Download New studies of optical pumping, spin resonances, and spin exchange in mixtures of inert gases and alkali-metal vapors

NEW STUDIES OF OPTICAL PUMPING, SPIN
RESONANCES, AND SPIN EXCHANGE IN
MIXTURES OF INERT GASES AND
ALKALI-METAL VAPORS
Yuan-Yu Jau
A DISSERTATION
PRESENTED TO THE FACULTY
OF PRINCETON UNIVERSITY
IN CANDIDACY FOR THE DEGREE
OF DOCTOR OF PHILOSOPHY
RECOMMENDED FOR ACCEPTANCE
BY THE DEPARTMENT OF
PHYSICS
JANUARY, 2005
c Copyright by Yuan-Yu Jau, 2005. All rights reserved.
°
Abstract
In this thesis, we present new studies of alkali-hyperfine resonances, new optical pumping
of alkali-metal atoms, and the new measurements of binary spin-exchange cross-section
between alkali-metal atoms and xenon atoms.
We report a large light narrowing effect of the hyperfine end-resonance signals, which
was predicted from our theory and observed in our experiments. By increasing the intensity
of the circularly polarized pumping beam, alkali-metal atoms are optically pumped into
a state of static polarization, and trapped into the hyperfine end-state. Spin exchange
between alkali-metal atoms has minimal effect on the end-resonance of the highly spinpolarized atoms. This new result will possibly benefit the design of atomic clocks and
magnetometer. We also studied the pressure dependence of the atomic-clock resonance
linewidth and pointed out that the linewidth was overestimated by people in the community
of atomic clock.
Next, we present a series study of coherent population trapping (CPT), which is a
promising technique with the same or better performance compared to the traditional microwave spectroscopy. For miniature atomic clocks, CPT method is thought to be particularly advantages. From our studies, we invented a new optical-pumping method, push-pull
optical pumping, which can pump atoms into nearly pure 0-0 superposition state, the superposition state of the two ground-state hyperfine sublevels with azimuthal quantum number
m = 0. We believe this new invention will bring a big advantage to CPT frequency standards, the quantum state preparation for cold atoms or hot vapor, etc. We also investigated
the pressure dependence of CPT excitation and the line shape of the CPT resonance theoiii
retically and experimentally. These two properties are important for CPT applications. A
theoretical study of “photon cost” of optical pumping is also presented.
Finally, we switch our attention to the problem of spin exchange between alkali-metal
atoms and xenon gas. This mechanism is very important to the spin-exchange optical
pumping. We report the first measurements of binary spin-exchange rate at high magnetic
field (9.4 tesla). We present the first calculation of the magnetic decoupling curves of the
binary spin-exchange rates by using two different methods, semi-classical approach (SCA)
and distorted-wave Born approximation (DWBA).
iv
Acknowledgements
Life is a road of learning. Sincerely, I would like to thank my advisor, William Happer, a
brilliant and respected scientist. With his help, I learned more physics and the attitude of
study. Because of his knowledge and intuition, my sight of physics became wider. It is a
great privilege to study physics with him.
I want to thank Nick Kuzma, our postdoc, for helping me on writing paper, doing
experiments, and generously providing his assistance on everything. Some of his interesting
stories also delighted me during some dry hours in the office.
I am very grateful to Eli Miron, a scientist from Israel. He joined our lab during his
sabbatical. His was a very kind, responsible, and work-hard person. Because of him, many
experiments can be carried out. He cheered people up and animated everything in the lab.
I was very fortunate to work with him.
I am grateful to Dan Walter and Warren Griffith for their company of my first two years
in Princeton. With their help, I was able to catch up on the lab works and get use to the
new environment.
It is my fortune to join the atomic physics group with such good people as Mike Romalis,
Brain Patton, Amber Post, Andrei Baranga, Tom Kornack, Micah Ledbetter, Igor Savukov,
and back to the early time, Ioannis Kominis and Kumar Raman. I learned lots of things
from them, and they made lab an interesting place to work. I especially appreciate Mike
Romalis, a knowledgeable person, who shared great knowledge and experience with me.
The staff of the Physics Department have been unfailing helpful. I would like to thank
Laurel Lerner, Mary De Lorenzo, Claude Champagne, John Washington, Kathy Warren,
v
Ellen Webster, Mike Peloso, Helen Ju, and Angela Qualls. They constantly helped whenever
I encountered problems in our department. I am also indebted to Charles Sule of Physics
Department and Mike Souza of the Chemistry Department. Without their expertise, many
experiments this thesis can not be carried out.
It has been very nice to correspond with Jacque Vanier, an expert of atomic clocks. He
is a professor in Montreal University in Canada. I learned many new things through the
discussions with him.
Finally, I want to thank my parents. Because of them, I entered college and then
obtained Ph.D. education in Princeton University.
vi
Contents
Abstract
iii
Acknowledgements
v
Contents
vii
List of Figures
x
List of Tables
xx
1 Introduction
1
1.1
Advantage of Alkali-Metal Atoms . . . . . . . . . . . . . . . . . . . . . . . .
2
1.2
Content Brief . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
2 Strong Resonances of Alkali Hyperfine Sublevels from Optical Pumping
2.1
2.2
Density Matrix Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
9
2.1.1
Evolution in Liouville Space . . . . . . . . . . . . . . . . . . . . . . .
12
2.1.2
Ground-State Relaxation . . . . . . . . . . . . . . . . . . . . . . . .
13
2.1.3
Excited-State Relaxation . . . . . . . . . . . . . . . . . . . . . . . .
16
2.1.4
Optical Pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
2.1.5
Microwave and RF Fields . . . . . . . . . . . . . . . . . . . . . . . .
21
2.1.6
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
Microwave Spectroscopy and CPT Spectroscopy
vii
. . . . . . . . . . . . . . .
23
2.3
2.4
2.5
2.2.1
Coherence Induced by Oscillating Fields . . . . . . . . . . . . . . . .
23
2.2.2
Optically Induced Coherence . . . . . . . . . . . . . . . . . . . . . .
24
2.2.3
Inhomogeneous Linewidth Broadening . . . . . . . . . . . . . . . . .
28
Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
2.3.1
Microwave Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
2.3.2
CPT Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
Results and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
2.4.1
Light Narrowing Effect . . . . . . . . . . . . . . . . . . . . . . . . . .
37
2.4.2
Transient Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
2.4.3
Pressure Broadening of Microwave Resonance . . . . . . . . . . . . .
47
2.4.4
Push-Pull Optical Pumping . . . . . . . . . . . . . . . . . . . . . . .
51
2.4.5
Pressure Dependent CPT Excitation . . . . . . . . . . . . . . . . . .
61
2.4.6
Line Shape of CPT Resonance . . . . . . . . . . . . . . . . . . . . .
65
2.4.7
Photon Cost of Optical Pumping . . . . . . . . . . . . . . . . . . . .
72
New Simple-Compact Frequency Standard . . . . . . . . . . . . . . . . . . .
83
3 High-Field
3.1
3.2
3.3
3.4
129 Xe
129 Xe-Alkali-Metal
Spin Exchange
88
Spin Relaxation Rates Equation . . . . . . . . . . . . . . . . . . . . .
89
3.1.1
Detailed Balancing . . . . . . . . . . . . . . . . . . . . . . . . . . . .
89
3.1.2
Rate Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
91
Magnetic Decoupling of
129 Xe-Alkali-metal
Spin Exchange . . . . . . . . . .
93
3.2.1
Semi-Classical Approach (SCA) . . . . . . . . . . . . . . . . . . . . .
93
3.2.2
Distorted-Wave Born Approximation (DWBA) . . . . . . . . . . . .
95
Experiments and Calculations . . . . . . . . . . . . . . . . . . . . . . . . . .
100
3.3.1
NMR Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . .
101
3.3.2
Faraday Rotation Measurements . . . . . . . . . . . . . . . . . . . .
105
Results and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
111
3.4.1
Binary Spin-Exchange Rate Coefficients . . . . . . . . . . . . . . . .
111
3.4.2
High-Field Contribution of van der Waals Molecules . . . . . . . . .
116
viii
A Evolution of the Density Matrix of Alkali-Metal Atoms
119
A.1 Ground-State Relaxation Due to Weak Collisions . . . . . . . . . . . . . . .
120
A.1.1 S-Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
122
A.1.2 Carver Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
122
A.2 Ground-State Relaxation Due to Strong Collisions and Other Mechanism .
123
A.2.1 Spin Exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
123
A.2.2 Spin Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
124
A.3 Excited-State Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
125
A.3.1 J-Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
125
A.3.2 Spontaneous Decay and Quenching . . . . . . . . . . . . . . . . . . .
126
A.4 Microwave and RF Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . .
127
A.5 Optical Pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
128
A.5.1 D1 Depopulation Pumping . . . . . . . . . . . . . . . . . . . . . . .
131
A.5.2 D1 Repopulation Pumping . . . . . . . . . . . . . . . . . . . . . . .
134
B Circuit Diagram of Signal Divider
137
References
138
ix
List of Figures
2.1
In our modelling, we calculated the ensemble behaviors of the alkali-hyperfine sublevels with D1 pumping and relaxation mechanisms, such as S-damping, spin exchange, Carver damping, diffusion, spontaneous decay, quenching, and J-damping.
2.2
12
The population of the sublevel b is pumped out. When a oscillating field turns on at
the resonant frequency, the coherence is generated. Some population transfers from
the sublevel a to b, and therefore it increases the light absorption or decrease the
light transmission. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3
23
Two monochromatic fields with frequencies Ω1 and Ω2 couple between two groundstate sublevels and the excited state. Two optical coherences between the ground
state and the excited state are generated. When the difference of two optical frequencies is equal to ground-state sublevel splitting, a strong ground-state coherence
can be produced. Under the same condition, photons begin scattering between
two monochromatic fields, and are less absorbed by atoms. The light absorption is
therefore reduced and observed as a transmission peak. When the optical detuning
is present, it is equivalent to stimulated Raman scattering.
x
. . . . . . . . . . . .
25
2.4
We use a hypothetic atom, which has a nuclear spin I = 1/2, to illustrate the CPT
coherence in the picture of spin oscillation. Different Λ schemes generate different
ground-state coherences. The coherence of 0-0 CPT is equivalent to the electron spin
oscillating along the z-direction. The end-state coherence is equal to spin precessing
on the x-y plane. The dotted circles represent the dark states for different CPT
schemes. The 0-0 CPT excitations from σ+ and σ− pumpings have 180◦ phase
difference.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
. . . . . . . . . . . . . . . . . . . . .
30
. . . . . . . . . . . . . . . . .
31
2.5
Lollipop (left) and rectangle (right) cells.
2.6
Experimental setup for microwave spectroscopy.
2.7
The noise spectrum of the transmission light in the experiments. Our homemade
divider circuit can suppress the background noise level by 15dB, and can increase
the SNR by a factor of 6 ∼ 7. This divider circuit was especially useful when we
used the ring laser as the pumping source. Because the laser was conducted by a
long fiber (20 m), lots of mechanical vibration noise was converted into the intensity
fluctuations of the output light.
2.8
. . . . . . . . . . . . . . . . . . . . . . . . . .
33
The top panel shows the transient of the transmission light when the microwave
field was on and off. The second panel is the zoom-in of the circle on the top panel.
The third one is the Fourier’s spectrum of the damping oscillation. The bottom one
shows the same resonance measured by scanning the microwave detuning (two sec
for each point). Its linewidth is broader due to the microwave power broadening. .
2.9
34
The probing beam for CPT experiments was generated by a Mach-Zenhder Modulator. By modulating the effective light propagation lengths from two crystals,
the output beam can be intensity modulated. The optical sidebands of the probing
beam can be measured from a Fabry-Perot spectrometer. . . . . . . . . . . . . .
xi
35
2.10 Two example CPT signals were found by setting the central frequencies of the modulation to their resonant frequencies. The scanning range was ±50 kHz. The ripple
tails were due to the fast scanning, which can be eliminate by slowing down the
sweeping speed or shorten the scanning range. The end-resonance signal was obtained by setting the probing beam with 45◦ to the magnetic field. . . . . . . . .
36
2.11 The light narrowing effect is shown by two different end resonances. The linewidth
gets narrower by increasing pumping power at beginning, because the optical pumping traps more atoms into the end state. The minimum point is where the spinexchange broadening is comparable to the pumping broadening. The linewidths
start increasing after the minimum point, because the optical pumping start dominating the linewidth broadening. However, the circularly polarized pumping can
only make the linewidth of 0-0 resonance worse. . . . . . . . . . . . . . . . . . .
2.12 The first light narrowing data of
87
40
Rb hyperfine end-resonance was found from the
high pressure lollipop cell. The data agrees with the theory very well. Two insets
show the resonance signals from the detuning scan. The scanning step was 400 Hz.
The linewidth decreases with increasing pumping power to a minimum value. After
this point, the linewidth begins to be dominated by the pumping rate. However, the
signal amplitude is always improved by the pumping power. . . . . . . . . . . . .
41
2.13 The top panel shows the calculated transient signal with frequency detuing = 7 kHz,
and its multi-exponential background has been subtracted. The middle and bottom
panels show the computational results of the complex damping rate at different
frequency detunings by solving the evolution equation and using Eq. (2.59). The
unit in their vertical axes is 1000/sec. The calculations show the distinction between
two different methods is only a few parts per thousand. . . . . . . . . . . . . . .
2.14 An experimental data of the end-transient damping rates from a
87
44
Rb cell with 1
atm N2 buffer gas. The data was fit into Eq. (2.59). The Rabi frequency can be
extracted from the fitting of the imaginary decay rates. . . . . . . . . . . . . . .
xii
46
2.15 The experimental data of microwave and Zeeman resonance damping rates as the
number density of nitrogen. Our experimental data are consistent with D.K. Walter’s
measurements, which is plotted as two shadowed areas. The hyperfine resonance
data is affected by both S-damping rate and Carver rate. The Zeeman resonance
data is only affected by S-damping rate. J. Vanier’s data predicts a much larger
values, which is plotted as a dark-gray curves on the top left corner.
2.16 Absolute hyperfine frequency shifts of
85
. . . . . . .
49
Rb induced by the buffer gases at a con-
stant pressure PB = 760 torr, inferred from the data of Bean and Lambert [7]. Also
shown are the spreads in frequency ∆νp that would be caused by a spread in temperature ∆T = 20 ◦ C centered at 30 ◦ C. The line broadening due to temperature
inhomogeneities in Ar is about ten times less than that in He, Ne, or N2 . . . . . .
50
2.17 The oscillation of the electron spin produces time dependent absorption cross sections for different circularly polarized pumping lights. The interlacing σ+ and σ−
light pulses repeated every 2π/ω00 can maximize the spin oscillation, and therefore
trap the atoms to the 0-0 superposition state. At this point, the transmission light
has the maximum transparency. . . . . . . . . . . . . . . . . . . . . . . . . . .
52
2.18 Alkali hyperfine sublevels with a hypothetical nuclear spin quantum number I = 1/2.
0-0 state is the only dark state by using a complementary Λ scheme, which has two
Λ pumpings with a 180◦ modulation phase difference between each other. Atoms
will be eventually trapped into the 0-0 state, the dark state. . . . . . . . . . . . .
53
2.19 The optical sidebands of a pulsed light is like a comb in the spectrum. With high
buffer-gas pressure, atoms see all optical sidebands. With low buffer-gas pressure,
only two sidebands are seen by atoms. . . . . . . . . . . . . . . . . . . . . . . .
54
2.20 The Michelson type interferometer converts an intensity modulated laser beam to
a polarization modulated laser beam. A λ/4 wave plate was needed in one of the
arms. The difference of the arms’ length is equal to c/(4ν00 ), where ν00 is the 0-0
resonance frequency. The details of other parts can be found in Fig. 2.6 and Fig. 2.9. 55
xiii
2.21 Comparison of the signal contrasts of conventional CPT and push-pull CPT. The
inset shows the modulation spectrum from a Fabry-Perot spectrometer. At the same
pumping rate, the CPT signal from push-pull pumping has a better signal contrast,
which is about 77 times larger than the conventional CPT signal. The conventional
CPT signal on the top panel was taken with 16 times more averages than the pushpull CPT for a better SNR. This experimental data was taken from a spherical cell
with diameter = 1.9 cm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
2.22 Experimental data and theoretical curves from our density-matrix modelling agree
very well to each other. The conventional CPT has an optimized value of pumping
rate for maximizing the signal contrast. Higher pumping rate degrades signal of
the conventional CPT due to the population trapped to the end state. However,
push-pull pumping has no such problem. The right diagram shows a very strong
CPT signal by using push-pull pumping of one circled data point on the left diagram. 57
2.23 The relative linewidth contributed from spin-exchange broadening. As the prediction
from Eq. (2.49), push-pull optical pumping can reduce the linewidth broadening due
the spin-exchange collisions, but the optical pumping used in conventional CPT can
not. However, in push-pull pumping, the decrease of the spin-exchange broadening
can not compensate the linewidth broadening due to the optical pumping. This tiny
light narrowing effect is hardly to be observed. . . . . . . . . . . . . . . . . . . .
58
2.24 The FOMs of the plots in Fig. 2.22. Push-pull CPT has larger FOM than conventional CPT. Here, FOM=contrast(%)/linewidth(kHz)×10. . . . . . . . . . . . . .
60
2.25 By changing the ratios of pumping intensities of σ+, σ−, I1 , and I2 with a selected
modulation frequency, we can put lots of population into some hyperfine sublevels.
The top-left panel shows 50% on |2, 0i and 50% on |1, 0i states. The top-right panel
shows 99% on |2, 0i state. The bottom-left panel shows 95% on |1, 1i state. The
bottom-right panel shows 100% on |2, 2i state. . . . . . . . . . . . . . . . . . . .
xiv
61
2.26 The amplitude of the optical coherence ρge is like a FM signal passing through a
resonator filter g(Ω) =
1
i(Ωge −Ω)+γop ,
where Ω is the optical angular frequency. Fre-
quencies of the signal E(t) higher or lower than the resonance frequency have lower
output amplitudes. The lower buffer gas pressure has a narrower filter bandwidth,
and can produce a stronger AC component in pumping rate to excite a larger CPT
signal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
62
2.27 The ground-state CPT excitation from the three correlated quantum levels can be
approximately analogous to an electronic-mechanical system. The CPT excitation
is equivalent to the mechanical oscillation of the two metallic plates on the right
diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
2.28 Calculated excitation efficiency of Cs 0-0 CPT resonance in the nitrogen buffer gas.
Here, the modulation index m is defined by E(t) = E0 exp(−iΩ0 t + im sin(ωm t)),
where Ω0 is the carrier frequency.
. . . . . . . . . . . . . . . . . . . . . . . . .
65
2.29 The left and the middle panel show the probing lights along two orthogonal directions, parallel or perpendicular to the spin at zero detuning δω = 0. The signal
amplitudes plotted as functions of δω. The in phase signal shows a symmetric curve,
and the out phase signal shows a completely asymmetric curve. By using a modulated pumping light, coherence can be excited as a spin along the x-direction by the
coherence optical pumping rate RΓ . If there is a light shift (δEv 6= 0), an effective
field Rv is generated. This emulated field produces a spin component along the
y-direction. Therefore the probing light both probes in phase and out phase signals.
The detuning curve becomes asymmetric. . . . . . . . . . . . . . . . . . . . . .
2.30 Numerical calculation of the asymmetry of the resonance of
87
67
Rb at 0.1 atm as a
function of Ω−Ω0 . The subplot shows the intensity spectrum of the optical sidebands
of the modulated light and the curve of the absorption cross section. The ground
state hyperfine splitting is resolved. When the optical detuning is small, only two
strong optical sidebands are important. Hence, it can be approximated by two-wave
pumping at low optical detuning.
. . . . . . . . . . . . . . . . . . . . . . . . .
xv
69
2.31 Experimental data shows the line shapes of the end CPT resonances as a function
of the laser frequency from a
87
Rb cell with 0.96 atm N2 buffer gas. By using
the numerical model, we can extract the parameter of optical pressure broadening
and the pressure shift. The two wave model can not fit to data, because in the
high pressure cell, atoms can see not only two optical sidebands but also high order
sidebands. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
2.32 The calculation result shows a smaller dδA /d∆ around the point of Ω−Ω0 ∼ 0 by using a different pumping spectrum. The inset shows the modulation spectrum, which
is generated by a Mach-Zehnder type modulator. The amplitude of the pumping
electric field is proportional to sin(φ0 + φ1 sin ωm t). . . . . . . . . . . . . . . . .
71
2.33 The left and middle diagrams show the σ+ pumping with different deexcitation
mechanisms. Quenching deexcitation causes higher pumping efficiency than spontaneous decay. The right diagram shows the hyperfine sublevels with an assumption of
nuclear spin I = 1/2. The electron spin has chance to be transferred to the nuclear
spin through the hyperfine coupling in the excited state for the pumping transitions
of region (1), but not of region (2). . . . . . . . . . . . . . . . . . . . . . . . . .
73
2.34 Photon cost for pumping potassium-39 to the end state, which is plotted as a function
of relative quenching rate and J-damping by using a contour diagram. The difference
of values between each contour is 0.2. Darker color represents lower value, and lighter
color means higher value. The value of the contour is equal to the photon cost. . .
76
2.35 Photon cost for pumping rubidium-85 to the end state, which is plotted as a function
of relative quenching rate and J-damping by using a contour diagram. The difference
of values between each contour is 0.2. Darker color represents lower value, and lighter
color means higher value. The value of the contour is equal to the photon cost. . .
77
2.36 Photon cost for pumping rubidium-87 to the end state, which is plotted as a function
of relative quenching rate and J-damping by using a contour diagram. The difference
of values between each contour is 0.2. Darker color represents lower value, and lighter
color means higher value. The value of the contour is equal to the photon cost. . .
xvi
78
2.37 Photon cost for pumping cesium-133 to the end state, which is plotted as a function
of relative quenching rate and J-damping by using a contour diagram. The difference
of values between each contour is 0.2. Darker color represents lower value, and lighter
color means higher value. The value of the contour is equal to the photon cost. . .
79
2.38 . Photon costs for two different optical pumpings and nuclear spins with different
relaxation mechanisms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
80
2.39 Two different cavity configurations of hyperfine-modulated lasers for two different
types of laser diodes. The regular edge emitting laser diode outputs linearly polarized laser light, and therefore it needs two λ/4 wave plates to produce alternating
circular polarization inside the cavity. The VCSEL diode is designed to have almost no preference of the light polarization, so the light polarization can be directly
modulated by the vapor cell. The steady lasing point of the two lasers is when the
modulation period of the output light is equal to 2π/ω00 . . . . . . . . . . . . . .
84
2.40 An illustration of the spectral responses due different causes. The optical comb in
the lasing spectrum is grown to produce the maximum transparency of the vapor
cell, and therefore obtain the maximum gain of photons. . . . . . . . . . . . . . .
85
2.41 The normalized amplitudes of the maximum 0-0 coherence with different nuclear
spins by using different fast push-pull pumping. D1 push-pull pumping can excited a
much stronger CPT coherence than D2, and also make the vapor cell more transparent. 86
3.1
An example of
129
Xe-Rb using SCA and DWBA numerical calculations. The SCA
curve behaves like the mean value of the DWBA curve. . . . . . . . . . . . . . .
3.2
A block diagram shows the total setup of the
xvii
129
100
Xe spin-exchange rate measurements. 101
3.3
The NMR probe was designed to fit into a 9.4-T Oxford superconducting magnet
with bore diameter ∼ 3.5”. The internal space was heated resistively with noninductive wiring. The heat conducting copper tube suppressed the thermal gradient
to about 1◦ C per inch. The temperature was stabilized by the feedback of two
thermal sensors. The interlayer between inner and outer tubes was stuffed by a
porous foam for good heat insulation. The sample cell was held by the NMR coil
and hung by two supports. The NMR coil was set to the resonance frequency
by adjusting a variable capacitor. The tuning work was done by using a sweep
generator (WAVETEK 1062). A probing beam for Faraday rotation measurement
passed through the two windows and the center of the cell. The field inhomogeneity
inside the 1 inch cell was about 0.3 ppm.
3.4
. . . . . . . . . . . . . . . . . . . . .
102
Arbitrary unit for y-axes. The top panel shows two orthogonal FID signals. The
bottom panel is the Fourier’s spectrum of the FID signal. This example signal was
the result of eight times average FID from a 4 amagat Xe cell. . . . . . . . . . . .
3.5
The amplitude of the FID signal (filled circles) is proportional to the
129
103
Xe nuclear
spin polarization hKz i as it recovers towards its thermal-equilibrium value hKz iT =
µK B/2kT . The relaxation time T1 is extracted from the fit of Eq. (3.57), solid curve.
Each data point was taken over 16 – 96 averages of the FID signal at the same delay
time τ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6
104
Faraday rotation signal sin2θ (open circles), fit to the calculation of Eqs. (3.63–3.65)
(solid line) using the fitting parameters [Cs], θ0 , and ωJ . The optical path length
is 2.5 cm and B = 9.4 T. The dashed line shows the mean photon spin s measured
from the first harmonic of the photodetector output. The residual non-zero photon
spin at high laser frequency (far detuning from the D2 line) is ascribed to the effect
of optical guiding mirrors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xviii
108
3.7
Measured longitudinal nuclear spin-relaxation rates 1/T1 plotted versus measured
Rb vapor number densities [Rb] for the four cells listed in Tab. 3.1. The experimental
temperature were between 160 to 200◦ C. The spin-exchange rate coefficients κ are
the slopes of the straight-line fits to the data. . . . . . . . . . . . . . . . . . . .
3.8
112
Measured longitudinal nuclear spin-relaxation rates 1/T1 plotted versus measured
Cs vapor number densities [Cs] for the five cells listed in Tab. 3.2. The experimental
temperature were between 110 to 150◦ C. . . . . . . . . . . . . . . . . . . . . . .
3.9
113
Theoretical magnetic decoupling curves used to extrapolate our from B=9.4T to
B=0T. Since the experimentally determined potential V had about 5% error, we
made 3 to 4% adjustments to the position of the minimum of the interatomic potential V can bring a good agreement between experimental data and calculations.
The shape of decoupling curves changes less than 0.1% by applying adjustments on V . 114
B.1 Because of this divider circuit, we were able to obtain the first light narrowing data
from the fiber-coupled and noisy Ti-sapphire laser light. . . . . . . . . . . . . . .
xix
137
List of Tables
2.1
Experimental and theoretical results of the key parameter of S-damping rate
and Carver rate. All numbers listed above have about 10 % errors. For
helium gas, the values measured by D.K. Walter were for 3 He. The Carver
rates from Vanier’s measurements, which are tagged by “∗”, are much larger
than other values in the same block. The numbers from our work are marked
by parentheses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1
47
Experimentally measured values of κ for the four Rb cells used in our experiment. (Note: 1 amg=2.69 × 1019 cm−3 , is equal to the number density of an
ideal gas at 0◦ C and 760 torr) . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2
Experimentally measured values of κ for the five Cs cells used in our experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3
111
111
A summary of binary spin-exchange rate coefficients at different magnetic
fields obtained by different groups. The zero-field rates quoted for our work
are slightly larger than the values measured at B = 9.4 T because of adjustments for the magnetic decoupling discussed in connection with Fig. 3.9.
xx
.
115
Chapter 1
Introduction
In 1950, A. Kastler proposed the idea of using photons to transfer order to atoms by scattering the resonant light [32]. By appropriately choosing the frequency and the polarization of
the pumping light, it is possible to generate a desired distribution of populations in atomic
energy sublevels of the atoms. In the following decade, this basic idea of optical pumping
[22] had been quickly developed with many breakthroughs. The first spin-polarized optical
pumping was succeeded by Brossel et al. [14] and Barrat et al. [6] between 1952 to 1954,
which were done with sodium atomic beam and sodium vapor cells respectively. The spinexchange optical pumping was first proposed by H.G. Dehmelt in 1957, and was observed
by M.A. Bouchiat in 1960 [13]. In 1960, T.H. Maiman used optical pumping to produce
the population inversion, and therefore successfully achieved the first laser, ruby laser [37].
In 1961, Bell and Bloom used intensity-modulated light to pump Rb and Cs atoms into a
state of continuous spin precession [8]. This was the first demonstration of optically induced
Zeeman coherences. In the intervening decades, these important achievements continued
to be improved, and have been extended to many scientific studies. Nowadays, to study
the atomic energy levels, optical pumping is not just used in microwave or radio frequency
(RF) spectroscopy. Optical pumping with multi-coherent lights has become an important
technique in the studies of atomic spectroscopies and other atomic physics. CPT (coherent
population trapping) spectroscopy [4] is one example. In this thesis, we probe into details
1
2
of hyperfine evolution of alkali-metal atoms through the density-matrix modelling and hyperfine spectroscopy. We investigate CPT phenomena including important details, which
were ignored in many previous studies. From the studies, we have developed a new optical
pumping technique, which can pump the atoms into nearly pure 0-0 superposition state.
Besides, we also study the problem of spin-dependent collisions between alkali-metal and
noble-gas atoms. This mechanism plays an important role in spin-exchange optical pumping to produce hyper-polarized noble gases, which are very useful in studies of physics and
biomedical applications.
1.1
Advantage of Alkali-Metal Atoms
Alkali-metal atoms are useful for studying atomic physics, because they are relatively simple,
and they are more controllable in experiments. These hydrogen-like atoms have only one
valence electron. Their simpler quantum properties provide a good connection between
quantum theories and experiments. For experiments, probing sources, such as light sources
and microwaves, for alkali-metal atoms are readily available and convenient. Besides, alkalimetals are relatively volatile. Therefore they are easy to be optically probed in the gas
phase. Because of the reasons above, different types of experiments with alkali-metal atoms
have been constructed for testing of fundamental physics, and many useful inventions are
based on alkali-metal atoms. For example, hyperfine and Zeeman spectroscopies of alkalimetal atoms provide ways to measure many fundamental constants; spin-polarized and
spin-exchange optical pumping offer a way to study the magnetic properties of atoms. In
the past decade, atom cooling and trapping techniques provide a cleaner environment to
study many quantum behaviors, such as Bose-Einstein condensation, quantum gas in optical
lattice, cavity quantum electrodynamics, quantum computation, and quantum information.
In addition, the important inventions related to alkali-metal atoms are atomic clocks, vaporcell magnetometers, noble-gas spin polarizers, etc.
3
1.2
Content Brief
In chapter 2, we discuss our detailed studies of the atomic-clock resonances and the optical
pumping with multi-coherent light. For deeper investigation, we developed a complete
density matrix model to theoretically calculate the relevant phenomena. Our modelling was
capable of calculating the full evolution of ground-state and excited-state density matrices.
This enabled us to have a completed picture of microwave and CPT spectroscopy. In the
past, people have often used three-level or four-level models to describe the CPT phenomena.
Such simplified pictures can give a good qualitative explanation for the experimental results,
but they are not precise. Moreover, because the several-level modelling is too simplified,
it can miss some important phenomena. In order to examine the idea and the theory, we
set up the experimental system for microwave spectroscopy and CPT spectroscopy. We
used rubidium and cesium vapor cells as the probing targets, and mainly focused on D1
pumping. For atomic-clock resonances, the hyperfine spin relaxations plays a critical part
in the quality of the resonance signal. Many of those key relaxation rates [48] have been
measured by J. Vanier et al., who did lots of pioneering works related to vapor-cell atomic
clocks. However, from our studies, we discovered that there seem to have been systematic
errors in some of Vanier’s measurements. We believe those errors were due to temperature
gradients in the alkali vapors [29, 38]. For the measurements of the pressure-dependent
