Optically Pumped Atoms
William Happer, Yuan-Yu Jau, and Thad Walker
WILEY-VCH Verlag GmbH & Co.
William Happer, Yuan-Yu Jau, and Thad Walker
Optically Pumped Atoms
WILEY-VCH Verlag GmbH & Co. KGaA
William Happer, Yuan-Yu Jau, and Thad Walker Optically Pumped Atoms
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William Happer, Yuan-Yu Jau, and Thad Walker
Optically Pumped Atoms
WILEY-VCH Verlag GmbH & Co. KGaA
The Author Prof. William Happer Dept. of Physics Princeton University Princeton, USA Dr. Yuan-Yu Jau Department of Physics Princeton University Princeton, USA Prof. Thad Walker Department of Physics University of Wisconsin Madison, USA
All books published by Wiley-VCH are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate. Library of Congress Card No.: applied for British Library Cataloguing-in-Publication Data: A catalogue record for this book is available from the British Library. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.d-nb.de. © 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form – by photoprinting, microfilm, or any other means – nor transmitted or translated into a machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law. Cover Design Adam-Design, Weinheim Typesetting le-tex publishing services GmbH, Leipzig Printing and Binding Strauss GmbH, Mörlenbach Printed in the Federal Republic of Germany Printed on acid-free paper ISBN 978-3-527-40707-1
V
Contents Preface IX Index to Codes XI 1
Introduction 1
2 2.1 2.2 2.3
Alkali-Metal Atoms 15 Electronic Energies 15 Valence-Electron Wave Functions 18 Hyperfine Structure 20
3 3.1 3.1.1 3.1.2 3.2 3.3
Wave Functions and Schrödinger Space Uncoupled States 25 Kronecker Products 25 Angular Momentum Matrices 27 Energy States 28 Zero-Field States 30
4 4.1 4.2 4.3 4.3.1 4.3.2 4.3.3 4.4 4.4.1 4.4.2 4.5 4.5.1 4.5.2 4.6 4.6.1 4.6.2 4.6.3
Density Matrix and Liouville Space 33 Purity and Entropy 35 Ground State, Excited State, and Optical Coherence 35 Column-Vector and Row-Vector Transforms 36 Column-Vector Transforms 36 Row-Vector Transforms 37 Expectation Values 38 Superoperators 39 Transposition Matrix 39 Evolution Matrices 40 Eigendecomposition of G 41 Nullspace 42 Critical Damping 43 Matrix Transformations from Schrödinger Space to Liouville Space Flat and Sharp Superoperators 44 Square Matrices 46 Commutator Superoperators 47
25
Optically Pumped Atoms. William Happer, Yuan-Yu Jau, and Thad Walker Copyright © 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-40707-1
44
VI
Contents
4.6.4
O-Dot Superoperators
47
5 5.1 5.2 5.2.1 5.2.2 5.2.3 5.2.4 5.2.5 5.2.6 5.2.7 5.3 5.4 5.5 5.6 5.6.1 5.6.2 5.6.3 5.7 5.8 5.8.1 5.8.2 5.8.3
Optical Pumping of Atoms 49 The Electric Field of Light 49 The Electric Dipole Moment of Atoms 50 Spherical Tensors 50 Hermitian Conjugates 51 Addition of Angular Momentum 52 Spherical Basis Tensors 53 Identities for Δ and Δ † 53 Amplitude D 56 Energy Basis 56 Spontaneous Emission 57 Electric Dipole Interaction 58 Rotating Coordinate System 59 Net Evolution 62 The Amagat Unit of Density 64 Normalization 64 Notation and Coding 65 Optical Bloch Equations 65 Liouville Space 66 Transients 68 Steady State 69 Steady State Versus Detuning 70
6 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.8.1 6.9 6.9.1 6.9.2 6.10
Quasi-Steady-State Optical Pumping 73 Ground-State Evolution 74 Excited-State Evolution 76 Collisions 77 Saturation 78 Identities 78 Net Evolution 80 Negligible Stimulated Emission 81 High-Pressure Pumping 82 Liouville Space 84 Spectral Width of Pumping Light 87 Gaussian Spectral Profiles 88 Plasma Dispersion Function 89 Doppler Broadening 90
7 7.1 7.2 7.2.1 7.2.2 7.2.3
Modulation 93 Magnetic Resonance 94 Modulated Light 94 High Pressure 95 Lower Pressure 95 Modulated Optical Pumping Matrices
96
Contents
7.3 7.4 7.5 7.5.1 7.5.2 7.5.3
Secular Approximation 97 Attenuation of Modulated Coherence in Passing through the Excited State 100 Examples 102 Isolated Magnetic Resonances 102 Zeeman Magnetic Resonances 103 Push–Pull Pumping 106
8 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9
Light Propagation 109 Induced Electric Dipole Moment 109 Absorption Cross Section 111 Small Magnetic Fields 112 Evolution of a Beam in Space and Time 114 First-Order Propagation Equation 115 Propagation of Weak Probe Light 116 Faraday Rotation 117 Specific Absorption 118 Fluorescent Light 119
9 9.1 9.2 9.3 9.3.1 9.4 9.5 9.6 9.7 9.8 9.8.1 9.8.2 9.9 9.9.1 9.9.2 9.10 9.11 9.12
Radiation Forces 121 Mean Force 121 Forces from Monochromatic Light 123 Forces in Magneto-Optical Traps 124 Repump Lasers 130 Pointing Probability 132 Momentum Space 135 Evolution in Spin-Momentum Space 138 Liouville Space 140 Compactification 140 Compactified p q Space 141 Compactification within a Tile 143 Displays 147 Momentum-Space Displays 147 Position-Space Displays 149 Momentum Diffusion 151 Momentum Diffusion Due to Spontaneous Emission 152 Momentum Diffusion from Pumping 153
10 10.1 10.2 10.3 10.4 10.5 10.6 10.7
Relaxation of Polarized Atoms 159 S-Matrix 160 Collisions in the Gas 163 Weak Collisions 164 Relative Power Spectrum 166 Sudden Collisions 167 Strong Collisions 168 Hyperfine-Shift Interaction 171
VII
VIII
Contents
10.8 10.8.1 10.8.2 10.9 10.9.1 10.9.2 10.9.3 10.10 10.11 10.12 10.12.1 10.12.2 10.13 10.14
Spin–Rotation Interaction 175 Binary Collisions 176 Experimental Measurements 178 Spin Exchange between Alkali-Metal Atoms and Noble Gas Atoms Binary Collisions 181 Spin Temperature 184 Experimental Measurements 186 Spatial Diffusion 189 Adsorption on the Walls 194 Spin Exchange between Pairs of Alkali-Metal Atoms 198 Partial-Wave Analysis 198 Semiclassical Calculation of Partial-Wave Cross Sections 207 Pressure Dependence of Relaxation in the Dark 208 Collisions of Excited Atoms 212
11 11.1 11.2 11.3
Mathematical Appendix 219 Electronic Multipoles 219 Projection Operators in Terms of S or J Recoupling Example 223 References 225 Index 231
221
178
IX
Preface The authors of this book have done experimental work in many areas related to optical pumping, for example, precision spectroscopy, atomic clocks, magnetometers, hyperpolarized noble gas nuclei, and trapped atoms. One might legitimately wonder why we would write a book with so little discussion of experimental detail and with so many equations and computer codes. The reason is that few professional theoreticians work in areas related to optical pumping, and experimentalists are usually left to their own devices. No special expertise is needed to model optically pumped systems as two- or three-level atoms, and these simplified models often give good qualitative insight into observed phenomena. But such models are often inadequate for quantitative analysis, they have limited predictive value, and they can lead to very serious errors in the determination of resonance lineshapes, transient responses, key experimentally determined parameters that account for relaxation phenomena, and so on. In this book, we show that modern computer software makes it possible to model real, multilevel atoms in much the same way as two-level atoms. Throughout the book, the presentation is structured to allow translation of the key formulas into practical, concise, and readable computer codes. Included are many examples of practical computer codes that have been used with success in our research. With minor changes, these codes can be modified to model very sophisticated modern experiments using optically pumped atoms. For the convenience of the readers, we have set up an open-source Web site (http://minds.wisconsin.edu/handle/1793/35675) to archive the computer codes contained in this book. Also included are additional codes that can be used to model a wide range of optical-pumping and spin-relaxation phenomena We are grateful to our families for patience and support while we were writing this book. Special thanks are due to Prof. Kiyoshi Ishikawa and Mr. Ben Olsen for reading drafts of the book and for providing many helpful suggestions. We thank Princeton University and the University of Wisconsin for giving us the opportunity to complete this book. We hope that the book falls into the traditional three missions of universities, the creation, transmission, and preservation of knowledge. Special thanks are due to the University of Wisconsin for maintaining the permanent Web site. Finally, we could not have acquired the background knowledge to write this book without the generous, long-term support of the Air Force Office Optically Pumped Atoms. William Happer, Yuan-Yu Jau, and Thad Walker Copyright © 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-40707-1
X
Preface
of Scientific Research, the Army Research Office, the Defense Advance Research Projects Agency, the National Science Foundation, the Department of Energy, and the National Institutes of Health. Princeton and Madison, October 2009
William Happer Yuan-Yu Jau Thad G. Walker
XI
Index to Codes
1.1: 1.2: 1.3: 1.4: 1.5: 1.6: 1.7: 2.1: 3.1: 3.2: 3.3: 3.4: 3.5: 3.6: 4.1: 4.2: 4.3: 4.4: 5.1: 5.2: 5.3: 5.4: 5.5: 5.6: 5.7: 5.8: 5.9: 5.10: 6.1: 6.2: 6.3: 7.1:
Schroedinger matrix conversion to Liouville vector . . . . . . . . . . . . . . . . . . . . . . 4 Liouville vector to Schroedinger matrix conversion . . . . . . . . . . . . . . . . . . . . . . 4 Transient response by matrix exponentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Use of nullfast() to speed null space computation . . . . . . . . . . . . . . . . . . . . . . 10 Evaluate and plot steady-state spin polarizations . . . . . . . . . . . . . . . . . . . . . . . . 10 Logical column vector for populations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Atomic parameter values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Spin matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Schroedinger energies and unitary matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Dot product of matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Clebsch-Gordan coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Delta symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Zero-field energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Projection operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 Liouville energy basis states and transposition operator . . . . . . . . . . . . . . . . . 39 Flat matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Sharp matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 6-j symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 Spherical and Cartesian components of dimensionless dipole operator . . 56 Energy basis matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 Coupling matrix for spontaneous emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 Light-atom interaction matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 Resonant frequency shift matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Uniform relaxation matrix and damping operator . . . . . . . . . . . . . . . . . . . . . . 67 Transient response of Optical Bloch Equations . . . . . . . . . . . . . . . . . . . . . . . . . 68 Evaluate and plot steady-state populations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 State populations vs detuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 Transients and steady state of high pressure optical pumping . . . . . . . . . . . 85 Cross product of matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 Fadeeva function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 Sublevel populations for microwave magnetic resonance . . . . . . . . . . . . . . 103
Optically Pumped Atoms. William Happer, Yuan-Yu Jau, and Thad Walker Copyright © 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-40707-1
XII
Index to Codes
7.2: 7.3: 8.1: 9.1: 9.2: 9.3: 9.4: 9.5: 9.6: 9.7: 9.8: 9.9: 9.10: 9.11: 9.12:
Sublevel populations for Zeeman magnetic resonance . . . . . . . . . . . . . . . . . 105 Sublevel populations for push-pull pumping . . . . . . . . . . . . . . . . . . . . . . . . . . 107 Absorption cross sections for unpolarized atoms . . . . . . . . . . . . . . . . . . . . . . 113 Matrices for quasi steady-state analysis of a MOT . . . . . . . . . . . . . . . . . . . . . . 128 MOT sublevel populations and forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 MOT forces including repumper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 Matrices assuming perfect repumping, cycling transition . . . . . . . . . . . . . . 136 Momentum and azimuthal labels for compactified momentum space . . 144 Left and right momentum and spin quantum numbers . . . . . . . . . . . . . . . . 144 Coupling coefficients for momentum space optical pumping . . . . . . . . . . 145 Optical pumping matrix for momentum space . . . . . . . . . . . . . . . . . . . . . . . . 145 Steady-state of momentum space optical pumping . . . . . . . . . . . . . . . . . . . . 146 Momentum-space displays. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 Pointing probability plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Momentum diffusion coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
1
1 Introduction The basic idea of optical pumping [1], for which Kastler [2] received the 1966 Nobel Prize in physics, is that the photons in a beam of light can transfer order to atoms or molecules by scattering. Originally the term “optical pumping” referred to ordering the spin degrees of freedom. More recently, optical pumping has been widely used to order the translational as well as the spin degrees of freedom for cooling and trapping experiments. Optical pumping experiments can be quite simple and inexpensive and still yield precise spectroscopic information. They can also be quite intricate and require substantial resources. Optical pumping is a fascinating area of study in its own right, but in addition it has many applications (clocks, magnetometers, quantum optics, spin-polarized nuclei) where the central focus is elsewhere. We will assume that the reader of this book already has a basic understanding of quantum mechanics, atomic physics, optics, and magnetic resonance. There are many excellent books that discuss these issues. Familiarity with linear algebra, complex analysis, Fourier and Laplace transforms, and the quantum theory of angular momentum would be helpful for following some of the mathematics in this book. Because real atoms can have a large number of spin sublevels, because the optical interactions, especially at low buffer-gas pressures, depend in a detailed way on the spectral profile and polarization of the light, and because of the complicated collisional relaxation mechanisms, realistic numerical modeling of optically pumped atoms is a computational challenge. Consequently, the system is often represented with models [3–5] that consider only a few spin sublevels and greatly simplify the physics of optical pumping and spin relaxation. Such simplified models can be valuable for conceptual insights and can give results in qualitative agreement with observations. A major aim of this book is to show that modern scientific computing software makes it practical to analyze the full, multilevel system of optically pumped atoms under most experimental conditions, for example, magnetic resonance with one or more oscillating magnetic fields, coherent-Ramanscattering (coherent population trapping) resonances induced by modulated light, magneto-optical forces on multilevel atoms, and various spin-relaxation processes. To make most effective use of contemporary mathematical software, it is especially useful to analyze optical pumping situations in the Liouville space of density matrices, , rather than in the traditional Schrödinger (Hilbert) space of wave functions. Optically Pumped Atoms. William Happer, Yuan-Yu Jau, and Thad Walker Copyright © 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-40707-1
2
1 Introduction
Liouville space has long been used, either explicitly or implicitly, to analyze physics like that of optical pumping. A good general introduction to Liouville space from a viewpoint similar to ours can be found in the book Principles of Magnetic Resonance in One and Two Dimensions, by Ernst et al. [6]. Another good introduction with more emphasis on lasers is The Liouville Space Formalism in Atomic Spectroscopy, by Schuller [7]. The focus of our book on aspects of Liouville space that are particularly well adapted to computer programming appears to be unique. To make the discussion of the book less abstract, we have illustrated key points with sections of MATLAB code, which can be assembled to generate fairly sophisticated programs. We have tried to write the codes for brevity and maximum clarity, not for maximum speed. Many sections of code will run much faster with minor modifications. Analogous code can be written for other scientific-computing software, for example, Mathematica. Some familiarity with scientific-computing software would be helpful to a reader interested in writing specialized programs based on these illustrative codes. The codes can be downloaded from http://minds. wisconsin.edu/handle/1793/35675 Wherever possible, we have tried to use traditional symbols for familiar physical quantities. Regrettably, this means that we use the same symbols to represent quite different physical quantities, for example, E for the energy or for the amplitude of the electric field, for electric charge density or for the density matrix of quantum mechanics, and S for the electric spin quantum number or for the Poynting vector (electromagnetic energy flux). Wherever the context is sufficient to make the meaning of the symbol clear, we have avoided introducing new fonts, superscripts, or subscripts to resolve the ambiguity. An Example Some of the most important ideas of Liouville space can be illustrated by the familiar example of the hypothetical spin-1/2 atom shown in Figure 1.1. The probability of finding the atom in sublevel jαi with azimuthal quantum number m S D α D C1/2 is α α D 1/2 C hS z i and the probability of finding the atom in sublevel jβi with m S D β D 1/2 is β β D 1/2 hS z i. Here hS z i is the expectation value of the longitudinal spin of the atom. The off-diagonal elements (coherences) of the density matrix, α β D hS x iSy i and β α D hS x C iSy i, describe transverse components of the spin. For the conditions outlined in the caption of Figure 1.1, the elements of the density matrix change at the rates
d Γc ω α α D 2 f Γp β β ( α α β β ) β α C α β , dt 2 2 d ω β α D (Γp C Γc ) β α C α α β β , dt 2 d ω α α β β , α β D (Γp C Γc ) α β C dt 2 d Γc ω β β D 2 f Γp β β C ( α α β β ) C β α C α β . dt 2 2
(1.1)
1 Introduction
Figure 1.1 Pumping of a hypothetical alkalimetal atom with no nuclear spin. The quantization (z) axis is defined by the propagation direction of a circularly polarized light beam that pumps atoms out of the ground-state sublevel with m S D β D 1/2 at a rate 2Γp . The light is weak enough for stimulated emission of excited atoms to be neglected. Owing to spontaneous radiative emission and quenching collisions, a fraction f
of excited atoms decays to the sublevel with m S D α D C1/2, and the remaining fraction returns to the initial sublevel. For pure spontaneous emission we would have f D 1/3. Spin-changing collisions damp the groundstate spin polarization at the rate Γc . An externally applied magnetic field causes the ground-state atoms to rotate about the y axis at rate ω.
We can group the elements μ ν as a 2 2 matrix, , in Schrödinger space D
α α β α
α β β β
D
X
jμihνj μ ν ,
(1.2)
μν
or as a 4 1 column vector, j), in Liouville space 2
3 α α 6 β α 7 X 7 j) D 6 jμ ν) μ ν . 4 α β 5 D μν β β
(1.3)
The column vector j) of (1.3) is formed from the matrix of (1.2) by placing each column of below the one to its left. Bases
The basis matrices jμihνj of the expansion (1.2) are jαihαj D jαihβj D
1 0
0 0
0 0
1 0
,
jβihαj D
,
jβihβj D
0 1
0 0
0 0
0 1
, ,
(1.4)
3
4
1 Introduction
and the basis vectors of the expansion (1.3) are 3 1 6 0 7 7 jα α) D 6 4 0 5, 0 3 2 0 6 0 7 7 jα β) D 6 4 1 5, 2
3 0 6 1 7 7 jβ α) D 6 4 0 5, 0 3 2 0 6 0 7 7 jβ β) D 6 4 0 5.
0
2
(1.5)
1
In accordance with the ordering convention of (1.2) and (1.3) we can denote the transformation of a basis matrix to a basis vector by jμ ν) D jjμihνj) .
(1.6)
Expansion Amplitudes The expansion amplitudes of (1.2) can be written as h i μ ν D Tr (jμi hνj)† D hμj jνi . (1.7)
Similarly, the expansion amplitudes of (1.3) can be written as μ ν D (μ νj) ,
(1.8)
where the row vectors, (μ νj D jμ ν)† , are (α αj D [1
0 0 0] ,
(β αj D [0
1 0 0] ,
(α βj D [0
0 1 0] ,
(β βj D [0
0 0 1] .
(1.9)
In MATLAB, the coding statement to generate the column vector crho, representing j) of (1.3), from rho, representing of (1.2), is Code 1.1 crho = rho(:)
The inverse coding statement, to convert the column vector crho back to a 2 2 matrix, is Code 1.2 rho = reshape(crho,2,2)
1 Introduction
We can write (1.1) as the matrix equation
The Damping Matrix
d j) D G j) . dt
(1.10)
The damping matrix is G D Γp A p C Γc A c C iωSy© 2 Γc ω ω 16 ω 2Γ C 2Γ 0 p c D 6 4 ω 0 2Γ C 2Γc 2 p Γc ω ω
3 4 f Γp Γc 7 ω 7. 5 ω 4 f Γp C Γc
(1.11)
The symbols A p , A c , and Sy© represent “superoperators” for Liouville space. The superoperators that account for optical pumping and damping are 0 1 0 0
0 0 1 0
3 2 f 0 7 7 0 5 2f
1 16 0 6 Ac D 4 0 2 1
0 2 0 0
0 0 2 0
2
0 6 0 Ap D 6 4 0 0
(1.12)
and 2
3 1 0 7 7. 0 5 1
(1.13)
The y component of the Liouville-space spin operator, a “commutator superoperator” that we will discuss in more detail below, is 2
Sy©
0 1 6 1 D 6 2i 4 1 0
1 0 0 1
1 0 0 1
3 0 1 7 7. 1 5 0
(1.14)
In the body of this book, we will discuss systematic ways to generate superoperators such as A p , A c , and Sy© . Alternative Basis Instead of using the simple basis matrices of (1.4), we will often find it convenient to use other linearly independent (but not necessarily orthonormal) bases. For example, we can use unit matrix 1fSg and the spin operators,
1fSg D Sy D
1 2i
1 0
0 1
0 1
Sx D
, 1 0
,
1 2
Sz D
1 2
0 1
1 0 1 0
, 0 1
.
(1.15)
5
6
1 Introduction
In analogy to (1.2) and (1.3), the column-vector equivalents of the matrices in (1.15) are 3 3 2 2 0 1 7 6 0 7 16 7 6 1 7 j1fSg ) D 6 4 0 5 , jS x ) D 2 4 1 5 , 0 1 3 3 2 2 0 1 7 6 1 6 1 7 7 , jS z ) D 1 6 0 7 . jSy ) D 6 (1.16) 5 4 4 1 0 5 2i 2 0 1 The Hermitian conjugate row vectors corresponding to (1.16) are (1fSg j D j1fSg )† D [1 0 0 1] , 1 (S x j D jS x )† D [0 1 1 0] , 2 1 † (Sy j D jSy ) D [0 1 1 0] , 2i 1 † (S z j D jS z ) D [1 0 0 1] . 2
(1.17)
We see that the total probability of finding the atom in some ground-state sublevel is Tr[1fSg ] D α α C β β D (1fSg j).
(1.18)
In (1.18) we have assumed that the pumping rate Γp is so small compared with the spontaneous decay rate and collisional quenching rates of the excited atoms that the probability of finding atoms in the excited state is negligibly small, and we can write (1fSg j) D 1 .
(1.19)
The expectation value of the longitudinal spin S z of the atom is hS z i D Tr [S z ] D
1 ( α α β β ) D (S z j). 2
(1.20)
In like manner, we find hS x i D (S x j) and hSy i D (Sy j). The expectation value of an arbitrary (not necessarily Hermitian) operator X is h X i D ( X † j). Constraints
(1.21)
From inspection of (1.11)–(1.14) we see that
(1fSg jA p D 0 ,
(1fSg jA c D 0 ,
(1fSg jSy© D 0 ,
and
(1fSg jG D 0 .
(1.22)
1 Introduction
In view of (1.22) and (1.11), we can multiply (1.10) by (1fSg j to find d d fSg Tr[] D (1 j) D (1fSg jG j) D 0 . dt dt
(1.23)
We see that the constraints (1.22) ensure that the number of atoms neither increases nor decreases as a result of the various evolution mechanisms, pumping, collisional relaxation, or Larmor precession. The evolution described by (1.10) must also keep the density matrix Hermitian. This implies an additional constraint on G that we will discuss in more detail in the body of this book, but which we will outline here. Define a transposition matrix, T, a superoperator that transforms the basis states (1.5) of Liouville space as T jμ ν) D jνμ) .
(1.24)
For the example discussed here we have 3 2 1 0 0 0 6 0 0 1 0 7 7 T D6 4 0 1 0 0 5. 0 0 0 1
(1.25)
We define the Liouville conjugate, G ‡ , of a superoperator G by G‡ D T GT .
(1.26)
Here G is the complex conjugate of G. As we will discuss in more detail below, in order that the evolution equation (1.10) keep Hermitian we must have G D G‡ .
(1.27)
The constraints (1.22) and (1.27) on G, and additional constraints that we will mention later, are analogous to the constraint H D H † for the Hamiltonian H of Schrödinger space, which ensures that the wave function jψi changes in a way that keeps hψjψi D 1. Eigenvectors Except for unusual situations of “critical damping”, the damping matrix G of (1.11) will have four linearly independent right (column) eigenvectors, jγ j ), and a corresponding set of left (row) eigenvectors, which we denote by ((γ j j. These are defined, aside from normalization factors, by
G jγ j ) D γ j jγ j )
and ((γ j jG D ((γ j jγ j .
(1.28)
As long as the eigenvectors are linearly independent, we can use them as a basis for Liouville space and we can normalize them such that ((γ j jγ k ) D δ j k . Then we can write the eigenvector expansion of G as X γ j jγ j )((γ j j . GD j
(1.29)
(1.30)
7
8
1 Introduction
Transients
The formal solution of (1.10) is
j t ) D eG t j0 ) .
(1.31)
Here j0 ) is the value of j) at time t D 0, and j t ) is the value at time t 0. If the atoms are unpolarized at time t D 0, we would have 0 D
1 fSg , 1 2
or
j0 ) D
1 fSg j1 ) . 2
(1.32)
A MATLAB code that carries out the matrix exponentiation of (1.31) to evaluate the transient response of the model atom is Code 1.3 Gmp=input(’Gmp=’);%optical pumping rate Gmc=input(’Gmc=’);%collisional relaxation rate f=input(’f=’);%fractional transfer rate w=input(’w=’);%rotation rate about y axis PS=[1 0; 0 1];%unit operator for Schroedeinger space cPS=PS(:); rPS=cPS’;%column, row vectors from PS Ap=[0 0 0 -2*f;0 1 0 0;0 0 1 0; 0 0 0 2*f];%pumping operator Ac=[1 0 0 -1;0 2 0 0;0 0 2 0;-1 0 0 1]/2;%collision operator %Spin operators in Schroedinger space Sx=[0 1;1 0]/2; Sy=[0 1;-1 0]/(2*i); Sz=[1 0; 0 -1]/2; cSx=Sx(:);cSz=Sz(:); rSx=cSx’;rSz=cSz’;%column, row vectors from Sj SyC=[0 1 1 0;-1 0 0 1;-1 0 0 1;0 -1 -1 0]/(2*i);%spin superoperator G=Gmp*Ap+Gmc*Ac+i*w*SyC;%damping superoperator nt=100; t=linspace(0,5,nt);%sample times rho=zeros(4,nt); for k=1:nt rho(:,k)=expm(-G*t(k))*cPS/2; end clf; plot(t,real(rSz*rho),’b-’); hold on; grid on; plot(t,real(rSx*rho),’r:’); xlabel(’time t’); legend(’\langle S_z \rangle’, ’\langle S_x \rangle’);
Figure 1.2 shows a representative transient calculated with Code 1.3. Eigenvector Expansions For situations where G can be written as an eigenvector expansion, we can substitute (1.30) into (1.31) to find
j t ) D
X
jγ j )((γ j j0 )eγ j t .
(1.33)
j
Here ((γ j j0 ) is the “weight” of the jth mode at time t D 0. The solution is the sum of decaying exponentials with damping rates γ j that may be complex. For critical damping, when the particular value of some parameter, such as the damping rate or the magnetic field, causes two eigenvalues of G to converge to a common value, γc ,
1 Introduction
– Figure 1.2 Transient calculated with Code 1.3 on page 8 for the parameters Γp D 1, Γc D 0.1, f D 0.4, and ω D 5. The horizontal lines are the steady-state spin polarizations calculated with Code 1.5 on page 10.
and when the corresponding right eigenvectors also converge to the same vector, jγc ), there are not enough independent eigenvectors of G to span Liouville space, and the eigenvector expansion (1.30) of G does not exist. This makes no difference if transients are calculated with (1.31), but the expression (1.33) must be revised to include not only a term with the simple exponential time dependence, eγc t , but also a term with the “critically damped time dependence”, teγc t . We will discuss critical damping in more detail below. Steady State For time evolution governed by (1.10), the steady-state solution, j1 ), is defined by
G j1 ) D 0 ,
(1.34)
(1fSg j1 ) D 1 .
(1.35)
with
In linear algebra, (1.34) is said to define the null space, j1 ), of G. With the exception of unusual, multistable conditions, there is a unique steady state for optical pumping problems, so the null space is one-dimensional. Modern mathematical software finds null spaces very efficiently, and when the null space is known to be one-dimensional, even faster algorithms are available. For example, the following MATLAB code nullfast() can find the one-dimensional null space more than 10 times faster than the conventional MATLAB built-in routine null().
9
10
1 Introduction
Code 1.4 function [NV invM]= nullfast(M) K=max(max(abs(M)))*1e-15; warning off;%search the largest matrix element invM=K*inv(M+K*eye(length(M))); [i j]=max(sum(invM));%find the dominant matrix column NV=invM(:,j);%pick up the matrix column as the null vector NV=NV/sqrt((NV’*NV));%normalize the null vector
If the following MATLAB statements are appended to Code 1.3 on page 8 the resulting program will evaluate and plot the steady-state spin polarizations. Code 1.5 rhoin = null(G); rhoin = rhoin/(rPS*rhoin); plot(t,real(rSz*rhoin)*ones(1,nt),’b-’); plot(t,real(rSx*rhoin)*ones(1,nt),’r-.’);
Analytic Formulas for Eigenvectors From inspection of (1.34) and (1.28) we see that we can interpret the steady-state solution j1 ) as the right eigenvector, jγ1 ) D j1 ), of G with zero eigenvalue, γ1 D 0. From inspection of (1.22) we see that the corresponding left eigenvector is ((γ1 j D (1fSg j. Solving (1.34) with the matrix G from (1.11), we find that the steady-state mode has the properties
3 1 C 2b 7 1 ˇˇ 1 6 2a 7 , jγ1 ) D ˇ1fSg C 4aS x C 4b S z D 6 5 4 2a 2 2 1 2b 2
γ1 D 0 ,
((γ1 j D (1fSg j D [1 0 0 1] .
(1.36)
The coefficients a and b are a D hS x i D
f Γp ω , (2 f Γp C Γc )(Γp C Γc ) C ω 2
b D hS z i D
f Γp (Γp C Γc ) . (2 f Γp C Γc )(Γp C Γc ) C ω 2
(1.37)
The steady-state solution jγ1 ) represents an ensemble of atoms with a spin polarization in the z x plane, perpendicular to the axis of rotation (the y axis) of the magnetic field. Field-Aligned Mode By symmetry it is clear that another mode (eigenvector) must represent spins polarized parallel to the magnetic field. From inspection of (1.11)
1 Introduction
we see that 3 0 1 6 1 7 7 , jγ2 ) D jSy ) D 6 2i 4 1 5 0 2
γ2 D Γp C Γc ,
((γ2 j D 2(Sy j D [0
i
i 0] .
(1.38)
Critical-Damping Modes Like the steady-state mode (1.36), the remaining two modes, jγ ) D jγ3 ) and jγC ) D jγ4 ), represent ensembles of atoms with spin polarization in the z x plane, perpendicular to the axis of rotation (the y axis) of the magnetic field. They can be written as
2
3 d˙ 1 6 c˙ 7 7. jγ˙ ) D c ˙ jS x ) C d˙ jS z ) D 6 2 4 c˙ 5 d˙
(1.39)
Using (1.39) with (1.28), we find that the eigenvalues are q γ˙ D γc ˙ ω 2c ω 2 .
(1.40)
We can write the critical damping frequency and the critical damping rate γc as ω c D (1/2 f )Γp
and
γc D (1/2 C f )Γp C Γc ,
and with the relative value of the coefficients given by p c˙ ω c ˙ ω 2c ω 2 D . d˙ ω
(1.41)
(1.42)
The eigenvalues γ˙ are real and distinct for ω 2 < ω 2c , and they are complexconjugate pairs if ω 2 > ω 2c . At the critical damping frequency, when ω 2 D ω 2c , the rates γC and γ become identical, and from (1.42) we see that there is a single eigenvector for the degenerate damping rate γc . Compactification If the transverse magnetic field is zero (if ω D 0), the system has symmetry about the z axis and (1.11) reduces to
G D Γp A p C Γc A c 2 Γc 0 16 0 2Γ C 2Γc p D 6 0 24 0 Γc 0
0 0 2Γp C 2Γc 0
3 4 f Γp Γc 7 0 7. 5 0 4 f Γp C Γc
(1.43)
From inspection of (1.10) and (1.43) we see that for zero transverse field, the population α α couples only to the other population β β and vice versa. The coherences α β and β α couple only to themselves. In such cases it is useful to partition the
11
12
1 Introduction
Liouville space of the density matrix into “compactified” subspaces, that is, we drop elements of the density matrix that are guaranteed to have a negligible effect on other elements for some fundamental reason. Compactification can be done conveniently with logical variables. In the case described above, we will introduce the logical variable for populations 3 2 1 6 0 7 1 0 7 (1.44) LD and jL) D 6 4 0 5. 0 1 1 The definitions (1.15) for 1fSg and (1.16) for j1fSg ) appear to be identical to the definition (1.44) for L and jL), but in (1.15) and (1.16) the 1’s and 0’s represent real numbers, but in (1.44) 1 denotes the logical (Boolean) value true and 0 denotes the logical value false. A simple MATLAB statement to create the logical column vector L, representing jL), from the nonlogical column vector cPS, representing j1fSg ) and evaluated by Code 1.3 on page 8, is Code 1.6 L=cPS>0;
In this example, populations are true and coherences are false. Slightly more complicated statements are needed to define a logical variable L that is true for coherences that are generated by magnetic resonance or modulated light, but is false for off-resonant coherences that are sure to be negligibly small. Logical variables are a powerful tool to increase the efficiency of computer calculations. For example, if crho is the computer variable denoting the column vector j), then crhoc=crho(L) is the shorter column vector representing the compactified density matrix of populations, for example,
if
0.8 0.1 crho D , 0.1 0.2
then
crho(L) D
0.8 . 0.2
(1.45)
Logical variables can also be used to compactify the evolution matrix (1.43). For example, the MATLAB matrix G can be compactified to the matrix Gc with the statement, Gc=G(L,L), for example,
if
0.05 0 GD 0 -0.05
G(L,L) D
0.05 -0.05
0 1.1 0 0
0 0 1.1 0
-0.85 . 0.85
-0.85 0 , 0 0.85
then
(1.46)
1 Introduction
As we shall show in subsequent examples, it is not necessary to define new compactified arrays like crhoc and cG above. One can simply enter statements like chro(L) and G(L,L) in lines of code where they are automatically interpreted as compactified quantities. Computer calculations are faster with the smaller, compactified vectors and matrices than with the original, often much larger ones. If the logical indices are chosen with good physical insight, there will be negligible loss of accuracy owing to the neglected terms. Summary The hypothetical, spin-1/2 alkali-metal atom illustrates themes that we will discuss in more detail for the more complicated situations of real atoms. Some important points are:
As illustrated in (1.3), the Liouville-space density matrix is conveniently represented as a column vector, j), formed by placing each successive column of the Schrödinger-space density matrix, , below the one to its left. In an analogous way, other Schrödinger-space operators like those of (1.16) can be transformed to column vectors. As indicated in (1.21), the expectation value h X i of a Schrödinger operator X is h X i D ( X †j). The time evolution of the density matrix j) is given by the first-order, matrix differential equation (1.10), j) P D G j). In simple cases – like the one we have discussed in this introduction – the evolution operator G of (1.10) is independent of time, t, or of . For spin exchange between identical alkali-metal atoms, the damping operator G depends on and (1.10) is nonlinear. When the atom is subjected to modulated magnetic fields or modulated light, G will be time-dependent. In these cases, it is often possible to define a “rotating coordinate system” where the evolution of the rotating-frame density matrix is given by an equation like (1.10), but with a time-independent G. As illustrated in (1.11) the damping operator G is a sum of terms representing various evolution mechanisms, for example, pumping, collisional relaxation, or spin interactions with external fields. Each process has a characteristic rate, such as the mean optical pumping rate Γp , the collisional rate Γc , or the rotation rate ω. The rate multiplies a dimensionless superoperator such as A p , A c , or Sy© . The evolution operator G plays much the same role as the Hamiltonian operator H (or more precisely iH/„) in Schrödinger space. There is no need for the operator G to be Hermitian or anti-Hermitian, and some of its eigenvalues may occur as complex-conjugate pairs. Unlike in Schrödinger space, for a given eigenvector γ j of the evolution operator, G, the left eigenvectors ((γ j j can differ from the Hermitian conjugate, (γ j j D jγ j )† , of right eigenvectors jγ j ). In rare cases of critical damping, the eigenvectors, jγ j ), no longer form a complete basis for Liouville space, in contrast to Schrödinger space, where the Hermitian Hamiltonian, H, always has eigenvectors that form a complete basis. For relaxation and pumping processes that conserve atoms and keep the density matrix Hermitian, the damping operator G must satisfy the constraints
13
14
1 Introduction
(1fSg jG D 0 and G D G ‡ . These, and analogous identities that we will discuss later, are useful for validating computer-generated superoperators, which can have very large dimensions for real atoms, and which are not easy to check for coding errors. As illustrated by (1.45) and (1.46), one can omit elements of the density matrix that have a negligibly small effect on the physics for some fundamental reason. This compactification process is conveniently done with logical variables, and it can substantially speed up computer calculations with negligible loss of accuracy. For example, the Liouville space of ground-state 133 Cs atoms has 256 dimensions. If the experimental conditions are such that the coherence amplitudes are sure to be negligible, compactifying the full space to only the populations reduces the size of the Liouville space to only 16 dimensions. Adding in the two coherences for a “0–0” atomic clock transition still leaves a relatively small space of 18 dimensions. For very simple systems like the two-level atom discussed in this introduction, an analysis in Liouville space has no particular advantage over other methods. However, the systematic use of Liouville space is substantially more efficient than use of traditional methods for real atoms with many sublevels. This is partly because Liouville methods are better adapted to computer implementation of linear algebra, and partly because the computer codes are simpler to write and therefore less prone to errors. Coding statements often look almost the same as formal statements; for example, compare the line of code in Code 1.3 on page 8 that represents (1.31).
Units For the computer programs outlined in this book we will use cgs units, but we will use other units to enter data, for example megahertz for frequency or amagat for number density. A typical MATLAB program for real atoms will open with statements that set the values of fundamental physical constants, for example,
Code 1.7 clear; kB=1.380e-16;%Boltzmann’s constant in erg/K NA=6.022e23;%Avogadro’s number c=2.9979e10;%speed of light in cm/s muB=9.2741e-21;%Bohr magneton in erg/G muN=5.051e-24;%nuclear magneton in erg/G hP=6.6262e-27;%Planck’s constant in erg/Hz hbar=hP/(2*pi);%in erg s re=2.816e-13;%classical electron radius in cm amg=2.6868e19;%Loschmidt constant (amagat) in cm^f-3g
15
2 Alkali-Metal Atoms The analysis outlined in this book is applicable to any optically pumped atom or ion, but we will most often use the alkali-metal atoms to illustrate key issues. The alkali-metal atoms are convenient choices for studies and applications of optical pumping because: Their resonance wavelengths in the visible and near-infrared are compatible with inexpensive and powerful laser sources. Their relatively high volatility allows dense vapors and atomic beams to be produced without severe difficulties. Their ground states have no orbital angular momentum, making them relatively immune to depolarizing collisions with other S-state atoms and with specially coated glass surfaces. Their effective one-electron electronic structure facilitates quantitative comparison between theory and experiment. In this chapter we summarize the properties of the alkali-metal atoms that are most relevant for optical pumping experiments. Similar considerations apply to other atoms or ions.
2.1 Electronic Energies
The energy-level structure of a typical alkali-metal atom (Rb) is shown in Figure 2.1. The ground-state configuration is [X]ns, where [X] is a noble-gas-like core, and the ground-state term is designated by 2 S1/2 . The binding energies vary from about 3.5 eV for Cs to 5.1 eV for Na. The lowest excited states have the configuration [X]np1 and the two terms that result from the spin–orbit interaction are designated by 2 P J , with J D 1/2 for the lower term and J D 3/2 for the upper term. The fine-structure splitting between the 2 P3/2 and 2 P1/2 terms ranges from 0.337 cm1 for Li to 1687 cm1 for Fr, a factor of about 5000. The resonance wavelengths for the strongest resonance lines are all in the visible or near-infrared portion of the
Optically Pumped Atoms. William Happer, Yuan-Yu Jau, and Thad Walker Copyright © 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-40707-1
16
2 Alkali-Metal Atoms I+3/2 I+1/2 I+1/2 I-3/2
P3/2
n2P
P1/2
I+1/2 I-1/2
F=I+1/2
2
n S1/2 F=I-1/2
Figure 2.1 Atomic structure of an alkali-metal atom with nuclear spin I D 3/2.
spectrum. The primary parameters needed to describe the electronic structure of the alkali-metal atoms are summarized in Table 2.1. The electronic energies of an alkali-metal atom where only the valence electron is excited are well described by the quantum defect formula (in atomic units or hartrees): En l D
1 . 2n 2
(2.1)
Here n D 1, 2, . . . corresponds to the principal quantum number of an electron in a hydrogen atom, and l D 0, 1, 2 is the orbital angular momentum quantum number. The effective principal quantum number, n D n μ n l ,
(2.2)
is smaller than n by the quantum defect μ n l . For a given l, the energy-dependence of the quantum defect is so slight that it can be can be well parameterized by the first two terms of a power series in the energy E, μ n l D μ 1l C E
d μl . dE
(2.3)
This simple hydrogen-like description of the alkali-metal atoms works because the weakly bound valence electron spends most of its time outside the closed-shell ion core of the atom. When the valence electron does penetrate the ion core, its kinetic energy is much larger than its binding energy, so the wave function inside the core only weakly depends on the binding energy. The quantum-defect parameters listed in Table 2.1 come from linear fits to the energies of the low-lying states and differ somewhat from formulas derived from high-n Rydberg studies.
16.28 [9] 0.320 16.23 [9] 0.641 1.348 0.132E 0.854 0.258E 0.015 C 0.081E
6709.77
6709.62 0.34
27.10 [8]
0.249 27.10 [8]
0.497 0.399 0.061E
0.047 C 0.050E
0.002 C 0.009E
λ(S-P1/2 ) (Å)
λ(S-P3/2 ) (Å) ΔEFS (cm1 )
τ(nP1/2 ) (ns)
f 1/2 τ(nP3/2 ) (ns)
f 3/2 μ ns
μnp
μ nd
5891.58 17.20
5897.56
[Ne]3s 5.1391
[He]2s 5.3917
11
3
Configuration IP (eV)
Na
Z
Li
0.278 C 2.137E
1.711 0.542E
0.669 2.180 0.314E
0.333 26.34 [10]
26.69 [10]
7667.01 57.71
7701.08
[Ar]4s 4.3407
19
K
1.353 C 1.760E
2.645 0.681E
0.695 3.130 0.419E
0.341 26.25 [11]
27.75 [11]
7802.41 237.60
7949.78
[Kr]5s 4.1771
37
Rb
2.476 C 0.372E
3.568 0.889E
0.715 4.048 0.573E
0.344 30.462 [12]
34.88 [12]
8523.47 554.04
8945.93
[Xe]6s 3.8939
55
Cs
3.41 0.095E
0.734 5.07 0.556E
0.339 21.0 [14]
29.45 [13]
7181.85 1687
8171.66
[Rn]7s 4.073
87
Fr
Table 2.1 Basic parameters governing the electronic and fine-structure states of the alkali-metal atoms. IP is the ns ionization potential, ΔEFS D nP3/2 nP1/2 is the fine-structure splitting, τ is the radiative lifetime, f is the oscillator strength, μ nl is the quantum defect for low-lying states nl, which depends on the binding energy E nl in hartrees. Many of these numbers are known to greater accuracy than given here.
2.1 Electronic Energies 17
18
2 Alkali-Metal Atoms
The lifetimes and oscillator strengths of the resonance transitions of the alkalimetal atoms have been extensively studied and are summarized in Table 2.1. The most accurate values for these properties have been obtained from precision molecular spectroscopy of alkali-metal dimers. Note that the oscillator strengths add up to a value slightly greater than 1 for the heaver alkali metals, presumably owing to the presence of small amounts of core mixing.
2.2 Valence-Electron Wave Functions
For most applications of optical pumping, the basic optical properties of the transitions involved are well known from experiment, in terms of oscillator strengths and/or radiative lifetimes and branching ratios. When such information is not available, the simple one-electron structure of the alkali-metal atoms makes obtaining relatively accurate valence-electron wave functions fairly straightforward. Since the energy required to excite a core electron is 15 eV, in contrast to the less than 5 eV binding energy of the valence electron, it is often an excellent approximation to assume that the valence electron moves in a static, spherically symmetric potential Vc (r) due to the core. Then the valence electron wave function can be written ψ n l (r) D
P n l (r) Yl m (θ , φ) . r
(2.4)
For details on the quantum theory of angular momentum, for example, the spherical harmonics, Yl m (θ , φ) of (2.4) (see Chapter 5 in [15]), we will refer to the very comprehensive work of Varshalovich et al., Quantum Theory of Angular Momentum [15], which is unmatched as a reference for those already familiar with the basics of angular momentum. For those who would like to become familiar, we recommend the classic book by Rose, Elementary Theory of Angular Momentum [16]. The radial wave functions P n l (r) give the dependence on the radial displacement r. The charge distribution of the core is well localized inside a radius rc , which is a few Bohr radii for the alkali-metal atoms. Thus, for r > rc , the potential experienced by a valence electron is very nearly the Coulomb potential, Vc (r) e 2 /r. In this Coulomb approximation [17], the bound-state solutions to the Schrödinger equation are determined from the experimental binding energies. The Coulombapproximation wave functions are Whittaker functions [18]: 2r , (2.5) P n l (r) D N n l Wn ,lC1/2 n with r in atomic units. The properties of Whittaker functions are summarized in Chapter 13 in [19]. The normalizing factor N n l is [18, 20] Nn l D p
1 n 2 ζ(n )Γ (n
C l C 1)Γ (n l)
,
(2.6)
2.2 Valence-Electron Wave Functions
where ζ(n ) D 1 C
dμ 1 dμ D 1 C 3 . d n n dE
(2.7)
The Whittaker and gamma functions are easily evaluated using MATLAB. Alternatively, a simple recurrence formula can also be used to evaluate the wave functions [17, 21]. For most radiative properties of alkali-metal atoms, matrix elements calculated with Coulomb-approximation wave functions generally reproduce experimental results to an accuracy of a few percent or better. The Coulomb-approximation wave functions are easy to evaluate but do have limitations – they are not valid for r < rc (e.g., they do not have the correct number of nodes), and for r > rc they do not account for the core polarization. The latter effect can be mitigated using less-convenient Mathieu functions (see Section 6.9 in [20] for more information). An alternative approach is to numerically integrate the Schrödinger equation with a model potential that is designed to accurately reproduce the experimental energies and accounts in an approximate manner for the core polarization. Such model potentials are given by Marinescu et al. [22], in the form z l (r) αc 6 Vl (r) D (2.8) 4 1 e(rrc) , r 2r where α c is the static dipole polarizability of the core and the shielded charge is z l (r) D 1 C (Z 1)ea 1 r r(a 3 C a 4 r)ea 2 r ,
(2.9)
Figure 2.2 Rb radial wave functions as calculated using the Coulomb approximation and model potentials.
19
20
2 Alkali-Metal Atoms
Figure 2.3 The spatial distribution of m J D 1/2 Rb levels (left to right 5s 1/2 , 5p 1/2 , 5p 3/2 ). The p quantity plotted is x 2 C z 2 jψ(x, 0, z)j2 .
where Z is the charge on the nucleus, and the parameters rc and a n are adjusted to agree with experimental energies. The numerical values of the various parameters are found in [22]. A comparison of Rb 5s and 5p wave functions generated by the Coulomb approximation and the model potential is shown in Figure 2.2. The wave functions agree well in the large-r region that dominates the radiative and collisional properties of the alkali-metal atoms. Plots of the spatial distribution of the 5s 1/2 , 5p 1/2 , and 5p 3/2 states of Rb are shown in Figure 2.3.
2.3 Hyperfine Structure
The naturally occurring isotopes of the alkali-metal atoms have nonzero nuclear spins, so all the states of the alkali-metal atoms are further split by hyperfine interactions. These splittings have been measured with tremendous precision using atomic beam magnetic resonance or optical pumping methods. Indeed, the hyperfine splitting of the 2 S1/2 state of the 133 Cs atom serves as the international frequency standard [23, 24]. We write the ground-state spin Hamiltonian of the atom as H fgg D Afgg I S μ fgg B ,
(2.10)
where Afgg is the coupling coefficient for the magnetic dipole hyperfine interaction of the nuclear spin I with the electron spin S of the atom. Also included in (2.10) is the interaction of the total magnetic dipole moment of the atom, μ fgg D g S μ B S C
μI I, I
(2.11)
with an externally applied magnetic field B. The Landé factor is g S D 2.0023, the Bohr magneton is μ B D 9.274 1021 erg G1 , the nuclear magnetic moment is μ I , and the nuclear spin quantum number is I.
2.3 Hyperfine Structure
In analogy with (2.10), we write the spin Hamiltonian of the 2 P J excited state as H feg D Afeg I J C B feg
3(I J)2 C (3/2)I J I(I C 1) J( J C 1) μ feg B . 2I(2I 1) J(2 J 1) (2.12)
In (2.12) the electronic angular momentum operator J D L C S of the excited atom is the sum of the contribution from the electron orbital angular momentum L and the electron spin S. The magnetic moment operator of the excited atom has a form analogous to (2.11), μI μ feg D g J μ B J C I, (2.13) I The Landé factor is very nearly g J D [ J]/3, with J D 1/2 or J D 3/2. Here and subsequently we denote the statistical weight (or degeneracy) of an angular momentum quantum number by [ J] D 2 J C 1 .
(2.14)
2
For atoms in the P3/2 excited state, the electrons produce an electric field gradient at the center of the atom. Nuclei with spin quantum numbers I > 1/2 have an electric quadrupole moment that can interact with this field gradient, and this interaction is represented in (2.12) by the term proportional to the quadrupole coupling coefficient B feg . The spherical symmetry of the electron charge density of the 2 P1/2 state guarantees that there is no electric-field gradient at the center of the atom, and that there is no quadrupole-interaction term in the spin Hamiltonian (2.12). Hyperfine interactions of higher multipolarity, magnetic octupole, electric hexadecapole, and so on, even when the angular momentum quantum numbers, I, J, or S, are large enough to allow them to be nonzero, are almost always negligibly small in alkali-metal or other atoms. The part of the static Hamiltonian that includes both the ground state and the excited state is H0 D „ω fe gg Q C H feg C H fgg .
(2.15)
Here „ω fe gg is the difference between the “center-of gravity” energies of the excited-state sublevels and the ground-state sublevels. We have introduced the “optical spin operator”, 1 feg (2.16) (1 1fgg ), 2 which plays much the same role in the analysis of optical transitions between the multilevel ground states and excited states of a real atom as the longitudinal electron spin S z does for a spin-1/2 atom. Schuller [7] uses the operator M D ωQ in much the same way that we will use Q. The excited-state and ground-state projection operators, 1feg and 1fgg , in terms of which Q is defined are X X 1feg D j μih N μj N and 1fgg D jμihμj . (2.17) QD
μN
μ
Here the sums extends over the eigenstates j μi N of H feg or jμi of H fgg .
21
3.25643 803.504087
46.01 [26]
2.96 [32] 0 [32]
0.82205 228.205259
17.39 [26]
1.12 [32] 0 [32, 33]
μ I (μ N ) [25] δν (nS1/2 )
A (nP1/2 )
A (nP3/2 ) B (nP3/2 )
92.5 3/2
7.5 1
7 Li
FI (%) I
6 Li
18.53 [34] 2.72 [34]
94.44 [27]
2.21752 1771.626128
100 3/2
23 Na
6.09 [28] 2.79 [28]
27.78 [28]
0.39147 461.719720
93.26 3/2
39 K
3.36 [28] 3.35 [28]
15.24 [28]
0.21487 254.013870
6.73 3/2
41 K
24.99 [35] 25.69 [35]
120.65 [29]
1.35335 3035.73244
72.17 5/2
85 Rb
84.72 [36] 12.50 [36]
406.12 [29]
2.75182 6834.682610
27.83 3/2
87 Rb
50.29 [37] 0.49 [37]
291.91 [30]
2.58203 9192.631770
100 7/2
133 Cs
78.0 [38] 51 [38]
946.3 [31]
4.40 46768.2
— 6
210 Fr
Table 2.2 Hyperfine structure parameters of the alkali-metal atoms. Frequencies are in megahertz. Most of these numbers are known to greater accuracy than given here. FI is the fractional abundance.
22
2 Alkali-Metal Atoms
2.3 Hyperfine Structure
Collisions will momentarily shift the values of the parameters in the Hamiltionian, for example, the values of Afgg and ω fe gg . In these cases it is often convenient to think of the mean values of Afgg , ω fe gg , and so on as “thermodynamic” quantities that depend slightly on the buffer-gas composition, pressure, and temperature. This leads to shifts in the observed resonance frequencies of the atoms. The fluctuations of the Hamiltonian parameters during collisions can be potent sources of spin relaxation or optical line broadening. Some more important parameters that describe the hyperfine structure of alkalimetal atoms are summarized in Table 2.2. In most cases the numerical values are known with much greater precision than given there. In a typical program, once the values of the basic physical constants have been set by statements like Code 1.7 on page 14, the parameters for the specific atom of interest are read in. For example, if appended to the program ending with Code 1.7 on page 14, the following MATLAB statements will set basic parameters, taken from Tables 2.1 and 2.2, for the 87 Rb atom. Code 2.1 I=input(’I = ’); S=input(’S = ’); J=input(’J = ’); %statistical weights gI=2*I+1; gS=2*S+1; gJ=2*J+1;gg=gI*gS; ge=gI*gJ; a=I+.5; b=I-.5;%ground-state ang. mom. quant. numbs. MW=86.9;%grams/mole LgS=2.00231;%Lande g-value of S1/2 state LgJ=gJ/3;%approximate Lande g-value of PJ state muI=2.751*muN;%nuclear moment in erg/G Ag=hP*3417.35e6;%S1/2 dipole coupling coefficient in erg if J==1.5 lamJ=7800e-8;%D2 wavelength in cm Ae=hP*84.852e6;%P3/2 dipole coefficient in erg Be=hP*12.611e6;%P3/2 quadrupole coupling coefficient in erg te=25.5e-9;%spontaneous P1/2 lifetime in s elseif J==0.5 lamJ=7947e-8;%D1 wavelength in cm Ae=hP*409e6;%P1/2 dipole coupling coefficient in erg te=28.5e-9;%spontaneous P1/2 lifetime in s end keg=2*pi/lamJ; weg=c*keg;%nominal spatial and temporal frequencies feg=c*gJ/(4*weg^2*re*te);%oscillator strength D=sqrt(gS*hbar*re*c^2*feg/(2*weg));%dipole moment in esu cm
For illustrative purposes, we have let the spin quantum numbers I, S, and J be arbitrary. For a real 87 Rb atom, we would have I D 3/2, S D 1/2, and J D 1/2 for D1 pumping or J D 3/2 for D2 pumping. To simplify subsequent discussions, we will sometimes illustrate important points for hypothetical Rb atoms with I ¤ 3/2 but with the same coupling coefficients, Afgg , ω fe gg , and so on, as 87 Rb.
