Table Of ContentAdvances in
Quantum Electronics
Edited by
D. W. GOODWIN
Department of Physics, University of York, England
VOLUME 3
1975
ACADEMIC PRESS
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ISBN: 0-12-035003-3
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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.
