Download New Physics And Technology For Spin-polarized Alkali-metal Atoms

New Physics And Technology For
Spin-polarized Alkali-metal Atoms
Fei Gong
a dissertation
presented to the faculty
of princeton university
in candidacy for the degree
of doctor of philosophy
advisor: William Happer
recommended for acceptance
by the department of
physics
August 2008
c Copyright 2008 by Fei Gong.
All rights reserved.
Abstract
In this theses, we present two important new physics phenomena of spin-polarized
alkali-metal atoms as well as one new technology for miniature atomic clocks based
on spin-polarized alkali-metal atoms.
We report that the hyperfine resonance frequencies of ground-state 87 Rb and 133 Cs
depend on the pressure of buffer gases Ar and Kr in a nonlinear way within some
pressure range[1]. We also show that for buffer gases He and N2 , no nonlinear dependence was observed. The experimental results suggest that the formation of van der
Waals molecules in Ar or Kr (e.g. RbAr or CsKr) contribute to the nonlinear pressure
dependence of the hyperfine resonance frequencies of Rb or Cs. We demonstrate that
a simple function can be used to fit the experimental data and the fitting results provide important information about the poorly-known interaction coefficients between
the alkali-metal atoms and the buffer-gas atoms.
Next we present a novel phenomenon which was reported for the first time[2].
For alkali-metal atoms that are optically pumped with D1 circularly-polarized laser
light, the microwave resonance signals will be reversed when the laser frequency is
close to the transition from the lower hyperfine multiplet of the ground state to
the excited state and when the laser intensity is sufficiently low. This counterintuitive phenomenon can be understood qualitatively with the help of a picture of
iii
spin-temperature distribution. Detailed density matrix calculation gave us results
consistent with the experimental observation.
Finally we demonstrate a new method to fill some alkali-metal vapor into a miniature vapor cell (with the volume of several mm3 )[3]. This method uses the electrolysis
of a specially-made borate glass which contains the target alkali-metal atoms. The
electrolysis can be done because the mobility of the alkali-metal atoms in the borate
glass is increased greatly at higher temperatures (e.g. 500 ◦ C) and in electric fields.
With this method, we are able to fill a well-controlled amount of alkali-metal atoms
into miniature vapor cells which are made with silicon and Pyrex glass by anodic
bonding. This method has the potential to be scaled to mass production of such
miniature vapor cells for miniature atomic clocks.
iv
Acknowledgments
This thesis could never be finished without the help and support of many people
whom I gratefully acknowledge here.
First of all, I would like to express my sincere thank and respect to my advisor,
Prof. William Happer. As a great physicist, he showed me how to explore the world
of science with enthusiasm and persistence. With his inspiration and great efforts to
explain things clearly and simply, he helped to make my thesis work more fun to me.
Whenever I encountered difficulties, it was his encouragement and advice that helped
me overcome them. It is really difficult to overstate my gratitude to Prof. Happer.
I would like to thank Dr. Yuan-Yu Jau for his generous help during my experiments, paper writing and so on. His wide knowledge in both theory and experiment
made my thesis work easier than it should have been. I really learned a lot from him.
I am also grateful to Prof. Michael Romalis for his smart ideas and wise advices.
His suggestions on my postdoc application and proposal writing helped me to know
more about how to be a good scientist. And I really appreciate his offer to be my
thesis reader.
I am honored to work with my colleagues in Prof. Happer’s group, Nick Kuzma,
Kiyoshi Ishikawa, Steve Morgan, Brian Patton, Amber Post, Tian Xia, Ben Olsen,
Bart McGuyer, people in Prof. Romalis’s group, and our assistant Ellen Webster and
v
Regina Savadge. They made my time here at Princeton full of happiness and laugh.
The experience with them will always be my precious memory.
I would also like to thank Michael Souza for his help on glass blowing. Without
those great cells and strings he made, I could not finish my thesis study. I also want to
express my gratitude to the staff members of the physics department, Laurel Lerner,
John Washington, Barbara Grunwerg, Claude Champagne, Mary De Lorenzo, Kathy
Warren, Mike Peloso, Helen Ju, Angela Qualls and many other people. I would like
to thank Laurel Lerner especially for her kind help throughout my whole graduate
time here. She was always patient and ready to help with all kinds of problems I
encountered.
It is also my honor to correspond with Dr. James Camparo. I appreciate the
discussions with him through emails and at conferences.
Finally, and the most importantly, I would like to thank my parents, Yuqin Ma
and Jianguo Gong. They brought me to this wonderful world, raised me with a happy
life, and sent me to college with all their efforts. They are always supportive with
their love. My husband, Lu Li, deserves my gratitude for his love, patience, support,
and company. Without him, I could not finish my PHD study so smoothly. I would
also like to thank my cat Tintin. She brings so much laugh to our life. To them I
dedicate this thesis.
vi
Contents
Abstract
iii
Acknowledgments
v
Table of Contents
vii
List of Figures
x
List of Tables
xiii
1 Introduction
1
2 Basic concepts in optical pumping
7
2.1
About optical pumping . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2.2
Free atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.3
The density matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
2.4
Ground-state relaxation . . . . . . . . . . . . . . . . . . . . . . . . .
17
2.4.1
Spin-rotation interaction . . . . . . . . . . . . . . . . . . . . .
18
2.4.2
Hyperfine-shift interaction . . . . . . . . . . . . . . . . . . . .
20
2.4.3
Spin-exchange interaction . . . . . . . . . . . . . . . . . . . .
23
vii
2.4.4
2.5
Relaxation due to spatial diffusion
. . . . . . . . . . . . . . .
28
Optical pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
2.5.1
Depopulation pumping . . . . . . . . . . . . . . . . . . . . . .
31
2.5.2
Repopulation pumping . . . . . . . . . . . . . . . . . . . . . .
31
3 Nonlinear pressure shifts
33
3.1
About nonlinear pressure shifts . . . . . . . . . . . . . . . . . . . . .
33
3.2
Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
3.3
The result of Rb and the physical picture . . . . . . . . . . . . . . . .
45
3.4
Summary of the results . . . . . . . . . . . . . . . . . . . . . . . . . .
52
3.5
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
4 Magnetic resonance reversals
57
4.1
About magnetic resonance reversals . . . . . . . . . . . . . . . . . . .
57
4.2
Hypothetical atoms with I=0 . . . . . . . . . . . . . . . . . . . . . .
57
4.3
Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
4.4
Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
4.5
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68
4.6
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
5 Electrolytic fabrication of miniature atomic clock vapor cells
78
5.1
About miniature vapor cells . . . . . . . . . . . . . . . . . . . . . . .
78
5.2
Review of existing fabrication methods of miniature vapor cells . . . .
79
5.2.1
Direct injection . . . . . . . . . . . . . . . . . . . . . . . . . .
79
5.2.2
Chemical reaction . . . . . . . . . . . . . . . . . . . . . . . . .
80
5.2.3
Wax micropackets . . . . . . . . . . . . . . . . . . . . . . . . .
80
viii
5.3
Anodic bonding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
80
5.4
Electrolysis in all-glass cells . . . . . . . . . . . . . . . . . . . . . . .
82
5.4.1
Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . .
82
5.4.2
Visual demonstration . . . . . . . . . . . . . . . . . . . . . . .
85
5.4.3
Physical demonstration . . . . . . . . . . . . . . . . . . . . . .
85
Electrolysis in silicon-based miniature cells . . . . . . . . . . . . . . .
87
5.5.1
Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . .
87
5.5.2
Visual demonstration . . . . . . . . . . . . . . . . . . . . . . .
90
5.5.3
Physical demonstration . . . . . . . . . . . . . . . . . . . . . .
92
5.5.4
Calculation of electrolytic current . . . . . . . . . . . . . . . .
94
5.5
6 Summary
100
Bibliography
101
ix
List of Figures
2.1
A simple example of optical pumping. . . . . . . . . . . . . . . . . . .
2.2
Energy levels of
2.3
Pressure shifts of the hyperfine resonance frequencies of
2.4
Spin-exchange interaction potentials for a colliding pair of alkali-metal
87
Rb and D1, D2 optical transitions. . . . . . . . . .
133
Cs. . . . .
8
13
23
atoms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
2.5
Spin-exchange distribution. . . . . . . . . . . . . . . . . . . . . . . . .
29
3.1
Simplified calculation of the effects of van der Waals molecules.
. . .
37
3.2
The specially designed alkali-metal vapor cell. . . . . . . . . . . . . .
39
3.3
Experimental apparatus of measuring nonlinear pressure shifts. . . . .
40
3.4
An example of the microwave resonance. . . . . . . . . . . . . . . . .
42
3.5
An example of the generation of an error signal for the microwave
resonance frequency feedback loop. . . . . . . . . . . . . . . . . . . .
87
3.6
Nonlinear pressure shifts of
Rb in Ar.
3.7
Physical picture of the nonlinear pressure shifts.
3.8
Summary of the pressure shifts of
x
87
. . . . . . . . . . . . . . . .
Rb and
133
. . . . . . . . . . .
43
46
48
Cs in Ar, Kr, He and N2 . 53
4.1
D1 optical pumping with circularly-polarized σ + light of a hypothetical
alkali-metal atom with no nuclear spin I = 0.
4.2
. . . . . . . . . . . . .
58
D2 optical pumping with circularly-polarized σ + light of a hypothetical
alkali-metal atom with no nuclear spin I = 0.
. . . . . . . . . . . . .
59
4.3
Apparatus of measuring the magnetic resonance reversals. . . . . . .
62
4.4
Specific absorption hAi versus longitudinal magnetic field B curves
when the oscillating magnetic field is perpendicular to the static
longitudinal field and the laser beam. . . . . . . . . . . . . . . . . . .
4.5
64
Specific absorption hAi versus longitudinal magnetic field B curves
when the oscillating magnetic field is parallel to the static longitudinal
field and the laser beam. . . . . . . . . . . . . . . . . . . . . . . . . .
4.6
65
hf m|A|f mi (narrower bars) and hf m|ρ|f mi (wider bars) for each groundstate sublevel at the two different laser detunings. . . . . . . . . . . .
69
4.7
Summary of the results of magnetic resonance reversals. . . . . . . . .
71
4.8
Specific absorption hAi versus static magnetic field B curves at different pumping intensities. . . . . . . . . . . . . . . . . . . . . . . . . .
4.9
71
Experimental results and simulations for a different cell with buffer-gas
mixture of 50 torr N2 and 480 torr Ar. . . . . . . . . . . . . . . . . .
75
5.1
Anodic bonding set-up. . . . . . . . . . . . . . . . . . . . . . . . . . .
81
5.2
Set-up of doing electrolysis by discharge of Xe in a conventional allglass cell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
83
5.3
Discharge of Xe gas and the cell with electrolyzed metal. . . . . . . .
84
5.4
Transmission intensity vs. laser frequency at different temperatures of
the conventional cell.
. . . . . . . . . . . . . . . . . . . . . . . . . .
xi
86
5.5
Schematic of the experimental process for electrolytic filling of siliconbased miniature cells. . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
5.6
Photographs of the top surface of two silicon-based cells. . . . . . . .
91
5.7
Number density of Cs atoms in a silicon-based cell at different temperatures.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
5.8
Microwave end resonance in a silicon-based cell. . . . . . . . . . . . .
93
5.9
Distribution of current from an ion anode to a silicon cathode through
a glass plate.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xii
98
List of Tables
3.1
Fitting parameters of the residual shifts for
87
Rb at 40 o C and
133
Cs
at 35 o C in Kr, Ar, He and N2 buffer gas. . . . . . . . . . . . . . . . .
4.1
54
Some important relaxation mechanisms for polarized alkali-metal atoms. 67
xiii
Chapter 1
Introduction
Ever since Isidor Rabi proposed the idea of making a clock using the atomic beam
magnetic resonance[4], tremendous effort has been put into the studies of making
and improving atomic clocks. Using Rabi’s technique, NIST (was National Bureau of
Standards at that time) announced the first atomic clock using ammonia molecules as
the source of frequency signals in 1948[5]. In 1952, NIST announced the first cesium
beam atomic clock. After about 10 years, Markowitz et. al.[6] suggested that the
atomic second was defined as 9,192,631,770 periods of the vibration of the ground state
hyperfine transition in cesium. Finally in 1967, the International System of Units (SI)
second was defined on the basis of the cesium atom at the 13th General Conference
on Weights and Measures after more than ten years of astronomical definition were
used. So far, four types of atomic clocks have been used and studied. Compared with
the atomic beam clocks, the atomic fountain clocks and the atomic masers, the gascell atomic clocks are less expensive and more compact. Besides, the gas-cell atomic
clocks also have good short-term stability.
1
2
In the operation of a gas-cell atomic clock, optical pumping and microwave resonance are the two key concepts. The idea of optical pumping was proposed by Alfred
Kastler in 1950[7] who won the 1966 Nobel Prize for this work. He pointed out that
by using light with appropriate frequency and intensity to shine on atoms, large population imbalances in the atomic ground- or excited-states could be achieved because
of the resonant absorption and scattering of the photons from the light. The first
optical pumping experiments, where ground-state polarization was observed, were
reported by Brossel et. al.[8] and by Hawkins and Dicke[9] in the system of sodium
atomic beams in 1952-1953. In 1954, Barrat[10] reported successful optical pumping
of sodium vapor in a quartz vapor cell. In a gas-cell atomic clock, atoms contained
in a vapor cell are optically pumped to desired states and the microwave resonances
are excited by an externally applied microwave field. Buffer gases are always used in
a vapor cell to slow down the diffusion of clock atoms to the cell walls with which the
collisions could destroy the polarization of the clock atoms.
The work of this thesis focuses on several of the most important issues of gas-cell
atomic clocks.
One is the effect of the buffer gases on the performance of the atomic clocks[1].
The microwave resonance frequencies of alkali-metal atoms in buffer gases are shifted
depending on the buffer-gas pressure and the cell temperature. This is why gascell atomic clocks are secondary frequency standards[11]. For a fixed temperature,
people have been assuming that the shifts caused by the buffer gases depend on the
buffer gas pressure linearly. This linear pressure shifts are believed to come from the
effect of binary collisions between the alkali-metal atoms and the buffer-gas atoms.
For a qualitative understanding of the linear pressure shift, one could think of the
3
amplitude of the shifts to be proportional to the collision time and frequency (inversely
proportional to the time between collisions). The collision time of a binary collision is
typically in the order of nano second (10−12 ) and the collision frequency is proportional
to the buffer gas pressure. Therefore, the pressure shifts are proportional to the buffer
gas pressure. However, if we consider the existence of van der Waals molecules in the
vapor cell, which has been proved by Bouchiat et. al.[12], it would be natural to think
that the pressure shifts are no longer linearly dependent on the buffer gas pressure.
That is because the formation of van der Waals molecules can be thought of as sticking
collisions. For such collisions, the collision frequency is proportional to the square of
the buffer gas pressure because a three-body collision is needed for a van der Waals
molecule to form. On the other hand, the collision time (the lifetime of the molecules)
is inversely proportional to the buffer gas pressure simply because a two-body collision
is enough to break up a molecule. As a result, the hyperfine frequency shifts caused
by the van der Waals molecules depend on the buffer gas pressure in a much more
complicated way. We developed a simplified model to characterize the pressure shifts
induced by van der Waals molecules. For very low buffer gas pressures (typically
below a few torr), the molecular pressure shifts are linear. For very high buffer gas
pressures (typically above a few tens of torr), the pressure shifts are also linear. In
between, the pressure shifts show a nonlinear dependence on the buffer gas pressure.
When the total pressure shifts are considered, the effect would be to superimpose a
nonlinear pressure shifts curve on a straight line. We set up an apparatus to measure
the precise clock frequency shifts of alkali-metal atoms in different buffer gases and
achieved the accuracy of roughly 1 Hz. Because the contribution from the van der
Waals molecules is much smaller than that from the binary collisions, we had to
4
subtract the big linear contribution from the measured frequency shift to make the
nonlinearities clearer. This is what we called “the residual shift”. We used our simple
model to fit the data and got important information of the interaction coefficients such
as the van der Waals potential, the hyperfine shift coefficient and the spin-rotation
coupling coefficient. This experimental technique proved to be a very useful way to
extract information about the physics of the van der Waals molecules that form in gas
cells. Furthermore, the frequency shifts in Ar have practical interest, since mixtures
of Ar and N2 gases are often used to minimize the temperature coefficient of pressure
shift for gas-cell atomic clocks[11].
The second aspect is the effect of the pumping light on the clock frequency[2]. It
is well understood that for D1 optical pumping of alkali-metal atoms with circularly
polarized laser light, the transmission of the pumping light decreases when a resonant
microwave field is applied. This is because the applied resonant microwave field
decreases the spin polarization of the atoms and lower polarization corresponds to
more absorption and less transmission. However, this is only true when the rule
of lower polarization corresponding to more absorption is valid. We discovered that
under some special conditions, lower polarization corresponds to less absorption. This
means that at resonance, the transmission of the pumping light is more than that
in the case where the applied microwave field is off resonance. If we look at the
transmission versus microwave frequency plot, in some cases the resonance signals
show as peaks and in some other cases they show as dips. We call this phenomenon
the reversals of the magnetic resonances. The conditions for a reversal to take place
include a laser frequency appropriately detuned to pump atoms mainly from the lower
hyperfine multiplet of the ground state to the excited state, sufficiently low pumping
5
intensity, at least partially resolved hyperfine splitting and a circular laser polarization.
On one hand, the reversals can be explained by solving the evolution equation of
the density matrix numerically using a computer program. On the other hand, this
phenomenon can also be understood qualitatively considering the limiting case of spintemperature distribution. The spin-temperature distribution prevails when the spinexchange relaxation mechanism dominates, that is to say, when the spin-exchange
rate is much higher than any other characteristic rate in the system such as the
optical pumping rate or the S-damping rate. When a spin-temperature distribution
is achieved, the absorption of resonant light from the atomic vapor increases with
the polarization of the vapor and the magnetic resonance signals are transmission
peaks. When people try to build miniature/chip-scale atomic clocks, they tend to use
relatively high vapor densities (high spin-exchange rate) and diode lasers with narrow
lines. Therefore, care should be exercised not to operate clocks with parameters close
to those that reduce or reverse the resonance signals.
