th round-trip). The initially uncoupled systems evolve in time starting from arbitrary initial conditions, and after 100 round trips, when their unsynchronized chaotic dynamics is fully developed, coupling is switched on, which results in a rapid synchronization. The synchronization efficiency can be quantified by means of the spatially-averaged synchronization error defined by Kocarev et al. (1997): ) *1 * En(1) (x, ) − En(2) (x, )2 dx, en = + (53) S S
where S is the size of the system. The temporal evolution of this quantity is shown in Fig. 38(c) for three values of the coupling coefficient c. Model Eqs. (50) and (51) have been numerically integrated in a 1-d lattice of 1000 cells of size dx = 0.1 spatial units, using a pseudospectral code for the propagation in Eq. (50). The results indicate that, for large enough coupling coefficient c, the synchronization error decreases exponentially as soon as coupling is switched on, with a characteristic time that increases with c. In order to encode and decode information in space and time, one can modify the scheme of Eqs. (52) according to Fig. 37, which leads to: (1) En(1) (x, 0) = F (1) En−1 (x, ) + Mn−1 (x) , (54) (1) (2) En(2) (x, 0) = F (2) (1 − c)En−1 (x, ) + c En−1 (x, ) + Mn−1 (x) . Upon synchronization between transmitter and receiver, the message can be decoded by simply subtracting the transmitted signal and the one coming from the "n (x) = En(1) (x, ) + Mn (x) − En(2) (x, ). For identical parameters of receiver: M the transmitter and the receiver (the situation considered so far), it can be seen analytically in a straightforward way that, as the coupling coefficient c tends to 1,
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F IG . 39. Transmission of a 2-d spatio-temporal static image. (a) Input image; (b) real part of the transmitted signal at a certain time; (c) recovered data. Parameters are those of Fig. 38, plus c = 0.7. From García-Ojalvo and Roy (2001b).
the difference |En(1) − En(2) | → 0 ∀x, which corresponds to perfect synchronization, and hence to perfect message recovery. It should be noted that the message is not merely added to the chaotic carrier, but rather the former is driving the nonlinear transmitter itself (i.e., chaos modulation). Therefore, the amplitude of the message need not be much smaller than that of the chaotic signal to provide good masking of the information. Figure 39 shows an example of data encoded and decoded using the scheme described above, where a static 2-d image has been transmitted in space and time with a coupling coefficient c = 0.7. The left plot depicts the input image, the middle plot the real part of the transmitted signal (a snapshot of it, in this case), and the right plot the recovered data. The message amplitude maximum is 0.01 (this value should be compared to the maximum intensity of the chaotic carrier, which oscillates between 1 and 10, approximately, for the parameters chosen). Simulations in this case were performed on a square array with 256 × 256 pixels of width dx = 1.0. The image is clearly recognizable even though the coupling coefficient is now as low as 0.7. Figure 39 shows qualitatively that, even though coupling between transmitter and receiver is not complete, information varying in time and space can be successfully transmitted and recovered with the setup described in Fig. 37. In order to have a quantitative measure of this effect, one can estimate the mutual information between the input and output message signals, and its dependence on several system parameters. To that end, one defines the entropy H of the message and the mutual information I between the original and recovered messages as Hinput = −
x
p(x) ln p(x),
I =−
x,y
p(x, y) ln
p(x)p(y) , p(x, y)
(55)
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F IG . 40. Information-theory characterization of 1-d message transmission. Full circles: mutual information I ; empty squares: entropy H of the recovered data; horizontal dashed line: entropy of the original image. Empty diamonds are the values of I in the presence of noise (see text). Parameters are those of Fig. 38. From García-Ojalvo and Roy (2001b).
" discretized in space and time, p(x) where x and y are the values of M and M and p(y) are their corresponding probability distributions, and p(x, y) is the joint probability. The entropy Houtput of the recovered message is defined analogously to Hinput . Note that, according to its definition, the mutual information is I = 0 for completely independent data sets, and I = Hinput when the two messages are identical. Figure 40(a) shows the value of the mutual information I , versus the coupling coefficient c, for the transmission of a 1-d message (García-Ojalvo and Roy, 2001b). As c increases, I grows from 0 to 4.0 (perfect recovery), corresponding to the entropy of the input image, given by the horizontal dashed line in the figure. This result shows that, even though good synchronization appears for c 0.4, satisfactory message recovery requires coupling coefficients closer to unity. This can also be seen by examining the behavior of the entropy H of the recovered image, plotted as empty squares in Fig. 40(a): for values of c substantially smaller than 1, the entropy of the recovered data is appreciably larger than that of the input message, indicating a higher degree of randomness in the former. Finally, the behavior of the mutual information in the presence of noise is shown as empty diamonds in Fig. 40(a). Uncorrelated, uniformly distributed noise is added continuously to the communication channel, with an amplitude 1% that of the message (García-Ojalvo and Roy, 2001b). The discussion above has considered so far identical parameters between emitter and receiver. In a realistic implementation, however, parameter mismatches will exist between both devices. A systematic study of the synchronization error between mismatched cavities shows that the most sensitive parameter in this respect is the amplitude A of the injected signal (García-Ojalvo and Roy, 2001b). The data plotted in Fig. 40(b) indicate that a slight mismatch in the value of A
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F IG . 41. The two states of the switch. (A) The off state. Weak symmetry breaking results in a two-spot output pattern. (B) The on state. A weak switching beam (light gray), directed along the cone, causes the output pattern to rotate. The state of polarization of the switching beam (light gray) is linear and orthogonal to that of the pump beams (dark gray). From Dawes et al. (2005).
degrades message recovery, by leading to values of I much smaller that the entropy of the input message, and to a recovered message with substantially larger entropy than the original.
6.2. A LL -O PTICAL S WITCHING As discussed in the previous section, complex spatial patterns are very sensitive to tiny perturbations and can be readily synchronized. Recently, Dawes et al. (2005) reported the realization of an all-optical switch that relies on this extreme sensitivity, allowing it to operate at extremely low light levels. It consists of laser beams counterpropagating through a warm rubidium vapor that induce an off-axis optical pattern. A switching laser beam perturbs the pattern, causing it to rotate even when the power in the switching beam is much lower than the power in the pattern, as shown schematically in Fig. 41. The observed switching energy density is very low, suggesting the exciting possibility that the switch might operate at the single-photon level with system optimization. Potential applications for a single-photon device are as a router in quantum information networks, as a highly efficient single-photon detector, or as a ‘photonnumber detector’. Unfortunately, the nonlinear optical interaction strength of most materials is so small that achieving single-photon switching is exceedingly difficult. This problem may be solved through modern quantum-interference methods,
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where the nonlinear interaction strength can be increased by many orders-ofmagnitude (Lukin, 2003; Braje et al., 2003). Another desirable property of alloptical switches is that the output beams are controlled by a weaker switching beam so they can be used as cascaded classical or quantum computational elements (Keys, 1970). Current switches, however, tend to control a weak beam with a strong one. The experimental setup of Dawes et al. (2005) consisted of a weak switching beam that controls the direction of laser beams emerging from a warm laserpumped rubidium vapor. Two pump laser beams counterpropagated through the vapor and induced an instability that generated new beams of light (i.e., a transverse optical pattern) when the power of the pump beams was above a critical level. The instability arises from mirror-less parametric self-oscillation (Lugiato, 1994; Silberberg and Bar-Joseph, 1984; Khitrova et al., 1988; Maître et al., 1995; Firth et al., 1990; Gauthier et al., 1990; Zibrov et al., 1999) due to the strong nonlinear coupling between the laser beams and atoms. Mirror-less self-oscillation occurs when the parametric gain due to nonlinear wave-mixing processes becomes infinite. Under this condition, infinitesimal fluctuations in the electromagnetic field strength trigger the generation of new beams of light. The threshold for this instability is lowest (and the parametric gain enhanced) when the frequency of the pump beams is set near the 87 Rb 5S1/2 ↔ 5P3/2 resonance (transition wavelength λ = 780 nm). The setup is extremely simple in comparison to most other low-light-level all-optical switching methods (Braje et al., 2003) and the spectral characteristics of the switching and output light match well with recently demonstrated single-photon sources and storage media (van der Wal et al., 2003; McKeever et al., 2004). For a perfectly symmetric experimental setup, the instability-generated light (referred to henceforth as the ‘output’ light) would be emitted both forward and backward along cones centered on the pump beams. The angle between the pumpbeam axis and the cone was of the order of ∼5 mrad and is determined by competition between two different nonlinear optical processes: backward four-wave mixing in the phase-conjugation geometry and forward four-wave mixing (Maître et al., 1995; Firth et al., 1990). The generated light has a state-of-polarization that is linear and orthogonal to that of the linearly co-polarized pump beams (Gauthier et al., 1990); hence, it was easy to separate the output and pump light using polarizing elements. Once separated, the output light propagating in one direction (say, the forward direction) can appear as a ring on a measurement screen that is perpendicular to the propagation direction and in the far field. This ring is known as a transverse optical pattern (Lugiato, 1994). Weak symmetry breaking caused by slight imperfections in the experimental setup reduced the symmetry of the optical pattern and selects its orientation (Maître et al., 1995). For high pump powers, Dawes et al. (2005) observed that the pattern consists of 6 spots with 6-fold symmetry. For lower powers near the instability threshold, only two
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F IG . 42. Low-light-level all-optical switching. (A) The off state with Ps = 0. The output light forms a two-spot transverse optical pattern. (B) The on state with Ps = 2.5 nW. The two-spot output pattern is rotated by −60◦ . (C) The on state with Ps = 230 pW. Approximately half the output power is rotated by −60◦ . From Dawes et al. (2005).
spots appeared in the far field in both directions (forward and backward). The azimuthal angle of the spots (and the corresponding beams) was dictated by the system asymmetry, which could be adjusted by slight misalignment of the pump beams or application of a weak magnetic field. The orientation of the spots was stable for several minutes in the absence of a switching beam; this is in contrast to the single-feedback-mirror experiments analyzed by Le Berre et al. (1990). In the all-optical switch, the direction of the bright output beams (total power Pout ) was controlled by applying a weak switching laser beam whose state of polarization was linear and orthogonal to that of the pump-beams (Fig. 41). The azimuthal angle of the output beams was extremely sensitive to tiny perturbations because the symmetry breaking of their setup was so small. Directing the switching beam along the conical surface at a different azimuthal angle (see Fig. 41(B)) causes the output beams to rotate to a new angle, while Pout remained essentially unchanged. Typically, the orientation of the output beams rotated to the direction of the switching beam and they found that the pattern was most easily perturbed when the switching beam is injected at azimuthal angles of ±60◦ , thereby preserving the 6-fold symmetry of the pattern observed for higher pump powers. Dawes et al. (2005) investigate the behavior of the switch for two values of the peak power Ps of the switching beam, where the spots rotate by −60◦ in the presence of a switching beam. In the absence of a switching beam (Ps = 0), they observed the pattern shown in Fig. 42(A), where Pout = 1.5 µW and the total power emitted from the vapor cell in the forward direction in both states of polarization was 19 µW. For the higher power switching beam (Ps = 2.5 nW), they observed complete rotation of the output beams (Fig. 42(B)), whereas they found that only approximately half the power in the beams rotated to the new azimuthal angle (Fig. 42(C)) at the lower switching power (Ps = 230 pW). However, they observed high-contrast switching in both cases.
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Under the conditions shown in Fig. 42(B), the total power in the output beams is much larger than the switching power: at least a factor of 600 larger. Furthermore, based on the response time τ of the switch (found to be ∼4 µs), the number of photons needed to change its state is given by Np = τ Ps /Ep = 40,000, where Ep = 2.55 × 10−19 J is the photon energy, and the switching energy is equal to Np Ep = 10 femtoJ. Another metric to characterize low-light-level switches is the energy density E, given in units of photons per (λ2 /2π). This metric gives the number of photons needed to actuate a switch whose transverse dimension has been reduced as small as possible—the diffraction limit of the interacting beams (Keys, 1970). For the spot size of the switching beam used in our experiment (1/e intensity radius of 166 µm), Dawes et al. (2005) found that E ∼ 4.4 × 10−2 photons/(λ2 /2π), corresponding to 11 zeptoJ/(λ2 /2π). They observed similar behavior for the lower switching power, as shown in Fig. 42(C). In this case, a weak switching beam controlled output beams that were 6,500 times stronger. The switching time was observed to be somewhat faster (τ ∼ 3 µs), possibly due to the fact that only part of the output light rotates to the new angle. Using this response time, they found Np = 2,700 photons, Np Ep = 690 attoJ, and E ∼ 3 × 10−3 photons/(λ2 /2π) [770 yoctoJ/(λ2 /2π)]. These results demonstrate that a switch based on transverse optical patterns is a promising candidate for low-level, possibly single-photon computation and communication systems. It exhibits much higher sensitivity than other devices and can be realized using a simple experimental setup. It is cascadable in that a very weak beam switches a stronger one so that the output of one switch could be used to drive the input of another—a required characteristic of switches in general purpose computers. Additionally, this switch could be used as a router if information is impressed on the output light (e.g., by modulating the pump beams).
7. Outlook Research on controlling optical chaos, spatio-temporal dynamics, and patterns is a vibrant field and we anticipate that the fundamental research results described in this chapter will begin to appear in practical devices and systems. We anticipate that the focus of much of the future fundamental work will be directed toward spatio-temporal systems where our understanding of the underlying orderly structure is not as complete as in lower-dimension dynamical systems.
8. Acknowledgement L.I. and D.J.G. gratefully acknowledge the long-term financial support of the US Army Research Office, most recently supported by grant # W911NF-05-1-0228
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and fruitful collaborations with many members of the Duke Center for Nonlinear and Complex Systems. R.R. acknowledges the financial support of the NSF and ONR, and is most grateful to his students and collaborators for making research such an enjoyable experience over the years.
9. References Abarbanel, H.D.I., Kennel, M.B. (1998). Synchronizing high-dimensional chaotic optical ring dynamics. Phys. Rev. Lett. 80, 3153–3156. Abarbanel, H.D.I., Kennel, M.B., Buhl, M., Lewis, C.T. (1999). Chaotic dynamics in erbium-doped fiber ring lasers. Phys. Rev. A 60, 2360–2374. Abarbanel, H.D.I., Kennel, M.B., Illing, L., Tang, S., Chen, H.F., Liu, J.M. (2001). Synchronization and communication using semiconductor lasers with optoelectronic feedback. IEEE J. Quantum Electron. 37, 1301–1311. Abarbanel, H.D.I., Rulkov, N.F., Sushchik, M.M. (1996). Generalized synchronization of chaos: The auxiliary system approach. Phys. Rev. E 53, 4528–4535. Afraimovich, V.S., Verichev, N.N., Rabinovich, M.I. (1986). Stochastic synchronization of oscillation in dissipative systems. Radiophys. and Quantum Electronics 29, 795–803. Ahlborn, A., Parlitz, U. (2004). Stabilizing unstable steady states using multiple delay feedback control. Phys. Rev. Lett. 93 , 264101/1–4. Amengual, A., Hernández-García, E., Montagne, R., Miguel, M.S. (1997). Synchronization of spatiotemporal chaos: The regime of coupled spatio-temporal intermittency. Phys. Rev. Lett. 78, 4379– 4382. Anthon, D., Sipes, D., Pier, T., Ressl, M. (1992). Intracavity doubling of CW diode-pumped Nd:YAG lasers with KTP. IEEE J. Quantum Electron. 28, 1148–1157. Argyris, A., Syvridis, D., Larger, L., Annovazzi-Lodi, V., Colet, P., Fischer, I., García-Ojalvo, J., Mirasso, C.R., Pesquera, L., Shore, K.A. (2005). Chaos-based communications at high bit rates using commercial fibre-optic links. Nature 437, 343–346. Ashwin, P., Buescu, J., Stewart, I. (1994). Bubbling of attractors and synchronization of chaotic oscillators. Phys. Lett. A 193, 126–139. Ashwin, P., Buescu, J., Stewart, I. (1996). From attractor to chaotic saddle: A tale of transverse instability. Nonlinearity 9, 703–737. Baer, T. (1986). Large amplitude fluctuations due to longitudinal mode coupling in diode-pumped intracavity-doubled Nd:YAG lasers. J. Opt. Soc. Amer. A 83, 1175–1180. Barland, S., Tredicce, J., Brambilla, M., Lugiato, L., Balle, S., Giudicl, M., Maggipinto, T., Spinelli, L., Tissoni, G., Knodl, T., Miller, M., Jager, R. (2002). Cavity solitons as pixels in semiconductor microcavities. Nature 419, 699–702. Benettin, G., Galgani, L., Giorgilli, A., Strelcyn, J.-M. (1980). Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems: A method for computing all of them. Part 2: Numerical application. Meccanica 15, 21–30. Bielawski, S., Bouazaoui, M., Derozier, D., Glorieux, P. (1993). Stabilization and characterization of unstable steady states in a laser. Phys. Rev. A 47, 3276–3279. Blakely, J.N., Gauthier, D.J. (2000). Attractor bubbling in coupled hyperchaotic oscillators. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 10, 835–847. Blakely, J.N., Illing, L., Gauthier, D.J. (2004a). Controlling fast chaos in delay dynamical systems. Phys. Rev. Lett. 92, 193901/1–4. Blakely, J.N., Illing, L., Gauthier, D.J. (2004b). High-speed chaos in an optical feedback system with flexible timescales. IEEE J. Quantum Electron. 40, 299–305.
