Atomic physics: An exploration through problems and solutions [2 ed.] 9780199532414

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ATOMIC PHYSICS An Exploration through Problems and Solutions

ATOMIC PHYSICS An Exploration through Problems and Solutions

Second Edition

Dmitry Budker Department of Physics, University of California, Berkeley, USA

Derek F. Kimball Department of Physics, California State University, East Bay, USA

David P. DeMille Department of Physics, Yale University, New Haven, USA

OXFORD UNIVERSITY

PRESS

OXFORD UNIVERSITY

PRESS

Great Clarendon Street, Oxford OX2 6DP Oxford University Press is a department of the University of Oxford. It furthers the University's objective of excellence in research, scholarship, and education by publishing worldwide in Ox ford New York Auckland Cape Town Dar cs Salaam Hong Kong Karachi Kuala Lumpur Madrid Melbourne Mexico City Nairobi New Delhi Shanghai Taipei Toronto With offices in Argentina Austria Brazil Chile Czech Republic France Greece Guatemala Hungary Italy Japan South Korea Poland Portugal Singapore Switzerland Thailand Turkey Ukraine Vietnam

Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Published in the United States by Oxford University Press Inc .. New York © Oxford University Press 2008

The moral rights of the author have been asserted Database right Oxford University Press (maker) Reprinted with corrections 201 O All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means. without the prior permission in writing of Oxford University Press. or as expressly permitted by law, or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department. Oxford University Press. at the address above You must not circulate this book in any other binding or cover And you must impose this same condition on any acquirer ISBN 978-0-19-953241-4

Printed in the United Kingdom by Lightning Source UK Ltd.. Milton Keynes

To our teachers

ACKNOWLEDGMENTS This book would not have been possible without our mentors, colleagues, and students whose ideas inspired many of the problems and solutions found in this book. Their suggestions, guidance, and readings of countless drafts have been invaluable. In particular, we would like to acknowledge the inspiration from and contributions of Victor Acosta, Evgeniy B. Alexandrov, Marcis Auzinsh, Lev M. Barkov, Ilya Bezel, Sarah Bickman, Chris J. Bowers, G. A. Brooker, Sid B. Cahn, Arman Cingoz, Eugene D. Commins, Andrew Dawes, Damon English, Victor Flambaum, Daniel Gauthier, Wojciech Gawlik, Jennie Guzman, Erwin Hahn, Theodor W. Hansch, Robert A. Harris, Chris Hovde, Larry Hunter, J. D. Jackson, losif B. Khriplovich, Gleb L. Kotkin, Mikhail G. Kozlov, Vitaliy V. Kresin, Chih-Hao Li, Yongmin Liu, Steve K. Lamoreaux, Alain Lapierre, Jon M. Leinaas, Robert Littlejohn, Richard Marrus, Jeff Moffitt, Hitoshi Murayama, Frank A. Narducci, Anh-Tuan D. Nguyen, A. I. Okunevich, J. B. Pendry, Simon M. Rochester, Michael V. Romalis, Yaniv Rosen, Neil Schafer-Ray, Gennady Shvets, Jason E. Stalnaker, Herbert Steiner, Mark Strovink, Alexander 0. Sushkov, Oleg P. Sushkov, Falguni Suthar, Mahiko Suzuki, Jeff T. Urban, Arkady I. Vainshtein, Louis A. Villanueva, Ronald Walsworth, David S. Weiss, Eric Williams, Valeriy V. Yashchuk, Jun Ye, Jerzy Zachorowski, Vladimir G. Zelevinsky, and Max Zolotorev. We would like to extend a special thanks to Damon English for providing the cover illustration. The authors also acknowledge the support of their research, that motivated many of the problems in this book, by the National Science Foundation, the Office of Naval Research, and the Miller Institute for Basic Research in Science.

CONTENTS

Preface to the Second Edition Preface to the First Edition Notation

xv

XVII

XIX

1

Atomic structure 1.1 Ground state of phosphorus 1.2 Exchange interaction 1.3 Spin-orbit interaction 1.4 Hyperfine structure and Zeeman effect in hydrogen 1.5 Hydrogenic ions 1.6 Geonium 1.7 The Thomas-Fermi model (T) 1.8 Electrons in a shell 1.9 Isotope shifts and the King plot 1.10 Crude model of a negative ion 1.11 Hyperfine-interaction-induced mixing of states of different J I .12 Electron density inside the nucleus (T) 1.13 Parity nonconservation in atoms 1.14 Parity nonconservation in anti-atoms 1.15 The anapole moment (T)

1 1 7 10 13 18 21 30 33 37 41 42 46 51 61 65

2

Atoms in external fields 2.1 Electric polarizability of the hydrogen ground state 2.2 Polarizabilities for highly excited atomic states 2.3 Using Stark shifts to measure electric fields 2.4 Larmor precession frequencies for alkali atoms 2.5 Magnetic field inside a magnetized sphere 2.6 Classical model of magnetic resonance 2.7 Energy level shifts due to oscillating fields (T) 2.8 Spin relaxation due to magnetic field inhomogeneity 2.9 The E x v effect in vapor cells 2.10 Field ionization of hydrogenic ions 2.11 Electric-field shifts of magnetically split Zeeman sublevels

75 75 78 79 81 84 85 90 I02 107 110 110

CONTENTS

X

3

4

2.12 Geometric (Berry's) phase 2.13 Nuclear dipole-dipole relaxation 2.14 Magnetic spin precession of a free magnet

112 116 118

Interaction of atoms with light 3.1 Two-level system under periodic perturbation (T) 3.2 Quantization of the electromagnetic field (T) 3.3 Emission of light by atoms (T) 3.4 Absorption of light by atoms 3 .5 Resonant absorption cross-section 3.6 Absorption cross-section for a Doppler-broadened line 3.7 Saturation parameters (T) 3.8 Angular distribution and polarization of atomic fluorescence 3.9 Change in absorption due to optical pumping 3 .10 Optical pumping and the density matrix 3.11 Cascade decay 3 .12 Coherent laser excitation 3.13 Transit-time broadening 3.14 A quiz on fluorescence and light scattering 3 .15 Two-photon transition probability 3.16 Vanishing Raman scattering 3.17 Excitation of atoms by off-resonant laser pulses 3.18 Hyperfine-interaction-induced magnetic dipole (Ml) transitions 3.19 Transitions with unresolved hyperfine structure 3.20 Optical pumping and quantum beats in Mercury 3.21 Thomson scattering 3.22 Classical mcxlel for a magnetic-dipole transition 3.23 Nonlinear three-wave mixing in isotropic chiral media 3.24 A negatively refracting atomic vapor? 3.25 Light propagation in anisotropic crystals 3.26 Electromagnetically induced transparency (EIT)

121 121 128 134 144 147 149 151 158 162 168 172 175 176 179 183 185 187

Interaction of light with atoms in external fields 4.1 Resonant Faraday rotation 4.2 Kerr effect in an atomic medium 4.3 The Hanle effect 4.4 Electric-field-induced decay of the hydrogen 2 2 S 1; 2 state 4.5 Stark-induced transitions (T) 4.6 Magnetic deflection of light 4.7 Classical model of an optical-pumping magnetometer

190

193 195 199 201 204 207 212 215 223 223 227 233 236 238 244 249

CONTENTS

xi

4.8 Searches for permanent electric dipole moments (T) 4.9 Sensitivity to electric dipole moments 4.10 Absorption, dispersion, optical rotation, and induced ellipticity 4.11 Optical rotation in a gas of polarized neutrons

253 264

5

Atomic collisions 5.1 Collisions in a buffer gas 5.2 Spectral line broadening due to phase diffusion 5.3 Dicke narrowing 5.4 Basic concepts in spin exchange 5.5 The spin-temperature limit 5.6 Electron-randomization collisions 5.7 Larmor precession under conditions of rapid spin exchange 5.8 Penning ionization of metastable helium atoms

273 273 274 277 281 285 287 288 290

6

Cold atoms 6.1 Laser cooling: basic ideas (T) 6.2 Magneto-optical traps 6.3 Zeeman slower 6.4 Bose-Einstein condensation (T) 6.5 Bose-Einstein condensation from an optical lattice 6.6 Cavity cooling 6.7 Cavity cooling for many particles: stochastic cooling 6.8 Fermi energy for a hannonic trap

295 295 302 306 311 322 324 329 331

7

Molecules 7 .1 Amplitude of molecular vibrations 7 .2 Vibrational constants for the Morse potential 7 .3 Centrifugal distortion 7.4 Relative densities of atoms and molecules in a vapor 7 .5 Isotope shifts in molecular transitions 7 .6 Electric dipole moments of polar molecules 7 .7 Scalar coupling of nuclear spins in molecules 7 .8 Zeeman effect in diatomic molecules 7 .9 Omega-type doubling

335 335 336 338 341 346 351 355 359 363

8

Experimental methods 8.1 Reflection of light from a moving mirror 8.2 Laser heating of a small particle

367 367 369

267 270

xii

CONTENTS

8.3 Spectrum of frequency-modulated light 8.4 Frequency doubling of modulated light 8.5 Ring-down of a detuned cavity 8.6 Transmission through a light guide 8.7 Quantum fluctuations in light fields 8.8 Noise of a beamsplitter 8.9 Photon shot noise in polarimetry 8.10 Light-polarization control with a variable retarder 8.11 Pile-up in photon counting 8.12 Photons per mode in a laser beam 8.13 Tuning dye lasers 8.14 Matter-wave vs. optical Sagnac gyroscopes 8.15 Femtosecond laser pulses and frequency combs 8.16 Magnetic field fluctuations due to random thermal currents 8.17 Photodiodes and circuits (T)

372 374 376 377 378 382 384 386 390 391 392 395 398 403 406

9

Miscellaneous topics 9.1 Precession of a compass needle? 9.2 Ultracold neutron polarizer 9.3 Exponentially growing/decaying harmonic field 9.4 The magic angle 9.5 Understanding a Clebsch-Gordan coefficient selection rule 9.6 The Kapitsa pendulum 9.7 Visualization of atomic polarization 9.8 Estimate of elasticity and tensile strength of materials 9.9 The Casimir force

415 415 417 418 420 426 428 431 438 440

A

Units, conversion factors, and typical values

443

B

Reference data for hydrogen and alkali atoms

449

C

Spectroscopic notation for atoms and diatomic molecules

451

D

Description of polarization states of light D.I The Stokes parameters D.2 The Jones calculus

455 455 456

E Euler angles and rotation matrices

459

CONTENTS

F

The Wigner-Eckart theorem and irreducible tensors F. I Wigner-Eckart theorem F.2 Irreducible tensors

xiii

461 461 467

G The density matrix G. I Connection between the density matrix and the wavefunction G.2 Ensemble-averaged density matrix G.3 Time evolution of the density matrix: the Liouville equation G.4 Atomic polarization moments

469 469 472 474 476

H Elements of the Feynman diagram technique

481

I

485 485 488

The 3-J and 6-J symbols I. I 3-J symbols 1.2 6-J symbols

Bibliography

491

Index

511

PREFACE TO THE SECOND EDITION

The first edition of Atomic Physics: an exploration through problems and solutions appeared in 2003. We are truly delighted with the enthusiasm the book has been met with by students and colleagues, some of whom use the book as a supplementary resource for their classes, as well as a research reference. For a while after the book appeared, we were pleasantly surprised with the relatively small number of errors and misprints found in the text. As time went on, however, a number of issues have been uncovered, ranging from benign misprints to, we are sorry to say, a few significant conceptual errors. We found some of these problems by ourselves (yes, we are among the ones who use the book as a research and teaching reference); others were found by our readers from near and far who were kind enough to e-mail us their criticisms. We are deeply indebted to everyone who has contributed. The second edition, presently offered to the reader, has been prepared by two of the three authors of the original Atomic Physics, for which "error correction" is one, but not the only motivation. New teaching and research experiences have stimulated us to come up with significant amount (over 70 pages) of new material: about twenty additional problems in various chapters, and a new Appendix. All the new material has been placed at the end of appropriate chapters, so the numbering of problems and appendices is compatible with the first edition. As we have done with the first edition, we plan to keep a record of errors on one of the author's web page:

http://budker.berkeley.edu/

We greatly appreciate further feedback from our readers!

Dmitry Budker ([email protected]) Berkeley, California November 2007

and

Derek F. Kimball ([email protected])

PREFACE TO THE FIRST EDITION

We have found that usually the best way to learn something new is to ask concrete questions and try to work out the answers. Often some of the simplest questions have surprising and unexpected answers, and some seemingly complex problems can be solved in a simple way. In this book we have collected some of these problems and our solutions to them. The book encompasses many issues we faced as we ourselves made the transition from undergraduate students to practicing experimental atomic physicists and instructors. However, the text is not intended to be comprehensive, but rather addresses various aspects of atomic physics which we have found interesting and important. In the course of doing atomic physics, we always find ourselves crossing boundaries into other subfields; the selection of problems reflects this gray area. It also reflects our specific interests, with several problems about symmetry violation, etc. that would not appear in more "standard" textbooks. It is our philosophy that working on specific problems usually helps with understanding of more general issues, and indeed may be the most useful way to really learn anything. It is our hope that some selection of the broad range of problems given here will pique the interest of any reader, and thus initiate this process. Where possible, we try to emphasize approximation methods, dimensional considerations, limiting cases, and symmetry arguments as opposed to fonnal mathematics. We often appeal to pictures, tables, and graphs. This problemsolving approach is aimed at developing intuition about physical principles and fosters the important ability to perfonn "back-of-the-envelope" calculations. These are the tools we find most useful when trying to solve the types of problems we commonly encounter in the laboratory. Of course, on occasion a formal mathematical approach (as painful as it could be) can lead to important insights. Generally, in order to deeply understand various aspects of physics, it is good to have both an intuitive picture as well as the appropriate mathematical tools. This book is intended for advanced undergraduates and beginning graduate students interested in atomic, molecular and optical physics, and we assume that readers possess basic knowledge of quantum mechanics [at the level of Griffiths ( 1995), Bransden and Joachain ( 1989), or similar texts], electrodynamics [at the level of Griffiths ( 1999), Purcell ( 1985), or similar texts], and thennodynamics [at the level of Reif ( 1965), Kittel and Kroemer ( 1980), or similar texts]. However, we hope that many of the problems will also be of interest to professional scientists.

PREFACE TO THE FIRST EDITION

xviii

In physics, there continues to be a raging debate over what is the best system of units to use and whether or not units should be standardized. We feel that the choice of units is a personal one, especially since converting between different systems is relatively straightforward. That said, in this book we have a tendency to use CGS units, since we find them most convenient (especially in problems involving electromagnetism). We also set Ii = 1 when it is convenient to do so and measure energies in frequency units, as is common practice in atomic physics (since energy measurements are typically performed by measuring frequencies). Each problem in the book is intended to stand on its own. If there is a particular subject in atomic physics that one is interested in learning about, there may be a problem about it in this book. We envision the reader turning right to that page and starting to try to figure it out. Hopefully, at the end of this exercise, one will have gained some familiarity with the topic, enabling her or him to understand more advanced, specialized literature on the subject, or go straight to the lab and get to work!