23
25
3 Wave Functions and Schrödinger Space Optically pumped atoms can have many energy sublevels. A primary aim of this book is to show how modern mathematical software makes it possible to handle multilevel atoms in much the same way as two-level atoms. The spin state of the polarized atoms is determined by the populations of the sublevels and by the coherences between them. These can be conveniently summarized as elements of the density matrix, the subject of this chapter. We begin by reviewing the basis states most commonly used to represent the density matrix.
3.1 Uncoupled States
For the 2 S1/2 ground states of alkali-metal atoms, we denote the electron spin basis states by jm S i D jS m S i. For the 2 P J excited-state atoms the basis states will be denoted by jm J i D j J m J i. The nuclear spin basis states will be denoted by jm I i D jI m I i. We order these states such that jm S i comes before jm 0S i if m S > m 0S , with an analogous convention for ordering the states jm J i and jm I i. For computer calculations, we will represent the basis vectors as column vectors in their respective spin spaces, for example, 2 2 3 3 1 0 6 0 7 6 1 7 6 6 7 7 6 7 6 7 0 7 , or jm i D jI 1i D 6 0 7 . jm I i D jI i D 6 (3.1) I 6 . 7 6 . 7 6 . 7 6 . 7 4 . 5 4 . 5 0
0
3.1.1 Kronecker Products
We can expand the total wave function of a 2 S1/2 atom on the uncoupled basis states, jm S m I i D jm I i ˝ jm S i . Optically Pumped Atoms. William Happer, Yuan-Yu Jau, and Thad Walker Copyright © 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-40707-1
(3.2)
26
3 Wave Functions and Schrödinger Space
Recall that the Kronecker product of a matrix A of m rows and n columns with a matrix B is defined to be the block matrix 2 3 A 11 B A 1n B 6 7 .. .. .. A˝B D4 (3.3) 5. . . . A m1 B A m n B For example, for S D 1/2 and I D 1/2, and with jm I i D j1/2i D
1 0
,
jm S i D j 1/2i D
0 1
,
(3.4)
(3.2) becomes 3 0 6 1 7 7 jm S m I i D jm I i ˝ jm S i D 6 4 0 5. 0 2
(3.5)
The nth element from the top of the column vector representing jm S m I i has the value 1 and all the other elements have the value 0. In the example (3.5), n D 2. We can use the integers n D n(m S m I ) to order the basis vectors jm S m I i. We will think of the index m S as labeling a row and m I as labeling a column of the rectangular array, jm S m I i. If we rearrange the basis vectors of this array as a single column vector, with each column of the rectangular array placed below the one on its left, the ordering indices n(m S m I ) will be in the same order as the integers, 1, 2, 3, . . . . With this ordering convention we can define the elements of the Kronecker product as hβ αjA ˝ Bjβ 0 α 0 i D B β β 0 A α α 0 .
(3.6)
Consider the [S ] [S ] matrix X with elements hm S j X jm 0S i that represents an operator that couples only the electron sublevels jm S i and jm 0S i. Multiplying the uncoupled basis state (3.2) by X and using the mixed-product rule [39], (A ˝ B)(C ˝ D) D (AC ) ˝ (B D) ,
(3.7)
we find jm I i ˝ X jm S i D XJ jm S m I i ,
(3.8)
where the “grave” matrix is XJ D 1fI g ˝ X .
(3.9)
Here 1fI g denotes the [I ] [I ] unit matrix with elements hm I j1fI g jm 0I i D δ m I ,m 0I .
(3.10)
3.1 Uncoupled States
Similarly, multiplying an uncoupled basis state (3.2) by an [I ] [I ] matrix Y that represents an operator that only couples nuclear sublevels, we find Y jm I i ˝ jm S i D YK jm S m I i ,
(3.11)
where the “acute” matrix is YK D Y ˝ 1fSg .
(3.12)
Here 1fSg denotes the [S ] [S ] unit matrix with elements hm S j1fSg jm 0S i D δ m S ,m 0S .
(3.13)
When no confusion is likely, we will drop the acute (K) and grave (J) accent superscripts that we have used to distinguish operators in the electron spin or nuclear spin spaces from their larger-dimension representations in the uncoupled product space. For example, the factor I S in the ground-state Hamiltonian of (2.10) can be interpreted as the [I ][S ] [I ][S ] matrix IK SJ D IKx SJ x C IKy SJy C IKz SJ z D I x ˝ S x C Iy ˝ Sy C I z ˝ S z . 3.1.2 Angular Momentum Matrices
When appended to the program ending with Code 2.1 on page 23, the following MATLAB statements will evaluate the various spin matrices we discussed above. Code 3.1 %operators in their own space sIp=diag(sqrt((1:2*I).*(2*I:-1:1)),1); sIj(:,:,1)=(sIp+sIp’)/2; sIj(:,:,2)=(sIp-sIp’)/(2*i); sIj(:,:,3)=diag(I:-1:-I); sSp=diag(sqrt((1:2*S).*(2*S:-1:1)),1); sSj(:,:,1)=(sSp+sSp’)/2; sSj(:,:,2)=(sSp-sSp’)/(2*i); sSj(:,:,3)=diag(S:-1:-S); sJp=diag(sqrt((1:2*J).*(2*J:-1:1)),1); sJj(:,:,1)=(sJp+sJp’)/2; sJj(:,:,2)=(sJp-sJp’)/(2*i); sJj(:,:,3)=diag(J:-1:-J); %operators in uncoupled space for k=1:3 aIjg(:,:,k)=kron(sIj(:,:,k),eye(gS)); gSj(:,:,k)=kron(eye(gI),sSj(:,:,k)); aIje(:,:,k)=kron(sIj(:,:,k),eye(gJ)); gJj(:,:,k)=kron(eye(gI),sJj(:,:,k)); end
27
28
3 Wave Functions and Schrödinger Space
Expanded “grave” and “acute” versions of the matrices for the full Schrödinger space of the atom are denoted with an initial letter g or a. Angular momentum operators in their own (“self”) space, for example, the space spanned by the simultaneous eigenvectors of I I and I z , are denoted with an initial letter s. The nuclear spin matrices, aIje and aIjg, will be identical if J D S , as in D1 pumping, but they will have different dimensions if J ¤ S as in D2 pumping.
3.2 Energy States
It is often most convenient to calculate various optical pumping and resonance phenomena with energy basis states rather than with the uncoupled basis states jm S m I i of (3.2). We assume that the ground-state Hamiltonian of (2.10) is independent of time, and that the magnetic field is directed along the z axis of a Cartesian coordinate system. The time-independent Schrödinger equation, H fgg jμi D E μ jμi ,
(3.14)
gives the eigenvalues, E μ , the shift of the energy of the sublevels jμi from the mean ground-state energy W fgg . We can label the ground-state sublevels with an integer, μ D 1, 2, . . . , g fgg , where the number of ground-state sublevels is g fgg D [I ][S ] .
(3.15)
Any well-defined ordering of the basis states jμi is acceptable. The energies, E μ , are often used for ordering, with jμi D j1i having the highest energy, E1 , with jμi D j2i having the next-highest energy, E2 , and so on. When there are energy degeneracies, for example, when the magnetic field is exactly zero, one can list the degenerate states in the same, decreasing, order as their azimuthal quantum numbers, m, along some direction. Similar ordering schemes can be chosen for “accidental” level crossings at specific, nonzero values of the magnetic fields. These can occur for the hyperfine sublevels of atoms in the 2 P3/2 excited states of alkalimetal atoms, and in appropriate electronic states of other atoms. Expanding jμi on the uncoupled basis vectors (3.2), we find X jμi D jm S m I ihm S m I jμi . (3.16) mS mI
We will write the expansion coefficients as the elements of an [I ][S ] [I ][S ] unitary matrix, U fgg , hm S m I jU fgg jμi D hm S m I jμi . Explicitly, the eigenvalue equation (3.14) is X hm S m I jH fgg jm 0S m 0I ihm 0S m 0I jU fgg jμi D hm S m I jU fgg jμiE μ . m 0S m 0I
(3.17)
(3.18)
3.2 Energy States
The energy basis vectors j μi N and energy splittings EN μN for the excited-state Hamiltonians H feg of (2.12) can be found in like manner, N D E μN j μi N . H feg j μi
(3.19)
We can label the excited-state sublevels with an integer, μN D 1, 2, . . . , g feg , where the number of excited-state sublevels is g feg D [I ][ J ] .
(3.20)
The unitary transformation from the uncoupled basis to the energy basis, analogous to (3.17), is hm J m I jU feg j μi N D hm J m I j μi N .
(3.21)
For high intensities of pumping light, where the amplitudes of the excited-state density matrix are comparable to those of the ground state, it is often convenient to make no distinction between the labels of excited-state and ground-state sublevels and to write both as jμi, with μ D 1, 2, . . . , g feg for excited-state sublevels and μ D g feg C 1, g feg C 2, . . . , g feg C g fgg , for ground-state sublevels. Unless we explicitly state otherwise – for example, for the simple example in Chapter 1 – the energy basis will be used in our discussions of optical pumping. The wave function jψi of an individual atom will be characterized by its projections hμjψi and h μjψi N on the energy basis states jμi of (3.14) and j μi N of (3.19). A groundstate observable Z will be characterized by the elements of the [I ][S ] [I ][S ] matrix Z fcg with elements Z μfcg ν D hμjZ jνi, where jμi and jνi are solutions of (3.14). When we need to avoid ambiguity, we use the superscript fcg (for “coupled”) to denote a matrix in the energy basis. It will often be convenient to first evaluate the matrix Z mfugm Im 0 m 0 D hm S m I jZ jm 0S m 0I i that represents Z in the uncoupled basis S
I
S
I
(3.1). The matrix Z fcg is related to the matrix Z fug by Z fcg D U fgg† Z fug U fgg ,
(3.22)
where the unitary matrix U fgg was given by (3.17). The energy-basis versions of excited-state operators can be defined in like manner. In subsequent discussions, we will drop complicating superscripts or subscripts from symbols denoting matrices whenever the context makes the interpretation clear. For example, three possible interpretations of S z are: 1. an [S ] [S ] matrix S z operating on the vectors jm S i. 2. an [I ][S ] [I ][S ] matrix SJ z D 1fI g ˝ S z operating on the vectors jm S m I i. 3. an [I ][S ] [I ][S ] matrix U fgg† SJ z U fgg , operating on the vectors jμi. When appended to the program ending with Code 3.1 on page 27, the following MATLAB statements will evaluate the energies E μ and E μN of (3.14) and (3.19) along with the unitary operators U fgg and U feg of (3.17) and (3.21).
29
30
3 Wave Functions and Schrödinger Space
Code 3.2 for k=1:3;%uncoupled magnetic moment operators umug(:,:,k)=-LgS*muB*gSj(:,:,k)+(muI/(I+eps))*aIjg(:,:,k); umue(:,:,k)=-LgJ*muB*gJj(:,:,k)+(muI/(I+eps))*aIje(:,:,k); end uIS=matdot(aIjg,gSj);%uncoupled I.S uIJ=matdot(aIje,gJj);%uncoupled I.J B=input(’Static magnetic field in Gauss, B = ’); uHg=Ag*uIS-umug(:,:,3)*B;%uncoupled Hamiltonian [Ug,Eg]=eig(uHg);%unsorted eigenvectors and energies [x,n]=sort(-diag(Eg));%sort energies in descending order Hg=Eg(n,n); Eg=diag(Hg); Ug=Ug(:,n);%sorted Hg, Eg and Ug uHe=Ae*uIJ - umue(:,:,3)*B;%Hamiltonian without quadrupole interaction if J>1/2&I>1/2 uHe = uHe+Be*(3*uIJ^2+1.5*uIJ-I*(I+1)*J*(J+1)*eye(ge))... /(2*I*(2*I-1)*J*(2*J-1));%add quadrupole interaction end [Ue,Ee]=eig(uHe);%unsorted eigenvectors and energies [x,n]=sort(-diag(Ee));%sort energies in descending order He=Ee(n,n); Ee=diag(He); Ue=Ue(:,n);%sorted He, Ee, and Ue
The following simple subroutine was used in Code 3.2 to evaluate the dot product of vectors, the elements of which are rectangular matrices, with rows and columns given by the first two indices, and with components along the Cartesian axes x, y, and z specified with the third index. Code 3.3 function C=matdot(A,B) C(:,:)=A(:,:,1)*B(:,:,1)+A(:,:,2)*B(:,:,2)+A(:,:,3)*B(:,:,3);
3.3 Zero-Field States
In the limit that the externally applied magnetic field vanishes, the spin Hamiltonians H D H fgg of (2.10) or H D H feg of (2.12) must commute with each Cartesian component of the total angular momentum operator FDICJ,
(3.23)
and also with its square F F D I(I C 1) C 2I J C J( J C 1) .
(3.24)
For the rest of this section, we will denote the total electronic angular momentum operator by J for either the ground state or the excited state of the atom. At ze-
3.3 Zero-Field States
ro magnetic field, the energy states of the atom can be chosen to be the angular momentum eigenstates, X fm C I m I J m J jm I i ˝ jm J i . (3.25) j f mi D mI mJ fm
Here C I m I J m J is a Clebsch–Gordan coefficient (see Chapter 8 in [15]). A MATLAB cγ subroutine to evaluate Clebsch–Gordan coefficients C a α b β , based on Van der Waerden’s formula (see (3) in Section 8.2.1 in [15]) is Code 3.4 function f=cg(a,al,b,be,c,ga) if al+be-ga ~= 0|c
a+b f=0; else w=prod(1:a+al)*prod(1:a-al)*prod(1:b+be)*prod(1:b-be)*... prod(1:c+ga)*prod(1:c-ga)*(2*c+1); w=sqrt(w); z1=max([0 b-c-al a+be-c]); %greatest lower bound of z z2=min([a+b-c a-al b+be]); %least upper bound of z f=0; for z=z1:z2; f=f+(-1)^z/(prod(1:z)*prod(1:a+b-c-z)*prod(1:a-al-z)*... prod(1:b+be-z)*prod(1:c-b+al+z)*prod(1:c-a-be+z)); end f=del(a,b,c)*w*f; end
Code 3.4 makes use of the Δ symbol (see (1) in Section 8.2.1 in [15]), (a C b c)!(a b C c)!(a C b C c)! . Δ(a, b, c) D (a C b C c C 1)!
(3.26)
A simple MATLAB function, del(a,b,c), to evaluate Δ is Code 3.5 function y=del(a,b,c) y=sqrt(prod(1:a+b-c)*prod(1:a-b+c)*prod(1:-a+b+c)/prod(1:a+b+c+1));
The zero-field states (3.25) are simultaneously eigenvectors of F F , F z , and of the zero-field Hamiltonian H, F Fj f mi D f ( f C 1)j f mi ,
(3.27)
F z j f mi D mj f mi ,
(3.28)
H j f mi D f j f mi .
(3.29)
31
32
3 Wave Functions and Schrödinger Space
As discussed in connection with (2.12), for an atomic level with nuclear spin quantum number I > 1/2 and total electronic spin quantum number J > 1/2 the zero-field energy of the hyperfine multiplet f is f D A
3C 2f C 3C f 4I(I C 1) J( J C 1) Cf CB . 2 8I(2I 1)2 J(2 J 1)
(3.30)
Here A and B are the magnetic-dipole and electric-quadrupole hyperfine coupling coefficients, and the coefficient is C f D f ( f C 1) I(I C 1) J( J C 1) .
(3.31)
If I 1/2 or J 1/2, the term proportional to B is omitted from (3.30) since there is no contribution of the nuclear quadrupole to the energy splittings in these cases. When appended to the program ending with Code 3.2 on page 30, the following MATLAB statements will calculate the energies of the zero-field multiplet from (3.30) and store them in the column vectors Eef and Egf for the excited state and the ground state. They will be ordered with decreasing values of the quantum number f, which will also be in order of decreasing energies if Afeg > 0 and Afgg > 0, and if jB feg j is not large enough to change the order (large to small) the zero-field energy states would have if B feg were zero. This is true for all of the stable alkalimetal atoms. Code 3.6 f=(I+S:-1:abs(I-S))’; C=(f.*(f+1)-I*(I+1)-S*(S+1))/2; Egf=Ag*C;%ground state f=(I+J:-1:abs(I-J))’; C=(f.*(f+1)-I*(I+1)-J*(J+1))/2; Eef=Ae*C;%excited state if I>.5&J>.5 Eef=Eef+Be*(3*C.^2+1.5*C-I*(I+1)*J*(J+1))/(2*I*(2*I-1)*J*(2*J-1)); end
Low-Field Labels We will often denote the energy states jμi with the same symbol, j f mi, as the zero-field states of (3.25) when the Zeeman interactions with the externally applied magnetic field B are much smaller than the internal hyperfine interactions. Then (3.27) remains approximately valid, so f is almost, but not quite, the total angular momentum quantum number. The azimuthal quantum number m is always exact if the z axis is chosen to point along the magnetic field. Finally, (3.29) remains valid if we make the replacement f ! E f m , where E f m is the true energy eigenvalue of H, given by (3.14) or (3.19) and not the zero-field limit (3.30).
33
4 Density Matrix and Liouville Space The number densities of optically pumped atoms are normally very large, typically 1011 cm3 or greater, so the observable quantities will be averaged over many atoms. As reviewed by Fano [40], one of the most convenient ways to describe spinpolarized atoms under such conditions is with the density matrix, . The atoms will be subject to many interactions, for example, the absorption and emission of light, collisions with buffer gas atoms such as He or N2 , collisions with each other, and collisions with the walls. In spite of the many interactions, we assume that the nth atom can be described by a single-atom wave function jψ n i D jψ n (t)i. Let X be a single-atom observable. For example, we might have X D S z , where S z is the longitudinal electron spin operator for ground-state atoms. The mean value of X for the ΔN atoms in the small volume ΔV is 1 X hX i D hψ n j X jψ n i . (4.1) ΔN n The wave function of the nth atom in ΔV is jψ n i. Because the number density of the atoms is so large, ΔN will be a very large number, even for quite small volume elements ΔV . We see that we can write (4.1) as h X i D Tr[ X ] .
(4.2)
The trace in (4.2) extends over the g D g feg C g fgg basis states of the atom. The g g density matrix D (r, t) is D
1 X jψ n ihψ n j . ΔN n
(4.3)
From the definition (4.3), we see that the density matrix is Hermitian: † D .
(4.4)
Consequently, will have g real eigenvalues, λ, and corresponding eigenvectors jλi, defined by jλi D λjλi . Optically Pumped Atoms. William Happer, Yuan-Yu Jau, and Thad Walker Copyright © 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-40707-1
(4.5)
34
4 Density Matrix and Liouville Space
The density matrix can therefore be written as D
X
λjλihλj .
(4.6)
λ
The eigenvectors jλi are not necessarily the same as the energy eigenvectors (3.16) of the atom but they can be chosen to be basis vectors for the individual atomic wave functions jψ n i. The eigenvalues λ are also nonnegative since λ D hλjjλi D
1 X 1 X hλjψ n ihψ n jλi D jhλjψ n ij2 0 . ΔN n ΔN n
(4.7)
P Summing (4.7) over λ, noting that λ hψ n jλihλjψ n i D 1 by completeness, and P that n 1 D ΔN , we see that the eigenvalues λ sum to unity, so the density matrix has unit trace: X λ D Tr[] D 1 . (4.8) λ
The eigenvalues λ must therefore satisfy the constraint 0λ1.
(4.9)
We can think of λ as the probability that the eigenstate jλi of the density matrix will be occupied. Normally, both λ and jλi will be time-dependent. Populations Let jμi be an energy eigenstate, for example, one of those defined by (3.14). Taking the matrix elements of (4.6), we readily see that μ μ D hμjjμi is a real number, bounded by
0 μ μ 1 .
(4.10)
We interpret μ μ as the probability that an atom, selected at random from an ensemble, will have energy E μ . Coherences The matrix element, hμjjνi D μ ν , between two different energy states jμi and jνi is called the coherence between the states. The coherence can be a complex number, but since is Hermitian, we must have
μ ν D ν μ .
(4.11)
Coherences are often produced by resonant radio-frequency, microwave, or optical fields. Oscillating parts of atomic observables, for example, precessing transverse spins, are proportional to coherences. Large, time-independent coherences can be produced near “level crossings”, where the variation of some parameter of the Hamiltonian causes two or more sublevels to have the same energy.
4.1 Purity and Entropy
4.1 Purity and Entropy
Following Fano, we define the purity of a density matrix as
X 2 λ . hi D Tr 2 D
(4.12)
λ
For a density matrix of maximum purity, one of the eigenvectors jλi of (4.5), say, jλi D jψi, will have the eigenvalue λ D 1, and all other eigenvalues will be zero. For a pure state, D jψihψj ,
2 D
hi D 1 .
and
(4.13)
If the ensemble (4.3) defines a density matrix that is a pure state, the individual atomic wave functions must be jψ n i D eiφ n jψi, where φ n is a real phase angle, characteristic of the nth atom. The purity reaches its minimum value for the unpolarized state, D
1X jλihλj g
and
hi D
λ
1 . g
(4.14)
The bounds on the purity are therefore 1 hi 1 . g
(4.15)
The entropy per atom is S D hln i D
X
λ ln λ .
(4.16)
λ
Density matrices of maximum purity with hi D 1 have zero entropy, S D 0. For the state of minimum purity, the unpolarized state of (4.14), with hi D 1/g, we have the maximum possible entropy, S D ln g.
4.2 Ground State, Excited State, and Optical Coherence
For optical pumping experiments it is usually adequate to write the density matrix in block form as fe eg fe gg . (4.17) D fg eg fg gg The ground-state density matrix, fg gg , can be written in terms of the energy basis states of (3.14) as X fg gg jμi μ ν hνj . (4.18) fg gg D μν
35
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4 Density Matrix and Liouville Space
The excited-state component, fe eg , which is produced by optical excitation of ground-state atoms by resonant light, can be written in terms of the energy basis states of (3.19) as X eg fe eg D j μi N fe Nj . (4.19) μN νN h ν μN νN
The optical coherences fe gg and fg eg are often proportional to the electric field amplitude of the light. They can be written as X X fe gg fg eg fe gg D j μi N μN ν hνj and fg eg D jμi μ νN h νN j . (4.20) μN νN
μ νN
Where collisions transfer appreciable numbers of atoms to states that are not coupled by optical pumping, additional components of the density matrix must be added to (4.17). An example of this would be collisional population of 2 P1/2 states when atoms are optically pumped into the 2 P3/2 state of alkali-metal atoms. For the analysis of cooling and trapping experiments, additional quantum numbers describing the center-of-mass momentum of the atoms and the numbers of photons in the light beam may also be needed.
4.3 Column-Vector and Row-Vector Transforms
For many situations it is more convenient to represent the density matrix as a g 2 1 column vector j) in Liouville space rather than a g g matrix in Schrödinger space. Liouville space has several important advantages: It is more readily adapted to modern computer software programs such as MATLAB and Mathematica than is Schrödinger space. Pumping and relaxation processes can be more readily described with Liouville space than with Schrödinger space. The “superoperators” that describe evolution in Liouville space have important symmetries and identities that are useful for validating computer codes. We write the density matrix (4.17) for Liouville space as the column vector 2 fe eg 3 j ) 6 jfg eg ) 7 7 j) D 6 4 jfe gg ) 5 . j
fg gg
(4.21)
)
4.3.1 Column-Vector Transforms
Each of the four elements jfm ng ) of the column vector (4.21) is itself a column vector, which we construct from the corresponding Schrödinger-space rectangular
4.3 Column-Vector and Row-Vector Transforms
matrix fm ng with g fmg rows and g fng columns by placing each successive column of the matrix below the one to its left, as illustrated in (1.2) and (1.3). If we omit the superscripts for simplicity and set fm ng ! , we can write X X jμ ν)(μ νj) D jk)(kj) . (4.22) j) D μν
k
As indicated in (4.22), there is a one-to-one correspondence between the basis operators jμihνj of Schrödinger space and the basis vectors jμ ν) D jk) of Liouville space – and there is a similar correspondence of the Hermitian conjugates, jμihνj $ jμ ν) D jjμihνj) D jk)
and
(jμihνj)† $ (μ νj D (kj .
(4.23)
We will refer to μ as the left label and ν as the right label. It will sometimes be convenient to represent μ ν with a single index k as in (4.22). Some simple examples of (4.23) were discussed in connection with (1.5). The Liouville-space and Schrödinger-space amplitudes of the density matrix are given by μ ν D Tr[(jμihνj)† ] D (μ νj) D (kj) .
(4.24)
As discussed in connection with (1.16), we use the same column-vector transform (4.22) to convert an arbitrary Schrödinger-space matrix X, not just the density matrix, to a column vector, X jX ) D jμ ν)(μ νj X ) . (4.25) μν
One can readily verify that (μ νj X †) D (νμj X ) .
(4.26)
4.3.2 Row-Vector Transforms
We define the row vector ( X j as the Hermitian conjugate of the column vector j X ). The elements of ( X j D j X )† are given by ( X jμ ν) D (μ νj X ) .
(4.27)
The elements of the row vector ( X j are found by placing each successive row of X † to the right of the row above it. Some simple row-vector transforms were shown in (1.17). The column-vector and row-vector transforms of the unit operators are frequently encountered. For the full Liouville space of the atom (ground state plus excited state plus optical coherences) we have 2 ˇˇ feg 3 ˇ1 6 ˇ0fg eg 7 ˇ fe gg 7 . j1) D 6 (4.28) 4 ˇ0 5 ˇ fgg ˇ1
37
38
4 Density Matrix and Liouville Space
The elements of the four column vectors that make up j1) are ( μN νN j1feg ) D δ μN νN ,
(μ νN j0fg eg ) D 0 ,
fe gg ) D 0 and ( μνj0 N
(μ νj1fgg ) D δ μ ν .
(4.29)
4.3.3 Expectation Values
We will denote the expectation value of an atomic observable X by h X i D Tr [ X] D X † j .
(4.30)
Note in particular the identity h1i D Tr [] D (1j) D 1 .
(4.31)
Here the row vector (1j is the Hermitian conjugate of the column vector j1) of (4.28). The probabilities hN feg i and hN fgg i for finding the atom in the excited state or the ground state are hN feg i D (N feg j)
and
hN fgg i D (N fggj) ,
(4.32)
where 3 j1feg ) 6 j0fg eg ) 7 7 jN feg ) D 6 4 j0fe gg ) 5 j0fg gg ) 2
3 j0fe eg ) 6 j0fg eg ) 7 7 and jN fgg ) D j1) jN feg) D 6 4 j0fe gg ) 5 . (4.33) j1fgg ) 2
When appended to the program ending with Code 3.6 on page 32, the following MATLAB statements will evaluate the excited-state and ground-state projection operators, 1feg and 1fgg , denoted by Pe and Pg, the column vectors j1feg ), j1fgg ), and j1), denoted by cPe, cPg, and cP, and the row vectors (1feg j, (1fgg j, and (1j, denoted by rPe, rPg, and rP. Also evaluated are the column vectors jN feg ), denoted by cNe, and jN fgg ), denoted by cNg, of (4.33) and their row-vector conjugates, denoted by rNe and rNg. For use in examining or plotting computational results, we have also included LrNe and LcNg, logical variables for the populations of excited-state and ground-state sublevels. Code 4.1 Pg=eye(gg); Pe=eye(ge);%projection operators cPe=Pe(:); rPe=cPe’; cPg=Pg(:); rPg=cPg(:)’; gt=(gg+ge)^2;%dimension of full Liouville space cNe=[cPe;zeros(gt-ge*ge,1)]; rNe=cNe’; LrNe=logical(rNe); cNg=[zeros(gt-gg*gg,1); cPg];rNg=cNg’; LrNg=logical(rNg); cP=cNe+cNg; rP=rNe+rNg;
4.4 Superoperators
4.4 Superoperators
Operations on the density matrix j) of Liouville space are described by g 2 g 2 “superoperators” [6]. In this section we will discuss some of the most important properties of superoperators, and how to generate a superoperator for Liouville space from the ordinary operators of Schrödinger space. 4.4.1 Transposition Matrix
As a first, simple example of a superoperator, we discuss the transposition matrix T. Recall that the density matrix must be Hermitian at all times. To discuss the implications of the fact that D † , it is convenient to introduce a g 2 g 2 transposition matrix, T, defined by T jμ ν) D jνμ) .
(4.34)
From inspection of (4.34) we see that the elements of T are either 0 or 1, so T is real. Also T is symmetric, so it is Hermitian. Finally, T is clearly its own inverse. In summary, the simple symmetries of T are T D T ,
T D T†
and
T2 D 1 .
(4.35)
A simple example of a transposition matrix was shown in (1.6). When appended to a program ending with Code 4.1 on page 38, the following MATLAB statements will generate column vectors mu and nu of the left and right labels for the basis states jμ ν) for ground-state atoms with gg sublevels, and they will also evaluate the transposition operator T of (4.34). Code 4.2 mu=(1:gg)’*ones(1,gg); mu=mu(:); nu=ones(gg,1)*(1:gg); nu=nu(:); T=zeros(gg*gg,gg*gg); for j=1:gg*gg T(:,j)=mu(j)==nu & nu(j)==mu; end
The column-vector transforms of a Hermitian-conjugate pair of Schrödingerspace operators, X and X † , satisfy the identity j X † ) D T j X ) ,
or
T j X †) D j X ) .
(4.36)
Here j X ) D j X ) , the column-vector transform of X , denotes a column vector with elements that are the complex conjugate of the elements of j X ). Hermitian operators like the density matrix, D † , therefore satisfy the identity j) D T j ) ,
or j ) D T j) .
(4.37)
39
40
4 Density Matrix and Liouville Space
4.4.2 Evolution Matrices
As discussed in connection with (1.10), optical pumping and atomic collisions usually cause j) to evolve in time as described by j) P D G j) .
(4.38)
We can use (4.36) and (4.37) to write the Hermitian conjugate of (4.38) as jP† ) D T jP ) D T G j ) D T G T T j ) D T G T j) .
(4.39)
As j) evolves in time according to (4.38), it must remain Hermitian, that is, we must have j) P D jP† ) .
(4.40)
From inspection of (4.38) and (4.39) we see that for (4.40) to be true we must have T G T D G . We define the Liouvillian conjugate G ‡ of a Liouville-space matrix G by G‡ D T GT .
(4.41)
Then (4.40) will be valid for all j) if G is “Liouvillian”, that is, G D G‡ .
(4.42)
The constraint that the Hamiltonian be Hermitian, H D H † , ensures that the wave function jψi of a single atom remains normalized in Schrödinger space. The constraint that the evolution matrix be Liouvillian, G D G ‡ , ensures that j) remains Hermitian as it evolves according to (4.38). The matrix elements of (4.41) are
ˇ ˇ μ ν ˇ G ‡ ˇ μ 0 ν 0 D νμ jG j ν 0 μ 0 .
(4.43)
From (4.43) we see that if G D G ‡ , (μ μ jG j νν) D (μ μ jG j νν) .
(4.44)
The elements of a Liouvillian matrix connecting populations are real. Besides being Liouvillian, G must satisfy additional constraints. For example, to ensure that the populations of any sublevel jμi remain bounded between 0 and 1 as the atoms evolve according to (4.38) we must have (μ μjG jμ μ) 0 ,
(4.45)
and (μ μjG jνν) 0 ,
if
μ¤ν.
(4.46)
4.5 Eigendecomposition of G
Differentiating Tr[] D (1j) D 1 with respect to time, where j1) was given by (4.28), and using (4.38) for j), P we find (1jG j) D 0. Since this must be true for any j), we must have (1jG D 0 .
(4.47)
The constraint (4.47) ensures that as j) evolves according to (4.38), we will always have Tr[] D 1. For evolution governed by (4.38), the rate of change of the purity (4.12) is given by d hi D (j) P C (j) P D (jG † C G j) . dt
(4.48)
The purity will be conserved if G † D G , that is, if G is anti-Hermitian.
4.5 Eigendecomposition of G
The matrix G of (4.38) will have eigenvalues γ , right eigenvectors jγ ), and left eigenvectors ((γ j, which are given by G jγ ) D γ jγ ) γ j X γ )
and ((γ jG D ((γ jγ .
(4.49)
Here we are using the (possibly complex) eigenvalue γ as a label of the eigenvector jγ ), which is the column-vector transform of the Schrödinger-space matrix X γ , that is, jγ ) j X γ ). Multiplying (4.49) on the left by (1j, where j1) was given by (4.28), and using (4.47), we find that γ (1jγ ) D 0 .
(4.50)
From (4.50) we see that the eigenvectors jγ ) with γ ¤ 0 must have (1jγ ) D Tr[ X γ ] D 0. Eigenvectors with γ ¤ 0 have no net population and they do not affect the normalization of the density matrix as they relax. If G represents optical pumping or other processes like spin exchange that can polarize the atoms, the left eigenvectors are not the same as the Hermitian conjugates of the right eigenvectors; that is, in general we will find that ((γ j ¤ (γ j jγ )† .
(4.51)
With the exception of unusual special cases corresponding to “critical damping”, the eigenvectors of G form a complete set and can be normalized such that ((γ jγ 0) D δ γ γ 0 . In such cases we can write the damping operator as X GD γ jγ )((γ j . γ
(4.52)
(4.53)
41
42
4 Density Matrix and Liouville Space
From inspection of the transient solution (1.33), we see that for the density matrix to remain finite with increasing time we must have Re γ 0 .
(4.54)
Here Re γ is the real part of the complex eigenvalue γ . If we take the complex conjugate of (4.49), and note that G ‡ D G , we find G jγ ) D γ jγ ) , or
or
T G T jγ ) D γ jγ ) ,
G T jγ ) D γ T jγ ) .
(4.55)
Equation 4.55 implies that if jγ ) is an eigenvector, then so is jγ ) D T jγ ) ,
(4.56)
which has the eigenvalue γ , that is, G jγ ) D γ jγ ) .
(4.57)
One can readily show that the left eigenvector corresponding to (4.56) is ((γ j D ((γ j T .
(4.58)
Here the elements of the row vector ((γ j are the complex conjugates of those of the row vector ((γ j. According to (4.56) and (4.36), the two Schrödinger-space matrices, X γ and X γ , corresponding the two Liouville-space column vectors, jγ ) j X γ ) and jγ ) j X γ ), are Hermitian conjugates of each other, X γ D X γ† .
(4.59)
For nondegenerate, real γ , (4.56) implies that we can normalize jγ ) such that T jγ ) D jγ ) .
(4.60)
Comparing (4.60) with (4.36), we see that for real γ , the Schrödinger-space matrix X γ corresponding to the Liouville-space column vector, jγ ) j X γ ), is Hermitian, X γ D X γ† ,
if
Im γ D 0 .
(4.61)
In later parts of this book, we will encounter many concrete examples of the abstract concepts summarized in this section. 4.5.1 Nullspace
For time-independent G, we will be particularly interested in eigenvalues of G with γ D 0. These correspond to steady-state solutions of the density matrix, which we denote by j X 0 ) j1 ), the value of the density matrix at infinite time, after all
4.5 Eigendecomposition of G
initial transients have damped away. Usually, there is a unique steady state, defined, aside from normalization, by G j X 0) D 0 .
(4.62)
In linear algebra, j X 0 ) is called the nullspace of G, and it is defined to within an overall normalization factor by (4.62). The normalization of j X 0 ) is defined by (1j X 0 ) D Tr[ X 0 ] D 1 ,
(4.63)
where j1) was given by (4.28). Equation 4.63 ensures that the population probabilities of the steady-state X 0 sum to unity. 4.5.2 Critical Damping
For exceptional values of experimental parameters, situations analogous to critical damping of a simple oscillator occur. Then there are not enough independent eigenvectors jγ ) to make a complete set. This is not a very common occurrence, but we will discuss it briefly, since critical damping does not occur at all in Schrödinger space. Taking the Laplace transform of (1.10), we find (G C s)jO s ) D j0 ) ,
(4.64)
where the Laplace transform of the density matrix is jOs ) D
Z1 j t )es t d t .
(4.65)
0
The inverse Laplace transform is Z 1 jOs )e s t d t . jt ) D 2πi
(4.66)
C
The integration contour, C, is a vertical path in the complex s plane, cutting the real axis at s D a, starting at a i1 and ending at a C i1. The value of a is not important as long as a > Re(γ j ) for each eigenvalue γ j of G. From inspection of (1.31) we see that we can write the time evolution operator as Z 1 (G C s)1 e s t d t . (4.67) eG t D 2πi C
The matrix elements of the resolvent, (G C s)1 , the inverse of G C s, are analytic functions of s except at the singularities s D γ j where G will have poles. We can expand (G C s)1 in partial fractions. The partial-fraction expansion is particularly simple when there is no critical damping and the right eigenvalues jγ j ) are linearly
43
44
4 Density Matrix and Liouville Space
independent. Then we find (G C s)1 D
X jγ j )((γ j j j
γj C s
.
(4.68)
Substituting (4.68) into (4.67) and evaluating the contour integral by the method of residues, we find X jγ j )((γ j jeγ j t , (4.69) eG t D j
in agreement with (1.33). If G has n ordinary eigenvalues γ1 , γ2 , . . . , γ n , each with its own, linearly independent eigenvector jγ1 ), jγ2 ), . . . , jγ n ), and a doubly degenerate eigenvalue γc that corresponds to a single eigenvector jγc ), then the resolvent is (G C s)1 D
n X j D1
Rj Ac Bc . C C γj C s γc C s (γc C s)2
(4.70)
Substituting (4.70) into (4.67) and evaluating the contour integral with the method of residues, we find eG t D
n X
R j eγ j t C A c eγc t C Bc teγc t .
(4.71)
j D1
The sum extends over the normal (noncritical) eigenvalues, some of which may be degenerate but still have independent eigenvectors. The matrices R j , A c , and Bc can be found as limiting cases of eigenvector products jγ j )((γ j j as some system parameter (e.g., the magnetic field, a collisional damping rate) approaches the value that causes critical damping.
4.6 Matrix Transformations from Schrödinger Space to Liouville Space
Here we discuss how the most important matrix operations in Schrödinger space are represented in Liouville space. 4.6.1 Flat and Sharp Superoperators
In calculating optical pumping and spin relaxation processes, we frequently need to transform Schrödinger-space matrix products of the form AB to Liouville space. Suppose that D fm ng is a block of the density matrix (4.17) with g fmg rows and g fng columns, where m and n can take on the values e for the excited state or g for the ground state. Then the matrix A must have the form A D Afp mg with g fp g rows and g fmg columns, and the matrix B must have the form B D B fn qg with g fng rows
4.6 Matrix Transformations from Schrödinger Space to Liouville Space
and g fqg columns. Then we can write hμjABjνi D A μ μ 0 μ 0 ν 0 B ν 0 ν D A μ μ 0 B νTν 0 μ 0 ν 0 .
(4.72)
Here B T denotes the transpose of B. In (4.72) and in many subsequent expressions, we use the Einstein convention that repeated labels like μ 0 and ν 0 are to be summed over. The Liouville-space version of (4.72) is (μ νjAB) D (μ νjB T ˝ Ajμ 0 ν 0 )(μ 0 ν 0 j) .
(4.73)
In ordering the Kronecker product of (4.73), we have assumed that (μ 0 ν 0 j) and (μ νjAB) are elements of column-vector transforms of the matrices and AB, analogous to (1.2) and (1.3). We can write (4.73) as the identity jAB) D B T ˝ Aj) .
(4.74)
We will frequently encounter the special case of the identity (4.74) for which B D 1fng and jAB) D jA). Then (4.74) becomes jA) D A[ j) ,
(4.75)
where the “flat” matrix is defined by A[ A[fng D 1fng ˝ A .
(4.76)
In like manner, a “sharp” matrix is defined by B ] j) D jB) ,
(4.77)
B ] B ]fmg D B T ˝ 1fmg .
(4.78)
where
If we think of AB as three successive notes on a piano keyboard, then A is the flat note for and B is the sharp note. Ernst et al. [6] call the flat and sharp matrices “left-translation superoperators” and “right-translation superoperators”. From the definition (4.75) we find μ νjA[ jμ 0 ν 0 D A μ μ 0 δ ν ν 0 , (4.79) and from (4.77) we find μ νjB ] jμ 0 ν 0 D δ μ μ 0 B ν 0 ν .
(4.80)
Evidently A[ is diagonal in the right indices, ν and ν 0 , and B ] is diagonal in the left indices, μ and μ 0 . One can readily verify that the Hermitian conjugates of flat and sharp square matrices are A[† D A†[
and
B ]† D B †] .
(4.81)
45
46
4 Density Matrix and Liouville Space
The Liouville conjugates are A[‡ D A†]
and
B ]‡ D B †[ .
(4.82)
Hermitian conjugation “commutes” with sharp and flat transformations. Liouvillian conjugation replaces a sharp transform of A with the flat transform of A† and vice versa. As indicated in (4.76) and (4.78), the flat matrix transformation of A depends on the number of columns of – or of any other matrix in the same position as in the symbol jA). The sharp transformation of B depends on the number of rows of or of any other matrix in the same position as in the symbol jB). When does not have the same number of rows as columns – which can happen for optical coherences D fe gg where the dimension g fgg of the excited state may differ from dimension g fgg of the ground state – we will use a dimension index, like fng in (4.76) or fmg in (4.78), to indicate the dimension of the unit matrices of the Kronecker products. 4.6.2 Square Matrices
Many applications of flat and sharp matrices involve only square matrices of the same dimensions in Schrödinger space. An example is the analysis of ground-state relaxation mechanisms. For such situations, the dimension superscripts are always the same and can be omitted. Evaluating (4.74) for square matrices, we readily find A[ B ] D B ] A[ .
(4.83)
Flat and sharp square matrices commute. From the definitions (4.72) and (4.73) we see that for square matrices (AB)[ D A[ B [ ,
but (AB)] D B ] A] .
(4.84)
Simple MATLAB functions to evaluate flat and sharp transforms for square matrices are Code 4.3 function M=flat(S); s=size(S); M=kron(eye(s(1)),S);
and Code 4.4 function M=sharp(S); s=size(S); M=kron(S’,eye(s(1)));
4.6 Matrix Transformations from Schrödinger Space to Liouville Space
4.6.3 Commutator Superoperators
We often encounter situations where , P the rate of change of the density matrix , is proportional to a matrix like X X † . For example, we might have X D H , where H D H † is a Hermitian Hamiltonian that is the same for all atoms of the ensemble. To deal with such situations, it is convenient to introduce the transform ˇ X © j) D ˇ X X † , or X © D X [ X †] . (4.85) We see from (4.85) that X © is the transformation to Liouville space of a commutator Schrödinger space. Ernst et al. [6] refer to X © for Hermitian operators X as a“commutator superoperator” or “derivative superoperator”. In most cases of interest to us, X and will have the same, square dimensions, as indicated by the simplified notation for the flat and sharp matrices of (4.85). We use the superscript © to denote the commutator transform, since the c inside the circle is the first letter in“commutator”. One can use (4.82) to show that X ©† D X †© .
(4.86)
From (4.82) we find X ©‡ D X © . It is straightforward to verify two other useful identities, ˇ ˇ X © j1) D ˇ X X † and (1j X © D X † X ˇ .
(4.87)
(4.88)
Here j1) and (1j are column and row vectors for populations with (μ νj1) D δ μ ν . 4.6.4 O-Dot Superoperators
Following the coding convention of MATLAB, we define a dot–star product between two matrices A and B of the same dimensions as hμjA. Bjνi D hμjAjνihμjBjνi .
(4.89)
There is no sum over the repeated indices μ and ν on the right of (4.89). For Liouville space (4.89) becomes jA. ) D Aˇ j) .
(4.90)
As shown in (4.90), we use the o-dot superscript (ˇ) to denote the Liouville-space transform of a dot–star matrix product in Schrödinger space. O-dot matrices are always square and diagonal, with the elements (μ νjAˇ jμ 0 ν 0 ) D δ μ μ 0 δ ν ν 0 hμ jAj νi .
(4.91)
No sum is implied over the repeated indices of (4.91). The diagonal elements of the matrix Aˇ are the same as the elements of the column vector, jA).
47
49
5 Optical Pumping of Atoms We turn now to the heart of optical pumping physics, the interaction of light with atoms. The basic interaction, D E, is the quantum-mechanical equivalent of the classical orientation energy of the electric dipole moment, D, of the atom in the electric field, E, of the light. In this chapter, we briefly discuss the parts of the electric field E that are responsible for the absorption or emission of photons from the pumping beam. We then introduce the dimensionless vector operator Δ and its Hermitian conjugate Δ † , such that Δ C Δ † is proportional to the dipole moment D. The dimensionless dipole operators have a number of important symmetries and they are particularly useful for analyzing optical pumping of mutilevel atoms. Finally, we show how to use a generalized“rotating coordinate system” to model the interaction D E with optical Bloch equations that are adapted to multilevel atoms.
5.1 The Electric Field of Light
It will be convenient to write the time dependence of the classical electric field of light at some position r in space as the Fourier integral Z 1 iω t M ED dω D E C E . (5.1) E(ω)e 2π Here and subsequently, the limits of integration are from 1 to 1 unless otherwise specified. As indicated in (5.1), we can write the classical field E as the sum of two components E and E . The component E has only negative Fourier frequencies and can be written as 1 E(t) D 2π
Z
1 iω t M E(ω)e dω D 2π
Z1 iω t M E(ω)e dω .
(5.2)
0
The inverse of (5.2) is Z M E(ω) D E (t)eiω t d t . Optically Pumped Atoms. William Happer, Yuan-Yu Jau, and Thad Walker Copyright © 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-40707-1
(5.3)
50
5 Optical Pumping of Atoms
In the signal-processing literature, the negative-frequency field E is often called the analytic field or the preenvelope [41]. From (5.2) we see that ( M E(ω) , if ω > 0 I M E(ω) D (5.4) 0, otherwise . If we were to quantize the electromagnetic field, the complex field E of (5.1) would be represented by a superposition of photon-annihilation operators. Similarly, E ! E † would be represented by a superposition of photon-creation operators.
5.2 The Electric Dipole Moment of Atoms
The interaction between light and atoms is given by the negative dot product, DE, of the electric dipole moment operator D of the atom and the optical electric field E of (5.1). The electric dipole operator D has nonzero matrix elements only between states of different parity, for example, between the 2 S1/2 ground state and the 2 P J excited state of an alkali-metal atom. We therefore write the electric dipole moment operator as D D D C D† , where DD
X
jS m S ihS m S jDj J m J ih J m J j D D Δ .
(5.5)
(5.6)
mS mJ
The Hermitian conjugate is X j J m J ih J m J jDjS m S ihS m S j D D Δ † . D† D
(5.7)
mS mJ
The sums extend over all electronic azimuthal quantum numbers m S and m J of the ground state and excited state of the atom, which have total electronic angular momentum quantum numbers S and J, respectively. As indicated in (5.6) and (5.7), the operators D and D † operate only on the electronic parts of the atomic wave functions, and they have no direct influence on the nuclear parts. We will discuss the dimensionless dipole moment operator Δ, and its amplitude D in the following sections. 5.2.1 Spherical Tensors
Calculations of matrix elements of the dipole operators (5.6) and (5.7) can be facilitated by writing them as spherical tensors. We will briefly review the most important properties of spherical tensors here. More information can be found in the literature [40].
5.2 The Electric Dipole Moment of Atoms
Recall that an irreducible spherical tensor Tl m of angular momentum l has 2l C1 elements, corresponding to each possible value of the azimuthal quantum number m D l, l 1, . . . , l. The elements of the tensor refer to various physical quantities that are referenced to a spatial coordinate system. The elements can be simple scalar quantities, or vectors, or matrices, or some more complicated structures. The elements of the rotated spherical tensor R Tl m are given by R Tl m D
X
l Tl m 0 D m 0m .
(5.8)
m0 l l The Wigner D function, D m 0 m D D m 0 m (α β γ ), can be parameterized by the three Euler angles, α, β, and γ , that define the rotation R. A simple example of a spherical tensor is the set of three spherical basis vectors, T1m D ξ m , where
x C iy x iy , p , ξ 0 D z , ξ 1 D p 2 2 with ξ †m D ξ m D (1) m ξ m . ξ1 D
(5.9)
Here x, y, and z are unit vectors along the three Cartesian axes of a right-handed coordinate system. The spherical basis vectors ξ m have angular momentum quantum number l D 1 (there are 2l C 1 D 3 of them), they are orthonormal and complete, that is X ξ †m ξ n D δ m n and ξ m ξ †m D 1 D x x C yy C z z . (5.10) m
We will sometimes think of the ξ m as the spin basis states of the photon. Other examples of irreducible spherical tensors are the basis states jS m S i and j J m J i that occur in (5.6) and (5.7). 5.2.2 Hermitian Conjugates †
We will often encounter expressions that contain the Hermitian conjugate Tl m of some spherical tensor Tl m . The Hermitian conjugate of (5.8) is (R Tl m )† D
X
†
l Tl m 0 D m 0m .
(5.11)
m0
When rotated, the Hermitian conjugates of spherical tensors transform with the complex conjugates of the Wigner D functions. Since the powerful recoupling identities of angular momentum algebra are valid for products of spherical tensors, all of which transform with the D functions and not their complex conjugates, it is useful to introduce versions of the Hermitian conjugates that transform as (5.8) instead of as (5.11). We will denote these as † TO l m D Tlm (1) lCm .
(5.12)
51
52
5 Optical Pumping of Atoms 0
l m m l Noting that D m Dm 0 ,m (see (2) in Section 4.4 in [15]), we see that 0 ,m D (1) l O the tensors Tm transform under rotations as (5.8). A simple example of (5.12) is 1Cm D ξ m . ξO m D ξ m (1)
(5.13)
For tensors that are ket vectors, for example, TS m S D jS m S i, it will be convenient to denote TO S m S fS m S j, where fS m S j D hS, m S j(1) SCm S .
(5.14)
5.2.3 Addition of Angular Momentum
In analogy to (3.25) we define a coupled irreducible tensor as fTl Tl 0 g LM D
X
0 0 C lLM m,l 0 m 0 Tl m Tl m .
(5.15)
m m0
Depending on the context, the product symbol Tl m Tl 0 m 0 on the right of (5.15) could represent a set of [l][l 0 ] scalars, column vectors, row vectors, matrices, dyadics, or some more complicated quantities. For example, important special cases of (5.15) are the dimensionless dipole operators of (5.6) and (5.7). Using (5.9) and (5.10), we can write ΔD
X
p p Δ m ξ m (1) m D 3fΔξ g00 D 3fΔ ξO g00 .
(5.16)
m
Here fS J g
Δ m D Δ ξ m D T1m
.
(5.17)
In writing the last two terms of (5.16), we noted that for any angular momentum quantum number j, C 00 j mI j,m D
(1) j m p . [j]
(5.18) fS J g
We will discuss the spherical basis tensor T1m In analogy to (5.16) we write Δ† D
of (5.17) in more detail below.
X p p (Δ † ) m ξ m (1) m D 3fΔ † ξ g00 D 3fΔ † ξO g00 .
(5.19)
m
Here f J Sg
(Δ † ) m D Δ † ξ m D (1) J S T1m
.
(5.20)
5.2 The Electric Dipole Moment of Atoms
5.2.4 Spherical Basis Tensors
A spherical basis tensor, operating between the states jS m S i on the right and the states h J m J j on the left, can be defined by f J Sg
TLM
D fj J ifS jgLM D
X
j J m J ifS, M m J jC JLM m J S Mm J .
(5.21)
mJ
Following the convention of equation (2) in Section 13.1 in [15], the matrix elements of the tensor operators (5.21) are given by the Wigner–Eckart theorem: D
f J S)g
J m F jTLM
E E D 1 Jm f J S)g jS m S D p C S m SJ LM JkTL kS . [ J]
The reduced matrix elements of the spherical tensors (5.21) are E p D f J Sg J kTL )kS D [L] .
(5.22)
(5.23)
The Hermitian conjugate of a spherical basis tensor is f J Sg†
TLM
fS J g
D (1) MC J S TL,M .
(5.24)
The basis tensors are orthonormal, h i f J Sg† f J 0 S 0 g Tr TLM TL0 M 0 D δ LL0 δ M M 0 δ J J 0 δ S S 0 ,
(5.25)
or in Liouville space are f J Sg f J0S0g TLM jTL0 M 0 D δ LL0 δ M M 0 δ J J 0 δ S S 0 .
(5.26)
f J0S0g
f J 0 S0g
The elements, (μ νjTL0 M 0 ), of the column vector, jTL0 M 0 ), are given by (4.24) and f J S)g
the elements, (TLM
f J Sg
jμ ν), of the row vector, (TLM j, are given by (4.27).
5.2.5 Identities for Δ and Δ †
For the dimensionless dipole operators of (5.17) and (5.20), one can readily verify that (Δ m )† D (1) m (Δ † )m ,
where
m D 0, ˙1 .
(5.27)
We will also use projections of Δ and Δ † on the Cartesian axes, Δj D Δ xj
and
(Δ † ) j D Δ † x j ,
(5.28)
53
54
5 Optical Pumping of Atoms
Here x j is a unit vector along the jth axis of a right-handed Cartesian coordinate system, with j D 1, 2, 3 for the x, y, and z axes. The Hermitian conjugates of the Cartesian projections of Δ are simpler than those of the spherical projections, †
(Δ j )† D (Δ † ) j D Δ j ,
with
j D 1, 2, 3 .
(5.29)
From (5.25) we find
Tr (Δ m )† Δ m 0 D (Δ m jΔ m 0 ) D δ m,m 0 ,
m D 0, ˙1 ,
where
(5.30)
and
Tr (Δ j )† Δ k D Δ j jΔ k D δ j k .
(5.31)
As outlined in Chapter 11, we can use Racah algebra to derive the dyadic identities, X 11L † SC J C1 fξ ξO g L TLfS Sg (5.32) ΔΔ D 3(1) SS J L
and Δ † Δ D 3(1) J CSC1
X 11L L
J JS
fξ ξO g L TL
f J Jg
.
