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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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).
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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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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.
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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.
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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.
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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 num