In the introduction to most problems there is a brief discussion of the relevance of the topic to modem atomic physics with references to research literature on the subject. The cited references are not intended to be comprehensive, but merely provide a starting point in a search for more information about the subject of the problem. We apologize in advance to the innumerable scientists whose important contributions are not mentioned. Also, for a few problems, especially in subfields of atomic physics dear to our hearts, there are some historical remarks. Of course, there is a great deal of history surrounding almost all of the topics covered in this book, and we could not tell all of it. Nonetheless, we thought a few, not widely known stories might be enjoyable. Some of the problems are written as tutorials on various subjects in atomic physics [they are marked with a (T)]. In such problems, there are a series of short questions that are intended to guide the reader through some important material. Hopefully the reader will find this more entertaining and interactive than just reading the explanation straight through. We hope you enjoy reading and using the book as much as we have enjoyed writing it!

D.B. D. F. K. D. P. D. Berkeley, Calif omia May 2003

NOTATION

The following is a table of symbols commonly used throughout the book, their meaning, and their value where appropriate.

Symbol

Meaning

Value

m, me

electron mass

9.1094

X

10- 28 g

0.5111\foV /c 2

proton mass

mp

1.6726

X

10- 24 g

938.27 MeV /c 2

neutron mass

mn

1.6750

X

10- 24 g

939.57 MeV /c -mp

fin

e

Ii

1.293 MeV /c 2

electron charge magnitude

4.8032 x 10-

10

esu

6.6261 x 10-

27

erg · s

1.0546 x 10-

27

erg· s

= h/(21r)

C

o:

difference between nucleon masses Planck's constant

h

= e 2 /(lie)

= 1i2 /(me 2 ) µo = e1i/(2mc)

ao

2

10

speed of Iight

2.99792458 x 10 cm/s

fine structure constant

1/137.036

Bohr radius

5.292 x 10- 9 cm

Bohr magneton

0.93 x 10- 20 erg/G 1.40 MHz/G 5.79 x 10- 9 eV /G

µN

= e1i/(2m,,c)

nuclear magneton

5.05 x 10- 24 erg/G 762 Hz/G

Rx

= me

4

/(41r1i3 c)

Rydberg constant

109,737 cm-

1

3.2898 x 1015 Hz

kB

Boltzmann's constant

1.38066 x 10-

16 5

erg/K

8.61735 x 10- eV /K L, l

S,s J,j

I F

orbital angular momentum electron spin total electronic angular momentum nuclear spin total atomic angularmomentum

units of Ii units of Ii units of Ii units of Ii units of Ii

NOTATION

xx

In most locations, we remind the reader of the meaning of the symbols when they appear. Also see Appendix A for practical units, conversion factors, and typical values of various parameters. When we deal with spin-1/2 systems, we will commonly employ the notation I+) and I-) to denote the spin up (m = + 1/2) and spin down (m = -1 /2) states, respectively. Here m is the projection of the spin along the quantization axis in units of h. The ubiquitous Clebsch-Gordan coefficients 1 describe the connection between the coupled basis IJ, M) and the uncoupled basis IJ 1, M 1 )IJ2, M2) (where J, J1, J2 are angular momenta and M, M 1 , M 2 are the projections of the respective angular momenta on the quantization axis):

IJ,M)

=

L

C(J1,hJ;M1,M2,.M)IJ1,M1)lhM2)

M1,~h

IJ1,M1)lh M2)

= L C(Ji, h

J; Mi, M2, M)IJ, M).

J,M

In the text we consistently use the notation:

C(J1, J2, J; M1, M2, M)

= (J1, M1, J2, M2IJ, M)

,

and employ the commonly used phase convention of Condon and Shortley ( 1970), Edmonds ( 1996), and Sobelman ( 1992).

1

The Clebsch-Gordan coefficients are also referred to as vector-coupling coefficients, vector-

addition coefficients, and Wignercoefficients in the literature.

1 ATOMIC STRUCTURE

1.1 Ground state of phosphorus One of the most important topics in atomic physics is the description of atomic energy levels. The study of atomic structure continues to be an exciting field, with increasingly precise measurements and improved calculational tools allowing ever more detailed comparisons between experiment and theory. The first few problems in this chapter deal with some of the basic features of atomic energy levels in multi-electron atoms. In the simplest atom, hydrogen, most of the splitting between energy levels comes from the difference between the principal quantum numbers n for different states - the energy En of an electron in the hydrogen atom is given, approximately, by the famous Bohr fonnula,

En~----

me 4 1

(I.I)

2!i2 n2 '

where m is the mass of the electron and e is the absolute value of its charge. As a first approximation, in more complex atoms we can consider the states of individual electrons as if they were moving in an effective centrally symmetric field created by the nucleus and the other electrons (the central field approximation). In this case we can assign a principal quantum number n and an orbital angular momentum l to each electron (the distribution of the electrons among the states with various n's and l's is known as the electron configuration), and, as in hydrogen, the differences between the principal quantum numbers for different configurations are an important source of energy level splittings. In contrast to hydrogen, however, the energy of a particular configuration for a multi-electron atom is also dependent on l. This is because electrons with larger l values are, on average, further from the nucleus due to the centrifugal barrier, and the other electrons screen the nuclear charge. Based on these two general ideas, one expects that configurations for which electrons have the lowest possible values of n and l have the lowest energies. For s and p orbitals, it is most important to have the smallest value of n ("regular" configurations). However, one finds that in some cases, for

ATOMIC STRUCTURE

2

d and / orbitals it can be energetically favorable for electrons to have higher n so 1

that they can occupy states with lower l ("irregular" configurations ). Next one must consider in more detail the mutual electrostatic repulsion between the electrons. Already, the central field approximation has accounted for the spherically symmetric part of the potential due to the electron-electron interaction which is responsible for the screening of the nucleus. There is also a nonspherically symmetric part of the electron-electron potential related to the fact that it is energetically favorable for electrons to be as far apart as possible. For a given configuration, the atomic states can be specified by the total orbital angular momentum l = L ~ and the total spin S = E Bi (this characterization of atomic states, valid for low to intermediate Z atoms, is known as the Russell-Saunders or L-S coupling scheme 2 ). Since the average separation between electrons are different for states with different L and S, these states are split in energy (see Problem 1.2). The states of a given configuration identified with particular L and S are known as the term. There are empirical rules, known as Hund's rules [see, for example, Bransden and Joachain (2003), Landau and Lifshitz (1977), or Herzberg (1944)], for determining which term has the lowest energy for a given configuration (in the Russell-Saunders coupling scheme). Hund's rules state that for the ground term with the lowest energy • the term with the largest S has the lowest energy, and • for a given S, the larger the total orbital angular momentum L the lower the energy (as long as there is only a single unfilled shell). These rules ensure that the ground state electrons are, on average, as far apart as possible, which minimizes the electrostatic repulsion between them. For both single and multi-electron atoms, there is also the spin-orbit interaction (see Problem 1.3) which causes splitting of states with different values of the total electronic angular momentum J (spin S coupled to orbital L). Since this splitting is typically considerably smaller than the energy differences from the previously discussed mechanisms, it is known as the fine-structure splitting.

1

An examr,le of such an irre§ular confi§uration is the ground state of potassium, which is ls 2 2s 2 2p6 3s 3p6 4s instead of ls 2s 2 2p6 3s 3p 6 3d. 2 In atoms with large Z, the spin-orbit energy (arising from relativistic effects) can become more imponant than the residual, nonspherical electrostatic interaction between the electrons. In such a case, it is useful to specify individual electron states by their total angular momentum J = f + i. This is known as the j-j coupling scheme. In each coupling scheme, we are trying to begin our considerations with what are nearly the energy eigenfunctions for the atom, but in the general case neither L-S nor j-j coupling give the correct energy eigenfunctions.

GROUND STATE OF PHOSPHORUS

3

Typically, the L, S, and J for a particular state are designated in spectroscopic notation (see, for example, Appendix C): 2S+lL

( 1.2)

J·

In this problem, we consider all of the aforementioned interactions in order to detennine the energy level structure for the terms corresponding to the ground state configuration of phosphorus (P). (a) What is the ground state configuration of P, Z

= 15?

(b) What is the ground tenn and J value for P according to Hund's rules?

(c) What other terms are possible for the ground state configuration? (d) For the term with the highest energy corresponding to the ground state configuration, identified in part (c), can one say what value of J has the highest energy using first-order perturbation theory?

Hints In part (b ), we can make use of the fact that electrons in the filled subshells (ls, 2s, 2p, 3s) have total spin and total orbital angular momentum equal to zero, and so we need only consider the three outer electrons in the 3p orbital to determine the ground tenn. For the ground term, first consider the maximum spin and maximum projection of the spin along the quantization axis for three electrons (according to Hund's first rule, this should be S for the ground term). What must be the orbital wavefunction in this case? Part (d) is tricky! One can think of a subshell (which can hold a total of 2(2l + 1) electrons) containing N electrons as consisting of N electrons or 2(2l + 1) - N holes. The spin-orbit splitting (to first order) has opposite sign for electrons and holes.

Solution (a) Phosphorus has sufficiently low Z that each shell fills regularly (there are no d or / states involved). A subshell can contain 2(2l + 1) electrons, where l is the orbital angular momentum quantum number for the individual electrons.

ATOMICSTRUCTURE

4

Therefore, the ground state configuration is

( 1.3)

(b) The maximum total spin that three electrons can possess is S

=

3/2. The S = 3/2 spin wavefunctions are symmetric under particle interchange. This can be seen from the fact that the stretched state IS,Als = S) is obviously symmetric ( 1.4) and all the other S = 3/2 states can be obtained by application of the lowering operators_ = S 1_ + S 2 _ + S 3 _, which is symmetric and so cannot change the exchange symmetry of the spin states. To satisfy the spin-statisticstheorem, 3 we must choose a spatial wavefunction which is totally antisymmetric with respect to particle interchange. With three p electrons, we have l 1 = 1, l 2 = 1, and l 3 = 1, in principle allowing L = 3, 2, 1 and 0, where Lis the total angular momentum of all three valence electrons. However, in order to construct a totally antisymmetric wavefunction, one needs at least as many states as there are particles. This can be seen by considering the Slater determinant [see, for example, Bransden and Joachain (2003)), which provides a simple method for finding a totally antisymmetric wavef unction for a system of particles. In this case we have three particles in, let us say, states o, /3, and , . Then the totally antisymmetric wavefunction is given by: \JI

-

AS -

1

v'fil

o(l) /3(1) ,(1) o:(2) /3(2) ,(2) o(3) /3(3) ,(3)

(l.5)

This wavefunction is clearly exchange antisymmetric, since interchanging two particles is equivalent to interchanging two rows of the matrix, which changes the sign of the determinant. Furthermore, the determinant vanishes if two columns are the same, which occurs if any two of the states o, {3,or 1 are the same. Thus the Pauli exclusion principle is seen to be a consequence of the fact that electrons must be in exchange antisymmetric states. must involve Therefore, for S = 3/2, the allowed spatial wavefunction l·l/Jspace) a superposition of electrons in the m 1 = 1, m 1 = 0, and m 1 = -1 states (denoted 3

There is an important distinction between the symmetrizationpostulate and the spin-statistics theorem. The symmetrization postulate states that any wavefunction for a system of identical panicles must be either exchange symmetric or exchange antisymmetric (i.e., the wavefunction must be an eigenstate of the permutation operator). The spin-statistics theorem states that integer spin panicles (bosons) must be in an exchange symmetric state and half-integer spin particles (fermions) are in exchange antisymmetric states.

GROUND STATE OF PHOSPHORUS

,'---m' ! 1 ms',.!

0

-1

•

•

•

+1/2

i

I

-1/2

I I

i

FIG.

5

i

1.1 A simple way to figure out the ground state term according to Hund's rules for the case of

phosphorus.

II), IO),l-1)), namely [using (1.5)],

l'l/-'space) = ~ [ll)alO)bl-l)c

+ I0) l-l)bll)c + 1-l)allhlO)c 0

(1.6)

- 1-l)alO)bll)c -11) 0 1-l)blO)c- IO)all)bl-l)c] ' so that the total projection of the orbital angular momentum along the z-axis is ML = 0. Thus the only possible value for the orbital angular momentum is L = 0. Here is another simple, graphical way to arrive at this conclusion. Consider the chart shown in Fig. 1.1. Each box corresponds to an individual electron state with unique quantum numbers. We want to think about the stretched state with maximum S. The Pauli exclusion principle demands that we can only put one electron in each box (since otherwise we could not form an exchange antisymmetric state). We put all electrons in m 8 = 1/2 boxes to maximize Ms (and hence S). Then the maximum projection of ML consistent with Ms = 3/2 is zero. This tells us that L = 0 for the ground state of phosphorus. The final step is to figure out what J should be. Luckily, there is only one choice in this case, J = 3/2. Consequently, the ground state for phosphorus is

~

(1.7)

(c) As we have seen in part (b), 4 S3 ; 2 is the only possible term when the electrons have total spin S = 3/2. But three electrons can also have total spin S = 1/2, so there are additional, higher-energy tenns possible for the given configuration. Unlike the S = 3/2 spin wavefunctions which are exchange symmetric, the S = 1/2 spin wavefunctions possess no definite exchange symmetry, so the approach we used in part (b) in which we first considered the exchange symmetry of the spin wavefunctions and then the exchange symmetry of the orbital wavefunctions fails. In order to construct totally antisymmetric wavefunctions for the three electrons

ATOMIC STRUCTURE

6 TABLE 1.1

All possible single particle states for the ground state configuration of phosphorus grouped according to Ah and Ms. For completeness, we have written out all the states, but because of the symmetry between the AfL and -AJ L states and the Ms and - Ms states, only one comer of the chart is actually necessary for specifying all the states.

Ms

Ah

=2

M1., = 1

+!