The third one is the miniaturization of the vapor cells for chip-scale atomic clocks
which is the focus of a project DARPA (Defense Advanced Research Project Agency)
started several years ago. In this project, people have been working to find a way
to mass-produce miniature atomic frequency standards at a cost, size, and power
requirement that would permit wider applications. As a key part of an atomic clock,
a miniature vapor cell is necessary for the miniaturization of the clock. The method
to build an empty miniature vapor cell is quite straightforward. It uses the anodic
bonding of a silicon wafer with a hole and two pieces of Pyrex glass as windows. The
challenging part is how to fill alkali-metal atoms into such a miniature vapor cell.
We developed an electrolytic method to fill controlled amount of alkali-metal vapor
6
into a miniature cell made by anodic bonding[3]. This method is based on the fact
that large amounts of alkali metal can be released by passing an electrolytic current
through hot glass. The material for the electrolysis is a specially made borate glass.
It is made from boron oxide and cesium oxide so that the mobile ions in the glass are
cesium ions. When this special glass is heated and applied with an external electric
field, cesium atoms can be released from the glass. If we put some powder of this
glass in a miniature vapor cell before it is sealed, we then can use the silicon wafer
as the cathode of the electrolysis. The Pyrex window together with some molten
sodium nitrate is used as the anode of the electrolysis. The use of sodium nitrate
is necessary here because it supplies for sodium ions into the Pyrex glass to keep
the electrolysis current continuous. The molten sodium nitrate also make the area
around the electrolysis position hot enough so that the ions in this area have enough
mobility. The electrolytic method of filling miniature vapor cells is relatively simple
and has the advantage of a good control of the amount of the alkali metal filled into
the cell. It also has the potential to be scaled to mass production of miniature vapor
cells for chip scale atomic clocks.
Chapter 2
Basic concepts in optical pumping
2.1
About optical pumping
The basic idea of optical pumping[13] for which A. Kastler[7] received the Nobel Prize
in 1966, is based on the interaction of resonance light with atoms or molecules. The
resonant scattering and absorption of resonance light could lead to large population
imbalances in atomic ground states and excited states. Thus, the polarization could
be transferred from the resonance light to ground-state or excited-state atoms or
molecules. Ground-state polarization generated by optical pumping was first observed
experimentally by Brossel, Kastler, and Winter[8] and by Hawkins and Dicke[9].
A basic optical pumping experiment consists of a light source such as a lamp or a
laser, an atomic vapor which scatters and absorbs the resonant photons, and a method
to detect the degree of polarization that has been attained in the pumped atomic vapor. A simple example of optical pumping is illustrated in Fig. 2.1. Hypothetical
7
2.1. About optical pumping
Laser
8
Quarter
wave plate
Transmission
P1/2
B
1/3
2/3
S1/2
Fluorescence
-1/2
+1/2
Figure 2.1: A simple example of optical pumping. Circularly polarized laser light is used
to illuminate hypothetical alkali-metal atoms with no nuclear spin. A small magnetic field
B is applied along the propagation direction of the pumping light. The spin polarization
of the pumped atoms can be detected by monitoring either the transmission light or the
fluorescence light.
alkali-metal atoms with no nuclear spin are illuminated by circularly polarized resonance light. A small magnetic field B is applied along the propagation direction of
the light. Atoms in the − 12 sublevel of the 2 S1/2 ground state can be excited to the
+ 12 sublevel of the 2 P1/2 excited state by absorbing the circularly polarized photons.
However, atoms in the + 12 sublevel of the ground state can not absorb photons because there is no sublevel in the excited state that can accommodate the additional
angular momentum of the photons. Atoms in the + 12 excited state will decay very
quickly back to the ground-state sublevels with corresponding branching ratios. In
the case shown in Fig. 2.1, atoms are twice as likely to fall to the − 12 sublevel as to
the + 21 sublevel. Given that there is no relaxation between sublevels of the ground
state, eventually all atoms will be pumped into the + 12 sublevel of the ground state
and no further absorption will occur.
2.2. Free atoms
9
There are basically two ways to detect the polarization of the pumped atomic
vapor, transmission monitoring or fluorescence monitoring. The former method monitors the transmitted light whereas the latter one monitors the scattered light. In real
cases, spin relaxation due to different mechanisms affect the ultimate polarization of
the atoms that can be achieved by optical pumping. For example, collisions between
the atoms and uncoated container walls will completely destroy the polarization of
the atoms. One way to overcome this problem is to mix the pumped atoms with
buffer gases such as nitrogen or noble gases. On one hand, the buffer gas molecules
or atoms can effectively slow down the diffusion of spin polarized atoms to the walls
of the container. On the other hand, collisions of the pumped atoms with the buffer
gas molecules or atoms only cause very slow spin relaxation. Therefore, much longer
spin-relaxation times can be obtained with the presence of buffer gases.
In this chapter, we will review the basic concepts and principles of optical pumping,
as well as relaxation mechanisms. We will focus on the systems of alkali-metal atoms.
2.2
Free atoms
The spin wave function |ψi of a free atom evolves according to the Schrödinger equation
ih̄
d
|ψi = H|ψi.
dt
(2.1)
The ground-state spin Hamiltonian of a free alkali-metal atom in an external
magnetic field of strength Bz along the z axis is
Hg = Ag I · S + gS µB Sz Bz −
µI
Iz Bz .
I
(2.2)
2.2. Free atoms
10
Here, Ag I·S is the hyperfine coupling between the nuclear spin I and the electron spin
S with the coupling coefficient Ag . The second term gS µB Sz Bz describes the Zeeman
interaction between the electron spin and the magnetic field, where gS = 2.00232 is
the g value of the electron and µB = 9.2741 × 10−24 JT −1 is the Bohr magneton. The
third term
µI
IB
I z z
represents the Zeeman interaction between the nuclear spin and
the magnetic field, where µI is the nuclear magnetic moment(in units of the nuclear
magneton µN = µB /1836) and I is the nuclear-spin quantum number.
We label the eigenstates of Hg with f , the total angular momentum quantum
number of the state in the limit Bz −→ 0, and m, the quantum number of the
projection of the total angular momentum on z so that
Hg |f mi = Efm |f mi.
(2.3)
f is the eigenvalue of F = I + S and m is the eigenvalue of Fz = Iz + Sz . The possible
values of f are


 I+
f=

 I−
1
2
= a upper hyperfine multiplets;
1
2
= b lower hyperfine multiplets.
(2.4)
To get the eigenvalues Efm of Hg , we use the time-independent perturbation theory(also known as the Rayleigh-Schrödinger perturbation theory) for nondegenerate
cases. We split the Hamiltonian Hg into two parts,
Hg = H0 + V,
(2.5)
H0 = Ag I · S,
(2.6)
where
and
V = gS µB Sz Bz −
µI
Iz Bz .
I
(2.7)
2.2. Free atoms
11
The V = 0 problem can be easily solved and the eigenvalues Ef0m of H0 are simply
Ef0m =
Ag
[f (f + 1) − I(I + 1) − S(S + 1)].
2
(2.8)
From perturbation theory,
∆f m = Ef m − Ef0m
X |Vnk |2
= Vnn +
.
En0 − Ek0
k6=n
(2.9)
Here,
Vnn = hf m|V |f mi
= gS µB hf m|Sz |f miBz −
µI
hf m|Iz |f miBz ,
I
(2.10)
µI
hf m|Iz |f 0 m0 iBz ,
I
(2.11)
and similarly,
Vnk = hf m|V |f 0 m0 i
= gS µB hf m|Sz |f 0 m0 iBz −
Following is a list of the necessary elements for the calculation expressed in terms
fm
of the Clebsch-Gordan coefficients CSm
.
S ImI
hf m|Sz |f mi =
i2 1 h
i2
1 h fm
C 1 1 I(m− 1 ) −
C f1 m
;
1
1
(− 2 )I(m+ 2 )
22
2
2
2
2
(2.12)
i2
i2
1 h fm
1 h fm
hf m|Iz |f mi = (m − ) C 1 1 I(m− 1 ) + (m + ) C 1 (− 1 )I(m+ 1 ) ;
22
2
2
2
2
2
2
(2.13)
ham|Sz |f 0 m0 i = ham|Sz |(a − 1)mi
(2.14)
=
1 am
1 am
bm
C 1 1 I(m− 1 ) C bm
C1 1
11
1 −
1 C1
1
1 ;
I(m−
)
2
22
2
2 22
2 2 (− 2 )I(m+ 2 ) 2 (− 2 )I(m+ 2 )
2.2. Free atoms
12
hbm|Sz |f 0 m0 i = hbm|Sz |(b + 1)mi
=
(2.15)
1 bm
1 bm
am
C 1 1 I(m− 1 ) C am
C1 1
11
1 −
1 C1
1
1 ;
I(m−
)
2
22
2
2 22
2 2 (− 2 )I(m+ 2 ) 2 (− 2 )I(m+ 2 )
ham|Iz |f 0 m0 i = ham|Iz |(a − 1)mi
(2.16)
1
1 am
bm
bm
)C 1 1
= (m − )C am
11
1 C1 1
1 + (m +
1 C1
1
1 ;
I(m−
)
I(m−
)
2
22
2
2 22
2 2 (− 2 )I(m+ 2 ) 2 (− 2 )I(m+ 2 )
hbm|Iz |f 0 m0 i = hbm|Iz |(b + 1)mi
(2.17)
1
1
= (m − )C bm
C am
+ (m + )C bm
C am
;
11
1
I(m− 21 ) 12 12 I(m− 12 )
(− 12 )I(m+ 12 ) 12 (− 12 )I(m+ 12 )
2
2
2
2
2
Inserting proper Clebsch-Gordan coefficients into the equations, we get
ham|Sz |ami =
m
;
2I + 1
(2.18)
hbm|Sz |bmi =
−m
;
2I + 1
(2.19)
ham|Iz |ami = m
2I
;
2I + 1
(2.20)
hbm|Iz |bmi = m
2I + 2
;
2I + 1
(2.21)
Therefore, the energy eigenvalues of Hg to second order in Bz are
Ag
[a(a + 1) − I(I + 1) − S(S + 1)]
2
m(gS µB − 2µI )
+
Bz
[I]
([I]2 − 4m2 )(gS µB + µI /I)2 2
+
Bz ;
2Ag [I]3
Eam =
(2.22)
2.2. Free atoms
13
m= -3
-2
-1
0
1
2
3
2
5 P3/2
52P1/2
D2
D1
52S1/2
Hyperfine
Figure 2.2: Energy levels of 87 Rb together with the D1 and D2 optical transitions. Transitions between ground-state sublevels are hyperfine transitions.
and
Ag
[b(b + 1) − I(I + 1) − S(S + 1)]
2
m[gS µB + 2µI (I + 1)/I]
−
Bz
[I]
([I]2 − 4m2 )(gS µB + µI /I)2 2
−
Bz ;
2Ag [I]3
Ebm =
(2.23)
Here [I] = 2I + 1 is the statistical weight of a spin quantum number.
A diagram of the energy levels of
87
Rb is shown in Fig. 2.2. D1 transition is from
the ground state to the first excited state, where D2 transition is from the ground state
to the second excited state. Transitions between ground-state hyperfine sublevels are
hyperfine transitions which are used in making atomic clocks.
2.3. The density matrix
2.3
14
The density matrix
It is often convenient to describe a system of N identical atoms using the density
matrix. Suppose that each atom in the system is described by a wave function |ψn i,
n = 1, 2, ..., N , then the density operator is
N
1 X
ρ=
|ψn ihψn |.
N n=1
(2.24)
If we expand the wave function of an individual atom in terms of |f mi
|ψn i =
X
cn (f m)|f mi,
(2.25)
fm
the elements of the matrix representation of the density operator ρij can be written
as
ρij = hi|ρ|ji =
N
N
1 X
1 X
hi|ψn ihψn |ji =
cn (i)c∗n (j),
N n=1
N n=1
(2.26)
where |ii and |ji are two eigenstates |f mi and |f 0 m0 i of the ground-state Hamiltonian
Hg . The real diagonal elements ρii of the density matrix are the occupation probabilities (or populations) of state |ii, and the off-diagonal elements ρij (i 6= j) are the
coherences between the two states |ii and |ji. The expectation value of some atomic
2.3. The density matrix
15
observable O is
N
N
X
1 XX ∗
1 X
hψn |O|ψn i =
hOi =
cn (f m)hf m|O|
cn (f 0 m0 )|f 0 m0 i
N n=1
N n=1 f m
f 0 m0
N
1 XXX
hψn |f mihf m|O|f 0 m0 ihf 0 m0 |ψn i
=
N n=1 f m f 0 m0
N
1 XXX
hf m|O|f 0 m0 ihf 0 m0 |ψn ihψn |f mi
N n=1 f m f 0 m0
"
#
N
XX
X
1
=
hf m|O|f 0 m0 ihf 0 m0 |
|ψn ihψn | |f mi
N n=1
f m f 0 m0
XX
hf m|O|f 0 m0 ihf 0 m0 |ρ|f mi
=
=
f m f 0 m0
= T r(Oρ).
(2.27)
Starting from the Schrödinger equation Eq. 2.1, the evolution of the density
matrix is
!
N
1 X
|ψn ihψn |
N n=1
N d
d
1 X
|ψn i hψn | + |ψn i
hψn |
=
N n=1
dt
dt
"
!
! #
N
N
1
1 X
1 X
=
H
|ψn ihψn | −
|ψn ihψn | H
ih̄
N n=1
N n=1
d
d
ρ =
dt
dt
=
1
[H, ρ].
ih̄
(2.28)
It is simpler to do the calculations of optical pumping or spin relaxation in the socalled “Liouville space” than in the customary “Schrödinger space”. In Schrödinger
space, the density operator ρ is a square, Hermitian 2[I] × 2[I] matrix. In Liouville
space, the density operator is represented by a state vector
|ρ) =
X
ij
|ij)(ij|ρ),
(2.29)
2.3. The density matrix
16
of length (2[I])2 . The basis vectors |ij) are defined in the following way
|ij) = |iihj|.
(2.30)
(ij|ρ) = T r[(|iihj|)† ρ] = ρij .
(2.31)
The amplitudes (ij|ρ) are
Generalizing Eq. 2.29 to other operators in Schrödinger space we get
|A) =
X
|ij)(ij|A).
(2.32)
ij
Similarly, the scalar product of any two Liouville space vectors |A) and |B) is
(A|B) = T r(A† B) = (B|A)∗ .
(2.33)
Then the expectation value of a Hermitian operator O is
hOi = T r(Oρ) = T r(O† ρ) = (O|ρ).
(2.34)
A useful representation in analyzing optical pumping experiments is the interaction representation. The interaction-picture wave function is defined as
iHg t/h̄
g
|ψ
|ψn i.
ni = e
(2.35)
The interaction-picture density operator is then
ρe = eiHg t/h̄ ρe−iHg t/h̄ .
(2.36)
If we denote the relaxation and optical pumping mechanisms by L(ρ), the Liouville
equation which describes the rate of change of the density operator is
d
1
ρ = [Hg , ρ] + L(ρ).
dt
ih̄
(2.37)
In the interaction picture, the Liouville equation becomes
d
e
ρe = L(ρ).
dt
(2.38)
2.4. Ground-state relaxation
2.4
17
Ground-state relaxation
Buffer gas atoms or molecules slow down the diffusion speed of the alkali-metal atoms
to the container walls by colliding with them many times without depolarizing them.
However, there are some spin relaxation mechanisms existing in the collisions between
the buffer gas atoms or molecules and the alkali-metal atoms. One is called the “spinrotation interaction”[14, 15]. This interaction couples the electron spin S of the alkalimetal atom to the rotational angular momentum N of the colliding pair and transfers
spin to the translational angular momentum of the buffer gas. The second one is
the “hyperfine-shift interaction”[15, 11, 16]. The hyperfine coupling coefficient Ag
changes during a collision of an alkali-metal atom with a buffer-gas atom or molecule,
therefore causes the pressure shifts of the clock frequencies of gas-cell atomic clocks.
Another important spin relaxation mechanism is the “spin-exchange interaction”[17,
18]. This interaction happens in collisions between pairs of alkali-metal atoms and
conserves the total angular momentum of the colliding pair.
Although the existence of buffer gas molecules or atoms in the vapor container can
hinder the diffusion of spin polarized alkali-metal atoms to the walls of the container,
some can still diffuse through the buffer gas to the walls where most of the spin
polarization is lost.
We will discuss these spin relaxation mechanisms in the following subsections
respectively.
2.4. Ground-state relaxation
2.4.1
18
Spin-rotation interaction
The spin-rotation interaction between the electron spin S of the alkali-metal atom
and the rotational angular momentum N of the colliding pair is
Vsr = γS · N
(2.39)
with γ = γ(r) being the coupling coefficient which depends on the internuclear separation r.
If we name the wave function of the alkali-metal atom before and after the collision
|ψi i and |ψf i respectively, then
|ψf i = e
− h̄i
R
Vsr dt
|ψi i = e−iφ·S |ψi i,
(2.40)
where the electron spin rotation angle is
N
φ=
h̄
Z
∞
γdt.
(2.41)
−∞
Here we have approximated the rotational angular momentum with the constant classical vector N. It has been shown in earlier experiments that the order of magnitude
of γ/h is less than 106 Hz. Considering the fact that the duration of a binary collision
is roughly 10−12 sec and N ∼ 100, the rotation angle φ is on the order of 10−4 radians.