9]
CONTROLLING OPTICAL CHAOS
693
Bleich, M.E., Socolar, J.E.S. (1996). Stability of periodic orbits controlled by time-delay feedback. Phys. Lett. A 210, 87–94. Boccaletti, S., Kurths, J., Osipov, G., Valladares, D.L., Zhou, C.S. (2002). The synchronization of chaotic systems. Phys. Rep. 366, 1–101. Born, M., Wolf, E. (1999). “Principles of Optics”, seventh ed. Cambridge University Press, New York. Bracikowski, C., Roy, R. (1991). Chaos in a multimode solid-state laser system. Chaos 1, 49–64. Braje, D.A., Bali´c, B., Yin, G.Y., Harris, S.E. (2003). Low-light-level nonlinear optics with slow light. Phys. Rev. Lett. 68, 041801/1–4. Brown, R. (1998). Approximating the mapping between systems exhibiting generalized synchronization. Phys. Rev. Lett. 81, 4835–4838. Brown, R., Rulkov, N. (1997). Synchronization of chaotic systems: Transverse stability of trajectories in invariant manifolds. Chaos 7, 395–413. Buldú, J., García-Ojalvo, J., Torrent, M. (2004). Multimode synchronization and communication using unidirectionally coupled semiconductor lasers. IEEE J. Quantum Electron. 40, 640–650. Carroll, T., Triandaf, I., Schwartz, I., Pecora, L. (1992). Tracking unstable orbits in an experiment. Phys. Rev. A 46, 6189–6192. Carroll, T.L., Pecora, L.M. (1993). Cascading synchronized chaotic systems. Phys. D 67, 126–140. Chang, A., Bienfang, J., Hall, G., Gardner, J., Gauthier, D. (1998). Stabilizing unstable steady states using extended time-delay autosynchronization. Chaos 8, 782–790. Colet, P., Roy, R. (1994). Digital communication with synchronized chaotic lasers. Opt. Lett. 19, 2056–2058. Collins, R.J., Nelson, D.F., Schawlow, A.L., Bond, W., Garrett, C.G.B., Kaiser, W. (1960). Coherence, narrowing, directionality, and relaxation oscillations in the light emission from ruby. Phys. Rev. Lett. 5, 303–305. Corron, N., Hopper, B., Pethel, S. (2003). Limiter control of a chaotic RF transistor oscillator. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 13, 957–961. Corron, N.J., Pethel, S.D., Myneni, K. (2002). Synchronizing the information content of a chaotic map and flow via symbolic dynamics. Phys. Rev. E 66, 036204/1–5. Cuomo, K.M., Oppenheim, A.V. (1993). Circuit implementation of synchronized chaos with applications to communications. Phys. Rev. Lett. 71, 65–68. Dawes, A.M.C., Illing, L., Clark, S.M., Gauthier, D.J. (2005). All-optical switching in rubidium vapor. Science 308, 672–674. Dedieu, H., Kennedy, M.P., Hasler, M. (1993). Chaos shift keying: Modulation and demodulation of a chaotic carrier using self-synchronizing Chua’s circuits. IEEE Trans. Circuits Syst. 40, 634–642. Deshazer, D., Breban, R., Ott, E., Roy, R. (2001). Detecting phase synchronization in a chaotic laser array. Phys. Rev. Lett. 87, 044101/1–4. Donati, S., Mirasso, C.R. (2002). Introduction to the feature section on optical chaos and applications to cryptography. IEEE J. Quantum Electron. 38, 1138–1140. Eckmann, J.-P., Ruelle, D. (1985). Ergodic theory of chaos and strange attractors. Rev. Mod. Phys. 57, 617–656. Feldmann, U., Hasler, M., Schwarz, W. (1996). Communication by chaotic signals: The inverse system approach. Internat. J. Ciruit Theory Appl. 24, 551–579. Firth, W.J., Fitzgerald, A., Paré, C. (1990). Transverse instabilities due to counterpropagation in Kerr media. J. Opt. Soc. Amer. B 7, 1087–1097. Frey, D.R. (1993). Chaotic digital encoding: An approach to secure communication. IEEE Trans. Circuits Syst. 40, 660–666. Fujisaka, H., Yamada, T. (1983). Stability theory of synchronized motion in coupled-oscillator systems. Prog. Theor. Phys. 69, 32–47. Gabor, D. (1946). Theory of communication. J. IEE (London) 93, 429–457. García-Ojalvo, J., Roy, R. (2001a). Parallel communication with optical spatio-temporal chaos. IEEE Trans. Circuits Syst. Part I 48, 1491–1497.
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L. Illing et al.
[9
García-Ojalvo, J., Roy, R. (2001b). Spatiotemporal communication with synchronized optical chaos. Phys. Rev. Lett. 86, 5204–5207. Gastaud, N., Poinsot, S., Larger, L., Merolla, J.-M., Hanna, M., Goedgebuer, J.-P., Malassenet, F. (2004). Electro-optical chaos for multi-10 Gbit/s optical transmissions. Electron. Lett. 40, 898– 899. Gauthier, D. (1998). Controlling lasers by use of extended time-delay autosynchronization. Opt. Lett. 23, 703–705. Gauthier, D., Sukow, D., Concannon, H., Socolar, J. (1994). Stabilizing unstable periodic orbits in a fast diode resonator using continuous time-delay autosynchronization. Phys. Rev. E 50, 2343– 2346. Gauthier, D.J. (2003). Resource letter: CC-1: Controlling chaos. Amer. J. Phys. 71, 750–759. Gauthier, D.J., Bienfang, J.C. (1996). Intermittent loss of synchronization in coupled chaotic oscillators: Toward a new criterion for high-quality synchronization. Phys. Rev. Lett. 77, 1751–1754. Gauthier, D.J., Malcuit, M.S., Gaeta, A.L., Boyd, R.W. (1990). Polarization bistability of counterpropagating laser beams. Phys. Rev. Lett. 64, 1721–1724. Gavrielides, A.T., Lenstra, D., Simpson, T.B., Othsubo, J. (2004). Introduction to the special issue on nonlinear optics. IEEE J. Selected Topics Quantum Electron. 10, 859–861. Gills, Z., Iwata, C., Roy, R., Schwartz, I.B., Triandaf, I. (1992). Tracking unstable steady states: Extending the stability regime of a multimode laser system. Phys. Rev. Lett. 69, 3169–3172. Goedgebuer, J.P., Larger, L., Porte, H. (1998). Optical cryptosystem based on synchronization of hyperchaos generated by a delayed feedback tunable laser diode. Phys. Rev. Lett. 80, 2249–2252. Goedgebuer, J.-P., Levy, P., Larger, L., Chen, C.-C., Rhodes, W.T. (2002). Optical communication with synchronized hyperchaos generated electrooptically. IEEE J. Quantum Electron. 38, 1178– 1183. Haken, H. (1975). Analogy between higher instabilities in fluids and lasers. Phys. Lett. A 53A, 77–78. Halle, K.S., Wu, C.W., Itoh, M., Chua, L.O. (1993). Spread spectrum communication through modulation of chaos. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 3, 469–477. Hayes, S., Grebogi, C., Ott, E. (1993). Communication with chaos. Phys. Rev. Lett. 70, 3031–3034. Hayes, S., Grebogi, C., Ott, E., Mark, A. (1994). Experimental control of chaos for communication. Phys. Rev. Lett. 73, 1781–1784. Hong, Y., Lee, M.W., Spencer, P.S., Shore, K.A. (2004). Synchronization of chaos in unidirectionally coupled vertical-cavity surface-emitting semiconductor lasers. Opt. Lett. 29, 1215–1217. Hunt, B., Ott, E. (1996). Optimal periodic orbits of chaotic systems. Phys. Rev. Lett. 76, 2254–2257. Hunt, B., Ott, E. (1998). A reply to the comment by Scott M. Zoldi and Henry S. Greenside. Phys. Rev. Lett. 80, 1791. Hunt, E.R. (1991). Stabilizing high-period orbits in a chaotic system: The diode resonator. Phys. Rev. Lett. 67, 1953–1955. Hunt, E.R., Johnson, G. (1993). Keeping chaos at bay. IEEE Spectrum 30, 32–36. Ikeda, K. (1979). Multiple-valued stationary state and its instability of the transmitted light by a ring cavity system. Opt. Commun. 30, 257–261. Ikeda, K., Kondo, K., Akimoto, O. (1982). Successive higher-harmonic bifurcations in systems with delayed feedback. Phys. Rev. Lett. 49, 1467–1470. Ilday, F., Buckley, J., Lim, H., Wise, F., Clark, W. (2003). Generation of 50-fs, 5-nJ pulses at 1.03 µm from a wave-breaking-free fiber laser. Opt. Lett. 28, 1365–1367. Illing, L., Bröcker, J., Kocarev, L., Parlitz, U., Abarbanel, H.D.I. (2002). When are synchronization errors small? Phys. Rev. E 66, 036299/1–8. James, G., Harrell II, E.M., Bracikowski, C., Wiesenfeld, K., Roy, R. (1990). Elimination of chaos in an intracavity-doubled Nd:YAG laser. Opt. Lett. 15, 1141–1143. Johnson, G., Hunt, E.R. (1993). Maintaining stability in Chua’s circuit driven into regions of oscillation and chaos. J. Circuits Syst. Comput. 3, 119–123.
9]
CONTROLLING OPTICAL CHAOS
695
Junge, L., Parlitz, U. (2000). Synchronization and control of coupled Ginzburg–Landau equations using local coupling. Phys. Rev. E 61, 3736–3742. Just, W., Bernard, T., Ostheimer, M., Reibold, E., Benner, H. (1997). Mechanism of time-delayed feedback control. Phys. Rev. Lett. 78, 203–206. Keys, R. (1970). Power dissipation in information processing. Science 168, 796–801. Khitrova, G., Valley, J.F., Gibbs, H.M. (1988). Gain-feedback approach to optical instabilities in sodium vapor. Phys. Rev. Lett. 60, 1126–1129. Kocarev, L., Halle, K.S., Eckert, K., Chua, L.O., Parlitz, U. (1992). Experimental demonstration of secure communications via chaotic synchronization. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 2, 709–713. Kocarev, L., Parlitz, U. (1995). General approach for chaotic synchronization with applications to communication. Phys. Rev. Lett. 74, 5028–5031. Kocarev, L., Parlitz, U. (1996a). Generalized synchronization, predictability, and equivalence of unidirectionally coupled dynamical systems. Phys. Rev. Lett. 76, 1816–1819. Kocarev, L., Parlitz, U. (1996b). Synchronizing spatio-temporal chaos in coupled nonlinear oscillators. Phys. Rev. Lett. 77, 2206–2209. Kocarev, L., Tasev, Z., Parlitz, U. (1997). Synchronizing spatio-temporal chaos of partial differential equations. Phys. Rev. Lett. 79, 5–54. Kolumban, G., Kis, G., Jákó, Z., Kennedy, M.P. (1998). FM-DCSK: A robust modulation scheme for chaotic communications. IEICE Trans. Fund. Electron. Commun. Comput. Sci. E81-A (9), 1798–1802. Larger, L., Goedgebuer, J.-P. (2004). Encryption using chaotic dynamics for optical telecommunications. C.R. Phys. 5, 609–611. Le Berre, M., Patrascu, A., Ressayre, E., Tallet, A. (1997). Localized structures in chaotic patterns: From disorder to ordering. Phys. Rev. A 56, 3150–3160. Le Berre, M., Ressayre, E., Tallet, A., Zondy, J.-J. (1990). Linear analysis of single-feedback-mirror spatio-temporal instabilities. J. Opt. Soc. Amer. B 7, 1346–1360. Lewis, C.T., Abarbanel, H.D.I., Kennel, M.B., Buhl, M., Illing, L. (2000). Synchronization of chaotic oscillations in doped fiber ring lasers. Phys. Rev. E 63, 016215/1–15. Lorenz, E.N. (1963). Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130–141. Lorenz, E.N. (1993). “The Essence of Chaos”. University of Washington Press, Seattle, WA. Lugiato, L.A. (1994). Transverse nonlinear optics: Introduction and review. Chaos, Solitons & Fractals 4, 1251–1258. Lukin, M.D. (2003). Colloquium: Trapping and manipulating photon states in atomic ensembles. Rev. Mod. Phys. 75, 457–472. Maître, A., Petrossian, A., Blouin, A., Pinard, M., Grynberg, G. (1995). Spatio-temporal instability for counterpropagating beams in rubidium vapor. Opt. Commun. 116, 153–158. McAllister, R., Uchida, A., Meucci, R., Roy, R. (2004). Generalized synchronization of chaos: Experiments on a two-mode microchip laser with optoelectronic feedback. Phys. D 195, 244–262. McKeever, J., Boca, A., Boozer, A.D., Miller, R., Buck, J.R., Kuzmich, A., Kimble, H.J. (2004). Deterministic generation of single photons from one atom trapped in a cavity. Science 303, 1992– 1994. McLaughlin, D.W., Moloney, J.V., Newell, A.C. (1983). Solitary waves as fixed points of infinitedimensional maps in an optical bistable ring cavity. Phys. Rev. Lett. 51, 75–78. Mulet, J., Mirasso, C., Heil, T., Fischer, I. (2004). Synchronization scenario of two distant mutually coupled semiconductor lasers. J. Opt. B, Quantum Semiclassical Opt. 6, 97–105. Murali, K., Lakshmanan, M. (1993). Transmission of signals by synchronization in a chaotic Van der Pol–Duffing oscillator. Phys. Rev. E 48, 1624–1626. Nakajima, H. (1997). On analytical properties of delayed feedback control of chaos. Phys. Lett. A 232, 207–210.
696
L. Illing et al.
[9
Nelson, D.F., Boyle, W.S. (1962). A continuously operating ruby optical maser. Appl. Opt. 1, 181–183. Nijmeijer, H. (2001). A dynamical control view on synchronization. Phys. D 154, 219–228. Ogata, K. (1990). “Modern Control Engineering”, second ed. Prentice-Hall, Englewood Cliffs, NJ. Ohtsubo, J. (2005). “Semiconductor Lasers: Stability, Instability and Chaos”. In: Springer Series in Optical Sciences, vol. 111. Springer, Heidelberg. Oka, M., Kubota, S. (1988). Stable intracavity doubling of orthogonal linearly polarized modes in diode-pumped Nd:YAG lasers. Opt. Lett. 13, 805–807. Ott, E., Grebogi, C., Yorke, J.A. (1990). Controlling chaos. Phys. Rev. Lett. 64, 1196–1199. Parlitz, U., Kocarev, L., Stojanovski, T., Preckel, H. (1996). Encoding messages using chaotic synchronization. Phys. Rev. E 53, 4351–4361. Parmananda, P., Rhode, M., Johnson, G., Rollins, R., Dewald, H., Markworth, A. (1994). Stabilization of unstable steady-states in an electrochemical system using derivative control. Phys. Rev. E 49, 5007–5013. Pastur, L., Gostiaux, L., Bortolozzo, U., Boccaletti, S., Ramazza, P. (2004). Experimental targeting and control of spatio-temporal chaos in nonlinear optics. Phys. Rev. Lett. 93, 063902/1–4. Pecora, L.M., Carroll, T.L. (1990). Synchronization in chaotic systems. Phys. Rev. Lett. 64, 821–824. Pecora, L.M., Carroll, T.L., Heagy, J.F. (1997a). Statistics for continuity and differentiability: An application to attractor reconstruction from time series. Fields Inst. Commun. 11, 49–62. Pecora, L.M., Carroll, T.L., Johnson, G.A., Mar, D.J., Heagy, J.F. (1997b). Fundamentals of synchronization in chaotic systems, concepts, and applications. Chaos 7, 520–543. Pikovsky, A., Rosenblum, M., Kurths, J. (2001). “Synchronization”. Cambridge University Press, Cambridge. Pyragas, K. (1992). Continuous control of chaos by self-controlling feedback. Phys. Lett. A 170, 421– 428. Pyragas, K. (1995). Control of chaos via extended delay feedback. Phys. Lett. A 206, 323–330. Pyragas, K., Tamasevicius, A. (1993). Experimental control of chaos by delayed self-controlling feedback. Phys. Lett. A 180, 99–102. Qu, Z., Hu, G., Ma, B. (1993). Controlling chaos via continuous feedback. Phys. Lett. A 178, 265–270. Romeiras, F.J., Grebogi, C., Ott, E., Dayawansa, W.P. (1992). Controlling chaotic dynamical systems. Phys. D 58, 165–192. Rosenblum, M.G., Pikovsky, A.S., Kurths, J. (1996). Phase synchronization of chaotic oscillators. Phys. Rev. Lett. 76, 1804–1807. Roy, R., Gills, Z., Thornburg Jr., K.S. (1994). Controlling chaotic lasers. Opt. Photonics News 5, 8–15. Roy, R., Murphy Jr., T.W., Maier, T.D., Gills, Z., Hunt, E.R. (1992). Dynamical control of a chaotic laser: Experimental stabilization of a globally coupled system. Phys. Rev. Lett. 68, 1259–1262. Roy, R., Thornburg Jr., K.S. (1994). Experimental synchronization of chaotic lasers. Phys. Rev. Lett. 72, 2009–2012. Rulkov, N., Afraimovich, V., Lewis, C.T., Chazottes, J.R., Cordonet, A. (2001). Multivalued mappings in generalized chaos synchronization. Phys. Rev. E 64, 016217/1–11. Rulkov, N.F., Sushchik, M.M., Tsimring, L.S., Abarbanel, H.D.I. (1995). Generalized synchronization of chaos in directionally coupled chaotic systems. Phys. Rev. E 51, 980–994. Rulkov, N.F., Volkovskii, A.R. (1993). Synchronization of pulse-coupled chaotic oscillators. In: “Proceedings of the 2nd Experimental Chaos Conference”, pp. 106–115. Rulkov, N.F., Vorontsov, M.A., Illing, L. (2002). Chaotic free-space laser communication over a turbulent channel. Phys. Rev. Lett. 89, 277905/1–4. Sargent III, M., Scully, M.O., Lamb Jr., W.E. (1974). “Laser Physics”. Addison-Wesley, Reading, MA, p. 52. Sauer, M., Kaiser, F. (1996). Synchronized spatio-temporal chaos and spatio-temporal on-off intermittency in a nonlinear ring cavity. Phys. Rev. E 54, 2468–2473. Schwarz, I.B., Triandaf, I. (1992). Tracking unstable orbits in experiments. Phys. Rev. A 46, 7439– 7444.