M 1. = 0

M1., = -2

M1, = -1

(l+)(o+)(-1+)

(1 + )(o+ )( 1-)

+½

(l+)(-1+)(1-)

(-1+)(0+)(1-)

(1 + )(o+ }(o-)

(-1+)(1+)(0-)

(l+)(-l+)(-1-)

(-1+)(0+)(-1-)

( -1 + )(O+ )(0-)

(1 + )(o+ )(-1 - ) (l+)(o-

-½

)(1-)

(l+)(-1-)(1-) co+

H1-

(-1+)(0-)(1-) }(o- >

(l+)(-1-)(0-)

(-1+)(1-)(-1-)

c-1

+ )(o- )( -1 - >

(0+)(-1-)(0-)

(0+)(1-)(-1-)

-~3

(1-)(0-)(-1-)

for these higher-energy terms, we must consider products of spatial and spin states for each individual electron. As pointed out in the discussion surrounding Eq. ( 1.5), in order to construct a totally antisymmetric wavefunction, one needs at least as many individual electron states as there are electrons. To determine the additional terms, we can write out all the possible states using the shorthand notation

(m?1~)a(mf1)b(m?1~)c, 8

which actually refers to the completely antisymmetric wavefunction built from the three states according to Eq. ( 1.5). To satisfy the Pauli exclusion principle, we require that no two of the states for the three electrons are the same. This notation is equivalent to the chart representation described in part (b) - the chart shown in Fig. I.I corresponds to (1+)(o+)(-1 +). Now we can make a table of all possible states grouped together by the projection of their total orbital angular momentum ML and the projection of their total spin Ms (Table l . l ). The states described in Table l. l are eigenstates of the operators Lz and Sz, and by forming appropriate linear combinations one can construct eigenstates of L 2 , S2 , Lz, and Sz. The states corresponding to a given term are eigenstates of the operators { L 2 , S2 , J 2 , Jz}. Since both sets of eigenstates form a complete basis for our system, we know that there must be the same number of eigenstates in each set. To figure out the additional terms possible for the ground state configuration, we begin with a stretched state (ML = Land Ms = S) for a particular term, and count off how many states are in that term. We continue this process until we account for all 20 states in Table 1.1. Let us begin with the term we already know about, 4 S. There are four states in this term (since J = 3/2), all with ML = 0, so that accounts for four states in the

EXCHANGE INTERACTION

7

= 0 column of our table. Next, we see that there is one state [( 1+) (o+) ( 1- ) ] with ML = 2 and Ms = 1/2. This is the stretched state of a 2 D term. The possible values of the total electronic angular momentum for the 2 D term are J = 5/2 and J = 3/2, so the 2 D term accounts for IO states (six with J = 5/2 and four with J = 3/2), two in each ML column, one corresponding to Ms = 1/2 and the other ML

with Ms = -1 /2 (since S = 1/2 for the 2 D term). There is another state with ML = l and Ms = 1/2 not accounted for by the 4 S or 2 D terms. This is the stretched state (the state with the maximum possible projection of all angular momenta along the quantization axis) for a 2 P term, which accounts for the six remaining states (for the 2 P term we have J = 3/2 and J = l /2). This covers all the terms:

14s,D, and 2

2

p -

I

( 1.8)

(d) As mentioned in the hint, to first order the spin-orbit interaction has opposite sign for electrons and holes. Since the p 3 configuration can be thought of as consisting of three holes or three electrons, we have that the energy splitting due to the spin-orbit interaction ~ELs satisfies

( 1.9) so

(1.10) To first order, the 2 P 1; 2 and the 2 P 3; 2 states have the same energy.

1.2 Exchange interaction In the nonrelativistic limit, the Hamiltonian describing the interaction of electrons with the nucleus is independent of the electron and nuclear spins. Nonetheless, as we have seen in Problem 1.1, energy levels in multi-electron atoms do depend on the spin state of the electrons. Consider a two-electron atom. The nonrelativistic Hamiltonian H is given by:

( I. 11) where

ze2)

2

Ho =

-

L i=l

(

li2

-V;.

2me

'

+-

Ti

( 1.12)

ATOMIC STRUCTURE

8

and e2

H1=1-r1

-1·

(1.13)

- r2

Here Ho contains the Coulomb attraction of the electrons to the nucleus and H1 describes the Coulomb repulsion between the electrons, where Ti is the position of the i-th electron, and lr1 - r2 I is the distance between the electrons. The overall electron wavefunction \JI is the product of a spatial wavefunction ·l/Jand a spin function X· Fermi statistics demands that \JI must be antisymmetric with respect to particle interchange for identical particles with half-integer spin. Thus if x is a triplet state (symmetric), then ,,µmust be antisymmetric; if X is a singlet state (antisymmetric), then 1/Jmust be symmetric. Thus the electron spin state dictates the symmetry of the spatial part of the wavefunction. It turns out that due to the Coulomb repulsion between the electrons (H 1), symmetric and antisymmetric spatial wavefunctions have different energies - this is called the exchange interaction [see, for example, Griffiths ( 1995) or Landau and Lifshitz (1977)]. (a) For the case where one electron is in the ground state and another electron is in an excited state with quantum numbers (n, l, m,), which spin state, the triplet or singlet, has higher energy? (b) As a simple illustration of the exchange interaction, consider two electrons in

a one-dimensional simple harmonic oscillator (SHO) potential. For the case where one of the electrons is in the ground state and the other is in the first excited state, calculate ((x2 - x1)2) for the triplet and singlet spin states, where x1 and x2 are the positions of the two electrons.

Solution (a) Consider the case where one electron is in the ground state ·l/)100 (ri) and (r 2 ). The symmetric 1/Js and antisymanother electron is in an excited state "Pnlm, metric 1Pa spatial wavefunctions are given by:

1

1/Js = y2[1P1oo{f1)· ·t/Jn1m,(f'2) + 1/J1oo(T2) · 1/Jnlm,(f'1)] 1

1Pa= J2 [1/J1oo{f1) · ·t/Jnlm, Vi) - 't/J1oo(f'2) · ·t/Jnlm, (1--:-i)] •

(1.14)

If the electrons are in 1Pa, they can never occupy the same position since if r1= 'r2

then 1Pa = 0. However, 1/Js# 0 if r 1 = r2 , so it is possible for the two electrons to be located at the same point. Thus it turns out that, on average, electrons in 'l/Jsare closer to one another than electrons in 1Pa• Therefore H 1 causes a state

EXCHANGE INTERACTION

9

with a symmetric spatial wavefunction to have a higher energy than one with an antisymmetric wavefunction, so the spin singlet states (x antisymmetric ---+ ·t/Jsymmetric) have higher energies than the triplet ones (x symmetric ---+ 't/Jantisymmetric). Based on this reasoning, we can plausibly argue that, in general, terms with higher total spin S have lower energy since they have more "antisymmetrical" spatial wavefunctions, leading to less spatial overlap of the electron wavefunctions. (b) We can illustrate this principle more explicitly by considering the expectation value ( (x2 - x 1)2) for two electrons in a ID simple harmonic oscillator potential. For the ID SHO we have energy eigenstates In)with energy eigenvalues /iw(n + 1/2). Suppose that one electron is in the ground state IO)and the other electron is in the first excited state 11).For electrons in the triplet spin states, the spatial wavefunction must be exchange antisymmetric ( 1.15) while for the singlet spin state, the spatial wavefunction must be exchange symmetric ( 1.16) We can evaluate the expectation value of the distance between the electrons by expressing the position operators in terms of the raising and lowering operators for the two electrons [see, for example, Griffiths (1995)]: ( 1.17) where

ailn\ = v'nIn- 1\ , a11n)i = Jn + 1 In+ l)i

.

( 1.18)

For the operator describing the distance between the electrons, we find

(xi - x2)2 = x~- 2x1x2 + x~,

( 1.19)

where, by using Eq. ( 1.17), we obtain ( 1.20) ( 1.21)

ATOMICSTRUCTURE

10

Using the above relations, it is straightforward to show that

((x2 - xi)

)triplet

31i = -rnw ,

(1.22)

)singlet

Ii = -rnw ·

( 1.23)

2

while

({x2 - xi)

2

Thus we see that, indeed, electrons in the triplet spin states are, on average, significantly further apart than those in the singlet states.

1.3 Spin-orbit interaction Beyond the electrostatic attraction between the electrons and the nucleus and the electrostatic repulsion between the electrons (Problems 1.1 and 1.2), the next most important cause of energy level splittings in low Z atoms is the relativistic effects, which are responsible for what is known as the fine structure of atomic spectra. For low-lying states in hydrogen, the fine structure splitting of the energy levels is a factor of rv o:2 smaller than the Bohr energies [Eq. ( 1. 1)]. There are two causes of this splitting: ( 1) the interaction of the magnetic moment of the electrons with the effective magnetic field the electrons see due to their motion around the nucleus, and (2) relativistic corrections to the kinetic and potential energies of the electrons. In the following problem we consider only the splitting due to the spin-orbit interaction (I), which is the principal cause of fine-structure splitting for heavier atoms. Consider the term 5 D for a multi-electron atom. (a) What are the possible values of J?

(b) Show the splitting of the 5 D term due to the spin-orbit interaction on an energy level diagram. Indicate the value of J and the energy for each level in tenns of A, where the spin-orbit interaction is described by the Hamiltonian 4

H

I

= AL.... · S.... .

( 1.24)

4

In a multi-electron atom, the ..single electron" spin-orbit coupling (meaning that the interaction between one electron's spin and another's orbital motion is ignored, which is a good approximation for elements located in the middle and end of the periodic system) is given by

Assuming that the spin-orbit coupling is small compared to the electrostatic repulsion between the electrons, the Russell-Saunders coupling scheme (see Problem 1.1) is still valid and, to first order, we

11

SPIN-ORBIT INTERACTION

Assume A is positive, as is the case for subshells which are less than half-filled. (c) What effect does the spin-orbit interaction have on the "center of gravity" (the mean perturbation of all the states of the term)?

Hints For part (c), you may find useful the following formulae for summations (which can be proved by induction): ( 1.25)

N

LJ

2

= !N(N + 1)(2N + 1),

J=O

( 1.26)

6

and N

L13= ~N2(N

+ 1)2.

( 1.27)

J=O

Solution (a) Here we use the general rule for addition of angular momenta

IL - SI< J < L + s '

l + S = f: ( 1.28)

which is known as the triangle inequality. Using ( 1.28), we find that the possible values of J are:

I

5

D: S

= 2,

L

=2

--+

J

= 4, 3, 2, 1, 0.

( 1.29)

(b) For the total angular momentum J we have the relation:

( 1.30) need only calculate the diagonal matrix elements (L, S, J, MJIH'IL, S, J, !v/J). Seeking the overall shift of an atomic state with total orbital angular momentum l and total spin S, we note that the average orbital angular momentum f of an electron in such a state is ex l, and the average spin sis ex § (a consequence of the Wigner-Eckart theorem, see Appendix F), allowing us to write H' as is done in Eq. ( 1.24) [see, for example, Condon and Shortley ( 1970) or Landau and Lifshitz ( 1977)].

ATOMIC STRUCTURE

12

4A

----D

s 4

0 -----------------------------JA ''

\

'

-5A -6A

'D2 'D 1 SD 0

FIG. 1.2 Splitting of the 5 D term due to the spin-orbit interaction H'

set Ii

= Al · S, with A > 0. We

= 1 for convenience.

and therefore, by squaring both sides, we have

J2

= S2 + L 2 + 2£ · S .

(1.31)

Therefore the spin-orbit Hamiltonian ( 1.24) is

H'

= A (J2 2

52 - L2) .

(1.32)

The states described by the spectral terms 28 + 1LJ are eigenstates of J 2 , S2 , and L 2 • Therefore, they are also eigenstates of H' with eigenvalues

AE

=

Ali 2 T[J(J

+ 1) -

S(S

+ 1) - L(L + 1)].

(1.33)

Using Eq. ( 1.33) and setting h = 1 we can evaluate the energy shifts for states with various J's with S = 2, L = 2:

~E(J

= 4) = 4A = 3) = 0

( 1.34)

~E(J ~E(J = 2) = -3A

(1.35)

= 1) = -5A

(1.37)

~E(J

~E(J = 0) = -6A.

(1.36) (1.38)

The energy level structure of this system is shown in Fig. 1.2. Note that the energy difference between adjacent components is given by

tlE(J)

- ~E(J - 1) = AJ.

This formula is known as the Lande interval rule.

(1.39)

HYPERFINE STRUCTURE AND ZEEMAN EFFECT IN HYDR 1) due to the larger nuclear charge. Hydrogenic ions are of interest for precision experiments testing quantum electrodynamics (Silver 200 I), measuring the mass of the electron (Quint 200 I ), detennining the fine structure constant (Quint 200 I), and testing the Standard Model of electroweak interactions (Zolotorev and Budker 1997), to name a few. For hydrogenic ions with nuclear charge Z, find the scaling with Z of: (a) the expectation values of r, I/ r, and I/ r 3 , where r is the distance of the electron from the nucleus, (b) the expectation value of the potential energy V,

(c) the total energy E, (d) the probability to find the electron at the origin, (e)

l1P(r= 0)12 ,

l¾l/i(r = 0)12,

(f) the fine structure energy splitting (see Problem 1.3), and

(g) the hyperfine structure energy splitting due to the magnetic dipole moment of the nucleus (see Problem I .4). In this part, neglect the nonsystematic dependence of the nuclear dipole moments on Z.

Hints You should not have to use any explicit wavefunctions - just consider the dimensions of the quantities of interest.

19

HYDROGENIC IONS

Solution First, let us consider how the natural length scale of hydrogen, the Bohr radius ao = h 2 / me 2 , compares to the natural length scale of a hydrogenic ion. The Hamiltonian for a hydrogenic ion is

H

=-

n2 v'2 - ze2 .

If we replace r with p

=r/Z

(1.71)

r

2m

in the Hamiltonian (I. 71 ), taking into account that ( 1.72)

we have

H

= z2(-

( 1.73)

/i2 v'2 - e2 ) . P p

2m,

Thus the Hamiltonian for a hydrogenic ion can be put into one-to-one correspon1 dence with the Hamiltonian for hydrogen by rescaling r by a factor of , so the natural length scale for a hydrogenic ion is

z-

a=

ao

( 1.74)

z.

One can also see from Eq. (l.73) that the total energy scales as Z 2 compared to hydrogen, which is the answer to part (c). (a) As mentioned above, a

=

/Z

is the only length scale for the system. Therefore any quantity with units [length]n must scale as Thus, for any n, a0

z-n.

( I. 75)

(b) The potential energy of an electron in a hydrogenic ion is

V(r) and, based on part (a), since r-

1

=

Ze 2 -r

,

( 1.76)

ex Z, it is clear that ( I. 77)

(c) Although we have already seen from Eq. (l.73) that (E) ex Z 2 , we can also obtain this result by relating the total energy to the potential energy. According to

ATOMIC STRUCTURE

20

the virial theorem, for two particles interacting via a central, conservative potential V (r) = Crn, the expectation value of the kinetic energy (T) is given by

(T)

= n (V).

( 1.78)

2

For the electrostatic attraction between the nucleus and the electron, n Therefore

1

= (T) + (V) = 2(V)

(E)

,

=

-1.

( 1.79)

so we find ( 1.80)

(d) Since the hydrogenic wavefunctions are nonnalized, 0 [

and it is apparent that

2 l1P(r)l d3r= 1 ,

l1P( r = 0) 12 has the dimensions [length)- 3 • Thus

I11P(r= 0)1 2

(e)

( 1.81)

l¾1P(r= 0)12has units of (lengthJ-

a

5,

3

•

I

(1.82)

so evidently

= 0)

-'lp(r 8r

ex:Z

2

ex Z 5

( 1.83)

•

(0 From the point-of-view of the electron, the nucleus of charge Z is orbiting around it with velocity v ~ Zoe (see Appendix A). Since the electric field f, due to the nucleus is

=

f,

Ze r2

( 1.84)

'

the magnetic field B due to the relative motion of the electron and nucleus is

B=

Vx C

£

Z 2 oe ,..._, 2 r

""-J

__

.

( 1.85)

GEONIUM

21

This magnetic field interacts with the spin magnetic dipole moment of the electron to induce an energy shift ( 1.86)

Since the expectation value of 1/ r 2 scales as Z 2 , the fine-structure splitting scales as ( 1.87) (g) In part (a) of Problem 1.4, we saw that the hyperfine energy splitting ~Ehr for an s state is ao '

(1.204)


and choosing the integration surface outside the boundaries of the system (where the current is zero), we see that the integral ( I .293) vanishes. On the other hand~ we also have 0

=JV

m(rirk}m)d

3

r

=

J(

t5mirk}m

+ rit5mk}m + Tirk VJ)d

3

r.

