Charged particle traps II: Applications

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Author: Günther Werth | Viorica N. Gheorghe | Fouad G. Major

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Springer Series on

atomic, optical, and plasma physics

54

Springer Series on

atomic, optical, and plasma physics The Springer Series on Atomic, Optical, and Plasma Physics covers in a comprehensive manner theory and experiment in the entire f ield of atoms and molecules and their interaction with electromagnetic radiation. Books in the series provide a rich source of new ideas and techniques with wide applications in f ields such as chemistry, materials science, astrophysics, surface science, plasma technology, advanced optics, aeronomy, and engineering. Laser physics is a particular connecting theme that has provided much of the continuing impetus for new developments in the f ield. The purpose of the series is to cover the gap between standard undergraduate textbooks and the research literature with emphasis on the fundamental ideas, methods, techniques, and results in the f ield. Please view available titles in Springer Series on Atomic, Optical, and Plasma Physics on series homepage http://www.springer.com/series/411

G. Werth

V.N. Gheorghe

F.G. Major

Charged Particle Traps II Applications

With 200 Figures

123

¨ Professor Dr. Gunter Werth Professor Dr. Viorica N. Gheorghe Johannes Gutenberg Universit¨at, Fachbereich Physik (18), Institut f¨ur Physik Staudingerweg 7, 55099 Mainz, Germany E-mail: [email protected], [email protected]

Dr. Fouad G. Major 284 Michener Court E., Severna Park, MD, USA E-mail: [email protected]

Springer Series on Atomic, Optical, and Plasma Physics ISBN 978-3-540-92260-5

ISSN 1615-5653

e-ISBN 978-3-540-92261-2

DOI 10.1007/978-3-540-92261-2 Library of Congress Control Number: 2009929168 © Springer-Verlag Berlin Heidelberg 2009 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specif ically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microf ilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specif ic statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting and production: SPi Cover concept: eStudio Calmar Steinen Cover design: WMX Design GmbH, Heidelberg SPIN 11959526 57/3180/spi Printed on acid-free paper 987654321 springer.com

Preface

This second volume of the Charged Particle Traps deals with the rapidly expanding body of research exploiting the electromagnetic conﬁnement of ions, whose principles and techniques were the subject of volume I. These applications include revolutionary advances in diverse ﬁelds, ranging from such practical ﬁelds as mass spectrometry, to the establishment of an ultrastable standard of frequency and the emergent ﬁeld of quantum computing made possible by the observation of the quantum behavior of laser-cooled conﬁned ions. Both experimental and theoretical activity in these applications has proliferated widely, and the number of diverse articles in the literature on its many facets has reached the point where it is useful to distill and organize the published work in a uniﬁed volume that deﬁnes the current status of the ﬁeld. As explained in volume I, the technique of conﬁning charged particles in suitable electromagnetic ﬁelds was initially conceived by W. Paul as a threedimensional version of his rf quadrupole mass ﬁlter. Its ﬁrst application to rf spectroscopy on atomic ions was completed in H.G. Dehmelt’s laboratory where notable work was later done on the free electron using the Penning trap. The further exploitation of these devices has followed more or less independently along the two initial broad areas: mass spectrometry and high resolution spectroscopy. In volume I a detailed account is given of the theory of operation and experimental techniques of the various forms of Paul and Penning ion traps. Of crucial signiﬁcance in the application to spectroscopy is the possibility of using laser ﬂuorescence and manipulation of internal quantum state populations, to cool the conﬁned ions to temperatures approaching absolute zero. The description of the ion behavior under these extreme conditions requires the use of quantum theory; and this is given in volume I for particles conﬁned in harmonic ﬁelds subject to laser excitation. At the lowest attainable temperatures, the ion has a high probability of being in the lowest vibrational state, the quantum ground state. When many ions are simultaneously conﬁned in the same trap, suﬃcient cooling can result in the ions

VI

Preface

freezing into a crystalline array, a phenomenon also treated at some length in volume I. In this volume, an attempt is made to present a consolidated view of the present status of representative laboratory activities aimed at advancing applications of ion traps. Among the principal applications are the following: Mass spectrometry in the context of molecular analysis, precise nuclear isotope masses, and precise elementary particle masses; precise magnetic resonance spectroscopy on elementary particles, the g-factors of multiply charged ions, and precise magnetic hyperﬁne splittings, with application to microwave atomic frequency standards; laser spectroscopy, including long radiative lifetime determination, ultra-high resolution optical spectroscopy and optical frequency standards and measurement; ﬁnally, the exciting new ﬁeld of quantum computing using ion crystals in which internal states and vibrational states are entangled through laser excitations based on the quantum eﬀects in ion traps, including the direct observation of “quantum jumps,” the quantum Zeno eﬀect, entanglement of quantum states between ions, and quantum teleportation. The literature relating to applications of trapped ions manipulated by laser ﬁelds has exploded in recent years and it is hoped that this volume provides a timely and useful addition to the literature. Mainz April 2009

G¨ unter Werth Viorica N. Gheorghe Fouad G. Major

Contents

Part I Electromagnetic Trap Properties 1

Summary of Trap Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Trapping Principles in Paul Traps . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 General Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Potential Depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Motional Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.4 Optimum Trapping Conditions . . . . . . . . . . . . . . . . . . . . . . 1.1.5 Storage Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.6 Ion Density Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.7 Storage Capability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.8 Paul Trap Imperfections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Trapping Principles in Penning Traps . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Theory of the Ideal Penning Trap . . . . . . . . . . . . . . . . . . . 1.2.2 Motional Spectrum in Penning Traps . . . . . . . . . . . . . . . . 1.2.3 Penning Trap Imperfections . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 Storage Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.5 Storage Capability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.6 Spatial Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Trap Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Trap Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Trapped Particle Detection . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Ion Cooling Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Buﬀer Gas Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Resistive Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3 Laser Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.4 Radiative Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3 5 7 8 8 9 10 10 11 13 13 15 16 18 20 20 21 21 23 28 28 29 30 33

VIII

Contents

Part II Mass Spectrometry 2

Mass Spectrometry Using Paul Traps . . . . . . . . . . . . . . . . . . . . . . 2.1 The Quadrupole Ion Trap as a Mass Spectrometer . . . . . . . . . . . 2.2 The “Mass Instability Method” of Detection . . . . . . . . . . . . . . . . 2.3 Sources of Mass Error in Ion Ejection Methods . . . . . . . . . . . . . . 2.4 Nonlinear Resonances in Imperfect Quadrupole Trap . . . . . . . . . 2.5 Quadrupole Time-of-Flight Spectrometer . . . . . . . . . . . . . . . . . . . 2.6 Tandem Quadrupole Mass Spectrometers . . . . . . . . . . . . . . . . . . . 2.7 Tandem Quadrupole Fourier Transform Spectrometer . . . . . . . . 2.8 Silicon-Based Quadrupole Mass Spectrometers . . . . . . . . . . . . . .

37 40 41 44 44 46 48 50 52

3

Mass Spectroscopy in Penning Trap . . . . . . . . . . . . . . . . . . . . . . . 3.1 Systematic Frequency Shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Electric Field Imperfections . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Magnetic Field Imperfections . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Misalignements and Trap Ellipticity . . . . . . . . . . . . . . . . . 3.1.4 Image Charges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.5 Magnetic Field Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Observation of Motional Resonances . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Nondestructive Observation . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Destructive Observation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Line Shape of Motional Resonances . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Nondestructive Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Destructive Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Experimental Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Reference Ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Selected Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Stable and Long Lived Isotopes . . . . . . . . . . . . . . . . . . . . . 3.5.2 Short-Lived Isotopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55 55 55 57 57 58 58 60 60 63 66 66 68 72 73 76 77 79

Part III Spectroscopy with Trapped Charged Particles 4

Microwave Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.1 Zeeman Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.1.1 g-Factor of the Free Electron . . . . . . . . . . . . . . . . . . . . . . . 86 4.1.2 g-Factor of the Bound Electron . . . . . . . . . . . . . . . . . . . . . 95 4.1.3 Atomic g-Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.1.4 Nuclear gI -Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.2 Hyperﬁne Structures in the Ground States . . . . . . . . . . . . . . . . . . 105 4.2.1 Summary of HFS Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 4.2.2 Early Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 4.2.3 Laser Microwave Double Resonance Spectroscopy . . . . . . 113

Contents

IX

4.3 Microwave Atomic Clocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 4.3.1 Deﬁnition of the Unit of Time . . . . . . . . . . . . . . . . . . . . . . 118 4.3.2 Trapped Ion Microwave Standards . . . . . . . . . . . . . . . . . . . 121 5

Optical Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 5.1 Optical Frequency Standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 5.1.1 Theoretical Limit to Laser Spectral Purity . . . . . . . . . . . . 129 5.1.2 Laser Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 5.1.3 Single Ion Optical Frequency Standards . . . . . . . . . . . . . . 133 5.1.4 Correction of Systematic Errors . . . . . . . . . . . . . . . . . . . . . 147 5.1.5 Optical Frequency Measurement . . . . . . . . . . . . . . . . . . . . . 152 5.2 Progress in Standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

6

Lifetime Studies in Traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 6.1 Radiative Lifetimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 6.1.1 Experimental Methods of Lifetime Measurement . . . . . . 162 6.1.2 Systematic Eﬀects on the Lifetimes . . . . . . . . . . . . . . . . . . 172 6.1.3 Quenching Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

Part IV Quantum Topics 7

Quantum Eﬀects in Charged Particle Traps . . . . . . . . . . . . . . . . 179 7.1 Quantum Jumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 7.2 The Quantum Zeno Eﬀect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 7.3 Entanglement of Trapped Ion States . . . . . . . . . . . . . . . . . . . . . . . 183 7.3.1 Entanglement of Two-Trapped Ions . . . . . . . . . . . . . . . . . . 184 7.3.2 Entanglement of Three-Trapped Ions . . . . . . . . . . . . . . . . 186 7.3.3 Multi-ion Entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 7.3.4 Trapped Ion–Photon Entanglement . . . . . . . . . . . . . . . . . . 189 7.3.5 Lifetime of Entangled States . . . . . . . . . . . . . . . . . . . . . . . . 190 7.4 Quantum Teleportation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 7.5 Sources of Decoherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 7.5.1 Decoherence Reservoirs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 7.5.2 Motional Decoherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 7.5.3 Collisions with Background Gas . . . . . . . . . . . . . . . . . . . . . 199 7.5.4 Internal State Decoherence . . . . . . . . . . . . . . . . . . . . . . . . . 200 7.5.5 Induced Decoherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 7.5.6 Control of Thermal Decoherence . . . . . . . . . . . . . . . . . . . . 203

8

Quantum Computing with Trapped Charged Particles . . . . . 207 8.1 Background Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 8.1.1 Quantum Bits: Qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 8.1.2 Some History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 8.1.3 Possible Alternatives: The DiVincenzo Criteria . . . . . . . . 212

X

Contents

8.2 Ion Traps for Quantum Computing . . . . . . . . . . . . . . . . . . . . . . . . 215 8.2.1 Trap Electrode Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 8.2.2 Choice of Ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 8.3 Qubits with Trapped Ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 8.4 Quantum Registers: Qregister . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 8.4.1 Initialisation of the Qubits . . . . . . . . . . . . . . . . . . . . . . . . . . 223 8.5 Creation of Nonclassical States . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 8.5.1 Fock States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 8.5.2 Coherent States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 8.5.3 Schr¨ odinger Cat States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 8.6 Quantum Logic Gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 8.7 Qubit Entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 8.8 Quantum Information Processing . . . . . . . . . . . . . . . . . . . . . . . . . . 232 8.8.1 Speed of Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 8.8.2 Nonclassical State Reconstruction . . . . . . . . . . . . . . . . . . . 235 8.9 Qubit Decoherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 8.10 Scalability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 8.11 Penning Trap as Quantum Information Processor . . . . . . . . . . . . 245 8.11.1 Computing with Electrons . . . . . . . . . . . . . . . . . . . . . . . . . . 245 8.11.2 Linear Multi-trap Processor . . . . . . . . . . . . . . . . . . . . . . . . 245 8.11.3 Planar Multi-trap Processor . . . . . . . . . . . . . . . . . . . . . . . . 247 8.11.4 Expected Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 8.12 Future Developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

1 Summary of Trap Properties

1.1 Trapping Principles in Paul Traps Three-dimensional conﬁnement of charged particles requires a potential energy minimum at some region in space, in order that the corresponding force is directed toward that region in all three dimensions. In general, the dependence of the magnitude of this force on the coordinates can have an arbitrary form; however, it is convenient to have a binding force that is harmonic, since this simpliﬁes the analytical description of the particle motion. Thus, we assume Fu = −ku u, u = x, y, z. (1.1) It follows from F = −∇ U ,

(1.2)

where U = QΦ is the potential energy, that in general the required function Φ is a quadratic form in the Cartesian coordinates x, y, z: Φ=

Φ0 (Ax2 + By 2 + Cz 2 ), d2

(1.3)

where A, B, C are some constants, d a normalizing factor, and Φ0 can be a time-dependent function. If we attempt to achieve such conﬁnement using an electrostatic ﬁeld acting on an ion of charge Q we ﬁnd that to satisfy Laplace’s equation ΔΦ = 0, the coeﬃcients must satisfy A + B + C = 0. For the interesting case of rotational symmetry around the z-axis, this leads to A = B = 1 and C = −2 giving the quadrupolar form Φ=

Φ0 2 Φ0 (x + y 2 − 2z 2 ) = 2 (ρ2 − 2z 2 ), d2 d

(1.4)

with ρ2 = x2 +y 2 . Equipotential surfaces are hyperboloids of revolution. If the radial distance from the center (ρ = z = 0) of a hyperbolic trap to the ring electrode is called r0 , and the axial distance to an end cap is z0 , the equations

4

1 Summary of Trap Properties

for the hyperbolic electrode surfaces are ρ2 − 2z 2 = r02 , ρ2 − 2z 2 = −2z02 .

(1.5)

If the potential diﬀerence between the ring and end caps is taken to be Φ0 , then d2 = r02 + 2z02 . (1.6) From the diﬀerence in signs between the radial and axial terms, we see that the potential has a saddle point at the origin, having a minimum along one coordinate but a maximum along the other. Earnshaw’s theorem states that it is not possible to generate a minimum of the electrostatic potential in free space. Nevertheless, it is possible to circumvent Earnshaw’s theorem by superimposing a magnetic ﬁeld along the z-axis to create what is called the Penning trap or to use a time-dependent electric ﬁeld, leading to the Paul trap [1, 2]. The electrodes which create a quadrupole potential consist of three hyperbolic sheets of revolution: A ring electrode and two end caps (Fig. 1.1) which share the same asymptotic cone. The size of the device ranges, in diﬀerent applications, from several centimeters for the characteristic dimension d to fractions of a millimeter. The trapped charged particles are constrained to a very small central region of the trap. In recent years, diﬀerent trap geometries that are easier to manufacture and align and in addition allow better optical access to the trapped particles without further modiﬁcation have become common; for example, the linear Paul trap which uses four parallel rods as electrodes (Fig. 1.2a). An ac voltage applied between adjacent electrodes leads to dynamical conﬁnement similar to the three-dimensional case, while axial conﬁnement is provided by a static voltage on the end electrodes. For the static Penning trap cylindrical electrodes are used (Fig. 1.2b). These electrode geometries produce a harmonic binding force near the center precisely of the classical form. Further away z

U x (a)

upper end cap

z 0 r0

y ring lower end cap

(b)

Fig. 1.1. Basic arrangement for Paul and Penning traps. The inner electrode surfaces are hyperboloids. The dynamic stabilization in the Paul trap is given by an ac voltage V0 cos Ωt. The static stabilization in the Penning trap is given by a dc voltage U = U0 and an axial magnetic ﬁeld. On the right is a photograph of a trap with ρ = 1 cm. One of the end caps is a mesh to allow optical access to trapped ions

1.1 Trapping Principles in Paul Traps

5

Fig. 1.2. Linear Paul trap (a) and open end cap cylindrical Penning trap (b). A radio-frequency (rf) ﬁeld applied to the rods of the linear Paul trap conﬁnes charged particles in the radial direction, and a dc voltage at the end segments serves for axial trapping. The Penning trap has guard electrodes between the central ring and the end caps to compensate partly for deviations from the ideal quadrupole potential near the trap center. Ion excitation can be performed by rf ﬁelds applied to segments of the electrodes

from the origin higher order terms in the expansion of the potential will become signiﬁcant. For cylindrical Penning traps these can be partially reduced by additional compensation electrodes placed between the ring and end caps as shown in Fig. 1.2b. 1.1.1 General Principles In an ideal Paul trap an oscillating electric potential usually in combination with a static component, U0 + V0 cos Ωt, is applied between the ring and the pair of end cap electrodes. It creates a potential of the form Φ=

U0 + V0 cos Ωt 2 (ρ − 2z 2 ). 2d2

(1.7)

Since the trapping ﬁeld is inhomogeneous, the average force acting on the particle, taken over many oscillations of the ﬁeld, is not zero. Depending on the amplitude and frequency of the ﬁeld, the net force may be convergent toward the center of the trap leading to conﬁnement, or divergent leading to the loss of the particle. Thus, although the electric force alternately causes convergent and divergent motions of the particle in any given direction it is possible by appropriate choice of ﬁeld amplitude and frequency, to have a time-averaged restoring force in all three dimensions toward the center of the trap as required for conﬁnement [3]. The conditions for stable conﬁnement of an ion with mass M and charge Q in the Paul ﬁeld may be derived by solving the equation of motion d2 u Q = (U0 + V0 cos Ωt)u, u = x, y, z. dt2 M d2

(1.8)

Using the dimensionless parameters ax = ay = −2az = −4QU0/(M d2 Ω2 ) and qx = qy = 2qz = 2QV0 /(M d2 Ω2 ), we obtain a system of three diﬀerential

6

1 Summary of Trap Properties

Fig. 1.3. The stability domains for the ideal Paul trap. Light gray: z-direction; dark gray: r-direction. Three-dimensional stability is assured in the overlapping regions

equations of the homogeneous Mathieu type [4, 5] d2 u + (a − 2q cos 2τ )u = 0, dτ 2

(1.9)

where τ = Ωt/2. The values of a and q for which the solutions are stable simultaneously for both directions, an obvious requirement for three-dimensional conﬁnement, are found by using the relationships az = −2ar , qz = 2qr to make a composite plot of the boundaries of stability for both directions on the same set of axes: The overlap regions lead to three-dimensional conﬁnement (Fig. 1.3). The most important for practical purposes is the stable region near the origin which has been exclusively used for ion conﬁnement. The stable solutions of the Mathieu equation can be expressed in the form of a Fourier series, thus ui (τ ) = Ai

∞ n=−∞

c2n cos(βi + 2n)τ + Bi

∞

c2n sin(βi + 2n)τ,

(1.10)

n=−∞

where Ai and Bi are constants depending on the initial conditions. The stability parameters βi are functions of ai and qi . The coeﬃcients c2n , which are the amplitudes of the Fourier components of the particle motion, decrease with increasing n. For small values of ai , qi 1, we can approximate the stability parameter βi by q2 (1.11) βi2 ai + i , i = ρ, z. 2 In this so-called adiabatic approximation only the coeﬃcients c2n with n = 1 giving c−2 = c+2 = −(qi /4)c0 are considered, and the other with n ≥ 2 are neglected. The ion motion simpliﬁes to qi ui (t) = A 1 − cos Ωt cos ωi t, (1.12) 2

1.1 Trapping Principles in Paul Traps

7

Fig. 1.4. Observed trajectory of a microparticle in a Paul-type trap [6]

with ωi = βi Ω/2.

(1.13)

This can be considered as the motion of an oscillator of frequency ω whose amplitude is modulated with the trap’s driving frequency Ω. Since it is assumed that β 1, the oscillation at ω, usually called the secular motion or macromotion, is slow compared with the superimposed fast micromotion at Ω. Because of the large diﬀerence in the frequencies ω and Ω, the ion motion can be well separated into two components and the behavior of the slow motion at frequency ω can be considered as separate, while time averaging over the fast oscillation at Ω. Wuerker et al. [6] have taken photographs of single particle trajectories of microparticles in a Paul-type trap at low frequencies of the trapping ﬁeld demonstrating the validity of this approximation (Fig. 1.4).

1.1.2 Potential Depth In the adiabatic approximation, an expression for the depth of the conﬁning potential can be derived by considering the secular motion only. The ion behaves in the axial direction as a harmonic oscillator of frequency ωz . For no dc voltage (a = 0) we have ωz = √

QV0 . 2M z02 Ω

(1.14)

This corresponds to a time-averaged (pseudo)potential of depth in the axial direction of 2 ¯ z = QV0 , D (1.15) 2 4M z0 Ω2 and similarly in the radial direction ¯r = D

QV02 . 4M r02 Ω2

(1.16)

¯r = D ¯ z /2. For r02 = 2z02 we have D The eﬀect of an additional dc voltage on the trap electrodes is to alter the depth of the potential in the ﬁeld direction. If the voltage U0 is applied symmetrically to the trap electrodes (that is, the trap center is at zero potential),

8

1 Summary of Trap Properties

we have ¯ z = D ¯ z + U0 /2, D ¯ r − U0 /2. ¯ = D D r

(1.17)

A typical value for a trap of 1 cm radius r0 driven by 500 V at a frequency of Ω/2π = 1 MHz and an ion atomic mass of 100 is 25 eV for the radial potential depth. 1.1.3 Motional Spectrum Equation (1.10) shows that the motional spectrum contains the frequencies (βi + 2n)Ω/2, where n is an integer having the fundamental frequencies given by n = 0. To experimentally demonstrate this spectrum, the motion can be excited by an additional (weak) radio-frequency (rf) ﬁeld applied to the electrodes. When resonance occurs between the detection ﬁeld frequency and one of the frequencies in the ion spectrum, the motion becomes excited and some ions may leave the trap, providing a signal commensurate with the number of trapped ions. Figure 1.5 shows an example where a cloud of stored N2 + ions is excited at the motional resonances occurring at the frequencies predicted by theory. 1.1.4 Optimum Trapping Conditions If we deﬁne the optimum trapping conditions for a Paul trap as those which yield the highest density of trapped particles, we are faced with conﬂicting requirements: On one hand, the maximum trapped ion number increases with

Fig. 1.5. Motional resonances of a N2 + cloud in a quadrupole Paul trap [7]

1.1 Trapping Principles in Paul Traps

9

Fig. 1.6. Optimum trapping conditions. (a) Computed lines of equal ion density within the stability diagram. The numbers give relative densities. (b) Experimental lines of equal ion density from laser-induced ﬂuorescence [8]

the potential depth D, since the number is limited by the condition that the space charge potential of the ions not exceed D. On the other hand, the oscillation amplitudes also increase at the same time, resulting in increasing ion loss for higher q in a trap of a given size. Consequently the maximum ion number is expected in a region around the center of the stability diagram. This has been conﬁrmed experimentally by systematic variation of the trapping parameters and a measurement of the relative trapped ion number by laser induced ﬂuorescence (Fig. 1.6) [8]. 1.1.5 Storage Time Once the ions are stored in a Paul trap they would remain there in the ideal case for an inﬁnitely long time. In practice, the storage time may be limited by trap imperfections as discussed later. Their eﬀects can be avoided by proper choice of the operating conditions. The limiting factor then would be collisions with neutral background molecules. Thus, operation of the traps in ultra-high vacuum is necessary; in fact at pressures around 10−8 Pa storage times of many hours are routinely obtained. Under certain conditions, however, higher background pressure may have a beneﬁcial eﬀect on the storage time as well. Major and Dehmelt [9] have pointed out that ion–neutral collisions lead to damping of the ion motion and thus to increased storage times when the ion mass exceeds the mass of the neutral atom or molecule. This has been experimentally veriﬁed in many cases

10

1 Summary of Trap Properties 4 Ion density (104 mm–3)

0

IF (a.u.)

40 30 20 10 0 (a)

–8

2 1 0

0 8 8 0 –8 x (mm) z (mm) Laser scanning direction

(b)

T=10 K 30 K 100 K 300 K 1000 K 3000 K 10000 K

3

0

1

2

3

4

5

6

Radial distance (mm)

Fig. 1.7. (a) Fluorescence IF of Ba ions in axial and radial direction in a Paul trap of 4 cm ring radius showing a Gaussian density distribution [10]. (b) Calculated density distributions for diﬀerent ion temperatures [11] +

and storage times of many days for heavy ions such as Ba+ or Pb+ have been obtained when operating with light buﬀer gases (He, N2 , Ne) at pressures as high as 10−2 Pa. 1.1.6 Ion Density Distribution In thermal equilibrium at high temperatures a cloud of trapped ions assumes a Gaussian density distribution, averaged over a period of the micromotion. This has been experimentally conﬁrmed by scanning a laser spatially through the trap and observing the ﬂuorescence light induced by laser exitation (Fig. 1.7). When the ion temperature is lowered, the cloud diameter shrinks until a homogeneously charged sphere is obtained. From the width of the distribution the ions’ average kinetic energy can be derived [10]. As a rule of thumb, it amounts to 1/10 of the potential depth. 1.1.7 Storage Capability The maximum density n of ions that can be stored occurs when the space charge potential created by the ion cloud, given by ∇2 Vsc = Qn/ε0 with ε0 the permittivity of free space, equals the trap potential depth. For a spherical potential shape n is given by n=

¯ 3ε0 D . 2 Qr0

(1.18)

The total stored ion number N follows from integration over the active trap volume. Taking a density distribution as shown in Fig. 1.7 we arrive at a maximum ion number N 106 in fair agreement with observations.

1.1 Trapping Principles in Paul Traps

11

1.1.8 Paul Trap Imperfections A single ion in a perfect quadrupole potential does not describe a real experimental situation. Truncations of the electrodes, misalignments, or machining errors change the shape of the potential ﬁeld. The equations of motion as discussed earlier are valid only for a single conﬁned particle. Simultaneous conﬁnement of several particles requires the consideration of space charge effects. For a low density cloud it seems reasonable to assume that the particles move in somewhat modiﬁed orbits, mainly independent of each other except for rare Coulomb scattering events. These collisions ultimately serve to establish a thermal equilibrium between the particles. They may be considered as small perturbations to the particle motion, provided the time average of the Coulomb interaction potential is small compared to the average energy of the individual particle. Then the particle cloud may be described as ideal gas of noninteracting particles in thermal equilibrium. Deviations of the trap potential from the ideal quadrupolar form can be treated by a series expansion in spherical harmonics, thus Φ(ρ, θ) = (U0 + V0 cos Ωt)

∞

cn

ρ n

n=2

d

Pn (cos θ),

(1.19)

where Pn (cos θ) are the Legendre polynomials of order n. For rotational and mirror symmetry the odd coeﬃcients cn vanish. The terms beyond the quadrupole (c2 ) may be looked on as perturbing potentials the lowest of which is the octupole (c4 ), followed by the dodecapole (c6 ). The equations of motion for a single particle in an imperfect Paul trap now become coupled inhomogeneous diﬀerential equations which cannot be solved analytically. It has been shown, however, that, under certain conditions that would otherwise give stability in a perfect quadrupole ﬁeld, the motion becomes unstable [12, 13]. These conditions can be expressed in terms of the stability parameters βr and βz , thus nr βr + nz βz = 2k, (1.20) or equivalently nr ωr + nz ωz = kΩ,

(1.21)

where nr , nz , k are integers. This relationship states that if a linear combination of harmonics of the ion macrofrequencies coincides with a harmonic of the high-frequency trapping ﬁeld, an ion will gain energy from that ﬁeld until it gets lost from the trap. Experimental proof of the instabilities has been obtained by measurements of the number of trapped ions at diﬀerent operating points. A high-resolution scan of the stability diagram was given by Alheit et al. [14], who observed instabilities resulting from very high orders of perturbing potentials (Fig. 1.8). From this ﬁgure it is evident that strong instabilities occur at the high-q region of the stability diagram, due to hexapole and octupole terms in the expansion

12

1 Summary of Trap Properties

Fig. 1.8. Experimentally observed lines of instabilities in the ﬁrst region of stability of a real Paul trap taken with H2 + ions. The unstable lines are assigned according to (1.21). The intensity in gray is proportional to the trapped ion number [14]

of the potential, which are the highest orders expected in a reasonably well machined trap. This makes it very diﬃcult to obtain long storage times at high amplitudes of the trapping voltage (for a given frequency). In fact, the most stable conditions are obtained for small q values near the a = 0 axis; here the instability condition on the frequencies is met only for a very high-resonance order which would occur only if the ﬁeld was highly imperfect. As another consequence of the presence of the higher order terms in the trapping potential, the motional eigenfrequencies are shifted with respect to the pure quadrupole ﬁeld, and moreover in an amplitude dependent way.

1.2 Trapping Principles in Penning Traps

13

These shifts are of particular importance in the case of Penning traps, when they are used for very high-resolution mass spectrometry, and will be discussed later. While for the ideal harmonic potential the line shape of the resonances is a Lorentzian, it now becomes asymmetric. From a theoretical ﬁt to the line shape the size of the higher order coeﬃcients in the series expansion of the potential (1.20) can be determined [15]. In addition to the ideal case the motional spectrum now contains not only the eigenfrequencies as calculated from (1.10) but also the sum and diﬀerence frequencies.

1.2 Trapping Principles in Penning Traps 1.2.1 Theory of the Ideal Penning Trap The Penning trap uses static electric and magnetic ﬁelds to conﬁne charged particles. The ideal Penning trap is formed by the superposition of a homogeneous magnetic ﬁeld B = (0, 0, B0 ) and an electric ﬁeld E = ∇Φ derived from the quadrupole potential as given in (1.4). A particle of mass M , charge Q, and velocity v = (vx , vy , vz ) moving in the ﬁelds E and B experiences a force F = −Q∇Φ + Q(v × B).

(1.22)

Since the magnetic ﬁeld is along the z-axis, the z component of the force is purely electrostatic, and therefore to conﬁne the particle in the z-direction we must have QU0 > 0. The x- and y-components of F are a combination of a dominant restraining force due to the magnetic ﬁeld, characterized by the cyclotron frequency |QB0 | ωc = , (1.23) M and a repulsive electrostatic force that tries to push the particle out of the trap in the radial direction. Newton’s equations of motion in Cartesian coordinates are as follows: d2 x dy 1 2 − ωz x = 0, − ωc 2 dt dt 2 d2 y dx 1 2 − ωz y = 0, + ωc dt2 dt 2 d2 z + ωz2 z = 0, dt2

where

(1.24)

2QU0 . (1.25) M d2 The motion in z-direction is a simple harmonic oscillation with an axial frequency ωz decoupled from the transverse motion in the x- and y-directions. ωz =

14

1 Summary of Trap Properties

To describe the motion in the x, y-plane we introduce the complex variable u = x + iy. The radial equations of motion then reduce to d2 u du 1 2 − ωz u = 0. + iωc dt2 dt 2

(1.26)

The general solution is found by setting u = exp(−iωt) to obtain the algebraic condition 1 ω 2 − ωc ω + ωz2 = 0. (1.27) 2 Then the general solution is given by u(t) = R+ exp[−i(ω+ t + α+ )] + R− exp[i(ω− t + α− )],

(1.28)

with ω+ =

1 (ωc + ωc2 − 2ωz2 ), 2

ω− =

1 (ωc − ωc2 − 2ωz2 ), 2

the modiﬁed cyclotron frequency and the magnetron frequency, respectively. Here R+ , R− , α+ , and α− are the radii and the phases of the respective motions, determined by the initial conditions. The relation (1.28) is the equation of an epicycloid. Figure 1.9 shows a sketch of the ion trajectory in all three dimensions. It describes a trajectory around the trap center where the particle moves within a circular strip between an outer radius R+ + R− and an inner radius |R+ − R− |. In order that the motion be bounded the roots of (1.29) must be real, leading to the trapping condition ωc2 − 2ωz2 > 0, or equivalently

(1.29)

|Q| 2 4|U0 | B > , QU0 > 0. (1.30) M 0 d2 Several useful relations exist between the eigenfrequencies of the trapped particle:

Fig. 1.9. Sketch of the ion trajectory in a Penning trap

1.2 Trapping Principles in Penning Traps

2 ω+

15

ω+ + ω− = ωc ,

(1.31)

ωz2 , ωc2 .

(1.32) (1.33)

2ω+ ω− = 2 + ω− + ωz2 =

The last is known as “invariance theorem” since the ﬁrst-order frequency shifts by trap imperfections cancel [16]. 1.2.2 Motional Spectrum in Penning Traps The motional spectrum of an ion in a Penning trap contains the fundamental frequencies ω+ , ω− and ωz as shown in the solution of the equations of motion. These frequencies can be measured using resonant excitation of the ion motion by applying an additional dipole rf ﬁeld to the trap electrodes. An indication of resonant excitation is a rapid loss of stored ions due to the gain in energy from the rf ﬁeld, which enables them to escape. The electrostatic ﬁeld inside the trap is expected to be a superposition of many multipole components, hence many combinations of the fundamental frequencies and their harmonics become visible, depending on the amplitude of the exciting ﬁeld. Figure 1.10 shows the number of trapped electrons as a function of the frequency of an excitation rf ﬁeld which enters the apparatus by an antenna placed near the trap. Identiﬁcation of the resonances is through their diﬀerent dependence on

Fig. 1.10. Observed motional resonances of electrons in a Penning trap taken at diﬀerent amplitudes of an exciting rf ﬁeld

16

1 Summary of Trap Properties

the electrostatic ﬁeld strength. The axial frequency ωz depends on the square root of the trapping voltage while ω+ and ω− vary linearly with the voltage. Of particular interest is the combination ω+ + ω− , since it is independent of the trapping voltage, and equals the cyclotron frequency of the free ion ωc whose measurement can serve to calibrate the magnetic ﬁeld at the ion position. The frequency ωc can be obtained in three diﬀerent ways: Direct excitation of the sideband ω+ + ω− , measurement of the fundamental frequency ω+ at diﬀerent trapping voltages U0 and extrapolating to U0 = 0, or using the relation (1.33). The last is particularly useful since it is, to ﬁrst-order, independent of perturbations arising from ﬁeld imperfections, as stated earlier [16]. 1.2.3 Penning Trap Imperfections As in the case of Paul traps, in real Penning traps the conditions are considerably more complicated than the ideal description in Sect. 1.2.2. The hyperbolic surfaces are truncated and may deviate from the ideal shape, the trap axis may be tilted with respect to the direction of the magnetic ﬁeld, and the long range Coulomb potential of ions adds to the trapping potential when more than a single ion is trapped. The departures of the ﬁeld from the pure quadrupole introduce nonlinearity in the equations of motion, and coupling between the degrees of freedom. The main eﬀects of these imperfections on the behavior of the trapped ions are a shift in the eigenfrequencies and reduction in the storage capability of the trap. Electric ﬁeld imperfections. As in the case of the previously discussed Paul trap, imperfections in the electrostatic ﬁeld can be treated by a multipole expansion of the potential with coeﬃcients cn characterizing the strength of higher order components in the trapping potential as stated in (1.19). Several authors [17–20] have calculated the resulting shift in the ion oscillation frequency. Assuming that due to rotation and mirror symmetry of the trap only even orders contribute and regarding the octupole term (c4 ) as the most important contribution the results are c4 2 2 ωz [Rz2 − 2(R+ + R− )], d2 ωz2 c4 2 (R2 + 2R− − 2Rz2 ), d2 ω+ − ω− + ωz2 3 c4 2 (R2 + 2R+ − 2Rz2 ). Δω− = − 2 d2 ω+ − ω− − 3 4 3 Δω+ = 4 Δωz =

(1.34)

To minimize these shifts, it is obviously necessary not only to make the trap as perfect as possible but also to reduce the oscillation amplitude R by some ion cooling method. Magnetic ﬁeld inhomogeneities. An inhomogeneity in the superimposed magnetic ﬁeld B0 of the Penning trap also leads to a shift of the ﬁelddependent frequencies. As in the electric ﬁeld case, a magnetostatic ﬁeld in

1.2 Trapping Principles in Penning Traps

17

a current-free region can be derived from a potential function Φm , which can be expanded in the following multipole series: Φm = B0

∞

bn ρn Pn (cos θ).

(1.35)

n=0

In the case of mirror symmetry in the trap’s mid plane, the odd coeﬃcients vanish. If we retain only the b2 term in the expansion, the solution of the equations of motion leads to a frequency shift for the most interesting case of the sideband ω+ + ω− = ωc given by ω− R+ − ω+ R− b2 Rz + Δωc = ωc . (1.36) 2 ω+ − ω−

Axial frequency (kHz)

Higher orders are treated in Sect. 3.12. Other imperfections such as an ellipticity of the trap or a tilt angle between the trap axis and the magnetic ﬁeld direction also shift the eigenfrequencies. It should be noted, however, that these shifts do not aﬀect the free ion cyclotron 2 2 frequency ωc in ﬁrst-order when using the invariance theorem ωc2 = ω+ + ω− + 2 ωz [16]. Space charge shift. The space charge potential of a trapped ion cloud also shifts the eigenfrequencies of individual ions. From a simple model of a homogeneous charge density in the cloud, we are led to a decrease in the axial frequency proportional to the square root of the ion number. This has been veriﬁed experimentally (Fig. 1.11). More important for high-precision spectroscopy is the space charge shift in the frequencies when we are dealing with small numbers of ions, where the simple model is no longer adequate. The observed shift depends on the mean inter-ion distance and consequently on the ion temperature. For ions cooled to 4 K, Van Dyck et al. [21] have observed shifts of opposite signs in the perturbed cyclotron and magnetron frequencies, amounting to 0.5 ppb per stored ion. The shift increases approximately linearly with the charge of the ions. The electrostatic origin of the shifts suggests that the sum of the

860 840 820 800 780 0

50 100 150 200 250 300 350 Ion number (x 104)

Fig. 1.11. Shift of the axial frequency as function of the trapped ion number

18

1 Summary of Trap Properties

perturbed cyclotron and magnetron frequencies, which equals the free ion cyclotron frequency, should be independent of the ion number. In fact, the shifts observed in the sideband ω+ + ω− = ωc are consistent with zero. Image charges. An ion oscillating with amplitude r induces in the trap electrodes image charges which create an electric ﬁeld that reacts on the stored ion and shifts its motional frequencies. The shift has been calculated by Van Dyck et al. [21] using a simple model. If the trap is replaced by a conducting spherical shell of radius a, the image charge creates an electric ﬁeld given by E=

1 Qa r. 4πε0 (a2 − r2 )2

(1.37)

Since this ﬁeld is added to the trap ﬁeld, it causes a shift in the axial frequency amounting to Q2 1 Δωz = − , (1.38) 4πε0 2M a3 ωz and like E scales linearly with the ion number n, being signiﬁcant only for small trap sizes. Since it is of purely electrostatic origin, the electric ﬁeldindependent combination frequency ω+ + ω− = ωc is not aﬀected. Instabilities of the ion motion. Another consequence of the presence of higher order multipoles in the ﬁeld is that ion orbit instability can occur at certain operating points, where it would otherwise be stable in a perfect quadrupole ﬁeld. Using perturbation theory to solve the equations of motions when the trap potential is written as a series expansion in spherical harmonics, Kretzschmar [17] has shown that the solution exhibits singularities for operating points at which n+ ω+ + n− ω− + nz ωz = 0,

(1.39)

where n+ , n− , nz are integers. The ion trajectory at such points is unstable and the ion is lost from the trap. In Fig. 1.12 the results of measurements obtained for diﬀerent trap voltages at a ﬁxed value of the magnetic ﬁeld, on a cloud of trapped electrons are shown. The combined eﬀect of space charge and trap imperfections leads to the loss of the particles [22]. The observed instabilities which become more discernible for extended storage times, can be assigned to operating conditions predicted from (1.39). These results show that in practice it is very diﬃcult to obtain stable operating conditions when the trapping voltage exceeds about half the value allowed by the stability criterion. 1.2.4 Storage Time Ideally a stored ion would remain in a Penning trap for unlimited time. As discussed earlier, trap imperfections, however, may lead to ion loss. Proper choice of the operating conditions and reduced ion oscillation amplitude by ion cooling makes this negligible. The limiting factor then would be collisions

1.2 Trapping Principles in Penning Traps

19

Fig. 1.12. Observed number of trapped electrons vs. trapping potential at diﬀerent magnetic ﬁeld strengths. The trap voltage is given in units of the maximum voltage as derived from (1.29). The combinations of eigenfrequencies leading to trap instabilities according to (1.39) is indicated [22]

with neutral background molecules, since they cause an increase in the magnetron orbit of the ion which will eventually hit the trap boundary. Typical storage times at 10−8 Pa are several minutes and at extremely low pressures of below 10−14 Pa, obtained by cryopumping, storage times of many months are obtained. The problem of collision-induced increase of the magnetron orbit can be overcome by an additional rf ﬁeld at the sum frequency of the perturbed cyclotron and the magnetron motions [23]. The ﬁeld is applied between adjacent segments of the ring electrode of the Penning trap, which is split into four quadrants. The eﬀect of this ﬁeld is to couple the two motions. The cyclotron motion is damped by collisions with background atoms and the coupling transfers the damping to the magnetron motion. The net eﬀect is that the ions aggregate near the trap center and consequently the storage time is extended [24].

20

1 Summary of Trap Properties

1.2.5 Storage Capability The limiting ability to trap charged particles in a Penning type of trap is determined by their mutual electrostatic repulsion. From the condition for stable conﬁnement which gives in eﬀect the magnetic ﬁeld required to balance the radial component of the electric ﬁeld, we can ﬁnd a value for the limit on the number density of ions that can be trapped at a given magnetic ﬁeld intensity. This limit, sometimes called the Brillouin limit, is given by nlim =

B2 , 2μ0 M c2

(1.40)

where μ0 is the permeability of free space. Naturally, as this limit is approached, the individual particle picture is no longer valid; one is dealing with a plasma. 1.2.6 Spatial Distribution As in the Paul trap, an ion cloud in thermal equilibrium in a Penning trap assumes a Gaussian density distribution. The width of the distribution, however, can be changed by a technique introduced in 1977 at the University of California at San Diego and called the “rotating wall”: The crossed electric and magnetic ﬁelds of a Penning trap causes a charged particle to rotate around the trap center at the magnetron frequency [25]. In addition, the cloud may be forced to rotate at diﬀerent frequencies by an external torque from an electric ﬁeld rotating around the symmetry axis. The ring electrode of a Penning trap is split into diﬀerent segments. A sinusoidal voltage Vwj of frequency ωw is applied to the segments at θj = 2πj/n, where n is the number of segments, with Vwj = Aw cos[m(θj − ωw t)]. Figure 1.13 illustrates an eight-segment conﬁguration. The additional centrifugal force from the plasma rotation leads to a change in plasma density and spatial proﬁle. Thus, variation of the rotating wall frequency allows the plasma to be compressed or expanded if the ﬁeld is rotating with or against the rotation of the plasma, respectively (Fig. 1.14).

Fig. 1.13. Ring electrode of a trap in an eight-segment conﬁguration for application of a rotating electric ﬁeld

1.3 Trap Techniques 2

C n (108 cm–3)

n (108 cm–3)

8 6 4

B

2

A

0 –1

–0.5

(a)

0 0.5 r (cm)

21

F

1 E 0

1

–1 (b)

–0.5

D

0 0.5 r (cm)

1

Fig. 1.14. Measured change of density by application of rotating wall ﬁeld on an electron (a) and Be+ (b) plasma. B, E: No rotating ﬁeld, A, D: Rotating ﬁeld opposite to magnetron motion, C, F: Field rotating in direction of the magnetron motion [26]

1.3 Trap Techniques 1.3.1 Trap Loading In-trap Ion Creation The easiest way to load ions into the trap is to create them inside the trapping volume by photon or electron ionisation of an atomic beam or the neutral background gas. At room temperature the energy of the ions is in general signiﬁcantly smaller than the trap potential depth and all ions are conﬁned. Alternatively the ions can be created by surface ionisation from a ﬁlament placed at the edge of the trapping volume (Fig. 1.15). Atoms of interest can be deposited on the ﬁlament’s surface or injected as ions into the material. Heating the ﬁlament releases ions from the surface. The eﬃciency depends on the work function of the ﬁlament’s material and the ionisation potential of the atom under investigation. Using ﬁlaments of Pt, W, or Rh with high workfunctions and atoms like Ba, Sr, Ca with low ionisation potentials, typical eﬃciencies are of the order of 10−5 . Ion Injection from Outside An ion injected into a Penning trap from an outside source under high-vacuum condition may be captured by proper switching of the trap electrodes: When the ion travels along the magnetic ﬁeld lines and approaches the ﬁrst end cap electrode, its potential is set to zero while the second end cap is held at some retarding potential. If the axial energy of the ion is smaller than the retarding potential, it will be reﬂected. Before the ion leaves the trap through the ﬁrst end cap, its potential is raised and prevents the ion from escaping (Fig. 1.16). This simple method requires that the arrival time of the ion at the trap is known; this is achieved by pulsing the ion source. Moreover, the switching has to be performed in a time shorter than twice the transit time of the ions through the trap. For ion kinetic energies of a few 100 eV and a trap size of

22

1 Summary of Trap Properties

Fig. 1.15. Filaments for ion production by surface ionisation placed in a slot of a traps end cap electrode

Fig. 1.16. Simple model describing the ion capture in a Penning trap

1 cm this time is of the order of 100 ns. Successful capture of ions from a pulsed source is routinely used at the ISOLDE-facility at CERN, where nearly 100% eﬃciencies are achieved [27]. Injection into a Paul trap under similar conditions is more diﬃcult, since the time varying trapping potential of typically a megaherz frequency cannot be switched from zero to full amplitude in a time of the order of less than a microsecond. The ion longitudinal kinetic energy, however, may be transferred into transverse components by the inhomogeneous electric trapping ﬁeld and thus the ions may be conﬁned for some ﬁnite time. Schuessler and Chun-sing [28] have made extensive simulations and phase considerations and have found that ions injected at low energy during a short interval when the ac trapping ﬁeld has zero amplitude may remain in the trap for some ﬁnite time. The situation is diﬀerent when the ions undergo a loss in kinetic energy while they pass through the trap. This energy loss is most easily obtained by collisions with a light buﬀer gas. The density of the buﬀer gas has to be at least of such a value that the mean free path of the ions between collisions is of the trap’s size. Coutandin et al. [29] have shown that the trap is ﬁlled up to its maximum capacity in a short time at pressures around 10−3 Pa in a 1 cm size Paul trap when ions are injected along the trap axis with a few kiloelectronvolts kinetic energy (Fig. 1.17). The same method is used routinely in many experiments where a high-energy ion beam is delivered from an accelerator. It is stopped and cooled by buﬀer gas collisions in a gas ﬁlled linear rf trap and then ejected at low energies with small energy spread into a trap operated at ultra-high vacuum for further experiments [30]. Injection into a Penning trap using collisions to entrap them requires some care. The ion’s

1.3 Trap Techniques

Ion number (a.u.)

1.0

23

H2 He N2

0.5

0 10–6 10–5 10–4 10–7 Buffer gas pressure (mbar)

Fig. 1.17. Trapped Ba+ ion number vs. buﬀer gas pressure for He, H2 , and N2 when injected at a few kiloelectronvolts kinetic energy. The maximum ion number for H2 is arbitrarily set to 1 [29]

motion becomes unstable since collisions lead to an increase of the magnetron radius. This can be overcome, as described earlier, if the trap ring electrode is split into four segments and an additional rf ﬁeld is applied between adjacent parts to create a quadrupolar rf ﬁeld in the radial plane. At the sum of the perturbed cyclotron frequency ω+ and the magnetron frequency ω− this ﬁeld couples the two oscillations. The damping of the cyclotron motion by collisions with the background atoms, overcomes the increase of the magnetron radius and as a result the ions aggregate near the trap’s center [23]. Since ω+ + ω− = ωc , the free ion cyclotron frequency depends on the ion mass, it is possible to stabilize a particular isotope or even isobar in the trap by proper choice of the rf frequency [31] while ions of unwanted mass do not remain in the trap. 1.3.2 Trapped Particle Detection Destructive Detection Paul traps. Ions conﬁned in a Paul trap may leave the trap either by lowering the potential of one end cap electrode or by application of a voltage pulse of high amplitude. They can be counted by suitable detectors such as ion multiplier tubes or channel plate detectors. By diﬀerent arrival times to the detector, ions of diﬀerent charge-to-mass ratio can be distinguished (Fig. 1.18). The amplitude of the detector pulse depends on the phase of the leading edge of the ejection pulse with respect to the phase of the rf trapping voltage. When several ion species are simultaneously trapped, a particle of speciﬁc charge-to-mass ratio can be selectively ejected from the trap and counted by a detector if an rf ﬁeld resonant with the axial oscillation frequency of the ion of interest is applied in a dipolar mode between the trap’s end cap electrodes [32]. Penning traps. Ions can be released from the Penning trap, generally placed in the most homogeneous part of a (superconducting) solenoid, by switching

24

1 Summary of Trap Properties

+ Fig. 1.18. Ejection of simultaneously trapped H+ , H+ 2 , and H3 ions from a Paul trap and detection by an ion multiplier located 5 cm from the traps end cap [14]

Fig. 1.19. Time-of-ﬂight detection of ions released from a Penning trap. The magnetic ﬁeld points in the vertical direction

one of the end cap potentials to zero. The particles then travel along the magnetic ﬁeld lines until they arrive at a detector outside the magnetic ﬁeld region (Fig. 1.19). With ﬁnite radial kinetic energies the ions have some angular momentum which has associated with it an orbital magnetic moment μ. In the fringing ﬁeld of the magnet a force F = ∇(μB) acts upon the ions and accelerates them onto the detector. The time-of-ﬂight for a given angular momentum is determined by the ion’s mass and thus the diﬀerent masses of simultaneously trapped ions can be distinguished. We note that the excitation of the radial motion by an additional rf ﬁeld increases the angular momentum and thus leads to a reduced time-of-ﬂight. This method of detecting ion oscillation frequencies, particularly the cyclotron frequency, serves as the basis for many high-precision mass spectrometers [33]. Nondestructive Detection Tank circuit damping. The mass dependent oscillation frequencies of ions in a trap can be used for detection without ion loss. A tank circuit consisting of an inductance L connected in parallel with the trap electrodes as capacitance C is weakly excited at its resonance frequency ωLC (Fig. 1.20). The ions’ axial oscillation frequency ωz can be changed by variation of the electric trapping ﬁeld. When the frequencies are tuned to resonance, energy is transferred from the LC circuit to the ions leading to a damping of the circuit, and a decrease

1.3 Trap Techniques I

Detection resonant circuit

R

C

Ion trap I

25

z L

Un

Amplifier

Fig. 1.20. Electronic circuit for nondestructive detection of trapped ion clouds

of the voltage across it. Modulation of the trap voltage around the operating point at which resonance occurs and rectiﬁcation of the voltage across the circuit lead to a repeated voltage drop, whose amplitude is proportional to the number of trapped ions. When diﬀerent ions are simultaneously conﬁned signals appear at diﬀerent values of the modulated trapping voltage. The ˜ of the resonance sensitivity of the method depends on the quality factor Q ˜ circuit. With moderate values of the order of Q = 50 about 1,000 trapped ions lead to an observable signal. Bolometric detection. In the bolometric detection of trapped ions, ﬁrst proposed and realised by Dehmelt and co-workers [34–36] the ions are kept in resonance with a tuned LC circuit connected to the trap electrodes in a way similar to what was discussed in the previous section. An ion of charge Q oscillating between the end cap electrodes of a trap induces a current in the external circuit given by I = Γ Qz/(2z ˙ (1.41) 0 ), where 2z0 is the separation of the end caps, and Γ is a correction factor, which accounts for the approximation of the trap electrodes by parallel plates of inﬁnite dimension. For hyperbolically shaped electrodes Γ 0.75. The electromagnetic energy associated with this current will be dissipated as thermal energy in the parallel resonance resistance of the LC circuit, R. The increased temperature T of that resistance results in an increased thermal noise voltage Unoise in a bandwidth δν: √ Unoise = 4kT Rδν. (1.42) If the number of stored ions is known, the noise voltage measured by a narrow band ampliﬁer can serve as a measure of the ion temperature. When ions of diﬀerent charge-to-mass ratios are stored simultaneously their oscillation frequencies can be brought into resonance with the circuit by sweeping the trap voltage. Each time an ion species is resonant with the circuit, the noise amplitude increases and can be detected. An example is shown in Fig. 1.21 where the increased noise of the axial resonances of diﬀerent highly charged ions in a Penning trap is recorded. For high sensitivity the thermal noise power of the circuit has to be kept as low as possible. When superconducting circuits at temperatures around 4 K are used, single ions can be well observed.

26

1 Summary of Trap Properties

Fig. 1.21. Thermal noise induced in an LC circuit by simultaneously trapped ions of diﬀerent charge-to-mass ratios in a Penning trap [37]

Fig. 1.22. Fourier transform of the noise from C 5+ ions induced in a tank circuit ˜ value of attached to two segments of the ring electrode of a Penning trap. The Q the circuit was 4,000. The magnetic ﬁeld was inhomogeneous and ions located at diﬀerent places in the trap have slightly diﬀerent cyclotron frequencies [37]

Fourier transform detection. A Fourier transform of the noise induced by the oscillating ions in a tank circuit shows maxima at the ion oscillation fre˜ of several quencies. When superconducting circuits with quality factors Q thousands are used the signal strength from a single ion is suﬃcient for detection. Figure 1.22 shows the noise induced by six ions in a circuit attached to two segments of the ring electrode of a Penning trap. The magnetic ﬁeld was inhomogeneous and ions located at diﬀerent places in the trap have slightly diﬀerent cyclotron frequencies. When the ions are kept continuously in resonance with the tank circuit, the increased thermal energy of the detection circuit, due to the induced ion currents, will be dissipated [38]. Consequently, the ion oscillation will be exponentially damped with a time constant τ given by τ=

(2z0 )2 M , Γ 2 R Q2

(1.43)

1.3 Trap Techniques

27

Fig. 1.23. Fourier transform of the noise of a superconducting axial resonance circuit in the presence of a single trapped C5+ ion in thermal equilibrium with the circuit. The sum of the circuit’s thermal noise and the induced noise from the oscillating ion leads to a minimum at the ion’s oscillation frequency [37]

where R is the impedance of the circuit. When the ions reach thermal equilibrium with the circuit, excess noise can no longer be detected; nevertheless, the presence of ions can be detected by a spectral analysis of the circuit noise. The voltage that the ion induces with its remaining oscillation amplitude adds to the thermal noise of the circuit, however with opposite phase. As a result, the total noise voltage at the ion oscillation frequency is reduced, and the spectral distribution of the noise shows a minimum at this frequency when ions are present in the trap. This is shown in Fig. 1.23 for a single C5+ ion. Optical detection. A very eﬃcient way to detect the presence of ions in the trap is to monitor their laser induced ﬂuorescence. This method is, of course, restricted to ions which have an energy-level scheme which allows excitation by available lasers. It is based on the fact that the lifetime of an excited ionic energy level is of the order of 10−7 s when it decays by electric dipole radiation. Repetitive excitation of the same ion by a laser at saturation intensity then leads to a ﬂuorescence count rate of 107 photons per second. Of those a fraction of the order of 10−3 can be detected if we assume a solid angle of 4π/10, a photomultiplier detection eﬃciency of 10%, and ﬁlter and transmission losses of 90%, leading to an easily observable signal (Fig. 1.24). The method is most eﬀective when the ion under consideration has a large transition probability for the excitation from the electronic ground state as in alkali-like conﬁgurations. It becomes particularly easy when the excited state ion falls back directly into the ground state. Such two-level systems are available in Be+ and Mg+ and consequently these ions are preferred subjects when optical detection of single stored particles is studied. For other ions of alkalilike structure such as Ca+ , Sr+ , or Ba+ it becomes slightly more complicated since the excited state may decay into a long-lived low lying metastable state which prevents fast return of the ion into its ground state; then an additional laser is required to pump the ion out of the metastable state. Signals of the expected strength have been obtained in all those ions. They allow even the visual observation of a single stored ion Ba+ as demonstrated in a pioneering experiment at the University of Heidelberg [40].

28

1 Summary of Trap Properties

Fig. 1.24. The ﬁrst observation of an atomic particle, as a single Ba+ ion (a) trapped in a miniaturized Paul trap (b) Reprinted with permission from [39]

1.4 Ion Cooling Techniques The mean kinetic energy of trapped ions depends on the operating conditions of the trap and the initial conditions of the ion motion; in thermal equilibrium a typical value is 1/10 of the maximum potential depth. This is in general much higher than room temperature and might cause signiﬁcant line shifts and broadening in high-precision spectroscopy. Therefore, a number of methods have been developed to reduce the temperature of the trapped particles. 1.4.1 Buﬀer Gas Cooling Paul traps. The inﬂuence of ion collisions with neutral background molecules can be treated by the introduction of a damping term in the equations of motion. Major and Dehmelt [9] have shown that ion cooling appears when the ion mass is larger than that of the background molecule. The equilibrium ion temperature results from a balance between collisional cooling and energy gain from the rf trapping ﬁeld; it depends on the buﬀer gas pressure (Fig. 1.25). The thermalization process in a Paul trap applies to the macromotion of the ions, while the amplitude of the micromotion remains ﬁxed, determined by the amplitude of the electric trapping ﬁeld at the ion’s position. Cutler et al. [11] found that the calculated radial density distribution, averaged over the micromotion, changes from a Gaussian at high temperatures to a spherical one with constant density when the temperature approaches zero (see Fig. 1.7). Penning traps. As in the Paul traps the axial motion of an ion in a Penning trap as well as the cyclotron motion are damped by collisions with neutral molecules. The magnetron oscillation, however, is a motion around an electric

Temperature (103 K)

1.4 Ion Cooling Techniques

29

12 10 8 6 4 2 0

0

50 100 150 Pressure (10-5 Pa)

200

Fig. 1.25. Average kinetic energy of Ca+ vs. N2 background gas pressures [41]

Fig. 1.26. Simulation of an ion radial trajectory in a Penning trap under the inﬂuence of collisions with neutral molecules (a) without and (b) with a quadrupolar rf coupling ﬁelds at ω = ω+ + ω−

potential hill and collisions tend to increase the radius of the magnetron orbit until the particle gets lost from the trap. This problem can be overcome by the introduction of an additional transverse quadrupole rf ﬁeld at the sum of the magnetron and reduced cyclotron frequencies applied between four segments of the ring electrode. It couples the two motions and the leads to an aggregation of the ions near the trap center [19, 23]. Figure 1.26 shows a simulation of the ion trajectory in the radial plane of a Penning trap under the inﬂuence of ion–neutral collisions with and without the coupling ﬁeld. 0 0 For initial radial amplitudes R+ and R− of the two motions, the amplitudes change as 0 R± (t) = R± exp(∓α± t), (1.44) with α± =

ω± γ . M ω+ − ω−

(1.45)

1.4.2 Resistive Cooling When an LC-circuit resonant with the ion oscillation frequency is attached between trap electrodes the image currents induced by the ion’s motion through the external circuit will increase its thermal energy which will be dissipated to the environment [34–36, 38]. It leads to an exponential energy loss of the ions with a time constant

30

1 Summary of Trap Properties

Fig. 1.27. Resistive cooling of an ion cloud in a Penning trap monitored by the induced image noise in an attached tank circuit. The center-of-mass (CM) energy is reduced in a short time of about 100 ms while the individual ion oscillation is damped with a much longer time constant (5 s)

τ=

2z0 Q

2

M 1 , ˜ ωL Q

(1.46)

˜ is the quality factor of the LC circuit. For an electron conﬁned in a where Q ˜ = 100 with L = 1 μH, oscillating at 300 MHZ trap with z0 = 1 cm, a circuit Q we obtain τ 5 ms. The increased time constant for heavier ions can be partly ˜ compensated by choosing superconducting circuits with Q-values of several thousands. The ﬁnal ion temperature will be the same as the temperature of the circuit which may be kept in contact with a liquid helium bath at 4 K. Resistive cooling is particularly well suited for Penning traps. In Paul traps the high amplitude of the rf trapping ﬁeld will in general cause currents through the attached tank circuit which prevents reaching low temperatures. We note that resistive cooling of an ion cloud aﬀects the center-of-mass (CM) energy only. Coulomb interaction transfers energy from the individual ion oscillation into the CM mode. The rate of this energy transfer depends on the ion number. Thus, two time constants are observed (Fig. 1.27). 1.4.3 Laser Cooling Laser cooling is the most eﬀective way to reduce the ion’s kinetic energy. It is based on the conservation of linear momentum in the scattering of many photons from the same ion by repetitive laser excitation. Thus, the basic requirement is that the ion can be excited by electric dipole radiation and the excited state decays rapidly back into the ground state. This is the case in singly ionized Mg and Be ions, which are most often used in laser cooling experiments. Other ions of alkali-like structure such as Ca+ , Sr+ , Ba+ , and

1.4 Ion Cooling Techniques

31

Hg+ have more complex-level schemes but can provide an eﬀective two-level scheme by additional lasers. Doppler cooling. When a laser is tuned in frequency slightly below a resonance transition of an ion only those ions absorb laser light with high probability which move against the direction of the laser beam, because the laser’s frequency is Doppler shifted into resonance. During photon absorption, momentum conservation requires the ion to lose momentum. However, the absorbed photon is reemitted with an angular distribution having zero average photon recoil momentum. The net eﬀect is a reduction of ion momentum. To cool stored ions, the laser frequency is swept from low frequency toward the resonance center. As soon as the laser frequency is higher than the ion’s resonace frequency the opposite eﬀect occurs: Ions moving in the direction of the laser beam absorb photons and gain energy. As a result the ion cloud expands and the spatial overlap with the laser is reduced. When the ﬂuorescence from the ion cloud is monitored the resonance lineshape becomes asymmetric showing a sharp edge at the high-frequency side (Fig. 1.28a). The rate at which energy is lost depends on the laser detuning from resonance δω, the transition matrix element P , the ion’s mass M , and the laser intensity. The lowest temperature Tmin is reached when the Doppler width of the transition is equal to the natural line width (“Doppler limit”): Tmin =

γ , 2kB

(1.47)

where γ is the decay rate of the excited state and kB is the Boltzmann constant. Typical values for Tmin are in the millikelvin range. When during the cooling process the temperature becomes so low that the ratio Γ between the Coulomb repulsion energy between ions of average distance a and the thermal energy kB T /2

Fig. 1.28. Fluorescence from a small cloud of Ca+ ions in a Paul trap when the laser frequency is slowly swept across the 4S1/2 − 4P1/2 resonance. At laser frequencies below resonance (Δν < 600 MHz) ions are cooled, above resonance they are heated and the cloud expands. (a) Low laser power, (b) high-laser power

32

1 Summary of Trap Properties

Fig. 1.29. Crystalline structures of small Ca+ ion clouds observed by a CCD camera [42]

Γ =

Q2 1 4πε0 akB T

(1.48)

becomes larger than 175, a phase transition to a crystalline structure appears in the ion cloud. It manifests itself by a kink in the ﬂuorescence lineshape, caused by a sudden reduction of the Doppler broadening (Fig. 1.28b). The crystalline structures can be observed directly by a CCD camera (Fig. 1.29). Sideband cooling. When the amplitude of the ion oscillation in the harmonic trap potential becomes smaller than the wavelength of the exciting laser no Doppler broadening appears. Instead distinct sidebands in the excitation spectrum at multiples of the oscillation frequency show up (“Dicke eﬀect”). The requirement is expressed by the condition η << 1, where η is the Lamb–Dicke parameter η=k

2M ω

1/2 =

ω

rec

ω

1/2 .

(1.49)

Here k is the wave number of the laser radiation, ω the ion oscillation frequency, and ωrec is the photon recoil energy. The sidebands are resolved when the spontaneous transition rate γ is small compared to ω/2π. This is in general the case only when transitions to long-lived metastable states are excited. An example is shown in Fig. 1.30. If one irradiates the ion with a narrow band laser tuned to the ﬁrst lower sideband at ω0 − ων , where ω0 is the resonance frequency of the ion at rest and ων is the oscillation frequency, it absorbs photons of energy (ω0 − ων ). The reemitted energy is symmetrically distributed among carrier and sidebands, thus amounts on average to ω0 . Hence, on the average each scattered photon reduced the ions vibrational energy by ων or, in a quantum mechanical picture, reduced the vibrational quantum number n by 1. Continuous excitation on the lower sideband ﬁnally drives the ion into the ground state of the conﬁning potential. For a vibrational frequency of 1 MHz this corresponds to a temperature of 50 μK. It is indicated by the disappearance of the lower sideband in the excitation

1.4 Ion Cooling Techniques

33

Fig. 1.30. Excitation of the 6S1/2 − 5D5/2 quadrupole transition on a single laser cooled 198 Hg+ ion at 282 nm showing a carrier frequency at zero laser detuning from resonance and sidebands at the ions micromotion frequency. Reprinted with permission from [43]

Fig. 1.31. Sideband cooling of a single Ca+ ion into the vibrational ground state in a Paul trap. The occupation probability PD of the D-level in the S1/2 − D5/2 quadrupole transition is plotted vs. the laser detuning δω after sideband cooling (full circles). The lower sideband after Doppler cooling (open circles) is shown for comparison. (a) First lower sideband; (b) ﬁrst upper sideband [44]

spectrum, because for n = 0 no state with n = −1 exists in the excited optical level (Fig. 1.31). 1.4.4 Radiative Cooling Accelerated charged particles lose energy by emission of radiation proportional to the square of their acceleration a: dE Q2 =− a2 . dt 4πε0 c3

(1.50)

For a particle oscillating with frequency ω the mean energy loss is described by dE = −γE, dt

(1.51)

34

1 Summary of Trap Properties

Fig. 1.32. Noise power induced in an electrode from a single electron conﬁned in a Penning trap showing the occupation of the lowest quantum states of the cyclotron harmonic oscillator. At T < 0.1 K the electron remains in the lowest quantum level for nearly inﬁnitely long times [45]

with the solution

E = E0 e−γt ,

where γ=

1 Q2 ω 2 . 6πε0 M c3

(1.52) (1.53)

Because of the low mass and the high frequency this energy loss is particularly signiﬁcant for the cyclotron motion of electrons conﬁned in a Penning trap at high magnetic ﬁelds while for atomic ions it can be totally neglected. In a ﬁeld of 5 T the time constant for electrons is calculated to be 12 s−1 . The ﬁnal temperature is given by the equilibrium with the environment because of excitation by blackbody photons. At cryogenic temperatures the quantum nature of the cyclotron oscillation becomes visible: The probability Pn of the population of a state with quantum number n is given by the Boltzman distribution Pn = A exp(−nωc /kB T ). (1.54) At B = 5 T and a temperature of 100 mK the electron remains almost exclusively in the ground level n = 0 of the cyclotron oscillation as experimentally demonstrated Peil and Gabrielse (Fig. 1.32) [45].

2 Mass Spectrometry Using Paul Traps

Mass is one of the basic quantities to characterize any material object, whether an atom, molecule, nucleus, or elementary particle. The measurement of mass therefore serves to detect and identify atomic, molecular, and nuclear species, and can help determine their structure and binding energy. For example, a precise determination of the mass of a nucleus is of importance through its binding energy, not only for various aspects of nuclear physics but also for other branches of physics, e.g. tests of the weak interaction, of quantum electrodynamics, and of the standard model [46]. Also in astrophysics the masses of unstable isotopes involved in stellar nucleosynthesis, especially the r process, are of signiﬁcance. On a more practical plane, modern mass spectrometers, as sophisticated analytical instruments, are indispensable in the ﬁelds of biochemistry and the development of new drugs. Mass spectrometry has been an active ﬁeld since the early days of atomic and nuclear physics, with continuous improvement in mass resolution and accuracy. All mass spectrometers rely on the dependence of the motion of charged particles in electric and magnetic ﬁelds on their charge-to-mass ratio. Traditional mass spectrometry [47–52] used the deﬂection of charged particles in such ﬁelds. These devices allowed mass measurements with a fractional uncertainty up to 10−7 . A diﬀerent class of mass analysers that avoided the use of ion beams in contoured magnetic ﬁelds appeared ﬁrst with the use of multipole electric ﬁelds to focus and separate charged particles in linear devices [53, 54]. Paul and coworkers developed what is now called the Paul trap for three-dimensional conﬁnement of charged particles [55]. In parallel, the Penning trap for conﬁnement of charged particles by static electric and magnetic ﬁeld was developed [3, 56] based on early ideas by Penning [57] and Pierce [58]. The requirements of high-mass resolution and accuracy clearly depend on the problem to be investigated: While for identiﬁcation of molecules in chemistry a mass resolution of 104 –105 is suﬃcient, the determination of fundamental constants at the present state-of-the-art requires a resolving power exceeding 1010 . Table 2.1 lists the requirements for diﬀerent applications.