9]
CONTROLLING OPTICAL CHAOS
697
Shinbrot, T., Grebogi, C., Ott, E., Yorke, J.A. (1993). Using small perturbations to control chaos. Nature 363, 411–417. Silberberg, Y., Bar-Joseph, I. (1984). Optical instabilities in a nonlinear Kerr medium. J. Opt. Soc. Amer. B 1, 662–670. Socolar, J., Sukow, D., Gauthier, D. (1994). Stabilizing unstable periodic orbits in fast dynamical systems. Phys. Rev. E 50, 3245–3248. Sugawara, T., Tachikawa, M., Tsukamoto, T., Shimizu, T. (1994). Observation of synchronization in laser chaos. Phys. Rev. Lett. 72, 3502–3505. Sukow, D.W., Bleich, M.E., Gauthier, D.J., Socolar, J.E.S. (1997). Controlling chaos in fast dynamical systems: Experimental results and theoretical analysis. Chaos 7, 560–576. Sushchik Jr., M., Rulkov, N.F., Larson, L., Tsimring, L., Abarbanel, H.D.I., Yao, K., Volkovskii, A. (2000). Chaotic pulse position modulation: A robust method of communicating with chaos. IEEE Commun. Lett. 4, 128–130. Tang, S., Liu, J.M. (2001a). Message encoding-decoding at 2.5 Gbits/s through synchronization of chaotic pulsing semiconductor lasers. Opt. Lett. 26, 1843–1845. Tang, S., Liu, J.M. (2001b). Synchronization of high-frequency chaotic optical pulses. Opt. Lett. 26, 596–598. Uchida, A., McAllister, R., Meucci, R., Roy, R. (2003). Generalized synchronization of chaos in identical systems with hidden degrees of freedom. Phys. Rev. Lett. 91, 174101/1–4. Uchida, A., Rogister, F., García-Ojalvo, J., Roy, R. (2005). Synchronization and communication with chaotic laser systems. In: Progress in Optics, vol. 48. North-Holland, Amsterdam, pp. 203–342. Ushio, T. (1996). Limitations of delayed feedback control in nonlinear discrete-time systems. IEEE Trans. Circuits Syst. 43, 815–816. van der Wal, C.H., Eisaman, M.D., Andre, A., Walsworth, R.L., Phillips, D.F., Zibrov, A.S., Lukin, M.D. (2003). Atomic memory for correlated photon states. Science 301, 196–200. VanWiggeren, G., Roy, R. (1998a). Communication with chaotic lasers. Science 279, 1198–1200. VanWiggeren, G., Roy, R. (1998b). Optical communication with chaotic waveforms. Phys. Rev. Lett. 81, 3547–3550. VanWiggeren, G.D., Roy, R. (1999). Chaotic communication using time-delayed optical systems. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 9, 2129–2156. Volkovskii, A.R., Rulkov, N.F. (1993). Synchronous chaotic response of a nonlinear oscillator system as a principle for the detection of the information component of chaos. Tech. Phys. Lett. 19, 97–99. White, J., Moloney, J. (1999). Multichannel communication using an infinite dimensional spatiotemporal chaotic system. Phys. Rev. A 59, 2422–2426. Williams, Q.L., García-Ojalvo, J., Roy, R. (1997). Fast intracavity polarization dynamics of an erbiumdoped fiber ring laser: Inclusion of stochastic effects. Phys. Rev. A 55, 2376–2386. Williams, Q.L., Roy, R. (1996). Fast polarization dynamics of an erbium-doped fiber ring laser. Opt. Lett. 21, 1478–1480. Winful, H.G., Rahman, L. (1990). Synchronized chaos and spatio-temporal chaos in arrays of coupled lasers. Phys. Rev. Lett. 65, 1575–1578. Yang, T.-H., Hunt, B.R., Ott, E. (2000). Optimal periodic orbits of continuous time chaotic systems. Phys. Rev. E 62, 1950–1959. Yu, Y.H., Kwak, K., Lim, T.K. (1995). Secure communication using small time continuous feedback. Phys. Lett. A 197, 311–316. Zibrov, S., Lukin, M.D., Scully, M.O. (1999). Nondegenerate parametric self-oscillation via multiwave mixing in coherent atomic media. Phys. Rev. Lett. 83, 4049–4052. Zoldi, S., Greenside, H. (1998). Comment on ‘optimal periodic orbits of chaotic systems’. Phys. Rev. Lett. 80, 1790.
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Index Above threshold ionization (ATI), 534 AC-Stark shifts, 27, 102 Acousto-optic deflectors (AOD), 572–4, 576 – beam waist at, 574 – lens effect, 576, 579 – time-bandwidth product, 580 Acousto-optic modulators (AOM), 663 Acousto-optic spectral analyzers (AOS), 552–4, 571 ADC, 553 ADK tunneling model, 515–6 Advanced Light Source (Berkeley), 272 Alkali BECs, 4 All-optical switching, 688–91 – energy density metric, 691 α-parameter, 97–8 Analog autocorrelators, 553–4 Analog to digital converters (ADC), 553 Angular momentum – collective total, 87 – of single atom, 87 – torsional, 436–7 Anisotropic expansion, 38–9 – as signature of superfluidity, 39 Anticrossings – barrier hyperfine, 456, 458–60 – rotational hyperfine, 456, 459–60 – rotational Stark, 456, 458–9 Antihydrogen research, 134, 176 Anti-synchronization, 622 AOD see Acousto-optic deflectors AOM see Acousto-optic modulators AOS, 552–4, 571 Apparent excitation cross sections, 326, 339, 340, 417–8 – comparison between rare gases, 413–5 – see also individual rare gases Argon – electronic structure, 322, 324 – ellipticity dependence of ionization yield, 525–6 – excitation cross sections – – apparent, 350–1, 367, 410, 414
– – direct, 350–1, 359, 364, 410 – excitation out of ground level, 349–65, 417 – – multipole field model, 346–7 – – pressure effects, 349–52, 385 – excitation out of metastable levels, 365–71, 417 – – multipole field model, 348 – – relation of cross sections to J , 368 – – relation of cross sections to optical oscillator strengths, 368–71 – harmonic generation, 297–8 – laser induced degenerate states (LIDS), 299–300 – optical emission cross sections, 349 – resonance lines, 354 – – absorption coefficients, 373 – time-resolved measurements, 351, 360, 365 Array antennas, 551 Aspect ratio, 38 Astrophysics, molecular line observations, 551 Atom–atom teleportation, 101 Atom–light interaction, 85–93 – atomic spin operators, 86–7 – effective Hamiltonian, 88 – off-resonant coupling, 88–9 – polarization states of light, 87–8 – – detection, 104 – propagation equations, 90–1 – – in canonical variables, 93–4 – – in rotating frame coordinates, 91 – – for two atomic samples, 92 – rotating frame, 91–2 – two oppositely oriented spin samples, 92–3 Atom–light quantum interfaces – deterministic – – atomic motion effect, 122–5 – – experimental methods, 103–7 – – future directions, 121–2 – – light induced Stark shifts, 125–7 – – see also Decoherence; Entanglement; Projection noise; Quantum information protocols; Quantum memory 699
700 – laser noise influence, 127–8 – probabilistic schemes, 84–5 Atom trapping, 135–7 Atomic alignment effects, 89 Atomic clocks, 159 Atomic density matrix equation, 565 Atomic motion – modeling, 122–4 – as source of decoherence, 124–5 Atomic nuclei, dynamics measurements, 514 Atomic photoionization, 271–5, 521–2 – differential cross section, 271 – Opacity Project, 273–5, 296 – reduced radial wave functions, 272 Atomic sphere approximation, 309 Atomic spin operators, 86–7 Atomic structure programs, 249–50 Attosecond electron pulses – applications, 514 – measurement, 523–35 – – correlation importance, 532–3 – – launching wave packets, 523–4 – – reading molecular clock, 528–31 – – spatial distribution of re-collision electron wave packet, 524–6 – – time structure confirmation, 531–2 – – time structure of re-collision electron, 527–8 – single pulse production, 533–4 Attosecond electron wave packet motion, 540–5 Attosecond imaging, 534–9 – controlling and imaging vibrational wave packet, 538–9 – laser induced electron diffraction, 536–8 – observing vibrational wave packet motion of D+ 2 , 534–6 Attosecond optical pulses, 520–2 – applications, 513 – measurement, 521–2 – single pulse production, 521 Attosecond science – applications, 513–4 – relation to collision science, 546 Attosecond streak camera, 514, 522 Attractor bubbling, 671 Auger decay – participator, 279 – spectator, 279–80 Auto-correlation function
Index – detected intensity, 603 – frequency noise, 603 – incident intensity, 601 AUTOSTRUCTURE, 250 Auxiliary-system method, 663 Avoided crossing molecular beam method, 432, 455–60 – see also Anticrossings B-spline bases, 246 Back action, 91, 118 Backward four-wave mixing, 689 Bardeen–Cooper–Schrieffer (BCS) theory, 4, 12 Bare molecule state, 18 Barrier height (V3 ), 424–6 Barrier Suppression Ionization (BSI), 515 BCS-BEC crossover, 7 – future directions, 71–2 – momentum distribution in, 64–71 – – comparison of kinetic energy to theory, 68–70 – – kinetic energy extraction, 67–8 – – measurement, 65–7 – – temperature dependence, 70–1 – phase transition in, 63–4 – physics of, 10–7 – – modern challenges, 17 – – strongly interacting pairs, 12 – temperature measurement in, 35–6 – see also Fermi gas; Feshbach molecules; Feshbach resonance(s) BCS theory see Bardeen–Cooper–Schrieffer (BCS) theory BEC see Bose–Einstein condensation Beryllium – electron-impact excitation and ionization, 214, 218–21 – uses, 218 Bethe plots, 368–9, 375–6 Bifurcation parameter, 631 Bit error rate (BER), 624, 678–9, 681 Blackbody radiation, 155 Bleaching, by probe, 576 Bloch operator, 242–3 Boltzmann factor, 402 Born approximation, 292, 365, 376 Born–Bethe approximation, 343, 354, 368, 371 Born–Ochkur approximation, 343, 360
Index Born–Oppenheimer approximation, 180 Born–Oppenheimer potential surfaces, 177 Born–Oppenheimer separation, 192 Born–Rudge approximation, 343 Boron, electron-impact ionization, results, 227, 229 Bose–Einstein condensation (BEC), 3–4 – connection with superfluidity, 5 – see also Molecular condensates Bragg angle tuning, 592 Breit–Pauli Hamiltonian, 250, 265, 280 Breit–Wigner resonance profile, 300 Bremsstrahlung see Free–free scattering Broad-area nonlinear optical cavity, 683–8 Buttle correction, 210, 245 C2 D6 – effect of vibrational motion on barrier height, 502 – measurements analyzed for, 430 – molecular parameters, 500–1 – ν9 band, 495–7 – ν9 + ν4 − ν4 band, 497–8 – pressure broadening parameter, 469, 476 – quartic parameters, 496–7 – torsion–rotation spectroscopy, 473–6 – torsional dipole moments, 469, 476 – torsional energy levels, 491, 496 – torsional spectrum, 473–4 – torsional splittings, 495 – vibration–torsion–rotation spectroscopy, 495–8 C2 H6 – abundance measurements, 426 – Coriolis coupling operator, 492 – effect of vibrational motion on barrier height, 502 – energy level scheme, 428 – Fermi coupling operator, 495 – measurements analyzed for, 429 – molecular parameters, 440–1, 499–501 – – for gs, 468 – – theoretical versus observed values, 499– 500 – ν3 band, 491, 494–5 – ν9 band, 490–4 – ν9 + ν4 − ν4 band, 435, 490–1, 503 – operators, 440–1 – partition functions, 471, 472 – permanent dipole moment, 453
701 – pressure broadening parameter, 469, 476 – quartic parameters, 497 – Raman studies, 432 – torsion–rotation spectroscopy, 467–73 – torsional dipole moments, 469, 476 – torsional energy levels, 491–2 – torsional spectrum, 468–70, 472–3 – torsional splittings, 491–3 – transition moment matrix element, 453 – vibration–torsion–rotation spectroscopy, 490–5 CADW method see Configuration-average distorted-wave (CADW) method Carbon, electron-impact ionization, 261–4 Carrier-envelope phase stabilization, 541 Cascades, 326 Cavity Quantum Electrodynamics (Cavity QED), 82–3 – interaction with vacuum modes, 83 Cavity solitons, 627–8 CCC method see Converged close-coupling (CCC) method Centrifugal distortion, 465 – coefficients, 461 Cesium, ground states, 86 CH3 CCCD3 , 503 CH3 CCCH3 , 503 CH3 CCSiH3 , 503 CH3 CD3 – avoided crossing studies, 459 – energy level scheme, 427–8 – measurements analyzed for, 430 – molecular parameters, 500–1 – pressure broadening parameter, 469, 476 – torsion–rotation spectroscopy, 475–6 – torsional dipole moments, 469, 476 – torsional splittings, 451, 503 – vibration–torsion–rotation Hamiltonian, diagonalization, 448 CH3 CF3 – avoided crossing studies, 459 – distortion moment Q-branch, 454 – measurements analyzed for, 430 – ν5 band, 498 – torsional splittings, 451 – vibration–torsion–rotation Hamiltonian, diagonalization, 447–8 CH3 CH2 D, 501 CH3 CH3 see C2 H6 CH3 CHD2 , 501
702 CH3 SiD3 – avoided crossing studies, 459 – measurements analyzed for, 429 – torsion–rotation spectroscopy, 466–7 – torsional splittings, 451 CH3 SiF3 – avoided crossing studies, 456–60 – distortion moment Q-branch, 454 – fluorine-fluorine dipolar interaction, 505 – measurements analyzed for, 431 – torsional splittings, 451 CH3 SiH3 – avoided crossing studies, 459 – measurements analyzed for, 429 – molecular parameters, 440–1, 500–1 – ν5 band, 476, 484–9 – – interactions contributing to Fermi-like shifts, 485–7 – – observed spectrum, 485, 487–8 – – probability distribution, 434–5, 488 – – resonance-enhanced tunneling in, 434, 488 – – synthesized spectrum, 485, 487–8 – – vibrational quanta, 489 – ν12 band, 476–84, 488–9 – – Coriolis-like shifts, 479–81 – – interactions contributing to Coriolis-like shifts, 481–3 – – p P -branch, 477, 479 – – p Q-branch, 481 – – r R-branch, 477–8 – – torsional angular momentum, 483 – – vibrational quanta, 489 – – zeroth-order torsion–rotation PAM Hamiltonian, 483 – operators, 440–1 – torsion–rotation spectroscopy, 462–6 – torsional splittings, 450–3 – vibration–torsion–rotation spectroscopy, 476–89 CH4 , ortho-para transitions, 460 Channel distortion effects, 678–80 – minimization, 680–1 Channel electron multipliers, 331 Chaos, 617 – route to, 618 – ‘type-II’, 620, 648 – see also Spatio-temporal chaos Chaos communication, 618, 672–82 – approaches, 672–4
Index – channel distortion effects, 678–80 – – minimization, 680–1 – recent examples, 623–6 – summary of research, 681–2 – using fiber lasers, 674–8 Chaos control, 618, 628–55 – limitation of controllability, 651 – non-feedback (open-loop) schemes, 628–9 – notation definition, 634 – OGY closed-loop feedback scheme, 629– 31, 635 – in one-dimensional logistic map, 631–4 – outlook, 691 – stabilization technique categories, 628 – summary of research, 655 – of UPOs see Unstable periodic orbits (UPOs), control – of USSs see Unstable steady states (USSs), control Chaos masking, 672–3 Chaos modulation, 673–5, 686 Chaos synchronization, 618, 656–72 – communication based on see Chaos communication – connection to control, 656–7 – early work, 618–9 – errors, 668–71 – – between mismatched cavities, 687–8 – – spatially-averaged, 685 – generalized synchronization, 661–5 – identical synchronization, 657–61 – phase synchronization, 665–8 – summary of research, 672 – in VCSELs, 621–3 Chaotic attractor, 657, 671, 679 Chaotic dynamics, 617 Chaotic impulse generators, 680 Chaotic Pulse Position Modulation method, 680–1 Charge-exchange fast beam sources, 336–7 Chirp transform see Photon echo chirp transform Chirped-pulse-amplification (CPA) scheme, 293 Chlorofluorocarbon compounds, 426 CIV3, 250 Classical trajectory Monte Carlo method, excited state ionization, 227–9 Clocks, synchronization of, 618, 656 Coherent control, 654–5
Index Coherent spin state (CSS), 95 Cold plasmas, 153, 415 – “boil-off” from, 154 – cold Rydberg-atom gases and, 153–5 – hydrogen, 288 Cold Rydberg atoms, 131–97 – auto-ionization, 195, 196 – coherent control of interactions, 159–76 – – coherent Rydberg excitations in manybody systems, 160–1 – – excitation-statistics measurement implementation, 169 – – excitation-statistics measurement results, 170–3 – – motivation, 159–60 – – see also Rydberg excitation blockade; Rydberg–Rydberg interactions – cold plasmas and gases of, 153–5 – decoherence sources, 159–60 – excitation-hopping effects, 176 – field ionization in strong electric and magnetic fields, 192–6 – Hamiltonian in strong magnetic field, 192 – long-lived see Guiding-center drift atoms – photo-ionization in strong magnetic field, 192–6 – preparation – – atom trapping, 135–7 – – detection of Rydberg atoms, 143–6 – – excitation in zero magnetic field, 137–9 – – laser cooling in strong magnetic field, 146–9 – – magnetic trapping in strong magnetic field, 146–9 – – STIRAP excitation into Rydberg states, 139–43 – radiative lifetimes, 159 – state-mixing collisions in gases of, 150–3 – state mixing in strong magnetic fields, 193– 6 – trapping, 176–86 – – benefits, 176 – – electrostatic, 177–9 – – experimental realization, 186–91 – – with ponderomotive optical lattices, 180– 2 – – in strong magnetic fields, 146–9, 182–6, 191 – – in weak magnetic fields, 179, 191 – – see also Trapped Rydberg atoms