(5.33)
Here fξ ξO g L TLfS Sg is a scalar product (see (30) in Section 3.1.8 in [15]) defined for a pair of spherical tensors with elements ULM and VLM by UL VL D
X
(1) M ULM VLM .
(5.34)
M
In accordance with (5.15), the spherical basis dyadics are defined by fξ ξO g LM D
X
LM ξ m ξO Mm C1mI1,Mm .
(5.35)
m
For example, evaluating (5.35) for LM D 00, we find 1 x x C yy C z z p fξ ξO g00 D p D . (5.36) 3 3
ab c The 6j symbols, , that occur in (5.32) and (5.33) can be evaluated with the de f following MATLAB subroutine, which is based on equation (1) in Section 9.2.1 in [15].
5.2 The Electric Dipole Moment of Atoms
Code 5.1 function y=sixj(a,b,c,d,e,f) y=del(a,b,c)*del(c,d,e)*del(a,e,f)*del(b,d,f); n1=max([-1 a+b+c c+d+e a+e+f b+d+f]);%greatest lower bound of n n2=min([a+b+d+e a+c+d+f b+c+e+f]);%least upper bound of n z=0; for n=n1:n2; z=z+(-1)^n *prod(1:n+1)/(prod(1:n-a-b-c)*prod(1:n-c-d-e)*... prod(1:n-a-e-f)*prod(1:n-b-d-f)*prod(1:a+b+d+e-n)*... prod(1:a+c+d+f-n)*prod(1:b+c+e+f-n)); end y=y*z;
The code uses the function del of Code 3.5 on page 31. From (5.32) and (5.33) we see that X 3 fgg x j ΔΔ † x j D Δ Δ† D 1 [S ]
(5.37)
j
and Δ† Δ D
X
x j Δ† Δ x j D
j
3 feg 1 , [ J]
(5.38)
where the unit operators 1fgg and 1feg were given by (2.17). One can use (5.31) to show that
Tr ΔΔ † D Tr Δ † Δ D 1 , (5.39) where the unit dyadic was given by (5.10). For the analysis of spontaneous emission a useful identity is
1JS f J Jg Δ TLM Δ † D 3 T fS Sg (1) J CSC1CL . LS J LM
(5.40)
For alkali-metal atoms with S D 1/2 and J D 1/2 or J D 3/2, the Cartesian matrix elements of (5.32) are †
Δk Δl D
1 2i(1) J S δkl C k l m Sm . 2 [ J]
(5.41)
Here Δ k D Δ x k and S m D S x m . A sum is understood over the repeated subscript m. The antisymmetric unit tensor is k l m D (x k x l ) x m .
(5.42)
Therefore, k l m D 0 if any pair of indices are equal, k l m D 1 if k l m is an even permutation of 123, and k l m D 1 if k l m is an odd permutation of 123. When appended to the program ending with Code 4.1 on page 38, the following MATLAB statements will evaluate the spherical component, Δ m of (5.17), as well J j as the Cartesian component Δ j D Δ x j , of Δ, along with the grave version Δ for use in the uncoupled, nuclear-electronic Schrödinger space of the atom.
55
56
5 Optical Pumping of Atoms
Code 5.2 sDs=zeros(gS,gJ,3);%spherical projections in electronic space for k=J:-1:-J for l=1:-1:-1 for m=S:-1:-S if k+l==m sDs(S-m+1,J-k+1,2-l)=sqrt(3/gS)*cg(J,k,1,l,S,m); end end end end sDj(:,:,1)=(-sDs(:,:,1)+sDs(:,:,3))/sqrt(2); sDj(:,:,2)=(-sDs(:,:,1)-sDs(:,:,3))/(i*sqrt(2)); sDj(:,:,3)=sDs(:,:,2);%Cartesian projections in electronic space for k=1:3 gDj(:,:,k)=kron(eye(gI),sDj(:,:,k));%grave matrix; end
5.2.6 Amplitude D
We will write the amplitude D of (5.6) in terms of the oscillator strength f fg eg [42], which can be defined as 2ω fe gg X h J m J jDjS m S i hS m S jDj J m J i . (5.43) f fg eg D 3[S ]„re c 2 m m J
S
Here the classical electron radius is re , the speed of light is c, and the nominal resonance frequency, ω fe gg , of the optical transition was discussed in connection with (2.15). fg eg The spontaneous emission rate Γs from the excited state e to the ground state g is related to the oscillator strength f fg eg by fg eg
Γs
D
2[S ]ω fe gg2 re f fg eg . c[ J ]
(5.44)
If there are additional lower electronic states to which the atom can decay, the rate fg eg Γs is only the partial spontaneous decay rate to state g and not the total spontaneous decay rate. Substituting (5.5)–(5.7) into (5.43) and using (5.37), we find that fg eg
j D j2 D
„re c 2 f fg eg [S ] „c 3 [ J]Γs D fe gg 2ω 4ω fe gg3
.
(5.45)
5.2.7 Energy Basis
So far, we have displayed code to evaluate matrices in the uncoupled basis of the atom, that is, as “acute” nuclear matrices of (3.12) and“grave” electronic matrices
5.3 Spontaneous Emission
of (3.9). For subsequent calculations we will find it most convenient to use the representations of these matrices in the energy basis of (3.14). When appended to the program ending with Code 5.2 on page 56, the following MATLAB statements will evaluate matrices in the energy basis. They also evaluate the low-field labels f and m for the energy basis state of Schrödinger space, and the left and right versions of these labels for Liouville space. The left and right labels of the excited state will not be needed in subsequent segments of code, so they are not evaluated, although the evaluation can be done in like manner. Code 5.3 for j=1:3; mug(:,:,j)=Ug’*umug(:,:,j)*Ug; Ijg(:,:,j)=Ug’*aIjg(:,:,j)*Ug; Ije(:,:,j)=Ue’*aIje(:,:,j)*Ue; Sj(:,:,j)=Ug’*gSj(:,:,j)*Ug; Jj(:,:,j)=Ue’*gJj(:,:,j)*Ue; Dj(:,:,j)=Ug’*gDj(:,:,j)*Ue; end fg=round(-1+sqrt(1+4*(2*diag(Ug’*uIS*Ug)+I*(I+1)+S*(S+1))))/2; fe=round(-1+sqrt(1+4*(2*diag(Ue’*uIJ*Ue)+I*(I+1)+J*(J+1))))/2; mg=round(2*diag(Ijg(:,:,3)+Sj(:,:,3)))/2; Fzg=diag(mg); me=round(2*diag(Ije(:,:,3)+Jj(:,:,3)))/2; Fze=diag(me); x=kron(fg,ones(1,gg));fgl=x(:);%left f label x=x’; fgr=x(:);%right f label x=kron(mg,ones(1,gg)); mgl=x(:);%left m label x=x’; mgr=x(:);%right m label
The low-field state labels f will only be correct for magnetic fields small enough that the spins are still well coupled by the hyperfine interactions and the energy shifts are well approximated by (3.30). For simplicity, we did not write Code 3.2 on page 30 to ensure that the energy eigenvectors (the columns of the matrices U fgg D Ug and U feg D Ue) will have good azimuthal quantum numbers m when B D 0. Thus, the values of m generated by Code 5.3 may not be correct if B D 0.
5.3 Spontaneous Emission fg eg
The natural radiative decay rate Γs of (5.44) can be used to write the damping of the excited-state density matrix associated with radiative decay, fg eg fe eg
Psfe eg D Γs
.
(5.46)
Spontaneous emission populates the ground state at the rate [1] fg gg
Ps
fg eg
D
4ω fe gg3 [ J]Γs D fe eg D † D 3„c 3 3
Δ fe eg Δ † .
(5.47)
57
58
5 Optical Pumping of Atoms
The electric dipole moment operators, D and D † , were defined by (5.6) and (5.7), and their dimensionless versions, Δ and Δ † , were defined by by (5.17) and (5.20). Using (5.37) with (5.47), we find fg gg
Tr[Ps
Liouville Space fg gg
jPs
fg eg
] D Γs
Tr[Qfe eg ] .
(5.48)
The Liouville-space version of (5.47) is fg eg
) D Γs
fg eg
As
jfe eg ) .
(5.49) fg eg
The dimensionless g fgg2 g feg2 matrix A s fg eg
As
D
follows from (5.47) and (4.74) and is
[ J] X Δj ˝ Δj . 3
(5.50)
j
The sum extends over the three Cartesian projections, Δ j D Δ x j , of the dimensionless dipole operator of (5.6). When appended to the program ending with Code 5.3 on page 57, the following MATLAB statements will evaluate the coupling matrix (5.50) for spontaneous emission. Code 5.4 Asge=zeros(gg*gg,ge*ge); for j=1:3%sum over three Cartesian axes Asge=Asge+(gJ/3)*kron(conj(Dj(:,:,j)),Dj(:,:,j)); end
5.4 Electric Dipole Interaction
The electric dipole interaction between light and atoms is D E D D † E D E D † E D E .
(5.51)
The term D † E of (5.51) represents simultaneous emission of a photon and excitation of an atom, and D E , represents simultaneous absorption of a photon and deexcitation of an atom. These virtual transitions do not conserve energy and they cause “light shifts”, typically on the order of 106 Hz. These are orders of magnitude smaller than the familiar light shifts [1, 43] that occur when an atom is virtually excited through absorption of a slightly off resonance photon, and which are of primary concern for applications of optically pumped alkali-metal atoms in clocks and magnetometers. We will therefore ignore the last two terms of (5.51) from now on, and write the optical interaction as D E D V C V † .
(5.52)
5.5 Rotating Coordinate System
The part V D D † E D D Δ † E
(5.53)
describes the absorption of a photon from the pumping beam and the conversion of a ground-state atom to an excited atom. The part V † D D E D D Δ E
(5.54)
describes the emission of a photon into the pumping beam and the conversion of an excited-state atom to a ground-state atom. Both V and V † are purely electronic operators that commute with the nuclear basis states jI m I i.
5.5 Rotating Coordinate System
As the simplest example of optical pumping let us consider an atom interacting with monochromatic light with electric field Q i(krω t) . E D Ee
(5.55)
We assume that the atom follows a classical, straight-line path in the intervals between collisions, so the position of its nucleus at time t is r D r0 C v t .
(5.56)
The velocity, v , and the position, r 0 , of the atom, extrapolated to time t D 0, will be time-independent between collisions, but v and r 0 will change abruptly at each collision. In Chapter 9 we will explicitly include the atomic momentum as one of the quantum numbers of the atom, but for many important situations, the classicalpath approximation (5.56) is convenient and accurate. We assume the light is propagating along the unit vector n D k/ k D sin θ cos φ x C sin θ sin φy C cos θ z ,
(5.57)
with colatitude angle θ and azimuthal angle φ. We take E to be transverse so we can write EQ D θ EQ θ C φ EQ φ .
(5.58)
The amplitudes EQ θ D θ EQ and EQ φ D φ EQ are projections along unit vectors in the θ and φ directions: @n D cos θ cos φ x C cos θ sin φy sin θ z , @θ @n 1 D sin φ x C cos φy . φD sin θ @φ θ D
The unit vectors are shown in Figure 5.1.
(5.59) (5.60)
59
60
5 Optical Pumping of Atoms
Figure 5.1 The light propagates along the unit vector n. Since the electric field is transverse, it can be expanded on the unit vectors θ and φ, which, together with n, form an orthonormal coordinate system.
The mean energy flux S (erg cm2 s1 ) of the light is directed along n and has magnitude SD
c Q 2 j Ej . 2π
(5.61)
With a field of the form (5.55) and the classical trajectory (5.56), the interaction (5.53) becomes V D VQ ei(kvω)t ,
Q ikr 0 . where VQ D D † Ee
(5.62)
When appended to the program ending with Code 5.4 on page 58, the following MATLAB statements will evaluate tV, the matrix representation of VQ . Code 5.5 Sl=1e4*input(’flux in mW/cm^2, Sl = ’);%convert to erg/(s cm^2) Dw=2*pi*1e6*input(’Detuning in MHz, Dnu = ’); %colatitude and azimuthal angles of beam direction in degrees thetaD = input(’thetaD = ’);%beam colatitude angle in degrees phiD = input(’phiD = ’);%beam azimuthal angle in degrees theta=thetaD*pi/180; phi=phiD*pi/180;%angles in radians Etheta = input(’Etheta = ’);%relative field along theta Ephi= input(’Ephi = ’);%relative field along phi Ej(1)=Etheta*cos(theta)*cos(phi)-Ephi*sin(phi); Ej(2)=Etheta*cos(theta)*sin(phi)+Ephi*cos(phi); Ej(3)=-Etheta*sin(theta);%Cartesian projections tEj=sqrt(2*pi*Sl/c)*Ej/norm(Ej);%field in esu/cm^2 tV=zeros(ge,gg); for j=1:3%sum over three Cartesian axes tV=tV-D*Dj(:,:,j)’*tEj(j); end
5.5 Rotating Coordinate System
The projection of the optical electric field amplitude, x j EQ , along the jth Cartesian axis is denoted by tEj(j). The field amplitudes are determined by the laser power, in accordance with (5.61). The unperturbed Hamiltonian (2.15) and the interaction (5.52) will cause the density matrix to evolve at the rate d 1
1 D [H0 , ] C (V C V † ), . dt i„ i„
(5.63)
To solve (5.63), it is convenient to introduce the phase-angle operator, Φ D (ω k v ) Qt .
(5.64)
where the “optical spin operator”, Q, was defined by (2.16). We will encounter more complicated phase matrices Φ in our discussion of magneto-optical traps. Using (5.64), we can write (5.63) as d 1 iΦ (H0 C VQ C VQ † )eiΦ , . D e dt i„
(5.65)
The “rotation operator”, eiΦ , can be written as eiΦ D ei(ωkv)t/21feg C ei(ωkv )t/21fgg .
(5.66)
In deriving (5.65) from (5.63), we noted that eiΦ H0 eiΦ D H0 ,
(5.67)
eiΦ VQ eiΦ D V ,
(5.68)
eiΦ VQ † eiΦ D V † .
(5.69)
Let the density matrix in the “rotating frame” be Q D eiΦ eiΦ .
(5.70)
Differentiating (5.70) with respect to time and using (5.65), we find d Q 1 [K, ] D Q . dt i„
(5.71)
The rotating-frame Hamiltonian, K D K0 C K1 ,
(5.72)
has a “dark part”, that is, a part that is independent of the optical electric field, P , K0 D H0 „ Φ
(5.73)
and a part that is proportional to the optical electric field, K1 D VQ C VQ † .
(5.74)
61
62
5 Optical Pumping of Atoms
Both K0 and K1 are time-independent in the intervals between collisions, which causes abrupt changes in v, r 0 , VQ , and VQ † . For the simple phase operator, (5.64), we find K0 D H feg C H fgg „(Δω k v )Q D K feg C K fgg ,
(5.75)
where „ (Δω k v ) and 2 „ D H fgg C (Δω k v ) . 2
K feg D H feg K fgg
(5.76)
The nominal detuning is Δω D ω ω fe gg .
(5.77)
The nominal optical resonance frequency ω fe gg was discussed in connection with (2.15).
5.6 Net Evolution
Writing out the components of (5.71), and adding in the effects (5.46) and (5.47) of spontaneous emission, we find d Qfe eg dt d Qfg eg i„ dt d Qfe gg i„ dt d Qfg gg i„ dt
i„
D E fe eg . Qfe eg C VQ Qfg eg Qfe gg VQ † ,
(5.78)
D VQ † Qfe eg C E fg eg. Qfg eg Qfg gg VQ † ,
(5.79)
D Qfe eg VQ C E fe gg . Qfe gg C VQ Qfg gg ,
(5.80)
fg gg
D i„Ps
Qfg eg VQ C VQ † Qfe gg C E fg gg. Qfg gg .
(5.81)
The elements of the complex energy-difference matrices are fe eg E μNfeμNeg 0 D Kμ N μN 0 i„γ1 , fg eg
D K μ μN i„γ2 ,
fe gg
D K μN μ i„γ2 ,
E μ μN E μN μ
fg gg
fg eg fe gg
fg gg
Eμ μ0 D Kμ μ0 .
(5.82) (5.83) (5.84) (5.85)
5.6 Net Evolution
The elements of the real energy-difference matrices are N 0 j μi N h μN 0 jK0 j μN 0 i D H μNfeμNeg K μNfeμNeg 0 D h μjK 0 , fg eg
D hμjK0 jμi h μjK N 0 j μi N D H μ μN C „(Δω k v) ,
fe gg
D h μjK N 0 j μi N hμjK0 jμi D H μN μ „(Δω k v ) ,
K μ μN K μN μ
fg eg fe gg
fg gg
fg gg
K μ μ 0 D hμjK0 jμi hμ 0 jK0 jμ 0 i D H μ μ 0 .
(5.86) (5.87) (5.88) (5.89)
The resonance-frequency shifts due to hyperfine splittings and applied external fields are accounted for by the matrices N feg j μi N h μN 0 jH feg j μN 0 i D E μN E μN 0 , H μNfeμNeg 0 D h μjH
(5.90)
fg eg
D hμjH fgg jμi h μjH N feg j μi N D E μ E μN ,
(5.91)
fe gg
D h μjH N feg j μi N hμjH fgg jμi D E μN E μ ,
(5.92)
H μ μN H μN μ
fg gg
H μ μ 0 D hμjH fgg jμi hμ 0 jH fgg jμ 0 i D E μ E μ 0 .
(5.93)
The matrices (5.90)–(5.93), when divided by Planck’s constant „, are the residual Bohr frequencies associated with each element of the density matrix after any optical component, ˙ω, of the frequency has been removed. For Hermitian Schrödinger-space operators like H feg , H fgg , K feg , and K fgg , which are diagonal in the energy basis, and from which we can define the Schrödinger-space operators H fg gg of (5.90)–(5.93), or K fg gg of (5.86)–(5.89), we will occasionally use identities like H fg ggˇ D H fgg©
and
H fe egˇ D H feg© .
(5.94)
These can be verified in a straightforward way from the definition (4.85) of commutator transforms and (4.91) for o-dot transforms. The “longitudinal” and “transverse” relaxation rates of (5.82)–(5.84) are fg eg
γ1 D Γs
and
γ2 D
1 fg eg C Γc α fe gg . Γs 2
(5.95)
The nominal collision rate is Γc , and we use a dimensionless, complex coefficient α fe gg D α fg eg to describe the collision efficiency. The collisional broadening is proportional to Re α fe gg and the collisional shift in the optical resonance frequency is proportional to Im α fe gg . Later, we will include additional terms proportional to Γc in (5.78) and (5.81) to account for spin relaxation in the ground state and the excited state, and for quenching collisions that cause nonradiative transfer of excited atoms to the ground state. Representative values of the contributions of various buffer gases to γ2 are shown in Table 5.1.
63
64
5 Optical Pumping of Atoms Table 5.1 Pressure broadening coefficients, (d γ2 /d N)/π (GHz amg1 ) for the alkali-metal (D1, D2) resonance lines in noble gases and N2 . Largely adapted from [44], except for Rb–He, Rb– N2 , and Rb–Xe from [45] and Cs–He and Cs–N2 from [46]. He
Ne
Ar
Kr
Xe
N2
Li
15
12
18
24
27
–
Na
(16, 19)
(8, 12)
(21, 18)
(23, 21)
(26, 24)
–
K Rb
(13, 18) (18, 18)
(7, 10) (8, 9)
(21, 16) (16, 16)
(19, 19) (16, 14)
(24, 24) (18, 18)
(21, 21) (19, 19)
Cs
(23, 19)
(8, 8)
(16, 16)
(16, 16)
(17, 18)
(16, 25)
5.6.1 The Amagat Unit of Density
In quoting experimental values of pressure-dependent quantities, we will use a practical unit of number density, the amagat. The amagat is the density of an ideal gas at a pressure of 1 atm and a temperature of 0 ı C. The abbreviation for an amagat is amg, and its value (the Loschmidt constant) in the cgs system of units used in this book is 1 amg D 2.6868 1019 cm3 .
(5.96)
Most experiments with buffer gases are done with sealed-off cells for a range of temperatures. The density of a nonreacting and noncondensing buffer gas is very nearly constant in such experiments, since there is little change in cell volume with temperature. However, the internal pressure of a sealed-off cell can change substantially with temperature. So it is convenient to quote experimental measurements for a specific density and temperature, rather than pressure and temperature. This is why the amagat unit is so commonly used in the optical pumping literature. 5.6.2 Normalization
With (5.78)–(5.81) one can verify that Tr [d /d Q t] (1jd /d Q t) D 0 .
(5.97)
We can therefore pick the normalization of Q such that Tr [] Q (1j) Q D1.
(5.98)
When appended to the program ending with Code 5.5 on page 60, the following MATLAB statements will evaluate (5.90)–(5.93).
5.7 Optical Bloch Equations
Code 5.6 Hee=Ee*ones(1,ge)-ones(ge,1)*Ee’; Hge=Eg*ones(1,ge)-ones(gg,1)*Ee’; Heg=Ee*ones(1,gg)-ones(ge,1)*Eg’; Hgg=Eg*ones(1,gg)-ones(gg,1)*Eg’;
5.6.3 Notation and Coding
We will often write expressions like (5.88) symbolically or in computer codes as K fe gg D H fe gg „ (Δω k v ) .
(5.99)
As indicated by (5.88), the expression (5.99) means that to get the elements of the matrix represented by K fe gg , one subtracts the number, „(Δω k v ), from each element of the matrix, H fe gg , and not only from the diagonal elements of H fe gg , as expressions like (5.99) sometimes mean. In fact, H fe gg can have different numbers of rows and columns, so it would not be clear what its “diagonal elements” are.
5.7 Optical Bloch Equations
We will call (5.78)–(5.81) together with (5.98) the “optical Bloch equations”. To see the connection to the classical Bloch equation of nuclear magnetic resonance, consider the limit where the there is only one sublevel of the excited state and one sublevel of the ground state. Of course this does not occur for real atoms, where one sublevel in the ground state and one in the excited state would correspond to S D 0 and J D 0, for which single-photon transitions are forbidden. Then the operator Q of (2.16) can be written as 1 1 0 Q D Sz D . (5.100) 2 0 1 Here S z D SQ z denotes the longitudinal spin operator of an effective spin-1/2 particle, for which spin-up is the excited state and spin-down is the ground state. In view of (5.98) we can write the diagonal elements of the density matrix as 1 ˝Q ˛ 1 ˝ ˛ and Qfg gg D SQ z . C Sz 2 2 In the same spirit we write ˛ ˛ ˝ ˛ ˝ ˝ ˛ ˝ Qfg eg D SQC D SQ x C i SQy and Qfe gg D SQ D SQ x i SQy . Qfe eg D
(5.101)
(5.102)
The coupling matrix VQ reduces to a single complex number which we write in terms of its real and imaginary parts as VQ D VQ 0 C iVQ 00 .
(5.103)
65
66
5 Optical Pumping of Atoms
The energy-difference matrices, (5.86) and (5.89), are both zero. Finally, (5.48) implies that fg gg
Ps
fg eg fe eg
D Γs
Q
.
(5.104)
Using this notation, one can readily show that (5.78)–(5.81) can be summarized by the vector equation d Q z h Si D γ1 z z γ20 [x x C yy] CΩ h SQ i γ1 . dt 2
(5.105)
The Larmor frequency in the rotating frame has the Cartesian components 2VQ 0 2VQ 00 , Ωy D and „ „ Ωz D Δω C γ200 D ω fe gg C γ200 ω . Ωx D
(5.106)
The Rabi frequency is Ω? D
q
Ωx2 C Ωy2 D
2jVQ j . „
(5.107)
We have ignored any Doppler shift. Equations (5.105)–(5.107) are the classic forms of the Bloch equations used to analyze magnetic resonance of a spin-1/2 particle. Spins are created at a rate γ1 with an initial value of h SQ i D z/2. Once created, the spins rotate at the apparent Larmor rate, Ω , of the rotating coordinate system. The longitudinal spin polarization relaxes at the real rate γ1 . The transverse spin polarization relaxes at the rate γ20 . The resonance frequency is shifted from ω fe gg to ω fe gg C γ200 . For real atoms with multiple sublevels in the ground state and excited state, it is necessary to use the generalized optical Bloch equations (5.78)–(5.81). There is a matrix of optical Rabi frequencies, 2VQ /„, and the positive occupation probabilities of the upper and lower sublevels become excited-state and ground-state density matrices Qfe eg and Qfg gg . These complications can be readily handled by modern mathematics packages for fast electronic computers.
5.8 Liouville Space
For efficient computer calculations, it is convenient to transform the optical Bloch equations to Liouville space. In analogy to (4.21), we define the Liouville-space density matrix as the column vector 2 ˇˇ fe eg 3 ˇQ 6 ˇQfg eg 7 ˇ fe gg 7 . j) Q D6 (5.108) 4 ˇQ 5 ˇ fg gg ˇQ
5.8 Liouville Space
Each element, jQfm ng ), of (5.108) is a column vector, assembled by placing each column of the g fmg g fng matrix Qfm ng below the one to its left. The Liouvillespace version of (5.78)–(5.81) is d j) Q D G j) Q . dt The evolution operator is 2 E fe egˇ VQ †[feg i 6 6 GD 6 VQ ]feg „4 fg eg fg eg i„Γs A s
(5.109)
VQ [feg E fg egˇ 0 Q V ]fgg
VQ †]feg 0 E fe ggˇ VQ †[fgg
0
VQ †]fgg VQ [fgg E fg ggˇ i„Γc A c
3 7 7 7 . (5.110) 5
The “o-dot” (ˇ), “flat”([), and “sharp” (]) matrix transformations from Schrödinger to Liouville space are defined in Section 4.6. The block-0’s in (5.110) have the appropriate numbers of rows and columns to fill out the matrix. We have assumed that collisions are slow enough that they need only be taken into account for the ground-state density matrix fg gg , with the factor Γc A c . We will discuss more realistic types of collisional relaxation in later sections, but for illustrative purposes we will let A c represent “uniform relaxation”, an example of which we already used in (1.13), and which we generalize to A c D 1fg gg
1 j1fgg )(1fgg j. g fgg
(5.111)
The physical meaning of (5.111) is that the collision destroys the entire (possibly polarized) density matrix, as represented by the unit matrix 1fg gg D 1fgg[ D 1fgg] D
X
jμ ν)(μ νj ,
(5.112)
μν
for the ground state of Liouville space. The term j1fgg )(1fgg j/g fgg replaces the polarized atoms with a completely unpolarized density matrix j1fgg )/g fgg . When appended to the program ending with Code 5.6 on page 65, the following MATLAB statements will evaluate the damping operator A c of (5.111) and the damping operator G of (5.110) for atomic velocity v D 0. Code 5.7 Gmc=input(’Collision rate Gmc = ’); Acgg=eye(gg*gg)-cPg*rPg/gg;%uniform-relaxation matrix %G0=dark, G1=pumping, G2=collisions; dGdw=dG/dw G0=zeros(gt,gt); G1=G0; G2=0;dGdw=G0; %index ranges for blocks n1=1:ge^2; n2=ge^2+1:ge^2+ge*gg; n3=ge^2+ge*gg+1:ge^2+2*ge*gg; ...
67
68
5 Optical Pumping of Atoms
Code 5.7 Continued the previous code. n4=ge^2+2*ge*gg+1:(ge+gg)^2; G1(n1,n2)=kron(Pe,tV)*i/hbar;%upper off diagonal elements G1(n1,n3)=-kron(conj(tV),Pe)*i/hbar; G1(n2,n4)=-kron(conj(tV),Pg)*i/hbar; G1(n3,n4)=kron(Pg,tV)*i/hbar; G1=G1-G1’;%add antihermitian conjugate G2(n4,n4)=Gmc*Acgg; dGdw(n2,n2)=i*eye(ge*gg);%dG/dw dGdw(n3,n3)=-dGdw(n2,n2); G0(n1,n1)=diag(Hee(:)*i/hbar+1/te);%diagonal elements G0(n2,n2)=diag(Hge(:)*i/hbar+1/(2*te)); G0(n3,n3)=diag(Heg(:)*i/hbar+1/(2*te)); G0(n4,n4)=diag(Hgg(:)*i/hbar); G0(n4,n1)=-Asge/te;%repopulation by stimulated emission G=G0+G1+G2+Dw*dGdw;%total damping matrix
5.8.1 Transients
One can use (5.109) to calculate transients induced by intense optical pulses. For example, suppose that at time t D 0 the atoms are unpolarized and in their ground state with the density matrix 2 ˇˇ fe eg 3 ˇ0fg eg ˇ0 7 1 6 ˇ 7 jQ0 ) D fgg 6 (5.113) 4 ˇ0fe gg 5 . g ˇ fgg ˇ1 The solution of (5.109) for t 0 with the initial condition (5.113) is j) Q D eG t jQ0 ) .
(5.114)
When appended to the program ending with Code 5.7, the following MATLAB statements will evaluate and plot the transient response described by (5.109). Code 5.8 rt=input(’relative pulse length, rt=tm/te = ’); tm=rt*te; nt=101;%number of time samples th=linspace(0,tm,nt);%time samples rhot=zeros(gt,nt);%initialize density matrix rho0=[zeros(ge^2+2*ge*gg,1);cPg]/gg;%density matrix at t=0 for k=1:nt rhot(:,k)=expm(-G*th(k))*rho0; end
5.8 Liouville Space
Code 5.8 Continued the previous code. Ne=rNe*rhot;%excited-state probability Ng=rNg*rhot;%ground-state probability clf; plot(th/te,real(Ng),’b-’); hold on; plot(th/te,real(Ne)x,’r-.’);grid on; xlabel(’Time, \Gammaˆf\fge\gg_f\rm sgt’); ylabel(’Populations’); legend(’Ng’,’Ne’)
Energy (not to scale)
P1/2
f=1 f=0 m = –1
0
1
ω
S1/2
ω{eg}
f=1 f=0 m = –1
0
1
Figure 5.2 Energy levels for optical pumping of a hypothetical Rb atom with nuclear spin quantum number I D 1/2, but with the same hyperfine coupling parameters Afgg and Afeg as 87 Rb.
A typical transient calculated with Code 5.8 on page 68 is shown in Figure 5.3. Note the Rabi oscillations that occur as the atom builds up to a final steady state. The excited-state and ground-state probabilities, Ne and Ng, sum to 1. 5.8.2 Steady State
The steady-state populations are indicated by the horizontal lines in Figure 5.3. In accordance with (1.34) and (1.35), the steady-state populations Neinf D (N feg j1 ) and Nginf D (N fgg j1 ), where jN feg ) and jN fgg ) were given by (4.33), can be evaluated by appending the following statements to the MATLAB code ending with Code 5.8 on page 68. Code 5.9 rhoinf=null(G);rhoinf=rhoinf/(rP*rhoinf); Nginf=rNg*rhoinf; Neinf=rNe*rhoinf; plot(th/te,real(Nginf)*ones(1,nt),’b-’) plot(th/te,real(Neinf)*ones(1,nt),’r-.’)
69
70
5 Optical Pumping of Atoms
Figure 5.3 Transient response of ground-state and excited-state populations calculated with the code ending with Code 5.8 on page 68. The parameters were I = 0.5, S = 0.5, J = 0.5, B = 1, Sl = 40, Dnu = -752, thetaD = 45, phiD = 0, Etheta = 1 Ephi = i, Gmc = .1/te, rt = 10. For a 752-MHz de-
tuning the light can resonantly couple atoms from the ground-state multiplet with f D 1 to the excited-state multiplet with f D 1. The horizontal lines are the steady-state limits, calculated with the code that ends with Code 5.9 on page 69.
5.8.3 Steady State Versus Detuning
When appended to the program ending with Code 5.7 on page 67, the following MATLAB statements will evaluate and plot the steady-state populations as a function of the optical frequency detuning for frequencies close to those of Figure 5.3. Code 5.10 nw=21; dw=linspace(Dw-3/te,Dw+3/te,nw); rhow = zeros(gt,nw); for k=1:nw G=G0+G1+G2+dw(k)*dGdw; rhow(:,k)=null(G);rhow(:,k)=rhow(:,k)/(rP*rhow(:,k)); end figure (2); clf; subplot(2,1,1) plot(dw/(2*pi*1e6),real(rNg*rhow),’b-’); hold on;%state populations plot(dw/(2*pi*1e6),real(rNe*rhow),’r-.’);grid on; ylabel(’State Populations’);
5.8 Liouville Space
Code 5.10 Continued the previous code. legend(’Ng’,’Ne’) subplot(2,1,2);%sublevel populations plot(dw/(2*pi*1e6),real(real(rhow(LrNg,:))),’b-’);hold on; plot(dw/(2*pi*1e6),real(real(rhow(LrNe,:))),’r-.’);grid on; xlabel(’Detuning, \Delta\nu in MHz’); ylabel(’Sublevel Populations’);
(a)
(b) Figure 5.4 State populations (a) and sublevel populations (b) for the same parameters as those for Figure 5.3 in the steady state and with variable optical detuning. The figure was produced with the code ending with Code 5.10 on page 70. The excited-state populations are suppressed in comparison to the transient
response because ground-state atoms are optically pumped into the most weakly absorbing sublevels. Using zero-field labels for the states, the ground-state populations for the most negative detuning, Δν D 769 MHz, are N(1, 1) D 0.3051, N(1, 0) D 0.1702, N(1, 1) D 0.1365, and N(0, 0) D 0.3469.
The results of running the code ending with Code 5.10 on page 70 are shown in Figure 5.4.
71
73
6 Quasi-Steady-State Optical Pumping For alkali-metal atoms, the number of rows or columns of the damping matrix G of (5.110) is n D f[I ]([S ] C [ J ])g2. So for D2 pumping of 133 Cs, with I D 7/2, S D 1/2, and J D 3/2, we would have n D 2304. Exponentiating such large matrices, as in (5.114), to find transient solutions, or using (1.34) and (1.35) to find steady-state solutions, is time-consuming although possible with fast computers. However, we are often interested in “quasi-steady-state” conditions where the relative change of the density matrix over one radiative lifetime is very small. For quasi-steady-state conditions, it is often possible to work with much smaller matrices, which require much less computing power. Quasi-steady-state analysis is also needed to discuss familiar concepts such as optical absorption cross sections, stimulated emission cross sections, or the index of refraction of an atomic vapor. The formal solution of (5.80) is fe gg Q μN ν (t)
1 D i„
Zt
fegg fg gg iEμN ν (tt 0)/„ eg 0 Q d t 0 VQ μN μ Q μ ν (t 0 ) Qfe . μN νN (t )V νN ν e
(6.1)
1
The biggest contributions to the integral on the right of (6.1) come from times t 0 that are not very much retarded from the time t, since the exponential, in addition 0 to oscillatory terms, contains a damping factor e(tt ) Re γ2 . According to (5.95), the optical-coherence damping rate Re γ2 is at least as fast as half the radiative decay fg eg rate, Γs . For such short time delays, t t 0 , we will assume that the time evolution of Q can be well approximated by that of an “atom in the dark”, fg gg
fg gg
fg gg
Q μ ν (t 0 ) D Q μ ν (t)eiKμν eg 0 Qfe μ N νN (t )
D
(t 0 t)/„
feeg 0 fg gg Q μN νN (t)eiKμN νN (t t)/„
and .
(6.2)
Substituting (6.2) into (6.1) and noting from (5.84) that fg gg
E μN ν
fg gg
Kμ ν
fg gg
D E μN μ
,
Optically Pumped Atoms. William Happer, Yuan-Yu Jau, and Thad Walker Copyright © 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-40707-1
(6.3)
74
6 Quasi-Steady-State Optical Pumping
we find
0 fe gg
Q μN ν
1 D@ i„
Z1 0
1 d τ VQ μN μ e
0
eg Qfe μN νN
@1 i„
Z1
fegg iEμN μ τ/„
gg A Qfg μν
1 d τ VQ νN ν e
fegg
iEνN ν τ/„ A
0 gg fe eg Q Q μN μ Qfg D W μ ν C Q μN νN W νN ν .
(6.4)
Q are The elements of the time-independent, rotating-frame operator W Q D VQ ./E fe gg . W
(6.5)
Following the coding convention of MATLAB, we define the dot–slash product between two matrices A and B of the same dimensions as hμjA./Bjνi D
hμjAjνi . hμjBjνi
(6.6)
In the laboratory frame, (6.4) becomes fe gg D W fg gg C fe eg W .
(6.7)
The time-dependent operator W is Q eiΦ D V ./E fe gg . W D eiΦ W
(6.8)
Range of Validity The approximation (6.2) is excellent for many optical pumping situations, especially for pumping with weak light where stimulated emission rates can be neglected or where there is no atomic coherence. An example where (6.3) would not be a good approximation is an atom responding to such intense light that the magnitudes of the Rabi frequencies, VQ /„, are comparable to or larger than the fe gg magnitudes of the self-evolution frequencies, E μN ν /„, of (5.84). Such a situation was illustrated in Figure 5.3. In situations where the validity of the quasi-steadystate approximation is in doubt, one can use the full optical Bloch equations, (5.109) and (5.110).
6.1 Ground-State Evolution
The part of (5.81) due to optical pumping (the terms proportional to VQ and VQ † ) is i„
d fg gg D VQ † Qfe gg Qfg eg VQ Q dt Q W Q † )VQ Q Qfg gg ) (W Q † Qfe eg Qfg gg W D VQ † (Qfe eg W d fg gg d fg gg . D i„ Q g g C Q g e dt dt
(6.9)
6.1 Ground-State Evolution
The transfer of atoms from the excited state to the ground state accompanied by stimulated emission of light is i„
d fg gg Q W Q † Qfe eg VQ . Q g e D VQ † Qfe eg W dt
(6.10)
The transfer of atoms from the ground state to the excited state accompanied by absorption of light is given by i„
d fg gg Q g g D δ HQ fgg Qfg gg Qfg gg δ HQ fgg† . dt
(6.11)
The effective Hamiltonian, δ H fgg , is Q D δ EQ fgg δ HQ fgg D VQ † W
i„ Q fgg . δΓ 2
(6.12)
The Hermitian light-shift operator is 1 δ EQ fgg D (δ HQ fgg C δ HQ fgg† ), 2
(6.13)
and the Hermitian light-damping operator is δ ΓQ fgg D
i (δ HQ fgg δ HQ fgg†). „
(6.14)
From inspection of (6.14) and (6.11) we see that d fg gg D Tr[Qfg gg δ ΓQ fgg ] D hδ Γ fgg i . Q g g Tr dt
(6.15)
The rate of depletion of ground-state atoms by depopulation pumping is the expectation value of the light-damping operator δ Γ fgg . Liouville Space
The Liouville-space version of (6.11) is
ˇ d ˇˇ fg gg fgg fg gg ˇ D Γp AQ p ˇQfg gg . ˇQ g g dt
(6.16)
The mean pumping rate for unpolarized, ground-state atoms is fgg
Γp
D
1 (δ Γ fgg j1fgg ) . g fgg
(6.17) fg gg
According to (4.75) and (4.77), the dimensionless operator A p iδ HQ fgg© fg gg . AQ p D fgg „Γp
is (6.18)
The Liouville-space version of (6.10) is d ˇˇ fg gg fg eg D Γpfeg AQ p jQfe eg ) , ˇQ g e dt
(6.19)
75
76
6 Quasi-Steady-State Optical Pumping
where the mean pumping rate of unpolarized excited-state atoms can be defined in terms of the rate (6.17) for ground-state atoms by fgg
g feg Γpfeg D g fgg Γp
.
(6.20) fg eg
According to (6.10) and (4.74), the dimensionless operator AQ p
is defined by
fg eg Q T ˝ VQ † VQ T ˝ W Q†. i„Γpfeg AQ p D W
(6.21)
6.2 Excited-State Evolution
The part of (5.78) due to optical pumping is i„
d fe eg D VQ Qfg eg Qfe gg VQ † Q dt Q † ) (Q fe eg W Q W Q † Qfe eg Qfg gg W Q Qfg gg )VQ † D VQ (W d fe eg d fe eg . D i„ Q e e C Q dt d t eg
(6.22)
The part of (6.22) that describes the transfer of atoms from the ground state to the excited state accompanied by the absorption of light is i„
d fe eg Q†. Q Qfg gg VQ † VQ Qfg gg W DW Q d t eg
(6.23)
The part of (6.22) that describes the transfer of atoms from the excited state to the ground state accompanied by stimulated emission of light is i„
d fe eg D δ HQ feg Qfe eg Qfe eg δ HQ feg† . Q d t ee
(6.24)
The effective Hamiltonian, δ H feg , of (6.24) is Q † D δ EQ feg δ HQ feg D VQ W
i„ Q feg . δΓ 2
(6.25)
The Hermitian light-shift operator is 1 δ EQ feg D (δ HQ feg C δ HQ feg† ), 2
(6.26)
and the Hermitian light-damping operator is δ ΓQ feg D
i (δ HQ feg δ HQ feg† ). „
From inspection of (6.24) and (6.27) we see that d fe eg D Tr[Qfe eg δ ΓQ feg ] D hδ Γ feg i . Q e e Tr dt
(6.27)
(6.28)
6.3 Collisions
The rate of depletion of excited-state atoms by stimulated emission is the expectation value of the light-damping operator δ Γ feg . Comparing the traces of (6.11) and (6.23) or (6.10) and (6.24), we find d fe eg d fg gg Tr D Tr , (6.29) Q e g Q g g dt dt d fe eg d fg gg D Tr . (6.30) Tr Q g e Q dt d t ee From (6.29) we see that for every atom removed from the ground state by absorption of a photon, another atom is created in the excited state; from (6.30) we see that for every atom removed from the excited state by simulated emission of a photon, another atom is created in the ground state. Liouville Space
The Liouville-space version of (6.24) is
d fe eg eg fe eg ) D Γpfeg AQfe ). jQ p jQ d t ee
(6.31)
The mean pumping rate for unpolarized, excited-state atoms is Γpfeg D
1 (δ ΓQ feg j1feg ) . g feg
(6.32)
One can readily verify that (6.32) is consistent with (6.20). The dimensionless opereg ator AQfe is defined by p iδ HQ feg© eg AQfe D . p „Γpfeg
(6.33)
The Liouville-space version of (6.23) is d fe eg fgg fe gg ) D Γp AQ p jQfg gg ) . jQ d t eg fe gg
The dimensionless operator AQ p fgg
i„Γp
(6.34)
is defined by
fe gg Q W Q ˝ VQ . AQ p D VQ ˝ W
(6.35)
6.3 Collisions
Let the atoms have a nominal collision rate Γc with buffer-gas atoms or molecules. As we shall discuss more extensively in subsequent sections, the collisions will cause the density matrix to evolve at the rate eg fe eg jPcfe eg ) D Γc Afe j ), c i h fg gg fg eg fg gg jPc ) D Γc A c jfe eg ) A c jfg gg ) .
(6.36) (6.37)
77
78
6 Quasi-Steady-State Optical Pumping eg The dimensionless matrix Afe describes the quenching of excited atoms due to c collisions with molecular buffer-gas atoms such as N2 and also transfer from one fg eg sublevel of the exited-state atom to another. The dimensionless matrix A c describes the number and spin polarization of ground-state atoms created by quenchfg gg ing collisions, and A c describes the collisional transfer of atoms between sublevels of the ground state. We assume the energies of the excited atoms are so great we can neglect endothermic collisions that produce excited-state atoms from ground-state atoms. The total number of atoms must be conserved by collisions, so we must have fg gg
(1feg jPcfe eg ) C (1fgg jPc
)D0.
(6.38)
6.4 Saturation
We will define a fully saturated density matrix as one for which each sublevel of the ground state is equally likely to be populated, and there are no coherences. Then the nonzero parts of the density matrix are fg gg
st
D
1fgg g
and
stfe eg D
1feg , g
or
jst ) D
1 j1) , g
(6.39)
where j1) was given by (4.28). The total number of subevels in the ground state and excited state is g D g fgg C g feg .
(6.40)
One can readily verify that for the saturated density matrix (6.39), the left sides of (6.9) and (6.22) are zero, d fg gg Qgg C dt d fe eg C Q d t ee
d fg gg Qge D 0 , dt d fe eg D0. Q d t eg
(6.41) (6.42)
Atoms with equal populations of all ground-state and excited-state sublevels, as expressed by (6.39), almost never occur in practice, since polarized or directional laser light of sufficient intensity to equalize the sublevel populations of the groundstate and the excited states is also likely to produce strong spin polarization in both states.
6.5 Identities
One of the easiest ways to check the numerical values of the superoperators that govern quasi-steady-state pumping is to be sure they satisfy various identities that
6.5 Identities
we have mentioned explicitly or are implicit in the discussion above. We summarize some of these identities here. The Liouville-space versions of the pumping identities (6.29) and (6.30) are fe gg
D (1fgg jA p
,
(6.43)
fg eg
eg D (1feg jAfe . p
(6.44)
(1feg jA p
(1fgg jA p
fg gg
The Liouville-space versions of the saturation identities (6.41) and (6.42) are fg gg
Ap
fg eg
j1fgg )/g fgg D A p
fe gg
eg feg Afe )/g feg D A p p j1
j1feg )/g feg ,
(6.45)
j1fgg )/g fgg .
(6.46)
Multiplying (6.16) on the left by (1fgg j, and noting that if jfg gg ) D j1fgg )/g fgg we fg gg fgg will have (1fgg jPdp ) D Γp , and carrying out analogous steps with (6.31), we find fg gg
(1fgg jA p
j1fgg ) D g fgg
eg feg and (1feg jAfe ) D g feg . p j1
(6.47)
Multiplying (6.43) and (6.44) on the right by j1fgg ) and j1feg ), respectively, and using (6.47), we find fe gg
(1feg jA p
j1fgg ) D g fgg
fg eg
and (1fgg jA p
j1feg ) D g feg .
(6.48)
The Liouville-space version of the spontaneous-emission identity (5.48) is fg eg
(1fgg jA s
D (1feg j ,
fg eg
which gives (1fgg jA s
j1feg ) D g feg .
(6.49)
Setting fe eg D 1feg in (5.47), evaluating the result with (5.37), and transforming to Liouville space, we find a second spontaneous-emission identity fg eg
As
j1feg )/g feg D j1fgg )/g fgg .
(6.50)
Using (6.38) with (6.36) and (6.37), we find the collisional-relaxation identities fg eg
eg D (1feg jAfe , c
(6.51)
fg gg
D0.
(6.52)
(1fgg jA c (1fgg jA c
When using identities like those in this section to check the validity of large matrices, one might, for example, execute code that evaluates the column vector fe gg eg feg Afe )/g feg A p j1fgg )/g fgg , which should be a column vector whose elp j1 ements are all zero according to (6.46). Some of the elements of the numerical column vector that is generated may be nonzero, with magnitudes on the order of the round-off error of the computation. This does not mean there is something fe gg eg wrong with the matrices Afe and A p . p
79
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6 Quasi-Steady-State Optical Pumping
From inspection of (5.86) and (5.89) we see that ( μN μjK N fe egˇ j μN μ) N D 0 and fg ggˇ (μ μjK jμ μ) D 0, so we must have (1feg jK fe egˇ D 0
and
K fe egˇ j1feg ) D 0 .
(6.53)
(1fgg jK fg ggˇ D 0
and
K fg ggˇj1fgg ) D 0 .
(6.54)
There are identities analogous to (6.53) and (6.54) for H fe eg and H fg gg, that is, (5.90) and (5.93).
6.6 Net Evolution
It will be convenient to write the excited-state and ground-state components of the density matrix as the single column vector fe eg ) jQ . (6.55) Q D j ρ) jQfg gg ) Combining the results of the discussion above, we find that the quasi-steady-state evolution of the excited-state and ground-state density matrices due to optical pumping, collisions, hyperfine couplings, and radiative decay is given by d Q ρ) j ρ) Q D Gj Q . dt Here GQ has the block elements, fe eg GQ GQ fe gg GQ D , GQ fg eg GQ fg gg
(6.56)
(6.57)
which couple the components jQfe eg ) and jQfg gg ) of (6.55): eg eg GQ fe eg D iE fe egˇ /„ C Γpfeg AQfe C Γc Afe , p c
(6.58)
fg eg fg eg fg eg fg eg GQ fg eg D Γs A s Γpfeg AQ p Γc A c ,
(6.59)
fgg fe gg GQ fe gg D Γp AQ p ,
(6.60)
fgg fg gg fg gg GQ fg gg D iE fg ggˇ/„ C Γp AQ p C Γc A c .
(6.61)
The steady-state solution of (6.56) is jρ 1 ), given by Q ρQ 1 ) D 0 , Gj where G is the damping matrix (6.57), and the normalization is given by fg gg fe eg C 1fgg jQ1 D 1 . 1feg jQ1
(6.62)
(6.63)
6.7 Negligible Stimulated Emission
6.7 Negligible Stimulated Emission fgg
We often encounter situations where the optical pumping rates Γpfeg and Γp
of
fg eg Γs .
Under these the atom are much slower than the spontaneous decay rate conditions, the optically pumped atoms will reach a quasi steady state after a few radiative lifetimes, and this will be followed by a much slower transient approach to equilibrium, with time constants on the order of the optical pumping rates and various ground-state collisional relaxation rates. The first “row” of (6.56) is d fe eg ) D GQ fe eg jQfe eg ) GQ fe gg jQfg gg ) . jQ dt
(6.64)
The formal solution of (6.64) is eg ) jQfe t
Zt D
Q feeg (tt 0)
d t 0 e G
fg gg GQ fe gg jQ t 0 ) ,
(6.65)
1 fg gg
eg where jQfe ) is the value of jQfe eg ) at time t and j t 0 ) is the value of jfg gg ) at an t earlier time t 0 . The exponential damping factor of (6.65) will be negligibly small for retarded times t 0 more than a few radiative lifetimes earlier than t. If we assume the damping times of the ground-state atoms and the oscillation periods of any ground-state coherences are much longer than the excited-state damping time, we can make the approximation fg gg
jQ t 0
fg gg
) D jQ t
),
(6.66)
jQfe eg ) D GQ fe eg1 GQ fe gg jQfg gg ) .
(6.67)
in (6.65) to find
As we will discuss in Chapter 7 on modulation, the approximation (6.66) needs to be modified if either jQfg gg ) or G fe gg or both have components oscillating with periods that are comparable to or shorter than the damping times of the excited state. This is often an issue in optically pumped atomic clocks, where the hyperfine clock frequency can be much larger than the radiative decay rate of excited atoms. Substituting (6.67) into the second “row” of (6.56), we find that the relatively slow evolution of the ground-state density matrix is described by d fg gg ) D f GQ fg gg GQ fg eg GQ fe eg1 GQ fe gg gjQfg gg ) . jQ dt
(6.68)
Since we have assumed that the stimulated emission of excited atoms is negligibly small compared with the much faster radiative decay and quenching, we can set Γpfeg D 0 in (6.59), so (6.68) becomes d fg gg fg gg fgg ) D fiH fg ggˇ/„ C Γc A c C Γp ( AQ dp AQ rp )gjQfg gg ) . jQ dt
(6.69)
81
82
6 Quasi-Steady-State Optical Pumping
The depopulation-pumping matrix is fg gg AQ dp D AQ p ,
(6.70)
and the repopulation-pumping matrix is AQ rp D
1
GQ fg eg GQ fe eg1 GQ fe gg fgg Γp n o fg eg fg eg fg eg D Γs AQ s C Γc AQ c
1 fg eg ˇ fe gg eg iH fe eg /„ C Γs AQ p . C Γc AQfe c
(6.71)
One can use the identities in Section 6.5 to show that (1fgg j( AQ dp AQ rp ) D 0 .
(6.72)
6.8 High-Pressure Pumping
The preceding discussion is particularly pertinent for nearly collision free conditions or low buffer-gas pressures where there is little collisional broadening of the optical absorption lines. We will give some specific examples of this analysis in Section 9.3. Here we discuss a simpler situation, the optical pumping of alkali-metal atoms at high gas pressures, a situation that frequently prevails for spin-exchange optical pumping of the nuclei of noble gas atoms. Saturation of the optical pumping transition is seldom an issue for high-pressure pumping, so we will assume unsaturated conditions (negligible stimulated emission) in the following discussions, and we will set fg gg D . For sufficiently high buffer-gas pressure, the damping rate of optical coherence γ2 can greatly exceed the hyperfine splittings or Doppler shifts, and it will be a good approximation to set E fe gg D „ ω ω fe gg C i γ2 in (5.84). Then (6.12) becomes δ H fgg D
jE j2 jDj2 e Δ Δ † e . „ ω ω fe gg C iγ2
(6.73)
The polarization unit vector of the light is eD
E . jEj
(6.74)
For alkali-metal atoms, with S D 1/2, and with J D 1/2 or J D 3/2 we can use (5.41) to write e Δ Δ† e D
2 1 (1) J S sS , 2 [ J]
(6.75)
6.8 High-Pressure Pumping
where the mean photon spin is s D ie e .
(6.76)
Substituting (6.75) into (6.73) and using(5.45), we find i„ fgg fgg (1 4K S ) , δ H fgg D Ep Γp 2
(6.77)
where fgg
Ep
π re c f eg S i„ fgg . Γp D fe gg
2 ω ω fe gg C iγ2 ω
(6.78)
Here S, the mean energy flux of the light, was given by (5.61). It will turn out that high-pressure optical pumping is described by precisely the fgg same evolution equations as spin exchange at the rate Γp with fictitious alkalimetal atoms. The electron spin of the fictitious atoms has the expectation value KD
(1) J S s. [ J]
(6.79)
This is half the mean photon spin s for D1 pumping with J D 1/2 and minus one quarter of the photon spin for D2 pumping with J D 3/2. Substituting (6.77) into (6.11), we find that the depopulation pumping rate is fgg
Pdp D
4Ep i„
fgg
[K S , ] Γp
( 2 fK S , g) .