38

2 Mass Spectrometry Using Paul Traps Table 2.1. Required resolving power for diﬀerent applications [33] Application

δM/M

General physics and chemistry <10−5 – separation of isobars Nuclear structure physics 10−6 Astrophysics <10−6 – separation of isomers Weak interaction studies ≤10−8 Metrology – fundamental constants <10−9 CPS Tests <10−10 QED in highly-charged ions <10−11 – separation of atomic states

Required resolving power 104 105 106 107 108 109 1010

Since the diﬀerent types of spectrometers have inherent advantages and disadvantages, the requirements of the particular application clearly determines the instrument to be used. The diﬃculty of determining, with high accuracy, the amplitude of the radio-frequency (rf) ﬁeld in a Paul trap and achieving a true quadrupole ﬁeld limits the accuracy and resolving power to about 105 . This is compensated by the relative simplicity and compactness of the instrument, resulting in a widespread use in analytical chemistry. On the other hand, the magnetic ﬁeld strength in Penning traps can be determined with high accuracy. This leads to very low uncertainties in the mass-dependent ion oscillation frequencies which serve for mass determination. Mass resolution of 109 is almost routinely obtained now in Penning trap spectrometers, and in some cases it approaches 1011 . Due to the introduction of the high-frequency quadrupole mass ﬁlter and ion trap by Paul and Steinwedel [53] in 1953, their applications to mass spectrometry in the areas of molecular biology and chemistry have widely proliferated becoming an integral part of the design of many sophisticated analytical instruments. The techniques of mass analysis of gaseous ions according to their M/z resolve into ﬁrst the preparation of the source of gaseous ions, their possible subsequent fragmentation, and ﬁnally their mass analysis and data reduction. The state of commercial development of mass spectrometers is so advanced that the principal concern of practitioners in the ﬁeld is the preparation of the sample, with its great diversity of forms of analyte atoms or molecules, and the interpretation of complex spectra obtained. Our discussion, however, will be limited to the physical design of the instrument itself in its various embodiments, and only a brief mention is made of the methods of sample preparation for the sake of completeness will be made. The reader is referred to the very extensive literature on the application of mass spectrometers we describe, to the ﬁelds of biochemical studies, particularly the important ﬁeld of proteomics; see for example the book by March and Todd [59].

2 Mass Spectrometry Using Paul Traps

39

The ﬁrst problem then in obtaining a mass spectrum is to convert analyte molecules (or atoms) into gas phase ions. The least complicated methods, where appropriate, are electron or UV photon ionization of the parent atom or molecule introduced into the trap for that purpose. For complex biochemical samples several other specialized techniques have been developed: One such technique that operates at atmospheric pressure is electrospray ionization (ESI) [60] in which the analyte is dissolved in a liquid and sprayed through a ﬁne needle resulting in the formation of ﬁne charged droplets of solution containing the analyte molecules. The solvent evaporates leaving the analyte in the form of gas phase ions at atmospheric pressure, and their introduction into a mass spectrometer requires diﬀerential pumping. Very large molecular ions can be formed this way. The application of this technique as a source for mass spectrometry was introduced in 1984 [61], and constitutes a standard adjunct to many commercial instruments. Another important technique is matrix-assisted laser desorption/ionization (MALDI) [62,63]. In this, a pulse of laser radiation is used to bombard a composite solid that consists of the analyte molecules imbedded in a substance, the matrix, that strongly absorbs at the wavelength of the laser radiation. In general, a high-power UV laser such as N2 is used in conjunction with an aromatic compound as the absorbing substance. The energy of the pulse delivered by the laser causes desorption of material containing the desired ions from the surface. The pulsed nature of the technique makes it a natural adjunct to time-of-ﬂight mass spectrometers. Molecular weights ranging up to 100,000 amu have been observed. These ionization methods are sometimes referred to as “soft methods,” which means that at least some analyte molecules are ionized without fragmentation. In many applications, where mixtures are involved, mass spectrometer inputs follow a stage of chromatographic separation, whether in the form of liquid or gas chromatography designated as LC/MS or GC/MS. The performance criteria of a mass spectrometer fall into several categories, essential attributes that determine its quality: Sensitivity, speed, mass resolution, mass range, accuracy of the mass scale, and linearity of the ion signal height. In addition to their compactness, mechanical simplicity, high sensitivity and potentially high-mass resolution, quadrupole ion traps allow sequential mass analysis and ion/molecule reaction studies, as well as nondestructive measurements on biochemical mixtures. Their mass selective ion accumulation properties give them the truly unique ability of performing multiple stages of mass spectrometry and ion–molecule interaction studies at low energy within a conﬁned space over protracted periods of time. This means that sequential tandem mass spectrometry can be performed in time rather than the usual space-sequential arrangement. The mass-selected precursor ion species is ﬁrst conﬁned and cooled, then fragmented through low-energy collisions with a background gas and the daughter ions further mass selected and fragmented multiple times before the spectrum is ﬁnally analyzed. As many as 12 stages of tandem mass spectrometry have been reported.

40

2 Mass Spectrometry Using Paul Traps

2.1 The Quadrupole Ion Trap as a Mass Spectrometer One of the earliest applications of the Ionenk¨ aﬁg, the three-dimensional quadrupole ion trap, was as a vacuum leak detector and residual gas analyzer; a sensitive, compact, instrument requiring no magnet. As a leak detector using He probe gas, it could be optimized to store and detect He+ ions. The theory of its operation has been treated elsewhere at great length in volume I [1] and in summary in the last chapter. It is its exceptionally high sensitivity to the presence of a given species of ion in the presence of far more numerous other ions that makes the Paul quadrupole trap an eﬀective leak detector. The original technique for ion detection used the resonant rf absorption ˜ tank circuit connected between the end caps. The mass damping of a high-Q resolution depends on the optimization of the trapping ﬁeld geometry, the background pressure and the scan rate; values approaching M/ΔM = 106 have been claimed with compensated rf ﬁeld distribution and a slow scan rate. Ideally a linear mass scale can be realized if the values of the dc and rf amplitudes are scanned together, keeping their ratio constant, so that the operating point on the a − q stability diagram follows a straight line called the ˜ tank circuit tuned to resonance with the secular scan line. For a given high-Q z-motion of ions represented by points along a particular iso-βz line, as the voltages are scanned, successive masses would pass through resonance where the scan line intersects the iso-βz line, yielding a mass spectrum. However, resonant damping of the tank circuit yields a signal amplitude that depends on too many variables to be simply proportional to the ion number. In mass spectrometry the Paul trap is seldom operated solely in its original form; in what follows we will discuss the ways in which the unique capabilities of the device have been exploited in the development of high-performance commercial mass spectrometers. When a scan line is chosen which crosses the stable region of a Paul trap near its upper corner, only a small rang of masses remains stable in the trap (Fig. 2.1). The closer the scan line is to the apex of the stable area the higher is the resolving power of the trap. Linearity in the measure of the ion number can be obtained by replacing ˜ tank circuit with the extraction of the ions from the the damping of a high-Q trap and using an electron multiplier to count them. In the ﬁrst application of this approach [64] a resonant dipole excitation ﬁeld drives the ions to the outer reaches of the trap where an extraction electric ﬁeld accelerates them onto an electron multiplier to be counted. For a narrow scan of diﬀerent ion masses a spectrum may be achieved by scanning the dipole excitation frequency while holding the primary rf amplitude ﬁxed. However, for a wide spectrum, it is necessary to hold the dipole frequency ﬁxed, creating what has been called a “hole,” actually more like a crevasse, in the a − q stability diagram, so that as the dc and rf ﬁeld amplitudes are scanned, ions whose representative point

2.2 The “Mass Instability Method” of Detection

41

(a0,q0)

0.237 0.2

0.1

0.706

a x,y

mass scan line x stable y unstable

x unstable y stable

x stable y stable 0.2

0.6

0.4

0.8

1.0

q x,y

Fig. 2.1. Scan line with constant ratio a/q for operation of a Paul trap as mass ﬁlter

passes through resonance will gain energy, their orbit becomes divergent and will leave the trap to be counted.

2.2 The “Mass Instability Method” of Detection A signiﬁcant advance in the performance of the Paul trap as a mass spectrometer and hence its commercial acceptance is attributable to the publication in 1984 of work by Staﬀord et al. [65] in which it was shown that the resolution and mass range are greatly improved by the use of a light buﬀer gas to cool the ions and to scan the spectrum in what has been called the mass instability mode. The fact of the cooling of massive ions by collisions with a lighter inert gas such as He, had been known for a long time [9]; what is new is the extent to which the mass resolution of a Paul trap beneﬁts from it as a spectrometer. This can be understood by noting that a real quadrupole inevitably has a ﬁeld distribution that deviates from the true quadrupole and therefore the ion motion is governed by slightly nonlinear equations. This means that the secular frequencies are somewhat amplitude dependent, and it is by damping the motion of the ions by collisions with an inert buﬀer gas that the ions congregate around the center where the ﬁeld more nearly approximates a true quadrupole. This reduces the spread of oscillation amplitude and therefore also the spread in the conditions of stability and frequency spectrum in the trap. The ions under study are produced either by electron impact ionization of the parent molecule in the system by an electron beam entering through an aperture in an end cap or injected from one of specialized ion sources that have been developed for complex molecules, such as MALDI. In the mass instability mode, after holding the amplitude of the trapping rf constant for a short

42

2 Mass Spectrometry Using Paul Traps

time during which the trap is ﬁlled with ions, the amplitude (or frequency) is ramped linearly to bring the (a, q) point of successively heavier ions to the edge and beyond the stability region in the a−q plane, corresponding to βz = 1. On (a, q) reaching that limit of stability, the ion trajectory becomes exponentially divergent, allowing it to escape through an aperture in the other end cap and enter a strong extraction electric ﬁeld along the axis of the spectrometer, which drives it onto the cathode of an electron multiplier. The mass spectrum can be displayed with a digital oscilloscope or multichannel scaler. The theoretical advantage of this technique over the resonant excitation-extraction method described earlier is that in crossing the stability boundary the ion trajectory diverges exponentially, rather than the linear expansion of the orbit in the case of resonant excitation, giving a sharper rise in ion signal which translates into a higher mass resolution. The rise time of the ejected ion signal can be further shortened by applying, just within microseconds of reaching the critical βz = 1, a dipole rf ﬁeld between the end plates that excites the secular oscillation away from the axis to where the quadrupole ﬁeld is stronger, driving the ions rapidly out of the trap. A recent example of an instrument [66] designed as a portable, rugged lowpower instrument, useful for environmental monitoring is shown schematically in Fig. 2.2. The hyperboloidal electrodes of pure Ti have r0 = 10 mm and are machined to close tolerance (±0.02 mm). In the absence of a dc component applied to the quadrupole, (a = 0), the critical point for the onset of instability is given in the axial direction by qz = 0.908, from which we obtain: 4V0 M = . Q 0.908r02 Ω20

(2.1)

For example at an amplitude of V0 = 500 V and Ω/(2π) = 1 MHz, the critical mass is M = 56 amu below which ions are unstable. By choosing diﬀerent frequencies diﬀerent ranges of mass can be covered. The reported

Fig. 2.2. A portable rugged low-power mass spectrometer used in mass instability mode. A – accelerating electrode, C – cathode, D – channelotron, E – extraction electrode, L – focusing electrode

2.2 The “Mass Instability Method” of Detection

43

mass resolution in a single sweep is reported as M/ΔM = 324, limited by the machine tolerance of the electrodes and their alignment, which leaves room for improvement. Using Ω/(2π) = 3.14 MHz, the mass resolution of the system was tested over diﬀerent mass ranges, using natural Xe gas over the range 123–137 amu, and the test substance with a wide mass spectrum, C6 F12 (dodecaﬂuorodimethylcyclo-butane) over the range 40–300 amu. The Xe isotopes are well resolved; with closer tolerance in the fabrication of the electrodes and stabilization of the rf amplitude, it should be possible to reach a resolution of M/ΔM = 1, 000. An equally important ﬁgure of merit is the sensitivity; a measure of this is the electron count rate (above background) at a particular mass number divided by the partial pressure of the sample gas; for N2 it is reported to be 2 × 1012 counts/(Pa s). An interesting miniature (r0 = 0.5 mm) spectrometer operating in the mass-instability mode has been described by Kornienko et al. of Oak Ridge national Laboratory [67]. It also uses electron impact ionization of gaseous samples, although photoionization using pulsed UV laser radiation had also been used in the past. The trap shown schematically in Fig. 2.3 has planar geometry consisting of a stack of three stainless steel plates, the center “ring” consists of a 0.9 mm thick plate with a 1.0 mm diameter circular hole centered on the axis, and the “end” plates spaced by Teﬂon spacers 0.1 mm thick that have a 0.45 mm diameter hole centered on the axis: One to admit the ionizing electron beam and the other to extract the unstable ions and accelerate then onto the electron multiplier. The sequence of operations follows the order described earlier. Because of the miniature size, it was found necessary to take special precautions to prevent dc ﬁelds from developing in the trap due to charges accumulating on the electrodes. A low-pass ﬁlter was applied to the “ring” electrode to short out dc and low-frequency noise. Operating frequencies ranged from 3 to 7.5 MHz with the potential amplitude ramped linearly from 58 to 185 V. The quoted mass range is 40–400 amu although operating at a frequency of 3 MHz, instability theoretically begins at a mass of about 925 amu for V0 = 185 V. The instrument was calibrated using

Fig. 2.3. Microquadrupole mass spectrometer. C – cathode; F – focusing electrode; MP – mounting plate; Q – quadrupole; GP – ground plate; EM – electron multiplier

44

2 Mass Spectrometry Using Paul Traps

the isotopes of Xe and the complex molecule NF27 (Perﬂuorotributylamine). Nonlinearity in the rf modulator/ampliﬁer supplying the trap rf potential is expected to be the main source of nonlinearity in the mass scale, unless the spectrum is displayed with respect to actually measured rf amplitude. A mass resolution of 0.2 amu is reported for a single sweep, broadened by jitter on multiple sweeps.

2.3 Sources of Mass Error in Ion Ejection Methods As already pointed out, mass spectrometers based on the quadrupole ion trap excel in sensitivity and mass range, but high resolution and mass accuracy require reﬁnements in design to mitigate the eﬀects of the aberrations in rf ﬁeld distribution due to imperfect electrode geometry and space charge. Errors in mass/charge ratios amounting to as much as 2% may well be incurred due to departures from the true quadrupole ﬁeld. The inevitable truncation of the electrodes and apertures in the end caps, no matter how closely the surfaces conform to the hyperboloid geometry, will produce a ﬁeld fault. This obvious fact was noted when the quadrupole was ﬁrst seriously designed as a mass spectrometer. Remedies include interposing compensating rings between the main ring and end caps. However, a simpler solution was found empirically: Simply increase the separation of the end caps by 10% beyond the ideal separation [65]; thus for a quadrupole with r0 = 1 cm the optimum value of z0 used in commercial instruments is 0.783 cm (rather than 0.707). This approach adopted by the Finnegan group cannot of course totally produce a perfect ﬁeld, but gives a signiﬁcant correction. An alternative approach favored by the Dalton–Franzen group modiﬁes the angle of the asymptotes of the quadrupole [68]. In addition to ﬁeld faults, several other possible sources of error in the determination of ion mass have been explored [69]. Among these are permanent electric dipole moment [70] or polarization of the molecular ion [71], and the space charge ﬁeld of the ions that clearly depends on the ion number density and their distribution. Other relevant processes include collisions occurring during the ejection phase aﬀecting the ejection delay. These include abrupt changes in the phase of the ion secular motion relative to the quadrupole rf ﬁeld and/or fragmentation of the molecular ion.

2.4 Nonlinear Resonances in Imperfect Quadrupole Trap The stability boundary in the a − q plane at βz = 1 is not the only line that may be exploited for mass selective expulsion of ions from the trap: There are also lines along which singularities occur in the solutions of the (nonlinear) equations of motion due to departures of the ﬁeld distribution from the pure quadrupole, which in itself would cause uncoupled linear equations of motion

2.4 Nonlinear Resonances in Imperfect Quadrupole Trap

45

in the r and z directions as already treated in Sect. 1.1.8. These additional lines of instability are classiﬁed on the basis of the order of terms in a multipole expansion of the ﬁeld. Any small deviations from the ideal quadrupole ﬁeld distribution, whether because of imperfect construction and truncation or intentionally incorporated in the design by reshaping the electrodes or modifying the electrode spacing, can be expressed in terms of a series expansion in spherical harmonics, with the dominant term being, by design, the (n = 2) second-order quadrupole, followed by the even-order harmonics (n = 4, 6, ...), the octopole, etc. If the system is rotationally symmetric the odd harmonics (n = 3, 5, ...), the hexapole, etc., are zero. In the presence of these higher harmonics the equations of motion of an ion acted on by such a ﬁeld are no longer linear nor are the r and z coordinate equations decoupled. The presence of higher order harmonics, even if small, can have signiﬁcant eﬀect on the motion of an ion. One very important consequence, not realized in the early days of the Paul trap, is that the ion motion is unstable if the axial and radial secular frequencies satisfy the following resonance condition [13, 72]: nr ωr + nz ωz = kΩ,

(2.2)

nr βr + nz βz = 2k,

(2.3)

which may be written where nr , nz , k are integers. It is evident from the form of these conditions that they deﬁne nonlinear resonances in which a linear combination of radial and axial secular frequencies resonates with the strong trapping ﬁeld frequency, thereby driving the ions to diverging trajectories rather than converging, as would be the case for other values of βr and βz in the stable regions of the a − q plane. This instability is to be distinguished from the crossing of the boundary values of βr and βz that separate regions of stability from those of instability. The instability is said to be of order N deﬁned as N = |nr | + |nz |. For the nth spherical harmonic strong resonances will occur for N = n, n−1, n−2, . . . . It can be shown that if nr nz ≥ 0, the ion will gain energy from the trapping ﬁeld and the amplitude of its secular oscillation will increase. However, since the ﬁeld is not a pure quadrupole, the ion frequencies are dependent on the amplitude and the resonance condition is not maintained as the ion gains energy. As the ion frequencies deviate from the resonance condition this leads to periodic shift in relative phase similar to beats whose amplitude will determine whether the ion will be lost. Douglas and coworkers at the University of British Columbia have deliberately introduced higher order multipoles into the trapping potential by changing the geometry of the trap electrodes and have shown that the mass resolution can be signiﬁcantly increased [73–75]. In Fig. 2.4 are sketched typical lines in the ﬁrst stable region in the a − q plane of the quadrupole ﬁeld along which resonances occur due to hexapole term (βz = 2/3) and octopole term (βr + βz = 1 and βz = 1/2).

46

2 Mass Spectrometry Using Paul Traps

Fig. 2.4. A plot of the boundaries of the ﬁrst stability region in the a − q plane showing some lines on which nonlinear resonances occur

2.5 Quadrupole Time-of-Flight Spectrometer By combining the Paul quadrupole trap with a time-of-ﬂight mass analyzer the mass selective ion accumulation and cooling capabilities of the trap as an ion source are retained while providing good spectral resolution over a wide range and linear particle counting through time-of-ﬂight analysis. This development and other more complicated tandem combinations of Paul quadrupole traps and mass ﬁlters have changed the whole aspect of mass spectrometry from one based on high-energy ion optics in contoured magnetic ﬁelds to multistage instruments permitting the low-energy study of various stages of fragmentation of complex biochemical molecules, with variable dwell time in each phase. In a recent reﬁnement [76] in the design of quadrupole time-of-ﬂight spectrometers (Fig. 2.5), particular attention is paid to the optics of the ion extraction process to minimize the “turn around time” of the ions in the trap. Naturally even after cooling through collisions with the light buﬀer gas, the ions will still have a spread in their velocity distribution, and therefore to attain the highest possible time-of-ﬂight resolution for any given ion species the eﬀect of that spread must be minimized. This is achieved during extraction by the simultaneous application of a high-negative potential (for positive ions) on the extraction side end cap and a calibrated lower positive potential on the other end cap. A slight curvature on the equipotential surfaces in the neighborhood of the extraction end cap smoothly accelerates ions through the circular hole with slightly convergent trajectories, however, the termination of the ﬁeld lines around the hole give a slightly divergent eﬀect resulting in a more nearly parallel beam outside the quadrupole.

2.5 Quadrupole Time-of-Flight Spectrometer

47

Fig. 2.5. Quadrupole – time-of-ﬂight mass spectrometer

The spectrometer layout shown schematically in Fig. 2.5, comprises a hyperboloid quadrupole ion trap Q ﬁlled with He buﬀer gas for cooling, a drift tube I deﬁning a ﬁeld-free drift space, an ion retro-reﬂector M at the end of it, followed by an ion detector D. The source end cap has a circular hole centered on the axis through which either ions from an external source, such as MALDI, enter the trap, or an electron beam, if electron impact ionization is to be used. The rf trapping potential is on for the duration of the loading period after which it is turned oﬀ and simultaneously the extraction dc potentials of −10 and +6 kV applied to the end caps. Of particular concern from the point of view of time-of-ﬂight resolution is the question of the eﬀect of the precise waveforms of the buildup of the dc potentials in time, and the timing of the switching of the rf ﬁeld. Another spectrometer design using a combination of rf and constant magnetic ﬁelds provides for enhanced dissociation of complex molecules by the process of electron capture (ECD). This means of fragmenting a complex molecule provides structure data complementary to that obtained through other processes such as collision-induced dissociation (CID). To achieve this requires the eﬃcient trapping of incident preselected precursor ions for extended periods of time while maintaining a high ﬂux of low-energy (<1 eV) electrons to interact with them. The instrument of choice for this is based on the powerful Fourier Transform-Ion Cyclotron Resonance (FT-ICR) technique, which will be discussed in Chap. 3. However, Baba and Hashimoto [77] have designed a simpler low-cost instrument, based on a combination of a linear quadrupole ion trap and weak axial magnetic ﬁeld. The precursor ions produced in an external mass separator are injected and trapped in the linear Paul trap and through collisions with a light background gas are cooled converging along the axis. The injection of the axial ion beam into the trap is eﬃcient because the rf ﬁeld along

48

2 Mass Spectrometry Using Paul Traps

the axis is zero, and has only a weak axial component at the ends of the trap. Further, an electron beam that is injected into the Paul trap to produce ECD is conﬁned along the axis, where the rf ﬁeld is near zero, by a weak axial magnetic ﬁeld (0.05 T) and there the electrons are not energized by an rf ﬁeld. The circular motion of the electrons about the magnetic ﬁeld lines increases their path among the ions to further increase the eﬃciency of the ion dissociation.

2.6 Tandem Quadrupole Mass Spectrometers In the usual understanding of the term, tandem mass spectrometers refers to those designed to mass select usually complex parent ions in one spectrometer stage, ions which are then subjected to disintegrating collisions in an intermediate stage and the daughter ions analyzed in a ﬁnal stage. We consider here a tandem mass spectrometer consisting of three Paul quadrupole mass ﬁlters that permit the study of selected ion fragmentation studies. It is included here in spite of the fact that it does not use three-dimensional trapping, but rather only transverse focusing of ions. The salient point in the design of the Paul mass ﬁlter is the dependence of the mass resolution on the dimensions of the electrodes and the ion energy. It is generally accepted that if the number of oscillations of the rf ﬁeld an ion experiences as it traverses the ﬁlter of length L is n then [78] M n2 = . ΔM 20 But if the axial ion energy is QV z then n is given by M Ω L . n= 2π 2QVz It follows that ΔM = 40

QVz , (f L)2

(2.4)

(2.5)

(2.6)

where f is the frequency of the rf ﬁeld. The maximum required rf amplitude is determined by the characteristic radius r0 and the maximum ion mass to be ﬁltered Mmax (amu) thus: Vmax = 14.5 × 10−8 Mmax f 2 r02 .

(2.7)

Hence by reducing the size of r0 , the required maximum rf amplitude is reduced for a given maximum mass, provided the product f r0 < 1. For example, a reduction of the radius by a factor of 10 will allow the frequency to be raised for example by a factor of 3 and the amplitude reduced by a factor of 11.

2.6 Tandem Quadrupole Mass Spectrometers

49

Tandem mass spectrometry with multiple quadrupole ﬁlters often referred to as triple quads have been known for almost 25 years. One such design uses two Paul quadrupole mass ﬁlters, operated with an rf voltage superimposed on a dc voltage chosen to pass a narrow selected mass range, separated by a linear quadrupole containing an ion-interacting gas and operated with only an rf voltage without a dc component. This gives it a broad focusing range, acting principally to guide the ions and permitting it to eﬃciently transmit with little ion loss a broad range of fragments resulting from collisions with the background neutral atoms/molecules introduced for the purpose. The presence of the background gas has the added beneﬁt of slowing down the ions and causing them to concentrate near the axis. This reduces the volume of phase space occupied by the ions and improves the ion optics and hence the ion signal. The ion motion in the high-frequency ﬁeld of the fragmentation stage results in a high eﬃciency of dissociation with ion energy typically less than 100 eV, unlike the rectilinear paths of ions at kiloelectronvolt energy typical of the old magnetic deﬂection spectrometers. This further improves the sensitivity. The third quadrupole is again operated as a narrow ﬁlter, and by scanning the applied voltages a spectrum of the daughter ions is produced. This type of instrument lends itself to operation in a high-transmission mode in which the ﬁrst and last quadrupole ﬁlters are set for a particular combination of masses as signature to detect the presence of a given unknown component in a mixture. The components of the mixture are initially separated using gas or liquid chromatography. This combines high transmission through the instrument with mass selectivity of the substance sought. The main weakness in the triple quads spectrometer is in the design of the fragmentation stage which must compromise between a high density of the interaction gas to increase the yield of the fragmentation and a low density to minimize the scattering of the daughter ions which reduces the transmission to the third stage and also reduces the mass resolution. In an attempt to circumvent this compromise, a recent advance has been introduced [79] in which the quadrupole of the disintegration stage has been replaced by a conical ion guide in a novel design. It consists of a double helix of wire wound on a conical form. A high-frequency voltage is applied between the two helices creating a pseudopotential that has a minimum on the axis of the cone and increases rapidly only near the cone. This has the eﬀect of deﬂecting the ions toward the axis without subjecting them to a strong rf ﬁeld throughout the volume counteracting the cooling eﬀect of the background gas. This concentrates low-energy ions along the axis and must be directed along the axis toward the end of the cone and there be extracted and injected into the analyzer. This achieves the ultimate aim of narrowing the velocity spread of the ions as they enter the analyzer thereby substantially improving the ion transmission and mass resolution. Again the use of a pure rf electric ﬁeld without a dc component ensures that a wide range of ion masses among the disintegration products will experience the focusing pseudo-potential.

50

2 Mass Spectrometry Using Paul Traps

Fig. 2.6. Tandem mass spectrometer. 1 – Ion source; 2 – capillary; 3 – ion guide; 4 – quadrupole mass ﬁlter; 5 – ion lens; 6 – conical double helix; 7 – quadrupole mass analyser; 8 – detector; 9 – vacuum pumps [79]

The Fig. 2.6 is a schematical representation showing the general layout of the multichamber vacuum system of the tandem quadrupole mass spectrometer described by Franzen. Since the quadrupole mass ﬁlter (4) and analyzer (7) require high vacuum, while the ion disintegration chamber requires a relatively higher pressure (0.1–10 Pa) diﬀerential pumping is required with small aperture ion lenses (5) separating those chambers. A multipole rod ion guide system (3) leads from the ion source to the ﬁrst quadrupole mass selector where the desired parent ion is singled out. This ion guide consists of parallel rods between which an rf voltage is applied to produce a cylindrical pseudopotential ﬁeld that acts as a pipe conducting the ions. Some acceleration occurs at the ﬁrst ion lens (5) followed by injection into the conical double helix with an energy 10–30 eV. Unlike the quadrupole ion trap, where the duration of the ion–molecule interaction can be varied allowing the kinetics of ion–molecules to be studied, the tandem machines do not have that ﬂexibility. On the other hand, the tandem quadrupole instrument has the advantage of allowing the kinetic energy of the ions to be varied. Actual performance comparisons have been reported for the Finnigan LCQ ion trap spectrometer and the PE/SciexAPI 300 triple quadrupole instrument. The results do not indicate suﬃcient superiority in the triple quadrupole instrument over the ion trap spectrometer to oﬀset the advantages of small size and low cost of the latter.

2.7 Tandem Quadrupole Fourier Transform Spectrometer Like the tandem time-of-ﬂight spectrometer, the quadrupoles in the tandem Fourier transform spectrometer play the roles ﬁrst as an initial mass ﬁlter to select a precursor ion and second to provide the optimal environment for CID into daughter ions and to moderate the ion energy and reduce the phase space occupied by them to achieve improved ion optics. However, what distinguishes the Fourier transform spectrometer is the use of wideband rf pulse excitation producing phase coherence in the periodic ion

2.7 Tandem Quadrupole Fourier Transform Spectrometer

51

motion in the ﬁeld conﬁning the ions. This coherent motion actively induces a signal in the electrodes system surrounding the ions, a signal whose Fourier transform gives directly the frequency spectrum of the ion motion and hence the related mass spectrum. Generally, Fourier transform mass analysis is assumed to mean the common method using the measurement of the cyclotron resonance frequency of the ions in a strong uniform magnetic ﬁeld [80, 81]. In its modern form, a cyclotron resonance mass spectrometer would incorporate a superconducting magnet to provide the most intense uniform magnetic ﬁeld ensuring the highest resolution and least perturbation from electric ﬁelds. A more detailed discussion of cyclotron resonance will be given in Chap. 3; here we simply mention it by way of drawing a parallel with an interesting new method of mass analysis, which we will be discussing. The standard cyclotron resonance cell generally consists of four electrodes arranged parallel to the magnetic ﬁeld as axis, with front and rear apertured (“trapping”) plates perpendicular to the axis, completing the cell. The four axial electrodes may be rectangular plates or parallel cylinders. The resonance signal is obtained in the inductive mode in which a burst of rf is applied to an opposing pair of electrodes and the signal induced by the phased cyclotron motion of the ions driven at resonance is picked up on the other opposing pair of electrodes. A deleterious consequence of the application of the electrostatic potential on the end plates is that it complicates the ion motion, introducing other electric ﬁeld dependent frequencies in the ion motion (magnetron and axial modes). This can broaden the ion resonance and hence reduce the mass resolution; therefore the magnetic ﬁeld must be as intense and uniform as possible and the dc potential on the trapping electrodes must be kept very low (<1 V). But this imposes a severe limit on the tolerable spread in the energy of the ion beam transmitted through the spectrometer, and on the density of ions that can be trapped. It has been demonstrated [82] that by interposing compensating electrodes between the end plates and the axial plates the cyclotron frequency sensitivity to the trapping potential is greatly reduced. In one typical instrument [83] the ﬁrst quadrupole was 25 cm long and the second 86.25 cm. The ICR cell was constructed from 7.75 × 2.8 × 2.8 cm polished stainless steel plates in the ﬁeld of a 7.0 T superconducting magnet. The vacuum pressure in the ICR cell was maintained at around 10−7 Pa using three cryogenic pumps: One at the ion source, one at the ﬁrst quadrupole and the third near the entrance to the third quadrupole. Each pump had a speed of about 680 l of air per minute. The mass resolution attainable for example on large ion clusters of CsI, such as (CsI)22 Cs+ whose induction signal persists for seconds, is given as in excess of 30,000 at M/z = 5,848 amu. In a recent development [84] an attempt has been made to mitigate the design problems associated with the operation of the cyclotron resonance cell. It departs from all previous designs in that it observes in a nondestructive way the induction signal from a stimulated phased rotation of ions about the axis of a modiﬁed linear quadrupole ion trap. In this design (Fig. 2.7) the electrodes of

52

2 Mass Spectrometry Using Paul Traps

Fig. 2.7. Cylindrical double quadrupole rf ion trap for Fourier transform analysis of induction signal following pulsed ion excitation [84]

the linear quadrupole ion trap are formed by cutting a circular cylinder parallel to the axis eight regularly spaced intervals around the cylinder creating eight electrodes grouped as four parallel electrodes of a standard linear quadrupole trap, and four additional parallel electrodes as part of the same geometric cylinder in the spaces between the electrodes of the ﬁrst quadrupole. Such an arrangement of electrodes had previously been described [85] as a way of approximating to higher order a true quadrupole ﬁeld using simple cylindrical electrode geometry as described. In the design of the mass analyzer described in [84] the ﬁrst set of four electrodes (“trapping electrodes”) are energized as in a linear quadrupole trap by an rf potential applied to opposite pairs, while an induction signal is detected between the two pairs of the second set (“detection electrodes”) which are connected to ground through a high resistance. To stimulate the phased response of the ions to produce an induction signal, a burst of rf is applied to the trapping set of electrodes.

2.8 Silicon-Based Quadrupole Mass Spectrometers There is great impetus to develop miniature instruments for space craft environments where such considerations as low weight and power consumption are of principal concern. This includes the development of instrumentation for mass spectrometry. Miniature versions of magnetic sector, time-of-ﬂight, cyclotron resonance as well as quadrupole ion traps have been constructed [86]. Miniature spectrometers in the form of micro-electro-mechanical systems (MEMS) based on integrated circuit fabrication techniques may be made an order-of-magnitude smaller. Currently, linear quadrupole mass spectrometers are relatively large requiring high-rf amplitudes for a wide spectral range. There are several advantages to the radical size reduction that may be possible using MEMS techniques: 1. A lower cost of manufacture since it may be possible to make several spectrometers at once in a batch from a single wafer.

2.8 Silicon-Based Quadrupole Mass Spectrometers

53

Fig. 2.8. End view of the four metal coated glass ﬁbers 0.5 mm in diameter located in V-grooves etched in silicon

2. The small dimensions allow the mean free path of the ions to be smaller and therefore a higher gas pressure is tolerable, requiring smaller (and less expensive) pumps. 3. The rf power required is lower and therefore less drain on the power supply. It oﬀers the possibility of fully integrating the spectrometer with other electronics on the same chip. Following Taylor et al. [87] we describe brieﬂy the fabrication of a linear Paul mass ﬁlter, with the conﬁdence that similar techniques can in the future be adapted to construct microscopic threedimensional ion traps. Indeed there is a broader interest in the latter in the form of arrays of ion traps in the context of quantum computing. The four parallel cylindrical electrodes (Fig. 2.8) are borosilicate optical ﬁbers of circular cross section 0.5 mm in diameter with a metallic coating. The characteristic radius of the quadrupole r0 = 0.435 mm corresponding to the optimum ratio of 1.148 for the two radii. The glass electrodes are bonded in position in parallel V-shaped grooves etched in two Si wafers. The correct position and alignment of the electrodes is ensured by each substrate having a rod that matches a V-groove in the other substrate. Lithography was used for precise etching of the crystal planes of the Si substrate. A process for batch fabrication has been developed based on 3 in. diameter (100)p-type Si wafers. For the purposes of testing the operation of such a microquadrupole it was mounted on a conventional vacuum feedthrough and equipped with a hot ﬁlament ion source and optics. It was operated at 6 MHz, but the low impedance the electrodes presented and relatively large leak currents dictated that a source matching that low impedance be provided. This is greatly simpliﬁed by the reduced amplitude required to operate a quadrupole of microdimensions. The reduced rf power requirement further reduces the weight and cost of the rf electronics, and makes it practicable to integrate the supply of the rf directly in the Si wafer. Taylor et al. experimented with diﬀerent prototypes and modes of operation, using as test gases He, N2 , and Ar. Mass spectra were obtained showing + peaks at He+ , N+ Ar2+ , N+ 2 , and Ar . While these early experiments yielded rather poor resonance line shapes with broad asymmetric bases, and hence mass resolution at best of about 31 at mass 40 amu with a peak width of

54

2 Mass Spectrometry Using Paul Traps

1.1 amu measured at 10% of peak height, they nevertheless successfully demonstrated that such micromachined quadrupoles can be fabricated in Si. It shows the same trade-oﬀ between resolution and sensitivity that ordinary scale devices exhibit. There is clearly considerable room for improvement in the line shape and mass resolution. In fact theoretically mass 40 amu ions entering the quadrupole operating at f = 6 MHz with an energy of 5 eV experience about 36 oscillations of the rf ﬁeld before exiting, corresponding to a resolution of about 65. The experimental microquadrupoles were also tested at higher pressures + in the range 10−4 –2 Pa of N2 gas. The signal ratio N+ 2 /N tends to increase with pressure reﬂecting the larger ionization cross-section for N2+ , but the ion transmission signal begins to fall oﬀ beyond a pressure of 10−1 Pa, no doubt due to increased ion–neutral collisions.

3 Mass Spectroscopy in Penning Trap

While mass spectrometry in Paul traps serves well mainly for molecular analysis in chemistry, Penning traps provide high accuracy and precision. The technique is based on the fact that the ratio of cyclotron frequencies ωc = (Q/M )B of two ions in the same magnetic ﬁeld B gives directly the ratio of their masses ωc (1)/ωc(2) = M (2)/M (1). If Carbon-12, the standard of the atomic mass scale, is used as reference, the mass of the ion of interest is obtained directly in atomic units. Although the cyclotron frequency is not an eigenfrequency of the Penning trap, it can be obtained from combinations of ω+ , ω− , and ωz as evident from the set of equations (1.31)–(1.33). In the ideal case of a perfect quadrupole trap and constant and homogeneous magnetic ﬁeld B no further correction has to be applied. In reality, however, the trap potential will deviate from the ideal shape leading to shifts of the motional frequencies. In the case of several simultaneously stored ions, space charge potential also modiﬁes the ion oscillation as well as image charges induced in the trap electrodes by the oscillating ions. Finally, since the comparison of frequencies has to be made at diﬀerent times, ﬂuctuations of the magnetic ﬁeld in time have to be considered.

3.1 Systematic Frequency Shifts 3.1.1 Electric Field Imperfections The eﬀect of deviations from the ideal quadrupole trap potential can be treated by a multipole expansion Φ = U0

∞ c2n H (ρ), 2n 2n d n=2

(3.1)

where ρ = (r2 + z 2 )1/2 , and d is the characteristic dimension of the trap. Here n = 2 represents the quadrupole part. The lowest order polynomials Hn are

56

3 Mass Spectroscopy in Penning Trap

given by H2 (r, z) = H3 (r, z) = H4 (r, z) = H5 (r, z) = H6 (r, z) =

1 (−r2 + 2z 2 ), 2 1 (−3r2 z + 2z 3 ), 2 1 4 (3r − 24r2 z 2 + 8z 4 ), 8 1 (15r4 z − 10r2 z 2 + 8z 5 ), 8 1 (−5r6 + 90r4 z 2 − 120r2 z 4 + 16z 6 ). 16

(3.2) (3.3) (3.4) (3.5) (3.6)

If rotational and mirror symmetries of the trap are preserved, only even orders of the multipole expansion contribute. The shift of the eigenfrequencies caused by the presence of higher order potentials has been calculated by several authors [1, 19, 88, 89]. Considering only octupole (n = 4), dodecapole (n = 6), and hexadecapole (n = 8) contributions, the shifts are: 3c4 ωz2 2 2 R± + 2R∓ − 2Rz2 , 2 4d ω1 3c4 ωz 2 2 2 Rz − 2R+ , − 2R− 2 4d 2 15c6ωz 4 2 2 4 2 2 ∓ [R + 6R+ R− + 3R∓ − 6Rz2 (R± + 2R∓ ) + 3Rz4 ], 16d4 ω1 ±

15c6 ωz 4 2 2 4 2 2 3(R+ + 4R+ R− + R− ) − 6Rz2 (R+ + R− ) + Rz4 , 4 16d 35c8ωz2 6 4 2 2 4 6 R± + 18R± ± R∓ + 18R± R∓ + 4R∓ − 4Rz6 32d6 ω1

2 2 4 2 2 4 + 2R∓ ) + 12Rz2 (R± + 6R+ R− + 3R∓ ) , +18Rz4(R± 35c8 ωz 6 4 2 2 4 6 4(R± + 9R± R∓ + 9R± R∓ + R∓ ) 32d6

2 2 4 2 2 4 +12Rz4(R± + R∓ ) − 18Rz2 (R± + 4R+ R− + R∓ ) − Rz6 .

Δ(4) ω± = ±

(3.7)

Δ(4) ωz =

(3.8)

Δ(6) ω± = Δ(6) ωz = Δ(8) ω± =

Δ(8) ωz =

(3.9) (3.10) (3.11)

(3.12)

The coeﬃcients cn represent the strength of the perturbation of order n, R± are the radii of the perturbed cyclotron and magnetron motions, respectively, and Rz is the amplitude of the axial oscillation. It is obvious that to keep these shifts small, a high-mechanical precision for the trap electrodes is required. Moreover, the radii and amplitudes of the oscillations have to be kept as small as possible. In the case of the axial and cyclotron orbits this can be achieved by cooling of the motions. Magnetron orbits, not initially close to the trap center, must be driven there by coupling to the cooled axial and cyclotron motions by a ﬁeld at the sum of their frequencies.

3.1 Systematic Frequency Shifts

57

3.1.2 Magnetic Field Imperfections The magnetic ﬁeld of the trap, ideally perfectly homogeneous, may suﬀer from inhomogeneities. Similar to the electric case it can be expanded in a power series to account for these imperfections: B = B0

∞ b2n H2n (ρ). 2n d n=1

(3.13)

Odd terms in the expansion will not lead to frequency shifts but cancel out by the ion’s oscillation. For the even terms we obtain in lowest order [1] Δ(2) ωz =

ωc b2

2 2 ω+ R+ , + ω− R− 2 2ω1 d

(3.14)

ωc b2

2 2 (3.15) ω± Rz2 − ω± R± , − ωc R∓ 2 2ω1 d 2

3ωc b4

2 2 2 2 2 2 ω+ R+ Rz − ω+ R+ Δ(4) ωz = + ω− R− − ω− R− − 2ωc R+ R− , 4 4ωz d (3.16) Δ(2) ω± =

Δ(4) ω± =

3ωc b4 4 4 2 2 ω± R± + (ω∓ + ωc ) R∓ + 2 (ω± + ωc ) R+ R− 4 16ω1 d 3ωc b4 2

2 2 (3.17) R 4ω± R± + ωc R∓ − ω± Rz2 . − 16ω1 d4 z

As in the electric case, apart from homogenization of the magnetic ﬁeld, reduction of the ion’s motional amplitudes reduces the size of the shifts. 3.1.3 Misalignements and Trap Ellipticity A misalignment of the magnetic ﬁeld direction with respect to the trap’s axis by an angle θ, an ellipticity ε of the ring electrode and a corresponding angle ϕ of the B-ﬁeld with respect to the principal axis of the ellipse also causes shifts of the motional frequencies as follows [1, 88]: 1 ≈ ω± + ω− sin2 θ(3 + ε cos 2ϕ), ω± 2 1 ωz ≈ ωz − ωz sin2 θ(3 + ε cos 2ϕ). 4

(3.18)

However, these shifts are of no signiﬁcance in ﬁrst order when the cyclotron frequency, as required for mass comparison, is obtained by the relation (“invariance theorem”) 2 2 ωc2 = ω+ + ω− + ωz2 , (3.19) where ωi with i = +, −, z represent the actually measured frequencies in a perturbed trap.

58

3 Mass Spectroscopy in Penning Trap

3.1.4 Image Charges Ions oscillating in the trap induce image charges in the trap electrodes which in turn produce a potential which adds to the applied trapping potential, causing corresponding shifts in oscillation frequencies. In a simple model where the electrodes are replaced by a conducting shell of radius a the corresponding shift for a single ion amounts to [90] Q2 1 , 4πε0 2M a3 ωz 2Q2 1 Δω+ = , 4πε0 M a3 ωc 2Q2 1 . Δω− = − 4πε0 M a3 ωc Δωz = −

(3.20) (3.21) (3.22)

The shifts scale linearly with the ion number. The cyclotron frequency, as the sum of ω+ and ω− is not aﬀected because of its independence of the electric trapping potential. Van Dyck et al. [21] have observed linear frequency shifts in ω+ and ω− , which are consistent with expectations of the model, for small numbers of ions in a trap of 0.158 cm ring radius placed in a 5 T magnetic ﬁeld. A residual shift in the cyclotron frequency, however, has also been observed which ranges from 0.23 ppb/ion for H+ to 3.33 ppb/ion for 12 C4+ . This is attributed to magnetic ﬁeld variations, during the course of the experiment. 3.1.5 Magnetic Field Fluctuations Temporal ﬂuctuations of the magnetic ﬁeld of Penning traps limit the precision of mass comparison of diﬀerent ions. Averaging over constant time intervals between measurements would eliminate linear drifts of the ﬁeld, but higher order variations would remain. The most stable magnetic ﬁelds are produced by superconducting coils. Flux jumps [91] lead to ﬂuctuations which typically are of the order of 10−8 –10−9 in time intervals of about 1 h. Typically, if uncontrolled, the magnetic ﬁeld of a superconducting solenoid will wander, as monitored by the cyclotron resonance of a single-trapped ion [92]. The variations are caused by temperature and pressure ﬂuctuations as demonstrated by correlations of the ﬁeld strength with these parameters (Figs. 3.1 and 3.2). Stabilization of temperature and pressure can reduce the remaining fractional shifts of the magnetic ﬁeld. For ΔT = ±5 mK and Δp 5 × 103 Pa fractional frequency shifts of 2.7×10−9 and 1.5×10−10 , respectively, have been achieved [93]. Similar results are obtained by other groups [92]. The eﬀect of ﬂuctuations of magnetic ﬁelds outside the solenoid on the ﬁeld strength at the trap center can be minimized by specially designed superconducting coils which provide shielding of the internal bore region against external ﬁeld variations [95]. Figure 3.3 shows the shielding factor of such a coil at various positions of the magnet’s bore and Fig. 3.4 shows the eﬀect on

3.1 Systematic Frequency Shifts

59

Fig. 3.1. Correlation between magnetic ﬁeld strength (dots), measured by the cyclotron frequency of a single-stored ion, and temperature at the center of a superconducting solenoid, measured by the resistance of a temperature sensor (solid line). The cyclotron frequency is corrected for a linear drift of the magnetic ﬁeld [94]

Fig. 3.2. Correlation between the magnetic ﬁeld strength and the ambient atmospheric pressure [92]

Compensation factor

180 140 100 60 20 112 114 116 118 Depth from top-plate (cm)

Fig. 3.3. Shielding factor of external ﬁeld ﬂuctuations by a superconducting stabilizing coil at various positions in a bore of a superconducting solenoid. The position at maximum shielding factor corresponds to the magnet’s center [92]

60

3 Mass Spectroscopy in Penning Trap

Fig. 3.4. Variation of the magnetic ﬁeld strength at the center of a superconducting solenoid with and without a stabilizing coil [92]

the stability of the magnetic ﬁeld strength. In spite of the progress in reducing magnetic ﬁeld variations the comparison of cyclotron frequencies of diﬀerent ions at diﬀerent times remains a problem when attempting to increase the precision of mass comparison. In any case it is desirable to perform a switch between diﬀerent ions under investigation as fast as possible.

3.2 Observation of Motional Resonances To determine the cyclotron frequency of a trapped ion the frequencies of the perturbed cyclotron, the axial, and the magnetron oscillations have to be measured. Of particular importance is the perturbed cyclotron motion since it represents the major part in (3.19). Observation can be performed either nondestructively in which the ion remains in the trap or destructively in which it is lost from the trap in the observation process. 3.2.1 Nondestructive Observation In nondestructive detection techniques the image charges induced in the trap electrodes by the ion’s oscillation are picked up by sensitive electronics. The advantage of this technique is that the particle remains in the trap for virtually inﬁnitely long times and the eﬀect of ﬁnite observation times is completely eliminated. If its kinetic energy is reduced by resistive cooling (Sect. 1.4.2) it oscillates in a small volume near the trap center, and trap imperfections play a very small role. Consequently, the highest accuracy in mass determination can be obtained. Let us consider the axial oscillation of the charged particle in a trap, where the end caps of distance 2z0 apart are connected by a resistor R. If it moves with velocity v toward an electrode it causes a current I through the resistor

3.2 Observation of Motional Resonances

I=k

Q v. 2z0

61

(3.23)

Here, k is a dimensionless factor which takes into account the shape of the electrodes; k = 1 holds for plates of inﬁnite dimension, and for a hyperbolic trap as well as for a cylindrical trap we have k = 0.9 [88]. Numerical example: An ion of mass 100 oscillating with a mean kinetic energy of 1 eV in a trap with z0 = 0.5 cm induces a current of 2 × 10−14 A in the end caps. This causes a voltage drop of 2 × 10−8 V, if we assume R = 1 MΩ. To detect such small currents or voltages the thermal noise of the detector has to be suﬃciently small. If the resistance R takes the form of a resonant circuit with a quality ˜ = ν0 /δν tuned to the ion’s oscillation frequency, the thermal noise factor Q voltage VN is given by (3.24) VN = 4kB RT δν, where kB is the Boltzmann constant and δν the bandwidth of the circuit at ˜ = 1, 000 at a frequency f0 of the resonance frequency ν0 . If we assume Q 100 kHz we arrive at a total noise voltage equal to the induced voltage when the temperature T of the circuit is at 4 K. In fact all nondestructive detection schemes for single-trapped ions operate at cryogenic temperatures. Figure 3.5 shows a detection scheme for the axial oscillation of trapped ions and Fig. 3.6 shows a signal arising from a single-trapped C5+ ion. Instead of measuring the total induced noise in the detection circuit, a Fourier transform of the noise allows one to determine the oscillation frequency without excitation of the oscillation. Figure 3.7 shows a signal of the perturbed cyclotron motion of a single ion. The statistical uncertainty of the line center UDrain

T = 300 K RD

UOut T = 4.2 K T1 R

UG1

R

UG2

R

UElectr

C C’

L C

C

Fig. 3.5. Example of detection electronics for the axial oscillation of a single-trapped ion. The arrow indicates the oscillating ion [96]

62

3 Mass Spectroscopy in Penning Trap

Power density (a.u.)

Fig. 3.6. Induced signal from the axial oscillation of a single-trapped O7+ ion using a superconducting resonance circuit with quality factor 1,500 and an impedance of 11 MΩ. The averaging time is 5 s [37] 40 30 20 10 0 –0.6 –0.4 –0.2 0.0 0.2 0.4 0.6 ω+ (Hz) – 24 075 552.802 6 Hz 2π

Fig. 3.7. Fourier transform of the noise in a resonance circuit attached to two segments of the ring electrode of a Penning trap in the presence of a single C5+ ion. The line centre has a statistical fractional uncertainty of 10−10 [37]

is about 1 part in 1010 which indicates the high precision of mass spectroscopy based on comparing cyclotron frequencies of diﬀerent ions. When the ion is resistively cooled to the environmental temperature no excess noise in the attached electronics appears; it can, however, still be detected by Fourier transform of the induced noise as described in Sect. 3.3. The noise induced by the oscillating ion adds to the thermal noise of the circuit at the oscillation frequency, however with opposite phase, resulting in a reduction of the total noise amplitude. An alternative method to measure the cyclotron frequency of a single ion has been developed at MIT, called the “pulse and phase” method [97]: The ion’s cyclotron motion is excited to a large amplitude. When the excitation is stopped the phase of the oscillation increases in time. After a given time Tevol an rf pulse at the sum of the cyclotron and axial frequencies is applied which couples the two motions, and the induced axial signal is recorded. The length of the pulse is chosen such that the energy between the modes is completely exchanged (π-pulse). The phase of the axial signal immediately after the π-pulse is measured with a Superconducting Quantum Interference Device

3.2 Observation of Motional Resonances

63

Fig. 3.8. Broadband Fourier spectrum of trapped organic molecules (Bovine GD1b Ganglioside) showing the charge-to-mass distribution [101]

(DC-SQUID) [98]. The ion’s cyclotron frequency is obtained from a measure of the accumulated phase versus Tevol . The detection of the induced noise from a single-trapped ion requires a narrow band detection scheme to suppress ampliﬁer and circuit noise. When, however, a large number of ions is simultaneously conﬁned, broadband detection with a subsequent Fourier analysis of the induced voltage allows identiﬁcation of the diﬀerent trapped ions by their diﬀerent cyclotron frequencies. This method, called Fourier transform-ion cyclotron resonance (FT-ICR) is widely practiced in chemical analysis [99,100]. Figure 3.8 shows an example of a broad band Fourier spectrum for mass analysis of an organic molecule. 3.2.2 Destructive Observation In the destructive detection scheme, ﬁrst described in [102] the ions are ejected from the trap and are counted outside the trapping region (see Fig. 1.19). In the inhomogeneous part of the magnetic ﬁeld they are accelerated by the force acting on their orbital magnetic moment. Thus, the time-of-ﬂight from the trap, placed at z = 0, to the detector at z = zmax depends on the initial axial and radial energies Ez and Er , respectively, and on the distribution of the magnetic ﬁeld strength B(z) along the ion’s path: zmax dz t= (3.25) . 0 B(z) 2 M Ez + Er − Er B(0) Figure 3.9 shows the calculated ﬂight times for ions with atomic mass M = 2 in a ﬁeld which decays from 5 T at z = 0 to zero at a distance of 50 cm for various radial and axial energies. When the perturbed cyclotron resonance is excited the radial energy of the ion increases and the ﬂight time to the detector is reduced as shown in Fig. 3.10. Variation of the excitation frequency shows the minimum ﬂight time at resonant excitation. As an alternative to the excitation of the perturbed cyclotron oscillation a radiofrequency (rf) ﬁeld at the pure cyclotron frequency

64

3 Mass Spectroscopy in Penning Trap

Fig. 3.9. Calculated time-of-ﬂight of ions with atomic mass 3 for various axial energies as function of the radial energy. The assumed distance from the trap center to the detector is 50 cm

Fig. 3.10. Time-of-ﬂight of 39 K+ ions without (above) and with (below ) excitation of the perturbed cyclotron oscillation. The width of the distribution reﬂects the scatter of initial energies over many repetitions of single-ion detection [33]

ω+ + ω− = ωc

(3.26)

can be used for mass determination. It has been shown, that this relation gives the true cyclotron frequency with high precision even in the case of a perturbed trap similar at the “invariance theorem” (3.19) [G. Gabrielse, Phys. Rev. Lett. 102, 172501 (2009)]. If, in a quadrupole trap, a ﬁeld is applied in the symmetry plane of the trap, e.g. by using adjacent pairs of segments in the ring electrode as shown in Fig. 3.11, it couples the perturbed cyclotron and the magnetron motion. The result of the coupling is an alternation between the magnetron and cyclotron motions. When the ion starts with a large magnetron amplitude and a small cyclotron orbit the magnetron radius will shrink and the cyclotron orbit increase in time as shown in Fig. 3.12. If Vrf is the amplitude of the coupling ﬁeld a full conversion is obtained after a time

3.2 Observation of Motional Resonances

65

Fig. 3.11. Potentials applied to segments of the ring electrode of a Penning trap to create a quadrupole ﬁeld in the radial plane [33]

Fig. 3.12. Calculated radial trajectory of an ion under the inﬂuence of a radial quadrupole rf ﬁeld, tuned to the sum of the perturbed cyclotron and the magnetron oscillations. The ion starts with a small cyclotron orbit which increases in time while the magnetron radius shrinks. The calculation stops when a full conversion is obtained [33]

Tconv =

2 4πR− B . Vrf

(3.27)

For a voltage Vrf = 1 mV and a magnetron radius R− of 1 mm, we obtain a conversion time of 0.2 s in a ﬁeld of B = 5 T. If the conversion is continued the process is reversed and the magnetron radius increases again while the cyclotron radius shrinks. Since the radial energy is mainly determined by the energy in the cyclotron mode, the increase in the cyclotron radius after a full conversion of the magnetron motion leads to a decrease in the ﬂight time to the detector as in the previously discussed excitation of the perturbed cyclotron oscillation. The conversion of the cyclotron to magnetron energy can also be used to purify the trap from unwanted ions: Since the coupling frequency ωc is inversely proportional to the mass, the magnetron radius of unwanted species can be selectively excited. When their magnetron radius becomes larger than a hole in the end cap electrode through which the ions are ejected, they

66

3 Mass Spectroscopy in Penning Trap

cannot leave the trap and only ions under investigation proceed. Such a puriﬁcation trap is generally used at facilities where ions of diﬀerent mass are produced and stored simultaneously [103–106]. Subsequently ions which have left the puriﬁcation trap are conﬁned in a second trap, where their masses are determined.

3.3 Line Shape of Motional Resonances 3.3.1 Nondestructive Detection In the case of a perfect quadrupole trap the response function I(ω) of the weakly excited ion oscillation is that of an harmonic oscillator and has a Lorentzian shape A I(ω) = . (3.28) (ω − ω0 )2 + γ 2 The lower bound to the half width is set by the δω1/2 damping γ from the coupling to the external detection circuit. For the case of the axial oscillation we have 2 ω0 L ˜ Q γz = (3.29) Q. 2z0 M ˜ is its quality factor. For small Here L is the inductance of the circuit and Q amplitudes of the excitation ﬁeld this is a good approximation when small anharmonicities are present as illustrated in Fig. 3.7. For a larger drive voltage, imperfections of the trapping potential become signiﬁcant. If we consider only an octupole term in the trapping potential, characterized by an amplitude c4 given by (3.1) the particle behaves as an anharmonic oscillator. The corresponding equation of motion 2 d d 2z 2 QV˜ 2 + ω 1 + z(t) = + γ cos ωt (3.30) z z dt2 dt z02 2z0 has been treated extensively in the literature (see e.g. [107]). Its solution depends on a parameter λ, given by 2 3 ωz zmax λ = c4 , (3.31) 4 γz z0 where zmax is the maximum amplitude of the oscillation. Figure 3.13 illustrates the solution for various parameters λ. Corresponding equations hold for the radial motion. The shifts δω = γ 2 /ωi2 , i = z, r of the maximum associated with the damping are negligible: Typical values of damping constants are of the order of 10 Hz for the axial oscillation and 0.01 Hz for the cyclotron motion at 1 and 25 MHz, respectively, using high quality detection circuits [108].

3.3 Line Shape of Motional Resonances 1.0

λ=0

a2/ a2max

0.8

67

λ=1 λ = 10

0.6 0.4 0.2 0

–5

0 5 2(ω–ωz)/γz

15

Fig. 3.13. Solution of the equation of motion of an anharmonic oscillator for various anharmonicity parameters λ Detection circuit l c

L

C

R

i

uA

r

Fig. 3.14. Equivalent circuit of a trapped ion with impedance l, capacitance c, and resistance r coupled to a detection circuit

For the case of no excitation the ion will assume thermal equilibrium with the detection circuit if its oscillation frequency matches the resonance frequency of the detection circuit. The noise voltage across the circuit is proportional to the total impedance Ztot , which is the sum of the impedance of the detection circuit ZLC and the equivalent circuit of the oscillating ion ZLC . As outlined in vol. 1, Sect. 6.2.2, the ion can be replaced by an equivalent LC circuit (Fig. 3.14). The resistance r takes the coupling to the circuit into account [38]. The noise voltage uA in a bandwidth δν is proportional to the total impedance Ztot . Using standard circuit theory we obtain: R 4kB T Rδν 1 + 2 r 1+Q2 (δ− 1δ ) uA = 2 . (3.32) 2 1

˜ Ω ¯ − 1¯ − QR(δ− δ )2 2 1 + r 1+Q2R(δ− 1 )2 + Q Ω [ ] r 1+Q2 (δ− 1δ ) δ ¯ = ω/ωLC for normalization. Figure 3.15 We have used δ = ω/ωz and Ω shows calculated noise amplitudes for a slight detunings of the ions oscillation frequency ωz and the circuit’s resonance frequency ωLC . For ωz = ωLC we obtain a Lorentzian line shape for the central minimum superimposed on the Lorentzian thermal noise spectrum of the circuit.

3 Mass Spectroscopy in Penning Trap

Noise voltage U

1.0

without ion

0.8

with ion

0.6 0.4 0.2

1.0 Noise voltage U

68

0.6 0.4 0.2 0.0

0.0 0.99990 (a)

0.8

1.00000 Frequency

0.99990

1.00010 (b)

1.00000 Frequency

1.00010

Fig. 3.15. Calculated noise spectra of a single-trapped ion in thermal equilibrium with an outer tank circuit for slight (a) and large (b) detunings of the ions oscillation ωz frequency and the circuit’s resonance frequency ωLC . The noise voltage is normalized to 1 and the frequency is given in units of ωLC

Fig. 3.16. Dipole (left) and quadrupole(right) excitation of the ion motion [105]

3.3.2 Destructive Detection In the destructive detection scheme the ion oscillation is excited by an rf ﬁeld of constant amplitude for a given period of time Δt. The associated energy increase is detected by a reduction in time-of-ﬂight to a distant detector outside the magnetic ﬁeld of the Penning trap when the ions are ejected from the trap as discussed in Sect. 3.2.2. We assume that the reduction in time-ofﬂight is proportional to the radial energy increase. Various excitation modes can be employed leading to diﬀerent line shapes. Dipole Excitation The motion of an ion under the inﬂuence of an oscillating dipole ﬁeld of amplitude A and frequency ω/(2π) (Fig. 3.16) follows from the equation of motion

3.3 Line Shape of Motional Resonances

69

Fig. 3.17. Perturbed cyclotron resonance line of 39 K+ detected by time-of-ﬂight method. The line is a least squares ﬁt of (3.36) to data points. The full width of 0.9 Hz corresponds to the expected width for a 1 s excitation time [109]

d2 u du ωz2 − u = Ae−iωt , + iωc 2 dt dt 2 where u = r, z. The general solution of this equation is given by u(t) = u0 e−iω+ t −

A ω2

− ωωc +

ωz2 2

eiωt .

(3.33)

(3.34)

When we consider only the frequency range close to resonance (ω+ = ω + Δω) and use a series expansion, we obtain in the ﬁrst approximation u(t) = −

A sin(Δω/2) −iωt e . ω Δω/2

(3.35)

The radial energy is proportional to the square of the amplitude u(t). Then the line shape is given by

sin(Δω/2) u Δω/2 2

2 .

(3.36)

This so-called Rabi-line shape has a full width at half maximum of δν = 0.89/Δt. This has been experimentally veriﬁed (Fig. 3.17). Quadrupole Excitation An rf ﬁeld in quadrupolar geometry (Fig. 3.16) excites ion resonances at the sum or diﬀerence of two eigenfrequencies. Of particular interest is the excitation at ω = ω+ + ω− = ωc . It avoids the need to measure several resonances for the comparison of cyclotron frequencies of two ions. The eﬀect of the quadrupolar rf ﬁeld is to couple the perturbed cyclotron and the magnetron

70

3 Mass Spectroscopy in Penning Trap

Fig. 3.18. Radial energy gain (in units of the initial energy) of the cyclotron oscillation for quadrupolar excitation near ω = ω+ + ω− for diﬀerent values of k0 Tcon : (a) π, (b) 1.5π, (c) 2π [109]

motions and to transfer radial energy between the two modes. The time Tcon for a full conversion is determined by the amplitude of the rf ﬁeld. The change in energy of the perturbed cyclotron mode can be detected by a change in the time-of-ﬂight from the trap to the detector, similar to the case of the dipole excitation. For a given interaction time, however, the shape of the response function depends on the rf amplitude k0 . If the amplitude for a full conversion from magnetron to cyclotron motion is given, the energy gain can be calculated for diﬀerent values of k0 Tcon (Fig. 3.18) [109, 110]. The inverse of the energy gain is the observed change in time-of-ﬂight. For k0 Tcon = π a Rabi line shape similar as in the case of dipole excitation with a half width of δν = 0.89/Tcon is obtained. Ramsey Excitation In 1949, Ramsey applied spatially separated oscillating ﬁelds to a molecular beam to induce transitions between molecular states [111, 112]. It proved to be extremely successful, leading to reduced transition line width and consequently to higher accuracy. In essence the molecular beam enters a ﬁrst region where a weak rf ﬁeld at a frequency close to a molecular resonance is applied, followed by a region with no oscillatory ﬁeld, and a third region identical with the ﬁrst one. It is essential that the oscillating ﬁelds in the ﬁrst and third regions have a precisely deﬁned phase relation. This has since become a standard technique in atomic and molecular beam spectroscopy. In 1992, Bollen et al. [113] suggested to use the Ramsey technique to improve the accuracy of Penning trap mass spectrometry. The ﬁrst experiments were performed by Bollen et al. at the ISOLTRAP facility CERN [113] and later on by Bergstr¨om et al. at SMILETRAP in Stockholm [106]. The basic idea is illustrated in Fig. 3.19a: An oscillatory ﬁeld is applied for two time intervals τ1 separated by the time τ0 . The ﬁeld can be applied in dipolar or quadrupolar geometry as discussed earlier. The respective frequencies are near the perturbed cyclotron or at the sum of perturbed cyclotron and magnetron oscillations.

3.3 Line Shape of Motional Resonances

71

Fig. 3.19. Schemes for Ramsey excitation. Oscillatory ﬁelds are applied to the trap during periods τ1 1 0.8 0.6 0.4 0.2 –4

–2

0 η

2

4

Fig. 3.20. Line shape for excitation of the cyclotron motion for pulse duration τ0 = 1.5τ1 (solid line). For comparison, the line shape for excitation by a single pulse of length τ0 + 2τ1 is shown (dashed line), illustrating the reduction in line width by Ramsey excitation. η is deﬁned as η = δ/(2g) [110]

Kretzschmar [110] has calculated the transition probability F for Ramsey excitation in a quantum mechanical framework. If we consider quadrupole excitation near the sum of perturbed cyclotron and magnetron frequency ωc = ω+ + ω− and call ωR = (2g)2 + (ωd − ωc )2 = (2g)2 + δ 2 the Rabi frequency, where ωd is the frequency of the driving ﬁeld of amplitude g, we obtain: 2 δτ0 (2g)2 δ δτ0 F (δ, τ1 , g) = sin(ω [cos(ω cos τ ) + sin τ ) − 1] . R 1 R 1 2 ωR 2 ωR 2 (3.37) Figure 3.20 displays the excitation probability as function of the detuning δ. In [110] more general excitation schemes involving several excitation periods as e.g. illustrated in Fig. 3.19b and c are also discussed, leading to more complex excitation functions. Figure 3.21 gives some examples. The predicted lineshape for two-pulse excitation has been experimentally veriﬁed. Figure 3.22 shows the result of an excitation of the cyclotron resonance of 39 Ca19 F+ , destructively detected by time-of-ﬂight.

72

3 Mass Spectroscopy in Penning Trap

Mean time of flight (μs)

Fig. 3.21. Theoretical resonance curves for normal excitation and diﬀerent Ramsey patterns with four, three, and two pulses [106] 39Ca19F

320 300 280 260 240 –3

–2

–1 0 1 2 nc – 1567016.02 (Hz)

3

Fig. 3.22. Time-of-ﬂight spectrum after two-pulse Ramsey excitation of the cyclotron resonance of 39 Ca19 F+ . The length of the excitation periods was τ1 = 100 ms and τ0 = 1 s. The solid line is the theoretically expected line shape [114]

3.4 Experimental Procedures All high precision Penning trap mass spectrometers operate with single stored ions. They diﬀer mainly in the way the motional resonances, particularly the perturbed cyclotron resonance of the stored ion, are detected. Nondestructive detection methods allow the cooling of the ions to low temperatures and to observe them for very long times. When unstable isotopes are investigated they have to be produced outside the trapping region and injected into the trap. Often their lifetime is very short and does not allow cooling. Then a destructive detection method is applied with rather simple ways of detecting ejected ions. In both cases the challenge is to keep the time between comparison of two diﬀerent masses as short as possible to reduce uncertainty caused by the temporal ﬂuctuations of the magnetic ﬁeld.

3.4 Experimental Procedures

73

3.4.1 Reference Ions The mass of a trapped ion is determined by a comparison of its cyclotron frequency to the one of a reference ion of known mass. Under ideal conditions the choice of the reference ion is arbitrary and it would be natural to take carbon ions as reference since the carbon isotope of mass 12 represents the deﬁnition of the atomic mass scale. Shifts of the cyclotron frequency, that appear from imperfections of the trapping potential, magnetic ﬁeld inhomogeneities, or by image charges depend on the ion’s mass. The inﬂuence of these shifts on the measured frequency ratios will be minimized when the mass of the reference ion is as close as possible to the ion under investigation. While in the low-mass region of the periodic table several masses are known with very high precision which can serve for reference this is not the case in the high mass region. Blaum et al. [115] have found a general way to reduce the mass diﬀerence of an unknown ion from a reference ion by using carbon clusters C+ n as reference. The only remaining uncertainty then is the binding energy of the carbon clusters which is of the order of a few electronvolts and known to some percent. They can be produced by laser ablation from a carbon surface and injected into the trap [116]. As evident from Fig. 3.23, the diﬀerence from any unknown isotope is at most six atomic mass units. Even smaller mass differences can be obtained when carbon–hydrogen compound clusters are used as in [117].