703 Collision cross section measurement, 36–7 Collision length, 328 Collision strength, 249 – effective, 249 Collisional coupling potential, 346–7 Collisional quenching of vibrations, 52 Complete synchronization see Identical synchronization Computer Physics Communications (CPC), 250 Configuration-average cross-section, 205 Configuration-average distorted-wave (CADW) method, 204–6 – beryllium ionization, 218–19 – neon ionization, 225–7 Configuration interaction basis, 242 Conjugate momenta operators, 439 Control loop latency, 638, 642 – controlling UPOs with large latency, 642–7 Controlled fusion research, 222 Converged close-coupling (CCC) method, 204, 259–61 – beryllium ionization, 219 – excited state ionization, 228 Cooper pairing, 11 Coriolis coupling constants, 470, 472, 502 Coulomb–Born exchange approximation, 265 Coulomb focusing, 518, 528 Coulomb four-body breakup, 215, 230 Cross-dimensional rethermalization, 36–7 Crossed dipole trap configuration, 66 Cryocoolers, 608 Cryostat windows, 580 Crystal field Hamiltonian, 309 Crystal field potential, 308 CTMC method see Classical trajectory Monte Carlo method Cuprate superconductors, 71 Cyclotron oscillation, 183 DAC, 552–4 Decoherence – atomic motion as source of, 124–5 – entanglement with, 98–9 – experimental results, 113–15, 120–1 – quantum memory mapping with, 100 – sources in Rydberg systems, 159–60 Delay dynamical systems, 623 Derivative control, 649
704 Deterministic chaos, 617 Deuterium – kinetic energy distribution of D+ dissociation – – at different laser wavelengths, 535–6 – – calculated, 532–3 – – observed, 530–1 – vibrational wave packet – – controlling and imaging, 538–9 – – observing motion, 534–6 Dielectronic recombination (DR), 227, 275– 6, 278 Digital autocorrelators (DAC), 552–4 Diode resonator, 640–1 Dipole acceleration operator, 271 Dipole blockade see Rydberg excitation blockade Dipole–dipole interactions – means of achieving, 165–7 – versus van der Waals interactions, 161, 163–5 Dipole length operator, 271 Dipole moment operator, 460–1 – torsional dependence, 433–4 – – for XY3 AB3 , 460–2 – see also Torsional dipole moments Dipole velocity operator, 271 Dirac Hamiltonian, 250 Direct excitation cross sections, 326, 340, 417–8 – comparison between rare gases, 410–3 – measurement, 340–1 – see also individual rare gases Direct mapping protocol, 99–100 Direct potential, CADW method, 206 Dissipative chaotic systems, 657 Dissociation, 283, 285 Dissociation spectra, 46 Dissociative attachment, 283, 285 Distorted-wave exchange approximation, 265 Distorted-wave (DW) method, 407–9 – excited-state ionization, 227–9 – see also Configuration-average distortedwave (CADW) method Distorted-wave R-matrix (DWRM) method, 265–8 – second-order effect inclusion, 269 Distortion dipole moment, 432, 453–4 Domain of control, 652 – coherent control, 655
Index – incoherent control, 654 Domain decomposition, 213 Double-differential cross sections (DDCS), 268 Double sequential ionization, 530 Dressed molecule, 19 DW method see Distorted-wave (DW) method DWRM method see Distorted-wave R-matrix (DWRM) method ECDL see External cavity diode laser ECS see Exterior complex scaling (ECS) method EDFRL, 674, 677–8 – see also Erbium-doped fiber amplifier EELS, 307 – LE-EELS, 307–11 Effective cross section see Apparent excitation cross sections Einstein–Podolsky–Rosen entangled state, 95 – see also Two mode squeezing protocol Electron atom scattering, laser-assisted, 293 Electron atom scattering at intermediate energies, 256–71 – Born-series methods, 265, 269–70 – computer programs, 268 – distorted wave methods, 265–70 – – see also Configuration-average distortedwave (CADW) method; Distortedwave R-matrix (DWRM) method – pseudostate methods, 256–64 – – see also R-matrix with pseudo states (RMPS) method Electron atom scattering at low energies, 241–56 – computer programs, 249–51 – illustrative results, 251–6 – R-matrix theory for, 241–9 – – arbitrary boundary condition methods, 246 – – asymptotic region solution, 248–9 – – collision strength, 249 – – external region solution, 246–8 – – internal region solution, 241–3 – – pseudo-orbitals, 245 – – radial continuum basis orbitals, 242, 244– 6 – – time-independent Schrödinger equation, 241
Index – – total cross section, 248–9 Electron-beam production of metastable atoms, 337–8 Electron energy distribution function (EEDF), 415 Electron energy-loss spectroscopy (EELS), 307 – low-energy (LE-EELS), 307–11 Electron exchange, effects on cross sections, 347–8 Electron-impact excitation of rare-gas atoms – analogy with optical excitation, 345 – applications, 415–6 – experimental methods, 325–41 – – time-resolved measurements, 340–1 – systematic trends in rare-gas cross sections, 410–5 – see also Argon; Electronic structure; Helium; Krypton; Neon; Xenon Electron interferometry, 523, 534, 540 Electron molecule scattering, 282–8 – from H+ 3 molecules, 288 – low-energy, by oxygen molecules, 285–8 Electron motion, in intense laser field, 516–8 Electron orbital, tomographic imaging, 540 Electron re-collision, 513, 518–9 – processes induced, 518–9 – quantum perspective, 519–20 Electron-Rydberg collisions, 151–3 Electron tunneling, ADK model, 296–7 Electron wave packet motion see Attosecond electron wave packet motion Electronic structure, 321–5 – argon, 322–4 – helium, 325 – krypton, 323–4, 384 – neon, 321–2, 324 – xenon, 324 Electro-optic feedback, 623, 625 Engraving chirps, 588 Engraving submodulation, 569 Entanglement, 95–9 – coherent spin state, 95 – conditional, 115–7 – criterion for, 95, 97 – experimental results, 115–8 – Gaussian state modeling, 99 – generation and verification, 95–7 – model with decoherence, 98–9 – theoretical modeling, 97–8
705 – unconditional, 117–8 Entangling pulse, 96 Entropy – message, 686–7 – translation to temperature, 35 Equivalent current densities, 527 Er3+ :YSO, in photon echo chirp transform spectrum analyzer, 589–94 Erbium – energy levels of ions, 589 – in fiber lasers, 659, 661 – see also Er3+ :YSO Erbium–doped fiber amplifier (EDFA), 661, 674–5 – see also EDFRL ETDAS see Extended time-delay autosynchronization Ethane isotopomers see C2 D6 ; C2 H6 ; CH3 CD3 Evaporative cooling, 25–6 Exchange potential, CADW method, 206 Excitation-autoionization (EA), 262, 265 Excitation gap, 8, 15 Excited-state ionization, 227–9 Exponential decay law, modification, 300 Extended time-delay autosynchronization (ETDAS), 637–42 – all-optical implementation, 641–2 – controlling USSs using, 649–55 – – in lasers, 651–5 – diode resonator, 640–1 – transfer function for feedback, 639–40 Exterior complex scaling (ECS) method, 204, 261 External cavity diode laser (ECDL), 558–9, 577, 596–8, 607 Far-off-resonance optical dipole trap (FORT), 25 Faraday rotation, 89 FARM, 251 Fast Rydberg atoms – collision-induced production, 155–9 – – time-of-flight data, 157 – – velocity distribution, 157–8 Feedback gain, 638 Fermi–Dirac distribution functions, 28–9 Fermi energy, 28 Fermi gas – cooling, 24–6
706 – creation, 6 – momentum distribution in BCS-BEC crossover, 64–71 – – comparison of kinetic energy to theory, 68–70 – – kinetic energy extraction, 67–8 – – measurement, 65–7 – – temperature dependence, 70–1 – in optical lattice potentials, 72 – optically trapped, 26–9 – pairing in, 11–2 – temperature measurement, 26, 29–36 – – in BCS-BEC crossover, 35–6 – – from momentum distribution, 29–32 – – using impurity spin state, 32–5 Feshbach molecules – condensation of see Molecular condensates – creation from Fermi gas of atoms, 42–50 – – magnetic-field association, 43–4 – – molecule conversion efficiency, 47–9 – – precise measurement of B0 , 50 – – rf spectroscopy, 44–7 – lifetime, 52–4 Feshbach resonance(s), 6, 18–24 – description, 18–9 – elastic scattering near, 36–42 – in electron scattering by Si III, 252 – inelastic collisions near, 51–5 – – in 6 Li systems, 55 – 40 K system, 19–24 – – summary, 42 – location, 7 – position determination via molecule dissociation, 50 – universal properties, 24 Fiber lasers, 590, 659–62 – chaos communication using, 674–8 Filter bank analyzers, 553–4 Floquet–Fourier expansion, 277, 294, 302 Floquet multiplier, of uncontrolled map, 632 Fluid convection, 617 FORT, 25 Four-wave mixing – backward, 689 – forward, 689 – time-delayed, 567–9 Fourier transform microwave (FTMW) spectrometer, 451–4 Fourier transform spectrometer (FTS), 332– 3, 378–9
Index Franck–Condon factor, 46 Free–free scattering, 276, 300 Free Rotor (FR) Model, 443–4, 503 Frequency agile laser oscillator (FALO), 582 Frequency agile laser technology, 595–607 – electro-optic tuning of diode laser extended cavity, 596–8 – laser chirp spectral purity characterization, 598–607 – – deterministic error detection, 600–1 – – stochastic error detection, 601–6 – perspectives, 607 – requirements, 595–6 Frequency locking loops, 607 Frequency noise – low-frequency technical, 601–5 – white, 601–2, 604–5 FTMW spectrometer see Fourier transform microwave (FTMW) spectrometer FTS see Fourier transform spectrometer Gabor analytic signal, 667 Gap equation, 14 – renormalized, 15 Gap parameter, 15 Gas-discharge lasers, 415 Gaussian state modeling, 99 Ge detectors, 333 Generalized synchronization, 661–5 Global warming reduction, 426 Globar, 467 Gram–Schmidt orthogonalization, 210 GRASP, 250 GRASP92, 250 ‘Green problem’, 620–1 – OPF control, 635–7, 647–8 gs barrier height parameter (V0,3 ), 499–502 gs barrier shape parameter (V0,6 ), 499–501 Guiding-center drift atoms, 183–4 – decay, 187–9 – production, 186–7 – trajectories, 190 – see also Trapped Rydberg atoms Harmonic generation, 293, 295–6 – in argon, 297–8 Hartree–Fock equations, 214 Hartree potential, direct, 214 Hartree term, 67 Heisenberg evolution, 90
Index Heisenberg uncertainty relation, 87 – for canonical atomic operators, 95 Helium – DDCS for ionization, 268 – distorted wave methods for ionization studies, 268–70 – electron-impact double ionization partial collision probability, 217 – – results, 230–2 – – total cross section, 215, 218 – electron-impact single ionization partial collision probability, 218 – – results, 230–2 – – total cross section, 218 – electronic structure, 325 – excitation out of ground level, 342–4 – excitation out of metastable levels, 344–5 – LS-coupling, 325 – multiphoton ionization, 296–7 – positronium fragmentation in collision with atoms of, 292 – second-Born RMPS results, 269 – SEPE process in electron scattering, 300–1 – TDCS for ionization, 268–9 HERSCHEL mission, 553 High-B atom traps, 146–9, 192 High Barrier (HB) Model, 439, 443–4, 503 High harmonics – intensity map of calculated spectrum, 544– 5 – photon energy to time relation, 543 – spatial coherence measurement, 513 HITRAN database, 426 Hollow-atom–hollow-ion decay paths, 272–3 Hollow cathode discharge, 334–6 Hopf bifurcation, 652 Huygens–Fresnel principle Fraunhofer approximation, 575 Hydrofluorocarbons, 426 Hydrogen – elastic scattering phase shifts, 210 – electron-impact ionization – – methods, 211, 261 – – partial collision probability, 213 – – results, 227–9 – – total cross section, 211, 213 – electron molecule scattering from H+ 3 molecules, 288 – ellipticity dependence of re-collision yield, 525–6
707 – kinetic energy distribution of H+ dissociation – – calculated, 531–2 – – observed, 528–9 – Lyman-α emission line, 354 – multiphoton ionization of H2 molecules, 303–5 – potential energy surfaces, 523–4 – SEPE process in electron scattering, 300 Hydrogenic manifolds, 152 Hyperconjugation, 425 Hyperfine Hamiltonian, 456, 503, 505 Hyperspherical close-coupling method, 204, 261 IAM see Internal Axis Method Identical synchronization, 657–61 IERM see Intermediate Energy R-matrix Method Impurity spin-state thermometry, 32–6 Incoherent control, 652–4 Inelastic decay processes, 51–2 Information loss, in chaos communications, 678–80 Infrared torsional bands, 432 Infrared transitions, measurement, 332–3 Interchannel coupling parameter, 22 Intermediate Energy R-matrix Method (IERM), 258, 261 Intermediate value theorem, 651 Internal Axis Method (IAM), 433–4, 436–8 – advantages and disadvantages, 437 Internal rotation in symmetric tops, 424–505 – avoided crossing molecular beam method, 432, 455–60 – – see also Anticrossings – discussion of results, 498–505 – future perspective, 503 – hybrid model, 432, 437–8 – – advantages, 438 – one-band model, 433 – rotational spectroscopy, 449–54 – simplified model, 449 – spectra types, 432 – three-band model, 433 – two-band model, 433 – zeroth-order torsion–rotation Hamiltonian, 436–7 – see also Torsion–rotation spectroscopy; Vibration–torsion–rotation Hamil-
708 tonian; Vibration–torsion–rotation spectroscopy Internal rotor Hamiltonian – zeroth-order – – X2 Y6 symmetric top, 438 – – XY3 AB3 symmetric top, 437 Intracavity intensity modulator, 675 Intramolecular vibrational redistribution (IVR), 426, 488 Ioffe–Pritchard trap, 146, 148 Ionic motion, 185 Ionization avalanche, 155 Ionization energies, rare gases, 415 Ionization spin symmetry, 259–60 Iron – R-matrix calculation results – – electron-impact excitation of Fe II (peak element), 254–6 – – electron-impact excitation of Fe XXVI, 280 – – photorecombination of electrons incident on Fe17+ , 281–2 – radiative transmissions, 274–5 IRON Project, 254 IVR see Intramolecular vibrational redistribution JAJOM, 250 Jaynes–Cummings approach, 82–4 K-matrix, 248 – transformation to pair coupling scheme, 250 – unphysical, 251 Kapitza–Dirac effect, 180 Keldysh parameter, 516 Kramers degeneracy, 589 Kramers–Henneberger transformation, 300 Krypton – Auger decay dynamics, 513 – core angular momentum, 323, 384, 391–2 – electronic structure, 323–4, 384 – excitation cross sections – – apparent, 393, 414 – – direct, 389, 410 – excitation out of ground level, 385–91 – excitation out of metastable levels, 391–7 – – theoretical calculations, 407–8 – number density ratio, 336 KTP crystal, 634–5
Index l-doubling, 451–2, 504 – Hamiltonian, 449 Ladder operators, 439 Lagrange mesh bases, 246 Landau energy levels, free-electron, 184 Landau quantization, in Rydberg atoms, 193–4 Landau–Zener formula, 48 Larmor frequency, 91 – ratio with Stark frequency, 179 Laser-assisted electron atom scattering, 293 Laser cavity mode frequency shift, 596 Laser cooling, 25 – in strong magnetic field, 146–9 Laser external cavity feedback, 625 Laser-induced-continuum-structure (LICS), 299 Laser induced degenerate states (LIDS), 298– 300 – in argon, 299–300 Laser induced electron diffraction, 536–8 Laser-induced-fluorescence (LIF), 326, 335, 339–40 Laser noise, influence on atom–light quantum interfaces, 127–8 Laser quenching, 365–6 Laser thresholds – first, 652 – second, 652 Lasers – broadband pulsed dye, 137–8 – narrowband diode, 138 LCLV see Liquid crystal light valve LE-EELS, 307–11 Level cross section see Direct excitation cross sections LIDS see Laser induced degenerate states Light–atom teleportation, 102 Light damping/loss, 100, 120–1 Light/matter interaction strength, 94 Light polarization, and Stark shifts, 125–7 Linear stability analysis, 650, 654 Liquid crystal light valve (LCLV), 626 LiTaO3 crystal wedge, 597 Lithium – electron-impact ionization, 214 – – results, 227–9 – hollow-atom–hollow-ion decay paths, 272– 3
Index – K-shell photodetachment, 280 Littrow wavelength shift, 596 Logistic map, chaos control in, 631–4 Low Velocity Intense Source (LVIS), 136 – pyramidal, 136, 146 Lyapunov dimension, 679 Lyapunov exponents, 642, 657–8 – transverse/conditional, 658–9, 670–1 m-mixing effects, in Rydberg atoms, 195–6 Mach–Zehnder interferometer, unbalanced, 599–607 – detected intensity, 599 – deterministic error detection, 600–1 – quadrature signals at output, 606–7 – stochastic error detection, 601–6 Mach–Zehnder modulator (MZM), 555, 570– 1, 578, 582 – in electro-optic feedback device, 623, 624 Macroscopic spin, measurement, 108–9 Madelung potential, 308 “Magic angle”, 332 Magnetic-bottle force, 184 Magnetic fields – association, 43–4 – in atom–light quantum interfaces, 105–7 – interaction of atoms with, 105 Magneto–Optical Resonance method (MORS), 105–7 Magneto-optical traps see MOTs Magnetron drift, 183–4 Magnetron magnetic moment, 184 Mandel Q parameter, 170 Many-body adiabaticity, time-scale for, 59– 60 Maxwell–Bloch equations, 90 MC scheme, 581, 583 MCDF, 250 MCHF, 250 MCM scheme, 581 Mean-field interaction energy measurement, 39–42 Methane see CH4 Methyl silane isotopomers see CH3 SiD3 ; CH3 SiH3 Microchannel plate detectors (MCP), 145–6, 148 Microwave spectroscopy, on trapped Rydberg atoms, 176 Mirror-less self-oscillation, 689