(6.80)
Collisions of the polarized excited atoms generated by optical pumping play a central role in high-pressure optical pumping. A more detailed discussion of these issues is given in Section 10.14, but we will briefly summarize the key physics here. The electronic polarization of excited atoms is efficiently destroyed in collisions with buffer-gas atoms or molecules. But a collision has little direct effect on the nuclear spin polarization, since the direct interaction must occur through a modification of the hyperfine coupling interaction during the collisional time. The hyperfine coupling strengths are too small to have any appreciable effect on the nuclear polarization during the few-picosecond duration of a collision. If the collisions occur at frequent enough intervals, which will be very nearly the case for high-pressure optical pumping, the hyperfine coupling interactions of the nucleus with the electrons will fluctuate so rapidly in sign and orientation that there will be negligible modification of the nuclear polarization, but nearly complete destruction of the electronic polarization, before the atom radiatively decays or undergoes a quenching collision. We will therefore assume that at high gas pressures the electronic polarization of the excited atoms will be destroyed, but the nuclear polarization will be conserved. As we will discuss in more detail in connection with (11.20), the part of the density matrix with only nuclear polarization is /4 C S S . We have assumed that
83
84
6 Quasi-Steady-State Optical Pumping
the atoms lost by depopulation pumping will be returned from the excited state with the original nuclear polarization intact, so the repopulation pumping rate will be 1 Prp D Pdp S Pdp S . 4
(6.81)
The total optical pumping rate is therefore Pop D Pdp C Prp D
3 Pdp S Pdp S . 4
(6.82)
Substituting the depopulation pumping rate (6.80) into (6.82) and simplifying the result with the aid of the identity δi j i C i j k Sk , 4 2
Si S j D
for the angular momentum operators of a spin-1/2 particle, we find fgg 3 Pop D Γp S S K [fS , g 2iS S ] 4 2i [K S, ] .
(6.83)
(6.84)
The shift parameter is fgg
D
2Ep
fgg
„Γp
.
(6.85)
Equation 6.84 is formally identical to the basic evolution equation for spin exfgg change, (10.253). So (6.84) describes spin-exchange collisions (at the rate Γp and with a frequency-shift parameter ) of the real alkali-metal atoms with fictitious alkali-metal atoms of electron spin polarization K. The fictitious atoms are the photons of the pumping light. 6.8.1 Liouville Space
Adding in the evolution due to the ground-state Hamiltonian to the high-pressure optical pumping of (6.84), we find that the damping operator G of (1.10) becomes GD
i fgg© fgg C Γp A op , H „
(6.86)
where the optical pumping matrix is A op D A sd K Aex 2i K S © .
(6.87)
6.8 High-Pressure Pumping
Here H fgg© and S © are commutator superoperators, generated in accordance with (4.85) for the ground-state Hamiltonian H fgg of (2.10) and the ground-state electronic spin operator S . The S-damping operator is A sd D
3 fg gg 1 S[ S] D S© S© . 1 4 2
(6.88)
For (6.88), the unit matrix, 1fg gg , for the ground state of Liouville space was given by (5.112), and we noted that S[ S[ D S] S] D
3 fg gg . 1 4
(6.89)
The spin-exchange operator is Aex D S [ C S ] 2iS [ S ] .
(6.90)
When appended to the program ending with Code 5.3 on page 57, the following MATLAB statements will evaluate the transients and steady state for high-pressure optical pumping. Code 6.1 Gmp=input(’Mean pumping rate, Gmp = ’); kappa=input(’Shift parameter, kappa = ’); Kj=input(’Fictitious spin, [Kx Ky Kz] = ’); for k=1:3 sharpS(:,:,k)=sharp(Sj(:,:,k));%sharp electron spin matrices flatS(:,:,k)=flat(Sj(:,:,k));%flat electron spin matrices end SC=flatS-sharpS;%Liouville-space spin matrices Asd=matdot(SC,SC)/2;%S-damping matrix Aex=flatS+sharpS-2*i*matcross(flatS,sharpS);%exchange matrix %high-pressure optical pumping matrix Aop=Asd-Kj(1)*Aex(:,:,1)-Kj(2)*Aex(:,:,2)-Kj(3)*Aex(:,:,3)... -2*i*kappa*(Kj(1)*SC(:,:,1)+Kj(2)*SC(:,:,2)+Kj(3)*SC(:,:,3)); HgC=flat(Hg)-sharp(Hg);%Liouville-space Hamiltonian G=i*HgC/hbar+Gmp*Aop;%static damping operator Lp=logical(cPg);%logical variable for populations nt=100;%number of sample points t=linspace(0,40,nt);%sample times in units of 1/Gmp rhoc=zeros(gg,nt);%initialize compactified density matrix for k=1:nt%evaluate transient rhoc(:,k)=expm(-t(k)*G(Lp,Lp)/Gmp)*cPg(Lp)/gg; end clf; plot(t,rhoc); grid on; hold on rhocin=null(G(Lp,Lp));rhocin=rhocin/sum(rhocin);%steady state plot(t,rhocin*ones(1,nt), ’-.’) xlabel(’Time in units of 1/\Gamma_p^\fg\gg’) ylabel(’Sublevel populations’)
85
86
6 Quasi-Steady-State Optical Pumping
Subroutines to generate flat and sharp supermatrices were given by Code 4.3 and Code 4.4 on page 46. The cross product of the spin matrices was evaluated with the following simple subroutine. Code 6.2 function C=matcross(A,B) C(:,:,1)=A(:,:,2)*B(:,:,3)-A(:,:,3)*B(:,:,2); C(:,:,2)=A(:,:,3)*B(:,:,1)-A(:,:,1)*B(:,:,3); C(:,:,3)=A(:,:,1)*B(:,:,2)-A(:,:,2)*B(:,:,1);
In Figure 6.1 we show an optical pumping transient calculated with code that ends with Code 6.1 on page 85. Except at very small magnetic fields, the coherences of the density matrix will be neglibibly small, and the problem has been compactified to populations. High-pressure pumping is equivalent to spin exchange with a fictitious second species of alkali-metal atom with mean spin polarization hS i D K. The equilibrium populations of sublevels of azimuthal quantum num-
Figure 6.1 Transient approach to equilibrium as evaluated with the code ending with Code 6.1 on page 85. The equilibrium populations are indicated by the horizontal lines. The
input parameters were I = 1.5, S = 0.5, J = 0.5, B = 1, Gmp = 1000, kappa = 0, and [Kx Ky Kz] = [0 0 0.25].
6.9 Spectral Width of Pumping Light
ber m, the late-time asymptotes of the figure, are proportional to e β m . Here β D ln([1C K z ]/[1 K z ]) is the spin-temperature parameter. Since we used K z D 0.25 to generate Figure 6.1, the steady state corresponds to β D ln 3, or a population ratio e β D 3. Each late-time asymptote of Figure 6.1 is a factor of 3 larger than the next-smaller one. We will discuss issues related to spin exchange in more detail in Section 10.12.
6.9 Spectral Width of Pumping Light
So far we have discussed monochromatic light that pumps atoms with a welldefined velocity v. As shown in (5.95), we have modeled collisions with bufferfe gg gas atoms by adding a collisional contribution, Γc α c , to the damping rate, γ2 , of the optical coherence. The homogeneous linewidth resulting from γ2 can be power-broadened at sufficiently high light intensities. Here we discuss two potentially important inhomogeneous line-broadening mechanisms, the spectral width of the light and Doppler broadening. Suppose that the light source has a wellcharacterized spectral width, ΔΩ . Then we will write the optical interaction amplitude of (5.63) as the time-dependent operator X VQ D VQ (n) eiΩn t . (6.91) n
We think of the amplitudes VQ (n) as variables that have the same stationary statistics as the pumping light. The amplitude products have the ensemble averages ˝ †(m) (n) ˛ D δ m n f n VQ † VQ . VQ VQ (6.92) Here VQ † and VQ are the amplitudes for purely monochromatic light of the same average power as the broadband light, and f n is the probability that the light will have frequency ω C Ωn . As indicated by the factor δ m n in (6.92), we assume that the amplitudes of the different frequencies are uncorrelated, so they are unable to drive coherent population trapping resonances in the atom. In Chapter 7 we will discuss light that has been intentionally modulated to drive coherent population trapping resonances. We assume that X fn D 1 . (6.93) n
We can think of (6.91) as a discrete Fourier transform with frequencies Ωn D
2π n , T
We assume that 2π ΔΩ , T
with
n D 0, ˙1, ˙2, . . .
(6.94)
(6.95)
so there are many sampling waves within the spectral bandwidth of the light.
87
88
6 Quasi-Steady-State Optical Pumping
For steady-state conditions, we can use (6.91) in (6.4) to find Q μN μ D W
iΩn t X VQ μ(n) Nμe fe gg
E μN μ „Ωn
n
.
(6.96)
fe gg
The energy shifts, E μN μ , due to hyperfine structure, magnetic fields, and the Doppler shift, were given by (5.84). Then we can account for the spectral width of the light in coupling terms that contain products like V † W , for example, the effective Hamiltonian (6.12) of ground-state atoms, 1 D E X † fgg fe gg , (6.97) VQ ν μN VQ μN μ E μN μ „Ω δ Hν μ D μN
fe gg
by replacing the complex energy-difference matrix E μN μ averaged matrix 1 X fn fe gg D E μN μ „Ω . fe gg n E μN μ „Ωn
„Ω by the spectrally
(6.98)
6.9.1 Gaussian Spectral Profiles
The spectral widths of lasers can often be modeled adequately with Gaussian spectral profiles. Then the probability that the laser frequency is displaced from the mean frequency ω by an amount between Ω and Ω C d Ω is f (Ω )d Ω D
dΩ 2 2 p eΩ /2σ ω . σ ω 2π
(6.99)
Letting f n ! f (Ω )d Ω in (6.98) and using (6.99), we find 1 fe gg D E μN μ „Ω
1 p σ ω 2π
Z
eΩ
2 /2σ 2 ω
fe gg E μN μ
dΩ
C „Ω
.
(6.100)
p Making the change of variable Ω D 2σ ω t, we find 1 fe gg D E μN μ „Ω
1 p
„σ ω 2π
Z
2
Z(z μN μ ) et d t D p . t z μN μ „σ ω 2
(6.101)
The plasma dispersion function [47], Z(z μN μ ) of (6.101), has the complex argument fe gg
z μN μ
E μN μ D p . „σ ω 2
(6.102)
6.9 Spectral Width of Pumping Light
6.9.2 Plasma Dispersion Function
We conclude this section by summarizing a few useful properties of the plasma dispersion function 1 Z(z) D p π
Z1 1
2
et d t tz
Im(z) 0 .
for
(6.103)
Let the argument z D x C iy have a real part x and an imaginary part iy. The “area” of Z(z) is Z1 Z(x C iy)d x D iπ .
(6.104)
1
A useful symmetry is Z (x C iy) D Z(x C iy) .
(6.105)
For jzj 1, it is often convenient to use the asymptotic expression Z(z)
1 . z
(6.106)
The plasma dispersion function Z(z) is a simple multiple of the convenient Faddeeva function, w (z), where p Z(z) D i πw (z) .
(6.107)
An expression for the Faddeeva function, valid for all finite z, is w (z) D e
z 2
1 erfc(iz) D p π
Z1 2 eiz uu /4 d u .
(6.108)
0
The integral on d u in (6.108) extends over all real nonnegative values of u. The complementary error function is defined for the complex number z by 2 erfc(z) D p π
Z1 2 et d t .
(6.109)
z
The integral starts from t D z in the complex t plane and extends to t D 1 along the positive real axis.
89
90
6 Quasi-Steady-State Optical Pumping
An efficient MATLAB code due to Weideman et al. [48] to evaluate the Faddeeva function is Code 6.3 function f=w(z); %Computes the function w(z)=exp(-z^2)erfc(-iz) using a rational %series with N terms. It is assumed that Im(z)>0 or Im(z)=0. %We use N=10, but higher accuracy can be obtained with larger N. N=10; M = 2*N; M2 = 2*M; k = [-M+1:1:M-1]’;%M2 = no. of sampling points. L = sqrt(N/sqrt(2));%Optimal choice of L. theta = k*pi/M; t = L*tan(theta/2);%Variables theta and t. f = exp(-t.^2).*(L^2+t.^2);f=[0;f];%Function to be transformed. a = real(fft(fftshift(f)))/M2;%Coefficients of transform. a = flipud(a(2:N+1));%Reorder coefficients. Z = (L+i*z)./(L-i*z); p = polyval(a,Z);%Polynomial evaluation. f = 2*p./(L-i*z).^2+(1/sqrt(pi))./(L-i*z);%Evaluate w(z).
6.10 Doppler Broadening
Let the atoms have a Maxwell–Boltzmann distribution of velocities, characterized by an absolute temperature T, and let v be the component of the atomic velocity along the direction of propagation of the light. The probability of finding the velocity between v and v C d v is Gaussian, and is given by W(v )d v D
dv 2 2 p ev /2σ v , σ v 2π
(6.110)
where the velocity variance is σ 2v D
kB T . M
(6.111)
Here kB is Boltzmann’s constant and M is the atomic mass. Assuming broadening from both the velocity distribution of the atoms and the spectral width of the pumping light, we can write the averaged resonance factor as 1
1 D E D fe gg σ σ ω v 2π E μN μ
Z
e( Ω
2 /2σ 2 Cv 2 /2σ 2 ω v
fe gg
)dΩ dv
H μN μ „(ω ω fe gg k v Ω ) i„γ2
Introducing the new variables p p x D v /σ v 2 and y D Ω /σ ω 2 ,
.
(6.112)
(6.113)
6.10 Doppler Broadening
we rewrite (6.112) as D
1 fe gg E μN μ
E D
1 π
Z fe gg H μN μ
„ ω
e(x ω fe gg
2 Cy2 )
d x dy . p 2 [k σ v x C σ ω y] i„γ2 (6.114)
Changing variables once more to t D [k σ v x C σ ω y]/σ and s D [σ ω x C k σ v y]/σ, where the net frequency variance is σ 2 D (k σ v )2 C σ 2ω ,
(6.115)
we find in analogy to (6.101) 1 fe gg hE μN μ i
D
Z(z μN μ ) p . σ„ 2
(6.116)
In analogy to (6.102), the argument of the plasma dispersion function is fe gg
z μN μ D
E μN μ p . „σ 2
(6.117) fe gg
In (6.117) it is to be understood that the matrix E μN μ of (5.84) is to be evaluated at v D 0. For atoms with a Maxwell–Boltzmann distribution of velocities, illuminated by light with a Gaussian spectral profile, the Doppler widths and laser widths add in quadrature, as shown in (6.115).
91
93
7 Modulation Resonances are often induced in optically pumped systems (e.g., atomic clocks or magnetometers) by subjecting them to oscillating (modulated) magnetic fields or to modulated light. If the oscillation frequency is nearly equal to the Bohr frequency for a pair of sublevels, transitions of populations between the sublevels and large modulated coherences can be induced. For clarity, we will focus on the ground state and set fg gg D and H fgg D H , but excited atoms respond to modulation in an analogous way. We will write the density matrix as the generalized Fourier series, X X (m) eimΩ t , or j) D j(m) )eimΩ t . (7.1) D m
m
To account for several different drive frequencies we introduce a drive-frequency vector, Ω , and a set of harmonic-index vectors m. For example, if an alkali-metal atom is subject to one magnetic resonance field oscillating at a radio frequency, Ω f1g , that is nearly resonant for low-field Zeeman resonances, and a second magnetic resonance field oscillating at a microwave frequency, Ω f2g , that is nearly resonant for the 0–0 microwave clock transition, we can write Ω f1g Ω D . (7.2) Ω f2g A typical index vector for this case would be h i m D m f1g , m f2g D [2, 1] .
(7.3)
These magnetic resonance fields could induce the coherence (kj) D ha2jjb0i to oscillate at the frequency m Ω D 2Ω f1g C Ω f2g , which is close to the Bohr frequency, ω k of the coherence. Here we are using low-field labels for the groundstate sublevels, j f mi, of an alkali-metal atom, with a D I C 1/2 and b D I 1/2. Alternatively, we use the single label k, such that jk) D ja2b0). The coherence is created when the atom simultaneously absorbs (or emits) m f1g D 2 photons of frequency Ω f1g and m f2g D 1 photon of frequency Ω f2g . In (7.1) and elsewhere, a quantity with the superscript (m) denotes the amplitude of a term that oscillates as eimΩ t . Since is Hermitian, we must have (m)† D (m) . Optically Pumped Atoms. William Happer, Yuan-Yu Jau, and Thad Walker Copyright © 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-40707-1
(7.4)
94
7 Modulation
Anticipating a point we will discuss later, in many cases the Fourier series for the element (kj(m) ) will contain only one potentially large term with m D n k , where n k , is the “multiple quantum index” of the basis state jk). Neglecting all Fourier amplitudes, (kj(m) ), except for (kj(n k ) ) is called the secular approximation.
7.1 Magnetic Resonance
A common way to create coherence is to apply a magnetic field oscillating at a frequency or frequencies close to that of one or more Bohr frequencies of the atom. We write the field as X BD B (m) eimΩ t . (7.5) m
Since B is real, we must have B (m) D B (m) .
(7.6)
The Schrödinger-space Hamiltonian has a Fourier series analogous to (7.1), X H (m) eimΩ t . (7.7) HD m
The Fourier amplitudes of (7.7), with m ¤ 0, are H (m) D μ fgg B (m) ,
(7.8)
where the magnetic-moment operator, μ fgg , was defined by (2.11). The Hermitian conjugates are H (m)† D H (m) . The Liouville-space versions of (7.7)–(7.9) are X H (m)© eimΩ t , where H (m)© D H (m)[ H (m)] . H© D
(7.9)
(7.10)
m
We will give examples of how to use the magnetic resonance interaction (7.10) to calculate single- and multiple-quantum resonance spectra at the end of this chapter.
7.2 Modulated Light
Light that is modulated in amplitude, frequency, or polarization is often used to excite resonances in optically pumped atoms. These are often called “coherent population trapping” (CPT) resonances.
7.2 Modulated Light
7.2.1 High Pressure
For the simple case of high-pressure optical pumping that we discussed in Section 6.8, we can represent intensity modulation of the light by writing the mean pumping rate as fgg
Γp
f (t) ,
(7.11)
fgg
where Γp is time-independent and where the modulation factor f (t) has the Fourier series X f (t) D f (m) eimΩ t , with f (0) D 1 and f (m) D f (m) . (7.12) m
The amplitudes f (m) must be such that f is never negative. For a single modulation frequency, such that m Ω D m Ω , a useful model for f is f D
e α cos Ω t , J0 (iα)
with
f (m) D
J m (iα) . i m J0 (iα)
(7.13)
Here J m (z) is the mth-order Bessel function of the complex argument z. The Fourier amplitudes, f (m) , of f can be found with the well-known generating function [19], sometimes used to define Bessel functions, X eiz sin φ D eim φ J m (z) . (7.14) m
The individual peaks of the pulse sequence (7.12) get narrower as the positive exponential coefficient, α, gets bigger. The Fourier amplitudes, f f(m) , of practical devices for light modulation, for example, Mach–Zehnder electro-optical modulators, depend on m in a way that is qualitatively similar to the simple model of (7.13), but the quantitative details are different. For the situation described by (7.11) the Fourier amplitudes of the optical pumping matrix are (m)
A op D A op f (m) ,
(7.15)
with A op given by (6.87). At the end of this chapter, we will give an example of CPT resonances excited by intensity-modulated light for high-pressure optical pumping. 7.2.2 Lower Pressure
At lower buffer-gas pressures, the more detailed features of the optical absorption line must be taken into account. As an example of how to analyze this situation, consider a superposition of plane waves with different frequencies but all propagating along the unit vector n of (5.57), X ( p) EQ ei(k p rω p t) . ED (7.16) p
95
96
7 Modulation
This is analogous to the monochromatic wave (5.55). The temporal frequencies ω p and the spatial frequencies k p for the optical sideband p are ωp D ω C p Ω
and
kp D
ωp n. c
(7.17)
In analogy to (7.3), we write the modulation indices as p D [p f1g , p f2g , . . .].
(7.18)
The elements p f j g will normally be integers, as in (7.3). With (7.16) we can generalize (5.62) to X VD V(p) . (7.19) p
The Fourier components are V ( p) D VQ ( p) ei(k p v ω p )t ,
where
( p) VQ ( p) D D † EQ eik p r 0 .
We can also generalize (6.8) to X WD W ( p) .
(7.20)
(7.21)
p
The Fourier components are Q ( p ) ei(k p v ω p )t , W ( p) D W
where
Q ( p) D VQ ( p) ./E pfe gg . W
(7.22)
In analogy to (5.84), the energy-difference matrix is fe gg
Ep
D H fe gg C „ Δω p k p v iγ2 ,
Δω p D ω p ω
fe gg
.
where (7.23)
7.2.3 Modulated Optical Pumping Matrices
Products of V † and W or of V and W † are used to construct the coupling matrices for optical pumping in Liouville space. Therefore, the coupling matrices will have terms oscillating at frequencies m Ω , where m D p p 0 is the difference in two light modulation indices, p and p 0 . To account for Doppler shifts of atoms moving with velocity v , the drive frequency vector must be modified to Ω ! Ω (1 n v /c) .
(7.24)
The light propagates along the unit vector n of (5.57). For example, the effective Hamiltonian of the ground state, (6.12), will become X δ HQ fgg(m) eimΩ t , (7.25) δ HQ fgg D m
7.3 Secular Approximation
where δ HQ fgg(m) D
X
Q (mCp ) . VQ ( p )† W
(7.26)
p
In analogy to (6.17) and (6.14), we write the mean pumping rate for ground-state atoms as 1 (δ Γ fgg(0) j1fgg ) , where g fgg i δ Γ fgg(0) D (δ H fgg(0) δ H fgg(0)†). „ fgg
Γp
D
(7.27)
We can readily use (7.19) and (7.21) to construct Fourier series analogous to (7.25) for the other coupling matrices. For example, the coupling coefficient of (6.34) becomes X fe gg(m) fe gg Ap eimΩ t , (7.28) Ap D m
where in analogy to (6.35), the Fourier amplitudes are given by X fe gg(m) Q (mCp) W Q ( p) ˝ VQ (mCp) ). D (VQ ( p) ˝ W i„Γpfeg A p
(7.29)
p
The analysis of low-pressure optical pumping has much in common with the analysis of the physics of magneto-optical traps, which we discuss in Section 9.3.
7.3 Secular Approximation
Having discussed the two most important ways to induce modulated coherence in optically pumped atoms, magnetic resonance and pumping with modulated light, we turn to the most convenient methods for analyzing this situation. In analogy to (4.38), the rate of change of the density matrix is given by d j) D iH (0)© /„ C G j) . dt
(7.30)
Here H (0)© is the Fourier amplitude (7.10) with m D 0, that is, the unmodulated part of the Hamiltonian. The energy basis states of Liouville space are eigenvectors of H (0)©, and the eigenvalues are proportional to the Bohr frequencies H (0)© jk) D „ω k jk) , jk) D jμ ν) and
where
ωk D ωμν D
Eμ Eν . „
(7.31)
The remaining part of the damping operator of (7.30) has the Fourier series X G (m) eimΩ t . (7.32) GD m
97
98
7 Modulation
The Fourier amplitudes are G (0) D
X
(0)
Γx A x
and
G (m) D
x
for
i (m)© X C Γx A(m) H x , „ x
m¤0.
(7.33)
The relaxation process x (e.g., optical pumping or some form of collisional spin relaxation) has a characteristic relaxation rate Γx and a dimensionless coupling matrix X (m) A x eimΩ t . (7.34) Ax D q
Most collisional relaxation processes are not modulated and have only terms with m D 0. However, if the light is modulated at the frequency m Ω , we would expect (m) some Fourier amplitudes, A op , of the optical pumping matrix to be nonzero for m ¤ 0. Multiplying (7.30) on the left by the row vector (kj, we find d (kj) D iω k (kj) (kjG j) . dt
(7.35)
We will restrict our discussion of modulated excitation to situations where the Bohr frequencies, ω k , of coherences are much bigger than any relaxation rates or magnetic resonance coupling rates. Then the first term on the right of (7.35) will be much bigger than the second, which represents relaxation mechanisms or magnetic resonance couplings. If we ignored the second term on the right of (7.35) entirely, the solution would be (kj(t)) D (kj(0)) eiω k t . This is the basic idea of the secular equation which consists in setting (kj(m) ) D 0 ,
unless
ωk m Ω ,
for some
m D nk .
(7.36)
Equation 7.36 defines a (possibly vector) “multiple-quantum number”, n k , and a secular-approximation column vector, jl), with elements given by the logical equation (kjl) D (ω k m Ω ,
for some
m D nk) .
(7.37)
We will always have (kjl) D 1 (true) for populations, since ω k D 0 and therefore n k D 0 for populations. If (kjl) D 0 (false), we will assume that the corresponding coherence (kj) is negligibly small, and we will drop it from the density matrix, which we compactify to a secular subspace, as shown in Figure 7.1. In evaluating (7.37), one should interpret the symbol to mean “equal to within a few natural linewidths”. The results are not very sensitive to how many linewidths are specified, but the computer codes will run faster the fewer “true” elements jl) contains. For states jk) for which (7.37) is true we define the element of the density matrix in the “rotating” frame by (kj) Q D kj(n k ) . (7.38)
7.3 Secular Approximation
Figure 7.1 Omitting negligibly small coherences from the density matrix, , can substantially speed up numerical calculations, with negligible loss in accuracy. Here we assume that an oscillating magnetic resonance field has excited 24 and 42 , and all other coherences are negligibly small. Retaining only
the populations and the nonzero coherences compactifies Liouville space from 16 to only six dimensions. For computer coding, the compactification is conveniently done with a logical compactification matrix, l, with 1’s at the positions of nonzero elements of and 0’s elsewhere.
We also define a “multiple-quantum” operator X N© D n k jk)(kj .
(7.39)
k
The sum extends over all basis states jk) for which (7.37) is true. Then the density matrix, j), in the “laboratory” frame is related to the density matrix j) Q in the “rotating” frame by j) D eiN
© Ω t
j), Q
or
j) Q D eiN
© Ω t
j) .
Substituting (7.39) into (7.30), we find d © © eiN Ω t Q D iH (0)©/„ G eiN Ω t j) Q . iN © Ω j) dt
(7.40)
(7.41)
©
Multiplying both sides of (7.41) on the left by eiN Ω t and dropping rapidly oscillating terms, in accordance with the secular approximation, we find the evolution equation for the rotating-frame density matrix, d Q . j) Q D i[H (0)© /„ N © Ω ] C GQ j) dt
(7.42)
The coupling terms due to the relaxation processes are described by the timeindependent matrix X GQ D G (m) . M fmg . (7.43) m
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100
7 Modulation
The secular-approximation matrices M fmg are arrays of logical variables, designed to eliminate rapidly oscillating coupling terms from (7.42) and defined by (kjM fmg j j ) D δ n k n j ,m .
(7.44)
In the steady state, (7.42) becomes i[H (0)© /„ N © Ω ] C GQ j) Q D0.
Resonance
(7.45)
In practice, we would solve (7.45) by letting j) Q be the null vector of i H (0)©/„ © Q normalized so the sum of the populations is 1. However, we can get N Ω ] C G, some insight by noting that a formal solution of (7.45) is kj QQ (kj) Q D . (7.46) γ k C i (ω k n k Ω ) Here the source rate of the element (kj) Q is X kj GQ j j ( j j) Q , kj QQ D
(7.47)
j ¤k
and the nominal damping rate is Q . γ k D kj Gjk
(7.48)
From (7.46) we see that the coherence (kj) Q will be resonantly enhanced when f1g C its Bohr frequency, ω k , is equal to the “drive frequency”, n k Ω D n f1g k Ω f2g f2g nk Ω C . The energy to make the atom jump from the initial to the final state of the coherence comes from n f1g quanta of frequency Ω f1g , n f2g quanta of k k f jg
frequency Ω f2g , and so on. The multiple-quantum indices, n k , are positive or negative integers, or zero. For populations we have both ω k D 0 and n k D 0, so populations are always “resonantly enhanced” and are always part of the compactified, secular subspace.
7.4 Attenuation of Modulated Coherence in Passing through the Excited State
With this introduction to the physics of modulation, we return to the discussion of quasi-steady-state optical pumping with negligible stimulated emission. The coupling matrix GQ fe eg of (6.56) represents evolution under the influence of the excited-state Hamiltonian, collisions, spontaneous radiative decay, and stimulated emission. All of these evolution mechanisms except stimulated emission will be unmodulated, so if we neglect stimulated emission, the only nonzero Fourier amplitude of GQ fe eg will be the unmodulated amplitude and we can set GQ fe eg D GQ fe eg(0) . By analogous arguments we can set GQ fg eg D GQ fg eg(0) , since without stim-
7.4 Attenuation of Modulated Coherence in Passing through the Excited State
ulated emission, only spontaneous radiative decay and quenching collisions, both unmodulated, contribute to to GQ fg eg . However, the coupling matrices GQ fg gg and GQ fe gg , both containing terms proportional to the potentially modulated absorption of light, should be written as the Fourier series X X GQ fg gg(m) eimΩ t and G fe gg D GQ fe gg(m) eimΩ t . (7.49) G fg gg D m
m
Then (6.64) must be modified to read X d X fe eg(m) imΩ t jQ )e D GQ fe eg jQfe eg(m) )eimΩ t dt m m X GQ fe gg( p) jQfg gg(n) )ei( pCn)Ω t .
(7.50)
p ,n
Equating terms of (7.50) with the same modulation factors, we find X d fe eg(m) GQ fe gg(mn) jQfg gg(n) ) . (7.51) ) D ( GQ fe eg im Ω )jQfe eg(m) ) jQ dt n The formal solution of (7.51) is fe eg(m) ) jQ t
Zt D
Q feeg imΩ )(tt 0)
d t 0 e( G
fg gg(n) GQ fe gg(mn) jQ t 0 ).
(7.52)
1
Assuming that the Fourier amplitudes are nearly comparable to constant for times the damping rates (radiative and collisional) of GQ fe eg im Ω , we can set fg gg(n)
jQ t 0
fg gg(n)
) D jQ t
)
(7.53)
in (7.52) to find jQfe eg(m) ) D ( GQ fe eg im Ω )1
X
GQ fe gg(mn) jQfg gg(n) ) .
(7.54)
n
Substituting (7.54) into the second row of (6.56) and equating terms with the same frequency factors, we find X d fg gg(m) GQ fg gg(mn) jQfg gg(n) ) ) D im Ω jQfg gg(m) ) jQ dt n X fg eg Q fe eg Q GQ fe gg(mn) jQfg gg(n) ). (G im Ω )1 CG
(7.55)
n
From inspection of (7.56) and (6.69)–(6.71) we conclude that for situations where the density matrix has modulated components, the repopulation matrix becomes AQ rp jk) D
1 fgg
Γp
GQ fg eg ( GQ fe eg in j Ω )1 GQ fe gg(n j n k ) jk) .
(7.56)
101
102
7 Modulation
7.5 Examples 7.5.1 Isolated Magnetic Resonances
The simplest resonances are those where only one coherence, and its Hermitian conjugate, can be excited. We assume that the Bohr frequencies, ˙ω k , of the coherences differ from other Bohr frequencies of the atom by many times the resonance linewidth. The linewidth is determined by the relaxation processes, for example, the γ k of (7.48), and in the case of magnetic resonance by the amplitude of the oscillating magnetic field. An example of an isolated resonance would be the microwave end resonance between sublevels j f mi D jb bi and j f mi D jaai, where a D I C 1/2 and b D I 1/2. Assume a microwave magnetic field oscillating along the x axis with a frequency Ω , close to the Bohr frequency, ω e , of the end resonance B D B z C B x x cos Ω t ,
or
B (0) D B z ,
B (1) D B (1) D
Bx x . (7.57) 2
Then the modulated amplitudes of the Hamiltonian, (7.8), are fgg
H (1) D H (1) D B x μ x /2 .
(7.58)
The magnetic dipole moment matrix μ fgg was specified by (2.11). The coupled (energy-basis) version of (2.11) should be used in (7.58). The coherent basis states (end states) that can be excited by the resonant microwave field are je) D jjaaihb bj)
and j eN ) D jjb bihaaj) .
(7.59)
The multiple-quantum numbers n k of (7.37) become n e D 1 and
n Ne D 1 .
(7.60)
Since there is only one drive frequency, we can simplify the vector index n k to the scalar index n k . No other coherences will be excited, and the compactified density matrix will have only populations with n k D 0 and the coherences je) and j eN ). The effective spin operator of (7.39) is therefore N © D je)(ej j eN )( eNj . The damping operator of (7.33) and (7.43) is GQ D A op . M f0g C iH (1)© /„ . M f1g C M f1g ,
(7.61)
(7.62)
where the optical pumping operator, A op , was given by (6.87) and the secularapproximation matrices, M fng , were given by (7.44).
7.5 Examples
The unmodulated part of the Hamiltonian, which appears in the damping operator of (7.45), is H (0)© D H fgg© .
(7.63)
When appended to the program ending with Code 6.1 on page 85, the following MATLAB statements will evaluate how the sublevel populations change when the frequency of a microwave field is tuned through the resonance frequency of the end resonance. Code 7.1 Bx=input(’Peak oscillating field in Gauss. Bx = ’); Hp1=-Bx*mug(:,:,1)/2;%Fourier amplitude of H Hp1C=flat(Hp1)-sharp(Hp1);%Liouville version Lcp1=fgl==a&mgl==a&fgr==b&mgr==b;%logical var. of |aa>
The results of running the code ending with Code 7.1 are shown in Figure 7.2a. For frequencies tuned well off resonance, the populations are in spin-temperature equilibrium, with a population ratio of 3, as in Figure 6.1. Although the populations of the two end sublevels are nearly equalized on resonance, the atoms remain partially spin polarized, since not all sublevels are equally populated. 7.5.2 Zeeman Magnetic Resonances
Low-field Zeeman resonances for alkali-metal atoms are complicated since the radio-frequency field easily drives multiple-quantum transitions in both the upper
103
104
7 Modulation
(a)
(b) Figure 7.2 (a) An isolated “end resonance” driven by microwaves between sublevels j f mi D jaai and j f mi D jbbi. The curves were evaluated with the code ending with Code 7.1 on page 103. (b) Zeeman resonances driven by a magnetic field oscillating along the x axis. The input parameters were I = 0.5, S = 0.5, J = 0.5, B = 10, Gmp
= 1000, kappa = 0, [Kx Ky Kz] = [0 0 0.25], and Bx = 0.02. The curves were evaluated with the code ending with Code 7.2 on page 105. Off resonance, the atoms have a spin temperature and the populations of sublevels with azimuthal quantum number m are proportional to e βm .
and the lower Zeeman multiplets. We will assume the same oscillating field (7.57) is applied to the atoms, but that the oscillation frequency is close to the nominal Zeeman resonance frequency, g S μ B G/(h[I ]) D 2.8/[I ] MHz G1 . It is clear that no hyperfine coherences will be excited, so the secular density matrix will include all populations and Zeeman coherences, that is, the Liouville basis states, jk) D jj f m l ih f m r j) .
(7.64)
States jk) for which the left and right azimuthal quantum numbers are equal, m l D m r , are basis states for populations. The frequency indices n k of (7.37)
7.5 Examples
become n k D (2 f I )(m l m r ) .
(7.65)
The factor (2 f I ) D ˙1 accounts for the different precession directions of angular momentum in the upper and lower hyperfine multiplets. So the analysis of Zeeman resonances differs from that of the isolated end resonance by having many more coherences included in the secular space, and in having multiple quantum indices as large as jn k j D [I ], corresponding to a transition between the sublevels jaai and ja, ai. When appended to the program ending with Code 7.1 on page 103, the following MATLAB statements will evaluate the steady-state values of the sublevel populations as the frequency of a radio-frequency field is tuned through the nominal Zeeman resonance frequency, the frequency of the [I ]-quantum transition of the upper Zeeman multiplet. This frequency is strictly proportional to the value of the magnetic field B. Unlike the single-quantum transitions, it has no “quadratic shifts”. Code 7.2 L=fgl==fgr;%logical var. for Zeeman coherences and populations nk=2*(fgl-I).*mgl-2.*(fgr-I).*mgr;NL=diag(nk);%"spin" Mp1=nk*ones(1,gg*gg)-ones(gg*gg,1)*nk’==1;%secular approx. matrices M0=nk*ones(1,gg*gg)-ones(gg*gg,1)*nk’==0; Mm1=nk*ones(1,gg*gg)-ones(gg*gg,1)*nk’==-1; tHmrC=Hp1C.*(Mp1+Mm1);%secular part of mag. res. interaction G0s=i*(HgC(L,L)+tHmrC(L,L))/hbar+Gmp*tAop(L,L);%no tuning om0=(Eg(1)-Eg(2*a+1))/(2*a*hbar);%resonance frequency rhosin=zeros(sum(L),nnu);%initialize secular density matrix for k=1:nnu Gs=G0s-i*(om0+om(k))*NL(L,L);%include tuning rhosin(:,k)=null(Gs);%evaluate steady state rhosin(:,k)=rhosin(:,k)/(Lp(L)’*rhosin(:,k)); end subplot(2,1,2); plot(om/(2*pi*1e3),real(rhosin(Lp(L),:))’); grid on xlabel(’Radiofrequency detuning in kHz’); ylabel(’Sublevel populations’);
The results of running the code ending with Code 7.2 are shown in Figure 7.2b. The static field B D 10 G is large enough to cause substantial quadratic splitting of the Zeeman multiplet with f D 1, so the two possible 1-quantum resonances, at approximate detunings of ˙58 kHz, are well resolved. The peak value of the transverse oscillating field, B x D 0.02 G, is large enough to nearly saturate the 1-quantum transitions, equalizing the populations of the pair of coupled sublevels. Some spin polarization is left at the 1-quantum resonance peaks, since all four sublevels are not equally populated. There is a much sharper 2-quantum transition at zero detuning. The field B x is large enough to saturate the 2-quantum transition and also to nearly destroy the spin polarization of the atom, leaving each of the four sublevels with about one quarter of the population. Most of the resonance
105
7 Modulation
linewidth is due to radio-frequency power broadening, but from (7.46) we see that in the limit of negligible power broadening, the natural width of a resonance is γ k /jn k j. The “natural” frequency width is inversely proportional to the number jn k j of absorbed or emitted quanta. 7.5.3 Push–Pull Pumping
0.5
0.5 Sublevel populations
Sublevel Populations
When modulated light is used to excite CPT resonances of the “0–0 clock transition”, the size of the resonance signal is limited, because the circularly polarized light tends to pump the atoms into the end states, of maximum azimuthal angular momentum, leaving few atoms in the initial and final states, ja0i and jb0i, of the 0–0 resonance. Jau et al. [49] pointed out that one can get around this problem with push–pull pumping, that is, pumping with light for which the circular polarization is modulated at the microwave frequency. As a simple model of push–pull pumping, we assume the light is the sum of two modulated beams, both propagating along the z axis, but with pure right and left
0.4 0.3 0.2 0.1 0 −1
(a)
−0.5 0 0.5 1 Microwave Detuning in kHz
0.4 0.3 0.2 0.1 0 −1
(b)
2 1.5 1 0.5 0 −0.5
(c)
−0.5 0 0.5 Microwave detuning in kHz
1
40 Modulatation factor, f
2.5 Modulatation factor, f
106
0 0.5 1 1.5 Time in oscillation periods
Figure 7.3 (a) shows the 0–0 resonance induced by push–pull coherent population trapping pumping. The top curve shows the populations of the initial and final sublevels, ja0i and jb0i, of the 0–0 transition. The bottom curve shows the population of each of the other sublevels. The resonance was induced by the pulse train of the form shown in (c),
30 20 10 0 −0.5
(d)
0 0.5 1 1.5 Time in oscillation periods
modeled by (7.13) with α D 1. (b) and (d) show the analogous quantities for a very narrow pulse train with α D 200. As the width of the pulses approaches zero, the density matrix approaches a pure superposition state of ja0i and jb0i. Relevant input parameters were I = 1.5, S = 0.5, J = 0.5, B = 10, Gmp = 1000, and kappa = 0.
7.5 Examples
circular polarization, respectively. The modulation peaks of one beam are delayed by half a modulation period, T D 2π/Ω , with respect to the other beam. We can model this situation by adding the high-pressure optical pumping produced by two beams of opposite circular polarization. According to (6.87), the combined optical pumping operator for the two beams is given by 2A op D (A sd A ex,z /2) f 1 (t) C (A sd C A ex,z /2) f 2 (t) .
(7.66)
We have assumed D1 optical pumping with K D z/2 for beam 1 and K D z/2 for beam 2. Beam 2 is delayed by half a coherence period with respect to beam 1, so modulation factors are related by f 2 (t) D f 1 (t C T/2). If we use the simple model (7.13) for f 1 , we conclude that the Fourier components of the modulation factors are (m)
f1
D f (m) D
J m (iα) (m) D (1) m f 2 . i m J0 (iα)
(7.67)
Using (7.67) in (7.66), we find (m) A op D A sd f (m) [1 C (1) m ]/2 A ex,z C 2iS z© f (m) [1 (1) m ]/4 . (7.68) The indices n k for this problem have the values n k D 0 for populations, j f mih f mj, and n c D 1 for jc) D ja0ihb0j and n cN D 1 for j cN ) D jb0iha0j, the only nonnegli(m) gible coherences. So we need only retain terms AQ op with jmj 2 and we find h i AQ op D A sd . M f0g C f (2) (M (f2g C M f2g ) A ex,z C 2iS z© . f (1) (M f1g C M (f1g )/2 . (7.69) When appended to the program ending with Code 7.2 on page 105, the following MATLAB statements will evaluate how the sublevel populations change when the light-modulation frequency of the push–pull pumping beam will induce a resonance between sublevels ja0i and jb0i of an alkali-metal atom (with half-integer nuclear spin quantum number I); see Figure 7.3. Code 7.3 Lcp1=fgl==a&mgl==0&fgr==b&mgr==0;%logical var. of |a0>
107
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7 Modulation
Code 7.3 Continued the previous code. y=exp(alpha*cos(2*pi*x))/besselj(0,i*alpha); figure (3);clf; subplot(2,1,2); plot (x,y); grid on xlabel(’Time in oscillation periods’); ylabel (’Modulatation factor, f’) tAop=Asd.*(M0+fm(2)*(Mp2+Mm2))-(Aex(:,:,3)-2*i*kappa*SC(:,:,3))/2 ... .*(fm(1)*(Mp1+Mm1));%secular optical pumping matrix G0s=i*HgC(L,L)/hbar+Gmp*tAop(L,L);%Gs no tuning om0=(Eg(a+1)-Eg(3*a+1))/hbar;%resonance frequency num=1e3; nnu=101;%maximum detuning and number of samples om=2*pi*linspace(-num,num,nnu);%sample detunings rhosin=zeros(sum(L),nnu);%initialize secular density matrix for k=1:nnu Gs=G0s-i*(om0+om(k))*NL(L,L);%include tuning rhosin(:,k)=null(Gs);%evaluate steady state rhosin(:,k)=rhosin(:,k)/(Lp(L)’*rhosin(:,k)); end subplot (2,1,1); plot(om/(2*pi*1e3),real(rhosin(Lp(L),:))’); grid on; xlabel(’Microwave Detuning in kHz’); ylabel(’Sublevel Populations’);
109
8 Light Propagation In this chapter we review some of the physics of light propagation in spin-polarized atomic vapors. For unpolarized vapors, the ideas of light propagation go back to pre-quantum-mechanical models of atoms as elastically bound electrons. Many of these early ideas, for example, oscillator strengths, were adapted and reinterpreted by quantum mechanics. Just as in Maxwell’s equations for classical optics, we will account for the interaction of light with atoms by introducing an electric dipole moment per unit volume (or per atom) that responds to the electric field of the light wave.
8.1 Induced Electric Dipole Moment
The pumping light will generate an oscillating electric dipole moment hDi in the atom. According to (5.5) this can be written as the sum of positive and negative frequency parts, hDi D hDi C hD † i .
(8.1)
Using (6.7), we find hDi D Tr[fe gg D] D Tr[(fe eg W W fg gg )D] D hW D D W i ,
(8.2)
or hD † i D Tr[D † fg eg ] D Tr[D † (W † fe eg fg gg W † )] D hD † W † W † D † i .
(8.3)
We can also write (8.2) and (8.3) as hDi D E hα feg iChα fgg i E
and
hD † i D hα feg† i E C E hα fgg† i . (8.4)
Optically Pumped Atoms. William Happer, Yuan-Yu Jau, and Thad Walker Copyright © 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-40707-1
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8 Light Propagation
In deriving (8.4) from (8.2) and (8.3), we used (5.53) and (6.8) to write the groundstate and excited-state polarizability dyadics as α fgg D D[D † ./E fe gg ] and α fgg† D [D./E fe gg† ]D † , h i h i α feg D D † ./E fe gg D and α feg† D D † D./E fe gg† .
(8.5) (8.6)
The polarizability dyadics, α fgg and α feg , play a central role in optical pumping and light propagation, since they operate on both the spin states of the atoms and the polarization states of the light. It will prove useful to write the polarizability operators as the sums of Hermitian and anti-Hermitian parts, 0
00
α fgg D α fgg C iα fgg
0
0
00
α feg D α feg C iα feg .
and
00
0
(8.7)
00
The Hermitian operators, α fgg , α fgg , α feg , and α feg are 0
1 fgg 00 (α C α fgg† ) and α fgg D 2 1 feg 00 D (α C α feg† ) and α feg D 2
α fgg D 0
α feg
1 fgg (α α fgg† ) , 2i 1 fgg (α α feg† ). 2i
(8.8) (8.9)
Relation of Polarizability to Effective Hamiltonians From inspection of (8.5) and (8.6), we see that for monochromatic pumping light, the effective Hamiltonians (6.12) and (6.25) can be written as
δ H fgg D E α fgg E
and
δ H feg† D E α feg E .
and
δ E feg D E α feg E .
(8.10)
The light-shift operators become 0
δ E fgg D E α fgg E
0
(8.11)
The damping operators are δ Γ fgg D
2 fgg00 E E α „
and
2 00 δ Γ feg D E α feg E . „
(8.12)
Saturation The expectation values for the polarizability dyadics (8.5) and (8.6) for the saturated density matrix (6.39) are
hα fgg i D
† 1 X D μ μN D μN μ fe gg g E μN μ μ μN
†
and hα feg i D
1 X D μN μ D μ μN . fe gg g E μN μ μ μN
(8.13)
Using (8.13) in (8.4), we see that for saturated atomic populations, hDi D 0 .
(8.14)
The light does not induce an electric dipole moment, because the dipole moment induced in the ground-state atoms is equal and opposite to that induced in the excited-state atoms.
8.2 Absorption Cross Section
8.2 Absorption Cross Section
Assuming that the time dependence of hDi is given by ei(kv ω)t , we find that the average power dissipated in the atom by the optical electric field is d d hDi D E hDi C c.c. dt dt D iω fe gg [E hα fgg i E C E hα feg i E ] C c.c.
hP i D E
D „ω fe gg [hδ Γ fgg i hδ Γ feg i] .
(8.15)
In writing the last line of (8.15), we used (8.11) and (8.12). We define the photon absorption cross section as hσi D
„ω fe gg [hδ Γ fgg i hδ Γ feg i] hP i D . S S
(8.16)
The energy flux S D cjEj2 /2π was defined by (5.61). From (8.15) we see that the power, hP i, absorbed from the beam has positive contributions owing to the absorption of photons of energy „ω fe gg by ground-state atoms at the rate hδ Γ fgg i and negative contributions from stimulated emission of photons at the rate hδ Γ feg i. Normally, absorption prevails over stimulated emission, so the cross section is positive. Unpolarized Ground-State Atoms If the atoms are equally likely to be in any one of the g fgg sublevels of the ground state, the absorption cross section (8.16) will be
σ fgg D
„ω fe gg hδ Γ fgg i 4π ω fe gg X fgg00 e α e. D S g fgg c μ
(8.17)
Noting that fe gg
E μN μ
D „(ω μN μ ω)Ciγ2 ,
with
ω μN μ D ω fe gg C(E μN E μ )/„Ckv , (8.18)
and taking γ2 to be real for simplicity, we find that the matrix elements of the polarization dyadic of (8.8) are fgg00
αμμ
D
† D μ μN D μN μ γ2 X . „ γ22 C (ω ω μN μ )2
(8.19)
μN
Substituting (8.19) into (8.12) and (8.16) we find σ fgg D
† 4π ω fe gg γ2 X e D μ μN D μN μ e g fgg „c γ22 C (ω ω μN μ )2 μN μ
† 2π r e c f fg eg γ2 X e Δ μ μN Δ μN μ e D . [I ] γ22 C (ω ω μN μ )2 μN μ
(8.20)
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112
8 Light Propagation
The cross section (8.20) has the familiar “area”, Z
σ fgg d ν D
π r e c f fg eg Tr e Δ Δ † e D π r e c f fg eg , [I ]
(8.21)
where ν D ω/2π. We used (5.32) to evaluate Δ Δ † .
8.3 Small Magnetic Fields
In the limit of vanishing magnetic field (spherical symmetry) we can replace the optical Bohr frequencies ω μN μ of (8.18) by ω fN f , the Bohr frequencies for transitions from the excited-state Zeeman multiplet fN to the ground-state Zeeman multiplet f. Setting μN ! fN m N and μ ! f m, we find that the sum in (8.20) becomes †
†
X Δ f mI fNmN Δ fNmI N fm fNm N fm
1 X Δ f mI fNmN Δ fNmI N fm D . 2 2 2 3 γ2 C (ω ω fN f ) γ2 C (ω ω fN f )2
(8.22)
fNm N fm
By symmetry, we see that the sum over azimuthal quantum numbers for the expression on the left of (8.22) produces a multiple of the unit dyadic 1. The right side of (8.22) comes from noting that for a dyadic of the form M D M 1, where M P is a scalar, we have 3M D i x i M x i . We can use angular momentum recoupling algebra, similar to that discussed in Section 11.3, to sum the matrix elements over the azimuthal quantum numbers to find
2 1X † N] J S1 Δ f mI fNmN Δ fNmI D [ f ][ f . N fm f fN I 3 0
(8.23)
mm
A simple code to evaluate the 6j symbol was given by Code 5.1 on page 55. Substituting (8.22) and (8.23) into (8.20), we find σ fgg D
X
σf
fN
.
(8.24)
f fN
The contribution to the absorption cross section of transitions from the groundstate Zeeman multiplet f to the excited-state multiplet fN is σf
fN D
2 γ2 2π r e c f fg eg [ f ][ fN] J S1 . f fN I γ22 C (ω ω fN f )2 [I ]
(8.25)
High-Pressure Limit At high buffer-gas pressure, the coherence damping rate γ2 will be so large compared with differences in the optical Bohr frequencies and
8.3 Small Magnetic Fields
Doppler shifts that we can set ω μN μ ! ω fe gg in the resonance denominator of (8.19). Then we can use (5.32) with (8.19) to find the absorption cross section for unpolarized atoms at high pressure, σ fgg D
2π r e c f fg eg γ2 . C (ω ω fe gg )2
γ22
(8.26)
When appended to the MATLAB Code 5.6 on page 65, the following statements will evaluate absorption cross sections for unpolarized atoms in their ground state, with and without Doppler broadening. Code 8.1 TK=input(’Kelvin temperature, TK = ’); sigv=(2*pi/lamJ)*sqrt(kB*TK*NA/MW);%Doppler variance Dnu1= min(min(Heg/hP))-sigv/2; Dnu2= max(max(Heg/hP))+sigv/2; ns=2000;%number of sample frequencies Dnu=linspace(Dnu1,Dnu2,ns);%sample frequencies sigT=zeros(1,ns); sig0=zeros(1,ns); for k=1:ns %Doppler broadened cross section z=-(Heg-hP*Dnu(k)-i*hbar/(2*te))/(hbar*sigv*sqrt(2)); tWT=tV.*w(z)*i*sqrt(pi/2)/(hbar*sigv); dHgT=-tV’*tWT; sigT(k)=-(2*weg/Sl)*imag(trace(dHgT)/gg); %cross section for atoms at rest tW0=tV./(Heg-hP*Dnu(k)-i*hbar/(2*te)); dHg0=-tV’*tW0; sig0(k)=-(2*weg/Sl)*imag(trace(dHg0)/gg); end figure (1); clf subplot(2,1,1); plot(Dnu,sigT); grid on; ylabel(’\sigma in cm^2’) subplot(2,1,2); plot(Dnu,sig0); grid on; ylabel(’\sigma in cm^2’) xlabel(’Detuning in GHz’)
As illustrated in the code, we will think of the symbol w as a matrix of values of the Faddeeva function, with element w μN μ D w (z μN μ ) equal to the value of w for the corresponding value, z μN μ , of the “dimensionless detuning” matrix, (6.117). Then the dot–slash products with the matrix of energy denominators E fe gg are replaced by dot–star products with the matrix of values of the Faddeeva function: Q D VQ ./E fe gg ! W
p i π p VQ . w . σ„ 2
(8.27)
Representative absorption cross sections calculated with Code 8.1 are shown in Figure 8.1.
113
114
8 Light Propagation –
–
–
(a) –
–
–
(b) Figure 8.1 Absorption cross sections of unpolarized ground-state atoms with the same parameters as those for Figure 5.3. (a) For a temperature of 300 K; (b) for atoms at rest.
8.4 Evolution of a Beam in Space and Time
The light used to optically pump atoms is itself modified as it passes through the spin-polarized atomic medium. If there is oscillating coherence in the atoms, the light can become modulated at the coherence frequency and its harmonics. For a medium with no free charge or current density, the Maxwell equations that couple the classical optical fields E and D with B D H are 1 @B (8.28) , c @t 1 @D (8.29) . r BD c @t The Maxwell displacement vector D can be written in terms of the polarization per unit volume P as r ED
D D E C 4πP .
(8.30)
We consider plane waves of the form (7.16), propagating along the unit vector n, so that the electric-field amplitude is X (p) ED EQ ei(k p ζω p t) . (8.31) p
8.5 First-Order Propagation Equation
The distance from the origin along the direction of propagation is ζ Dnr .
(8.32)
The temporal and spatial frequencies are ω p D ω0 C p Ω
and
k p D ω p /c ,
with
p D 0, ˙1, ˙2, . . .