Fig. 3.23. Nuclear chart with stable (full squares) and unstable (open squares) isotopes (lifetimes >100 ms). The diagonal lines represent masses isobaric with 12 C+ n clusters. The dashed lines represent nuclear shell closures [115]

74

3 Mass Spectroscopy in Penning Trap

Another advantage of carbon clusters is the possibility of checking for systematic errors: A mass comparison of an isotope with carbon clusters of diﬀerent masses might reveal any remaining mass dependent shift of the cyclotron frequencies [116]. The comparison of cyclotron frequencies of the reference ion with the ion of interest is performed at diﬀerent times. Trap parameters like magnetic or electric ﬁeld strength will vary in time because of temperature or pressure changes in the laboratory or spontaneous ﬂux jumps in superconducting coils. It is therefore required to minimize the time delay between the measurements as much as possible. In the destructive detection scheme, mainly used for unstable isotopes, ions are kept for about 1 s in the trap and then ejected to measure the time-of-ﬂight to a distant detector. After each data point a reference ion may then be injected into the trap from a second source. Figure 3.24 shows the setup at the ISOLDE facility, CERN, as an example. Linear drifts of the ﬁelds will then be cancelled when the cyclotron frequencies from the full resonance lines are evaluated. Higher order drifts, however, will remain. For stable isotopes and nondestructive detection of the cyclotron resonance diﬀerent methods are developed to keep the exchange time between the ion of interest and the reference ion small. One way is to create several potential minima and shift the ions between them. An example is shown in Fig. 3.25: The measurement of the cyclotron frequency takes place in the central minimum (“precision trap”) while the second ion is placed in one of the “preparation traps.” By switching the voltages at the trap electrodes the second ions is transferred into the precision trap for comparison of the cyclotron frequencies. The change takes place in a fraction of a second and repetitive measurements can be performed in a short time to average about time variations of the magnetic ﬁeld. A diﬀerent approach to minimize the eﬀect of magnetic ﬁeld ﬂuctuations has been developed by a group at MIT [97, 118]: two ions are brought on opposite sides of the same magnetron orbit of diameter ∼1 mm. The way in which this is performed is to start with a single ion of one species and drive it to a large magnetron radius using a short resonant pulse at the magnetron frequency. Then an ion of the second species is loaded into the trap. The small Coulomb interaction mixes the frequency-degenerate magnetron modes into two new collective modes: the common mode and the separation mode, with constant mode amplitudes ρcom and ρs , respectively. To determine the ion–ion separation ρs , the beat frequency Ωm between the collective magnetron modes is measured: Q Ωm = . (3.38) 2πε0 B0 ρ3s A ﬁxed-frequency axial drive applied just below the axial resonance of one of the ions couples both axial motions. In the presence of electrostatic anharmonicities, as always present in the trap, the detuning of the axial frequency of the ion from the ﬁxed frequency drive is modulated at the beat frequency

3.4 Experimental Procedures

75

Fig. 3.24. Part of the setup at the ISOLDE facility, CERN, where ions under investigation are injected into the Penning trap from the production line (RICB) and reference ions (Carbon clusters) are alternately produced in a separate source [116]

Ωm because of the changing radial position in the trap. Thus, the axial frequency modulation is converted into axial amplitude modulation. The axial amplitude modulation combined with an electrostatic anharmonicity generates a modulation of the magnetron frequency of that ion. As the ions pass through equal magnetron radii, the magnetron frequency modulation of the driven ion creates a small phase advance or lag of its magnetron position with respect to the other ion. This magnetron phase advance or lag is modulated at Ωm , so that the relative phase shifts coherently add. The result is that the ions slowly “walk” away from one another; that is, ρs increases.

76

3 Mass Spectroscopy in Penning Trap

Fig. 3.25. Stack of cyclindrical electrodes to create several potential minima. The precise measurement of the cyclotron frequency of an ion takes place in the central “precision trap” while the reference ion is stored in an adjacent “preparation trap”. Ions can be exchanged in a short time (Courtesy K. Blaum)

The common mode amplitude ρcom can be determined from the amplitude of the axial frequency modulation and knowledge of the electrostatic anharmonicity. It needs to be set at zero to place the ion pair on the desired magnetron orbit. This is enabled by a nonlinear coupling technique that resonantly and reversibly transfers canonical angular momentum between the common and separation magnetron modes. The coupling is nonlinear in the amplitudes ρs and ρcom , because it is driven by the modulation of the radial position of the ions, which goes to zero as ρcom goes to zero. As a result, the system will exponentially relax to the desired conﬁguration with ρcom = 0. This is illustrated in Fig. 3.26. When ρs is of the order of 1 mm the ion–ion Coulomb interaction perturbs the measured cyclotron frequency ratio by less than 10−11 . Because the ions move on a shared magnetron orbit, they spatially average magnetic ﬁeld inhomogeneities and electrostatic anharmonicities.

3.5 Selected Results Numerous atomic and molecular masses have been determined by Penning trap mass spectrometers. As examples we select a few of them which are of particular importance for fundamental aspects. For stable isotopes mass uncertainties of the order of 10−10 or below have been obtained in many cases. For short-lived isotopes the accuracy is in general lower, particularly in case of very short-lived species, due to the limited observation time. The mass of the neutral atom is usually required; it is determined from the ions mass by νref Qion matom = · mref + Qion · me − E. (3.39) νion Qref

3.5 Selected Results

77

Fig. 3.26. (a) Power spectra of the instantaneous axial frequency of one ion species for a sliding time window of 100 s. By placing a constant axial drive below the axial resonance and introducing an electrostatic anharmonicity C4 , canonical angular momentum in the common mode can be transfered to the separation mode, causing the ion–ion separation ρs to increase and Ωm to decrease, until ρcom 0; (b) initial magnetron motion; (c) ﬁnal magnetron motion, which approximates the ideal orbit conﬁguration [118]

Here νion and νref are the measured cyclotron frequencies of the ion under investigation and the reference ion with charge state Qion and Qref , respectively, me is the electron mass, and E is the binding energy of the electrons. The electron mass is known very accurately and does not contribute to the uncertainty of the atom’s mass. The uncertainty of the binding energy contributes less than 10−10 for singly charged ions; for highly charged heavy ions few experimental values are available and the uncertainty of calculated values is estimated to the order of 10−9 . 3.5.1 Stable and Long Lived Isotopes 3

H–3 He Mass Diﬀerence

The β-decay spectrum of tritium is used to determine the mass of the electron neutrino. A ﬁnite neutrino mass would change the shape of the decay spectrum

78

3 Mass Spectroscopy in Penning Trap

near the endpoint energy which is given by the mass diﬀerence between 3 H and 3 He. This quantity therefore is an important input parameter to ﬁt the observed spectrum. Presently an upper limit for the neutrino mass of 2.3 eV (95% conﬁdence level) [119] has been obtained. A new experiment [120] aims at a sensitivity of 0.2 eV. The accuracy of the mass diﬀerence between 3 H and 3 He as a signiﬁcant input parameter would then be about 1 ppm. This requires a fractional uncertainty in the individual masses of about 10−11 . Presently the most precise value of the respective masses in atomic units are m (3 H) = 3.0160492787(25), m (3 He) = 3.0160293217(26) [121]. From these values the 3 H–3 Het mass diﬀerence is determined as 18.5898(12) keV. Proton/Electron Mass Ratio The electron mass and the proton/electron mass ratio are key parameters for the least squares adjustment of the fundamental constants [122] and provide a necessary link between tests of physical theories through the analysis of experimental data. In Penning trap experiments performed at the University of Washington the cyclotron frequency of single-trapped electrons was compared to that of a single C6+ ion. Comparison to neutral carbon as standard of the atomic mass scale was provided by correcting the ion mass for the removed electrons and their binding energies. It results in a value for the electrons mass me of 0.0005485799111(12)u [123]. The main contributions to the uncertainty arise from temporal magnetic ﬁeld instabilities between the measurements and relativistic changes of the axial and radial energy of the electron which was held in thermal equilibrium with the 4 K environment. With the protons mass taken from the CODATA table of fundamental constants the value for the proton/electron mass ratio is calculated to be mp /me = 1836.1526665(40). An indirect determination of the electron mass from measurements of g factor of the electron bound in hydrogen-like C5+ [124] and O6+ [125] ions in a Penning trap and based on the theory of bound-state quantum electrodynamics improves the values to me = 0.00054857990945(24) and mp /me = 1836.15267261(84) [122]. Proton/Antiproton Mass Ratio The proton/antiproton mass ratio is one of the basic tests of the CPT invariance. In a series of experiments at the antiproton facility at CERN single H− ions and antiprotons are stored simultaneously in a Penning trap and their cyclotron frequencies have been measured by Fourier transform of the induced noise in the trap electrodes. The ratio of p and p ¯ is determined to 0.99999999991(9) corresponding to a fractional uncertainty of 9 × 10−11 [126]. Cs Mass and the Fine Structure Constant The ﬁne structure constant α is one of the most important physical constants and is used in particular for comparison of calculations and experimental

3.5 Selected Results

79

data in quantum electrodynamics. The most accurate value is α−1 = 137.035999070(98)(0.71)ppb [127, 128]. It is based on a comparison of the experimental value of the g factor of the free electron [129] and a high order QED calculation of this quantity [130, 131]. It is desirable to have a value for α which is independent of theoretical calculations and which matches the experimental accuracy obtained in the g factor experiments on the free electron. One way toward this goal is to use the relation, α2 =

h 2R∞ h 2R∞ mCs mp · · = · · , c me c mCs mp me

(3.40)

where R∞ is the Rydberg constant, known to a fractional error of 6.6 × 10−12 [122]. The quantity h/mCs has been determined from photon recoil measurements on Cs atoms [132] to 3.2×10−9 . Apart from the proton/electron mass ratio as discussed above, only the mass of the Cs atom remains. The most precise values to date, obtained on single Cs ions in a Penning trap are mCs = 132.905451931(27)(0.20 × 10−9) [117] and mCs = 132.90545159(41)(3 × 10−9) [133]. SI Mass and the Kilogram The present deﬁnition of the kilogram as one of the basic elements in the international systems of units (SI) is based on an artefact, a Pt–Ir piece kept in Paris. Because this may be subject to change in time attempts are under way to replace the artefact by an atomic standard. The AVOGADRO project [134] has the aim of replacing the kilogram artefact by a high-purity, perfect single crystal of natural or isotope-enriched silicon. The Avogadro constant NA = MMSi a3V/8 with MSi the atomic mass of silicon, M and V the mass and volume of the crystal, and a the lattice spacing of the silicon crystal, links the atomic mass of silicon to the mass of the crystal when the number of atoms in the crystal, the isotopic composition, and the crystal volume are known. Mass spectrometry in Penning traps has provided accurate values for the masses of the stable Si isotopes. For the most abundant isotope 28 Si a group at MIT obtained M(28 Si) = 27.976926 532 4 (20) [135]. This value has been conﬁrmed with similar accuracy at the SMILETRAP facility, Stockholm, using highly charged ions of diﬀerent charge states [136]. 3.5.2 Short-Lived Isotopes The main interest in the determination of masses of unstable isotopes arises from nuclear physics: The mass of a nucleus M(Z, N) with Z protons and N neutrons is determined by the masses of the constituents mP and mN and the binding energy B(N, Z): B(Z, N ) = [ZmP + N mN − M (Z, N )]c2 .

(3.41)

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3 Mass Spectroscopy in Penning Trap

The binding energy is a fundamental quantity characterizing each nucleus. The comparison of experimental value with theoretical predictions is a stringent test of nuclear models. The large quantity of experimental data accumulated over the past years has contributed substantially to our understanding of nuclear structure. Some speciﬁc questions are concerned with the astrophysical production mechanism of nuclei with a mass larger than iron. For example, a theoretical explanation for the creation of 92 Mo, 94 Mo and 96 Ru, 98 Ru by neutrino wind [137] requires the knowledge of proton separation energies or Q-values which can be obtained from mass measurements. Halo nuclei such as 11 Li, 11 Be, 19 C, or 8 He are of particular interest in nuclear systems and an input parameter of the theory, the two-neutron separation energy, is a critical test. For its determination the mass of the nuclei has to be determined with high precision. A recent example is the mass of 8 He (T1/2 = 119 ms) which has been determined with a relative uncertainty of 5 × 10−8 at the TITAN facility [138]. Far from the valley of stability shell closure eﬀects are expected to disappear at the classical magic numbers and to appear at new magical numbers. This phenomenon can be observed in the trend of neutron and proton separation energies. To constrain nuclear models describing these structures, mass measurements of very neutron rich isotopes are required. A number of dedicated facilities produce unstable isotopes by high-energy reactions. They are extracted from the production area, mass separated by conventional mass spectrometers and then injected into a Penning trap where their cyclotron frequency is measured and compared to the cyclotron frequency of an ion of known mass. The requirement for accuracy is less stringent as compared to the stable isotopes discussed in the previous section: In general uncertainties of the order of 10−8 are suﬃcient to allow diﬀerentiating between diﬀerent nuclear models. The main emphasis of the diﬀerent facilities is to reduce the time for a complete measurement cycle, that is the time between ion production, ion injection into the trap, cyclotron excitation, and detection to extend the range of nuclides far away from the stable area. The set up of the existing facilities follows in general the one shown in Fig. 3.24 for the CERN/ISOLDE Table 3.1. Mass uncertainty δm/m and shortest half life T1/2 of isotopes obtained at facilities for mass determination of unstable nuclides Name ISOLTRAP JYFLTRAP SHIPTRAP LEBIT TRIUMF CPT MLL-TRAP TRIGATRAP

δm/m

T1/2

8 × 10−9 65 ms 10−8 116 ms <1 s 10−8 10−8 <100 ms 100 ms 10−8 10−8 50 ms 10−7 10−7 100 ms

3.5 Selected Results

81

mass spectrometer [105] which is the ﬁrst one of several similar projects: CPT in Argonne [139], SMILETRAP in Stockholm [140], JYFLTRAP in Jyv¨ askyl¨ a [141], LEBIT in Michigan [142], SHIPTRAP at GSI Darmstadt [143], TRIUMF in Vancouver [144]. New facilities based on the same principle will come up soon: TRIGATRAP in Mainz [J. Ketelaer et al., Nucl. Instr. Meth. A 594, 162 (2008)], MLL-TRAP [145] in Munich and RIKEN-trap in Japan [146]. Technical details of these facilities can be found under the given references. Table 3.1 lists the obtained mass uncertainty δm/m and shortest half life T1/2 of isotopes at the existing facilities.

4 Microwave Spectroscopy

4.1 Zeeman Spectroscopy The magnetic moment associated with the spin of a particle or an atom has played a signiﬁcant role in the development of atomic theory. The concept of the spin 1/2 (in units of ) for the electron introduced by Goudsmit and Uhlenbeck in 1925 [147] led to the understanding of the anomalous Zeeman eﬀect and the ﬁne structure doubling of spectral lines. The Dirac theory of the electron [148] showed that the spin is a purely relativistic eﬀect and that the magnetic moment associated with it is the Bohr magneton μB = e/2m. If we deﬁne a dimensionless proportionality constant g between the magnetic moment μ and the spin s (in units of ) μ = gμB s,

(4.1)

we arrive at g = 2 for the electron. A similar expression holds when we replace the spin by the total angular momentum J of an atom as the vector sum of the total spin S and the total orbital momentum L, provided J is a “good” quantum number (I = 0 and B not extremely large). The g-factor then is labelled gJ . To determine experimentally the value of a magnetic moment, it is necessary to measure the energy diﬀerence ΔE between magnetic sublevels of the system under investigation in a known magnetic ﬁeld B: ΔEJ = gJ μB BΔmJ ,

(4.2)

where mJ is the orientation quantum number of the angular momentum J. Measurements of magnetic moments often refer to measurements of the g-factor, and g − 2 is a measure of the deviation of the magnetic moment from the Dirac value for a free electron. Numerous measurements of magnetic moments of neutral atoms have been performed using the atomic beam or Rabi-technique: A beam of atoms is sent through an inhomogeneous magnetic ﬁeld. Zeeman substates of the electronic

86

4 Microwave Spectroscopy

ground level of diﬀerent mJ are spatially separated by the Stern–Gerlach force F = ∇(μB) = gmJ μB ∇B, under the assumed conditions. They are deﬂected by a second inhomogeneous ﬁeld onto a detector. In a homogeneous magnetic ﬁeld placed between the two inhomogeneous ﬁelds the atoms are subject to an rf ﬁeld which attempts to induce transitions between diﬀerent mJ states. A successful transition at the frequency ω = ΔE/ is indicated by a change of the beam intensity at the detector. After proper calibration of the B-ﬁeld a value for g is obtained. For charged particles this method is not applicable since the Lorentz force F = Q(v × B) acting on the moving charge in a magnetic ﬁeld supersedes the Stern–Gerlach force by many orders-of-magnitude. In fact it has been argued [149, 150] that it is impossible to perform g-factor measurements of the Rabi-type on charged particles since the uncertainty in the beam position from Heisenberg’s uncertainty relation will always be larger than the change in position due to action of the inhomogeneous magnetic ﬁeld. The instrument of choice for g-factor measurements on charged particles is the Penning trap. As will be shown in the following paragraphs, Zeeman energy diﬀerences of stored charged particles can be measured by various methods with high precision. The calibration of the magnetic ﬁeld can be performed by measurements of the cyclotron frequency ωc = (e/m)B of the same particles. This has the advantage that no uncertainly exists as to whether the calibration is taken at the same position in the trap at which Zeeman transitions are induced. Moreover, the small volume which a stored ion or even a single particle occupies makes it rather insensitive to magnetic-ﬁeld inhomogeneities. As a result, to date the most precise values of magnetic moments are obtained using Penning traps. 4.1.1 g-Factor of the Free Electron The ﬁrst suggestion that the g-factor of the electron might vary from 2 was made by Breit in 1947 [151], and arose from a disagreement between experiment and theory in the hyperﬁne structure of hydrogen [152, 153]. This initiated a reshaping of the theory of the interaction between electrons and electromagnetic radiation, the so-called quantum electrodynamics. The ﬁrst theoretical calculation of the deviation of the free electron g-factor from the Dirac value g = 2, often expressed as g-factor anomaly a = (g − 2)/2, was performed by Schwinger [154] by the evaluation of the ﬁrst-order Feyman diagram, called the vertex correction (Fig. 4.1). The result was (g − 2)/2 = α/(2π) 1/800, α = e2 /c 1/137 being the ﬁne structure constant. The inclusion of higher order Feynman diagrams forms a series expansion with α/π as expansion parameter: a = 0.5α/π + C2 (α/π)2 + C3 (α/π)3 + C4 (α/π)4 + 1.701 (19) × 10−12 . (4.3) The size of the coeﬃcients Cn has been calculated by several authors. Presently, the best values are:

4.1 Zeeman Spectroscopy

87

Fig. 4.1. First-order Feynman diagram (vertex correction) for the g-factor anomaly of the free electron

C2 = −0.328 478 965 579 . . . [155],

(4.4)

C3 = +1.181 241 456 587 . . . [156, 157],

(4.5)

C4 = −1.9144 (35) [128]. C2 and C3 are known analytically while the value of C4 is based on a numerical evaluation. To illustrate the diﬃculty of the calculation, the fourth order requires the evaluation of 891 four-loop Feynman diagrams and took a total time of more than 15 years. The last term in (4.3) consists of contributions from vacuum-polarization loops involving muons, and from hadronic and weak interactions [158]. On the experimental side the ﬁrst determination of the free electron g-factor was performed by Crane and coworkers at the University of Michigan [159]. Electrons, polarized by Mott scattering, were injected into a magnetic bottle ﬁeld. They oscillated in this ﬁeld for a given time and then their direction of polarisation was determined. For a Dirac value of g = 2 the spin precession (Larmor) frequency ωL = gμB B/h in the magnetic ﬁeld would be identical to the cyclotron frequency ωc = (e/m)B and the polarisation with respect to the direction of electron momentum would not change in time. A deviation from g = 2 leads to an oscillation of the polarisation as function of time at the diﬀerence frequency of ωL − ωc = (1/2)(g − 2)(e/m)B. These experiments have been further reﬁned by Rich and coworkers and their ﬁnal result was (1/2)(g − 2) = 0.001 159 657 7 (35) [160]. In a similar way, the g-factor anomaly for the positron has been determined to be (1/2)(g − 2) = 0.001 160 3 (12) [161]. The uncertainties in these experiments arose from the limited knowledge of the average value of the inhomogeneous magnetic bottle ﬁeld along the path of the electrons. A detailed review of the Michigan experiments can be found in [162]. A variation of this precession experiment was performed at the University of Edinburgh [163], where a uniform magnetic ﬁeld was used and a weak electric ﬁeld E was superimposed perpendicular to B. This causes a drift of the electron cyclotron orbit in the direction of the E × B force. The electron polarisation was measured as a function of time as in the Michigan experiment. Due to the small number of electron turns in the ﬁeld, the accuracy was limited to 2% in the electron anomaly. Penning traps were introduced to g-factor experiments on free electrons by Graeﬀ et al. [56] and Dehmelt et al. [35]. The g-factor was determined

88

4 Microwave Spectroscopy 6-Pole magnet (Polarizer) Electron gun

Oven

Electron trap Homogeneous field region Na beam

Fig. 4.2. Sketch of the experimental setup for g-factor measurements on an electron cloud [56]

by induced resonance transitions between the two spin states of the electron according to (4.3) in a B-ﬁeld whose value was measured by the cyclotron resonance of the stored particle. The experiment by Graeﬀ et al. used a cloud of about 105 stored electrons in a trap of a few electronvolts depth. They were spin polarized by spin exchange collisions with a polarized Na atomic beam prepared by state selection in a hexapole magnetic ﬁeld. The Na beam passes through the mid plane of the trap through holes in the ring electrode (Fig. 4.2). For an atomic ﬂux of 1013 atoms cm−2 s−1 the electrons are polarized in about 10 ms. In addition to spin exchange inelastic collisions between Na and the electrons take place at a slower rate. Because electrons and Na atoms have the same polarisation, only triplet collisions take place: Na (↑) + e− (↑) −→ Na∗ (↑) + e− (↑).

(4.6)

When the spin direction of the electron is changed by an rf induced transition also singlet transitions become possible: Na (↑) + e− (↓) −→ Na∗ (↑) + e− (↓).

(4.7)

Inelastic collisions determine the mean kinetic energy of the stored electrons at a given time after trap loading. The cross section for singlet and triplet inelastic collisions diﬀers by a small amount. Analysing the energy distribution of the electron cloud at a ﬁxed time after loading (typically 1–2 s) the direction of the electron spin can be determined. On variation of the frequency of the excitation ﬁeld a resonance curve is obtained. Spin ﬂip transitions Δms = 1 are of magnetic dipole type and can be induced by a uniform magnetic ﬁeld B = B1 x ˆ cos ωL t, while cyclotron orbit transitions are electric dipole transitions and require a ﬁeld E = E1 x ˆ cos ωc t. A simultaneous combination of both transitions can be induced by an rf ﬁeld B = (∂B/∂x)(xˆ x + y yˆ − 2z zˆ) cos(ωL − ωc )t. Since the fractional diﬀerence between ωL and ωc is only of the order of 0.1%, one gains three orders of magnitude in precision of the g-factor if the diﬀerence frequency can be directly observed. This has in fact

4.1 Zeeman Spectroscopy

89

Fig. 4.3. Signal corresponding to the mean kinetic energy of a stored electron as function of frequency of an rf quadrupole ﬁeld. The center frequency corresponds to the diﬀerence between Larmor precession and cyclotron frequency [164]

been obtained (Fig. 4.3) and the ﬁnal result was [164] a = 0.001 159 66 (30).

(4.8)

The calibration of the B-ﬁeld in these experiments was by resonant excitation of the stored electrons at the cyclotron frequency. The energy gain on excitation leads to a loss of particles from the trap. The main limitation in this experiment was caused by the fact that the diﬀerence in cross section between triplet and singlet inelastic e− − Na collisions is small and an averaging time of several hours was required to obtain a resonance curve. This made it diﬃcult to keep the trap parameters suﬃciently constant. Also the Coulomb ﬁeld between electrons in the cloud adds to the Penning trapping ﬁeld and causes ﬂuctuations in the electron detection signals requiring long averaging times. The latter problem was avoided in a series of experiments performed at the University of Washington in which a single trapped electron was used. During this project, a number of novel techniques have been developed which have become very signiﬁcant for many experiments on single stored particles in traps. Figure 4.4 shows the trap used in these experiments. The detection of a single electron is made possible by the so-called “bolometric” method [35,38]: The end caps of the Penning trap are connected by an inductance to form a tank circuit of resonance frequency ω0 . The thermal noise voltage across this circuit is ampliﬁed and detected by a square wave detector. An electron oscillating in the axial direction at frequency ωz = [eU/(md2 )]1/2 , given by the trap size d and the applied dc trapping voltage U , can be tuned into resonance with the circuit and the additional noise induced by the oscillating electron in the trap electrodes can be observed if the thermal noise of the circuit is suﬃciently small. This is provided by keeping the trap and the circuit in thermal contact to a liquid He bath. Keeping the electron’s

90

4 Microwave Spectroscopy

Fig. 4.4. Penning trap for g-factor measurements on electrons at the University of Washington. It shows the hyperbolic shaped trap electrodes with additional guard electrodes between ring and end caps. Electrons are created by ﬁeld emission above the upper end cap and injected into the trap through a small hole. The apparatus is held in thermal contact to a liquid He bath which provides ultra-high vacuum by cryo-pumping. The trap is placed at the center of a superconducting magnet at a ﬁeld strength of about 5 T Reprinted with permission from [165]

oscillation in resonance with the circuit will eventually lead to thermal equilibrium between the electron’s kinetic energy and the environment. This is called “resistive cooling,” and has been treated in Vol. 1 [1]. See also Sect. 1.4.2. The time constant for reaching thermal equilibrium is given by τ −1 = (d/e)2 (m/R),

(4.9)

where R is the resonance impedance of the circuit. In case of the Washington experiments τ was below 1 ms. The radial motion of the electron is a superposition of two oscillations: The perturbed cyclotron oscillation at frequency ω+ = ωc /2 + (ωc2 /4 − ωz2 /2)1/2 , slightly shifted from the free electrons cyclotron frequency ωc = (e/m)B, and the magnetron motion, a slow drift of the cyclotron orbits around the trap center at frequency ω− = ωc /2 − (ωc2 /4 − ωz2 /2)1/2 . The perturbed cyclotron oscillation is cooled by emission of synchrotron radiation in the strong magnetic ﬁeld. The time constant for radial energy loss is given by τ −1 =

4 e4 2 B . 3c2 m3

(4.10)

4.1 Zeeman Spectroscopy

91

When thermal equilibrium with the 4 K thermal bath is reached, the cyclotron motion can no longer be treated as classical motion, but the quantized nature of cyclotron motion as of a harmonic oscillation has to be considered. An electron at 4 K spends about 80% of its time in the quantum mechanical ground state. The radius of the magnetron motion is reduced by “sideband cooling”: Excitation of the radial motion can be performed at the sum frequency of the perturbed cyclotron and the magnetron oscillations, which is a high-frequency sideband of ω+ . Continuous dissipation of the cyclotron energy by synchrotron radiation then leads to simultaneous cooling of the magnetron motion. The solution of the Schroedinger equation for a charged particle in the Penning trap potential gives for the total energy [166, 167] E = (n + 1/2)ω+ + (m + 1/2)ωz − (l + 1/2)ω− + ms ωL .

(4.11)

In addition to the motional energy, we have included the energy of the magnetic moment associated with the electron spin. The negative sign in front of the magnetron energy indicates that this part of the motion takes place on a radial potential hill and an increase of the magnetron quantum number lowers the total energy. An energy level diagram corresponding to (4.11) is shown in Fig. 4.5. Diﬀerent transitions between the energy levels can be induced: Δn = 1 transitions excite the perturbed cyclotron motion and serve (after a small correction provided by the magnetron frequency) for calibration of the magnetic ﬁeld strength. The transitions having Δms = 1 change the spin direction

Fig. 4.5. Energy level diagram of an electron in a Penning trap (not to scale). The ladder at the left side represents the cyclotron energy levels. Each level is split by the two spin states. The axial harmonic oscillator levels are added as well as those of the magnetron motion. The magnetron levels are inverted since the motion takes place on a harmonic potential hill

92

4 Microwave Spectroscopy

and give the Larmor precession frequency ωL . The g-factor then could be determined from g ωL . (4.12) = 2 ωc From this the g-factor anomaly a = (g − 2)/2 is derived as a=

ωL − ωc . ωc

(4.13)

Since ωL − ωc 10−3 ωc , large errors in a occur when ωL and ωc are measured separately and then subtracted. It is preferable to measure the diﬀerence frequency ωL − ωc = ωa directly. This corresponds to a simultaneous change in the quantum numbers n and ms . It can be induced, as stated above in the discussion of the experiment by Graeﬀ et al., by an rf magnetic ﬁeld gradient. In the experiment of Dehmelt et al., this was provided by current ﬂowing in opposite directions through the guard electrodes of the trap which are split on one side for this purpose. All information on the trapped electron is taken from measurements on the axial tank circuit, either by the noise amplitude or by the electron’s oscillation frequency. Small changes in this frequency are detected by a “frequency shift detector”: The axial frequency is locked to a stable frequency synthesizer and a correction voltage is sent to the trap when the oscillation frequency is changed. This voltage serves as frequency shift signal and fractional changes in the axial oscillation frequency as small as a few parts in 108 could be detected [168]. The observation of the spin direction is made possible by the “continuous Stern–Gerlach eﬀect”: A ferromagnetic wire is wound in the center plane around the ring electrode. It distorts the magnetic ﬁeld in a bottle-shaped manner. Because of rotation and mirror symmetry a series expansion of the ﬁeld strength shows only even exponents: B = B0 + B2 z 2 + · · ·

(4.14)

The inhomogeneous ﬁeld acts upon the spin magnetic moment of the electron: F = −∇(μB).

(4.15)

Depending on the orientation of μ with respect to B this force adds or subtracts to the force of the electric trapping ﬁeld on the charge. It is important to note that the extra force has a square dependence on the coordinates as does the electric quadrupole ﬁeld. Thus the ion motion remains a harmonic oscillation, but at diﬀerent frequencies for spin up and spin down. The diﬀerence depends on the strength of B2 : δωz = gμB

B2 . mωz

(4.16)

4.1 Zeeman Spectroscopy

93

Fig. 4.6. Spin ﬂips recorded by means of the continuous Stern–Gerlach eﬀect. The radom jumps in the base line indicate induced transitions at a rate of about 1/min. The upward spikes reﬂect the random excitations and spontaneous decay of the cyclotron levels by blackbody radiation at 4 K. Reprinted with permission from [169]

Number of spin flips

12 10ppb

8

4

0 72

v’00 76 80 vrf - 163, 918, 900 (Hz)

84

Fig. 4.7. Single-electron anomaly resonance showing the number of induced spin ﬂips per unit time. The line shape reﬂects the Boltzman distribution of the thermally ﬂuctuating energy changing the resonance frequency in the inhomogeneous magnetic bottle ﬁeld. The dashed line indicates the resonance frequency derived from a least squares ﬁt by the proper line shape formula. Its uncertainty is 0.2 Hz. Reprinted with permission from [169]

To make changes in the spin direction easily observable one may want to choose a value of B2 as large as possible. On the other hand, a magneticﬁeld inhomogeneity prevents accurate measurements of the ﬁeld-dependent motional frequency. As compromise between these conﬂicting requirements a value of B2 = 120 T m2 has been chosen. The expected axial frequency change then is δωz /2π = 0.1 Hz, which is what has been observed, as seen in Fig. 4.6. Superimposed are the ﬂuctuations of the cyclotron orbits induced by the thermal background radiation at 4 K. A variation of the microwave frequency around the expected anomaly frequency (ωL − ωc )/2π (ca. 164 MHz at B0 = 5 T) shows a corresponding rate of induced spin ﬂips from which a resonance curve is obtained. The ﬁnal results of the Washington experiments are [169] (g − 2)/2(e− ) = 1 159 652 188.4 (4.3) × 10−12 (4 ppb) (g − 2)/2(e ) = 1 159 652 187.9 (4.3) × 10 +

−12

(4 ppb).

(4.17)

94

4 Microwave Spectroscopy

Fig. 4.8. Cylindrical Penning trap as used in the Harvard g − 2 experiment. Reprinted with permission from [174]

This agrees within 1.3 standard deviations with the theoretical value obtained from (4.3): (g − 2)/2 = 1 159 652 153.5 (28.0) × 10−12 . (4.18) Here a value of the ﬁne structure constant α−1 = 137.036 003 7 (33) from the quantum Hall eﬀect measurements is taken for the evaluation. The number within brackets reﬂects the uncertainty of this measurement [170]. The uncertainty in the experimental value arises nearly exclusively from a potential cavity-mode shift. The cyclotron frequency of the electron is shifted by image charges induced in the trap electrodes, an eﬀect depending on the cavity frequency modes [171, 172]. These are diﬃcult to compute for a hyperbolic shaped trap. Cylindrical cavities are simpler to calculate, and Gabrielse and Tan [173] have proposed to use a compensated open end cap cylindrical trap for improved g − 2 measurements. Figure 4.8 shows the trap as used for improved g-factor measurements. An additional improvement is the reduction of the ambient temperature to below 100 mK. At this temperature, the electron spends almost all of its time in the quantum mechanical ground state of the cyclotron harmonic oscillator. This eliminates completely any uncertainty of the electrons energy. Similar to the Washington experiment, the induced spin ﬂips are detected via a change in the axial frequency due to the coupling of spin and cyclotron motions in a bottle-like inhomogeneous magnetic ﬁeld. Figure 4.9 shows results from the experiment performed at Harvard University. Along with technical improvements in trap material and cryo-electronics a result was obtained [129] which improves the previous measurements by a factor of 18 and leads to (g − 2)/2 = 1 159 652 180 85 (76) × 10−12 (0.66 ppb).

(4.19)

A better measurement and understanding of the electron–cavity interaction removes cavity shifts as a major uncertainty and has led to an even more

4.1 Zeeman Spectroscopy

95

Fig. 4.9. Results from the Harvard g − 2 experiment on the free electron. (a): Shift of the axial frequency upon an induced spin ﬂip; (b) excitation of the cyclotron motion from the quantum mechanical ground state to the ﬁrst excited state; (c) resonance line for the excitation of the anomaly frequency; (d) excitation of the cyclotron frequency. The solid lines in (c) and (d) are ﬁts to calculated line shapes, the bands indicate 68% conﬁdence limits for distributions of measurements about the ﬁt values. Reprinted with permission from [129]

precise value [174]: (g − 2)/2 = 1 159 652 180 73 (28) × 10−12 (0.24 ppb).

(4.20)

This represents the most precise number of any elementary particle. Assuming the theoretical calculation for the g-factor are correct, a value for the ﬁne structure constant α can be derived from a comparison of experimental and theoretical value [174]: α−1 = 137.035 999 084 (51) (0.37 ppb).

(4.21)

4.1.2 g-Factor of the Bound Electron A single electron bound to a nucleus changes its magnetic moment due to the presence of the nucleus. From the solution of the Dirac equation for a point like nucleus of charge Z, Breit [175] obtained an analytical solution for the g-factor: 1 2 2 g=2 + 1 − (Zα) . (4.22) 3 3 In addition to the eﬀect of binding, further changes to the g-factor occur. Most important are bound state quantum electrodynamic (BS-QED) eﬀects. The interaction of the electron with the nucleus is modiﬁed by vertex corrections and vacuum polarisation similar to the case of a free electron. The

96

4 Microwave Spectroscopy

mathematical treatment, however, is much more diﬃcult: While in the free electron case the wave function is given by a simple plane wave, we now have as wave function the solution of the Dirac equation. An additional problem arises from a perturbative treatment of the QED contributions: As in the case of the free electron a series expansion for the diﬀerent orders of the BS-QED eﬀect can be obtained, however, the series now has (Zα) as expansion parameter. Particularly for large values of Z this is no longer 1, the series converges much less rapidly and higher orders will contribute signiﬁcantly. A non-perturbative treatment is required for large Z, and in fact remarkable progress in the theory of the g-factor in hydrogenic systems has been made. The presently quoted level of accuracy is below 10−9 [176]. The calculations include smaller eﬀects on the g-factor from nuclear structure and the ﬁnite mass of the nucleus. Figure 4.10 illustrates the diﬀerent contributions to the g-factor for various values of the nuclear charge Z. Experimentally the determination of the g-factor follows the route given for the g-factor of the free electron: A single hydrogen-like ion of charge qe and mass M is produced by consecutive ionisation of lower charged ions by continuous electron bombardment inside the trap. By resonance circuits attached to the end cap electrodes and two segments of the split ring electrode of a Penning trap it is cooled to the ambient temperature of 4 K if the ion’s oscillation frequencies in the radial and axial directions are kept equal to the resonance frequencies of the circuit. The g-factor of the bound electron is obtained from the ratio of the spin precession frequency ωL and the cyclotron frequency ωc of the ion: relativistic L-S-coupling

Contribution to the g-factor

10–1 BS-QED corrections 1-st order in α

10–2 –3

10

10–4

nucleus volume nuclear recoil

10–5 10–6 10–7

BS-QED corrections 2-nd order in α

10–8 10–9

10–10 10–11 0

10

20

30 40 50 60 70 80 Nuclear charge number Z

90 100

Fig. 4.10. Contributions to the electron g-factor in hydrogen-like ions for diﬀerent values of the nuclear charge Z

4.1 Zeeman Spectroscopy

97

Fig. 4.11. Fourier transform of the induced noise in the ring electrode of a Penning trap from a single O7+ ion at the perturbed cyclotron frequency in a B-ﬁeld of 3.7 T. The statistical uncertainty of the center frequency is about 1 part in 1010

Fig. 4.12. Fourier transform of the noise induced in the Penning trap end cap electrodes by the axial oscillation of a single O7+ ion in temperature equilibrium with the 4 K environment

g=2

ωL q m . ωc e M

(4.23)

The actual experiments take place in a cylindrical trap. Correction voltages applied to guard electrodes between ring and end caps serve to make the trapping potential harmonic in the small volume which the oscillating ion occupies. The cyclotron frequency as well as the axial frequency are measured by a Fourier transform of the noise induced in the ring and end cap electrodes, respectively (Figs. 4.11 and 4.12). A microwave-induced change in the spin direction is observed, as in the free electron case, by the “continuous Stern-Gerlach eﬀect”: The ring electrode of the trap is made of ferromagnetic nickel, distorting the B-ﬁeld in a bottle like manner. The additional force of the inhomogeneous B-ﬁeld

98

4 Microwave Spectroscopy

Fig. 4.13. Axial frequency change upon a spin ﬂip for various hydrogen-like ions. Parameters: ωz /2π = 360 kHz, B2 = 8.2 T m−2

Fig. 4.14. Change in the axial oscillation frequency of a single O7+ ion when spin ﬂips are induced

changes the axial oscillation frequency by a small amount given by (4.24), which can be detected. The amount of frequency change depends on the size of B2 and the mass of the hydrogenic ion (Fig. 4.13). δωz = gμB

B2 . M ωz

(4.24)

In the case of O7+ and for the experimentally realized parameters ωz /2π = 360 kHz and B2 = 8.2 T m−2 we have δωz /2π = 0.46 Hz. To detect such a small frequency jump and to distinguish it from accidental frequency ﬂuctuations is an experimental challenge requiring extremely stable operating conditions. Figure 4.14 shows that the two-spin directions can be clearly distinguished. An important improvement compared to the experiments on the free electron has been introduced by the use of a double Penning trap: In addition to the trap having a ferromagnetic ring to produce the magnetic-ﬁeld inhomogeneity required for detecting spin ﬂips a second trap has been placed a few centimeters apart from the ﬁrst trap (Fig. 4.15). This second trap is identical in geometry with the ﬁrst one, however the ring electrode has the same material (Cu) as the remaining electrodes. Thus, the magnetic ﬁeld is not

4.1 Zeeman Spectroscopy

99

Fig. 4.15. Double Penning trap arrangement for g-factor measurements on highly charged ions. The upper trap with inhomogeneous B-ﬁeld is called “analysis trap”, the lower one “precision trap”

Fig. 4.16. Induced spin ﬂip probability at diﬀerent frequencies of a microwave ﬁeld. The frequencies are divided by the simultaneously measured cyclotron frequency. This eliminates the inﬂuence of magnetic-ﬁeld ﬂuctuations

distorted in this trap. This allows separating in space the regions where the spin direction is determined requiring a strong inhomogeneous B-ﬁeld, from the region where spin ﬂips are induced and the eigenfrequencies are measured requiring a homogeneous ﬁeld. The sequence of a spin ﬂip measurement thus starts with a determination of the spin direction in the “analysis trap.” Then the ion is transferred to the (“precision trap”) by a slow change of the voltages to the electrodes. After irradiation with a microwave ﬁeld (νMW ), near the Larmor precession frequency, the ion is transferred back to the analysis trap. By a measurement of the axial oscillation frequency it is determined whether a spin ﬂip has occurred in the precision trap or not. The spin ﬂip rate shows a maximum at the Larmor frequency (Fig. 4.16).

100

4 Microwave Spectroscopy

A second improvement is that the induced noise of the oscillating ion which serves for determination of the eigenfrequencies for accurate B-ﬁeld calibration is taken at the same time as the spin ﬂips are induced. This reduces the inﬂuence of magnetic ﬁeld drifts and ﬂuctuations on the resonance line shape. They occur in typical superconducting magnets at rates of about 10−8 /h. Results have been obtained for hydrogen-like carbon 12 C5+ and oxygen 16 7+ O [124, 125]. Both results agree with the BS-QED calculations on the level of a few parts in 1010 . Test of BS-QED would be even more stringent when working with hydrogen-like ions of higher nuclear charge since the BS-QED contribution scales approximately with Z 2 . However, as evident from Fig. 4.13, the change in the axial frequency upon a spin ﬂip is reduced for high-Z ions. This represents a serious experimental challenge which may be overcome by detecting the phase change Δϕ(t) = Δω t associated with a frequency change Δω in the axial oscillation (Fig. 4.17). The phase diﬀerence increases linear in time (Fig. 4.18) and thus small frequency diﬀerences can be detected after a sufﬁciently long waiting period provided no phase ﬂuctuations occur, e.g. by instabilities of electric or magnetic ﬁelds. We may note that the possibility of improving the accuracy of the measurement by inducing combined Larmor- and cyclotron-transitions at the

Fig. 4.17. Phase diﬀerence of the axial oscillations for the two spin directions of an ion when an inhomogeneous B-ﬁeld is superimposed

Fig. 4.18. Experimentally observed phase diﬀerence in the axial oscillation for the two electron spin directions of a single trapped ion in an inhomogeneous magnetic ﬁeld [177]

4.1 Zeeman Spectroscopy

101

diﬀerence (g − 2) frequency as in the case of the free electron is not available for heavy ions, since the Larmor frequency ωL = g(e/2m)B diﬀers from the cyclotron frequency ωc = (Q/M )B by several orders of magnitude due to the charge state q and the mass ration m/M of electron and ion. 4.1.3 Atomic g-Factor In atomic ions, the electrons have both spin S and orbital momentum L and their quantum state is characterized by the total angular momentum J = L + S. From nonrelativistic quantum theory of atoms in a magnetic ﬁeld, it follows that the g-factor can be calculated from the Land´ e-formula g = 1+

J(J + 1) − L(L + 1) + S(S + 1) , 2J(J + 1)

(4.25)

which leads to g = 2 for pure spin states. For relativistic wave functions diﬀerent corrections to the g-factor appear. For light atoms or ions relativistic eﬀects dominate while for heavy systems many body eﬀects mainly contribute [178]. Thus, a measurement of the g-factor and a comparison to theoretical values serves as a test of relativistic wave functions. To determine experimentally the value of a magnetic moment, it is necessary to measure the energy diﬀerence ΔE between magnetic sublevels of the system under investigation in a known magnetic ﬁeld B: ΔEJ = gμB BΔmJ .

(4.26)

Here mJ is the orientation quantum number of the angular momentum J. A typical experimental setup is shown in Fig. 4.19: The Penning trap is located at the center of a superconducting solenoid. Ions are injected along the magnetic-ﬁeld lines and stopped by buﬀer gas collisions inside the trap. Alternatively, they are produced by surface ionisation from a ﬁlament placed at one of the end cap electrodes. A laser beam enters the trap through holes in the ring electrode and excites the ions. Excitation is monitored by ﬂuorescence detection perpendicular to the laser beam through an end cap electrode which is made from a mesh. In a magnetic ﬁeld of a few Tesla the Zeeman splitting of the ground state is of the order of many gigahertz, larger than the bandwidth of a laser and the ﬁrst-order Doppler width of the optical transition. This allows selective excitation of an individual Zeeman sublevel in the ground state. This state will then be depleted after a short time by optical pumping, indicated by the vanishing of the ﬂuorescence. When a transition to another Zeeman level is induced by microwaves, blown into the trap through a microwave guide (Fig. 4.19), the population of the depleted state increases again. The ﬂuorescence intensity shows a maximum at the resonant Zeeman transition frequency. The induced transitions show no ﬁrst-order Doppler eﬀect which might broaden or shift the transition frequency. This is because of the Dicke-eﬀect

102

4 Microwave Spectroscopy

Fig. 4.19. Setup for Zeeman spectroscopy in a Penning trap

Fig. 4.20. Induced transition between Zeeman levels in the ground state of 138 Ba+ in a ﬁeld of 2.9 T. An unshifted narrow carrier and symmetric sideband at the frequencies nωc /(2π), n = 1, 2, 3 are observed [180]

[179]: When the amplitude of oscillation is smaller than the wavelength of the radiation the ion oscillation leads to an unshifted and unbroadened carrier at the frequency of the ion at rest and sidebands at the oscillation frequencies. This criterion is in general easily fulﬁlled in Zeeman spectroscopy: Transition wavelength is typically between 1 and 100 cm, larger than the ion oscillation amplitude of at most a few millimeters even for uncooled ions. The sidebands are in general well resolved as shown in Fig. 4.20. The linewidth of the carrier transitions are limited in resolution by residual B-ﬁeld inhomogeneities and

4.1 Zeeman Spectroscopy

103

Fig. 4.21. High-resolution microwave scan of the carrier transition in the ms = −1/2 → ms = −1/2 transition in the ground state of 138 Ba+ [180] Table 4.1. Results of g electronic g-factor measurements in Penning traps Ion +

Be Mg+ Ca+ Ca+ Ba+ Ba+ Hg+ Hg+ H+ 2

State

g-Factor

2S1/2 3S1/2 4S1/2 3D5/2 6S1/2 5D3/2 6S1/2 6P1/2 1Σ1/2

2.002 262 63 (33) 2.002 254 (3) 2.002 256 64 (9) 1.200 334 0 (3) 2.002 491 92 (3) 0.799 327 8 (3) 2.003 174 5 (74) 0.6652 (20) 2.002 283 7 (18)

Reference [194] [548] [549] [564] [180] [550] [551] [551] [552]

by ﬂuctuations of the ﬁeld during the period of measurement. The fractional uncertainty of the center frequency is generally below 10−8 (Fig. 4.21) which at present is below the uncertainty of theoretical calculations. Table 4.1 shows results of electronic g-factor measurements on ions performed in Penning traps. 4.1.4 Nuclear gI -Factor Similar to the case of the electron, the nuclear gI -factor is deﬁned by μI = gI μN I,

(4.27)

where I is the spin of the nucleus and μN = e/(2M ) the nuclear magneton with M the proton mass. The nuclear magnetic dipole moment is composed of the spin and orbital moments of the constituent particles and values of μI

104

4 Microwave Spectroscopy

are predicted based on nuclear models. Thus, measurements of gI provides information on the validity of nuclear models. Numerous values of nuclear magnetic moments have been obtained by radiofrequency (rf) spectroscopy on atomic beams [181]. The measured values, however, have to be corrected since the external magnetic ﬁeld B0 induces a diamagnetic current in the electron cloud around the nucleus. It leads to an induced magnetic ﬁeld which opposes the applied ﬁeld and reduces the ﬁeld strength B seen by the nucleus. This can be taken into account by a shielding factor σ: B = B0 (1 − σ) .

(4.28)

Shielding factors have been calculated by diﬀerent approximate methods. They range from 10−5 for light atoms to about 3 × 10−2 at the end of the periodic system of elements [181]. The results of diﬀerent calculations, however, diﬀer sometimes substantially [182]. Because of the proportionality of B and B0 one cannot determine σ by variation of B0 . A comparison of experimental values for μI determined in neutral atoms and in ions may shed some light on the validity of the calculations. Experimental determinations of nuclear dipole moments using ion traps have been performed in a few cases. The method is similar to the case of the electron moment: Laser excitation depletes a Zeeman substate of the electronic ground state by optical pumping and an induced transition between diﬀerent Zeeman states is monitored by the change in ﬂuorescence intensity. Because of the small energy spacing between nuclear Zeeman states it is in general not possible to separate out an individual substate. Then a triple resonance detection scheme may be used as shown in Fig. 4.22 for the case of an S1/2 electronic state and gI = 3/2.

Fig. 4.22. Pumping scheme for triple resonance experiments to determine nuclear dipole moments. A broad band laser depletes the mJ = +1/2 manifold of nuclear Zeeman substates. A ΔmJ = 0 microwave transition depletes one of the nuclear Zeeman states in the mJ = −1/2 manifold and a radiofrequency (rf) ﬁeld induces ΔmI = ±1 transitions between adjacent nuclear Zeeman states

4.2 Hyperﬁne Structures in the Ground States

105

Table 4.2. Results of nuclear g-factor of ions measured in Penning traps. Results from neutral atom experiments are given for comparison [183] Isotope 25

+

Mg Ba+ 151 Eu+ 137

gI (ion) −6

3.419 804 (27) (8 · 10 ) 0.623 876 (3) (4.8 · 10−6 ) 1.377 34 (6) (4.4 · 10−5 )

Reference

gI (atom)

[193] [180] [564]

3.4218 (3) (1 · 10−4 ) 0.6249 (1) (2 · 10−5 ) 1.388 7 (3) (2 · 10−4 )

The energy W of a particular Zeeman substate for arbitrary values of nuclear spin I and electron angular momentum J can be calculated using perturbation theory. For the case that either J or I is 1/2, it is analytically given by the Breit–Rabi formula. In the weak ﬁeld approximation, where J and I couple to F = J + I, it reads 1 ΔE0 ΔE0 4mF W (F = I ± , mF ) = − + mF gI μN B ± x + x2 , 1+ 2 2(2I + 1) 2 2I + 1 gJ μB − gI μN B. (4.29) x= ΔE0 Here ΔE0 is the hyperﬁne splitting energy at zero magnetic ﬁeld. The sign before the third term in the Breit–Rabi equation is positive for the larger of the two values of F , and negative for the smaller one. The transition energy depends on both electronic and nuclear g-factors. Since the magnetic moment of the electron is about three orders-of-magnitudes larger than the nuclear dipole moment the electron part dominates and the accuracy of the nuclear g-factors is signiﬁcantly lower than that for the electron. A few results have been obtained for alkali-like ions (Table 4.2). They are by about 1 order-of-magnitude more accurate than the corresponding values from atomic beam experiments, demonstrating the potential for high-accuracy measurements in Penning traps.

4.2 Hyperﬁne Structures in the Ground States 4.2.1 Summary of HFS Theory The hyperﬁne energy of an atomic system including both the nuclear magnetic dipole and electric quadrupole interactions, deﬁned over the {J, I, F, mF } eigenstates, where F = I + J is the total angular momentum, is given by [183] 3 K(K + 1) − 2I(I − 1)J(J + 1) 1 WF = hAK + hB 2 , (4.30) 2 2I(2I − 1)2J(2J − 1) where K = F (F + 1) − I(I + 1) − J(J + 1). The ﬁrst term is the magnetic dipole contribution, while the second is the electric quadrupole one, which is

106

4 Microwave Spectroscopy

nonzero only if I, J ≥ 1. For an s-electron, such as in alkali-like ions, only the contact magnetic dipole interaction is nonzero and the magnetic dipole constant A is given by the Fermi-Segr´e formula As = −

1 16π μ0 2 μ gI | Ψs (0) |2 , h 3 4π B

(4.31)

whereas, if the electron has orbital angular momentum l > 0 there is a contribution to A given by Al = −

1 μ0 2 2l(l + 1) μ gI r−3 nl , h 4π B J(J + 1)

(4.32)

where r−3 nl is the average taken over the electron state (n, l). The electric quadrupole interaction constant B is given by B=

1 e2 2J − 1 < r−3 >nl Q. h 4πε0 2J + 2

(4.33)

These results are based on a nonrelativistic central-ﬁeld model with the closed shells not contributing to the above interactions. That is, the hyperﬁne structure is due solely to the one valence electron. A more realistic model must include an interaction of the outer electron with the core called exchange polarization, which will aﬀect both the dipole and quadrupole coupling constants. Another correction to the central-ﬁeld model is made necessary by the correlation between the motions of diﬀerent electrons leading to mutual polarization of electrons in diﬀerent shells. In applying these formulas, we note that the magnetic dipole interaction constant As for an s-electron is derived on the basis of a point nucleus, and that a realistic model requires a correction for the ﬁnite distribution of nuclear magnetic moment. However in a ﬁrst approximation As is proportional to gI and hence the ratio of the dipole interaction constants of two isotopes would be proportional to the ratio of their nuclear g-factors, as ﬁrst recognized by Fermi. The extent that the ratio departs from this rule is called a hyperﬁne anomaly Δ12 deﬁned as A1 gI2 Δ12 = − 1, (4.34) A2 gI1 where the subscripts 1, 2 refer to two isotopes of the same element. It is expected that Δ12 is a small quantity. It reﬂects the spatial distribution of magnetization in the nucleus on the hyperﬁne interaction called the BohrWeisskopf eﬀect. It is by precise measurement of the hyperﬁne anomaly for ions of a series of isotopes of the same element such as Hg+ or Eu+ ions that sensitive data are obtained on nuclear wavefunctions, particularly those of neutrons, which are not easily obtainable from other data. Such neutron wavefunctions are useful in certain studies in parity nonconservation [184,185].

4.2 Hyperﬁne Structures in the Ground States

107

4.2.2 Early Experiments From the time Paul’s group ﬁrst published its work on the RF-ﬁeld conﬁnement of ions, it was recognized that if ions could be conﬁned free of perturbation for extended periods of time, unprecedented degrees of spectral resolution would be possible on weak quantum transitions, such as the magnetic dipole hyperﬁne transitions in atomic systems. This was expected on the basis of two arguments: First, unlike optical electric dipole transitions, magnetic dipole (and electric quadrupole) transitions have extremely long mean lifetime against spontaneous emission, and therefore in the absence of relaxation due to collisions, the resonance line widths attainable are determined by the length of time the system is free to be observed without perturbation. Second, for transitions in the microwave region of the spectrum, for which the wavelength is in the centimeter range, the ions may be conﬁned in a space small compared with the wavelength, with the result that the ﬁrstorder Doppler eﬀect gives a Dicke pattern consisting of a central undisplaced line, broadened by only the second-order Doppler eﬀect, with resolved sidebands rapidly falling in amplitude away from the center. With the advent of the technique of laser cooling of ions, even the second-order Doppler broadening of the central line can be radically reduced. This realization shows the remarkable synergy between lasers and ion traps; indeed one can now observe with exquisite resolution the spectrum of a single isolated ion having zero point energy. The extraordinary degree of isolation in a trap can be exploited for hyperﬁne spectroscopy, however, only if methods can be devised that manipulate the populations of the hyperﬁne states of the stored ions, and monitor transitions between those states. This requirement is a consequence of the fundamental quantum fact that the transition probabilities for absorption and stimulated emission are equal, and that for all but the lowest temperatures the hyperﬁne states have almost identical populations. The application of the Paul ion trap to rf spectroscopy, rather than mass spectrometry, was ﬁrst conceived in the mid 1950s, soon after Paul’s original publication. It predated the invention of the laser and relied on specialized techniques: spin-dependent collision processes, involving the ions and spinoriented atoms, or polarized photons to achieve unequal populations among the Zeeman sublevels of the ground state. The ﬁrst experiment was designed to demonstrate magnetic resonance on the ground state of 4 He+ [9], leading to the measurement of the hyperﬁne splitting in 3 He+ . The method developed to produce spin polarization was to use spin exchange collisions with spin polarized Cs atoms that passed through the Paul ion trap in the form of a beam. At the time, spin exchange between free hydrogenic atomic systems had become familiar and applied to a variety of atomic species. The collision process can be represented as A(↑) + B(↓) −→ A(↓) + B(↑) ,

(4.35)

108

4 Microwave Spectroscopy

where the arrows represent the spin directions of the single outer electrons in A and B. The physical origin of this reaction is not the direct magnetic interaction between the two electrons but rather the quantum antisymmetry requirement of the total two-electron wavefunction (including spin) during the collision. This requires that the initial two-electron spin state during the collision be expressed as an equal mixture of singlet S = 0 and triplet S = 1 states; but the total wavefunction must be antisymmetric, hence the coordinate parts of the wavefunction must have opposite symmetry to the spin parts. This results in the scattered amplitudes of the two spin states being diﬀerent, and the appearance of a spin ﬂip depending on the extent of the diﬀerence. However, the challenge was to ﬁnd a way to detect transitions between the magnetic sublevels when a resonant RF or microwave ﬁeld is applied to observe the spectrum. Direct optical detection is totally impractical since the ﬁrst resonance line (corresponding to the ﬁrst Lyman line in hydrogen atoms) has a wavelength around 30.4 nm! There are two observables that may possibly indicate the magnetic polarization state of the ion, namely, ion kinetic energy and ion number, since they both may be aﬀected in a spin-dependent way if the cross sections for inelastic collisions or charge transfer collisions between the ions and Cs atoms are spin dependent. The cross section for charge transfer in which the He+ is neutralized depends very strongly on the energy defect in the reaction, and hence the spin state of the He+ , thus He+ (↑) + Cs(↑) −→ He∗ (↑↑) + ΔE3 ,

(4.36)

∗

He (↑) + Cs(↓) −→ He (↑↓ + ↓↑) + ΔE3 , +

He∗ (↑↓ − ↓↑) + ΔE1 , and it happens that ΔE1 < ΔE3 . It follows that the spin polarization of the Cs atoms will result in the more rapid neutralization of He+ ions when transitions are induced in them to the opposite spin direction. This method of detection was shown to work as theoretically predicted in observing magnetic resonance on the ground state of 4 He+ ions, and subsequently several transitions involving the magnetic hyperﬁne states of 3 He+ were measured and the hyperﬁne interval found to be Δν = 8,665,649,867 ± 10 Hz [186]. The other pioneering spectroscopic experiment in a Paul ion trap [64] was directed at the hyperﬁne spectrum of the molecular hydrogen ion H+ 2 . In this case the molecular ion was aligned by selective photodissociation. A powerful beam of UV radiation was linearly polarized and passed through a Paul trap containing H+ 2 ions. The rate of the reaction + hν + H+ 2 −→ H + H + ΔEk

(4.37)

is proportional to cos2 (E , R), where E and R are vectors referring to the electric light vector and the molecular axis, respectively. This follows from the fact that the dissociation results from an electric dipole transition between the ground state and the ﬁrst (repulsive) excited state of the molecular ion.

4.2 Hyperﬁne Structures in the Ground States

109

The angular momentum of the molecule F is made up of the rotation of the two protons K, their total spin I and the electron spin S. To compute the dissociation rate for each quantum state therefore requires the evaluation of the quantum mechanical average of sin2 (K , E ). This can be written as R = A K, F2 , F, mF | K 2 − Kz2 | K, F2 , F, mF ,

(4.38)

where F2 = I + S and I = I1 + I2 is the total spin of the protons. The use of polarized UV alone produces only alignment of angular momentum with populations in ±mF states remaining equal. By saturating transitions between pairs of states, the dissociation rate is measurably aﬀected, leading to an observable resonance spectrum. An important general method of observing hyperﬁne transitions in stored ions was ﬁrst demonstrated in the optical hyperﬁne pumping of 199 Hg+ [187]. The motivation was to exploit the extraordinary potential for realizing an ultrastable frequency standard in a light compact device suitable for aerospace applications. It was predicted that a microwave resonance of unprecedented spectral resolution was possible at around 40.5 GHz in a perturbation-free environment for extended periods of time. Further, the relatively large mass of the mercury ion has the advantage of smaller (second-order) Doppler broadening for a given energy spread. The frequency is at the high end of the microwave range implying for a given line width a high Q, yet falls in a range conveniently reached by common frequency synthesis techniques. Finally, the short wavelength of the resonant microwave ﬁeld (7.4 mm) permits the physical size of the microwave components to be correspondingly small. For all these reasons it was argued that 199 Hg+ is particularly suitable for a spacecraft clock. The mercury atom and ion are obviously complicated systems with, respectively, 80 and 79 electrons around the nucleus. The ion in its ground state has one unpaired 6s electron around closed shells; hence the ground state is spectroscopically given as a 2 S1/2 , the same as an alkali such as Cs. In such an S−state an electron has a ﬁnite probability of being at the nucleus and therefore those isotopes of mercury that have a nuclear moment (odd mass number) will have a magnetic hyperﬁne structure. Fortunately, 199 Hg+ has a nuclear spin of only I = 1/2, and therefore the hyperﬁne structure is simple, with F = 1 and F = 0. The hyperﬁne structure in the ground state as a function of an external magnetic ﬁeld is shown in Fig. 4.23. To produce unequal populations between these states and monitor transitions between them, the technique of optical hyperﬁne pumping was used. This was known from the classic work of Beatty et al. [188] on the Rb frequency standard, still very much in use today. Since this early work precedes the coming of the laser, the technique depended on the availability of a monochromatic light source of the only type then available, namely resonance lamps. In the case of the Rb clock hyperﬁne optical pumping was made possible by a fortuitous matching of wavelengths (due to the isotope shift) between an emission line in the spectrum of the isotopes 85 Rb and a hyperﬁne line in the ground state

110

4 Microwave Spectroscopy

Fig. 4.23. The ground state HFS in energy levels in the mass 199 mercury ion, showing the dependence on an external magnetic ﬁeld B

Fig. 4.24. (a) Relevant energy levels and (b) structure of resonance emission line at 194.2 nm of the two mercury ion isotopes

of 87 Rb. Today of course, lasers are available for optically pumping Rb. In the case of 199 Hg+ an interpolation of the limited data on the UV spectrum of the ions of Hg isotopes showed that a hyperﬁne emission line in the spectrum of 202 Hg+ nearly matches a hyperﬁne component in the 199 Hg+ spectrum. Thus, a crucial experimental component was a UHF driven lamp containing enriched 202 Hg vapor operating in a vacuum arc mode to enhance the emission of the UV ion resonance line at 194.2 nm. Figure 4.24 shows the coincidence of one of the hyperﬁne components of the 194.2 nm line in the spectrum of mass 199 mercury ions with the single line from the mass 202 mercury ion. We see that radiation from the 202 Hg lamp induces only the transitions originating from the (F = 1) hyperﬁne level and not (F = 0), with the result that in the absence of relaxation all the ions will be pumped into the nonabsorbing (F = 0) level, and ﬂuorescence would eventually cease. If a microwave ﬁeld with the appropriate polarization and resonant frequency induces transitions between the (F = 0, mF = 0) and (F = 1, mF = ±1, 0) sublevels, this is indicated by the reappearence of ﬂuorescence. Unfortunately, the light from a conventional UHF lamp, in contrast with a laser source, has limited brightness, and a diﬀuse cloud of ions in the trap at normal temperatures produces an extremely weak ﬂuorescence signal, making

4.2 Hyperﬁne Structures in the Ground States

111

the initial experiments monstrously diﬃcult, particularly since the hyperﬁne frequency was only approximately deduced from UV spectroscopic data [189]. The experimental diﬃculties are aggravated by the strong emission of the neutral mercury line at 253.7 nm, necessitating a narrow band ﬁlter; but the greatest challenge was to discriminate against stray light scattered from diﬀerent parts of the apparatus. In this regard it is fortunate that the ﬂuorescence is in the UV part of the spectrum, enabling the use of a solar blind photomultiplier detector, that is insensitive to ambient room lights and cathode glow from the electron source. To estimate the expected ﬂuorescence signal intensity, assume the ions are irradiated by a parallel beam of resonant UV light of intensity Iν so that jν = Iν /hν is the spectral ﬂux density of photons, then the probability per unit time of an ion absorbing a photon can be estimated from the following useful result: 1 λ2 jν Δνn . = (4.39) τp 4 The form of this result suggests that λ2 /4 is the cross section that an ion presents to the beam in its resonant response, a cross section that curiously reﬂects the wavelength of the radiation rather than the physical size of the ion. Resonance scattering can have a far greater cross section as a means of probing the state of other particles than for example spin exchange. The ability to project the output of the lamp into the trapping region is fundamentally limited because the lamp output comes from an extended source, and any attempt to reduce the beam diameter by some form of optics will inevitably increase the angular divergence of the beam. This is a consequence of the brightness theorem. To show the severity of the problem of stray light, assume N ions contribute to the ﬂuorescence signal so that N/τp is the average number of photons radiated per second into a 4π solid angle, so that if Ω is the acceptance solid angle of the optics, then the number of photons reaching the detector would be (Ω/4π)(N/τp ). An important experimental quantity is the fraction of the total ﬂuorescence photons actually reaching the detector. If the spectral width of the 194.2 nm line from the lamp is ΔνL and A the cross sectional area of the beam, then Id λ2 Δνn Ω . =N I0 4A ΔνL 4π

(4.40)

If the typical numerical values N = 106 , Δνn /ΔνL = 0.1, Ω/(4π) = 0.01 are substituted, one ﬁnds an extremely small ratio, on the order of 10−7 , showing the extent to which stray light must be suppressed if its accompanying noise is not to swamp the signal. The original apparatus is shown in Fig. 4.25. Since the ions are conﬁned in a small space compared with the wavelength of the microwaves, and the mean center of the ion population is ﬁxed, the microwave ﬁeld inducing resonant transitions can be applied in the form of a running wave without introducing a ﬁrst-order Doppler shift. Thus a microwave horn was used to propagate the microwaves through the glass vacuum

112

4 Microwave Spectroscopy

Fig. 4.25. The original NASA

199

Hg+ ion microwave resonance apparatus [190]

envelope of the ion trap. The source was a klystron whose output frequency is phase locked by a commercial frequency synchronizer to a 180 MHz reference. This frequency is obtained through multiplication by six the output of a 30 MHz frequency synthesizer, referenced to a Cs frequency standard. The ﬁeld independent (F = 0, mF = 0) −→ (F = 1, mF = 0) transition frequency was observed at ν = 40.507348 GHz with a rather poor signal to noise ratio, however a line width of only 3 Hz was demonstrated corresponding to an unprecedented line-Q greater than 1010 ! There are ﬁve principal corrections that must be applied to the observed resonance frequencies: 1. Magnetic Field: From the Breit–Rabi formula we have for the 199 Hg+ ion: ν0↔0 = ν0 + 9.7B 2 , and νm=±1 = ν0 ±1.4 × 1010B + 9.7B 2 in the SI system of units. The frequencies of the ﬁeld-dependent transitions are measured to provide the intensity of the magnetic ﬁeld at the position of the ions. This dictates that the environment of the ions do not have stray magnetic noise or ﬁeld anomalies that can broaden those transitions. In the case of the 199 Hg+ ion the hyperﬁne frequency is so much higher than other ions, that greater magnetic perturbation can be tolerated. 2. Doppler Eﬀect: The ﬁrst-order eﬀect should be totally absent unless there is a curious asymmetry in the direction in which ions escape from the trap. The second-order eﬀect is simply Emean /M c2 , and for Hg+ ions assuming Emean 1 eV the fractional shift would be about 5 × 10−12 . 3. Light Shift: The possibility of light shift dictates that resonance be observed “in the dark,” that is, the pumping light must be interrupted during the microwave probing period. 4. Pressure Shift: Collisions with buﬀer He atoms, if used to cool the ions, could give rise to a slight blue shift in the frequency, depending on the pressure, easily kept below 10−2 Hz. 5. Stark Shift: A small shift is expected due to the trapping electric ﬁelds, however based on available data on neutral atoms of comparable size and

4.2 Hyperﬁne Structures in the Ground States

113

term structure, probably could be kept below 10−1 Hz. Experimentally, since the amplitude of the trapping ﬁeld aﬀects the ion energy distribution, the separation of the Stark eﬀect from the second-order Doppler eﬀect would require detailed measurement and analysis. 4.2.3 Laser Microwave Double Resonance Spectroscopy The advent of the laser and the advancement of optical frequency synthesis techniques have completely transformed the whole aspect of microwave spectroscopy of ions in Paul traps. In one stroke the original drawbacks to the application of ion traps to spectroscopy, namely the poor signal to noise ratio of the ﬂuorescence and the relatively high temperature of the conﬁned ions, were so completely removed by the laser that a radically higher level of precision and accuracy became the norm. The laser optical-microwave double resonance technique is broadly applicable to ions for which a suitable laser source is practicable. Unfortunately, the wavelength of resonance radiation of ions tends to be in the near ultraviolet region of the spectrum and beyond: for example even the heavy alkaline earth ion Ba+ has λres = 493.4 nm, while the lighter element Mg+ has λres = 279.6 nm, and as we saw Hg+ has λres = 194.2 nm. In the hyperﬁne spectroscopy of the various ionic species, the design of experiments diﬀers mainly in the laser sources available to produce the required sharp wavelengths. Since laser optics and technology continue to advance relentlessly, experimental designs have evolved rapidly and future designs may very well bear little resemblance to what has been published in the past. Currently, there is far greater reliance on solid state lasers than liquid dye or gas lasers; but of course the principles of optical-microwave double resonance remain the same. One of the early applications of laser sources to hyperﬁne optical pumping of stored ions was carried out by Blatt and Werth on 137 Ba+ [191]. In this case a dye is available to operate a dye laser at the resonance wavelength 493 nm (Fig. 4.26): Ba+ is one of the few interesting ions whose resonance wavelength can be reached directly with a dye laser. In the cited reference the laser source was a pulsed, pressure-tuned dye laser; the ion trap a classic

Fig. 4.26. Partial energy level diagram of Ba+ showing the optical pumping transitions and the two-channel branching of the emission from the P-state

114

4 Microwave Spectroscopy

hyperboloid quadrupole with r0 = 20 mm, operated at Ω/(2π) = 490 kHz with an amplitude of V0 = 756 V and U0 = 8 V. The 137 Ba+ ion source was a heated Pt ﬁlament with a thin coating of 137-isotope enriched Ba metal. It was estimated that this source yielded about 105 trapped ions in a few seconds. The hyperﬁne optical pumping technique clearly requires ﬁrst that the (ﬁrst-order) Doppler broadening of the spectral width of the ion absorption be smaller than the known 8 GHz hyperﬁne splitting of the 137 Ba+ ground state, and second that the spectral width of the laser light meet the same requirement. Furthermore, as evident from Fig. 4.26 ions in the excited 2 P1/2 state radiate along two channels, one of which leads to the 52 D3/2 state, which is metastable because an electric dipole transition to the ground state is forbidden. The radiative lifetime of the metastable D-state is so long that the ions making a transition to it would no longer participate in producing a signal. To ensure that this state is quenched, a light buﬀer gas, He, was used at a pressure of about 10−5 Pa; this had the additional salutary eﬀect of moderating the temperature of the ions. The collisions with He are expected to blue shift the hyperﬁne frequency, but only a negligible fractional change (in the present context) of less than 10−15 [192]. The transition of the D-state to the ground state is a weak electric quadrupole transition, whose potential use as an optical frequency standard transition will be discussed in Chap. 5. The source of the microwaves to induce the hyperﬁne transitions was similar in principle to the klystron system described above for the 199 Hg+ experiment, but introduces the microwaves into the trap by a hairpin antenna between one end cap and the cylinder. Resonance was monitored by the intensity of ﬂuorescent emission arising from the transition down to the D-state at λ = 649 nm, thereby allowing ﬁlters to totally eliminate stray pumping light from reaching the detector. This removed the stray light suppression problem that plagued the original 199 Hg+ experiment. Since the applied microwave ﬁeld did not have a well-deﬁned polarization, all transitions allowed by the selection rules ΔF = ±1, ΔmF = 0, ±1 are reported to have been observed. The line shape of the microwave transition follows from the fact that the ions interact freely with the microwave ﬁeld in the time T between two consecutive laser pulses while the short and intense laser pulse destroys coherence. For constant microwave amplitude the line shape is then given by

γ2 T 2 2 2 1/2 P = (ω − ω0 ) + γ , sin (4.41) (ω − ω0 )2 + γ 2 2 where γ is proportional of the microwave amplitude. For small γ the full width at half maximum is given by 1/T , the laser repetition frequency. If the ion cloud size is larger than the laser beam diameter some ions will escape the laser pulse which extend their coherence time to multiples of T . Accordingly the transition probability is given by

4.2 Hyperﬁne Structures in the Ground States

115

Fig. 4.27. Line shape of the “0 ↔ 0” (F = 1, mF = 0) −→ (F = 0, mF = 0) transition in 137 Ba+ observed with pulsed laser excitation at regular intervals. The full line is a ﬁt to the date according to (4.42) [191]

P =

n

kT γ2 2 2 2 1/2 (ω − ω . a sin ) + γ k 0 (ω − ω0 )2 + γ 2 2

(4.42)

k=1

The coeﬃcients ak are determined purely by geometry. A least squares ﬁt of the observed data to (4.42) (Fig. 4.27) contains information of the ion cloud size when the laser beam diameter is known. Alternatively, the Ramsey method of separated oscillatory ﬁeld can be applied. The exciting laser intensity is chosen to obtain 50% optical transition probability in a pulse of length τ (π-pulse). A second pulse of same amplitude and length, coherent in phase, is applied after a time T . The line shape is given by 2 16b2 Ωτ δωT Ωτ δω δωT Ωτ P = 2 sin2 cos cos − sin sin . (4.43) Ω 2 2 2 Ω 2 2 The constant b is the Rabi frequency, proportional to the microwave amplitude, δω = ω − ω0 is the detuning from the resonance frequency ω0 , and Ω2 = (δω)2 + b2 . For T τ this can be approximated by P =

ΩT 16b2 . sin2 2 Ω 2

(4.44)

This method has been successfully applied in hyperﬁne spectroscopy on the ground state of Mg+ [193] and Be+ [194] ions. Waiting times T as long as 41 s have been applied demonstrating the long coherence time achievable in trapped ion spectroscopy (Fig. 4.28). Continuous laser excitation and microwave transitions lead to a Lorentzian lineshape (Fig. 4.29). The width often is limited by the stability of the available frequency reference.