resonance linewidth, we employed the measurement of a damped transient oscillation of the
transmission light, and used thin cells to avoid possible temperature gradient contributions
to the measurements. Our results agreed with theoretical calculations and with another
experimental measurement done by D.K. Walter [53], who acquired those key parameters by
different means. For the optical pumping with modulated light, we present the calculation
and experimental data regarding CPT excitation under different buffer gas pressures, CPT
line shapes, and the ”photon cost” for optical pumping. The CPT signal is excited by
a modulated light beam. The pumped atoms respond differently to the modulated light
at different buffer-gas pressure, because of pressure broadening of the optical absorption
lines. Therefore the CPT excitation efficiency is not the same for different modulation
4
schemes, such as AM (amplitude modulation) or PM (phase modulation), and for different
gas pressures. Besides, the population of ground-state multiplets is also affected by the
buffer gas pressure. The common several-level model can not reflect those important details.
The CPT line shape is affected by light shifts, which include virtual and real transitions.
The virtual shift is controlled by the optical detuning and the pressure broadening. The real
shift is affected by the evolution in the excited state. Again, the common simple CPT model
can not really calculate those detailed features. In addition, we further discuss some details
of the evolution of the excited state, which was mostly ignored in previous CPT studies,
through the calculations of ”photon cost”, the mean number of photons per atom that
must be absorbed for pumping to a certain quantum state. The photon cost is affected by
the excited-state relaxation mechanisms, such as J-damping, quenching, and spontaneous
decay.
One of the important results from this research is that we confirmed the theoretically
predicted “light narrowing effect” of the hyperfine end resonances with our experiments [29].
This effect is due to the conservation of angular momentum. Spin exchange collisions, the
dominant relaxation mechanism in dense alkali-metal vapors, conserve angular momentum.
Since atoms in the end state have more angular momentum than any potential final state
of the collision, spin exchange collisions do not affect end state atoms. To obtain light
narrowing, the atoms have to be pumped by D1 circularly polarized light. By increasing
the pumping power, we can increase the fraction of atoms in the end state, and therefore
suppress the spin-exchange broadening. The light narrowing effect promises to generate a
end-resonance signal with a narrower linewidth and stronger amplitude compared to the
traditional 0-0 resonance used by most atomic clocks. In general, the performance of an
atomic clock is proportional to SNR/∆ν (the signal to noise ratio divided by the linewidth).
By using light narrowing effect, we might be able to build a miniature atomic clock or
magnetometer with a good performance. The miniature device requires a vapor cell with a
small size down to millimeter or sub-millimeter scale. Hence, a high working temperature
is necessary to increase the saturated vapor pressure of alkali-metal atoms and to produce
5
sufficient light absorption. However, the spin exchange collision rate between alkali-metal
atoms increase at higher atomic densities. The spin-exchange broadening increases the
linewidth and decreases the amplitude of the conventional resonance signal, but it has much
less influence on the end resonance of a highly spin-polarized atom. The end resonance can
be a potential atomic-clock resonance.
Besides using the static spin-polarized optical pumping to enhance the signal of the
end resonance, we report a new optical pumping method, “push-pull optical pumping”,
to greatly improve the 0-0 resonance signal [28]. This is the other significant result from
our research. Push-pull optical pumping is a way to optically pump atoms into a pure
coherent atomic state. The key requirement is to use D1 pumping light, which alternates
between two orthogonal circular polarizations at the Bohr frequency of 0-0 state. This
oscillating spin-polarized optical pumping can concentrate the nearly all the atoms into a
pure 0-0 superposition state, analogous to the static spin-polarized optical pumping, which
concentrates the entire population to the end state with no pressure dependence. The idea
of push-pull optical pumping was inspired by Bell-Bloom experiment and the experiments
of coherence population trapping. The basic picture of push-pull pumping is to use the
alternating photon spin to drive the electron spin, analogous to “push” and “pull”. In the
CPT language, push-pull pumping is a special CPT excitation, which has only one dark state
(un-pumped state). Therefore, the population will eventually be trapped into the only dark
state. In conventional 0-0 CPT, the ground-state 0-0 coherence is excited from a modulated
beam with a fixed circular polarization. However, the hyperfine population prefers to be
pumped into end state by the circularly polarized light. Therefore its 0-0 resonance signal
is weaker. Because of the exceptional capability of push-pull optical pumping, we believe
that push-pull optical pumping can enhance the performance of all kinds of CPT frequency
standards, and will have further applications of alkali masers, state preparation for studies
of cold atoms, quantum computation, and quantum information. In the end of this chapter,
we talk about a new possible way to build a compact and simple frequency standard, which
is the extension of using the concept of push-pull pumping and combining the technology
6
of semiconductor lasers.
In chapter 3, we switch our attention to the spin-exchange collisions between alkalimetal atoms and xenon gas. We designed a high-field experiment to determine the binary
spin-exchange rate coefficient of
129 Xe-Rb
[25] and
129 Xe-Cs
[27], which has been measured
by some other groups previously with results that disagreed by large factors due to various systematic errors. The previous measurements were flawed by misestimates of the
atomic number density and by misestimates of non-binary relaxation due to van der Waals
molecules. People mostly used empirical vapor pressure formulas to calculate the atomic
number density. Such estimates can be incorrect for many reasons, such as impurities in
the alkali metal. More and more experimental data tell us that we do not understand the
spin exchange between xenon and alkali-metal atoms through van der Waals molecules at
high pressure (≥ 1 atm) and low field. It is hard to extrapolate the binary spin-exchange
rate from the spin exchange in the presence of van der Waals molecules. We solved these
two problems by using Faraday rotation measurement to determine the number density of
rubidium or cesium experimentally and by using high magnetic fields to eliminate the contribution from van der Waals molecules. Although we artfully avoided the trouble of van der
Waals molecules by using high field, the experimental evidence shows that the three-body
model is incomplete. Further investigation has to be carried out. A better understanding
of the spin exchange will ultimately improve the performance of xenon polarization system.
Our measurements were carried out by using NMR techniques to measure thermally polarized 129 Xe nuclear spin. We discuss the rate equations of 129 Xe nuclear spin with detailed
balancing in the first section. Since the experiment was done in a very high field, 9.4T,
magnetic decoupling of the binary spin-exchange become non-negligible. We compare two
different methods, semi-classical approach (SCA) and distorted-wave Born approximation
(DWBA), for calculating the magnetic decoupling [26]. These two methods show a good
agreement with each other for conditions of experimental interest. The same calculations
can be applied to many other spin relaxation processes. In the last section, we summarize
our experimental result. We conclude that there was a negligible van der Waals molecules
7
contribution in our measurement. Cesium has twice larger spin-exchange rate coefficient
than rubidium, which agrees with our theoretical calculations. Therefore, cesium is a better
candidate for xenon polarizer, as long as a high-power Cs pumping laser is available.
Chapter 2
Strong Resonances of Alkali
Hyperfine Sublevels from Optical
Pumping
The study of the ground-state resonances of alkali-metal atoms is still an interesting topic
today. It directly benefits the design of atomic clocks and magnetometers. These two different devices tell us how precise we can measure “time” and the strength of “spin-coupled
fields”. By pushing these two devices to a better precision, we can have a better understanding of the theoretical structure of the fundamental physics. Besides, many advantages
can be produced from those two devices to our daily life, such as GPS (global positioning
system), high-speed digital systems, biological or medical diagnostics, etc. We have implemented a series of studies of resonances of hyperfine sublevels. We basically focused on the
atomic-clock resonances, and several important results were found from our studies.
For the atomic-clock resonances, the spin-relaxation mechanisms play a crucial role,
and affect the performance of the atomic clocks. From our experiments, one important
relaxation mechanism, the Carver rate, which is due to the collisional hyperfine modulation,
was found to have been overestimated by J. Vanier, who has made significant contributions
to the development of vapor-cell atomic clocks. Our new results of pressure-dependent
8
9
linewidth of the hyperfine resonances might change the idea of the pressure working range
of the vapor-cell atomic clocks.
To enhance the signal of the atomic-clock resonances, we developed two procedures for
different clock resonances. We took the advantage of light narrowing effect to obtain intense
end resonances. The light narrowing effect was achieved by using a fixed circularly polarized
pumping light. The increase of the atomic polarization suppressed the spin-exchange broadening and concentrated the population to the end-state. The linewidth became narrower,
and the signal amplitude became larger. This effect is especially useful at high temperature,
where the alkali vapor is dense and the spin exchange dominates the linewidth broadening. For the traditional 0-0 atomic-clock resonance, we contrived a new pumping method,
push-pull optical pumping. This new pumping method is capable of concentrating the population into the 0-0 states and producing a tiny suppression of the spin-exchange broadening.
Therefore it can definitely improve the performance of conventional atomic clocks, which
are based on the 0-0 transition of the alkali-metal atom. Push-pull optical pumping also
has other applications, such as for alkali masers and the state preparation for alkali-metal
atoms. Its capability of state preparation can be useful for experiments ranging from cold
atoms to hot vapor. The detailed studies required the methods of microwave and CPT
spectroscopies. In order to understand the relative phenomena, we used density-matrix
modelling to calculate the effects of important relaxation mechanisms on the evolution of
the ground-state and the excited-state atoms.
2.1
Density Matrix Equations
In many experiments of atomic physics, instead of one atom, we look at an ensemble of
many atoms. A well-developed method, density matrix, gives an easier and a beautiful way
for calculating the ensemble behaviors. For our studies, we took advantage of the density
matrix to calculate the relevant phenomena of the experimental observations. The density
10
matrix of an ensemble is given by
N
1 X
|ψn ihψn |,
ρ=
N
(2.1)
n=1
where N is the number of alkali-metal atoms, and |ψn i denotes the quantum state of each
atom. Hence, we have Tr(ρ) = 1. The dynamic equation of the density matrix is basically
P
described by ρ̇ = n [Hn , ρ]/ih̄, where Hn is the total Hamiltonian of the nth atom. We can
verify that Tr(ρ̇) = 0, which gives the conservation of the atom number. For an operator
P
or an observable O, the expectation value hOi = N −1 n hψn |O|ψn i is equal to Tr(Oρ).
By calculating the evolution of the density matrix, we can obtain different observables as
functions of time, which can be measured with experiments.
For the total Hamiltonian, we separate it into three parts, Hn = H
H
(0)
is the time-independent “unperturbed” Hamiltonian, H
to pumping lights and oscillating magnetic fields. Here, H
(1)
(0)
(0)
+H
(1)
(2)
+Hn , where
is the interaction term due
and H
(1)
are identical for all
(2)
atoms. The interaction of each atom due to random collisions is described by Hn . For
(0)
the ground state, S-state, the spin-related unperturbed Hamiltonian, Hg , of the valence
electron of an alkali-metal atom can be expressed by
(0)
Hg = AI · S + (µB gS Sz − µI Iz /I)B0 ,
(2.2)
where A is the hyperfine coupling coefficient, I = Ix x+Iy y+Iz z is the nuclear spin operator,
S = Sx x+Sy y +Sz z is the electron spin operator, gS is the electron g-factor, µB is the Bohr
magneton, µI is the nuclear magneton, I is the nuclear spin quantum number, and B0 is the
background DC magnetic field. The ground state has 2[I] = 2×(2I +1) hyperfine sublevels.
We usually use the vector |f, mi, which is the eigenstate of the hyperfine interaction, as
the bases to compose the density matrix, where f = I ± 2 and m = −f, · · · , f . Those
(0)
eigenstates remain good for Hg in a weak magnetic field (µB B0 ¿ A).
Ground-state relaxation is mainly caused by random interactions between atoms. Con(2)
sidering a few important mechanisms, each atom has its own Hamiltonian, Hg , which has
a form
(2)
Hg = γN · S + JS0 · S + δAI · S.
(2.3)
11
Here, the collisional Hamiltonian describes S-damping, spin exchange, and Carver damping
interactions.
For the excited state, we only consider the P1/2 excited state because of our experimental
(0)
interests. Therefore we find He
as
(0)
He = Aj I · J + (µB gj Jz − µI Iz /I)B0
(2.4)
where Aj is the hyperfine coupling coefficient, J is the total electron angular momentum,
gj = 2/3 is the Landé g-factor. If we count only spontaneous decay, quenching, and Jdamping, we can write the Hamiltonian for the random interactions as
(2)
He = γj N · J + H d ,
(2.5)
where γj is the coupling coefficient of the angular momentum interaction. The last term
H d appeared in Eq. (2.5) is the Hamiltonian for deexcitation, which includes spontaneous
decay and quenching mechanisms.
For atoms interacting with lights and oscillating fields, we write the interaction Hamiltonian as
H
(1)
= −D · Eop + (µB gS S + µB gj J − µI I/I) · B.
(2.6)
Here, D is the electric dipole operator, Eop = Eop (t) is the electric field of the pumping
light, and B = B(t) is the oscillating magnetic field. The transition between the S and the
P states is only caused by the electric dipole interaction.
In our calculations, the evolution of the density matrix is determined by the Hamiltonian
described above. Figure 2.1 sketches an overall picture of our density-matrix modelling. In
order to reduce the complexity, the numerical calculations were carried out in Liouville space
by using MATLAB programming. Our full model is capable of calculating the evolution
of the atomic ensemble with D1 optical pumping. With several modifications, it can be
extended for the calculation of the density matrix with D2 pumping. The main equations
of the density matrix are stated in the following subsections. As for the optical pumping,
we mainly discuss the D1 transition, since our experiments were carried out by using D1
12
Excited-state hyperfine sublevels
n
J-damping
P1/2
Quenching
+
Spontaneous
decay
Repopulation
and
D1 pumping
a
f = a = I+1/2
-a
n
S1/2
f = b = I-1/2
-b
-1
0
-1
0
1
1
b
Ground-state hyperfine sublevels
S-damping
Spin exchange
Diffusion
Carver rate
Optical pumping
Repopulation
Figure 2.1: In our modelling, we calculated the ensemble behaviors of the alkali-hyperfine
sublevels with D1 pumping and relaxation mechanisms, such as S-damping, spin exchange,
Carver damping, diffusion, spontaneous decay, quenching, and J-damping.
pumping. Detailed derivations of the evolution equations of the density matrix is discussed
in appendix A.
2.1.1
Evolution in Liouville Space
For calculations of the density matrix in Lioville space, each element of the density matrix
becomes an orthogonal vector. Therefore, an N by N density matrix can be represented as
a vector with N2 components. To do this, we find
ρ −→ |ρ),
ρ̇ −→
d
|ρ) = −R|ρ),
dt
(2.7)
where |ρ) is the Liouville vector, and R is the relaxation matrix, which has a dimension
equal to N2 by N2 . The elements of R can be found by calculating
Rij = (i|R|j) = −hα|ρ̇(|µihν|)|βi = Rαβ,µν ,
(2.8)
where |j) = ρµν and |i) = ραβ . In practice, the number of elements of R is much less than
N4 , because most off-diagonal elements are zeros. For example, considering a simplest case
13
with only one coherence in the ground state. The size of the ground-state relaxation matrix
R is only equal to g + 2 by g + 2, where g = 2[I].
To write the relaxation equation into a more general form, we have
d
|ρ) + R|ρ) = |Υ),
dt
(2.9)
where |Υ) = |Υ(t)) represents the possible source term. For a steady solution, if |Υ) is
independent of time, we find
|ρ) = R−1 |Υ).
For a dynamic solution, we find
XZ t
0
|ρ(t)) =
|ρk )e−γk (t−t ) (ρk |Υ(t0 ) + ρ(0)δ(t0 ))dt0 ,
k
(2.10)
(2.11)
0
where |ρk ) is the normalized eigenvector of R, and γk is the corresponding eigenvalue of R.
When spin-exchange rate is not equal to zero, R is a function of the electronic polarization,
and Eq. (2.9) becomes nonlinear. Equation (2.11) is only approximately accurate if spin
exchange dominates the relaxation process. However, if |Υ) = 0, we can use an iteration
method to find the exact steady solution of Eq. (2.9).
In Liouville space, the evolution equation of the density matrix has a simpler equation
formalism. Therefore, it is easier to be applied in the numerical calculations.
2.1.2
Ground-State Relaxation
S-Damping or Spin Destruction
The spin-rotation interaction, γ(r)N · S, causes atoms lose their electronic spin-polarization
to the angular momentum of the colliding atom pairs. Here, N is the angular momentum of
the colliding pair and S is the electron spin operator. We find the corresponding equation
of the density matrix due to the spin-rotation interaction is
ρ̇sd = Γsd (ϕ − ρ),
(2.12)
where ϕ = ρ/4 + S · ρS is the density matrix without electronic polarization. Here, Γsd is
called S-damping rate or spin destruction rate, because this interaction destroys the spin
14
polarization; and Γsd is proportional to the number density of buffer gas. By looking at the
time derivative of total atomic polarization dhFi/dt, where F = I + S, we find
d
hFisd = Tr(Fρ̇sd ) = −Γsd hSi.
dt
(2.13)
The result here is a little bit subtle. When a spin-rotation collision occurs, a small amount
of the hyperfine coherence is generated. Due to the hyperfine splitting and Zeeman splitting,
these zero-frequency coherences quickly decay because of the spin precession in the different
phases. Therefore, the off-diagonal elements usually remain zero through the S-damping
process. It can be comprehended as that the electron spin transfers its polarization to the
nuclear spin by precessing with the nuclear spin through the I · S interaction. The nuclear
spin is just like a “spin sink” or “spin reservoir”, which stores the angular momentum
acquired from the electron [3].
Spin Exchange
The spin-spin coupling interaction, J(r)S0 · S, accounts for the spin exchange between the
two colliding atoms of electron spins S0 and S. Its corresponding equation of the density
matrix is
ρ̇ex =
1
[ δEex , ρ ] + Γex [ϕ(1 + 4hSi · S) − ρ] ,
ih̄
(2.14)
where δEex is the energy shift operator due to the spin-exchange collisions. The spinexchange rate Γex is proportional to the number density of alkali-metal atoms and therefore
increases with the environmental temperature. By checking Tr(Fρ̇ex ), we can verify the
conservation of the spin polarization from spin exchange.
d
hFiex = Tr(Fρ̇ex ) = 0.
dt
(2.15)
Carver Rate
The coupling coefficient, A, of the hyperfine interaction, AI · S, can be perturbed by the
atomic collisions. We separate the perturbation term, δA(r)I · S, from the unperturbed interaction. The corresponding equation of the density matrix due to the random modulation
15
of the hyperfine interaction is
ρ̇C =
η 2 [I]2
1
[ δEC , ρ(m) ] − I
ΓC ρ(m) ,
ih̄
8
(2.16)
where δEC denotes the pressure shift of the hyperfine splitting frequency, and ρ(m) represents
the density matrix with only off-diagonal elements of the coherences between upper f =
I + 1/2 = a and lower f = I − 1/2 = b hyperfine multiplets. We use letter “m” to label it,
because the hyperfine transitions are mostly in microwave frequencies. The Carver rate ΓC
was first introduced by D.K. Walter [53], which is proportional to the number density of
the buffer gas. The Carver-rate relaxation can cause the dephasing of hyperfine coherences,
but can not affect the hyperfine population and the coherence of Zeeman sublevels for the
relatively small magnetic fields.
Diffusion
In experiments, instead of tracking a cluster of atoms, we probe the atoms from a fixed
volume. The atoms can randomly move from one place to others. This spatial diffusion of
atoms is equivalent to the diffusion of atomic spins. When the atom diffuses to the wall,
both nuclear and electron spin can be completely destroyed. Therefore the population of
the hyperfine ground state is relaxed toward equilibrium. The corresponding equation of
the density matrix for diffusion mechanism is
ρ̇d = Γd (1/g − ρ),
(2.17)
where 1 is the unit matrix, and the relaxation rate Γd = 3D(π/l)2 if we assume a cubic
cell with side length l and the pumping light covers the entire cell. Here D is the diffusion
coefficient, which is inversely proportional to the buffer gas pressure. We can check that
d
hFid = Tr(Fρ̇d ) = −Γd hFi.
dt
(2.18)
16
2.1.3
Excited-State Relaxation
J-Damping
The J-damping mechanism occurred in the P1/2 state is very similar to the S-damping in
ground state. The relaxation is dominated by the interaction γj N · J, which causes angular
momentum transferring between N and J. Here, J = L + S is the operator of the total
electron angular momentum. We can find the corresponding equation of the excited-state
density matrix ρ(e) is
1
(e)
ρ̇jd = Γjd (J · ρ(e) J − {J · J, ρ(e) }),
2
(2.19)
where Γjd is the J-damping rate, and { } denotes the anti-commutator. J-damping is an
important depolarization mechanism in the excited state. Here, we use the superscript (e)
to distinguish from the part of the ground-state density matrix.
Spontaneous Decay & Quenching
Atoms return to the ground state from the excited state by either spontaneously emitting
photons or colliding with quenching atoms. The corresponding relaxation of the density
matrix of the excited state is
(e)
ρ̇(e)
− Γq ρ(e) ,
sq = −Γs ρ
(2.20)
where Γs is the spontaneous decay rate and Γq is the quenching rate. More detailed discussions about this subject is in appendix A.
2.1.4
Optical Pumping
The optical pumping includes depopulation and repopulation mechanisms. The depopulation pumping transfers some population from the ground state to the excited state. The
repopulation pumping sends the population back from the excited state to the ground state
due to the decay mechanisms (quenching and spontaneous decay).
17
D1 Depopulation Pumping
(1)
The depopulation pumping is caused by the interaction Hop = −D · Eop . For the buffergas pressure above a few tens of torr, the hyperfine splitting of the excited state becomes
unresolved due to the optical pressure broadening. Therefore we can approximate energies
of all excited hyperfine sublevels to their center of gravity. For unsaturated optical pumping,
we find the effective Hamiltonian of D1 depopulation pumping for the ground state as
δHop =
(op)
Here, the pumping rate, Γ
h̄
(op)
(1 − 2s · S)Γ .
2i
(2.21)
(op)
= Γαβ δα,β , is a time-dependent complex matrix with only
(op)
diagonal elements. Taking the matrix elements of Γ , we find
Z
Z ∞
∗ (Ω − ω)Ē (Ω)
|D|2 Ēop
(op)
op
−iωt
Γαα = dω e
,
dΩ 2
2h̄ i(Ωα − Ω) + γop
0
(2.22)
where D is the dipole strength between P1/2 and S1/2 , Ēop (Ω) is the Fourier’s spectrum
of the pumping electric field, Ωα is the optical transition frequency from the ground-state
sublevel α to the excited state, and γop /π is the optical linewidth. The time-dependent
pumping rate can be produced by modulating a CW pumping light, and its dependence
of time is determined by the spectrum of the pumping field, optical detuning of the main
carrier, optical linewidth, etc. The detailed derivation of the pumping rate can be found in
appendix A. The vector s denotes the photon spin of the pumping light, which has values
|s| ≤ 1. The effective Hamiltonian δHop is not Hermitian, which can be written as the
sum of a Hermitian and an anti-Hermitian operators. Then, δHop = δEv −
δEv =
1
2 (δHop
†
+ δHop
) and δΓ =
i
h̄ (δHop
ih̄
2 δΓ,
where
†
− δHop
). We find the equation of the density
matrix for depopulation pumping of ground state as
ρ̇dp =
1
1
1
†
[δHop ρ − ρδHop
] = [δEv , ρ] − {δΓ, ρ}.
ih̄
ih̄
2
(2.23)
Here, we can define δEv as the light shift operator due to the virtual transition, since it has
nonzero values only when there is an optical detuning between ground-state and excitedstate sublevels. The operator δΓ is the absorption operator. The averaged light absorption
rate of an atom is equal to the total depopulation rate, −Tr(ρ̇dp )=Tr(δΓρ) = hδΓi.
18
For a simpler case, if all hyperfine sublevels see the same pumping rate, the pumping(op)
rate matrix Γ
h̄
2 (1
= Γop becomes a pure time-dependent complex number. Therefore, δEv =
− 2s · S)Im(Γop ), and δΓ = (1 − 2s · S)Re(Γop ).
D1 Repopulation Pumping
There is no equation in a simple form for repopulation pumping by using the regular densitymatrix formalism. The simplest expression is in Liouville space as
d
|ρ)rp = R(rp) |ρ),
dt
(2.24)
where R(rp) is the repopulation matrix in Liouville space, and its detail can be found in
appendix A.
For a common experimental condition, repopulation pumping is dominated by quenching. Assuming that all ground-state hyperfine sublevels see the same pumping rate, and
only if the quenching rate is much larger than the excited-state hyperfine frequency, the
change of the ground-state population due to the quenching dominated D1 repopulation
pumping is
µ
ρ̇rp = Re(Γop )
¶
1
{1 − 4s · S, ρ} + (S · ρS − iS × ρS) .
8
(2.25)
It is understood that Eq. (2.25) is only valid for the calculation in the manner of ρ̇αα =
(rp)
Rαα,µν ρµν .
Full Optical Pumping Equations for Ground-State Population
In our experiments, we use nitrogen as the buffer gas, which is very good for quenching.
With enough amount of quenching gas, we can assume the decay from the excited state
happens much faster than spin relaxation and spin precession in the excited state. Therefore,
the nuclear spin is conserved in the excited state. Under this condition, we find the evolution
equations of the complete optical pumping (depopulation+repopluation) for the groundstate population, ραα , as the summarization in Eq. (2.26) to Eq. (2.29). Here, we also include
(op)
D2 = − ih̄ (1 + s · S)Γ
two cases of D2 pumping. The effective Hamiltonian for D2 is δHop
2
.
19
Assuming all hyperfine sublevels see the same pumping rate, from Eq. (2.23) and Eq. (2.25),
we find
D1 pumping + strong quenching:
ρ̇op = Re(Γop ) [ϕ(1 + 2s · S) − ρ] .
(2.26)
D1 pumping + pure spontaneous decay: (assuming the excited hyperfine frequency = 0)
µ
ρ̇op = Re(Γop )
¶
2
1
2
[ϕ(1 + 2s · S) − ρ] + Sz ρSz − ρ .
3
3
6
(2.27)
D2 pumping + strong quenching:
ρ̇op = Re(Γop ) [ϕ(1 − s · S) − ρ] .
(2.28)
D2 pumping + pure spontaneous decay: (assuming the excited hyperfine frequency = 0)
µ
ρ̇op = Re(Γop )
¶
1
1
1
[ϕ(1 + 2s · S) − ρ] + Sz ρSz − ρ .
3
3
12
(2.29)
We have to notice that Eq. (2.27) and Eq. (2.29) are only qualitatively correct in the
real cases, because the excited hyperfine splitting is usually larger than the spontaneous
decay rate. Therefore, the electron has enough time to transfer its polarization to the
nuclear spin through the precession with the nuclear spin in the excited state before the
spontaneous decay. The polarization transfer between electron and nuclear spin cause the
optical pumping to polarize atoms more efficiently. Equation (2.26) and (2.28) are good
by using the quenching gas such as nitrogen in the pressure near or above an atmosphere,
where the quenching rate much exceeds the excited-state hyperfine splitting frequency. Some
buffer gases, like helium or argon, are not good for quenching but causing larger J-damping
relaxation in the excited state. In this situation, the evolution equation of the density
matrix for optical pumping is more complicated and can not be described by the equations
listed above. Basically, the population changed by optical pumping are affected by ways of
depopulation, repopulation, and the evolution in the excited state. More detail of optical
pumping will be discussed in section 2.4.7.
20
There is an interesting feature of D2 pumping. By looking at Eq. (2.28) and Eq. (2.29),
we find that the different repopulation mechanisms have different pumping polarizations by
using the same photon spin. We can check Tr(Fρ̇op ) from D2 pumpings.
´
dhFiop
Re(Γop ) ³ s
=
− hSi .
dt
3
2
³
´
dhFiop
s
= Re(Γop ) − − hSi .
Strong Quenching:
dt
4
Spontaneous Decay:
(2.30)
(2.31)
Since the quenching rate is proportional to the pressure of the quenching gas and the
spontaneous decay rate is nearly constant, it implies that there is a magic pressure where has
no spin-polarized optical pumping. At that pressure, we can not use circularly polarized D2
light to spin-polarize atoms. Although, Eq. (2.30) underestimates the polarization pumping
rate for many real cases, but it does not change the direction of the spin polarization.
Ground-State Coherence Created by Optical Pumping
The ground-state coherence can be generated through ρ̇dp + ρ̇rp . The quasi-steady solution
of the ground-state coherence can be obtained by using the secular approximation (rotation
wave approximation) and the slowly varying approximation. We find
ρeµν = −
(1)
1
2 (δΓµν ρνν
(1)
(1)
(1)
+ ρµµ δΓµν ) + h̄i (δEv,µν ρνν − ρµµ δEv,µν )
(1)
+ R(rp)
µν,αα ραα .
i(ωµν − ω) + γµν
(2.32)
where ωµν and γµν are the Bohr frequency and the relaxation rate of the coherence. Here,
δΓ
(1)
(0)
is defined by δΓ = δΓ
(1)
+ (δΓ e−iωt + δΓ
(−1)
(1)
eiωt ) + · · · , and the same for δEv and
(1)
R(rp) . We define the amplitude of the coherence, ρeµν , by ρµν = ρeµν (t)exp(−iωt), and ω
(1)
is the frequency near to ωµν . In Eq. (2.32), the light shift term Ev introduces a quadrature
complex phase into the coherence. This causes the phenomenon of the asymmetric line
(1)
shape, which will be discussed in section 2.4.6. The repopulation term R(rp)
can also
introduce a quadrature complex phase into the coherence from the “real ”-transition light
shift. The difference between these two light shifts to the coherence is that the virtual
light shift requires a population difference (ρµµ − ρνν 6= 0), but the real light shift does
not need the population difference. Optically induced atomic coherence is the central part
21
of coherent population trapping (CPT) and the electromagnetically induced transparency
(EIT). More detail will be discussed in later sections.
2.1.5
Microwave and RF Fields
Traditionally, we use microwave or RF (radio frequency) fields to excite the coherences of
hyperfine sublevels if the frequencies of the oscillating fields match to the hyperfine splitting
or Zeeman splitting. Assuming the oscillating field B(t) = Bof (eiωt + e−iωt ), we find the
interaction Hamiltonian,
(1)
Hof = B(t) · (µB gS S − µI I/I).
(2.33)
For two different hyperfine sublevels |µi and |νi, the Rabi frequency produced by the oscil(1)
lating field is equal to ωR = h̄2 hµ|Hof |νi. For a quasi-steady solution, we find the amplitude
of the coherence due to the oscillating field is given by
ρeµν (t) =
iωR /2(ρµµ (t) − ρνν (t))
.
i(ωµν − ω) + γµν
(2.34)
The rate equation of the density matrix due to the oscillating field in the quasi-steady state
is described by
½
ρ̇of
=
¾
δEof
[(ρµσ |µihσ| − ρνσ |νihσ|) − (ρσµ |σihµ| − ρσν |σihν|)] +
ih̄
{−Γof [(ρµµ − ρνν )|µihµ| + (ρνν − ρµµ )|νihν|]} ,
(2.35)
where σ 6= µ and σ 6= ν. The AC Stark shift energy Eof and the effective pumping rate Γof
are defined by
δEof =
2 /4(ω
h̄ωR
µν − ω)
,
2
2
(ωµν − ω) + γσ(ν,µ)
Γof =
2 /2γ
ωR
µν
.
2
(ωµν − ω)2 + γµν
(2.36)
From Eq. (A.37), one can see that the oscillating field tends to equalized the population of
|µi and |νi.
22
2.1.6
Summary
A complete evolution equation of the density matrix is summarized by
ρ̇ = ρ̇d + ρ̇sd + ρ̇ex + ρ̇C + ρ̇of + ρ̇dp + ρ̇rp
=
(2.37)
1
1
1
1
1
[ H (0) , ρ ] + [ δEex , ρ ] + [ δEC , ρ ] + [ δEof , ρ ] + [ δEv , ρ ]
ih̄
ih̄
ih̄
ih̄
ih̄
η 2 [I]2 (m)
+Γd (1/g − ρ) + Γsd (ϕ − ρ) + Γex [ϕ(1 + 4hSi · S) − ρ] − ΓC I
ρ
8
n
o
1
(rp)
+ {−Γof [(ρµµ − ρνν )|µihµ| + (ρνν − ρµµ )|νihν|]} − {δΓ, ρ} + Rαβ,µν ρµν |αihβ| .
2
In the quenching dominated repopulation, we find the longitudinal evolution equation
(ρ̇αα = Rαα,µν ρµν ) as
ρ̇L = ρ̇d + ρ̇sd + ρ̇ex + ρ̇op + ρ̇of
= Γd (1/g − ρ) + Γsd (ϕ − ρ) + Γex [ϕ(1 + 4hSi · S) − ρ] + Re(Γop ) [ϕ(1 + 2s · S) − ρ]
+ {−Γof [(ρµµ − ρνν )|µihµ| + (ρνν − ρµµ )|νihν|]} .
(2.38)
Note: The density matrix evolution due to the oscillating field in Eq. (2.37) and Eq. (2.38)
is only correct for the late-time evolution after the field turns on.
23
Excited state
Turn on the oscillating field
a
coherence
pumping
light
b
Ground state
oscillating
fields
b
Ground state
Transmission
a
0
Frequency detuning of the oscillating field
Figure 2.2: The population of the sublevel b is pumped out. When a oscillating field turns
on at the resonant frequency, the coherence is generated. Some population transfers from the
sublevel a to b, and therefore it increases the light absorption or decrease the light transmission.
2.2
Microwave Spectroscopy and CPT Spectroscopy
Optical hyperfine spectroscopy uses a probing or pumping light to detect the atomic coherence. The on-resonance signal is observed as a transmission dip or a transmission peak of
the probing light.
2.2.1
Coherence Induced by Oscillating Fields
According to Eq. (2.34), the strongest ground-state coherence can be generated when a
oscillating magnetic field couples to the two of hyperfine sublevels, and its oscillating frequency matches to the splitting frequency of the two sublevels. For the steady solution, the
oscillating field not only produces the coherence, but also change the relative population
of the two sublevels. From the pseudo-spin picture, the oscillating field rotates the pseudo
spin from the longitudinal direction to the transverse plane. The coherence is equivalent of
the component of the pseudo spin on the transverse plane. More discussions of pseudo-spin
picture is in section 2.4.6. To simplify the discussion, we approximate a multi-level problem
to a two-level system. Assuming that we label the two selected sublevels as “a” and “b”,
24
from Eq. (2.38), we find