The density matrix after the collision is then
ρf = he−iφ·S ρi eiφ·S iφ ,
(2.42)
where h...iφ denotes an average over all possible collisional rotation angles φ. Since φ
is so small, we can expand Eq. 2.42 to second power in φ
ρf = ρi − i h[φ · S, ρi ]iφ −
1 (φ · S)2 ρi + ρi (φ · S)2 − 2 (φ · S) ρi (φ · S) φ . (2.43)
2
2.4. Ground-state relaxation
19
And the rate of change of the density matrix is simply the average change (ρf − ρi )
per collision times the collision rate 1/T :
1
d
ρ = (ρf − ρi ).
dt
T
(2.44)
We assume isotropic collisions (φ is equally likely to point in any directon) in this
case and that gives us
hφi i = 0,
(2.45)
1
hφi φj i = δij hφ2 i,
3
(2.46)
and
Here we have taken away the average subscript from h...iφ . Therefore,
ρf − ρi = −
hφ2 i
[S(S + 1)ρi − S · ρi S] ,
3
(2.47)
where S is the electron spin quantum number. For alkali-metal atoms that we are
interested in, S =
1
2
and from Eq. 2.44 we get
hφ2 i 3
d
ρ=−
ρ − S · ρS = Γsr (ϕ − ρ).
dt
3T 4
(2.48)
Here
Γsr =
hφ2 i
3T
(2.49)
is called the spin-rotation rate (it is called the S-damping rate Γsd in [15]).
1
ϕ = ρ + S · ρS
4
(2.50)
is the part of the density matrix without electron polarization. Then the rest part of
the density matrix which contains electron polarization is
3
Θ · S = ρ − S · ρS.
4
(2.51)
2.4. Ground-state relaxation
20
Now let’s calculate the relaxation rate of the longitudinal spin polarization
d
d
d
Fz = Tr(Fz ρ) = Tr(Fz ρ).
dt
dt
dt
(2.52)
d
3
Fz = − Γsr Tr(Fz ρ) + Γsr Tr(Fz S · ρS).
dt
4
(2.53)
Inserting Eq. 2.48 we get
Expanding the second half of the right side we get
Tr(Fz S · ρS) = Tr(Fz Sx ρSx + Fz Sy ρSy + Fz Sz ρSz )
= Tr[(Sx Fz Sx + Sy Fz Sy + Sz Fz Sz )ρ]
= Tr(Fz S 2 ρ − Sz ρ)
=
3
hFz i − hSz i.
4
(2.54)
Here we use the property of trace that Tr(ABCD) = Tr(DABC). Therefore,
d
hFz i = −Γsr hSz i.
dt
(2.55)
This interaction is also called ”S-damping” in a lot of references.
2.4.2
Hyperfine-shift interaction
During the time of a collision between an alkali-metal atom and a buffer-gas atom,
the wave function of the valence electron of the alkali-metal atom will be slightly
perturbed by the approaching buffer-gas atom. This perturbation causes the amplitude of the valence-electron wave function at the nucleus of the alkali-metal atom to
change a little bit, thereby modifying the ground-state hyperfine interaction Ag I · S.
This effect can be written as
Vhs = δAg (r)I · S.
(2.56)
2.4. Ground-state relaxation
21
At large r, the term δA comes from the electrostatic Van der Waals forces which
tend to pull the valence electron away from the nucleus of the alkali atom and therefore
to decrease the contact interaction. Hence δA(r) is negative at large interatomic
separations. At very small r, exchange forces between the alkali valence electron and
the buffer-gas electrons tend to squeeze the valence electron strongly at the center of
the alkali atom and thereby to increase the contact interaction. So δA(r) is positive
at small interatomic separations. The hyperfine-shift interaction δAg (r)I · S is the
reason for the well-known pressure shifts of the 0-0 hyperfine transition of alkali-metal
atoms in buffer gases.
The time-evolution operator for the pressure-shift interaction Eq. 2.56 is[19]
Uhs = e−iηφC I·S .
(2.57)
where η is a dimensionless number which is defined as
η=
µI
,
2IµN
(2.58)
and φC is
1
φC =
h̄
Z
+∞
δ Ā(t0 )dt0
(2.59)
−∞
Here δ Ā is the hyperfine shift coefficient of a hypothetical Rb isotope with nuclear
spin 1/2 and nuclear moment of one nuclear magneton µN .
In the case of binary collisions (∼ 10−12 sec), the phase φC is very small and the
time-evolution operator can be expanded in powers of ηφC as
Uhs ≈ 1 − iηφC I · S −
η 2 φ2C
(I · S)2 + · · · .
2
(2.60)
Then we have
ρf = U ρU † = ρ − iηφC [I · S, ρ] −
η 2 φ2C
{(I · S)2 , ρ} + η 2 φ2C I · SρI · S.
2
(2.61)
2.4. Ground-state relaxation
22
Here, the second term which is the first order in φC causes the pressure shift of the
clock frequency. Assuming that the collision rate is 1/T and the average phase during
a collision is hφC i, then the time rate of change of the density matrix caused by this
term is
1
d
ρ=
[δH, ρ] ,
dt
ih̄
(2.62)
ηhφC i
I · S.
T
(2.63)
where
δH = h̄
Therefore the shift in the ground-state hyperfine frequency due to the interaction
discussed above is
ηhφC i
δν =
2πT
1
I+
.
2
(2.64)
Since 1/T is proportional to the buffer-gas pressure, the frequency shift due to binary
collisions is also proportional to the buffer-gas pressure.
The pressure shifts of the hyperfine resonance frequencies of
133
Cs in different
buffer gases are shown in Fig. 2.3, from Ref. [20]. We can see from the figure that
lighter buffer-gas atoms cause positive pressure shifts and heavier ones cause negative
shifts. This could be understood in a qualitative way considering the competition
between the exchange forces and the Van der Waals forces on the valence electrons
of the alkali-metal atoms. Heavier hence larger buffer-gas atoms such as Kr and Xe
are highly polarizable thus inducing larger Van der Waals interactions. Lighter and
smaller buffer-gas atoms such as He and Neare not very polarizable therefore exchange
forces dominate during the collisions.
The third and forth terms in Eq. 2.61 which are the second order in φC account
for the relaxation. This effect can be parametrized by the isotope-independent Carver
2.4. Ground-state relaxation
23
Figure 2.3: Pressure shifts of the hyperfine resonance frequencies of 133 Cs, from Ref. [20].
rate ΓC = hφ2C i/T in honor of Carver’s pioneering work in the studies of the hyperfine
pressure-shift effect.
2.4.3
Spin-exchange interaction
There are two kinds of spin-exchange interaction in a system of alkali-metal atoms
and buffer-gas atoms. One is the electron-spin exchange interaction between colliding
alkali-metal atoms. The other couples the nuclear spin of buffer-gas atoms to the
electron spin of alkali-metal atoms. The spin-exchange interaction is a very important
relaxation mechanism which can transfer spin polarization from atoms of one kind
to another. This makes it possible to polarize those atoms such as 3 He or
129
Xe
which cannot be optically pumped easily. The idea would be to spin polarize some
2.4. Ground-state relaxation
24
atoms (such as alkali-metal atoms) which can be optically pumped easily and then let
the spin-exchange interaction transfer the spin polarization. Those atoms cannot be
easily polarized by optical pumping because their resonance absorption lines line in the
vacuum ultraviolet region where good light sources and optical components are hard
to find, expensive or even unobtainable. We will discuss the electron-spin exchange
interaction between colliding alkali-metal atoms, but similar consideration is valid
for the nuclear-electron spin-exchange interaction. The most remarkable property of
spin-exchange collisions is that even though the spin of an individual atom may flip
during a collision, the total spin of a colliding pair of atoms is conserved.
A spin-exchange collision between two alkali-metal atoms A1 and A2 can be represented by
A1 (↑) + A2 (↓) → A1 (↓) + A2 (↑).
(2.65)
Here the arrows denote the direction of the electron spins. Before the collision, the
spin of atom A1 is up, while the spin of atom A2 is down. After the collision, the spin
orientations of the two atoms are exchanged with spin down for A1 and spin up for
A2 .
When two atoms come close to each other, the interaction between their electron
spins S1 and S2 causes the interaction potential curve to split into two curves, corresponding to the singlet (total electron spin is 0) and triplet (total electron spin is 1)
state respectively. The interaction potential curves as a function of the internuclear
separation r for Rb-Rb colliding pair are shown in Fig. 2.4. In this case, the triplet
potential is repulsive, while a part of the singlet potential is attractive. The total interaction potential is the sum of a spin-independent part and a spin-dependent part,
2.4. Ground-state relaxation
25
Figure 2.4: Spin-exchange interaction potentials for a colliding pair of alkali-metal atoms,
from Ref. [21].
Vse (r) = V0 (r) + J(r)S1 · S2 .
(2.66)
Taking the expectation value of S1 · S2 , the interatomic potentials of the singlet and
triplet states are then
3
Vs (r) = V0 (r) − J(r)
4
1
Vt (r) = V0 (r) + J(r).
4
(2.67)
(2.68)
Right before the collision, the wave function of the colliding pair can be written
as
|ψi i = |ψ1 i|ψ2 i.
(2.69)
2.4. Ground-state relaxation
26
This wave function contains both singlet and triplet spin components. By using the
projection operators
1
− S1 · S2
4
3
=
+ S1 · S2 ,
4
Ps =
(2.70)
Pt
(2.71)
we can project the two components out of the total wave function. Therefore, the
final wave function after the collision will be
|ψf i = Use |ψi i,
(2.72)
where the unitary time-evolution operator Use associated with the spin-exchange process is
Use = e−iφs Ps + e−iφt Pt
and the phase evolution angles can be written semiclassically
Z
1 +∞
Vs (t0 )dt0
φs =
h̄ −∞
Z
1 +∞
φt =
Vt (t0 )dt0 .
h̄ −∞
(2.73)
(2.74)
(2.75)
For uncorrelated colliding pair of atoms before the collision, the density operator is
simply
(ρ12 )i = ρ1 ρ2 .
(2.76)
Then the density operator of the colliding pair after the collision is
(ρ12 )f = Use ρ1 ρ2 Use† .
(2.77)
Substituting Eq. 2.73 into Eq. 2.77 we get
(ρ12 )f = ρ1 ρ2 + i sin(∆φ)[ρ1 ρ2 , S1 · S2 ]
3
∆φ
2
+ sin
− ρ1 ρ2 + {ρ1 ρ2 , S1 · S2 } + 4S1 · S2 ρ1 ρ2 S1 · S2 ,(2.78)
2
4
2.4. Ground-state relaxation
27
where the phase difference is
1
∆φ = φt − φs =
h̄
Z
+∞
J(t0 )dt0 .
(2.79)
−∞
We assume that the density operators of the colliding atoms remain uncorrelated after
the spin-exchange collisions due to the random nature of collisions in a gas cell. Then
the time rate of change of the density matrix due to spin-exchange collisions is
d
1
(ρ1 ) = [T r2 (ρ12 )f − ρ1 ].
dt se
T
(2.80)
where 1/T is the collision rate. Substituting Eq. 2.78 into Eq. 2.80 we find
1
3
d
(ρ1 ) =
[− ρ1 + S1 · ρ1 S1 + hS2 i · S1 ρ1 + ρ1 S1 · hS2 i
dt se
Tse 4
− 2ihS2 i · (S1 × ρ1 S1 )] − i[δω · S1 , ρ1 ],
(2.81)
where the spin exchange rate is
1
1
=
Tse
T
∆φ
sin
2
2
(2.82)
and the rotation rate is
δω =
The spin exchange rate
1
Tse
hsin ∆φi
hS2 i.
T
(2.83)
is proportional to the number density of the alkali-metal
atoms.
Considering the limiting case where the spin exchange rate is much faster than
other relaxation rates, the evolution equation for the spin polarized atoms is
d
d
1
ρ1 =
(ρ1 ) + [Hg , ρ1 ].
dt
dt se
ih̄
(2.84)
There is only one possible equilibrium solution to Eq. 2.84, as was first pointed out
by Anderson et al.[22]. The solution is[23]
ρ=
eβFz
eβIz eβSz
=
.
Z
ZI ZS
(2.85)
2.4. Ground-state relaxation
28
The partition function (Zustandssumme) Z is the product of a nuclear part ZI and
an electronic part ZS with the general form for a spin quantum number J
ZJ =
J
X
m=−J
eβm =
(1 + P )[J] − (1 − P )[J]
sinh(β[J]/2)
=
,
sinh(β/2)
2P (1 − P 2 )J
(2.86)
where P is the overall spin polarization which is defined in terms of the mean electron
spin. The constant β is called the spin-temperature parameter which is related to the
spin polarization P by
P = 2hSz i = tanh
β
2
or
β = ln
1+P
.
1−P
(2.87)
In spin-temperature equilibrium, the diagonal elements of the atomic density operator
ρ (also the populations of the ground-state sublevels) are
hf m|ρ|f mi =
eβm
.
Z
(2.88)
The spin-temperature distribution is sketched in Fig. 2.5. For sufficiently dense
alkali-metal vapors the spin-temperature distribution is a good description of the spin
polarization.
2.4.4
Relaxation due to spatial diffusion
In a sealed volume where the alkali-metal atoms and the buffer-gas atoms are contained, the alkali-metal atoms will diffuse through the buffer-gas atoms to the walls of
the container. These collisions with the walls will totally destroy the spin polarization
of the alkali-metal atoms. The diffusion equation for the density operator ρ is
∂
ρ = D∇2 ρ,
∂t
(2.89)
2.4. Ground-state relaxation
29
Figure 2.5: Spin-exchange distribution, from Ref. [24].
where D = λv/3 is the diffusion coefficient of the alkali-metal atoms in the gas, λ
is the mean free path of the alkali atoms and v is the man thermal velocity. D is
inversely proportional to the buffer-gas pressure.
Assuming that the alkali atoms diffuse into and out of a cubic volume with dimension l × l × l where the atoms are polarized and outside atoms are unpolarized, then
the effects of the diffusion can be parameterized with a relaxation rate Γd = 3D(π/l)2 .
Similarly, if the volume can be treated as a 2D space, then the corresponding relaxation rate is Γd = 2D(π/l)2 and in a 1D case it is Γd = D(π/l)2 .
2.5. Optical pumping
2.5
30
Optical pumping
Suppose the applied pumping light is a monochromatic plane wave traveling along
the ẑ direction with the transverse electric field being
E = E0 ei(k·r−ωt) + c.c.,
(2.90)
where E0 is the complex amplitude of the electric field. This oscillating electric field
induces an oscillating electric dipole moment in the atom being pumped
hPi = hαi · E0 ei(k·r−ωt) + c.c.,
(2.91)
where hαi is the dielectric polarizability tensor. Then the mean optical power absorbed by the atom is
P = −iωE∗0 · hαi · E0 + c.c..
(2.92)
For D1 optical pumping, the dielectric polarizability tensor is α = α(1 − 2iS×)
while for D2 optical pumping, α· = α(1 · +iS×)[23]. Here α = α0 + iα00 is a complex
number. Inserting the dielectric polarizability into Eq. 2.92 we get
P = hσihνΦdν,
(2.93)
cE02
,
2πhν
(2.94)
where
Φ(ν)dν =
is the photon flux of the light wave.
For D1 optical pumping,
hσi = σop (1 − 2s · hSi),
(2.95)
2.5. Optical pumping
31
and for D2 optical pumping,
hσi = σop (1 + s · hSi),
(2.96)
where σop = 4πkα00 is the photon absorption cross section for unpolarized atoms. The
mean photon spin is
s=
2.5.1
1 ∗
E × E0 ,
iE02 0
(2.97)
Depopulation pumping
Depopulation pumping occurs when atoms on some ground-state sublevels absorb
light more than those on other ground-state sublevels. Therefore, atoms tend to
accumulate on the sublevels where they absorb less light. In the case of alkali-metal
atoms, circularly-polarized light is always used for depopulation pumping to generate
population imbalances between the hyperfine multiplets. Fig. 2.1 shows an example
of depopulation pumping. This is a limiting case where atoms on the spin-up state
will not absorb any light. Atoms from the excited state can decay spontaneously to
either of the two ground-state levels. If there is no relaxation happening between
the ground-state sublevels, eventually all atoms will be pumped to the spin-up state
and no further absorption will happen. Actually, in depopulation pumping, it does
not matter much whether the excited atoms decay by spontaneous radiation or by
non-radiative quenching.
2.5.2
Repopulation pumping
Repopulation pumping happens when atoms from the excited state decay to the
ground state by spontaneous radiation which can be seen also in Fig. 2.1. Atoms
2.5. Optical pumping
32
from the spin-up excited-state sublevel decay to the spin-up and spin-down groundstate sublevels with different branching ratios ( 31 and
2
3
respectively). In this way,
polarization of the excited state is transferred to the ground state (but in an opposite
direction in the case of Fig. 2.1). Accordingly, repopulation pumping will lose its
efficiency if atoms on the excited state collide with buffer-gas atoms and lose their
polarization. However, this will increase the optical pumping efficiency. In the case
of no excited-state mixing, 1/3 of the excited atoms come back to the |1/2i groundstate sublevel, but in the case of complete excited-state mixing due to collisions with
buffer-gas atoms, 1/2 of the excited atoms decay to the |1/2i ground-state sublevel.
Chapter 3
Nonlinear pressure shifts
3.1
About nonlinear pressure shifts
About forty years ago, Bouchiat [12, 25] and her colleagues proved the existence
of Rb-rare-gas Van der Waals molecules with their pioneering work in the study of
the spin relaxation of optically polarized Rb atoms in the presence of a rare gas.