116

4 Microwave Spectroscopy

Fig. 4.28. Ramsey optical-microwave double resonance spectroscopy on the ground state of Mg+ ions. The laser pulse duration τ was 1.02 s and the waiting time T = 41.4 s. The data points were ﬁtted to the approximate line shape of (4.44). The vertical arrow marks the resonance frequency ω0 [193]

Fig. 4.29. Magnetic ﬁeld “independent” F = 3, m = 0 ↔ F = 2, m = 0 transition in 173 Yb+ . The solid line corresponds to a Lorentzian line shape and a linear decreasing background [195]

The examples mentioned above refer to ions with alkali-like level structures that exhibit strong resonance transitions for optical excitation. The electronic ground state is S1/2 , leading to a two-level hyperﬁne structure if we neglect the Zeeman splitting. Some examples exist where more complex ions have been investigated, demonstrating the more general applicability of hyperﬁne spectroscopy in ion traps. A series of experiments on Eu+ ions has been started at the University of Mainz. They were motivated by the possible study of the diﬀerential hyperﬁne anomaly (Bohr–Weisskopf eﬀect): 1

Δ2 =

A(1)gI (2) − 1. A(2)gI (1)

(4.45)

Here A and gI are the hyperﬁne coupling constants and the nuclear g-factor of two isotopes 1 and 2, and 1 Δ2 describes the change in distribution

4.2 Hyperﬁne Structures in the Ground States

117

Fig. 4.30. The energy terms of 151,153 Eu+ relevant to optical pumping of hyperﬁne levels and detection of transitions between them

of magnetization over an extended nuclear volume for the two isotopes. Europium oﬀers an expanded chain of long-lived isotopes to be studied, two of which are stable 151 Eu (47%) and 153 Eu (52.2%) and 13 unstable, with half lives ranging from minutes to years. It has also an optical wavelength that can be supplied by standard laser techniques. The electronic ground state of Eu+ is 9 S4 . The relevant energy levels are shown in Fig. 4.30. The nuclear spins vary between 5 and 5/2 for diﬀerent isotopes. Coupling to J = 4 leads to a very complex hyperﬁne structure of the electronic ground state. In particular the mF states have half integral values and have a linear dependence on the magnetic ﬁeld, thus imposing a greater demand on its uniformity. The experiments were carried out on a cloud of Eu+ ions numbering on the order of 105 –106 conﬁned in a r0 = 2 cm Paul trap. A buﬀer gas consisting of N2 at a pressure of about 2×10−2 Pa was used to dampen the ion motion and extend the storage time to about one day. Although the ion spectrum is Doppler broadened to a line width of about 5 GHz, several hyperﬁne components of the ground state could be resolved and their population depleted by optical pumping. Transitions induced into such hyperﬁne states by a microwave ﬁeld are detected by the increase in the optical ﬂuorescence intensity. By lowering the microwave power it was possible to resolve the Zeeman components in the hyperﬁne transitions of the ground state in a small magnetic ﬁeld. The frequencies of ﬁve hyperﬁne transitions in each isotope were determined with fractional uncertainties ranging from 1×10−9 to 2×10−8. Figure 4.31 shows an example. In the work on the radioactive isotopes, samples were collected in quantities on the order of 1012 atoms on Pt or Re ﬁlaments at the ISOLDE-CERN facility. The ﬁlament was mounted in a small slot in the lower end cap of the Paul trap and the radioactive ions were emitted from the heated ﬁlament by surface ionization.

118

4 Microwave Spectroscopy

Fig. 4.31. ΔmF = 0 component of the Zeeman transitions between the F = 13/2 and F = 11/2 ground-state hyperﬁne level in 151 Eu+ . The residual magnetic ﬁeld at the ion position was 270 mG [196]

Table 4.3 lists the hyperﬁne structure coupling constants measured to date in various ionic states using ion traps.

4.3 Microwave Atomic Clocks 4.3.1 Deﬁnition of the Unit of Time Historically the very concept of time was identiﬁed with the rhythm of celestial bodies (“the music of the spheres”), and time was measured in terms of the apparent cyclical motions of the sun and moon. The basic unit of time, the second, was deﬁned for centuries in terms of the solar day and deﬁned until relatively recently as the fraction 1/86,400 of the mean solar day. The mean solar day was generally taken to mean the average duration of a solar day taken over a year, without further speciﬁcation. However, with the emergence of more precise electronic time keeping, irregularities in the rotation of the Earth became apparent, making that deﬁnition unsatisfactory. In 1960 the 11th Conf´erence G´en´eral des Poids et Mesures (CGPM) adopted a deﬁnition of the second (the ephemeris second ) based on the period of revolution of the

4.3 Microwave Atomic Clocks

119

Table 4.3. Hyperﬁne structure coupling constants of ions measured in traps Ion 3

Nucl. spin State +

He Be+ 25 Mg+ 43 Ca+ 43 Ca+ 43 Ca+ 87 Sr+ 87 Sr+ 113 Cd+ 131 Ba+ 133 Ba+ 135 Ba+ 137 Ba+ 148 Eu+ 148 Eu+ 149 Eu+ 150 Eu+ 150 Eu+ 151 Eu+ 151 Eu+ 153 Eu+ 153 Eu+ 171 Yb+ 173 Yb+ 199 Hg+ 207 Pb+ 207 Pb+ 9

1/2 3/2 5/2 7/2 7/2 7/2 9/2 9/2 1/2 1/2 1/2 3/2 3/2 5 5 5/2 5 5 5/2 5 5/2 5 1/2 5/2 1/2 1/2 1/2

1S1/2 2S1/2 3S1/2 4S1/2 4P1/2 3D3/2 5S1/2 4D5/2 5S1/2 6S1/2 6S1/2 6S1/2 6S1/2 6 9 S4 6 7 S3 6 9 S4 6 9 S4 6 7 S3 6 9 S4 6 7 S3 6 9 S4 6 7 S3 S1/2 S1/2 S1/2 6 P1/2 6 P3/2

A[MHz]

B[kHz]

References

8 665.649 867 (10) 625.008 837.048 (4) 596.254 376 (54) 813.902 071 6 (1) −142 (8) −48.3 (1.6) 1 000.473 673 (11) 2.1743 (14) 15 199.862 858 (2) 9 107.913 698 97 (50) 9 925.453 554 59 (10) 3 591.670 117 45 (29) 4 018.870 833 85 (18) 517.281 950 (150) −561.647 (100) 1 585.450 570 (250) 599.010 680 (200) −650.334 (2) 1 540.297 394 (13) −1 672.457 109 (266) 684.565 993 (9) −743.183 577 (82) 12 642.812 118 471 (6) 3 497.240 079 85 (3) 40 507.347 997 8 (10) 12 968.180 601 61 (22) 580 (3)

[186] [194] [193] [197] [198] [198] [199] 49 110 (60) [200] [201] [202] [202] [203] [191] −292.63 (1.00) [204] [205] −534.85 (1.90) [204] −839.73 (3.00) [204] [205] −660.862 (2.31) [196] −599.4 (3.1) [204] −1 752.868 (84) [196] 2 448.35 (8) [205] [206] [195] [187, 207, 208] [209] [210]

Earth around the Sun with respect to the vernal equinox, that is the tropical year, 1900. Even this deﬁnition suﬀers from at least two drawbacks: First, at the precision attainable with atomic clocks the tropical year is not constant; even allowing for the precession of the Earth’s axis which forms a predictable component to its variation. Second, for navigators it was important (at least before the advent of satellite-based global navigation) to have time based on the solar day available, and not ephemeris time. In the meantime the development of atomic standards had advanced so far that they were not only necessary to implement the astronomical standard in the near term, but actually became the de facto reference against which astronomical observations were compared. These standards, based on the frequency of radiation resonantly inducing quantum transitions in molecules and atoms could be far more accurately realized and widely reproduced. Moreover,

120

4 Microwave Spectroscopy

new applications of science and technology were rapidly creating a demand for very precise and stable frequency standards that are readily accessible. Therefore, the 13th CGPM (1967/68) replaced the deﬁnition of the second by the following: The second is the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperﬁne levels of the ground state of the cesium 133 atom. This number of periods was chosen of course to make the new atomic second as close as possible to the old ephemeris deﬁnition. It follows that the hyperﬁne splitting of the 133 Cs atom is by deﬁnition exactly 9,192,631,770 Hz. At the 1997 meeting of CIPM (Comit´e International des Poids et Mesures), the following was added to the deﬁnition: This deﬁnition refers to a cesium atom at rest at a temperature of 0 K. This addendum reaﬃrms that the deﬁnition is intended to apply to atoms unaﬀected by the (relativistic) Doppler eﬀect or black body radiation. This means of course that all physical realizations which in practice do not fulﬁll that requirement will have to make the necessary corrections. However, the deﬁnition as adopted takes no account of possible corrections to time scales in accordance with general relativity. If general relativistic eﬀects are to be taken into account, then the deﬁnition of the unit of time applies to only a small spatial domain around the reference atom, and sharing its motion. This is termed the proper unit of time in which the only relativistic eﬀect that needs to be taken into account is that of Special Relativity. Unlike other proper units, realizations of the unit of time can be compared locally or remotely using radio waves. In the latter case, where the standards are far apart, the general theory of relativity must be used to correct for the possible diﬀerence in gravitational potential and for the Sagnac eﬀect due to the rotation of the Earth. The theory predicts a gravitational red shift in the frequency of about 1 part in 1016 per meter of altitude diﬀerence at the Earth’s surface between the two standards. Diﬀerences of this magnitude are signiﬁcant when comparing present day national standards. Since 1983, the unit of time has taken on added signiﬁcance. In that year the 17th CGPM deﬁned the unit of length, the meter, as follows: The meter is the length of path travelled by light in vacuum during the time interval of 1/299,792,458 of a second. This transfers the burden of maintaining a standard of length to that of time, which had been shown to be far more accurate. It ﬁxes the speed of light as being by deﬁnition 299,792,458 ms−1 , the number being chosen of course to make the new meter as close as possible to the old. This deﬁnition replaced the 1889 platinum–iridium prototype bar and the 1960 krypton-86 radiation wavelength, showing a progression toward fundamental quantum properties of matter. The new deﬁnition recognizes as a matter of principle the relativistic point of view that space and time are not absolute and separate, and that the velocity of light is merely a scale factor between them. Moreover as a practical matter all distance measurements based on radar, and this includes all aerospace and much of interplanetary measurements are in fact propagation times for electromagnetic waves. With the new deﬁnition of the meter the measurement of distance between spatially

4.3 Microwave Atomic Clocks

121

separated points is reduced to measuring the time delay between the sending of an electromagnetic impulse from the ﬁrst point and receiving a reﬂection from the other. This, incidentally provides the operational deﬁnition, ﬁrst propounded by Poincar´e, of the synchronization of clocks throughout a spatial region [211]. 4.3.2 Trapped Ion Microwave Standards Initially the physical embodiment of the standard second naturally took the form of the existing Cs beam resonance apparatus that had prompted the deﬁnition in the ﬁrst place. The microwave transition between the ground state hyperﬁne states in 133 Cs were observed initially on thermal atoms in a beam using state selection and analysis with Stern–Gerlach magnets, and a Ramsey microwave cavity to provide the probing microwave ﬁeld. More recent embodiments replace the magnets with laser optical pumping, with some reﬁnements in the Ramsey cavity design to further minimize ﬁrst-order Doppler shifts, but otherwise unchanged. Finally with the development of laser cooling of atoms and the formation of magneto-optical traps, atomic fountain standards became a reality in which a succession of bunches of cold Cs atoms are projected vertically up and fall back along very narrow parabolae providing a prolonged observation time in the Ramsey cavity at the top of their trajectory. Since these realizations of the primary frequency standard do not involve trapping of charged particles but of neutral atoms in electromagnetic ﬁelds, we will not discuss the Cs standard per se, but refer the interested reader to the extensive literature on the subject [212]. In what follow, therefore, we will describe the application of charged particle trapping to what we must call secondary standards, in spite of the fact that some of the ion standards already match the accuracy of the primary standard, and indeed since 2005 have surpassed it. It is obvious that since the deﬁnition of the standard second is essentially a spectroscopic one, the design objectives of a realization of it should be the same as those of high-resolution spectroscopic measurements. Therefore, much of what was described in Sect. 4.2 on hyperﬁne spectroscopy is pertinent to the present subject. Indeed it was the demonstrated exactitude with which the frequencies of spectral lines could be determined that led to the adoption of the atomic second. As we saw in Sect. 4.2 the combination of the techniques of ion storage and laser pumping has pushed the resolution and accuracy of both microwave and optical spectroscopy to new heights. In fact it made possible in one revolutionary advance the ideal goal of observing the spectrum of individual ions at the quantum zero point of energy. In volume I of this series the principles of these techniques are discussed at length and in Sect. 4.2 their implementation was introduced in the context of ion spectroscopy. As these techniques became widely used, it became apparent that there are many possible species of ion and types of transitions that could serve as a reference, both in the microwave and optical regions of the spectrum. Since all standards

122

4 Microwave Spectroscopy

based on trapped ions are for the foreseeable future all passive, a microwave or optical ﬁeld must be synthesized with suﬃcient spectral purity to lock on to the standard transition frequency to transfer its stability without degradation. The degree of complexity involved in synthesizing such a ﬁeld may in the end decide which ion species will be favored for universal use. In discussing the relative merits of diﬀerent types of atomic standards we need to recall their design principles particularly as they apply to stored ions. We limit ourselves to standards in which the resonant response of an ion is detected by counting emitted photons. We must distinguish between standards in which a large ensemble of ions contributes to the photon count with a small probability per ion, as compared with standards in which a small number of ions contribute very many photons per ion. An example of the ﬁrst category is the 199 Hg+ standard using a conventional resonance mercury lamp in which as many as 106 ions contribute to a weak photon signal. In this case the number of scattered photons is very much smaller than the number of ions and hence the photon fractional shot noise far exceeds that of the ion population. This assumes, of course that the ion population is free of any deterministic instabilities. The conventional deﬁnition of the stability of a frequency standard is a measure of the statistical frequency ﬂuctuation as determined by the twosample Allan variance deﬁned as follows [213]: σy2 (τ )

N −1 [ ωkτ − ωk+1τ ]2 1 = , 2(N − 1) ω02

(4.46)

k=1

where ω0 is the angular frequency and ωk τ is the kth frequency measurement averaged over time τ. When applied to the resonance lamp mercury ion standard one ﬁnds Tc Δν σy (τ ) = , (4.47) πν0 (S/N ) τ where ν0 is the transition frequency. Δν the line width of the transition, Tc the duration of one measurement cycle, and S/N is the signal to noise ratio, given by S/N = I1/2 /(I1/2 + Ib ), where I1/2 is the photon count at half maximum and Ib is the background count. The measurement cycle time is the sum of the time of interaction with the ion Tr and the dead time during which optical pumping of the state populations occurs, Td . The interaction time determines the resonance line width: for a Ramsey pattern Δν = 1/(2Tr). If the measuring cycle is repeated√ M times then τ = M Tc = M (Tr + Td ) and the frequency stability σy (τ ) ∝ 1/ M . This inverse square root law presumes stability limited by only fundamental shot noise. The analysis of the stability of oscillators controlled by an intermittent periodic reference to an atomic resonance in the presence of noise whose spectrum is governed by diﬀerent power laws has been extensively discussed in the literature [214]. For laser-pumped ion standards with only one or at most a few ions strongly emitting a large number of photons, it is not the shot noise of the

4.3 Microwave Atomic Clocks

123

photons that is dominant, but quantum projection noise. This has to do with the ﬂuctuation in the number of ions, in a mixed state as a result of the Ramsey pulses, that have been observed to make a clock transition by the photon emission. In the absence of other sources of instability, the stability of such an ion standard is given by Tr Δν √ σy (τ ) = , (4.48) τ πν0 Ni where it is assumed that Td << Tr and as before Δν = 1/2Tr. Substituting for Δν we have the slightly more compact expression: σy (τ ) =

1 √ τ −1/2 , 2πν0 Ni Tr

(4.49)

showing the τ −1/2 dependence. Since ion standards are all essentially passive frequency discriminators a local oscillator is necessary to probe the ion resonance line. The short-term stability values quoted above assume that the stability of the local oscillator is much better than a single measurement stability of the ion. In practice, there must be a dead time in each measurement cycle for pumping and detection during which the local oscillator may drift, degrading the performance of the standard. This degradation has been the subject of extensive study since the situation it describes has wider application [215]. The JPL

199

Hg+ Standard

The direct exploitation of the original 1973 NASA 199 Hg+ hyperﬁne frequency measurement has led to the development more recently (1996) of an ion standard for deployment at Deep Space Network (DSN) stations operated by the Jet Propulsion Laboratory (JPL). It is based on the ground state hyperﬁne splitting in the 199 Hg+ ion, between which transitions induced by a microwave ﬁeld are made observable by ﬂuorescent emission using light from a 202 Hg+ vapor lamp. Some details of the original experiment are given in Sect. 4.2. The JPL design diﬀers from the original in using a linear multipole ion trap, and a He buﬀer gas to cool the ions to room temperature. Its relative portability and ruggedness are due to the use of the 202 Hg+ resonance lamp for hyperﬁne pumping and resonance detection. However, a conventional lamp does not emit a suﬃciently intense, sharp spectral line to cool the trapped ions, and on that account the JPL unit is not expected to be in the same performance class as the laser-cooled laboratory systems. Nevertheless, with the advancement of laser techniques a simpler laser-based standard may become a reality for space borne applications in the future. The new and improved JPL linear ion trap standard, designated as LITS, is based on an extended form of multipole linear ion trap [216] reminiscent of those used in tandem mass spectrometers. It is composed of two sections,

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4 Microwave Spectroscopy

one for optical pumping and transition detection, while the other is where the microwave clock transition is induced. By using a higher multipole (dodecapole) trap for the second section, with its reduced rf-ﬁeld amplitude in the vicinity of the axis compared to the quadrupole, the distribution of ions is more spread out and they generally have lower kinetic energy. Further, the microwave resonance can be better shielded from external magnetic ﬁelds and cooled by liquid nitrogen to keep neutral mercury background pressure below 10−7 Pa. The system is ﬁlled with about 10−3 Pa of pure He gas to moderate the temperature of the ions. Figure 4.32 shows a schematic drawing of the extended linear ion trap enclosed in its vacuum shell. The overall dimensions are 10 by 50 cm. The optical pumping and transition detection are carried out in the upper section of the trap, which shares the same rf as the lower section but is insulated from it to enable an axial dc ﬁeld to manipulate the placement of the ions along the axis, moving them from one section to the other with little loss. The sequence of operations in a measurement cycle is ﬁrst to create, cool and optically pump the ions in the upper section, transfer them to the lower section by the appropriate application of a dc bias, interrogate them with a Ramsey sequence of microwave pulses, and ﬁnally transfer them back to the upper section to optically detect the transition. The microwave frequency is switched symmetrically between the inﬂection points on the two sides of the resonance curve. Its source must be of exceptional spectral purity and low-phase noise. The engineering prototype was designed to interface with any one of the common stable frequency sources: the H-maser, the cryogenic

Fig. 4.32. Two-section extended linear quadrupole ion trap design of the JPL 199 Hg+ frequency standard [216]

4.3 Microwave Atomic Clocks

125

dielectric resonator oscillator (DRO) and ultra-low noise quartz crystal oscillator (VCXO). With these as reference the desired resonant 40.507,347... GHz signal must be coherently synthesized involving frequency multiplication and phase locking techniques. The source used in stand-alone operation of the standard in the (NASA) Deep Space Network is an electrodeless BVA design [217] SC-cut quartz crystal oscillator with a fractional frequency stability of 1.2 × 10−13 for sampling times of 1–100 s. On the basis of the measured signal/noise ratio and the atomic resonance Q, the unit designated LITS-4 will have a short term frequency stability of σy (τ ) = 10−14 /τ 1/2 . This corresponds to a remarkable stability of about 10−16 averaged over one day [218]. The NIST

199

Hg+ Standard

Interest in the use of 199 Hg+ as the basis for both a microwave and optical standard has been most notably pursued by Wineland and his group at the (US) National Institutes of Standards and Technology (NIST) [219]. For the microwave standard at 40.5 GHz, the linear form of the Paul trap was used to laser cool several ions to the point of forming a crystal along the axis where the high-frequency electric ﬁeld is zero. The absence of the ﬁeld driven micromotion there prevents the ions from gaining kinetic energy through collisions. This allows the cooling laser to be turned oﬀ during the interrogation to prevent any light shift in the hyperﬁne frequency. The relevant energy levels in 199 Hg+ showing the transitions for cooling, repumping and interrogating the ions are shown in Fig. 4.33. The ions are Doppler cooled using a laser beam tuned below the frequency of the transition 2 S1/2 (F = 1) −→ 2 P1/2 (F = 0) at λ =194 nm. Since F = 0 ↔ F = 0 transitions are forbidden, the cooling transition is ideally a cycling one, in which the ion continuously absorbs and re-emits the same wavelength. However in practice the cooling laser spectrum slightly overlaps the other hyperﬁne component 2 P1/2 (F = 1) from which the ion may decay into the 2 S1/2 (F = 0) in eﬀect transferring ions out of the 2 S1/2 (F = 1) level, which is required for the

Fig. 4.33. Energy level diagram for repumping and interrogating the ions

199

Hg+ showing the transitions for cooling,

126

4 Microwave Spectroscopy

cooling transition. Hence the need for a repumping beam tuned to 2 S1/2 (F = 0) −→ 2 P1/2 (F = 1) to counter this transfer, thereby allowing the cooling to continue unabated. In the Doppler cooling process it is necessary that the ions not be pumped into a dark nonabsorbing state. Recall a discussion in Sect. 4.2 of the eﬀect of coherence between polarization states in the laser beam in the limit of overlapping magnetic sublevels. The remedies adopted in this work are ﬁrst the degeneracy of the ground state 2 S1/2 (F = 1) is broken by a magnetic ﬁeld on the order of (1–5)×10−5 T, and second to split the cooling and repumping beams into two components, one of which is linearly polarized and the other circularly polarized with the sense of polarization modulated between right and left handed polarization. The ions are conﬁned in a linear quadrupole trap shown schematically in Fig. 4.34. The four rods were 1.7 mm apart and the two end rings 4 mm apart. By applying an rf voltage of amplitude 150 V at a frequency of 8.5 MHz radial conﬁnement was achieved, characterized by a radial secular frequency of 230 kHz. Axial conﬁnement was achieved by applying to the rings a dc potential of about 10 V with respect to the rods. The trap assembly is at the base of a liquid He Dewar to cryogenically maintain the highest possible vacuum by freezing out all but a low-residual He pressure. This lengthens the lifetime of the ions and eliminates frequency shifts arising from collisions with background particles. The measurement cycle is basically similar to that described for the JPL unit: First the pumping and recycling beams are on for typically 300 ms, next the recycling beam is cut for about 60 ms allowing the ions to be pumped into the 2 S1/2 (F = 0) level. During the interrogation period that follows both laser beams are blocked, while a Ramsey sequence of two π/2 microwave pulses are applied separated by a free precession period τF which may be set between 2 and 100 s. The clock transitions induced by the microwave ﬁeld to the F = 1 level are detected by reapplying the pumping beam and counting the number of ﬂuorescent photons to restore the population of the F = 0 state. Critical to the performance of an ion standard is the realization of a local oscillator to match the extraordinary sharpness of the microwave resonance. For the single ion 199 Hg+ standard this requires a fractional frequency stability in the range of 10−15 averaged over 100 s. For the NIST standard a

Fig. 4.34. Schematic drawing of the linear ion trap used in the laser-cooled NIST 199 Hg+ frequency standard

4.3 Microwave Atomic Clocks

127

prototype synthesizer has been described [220] that uses an active hydrogen maser to provide a reference with suﬃcient stability; however, other possibilities include the highly stable BVA quartz crystal VCXO and the cryogenic DRO. A 100 MHz oscillator is phase locked to the 5 MHz reference source and a commercial digital synthesizer is used to oﬀ-set lock a 100.018 MHz oscillator to the 100 MHz signal. Frequency scan resolution and modulation are provided by a digital synthesizer. The 100.018 MHz is multiplied by 5, ﬁltered, ampliﬁed and the 81st harmonic generated in a step recovery diode, giving the desired transition frequency at 40.50729 GHz. The microwave frequency ν is periodically stepped ±Δν about a mean frequency, a binary modulation equal to one half of the resonance line width, corresponding to switching between points that are symmetrically at half maximum points on the central Ramsey fringe. A complete Ramsey measurement cycle is completed at each sign of Δν. The diﬀerence between the +Δν and −Δν signals constitutes the error signal εn for the digital servo to lock on to the center of the resonance ν0 according to the following [219]: νM+1 = ν0 + gp εM + gi

N

εn ,

(4.50)

n=1

where gp and gi are the proportional gain and the integral gain and M is the number of iterations. For N = 7 ions in the trap and a free Ramsey precession period of 100 s these authors ﬁnd σy (τ ) ≈ 3.3×10−13τ −1/2 which is comparable to the advanced Cs fountain standard. In Fig. 4.35 the Allan variances for the mercury ion standard operated with Ramsey free times of 10 and 100 s are compared with a hydrogen maser and a Cs fountain standard. The mercury values are two times larger than

Fig. 4.35. Stability of the NIST 199 Hg+ microwave standard using an array of seven trapped ions

128

4 Microwave Spectroscopy

the theoretical value based on the number of ions and the Ramsey free time, a discrepancy thought to result from residual ﬂuctuation in the laser intensity at the site of the ions. To obtain an accurate value for the 199 Hg+ ground state hyperﬁne splitting to this degree of resolution requires of course a commensurate analysis of all possible mechanisms that may displace the frequency. The largest among these is the quadratic Zeeman eﬀect (∼ 10−15 ), resulting from not only the uniform static magnetic ﬁeld purposely applied, but also residual ac and rf ﬁelds at the site of the ion. Next in order of signiﬁcance in practice have their origin in the stability and spectral symmetry of the interrogating microwave ﬁeld (8×10−16 ). Below that several weaker eﬀects become signiﬁcant, including blackbody ac Stark shifts, light shift from stray 194 nm radiation (3×10−16), second-order Doppler eﬀect, neighboring transitions and blackbody ac Zeeman (∼3 × 10−17 ). Since the standard was operated at the liquid He temperature of 4 K, the residual background gas consisting almost totally of He at low pressure is estimated to cause a fractional frequency shift of less than 10−19 , entirely negligible. By referencing the H-maser used in the synthesizer to the primary standard of frequency through International Atomic Time (TAI) the corrected 199 Hg+ frequency was obtained with unprecedented accuracy as 40 507 347 996.84159 (41) Hz where only the largest source of uncertainty is quoted, namely that due to the frequency comparison with TAI. Other Possible Ion Microwave Standards It is clear from Sect. 4.2 that there are a number of diﬀerent species of ions having isotopes with a ground state hyperﬁne structure that may be measured with suﬃcient accuracy to qualify if not as a primary standard, at least as a secondary one. The earliest among these to be studied [191] is the 137 Ba+ ion, which has a ground state hyperﬁne splitting of around 8.038 GHz. This ion has the practical advantage that the wavelength of the optical resonance radiation 493 nm is readily accessible using laser sources; however, the microwave frequency is only 1/5 that of the 199 Hg+ ion, and therefore the quality factor of the microwave transition in the barium ion is a factor of 5 smaller, all other things being equal. Another ion species that has attracted a good deal of interest both as a microwave and optical frequency standard is 171 Yb+ [221]. This ion has a hyperﬁne splitting in the ground state of 12.6 GHz and an optical resonance wavelength of 369 nm, obtainable by frequency doubling the output frequency of a Ti:Sa laser. Again the clock frequency is only 1/3 the value for the mercury ion, but the optical resonance wavelength somewhat more accessible than the mercury ion.

5 Optical Spectroscopy

5.1 Optical Frequency Standards 5.1.1 Theoretical Limit to Laser Spectral Purity From the earliest time in the development of lasers, eﬀorts were made to stabilize their frequency and narrow their spectral line width to serve as standards of length. This was driven by the knowledge that the spectrum of a laser executing multimode oscillation in a cavity, and subject to “technical” sources of ﬂuctuation such as vibration and the environment, left considerable room for improvement in spectral purity. The fundamental quantum limit on spectral purity was known to far exceed that which was achieved in any common laser. To understand the origin of this inherent limit on the spectral purity of a laser, that is, the limit on the short-term phase stability of the laser, we must go back to the fundamental processes involved in its operation. There are two light-emission processes that atoms of the laser medium undergo: spontaneous and stimulated emission. In spontaneous emission, which occurs with a probability independent of the prior presence or absence of photons, the photons emitted by diﬀerent atoms bear no phase relationship to each other, nor do photons emitted by the same atom at diﬀerent times. In contrast, the stimulated emission of photons occurs with a probability proportional to the number of interacting photons already present, and the phases of photons emitted by diﬀerent atoms, or by the same atom at diﬀerent times, have a deﬁnite relationship; that is, they are coherent. It is the inevitable presence of spontaneously emitted, incoherent photons in the otherwise coherent stream of photons constituting the laser output beam that sets the limit on spectral purity mentioned above. Quantitatively, it can be shown that the mean square deviation in phase Δφ2 is given by

Δφ2 =

Nspont , Ntotal

(5.1)

where the average is taken over a time during which N spont photons are spontaneously emitted, and Ntotal is the total number of photons in the given

130

5 Optical Spectroscopy

Fig. 5.1. The phasor representation of the stimulated and spontaneous optical ﬁelds

ﬁeld mode. This result can be made plausible by noting that the ratio of photon numbers is proportional to (E spont E total )2 , where E spont and E total are the corresponding optical ﬁeld amplitudes. That is, in terms of a ﬁeld picture we have an oscillating optical ﬁeld vector with its phase randomly ﬂuctuating over a narrow range because of the addition of a (small) phaseincoherent ﬁeld component (the spontaneous photon). From the size of the phase shift we can deduce by vector addition of the two rotating vectors representing them in a phasor diagram, as shown in Fig. 5.1. Now, under steady oscillating conditions where the population inversion is sustained by pumping at a constant rate, the mean optical ﬁeld amplitude E total remains constant, and in cases of practical interest E spont E total . From the ﬁgure, it is clear that the maximum change Δϕ in the phase of the optical ﬁeld due to the addition of a small vector increment occurs when the phase of the latter is at π/2 to the main ﬁeld vector. It follows that small ﬂuctuations in the amplitude E spont can produce a maximum phase change (in radians) given by Δϕ =E spont /E total . Of course, the eﬀect of the spontaneous component on the phase varies randomly, which we recognized as a random walk problem. It can be shown [222] that this leads to a laser output with a Lorentzian spectral intensity distribution and a spectral line width Δν given by Δν =

4πhν (Δνc )2 , P

(5.2)

where Δνc is the passive cavity resonance line width (in the absence of the lasing medium) and P is the power in the cavity mode. It was on the basis of an expression of this form, ﬁrst derived in 1958, prior to the realization of a working laser, that the extraordinary potential spectral purity of lasers was predicted. Assume, for example, that we have a laser operating at 633 nm with an ideal cavity of length L = 1 m and an output mirror with 1% transmission. The cavity resonance line width can be obtained from the average lifetime of a photon in the cavity. Thus, a given photon has a 1% chance of leaving the cavity in the time required to traverse the cavity in both directions, and

5.1 Optical Frequency Standards

131

will therefore spend on the average 200 L/c s before leaving the cavity. The corresponding (full) spectral width of the cavity resonance is then Δνc = c/(2π 200 L); that is, in this case ≈0.24 MHz. For a laser output power of 1 mW, we ﬁnd, on substituting into the expression for the laser spectral line width the result Δν/ν = 4.7×10−19 Hz! This quantum limit is so small that it was thought, at the time it was ﬁrst calculated, that it was of no practical consequence; however, as we shall see, recent work on laser stabilization has led to claims of extraordinary spectral purity, making this quantum limit relevant. 5.1.2 Laser Stabilization From the point of view of a frequency standard, a laser oscillator serves to provide a stable, spectrally pure source of radiation, much as a dielectric resonator oscillator might do for a passive microwave atomic standard. Just as in the microwave case, long-term stability and reproducibility are achieved by locking the laser frequency on resonance with a suitable reference quantum transition, free of Doppler and other sources of spectral line distortion. In this role the essential attribute of the laser is low-phase instability, whose spectrum is well within the bandwidth of the servo loop stabilizing it. We distinguish of course between “technical” or “artiﬁcial” sources of phase/frequency noise and the fundamental noise, described in Sect. 5.1.1. In the years before the laser cooling of individual trapped ions became a reality, the principal challenge was to achieve ultra-narrow resonances in atomic or molecular ensembles at room temperature, and hence the need to eliminate the ever present Doppler broadening due to thermal motion. To provide some historical perspective, we summarize here these early techniques of laser stabilization, even though ﬁeld conﬁnement of particles is not involved. Two common techniques were developed that eliminated the ﬁrst-order Doppler broadening: The ﬁrst is saturated absorption [223], and the second is two-photon excitation [224]. The ﬁrst depended on having two identical laser beams counterpropagating through the reference gas or vapor causing only molecules with zero velocity along the beam axis to be resonant with both at the same time. All other molecules can be resonant with only one beam because of the Doppler eﬀect. The greater intensity experienced by the zero velocity molecules leads to saturation and, as ﬁrst predicted theoretically by Lamb [225], “hole burning” in the absorption proﬁle, that is a narrow range of reduced absorption. Lamb showed that the width of the “hole” is the natural line width of the transition. It should be noted that as long as there is no gas convection, even if the counterpropagating beams have unequal intensity there will be no systematic shift of the Lamb feature. In the case of the two-photon excitation method, again counterpropagating laser beams of equal wave length are used. In this case, the ﬁrst-order Doppler shifts of the moving atoms are of equal size but opposite sign for the two directions and cancel in the simultaneous absorption of the two photons. The normally low probability

132

5 Optical Spectroscopy

of this process is favored by the presence of another intermediate quantum state between the two states and between which the reference transition occurs. These methods of eliminating thermal Doppler broadening have been supplanted by the technique of laser cooling, which has changed the whole aspect of high-resolution optical spectroscopy and frequency measurement. The initial observation of saturated absorption in N2 gas of the λ = 3.39μm He–Ne laser line led Barger and Hall [226] and others [227] to exploit the phenomenon in the stabilization of lasers using narrow molecular absorption lines that fortuitously matched their wavelengths. The primary example of this is the stabilization of the He–Ne laser infrared line using what is designated as the P(7) line of the ν3 band of CH4 . It is a transition from a level that is well populated even at liquid nitrogen temperatures, has a natural lifetime estimated at about 10 ms and a large absorption coeﬃcient. The frequency match is not perfect, however, there being an oﬀset of about 100 MHz which must be closed, for example by pressure shifting the laser frequency. As a frequency standard (or length standard, as originally conceived) the salient ﬁgure, namely, stability and therefore line width, is limited in practice by the transit time of a molecule across the laser ﬁeld, which of course is determined by the diameter of the beam and the temperature of CH4 . The line width reported by Barger and Hall is about 100 kHz corresponding to Q = 109 . The diﬀerence between two lasers independently locked to CH4 was found to be less than an experimental uncertainty of ±1 kHz corresponding to reproducibility of 1 in 1011 . More recently [228] a frequency chain was constructed at PTB (Germany) to measure the frequency of a hyperﬁne line of the ν3 P(7) CH4 multiplet at 88 THz to within ±1 kHz. Other important examples are the stabilization of the λ = 633 nm output of an He–Ne laser using saturated absorption in I2 vapor and the absorption of the λ = 10.6 μm output of a CO2 laser in SF6 gas. In the latter case, the power output is so high that saturated absorption is observable in a cell outside the laser cavity. The results of absolute frequency measurements over a 3-year period were recently reported [229] by NRC (Canada) on the 633 nm iodine-stabilized He–Ne laser using both a frequency chain and the wideband frequency comb technique to be described later. The results indicate a reproducibility better than one part in 1012 in these stabilized lasers. As the types of stabilized lasers proliferated, each suﬃciently phase coherent to produce sum and diﬀerence frequencies in nonlinear devices, frequency chains could be constructed bridging the wide frequency gap that separates optical frequencies from the microwave frequency of the primary Cs standard, with these serving as secondary standards. At the same time, as techniques of laser frequency comparison developed involving harmonic generation and phase lock loops (PLL), it became common to speak of the frequency rather than wavelength of optical radiation, a description that was devoid of any functional signiﬁcance prior to the laser. Secondary standards consisting of lasers stabilized with reference to various atomic and molecular resonances, have frequencies spanning the spectral range from the far infrared to the

5.1 Optical Frequency Standards

133

Fig. 5.2. Stabilized lasers extending from the far infrared to the visible

optical ends of the spectrum, as shown in Fig. 5.2, on a logarithmic frequency scale. The advent of the extremely wide-band optical frequency comb technique has introduced a new direction in optical frequency measurement rendering the harmonic chain method obsolete by making it possible to cover the full spectral range coherently with a single frequency standard. This development has greatly simpliﬁed frequency measurement in the optical region of the spectrum and is treated in Sect. 5.1.3. 5.1.3 Single Ion Optical Frequency Standards As explained in Sect. 4.4.2 the stability of a frequency standard, as measured ˜ of the atomic by its Allan variance, is determined by the quality factor, Q, resonance used as reference to stabilize it. Since the natural line width of optical quadrupole (and higher order) transitions can be less than 1 Hz with a center frequency 4 or 5 orders-of-magnitude larger than the Cs standard, po˜ tential Q-values for optical resonances in the order of 1015 are achievable. For this reason, there have been world-wide eﬀorts, mainly by national standards laboratories naturally, to build and test trapped ion standards based on optical quadrupole transitions in a variety of ions. Thus advanced studies have been reported on the ion species shown in Table 5.1, and in outlining these various choices it will be evident that they share the same general experimental procedures and therefore only one ion species need be described in some detail secure in the knowledge that the others diﬀer only in some details of laser system design. As laser technology advances, there will be in future less reliance on experimental conﬁgurations and more on commercially integrated solid state laser systems. Servo-Related Limit on Stability: The Dick Eﬀect All of the optical standards under consideration are built around a single ion optical transition which must be probed intermittently to allow periods when laser pumping and cooling can take place without incurring light shifts in the reference frequency. This means that the servo locking the laser frequency to the reference line center receives an error signal only during part of the operating cycle and the laser frequency is corrected only at the end of each cycle. To study the consequent degradation in the frequency noise of the locked laser,

134

5 Optical Spectroscopy

Table 5.1. The principal contenders for single ion optical frequency standard with presently experimentally achieved uncertainties of the clock transition Ion

Clock transition

171

S1/2 −2 F7/2 S1/2 −2 D5/2 2 S1/2 −2 D5/2 1 S0 −3 P0 2 S1/2 −2 D5/2 2 S1/2 −2 D5/2 1 S0 −3 P0

Yb+ Hg+ 88 Sr+ 115 In+ 171 Yb+ 43 Ca+ 27 Al+ 199

2

2

˜ λ Q Uncertainty Laboratory (nm) (theoretical) (Hz) 467 282 674 238 435 729 267

1.28×1024 6×1014 1×1015 1.5×1015 2×1014 2.5×1015 2×1017

600 1.5 1.5 230 6 1.0 6

NPL NIST NPL, NRC M-PQ PTB , NPL U. Inns, U. Prov. NIST

the theory draws on the prior work of Dick [230] on the aliasing of oscillator noise in passive atomic standards, in which he introduced a sensitivity function g(t). In the present case of locking a laser to an optical transition, it may be deﬁned as follows: 1 δP = g(t)δω(t)dt, (5.3) 2 Ti where δP is the change in the probability that an ion transition has occurred during the interaction time Ti with the laser ﬁeld, and δω(t) is the assumed ﬂuctuation in the laser frequency during that time. A generalized deﬁnition of the function g(t) has been derived [215] assuming a quantum mechanical description of the transition applicable to diﬀerent interrogation sequences, particularly the standard Ramsey pulsed mode. In the case of the favored Ramsey scheme, using square pulse modulated interrogating laser ﬁeld, an analytical solution is possible. To simplify, the resulting expressions assume the duration of each Ramsey pulse ti is short compared with (Tc − 2ti ) the period of free evolution of the ion system between pulses. If we further assume that the frequency detuning of the probe laser δω = ω − ω0 is small compared with the Rabi frequency b during ti , then g(t) is given by the following: ⎧ if 0 ≤ t ≤ ti , ⎨ − sin bt sin δωTc sin bti (5.4) g if ti ≤ t ≤ Tc /2, − sin2 bti sin δωTc ⎩ g(Tc − t) all t. To this approximation, which neglects the time during which pumping and cooling of the ion takes place, the sensitivity function is essentially constant and any laser noise will not degrade the long-term stability; only noise on the ion signal does. However in general the low-frequency noise of the locked laser resulting from aliasing has a power spectral density given by [231]: σy2 lim (τ )

∞ c2 s2 1 gm gm = + 2 Syf (m/Tc ), τ m=1 g02 g0

(5.5)

5.1 Optical Frequency Standards

135

where s = gm

Tc

g(t) sin 0

2πmt dt, Tc

c gm =

Tc

g(t) cos 0

2πmt dt. Tc

(5.6)

This shows explicitly the extent to which the intermittent sampling of the ion resonance can lead to the laser noise Syf (m/Tc ) at Fourier frequencies near multiples of 1/Tc being down-converted, degrading the long-term stability of the standard. It is also shown that the same function g(t) can be used to calculate the eﬀect of phase noise originating in the ion signal as well as in the laser. The former is manifested as an average weighted by the sensitivity function. However, as pointed out by Lemonde et al. [215], not all frequency shifts can be taken into account by this theory. This includes all eﬀects that cannot be treated as arising from ﬂuctuation in the atomic frequency or laser ﬁeld phase, such as those involving more than two quantum levels or asymmetry in the spectrum of the interrogating ﬁeld. Quantum Projection Noise In single ion standards the detection of transitions is made with such large numbers of photons that photon shot noise and thermal noise in the photon detector are negligible compared with the quantum projection noise arising from the single ion clock transition. The ﬂuctuation in the probability ΔP that the ion has made a transition is [232]: ΔP = p(1 − p), (5.7) where p is the transition probability. If np is the mean number of photons scattered and detected in a single measurement, then the signal S is S = np P , and the projection noise is given by: N = np p(1 − p). (5.8) If the Ramsey scheme is used to probe the ion resonance, then the signal observed at a frequency detuned from resonance by δω is S = np [1 + cos(δωT )]/2,

(5.9)

where T is the time between Ramsey pulses. If it is assumed that the modulation is symmetrical between the half maximum points, then for an error we have δω = δω0 ± , where δω0 = π/(2T ). Then the signals on the two sides of resonance are S± = np [1 ± sin (T )]/2. (5.10) It follows that:

∂ (S+ − S− ) = T np cos (T ), ∂

(5.11)

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5 Optical Spectroscopy

and the total noise is Ntot = np

p+ (1 − p+ ) + p− (1 − p− ).

(5.12)

Hence, the rms frequency error in one measurement of both sides is −1 1 1 ∂(S+ − S− ) Ntot δνRMS = ≈ [p+ (1 − p+ ) + p− (1 − p− )] T −1 . 2π ∂ 2π (5.13) Finally to put it in the usual Allan variance form, let the measurement be repeated τ /Tc times with p ≈ 1/2, where τ is the total averaging time and Tc is the time to complete a cycle, and write ΔνR for the Ramsey line width. Then, we have: ΔνR Tc σy (τ ) = . (5.14) πν0 2τ The

199

Hg+ Optical Standard

Of all the ion species in which quadrupole transitions are currently under study as prospective optical frequency standards, 199 Hg+ has currently the ˜ at 6×1014, having an observed line width of 6.7 Hz [233] highest measured Q at a frequency of 1.06×1015 Hz. The relevant energy levels in 199 Hg+ have already been described in Sect. 4.4.4, and are reproduced here in Fig. 5.3. The reference transition for the optical standard is taken to be the electric quadrupole transition 2 S1/2 (F = 0, mF =0) ←→ 2 D5/2 (F = 2, mF = 0) at a wavelength of 281.5 nm, with a natural line width of 1.7 Hz. The ﬁrst experimental observation of this transition on a single laser cooled ion [234] demonstrated a fractional resolution of 3×10−11 limited by the 15 kHz line width of the probing laser. This was, however, suﬃcient to resolve nicely the Doppler sidebands, and continue laser cooling on the ﬁrst lower side band.

Fig. 5.3. Partial energy level diagram of

199

Hg+ showing the relevant transitions

5.1 Optical Frequency Standards

137

The physical design of the ion traps and their auxiliary equipment used by NIST have evolved over time as diﬀerent improvements have been pursued in the implementation of the trapped ion technique [235]. It has become a common practice to produce the mercury ions by electron impact ionization of atoms produced by the thermal decomposition of isotope-enriched HgO. Linear as well as spherical traps, operating at rf frequencies of a few megahertz, are used for ion storage. Since experience with residual mercury vapor in a conventional ultra-high vacuum system (10−7 Pa) showed that the resonant charge exchange of the trapped ions with background mercury signiﬁcantly reduced the ion lifetime, the NIST group found it expedient to use cryogenic pumping. Accordingly the trap assembly, housed in a pure copper vacuum can, was mounted at the base of a liquid helium cryostat. The copper can is electroplated on the inside with a 5 μm coating of lead, to provide a superconducting shield. Maintaining the temperature of the trap housing at around 4 K cryopumps the system down to the 10−14 Pa range, totally eliminating collisions with background particles, except possibly residual helium, thereby extending the lifetime of the ion in the trap to at least several days. The superconducting inner lining of the copper vacuum can provides eﬀective magnetic shielding through the Meissner eﬀect. Once a small number of 199 Hg+ ions, ultimately reduced to a single ion, is loaded into the trap, its quantum state populations must be pumped and its secular motion cooled to prepare for the probing of the quadrupole transition. The same transitions for cooling and pumping are used as previously described in Sect. 4.4.4 pertaining to the mercury ion microwave standard. We recall that the Doppler cooling uses a laser beam tuned below the hyperﬁne component 2 S1/2 (F = 1) −→ 2 P1/2 (F = 0) of the strong resonance line at 194 nm, taking the ion near the limiting temperature of 1.7 mK. In a “strong” miniature trap it is possible to resolve the Doppler side bands and continue the cooling in the Lamb–Dicke regime to nearly the zero point energy in the microkelvin range, making the Doppler correction entirely negligible. However the main challenges in realizing an optical frequency standard are ﬁrst the detection of a very weak optical transition, and second the development of a radiation source of suﬃciently high-spectral purity, commensurate with the sharp quadrupole transition. The ﬁrst is achieved by the double resonance technique, that takes on an on–oﬀ aspect when applied to a single ion; it is known metaphorically as the “electron shelving” or quantum jump technique [236, 237]. It is practicable because of the extraordinary spectral brightness of a laser and the large signal-to-noise ratio attainable with laser scattering from a single ion. The second requirement has seen such extraordinary progress in recent years that it has brought within reach a frequency standard in the optical range, that outperforms the microwave standards. The central component of the probe laser system is an ultra-high ﬁnesse (F > 150,000) Fabry–P´erot cavity [238], constructed with materials having low-intrinsic thermal expansion, and isolated from external sources of mechanical and thermal ﬂuctuations. The stable 281.5 nm radiation to probe the

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5 Optical Spectroscopy

clock transition is obtained by doubling the frequency, in an AD∗ P crystal, the output of a dye laser tuned to 563 nm, which is ultimately locked to the high-ﬁnesse cavity. This is not done directly; however, the laser is prestabilized and its line width narrowed ﬁrst by using a Pound–Drever–Hall lock (Sect. 4.1.3) on a resonance in a cavity having a lower ﬁnesse of approximately 800. This prestabilization narrows the dye laser short-term linewidth to about 1 kHz. An optical ﬁber delivers light from the dye laser optical table to the table that supports the high-ﬁnesse cavity, where an acousto-optic modulator (AOM) driven by a precision rf source is used to shift the dye laser frequency into resonance with a mode of the high-ﬁnesse cavity. With the latter acting as reference, a Pound–Drever–Hall lock is again used to stabilize the prestabilization cavity both at high frequency (up to 90 kHz) through the AOM drive frequency, and low frequency through a PZT controlling the cavity length. Two AOMs are used to precisely ﬁne-tune and square-wave modulate the stabilized 563 nm radiation in order that the AD∗ P crystal output probes the quadrupole resonance frequency. A remarkable spectral width less than 1 Hz is reported for the stabilized output at 281.5 nm [239]. Having achieved such high degree of spectral purity it remains to impose on the laser the long-term stability and calibration of a frequency-lock on the center (recoil-free) resonance of the ion quadrupole transition. The measurement cycle involving ion cooling and quantum state preparation is similar to that described for the 199 Hg+ microwave standard, except of course here the probe ﬁeld is at optical frequency inducing quadrupole electric transitions. To achieve the frequency lock, the 563 nm laser beam is square-wave frequency modulated by one of the AOMs to probe alternately the two sides of resonance. Photon counts are taken repeatedly many times on each side of resonance and the mean diﬀerence between them is used as the tuning error between the probe laser and the center of the ion resonance. This error signal is used in a digital servo loop to steer the average frequency of the probe laser by means of the other AOM to the center of the resonance. The fractional frequency instability of the stabilized probe radiation (the Allan deviation) is given as 7 × 10−15 τ −1/2 [240], a ﬁgure stated to be in good agreement with the quantum-limited instability to be expected under the given conditions. Optical Frequency Standards Based on Alkaline Earth Ions Among the ion species in which ultra-narrow optical transitions have been studied are those of the alkaline earth elements, namely 40 Ca+ , 88,87 Sr+ , and 137 Ba+ . The singly charged alkaline earth ions have a single electron outside closed shells with a 2 S1/2 ground state. The odd mass isotopes 87 Sr and 137 Ba have nuclear spins I = 9/2 and 3/2, respectively, while the even mass isotope 40 Ca has I = 0. These ions have the practical advantage of having the ﬁrst resonance wavelength within reach of simpler laser systems than the ultraviolet wavelengths of other ions.

5.1 Optical Frequency Standards

Fig. 5.4. The relevant energy levels in transition

137

139

Ba+ showing the quadrupole clock

Studies, both theoretical and experimental, have been conducted on the Ca+ ionic species at NICT (Japan) [241], University of Provence (France) [242] and at the University of Innsbruck [243]. The quadrupole transition 2 S1/2 −→ 2 D5/2 at 729 nm with a natural linewidth of 0.13 Hz has a theo˜ ≈ 1015 . The ultra-narrow quadrupole transition 2 S1/2 −→ 2 D3/2 retical line Q 137 + ˜ ≈1016 ) as a possible basis for an optical frein Ba at 2,051 nm (line Q quency standard has been pursued at the University of Washington [244,245]. Interest in this work is motivated mainly as high-resolution spectroscopy to test fundamental theory. The relevant energy levels of the barium ion are shown in Fig. 5.4. The quadrupole transition between the ground 2 S1/2 (F = 2) hyperﬁne state and the metastable 2 D3/2 (F = 0) hyperﬁne state occurs at λ = 2, 051 nm in the infrared region of the spectrum. The upper state radiative ˜ ≈1016 , higher than the other ion lifetime is about 80 s, implying a line Q candidates. The radiation for cooling and repumping at around the ﬁrst resonance wavelength λ = 493 nm, as well as the probe wavelength can be readily generated using solid state devices. The probe wavelength is generated by a diode-pumped Tm,Ho:YLF solid-state laser. Interest in the isotopes 88 Sr+ and 87 Sr+ as potential optical frequency standards follows from the high resolution achieved in the hyperﬁne spectroscopy of the quadrupole transition in these ions by the group at NPL, described earlier in Sect. 4.3.3. The development of an optical clock based on these ions has reached an advanced stage by standards laboratory groups, not only at NPL, but also at NRC (Canada) [246]. Initially studies were conducted on the even isotope 88 Sr presumably because it has I = 0 and its ground state is devoid of hyperﬁne structure, making it easier to cool. It is 40,43

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5 Optical Spectroscopy

Fig. 5.5. Relevant transitions in

87

Sr + as basis for an optical standard

also 12 times more abundant than 87 Sr. However, as pointed out earlier, the odd isotope has the advantage, important for a frequency standard, of having a half-integral nuclear spin I = 9/2 and therefore the total angular momentum in both the ground state and the metastable state will have integral values F = 5, 4 and F = 7, . . . , 2, and therefore there are magnetic sublevels mF = 0 between which the transition frequency has zero ﬁrst-order Zeeman shift. A partial energy diagram for 87 Sr+ is reproduced here for convenience in Fig. 5.5. The optical clock transition is chosen to be the hyperﬁne component 2 S1/2 (F = 5) −→ 2 D5/2 (F = 7) of wavelength λ = 674 nm, whose natural spectral width is about 0.4 Hz. The NPL 87 Sr+ optical clock project [200] is based on the same experimental set up as that described earlier in Sect. 4.3.3 dealing with the high-resolution spectroscopy of that (dipole) forbidden transition. As previously described, two 844 nm ECDL’s, frequency oﬀset-locked at half the 5 GHz hyperﬁne splitting of the ground state, have their outputs independently frequency doubled to provide the cooling and repumping radiation at 422 nm. The laser driving the cooling transition 2 S1/2 (F = 5) −→ 2 P1/2 (F = 5) avoids the now familiar phenomenon of pumping into a dark state by resolving the beam into a mix of π− and σ− polarization components oﬀset in frequency by about 7 MHz. The branching of the decay of the 2 P1/2 state by dipole radiation to the ground state and the metastable 2 D3/2 state necessitates the emptying of the latter prior to probing the clock transition. This is done by applying another laser beam at 1,092 nm from a DBR laser, one modulated to spread its frequency to cover the frequency range of the hyperﬁne structure of the 2 D3/2 state. To avoid optical pumping into a dark state, this

5.1 Optical Frequency Standards

141

Fig. 5.6. Layout of the optical system for the NPL proposed 87 Sr+ optical standard

time the 1,092 nm radiation is polarization modulated at a rate of 0.8 MHz. The layout of the optical system is reproduced in Fig. 5.6. The narrow linewidth clock transition at 674 nm is probed with a stabilized Al–Ga–In–P diode laser, whose output linewidth is ﬁrst narrowed by ˜ optical cavity, then stabilized against optical feedback locking to a medium-Q an ultra-high ﬁnesse (F ≈ 200, 000) cavity using the Pound-Drever Hall technique. The arrangement is illustrated in Fig. 5.7, showing the two feedback ˜ resonator PZT, and the loops: the low frequency one acting on the medium-Q high frequency one acting on the diode current. Like the NIST 199 Hg+ standard, the high-ﬁnesse cavity is constructed using ULE material and extreme care taken in achieving seismic and thermal isolation. AOM’s are used to bridge the frequency gap between the nearest mode of the cavity and the strontium transition frequency. Optical Frequency Standard Based on

171

Yb+ Ion

The transition 2 S1/2 −→ 2 D5/2 in the ytterbium ion 171 Yb+ was ﬁrst suggested by Werth [247] as a candidate for an optical frequency standard. It has been under development for about a decade by both NPL and PTB [248]. As a subject of high-resolution spectroscopic study it was discussed in Sect. 4.3.3; we reproduce the relevant energy levels for convenience in Fig. 5.8. The ytterbium ion has the advantage over mercury of a longer resonance wavelength at 369 nm, making it directly accessible by frequency doubling the output of solid-state lasers. The term structure of 171 Yb+ is similar to 199 Hg+ with a nuclear spin I = 1/2 and a 2 S1/2 ground state having a hyperﬁne splitting of 12.6 GHz.

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5 Optical Spectroscopy

Fig. 5.7. Stabilization of 674 nm diode laser for probing

Fig. 5.8. Relevant transitions in

171

87

Sr+ clock transition

Yb+ as basis for an optical standard

The ion is distinctive in having several potential reference transitions in the microwave, infrared and optical regions of the spectrum. The clock transition chosen by the PTB is the hyperﬁne component 2 S1/2 (F = 0) −→ 2 D3/2 (F = 2) at 435.5 nm with a natural linewidth of 3.1 Hz, whereas the NPL choice is 2 S1/2 (F=0) −→ 2 D5/2 (F = 2) at 411 nm. The laser systems developed at the PTB for the high-resolution spectroscopic study of the 171 Yb+ hyperﬁne structure were outlined in Sect. 4.3.3; the completion of an optical frequency standard further requires the servo-lock of the probe laser to the center of the ion resonance. The general design of the observation cycle and the steering of the probe laser to lock onto the ion resonance follow similar lines to the 199 Hg+ optical clock, described earlier.