ρ̇a = Υa − Γa ρa − Γof (ρa − ρb )


ρ̇b = Υb − Γb ρb − Γof (ρb − ρa )
,
(2.39)
where Υa and Υb are the equivalent transfer rate from other states to state a and b; Γa
and Γb represent the total relaxation rates of state a and b. By using the definition of Γof
(the equivalent pumping rate of the oscillating field) in Eq. (A.39), we can find the steady
solution by solving ρ̇a = ρ̇b = 0. Therefore

2
−1


b )−(1+Γa /Γb )Υa ](Γa Γb )
ρa = ωR γab [(Υa +Υ
+
(ω −ω)2 +γ 2 +(Γ−1 +Γ−1 )γ ω 2
ab


ρb =
ab
a
b
ab R
2 γ [(Υ +Υ )−(1+Γ /Γ )Υ ](Γ Γ )−1
ωR
a
a
a b
ab
b
b
b
2 +(Γ−1 +Γ−1 )γ ω 2
(ωab −ω)2 +γab
a
ab
R
b
+
Υa
Γa
,
(2.40)
Υb
Γb
where ωab and γab are the Bohr frequency and decay rate of the coherence. Equation (2.40)
can be written into a more general form
(ρa or ρb ) =
2A
ωR
2 + T /T ω 2 + B,
(ωab − ω)2 + γab
1
2 R
(2.41)
−1
where A and B are some numbers; T1 = Γ−1
a + Γb and T2 = 1/γab .
RF or microwave spectroscopy is to selectively probe(or pump out) the sublevel a or
b, the light absorption is proportional to the population on the selected sublevel. An
illustration is shown in Fig. 2.2. According to Eq. (2.41), the absorption curve is a Lorentzian
function. The peak absorption occurs when ω = ωab , and the linewidth is equal to ∆ν =
q
2 + T /T ω 2 (FWHM). The second term inside the square root is called the power
π −1 γab
1
2 R
2 is proportional to the power of the oscillating field.
broadening effect, since ωR
Although Eq. (2.41) is only an approximated result, it has a good physics picture. For
most of cases, it is a good approximation. For a precise calculation, the absorption is
proportional to Tr(δΓρ), where ρ is the steady solution of Eq. (2.38). The calculation can
be easily carried out numerically.
2.2.2
Optically Induced Coherence
Unlike microwave spectroscopy, CPT spectroscopy uses modulated light to excite the coherence. To simplify the discussion, we consider two monochromatic optical fields, which
25
Figure 2.3: Two monochromatic fields with frequencies Ω1 and Ω2 couple between two groundstate sublevels and the excited state. Two optical coherences between the ground state and the
excited state are generated. When the difference of two optical frequencies is equal to groundstate sublevel splitting, a strong ground-state coherence can be produced. Under the same
condition, photons begin scattering between two monochromatic fields, and are less absorbed
by atoms. The light absorption is therefore reduced and observed as a transmission peak. When
the optical detuning is present, it is equivalent to stimulated Raman scattering.
couple the ground-state sublevels to the excited-state sublevels, each optical field can induce the coherence between the corresponding ground-state sublevels and the excited state.
When the frequency difference of the two monochromatic fields matches to the splitting
frequency of the two ground-state sublevels, a ground-state coherence can be produced
through a process of two-photon transitions. The CPT phenomena can also be represented
by the picture of pseudo spin, which is described in section 2.4.6.
Assuming there is no optical detuning and the two sublevels are optically resolved, from
Eq. (2.22), we can find the relevant time-dependent pumping rate to be
(0)
(1)
Γop (t) = Γop + Γop (eiωt + e−iωt ),
(2.42)
(0)
(1)
where ω = Ω1 − Ω2 , the difference of two optical frequencies, and Γop = Γop = Dop is a real
number. We can therefore find the light absorption rate for D1 pumping, which is equal to
the total depopulation rate.
(0)
(0)
(1)
(1)
Tr(δΓρ) = Tr(δΓ ρ ) + Tr(δΓ ρ )
(0)
(1)
(0)
= Dop (1 − 2s · hS i − 2s · hS i) = Dop (1 − s · P
(1)
− s · P ).
(2.43)
26
(0)
Here, we use superscript (0) to label the static components, where ρ
(0)
sity matrix with only diagonal elements, and P
represents the den-
(0)
= 2hS i represents the static spin(1)
polarization. We use superscript (1) to label the oscillating components, where ρ
(1)
sents the density matrix with only off-diagonal elements, and P
repre-
(1)
= 2hS i represents the
amplitude of the oscillating spin-polarization. The oscillating spin comes from the atomic
coherence. From Eq. (2.32), we find the amplitude of the oscillation spin-polarization to be
(1)
P
=
γµν N Dop
(|s · Pµν | ŝ),
2
(ωµν − ω)2 + γµν
(2.44)
where N = ρµµ + ρνν is the total population of the two sublevels, Pµν = 2hµ|S|νi is the
equivalent spin-polarization of the coherence, and ŝ is the unit vector along the direction
of the photon spin. D1 optical pumping tends to generate the electron spin aligned to the
photon spin. The expression (1 − s · P
(0)
(1)
− s · P ) appeared in Eq. (2.43) is usually called
the “specific absorption”. Therefore, the intensity of the transmission light affected by the
(1)
oscillation polarization is proportional to s · P . By checking Eq. (2.44), we find there is
a maximum transparency when ω = ωµν . This is the basic principle of CPT spectroscopy.
An illustration is shown in Fig. 2.3. People usually like to call this phenomena as coherent
population trapping (CPT), because when the probing light reach the peak transmission,
lots of atoms tend to be trapped into the superposition state, the dark state or non-coupled
state. The two-wave optical pumping is called Λ configuration, because of its look. In
Fig. 2.3, the dark superposition state is written by α|1i + β|2i, and we find R1 α + R2 β = 0
(The transition amplitude to the excited state is zero). Here, R1 and R2 are the Rabi
frequencies or the coupling strengths of the two optical transitions Ω1 and Ω2 . The socalled bright state or coupled state therefore is β|1i − α|2i. The two-wave pumping light
pumping out the atoms in bright state and produce atoms in dark state. Figure 2.4 shows
the spin oscillation picture to represent the CPT excitation. The 0-0 CPT coherence is
equal to the spin oscillation along the z-direction. The end CPT coherence is equal to
the spin precession on the x-y plane. The period of the spin oscillation or precession is
equal to the inverse of the splitting frequency of the two sublevels. For the conventional
0-0 CPT, there are two dark states, 0-0 superposition state and the end state. When the
)
(
'
&
!
"
.
/
1
0
)
#
(
'
!
"
&
)
,
'
&
$
+& %!"'
)
*
27
Figure 2.4: We use a hypothetic atom, which has a nuclear spin I = 1/2, to illustrate the CPT
coherence in the picture of spin oscillation. Different Λ schemes generate different ground-state
coherences. The coherence of 0-0 CPT is equivalent to the electron spin oscillating along the
z-direction. The end-state coherence is equal to spin precessing on the x-y plane. The dotted
circles represent the dark states for different CPT schemes. The 0-0 CPT excitations from σ+
and σ− pumpings have 180◦ phase difference.
pumping intensity increases, atoms prefer to be trapped into the end state. This causes 0-0
CPT signal to be weak. For the end-resonance CPT, there is only one dark state, the end
state. Therefore, we can obtain a stronger signal from end-resonance CPT. A new pumping
method, push-pull optical pumping, can solve the problem of the conventional 0-0 CPT,
which will be discussed in section 2.4.4.
In the discussion above, we ignore the effect of the optical detuning, optical pressure
broadening, multiple optical sidebands from the modulated light, and the repopulation
pumping. The optical detuning causes virtual light shift and therefore introduce an extra
phase to the coherence. It turns out the line shape of the CPT signal to be asymmetric. The
optical pressure broadening and the optical sidebands can affect both the line shape and
the strength of CPT excitation. The repopulation pumping has only negligible influence to
the ground-state microwave coherence when the buffer gas pressure is lower than 10 atm
28
(depends on the real quenching and J-damping rate). The detailed discussions will be in
the later sections. For real cases, a numerical calculation is required to precisely calculate
the CPT phenomena.
2.2.3
Inhomogeneous Linewidth Broadening
Assuming the line shape of the resonance signal from microwave or CPT spectroscopy is
described by a Lorentzian function, and the linewidth broadening is due to the intrinsic
decay rate, we find
Resonance Signal =
(ωµν
Aγµν
+ B,
2
− ω)2 + γµν
(2.45)
where A and B are two constants. We usually measure the linewidth in FWHM (full width
in half maximum). Therefore, the linewidth ∆ν = γµν /π in Hz. From Eq. (2.37), we know
that the resonant frequency can be shifted by different influences, such as magnetic field,
temperature, pressure, light intensity, etc. Atoms in the different positions of the light path
can have different frequency shifts due to the inhomogeneities from different causes. The
real resonance signal is a convolution of the Lorentzian function with the distribution of the
central frequency. Thereby, the effective linewidth can be much broader. This is called the
inhomogeneous broadening.
We can consider a 1st order approximation. Assuming the central frequency of the
resonance is spread uniformly from −∆/2 to ∆/2, the new line shape becomes
·
µ
¶
µ
¶¸
(ωµν − ω) + ∆/2
(ωµν − ω) − ∆/2
A
arctan
− arctan
+B.
Resonance Signal =
γµν ∆
γµν
γµν
(2.46)
We find that the FWHM linewidth ∆ν = (γµν + ∆)/π for this inhomogeneous broadening.
When ∆ −→ 0, Eq. (2.46) turns back to Eq. (2.45). Therefore, if there is a magnetic
field gradient, a temperature gradient, and a light intensity gradient, we can find that the
inhomogeneous broadening caused by these three different sources is given by
∆ν 0 =
1
π
µ
¶
∂ωµν
∂ωµν
∂ωµν
∆B +
∆T +
∆I ,
∂B
∂T
∂I
(2.47)
29
where ∆B, ∆T , and ∆I represent the total difference of the magnetic field, the temperature,
and the light-intensity. The total line-width becomes ∆ν = γµν /π + ∆ν 0 . The temperature
gradient can cause a inhomogeneous broadening, because the pressure shift is temperature
dependent. The related subject is in section 2.4.3.
30
Figure 2.5: Lollipop (left) and rectangle (right) cells.
2.3
Experiments
In order to do the microwave and CPT spectroscopy of the alkali-metal atoms, we prepared
cells filled with different alkali-metal and the desired gas mixture. The alkali metal was
released by using a gas-oxygen torch to heat alkali-chloride salt, mixed with small chips
of calcium metal, in an evacuated glass retort that was attached to a glass tube, which
connected with glass cells. A torch was used to distill the free metal into the cells. After
the metal had been distilled into the cells the buffer gas, for example, nitrogen and argon,
was first mixed and then condensed into the sample cells. Details of cell preparation method
is described in Ref.[25]. By using that method, we were able to make cells with pressures
from zero to several atmospheres. We used spherical cells and two different flat cells, the
“lollipop” type and the rectangle type. Lollipop cells were assembled by two circular Pyrex
glass plates in parallel. The separation of two glass plate was about 2 mm. Rectangle cells
were made from a piece of rectangular Pyrex tubing, which was 1 cm wide and 1 mm thick,
and connected with a ballast compartment. Figure 2.5 shows a picture of the two different
M]N^OdicPkh_eQP̀RoljfgnmSTUaVWX>YZ;[R<JI?\K
F:@AIG@HL(7)E$89%&'#=AB":!;D*4<+5,Cp-.bq6rs/0123
31
Figure 2.6: Experimental setup for microwave spectroscopy.
types of cells.
The prepared cells were eventually placed into an oven, which was made by using O-10
plastic material as a frame and 1 cm thick glass plates as window walls. The internal space
of the oven was heated by the hot air flow. A feedback loop constructed from a thermal
sensor and an OMEGA temperature controller (CN76000) stabilized the oven temperature.
We could adjust the oven temperature from room temperature to 160◦ C to obtain different
vapor density inside the cell.
2.3.1
Microwave Setup
Figure 2.6 shows the apparatus for the experiments with microwave spectroscopy. We used
three Helmholtz coils to compensate the external field in x- and y-direction and also produce
the Bz field. Each coil was about 70 cm in diameter and had about 200 turns of enamelled
wires. The Bz field was adjustable to ±15 G. On the z coil, we added two extra coils, not
shown in Fig. 2.6. One is for compensating the field gradient along the z-direction, and the
32
other was to cancel the 60-Hz field. A commercial 60-Hz cancellation system suppressed
the low frequency AC field actively by using an inductive coil, which was wrapped on the
two sides of the oven, to sense the field strength and feedback to its controller circuit. The
controller then generated a compensated field by sending currents to the cancelling coil.
In order to reduce the influence of the 60-Hz field, we also shut down the electrical power
for nearby room outlets. This minimized the amount of the AC current flowing around
the room. After these two steps, we reduced the AC field amplitude to a value less than
30 µG. The microwave field was generated by a microwave horn. Two horns, rated at 6.8
GHz and 9.2 GHz, were used in the experiments. The horn was connected to a frequency
synthesizer (HP83752B) with a high frequency coaxial cable. We used two laser systems
as the pumping or probing sources, a Ti-sapphire ring laser system (Coherent 899-29) and
a diode laser system (Toptica DL100). The ring laser was fiber coupled from the next
room. It had higher optical power (up to 1 watt) and wider tuning range (from 785 nm
to 910 nm). The diode laser has lower power (up to 50 mW), shorter tuning range (∼ 50
GHz), narrow linewidth, and much less optical noise. We had two diode laser heads for 795
and 894 nm experiments. The precise laser frequency was measured by Burleigh WA-1000
wavemeter. The probing light was linearly polarized first, and we used a quarter-wave plate
to switch the light polarization between linear and circular. A small fraction of the light was
sent to a background photodetector before the quarter-wave plate. The probe beam passed
through the cell and was focused on the transmission photodetector.
A divider circuit
was made for reducing the background noise level, which is mostly due to the intensity
fluctuation of the probing light. The circuit was composed of two commercial ICs, TL082
and AD734. Two photodetectors detected the light intensities before and after propagation
through the vapor cell. These two signals were pre-amplified by OP-Amps TL082 and then
sent to the analog divider AD734, which generated an output a voltage signal proportional
to transmission/background. In this way, the common mode noise of the probing light can
be cancelled out to a certain degree, as shown in Fig. 2.7. The output signal from the
divider circuit was connected to a voltage amplifier (SRS SR560). Its output was linked
33
10
-3
8
6
Raw transmission signal
Divider output
4
Noise level (A.U.)
2
10
-4
8
6
4
2
10
-5
8
6
4
0
100
200
300
Frequency (Hz)
400
500
Figure 2.7: The noise spectrum of the transmission light in the experiments. Our homemade
divider circuit can suppress the background noise level by 15dB, and can increase the SNR by
a factor of 6 ∼ 7. This divider circuit was especially useful when we used the ring laser as the
pumping source. Because the laser was conducted by a long fiber (20 m), lots of mechanical
vibration noise was converted into the intensity fluctuations of the output light.
to a lock-in amplifier (SRS SR530) and to the computer’s A/D converter (NIDAQ 6052E).
The output of the lock-in amplifier then connected to a digital oscilloscope (TDS 3304). All
of instruments with available GPIB interfaces were linked together with a PC computer.
The job of instruments control, data acquisition, and data processing was done by IGOR
program installed in the computer.
Signal Measurement of Microwave Spectroscopy
We chopped the microwave field at a frequency between 25 to 100 Hz and used the lock-in
amplifier to read out the steady intensity of the transmission light at different microwave
detuning. Because of the low chopping frequency, we chose stepping the frequency instead
of continuously scanning. The computer A/D interface was used for reading the transient
signal when the microwave field was switching on and off. We applied a differential input method on the A/D interface to reduce the electrical noise. The transient signal was
34
microwave OFF
microwave ON
A.U.
10
Damping oscillation
0
-10
Transient trace
0
10
20
30
40
msec
Raw transient data
Baseline fitting curve
After subtraction of the baseline
A.U.
20
10
0
512 Average
0
A.U.
3.0
1
2
3
fc = 6.825397600 GHz
Offset = 16.7 ± 0.3 kHz
Linewidth = 2.2 ± 0.1 kHz
2.0
4
msec
Fourier's spectrum of the
modified transient trace
Lorentzian fitting curve
1.0
0.0
A.U.
8
2.0
1.5
1.0
0.5
0.0
10
12
14
16
18
20
22
24
fc = 6825381.500 kHz
Offset = 0.67 kHz
Width = 2.96 kHz
-10
-5
26 kHz
Lock-in signal
Lorentzian fit
0
Microwave detuning (kHz)
5
10
Figure 2.8: The top panel shows the transient of the transmission light when the microwave
field was on and off. The second panel is the zoom-in of the circle on the top panel. The
third one is the Fourier’s spectrum of the damping oscillation. The bottom one shows the same
resonance measured by scanning the microwave detuning (two sec for each point). Its linewidth
is broader due to the microwave power broadening.
triggered in synchronism with the chopping frequency. Therefore, we were able to do the
signal averaging to increase the SNR. To measure the equivalent resonance linewidth, we
subtracted baseline from the transient data and then Fourier’s transformed it to a spectrum.
In Fig. 2.8, the transient method took about 10 second, but the detuning scanning took
about 2 minutes. For the linewidth measurements, we prefer to use the transient method.
It needs less data acquisition time and had no problem of microwave power broadening.
2.3.2
CPT Setup
For CPT spectroscopy, we kept the same experimental setup and added a few parts in
addition to those sketched in Fig. 2.6. In order to do the light modulation, we used a
4
)
+
,
(
*
2
.
/
0
1
3
%&'
$
!"#
35
Figure 2.9: The probing beam for CPT experiments was generated by a Mach-Zenhder Mod-
ulator. By modulating the effective light propagation lengths from two crystals, the output
beam can be intensity modulated. The optical sidebands of the probing beam can be measured
from a Fabry-Perot spectrometer.
Mach-Zehnder type intensity modulator (EOspace, Model AZ 0K1-12-PFU-SFU-800). Its
modulation frequency can reach up to 20 GHz. The modulator worked by splitting the
input beam and sending each sub-beam through parallel electrooptic crystals (LiNbO3 ).
The refraction indices of the crystals were controlled by a DC-biased electric field and
the microwave electric fields. The control signal was arranged to generate an opposite
optical phase retardation from each crystal. Two sub-beams then recombined to produce
the intensity modulation from the interference. The modulation envelope is proportional
√
to sin(φ0 + φ1 sin ωm t), where φ0 ∝ V0 (bias voltage), and φ1 ∝ Pm (square root of
microwave power). Here, ωm is the modulation frequency. We used an advanced microwave
source (Agilent E8257C) for light modulation. We first coupled the laser beam into a singlemode, polarization-maintaining (PM) fiber, which connected to the modulator. The input
laser beam was linearly polarized. The orientation of linear polarization should align to
the main optical axis of the PM fiber, because the modulation device was only designed
36
60
40
AU
20
0
-20
Ramp signal
End resonance CPT signal
0-0 resonance CPT signal
-40
-60
0
2
4
6
Scanning time (sec)
8
-3
10x10
Figure 2.10: Two example CPT signals were found by setting the central frequencies of the
modulation to their resonant frequencies. The scanning range was ±50 kHz. The ripple tails
were due to the fast scanning, which can be eliminate by slowing down the sweeping speed
or shorten the scanning range. The end-resonance signal was obtained by setting the probing
beam with 45◦ to the magnetic field.
for the selected polarization. We used a single-mode fiber to couple the light out from the
modulator. A fraction of the output beam was sent to a Fabry-Perot spectrometer (Coherent
Model 240), which had 30 GHz free spectral range, for monitoring the modulation pattern.
Finally, the modulated beam was guided passing through the sample cell. The detailed
schematics is shown in Fig. 2.9.
To measure CPT signals, we directly used the nice internal function of frequency scanning of the synthesizer. The CPT resonance signal was observed as a transmission peak
when the modulation frequency or its integer multiple swept through the resonant frequency. The example results from a fast ramping rate is shown in Fig. 2.10. To record
the signal traces, the data acquisition interface was triggered by the ramping signal. The
trace average was required if the SNR is poor. Unlike the microwave spectroscopy, for the
end-resonance CPT, a angle between the probing beam and magnetic field was needed to
excite the coherence.
37
2.4
Results and Analysis
We carried out a series of experiments, including the examination of light narrowing effect,
the measurements of the pressure-dependent hyperfine relaxation rates, the idea of push-pull
optical pumping, and the line shape of CPT resonances. The theoretical calculations for
the evolution of the density matrix were carried out by the MATLAB programming. We
compiled a few individual MATLAB codes to numerically compute the results of microwave
and CPT spectroscopies, including all experimental observations. Some further studies
were also included, such as the transient analysis, CPT excitation with different modulation
schemes and at different buffer gas pressure, and the photon costs for optical pumping.
2.4.1
Light Narrowing Effect
The light narrowing effect is a narrowing of the magnetic resonance or CPT resonance
linewidth caused by increasing the intensity of pumping or probing lights. The basic idea is
to use the optical pumping to change the hyperfine population and therefore suppress the
spin-exchange broadening. Assuming a magnetic field is present to define the z-direction,
we can calculate the total relaxation rate with D1 pumping by using Eq. (2.37).
γµν
η 2 [I]2
|∆f | + (Γsd + Γex )
= Γd + ΓC I
8
µ
¶
3
− hµ|Sz |µihν|Sz |νi + Dop − Rop
4
(0)
−(sz (Dop − 2Rop ) + hSz iΓex )(hµ|Sz |µi + hν|Sz |νi) − 4Rop hµ|Sz |µihν|Sz |νi
!
Ã
X
(2.48)
hµ|S|αi × hα|S|νiραα ,
−Γex hµ|S|νi · hµ|S|νi(ρµµ + ρνν ) − 2i
α
(0)
where the effective pumping rate for depopulation and repopulation are Dop = Re(Γop )
and Rop =
(0)
Γ2q
1
4 ∆ω 2 +Γ2q Re(Γop ),
(g)
(e)
where ∆ω = ωhf − ωhf , and Γq is the quenching rate. Here,
(0)
hSz i denotes the static part of the longitudinal electronic spin. By definition, the intrinsic
linewidth is equal to γµν /π. According to the Eq. (2.48), the relaxation rate due to the
spin-exchange mechanism is a function of the population of the hyperfine sublevels. Since
the population of hyperfine sublevels is affected by the optical pumping, when the spin
exchange dominates the linewidth broadening, the linewidth is no longer a linear function
38
of the optical pumping rate. For further discussions, we focus on three selected resonances:
0-0 resonance, |a, 0i ←→ |b, 0i and |∆f | = 1. Therefore,
η 2 [I]2 3
Γ00 = γa0b0 = Γd + ΓC I
+ (Γsd + Γex ) + Dop − Rop
(2.49)
8
4
·
¸
1
[I] + 2
[I] − 2
−Γex (ρa0a0 + ρb0b0 ) +
(ρa1a1 + ρa−1a−1 ) +
(ρb1b1 + ρb−1b−1 ) .
4
8[I]
8[I]
End resonance, |a, ai ←→ |b, bi and |∆f | = 1. Therefore,
µ
¶
η 2 [I]2
3 ([I] − 1)2 − 1
Γend = γaabb = Γd + ΓC I
+ (Γsd + Γex )
+
+ Dop − Rop
8
4
4[I]2
P Γex /2 + sz (Dop − 2Rop )
([I] − 1)2 − 1
−
+ 4Rop
[I]
4[I]2
·
¸
(2[I] − 1)2 − 1)
1
2[I] − 1
−Γex
ρ
+
ρ ) .
(2.50)
(ρaaaa +
4[I]2
2[I] bbbb 2[I] abab
Zeeman end resonance, |a, ai ←→ |a, bi and |∆f | = 0 . Therefore,
µ
¶
3 ([I] − 1)2 − 1
ΓZend = γaaab = Γd + (Γsd + Γex )
−
+ Dop − Rop
4
4[I]2
(P Γex /2 + sz (Dop − 2Rop ))([I] − 1)
([I] − 1)2 − 1
− 4Rop
−
[I]
4[I]2
¸
·
2[I] − 1
1
1
(ρaaaa +
ρbbbb +
ρ ) .
(2.51)
−Γex
[I]
2[I]
2[I] abab
(0)
In Eq. (2.50) and Eq. (2.51), P = 2hSz i represents the static electronic polarization.
When there is no or only a weak oscillating field, the steady electronic polarization can be
calculated by solving Tr(Fρ̇) = 0. Hence, using Eq. (2.38), we got
d
sz
hFz i = −Γd hFz i − Γsd hSz i + Dop ( − hSz i) = 0.
dt
2
(2.52)
By using the spin-temperature approximation [3], we find hFz i/hSz i = 1 + ²(I, P ), where
²(I, P ) = hIz i/hSz i is a function of the nuclear spin and the electronic polarization. Therefore,
P =
sz Dop
.
Dop + Γsd + (1 + ²)Γd
(2.53)
Equation (2.53) is a very good approximated result for Γd 6= 0. When there is no diffusion
relaxation and the oscillating field (Γd = 0 and Γof = 0), the spin-temperature distribution
is the exact solution of Eq. (2.38), and therefore Eq. (2.53) is exact.
39
The spin-temperature distribution describes the population of the hyperfine sublevels
by
ρ=
eβIz eβSz
eβFz
=
.
Z
ZI ZS
(2.54)
Here β is the spin temperature parameter, and it can be determined by β = ln( 1+P
1−P ). For
P
any spin number J, ZJ = Jm=−J eβm . By carefully inspecting Eq. (2.50) and Eq. (2.51),
we also find that the repopulation pumping rate Rop actually plays no role if the photon spin
sz = 1. In this situation, the state |a, ai is a dark state of D1 pumping, and ground states
|a, bi and |b, bi are pumped to |f = a, m = ai of P1/2 excited state. No end-state coherence
can be generated in the excited state, and therefore no coherence with the right phase can
be repopulated to the ground state. If sz = 1, and using spin-temperature distribution, we
can further reduce Eq. (2.50) and Eq. (2.51) into
Γend
µ
¶
ηI2 [I]2
1
+ (Γsd + Γex ) 1 −
+ Dop
= Γd + ΓC
8
2[I]
P Γex /2 + Dop
(2[I] − 1)2 − 1)
− Γex
−
Qm̄ (P ),
[I]
4[I]2
(2.55)
and
µ
ΓZend
¶
1
1
= Γd + (Γsd + Γex )
+
+ Dop
2 2[I]
(P Γex /2 + Dop )([I] − 1)
1
−
− Γex Qm̄ (P ).
[I]
[I]
(2.56)
Here Qm̄ = eβ m̄ /ZI , and m̄ = I for the end state.
Considering the spin-exchange broadening, we find that when the population is uniform
in the ground state, the total relaxation rates contributed by the spin-exchange rate are 58 Γex
for 0-0 resonance, (1 −
3
2[I]
+
1
)Γex
[I]2
for end resonance, and ( 12 +
1
2[I]
−
1
)Γex
[I]2
for Zeeman
end resonance. We also find the minimum values of the spin-exchange rate. They are 12 Γex
for 0-0 resonance when all population is in 0-0 states, and no spin-exchange broadening for
both end resonance and Zeeman end resonance when all population is in |a, ai state, which
is the same as P = 1. By using Eq. (2.53), Eq. (2.49), Eq. (2.55) and Eq. (2.56), we plot the
relative linewidths of three different resonances as a function of the relative pumping rate
in Fig. 2.11. For comparison, we use D1 pumping with sz = 1 and assume that I = 3/2,
40
Relative linewidth ( Γ / Γsd )
120
D1 pumping, and sz = 1
Nuclear spin I = 3/2, and Γex/Γsd = 100
100
80
60
0-0 resonance
Hyperfine end resonance
Zeeman end resonance
40
20
0
0
5
10
15
Relative pumping rate ( Dop / Γsd )
20
Figure 2.11: The light narrowing effect is shown by two different end resonances. The linewidth
gets narrower by increasing pumping power at beginning, because the optical pumping traps
more atoms into the end state. The minimum point is where the spin-exchange broadening is
comparable to the pumping broadening. The linewidths start increasing after the minimum
point, because the optical pumping start dominating the linewidth broadening. However, the
circularly polarized pumping can only make the linewidth of 0-0 resonance worse.
Γex = 100Γsd , and there are no other relaxation mechanisms.
We can see a strong light
narrowing effect for two different end resonances. Optical pumping increase the atomic
polarization and traps more population into the end state. Therefore, it suppresses the
spin-exchange broadening to end resonances. For 0-0 resonance, the spin-polarized optical
pumping can only make the linewidth broader, because the optical pumping evacuates the
population from 0-0 states and its adjacent states. This disadvantages the conventional 0-0
CPT atomic clocks, which use a fixed circularly polarized laser beam to excited the CPT
resonance. Because the 0-0 state population is also evacuated, the signal is much weaker.
However, if we use push-pull optical pumping, which will be discussed in the later section,
the population can be concentrated into 0-0 state. We can obtain a strong CPT resonance
signal, and the maximum suppression of the spin-exchange broadening is from 58 Γex to 12 Γex
(20% suppression).
41
87
12
10
Experimental data,
-3
8
x10
End resonance linewidth (kHz)
o
Cell T108 (N2 700 torr + Rb ) at 140 C, thickness 2 mm
Laser frequency 377104 GHz,
Beam DIA ~ 6mm
Microwave fc = 6.8253815 GHz, Rabi freq. ~ 110 Hz
Bz ~ 4.6 Gauss
14
6
Fitting curve
10
5
0
-5
-10
60
40
20
0
-10
-5
0
kHz
5
-10
-5
4
0
kHz
5
2
0
1
2
3
4
5
6
7
2
Laser power ( mW/cm )
Figure 2.12: The first light narrowing data of
87
Rb hyperfine end-resonance was found from
the high pressure lollipop cell. The data agrees with the theory very well. Two insets show
the resonance signals from the detuning scan. The scanning step was 400 Hz. The linewidth
decreases with increasing pumping power to a minimum value. After this point, the linewidth
begins to be dominated by the pumping rate. However, the signal amplitude is always improved
by the pumping power.
Figure 2.12 shows an experimental result of light narrowing effect by using microwave
spectroscopy. By using Eq. (2.55), we can fit data to the theoretical curve very well.
At low laser power, the pumping was too weak to pumping atoms to the end state, the
spin-exchange broadening dominates the linewidth. By increasing the pumping power,
we observed a strong narrowing effect and also more intense signals. From examination of
p
Eq. (2.55), one can show that the minimum linewidth is given by ∆ν = c Γex Γ∗sd +constant,
where c ≈ 0.4 slightly depends on P and I, and Γ∗sd = Γsd + (1 + ²)Γd . Here, the constant
contribution to linewidth broadening are due to other relaxation mechanisms and also the
inhomogeneous broadening. Comparing the data point with the narrowest linewidth to the
initial value of the linewidth, we obtained about 1/5 factor of the narrowing, and also the
42
SNR was increased by a factor more than 3000. The equivalent FOM (figure of merit) =
SNR/∆ν is increased by a factor more than 15000. These enhancing factors are not absolute
but depend on the spin-exchange rate. Higher temperature, which has higher spin-exchange
rate, has larger enhancing factors.
The light narrowing of the end resonances is a result of the conservation of the angular
momentum. This special effect will possibly benefit the design of small vapor-cell atomic
clocks. For a miniature cell atomic clock, higher working temperature is required to obtain
more light absorption. The spin-exchange rate is increased by the temperature. By using
spin-polarized optical pumping, the end resonances not only eliminate the spin-exchange
broadening, but also have stronger resonance signal, because the population is trapped to the
end state. The resonance frequency of the end resonance is more sensitive to the magnetic
field than the 0-0 atomic-clock resonance. For constructing an end-resonance atomic clock,
an additional field-stabilizing feedback loop is required. We propose a “double lock” method.
By using both hyperfine and Zeeman end resonances, we will be able to lock the hyperfine
frequency and the magnetic field [29, 56]. Our collaborators in Sarnoff corporation have
successfully demonstrated the double-lock technique.
2.4.2
Transient Analysis
In microwave spectroscopy, we measure the steady intensity of the transmission light as
a function of the frequency detuning of the microwave field. If we suddenly turn on the
microwave field and look at the transient of the transmission light, we would expect to
see the damping oscillation. The damping rate of the oscillation is determined by the
longitudinal and the transverse relaxation rates. Equation (2.38) is a quasi-steady solution
for the presence of the oscillating field. We can obtain a correct result of late-time evolution
of the density matrix from Eq. (2.38), but not the transient at the beginning. To calculate
the transient signal, we can solve the evolution equation in Liouville space. Since there is
only one coherence excited by the oscillating field, for the ground-state evolution, we only
43
need to consider a Liouville vector with g + 2 components, where g = 2[I]. Hence,


ρ11
..
.





|ρ) = 
 ρgg


 ρ̃µν

ρ̃νµ





.





(2.57)
Here, the first g components are the diagonal elements of the ground-state density matrix,
and the last two components are the selected off-diagonal elements in the rotation coordinate
of frequency ω. We define ρµν = ρ̃µν e−iωt , and we know that ρνµ = ρ∗µν . The order of the
vector components is not important. The evolution equation is simply as |ρ̇) = −R|ρ), and
we find

..
.


 ···


R=
 ···


 0

0

..
.
..
.
0
Γµµ
Γµν
−iωR /2
0


iωR /2 


Γνµ
Γνν
iωR /2
−iωR /2 
.