The effect of the formation of Van der Waals molecules on the relaxation rate of
Rb atoms demonstrated in their experiments came from the spin-rotation interaction
γS · N between the rotational angular momentum N of the molecule and the electron
spin S of the alkali-metal atom. Recalling our discussion about the spin-relaxations
mechanisms in Chapter 2, when an alkali-metal atom interact with a buffer-gas atom,
there exists another interaction which is called the “hyperfine-shift interaction”. It is
often written as δAI·S, which means the modification of the Fermi contact interaction
between the nuclear spin I and the electron spin S. Therefore, when a Van der waals
molecule forms, there should be two interactions, the spin-rotation interaction γS · N
33
3.1. About nonlinear pressure shifts
34
and the hyperfine-shift interaction δAI · S. In the following simplified model, we
will show that the interaction δAI · S in molecules induces nonlinear pressure shifts
to the linear pressure shifts of the hyperfine transition frequencies caused by binary
collisions of the alkali-metal atoms with the buffer-gas atoms.
Assuming that the buffer gas is composed of one component such as Ar, Kr or
Xe. Consider the case where the alkali-metal atoms are in a coherent superposition
state between a sublevel |αi of the upper hyperfine multiplet with the total angular
momentum quantum number f = a = I +1/2 and a sublevel |βi of the lower hyperfine
multiplet with f = b = I − 1/2. Then the coherence for the microwave resonance
between |αi and |βi represented by the density operator ρ can be written as
hα|ρ|βi = ρ1 + ρ2 .
(3.1)
Here ρ1 is for free alkali-metal atoms and ρ2 is for atoms in Van der Waals molecules.
Accordingly, the hyperfine resonance frequency for free alkali-metal atoms is ω1 and
for atoms in Van der Waals molecules is ω2 . Let the difference between ω1 and ω2
be Ω. Suppose that the formation rate of Van der Waals molecules between free
alkali-metal atoms and the buffer-gas atoms is
1
= κf p2 ,
τ1
(3.2)
and the break-up rate of the molecules is
p
1
=
.
τ2
κm
(3.3)
Here, p is the pressure of the buffer gas. κf and κm are two pressure-independent
coefficients which depend slightly on the temperature of the mixture. The reason for
the formation rate to the proportional to the square of the buffer-gas pressure is that
3.1. About nonlinear pressure shifts
35
the formation process is a three-body collision. Because the binding energy between
an alkali-metal atom and a buffer-gas atom due to the Van der Waals attraction is
comparable to kT , a three-body collision is necessary for taking away binding energy
of the newly formed molecule. And also because of the shallow attractive potential,
any collision of the molecule with another atom will break the molecule. This is
why the break-up rate is proportional to the buffer-gas pressure. The inverse of the
break-up rate is also the lifetime of the molecule which is inversely proportional to
the buffer-gas pressure. The typical lifetime of a binary collision is roughly 10−12 sec
while the typical duration of a Van der Waals molecule is on the order of 10−9 sec.
For simplicity, we consider only the hyperfine coupling of the alkali-metal atoms
and the effects of the Van der Waals molecules. Then the time evolution of the
coherence amplitudes can be written as
1
1
dρ1
= −(iω1 + )ρ1 + ρ2 ,
dt
τ1
τ2
(3.4)
1
1
dρ2
= ρ1 − (iω2 + )ρ2 .
dt
τ1
τ2
(3.5)
and
The coherence amplitude ρ1 for free alkali-metal atoms evolves at the frequency ω1
and damps at the rate 1/τ1 due to the formation of Van der Waals molecules. But
at the same time, the break-up of Van der Waals molecules contribute
1
ρ
τ2 2
to ρ1 .
The interpretation of Eq. 3.5 is similar. We have neglected the mechanisms such as
optical pumping, spin-rotation and so on.
To solve the equations, we write the coherence amplitudes as


 ρ1 
|ρ) = 
.
ρ2
3.1. About nonlinear pressure shifts
36
Then Eq. 3.4 and Eq. 3.5 can be written as
d
|ρ) = −Γρ,
dt
(3.6)
where the evolution matrix Γ is


 iω1 +
Γ=
− τ11
1
τ1
− τ12
iω2 +
1
τ2

.
We look for the solution of the form
|ρ) = e−iω |λ),
(3.7)
where |λ) is a time-independent eigenvector, and λ is a complex eigenvalue. Substituting Eq. 3.7 into Eq. 3.6 we get the eigenvalue equation
(Γ − iω)|λ) = 0.
(3.8)
In order for Eq. 3.8 to have nontrivial solutions, we need
det(Γ − iω) = 0.
(3.9)
Suppose that the solutions of the above equation are ω± , the shift in the hyperfine
resonance frequency due to Van der Waals molecules is
∆ν =
1
<(ω− − ω1 ),
2π
(3.10)
and the damping rate caused by the Van der Waals molecules is
γ = −=(ω− − ω1 ).
(3.11)
Here < and = represent the real and imaginary parts of a complex number. The
damping rate γ contribute γ/π to the FWHM (full width at half maximum) of the
resonance.
3.1. About nonlinear pressure shifts
37
0
Re(ω--ω1)/(2π) (Hz)
-2000
-4000
-6000
-8000
-10000
-12000
-14000
0
10
20
30
40
50
Buffer gas pressure (torr)
Re(i[ω--ω1]) (sec-1)
25000
20000
15000
10000
5000
0
0
10
20
30
40
50
Buffer gas pressure (torr)
Figure 3.1: An example of the real part (upper panel) and imaginary part (lower panel)
of (ω− − ω1 ) with corresponding parameters of Ω = ω2 − ω1 = 5 × 108 sec−1 , κf = 1000
torr−2 sec−1 , κm = 1 × 10−8 torr sec, and the contribution from binary collision to the shift
−1100 Hz/torr.
3.2. Apparatus
38
At very low buffer-gas pressure, ω− = ω1 − i/τ1 which means that free alkalimetal atoms oscillate at the angular frequency ω1 for a long period of time before
they form Van der Waals molecules. ω+ = ω2 − i/τ2 , which means that alkali-metal
atoms bound in Van der Waals molecules oscillate at ω2 for a long time before they
are released to be free atoms because the break-up rate is so low at low buffer-gas
pressures. However, we would expect that at low pressure, it is almost impossible
to observe the signal from alkali-metal atoms in Van der Waals molecules because
the formation rate is so low and the linewidth is on the order of MHz. At very high
buffer-gas pressure, ω+ = (ω1 + Ω/2) − i/τ2 which represents a rapidly damping (on
the order of 107 sec−1 ) resonance. ω− = (ω1 + Ωτ2 /τ1 ) − i(Ωτ2 )2 /τ1 which represents
a linear pressure shift Ωτ2 /τ1 in the resonance frequency and a pressure-independent
damping rate (Ωτ2 )2 /τ1 . The above discussion is for high- and low-pressure limits.
For pressures in between and with appropriate simplifications, we can calculate the
pressure shift to be
∆ω = <(ω− − ω1 ) =
κf κm Ωp3
.
p2 + κ2m Ω2
(3.12)
Considering the linear pressure shift contributed by the binary collisions, the total
pressure shift should depend on the buffer-gas pressure nonlinearly due to the contribution from the Van der Waals molecules as represented by Eq. 3.12. In both low
and high buffer-gas pressure limits, the pressure shift approaches a straight line.
3.2
Apparatus
As shown in Fig. 3.2, the experimental data were taken from a specially designed cell
connected to a string. The glass part includes the cylindrical cell (25 cm long and 17
3.2. Apparatus
39
buffer-gas reservoir
3
1
pressure gauge
2
glass part
metal part
cell
Figure 3.2: The specially designed alkali-metal vapor cell.
mm in diameter) and part of the string. The metal part consists the main string, a
buffer-gas reservoir and a pressure gauge. The main string is made of VCR fittings.
The buffer-gas reservoir is made of stainless steel and has the volume of 50 ml. The
pressure gauge is a Baratron 626A capacitance manometer from MKS Instruments.
The full scale pressure range of the gauge is 100 torr. The accuracy is 0.25% of reading
for pressures above 4 torr and 0.01 torr for pressures below 4 torr. The whole string is
shut off from the atmosphere with a shut-off valve 1. The string can be connected to
our vacuum system to be pumped and refilled with the designated buffer gas. When
the string is filled, for example, with Ar gas, it is actually the buffer-gas reservoir
which is filled with a few hundred torr of Ar. Other parts are kept under vacuum
(typically 10−7 torr). The buffer-gas reservoir is closed with a shut-off valve 3 as
3.2. Apparatus
40
PE
DL
FR
LCW
BS
PO1 PO2
BE
I
O
HC
PD
L
NDF
CP
H
FG 2 Hz
FC
PID2
FS
LA2
400 Hz
LA1
PID1
Figure 3.3: Experimental apparatus. DL, diode laser; FR, Faraday rotator; PE, pellicle;
PO, polarizer; LCW, liquid crystal waveplate; BS, beam shaper; NDF, neutral density
filter; BE, beam expander; I, iris; O, oven; H, horn; HC, Helmholtz coils; L, lens; PD,
photodetector; CP, current preamplifier; LA, lock-in amplifier; PID, PID controller; FS,
frequency synthesizer; FC, frequency counter; FG, function generator.
well as a needle valve 2. During the experiments, the buffer gas is first released from
the reservoir through valve 3 to the volume between valve 2 and 3. Then valve 3 is
closed and the buffer gas is released into the cell through the needle valve 2. After
the system comes to an equilibrium, we read the pressure of the buffer gas using the
pressure gauge supposing that the pressure at the position close to the gauge is the
same as that in the vapor cell. By doing all these, we can change and measure the
buffer gas in the vapor cell in the step as small as 0.01 torr.
3.2. Apparatus
41
Now let’s look at the experimental setup which is shown in Fig. 3.3. During the
measurement, the cylindrical vapor cell was mounted in a oven, the temperature of
which is controlled with hot air flow together with a Omega temperature controller.
For the data we presented here, the temperature in the oven was stabilized at 40
◦
C for
87
Rb and 35 ◦ C for
133
Cs. Three sets of Helmholtz coils were used to cancel
transverse components of the ambient magnetic field and to produce a longitudinal
static field B on the order of 1 G, parallel to the pumping laser light. A microwave
horn antenna was pointed to the vapor cell from a direction perpendicular to the
laser beam. It was connected to a microwave synthesizer to apply an oscillating
magnetic field in order to drive the hyperfine magnetic resonances of the alkali-metal
atoms. The oscillating direction of the microwave field was along the beam direction
while the propagation direction of the microwave field was perpendicular to the beam
direction. The laser light was from a diode laser (Toptica DL100) stablized by an
external cavity. For
87
Rb, the wavelength of the laser light is 795-nm and for
133
Cs
it is 895-nm. A linear polarizer was used to eliminate slight imperfections in the
natural linear polarization of the laser light. A Faraday-rotation isolator was used
to keep light from being reflected back into the laser head from downstream optics
because these back-reflections will greatly affect the stability of the output spectrum
of the laser. A small fraction of the light was deflected by a pellicle to an optical
spectrum analyzer and a wave meter (not shown). A function-generator-controlled
liquid crystal wave plate together with a linear polarizer were used to modulate the
intensity of the laser light by 30-40% at the speed of a few Hz. At the same time,
the mean beam intensity could be changed with a rotatable neutral density filter. To
illuminate the vapor in the cell volume uniformly with light, we used a beam expander
3.2. Apparatus
42
Photo detector signal (V)
1.240
1.235
1.230
1.225
9.192627
9.192628
9.192629
9.192630
9.192631
Microwave frequency (GHz)
Figure 3.4: The 0-0 microwave resonance signal of
FWHM is 651 Hz.
133 Cs
in Kr of 3 torr at 40 ◦ C. The
together with a iris. A photodetector was used to detect the transmission light from
the vapor cell to supply for the error signals of the two feedback loops.
We used two feedback loops to lock the output frequency of the microwave synthesizer to the intrinsic 0-0 hyperfine frequency of the alkali-metal atoms which was
independent on the laser light intensity. The first loop was supposed to lock the
output frequency of our frequency synthesizer to the 0-0 resonance frequency. An
example of the 0-0 microwave resonance signal is shown in Fig. 3.4. The FWHM is
651 Hz. A schematic of how we generate the error signal for this feedback loop is
shown in Fig. 3.5. To generate the error signal, we modulated the output microwave
frequency of the synthesizer at 400 Hz and detected the 400-Hz signal from the output
of the photodetector with a lock-in amplifier (SR830 from SRS) with a time constant
of 10 ms. This is shown in panel (a) of Fig. 3.5. The output signal from the lock-in
3.2. Apparatus
43
Transmission signal (a.u.)
(a)
-50
-40
-30
-20
-10
0
10
20
30
40
50
0
10
20
30
40
50
Error signal (a.u.)
(b)
zero-crossing point
0
-50
-40
-30
-20
-10
Frequency detuning (a.u.)
Figure 3.5: An example of the generation of an error signal for the first feedback loop.
(a), the transmission of light recorded by the photodetector when the microwave frequency
is tuned across the resonance frequency; (b), the error signal generated by modulating the
microwave frequency and recording the corresponding oscillation amplitude of the transmission signal with a lock-in amplifier.
3.2. Apparatus
44
amplifier is like the one shown in panel (b) of Fig. 3.5 which is the error signal for the
feedback loop. The zero-crossing point at the center of the figure is where we hope
to lock the system. The error signal from the lock-in amplifier was sent to a PID
controller (SIM960 from SRS) with appropriate parameters. The PID controller then
sent a feedback signal to the synthesizer. The locked output frequency of the synthesizer was measured with a frequency counter which used a Rb frequency standard as
an external time reference.
The second loop was used to lock the laser frequency to the “sweet spot” (zeroshift frequency) where there was no light shift[26]. Laser frequencies higher than
the zero-shift frequency cause the microwave resonance frequency to be smaller and
those lower than the zero-shift frequency cause the microwave resonance frequency
to be larger. This feature is usually represented as a curve of light shift (shift in
the microwave resonance frequency by unit intensity change of the laser) versus laser
frequency detuning. If we take
87
Rb for example, this curve around the zero-crossing
point is similar to the one shown in Fig. 3.5. To eliminate this light-shift effect, we
modulated the laser light intensity at 2 Hz. If the laser frequency was not exactly
at the zero-crossing spot, the 2 Hz modulation in the laser intensity would induce
a 2 Hz modulation in the microwave resonance frequency. What the first feedback
loop (discussed in the previous paragraph) did was to make the output frequency
of the microwave synthesizer to follow the 0-0 hyperfine resonance frequency of the
alkali-metal atoms. Therefore, the modulation in the microwave resonance frequency
would appear as an oscillation in the feedback signal of the first feedback loop. We
then used the feedback signal from the PID controller of the first loop as the input
into the lock-in amplifier of the second feedback loop and the output signal from this
3.3. The result of Rb and the physical picture
45
lock-in amplifier would be the error signal for the second feedback loop. The time
constant of this lock-in amplifier was 3 sec. Similarly, the PID controller of this loop
always tried to pull the laser frequency to the point where the error signal was zero
which was the point where there was no light shift. The feedback signal from the
second PID controller was sent to the piezo-actuator mounted in the laser head.
By doing all the above, we were able to achieve the level of accuracy of about 1
Hz.
3.3
The result of Rb and the physical picture
The frequency measurement result of
87
Rb in Ar is shown in Fig. 3.6. Panel (a)
shows the measured 0-0 hyperfine resonance frequencies of
87
Rb in Ar with different
pressures. The temperature of the experiment was 40 ◦ C and the static magnetic
field was 2 G along the light beam. The solid circles are the experimental data. The
solid line is the high-pressure limit ν0 + sp where ν0 is the zero-pressure frequency
and s is the coefficient of the frequency shift. The dashed line is the low-pressure
limit (s − sm )p = sb p which includes only the contribution from binary collisions.
From panel (a), we can see that the data points actually do not line up to because
the low-pressure limit and the high-pressure limit do not have the same slope. The
nonlinearity is so small that it is almost invisible if we do not put the two limiting
lines in the figure. To make this nonlinear feature more visible, we subtracted the
high-pressure limit line from the measured data and got the result shown in the panel
(b) of Fig. 3.6, which is called the residual shift ∆2 ν = ν − ν0 − sp. It is easy to
see that the high-pressure data points approach a horizontal line. And the deviation
(a)
ν-6,834,680,000 (Hz)
3.3. The result of Rb and the physical picture
46
Experimental data
4600
high-pressure limit
4300
low-pressure limit
4000
3700
3400
0
∆2 ν (Hz)
(b)
5
10
15
20
1
0
-1
-2
-3
-4
-5
-6
25
30
residual shift
fitting
0
5
10
15
20
25
30
Ar pressure p (torr)
Figure 3.6: (a), the measured 0-0 resonance frequencies ν of 87 Rb in Ar at a temperature
of 40 ◦ C and a magnetic field B=2 G. The error bars of the data points are too small to
display. The solid line in this panel is the linear, limiting shift, ν0 + sp. The dashed line is
the contribution from only binary collisions in the low pressure limit, (s − sm )p = sb p; (b),
the residual shift ∆2 ν = ν − ν0 − sp. The solid curve is the fitting from Eq. 3.13.
3.3. The result of Rb and the physical picture
47
of the points at lower pressures from the horizontal line is the nonlinear effect. The
solid line in panel (b) is the fitting to the data with the function
φ3
1
2
.
∆ ν=−
2πT 1 + φ2
(3.13)
Here 1/T is the effective three-body formation rate and φ is the effective hyperfine
phase shift. Both T p2 and φp are pressure independent but weakly dependent on
temperature. We will discuss the physics that leads to the nonlinear pressure shift
and derive the fitting function in the following paragraphs.
Fig. 3.7 shows the formation and breaking-up of a Van der Waals molecule RbAr
as well as the phase shift induced to the phase of the microwave coherence of a bound
atom. At the time t − τ , a three-body collision Rb + Ar + Ar → RbAr + Ar takes
place and a Van der Waals molecule RbAr is formed in the potential well Vm . This
molecule evolves for a period of τ before it experiences another collision RbAr + Ar
→ Rb + Ar + Ar at time t. The Van der Waals potential Vm is interpolated from
experimentally inferred potentials for KAr and CsAr[27]. During the lifetime of a
Van der Waals molecule form of a Rb atom and an Ar atom, there are two important
perturbations, the hyperfine-shift interaction
Vhs = δAg I · S,
(3.14)
Vsr = γN · S.