( 1.294)

The last term in the integrand of ( I .294) vanishes because the divergence of the current is zero in a steady state. The other two terms correspond to the symmetric combination in Eq. ( 1.291). We now consider the antisymmetric part of the integral in Eq. ( I .291 ): (1.295)

Because (1.296)

we recognize in Eq.( I .295) the familiar expression for the vector potential from a magnetic dipole m A~I) i

(ii)= 1TI,R3 ii, X

( 1.297)

where the magnetic dipole moment is defined as

( I .298) (d) Finally, we tum to the second-order terms in the muJtipole expansion of the vector potential ( I .288) that will yield us the anapole moment: ( 1.299) Show that all symmetric terms in Eq. ( I .299) vanish.

Solution The integrand in Eq. ( 1.299) is a third-order tensor (i.e., a tensor resulting from combining three vectors) that can generally be decomposed into irreducible tensors of ranks 3, 2, I , and 0, the first being totally symmetric with respect to the

69

THE ANAPOLE MOMENT (T)

components of the constituent vectors and the last being completely antisymmetric. 23 We now use a trick similar to that used in Eq. ( J .294) to show that the symmetric part of the tensor integrates to zero: 0=

J

Vm(rirkrdm)d

3

r

J =J =

3

(tSmiTkTdm+ ritSmkTdm+ TirktSmdm)d r

(rkrdi

+ rirdk + rirk}1)d3r.

(J.300)

The remaining parts of the tensor in the integrand of Eq.( J .299) are antisymmetric rank-two tensors and rank-one tensors (i.e., vectors). 24 (e) Show that the antisymmetric rank-two part of the second order tenn in the vector potential expansion violates T-invariance.

Solution Note that aJJ of the remaining parts of the tensor in the integrand of ~q.( l .29~) change sign under both spatial inversion (transforming --+ -rand j --+ - j) and time reversal (transforming --+ rand --+ -]). By the Wigner-Eckart theorem (Appendix F), any second-rank tensor characteristic of the system (in this case, nucleus) with total angular momentum I has to be proportional to the onlX irreducible second-rank tensor that can be constructed out of the components of/:

r

J

r

(J.301) which is even under both spatial inversion and time reversal. Therefore, the existence of the second-rank moment (called the magneticquadrupolemoment) would violate both parity (P) and time-reversal (T) invariance. In this problem, we restrict ourselves to T-conserving moments and wilJ not further consider the magnetic quadrupole. 23

In order to see that the irreducible rank-three tensor is symmetric, it is possible to think of each of the constituent vectors (each described by three independent coordinates) as spin-one objects, and the irreducible rank-three tensor as a spin-three object. It is clear that to get a spin-three by combining three spin-ones, it is necessary to construct a totally symmetric wavefunction. For rank-zero, the only scalar that can be built out of the three vectors v1-a is their mixed product v1· (v2 x va) = (vt)i(v2) 1 (va)kfiJk, where fiJk is the totally antisymmetric (Levi-Civita) tensor. With two of the constituent vectors being the same, the rank-zero term vanishes. 24 A general third-order tensor (27 independent components) is decomposed into one irreducible rank-three tensor (7 components), two irreducible rank-two tensors (2 x 5 components), three vectors (3 x 3 components), and a scalar (one component; see, for example, Varshalovich et al. ( 1988), Chapter 3.2.2); however in the present case the number of possible tensors is reduced because two of the constituent vectors are the same and only structures symmetric in the corresponding components are allowed.

ATOMIC STRUCfURE

70

(f) Show that the remaining rank-one tensor in the second-order term of the vector potential expansion is a vector proportional to r 2

J.

Solution Let us now tum to the vectors that can be constructed out of TiTk)L· The only possibilities are: r 2 and r(r •J).It turns out that integrals over these two vectors are not independent. To see this, note that there is no summation over indices in Eq. ( 1.300), so the identity is valid for any set of components. If so, it is also true for sums of such components. In particular, the sum (called the contraction of the tensor over two indices) gives

J

Thus, it is sufficient to only consider the r 2J structure. (g) Using the above results, prove that the second order term in the expansion the vector potential is given by

of

(1.303) where Vp is a dual vector for the antisymmetric second-order tensor (see, for example, Arfken (1985), Chapter 3.4):

Jkri

-

Jirk

(1.304) This form explicitly accounts for the antisymmetry. The explicit expression of the components of V is found by contracting both sides of Eq. ( 1.304) with Ekiq and taking into account that

( 1.305)

( 1.306)

Solution Let us return to Eq. ( 1.299) and transform it taking into account Eq. ( 1.300) and the fact that the tensor (v'k V,-k)is symmetric in indices k and l. First, from Eq.

71

THE ANAPOLE MOMENT (T)

( 1.300) we have: ( J .307)

Substituting this into Eq. ( 1.299), we have:

( 1.308)

k)

where in the last expression the symmetry of (Vk "v1 is used. One can combine Eqs. ( 1.299) and ( 1.308) (by adding ( 1.299) and ( 1.308) divided by two) to obtain a combination in the integrand that is antisymmetric in indices i and k: ( 1.309) Using Eq. ( 1.304) in Eq. ( 1.309), we obtain the desired result:

( 1.3 J0)

(h) The term

.J;2>( R)corresponds

to the vector potential due to the anapole

moment. Use Eq. ( 1.310) to prove Eq. ( 1.287).

Solution In order to complete the expansion of the integrand into irreducible structures, we can split the tensor Vpr,into its symmetric and antisymmetric parts. The symmetric part is a tensor of rank two and corresponds to the magnetic quadrupole moment. As we have discussed above, this moment violates time-reversal invariance and we do not consider it in this problem. We now tum to the antisymmetric part of Vpr, which we write, in analogy with Eq. ( 1.304), as

1 1 2(Vpr1 - Vtrp) = fptqWq, 2

(l.311)

Substituting this into Eq. ( 1.310), and using the identity (l.312)

ATOMIC STRUCTURE

72

we write (1.313)

We see that we have reduced the expression for the vector potential to the f onn that depends only on a single vector characteristic of the system, the integral of W.It is clear that it must be proportional to the anapole moment ( J .286). In order to see this explicitly, we use Eq. ( J .306) and its analog for the components of W and write:

j

~d 3r

= ~fpli = ¾fpli

J J

3 (Vpr, - Vtrp)d r

3 [Ekqp(jkrq- }qrk)r, - Ekql{jkrq - }qrk)rpJd r ( 1.314)

where once again we have used the identity (1.312). The expression (1.314) can be simplified with the help of Eq. ( 1.302): (1.315)

where in the latter equality we have substituted the definition ( 1.286). We are now ready to substitute the result ( 1.315) into the expression for the vector potential ( 1.313):

(1.316) The first term in Eq. ( 1.316) contains

(1.317) The last equality is just the statement of the Laplace equation for the scalar potential of a point charge. Let us now discuss the second term in Eq. ( 1.316). It is proportional to

( vkv,~}5ilak = vi(a-V ~),

( 1.318)

i.e., a gradient of a scalar function. Recall, however, that the vector potential is not unique: it is defined up to a gradient of an arbitrary scalar function; addition of such a function is called gauge transformation [see, for example, Landau and

THE ANAPOLE MOMENT (T)

73

Lifshitz ( 1987)]. 25 In fact, our starting point, Eq. ( 1.288), is written in a particular Coulomb gauge for which V •A = 0. Skipping the second term in Eq. ( 1.316) because it can be eliminated by a gauge transformation and employing Eq. ( 1.317), we obtain the sought-for Eq. ( 1.287).

25

This is because the observables are fields rather than potentials. To find the magnetic field, we take the curl of A. Because the curl of a gradient is zero, the field is invariant with respect to a gauge transformation.

2 ATOMS IN EXTERNAL FIELDS 2.1

Electric polarizability of the hydrogen ground state

This is a classic problem concerning a hydrogen atom in the ground state immersed a uniform electric field l. To second order in the electric field, the shift of the ground-state energy is (2.1)

where o is the polarizability. Estimate the polarizability o for the hydrogen ground state. For the purposes of this estimate, one may neglect the fine (Problem I .3) and hyperfine (Problem 1.4) structure, i.e., ignore effects associated with the electron spin. Compare this result to the polarizability of a classical conducting sphere of radius ao. Measurements of electric polarizabilities have been used extensively to elucidate atomic and molecular structure since they are sensitive probes of the electronic wavefunctions far from the nucleus.

Hint For an estimate of the polarizability, we can say that Ef) - E~o)~ EI where

Et>are the unperturbed

0

0

) -

E~ l,

hydrogen energies.

Solution The Hamiltonian describing this system is given by H=

(2.2)

Ho+H1,

where p2 Ho=--2m

e2

r

(2.3)

ATOMS IN EXTERNAL FIELDS

76

is the Hamiltonian for a free hydrogen atom, and H1 is the perturbing Hamiltonian due to the appJied electric field. Orienting the quantization axis (z) along the electric field gives

H1

= -l- l = e£z = e£rcos8.

(2.4)

Since H 1 only connects states of opposite parity, there is no first-order

shiF't

(~E(I)),

~Ej

1

)

= (1, 0,

0IHi/1, 0, 0) = 0,

(2.5)

where In, l, m) denotes the unperturbed state with principal quantum number n't orbital angular momentum quantum number l, and projection of orbital angular momentum along the z-axis mli. Therefore we must use second-order perturbation theory, where the energy shifts are given by 1

L

LlEf) =

l(n, l, mlH1 II, 0, 0)/

2

(2.6)

EiO) - EAO)

n,l,m=,61,0,0

Therefore, based on Eqs. (2. I) and (2.6), the polarizability a can be found from

/(n, l, m/H1 /1,0, 0) /2 E[°) - E~o)

(2.7)

Using the approximation suggested in the hint, namely that Ef 0>- E~o) ~ E}°) _ (0)

E2

,

we have 1

2a£

2

~

1

(0) E2 -

~

(0) ~ E1 n,l,m

/(n,l,m/H1/l,0,0)/

2

,

(2.8)

where (in order to make later use of the completeness relation) we have included in the sum the n = 1, l = 0, m = 0 state, allowed since, as noted in Eq. (2.5), (1, 0, 0IH1/l, 0, 0) = 0. Now consider the sum

L

/(n,l,m/Hi/1,0,0)/

n,l,m

2

= (l,0,0/H1

(L

/n,l,m)(n,l,m/)H1/1,0,0)

n,l,m

= (1, 0, 0/H;/1, 0, 0)

,

(2.9)

Although the electric dipole selection rules for a z-oriented electric field (~l = ± 1, ~m = 0) imply that many terms in the sum (2.6) are zero, we keep them in order to make explicit use of the completeness relation later in the calculation. .

1

ELECTRIC POLARIZABILITY OF THE HYDROGEN GROUND STATE

77

where we have made use of the completeness relation

,E /n, l, m)(n, l, m/ = 1 .

(2. 10)

n.l.m

Based on Eqs. (2.8) and (2.9), we have 1 "2

-oc. 2

(l,0,0/Hf/1,0,0) (0) {O) E2 - El

::::::

.

(2. I J)

The ground state wavefunction for hydrogen is .,/, ( ) _ 'PIOO r -

1 r=

_

v1ra

1 3/2 0

e

-r/ao

'

(2. J2)

so the matrix element in (2.1 I) is given by the integral

The difference in energy between the ground and first excited states of hydrogen is

Et> - E(O) = 1

-~

(~ -

2a0 4

1)= Bao

3e2 .

(2.14)

Employing Eqs. (2. J3) and (2. I 4) in (2. J J), we find an estimate for the polarizability of the ground state of hydrogen:

o ~

16 -a3 ~ 0. 79 x 3

10-

24

cm 3

•

(2. I 5)

Because of the approximation made in Eq. (2.8), this is in fact an upper limit on the polarizability of the hydrogen ground state. It turns out that this is close to the exact value [see, for example, Bethe and SaJpeter (J 977)]:

a

9 = -a3 ~ 0.67 x 2

10- 24 cm 3

.

(2. I 6)

It is also interesting to compare a to the polarizability of a classical conducting sphere of radius a 0 • The electric dipole moment i of the sphere is related to the

78

ATOMS IN EXTERNAL AELDS

applied electric field f. by the poJarizabiJity 2 (2. J7)

i=al.

The electrons on the conductor wiJJ arrange themselves so that the electric field on the sphere is perpendicular to the surf ace (otherwise there would be a tangential force on the free charges). Outside the sphere, we can assume that t~e field due to the conductor is a dipole field. 3 The electric field f.d from a dipole d is (2.18)

At the surface of the sphere (r = a 0 ) the component of l + ld not along f must vanish, so, for example, on the equator of the sphere (where J. f = 0) we have (2.19)

Thus the poJarizability of a classical conducting sphere is

(2.20)

2.2

Polarizabilities for highly excited atomic states

Discuss the general scaling of electric polarizabilities with the principal quantum number n (for highly excited states).

Solution In the presence of an external electric field, an atomic energy level k shifts according to

(2.21) 2

The connection between Eq. (2.17) and (2. I) follows from

E=-

r.d·de - -, Lr.aede, ,=--a:e L 0

3

1

=-

0

2

.

2

This_can be guessed from the high symmetry of the configuration: the conductor is spherically symmet,:ic and t~ electric field has only a direction. This guess indeed satisfies Poisson's equation, so by umqueness 1t must be the correct solution [see, for example, Jackson ( 1975)].

79

USING STARK SHIFTS 10 MEASURE ELECTRIC FIELDS

where the sum is taken over all states i coupled to k by the electric dipole operator d, and the dipole matrix element dik (with quantization axis z) is given by dik

= (ifdfk) = -(ifezfk) = -(ifercos8lk).

(2.22)

One can expect that the sum (2.21) is dominated by tenns corresponding to levels with close values of the principal quantum number n: ni ~ nk, since for these levels the energy denominators are smallest and dipole matrix elements are large because of good overlap between wavefunctions. The dipole matrix elements scale as the radius of the electron's orbits which go as n 2 • This can be seen by comparing the expectation value of the energy determined by the virial theorem [Eq. ( 1.79)]:

(En)

e

2

/1)

= -2 \ ;:- ,

(2.23)

with the Bohr formula [Eq. ( J. J)]

(2.24) fhe scaling of the energy denominators can be estimated from the fact that En {h

and sweeps the field to

CLASSICAL MODEL OF MAGNETIC RESONANCE

The ref ore, from Eqs. (2.57) and (2.58), we find that the frequency

J and 11precess

87

around

Boat

(2.59)

v in the lab frame is related to its evolution in a velocity waccording to the classical formula [see, for

(b) The evolution of a vector

frame rotating with angular example, Marion and Thornton (1995)]:

dvl

-

dt

=-

lab

dvl dt

rotating

_ _

(2.60)

+wxv.

In the lab frame, we know that J evolves according to Eq. (2.56), so using Eq. (2.60), we find that in the rotating frame

d]I dt

rotating

= ,J x

[Bo + ~] .

