Advances in Quantum Electronics Edited by D. W. GOODWIN Department of Physics, University of York, England VOLUME 3 1975 ACADEMIC PRESS L O N D ON A ND N EW YORK A Subsidiary of Harcourt Brace Jovanovich, Publishers ACADEMIC PRESS INC. (LONDON) LTD. 24/28 Oval Road, London NWl United States Edition published by ACADEMIC PRESS INC. Ill Fifth Avenue New York, New York 10003 Copyright © 1975 by ACADEMIC PRESS INC. (LONDON) LTD. All Rights Reserved No part of this book may be reproduced in any form by photostat, microfilm, or any other means, without written permission from the publishers Library of Congress Catalog Card Number: 72-12267 ISBN: 0-12-035003-3 Printed in Great Britain by William Clowes & Sons, Limited London, Colchester and Beccles LIST OF CONTRIBUTORS L. C. BALLING, Department of Physics, University of New Hampshire, Durham^ New Hampshire, U.S.A, C. C. DAVIS, Physics Department, Schuster Laboratory, University of Man­ chester, Manchester, England. T. A. KING, Physics Department, Schuster Laboratory, University of Manches­ ter, Manchester, England. PREFACE This, the third volume of Advances in Quantum Electronics contains two major review articles considered to be of complementary interest to laser physicists. First, an article by Balling reviews theoretical and experimental work undertaken in the field of optical pumping. Paramagnetic atoms can be orien­ tated in their ground state by illumination with circularly polarised resonance radiation and their hyperfine splittings and magnetic moments studied. This has led to both an understanding in depth and the development of frequency standards, magnetometers and masers. Secondly, Davis and King review the field of gaseous ion lasers. Because of their technological significance attention is concentrated on the noble gas and metal vapour ion lasers and their practical applications. These two review articles should prove of value to those engaged in theoretical and experimental studies in the field of quantum electronics. February, 1975 D. W. GOODWIN OPTICAL PUMPING L. C. BALLING Department of Physics ^ University of New Hampshire, Durham^ New Hampshire, USA I. Introduction 2 II. Optical Pumping: An Overview 3 A. Introduction 3 B. Magnetic Resonance 4 C. Optical Pumping 8 D. Optical Pumping of Alkali Atoms 13 E. Spin-exchange Optical Pumping 18 F. Optical Pumping of Mercury and Other ^SQ Atoms 20 G. Optical Pumping of Helium 22 H. Frequency Shifts in Optical-pumping Experiments 24 I. Crossed Beam Detection 27 J. Spin Relaxation . 28 III. Density Matrix Methods 30 A. The Density Operator 30 B. The Density Matrix 33 C. The Density Matrix for a Spin-i System 35 D. Spin-exchange Collisions 36 E. Spin-relaxation Times 38 IV. Optical Pumping of a Spin-i System 39 A. The Optical-pumping Process 39 B. Optical Pumping of a Spin-i Atom 50 C. The Equilibrium Transmission Signal 57 D. The Spin-i Transient Transmission Signal 60 E. The Equilibrium Crossed-beam Signal 66 V. Optical Pumping of Alkah Atoms 67 A. Effective Hamiltonian for an Alkali Atom in a Weak Magnetic Field . 67 B. Density Matrix for the Alkali Atom Ground State 69 C. Magnetic Resonance in a Weak Field 71 D. The Optical-pumping Cycle 73 E. Absorption of the Pumping Light 78 F. Spin Relaxation 79 G. The Low-Field Optical-pumping Signal 81 H. Alkali Atoms in a Magnetic Field of Intermediate Strength: Resolved Zeeman Transitions 84 I. Optical-pumping Signals under Varying Pumping Light Conditions . 88 J. Hyperfine Transitions 90 VI. Spin-exchange Optical Pumping 91 A. Spin Exchange Between Two Species of Spin-i Particles . . .. 92 B. The Spin-exchange Optical-pumping Signal for the Spin-i System . 98 C. The Effect of Nuclear Spin on Electron-Alkali Atom Spin-exchange Collisions 102 1 2 L. C. BALLING D. The Spin-exchange Electron Resonance Signal when the Effects of Nuclear Spin are Considered .109 E. The Effect of Nuclear Spin on Spin-exchange Collisions between Alkali Atoms Ill F. Application of Spin-exchange Results to the Relaxation of the Alkali Spin by Spin-randomizing Collisions .115 VII. Optical-Pumping Experiments 116 A. Alkali Optical Pumping at High and Low Temperatures . . , .116 B. Precision Measurements 117 C. Hyperfine Pressure (Density) Shifts 126 D. Electron-Alkali Atom Spin-exchange Collisions 130 E. Spin-Exchange between Alkali Atoms 134 F. Spin-relaxation Times 135 G. Optical-pumping Orientation of Ions 144 H. Optical Pumping of Atomic Ρ States 146 I. ^-Factor Shifts due to Resonant and Nonresonant r.f. Fields . . .148 VUI. The Construction and Operation of an Alkali Optical-pumping Apparatus . 