(8.33)
For simplicity, we consider a single modulation frequency Ω , but one can readily generalize these methods to more complicated modulation schemes. For unmod(0) ulated light, the only nonzero amplitude is EQ . We assume that the amplitudes B, D, and P have expansions analogous to (8.31). Then we can replace the classical (p ) (p ) fields E, B, and so on by their Fourier amplitudes, EQ , BQ , and so on in (8.28) and (8.29), provided that we make the replacements @ @ @ (8.34) rD C ik p n and D iω p , @ζ @t @t and we find @ 1 @ (p) (p ) C ik p n EQ D iω p BQ , @ζ c @t @ 1 @ (p) Q (p ) . C ik p n BQ D iωp D @ζ c @t
(8.35) (8.36)
Multiplying both sides of (8.35) on the left by r, and using (8.36), we find 2 2 @ @ 1 (p) Q (p ) . (8.37) C ik p T EQ D 2 iω p D @ζ c @t The transverse projection operator is T D 1 nn ,
(8.38)
and the unit dyadic is 1 D x x C yy C z z. Multiplying (8.37) on the left by T and noting that T T D T , we find the exact wave equation " 2 2 # @ @ 1 (p ) T EQ C ik p 2 iω p @ζ c @t 2 4π @ (p) D 2 (8.39) iω p T PQ . c @t
8.5 First-Order Propagation Equation
For the conditions of optical pumping it is a good approximation to assume that ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ @ ( p )ˇ ˇ @ ( p )ˇ ( p )ˇ ˇ ˇ Q (p ) ˇ ˇ ˇ ˇ Q Q Q ˇk p E ˇ ˇ E ˇ , ˇω p E ˇ ˇ E ˇˇ and @ζ @t ˇ ˇ ˇ ˇ ˇ ˇ @ (p ) ˇ ˇ (8.40) ˇω p PQ ˇ ˇˇ P ( p ) ˇˇ . @t
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8 Light Propagation
Using (8.40) and equating the largest terms on the left and right of (8.39), we find @ 1 @ Q (p) (p) E D i2π k p PQ . (8.41) C @ζ c @t The approximation (8.41) is analogous to the semiclassical Wentzel–Kramers– Brillouin approximation of quantum mechanics. As in (8.41), we will omit the (p ) (p ) (p ) (p ) transverse projection operator and let T EQ ! EQ , T PQ ! PQ when no confusion is likely. There can be longitudinal components of the fields for anisotropically polarized vapor, but the longitudinal components are normally many orders of magnitude smaller than the transverse parts. For simplicity, we assume that the light is weak enough for us to ignore contributions to the polarizability from excited atoms. Then the expectation value of the atomic polarizability (8.5) for light of frequency ω p is X D (m) E hα(ω p )i D (8.42) α p eim Ω t . m
The amplitude is E D
(m) α p D Tr α(ω p )(m) .
(8.43)
Here (m) is the Fourier amplitude of the modulated ground-state density matrix (7.1). If the number density of polarized atoms is N, the dielectric polarization per unit volume is X (p ) X D (m) E ( p ) PQ ei(k p ζω p t) . (8.44) PDN α p EQ ei(k p ζω p Cm t) D pm
p
The Fourier amplitude is X D ( p q)E (q) (p) EQ eik0 (qp )ζ . PQ D N αq
(8.45)
m
Then for the steady state, the propagation equation (8.41) becomes @ EQ @ζ
(p)
D i2πN
X
D E (q) ( p q) kp αq EQ eik0 (qp )ζ .
(8.46)
p0
The oscillatory phase-matching factor eik0 (qp )ζ is important for propagation distances comparable to or greater than the wavelength of the modulation, when the momentum conservation of the Raman scattering begins to be important.
8.6 Propagation of Weak Probe Light
In the next three sections we discuss three straightforward ways to detect the effects associated with optical pumping: (1) to observe changes in the attenuation of the
8.7 Faraday Rotation
pumping light, (2) to observe changes in the polarization of the transmitted light, or (3) to observe changes in the fluorescent light scattered out of the pumping beam. A common method for monitoring optical pumping is to send a weak “probe” beam of light through the sample of atoms, and monitor its change in amplitude or polarization due to optical pumping. We shall assume that this monitoring beam has a negligible effect on the atoms. We shall also assume that the polarizability is quasistatic, so it does not change substantially over the propagation time through the sample of atoms. Then the probe propagation obeys @ EQ D i2πN khα ? i EQ , @ζ
(8.47)
where the transverse polarizability is hα ? i D T hαi T . The transverse polarizability has two spatial eigenvectors: hα ? ijα λ ) D α λ jα λ ) .
(8.48)
If we assume that the polarizability is spatially uniform, these modes provide a simple means of following the light propagation through the vapor. After propagation of a distance l, a probe electric field EQ 0 entering the vapor will become EQ D
X
jα λ )((α λ j EQ 0 )ei2πN k α λ l .
(8.49)
λ
The imaginary parts of the eigenvalues α λ produce attenuation of the probe beam, whereas differences between the real parts rotate the polarization axis. Both the real and the imaginary parts are quite sensitive to the atomic density matrix, so, in general, the polarization state of the light can be substantially altered by the optically pumped atoms.
8.7 Faraday Rotation
If the probe light is tuned far from the atomic resonance frequencies, so that the atomic hyperfine splittings, Zeeman shifts, and linewidths are negligible, the polarizability for alkali-metal atoms takes the simple form, analogous to (5.41), 2(1) J 1/2 hαi D α 0 1 C ihS i D α 0 C iα g t hS i , [ J]
(8.50)
where α 0 jDj2 /„Δω is the scalar polarizability at the probe detuning Δω, and α g t is termed the gyrotropic polarizability. Then the eigenmodes of the transverse polarizability are ˇ ˇ α 0 ˙ α g t h Jz 0 i D ξ 0 ˙1
(8.51)
117
118
8 Light Propagation
in a coordinate system whose zO 0 direction is oriented along n. The probe electric field (8.49) becomes EQ D ei2πN k α 0 l ξ 01 ξ 01 ei2πN k α g t hSz 0 il C ξ 01 ξ 01 ei2πN k α g t hSz 0 il EQ 0 . (8.52) This corresponds to a rotation of the polarization axis of the light by an angle θFR D 2πN k α g t lhn S i D (1) J C1/2
2„r e c f fg eg N l hn S i , [ J]Δω/2π
(8.53)
where we have used (5.45) to express the polarizability in terms of atomic constants. This paramagnetic Faraday rotation of the polarization axis is proportional to the component of the electron spin polarization along the probe propagation axis. Note that the rotation angle in this large detuning limit is insensitive to the atomic lineshape, which makes Faraday rotation angle measurements particularly attractive for making accurate absolute measurements of the atom density and/or polarization.
8.8 Specific Absorption
A convenient, dimensionless measure of the absorption of light is the specific absorption hηi, which we define by ˝ fgg ˛ ˝ feg ˛ δΓ δΓ hσi hηi D D fgg . (8.54) fgg σ Γp From (8.54) we see that the specific absorption is the ratio of the amount of light absorbed by a polarized atom to the amount of light that would be absorbed if the atom were in its ground state and completely unpolarized. The specific absorption can be greater than 1 if most of the atoms are pumped to the most strongly absorbing sublevels of the ground state. The specific absorption can be negative if there is a population inversion and the atoms amplify rather than attenuate the pumping light. The specific absorption is also the ratio of hσi, the absorption cross section (8.16) of optically pumped atoms, to σ fgg , the absorption cross section (8.17) of unpolarized atoms in their ground state. In accordance with (6.15) and (6.28) we find h i fg gg fe eg Tr Pdp Pdp hηi D D (ηjρ) , (8.55) fgg Γp where the density matrix jρ) without optical coherences was given by (6.55), and where the row vector for specific absorption is h i fgg feg fg gg eg (ηj D Γpfeg (1feg jAfe jA p . (8.56) p , Γp (1
8.9 Fluorescent Light fg gg
eg For pumping with modulated light, the pumping matrices Afe and A p will p be time-dependent. Also, for either magnetic resonance or coherent population trapping resonance experiments, the steady-state density matrix jρ) may have components oscillating at the drive frequencies or their harmonics.
8.9 Fluorescent Light
As discussed extensively in the literature [1], the rate of emission of fluorescent photons of polarization vector e λ into the solid angle ΔΩ is h i hΔΦλ i D Tr fe eg ΔΦλ . (8.57) The emission operator is ΔΦλ D
ΔΩ ω fe gg3 † ΔΩ Γ fg eg [ J] † D D W e λ e λ D Δ Δ W e λ e λ . 3 2π„c 8π
(8.58)
To simplify the notation of (8.58) and subsequent formulas we will define the double-dot product of dyadics A and B by X A W B D Tr [A B] D Ai j Bji . (8.59) ij
The double-dot product is a scalar with respect to spatial rotations. The electric dipole moment operators, D and D † , were defined by (5.6) and (5.7). Their dimensionless versions, Δ and Δ † , were defined by (5.17) and (5.20). For fluorescent photons propagating along the unit vector n, the two independent polarization vectors, e 1 and e 2 , can be chosen to be orthogonal to each other and to n, such that the three unit vectors form an orthonormal basis and the unit dyadic is e 1 e 1 C e 2 e 2 C nn D 1 .
(8.60)
The operator for the total emission rate of photons per solid angle is Γ fg eg [ J] † 1 X dΦ ΔΦλ D D Δ Δ W (1 nn) dΩ ΔΩ 8π λ ! r
6π Γ fg eg 112 f J Jg J CS D . 1 (1) [ J] Y2 T2 J JS 4π 5 f J Jg
(8.61)
is a scalar product like that of (5.34) between the spherical harmonic Here Y2 T2 f J Jg Y2m (n) and the spherical basis tensor T2m for excited atoms, with the electronic angular momentum quantum number J. The atom decays to a ground state of electronic angular momentum quantum number S. To evaluate the second line of (8.61) we used the multipole expansion of Δ † Δ given by (5.33), and we noted that
119
120
8 Light Propagation
1 n fξ ξO g00 n D p , 3 n fξ ξO g1m n D 0 , r 8π n fξ ξO g2m n D Y2m (n) . 15
(8.62) (8.63) (8.64)
121
9 Radiation Forces An atom experiences forces from the pumping light. The force has a mean value associated with the mean scattering rate of photons and with gradients in the light shift. There is also a fluctuating component of the force that comes from the irregular recoils of the atom as it absorbs and emits photons propagating in various directions. This causes the atom to make a random walk in momentum space. The force depends strongly on the velocity of the atom with respect to the laser beam, and this velocity dependence plays an important role for laser cooling of atoms. The force also depends strongly on the applied electric and magnetic fields, which can be used to provide a controlled spatial dependence to the light force, allowing for stable confinement, or trapping, of the atoms. As we illustrate in this chapter, the cooling and confinement forces, as well as the fluctuations in the force, are dramatically enhanced by optical pumping processes. The discovery and exploitation of optical pumping effects in laser cooling resulted in the award of the 1997 Nobel Prize to Steven Chu [50], Claude Cohen-Tannoudji [51], and William Phillips [52].
9.1 Mean Force
A classical electric field E and a classical magnetic field B will exert a classical force hFi on the electric charge distribution, hi, and the current distribution h j i, Z 1 hFi D d 3 rhiE C h j i B . (9.1) c Expanding the fields E and B as a power series around center of the atom and retaining only the first, nonvanishing terms of the expansion, we find that the force is 1 P hFi D hDi rE C hDi B C c The electric dipole moment and its derivative are Z hDi D rhid 3 r Optically Pumped Atoms. William Happer, Yuan-Yu Jau, and Thad Walker Copyright © 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-40707-1
(9.2)
(9.3)
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9 Radiation Forces
and P D hDi
d hDi D dt
Z h j id 3 r .
(9.4)
The first term on the right of (9.2) arises because an inhomogeneous electric field can pull harder on the positive end of the electric dipole moment hDi than it pushes on the negative end. The second term of (9.2) is the net magnetic force on the currents inside the atom. The first and second terms can have comparable magnitudes for resonant optical excitation of an atom. It is convenient to use Faraday’s law, r ED
1 @ B, c @t
(9.5)
to write the force due to the first two terms on the right of (9.2) as hFi D hDi rE C hDi (r E) C
1 @ [hDi B] . c @t
(9.6)
The third term of (9.6), being the time derivative of a bounded function, will average to zero after a few optical cycles. The sum of the first two terms gives the time-averaged force, hFi D
X ij
xi
@E j hD j i . @x i
(9.7)
Here x i is a unit vector along the ith axis of a Cartesian coordinate system, D j D D x j and E j D E x j . Using (5.54) and (8.1)–(8.3) with (9.7), and dropping terms that oscillate at twice the laser frequency ω, we find that the mean force can be written as hFi D hF fgg i C hF feg i .
(9.8)
The ground-state and excited-state contributions to the mean force are ˛ ˝ hF fgg i D fr V g† W C c.c.
(9.9)
and ˛ ˝ hF feg i D W fr V g† C c.c.
(9.10)
One can readily verify that for the saturated density matrix of (6.39), the forces from the excited-state and ground-state components of the density matrix cancel and the total force is hFi D 0. For the weak optical pumping often used for optical traps, the force (9.10) from excited atoms can be negligibly small compared with the force (9.9) from ground-state atoms.
9.2 Forces from Monochromatic Light
9.2 Forces from Monochromatic Light
Consider monochromatic light of the form X (k) EQ ei(krω t) C c.c. ED
(9.11)
k
This represents light of temporal frequency ω, propagating with spatial frequencies k D ωn/c, along the directions of various unit vectors n. Substituting (9.11) into (9.7) and using (8.13), we find that the contribution to the force from ground-state atoms is hF fgg i D rhδ E fggi C
0 0 „ X k C k 0 hδ Γ fgg(k k) iei(kk )r . 2 0
(9.12)
k k
The first term on the right of (9.12) is a conservative force, the negative spatial gradient of a potential. The potential is the light shift, X 0 0 hδ E fgg i D hδ E fgg(k k) iei(kk )r , (9.13) k0 k
with the Fourier amplitudes 0
hδ E fgg(k k) i D EQ
(k 0 )
0
hα fgg i EQ
(k)
.
(9.14)
The second term on the right of (9.12) is a dissipative force due to the scattering of photons. It is proportional to elements of the light-absorption operator 0
hδ Γ fgg(k k) i D
2 Q (k 0 ) 00 (k) E hα fgg i EQ . „
(9.15)
The dissipative force is the net result of momentum impulses of magnitude „k, acquired or lost when the atom absorbs or emits a photon of spatial frequency k. The contribution to the force from excited-state atoms is 0 0 „ X k C k 0 hδ Γ feg(k k) iei(kk )r . (9.16) hF feg i D rhδ E feg i 2 0 k k
The light shift is hδ E feg i D
X 0 0 hδ E feg(k k) iei(kk )r ,
(9.17)
k0 k
with the Fourier amplitude 0
hδ E feg(k k) i D EQ
(k 0 )
0
hα feg i EQ
(k)
.
(9.18)
The elements of the light-emission operator are 0
hδ Γ feg(k k) i D
2 Q (k 0 ) 00 (k) E hα feg i EQ . „
(9.19)
123
124
9 Radiation Forces
Optical Earnshaw Theorem
If we take the spatial divergence of (9.12), we find
r hF fgg i D r 2 hδ E fggiC
i 0 0 „ X h k C k 0 rhδ Γ fgg(k k) i ei(kk )r . (9.20) 2 0 k k
We see that the divergence of the scattering forces, represented by the second 0 term of (9.20), vanishes if rhδ Γ fgg(k k) i D 0, that is, if the Fourier components 0 hδ Γ fgg(k k) i of the light-absorption operator are independent of position. Position independence is guaranteed if the ground state has only one sublevel, that is, if 0 the atom has angular momentum quantum number f D 0. Then hδ Γ fgg(k k) i D 0 (k ) (k) EQ /„ is a position-independent scalar, and we will have r F D 0 2α 00 EQ for the part of the force due to photon scattering. A point of equilibrium, where F D 0, can be stable only if there are positive restoring coefficients along all principal axes, that is, only if @F x /@x < 0, @Fy /@y < 0, and @F z /@z < 0. But this is impossible for a force with zero divergence. This is the essence of the optical Earnshaw theorem [53], which states that photon-scattering forces cannot stably confine an atom in three dimensions if the ground state of the atom is nondegenerate and the dielectric polarizability α 00 is a position-independent scalar. An atom with many ground-state sublevels can be optically pumped in such a way that its spin polarization will vary with position to make r F < 0 and make it possible to optically trap atoms in three dimensions [53–55].
9.3 Forces in Magneto-Optical Traps
As an example of radiation forces, and of nearly collision free optical pumping, we will discuss the physics of magneto-optical traps (MOTs) [55]. To provide threedimensional confinement, a MOT usually has three pairs of monochromatic, counterpropagating beams, for example, beams along the x, y, and z axes of a Cartesian coordinate system. In most MOT situations, the lasers are intentionally kept weak enough to neglect stimulated emission, and to permit maximum cooling of the trapped atoms. Figure 9.1 shows the energy levels of a 87 Rb atom, with I D 3/2, in a small magnetic field. Also shown are the counterpropagating beams of a strong trap laser, which confines the atoms along the z direction. A much weaker repump laser is normally present as well. The frequency ω of the trap laser is tuned a few linewidths less than the zero-field resonance frequency for the cycling transition from the 2 S1/2 ground-state multiplet with spin quantum number f D I C1/2 D 2 to the 2 P3/2 excited-state multiplet with fN D I C 3/2 D 3. The trap laser will weakly drive off-resonant transitions to excited-state multiplets with fN D I C 1/2 D 2 and fN D I 1/2 D 1, both of which can decay to the lower ground-state multiplet. To cope with these losses, the weak repump laser is frequency-tuned to cause maximum pumping from the lower ground-state multiplet with f D I 1/2 to the excited-state hyperfine multiplet with fN D I C 1/2 D 2, where a large fraction
9.3 Forces in Magneto-Optical Traps
(a)
(b)
Figure 9.1 (a) Optical pumping in a magneto-optical trap (MOT). The helicity, η D ˙1, has the same sign as the field gradient, @B z /@z. Relative pumping rates of the trap laser are indicated. (b) Zeeman shifts of the ground-state and excited-state sublevels of the cycling transition.
of the atoms radiate back to the ground-state multiplet with f D I C 1/2 D 2, where they can again scatter photons from the trap laser. Thus, atoms spend most of their time in the multiplet with f C I C 1/2 D 2, where they strongly scatter the light of the trap laser. The trap laser keeps the atoms close to the origin of the coordinate system as a result of its mean force, and it also cools the atoms through the fluctuating components of its force. Let the center of the trap be at the origin of a coordinate system, where z D 0. “Anti-Helmholtz” coils produce a magnetic field. For small displacements along the z axis, the field is well described by B D βzz ,
where
β D @B z /@z .
(9.21)
The field-gradient coefficient, β, which can be taken to be a constant for small displacements z, is proportional to the current in the coils, to the number of windings, and to geometrical factors. At the center of the trap, where B D 0, the atoms will be equally likely to scatter up-going or down-going photons. Let β > 0 and assume that the atoms move down into a region of negative magnetic field. Then there will be Zeeman shifts of the ground-state and excited-state sublevels of the cycling transition like those indicated in Figure 9.1b, such that σC transitions are more in tune with the “redshifted” trap laser than σ transitions. Since we are assuming light of helicity η D 1, the σC transitions are associated with up-going photons, which carry positive momentum and therefore impart an upward (restoring) force to the atoms as they are scattered. This upward force is further enhanced by optical pumping, which will try to transfer the atoms to the right end state with j f mi D j2, 2i, which has the maximum scattering rate for σC light. For negative
125
126
9 Radiation Forces
helicity, the momentum from the most strongly scattered photons will “detrap” the atoms. We see that for a trapping situation, the helicity η must have the same sign as β D @B z /@z. MOTs also provide strong velocity damping, which narrows the momentum distribution of the atoms and cools them. If the atom has a positive velocity, v > 0, along the z axis, it will be more in resonance with the down-going beam, and the scattering of these down-going photons will tend to drive the atomic velocity back toward zero velocity. Because of the circular polarization of the down-going beam, the atoms will be pumped into a strongly absorbing end state, and this enhances the “viscous damping force”. The force would reverse if v < 0, so the force always tries to drive the atomic velocity to zero. For zero magnetic field, this “viscous damping force” is independent of the helicity. As we will discuss in more detail below, strong transverse spin coherences are generated in atoms with nearly zero velocity, and these coherences are associated with even more rapid changes of the force with velocity [50–52]. Turning to a detailed analysis of the mean forces in a MOT, we consider the simple case of beams counterpropagating along the z axis. The three-dimensional case can be analyzed in a similar way, and to a first approximation, one can vectorially add the forces from three MOT beams counterpropagating along the three coordinate axes. The electric-field amplitude of the light at position z is X (r) ED EQ ei(r k zω t) r
o EQ n [x C iηy] ei(k zω t) C [x iηy] ei(k zω t) D 2 D EQ [x cos ψ C y sin ψ] eiω t .
(9.22)
Here k D ω/c is the spatial frequency of the light, the up beam has the direction quantum number r D 1 and the down beam has r D 1. The laser frequency ω is often “red-detuned” to be a few radiative linewidths smaller than the resonance frequency of the atomic transition that is used for cooling or trapping. Equation 9.22 represents beams with perfect circular polarization and with the same helicity η, which we take to be η D 1 or η D 1. At any given instant of time, t, the tip of the Q makes a “corkscrew” about the z axis. The rotation angle vector E, of amplitude E, is proportional to z and is given by ψ D η k z D η k v t .
(9.23)
From inspection of (9.22) we see that the electric-field amplitudes for the up and down beams can be written as r η EQ (r) EQ D p ξ r η . (9.24) 2 The spherical basis vectors ξ m were defined by (5.9). Using (9.22), we find that the interaction V of (5.53) becomes V D E D † D
X r
VQ (r) ei(r k zω t) ,
r η EQ with VQ (r) D p ξ r η D † . 2
(9.25)
9.3 Forces in Magneto-Optical Traps
The analysis of the simplest MOT pumping situation that we are considering here is facilitated by using the total azimuthal angular momentum operator of the atoms, fgg
Fz D Fz
C F zfeg .
(9.26)
Since we assume that the externally applied magnetic field is directed along the fgg z axis, we must have [F z , H fgg ] D 0 and [F zfeg , H feg ] D 0, where the groundstate and excited-state hyperfine Hamiltonians H fgg and H feg were given by (2.10) and (2.12). The Hamiltonians are implicit functions of the position z, because of the position dependence of the magnetic field, (9.21). The energy basis states, jμi of (3.14) and j μi N of (3.19), will therefore be eigenstates of F z with integer or halfinteger azimuthal eigenvalues m μ and m μN , that is, fgg
F z jμi D m μ jμi
and
F zfeg j μi N D m μN j μi N .
We can write the angular momentum operators as X X fgg m μN j μih N μj N and F z D m μ jμihμj . F zfeg D μN
(9.27)
(9.28)
μ
Because of the azimuthal symmetry, the matrix elements of (9.25) have the selection rule N VQ (r) jνiδ m μN ,r ηCm ν . h μj N VQ (r) jνi D h μj
(9.29)
Using (9.27) and (9.29), we see that N VQ (r) jνieiψ(m μN m ν ) D h μj N VQ (r) jνieir k z . h μje N iψ Fz VQ (r) eiψ F z jνi D h μj
(9.30)
Then the time-dependent interaction (9.25) becomes V D eiΦ VQ eiΦ ,
(9.31)
where the phase operator, the analog of (5.64), is Φ D ωt Q C ψ F z .
(9.32)
The optical spin operator Q was defined by (2.16). The time-independent amplitude of (9.31) is X VQ D VQ (r) D E x D † . (9.33) r
Not surprisingly, (9.31) shows that the interaction (9.25) corkscrews around the z axis, just like the electric field E of (9.22). The dark part, (5.73), of the rotating-frame Hamiltonian becomes K0 D H feg C H fgg „(ΔωQ η k v F z ) .
(9.34)
127
128
9 Radiation Forces
The expression (9.34) for the counterpropagating MOT light beams differs from the expression (5.76) for a single light beam by having the Doppler shift term, η k v F z , rather than k v Q and variants of the subsequent code intended to model single pump beans should be modified accordingly. As in (6.8), we can define the operator W D V ./EQ fe gg ,
where
E fe gg D H fe gg „ Δω η k v F fe gg C iγ2 . (9.35)
The matrix F fe gg , which accounts for opposite Doppler shifts for the two beams, has elements analogous to those of H fe gg of (5.90)–(5.93), for example, fe gg
F μN μ
fgg
D h μjF N zfeg j μi N hμjF z jμi D m μN m μ .
(9.36)
Q can be used to calculate the various optical pumping The matrices VQ and W matrices given in Chapter 6, and the steady-state density matrix of the atoms in the MOT trap will be given by (1.34) and (1.35), with the damping operator given by fgg© fgg G D i H fgg© /„ C η k fe gg v F z (9.37) C Γp A op . We can substitute (9.31) into (9.9) to find the magneto-optical force ˛ ˝ F D (r V )† W C c.c. Q eiΦ i C c.c. D h(reiΦ VQ eiΦ )† eiΦ W ˛ ˝ Q C c.c. D i [r Φ , VQ ]† W ˛ ˝ Q C c.c. D iη k z(W Q † [F z , VQ ]j) C c.c. D iη k z [F z , VQ ]† W
(9.38)
When appended to the program ending with Code 5.4 on page 58, the following MATLAB statements will evaluate the quantities needed for the quasi-steady-state analysis given in Chapter 6: the effective Hamiltonians δ H feg and δ H fgg , denoted by dHe and dHg, the light-damping operators δ Γ feg and δ Γ fgg , denoted by dGme fgg and dGmg, the mean pumping rates Γpfeg and Γp , denoted by Gmpe and Gmpg, the fe gg
fg eg
fg gg
eg , denoted by Apee, Apgg, Apeg, and Apge, matrices Afe p , A p , A p , and A p and the matrices A dp and A rp , denoted by Adp and Arp.
Code 9.1 St=1e4*input(’one-way trap flux in mW/cm^2, St = ’);%to erg/(s cm^2) eta=input(’helicity, eta = ’); v=input(’velocity in cm/s, v = ’); tE=sqrt(4*pi*St/c);%field amplitude in esu/cm^2 tV=-D*Dj(:,:,1)’*tE;%interaction for field along x axis Dnut=1e6*input(’trap detuning in MHz, Dnut = ’); Dw=(Eef(1)-Egf(1))/hbar+2*pi*Dnut; Hee=Ee*ones(1,ge)-ones(ge,1)*Ee’; Heg=Ee*ones(1,gg)-ones(ge,1)*Eg’;
9.3 Forces in Magneto-Optical Traps
Code 9.1 Continued the previous code. Fee=me*ones(1,ge)-ones(ge,1)*me’; Feg=me*ones(1,gg)-ones(ge,1)*mg’; Eeg=Heg-hbar*(Dw-keg*v*eta*Feg+0.5*i/te);%complex energy differences tW=tV./Eeg; dHg=-tV’*tW; dHe=tV*tW’; dGmg=(dHg-dHg’)*i/hbar; dGme=(dHe-dHe’)*i/hbar; cdGmg=dGmg(:); rdGmg=cdGmg’;%row and column vectors cdGme=dGme(:); rdGme=cdGme’; Gmpg=rdGmg*cPg/gg; Gmpe=rdGme*cPe/ge; Apgg=(flat(dHg)-sharp(dHg)’)/(-i*hbar*Gmpg); Apee=(flat(dHe)-sharp(dHe)’)/(-i*hbar*Gmpe); Apeg=(kron(conj(tV),tW)-kron(conj(tW),tV))/(i*hbar*Gmpg); Apge=(kron(tW.’,tV’)-kron(tV.’,tW’))/(i*hbar*Gmpe); Adp=Apgg; Arp=(Asge/te)*(diag(i*Hee(:)/hbar+i*eta*keg*v*Fee(:)+1/te))^(-1)*Apeg; Aop=Adp-Arp; HgC=flat(Hg)-sharp(Hg); FzgC=flat(Fzg)-sharp(Fzg); G=i*(HgC/hbar+eta*keg*v*FzgC)+Gmpg*Aop; L=fgl==fgr;%log. var. to neglect hfs coherences Lp=fgl==fgr&mgl==mgr;%log. var. for populations cLp=Lp(L);%log. var. for pops. in compactified space t=10/Gmpg;%10 optical pumping cycles rhoct=expm(-t*G(L,L))*cPg(L)/gg; rhocin=null(G(L,L)); rhocin=rhocin/(rPg(L)*rhocin);%steady state Fmot=i*eta*keg*tW’*(Fze*tV-tV*Fzg)/kB; cFmot=Fmot(:); rFmot=cFmot’; Ft=rFmot(L)*rhoct; Ft=Ft+conj(Ft);%early-time force Fin=rFmot(L)*rhocin; Fin=Fin+conj(Fin);%steady-state force
We made use of the identities (5.94) to change some of the o-dot transforms to commutator transforms. In evaluating these coupling matrices, we used (9.33) for Q . The MOT forces were evaluated with (9.38). VQ and (9.35) for W The sublevel populations and forces evaluated by the code ending with Code 9.1 on page 128 can be plotted with the following statements to generate Figure 9.2. Code 9.2 figure(1); clf; mm=-a:1:a; ma=mg(1:2*a+1); mb=mg(2*a+2:gg); [xa,ja]=sort(ma); [xb,jb]=sort(mb); rho=real(rhoct(cLp));%early populations rhoa=rho(1:2*a+1);rhob=rho(2*a+2:gg); subplot(2,1,1) P=[rhoa(ja),[0;rhob(jb);0]]; bar(mm,P,0.5) ylabel(’\rho(t)’) grid on; hold on; title([’F = ’ num2str(Ft) ’ k_B K/cm’] ) rho=real(rhocin(cLp));%steady-state populations rhoa=rho(1:2*a+1);rhob=rho(2*a+2:gg); subplot(2,1,2)
129
9 Radiation Forces
Code 9.2 Continued the previous code. P=[rhoa(ja),[0;rhob(jb);0]];bar(mm,P,0.5);grid on; ylabel(’\rho(\infty)’); xlabel(’Azimuthal quantum number, m’) title([’F = ’ num2str(Fin) ’ k_B K/cm’] )
F = 3.0736 k K/cm B
ρ(t)
1
(a)
0.5 0
−2
−1
0
1
2
F = 0.030469 k K/cm B 1
ρ(∞)
130
0.5 0
−2
(b)
−1 0 1 2 Azimuthal quantum number, m
Figure 9.2 (a) Population distribution of 87 Rb fgg
fgg
atoms at time t D 10/Γp , where Γp is the mean optical pumping rate, (6.17), of the trapping beam. The other parameters were J = 1.5, B = -3, Sl = 1, eta = 1, v = 0, and Dnut = -5. The atoms in the upper hyperfine multiplet with total angular momentum quantum number f D I C 1/2 D 2 have been pumped into a quasi steady state. At this
early time, approximately three eighths of the atoms remain equally distributed in the lower hyperfine multiplet, just as they were at time t D 0. (b) Population distribution for the true steady state of 87 Rb atoms. Most of the atoms in the upper hyperfine multiplet have been “burned out” by the trap laser, and the force is a tiny fraction of its value for the early-time, quasi steady state of (a).
9.3.1 Repump Lasers
For MOTs, the atoms are normally pumped with D2 light, detuned a few megahertz below the zero-field resonance frequency of the transition from the ground-state multiplet of total angular momentum quantum number f D I C1/2 of the ground state to the multiplet with total angular momentum quantum number fN D I C 3/2 of the excited state. This type of transition is often referred to as a “cycling transition” since most of the atoms return to the initial multiplet with f D I C 1/2 by spontaneous or simulated emission. The atomic populations quickly reach a quasi steady state, similar to that shown in Figure 9.2a. However, the trap laser weakly excites atoms into excited-state multiplets with fN D I C 1/2 and fN D I 1/2 because there is some overlap of the laser frequency with the edges of the corresponding absorption lines. These excited multiplets can decay to the groundstate multiplet with f D I 1/2. As a result of this “Raman scattering”, the trap
9.3 Forces in Magneto-Optical Traps
laser slowly pumps atoms into the lower multiplet with f D I 1/2. If no repump laser is used to reverse this, most of the atoms will eventually be pumped into the lower multiplet, where they scatter light very weakly since the absorption lines from the lower multiplet are very far from resonance with the frequency of the trap laser. This true steady state of the atoms with no repump laser is shown in Figure 9.2a. The effect of adding a weak repump laser is shown in Figure 9.3, which was generated by adding the following segment of code to that ending with Code 9.2 on page 129, followed by another copy of Code 9.2 for displaying the results. Code 9.3 fr=input(’fractional intensity of repump laser, fr = ’); tV=sqrt(fr)*tV;%attenuate interaction Dw=(Eef(2)-Egf(2))/hbar;%repump detuning Eeg=Heg-hbar*(Dw-keg*v*eta*Feg+0.5*i/te); tW=tV./Eeg; dHg=-tV’*tW; dHe=tV*tW’; dGmg=(dHg-dHg’)*i/hbar; dGme=(dHe-dHe’)*i/hbar; dGmg=(dHg-dHg’)*i/hbar; dGme=(dHe-dHe’)*i/hbar; dGmg=(dHg-dHg’)*i/hbar; dGme=(dHe-dHe’)*i/hbar; cdGmg=dGmg(:); rdGmg=cdGmg’;%row and column vectors Gmpg=rdGmg*cPg/gg; Gmpe=rdGme*cPe/ge; Apgg=(flat(dHg)-sharp(dHg)’)/(-i*hbar*Gmpg); Apee=(flat(dHe)-sharp(dHe)’)/(-i*hbar*Gmpe); Apeg=(kron(conj(tV),tW)-kron(conj(tW),tV))/(i*hbar*Gmpg); Apge=(kron(tW.’,tV’)-kron(tV.’,tW’))/(i*hbar*Gmpe); Adp=Apgg; Arp=(Asge/te)*(diag(i*Hee(:)/hbar+i*eta*keg*v*Fee(:)+1/te))^(-1)*Apeg; Aop=Adp-Arp; G=G+Gmpg*Aop;%add repumping rhoct=expm(-t*G(L,L))*cPg(L)/gg; rhocin=null(G(L,L)); rhocin=rhocin/(rPg(L)*rhocin);%steady state Fmot=Fmot+i*eta*keg*tW’*(Fze*tV-tV*Fzg)/kB; cFmot=Fmot(:); rFmot=cFmot’; Ft=rFmot(L)*rhoct; Ft=Ft+conj(Ft);%early-time force Fin=rFmot(L)*rhocin; Fin=Fin+conj(Fin);%steady-state force
The code ending with Code 9.3 can be used to calculate how the force depends on the atomic velocity or on the magnetic field, by adding appropriate looping statements for the velocity or the field. For example, Figure 9.4 shows a rapid change of force with velocity for jv j < 5 cm s1 . This is associated with the transverse coherence that can be created at very low velocities when the light appears to be linearly polarized to the slowly moving atoms. The transverse coherence has the same corkscrew pattern along the z axis as the electric field of the pumping light. For larger values of the velocity, there is little transverse coherence, and the force is less sensitive to changes in the atom velocity. The range of velocities where the coherence is important is proportional to the intensity of the trap laser. The scattering rate for the polarized atoms is nearly twice as large as that of unpolarized atoms because the repump laser has pumped most of the atoms from the lower Zeeman multiplet of the ground state to the upper multiplet, and because the trap laser has
131
9 Radiation Forces F = 3.1232 k K/cm B
ρ(t)
1
(a)
0.5 0
−2
−1
0
1
2
F = 5.8843 kB K/cm 1
ρ(∞)
132
0.5 0
−2
(b)
−1 0 1 2 Azimuthal quantum number, m
Figure 9.3 This figure was generated with the same parameters as used for generation of Figure 9.2, except that a repump laser was used to prevent accumulation of atoms in the ground-state multiplet with f D I 1/2. The intensity of the repump laser is 1% that of the trap laser and the frequency is that of the zero-field transition from the ground-state multiplet with f D I 1/2 to the excited state multiplet with fN D I C 3/2. The late-
time distribution shown in (b) has most of the atoms in the upper hyperfine multiplet with f D I C 1/2. Since most atoms are in the upper multiplet, where they strongly scatter the beam of the trap laser, the force is nearly twice as large for the true steady state as for the quasi steady state in (a), where the three eighths of the atoms in the lower multiplet hardly interact with the strong trap laser.
pumped most of the atoms in the upper multiplet to the strongly scattering end states, with distributions of sublevel populations similar to those in Figure 9.3. In Figure 9.5 we show how the force on atoms with zero velocity depends on the magnetic field B. As for the velocity dependence, the force changes very rapidly with magnetic field near zero velocity because of the generation of transverse atomic coherence at fields close to zero.
9.4 Pointing Probability
Traditional plots of sublevel populations, like those in Figures 9.2 and 9.3, do not characterize the density matrix very well for small velocities v or for small fields B, where the atoms can have large transverse coherences. To get more insight into the nature of the spin polarization in such cases, it is useful to introduce the “pointing probability”, hW i D
[a]n hnjjni, 4π
or
W D
[a]n jnihnj . 4π
(9.39)
9.4 Pointing Probability
Force (kB K/cm)
0.5
0
−0.5
(a) −60
−40
−20
0
20
40
60
5
Rate (s−1)
15
x 10
10 5 Total polarized
0 −60
−40
−20
(b)
Trap unpolarized
0 v in (cm/s)
Figure 9.4 (a) Variation of the force on a 87 Rb atom in a MOT with atomic velocity near the center of a MOT where the magnetic field is nearly zero. The computational parameters that differ from used to generate Figure 9.3 are B=0.00001 and Dnut = -10. (b) The
Repump unpolarized
20
40
60 fgg
mean pumping rates, Γp (the scattering rate of unpolarized atoms), for the trap and repump laser, along with the total scattering rate from both lasers, hδΓ i, for the polarized atoms.
Force (kB K/cm)
8 4 0 −4 −8
(a) −10
−5
0
5
10
6
15
x 10
−1
Rate (s )
Total polarized
Repump unpolarized
10 5 0 −10
(b)
Trap unpolarized
−5
0 B in (G)
Figure 9.5 (a) Variation of the force on a 87 Rb atom in a MOT with magnetic field for zerovelocity atoms. The computational parameters are exactly the same as those used to generate Figure 9.3 except that there is a range
5
10
of magnetic fields. (b) The mean pumping fgg
rates, Γp (the scattering rate of unpolarized atoms), for the trap and repump lasers, along with the total scattering rate from both lasers, hδΓ i, for the polarized atoms.
133
134
9 Radiation Forces
The unit vector, n D x sin θ cos φ C y sin θ sin φ C z cos θ ,
(9.40)
is parameterized by the colatitude angle θ and the azimuthal angle φ. The ket vector jni is defined by jni D eiφ Fz eiθ Fy jzi,
or jni D
X
a jamiD m a .
(9.41)
m fgg
fgg
Here F z D F z and Fy D Fy are z and y projections of the ground-state ana a gular momentum operator. The coefficients D m a D D m a (φ, θ , 0) are Wigner D functions. The ket jami is a zero-field basis state j f mi of (3.25) with f D a D I C 1/2. Sublevel jzi D jaai has the maximum possible azimuthal quantum number m D a and the maximum possible total spin quantum number f D a, that is, jzi D jaai,
where
F F jaai D a(a C 1)jaai
F z jaai D ajaai .
and (9.42)
The ket jni represents an atom with maximum possible spin along the unit vector n, and one can readily show that F Fjni D a(a C 1)jni and
n Fjni D ajni .
(9.43)
The probability that atoms described by the density matrix will be in the hyperfine multiplet a and will have their spins pointing into a small solid-angle increment d Ω D n sin θ d θ d φ is hW i d Ω . One can use the orthogonality properties of the
Figure 9.6 The pointing probability hW i, defined by (9.39): at the center of the capture range of Figure 9.4 with v D 0 (a), and outside the capture range for v D 60 cm s1 (b). These pointing probabilities are plotted for
t D 0 in the laboratory frame. In accordance with the phase (9.32) of the rotating frame, the distributions rotate around the z axis at angular velocity d ψ/d t D ηk v.
9.5 Momentum Space
D functions to show that Z
W d Ω D 1fag D
X
Z jamihamj ,
or
hW i d Ω D Tr[1fag ] .
m
(9.44) Representative pointing probabilities for Figure 9.4 are shown in Figure 9.6. At zero velocity, the atoms are polarized along the electric field vector of the light, E D EQ x. For a velocity outside the capture range, v D 60 cm s1 , the density matrix has little transverse coherence. The largest fraction (58%) of the atoms is in the end state ja, ai D j2, 2i, and these atoms make the strong, downward lobe of the pointing probability. But the next largest fraction (18%) of atoms is in the other end state, jaai D j22i, and this produces the smaller, upward lobe. The large damping coefficient, @F z /@z, at small velocity is associated with the transformation of the polarization from transverse to longitudinal.
9.5 Momentum Space
We turn now to the combined effects of the mean and fluctuation forces on the atom. The random nature of the absorption and emission of photons, and the motion of the atom through gradients in the light shift leads to fluctuating forces that change the momentum distribution and heat or cool the atoms. To analyze the effects of these fluctuating forces, we include the atomic momentum as a quantum number and we introduce the spin-momentum basis states jμ p i D jp i ˝ jμi .
(9.45)
We already discussed the spin basis state jμi in connection with (3.16). At position z along the light beam the atomic momentum basis state has amplitude eip k z hzjp i D p , V
(9.46)
p D pz
(9.47)
so
is the atomic momentum in units of „k. For clarity, we assume the momentum states (9.46) are confined – or have periodic boundary conditions – inside a large volume V. The precise value of V is not important. The total energy of the state (9.45) is 1 E μ p D „ω fe gg C E μ C E p . 2
(9.48)
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9 Radiation Forces
The mean energy, „ω fe gg /2, of the ground-state sublevels was discussed in connection with (2.15), and the energy shift, E μ , of the sublevel, jμi, was given by (3.14). The kinetic energy of the atom is Ep D
p2 („k p )2 D „δ ω R , 2M 2
(9.49)
where the recoil frequency is δ ω R D „k 2 /M .
(9.50)
For an excited atom, the spin-momentum basis state j μN pN i and its energy E μN pN can be defined in an analogous way. We write the spin-momentum density matrix in Schrödinger space as D
X
ffg gg( p q) C fg eg( p q) C fe gg( p q) C fe eg( p q) gjp ihqj .
(9.51)
pq
We will call the g fgg g fgg matrix fg gg( p q) with the elements hμ p jfg ggjνqi D fg gg(p q) μ ν the p q-th “tile” of the ground-state density matrix. There are analogous tiles for the optical coherences and the excited state. For the counterpropagating beams, the interaction operator V of (5.62) in spinlocation space is VQ D
X
VQ (r) eir k z ,
(r) with VQ (r) D D † EQ .
(9.52)
r
The factor VQ (r) of (9.52) acts on the spin portion of the wave function. The photon momentum quantum number is r D ˙1. To simplify the subsequent discussion, we will assume perfect repumping, with all of the alkali-metal atoms in the ground-state multiplet with total spin quantum number f D a D I C 1/2. When appended to the program ending with Code 1.7 on page 14, the following MATLAB statements will evaluate the matrices VQ (r) for the cycling transition from f D fg to fN D fe D fg C 1. Code 9.4 fg=input(’ground-state spin, fg = ’); fe=fg+1; Mg=input(’atomic mass in Daltons, Mg = ’); Mg=Mg/NA;%to gram lam=input(’wavelength in nm, lam = ’); lam=lam*1e-7;%to cm rpg=input(’pumping rate/recoil shift, Gpg/dwR= ’); xi=input(’detuning in units of 1/te, xi = ’); eta=input(’helicity, +/- 1; eta = ’); te=input(’spontaneous lifetime in nsec, te = ’); te=te*1e-9;%to s ns=input(’number of positive (or negative) momentum sidebands, ns = ’); gg=2*fg+1; ge=2*fe+1;%statistical weights keg=2*pi/lam; weg=keg*c;%nominal spatial and temporal frequencies D=sqrt(hbar*c^3*ge/(4*weg^3*te));%nominal electric dipole amplitude
9.5 Momentum Space
Code 9.4 Continued the previous code. dwR=hbar*keg^2/Mg;%recoil shift in rad/s kap=dwR*te;%dimensionless recoil shift Gpg=rpg*dwR;%mean pumping rate of ground-state atoms at rest E0=-hbar*(xi+i/2)/te;%energy denom. for atoms at rest Ek=sqrt(Gpg*gg*abs(E0)^2*te/(2*D^2));%electric field amplitude ek=[1 1; i*eta -i*eta;0 0]/sqrt(1+eta^2);%polarization vectors for k=fe:-1:-fe;%spherical components of dimensionless dipole op. for l=1:-1:-1 for m=fg:-1:-fg if k+l==m Ds(fg-m+1,fe-k+1,2-l)=sqrt(3/gg)*cg(fe,k,1,l,fg,m); end end end end Dj(:,:,1)=(-Ds(:,:,1)+Ds(:,:,3))/sqrt(2); Dj(:,:,2)=(-Ds(:,:,1)-Ds(:,:,3))/(i*sqrt(2)); Dj(:,:,3)=Ds(:,:,2);%Cartesian components tV=zeros(ge,gg,2);%interaction; kr=1 or 2 for up or down for kr=1:2 for j=1:3%sum on Cartesian indices j tV(:,:,kr)=tV(:,:,kr)-D*Dj(:,:,j)’*Ek*ek(j,kr); end end Asge=zeros(gg*gg,ge*ge);%spontaneous emission for j=1:3%sum over three Cartesian axes Asge=Asge+(ge/3)*kron(conj(Dj(:,:,j)),Dj(:,:,j)); end
The matrix elements of (9.52) between the basis states (9.45) are h μN pN jVQ jμ p i D
X
(r) δ p,rCp VQ μN μ . N
(9.53)
r
In spin-momentum space, the interaction (9.52) can therefore be written as VQ D
X
jp C rihp j ˝ VQ (r) .
(9.54)
pr
Q p(r) , with the matrix elements In analogy to (6.8) we define the operator, W Q p(r) jμ p i D h μN pN jW
(r) VQ μN μ
h μN pN jE (r) jμ p i
.
(9.55)
The energy denominator is h μN pN jE (r)jμ p i D
fg eg „δ ω R 2 i„Γs pN p 2 C„(ω fe gg ω)CE μN E μ . (9.56) 2 2
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9 Radiation Forces
For the remainder of this chapter, we will focus on cycling transitions between a ground-state hyperfine multiplet of angular momentum quantum number f and an excited state multiplet of angular momentum quantum number fN. We will assume that the Zeeman energy splittings are negligibly small, so we can set E μN D E fN and E μ D E f , where E fN and E f are the energies of the excited-state and ground-state hyperfine multiplets at zero magnetic field. We must have pN D p C r for the nonzero matrix elements of h μN pN jVQ (r) jμ p i. The energy denominator (9.56) depends only on the photon momentum index r and on the right momentum index p of the atom, and we write it as (r)
h μN pN jE (r)jμ p i D E p D E0 C „δ ω R p r .
(9.57)
The zero-momentum energy denominator is fg eg
(˙1)
E0 D E0
D „(ω 0 ω)
i„Γs 2
,
(9.58)
and the zero-momentum resonance frequency is ω 0 D ω fe gg C
E fN E f „
C
δωR . 2
(9.59)
Q becomes In summary, for spin-momentum space, the operator W Q D W
X
Qp , jp C rihp j ˝ W (r)
pr
Q (r) Q p(r) D V . where W (r) Ep
(9.60)
9.6 Evolution in Spin-Momentum Space
Let the atoms be pumped to a single hyperfine multiplet of the excited state, with angular momentum quantum number fN. We assume that the magnetic field is negligibly small, so we can ignore any Zeeman precession of the excited atoms. Then we can simply add a term representing spontaneous radiative decay to (6.23) to find the total evolution equation: i„
d fe eg eg Q † i„Γsfg eg fe Q fg gg VQ † VQ fg gg W DW . eg dt
(9.61)
This has the steady-state solution fe eg D
1 fe gg
i„Γs
Q †] . Q fg gg VQ † VQ fg gg W [W
(9.62)
Equation 9.63 implies that the steady-state value of the tile fe eg( p q) is fe eg( p q) D
1 fe gg
i„Γs " # X VQ (r) fg gg( p r, qs)VQ (s)† VQ (r) fg gg( p r,qs)VQ (s)† . (9.63) (r) (s) E p r E qs rs
9.6 Evolution in Spin-Momentum Space
Substituting (9.63) back into (5.47), adding (5.47) to (6.9), and neglecting stimulated emission in comparison with spontaneous emission, we find i„
d fg gg Q † VQ VQ † W Q fg gg D [H fgg , fg gg ] C fg gg W dt [ fN] Q † ]Δ † . Q fg gg VQ † VQ fg gg W C Δ [W 3
(9.64)
Finally, with the understanding that ( p q) D fg gg( p q) in the following equation, we see that optical pumping will cause the density matrices of the ground-state tiles to evolve at the rate 2 3 N X VQ (s)† VQ (r) ( p rCs,q) ( p ,qrCs)VQ (r)† VQ (s) d (p q) 4 5C [f] i„ op D (r) (r) dt 3 E p rCs E qrCs rs " # X VQ (r) ( p r,qs)VQ (s)† VQ (r) ( p r,qs)VQ (s)† Δ Δ† . (r) (s) E p r E qs rs (9.65) For simplicity, in the last line of (9.65) we have ignored the momentum variance due to the random directions of spontaneously emitted photons. As we will show in the next section, these contributions are very small compared with the contributions from optical pumping – especially for high-spin atoms. fgg
Mean Pumping Rate Generalizing (6.17), we define the mean pumping rate, Γp , of ground-state atoms as the rate of photon absorption by unpolarized atoms with zero momentum. Then the only nonzero tile of the ground state is (00) D 1fgg /[ f ], and we find from (9.65) fgg
Γp
D
1 X Q (r)† Q (r) 1 d (00) 1 D Tr V V Tr dt i„[ f ] r E0 E0 fe gg
D
2jDj2 jE (1) j2 Γs [ f ]jE0 j2
D 2Φ (1) σ fgg .
(9.66)
The energy denominator, E0 , for zero-momentum atoms was given by (9.58). We evaluated the trace in (9.66) with the aid of (9.52), (5.6), (5.7), and (5.30). The absorption cross section for unpolarized atoms at rest is σ
fgg
D
λ 2 [ fN] 8π[ f ]
!
fe gg2
Γs fe gg2
Γs
/4 C (ω ω 0 )2
.
(9.67)
The unidirectional photon flux is Φ (1) D
cjE (1) j2 . 2π„ω fe gg
(9.68)
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9 Radiation Forces
9.7 Liouville Space
The Liouville-space version of (9.64) is d j) D G j) , dt
where
GD
i © fgg H C Γp A op , „
(9.69)
or X i ˇ ( p q) d ˇˇ ( p 0 q 0 ) fgg 0 0 0 0 © ˇ p p . q jH jp q C Γ q jA jp q D ˇ p op dt „ pq (9.70) Here j( p q) ) is an [ f ]2 1 column vector for each pair of momentum quantum numbers, p q. We will often compactify j( p q) ) substantially, and by an amount that depends on the momentum quantum numbers p q. Since we are assuming negligibly small magnetic fields, the matrix elements of the Hamiltonian are determined only by the momentum quantum numbers and are
„δ ω R 2 p 0 q 0 jH © jp q D δ p 0 p δ q 0 q p q2 . 2
(9.71)
Comparing (9.69) and (9.65), we see that the matrix elements of the optical pumping operator are given by (p 0 q 0 jA op jp q) 1
D
X
"
[ VQ (s)† VQ (r) δ p 0,p Crs δ q q 0
rs
Ep
fgg
i„Γp
fg eg
A s
(r)
δ p 0,p Cr δ q 0 ,qCs VQ (s) ˝ VQ (r)
1
(r) Ep
] δ p 0 p δ q 0,qCsr VQ (s)† VQ (r) (s)
1
Eq !#
(s) Eq
. (9.72)
fg eg
The spontaneous emission matrix, A s
, was given by (5.50).
The steady-state solution of (9.69) is X X h ( p p )i hμ p j1jμ p i D Tr 1 D 1 . G j1 ) D 0 , with
Steady State
μp
(9.73)
p
9.8 Compactification
We now turn to the technical details of how to code for the complicated spinmomentum space which we use to describe laser cooling. Two types of energy de(r) (s) nominators, E p and E q occur in (9.72). Each term with an energy denominator
9.8 Compactification
Figure 9.7 Optical pumping can couple an initial tile, (0,0) in this case, to itself and to eight other tiles, located to the “east, northeast, north, northwest, west, southwest, south and
southeast”. The tiles labeled with (p q) are optical coherences through which the coupling takes place.
(s)
E q is the Liouville conjugate of a corresponding term with the energy denomina(r) tor E p . It is therefore sufficient to evaluate all the matrix elements with the energy (r) denominators E p to find the partial matrix AQ op . The full optical pumping matrix ‡ is simply A op D AQ op C AQ op . The matrix elements of (9.72) correspond to displacements (including the null displacement) in the p q plane. These are sketched in Figure 9.7. 9.8.1 Compactified p q Space
In most situations of experimental interest, the density matrix is nonzero in only a finite number of tiles that are relatively close to the origin of p q space. For counterpropagating, circularly polarized beams of the same helicity, absorbing a photon increases the azimuthal spin quantum m of a ket j f mi by the helicity, η D ˙1. If the atoms start in noncoherent momentum states with p D q before they are exposed to the pumping light, the maximum momentum difference that can be generated is bounded by jp qj 2 f for integer quantum numbers f D 0, 1, 2, . . . , and by jp qj 2 f 1 for half-integer quantum numbers with f D 1/2, 3/2, 5/2, . . . . Larger values of jp qj would correspond to giving the atom an azimuthal spin quantum number m > f . Since the photons are absorbed or emitted in pairs, the differences in the momentum quantum numbers must be even, that is, p q D 0, ˙2, ˙4, . . . . For the “lin-perp-lin” polarizations
141
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9 Radiation Forces
(c)
(b)
(a) (d)
Figure 9.8 A compactified p q space for atoms with ground-state spin quantum number f D 2 and MOT pumping. Optical pumping couples a black tile to itself and to the eight nearest-neighbor black tiles. Couplings to tiles with jp qj > 2 f D 4 are zero and are
dropped. Couplings across the skew-diagonal boundaries where jp C qj D 2n s D 8 are “folded” to make periodic boundary conditions in momentum space. The tile labeled 23 has p q D 00.
of the counterpropagating beams used in Sisyphus cooling, p q must still be an even integer, but there is no restriction on the maximum value of jp qj, and the density matrix extends into a larger region of p q space than for one-dimensional MOTs. Cooling experiments are designed to limit the values of jp j and jqj. It is therefore possible to compactify p q space sufficiently to permit relatively speedy computations in the compactified space while obtaining answers that hardly differ from those one would obtain with the full, infinite-dimensional Liouville space. An example of a compactified p q space for MOT beams is shown in Figure 9.8. The diagonal boundaries where jp qj D 2 f D 4 are well defined and set by conservation laws of linear and angular momentum. The tile δ on the top skew boundary (label 10) couples directly to the tiles with the labels 1, 2, 10, 11, 19, and 20. It has folded couplings to the tiles labeled 9, 18, and 27 at the opposite skew boundary, as though there were a “virtual” tile δ just outside the lower boundary. This “folding” of the couplings at the skew-diagonal boundaries ensures that there will be periodic boundary condions in momentum space, and that atoms will be conserved in the compatified space. The two skew-diagonal boundaries can be chosen by trial and error so that the value of the density matrix in tiles along the skew boundaries is negligibly small. If doubling the size of the compactified space makes a negligible difference to the answer, the original compactified space was large enough.