(2.61)

1

If we compare Eq. (2.61) to Eq. (2.56), we see ~hat in the rot~ting f_!ame,the apparent magnetic field acting on the dipole is not Bo but rather Bo

+ Bf,

where

def>

FIG.

2.2 The torque on a magnetic dipole ji, precession of the magnetic dipole (see text).

= ,J

due to a static magnetic field

Boleads

to

ATOMS IN EXTERNAL FIELDS

88

Lab Frame

Rotating Frame

I

I

+ro FIG.

2.3 Magnetic fields in the lab frame and a frame rotating with

BI is a "fictitious"

B.L ( t) •

field given by (2.62)

Note that if we choose

w= - 1 Bo,the dipole

appears to be stationary.

(c) If we go into a frame which rotates so that R.1 appears to be stationary, as we have seen in part (b), the apparent magnetic field in the z-direction has magnitude Bo - w / 1 (Fig. 2.3). Thus in the rotating frame, the dipole "sees" an effective with magnitude static magnetic field Reff

Beff

The angle 'P between

= ✓ Bl + ( Bo -

Reff and the z-axis

2

~)

.

(2.63)

is found from

(2.64) The dipole ji, precesses about

Reff at a frequency (2.65)

As can be see~ from the diagram in Fig. 2.4, the amplitude of the change in the projection of J on the z-axis is given by

(2.66)

CLASSICAL MODEL OF MAGNETIC RESONANCE

89

z

l --------

2~J~.. 1 ·------

FIG.

2.4 Precession about the effective magnetic field in the rotating frame.

Thus (after a few trigonometric substitutions and some algebra) the overall timedependence of Jz is found to be

(2.67)

where n is given by Eq. (2.65) and sin 2 c.pis given by Eq. (2.66). We can see from Eqs. (2.66) and (2.65) that when w = nL, sin 2 cp = 1 and n = ,B.1. According to Eq. (2.67), this means that Jz oscillates between ±J at frequency n. This is because when B.1 rotates at nL, in the rotating frame it appears that there is no field along z, so the dipole precesses about B .1 at the appropriate Larmor frequency , B .l · This is the essence of magnetic resonance. (d) Once again, let us work in a frame co-rotating with

B.1(t). In this frame,

the magnetic dipoles see an effective field Reff with magnitude given by Eq. (2.63) pointing at an angle c.p[given by Eq. (2.64)] from the direction of Bo(see Fig. 2.3). If one slowly sweeps the frequency w, the magnetic dipoles adiabatically follow the effective magnetic field. The effective magnetic field, sketched as a function of time in Fig. 2.5, rotates from pointing in the direction of Boto pointing opposite to Bo. This tells us that the procedure flips all the spins. What are the conditions necessary to ensure adiabaticity? ~

ATOMSIN EXTERNAL FIELDS

90

t 'y

i .······►

1·····►

~

B.l.

•

•

Time

(t)>>!l

(1) ~ 8t Beff

Assuming a constant sweep rate of the frequency satisfied as long as

.

w, the

(2.68) adiabatic condition is

(2.69)

2.7

Energy level shifts due to oscillating fields (T)

In this tutorial we analyze the shift of atomic energy levels due to an electric field which varies in time, known as the AC Stark effect. The AC Stark effect is an invaluable tool for laser trapping and cooling and is a basic mechanism behind many nonlinear optical effects. We will also see that the case of a time-varying magnetic field (AC Zeeman effect) is completely analogous to the AC Stark effect when there is a strong, static leading field Boand a weak, oscillating field transverse to Boapplied to an atomic system. 5 5

There is a fundamental difference between the Zeeman and Stark effects in that magnetic fields lead to first-order energy shifts while electric fields cause shifts only in second order. However, in the presence of a strong leading magnetic field, additional weak transverse fields only lead to second-order energy shifts, making this case fully analogous to the Stark effect.

ENERGY LEVEL SHIFfS DUE TO OSCILLATING FIELDS (T)

91

Consider an atom with two states la) and lb) separated in energy by wo(in this problem we use units where Ii = 1). The states are coupled by the electric dipole operator d and a sinusoidally varying electric field 0 sin Wm t is applied to the atom. Throughout this problem, we neglect the linewidths of the energy levels, i.e., the relaxation rate for the states is much less than all other relevant frequencies in the problem ( lwo- Wm I,Wm, w 0 ). The amplitude eoof the applied field is sufficiently small so that the effect of the electric field may be treated as a weak perturbation, i.e., deo OL and w < OL [see Problem 2.7, Eqs. (2.102) and (2.103)]. The spectral distribution of the motional field is symmetric with respect to w = 0, not OL, thus it causes AC Zeeman shifts between the two sublevels due to the "uncompensated" part of the spectrum shown in Figure 2.7. The uncompensated portion of the spectrum contains"' 20L/(21r /re) of the total power contained in the whole spectrum. The AC Zeeman shift in this case can be estimated from the square of the effective Zeeman shift due to the uncompensated field and from the effective detuning (1r /re), using the results of Problem 2.7:

(2.144)

A more precise calculation based on the density matrix formalism (Appendix G) gives a result that is a factor of 1r2 /9 greater than our estimate [Eq. (2.144)], see Khriplovich and Lamoreaux (1997). Note that the sign of the shift is the same in both limiting cases. Also note the opposite dependence on Bo in Eqs. (2.143) and (2.144). In the limit where nL re

___

v

T;,ump I

19> F IG. 3 . 5 Level

diagram for the two-level system considered in part ( a).

condition (3.157) ensures that the intensity of the light field does not significantly change as the light propagates through the sam~le and, ~s long. as all dimensions of the atomic sample are similarly small, that high atomic density effects such as radiation trappingg are not important. Additionally we assume that the average spacing between the atoms n - 1/ 3 is considerably larger than the wavelength of the light A. This allows us to ignore effects that involve cooperative behavior of the atoms [such as Dicke superradiance (Dicke 1954), see Problem 3.14]. (a) Consider two-level stationary atoms for which the only source of line broadening is the spontaneous decay of the upper state le) back to the lower state lg) (Fig. 3.5). Calculate the saturation parameter "' for the lg) __,..le) transition for narrow-band (monochromatic) incident light, and find the dependence of the fluorescence intensity on ""·

Solution !he excitation rater pump (we can think of the light effectively ''pumping" the atoms mto the excited state) is given by Eq. (3. I 32) from Problem 3.4:

r

-

pump -

d2£2 --

0

10

(3.158)

where~ is the di~ole matrix element (eldlg) between the states, £ 0 is the amplitude is the spontaneous decay rate of le) to lg), and we have of the hght electric field,

,o

9

If the atomic density is sufficiently high, there can be a significant probability that spontaneous) emitted photons are re-absorbed. Thus the photons must diffuse out of the atomic sample h" ~ affects, for example, measurements of excited state Iifeti mes. See, for example, Corney ( tc _

1988;

153

SATURATIONPARAMETERS(T)

set h

= l. The relaxation

rate in this problem is ,o, so from (3.156) we have

(3.159) The fluorescence intensity IF is proportional to the number of atoms in the excited state Ne multiplied by the spontaneous decay rate , 0 • To find the population of the upper state we can write rate equations for the number of atoms in the excited state Ne and the number of atoms in the ground state Ng:

d:g= -r pumpNg+ho+

rpump)Ne,

(3. 160)

d:e= +rpumpNg- ho+

r pump)Ne.

(3.161)

We also know that Ne + Ng = N, 01 where N, 01 is the total number of atoms in the sample. We have included the pumping rate for both the lg) -+ le) transition and the le) --+ lg) transition because at sufficiently high light powers (K ~ 1), stimulated emission from the upper state becomes important compared to spontaneous emission. It is clear that the stimulated emission and absorption rates should be the same from time-reversal symmetry [this can also be seen from Einstein's famous argument involving an atomic gas in thermal equilibrium with a photon gas, which was used to derive the A and B coefficients; see, for example, Griffiths ( 1995) or Bransden and Joachain (1989)). In equilibrium, dN 9 /dt and dNe/dt are zero, and we find that K,

Ne

(3.162)

= l + 2K N101 ,

so the fluorescence intensity is proportional to K/(1

+ 2K) (Fig.

3.6).

(b) Now suppose we have a three-level system as shown in Fig. 3.7. The incident light is resonant with the lg) --+ le) transition and the excited state le) primarily decays to a metastable level Im,)at a rate , 0 • There is a slow relaxation rate ,rel (o) oc 10. z1= l(cosO X -

(3.182)

z12 = sin 2 0.

sinO Z).

Thus the overall intensity is J 10/1r>(O)oc sin 2 8. Now consider the u + emission. In a classical picture, such emission is produced by a dipole rotating (rather than oscillating) in the xy-plane. We can therefore ex~ct that the largest emission intensity is along z (8 = 0, 1r), while in the equatonal plane (8 = 1r /2), the emission should have one half of the intensity. (This is because the dipole's rotation can be decomposed into two orthogonal oscillations, and onl_yone of them is "seen" from the equatorial plane.) These expectations are_a~am _confirmed by the calculation. Using the polarization vector for the u + em1ss1onm the fonn £+ A

1 (x+iy = - v'2 A

O ")

(3.183)

,

we have

J~CT>(o) oc IY.i+l2

ieCT)( 8) oc ,8-i+ ...

2

1

= Y. _I_(x + iy") 2 - ! .

v'2

=

(3.184)

2 '

(cos8x-sin8z)·-(x+iy) 1

v'2

2

---

COS

2

2

8

(3.185)

ANGULAR DISTRIBUTION AND POLARIZATION OF ATOMIC FLUORESCENCE

,,,,

0.75

,/

'

I

''

\

I

\

I

\

I

\

I

0.5 0.25

I I

I I I

I

I

,I

0

i I

- 0.25

161

•

,,,,. -- ·- . \ ·

-·-·/

f

/

---·-

I

I

"}' /1

'I

- -- ·

/

I I

I I

-0. 5

I I

- 0.75

I

/

-0.75-0 .5-0.25

0

0.25 0.5 0.75

X

FIG.

3.12

Normalized angular distributions of fluorescence intensity for

1r

emission (dot-dashed

line), u emission (dashed line), and the overall distribution (solid line), which is isotropic in the present case.

The total intensity of the a-emission is

(3.186) To find the overall intensity of radiation in a given direction, we need to add the two contributions, lto/'r) (0) and lto/o-) ( 0), weighted by the total probability of the corresponding emissions ( 1/3 and 2/3, respectively). The result is independent of (}, meaning that the total light intensity is emitted isotropically. The normalized angular distributions for the 1r and a light and the overall isotropic distributions are shown in Fig. 3.12 (b) Even before any calculations, it is clear that for 0

= 0 or 1r, only the a light is

seen, and therefore light is completely circularly polarized. In the equatorial plane (8 = 1r /2), equal amounts of independent contributions of the vertically polarized 1r light and horizontally polarized a light are seen, so the light is unpolarized. For a general value of 0, a horizontally oriented polarizer (along y) will transmjt an intensity ex 1/2 from the a light, while a vertically oriented polarizer (along 0) will transmit an intensity oc (cos2 0)/2 from the a light and oc (sin 2 0)/2 from the

INTERACTIONOF ATOMSWITH LIGHT

162

1r light. Thus for the first Stokes parameter (Appendix D) we have: 2

2

= Ix - Iy _ 1 - cos (J - sin 8 = O. 81 Io 2

(3.187)

From symmetry, it is clear that S2 = 0 also. The linearly polarized 1r light cannot contribute to S3 • From the discussion in part (a) [in particular Eqs. (3.184) and (3.185) ], we also see that the vector amplitude of the u + light emitted in a given direction can be written in the form

i

cos (J A

1 (

- ../2fl- ../2 8 ex ../2 fl-

A)

icosfJ 8 ,

(3.188)

where we have removed the overall phase factor. By taking the scalar product of this amplitude with the amplitude vectors

,_,

€+

for the left-circular polarization

1 ( •(J") = - ../2 y +i

12

and

,_,

€_

(3.189)

A

1 (A = ../2 y -

"IJA)

(3.190)

1,

for the right-circular polarization, we find that 2

= I+ - [ _ __ (l_+_cos_lJ_)2_-_(_1_-_c_o_s_lJ)_ = cosfJ. 83 Io 4

(3.191)

This shows that light emitted with fJ = 0 is left-circularly polarized, while light emitted with fJ = 1r is right-circularly polarized. The degree of polarization is P = Icos IJIin agreement with our qualitative argument above.

3.9

Change in absorption due to optical pumping

For J = 1 -+ J' electric dipole (El) transitions (with J' = 0, 1, 2), find the relative changes in populations of the J = 1 Zeeman sublevels as a result of optical pumping with linearly polarized light (assume the quantization axis along the axis of light polarization, i.e., 1r-polarization). Assume "closed" transitions (i.e., atoms excited to the upper level can only decay back to the lower level), that the excitation light is on resonance, the medium is optically thin, and that, in order to simplify The circular polarization vectors i~ and i'_ are defined to be orthogonal to k, and we identify fl as the horizontal direction and -8 as the vertical direction to form an appropriate right-handed coordinate system [compare with Eq. (3.183)]. 12

CHANGE IN ABSORPTION DUE TO OPTICAL PUMPING

163

calculations, the optical pumping saturation parameter K (see Problem 3.7) is much less than one [assume a relaxation rate 1'rel

This result can be derived by expanding the initial and final states into the IJ, MJ)IJ, M 1 ) basis and applying rules for summation of combinations of the angular coefficients. Of course, the present problem can also be solved by applying the Clebsch-Gordan expansion without explicitly using the result (3.272).

Solution (a) From Eqs. (3.272) and (3.132),

W

_ ll(e,J'lldllg,J)j

eg -

r

2

e~(J,M1,l,OIJ',M~)2 2J'

fi2

+1

'

we find the following relative absorption rates:

lv,(0)) : F F

= 1 --+ F' = 0, = 1 --+ F' = 1,

W oc 2/3, W oc0. (3.273)

lv,(t)) : F F

= 1 --+ F' = 0, = 1 --+ F' = 1,

W oc 0, W oc 2/3.

Note that for the given choice of light polarization, lv,(0)) is a dark state for the 1 --+ 1 transition and Iv,(t)) is a dark state for the 1 -+ 0 transition.