148 A. Light Sources 149 B. Signal Detection 151 C. The Magnetic Field 152 D. R.F. Generation and Measurement 155 E. Sample Preparation 157 F. Optical Pumping at High and Low Temperatures 159 G. Obtaining the Signal 160 Acknowledgements 162 Review Articles and Books 162 Bibliography 162 I. INTRODUCTION This chapter is primarily intended to provide the reader with an introduction to optical pumping sufficient to enable him to undertake research in the field. I have attempted to strike a balance between a superficial review of all aspects of optical pumping and a detailed discussion of a limited number of topics. I have placed an emphasis on the optical pumping of alkali atoms because of the great variety of experiments which have been and can be performed with alkah optical pumping techniques. Section II presents an overview of the optical-pumping field on an elementary level; Section III contains a brief review of the use and properties of the density matrix as applied to the statistical behavior of assemblages of atoms or ions. In Sections IV-VI, the density matrix approach is systematically applied to the theory of optical-pumping r.f. spectroscopy and spin-exchange optical pumping. The theoretical discussion is at a level which should be readily under­ standable to a student who has taken two or three semesters of graduate non- relativistic quantum mechanics. Because the sections on the theory of optical- pumping experiments contain a straightforward application of nonrelativistic quantum mechanics to the analysis of the behavior of atoms interacting with OPTICAL PUMPING 3 each other and electromagnetic fields, they might well be of interest to graduate students who are not interested in optical pumping per se. This chapter has been written on the assumption that the sections will be read consecutively. This is particularly true of the first five sections in which the theoretical discussion builds steadily on the development of preceding sections and chapters. Sections VI and VII deal with the experimental side of optical pumping. Section VII is a review of optical-pumping experiments and contains numerous tables of physical data such as atomic g-factors, hyperfine splittings, hyperfine pressure shifts, spin-exchange cross sections, relaxation times, etc. These tables include data obtained by optical-pumping methods and by other experimental techniques as well, and in a number of cases the data is compared with theoretical calculations. Section VIII is intended to aid a newcomer to the field in the construction and operation of an alkah optical-pumping apparatus. Although I have attempted to present a reasonably broad view of optical pumping, my choice of topics and the emphasis I have placed on them tends to reflect my own research background and interests. For different views of the subject, the reader is invited to consult the review articles and books which are listed at the end of this chapter on page 162. The bibliography contains only those articles and books which are referred to in the text. II. OPTICAL PUMPING: AN OVERVIEW A. INTRODUCTION In 1950, Kastler proposed a method for orienting paramagnetic atoms in their ground state by illuminating them with circularly-polarized optical resonance radiation. He called this process "optical pumping". His proposal introduced a new and powerful technique for studying the properties of atoms and ions by means of r.f. spectroscopy. In the succeeding twenty years, optical pumping has been used to measure the hyperfine splittings and magnetic moments of an impressive variety of atoms and ions. In terms of precision and reliability, optical pumping competed favorably as a technique with the far more expensive atomic beam method. In addition, many kinds of interatomic interactions have been studied in optical-pumping experiments. The optical-pumping process itself has been the subject of considerable study, providing as it does the opportunity to investigate in detail the interaction of atoms with resonant and off-resonant light. Optical pumping has also been applied to the construction of frequency standards, magnetometers and masers. In short, optical pumping is an important and well established experimental technique in atomic physics. A surprising number of different types of experi- 4 L. C. BALLING ments can be performed at relatively low cost, because the apparatus is basically quite simple. The theoretical analysis of optical-pumping experiments can be a challenging and satisfying application of non-relativistic quantum mechanics. Despite this, comparatively few physicists have worked in this area. It is the purpose of this article to introduce the reader to the field of optical pumping. This section is designed to give an overview of the subject on an elementary level. The theory of optical-pumping r.f. spectroscopy and descrip­ tions of optical-pumping experiments and techniques will be treated in detail in succeeding chapters. We will primarily, though not exclusively, be concerned with optical pumping as a means of producing population differences in the ground state sublevéis of paramagnetic atoms in order to detect radio frequency transitions between these levels. That is, we will be dealing with magnetic resonance in atoms, and we shall begin our discussion of optical-pumping with a simplified treatment of magnetic resonance familiar to students of NMR and EPR. Β. MAGNETIC RESONANCE If we place an atom with total angular momentum ÄF in a weak magnetic field HQÍC in the z-direction, it will interact with the field through its magnetic dipole moment μ. The Hamiltonian Jif for the interaction is 3^ = -μ.