(3.15)
and the spin-rotation interaction
In the time between collisions, the free Rb atoms evolve under the influence of a
common ground-state Hamiltonian Hg = Ag I · S + gS µB Sz Bz −
µI
IB.
I z z
The phase of
the density-matrix element ραβ of the free Rb atoms evolves at the Bohr frequency ωαβ
3.3. The result of Rb and the physical picture
48
Energy (meV)
(a)
10
8
6
4
2
0
-2
-4
-6
Ar
Ar
Ar
t
t-τ
Ar
Vm
RbAr
5
Rb
Rb
δA∗107
6
7
8
o
9
10
Internuclear seperation R (A)
(b)
Re ραβ
Collision
φ=π
No collision
t-τ
t
Time
Figure 3.7: (a), the formation of a RbAr Van der Waals molecule in the potential well Vm
by a three-body collision Rb + Ar + Ar → RbAr + Ar at the time t − τ . The molecule
is broken up when experiencing another collision RbAr + Ar → Rb + Ar + Ar at time t;
(b) a greatly exaggerated shift in the phase φ of the microwave coherence ραβ for a bound
atom due to the shift hv|δA|vi of the hyperfine coupling in the molecule where |vi is some
vibration-rotation state of the molecule. The Van der Waals potential Vm is interpolated
from experimentally inferred potentials for KAr and CsAr[27].
3.3. The result of Rb and the physical picture
49
where |αi and |βi are the upper and lower hyperfine sublevels with azimuthal angular
momentum quantum number m = 0. For the bound Rb atoms in Van der Waals
molecules, the Rb spins evolves under the influence of the perturbed Hamiltonian
H̄ = Hg + hv|(Vhs + Vsr )|vi,
(3.16)
where |vi is some vibration-rotation state of the molecule. For a bound Rb atom, the
corresponding Bohr frequencies ωᾱβ̄ are slightly different from ωαβ of free Rb atoms.
Therefore, the phase of a Rb atom bound in a molecule for a time τ will be shifted
by the amount
φ = (ωᾱβ̄ − ωαβ )τ,
(3.17)
with respect to that of a free Rb atom. This effect is shown in the panel (b) of Fig.
3.7. The states |ᾱ and β̄ are upper and lower energy sublevels of ground-state Rb
atoms with azimuthal quantum number m̄ = 0 along the effective magnetic field in
the molecule. This effective field includes the magnetic field induced by the spinrotation interaction Vsr = γN · S so it may differ in magnitude and direction from the
externally applied static field.
If we write the wave function of a free Rb atom at time t to be |ψi if the atom did
not experience any collision, then the wave function of the free atom at time t − τ is
eiHτ /h̄ |ψi. For a Rb atom which forms a Van der Waals molecule at the time t − τ
and is released from the molecule at time t, its wave function at time t would be
e−iH̄τ /h̄ eiHτ /h̄ |ψi. Suppose that the collisional formation rate for the Van der Waals
molecules is 1/Tv , then the evolution equation of the density operator ρ is
1
∂
ρ = (e−iH̄τ /h̄ eiHτ /h̄ ρe−iHτ /h̄ eiH̄τ /h̄ − ρ),
∂t
Tv
(3.18)
3.3. The result of Rb and the physical picture
50
Taking matrix elements of Eq. 3.18 and averaging over an exponential distribution of
molecular lifetimes, with the mean lifetime τv for state |vi, we find that ραβ = hα|ρ|βi
evolves at the rate
1
∂
ραβ = −
∂t
Tv
1−
X
µ̄ν̄
|hα|µ̄ihν̄|βi|2
1 + i(ωµ̄ν̄ − ωαβ )τv
!
ραβ
= −(∆γv + i2π∆νv )ραβ ,
(3.19)
where the free-atom Bohr frequency is ωαβ = (Eα − Eβ )/h̄, and the bound-atom Bohr
frequencies are ωµ̄ν̄ = (Eµ̄ − Eν̄ )/h̄. The energies Eµ and eigenvectors of the free atom
are given by H|µi = Eµ |µi, while the energies and eigenvalues of the bound atom are
given by H̄|µ̄i = Eµ̄ |µ̄i. And the frequency shift induced by molecules is
∆ν =
1 X |hα|µ̄ihν̄|βi|2 (ωµ̄ν̄ − ωαβ )τv
.
2πTv µ̄ν̄
1 + (ωµ̄ν̄ − ωαβ )2 τv2
(3.20)
For any Ar pressure p, the three-body formation rate is proportional to p2 (which
is 1/Tv ∝ p2 ) and the molecular lifetime is inversely proportional to the pressure p
(which is τv ∝ 1/p). As a result, at very high Ar pressures, the frequency shift of Eq.
3.20 will approach a limit which is linear in p and we will name this linear slope to
be smv because it comes from the contribution of molecules. The high-pressure limit
contributed by molecules is
∆ν∞ =
1 X
|hα|µ̄ihν̄|βi|2 (ωµ̄ν̄ − ωαβ )τv = smv p.
2πTv µ̄ν̄
(3.21)
Now looking at the experimental residual shift, it is obtained as ∆2 ν = ν − ν0 − sp,
the difference between the measured frequency ν and the high-pressure frequency
limit ν0 + sp. The linear part of the high-pressure limit consists two parts, one
is the contribution from binary collisions sb p, which actually is true for the whole
3.3. The result of Rb and the physical picture
51
pressure range, and the other is the contribution from molecules sm p. That is to say,
sp = sb p + sm p. The calculated frequency shift in Eq. 3.21 is the result of molecules,
∆ν = ν − ν0 − sb p. To get the residual shift from this equation, we need to subtract
the high-pressure limit for the molecular part which is smv p. Then the theoretical
residual shift is ∆2 ν = ∆ν − smv p = ν − ν0 − sb p − smv p = ν − ν0 − sp which has the
same meaning as the experimentally obtained value. From Eq. 3.20 and 3.21 we get
the theoretical residual shift
2
∆ ν = ∆ν − ∆ν∞
1 X |hα|µ̄ihν̄|βi|2 (ωµ̄ν̄ − ωαβ )3 τv3
.
=−
2πTv µ̄ν̄
1 + (ωµ̄ν̄ − ωαβ )2 τv2
(3.22)
For simplicity, we suppose a hypothetical Van der Waals molecule with hv|γN |vi = 0,
and with only one vibration-rotation state |vi. Then Eq. 3.22 can be written as
1
φ3
2
,
(3.23)
∆ ν = ∆ν − ∆ν∞ = −
2πT 1 + φ2
which is the same as Eq. 3.13 used to fit the experimental data. Here
φ = (ωᾱβ̄ − ωαβ )τv =
hδAi(I + 1/2)τv
h̄
(3.24)
is the phase shift of the microwave coherence of atoms in molecules. The high-pressure
residual shift is
sm = smv =
φ
.
2πT p
(3.25)
Note that if we substitute φ with Ωτ = Ωκm /p and 1/T with κf p2 , we will get the
same result as shown in Eq. 3.12.
We evaluated the more complicated expression Eq. 3.22 for plausible values of the
perturbation coefficients hv|δA|vi and hv|γN |vi. For values of hv|γN |vi comparable
to those reported by Bouchiat et al. [12], the result of Eq. 3.22 differed only slightly
from from the results obtained from Eq. 3.13, which is the case when hv|γN |vi = 0.
3.4. Summary of the results
52
In our experiments, the static magnetic field was changed in the range of 0.03 to
2 G. We found that the fitting parameters T p2 and φp hardly changed for fields in
this range. We also used Eq. 3.22 to model the effects of similar magnetic fields and
found that the field had negligible effect.
Our data also show that for RbAr, RbKr, CsAr, and CsKr, the Van der Waals
molecules produce a small positive frequency shift, while the overall shift, mostly due
to binary collisions, is negative. This suggests that the dependence of δA = δA(R) on
internuclear separation R is similar to the dotted line in Fig. 3.7. A radial dependence
similar to this would be consistent with the temperature dependence of the pressure
shift measured by Bean and Lambert [28]. Another similar radial dependence was
predicted by Camparo [29].
3.4
Summary of the results
We have measured the pressure shifts of both
87
Rb and
133
Cs in the buffer gases Ar,
Kr, He and N2 . The results are summarized in Fig. 3.8. As is seen from the figure,
Ar and Kr cause nonlinear pressure shifts for both
87
Rb and
133
Cs atoms. However,
to the limit of our experimental accuracy, the pressure shifts data for He and N2 are
linear for both
87
Rb and
133
Cs. This is not surprising considering that the lighter He
and the diatomic molecules N2 are not expected to form Van der Waals molecules
with the alkali-metal atoms. We fit the data for Ar and Kr using Eq. 3.13 with T p2 ,
φp, s and ν0 as free fitting parameters. Therefore Eq. 3.13 should be rewritten as
p
(φp)3
+ ν0 + sp.
ν = ∆ ν + ν0 + sp = −
2
2
2π(T p ) [p + (φp)2 ]
2
(3.26)
3.4. Summary of the results
53
(a)
∆2 ν (Hz)
0
Rb-Ar
Rb-Kr
Rb-He
Rb-N2
-10
-20
-30
0
(b)
10
20
30
40
0
Cs-Ar
Cs-Kr
Cs-He
Cs-N2
-20
∆2 ν (Hz)
50
-40
-60
-80
-100
-120
0
10
20
30
Buffer gas pressure p (torr)
40
50
Figure 3.8: (a), residual shifts ∆2 ν measured for 87 Rb in Kr, Ar, He and N2 buffer gas at
the temperature of 40 ◦ C; (b), residual shifts measured for 133 Cs at 35 ◦ C. The points with
error bars are experimental data. The solid lines through the points of Ar and Kr are the
fitting with Eq. 3.26, while those through the points of He and N2 are linear fitting. The
fitting parameters are summarized in Table 3.1.
3.4. Summary of the results
54
The best-fit values of ν0 are usually within a few Hz of the value expected for a free
87
Rb or
133
Cs atom in the static magnetic field B of our experiment. In the case of
He or N2 , we compared the χ2 values [30] of fitting the residual shifts using either
our model or a straight line. The result turns out that χ2 were only slightly smaller
(a factor of 1.2 or less) when our model was used instead of a straight line. For Ar or
Kr, χ2 could be 30 times smaller if we used our model instead of a straight line.
Table 3.1: Fitting parameters of the residual shifts for
in Kr, Ar, He and N2 buffer gas.
Metal
87
Rb
133
Cs
87 Rb
at 40 o C and
Gas
T p2
(sec·torr2 )
φp
(rad·torr)
s
(Hz torr−1 )
Ar
Kr
He
N2
0.094±0.025
1.287±0.096
2.69±0.22
25±2.13
-54.26±0.047
-559.58±0.172
714.3±0.018
518.4±0.014
Ar
Kr
He
N2
0.05±0.01
0.153±0.015
3.90±0.44
14.82±0.74
-194.3±0.133
-1123±0.732
1132.86±0.046
824.6±0.034
133 Cs
at 35 o C
The fitting parameters are summarized in Table 3.1. Since there are no T p2 and
φp for He and N2 , we simply leave the corresponding space blank.
3.5. Conclusion
3.5
55
Conclusion
The important role of Van der Waals molecules in the spin relaxation of alkali-metal
atoms was first pointed out in the pioneering work of Bouchiat, Brossel, and Pottier [12]. In their paper, they measured a parameter Tf p2 that is analogous to our
parameter T p2 . Their reported Tf p2 for Rb were (1.6-2.2)×10−2 s torr2 in Ar and
1.06×10−2 s torr2 in Kr. The corresponding values of T p2 from Table 3.1 are much
larger, 9.7 ×10−2 s torr2 and 129 ×10−2 s torr2 . As we discussed in section 3 of
this chapter, a more theoretically justified fitting function should take into account
the effects of all the vibration-rotation states |vi of the molecule, which is Eq. 3.22.
However, we do not have nearly enough information about the essential parameters
hv|δA|vi, hv|γN |vi, 1/Tv , τv , etc., to make a reliable evaluation of Eq. 3.22. As we
mentioned at the end of section 3, δA must change sign at least once as the internuclear separation R increases. Therefore, hv|δA|vi is likely to be positive for some
vibration-rotation states and negative for others. As a result, contributions to ∆2 ν
from different vibration-rotation states could be significantly canceled and thus the
parameter T p2 could be much larger than the analogous parameter Tf p2 reported by
Bouchiat et al. [12]. In their case, hv|γN |vi is expected to have the same sign for the
experimentally important values R [31] and the relaxation rates are independent of
the sign of hv|γN |vi.
In conclusion, we report here, for the first time, the nonlinear pressure shifts. Besides the spin relaxation method discovered and investigated by Bouchiat et al., our
method appear to be the only one that precisely measures a physical phenomenon
which provides information about the physics of Van der Waals molecules, like RbAr,
3.5. Conclusion
56
in gas cells. The shifts can provide important constraints on the poorly known interaction coefficients Vm , δA, and γ. The shifts in Ar are of practical interest, since
mixtures of Ar and N2 gas are often used to minimize the temperature coefficient of
pressure shifts for atomic clocks [11].
Chapter 4
Magnetic resonance reversals
4.1
About magnetic resonance reversals
It has been believed and reported that circularly-polarized resonant D1 light will
increase the transparency of the alkali-metal vapor. In this chapter we show that
under some conditions, more resonant light will decrease the transparency of the
alkali-metal vapor. The conditions include specific laser frequency (closer to the
transition from the lower hyperfine multiplet to the first excited state), pumping
intensity (weak) and polarization (circular).
4.2
Hypothetical atoms with I=0
We will start the discussion from a hypothetical alkali-metal atom which has no
nuclear spin I = 0. The process of optical pumping with D1 circularly-polarized
resonance light is shown in Fig. 4.1, and with D2 circularly-polarized resonance light,
it is shown in Fig. 4.2.
57
4.2. Hypothetical atoms with I=0
58
(a)
D1
S1/2
σ+
2
3/2
1.0
σop
0.8
2
1
A
P3/2
P1/2
(b)
1/2
-1/2
m= -3/2
0.6
ν
0.4
0.2
0
500
1000
(a.u.)
1500
2000
Figure 4.1: (a) Energy levels and D1 optical pumping with circularly-polarized σ + light
of a hypothetical alkali-metal atom with no nuclear spin I = 0. Only atoms from the −1/2
sublevel can absorb the σ + photons and be excited to the 1/2 excited-state sublevel, denoted
by the arrow pointing up. Excited atoms on this sublevel then decay spontaneously back to
the ground-state sublevels with the relative branching ratios, denoted by the numbers beside
the wavy arrows pointing down. Ground-state spin relaxation is denoted by the curved
double-end arrow. The bars on each ground-state sublevel are steady-state occupation
probabilities. (b) Dependence of the specific absorption hAi on the optical pumping flux
S. The inset shows the optical cross section and different detunings of the laser frequency.
Detuning the laser from the maximum of the cross section to half maximum simply stretches
out the curves.
4.2. Hypothetical atoms with I=0
59
(a)
(b)
1/2
D2
1
σ+
1
2
3/2
3
σ+
1.4
σop
1.3
3
A
-1/2
P3/2 m= -3/2
P1/2
1.2
ν
1.1
1.0
S1/2
0
500
1000
(a.u.)
1500
2000
Figure 4.2: (a) Energy levels and D2 optical pumping with circularly-polarized σ + light
of a hypothetical alkali-metal atom with no nuclear spin I = 0. Only atoms from the
−1/2 sublevel can absorb the σ + photons and be excited to the 1/2 excited-state sublevel,
denoted by the arrow pointing up. Excited atoms on this sublevel then decay spontaneously
back to the ground-state sublevels with the indicated branching ratios, denoted by the
numbers beside the wavy arrows pointing down. Ground-state spin relaxation is denoted
by the curved double-end arrow. The bars on each ground-state sublevel are steady-state
occupation probabilities. (b) Dependence of the specific absorption hAi on the optical
pumping flux S. The inset shows the optical cross section and different detunings of the
laser frequency. Detuning the laser from the maximum of the cross section to half maximum
simply stretches out the curves.
Circularly polarized light pumps alkali-metal atoms from states with lower azimuthal quantum numbers to those with higher azimuthal quantum numbers. When
optical pumping rate caused by the pumping light greatly exceeds the spin relaxation
rate of the ground state, nearly all atoms will be pumped to the state with the highest
azimuthal quantum number eventually. Here we have assumed that the repopulation
pumping comes mainly from spontaneous decay where collisional depolarization or
quenching of the excited atoms can be neglected.
Assuming that the photon absorption cross section for photons of frequency ν is
σ = σop (ν), then the photon absorption rate per unpolarized atom is
Γop = σop
S
,
hν
(4.1)
4.2. Hypothetical atoms with I=0
60
where S is the optical energy flux (erg cm−2 s−1 ). This is the result for unpolarized
atoms. However, alkali-metal atoms can be polarized by optical pumping. When this
happens, the actual absorption rate could be different from that of the unpolarized
atoms. We define the actual photon absorption rate to be hδΓi, where δΓ is the
photon absorption operator[13]. The relation between hδΓi and Γop is given as
hδΓi = hAiΓop .
(4.2)
Here we define another important ground-state spin operator A which is called the
specific absorption operator. For unpolarized atoms, it is straightforward to see that
hAi = 1. For D1 optical pumping with right (σ + ) circularly-polarized light , as we
discussed in Chapter 2, A = 1 − 2Sz , while for D2 optical pumping, A = 1 + Sz .