OPTICAL PUMPING AND QUANTUM BEATS IN MERCURY

195

(b) If the final-state hyperfine structure is unresolved, in order to find the total

absorption rate, we need to add the absorption rates for the 1 ~ 1 and 1 ~ 0 transitions. We see that, in this case, the absorption does not change when the wavefunction is rotated by 1r /2 around the quantization axis. In fact, it is also straightforward to see from this result that the absorption rate is invariant with respect to an arbitrary rotation of the initial state around z! (c) The reason that absorption is invariant with respect to rotations of the atomic state when hyperfine structure is not resolved is that the total electronic angular momentum of the initial state is J = 1/2 and such a state cannot support polarization moments (Problem 9.7 and Appendix G) with rank K higher than 2J = 1. The polarization moments corresponding to the states l'l/J{O))and l'l/1(t))are of rank K = 0 (population) and K = 2 (alignment). When hyperfine structure is not resolved, we can neglect the nuclear-spin part of the wavefunction. However, since the electronic state cannot support alignment, it is impossible to detect the alignment of the initial state using light absorption. The general theorem can be formulated as follows. Suppose we have a state with electronic angular momentum J, nuclear spin I and total angular momentum F. We prepare any kind of polarization in this state and want to probe it using weak probe light tuned to a transition to some other state with J', for which the hyperfine structure is unresolved. Then, the signal is only sensitive to the initialstate polarization moments of rank O < K < min{2, 2J}. Here we have taken into account the fact that one-photon transitions are insensitive to polarization moments with "" greater than 2 (alignment) because photons have spin I. In order to detect polarization moments of rank K, the atomic states involved in the transition as well as the light field must be able to support polarization moments of rank "'· Because the spatial symmetry of an angular-momentum state is directly related to polarization moments (Problem 9. 7), there is a consequence of the theorem for aeman beats. In general, if an atom is undergoing Larmor precession in a magnetic field, in the probe signal, one can observe contributions (beats) at the Larmor frequency (due to orientation) and twice the Larmor frequency (due to alignment). However, when the final-state hyperfine structure is unresolved, no beats can be observed for J = 0 states and only beats at the Larmor frequency (but not twice the Larmor frequency) can be observed for J = 1/2 states.

3.20

Optical pumping and quantum beats in Mercury

This problem will provide an illustration of how nuclear spins can be optically polarized, even if the transitions' hyperfine structure is not spectrally resolved. It will also provide a nice demonstration of the phenomenon of hyperfine quantum beats [see Haroche (1976), for example]. Quantum beats occur when an atom is

196

INTERACfION OF ATOMSWITH LIGHT

put into a superposition of different eige~states ~f the_atomic Hamiltonian with distinct energies. In such a situation. light mteractmg with the atom exhibits oscillatory behavior at the frequency corresponding to the energy splitting between the eigenstates. The oscillatory behavior can be observed, for example, in the intensity of fluorescence emitted in a particular direction or in the polarization of light absorbed or emitted. The ground state of the mercury atom has total electronic angular momentum J = O, while the nuclear spin for 199 Hg is I = 1/2. Suppose we initially have unpolarized atoms, and we apply a short pulse of circularly polarized light tuned to resonance with a transition to an excited J' = 1 state. (a) What are the hyperfine-structure levels in the ground and the excited state? (b) What is the nuclear polarization (the average ~roject~on of the nuclear spin

on the direction of the circular polarization of the hght) nght after the excitation pulse? Give a qualitative explanation to this result. (c) What is the nuclear polarization as a function of time? (d) Explain how the results above pertain to optical pumping of nuclear spins. In parts (b)-(c), only consider the atoms in the excited electronic state.

Solution (a) The hyperfine structure of the states involved is shown in Fig. 3.27 along with the laser-induced transitions. Obviously, there is no hyperfine splitting in the ground state because J = 0. (b) Before the excitation pulse, we have an unpolarized ground state, i.e., an incoherent mixture of the ground state Zeeman sublevels. With left-circularly polarized light, the atoms in the IF= 1/2, MF = 1/2) sublevel can only be excited to the IF' = 3/2, MF' = 3/2) sublevel, which is an eigenstate of the Hamiltonian for the atom. Therefore, upon excitation, there are no quantum beats associated with this state since the atom is not in a superposition state. Note that the polarization of the nucleus is not affected by the transition - the nucleus is just a "spectator" to the process. Let us now consider the fate of the atoms that are initially in the IF = 1/2, MF = -1/2} sublevel. Such atoms are coherently excited to a superposition of IF' = 3/2, MF, = 1/2) and IF' = 1/2, MF, = 1/2). Let us evaluate the corresponding amplitudes in terms of the reduced matrix element of the transition (J'lldllJ}.To do this, we find the values of the reduced matrix elements (F'lldllF}

OPTICAL PUMPING AND QUANTUM BEATS IN MERCURY

197

F'= 3/:! 1/2

F = 1/2 -l/2

FIG.

3.27 199

=

Hyperfine structure and laser-induced transitions in a J

Hg (nuclear spin /

0 ~ J'

=

1 transition in

= 1/2).

according to Eq. (3.272) (see Problem 3.19),

((J' J)F'lldll(J I)F)

= (-l)J'+l+F+l

J(2F

(3.274)

+ 1)(2F' + 1) {

J' F

F' J

Il } (J'lldllJ),

which are -v'2(J'lldllJ)/3 for the case of F' = 1/2 and 2(J'lldllJ)/3 for the case of F' = 3/2. Using the Wigner-Eckart theorem [see Appendix F and Appendix I, Eq. (1.18)]

(F'MF'ldqlFJ\;fF)

= (-l)F'-M,,,,

MF'

( -

F'

l q

we find that the following superposition of the MF' the F = 1/2, MF= -1/2 state (see Problem 3.4):

11/J) = ~IF'=

3/2, MF' = 1/2)

+ ~IF'=

MF) (F'lldllF),

(3.275)

F

=

1/2 states is excited from

1/2, MF' = 1/2).

(3.276)

For further reference, we also write the wavefunction of the atoms excited from the IF = 1/2, MF = 1/2) state:

I~) = IF'= 3/2, AJF' = 3/2).

(3.277)

and Icp)are also Because the initial sublevels are incoherent, the wavefunctions 11/J) mutually incoherent.

198

INTERACl1ON OF ATOMS WITH LIGHT

Expression (3.276) can ~ decomposed into the uncoupled basis IMJ,, Mi) using Clebsch-Gordan coefficients

11/J)=

G+ D

11,-1/2)

+(

f -f)

10,1/2)

= 11,-1/2) ,

(3.278)

which shows explicitly that only the MJ, = 1 state for the electron is excited b the laser pulse, while the nucleus just "goes for the ride." y A similar decomposition for the wavefunction (3.277) is trivial:

lip) = 11,1/2).

(3.279)

From Eqs. (3.278) and (3.279), we see that the nucleus, being in an incoherent mixture of the M 1 = ±1/2 states with equal weight, remains unpolarized upon short-pulse excitation. (c) In order to find the time dependence of nuclear polarization, we go back to Eq. (3.276), and add time dependence of the eigenfunctions by introducing the hyperfine frequency w = (EF'=l/ 2 - EF'=I/2)/li corresponding to the energy interval between the two hyperfine states:

1

11/J(t)) = (- IF'= 3/2, MF' = 1/2)e-iwt

v'3

Ji = +D +

IF' = 1/2, MF,

where r is the lifetime of the J'

11/J(t))= [ Ge-iwt

(3.280) 2

= 1/2)] e-t/

T,

1 state. Correspondingly, Eq. (3.278) becomes

11,-1/2)

+(

f

e-iwt -

f)

IO,1/2)] e-t/2T,

(3.28 I) so that the average nuclear projection on the direction of the light circular polarization (taking into account also the atoms excited from the MF = 1/2 sublevel and assuming an initially equal distribution in the two ground state sublevels) for atoms in the excited state is:

~ (-½) + f e-iwt 2

(Mi)~ [ 1e-iwt

+

•

f G)+ ½] 2

•

e-t/T

(3.282) From this expression we see that nuclear polarization, while zero at t = O, is generally non-zero at later times. The oscillations of the polarization (quantum beats) are superimposed on the exponential decay of the upper state.

THOMSON SCATTERING

(d) This discussion shows that an optical-pumping cycle consisting of a h

199

excitation followed by spontaneous decay back to the ground state . s on-pulse increase population of the Af F = Ah = 1/ 2 state. Once there, the atoms (or · s the cisely, their nuclei) are "trapped" as subsequent additional cycles ca ' rnore Pre. nnot rern 0 them from the !vi1 = 1/ 2 state. Of course, th 1s conclusion is also tr ve 199 H ~ h · ue for co · · d h" · · g nuclei are optically ntinuous exc1tat1on, an t 1s 1s, m 1act, ow purnped • practice [see Happer (1972), for example]. Note that, while resolvin h •n structure is not necessary for efficient pumping, it is necessary that th! /1:r~ne frequency w is not much smaller than the spontaneous relaxation rate / YP rfine 1 Note that, considering the results of Problem 3.19, even though we h '· . . ave acco Illplished optical pumping of the ground J = 0 state, as long as the u hyperfine structure is unresolved, we can only probe total population of t~per- st ate state. In other words, we cannot optically detect that the ground state heground . as been polanzed.

3.21

Thomson scattering

Consider a free electron interacting with a light wave. Based on the cl . . wh1c . h 1· . 1s . d ue to t he ra d"1atton . of the acceleratingass1caJ model m 1gh t scattenng 1 tron derive an expression for the total scattering cross-section Neglect the e e~-I ' . . · reco 1 effects and the effects of the optical magnetic field. Formulate the conditions f · · to be va l"d or such approx1mat1on 1.

Solution The electric force on the electron is

F'= -el

cos wt,

(3.283)

where f. is the optical electric-field amplitude, w is the the frequency of the light, and we have chosen the phase of the wave arbitrarily. The ele~ron accelerates upon the action of this force and so produces an oscillating dipole d with the second time derivative ••

-

e

2~ L

d=-coswt m.

(3.284)

where m is the electron mass. According to the classical dipole-radiation formulae, the total power radiated by the oscillating dipole is given by [Panofsky and Phillips

200

INTERACTIONOF ATOMSWITH LIGHT

( 1962), Sect. 20-2; Landau and Lifshitz, ( 1987), Sect. 67] 2

p

:.:. 2

= 3c3 (d) ,

(3.285)

which, upon the substitution of Eq. (3.284) and integration over one period of the light oscillation gives

(3.286) By definition, the number of photons scattered per unit time by the electron is given by ~u, where ~ is the incident photon flux, and u is the sought-after scattering cross-section. In order to connect Eq. (3.286) with these quantities, we replace P with the number of photons scattered per unit time times the energy of a scattered photon, 1iw(which is the same as that of an incident photon, as long as we neglect recoil). Also, the average incident light intensity, liw,is given by the average Poynting vector

(3.287)

e

Solving Eq. (3.287) for 2 and making the corresponding substitutions into Eq. (3.286), we see that the Ii, factor cancels (after all, our derivation is classical) along with w, and we get

(3.288) where r 0 ~ 2.8 • 10- 13 cm is the classical radius of the electron. In the derivation of Eq. (3.288) we have neglected photon recoil. A char~cteristic magnitude of the electron's momentum change in a scattering process IS

(3.289) which leads to a change in the electron's kinetic energy of t::.p2

t::,.E= 2m

fi2w2

~ mc 2

·

(3.290)

As long as 1iw w 0 (i.e., A > O), and me(wo + w)Ll N > 2 . (3.301) 41re From Eq. (3.301 ), it is seen that for large~ on the order of the atomic frequency w 0 , the condition on the density can be re-written as (3.302) i.e., the density should be greater than the density of condensed matter, about 1024 atoms/cm 3 , which would mean that the system was no longer a vapor.20 Here we have used the fact that the energy difference between the atomic states !u.,;0 should be on the order of the Bohr energy e 2 / ao (see Appendix A). On the other hand, the density requirement [Eq. (3.301 )] is significantly relaxed as the frequency w approaches the resonance frequency wo. From Eqs. (3.301) and (3.302), we find that closer to resonance we require

N>]_~ rv

a~ wo .

(3.303)

Recall that for the refractive index to remain real we cannot operate too close to resonance, so there is a lower limit on ~ of several times the resonance width ,. At the densities N necessary to satisfy Eq. (3.303), , is dominated by pressure broadening, i.e., (3.304) where up is the pressure broadening cross-section and v is the thermal velocity of the atoms (for a typical room-temperature gas, pressure broadening dominates 20

Incidentally, there do exist solids with negative electric pennittivity. In addition, dense plasmas have been also produced that have e < 0.

210

INTERACflON OF ATOMS WITH LIGHT

when N apv ~ 21rx 1 GHz, where 1 GHz is th~ typical value of the D~ppler width for an optical transition). Re-writing the requirement (3.303) by settmg il rv ""f and using the expression (3.304) for 1 , we discover that the condition becomes independent of N:

1 UpV 1 > --"' a~ wo ·

(3.305)

The question now becomes whether or not the inequality (3.305) can be satisfied by typical atoms. Atomic pressure broadening cross-sections are usually in the range of 10- 15 - 10- 14 cm 2 (see Appendix A), so in fact it is possible for a room temperature gas (where v "' 104 cm/s) to achieve negative permittivities near resonance. For these conditions, the atomic density will be N '"" 1018 cm- 3 , six orders of magnitude smaller than in the far-detuned case, and c reaches negative values of order unity. Note, as Eq. (3.305) implies, that increasing the density past this point does not improve the situation. However, note that the condition (3.305) does depend on the atoms' velocity v, so by using laser cooling (where thermal velocities can reach ;S 1 cm/s) or cryogenic techniques, the required density to achieve negative permittivities can be further reduced. With the realization of the difficulties that arise when one is attempting to achieve an atomic vapor with negative electric permittivity, we now tum to the discussion of the magnetic permeability. Again, we will use a two-level model; however, in this case, the levels must be coupled by magnetic-dipole, rather than electric-dipole interaction. We can obtain the expression for the magnetic permittivity by analogy with Eq. (3.298) if we first rewrite it to introduce the transition dipole moment d as

c=l+

41rNd2 mea 02

1 2

w0

-

w2 - i 1 w

(3.306)

·

Then, it is clear that the permeability is of the form

41rNµ5 1 µ= 1 + mea 2 w~2 - w2 - i 'w 1 0

'

(3.307)

where µo is the Bohr magneton, w' and ,' are the frequency and width of the corresponding magnetic-dipole transition. 21 We can now easily obtain the conditions on the density necessary to make the permeability negative by following the same line of reasoning employed for the electric permittivity. The required density is thus the same as for the case of the 21

Here, we treat µ on the same footing as e. Some authors [e.g., Landau and Lifshitz (1995), Section 103] prefer, instead of treating electric and magnetic effects symmetrically, to take advantage of the relation between the electric and magnetic fields in an electromagnetic wave and introduce the magnetic effects via spatial dispersion of c, i.e., the dependence of e on the wave vector.

A NEGATIVELY REFRACTING ATOMIC VAPOR?