Ηο11, ...(B.l) The magnetic moment is related to the total angular momentum by μ = gFμo'P, ...(B.2) where μο is the Bohr magneton and gp is the g-factor. The g-factor can be positive or negative, depending on the atom's ground state configuration. The eigenstates of are just the eigenstates of F^, The energy eigenvalues are E^ = -g,μoHoM, „.(B.3) where Mis the eigenvalue of F^. The energy difference /dF"between two adjacent levels is ΔE = gpμoHo. ...(B.4) If an oscillating magnetic field ΙΗ^ιζο^ωί is applied in the x-direction, the Hamiltonian 3^ becomes ^ = -gF μο Ho Fz - gf μο 2//i cos ω(Γ^, .. .(B.5) First-order perturbation theory for a harmonic perturbation tells us that the atoms will undergo transitions between adjacent magnetic sublevéis and that the transition probabilities ΓΜ-,Μ+Ι and Γ^+Ι-,Μ are given by ΓM^M-,^ = Ji8Fμo^for\<M\F,\M+ly\'δ{AE-^fiωl ,..(B.6) OPTICAL PUMPING 5 and r^,,^^ = ^{g,ßoHoy\<M+l\F,\My\'S{-AE+hw). ...(B.7) The delta functions in these equations show that transitions will occur only when the resonance condition /ϊω=\ΑΕΐ ...(B.8) is satisfied. Because the energy levels are equally spaced, transitions between all adjacent sublevéis will occur simultaneously. Implicit in these equations is the assumption that the state |M-f-1> is higher in energy than the state |M> which is only true if gp is negative. Besides the resonance condition, the important point to notice is that r^^+i->M = ^M-^M+I- This means that if there are equal numbers of atoms in the two energy levels, the number of atoms undergoing transitions from |M> to |M + 1> will equal the number of atoms going from |M + 1> to |M>. If we wish to observe a macroscopic change in the magnetization of an ensemble of atoms, there must be an initial difference in the populations of the two levels. In NMR and EPR experiments, the Boltzman distribution of the popula­ tions of the energy levels of spins in a bulk sample in thermal equilibrium is relied upon to produce the desired population differences. When working with orders of magnitude fewer free atoms, however, one must develop artificial means for producing large population differences and sensitive detection schemes in order to observe magnetic resonance transitions. The optical- pumping process provides the means for achieving the necessary population differences and also the means for detecting the transitions. Before going on to a discussion of the optical-pumping process, we will look at magnetic resonance from a classical point of view. The classical approach is often more useful for a qualitative understanding of the signals one observes. A classical analysis is possible because, as we shall see in later chapters, h does not appear in the quantum mechanical equations of motion for the operator F. The classical equation of motion for the atomic angular momentum is dF η— = μχΗοΚ ...(B.9) or f = ^ . ^ F x ^. ...(B.10) Since dFjat is orthogonal to F, the torque produced by the field HQIC causes a precession of F about the z-axis with angular velocity COQ. That is, ^ = 0)0^ xF, ...(B.ll) 6 L. C. BALLING with ωο = - ...(B.12) If we view the precessing magnetic moment from a reference frame rotating in the same sense about the z-axis with angular velocity ω, the time derivative 9F/9i in the rotating frame is related to dF/di in the laboratory frame by dF r ^ — = ω/ο X F + — · ...(B.13) át dt Using equation (B.ll), we see that the time dependence of F in the rotating frame is ^ = (ωο - ω) fe X F. ...(B.14) ot FIG. 1. The superposition of two counter-rotating magnetic fields of equal amplitude. If we apply a rotating magnetic field fixed along the x-axis of the rotating coordinate system, the equation of motion in the rotating frame is ^ = (ωο - ω) fc X F + ωι f X F, ,..(B.15) ot where ...(B.16) η The rotating field is equivalent to the linearly oscillating field IH^ icoswt in the laboratory frame which was considered above. To see this we note that 2//i feos ωί = //i(f cos ωt +7 sin ω/) + //i(fcos ωt -7 sin ω/). That is, the oscillating field is the superposition of two fields rotating in op­ posite directions as shown in Fig. 1.