5.1 Optical Frequency Standards

143

Of particular interest in the term structure of 171 Yb+ is an extremely long-lived metastable 2 F7/2 state from which the lowest order transition to the ground state is the (electric) octupole, with a radiative lifetime on the order of 6 years [249]. This would mean of course that the transition linewidth, like that of magnetic dipole transitions, is not limited in practice by the natural linewidth, but by the free observation time, potentially giving a resonance linewidth even narrower than the quadrupole transition. To exploit such an ultra-narrow resonance presumes of course the availability of a laser source with commensurate spectral purity. To achieve a comparable increase in accuracy would naturally require the ability to correct for even more subtle sources of systematic frequency shifts. An optical standard based on this octupole transition has been pursued at NPL [250]. Their initial observation of this weak 467 nm transition was achieved on the even isotope 172 Yb+ , which, lacking a hyperﬁne structure is easier to cool and detect its transitions. The ion trap design favored by the NPL group is what has been referred to as an end cap trap [251], shown schematically in Fig. 5.9. Like other miniature variants of the Paul trap it seeks to maximize the optical access to the trapped ion, allowing freedom in the orientation of the laser beams, and thus the ion motion to be controlled and monitored in all three dimensions. Laser cooling is carried out on the resonance transition 2 S1/2 (F = 1) −→ 2 P1/2 (F = 0) at 369 nm using the frequency-doubled output of a Ti:Sa laser. The ion is also irradiated with microwaves at 12.6 GHz, the hyperﬁne splitting of the ground state, to oﬀset the optical pumping from F = 1 to F = 0 in the ground state through oﬀ-resonance excitation of the 2 P1/2 (F = 1) state by the cooling laser. Also there is branching in the decay of the 2 P1/2 state with a signiﬁcant probability of transitions to the 2 D3/2 (F = 1) state, necessitating the application of a 935 nm diode laser beam to return the ion to the ground state via the fast decaying 3 D[3/2]1/2 (F = 0)

Fig. 5.9. Schematic diagram of the NPL “end cap” high-frequency ion trap

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5 Optical Spectroscopy

state. The source of the narrow linewidth clock radiation is a frequencydoubled Ti:Sa laser locked to a high-ﬁnesse optical cavity made of ULE glass, thermally and acoustically isolated. Stabilized laser linewidths on the order of ˜ of 3×1012 . 0.2 kHz have been achieved at a frequency of 642 THz, or a line Q The much narrower Fourier transform width of the clock transition compared with the probe laser width means that relatively high power is needed to observe a reasonable rate of quantum jumps. This is achieved by a tight focussing of the laser beam onto the ion. The extreme radiative lifetime of the 2 F7/2 state introduces a complication which was not present in the diﬀerent optical clocks so far proposed: the ion would remain in the upper clock state an inordinate length of time unless a faster transition to the ground state is induced for example via the 1 D[5/2]5/2 state, which requires irradiation at 638 nm. This is provided by an additional diode laser. Optical Frequency Standard based on

115

In+ Ion

The indium ion 115 In+ is unique among the ions so far considered in that it is a two-electron alkaline earth system, and the transition chosen as reference is the forbidden transition 5s2 1 S0 −→ 5s5p 3 P0 at a wavelength of 236.5 nm between terms that have vanishing total electronic angular momentum. It is expected that this transition between J = 0 states should be immune from shifts due to coupling to the electric ﬁeld of the trap. The experimental realization of an indium ion optical standard was initiated some years ago by Walter at the Max Planck Institute for Quantum Optics (Garching) and the work has continued to reach new levels of precision and accuracy. In a recent paper [252] dedicated to the memory of Walter, an absolute frequency measurement with respect to the Cs standard is reported with a fractional uncertainty of 1.4×10−14. The energy levels of 115 In+ relevant to its use as an optical frequency standard are shown in Fig. 5.10. The experimental set up consists essentially of a laser-cooled single 115 In+ ion conﬁned in a Paul-Straubel trap, which at its center is nothing but a 1 mm diameter elliptical ring of beryllium–copper. The trap is driven at a frequency of 10 MHz and amplitude of 1 kV, resulting in strong conﬁnement with the

Fig. 5.10. Simpliﬁed energy level diagram of

115

In+ showing the relevant levels

5.1 Optical Frequency Standards

145

three secular frequencies of 1.52, 0.87, and 0.93 MHz. The formation of the indium ion in the trap is somewhat unusual: it is achieved by two-stage laser excitation at 410 nm, which is resonant with the 5s2 5p 2 P1/2 −→ 5s2 6s 2 S1/2 transition in neutral indium atoms as well as the subsequent transition from 5s2 6s 2 S1/2 to the autoionizing 5s5p2 2D state. Cooling of the ion involves Doppler cooling followed by direct side-band cooling using the narrow intercombination line 1 S0 −→ 3 P1 at 230.6 nm having a natural linewidth of 0.36 MHz. The cooling laser source is the quadrupled output of a stabilized 922 nm diode laser. Two stages of frequency doubling enhanced in cavities ultimately produce 1 mW at the desired ultra-violet wavelength of 230.6 nm. An electro-optical modulator produces sidebands that are ﬁnely tunable to the desired wavelength. Cooling is executed simultaneously on two wavelengths, one red-tuned to higher order micromotion sidebands, and the other to the lower ﬁrst-order secular frequency sideband. The cooling beams are circularly polarized so that the ion is pumped into the extreme Zeeman sublevels (mF = 9/2 ←→ mF = 11/2) eﬀectively reducing the transition to one between two states. By switching the sense of the circular polarization, while adjusting the oﬀ-set magnetic ﬁeld intensity, the resulting switch in the Zeeman frequency shift in the cooling transition is used to annul the magnetic ﬁeld. The source of the clock radiation is an Nd:YAG nonplanar ring oscillator (NPRO) (also referred to as a monolithic isolated single-mode end-pumped ring (MISER) laser) operating at 946 nm. The Nd:YAG crystal is pumped by two diode lasers at 808 nm with a power of 2 W each, resulting in an output power of 250 mW. The frequency of the MISER laser is stabilized against a high-ﬁnesse ultra-stable monolithic Zerodur cavity. The stabilized MISER laser wavelength is quadrupled using periodically poled nonlinear crystals of KTiOPO4 (KTP) and beta-BaB2 O4 (BBO), each placed in an enhancement cavity to produce the desired 237 nm radiation. The performance of this clock laser system was checked by measuring the beat frequency between two independently stabilized frequencies one using a ULE cavity and the other Zerodur. A relative laser linewidth of 1.34 Hz was found in an integration time up to 3 s. However, there is a measurable drift in the resonance frequency of the high-ﬁnesse cavity. An average drift rate over a period of 4 months was extracted by linear ﬁtting to be 0.679 Hz/s. Optical Frequency Standard Based on

27

Al+ Ion

The theoretical advantages of the ions of group IIIA atoms, such as 26 Al+ , as candidates for optical frequency standards were recognized some time ago [253]. The forbiden transition 1 S0 −→ 3 P0 in 27 Al+ , similar to that in 115 In+ , has recently been successfuly observed at high resolution [254], by circumventing the lack of a suitable transition for directly laser cooling the ion. The breakthrough came with the application of quantum logic techniques [255] to transfer the functions of cooling and detection to an auxilliary

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5 Optical Spectroscopy

Fig. 5.11. Partial energy level diagram for 27 Al+ and 9 Be+ showing the clock reference, Doppler cooling, detection, and Raman transitions

“logic” ion (chosen to be 9 Be+ ) trapped simultaneously with the 27 Al+ ion. Unlike the radioactive 26 Al+ described in reference [253] which has I = 0, the 27 Al+ ion has I = 5/2 leading to half integral values of mF and mF , and therefore no (ﬁrst-order) magnetic ﬁeld-independent transitions. However, as has been demonstrated on 88 Sr+ [256], the Zeeman shifts of the transitions (mF = −5/2) −→ (mF = −5/2) and (mF = +5/2) −→ (mF = +5/2) are equal and opposite, and hence by alternating between them, and taking the mean, the ﬁeld dependence cancels out. The measurement of these Zeeman shifts automatically provides a real-time determination of the magnetic ﬁeld intensity to compute the quadratic Zeeman correction. The relevant energy levels in 27 Al+ and 9 Be+ are shown in Fig. 5.11. The strong resonance transition 1 S0 → 1 P1 in 27 Al+ is at λ = 167 nm and would be diﬃcult to generate laser radiation with suﬃcient intensity to use it for Doppler cooling, a reality which inspired the application of sympathetic cooling with 9 Be+ ions. These berylium ions had been used by the NIST group in studies of quantum logic techniques using trapped ions [257], in which the initialization of the ion quantum state required cooling them to the zero point using stimulated side-band Raman cooling (see [1], Sect. 4.6). Further, the functions of preparation of the quantum state in 27 Al+ and the detection of clock transitions are transferred to the 9 Be+ ion according to the following protocol. Assume the two species of ion are trapped simultaneously along the axis of a linear trap, their motion being coupled by the Coulomb interaction leading to two normal modes, assumed to be initially cooled to the zero point of energy |0m . Let the initial wavefunction representing the two-ion system be Ψ0 = |↓Al |↓Be |0m , and assume a coherent optical ﬁeld induces the clock transition leading to Ψ1 = [α |↓Al |0m + β |↑Al |0m ] |↓Be . If this is followed by a red side-band π−pulse driving the |↓Al → |↑Al transition, we get Ψ2 = [α |↓Al |0m + β |↓Al |1m ] |↓Be = [α |0m + β |1m ] |↓Al |↓Be . That is, the

5.1 Optical Frequency Standards

147

Fig. 5.12. Internal state transfer sequence from a 27 Al+ to a 9 Be+ ion sharing a common normal mode of oscillation in a Paul trap. (a) Initialization (b) interrogation (c) transfer of internal state of 27 Al+ to motional superposition state using red side band (RSB) π-pulse and (d) transfer of motional superposition state into internal superposition state in 9 Be+ ion using RSB π-pulse 27

Al+ internal state has been transferred to the normal mode shared by both ions. The application of another π−pulse on the red side-band of the |↓Be → |↑Be transition will ﬁnally lead to Ψﬁnal = |↓Al [α |↓Be + β |↑Be ] |0m , showing that the internal states of the two ions have been exchanged. The sequence of steps is illustrated in Fig. 5.12. The physical implementation of this approach was carried out by the NIST group on a two-ion “crystal” along the axis of a linear Paul trap with an inphase axial normal mode frequency of 2.62 MHz. The 9 Be+ ion is irradiated by laser beams of 313 nm wavelength for Doppler and stimulated Raman sideband cooling as well as repumping. A frequency quadrupled ﬁber laser at 267.4 nm having approximately 3 Hz linewidth is used to excite the clock transition in the 27 Al+ ion. Detection is achieved through the 1 S0 → 3 P1 transition, whose incidence is transferred to the 9 Be+ using the quantum logic method described above. The clock transition frequency was measured to be ν = 1 121 015 393 207 851 (6) Hz. 5.1.4 Correction of Systematic Errors Of equal importance to the question of the frequency stability and precision of a standard of frequency, is the question of accuracy, that is, the extent to which the measured frequency is in fact the quantity to be deﬁned as the standard, namely in the above examples, the quadrupole transition frequency in the rest frame of the ion, in empty space free from any perturbing ﬁelds. Table 5.2 summarizes the main contributions to systematic uncertainties for the NIST 199 Hg+ optical clock [258] arranged in decreasing uncertainty. Needless to say a correction such as the quadratic Zeeman shift may be relatively large, but to the extent that the magnetic ﬁeld at the site of the ion can be measured accurately the uncertainty it introduces will be small. The same applies to the gravitational red shift.

148

5 Optical Spectroscopy Table 5.2. Main contributions to systematic errors in

199

Hg+ optical clock

Phys. eﬀect

Correction (Hz)

Fract. uncert. (10−17 )

Quadrupole shift Servo Error Zeeman (ac, trap) Zeeman (dc) Micromotion Gravitation Doppler(thermal) Other

0 0 0 1.203 0 0.524 0 0

5 3 2 2 2 1 1 2

TOTAL

1.727

7.2

The Electric Quadrupole Shift It is evident from Table 5.2 that the source of the largest uncertainty in the frequency of the 199 Hg+ optical standard is the shift due to the quadrupole interaction with electric ﬁeld gradients in the ion trap. This applies equally to the other contenders that are also based on a quadrupole transition in ions with a similar term structure. Other electric ﬁeld induced perturbations such as the quadratic Stark eﬀect due to trapping ﬁelds and black body radiation are also present, but their magnitudes are generally much smaller. Ideally a single ion cooled to the point of occupying the center of a perfect Paul trap is not acted on by any trapping electric ﬁeld. However, in practice microscopic traps are plagued with spurious “patch” surface charges whose perturbing ﬁelds must be annulled. This can be more or less successfully done by small potentials applied to the electrodes or larger potentials applied to external compensating electrodes. The electric quadrupole interaction arises from a nonspherical electron charge distribution in an atomic system, characterized by a quadrupole moment. The perturbation of hyperﬁne states caused by a nuclear quadrupole moment in free atoms and molecules was ﬁrst studied in the context of molecular beam resonance methods [259]. A similar theoretical treatment is applicable to the present case of an atomic state such as 2 D5/2 having a quadrupole moment aﬀecting the hyperﬁne structure of that state. A detailed analysis of the spectral frequency shifts caused by external electric ﬁelds acting on an ion in a Paul trap, with particular reference to the 199 Hg+ optical standard has been published by Itano [260]. The computations rely on a computer code implementing the Hartree–Fock method, assuming ﬁrst-order perturbation theory in which the quadrupole perturbation is small compared with the Zeeman splitting. For an atomic state written as |γ, I, J, F, mF , where γ speciﬁes the electron conﬁguration, and I, J, F, mF have their usual meaning, the reduced matrix of the electric quadrupole moment operator is

5.1 Optical Frequency Standards

149

written as γIJF Θ(2) γIJF . The expression given by Itano for the electric quadrupole energy shift HQ is as follows:

γIJF mF | HQ | γIJF mF = −2A γIJF mF Θ(2) γIJF mF 3m2F − F (F + 1) × [(3 cos2 β − 1) − sin2 β cos (2α)], {(2F + 3)(2F +2)(2F +1)2F (2F −1)}1/2 (5.15) where the angles α and β are two of the Euler rotation angles between the principal axes of the ﬁeld gradient tensor and the quantization axis deﬁned by the magnetic ﬁeld. The quantity is an asymmetry parameter of the electric potential function, usually negligible in ion traps [229]. Since the ion optical clocks under consideration have an S lower state, it is only the upper D or F state which can have a quadrupole moment. An interesting method of cancelling out the quadrupole shift based on the dependence of the shift on the Zeeman sublevels has been given by the NRC group [261]. They note that the quadrupole shift depends on mF according to the factor [3m2 F − F (F + 1)] so that by measuring the clock frequency for diﬀerent sublevels mF it is possible to interpolate the frequency corresponding to zero quadrupole shift. It is also noted that for a half integral F , the sum of the shifts over all sublevels mF is exactly zero. The method was applied to their 88 Sr+ ion standard and it was veriﬁed that indeed plots of frequency vs. m2F yielded linear graphs having slopes that depend on the direction of the magnetic ﬁeld, but which intersect at one point. Since the hyperﬁne component F = 5/2 is used, the shift is expected to vanish for m2 F = 5/12, as conﬁrmed by the experimental point of intersection. The Quadratic Zeeman Shift With the exception of 88 Sr+ which has I = 0, the proposed transitions for an optical standard are chosen to be between states that have J = 12 and a half-integral I so that the hyperﬁne levels will have integral values, and transitions between sublevels with mF = 0 exist. For such 0 − 0 transitions the linear dependence on an external magnetic ﬁeld is zero and the lowest order dependence is the quadratic Zeeman shift. To calculate the 0 − 0 shift, a knowledge of the hyperﬁne constants of the upper 2 D5/2 is suﬃcient since the eﬀect is expected to be much smaller in the lower 2 S1/2 ground state. Using second-order perturbation theory the frequency shift Δν in the hyperﬁne component F of the upper state subject to a magnetic ﬁeld of intensity B is given by [262]: Δν(F ) =

gJ μB B h

2 | IJF mF | Jz | IJF ”mF |2 , (νF − νF ” ) F ”=F

(5.16)

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5 Optical Spectroscopy

where gJ is the Land´e factor for the upper state 6/5. The shift for the mF = 0 −→ mF = 0 transition to the F = 7 hyperﬁne component in the 2 D5/2 upper state of 87 Sr+ (I = 9/2) is for example [200]: 2 gJ μB B 0.8654 , (5.17) Δν(F = 7) = 7A + (7/15)B h where A and B are the two 2 D5/2 hyperﬁne constants. Relativistic Doppler Shift In the context of frequency standards even at the low velocities of cooled ions a signiﬁcant correction to the observed frequency is the relativistic Doppler eﬀect by which is meant the second-order term in the expansion of the relativistic Doppler formula in powers of v/c: 2 1 V V ν = ν0 1 − cos θ + + ··· , (5.18) c 2 c In a “strongly” conﬁned cooled ion the ﬁrst-order Doppler eﬀect merely gives rise to a strong undisplaced carrier frequency with sidebands of rapidly decreasing amplitude: the Lamb–Dicke regime [263]. To correct for the secondorder Doppler shift requires a knowledge of the mean kinetic energy of the ion. This can be deduced from the ratio of the intensity of the ﬁrst Doppler side band to the intensity of the carrier [234, 264]). The required theory is available in a classic paper on ion cooling [265] in which this ratio for an ion in thermal equilibrium is given for the ﬁrst side band approximately as I(1) ωs I1 (u) 4π 2 kB T 1 1+ , u 2 , − (5.19) I(0) 4πkB T I0 (u) λ M ωs ωs 2 where I0 and I1 are modiﬁed Bessel functions, ωs is the the secular (angular) frequency, λ is the optical probe wavelength, M the ion mass, and T the mean ion temperature, typically in the 1–10 mK range. Assuming the mean temperature is given by 3kB T /2 = M < V 2 > /2, we obtain ﬁnally for the fractional shift in frequency due to the secular motion: Δν 3kB T = . ν0 2M c2

(5.20)

To achieve this, thermal Doppler limit presumes that the micromotion at the trapping ﬁeld frequency has been suﬃciently annulled. Because of its higher frequency and hence smaller modulation index, the sideband amplitudes for the micromotion are relatively weaker than the secular motion ones. In practice, the micromotion sets a higher limit on the Doppler shift than the secular (thermal) motion. The Doppler limit for the 199 Hg+ optical standard is on the order of 2×10−18.

5.1 Optical Frequency Standards

151

Quadratic Stark Shifts The displacement of the ion from the center of the trap due to its secular motion and micromotion brings into play the high-frequency trapping electric ﬁeld, which can induce ac Stark shifts in the ground and excited levels. To a lesser extent the blackbody radiation characteristic of the temperature of the trap, as well as stray laser pumping or repumping radiation will also produce ac Stark shifts. Even the probing radiation itself causes in principle a light shift. Unlike the quadrupole shift, the Stark shift depends quadratically on the electric ﬁeld and is therefore generally weaker. To compute the size of the shift requires a knowledge of the scalar and tensor polarizabilities α0 and α2 of the 2 S1/2 lower level, and the upper 2 D5/2 level. The 2 S1/2 level has α2 = 0. In terms of these, the Stark shifts can be computed from the following [266]: 1 1 2 1 3m2 − J(J + 1) 2 , Δνs = E − α0 − α2 (3 cos θ − 1) (5.21) h 2 4 J(2J − 1) where θ is the angle between the electric ﬁeld and the axis of quantization. Gravitational Red Shift Einstein’s General Theory of Relativity predicts that in a static gravitational ﬁeld such as that of the Earth (if we neglect its relatively slow rotation), the particular proper time that attaches to a given point in the ﬁeld diﬀers from the coordinate time scale which belongs to a general frame of reference deﬁned far from the ﬁeld region. Moreover, quantitatively the theory predicts that if two identical clocks are placed at diﬀerent points in a gravitational ﬁeld where the gravitational potentials are Φ1 and Φ2 then they will run at diﬀerent frequencies, thus: Δν =

(Φ1 − Φ2 ) ν. c2

(5.22)

This eﬀect will naturally be important for frequency standards aboard spacecraft, such as those aboard GPS satellites. Other Systematic Biases As the level of accuracy contemplated is pushed higher and higher there will inevitably emerge subtle processes that become signiﬁcant. But even at the current level there are some biases whose existence may be diﬃcult to identify and correct. This applies particularly to the data collecting and handling electronics, ground loops, asymmetry in the laser spectrum, the eﬀect of neighboring ion levels.

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5 Optical Spectroscopy

5.1.5 Optical Frequency Measurement The central problem in optical frequency measurement is the coherent comparison of diﬀerent frequencies bridging the wide range from the optical down to the microwave region of the spectrum. The classical approach was to mix one coherent source of radiation with another in a nonlinear device, such as a point contact metal-insulator-metal (MIM) diode and measure the beat frequency between the higher frequency and a harmonic of the lower, thus: νbeat = ν2 − nν1 .

(5.23)

To complete a chain of frequency comparisons based on frequency multiplication is complicated by the wide gaps between available reference lasers and their limited tuning ranges. A more recent method relies on a frequency division method in which at each stage the diﬀerence between two frequencies is divided by 2. If the two frequencies are ν1 and ν2 , ﬁrst the frequency ν3 = (ν1 + ν2 )/2 is generated and then ν3 is mixed with ν2 to obtain (ν3 − ν2 ) = (ν1 − ν2 )/2. The process is continued until the frequency difference is in the microwave region of the spectrum. Another technique worth mentioning is the method of parametric division [267], in which an optical parametric oscillator consisting of a biaxial KTP crystal is placed in a doubly resonant cavity where the signal and idler signals are simultaneously resonant at slightly diﬀerent frequencies ν1 and ν2 . The beat frequency between these two νμ = ν2 − ν1 , which is in the microwave range, is phase locked to a standard, so that since the pump frequency νp = ν1 + ν2 , we have ν1,2 = (νp ± νμ)/2, again producing division by 2. The whole approach of establishing frequency chains by multiplication or division was rendered obsolete by a technique ﬁrst introduced back in 1978 by the H¨ ansch group [224] involving frequency combs. The term applies to a regular sequence of coherent equally spaced optical frequencies which serves as a “ruler” for the measurement of an unknown frequency or for the synthesis of other frequencies. There are two methods that have been applied in producing such a frequency comb: the ﬁrst is based on the use of an electro-optic modulator (EOM) inside a cavity resonantly locked to an external laser. The driving frequency of the EOM is tuned to one of the mode frequencies in the cavity, as shown in Fig. 5.13.

Fig. 5.13. Intracavity EOM frequency comb generator

5.1 Optical Frequency Standards

153

It was not until 1991, however, that a transforming development in the ﬁeld occurred with the demonstration at St Andrews University by Sibbett’s group [268] of the Kerr-lens mode-locked (KLM) Ti:Sa laser. The choice of the crystal Ti:Sa is due to its extremely wide gain bandwidth allowing it to support ultra-short pulses with a broad Fourier spectrum. Before discussing the workings of the KLM Ti:Sa laser it may be useful to recall the properties of the optical Kerr eﬀect. First, one must distinguish between the optical (or AC) Kerr eﬀect from the static ﬁeld eﬀect in which a birefringence is induced in a material, a property long used to construct an optical shutter. The optical Kerr eﬀect on which the mode locking of the Ti:Sa laser depends is a nonlinear phenomenon in the crystalline medium. The essential property is that in the Ti:Sa crystal the refractive index is a function of the intensity of the laser ﬁeld according to the following: n = n0

3χ(3) | Eω |2 . 8n0

(5.24)

We note that in spite of the quadratic dependence on Eω it is the thirdorder susceptibility that is involved, and therefore the eﬀect is not restricted to crystals lacking inversion symmetry. Since the optical ﬁeld amplitude in a laser cavity is maximum along the axis, falling oﬀ rapidly away from it as a Gaussian function, it follows that the wavefront will travel slower along the axis making the wavefront converge toward the axis. Since the change in refractive index increases with light intensity, the tighter the convergence the more it is reinforced. This can lead to mode locking under two circumstances: (1) the presence of an eﬀective “soft” aperture or (2) the presence of an actual “hard” aperture. The ﬁrst can occur when the Ti:Sa crystal is optically pumped by a laser beam narrowly concentrated along the axis, in which case the convergence produced by the Kerr eﬀect increases the overlap with the pumping beam and hence the optical gain. On the other hand, a small physical aperture appropriately placed in the cavity to absorb all radiation beyond a ﬁxed radius would again favor the build up of high-peak intensity by phase locking the longitudinal modes. The result is an output consisting of a train of equally spaced, extremely sharp, pulses. The pulse width and repetition rate are determined by the highest longitudinal mode the laser amplifying medium can support and the free spectral range of the cavity. Thus if N is the maximum order of longitudinal modes and the mode separation is fr , then the requisite bandwidth of the laser medium is N fr if all the modes share a common phase ϕ0 we have for the laser ﬁeld: n=(N −1)/2

E(t) = e

iϕ0

E0 exp[ 2πi(f0 + nfr )t],

(5.25)

n=−(N −1)/2

and the desired time dependence of the intensity is given by E(t) · E ∗ (t) averaged over a time long compared with 1/f0 but short compared with 1/fr , which yields the following:

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5 Optical Spectroscopy

I(t) =

I0 sin2 (πN tfr t) . N sin2 (πfr t)

(5.26)

This function has a large amplitude whenever t = n/fr , where n is any integer, but otherwise oscillates with small amplitude. It is important to note that the frequencies in the comb are not simple harmonics, but contain an oﬀset frequency f0 ; thus we have fn = f0 + nfr .

(5.27)

This mode structure is more easily understood in the time domain: a sharp ﬁeld pulse bounces back and forth with the group velocity in the laser cavity producing an output series of pulses that repeat with the frequency fr . On the other hand, because of optical dispersion, the phase and group velocities are diﬀerent inside the cavity, which leads to a pulse-to-pulse shift in the phase between the carrier frequency and the pulse envelope. The result is a shift in the frequencies of all the lines in the comb by the same amount: the frequency oﬀset f0 as illustrated in Fig. 5.14. The basic elements of a KLM Ti:sapphire laser are shown schematically in Fig. 5.15. The design depicted in Fig. 5.15 is based on a standard Z-folded fourmirror resonator with two focussing spherical mirrors and two arms bounded at one end by a chirped plane mirror, and the other one partially transmitting, is labelled OC, the output coupler. If the eﬀective distance D between the two focusing mirrors M1 and M2 is written D = F1 + F2 + δ, then δ is a design parameter that determines stability. Kerr lens mode locking usually occurs in a particular range of δ. The Ti:Sa crystal is placed between the focussing mirrors and generally pumped by a diode pumped solid-state (DPSS) laser at λ = 532 nm delivering 4–5 W of pump power.

Fig. 5.14. (a) Relative phase of carrier and envelope slips Δϕ per interval fr−1 in the time domain; (b) Resulting frequency oﬀset f0 in the frequency domain

5.1 Optical Frequency Standards

155

Fig. 5.15. Schematic diagram of the layout of a mode-locked Ti:Sa laser

Fig. 5.16. Cross-section of air-silica microstructure optical ﬁber

To attain phase lock over the broadest possible frequency range, it is necessary to compensate for optical dispersion. This is accomplished in a Z-folded cavity through the use of the chirped multilayer dielectric mirror and the dispersive property of prisms. The design calls for two prisms arranged to produce a negative dispersion, in which red wavefronts are delayed more than the blue, opposite to the “normal” positive dispersion. The distance between the prisms must be critically adjusted to compensate for group delay dispersion. The prisms are set at minimum deviation with the beam incident at Brewster’s angle for minimum loss. What really transformed the measurement of optical frequencies was an advancement in ﬁber optic technology by a group at Bell Labs [269] who analysed the waveguide properties of ﬁbers, in which the silica core is surrounded by cladding with an array of capillary air holes running parallel to the axis. An essential feature of this microstructure optical ﬁber, also called photonic crystal ﬁber (PCF), is the ability to achieve zero dispersion at around a speciﬁed wavelength by balancing the normal material dispersion against the anomalous waveguide dispersion. This, combined with a small eﬀective core cross section, has made possible a number of nonlinear functions including broadband continuum generation in the visible part of the spectrum. In Fig. 5.16 the cross-section of such an air-silica microstructure ﬁber is shown schematically. The properties that these ﬁbers have which make it possible to directly compare optical and microwave frequencies are: (1) the extreme peak intensity of the pulsed laser ﬁeld along the core, raising the eﬃciency of nonlinear eﬀects, such as harmonic generation, frequency mixing, Raman scattering etc., (2) the possibility of designing the air-silica microstructure so that only one mode is

156

5 Optical Spectroscopy

Fig. 5.17. The dotted curve is the power spectrum direct from a Ti:Sa laser; the solid curve is the ﬁber output spectrum

propagated, and (3) being able to shift the wavelength at which the dispersion is zero, to an optimum value. When a laser beam passes through such an optical ﬁber the output spectrum is extremely widened, as shown in Fig. 5.17. When combined with the KLM Ti:Sa laser the microstructure optical ﬁber has been shown to be capable of generating a frequency comb spanning a full octave. This was an important threshold: it marked the possibility of a radically simpliﬁed approach to frequency synthesis and measurement throughout the entire spectrum. Such a comb provides a ruler whose frequency “units” can be directly and coherently related to a microwave standard, a ruler that can span the full range from one frequency to its second harmonic, as produced in a nonlinear element. It is no longer necessary to build a frequency chain consisting of a room full of lasers and electronics for each particular frequency comparison. In 2000 at the Max Planck Inst. (Garching) Holtzwarth [270] introduced the use of a PCF in conjunction with a femtosecond Ti:Sa laser to set up an optical frequency synthesizer, and used it to measure directly the frequency of an iodine stabilized Nd:YAG laser at 282 THz (λ = 1, 064 nm) in terms of the microwave separation of the comb frequencies. The Ti:Sa laser used had a bandwidth supporting 25 fs pulses at a repetition rate of 625 MHz. We recall that the spectrum of the frequency comb contains an oﬀset frequency which must be determined before the comb can be used to measure absolute frequencies. The ability to construct a comb that spans over a full octave makes it possible to determine the oﬀset frequency in a “self referencing” manner; that is, it does not rely on any outside frequency. That is the crucial breakthrough in the birth of the new optical frequency metrology. In the Garching work alluded to, the procedure was as follows. First, lock the comb repetition frequency to a reference 10 MHz standard by controlling the Ti:Sa laser cavity length with a piezo-mounted mirror. Second, lock a stabilized Nd:YAG laser (λ = 1, 064) to one of the modes of the frequency comb by forcing the beat to be phase locked to a local oscillator LO1064 . Then double the frequency f1,064 and observe a beat note between 2 × f1,064 and the nearest mode frequency in the comb, that is: f0 + 2nfr .

5.2 Progress in Standards

157

The beat frequency must equal f0 − 2LO1064 . Lastly, the beat frequency f0 − 2LO1,064 is locked to another local oscillator LO532 so that f0 − 2LO1,064 = −LO532 by adjusting the power of the pump laser driving the Ti:Sa laser, thereby changing the optical length in the cavity through the optical Kerr eﬀect. The two adjustments used to achieve the described locks, that is of power and cavity mirror, are not “orthogonal” but do aﬀect the phase and group velocities in the laser cavity diﬀerently. The ﬁnal result for f0 is then (2LO1,064 − LO532 ). A recent signiﬁcant measurement at NIST [271] uses this new technology to carry out an absolute frequency measurement of the 199 Hg+ optical standard in terms of the SI second, as realized by the NIST-F1 cesium fountain standard, with a total fractional uncertainty of about 1×10−15, limited by the statistical measurement uncertainty. This uncertainty is within a factor of only about 2.3 of the current uncertainty of the Cs fountain standard itself.

5.2 Progress in Standards Over the course of evolution of time-keeping methods, there has been a relentless acceleration in the pace of technological change. After centuries of sundials and water clocks, tower clocks and pendulums, we have, within just the last

Fig. 5.18. The evolution of clock accuracy over the centuries

158

5 Optical Spectroscopy

century the appearance of electronic time keeping rf quartz oscillators, and microwave atomic clocks. This culminated toward the end of the century and the beginning of the present one with the realization of optical clocks based on single ions, thereby coming close to the ultimate time-keeping mechanism: the internal quantum transition of an isolated atomic ion frozen in space. The historical progression toward greater and greater stability and accuracy of mechanical (nonastronomical) devices for time keeping parallels operation at higher and higher frequency on systems less susceptible to environmental factors. Figure 5.18 illustrates the steeply increasing rate of progress in accurate time keeping over the last 100 years compared to earlier times. If the progression of the microwave standard uncertainty and that of optical frequency standards over the last half century are plotted against time, an interesting exponential decrease is revealed, as shown in Fig. 5.19. The microwave standards uncertainty decreases on the average by a factor of 10 every decade, while the optical standards are improving on the average a factor of 10 in only about 3.5 years! Not only does the apparent exponential dependence of uncertainty on time itself deserve explanation, but also the fact that the exponential coeﬃcient is so diﬀerent for the two types of standards. These are questions that organizations that support research would do well to ponder. Presumably the more rapid development of optical standards is a result of breakthroughs in ion trapping and laser cooling that brought immediate beneﬁts in terms of accuracy.

Fig. 5.19. The exponential rates of progress in accuracy of microwave standards vs. the optical standards

5.2 Progress in Standards

159

This is in contrast to the relatively slow and methodical evolution over time of Cs beam machines culminating in the ultimate vertical “fountain” machine that also beneﬁtted from laser cooling. It is remarkable that at least, as far as the data extend, the rate of improvement of neither type of standard shows any sign of further improvements slowing down.

6 Lifetime Studies in Traps

6.1 Radiative Lifetimes A knowledge of the radiative lifetimes of excited atomic states is of wide interest, not only in the detailed understanding of intrinsic atomic structure and dynamics but also in the ﬁelds of plasma diagnostics and astrophysics. The transition rates of intercombination and electric-dipole forbidden lines are of particular importance, since their low transition probability gives them long optical depths in plasmas and astrophysical environments. The measurement of radiative lifetimes of ions requires their isolation, free from disturbing collisions with other particles, at least for times long compared to their lifetime. For short nanosecond-range lifetimes, many techniques are available; however, for highly metastable states, ion trapping electrodynamically in ultra-high vacuum has a unique advantage; using it, the range of accessible lifetimes starts in the millisecond region and extends to the almost limitless storage time in the trap [272]. By using ultra-high vacuum techniques such as cryopumping, collisions involving other particles, which may cause nonradiative de-excitation, can be made rare indeed. Thus, by using cooled single ion conﬁnement techniques it has been possible to accurately measure many forbidden higher multipole transitions, providing important tests of the theory of atomic structure. The mean lifetime τe of an excited level e is related to the transition rates Ane to N lower levels by the following: τe−1 =

N

Ane .

(6.1)

n=1

The spontaneous transition probability, or Einstein A-coeﬃcient, for electric dipole transitions is given by Ane =

8πω 3 n

3c3 h

| φn | D | φe |2

(electric dipole),

(6.2)

162

6 Lifetime Studies in Traps

where D is the electric dipole operator given by D = −e relec . Note that this spontaneous dipole transition probability is of the order of α3 , in atomic units. For the magnetic dipole transition probability, D is replaced by the e operator μ = − 2mc (L+2S), while for an electric quadrupole transition the transition probability is given by Ane =

πω 5 | φn | Dij | φe |2 5h 45c n

(electric quadrupole) ,

where Dij is the quadrupole moment operator deﬁned by qr2 (3ri rj − δij ). Dij =

(6.3)

(6.4)

q

The magnetic dipole and electric quadrupole transition probabilities are of the order of α5 , making them about 1372 times smaller than the electric dipole value. As with the measurement of any physical quantity, there are two aspects to the merit of the experimental method chosen: precision and accuracy. The ﬁrst is determined by the statistical error in measurement, which may be due to uncontrolled “random” or fundamental quantum ﬂuctuations. The reduction of statistical error requires a large number of repetitions of the measurement cycle, and therefore, the method chosen should have the measurement cycle occupy the shortest possible length of time. Accuracy, on the other hand, relates to the question of whether the measured quantity is in fact what is intended to be measured, that is, corrections must be determined and applied for all relevant systematic sources of error. 6.1.1 Experimental Methods of Lifetime Measurement In what follows, experimental design and procedures will be described in terms of a speciﬁc system: the metastable 3D states of the singly charged calcium ion, Ca+ , with the understanding that the treatment is typical of a number of systems, such as Sr+ , Ba+ , Ra+ , Hg+ , Yb+ , which have similar level structures. The mean radiative lifetime of the D states in these ions ranges between a few 10 ms to 80 s. The relevant energy levels of Ca+ are shown in Fig. 6.1. As a system with one electron outside a closed shell structure it has an alkali-like term structure with a 42 S1/2 ground state with a strong (“ﬁrst resonance”) electric dipole transitions allowed to the 42 P1/2 , 42 P3/2 states at λ = 397, 393 nm and weak electric quadrupole transitions to the 32 D3/2 , 32 D5/2 states at λ = 733, 729 nm, respectively. Direct Decay Method In this direct technique involving one laser [41], a cloud of ions stored in a Paul trap, possibly cooled by a buﬀer gas, is irradiated with a laser pulse of

6.1 Radiative Lifetimes

163

Fig. 6.1. A partial energy level diagram of Ca+ showing the 3D metastable states

a few nanoseconds duration to excite the ion from the 42 S1/2 ground state to the 42 P3/2 state from which its decay branches between the ground state and the metastable 3D-states. The exponential decay of the quadrupole radiation intensity at λ = 729, 733 nm emitted by the ions as they return to the ground state is measured by a photomultiplier using appropriate narrow-band ﬁlters. An approximate estimate of the expected signal/noise ratio under typical experimental conditions reveals that an extremely low dark current photomultiplier is required. Thus, assuming a typical 105 ion population in a Paul trap with 2r0 = 1 cm, and taking into account the excitation eﬃciency and branching ratio in the emission, the acceptance solid angle of the detector, the quantum eﬃciency of the photomultiplier and ﬁlter, and other losses, the number of detected photons per laser shot is perhaps on the order of 5. Since the time between consecutive laser pulses must exceed the mean lifetime of the D-states, typically 5 s, we are left with a data accumulation rate of 1 count per second! This imposes extreme demands on low dark current and the suppression of stray light reaching the photomultiplier, as well as very long averaging times to reduce the statistical scatter of experimental points around an exponentially decreasing decay curve. The long containment time possible in a Paul trap permits such long observation times on the same ion ensemble; however, the data shown in Fig. 6.2 were taken at a background pressure of about 10−3 Pa, and the observed decay is dominated by collisional quenching. A recent development in this technique [273] takes advantage of picosecond laser pulse technology. Again only one laser is used, but one producing picosecond pulses, which selectively excite a single trapped ion to a particular excited state. But because the time resolution is so high, it is able to achieve satisfactory accuracy on short lifetimes where the excitations can be repeated at high frequency, and the statistics built up in a reasonable time. Thus the measurement cycle, which is repeated at high frequency, consists simply of the following: an ultra-sharp laser pulse selectively excites the ion in a Paul trap to the desired quantum level, followed by the detection of the spontaneously

164

6 Lifetime Studies in Traps

Fig. 6.2. Time resolved ﬂuorescence decay of the 3D3/2 state of Ca+ ion at 10−5 mbar buﬀer gas pressure. The averaging time was 1 h [41]

emitted photon, whose arrival at a photomultiplier is correlated in time with the excitation pulse. The lifetime is extracted from the distribution of time delays from a large number of such cycles. Only one decay photon is emitted per cycle; nevertheless, since it is assumed that the lifetime is in the nanosecond range, a high-repetition rate can be used, easing the requirement on low dark current and background radiation in the detector. The method was applied by the group at the University of Michigan to the 5p2 P1/2 and 5p2 P3/2 levels in Cd+ . The pulsed excitation source was a picosecond mode-locked Ti:Sa laser operating at λ = 906 nm, whose output was quadrupled in frequency to λ = 226.5 nm through phase-matched LBO and BBO nonlinear crystals. The laser pulse energy was about 10 pJ and the bandwidth was approximately 0.4 THz corresponding to a pulse length tp = 1.0 ps. The probability of excitation Pex of the ion into the P-states can be estimated from the following expression: Pex = sin2 (γ 2 /4πIs )(Ep tp /w2 ), (6.5) where γ is the atomic linewidth, Is the saturation intensity, Ep , tp the pulse energy and duration, and w is the laser beam waist at the ion. The estimated excitation probability in the experiment was 10%. By tuning the pulsed laser to the 1 S0 → 1 P1 resonance transition at λ = 228.8 nm in neutral cadmium atoms, it was also used to load ions in the trap by photoionization. The Doppler cooling of the ion is produced by a CW laser tuned about one linewidth below the resonance wavelength at λ = 214.5 nm, and results in the ion micromotion in the trap being reduced to have a kinetic energy under 1 K. To record digitally the time delay between the excitation pulse and emitted photon, a time-to-digital converter is used in a time reversed mode in which the start pulse is the output signal from the photomultiplier and the stop pulse is synchronized with the reference clock of the mode-locked laser providing the excitation pulse. Following the Doppler cooling of the ion for 500 ns, a reference clock synchronized with the laser pulse train triggers an AOM to allow about

6.1 Radiative Lifetimes

165

15 laser pulses separated by intervals of 12.5 ns (the longest lifetime designed for) to irradiate the ion. The repetition rate of this cycle was limited to 1 MHz. The probability of detecting the emitted photon is estimated at about 2×10−4 giving an average photon count rate of 3,000 counts per √ second corresponding to a statistical fractional error of Δτ /τ 2.5×10−3/ T , where T is the data collection time in minutes. The eﬀective isolation in ultra-high vacuum and freezing of the subject ion at the center of a Paul trap, where the rf ﬁeld is zero, and the radiative decay “in the dark,” guarantee the absence of common sources of error in the measurement of radiative lifetimes. However in the present case, involving high-speed pulse electronics, a signiﬁcant source of error is instrumental, principally the time-to-digital converter. Correction for this instrumental error requires the normalized response function of the pulse handling system. The lifetime of the 5p2 P1/2 state extracted from ﬁtting the data taken with this system to a single exponential is expected to have a systematic bias of 3–5% as a consequence of the convolution of the timing system response function. This is aggravated by t = 0 events arising from scattering of the excitation pulse from the apparatus. An attempt is made to take this into account by an additional convolution of a δ-function at t = 0. The ﬁnal results are 3.148±0.011 ns for the 2 P1/2 state and 2.647 ± 0.010 ns for the 2 P3/2 state. Sequential Pulsed Laser Methods In the determination of long lifetimes of metastable states, the severely low count rate of decay photons in the direct single laser excitation method can be circumvented, for example in the case of the D3/2 level in Ca+ , by using two sequentially pulsed lasers [274]: one laser tuned to the S1/2 –P1/2 resonance transition and the other tuned to the D3/2 –P1/2 transition. The ﬁrst laser excites the ions into the P1/2 -state, where they undergo a branching decay into the S1/2 and D3/2 levels. Ions that decay into the metastable D3/2 state can leave it only through the weak quadrupole transition to the ground state, and therefore, initially all will accumulate in that state. The second laser is applied at a precisely variable interval after the ﬁrst laser pulse. Since it induces an allowed electric dipole transition to the P-state, which itself undergoes a strong dipole transition to the ground state, the eﬀect of the second-laser pulse is to transfer all the ions remaining in the D3/2 state to the ground state at the end of the delay. In doing so, those ions will emit ﬂuorescent photons which are detected by a photomultiplier. The time required for this process, and thereby the measurement of the D3/2 -state population, is set by the short radiative lifetime of the P1/2 -state which is about 108 times shorter than for the quadrupole transition under study. This means that the photon count rate is correspondingly higher, and therefore, not only is the need for low photomultiplier dark current relaxed, but also a good signal-to-noise ratio can be obtained in a much shorter time. By varying the time delay between

166

6 Lifetime Studies in Traps

Fig. 6.3. Time decay of the 32 D3/2 state in Ca+ using the two laser pump-probe method. The averaging time was 10 min [274]

the two laser pulses the exponential decay of the D3/2 -state population can be derived and its mean lifetime computed. In the case of the 32 D5/2 metastable state, another method involving three lasers allows all the ions to be pumped (“shelved”) into that state and by blocking one laser, to eﬀectively make each ion that undergoes a quadrupole transition to the ground state cause strong resonance ﬂuorescence to be emitted [275]. The measurement sequence is as follows: the ﬁrst laser continuously excites the ions from the ground state into the 4P1/2 state, where they quickly undergo a branching decay to the ground state and the D3/2 , the second CW laser is tuned to drive the transition P1/2 → D3/2 to saturation. If the intensities of the two lasers are such that their respective transitions are saturated, the rate of ﬂuorescent photon emission is determined by the lifetime of the P1/2 state, which is about 7 ns. This corresponds to a ﬂuorescence rate on the order of 107 s−1 from a single ion. Even taking into account the severely lowdetection eﬃciency and the fact that the lasers may illuminate only a fraction of the ion cloud, nevertheless an acceptable signal-to-noise ratio is possible for the ﬂuorescence signal. The placing of the ions in the metastable D5/2 state is achieved simply by applying another laser tuned to the D3/2 → P3/2 transition which allows spontaneous transitions P3/2 → D5/2 to occur, leading to an accumulation of ions in the D5/2 state, from which there is no escape except by possible quenching collisions with other particles or quadrupole transitions to the ground state. Therefore, the ﬂuorescence will cease and the signal falls to the background scatter level. To measure the lifetime of the D5/2 state, it is now necessary to block the third laser so that no ions are able to enter that state, and as ions decay from it to the ground state they produce a commensurate increase in the intensity of ﬂuorescence induced by the other lasers. All that remains is to record the ﬂuorescent photon count rate as a function of time following the blockage of the third laser to obtain the lifetime of the D5/2 state. Since, the

6.1 Radiative Lifetimes

167

Fig. 6.4. (a) Relevant transitions in Ca+ . (b) Fluorescent intensity as a function of time. At t0 the laser L3 is on, and at t1 the laser L3 is blocked [275]

radiative decay of the D5/2 level is an exponential function of time it follows that the intensity of the ﬂuorescence should increase as I = I0 [1 − exp(−t/T )],

(6.6)

where T is the mean lifetime of the state. Referring to Fig. 6.4 the cooling (L1 ) and repumping laser (L2 ) are running continuously. At t = t0 laser L3 is turned on to cause ions to accumulate in the 3 2 D5/2 state. At t = t1 the laser beam L3 is blocked allowing only the radiative decay of 3 2 D5/2 population to the ground state and the reappearance of the ﬂuorescence. By tuning laser L1 below the frequency of the transition 2 S1/2 →2 P1/2 , Doppler cooling of the ion cloud will take place and its diameter will shrink, providing a better spatial overlap with the laser beams. Moreover, the Doppler width of the transitions will be reduced, further enhancing the laser-induced transitions. The method has a clear advantage in signal-to-noise ratio: in fact a single measurement lasting perhaps a few seconds is suﬃcient to yield a measure of the lifetime, making it possible to rapidly vary parameters to study any possible systematic eﬀects. A possible source of systematic error could arise when the ions, having been pumped into the D5/2 state are decoupled from the cooling laser, and their distribution expands as their temperature rises. Potentially a much “cleaner” way to measure the lifetime of a long lived excited state is on a single trapped ion in ultra-high vacuum. In the case of the Ca+ ion, the same method can be applied to a single ion as discussed for an ensemble in Sect. 6.1.1. This method, when applied to a single ion takes on a binary aspect, the ﬂuorescence being either “on” or “oﬀ” accordingly as the ion has or has not made a quantum “jump” (Chap. 7). Its application to the Ba+ was ﬁrst published practically simultaneously by Dehmelt [237] and Toschek group [236] in 1986. As described in the last section, the ion is continuously excited by two lasers tuned to the 4S1/2 → 4P1/2 and 4P1/2 → 3D3/2 transitions. Detection of transitions is monitored by the ﬂuorescence from the 4P1/2 state. The

168

6 Lifetime Studies in Traps

Fig. 6.5. Typical photomultiplier output of single Ba+ ion ﬂuorescence in diﬀerent buﬀer gas pressure. Reprinted with Permission from [276]

application of a third laser tuned to excite the ion into the 4P3/2 state, causes it to end up in the metastable 3D5/2 state, and the ﬂuorescence vanishes completely until the ion decays back into the ground state. Thus, the ﬂuorescence discretely jumps repeatedly between bright and dark periods. The periods of zero ﬂuorescence are the lifetimes determined by radiative decay of the ion to the ground state and are distributed according to the exponential law given earlier. Figure 6.5 shows the typical output of the photomultiplier, showing the bright and dark periods corresponding to the state of the ion. A histogram of the dark periods should follow a decaying exponential function of time, from which the mean lifetime may be computed. A disadvantage of this method is that each dark period, which may last several seconds, provides only one data point, and many thousands of points may be required to achieve a desired low level of statistical error. However, it does have the advantage that the measurement is being carried out in total isolation at the center of the trap, and all possible sources of systematic errors are either absent or well controlled. Figure 6.6 shows a typical histogram of dark periods taken from several thousands of CCD images of Ba+ ions in a linear Paul trap [277]. Experiments on single Ca+ ions in a linear trap, using similar laser excitation and detection have been reported by the Mainz [278] and Oxford group [277], with a meticulous discussion of errors, including the statistics of the photons emitted in the quantum “jumps.” In a more recent article by the group at the University of Innsbruck [279], a somewhat diﬀerent procedure of measuring the D5/2 lifetime is described in which that state is populated by coherent excitation from the ground state. Thus, after applying the cooling and repumping lasers, a coherent laser π-pulse is applied at λ = 729 nm, the S1/2 → D5/2 transition frequency, to transfer

6.1 Radiative Lifetimes

169

Fig. 6.6. Histogram of dark periods recorded by CCD camera of Ca+ ion ﬂuorescence. Reprinted with permission from [277]

the ion to the metastable state, where it is left to decay spontaneously to the ground state in the absence of any laser light. The detection of the ion’s transition to the ground state is through the appearance of ﬂuorescence at λ = 393 nm due to P1/2 → S1/2 transitions. After a variable measured waiting time Δt, the ﬂuorescence is again measured, from which the decay probability of the D5/2 -level in time Δt is deduced. Thus, the sequence of operations is as follows: 1. Ca+ cooling and state preparation consisting of 2 ms of Doppler cooling using 397 and 866 nm lasers, repumping from the D5/2 level using 854 nm laser and optical pumping into the S1/2 (m = −1/2) Zeeman sublevel using circularly polarized 397 nm laser light. 2. Coherent excitation at 729 nm, with pulse length and intensity corresponding to a π-pulse, to produce nearly unity excitation of the D5/2 (m = −5/2) Zeeman sublevel. 3. Detection for 3.5 ms of the possible ﬂuorescence at λ = 393 nm by the ion at the beginning and end of a predetermined delay Δt to establish whether it has decayed from the D5/2 level during that time. The mean decay probability during Δt is computed as the ratio of the number of ﬂuorescent signals to the total number of repetitions of the cycle. This measurement cycle must be repeated typically thousands of times to narrow suﬃciently the statistical spread of data points. If p is the probability of decay,

170

6 Lifetime Studies in Traps

then the data is ﬁtted to the following exponential function: 1 − p = exp(−Δt/τ ).

(6.7)

The determination of the lifetime of the D3/2 level is complicated by the fact that the laser inducing the detection transition S1/2 −→ P1/2 leads to branching decay into that level, violating its free decay. The method adopted by the Innsbruck group of circumventing this is to transfer to the D5/2 level the ion population that decays from the D3/2 level to the ground state. This is done by a coherent π-pulses at λ = 729 nm. Since the D3/2 level may decay into both of the two Zeeman sublevels of the ground S1/2 level, two π-pulses are required to transfer the entire ion population to the D5/2 level. Methods Using the Static (Kingdon) Ion Trap Another approach to the measurement of radiative lifetimes in ions uses conﬁnement in a static trap with coaxial cylinders and end caps, known as a Kingdon trap as discussed in [1]. Used in conjunction with high-energy ion sources [280] it is particularly adapted to the study of the lifetimes of metastable states in multiple charged ions [281]. These are of particular interest in the study of astrophysical sources, such as the solar corona, interstellar media, and active galactic nuclei (AGNs) where a knowledge of transition rates is important. It is a very active ﬁeld of study with an extensive literature [282] dealing with the many ion species of interest; however since our main interest lies at the other end of the energy regime, only the essential elements of the experimental aspects will be given here. The sources of the metastable ions injected into the Kingdon trap are typically machines such as the electron cyclotron resonance ion source (ECRIS) [283], the electron beam ion source (EBIS) [284] and the electron beam ion trap (EBIT) [285]. As a representative example of the implementation of this technique and what can be accomplished using it, we will outline the work at the Jet Propulsion Laboratory [286]. It consists of a series of measurements of lifetimes of metastable levels in multiple charged ions [287]. The work reported in the 2004 paper addresses the lifetime of the 2s2 2p2 1 S0 level in O+2 decaying by magnetic-dipole (M1) transitions to the 3 P1 level at λ =232.17 nm. Figure 6.7 shows the relevant energy levels of O+2 and the M1 transition to the 3 P1 level. The experiments were carried out using the 14 GHz ECRIS at the JPL Highly Charged Ion Facility [288]. The O+2 ions were generated using CS2 and CO or O2 feed gas, respectively, into the ECR, and extracted with energies of 5.2 and 2×6.4 keV, then they are mass/charge selected using a double focusing 90◦ selection magnet. On emerging from a three-way electrostatic beam switcher the ion beam is focused into the Kingdon trap, where the radiation resulting from the decay of metastable states of lifetimes exceeding about 50 μs is detected. No other ions of M/Q = 8 such as 56 Fe7+ or 32 S4+ were detected in the trap. The proportion of ions in metastable states in the

6.1 Radiative Lifetimes

171

Fig. 6.7. Partial energy level diagram of O+2 showing the 2s2 2p2 1 S0 →3 P1 transition

Fig. 6.8. Schematic diagram of the Kingdon ion trap system for measuring lifetimes in highly charged ions

ECR is found experimentally to depend on the feed gas pressure, the microwave power and the strength of the magnetic ﬁeld. In Fig. 6.8, a schematic diagram of the Kingdon trap-based set up for measuring lifetimes in highly charged ions is given. The Kingdon trap used in these experiments consisted of an aluminum cylinder, 15 cm long and 10 cm in diameter with four equally spaced 2.6 cm diameter holes in its mid-plane to allow the entry and exit of the ion beam, the emergence of the radiated photons, and ﬁnally the pulsed ejection of the trapped ions from the trap onto a microchannel plate. The intensity of the exit beam is monitored using a Faraday cup. The ends of the cylinder are covered by insulated circular plates held at a positive potential relative to the cylinder. Radial conﬁnement is achieved by having along the axis of the cylinder a 0.1 mm diameter wire, whose potential can be suddenly switched from having the same potential as the cylinder at around 5.1 KV for O+

172

6 Lifetime Studies in Traps

during the injection phase, to a potential of 2.5 KV during the conﬁnement phase. A thyratron pulser is used to switch the potential on the wire in less than 100 ns, and it is estimated that 6×106 ions/particle microampere are trapped and stabilized in about 2 ms. During the time the radiative decay is being observed the ion beam is cut oﬀ by applying a potential to a deﬂector electrode upstream to prevent additional ions from entering the trap. At an operating vacuum pressure of 4×10−8 Pa the mean storage time of the ions in the trap was determined by collisions with neutral background particles; hence the range of radiative lifetimes that can be measured extends from the equilibration time of the ions in the trap (≈1–2 ms) to the survival time in the trap (≈1–2 s). The UV optics for collection, ﬁltering and counting of the decay photons conformed to standard practice. To deduce the desired radiative lifetime, the actual observed photon count accumulated and stored in the multichannel scaler must be corrected for a number of sources of spurious eﬀects: ﬁrst of course is the decay of the ion population which has an initial rapid settling period, also there will be ions in other quantum states that on collision with parts of the apparatus may emit photons into the passband of the detection system. 6.1.2 Systematic Eﬀects on the Lifetimes The decay time of a metastable state actually observed may not result from radiative transitions of a free ion, but may be subject to instrumental and environmental shifts that must be taken into account. The most important among these, especially for long-lived metastable states, are de-excitation by collisions with background particles, spurious light scattering, and mixing of the state under investigation with other states by the electric ﬁeld of the trap. Thus, we may write: γtot = γrad + γcoll + γlight + γmix .

(6.8)

Mixing of States If the trapped ion is subjected to a residual electric ﬁeld, its metastable state is perturbed, resulting in an admixture of states of diﬀerent parity, If the admixed state has a fast decay rate, even a small amount of admixture may signiﬁcantly change the observed decay rate. For example in the case of Ca+ the closest states that are mixed with the metastable 3D states are the 4P states, whose lifetime is only about 7 ns. The eﬀect of mixing can be roughly estimated; thus if the electric ﬁeld strength is E0 , the D-state wave function ψD would be replaced by ψD as follows: ψD = aψD + bψP ,

with a2 + b2 = 1. The amount of admixture of the P-state is given by

(6.9)

6.1 Radiative Lifetimes

b=

ea0 E0 , wD − wP

173

(6.10)

where e is the electronic charge, wD and wP are the energies of the two states and a0 is a characteristic length. The decay rate of the mixed state is given by γmix = a2 γD + b2 γP . (6.11) As a numerical example to convey the relative importance of this eﬀect, assume a0 is equal to the Bohr radius (∼0.5 × 10−10 m) and take the case of Ca+ for which wD − wP 1.5 eV and γP 107 s−1 . Assuming an average electric ﬁeld strength E 2 103 V m−1 we ﬁnd for b2 γP a value of 10−8 s−1 , which is negligible compared with a2 γD , which is 1 s−1 . The result would have been quite diﬀerent of course if the energy diﬀerence between the levels happens to be much smaller. Light Scattering Eﬀects Spurious light originating for example from ampliﬁed spontaneous emission (ASE) in diode or ﬁber lasers may lead to resonant transitions from the metastable state under study to other fast decaying states. Also oﬀ-resonance excitation by a strong light source may have the same eﬀect. For example, it has been reported [278] that in lifetime measurements on a single Ca+ ion, the observed lifetime of the 3D5/2 metastable state depended on the power of the 3D3/2 → 4P1/2 repumping laser. This may have resulted from an oﬀ-resonance excitation of the 3D5/2 → 4P3/2 transition or possibly ASE from the same laser. Correction for this eﬀect requires that the lifetimes be determined for diﬀerent values of laser power, and extrapolated to zero value. An interesting eﬀect has been observed [278] in the spontaneous decays of a string of several ions trapped along the axis of a Paul trap several millimeter apart, an eﬀect which may be explained as resulting from the same eﬀect as the laser power dependence of the lifetime. The ions in the chain undergo quantum jumps which are simultaneous much more often than expected of random events. Eﬀect of Collisions In single ion lifetime measurements, collisions with background particles, though rare, still impose the most important limitation on the accuracy of long lifetime measurements, even under ultra-high vacuum conditions. Since the most important eﬀect is collisional de-excitation through binary collisions, the rate is expected to depend linearly on the background pressure, independent of the nature of the residual background gas. Lifetime measurements must be carried out at a number of pressures and an extrapolation to zero pressure carried out. In Fig. 6.9 the data taken on the Ba+ in an atmosphere of hydrogen at diﬀerent pressures plotted on the horizontal axis are shown [276].

174

6 Lifetime Studies in Traps

Fig. 6.9. Pressure dependence of the decay rate of the 2 D5/2 state in Ba+ with H2 as buﬀer gas. The value at p = 0 gives the natural decay rate (τ0 = 34.5±3.4 s) [276] Table 6.1. Natural lifetimes of metastable states in single ionized atoms measured in ion traps (SIQJ: single ion quantum jumps) Ion +

He Li+ C+ C+ C+ Ca+ Ca+ Sr+ Sr+ Yb+ Yb+ Yb+ Ba+ Ba+ Li+ Hg+ Hg+ In+ Pb+

State

Exp. lifetime

Method

References

2S1/2 23 S1 2P1/2 2P3/2 2P5/2 3D3/2 3D5/2 4D5/2 4D3/2 5D3/2 5D5/2 4F7/2 5D3/2 5D5/2 23 S1 6D3/2 6D3/2 53 P0 6P3/2

1 922 (82) ms 58.6 (12.9) ms 6.8 (4) ms 86 (12) ms 19.5 (1.2) ms 1 176 (11) ms 1 168 (7) ms 390.8 (1.6) ms 395 (38) ms 52.15 (1.00) ms 7.0 (0.4) ms 5.4 y 78.8 (4.6) s 34.5 (3.5) s 58.6 (12.9) s 9.17 (4.2) ms 86.2 (3.0) ms 140 (20) ms 41.2 (0.7) ms

Direct Direct Direct Direct Direct SIQJ SIQJ SIQJ Direct Direct SIQJ SIQJ SIQJ SIQJ Direct SIQJ SIQJ SIQJ Direct

[553] [554] [555] [555] [555] [279] [277] [556] [557] [292] [558] [249] [559] [276] [554] [560] [560] [561] [562]

decay decay decay decay decay

decay decay

decay

decay

Table 6.1 summarises lifetimes of metastable states in single-ionized atoms, experimentally determined in traps. While the achievement of accuracy in radiative lifetime measurements clearly requires experimental data on the eﬀect of collisions, such as quenching

[289]

1.05 (0.40)

0.9 (0.7) 29.5 (17.0)

Ne Ar

[289] [289]

[289]

[289]

54+91 −17

170 (20)

N2

[289]

References

H2 O CH4 CO2 He

37 (14)

Ca+

H2

Buﬀer gas 350 (50) 37 (20) 230 (30) 44 (3) 600 (200) 600 (80) 270 (50) 10 (3) 0.3 (0.2) 0.5 (0.2) 7 (5)

Ba+ [276] [290] [276] [290] [276] [276] [276] [276] [290] [290] [276]

References

0.83

0.27

2.3

Sr+

[291]

[291]

[291]

References

1780 (190) 150 (50)

1020 (100)

Yb+

[292] [293]

[292]

References

Table 6.2. Quenching rates of metastable ionic D-state for various buﬀer gases. All numbers are given in 10−12 cm3 s−1

6.1 Radiative Lifetimes 175

176

6 Lifetime Studies in Traps

involving background gases, the data can also be of interest from the point of view of the collision processes themselves. 6.1.3 Quenching Collisions The experimental techniques thus far described for the measurement of natural decay times of metastable states can be extended to study systematically the quenching eﬀect itself of diﬀerent speciﬁc gases on the radiative process. For this purpose, the ultra-high vacuum is replaced by a controlled atmosphere of precisely determined composition and density. There are a number of possible collision processes involving the ion and background particles which can cause quenching of the radiation, that is, inhibiting a particular radiative transition from taking place. This can happen commonly, for example, when the ion collides with a molecule and the ion excitation energy is transferred to the molecule raising it to one of its many excited states, or even dissociating it. Also, depending on the relative positions of the quantum levels of the ion and the colliding partner there may be a signiﬁcant cross section for electron transfer to the ion. More generally the perturbation of the metastable state during the collision with another particle will alter the transition probability out of the metastable state by inducing diﬀerent parity states. Quenching rates have been determined for the metastable states of ions in diﬀerent buﬀer gases. The quenching rate coeﬃcients were determined by measuring the lifetime of the metastable state as a function of the background gas pressure and composition, using the distribution of dark periods in the manner described in previous sections for the measurement of lifetimes. For an exponential distribution of dark times f (T ) = R exp (−RT ), the mean of values of T larger than the time constant τ is given by [276] < T >τ =

1 + τ, R

(6.12)

where R is rate of decay out of the metastable state. That is, the distribution of T remains exponential but the mean is shifted by τ. The decay rate is the sum of the natural radiative rate (1/τR ) and the eﬀect of the individual quenching gases, thus: 1 R= + Ri Pi , (6.13) τR i where Pi are the individual partial pressures and Ri the corresponding quenching rates. These partial pressures are measured and a full multivariate linear regression ﬁt performed using the quenching coeﬃcients of the diﬀerent components in the buﬀer gas as free variables. In Table 6.2 are presented the measured quenching rates for a number of ion species in diﬀerent gases. The large discrepancies between some results arise from the diﬃculties of determining the buﬀer gas density correctly.

7 Quantum Eﬀects in Charged Particle Traps

It is a fundamental feature of quantum mechanics that a group of particles can be in a state described by one common wavefunction which cannot be factored into individual particle wavefunctions; they are then said to be in an entangled state [294–296]. A measurement of the state of a constituent part of the entangled system determines the state of all the others. In a system that is not entangled, the states of the individual particles are determined independently. Ions isolated and trapped in vacuo in electromagnetic ﬁelds provide an unparalleled means of realizing long-lived entangled quantum states [297] through the coupling of the normal modes of oscillation in the trap by the long range Coulomb interaction [298–300]. The realization of entangled states among many ions [301,302] has enabled quantum applications ranging from precision metrology [255, 303] to quantum information processing [304, 305]. The main obstacle to maintaining entanglement and implementing these applications is the loss of coherence due to interactions with the environment. Among the important causes of decoherence are spontaneous emission [306–308], ﬂuctuations in the applied trapping and cooling ﬁelds, and collisions with the residual background gas particles. Great progress has been achieved in recent years to overcome these diﬃculties and demonstrate quantum entanglement phenomena in trapped ions. The most exciting recent developments in applying entangled states is in the emerging ﬁeld of quantum computing. This has been motivated by the realization that semiconductor miniaturization is approaching a physical limit set by the dissipation of heat [309], and electron tunneling. The transition to a technology based on the quantum entanglement of the states of individual ions to encode information introduces parallelism in handling information, which not only enables traditional tasks to be processed faster but also to accomplish computations not accessible to semiconductor computers. In 1985, Deutsch [310] introduced the universal quantum computer based on entanglement in the computational process, thus initiating the era of quantum information. Of the physical systems that have been proposed for implementing the operations of a quantum computer, ions frozen and isolated in ion traps have

180

7 Quantum Eﬀects in Charged Particle Traps

a clear advantage with respect to scalability and the long coherence time of entangled states so essential for parallel operation. The system proposed by Cirac and Zoller [311] which is based on such an ion system has already been implemented in the form of ions in a linear Paul trap. Other proposals involve electrons stored in a Penning trap [312, 313]. Looking to the future, quantum information handling will include teleportation of quantum states from one laboratory to another, as well as advanced cryptography with secure quantum key distribution extending over wide areas made possible by relay stations involving isolated ions in quantum cavities.

7.1 Quantum Jumps When transitions occur between quantum states of an individual atomic system, they have come to be described as quantum jumps, unlike transitions in ensembles of atoms where some atoms are in the initial state, while others are in the ﬁnal state. Since individual ions have become observable using ﬁeld conﬁnement, transitions that qualify to be called quantum jumps have been observed on them using a technique that, in this context, is called electron shelving. A description of such transitions in individual ions and their application to particular atomic systems has been given at some length in a prior chapter on the measurement of radiative lifetimes of metastable states in alkaline-earth ions, such as Ba+ (Fig. 3.5).

7.2 The Quantum Zeno Eﬀect It has been shown [314] that the time evolution of an unstable quantum state inevitably deviates from the simple exponential decay near t = 0 becoming exponential over an intermediate time period and tending to a power law dependence for very long times. For suﬃciently short times the probability of the ion remaining in its initial state falls oﬀ slower than would be expected on the basis of the exponential decay law. It follows that if the decaying quantum state is repeatedly “collapsed” back to the initial state at suﬃciently small intervals of time by observing it, it would be longer lived than if it were observed at longer intervals where the decay law is exponential. In the limit of continuous observation the atom will be found not to decay at all: This is the so-called quantum Zeno eﬀect, named for the ﬁfth century BC Greek philosopher because it is reminiscent of (but not logically analogous to) one of his famous paradoxes. Itano et al. [315] have demonstrated the quantum Zeno eﬀect on an ensemble of 5,000 trapped 9 Be+ ions using an rf transition induced between the hyperﬁne sublevels in the 2s 2 S1/2 ground state (mI , mJ ) = ( 32 , 12 ) and ( 12 , 12 ) in high-magnetic ﬁeld. These two sublevels are designated as levels 1 and level 2

7.2 The Quantum Zeno Eﬀect

181

Fig. 7.1. Diagram of the energy levels of 9 Be+ in a magnetic ﬁeld B = 0.8194 T. Reprinted with permission from [330]

in Fig. 7.1, while a sublevel of the excited 2p2 P3/2 state (mI , mJ ) =( 23 , 32 ) is designated level 3, which can decay only to level 1. The 9 Be+ ions are conﬁned in a cylindrical Penning trap and are optically pumped, cooled, and detected using 313 nm laser radiation with polarization set perpendicular to the magnetic ﬁeld. Before the rf ﬁeld resonant with the 1 ↔ 2 transition is applied, the 313 nm radiation optically pumps the ions so that about 16 17 of them end up in level 1. At this point the 313 nm beam is blocked and the rf ﬁeld, tuned to resonance with the 1↔ 2 transition, is turned on for T = 256 ms and the amplitude adjusted to make it a π-pulse. During the rf pulse, a series of n regularly spaced pulses of 313 nm radiation were applied, where n = 1,2,4,8,16,32 or 64. The time between the beginning of one laser pulse and the beginning of the next was therefore T /n. The number of ions in level 1 was observed by counting the number of ﬂuorescent photons induced by the 313 nm radiation in the ﬁrst 100 ms at the end of the rf π-pulse. From this photon count the population of level 1 was deduced after calibrating the system to allow for background counts, counter deadtime and optical pumping. If we let P2 (t) represent the probability of an ion being in level 2, then in the absence of the laser detection pulses, its value at the end of the π − pulse would be 1. Now let n detection laser pulses be applied at times tn = kT /n where k = 1,2,3,. . . n. It is shown in [315] that P2 (T ) is given by: P2 (T ) =

1 [1 − cosn (π/n)] , 2

(7.1)

and that for large n, the probability P2 (T ) decreases monotonically toward zero as n goes to inﬁnity, thus 1 1 1 − exp − π 2 /n . P2 (T )n→∞ = (7.2) 2 2 In Fig. 7.2 the theoretical and experimental probabilities for the 1 → 2 and 2 → 1 transitions showing graphically the decrease in the transition probabilities with increasing n, that is, the quantum Zeno eﬀect are plotted. The use of an ensemble of 9 Be+ ions is subject to the objection that collisions may also mimic the observed quantum behavior through a loss of

182

7 Quantum Eﬀects in Charged Particle Traps

Fig. 7.2. The decrease of the experimental and calculated transition probabilities between the states (a) 1 → 2, and (b) 2 → 1 with increasing of the number of measurement pulses n demonstrating the quantum Zeno eﬀect. Reprinted with permission from [315]

Fig. 7.3. The relevant energy levels of Yb+

coherence, which argues for doing the experiment on a single ion as originally suggested by Cook [316]. To that end, Wunderlich et al. [317] studied the Zeno eﬀect in a single 171 Yb+ ion using the hyperﬁne transition F = 0 ↔ F = 1 at 12.6 GHz in the 2 S1/2 ground state. The relevant transitions are shown in a simpliﬁed diagram in Fig. 7.3. The sequence of operations is similar to that described for 9 Be+ : Initially the ions are prepared in state |F = 0 by pumping with the 369 nm laser radiation, followed by a microwave π-pulse to place the ion in the |F = 1 state. Then, unlike the 9 Be+ case, the ion is subject to a resonant microwave driving π-pulse that is “fractionated” into a sequence of π/n-pulses alternating with the measurement laser pulses to separate their action, as shown in Fig. 7.4. A photon counter registers the presence or absence of ﬂuorescence, thereby indicating whether a transition has occurred. The observation of sequences in which all photon counts are zero would indicate the suppression of transitions out of the initially prepared state. To obtain the probability of survival in the initial state as a function of n, the number of such null photon count sequences is divided by the total number. This probability is plotted in Fig. 7.5, which shows that the probability increases to 77% for n = 10, proof the quantum Zeno eﬀect.

7.3 Entanglement of Trapped Ion States

183

Fig. 7.4. Excitation of the hyperﬁne transition in 171 Yb+ by applying (a) a microwave π-pulse; (b) a fractionated π-pulse without intermediate probing; (c) with intermediate probing using light at 369 nm and simultaneous detection of the scattered photons with a photon counter (PC); (“prep” means initial preparation) [317]

Fig. 7.5. Transition probability vs. the number of probe applications. The gray, black, and white bars indicate, respectively, the measured, quantum calculated, and classically calculated survival values [317]

The repeated measurement of decaying quantum states can also yield an inverse eﬀect in which the lifetime is shortened, an eﬀect naturally given the name anti-Zeno eﬀect [318–321].