−iωR /2 iωR /2 γ2 + iδω
0


iωR /2 −iωR /2
0
γ2 − iδω
(2.58)
Here, Γαβ denotes the transition rate from ραα to ρββ . The transverse decay rate γ2 = γµν ,
and the frequency detuning δω = (ωµν − ω), where ω is the frequency of the oscillating field.
Both Γαβ and γ2 can be obtained from Eq. (2.38) and Eq. (2.37) by setting Γof = 0. The
spin exchange is the only nonlinear term in the evolution equation. If the temperature is
sufficiently low, the spin exchange has small contribution to the relaxation rate. We can
treat it as a linear system. Therefore. the evolution the of the vector |ρ) can be acquired
by solving the eigenvalues and eigenvectors of matrix R. The complete solution can be
obtained by using Eq. (2.11). The analytical result shows that there are g real eigenvalues
and 2 complex eigenvalues. These two complex eigenvalues are complex conjugate and
govern the damping oscillation on the transient signal. Further analysis shows that when
the Rabi frequency is larger than all the relaxation rates, ωR /Γαβ > 1 and ωR /γ2 > 1, the
AU
44
Solution of the equation
Solution by using eigenvectors
10
5
0
-5
-10
Re(Γ)/π
1.1
1.0
0.9
0.8
Im(Γ)/2π
0
12
8
4
1
2
3
ms
4
5
Solution of the equation
Analytical approximation
Solution of the equation
Analytical approximation
-10
-5
0
Detuing (kHz)
5
10
15
Figure 2.13: The top panel shows the calculated transient signal with frequency detuing = 7
kHz, and its multi-exponential background has been subtracted. The middle and bottom panels
show the computational results of the complex damping rate at different frequency detunings by
solving the evolution equation and using Eq. (2.59). The unit in their vertical axes is 1000/sec.
The calculations show the distinction between two different methods is only a few parts per
thousand.
solution of the complex eigenvalue can be very good approximated by
r
2
γ2 (δω)2 + 21 (γ2 + γ1 )ωR
2 − 1 (γ − γ )2 ,
Γ=
± i (δω)2 + ωR
2
1
2
2
4
(δω) + ωR
(2.59)
where γ1 = 12 (Γµµ + Γνν + Γµν + Γνµ ). This approximation has a better precision when the
ratio between Rabi frequency and the relaxation rates becomes larger. For the hyperfine
end resonance, γ2 is given by Eq. (2.55), and γ1 is given by
µ
γ1 = (Γex + Γsd + Γop ) 1 −
1
2[I]2
¶
−
P Γex + Γop
+ Γd .
2[I]
(2.60)
In order to understand how accurate the Eq. (2.59) is, we consider a real case of 87 Rb. We
choose similar ωR that we used in the lab for transient measurements. We try to make γ1
and γ2 close to ωR to examine the limit of Eq. (2.59). Assuming we have a 4 amagat N2
87 Rb
cell at 60◦ C. The approximate rates are Γop = 500s−1 , Γsd = 550s−1 , Γex = 300s−1 ,
45
ΓC = 1580s−1 , Γd = 31s−1 , and the Rabi frequency ωR = 2π · 2000s−1 . According to the
rates above, we find ωR /γ2 ∼ 3.5 and ωR /γ1 ∼ 5.2 for the hyperfine end resonance. For
comparison, computational results are shown in Fig. 2.13 obtained by numerically solving
the differential equation and by using the analytical approximation in Eq. (2.59). One can
see the results from Eq. (2.59), which is shown as the solid curve, go through the circles,
which are the numerical solutions of the evolution equation. The relative error between
two different methods is only a few parts per thousand. Usually the experimental errors of
the measurements of relaxation rates are more than a few percent. Therefore, Eq. (2.59)
provides a quick way to measure the transient damping oscillation. Some relaxation key
parameters can be measured by using the transient method if we properly change some
relative variables to the relaxation rates. Those key parameters play a crucial part of
making an atomic clock.
Considering buffer-gas pressure-dependent relaxation rates, Γsd , ΓC , and Γd . We can
define the key parameters as relaxation rate per unit atomic number density of the buffer
gas. Therefore,
Γsd = γsd N, ΓC = γC N, and Γd = γd /N,
(2.61)
where N is the atomic number density of the buffer gas, which is in unit, amagat [1]. By
using Eq. (2.55), Eq. (2.60), and Eq. (2.59), we can find the decay rate as a function of the
buffer-gas number density at two frequency detunings for the hyperfine end resonance.
Γ(δω=0)
Γ(δωÀ0)
µ
¶
ηI2 [I]2
1
1
= γd /N +
γC N + 1 −
−
γsd N + const.
16
4[I] 4[I]2
µ
¶
ηI2 [I]2
1
= γd /N +
γC N + 1 −
γsd N + const.
8
2[I]
(2.62)
Here, we assume that other contributions to the relaxation rates are independent of N . If
the diffusion decay parameter γd is known, we can extract γsd and γC from experiments
by fitting the data of pressure-dependent damping rates Γ into Eq. (2.62) at two different
detuning frequencies .
In practice, we prefer using end-resonances for measuring the S-damping rate and Carver
rate. Since these two relaxation rates linearly depend on the buffer gas pressure, we can
Re(Γ/π) (Hz)
46
600
550
500
450
400
350
Im(Γ/2π) (Hz)
6.825360
6.825362
6000
5000
4000
3000
2000
6.825364
GHz
6.825366
6.825368
6.825370
6.825368
6.825370
fc = 6.825364474 GHz
ωR = 1997 Hz
6.825360
6.825362
6.825364
GHz
6.825366
Figure 2.14: An experimental data of the end-transient damping rates from a
87
Rb cell with
1 atm N2 buffer gas. The data was fit into Eq. (2.59). The Rabi frequency can be extracted
from the fitting of the imaginary decay rates.
reduce the uncertainties of the relaxation measurements by using high pressure cells. The
optical pumping efficiency for end resonances is not affected by the buffer gas pressure. For
0-0 resonance, when the buffer gas pressure is high enough that the splitting of the optical
absorption line due to the ground-state hyperfine structure is no longer resolved, the optical
pumping becomes poor and can not produce a good signal. The transient method with end
resonances can provide a good measurement of the relaxation key parameters within the
experimental precision. We applied this method to measure S-damping and Carver rate key
parameters of Rb and Cs, which will be discussed in the next subsection. Figure 2.8 shows
the real signal of the transient measurement. By using the Fourier’s analysis, we can obtain
the real and imaginary damping rates from the linewidth and the offset of the resonance
peak on the spectrum. Figure 2.14 shows an example result from a
87
Rb cell with about
1 atm N2 buffer gas. We used Eq. (2.62) to approximately retrieve two key parameters
first. After that, we assigned previous obtained numbers into the numerical code to have a
better data fitting by solving the evolution equation. Usually, this refinement step did not
improve the experimental precision, since we know Eq. (2.59) already has a precision better
than 1 % for all our experimental conditions. The only requirement of using Eq. (2.59) and
47
Eq. (2.62) is that the Rabi frequency should exceed all relaxation rates.
2.4.3
Pressure Broadening of Microwave Resonance
Gas
&
Worked by
He:
J. Vanier et al. [48]
D.K. Walter et al.[53]
N. Beverini et al. [9]
T.G. Walker [51]
Our work [38]
Ne:
J. Vanier et al.
N. Beverini et al.
T.G. Walker
Ar:
J. Vanier et al.
D.K. Walter et al.
N. Beverini et al.
T.G. Walker
Kr:
N. Beverini et al.
M.A. Bouchiat et al.[11]
T.G. Walker
N2 :
J. Vanier et al.
D.K. Walter et al.
N. Beverini et al.
Our work [29]
(s−1
γsd
Rb
Exp.
Thy.
Key Parameters (at 27◦ C)
amg−1 )
γC (s−1 amg−1 )
Cs
Rb
Cs
Exp.
Thy.
Exp. Thy.
Exp.
Thy.
3093∗
191
≈0
8.5
96
26
208
(135)
(332)
2828∗
≈0
151
30
32
637
610
14
≈0
1270
687
964
21244
28108
24985
27382
3062∗
395
118
115
(122)
884
(820)
(430)
(1632)
Table 2.1: Experimental and theoretical results of the key parameter of S-damping rate and
Carver rate. All numbers listed above have about 10 % errors. For helium gas, the values
measured by D.K. Walter were for 3 He. The Carver rates from Vanier’s measurements,
which are tagged by “∗”, are much larger than other values in the same block. The numbers
from our work are marked by parentheses.
For atomic-clock resonances, buffer gas causes frequency shift and ground-state spin
relaxation. Both results affect the performance of atomic clocks. By carefully selecting the
mixture of the buffer-gas, we can optimize the performance of atomic clocks. There are two
48
relaxation mechanisms, S-damping and Carver rate, causing the pressure broadening of the
atomic-clock linewidth. Some prior measurements of these two rates were done by different
experiments. Most of measured S-damping rates in different buffer gases are consistent. For
Carver rates, however, they were seldom measured and had an inconsistent problem. Before
our work, the only two experimental data, which give the information of Carver rates of Rb,
were done by J. Vanier and D.K. Walter. Two results have a large discrepancy. We believe
that Carver rates measured by D.K. Walter are reliable, but Vanier’s results are flawed. In
order to have more comparisons, we carried out the theoretical calculations for He buffer
gas and also the experimental measurement for nitrogen buffer gas. We obtained consistent
results with Walter. Figure 2.15 shows the transient data of hyperfine and Zeeman end
resonances of
87 Rb
with different N2 pressures. In Table 2.1, we summarize γsd and γC of
Rb and Cs in different buffer gases obtained by different groups. We would expect close
values in the same blocks on Table 2.1. However, we find a big difference of Rb Carver
rates from Vanier’s measurements. From our further study, we believe that unusual large
damping rates measured by Vanier were due to the inhomogeneous broadening caused by
the temperature gradients.
The temperature shift of the hyperfine frequencies are related to the pressure shift. The
perturbation to the hyperfine interactions is a function of both temperature and buffer-gas
pressure. Bean and Lambert [7] have studied the temperature-dependent pressure shifts of
23 Na, 39 K,
and
85 Rb
in the buffer gases He, Ne, Ar, and N2 . According to their results, the
frequency shift can be expressed by
∆νs = (ν − ν0 ) = β(T )NB ,
(2.63)
where, ν0 is the zero pressure-shift frequency, ν is the shifted frequency, β is a temperaturedependent coefficient, and NB is the number density of the buffer gas. If we assume the
buffer-gas pressure is equal inside a cell everywhere, we find the local number density as
NB = PB /kT , where PB is the buffer-gas pressure. There the hyperfine frequency shift due
49
Damping rate, Γ/π (1000/sec)
2.0
Hyperfine end resonance data fitted with:
∆ν = (272 ± 16)⋅[Ν2] + (185 ± 40) + 42/[Ν2] Hz
Zeeman end resonance data fitted with:
∆ν = (30 ± 2)⋅[Ν2] + (127 ± 5) + 42/[Ν2] Hz
1.5
Hyperfine end resonance (J. Vanier)
Hyperfine end resonance (D. K. Walter)
1.0
0.5
0.0
0
1
2
3
Nitrogen buffer-gas number density [N2] (amg)
4
Figure 2.15: The experimental data of microwave and Zeeman resonance damping rates as
the number density of nitrogen. Our experimental data are consistent with D.K. Walter’s
measurements, which is plotted as two shadowed areas. The hyperfine resonance data is affected
by both S-damping rate and Carver rate. The Zeeman resonance data is only affected by Sdamping rate. J. Vanier’s data predicts a much larger values, which is plotted as a dark-gray
curves on the top left corner.
to the local temperature becomes
∆νs =
β(T )PB
.
kT
(2.64)
Here, PB acts as a constant, because of the uniform pressure inside the cell. The buffer gas
R
pressure can be determined from NT = PB (kT (r))−1 dV , where NT is total atom number,
T (r) is the local temperature, and dV is the local volume. By using linear approximation,
PB = NT k T̄ /V , where T̄ is the average temperature. By using Eq. (2.47), we know the inhomogeneous broadening cause by the temperature gradient is ∂(∆νs )/∂T ·∆T . Figure 2.16
shows the linewidth broadening of 85 Rb in different buffer gases caused by the temperature
gradient.
50
Figure 2.16: Absolute hyperfine frequency shifts of
85
Rb induced by the buffer gases at a
constant pressure PB = 760 torr, inferred from the data of Bean and Lambert [7]. Also shown
are the spreads in frequency ∆νp that would be caused by a spread in temperature ∆T = 20 ◦ C
centered at 30 ◦ C. The line broadening due to temperature inhomogeneities in Ar is about ten
times less than that in He, Ne, or N2 .
For Vanier’s results, we found that the required temperature gradients for producing
their linewidth data were only from 0.6 to 0.8◦ C. By carefully inspecting Vanier’s experiment, they used a big cell about 7 cm long. The cell was heated from its bottom. It was
not hard to have such a small temperature gradient in their cell. From Fig. 2.16, we know
Ar has a much smaller temperature shift. This explains the consistent results of Ar from
Vanier’s and Walter’s measurements.
There are still many key parameters have not been determined. For designing vapor-cell
atomic clocks, it is important to fill out Table 2.1. This will require future experiments or
calculations.
51
2.4.4
Push-Pull Optical Pumping
For regular spin-polarized optical pumping, we are able to concentrate the ground-state
population to the end-states, |a, ai or |a, −ai, by using a pure circularly polarized pumping
light. For concentrating the population to other ground-state hyperfine sublevels, such
as m = 0 states, some techniques have been developed, but they are complicated. For
example, while the hyperfine sublevels are optically unresolved, we can use regular optical
pumping combining with the microwave or RF fields to prepare the population to the
selected sublevel, but the static magnetic field must be large enough to resolve the Zeeman
resonance between different pairs of sublevels. The oscillating field swaps the population
between the sublevels. However, this kind of state preparation is laborious especially when
we need to move the population from the far end state to the 0-0 state. Without the
oscillating magnetic field, we would be able to concentrate the population to the 0 states,
but it requires a resolved excited hyperfine splitting of P1/2 . One can use two linearly
polarized pumping beams, which are perpendicular to the magnetic field. The pumping
lights connect the ground-state hyperfine sublevels to one of the excited hyperfine multiplet.
The transitions, |a, 0i(g) ↔ |a, 0i(e) and |b, 0i(g) ↔ |b, 0i(e) , are forbidden. Therefore, |a, 0i(g)
or |b, 0i(g) can become a dark state (unpumped state) if the optical transition occurs only
between f = a or f = b of the excited state and the ground state. The population will
eventually be trapped into one of the 0 states. Obviously, this method can only be applied
in the low buffer gas pressure.
We report a new method, “push-pull optical pumping”, which can pump atoms into
nearly pure 0-0 superposition state in the very low to very high buffer gas pressure. There
are only two essential requirements: (i) the time-averaged optical pumping rate must greatly
exceed the ground-state relaxation rates and (ii) the use of alternating polarization of D1
pumping light. This new idea can be understood through the following discussion. Let the
0-0 superposition state be described at time t by
|ψi =
|a, 0ie−iEa0 t/h̄ + |b, 0ie−iEb0 t/h̄
√
.
2
(2.65)
52
Sz
0
Intensity of σ+
Absorption probability of σ−,
Intensity of σ−
AU
Absorption probability of σ+,
0
0
0
1
2
Period (2π/ω00)
3
4
Figure 2.17: The oscillation of the electron spin produces time dependent absorption cross
sections for different circularly polarized pumping lights. The interlacing σ+ and σ− light pulses
repeated every 2π/ω00 can maximize the spin oscillation, and therefore trap the atoms to the
0-0 superposition state. At this point, the transmission light has the maximum transparency.
Here Ea0 and Eb0 are the energies of the basis states |a, 0i and |b, 0i. The phases of the
basis states can be chosen such that ha, 0|Sz |b, 0i = 1/2. Then the expectation value of the
longitudinal electron spin is
hSz i = hψ|Sz |ψi =
1
cos ω00 t.
2
(2.66)
The Bohr frequency is ω00 = (Ea0 − Eb0 )/h̄ = 2π/T00 . The 0-0 superposition state is
equivalent to the oscillation of the electron spin along the z-direction at Bohr frequency,
which has been shown in Fig. 2.4. In the high-pressure limit, the probability for a spinpolarized alkali-metal atom to be excited by a light pulse is proportional to (1 − 2s · hSi),
where s is the expectation value of the photon spin [3]. Then the relative probabilities p+
and p− of absorbing RCP(right circular polarization σ+) and LCP(left circular polarization
σ−) pulses by the superposition state of Eq. 2.65 are
p+ = sin2
ω00 t
2
and p− = cos2
ω00 t
.
2
(2.67)
As sketched in Fig. 2.17, we assume that RCP pulses hit the atoms at the times t = 0, T00 ,
"
#
!
53
Figure 2.18: Alkali hyperfine sublevels with a hypothetical nuclear spin quantum number
I = 1/2. 0-0 state is the only dark state by using a complementary Λ scheme, which has
two Λ pumpings with a 180◦ modulation phase difference between each other. Atoms will be
eventually trapped into the 0-0 state, the dark state.
2T00 , etc., when p+ = 0. LCP pulses hit the atoms at the times t = 21 T00 , 32 T00 , 52 T00 , etc.,
when p− = 0. Atoms in the superposition state |ψi of Eq. (2.65) will absorb very little light
from pulses of either polarization. Atoms in any other state will be excited by the pulse
train. No matter what the initial state of the atoms is, nearly all of them will eventually
be pumped into the superposition state of Eq. (2.65), the “dark state” of the system. The
pumping light with alternating circular polarization forces electron spin to oscillate at the
alternation frequency along the direction of the light. This is an analogous to “push” and
“pull” the electron spin.
For low buffer-gas pressures, where the hyperfine structure of the optical absorption
lines is well resolved, it is sufficient to use a simple electro-optic modulation of linearlypolarized monochromatic laser light to produce two equal-intensity spectral sidebands with
a frequency difference of ω00 . One sideband should be approximately resonant for pumping
out of the lower hyperfine multiplet and the second should be approximately resonant for
pumping out of the upper hyperfine multiplet. If the resulting “two-wave” light is separated
into right- and left-circularly polarized sub-beams, and if one sub-beam is delayed by T00 /2
with respect to the other, the combined beam can pump nearly all of the atoms into the
0-0 superposition state |ψi of Eq. (2.65). In the language commonly used in CPT research,
54
Intensity (AU)
Light pulses
Time
Intensity (AU)
Spectrum of light pulses
Normailized absorption
cross section of
87Rb with 5 atm N2
Normalized absorption cross section
of 87Rb with 0.05 atm N2
Optical comb
Optical frequency
Figure 2.19: The optical sidebands of a pulsed light is like a comb in the spectrum. With
high buffer-gas pressure, atoms see all optical sidebands. With low buffer-gas pressure, only
two sidebands are seen by atoms.
this is equivalent to a combination of two Λ-pumping schemes, one with σ + (RCP) and the
other with σ − (LCP) light, where a 180◦ modulation phase difference is maintained between
the two as shown in Fig. 2.18.
By using pulsed light, push-pull optical pumping can work at any gas pressure. In the
optical spectrum, the pulsed train produces widely-spread optical sidebands (as the optical
comb shown in Fig.2.19), which is separated by the inverse of the pulsing period. When the
buffer gas pressure is low, only two optical sidebands can be seen by the atoms. Therefore,
two-wave pumping is sufficient for low pressure regime. Whatever the gas pressure is, the
alternating circular polarization of the pumping light is needed. In order to generate the
alternating circular polarization, we employed a Michelson type interferometer to convert
an intensity modulated laser beam to a polarization modulated probing beam. The detail
is shown in Fig. 2.20.
The intensity modulated laser beam was linearly polarized before
entering the Michelson interferometer. A quarter-wave plate was placed in one of the
= =
2
3
79
$(!%'0%)#/.
:
1"#!,)+*&-
6>
6
6
8
6
7
5
4
;
<
3
? 55
+
−
+
/
−
/2
Figure 2.20: The Michelson type interferometer converts an intensity modulated laser beam
to a polarization modulated laser beam. A λ/4 wave plate was needed in one of the arms. The
difference of the arms’ length is equal to c/(4ν00 ), where ν00 is the 0-0 resonance frequency.
The details of other parts can be found in Fig. 2.6 and Fig. 2.9.
arms, which converted the horizontal linear polarization to vertical linear polarization of
the sub-beam. The length difference of the two arms of the interferometer was adjusted to
produce a correct retardation between two sub-beams. The shortest time delay from one
sub-beam to the other must be equal to the half of the oscillation period of the hyperfine
frequency. For
87 Rb,
the ground-state hyperfine frequency is about 6.8 GHz, hence, the
path difference of two arms is c/(6.8 × 109 )/4 ≈ 1.1 cm, where c is the speed of light in
air. Two sub-beams was therefore recombined into one beam with alternation between two
linear polarization states on the output of the Michelson interferometer. The intermediate
polarized state between the two orthogonal linear polarizations can be circular polarization,
linear polarization, or elliptical polarization. The result depends on the difference of the
optical phases of two combining sub-beams. The wavelength of the laser beam was about
1 micron. The stability of the optical phase relied on mechanical stability of the Michelson
56
Figure 2.21: Comparison of the signal contrasts of conventional CPT and push-pull CPT. The
inset shows the modulation spectrum from a Fabry-Perot spectrometer. At the same pumping
rate, the CPT signal from push-pull pumping has a better signal contrast, which is about
77 times larger than the conventional CPT signal. The conventional CPT signal on the top
panel was taken with 16 times more averages than the push-pull CPT for a better SNR. This
experimental data was taken from a spherical cell with diameter = 1.9 cm.
interferometer. In order to guarantee the production of a probing beam with alternating
circular polarization, another quarter-wave plate, placed on the output beam, was required.
Finally, this probing beam was guided to pass through the sample cell.
Figure 2.21 shows the very large signal enhancement of the 0-0 CPT resonance produced by push-pull pumping in comparison to the CPT resonance induced by intensity
modulated light of fixed circular polarization. The signals were obtained from a cell containing isotopically-enriched
87 Rb
and nitrogen gas at a pressure of 730 torr (defined at
room temperature). The very small conventional CPT signal of Fig. 2.21 is mostly due to
the optical pumping of atoms into the end state |a, ai, or |a, −ai, depending on the sign of
the fixed circular polarization. Push-pull pumping eliminates this cause of signal degradation, because the time averaged photon spin is equal to zero. One can see the baseline of
57
Figure 2.22: Experimental data and theoretical curves from our density-matrix modelling
agree very well to each other. The conventional CPT has an optimized value of pumping rate
for maximizing the signal contrast. Higher pumping rate degrades signal of the conventional
CPT due to the population trapped to the end state. However, push-pull pumping has no such
problem. The right diagram shows a very strong CPT signal by using push-pull pumping of
one circled data point on the left diagram.
the conventional CPT signal is higher than the push-pull CPT in Fig. 2.21, because atoms
in conventional CPT were polarized by using fixed circular polarization and became more
transparent.
More systematic studies of push-pull and conventional CPT resonances are summarized
by the experimental points in Fig. 2.22. The measurements were done with Rb cells containing much lower gas pressures than the cell of Fig. 2.21. Modelling calculations of the
expected CPT signals, shown by the continuous lines in Fig. 2.22, are in good agreement
with the experiments. Typical population distributions from the modelling are shown as
insets. Conventional CPT signals first grow with increasing light intensity, as faster pumping overcomes relaxation processes. After reaching a peak, the signals decrease as the light
pumps more and more atoms into one of the end states, the dark states for the light of
fixed circular polarization [47]. Faster push-pull pumping, however, always increases the
58
contributed from spin exchange
Relative linewidth ( Γ
(ex)
/ Γsd )
80
75
70
D1 pumping, and sz = 1
Nuclear spin I = 3/2, and Γex/Γsd = 100
65
Conventional CPT
Push-Pull CPT
60
55
50
0
100
200
300
400
Relative pumping rate ( Dop / Γsd )
500
Figure 2.23: The relative linewidth contributed from spin-exchange broadening. As the prediction from Eq. (2.49), push-pull optical pumping can reduce the linewidth broadening due the
spin-exchange collisions, but the optical pumping used in conventional CPT can not. However,
in push-pull pumping, the decrease of the spin-exchange broadening can not compensate the
linewidth broadening due to the optical pumping. This tiny light narrowing effect is hardly to
be observed.
concentration of atoms in the superposition state |ψi of Eq. (2.65), the only dark state for
the light of alternating circular polarization. As shown in the left diagram and top panel
of Fig. 2.22, a 30% contrast was measured for the push-pull CPT resonance, a substantially larger figure than has ever been previously reported with similar buffer gas pressure,
temperature, thickness of the cell, etc.
Because of birefringence in the glass windows of our oven and the glass walls of the
vapor cell, the circular polarization was degraded and the peak spin of absorbed photons
was measured to be |sz | = 0.87. Taking this into account, our modelling calculations predict
that 51% of the atoms were concentrated in the superposition state |ψi of Eq. (2.65), for
the left diagram, top-right experimental point in Fig. 2.22. The predicted concentration for
fast pumping and for peak |sz | = 1 at these conditions is 100%. Analogous to Eq. (2.53),
we can find an equation for the on-resonance population concentration, P0−0 , of the two 0
59
states to be
(0)
P0−0 =
Dop + γ10
¤
£ (0)
,
(0)
(1)
(0)
([I] − 1) Dop + γ10 − (|sz |Dop )2 /(Dop + γ20 ) + Dop + γ10
(2.68)
where γ10 and γ20 are the longitudinal relaxation rate and the coherence decay rate of the
(0)
(1)
two 0 states without optical pumping. Here, Dop and Dop are DC and AC components of
(1)
(0)
the depopulation pumping rate of push-pull pumping, where |Dop /Dop | ≤ 1. The absolute
value of the z-direction photon spin |sz | ≤ 1. One can check Eq. (2.68), without the light
(1)
modulation, Dop = 0, or without circular polarization of the pumping light |sz | = 0, we
find P0−0 = 1/[I]. The maximum value P0−0 = 1, which represents 100% population in the
two 0 states. The signal contrast is basically proportional to the strength of the coherence
(1)
ρ̃00 . From Eq. (2.32), we find that |ρ̃00 | =
1 |sz |Dop
2 D(0) +γ 0 P0−0 .
op
2
Push-pull optical pumping is a special CPT excitation, which has only one dark state.
Its exceptional ability of trapping atoms into 0-0 state can greatly improve the 0-0 CPT
signal, and it can therefore permit improved design for 0-0 CPT atomic clocks. In addition,
since the population is trapped to the 0-0 state, the spin-exchange broadening can be
further suppressed according to Eq. (2.49). Figure 2.23 shows the relative spin-exchange
broadening in conventional CPT and push-pull CPT as function of the relative pumping
rate. The plot is based on numerical calculation with the same conditions used in Fig. 2.11.
As we can see, the conventional 0-0 CPT gains more spin-exchange broadening by increasing
the pumping intensity, because the population is evacuated from the m = 0, ±1 states as
the prediction from Eq. (2.49). By increasing the pumping power, push-pull CPT has less
linewidth broadening from spin-exchange collisions because of the population concentration
to the 0-0 state. However, this light narrowing effect is not strong enough to be noticed,
because the decrease of the spin-exchange broadening is not fast enough to catch up the
optical pumping broadening.
For atomic clocks, instead of regarding the signal contrast of the resonance, we like to
optimize the figure of merit (FOM), which is proportional to the signal contrast dividing
by the linewidth. Figure 2.24 shows FOMs as functions of the pumping rate from our
experimental data, shown in Fig. 2.22. One can see that push-pull CPT is better than
60
140
500
87
300
87
120
200
o
Rb with 15 torr N2 at 45 C
Push-Pull CPT
Conventional CPT
100
FOM
FOM
400
o
Rb with 90 torr N2 at 60 C
Push-Pull CPT
Conventional CPT
80
60
40
100
20
0
0
0
5
10
15
-1
Optical pumping rate (sec )
3
20x10
0
5
10
15
3
20x10
-1
Optical pumping rate (sec )
Figure 2.24: The FOMs of the plots in Fig. 2.22. Push-pull CPT has larger FOM than
conventional CPT. Here, FOM=contrast(%)/linewidth(kHz)×10.
conventional CPT. In our experimental condition, the photon spin was only about 0.87.
For a 100% circularly polarized pumping light, the enhancement of FOM of push-pull CPT
will be more pronounced. Another factor, which degrades the performance of push-pull CPT
but improves conventional CPT, is the diffusion relaxation. The diffusion relaxation resists
push-pull optical pumping to the m = 0 states and also conventional optical pumping to the
end state. Therefore, it benefits conventional 0-0 CPT, but disadvantages push-pull CPT.
As we can see from Fig. 2.24, low pressure cell has a smaller difference of the maximum
FOMs of two different CPT signals, which is due to the larger diffusion at lower pressure.
Push-pull optical pumping can improve not only the performance of the CPT atomic
clocks, but others like CPT masers, the state preparation for cold alkali-metal atoms or hot
vapors. The state preparation of push-pull pumping can be used for atomic-beam atomic
clocks, cesium fountain atomic clock, quantum computation, quantum information, etc.
Considering that ground-state hyperfine splitting is resolved, we are able to concentrate the
population to different hyperfine sublevels by adjusting the pumping intensity for upper
and lower hyperfine multiplets and the ratio of the intensity of the RCP(σ+) and LCP(σ−)
components. Figure 2.25 shows an example of
87 Rb
from theoretical calculation. The
end state concentration can be achieved by using conventional optical pumping. For other
sublevels, besides the adjustments of the amplitudes of the two related optical sidebands
and the σ + /σ − ratio, we have to adjust the modulation frequency to match the resonance
frequency of the two related sublevels. For 0 states and end states, we can achieve nearly
61
Figure 2.25: By changing the ratios of pumping intensities of σ+, σ−, I1 , and I2 with a
selected modulation frequency, we can put lots of population into some hyperfine sublevels.
The top-left panel shows 50% on |2, 0i and 50% on |1, 0i states. The top-right panel shows 99%
on |2, 0i state. The bottom-left panel shows 95% on |1, 1i state. The bottom-right panel shows
100% on |2, 2i state.
100% population trapping. For other states, we can still have high degree of population
trapping as shown in Fig. 2.25.
2.4.5
Pressure Dependent CPT Excitation
To excite CPT resonances, at least two coherent lights need to be used. A simple way to
do this is to modulate a CW laser light. Due to the limitation of the experimental devices,
it was easier to produce a phase or frequency modulated (PM or FM) light to GHz range
in the past, such as using EOM (electro-optic modulator) to modulate the laser light or
modulating the supplied current to the laser diode. However, experiments shows that the
CPT signals from PM lights vanish when the buffer gas pressure exceeds a few hundred
torr [4, 19]. Because more optical sidebands, which are seen by atoms, can interfere with
62
Figure 2.26: The amplitude of the optical coherence ρge is like a FM signal passing through
1
a resonator filter g(Ω) = i(Ωge −Ω)+γ
, where Ω is the optical angular frequency. Frequencies
op
of the signal E(t) higher or lower than the resonance frequency have lower output amplitudes.
The lower buffer gas pressure has a narrower filter bandwidth, and can produce a stronger AC
component in pumping rate to excite a larger CPT signal.
each other, the CPT excitation degrades when the light is phase- or frequency-modulated.
For amplitude modulated light, the degradation of CPT excitation at high gas pressure is
minimal. Therefore, we chose intensity modulated laser beam to do the CPT spectroscopy.
Ideally, pulsed light is the best choice for CPT pumping at high buffer-gas pressure.
According to Eq. (2.32), the strength of CPT excitation is proportional to the AC component of the depopulation pumping rate, whose frequency matches to the resonance frequency.
Mathematically, the time-dependent pumping rate can be calculated from Eq. (2.22). For
a better picture to understand, we compare the pumping rate at two different pressures in
time domain as the plots shown in Fig. 2.26.
Firstly, the optical coherence ρge between the ground-state sublevel and the excited-state
sublevel is excited by the pumping electric field. Atoms are optical resonators. The time
dependence of the coherence ρge is proportional to the pumping electric field passing through
the atomic-resonator filter, which has the resonance frequency Ωge and the linewidth γop
63
CPT on Three Correlated Levels
(ground-state resonance excited by optical fields)
Excited state
Electronic-Mechanical Analogy of CPT Phenomena
(mechanical oscillation excited by electric fields)
Electronic resonance frequency
and damping :
iΩ1 + γop
iΩ2+ γop
C1
C2
L
L
Z0(Ω)
R
iω0 + γ2
Ground state
m
C
+q
d
k
E(t)
R
m
Ω1 ~ (LC1)-1/2
)-1/2
Ω2 ~ (LC2
γop ~ R/L
Ω2 - Ω1 = ω0
-q
Mechanical oscillation
ω0 ~ (2k/m)1/2
γ2 ~ mechanical damping
Figure 2.27: The ground-state CPT excitation from the three correlated quantum levels can be
approximately analogous to an electronic-mechanical system. The CPT excitation is equivalent
to the mechanical oscillation of the two metallic plates on the right diagram.
(proportional to the buffer-gas pressure). The interaction between the optical coherence and
the pumping field produces the effective ground-state pumping rate. The time-dependent
pumping rate Γop (t) is proportional to the multiplication of the pumping field and the
optical coherence. Detailed calculations can be found in Appendix A. Figure 2.26 explains
why there is poor CPT excitation at high gas pressure. When the buffer-gas pressure is
higher, the filter bandwidth is broader. The optical coherence has a waveform pattern closer
to the pumping field. Since the pumping field is phase modulated, its amplitude is constant.
Therefore, the pumping rate has weaker AC component at high gas pressure. When the gas
pressure is lower, it produces more amplitude modulation on ρge , and therefore it generates
a higher AC pumping rate. If the gas pressure is further getting lower and the bandwidth
of the resonant filter is narrower than the separation of the optical sidebands, ρge becomes
constant amplitude with a single oscillation frequency. The interaction between E(t) and
ρge can generate the maximum AC pumping rate.
An electronic-mechanical analogy of
the CPT phenomena is sketched in Fig. 2.27. The modulated optical field to the atoms is
equivalent to the modulated electric field E(t) to the system on the right of Fig. 2.27. The
two LCR circuits define the peak response frequencies, Ω1 and Ω2 , which mimic the two
64
optical transition frequencies. The field-induced charge current can flow through the LCR
circuits. The electric field E(t) induces a charge difference, 2q, between two metallic plates.
Assuming the capacitance, C, of the two plates are large enough, so |Ω̄Z0 C| À 1, where
Ω̄ = (Ω1 + Ω2 )/2, and Z0 = Z0 (Ω) is the total impedance of the two LCR circuits. The
R
iΩt dΩ, which interacts with
induced charge q on the one of the plates is q(t) = i1Ω̄ Ẽ(Ω)d
Z0 (Ω) e
the electric field, and then produces force F (t) = q(t)E(t) to excite the mechanical motion.
These two plates have mass m respectively, and they are connected by a spring and have
p
the resonance frequency ω0 = 2k/m = Ω2 − Ω1 . When E(t) is modulated at frequency
ω0 /n, where n is an integer, the oscillation of the plates can be excited. One can see that
the resistance R, which serves electronic damping, is equivalent to the buffer-gas pressure.
If E(t) has only two frequency components Ω1 and Ω2 , a strong mechanical oscillation
of the two plates can be produced. If E(t) is frequency-modulated and R/L À ω0 , the
mechanical oscillation of the two plates is hardly to be induced. This is equivalent to the
pressure-dependent CPT excitation.
For amplitude modulated pumping light, the amplitude modulation is always present
on the optical pumping rate. Hence, there is no high pressure degradation of CPT signals
from AM pumping lights. Figure 2.28 sketches the CPT excitation efficiency by using
different pumping schemes at different buffer gas pressures. For simplifying the problem, we
consider zero optical-detuned carrier frequency. To excite the CPT resonance, the multiple
of the modulation frequency should match the resonance frequency. When the frequency is
matched, the time-dependent pumping rate can be therefore described by
(0)
Re(Γop (t)) = Re(Γop ) +
´
X³
(n)
n
Re(Γopm )ei m ωµν t + c.c. ,
(2.69)
n
where n and m = ωµν /ωm are integers. Here, ωm is the modulation frequency. Below a few
atmospheres of the buffer gas pressure, the repopulation pumping has negligible influence
to the ground-state coherence. At high very pumping rate, the ground-state decay rate
(0)
γµν = Re(Γop ). From Eq. (2.32), the CPT coherence is
(1)
Kµν Re(Γop )
|ρµν | =
(ρµµ + ρνν ),
2 Re(Γ(0)
op )
(2.70)
(1)
(0)
Excitation efficiency ( Re(Γop )/ Re(Γop ) )
65
1.2
1.0
0.8
0.6
Ideal δ pulse
Two-wave pumping (sinusoidal AM)
PM (modulation index m = 2.45)
0.4
0.2
0.0
0
1
2
3
Number density of buffer gas (amagat)
4
5
Figure 2.28: Calculated excitation efficiency of Cs 0-0 CPT resonance in the nitrogen buffer
gas. Here, the modulation index m is defined by E(t) = E0 exp(−iΩ0 t + im sin(ωm t)), where
Ω0 is the carrier frequency.
where Kµν is a positive constant, which only depends on the angle between probing beam
and the magnetic field and the absolute value of the photon spin |s|. It can be proved the
(1)
(0)
maximum of Re(Γop )/ Re(Γop ) ratio is equal to one. For 0-0 coherence, if |s| = 1, and the
probing beam is along the magnetic field, we find that Kµν = 1. So the maximum |ρµν |
(1)
(0)
is 0.5. We define the excitation amplitude as Re(Γop )/ Re(Γop ). In Fig. 2.28, one can see
that the excitation efficiency of PM or FM pumping decays quickly with increase of the
buffer gas pressure, but for AM pumping, it only decays initially and then approaches to
a constant. If the pumping light is modulated as a δ-pulse train, the excitation efficiency
is not affected by the buffer gas pressure. For PM-FM pumping, smaller modulation index
leads to a quicker degradation of the excitation efficiency by increasing gas pressure.
2.4.6
Line Shape of CPT Resonance
In the CPT spectroscopy, we usually find the resonance signals with asymmetric line shapes,
but we seldom see such phenomena from microwave resonance signals. This asymmetric line
shape is more pronounced in end-resonance CPT signals when the light shift is present. A
66
simple explanation for this phenomena is that the light shift mimics a magnetic field, which
is similar to the microwave field used for microwave spectroscopy, and therefore introduces
an extra component of the coherence, which has 90 degree phase difference to the CPT
coherence when there is no light shift. So, the total coherence has a phase difference to the
oscillating probing light when the light shift is present. The line shape therefore changes
from a pure Lorentzian curve to a combination of a Lorentzian and dispersion curve, and
becomes asymmetric. For better understanding, we use a spin precession picture to explain
the result. For 0-0 CPT, we can use a pseudo-spin system to understand the asymmetric
phenomena. For end-resonance CPT, the pseudo-spin picture is very nearly the same as a
real spin picture.
Considering that there is a coherence generated between state |µi and state |νi due to
optical pumping of modulated light, we find the corresponding rate equations regarding
these two sublevels to be
ρ̇µµ =
1
γ1 (ρνν − ρµµ ) − iRv /2(ρ̃νµ − ρ̃µν ) − RΓ /2(ρ̃νµ + ρ̃µν ) + Υµ
2
1
γ1 (ρµµ − ρνν ) − iRv /2(ρ̃µν − ρ̃νµ ) − RΓ /2(ρ̃µν + ρ̃νµ ) + Υν
2
ρ̇νν
=
ρ̇µν
= (−iωµν − γ2 )ρµν + iRv /2(ρµµ − ρνν )e−iωt − RΓ /2(ρµµ + ρνν )e−iωt
ρ̇νµ = (iωµν − γ2 )ρνµ − iRv /2(ρµµ − ρνν )eiωt − RΓ /2(ρµµ + ρνν )eiωt ,
(2.71)
where γ1 is the longitudinal relaxation rate, γ2 is the transverse relaxation rate, and Υσ
is the equivalent pumping rate to state σ. Here, we define two numbers: the coherence
(1)
pumping rate RΓ = hµ|δΓ |νi, and the Rabi frequency of the pseudo-magnetic field Rv =
(1)
2hµ|δEv |νi/h̄. In a common two-level system coupled to an effective field, we can use
Eq. (2.71) with substitutions of RΓ = 0 and Rv = ωR . For convenience, we can map these
two levels into a pseudo-spin representation. Therefore
1
i
1
Ux = (ρ̃µν + ρ̃νµ ), Uy = (ρ̃µν − ρ̃νµ ), Uz = (ρµµ − ρνν ),
2
2
2
(2.72)
where Ux , Uy , and Uz are the pseudo-spin expectation values in the rotation frame at
frequency ω. The coherence ρµν is defined by ρµν = ρ̃µν exp(−iωt), and we know that
"'(%$)&')$%"!
!
"$#&%"'()#$%
67
Figure 2.29: The left and the middle panel show the probing lights along two orthogonal
directions, parallel or perpendicular to the spin at zero detuning δω = 0. The signal amplitudes
plotted as functions of δω. The in phase signal shows a symmetric curve, and the out phase
signal shows a completely asymmetric curve. By using a modulated pumping light, coherence
can be excited as a spin along the x-direction by the coherence optical pumping rate RΓ . If
there is a light shift (δEv 6= 0), an effective field Rv is generated. This emulated field produces
a spin component along the y-direction. Therefore the probing light both probes in phase and
out phase signals. The detuning curve becomes asymmetric.
ρµν = ρ∗νµ . If we only consider the coherence generated through the depopulation pumping,
we can find the Bloch equation for this two-level system from Eq. (2.71).
U̇x = −δωUy − γ2 Ux − RΓ N /2
U̇y = δωUx − γ2 Uy − Rv Uz
U̇z = γ1 (Uz0 − Uz ) + Rv Uy ,
(2.73)
and its vector form
U̇ = (−RΓ N /2 − γ2 Ux )x − γ2 Uy y + γ1 (Uz0 − Uz )z + R × U,
(2.74)
where R = Rv x + δωz is the vectored angular frequency in the rotating coordinate system.
Here, Uz0 = (Υµ − Υν )/(2γ1 ) is the equilibrium value of Uz when Rv = 0, N = ρµµ + ρνν is
the total population of two states, and δω = (ωµν −ω) is the frequency detuning. We have to
68
notice that the splitting frequency ωµν = ω0 +∆ωv is shifted by the light shift ∆ωv , and both
γ1 and γ2 linearly depend on the DC optical pumping rate if we ignore the spin-exchange
mechanism. The CPT signal can be thought as an optically probed signal of a pseudo-spin in
a rotation frame. Figure 2.29 shows the probing signal as a function of frequency detuning.
A symmetric line shape, in phase signal, is found when the probing light is parallel to the
spin at zero detuning. A pure dispersive Lorentzian line shape, quadrature signal, is found
if the probing light is perpendicular to the spin at zero detuning. A modulated light is