709 Mode hop free frequency tuning, 597 Molecular clock, 523 – reading, 528–31 Molecular condensates – emergence from Fermi gas of atoms, 56–9 – observation in BCS-BEC crossover, 60–3 – phase diagram measurement, 63–4 Molecule conversion efficiency – dependence on magnetic-field ramp rate, 47–8 – dependence on phase space density, 49 – understanding, 47–9 Molecule decay rate – dependence on atom–atom scattering length, 53–4 – – comparison of 40 K and 6 Li, 54–5 Molecule dissociation, via rf spectroscopy, 47 Molecule momentum distribution, 57–8 MOTs – double, 136 – LVIS type, 136 – – pyramidal, 136, 146 – magneto-optic compression phase, 137 – vapor-cell, 135–7 Mott insulator state, 5 MQDT see Multichannel quantum defect theory Muffin-tin approximation, 309 Multichannel bound state, of Feshbach resonance, 22 Multichannel potential, 18–9 Multichannel quantum defect theory (MQDT), 251, 276, 279, 280 Multi-electron atomic systems, approximate solutions, 214–5 Multimode semiconductor lasers, 682–3 Multiphoton ionization, 293, 295–6, 299, 307, 516 – in helium atoms, 296–7 – in hydrogen molecules, 303–5 Multiphoton processes, 293–307 – time-dependent R-matrix theory (TDRM), 305–7 – see also Harmonic generation; Laserassisted electron atom scattering; Multiphoton ionization; R-matrix–Floquet (RMF) theory Multiple delayed feedback control, 620 Multipole field model – applied to heavy rare gases, 346–8
710
Index
– – excitation out of ground level, 346–8 – – excitation out of metastable levels, 348 – helium, 342–5 Mutual information, 686–7
NSR ground state, 70 NSR theory, 14, 17 Nuclear reactions, R-matrix theory for, 206, 237
Narrow resonance, 24 Nd:YAG lasers – generalized synchronization, 663–5 – in ‘green problem’, 620–1, 634–5 – identical synchronization, 659 – OPF control, 635–7 – phase synchronization, 667–8 Neodymium, in fiber lasers, 659 Neon – double ionization, 514 – electron-impact excitation and ionization, 221–7 – – R-matrix calculation results, 251–2 – electronic structure, 321–2, 324 – excitation cross sections – – apparent, 383, 414 – – direct, 376, 378, 410 – excitation out of ground level, 372–82 – excitation out of metastable levels, 382–4 – gas-puff experiments, 222 – inner shell photoexcited resonances, 280 – LS-coupling, 321 – optical emission cross sections, 377, 382 – resonance lines, absorption coefficients, 373–4 – time-resolved measurements, 379 NiO, 3d–3d excitations in, 307–11 Nitrogen, double ionization, 514 Noise, high quality synchronization prevention by, 670–1 Nonlinear optical cavity, broad-area, 683–8 Non-perturbative close-coupling methods, 204, 206–32 – results, 218–32 – – electron-impact excitation and ionization of beryllium, 218–21 – – electron-impact excitation and ionization of neon, 221–7 – – electron-impact single and double ionization of helium, 230–2 – – ionization out of excited states of H-like ions, 227–9 – see also R-matrix with pseudo states (RMPS) method; Time-dependent close-coupling (TDCC) method
Occasional Proportional Feedback (OPF), 635 – application to ‘green problem’, 635–7, 647–8 – stabilizing USSs using, 647–8 ODIN satellite, 553 Off-resonant coupling, 88–9 OGY control algorithm, 629–31, 635 One-dimensional logistic map, chaos control in, 631–4 One-electron atomic systems, exact solutions, 211–4 Opacity Project, 273–5, 296 OPA, 528 OPCPA, 541 Open channel, 18 Open loop configuration, 660–1, 670 Open-loop devices, 624 Optical Bloch equation, 566 Optical chaos see Chaos Optical dipole trap, 27 – harmonic approximation, 28 – see also FORT Optical emission cross section, 325–6, 328 – dependence on target gas pressure, 327, 349–51 – systematic uncertainty of measurements, 331 – in terms of experimentally measurable quantities, 329 Optical memory, 550 Optical oscillator strengths – relation of cross sections to – – argon, 368–71 – – krypton, 393, 395–7 – – neon, 383–4 – – xenon, 405–6 Optical parameter amplifier (OPA), 528 Optical pulses – pulse duration decreases, 512 – see also Attosecond optical pulses Optically carried RF signals see RF spectrum analysis Opto-electronic feedback, 623, 668 Orbital tomography, 540
Index Ortho-para transitions, 460 Oxygen – inner shell photoexcited resonances, 280 – low-energy electron scattering by O2 molecules, 285–8 – photorecombination cross sections for O VIII, 280 Ozone layer depletion, 426 Page-oriented storage, 550 Pair dissociation temperature, 17 Pairing gap, 17 Pairing temperature, 17 PAM see Principal Axis Method Partial wave pairing, higher, 71 Partition functions, ethane, 471–2 Paschen–Back regime, 148 Patterns, 688–91 – outlook, 691 Pauli blocking, 11, 26 Pauli exclusion principle, 26 Penning ionization, 156–8 Penning-ionizing collisions, 153, 155, 158–9, 187 Permanent dipole moment, 428, 453 Perturbation sensitivity, of uncontrolled map, 632 Perturbation theory, 516 – see also Multiphoton ionization Phase jitter, 601 Phase synchronization, 665–8 Phase transition temperature, 16 Photodiode array (PDA), in rainbow analyzer, 578 Photoexcitation of the core (PEC), 275 Photoionization see Atomic photoionization Photon echo chirp transform, 554, 581–8 – bandwidth condition, 585 – echo duration condition, 585 – echo exponential decay, 587 – engraving chirps, 588 – expected features, 585–6 – material finite coherence time influence, 586–7 – physical picture, 583–5 – RF modulation, 584–5 – single side band analysis, 586 – time separation condition, 585 Photon echo chirp transform spectrum analyzer, 581–94
711 – advantages, 594 – discussion of results, 594 – experimental demonstration in Er3+ :YSO, 589–94 – – high resolution testing, 593 – – multi-frequency RF signal input, 591–2 – – signal decay, 593–4 – – square signal input, 591 – laser requirements, 595–6 – stimulated photon echo signal, 582 – technological challenge, 594 Photon echo configuration, 569–70, 575–6 Photonic device, chaotic, 643–6 Photorecombination, 275–82 – cross section, 275, 279 – dielectronic recombination (DR), 227, 275– 6, 278 – examples, 280–2 – free–free scattering, 276, 300 – radiative recombination (RR), 275–6 π -pulses, 140 Pitchfork bifurcation, 652 Planetary aurora, 288 Plasmas, low-temperature see Cold plasmas Polarization density, 565, 569 Polarization-free geometry, 331–2 Polarization states of light, 87–8 – detection, 104 Pole approximation, 21 Poly-Logarithmic function of order n, 29 Ponderomotive energy, 516 Ponderomotive force, 180 Ponderomotive optical lattices, 180–2 Ponderomotive potential, 180 Ponderomotive shift, of ionization threshold, 304 Positron atom scattering, 289–92 – by atomic hydrogen, 290–1 Positronium (Ps), 289 – formation, 289–90 – fragmentation in collision with He atoms, 292 – scattering, 291 – states, 291 Pound–Drever–Hall technique, 598 PRECUPD, 211 Preformed pairs, 17, 65 Pressure broadening parameter, ethane isotopomers, 469, 476 Principal Axis Method (PAM), 433, 436–8
712 – advantages and disadvantages, 437 Probe field, 103 – spectral shape, 565 Probing pulse, 583 Programmable filters, 564–70 – coherent combination of local response, 569–70 – linear response, 565–7 – reconfigurable filtering, 564–5 – time-delayed four-wave mixing, 567–9 Projection noise (PN), 95, 96, 108–13 – experimental investigation, 110–1 – level prediction, 109–10 – linearity, 113 – macroscopic spin measurement, 108–9 – thermal spin noise, 111–2 Propagation equations – for atom–light interaction, 90–1 – – in canonical variables, 93–4 – – in rotating frame coordinates, 91 – – for two atomic samples, 92 Pseudo-orbitals, 210, 245 – non-orthogonal Laguerre, 210 Pseudo potentials, 214 PSTG1, 211 PSTG2, 211 PSTG3, 211 PSTGF, 211 Pulsed plasma afterglows, 338–40 Pump-probe approaches, 534, 543–5 Q-branch, 432 Quadrupole sum rule, 345 Quantum cryptography, secure, 159 Quantum feedback, 84 Quantum gates, realization, 175 Quantum information processing – entangled sample control in, 159 – Rydberg-atom trapping benefit, 176 Quantum Information Processing and Communication (QIPC), 82 Quantum information protocols, 93–102 – deterministic generation of entanglement see Entanglement – direct mapping, 99–100 Quantum memory, 99–102 – alternative proposals, 101–2 – direct mapping protocol, 99–100 – experimental results, 118–21 – fidelity, 120
Index – mapping with decoherence, 100 – multipass protocols, 121 – retrieval, 101 Quantum non-demolition (QND)–Faraday – – interaction, 84, 101–2 Qubit state manipulation, deterministic, 122 R-branch, 432 R-matrix basis functions, 207 R-matrix–Floquet (RMF) theory – atomic, 293–301, 305 – – laser field vector potential, 293 – – recent applications, 300–1 – – results, 296–300 – – total electron momentum operator, 294 – molecular, 301–5 – – diatomic molecules in fixed-nuclei approximation, 301–2 – – laser field vector potential, 302 R-matrix methods see R-matrix theory R-matrix programs, 211, 249–51, 268 – parallel, 251, 256 R-matrix propagator methods, 247–8, 302, 307 R-matrix with pseudo states (RMPS) method, 204, 206, 209–11, 257–8 – beryllium ionization, 218–21 – carbon ionization, 261–4 – excited state ionization, 227–9 – future perspective, 233 – hydrogen H+ 3 electron molecule scattering, 288 – hydrogen ionization, 259–61 – hydrogen positron scattering, 290–1 – neon excitation, 222–4, 408 – neon ionization, 226–7 – second-Born, 269 R-matrix theory, 206–9, 237–312 – applications, 238, 311, 407–9 – atomic photoionization, 271–5 – beryllium ionization, 219–21 – configuration space partitioning, 238–9, 242, 284, 294, 302 – early work, 237–8, 239 – electron atom scattering, 238–9 – – see also Electron atom scattering at intermediate energies; Electron atom scattering at low energies – electron energy loss from transition metal oxides, 307–11
Index – electron molecule scattering, 283–8 – – fixed-nuclei approximation, 283–4 – – see also Electron molecule scattering – with extended basis set (RMEB), 408–9 – future perspective, 311–2 – (N + 1)-electron problem, 407 – neon excitation, 222–4 – non-adiabatic, 284, 305 – optical potential extension, 279–80 – photorecombination and radiation damping, 276–82 – – see also Photorecombination; Radiation damping – positron atom scattering, 290–2 – – see also Positron atom scattering – time-dependent (TDRM), 305–7 – see also Intermediate Energy R-matrix Method (IERM); R-matrix–Floquet (RMF) theory Racah notation, 324 RADAR, 551 Radial Slater integrals, 205 Radiation damping, 275–82 – effect on electron impact excitation of Fe XXVI ion, 280 – potential, 278 Radiation trapping, 351, 373, 381, 418 Radiative recombination (RR), 275, 276 Radio frequency spectral analyzers see RF spectrum analysis Rainbow analyzer, 570–80 – bandwidth flexibility, 577 – box configuration, 572 – dynamic range, 577, 580 – experimental results, 578–9 – experimental setup, 577–8 – future improvements, 579–80 – laser requirements, 595 – principle of operation, 570–2 – programming stage, 572–5 – spectral resolution, 575–7 Rare earth ion-doped crystals (REIDC), 556– 7 Rare gases – technological importance, 325 – see also Argon; Electron-impact excitation of rare-gas atoms; Helium; Krypton; Neon; Xenon Reading pulse see Probing pulse Relaxation oscillations, 635
713 – destabilization, 635 REMPI peaks, 296, 304 Resonance-enhanced-multiphoton-ionization (REMPI) peaks, 296, 304 Resonance levels, 327 Resonance radiation trapping, 326–9 Resonance superfluids, 7 Resonance width parameter, 22 Resonant coupling strength, 167 Resonant-excitation-auto-double-ionization (READI), 263 Resonant-excitation-double-autoionization (REDA), 262–3 Resonant levels, 327 RF spectroscopy, 44–7 – technique extension, 47 RF spectrum analysis – conventional analyzers, 552–4 – integrated cryogenics requirement, 608 – see also Frequency agile laser technology; Photon echo chirp transform spectrum analyzer; Programmable filters; Rainbow analyzer; Spectrum photography architecture Rigid rotor Hamiltonian, 437 RMATRIX I programs, 211 RMATRX-ION, 268 RMEB method, 408–9 RMF theory, see R-matrix–Floquet (RMF) theory RMPS method see R-matrix with pseudo states (RMPS) method Rössler oscillator, periodically driven, 665–7 Rotating frame, in atom–light interaction, 91–2 Rotating wave approximation, 566 Rotational constants, effective, 450 Rotational excitation, 283 Rotational spectroscopy, 449–54 Routh–Hurwitz stability criterion, 651, 654 Rubidium, two-photon transition, 137 Rydberg atom spectroscopy, 182 Rydberg atoms, 135 – see also Cold Rydberg atoms; Fast Rydberg atoms; Trapped Rydberg atoms Rydberg electron, 180 – potential in uniform external electric field, 177 – Schrödinger equation, 177 Rydberg excitation blockade, 161
714 – effect measurement – – methods, 168–9 – – results, 171–3 – effect on transition linewidth, 173–4 – experiments in progress, 175 – planned research, 175–6 – saturation effect, 174 – state-mixing collisions importance evaluation, 162–3 Rydberg–Rydberg interactions, 161–5 – control options, 165–7 – coupling strength, 164 – energy mismatch, 164 Rydberg–Rydberg transition frequencies, 182 Rydberg states – negative-m, 185–6 – positive-m, 185 S-matrix, 248 s-wave pairing, 6 Scattered electron radial wave-function, 208 Scattering length – measured through mean-field interaction, 41 – s-wave, 12–3 Scattering potentials, CADW method, 206 Schawlow–Townes line-width, 601–2 Semiconductor chip manufacturing, 415 SEPE, 300–1 Sextic splittings, 504–5 SHB see Spectral hole burning Silicon, electron-impact excitation, R-matrix calculation results, 252–4 Simultaneous electron-photon excitation (SEPE), 300–1 Single cell atomic memory, 102 Single-photon devices, 688, 691 Slater–Condon parameters, 413 Slew rate, 144 SOFIA, 554 Solar spectra analysis, 252 Spatio-temporal chaos, 619, 682–8 – outlook, 691 – recent examples, 626–8 – synchronization efficiency, 685 Spectral addressing, 550 Spectral hole burning (SHB), 550 – applications, 550–2 – – see also Frequency agile laser technology; Photon echo chirp transform spectrum
Index analyzer; Programmable filters; Rainbow analyzer; RF spectrum analysis; Spectrum photography architecture Spectrum photography architecture, 555–64 – basic spectroscopic properties of Tm3+ :YAG, 556–8 – collinear experiment, 558–62 – – detected intensity, 562 – dynamic range of spectrum analyzer, 561 – laser requirements, 595 – non-collinear experiment, 562–4 – – detected intensity, 563 – principle of operation, 555–6 SPIDER, 528, 540 Spin–orbit interactions – in rare gases, 413 – in Rydberg atoms, 195–6 Spin squeezing, in atomic ensemble, 84, 117 Spin state – characterization, 105–7 – manipulation, 107 Squint effect, 551 SSFI, 144–6 Staircase frequency scan, 578–9 Stark effect, state dependence of, 462 Stark maps, 144, 165, 166 Stark shifts, 177, 539 – AC-, 27, 102 – light polarization and, 125–7 Stark spectra, 166 Stark state, energy, 177 Stark time, 178 State-selective field ionization (SSFI), 144–6 Stern–Gerlach imaging, 44–5 STGF, 250 Stimulated Raman Adiabatic Passage (STIRAP) excitation, 139–43 – excitation efficiency, 143 Stokes operators, 87–8 Stokes pulse, 140 Stopping power of matter, for charged particles, 344 Stratospheric Observatory For Infrared Astronomy (SOFIA), 554 Streaking, 514 Strong field approximation, 517 Sturmian orbitals, 210 Sun, elemental abundances, 275 Superconducting order parameter, 17 Superconductivity, discovery, 2