7.3 Entanglement of Trapped Ion States The preparation and study of entangled atomic states is not only of intrinsic interest but also for quantum computing and transmission of quantum states between distant sites [322–326]. Communication over long optical channels is limited in practice because losses cause an exponential decrease of communication ﬁdelity with channel length. Fidelity in this context is deﬁned as the probability that if a transmitted pure state Ψ is received as a mixed state represented by a density matrix, then a measure of ﬁdelity is F = Ψ |ρ| Ψ. The

184

7 Quantum Eﬀects in Charged Particle Traps

Fig. 7.6. Generation of the entangled state of the trapped ions by exchange of phonons. Here |↑ or |↓ are the internal electronic states of the ion, and |0 and |1 are CM motion states

entanglement of states provides a means of achieving long distance quantum communication using low-loss linear optical channels. The preparation of atomic systems in an entangled state generally requires that they be brought into an electromagnetic interaction, under suitable environmental conditions free of perturbations that preferentially act on the particles to destroy the entanglement. That is precisely the reason that there is such a strong interest in ions isolated and cooled in an electrodynamic trap for engineering of entangled states. A description of the entanglement of quantum states in the context of ions in a trap begins with the quantum description of the ion motion near its zero point energy. Since the amplitude of oscillation is small, the trapped ions may be regarded as anisotropic simple harmonic oscillators with frequencies νi , (i = x, y, z) relative to their center of mass (CM), plus the motion of the CM itself. Under the presumed low temperature conditions, the interaction of the ions with an optical laser beam will be in the Lamb–Dicke regime. In the case of two ions, there are just two modes: the CM mode in which the two particles oscillate in phase, and the “stretch” mode in which they move in opposite phase. The two √ modes of oscillation along the x-axis for example have frequencies νx and 3νx , respectively. By irradiating the ions with laser radiation appropriately tuned to a motional sideband of the resonance spectrum they can undergo transitions in their oscillation mode simultaneous with those in the internal electronic states. When combined with the exchange of phonons [311, 327] mediated by the Coulomb interaction this can lead to an entanglement of ionic states, as illustrated in Fig. 7.6. The quantum eﬀect of entanglement has also been extended to the coherent coupling of an entangled state in a single trapped ion (Ca+ ) with a photon ﬁeld in a high-ﬁnesse optical cavity [328]. A scheme has been proposed [327] for using coherent photons for generating entanglement between remote atomic ensembles without the need for high-ﬁnesse cavities or nonclassical light ﬁelds. 7.3.1 Entanglement of Two-Trapped Ions An example of the deterministic entanglement of states in two trapped ions is found in the work of Turchette et al. [329]. The states chosen are the two

7.3 Entanglement of Trapped Ion States

185

Fig. 7.7. (a) Relevant 9 Be+ energy levels. R1–R3: Raman beams, D1–D3: Doppler cooling, optical pumping, and detection beams; (b) the internal basis states of two spins shown with the vibrational levels coupled by the red motional sideband; n is the motional state quantum number, and Ωi,± , i = 1, 2 are the Rabi frequencies connecting the indicated states [329]

hyperﬁne levels in the 2s2 S1/2 ground state of 9 Be+ ions |F = 2, mF = 2, and |F = 1, mF = 1, represented as |↑ and |↓, respectively. The ions are conﬁned in an elliptical Paul trap with major axis 0.5 mm operated at a frequency of Ω = 238 MHz. Figure 7.7 shows schematically the transitions involved in the cooling and preparation of the desired states. The ﬂuorescence induced by the beam D2 serves to monitor with nearly 100% eﬃciency whether an ion is in the |↑ or |↓ state. The beams R1,3 and R2 are tuned to induce Raman transitions in which transitions between internal states of the ions are made to be accompanied by transitions in their oscillator states by appropriate tuning relative to the side-band spectrum. The initial two-ion state from which the process of creating entanglement begins is represented by |Ψ(0) = |↓↑. By tuning the Raman beams to drive the red side-band of the stretch-mode vibration it has been shown [330] that the following entangled state can be generated: |Ψe (Φ) =

4 3 |↓↑ − eiΦ |↑↓ , 5 5

(7.3)

where Φ is a phase function that depends on the spatial separation of the ions; √ the value Φ = 0 corresponds to the singlet Bell state: Ψe (Φ) (|↓↑−|↑↓)/ 2 √ while Φ = π corresponds to the triplet Ψe (Φ) (|↓↑ + |↑↓)/ 2. To measure the extent of the entanglement achieved, as deﬁned by the ﬁdelity, the state-dependent ﬂuorescence photons produced by beam D2 were counted for durations of 500 μs, repeated 104 times and a histogram of the collected photons built up to estimate the probability of the entangled state. The results expressed in terms of the density operator ± , whose diagonal elements are the populations, are given [155] as P↓↑ P↑↓ 0.4, P↓↓ 0.15, P↑↑ 0.05, leading to a ﬁdelity of Ψe (π, 0) |± | Ψe (π, 0) 0.7.

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7 Quantum Eﬀects in Charged Particle Traps

7.3.2 Entanglement of Three-Trapped Ions The deterministic entanglement of three trapped ions has been achieved by Roos et al. [331] on 40 Ca+ ions in a linear Paul trap. Their motional state was brought to nearly the zero point by Doppler followed by side-band laser cooling, and their internal states initialized in the 2 S1/2 state by optical pumping, leaving the other 2 D5/2 state involved in the entanglement unoccupied. It can be shown that the three ion states constructed from superpositions of the S1/2 and D5/2 states can be entangled in only two diﬀerent ways: the maximally entangled Greenberger–Horne–Zeilinger (GHZ) state, namely √ |GHZ = (|SSS + |DDD)/ 2, or what is denoted as the W-state: |W = √ (|DDS+|DSD+|SDD)/ 3 state, which retains bilateral entanglement even when one of the states is measured in the {|S + |D} basis. In contrast the GHZ state would be destroyed by such a measurement. By appropriately designed laser optics, individual trapped ions in a string can be individually addressed, which allows the state of one ion to be determined without perturbing the other two. The degree of three-ion entanglement achieved was demonstrated by measuring the density matrix elements of the GHZ- and W-states to have a ﬁdelity of 0.72 and 0.83, respectively (Fig. 7.8) [331]. The observed coherence times for the GHZ state

Fig. 7.8. Absolute experimental values of the density matrix elements of the (a) GHZ quantum state, (b) W-state, (c) GHZ state after measuring the state of the ﬁrst ion only. The coherences have disappeared and the state is described by a classical mixture of |SSS and |DDD, (d) W-state remains partially entangled after measuring the state of the ﬁrst ion. Reprinted with permission from [346]

7.3 Entanglement of Trapped Ion States

187

were relatively short (1 ms) due to sensitivity to magnetic ﬁeld ﬂuctuations, whereas for the W-state longer coherence times (200 ms) were observed. Because of statistical experimental errors a description of the entangled quantum state must be given in terms of the density matrix operator ρ. For its experimental reconstruction it is expressed as a superposition of a set of orthogonal operators whose expectation values can be measured by subjecting the ions to a series of appropriately tuned and timed laser pulses. This in essence is the basic technique of state tomography [331, 332].

7.3.3 Multi-ion Entanglement Any practical implementation of quantum state entanglement in quantum computing will require such entanglement in multiparticle arrays if an interesting level of computing power is to be reached. Unfortunately, the probability of detecting a multi-ion entangled state falls rapidly with increasing number of ions [333]. The realization of such multi-ion systems is fraught with diﬃculties: ﬁrst is not only the preparation and detection of the states of individual ions but also the control of their mutual interactions. An interesting class of multi-ion entangled states is the “Schr¨ odinger cat state” consisting of an entanglement between a number of ions all of which are in the same state and an equal number in another [334]. Using the usual representation of a two-state system as formally equivalent to the states of a spin 1/2 particle with the symbols |↑ and |↓, a Cat state of N particles may be written as follows: 1

|N Cat = √ |↑, N + eiθ |↓, N , 2

(7.4)

where |↑, N = |↑1 |↑2 . . . |↑N and similary for |↓, N . One may attempt to prepare |N Cat by placing the ions in the initial state |↓, N ﬁrst, and then applying the unitary operator [335] π π ξπ ξπ UN = exp i Jx exp i Jz exp i Jz2 exp i Jx , ξ = 1, 0, 2 2 2 2 (7.5) N where J = j=1 Sj is the global spin operator, and the parameter ξ = 1 for N odd and ξ = 0 for N even [336]. The resulting entangled state |ΨN , however, diﬀers from the ideal entangled state |N Cat as measured by its ﬁdelity, deﬁned as [301]: 2

FN Cat = | ΨN | N Cat | =

1 (P↑N + P↓N ) + |C↓N ;↑N | ≥ 2 |C↓N ;↑N | , 2

(7.6)

where P↑N and P↓N are the probabilites of being in the |↑, N or |↓, N state, respectively, and the coeﬃcient C↓N,↑N is the far oﬀ-diagonal matrix element of the density matrix.

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7 Quantum Eﬀects in Charged Particle Traps

The preparation of Cat states is made diﬃcult by their sensitivity to perturbations that cause decoherence; nevertheless H¨ aﬀner, et al. have reported the successful preparation of entangled states of the GHZ and W-types in up to six Ca+ ions in a linear Paul trap [337]. The linear ensemble of ions is cooled by a combination of Doppler cooling on the resonance S–P transition followed by sideband Raman cooling [304]. A series of narrow laser pulses are applied individually to the ions on the 2 S1/2 (mj = −1/2) ↔2 D5/2 (mj = −1/2) quadrupole transition to create the initial state |0, D, D, D.... For the subsequent entanglement procedure to achieve the |WN state, and the density matrix construction to analyze the ﬁdelity as shown in Fig. 7.9, the reader is referred to the original article [337]. In the experiments discussed so far it has been assumed that to produce the entangled states it is necessary to begin with the ions cooled to the zero point

Fig. 7.9. Absolute values ρ of the reconstructed density matrix of a |W6 state [338], and a |W8 state as obtained from quantum state tomography [337]

7.3 Entanglement of Trapped Ion States

189

and thermally isolated to prevent decoherence; but Mølmer and Sørensen [299] have proposed a procedure generally applicable to any number of ions having uncontrolled thermal CM motion without deleterious eﬀect on the internal states of the ions. It does not even require access to individual ions. The method seeks to construct states for N ions of the form:

1 |Ψ = √ eiΦg |gg...g + eiΦe |ee...e , 2

(7.7)

where the states |gg...g and |ee...e are product states describing N ions that are all in the same g or e state and Φg , Φe are phase functions. We note that for N = 3 these states reduce to the GHZ states. The essential idea is that states of the form |Ψ can be obtained by the application of one simple interaction Hamiltonian to an ensemble of ions in a trap initially all in the |g state, an interaction Hamiltonian which can be realized in an ion trap simply by illuminating all the ions with laser beams of two diﬀerent wavelengths. These wavelengths can be chosen so that neither can be absorbed by individual ions but will induce only transitions in which two ions undergo |gg ↔ |ee simultaneously, and does so resonantly, that is ω1 + ω2 = 2ωe↔g . By choosing ω1 = ωe↔g − δ and ω2 = ωe↔g + δ, where δ is close to, but not resonant with the CM vibrational frequency, the perturbative analysis will involve intermediate states with one excited ion and a vibrational quantum number raised or lowered by unity. The lack of degeneracy in the frequency of the vibrational modes is crucial because it prevents the interference of modes, aﬀecting collective transitions in the multi-ion system. On the basis of this method, Sackett et al. of the NIST group were able to successfully engineer entangled states in four ions using a single laser [301]. 7.3.4 Trapped Ion–Photon Entanglement The coherent coupling of the states of a single-trapped ion with one mode of the electromagnetic ﬁeld in a high-ﬁnesse optical resonator is of great interest in the implementation of quantum computing and information processing schemes. It can be accomplished by the resonance laser excitation of an ion trapped inside an optical cavity in such a manner that the ﬂuorescence sidebands at ν = νL ±νz coincide with two cavity modes, as illustrated in Fig. 7.10. The importance of ion–photon entanglement rests on providing an interface between information localized on a material particle and that transmitted by a photon. Following Blinov et al. [340], Becher et al. [338] have reported the successful entanglement of hyperﬁne states in Ca+ ion and the polarization state of a spontaneously emitted photon. This was accomplished using a linear trap mounted concentrically with a high-ﬁnesse optical cavity. The laser excitation of the Ca+ resonance transition 2 S1/2 ↔ 2 P3/2 in a cavity tuned to the 2 P3/2 ↔ 2 D5/2 results in the emission of a single photon by the adiabatic Raman passage process [341]. This process has also been applied to create

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7 Quantum Eﬀects in Charged Particle Traps

Fig. 7.10. (a) Einstein–Podolsky–Rosen (EPR)-entangled light from a singletrapped cooled ion. (b) A miniature linear ion trap coupled to an optical cavity having mirror separation smaller than 0.5 mm [339]

superpositions of S1/2 , D5/2 states as well as of Zeeman sublevels of the S1/2 state. A similar adiabatic Raman passage can then be used to transfer part of such entanglement to the excitation of a cavity mode; that is, from a basis of {|S |D} or {|S |S } to a photon Fock basis {|0 |1}. The absence of spontaneous emission and its attendant decoherence in the adiabatic Raman process makes it easier to achieve entanglement between ion and photon states. To measure this entanglement a method has been developed based on converting the Fock photon basis to a time-bin basis [342]. 7.3.5 Lifetime of Entangled States The most crucial requirement for entanglement is its durability or immunity from decoherence, allowing it to persist suﬃciently long for the needs of quantum communication and computer functions. Considerable theoretical work has been published on quantifying what is generally referred to as the robustness of entanglement [343,344]. These studies consider the disappearance of entanglement (appearance of separability) when pure entangled states are mixed with other states. In particular cases, the robustness of entanglement can be estimated by observing the lifetime of the Bell states. Yu and Eberly [345] have discussed robustness and fragility of states using solvable models; in particular the dynamics of robust and entangled states in a two-qubit model. They ﬁnd that for the assumed coupling, the entanglement decays faster than local dephasing, and that for increasing number of qubits the entanglement decoherence time would be expected to become exceedingly small. Nevertheless it has been shown recently that it is possible to realize a robust W-type entanglement between magnetic sublevels of two Ca+ ions conﬁned in a trap [302,337,346]. For that purpose the two 40 Ca+ ions were separately √ excited by a sequence of laser pulses to produce the Bell state (|SD + |DS)/ 2 having a coherence time limited only by the (long) decay time of the metastable D5/2 state. By transferring the entanglement to the Zeeman or hyperﬁne sublevels

7.4 Quantum Teleportation

191

Fig. 7.11. The entanglement of the levels S1/2 , mJ = −1/2 and S1/2 , mJ = +1/2 in 40 Ca+ ion produced by a π-pulse which transfers D-state population to the S-state manifold [337]

Fig. 7.12. Minimum ﬁdelity of the Bell state as a function of time. The value of ﬁdelity greater than 0.5 indicates entanglement. The inset shows the ﬂuorescene image of two entangled Ca+ ions [337]

of the ground state, as shown in Fig. 7.11, a coherence time was achieved extending to about 20 s, as shown in Fig. 7.12. Several factors contribute to the actual decay of the ﬁdelity including magnetic ﬁeld ﬂuctuations, scattered radiation and collisions.

7.4 Quantum Teleportation Teleportation is a term borrowed from science ﬁction where it is conceived as a process in which a physical object could disintegrate at one location and reappear at another, with the destruction of the original (unlike the fax transmission). For that to really happen, the structure of the object would have to be completely characterized through measurement down to the quantum level, and that information must be sent to another location without corruption, and then the object must be reconstructed according to that information. Of course, this is impossible because Heisenberg’s Uncertainty Principle prevents classical measurements from determining a complete description of an object at the quantum level. The term quantum teleportation is deﬁned as the disassembly and reconstruction of a quantum state over arbitrary distances, without the physical transfer of the material carrier [322,347,348]. It was in 1993, that Bennett et al.

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7 Quantum Eﬀects in Charged Particle Traps

[348] proposed that teleportation might be realized by quantum Einstein– Podolsky–Rosen (EPR) entanglement combined with the transmission of purely classical information. EPR entanglement simply refers to the quantum correlation arising from superposition of two quantum states. Thus, according to this procedure, the teleportation of one quantum state to another quantum system requires two functions: the ﬁrst is the establishment of entanglement between the states of an ancillary qubit spanning the intervening distance, and the second is the classical transmission of data which serve as corrections to the received information. In the course of transferring the quantum state of a particle, no measurement may be made in transit, moreover the transfer destroys the original quantum state. Ions isolated in traps oﬀer special advantages for the demonstration of teleportation, allowing as they do durable entanglement of states and the possibility of high ﬁdelity. The NIST group has reported [349] the successful teleportation of a quantum state from one Be+ ion to another in a special multi-zone trap (Fig. 7.13) using laser manipulation on three ions. To explain the process, we will describe the protocol as applied to a simple two-state qubit transport. We follow common practice and refer to “Alice and Bob” as the parties between whom the transfer takes place. Following Barrett et al. [349], we consider two entangled qubits (#1 and #3) taken to be in the singlet state |Ψ13 = |↑1 |↓3 − |↓1 |↑3 being placed one each at Alice and Bob, and qubit #2 of unknown state |ψ2 = α |↑2 + β |↓2 at Alice to be teleported to Bob. Alice entangles her unknown particle with her half of the EPR pair by performing a complete set of measurements in the Bell operator basis; thus the initial joint state of all three particles is rewritten using Bell states as basis, as follows: |Φ =

k=4

|Ψk 1,2 Uk |ψ3 ,

(7.8)

k=1

Fig. 7.13. The multi-zone ion trap used to demonstrate for the ﬁrst time the teleportation of an ion state between two locations in the trap. The trapping space is the horizontal opening near the center of the image. Potentials on tiny electrodes are used to control the positions of the three Be+ ions, either individually or in groups of two or three [350]

7.4 Quantum Teleportation

193

Fig. 7.14. Quantum teleportation of a photon polarization using entanglement

where |ψ3 = α |↑3 + β |↓3 . Once Bob learns from Alice through classical communication the results of measurements in the Bell state basis, he is able to apply the appropriate unitary operator Uk−1 to his state Uk |ψ3 to retrieve the transported state |ψ. The scheme for teleporting particle states applies of course equally to the states of a photon, as illustrated in Fig. 7.14. Assume, for example, that the polarization state Ψin of a photon at Alice’s location is to be teleported to Bob at another location. She would entangle the photon with one of a pair of EPR entangled photons and observe the resulting entangled state with respect to Bell states as basis. This observation on one member of an entangle pair determines the state of the other member at Bob’s location, to within a unitary operation. By sending Bob the result of her observation through a classical communication channel it is possible for him to apply the appropriate unitary operation to recover the state of Alice’s photon. Quantum teleportation is not limited to two-valued (“dichotomic”) variables but has also been shown to be possible for dynamical quantities with continuous spectra for example the electromagnetic ﬁeld [351] and the motion of massive particles [352]. Braunstein and Kimble [351] have analyzed and proposed a protocol for the teleportation of the wavefunction of a single mode of the electromagnetic ﬁeld. This is based on prior work on the EPR paradox for variables with continuous spectra, and uses a highly squeezed state of the electromagnetic ﬁeld as the entangled shared state between Alice and Bob. The scheme is illustrated in Fig. 7.15, and the reader is referred to the original paper for details. The application of teleportation to quantum communication requires that the degradation in the quality of entanglement of EPR pairs due to a lossy quantum channel be overcome. Protocols such as entanglement puriﬁcation for doing this have been developed [353], making it possible to teleport the quantum state of a particle over noisy channels [322, 353–355]. The process of puriﬁcation involves the entangled pairs that are shared between Alice and Bob. Brieﬂy, both parties Alice and Bob perform local unitary operations and measurements on their part of the shared entangled pairs, coordinating their actions through a classical communications channel and eliminating some of the entangled pairs to increase the purity of what remains. In this way high-ﬁdelity

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7 Quantum Eﬀects in Charged Particle Traps

Fig. 7.15. Scheme showing quantum teleportation of an unknown input state Win (αin ) from the station S to the station R as the output state Wout (αout ). EPR – source to produce an ancillary entangled pair of states (1 and 2); Da , Db – detectors; a, b - ways; ixa , ipb – classical outcomes of the entanglement between Win (αin ) and the state 1; Ax,p – the element performing linear displacements of the real and imaginary components of the complex amplitude α2 to produce αout [351]

Fig. 7.16. Quantum repeater to regain overall ﬁdelity F = Ψ |ρ| Ψ 1, with L the length of communication channel [357]

teleportation can be achieved over extended distances circumventing the limitations imposed by a lossy line of communication. There are limits on the useful range of ﬁdelity attainable by puriﬁcation: both an Fmin for the method to be eﬀective, and an Fmax that can be attained by it. The projected limit to the distance set by losses in the communication link to 10 km [356]. To extend the distance that quantum information may be transmitted requires overcoming the attenuation of the signal by the use of “quantum repeaters” (Fig. 7.16), by analogy with the classical case. These consist of auxiliary particles at intermediate connecting points. Figure 7.17a shows schematically such repeaters as proposed by Briegel et al. [356], in which the communication between two distant nodes A and B is divided into a number of segments with elementary EPR pairs created between the auxiliary nodes at which the segments terminate. These EPR pairs are then connected by making Bell measurements at the auxiliary nodes and classically communicating the result between them, as in teleportation. Briegel, et al. have also introduced the concept of nested puriﬁcation protocol [356], as shown in Fig. 7.17b. This is an iterative process involving the creation of parallel copies of EPR pairs. Such a scheme would permit the creation of an entangled pair over arbitrarily large distances with tolerable error. Currently, the entanglement of atomic

7.5 Sources of Decoherence

195

Fig. 7.17. (a) Connection of a sequence of N EPR pairs; (b) nested puriﬁcation with repeated creation of auxiliary pairs. Reprinted with permission from [356]

states has been realized over distances on the order of half a meter [358–360]. This advancement in the extension of entanglement impacts directly the capability of teleportation, which opens up new directions of study in fundamental quantum theory.

7.5 Sources of Decoherence 7.5.1 Decoherence Reservoirs In describing physical reality, a distinction is usually drawn between the system under study and its environment with which it interacts, which may include some measurement apparatus. In the context of trapped ions, the environment generally consists of a background of interacting atoms and photons acting as a thermal reservoir. Interaction with this reservoir leads to loss of coherence by the ion of its quantum state superpositions that are of the essence of quantum behavior and the degeneration to classical probability distributions requiring a density matrix treatment [361–363]. The question of how long coherent quantum superpositions of states can be preserved (decoherence time) is of course crucial to applications such as quantum computers. Numerous studies have been conducted on decoherence resulting from the various types of interaction between trapped ions and the atomic or ﬁeld environments [362, 364, 365]. For example, the decoherence eﬀect of a ﬂuctuating electric ﬁeld amplitude, to simulate Johnson noise, has been studied on a superposition state in a trapped ion [366]. Of fundamental importance, as a source of decoherence, is spontaneous photon emission; this has been studied using two laser frequencies to induce transitions to internal states in the ion that can decay spontaneously to the ground state.

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7 Quantum Eﬀects in Charged Particle Traps

Fig. 7.18. Double needle rf ion trap allowing the electrode–ion distance to be adjustable in situ [297]

7.5.2 Motional Decoherence The principal form of decoherence of trapped ion states is the motional state arising from trapping ﬁeld ﬂuctuations, coupling of ion motion to ambient black body radiation, currents induced by the ions, and rf heating [367]. Uncontrolled electric ﬁeld at the position of the ion can excite the ion string from the ground state. The heating rate from a quantum state |n for an ion oscillating at ωz in a trap driven at the frequency Ω is given by ωz2 e2 ˙n SE (ωz ) + SE (Ω ± ωz ) , (7.9) ¯= 4mωz 2Ω2 where SE is the spectral noise density. Johnson noise is a potential source of decoherence scaled as SE ∼ d−2 with the characteristic trap size d. The statistically ﬂuctuating electric ﬁelds at the position of the ion (Patch eﬀect) are related to the electrode surface material/structure. Scaling of SE with trap size has been investigated using movable trap electrodes (Fig. 7.18) and a size dependence of the heating rate has been found in fair agreement with the expectation [297]. Observed single-quantum excitation times typically lie between 100 μs and 100 ms for room temperature. The mobility of the surface patches and thus the power density of the perturbing electric ﬁelds are expected to be reduced at lower temperatures. In fact, a reduction of the heating time constant of more than 1 order-of-magnitude has been found when working at 150 K (Fig. 7.19) [297]. A further reduction of heating rates has been found when the trap electrodes are carefully annealed. In situ cleaning of trap electrodes and the use of diﬀerent electrode materials may also lead to improvements. If the motional heating can be reduced suﬃciently, this will allow the use of smaller traps thereby increasing gate speed and facilitating ion separation in multiplexed trap scheme.

7.5 Sources of Decoherence

197

Fig. 7.19. (a) Average thermal occupation number n ¯ measured after diﬀerent delay time τ indicates a motional decoherence (heating rate) of n ¯˙ = 2, 380 ± ¯˙ as a function of distance z0 from a sin440 quanta s−1 ; (b) axial heating rate n gle ion to a trap electrode at a ﬁxed axial oscillation frequency ωz /2π = 2.07 MHz. Solid points: 300 K electrodes. Open points: Trap electrodes cooled to approximately 150 K. The shaded band at bottom, scaling as z0 −2 , is the expected range of heating from thermal Johnson noise in the trap circuitry [297]

Finally, we have ﬁeld emission of electrons from sharp points on the surface of the electrodes, electrons which can cause ion heating either by direct Coulomb interaction with the ions or through ﬂuctuations that may be caused in the electrode potentials. The conditions of intense rf ﬁelds at the surface of micro-electrodes increase the probability of ﬁeld emission; however, the momentary application of very high-negative potentials will “burn oﬀ” such points. Residual slow instabilities in the ion trap ﬁeld parameters will also cause dephasing of the motion and consequent loss of coherence. For a linear ion trap having Mathieu parameters a q, small adiabatic ﬂuctuations δV0 , δU0 and δΩ in the applied voltages and driving frequency, and δR in the radius of the trap, will change the motional frequency as follows: δωx,y /ωx,y = δV0 /V0 − δΩ/Ω − 2δR/R, δωz /ωz = (δk/k − δU0 /U0 )/2,

(7.10)

where k is a geometric factor characterizing the static harmonic potential well Φs near the trap center according to 1 Φs = kU0 z 2 − (x2 + y 2 ) . (7.11) 2 Radiative decoherence refers to the loss of coherence due to the radiative coupling of the thermal ﬁelds of the environment at temperature T to the dipole associated with the oscillating trapped ion, and is the most important for the CM modes of ion motion [368–371]. If n is the mean number of motional quanta then in the limit n 1 the corresponding decoherence time can be written as t∗ = 1/( nγ) where γ is the relaxation rate of the ion energy to thermal equilibrium with the environment.

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7 Quantum Eﬀects in Charged Particle Traps

Fig. 7.20. The ion trap and its equivalent lumped circuit. (a) Hyperbolic Paul trap electrodes; (b) the equivalent circuit: CT interelectrode capacitance, r the resistive losses, and lL (cL ) the inductance (capacitance) for ion CM mode along z-axis [367]

Since the ion oscillation frequencies correspond to radiation wavelengths very much larger than the trap dimensions, the dipole interaction with the environmental radiation can be modeled using lumped circuit elements [367]. The ion oscillation induces currents in the trap electrodes, currents that ﬂow in outside circuit elements. The ion CM motion, can be modeled as a series lL cL circuit in parallel with the capacitance between the trap electrodes CT (Fig. 7.20b) [38, 372]. The losses in the electrodes and associated conductors are taken into account by including the resistance r, which not only make a damping contribution to the ion oscillation but also, through the Johnson noise, has a positive contribution to the heating of the ion. The theoretical decoherence time t∗hyp is given by [373]: t∗hyp =

4M ωz , Q2 SE (ωz )

(7.12)

where SE is the spectral density of the ﬂuctuations in E 2 acting on the ion. Typical values for t∗hyp are several seconds. If this E-ﬁeld ﬂuctuation is attributable to the Johnson noise in an external resistance r, then one may write SE (ωz ) = 4akB T r, where a is a geometrical factor. Taking the experimental values of lL = 6 × 104 H and r = 0.0415 W and a temperature of 300 K for a hyperbolic trap, the computed value of t∗ was “considerably longer” than the observed ∼1 ms [367]. In the case of a miniature linear ion trap, SE (ωz ) = (Es /U0 )2 SU0 (ωz ), where SU0 is the spectral density of ﬂuctuations in the potential and ES the stray static electric ﬁeld, such that from (7.12) it follows [367]: 2 U0 4M ωz ∗ tlin = 2 . (7.13) Q SU0 (ωz ) Es Typically t∗lin is 2 orders-of-magnitude larger than t∗hyp . The value achieved for t∗lin by the NIST group in 1989 in the 198 Hg+ experiments was about 150 ms [374], while in the 9 Be+ ion work the value reported in 1995 was about 1 ms [375]. Another source of decoherence in a Paul trap is the rf micromotion of the ion at the trapping frequency [1]. Since the rf trapping voltage V0 at the

7.5 Sources of Decoherence

199

frequency Ω is usually applied from a resonant stepup rf transformer between the end caps and the ring, the ion’s micromotion can induce a current at the frequencies Ω ± ωz in the tuned circuit if the ion motion is not symmetric between the end caps. The parallel impedance of the resonant circuit is given by RT Z(ω) = = rs (ω) + iX(ω), (7.14) ˜ ω−Ω 1 + 2iQ Ω ˜ is the quality factor. However, the Johnson noise associated with the where Q resistance rs (ω) is reported to be negligible in the NIST experiments on single 9 Be+ ions [367]. Another source of ion heating in a linear trap is the Coulomb interaction between the ions, that causes all the modes of oscillation, except that of the CM, to become anharmonic. This results in a form of chaotic ion motion that in the case of two interacting ions has been widely studied [376]. Equally important are the eﬀects of terms higher order than quadratic in the trapping ﬁeld: these include coupling between the CM mode and other modes, plus what is more pernicious, instabilities in the ion motion. These occur for example in linear traps whenever i ni ωi = Ω, where ωi is the normal mode frequency. These instabilities can be avoided by proper choice of the operating parameters a and q of the trap [1]. 7.5.3 Collisions with Background Gas The advanced state of ultra-high vacuum technology makes it possible to reach pressures below 10−8 Pa; nevertheless, collisions of the trapped ions with residual background gas particles [377] can be an important source of decoherence. During the early phase of loading the trap with ions, elastic collisions with background particles perform the useful task of relaxing the ion energy distribution to that of thermal equilibrium at the temperature of the surrounding gas. For a background gas of H2 at 10−8 Pa the relaxation rate has been given as γelastic = 0.03 s−1 [367]. The cross section for other inelastic collision processes in which the internal state of the ion is changed will depend on the energy level structures of the colliding partners, but are expected to be very small unless there is a near resonance between the energy levels of the colliding partners. Furthermore they must penetrate the angular momentum barrier suﬃciently for an overlap of their wavefunctions. In collisions involving an ion with polarizable particles, the ion polarizes the neutral collision partner leading to a classical upper limit to the rate given by the Langevin formula [367]: πα γLangevin = nkLangevin = nQ , (7.15) 0 μ where μ is the reduced mass, and n is the density of the background gas. Several examples have been reported of the formation by the subject ion of

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7 Quantum Eﬀects in Charged Particle Traps

Fig. 7.21. A trapped 9 Be+ ion is scattered by the oscillations of the polarizable background gas [382]

Fig. 7.22. Damping of Rabi oscillations of trapped 9 Be+ with a background gas, for diﬀerent initial prepared motional state. (a) |n = 0, ↓; (b) |n = 1, ↓. Full lines computet [382], dotted lines: heuristic relation to ﬁt the data [383]. Reprinted with permission.

molecular ions with background atoms: HgHg+ [378], BeH+ [334, 379, 380] and YbH+ [381]. An interesting theoretical analysis of the decoherence caused by the interaction of the ions with a polarizable background gas has been given by Serra et al. [382] based on a model of the gas as a continuous polarizable medium with the trapped ions being scattered by phonons in this medium, in analogy with the interaction of electrons in liquid helium, as illustrated by Fig. 7.21. The interaction of a trapped ion with this medium acting as a thermal reservoir clearly produces decoherence on motional states of the ion. It is further shown by Serra et al. that the decay of Rabi oscillations observed in experiments on Be+ ions at NIST, illustrated in Figs. 7.22 and 7.23 can be explained in terms of such a model. 7.5.4 Internal State Decoherence Environmental causes of decoherence in internal states of trapped ions, such as external ﬁelds, are generally small; Stark shifts are small and tightly controlled, as are Zeeman shifts. Spontaneous transitions between internal states place a fundamental limit on the coherence time of entanglements involving them. The incidence of just one spontaneous transition in a coherent ensemble can destroy the coherence. The extraordinary spectral resolution that has been achieved with transitions from long-lived metastable states attests to the advantages of ion trapping in this respect [378, 384–386]. Thus, by taking

7.5 Sources of Decoherence

201

Fig. 7.23. The damping of the distribution function of coherent motional states of trapped ions at times Γ t = 0.2 and Γ t = 0.9, with Γ the damping constant. Reprinted with permission from [382]

advantage of the cold isolation of trapped ions, one can use the metastability of some internal states to extend the lifetime limit on the decoherence time of entanglements involving that state. Transitions between internal states such as the magnetic hyperﬁne transitions in the ground electronic state have such long radiative lifetimes that they may be neglected as sources of decoherence; it is otherwise with allowed electric dipole transitions, however, in a number of interesting ions (Ca+ , Ba+ , Hg+ ) a long-lived metastable state exists from which transitions to the ground state are electric quadrupole to the lowest order. Residual ﬂuctuations in the external magnetic ﬁeld may cause ﬂuctuations in the energy separation between ion levels of interest thereby aﬀecting adversely the coherence time. The change in the transition frequency between the levels may be expanded in a Taylor series thus ∂ω 1 ∂2ω ω0 + δωB = ω0 + (B − B0 ) + (B − B0 )2 , (7.16) ∂B ω0 2 ∂B 2 ω0 where B0 is the average ﬁeld intensity and ω0 is the frequency for B = B0 . Clearly to minimize the eﬀect of external magnetic ﬁelds eﬀort is required on two fronts: proper choice of ﬁeld-insensitive quantum levels [378,379,387] and the eﬀective shielding of the ions either through layers of magnetic material or through the Meissner eﬀect using a superconducting enclosure. In the case of electric ﬁeld perturbation of atomic levels, aside from the very special case of atomic hydrogen there is no linear Stark eﬀect, and the lowest order eﬀect is quadratic in the electric ﬁeld strength E, thus we have: 1 ∂2ω ω0 + δωE = ω0 + E2 + · · · (7.17) 2 ∂E 2 E=0 As already indicated, in miniature ion traps the “patches” that form on the electrodes produce stray electric ﬁelds that may shift the center of oscillation

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7 Quantum Eﬀects in Charged Particle Traps

and thereby subject the ion to the high-frequency trapping ﬁeld, leading to a Stark shift [388]. The elimination of the many sources of decoherence is a daunting task almost certainly not achievable entirely. To ensure the prolonged coherence time that quantum computation demands, two approaches are possible: develop “active” coherence-correcting codes or the “passive” use of a subspace immune to decoherence eﬀects. It has been shown [389] that in the prevalent case where all entangled states (qubits) are coupled to a common environment, a subspace exists in terms of which encoding the states reduces the eﬀect of the environment at most to a negligible common phase factor. It has been pointed out by Aolita et al. [390, 391] that the decohering action of a randomly ﬂuctuating ﬁeld is equivalent to the of the z-component of action N the total angular momentum operator Sz = i Szi . Kielpinski et al. [392] have successfully demonstrated experimentally using trapped ions a quantum memory that is immune from environmental decoherence. A one qubit memory was encoded into a decoherence-free subspace embodied in a pair of trapped 9 Be ions. Denoting the hyperﬁne states corresponding to the sublevels |F = 2, mF = −2 and |F = 1, mF = −1 of the ground 2 S1/2 state of 9 Be+ as |↓ and |↑ and entangled states of both ions as |↓↓, |↑↑, |↓↑ and |↑↓, the decoherence-free subspace is spanned by: √ |Ψ− = (|↓↑ − i |↑↓)/ 2, (7.18) √ |Ψ+ = (|↓↑ + i |↑↓)/ 2. 7.5.5 Induced Decoherence In the coherent manipulation of ions between diﬀerent quantum states by means of laser pulses, any errors in the tuning of the laser can accumulate to produce a phase shift, causing a loss of coherence. If δj (t) is the detuning of consecutive laser pulses of constant amplitude and duration, then the accumulated phase shift Φ will amount to: tk Φk =

δj (t)dt.

(7.19)

t0

Methods of obviating this problem include the use of adiabatic fast passage [393] or transfer through dark states [394–396]. However, the principal source of induced decoherence are ﬂuctuations in laser intensity at the site of the ion. This clearly can result either from ﬂuctuations in the intensity of the laser itself, or from instability in the illumination of the ion. The fundamental shot noise of a photon beam of average power P0 is given by [367]: ωl δP , (7.20) = P0 P0 Δt

7.5 Sources of Decoherence

203

where ωl is the laser frequency and Δt is the measurement time. The actual laser noise in practice is considerably above this quantum limit because of mechanical and instrumental instabilities in the optical system. It can be reduced to approach the shot noise limit by eﬀective elimination of sources of instability including mechanical isolation to minimize vibration and by a well designed wideband servo feedback using a beam splitter to sample the actual output intensity for the error signal. 7.5.6 Control of Thermal Decoherence The description thermal is applied to decoherence when it arises from transitions induced by thermal radiation between the subject entangled states and a third state. Figure 7.24 illustrates a model consisting of an ion with entangled states |1 and |2 subject to interaction with thermal radiation characterized by temperature T that induces transitions between |1 and a third state |3. An analysis of the evolution of this model using the equations of motion of the density matrix σ in the van Hove limit yield ultimately the results for the 2 2 coherence η12 of the states |1 and |2 deﬁned as η ≡ σ11 + σ22 + 2 | σ12 |2 summarized in Fig. 7.25 [397]. Two methods have been published for inhibiting the process [397]: 1. Dynamical decoupling. This entails exciting Rabi oscillations to a fourth level, as ﬁrst proposed by Viola and Lloyd [398–405]. 2. Quantum Zeno control. Here, the excited state is subjected to repeated measurement at suﬃciently short intervals; however, the repetition rate must exceed a certain minimum to avoid the inverse Zeno eﬀect.

Fig. 7.24. A simpliﬁed system scheme showing transitions involved in thermal decoherence. The levels |1, |2 are the lower hyperﬁne states and |3 the excited state [397]

Fig. 7.25. The decoherence of the subject state as measured by the decrease in time of the purity η. The unit of time on the horizontal axis is the decoherence time γd−1 [397]

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7 Quantum Eﬀects in Charged Particle Traps

Dynamical Decoupling The principle of this method, as described initially by Viola and Lloyd is to prevent the decoherence of states |1 and |2 caused by thermally induced transitions between states |1 and |3, by in eﬀect decoupling |3 from the evolution of the states |1 and |2. This is accomplished by a rapid sequence of pulses inducing Rabi oscillation between |3 and another level |4 (Fig. 7.26). Tasaki et al. [397] have shown in the context of the decoherence of hyperﬁne states, that, ideally, complete decoupling is approached in the limit of an inﬁnite pulse rate [398, 399]. In the limit of strong coupling, this method is equivalent to the dynamical Zeno eﬀect [315, 316, 406–413]. The results of a simpliﬁed density matrix analysis of the four level system in Fig. 7.26 are summarized in Fig. 7.27 [397]. The purity η of the state is plotted vs. time and the decoherence rate as function of Tc ω3 , where Tc is the control time, ω3 = ω3 − λδ, ω3 the energy of the state |3, λ the strength of interaction and δ the detuning. We note in Fig. 7.27b showing the decoherence rate γdR that it actually increases when the |3 ↔ |4 transition is driven insuﬃciently hard.

Fig. 7.26. The scheme for quantum dynamical decoupling of the ion from thermal reservoir [397]

Fig. 7.27. (a) Evolution of the purity η of the system state. The time unit along the Control horizontal axis is the decoherence time γd−1 in the absence of intervention. frequency (1) |ω− | = 150ω3 ; (2) |ω− | = 0.5ω3 . Here ω− = (−ξ − ξ 2 + 4Ω2 )/2 with −ξ the energy of |4, whose coupling constant to |3 is Ω. The dotted line is without control; (b) decoherence rate versus control frequency [397]

7.5 Sources of Decoherence

205

Fig. 7.28. (a) Evolution of the purity η of the target states. The time unit is the decoherence time γd−1 for the uncontrolled case. (1) Control frequency 2π/Tc = 5 × 106 ω3 ; (2) control frequency 2π/Tc = 0.5ω3 . Dashed line shows the behavior of η without control; (b) decoherence rate vs. control frequency 2π/(Tc ω3 ) [397]

Quantum Zeno Control The quantum Zeno eﬀect [315, 407, 409] in which rapidly repeated measurements on a state inhibits transitions from it, can also be applied to the suppression of thermal decoherence. In the limit of inﬁnitely high-repetition frequency of measurement, the coupling of the ion system with the thermal reservoir vanishes [397]. Figure 7.28 shows the results of a simpliﬁed density matrix analysis similar to the dynamical decoupling method. Again, although the detailed dependence of the decoherence rate on the repetition frequency diﬀers for the two methods, the quantum Zeno control also shows an increase at lower frequency.

8 Quantum Computing with Trapped Charged Particles

The concept of quantum computing has no clear cut origin. It emerged from combinations of information theory and quantum mechanical concepts. A decisive step was taken by Feynman [414, 415] who considered the possibility of universal simulation, a quantum system which could simulate the physical behavior of any other. Feynman gave arguments which suggested that quantum evolution could be used to compute certain problems more eﬃciently than any classical computer. His device may be considered as not suﬃciently speciﬁed to be called a computer. The next important step was taken in 1985 by Deutsch [310]. His proposal is generally considered to represent the ﬁrst blueprint for a quantum computer. It is suﬃciently speciﬁc and simple to allow real machines to be contemplated, but suﬃciently versatile to be an universal quantum simulator. Deutsch’s system is essentially an ensamble of two-state systems building a register. He proved that if the two-state systems could be made to evolve by means of a speciﬁc small set of simple operations, then any unitary evolution could be produced, and therefore their evolution could be made to simulate that of any physical system. Deutsch’s simple operations are now called quantum “gates,” since they play a role analogous to that of binary logic gates in classical computer. Unfortunately, all that could be found were a few mathematical problems, until Shor elaborated in 1994 a method for using quantum computers to crack an important problem in number theory, namely factorization of huge numbers [416]. He showed how an ensemble of mathematical operations, designed speciﬁcally for a quantum computer, could be organized to enable such a machine to factor huge numbers extremely rapidly, much faster than is possible on conventional computers. Various authors have reﬁned these basic concepts and since then many new ideas and proposals have emerged. Various physical systems are presently considered to implement quantum gates and perform logical operations such as Nuclear Magnetic Resonance, Quantum Dots, Optical Lattices, Cavity Quantum Electrodynamics, and others. Among them strings of ions conﬁned in linear Paul traps are presently regarded as the most promising route toward realization of a quantum computer. In this chapter, we will outline the basic concept of ion

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8 Quantum Computing with Trapped Charged Particles

trap quantum computing and mention various experimental attempts to perform quantum logical operations. A number of general reviews on this topic have been published in recent years [305, 417–424] and the reader is referred to these articles for further information.

8.1 Background Fundamentals 8.1.1 Quantum Bits: Qubits The fundamental unit of quantum information is the quantum bit (or qubit ), the analog of the classical bit which represents the binary Boolean values 0 and 1. The bit is realized in practice as two distinguishable states of a classical system such as the voltage level in an electronic circuit. A string of N binary units or bits can represent anyone of 2N numbers; however, such an N -bit register can store only one number at any given moment in time. In contrast, a qubit is embodied in a quantum system such as two polarization states of a photon, or two electronic states of an atom, or the spin direction of an electron in a magnetic ﬁeld. These prescribed pairs of normalized and mutually orthogonal quantum states are labeled as {|0 , |1} and correspond to the Boolean states 0 and 1. A system of N qubits requires as basis a Hilbert space of 2N dimensions. The two states of a qubit form a computational basis and any other pure state of the qubit can be written as a linear superposition as follows: |Ψ = α |0 + β |1 , (8.1) 2

2

where α, β are complex numbers such that |α| + |β| = 1. Thus a qubit can be represented as a vector of unit length in a twodimensional complex vector space, in which the special values |0 and |1 are orthonormal basis states having (within an arbitrary phase factor) the general form: θ θ |Ψ = cos |0 + eiϕ sin |1 , (8.2) 2 2 where θ and ϕ are real numbers. The vector whose Cartesian components are (cos ϕ sin θ, sin ϕ sin θ, cos θ), called the Bloch vector, can be visualized on the unit sphere as a point of spherical coordinates (θ, ϕ) (Fig. 8.1). A qubit can assume any of a continuum of states given by the linear superposition deﬁned by (8.1), until it is observed; then it “collapses” into a single quantum state: either |0 or |1. A measurement to determine the state 2 of a qubit gives either the result |0 with a probability |α| or the result |1 2 with the probability |β| . Although a measurement can lead to the destruction of a qubit, it nevertheless has an identity embodied for example in the electron states of an atom, whose value can be manipulated by pulsed resonant optical excitations. In a two-qubit register, there are four possible basic states |00, |01, |10, |11, which can also exist in superposition states as follows:

8.1 Background Fundamentals

209

Fig. 8.1. The geometric representation of a qubit on the Bloch sphere

|Ψ = c00 |00 + c01 |01 + c10 |10 + c11 |11 ,

(8.3)

2

where |cij | = 1. A measurement on Ψ will yield the basic states | ij with relative probability |cij |2 . If just the ﬁrst qubit is measured the result is 0 2 2 with a probability |c00 | + |c01 | , and the post-measurement state [425]: |Φ =

c00 |00 + c01 |01 . c200 + c201

(8.4)

A special two-qubit state is the Bell state or Einstein–Podolsky–Rosen (EPR) pair |00 + |11 √ |χ = (8.5) . 2 It is a distinguishing property of such an EPR pair that, unlike classical bits, the outcome of a measurement of one qubit determines the value of the other, that is, there is a strong correlation between them. A three-qubit register can simultaneously store up to eight diﬀerent numbers (000, 001, 010, ...., 111) in a quantum superposition, whereas a classical three-bit register can store only one number at a time. In general an N -qubit register can store up to 2N numbers at once, and if the register is prepared as a superposition of these qubits, then operations can be performed on all of them simultaneously. Contrast this with the classical case of a bit-register, where the operations would have to be repeated 2N times, or use 2N processors working in parallel. Thus, a quantum-based processor achieves through superposition of states a parallelism of operation having an enormous advantage in speed and memory capacity. The idea that an atomic system can simultaneously exist in two states at once, of course runs counter to intuition based on everyday experience with macroscopic objects. Indeed, historically it has commanded considerable attention, both from the point of view of epistemology and physical theory. Thus, we have the famous Schr¨odinger cat that is half alive and half dead, and the Einstein–Podolski–Rosen paradox. The classical two-slit experiment

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8 Quantum Computing with Trapped Charged Particles

Fig. 8.2. In a double-slit experiment performed even with a single particle, an interference pattern arises behind the slits

Fig. 8.3. Single-particle interference. 1(3) – half-silvered mirrors; 2(4) – fully silvered mirrors; T(R) – transmitted (reﬂected) beam; 5 – a glass sliver destroying interference

of Young illustrated in Fig. 8.2 proves the interference resulting from quantum superposition of states of a single photon. Another simple experiment that brings out the essential quantum behavior of a photon in a superposition state is illustrated in Fig. 8.3. A photon enters the interferometer at the half-silvered mirror (1) and has two possible paths R and T that are reﬂected by fully silvered mirrors (2) and (4). They converge on another half-silvered mirror (3) and then to the detectors D1 and D2 . It might be thought that the photon will reach either detector with equal probability. But, in fact, it is observed that the photon always reaches detector D1 and never detector D2 ! One is led to conclude that the photon travels both paths simultaneously, and self interferes at the point of convergence at mirror (3) annulling the probability of reaching D2 . If one of the optical paths is obstructed, then the two detectors register with equal probability; thus action on one path has aﬀected both, a crucial property in quantum computing. 8.1.2 Some History The emerging computer technology based on quantum phenomena represents a radical departure from what has gone on before and promises far superior

8.1 Background Fundamentals

211

Fig. 8.4. Transition from an early computing machine built from mechanical gears (a), to microtechnology (an IBM chip with 0.25 μm features) (b); (c) the nanostructure of a single-electron transistor as computer’s basic element. The electron can tunnel (a quantum eﬀect) through the barriers between “source” and “drain” ceasing the transistor operation; (d) the atom as the cell of the quantum computer

capabilities in the future. It is well to pause to consider how computational technology has evolved into a child of quantum physics (Fig. 8.4). Some of the interesting events along that path began as early as 1935 when Schr¨ odinger recognized the special property of composite quantum states, later termed entanglement, and in the same year Einstein et al. [294] published what came to be known as the Einstein–Podolski–Rosen (EPR) paradox. They sought to show that quantum theory must be incomplete because it predicts that measurements on correlated quantities in an entangled quantum system yield results that diﬀer from what would be expected if the properties of the measured system are the same prior to, and independent of the act of observation. This was taken by them to violate a fundamental requirement of any theory that it correspond to physical reality, and the result was therefore described as a paradox and put forward as proof that quantum mechanics is an incomplete theory. In 1964, Bell [426] deduced for the EPR model an inequality between correlations that could be tested experimentally to verify the quantum mechanical predictions about entanglement [427]. In 1981 Feynman [428] made a seminal presentation on the topic of simulating physics with computers, in which he addressed in particular the challenges of simulating the probabilistic nature of quantum measurements. He posits the question as to whether “intersimulable” quantum systems might not be solved by a universal quantum computer which can imitate any system including the physical world. He gives as a possible example a lattice of points at every point

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8 Quantum Computing with Trapped Charged Particles

of which there are two base states: either the point is occupied or not occupied. The mathematics of quantum operators associated with lattice points could be the Hermitian Pauli σ-matrices, familiar from the theory of spin 1/2 particles. The output of a probabilistic computer would not be a deterministic function of the input, but would seek rather to simulate nature only in the sense that the computer goes from the same initial state to a ﬁnal state with the same probability as the physical system goes from initial to ﬁnal state. In 1985 Deutsch [310] achieved a landmark in the theory of a universal quantum computer, one that is able to simulate the action of any other computer. He showed that if the evolution of a two-state system can be described through a set of simple operations, called quantum gates, the evolution of any quantum system can be simulated, thereby modeling it. Other important events followed in quick succession: In 1993, the possibility of quantum teleportation, in which an unknown quantum state may be “teleported” between two separated particles prepared in an entangled quantum state, was proposed by several authors [348]. In 1994, Shor [416] elaborated an algorithm using qubit entanglement to ﬁnd the prime factors of an integer much faster than is possible on classical computers, spurring commercial interest in quantum computers. In 1995, DiVincenzo [429] formulated the criteria which a quantum system has to fulﬁll to be used for large scale quantum information processing. In 1995, Cirac and Zoller [430] proposed a quantum computer based on ions in a trap. Wineland et al. at NIST, Boulder [305], and Blatt et al. at the University of Insbruck [431] have made a number of notable advances in the application of trapped ions to the realization of a quantum computer, including the ﬁrst quantum logic gate [375]. In 1996, Preskill et al. [432] published the complete set of quantum programing instructions required to evaluate Shor’s factoring algorithm. In the period 1996-1998 entangled states involving a few trapped ions cooled to the vibrational ground state were created at NIST, at the California Institute of Technology, and at Innsbruck, but their decoherence time was quite short, prompting suggestions on means of quantum error correction. In 2004, the NIST group reported the successful demonstration of deterministic teleportation of qubits involving entanglement between three ions in a trap [350]. Starting in 2003, feasibility analyses were initiated for a universal scalable multi-electron quantum processor, based on Penning ion trap technology, with the aim of producing a prototype quantum computer [312, 313]. 8.1.3 Possible Alternatives: The DiVincenzo Criteria In order for a system to function as a quantum computer it must meet certain requirements, conveniently summarized by DiVincenzo. They are essentially: (1) Scalability, assuring a suﬃciently long qubit register for any given application. (2) It must be possible to initialize the qubits in a well-deﬁned

8.1 Background Fundamentals

213

state. (3) The system must allow one- and two-qubit coherent manipulations. (4) The phase coherence time must be suﬃciently long to allow logic operations. (5) The system must allow a reliable read-out of the postcalculation qubit states [429]. The speciﬁc requirements that must be met clearly depend on the application to be performed; that is why diﬀerent equivalent embodiments of the computer may be required in practice. For example, for high precision but small size, optical methods (cavity QED) may be appropriate [433–436]. Unfortunately photons are diﬃcult to store and the available media for nonlinear photon interactions are not very eﬃcient, so it is unlikely that photon states will form the basis of large-scale quantum computers. However, laser beam interactions are likely to play a critical role in the functioning of ion trap technologies for quantum information processing. There are currently a number of diﬀerent experimental approaches under consideration, most of which are at the proof-of-principle stage with respect to the DiVincenzo criteria [437–441]. Only two of these candidate systems accommodate computation on 10–40 qubits and the implementation of simple operations of quantum logic. These are the Cirac–Zoller model using a linear array of trapped ions [430, 442–444], and the proposal to use NMR in the liquid state [445, 446]. However, it is doubtful that it will be possible using NMR to realize a computer handling hundreds of qubits. Aside from the last two, the other systems are far from reduction to practice even in prototype form, and no analysis has yet been published of how to integrate several of them into processing systems. In Sect. 8.2, we will brieﬂy present the status of quantum computing with trapped ions, the experimental techniques involved, the physical process by which quantum logic gates may be implemented, and the practical limitations of the method set by the yet small number of qubits, the ﬁnite decoherence time of entangled states and the speed of operation. The linear ion trap is a system which fulﬁlls the general requirements for a quantum computer using a universal set of gates (unitary transformations). In this a string of ions is conﬁned in ultrahigh vacuum along the axis of a potential created by four electrodes of a linear Paul trap (Fig. 8.5a). Each ion in the string can be individually addressed by a laser beam (Fig. 8.5b) since the inter-ion separation (about 20 μm) is larger than the wavelength of the exciting radiation generating the qubit (Fig. 8.5c). Laser cooling is used to bring the ions to the zero point of vibrational energy, while optical pumping puts them in the desired initial state and further manipulates their internal states. Unlike the hyperbolic Paul trap in which only a single ion can be cooled to the zero-point energy at the center of the trap where the rf ﬁeld is zero, in a linear trap the rf ﬁeld is zero along most of the symmetry axis and therefore multiple ions can be frozen in a linear array along the axis with small inter-particle separations. The ions vibrate as quantum oscillators with their motion coupled by the Coulomb interaction, inducing various collective motions known as phonons. By applying an

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8 Quantum Computing with Trapped Charged Particles

Fig. 8.5. (a) Scheme of an ion trap quantum information processor. A single laser beam is split by beam splitters and acousto-optic modulators into many beam pairs, one pair addressing each ion from a string of ions conﬁned along the axis of a linear Paul trap; (b) adressing individual ions in a linear 3-ion string; (c) the observed qubit in Ca+ [447]

appropriate laser pulse it is possible to couple the vibrational state to an internal state representing one qubit, while another laser pulse can induce transitions in the state-of-motion of the ion string, generating another qubit called a bus, since its function is to transfer quantum information among the ions. Coherence times of suﬃcient length have been achieved for both the ion internal states and the vibrational modes to apply a single-qubit gate to any ion in the string. Quantum logic operations and processing involving a few qubits are possible in a linear ion trap; however, the implementation of the Shor factorization algorithm to break classical cryptographic codes requires at least 400 qubits [448]! Passive stabilization will allow about 200 computing operations on ten ions, but with quantum error corrections to reduce errors due to system imperfections in the operators and isolation from the environment, this number is expected to be greatly increased in the future [417]. Read-out of the qubit states is accomplished with almost 100% eﬃciency using a strong resonance ﬂuorescence transition which yields a high signal-to-noise ratio even for individual ions. This kind of quantum computation is already currently implemented. By coupling a trapped ion with the optical ﬁeld in a cavity subject to quantum electrodynamics (QED) to form a hybrid processor it becomes an interesting candidate not only as a computer but also for quantum communications. The strong coupling between a trapped ion and a cavity QED mode can be used to transfer a qubit from the ion to the polarization state of a photon, which would leave the system in a chosen direction as a ﬂying qubit [449]. However, for the purposes of quantum communication there are other areas of development that deserve to be considered, namely, optical lattices, Bose condensates, and quantum dots.

8.2 Ion Traps for Quantum Computing

215

8.2 Ion Traps for Quantum Computing 8.2.1 Trap Electrode Design The general electrode geometry of a linear ion trap and a treatment of the dynamics of trapped charge particles in such a trap was given in [1]. Therefore, only a brief recapitulation of the theory is given here. The trapping rf voltage V0 cos Ωt is applied between diagonally connected pairs of electrodes mounted parallel and equally spaced around the central axis. In the adiabatic approximation [1], with U0 = 0, the conﬁned ion oscillates in a harmonic pseudopotential Ψ in the radial direction with the frequency ωr given by QV0 . ωr √ 2M Ωr02

(8.6)

Along the trap axis, conﬁnement is achieved with a weak-static ﬁeld created by a dc voltage U0 applied to electrodes at the two ends of the cylinders. This axial conﬁning potential is approximately parabolic in the neighborhood of the trap center, and has depth given by M ωz2 z02 /2 = kQU0 , where Q is the trapped ion charge, ωz its axial frequency, z0 the distance from the center to an end electrode and k is a geometric correction factor. Many variants of the original Paul mass ﬁlter [450] (Fig. 8.6a–c) have been used in which end electrodes have been introduced in the form of rings, cones, or in-line cylinders to produce three-dimensional conﬁnement [451–458]. A more radical departure in geometry is shown in Fig. 8.6d which is essentially a cylindrical trap stretched along one radius resulting in an elliptical “ring” and elongated “end caps.” This geometry has the virtue of allowing the conﬁnement of strings of two or three ions oriented along the long axis of the elliptical ring with greater optical access through a larger solid angle and improved optics for laser–ion interaction [257]. Another trap design that has been used is the storage ring trap, a form of re-entrant linear trap shown in Fig. 8.7 [459–463]. The four electrodes to which the rf potential is applied are in the form of circles equally spaced on the surface of a toroid, such that the quadrupole potential provides conﬁnement perpendicular to the circular axis. Small stopping potentials arising from surface charges on the trap electrodes impede trapped ions to freely move along

Fig. 8.6. Diﬀerent realizations of the linear ion trap. (a) Linear quadrupole trap; (b) linear knife edge trap; (c) four rod trap; (d) elliptical trap [451]

216

8 Quantum Computing with Trapped Charged Particles detector laser cooling

2:1 115

mm

oven ionization

5mm electron gun

Fig. 8.7. Quadrupole storage ring. The cross section of the electrode conﬁguration is shown on the insert on the righthand side of the ﬁgure [459, 460]

Fig. 8.8. Planar photolithographic double wafer–stack trap [464]

the circular axis. The trapped ions are distributed in such a way that the average distance between them is constant and their energy minimal. The adaptation of ion trap electrode design to the demands of an integrated quantum computer has been realized in the form of miniaturized geometries suitable for lithographic fabrication (Fig. 8.8) [464]. The wafer separation is on the order of 400 μm, and an rf voltage of peak amplitude about 500 V at 230 MHz can be applied for stable ion conﬁnement in this trap. Of particular interest are the ion traps that have been constructed, using integrated circuit technology, in the form of a single chip containing the trap and associated electronics (Fig. 8.9). This is particularly well suited for scalability of quantum computers as discussed later. Monolitic structures with electrode spacings below 100 μm using standard Ga–Ar [466] or other technologies [467, 468] can be fabricated. Furthermore, they can be integrated with on-chip optics and electrodes. They can host a large number of qubits, and are considered as a possible route toward scalable quantum computers. 8.2.2 Choice of Ion The choice of ion species is dictated by the requirement of an energy level structure that includes a two-level system immune from decoherence through spontaneous emission or other interactions and allowing convenient optical cooling and detection; this is usually found in ions having one electron in

8.2 Ion Traps for Quantum Computing

217

Fig. 8.9. Micrographs of planar trap chip (gold on fused silica) [465]

Fig. 8.10. Level scheme and the corresponding transitions for two ions used in quantum computing experiments. (a) 40 Ca+ ; (b) 9 Be+

the outermost shell. The two-level system can consist of the ground and a long-lived metastable state (Fig. 8.10a), or two hyperﬁne components of the electronic ground state (Fig. 8.10b). The choice of ions depend on the availability of laser sources for ion cooling and information processing. The following ions, shown in Table 8.1, have operating frequencies that are readily generated and have therefore been the subject of intensive study as candidates for a quantum computer: 9 Be+ [257], 199 Hg+ [469], 25 Mg+ [467], 40 Ca+ [431,470], 138 Ba+ [471], 171 Yb+ [471]. We consider the ion Ca+ as example. Its low-lying energy levels are shown in Fig. 8.11. It is a particularly attractive candidate because the laser wavelengths it needs for cooling and repumping, at 397 and 866 nm, or 850 nm, respectively, can be produced by diode lasers. A qubit can be encoded in the S1/2 –D5/2 transition (“optical qubit”). The lifetime of the D5/2 state is about 1 s. The wavelength of the qubit transition at 729 nm is readily commercially

218

8 Quantum Computing with Trapped Charged Particles Table 8.1. Some single-charged ions suitable for information processing

Ion (isotopes) 9

Be+ Mg+ 25 Mg+ 40 Ca+ 43 Ca+ 87 Sr+ 88 Sr+ 135 Ba+ 137 Ba+ 138 Ba+ 199 Hg+ 201 Hg+ 202 Hg+ 171 Yb+ 173 Yb+ 174 Yb+ 6 + Li 7 + Li 24

Natural abundance (%)

Nuclear spin ()

Hyperﬁne splitting (GHz)

λ S–P (nm)

100 79 10 97 0.14 7 83 6.6 11 72 17 13 30 14 16 32 7.5 92.5

3/2 0 5/2 0 7/2 9/2 0 3/2 3/2 0 1/2 3/2 0 1/2 5/2 0 1 3/2

1.25

313 280

λ S–D (nm)

1.78 397

730

422

674

493

1760

194

282

369

411

3.25 5.00 7.18 8.03 40.50 30.16 12.64 10.49 3.00 11.89

539

Fig. 8.11. Low-lying energy levels of the 43 Ca+ ion and relevant optical transitions for cooling, repumping, and qubit encoding

available by diode laser Similar transitions occur in Sr+ , Ba+ , Yb+ , and Hg+ ions. Alternatively hyperﬁne states of the electronic ground level can be used (“hyperﬁne qubit”). This requires the use of odd isotopes with nonzero nuclear

8.3 Qubits with Trapped Ions

219

spin. Taking 43 Ca+ with nuclear spin 7/2 as example, we can choose the |F = 4, mF = 0 ↔ |F = 3, mF = 0 transition. The qubit can be encoded by stimulated Raman transitions (Fig. 8.11) [470]. These levels are resistant to phase decoherence induced by laser radiation and, to ﬁrst order, external magnetic ﬁelds. Decoherence times up to 100 ms have been experimentally observed [470]. Among the candidates listed in Table 8.1, 9 Be+ has attracted most attention since no metastable electronic states exist between the ground and ﬁrst excited states. Thus, it requires only one (frequency doubled) laser to generate the required wavelength of 313 nm for cooling and qubit encoding. It has a nuclear spin of 3/2 and a hyperﬁne frequency of 1.25 GHz, accessible with acousto-optical modulators. At a magnetic ﬁeld of 0.01194 T, superimposed on the trap, the |F = 2, mF = 0 ↔ |F = 1, mF = 1 transition becomes independent of the magnetic ﬁeld to the ﬁrst order. Decoherence times for qubit states of about 15 s have been obtained by the NIST group [472].

8.3 Qubits with Trapped Ions As mentioned in the Chap. 7 qubits can be encoded in a two-level system which may be formed either by the electronic ground and a long-lived excited state of an ion, by hyperﬁne states of the electronic ground level or by Zeeman levels of the ground state. Resonant laser pulses of a given length and amplitude can be used to create arbitrary superposition states of a qubit by driving Rabi oscillations between the two-qubit states. In case of hyperﬁne states two laser beams are used to connect the qubit states by Raman transitions trough a virtual intermediate level as indicated in Fig. 8.11. Apart from qubits formed by internal ionic states the vibration of a trapped ion can also be used. The vibrational ground- and ﬁrst excited state of the ion oscillation also represent a two-level system suitable to implement a qubit (Fig. 8.12). Measuring the state of the qubit stored in an ion is quite simple. For that, a laser is applied to the ion coupling one of the qubit states. When the ion collapses into this state during the measurement, the laser will excite it, resulting in a photon being relased when the ion decays from the excited state.

Fig. 8.12. Ground- and ﬁrst-excited quantum state of the ion oscillation in the axial parabolic potential of a linear ion trap

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8 Quantum Computing with Trapped Charged Particles

After decay, the ion is continually excited by the laser and repeatedly emits photons, which can be collected by a photomultiplier or a CCD camera. If the ion collapses into the other qubit state, then it does not interact with the laser and no photon is emitted. By counting the number of collected photons, the state of the ion may be determined with a very high accuracy.

8.4 Quantum Registers: Qregister A quantum register (qregister) is the quantum mechanical analog of a classical processor register. An n qubit quantum register is described by an element |Ψ = |Ψ1 ⊗ |Ψ2 ⊗ · · · ⊗ |Ψn in the corresponding tensor product Hilbert space. Quantum registers need a row of trapped ions which interact with each other in a controlled way. Under the action of laser cooling they form a string along the axis of a linear Paul trap. N ions then represent (N + 1) qubits because the collective vibration of the whole string acts as the (N + 1)th qubit bus qubit. By solving the classical equations of motion for a string of N ions along the axis of a linear Paul trap one obtains the equilibrium positions (Fig. 8.13a). The separation distance between the ions is given by [417] zs =

Q2 4πε0 M ωz2

1/3 ,

(8.7)

and it can be adjusted by means of the axial dc voltage. For typical operating parameters, we ﬁnd zs 10–100 μm. The string can oscillate in diﬀerent modes. The lowest modes are the center-of-mass (CM) mode and the breathing mode (Fig. 8.14). The mode frequencies are nearly independent √ of N (Fig. 8.13b). For N = 1, 2, 3, 4, 5, 6, . . . ions they are given by 1, 3, 29/5, 3.051, 3.671, 4.272, . . . , in units of ωz . In the lowest mode (CM oscillation) the distance between the ions remains unchanged. This mode is generally used as bus qubit. To address each of the

Fig. 8.13. (a) Equilibrium position and (b) normal mode frequency in units of ωz for a line of point charges in a quadratic potential, as a function of N . The solid line shows the dependence of the inter-ion distance as N −0.57 on the ion number N [417]

8.4 Quantum Registers: Qregister

221

Fig. 8.14. Common mode excitation of a string of seven ions. (upstairs) CM-mode; (downstairs) the breathing mode [473]

ions individually with a diﬀerent laser beam their separation has to be large compared to the extent of their wavefunction. The bus qubit has to be treated quantum mechanically if the ion temperature is reduced to the quantum limit (kB T << ωz ), a temperature limit that has already been reached in a single ion. In a quantum treatment [474–477], the Hamiltonian for a system of N identical charged particles of mass M and electric charge Q conﬁned in general axially symmetric electric and magnetic ﬁelds, with the Paul and Penning ﬁelds as special cases, is given by: H=

N

Hα + V.