9.8 Compactification
9.8.2 Compactification within a Tile
Absorption of a photon with direction quantum number r changes the momentum quantum number p to p C r and it changes the spin quantum number m to m C r η. Because of this correlated change of momentum and spin indices for MOT pumping beams, if the atom has neither spin nor momentum coherence when the pumping light is first turned on, absorption and emission of pairs of photons will create correlated spin and momentum coherences. A tile on the main diagonal of p q space for which only the populations are nonzero is coupled to spin coherences of a nearest-neighbor northeast tile, such that the coupled spin coherences are displaced by η units from the main diagonal within the tile. Other couplings have similar selection rules. Thus, most of the density-matrix elements within any tile will be zero, and the density matrix will have the form (p q)
( p q)
μ ν D δ p q,η(μν) μ ν .
(9.74)
In Figure 9.9 we illustrate some of the logical variables, δ p q,η(μν), for potentially nonzero elements of the density matrix in the case of an atom with groundstate spin quantum number f D 2. When appended to the program ending with Code 9.4 on pages 136 and 137, the following MATLAB statements will generate momentum labels p q of the tiles in the compactified space, and also the azimuthal quantum numbers m μ and m ν of the spin states within each tile.
Figure 9.9 Logical variables, δ p q,η(μν) , for intratile compactification. In this example, the ground-state atom has spin quantum number f D 2, and the helicity is η D 1. All logically false elements, labeled with 0’s, are negligibly small, and can be neglected.
143
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9 Radiation Forces
Code 9.5 mug=kron(ones(gg,1),(fg:-1:-fg)’);%left spin indices of ground-state tile nug=kron((fg:-1:-fg)’,ones(gg,1));%right spin indices of ground-state tile lpg=mug==nug;%logical variable for ground-state populations mue=kron(ones(ge,1),(fe:-1:-fe)’);%left spin indices of excited-state tile nue=kron((fe:-1:-fe)’,ones(ge,1));%right spin indices of excited-state tile np=2*ns+1;%number of tiles along diagonal of pq space p=[]; q=[];%initialize tile indices maxpmq=2*floor(fg);%maximum |p-q| for MOT for km=1:maxpmq+1 lpmq(:,km)=mug-nug==maxpmq+2-2*km;%log. var. of allowed coherences for kp=1:np%generate list of momentum indices p=[p;ns+maxpmq/2+2-km-kp];%left momentum index q=[q;ns-maxpmq/2+km-kp];%right momentum index end end gpq=length(p);%number of pq tiles
When appended to program ending with Code 9.5, the following MATLAB statements will evaluate the left and right momentum quantum numbers p and q and the left and right spin azimuthal quantum numbers m μ and m ν , denoted by pp, qq, mmu, and nmu. Code 9.6 pmq=eta*(p-q);%azimuthal qn of spin coherences gmunu=2*fg-abs(pmq)+1;%number of spin coherences in pq tile km=maxpmq/2+1-pmq/2; jf=0;%azimuth index, initial spin index for k=1:gpq ji=jf+1;%initial coherence index of tile k jf=jf+gmunu(k);%final coherence index of tile k pp(ji:jf)=p(k);%replicate p for spin states qq(ji:jf)=q(k);%replicate q for spin states l=lpmq(:,km(k));%l.v. for spin states in tile k mmu(ji:jf)=mug(l);%mu for coherence in tile k nnu(ji:jf)=nug(l);%nu for coherence in tile k lmunu(:,k)=l;%save l.v. for cohrence location in tile k infkg=(ji:jf)’;%save coherence indices for each tile k end pp=pp’; qq=qq’; mmu=mmu’; nnu=nnu’; in=in’;%rows to columns lp=pp==qq & mmu==nnu;%logical variable of populations gsm=length(pp);%number of spin-momentum basis states for km=1:maxpmq+1 lm(:,km)=pp==qq+maxpmq+2-2*km;%logical variables for coherences end
9.8 Compactification
When appended to the program ending with Code 9.6, the following MATLAB statements will evaluate the numerators of the coupling coefficients of (9.72) that are illustrated in Figure 9.7. Code 9.7 Nuu=flat(tV(:,:,1)’*tV(:,:,1))/(i*Gpg*hbar);%self coupling up Ndd=flat(tV(:,:,2)’*tV(:,:,2))/(i*Gpg*hbar);%self coupling down Nnw=-Asge*kron(conj(tV(:,:,1)),tV(:,:,1))/(i*hbar*Gpg);%nw Nse=-Asge*kron(conj(tV(:,:,2)),tV(:,:,2))/(i*hbar*Gpg);%se Nne=-Asge*kron(conj(tV(:,:,2)),tV(:,:,1))/(i*hbar*Gpg);%ne Nsw=-Asge*kron(conj(tV(:,:,1)),tV(:,:,2))/(i*hbar*Gpg);%se Nnn=flat(tV(:,:,2)’*tV(:,:,1))/(i*Gpg*hbar);%nn Nss=flat(tV(:,:,1)’*tV(:,:,2))/(i*Gpg*hbar);%ss
When appended to the program ending with Code 9.7, the following MATLAB statements will evaluate the elements of the optical pumping matrix A op (9.72). Code 9.8 tAop=zeros(gpq,gpq);%initialize for kcc=1:gpq%run through columns of matrix tAop pcc=p(kcc); qcc=q(kcc); lcc=lmunu(:,kcc);%pq and spin lv of starting tile tAop(infkccg,infkccg)=Nuu(lcc,lcc)/(E0+hbar*kap*pcc/te) ... +Ndd(lcc,lcc)/(E0-hbar*kap*pcc/te); pnw=pcc+1; qnw=qcc+1;%pq for northwest tile if pnw+qnw>2*ns;%in skew boundaries? pnw=pnw-(2*ns+1); qnw=qnw-(2*ns+1);%wrap end knw=find(pnw==p&qnw==q); lnw=lmunu(:,knw);%indices, lv’s of coupled tiles tAop(infknwg,infkccg)=Nnw(lnw,lcc)/(E0+hbar*kap*pcc/te); pse=pcc-1; qse=qcc-1;%pq for southeast tile if abs(pse+qse)>2*ns;%in skew boundaries? pse=pse+(2*ns+1); qse=qse+(2*ns+1);%wrap end kse=find(pse==p&qse==q); lse=lmunu(:,kse);%indices, lv’s of coupled tiles tAop(infkseg,infkccg)=Nse(lse,lcc)/(E0-hbar*kap*pcc/te); if fg>.5%coupling in off-diagonal direction pne=pcc+1; qne=qcc-1;%pq for northeast tile if abs(pne-qne)<=maxpmq;%in diagonal boundaries? kne=find(pne==p&qne==q); lne=lmunu(:,kne); tAop(infkneg,infkccg)=Nne(lne,lcc)/(E0+hbar*kap*pcc/te); end
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9 Radiation Forces
Code 9.8 Continued the previous code. psw=pcc-1; qsw=qcc+1;%pq for southwest tile if abs(psw-qsw)<=maxpmq;%in diagonal boundaries? ksw=find(psw==p&qsw==q); lsw=lmunu(:,ksw); tAop(infkswg,infkccg)=Nsw(lsw,lcc)/(E0-hbar*kap*pcc/te); end pnn=pcc+2; qnn=qcc;%pq for north tile if abs(pnn-qnn)<=maxpmq%in diagonal boundaries? if abs(pnn+qnn)>2*ns;%in skew boundaries? pnn=pnn-2*ns-1; qnn=qnn-2*ns-1; end knn=find(pnn==p&qnn==q) ;lnn=lmunu(:,knn); tAop(infknng,infkccg)=Nnn(lnn,lcc)/(E0+hbar*kap*pcc/te); end pss=pcc-2; qss=qcc;%pq for south tile if abs(pss-qss)<=maxpmq%in diagonal boundaries? if abs(pss+qss)>2*ns%outside skew boundaries? pss=pss+2*ns+1; qss=qss+2*ns+1; end kss=find(pss==p&qss==q);lss=lmunu(:,kss); tAop(infkssg,infkccg)=Nss(lss,lcc)/(E0-hbar*kap*pcc/te); end end end for j=1:gsm%transposition matrix T(:,j)=pp(j)==qq&qq(j)==pp&mmu(j)==nnu&nnu(j)==mmu; end Aop=tAop+conj(T*tAop*T);%add Liouville conjugate
The preceding code runs quite slowly, and it was written for clarity, not for speed. When appended to the program ending with Code 9.8, the following MATLAB statements will evaluate the steady-state solution (9.73). Code 9.9 G=Aop+i*diag((pp.^2-qq.^2)/(2*rpg));%evolution operator rhoin=nullfast(G);%find steady state rhoin=rhoin/(lp’*rhoin);%normalize crhoin=zeros(gg^2,2*ns+1);%initialize rho versus momentum for k=1:maxpmq+1 x=rhoin(pp-qq==maxpmq+2-2*k); nx=length(x)/(2*ns+1); y=reshape(x,nx,2*ns+1); l=logical(lpmq(:,k)*ones(1,2*ns+1)); crhoin(l)=y; end
9.9 Displays
9.9 Displays
Here we briefly discuss ways to display the physics of the very large spin-momentum density matrix j1 ) that is the solution of (9.73). 9.9.1 Momentum-Space Displays
For each momentum quantum number p we assemble a g fgg g fgg matrix (p ) D ( p p ) C ( p C1,p 1) C ( p 1,p C1) C ( p C2,p 2) C ( p 2,p C2) C (9.75) from the elements ( p q) of the steady-state solution (9.73). The column-vector transforms j(p ) ) of the matrices (9.75) are stored as the column of the array crhoin by the code ending with Code 9.9 on page 146. When appended to the program ending with Code 9.9, the following MATLAB statements will plot, versus the momentum quantum number p, the total populations Tr[(p ) ] and the sublevel populations hμj( p )jμi of the atoms. Code 9.10 m=ns:-1:-ns;%list momentum samples Nin=(real(lpg’*crhoin))’;%total population per momentum state figure(1); clf subplot(2,1,1) plot(m,Nin,’r-’); grid on ;hold on; ylabel (’Tr[\rho^f(p)g]’) rhomu=real(crhoin(lpg,:));%sublevel populations versus p subplot(2,1,2); plot(m,rhomu); grid on xlabel (’Momentum, p, in units of h/\lambda’) legend(’Populations of sublevels \langle \mu|\rho^f(p)g|\mu \rangle’)
From Figure 9.10 we see that the equilibrium density matrix has strong spin polarization that is correlated in a complicated way with the momentum. To better understand the coherence of the density matrix, we can plot the amplitude of the transverse pointing probability, hW (φ)i, which is given by (9.39) with θ D π/2, so n D x cos φ C y sin φ .
(9.76)
Figure 9.11 shows the transverse pointing probabilities for p D 0, ˙8 and with other parameters the same as for Figure 9.10. The transverse pointing probabilities decrease as jp j increases, and they rotate slightly to the right or left, depending on the sign of p. When appended to the program ending with Code 9.10 on page 147, the following MATLAB statements will generate Figure 9.11.
147
148
9 Radiation Forces
(a)
(b) Figure 9.10 Momentum distribution of 87 Rb atom from the solution of (9.73) with the code ending with Code 9.10 on page 147. The parameters were fg = 2, Gpg/dwR = 10, xi = -5, eta = 1, and ns = 30. (a) shows the total (b) the sublevel population.
Figure 9.11 Transverse pointing probability for the same parameters as those used for Figure 9.10 and for the momenta p D 0, ˙8.
9.9 Displays
Code 9.11 Fp=diag(sqrt((1:2*fg).*(2*fg:-1:1)),1); Fj(:,:,1)=(Fp+Fp’)/2; Fj(:,:,2)=(Fp-Fp’)/(2*i); Fj(:,:,3)=diag(fg:-1:-fg); cff=[1;zeros(gg-1,1)];%ket vector |ff> zz=cff*cff’;%rho for spin pol along z Rtheta=expm(-i*pi*Fj(:,:,2)/2);%90 degree rot. about y xx=Rtheta*zz*Rtheta’;%rho for spin pol. along x nphi=50;%number of azimuthal sample points phi=linspace(0,2*pi,nphi); Wn=zeros(nphi,gg^2); for k=1:nphi Rphi=expm(-i*phi(k)*Fj(:,:,3)); nn=Rphi*xx*Rphi’;%rho for spin parallel to n Wn(k,:)=(gg/4/pi)*nn(:)’;%prob. to point along n end figure(2); clf;%momentum space polar(phi,real(Wn*crhoin(:,ns+1))’) hold on; polar(phi,real(Wn*crhoin(:,ns+1+round(ns/4)))’,’r’) polar(phi,real(Wn*crhoin(:,ns+1-round(ns/4)))’,’g’) figure(3);clf;%position space rho0=sum(crhoin,2);%rho summed over all p polar(phi,real(Wn*rho0)’) title(’ for z = 0’)
9.9.2 Position-Space Displays
For MOT beams, we see from (9.74) that the position-space density matrix (i.e., the part diagonal in z) is fzg D hzjjzi D
X
eip k z ( p q) eiq k z D eiη k z Fz f0g eiη k z Fz ,
(9.77)
pq
where f0g D
X pq
( p q) D
X
( p ) .
(9.78)
p
Combining (9.78) with (9.39) and (9.76), we find that the expectation value of the transverse pointing probability is [f] hxjei(φCη k z)Fz f0g ei(φCη k z)Fz jxi 4π D hW (φ C η k z , 0)i .
hW (φ, z)i D Tr[W (φ)fzg ] D
(9.79)
149
150
9 Radiation Forces
Figure 9.12 Transverse pointing probability in position space for z D 0 and the same parameters as those used for Figure 9.10.
Figure 9.13 The transverse (θ D π/2) pointing probabilities hW i for the solutions of Figure 9.4. When plotted in position space (a), the transverse pointing probability spirals around the z axis, in accordance with the spiraling linear polarization of the MOT
pumping beams. When plotted in momentum space (b), the transverse coherence decreases rapidly as the magnitude of the momentum (velocity) increases. This is consistent with Figure 9.6, which shows the spin polarization for fixed nonquantized momenta (velocities).
We see that the transverse pointing probability spirals around the z axis, just like the linear polarization of the MOT pumping beams. The transverse pointing probability for z D 0, calculated with the code ending with Code 9.11 on page 149, is shown in Figure 9.12. A fuller display of the transverse pointing probability for all values of momentum and position is shown in Figure 9.13.
9.10 Momentum Diffusion
9.10 Momentum Diffusion
We now discuss “momentum diffusion”, the growth in the width of the momentum distribution due to the fluctuating parts of the forces from optical pumping and spontaneous emission of light. From considerations of the interplay between the mean force (9.7) and momentum diffusion, which we discuss in the next few sections, one can obtain much insight into the equilibrium momentum distributions of cooled atoms, though not with the quantitative detail that is provided by the use of spin-momentum space, which we have just discussed. Let m D „p be the longitudinal momentum of the atom and let N d m be the probability of finding the atom with a momentum between m and m C d m at time t. We first assume that a mean force F and the fluctuating recoils of the atoms cause the probability density to satisfy the drift-diffusion equation @N @F N @2 D N . D C @t @m @m 2
(9.80)
If the force F is independent of velocity (i.e., independent of m), the solution of (9.80) that evolves from the distribution N D δ(m) at time t D 0 is ND p
(m F t)2 . exp 4D t 4πD t 1
(9.81)
For times p t > 0, the momentum has the normal distribution with the variance σ m D 2D t. Figure 9.4 showed that for MOT beams the force F is not constant but depends on velocity in a complicated way. For sufficiently small velocities v (or sufficiently small momenta m) we can write the force as F D γ m D M γ v ,
(9.82)
where γ is the low-velocity viscous damping rate of the atom, which we take to be independent of m, and M is the atomic mass. Then in the steady state, (9.80) becomes γ mN C
dD N DcD0, dm
(9.83)
where the constant c must be zero to ensure that both the density N and the diffusion current D d N/d m approach zero as jmj ! 1. The normalized solution to (9.83) is r
1 γ γ m2 m2 D p . (9.84) ND exp exp 2πD 2D 2M k T 2πM k T The effective temperature of the optically cooled atoms is T D
D . kMγ
(9.85)
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152
9 Radiation Forces
To get the coldest possible atoms, we want a large viscous damping rate γ , that is, a large value of d F/d v at v D 0 in Figure 9.4, and we want the smallest possible momentum diffusion coefficient D. In the next two sections, we will show how to estimate values of D. It is customary to think of the momentum diffusion as the sum of diffusion due to spontaneous emission of light and the diffusion due to the absorption of photons going in various directions during optical pumping. As we will soon show, the diffusion due to spontaneous emission is very small compared with the diffusion due to optical pumping. Problems We conclude this section by summarizing a few problems with the force-diffusion methods outlined above:
The detailed spin-momentum distributions, like those in Figure 9.10, are not Gaussian, as implied by (9.84). As shown in Figure 9.4, the damping rate γ D d F/d m is not independent of m as assumed in the derivation of (9.84). Keeping track of the spin polarization of the atoms, and how that depends on momentum is a complicated task for the force–diffusion approach.
9.11 Momentum Diffusion Due to Spontaneous Emission
The momentum for an individual fluorescent photon propagating along the unit vector n is p D „k n .
(9.86)
The recoil momentum of the atom is equal and opposite to the photon momentum (9.86). The rate of growth of the atom’s mean momentum due to spontaneous emission of photons is therefore Z d dΦ D „k n dΩ hp i dt dΩ fg eg
D
„k Γs [ J] 8π
Z nΔ † Δ W (1 nn)d Ω D 0 .
(9.87)
The spontaneous emission rate, d Φ /d Ω , of photons per solid angle was given by (8.61). Since the integral over all solid angles of any odd number of factors n must be zero, the fluorescent photons transfer no average momentum to the atoms. The mean force comes entirely from absorption of light. Let N d 3 p be the probability of finding the atomic momentum in a volume element d 3 p of momentum space, centered on the momentum p at time t. We assume that the fluctuating recoils of the atoms cause the probability density to
9.12 Momentum Diffusion from Pumping
satisfy the anisotropic diffusion equation X @2 N @N Di j . D @t @p i @p j
(9.88)
ij
The diffusion coefficient (a dyadic) can be defined as i h d DppE . hDi D Tr Dfe eg D dt 2
(9.89)
From (9.89) and (9.86) we see that the diffusion-coefficient operator is „2 k 2 DD 2
Z fg eg
D
dΦ dΩ dΩ
112 f J Jg J CS 3[ J ] O 1 C (1) fξ ξ g2 T2 . J JS 5
nn
„2 k 2 Γs 6
(9.90)
To evaluate the second line of (9.90) we used the multipole expansion (8.61) for d Φ /d Ω and the identity 1 nn D 3
r
8π Y2 (n) fξ ξO g2 . 15
(9.91)
The expectation value, hDi, of the diffusion coefficient is a real, symmetric tensor with positive eigenvalues, which are the principal diffusion coefficients. The eigenvectors are the principal directions of diffusion. The diffusion coefficient will be anisotropic if the excited atoms have nonzero components of the “alignment”, hT2m i. This is possible if J > 1/2, but for J 1/2 the diffusion will always be isotropic. Longitudinal Diffusion Coefficient For comparison with expressions in the literature, we will often be interested in the longitudinal diffusion coefficient due to spontaneous emission. From (9.90) and (8.64) we see that this is
Dse D hz D zi fg eg
„2 k 2 Γs D 6
feg
1
(1)
J CS
! p
6[ J] 112 f J J g feg T j . J J S 20 5
(9.92)
9.12 Momentum Diffusion from Pumping
If spontaneous emission were the sole source of momentum diffusion, the large enhancement of the velocity-dependent force due to optical pumping would rapidly cool the atoms to nearly the recoil energy „2 k 2 /2M . However, optical pumping can greatly enhance the momentum diffusion, since the resulting spin polarization
153
154
9 Radiation Forces
leads to large fluctuations of momentum transfer from the pumping light to the atoms. This enhanced momentum diffusion limits the cooling. Here we discuss the simple but instructive model of Castin and Mølmer [56], who considered optical pumping of a cycling transition between an atom in a ground-state Zeeman multiplet with angular momentum quantum number f to an excited-state multiplet with quantum number fN D f, f ˙ 1. Counterpropagating, circularly polarized laser beams of the same helicity and intensity pump the atoms. The lasers are weak enough for us to ignore stimulated emission. The simplest Castin–Mølmer model assumes negligible coherence between either the spin or the momentum of the atomic ground state. We therefore expect the Castin– Mølmer model to provide a good estimate of momentum diffusion for velocities outside the “capture range”, for example, for jv j > 10 cm s1 for the parameters shown in Figure 9.4. With this assumption we need only consider terms in (9.70) with p D q and with r D s. For simplicity we denote the diagonal matrix ( p p ) by (p ) , and the matrix elements (p 0 p 0 jA op jp p ) by A p 0 p . Noting from (9.71) that H © jp p ) D 0, we see that (9.70) becomes h ˇ ˇ i @ ˇˇ ( p ) fgg D Γp A p ,p C1j( p C1) ) C A p p ˇ( p ) C A p ,p 1 ˇ( p 1) . @t
(9.93)
In the simple Castin–Mølmer model any dependence of the coupling matrices in (9.97) on the momentum index p is neglected. This would correspond to setting (r) E p D E0 in (9.72). Then we can write A p ,p C1 D A 1,0 ,
A p p D A 00
and
A p ,p 1 D A 1,0 .
(9.94)
The density matrix for the internal spin state of the atom, irrespective of the momentum, is Xˇ ˇ ( p ) . (9.95) j) D p
If we sum both sides of (9.97) over all p, we find @ fgg j) D Γp A op j) . @t
(9.96) fgg
This is the same as (1.10) with the relaxation operator G D Γp A op , determined solely by optical pumping. There is no need to include the Hamiltonian in G since,H © D 0 for populations. We also ignore collisional contributions to the damping operator. The total optical pumping operator is A op D A 1,0 C A 00 C A 1,0 .
(9.97)
The right and left eigenvectors of A op are defined by A op jα j ) D α j j α j ) ,
((α j jA op D ((α j jα j
and ((α j jα k ) D δ j k . (9.98)
9.12 Momentum Diffusion from Pumping
The eigenvalue index is j D 1, 2, . . . , g fgg , and we will assume that the eigenvalues α j ordered such that jα j j jα j C1 j. For j D 1 we have jα 1 ) D j1 ),
((α 1 j D (1fgg j
and
α1 D 0 .
(9.99)
Here j1 ) is the steady-state density matrix for optical pumping alone, without momentum diffusion. Since we have compactified the Liouville space to the populations of the ground-state sublevels, (1fgg j is a 1 g fgg row vector, all elements of which are 1. Continuum Limit At sufficiently late times, we may assume that the momentum amplitudes (9.97) will vary slowly with the momentum index p. Then the integer variable p can be replaced by a continuous real variable. To order @2 /@p 2 we write the terms of (9.97) as
j(p ˙1) ) D j( p ) ) ˙
@ (p) 1 @2 ( p ) j ) C j ) C @p 2 @p 2
Substituting (9.100) into (9.97), we find @ ˇˇ (p ) @ @2 ˇ 1 fgg D Γp A op δ A rp C A rp 2 ˇ( p ) . @t @p 2 @p
(9.100)
(9.101)
Here we have introduced the repopulation operator A rp D A 1,0 A 1,0
(9.102)
and the differential repopulation pumping operator δ A rp D A 1,0 A 1,0 .
(9.103)
Multiplying (9.101) on the left by ((α j j, we find @ fgg ((α j j( p ) ) D Γp α j ((α j j( p ) ) @t @ @2 ˇ 1 fgg C A rp 2 ˇ( p ) . C Γp ((α j j δ A rp @p 2 @p
(9.104)
At late times the left side of (9.104) will get smaller and smaller as the distribution of atoms widens in momentum space at an ever slower rate. The only way (9.104) can remain valid for late times is for the terms on the right side to nearly cancel. Making the extreme approximation that the right side of (9.104) is zero, we find (for j ¤ 1 and α j ¤ 0) ˇ 1 ˇˇ @ @2 ˇ 1 α j δ A rp (9.105) C A rp 2 ˇ( p ) . ((α j ˇ(p ) D αj @p 2 @p ˇ From inspection of (9.104) we see that any spin-momentum amplitudeˇ((α j ˇ( p ) with j > 1 and α j ¤ 0 will damp more rapidly than the amplitude ((α 1 ˇ( p ) with α 1 D 0. So at late times, we expect to have ((α j j(p ) ) ((α 1 j( p ) ) ,
for
j >1.
(9.106)
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156
9 Radiation Forces
That is, we expect each momentum state, j( p ) ), to have a a spin distribution that differs only slightly from the steady-state distribution, jα 1 ) D j1 ). As the momentum distribution gets smoother at late times, we expect to have @2 @ ((α j j( p ) ) ((α j j( p ) ) . 2 @p @p
(9.107)
In view of (9.105)–(9.107) we take the late-time approximation to (9.104) to be ((α j j( p ) ) D
1 @ ((α j jδ A rpjα 1 ) ((α 1 j( p ) ) . αj @p
(9.108)
Substituting (9.108) into (9.104) and retaining only the largest terms on the right, we find the diffusion equation in momentum space, Dop @2 @ ((α 1 j( p ) ) D 2 2 2 ((α 1 j( p ) ) , @t „ k @p where the momentum diffusion coefficient due to optical pumping is 9 8
(9.109)
(9.110)
j >1
The two terms in curly brackets are reminiscent of first-order and second-order corrections in perturbation theory. Noting that the equilibrium density matrix of the excited state, summed over all momentum states, is fgg
jfeg ) D
Γp
fg eg Γs
fe gg
Ap
jα 1 ) ,
(9.111)
we can write the contribution (9.92) of spontaneous emission to the momentum diffusion coefficient as ˇ p
ˇ 6[ J] 112 f J J g ˇ fe gg 2 2 fgg 1 feg J CS Dse D „ k Γp 1 (1) T20 ˇ A p jα 1 ) . ˇ J JS 6 5 (9.112) The pumping-rate parameter Γ s of Castin and Mølmer [56] is related to our fgg mean pumping rate Γp by fgg
Γp
D
(2 f e C 1) Γs . 3(2 f g C 1)
(9.113)
Taking (9.113) into account, we find for f D 1 and fN D 2, (Dop C Dse ) 5/9 D (225/88 C 23/220), the same as (2.7) of Castin and Mølmer [56], and for f D 0 and fN D 1, (Dop C Dse ) D (1/2 C 1/5), the same as (2.8) of Castin and Mølmer [56].
9.12 Momentum Diffusion from Pumping
The contribution (9.110) of optical pumping to momentum diffusion is larger than the contribution (9.92) of spontaneous emission. For larger values of f, the contribution from optical pumping completely dominates the contribution from spontaneous emission. For the cycling transition from f D 2 to fN D 3 in 87 Rb, the relative values are Dop /Dse D 100.4. For the cycling transition from f D 4 to fN D 5 in 133 Cs the relative values are Dop /Dse D 865.9. As pointed out by Castin and Mølmer [56], the great enhancement of the diffusion coefficient for optical pumping on the cycling transitions comes from the tendency of atoms to be pumped to sublevels with m f or m f , similar to those in Figure 9.1b. For these highjmj states, the atoms successively absorb many photons propagating in the same direction, photons with polarization that is strongly absorbed by atoms in states with large jmj. There are relatively long sequences of directed rather than random walks in momentum space, and this gives the large diffusion coefficients. This is the reason why we have ignored the additional momentum variance due to the random directions of spontaneously emitted photons in the last line of (9.65). When appended to the program ending with Code 9.7 on page 145, the following MATLAB statements will evaluate and display the diffusion coefficients Dop and Dse . Code 9.12 mue=kron(ones(ge,1),(fe:-1:-fe)’);%left spin indices of excited state nue=kron((fe:-1:-fe)’,ones(ge,1));%right spin indices of excited state lpe=mue==nue;%logical variable for excited-state populations Apeg=(kron(conj(tV(:,:,1)),tV(:,:,1))+kron(conj(tV(:,:,2)),tV(:,:,2)))... *(1/(i*hbar*Gpg))*(1/E0-1/conj(E0));%excitation matrix Arp=2*real((-Nnw(lpg,lpg)-Nse(lpg,lpg))/E0); dArp=2*real((-Nnw(lpg,lpg)+Nse(lpg,lpg))/E0); Aop=2*real((Nnw(lpg,lpg)+Nuu(lpg,lpg)+Ndd(lpg,lpg)+Nse(lpg,lpg))/E0); [raj aj]=eig(Aop);%eigenvalues and right eigenvectors of sAop [aj, nr]=sort(diag(real(aj))); raj=raj(:,nr);%sorted right eigenvectors raj(:,1)=raj(:,1)/sum(raj(:,1));%renormalize non-relaxing eigenvector laj=inv(raj);%find left eigenvectors piAop=zeros(gg,gg);%initialize for k=2:gg%pseudoinverse of sAop piAop=piAop+raj(:,k)*laj(k,:)/aj(k); end Dop=real(laj(1,:)*(Arp/2+dArp*piAop*dArp)*raj(:,1)); T20=zeros(ge,ge);%initialize multipole moment for me=fe:-1:-fe T20(fe-me+1,fe-me+1)=cg(fe,me,2,0,fe,me)*sqrt(5/ge); end rhoe=Apeg(lpe,lpg)*raj(:,1);%excited-state density matrix Dse=real(sum((eye(ge)-(-1)^(fe-fg)* ... (sqrt(6)*ge/5)*sixj(1,1,2,fe,fe,fg)*T20)*rhoe)/6); disp([’D_fopg = ’ num2str(Dop) ’; D_fseg= ’ num2str(Dse)])
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159
10 Relaxation of Polarized Atoms Optically pumped alkali-metal atoms can have their polarization destroyed or modified by various relaxation mechanisms. Particularly important are collisions with buffer-gas atoms, spin-exchange collisions with other alkali-metal atoms, spatial diffusion between regions with differing spin polarization, and spin relaxation during adsorption on the walls. We will focus on the relaxation of ground-state atoms. The depolarization rates of alkali-metal atoms in their ground state are so slow that relatively weak collisional interactions can have a noticeable effect on the relaxation. In contrast, the relaxation rates of excited atoms are always at least as fast as fg eg the spontaneous radiative decay rate, Γs , so only strong collisional interactions have an appreciable effect on the physics. We suppose that the atoms evolve under the influence of a common groundstate Hamiltonian H fgg D H of (2.10). The magnetic field may be spatially inhomogeneous and it may have oscillating components, intended to induce magnetic resonance. The common Hamiltonian H D H(r, t) may therefore depend on the spatial location, r, of the atom, and on the time t. The atom evolves under the combined influence of the common Hamiltonian H and a collisional spin interaction, V, which will have a different time history for each atom of the ensemble. If V represents the interaction between a polarized atom and a perturbing atom or molecule, it will at least depend on the internuclear separation, and possibly on the collision velocity and the internal state of the perturbing atom or molecule as well. For a single collision, the effects of V can be calculated with partial-wave scattering theory, or by assuming that the translational motion of the atom and its collision partner follow well-defined classical paths. These alternative ways of calculating collisional effects are sketched in Figure 10.1. The classical-path approximation works well for most collision processes of importance for optical pumping, with the notable exception of spin-exchange collisions between pairs of alkali-metal atoms, which are best treated with partial-wave scattering theory.
Optically Pumped Atoms. William Happer, Yuan-Yu Jau, and Thad Walker Copyright © 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-40707-1
160
10 Relaxation of Polarized Atoms
│ψ
(r)│2, ψ (r) is radical wave function
(a)
(b)
Figure 10.1 (a), (b) Spin relaxation due to collisions can be analyzed with classical-path calculations (a) or partial-wave calculations (b). With the notable exception of spin-exchange
collisions between alkali-metal atoms, these two methods give results that are within a few percent of each other. Classical-path calculations are less time-consuming.
10.1 S-Matrix
We describe the spin state of an individual alkali-metal atom with a wave function jψi. For our introductory discussion we assume that the partner’s internal state plays no role in the collision, so we ignore the wave function of the collision partner. In later sections on spin exchange, it will be necessary to include the wave function of the collision partner. We will also assume that the translational energies of the colliding partners are so large that we can ignore the changes in translational energies that are required to compensate for changes in the internal states of the colliding atoms during an inelastic collision. It will be convenient to call this the “infinite-temperature” approximation. For this situation the evolution of jψi is defined by the Schrödinger equation i„
@ jψi D HN jψi . @t
(10.1)
The total Hamiltonian HN acting during a collision is HN D H C V ,
(10.2)
where H is the Hamiltonian of the free atom and V is the collisional interaction. If the buffer-gas pressure does not exceed a few atmospheres, we expect to find that V D 0 for relatively long periods of free flight between collisions. Binary collision have a duration τ of a few picoseconds. Three-body collisions can lead to the
10.1 S-Matrix
formation of van der Waals molecules, for which the collision duration τ can be tens of nanoseconds at buffer-gas pressures of 1 Torr. When an atom hits the wall, the collision duration τ may also be relatively long, since the atom may hop along the surface for many nanoseconds before it is finally able to evaporate into the gas phase. It is convenient to describe collisions in the gas with an S-matrix. Suppose that an atom has a collision just before time tc . The collision duration is τ, in the sense that V(t) is negligibly small unless tc τ < t < tc . At time tc the atomic wave function has become jψc i as a result of the collision, whereas it would have been jψi if there were no collision. The S-matrix is defined by jψc i D S jψi .
(10.3)
It is convenient to write the S-matrix as S D U(τ)eiH τ/„ .
(10.4)
Note that eiH τ/„ jψ(tc )i D jψ(tc τ)i is the value of the wave function at the starting time of the collision. The evolution operator U D U(θ ) is a function of the time, θ D t tc C τ, after the start of the collision. From the Schrödinger equation we see that during the collision U is defined by i„
@U D HN U , @θ
with
U(0) D 1 .
(10.5)
Here HN D H C V(tc τ C θ ). For a three-body collision, resulting in the formation of a van der Waals molecule in a particular vibration–rotation state, we can take V and HN to be time-independent during the molecular lifetime, τ, and write the solution of (10.5), for 0 θ τ, as N
U(θ ) D ei H θ /„ .
(10.6)
Then the S-matrix (10.4) for van der Waals molecules of lifetime τ is N
S D ei H τ/„ eiH τ/„ .
(10.7)
For binary collisions, V will change gradually from negligibly small values at the beginning of the collision to large values near the time of closest approach to negligibly small values at the end of the collision. The integration of (10.5) to find U can be facilitated by writing Q ). U(θ ) D eiH θ /„ U(θ
(10.8)
Substituting (10.8) into (10.5), we find that UQ is the solution of i„
@UQ Q D VQ U, @θ
with
Q U(0) D1.
(10.9)
161
162
10 Relaxation of Polarized Atoms
The interaction-picture potential, VQ D VQ (θ ), is VQ D eiH θ /„ V eiH θ /„ ,
with
V D V(tc τ C θ ) .
(10.10)
Using (10.8) in (10.3), we find iH τ/„ Q . S D eiH τ/„ U(τ)e
(10.11)
For binary collisions, one can ignore the factors e˙iH τ/„ in (10.11) and set Q S D U(τ) .
(10.12)
The omitted operators e˙iH τ/„ simply translate the wave function jψi of a free atom along the time axis by ˙τ, so when we use (10.12) instead of (10.11), the collision appears to happen at the slightly earlier time, tc τ. The relaxation described by (10.12) is identical to that described by (10.11). For example, the probability jhνjS jμij2 for making a transition from the state jμi to the state jνi is the same for both expressions. In conclusion, we recall that since the Hamiltonian evolution operators H and V are Hermitian, the S-matrix is unitary: S † D S 1 .
Liouville Space
(10.13)
The Liouville-space version of (10.3) is
c D SS † ,
or
jc ) D S £ j) .
(10.14)
The Liouville-space S-matrix, S £ , is obtained from the Schrödinger-space S-matrix, S, by the transformation S £ D S [ S †] D S ˝ S .
(10.15)
Equation 2.1.83 in [6], which corresponds to (10.15), has the order of S and S reversed because the column vector j) in [6] is assembled from the rows of the matrix rather than from the columns, the convention we use in this book. One can use the identities (4.81) with (10.13) to show S £ of (10.15) is unitary:
S£
†
1 D S£ .
(10.16)
Two easily verified identities for S £ are (1fgg jS £ D (1fgg j and
S £ j1fgg ) D j1fgg ) .
(10.17)
ground state was given by (2.17). The The unit matrix 1fgg for the ˇ Schrödinger elements of the row vector, 1fgg ˇ, are 1 for populations and 0 for coherences. The left identity of (10.17) expresses the fact that the collision does not change the total population of the atom. The right identity expresses the fact that collisions with unpolarized atoms or molecules will leave an initially unpolarized atom unpolarized.
10.2 Collisions in the Gas
10.2 Collisions in the Gas
We introduce a nominal collision rate 1 D Nvσ , Tc
(10.18)
between an alkali-metal atom and buffer-gas atoms or molecules of number density N. The mean relative velocity between an alkali-metal atom and a buffer-gas atom or molecule is v, and σ is a cross section. For binary collisions we can set σ D π r 2, where r is an impact parameter that is large enough that the influence of V will be negligible during a collision. The choice of r is not critical, provided that it is large enough. For three-body collisions, σ will be proportional to N. It is clear from the discussion of the previous section that the change, Δ, in the density matrix, D jψihψj, in the small time interval Δ t due to collisions is Δ D
Δt hSS † i . Tc
(10.19)
Here the angle brackets h i represent an ensemble average of collisions with all possible impact parameters and orbital planes – or if van der Waals molecules are formed, over all vibration–rotation states. The overall evolution of the density matrix due to collisions and due to the influence of the free-atom Hamiltonian H is @ i 1 D [H, ] C hSS † i . @t „ Tc In Liouville space (10.20) becomes 1 i © @ A j) , j) D H C @t „ Tc
(10.20)
(10.21)
where the operator A is A D 1 hS £ i .
(10.22)
£
If we let S c denote the S-matrix that describes a collision with a particular impact parameter and orbital plane, indicated by the subscript c, the average value of the S-matrix for an ensemble of collisions is hS £ i D
1X £ Sc , g c
(10.23)
where c labels a particular collision of the ensemble and g is the total number of sample collisions. We will often call the operator A of (10.22) the accommodation operator because it (or more precisely each of its eigenvalues, α) is analogous to the thermal accommodation coefficient [57], α D (Ti Tf )/(Ti Tw ), of gas molecules of temperature
163
164
10 Relaxation of Polarized Atoms
Ti impinging on a wall of temperature Tw and being desorbed from the wall at some final temperature Tf . If the final temperature is the same as the wall temperature, the accommodation coefficient is α D 1, and the wall is maximally efficient in destroying the temperature difference, Ti Tw . If the final temperature of the molecules is the same as the initial temperature, the accommodation coefficient is α D 0, and the wall is completely unable to destroy the temperature difference. In view of (10.17), the matrix A satisfies the identities (1fgg jA D 0
and
Aj1fgg ) D 0 .
(10.24) ˇ AD From inspection of (10.21), we see that the equation on the left of (10.24), 1 0, ensures that @ 1feg j /@t D @Tr[]/@t D 0. This is a completely general property of accommodation operators describing processes that simply redistribute ˇ atoms between the energy sublevels. The equation on the right of (10.24), A ˇ1fgg D 0, ensures that if the atoms are unpolarized at time t D 0, the collision process described by A will leave the atoms unpolarized for all later times. This is not a general property of accommodation operators, and it comes from ignoring the internal states of the collision partner and from assuming the same collision rate 1/ Tc for both endothermic and exothermic inelastic collisions, an approximation that is not valid if the energy intervals between spin sublevels become comparable to k T .
fgg ˇ
10.3 Weak Collisions
Weak collisions are those for which the collision time τ is short enough and the interaction V is small enough that for any pair of sublevels, jμi and jνi, Ztc jhμjV jνij d t „ .
(10.25)
t c τ
Strong collisions are those for which the condition (10.25) is violated. In weak collisions, it is most likely that a collision will cause no change in the internal quantum numbers of the atom. With the notable exception of spin-exchange collisions between pairs of alkali-metal atoms, the most important interactions for binary collisions satisfy (10.25). It will be convenient to work directly in Liouville space, where the analog of (10.9) is @UQ £ i„ (10.26) D VQ © UQ £ , with UQ £ (0) D 1fg gg . @θ The unit operator for Liouville space, 1fg gg was given by (5.112). The Liouville-space time-evolution operator UQ £ and interaction VQ © are related to their Schrödingerspace equivalents by UQ £ D UQ [ UQ †] D UQ ˝ UQ
and
VQ © D VQ [ VQ ] .
(10.27)
10.3 Weak Collisions
For weak collisions, we can use the familiar perturbation series to solve (10.26), 1 S D UQ (1) D 1 C i„ £
Z1
£
Q ©0
V 1
1 dt C (i„)2
Z1
0
Q ©0
V
dt
1
Zt 0
0
00 VQ © d t 00 C
1
(10.28) 0 To simplify the form of the subsequent equations we have set VQ © D VQ © (t 0 ) and 00 VQ © D VQ © (t 00 ), with VQ © (t) given in analogy to (10.10) by © © VQ © D eiH θ /„ V © eiH θ /„ ,
with
V © D V © (tc τ C θ ) .
(10.29)
Since V © D 0 for times outside the collision interval, the time integrals can extend from1 to 1. We write the interaction V © D V © (t) in terms of its Fourier transform VM © D VM © (ω), 1 V (t) D 2π
Z1
©
M©
iω t
V (ω)e
dω
and
Z1
M©
V (ω) D
1
V © (t)eiω t d t . 1
(10.30) Since V © (t) D V ©† (t), we must have VM ©† (ω) D VM © (ω) .
(10.31)
Taking matrix elements of (10.28) between Liouville basis states ( j j and jl), we find £
Sjl D δ jl C C
1 M© V (ω j l ) i„ j l 1 X Z
1 (2πi„)2 Z1
Z1 dτ
0
k 1
d t0
Z1
0
d ω 0 VM j©k (ω 0 )ei(ω ω j k )t
0
1
d ω 00 VMk©l (ω 00 )ei(ω
00 ω
k l )(t
0τ)
C
(10.32)
1
Changing the order of integration, and carrying out the integrations over d t 0 and d ω 0 , we find £
1 M© V (ω j l ) i„ j l Z1 Z1 1 X 00 00 M © 00 M © 00 V ω V d ω ω (ω ) d τei( ω ω k l ) τ . jl jk kl 2π„2
Sjl D δ jl C
k 1
0
(10.33) According to the secular approximation, which we discussed in Section 7.3, the basis states j j ) and jl) will be coupled only if their Bohr frequencies are almost
165
166
10 Relaxation of Polarized Atoms
the same, ω j ω l . Any coupling terms that do not satisfy this criterion will have negligible influence on the evolution of the atoms. Therefore, we will set ω j D ω l in (10.33), and we will take its ensemble average to find £
1 M© hV (0)i i„ j l Z1 Z1 1 X 00 0 M© 00 M © 00 d ω h V (ω ) V (ω )i d τei(ω ω k l )τ . (10.34) jk kl 2π„2
hS j l i D δ j l C
k 1
0
The last integral over d τ can be evaluated with the identity Z1 d teiω t D π δ(ω) C 0
i} . ω
(10.35)
Here δ(ω) is the Dirac delta function and } denotes the principal part. One can prove (10.35) by inserting a convergence factor e t in the integrand, and taking the limit of the result when the small positive parameter goes to zero. Using (10.35) with (10.34) and (10.31), we find to order V 2 , £
1 M© hV (0)i i„ 8j l 1 X < M ©† hV (ω k l )VM k©l (ω k l )i 2 : jk 2„
hS j l i D δ j l C
k
i} C π
Z1
d ω 00
1
9 ©† hVM j k (ω 00 )VM k©l (ω 00 )i = ω 00 ω k l
;
.
(10.36)
The part of (10.36) that is first order in V © represents shifts in the Bohr frequencies of the atom due to the time-averaged value of V © , which acts like a small, timeindependent perturbation. The part of (10.36) that is second order in V © represents transfer of populations or coherences between the Liouville basis states jl) and j j ) as well as a second-order frequency shift. The second-order shifts are dominated by the absorption and reemission of collisional “phonons” of frequency ω 00 close to the frequency ω k l of the transition from jl) to the intermediate state jk).
10.4 Relative Power Spectrum
It is often a good approximation to assume that hVM © (ω)VM ©† (ω)i D hVM © (0)VM © (0)i J(ω) .
(10.37)
Equation 10.37 defines the real, relative power spectrum, which has the properties J(ω) D J(ω) and
J(0) D 1 .
(10.38)
10.5 Sudden Collisions
We will define the width of the power spectrum by Z1 Δω D
J(ω)d ω .
(10.39)
1
If the collisions are well described by the relative power spectrum of (10.37), we can use (10.36) with (10.22) to find the matrix elements of the accommodation operator,
i 1 X M© M© A j l D hVM j©l (0) i C 2 hV j k (0)Vk l (0)i J(ω k l ) iK(ω k l ) . (10.40) „ 2„ k
The real, frequency-shift function K(ω) is the Hilbert transform of the relative power spectrum, } K(ω) D π
Z1 1
J(ω 0 ) d ω0 . ω ω0
(10.41)
The frequency-shift function is antisymmetric, so K(ω) D K(ω) and
K(0) D 0 .
(10.42)
10.5 Sudden Collisions
Collisions of such short duration, τ 1/Δω, that jω k l j Δω
(10.43)
will be called sudden collisions. The spectral width, Δω, was given by (10.39). Since the duration of binary collisions at room temperature or above is measured in picoseconds, corresponding to Δω 1012 s1 , whereas the difference, ω k l , between the Bohr frequencies of the coherences jk) and jl) seldom exceed 1010 s1 for magnetic fields of less than about 1 kG, binary collisions are usually sudden. Therefore, we can consider the Bohr-frequency differences ω k l to be close enough to zero that in accordance with (10.38) and (10.42) we can set J(ω k l ) D 1 and
K(ω k l ) D 0 .
(10.44)
Then (10.40) simplifies to AD
i M© 1 hV (0)i C 2 hVM © (0)VM © (0)i . „ 2„
(10.45)
Substituting (10.45) into (10.21), noting that VM † (0) D VM (0), and therefore that VM © (0)j) D j[VM (0), ]), we find the Schrödinger-space version of (10.21) when A is given by (10.45), i @ i 1 i M 1 h (10.46) D [H, ] [V (0), ] 2 VM (0), [VM (0), ] . @t „ Tc „ 2„
167
168
10 Relaxation of Polarized Atoms
Noting that Aj1fgg ) D VM © j1fgg ) D j[VM (0), 1fgg ]) D 0, we see that the accommodation operator A of (10.45) satisfies the identities (10.24).
10.6 Strong Collisions
Several of the important collisional relaxation processes for alkali-metal atoms cannot be described with the theory of weak collisions. For sufficiently low buffer-gas pressures, a typical lifetime τ v of a van der Waals molecule in the vibration–rotation state v can be long enough that the criterion, (10.25), for weak collisions is violated. The important role played by van der Waals molecules in the spin relaxation of alkali-metal atoms in the heavy noble gases Ar, Kr, and Xe was first discovered and analyzed by Bouchiat et al. [58, 59]. Another important process that cannot be described by the theory of weak collisions is spin-exchange collisions between pairs of alkali-metal atoms, where the interaction potential V is so large that (10.25) is violated, even when τ c is only a few picoseconds. Van der Waals Molecules The heavier noble gases Ar, Kr, and Xe are known to form loosely bound van der Waals molecules with alkali-metal atoms. The physics involved in the formation and breakup of such molecules is sketched in Figure 10.2. For gases like these, say, Xe, there will be a density matrix for unbound alkalimetal atoms, say, Rb, of number density [Rb] and a density matrix σ v for alkalimetal atoms bound in the vibration–rotation state v of the RbXe molecule. Let [RbXe] v be the number density of molecules in the state v, so the total density of bound molecules is X [RbXe] D [RbXe] v . (10.47) v
Figure 10.2 (a) Van der Waals molecules are broken up by collisions, and they can be formed in three-body collisions. The third body is necessary to carry away the binding energy of the molecule. (b) Potential energy diagram.
10.6 Strong Collisions
We will normalize the density matrices such that 1 X [RbXe] Tr[σ v ] D Tr[] v [Rb]
and Tr[] C
X
Tr[σ v ] D 1 .
(10.48)
v
The number densities of free and bound alkali-metal atoms are related to the number density [Xe] of buffer-gas atoms by the law of mass action: [Rb][Xe] D d . [RbXe]
(10.49)
The dissociation coefficient d , which depends weakly on the temperature, can be calculated with elementary statistical mechanics if the binding potential of RbXe is known. The spin evolution of the free-atom density matrix is given by X i © 1 j) C j) P D γ v jσ v ) . (10.50) H C „ Tc v Here X 1 1 D , Tc Tv v
with
1 D λ v [Xe]2 , Tv
(10.51)
is the formation rate of molecules in all vibration–rotation states, and λ v is the coefficient for forming molecules in state v by three-body collisions. The fraction of atoms formed in state v is fv D
Tc . Tv
(10.52)
The two-body breakup rate of molecules in the state v is γv D
1 D v [Xe] . τv
(10.53)
In analogy to (10.50), σ v evolves at the rate j σP v ) D G v jσ v ) C
1 j) . Tv
(10.54)
The damping operator for the state v is Gv D
i N© H C γv . „ v
(10.55)
] Here HN v© D HN v[ HN v is the Liouville-space Hamiltonian (10.2) for the state v of the molecule. We assume that the rotational angular momentum quantum numbers, N v , for most of the states v are so large that we can consider the rotational angular momentum operator N v to be a time-independent classical vector. Then HN v©
169
170
10 Relaxation of Polarized Atoms
and G v can be taken to be time-independent spin operators during the molecular lifetime, and the exact solution to (10.54) is 1 Tv
jσ v ) D
Zt
d t 0 expfG v (t t 0 )gj0 ) ,
(10.56)
1
where σ v D σ v (t) and 0 D (t 0 ). The collisional breakup rate γ v of the molecule is so large compared with 1/ Tc that it is a very good approximation to write the factor j0 ) of (10.56) as j0 ) D expfiH © (t t 0 )/„gj) ,
(10.57)
where D (t). Substituting (10.57) into (10.56), we find 1 £ S v j) . γ v Tv
jσ v ) D
(10.58)
The S-matrix for van der Waals molecules formed in the state v, with a mean life τ v D 1/γ v , is £ Sv
Z1 D
γ v d t expfG v tg expfiH © t/„g 0
D
X kN k
j kN v )( kN v jk)(kj . 1 C i(ω kN v ω k )τ v
To evaluate (10.59) we noted that X ω k jk)(kj and H© D „
Gv D
(10.59)
X (iω kN v C γ v )j kN v )( kN v j .
(10.60)
kN
k
The Bohr frequencies and basis states of the free and bound atoms are most simply obtained from the Schrödinger-space eigenvalue equations, H jμi D E μ jμi
and
HN v j μv N i D E μN v j μv N i.
(10.61)
From (10.61) we find the Liouville-space quantities for (10.60) jk) $ jμihνj and j kN v ) $ j μv N ih νN v j
„ω k D E μ E ν I and „ω kN v D E μN v E νN v .
(10.62) (10.63)
Substituting (10.58) into (10.50), we find (10.21), with the accommodation operator X £ f v Sv . (10.64) AD 1 v
Substituting (10.59) into (10.64), we find that the matrix elements of the accommodation operator are A k0k D δ k0k
X kN v
f v (k 0 j kN v )( kN v jk) . 1 C i(ω kN v ω k )τ v
(10.65)
10.7 Hyperfine-Shift Interaction
10.7 Hyperfine-Shift Interaction
We recall that the coefficient Afgg for the magnetic dipole interaction Afgg I S in the 2 S1/2 ground state of an alkali-metal atom is given by the Fermi–Segré formula Afgg D
8πg S μ B μ I fgg 2 jψ (0)j , 3I
(10.66)
where jψ fgg (0)j2 is the probability density for the valence electron to be at the site of the nucleus. When a buffer-gas atom or molecule is a small distance R from an alkali-metal atom, it perturbs the value of jψ fgg (0)j2 , and makes it slightly larger or smaller than that for a free atom. From (10.66) we see that this will produce a corresponding perturbation, δ Afgg D δ Afgg(R), of the coupling coefficient Afgg . This leads to the hyperfine-shift interaction: V D δ Afgg I S .
(10.67)
Notable Aspects of the Hyperfine-Shift Interaction: 1. The interaction (10.67) conserves the total spin angular momentum, F D I CS, of the atoms, but it does not conserve S or I separately. 2. At small magnetic fields, the interaction (10.67) causes negligible relaxation of population differences. 3. At large magnetic fields, the interaction (10.67) tends to equalize the populations of sublevels jμi and jνi with the same azimuthal quantum numbers, m μ D m ν . The overall effect is similar to that of spin exchange between pairs of alkali-metal atoms. 4. The interaction (10.67) is responsible for almost all of the pressure shift of the microwave resonance frequencies. 5. The frequency shift from the interaction (10.67) is so large that atoms in parts of the cell with slightly different temperatures can have substantially different microwave resonance frequencies. The resulting line broadening from temperature inhomogeneities can be much larger than the homogeneous collisional broadening from the same interaction. 6. The interaction (10.67) causes negligible homogeneous or inhomogeneous line broadening for low-field Zeeman resonances.
Some of the key features of the hyperfine shift interaction are shown in Figure 10.3 Binary Collisions In practice, the shift coefficients δ Afgg of (10.67) are small enough that for binary collisions the criterion (10.25) for weak collisions is very well satisfied. For magnetic fields .1 T, the criterion (10.43) for sudden collisions is also satisfied. We will consider sudden, weak, binary collisions in this section. The zero-frequency Fourier amplitude (10.30) of (10.67) becomes
VM © (0) D „φ(I S)© ,
(10.68)
171
172
10 Relaxation of Polarized Atoms
(c) (a)
ρ37
(d)
(b) Figure 10.3 (a) Interaction of a buffer-gas atom and an alkali-metal atom. The hyperfineshift interaction shifts the frequency (c) and damps the Bohr frequency (d) of microwave
resonances with Δ f D 1. At low magnetic fields, this interaction has negligible effects on Zeeman resonances or sublevel populations (b).
where the phase angle is φD
1 „
Z1
δ Afgg d t .