211

electric permittivity, except increased by a factor of ( d / µ 0 ) 2 "' 105 • In the fardetuned regime, this would require densities larger than those of typical condensed matter, ruling out this possibility, while in the regime where w ---+ wo, typical levels of pressure broadening would prohibit negative values for µ for room temperature gases. For ultracold atomic gases, the required density (N "' 1019 cm- 3 ) is far too large for present technology. Not only are the conditions for negative permittivity and negative permeability extraordinarily difficult to achieve in an atomic vapor, the conditions must be simultaneously satisfied at a particular frequency. We are thus left with a pessimistic assessment of the prospects of "left-handed" atomic vapors. In our above considerations we have assumed that the electric dipole (E 1) and magnetic dipole (M 1) transitions are between the ground state of the atom and two distinct upper states, so the permittivity and permeability can be calculated independently. This seems a reasonable assumption, since E 1 transitions occur between states of opposite parity and M 1 transitions occur between states of the same parity. Recently, there has been a suggestion by Pendry (2004b) that the above requirements can be loosened somewhat in a chiral medium (molecular vapors, for example). The idea is similar to that employed in, for example, atomic parity violation experiments that take advantage of Stark-induced transitions (see discussion in Problem 4.5). In a chiral medium, the states have mixed parity, and thus transitions have both E 1 amplitudes (proportional to d) and M 1 amplitudes (proportional toµ). Interference between the Ml and El amplitudes enhances the magnetic dipole transition, and therefore the required density for negative permeability in a chiral medium is only a factor of"' d/µ 0 "' 2 x 137 larger than that for the permittivity. It is interesting to note that a similar situation can arise when nearly degenerate opposite parity levels are mixed with external fields, as in the case of hydrogen and dysprosium [Budker et al. (1994)]. Even so, achieving the vapor density ("' 1020 atoms/ cm 3 ) required for negative permeability in the small detuning regime would necessitate anomalously small pressure broadening. Using cold chiral molecules can reduce the required vapor density, but remains at present a technically difficult path to achieving a negatively refracting vapor. One might wonder if electromagnetically induced transparency (EIT) in a multi-level atomic system might offer a possible solution to this dilemma - permitting high densities while maintaining narrow linewidths and small absorption. Yet it appears that even with EIT the stringent requirements on the magnetic permeability will be difficult to achieve in atomic vapors [Oktel and Mustecaplioglu (2004)). The use of quantum interference effects similar to EIT are being explored as a way to generate chirality while minimizing absorption [see, for example, Kastel et al. (2007)].

212

INTERACTION OF ATOMS WITH LIGHT

3.25

Light propagation in anisotropic crystals

Recall the following results pertaining to linear plane-wave-light propagation in a transparent non-gyrotropic, anisotropic medium such as a uniaxial or biaxial crystal, for which the dielectric tensor eij is real and symmetric [these properties follow from energy conservation: see, for example, Fowles (1975), Section 6.7; Landau et al ( 1995), Section 97; Born and Wolf ( 1980), Chap. 14]: • For a given direction of the wave vector, there are two polarization eigenmodes, where the polarization is characterized by the direction of the electric displacement vector fJ.An eigenmode is a wave that propagates in the crystal maintaining its polarization. • The two eigenmodes are linearly polarized with mutually orthogonal directions of fJ. Now on to the questions: (a) Why are the eigenmodes characterized by

fJ rather than the electric

field

e?

(b) Why are there two eigenmodes?

(~) Why do the two eigenmodes correspond to mutually orthogonal directions of D? (d) Why are the eigenmodes linearly polarized?

(e) Comment on the analogies between this problem and quantum mechanics.

Solution (a) For the case of a monochromatic light wave (of frequency w, so all quantities have temporal dependence e-iwt), Maxwell's equations take on the form:

v- x e- = V

X

H

---

I

8ii

C

8t

iw -

= -H C

1 afJ iw = -= --D. C 8t C

'

-

(3.308) (3.309)

213

LIGHT PROPAGATION IN ANISITTROPIC CRYSTALS

Substituting the spatial dependence of all quantities in the form of eik-r (where is the wave vector), we find: k- x e = w-H C

k-

X

H-

k

(3.310)

'

w = --D.

(3.311)

C

Equation (3.310) indicates that k is perpendicular to ii. Also, Eq. (3.31 I) shows that fJ is perpendicular to both k and ii. Thus, the vectors k, D, and ii are all mutually perpendicular. On the other hand, while £ is perpendicular to ii [Eq. (3.310)), it is not, in general, perpendicular to k in an anisotropic medium (Fig. 3.32).

-

D

k

FIG.

3.32 Various vectors describing plane-wave propagation in an anisotropic medium.

(b-d) [Here we follow Bredov, et al. (1985), Section 29.]

First, let us eliminate

ii from Eqs. (3.310) and (3.311) to obtain the wave equation - = k- x k- x e

2

- 2w n-

(3.312)

C

which, using the "BAC-minus-CAB" vector identity, becomes 2

k(k.l) - lk = - w2 D. 2

C

(3.313)

214

INTERACflON OF ATOMS WITH LIGHT

This is a vector equation. We will choose a coordinate system with one of the axe~ along f, and look at the projections of each of the terms onto an axis (a) orthogonaJ to k.Since k has zero projection on this axis, the first term in Eq. (3.313) does not contribute, and we have:

(3.3)4) Until this point, we have only used Maxwell's equations, and have not used any information about the properties of the medium. All the relevant properties are contained in the material equations

Di=

(3.315)

CijCj.

These relations can be inverted to read

ei = c;/DJ.

(3.3 J6)

Since Eij is symmetric, so is its inverse, the inverse permittivity tensor c;; for example, Riley et al. (2002), Section 8.12]. Substituting Eq. (3.316) into Eq. (3.314 ), we have -1

2

c013D13k

W

1

[see,

2

= c2 D0 .

(3.317)

In this equation, since D is confined to a plane, there are only two possible values of the indices o and /3,and the relevant components of c;J correspond to a 2 x 2 symmetric matrix. Equation (3.317) can be regarded as an eigenvalue problem for the the 2 x 2 tensor c- 1, where the eigenvalues are n- 2 = w 2 /(k 2 c2 ), and n is the effective refractive index for a given eigenmode. We now recall that any symmetric tensor can be diagonalized, and the principal axes are orthogonal to each other [see, for example, Riley et al. (2002), Section 8:.}3]. Thus, the eigenpolarizations correspond to two orthogonal directions of D along these principal axes, so the two eigenpolariz.ations are linear. (e) The analogy with quantum mechanics can be made rather complete. In quantum mechanics, we often talk about temporal evolution of the wavefunction governed by the Hamiltonian. In this case, we deal with spatial propagation of an electromagnetic wave, and we were able to reduce the propagation problem, using the Maxwell's equations and the material relations, to the eigenvalue problem (3.317) analogous to that of finding eigenstates in a quantum-mechanical two-level system. It is interesting to note that "orthogonality" of eigenstates corresponding to distinct eigenvalues, which in quantum mechanics usually refers to Hilbert space, here can be understood literally - the two eigenpolarizations are orthogonal in real space.

ELECTROMAGNETICALLY

3.26

INDUCED TRANSPARENCY (EIT)

215

Electromagnetically induced transparency (EIT)

The basic phenomenon of electromagneticaJJy induced transparency involves three levels and two light fields [Fig. 3.33(a)J. Suppose that, in the absence of Jight., alJ atoms reside in state IA). If a weak field (the probe field) is applied on resonance with the IA) --+ IC)transition, in the absence of other light fields, the probe field experiences absorption. However, if a strong light field (the drive field) is applied on resonance with the adjacent transition between levels IB) and IC), absorption on the probe transition vanishes. [There is vast literature available on the subject of EIT. A few accessible references include papers by Vrijen et al. () 996), Harris ( 1997), and Kasa pi ( 1996). ]

IC)

fC)

(a)

(b)

--,

1B)

IA) FIG.

3.33

_______ IB_)

IA)

(a) Illustration of the basic phenomenon of electromagneticaJJy induced transparency

(EIT). Absorption on the probe (IA) ~ IC)) transition is suppressed when a strong drive field is applied on the adjacent transition between levels IB) and /C}. (b) EIT in the case where light frequencies are detuned from their respective one-photon resonances.

(a) Give a simple quantum-mechanical explanation of the EIT phenomenon. In particular, show that there is a coherent superposition of states /A) and /B) that does not absorb light from either probe or drive field (a dark state). Make the simplifying assumption that both light fields are monochromatic. (b) What role does spontaneous emission play in EIT? For example, is spontaneous emission necessary to put atoms originally prepared in state /A) into the dark coherent superposition of states IA) and I B)?

(c) In part (a), we considered the case where both the probe and the drive fields are tuned to their respective one-photon resonances. In this case, the two-photon (Raman) resonance condition is (3.3 I 8)

INTERACflON OF ATOMS WITH LIGHT

216

where wp and wd are the probe and drive frequencies, respectively, and EA, E 8 are the energies of the corresponding states. Now consider the same system, except that the light fields are not exactly on resonance with the corresponding one-photon transitions [Fig. 3.33(b)]. Since the strong drive light field is off-resonant, it causes ac-Stark shifts of the levels IB) and IC) (see Problem 2.7), so the two-photon (Raman) resonance condition is no longer given by Eq. (3.318). In this case, at what frequency of the probe light does absorption vanish due to EIT?

Solution (a) The basic EIT phenomenon can be understood by considering the interaction of an atomic system with the bichromatic light field consisting of the probe and drive light fields. The states IA) and 1B) are both coupled to the upper state IC) by the light field. In general, we can consider the interaction with IC) of any coherent superposition of states IA) and IB):

= alA) 11/J)

+ blB) ,

(3.319)

where a and b are some constant complex coefficients. The amplitude A for photon absorption in the dipole approximation is proportional to the prcxluct of the dipole transition matrix element and the optical electric field J. (see, for example, Problem 3.4). In the present case,

e

A oc (Cid·£Iv,) = a(CldlA)ep

+ b(CldlB)ed

= adcAep

+ bdcBed,

(3.320)

where ep is the amplitude of the probe field and ed is the amplitude of the drive field. In the above we neglect far off-resonant interactions such as the direct interaction of the drive light field with the IA) ~ IC) transition. Here an essential point is that we add transition amplitudes (not probabilities). The amplitude A in Eq. (3.320) vanishes when

b=

_ dcAf.p a . dcBed

(3.321)

When the condition (3.321) is satisfied, the atoms are in a dark state, 11/Jdark), where they do not interact with light (see also Problems 3.9 and 3.10). Note that choosing the ratio of the drive and probe intensities sufficiently high, we can have essentially all of the atomic population remaining in state IA). Let us recap the situation. The atoms are in state IA), the probe light is applied on resonance with the IA) ~ IC) transition, but there is no absorption, where there would have been absorption in the absence of the light driving the 1B) +-+ IC) transition. This is EIT.

ELECTROMAGNETICALLY INDUCED TRANSPARENCY (EIT)

217

But how does the atomic system end up in the dark state? It turns out that in practice preparing the system in the dark state is not difficult, since the system tends to self-adjust to be in the appropriate state for EIT to occur. The creation of a dark coherent superposition of atomic states is also known as coherent population trapping (see Problem 3.10). We briefly discuss this in the solution to part (b). Coherent population trapping is reviewed by Arimondo (1996) and coherent population transfer is reviewed by Bergmann et al. ( 1998). (b) This is indeed a somewhat tricky question. Clearly there must be spontaneous

emission in the system: we have level IC) that lies above levels IA) and 1B) and has non-zero electric-dipole matrix elements coupling to both IA) and 1B). Thus, if we populate level IC), there is necessarily spontaneous emission to the lower levels. While this is true, it does not answer the question of the role of spontaneous emission in the generation of EIT. It turns out that in some instances, spontaneous emission does indeed play a role in the creation of EIT. Imagine a situation where all atoms are initially in state IA), and the light fields are turned on abruptly. State IA) is a superposition of the dark state and the bright state (where the bright state is the coherent superposition of states IA) and IB) orthogonal to l"Pdark) ). The dark state component of IA) does not interact with light, but the bright component does. Therefore, after the light is turned on, some atoms will be excited. Following the excitation, an atom can spontaneously decay to levels IA) and 1B), which may land it either in the bright or the dark state. Since atoms are not re-excited from the dark state, eventually, all atoms are pumped into 11/Jdark)This scenario, however, does not mean that spontaneous emission is always necessary for establishing EIT. It turns out that it is possible to put the system into the dark state without having any spontaneous emission to the lower levels. Here is how it can be done. Let us first tum on the drive field alone. This light drives transitions between two empty levels (1B) and IC)) and there is no fluorescence. This means that the system happens to already be in the dark state [11/Jdark) = IA) for this particular light field, ep = 0, see Eq. (3.321 )]. Now we have to exercise some caution in turning on the probe field - if we tum it on abruptly, there will be fluorescence. However, if we tum this field on adiabatically (see Problems 2.6 and 3.17), the system will remain in the dark state. The atoms evolve into a superposition of the states IA) and IB), and no spontaneous decay ever occurs! This situation is directly analogous to many different adiabatic following problems, for example, a classic problem in polarization optics where the goal is to rotate the direction of linear polarization of a light beam if only (ideal) dichroic polarizers are available. The trick is to pass the light through a sequence of such polarizers, each rotated by a very small angle with respect to its input linear polarization. In such an arrangement, the polarization rotation angle per polarizer is

218

INTERACTION OF ATOMS WITH LIGHT

>

Wsp

~ 250V/cm.

(4.128)

2dsp

(b) One way to think about the consequences of an atomic EDM

d:is to consider

the Hamiltonian (4.129)

Show that the existence of such a dipole moment would violate parity (P) and time-reversal (T) invariance. (What vectors are available for to point along?) This violation is the reason that EDMs are of such considerable interest. From a macroscopic viewpoint, it seems evident that nature has an arrow of time,9 but

d:

9

The prime example of an ..arrow of time" is the second law of thermodynamics which says that entropy cannot decrease for an isolated system. This law is based on the fact that the more microscopic states (microstates) available to a system for a given macroscopic state (macrostate), the more likely it is that the system will be in that macrostate. However, such an arrow of time neither implies nor depends on the laws of physics governing the dynamics of the system violating T-invariance. Therefore, it is, in principle, physically possible for a system to go from a state of high entropy to a state of low entropy, it is just extraordinarily unlikely that this would occur for statistically large systems. Such issues are discussed at length in the book by Sachs ( 1987).

256

INTERACTION OF LIGHT WITH ATOMS IN EXTERNAL FIELDS

the only evidence of T-violation in a microscopic system comes from the observation of CP-violation in neutral K- and B-mesons (which, according to the widely held belief that nature respects the combined symmetry CPT, implies T-violation). CP-violation is incorporated into the Standard Model phenomenologically, and the Standard Model's prediction for the size of the electron EDM due to this CPviolation is many orders of magnitude below the current experimental sensitivity. The source of CP-violation remains, to a large degree, a mystery, and various proposals to explain this phenomenon (such as supersymmetry) often predict values for de that are presently experimentally accessible. Thus searches for ED Ms tum out to be a good way to test new theories in particle physics.

Solution > 1, what is the state of the atoms after they leave the laser interaction region? (The laser is linearly polarized along z, as shown by the double arrow in Fig. 4.11.) After interacting with the first laser beam, the atoms enter a region in which an rf magnetic field (orthogonal to the page), oscillating at the Larmor frequency 9FµoB, is applied. The strength of the rf field is chosen so that the axis of the atomic polarization is rotated by 1r /2 (this stage uses the technique of magnetic 11

For example, Purcell and Ramsey ( 1950) originally proposed to search for a neutron EDM using separated oscillatory fields, and indeed carried out the experiment (Smith et al. 1957). This is also the method used to obtain the present best limit on the electron EDM (Regan et al. 2002). 12 Instead of using separated fields, one could imagine employing the traditional Rabi technique [see discussion, for example, in the book by Ramsey ( 1985)] with a large interaction region comparable in size to the separation between the rf regions in Ramsey's method. This approach is unsatisfactory for the technical reason that it is difficult to maintain a homogeneous rf field over a large region and avoid broadening of the resonance due to gradients.