Index Superconductors, 3 – cuprate, 71 – high-Tc , 7 Superfluidity, connection with Bose–Einstein condensation, 5 Superfluids, 3 Superposition constants, 444, 449 SUPERSTRUCTURE, 250 Surface acoustic wave (SAW) devices, 554, 582 Surface acoustic wave (SAW) dispersion lines, 554 Symmetric tops, internal rotation in see Internal rotation in symmetric tops Synchronization – of chaotic systems see Chaos synchronization – of clocks, 618, 656 Synchronization manifold, 658 – asymptotic stability, 670 – unstable sets in, 671 Synchrotron radiation, 272 Synchrotrons, application, 467 Tgauss , 33 T-matrix, 248 Tang–Statz–deMars equations, 665 TDAS, 638–9, 645–6 TDCC method see Time-dependent closecoupling (TDCC) method TDCS see Triple-differential cross sections TDRM theory, 305–7 Teleportation – atom-atom, 101 – future perspectives, 121 – light-atom, 102 Test Storage Ring (Heidelberg), 280 Thermal spin noise, 111–2 Thomas–Fermi approximation, 29 Thomas–Fermi momentum distribution, 58 Thomas–Fermi statistical model potential, 244 Three-body recombination, 54, 133, 153, 154–5 Thulium, 557 – see also Tm3+ :YAG Time-delay autosynchronization (TDAS), 638–9, 645–6 Time-delay systems – control, 643–7
715 – – non-chaotic fast devices, 646–7 Time-delayed four-wave mixing, 567–9 Time-dependent close-coupling (TDCC) method, 204, 211–8, 261 – approximate solutions to multi-electron atomic systems, 214–5 – beryllium ionization, 218–20 – exact solutions to one-electron atomic systems, 211–4 – exact solutions to two-electron atomic systems, 215–8 – excited state ionization, 227–9 – future perspective, 233 – neon ionization, 224–7 Time evolution operator, unitary Cayley form, 306 Time-independent electron–atom scattering method, 212 Time lens, 581 Time-of-flight imaging, 26 Time-of-flight (TOF) mass spectrometry, 528–9 Tm3+ :YAG – basic spectroscopic properties, 556–8 – in photon echo chirp transform spectrum analyzer, 589 – in rainbow analyzer, 578 Topbase database, 274 Torsion distortion effects, 465 Torsion–rotation Hamiltonian, zeroth-order, symmetric top, 436–7 Torsion–rotation partition function, 471 Torsion–rotation spectroscopy, 460–76 – C2 D6 (CD3 CD3 ), 473–6 – C2 H6 (CH3 CH3 ), 467–73 – CH3 CD3 , 475–6 – CH3 SiD3 , 466–7 – CH3 SiH3 , 462–6 – torsional dependence of electric dipole moment for XY3 AB3 , 460–2 Torsion–vibration difference bands, 453 Torsional dipole moments, of ethane isotopomers, 469, 476 Torsional dipole operators, 460, 475 Torsional potential (V ), 424–5 Torsional satellite method, 450 Torsional splittings, 450–3 – intrinsic, 452 Trace rare gases optical emission spectroscopy (TRG-OES), 416
716 Transition metal oxides, electron energy loss from, 307–11 Transverse-beam instabilities, 626 Transverse nonlinear optics, 619 – see also Spatio-temporal chaos Transverse optical pattern, 689 Trapped Rydberg atoms – oscillations in clouds of, 189–90 – state analysis, 190–1 – see also Cold Rydberg atoms, trapping TRG-OES, 416 Triple-differential cross sections (TDCS), 268–9 Tunnel ionization, 513, 515–6 – quantum perspective, 519–20 Tunneling time, 516 Two-body adiabaticity, time-scale for, 48, 60 Two-body bound state, 11 Two-electron atomic systems, exact solutions, 215–8 Two mode squeezing protocol, 95–9 Unbalanced interferometer see Mach– Zehnder interferometer Unitarity, 14 Unstable periodic orbits (UPOs), 628 – control, 633–47 – – extended time-delay autosynchronization, 637–42 – – with large control-loop latency, 642–7 – – OPF control for ‘green problem’, 635–7 Unstable steady states (USSs), 628 – control, 647–55 – – using ETDAS, 649–55 – – using OPF, 647–8 UPOs see Unstable periodic orbits USSs see Unstable steady states V-type atomic level scheme, 83 Van der Waals interactions, 18 – versus dipole–dipole interactions, 161, 163–5 Vapor cells – paraffin coated, 103–4 – see also MOTs VCSELs – in cavity solitons, 627–8 – chaos synchronization in, 621–3 Verifying pulse, 96–7
Index Vertical-cavity surface-emitting semiconductor lasers see VCSELs Vibration–torsion–rotation energy, 449 Vibration–torsion–rotation Hamiltonian, 433, 439–44, 498 – basis functions, 445, 448–9 – diagonalization, 447 – eigenfunctions, 446–7 – eigenstates, 447 – eigenvalues, 445, 446–7 Vibration–torsion–rotation partition function, 471 Vibration–torsion–rotation spectroscopy, 476–98 – C2 D6 (CD3 CD3 ), 495–8 – C2 H6 (CH3 CH3 ), 490–5 – CH3 SiH3 , 476–89 Vibrational excitation, 283–5 Vibrational partition function, 471 Vibrational wave packet, 530 – controlling and imaging, 538–9 – observing motion, 534–6 VUV emissions, measurement, 331–2 Wiener–Khinchin theorem, 553, 601 X2 Y6 symmetric tops – degenerate fundamental hot band inclusion, 498 – eigenstates, symmetry properties, 446 – molecule-fixed co-ordinate system, 433 – Raman spectroscopy in, 432 – vibration–torsion–rotation Hamiltonian, matrix elements, 449 – see also C2 D6 ; C2 H6 Xenon – core angular momentum, 399, 402 – electronic structure, 324 – excitation cross sections – – apparent, 404, 414 – – direct, 399, 410 – – measurement comparisons, 338, 401–2 – excitation out of ground level, 397–402 – – pressure effects, 397–8 – excitation out of metastable levels, 402–6 XY3 AB3 symmetric tops – distortion dipole moment, 453–4 – eigenstates, symmetry properties, 444 – molecule-fixed co-ordinate system, 432–3 – non-degenerate fundamental study, 498
Index – torsional dependence of electric dipole moment, 460–2 – torsional splitting investigation methods, 504–5 – vibration–torsion–rotation Hamiltonian, matrix elements, 448–9
717 – see also CH3 CD3 ; CH3 CF3 ; CH3 SiD3 ; CH3 SiF3 ; CH3 SiH3 Ytterbium, in fiber lasers, 659 z-bounce motion, 183–5, 190, 192
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CONTENTS OF VOLUMES IN THIS SERIAL Volume 1 Molecular Orbital Theory of the Spin Properties of Conjugated Molecules, G.G. Hall and A.T. Amos Electron Affinities of Atoms and Molecules, B.L. Moiseiwitsch Atomic Rearrangement Collisions, B.H. Bransden The Production of Rotational and Vibrational Transitions in Encounters between Molecules, K. Takayanagi The Study of Intermolecular Potentials with Molecular Beams at Thermal Energies, H. Pauly and J.P. Toennies High-Intensity and High-Energy Molecular Beams, J.B. Anderson, R.P. Anders and J.B. Fen
Volume 2 The Calculation of van der Waals Interactions, A. Dalgarno and W.D. Davison Thermal Diffusion in Gases, E.A. Mason, R.J. Munn and Francis J. Smith Spectroscopy in the Vacuum Ultraviolet, W.R.S. Garton The Measurement of the Photoionization Cross Sections of the Atomic Gases, James A.R. Samson The Theory of Electron–Atom Collisions, R. Peterkop and V. Veldre Experimental Studies of Excitation in Collisions between Atomic and Ionic Systems, F.J. de Heer Mass Spectrometry of Free Radicals, S.N. Foner
Volume 3 The Quantal Calculation of Photoionization Cross Sections, A.L. Stewart Radiofrequency Spectroscopy of Stored Ions I: Storage, H.G. Dehmelt Optical Pumping Methods in Atomic Spectroscopy, B. Budick Energy Transfer in Organic Molecular Crystals: A Survey of Experiments, H.C. Wolf Atomic and Molecular Scattering from Solid Surfaces, Robert E. Stickney Quantum, Mechanics in Gas Crystal-Surface van der Waals Scattering, E. Chanoch Beder Reactive Collisions between Gas and Surface Atoms, Henry Wise and Bernard J. Wood
Volume 4 H.S.W. Massey—A Sixtieth Birthday Tribute, E.H.S. Burhop Electronic Eigenenergies of the Hydrogen Molecular Ion, D.R. Bates and R.H.G. Reid Applications of Quantum Theory to the Viscosity of Dilute Gases, R.A. Buckingham and E. Gal Positrons and Positronium in Gases, P.A. Fraser Classical Theory of Atomic Scattering, A. Burgess and I.C. Percival Born Expansions, A.R. Holt and B.L. Moiseiwitsch Resonances in Electron Scattering by Atoms and Molecules, P.G. Burke 719
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CONTENTS OF VOLUMES IN THIS SERIAL
Relativistic Inner Shell Ionizations, C.B.O. Mohr Recent Measurements on Charge Transfer, J.B. Hasted Measurements of Electron Excitation Functions, D.W.O. Heddle and R.G.W. Keesing Some New Experimental Methods in Collision Physics, R.F. Stebbings Atomic Collision Processes in Gaseous Nebulae, M.J. Seaton Collisions in the Ionosphere, A. Dalgarno The Direct Study of Ionization in Space, R.L.F. Boyd Volume 5 Flowing Afterglow Measurements of Ion-Neutral Reactions, E.E. Ferguson, F.C. Fehsenfeld and A.L. Schmeltekopf Experiments with Merging Beams, Roy H. Neynaber Radiofrequency Spectroscopy of Stored Ions II: Spectroscopy, H.G. Dehmelt The Spectra of Molecular Solids, O. Schnepp The Meaning of Collision Broadening of Spectral Lines: The Classical Oscillator Analog, A. Ben-Reuven The Calculation of Atomic Transition Probabilities, R.J.S. Crossley Tables of One- and Two-Particle Coefficients of Fractional Parentage for Configurations s λ s tu p q , C.D.H. Chisholm, A. Dalgarno and F.R. Innes Relativistic Z-Dependent Corrections to Atomic Energy Levels, Holly Thomis Doyle Volume 6 Dissociative Recombination, J.N. Bardsley and M.A. Biondi Analysis of the Velocity Field in Plasmas from the Doppler Broadening of Spectral Emission Lines, A.S. Kaufman
The Rotational Excitation of Molecules by Slow Electrons, Kazuo Takayanagi and Yukikazu Itikawa The Diffusion of Atoms and Molecules, E.A. Mason and T.R. Marrero Theory and Application of Sturmian Functions, Manuel Rotenberg Use of Classical Mechanics in the Treatment of Collisions between Massive Systems, D.R. Bates and A.E. Kingston
Volume 7 Physics of the Hydrogen Maser, C. Audoin, J.P. Schermann and P. Grivet Molecular Wave Functions: Calculations and Use in Atomic and Molecular Process, J.C. Browne Localized Molecular Orbitals, Harel Weinstein, Ruben Pauncz and Maurice Cohen General Theory of Spin-Coupled Wave Functions for Atoms and Molecules, J. Gerratt Diabatic States of Molecules—QuasiStationary Electronic States, Thomas F. O’Malley Selection Rules within Atomic Shells, B.R. Judd Green’s Function Technique in Atomic and Molecular Physics, Gy. Csanak, H.S. Taylor and Robert Yaris A Review of Pseudo-Potentials with Emphasis on Their Application to Liquid Metals, Nathan Wiser and A.J. Greenfield
Volume 8 Interstellar Molecules: Their Formation and Destruction, D. McNally Monte Carlo Trajectory Calculations of Atomic and Molecular Excitation in Thermal Systems, James C. Keck
CONTENTS OF VOLUMES IN THIS SERIAL Nonrelativistic Off-Shell Two-Body Coulomb Amplitudes, Joseph C.Y. Chen and Augustine C. Chen Photoionization with Molecular Beams, R.B. Cairns, Halstead Harrison and R.I. Schoen The Auger Effect, E.H.S. Burhop and W.N. Asaad Volume 9 Correlation in Excited States of Atoms, A.W. Weiss The Calculation of Electron–Atom Excitation Cross Section, M.R.H. Rudge Collision-Induced Transitions between Rotational Levels, Takeshi Oka The Differential Cross Section of Low-Energy Electron–Atom Collisions, D. Andrick Molecular Beam Electric Resonance Spectroscopy, Jens C. Zorn and Thomas C. English Atomic and Molecular Processes in the Martian Atmosphere, Michael B. McElroy Volume 10 Relativistic Effects in the Many-Electron Atom, Lloyd Armstrong Jr. and Serge Feneuille The First Born Approximation, K.L. Bell and A.E. Kingston Photoelectron Spectroscopy, W.C. Price Dye Lasers in Atomic Spectroscopy, W. Lange, J. Luther and A. Steudel Recent Progress in the Classification of the Spectra of Highly Ionized Atoms, B.C. Fawcett A Review of Jovian Ionospheric Chemistry, Wesley T. Huntress Jr. Volume 11 The Theory of Collisions between Charged Particles and Highly Excited Atoms, I.C. Percival and D. Richards
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Electron Impact Excitation of Positive Ions, M.J. Seaton The R-Matrix Theory of Atomic Process, P.G. Burke and W.D. Robb Role of Energy in Reactive Molecular Scattering: An Information-Theoretic Approach, R.B. Bernstein and R.D. Levine Inner Shell Ionization by Incident Nuclei, Johannes M. Hansteen Stark Broadening, Hans R. Griem Chemiluminescence in Gases, M.F. Golde and B.A. Thrush
Volume 12 Nonadiabatic Transitions between Ionic and Covalent States, R.K. Janev Recent Progress in the Theory of Atomic Isotope Shift, J. Bauche and R.-J. Champeau Topics on Multiphoton Processes in Atoms, P. Lambropoulos Optical Pumping of Molecules, M. Broyer, G. Goudedard, J.C. Lehmann and J. Vigué Highly Ionized Ions, Ivan A. Sellin Time-of-Flight Scattering Spectroscopy, Wilhelm Raith Ion Chemistry in the D Region, George C. Reid
Volume 13 Atomic and Molecular Polarizabilities— Review of Recent Advances, Thomas M. Miller and Benjamin Bederson Study of Collisions by Laser Spectroscopy, Paul R. Berman Collision Experiments with Laser-Excited Atoms in Crossed Beams, I.V. Hertel and W. Stoll Scattering Studies of Rotational and Vibrational Excitation of Molecules, Manfred Faubel and J. Peter Toennies
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CONTENTS OF VOLUMES IN THIS SERIAL
Low-Energy Electron Scattering by Complex Atoms: Theory and Calculations, R.K. Nesbet Microwave Transitions of Interstellar Atoms and Molecules, W.B. Somerville Volume 14 Resonances in Electron Atom and Molecule Scattering, D.E. Golden The Accurate Calculation of Atomic Properties by Numerical Methods, Brain C. Webster, Michael J. Jamieson and Ronald F. Stewart (e, 2e) Collisions, Erich Weigold and Ian E. McCarthy Forbidden Transitions in One- and Two-Electron Atoms, Richard Marrus and Peter J. Mohr Semiclassical Effects in Heavy-Particle Collisions, M.S. Child Atomic Physics Tests of the Basic Concepts in Quantum Mechanics, Francies M. Pipkin Quasi-Molecular Interference Effects in Ion–Atom Collisions, S.V. Bobashev Rydberg Atoms, S.A. Edelstein and T.F. Gallagher UV and X-Ray Spectroscopy in Astrophysics, A.K. Dupree Volume 15 Negative Ions, H.S.W. Massey Atomic Physics from Atmospheric and Astrophysical, A. Dalgarno Collisions of Highly Excited Atoms, R.F. Stebbings Theoretical Aspects of Positron Collisions in Gases, J.W. Humberston Experimental Aspects of Positron Collisions in Gases, T.C. Griffith Reactive Scattering: Recent Advances in Theory and Experiment, Richard B. Bernstein Ion–Atom Charge Transfer Collisions at Low Energies, J.B. Hasted
Aspects of Recombination, D.R. Bates The Theory of Fast Heavy Particle Collisions, B.H. Bransden Atomic Collision Processes in Controlled Thermonuclear Fusion Research, H.B. Gilbody Inner-Shell Ionization, E.H.S. Burhop Excitation of Atoms by Electron Impact, D.W.O. Heddle Coherence and Correlation in Atomic Collisions, H. Kleinpoppen Theory of Low Energy Electron–Molecule Collisions, P.O. Burke Volume 16 Atomic Hartree–Fock Theory, M. Cohen and R.P. McEachran Experiments and Model Calculations to Determine Interatomic Potentials, R. Düren Sources of Polarized Electrons, R.J. Celotta and D.T. Pierce Theory of Atomic Processes in Strong Resonant Electromagnetic Fields, S. Swain Spectroscopy of Laser-Produced Plasmas, M.H. Key and R.J. Hutcheon Relativistic Effects in Atomic Collisions Theory, B.L. Moiseiwitsch Parity Nonconservation in Atoms: Status of Theory and Experiment, E.N. Fortson and L. Wilets Volume 17 Collective Effects in Photoionization of Atoms, M.Ya. Amusia Nonadiabatic Charge Transfer, D.S.F. Crothers Atomic Rydberg States, Serge Feneuille and Pierre Jacquinot Superfluorescence, M.F.H. Schuurmans, Q.H.F. Vrehen, D. Polder and H.M. Gibbs Applications of Resonance Ionization Spectroscopy in Atomic and Molecular
CONTENTS OF VOLUMES IN THIS SERIAL Physics, M.G. Payne, C.H. Chen, G.S. Hurst and G.W. Foltz Inner-Shell Vacancy Production in Ion–Atom Collisions, C.D. Lin and Patrick Richard Atomic Processes in the Sun, P.L. Dufton and A.E. Kingston Volume 18 Theory of Electron–Atom Scattering in a Radiation Field, Leonard Rosenberg Positron–Gas Scattering Experiments, Talbert S. Stein and Walter E. Kaupplia Nonresonant Multiphoton Ionization of Atoms, J. Morellec, D. Normand and G. Petite Classical and Semiclassical Methods in Inelastic Heavy-Particle Collisions, A.S. Dickinson and D. Richards Recent Computational Developments in the Use of Complex Scaling in Resonance Phenomena, B.R. Junker Direct Excitation in Atomic Collisions: Studies of Quasi-One-Electron Systems, N. Andersen and S.E. Nielsen Model Potentials in Atomic Structure, A. Hibbert Recent Developments in the Theory of Electron Scattering by Highly Polar Molecules, D.W. Norcross and L.A. Collins Quantum Electrodynamic Effects in Few-Electron Atomic Systems, G.W.F. Drake Volume 19 Electron Capture in Collisions of Hydrogen Atoms with Fully Stripped Ions, B.H. Bransden and R.K. Janev Interactions of Simple Ion Atom Systems, J.T. Park High-Resolution Spectroscopy of Stored Ions, D.J. Wineland, Wayne M. Itano and R.S. Van Dyck Jr.