(10.69)
1
Using (10.68) in (10.45), we find the accommodation operator A D ihφi(I S)© C
hφ 2 i (I S)©2 . 2
(10.70)
Low Fields For the limit of vanishing magnetic field, B ! 0, the Liouville basis states j f m f 0 m 0 ) are eigenstates of A,
Aj f mI f 0 m 0 ) D α f mI f 0 m 0 j f mI f 0 m 0 ) .
(10.71)
The eigenvalues are α f mI f 0 m 0 D
i 1 hφi[I ]( f f 0 ) C hφ 2 i[I ]2 ( f f 0 )2 . 2 8
One can verify (10.72) by noting that ˇ
(I S )© j f mI f 0 m 0 ) D ˇ I S, j f mih f 0 m 0 j 1 D [I ]( f f 0 )j f mI f 0 m 0 ) . 2
(10.72)
(10.73)
From (10.71) and (10.72), we see that the hyperfine-shift interaction has no effect on populations j f mI f m) or Zeeman coherences, j f mI f m 0 ). The hyperfine-shift
10.7 Hyperfine-Shift Interaction
interaction shifts the resonance frequency of the hyperfine coherence jamI b m 0 ) by δ ω a mIb m 0 D
1 1 Im(α a mIb m 0 ) D hφi[I ] . Tc 2Tc
(10.74)
The contribution to the damping rate of the coherence is δ γ a mIb m 0 D
1 1 Re(α a mIb m 0 ) D hφ 2 i[I ]2 . Tc 8Tc
(10.75)
High Fields At higher magnetic fields, where the magnetic interaction μ fgg B of (2.10) becomes comparable to the hyperfine interaction Afgg I S, the hyperfineshift interaction couples populations with the same azimuthal quantum number m, and it has some of the same qualitative effects on the observed spin relaxation as spin-exchange collisions between pairs of alkali-metal atoms. By measuring spin relaxation at high magnetic fields, Walter and Griffith [60] were able to make some of the first accurate measurements of Carver rates by observing the “decoupling” of the relaxation rates in large magnetic fields of several thousand gauss. As has been pointed out by Oreto et al. [21], it is difficult to measure Carver rates at low field, since the homogeneous relaxation rate of the microwave coherence predicted by (10.76) can easily be confused with inhomogeneous relaxation of the coherence due to the frequency shift (10.74) in cells with small temperature gradients. Experimental Measurements From inspection of (10.66), we see that the phase angle φ of (10.69) should be proportional to μ I /I for different isotopes of the same chemical element, for example 39 K and 41 K or 85 Rb and 87 Rb. It is useful to quote measured values of quantities involving the hyperfine-shift interactions with an isotope-independent parameter. For spin relaxation, a convenient, isotope-independent parameter is the Carver rate ΓC , with the theoretical, isotopeindependent value
ΓC D
hφ 2 i , Tc η 2I
where
ηI D
μI . 2I μ N
(10.76)
The efficiency factor, η I , accounts for different isotopes of the same element. For a hypothetical isotope with nuclear spin I D 1/2 and with a nuclear magnetic moment μ I D μ N of one nuclear magneton, we would have η I D 1. Physically, ΓC is the rate at which the electron spin hS z i and the nuclear spin hI z i of the hypothetical atom would relax to a common value at magnetic fields large enough to decouple the hyperfine interaction. From inspection of (10.76), (10.70), and (10.21) we see that the spin relaxation rate of the density matrix due to the hyperfine-shift interaction is @ ΓC η 2I (I S)©2 j) . j) D @t 2
(10.77)
173
174
10 Relaxation of Polarized Atoms
A convenient parameter to characterize experimental measurements of relaxation due to the hyperfine-shift interaction is the coefficient C D
d ΓC . dN
(10.78)
Here N is the number density, usually quoted in amagats, of the perturbing gas. From (10.74) we see that the ratio of the hyperfine shift δ ω to the unperturbed hyperfine frequency ω D Afgg [I ]/(2„) should be the isotope-independent fraction, δω „hφi . D ω Tc Afgg
(10.79)
A convenient parameter to characterize experimental measurements of the frequency shift due to the hyperfine-shift interaction is the coefficient δω d λC D . (10.80) dN ω Table 10.1 C (s1 amg1 ) for the damping of hyperfine coherences. These are not expected to vary significantly with temperature [21]. Theoretical estimates from [62] are shown in parentheses, at 390, 158, 65, 60, and 30 ı C for Li through Cs respectively. He
Ne
Ar
Kr
Xe
N2
Li Na
(0.5) (7)
(0.4) (6)
(0.1) (1)
– (3)
– –
– –
K
(22)
(24)
(6)
(36)
–
–
Rb
191 110 ı C [60] (378)
(148)
0 100 ı C [60] (144)
–
–
–
–
395 100 ı C [60] 1632 50 ı C [61]
Cs
(487)
Table 10.2 Experimentally determined hyperfine frequency shift parameters λ C 106 (amg1 ) [64]. The temperature-shift coefficients, d λ C /d T , can be substantial [64]. He
Ne
Ar
Kr
Xe
N2
Li
54 390 ı C [63]
28 390 ı C [63]
3.7 390 ı C [63]
–
–
–
Na
39 100 ı C [64] 61 65 ı C [64]
23 100 ı C [64] 33 65 ı C [64]
0.9 100 ı C [64] 3.5 65 ı C [64]
–
–
56 65 ı C [65]
–
25 100 ı C [64] 43 65 ı C [64]
Rb
69 40 ı C [66]
39 40 ı C [64]
5.3 40 ı C [66]
54 40 ı C [66]
114 40 ı C [67]
50 40 ı C [66]
Cs
83 35 ı C [66]
48 35 ı C [68]
14 35 ı C [66]
82 35 ı C [66]
181 35 ı C [69]
60 35 ı C [66]
K
10.8 Spin–Rotation Interaction
To the extent that contributions from van der Waals molecules can be neglected, both C and λ C are independent of the gas pressure and have the same values for different isotopes of the same chemical element. However, λ C and C depend on temperature. Our current knowledge of the values of these parameters is summarized in Tables 10.1 and 10.2.
10.8 Spin–Rotation Interaction
As first pointed out by Bernheim [70], the spin–rotation interaction, V D γN S ,
(10.81)
couples the electron spin S to the relative angular momentum N of an alkali-metal atom and a colliding buffer-gas atom or molecule. For most experimental conditions in gas cells, representative values of N are very large compared with the values of S or I, typically N 100. Therefore, we can usually consider N to be a time-independent, classical vector, N D Nn .
(10.82)
We assume that the unit vector n is equally likely to point in any direction, so δi j . (10.83) 3 Here n i D n x i is the projection of n along the Cartesian basis vector x i . The angle brackets denote an average over the directions of n. hn i i D 0 and hn i n j i D
Notable Aspects of the Spin–Rotation Interaction 1. The interaction (10.81) is responsible for most of the longitudinal spin relaxation induced by collisions with buffer-gas atoms or molecules in low magnetic fields where the hyperfine-shift interaction of (10.67) has little effect. 2. The interaction (10.81) leads to multiexponential relaxation transients. To make accurate measurements, one must distinguish the contributions of the different transients to the observed relaxation signal. 3. The interaction (10.81) has an especially large effect on longitudinal relaxation rates at low magnetic fields and at low pressures for buffer-gas atoms or molecules that can form van der Waals molecules. 4. The interaction (10.81) is responsible for most of the homogeneous broadening of Zeeman resonance lines at low magnetic fields. Together with the hyperfineshift interaction (10.67), it also contributes to the homogeneous linewidths of microwave resonances. 5. In contrast to the hyperfine shift interaction (10.67), where the longitudinal relaxation rates increase with increasing magnetic field, many of the longitudinal rates for the interaction (10.81) decrease with increasing magnetic field.
Some of the key features of the spin–rotation interaction are shown in Figure 10.4.
175
176
10 Relaxation of Polarized Atoms
(b)
(a)
Figure 10.4 The spin–rotation interaction during a binary collision (a) can flip the electron spin S with negligible effect on the nuclear spin I . Such “S-damping collisions” lead to the relative collisional transition rates between low-field energy sublevels shown in (b).
10.8.1 Binary Collisions
In practice, the coupling coefficients γ N of (10.81) are small enough that the criterion (10.25) for weak collisions is well satisfied. For magnetic fields .1 T, the criterion (10.43) for sudden collisions is also satisfied. The zero-frequency Fourier amplitude (10.30) of (10.81) becomes VM © (0) D „φ n S © , where the electron spin operator in Liouville space is S angle, 1 φD „
(10.84) ©
[
]
D S S . The phase
Z1 γN dt ,
(10.85)
1
depends on the classical trajectory. Using (10.84) and (10.83) in (10.45), we find AD
hφ 2 i © © hφ 2 i S S D A sd , 6 3
(10.86)
where the S-damping operator, A sd D S © S © /2, was introduced in (6.88). Some useful identities involving A sd are (SjA sd D (S j ,
(I jA sd D 0 and (1fgg jA sd D 0 .
(10.87)
The 1 [I ]2 [S ]2 row vectors, (Sj, (I j, and (1fgg j, are defined by (4.27) and (4.25), with 1fgg given by (2.17). Combining (10.86) with (10.21), we find that the relaxation due to the spin– rotation interaction is @ (10.88) j) D iH © /„ C Γsd A sd j) , @t where the S-damping rate from the spin–rotation interaction is Γsd D
hφ 2 i . 3Tc
(10.89)
10.8 Spin–Rotation Interaction
Populations Using (6.88) with (4.85), we see that the nonzero matrix elements of (6.88) between population basis states are
1 μ μj S j , S j , jμ 0 ihμ 0 j 2
˛
1 D hμj S j , S j , jμ 0 ihμ 0 j jμ 2 3 D δ μ μ 0 hμjS jμ 0 i hμ 0 jS jμi . 4
(μ μjA sdjμ 0 μ 0 ) D
(10.90)
In (10.90) the repeated Cartesian indices j are summed over, but there is no sum over the repeated labels μ and μ 0 . The relative transfer rates, (μ μjA sdjμ 0 μ 0 ), of populations between different low-field state population basis states jμi and jμ 0 i are shown in Figure 10.4b. The projection of the accommodation operator in the population subspace can be expanded on the population eigenvectors j j ) of A sd , X AN sd D α j j j )(( j j . (10.91) j † Since AN sd D AN sd , we can take the eigenvectors j j ) of (10.91) to be orthonormal, with (( j j D ( j j, and ( j jk) D δ j k . In the limit that the externally applied magnetic field vanishes, the eigenvectors j j ) have a well-defined multipolarity L j . The normalized, nonrelaxing mode with p α j D 0 is simply j j ) D j1fgg )/ g fgg , where the atom is equally likely to be in any of the g sublevels jμi of the ground state. Along with the pure, unpolarized eigenstate, with α j D 0, Figure 10.5 shows the density matrices ˇ fgg ˇ1 (10.92) jL j , α j ) D fgg C c j j j ) , g
with maximum initial content of the relaxing mode j j ). For α j ¤ 0, some of the amplitudes (μ μj j ) must be negative, and one of the negative amplitudes, (μ j μ j j j ) (μ μj j ), will be most negative. The coefficient cj D
1 j μjj j)
g fgg (μ
(10.93)
is chosen to make the minimum sublevel population of jL j , α j ) zero. A special case is the unpolarized, nonrelaxing mode with α j D 0, where we take c j D 0 in (10.92). The relaxing modes, j j ), for atoms with no nuclear polarization are completely destroyed in each S-damping collision and they therefore have eigenvalues (accommodation coefficients) α j D 1. These are the modes shown along the bottomD of FigureE10.5. Modes with nuclear polarization can be written as (μ μj j ) D fI I g fI I g k j μjTL0 jμ , where k j is a constant coefficient and TL0 is a nuclear basis tensor, defined in analogy to (5.21). The nuclear polarization is conserved in an S-damping collision. It partially regenerates the polarization mode in the intervals between collisions, leading to accommodation coefficients α j < 1.
177
178
10 Relaxation of Polarized Atoms
Figure 10.5 The density matrices jL j , α j ) of (10.92), with the maximum possible content of the modes j j ) of (10.91). These are the modes for S-damping caused by the spin– rotation interaction (10.81) at low magnetic
fields for alkali-metal atoms with nuclear spin quantum number I D 3/2. There are two independent modes of different relaxation rates for the multipolarities L D 0, 1, 2 and one mode each for the multipolarities L D 3, 4 .
10.8.2 Experimental Measurements
By analyzing experimentally observed data on spin relaxation with models based on (10.88), one can deduce values for the characteristic rate Γsr . It is convenient to summarize these measurements with a cross section σ sr , defined by Γsr D N σ sr v ,
(10.94)
where N is the number density of the perturbing gas and v is the mean relative velocity of the colliding pairs. Experimental measurements and theoretical estimates of these parameters for various alkali-gas pairs are summarized in Table 10.3. There are striking, order-of-magnitude variations in the sizes of the cross sections for different alkali–noble gas pairs. For the heavy noble gases, the cross sections vary dramatically from Ne to Xe but show a much smaller variation as the alkali-metal partner is changed. In these cases, the spin–rotation interaction comes primarily from spin–orbit coupling in the core of the noble gas atom [71]. In contrast, the alkali-metal–He relaxation rates correlate well with the spin–orbit interaction in the alkali-metal core [72].
10.9 Spin Exchange between Alkali-Metal Atoms and Noble Gas Atoms
We turn now to spin-exchange processes that involve optically pumped alkalimetal atoms. Some of the most important exchange processes are sketched in Figure 10.6. The interaction primarily responsible for spin exchange between op-
10.9 Spin Exchange between Alkali-Metal Atoms and Noble Gas Atoms Table 10.3 Electron-spin randomization cross sections (1022 cm2 ). These cross sections are often strongly temperature dependent (e.g., T 3.7 for Rb–He [77]). Theoretical estimates at 100 ı C from [84] are given in parentheses. He
Ne
Ar
Kr
Xe
N2
Li
–
–
–
–
–
–
Na
0.0036 327 ı C [73]
(0.18)
(200)
(72)
1090 180 ı C [74]
–
K
0.005 150 ı C [75] 0.087 150 ı C [77]
(0.16)
6.3 117 ı C [76] 6.1 27 ı C [59]
(190)
(1800)
270 27 ı C [59]
2200 150 ı C [78]
0.75 150 ı C [83] 1.0 70 ı C [79]
0.24 0 ı C [82]
(0.20)
10.4 0 ı C [80]
260 ? ı C [81]
(2300)
Rb Cs
(0.18)
Figure 10.6 Spin exchange: (a) the exchange of electron spin polarization in a binary collision of a pair of alkali-metal atoms; (b) the exchange of the electron spin polarization of an alkali-metal atom with the nuclear spin
5.5 15 ı C [82]
polarization of a noble gas atom in a binary collision; (c) there is an especially high probability of spin exchange when a long-lived van der Waals molecule is formed in a three-body collision.
tically pumped alkali-metal atoms and noble gas nuclei is the isotropic, magnetic dipole hyperfine interaction, V D αS K ,
(10.95)
between the electron spin S of the alkali-metal atom and the nuclear spin K of the noble gas atom. We will take (10.95) to be the interaction potential of (10.2). The coupling coefficient α D α(R) D Afgg is given by the Fermi–Segré formula (10.66), with ψ fgg (0) being the amplitude of the valence electron wave function at the site of the noble gas nucleus. Consequently, α is a rapidly decreasing function of the internuclear separation R of the colliding pair. Other interactions that couple the spins of the two atoms, for example, the anisotropic magnetic dipole interaction, are much smaller than (10.95) and can be ignored. The translational motion of the
179
180
10 Relaxation of Polarized Atoms
colliding pair is hardly affected by spin interactions, so we can use a semiclassical analysis, where the orbital motion of the atoms about each other during a collision is treated classically, and the spin evolution of the two atoms is treated quantum mechanically. Before the collision, let the spin wave function of the ground-state alkali-metal atom be jψ f1g i D jψ fgg i and let the spin wave function of the noble gas atom be jψ f2g i. We will take the initial spin wave function of the colliding pair as the Kronecker product jψi D jψ f2g i ˝ jψ f1g i .
(10.96)
It will be convenient to expand the wave function (10.96) on the basis states jμ 1 μ 2 i D jμ 2 i ˝ jμ 1 i .
(10.97)
The common Hamiltonian of (10.2) becomes H D H f1g C H f2g .
(10.98)
The Hamiltonian, H f1g D H fgg , of the free alkali-metal atom was given by (2.10). The Hamiltonian of the free noble gas atom, with nuclear spin quantum number K and nuclear magnetic moment μ K , is H f2g D μ f2g B ,
where
μ f2g D
μK K. K
(10.99)
As we discussed in Section 3.1, free-atom Hamiltonians of (10.98) should be thought of as the Kronecker products H f1g ! 1f2g ˝ H f1g
and
H f2g ! H f2g ˝ 1f1g .
(10.100)
Here 1f1g D 1fgg and 1f2g are the unit operators for the Schrödinger spaces of the alkali-metal atom and the noble gas atom. Similarly, the interaction (10.95) can be identified with the interaction V of (10.3), which we write as X VDα Kj ˝ Sj , (10.101) j
where the Cartesian index j takes on the values x, y, and z. We will think of the coupling coefficient α D α(t) as that of a collision that occurs at time t D 0 so that α(t) is strongly peaked at t D 0 and approaches zero rapidly as t ! ˙1. In accordance with (10.20), the change in the density matrix caused by a single collision is Δ D SS † ,
(10.102)
where S D S c is the S-matrix, defined by (10.3) or (10.11). The initial density matrix of the colliding pair, before the collision, is D jψihψj D f2g ˝ f1g ,
(10.103)
10.9 Spin Exchange between Alkali-Metal Atoms and Noble Gas Atoms
where f1g D jψ f1g ihψ f1g j
and f2g D jψ f2g ihψ f2g j .
(10.104)
The postcollision density matrix, SS † , of (10.102) has correlations between the spin-observables of the alkali-metal atom and the noble gas atom. These correlations are normally lost when an ensemble average is taken over all types of collisions. We can therefore assume that the collisional contributions to the time evolution of the density matrices of the two species of atom are i h P1 D Γ f1g f1g Tr2 hSS † i and i h (10.105) P2 D Γ f2g f2g Tr1 hSS † i . In (10.105), the angle brackets denote an average over an ensemble of collisions. P The partial trace Tr2 [ X ] D m K hμ K j X jμ K i converts an operator X of dimension [I ][S ][K ] [I ][S ][K ] for the product spin space of the alkali-metal atom and the noble gas atom to a matrix of dimension [I ][S ] [I ][S ] operating within the spin P space of the alkali-metal atom. Similarly Tr1 [ X ] D m I m S hμ S m I j X jμ S m I i. The collision rates for the alkali-metal atoms and noble gas atoms are Γ f1g D N f1g v σ
and
Γ f2g D N f2g v σ ,
(10.106)
where N f1g is the number density of alkali-metal atoms and N f2g is the number density of noble gas atoms. 10.9.1 Binary Collisions
In practice, the coupling coefficients α of (10.95) are small enough that the criterion (10.25) for weak collisions is well satisfied. We will assume that the magnetic field is small enough, typically .1 T, that the criterion (10.43) for sudden collisions is also satisfied. The zero-frequency Fourier amplitude (10.30) of (10.95) becomes VM (0) D „φ S K .
(10.107)
The phase angle, φD
1 „
Z1 αdt ,
(10.108)
1
depends on the impact parameter of the classical trajectory. Substituting (10.107) into (10.46), and writing products of angular momentum operators as, Ki K j D
1 1 fK i , K j g C [K i , K j ] D fK i , K j g C i i j k K k , 2 2
(10.109)
181
182
10 Relaxation of Polarized Atoms
with an analogous expression for S i S j , we find that the rate of change of the density matrix 1 of alkali-metal atoms due to spin-exchange collisions with noble gas atoms is @ f1g D iΓ f1g hφi[hK i S , f1g ] @t o 1n Γ f1g hφ 2 i hK K i W 2S f1g S S S , f1g C 4 2 h n oi . (10.110) hKi 2iS f1g S S, f1g An analogous equation for @f2g /@t can be obtained from (10.110) by making the substitutions S $ K and f1g $ f2g. The symmetric, dyadic operators of (10.110) are constructed from the anticommutators of the angular momentum operators, X X KK D x i fK i , K j gx j and S S D x i fS i , S j gx j . (10.111) ij
ij
The double-dot product of two dyadics was defined by (8.59). The quantummechanical expectation values of (10.110) are h i h i and hK K i D Tr2 K Kf2g . (10.112) hKi D Tr2 Kf2g Noble Gases with K D 1/2 The noble gases 3 He and 129 Xe are often polarized by spin exchange with optically pumped alkali-metal atoms, and both have nuclear spin quantum numbers K D 1/2. Of course, the electron spin of the alkali-metal atom is S D 1/2. Then the dyadics in (10.110) simplify to K K D S S D 1/2, where the unit dyadic is 1 D x x C yy C z z, and we find @ f1g Γ f1g hφ 2 i 3 f1g D S f1g S hK i @t 4 4
[fS , f1g g 2iS f1g S ] i hφi h f1g . C4i 2 hK i S , hφ i
(10.113)
Conservation of Spin Multiplying both sides of (10.110) by S and taking the trace of the result, we find
h SP i D Tr2 [S Pf1g ] D Γ f1g hφihK i hS i Γ f1g hφ 2 i
hη S i hK i hη K i hS i . 2 The dyadic coefficients, C
(10.114)
1 1 K K and η S D S(S C 1)1 S S , (10.115) 2 2 determine how efficiently atoms with spin K depolarize atoms with spin S and vice versa. For K D 1/2 or S D 1/2, η K D η S D 1/2. η K D K(K C 1)1
10.9 Spin Exchange between Alkali-Metal Atoms and Noble Gas Atoms
To derive (10.114) from (10.110), one can use the identities, valid for the Cartesian projections S x , Sy , and S z of spin operators for arbitrary spin quantum number S, S i S l S i D [S(S C 1) 1]S l .
(10.116)
S i S l S j i j k D i fS l , S k g S(S C 1)δ l k .
(10.117)
[S i , S j ], S k D S i δ j k δ i k S j .
(10.118)
It is understood that one should sum over the repeated subscripts of (10.116) and (10.117). For noble gas atoms we can make the substitutions S $ K and f1g $ f2g in (10.114) to find h KP i D Γ f2g hφihS i hKi C
Γ f2g hφ 2 i
hη K i hSi hη S i hK i . (10.119) 2
Multiplying (10.114) by the number density N f1g and (10.119) by the number density N f2g , adding the results, and noting from (10.106) that N f1g Γ f1g D N f2g Γ f2g , we find N f1g h SP i C N f1g h KP i D 0 .
(10.120)
Spin-exchange relaxation conserves the total spin per unit volume, N f1g hS i C N f2g hK i, of the two species of atoms. Axial Symmetry For many practical situations, the coherences of f2g are negligibly small and we can write f2g as a sum of populations: X jm K ihm K jf2g jm K ihm K j . (10.121) f2g D mK
Then hK i D hK z iz ,
(10.122)
and ˝
˛
K K D K(K C 1) hK z2 i [x x C yy] C 2hK z2 iz z .
(10.123)
Substituting (10.122) and (10.123) into (10.110) and noting that 2iz (S f1g S) D S f1g SC SC f1g S and x x C yy D [x C x C x x C ] /2, we find @ f1g i D [H f1g , f1g ] iΓ f1g hφihK z i[S z , f1g ] @t „ Γ f1g hφ 2 i hη K C K z i[2SC f1g S fS SC , f1g g] C 8 C hη K K z i[2S f1g SC fSC S , f1g g] C 4hK z2 i[2S z f1g S z fS z2 , f1g g] .
(10.124)
183
184
10 Relaxation of Polarized Atoms
To simplify the notation, we have defined η J D z η J z D J( J C 1) J z2 ,
(10.125)
with J D K in (10.124). An equation analogous to (10.124) can be obtained for Pf2g by making the substitutions S $ K and f1g $ f2g. 10.9.2 Spin Temperature
The steady-state solutions to (10.124) and the analogous equation for @f2g /@t are the spin-temperature distributions f1g D
e β I z e β Sz Z I ZS
and f2g D
e β Kz . ZK
(10.126)
Here β is the spin-temperature parameter, and the partition function for a spin of quantum number J D S or J D K is Z J D Tr[e β J z ] D
X
eβ m J D
mJ
sinh β[ J]/2 . sinh β/2
(10.127)
There are three common situations where collisions or optical pumping produce spin-temperature distributions: 1. Spin-exchange collisions between pairs of alkali-metal atoms. 2. Spin exchange between the electron spins of alkali-metal atoms and the nuclear spins of noble gas atoms, a situation we discussed in the previous section. 3. Optical pumping at high gas pressures when there is negligible depolarization of the nuclear spin during the lifetime of the optically excited atom and nearly complete depolarization of the electron spin. For spin temperature distributions like (10.126) it is straightforward to show that for arbitrary spin quantum number J, h J z i D hη J iP .
(10.128)
The operator η J was defined by (10.125). The spin polarization P is related to the spin-temperature parameter β, and vice versa, by [85] P D tanh
β , 2
or
β D ln
1CP . 1P
(10.129)
The partition function (10.127) can be written in terms of P as ZJ D
(1 C P )[ J ] (1 P )[ J ] . 2P(1 P 2 ) J
(10.130)
10.9 Spin Exchange between Alkali-Metal Atoms and Noble Gas Atoms
From (10.114) or (10.119) we see that for longitudinal spin polarization we will have a steady state, that is, h SP z i D 0 and h KP z i D 0 if hK z i hS z i D DP. hη S i hη K i
(10.131)
For noble gas atoms with nuclear spin quantum number K D 1/2, such as 3 He or Xe, the polarization is simply P D 2hK z i, but for atoms with larger nuclear spin quantum numbers such as 21 Ne or 131 Xe, it is necessary to use the more general expression P D hK z i/hη K i, since hη K i depends on P for K > 1/2. In spin-temperature equilibrium for spins of arbitrary quantum number J 1/2 one finds the useful identities
129
hη J i D R J cosh(β/2) and
h J z i D R J sinh(β/2) ,
(10.132)
where the function R J D R J (β) is RJ D
X
eβ μ ( J C 1/2)2 μ 2 . ZJ μ
(10.133)
The summation index μ ranges in unit steps from μ D J 1/2 to μ D ( J 1/2). For the special case of J D 1/2, (10.132), (10.133), and (10.127) become hη J i D
1 P 1 , h Jz i D , RJ D , and Z J D 2 cosh(β/2) . 2 2 2 cosh(β/2) (10.134)
From (10.132) one can readily show that for arbitrary spin quantum number J 1/2, eβ D
hη J C J z i . hη J J z i
(10.135)
Steady State To show that f1g of (10.126) is the steady-state solution of (10.124) P we write e β Sz D m S e β m S jm S ihm S j to prove the identities for a spin of arbitrary quantum number S,
2SC e β Sz S fS SC , e β Sz g D X eβ/2 ,
(10.136)
2S e β Sz SC fSC S , e β Sz g D X eβ/2 ,
(10.137)
2S z e β Sz S z fS z2 , e β Sz g D 0 ,
(10.138)
where the operator X D X(β) is X D 4e β Sz fS z cosh(β/2) η S sinh(β/2)g .
(10.139)
185
186
10 Relaxation of Polarized Atoms
Here η S was defined by (10.125). Substituting f1g into (10.124), noting that [F z , H f1g ] D 0, and using the identities (10.136)–(10.138) with (10.135), we find @ f1g @t [H f1g , e β Fz ] Γ1 hφ 2 ie β I z hη K K z i X eβ/2 hη K C K z i β D C e i„Z I Z S 8Z I Z S hη K K z i D0. (10.140) 10.9.3 Experimental Measurements
Spin-exchange rate coefficients have been measured for a number of alkali metals and noble gases. We briefly review the most common methods of measurement here. It is convenient to define a spin-exchange rate coefficient: kse D
Γ f1g hφ 2 i Γ f2g hφ 2 i D . f2g 4N 4N f1g
Then, for axial symmetry (10.119) becomes h KP z i D N f1g kse 2hη K ihS z i N f1g kse C Γw hK z i .
(10.141)
(10.142)
In writing (10.142), we noted that hη S i D 1/2, and we added a relaxation rate Γw that is independent of the density N f1g of alkali-metal atoms and that is usually dominated by relaxation at the walls. For the steady state, (10.142) implies that hK z i D
2hη K ihS z ikse N f1g . kse N f1g C Γw
(10.143)
If hη K ihS z i can be considered time-independent once pumping light has been turned on, we see from (10.142) that the nuclear spin, hK z i, will build up to the equilibrium value (10.143) with an exponential rate: γ2 D N f1g kse C Γw .
(10.144)
The rate coefficient is most often determined by measuring the rate γ2 for a series of increasing cell temperatures. The cells usually contain condensed alkali metal, liquid or solid, and N f1g will therefore increase with increasing temperature. The number density, N f1g , of alkali-metal atoms is often several tens of percent smaller than what one would infer from the saturated vapor pressure curves of the metals, so N f1g must be measured directly in some way and not inferred from the temperature. If both kse and the wall relaxation rate Γw are known to have negligible dependences on temperature, the rate coefficient kse is simply the slope of a plot of experimentally measured values of γ2 versus N f1g . This method does not work well for 3 He because Γw is usually strongly temperature dependent for 3 He cells.
10.9 Spin Exchange between Alkali-Metal Atoms and Noble Gas Atoms
One way to cope with temperature-dependent quantities is to use the “rate balance method” [86]. If the rate γ2 of (10.144) as well as the steady-state values of the spin polarizations hS z i and hK z i, and the number density, N f1g , are measured with sufficient accuracy at a fixed temperature, it follows from (10.143) that the rate coefficient at that temperature is given by kse D
γ2 hK z i . 2hη K ihS z iN f1g
(10.145)
For the spin-1/2 nuclei 3 He and 129 Xe we have 2hη K i D 1. One can use the “repolarization method” [77] to measure kse , where the steadystate spin polarization hS z i of alkali metal is measured with very weak pumping light, so all the polarization is due to spin-exchange collisions with noble gas nuclei of spin polarization hK z i. For axial symmetry (10.114) becomes h SP z i D N f2g kse hK z i (2hη K ikse N f2g C Γs )hS z i .
(10.146)
In writing (10.144), we added a relaxation rate, Γs , that accounts for collisional spin relaxation mechanisms of hS z i other than spin exchange, usually the collisional spin–rotation interaction. For the steady-state (10.144) implies that hS z i D
kse N f2g hK z i . kse N f2g 2hη K i C Γs
(10.147)
In analogy to (10.143), the rate coefficient is given by kse D
γ1 hS z i . N f2g hK z i
(10.148)
The exponential rate coefficient for the polarization hS z i to build up owing to spinexchange collisions with noble gas nuclei is the denominator of (10.147), or γ1 D kse N f2g 2hη K i C Γs .
(10.149)
The number density N f2g of the noble gas is normally known to high accuracy, but one must make accurate measurements of hS z i, hK z i, and γ2 to determine kse with (10.148). A summary of measured spin-exchange rate coefficients is given in Table 10.4. From (10.110) we see that the spin-exchange collisions cause the electron spin of the alkali-metal atom to rotate at the rate δ ω D Γ f1g hφihK i, the same rotation rate that would be produced by a magnetic field shift δ B D „δ ω/(g S μ B ). The resulting shifts of the magnetic resonance frequencies of alkali-metal atoms can serve as a convenient and precise way to measure the spin polarization hK i of the noble gas nuclei. The frequency shift is conveniently parameterized by a dimensionless factor 0 D δ B/δ Bc that measures how much larger δ B is than the classical field δ B c D 8π μ K hK i/(3K ) produced by a noble gas with a spin polarization hK i inside a spherical container. If ν j is the frequency of some field-dependent resonance
187
188
10 Relaxation of Polarized Atoms Table 10.4 Rate coefficients (cm3 s1 ) for spin-exchange in binary collisions between alkalimetal atoms and certain noble gas isotopes. These are usually insensitive to temperature. Values in parentheses are theoretical estimates from [94]. 3 He
21 Ne
37 Ar
83 Kr
129 Xe
Li
–
–
–
–
–
Na
6.11020 [73]
(7 1021 )
(1.8 1019 )
(4.9 1020 )
(9.5 1018 )
K Rb
5.51020 [91] 6.81020 [86]
(2.3 1020 ) 4.71019 [87]
(7.1 1019 ) (1.2 1018 )
(2.3 1019 ) 2.11018 [92]
(6.2 1017 ) 2.21016 [93]
Cs
(8.1 1020 )
(7.3 1020 )
(2.6 1018 )
(9.1 1019 )
4.11016 [93]
Table 10.5 Values of the EPR frequency shift parameter 0 . These typically have a weak temperature dependence. Theoretical estimates from [84] are shown in parentheses. He
Ne
Ar
Kr
Xe
Li
–
–
–
–
–
Na
4.72 200 ı C [90] 5.99 200 ı C [90]
(27)
(29)
(120)
(310)
(34)
(56)
(230)
(660)
Rb
6.15 175 ı C [88]
32.0 128 ı C [89]
(61)
270 90 ı C [95]
644 80 ı C [95]
Cs
(10)
(44)
(83)
(340)
(880)
K
frequency of the alkali-metal atom, then the polarized noble gas will shift that frequency by δ ν j D 0
@ν j @B
δ B c D 0
@ν j @B
8πN f2g μ K hK z i . 3K
(10.150)
Here @ν j /@B is the rate of change of the magnetic resonance frequency of the transition j. For example, for low-field Zeeman transitions, @ν j /@B D g S μ B /(h[I ]). If the coefficient 0 is known, one can determine the spin polarization of the noble gases very precisely by simply measuring the shift, δ ν j , in the magnetic resonance frequency, ν j , when hK z i is driven to zero by a magnetic resonance field, or is reversed in sign by adiabatic fast passage. Romalis and Cates [88] measured 0 to high accuracy by studying the frequency shifts of polarized 3 He gas in a cylindrical cell at different orientations. This allowed 0 for Rb–3 He to be deduced to 1.5% accuracy without an accurate measurement of the 3 He polarization. Using cells with atom mixtures, Stoner and Walsworth [89] and Babcock et al. [90] used ratios of frequency shifts to extend this result to [Rb21 Ne], [K3 He], and [Na3 He]. Table 10.5 summarizes experimental measurements of 0 .
10.10 Spatial Diffusion
10.10 Spatial Diffusion
The density matrix D (r) will normally depend on the location r within the cell. Spatial diffusion and bulk relaxation, represented by the evolution operator G, will cause the density matrix, , to change at the rate @ j) D Dr 2 G j) . @t
(10.151)
Here DD
λv 3
(10.152)
is the classical diffusion coefficient. The mean free path of atoms in the gas is λ and mean velocity is v. A representative mean velocity for a heavy alkali-metal atom is v 3 104 cm s1 .
(10.153)
A representative value of the diffusion coefficient at a buffer-gas density of [B] D 1 amg is D 0.1 cm2 s1 , so a representative value of λ is λ 104 [B]1 cm amg .
(10.154)
The ratio of the cell volume V to the surface area S is a characteristic length l, lD
V . S
(10.155)
The diffusion equation (10.152) will be a good approximation to the spatial transport of atoms in a cell provided that λ l.
(10.156)
The criterion (10.156) is usually satisfied if the buffer-gas pressure exceeds a few tens of millitorr. Eigenmode Expansion The evolution operator G represents evolution due to the spin evolution, collisional relaxation, optical pumping, and any other bulk processes that would operate in the absence of spatial diffusion. We suppose that G is independent of position, and has eigenvectors jγ ) given by (4.49). Then we can substitute the expansion X j) D jγ )((γ j) (10.157) γ
into (10.151) to find the diffusion equation for the spatially dependent amplitude ((γ j), @ ((γ j) D Dr 2 γ ((γ j) . @t
(10.158)
189
190
10 Relaxation of Polarized Atoms
We assume we can find exponentially damping solutions, ((γ j D ((γ Γ j)eΓ t .
(10.159)
Substituting (10.159) into (10.158), we find the time-independent mode amplitude satisfies the Helmholtz equation (r 2 C k 2 )((γ Γ j) D 0 ,
(10.160)
where the spatial frequency is k2 D
Γ γ . D
(10.161)
Boundary Conditions To solve the Helmholtz equation (10.160), we need boundary conditions for the density matrix at the cell walls. On the scale of an atomic diameter, the surface at the walls may be very rough, but we will assume that there is a well-defined unit vector n, normal to the local “macroscopic” surface, and pointing into the material of the wall. According to kinetic theory [96], the current density of polarized atoms hitting the wall at a point with surface normal n is v vλ j JC ) D N n r j) . (10.162) 4 6
Here N is the number density of atoms in the volume element of the cell adjacent to the surface. The density matrix of the diffusing atoms is . As indicated in (10.162), it is convenient to discuss the physics in Liouville space. The current of atoms desorbed from the surface is j J ) D hS £ ij JC ) .
(10.163)
Here hS £ i is the ensemble-averaged S-matrix, defined in analogy to (10.23) for collisions with the wall. An expression for hS £ i is given by (10.189) in Section 10.11. The net current density into the wall is j JC ) j J ) D (1 hS £ i)j JC ) D Aj JC ) .
(10.164)
The accommodation operator A was defined by (10.22). Equating the current density (10.164) to the classical diffusion-current density at the wall, D N n rj), we find the boundary condition
3 Aj) D (2 A)n rj) , 2λ
(10.165)
or in terms of the normal gradient operator M, n rj) D M j) .
(10.166)
10.10 Spatial Diffusion
We can write the normal gradient operator as X 3A MD μjμ)((μj . D 2λ(2 A) μ
(10.167)
The eigenvalues and eigenvectors of A and M are related by μD
3α , 2λ(2 α)
with
jμ) D jα)
and
((μjμ 0) D δ μ μ 0 .
(10.168)
We assume that the eigenvectors can be normalized in analogy to (4.52). The boundary condition (10.166), often called a Robin boundary condition, was introduced by Masnou-Seeuws and Bouchiat [96] in connection with their studies of cells for which the walls caused little spin depolarization. To account for coherent interactions of spin-polarized atoms with the cell walls, Wu et al. [97] showed that it is necessary to think of M D M(r) as an operator in Liouville space that may be a function of position r along the surface. Weakly Depolarizing Walls A weakly depolarizing wall has jαj 1 for all eigenvalues α. Then (10.167) becomes X 3α M! jα)((αj . (10.169) 4λ α
Using (10.169) and(10.152), we write the current density of polarized atoms at a weakly depolarizing wall as Nv X αjα)((αj) . (10.170) j J) D N D n rj) D 4 α From kinetic theory, the current of polarized atoms hitting the walls is N v j)/4. According to (10.170) the current carried by the component jα)((αj) of the density matrix is suppressed by a factor α from the atomic current N v /4 that hits the wall. Physically, this occurs because a fraction, 1 α, of the atoms are desorbed in the same state jα) in which they were adsorbed, so the current into the wall is nearly canceled by the current out of the wall. Unconfined Polarization If the depolarization at the walls is so slow that the atoms can diffuse freely throughout the entire volume V of the cell before substantial relaxation occurs, the density matrix j) will be nearly independent of the position in the cell and almost equal to the volume-averaged value Z 1 j) N D j)dV . (10.171) V
Masnou-Seeuws and Bouchiat [96] call this the limit of “unconfined polarization”. Using (10.151), (10.166), and (10.169) with the divergence theorem to evaluate the time rate of change of j) N due to diffusion to the wall, we find Z XZ vα @ D j) N D M j)d S D jα)((αj)d S . (10.172) @t V 4V α
191
192
10 Relaxation of Polarized Atoms
Here d S is an element of surface area along the wall. Multiplying both sides of (10.172) on the left by ((αj, we find that adsorption on the wall causes the elements (( αj) N of the density matrix to relax at the rate @ N , ((αj) N D γN α ((αj) @t
(10.173)
where the damping coefficient is Z vα dS . γN α D 4V
(10.174)
For the condition of unconfined polarization to be valid, the diffusion rates in the gas, which are of order v λ/ l 2 , must be much faster than the wall-damping rates (10.174), which are of order jαjv / l. The criterion for unconfined polarization is therefore jαj
λ . l
(10.175)
We must have λ/ l 1 for the diffusion equation (10.151) to be valid, so the accommodation coefficients α must be extremely small for conditions of unconfined polarization. The bulk relaxation rates γ must also be exceptionally small. Such conditions can occur for alkali-metal atoms at low buffer-gas pressures with very carefully prepared wall coatings. Unconfined polarization is a much more common phenomenon for noble gases with spin-polarized nuclei, where all the relaxation processes are much slower than for alkali-metal atoms. Neumann Boundary Conditions In the limit of walls that cause no depolarization at all, α ! 0 for all jα), so M ! 0 and the Robin boundary condition (10.166) reduces to the Neumann boundary condition:
n rj) D 0 .
(10.176)
For walls that cause finite but small depolarization, one can treat the normal gradient operator M as a perturbation [97]. Strongly Depolarizing Walls X j) D jμ)((μj) .
We can expand j) on the eigenvectors of M to find (10.177)
μ
Substituting (10.177) into (10.166) and multiplying on the left by (μj, we find the amplitudes of at the cell walls are given by ((μj) D
1 ((μjn rj), μ
for
μ¤0.
(10.178)
Combining (10.178) with (10.177), we find that the density matrix at the wall is given by X 1 1 j) D jø) C (10.179) jμ)((μjn rj) , where jø) D fgg j1fgg . μ g μ¤0
10.10 Spatial Diffusion Table 10.6 Diffusion coefficients (cm2 s1 ) of alkali-metal atoms in common buffer gases. 4 He
Ne
Ar
Kr
Xe
N2
Li
–
–
–
–
–
–
Na
0.92 155 ı C [98] 0.45 52 ı C [99] 0.35 80 ı C [78]
0.5 155 ı C [98] 0.19 27 ı C [99] 0.16 27 ı C [100]
0.28 155 ı C [98] 0.31 117 ı C [76] 0.14 27 ı C [100]
–
–
–
–
0.54 155 ı C [98] –
0.12 27 ı C [59]
0.13 47 ı C [101]
0.159 60 ı C [102]
0.39 27 ı C [103]
0.15 0 ı C [104]
0.13 0 ı C [104]
0.14 27 ı C [105]
–
0.12 27 ı C [106]
K Rb Cs
For strongly depolarizing walls with jαj 1 – except for α D 0 for jø) – we see from (10.168) that 1/μ λ. For situations of practical interest we will usually have ((μjn rj) (μj)/ l, where the characteristic length l was given by (10.155). Then all terms in the sum of (10.179) will be smaller that the first term, jø), by a factor of order λ/ l 1. We can therefore ignore all but the first term in (10.179) and use the resulting equation, j) D jø) ,
(10.180)
as a Dirichlet boundary condition to replace the Robin boundary condition (10.166) for strongly depolarizing walls. Slowest Diffusion Mode For walls that are sufficiently depolarizing that (10.180) is an appropriate boundary condition, it is often adequate to represent the spatial variation of j) with the most slowly relaxing diffusion mode, of spatial frequency k π/ l, where l is the characteristic length (10.155). Then r 2 j) D k 2 fj) jø)g, and (10.151) reduces to
@ j) D Γd fjø) j)g . @t
(10.181)
The diffusional damping coefficient is Γd D k 2 D .
(10.182)
Relaxation like that described by (10.181) is often called uniform relaxation. In the case of an atom without coherence, it is the kind of relaxation one would get if atoms were collisionally removed from each sublevel jμi at the rate Γd , and transferred with equal likelihood (uniformly) to each of the g fgg sublevels of the atomic ground state. Some values of D are shown in Table 10.6.
193
194
10 Relaxation of Polarized Atoms
10.11 Adsorption on the Walls
During the time the atom is adsorbed on the wall, it will be subject to the perturbation V C F , the sum of a time-independent part, V, that plays much the same role as the time-independent potential V of a van der Waals molecule and a part, F, that fluctuates while the atom is hopping along the surface. In analogy to (10.50) we assume that the density matrix, σ, of atoms stuck to the walls evolves at the rate P D G jσ) C j σ)
1 j) . Tf
(10.183)
Free atoms close to the wall have a density matrix . They are adsorbed on the walls at the rate 1/ Tf . The damping operator for the adsorbed atoms is G D γd C
X i N© N kj N . H CΓ D (γd C γ kN )j k)(( „
(10.184)
kN
Atoms are desorbed from the wall at the rate γd D 1/τ d , where τ d is the mean lifetime on the wall. The static Hamiltonian for atoms on the wall is HN © D H © C V © .
(10.185)
As we will discuss below, the fluctuating perturbation, F, leads to polarization damping, described by the superoperator Γ , which we will discuss in more deN and eigenvalues γd C γ N of G are determined by tail below. The eigenvectors j k) k the eigenvalue equation, N D (γ d C γ N )j k) N . G j k) k
(10.186)
In analogy to (10.55), the exact solution to (10.184) is jσ) D
1 Tf
Zt
d t 0 expfG(t t 0 )gj0 ) .
(10.187)
1
Here σ D σ(t) is the density matrix of atoms adsorbed at some surface element of the wall at time t and 0 D (t 0 ) is the density matrix of nearby free atoms at an earlier time t 0 < t. We assume that the desorption rate, γd , is so large compared with 1/ Tf that we can use (10.57) for j0 ) in (10.186). As in (10.57), we find jσ) D
1 hS £ ij) . γd Tf
(10.188)
In analogy to (10.58), we find that the S-matrix for adsorbed atoms is Z1 hS £ i D γd
d t expfG tg expfi(H © /„)tg 0
D
X kN k
N kjk)((kj N j k)(( . 1 C (γ kN iω k )τ d
(10.189)
10.11 Adsorption on the Walls
The free-atom basis states jk) and Bohr frequencies ω k were given by (10.62). Using (10.189) with (10.22), we find the accommodation operator A D 1 hS £ i D
N kjk)((kj(γ N X j k)(( N iω k )τ d k
kN k
1 C (γ kN iω k )τ d
.
(10.190)
Populations We are often interested in the populations of the atomic subevels jk) $ jμihνj. For populations, the Bohr frequencies vanish, ω k D 0, so we can write (10.190) for the population subspace as
AD
X
N kj N , α kN j k)((
where
α kN D
kN k
γ kN τ d . 1 C γ kN τ d
(10.191)
N that is, The wall will be a weak depolarizer of the population eigenobservable j k), α kN 1, when the dwell time τ d is sufficiently short, or the wall-damping rate γ kN is sufficiently slow that γ kN τ d 1. In the other extreme of long dwell times and rapid wall-damping rates, when γ kN τ d 1, the accommodation coefficient will approach N kj), N its maximum value, α D 1, and the component, j k)(( of the density matrix will be destroyed at every collision with the wall. The Fluctuating Wall Interaction F We assume the mean value of F © is zero, and that we can define an autocorrelation function,
1 hF F i τ D lim T !1 T ©
ZT/2 F © (t τ)F © (t)d t D hF © F © i0 C(τ) .
©
(10.192)
T/2
The relative autocorrelation function, C(τ), is a scalar function of the delay, τ, and C(0) D 1 ,
C(τ) D C(τ) and
C(τ) ! 0 as
τ!1.
(10.193)
For the statistical analysis of F © , suppose that we define a finite sample of the fluctuating perturbation as FT© D F © (t) for jtj T/2 and FT© D 0 for jtj > T/2. Then we can write FT© as the Fourier transform Z 1 (10.194) FMT© (ω)eiω t . FT© (t) D 2π Substituting (10.194) into (10.192), we find Z 1 ©† eiω τ h FMT© (ω) FMT (ω)id ω hFT© FT© i τ D 2π T Z 1 ©† D h FMT© (0) FMT (0)i eiω τ J(ω)d ω 2π T Z 1 ©† D h FMT© (0) FMT (0)i J(ω)d ωC(τ) . 2π T
(10.195)
195
196
10 Relaxation of Polarized Atoms
In analogy to (10.37), we have defined the relative power spectrum, J(ω), by ©† ©† h FMT© (ω) FMT (ω)i D h FMT© (0) FMT (0)i J(ω) .
(10.196)
We note that J(0) D 1 and
J(ω) D J(ω) .
(10.197)
Equating the last two lines of (10.195), we find a version of the Wiener–Khintchine theorem, Z Z eiω τ J(ω)d ω D C(τ) J(ω)d ω . (10.198) Integrating both sides of (10.198) over τ and using (10.197), we find the “uncertainty principle”, Δτ Δω D 2π ,
(10.199)
where the full width of the autocorrelation function, Δτ, and the full width of the power spectrum, Δω, are Z Z Δτ D C(τ)d τ and Δω D J(ω)d ω . (10.200) Relaxation on the Wall The evolution of the density matrix, jσ) D jσ(t)), of a single adsorbed atom will be given by
@ 1 jσ) D (H © C F © )jσ) . @t i„
(10.201)
The small, time-independent part of the wall perturbation, V, can be ignored compared with H for this analysis. The interaction-picture density matrix, Q D j σ(t)) Q j σ) D eiH
© t/„
jσ) ,
(10.202)
evolves at the rate @ 1 Q , Q D FQ © j σ) j σ) @t i„
(10.203)
© © FQ © D eiH t/„ F © eiH t/„ .
(10.204)
where
The perturbation-series solution to (10.203) is *
1 Q D 1C j σ) i„
Zt
Q ©0
F 0
1 dt C (i„)2 0
Zt
Q ©0
F 0
dt
0
Zt 0
+ Q ©00
F 0
00
d t C j σQ 0 ) , (10.205)
10.11 Adsorption on the Walls 0 00 Q where j σQ 0 ) D j σ(0)), FQ © D FQ © (t 0 ), and FQ © D FQ © (t 00 ). The angle brackets Q to denote indicate an ensemble average. It is convenient to use the same symbol j σ) the density matrix of a single adsorbed atom in (10.203) and the ensemble average in (10.205). For small t, we find to second order in F,
Q D (1 X )j σQ 0) , j σ)
(10.206)
where *
1 X D 2 „
Zt
Q ©0
F
dt
0
0 00
Zt 0
+ Q ©00
F
dt
00
.
(10.207)
0
0
Letting t D t τ, we find that the matrix elements, X j l D ( j j X jl), of (10.207) are
X jl
1 D 2 „
Zt
t0
d t 0 eiω j l
t0
XZ k
0
D
1 „2
Zt
d t 0 eiω j l t
0
Zt D
0
XD
© F© j k Fk l
k
0
1 D Δω„2
E D © 0 0 d τ eiω k l τ F © j k (t )F k l (t τ)
Zt
0 iω j l t
dt e
E Z1 d τ eiω k l τ C(τ) 0
0
D 0 X k
0
© F© j k Fk l
Z1 E Z1 iω k l τ dτ e eiω τ J(ω)d ω 0
0
1
0
d t 0 eiω j l t Γ j l .
0
(10.208) The elements of the damping matrix of the last line are π X D © ©E
J(ω k l ) iK(ω k l ) F j k Fk l Γj l D 0 Δω„2 k E
1 X D M© F T I j k (0) FMT©Ik l (0) J(ω k l ) iK(ω k l ) . D 2 2„ T
(10.209)
k
Then we can let t 0 ! 1 in the upper limit of the integral of the second line. To derive the last line, we interchanged the order of integration over τ and ω in the preceding line, and used the identity (10.35). The frequency-shift coefficient K(ω k l ) was defined in terms of the relative power spectrum J(ω) by (10.41). Multiplying both sides of (10.206) on the left by ( j j, differentiating the resulting Q for small time, we find equation with respect to time, and letting j σQ 0 ) j σ) @ Q . Q D eiω j l t Γ j l (lj σ) ( j j σ) @t
(10.210)
197
198
10 Relaxation of Polarized Atoms
The matrix elements Γ j l of (10.209) can be used to construct the operator Γ : Γ D
X
Γ j l j j )(lj .
(10.211)
jl
Then we can substitute (10.202) into (10.210) to find i © @ jσ) D H C Γ jσ) , @t „
(10.212)
as anticipated by (10.183) and (10.184). In the analysis of (10.208) we have assumed that is possible to choose a time t that is small enough that j X j 1 but that is large enough that t Δ t. This is the familiar assumption used to derive Fermi’s golden rule, a slightly generalized example of which is (10.212). Comparing (10.212) with (10.21), we see that the nominal collision rate 1/ Tc for weak collisions in a gas plays the same role as the sampling rate 1/ T for a fluctuating perturbation on the wall, and T Γ plays the same role as the accommodation operator A. Comparing (10.209) with (10.40), we see that the potential V © for a weak binary collision plays the same role as the sampled, fluctuating potential FT© for the wall relaxation.
10.12 Spin Exchange between Pairs of Alkali-Metal Atoms
Binary collisions between alkali-metal atoms lead to very efficient exchange of the electron spins because of the large difference between the triplet and singlet potentials of the atomic pair. The potentials, similar to those shown in Figure 10.7, are much too large for the semiclassical analysis of the preceding sections to work. Both the translational and the spin states should be treated quantum mechanically with partial-wave scattering theory. 10.12.1 Partial-Wave Analysis
We consider two alkali-metal atoms, 1 and 2, with masses M1 and M2 , and with reduced mass M D M1 M2 /(M1 C M2 ). Let the nucleus of the first atom be located at r 1 and the nucleus of the second atom be located at r 2 . Atom 2 is displaced from atom 1 by r D r 2 r 1 . We write the wave function of the pair as the column vector jψi D
X
jμ 1 μ 2 ihμ 1 μ 2 jψi .
(10.213)
μ 1 ,μ 2
The basis states, jμ 1 μ 2 i D jμ 2 i ˝ jμ 1 i, are Kronecker products analogous to those of (10.97). The amplitude hμ 1 μ 2 jψi will depend on the interatomic separation r. The relative velocity v and the kinetic energy E of the atoms before the collision are
10.12 Spin Exchange between Pairs of Alkali-Metal Atoms
related to the spatial frequency k of the initial state by vD
„k M
and
ED
„2 k 2 . 2M
(10.214)
The pair wave function jψi obeys the Schrödinger equation: i„
„2 2 @ jψi D r jψi C V jψi . @t 2M
(10.215)
The Laplacian operator r 2 acts on the translational part of jψi, but has no effect on the spin parts. The interaction potential, V, acts on both the translational and the spin parts of jψi. Potentials
The interaction potential
V D Vse C Vsa ,
(10.216)
is the sum of a spin-exchange potential Vex and a spin-axis potential Vsa . We ignore other much smaller contributions to V, for example, the magnetic dipole interaction of the electronic spins. The exchange potential has singlet and triplet parts and can be written as Vse D Π f0g V f0g C Π f1g V f1g D Vd C Ve Πe .