(8.8)

α=1

Here Hα is the Hamiltonian for the particle α which may be written as Hα = −

3 2 ∂ M λr 2 M λa 2 ωc (xα1 + x2α2 ) + xα3 − Lα3 , + 2M j=1 ∂x2αj 2 2 2

(8.9)

where xα1 , xα2 , and xα3 are the coordinates of particle α and V is the interaction potential between the particles. The parameters of the trap λa and λr are given by Q 1 (8.10) λa = −4 A(t), λr = (ωc2 − 2λa ), M 4 where U0 and V0 are the static and time varying voltages applied to the trap electrodes of semi-axes r0 and z0 , and A(t) = (r02 + 2z02 )−1 (U0 + V0 cos Ωt). The function A(t) is either time-periodic with period T = 2π/Ω, for dynamical traps, or stationary for Penning traps. The Hamiltonian commutes with the z−axis angular momentum operator which for particle α is Lα3 , deﬁned as: ∂ ∂ Lα3 = i xα2 . − xα1 (8.11) ∂xα1 ∂xα2

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8 Quantum Computing with Trapped Charged Particles

To separate the bus motion, that is the CM motion, from other intrinsic collective motions we introduce the translation-invariant coordinates yαj thus: yαj = xαj − Xj ,

Xj =

N 1 xαj N α=1

(8.12)

with Xj the CM coordinates, and the translation-invariant diﬀerential operators Dαj : N ∂ 1 ∂ Dαj = − Dj , Dj = , (8.13) ∂xαj N α=1 ∂xαj where α = 1, . . . , N and j = 1, 2, 3. Then N

yαj = 0,

α=1 N

N

Dαj = 0,

α=1

x2αj = N Xj2 +

α=1

N

2 yαj ,

α=1

N

1 , Dβk (yαk ) = δkj δαβ − N

N N ∂2 2 2 = N D + Dαj , j 2 ∂x αj α=1 α=1

˜ 3, Lα3 = Lcm 3 + L

(8.15)

(8.16)

α=1

Lcm 3 = X1 P2 − X2 P1 ,

(8.14)

˜ 3 = i L

N

(yα2 Dα1 − yα1 Dα2 ),

α=1

˜ 3 is where Pj = −iN Dj , Lcm 3 is the center-of-mass angular momentum and L the intrinsic angular momentum of the N -particle system. Then (8.8) becomes ˜ + V, H = Hcm + H

(8.17)

where the center-of-mass Hamiltonian is given by [478] Hcm =

M 2

ωc 1 2 λr X1 + X22 + λa X32 − L3 (8.18) P1 + P22 + P32 + 2M 2 2

with M = N M , having exactly the form of the quantum Hamiltonian for a single trapped particle of mass M . The intrinsic Hamiltonian is deﬁned by ⎡ ⎤ N 3 2

1 1 ωc ˜ 2 2 2 2 ⎦ ˜ = ⎣− H + M λa yα3 Dαj + M λr yα1 + yα2 − L 3. 2M 2 2 2 α=1 j=1 (8.19) The complete and explicit analytical solution for the quantum motion of a single trapped particle was constructed in [479] on the basis of group representation theory. There it was shown that the group Sp(2,R) is the dynamical group for the Hamiltonian describing the quantum motion, and correspondingly, some coherent states of this group are (to within a phase factor) the

8.4 Quantum Registers: Qregister

223

Fig. 8.15. Periodic trajectories of the wave packet center in a combined trap for (a) dominant rf ﬁeld or (b) dominant magnetic ﬁeld

quantum solutions of the motion of a single ion. Periodic trajectories of the wave packet center corresponding to the CM motion in a combined trap follow the classical trajectories for a single ion, as can be seen in (Fig. 8.15). The classical result that the CM normal mode has frequency ωz remains valid even though the ion wavefunctions overlap, since all the internal interactions among the ions cancel when one calculates the CM motion. For the oscillator of mass N M and frequency ωz the eigenfunctions are as follows [417]: Ψn (zCM ) =

N M ωz π22n (n!)2

1/4 Hn (zCM

2 N M ωz / ) e−N Mωz zCM /2 ,

(8.20)

where zCM is the CM coordinate, and Hn is the Hermite polynomial. For the ground vibrational state the spatial extent of the Gaussian ground state probability distribution is given by [417] Δzcm = /(2N M ωz). (8.21) 8.4.1 Initialisation of the Qubits The preparation of the input state requires the register to be brought to the initial state by transferring all ions to one of the two basis states of the qubit as described later. This can be performed by optical pumping for the internal qubit and by laser cooling to the motional qubit. In optical pumping, a laser couples the ion to some excited states which eventually decay to one state which is not coupled to by the laser. Once the ion reaches that state, it has no excited levels to couple to in the presence of that laser and, therefore, remains in that state. If the ion decays to one of the other states, the laser will continue to excite the ion until it decays to the state that does not interact with laser. The arbitrary superposition states can be excited by resonant laser pulses. Assuming the Lamb–Dicke regime, by applying a laser π-pulse at the ﬁrst red sideband of the ion resonance frequency, the internal state of the addressed ion will be mapped to the vibrational state of the string (Fig. 8.16a). In this way, a superposition of the ground- and ﬁrst-excited states of the CM motion is transferred from a similar superposition of the internal states in the ion.

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8 Quantum Computing with Trapped Charged Particles

Fig. 8.16. CM-mode of an ion string as a quantum data bus. (a) The state of ion #1 is mapped to the CM-mode; (b) the state of ion #2 is changed conditional on the state of the CM-mode; (c) after the transfer of the quantum information to the ion #2, the initial state of the CM-mode and of the ion #1 are restored [451]

Since all ions in the string can share the quantum information through the CM motion, it is acting as a quantum bus linking the qubits along the string (Fig. 8.16b). At the end of the operation, the CM mode and the ion in question have to be returned to their initial state. Initialisation of the Bus Qubit At the start of a logic operation the motion of the ion string has to be completely frozen, or, in quantum mechanical terms, it has to be brought into the ground state of the corresponding harmonic oscillator. For ion cooling diﬀerent methods have been developed which have been discussed in detail in vol. 1 of this series [1]. Here, we restrict ourselves to the most frequently applied method for ground state cooling, so-called sideband cooling. Sideband cooling is a form of optical pumping of the ions to lower oscillator n-states, in which the ion loses ωv for every photon absorbed and re-emitted. A prerequisite is that the optical sideband spectrum of the oscillating ion be resolved deﬁning the Lamb–Dicke regime, where the Lamb–Dicke parameter given by η = (2πωrec /ωion )1/2 is smaller than 1, ωrec = 2 /(2M ) is the recoil frequency for a photon of momentum k and ωion is the ion oscillation frequency. The Lamb-Dicke regime can be reached by Dopplercooling, i.e. photon recoil from a laser slightly detuned to the low-frequency side of a resonant absorption line. In this regime the absorption spectra show an unshifted carrier at the resonance frequency of an ion at rest and symmetrical sidebands at the ion oscillation frequency (Fig. 8.17a) [480]. For sideband cooling, the laser absorption occurs at the ﬁrst lower sideband in the ion spectrum, that is, on the transition |S, n → |D, n − 1. Subsequent spontaneous emission appears predominantly at the carrier frequency, that is, |D, n − 1 → |S, n − 1 and thus the average energy of oscillation is lowered by one vibrational quantum. The accompanying recoil is not, on average, absorbed by the ion motion, but rather by the ion trap as a whole. As this laser-induced ﬂuorescence continues, the ion is ﬁnally cooled to the ground state |S, 0, at which point no further absorption can take place: the ion is effectively decoupled from the laser ﬁeld. This is indicated by the disappearance

8.4 Quantum Registers: Qregister

225

Fig. 8.17. (a) Quantum description of sideband cooling. A laser is tuned to lowfrequency sideband. Absorption is followed by spontaneous re-emission such that a net energy loss of ων results. (b) Absorption spectrum I(δ) vs. detuning δ (in units of the harmonic trapping frequency ν) of two trapped ions inside the Lamb–Dicke regime [480]

Fig. 8.18. Lower (a) and upper sideband (b) in the S1/2 -D5/2 absorption spectrum of a single Ca+ ion. Open circles: after Doppler cooling; Full circles: after resolved sideband cooling [431]

of the lower sideband in the absorption spectrum. It has been experimentally demonstrated that sideband cooling can be used to bring ions to their motional ground state [374, 481]. Figure 8.18 shows lower (a) and upper (b) sidebands of the absorption spectrum of a single trapped Be+ ion, oscillating with a frequency of 11.2 MHz, after Doppler precooling (solid points). The absorption spectrum becomes asymmetric, indicating a thermal average occupation number of the ground vibrational nz state of about 0.47. After sideband cooling (open points) the reduction of the lower sideband corresponds to nz = 0.014 (10) [431]. Tuning to the lower sideband leads to cooling of the corresponding mode in an ion string, as for a single ion. However, the other modes, that do not interact with the laser, may be heated due to spontaneous emission.

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8 Quantum Computing with Trapped Charged Particles

8.5 Creation of Nonclassical States The possibility of quantum computing with trapped ions relies on the ability to prepare nonclassical states of large strings of cooled trapped ions. Nonclassical states are those in which the quantum motion of ions in a harmonic oscillator potential is strongly coupled to their internal quantum states. A great number of diﬀerent families of nonclassical states belonging to the Schr¨odinger equation for the harmonic oscillator can be obtained by choosing diﬀerent algebras and basic states [478, 479, 482]. In the preparation of the nonclassical motional states of the ions, the internal levels of the ions cooled to the oscillator ground state are coupled to each other and to the external vibrational levels [334, 483, 484]. Some important nonclassical states that are prominent in the ﬁeld of quantum computing are: the number states (which are just the Fock states, if the oscillator is independent of time [482]), the thermal states (distributions) (which are deﬁned for a single ion in thermal equilibrium with an external reservoir at a determined temperature), and coherent states [485, 486]. A Schr¨ odinger-cat state is a superposition of two coherent states such that their maximum spatial separation is much larger than the single component wave packet extension. 8.5.1 Fock States A Fock state is any state with a well-deﬁned number of particles. For a harmonic oscillator, a Fock state is of the type |n with n an integer. A single ion cooled into the ground state |0 of its oscillation in the trap and prepared in a well-deﬁned quantum state, which we may call |↑ is in the Fock state |↑ |0. Applying a laser pulse for a time τ to the blue sideband of the absorption spectrum of a narrow magnetic dipole transition brings the ion into the state |↓ |1 state with probability P (τ ) = cos2 Ωτ where Ω = μB Bl /(4) is the Rabi frequency, proportional to the magnetic ﬁeld amplitude Bl of the laser. Figure 8.19 shows the time dependence of the probability of ﬁnding a single ion in the initially prepared Fock state after applying laser pulses at

Fig. 8.19. Probability of ﬁnding a single ion in the initially prepared Fock state after applying laser pulses at the blue sideband [487]

8.5 Creation of Nonclassical States

227

the blue sideband. The reduction in probability is caused by decoherence from interaction with uncontrolled time varying electromagnetic ﬁelds. 8.5.2 Coherent States A coherent state |α is the eigenstate of the annihilation operator [488], where α is a complex number. Its dynamic is most closely related to the behavior of a classical harmonic oscillator. A coherent state is left unchanged by the detection (or annihilation) of a particle. The eigenstate of the annihilation operator has a number distribution. This is a condition that all detections are statistically independent. In contrast, in a Fock state once one particle is detected, there is zero probability of detecting another. Coherent states can be produced in diﬀerent ways starting from the state |n = 0 of a single ion. It requires a spatially uniform force acting on the trapped ion. It can be realized by applying an electric ﬁeld which oscillates at the motional frequency of the trapped ion (harmonic oscillator). Starting from the |n = 0 state the force displaces an initial motional state |Ψ to D |Ψ, where the displacement D is proportional to the amplitude of the driving ﬁeld and the time of application. Wineland et al. [486] superimposed two traveling laser ﬁelds with a frequency diﬀerence Δω between them, tuned to the frequency (ω0 + δ), where ω0 is the optical resonance frequency of e.g. a S1/2 − P1/2 transition with the oﬀ-set δ. Since each of the laser is nearly resonant with this transition, an oscillating electric dipole force acts on the ion, enhancing Raman transitions. If Δω = ω0 a coherent displacement occurs in the motional wavefunction. Large coherent states produced with this method in a string of up to seven ions have been reported [473]. 8.5.3 Schr¨ odinger Cat States The term “Schr¨odinger cat states” refers to the superposition of two coherent states such that their maximum spatial separation is much larger than the single component wave packet extension. It allows distinguishing the positions of the two coherent states, and thus they are considered as “macroscopic” states. The creation of Schr¨ odinger cat states of a single ion has been performed by the NIST group by a series of polarized Raman laser pulses as illustrated in Fig. 8.20 [334]: Starting from the ground vibrational level (a), a π/2−pulse on the carrier transition splits the wave function into a superposition of the ground state |g, n = 0 and the |e, n = 0 with equal probability (b). The Raman beams excite only the motion correlated with state |e to a coherent state |α (c). Then a π-pulse on the carrier frequency interchanges the internal states |e and |g of the superposition (d). A second pulse of the Raman beams excites #the motion correlated with the new |e component to a $ second coherent state #αeiΦ (e). Finally a π/a−pulse on the carrier #transition $ combines the two states. The ﬁnal state has the form |g |α + |e #αeiΦ , in

228

8 Quantum Computing with Trapped Charged Particles

Fig. 8.20. Creation of Schr¨ odinger-Cat states with trapped ions for α = 3 and φ = π by entangling the ion internal states |g, n = 0 and |e, n = 0 with the motional states. (a) The initial wavepacket; (b) the wavepacket splitting; (c) only the motion correlated with the |e component is excited; (d) the internal state population between |g and |e states are interchanged; (e) the Schr¨ odinger’s-cat state [334]

which the coherent motional components have the same amplitude but diﬀerent phases. This combined state can be written for a relative phase Φ = π as follows [334]: # $ $ 1 # |Ψ = √ |e #αeiΦ + |g #αe−iΦ , (8.22) 2 Schr¨ odinger-cat states with α 2.97(6) have been measured corresponding to an average vibrational quantum number of n ¯ 9; the individual wave packets were spatially separated by 83(3) nm (a single wave-packet size was about 7.1(1) nm) [334]. Schr¨ odinger cat states including several ions have been produced by the NIST group [489] and at Innsbruck [302]. Cat states are very sensitive to decoherence, and as a result their preparation is challenging and can serve as a demonstration of good quantum control.

8.6 Quantum Logic Gates One of the requirements of universal quantum computing is to coherently change the state of qubits. For example, this can transform a qubit starting out in |0 into any arbitrary superposition of |0 and |1 deﬁned by the user. In a trapped ion system this is often done using stimulated Raman transitions for hyperﬁne qubits and electric quadrupole transitions for optical qubits. Quantum logic gates are elementary sets of quantum operations, each involving a small number of qubits, that together constitute a quantum computer capable of processing quantum information. Any quantum computation can be composed of a series of single-bit rotations and two-bit controlled-not operations. Single bit rotations of a qubit α |0 + β |1 can be characterized by a unitary transformation Uθ cos θ/2 sin θ/2 . Uθ = (8.23) − sin θ/2 cos θ/2

8.6 Quantum Logic Gates

229

One-Qubit Gate The most basic linear transformation of a single qubit is the Hadamard gate (H) described by the unitary matrix: i 1 1 H= √ , (8.24) 2 1 −1 which, acting on the basis states gives: 1 H |0 → √ (|0 + |1), 2

1 H |1 → √ (|0 − |1). 2

(8.25)

Another one-single qubit gate is the phase shifter gate Φ(ϕ) represented by 2 × 2 matrices of the general form [490] 1 0 Φ(ϕ) = , (8.26) 0 eiϕ where ϕ is the phase shift, thus: Φ |0 = eiϕ |0 ,

Φ |1 = |1 .

(8.27)

It introduces a phase shift ϕ between the two states |0 and |1 of the qubit. Two-Qubit Gates The quantum controlled Not (CNOT) gate, also known as the exclusive or XOR gate (Fig. 8.21) is the simplest two-qubit gate; it makes the evolution of one qubit (the target) conditional on the state of the other (the control). Thus the target qubit ﬂips only if the control qubit has the logical value 1 (Table 8.2). In a two-particle state basis written as the vectors: ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 0 0 0 ⎜0⎟ ⎜1⎟ ⎜0⎟ ⎜0⎟ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ |00 = ⎜ (8.28) ⎝ 0 ⎠ , |01 = ⎝ 0 ⎠ , |10 = ⎝ 1 ⎠ , |11 = ⎝ 0 ⎠ . 1 0 0 0

Fig. 8.21. The controlled NOT gate negates the logical value of the target qubit, if and only if the control qubit has the logical value 1 [490]

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8 Quantum Computing with Trapped Charged Particles

Table 8.2. The CNOT gate has two inputs. The second input is negated only if the ﬁrst input is true Input 1

Input 2

1 0

Output 1

0 1

Output 2

1 0

1 1

Fig. 8.22. Quantum circuit for CNOT gate swapping a pair of bits. The lowest bit |B is ﬂipped whenever the upper bit |A is set [490]

Table 8.3. The quantum Toﬀoli gate Input 1

Input 2

1 1

1 0

Input 3

Output 1

1 0

the CNOT gate may be represented as ⎛ 1 ⎜0 UCNOT = ⎜ ⎝0 0

1 1

Output 2 1 0

Output 3 0 0

the unitary operator [490]: ⎞ 0 0 0 1 0 0⎟ ⎟. 0 0 1⎠ 0 1 0

(8.29)

Here, the ﬁrst particle acts as a conditional gate to ﬂip the state of the second particle. The action of the CNOT gate is represented by the schematic diagram shown in (Fig. 8.22). Three-Qubit Gates Of the more complex gates, we mention brieﬂy the three-qubit Toﬀoli gate which has three inputs. The ﬁrst two inputs are control qubits, the third is a target qubit that is ﬂipped only if the ﬁrst two are both 1, as shown in Table 8.3. The Toﬀoli gate is reversible, having itself as inverse such that applying it twice to a set of qubits returns the qubits to their original states. The construction of a quantum Toﬀoli gate is based on a combination of twobit and a one-bit unitary operations such that the AND of the ﬁrst two inputs controls the third through a CNOT gate, as shown in Fig. 8.23. A Toﬀoli gate together with a Hadamard, phase and CNOT gates give a universal allowing to perform all the gates required by a quantum computer

8.7 Qubit Entanglement

231

Fig. 8.23. The circuit representation for the Toﬀoli gate [490]

with trapped ions. For further reading on this subject the reader is referred to Bouwmeester et al. [490].

8.7 Qubit Entanglement For quantum computing it is important to know the extent to which the quantum states of the ions are entangled, and how to manipulate those states [330]. A two-qubit entangled state has the form [490]: 1

|Ψ12 = √ |01 |12 + eiγ |11 |02 , (8.30) 2 where γ is a phase factor. An entangled state is never orthogonal to any of its component states. As soon as one qubit is measured, the other qubit will be found to carry the opposite value. It can be created by diﬀerent methods [490]. A possibility to generate two-qubit entanglement is to create a source in which a spin-0 system decays into two spin-1/2 particles; conservation of internal angular momentum then gives the state: 1 |Ψ12 = √ (|↑1 |↓2 − |↓1 |↑2 ) . (8.31) 2 The three qubit entangled GHZ state has the form: 1 (8.32) |ΨGHZ = √ (|01 |02 |03 + |11 |12 |13 ) . 2 Multi-qubit entanglement is fundamentally diﬀerent from the case of twoqubits requiring minimal control over interactions. Multi-qubit entanglement of trapped ions can be constructed using axial and radial linear systems for the dynamical symplectic group Sp(2,R). The axial generators Ka0 , Ka± and the radial generators Kr0 , Kr± are [478, 479] ∂ 1 ∂2 ∂2 ∂ 2 2 x + y + 2 + 2 ∓ 2x ∓ 2y ± 1, (8.33) Kr± = − 4 ∂x ∂y ∂x ∂y 1 1 ∂2 ∂ z 2 + 2 ∓ 2z ± , Ka± = − 4 ∂z ∂z 2 for [Kc− , Kc+ ] = 2Kc0 with c = a, r. Introducing the unitary operators Uc (zc ϕc ) = exp(zc Kc+ ) exp(ηc Kc0 ) exp(−¯ zc Kc− ) exp(−iKc0 ϕc ), where ηc = ln(1 − zc z¯c ), zc is a complex coordinate in the unit disc |zc | < 1, and the

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8 Quantum Computing with Trapped Charged Particles

global phase ϕc can be a function of zc , the symplectic coherent states Φ(za , zr ) = Ua (za )Ur (zr )Φ with Φ a ﬁxed weight vector can be deﬁned. In the case of a single ion stored in a Paul trap, the quasienergy solutions of the time-dependent Schr¨ odinger equation are symplectic coherent states for suitable phases ϕa and ϕr . The system of nonorthogonal symplectic states is overcomplete. Then the general entangled symplectic coherent states are Ψ = dμ(Z)f (Z)Φ(za1 , zr1 ) ⊗ ... ⊗ Φ(zan , zrn ), (8.34) where Z = (za1 , zr1 , . . . , zan , zrn ) and μ is the measure corresponding to the resolution of unity on the product of 2n unit discs. The degree of entanglement puriﬁcation can be characterized by the distribution f . The symplectic multi-cat states corresponding to the multi-qubit entanglement are obtained for n 1 + π f (Z) = n exp i(εj + σj ) δ(zaj − εj waj )δ(zrj − σj wrj ), 2 4 j=1

(8.35)

where the sum is over all εj = ±1 and σj = ±1.

8.8 Quantum Information Processing In 1995 Cirac and Zoller [311] proposed a procedure for using CNOT-gate operations between any two qubits to build a universal quantum computer. The basic idea was that the ion motion provides the degree of freedom which can be used to carry and convey information. The internal electronic state of any given ion can be linked to the quantum state of the whole string of ions. Cirac and Zoller proposed to use a sequence of such links to achieve a state change of a target qubit conditional on the state of a controlling qubit. Starting with the quantum mechanical ground state of the ion string oscillation, a laser pulse directed to the controlling qubit maps the internal state information of that particular qubit onto the center-of-mass motion (bus motion), whereas the internal state is put into its ground state. Thus, the new quantum state of the register is an entangled state of the internal and external states. With the laser subsequently directed to any other ion, serving as target ion, it is possible to manipulate the internal state of that target ion if and only if there is motion in the string. Finally, taking the motion out of the string by undoing the ﬁrst step to the controlling ion and thus restoring the controlling qubit, makes it possible to realize a truth table of the CNOT gate operation. The Cirac–Zoller gates have been realized till now by groups at NIST [375, 491], and Innsbruck [304]. We take the experiments on Ca+ ions of the Innsbruck group as example: Two ions are loaded into a linear Paul trap. The

8.8 Quantum Information Processing

233

Fig. 8.24. Lasers and wave packet conﬁgurations for a two-qubit gate at nonzero temperature. The dark (light) ﬁlling of the wave packets stands for the excited (ground) states [496]

ﬁrst one serves as control qubit and the second one represents the target qubit. With a laser pulse at a sideband frequency of the resonance transition directed at ion #1 the external and internal degrees of freedom were excited. Then a series of laser pulses was applied to ion #2 in such a way that the amplitude of the wavefunction is changed if and only if there was any motion in the two-ion string. Finally, a laser pulse applied to ion #1 remaps the motional state to the excited state of the controlling ion and the string is at rest as it was before the start of the operation (Fig. 8.16). Besides the controlled-NOT gate proposed by Cirac and Zoller, several equivalent, but more robust, schemes have been proposed [298, 492, 493] and implemented experimentally since [336, 494, 495]. The development of a quantum gate operable at nonzero temperature [496] has made it possible to avoid the complication of having to cool the ions to the vibrational ground state to prepare the initial state of the CM mode. It uses the movement of one ion’s wave packet under laser excitation to produce a conditional shift in the position of a second ion, as illustrated in Fig. 8.24. It was implemented according to the following sequence of operations: First, by means of a short laser pulse the control ion (1) is kicked left (or right) depending on its internal state, |0 (or|1). Because of Coulomb repulsion between the ions, the target ion (2) will experience a corresponding kick that is therefore in a direction conditional on the internal state of ion (1). The motional state of ion (2) will be split spatially into two possible wave packets. After a certain time, a laser beam is focused only on that half of the wave packet of ion (2) that corresponds to ion (1) having been in state |0 to induce the transition |02 ↔ |12 . Had ion (1) been in state |1 this second laser pulse would have no eﬀect on ion (2). Finally after another interval of time, a laser π−pulse is applied to ion (1), completing the two-qubit gate operation. It has been shown by Cirac et al. [496] that the action of this gate is independent of the temperature.

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8.8.1 Speed of Operation An important ﬁgure of merit of a computer is its switching rate, R, or clock rate. The switching rate of a processor using laser pulses is limited by the duration of these pulses. It is given in the Lamb–Dicke limit by: R

ηΩR √ , 2π N

(8.36)

where ΩR is the Rabi frequency for a free ion, η the Lamb–Dicke parameter for a single trapped ion, and N the number of ions in the string. It is apparent from this result that the switching rate decreases with the number of ions in the string. The switching rate is related to gate precision ε ΩR /(2ηωz ) or the gate ﬁdelity f = 1 − ε2 . To attain a given gate precision ε, the Rabi frequency must satisfy the relation ΩR ≤ 2εηωz. Experimentally, gate times of a few μs per ion have been obtained [497]. The maximum switching rate which holds for η 2 N is given by [417]: R2 <

1 ER ωz · , 20π N

(8.37)

where ER = (k)2 /(2M ) is the recoil energy of an ion of mass M after the emission of a single photon. Unfortunately, R cannot be made larger by increasing ωz , because as ωz is made larger the unwanted Δn = 0 transitions appear when Δn = ±1 transitions are driven to perform quantum gates between an ion and the phonon bus and thus the gate precision decreases. Cirac et al. [311, 498] have suggested that ωz be increased by positioning the ion in the node of a standing wave; however, it may not be technically feasible to achieve this for more than a few ions. The data transmission speeds with the bus channel are small (between 10 and 100 kHz); a greater speed (about 1 MHz) can be obtained with another type of quantum communications channel, one in which the transfer of the wavefunction from one atom to another is based on adiabatic passage via a dark state of the two atoms in a cavity, as shown in Fig. 8.25a. The atoms are assumed to have ground states that are split into two sublevels (a, b) and (a’, b’) which store the quantum information, and optically excited states. One

Fig. 8.25. (a) Quantum communication channel using cavity photons; (b) two atomic particles exchanging quantum information by a cavity photon communication channel [499]

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235

of the two optical transitions of each atomic particle is driven by an external laser, while the other is resonant with the cavity photon. The spontaneous emission process of the two atoms is greatly reduced as the photons exchanged by them are virtual photons [500,501]. The data transmission speeds with the cavity photon channels are greater than with the quantum bus, but decreases with the number of particles in the cavity. The limiting speed of this method is set by the vacuum Rabi frequency, a parameter inversely proportional to the square root of the number of atoms [437]; thus such quantum channels provide fast communication for relatively few qubits. It seems that there are no fundamental limitations to the speed of entangling gates [502], but gates in this impulsive regime (faster than 1 μs) have not yet been demonstrated experimentally. The ﬁdelity of these implementations has been greater than 97%. 8.8.2 Nonclassical State Reconstruction To quantify the accuracy of the gate operation and to determine potentially detrimental eﬀects like decoherence or motional heating it is required to reconstruct the state of any trapped ion with high ﬁdelity [482]. Several schemes, called quantum state tomography or endoscopy have been proposed [498,503– 506]. A complete description of the system can be obtained by construction of its density matrix [507], the Husimi-Kano Q function [508], the quadrature distribution function [509], or the Wigner function (analog to the classical phase space distribution function [510]). Experimentally quantum jump spectroscopy of each individual ion using excitation of the sidebands of the resonance transition is used to determine the internal and external state of the ion [337]. The reconstruction is characterized by the ﬁdelity, the diagonal elements of the density matrix of the entangled state: F = Ψ |ρ| Ψ. Figure 8.26 shows the result of such a measurement including the truth table (right) where the |S and |D states of the Ca+ ion are used as qubit [304].

Fig. 8.26. Experimentally observed truth table of a Cirac–Zoller CNOT operation using the |S and |D states of the Ca+ as qubit. The ﬁdelity of the gate operation is limited to about 70–80% [337]

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8 Quantum Computing with Trapped Charged Particles

H¨aﬀner et al. [337] have reconstructed the density matrix of two robustly entangled Ca+ ions using the method of full state tomography. Assuming F ≥ Fmin = 2 |ρ01,10 |, where Fmin is a lower bound of F, the ﬁdelity can be determined by measuring a single element of the density matrix, namely

01 |ρ| 10 [301]. An approach for recovering full information about the harmonic motional state is called phase space tomography which uses repeated measurement of a complete set of observables [511]. From the experimentally measured probabilities Q(α) =

1

α |ρ| α, π

P (x, θ) = x, θ |ρ| x, θ,

(8.38)

where |α = |α| eiθ is a coherent state, and |x, θ is a quadrature eigenstate, one can determine uniquely the density operator ρ=

∞

ρnm |n m|

(8.39)

n,m=0

represented in the Fock basis of the harmonic oscillator. To determine the Q(α) function and the quadrature distribution function P (x, θ), applied a unitary transformation U (|α| , θ) to transform the original density matrix ρ as ρ˜ = U (|α| , θ)ρU † (|α| , θ), D(α) = exp(αa† − α∗ a),

U † (|α| , θ) = R† (θ)D(|α|), R(θ) = exp(−iθa† a),

(8.40)

where D(α) is the displacement operator corresponding to a sudden displacement of the harmonic trap, R(θ) is the phase shifting operator that acts on vacuum (in a free evolution) to create the coherent state |α =| |α| eiθ , and a† and a are the creation and annihilation operators such that [a, a† ] = 1. The transformation U † has to be identiﬁed as a physical process applying to an ion trapped in a harmonic potential. Thus in order to measure the Q-function, one has to allow the ion motion to evolve freely for a time t in the trap to develop a phase shift θ, then displace suddenly the center of the trap potential to the right for a distance |α|, and ﬁnally to measure the probability of the ion being in the lowest motional state |0; this gives the function Q (Fig. 8.27a). This procedure can be implemented on an ion with an internal energy level structure consisting of three levels: |g, |e, and |r, with the transition |g ↔ |e an allowed transition and |g ↔ |r an electric dipole forbidden one. The ion is prepared initially in the internal ground state |g. After the displacement of the trap center, a laser beam is tuned to the lower ﬁrst sideband of the |g → |r transition to transfer completely the population of the ground state |n, g (with n = 1, 2, 3, . . .) to the excited state |n − 1, r [509]. After this population transfer, another laser is applied on resonance with the transition |g → |e . The appearance of ﬂuorescence indicates the presence of population in the |0, g state. For the ﬁnal reconstruction of the quantum

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237

Fig. 8.27. Phase space representation of the operations (phase shifting, displacement and squeezing) required to measure the Q(α) function (a) and the quadrature distribution (b) [508]

state from the experimental data of the Q-function, it can be assumed that ρnm = 0 for n, m greater than some maximum value nmax . The measurement of Q(αi ) for n2max independent values of αi would allow ρn,m to be computed by simple matrix inversion, however, the errors would be large [508]. A more precise way of tomographic reconstruction of the state of a quantum system is by measuring the quadrature distribution function P (x, θ) deﬁned in (8.38). The measuring scheme (Fig. 8.27b) is based on the property of squeezed states |x, θ, deﬁned as [508]: |x, θ = lim Nε |α, ε , |ε|→∞

Nε = exp [2 |ε| /(4π)]1/4 ,

(8.41)

2

|α, ε = D(α)S(ε) |0 , S(ε) = exp [(ε∗ a2 − εa† )/2], √ where α = xeiθ / 2 for 0 ≤ θ < π, ε = |ε| e2iθ is the squeezing parameter corresponding to the squeeze operator S(ε), and Nε is a constant of proportionality. Then P (x, θ) is given by P (x, θ) = lim |Nε |2 0 |˜ ρ| 0, |ε|→∞

ρ˜ = U ρU † ,

(8.42)

where U † = R† DS denotes the operation creating a squeezed state. Thus to measure the quadrature distribution one has to ﬁnd the physical processes corresponding to the unitary operators R, D, S. For that one has to wait a time t such that θ = (Ω/2π)t to perform a sudden displacement of the center of the trap to the right a distance |α|, then change the trap frequency instantaneously from Ω/(2π) to Ω /(2π) to create the squeezed states [512], and ﬁnally to determine the population of the motional ground state. Once the quadrature distribution is known one can construct the Wigner function W (x, p) by means of the inverse Radon transformation [511], or one can construct the density matrix directly from discrete measured data [513].

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Fig. 8.28. Real and imaginary part of the density matrix ρΨ+ that approximates √ (a) Ψ+ = (|10 + |01)/ 2 with the measured ﬁdelity FΨ+ = Ψ+ |ρΨ+ | Ψ+ = 0.91, √ and (b) Ψ− = (|10 − |01)/ 2 with ﬁdelity FΨ− = 0.90; density matrix elements of (c) ρΦ+ with ﬁdelity FΦ+ = 0.91 and (d) ρΦ− with ﬁdelity FΦ− = 0.88 [331]

Roos et al. [331] have tomographically reconstructed the density matrix of two deterministically entangled trapped ions (Fig. 8.28) using the Pauli matrices , - for the ions (1) and (2) and measuring the expectation values (1)

σi

(2)

⊗ σj

where i, j = 0, 1, 2, 3. The experiments were done on two 40 Ca+

ρ

ions, conﬁned in a linear Paul trap and laser cooled until the axial breathing mode reached the ground state. For quantum state manipulation a narrow

8.9 Qubit Decoherence

239

Fig. 8.29. Phase evolution of the Bell state Ψ+ [331]

band Ti:Sa laser was tightly focussed individually on either one of the two ions. By exciting the S1/2 (m = −1/2) → D5/2 (m = −1/2) quadrupole transition (denoted here as |0 → |1) a single ion was prepared in a superposition of these states. If the laser is tuned to the ﬁrst blue sideband of the transition, the electronic states |0 and |1 become entangled with the motional states of the CM mode. An acousto-optical modulator was used to switch between the carrier and sideband frequencies, to control the phase of the laser ﬁeld, and to switch the laser beam from one ion to the other, over a distance of 5.3 μm. By a sequence of laser pulses of appropriate length, frequency and phase the two ions are prepared in a Bell state. For detection of the internal quantum states, the S1/2 → P1/2 dipole transition near 397 nm was excited and the ﬂuorescence from each ion monitored. If ﬂuorescence appears, the ion is in the S1/2 state, otherwise it is in the D5/2 state. By repeating the experimental cycle 200 times Roos et al. found the average populations of all product basis states: |00, |01, |10, |11. The negative eigenvalues in Fig. 8.28 show that the mixed state ρ of the two ions is indeed entangled [514, 515]. The characteristics of all four created Bell states, that √ two-ion deterministically √ is Ψ± = (|10 ± |01)/ 2 and Φ± = (|11 ± |00)/ 2, which included their time evolution, their entanglement and decay were monitored by doing state tomography at time intervals (Fig. 8.29). The time evolution of the relative phase of the |01 and |10 components of Ψ± was found to vary linearly with time due to a magnetic ﬁeld gradient producing diﬀerent Zeeman shifts in the two ions. The decay of the Bell state into a mixed state can also be seen by plotting the entanglement as a function of time (Fig. 8.30); the maximum measured lifetime of the Ψ± state was 20 ms [331].

8.9 Qubit Decoherence To perform a certain number of gate operations as required for any algorithm it is necessary that the qubit states remain unperturbed for a time much longer than the switching time. This remains today a serious challenge particularly when a large number of ions is involved. A number of diﬀerent

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8 Quantum Computing with Trapped Charged Particles

Fig. 8.30. Decay in time of the Bell state Ψ+ into a statistical mixture. The insets show the density matrix elements measured after 2 and 8 ms, respectively [331]

sources for decoherence for the internal or external qubits have been identiﬁed (see Sect. 7.5). The decay times for hyperﬁne qubits can be extremely long. For example Fisk et al. [516] have observed coherence times greater than 10 min. Such qubits require the use of ﬁrst-order magnetic-ﬁeld “independent transitions”; that is, use of an ambient magnetic ﬁeld, where the spin qubit energy separation goes through an extremum with respect to magnetic ﬁeld. Spontaneous decay of hyperﬁne levels play no role since their natural decay times are typically of the order of several years. Coherence times of optical qubits from the ground to a long-lived excited state are limited by the natural decay time of the excited state which, e.g. in the case of the Ca+ ion is of the order of 1 s. This is substantially longer than experimentally achieved gate times. Insensitivity to ﬂuctuating ambient magnetic ﬁeld can be obtained by using Zeeman substates having the same g factor forming a decoherence-free subspace. During qubit operations laser intensity and phase ﬂuctuations will cause decoherence. Active laser stabilisation to the highest possible degree is therefore required.

8.10 Scalability Scalability is a desirable property of a system indicating its ability to either handle growing amounts of work in a graceful manner or to be readily enlarged. The performance of a scalable system improves after adding hardware proportional to the added capacity. A scalable processing system is one that can be upgraded to process more transactions by adding new processors, and which can be upgraded easily and transparently without shutting it down. There are two methods of scaling: Scale vertically (up) and scale horizontally (out). To scale up means to add resources to a single node in a system (e.g. the addition of memory to a single computer). To scale out means to add more nodes to a system, for example adding a new computer. It is typically cheaper to add a new node to a system to achieve improved performance than to partake in performance tuning to improve the capacity that each node can handle.

8.10 Scalability

241

For implementation of algorithms and operation of ion trap quantum computers scaling to large qubit numbers is a necessary requirement. This represents in fact one of the main hurdles in the construction of an ion trap quantum computer. Currently, most ion traps are bulky structures which require manual alignment of the electrodes. As traps become more complex, such an assembly technique becomes unmanageable. Therefore, the design and implementation of scalable ion trap fabrication methods is of key importance for ion trap quantum computing. Provided that the motional heating can be reduced suﬃciently, miniature size traps not only allow a large number of qubits to be stored on small areas but at the same time one beneﬁts from increasing gate speed and possible ion separation in multiplexed traps. Two diﬀerent routes are presently under investigation: Using arrays of interconnected traps, each loaded with a single ion, photon interconnection can be employed to transfer quantum superpositions from a qubit in one trap to a second one [517–519]. Connection of diﬀerent traps by wires is also being investigated: The image voltage induced by the oscillating ion in one trap is transferred to a second trap conveying information on the motional state of the ﬁrst ion [520]. Cirac and Zoller [521] have put forward a possible quantum computer based on an array of ion microtraps using conﬁnement ﬁelds at optical or radio frequencies (Fig. 8.31). The ions are adressed and interact by applying an external ﬁeld such that its motional wave packet is displaced by a small amount depending on its internal state (Fig. 8.31a). To implement a two-qubit phase gate of the type required for quantum computing, two neighboring ions are addressed inducing a diﬀerential energy shift, that depends on their internal states, due to the Coulomb interaction between ions. A major advantage of this model is that it is readily scalable since the motion of the ions is manipulated independently, and moreover the ions need not be initially brought

Fig. 8.31. Quantum gates in scalable quantum computer. (a) Ions are trapped in independent traps separated by the distance d large enough, so that the Coulomb repulsion is not able to excite the vibrational state of the ions. For the gate, they are pushed by an external force to a distance ao provided they are in state | 1 >; (b) A planar array of independent traps, and a diﬀerent ion (“head”) that moves above this plane, approaching any particular ion. By switching on a laser propagating in the perpendicular direction to the plane, the two-qubit gate between the target ion and the head can be performed. Swaping the state of the ion to the head, entanglement operations between distant ions can be performed [521]

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8 Quantum Computing with Trapped Charged Particles

Fig. 8.32. An elliptical planar ion trap (with axes of 300 μm and 200 μm) trapping a few ions to embody one register. The trap array is constructed from a single ﬂat conducting sheet in which elliptical holes are made by photolithography [499]

to the zero point of energy. Ions moving above the array plane can be brought very close together, and thus the gate operation time can be reduced to only some microseconds. The distance between the ions in the plane can be very large, since no interaction between them is required. In a US patent ﬁled in 1996, DeVoe [499] proposed parallel architectures based on large arrays of small ion traps consisting of elliptical micro-apertures in a conducting plane, with axes parallel to each other, such that each conﬁnes a linear crystal of only a few ions (Fig. 8.32). DeVoe [518] has shown that a planar elliptical trap with high excentricity can hold a large number of ions, and that the amplitude of the micromotion is a weak function (N 1/5 ) of the number of ions N . Each trap contains one register and has its own CM phonon as a quantum communication channel with phonon frequencies of about 1 MHz, allowing fast quantum processing. DeVoe’s architecture stipulates the use of short communication channels that operate in parallel to permit the use of algorithms to increase the speed of basic arithmetical operations, while a few larger channels are reserved for more complex infrequently performed operations. The short channels for coupling between registers are realized by photon coupling which oﬀers a greater channel speed [499]. This is accomplished with an optical cavity having a resonant optical mode that irradiates two ions belonging to diﬀerent elliptical traps. The optical cavity consists of two concave mirrors on one side of the plane of the traps and one ﬂat mirror on the other; each concave mirror has a curvature such that a Gaussian optical mode has a focal spot size small enough to illuminate only one ion in the string. To exchange quantum states

8.10 Scalability

243

Fig. 8.33. Two elliptical planar ion traps and a three-mirror optical resonant cavity forming a communication channel for transmitting quantum information from one trap to the other [499]

Fig. 8.34. Proposed multizone trap to separate diﬀerent logical operations in space [522]

between the two illuminated ions in the two traps, control laser beams with tailored overlapping pulse shapes are applied, as shown in Fig. 8.33. Alternatively ion qubits can be moved from one trap to another or to diﬀerent trapping zones by application of time varying voltages to diﬀerent electrodes (Fig. 8.34) [522]. Many proposals have been made for large scale architectures of quantum computers using sub-millimeter size traps. Planar structures on a chip are very likely the most suitable solution. The ﬁrst successful operation of such a trap was reported by a group at the University of Michigan (Fig. 8.35) [523] and at NIST (Fig. 8.36) [524]. The structures were grown by molecular beam epitaxy on a GaAs substrate (Michigan) or by photolithography and gold deposition on fused silica (NIST). More recent designs have been proposed and are being tested at MIT [525] and NPL [526]. To separate in space the regions of manipulation of qubits and readout it has been proposed to move the ions into diﬀerent zones of an array of linear traps [467, 527, 528]. It may signiﬁcantly reduce the overhead on laser-beam

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8 Quantum Computing with Trapped Charged Particles

Fig. 8.35. (a) Scanning electron microscope image of a monolithic GaAs semiconductor linear ion trap with several axial segments cantilevered over a through hole in the center. The bright sqares are gold bonding pads. The electrode distance across the gap is 60 μm. (b) Photograph of the trap [523]

Fig. 8.36. Central trapping region (marked by x) of a microtrap fabricated at NIST. The rf tapping voltage is applied at the indicated electrode, electrodes 1–5 serve for dc axial conﬁnement, correction of trap unharmonicities, and for excitation of the trapped ion oscillation [524]

Fig. 8.37. Segmented linear trap with diﬀerent zones for loading, qubit manipulation and readout (Courtesy F. Schmidt-Kaler, Ulm University)

control for performing single- and multiple-qubit operations on trapped ions. Figure 8.37 shows as example the trap designed and operated at Ulm. The group at Michigan has devised a trap in which the ions are moved around a corner (Fig. 8.38) [468]. Possible inﬂuence on decoherence has still to be explored.

8.11 Penning Trap as Quantum Information Processor

245

Fig. 8.38. Photograph of a microtrap, produced at the University of Michigan, having a junction between diﬀerent trap zones. Ions have been moved around the corner many times without loss

8.11 Penning Trap as Quantum Information Processor 8.11.1 Computing with Electrons A system based on individual electrons conﬁned in an array of Penning traps is a promising candidate for quantum information processing that would beneﬁt from techniques devised for NMR and Paul trap quantum computing [529, 530]. Unlike the Paul trap, a Penning trap uses only static ﬁelds for trapping, ﬁelds that are capable of greater stability and hence slower decoherence of the trapped particle states. A quantum processor based on electrons has the advantage over ions of being more stable and uniquely able to be encoded with up to three qubits: An internal qubit embodied in the up and down spin orientations in a magnetic ﬁeld, as well as two external qubits, the ground and ﬁrst excited states of the axial and cyclotron motions. Due to their smaller mass, electrons have higher resonance frequencies than ions, frequencies that lie in the radiofrequency (rf) and microwave parts of the spectrum, allowing the use of techniques developed for NMR experiments. Like ions, the manipulation of electron states between vibrational and cyclotron modes must be accomplished without excitation to higher states outside the computational space. For ions, this is achieved by using optical pulses on sideband transitions, and an equivalent method must be devised for electrons. As for the exchange of information in the electron case, it can be realized either by the Coulomb interaction (controllable through geometrical and electrical parameters of the trap), or by the image charge which an oscillating electron induces in the central electrode of the trap [38, 512, 531]. The electric ﬁeld produced by the image charge shifts the frequency of the axial motion of the trapped electron to a degree proportional to the electron number [21, 532], allowing qubit detection. 8.11.2 Linear Multi-trap Processor A linear Penning multi-trap, that is, a linear array of traps, may take the form of a set of coaxial ring electrodes held at diﬀerent potentials as in the example shown in Fig. 8.39 with ten traps [37,108]. Each trap in the array has a central

246

8 Quantum Computing with Trapped Charged Particles

Fig. 8.39. Quantum information processor with electrons trapped in a linear array of Penning traps. 1–10 – Penning traps; 11 – electron source; 12 – anode; 13 – extraction electrode; 14 – ﬁeld emission point; 15 – electrode for reﬂection of electron beam; 16 – vacuum ﬂange; 17 – microwave inlet; B – magnetic ﬁeld

ring electrode and one pair of compensation electrodes, with each pair of adjacent traps sharing a common end electrode. The central ring electrodes are made of a magnetic material to provide an inhomogeneous component to the magnetic ﬁeld for spin and cyclotron state detection, while the compensation and end electrodes are made of pure copper. This periodic arrangement of electrodes produces along the z-axis a periodic potential consisting of saddle points at the centers of the traps. As applied to quantum computing such a Penning multi-trap would store one electron in each of the axial potential minima, which together with an axial magnetic ﬁeld provide the same conﬁnement conﬁguration as a quadrupole Penning trap. The output of an ultra-stable dc power supply is connected through low pass ﬁlters to the trap electrodes to minimize external noise. The addressability of individual electrons requires that their characteristic frequencies, including the axial, cyclotron, and magnetron modes, be resolved. Control and separability of the frequencies of individual electrons is possible through gradients in the magnet ﬁeld and selection of static potentials on the electrodes. The inﬂuence of the Coulomb interaction between adjacent electrons on their frequencies becomes signiﬁcant only when they approach resonance, otherwise the electrons are independent. The extensive experience that has in the past been accumulated on the modes of behavior of a single charged particle in a Penning trap, and their manipulation, apply equally to the individual traps in the array, except when adjacent electrons are near a resonance. For example the experimental studies on the energy exchange between the cyclotron and magnetron motions done on individual ions at a temperature of 4K [37, 108, 533] could readily be extended to electrons by an appropriate change of ﬁeld strengths and increase in the frequencies. In prior single ion experiments [23], the cyclotron–magnetron mode coupling was achieved by an rf quadrupole ﬁeld in the radial plane created by dividing the ring electrode into four equal sectors, with opposite sectors connected together and an rf potential applied between the two pairs. The coupling of the cyclotron to the axial motion was realized through an inhomogeneous component in the magnetic ﬁeld [37]. All these methods are equally valid in the development of an electron-based system.

8.11 Penning Trap as Quantum Information Processor

247

When the axial frequencies of adjacent electrons are on resonance, they will experience signiﬁcant Coulomb interaction if their separation is small enough, otherwise an interaction may take place through the image charges induced in the trap electrodes. In the latter case, quantum information can therefore be transmitted from one trap to another in the array by connecting a suitable electrode of one trap to another by a wire or information exchange rail [534]. To extend the ability of any electron to interact with any other in the array, a part of the compensating ring of each trap is ac-coupled (via motional coupling capacitors) to the corresponding electrodes of all the others, so that if any two electrons in the array have, by adjustment of their trapping potentials, axial frequencies that are in resonance, they will interact. This type of interaction between particles allows scalability in a natural way, an important advantage over the vibrational mode coupling employed by the ion trap processor. 8.11.3 Planar Multi-trap Processor Recently a new electron-based, multi-Penning trap quantum processor has been proposed [534], that is predicted to be able to implement quantum operations with the precision prescribed for fault-tolerant computations. It is a planar array of Penning traps (Fig. 8.40d) obtained by embedding an arbitrary number of identical traps in a regular array in a common dielectric substrate plane, of alumina, or ceramic. Each planar trap has electrodes consisting of a central circular disc surrounded by two rings, the outermost one being split into quadrants lithographically deposited on the substrate

Fig. 8.40. Trapped electron processor. (a) Sketch of the simplest planar Penning trap structure (a disk electrode and a ring one) obtained by removing the upper end cap and embedding the other trap electrodes in the same plane; (b) trap with three active electrodes (black color ) plus an external grounded one (gray color ). The small hole from the center allows the electron loading from the rear side; (c) shape of the trap electric potential above the substrate surface; (d) planar array of traps providing single electron addressing and quantum scalability

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8 Quantum Computing with Trapped Charged Particles

Fig. 8.41. (a) A multi-electrode planar trap having a small hole in the center to allow trap loading from bottom. Voltages applied to additional rings are used to modify the trap potential shape; (b) electrode electrical connections [536]

(Fig. 8.40a,b). When a trapping potential is applied between the electrodes of a trap a potential saddle point occurs along the perpendicular axis above the plane (Fig. 8.40c) assuring axial conﬁnement of the electron. As with the conventional Penning trap, radial conﬁnement is provided by an uniform magnetic ﬁeld Bz perpendicular to the substrate plane. This open trap structure allows easy access to the electron and avoids perturbing cavity eﬀects such as damping of the cyclotron motion of the electron [45]. The distance of the potential minimum along the perpendicular axis from the substrate plane has been numerically calculated to be of the order of the radius of the central electrode of the trap. For a ring diameter of 2 mm, the stored electron freely ﬂoats at about 1 mm above the substrate surface. For a trap diameter of 2.5 mm and a substrate disc 2.5 cm in diameter, about 100 traps can be accommodated (Fig. 8.40d). A multielectrode planar trap and its electrical connections to a stable dc voltage supply are shown in Fig. 8.41a. Clearly with this proposed planar geometry quantum operations with single electrons can be naturally extended to many individual electrons, thus oﬀering an elegant solution to the fundamental requirement of scalability. The system is designed to operate at liquid He temperatures and high magnetic ﬁelds. Preliminary experiments have been conducted on ions in planar traps at room temperatures in magnetic ﬁelds around 2 T. The ultimate object is to trap single electrons in such a trap at temperatures of 100 mK or below to prolong the coherence time, and a magnetic ﬁeld of around 7 T for fast cyclotron radiation damping. The electrostatic trapping described by the ﬁeld of each planar trap is 2 potential function Φ(0, z) = i Vi [(1+Ri2 /z 2 )−1/2 −(1+Ri+1 /z 2 )−1/2 ], where Ri , for i = 1, 2, 3, are the radii of the ring electrodes. The minimum occurs at z = z0 , where ∂Φ(0, z)/∂z = 0;there it is anharmonic with the anharmonicity ∞ characterized by Δωz /ωz = )| where ci are the coeﬃcients i=1 |ci+2 /(2c 2∞ in the potential series expansion Φ(z) = n=0 cn (z − zm )n , and zm is the amplitude of the oscillation. The asymmetry of the trapping potential about z = z0 shifts and broadens the axial frequency and has to be minimized by appropriate choice of the applied voltages.

8.11 Penning Trap as Quantum Information Processor

249

Fig. 8.42. Overview of trap setup inside the 100 mK-vacuum chamber, in which the trapped electrons are driven to the ∼200th energy level of their axial motion for a trapped frequency of ∼10 MHz [535]

The loading of the multi-trap array with electrons may be accomplished using a conventional hyperbolic Penning trap to accumulate electrons from a ﬁeld emission point (F.E.P.) placed above the multi-trap substrate (Fig. 8.42). By applying an appropriate nanosecond voltage pulse on the ring of the loading trap and widening the electron magnetron radius, about 108 electrons will be ejected through the mesh of the lower end cap. It is estimated that about 10–100 of them will be captured in each trap of the array. This number has to be lowered ultimately to a single electron by a sequence of pulses. The ensemble of electrons in the trap array may be considered as an artiﬁcial molecule suitable for NMR-like quantum information processing [536]. The failure of a single trap will not require a new loading cycle, but only reduces the total number of available electrons by one, speeding up considerably the computing process. Qubit Initialization and Preparation In the quantum electron processor, qubits can be encoded in the spin system, the cyclotron motion and the axial oscillation. All three qubits have to be prepared in their quantum ground states. The ground state of the cyclotron motion is achieved by synchrotron radiation. As experimentally demonstrated by Peil and Gabrielse [45], electrons lose their cyclotron energy until they

250

8 Quantum Computing with Trapped Charged Particles

reach equilibrium with the environment. At a magnetic ﬁeld strength of a few Tesla and temperatures below 100 mK they reach the quantum ground state and remain there for virtually unlimited time. A well-deﬁned initial state |0 of the axial mode is assured by tightly coupling it to the superconducting resonator (Fig. 8.42) thereby damping the axial motion and resetting that state of the electron system to a well-deﬁned initial condition. Coupling of the axial mode to the cyclotron motion will, through the emission of synchrotron radiation, reduce both to the ground state, and remain there in a system entirely maintained (including the ﬁrststage ampliﬁer) at a temperature below 100 mK. The spin-up state of each electron can be ﬂipped by a transverse high frequency ﬁeld at the frequency diﬀerence between the cyclotron and spin resonance. This transfers the spin energy to the cyclotron motion which dissipates it to the environment as synchrotron radiation. At the cryogenic temperatures and strong magnetic ﬁeld in the trap, the thermal excitation between the electron spin states is entirely negligible, so that, after initialization, it will remain in the spin-down ground state. Single Electron Addressibility The requirement that the electrons in the array be individually and simultaneously selectable is met by introducing distinctive shifts in their characteristic frequencies using slight spatial variations in the electrostatic and magnetic trapping ﬁelds. The axial frequencies are diﬀerentiated by adjusting the distribution of voltage among the members of the array from the ultra-stable dc voltage supply. The spin qubits can be selectively addressed by varying the resonant frequency with a small magnetic ﬁeld gradient, and probing with a spectrally pure microwave pulse introduced into the cryogenic vacuum system above the multi-trap array (Fig. 8.42). The microwave ﬁeld at the site of an electron must be oﬀ-axis and have the proper polarization angle to provide the quadrupole interaction necessary for cyclotron-magnetron mode coupling [534]. Qubit Interactions The electrons can be made to interact pairwise in a controlled manner through image current coupling [512,520,531] which provides a rapid switching on and oﬀ and is compatible with straight forward scalability. An electron induces an oscillating image charge in the disc electrode of the trap as it executes its axial motion. The disc electrodes are capacitatively coupled to a switching element S (Fig. 8.43a). By coupling every disc electrode in the trap array to a corresponding array of cryogenic voltage switches beneath the substrate, forming an integral part of the structure, the image charge of every electron can be coupled to every other electron (Fig. 8.43b). When a switch between

8.11 Penning Trap as Quantum Information Processor

251

Fig. 8.43. Trap connection. (a) Two certain traps from the array, electrically connected through a cryogenic switch S; (b) switching matrix of N cryogenic switches for controlled electron-electron interaction by the image-current method [534]

two traps in the array is closed, the two axial harmonic oscillators become coupled creating entangled states of multiple qubits on demand. The spin motion of an electron can be coupled to the axial one and to the spin motion of other electrons in the array. Furthermore, tunable spin-axial motion coupling, as well as eﬀective spin–spin coupling between diﬀerent electrons can be achieved by the Coulomb interaction between electrons acting in the presence of a linear magnetic ﬁeld gradient [536]. This ﬁeld gradient is produced by making the ring electrodes of the linear multi-trap processor (Fig. 8.39) out of Ni, which however, causes a quadratic gradient, but by displacing the ring with respect to the trap axis, linear gradients will be seen by the electrons in addition to the quadratic ones. In such gradients, the electron resonance frequency suﬀers an easily detectable shifts on the order of 10 Hz depending on the spin state. Let the linear magnetic gradient with rotational symmetry with respect to the z-axis described by x y B1 = b z kˆ − ˆi − ˆj , (8.43) 2 2 with b 50 T/m, be added to the uniform trapping ﬁeld B , then the total ﬁeld acting on an electron situated a distance x0 from the center of the array is B = B z + B1 . In this ﬁeld the cyclotron frequency of an electron depends on the z-coordinate, thus: ωc (z) = Introducing

|e| (Bz + bz) . m

(8.44)

ωc2 − 2ωz2 , (8.45) where ωz = 2 |e| U0 /(md2 ) is the axial frequency, it follows that the magnetron frequency ω− and the modiﬁed cyclotron frequency ω+ deﬁned as: ω1 (z) =

ω− (z) =

ωc − ω1 , 2

ω+ (z) =

ωc + ω1 , 2

(8.46)

252

8 Quantum Computing with Trapped Charged Particles

also depend on the z-coordinate. Hence, a linear magnetic gradient introduces coupling between axial and cyclotron/magnetron motions. Moreover, the z-coordinate dependence extends to the spin part of the Hamiltonian, thus: g ge g |e| b Hs σ·B = ωc σ z − (σ x x + σ y y), (8.47) 4m 4 8m where g donates the gyromagnetic ratio and σi , i = x, y, z, are the Pauli matrices. It follows that the general Hamiltonian describing the motion of an electron in an electromagnetic ﬁeld, that is H=

(p − eA)2 ge + eV + σ · B, 2m 4m

(8.48)

where A, the vector potential of the total magnetic ﬁeld B, can be rewritten for the present case of a trapped electron as follows: g H −ωm0 a†m am +ωc0a†c ac +ωz a†z az + ωs0 σz + ωz ε(az +a†z )σz , (8.49) 2 4 in which

2 (py + ipx ) , mω1 1 mω1 2 (py − ipx ) , am = [(x − x0 ) + iy] − 2 2 mω1 mωz 1 |e| b z+i pz , ε = Δz, Δz = /(2mωz ), (8.50) az = 2 2mωz mωz 1 ac = 2

mω1 [(x − x0 ) − iy] + 2

showing explicitly the coupling between the axial and spin degrees of freedom. Here ac and am are the cyclotron and magnetron ladder operators satisfying the usual commutation relation, Δz is the amplitude of the electron axial motion in the ground state, ωc0 = |e| Bc /m, ωm0 = ωz2 /(2ωc0 ), ωs0 = gωc0 /2, and the parameter g |e| bΔz/(2m) is roughly the spin frequency variation due to the magnetic gradient when the axial motion is in the ground state. It must be assumed that this variation is enough to individually address the spin qubits via microwave radiation [536]. If the linear magnetic gradient is applied to a linear array of N Penning traps each containing a single electron, the Hamiltonian of the system becomes: H=

N (pi − eAi )2 i=1

2m

+ eVi +

N e2 ge σi · B i + , 4m 4πε0 |r i − rj | i>j

(8.51)

where the subscript i refers to the ith electron in the array. Following the same notation as previously introduced for a single electron, the Hamiltonian (1.73) can be separated into a part that involves only the magnetic interactions

8.11 Penning Trap as Quantum Information Processor

253

from the part (HC ) involving only the Coulomb interaction. If the oscillation amplitude of any two electrons in the array is much smaller than their average separation ri,j = |xi.0 − xj,0 | , the Hamiltonian HC can be rewritten as: HC −

e2 e2 2 (Δxi − Δxj ) − 2 3 (zi − zj ) , 4πε0 ri,j 8πε0 ri,j

(8.52)

where Δxi = xi − xi,0 . The ﬁrst term in (8.52) gives the displacement of the equilibrium position of the electron along the x−axis, which can be eﬀectively removed by redeﬁning the centers of the two traps. However, the second term contains a term proportional to zi zj representing the Coulomb coupling between the axial motions of the i−th and j−th electrons. If the following unitary transformation [537] H = eS He−S ,

S=

N g ε (a†z,i − az,i )σz,i , 4 i=1

(8.53)

is applied to the global Hamiltonian, terms of the form ∼ σz,i σzj appear, representing an eﬀective coupling between the spin motions of diﬀerent electrons. The spin part of the global Hamiltonian considered here has the form: Hs

N i=1

2

ωs0,i σz,i +

where ωs0,i

geBz 2m

N i>j

.

b2 x2i,0 1+ 8Bz2

2

πJi,j ωs0,i σz,i σz,j ,

(8.54)

b2 . ωz4 d3i,j

(8.55)

/ ,

Ji,j ∝

The Hamiltonian Hs is analogous to the nuclear spin Hamiltonian of molecules in NMR, with the nuclear spins of a molecule replaced by electron spins [538]. The coupling constants Ji,j can be modiﬁed by changing the gradient strength b and the axial frequency ωz , decreasing very rapidly with the electron distance di,j [538, 539]. Qubit Read-out At the end of a computation, the ﬁnal states of the individual qubits in the array can be determined by tuning the axial frequency of each electron to that of a superconducting resonance circuit of high quality factor forming the input of a cryogenic low noise preampliﬁer. This detects the image current induced in the central ring of each trap and its output is connected to a cryogenic switching matrix (Fig. 8.43). A Fourier transform analysis of the induced current [540] shows a peak at the electron axial frequency. The spin and cyclotron states are detected simultaneously and independently using the continuous Stern–Gerlach eﬀect that enables the detection of

254

8 Quantum Computing with Trapped Charged Particles

the spin/cyclotron state of each electron using only the frequency of its axial motion [37,108]. The axial motion is driven from a constant frequency source and the phase of the response is measured, giving a frequency resolution of less than 1 in 108 . The excitation to higher vibrational levels is detected by quantum jumps in the signals induced in the (segmented) ring electrode of the trap that is part of the (cyclotron) axial detection circuit. 8.11.4 Expected Performance Universal Quantum Processor Universal quantum processing using electrons in a multi-trap array such as we have been considering can be achieved using techniques similar to those developed in the ﬁeld of NMR. Unlike NMR, however, the information carriers in the present system are far more isolated from the environment, the number of qubits is neither limited by molecular size nor frequency range. and operates at the quantum level. To build up such a universal processor, the axial, cyclotron, and spin degrees of freedom have to be driven to the lowest quantum state and the information has to be appropriately encoded in all of them to realize a three qubit system. Also universal quantum gates have to be implemented by coupling diﬀerent motional states, creating superpositions of spin and vibrational states, and interconnecting diﬀerent electron traps in the array. Simple quantum operations can be accomplished with a Toﬀoli gate [530], however, Deutsch and Shor algorithms for at least ten electrons have to be realized, and error correction techniques [312, 313] have to be implemented to evaluate the fault tolerance of the device. Finally, the results of a computation must be read out solely through the axial degree of freedom. Fidelity and Clock Speed The achievable ﬁdelity of the universal gate operations and clock speed are the primary criteria for judging the performance of any experimental prototype quantum computer. Gate ﬁdelity is adversely aﬀected by the imperfections in the control ﬁelds by which the system is manipulated (coherent errors) as well as by coupling to uncontrollable degrees of freedom (incoherent errors). The errors aﬀecting single electron quantum gates are produced mainly by the electric potential anharmonicity in the trap, the magnetic ﬁeld inhomogeneity, and by ﬂuctuations in the phase and amplitude of the applied electromagnetic ﬁelds. The ﬁdelity of two-electron quantum gates is aﬀected through capacitative coupling to the thermal noise arising from the image charge coupling circuits. The gate ﬁdelity of operations on many electrons depends on interparticle distance, as well as all signiﬁcant parameters in the qubit detection process; moreover it is aﬀected by the residual interaction with neighboring nonresonant electrons. The eﬀect of all these errors on the ﬁdelity of the gate operations may be evaluated by applying appropriate perturbation analysis to

8.12 Future Developments

255

realistic models or by using numerical simulation. For a given ﬁdelity, there is a maximum speed of gate operations (clock speed) which has to be established by optimizing the values of the working parameters. A high clock speed (several megahertz) can be obtained by optimizing the central electrode radius of each trap in the array between 0.1 and 1 mm, and shifting the trapping potential minimum along the z-axis [534]. Decoherence Eﬀects A quantum computer based on electrons in a Penning multitrap array is expected to have weak decoherence compared with Paul traps, because the static electric and magnetic ﬁelds can have extremely low noise; for example, they may be stabilized to the point that for the magnetic ﬁeld ΔB/B ≤ 10−9 /h and the electric ﬁeld noise is less than 1 nV. However, there can still be small residual inhomogeneities on the order of μT mm−2 in the ﬁeld of the superconducting magnet, as well as the electric patch ﬁelds due to irregularities in the electrode surfaces, and other types of noise on the electrodes causing electric ﬁeld ﬂuctuations. Error Correction Codes A processor’s performance can be substantially improved by employing various error correction codes, requiring gate operations to be applied in a basis of highly entangled states of the whole quantum system. The possibility of using three degrees of freedom per electron and high dimensional bases for axial and cyclotron motion makes it easier to implement the encoding and decoding operations with fault tolerant extensions. The general fault tolerant codes are very diﬃcult to implement, since they require coherent manipulation of a large number of qubits. That is why as a ﬁrst approximation, it would be suﬃcient to estimate the threshold probability of error per quantum gate, below which the codes achieve fault tolerance. In spite of the fact that the model of a quantum computer, as we have described it, meets most of the DiVincenzo criteria, it can serve only as proof-of-principle of the application of trapped charged particles to large scale quantum computers.

8.12 Future Developments Theoretically, there seems no fundamental limitation toward quantum computation using trapped charged particles. The long coherence times achieved in ionic two-level systems provide a robust quantum memory. Moreover, the near-unity state detection and the availability of a universal set of gate operations have already allowed for small-scale quantum computation. In fact ion traps have passed all the DiVicenco criteria. Techniques to build large-scale ion trap quantum computers are presently being developed and ﬁrst steps

256

8 Quantum Computing with Trapped Charged Particles

have been successfully demonstrated. “Ion chips” are likely to represent the basic building blocks for scaling up present devices. To overcome the scaling problem in ion traps the ions are shuttled through multizone traps. In such a system, quantum gates are performed in entangling zones of the larger trap structure, and the ions are shuttled to other zones for further operations as necessary. Still, as the number of atoms grows and the dimensions of the traps shrink, preserving coherence may become exceedingly diﬃcult. Another approach to scaling up atomic quantum computation systems may be to use photon-mediated entanglement. Moehring et al. [541] have theoretically investigated this approach. Ions in diﬀerent trapping zones can be entangled without the need for cooling to the motional ground state or even localization within the Lamb–Dicke regime. Experiments have been performed which show coupling of qubits to photons at the single-photon level [542,543]. Error correction codes to correct for phase and spin ﬂip errors, and fault-tolerant operations with multiple qubits are important issues in future quantum computation [544]. It has already been realized in simple cases [545]. First algorithms have been implemented on an ion trap quantum computer showing the gain in eﬃciency with a quantum algorithm as proposed in the Deutsch–Jozsa algorithm [546]. More will be required to demonstrate the capability of quantum computation: the implementation of some simple quantum algorithms with at least seven qubits, such as prime factoring of 15, the Grover algorithm, as well as error correction codes to assure fault tolerance of the device. Other potential areas of development include multilevel quantum logic that takes advantage of multilevel axial and cyclotron degrees of freedom to enhance the memory space (more qubits per particle) and reduce computational time; the development of neural networks [547] consisting of chains of trapped particles with long range interaction, since they are able to store information distributed over the entire system (acting as basins of attraction in thermodynamics) and thus have associative memory. Such neural networks are remarkably robust in the sense that their performance is preserved even when any part of them is destroyed. In addition to developing speciﬁc technologies for quantum computing with trapped particles, it will be also necessary to realize interfaces between diﬀerent computing technologies complementing each other. Some of them are more suitable for quantum memories, some for quantum processing, and others for quantum communication. The hybrid technologies and interfaces between them are the long-term goals for the future.

References

1. F. Major, V.N. Gheorghe, G. Werth, Charged Particle Traps, Physics and Techniques of Charged Particle Field Conﬁnement (Springer, Berlin, 2005) 2. W. Paul, Rev. Mod. Phys. 62, 531 (1990) 3. H. Dehmelt, Adv. At. Mol. Phys. 3, 53 (1967) 4. J. Meixner, F.W. Sch¨ aﬀke, Mathieusche Funktionen und Sph¨ aroidfunktionen (Springer, Heidelberg, 1954) 5. N.W. McLachlan, Theory and Application of Mathieu Functions (Oxford University Press, Oxford, 1947) 6. R.F. Wuerker, H. Shelton, R.V. Langmuir, J. Appl. Phys. 30, 342 (1959) 7. X.Z. Chu et al., Int. J. Mass Spectrom. Ion Processes 173, 107 (1998) 8. R. Iﬄ¨ ander, G. Werth, Metrologia 13, 167 (1977) 9. F.G. Major, H.G. Dehmelt, Phys. Rev. 170, 91 (1968) 10. H. Schaaf, U. Schmeling, G. Werth, Appl. Phys. 25, 249 (1981) 11. L.S. Cutler et al., Appl. Phys. B 36, 137 (1985) 12. G. Kotowski, Z. Angew. Math. Mech. 230, 213 (1943) 13. Y. Wang, J. Franzen, K.P. Wanczek, Int. J. Mass Spectrom. Ion Processes 124, 125 (1993) 14. R. Alheit et al., Int. J. Mass Spectrom. 154, 155 (1996) 15. G. Tommaseo et al., Eur. Phys. J. D 28, 29 (2004) 16. L.S. Brown, G. Gabrielse, Phys. Rev. A25, 2423 (1982) 17. M. Kretschmar, Z. Naturf. 45a, 965 (1990) 18. L.S. Brown, G. Gabrielse, Rev. Mod. Phys. 58, 233 (1986) 19. G. Bollen et al., J. Appl. Phys. 68, 4355 (1990) 20. C. Gerz, D. Wilsdorf, G. Werth, Nucl. Instr. Meth. B 47, 453 (1990) 21. R.S. Van Dyck, Jr. et al., Phys. Rev. A 40, 6308 (1989) 22. P. Paasche et al., Eur. Phys. J. D 22, 183 (2003) 23. G. Savard et al., Phys. Lett. A 158, 247 (1991) 24. Ch. Lichtenberg et al., Eur. Phys. J. D 2, 29 (1998) 25. X.-P. Huang et al., Phys. Rev. Lett. 78, 875 (1997) 26. E.M. Hollmann, F. Anderegg, C.F. Driscoll, Physics of Plasmas 7, 2776 (2000) 27. G. Bollen et al., Nucl. Instrum. Meth. A 368, 675 (1996) 28. H.A. Schuessler, O. Chun-sing, Nucl. Instr. Meth. 186, 219 (1981) 29. J. Coutandin, G. Werth, Appl. Phys. B 29, 89 (1982) 30. F. Herfurth et al., Nucl. Instrum. Meth. A 469, 254 (2001)

258 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72.