For a hypothetical alkali-metal atom with the nuclear spin I = 0, the rate of
change of hSz i is
1
d
hSz i = Γop (1 − 2hSz i) − Γs hSz i.
dt
3
(4.3)
Here we have assumed that diffusion to the walls and collisions with buffer gas atoms
cause the longitudinal spin to relax at the rate
d
hSz i
dt
= −Γs hSz i. We also assume that
the repopulation pumping comes from spontaneous decay with negligible collisional
depolarization or quenching of the excited atoms. Since for D1 optical pumping,
hSz i = 12 (1 − hAi), we then get the rate of change of hAi to be
−
1d
Γop
1 − hAi
hAi =
hAi − Γs
.
2 dt
3
2
Solving for the steady-state solution of hAi where
hAi =
d
hAi
dt
3Γs
.
2Γop + 3Γs
(4.4)
= 0 we get
(4.5)
4.3. Apparatus
61
Similarly, in the case of D2 optical pumping,
d
1
hSz i = Γop (1 − 2hSz i) − Γs hSz i,
dt
6
(4.6)
and
hAi =
3Γop + 6Γs
.
2Γop + 6Γs
(4.7)
The relative probabilities for absorption of σ + light and spontaneous emission from
the excited state to the ground state are denoted in Fig. 4.1 and 4.2 for D1 and D2
optical pumping respectively. The panel (b) of each figure shows how hAi changes
with increasing optical energy flux S. The insets show the laser detuning. For D1
optical pumping, hAi always decreases monotonically from 1 to 0 with increasing
pumping intensity either the pumping light is on resonance or tuned away from the
resonance. For D2 optical pumping, hAi increases monotonically from 1 to 3/2.
4.3
Apparatus
The apparatus is shown in Fig. 4.3. A Pyrex glass cell containing s small amount of
Cs metal and buffer gas (50 torr N2 and 280 torr Ar) was used. The space between the
front and rear walls of the cell is l = 0.1 cm. The cell was mounted in a temperaturecontrolled, air-heated nonmagnetic oven. Three sets of Helmholtz coils were used to
cancel the ambient magnetic field (down to no more than 20 mG) and to produce a
small longitudinal static field B, parallel to the pumping light. To drive magnetic
resonances with ∆f = ±1, microwaves from a frequency synthesizer were beamed
onto the cell from a horn antenna located 10 cm away. The light from a diode laser,
stabilized by an external cavity, and operating at 895-nm Cs D1 line, was circularly
polarized. Neutral density filters were used to control the beam intensity. The light
4.3. Apparatus
62
x coil
x
y
z
ND filter
pellicle
y coil
λ
_
4
λ
_
2
lens
Laser
oven
iris
oscilloscope
pellicle
z coil
horn
Fabry-Perot
interferometer
wavemeter
cell
photo
diode
fiber
coupler
oscilloscope
synthesizer
Figure 4.3: Apparatus.
transmitted through the optically pumped Cs cell was continuously monitored using a
photodiode. The photodiode dark current was negligible. The output signal from the
photodiode was amplified, digitized, and recorded. To record the hyperfine magnetic
resonances, we swept B continuously from one direction to the other and recorded
the transmitted light intensity. A Fabry-Pérot interferometer and a wavemeter were
used to monitor the performance of the laser.
The voltage output of the photodiode of Fig. 4.3 is,
V = V0 exp(−[Cs]σop hAil),
(4.8)
where the parameter V0 is proportional to the laser light intensity and also includes
scattering and attenuation losses of the glass walls of the oven and the glass cell. The
number density of Cs atoms, [Cs], is mainly controlled by the temperature of the cell.
4.3. Apparatus
63
The experimental specific absorption hAi is defined by
Z l
Tr(Aρ)dz,
lhAi =
(4.9)
0
averaged along all length elements dz of the optical path through the vapor. The
density operator ρ = ρ(z) can vary along the optical path length through the vapor,
since the light can be attenuated substantially for [Cs]σop l > 1, especially for low
laser powers, and less so at higher powers where intense, circularly polarized light can
burn through the vapor by pumping most of the atoms into the nonabsorbing end
states.
The specific absorption was determined from four measured photodiode voltages.
(i) VC0 , the voltage at the photodiode produced by circularly polarized laser light
at room temperature when [Cs]σop hAil 1.
(ii) VC = VC0 exp(−[Cs]σop hAil), the voltage at the photodiode produced by circularly polarized laser light at the temperature, magnetic field, microwave power, and
polarization of the experiment.
(iii) VL0 , the voltage at the photodiode produced by linearly polarized light at
room temperature when [Cs]σop hAil 1.
(iv) VL = VL0 exp(−[Cs]σop l), the voltage at the photodiode produced by linearly
polarized laser light at the temperature of the experiment.
The experimentally determined specific absorption was then given by
VL
VC
hAi = ln
ln
.
VC0
VL0
(4.10)
The high temperature and the resulting high spin-exchange rates efficiently destroy
the hyperfine polarizations hI · Si and birefringent polarizations h3fz2 − f (f + 1)i that
can be produced by linearly polarized pumping light[32]. This ensured that taking
4.4. Result
64
f=4
m
- 27 - 25 - 23 - 21
3
2
1
2
5
2
7
2
f=3
1.10
3
2
1.05
A
1.00
335122.0 GHz
5
2
- 7
2
7
2
σop
-
5
2
0.90
335109.0 GHz
0.85
3
2
-0.8
-0.6
5
2
-0.4
3
2
ν
0.95
0.80
-
- 7
2
7
2
-0.2
0.0
0.2
Magnetic field B (G)
-
σop
ν
3
2
5
2
-
0.4
0.6
0.8
Figure 4.4: Specific absorption hAi versus longitudinal magnetic field B curves when the
oscillating magnetic field is perpendicular to the static longitudinal field and the laser
beam. The upper trace is for the laser frequency of 335 109.0 GHz. The lower trace is
for 335 122.0 GHz. The two insets in the figure show the optical absorption cross section
curve for unpolarized atoms and the frequency positions of the two laser detunings. The
inset above the figure show the label m̄ for the possible transitions that are excited by the
oscillating magnetic field.
hAi=1 for linearly polarized light was a good approximation. In general cases where
the spin-temperature distribution is not necessarily valid, the denominator in Eq.
4.10 should be the value when the pumping intensity is low enough not to polarize
the atoms.
4.4
Result
Fig. 4.4 and Fig. 4.5 show the observed specific absorption hAi versus magnetic field
B when the oscillating magnetic field is perpendicular or parallel to the laser beam
and the static magnetic field. Remember that for a hypothetical atom with I = 0
pumped with D1 resonance light, hAi ≤ 1 for any laser detuning as shown in Fig.
4.4. Result
65
f=4
m
-3
-2
0
-1
1
2
3
f=3
1.10
1.05
A
1.00
335122.0 GHz
σop
ν
0.95
0.90
335109.0 GHz
0.85
σop
ν
0.80
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
Magnetic field B (G)
Figure 4.5: Specific absorption hAi versus longitudinal magnetic field B curves when the
oscillating magnetic field is parallel to the static longitudinal field and the laser beam.
The upper trace is for the laser frequency of 335 109.0 GHz. The lower trace is for 335
122.0 GHz. The two insets in the figure show the optical absorption cross section curve for
unpolarized atoms and the frequency positions of the two laser detunings. The inset above
the figure show the label m̄ for the possible transitions that are excited by the oscillating
magnetic field.
4.1. In contrast, for a real Cs atom with I = 7/2, we observed hAi ≥ 1 for a laser
frequency of 335 122.0 GHz which pumps atoms predominantly out of ground-state
sublevels with total angular momentum quantum numbers f = I − 1/2 = 3. For a
laser frequency of 335 109.0 GHz which pumps atoms predominantly out of groundstate sublevels with total angular momentum quantum numbers f = I + 1/2 = 4,
we observed hAi ≤ 1. At the cell temperature of 130 ◦ C, the unpolarized Cs vapor
attenuated the 335 109.0 GHz light by 2.9 e-foldings and the 335 122.0 GHz light
by 1.3 e-foldings. The circularly polarized incident laser power density S was 0.6-0.9
mW cm−2 .
The results for an oscillating magnetic field perpendicular to the laser beam are
shown in Fig. 4.4 where the upper trace is for 335 122.0 GHz and the lower trace is
4.4. Result
66
for 335 109.0 GHz. In this case, the oscillating magnetic field excites transitions with
∆m = ±1 and ∆f = ±1, as shown in the inset above the figure. The transitions
are labeled with m̄, the average of mi (the azimuthal quantum number of the initial
state) and mf (the azimuthal quantum number of the final state). From the figure we
can see that transitions with m̄ = ±7/2, ±5/2 and ±3/2 were observed. The large
resonances at zero static magnetic field are caused by the depolarization induced by
the residual transverse static magnetic field which is not completely compensated with
the Helmholtz coils. They are usually called the ”zero-dips”, which can be broader
or narrower depending on how carefully we compensated for the transverse field with
the coils.
For an oscillating magnetic field parallel to the static magnetic field, the results
are shown in Fig. 4.5. In this case, transitions with ∆m = 0 should be excited by the
oscillating field. However, we did not observe any resonance peaks/dips corresponding to those transitions at either 335 109.0 or 335 122.0 GHz. The barely visible
resonances that can be seen in the lower trace of Fig. 4.5 are actually resonances
with ∆m = ±1 that result from a slight misalignment of the oscillating magnetic
field and the static magnetic field.
From Fig. 4.5 we can get an interesting and important conclusion that the two
ground-state sublevels |4mi and |3mi with the same azimuthal quantum number m
have very nearly the same populations, since the ∆m = 0 resonances are not observed.
This observation gave us the hint that the atomic distribution among the ground-state
sublevels should be similar to the so-called “spin-temperature distribution”[33]. As we
discussed in Chapter 2, under the conditions of “spin-temperature distribution”, the
sublevel populations, which are the diagonal elements of the atomic density operator
4.4. Result
67
ρ, are hf m|ρ|f mi = eβm /Z. Here β = ln(1 + P ) − ln(1 − P ) is the spin-temperature
parameter, and P = tanh(β/2) = 2hSz i is the electron spin polarization. The spin
P
partition function is Z = f m eβm . The spin-temperature distribution prevails when
the spin exchange collision rate Γse between pairs of alkali-metal atoms is larger
than any other relaxation rate of the system. We will discuss the conditions of our
experiments in more detail.
Table 4.1: Some important relaxation mechanisms for polarized alkali-metal atoms.
Relaxation mechanisms
Potential
Damping rate
···
Γop
Spin exchange
J(r)S1 · S2
Γse
Spin rotation
γ(r)S · N
Γsr
Hyperfine shift
δAg (r)I · S
ΓC
···
Γd
Optical pumping
Diffusion to the walls
Recalling our discussion in Chapter 2, the evolution equation of the density operator ρ is
ρ̇ =
X
1
[Hg , ρ] +
ρ̇x ,
ih̄
x
(4.11)
where Hg = Ag I · S + (gS µB Sz − µI Iz /I)Bz is the ground-state Hamiltonian for an
alkali-metal atom in a magnetic field. ρ̇x denotes the contribution to the evolution
from the spin relaxation mechanism x, such as the optical pumping, spin-exchange
4.5. Discussion
68
collisions, spin-rotation collisions, hyperfine-shift collisions and diffusion to the walls.
They are summarized in Table 4.1. For the experimental results shown in Fig. 4.4
and Fig. 4.5, we estimated that the spin-exchange rate was Γse = 4.6 × 104 s−1 ,
the spin-rotation rate was Γsr = 521 s−1 , The Carver rate was ΓC = 107 s−1 , the
diffusion rate Γd = 47 s−1 and the optical pumping rate was Γop = 500 s−1 . We can
obtain steady-state solutions to the evolution equation Eq. 4.11 with ρ̇ = 0 with
computers. And as we will show later, the results are in very good agreement with
our experimental observations.
4.5
Discussion
We can understand the signal reversals qualitatively in stead of using computer simulations. Let us consider the spin-temperature limit, when the spin-exchange rate Γse
is much larger than any other characteristic rate of the system. Considering that we
used weak pumping light in our experiments, the conditions of our experiments are
quite close to the spin-temperature limit.
In the spin-temperature limit, the atomic occupation of each ground-state sublevel
is determined by only one parameter, either β or P . In out discussion, we will
take P = 2hSz i to be the parameter because it can be directly determined from
the experimental data. Therefore, the specific absorption hAi will be a well-defined
function of P . It is mentioned in the second section of this chapter that for unpolarized
(P = 0) atoms where all the ground-state sublevels are equally populated, hAi = 1.
It is also easy to see that for completely polarized P = 1 atoms where all atoms are
4.5. Discussion
69
(a)
(b)
σop
σop
ν
f=4
ν
0
1
2
3
0
1
2
3
4
m= -4 -3 -2 -1
0
1
2
3
m= -3 -2 -1
0
1
2
3
4
f=3
m= -4 -3 -2 -1
m= -3 -2 -1
Figure 4.6: hf m|A|f mi (narrower bars) and hf m|ρ|f mi (wider bars) for each ground-state
sublevel at the two different laser detunings.
pumped into the nonabsorbing |44i sublevel, hAi = 0. Therefore the value of hAi
must change from 1 to 0 with increasing polarization P .
Fig. 4.6 shows the value of hf m|A|f mi (red, narrower bars) and hf m|ρ|f mi (blue,
wider bars) for each ground-state sublevel |f mh at the two different laser detunings.
We have assumed that the pumping light has a right circular polarization so sublevels
with higher azimuthal quantum number m has more atomic occupation and larger
hf m|ρ|f mi. A spin-temperature distribution is also assumed which means we are in
the range where the laser intensity is not so high as to destroy the spin-temperature
distribution.
In panel (a) of Fig. 4.6, the laser frequency is chosen to be close to the frequency
of the optical transition from the upper ground-state multiplet to the first excited
state. Therefore, atoms on the upper multiplet absorb more light than those on the
lower multiplet. This effect is shown with the higher bars on the upper multiplet
sublevels. In this case, when the laser intensity increases, more atoms are pumped
4.5. Discussion
70
to the sublevels with higher m and the polarization P increases. Considering the
fact that the specific absorption hAi is an average of the hf m|A|f mi values of all
the ground-state sublevels, the increase in P induces a decrease in hAi because more
atoms will be pumped to the sublevels where they can absorb less light (as shown
with shorter red narrower bars on sublevels with higher m).
Panel (b) of Fig. 4.6 shows the case where the laser is tuned to excite atoms
predominantly from the lower ground-state multiplet with f = 3. If the pumping
intensity is not too great, the spin-temperature distribution prevails. As shown in
the figure, atoms from the lower multiplet absorb more light due to the laser detuning.
When the laser intensity increases, similarly, more atoms are excited to the sublevels
with higher m and hence the polarization P of the vapor increases. However, in this
case atoms on the lower-multiplet sublevels with higher m absorb more light, as can be
seen from the longer red (narrower) bars on the lower-multiplet sublevels. Therefore,
an increase in P causes an increase in hAi. This is true when the laser intensity is
not too high and therefore the spin-temperature distribution prevails. Recalling our
discussion about the case where the laser intensity is very high, hAi will approaches
zero because sufficiently fast pumping with circularly polarized D1 light will excite
all atoms to the nonabsorbing “end state” |44i, and this is true irrespective of the
laser detuning. So for this laser detuning, roughly speaking, hAi increases first with
increasing pumping intensity in the range of low pumping intensity and then decreases
with increasing pumping intensity when the pumping intensity is sufficiently high.
The above discussion is qualitative and simply provides us a straightforward way
to understand the physical picture of the signal reversals. We will now start to look
at this effect in a quantitative way. In Fig. 4.7 we display both the experimental data
4.5. Discussion
71
(a)
A
1.0
0.8
0.6
σop
0.4
0.2
σop
ν
ν
0
5
10
15
20x10
Incident optical pumping rate (Hz)
4
(b)
A
1.0
0.8
0.6
0.4
0.2
0.0
0.0
B
A
σop
C
σop
ν
ν
0.2
0.4
0.6
0.8
1.0
Average polarization P
A
Figure 4.7: (a), specific absorption hAi versus optical pumping rate; (b), specific absorption hAi versus average polarization P . The upper trace is for the laser frequency 335
122.0 GHz, and the lower trace is for the laser frequency 335 109.0 GHz. Solid circles are
experimental data and lines are calculations.
1.08
(c)
1.05
(b) 1.00
(a)
1.04
0.95
1.06
1.03
0.90
B
1.04
A
C
1.02
0.85
1.02
1.01
0.80
1.00
1.00
-0.8 -0.4 0.0 0.4 0.8
-0.8 -0.4 0.0 0.4 0.8
-0.8 -0.4 0.0 0.4 0.8
Magnetic field B (G)
Figure 4.8: Specific absorption hAi versus static magnetic field B curves corresponding
to points A, B and C in panel (b) of Fig. 4.7.
4.5. Discussion
72
(solid circles) and the results of computer simulations (solid lines). Panel (a) shows
the curves of hAi versus the incident optical pumping rate Γop . Panel (b) shows
the same results but the horizontal axis is converted to the average polarization
P . The experimental hAi was got using Eq. 4.10. The calculation results were
obtained by finding the steady-state solutions of Eq. 4.11 with a computer program
written in Matlab which takes the most important relaxation processes mentioned
above into account. Their density matrix representations are the following[34]. For
spatial diffusion, ρ̇d = Γd (1/g − ρ). For spin-rotation interaction, ρ̇sr = Γsr (ϕ −
ρ). For spin-exchange interaction, ρ̇se =
hyperfine-shift interaction, ρ̇C =
microwave and RF fields, ρ̇of =
1
[δεC , ρ]
ih̄
1
[δεse , ρ]
ih̄
− ΓC
+ Γse [ϕ(1 + 4hSi · S) − ρ]. For
ηI2 [I]2
ρ.
8
For oscillating fields such as
1
[δεof , ρ] + {Γof [(ρµµ − ρνν )|µihµ| + (ρνν
ih̄
− ρµµ )|νihν|]}.