equivalent to a fixed-direction pumping or probing beam in the rotation frame. As shown
in Fig. 2.29, the pumping beam produces the atomic coherence as a pseudo-spin along the
x-direction. When the light shift is present, a mimic field Rv is generated along x-direction
and can rotate the spin from z-direction to y-direction, which is the same phenomenon of
spin-flip Raman scattering [23, 46]. Therefore, the spin projection on the x-y plane has
both x and y components. Since the probing beam is still along the x-direction, both in
phase and out phase signals can contribute to the line shape. It turns out that the line
shape becomes asymmetric. To generate the spin component along to y-direction from the
emulated field Rv requires the non-zero Uz originally. This means a population difference
of the two sublevels is needed to produce an asymmetric line shape. This explains why the
end-resonance CPT signal is more likely to be asymmetric. The end states have a larger
population difference when they are optically pumped.
The analytical result can be calculated by solving the steady light-absorption at different
detuning from Tr(δΓρ). Therefore, the absorption due to the oscillation part of the pumping
(1)
light is proportional to Tr(δΓ ρ) = RΓ Ux , where the steady solution
Ux = −
RΓ N /2(γ2 + Rv2 /γ1 ) − δωRv Uz0
.
δω 2 + γ22 + γ2 /γ1 Rv2
(2.75)
For an optically thin cell, the transmission signal is proportional to −ARΓ Ux + B, where A
and B are two constants. A transmission peak can be observed when the frequency detuning
δω ∼ 0. We can see that when Rv 6= 0 and Uz0 6= 0, the line shape becomes asymmetric.
For two-wave pumping, Rv /RΓ = −∆/γop , where ∆ = Ω0 − Ω is the optical detuning of
the carrier, and γop /π is the optical linewidth. Here, Ω0 is transition frequency from the
69
Pressure=0.1atm, Modulation frequency=3.4GHz,
Numerical
Two-wave model
Asymmetry factor (δA)
3
Excited state
2
1
γop=0.95GHz.
∆
Ω0
0
-1
Ground state
-2
-3
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Carrier frequency offset to Ω0/2π (GHz)
Figure 2.30: Numerical calculation of the asymmetry of the resonance of
87
Rb at 0.1 atm as
a function of Ω − Ω0 . The subplot shows the intensity spectrum of the optical sidebands of
the modulated light and the curve of the absorption cross section. The ground state hyperfine
splitting is resolved. When the optical detuning is small, only two strong optical sidebands are
important. Hence, it can be approximated by two-wave pumping at low optical detuning.
middle of the two ground-state sublevels to the excited state, and Ω is the optical carrier
frequency of the pumping light. When γ2 γ1 > Rv2 , the line shape can be described by
γ − δA (ωµν − ω)
,
(ωµν − ω)2 + γ 2
p
2R U
where the asymmetric factor δA = RvΓ Nz0 , and γ = γ22 + γ2 /γ1 Rv2 .
CP T resonance signal ∝
(2.76)
Since δEv = h̄2 (1−2s·S)Im(Γop ), and δΓ = (1−2s·S)Re(Γop ), we find that the light shift
generates as a real magnetic field along the direction of photon spin s, and the depopulation
operator δΓ pumps the electron spin along s. For the end-resonance CPT, the coherence is
equivalent to the spin precession on the x-y plane in the real space. The pseudo spin can
be mapped to the real spin.
In the real experiment, a modulated light can produce not only two optical sidebands.
Therefore the asymmetric factor δA is not proportional to −∆/γop but is a polynomial
function δA = δA (∆/γop ), because the high order optical sidebands can also affect the
CPT excitation. Here, δA (∆/γop ) depends on the pattern of the optical sidebands and the
70
Figure 2.31: Experimental data shows the line shapes of the end CPT resonances as a function
of the laser frequency from a 87 Rb cell with 0.96 atm N2 buffer gas. By using the numerical
model, we can extract the parameter of optical pressure broadening and the pressure shift. The
two wave model can not fit to data, because in the high pressure cell, atoms can see not only
two optical sidebands but also high order sidebands.
buffer gas pressure. Figure 2.30 shows an example of numerical calculation of the CPT
line shape as a function of the optical detuning, which assumes the unresolved excited-state
and 100 % population difference of the two ground-state sublevels.
Here, we choose the
same pumping spectrum with two strong peaks as we used for some of our experiments.
Because the ground-state hyperfine splitting is resolved, two-wave model agrees with the
numerical calculation very well at small optical detuning. However, it starts deviating when
∆ is larger. This is due to the contribution from high order optical sidebands. Figure 2.31
shows the real experimental data from the end-resonance CPT signals. Since the laser beam
was arranged as 45◦ to the magnetic field, there is no way to produce a 100% population
difference. Our numerical model was able to fit the data from the experiments. Two
parameters, optical pressure broadening and the pressure shift, can be extracted from the
fitting of data, which has a good agreement with a previous measurement [42] by other
71
Pressure=0.1atm, Modulation frequency=3.4GHz,
γop=0.95GHz.
Asymmetry factor ( δA )
1
Mach-Zehnder type AM modulator:
phase φ0 = π/2.7
modulation index φ1 = 3
0.5
0
-0.5
-1
-2
-1.5
-1
-0.5
0
0.5
1
Carrier frequency offset to Ω0/2π (GHz)
1.5
2
Figure 2.32: The calculation result shows a smaller dδA /d∆ around the point of Ω − Ω0 ∼ 0
by using a different pumping spectrum. The inset shows the modulation spectrum, which is
generated by a Mach-Zehnder type modulator. The amplitude of the pumping electric field is
proportional to sin(φ0 + φ1 sin ωm t).
means. The two wave model can not be used for high pressure cell, because atoms see more
than two optical sidebands at the same time.
As the results show, the CPT resonance signals is more sensitive to the light shift at
low gas pressure and is less affected by the light shift at higher gas pressure. However, by
changing the structure of the optical sidebands, we would be able to reduce the asymmetry
factor δA depending on the optical detuning ∆. Figure 2.32 shows a calculation result with a
very small dependence on ∆ around the point of Ω−Ω0 ∼ 0 compared to Fig. 2.30. However,
such pumping spectrum sacrifices CPT signal contrast, because the strong high order optical
sidebands are less absorbed by the vapor. For CPT atomic clocks, the stability of the laser
optical frequency is significant. The change of the resonant line shape and the light shift
can affect the determination of the resonant frequency. Therefore, more systematic studies
of the asymmetries of CPT line shapes are important, especially for the end-resonance CPT
clocks.
72
2.4.7
Photon Cost of Optical Pumping
In spin-polarized optical pumping, a photon transfers its angular momentum to an atom
through the depopulation pumping. The atom loses its spin polarization through the relaxation mechanisms in the excited state, deexcitation (or repopulation) process, and groundstate relaxation. When the optical pumping rate exceeds all the ground state relaxation
rates, only excited-state relaxation and deexcitation process are significant. In this subsection, we discuss how different excited-state evolutions and repopulation processes affect the
efficiency of optical pumping. We assume that the optical pumping rate is much larger than
all ground-state spin-relaxation rates. Therefore, the spin relaxation in the ground state
is virtually zero. We use “photon cost” to denote the average number of photons, which
need to be absorbed per atom, for completely polarizing an atom or pumping an atom to a
pure quantum state. Lower photon cost represents higher pumping efficiency. The qualitative results can be understood from simple physics principles. The detailed results require
numerical calculations.
Considering D1 pumping with circularly polarized light, if we ignore the hyperfine structure, the ground state and the excited state have only two levels respectively, as shown
in Fig. 2.33.
Without hyperfine coupling, Fig. 2.33 shows that optical pumping with
quenching-dominated deexcitation has a better pumping efficiency than if there is only
spontaneous decay, because by absorbing a photon, there is 50% chance for polarizing an
atom with pure quenching, but only one-third probability with spontaneous decay.
With hyperfine coupling, the electron spin can always transfer its angular momentum
to the nuclear spin through the ground-state hyperfine coupling. Because the groundstate relaxation rates are always smaller than the ground state hyperfine frequencies, the
electron spin and the nuclear spin have plenty of time to precess about each other. For the
excited state, the relaxation rate can either be larger or smaller than the excited hyperfine
frequency. The relaxation rates for excited atoms depend on the pressures of buffer gases.
Therefore, the efficiency of angular momentum transfer through the excited state depends
on the buffer-gas pressure, which can influence the efficiency of optical pumping.
$
*
)
+
&'(
&
!
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!
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#
%
'
,
,
.
,
/
-
73
Figure 2.33: The left and middle diagrams show the σ+ pumping with different deexcitation
mechanisms. Quenching deexcitation causes higher pumping efficiency than spontaneous decay.
The right diagram shows the hyperfine sublevels with an assumption of nuclear spin I = 1/2.
The electron spin has chance to be transferred to the nuclear spin through the hyperfine coupling
in the excited state for the pumping transitions of region (1), but not of region (2). .
For better understanding, we can consider three special cases to discuss the static spinpolarized optical pumping for unpolarized atoms. (1) Quenching dominates the deexcita(e)
tion, and the quenching rate is much larger than excited hyperfine frequency, Γq À ωhf .
Hence, there is no way to transfer the angular momentum between the electron and the
nuclear spin in the excited state. When an atom absorbs one photon, there is a 50% probability that the photon spin can be transferred to the atom. The spin of a fully polarized
atom is I + 1/2. Therefore, on average, (I + 1/2)/0.5 = 2I + 1 = [I] photons are needed to
fully polarize an atom. (2) Spontaneous decay dominates the deexcitation, and the spon(e)
taneous decay rate is much larger than the excited hyperfine frequency, Γs À ωhf . Hence,
by using very similar arguments of previous one, because, in average, only 1/3 photon spin
can be transferred to an atom from each absorption, the photon cost is equal to 1.5[I].
However, we have to realize that this result can not happen for real cases, because usually
(e)
ωhf À Γs for real atoms. (3) J-damping dominates the excited-state spin relaxation, and
(e)
ωhf À Γjd À (Γq + Γs ). Hence, J-damping can even out the population distribution in the
excited state before the atom decays back to the ground state. In this case, there is an equal
probability for all ground-state hyperfine sublevels to be repopulated from the excited state.
Assuming that we use σ+ pumping light, to fully spin polarize atoms means to pumping
74
the population to the end state |a, ai. There is a total of g = 2[I] sublevels. Therefore, on
average, for each photon absorption, 1/g of ground-state population, excluding the state
|a, ai, is transferred to the state |a, ai. Initially, there (g − 1)/g population for each atom
needing to be pumped to the end state, so the photon cost is (g − 1)/g × (1/g)−1 = 2[I] − 1.
For real cases, atoms with pure spontaneous decay have lower photon cost than 1.5[I].
(e)
Since ωhf is always larger than Γs , electron spin transfers part of its angular momentum to
the nuclear spin before decay back to the ground state. The deexcitation can destroy the
electron spin as the way described in Fig. 2.33, but conserves the nuclear spin. Therefore, if
there is no J-damping relaxation in the excited state, the spin transfer in the excited state
can always increase optical pumping efficiency. From Fig. 2.34 to Fig. 2.37, we show the
calculation results of photon costs of 39 K, 85 Rb, 87 Rb, 133 Cs, as a function of quenching and
J-damping rates for optical pumping to the end state. Here, we use the relative quenching
rate, γq = Γq /Γs , and relative J-damping rate, γjd = Γjd /Γs to label the axes. The results
of photon costs are shown as a contour diagram. If we track along two axes, there is
a minimum along the γq axis, and there is a maximum along the γjd axis. These two
phenomena can be comprehended with following explanations. By increasing γq , more
electron spin polarization can be transferred back to the ground state. Therefore, it reduces
the photon cost. However, larger quenching rate means less duration in the excited state.
Hence, after a certain point, the higher quenching rate can not further help the optical
pumping due to the less transfer of angular momentum from electron to nuclear spin in the
(e)
excited state. Eventually, the photon cost increases to [I] when Γq À ωhf . This is the same
result of case (1) described previously. By increasing the γjd , population in the excited state
can spread to more sublevels. Therefore, it increases the photon cost. However, larger Jdamping rate means that the spin precession time between each J-damping collision becomes
shorter. Hence, less angular momentum is transferred between the electron and the nuclear
spin. For J-damping, it means that the population spreads more slowly. Eventually, when
(e)
Γjd À ωhf , the J-damping evolution becomes independent of nuclear spin. The photon cost
become [I], which is the same as extremely high quenching rate. The result of case (3)
75
described previously can not occur in the real cases, because usually the excited hyperfine
frequency is not high enough. The closest result of case (3) is where the maximum on the
γjd axis.
76
−1
K−39, Γs=3.85e+007 s . Excited hyperfine frequency=57.8 MHz
3
10
2
10
4
3.6
γq
1
3.8
10
3.2
0
10
3.4
4.6
4.2
4.8
4.4
5
−1
10
−1
10
0
10
1
10
γ
2
10
3
10
jd
Figure 2.34: Photon cost for pumping potassium-39 to the end state, which is plotted as a
function of relative quenching rate and J-damping by using a contour diagram. The difference
of values between each contour is 0.2. Darker color represents lower value, and lighter color
means higher value. The value of the contour is equal to the photon cost.
77
−1
Rb−85, Γs=3.57e+007 s . Excited hyperfine frequency=363 MHz
6
3
10
6
2
4.6
γq
4.4
4.2
1
5
5.8
10
6.2
4.8
10
7
7.4
6.8
7.2
6.4
8.2
8.8
7.8
8.6 8
9
8.4
9.2
6.2
6.6 6.4
0
10
6.2
6.8
6.4
7
6.6
7.2
7.4
7.6
6.86.6
−1
10
9.4
7
−1
10
0
10
1
10
γjd
2
10
3
10
Figure 2.35: Photon cost for pumping rubidium-85 to the end state, which is plotted as a
function of relative quenching rate and J-damping by using a contour diagram. The difference
of values between each contour is 0.2. Darker color represents lower value, and lighter color
means higher value. The value of the contour is equal to the photon cost.
78
Rb−87, Γ =3.57e+007 s−1. Excited hyperfine frequency=818 MHz
s
4
4
3
10
4
8
3.
3.6
3.4
2
10
4
3
3.2
4.2
4.2
4.4
4.6 4.4
5.4
5.2 5
4.84.6
5.8 5.6
6
5.45.25 4.8
6.2
5
5.6 5.2
5.8 5.4
6
4.8
1
γq
4.2
4.4
4.6
10
5
0
10
5.6
6.6
6.2
6.4
5.8
−1
10
6
−1
10
0
10
1
10
γ
2
10
3
10
jd
Figure 2.36: Photon cost for pumping rubidium-87 to the end state, which is plotted as a
function of relative quenching rate and J-damping by using a contour diagram. The difference
of values between each contour is 0.2. Darker color represents lower value, and lighter color
means higher value. The value of the contour is equal to the photon cost.
79
Cs−133, Γs=2.94e+007 s−1. Excited hyperfine frequency=1120 MHz
8
3
7.
10
8
8
7.
6
7.
2
8
6.
86
5.4
6.
4
5.8
5.6
7
7.4
8
1
γq
8
.6
10
6.2 6
2
10
9
0
10
8.2
8.2
8.2
8.4
8.6 8.4
8.8
8.4
9 8.6
9.29.6 9.4 9.28.8 8.6
9 8.8
9.8 10
9.4
10.2
9.6
10.4
9.8 9.2 9
10.6
9.4
10.8
10 9.6
11
10.2
9.8
10.4
11.2 10.6
10
10.2
11.4 10.810.4
11.811.6 11 10.6
11.210.8
12
12.2
11.411
12.4
11.6
11.8
11.2
12
7.8
11.4
12.6
12.2 11.6
11.8
12
12.4
−1
10
12.8
−1
10
0
10
1
10
γjd
2
10
3
10
Figure 2.37: Photon cost for pumping cesium-133 to the end state, which is plotted as a
function of relative quenching rate and J-damping by using a contour diagram. The difference
of values between each contour is 0.2. Darker color represents lower value, and lighter color
means higher value. The value of the contour is equal to the photon cost.
80
Figure 2.4.7 summarizes the photon costs for optical pumping to the end state and
push-pull optical pumping to 0-0 state. We calculated three different cases, spontaneous
(e)
(e)
(e)
decay (ωhf À Γs ), quenching (Γq À ωhf ), and J-damping (ωhf À Γjd À (Γq + Γs )), for
different nuclear spin numbers.
Photon cost of optical pumping to the end state
20
Number of photons
Excited-state relaxation dominated by:
15
Quenching
Spontaneous decay
J-damping
10
5
0
0
1
2
Nuclear spin ( I )
3
4
Photon cost of push-pull optical pumping to the 0-0 state
40
Number of photons
Excited-state relaxation dominated by:
30
Quenching
Spontaneous decay
J-damping
20
10
0
0
1
2
Nuclear spin ( I )
3
4
Figure 2.38: . Photon costs for two different optical pumpings and nuclear spins with different
relaxation mechanisms.
81
The numerical results are listed as following with the sequence of the nuclear spin
numbers I = 12 , 23 , 25 , and 72 :
• Photon cost for pumping to the end state:
– Spontaneous decay: 2.25 (exact), 3.52, 4.88, 6.26
– Quenching: 2I + 1 −→ 2, 4, 6, 8
– J-damping: 4I + 1 −→ 3, 7, 11, 15
• Photon cost for pumping to the 0-0 state:
– Spontaneous decay: 3.5 (exact), 7.85, 13.3, 19.7
– Quenching: 3 (exact), 9.67, 19.4, 32.1
– J-damping: 4I + 1 −→ 3, 7, 11, 15
We can find that excited-state relaxation dominated by J-damping produces maximum
photon costs for optical pumping to the end state, but minimum photon costs for the
push-pull optical pumping to the 0-0 state. However, we have to understand that the
(e)
condition, ωhf À Γjd À (Γq + Γs ), used for calculations is not completely correct. As our
previous discussions, the closest results are on the maximum of γjd axes of Fig. 2.34 to 2.37.
Therefore, for real cases with relaxation dominated by J-damping, the photon costs are less
than the result shown in Fig. 2.38 for end-state pumping, and are more than the result
shown in Fig. 2.38 for 0-0 state pumping. Another interesting result is that if we compare
the photon costs of the spontaneous decay with the quenching, we find that the spontaneous
decay has higher photon cost than quenching when I = 1/2. We can understand this result
through the right sub-diagram in Fig. 2.33. The optical pumping in the region (1) has
higher efficiency, because of the transfer of angular momentum between the electron and
the nuclear spin in the excited state. But there is no such advantage in the region (2), which
means the spontaneous decay has poor pumping efficiency than the quenching in the region
(2). When I > 1/2, region (1) has more population, the total photon cost is dominated
by region (1). Therefore, there is less photon cost for spontaneous decay. When I = 1/2,
82
the region (2) has more population then region (1), the higher pumping efficiency in the
region (1) can not compensate the lower pumping efficiency in the region (2). Therefore,
the spontaneous decay has higher photon cost than the pure quenching.
Different buffer gases can affect optical pumping differently, such as N2 , which is a good
quenching gas; He and Ar, which are very bad for quenching but can introduce large Jdamping rate. According to the study of N. D. Bhaskar [10] in 1979, optical pumping can
generate a spin-polarized wave front inside the vapor cell. The propagation speed of this
wave front is inversely proportional to the photon cost. Therefore, it is possible to measure
the quenching and J-damping rate of different buffer gases for P1/2 state by measuring the
speed of wave front from a series of vapor cells with different buffer gas mixtures. We can
use those experimental data to fit with contour calculations shown in Fig. 2.34– 2.37 to
extract quenching and J-damping rates.
83
2.5
New Simple-Compact Frequency Standard
Besides the applications of push-pull optical pumping mentioned in section 2.4.4, one of the
most promising uses of push-pull optical pumping is to build a new small-size frequency
reference, which can have a precision comparable to the conventional atomic clocks but
without their complexity.
Comparing different types of frequency references, quartz-crystal oscillators are easily
miniaturized, but they have poor long-term (days to years) stability; atomic clocks have
excellent long-term stability, but they are hard to miniaturize. Today, the best quartz oscillator demonstrates a short-term (seconds to hours) frequency uncertainty as low as 10−14 ,
but the long-term uncertainty is only ∼ 10−9 ; the very best atomic clocks can provide both
short-term and long-term frequency uncertainty down to 10−17 . The fundamental frequency
of a quartz oscillator is determined by the mechanical property of the quartz crystal. The
“aging” problem of the crystal hinders its frequency stability from extending to a longer
period. The fundamental frequencies of atomic clocks are determined by the fundamental
atomic constants and by small perturbations of environmental parameters. Therefore, the
atomic clock is much more stable. Because of the fast development of digital technology,
such as high-speed digital communications, GPS system, etc., a precise, long-term, smallsize, time base becomes more and more important for every digital system. For example,
the basic requirement of the time base for a GPS system is to have a daily uncertainty as
low as 10−11 . The crystal oscillators are not good enough for those applications. Hence,
miniature atomic clocks become the only choice, and several high technology companies
develop such clocks.
So far, all types of atomic clocks output the time-base signals by passively locking to the
hyperfine frequencies of the alkali-metal atoms (usually
87 Rb
and
133 Cs).
They all require
frequency-tunable local oscillators with frequency-locking loops. The microwave circuits
increase the complexity of the atomic clocks and consume a large fraction of the operating
power. To overcome those difficulties, we propose a new idea to build a simple-compact
frequency standard, which is expected to greatly reduce the system complexity and the
84
(1)
Edge emitting
laser diode
(2)
Figure 2.39: Two different cavity configurations of hyperfine-modulated lasers for two different
types of laser diodes. The regular edge emitting laser diode outputs linearly polarized laser
light, and therefore it needs two λ/4 wave plates to produce alternating circular polarization
inside the cavity. The VCSEL diode is designed to have almost no preference of the light
polarization, so the light polarization can be directly modulated by the vapor cell. The steady
lasing point of the two lasers is when the modulation period of the output light is equal to
2π/ω00 .
power consumption. Our scheme combines the idea of push-pull optical pumping and the
technology of semiconductor lasers, which is shown in Fig.2.39. A laser is a positive feedback
amplifier of photons. An alkali-vapor cell inside the laser cavity plays like a photonic filter
and converter to generate a special lasing mode, which produces the light modulation.
Generally, a laser tends to lase in an optical mode, which has the maximum gain or the
minimum loss of photons from their round-trip inside the cavity. Without the vapor cell,
the lasing spectrum is determined by the characteristics of the laser cavity and the gain
profile. With a vapor cell inside the cavity, a steady lasing point is met while the lasing
spectrum produces the maximum efficiency of push-pull optical pumping, which makes the
vapor cell become the most transparent. At this point, the output laser light is modulated
85
Spectral response (not to scale)
External cavity
Gain medium in laser diode
Bragg mirror
Growth of
optical comb
Photons scatter between
each peak
Non-polarized
alkali vapor
ν00
Optical frequency
Figure 2.40: An illustration of the spectral responses due different causes. The optical comb
in the lasing spectrum is grown to produce the maximum transparency of the vapor cell, and
therefore obtain the maximum gain of photons.
at 0-0 hyperfine frequency ω00 , and its optical spectrum is like an optical comb as shown
in Fig. 2.40.
We have to notice that the vapor cell inside the cavity acts not only as a
filter of the photon spin and the frequency of photons, but also the frequency converter of
photons. When coherent population trapping is present, a scattered photon from an atom
can change its frequency from original ν to ν ± ν00 . Eventually, an optical comb (many
peaks separated by ν00 ) is established in the lasing spectrum. Conventional mode-locked
lasers can also generate comb spectrum. However, the spacing of the comb peaks is usually
determined by the free spectral range of the cavity.
In Fig. 2.39, there are two different types of cavity configuration. For edge emitting
diode, the output light is linearly polarized. In order to generate the condition of pushpull pumping, two quarter-wave plates placed on the two sides of the vapor cell inside the
cavity is needed. The round-trip time of the photon inside the cavity should be about
the multiple of 2π/ω00 . The vapor cell should be placed around the middle inside the
cavity. It is easy to see that the configuration shown in (1) of Fig. 2.39 is the same idea of
producing alternating circular polarization by using Michelson type interferometer as shown
86
1.0
D1 push-pull pumping CPT
D2 push-pull pumping CPT
0-0 CPT efficiency
0.8
0.6
0.4
0.2
0.0
0
1
2
3
4
Nuclear spin (I)
Figure 2.41: The normalized amplitudes of the maximum 0-0 coherence with different nuclear
spins by using different fast push-pull pumping. D1 push-pull pumping can excited a much
stronger CPT coherence than D2, and also make the vapor cell more transparent.
in Fig. 2.20. The only difference is that we employ the counter propagation of photons to
obtain the alternating polarization inside the cell. Because we do not want the cavity mode
to dominate the lasing spectrum, the Q value of the cavity should not be too high. In
this case, the length of the cavity and the position of the cell do not need to be exact,
and can have more tolerance. A simpler version of the hyperfine modulated laser can be
found in (2) of Fig. 2.39. The VCSEL diode can be designed to be no preference of output
light-polarization. The polarization of photons inside the cavity is controlled by the vapor
cell. A previous study of polarization modulation of VCSEL can be found in Ref.[40, 35].
In practice, the size of the cavity can be scaled down by using some transparent material
with large dielectric constant.
In order to prevent the diode from lasing at the wavelengths outside the absorption
line of the vapor, a bragg mirror is placed into the cavity to guarantee the correct lasing
wavelength and narrow the lasing linewidth. The vapor cell should have a high buffer-gas
pressure to ensure a broad enough optical absorption line. One can see that the push-pull
87
optical pumping is necessary for cavity modulation with a vapor cell. Conventional CPT
uses fixed circular polarization, which will bleach out the vapor inside cell and produce a
steady circular polarization without any modulation. In addition, it is better to use a laser
diode with an output wavelength equal to that of the alkali D1-line. As shown in Fig. 2.41,
D1 push-pull CPT has much better efficiency than D2 push-pull CPT.
This new idea has not been tested experimentally. However, many studies about the
external cavity modulation of the laser diode [57, 40, 35] and the demonstration of an
optoelectronic vapor cell oscillator [43] strongly support our idea. We believe this new type
of frequency standard can bring lots benefit to the industry of electronics and optics.
Chapter 3
High-Field 129Xe-Alkali-Metal Spin
Exchange
Spin-exchange optical pumping has been widely used in producing highly polarized 3 He
and
129 Xe.
The basic idea is to use alkali-metal vapor as an intermediary for transferring
the angular momentum of photon to the nuclei of noble gases. In practice, the alkali-metal
atoms are first polarized by scattering circularly polarized light. Through spin-exchange
collisions between noble-gas atoms and alkali-metal atoms, the nuclear spins of the noble-gas
atoms are eventually polarized to their maximum angular momentums.
Comparing to 3 He,
129 Xe
has some advantages for scientific applications, such as much
larger abundance on Earth, stronger interaction with chemical and biological compounds,
larger chemical shifts of the nuclear resonances, and a much higher liquefaction temperature.
Those features make hyperpolarized
129 Xe
a powerful probe in many experiments, and it
has played an important role in many areas of scientific research [41, 34] as well as in
biomedical applications [2, 44]. A better understanding of spin exchange between
and alkali-metal atoms can help us produce the hyperpolarized
129 Xe
129 Xe
more efficiently and
successfully use it in more scientific studies.
For 129 Xe and alkali-metal atoms, there are two dominant mechanisms of spin exchange.
One is binary collisions and the other is van der Waals molecules. For binary collisions, the
88
89
spin exchange occurs when two atoms collide with each other; the interaction time is about
a few picoseconds. There is a weakly attractive van der Waals force between alkali-metal
atoms and
129 Xe.
The binding energy is around 10 meV. At room temperature, the van
der Waals molecules can be generated through the process of three-body collisions. The
third particle, which can be another nearby atom or molecule, can carry away the extra
energy of the two colliding atoms by bumping with the colliding
129 Xe
and alkali-metal
atom. Two atoms are therefore trapped into their shallow potential wall. The duration
of the spin interaction in van der Waals molecules can be up to a few nanoseconds. This
short-lived molecule breaks into two atoms by colliding with another atom or molecule.
Since the formation rate and the spin interaction time of van der Waals molecules are
determined by the third particle collisions, the spin exchange contributed from the van
der Waals molecules has a complicated dependence on the xenon or buffer gas pressures.
People usually used to measure the pressure-dependent spin-exchange rate to extrapolate
the binary spin-exchange rate. But recent evidences show that the behavior of the spin
exchange from van der Waals molecules at high gas pressure is still poorly understood. We
believe previous results of binary spin-exchange are incorrect. Because of the longer spin
interaction time of van der Waals molecules, the spin exchange of van der Waals molecules
can be easily suppressed to negligible values at high magnetic field if the inverse of the
electron Larmor frequency is much shorter than the interaction duration. Therefore, the
high-field spin-exchange measurements can give spin-exchange rates that are dominated by
binary collisions with negligible contribution from van der Waals molecules.
3.1
3.1.1
129
Xe Spin Relaxation Rates Equation
Detailed Balancing
For binary collisions, spin relaxation is caused by random spin-flipping collisions. When a
magnetic field is applied to a spin system, a spin-flipping collision can cause a change of
the internal atomic energy and therefore cause a difference between incoming and outgoing
90
kinetic energies. According to the principles of quantum scattering, the collision cross
section σ can be described by:
σ21 (v1 ) =
v1 |f (v1 , v2 )|2
v2 |f (v1 , v2 )|2
and σ12 (v2 ) =
.
v1
v2
(3.1)
Here σ21 (v1 ) is the total cross section for scattering from an initial spin state 1 to a final spin
state 2 for which the relative speed of the atoms before the collision is v1 and the relative
speed after the collision is v2 . The total scattering amplitude is f (v1 , v2 ) = f (v2 , v1 ). Hence
we conclude the condition of its detailed balance:
σ21 (v1 )v12 = σ12 (v2 )v22 .
(3.2)
The spin-exchange rate is proportional to hσvi, the ensemble average of σv. Assuming
that the spin flipping energy in the magnetic field is equal to ∆E, due to the conservation
of the total energy, we know that ∆E is equal to the difference of kinetic energies. That is
1
1
∆E = ∆ES + ∆EK = gS µB B + 2µK B = M v22 − M v12 ,
2
2
(3.3)
where ∆ES is electron spin-flipping energy of the alkali-metal atom, ∆EK is nuclear spinflipping energy of
129 Xe,
gS is the g-factor of electron, µK is the nuclear magnetic moment,
B is the strength of magnetic field, and M is the reduced mass of a colliding pair. Assuming
the system is in a thermal equilibrium at temperature T , We define the spin-exchange rate
coefficients κ21 and κ12 by
Z
κ21 = hσ21 v1 i =
dv1 P (v1 )v1 σ21 (v1 )
µ
¶
µ
¶
Z
Z
M v12
M v22 ∆E
3
= C dv1 exp −
v1 σ21 (v1 ) = C dv2 exp −
+
v23 σ12 (v2 )
2kT
2kT
kT
Z
= e∆E/kT dv2 P (v2 )v2 σ12 (v2 ) = hσ12 v2 ie∆E/kT = κ12 ,
(3.4)
where
µ
¶
M v2
P (v) = Cv exp −
,
2kT
2
r µ ¶3/2
2 M
C=
.
π kT
(3.5)
Here k is the Boltzmann constant, and P (v) is the velocity probability function. Equation (3.4) shows the result of detailed balance in thermal equilibrium, and there is a factor
91
e∆E/kT between hσ21 v1 i and hσ12 v2 i. Here we define the geometric mean of the spinexchange rate coefficient
√
κ21 κ12 ≡ hσvi,
(3.6)
κ21 = κe∆E/2kT and κ12 = κe−∆E/2kT .
(3.7)
κ=
where
When ∆E > 0, we have “exothermic” rate coefficient κ21 and “endothermic” rate coefficient
κ12 based on the result of Eq. (3.4). Similarly, we can find the spin-relaxation rates for 129 Xe
and alkali-metal atom as following:
α21 = αe∆EK /2kT ,
α12 = αe−∆EK /2kT ,
γ21 = γe∆ES /2kT ,
γ12 = γe−∆ES /2kT ,
where α and γ denote the spin-relaxation rates of
129 Xe
(3.8)
and alkali-metal atom respectively
due to other causes.
3.1.2
Rate Equations
To obtain the spin-relaxation rate equations, we set the number density of alkali-metal
atoms [A] = [A+ ] + [A− ] and
[A± ] = [A](1/2 ± hSz i).
Similarly, for the number density of
(3.9)
129 Xe,
[Xe± ] = [129 Xe](1/2 ± hKz i).
(3.10)
Here we use “+” and “−” to label the spin up and spin down states. The expectation values
of electron and nuclear spins are hSz i and hKz i respectively. Thus,
d
1
1
[A+ ] = −κ21 [Xe− ][A+ ] + κ12 [Xe+ ][A− ] − γ21 [A+ ] + γ12 [A− ],
dt
2
2
d
1
1
[Xe+ ] = κ21 [Xe− ][A+ ] − κ12 [Xe+ ][A− ] + α21 [Xe− ] − α12 [Xe+ ].
dt
2
2
(3.11)
92
Substituting Eq. (3.9) and (3.10) into Eq. (3.11), we find
d
[129 Xe] d
hSz i = γ cosh(∆ES /2kT )(hSz iT − hSz i) −
hKz i,
dt
[A] dt
(3.12)
and also
d
hKz i = α cosh(∆EK /2kT )(hKz iT − hKz i) +
dt
[A]{κ21 (1/2 − hKz i)(1/2 + hSz i) − κ12 (1/2 + hKz i)(1/2 − hSz i)}, (3.13)
where the thermal equilibrium values
hKz iT =
1
1
tanh(∆EK /2kT ) and hSz iT = tanh(∆ES /2kT ).
2
2
(3.14)
Equation (3.12) and (3.13) are nonlinear and their steady solutions are hKz i = hKz iT and
hSz i = hSz iT . At a representive experimental temperature T = 400 K and a magnetic field
B = 9.4 T, we find the Xe nuclear thermal polarization |2hKz iT | ∼ 6.6 ppm and for alkalimetal electron, |2hSz iT | ∼ 1.6%. Typically, γ/[Xe] À κ, which implies that hSz i ∼
= hSz iT is
a very good approximation. Therefore we can simplify Eq.(3.13) into
1
d
hKz i = − (hKz i − hKz iT ) ,
dt
T1
(3.15)
where the longitudinal relaxation rate
cosh(∆EK /2kT )
1
= [A]κ
+ α cosh(∆EK /2kT ) ∼
= [A]κ + Γ0 .
T1
cosh(∆ES /2kT )
(3.16)
The ratio of the hyperbolic cosine functions in Eq. (3.16) gives a factor of 0.9995 in typical
conditions of our experiments.
An interesting result is that if we artificially make hSz i = 0 and assume α ∼ 0, we can
find the steady solution of Eq.(3.13) is
hKz i =
1 κ21 − κ12
' hSz iT .
2 κ21 + κ12
(3.17)
Here, the polarization is transferred from the electron spin to the xenon nuclear spin.
93
3.2
Magnetic Decoupling of 129 Xe-Alkali-metal Spin Exchange
The typical duration of binary collision is less than 10−11 seconds, so it is difficult to be
decoupled by magnetic field. However, when the field strength is larger than a few tesla,
the electron Larmor frequency becomes comparable to the inverse of the binary interaction
time. The binary spin-exchange can be suppressed. We calculate the spin-exchange rate
coefficient as a function of B field by using two different methods, the semi-classical approach
(SCA) and the distorted-wave Born approximation (DWBA) [25, 27].
3.2.1
Semi-Classical Approach (SCA)
The Hamiltonian, H, for the spin energy of a colliding pair can be written in the central
mass frame as
H = H (0) + H (1) ,
(3.18)
where
H (0) = gS µB Sz B − 2µK Kz B, and H (1) = α(r)K · S.
(3.19)
Here gS is the electron g-factor, µB is the Bohr magneton, µK is the nuclear magnetic
moment, and B is magnetic field along the z direction. α(r) denotes the coupling coefficient
for nuclear spin of
the spins for
129 Xe
129 Xe
and electron spin of the alkali-metal atom, K and S represent
and the alkali-metal atom respectively. The H (1) (r) is time dependent
because the internuclear separation r of the two colliding atoms depends on the time t.
For the spin state, |qi = |mK , mS i, there are four totally orthogonal combinations,
|1i = |mK = 1/2, mS = 1/2i,
|2i = |mK = −1/2, mS = 1/2i,
|3i = |mK = 1/2, mS = −1/2i,
|4i = |mK = −1/2, mS = −1/2i.
(3.20)
At magnetic field B and without the spin interaction term, H (1) , there are energies [²1 ²2 ²3 ²4 ]
for four spin states, where ²q = gS µB mS B − 2µK mK B. The spin states |1i and |4i are the
94
eigenstates of H (0) , and H (1) cause spin flipping between state |2i and |3i, since
1
K · S = Kz Sz + (K+ S− + K− S+ ).
2
(3.21)
In SCA, we use the laws of classical mechanics to calculate the trajectory of the two
colliding atoms. We ignore the small spin energy, and we express the relative velocity as a
function of the separation between two atoms.
s µ
¶
2
E0 b2
vr =
E0 − V (r) − 2
,
M
r
(3.22)
where vr = |dr/dt| is the relative radial speed, M is the reduced mass, E0 is the initial kinetic
energy, V (r) is the interaction potential, and b is impact parameter. Equation (3.22) follows
from the conservation of energy and angular momentum. From first-order time-dependent
perturbation theory, we find the transition probability between state |2i and |3i is
¯ Z ∞
¯2
¯1
¯
0
(1)
i∆ωt0 ¯
¯
W = W32 = W23 = ¯
dt h3|H |2ie
¯ ,
ih̄ −∞
(3.23)
where ∆ω = ±(²2 −²3 )/h̄ = ±(gS µB +2µK )B/h̄ = ±∆E/h̄ is the energy difference between
two states. The Fermi contact Hamiltonian, H (1) , is a function of r, and r is a function of
t. We can calculate the time is a function of the atomic separation, r, by using Eq. (3.22).
Z r
dr0
,
(3.24)
t(r) = ±
0
r0 vr (r )
where r0 is the shortest distance between a colliding pair. Therefore the probability W
becomes a function of initial conditions of the two colliding atoms and magnetic field. To
do the ensemble average of σv, we have to integrate all possible initial velocities and impact
parameters. Therefore the binary spin-exchange rate coefficient, κ, which represents the
spin-exchange rate for unit number density of the alkali-metal atoms is
Z Z
κ =
2πbdbdv vP (v)W (b, v)
¯ Z ∞
¯
Z ∞Z ∞
¯1
dr ¯¯2
¯
=
dbdv 2πbvP (v) ¯
α · cos(∆ωt) ¯ ,
ih̄ r0
vr
0
0
(3.25)
where α, t, and vr are functions of r, and P (v) is the Maxwell-Boltzmann distribution
p
appeared in Eq. (3.5). The integral variable v is equal to 2E0 /M . Since ∆ω is propor-
95
tional to the magnetic field, a stronger field makes the oscillation more rapid inside the
integral (3.25). Therefore, it decreases the value of κ and causes the magnetic decoupling.
The calculation in integral (3.25) ignores the small difference of incoming and outgoing
scattering velocities, which is due to the spin flipping at magnetic field. Consider the
detailed balancing, we can write the endothermic and exothermic rate coefficients as
κ32 = κ e∆E/2kT , κ23 = κ e−∆E/2kT .
3.2.2
(3.26)
Distorted-Wave Born Approximation (DWBA)
In DWBA method, we treat a colliding event completely quantum mechanically, so the
unperturbed Hamiltonian of Eq. (3.19) becomes
H (0) = −
h̄2 2
∇ + V (r) + gS µB Sz B − 2µK Kz B,
2M
(3.27)
where M is the reduced mass, V (r) denotes the atomic interaction potential between two
colliding atoms. Starting from the Eq. (3.18), we treat the Fermi contact term as a perturbation term. Hence we can find a Schrödinger equation
(H (0) − E)|Ψi = −λH (1) |Ψi,
(3.28)
where E is the total energy of the system and λ is the perturbation parameter. To solve
this equation, we can expand the wave function into a power series in λ.
|Ψi =
4
X
³
´
|qi ψq(0) + λψq(1) + λ2 ψq(2) + · · · ,
(3.29)
q=1
where |qi represents four spin states. Therefore we can rewrite Eq. (3.28) as
µ
¶
4
2M
2M X
2
2
(n)
∇ − 2 V (r) + kq ψq = 2
hq|H (1) |piψp(n−1) ,
h̄
h̄ p=1
(3.30)
where kq2 = 2M (E − ²q )/h̄2 . When n = 0, the righthand side of Eq. (3.30) is zero. For the
scattering process, we find the asymptotic form of the wave function,
|Ψi ∼ |pieikp ·r +
4
eikq r X
|qifqp (r̂),
r
q=1
(3.31)
96
where p and q denote the incoming and outgoing spin states, and fqp (r̂) is the scattering
amplitude, which is a function of the direction r̂ of the scattered wave. We can also expand
fqp into the power series in λ,
(0)
(1)
(2)
fqp = fqp
+ λfqp
+ λ2 fqp
+ ··· .
(0)
(3.32)
(1)
Here, fqp is the zeroth-order scattering amplitude without spin flipping, and fqp gives the
first-order scattering amplitude with spin-exchange. By comparing the Eq. (3.29), (3.31),
and (3.32), we find asymptotic forms as
eikq r (1)
eikq r (0)
fqq δqp , ψq(1) ∼
f .
r
r qp
ψq(0) ∼ eikq ·r +
(0)
To get the analytical solution of ψq
(3.33)
(1)
and ψq , we firstly rewrite the lefthand side of
Eq. (3.30) into the radial form. Then there are two equations should be solved.
1
D̂lq ulq = 0, D̂lq βlq = δ(r − r0 ),
r
(3.34)
where
D̂lq =
d2
l(l + 1) 2M
−
− 2 V (r) + kq2 ,
2
dr
r2
h̄
(3.35)
and where l is the quantum number of angular momentum. There are two linearly independent solutions for ulq in Eqs. (3.34). They are glq and hlq . When r = 0, glq ∝ rl+1
and hlq ∝ r−l . While r is outside the potential wall of V (r), they can be expressed into
asymptotic forms,
glq (r) ∼ sin(kq r − πl/2 + δlq ), and hlq (r) ∼ ei(kq r−πl/2+δlq ) .
(3.36)
(0)
The wave function ψq (r) can be represented by the products of glq (r)/r and angular wave
function Ylm (r̂).
ψq(0) (r) =
X
lm
alm
glq (r)
Ylm (r̂).
r
(3.37)
When r is large, Eq. (3.37) should reduce to the asymptotic form shown in Eqs. (3.33). By
comparing two equations, we can determine the coefficient alm , and therefore
ψq(0) =
4π X i(πl/2+δlq )
e
glq (r)Yl (r̂) · Yl (k̂q ),
kq r
l
(3.38)
97
where Yl (r̂) · Yl (k̂q ) =
P
m Ylm (r̂)
(1)
∗ (k̂ ). For ψ , the solution can be found by using a
· Ylm
q
q
method of Green’s function. The Green’s function is defined as
µ
¶
2M
∇2 − 2 V (r) + kq2 Gq (r, r0 ) = δ (3) (r − r0 ).
h̄
(3.39)
Therefore, from Eq. (3.30), we find
Z
ψq(1) (r) =
d3 r0 Gq (r, r0 )
4
2M X
hq|H (1) |piψp(0) (r0 ).
h̄2 p=1
(3.40)
The solution of Green’s function is
Gq (r, r0 ) =
X1
l
r
βlq (r, r0 )Yl (r̂) · Yl (r̂0 ),
(3.41)
where βlq (r, r0 ) is one of the solution in Eqs. (3.34), which can be constructed by glq and hlq .
In order to make Gq have asymptotic form as outgoing spherical wave, which is ∝ eikq r /r,
and also have a finite value when r = 0, we require