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Spin-Dependent Phenomena in Inelastic Electron–Atom Collisions, K. Blum and H. Kleinpoppen The Reduced Potential Curve Method for Diatomic Molecules and Its Applications, F. Jenˇc The Vibrational Excitation of Molecules by Electron Impact, D.G. Thompson Vibrational and Rotational Excitation in Molecular Collisions, Manfred Faubel Spin Polarization of Atomic and Molecular Photoelectrons, N.A. Cherepkov Volume 20 Ion–Ion Recombination in an Ambient Gas, D.R. Bates Atomic Charges within Molecules, G.G. Hall Experimental Studies on Cluster Ions, T.D. Mark and A.W. Castleman Jr. Nuclear Reaction Effects on Atomic Inner-Shell Ionization, W.E. Meyerhof and J.-F. Chemin Numerical Calculations on Electron-Impact Ionization, Christopher Bottcher Electron and Ion Mobilities, Gordon R. Freeman and David A. Armstrong On the Problem of Extreme UV and X-Ray Lasers, I.I. Sobel’man and A.V. Vinogradov Radiative Properties of Rydberg States in Resonant Cavities, S. Haroche and J.M. Raimond Rydberg Atoms: High-Resolution Spectroscopy and Radiation Interaction—Rydberg Molecules, J.A.C. Gallas, G. Leuchs, H. Walther, and H. Figger Volume 21 Subnatural Linewidths in Atomic Spectroscopy, Dennis P. O’Brien, Pierre Meystre and Herbert Walther Molecular Applications of Quantum Defect Theory, Chris H. Greene and Ch. Jungen
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CONTENTS OF VOLUMES IN THIS SERIAL
Theory of Dielectronic Recombination, Yukap Hahn Recent Developments in Semiclassical Floquet Theories for Intense-Field Multiphoton Processes, Shih-I Chu Scattering in Strong Magnetic Fields, M.R.C. McDowell and M. Zarcone Pressure Ionization, Resonances and the Continuity of Bound and Free States, R.M. More Volume 22 Positronium—Its Formation and Interaction with Simple Systems, J.W. Humberston Experimental Aspects of Positron and Positronium Physics, T.C. Griffith Doubly Excited States, Including New Classification Schemes, C.D. Lin Measurements of Charge Transfer and Ionization in Collisions Involving Hydrogen Atoms, H.B. Gilbody Electron Ion and Ion–Ion Collisions with Intersecting Beams, K. Dolder and B. Peart Electron Capture by Simple Ions, Edward Pollack and Yukap Hahn Relativistic Heavy-Ion–Atom Collisions, R. Anholt and Harvey Gould Continued-Fraction Methods in Atomic Physics, S. Swain Volume 23 Vacuum Ultraviolet Laser Spectroscopy of Small Molecules, C.R. Vidal Foundations of the Relativistic Theory of Atomic and Molecular Structure, Ian P. Grant and Harry M. Quiney Point-Charge Models for Molecules Derived from Least-Squares Fitting of the Electric Potential, D.E. Williams and Ji-Min Yan Transition Arrays in the Spectra of Ionized Atoms, J. Bauche, C. Bauche-Arnoult and M. Klapisch
Photoionization and Collisional Ionization of Excited Atoms Using Synchrotron and Laser Radiation, F.J. Wuilleumier, D.L. Ederer and J.L. Picqué Volume 24 The Selected Ion Flow Tube (SIDT): Studies of Ion-Neutral Reactions, D. Smith and N.G. Adams Near-Threshold Electron–Molecule Scattering, Michael A. Morrison Angular Correlation in Multiphoton Ionization of Atoms, S.J. Smith and G. Leuchs Optical Pumping and Spin Exchange in Gas Cells, R.J. Knize, Z. Wu and W. Happer Correlations in Electron–Atom Scattering, A. Crowe Volume 25 Alexander Dalgarno: Life and Personality, David R. Bates and George A. Victor Alexander Dalgarno: Contributions to Atomic and Molecular Physics, Neal Lane Alexander Dalgarno: Contributions to Aeronomy, Michael B. McElroy Alexander Dalgarno: Contributions to Astrophysics, David A. Williams Dipole Polarizability Measurements, Thomas M. Miller and Benjamin Bederson Flow Tube Studies of Ion–Molecule Reactions, Eldon Ferguson Differential Scattering in He–He and He+ –He Collisions at keV Energies, R.F. Stebbings Atomic Excitation in Dense Plasmas, Jon C. Weisheit Pressure Broadening and Laser-Induced Spectral Line Shapes, Kenneth M. Sando and Shih-I. Chu Model-Potential Methods, C. Laughlin and G.A. Victor
CONTENTS OF VOLUMES IN THIS SERIAL Z-Expansion Methods, M. Cohen Schwinger Variational Methods, Deborah Kay Watson Fine-Structure Transitions in Proton–Ion Collisions, R.H.G. Reid Electron Impact Excitation, R.J.W. Henry and A.E. Kingston Recent Advances in the Numerical Calculation of Ionization Amplitudes, Christopher Bottcher The Numerical Solution of the Equations of Molecular Scattering, A.C. Allison High Energy Charge Transfer, B.H. Bransden and D.P. Dewangan Relativistic Random-Phase Approximation, W.R. Johnson Relativistic Sturmian and Finite Basis Set Methods in Atomic Physics, G.W.F. Drake and S.P. Goldman Dissociation Dynamics of Polyatomic Molecules, T. Uzer Photodissociation Processes in Diatomic Molecules of Astrophysical Interest, Kate P. Kirby and Ewine F. van Dishoeck The Abundances and Excitation of Interstellar Molecules, John H. Black Volume 26 Comparisons of Positrons and Electron Scattering by Gases, Walter E. Kauppila and Talbert S. Stein Electron Capture at Relativistic Energies, B.L. Moiseiwitsch The Low-Energy, Heavy Particle Collisions—A Close-Coupling Treatment, Mineo Kimura and Neal F. Lane Vibronic Phenomena in Collisions of Atomic and Molecular Species, V. Sidis Associative Ionization: Experiments, Potentials and Dynamics, John Weiner Françoise Masnou-Seeuws and Annick Giusti-Suzor On the β Decay of 187 Re: An Interface of Atomic and Nuclear Physics and
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Cosmochronology, Zonghau Chen, Leonard Rosenberg and Larry Spruch Progress in Low Pressure Mercury-Rare Gas Discharge Research, J. Maya and R. Lagushenko Volume 27 Negative Ions: Structure and Spectra, David R. Bates Electron Polarization Phenomena in Electron–Atom Collisions, Joachim Kessler Electron–Atom Scattering, I.E. McCarthy and E. Weigold Electron–Atom Ionization, I.E. McCarthy and E. Weigold Role of Autoionizing States in Multiphoton Ionization of Complex Atoms, V.I. Lengyel and M.I. Haysak Multiphoton Ionization of Atomic Hydrogen Using Perturbation Theory, E. Karule Volume 28 The Theory of Fast Ion–Atom Collisions, J.S. Briggs and J.H. Macek Some Recent Developments in the Fundamental Theory of Light, Peter W. Milonni and Surendra Singh Squeezed States of the Radiation Field, Khalid Zaheer and M. Suhail Zubairy Cavity Quantum Electrodynamics, E.A. Hinds Volume 29 Studies of Electron Excitation of Rare-Gas Atoms into and out of Metastable Levels Using Optical and Laser Techniques, Chun C. Lin and L.W. Anderson Cross Sections for Direct Multiphoton Ionization of Atoms, M.V. Ammosov, N.B. Delone, M.Yu. Ivanov, I.I. Bandar and A.V. Masalov Collision-Induced Coherences in Optical Physics, G.S. Agarwal
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CONTENTS OF VOLUMES IN THIS SERIAL
Muon-Catalyzed Fusion, Johann Rafelski and Helga E. Rafelski Cooperative Effects in Atomic Physics, J.P. Connerade Multiple Electron Excitation, Ionization, and Transfer in High-Velocity Atomic and Molecular Collisions, J.H. McGuire Volume 30 Differential Cross Sections for Excitation of Helium Atoms and Helium-Like Ions by Electron Impact, Shinobu Nakazaki Cross-Section Measurements for Electron Impact on Excited Atomic Species, S. Trajmar and J.C. Nickel The Dissociative Ionization of Simple Molecules by Fast Ions, Colin J. Latimer Theory of Collisions between Laser Cooled Atoms, P.S. Julienne, A.M. Smith and K. Burnett Light-Induced Drift, E.R. Eliel Continuum Distorted Wave Methods in Ion–Atom Collisions, Derrick S.F. Crothers and Louis J. Dube Volume 31 Energies and Asymptotic Analysis for Helium Rydberg States, G.W.F. Drake Spectroscopy of Trapped Ions, R.C. Thompson Phase Transitions of Stored Laser-Cooled Ions, H. Walther Selection of Electronic States in Atomic Beams with Lasers, Jacques Baudon, Rudalf Dülren and Jacques Robert Atomic Physics and Non-Maxwellian Plasmas, Michèle Lamoureux Volume 32 Photoionization of Atomic Oxygen and Atomic Nitrogen, K.L. Bell and A.E. Kingston Positronium Formation by Positron Impact on Atoms at Intermediate Energies, B.H. Bransden and C.J. Noble
Electron–Atom Scattering Theory and Calculations, P.G. Burke Terrestrial and Extraterrestrial H+ 3, Alexander Dalgarno Indirect Ionization of Positive Atomic Ions, K. Dolder Quantum Defect Theory and Analysis of High-Precision Helium Term Energies, G.W.F. Drake Electron–Ion and Ion–Ion Recombination Processes, M.R. Flannery Studies of State-Selective Electron Capture in Atomic Hydrogen by Translational Energy Spectroscopy, H.B. Gilbody Relativistic Electronic Structure of Atoms and Molecules, I.P. Grant The Chemistry of Stellar Environments, D.A. Howe, J.M.C. Rawlings and D.A. Williams Positron and Positronium Scattering at Low Energies, J.W. Humberston How Perfect are Complete Atomic Collision Experiments?, H. Kleinpoppen and H. Handy Adiabatic Expansions and Nonadiabatic Effects, R. McCarroll and D.S.F. Crothers Electron Capture to the Continuum, B.L. Moiseiwitsch How Opaque Is a Star?, M.T. Seaton Studies of Electron Attachment at Thermal Energies Using the Flowing Afterglow–Langmuir Technique, David Smith and Patrik Španˇel Exact and Approximate Rate Equations in Atom–Field Interactions, S. Swain Atoms in Cavities and Traps, H. Walther Some Recent Advances in Electron-Impact Excitation of n = 3 States of Atomic Hydrogen and Helium, J.F. Williams and J.B. Wang Volume 33 Principles and Methods for Measurement of Electron Impact Excitation Cross
CONTENTS OF VOLUMES IN THIS SERIAL Sections for Atoms and Molecules by Optical Techniques, A.R. Filippelli, Chun C. Lin, L.W. Andersen and J.W. McConkey Benchmark Measurements of Cross Sections for Electron Collisions: Analysis of Scattered Electrons, S. Trajmar and J.W. McConkey Benchmark Measurements of Cross Sections for Electron Collisions: Electron Swarm Methods, R.W. Crompton Some Benchmark Measurements of Cross Sections for Collisions of Simple Heavy Particles, H.B. Gilbody The Role of Theory in the Evaluation and Interpretation of Cross-Section Data, Barry I. Schneider Analytic Representation of Cross-Section Data, Mitio Inokuti, Mineo Kimura, M.A. Dillon, Isao Shimamura Electron Collisions with N2 , O2 and O: What We Do and Do Not Know, Yukikazu Itikawa Need for Cross Sections in Fusion Plasma Research, Hugh P. Summers Need for Cross Sections in Plasma Chemistry, M. Capitelli, R. Celiberto and M. Cacciatore Guide for Users of Data Resources, Jean W. Gallagher Guide to Bibliographies, Books, Reviews and Compendia of Data on Atomic Collisions, E.W. McDaniel and E.J. Mansky Volume 34 Atom Interferometry, C.S. Adams, O. Carnal and J. Mlynek Optical Tests of Quantum Mechanics, R.Y. Chiao, P.G. Kwiat and A.M. Steinberg Classical and Quantum Chaos in Atomic Systems, Dominique Delande and Andreas Buchleitner
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Measurements of Collisions between Laser-Cooled Atoms, Thad Walker and Paul Feng The Measurement and Analysis of Electric Fields in Glow Discharge Plasmas, J.E. Lawler and D.A. Doughty Polarization and Orientation Phenomena in Photoionization of Molecules, N.A. Cherepkov Role of Two-Center Electron–Electron Interaction in Projectile Electron Excitation and Loss, E.C. Montenegro, W.E. Meyerhof and J.H. McGuire Indirect Processes in Electron Impact Ionization of Positive Ions, D.L. Moores and K.J. Reed Dissociative Recombination: Crossing and Tunneling Modes, David R. Bates Volume 35 Laser Manipulation of Atoms, K. Sengstock and W. Ertmer Advances in Ultracold Collisions: Experiment and Theory, J. Weiner Ionization Dynamics in Strong Laser Fields, L.F. DiMauro and P. Agostini Infrared Spectroscopy of Size Selected Molecular Clusters, U. Buck Fermosecond Spectroscopy of Molecules and Clusters, T. Baumer and G. Gerber Calculation of Electron Scattering on Hydrogenic Targets, I. Bray and A.T. Stelbovics Relativistic Calculations of Transition Amplitudes in the Helium Isoelectronic Sequence, W.R. Johnson, D.R. Plante and J. Sapirstein Rotational Energy Transfer in Small Polyatomic Molecules, H.O. Everitt and F.C. De Lucia Volume 36 Complete Experiments in Electron–Atom Collisions, Nils Overgaard Andersen and Klaus Bartschat
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CONTENTS OF VOLUMES IN THIS SERIAL
Stimulated Rayleigh Resonances and Recoil-Induced Effects, J.-Y. Courtois and G. Grynberg Precision Laser Spectroscopy Using Acousto-Optic Modulators, W.A. van Mijngaanden Highly Parallel Computational Techniques for Electron–Molecule Collisions, Carl Winstead and Vincent McKoy Quantum Field Theory of Atoms and Photons, Maciej Lewenstein and Li You Volume 37 Evanescent Light-Wave Atom Mirrors, Resonators, Waveguides, and Traps, Jonathan P. Dowling and Julio Gea-Banacloche Optical Lattices, P.S. Jessen and I.H. Deutsch Channeling Heavy Ions through Crystalline Lattices, Herbert F. Krause and Sheldon Datz Evaporative Cooling of Trapped Atoms, Wolfgang Ketterle and N.J. van Druten Nonclassical States of Motion in Ion Traps, J.I. Cirac, A.S. Parkins, R. Blatt and P. Zoller The Physics of Highly-Charged Heavy Ions Revealed by Storage/Cooler Rings, P.H. Mokler and Th. Stöhlker Volume 38 Electronic Wavepackets, Robert R. Jones and L.D. Noordam Chiral Effects in Electron Scattering by Molecules, K. Blum and D.G. Thompson Optical and Magneto-Optical Spectroscopy of Point Defects in Condensed Helium, Serguei I. Kanorsky and Antoine Weis Rydberg Ionization: From Field to Photon, G.M. Lankhuijzen and L.D. Noordam Studies of Negative Ions in Storage Rings, L.H. Andersen, T. Andersen and P. Hvelplund
Single-Molecule Spectroscopy and Quantum Optics in Solids, W.E. Moerner, R.M. Dickson and D.J. Norris