References H. Raimbault-Hartmann et al., Nucl. Instr. Meth. B 126, 378 (1997) F. Vedel, M. Vedel, Phys. Rev. A 41, 2348 (1990) K. Blaum, Phys. Rep. 425, 1 (2006) H.G. Dehmelt, Bull. Am. Phys. Soc. 7, 470 (1962) H.G. Dehmelt, F.L. Walls, Phys. Rev. Lett. 21, 127 (1968) D.A. Church, H.G. Dehmelt, J. Appl. Phys. 40, 342 (1969) G. Werth, H. H¨ aﬀner, W. Quint, Adv. At., Mol. and Optical Phys. 48, 191 (2002) D.J. Wineland, H.G. Dehmelt, J. Appl. Phys. 46, 919 (1975) W. Neuhauser et al., Phys. Rev. Lett. 41, 233 (1978); Appl. Phys. 17, 123 (1978) W. Neuhauser et al., Phys. Rev. A 22, 1137 (1980) F. Arbes et al., Z. Phys. D 25, 295 (1993) M. Block et al., J. Phys. B 33, L375 (2000) J. Bergquist, W. Itane, D.J. Wineland, Phys. Rev. A 36, 428 (1987) F. Schmidt-Kaler et al., J. Mod. Opt. 47, 1573 (2000) S. Peil, G. Gabrielse, Phys. Rev. Lett. 83, 1287 (1999) D. Lunney, J.M. Pearson, C. Thibault, Rev. Mod. Phys. 75, 1021 (2003) F.W. Aston, Nature 105, 617 (1920) E. Arnold, Mass Spectra and Isotopes (Edward Arnold, London, 1933) A.J. Dempster, Proc. Am. Phil. Soc. 75, 755 (1925) J.H.E. Mattauch, Phys. Rev. 50, 617 (1936) A.O. Nier, T.R. Roberts, Phys. Rev. 81, 507 (1951) C. Thibault et al., Phys. Rev. C 12, 644 (1975) W. Paul, H. Steinwedel, Naturforsch. A 8a, 448 (1953) W. Paul, M. Raether, Z. Phys. 140, 262 (1955) E. Fischer, Z. Phys. 156, 1 (1959) G. Graeﬀ et al., Phys. Rev. Lett. 21, 340 (1968) F.M. Penning, Physica 3, 873 (1936) J.R. Pierce, Theory and Design of Electron Beams, 2nd edn. (van Norstrand, Princeton, NJ, 1954) R.E. March, J.F.J. Todd, Quadrupole Ion Trap Mass Spectrometry, 2nd edn. (Wiley, New York, 2005) M. Dole et al., J. Chem. Phys. 49, 2240 (1968) M. Yamashita, J.B. Fenn, J. Phys. Chem. 88, 4671 (1984) K. Tanaka et al., Rapid Comm. Mass Spectrom. 2, 151 (1988) M. Karas, F. Hillenkamp Anal. Chem. 60, 2299 (1988) K.B. Jeﬀerts Phys. Rev. Lett. 20, 39 (1968) G.C. Staﬀord et al., Int. J. Mass Spectrom. Ion Processes 60, 85 (1984); See also US Patent No. 4540884 (1985) O.J. Orient, A. Chutjian, Rev. Sc. Instr. 73, 2157 (2002) O. Kornienko et al., Rev. Sc. Instr. 70, 3907 (1999) J. Franzen et al., in Practical Aspects of Ion Trap Mass Spectrometry, vol 1, R.E. March, J.F.J. Todd (eds) CRC, p. 49 (1995) W.R. Plass et al., J. Phys. Chem. 104, 5059 (2000) O. Bortolini, P. Traldi, in Practical Aspects of Ion Trap Mass Spectrometry, vol. 2, R. E. March, J.F.J. Todd. (eds) CRC, p. 145 (1995) C.D. Cleven et al., J. Phys. Chem. 100, 40 (1996) R. Alheit et al., Rapid Commun. Mass Spectrom. 10, 583 (1996)

References

259

73. M. Sudakov, D.J. Douglas, Rapid Commun. Mass Spectrometry 17, 2290 (2003) 74. D.J. Douglas, N.V. Konenko, Rapid Commun. Mass Spectrom. 16, 1425 (2002) 75. D.J. Douglas, A.J. Frank, D. Mao, Mass Spectrom. Rev. 24, 1 (2005) 76. E. Kawato, US Patent No. 6380666 (2002) 77. T. Baba, Y. Hashimoto, US Patent No 7166835 (2007) 78. Z. Du et al., J. Anal. Atm. Spectrom. 14, 1111 (1999) 79. J. Franzen, U.S. Patent No. 6559444 (2003) 80. M.B. Comisarov, Adv. Mass Spectrom. 8, 1698 (1980) 81. M.B. Comisarov, Int. J. Mass Spectrom. Ion Processes 37, 251 (1981) 82. V.H. Vartanian, F. Hadjarab, D.A. Laude, Int. J. Mass Spectrom. Ion Processes 151, 157 (1995) 83. D.F. Hunt et al., Proc. Nat. Acad. Sc. USA 84, 620 (1987) 84. M.A. Park, US Patent No.6784421 (2004) 85. J.D. Prestage, NASA Tech. Brief 23(5), 168 (1999) 86. C.B. Freidhoﬀ, J. Vac. Sci. Technol. A 17, 2300 (1999) 87. S. Taylor, R.F. Tyndall, R.R.A. Syms, J. Vac. Sci. Technol. B 19, 557 (2001) 88. L. Brown, G. Gabrielse, Rev. Mod. Phys. 58, 233 (1986) 89. Ch. Gerz, D. Wilsdorf, G. Werth, Nucl. Instr. Meth. 47, 453 (1990) 90. R.S.V. Dyck et al., Phys. Rev. A 40, 6308 (1989) 91. P.W. Andersen, Y.B. Kim, Rev. Mod. Phys. 36, 39 (1964) 92. R.S. van Dyck et al., Rev. Sci. Instr. 70, 1665 (1999) 93. M. Marie-Jeanne et al., Nucl. Instr. Meth. 587, 464 (2008) 94. K. Blaum et al., J.Phys. G, Nucl. Part. Phys. 31, 1775 (2005) 95. G. Gabrielse, J. Tan, J. Appl. Phys. 63, 5143 (1988) 96. K. Hermanspahn, “Das magnetische Moment des gebundenen Kohlenstoﬀ C5+ ”, Ph.D. thesis, Johannes Gutenberg University, Mainz, 1999 97. E.A. Cornell et al., Phys. Rev. A 45, 3049 (1992) 98. S. Rainville et al., Hyperﬁne Interactions 132, 177 (2001) 99. M.B. Comisarow, A.G. Marshall, Chem. Phys. Lett. 25, 282 (1974) 100. A.G. Marshall, Int. J. Mass. Spectrom. 200, 331 (2000) 101. C.L. Nilsson et al., “Structural characterization of the brain gangliosides GM1 and GD1 by high Resolution FT-ICR tandem mass spectrometry”, 54th ASMS Conference on Mass Spectrometry, Seattle, Washington, DC, USA, May 28– June 1 (2006) 102. G. Gr¨ aﬀ, H. Kalinowski, J. Traut, Z. Phys. A 297, 35 (1980) 103. V. Kolhinen et al., Nucl. Instr. Meth. in Phys. Res. A 528, 776 (2004) 104. M. Block et al., Eur. Phys. J. A 25, 49 (2005) 105. G. Bollen et al., Nucl. Instr. Meth. A 368, 675 (1996) 106. I. Bergstr¨ om et al., Nucl. Instr. Meth. Phys. Res. A 487, 618 (2002) 107. L.D. Landau, E.M. Lifshitz, Mechanics, Ch. 29 (3rd edn., Pergamon, Oxford, 1976) 108. H. H¨ aﬀner et al., Eur. Phys. J D 22, 163 (2003) 109. M. Koenig et al., Int. J. Mass Spectrom. Ion. Process 142, 95 (1995) 110. M. Kretschmar, Int. J. Mass Spectrom. 264, 122 (2007) 111. N.F. Ramsey, Phys. Rev. 76, 996 (1949) 112. N.F. Ramsey, Rev. Mod. Phys. 62, 541 (1990) 113. G. Bollen et al., Nucl. Instrum. Methods B 70, 490 (1992) 114. S. George et al., Int. J. Mass Spectrom. 264, 110 (2007)

260 115. 116. 117. 118. 119. 120. 121. 122. 123. 124. 125. 126. 127. 128. 129. 130. 131. 132. 133. 134. 135. 136. 137. 138. 139. 140. 141. 142. 143. 144. 145. 146. 147. 148. 149. 150. 151. 152. 153. 154. 155. 156. 157. 158. 159. 160. 161.

References K. Blaum et al., EPJ A 15, 245 (2002) A. Kellerbauer et al., EPJ D 22, 53 (2003) M.P. Bradley et al., PRL 83, 4510 (1999) S. Rainville, J.K. Thompson, D.E. Pritchard, Science 16, 334 (2004) Ch. Kraus et al., Eur. Phys. J. C40, 447 (2005) J. Angrik et al., FZKA Scientiﬁc Report 7090 (2005) Sz. Nagy et al., Europhys. Lett., 74, 404 (2006) P. Mohr, B. Taylor, Rev. Mod. Phys. 77, 1 (2005) P.B. Schwinberg, Phys. Rev. Lett. 75, 3598 (1995) H. H¨ aﬀner et al., Phys. Rev. Lett. 85, 5308 (2000) J. Verdu et al., Phys. Rev. Lett. 92, 093002 (2004) G. Gabrielse et al., Phys. Rev. Lett. 82, 3198 (1990) G. Gabrielse et al., Phys. Rev. Lett. 97, 030802 (2006) G. Gabrielse et al., Phys. Rev. Lett. 99, 039902 (2007) B. Odom et al., Phys. Rev. Lett. 97, 030801 (2006) T. Kinoshita, M. Nio, Phys. Rev. D 73, 013003 (2006) T. Aoyama et al., Phys. Rev. D 77, 053012 (2008) A. Wicht et al., Phys. Scr. T102, 82 (2002) C. Carlberg, T. Fritioﬀ, I. Bergstr¨ om, Phys. Rev. Lett. 83, 4506 (1999) A. Paul et al., Hyperﬁne Interactions 132, 189 (2001) F. DiFilippo et al., Phys. Rev. Lett. 73, 1481 (1993) I. Bergst¨ om et al., Phys. Scripta 66, 201 (2002) C. Fr¨ ohlich et al., Phys. Rev. Lett. 96, 142502 (2006) L. Ryjkov et al., Phys. Rev. Lett. 96, 033002 (2006) G. Savard et al., Hyperﬁne Interactions 132, 221 (2001) I. Bergstr¨ om et al., Nucl. Instr. Meth. A 532, 203 (2004) V.S. Kolhinen et al., Nucl. Instr. Meth. A 487, 618 (2002) G. Bollen et al., Nucl. Instr. Meth. A 532, 203 (2004) G. Marx et al., Hyperﬁne Interact. 132, 459 (2001) J. Dilling et al., Int. J. Mass Spectrom. 251, 198 (2006) J. Szerypo et al., Nucl. Instr. Meth. B 204, 512 (2003) M. Wada et al., Nucl. Instr. Meth. B 204, 570 (2003) G.E. Uhlenbeck, S. Goudsmit, Naturwissenschaften 47, 953 (1925) P.A.M. Dirac, Proc. R. Soc. A 117, 610 (1928) N. Bohr, Collected Works of N. Bohr, (J. Kalkar ed.), vol.6, p.333, NorthHolland, Amsterdam, 1996) W. Pauli, Handbuch der Physik (S. Flgge ed.), vol.5, p. 167, Springer, Heidelberg, 1933 G. Breit, Phys. Rev. 72, 984 (1947) J.E. Nafe et al., Phys. Rev. 71, 914 (1947) P. Kusch, H.M. Foley, Phys. Rev. 74, 250 (1947) J. Schwinger, Phys. Rev. 73, 416 (1948) C.M. Sommerﬁeld, Phys. Rev. 107, 328 (1957); Ann. Phys. (N.Y.) 5, 26 (1958) T. Kinoshita, Rep. Progr. Phys. 591, 459 (1996) S. Laporta, E. Remiddi, Phys. Lett. B 379, 283 (1996) A. Czarnecki et al., Phys. Rev. Lett. 76, 3267 (1996) A.A. Schupp, R.W. Pidd, H.R. Crane, Phys. Rev. 121, 1 (1961) J.C. Wesley, A. Rich, Phys. Rev. A 4, 1341 (1971) J. Gilleland, A. Rich, Phys. Rev. Lett. 23, 1130 (1069)

References 162. 163. 164. 165. 166. 167. 168. 169. 170. 171. 172. 173. 174. 175. 176. 177. 178. 179. 180. 181. 182. 183. 184. 185. 186. 187. 188. 189. 190. 191. 192. 193. 194. 195. 196. 197. 198. 199. 200. 201. 202. 203. 204. 205. 206. 207.

261

A. Rich, J.C. Wesley, Rev. Mod. Phys. 44, 250 (1972) P. Farago et al., Proc. Phys. Soc (London) 3, 82 (1963) G. Graeﬀ, E. Klempt, G. Werth, Z. Phys. 222, 201 (1969) G. Gabrielse, Phys. Rev. A 29, 462 (1984) A.A. Sokolov, Y.G. Pavlenko, Opt. Spectrom. 22, 1 (1965) G. Graeﬀ, E. Klempt, Z. Naturf. 229, 1960 (1967) R.S. van Dyck, P.B. Schwinberg, H. G. Dehmelt, Phys. Rev. D 34, 722 (1986) Van Dyck, R.S., Jr., P.B. Schwinberg, H.G. Dehmelt, Phys. Rev. Lett. 59, 26 (1987) E. Krueger, W. Nistler, W. Weirauch, Metrologia 32, 117 (1995) L.S. Brown et al., Phys. Rev. A 32, 3204 (1985) L.S. Brown, L.S. et al., Phys. Rev. Lett. 55, 44 (1985) G. Gabrielse, J. Tan, Adv. At. Mol., Opt. Phys. Suppl. 2, 267 (1994) D. Hanneke, S. Fogwell, G. Gabrielse, Phys. Rev. Lett., 100, 120801 (2008) G. Breit, Nature 122, 649 (1928) K. Pachucki et al., Phys. Rev. A 72, 022108 (2005) S. Stahl et al., J. Phys. B 38, 297 (2005) V. Flambaum et al., Zh. Eksp.Teor. Fiz. (Sov. Phys. JETP) 75, 75 (1978); Phys. Lett. A 67, 177 (1978) R.H. Dicke, Phys. Rev. 89, 472 (1953) G. Marx et al., Eur. Phys. J. D 4, 1279 (1998) P. Raghavan, Atomic Data and Nuclear Data Tables 42, 189 (1989) M. Gustavsson, A.-M. Martensson-Pendrill, Phys. Rev. A 58, 3611 (1998) E. Arimondo, M. Inguscio, P. Violino, Rev. Mod. Phys. 49, 31 (1977) H.H. Stroke, H.T. Duong, J. Pinard, Hyperﬁne Interactions 129, 319 (2000) S. Trapp et al., Hyperﬁne Interactions 127, 57 (2000) E.N. Fortson, F.G. Major, H.G. Dehmelt, Phys. Rev. Lett. 16, 221 (1966) F.G. Major, G. Werth, Phys. Rev. Lett. 30, 1155 (1973) E.C. Beatty, P.L. Bender, A.R. Chi, Phys. Rev. 112, 450 (1958) S. Mrozowski, Phys. Rev. 57, 207 (1940) F.G. Major, The Quantum Beat (Springer, New York, 1998) p. 298 R. Blatt, G. Werth, Phys. Rev. 25, 1476 (1982) M. Arditi, T.R. Carver, Phys. Rev. 112, 449 (1958) W.M. Itano, D.J. Wineland, Phys. Rev. A 24, 1364 (1981) J.J. Bollinger et al., Laser Spectroscopy VI, ed. by H.P. Weber and W. Luthy (Springer, Berlin, 1983) A. M¨ unch et al., Phys. Rev. A 35, 4147 (1987) O. Becker et al., Phys. Rev. A 48, 3546 (1993) F. Arbes et al., Z. Phys. D 31, 27 (1994) F. Kurth et al., Z. Phys. D 34, 227 (1995) H. Sunaoshi et al., Hyperﬁne Interactions 78, 241 (1993) G.P. Barwood et al., Phys. Rev. A 67, 013402 (2003) U. Tanaka et al., Phys. Rev. A 53, 3982 (1996) H. Knab et al., Europhys. Lett. 4, 1361 (1987) W. Becker, G. Werth, Z. Phys. A 311, 41 (1983) K. Enders et al., Phys. Rev. A 56, 265 (1997) K. Enders et al., Z. Phys. D 42, 171 (1997) R. Blatt et al., Z. Phys. A 312, 143 (1983) M.D. McGuire et al., Phys. Rev. A 17, 1999 (1978)

262 208. 209. 210. 211. 212. 213. 214. 215.

216.

217. 218.

219. 220.

221.

222. 223. 224. 225. 226. 227. 228. 229. 230. 231.

232. 233. 234. 235. 236. 237. 238. 239. 240.

References J.D. Prestage et al., J. Mod. Opt. 39, 221 (1992) X. Feng et al., Phys. Rev. A 46, 2958 (1992) X. Feng et al., Phys. Rev. A 46, 327 (1992) P. Galison, Einstein’s Clocks, Poincare’s Maps (Scepter, London, 2004) S.R. Jeﬀerts, T.P. Heavner, E.A. Donley, Jpn. J. Appl. Phys. 43, 2803 (2004) P. Lesage, T. Ayi, IEEE Trans. Instrum. Meas. 20, 105 (1971) G. Santarelli et al., IEEE Trans UFFC 45, 887 (1998) P. Lemonde et al., The Sensitivity Function, a New Tool for the Evaluation of Frequency Shifts in Atomic Spectroscopy, in Proceedings of the IEEE International Frequency Control Symposium, Pasadena, CA, 1998, p. 110 J.D. Prestage et al., Progress Report on Improved Linear Ion Trap Physics Package, in Proceedings of the IEEE International Frequency Control Symposium, San Francisco, CA, 1995, p. 82 R. Besson, A New “Electrodeless” Resonator Design, in Proceedings of Annual Frequency Control Symposium, Fort Monmouth, NJ, 1977, p. 147 R.L. Tjoelker et al., A Mercury Ion Frequency Standard Engineering Prototype for NASA Deep Space Network, in Proceedings of IEEE International Frequency Control Symposium, Honolulu, HI, 1996, p. 1073 D.J. Berkeland et al., Phys. Rev. Lett. 80, 2089 (1998) C.W. Nelson et al., Progress on Prototype Synthesizer Electronics for 199 Hg + at 40.5 GHz, in Proceedings of IEEE International Frequency Control Symposium, Hershey, PA, 1992, p. 64 R.B. Warrington et al., Development of a 171 Yb+ Microwave Frequency Standard at the National Measurement Institute Australia, in Proceedings of IEEE International Frequency Control Symposium, Vancouver, B.C., p. 611, 2005; D. Schnier et al., ESA Freq. Time Forum, 1992, p. 415 A.L. Schawlow, C.H. Townes, Phys. Rev. 112, 1940 (1958) P.L. Lee, M.L. Skolnick, Appl. Phys Lett. 10, 303 (1967) J.N. Eckstein, A.I. Ferguson, T.W. H¨ ansch, Phys. Rev. Lett 40, 847 (1978) W.E. Lamb, Jr., Phys. Rev. A 134, 1429 (1964) R.L. Barger, J.L. Hall, Phys. Rev. Lett. 22, 4 (1969) A. Brillet, P. Cerez, H. Clergeot, IEEE J. Quant. Elec. 10, 526 (1974) C.O. Weiss et al., IEEE J. Quant. Elec. 24, 1970 (1988) A.A. Madej et al., Metrologia 41, 152 (2004) C.A. Greenhall, G.J, Dick, IEEE Trans. Ultr. Fer. Fr. Contrl. 471593 (2000) G. Santarelli et al., Theoretical description and experimental evaluation of the eﬀect of interrogation oscillator frequency noise on the stability of pulsed atomic frequency standards, in 10th European Frequency and Time Forum, Brighton, 5–7 March 1996, p. 66 L. Hollberg et al., IEEE J. Quant. Elec. 37, 1502 (2001) R.J. Rafac et al., Phys. Rev. Lett. 85, 2462 (2000) J.C. Berquist, W.M. Itano, D.J. Wineland, Phys. Rev. 36, 428 (1987) U. Tanaka et al., J. Phys. B At. Mol. Opt. Phys. 36, 545 (2003) Th. Sauter et al., Phys. Rev. Lett. 57, 1696 (1986) W. Nagourney, J. Sandberg, H. Dehmelt, Phys. Rev. Lett. 56, 2797 (1986) B.C. Young et al., Phys. Rev. Lett. 82, 3799 (1999) S. Bize et al., Phys. Rev. Lett. 90, 150802 (2003) S.A. Diddams et al., Science 293, 825 (2001)

References

263

241. K. Matsubara et al., Single Ca+ ion trapping toward precise frequency measurement of 42 S1 /2 − 32 D5 /2 transition, in Joint 2005 IEEE International Frequency Control Symposium and Precise Time and Time Interval, Vancouver, 29–31 August, p. 616 242. C. Champenois et al., Phys. Lett. A 331/5, 298 (2004) 243. C.F. Roos et al., Nature 443, 316 (2006) 244. N. Yu, W. Nagourney, H.G. Dhmelt, Phys. Rev. Lett. 78, 4898 (1997) 245. J.A. Sherman et al., Phys. Rev. Lett. 94, 243001 (2005) 246. P. Dube et al., 88 Sr+ single ion optical frequency standard, in Proceedings of the IEEE International Frequency Control Symposium, Miami, FL, 5–7 June 2006, p. 409 247. G. Werth, Metrologia 22, 190 (1986) 248. Chr. Tamm, D. Engelke, V. B¨ uhner, Phys. Rev. A 61, 053405 (2000) 249. M. Roberts et al., Phys. Rev. A 62, 020501 (R (2000) 250. P.J. Blythe et al., Phys. Rev. A 67, 020501)(R (2003) 251. C.A. Schrama et al., Opt. Commun. 101, 32 (1993) 252. Y.H. Wang et al., Opt. Commun. 273, 526 (2007) 253. N. Yu, H.G.Dehmelt, W. Nagourney, Proc. Natl. Acad. Sci. USA 89, 7289 (1992) 254. T. Rosenband et al., Phys. Rev. Lett. 98, 220801 (2007) 255. P.O. Schmidt et al., Science 309, 749 (2005) 256. J.E. Bernard, L. Marmet, A.A. Madej, Opt. Commun. 150, 170 (1998) 257. B.E. King et al., Phys. Rev. Lett. 81, 1525 (1998) 258. W.H. Oskay, Phys. Rev. Lett. 97, 020801 (2006) 259. N.F. Ramsey, Molecular Beams (University Press London, Oxford, 1956) 260. W.M. Itano, J. Res. NIST 105, 829 (2000) 261. P. Dube et al., Phys. Rev. Lett. 95, 033001 (2005) 262. I.I. Sobel’man et al., Excitation of Atoms and Broadening of Spectral Lines, second ed. (Heidelberg, Springer, 1995) 263. R.H. Dicke, Phys. Rev. 89, 472 (1953) 264. S. Urabe et al., Appl. Phys B, Lasers Opt. 67, 223 (1998) 265. D.J. Wineland, W.M. Itano, Phys. Rev. A 20, 1521 (1979) 266. A.A. Madej et al., Phys. Rev. A 70, 012507 (2004) 267. N.C. Wong, D. Lee, Optical parametric division,in Proceedings of the IEEE Frequency Control Symposium, Hershey, PA 27–29 May 1992, p. 32 268. D.E. Spence et al., Opt. Lett. 16, 42 (1991) 269. S.A. Diddams et al., Phys. Rev. Lett. 84, 5102 (2000) 270. R. Holtzwarth et al., Phys. Rev. Lett. 85, 2264 (2000) 271. J.E. Stalnaker et al., Absolute optical frequency measurement with fractional frequency uncertainty at 10−15 , in Proceedings of the IEEE Frequency Control Symposium, Miami, FL 5–7 June 2006, p. 462 272. E. Tr¨ abert, Phys. Scr 61, 257 (2000) 273. D. L. Moehring et al., Phys Rev A 73, 023413 (2006) 274. F. Arbes et al., Z. Phys D 29, 159 (1994) 275. T. Gudjons et al., Europhys. Lett. 33, (8) 595 (1996) 276. A.A. Madej, J.D. Sankey, Phys. Rev. A 41, 2621 (1990) 277. A. Barton et al., Phys. Rev. A 62, 032503 (2000) 278. M. Block et al., Europhys. J. D 7, 461 (1999) 279. A. Kreuter et al., Phys. Rev. A 71, 032504 (2005)

264 280. 281. 282. 283. 284. 285. 286. 287. 288. 289. 290. 291. 292. 293. 294. 295. 296. 297. 298. 299. 300. 301. 302. 303. 304. 305. 306. 307. 308. 309. 310. 311. 312. 313. 314. 315. 316. 317. 318. 319. 320. 321. 322. 323. 324.

References D.P. Moehs, D.A. Church, R.A. Phaneuf, Rev. Sci. Instr. 69, 1991 (1998) D.A. Church et al., Phys. Scr. T 80A, 148 (1999) E. Tr¨ abert, Canadian J. Phys. 80, 1481 (2002) C.M. Lyneis et al., Rev. Sci. Instr. 75, 1389 (2004) N. Nakamura et al., Rev. Sci. Instr. 75, 3034 (2004) K. Fahy et al., Phys. Rev. A 75, 032520 (2007) S.J. Smith et al., Astrophys. J. 602, 1075 (2004) S.J. Smith et al., Phys. Rev. A 60, 3569 (1999) A. Chutjian et al., In Applications of Accelerators in Research and Industry, ed. by J.L. Duggan, I.L. Morgan (AIP, New York), p. 881 (1999) M. Knoop et al., Phys. Rev. A 58, 264 (1998) A. Hermanni, G. Werth, Z. Phys. D 11, 301 (1989) J. Yoda et al., Jpn. J. Appl. Phys. 38, 3747 (1999) Ch. Gerz et al., Z. Phys. D 8, 235 (1988) D.J. Seidel, L. Maleki, Phys. Rev. A 51, R2699 (1995) A. Einstein, N. Rosen, B. Podolsky, Phys. Rev. 47, 777 (1935) E. Schr¨ odinger, Naturwissenschaften 23, 807 (1935) J.S. Bell, Speakable and Unspeakable in Quantum Mechanics (Cambridge University Press, Cambridge, England, 1988), p. 196 L. Deslauriers et al., Phys. Rev. Lett. 97, 103007 (2006) G.J. Milburn, S. Schneider, D.F.V. James, Fortschr. Phys. 48, 801 (2000) K. Mølmer, A. Sørensen, Phys. Rev. Lett. 82, 1835 (1999) J.I. Cirac, P. Zoller, Physics Today 38-40 (March 2004) C.A. Sackett et al., Nature 404, 256 (2000) H. H¨ aﬀner et al., Nature 438, 643 (2005) V. Meyer et al., Phys. Rev. Lett. 86, 5870 (2001) F. Schmidt-Kaler et al., Nature 422, 408 (2003) D.J. Wineland et al., J. Res. Natl. Inst. Stand. Technol. 103, 259 (1998) D.S. James, Elementary Information Theory (Clarendon, Oxford 1998) R.J. Hughes et al., Phys. Rev. Lett. 77, 3240 (1996) M.B. Plenio, P.L. Knight, Proc. R. Soc. London A 453, 2017 (1997) R. Landauer, Phys. Today, May 23 (1991) D. Deutsch, Proc. R. Soc. Lond. A 400, 97 (1985) J.I. Cirac, P. Zoller, Phys. Rev. Lett. 74, 4091 (1995) S. Mancini, A. Martins, P. Tombesi, Phys. Rev. A 61, 012303 (2000) G. Ciaramicoli, I. Marzoli, P. Tombesi, Phys. Rev. A 63, 052307 (2001) C.B. Chiu, E.C.G. Sudarshan, B. Misra, Phys. Rev. D 16, 520 (1977) W.M. Itano et al., Phys. Rev. A 41, 2295 (1990) R.J. Cook, Phys. Scr. T 21, 49 (1988) Ch. Wunderlich, Ch. Balzer, arXiv,quant-ph/0305129 vI 22 May 2003 A.G. Kofman, G. Kurizki, Nature 405, 546 (2000) P. Facchi, H. Nakazato, S. Pascazio, Phys. Rev. Lett. 86, 2699 (2001) P. Facchi, S. Pascazio, Progress in Optics, ed. E. Wolf (Elsevier, Amsterdam, 2001), vol. 42, Ch. 3, p. 147 M. Fischer, B. Gutierrez-Medina, M.G. Raizen, Phys. Rev. Lett. 87, 040402 (2001) D. Bouwmeester et al., Nature 390, 575 (1997) A. Furusawa et al., Science 282, 706 (1998) D. Boschi et al., Phys. Rev. Lett. 80, 1121 (1998)

References 325. 326. 327. 328. 329. 330. 331. 332. 333. 334. 335. 336. 337. 338.

339. 340. 341. 342. 343. 344. 345. 346. 347. 348. 349. 350. 351. 352. 353. 354. 355. 356. 357. 358. 359. 360. 361. 362. 363. 364. 365. 366. 367. 368. 369.

265

L.M. Duan et al., Nature 414, 413 (2001) A.K. Ekert, Phys. Rev. Lett. 67(6), 661 (1991) L.M. Duan et al., Phys. Rev. Lett. 85, 5643 (2000) A.B. Mundt et al., Phys. Rev. Lett. 89, 103001 (2002) Q.A. Turchette et al., Phys. Rev. Lett. 81, 3631 (1998) C.H. Bennett et al., Phys. Rev. A 53, 2046 (1996) C. Roos et al., Phys. Rev. Lett. 92, 220402 (2004) D.F.V. James et al., Phys. Rev. A 64, 052312 (2001) R. Blatt, Nature 404, 231 (2000) C. Monroe et al., Science 272, 1131 (1996) D. Leibfried et al., Science 304, 1478 (2004) D. Leibfried et al., Nature 422, 412 (2003) H. H¨ aﬀner et al., Appl. Phys. B 81, 151 (2005) C. Becher et al., “Entanglement of trapped ions” in Proceedings of the 17th International Conference on Laser Spectroscopy, Cairngorms National Park, Scotland, 2005, ed. by E.A. Hinds, A. Ferguson, E. Ries (World Scientiﬁc) M. Keller et al., Nature 431, 1075 (2004) B.B. Blinov et al., Nature 428, 153 (2004) C. Maurer et al., New J. Phys. 6, 94 (2004) J. Brendel et al., Phys. Rev. Lett. 82, 2594 (1999) G. Vidal, R. Tarrach, Phys. Rev. A 59, 141 (1999) M. Steiner, Phys. Rev. A 67, 054305 (2003) T. Yu, J.H. Eberly, Phys. Rev. B 66, 193306 (2002) C. Roos et al., Science 304, 1478 (2004) C.H. Bennett, Physics Today 48, 24 (1995) C.H. Bennett et al., Phys. Rev. Lett. 70, 1895 (1993) M.D. Barrett et al., Nature 429, 737 (2004) D.J. Wineland, Lecture notes of the Les Houches Summer School, ed. by D. Estere, J.-M. Raimond, J. Dalibard (Elsevier, 2003) S.L. Braunstein, H.J. Kimble, Phys. Rev. Lett. 80, 869 (1998) L. Vaidman, Phys. Rev. Lett. 49, 1473 (1994) C.H. Bennett et al., Phys. Rev. Lett. 76, 722 (1996) N. Gisin, Phys. Lett. A 210, 151 (1996) D. Deutsch et al., Phys. Rev. Lett. 77, 2818 (1996) H. Briegel et al., Phys. Rev. Lett. 81, 5932 (1998) P. Zoller (Editing author), Quantum Information Processing and Communication QIST Era-Pilot Project, Version 1.1, June 2005 E.S. Fry, T. Walther, Phys. Rev. A 52, 4381 (1995) E. Hagley et al., Phys. Rev. Lett. 79, 1 (1997) B. Julsgaard, A. Kozhekin, E.S. Polzik, Nature 4130, 400 (2001) W.P. Schleich, J.A. Wheeler, Nature 326, 574 (1987) W.H. Zurek, Physics Today 44, 36 (1991) W.P. Schleich, Nature 4036, 256 (2000) W.H. Zurek, Progress of Theoretical Physics 89, 281 (1993) W.H. Zurek, Vistas in Astronomy 37, 185 (1993) R. Bonifacio, Nuovo Cimento B 114, 473 (1999) D.J. Wineland et al., J. Res. Natl. Inst. Stand. Technol. 103(3) 259 (1998) C.M. Savage, D.F. Walls, Phys. Rev. A 32, 2316 (1985) M.S. Kim, V. Buˇ z ek, Phys. Rev. A 46, 4239 (1992)

266 370. 371. 372. 373. 374. 375. 376. 377. 378. 379. 380. 381. 382. 383. 384. 385. 386. 387. 388. 389. 390. 391. 392. 393. 394. 395. 396. 397. 398. 399. 400. 401. 402. 403. 404. 405. 406. 407. 408. 409. 410. 411. 412. 413.

References P. Goetsch, R. Graham, F. Haake, Phys. Rev. A 51, 3112 (1995) D.F. Walls, G.J. Milburn, Phys. Rev. A 31, 2403 (1985) D.J. Wineland et al., Phys. Rev. A 50, 67 (1994) S.K. Lamoreaux, Phys. Rev. A 56, 4970 (1997) F. Diedrich et al., Phys. Rev. Lett. 62, 403 (1989) C. Monroe et al., Phys. Rev. Lett. 75, 4714 (1995) H. Walther, Adv. At. Mol. Phys. 31, 137 (1993) C.M. Niculae, V.N. Gheorghe, Rom. J. Physics 43, 9 (1998) J.C. Bergquist, W.M. Itano, D.J. Wineland, in Frontiers in Laser Spectroscopy, ed. by T.W. H¨ ansch, M. Inguscio (North Holland, Amsterdam, 1994) p. 359 J.J. Bollinger et al., IEEE Trans. Instrum. Measure. 40, 126 (1991) S.R. Jeﬀerts et al., Phys. Rev. A 51, 3112 (1995) K. Sugiyama, J. Yoda, Phys. Rev. A 55, R10 (1997) R.M. Serra et al., Phys. Rev. A 64, 033419 (2001) D.M. Meekhof et al., Phys. Rev. Lett. 76, 1796 (1996) M. Zhu, J.L. Hall, J.Opt. Soc. Am. B 10, 802 (1993) J.L. Hall, in Frequency-Stabilized Lasers and Their Applications, SPIE Proceedings, Vol. 1837, 1993, p. 2 M. Zhu, J.L. Hall, Experimental Methods in the Physical Sciences, ed. by F.B. Dunning, R. Hulet (Academic, San Diego, 1997) p. 103 D.J. Wineland et al., IEEE Trans. on Ultrasonics, Ferroelectrics, and Frequency Control 37, 515 (1990) W.M. Itano, L.L. Lewis, D.J. Wineland, Phys. Rev. A 25, 1233 (1982) P. Zanardi, M. Rasetti, Phys. Rev. Lett. 79, 3306 (1997) L. Aolita et al., Phys. Rev. A 75, 052337 (2007) L. Aolita et al., Phys. Rev. A 76, 040303(R (2007) D. Kielpinski et al., Science 291, 1013 (2001) A. Messiah, Quantum Mechanics (North-Holland, Amsterdam, 1962) vol. II, p. 744 J. Lawall, M. Prentiss, Phys. Rev. Lett. 72, 993 (1994) L.S. Goldner et al., Phys. Rev. Lett. 72, 997 (1994) M. Weitz, B.C. Young, S. Chu, Phys. Rev. Lett. 73, 2563 (1994) S. Tasaki et al., Int. J. Quant. Chem. 98, 160 (2004) L. Viola, S. Lloyd, Phys. Rev. A 58, 2733 (1998) L. Viola, E. Knill, S. Lloyd, Phys. Rev. Lett. 82, 2417 (1999) L. Viola, S. Lloyd, E. Knill, Phys. Rev. Lett. 83, 4888 (1999) L. Viola, E. Knill, S. Lloyd, Phys. Rev. Lett. 85, 3520 (2000) P. Zanardi, Phys. Lett. A 258, 77 (1999) D. Vitali, P. Tombesi, Phys. Rev. A 59, 4178 (1999) D. Vitali, P. Tombesi, Phys. Rev. A 65, 012305 (2001) M.S. Byrd, D.A. Lidar, Quantum Information Processing 1, 19 (2002) T. Petrosky, S. Tasaki, I. Prigogine, Phys. Lett. A 151, 109 (1990) B. Misra, E.C. Sudarshan, J. Math. Phys. 18, 756 (1977) D. Home, M.A.B. Whitaker, Ann. Phys. 258, 237 (1997) T. Petrosky, S. Tasaki, I. Prigogine, Physica A 170, 306 (1991) P. Facchi, H. Nakazato, S. Pescazio, Phys. Rev. Lett. 86, 2699 (2001) A. Sudbery, Ann. Phys. 157, 512 (1984) A. Peres, Am. J. Phys. 48, 931 (1980) K. Kraus, Found. Phys. 11, 547 (1981)

References

267

414. R.P. Feynman, Int. J. Theor. Phys. 21, 467 (1982) 415. R.P. Feynman, Found. Phys. 16, 507 (1986) 416. P.W. Shor, Proceedings of the 35th Annual Symposium on Foundations of Computer Science (IEEE Computer Society Press, Santa Fe, NM) Preprint quant-ph/9508027) 417. A. Steane, Appl. Phys. B 64, 623 (1997) 418. A.M. Steane, Rep. Prog. Phys. 61, 117 (1998) 419. D.J. Wineland et al., Fortschr. Phys. 46 (1998) 420. C. Monroe, Nature 416, 238 (2002) 421. D.F.V. James, Applied Physics B 66, 181 (1998) 422. D. Bowmeester, A. Ekert, A. Zeilinger, The Physics of Quantum Information, (Springer, Berlin, 2000) 423. M.A. Rowe et al., Quantum Information and Computation (Rinton, 2001) 424. R. Blatt, D.J. Wineland, Nature 453, 1008 (2008) 425. M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information (University Press, Cambridge, 2000) 426. J.S. Bell, Physics 1, 195 (1964) 427. J.F. Clauser, M.A. Horne, Phys. Rev. D 10, 526 (1974) 428. R.P. Feynman, R.B. Leighton, M. Sands, The Feynman Lectures of Physics, Vol. III, Quantum Mechanics, (Addison-Wesley, Reading, MA, 1965) 429. D.P. DiVincenzo, Phys. Rev. A 51,1015 (1995) 430. J.I. Cirac, P. Zoller, Phys. Rev. Lett. 74, 4091 (1995) 431. Ch. Roos et al., Phys. Rev. Lett. 83, 4713 (1999) 432. Preskill et al., Phys. Rev. A 54, 1034 (1996) 433. R.J. Hughes et al., Contemp. Phys. 36, 149 (1995) 434. K. Mattle et al., Phys. Rev. Lett. 76, 4656 (1996) 435. D. Boschi et al., Phys. Rev. Lett. 80, 1121 (1998) 436. D. Bouwmeester et al., Nature, 390(6660), 575 (1997) 437. T. Pellizzari et al., Phys. Rev. Lett. 75, 3788 (1995) 438. G.P. Berman et al., Phys. Lett. 193, 444 (1994) 439. S. Lloyd, Science 261, 1569 (1993) 440. D.P. DiVincenzo Quantum Computation, Science 270, 255 (1995) 441. A. Barenco et al., Phys. Rev. Lett. 74, 4083 (1995) 442. C. Monroe et al., Phys. Rev. A55, R2489 (1997) 443. J.F. Poyatos, J.I. Cirac, P. Zoller, Phys. Rev. Lett. 78(2), 390 (1997) 444. A.M. Steane, Phys. Rev. Lett. 78, 2252 (1997) 445. N.A. Gershenfeld, I.L Chuang, Science 275, 350 (1997) 446. D.G. Cory, A.F. Fahmy, T.F. Havel, Nuclear Magnetic Resonance Spectroscopy, an Experimentally Accessible Paradigm for Quantum Computing, in Proceedings of the 4th Workshop on Physics and Computation (Complex Systems Institute, Boston, New England), 1996 447. H.C. N¨ agerl et al., Phys. Rev. A 60, 145 (1999) 448. M.B. Plenio, P.L. Knight, Phys. Rev. A 53, 2986 (1996) 449. Q.A. Turchette et al., Phys. Rev. Lett. 75, 4710 (1995) 450. W. Paul, H. Steinwedel, Z. Naturforsch. A 8, 448 (1953) 451. H.C. N¨ agerl et al., “Linear Ion Traps for Quantum Computation”. In The Physics of Quantum Information, ed. by D. Bouwmeester, A. Ekert, A. Zeilinger (Springer, Berlin, 2000), p. 163 452. J.D. Prestage, G.J. Dick, L. Malecki, J. Appl. Phys. 66, 1013 (1989)

268 453. 454. 455. 456. 457. 458. 459. 460. 461. 462. 463. 464. 465. 466. 467. 468. 469. 470. 471. 472. 473. 474.

475. 476.

477. 478. 479. 480. 481. 482. 483. 484. 485. 486.

487. 488. 489. 490. 491.

References M.G. Raizen et al., J. Mod. Optics 39, 233 (1992) U. Tanaka et al., Opt. Lett. 22, 1353 (1997) W. Alt et al., Phys. Rev. A58, R23 (1998) M. Drewsen et al., Phys. Rev. Lett. 81, 2878 (1998) C.F. Roos et al., Phys. Rev. Lett. 85, 5547 (2000) K. Matsubara et al., Appl. Phys. B 76, 209 (2003) I. Waki et al., Phys. Rev. Lett. 68, 2007 (1992) G. Birkl, S. Kassner, H. Walther, Nature 357, 2007 (1992) T. Drees, W. Paul, Z. Phys. 180, 340 (1964) D.A. Church, J. Appl. Phys. 40, 3127 (1969) T. Sch¨ atz, U. Schramm, D. Habs, Nature 412, 717 (2001) M.A. Rowe et al., Quantum Information and Computation, vol. 1 (Rinton, 2001), p. 1 S. Seidelin et al., Phys. Rev. Lett. 96, 253003 (2006) Stick et al., Nature Physics 2, 36 (2006) M.A. Rowe et al., Quant. Inf. Comp. 2, 257 (2002) W.K. Hensinger et al., Appl. Phys. Lett. 88, 034101 (2006) G. Brassard et al., Proc. Natl. Acad. Sci. (US) 95, 11032 (1998) Ch. Roos et al., Phys. Rev. A 77, 062306 (2008) S. Olmschenk et al., Phys. Rev. A 76, 052314 (2007) C. Langer et al., Phys. Rev. Lett. 95, 060502 (2005) H.C. N¨ agerl et al., Opt. Exp. 3, 89 (1998) V.N. Gheorghe, P.E. Toschek, On the Coherent states of an ion in a quadrupole radiofrequency trap, in Proceedings of the 11th Internatioanl Conference on Atomic Physics, Paris, 4–8 July, 1988, p. 238 V.N. Gheorghe, F. Vedel, Pulsed quasienergy states of a trapped ion array, in Proceedings of the ECAMP’5, April 1995, Uppsala, Sweden, p. 467 V.N. Gheorghe, On the stability of ion clusters in quadrupole traps, Invited paper, in Proceedings of the 8th International Conference on Plasma Physics, May 94, Iassy, Romania, p. 7 V.N. Gheorghe, A. Gheorghe, On the coherent states of a stored particle, in Proceedings of the 22nd EGAS, 10–13 July, 1990, Edinburgh, p. 411 V.N. Gheorghe, G. Werth, EPJ D 10, 197 (2000) V.N. Gheorghe, F. Vedel, Phys. Rev. A 45, 4828 (1992) G. Morigi et al., Phys. Rev. A 59, 3797 (1999) C. Monroe et al., Phys. Rev. Lett. 75, 4011 (1995) D. Leibfried et al., Rev. Mod. Phys. 75, 281 (2003) D.J. Wineland et al., Ann. Phys. 9, 851 (2000) R. Blatt et al., Phys. Scripta T59, 294 (1995) P. Carruthers, M. Nieto, Phys. Rev. Lett. 14, 387 (1965) D.J. Wineland et al., in Laser Manipulation of Atoms and Ions, Proceedings of the International School of Physics “Enrico Fermi” Course 118. ed. by E. Arimondo, W.D. Phillips, F. Strumia (North-Holland, Amsterdam, 1992) D.M. Meekhoﬀ et al., Phys. Rev. Lett. 76, 1796 (1996) D.F. Walls, G.J. Milburn, Quantum Optics (Springer, Berlin, 1994) D. Leibfried et al., Nature 438, 639 (2005) D. Bouwmeester, A. Ekert, A. Zeilinger, The Physics of Quantum Information (Springer, Berlin, 2000) B. DeMarco et al., Physical Review Letters 89, 267901 (2002)

References 492. 493. 494. 495. 496.

497. 498. 499. 500. 501. 502. 503. 504. 505. 506. 507. 508. 509. 510. 511. 512. 513. 514. 515. 516. 517. 518. 519. 520. 521. 522. 523. 524. 525. 526. 527. 528. 529. 530. 531. 532. 533. 534. 535.

269

A. Sørensen, K. Molmer, Phys. Rev. Lett. 82, 1971 (1999) E. Solano et al., Phys. Rev. A 59, R2539 (1999) P.C. Haljan et al., Phys. Rev. A 72, 062316 (2005) J.-P. Home et al., New J. Phys. 8, 188 (2006) J.I. Cirac, P. Zoller, J.F. Poyatos, Quantum gates and quantum computation with trapped ions, in The Physics of Quantum Information, ed. by D. Bouwmeester, A. Ekert, A. Zeilinger (Springer, Berlin, 2000), p. 163 A. Steane et al., Phys. Rev. Lett. 62, 042305 (2000) J.I. Cirac et al., Phys. Rev. A 48, 1434 (1993); J.I. Cirac et al., Phys. Rev. Lett. 70, 762 (1993) R.G. DeVoe, United States Patent Nr. 5,793,091, August 11, 1998 P. Berman, Cavity Quantum Electrodynamics (Academic, New York, 1993) Parkins et. al., Phys. Rev. Lett. 71, 3095 (1993) Garcia-Ripoll, J.I. Cirac, P. Zoller, Phys. Rev. Lett. 91, 157901 (2003) C. D’Helon, G. J. Milburn, Phys. Rev. A 54, R25 (1996) D.T. Smithey et al., Phys. Rev. Lett. 70, 1244 (1993) P.J. Bardroﬀ et al., Acta Phys. Slov. 46, 1 (1996) J.I. Cirac et al., Phys. Rev. A 49, 421 (1994) S. Wallentowitz and W. Vogel, Phys. Rev. Lett. 75, 2932 (1995) J.F. Poyatos et al., Phys. Rev. A 53, R1966 (1996) J.I. Cirac, R. Blatt, P. Zoller, Phys. Rev. A 49, R3174 (1994); J.I. Cirac, A.S. Parkins, Phys. Rev. A 50, R4441 (1994) L.G. Lutterbach, L. Davidovich, Phys. Rev. Lett. 78, 2547 (1997) K. Vogel, H. Risken, Phys. Rev. A 40, R2847 (1989) D.J. Heinzen, D.J. Wineland, Phys. Rev. A 42, 2977 (1990) G.M. D’Ariano, C. Macciavello, M.G.A. Paris, Phys. Rev. A 50, 4298 (1994) M. Horodecki, P. Horodecki, R. Horodecki, Phys. Lett. A 223, 1 (1996) A. Peres, Phys. Rev. Lett. 77, 1413 (1996) P.T.H. Fisk et al., IEEE Trans. Instr. Meas. 44, 113 (1995) J.I. Cirac et al., Phys. Rev. Lett. 78, 3221 (1997) R.G. DeVoe, Phys. Rev. A 58, 910 (1998) T. Pellizzari, Phys. Rev. Lett. 79, 5242 (1997) J.R. Zurita-Sanchez, C. Henkel, Phys. Rev. A 73, 063825 (2006) J.I. Cirac, P.Zoller, Nature 404, 579 (2000) D. Kielpinski, C.R. Monroe, D.J. Wineland, Nature 417, 709 (2002) D. Sick et al., Nature Physics 2, 36 (2006) S. Seidelin et al., Phys. Rev. Lett. 96, 253003 (2006) J. Labaziewicz et al., Phys. Rev. Lett. 100, 013001 (2008) M. Brownut et al., New J. Phys. 8, 2328 (2006) M.D. Barrett et al., Nature 429, 737 (2005) D. Leibfried et al., Phys. Rev. A 76, 032324 (2007) G. Ciaramicoli, I. Marzoli, P. Tombesi, Phys. Rev. Lett. 91, 017901 (2003) G. Ciaramicoli, I. Marzoli, P. Tombesi, Phys. Rev. A 70, 032301 (2004) A.S. Sørensen et al., Phys. Rev. Lett. 92, 063601 (2004) J.V. Porto, Phys. Rev. A 64, 023403 (2001) J. Verd´ u et al., Phys. Rev. Lett. 92, 093002 (2004) S. Stahl et al., Eur. Phys. J. D 32, 139 (2005) I. Marzoli et al., J. Phys. B 42, 154010 (2009)

270 536. 537. 538. 539. 540. 541. 542. 543. 544. 545. 546. 547. 548. 549. 550. 551. 552. 553. 554. 555. 556. 557. 558. 559. 560. 561. 562. 563. 564.

References G. Ciaramicoli et al., Phys. Rev. A 72, 042323 (2005) F. Mintert, Ch. Wunderlich, Phys. Rev. Lett. 87, 257904 (2001) J.A. Jones, Prog. NMR Spectrosc. 38, 325 (2001) D.Mc. Hugh, J. Twamley, Phys. Rev. A 71, 012315 (2005) M. Diederich et al., Hyperﬁne Interact. 115, 185 (1998) D.L. Moehring et al., J. Opt. Soc. Am. B 24, 301 (2007) G.R. Guth¨ ohrlein et al., Nature 414, 49 (2001) J. Eschner et al., Nature 413, 495 (2001) A.M. Steane, Phys. Rev. Lett. 77, 793 (1996) J. Chiaverini, Nature 432, 602 (2004) S. Gulde et al., Nature 421, 48 (2003) D.J. Amit, Modeling Brain Function (Cambridge University Press, Cambridge, 1989) J.J. Bollinger et al., Bull. Am. Phys. Soc. 37, 1117 (1992) G. Tommaseo et al., Eur. Phys. J. D 25, 113 (2003) K.H. Knoell et al., Phys. Rev. A 54, 1199 (1996) W. Itano et al., J. Opt. Soc. Am. B 2, 1352 (1985) R. Loch, R. Stengler, G. Werth, Phys. Rev. A 38, 5484 (1988) M. Prior, Phys. Rev. Lett. 29, 611 (1972) R. Knight, M. Prior, Phys. Rev. A 21, 179 (1980) Z. Fang, et al., Phy. Rev. A 48, 1114 (1993) L. Letchumanan et al., Phys. Rev. A 72, 012509 (2005) Ch. Gerz et al., Z. Phys. D 5, 97 (1987) N. Yu, L. Maleki, Phys. Rev. A 61, 022507 (2000) N. Yu et al., Phys. Rev. Lett. 78, 4898 (1997) W. Itano et al., Phys. Rev. Lett. 59, 2732 (1987) E. Peik et al., Phys. Rev. A 49, 492 (1994) A. Roth et al., Z. Phys. D 11, 283 (1989) S. Trapp et al., Eur. Phys. J 26, 237 (2003) M. Chwalla et al., Phys. Rev. Lett. 102, 023002 (2009)

Index

absorption (saturated), 31, 131, 132, 224, 225 adiabatic approximation, 7, 215 Allan variances, 122, 127, 133 angular momentum, 24, 76, 77, 85, 101, 105, 106, 109, 140, 144, 199, 202, 221 anharmonicity, 67, 75, 77, 248, 254 axial motion, 28, 74, 245, 246, 249, 250, 252–254 background (buﬀer) gas, 7, 21–23, 28, 41, 46, 47, 123, 162, 164, 168, 174, 176 barium ion, 103, 105, 113, 114, 128, 138, 139, 162, 167, 168, 173, 174, 180, 218 beryllium ion, 103, 202 Bessel function, 150 black body radiation, 120, 128, 148, 196 Bloch vector, 208 Bohr-Weisskopf eﬀect, 106, 116 Bolometric ion detection, 89 Breit-Rabi formula, 105, 112 Brewster’s angle, 155 calcium ion, 119, 134, 162, 165–169, 172, 175, 184, 186, 188, 190, 214, 217, 219, 225 carrier frequency, 33, 150, 154, 224, 227 cavity microwave, 121 optical, 141, 144, 184, 189, 190, 242 Cirac–Zoller, 180, 212, 213, 232

coherence, 50, 114, 115, 126, 179, 180, 182, 186, 190, 197, 240, 248, 255 collisions, 9, 11, 18, 22, 23, 28, 29, 41, 44, 46–49, 54, 88, 101, 107, 108, 112, 114, 125, 126, 137, 161, 166, 172–174, 176, 179, 191, 199 charge exchange, 137 cross section, 53, 54, 88, 89, 108, 111, 155, 176, 199, 216 elastic/inelastic, 88, 89, 108 spin exchange, 88, 107, 111 communication classical, 193, 194 quantum, 184, 234 cooling of ions, 16, 18, 28, 138, 150, 217, 224 buﬀer gas in paul trap, 28 buﬀer gas in penning trap, 28 Doppler, 31, 33, 126, 137, 145, 146, 164, 167, 169, 185, 188, 224, 225 radiative, 33, 188 resistive, 29, 30, 60, 90 resolved sidebands, 32, 107, 225 sympathetic, 146 Coulomb interaction/scattering, 11, 30, 74, 76, 146, 179, 184, 197, 199, 241, 245–247, 251, 253 cyclotron motion, 19, 23, 28, 34, 56, 60–62, 64, 66, 70, 71, 91, 94, 95, 245, 248–250, 255 frequency, 13, 14, 16, 18, 23, 24, 51, 55, 58–60, 62, 63, 73, 74, 76, 78,

272

Index

80, 86, 87, 89, 90, 94, 97, 99, 101, 251 radius, 65 cylindrical ion trap, 4, 5, 50, 52, 53, 61, 94, 97, 215 dark states, 202 decoherence, 179, 188–190, 195, 197– 205, 212, 213, 216, 219, 227, 228, 235, 239, 240, 244, 255 anomalous motion, 196, 197 free subspace, 240 induced, 202 thermal, 203 density matrix, 183, 186, 188, 195, 203, 204, 235 detection ion, 40, 64, 145 axial oscillation, 23, 56, 60–62, 66, 92, 97–100, 197, 249 bolometric detection, 25 Fourier transform, 26, 27, 47, 50–52, 61–63, 78, 97, 144, 253 optical detection, 27, 108 resonant absorption, 224 single electron, 34, 89, 93, 95, 138, 211, 247–250, 252, 254 Dick eﬀect, 133 Dicke eﬀect, 101 diode laser, 141–145, 217, 218 distributed bragg reﬂector (DBR), 140 extended cavity (ECDL), 140 dipole excitation, 40, 68, 70 dispersion, 154–156 DiVincenzo criteria, 213, 255 Doppler broadening, 32, 114, 131, 132 cooling, 31, 33, 126, 137, 145, 146, 164, 167, 169, 185, 188, 224, 225 second order (relativistic) shift, 150 shift, 31, 111, 121, 131, 150 side bands, 137 double penning trap, 98, 99 eigenfrequencies, 12–14, 16, 17, 56, 69, 99, 100 Einstein–Podolsky–Rosen (EPR) correlations, 192, 209, 211 elastic/inelastic collisions, 16, 88

electric quadrupole interaction, 106, 148 electro-optic modulator (EOM), 152 electron beam ion trap (EBIT), 81, 170 electron cyclotron resonance (ECRIS), 170 electron mass, 77–79 energy defect, 108 entanglement, 179, 183–195, 200, 211, 212, 231, 232, 239, 241 error correction techniques/codes, 254–256 errors systematic, 74, 147, 168 europium ion, 117 excitation spectrum, 32, 33 Fabry-P´erot cavity, 137 Fermi-Segr´e formula, 106 ﬁeld electric/magnetic, 55, 57, 58, 74, 112 ﬂuorescence, 9, 31 Fourier spectrum, 63, 153 transform spectrometer, 50 Fourier transform detection, 26, 253 frequency shifts, 15, 55, 57, 58, 126, 135, 143, 148 g-factor anomaly, 86, 87 bound state, 78, 95 free electron, 86 gravitational red shift, 120 harmonic/anharmonic oscillator, 7, 34, 66, 67, 91, 94, 184, 224, 226, 227, 236 helium buﬀer gas, 112, 114 helium ion, 108 hydrogen-like atoms, 78, 96, 98 hyperﬁne anomaly, 106, 116 pumping, 104, 109, 110, 112–114, 117, 121, 123 qubit, 218 spectroscopy, 113 structure, 105, 109 transitions, 107, 109, 114, 117, 118 image charges, 18, 55, 58, 60, 73, 94, 247

Index inelastic collisions, 108 instabilities electric/magneticﬁeld, 78, 100 Paul trap, 12, 16, 20, 22 Penning trap, 16, 18 interference, 62, 210 International Atomic Time (TAI), 128 Invariance theorem, 15, 17, 57 ion chips, 256 conﬁnement, 3 density distribution, 10, 20, 28 detection, 40 injection, 21 motion, 6, 9, 15, 18, 28, 41, 45, 49, 51, 68, 92, 117, 143, 184, 196, 197, 199, 224, 232, 236 resonance, 51, 69, 110, 111, 123, 135, 138, 142, 223 temperature, 10, 17, 25, 28, 150, 221 trajectory, 14, 18, 29, 42 ion sources, 41, 44, 46, 170 (photo)ionization, 39, 41, 43, 47, 54, 117, 137, 164 ISOLDE-facility, 22, 74, 75 isotopes stable/unstable, 37, 43, 45, 72–74, 76, 77, 79, 80, 117, 120, 124, 127, 131 Johnson noise, 195, 197–199 Kingdon trap, 170, 171 klystron, 112, 114 Lamb–Dicke parameter/regime, 32, 137, 150, 184, 223–225, 234 Land´e factor, 150 Larmor frequency, 87, 99, 101 Laser diode, 142–145, 217, 218 Nd:YAG, 145, 156 stabilization, 131 Ti:Sa, 128, 143, 144, 153, 155–157, 164 lifetime entangled states, 190, 251 radiative, 114, 143, 144, 161, 162, 165, 170, 172, 174 light shift, 112, 125, 128, 133

273

line motional resonances, 8, 15, 60, 66, 72 shapes, 53, 66–71, 93, 95, 100, 114, 115 width, 164 linear ion trap, 123, 124, 126, 190, 198, 213–215, 219, 244 magnesium ion, 105, 119 magnetic dipole, 88, 103, 106, 143, 162, 226 ﬁeld corrections, 55, 86, 91, 111 ﬂuctuations, 58 resonance, 107, 207, 214 shielding, 137 magnetron frequency, 14, 20, 23, 71, 75, 91, 251 masers, 124, 127, 128 mass, 60 cesium, 78 comparison, 57, 74, 78, 79, 81, 101 neutrino, 78 proton, 103 proton/antiproton, 78 proton/electron, 78 reduced, 203 resolution, 37, 39, 41–43, 48, 49, 51, 53, 54 SI standard, 79 spectra, 53 uncertainty, 31 unstable isotopes, 80 mass ﬁlter Paul, 38, 41, 215 mass spectrometers, 48, 52, 72, 76 microquadrupole, 43, 53, 54 tandem, 39, 46, 48–50 time-of-ﬂight, 46 Mathieu equation, 6 parameters, 197 mean free path, 22, 53 mercury ion, 109, 110, 122, 127, 128, 137, 141, 217, 218 metastable states, 32, 161, 163, 165, 170, 172, 174, 176, 180, 200 micromotion, 7, 10, 28, 33, 145, 150, 151, 164, 198, 242 microwave transition, 104, 114, 121, 128 motional spectrum, 8, 13, 15

274

Index

natural linewidth, 142, 145 nitrogen, 132 noise induced by ions, 26, 63, 78, 97, 100 quantum projection, 123, 135 spectrum, 68 nuclear g-factor, 105, 106 nuclear magnetic resonance (NMR), 213 optical pumping, 101, 104, 109, 113, 114, 117, 121, 122, 124, 140, 143, 169, 181, 185, 186, 213, 223, 224 Paul trap, 198 imperfections, 11, 15 mass ﬁlter, 37 non-linear resonances, 45 operations, 9, 41, 43, 49, 51, 53 Penning trap cylindrical, 5, 94, 181 imperfections, 16–18, 55, 57, 60, 66, 73 non-linear resonances, 16 operation, 11 photodissociation, 108 potential saddle point, 248, 252 Pound–Drever–Hall method, 138 pressure shift, 112, 132 pseudopotential, 49, 50 QED eﬀects, 79, 95 quadrupole potential, 4, 5, 11, 13, 215 excitation, 65, 70, 71 transitions, 107, 133, 135–137, 162, 167, 228 quality-factor, 26, 30, 61, 62, 66, 128, 133, 199, 253 quantum error correction, 212, 214 jumps, 144, 173, 174, 180, 254 logic gates, 199, 213, 228 oscillator, 213 register, 220 zero point, 107, 121, 137, 146, 186, 188, 213 qubits, 190, 192, 202, 208, 209, 212–214, 216, 219, 220, 223, 224, 228,

230–232, 235, 240, 241, 243, 245, 249–256 quenching collisions, 166, 176 Rabi frequency, 115, 134, 226, 234, 235 radiative lifetime, 114, 143, 144, 161, 162, 165, 170, 172, 180, 201 radioactive isotopes, 117 Raman scattering, 155 Raman transitions, 146, 219, 227, 228 Ramsey excitation, 70–72 register capacity, 209, 210, 212 residual gas analyzer;, 40 resolution spectral, 46, 107, 109, 200 resonance ﬂuorescence, 10, 31 line shape, 13, 53, 66, 67, 69–72, 95, 100, 115 motional, 8 ring trap, 215 rotating wall, 20 saddle point, 4 saturation (intensity), 27, 131, 164, 166 scattering, 11, 30, 49, 87, 111, 137, 155, 165, 172, 173 Schr¨ odinger Cat state, 187, 188, 226, 227, 232 equation, 91, 226 secular motion, 7, 44, 126, 137, 150 selection rules, 114 semiconductor, 179, 244 servo control, 127, 131, 133, 138, 142, 148, 203 shot noise, 122, 135, 202 side-band ion cooling, 145 silicon-based mass spectrometer, 52, 53 space charge, 9–11, 17, 18, 44, 55 spectral purity, 122, 125, 129–131, 137, 138, 143 spontaneous emission, 107, 129, 173, 179, 189, 216, 224, 235 squeezed states, 237 stability parameters, 11 stability/instability ion motion, 6, 9, 11, 12, 18, 40–46, 60, 80, 122, 123, 128, 129, 131–135, 138, 147, 154, 203, 245 Stark shift, 113, 128, 151, 200, 202

Index Stern–Gerlach continuous, 86, 92, 93, 97, 121, 253 stimulated emission/force/eﬀect, 107, 129 storage capacity, 16, 20 time, 9, 12, 18, 19, 117, 161, 172 strong correlation, 209 strontium ion, 119, 134, 139, 141, 142, 162 superposition, 13, 15, 90, 147, 186, 192, 195, 208, 210, 219, 223, 226–228, 239, 241, 254 surface ionization, 101, 117, 190 synchrotron radiation, 90, 250 systematic frequency shifts, 143 Tandem mass spectrometer, 50 teleportation, 180, 191–194, 212 thermal, 26 equilibrium, 10, 11, 20, 27, 28, 67, 68, 78, 90, 91, 150, 197, 199, 226 noise, 25, 27, 61, 62, 67, 89, 135 time-of-ﬂight spectrometer, 46, 47, 65 trajectory of charged (micro)particles, 7, 29 transition probability, 27, 71, 114, 134, 161, 162, 176, 183 trap elliptical, 242 hyperbolic, 3, 4, 16, 25, 61, 90, 94, 198, 213, 249 miniature, 137

275

planar, 217, 247, 248 storage ring, 215 triple resonance experiment, 104 two-level atomic system, 27, 31, 116, 216, 219, 255 two-photon transition, 131 ultra-low expansion (ULE) material, 141, 144, 145 vacuum Rabi frequency, 235 vacuum system, 50, 126, 137 cryogenic pumping, 137 wavefunction, 106, 108, 179, 193, 223, 227, 234 waveguide, 155 wavelength, 32, 39, 107–109, 111, 113, 117, 125, 128, 132, 136, 138, 140, 141, 144, 145, 147, 150, 155, 164, 189, 198, 213, 217, 219 ytterbium ion, 119, 128, 134, 141–143, 162, 174, 175, 182, 183, 217, 218 Zeeman correction, 146 shifts, 140, 146, 147, 149, 200, 239 spectrum, 85, 102–104, 107, 151, 219, 240 Zeno eﬀect, 204 quantum control, 203, 205 zero point energy, 107, 137, 184, 213

Springer Series on

atomic, optical, and plasma physics Editors-in-Chief: Professor G.W.F. Drake Department of Physics, University of Windsor 401 Sunset, Windsor, Ontario N9B 3P4, Canada

Professor Dr. G. Ecker Ruhr-Universit¨at Bochum, Fakult¨at f¨ur Physik und Astronomie Lehrstuhl Theoretische Physik I Universit¨atsstrasse 150, 44801 Bochum, Germany

Professor Dr. H. Kleinpoppen, Emeritus Stirling University, Stirling, UK, and Fritz-Haber-Institut Max-Planck-Gesellschaft Faradayweg 4–6, 14195 Berlin, Germany

Editorial Board: Professor W.E. Baylis

Professor B.R. Judd

Department of Physics, University of Windsor 401 Sunset, Windsor, Ontario N9B 3P4, Canada

Department of Physics The Johns Hopkins University Baltimore, MD 21218, USA

Professor Uwe Becker Fritz-Haber-Institut Max-Planck-Gesellschaft Faradayweg 4–6, 14195 Berlin, Germany

Professor K.P. Kirby Harvard-Smithsonian Center for Astrophysics 60 Garden Street, Cambridge, MA 02138, USA

Professor Philip G. Burke

Professor P. Lambropoulos, Ph.D.

School of Mathematics and Physics Queen’s University David Bates Building, Belfast BT7 1NN, UK

Max-Planck-Institut f¨ur Quantenoptik 85748 Garching, Germany, and Foundation for Research and Technology – Hellas (F.O.R.T.H.), Institute of Electronic Structure and Laser (IESL), University of Crete, PO Box 1527 Heraklion, Crete 71110, Greece

Professor R.N. Compton Oak Ridge National Laboratory Building 4500S MS6125 Oak Ridge, TN 37831, USA

Professor M.R. Flannery

Professor G. Leuchs

School of Physics Georgia Institute of Technology Atlanta, GA 30332-0430, USA

Friedrich-Alexander-Universit¨at Erlangen-N¨urnberg Lehrstuhl f¨ur Optik, Physikalisches Institut Staudtstrasse 7/B2, 91058 Erlangen, Germany

Professor C.J. Joachain

Professor P. Meystre

Facult´e des Sciences Universit´e Libre Bruxelles Bvd du Triomphe, 1050 Bruxelles, Belgium

Optical Sciences Center The University of Arizona Tucson, AZ 85721, USA

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