A(r0 )glq (r), r < r0
0
βlq (r, r ) =


B(r0 )hlq (r), r ≥ r0
.
(3.42)
At the joint point, r = r0 , we ask βlq to be continuous but discontinuous of its 1st order
derivative. Hence



0
1/r0

=
−glq
(r0 )
0 (r 0 )
−glq
hlq
(r0 )


h0lq (r0 )
A(r0 )
B(r0 )

.
(3.43)
The solution of Eq. (3.43) is


A(r0 )
B(r0 )

=

hlq (r0 )

1 
,
r0 W
0
glq (r )
(3.44)
0 (r 0 ). From Eqs. (3.34), we can verify dW/dr 0 = 0, hence
where W = glq (r0 )h0lq (r0 ) − hlq (r0 )glq
W is a constant. So we can use the asymptotic forms of glq and hlq to calculate W . Hence
W = ikq sin θeiθ − kq cos θeiθ = −kq ,
(3.45)
98
where θ = kq r − πl/2 + δlq . The solution of Green’s function is
Gq (r, r0 ) = −
X
l
1
hlq (r> )glq (r< )Yl (r̂) · Yl (r̂0 ),
kq rr0
(3.46)
where r> is the larger of r or r0 , and r< is the smaller. The asymptotic form for r À r0 is
Gq ∼ −
ekq r X i(−πl/2+δlq ) 1
glq (r0 )Yl (r̂) · Yl (r̂0 ).
e
r
kq r 0
(3.47)
l
By substituting the Eq. (3.47) and Eq. (3.38) into Eq. (3.40) and comparing the asymptotic
form in Eq. (3.33), we obtain first-order scattering amplitude of spin flip for initial state |pi
to final state |qi,
(1)
fqp
(r̂) =
X
ll0
−
2M
h̄2
Z
d3 r0 {Yl0 (r̂) · Yl0 (r̂0 )eiδl0 q
1
4π
gl0 q (r0 )hq|H 0 |pieiδlp
glp (r0 )Yl (r̂0 ) · Yl (k̂0p )}.
0
kq r
kp r 0
(3.48)
For spin exchange, [qp] can only be [32] or [23]. Hence the total cross section
Z
(1)
dΩ|f32 (r̂)|2
σ̄ =
(3.49)
¯ Z ∞
¯2
X π
¯1
¯
0
0
0 ¯
0 2M
¯
=
(2l + 1) ¯
α(r )gl3 (r )gl2 (r )¯ ,
dr
2
h̄ 0
k̄
h̄k̄
l
where k̄ =
√
√
(1)
(1)
k3 k2 and f32 = f23 . We define σ̄ = σ32 σ23 , where
σ32 =
k3
k2
σ̄ and σ23 = σ̄,
k2
k3
(3.50)
which satisfy the detailed balancing, σ32 k22 = σ23 k32 , appeared in Eq (3.2). Finally, we
obtain the binary spin-exchange rate coefficient
Z
κ=
dv vP (v)σ̄(v).
(3.51)
The value κ32 and κ23 can be evaluated by replacing σ̄ by σ32 and σ23 into Eq. (3.51)
correspondingly.
99
Connection Between SCA and DWBA
To connect the SCA and the DWBA, we first look at the angular momentum, L ⇒ M vb ⇔
p
h̄ l(l + 1), of the two colliding atoms. We can find the first connection as
Z
Z
db 2πb =
dl
X π
πh̄2
(2l
+
1)
←→
(2l + 1).
p2
k̄ 2
(3.52)
l
Secondly, by using Wentznel-Kramers-Brillouin (WKB) approximation, we can approximate
glq (r) as
s
glq (r) ≈
q
2M (E − ²q )/h̄2 , θlq (r) =
where kq =
kq
0
θlq (r)
Rr
rlq
sin (θlq (r)) ,
(3.53)
0 (r 0 ) + π/4 (note: the phase π/4 can be
dr0 θlq
obtained by using Airy function when r → rlq ), and
s
µ
¶
h̄2 l(l + 1)
2M
0
θlq (r) =
E − ²q − V (r) −
.
2M r2
h̄2
It is easy to find vlq (r) =
h̄ 0
M θlq (r)
(3.54)
from classical analogy. Hence, we find the second
connection as
Z
dr
where v̄r =
0 2M
h̄k̄
Z
0
0
0
α(r )gl3 (r )gl2 (r ) ←→
dr0
α(r0 )
[cos(θl3 − θl2 ) − cos(θl3 + θl2 )],
v̄(r0 )
(3.55)
√
v3 v2 . When E À ∆²q , we find the third connection, which is
θl3 − θl2
(²2 − ²3 )
d
θlq (r) =
≈ (²3 − ²2 )
d²q
h̄
Z
r
r0
dr0
M
0 (r 0 ) = ∆ω · t.
h̄θlq
(3.56)
By using three connections, (3.52), (3.55), and (3.56), and discarding the rapidly oscillating
part, cos(θl2 +θl3 ) in Eq. (3.55), we can show the SCA and DWBA are equivalent if the total
energy E is much larger than the spin-flipping energy, and the total integration distance
is much larger than de Broglie wavelength. Figure 3.1 shows a typical result of SCA and
DWBA.
100
v=300 m/s, l=0, B=30T
0.04
1
0.02
0
0
-1
glq(r)
(SCA)
-5
V(r) (eV), −α (r) ( 10 eV)
0.06
(DWBA)
-0.02
0.5
1
Radial distance (nm)
1.5
Figure 3.1: An example of 129 Xe-Rb using SCA and DWBA numerical calculations. The SCA
curve behaves like the mean value of the DWBA curve.
3.3
Experiments and Calculations
At high temperature, where alkali vapor density is high enough to make spin-exchange
dominate the Xe relaxation, the spin-exchange rate coefficient κ of
129 Xe
is equal to the
slope of longitudinal spin-relaxation rate 1/T1 as a function of the alkali-metal number
density [A], namely κ = d(1/T1 )/d[A], (Eq. (3.16)). By independently measuring 1/T1
and [A] under the same conditions, we can retrieve κ. 1/T1 was measured by using NMR
method, and the technique of Faraday rotation measurement was used to determine the
value of [A]. The detailed information about the experimental apparatus is summarized in
Fig. 3.2. A number of spherical sample cells, made of aluminosilicate glass, were prepared
for the experiments[25]. Each cell was about 1 inch diameter, and contained Rb or Cs metal
Z
\+e)]^_`abPQR!&"#'(IJ$%K
[Tcd
UL),WMNLN*
T
Y,
WV.PXQRfTUWNLI.S
I
V
+
.
O
*
+
BF5G
C8?0DHE:721A6@
C9/;0:1<2=3>4?56789@
:A0481
101
Figure 3.2: A block diagram shows the total setup of the
ments.
with different mixture of enriched
129 Xe
129
Xe spin-exchange rate measure-
(71%) and pure nitrogen gas. The sample cell was
set into a probe, shown in Fig. 3.3. The experiments were carrying out in 9.4-T field, which
was produced by an Oxford superconducting magnet.
For comparisons of the spin-exchange rates from experiments to theories, C and MAT-
LAB programs were used for the theoretical rate calculations. Since MATLAB is much
more convenient for solving the differential equation, we used C to do the SCA calculations
and MATLAB to do the DWBA calculations. With the same accuracy, the SCA calculation
performed in C code was about 100 times faster than DWBA in MATLAB code.
3.3.1
NMR Measurements
A homemade NMR spectrometer was built for measuring the longitudinal decay time T1
by detecting the nuclear spin signal of
129 Xe.
IGOR software installed in a PC computer
102
Laser
beam
UP
B
(Scaled)
Oxford 9.4T
Superconducting
Magnet
Figure 3.3: The NMR probe was designed to fit into a 9.4-T Oxford superconducting magnet
with bore diameter ∼ 3.5”. The internal space was heated resistively with non-inductive wiring.
The heat conducting copper tube suppressed the thermal gradient to about 1◦ C per inch. The
temperature was stabilized by the feedback of two thermal sensors. The interlayer between
inner and outer tubes was stuffed by a porous foam for good heat insulation. The sample
cell was held by the NMR coil and hung by two supports. The NMR coil was set to the
resonance frequency by adjusting a variable capacitor. The tuning work was done by using a
sweep generator (WAVETEK 1062). A probing beam for Faraday rotation measurement passed
through the two windows and the center of the cell. The field inhomogeneity inside the 1 inch
cell was about 0.3 ppm.
with a Windows 2000 operation system took care of device communications, experimental
control, and the data analysis. A complete NMR measurement cycle was accomplished
by several stages: 1. The probing pulse sequence was originally arranged by the IGOR
program and then sent to a pulse generator board (Pulse Blaster). 2. The output of the
pulse generator board passed to an analog switch (Mini-Circuit ZASWA-2-50DR), which
acted as a shutter to turn on or off the RF field. We used a frequency synthesizer (PTS310)
as the RF source. It had a frequency range from 0 to 310MHz with an uncertainty less
than 10 ppb. It was set to near 110.48MHz, which was about a few kHz offset from the
precession frequency of 129 Xe nuclear spin at 9.4T. 3. After the analog switch, the RF pulse
sequence was then amplified by 30–70dB (AMT 3304C with attenuators) and finally sent
103
Figure 3.4: Arbitrary unit for y-axes. The top panel shows two orthogonal FID signals. The
bottom panel is the Fourier’s spectrum of the FID signal. This example signal was the result
of eight times average FID from a 4 amagat Xe cell.
to the NMR resonant coil inside the probe, Fig. 3.3. The probe was temperature stabilized
by two OEMGA CNi-3252-C24 controllers. 4. In the part of signal detection, a π/2 pulse
was sent to flip the nuclear spin to the transverse plane to excite a free-induction decay
(FID) signal. A pre-amp (MITEQ 1467) with 60dB gain magnified the FID signal. 5. The
magnified signal was then split into two channels and mixed with two orthogonal sine waves
from the RF source. This down converted hundred megahertz FID signal to the kilohertz
range. 6. Two channels connected with a low-pass filter (KROHN-HITE 3940) to get rid of
the high frequency noise that would be down converted to the low frequency noise by the
sampling process. 7. Digitized FID signals were finally acquired by a digital oscilloscope
(YOKOGAWA DL708E) then transmitted back to a computer to do the Fourier’s spectrum
analysis by IGOR. The example FID signal is shown in Fig. 3.4.
We used saturation-recovery NMR technique to measure T1 . Initially, a saturating pulse
train was sent to destroy the thermal equilibrium nuclear polarization hKz iT = µK B/2kT
104
Figure 3.5: The amplitude of the FID signal (filled circles) is proportional to the 129 Xe nuclear
spin polarization hKz i as it recovers towards its thermal-equilibrium value hKz iT = µK B/2kT .
The relaxation time T1 is extracted from the fit of Eq. (3.57), solid curve. Each data point was
taken over 16 – 96 averages of the FID signal at the same delay time τ .
(here µK is the nuclear magnetic moment of
129 Xe,
k is the Boltzmann constant, and T
is the absolute temperature). After a delay time τ , a single 90◦ tipping pulse was used to
excite an FID signal from the recovering nuclear spin polarization hKz i, described by
hKz i = hKz iT (1 − e−τ /T1 ) .
(3.57)
To evaluate the amplitude of the FID signal, we Fourier transformed the signal traces from
time domain to frequency domain. The FID signal appeared as a resonant peak on the
spectrum. This spectrum provided the information of the amplitude, the linewidth, and
the frequency offset. Using the signal spectrum, we can verify the strength of the RF field,
the homogeneity of the magnetic field, and the buffer gas pressure, which caused a chemical
shift. By recording the amplitude of the FID signal at different delays τ , the relaxation
rate 1/T1 can be retrieved as a parameter of the exponential fit, Eq. (3.57). A typical T1
measurement is shown in Fig. 3.5.
105
3.3.2
Faraday Rotation Measurements
To avoid the uncertainty in alkali-metal vapor density, [A], associated with using the empirical vapor-pressure formulas [45, 33], we employed a Faraday rotation technique [25] to
determine [A] at different temperatures and buffer-gas pressures. A Ti-sapphire ring laser
(model “Coherent 899-29”), pumped by a Nd-Vanadate “Verdi” laser (Coherent), was used
to generate the probing beam with a capability of wavelength scanning. Two optics sets
were used for the ring laser. The short-wave optics were for scanning from 700 nm to 830
nm, which covers D1 and D2 of rubidium. The mid-wave optics were for scanning from
785nm to 910nm, which covers D1 and D2 of cesium. The probe beam was guided by two
mirrors to redirect the beam vertically from horizontal setup.
Detail of the measurement
In our experiment, we can describe a light beam in a pure polarization state with two
parameters: (i) the orientation angle θ of the semimajor axis of the elliptically polarized
light relative to the x axis in a reference plane normal to the beam direction, and (ii) the
mean photon spin s, a real number ranging from −1 to +1, which describes the relative
circular polarization of the light. The ratio a of the minor to the major axis of the ellipse
√
is related to the mean spin s by a = (1 − 1 − s2 )/s.
In our experiment, the semimajor axis of the elliptically polarized light entering the cell
was oriented at an angle θ0 , which we could vary for experimental convenience. The mean
spin s0 of the light entering the cell was close to zero. By using a linear polarizer (LP)
shown in Fig. 3.2, the probe light had nearly perfect linear polarization. Stray birefringence
in lenses and windows and reflections from two turning mirrors, not shown in Fig. 3.2, could
change both θ and s, but we verified experimentally that these were negligible effects. The
orientation angle of the light emerging from the cell is θ = θ0 + ∆θ, where the Faraday
rotation angle ∆θ can be several radians because of large [A] and B. In principle, the
circular dichroism of the vapor can induce a corresponding change, ∆s, in the mean photon
spin, but unless the laser wavelength is very close to the center of the absorption lines, ∆s
106
will be negligible and the light emerging from the vapor will remain nearly linearly polarized
with s ≈ 0.
To measure the Faraday rotation angle ∆θ, we set the stress axis of a photo-elastic modulator (PEM) (HINDS PEM-80) along the x axis, at 45◦ to the axes of a linearly-polarizing
beam-splitter cube (BS-LP) [20]. The angle ψ of the relative phase retardation between
the fast and slow axes of the PEM was varied sinusoidally with time t at the frequency
ωp /2π = 83.4 kHz. That is, ψ = ψp sin ωp t, where the peak phase shift ψp is proportional
to the driving voltage of the PEM and inversely proportional to the laser wavelength. After passing through the BS-LP, the probe beam was detected with a photodiode, which
produced the output voltage,
VPD = gIB · (1 +
p
1 − s2 sin 2θ cos ψ + s sin ψ),
(3.58)
where g is the effective gain of the optical detection system. The laser power IB = IB (t)
was mechanically chopped at the frequency Ω/2π = 153 Hz.
The laser wavelength λ was scanned with time, and this caused large variations in the
angle θ, and also some variation of IB because of the wavelength dependence of the laser
output power. There were also noise fluctuations of IB .
We can decompose the signal (3.58) into different frequency harmonics in the familiar
way for the frequency modulation, VPD = V0 + Vωp sin ωp t + V2ωp cos 2ωp t + · · · , where
amplitudes
p
1 − s2 J0 (ψp ) sin 2θ) ,
V0
= gIB · (1 +
Vωp
= 2gIB · sJ1 (ψp ) ,
p
= 2gIB · 1 − s2 J2 (ψp ) sin 2θ
V2ωp
(3.59)
(3.60)
(3.61)
are chopped at the relatively low frequency Ω. Here Jn are Bessel functions of the first kind.
We empirically adjusted the PEM driving voltage to find the smallest amplitude ψp = ψ0
(that is, ψ0 = 2.405) for which J0 (ψp ) = 0. Then V0 = gIB became independent of the
linear polarization angle θ, and could be used to normalize the signals Vωp and V2ωp . A
107
small correction was made to the data to account for the slight variation of ψp ∝ 1/λ over
the 1% range of the wavelengths used in this work.
As sketched in Fig. 3.2, the signal V2ωp was detected with two lock-in amplifiers (LIA).
The voltage VPD was applied to the first LIA, which was referenced to the PEM drive
frequency 2ωp with a 1 ms integration time. The output of the first LIA was applied to a
second LIA, referenced to the probe laser chopping frequency Ω with a 0.1 s integration time.
For further analysis, the output voltage of the second LIA is processed with one channel of
a digitizing interface (not shown in Fig. 3.2) to obtain a digitized signal S2ωp ∝ V2ωp . This
was stored, along with the laser wavelength, in a computer (PC). The signal V0 was detected
by applying VPD to a third LIA, referenced to the chopper frequency Ω and with a 0.1 s
integration time, and acquired as described above to obtain the digitized signal S0 ∝ V0 . A
digitized record of the probe laser wavelength λ was stored along with each pair of values,
S0 and S2ωp .
For further analysis of the digitized data, we computed the ratio R = R(λ) = S2ωp /S0 .
According to (3.59) and (3.61) the ratio should be
p
R = S2ωp /S0 = G 1 − s2 sin 2(θ0 + ∆θ),
(3.62)
where the constant G accounts for the differences in LIA gains in the system. In order to
acquire the Faraday rotation data sin2θ, we first measured the mean photon spin s by using
a lock-in amplifier locked to the signal s sin ψ, contributed by the last term in Eq. (3.58),
at the first harmonic of ωp . A typical measured value of s as a function of laser detuning is
shown by a dashed line in Fig. 3.6. By locking to the second harmonic of ωp , we measured
√
1 − s2 sin 2θ that appears in the second term of Eq. (3.58). We used the measured mean
photon spin s to extract sin2θ in Eq. (3.62) from the second-harmonic signal.
108
Figure 3.6: Faraday rotation signal sin2θ (open circles), fit to the calculation of Eqs. (3.63–
3.65) (solid line) using the fitting parameters [Cs], θ0 , and ωJ . The optical path length is
2.5 cm and B = 9.4 T. The dashed line shows the mean photon spin s measured from the
first harmonic of the photodetector output. The residual non-zero photon spin at high laser
frequency (far detuning from the D2 line) is ascribed to the effect of optical guiding mirrors.
High-field Faraday rotation
From the well-known theory of Faraday rotation (e. g. see [55]), the rotation angle for the
light of wavelength λ = 2πc/ωλ close to the strong D1 and D2 resonance lines is
∆θ = l [A]
X
Jmσ
AJmσ (ωλ − ωJmσ )
,
[(ωλ − ωJmσ )2 + γ 2 /4]
(3.63)
where l is the optical path length of the laser beam inside the cell and the quantum numbers
Jmσ are discussed below. The coefficients AJmσ are
h
i2
J, m+σ
3π ωλ re c fJ σ C1/2,
ρm
m; 1, σ
AJmσ =
,
(2J + 1) ωJ
(3.64)
where re = e2 /me c2 = 2.82 × 10−13 cm is the classical electron radius, e and me are the
electron charge and mass in cgs units, c is the speed of light, fJ are the oscillator strengths
of absorption lines to the 2 PJ excited states. To a good approximation f1/2 = 1/3 and
109
f3/2 = 2/3. The resonant frequencies of the two D lines are ωJ = 2πc/λJ . These are
pressure shifted slightly from the free-atom values of λ1/2 = 7947 Å and λ3/2 = 7800 Å
for Rb and λ1/2 = 8943 Å and λ3/2 = 8521 Å for Cs. We assume pressure-broadened
Lorentzian lines with the same width at half maximum of γ, as indicated in Eq. (3.63).
The quantity in square brackets in Eq. (A.43) is a Clebsch-Gordan coefficient [49]. The
index σ = ±1 determines whether the absorbed light is left- or right-circularly polarized.
The index m = ±1/2 is the azimuthal electron spin quantum number of the ground-state
Rb atom, and ρm = Z −1 exp{−2mµB B/kT } is the probability of a given atom being in a
state with the quantum number m. Here Z = 2 cosh µB B/kT ; the cell temperature is T (in
kelvin), and µB is the Bohr magneton.
The resonant frequencies ωJmσ are
ωJmσ = ωJ +
µB B
[gJ (m + σ) − 2m],
h̄
(3.65)
where the Zeeman splitting of the excited-state azimuthal levels is determined by the gvalues, g1/2 = 2/3 and g3/2 = 4/3. Of the eight possible combinations for the summation
indices in Equation (3.63), Jmσ = (1 ± 1/2, ± 1/2, ±1), the triples (1/2, 1/2, 1) and
(1/2, −1/2, −1) are excluded since they refer to non-existent excited-state sublevels with
J = 1/2 and mJ = ±3/2. So there are only 6 non-zero terms in the sum (3.63).
From inspection of Eq. (3.62), we see that the experimentally-determined ratio R on
the left can be made equal to the theoretical expression on the right by setting the gain
parameter G to the maximum value of R, and then fitting the unknown number density
[A] along with the approximately known initial-orientation angle θ0 and the resonance-line
wavelengths λJ . The errors of such [A] measurements are about 3%, mostly due to small
uncertainties in l, γ, and ρm (T ).
With the relatively large values of magnetic field (B = 9.4 T) and atomic number densities
([A] ∼ 1014 cm−3 ) used in our experiments, we could observe a few oscillations of the
signal V2ωp , with negligible attenuation, as the laser was tuned toward the resonance line.
The resulting highly-structured signal could be fit very precisely (Fig. 3.6). To obtain the
maximum data with this limitation, the initial orientation angle θ0 was chosen to make
110
the signal from the second LIA slightly positive in the limit of large positive detunings,
ωλ − ω3/2 À γ.
111
3.4
3.4.1
Results and Analysis
Binary Spin-Exchange Rate Coefficients
We prepared four Rb cells and five Cs cells for the measurements of spin-exchange rate
coefficients. By changing the temperature, we obtained different number density of Rb
or Cs. So we were able to measure the nuclear spin relaxation rates depending on the
Rb or Cs number density. The measured values of κ are listed in Table 3.1 and 3.2.
Figure 3.7 and 3.8 summarize the experimental results of all data points. We obtained our
experimentally determined values of κ are κRbXe (180◦ C, 9.4T) = (1.75±0.12)×10−16 cm3 /s
and κCsXe (130◦ C, 9.4T) = (2.81 ± 0.20) × 10−16 cm3 /s.
Cell No.
T094
T095
T096
T097
Xe density (amg)
0.79
0.79
0.79
4.0
N2 density (amg)
3.0
1.0
0
0
κ (10−16 cm3 /s)
1.84 ± 0.17
1.60 ± 0.17
1.76 ± 0.16
1.78 ± 0.10
Table 3.1: Experimentally measured values of κ for the four Rb cells used in our experiment.
(Note: 1 amg=2.69 × 1019 cm−3 , is equal to the number density of an ideal gas at 0◦ C and
760 torr)
Cell No.
T114
T115
T116
T117
T118
Xe density (amg)
1.0
4.0
1.0
1.0
1.0
N2 density (amg)
3.5
0
0
1.0
2.0
κ (10−16 cm3 /s)
2.98 ± 0.14
3.05 ± 0.15
2.72 ± 0.20
2.72 ± 0.25
2.57 ± 0.13
Table 3.2: Experimentally measured values of κ for the five Cs cells used in our experiment.
Several previous measurements of spin-exchange rates were done at very low magnetic
field. In order to compare our results with the others, we used theoretical decoupling curve
to extrapolate the κ values from 9.4T to zero field. The decoupling curves were numerically
calculated by using SCA and DWBA methods. For the van der Waals potential V (r), we
chose the semiempirical potentials of Buck and Pauli[15], which are based on the results of
112
3
Spin-exchange rate coefficients (cm /s)
-1
Measured relexation rate 1/T1 (s )
0.12
T094 ( 1.84 ± 0.17)×10
0.10
T095 ( 1.60 ± 0.17)×10
T096 ( 1.76 ± 0.16)×10
0.08
T097 ( 1.78 ± 0.10)×10
-16
-16
-16
-16
0.06
0.04
0.02
0.00
0
100
200
300
400
500
600
12
-3
Measured Rb number density [Rb] (10 cm )
Figure 3.7: Measured longitudinal nuclear spin-relaxation rates 1/T1 plotted versus measured
Rb vapor number densities [Rb] for the four cells listed in Tab. 3.1. The experimental temperature were between 160 to 200◦ C. The spin-exchange rate coefficients κ are the slopes of the
straight-line fits to the data.
scattering experiments. They are parametrized as follows:

h¡ ¢
¡ ¢ i


 4² rm 11 − 11 rm 4 , r ≤ rm
7
r
4
r
V (r) =
h¡ ¢
¡
¢ i


 3² rm 14 − 7 rm 6 , r > rm
4
r
3
r
,
(3.66)
where ² is the depth of the minimum potential energy and rm is the distance where the
potential energy is equal to ². For Xe-Rb, ² = 13.68 meV and rm = 0.536 nm. For Xe-Cs,
² = 13.62 meV and rm = 0.547 nm.
The spin coupling coefficient α for a
129 Xe
atom, displaced a distance r from an alkali-
metal atom, is
α(r) =
8πgS µB µK
|ψ(r)|2 ,
3K
(3.67)
where the nuclear spin K = 1/2 and µK = −0.7725 × 5.05 × 10−27 J/G. We use the
orthogonalized wave (OW) approximation[54] to write ψ(r), the amplitude of the valence
113
3
Spin-exchange rate coefficients (cm /s)
-1
Measured relexation rate 1/T1 (s )
0.05
T114 ( 2.98 ± 0.14)×10
0.04
T115 ( 3.05 ± 0.15)×10
T116 ( 2.72 ± 0.20)×10
T117 ( 2.72 ± 0.25)×10
0.03
T118 ( 2.57 ± 0.13)×10
-16
-16
-16
-16
-16
0.02
0.01
0.00
0
20
40
60
80
100 120 140
12
-3
Measured Cs number density [Cs] (10 cm )
Figure 3.8: Measured longitudinal nuclear spin-relaxation rates 1/T1 plotted versus measured
Cs vapor number densities [Cs] for the five cells listed in Tab. 3.2. The experimental temperature
were between 110 to 150◦ C.
electron wave function of the alkal-metal atom at the nucleus of the Xe atom as
ψ(r) = φ(r) −
5
X
Z
χn (0)
d3 s χ∗n (s)φ(r + s).
(3.68)
n=1
Here φ(r) is the amplitude of the unperturbed valence electron orbital of an alkali-metal
atom (5s for Rb and 6s for Cs). The occupied s-wave orbitals of the Xe core are denoted by
χn (s) with principal quantum numbers n. We denote the displacement of an electron from
the Xe nucleus by s. For numerical evaluation of (3.68) we used the tabulated orbitals of
Clementi and Roetti [17]. The calculated V (r) and α(r) of
129 Xe-Rb
are shown in Fig. 3.1.
The numerical computation showed that there is less than 2% difference of the results
between SCA and DWBA [26]. Figure 3.9 shows the calculated decoupling curves by using
SCA method. We obtained the κ values at zero field to be κRbXe (180◦ C, 0T) = (2.2±0.15)×
10−16 cm3 /s and κCsXe (130◦ C, 0T) = (4.1 ± 0.30) × 10−16 cm3 /s from the extrapolations.
From the calculations, there was a slight temperature dependence of κ, which was about
κ−1 dκ/dT = 240ppm/◦ C around T = 100◦ C and B = 0 tesla. The magnetic decoupling in
3
Spin-exchange rate coefficient κ (10-16 cm /s)
114
5
CsXe theory
RbXe theory
4
Experiment at 9.4 T:
3
CsXe:
-16
3
(2.81 ± 0.20)×10 cm /s
2
RbXe:
-16
3
(1.75 ± 0.12)×10 cm /s
1
0
0
10
20
30
40
Magnetic field B (T)
50
60
Figure 3.9: Theoretical magnetic decoupling curves used to extrapolate our from B=9.4T to
B=0T. Since the experimentally determined potential V had about 5% error, we made 3 to
4% adjustments to the position of the minimum of the interatomic potential V can bring a
good agreement between experimental data and calculations. The shape of decoupling curves
changes less than 0.1% by applying adjustments on V .
Xe-Cs is more pronounced than in Xe-Rb, because of the slower relative velocity between
xenon and cesium.
In Table 3.3, we list κ-values of 129 Xe-Rb and 129 Xe-Cs obtained by different groups. In
out experiments, we eliminated two potentially serious sources of systematic error: (1) the
number density of alkali-metal atoms, and (2) the poorly-understood contribution of van der
Waals molecules to the spin exchange. One or both of these sources of error were present in
all of the earlier measurements listed in Table 3.3. We used Faraday rotation measurements
to determine the number density of alkali-metal atoms. Under different conditions, we found
that the real number density of alkali-metal atoms might be larger or smaller by a factor
of two than the prediction from the empirical vapor pressure formula. We chose a high
magnetic field to eliminate the contribution of van der Waals molecules to spin exchange.
The other groups either used saturated vapor pressure formula to determine the atomic
115
Name
Rb experiments:
Grover[21]
Volk[50]
Zeng[58]
Cates[16]
Augustine[5]
Rice[39]
Our work[25]
Rb calculations:
Walker[51]
Walter[54]
Our work[26, 27]
Cs experiments:
Zeng[59]
Liu[36]
Ishikawa[24]
Our work[27]
Cs calculations:
Walker[51]
Walter[54]
Our work[26, 27]
Temperature (◦ C)
B-field (Tesla)
κ (10−16 cm3 /s)
Year
∼ 70
∼ 65
∼ 90
∼ 100
∼ 85
∼ 110
∼ 180
2 × 10−7
2 × 10−7
5 × 10−4
1.1 × 10−5
2.35
4.7
9.4
< 87
< 41
4.1
3.7 ± 0.7
< 2.8
0.6 ± 0.1
1.75 ± 0.12
(1978)
(1980)
(1985)
(1992)
(1997)
(2002)
(2002)
100
100
100
0
0
0
6.3
1.2
2.1 ± 0.15
(1989)
(1998)
(2004)
∼ 50
∼ 60
∼ 25
∼ 130
1.88
4.7
2.2 × 10−4
9.4
< 5.1
< 6.3
< 10.2
2.81 ± 0.2
(1991)
(1992)
(2000)
(2003)
100
100
100
0
0
0
9.2
2.7
4.0 ± 0.3
(1989)
(1998)
(2004)
Table 3.3: A summary of binary spin-exchange rate coefficients at different magnetic fields
obtained by different groups. The zero-field rates quoted for our work are slightly larger
than the values measured at B = 9.4 T because of adjustments for the magnetic decoupling
discussed in connection with Fig. 3.9.
density from temperature [21, 50, 58, 16, 5, 39, 59, 36] or used extrapolation to eliminate
the contribution of van der Waals molecules [58, 16, 39]. The recent work of Kadlecek [30]
shows that we don’t really understand the spin-exchange mechanism from van der Waals
molecules when the pressure is larger than one atmosphere. Their result showed that at
high pressure, van der Waals molecules can contribute a substantially larger spin-relaxation
rates than we expected. Thus the extrapolation method used by authors, such as Cates [16]
and the others, might not be appropriate.
From our experimental data, we did not see any noticeable pressure dependence of the κ
values. The pressure independence is to be expected if the rates are due to binary collisions,
but not if the rates include a substantial contribution from the formation and breakup of
116
van der Waals molecules. We concluded there were no important contributions from van
der Waals molecules in our measurements. The results also show that Cs has a larger spinexchange rate coefficient, which agrees with some previous measurements. Therefore, once
the high power pumping source is ready for cesium, Cs can be a potential candidate in
spin-exchange optical pumping of xenon, not only because of its larger κ, but also its higher
vapor pressure.
3.4.2
High-Field Contribution of van der Waals Molecules
At high field, we estimated the contribution of van der Waals molecules to spin-exchange
rates. We use Xe-Rb as an example. Similarly arguments hold for Xe-Cs. For simplicity
of notation we assume 100% isotopic abundance of
129 Xe.
From the similar discussion in
section 3.2, we can write the spin Hamiltonian of a Xe-Rb molecule as
H = h̄ωSz + α S · K = H (0) + H (1) .
(3.69)
Here, S is the spin operator of the Rb valence electron, K is the spin operator of the
129 Xe
nucleus, a representative value of the hyperfine coupling coefficient averaged over
vibrational-rotational states of the molecule is ᾱ/h̄ ≈ 2.0 × 108 s−1 , and ω = 2µB B/h̄ =
1.65 × 1012 s−1 is the Larmor frequency of the electron in a magnetic field of 9.4 T. At this
large field, we can ignore the nuclear spin I of the Rb atom in calculating its spin exchange
rates with
129 Xe.
The zeroth-order part of the Hamiltonian (3.69) is H (0) = h̄ωSz + ᾱSz Kz
and the spin-exchange perturbation is H (1) = ᾱ (S+ K− + S− K+ )/2.
The
129 Xe-Rb
molecule will be formed in one of the four spin states |mS mK i, where
mS = ±1/2 and mK = ±1/2 are the azimuthal quantum numbers of the Rb electron
spin and the Xe nuclear spin, correspondingly. These are eigenstates of the zeroth-order
Hamiltonian, H (0) |mS mK i = (h̄ωmS + ᾱmS mK )|mS mK i ≈ h̄ωmS |mS mK i. We denote
with |1i for mS = 1/2, mK = −1/2 and |2i for mS = −1/2, mK = 1/2. One can
(1)
readily verify that H12 = h1|H (1) |2i = ᾱ/2. From first-order perturbation theory, the
probability that a molecule, formed in the state |1i at time t = 0, will have made a transition
to the state |2i at time t > 0 is W = (ᾱ/h̄ω)2 sin2 (ωt/2). Previous experimental work
117
[58, 12] has shown that Xe-Rb molecules are broken up by collisions with other xenon
atoms or nitrogen molecules. The mean lifetime τ was found to be very nearly 1/τ =
([N2 ] + 3.6 [Xe]) × 5.5 × 109 s−1 amg−1 . A Xe atom is approximately 3.6 times as likely to
break up a Xe-Rb molecule as an N2 molecule. Considering a high pressure cell, T094, we
find a shorter duration τ = 31 ps. Therefore τ ω = 51. The magnetic field is large enough
that the probability of a spin flip from state |1i to |2i is independent of the molecular
lifetime τ and is hW i = (ᾱ/h̄ω)2 /2 = 7.3 × 10−9 . The factor of 1/2 is from the mean value
of sin2 (ωt/2).
A Xe-Rb molecule can form in simultaneous collisions of a Xe atom, a Rb atom and a
third atom or molecule. The third body is another Xe atom or a nitrogen molecule N2 in
our cells. The number density of molecules will therefore grow at the rate
d
[XeRb] = ζ [Xe][Rb].
dt
(3.70)
where ζ = Zr ( [N2 ] + 3.6 [Xe] ). Three-body rate coefficients Zr have been determined from
previous experimental work. For nitrogen [58], Zr = 3.9 × 10−32 cm6 /s (after rescaling the
original data by the ratio of temperatures 349 K/453 K). For cell T094, this implies that
ζ = 6.2×10−12 cm3 /s. Denoting the number densities of spin-up and spin-down atoms with
the subscripts + and − respectively, we see the formation of Xe-Rb molecules will cause
[Xe+ ] to change at the rate
d
[Xe+ ] = ζ ([Xe− ][Rb+ ] − [Xe+ ][Rb− ])hW i.
dt
(3.71)
We have neglected the slight difference in the exothermic and endothermic collisions. In
keeping with this approximation we have neglected the thermal polarization of the Rb
atoms. From Eq (3.9) and Eq (3.10), We find
d
hKz i = −κ0 [Rb]hKz i,
dt
(3.72)
where the three-body rate coefficient for cell T094 is
κ0 = ζ hW i = 4.5 × 10−20 cm3 /s.
(3.73)
118
This is 3900 times smaller than the experimental result, κ = 1.75 × 10−16 cm3 /s. Similarly
small estimates are found for the other cells.
Although the spin exchange from van der Waals molecules is still not understood above
an atmosphere pressure. We believe the estimates shown above is valid based on following
arguments. The molecule spin-exchange rate is proportional to the molecule formation rate
and the transition probability. From the discussion above, we know that in a magnetic
field the transition probability can not be larger than (ᾱ/h̄ω)2 . From the calculations, it
is smaller than the transition probability of a binary collision in our experiments. Besides,
the molecule formation needs some mechanisms involved with other particles rather than
just binary collisions. In our experiments, the formation rate should be less than the binary
collision rate. Otherwise, we might be able to see a systemic pressure dependence on our
experimental data. We conclude that Xe-Rb and Xe-Cs molecules cannot contribute more
than 1% to the rates measured in these experiments. The uncertainties in the estimates of
the molecular formation and breakup rates are not large enough to change this conclusion.
Appendix A
Evolution of the Density Matrix of
Alkali-Metal Atoms
To derive the evolution equation of the density matrix, we can start from the Schröedinger
equation for the ith atom of an ensemble,
ih̄
d
|ψi i = Hi |ψi i,
dt
(A.1)
where |ψi i = |ψi (t)i represents the atomic quantum state of an alkali-metal atom, and
Hi = H
(0)
+H
(1)
(2)
+ Hi
is the total Hamiltonian for each atom. Here H
independent “unperturbed” Hamiltonian, and H
(1)
=H
(1)
(0)
is a time-
(t) is a time-dependent purtur-
bation representing the interaction of the atom with the optical pumping, microwave, and
RF fields. Both H
(0)
and H
(1)
are identical for all atoms. The collisional interaction of the
(2)
ith atom is represented by the time-dependent Hamiltonian Hi
(2)
= Hi (t). The density
matrix of an ensemble of N identical atoms can be described by
ρ=
1 X
|ψi ihψi |.
N
(A.2)
i
In the interaction representation, we can define the quantum state, |ψei i, in the interaction
frame by
|ψi i = e−iH
(0)
119
t/h̄
|ψei i.
(A.3)
120
Then,
ih̄
d e
e (1) + H
e i(2) )|ψei i,
|ψi i = (H
dt
(A.4)
where
e (n) = eiH
H
(0)
t/h̄
H
(n)
e−iH
(0)
t/h̄
.
(A.5)
The density matrix in the interaction frame, ρ̃ is defined by
1
ρ = e ih̄ H
A.1
(0)
t
1
ρ̃ e− ih̄ H
(0)
t
.
(A.6)
Ground-State Relaxation Due to Weak Collisions
(2)
A weak collision is one for which the peak value of hHi /h̄i is much smaller than the inverse
of the collisional duration δt. S-damping and Carver damping come from weak collisions.
But, spin exchange from a collision of the two alkali-metal atoms is a strong collision. For
each collision event, we can find the evolution for the quantum state |ψei i from Eq. (A.4).
ei |ψei (t)i,
|ψei (t + δt)i = U
(A.7)
where the evolution operator is
µ
ei = exp
U
1
ih̄
Z
t
t+δt
¶
(2)
0
0
e
Hi (t )dt .
(A.8)
In Schröedinger picture, the evolution operator for |ψi i becomes
Ui = e−iH
(0)
(t+δt)/h̄ e iH
Ui e
(0)
t/h̄
.
(A.9)
Ui is unitary so U U † = 1.
If we denote the density matrix of an atom just before the ith collision by ρ̃ and just
ei ρ̃U
e † . We obtain that the time derivative of
after the collision by ρ̃ + δ ρ̃, we find ρ̃ + δ ρ̃ = U
i
ρ due to H
(2)
is equal to the product of the collision frequency and the average of the small
change, δ ρ̃, from each collision. Therefore
·
dρ̃
dt
¸(2)
D
E
ei ρ̃ U
e † − ρ̃ ,
=N U
i
(A.10)
121
where N is the number of collisions occurred per unit time in an ensemble system, and
h i denotes the ensemble average over all possible types of collisions, for example, over all
ei into
impact parameters and orbital directions. We can expand U
Z
∞
e i(2) (t0 ) dt0 + 1
H
2
−∞
1 2
= 1 − iΘi − Θi + · · · .
2
ei = 1 + 1
U
ih̄
µ
1
ih̄
Z
∞
−∞
e (2) (t0 )i dt0
H
¶2
+ ···
(A.11)
Here, Θi is a Hermitian operator. For weak collisions, hΘi i ¿ 1. So we can approximate
e , hence
Eq. (A.10) to the second order of U
·
dρ̃
dt
¸(2)
·
¸
1
= N −i[ hΘi i, ρ̃ ] + hΘi ρ̃Θi i − { hΘi Θi i, ρ̃ } .
2
(A.12)
Considering the different initial state |νi and the final state |µi, we find
1
hµ|Θi |νi =
h̄
Z
∞
−∞
(2)
(2)
hµ|Hi (t)|νi eiωµν t dt = hµ|Hi (−ωµν )|νi/h̄,
(2)
(A.13)
(2)
where Hi (ω) is the Fourier’s spectrum of Hi (t). The result of Eq. (A.13) shows that
hµ|Θi |νi depends on the Bohr frequency ωµν . If we can change δtωµν substantially by
changing the external field, we can change the nature of the collisional relaxation. For
most cases, the effective duration of a collision, δt, is much shorter than 2π/ωµν , so Θi ∼
=
(2)
Hi (0)/h̄.
For our experimental interest, the collisional duration, δt, is so short compared to the
ei . Combining H (0) , H (1) , and H (2) , we find the total dynamic
inverse of ωµν . Hence, Ui ≈ U
equation of the density matrix to be
·
¸
dρ
1
1
1
(0)
(1)
= [ H , ρ ] + [ H , ρ ] + N −i[ hΘi, ρ ] + hΘρΘi − { hΘΘi, ρ } .
dt
ih̄
ih̄
2
(A.14)
For convenience, we use hΘi to denote hΘi i, and apply the same rule for all other ensemble
averages. The evolution equation of the density matrix should conserve the total number
of atoms. We can check that Tr(dρ/dt) = 0 from Eq. (A.14), because Tr(Oρ)=Tr(ρO),
where O is any operator. By using Eq. (A.14), we are able to calculate the evolution of all
observables analytically or numerically.
122
A.1.1
S-Damping
The S-damping or spin destruction rate is caused by the transferring of the angular momentum between the electron spin S and the angular momentum N of the two colliding atoms
(2)
through spin-rotation interaction, Hsd = γ(r)N · S. Here the γ is the coupling coefficient,
which is a function of the atomic separation. Assuming that the duration of the collision is
much shorter than all 2π/ωµν , we find
Θsd
Z
1
=
h̄
∞
γN · Sdt = φ · S,
(A.15)
−∞
where φ = (φx , φy , φz ). The statistical property of φ is that hφi = 0 and hφi φj i = δij hφ2i i.
The integration performed in Eq. (A.15) can be carried out by using the trajectory calculation. In this semi-classical approximation, the atomic separation r(t) is a function of time.
By using Eq. (A.12), we find the relaxation due to the S-damping as
ρ̇sd
¶
µ
¶
µ
3
1
= Γsd S · ρS − {S · S, ρ} = Γsd S · ρS − ρ ,
2
4
(A.16)
where the S-damping rate Γsd = Nsd hφ 2 i/3. We like to express the S-damping relaxation
as
ρ̇sd = Γsd (ϕ − ρ),
(A.17)
where ϕ = ρ/4 + S · ρS is the density matrix without the electronic polarization. We can
verify that Tr(ϕ) = 1 and Tr(Sϕ) = 0.
A.1.2
Carver Rate
The atomic hyperfine structure is ascribed to electron-nuclear spin coupling, AI · S. The
coupling strength A is perturbed by the collisions between atoms. We describe the per(2)
turbation Hamiltonian of the hyperfine interaction as HC = δAI · S. The perturbation
δA = δA(r) is due to the collisionally-induced change in the value of the valence-electron
wave function at the nuclei of the alkali-metal atom. Therefore, we can find
1
ΘC =
h̄
Z
∞
−∞
δAI · Sdt = φC I · S.
(A.18)
123
Similarly, we obtain the evolution of the density matrix due to the random hyperfine modulation by applying Eq. (A.12).
ρ̇C
µ
¶
hφ2C i
hφ2C i
= NC −i[ hφC iI · S, ρ ] + 2 I · SρI · S −
{I · SI · S, ρ} .
h̄
2h̄2
(A.19)
In the weak magnetic field, we can still use |f, mi to represent the eigenstate. We use
f = I +1/2 = a to label the upper hyperfine sublevels and f = I −1/2 = b to label the lower
hyperfine sublevels. So we have I · S|a, mi = 12 I|a, mi and I · S|b, mi = − 12 (I + 1)|b, mi. We
P
define the density matrix of hyperfine transition ρ(m) =
|fi , ma ihmb , fj |, where fi 6= fj .
We can rewrite Eq. (A.19) into
ρ̇C =
η 2 [I]2
1
[ δEC , ρ(m) ] − I
ΓC ρ(m) ,
ih̄
8
(A.20)
where [I] = 2I + 1. Here, the collision shift or pressure shift δEC = h̄NC hφC iI · S. The
”Carver rate” ΓC = NC hφ2C i/ηI2 , which was first introduced by D.K. Walter [53]. The
isotope coefficient is ηI = µI /(2IµN ), where µI and µN are the nuclear magnetic moment
and nuclear magneton. The random modulation of hyperfine splitting can only cause the
dephasing of the coherence between upper and lower hyperfine sublevels when the magnetic
field is weak. It does not transfer population from one sublevel to another.
A.2
Ground-State Relaxation Due to Strong Collisions and
Other Mechanism
A.2.1
Spin Exchange
Spin exchange occurs when two alkali-metal atoms collide with each other. Spin-exchange
collisions are the extreme opposite of weak collisions. Through the spin-spin coupling,
(2)
Hex = J(r)S0 · S, electrons can exchange their spins from each other. Usually the strength
of the spin-spin coupling between two alkali-metal atoms is comparable to the interatomic
potential V0 (r). Hence, we can not use Eq. (A.12). More detailed discussions can be found
124
in Ref [22, 52, 31], and we find the relaxation due to spin exchange as
ρ̇ex =
=
=
=
µ
¶
­ 0
® 3
00 0
0
0
0
[A] −i[ 2hvσ S i · S, ρ ] + hvσ i(4 S · SρS · S − ρ + {hS · Si, ρ})
4
µ
¶
1
3
[ δEex , ρ ] + Γex S · ρS + 2ihS0 i · iS × ρS − ρ + {hS0 i · S, ρ}
ih̄
4
¡
¢
1
[ δEex , ρ ] + Γex ϕ − ρ − 2hS0 i · S × ρS + {hS0 i · S, ρ}
ih̄
¡
¢
1
[ δEex , ρ ] + Γex ϕ − ρ + 4ϕhS0 i · S ,
(A.21)
ih̄
where [A] is the number density of the alkali-metal atoms, v is the relative velocity of the
two colliding atoms, and the spin-exchange cross section σ = σ 0 + iσ 00 is defined by [31]
σ0 =
σ 00 =
π X
(2l + 1) sin2 δl
k2
l
π X
(2l + 1) sin2 2δl .
2k 2
(A.22)
l
Here, δl = 3 δl −1 δl is the difference between phase shifts 3 δl and 1 δl for scattering of the
lth partial wave on the triplet and singlet potentials of the alkali-metal dimers. The spatial
frequency k, is proportion to the relative velocity v. Some discussions about the partial
wave calculations can be found in section 3.2.2. If the kinetic energy in the central mass
frame of the two colliding atoms is larger than the interaction strength hJi, we find that
δl ∝ hJi. The spin-exchange shift δEex = 2h̄[A]hvσ 00 S0 i · S, which produces an effective
magnetic field from the spin polarization, and the spin-exchange rate Γex = [A]hvσ 0 i. Since
both S0 and S belong to the same ensemble, we can express the spin-exchange relaxation as
ρ̇ex =
1
[ δEex , ρ ] + Γex [ϕ(1 + 4hSi · S) − ρ] .
ih̄
(A.23)
We can verify that Tr(Sρ̇ex ) = 0. The spin-exchange mechanism conserves the total electron
spin.
A.2.2
Spin Diffusion
In the alkali-metal vapor, atoms keep colliding and changing their position. This causes
atoms to diffuse spatially. The relaxation of the density matrix in a certain coordinate
125
caused by diffusion process can be described by
∂ρ(r)
= D∇2 ρ(r),
∂t
(A.24)
where D is the diffusion coefficient. Since each atom carries their own atomic spin, the
diffusion of atoms leads to the spin diffusion. When the spin flow hits a boundary, such as
the cell wall, a strong depolarization occurs and forces atoms to even out the population of
the spin states. (Note: it is very possible that the cell wall traps the incoming atoms and
releases new unpolarized atoms) We can write down the evolution of the density matrix due
to the spin diffusion as
ρ̇d = Γd (1/g − ρ),
(A.25)
where 1 is the unit matrix, and g = 2[I] is the total number of the ground-state sublevels.
We choose Γd to represent the relaxation rate due to the spin diffusion. Considering a
simplest case with a cubic cell, the length of each side is l. Hence we can find that
ρ(x, y, z) ∝
X
Aa sin(
a,b,c
aπ
bπ
cπ
x)Bb sin( y)Cc sin( z),
l
l
l
(A.26)
where a, b, c are integer numbers. If we only consider the lowest mode, which is a = b =
c = 1, the diffusion relaxation rate Γd = 3D(π/l)2 .
A.3
Excited-State Relaxation
For our experimental interest, we only discuss the P1/2 excited state of alkali-metal atoms.
A.3.1
J-Damping
The J-damping rate is very similar to the S-damping. Instead of depolarizing the electron
spin in S1/2 ground state, the expectation value of electron angular momentum in P1/2 is
(2)
diminished due to the interaction Hjd = γj (r)N · J. Here the γj is the coupling coefficient.
Similarly, the relaxation due to the J-damping in P1/2 state is
¶
µ
1
(e)
(e)
(e)
ρ̇jd = Γjd J · ρ J − {J · J, ρ } ,
2
where Γjd is the J-damping rate.
(A.27)
126
A.3.2
Spontaneous Decay and Quenching
There are two important deexcitation mechanisms causing atoms to decay from the excited
state to the ground state. Spontaneous decay is due to the electromagnetic perturbation
from the ground state of the quantized EM field to the atoms. Atoms return to the S1/2
states from the P1/2 states by randomly emitting photons. Quenching decay is caused by
atoms colliding with quenching gases. Atoms decay to ground states by transferring the
excited-state energy to the internal degrees of freedom of the quenching gases. Usually, the
decay process is dominated by the strong coupling with electron. It is fairly good to ignore
the tiny contribution from the nuclear spin. Therefore, spontaneous decay or quenching
partially or completely destroys the electronic polarization, but they conserved the nuclear
polarization in the decay process. The relaxation equation due to spontaneous decay and
quenching is
(e)
(e)
ρ̇sq = −Γs ρ
(e)
− Γq ρ ,
(A.28)
where Γs is the spontaneous decay rate and Γq is the quenching rate.
With optical pumping, some population was transferred from the ground state to the
excited state. Assuming that Υ is the population injection rate from the ground state by
optical pumping, and Γjd = 0, we find
(e)
ρ̇
=
1
(0)
(e)
(e)
[ H , ρ ] − (Γs + Γq )ρ + Υ.
ih̄ e
(A.29)
(0)
Here, He
is the unperturbed spin-dependent Hamiltonian of the P1/2 state. For each
P (en)
(e)
(e)
component of the density matrix, ρkl , we can expand to ρkl = n ρkl e−inωt . We can find
(en)
the steady solution of the amplitude ρkl
for each frequency component nω by solving
d (en)
(en)
ρkl = −(i(ωkl − nω) + Γs + Γq )ρkl + Υkl = 0.
dt
(A.30)
Hence,
(n)
(en)
ρkl =
(n)
(0)
Υkl
,
i(ωkl − nω) + Γs + Γq
(A.31)
(1)
where Υkl is defined by Υkl = Υkl + Υkl e−iωt + · · · . The result of Eq. (A.31) will be useful
in the discussion of the repopulation pumping with non-zero frequency component of the
127
density matrix. The repopulation pumping due to spontaneous decay and quenching will
be discussed in the section of optical pumping.
A.4
Microwave and RF Fields
Considering the ground-state hyperfine sublevels, the oscillating field interacts with alkalimetal atoms through
(1)
Hof = B(t) · (µB gS S − µI I/I),
(A.32)
where B(t) = Bof (eiωt + e−iωt ) is the oscillating magnetic field. Usually, we ignore the
interaction to the nuclear spin I, because it is much smaller than the interaction to the
electron spin S. Therefore, we find the time evolution equation due to the oscillating field
to be
ρ̇of =
1
(1)
[ Hof , ρ ].
ih̄
(A.33)
Considering two hyperfine sublevels |µi and |νi coupled by the oscillating field, we find that
(1)
the Rabi frequency produced by the oscillating field is equal to ωR = h̄2 hµ|Hof |νi. We can
also find the rate equation related to these two sublevels due to the oscillating field as