Volume 39 Author and Subject Cumulative Index Volumes 1–38 Author Index Subject Index Appendix: Tables of Contents of Volumes 1–38 and Supplements
Volume 40 Electric Dipole Moments of Leptons, Eugene D. Commins High-Precision Calculations for the Ground and Excited States of the Lithium Atom, Frederick W. King Storage Ring Laser Spectroscopy, Thomas U. Kühl Laser Cooling of Solids, Carl E. Mangan and Timothy R. Gosnell Optical Pattern Formation, L.A. Lugiato, M. Brambilla and A. Gatti
Volume 41 Two-Photon Entanglement and Quantum Reality, Yanhua Shih Quantum Chaos with Cold Atoms, Mark G. Raizen Study of the Spatial and Temporal Coherence of High-Order Harmonics, Pascal Salières, Ann L’Huillier, Philippe Antoine and Maciej Lewenstein Atom Optics in Quantized Light Fields, Matthias Freyburger, Alois M. Herkommer, Daniel S. Krähmer, Erwin Mayr and Wolfgang P. Schleich Atom Waveguides, Victor I. Balykin Atomic Matter Wave Amplification by Optical Pumping, Ulf Janicke and Martin Wikens
CONTENTS OF VOLUMES IN THIS SERIAL Volume 42 Fundamental Tests of Quantum Mechanics, Edward S. Fry and Thomas Walther Wave-Particle Duality in an Atom Interferometer, Stephan Dürr and Gerhard Rempe Atom Holography, Fujio Shimizu Optical Dipole Traps for Neutral Atoms, Rudolf Grimm, Matthias Weidemüller and Yurii B. Ovchinnikov Formation of Cold (T ≤ 1 K) Molecules, J.T. Bahns, P.L. Gould and W.C. Stwalley High-Intensity Laser-Atom Physics, C.J. Joachain, M. Dorr and N.J. Kylstra Coherent Control of Atomic, Molecular and Electronic Processes, Moshe Shapiro and Paul Brumer Resonant Nonlinear Optics in Phase Coherent Media, M.D. Lukin, P. Hemmer and M.O. Scully The Characterization of Liquid and Solid Surfaces with Metastable Helium Atoms, H. Morgner Quantum Communication with Entangled Photons, Herald Weinfurter
Volume 43 Plasma Processing of Materials and Atomic, Molecular, and Optical Physics: An Introduction, Hiroshi Tanaka and Mitio Inokuti The Boltzmann Equation and Transport Coefficients of Electrons in Weakly Ionized Plasmas, R. Winkler Electron Collision Data for Plasma Chemistry Modeling, W.L. Morgan Electron–Molecule Collisions in Low-Temperature Plasmas: The Role of Theory, Carl Winstead and Vincent McKoy Electron Impact Ionization of Organic Silicon Compounds, Ralf Basner, Kurt Becker, Hans Deutsch and Martin Schmidt
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Kinetic Energy Dependence of Ion–Molecule Reactions Related to Plasma Chemistry, P.B. Armentrout Physicochemical Aspects of Atomic and Molecular Processes in Reactive Plasmas, Yoshihiko Hatano Ion–Molecule Reactions, Werner Lindinger, Armin Hansel and Zdenek Herman Uses of High-Sensitivity White-Light Absorption Spectroscopy in Chemical Vapor Deposition and Plasma Processing, L.W. Anderson, A.N. Goyette and J.E. Lawler Fundamental Processes of Plasma–Surface Interactions, Rainer Hippler Recent Applications of Gaseous Discharges: Dusty Plasmas and Upward-Directed Lightning, Ara Chutjian Opportunities and Challenges for Atomic, Molecular and Optical Physics in Plasma Chemistry, Kurl Becker Hans Deutsch and Mitio Inokuti
Volume 44 Mechanisms of Electron Transport in Electrical Discharges and Electron Collision Cross Sections, Hiroshi Tanaka and Osamu Sueoka Theoretical Consideration of Plasma-Processing Processes, Mineo Kimura Electron Collision Data for Plasma-Processing Gases, Loucas G. Christophorou and James K. Olthoff Radical Measurements in Plasma Processing, Toshio Goto Radio-Frequency Plasma Modeling for Low-Temperature Processing, Toshiaki Makabe Electron Interactions with Excited Atoms and Molecules, Loucas G. Christophorou and James K. Olthoff
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CONTENTS OF VOLUMES IN THIS SERIAL
Volume 45 Comparing the Antiproton and Proton, and Opening the Way to Cold Antihydrogen, G. Gabrielse Medical Imaging with Laser-Polarized Noble Gases, Timothy Chupp and Scott Swanson Polarization and Coherence Analysis of the Optical Two-Photon Radiation from the Metastable 22 Si1/2 State of Atomic Hydrogen, Alan J. Duncan, Hans Kleinpoppen and Marian O. Scully Laser Spectroscopy of Small Molecules, W. Demtröder, M. Keil and H. Wenz Coulomb Explosion Imaging of Molecules, Z. Vager Volume 46 Femtosecond Quantum Control, T. Brixner, N.H. Damrauer and G. Gerber Coherent Manipulation of Atoms and Molecules by Sequential Laser Pulses, N.V. Vitanov, M. Fleischhauer, B.W. Shore and K. Bergmann Slow, Ultraslow, Stored, and Frozen Light, Andrey B. Matsko, Olga Kocharovskaya, Yuri Rostovtsev George R. Welch, Alexander S. Zibrov and Marlan O. Scully Longitudinal Interferometry with Atomic Beams, S. Gupta, D.A. Kokorowski, R.A. Rubenstein, and W.W. Smith Volume 47 Nonlinear Optics of de Broglie Waves, P. Meystre Formation of Ultracold Molecules (T ≤ 200 μK) via Photoassociation in a Gas of Laser-Cooled Atoms, Françoise Masnou-Seeuws and Pierre Pillet Molecular Emissions from the Atmospheres of Giant Planets and Comets: Needs for Spectroscopic and Collision Data, Yukikazu Itikawa, Sang Joon Kim, Yong Ha Kim and Y.C. Minh
Studies of Electron-Excited Targets Using Recoil Momentum Spectroscopy with Laser Probing of the Excited State, Andrew James Murray and Peter Hammond Quantum Noise of Small Lasers, J.P. Woerdman, N.J. van Druten and M.P. van Exter Volume 48 Multiple Ionization in Strong Laser Fields, R. Dörner Th. Weber, M. Weckenbrock, A. Staudte, M. Hattass, R. Moshammer, J. Ullrich and H. Schmidt-Böcking Above-Threshold Ionization: From Classical Features to Quantum Effects, W. Becker, F. Grasbon, R. Kapold, D.B. Miloševi´c, G.G. Paulus and H. Walther Dark Optical Traps for Cold Atoms, Nir Friedman, Ariel Kaplan and Nir Davidson Manipulation of Cold Atoms in Hollow Laser Beams, Heung-Ryoul Noh, Xenye Xu and Wonho Jhe Continuous Stern–Gerlach Effect on Atomic Ions, Günther Werth, Hartmut Haffner and Wolfgang Quint The Chirality of Biomolecules, Robert N. Compton and Richard M. Pagni Microscopic Atom Optics: From Wires to an Atom Chip, Ron Folman, Peter Krüger, Jörg Schmiedmayer, Johannes Denschlag and Carsten Henkel Methods of Measuring Electron–Atom Collision Cross Sections with an Atom Trap, R.S. Schappe, M.L. Keeler, T.A. Zimmerman, M. Larsen, P. Feng, R.C. Nesnidal, J.B. Boffard, T.G. Walker, L.W. Anderson and C.C. Lin Volume 49 Applications of Optical Cavities in Modern Atomic, Molecular, and Optical Physics, Jun Ye and Theresa W. Lynn
CONTENTS OF VOLUMES IN THIS SERIAL Resonance and Threshold Phenomena in Low-Energy Electron Collisions with Molecules and Clusters, H. Hotop, M.-W. Ruf, M. Allan and I.I. Fabrikant Coherence Analysis and Tensor Polarization Parameters of (γ , eγ ) Photoionization Processes in Atomic Coincidence Measurements, B. Lohmann, B. Zimmermann, H. Kleinpoppen and U. Becker Quantum Measurements and New Concepts for Experiments with Trapped Ions, Ch. Wunderlich and Ch. Balzer Scattering and Reaction Processes in Powerful Laser Fields, Dejan B. Miloševi´c and Fritz Ehlotzky Hot Atoms in the Terrestrial Atmosphere, Vijay Kumar and E. Krishnakumar Volume 50 Assessment of the Ozone Isotope Effect, K. Mauersberger, D. Krankowsky, C. Janssen and R. Schinke Atom Optics, Guided Atoms, and Atom Interferometry, J. Arlt, G. Birkl, E. Rasel and W. Ertmet Atom–Wall Interaction, D. Bloch and M. Ducloy Atoms Made Entirely of Antimatter: Two Methods Produce Slow Antihydrogen, G. Gabrielse Ultrafast Excitation, Ionization, and Fragmentation of C60 , I.V. Hertel, T. Laarmann and C.P. Schulz Volume 51 Introduction, Henry H. Stroke Appreciation of Ben Bederson as Editor of Advances in Atomic, Molecular, and Optical Physics Benjamin Bederson Curriculum Vitae Research Publications of Benjamin Bederson A Proper Homage to Our Ben, H. Lustig Benjamin Bederson in the Army, World War II, Val L. Fitch
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Physics Needs Heroes Too, C. Duncan Rice Two Civic Scientists—Benjamin Bederson and the other Benjamin, Neal Lane An Editor Par Excellence, Eugen Merzbacher Ben as APS Editor, Bernd Crasemann Ben Bederson: Physicist–Historian, Roger H. Stuewer Pedagogical Notes on Classical Casimir Effects, Larry Spruch Polarizabilities of 3 P Atoms and van der Waals Coefficients for Their Interaction with Helium Atoms, X. Chu and A. Dalgarno The Two Electron Molecular Bonds Revisited: From Bohr Orbits to Two-Center Orbitals, Goong Chen, Siu A. Chin, Yusheng Dou, Kishore T. Kapale, Moochan Kim, Anatoly A. Svidzinsky, Kerim Urtekin, Han Xiong and Marlan O. Scully Resonance Fluorescence of Two-Level Atoms, H. Walther Atomic Physics with Radioactive Atoms, Jacques Pinard and H. Henry Stroke Thermal Electron Attachment and Detachment in Gases, Thomas M. Miller Recent Developments in the Measurement of Static Electric Dipole Polarizabilities, Harvey Gould and Thomas M. Miller Trapping and Moving Atoms on Surfaces, Robert J. Celotta and Joseph A. Stroscio Electron-Impact Excitation Cross Sections of Sodium, Chun C. Lin and John B. Boffard Atomic and Ionic Collisions, Edward Pollack Atomic Interactions in Weakly Ionized Gas: Ionizing Shock Waves in Neon, Leposava Vuškovi´c and Svetozar Popovi´c Approaches to Perfect/Complete Scattering Experiments in Atomic and Molecular Physics, H. Kleinpoppen, B. Lohmann, A. Grum-Grzhimailo and U. Becker Reflections on Teaching, Richard E. Collins
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CONTENTS OF VOLUMES IN THIS SERIAL
Volume 52 Exploring Quantum Matter with Ultracold Atoms in Optical Lattices, Immanuel Bloch and Markus Greiner The Kicked Rydberg Atom, F.B. Dunning, J.C. Lancaster, C.O. Reinhold, S. Yoshida and J. Burgdörfer Photonic State Tomography, J.B. Altepeter, E.R. Jeffrey and P.G. Kwiat Fine Structure in High-L Rydberg States: A Path to Properties of Positive Ions, Stephen R. Lundeen A Storage Ring for Neutral Molecules, Floris M.H. Crompvoets, Hendrick L. Bethlem and Gerard Meijer Nonadiabatic Alignment by Intense Pulses. Concepts, Theory, and Directions, Tamar Seideman and Edward Hamilton Relativistic Nonlinear Optics, Donald Umstadter, Scott Sepke and Shouyuan Chen Coupled-State Treatment of Charge Transfer, Thomas G. Winter Volume 53 Non-Classical Light from Artificial Atoms, Thomas Aichele, Matthias Scholz, Sven Ramelow and Oliver Benson Quantum Chaos, Transport, and Control—in Quantum Optics, Javier Madroñero, Alexey Ponomarev, Andrí R.R. Carvalho, Sandro Wimberger, Carlos Viviescas, Andrey Kolovsky, Klaus Hornberger, Peter Schlagheck, Andreas Krug and Andreas Buchleitner
Manipulating Single Atoms, Dieter Meschede and Arno Rauschenbeutel Spatial Imaging with Wavefront Coding and Optical Coherence Tomography, Thomas Hellmuth The Quantum Properties of Multimode Optical Amplifiers Revisited, G. Leuchs, U.L. Andersen and C. Fabre Quantum Optics of Ultra-Cold Molecules, D. Meiser, T. Miyakawa, H. Uys and P. Meystre Atom Manipulation in Optical Lattices, Georg Raithel and Natalya Morrow Femtosecond Laser Interaction with Solid Surfaces: Explosive Ablation and Self-Assembly of Ordered Nanostructures, Juergen Reif and Florenta Costache Characterization of Single Photons Using Two-Photon Interference, T. Legero, T. Wilk, A. Kuhn and G. Rempe Fluctuations in Ideal and Interacting Bose–Einstein Condensates: From the Laser Phase Transition Analogy to Squeezed States and Bogoliubov Quasiparticles, Vitaly V. Kocharovsky, Vladimir V. Kocharovsky, Martin Holthaus, C.H. Raymond Ooi, Anatoly Svidzinsky, Wolfgang Ketterle and Marlan O. Scully LIDAR-Monitoring of the Air with Femtosecond Plasma Channels, Ludger Wöste, Steffen Frey and Jean-Pierre Wolf
Supplements Atoms in Intense Laser Fields, edited by Mihai Gavrila (1992) Multiphoton Ionization, H.G. Muller, P. Agostini and G. Petite Photoionization with Ultra-Short Laser Pulses, R.R. Freeman, P.H. Bucksbaum,
W.E. Cooke, G. Gibson, T.J. McIlrath and L.D. van Woerkom Rydberg Atoms in Strong Microwave Fields, T.F. Gallagher Muiltiphoton Ionization in Large Ponderomotive Potentials, P.B. Corkum, N.H. Burnett and F. Brunel
CONTENTS OF VOLUMES IN THIS SERIAL High Order Harmonic Generation in Rare Gases, Anne L’Huillier, Louis-André Lompré, Gerard Manfrey and Claude Manus Mechanisms of Short-Wavelength Generation, T.S. Luk, A. McPherson, K. Boyer and C.K. Rhodes Time-Dependent Studies of Multiphoton Processes, Kenneth C. Kulander, Kenneth J. Schafer and Jeffrey L. Krause Numerical Experiments in Strong and Super-Strong Fields, J.H. Eberly, R. Grobe, C.K. Law and Q. Su Resonances in Multiphoton Ionization, P. Lambropoulos and X. Tang Nonperturbative Treatment of Multiphoton Ionization within the Floquet Framework, R.M. Potvliege and Robin Shakeshaft Atomic Structure and Decay in High Frequency Fields, Mihai Gavrila Cavity Quantum Electrodynamics, edited by Paul R. Berman (1994) Perturbative Cavity Quantum Electrodynamics, E.A. Hinds
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The Micromaser: A Proving Ground for Quantum Physics, Georg Raithel, Christian Wagner, Herbert Walther, Lorenzo M. Narducci and Marlan O. Scully Manipulation of Nonclassical Field States in a Cavity by Atom Interferometry, S. Haroche and J.M. Raimond Quantum Optics of Driven Atoms in Colored Vacua, Thomas W. Mossberg and Maciej Lewenstein Structure and Dynamics in Cavity Quantum Electrodynamics, H.J. Kimble One Electron in a Cavity, G. Gabrielse and J. Tan Spontaneous Emission by Moving Atoms, Pierre Meystre and Martin Wilkens Single Atom Emission in an Optical Resonator, James J. Childs, Kyungwon An, Ramanchandra R. Dasari and Michael S. Feld Nonperturbative Atom–Photon Interactions in an Optical Cavity, H.J. Carmichael, L. Tian, W. Ren and P. Alsing New Aspects of the Casimir Effect: Fluctuations and Radiative Reaction, G. Barton
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