260

INTERACTION OF LIGHT WITH ATOMS IN EXTERNAL FIELDS

~Detector Laser beam ~

I RF B-field 2 ~

• • • • • • •

~

•

B

E

•

Atomic beam

_______I RF B-field Laserbeam

Oven

I

8 ◄

z

FIG. 4.11 Schematic setup for an EDM experiment using Ramsey's method of separated oscillatory fields (see text). Laser beams propagate into the page and are linearly polarized along z.

resonance, see Problem 2.6). What is the state of the atoms (expressed in the spinor representation) after they leave the first rf region? Next the atoms pass through the electric field region where E shifts the energy of the MF = O state away from that of the MF = ±l states due to the usual quadratic Stark effect (see, for example, Problem 2.1). The MF = ±1 states are split due to the magnetic field. If the atom possesses an EDM, there is also a small contribution from da to the splitting between the MF = ± l states.

SEARCHES FOR PERMANENT ELECTRIC DIPOLE MOMENTS (T)

261

After exiting the electric field region, the atoms pass through a second rf region identical to the first, except that the phase of the second rf magnetic field can be offset by some amount ¢ from that in the first region. The detected signal is the fluorescence from atoms excited in the second laser-beam-interaction region. The second beam is also resonant with the F = 1 ~ F' = 1 transition and linearly polarized along z. Show that for an appropriate choice of¢, the fluorescence signal is linear in d0 • (Assume that if ¢ = 0 and E = 0, the second rf region returns the atoms to the state that they were pumped into initially in the first laser interaction region.) Note that this basic setup suffers from a serious problem. The atoms moving through the electric field see a motional magnetic field (Problem 2.9) .... .... .... V (4.136) Bmot=EX - , C

which couples to the magnetic moment of the atoms and causes additional precession which is linear in E.In practice, this effect can be distinguished from an EDM signal by, for example, having two counter-propagating atomic beams (since the unlike an EDM signal). In addition to this E x effect, effect reverses sign with there are numerous other subtle effects [such as geometric phases (Problem 2.12), leakage currents, etc.] which must be understood and controlled.

v,

v

Solution In the first laser interaction region, atoms are optically pumped into the MF = 0 sublevel (optical pumping for F ~ F transitions is discussed in Problem 3.9). Since"' >>1, we may assume that the atoms are completely pumped into the state

,~1) =

(!)

(4.137)

in the spinor representation (Fig. 4.12). Note that with respect to the laser lighJ, 11/Ji) is a dark state (see Problem 3.9). Also, since the polarization axis is along B, the magnetic field has no effect on this state. The rf magnetic field in the first rf region rotates the atomic state by 1r /2 about y. Using the appropriate rotation matrix, 1)(0, 1r /2, 0), for such a transformation (Appendix E), we have

11/12) =

(-~~~ 1/2

11 '!21~~) ·(~) = ~ ( ~) .

-1//2

1/2

0

v 2 -1

(4.138)

The condition for such a rotation is easy to see in the frame rotating with the Larrnor frequency nL = gµ 0 B (Problem 2.6). The linearly polarized rf field Brr

262

INTERACTION OF LIGHT WITH ATOMS IN EXTERNAL FIELDS After RF 8-field 2 (no EDM)

After RF 8-field 2 (with EDM)

~ "'~.....·~

Just before RF B-field 2 (no EDM)

~

Just before RF B-field 2 ~~, :fl (with EDM)

After RF 8-field I

After first laser interaction

t

Unpolarized sample out of oven

X

FIG. 4.12 Probability surfaces (the distance to the origin is proportional to the probability of finding the projection M = F along this direction, see Problem 9.7) describing the atomic polarization at different stages of the model EDM experiment described in the text (see also Fig. 4.11 ).

can be represented as two circularly polarized fields. In the rotating frame, on resonance, one circular component appears to be a static transverse field of magnitude Bn/2. while the other appears to rotate at 2S'h. Making the rotating wave approximation (Problem 2.7), we neglect the fast rotating component. The atomic polarization precesses about the static transverse field with frequency gµ 0 Bn/2. If the atoms have an average transit time of T, then the magnitude of the rf field is adjusted so that

gµoBrt

T

= 1r

to produce the desired rotation. Now the atomic polarization axis is perpendicular to precesses at the Larmor frequency nL:

(4.139)

B, and the polarization (4.140)

SEARCHES FOR PERMANENT ELECTRIC DIPOLE MOMENTS (T)

2

63

If the atom possesses an EDM, there is an additional phase shift between the states: l (e-i(fli.+d,,E)t)

l'l/,,3(t))=

-

J2

o

-ei(fli.+d,,E)t

.

(4.141)

If we again go into the frame rotating with OL, we have

(4.142)

Finally, the atoms enter a second rf region. The rf field in this region is phase shifted by with respect to the first rf field. Thus in the rotating frame with quantization axis along the rf field in the second region, the state of the atom as it enters the region is

(4.143)

The rf field transforms the atomic polarization according to

l'l/J~) = 1>(0,1r ;2, o) l¢t

1/2 - ( -1//2 1/2

0

t\t))

1//2 0 -1//2

(4.144)

1/2) l (e-i(daEt+)) 1//2 · 0 1/2 J2 -ei(d ..Et+q,)

J2sin(d Et +

-

299

--+

1

0.75 0.5 ~

0.25

~

0

~

-0.25

/

v

-0.5 -0.75

-4

-2

0

2

4

p 2 ly c) FIG. 6.2

rD

Velocity-dependent force for a one-dimensional optical molasses in the regime where

;S -;-o.Note that atoms experience a force which opposes their motion, and in the range

lvl v.J ) h · L 0, t e presence of the atom m the cavity pulls the resonance frequency of the cavity tow~ds higher frequencies, i.e., reduces its effective length. On the other hand, if the mput frequency is tuned below the atomic resonance (wL < w ), the presence 0 of the atom in the cavity pulls the resonance frequency of the cavity towards lower frequencies and increases its effective length. (b) The mechanical potential cl>arises due to the AC Stark shift of the atomic ground state [see, for example, Problem 2. 7J:

d2e2 4/i(wL - wo)'

WQ. According to the argument of part (a), the pres;11nce of t~e atom near a crest pushes the cavity resonance freq~ency t~w~s higher it equencies, i.e., away from wL· This reduces the light_powe~c1rculatm~ m th~ cavJ\~·On ~heother hand, if an atom is near a node, the c1rculatmg power_1smaximal. corchng to Eq. (6.122), a node in this case also corresponds to a mimmum of th e ~echanical potential (because the light-induced energy shift is positive). Thus the circuJating · power is maximal when the atom 1s · at a mm1mum · · . I of the mec hanica Potential (J). · If w L < wo, the presence of the atom in the vicinity of a crest of the Sland1ngw ti · · ave pushes the cavity resonance frequency towards lower requencies, i.e., 8 . · · · h . Th e c1rcu. Itoward . W£. This increases the light power c1rculatmg m t e cavity. ating power is thus maximal when the atom is located at a crest of the standing ::e. !'ccording to Eq. (6.122), this correspo~ds _toa mi~imum of the mechanical . ntial (because the light-induced energy shift 1s negative). Thus, as before, the circulating power is maximal when the atom is at a minimum of the mechanical Potential (J). ~e) The atom moving along the axis of the cavity is constantly climbing and Th~en~i~g the peaks of the light-shift-induced m~hanical pote?tial cl>(Fig. 6.8). ~ngm of the slowing force can be understood m the followmg manner. If an atom is resting at a minimum of cl>then the circulating power and the height of the mech~nical potential are maximal '[as discussed in part (b)J; for an atom resting at ~ maximu~ of cl>,the height of the mechanical potential is minimal. Changes in fi e mechanical potential persist for a characteristic time 'Y;1, so as an atom travels rom a minimum of ~ to a maximum, the height of the potential is greater than for

COLD ATOMS

328

e .... C: ....

~ d)

0..

\ Po ition

FIG. 6.8 Illustration of the cavity cooling effect for w L well for

WL

< wo as discussed

> wo [note that the technique

in part (b)]. Atom moving with velocity

v in

works equally

the mechanical

potential 4>created by the standing light wave in an optical cavity. When the atom is at a node of the optical electric field, it has no effect on the cavity, and the circulating light power (and hence amplitude of 4>)is maximal (solid line in the plot of 4>).When the atom is at an antinode, the index of refraction is modified in such a way as to move the cavity resonance frequency away from the light frequency, thereby reducing the circulating light power and decreasing the amplitude of the potential 4>(dashed line). The cavity response is delayed by a characteristic time ,..,;

1

,

so as the atom moves along the cavity axis it always climbs a higher potential ..hill"

than it descends. Hence energy is transferred from the atom to the electromagnetic cavity, and the atom slows down.

field in the

when an atom is travelling from a maximum to a minimum of ell. The net result is that the atom always climbs a higher "hill" than it is descending; hence it slows down (Vuletic and Chu 2000). To estimate the slowing force, we first estimate the change of the power in the cavity and the height of the mechanical potential due to the atom. The circulating power goes from its maximum value to essentially zero when the cavity is tuned by ~ 1c from the laser frequency. When the laser is tuned to the slope of the cavity transmission peak, we have:

(6.124)

CAVITY COOLING FOR MANY PARTICLES: STOCHASTIC COOLING

329

Using Eqs. (6.121) and (6.122), one finds for the change in the height of the potential: (6.125) The average power P taken away from the atom can be estimated as the change of the potential given by Eq. (6.125) divided by the characteristic time,; 1 :

(6.126)

,e,

Although this expression does not explicitly depend on there is, of course, an implicit dependence through The average slowing force F can be found by dividing the power by the velocity of the atom (or, more precisely, its component along the cavity axis). Thus, the circulating power is maximal when the atom is at a minimum of the mechanical potential ~- If v «: :: of as resultmg from the Doppler shift upon reflection from a movmg mirror (. Problem 8.1). Light with wave vector k reflecting from a mirror (slowly) movtn~ with velocity Vm experiences a first-order change in the magnitude of the wav

vectorof (S.85) where in this case v = c is the speed of light. In the case of a nonrelativistic ~to: moving with velocity v « vm, elementary consideration of elastic scattering ro a movingmirroragainleads to the resultdescribedby Eq. (8.85).

MATfER-WAVE VS. OPTICAL SAGNACGYROSCOPES

397

Using the fact that the velocity of a mirror is given by vm= 0 x r,where f is the radius vector of the mirror with respect to the interferometer rotation axis, Eq. (8.85) yields:

... ... ... ... = Ak . n x r = _ n . Ak x r .

Ak

V

(8.86)

V

...

Note that no Doppler shift occurs when ~k is in the radial direction. A simple geometry for a Sagnac interferometer is the circular one shown in Fig. 8.14(c), where light is guided about a circle using, for example, a fiber-opticcable. In this setup, the magnitude of the wave vector is only Doppler-shiftedat the input and the output. The change in the wave vector magnitude for the part of the wave reflected by the mirror at the input is ...

Ak = _ n .

...

...,

(f'_f) x T = _ n · k

V

...

I

x r = _ k nr ~ _ knr '

V

V

(8.87)

V

where k is the wave vector for the incident light (which is parallel tor, with magnitude k) and k' is the wave vector for the reflected light (which is orthogonal to with magnitude k' ~ k + ilk), and we ignore tenns to second order in (rO/v). According to the expression (8.87), as the reflected light beam circulates around the loop, it accumulates a phase shift of magnitude

r,

21r

A =

f lo

...

...

4 Ak r d8 = 21rrAk = 2 kAO = 1rn . A v

(8.88)

.,\v

...

...

due to the rotation n, where fJis the angle between rand -k. The light trajectory in interferometersof more complex shape can be thought of in general as consisting of a series of infinitesimalcircular arcs and radial regions. If the light traces out an arbitrary path, we see that the resultant phase shift acquired by the light is given by

r21r

A=Jo Ak(r)r d8

(8.89)

where r is now a function of fJand ilk(r) = kOr/v as before, only in this case instead of resulting solely from the initial reflection, it is obtained from a series of reflectionsoff mirrors around the light path. Thus we have

Ac/> = kn v

/2'"r( 8)2 d8 = lAn

lo

'

v

so the resulting phase shift per round trip is still given by Eq. (8.88).

(8.90)

398

EXPERIMENTAL METHODS

Note that Eq. (8.90) can also be derived from a quite different perspective_ that of the Feynman path integrals, described in detail in the tutorial article by Storey and Cohen-Tannoudji (1994). (b) From Equation (8.88), the phase shift is inversely proportional to Av. The deBroglie wavelength Ade is (8.9))

so ~ in correspondence with the 2K + 1 components of an irreducible spherical tensor The relationship between the Cartesian tensor

TJ.

2

components ~~ > and the irreducible tensor components for example, Varshalovich et al. ( 1988)]:

TJare as follows

TJ= T;;),

Tl1=

(F.43)

~JI (TJ~>iTJ:>),

2 = fi(T(2 ) T±2 y 6 xx

[see,

-

±

(F.44)

2 2 r.( uu) ± 2iT(xu>).

(F.45)

The decomposition of higher-rank tensors becomes quite complicated: for instance, from the 27 independent Cartesian components of a third-rank tensor, one can construct seven irreducible tensors (one zero rank, three first rank, two second rank, and one third rank)! Not to mention that the decomposition is not umque. For more detailed discussions of tensors and tensor operators, see texts such as Fano and Racah (1959) and Zare (1988).

APPENDIX

G

THE DENSITY MATRIX The density matrix is a tool which makes it possible to describe ensembles 1 of quantum systems (e.g., atoms) that are more general than the ensembles which can be described by a wavefunction. In this appendix, we review the basic properties of the density matrix and offer a few examples to illustrate how it can be used. The examples are intended to be simple enough to be solved without the density matrix (usually just by thinking for a moment!), and are merely intended for illustration. For more detailed discussions, see, for example, the books by Stenholm ( 1984) and Blum ( 1996).

G.1

Connection between the density matrix and the wavefunction

The key point of the density matrix is that it is a more general description of an ensemble than a wavefunction. A wavefunction can only describe an ensemble that is fully coherent (or pure), while the density matrix can also describe partially coherent or incoherent ensembles. What does this mean? Let us consider a simple example. Suppose we have an ensemble of N spin- I/2 atoms and consider only their internal states. If all of the atoms are in the same state, e.g., (G. I)

then we say that the ensemble is in a pure state. In this case, the wavefunction l'l/J) is sufficient to describe the behavior of the entire ensemble. Now suppose that the atoms undergo collisions 2 which change the relative phase between the spin up and spin down components for each atom in a random 1

An ensemble can either be assembled spatially (e.g., atoms contained in a vapor cell) or consist of sequential measurements separated temporally (under proper circumstances, this could even be a single quantum system). 2 Relaxation processes, such as collisions and spontaneous emission, tend to destroy coherence, and it turns out that such decoherence is one of the primary reasons that quantum-mechanical behavior is so difficult to observe in macroscopic systems.

470

THE DENSITY MATRIX

way, so now, at a particular time, we have for the state of the i-th atom (G.2)

where