(10.217)
As can be seen from Figure 10.7, the 1 Σg singlet potential V f0g D V f0g (r) and the 3 Σu triplet potential V f1g D V f1g (r) differ by about 0.5 eV near the classical turning points. The projection operators of (10.217) are given in terms of the electronic spin matrices, written in analogy to (10.100) as S 1 ! 1f2g ˝ S 1 and S 2 ! S 2 ˝ 1f1g , of the two atoms: Π f0g D
1 1 Πe S1 S2 D 4 2
(10.218)
Π f1g D
3 1 C Πe C S1 S2 D . 4 2
(10.219)
and
The exchange operator is Πe D
1 C 2S 1 S 2 D Π f1g Π f0g . 2
(10.220)
Since Π f0g and Π f1g are projection operators we must have Π f0g Π f0g D Π f0g , Π
f0g
Π
f1g
DΠ
f1g
Π
Π f1g Π f1g D Π f1g f0g
D0.
and (10.221)
From (10.218)–(10.221) we see that Πe2 D Π f1g C Π f0g D 1 .
(10.222)
199
200
10 Relaxation of Polarized Atoms
Figure 10.7 Rb–Rb potentials, from [107]. The electronvolt-scale difference between the singlet and triplet potentials leads to efficient spin exchange even though the duration of a binary collision is only a few picoseconds.
The direct and exchange potentials of (10.217) are V f1g C V f0g V f1g V f0g and Ve D . (10.223) 2 2 The spin-axis potential Vsa of (10.216) is much smaller than the exchange potential Vex and is given by [108, 115] Vd D
2λ S (n 0 n 0 1) S . 3 The total electronic spin of the two atoms is Vsa D
(10.224)
S D S1 C S2 ,
(10.225)
and the unit vector pointing from atom 1 to atom 2 is r (10.226) n0 D . r For the time being, we will ignore the small spin-axis interaction (10.224). Its effects can be analyzed either by a classical-path calculation for an orbit governed by the triplet potential V f1g or with the aid of the distorted-wave Born approximation [115]. Scattering State To calculate the relaxation of the spins of atoms of species 1 with number density [N1 ] as a result of collisions with atoms of species 2 with number density [N2 ] we write the scattering state as ! p eik r ikr F jψi , (10.227) jψi [N2 ] e C r
10.12 Spin Exchange between Pairs of Alkali-Metal Atoms
where jψi, the initial spin state of the atomic pair is given by (10.96) with jψ1 i and jψ2 i now representing the spin states of two alkali-metal atoms rather than the spin states of an alkali-metal atom and the nuclear spin of a noble gas atom. Species 2 might be K and species 1 might be Rb. The scattering-amplitude operator, F D F (n, n 0 ), depends on the directions n and n 0 of the incident and scattered waves, and we write it as F D Π f0g f f0g C Π f1g f f1g
D f d C Πe f e .
(10.228)
The direct and exchange scattering amplitudes, f d and f e , are related to the triplet and singlet scattering amplitudes, f f1g and f f0g , by fd D
f f1g C f f0g 2
and
fe D
f f1g f f0g . 2
(10.229)
The singlet (s D 0) and triplet (s D 1) amplitudes are given by the partial-wave expansion f fsg D
2π X 2iδ fsg e l 1 Yl (n) Yl (n 0 ) . ik
(10.230)
l
The scattering amplitudes, f f0g , f f1g , f d , and f e , depend implicitly on the relative energy E (or spatial frequency k) of the colliding pair, and on the directions n and n 0 of the incident and scattered waves. The unit vector n 0 was defined by (10.226) and the unit vector n is nD
k . k
(10.231)
We have written the spherical harmonics as Yl m (n) D Yl m (θ , φ), where the colatitude angle θ and the azimuthal angle φ parameterize the unit vector n, as in (5.57). Similarly, Yl m (n 0 ) D Yl m (θ 0 , φ 0 ). The generalized dot product of the spherical harmonics is related to the Legendre polynomial P l by the addition theorem, Yl (n) Yl (n 0 ) D
X
Yl m (n)Ylm (n 0 ) D
m
2l C 1 P l (n n 0 ) . 4π
(10.232)
for the lth partial waves can be found by solving the radial The phase shifts, δ fsg l Schrödinger equations for singlet and triplet waves of orbital angular momentum l, d2 l(l C 1) 2M fsg 2 2 C g fsg C V k D0. (10.233) l dr r2 „2 The boundary conditions of the radial wave function for small and large r are g fsg l (r) ! 0 as r ! 0 and
fsg g fsg l (r) sin(k r π l/2 C δ l ) .
(10.234)
201
202
10 Relaxation of Polarized Atoms
From the Schrödinger equation (10.215) we see that @ 1 V, jψihψj , (10.235) jψihψj C r j D @t i„
Equation of Continuity
where the current operator is j D
„ (rjψi) hψj C h.c. 2iM
(10.236)
The equation of continuity (10.235) includes a source term on the right which represents the transfer of atoms between spin states by the potential V. We find the rate of change of the spin density matrix due to spin-exchange scattering by integrating the current density over the surface of a sphere of radius r 1/ k, where the scattered wave is well described by the asymptotic expression (10.227) and where V is negligibly small: Z P D r 2 d Ω 0 n 0 j 0 ( ) Z „k N2 r 2 eik r 0 0 ik r nn 0 0 0 D C n F jψi d Ω n njψie 2M r ) ( Z eik r „k N2 0 C h.c. D d Ω 0 F 0 F 0† hψjeik r nn C hψjF 0† r M Z h i „k r N2 0 0 d Ω 0 n 0 (n C n 0 ) eik r(1nn ) F 0† C eik r(1nn ) F 0 . C 2M (10.237) The element of the surface is r 2 n 0 d Ω 0 , where d Ω 0 D sin θ 0 d θ 0 d φ 0 is an increment of solid angle, centered on the direction n 0 of the scattered wave. We use h.c. to denote the Hermitian conjugate of the preceding expression. The current density at the surface increment is j 0 D j (r n 0 ), and the lateral-scattering amplitude is F 0 D F (n, n 0 ). The density matrix, , before scattering is related to the wave function, jψi, before scattering by (10.103). 0 For k r 1, the contributions to the integrals containing the factors e˙ik r(1nn ) 0 come almost entirely from the forward direction where n n, the phase is “stationary”, and we can write Z Z 0 0 0 0 ˙ik r(1nn 0 ) 0 N d Ω n (n C n )e d Ω 0 (1 C n 0 n)e˙ik r(1nn ) F F
˙
4πiFN . kr (10.238)
The forward-scattering amplitude is FN D fNd C fNe Πe ,
where FN D F (n, n), fNd D f d (n, n) and fNe D f e (n, n).
(10.239)
10.12 Spin Exchange between Pairs of Alkali-Metal Atoms
Optical Theorem Using (10.238) in (10.237), we find the fundamental equation for the evolution of the density matrix, Z „k[N2 ] 2πi„[N2 ] N † N F F C F 0 F 0† d Ω 0 . (10.240) P D M M
Scattering conserves the total number of atoms, so we must have Tr[] P D 0 in (10.240). This leads to the optical theorem, Z h i 2πi FN FN† C k F 0† F 0 d Ω 0 D 0 . (10.241) Substituting the scattering amplitude (10.228) into (10.241), and carrying out the integrations, we find the identities k(σ d d C σ e e ) D 4π fNd00 ,
(10.242)
k(σ e d C σ d e ) D 4π fNe00 .
(10.243)
We have use single and double primes to denote the real and imaginary parts of a complex numbers, for example, fNd D fNd0 C i fNd00 . The cross sections are defined by Z f i f j d Ω 0 D σ j i . (10.244) σi j D The indices i and j can take on the values d (direct) and e (exchange). The products of the lateral-scattering amplitudes f i D f i (n, n 0 ) are integrated over all possible directions n 0 of the scattered particle. Partial-Wave Cross Sections Using the optical-theorem identities (10.242) and (10.243) and the scattering amplitude (10.228) in (10.240), we find ˚ (10.245) P D v [N2 ] σ 0 (Πe Πe ) C iσ 00 [Πe , ] .
The relative velocity of the colliding pair is v D „k/M , and the number density of atoms of species 2 is [N2 ]. Following Kartoshkin [109], we have defined a complex, spin-exchange cross section, σ D σ 0 C iσ 00 D
π X (2l C 1)(1 e2iδ l ) . 2k 2
(10.246)
l
The difference between the triplet and the singlet phase shift for the lth partial wave, which we will call the differential phase shift, is δ l D δ f1g δ f0g . l l
(10.247)
The real part of the cross section (10.246) is σ0 D σ ee D
π X (2l C 1) sin2 δ l , k2 l
(10.248)
203
204
10 Relaxation of Polarized Atoms
and the imaginary part is σ 00 D
2π N0 π X f C σ 00d e D 2 (2l C 1) sin δ l cos δ l . k e k
(10.249)
l
We can write (10.245) in the same form as the collisional part of (10.20): o X n † Γl S l S l P D l
D
X
˚ Γl sin2 δ l (Πe Πe ) C i sin δ l cos δ l [Πe , ] .
(10.250)
l
The partial rates are Γl D N v
π(2l C 1) , k2
(10.251)
and the S-matrix for the lth partial wave is S l D eiδ l Πe D cos δ l C i sin δ l Πe .
(10.252)
In writing (10.252), we expanded the exponential as a power series in iδ l Πe , and we noted that 1 D Πe2 D Πe4 D and Πe D Πe3 D Πe5 D . Taking the partial trace of (10.245) over the spin coordinates of the second atom, in accordance with (10.105), we find 3 f1g P D Γ f1g Pf1g D Tr2 [] S 1 f1g S 1 4 h i h i hS 2 i ff1g , S 1 g 2i S 1 f1g S 1 2i hS 2 i S 1 , f1g . (10.253) Here the exchange rate Γ f1g and the frequency-shift parameter are given by Γ f1g D N f2g v σ 0
and D
σ 00 . σ0
(10.254)
An analogous equation for Pf2g can be obtained from (10.253) by interchanging the subscripts, f1g $ f2g. Like Atoms For a vapor of identical alkali-metal atoms, we have N f1g D N f2g D N , f1g D f2g D , Γ f1g D Γ f2g D Γse , and S 1 D S 2 D S , so for self-spinexchange, (10.253) becomes 3 Pse D Γse S S hS i [f, Sg 2i S S ] 4 2i [hSi S, ] . (10.255)
Unlike expressions leading up to (10.255), where the density matrix without a subscript denoted a pair of atoms, the density matrix of (10.255) is that of a single atom.
10.12 Spin Exchange between Pairs of Alkali-Metal Atoms
Spin Temperature The evolution equation (10.253) is identical to (10.113) if we make the replacements
hS 1 i ! hS i , ! 2
hS 2 i ! hK i ,
Γ f1g !
hφi . hφ 2 i
Γ f1g hφ 2 i 4
and (10.256)
We already demonstrated that the spin-exchange distribution, f1g , of (10.126) is the steady-state solution of (10.110), of which (10.113) is a special case. Therefore, the steady-state solutions of (10.253) and (10.255) are spin-temperature distributions like (10.126) [110]. For unlike alkali-metal atoms, the spin-temperature parameter β is the same for both species. Evolution in Liouville Space
The Liouville-space analog of (10.255) is
jPse ) D Γse hA seij) ,
(10.257)
where the dimensionless spin-exchange operator is hA se i D A sd hS i Aex 2ihSi S © .
(10.258)
The S-damping operator, A sd , was defined by (6.88), and the exchange operator, Aex , was defined by (6.90). From (4.85) we see that S © D S [ S ] . We note that Aex D S [ C S ] C 2iS [ S ] , †
(10.259)
†
(10.260)
and that Aex jM ) D jfM, Sg C 2iS M S) .
Some useful identities that follow from setting M D S i , M D I i , and M D 1fgg in (10.260) and taking the Hermitian conjugate of the result are (S i jA ex, j D δ i j (1fgg j ,
(I i jA ex, j D 0 and (1fgg jA ex, j D 0 .
(10.261)
Analogous identities for S © are (S i jS © j D i i j k (S k j ,
fgg (I i jS © jS © j D 0 and (1 j D 0.
(10.262)
There is an implicit sum over the repeated index k of (10.262). Multiplying (10.257) on the left by (Sj and (I j, respectively, and using the identities (10.261), (10.262), and (10.87), we find (S jPse ) D 0 and (I jPse ) D 0 .
(10.263)
Spin-exchange collisions conserve both hS i D (Sj) and hI i D (I j). The magnetic dipole hyperfine interaction Afgg I S , together with a nonzero, longitudinal
205
206
10 Relaxation of Polarized Atoms
magnetic field, must act in the intervals between spin-exchange collisions to drive the density matrix to the spin-temperature distribution (10.126). We can write (10.257) as the matrix Ricatti equation: jPse ) D Γse A sd j) C Γse Bse j) ˝ j) .
(10.264)
The bilinear exchange operator is 3 X Bse D (Sj ˝ Aex C 2i S © D (S j j ˝ A ex, j C 2iS © . j
(10.265)
j D1
Here Bse is an [I ]2 [S ]2 [I ]4 [S ]4 -dimensional matrix. But experimental conditions are often simple enough that it is possible to use much smaller, compactified versions of Bse . Spin-Temperature Limit of Relaxation Processes Let the evolution of the atoms be dominated by the common Hamiltonian H D H fgg of (2.10), spin exchange, and a linear relaxation process that destroys spin polarization. The evolution of the density matrix in such a situation is then given by
j) P D jPse ) C
1 © H j) Γx A x j) . i„
(10.266)
Here Γx is the characteristic rate of the relaxation process, and A x is the corresponding dimensionless operator that accounts for transitions between the various sublevels, dephasing collisions, and so on. Multiplying (10.266) on the left by (F z j, and noting that (F z jH © D ([H, F z ]j D 0, we find P D Γx (F z jA x j) D α x Γx (F z j) , (F z j)
(10.267)
Equation 10.267 says that hF z i D (F z j) decays at the rate α x Γx , where the slowing-down factor (the accommodation coefficient) is αx D
(F z jA x j) . (F z j)
(10.268)
If the spin-exchange ˇ collisions are sufficiently rapid, it will be a good approximation to write j) D ˇe β Fz /Z , where the spin-temperature parameter β will depend on time. Then the slowing-down factor, α x D α x (β), will be a well-defined function of the spin-temperature parameter β and can be written as F z jA x je β Fz αx D . (10.269) F z je β Fz We can readily evaluate the low-polarization limit of (10.269) the approx by making imation, e β Fz ! 1fgg C β F z , for β ! 0, and noting that F z j1fgg D Tr[F z ] D 0,
10.12 Spin Exchange between Pairs of Alkali-Metal Atoms
and A x j1fgg ) D 0 for relaxation processes that cannot create polarized atoms. Then we find (F z jA x jF z ) , for β ! 0 . (10.270) αx ! (F z jF z ) For example, suppose that the spin is lost by S-damping in the presence of much more rapid spin-exchange collisions. Then A x D A sd , where A sd was given by (6.88). Since (F z jA sd D (S z j from (10.87), we see that the low-polarization limit of the slowing-down factor is α sd D
(S z jF z ) 3 D . 2 (F z jF z ) [I ] C 2
(10.271)
10.12.2 Semiclassical Calculation of Partial-Wave Cross Sections
For optical pumping experiments at room temperature and higher, several hun(s) dred partial waves may have nonnegligible phase shifts, δ l . Finding these phase shifts from numerical solutions of the radial Schrödinger equation (10.233) is timeconsuming. The de Broglie wavelengths of the colliding atoms are short compared with the range of the potentials, so the phase shifts can found accurately and much more simply by semiclassical methods. In the Wentzel–Kramers–Brillouin approximation, the phase shifts are [111] δ fsg l
Z1 π D (q fsg k)d r C (l C 1/2) k r0fsg . 2
(10.272)
(s) r0
Here the superscript s D 0 for the singlet phase shift and s D 1 for the triplet phase shift. The classical radial component of the momentum (in units of „) is q (10.273) q fsg D 2M E V fsg /„2 (l C 1/2)2 /r 2 . Singlet and triplet potential curves V f0g and V f1g can be found in the literature [107]. The classical turning point, r0fsg , is the value of r for which q fsg D 0. For some angular momenta and energies, there are three turning points. The two smallest are the turning points of a potential well in which metastable dimer molecules can be formed. For these cases the outermost turning point is used in (10.272), since the effects of tunneling into the well can be neglected. Partial cross sections calculated from (10.248) and (10.249) with the Wentzel– Kramers–Brillouin phase shifts of (10.272) are shown in Figure 10.8. The differential phase shifts δ l D δ f1g δ f2g are hundreds of radians, and are mostly due to l l the singlet potential. They vary rapidly with l, producing oscillations in the partial cross sections, until the effective singlet potential Vlf0g D V f0g C „2 (l C 1/2)2 /2μ r 2 produces a barrier with a large-r turning point at the collision energy. This is illustrated in Figure 10.7. For angular momenta beyond this, the atoms no longer sample the strong attractive singlet well and the phase shifts become relatively small and rapidly approach zero.
207
208
10 Relaxation of Polarized Atoms
Figure 10.8 Partial cross sections for Rb–Rb spin exchange, at a collision energy of 35 meV. The straight line denotes the unitarity limit π(2l C 1)/ k 2 .
From Figure 10.8 we see that the imaginary part of the spin-exchange cross section, σ 00l , oscillates about zero with increasing l and that the real part, σ 0l , is never negative. Therefore, the spin-exchange frequency shifts, proportional to σ 00 D P 00 , are much smaller than the spin-exchange rates, which are proportional to l σ lP σ 0 D l σ 0l . Experimental Measurements We have collected some experimental measurements of spin-exchange cross sections in Table 10.7. While alkali–alkali collisions conserve spin to a first approximation, the spin-axis coupling (10.224) causes some loss of spin to the orbital angular momentum of the colliding pair. The coupling λ can exceed 10 GHz at typical turning points of alkali–alkali collisions. For the case of short-duration collisions, the effect of the spin–axis interaction can be accurately described as an S-damping process with a rate analogous to (10.94). It was recently discovered experimentally through magnetic decoupling experiments that the formation of triplet van der Waals molecules contributes to relaxation at a rate that is comparable to the electron randomization rate [108, 113]. The molecular contribution to the relaxation couples to the nuclear spins and therefore cannot properly be characterized as S-damping.
10.13 Pressure Dependence of Relaxation in the Dark
To illustrate some of the ideas of the previous sections, in this section we discuss a problem that is frequently encountered in practice: how spin-relaxation rates vary
10.13 Pressure Dependence of Relaxation in the Dark Table 10.7 Spin-exchange and spin-relaxation cross sections for collisions between alkalimetal atoms. The units are all cm2 . At magnetic fields that are larger than the hyperfine
splitting, the relaxation rates are reduced typically by a factor of 2 [115]. These cross sections are insensitive to temperature.
Spin exchange
Spin relaxation
Li
–
–
Na
1.0 1014 [112]
K Rb
1.5 1014
[112] [60]
1.0 1018 [115] 9.3 1018 [113]
Cs
2.1 1014 [112]
2.3 1016 [114]
1.9 1014
5 1019 [73]
with pressure when both the spatial diffusion to the cell walls and gas-phase collisions are important. There are two primary methods to prevent rapid depolarization of alkali-metal atoms by cell walls. The cell walls can be coated with hydrocarbons such as paraffin that allow for many wall collisions before relaxation occurs. This is quite effective for alkali-metal cells with no buffer gas. Alternatively, diffusion to the walls can be slowed down with a buffer gas. Increasing the buffer-gas pressure reduces the diffusion to the walls at the expense of increasing the spin-relaxation rates due to collisions with the buffer-gas atoms or molecules. Here we consider the trade-offs between these two methods of suppressing wall relaxation. We suppose that we have prepared a spherical cell of radius a, and that the cell has been filled with a small amount of alkali metal and a buffer gas. For simplicity we will assume that the density of the alkali-metal vapor is large enough that spin-temperature prevails, and that the polarization is small. Then the mode amplitude of (10.159) can be chosen to be ((γ j) D hF z i .
(10.274)
The mode amplitude must satisfy the Helmholtz equation (10.160), which is most conveniently written for a spherical cell as @ 2 @ 1 2 (10.275) r 2 L L C k hF z i D 0 . @r @r r Although (10.275) describes the spatial dependence of a diffusion mode, it is also the equation of motion for a free particle, and a solution can therefore be written as a linear combinations of the partial waves: hF z i D j l (k r)Yl m (θ , φ) .
(10.276)
Here Yl m (θ , φ) is a spherical harmonic function of the colatitude and azimuthal angles θ and φ of the position vector r, and j l (k r) is a spherical Bessel function. The possible values of the spatial frequency k are determined by the boundary con-
209
210
10 Relaxation of Polarized Atoms
dition (10.166), which we write at r D a as d j l D μ j l . dr
(10.277)
For l D 0, when j 0 (x) D sin x/x, the boundary condition becomes μ a D 1 k a cot k a .
(10.278)
There are an infinite number of positive spatial frequencies, k, that are solutions of (10.278). We denote them in order of increasing magnitude by k1 , k2 , k3 , . . . The first 10 values of k obtained by numerical solution of (10.278) are shown in Figure 10.9. From (10.161) we see that the relaxation rate of the spherically symmetric mode n is the sum of a diffusional contribution proportional to the square of the spatial frequency and a contribution γ due to collisional relaxation, Γn D k n2 D C γ .
(10.279)
For S-damping collisions at the rate Γsd we would have γ D α sd Γsd ,
(10.280)
where the slowing-down factor α sd for the spin-temperature equilibrium at low polarization was given by (10.271). Once the k n are found, the general solution for hF z i is X hF z i n eΓn t , (10.281) hF z i D n
where Za hF z i n D
hF z i(r, 0) j 0 (k n r)r 2 d r
(10.282)
0
gives the amplitude of the nth diffusion mode at time t D 0. Weakly Depolarizing Walls Let us focus on the lowest spherically symmetric mode in a cell where the wall has such a small accommodation coefficient, α 1, that there is a range of low pressures where μ a 1, but where the mean free path λ is still much smaller than the cell radius a so that the diffusion equation is a valid description of spatial transport of atomic polarization. For k a ! 0 one can readily verify that (10.278) gives
k12 !
3μ 9α ! . a 4λa
(10.283)
The relaxation rate (10.279) becomes Γ1 D
3α v C [X]sd α sd . 4a
(10.284)
10.13 Pressure Dependence of Relaxation in the Dark
Figure 10.9 Spatial frequencies k1 , k2 , k3 , . . . , k10 of the first 10 spherically symmetric diffusion modes for a cell of radius a as a function of the normal gradient parameter μ
at the wall. For the smallest spatial frequency, k12 ! 3μ/a as μ ! 0, and this corresponds to the uniform polarization limit of a cell with very little wall relaxation.
The first term of (10.284) is independent of gas pressure and it is the relaxation rate for unconfined polarization (10.174) for a sphere of radius a. The second term is the contribution of collisions in the gas. The S-damping rate coefficient is D sd D Γsd /[X], and the slowing-down factor was given by (10.271). The relaxation rate increases (slowly, assuming sr is small) with increasing buffer-gas density [X]. The pressure dependence of spin-polarized noble gas atoms, or of alkali-metal atoms in cells with very weakly relaxing walls is similar to that described by (10.284). Strongly Depolarizing Walls For walls with poor coatings or no coating at all, the accommodation coefficients α are not small enough to get the conditions of unconfined polarization mentioned above. From inspection of (10.278) or of Figure 10.9 we see that for strongly depolarizing walls we will have μ a 1 and the smallest spatial frequency is given by
k1 a D π .
(10.285)
At high pressure, the relaxation rate is Γ1 D
D0 π 2 [X0 ] C [X]sd α sd , a 2 [X]
(10.286)
211
212
10 Relaxation of Polarized Atoms
where [ X 0 ] is the reference number density, normally Loschmidt’s constant (2.69 1019 cm19 ), the number density of an ideal gas at 0 ı C and 1 atm. The minimum value of (10.286) occurs when s π D0 [X0 ] [X] D , (10.287) a sd α sd when we have Γ1 D
2π v a
r
σ sd α sd . 3σ k
(10.288)
We have expressed the diffusion coefficient in terms of an effective diffusion cross section, σ k , via D0 [X0 ] D v /3σ k . For a wall coating to give the same relaxation time as the optimum number density (10.286) of a buffer gas, we equate the first term on the right of (10.284) to the minimum rate (10.287) to find that the accommodation coefficient of the wall must be r 8π σ sd α sd 1 α , (10.289) 3 3σ k 4000 where the numerical value assumes σ sd 1022 cm2 and σ k 1015 cm2 , the approximate values for N2 buffer gas. Thus, the wall coating must allow several thousand collisions to give relaxation rates that are slower than cells filled with buffer gas. Note that although this conclusion is independent of the cell size, the minimum attainable relaxation rate is inversely proportional to the cell size. The density dependence of the lowest mode is shown in Figure 10.10.
10.14 Collisions of Excited Atoms
Binary depolarization and quenching cross sections of optically excited alkali-metal atoms in buffer gases are very much larger than those of the alkali-metal atoms in the ground state. For 2 P3/2 states, the spin depolarization cross sections can be larger than gas kinetic cross sections. For 2 P1/2 states, with their spherically symmetric charge distributions, as shown in Figure 2.3 the depolarization cross sections are large compared with those of the 2 S1/2 ground state, but they can be substantially smaller than gas kinetic cross sections, especially for Cs (and presumably Fr), where the fine structure splitting between 2 P3/2 and 2 P1/2 states is many times larger than the mean thermal energy k T . The depolarization cross sections for atoms in 2 P3/2 states, which have nonspherically symmetric charge distributions, are always large and comparable to gas-kinetic cross sections. Collisions of alkali-metal atoms in the 2 P J state with molecular buffer gases such as N2 and H2 are particularly efficient in transferring the atoms to other electronic states, for example, to the other member, 2 P J ˙1 , of the same fine-structure doublet, or to the 2 S1/2 ground state. Vibrational and rotational degrees of freedom of
10.14 Collisions of Excited Atoms
Figure 10.10 Relaxation rate as a function of buffer-gas pressure for weakly relaxing walls with differing probabilities α of relaxation per wall collision. The electron randomization rate was taken to be 1 1022 cm2 and the diffusion coefficient was 0.12 cm2 /s.
the molecules absorb or release the energy needed for the atomic transition. The fine-structure splittings for the lighter alkali-metal atoms K, Na, and Li, are small enough that collisions with monatomic noble gas atoms, especially He, can efficiently transfer atoms between the 2 P3/2 and 2 P1/2 states.
Notable Effects of Excited-State Collisions 1. A binary collision of an excited atom with a buffer-gas atom or molecule lasts for only a few picoseconds, and there is not enough time for the relatively weak magnetic dipole or electric quadrupole interactions to change the polarization of the nucleus appreciably. 2. The axis–orbit interactions are large enough to substantially change the electron polarization of an excited atom during a single collision. However, the collisions occur with all possible impact parameters and orbital planes. If the external magnetic field is zero, we see by symmetry that the collisions cannot change an electronic dipole polarization to a quadrupole polarization, and so on. For magnetic fields larger than a few teslas, there can be collisional mixing of the electronic multipolarities.
213
214
10 Relaxation of Polarized Atoms
Detailed Analysis
We write the collisionally induced relaxation of excited atoms, in accordance with (10.21), as @ j) D Γc A c j) . @t
(10.290)
The collision rate of excited atoms with buffer-gas atoms or molecules is Γc . In accordance with the two properties mentioned above, we write the accommodation operator as Ac D
X
f J J 0g
αl
f J J 0g
Πl
.
(10.291)
l J J0 f J J 0g
The multipole transfer operators Πl Using (11.12) with (10.290), we find
are defined by (11.4) in Chapter 11.
X f J J 0g @ Jf Jg f J Jg 0 0 0 ) D Γc α0 (1J f J g jf J J g ) . (1 j @t 0
(10.292)
J
f J Jg
We see that α 0 is the probability per unit time that an atom, initially in the electronic state J, has been collisionally removed from that state and transferred f J J 0g to another electronic state. Similarly, we can interpret α 0 as the probability per unit time that an atom initially in the electronic state J 0 ¤ J is transferred to the state J. We assume the collisions conserve the number of atoms, so the accommodation coefficients must satisfy the constraint X
f J J 0g
α0
D0.
(10.293)
J f J J 0g
The coefficients α l must satisfy additional constraints, for example, they must lead to a Boltzmann distribution of atoms among the energy sublevels. If the orbit–axis potentials are known, the multipole transfer coefficients, f J J 0g αl , can be calculated by a partial-wave analysis similar to that outlined for spin exchange. These parts of these potentials that are diagonal in J (the Born– Oppenheimer potentials) are fairly well known from calculations and experiment. However, little is known about the off-diagonal parts. We will simply regard the f J J 0g αl as parameters to be determined from experimental measurements. If the collisional transfer rate from the electronic state J to other states is negligif J J 0g f J Jg ble, we can set α l D δ J J 0 α l and Πl D Πl in (10.291) to find Ac D
X
α l Πl .
l
For (10.293) to remain valid, we must have α 0 D 0.
(10.294)
10.14 Collisions of Excited Atoms Table 10.8 Excited-state spin-relaxation cross sections (Å2 ) for the P1/2 level. He
Ne
Ar
Kr
Xe
N2
Li
–
–
–
–
–
–
Na
27 177 ı C [116]
29 177 ı C [116]
58 177 ı C [116]
126 177 ı C [116]
142 177 ı C [116]
–
K
190 115 ı C [117] 32 57 ı C [118]
145 115 ı C [117] 70 57 ı C [118]
160 115 ı C [117] 48 57 ı C [118]
–
–
–
68 57 ı C [118]
78 57 ı C [118]
64 57 ı C [118]
24 43 ı C [119]
10 43 ı C [119]
22 43 ı C [119]
76 43 ı C [119]
144 43 ı C [119]
–
Rb Cs
If the excited state is 2 P1/2 , the only nonzero term in the sum (10.294) has l D 1, and (10.290) becomes 3 @ (10.295) j) D Γ j d Π1 j) D Γ j d J [ J ] j) , @t 4 where Γ j d D Γc α 1 is the J-damping rate, and we used (11.22) to write the dipole projection operator Π1 in terms of the electronic angular momentum operator J of the excited state. Table 10.8 gives experimental results for J-damping cross sections, defined via Γ j d D N σ j d v . Fine-structure-changing collisions involve both l D 0 and l D 1 multipoles. These cross sections (Table 10.9) are quite small for collisions with noble gas atoms and the heavy alkali-metal atoms, but are a significant fraction of the total collision cross section for collisions with the light alkali-metal atoms and for collisions between the alkali-metal atoms and N2 molecules (Table 10.10). For the heavy alkali f3/2,1/2g
Table 10.9 Fine-structure-changing cross sections (Å2 ) σ 0 He
Ne
Ar
f3/2,1/2g
and σ 1
Kr
for P1/2 ! P3/2 . Xe
Li
–
–
–
–
–
Na
107, 30 177 ı C [116]
77.4, 16 177 ı C [116]
128.5, 38 177 ı C [116]
112.3, 27 177 ı C [116]
105.5, 20 177 ı C [116]
K
37, 0.9 107 ı C [120] 0.072, – 57 ı C [118]
6.4, – 69 ı C [122] 0.0014, – 57 ı C [118]
16, – 69 ı C [122] 0.00051, – 57 ı C [118]
24.9, – 69 ı C [122] 0.00036, – 57 ı C [118]
44.9, – 69 ı C [122] 0.00034, – 57 ı C [118]
5.7 105 , – 77 ı C [123]
1.9 105 , – 77 ı C [123]
1.6 105 , – 77 ı C [123]
8.3 105 , – 77 ı C [123]
7.2 105 , – 77 ı C [123]
Rb Cs
215
216
10 Relaxation of Polarized Atoms f3/2,1/2g
Table 10.10 Quenching and fine-structure-changing cross sections (σ 0 alkali-metal–N2 collisions. nP1/2 ing
quench-
nP3/2 ing
quench-
–
–
–
Na
23 640 ı C [124] 94 100 ı C [125]
23 640 ı C [124] 94 100 ı C [125]
–
58 57 ı C [118] 77 42 ı C [127]
43 57 ı C [118] 69 42 ı C [127]
13.2, 1.1 57 ı C [118] 4.7, – 42 ı C [127]
Rb Cs
) for
Fine structure
Li
K
f3/2,1/2g
, σ1
79, – 69 ı C [126]
metals the fine-structure splitting can be comparable to k T , so the cross sections are strongly temperature dependent [128] and obey the detailed balance relation σ f3/2,1/2g D 2σ f1/2,3/2g eΔ EFS /(k T ) .
(10.296)
Quenching of resonance fluorescence is essential for allowing optical pumping of optically thick vapors, since the resonance fluorescence is effectively unpolarized, so absorption of fluorescent light is an efficient depolarization mechanism. Without a quenching gas, the density of alkali-metal vapor that can be optically pumped is strongly limited, though it can be increased by using a large magnetic field [129]. The near resonance between rovibrational levels of N2 and the alkaliTable 10.11 Excited-state multipole cross sections (Å2 ) for the P3/2 level. He
Ne
Ar
Kr
Xe
N2
Li
–
–
–
–
–
–
Na
σ 1 D 58 σ 2 D 99 σ 3 D 81 177 ı C [116] 87 120 100 107 ı C [120]
60 90 70 177 ı C [116] 130 170 130 107 ı C [120]
130 160 120 177 ı C [116] 175 230 190 75 ı C [121]
58 190 230 200 ı C [116] 270 330 270 107 ı C [120]
210 250 81 220 ı C [116] –
–
–
Rb
240 280 200 100 ı C [118]
320 560 390 100 ı C [118]
–
610 940 700 100 ı C [118]
750 930 690 100 ı C [118]
200 260 260 100 ı C [118]
Cs
–
–
–
–
–
–
K
10.14 Collisions of Excited Atoms
metal resonance lines plus the nonreactivity of N2 with the alkali metals make N2 the usual choice for the quenching gas. The 1.4–2.1-eV excitation energy carried off by the N2 molecules can result in substantial heating of the cell under high-density pumping conditions [130]. The l D 0 cross sections for quenching by N2 are shown in Table 10.10. Relaxation within the P3/2 Zeeman levels involves l D 1, 2, 3 multipoles. Using strong magnetic fields to isolate the Zeeman sublevels and decouple the atomic hyperfine structure, one can measure state-to-state cross sections (defined via Γc α l D N σ l v ) and from them determine the multipole cross sections. A summary of the experimental knowledge of these processes is given in Table 10.11.
217
219
11 Mathematical Appendix In this chapter we discuss some specialized mathematical tools that are useful for analyzing optical pumping and collisional relaxation.
11.1 Electronic Multipoles
Many important processes in optical pumping have no direct effect on the nuclear spin polarization of the optically pumped atom and only affect the electronic part of the density matrix. Examples are spontaneous radiative emission, pumping with light of a broad spectral profile, many types of collisions, and so on. For discussing f Jg these processes, it is convenient to introduce the [I ]2 [ J]2 [I ]2 matrix N l m defined by ˇ X ˇˇ fI I g ˇ f Jg f J Jg (11.1) TλfIμ I g ˇ . Nl m D ˇTλ μ ˝ Tl m λμ
f J Jg
In analogy to (5.21), TλfIμ I g is a nuclear multipole operator and Tl m is an electronic multipole operator for an atomic state with electronic angular momentum J. We f J g† f J 0 g see that the products N l m N l 0 m 0 are orthonormal, f J g†
f J 0g
N l m N l 0 m 0 D δ l l 0 δ m m 0 δ J J 0 1fI I g . Here 1fI I g D
X ˇˇ fI I g fI I g ˇˇ Tλ μ ˇ ˇTλ μ
(11.2)
(11.3)
λμ
is the [I ]2 [I ]2 unit matrix for the nuclear part of Liouville space. We use the matrices (11.1) to define multipole transfer operators s [ J 0 ] X f J g f J 0 g† f J J 0g D N N Πl [ J] m l m l m s ˇ [ J 0 ] X ˇˇ fI I g f J Jg f J 0 J 0gˇ D TλfIμ I g ˝ Tl m ˇ . ˇTλ μ ˝ Tl m [ J] λ μIm
Optically Pumped Atoms. William Happer, Yuan-Yu Jau, and Thad Walker Copyright © 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-40707-1
(11.4)
220
11 Mathematical Appendix
For J D J 0 the transfer operators (11.4) become projection operators of electronic multipole polarization for atoms in the states of electronic angular momentum J, and one can readily verify that X f J Jg Πl D 1fI I g ˝ 1f J J g . (11.5) l
Here
X ˇˇ f J J g f J J g ˇˇ Tl m ˇ ˇTl m
1f J J g D
(11.6)
lm
is the [ J]2 [ J]2 unit matrix for the electronic part of Liouville space for atoms in a state with electronic angular momentum J. The products of the multipole transfer operators are s [ J 000 ] f J J 000 g f J J 0g f J 00 J 000 g Πl Πl 0 D δ l l 0 δ J 0 J 00 , (11.7) Π [ J] l and the Hermitian conjugates are f J J 0 g†
Πl
D
[ J 0] f J 0 J g . Π [ J] l
Note that ˇ j1J f J g ) ˇ fI I g f J Jg D p , ˇT00 ˝ T00 [I ][ J ]
(11.8)
(11.9)
where in accordance with (3.9) we write 1J f J g D 1fI g ˝ 1f J g .
(11.10)
With (11.1) and (11.9) we find ˇ p f Jg fI I g ˇ (1J f J g jN l m D δ l0 δ m0 [I ][ J] T00 ˇ ,
(11.11)
and using (11.9) with (11.4), we find f J J 0g
(1J f J g jΠl
0
D δ l0 (1J f J g j .
(11.12)
Multipole Expansions In accordance with (11.5) we can expand the projection, jf J J g ), of the density matrix in an atomic state of electronic angular momentum J in terms of the electronic multipole moments X f J Jg j l ), (11.13) jf J J g ) D l
where f J Jg
j l
f J Jg
) D Πl
jf J J g ) .
(11.14)
11.2 Projection Operators in Terms of S or J f J J 0g
f Jg
Energy Basis The operators Πl and N l m in Section 11.1 were represented in terms of the uncoupled basis states, jm I m S i or jm I m J i of Schrödinger space, and by the corresponding Liouville-space quantities. Computer calculations are usually done in the energy basis, which is related to the uncoupled basis by the unitary transformations U D U f J g of (3.17) for the ground state or (3.21) for the excited state. The Liouville-space version of the unitary operator is
U f J g£ D U f J g[ U f J g†] D U f J g ˝ U f J g .
(11.15)
f Jg
The transformation of the operator N l m from the uncoupled basis to the energy basis is then f Jg
f Jg
N l m ! U f J g£† N l m .
(11.16)
The corresponding transformation of the projection operators is f J J 0g
Πl
f J J 0g
! U f J g£† Πl
UfJ
0 g£
.
(11.17) f Jg
The context will normally make clear whether the operators N l m and Πl J J 0 are to be represented in the uncoupled basis or in the energy basis.
11.2 Projection Operators in Terms of S or J
It is sometimes more convenient to write the projections discussed above in terms of the electronic angular momentum operators S and J instead of spherical tensors as in the previous section. For the 2 S1/2 ground state the electronic tensors are p p fS Sg fS Sg T00 D 1/ 2 and Tp D S m 2, where the spherical components S m of S are 1m S˙1 D (S x ˙ iSy )/ 2 and S0 D S z . We can therefore write the multipole terms of the expansion, fS Sg D 0fS Sg C 1fS Sg , of (11.13) and (11.14) as 0fS Sg D A ˝ 1fSg
and 1fS Sg D
X
Bj ˝ Sj ,
(11.18)
j
where A, B x , By , and B z are [I ] [I ] nuclear operators. The symbol 1fSg in A ˝ 1fSg denotes the [S ] [S ] unit operator. Using the identity for a spin-1/2 angular momentum operator, X k
1 Sk S j Sk D S j , 4
(11.19)
we find 0fS Sg D
1 fS Sg C S fS Sg S 4
and 1fS Sg D
3 fS Sg S fS Sg S . (11.20) 4
221
222
11 Mathematical Appendix
In accordance with (11.13) and (11.14), we see from (11.20) that the projection operators ΠlfS Sg for the 2 S1/2 ground state are 1 1 C S[ S] D 1 S© S© , 4 2 3 1 D S[ S] D S© S© . 4 2
Π0fS Sg D
(11.21)
Π1fS Sg
(11.22)
Here S © D S [ S ] . In (11.21) and (11.22) it is understood that the factors of 1/4, 1, and 3/4 are multiplied by [S ]2 [I ]2 [S ]2 [I ]2 unit matrices. The S-damping operator of (6.88) is identical to the projection operator (11.22), that is, Π1fS Sg D A sd .
(11.23)
There are expressions analogous to (11.21) and (11.22) for the electronic projection operators of atoms in any electronic state of electronic angular momentum J. The expressions for the electronic projection operators for the 2 P1/2 state can be obtained from (11.21) and (11.22) by making the replacement S ! J. Here we outline a general way to find the projection operators analogous to (11.21) and (11.22) for arbitrary electronic quantum numbers J. We note that (11.19) is a special case of the identity X f J Jg f J Jg J k Tl m J k D f l Tl m , (11.24) k
where the coefficient is f l D J( J C 1)
1 l(l C 1) . 2
(11.25)
Using (11.24) with (11.13) and (11.14), we find 3 X
J k f J J g J k D
X
kD1
f J Jg
f l l
.
(11.26)
l
Transforming the k-fold repetition of the operation (11.26) to Liouville space, we find X k jf J J g ) D
2J X
f J Jg
M k l j l
)D
lD0
2J X
f J Jg
M k l Πl
jf J J g ) ,
(11.27)
lD0
where the operator X is X D J[ J] D
3 X
J kT ˝ J k .
(11.28)
iDk
The coefficients M k l of (11.27) are Mk l D ( f l )k .
(11.29)
11.3 Recoupling Example
We will consider only the indices k D 0, 1, . . . , 2 J and l D 0, 1, . . . , 2 J, so the M k l will be the elements of a [ J] [ J] matrix. Since (11.27) must be true for any density matrix jf J J g ), it is equivalent to the linear equations 2J X
f J Jg
M k l Πl
D Xk .
(11.30)
lD0
Inverting the matrix equation (11.30), we find the desired projection operators for arbitrary electron spin quantum number J, f J Jg
Πl
D
2J X
k M l1 k X .
(11.31)
kD0
One can readily verify that (11.21) and (11.22) are special cases of (11.31) for J D 1/2. For the 2 P3/2 excited state of an alkali-metal atom, with J D 3/2 we find from (11.31) f J Jg
Π0
f J Jg
Π1
f J Jg
Π2
f J Jg
Π3
1
297 372X 80X 2 C 64X 3 , 1152 1
405 C 468X C 144X 2 64X 3 , D 640 1
1485 276X 272X 2 C 64X 3 , D 1152 1
D 495 972X C 464X 2 64X 3 . 5760 D
(11.32) (11.33) (11.34) (11.35)
11.3 Recoupling Example
Here we outline representative steps to derive (5.32) and similar recoupling identities. Using (5.16) and (5.19), we find n n o o Δ Δ † D 3 fξ Δg00 ξO Δ † D 3 fξ Δg0 f ξO Δ † g0 00 00 9 8 110 = < n Xp ˚ o fξ ξO g L ΔΔ † L D 3 [0][0][L][L] 110 ; : 00 L LL0 X ˚ D (1) L fξ ξO g L ΔΔ † L . (11.36) L
To go from the first line to the second line of (11.36) we used the fundamental definition of the 9 j symbol as the coefficient for recoupling four angular momenta (see (5) in Section 10.1 in [15]). The last line comes from evaluating 9 j symbols in which some of the angular momentum arguments are zero (see (2) in Section 10.9 in [15]) and when some of the angular momentum arguments of the resulting 6 j
223
224
11 Mathematical Appendix
symbol are zero (see (1) in Section 9.5.1 in [15]). The scalar product of the two spherical tensors of the last line was defined by (5.34). Using (5.17), (5.20), and (5.21), we find ˚ fΔΔ † g LM D (1) J S fjS if J jg1 fj JifS jg1 LM 9 8 < J S1 = Xp D (1) J S [1][1][K ][N ] J S1 ; : KN KNL ˚ (f J jj Ji)K (jS ifS j) N LM .
(11.37)
Since jS i and f J j operate in different spin spaces, we used the identity (jS if Jj)1 D (1) SC J 1 (f J jjS i)1
(11.38)
in going from the first to the second line of (11.37). Since f Jj and j Ji operate in the same spin subspace, we find (f J jj J i)K M D
X
C JKMM0 I J,MM 0 f J M 0 j J, M M 0 i
M0
D δ M0
X
J CM C JK0 M 0 I J,M 0 (1)
M0
D δ M0 (1)2 J
0
p X K0 [ J] C J M 0 I J,M 0 C 00 J M 0 I J,M 0 M0
p D δ M0 δ K0 [ J](1)2 J .
(11.39)
We inserted the Clebsch–Gordan coefficient C 00 J M 0 I J,M 0 in the third line with the aid of (5.18). Substituting (11.39) back into (11.37), we find fΔΔ † g LM
9 8 < J S1= p D [1][1][ J][L] J S1 fjS ifS jgLM ; : 0LL
11L D3 T (S S )(1) SC J CL . S S J LM
Substituting (11.40) back into (11.36), we find (5.32).
(11.40)
225
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79 Walter, D.K., Griffith, W.M., and Happer, W. (2002) Magnetic slowing down of spin relaxation due to binary collisions of alkali-metal atoms with buffer-gas atoms. Phys. Rev. Lett., 88 (9), 093004. 80 Marshall, T.R., Boggy, R., and Franz, F.A. (1977) Spin relaxation of optically pumped cesium in high magnetic fields. Phys. Rev. A, 16 (2), 618. 81 Beverini, N., Violino, P., and Strumia, F. (1973) Optical pumping of caesium in the presence of heavy noble gases. Z. Phys. A, 265, 189. 82 Franz, F.A. and Sooriamoorthi, C.E. (1974) Spin relaxation within the 62 P1/2 and 62 S1/2 states of cesium measured by white-light optical pumping. Phys. Rev. A, 10 (1), 126. 83 Kadlecek, S. (2008) Private communication. 84 Walker, T.G. (1989) Estimates of spinexchange parameters for alkali-metal– noble-gas pairs. Phys. Rev. A, 40, 4959. 85 Appelt, S., Baranga, A.B., Erickson, C.J., Romalis, M.V., Young, A.R., and Happer, W. (1998) Theory of spin-exchange optical pumping of 3 He and 129 Xe. Phys. Rev. A, 58, 1412. 86 Chann, B., Babcock, E., Anderson, L.W., and Walker, T.G. (2002) Measurements of 3 He spin-exchange rates. Phys. Rev. A, 66, 32703. 87 Chupp, T.E. and Coulter, K.P. (1985) Polarization of 21 Ne by spin exchange with optically pumped Rb vapor. Phys. Rev. Lett., 55(10), 1074. 88 Romalis, M. and Cates, G. (1998) Accurate 3 He polarimetry using the Rb Zeeman frequency shift due to the Rb– 3 He spin-exchange collisions. Phys. Rev. A, 58, 3004. 89 Stoner, R. E. and Walsworth, R.L. (2002) Measurement of the 21 Ne Zeeman frequency shift due to Rb–21 Ne collisions. Phys. Rev. A, 66(3), 032704. 90 Babcock, E., Nelson, I.A., Kadlecek, S., and Walker, T.G. (2005) 3 He polarization-dependent EPR frequency shifts of alkali-metal–3 He pairs. Phys. Rev. A: At. Mol. Opt. Phys., 71 (1), 013414. 91 Babcock, E. (2005) Spin-Exchange Optical Pumping with Alkali-Metal Vapors,
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Index a absorption 76 – high-pressure limit 112 – small magnetic fields 112 – unpolarized ground-state atoms 111 absorption cross section 111 accommodation operator 163 – identities 164 acute matrix 27 adsorption on the walls 194 alkali-metal atoms 15 – basic parameters 17 – electronic energies 15 – lifetimes 18 – oscillator 18 – quantum defect 16 – valence-electron wave functions 18 alkali-noble gas spin exchange – K D 1/2 182 – axial symmetry 183 – binary collisions 181 – conservation of spin 182 – experimental measurements 186 – frequency shift 187 alkali spin exchange – experimental measurements 208 – like atoms 204 – Liouville space 205 – optical theorem 203 – partial-wave analysis 198 – partial-wave cross sections 203, 207 – potentials 199 – scattering amplitudes 201 – scattering state 200 – slowing-down factor 206 – spin-axis potential 200 Amagat 64 angular momentum 18 – addition of angular momentum 52
– angular momentum eigenstates 31 – angular momentum matrix 27 – dimensionless dipole operator 53 – recoupling 51 – spherical basis dyadics 54 – Wigner–Eckart theorem 53 antisymmetric unit tensor 55 atomic clock 81 atomic polarizability 116 attenuation 117 b Bohr frequencies 98 c Clebsch–Gordan coefficient 31 coherence 2, 34 – excited state 35 – ground state 35 coherent population trapping 94 collision 77, 83 collisions 59 ff collisions of excited atoms 212 commutator superoperator 47 commutator transform 63 compactification 11 critical damping 11, 43 d damping matrix 5 damping operator 41 density matrix 2, 33 – eigenvalues 34 – entropy 35 – purity 35, 41 depopulation-pumping 82 dimensionless dipole operator 49, 50, 52, 119 Doppler broadening 90 dot–slash product 74 drive-frequency vector 93
Optically Pumped Atoms. William Happer, Yuan-Yu Jau, and Thad Walker Copyright © 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-40707-1
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Index e effective Hamiltonian 75, 76 eigendecomposition 41 – left eigenvector 41, 42 – right eigenvector 41 electric dipole interaction 58 electric dipole moment 50 electric field 49, 59 electronic multipoles 219 energy basis 29, 56 energy states 28 evolution in space and time – first-order propagation 115 evolution matrix 40 evolution of a beam in space and time 114 excited-state collisions – detailed analysis 214 – detailed balance 216 – fine-structure-changing collisions 215 – multipole transfer coefficients 214 – notable effects 213 – quenching 216 excited-state evolution 76 expectation value 38 f flat superoperators 44 force 122 – conservative 123 – monochromatic light 123 – optical Earnshaw theorem 124 g grave matrix 26 h harmonic-index vector 93 high pressure 95 high-pressure pumping 82 – Liouville space 84 hyperfine shift – binary collisions 171 – experimental measurements 173 – high fields 173 – low fields 172 hyperfine structure 20, 22 – electric-quadrupole 32 – magnetic-dipole 32 i identities 78 induced electric dipole moment 109
k Kronecker product 25, 26, 45, 46 – mixed-product rule 26 l light-damping operator 75, 76 light-shift operator 75, 76 Liouville space 1, 33 – bases 3 – column-vector and row-vector transforms 36 – constraints 6 – eigenvector expansions 8 – eigenvectors 7 – example 2 – expansion amplitudes 4 – Liouvillian conjugate 40 – steady state 9 – transients 8 logical variables 12, 14 m magnetic moment operator 21 magnetic resonance 94 MATLAB code 2 matrix transformation 44 mean force 121 modulation 93 – attenuation 100 – isolated magnetic resonances 102 – light 94 – optical pumping matrices 96 – repopulation matrix 101 momentum diffusion 151 – Castin–Mølmer model 154 – continuum limit 155 – diffusion coefficient 153 – optical pumping 156 – pumping 153 – spontaneous emission 152 momentum space 121, 135 – compactification 140 – compactification within a tile 143 – compactified p q space 141 – displays 147 – evolution in spin-momentum space 138 – Liouville space 140 – mean pumping rate 139 – pointing probability 147 – spontaneous radiative 138 – tile 136 MOT 124 – magneto-optical force 128
Index – mean forces 126 – perfect repumping 136 – pointing probability 132 ff – repump laser 124, 130 – transverse coherence 131 – trap laser 124 multiple-quantum operator 99 multipole expansions 220 n null space 9, 42 o o-dot superoperator 47 o-dot transform 63 optical Bloch equations 65 ff – evolution operator 67 – Liouville space 66 – net evolution 62 – steady state 69 – transients 68 optical coherence 35 optical Rabi frequencies 66 optical spin operator 21, 61 oscillator strength 56 p paramagnetic Faraday rotation 118 plasma dispersion function 89 polarizability 110 – effective Hamiltonians 110 – gyrotropic polarizability 117 – scalar polarizability 117 – transverse polarizability 117 population 34 position-space 149 projection operators 221 propagation of weak probe light 116 push–pull pumping 106 q quasi-steady-state 73 ff – Liouville space 75, 77 – negligible stimulated emission 81 – net evolution 80 r radiation forces 121 recoupling 223 relaxation 159 repopulation-pumping 82 resonance 93, 100 rotating coordinate system 13, 59 ff rotating frame 74, 98
s S-matrix 160 – Liouville space 162 saturation 78, 79, 110 Schrödinger space 25 secular approximation 94, 97, 99 sharp superoperators 44 spatial diffusion 189 – boundary conditions 190 – eigenmode expansion 189 – normal gradient operator 191 – relaxation in the dark 208 – slowest diffusion mode 193 – strongly depolarizing walls 192, 211 – unconfined polarization 191 – weakly depolarizing walls 191, 210 specific absorption 118 spectral width of pumping light 87 ff – Gaussian profiles 88 spherical tensor 50 – Hermitian conjugate 51 – spherical basis tensor 53 spin-exchange 84, 85 – between alkali-metal atoms 198 – between alkali-metal atoms and noble-gas atoms 178 spin-rotation interaction 175 – binary collisions 176 – experimental measurement 178 – multipolarity 177 – populations 177 – S-damping operator 176 spin temperature 184, 205 – spin-temperature limit of relaxation processes 206 – steady state 185 spontaneous emission 55–57, 79 – Liouville space 58 stimulated emission 76 strong collisions 168 sudden collisions 167 superoperators 5, 36, 39 t transposition matrix trap laser 124
39
u uniform relaxation 67 unitary transformation 29 units 14 v valence-electron wave functions 18
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Index van der Waals molecules 168 – accomodation operator 170 – S-matrix 170 velocity damping 126
weak collisions 164 – relative power spectrum 166 – secular approximation 165 weakly depolarizing walls 191
w wall adsorption – fluctuating wall interaction 195 – populations 195 – relaxation on the wall 196
z Zeeman magnetic resonances 103 zero-field states 30