1
[δεv , ρ]
ih̄
− 12 {δΓ, ρ}. For D1 repopulation
pumping, ρ̇rp = <(Γop ) 18 {1 − 4s · S, ρ} + (S · ρS − iS × ρS) . All the above terms
P
should be included in the x ρ̇x term of Eq. 4.11. In all precious sections, we have
For D1 depopulation pumping, ρ̇dp =
assumed the atoms were at rest. For an atom moving with velocity v and making
a transition between the ground-state sublevel |νi to the excited state sublevel |ki,
the resonance frequency is ωkv + k · v where k is the wave vector of the moving
atom and ωkv is the Bohr frequency of the transition of an atom at rest. Considering
the Maxwell-Boltzmann distribution of the atomic thermal velocities, we define the
p
characteristic Doppler broadening frequency, ωD = ωo 2kB T /(M c2 ) where ωo is the
mean frequency of the fluorescent light from the excited atoms and M is the mass of
the atom. The Doppler broadening frequency is defined in such a way that the full
width at half maximum of an atomic absorption line with pure Doppler broadening
√
is 2 ln 2ωD .
4.5. Discussion
73
For the laser frequency which excites atoms predominantly from the upper groundstate multiplet, the specific absorption hAi decreases monotonically with increasing
optical pumping rate Γop , while for the laser frequency which pumps atoms predominantly from the lower ground-state multiplet, hAi increases first with increasing
pumping rate then decreases. The trend of the curves are similar for the hAi-P
curves. The relation between P and Γop is
P =
Γop
,
Γop + Γsr + Γd
(4.12)
where is defined as = hFz i/hSz i − 1 [23]. Considering the fact that Γop decreases
along the laser beam path inside the alkali-metal vapor, we also averaged the polarRl
ization P along the beam path as 0 P (z)dz/l where z is the distance from the front
window of the vapor cell along the beam path and l is the distance between the front
and rear window of the vapor cell.
The applied transverse oscillating magnetic field excites microwave resonance transitions with ∆m = ±1, which causes the electron spin polarization P of the vapor
to decrease. This is a useful rule to determine the sign of the resonance signal. For
example, in the case of the lower trace of Fig. 4.7 panel (a) where the laser is tuned
to excite atoms predominantly from the sublevels with f = 4, the decrease in P
induced by the applied oscillating field always causes an increase in the specific absorption hAi. This is because of the fact that at this laser detuning, hAi decreases
monotonically with increasing P . The situation is more complicated when the laser
is tuned to excite atoms mainly from the sublevels with f = 3 because in this case
hAi does not change with P monotonically. The sign of the resonance signal depends
on whether the hAi-P curve has a positive or a negative slope. In the region where
dhAi/dP < 0, the microwave resonances increase the specific absorption hAi, similar
4.5. Discussion
74
to the case of the other laser detuning. But in the region where dhAi/dP > 0, the
microwave resonances actually decrease the specific absorption hAi and this is what
we called the reversals of the resonance signals.
Let us now take the three points A, B and C which are marked in panel (b) of Fig.
4.7 for examples to discuss the features of the microwave resonance signals. Before
we look at the experimental data, we need to note that for the “zero-dip” where the
polarization is completely destroyed by the residual transverse static magnetic field,
the sign of the microwave resonance signal is determined by whether hAi is larger or
smaller than 1 because hAi = 1 for unpolarized atoms. The hAi-B curves for the three
points are shown in Fig. 4.8. Point A is in the region where hAi > 1 and dhAi/dP > 0.
Therefore both the microwave resonances and the zero-dip appear as absorption dips,
as shown in the panel (a) of Fig. 4.8. Moreover, since the optical pumping rate
is very low, the atoms are only weakly polarized and close to the spin-temperature
limit. Thus the population difference between the sublevels with +m and +(m − 1)
is about the same as that between the sublevels with −m and −(m − 1). This can
also be seen from the figure since microwave resonances appear symmetrically on
both side of the zero-dip. Point B is in the range where hAi > 1 and dhAi/dP < 0.
Note that magnetic resonances only cause small decreases in the polarization, while
the residual transverse static magnetic field can totally destroy the spin polarization.
Therefore the magnetic resonances show as absorption peaks while the zero-dip is
still an absorption dip. Because the optical pumping rate in this case is larger, more
atoms are excited to one side of the ground-state sublevels (either the m > 0 or m < 0
side depending on the polarization of the light). This is shown as the asymmetry of
the microwave resonance signals with respect to the zero-dip. Now let’s look at point
4.5. Discussion
75
(a)
1.02
335119.5 GHz
A
1.01
335116.8 GHz
1.00
335115.0 GHz
0.99
0.98
0.97
335112.3 GHz
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
(b)
1.02
A
1.01
1.00
0.99
0.98
0.97
Magnetic field B (G)
Figure 4.9: Experimental results and simulations for a different cell with buffer-gas mixture of 50 torr N2 and 480 torr Ar. (a), Measured hAi versus static magnetic field B curves
at four different laser frequencies; (b), simulation results.
C where hAi < 1 and dhAi/dP < 0. This is similar to the case of the lower curve
of panel (b), Fig. 4.7. Both the microwave resonances and the zero-dip appear as
absorption peaks. And for the same reason, we could see microwave resonance signals
on one side of the zero-dip but barely see them on the other side.
In the above discussions, we have talked about the case where the temperature of
the vapor cell is so high that the number density of Cs atoms are high enough for the
spin-exchange rate Γse to be much larger than other characteristic rates in the system
at low pumping rate. That is to say, the spin-temperature distribution is valid in the
4.5. Discussion
76
case of low enough pumping intensity. Under these conditions, we can understand
the reason of the signal reversals qualitatively with the help of the spin-temperature
distribution. By solving for the steady-state solutions of the density matrix, we can
also get a quantitative understanding of the features of the signal reversals. Now we
are going to consider a more general case where the spin-temperature distribution
does not always prevail. When this happens, we need to use the full density-matrix
calculation to understand the signs of the resonance signals.
We tested a different cell with the buffer gas mixture to be 50 torr N2 and 480
torr Ar. The cell has such a dimension that the distance between the front and rear
walls is 1.7 cm. The incident laser power density S was about 0.1 mW cm−2 for all
the laser frequencies. The experiments were done at the cell temperature of 90 ◦ C.
Compared with the 130 ◦ C temperature of all the above experiments, 90 ◦ C is so
low that it reduces the number density of Cs atoms to a level corresponding to a
much smaller spin-exchange rate. Therefore the system was no longer close to the
spin-temperature limit. However, reversals of the magnetic resonance signals were
still seen in this cell. Fig. 4.9 shows the comparison between the experimental data
(panel (a)) and the computer simulations (panel (b)) at four different laser frequencies
indicated in the figure. The values of the calculation parameters were: the optical
pumping rate Γop = 20 s−1 , the spin-exchange rate Γse = 4500 s−1 , the spin-rotation
rate Γsr = 919 s−1 , the Carver rate ΓC = 107 s−1 , and the diffusion rate Γd = 28 s−1 .
The simulation is strikingly consistent with the experimental data.
4.6. Conclusion
4.6
77
Conclusion
In summary, we have experimentally demonstrated unusual signal reversals of the
magnetic resonances of ground-state alkali-metal atoms pumped by D1 circularly polarized laser light. With D1 optical pumping, normal magnetic resonance signals
show as transmission dips while reversed magnetic resonance signals show as transmission peaks. To observe the signal reversals, the hyperfine splitting has to be at
lease partially optically resolved, and the laser frequency need to be tuned to be close
to transition from the lower ground-state hyperfine multiplet to the excited state.
This phenomenon cannot be explained by the conventional two- or three-level model,
or even a multilevel model with a universal relaxation rate. A valid theoretical explanation involves a density-matrix calculation with two important spin-relaxation
mechanisms, the spin-rotation interaction and the spin-exchange interaction. The signal reversals are more pronounced especially when a rapid spin-exchange rate induces
a spin-temperature distribution of the atomic population among the ground-state sublevels.
In the CSAC (chip scale atomic clock) project of DARPA (Defense Advanced
Research Projects Agency), laser-pumped miniature Cs or Rb vapor cells are supposed
to form the basis of a chip scale atomic clock. The relatively high densities of the alkalimetal vapors and the corresponding high spin-exchange rate might lead to conditions
similar to those reported here. So for such atomic clocks, people should pay attention
not to operate these clocks with parameters close to those that reduce or reverse the
resonance signals.
Chapter 5
Electrolytic fabrication of
miniature atomic clock vapor cells
5.1
About miniature vapor cells
An atomic clock is a type of device that depends for its operation on a local oscillator
regulated by the resonance frequency of an atomic system. Since the first atomic
clock was built in 1949, a lot of efforts have been made to improve the performance
of such clocks. Recently, there is increasing interest in developing very small atomic
clocks. In 2001, the Defense Advanced Research Projects Agency (DARPA) launched
the Chip-Scale Atomic Clock (CSAC) program. The goal of this program is to build
atomic frequency references with great reductions in size and power but still matching
performance relative to present devices[35]. The ability to build chip-scale atomic
clocks will make wristwatch-sized time/frequency references possible for high-security
ultra high frequency (UHF) communication systems and jam-resistant GPS receivers.
78
5.2. Review of existing fabrication methods of miniature vapor cells
79
At the same time, it is also hoped that such frequency references could be mass
produced so that they would have wider applications.
We demonstrated here a new electrolytic method to fabricate alkali-metal vapor
cells. This method is combined with anodic bonding technology so as to make siliconbased miniature vapor cells.
5.2
Review of existing fabrication methods of miniature vapor cells
In the past, three methods have been used to fill the anodically bonded cells with
alkali metal vapors. They are direct injection[36, 37], chemical reaction[38] and wax
micropackets[39]. In the following subsections, the three methods will be briefly
reviewed respectively.
5.2.1
Direct injection
The most straightforward way to fill a vapor cell is to inject a small amount of alkali
metal (mainly Rb or Cs) into a half-made cell, along with an appropriate mixture and
pressure of buffer gases. The half-made cell is then sealed by anodic bonding. The
alkali metal that is to be filled into the cell could be pure metal or could come from
chemical reactions. The chemical reactions mentioned here should not be confused
with the second method-chemical reaction. In this method, the chemical reaction
happens outside of the cell.
5.3. Anodic bonding
5.2.2
80
Chemical reaction
In this method, mixture of alkali-metal salt (usually chloride) and reducing agent (for
example, barium azide BaN6 ) is introduced into the half-made cell together with the
proper buffer gas. Then the half-made cell is sealed and heated to initiate the reaction
between the alkali-metal salt and the reducing agent. One example reaction[38] is:
BaN6 + CsCl → BaCl + 3N2 + Cs.
5.2.3
(5.1)
Wax micropackets
In this method, alkali metals are enclosed in a chemically inert wax to preform alkali
metal-wax micropackets. Then the micropacket is attached to the silicon nitride
membrane side of a cavity and the enclosed alkali metal is released into the cavity by
laser ablating the silicon nitride membrane through the other glass wafer side of the
cavity.
5.3
Anodic bonding
In 1969, Wallis and Pomerantz[40] first demonstrated that an external electric field
could help lower the temperature of metal to glass sealing to well below the softening
point of the glass. This method of doing metal to glass to metal sealing is known as
anodic bonding. It has been widely used in microelectromechanical systems (MEMS),
microelectronics, vacuum packaging, hermetic sealing and encapsulation.
The formation of a SiO2 layer between Si-glass interface is believed to be the
reason for the strong bond. The schematic of anodic bonding set-up is shown in Fig.
5.3. Anodic bonding
+
-
81
+
-
+
-
Px
Si
+
Figure 5.1: Anodic bonding set-up.
5.1. A piece of Pyrex glass plate and a piece of Si plate are pressed against each
other. Two electrodes (black in the figure) are connected to the two plates, cathode
to Pyrex and anode to Si. There are a few percent of Na+ ions in Pyrex. During the
anodic bonding process, a heater is usually used to heat the material and increase
the mobility of Na+ ions. When a high voltage is applied between the electrodes,
the migration of positive ions Na+ in the Pyrex toward the cathode causes a chargedepletion layer in the Pyrex glass at the Pyrex-Si interface. As a result, a large electric
field is built up in this region which brings the glass and Si to be close enough to
permit covalent oxygen bonds to form.
5.4. Electrolysis in all-glass cells
5.4
5.4.1
82
Electrolysis in all-glass cells
Experiments
We first demonstrated that electrolysis of specially made Cs borate glass can be used
to fill a sealed glass cell with Cs metal. This method is based on the fact that large
amounts of alkali metal can be released by passing an electrolytic current through
hot glass[41]. The schematic is shown in Fig. 5.2. The Cs borate glass is made by
heating the mixture of Cs2 CO3 and B2 O3 at 900 ◦ C for 30 minutes and then letting it
cool down. The Cs borate glass made in this way is clear and transparent. Then the
glass is ground into powder and put in the half-made Pyrex tube, in which a piece
of Mo (Molybdenum) wire is sealed. To form a good contact between the borate
glass and the tube, the bottom of the tube is heated with a torch so that the borate
glass can melt and form a thin layer at the bottom. This can be done because the
softening point of the borate glass is lower than that of the Pyrex glass. The space
between the tip of the Mo wire and the borate glass is 0.5 cm. The tube will then be
connected to a vacuum system, filled with a few torr of Xe (Xenon) gas and sealed.
The electrolysis is done by discharging the Xe gas. The cathode is the Mo wire and
the anode is molten NaNO3 contained in a copper container. A hot plate set at 500
◦
C is used to melt the salt. After the NaNO3 melts at 307 ◦ C, The bottom of the
tube is dipped in the salt and a voltage of about 1000 V is applied to generate the
discharge and build up a plasma channel between the wire tip and the borate glass.
Then the voltage was kept at 200 V.
5.4. Electrolysis in all-glass cells
83
Mo wire
Pyrex
Xe
Cs
borate
glass
NaNO3
+
Hot plate
Figure 5.2: Set-up of doing electrolysis by discharge of Xe in an all-glass cell. Cs borate
glass acts as the source for Cs atoms. A few torr of Xe gas is filled into the cell as both
buffer gas and discharge gas. A piece of Mo (Molybdenum) wire is sealed in the cell to be
the cathode. A copper container containing NaNO3 is the anode. A hot plate is used to
heat the copper container and melt the NaNO3 . The use of NaNO3 will be explained in the
text.
5.4. Electrolysis in all-glass cells
84
(a)
(b)
Figure 5.3: (a)The blue light comes from the discharge of Xe gas; (b) A layer of golden
metal appears on the inner wall of the tube after several minutes of discharge.
5.4. Electrolysis in all-glass cells
5.4.2
85
Visual demonstration
A picture of the discharge is shown in Fig. 5.3, (a). During the discharge, the current
is monitored with a current meter and is normally stable at 7 mA. The discharge is
stopped after around 10-15 minutes. A layer of golden metal can be seen on the inner
wall of the upper part of the tube. Using a flame of natural gas to treat the cell for
several minutes and tiny golden droplets can be seen, as shown in Fig. 5.3, (b). This
is the primitive evidence of the presence of Cs. The mechanism for the Cs atoms to
come out from the borate glass is believed to be the following. As the discharge of
Xe is generated by the externally applied electric field, a plasma channel between the
cathode and the Cs-enriched borate glass is built up. Electrons from the cathode can
get through this channel and combine with Cs+ ions in the borate glass to form Cs
atoms. As the first several layers of Cs+ ions get out of the borate glass, more inner
Cs+ ions will move toward the surface of the borate glass. This will be followed by
the moving of Na+ ions in the Pyrex glass across the interface between the borate
glass and the Pyrex glass and the moving of Na+ ions from the molten NaNO3 to
the Pyrex glass. This process is necessary for keeping the current continuous during
the electrolysis. The high temperature kept by the hot plate helps to increase the
mobility of the Cs+ ions in the borate glass and Na+ ions in the Pyrex glass.
5.4.3
Physical demonstration
To prove that the golden metal in the cell is Cs, optical resonances and the number
density of Cs are measured by looking at the transmission intensity curve of laser
light through the vapor using an external-cavity, single-mode diode laser (Toptica
DL100). If we denote the on-resonance transmission intensity as Ion when the laser
5.4. Electrolysis in all-glass cells
86
Transmission (V)
0.15
baseline
T=40C n=0.65
T=50C n=1.46
T=60C n=3.19
T=70C n=6.12
x1011cm-3
0.10
0.05
0.00
0
1
2
3
4
5
6
Laser frequency offset (ν−ν0) (GHz)
Figure 5.4: Transmission intensity vs. laser frequency at different temperatures. The
number density of Cs in the cell can be calculated by measuring the on- and off-resonance
transmission intensity, n = ln IIoff
/(σL). Here, σ is the cross section of the photons scattered
on
from the alkali metal atoms, n is the number density of the alkali metal atoms in the cell,
L is the optical depth of the laser beam through the vapor. The two-dip structure of the
curve is due to the two transitions from one of the two ground hyperfine multiplets to the
two hyperfine multiplets of the first excited state. The black baseline is the background
intensity level of the photo detector. The number density is expressed in the unit of cm−3 .
5.5. Electrolysis in silicon-based miniature cells
87
frequency matching that of the D1 light (from the alkali metal ground state to the
first excited state) and denote the off-resonance transmission intensity as Ioff when
the laser frequency is tuned away from the D1 frequency, Ion /Ioff = e−σnL . Here, σ
is the cross section of the photons scattered from the alkali metal atoms, n is the
number density of the alkali metal atoms in the cell, L is the optical depth of the
laser beam through the vapor. From the above equation we know that,
n=−
on
ln IIoff
σL
(5.2)
Therefore, by measuring the on- and off-resonance transmission intensity of the
laser light, we can get the number density of Cs in the cell at different temperature.
The result is shown in Fig. 5.4. The number density of Cs vapor in the cell is the
same as those in the conventional cells.
In summary, we demonstrated that electrolysis can be used to fill a traditional
all-glass cell with Cs vapor. In this method, molten NaNO3 is necessary for supplying
Na+ ions to the Pyrex glass to keep the current continuous.
5.5
5.5.1
Electrolysis in silicon-based miniature cells
Experiments
The steps of a proof-of-principle experimental illustration for the electrolytic filling
of silicon-based miniature cells are shown in Fig. 5.5. The Cs-enriched glass was
made in the same way discussed in the past section. A hole, 2.5 mm in diameter, was
drilled through a h100i silicon wafer, which was 2.5 mm thick and polished on both
sides. The wafer was p doped and had a resistivity ≥ 1Ωcm2 . A shallow well, also
5.5. Electrolysis in silicon-based miniature cells
(a)
88
anodic
bonding
silicon
Pyrex
Cs-enriched
borate glass
anodic
bonding
(b)
Pyrex
silicon
(c)
+
NaNO3
copper
hot plate
Figure 5.5: Schematic of the experimental process for electrolytic filling of silicon-based
miniature cells. (a) A silicon wafer was anodically bonded to a Pyrex wafer. A hole with
the diameter of 2.5 mm was drilled through the silicon wafer with a diamond drill. There
was a shallow well in the Pyrex wafer with about the same diameter. When the two wafers
were bonded together, the well was displaced from the hole as shown in the figure. Several
pieces of Cs-enriched borate glass were put into the well and molten to form a good contact
with the Pyrex wafer. (b) Another Pyrex wafer was anodically bonded to the other surface
of the silicon wafer under an argon cover gas. The cross section of the half-made cell is
shown on the right-hand side in the above two panels. (c) A copper container, containing
molten NaNO3 salt in a basin at the top, was placed on a hot plate with the set temperature
to be 540 ◦ C (the highest for the available model). The sealed cell was then put on the top
of this copper container with the well just above the molten NaNO3 . A potential of 700 V
was applied for a few minutes between the molten NaNO3 and the silicon with the silicon
to be the cathode.
5.5. Electrolysis in silicon-based miniature cells
89
2.5 mm in diameter, was drilled into the surface of a Pyrex wafer, which was 3 mm
thick. Both the hole and the shallow well were made with a hand-hold diamond drill.
Piece of Cs-enriched borate glass were put in the shallow well. As sketched in Fig.
5.5(a), the silicon wafer and the Pyrex wafer with Cs-enriched glass in the well were
then pressed against a graphite disk (not shown) and anodically bonded by heating
the assembly on a hot plate at 540 ◦ C and applying a potential difference of 1000V
between the silicon and the graphite.
A second Pyrex wafer was anodically bonded to the other surface of the silicon
wafer to make a sealed cell, as shown in Fig. 5.5(b). Normally this step should
be done in a glove box so that the gas sealed in the cell can be controlled both in
pressure and in composition. For the purpose of demonstration and simplicity, this
step in our experiment was done under the coverage of flowing argon gas. The cell
was surrounded with a Pyrex tube and argon gas then flowed into the tube to form a
coverage around the cell.The newly manufactured cell contained nearly pure argon gas.
Because this step was done at 540 ◦ C, the argon pressure in the finished cell at room
temperature was about 0.4 atm. For the application of making clocks, appropriate
mixtures of argon with nitrogen gas can be used to diminish the sensitivity of the clock
frequency to temperature fluctuations[42, 20] and to suppress radiation trapping. The
well region of the cell was heated with a gas flame. By doing this, the Cs-enriched
glass with a lower softening point would melt and form a layer on the bottom of the
well. Thus, we could make a good contact between the Cs-enriched glass and the
Pyrex well. Our Cs-enriched borate glass had a much larger coefficient of thermal
expansion than Pyrex so cracks developed in the Cs-enriched glass when the cell
cooled. This phenomenon would make the corresponding position of the Pyrex wafer
5.5. Electrolysis in silicon-based miniature cells
90
bad for optical purpose. This was why we display the shallow well from the hole when
we did the anodic bonding between the silicon wafer and the first Pyrex wafer. The
shallow well was displaced from the hole in the silicon by about 2 mm, so most of
the cell window was not destroyed by the crazed Cs glass. The final cell consisted of
a three-layered bonded structure with optically transparent Pyrex windows on either
side of the hole in the silicon.
As sketched in Fig. 5.5(c), the sealed cell was put on the top of a copper container.
A small basin at the top of the copper container contained a small amount of molten
NaNO3 salt. The copper container together with the molten salt served as anode
in the next electrolysis. We named the anode as ”ion anode” because it supplied
Na+ ions for injection into the Pyrex glass and maintained the continuity of the
electrolysis current. The ion anode was centered under the Cs-enriched glass. The
copper container was placed on a hot plate whose temperature was set at 540 ◦ C.
We waited a few minutes for thermal equilibrium to be established and then turned
on the high voltage power supply with its voltage gradually increased to 700 V. The
current was monitored with a multimeter in the circuit and was stabilized at several
milliamperes during the electrolysis.
5.5.2
Visual demonstration
The electrolysis was done for several minutes until a film of yellow Cs metal formed
on the top window of the cell and coalesced into droplets. This is much more Cs
than would be needed or desired in practice. A gas flame could be used to heat the
two windows so that most of the metal condensed on the silicon sidewalls of the cell.
Several such cells have been made successfully with this electrolytic method so it is
5.5. Electrolysis in silicon-based miniature cells
(a)
91
Cs
3 mm
Figure 5.6: Photographs of the top surface of two cells. (a) with a visible layger of metal;
(b) without visible metal.
5.5. Electrolysis in silicon-based miniature cells
92
proved to be reproducible. Photographs of two cells are shown in Fig. 5.6. Panel (a)
is for a cell with a visible layer of golden metal inside. Panel (b) is for a cell without
visible metal inside. The advantage of the first cell over the second one is the less
amount of Cs vapor. Too much Cs metal in the cell might condense on the windows
when the cell is heated and cooled repeatedly. The distribution of the metal film on
the windows will change over time. As a result, the transmission of laser light through
the vapor cell might be changed not because of the vapor itself but because of the
distribution of metal on the windows. This phenomenon will induce systematic errors
to the systems under investigation. So in a practical case, we would prefer to fill the
cell with an appropriate amount of alkali metal vapor by controlling the electrolysis
current and time.
5.5.3
Physical demonstration
To prove that there was free Cs metal in the cells made with this method, we measured
the absorption of Cs D1 light by the vapor cell. The laser frequency was tuned to the
peak of the D1 resonance line of Cs. We measured the peak absorption of the cell for
temperatures ranging from 90 to 130 ◦ C. The calculation of the Cs number density
from the absorption data was the same as discussed in the previous section. The
peak absorption cross section[43], σ = 4.0 × 10−13 cm2 , for Cs atoms in 0.4 amagat
of Ar is nearly independent of temperature. All the cells have been measured and all
of them have Cs inside. One of the results of these measurements is shown as points
in Fig. 5.7. The continuous curve is the number density of Cs vapor in equilibrium
with pure liquid Cs as tabulated by Nesmeyanov[44]. The density of Cs vapor in the
electrolytically filled cell is very nearly equal to the saturated number density.
5.5. Electrolysis in silicon-based miniature cells
93
15
10
[Cs] (cm-3)
Measured [Cs]
Reference
14
10
13
10
12
10
80
90
100
110
120
130
140
Temperature (C)
Figure 5.7: Number density of Cs atoms in our cell (red solid dots) and the standard
number density of the Cs vapor which is in thermal equilibrium with metal (blue curve).
Photodiode signal (mV)
135
130
125
4
2 3
0 1
-1
-2
m= -4 -3
120
115
end resonance
110
m= -3 -2
-1 0
1 2
3
105
100
-40
-20
0
20
MW detuning (kHz)
Figure 5.8: Microwave end resonance.
40
60
5.5. Electrolysis in silicon-based miniature cells
94
We also carried out optical pumping experiments with these electrolytically filled
miniature cells. For example, a microwave end resonance[45] for one of the cells
is shown in Fig. 5.8. It is a resonance between the two energy sublevels of the
hyperfine multiplets, each of which has the largest azimuthal quantum number in the
corresponding multiplets. The end resonance for 133 Cs is shown as an inset in Fig. 5.8.
The cell was pumped with 7.6 mW/cm2 of circularly polarized light from the same
diode laser used to make the density measurements of Fig. 5.7. The transmitted light
was measured with a photodiode. The microwaves came from a horn antenna. The
resonance peak centered at 9.19314 GHz with the FWHM (full with at half maximum)
being 12.3 kHz. On resonance the transmission decreased by 16.7%. The magnetic
field along the light path was 0.13 G and the temperature was 110 ◦ C.
5.5.4
Calculation of electrolytic current
It needs to be noted that the electrolysis current through all the glasses can not only
reduce Cs+ ions to Cs atoms in the cell, but also reduce Na+ ions to Na atoms. This
is because the presence of Na+ in the Pyrex glass. When the electrons from the
cathode move to the Cs-enriched borate glass and combine with Cs+ there, they can
also move to the Pyrex glass close to the silicon cathode and combine with Na+ there.
Too many Na atoms can ruin the bond between the silicon and the Pyrex[46, 47].
Thus it is important that most of the electrolytic current is focused through the Csenriched glass. In our experiments, two methods were used to concentrate most of
the electrolytic current through the Cs-enriched glass: (1) a small-diameter ion anode
was used, and (2) the molten salt was used. Method (1) will be discussed in detail
in the following paragraph. In method (2), the molten salt provided a good thermal
5.5. Electrolysis in silicon-based miniature cells
95
contact to the glass and permitted us to keep the Pyrex above the anode hotter and
more highly conducting than for the surrounding Pyrex glass.
In order to illustrate the necessity of designing the ion anode carefully, we show
in Fig. 5.9 calculated current distributions from an ion anode, through a glass plate,
to a silicon cathode. For simplicity, we assumed all three parts to be cylinders along
the same direction and constant conductivity σ in the glass. The electrical current
goes along the axis of the cylinder.
The current density is,
j = σE = −σ∇φ,
(5.3)
where E = −∇φ is the applied electric field and φ is the electrostatic potential. In a
system like this, we chose to use the cylindrical coordinate system with the coordinates
r, θ, z. The origin of the coordinate system is at the center of the interface between
the glass and the ion anode. The z axis is along the axis of the cylinder.
In order to find the current distribution, we solved the Laplace’s equation,
∇2 φ = 0,
(5.4)
and got the axially symmetric solution φ = φ(r, z) in cylindrical coordinates. The
boundary conditioins were φ = 0 for the glass-silicon interface and φ = V at the
glass-salt interface. We assumed no current flow into glass except through the ion
anode, so for the region of the bottom surface of the glass that were not in contact
with the ion anode we set ∂φ/∂z = 0. We also assumed that at the side wall of the
glass disk, the current flowed along the wall, i.e. no current flowed out of the wall
∂φ/∂r = 0.
5.5. Electrolysis in silicon-based miniature cells
96
In cylindrical coordinates,
∇2 φ =
1 ∂ 2φ ∂ 2φ
1 ∂ ∂φ
(r ) + 2 2 + 2 = 0,
r ∂r ∂r
r ∂θ
∂z
(5.5)
Let φ = u(θ)v(z)w(r) and separate Eq. 5.5 into
d2
u(θ) + m2 u(θ) = 0
dθ2
d2
v(z) − k 2 v(z) = 0
2
dz
1 d
m2
d2
2
w(r)
+
w(r)
+
(k
−
)w(r) = 0
dr2
r dr
r2
(5.6)
(5.7)
(5.8)
The general solution was chosen to be
φ=
X
[An sinh(kn z) + Bn cosh(kn z)]J0 (kn r),
(5.9)
n
and the coefficients were determined by the given boundary conditions,
φ(z = h) = 0
(5.10)
∂φ
(r = 1) = 0
∂r
(5.11)
φ(z = 0, r ≤ f ) = 0
(5.12)
∂φ
(z = 0, r > f ) = 0
∂z
(5.13)
Here the radius of the glass plate was set to be 1, the thickness of the glass plate
was h, and the radius of the ion anode was f . Jm (kn r) is the Bessel function of the
first kind. Because the solution had a axial symmetry, m = 0. So only J0 (kn r) was
included in the solution. For the same reason, θ did not appear in Eq. 5.9.
From Eq. 5.11, we got
X
n
(An sinh(kn z) + Bn cosh(kn z))
∂J0 (kn r)
(r = 1) = 0,
∂r
(5.14)
5.5. Electrolysis in silicon-based miniature cells
Since
dJ0 (x)
dx
97
= −J1 (x), Eq. 5.14 becomes
X
(An sinh(kn z) + Bn cosh(kn z))J1 (kn ) = 0,
(5.15)
n
It is valid for any z between 0 and h only when J1 (kn ) = 0 for each kn . A Matlab
code was written to find the zeros of J1 (x) considering the properties of the Bessel
function. J0 (x) has a zero in each interval of length π. For Jm (x)(m >
1
),
2
the
difference between two consecutive zeros is greater than π and approaches π as x
approaches ∞.
From Eq. 5.10 we got
X
(An sinh(kn h) + Bn cosh(kn h))J0 (kn r) = 0,
(5.16)
n
for any r. This gave the relation between An and Bn to be
An sinh(kn h) + Bn cosh(kn h) = 0,
(5.17)
Considering this condition, we rewrote the solution for φ as
φ=
X
An [cosh(kn z) − coth(kn h)sinh(kn z)]J0 (kn r).
(5.18)
n
Using the boundary conditions Eq. 5.12 and 5.13 on the new expression of φ we
got
X
An J0 (kn r)(r ≤ f ) = 0
(5.19)
−An kn coth(kn h)J0 (kn r)(r > f ) = 0
(5.20)
n
X
n
Coefficients An were solved from Eq. 5.19 and 5.20 with Matlab by considering
finite number of terms in the summation. The current density was then calculated
5.5. Electrolysis in silicon-based miniature cells
-4
-3
-2
-1
0
-4
-3
-2
-1
0
-4
-3
-2
-1
0
1
2
0.9
0.7
0.5
z
0.3
(a)
0
-5
Copper
NaNO3
0.1
1
98
3
4
5
3
4
5
3
4
5
z
0
-5
1
0.9
0.1
0.3
0.5
1
0.7
(b)
2
z
0
-5
x
1
0.9
0.1
0.3
0.5
1
0.7
(c)
2
Figure 5.9: Distribution of current from an ion anode to a silicon cathode through a glass
plate. The horizontal coordinate is the distance from the center of the glass. The vertical
coordinate is the height from the bottom of the glass. Both distances are given in units of
the plate thickness d. (a) The diameter of the ion anode is 8d. (b) The diameter of the ion
anode is d. (c) The diameter of the ion anode is negligibly small compared with d.
using Eq. 5.3. In Fig. 5.9, the labels 0.1,...,0.9 indicate surfaces of revolution containing fractions 0.1,...,0.9 of the electrolytic current. As shown in Fig. 5.9(a), when the
diameter of the anode is much larger than the glass thickness, most of the current
flows to a cathode area that is only slightly larger than the anode area. The current
collection area on the cathode can be diminished by diminishing the diameter of the
ion anode. Fig. 5.9(b) shows the current flow for an ion anode with a diameter equal
to the glass thickness. Fig. 5.9(c) shows that the collection area for a “point-source”
anode is only slightly smaller than that of the finite anode of Fig. 5.9(b).
5.5. Electrolysis in silicon-based miniature cells
99
As mentioned above, there will be further concentration of the current because of
higher conductivity of the hot glass above the anode. This calculation is a simplified
model of how the decrease of the ion anode’s size can help concentrate the electrolysis
current. In the real case, the glass consists of two parts. One is the Cs-enriched borate
glass with higher conductivity and the other is the Pyrex glass with lower conductivity.
The higher conductivity of the Cs-enriched borate glass can help concentrate the
current even more than the calculated result. Since the current is concentrated into
the Cs-enriched glass, most of the reduced metal is Cs rather than Na. The total
number of metal atoms is equal to the time integral of the current divided by the
elementary charge e.
Chapter 6
Summary
In summary, we have presented in this thesis some new physics and technology of
spin-polarized alkali-metal atoms.
• We have demonstrated that a new method of measuring the dependence of the
microwave resonance frequencies of alkali-metal atoms on buffer-gas pressure
can provide information about many important interaction terms such as the
molecular potential Vm and the hyperfine-shift coefficient δA. We presented
a nonlinear dependence of
87
Rb and
133
Cs on the pressure of the buffer gases
Ar and Kr. We also showed that there are no nonlinearities observed in the
gases He and N2 . These results strongly suggest that the nonlinearities come
from the contribution of Van der Waals molecules that form in Ar and Kr
(e.g. CsAr or RbKr), but not in He or N2 . We developed a simple model,
parameterized by the effective three-body formation rate of molecules and by
the effective product of the collisionally limited lifetime times the shift of the
100
101
hyperfine coupling coefficient in the molecule, and successfully explained the
nonlinear data.
• We also presented another unusual phenomenon that has not been reported
before. For alkali-metal atoms that are optically pumped with D1 circularly
polarized laser light, the microwave resonance signals will be reversed from
transmission dips to transmission peaks when the frequency and intensity of the
laser light are under some special condition. The first condition is that the laser
frequency should be close to that of the transitions from the lower hyperfine
multiplet of the ground state to the excited state. The second condition is
that the intensity of the laser light should be sufficiently low. And the third
condition is that the hyperfine splitting has to be at least partially resolved. This
counterintuitive phenomenon can be qualitatively understood with the help of
a model considering the spin-temperature distribution. Calculations with the
complete density matrix give results that are consistent with the experimental
data.
• The third and last part is a new technology which deals with the problem of
filling a small amount of alkali metal into a miniature vapor cell. We invented a
new method which uses electrolysis of a specially-made borate glass (containing
the target alkali metal) to fill the alkali metal into a miniature vapor cell made by
anodic bonding of Pyrex glass and silicon wafer. This method has the advantage
of being able to fill controlled amount of alkali metal into the cell. It is also
convenient and possible to be used in mass production of such cells for miniature
clocks.
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