ρ̇µµ = −iωR /2(ρ̃νµ − ρ̃µν )
ρ̇νν
= −iωR /2(ρ̃µν − ρ̃νµ )
ρ̃˙ µν
= (−i(ωµν − ω) − γµν )ρ̃µν + iωR /2(ρµµ − ρνν )
ρ̃˙ νµ = (i(ωµν − ω) − γµν )ρ̃νµ − iωR /2(ρµµ − ρνν ),
(A.34)
where ρ̃µν is define by ρµν = ρ̃µν e−iωt . Here, we drop the rapid oscillation term with the
frequency component ωµν +ω (secular approximation or rotation wave approximation). The
solution of ρ̃µν is
iωR
ρ̃µν (t) =
2
Z
t
t0
0
e−[i(ωµν −ω)+γµν ](t−t ) (ρµµ (t0 ) − ρνν (t0 ))dt0 .
(A.35)
Here, the integral shows that the oscillating field is turned on at t0 , and t ≥ t0 . For late-time
evolution, ρµµ (t) and ρνν (t) become slowly varying. We find the quasi-steady solution of
128
ρ̃µν to be
ρ̃µν (t) =
iωR /2(ρµµ (t) − ρνν (t))
,
i(ωµν − ω) + γµν
(A.36)
which is the exact solution if the ensemble reaches equilibrium. By substituting Eq. (A.36)
into Eq. (A.34), we find the evolution equation of the density matrix due to the oscillating
field in the quasi-steady state to be
ρ̇of = −Γof ((ρµµ − ρνν )|µihµ| + (ρνν − ρµµ )|νihν|) ,
(A.37)
and similarly, we find the equivalent light shift operation
δEof
1
[δEof , ρof ] =
((ρµσ |µihσ| − ρνσ |νihσ|) − (ρσµ |σihµ| − ρσν |σihν|)) ,
ih̄
ih̄
(A.38)
where σ 6= µ and σ 6= ν. The AC Stark shift energy Eof and the effective pumping rate Γof
are defined by
δEof =
2 /4(ω
h̄ωR
µν − ω)
,
2
2
(ωµν − ω) + γσ(ν,µ)
Γof =
2 /2γ
ωR
µν
.
2
(ωµν − ω)2 + γµν
(A.39)
The damping rate γσ(ν,µ) shown in Eq. (A.39) has a swapped subscript, which means γσν
for ρσµ and ρµσ ; also γσµ for ρσν and ρνσ . We have to notice that the results of Eq. (A.37)
−1 .
and Eq. (A.38) are correct only when t À γµν
A.5
Optical Pumping
For D1 pumping of the alkali-metal atoms, the S1/2 ground state and the P1/2 excited state
are coupled to the electromagnetic field of the pumping light. The transition between the
S and the P states is caused by the electric dipole interaction with the oscillating electric
field. Therefore, the interaction Hamiltonian of optical pumping is
(1)
Hop = −D · Eop = −D · (E + E∗ ),
(A.40)
where D = −er is the dipole operator, and Eop = Eop (t), a real-value function, is the
electric field of the pumping light. Here, E and E∗ are complex conjugate. We use E to
129
represent the negative-frequency field, and use E∗ to represent the positive-frequency field.
The complex field E = E(t) can be expressed by
E(t) =
1
2π
Z
∞
0
dΩĒop (Ω)e−iΩt ,
(A.41)
where Ēop (Ω) = Ē∗op (−Ω) is the spectrum of the pumping electric field, and Ω is the optical
frequency. For transitions between S1/2 and P1/2 , we use |S, mS i and |J, mJ i to represent
the angular-momentum quantum states of the valence electron of the ground state and the
excited state. We find that the dipole operator can be expressed by [22, 3]
D = D(ge) + D(eg) = DA + D∗ A† ,
(A.42)
where transition operators A† and A can be decomposed by
A†+ = |J, 21 ihS, − 12 |,
A†− = |J, − 12 ihS, 12 |,
A†z = 12 (|J, 12 ihS, 21 | − |J, − 12 ihS, − 12 |)
.
and
A+ = |S, 21 ihJ, − 12 |,
A− = |S, − 12 ihJ, 12 |,
Az = 12 (|S, 12 ihJ, 21 | − |S, − 12 ihJ, − 12 |)
(A.43)
In Cartesian coordinate, we find
A† = A†x x + A†y y + A†z z =
A†+ +A†−
x
2
+
A†+ −A†−
y
2i
+ A†z z
.
and
A = Ax x + Ay y + Az z =
A+ +A−
x
2
+
A+ −A−
y
2i
(A.44)
+ Az z
For D1 transition, S = 1/2 and J = 1/2. Therefore we know both A† and A have the same
matrix representation as the spin- 12 operator S. In Eq. (A.42), the dipole amplitude D is
defined by [18]
|D|2 =
2h̄re c2 feg
,
ω0
(A.45)
where re = 2.82 × 10−12 cm is the classical electron radius, c is the speed of light, feg ≈ 1/3
is the oscillation strength of D1 resonance, and ω0 = ωeg is the mean Bohr frequency of D1
transition. Equation (A.40) now can be written as
(1)
Hop = −DA · E∗ − D∗ A† · E.
(A.46)
130
Here, we only need to consider the processes of the excitation from the ground state to
the excited state by annihilating photons and the deexcitation from the excited state to
the ground state by creating photons. In Eq. (A.46), we throw out the terms of −DA · E
and −D∗ A† · E∗ , which make the excitation with photon creation and deexcitation with
photon destruction. Although, these virtual transitions can really happen, their effects are
extremely small compared to our experimental concerns.
The complex electric field E(t) = E(t)ε̂, where E(t) is the complex scalar field, and ε̂ is
a complex vector with unitary length to describe the polarization. The photon spin s can
be expressed by [3]
s=
iε̂ × ε̂∗
= iε̂ × ε̂∗ .
ε̂ · ε̂∗
(A.47)
Optical Coherence
The optical coherence is induced by the pumping field. In the interaction picture, we find
d
1 e (1)
ρ̃ = [ H
, ρ̃ ].
dt
ih̄ op
(A.48)
(1)
e op
Here, H
is defined by Eq. (A.5), and ρ̃ is defined by Eq. (A.6). For distinguishing the
ground state and the excited state, we use Greek letters to label the ground state, and
English letters to label the excited state. From Eq. (A.48), we find
d
1 e (1)
e (1)
ρ̃kν = [ H
kµ ρ̃µν − ρ̃kl Hlν ].
dt
ih̄
(A.49)
(1)
e (1) = hk|H
e op
Here, H
|µi. In Eq. (A.49), the two consecutive subindices represent the action
kµ
of summation. For calculation, we have to sum over the subindices µ and l in Eq. (A.49).
The same rule is applied for all following calculations. Typically, the pumping rate, Γop is
up to 104 s−1 , and the deexcitation rate, Γsq = Γs + Γq ≥ 108 s−1 . Under the condition of
unsaturated optical pumping (Γop ¿ Γsq ), the population in the excited state is negligible.
Therefore, we can confidently neglect the part of ρ̃kl in Eq. (A.49). With the introduction
of the optical coherence damping rate γop , we find the differential equations for optical
131
coherences ρ̃kµ as
d
dt ρ̃kµ
∗
= − Dih̄ E · A†kν eiωkν t ρ̃νµ − γop ρ̃kµ ,
.
and
d
dt ρ̃µk
(A.50)
D
= − ih̄
ρ̃µν E∗ · Aνk eiωνk t − γop ρ̃µk
The solution of Eqs. (A.50) can be found in the same manner of Eq. (A.35). Therefore
ρ̃kµ
D∗
=−
ih̄
Z
t
t0
0
0
e−γop (t−t ) eiωkν t E(t0 ) · A†kν ρ̃νµ dt0 .
(A.51)
Generally, the maximum changing rate of ρ̃νµ is a few kHz. Therefore, by substituting
Eq. (A.41) into Eq. (A.51), we can carry out the integration by using the slow varying
approximation, and find
ρ̃kµ = −
D∗
Ê(t, ωkν ) · A†kν ρ̃νµ eiωkν t ,
ih̄
where
1
Ê(t, ωkν ) =
2π
Z
∞
dΩ
0
Ēop (Ω)e−iΩt
.
i(ωkν − Ω) + γop
(A.52)
(A.53)
One can see that Ê(t, ωkν ) is equivalent to that the pumping field Eop (t) is filtered by
a resonator filter at frequency ωkν with the bandwidth 2γop . Similarly, we can integrate
Eq. (A.51) to find
ρ̃µk =
A.5.1
D
ρ̃µν Aνk · Ê∗ (t, ωkν )eiωνk t = ρ̃∗kµ .
ih̄
(A.54)
D1 Depopulation Pumping
To calculate the population, which is pumping out from the ground state to the excited
state, we take the matrix elements of Eq. (A.48) to find
d
ρ̃µν
dt
1 e (1)
e (1) ]
[ Hµk ρ̃kν − ρ̃µk H
(A.55)
kν
ih̄
i
|D|2 h
= − 2 E∗ · Aµk A†kσ · Ê(ωkσ )ρ̃σν eiωσν t + ρ̃µσ Ê∗ (ωσk ) · Akµ A†kν · Eeiωµσ t eiωµν t .
h̄
=
In the Schröedinger picture, Eq. (A.55) can be rewritten as
d (g)
1
1
(0)
†
ρ = [ H , ρ(g) ] + [ δHop ρ(g) − ρ(g) δHop
].
dt
ih̄
ih̄
(A.56)
132
Here, ρ(g) denotes the density matrix of the ground-state, and the matrix elements of the
effective Hamiltonian δHop are
hµ|δHop |νi = δHµν =
|D|2 ∗
2h̄ ∗
(op)
†
E · Aµk A†kν · Ê(ωkν ) =
ε̂ · Aµk Akν · ε̂ Γkν (t),
ih̄
i
(A.57)
where the time-dependent optical pumping rate between the ground-state sublevel ν and
the excited-state sublevel k is
Z
(op)
Γkν (t) =
Z
dω e
−iωt
∞
dΩ
0
∗ (Ω − ω)Ē (Ω)
|D|2 Ēop
op
.
2 i(Ω
2h̄
kν − Ω) + γop
(op)
(A.58)
(op)
From the calculation of Eq. (A.51), we find that Γνk = Γkν ∗ . The effective Hamiltonian
can be written as the sum of a Hermitian and an anti-Hermitian operators,
δHop = δEop −
ih̄
δΓ.
2
(A.59)
The Hermitian light-shift operator is
1
†
δEv = (δHop + δHop
),
2
(A.60)
and the Hermitian light-absorption operator is
δΓ =
i
†
(δHop − δHop
).
h̄
(A.61)
Here, δEv is called “virtual transition” light-shift operator. It has nonzero value only when
the optical detuning of the pumping light is performed. The equation for depopulation
pumping is written as
ρ̇dp =
1
1
1
†
[δHop ρ − ρδHop
] = [δEv , ρ] − {δΓ, ρ}.
ih̄
ih̄
2
(A.62)
In our experimental studies, the Zeeman splittings are much smaller than the hyperfine
(e)
splitting, and the excited hyperfine splitting is not optically resolved (γop > ωhf ). Under
these experimental conditions, the optical pumping rate is only determined by the upper
or lower ground-state hyperfine multiplets. Therefore, from Eq. (A.57), we find
1
†
ε̂∗ · Aµk Akν · ε̂ −→ ε̂∗ · AA† · ε̂ = ε̂∗i (AA† )ij ε̂j = (1 − 2s · S).
4
(A.63)
133
Here, we use one identity of D1 transition, which says (AA† )ij = (SS)ij = 41 (δij − iSk ²ijk ).
Hence, the Eq. (A.57) can be further simplified into
h̄
(op)
(1 − 2s · S)Γ ,
2i
δHop =
(A.64)
where the optical pumping rate is a diagonal matrix, and its diagonal matrix elements are
Z
(op)
Γµµ =
Z
dω e
∞
−iωt
0
∗ (Ω − ω)Ē (Ω)
|D|2 Ēop
op
.
dΩ 2
i(Ω
−
Ω)
+
γop
2h̄
µ
(A.65)
Here, Ωµ is the transition frequency from the selected ground-state sublevel µ to the central
gravity of the target of the excited hyperfine sublevels. For the CPT experiments, the
probing light is usually modulated at the half frequency of the hyperfine resonance. In this
way, the symmetric optical sidebands to the carrier frequency is produced. By calculating
Eq. (A.65), the complex pumping rate of the frequency component, which is equal to the
microwave resonance frequency, has the same value for both upper and lower hyperfine
multiplets. In our experiments, we intensity-modulated the laser to generate the equal
intensity two first-order sidebands and suppressed the carrier and high-order sidebands. In
this case, when the optical detuning is small, it is a very good approximation to write the
scalar pumping rate as
(0)
(1)
(−1)
Γop = Γop + Γop e−i2ωm t + Γop ei2ωm t ,
(−1)
where Γop
(A.66)
(1)
= Γop ∗ , and ωm is the modulation frequency, which is equal or slightly de-
tuned to the hyperfine resonance frequency. In this two-wave approximation, we find
(0)
(0)
(1)
(1)
that Im(Γop )/Re(Γop ) = Im(Γop )/Re(Γop ) = −∆/γop , where the optical detuning ∆ =
(Ω0 − Ωc ). Here, Ω0 is the transition frequency from the center of the two resonant hyperfine sublevels to the excited state, and Ωc is the carrier frequency of the probing beam. The
corresponding light-shift and light-absorption operators are δEv = h̄2 (1 − 2s · S)Im(Γop ) and
δΓ = (1 − 2s · S)Re(Γop ). We find the expectation value of the DC photon absorption rate
to be
(0)
(0)
Tr(δΓ ρ) = Re(Γop )(1 − 2s · hSi).
(A.67)
134
A.5.2
D1 Repopulation Pumping
Repopulation is a regeneration of ground-state atoms due to spontaneous decay or quenching
decay of excited atoms. The evolution equation of the density matrix of repopulation
pumping cause by spontaneous decay is [22]
4
4ω03
d (g)
(e)
(e)
DA · ρ D∗ A† = Γs A · ρ A† .
ρ =
dt
eh̄c3
3
(A.68)
The spontaneous decay rate Γs = 2ω02 re feg /c. For quenching decay, we assume that the electron polarization is completely destroyed. Comparing to ϕ, which is the ground-state density matrix without the electronic polarization, we find the evolution equation for quenching
process is given by
d (g)
ρ = Γq
dt
µ
¶
1 (e) †
(e)
†
Uρ U + A · ρ A .
4
(A.69)
Here, U = |S, 12 ihJ, 12 | + |S, − 12 ihJ, − 12 | is the mapping operator. To evaluate Eq. (A.68) and
Eq. (A.69), we need to find the steady solution of the density matrix of the excited state.
The injection population rate Υkl appeared in Eq. (A.30) can be calculated by applying
Eq. (A.48). Therefore, we find
1 e (1)
e (1) ] = d ρ̃kl
[ Hlν ρ̃νl − ρ̃kν H
(A.70)
νl
ih̄
dt
i
|D|2 h
†
†
∗
∗
=
Ê(ω
)
·
A
ρ̃
A
·
E
+
E
·
A
ρ̃
A
·
Ê
(ω
)
ei(ωkl −ωνµ )t
νµ
νµ
kν
µl
µl
lµ
kν
kν
h̄2
(op)
(op)
= 2(Γkν + Γµl )(ε̂ · A†kν ρ̃νµ Aµl · ε̂∗ )ei(ωkl −ωνµ )t .
e kl =
Υ
e kl is defined by Υkl = Υ
e kl e−iωkl t . For the modulated light, we only care about the
Here, Υ
pumping rate with frequency components of ω = 0 and ω = 2ωm ∼ ω0 , where ω0 is the
(op)
(0)
(1)
(−1)
selected ground-state resonance frequency. Hence, Γkν = Γkν +Γkν e−i2ωm t +Γkν ei2ωm t . In
our experiments, we used nitrogen as the buffer gas, which produced minimal J-damping relaxation in the excited state. From Eq. (A.31), we find the steady value of ρkl in Schröedinger
frame to be
ρkl =
X
µ,ν;n=(0,±1)
=
X
µ,ν;n=(0,±1)
(n)
i(ωkl − ωµν
Υkl
− 2nωm ) + Γs + Γq
2(Γkµ + Γνl )(ε̂ · A†kµ ρµν Aνl · ε̂∗ )e−i2nωm t
(n)
(n)
i(ωkl − ωµν − 2nωm ) + Γs + Γq
.
(A.71)
135
For our experimental condition, the buffer gas leads to a quenching-dominant deexcitation,
which is Γq À Γs . Therefore, by using Eq. (A.69), we find the regeneration of the groundstate population due to the repopulation pumping as
d
ραβ =
dt
X
µ,ν,n,k,l
µ
Γq
(n)
(n)
2(Γkµ + Γνl )e−i2nωm t
i(ωkl − ωµν − 2nωm ) + Γq
×
¶
1
†
†
∗ †
∗ †
Uαk ε̂ · Akµ ρµν Aνl · ε̂ Ulβ + Aαk · (ε̂ · Akµ ρµν Aνl · ε̂ Alβ ) .(A.72)
4
It is easier to write the repopulation equation in Liouville space,
(rp)
ρ̇αβ = Rαβ,µν ρµν ,
(A.73)
where the matrix elements of the repopulation operator R(rp) is
(rp)
Rij
=
(rp)
Rαβ,µν
=
X
(n)
(n)
2(Γkµ + Γνl )e−i2nωm t
×
i(ωkl − ωµν − 2nωm ) + Γq
n,k,l
µ
¶
1
†
†
∗ †
∗ †
Γq
Uαk ε̂ · Akµ Aνl · ε̂ Ulβ + Aαk · (ε̂ · Akµ Aνl · ε̂ Alβ ) .(A.74)
4
(rp)
Here, we find the “real transition” light shift operator ih̄Im(Rµν,µν ) for the density-matrix
element ρµν . The light shift here is due to the real transition. The ground-state coherence is optically pumped to the excited state. The difference of the coherence frequencies
between the ground-state and the excited state causes a difference of time-phase between
the repopulated coherence and the ground-state coherence. Hence, the effective coherence
oscillation is dragged or pushed by the repopulation pumping, which leads to a frequency
shift.
Typically, when the buffer-gas pressure is above hundred torr, the quenching rate is
(e)
larger than the excited hyperfine splitting (Γq > ωhf ). If all ground-state hyperfine sublevels
see the same pumping rate, we can simplified Eq. (A.72) for the diagonal elements of the
density matrix into
d (g)
ρ
dt
³
´
(g)
(g)
= 4Re(Γop ) U ε̂ · A† ρ A · ε̂∗ U † + A · (ε̂ · A† ρ A · ε̂∗ A† )
µ
¶
1
= Re(Γop )
{1 − 4s · S, ρ} + (S · ρS − is · (S × ρS)) .
8
(A.75)
136
(rp)
It is understood that Eq. (A.75) is only valid for ρ̇αα = Rαα,µν ρµν . Combining depopulation
pumping, we find the longitudinal evolution equation of the density matrix to be
d (g)
ρ
dt
µ
¶
1
1
= − {δΓ, ρ} + Re(Γop )
{1 − 4s · S, ρ} + (S · ρS − is · (S × ρS))
2
8
·
¸
1
1
= Re(Γop ) − {1 − 2s · S, ρ} + {1 − 4s · S, ρ} + (S · ρS − is · (S × ρS))
2
8
= Re(Γop ) [ϕ(1 + 2s · S) − ρ] .
(e)
(A.76)
(g)
When ωhf < Γq < ωhf , the result of Eq. (A.76) is good for the calculation of ρ̇αα =
(rp)
(g)
Rαα,µν ρµν . When Γq > ωhf , the result of Eq. (A.76) is valid for the calculation of all
(rp)
density-matrix elements, which is ρ̇αβ = Rαβ,µν ρµν .
In this subsection of repopulation pumping, we assume that the buffer gas, such as
nitrogen, has strong quenching ability but small J-damping in the excited state. However,
many buffer gases, such as helium and argon, can produce large J-damping rates in the
excited state. The equations shown above are on longer valid. For the additional J-damping
mechanism in the excited state, it is easier to numerically carry out the calculation in
Liouville space. There is no simple form that can be found by using the regular formalism
of the density matrix.
Appendix B
Circuit Diagram of Signal Divider
C3
1uF
R3 200
To B
4
+15V
2
U2A
TL082
U1 AD734
1
3
1
2
+
BNC
PD1 in
BNC
PD1 out
+VCC
DD
14
13
C1
1uF
8
To C
8
To A
3
BNC
PD2 out
2
W
Z1
Z2
U4A
TL082
+
1
-
3
4
5
R1
2M
10nF
6
7
4
C4
X1
X2
R4 20k
ERREF
U0
U1
U2
Y1
Y 2 -VCC
12
11
10
BNC
(PD1/PD2) x Gain out
9
C2 1uF
8
To D
U3B
TL082
BNC
Gain out
-15V
7
8
4
5
8
-
+
To C
To D
D1
20V
6
5
+
BNC
PD2 in
U5B
TL082
-
6
7
4
To B
R2
200k
To A
0.1uF
C9
A
B
C6 1uF
C
C5 1nF
+15V
-15V
D
C7 1uF
C8 1nF
+15V
-15V
R5
5k
+15V
Figure B.1: Because of this divider circuit, we were able to obtain the first light narrowing
data from the fiber-coupled and noisy Ti-sapphire laser light.
137
References
[1] 1 amagat (amg) = 2.687 × 1019 cm−3 .
[2] M. S. Albert, G. D. Cates, B. Driehuys, W. Happer, B. Saam, C. S. Springer, and
A. Wishnia. Biological magnetic-resonance imaging using laser polarized 129xe. Nature,
370(6486):199–201, 1994.
[3] S. Appelt, A. B.-A. Baranga, C. J. Erickson, M. V. Romalis, A. R. Young, and W. Happer. Theory of spin-exchange optical pumping of 3he and 129xe. Phys. Rev. A,
58(2):1412–1439, 1998.
[4] E. Arimondo. Coherent population trapping in laser spectroscopy, volume 35 of Progress
in Optics, pages 257–354. North-Holland Pub. Co., Amsterdam, 1996.
[5] M. P. Augustine and K. W. Zilm. Optical pumping magnetic resonance in high magnetic fields: Measurement of high field spin exchange cross sections. Chem. Phys. Lett.,
280(1-2):24–30, 1997.
[6] J. P. Barrat, J. Brossel, and A. Kastler. Production optique dune orientation atomique dans la vapeur saturante de sodium. Comp. Rend. Hebdom. Sean. Acad. Sci.,
239(19):1196–1198, 1954.
[7] B. L. Bean and R. H. Lambert. Temperature-dependence of hyperfine density shifts:
23na, 39k, and 85rb in he, ne, ar, and n2 at low temperatures. Phys. Rev. A, 13(1):492–
494, 1976.
138
139
[8] W. E. Bell and A. L. Bloom. Optically driven spin precession. Phys. Rev. Lett.,
6(6):280–281, 1961.
[9] N. Beverini, P. Minguzzi, and F. Strumia. Foreign-gas-induced cesium hyperfine relaxation. Phys. Rev. A, 4(2):550, 1971.
[10] N. D. Bhaskar, M. Hou, B. Suleman, and W. Happer. Propagating, optical-pumping
wave fronts. Phys. Rev. Lett., 43(7):519–521, 1979.
[11] M. A. Bouchiat, J. Brossel, and L. Pottier. Interpretation of experimental results
on relaxation of optically pumped rb in collisions with kr atoms. Phys. Rev. Lett.,
19(15):817, 1967.
[12] M. A. Bouchiat, J. Brossel, and L. C. Pottier. Evidence for rb rare-gas molecules from
relaxation of polarized rb-atoms in a rare-gas - experimental results. J. Chem. Phys.,
56(7):3703, 1972.
[13] M. A. Bouchiat, T. R. Carver, and C. M. Varnum. Nuclear polarization in he-3 gas
induced by optical pumping and dipolar exchange. Phys. Rev. Lett., 5(8):373–375,
1960.
[14] J. Brossel, A. Kastler, and J. Winter. Greation optique dune inegalite de population
entre les sous-niveaux zeeman de letat fondamental des atomes. J. Phys. Radium,
13(12):668–668, 1952.
[15] U. Buck and H. Pauly. Interferences in atomic collision processes and their interpretation using a modified lennard-jones potential. Z. Phys., 208(4):390, 1968.
[16] G. D. Cates, R. J. Fitzgerald, A. S. Barton, P. Bogorad, M. Gatzke, N. R. Newbury,
and B. Saam. Rb-129 xe spin-exchange rates due to binary and three-body collisions at
high xe pressures. Phys. Rev. A, 45(7):4631–4639, 1992.
[17] E. Clememnti and C. Roetti. Atom. Data Nucl. Data Tabl., 14:177, 1974.
140
[18] E. U. Condon and G. H. Shortley. The Theory of Atomic Spectra. Cambridge University
Press, 1959.
[19] D. E. Nikonov et al. Atomic coherence effects within the sodium d1 manifold: Coherent
optical-pumping. Quant. Opt., 6(4):p. 257, 1994.
[20] C. J. Erickson. Measurement of the magnetic field depence of the spin-relaxation rate
in alkali metal vapors. PhD thesis, Princeton University, May 2000.
[21] B. C. Grover. Noble-gas nmr detection through noble-gas - rubidium hyperfine contact
interaction. Phys. Rev. Lett., 40(6):391–392, 1978.
[22] W. Happer. Optical pumping. Rev. Mod. Phys., 44(2):169–249, 1972.
[23] P. Hu, S. Geschwind, and T. M. Jedju. Spin-flip raman echo in n-type cds. Phys. Rev.
Lett., 37(20):1357V1360, 1976.
[24] K. Ishikawa, S. Morii, K. Enomoto, and T. Yabuzaki. Visualization of foreign atoms by
opitcal magnetic-resonance imaging of cs atoms. J. Opt. Soc. Am. B, 17(2):182–187,
2000.
[25] Y.-Y. Jau, N. N. Kuzma, and W. Happer. High-field measurement of the 129xe-rb
spin-exchange rate due to binary collisions. Phys. Rev. A, 66:052710, 2002.
[26] Y.-Y. Jau, N. N. Kuzma, and W. Happer. Magnetic decoupling of 129xe-rb and 129xecs binary spin exchange. Phys. Rev. A, 67:022720, 2003.
[27] Y.-Y. Jau, N. N. Kuzma, and W. Happer. Measurement of 129xe-cs binary spinexchange rate due to binary collisions. Phys. Rev. A (RC), 69:061401, 2004.
[28] Y.-Y. Jau, E. Miron, A. B. Post, N. N. Kuzma, and W. Happer. Push-pull optical
pumping of pure superposition states. Phys. Rev. Lett., 93(16):160802, 2004.
[29] Y.-Y. Jau, A. B. Post, N. N. Kuzma, A. M. Braun, M. V. Romalis, and W. Happer.
Intense, narrow atomic-clock resonances. Phys. Rev. Lett., 92(11):110801, 2004.
141
[30] S. Kadlecek, L. W. Anderson, and T. G. Walker. Field dependence of spin relaxation
in a dense rb vapor. Phys. Rev. Lett., 80(25):5512–5515, 1998.
[31] V. A. Kartoshkin. Complex cross-sections for the spin-exchange between cs and rb
alkali atoms. Opt. Spektrosk., 79(1):26–31, 1995.
[32] A. Kastler. Quelques suggestions concernant la production optique et la detection
optique dune inegalite de population des niveaux de quantification spatiale des atomes
- application a lexperience de stern et gerlach et a la resonance magnetique. J. Phys.
Radium, 11(6):255–265, 1950.
[33] T. J. Killian. Thermionic phenomena caused by vapors of rubidium and potassium.
Phys. Rev., 27(5):578–587, 1926.
[34] M. P. Ledbetter and M. V. Romalis. Nonlinear effects from dipolar interactions in
hyperpolarized liquid 129xe. Phys. Rev. Lett., 89(28):287601, 2002.
[35] H. Li, A. Hohl, A. Gavrielides, H. Hou, and K. D. Choquette. Stable polarization selfmodulation in vertical-cavity surface-emitting lasers. Appl. Phys. Lett., 72(19):2355–
2357, 1998.
[36] Z. D. Liu, M. X. Zhao, C. J. Wu, L. Y. Li, S. L. Li, X. Z. Zeng, and W. Xiong.
Nmr-study of low-pressure 129xe gas. Chem. Phys. Lett., 194(4-6):440–445, 1992.
[37] T. H. Maiman. Optical and microwave-optical experiments in ruby. Phys. Rev. Lett.,
4(11):564–567, 1960.
[38] P. J. Oreto, Y.-Y. Jau, A. B. Post, N. N. Kuzma, and W. Happer. Buffer-gas-induced
shift and broadening of hyperfine resonances in alkali-metal vapors. Phys. Rev. A,
69(4):042716, 2004.
[39] C. V. Rice and D. Raftery. Rubidium-xenon spin exchange and relaxation rates measured at high pressure and high magnetic field. J. Chem. Phys., 117(12):5632–5641,
2002.
142
[40] F. Robert, P. Besnard, M. L. Chares, and G. M. Stephan. Polarization modulation
dynamics of vertical-cavity surface-emitting lasers with an extended cavity. IEEE J.
Quantum Electron., 33(12):2231–2239, 1997.
[41] M. V. Romalis and M. P. Ledbetter. Transverse spin relaxation in liquid 129xe in the
presence of large dipolar fields. Phys. Rev. Lett., 87(6):067601, 2001.
[42] M. V. Romalis, E. Miron, and G. D. Cates. Pressure broadening of rb d1 and d2 lines
by 3he, 4he, n2, and xe: Line cores and near wings. Phys. Rev. A, 56(6):4569–4578,
1997.
[43] D. Strekalov, D. Aveline, N. Yu, R. Thompson, A. B. Matsko, and L. Maleki. Stabilizing
an optoelectronic microwave oscillator with photonic filters. J. Lightwave Technol.,
21(12):3052–3061, 2003.
[44] S. D. Swanson, M. S. Rosen, B. W. Agranoff, K. P. Coulter, R. C. Welsh, and T. E.
Chupp. Brain mri with laser-polarized 129xe. Magn. Reson. Med., 38(5):695–698, 1997.
[45] J. B. Taylor and I. Langmuir. Vapor pressure of caesium by the positive ion method.
Phys. Rev., 51(9):753–760, 1937.
[46] D. G. Thomas and J. J. Hopfield. Spin-flip raman scattering in cadmium sulfide. Phys.
Rev., 175(3):1021V1032, 1968.
[47] J. Vanier, M. W. Levine, D. Janssen, and M. Delaney. Contrast and linewidth of the
coherent population trapping transmission hyperfine resonance line in 87rb: Effect of
optical pumping. Phys. Rev. A, 67(6):065801, 2003.
[48] J. Vanier, J.-F. Simard, and J.-S. Boulanger.
Relaxation and frequency-shifts in
ground-state of rb85. Phys. Rev. A, 9(3):1031–1040, 1974.
[49] D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii. Quantum Theory of
Angular Momentum. World Scientific, Singapore, 1988.
143
[50] C. H. Volk, T. M. Kwon, and J. G. Mark. Measurement of the 87rb-129xe spin-exchange
cross section. Phys. Rev. A, 21(5):1549–1555, 1980.
[51] T. G. Walker. Estimates of spin-exchange parameters for alkali-metal noble-gas pairs.
Phys. Rev. A, 40(9):4959–4963, 1989.
[52] D. K. Walter. Optical probes of angular momentum and energy transport in spinexchange optical pumping system. PhD thesis, Princeton University, 2001.
[53] D. K. Walter, W. M. Griffith, and W. Happer. Magnetic slowing down of spin relaxation
due to binary collisions of alkali-metal atoms with buffer-gas atoms. Phys. Rev. Lett.,
88(9):093004, 2002.
[54] D. K. Walter, W. Happer, and T. G. Walker. Estimates of the relative magnitudes
of the isotropic and anisotropic magnetic-dipole hyperfine interactions in alkali-metalnoble-gas systems. Phys. Rev. A, 58(5):3642–3653, 1998.
[55] Z. Wu, M. Kitano, W. Happer, M. Hou, and J. Daniels. Optical determination of alkalimetal vapor number density using faraday rotation. Appl. Optics, 25(23):4483–4492,
1986.
[56] R. Wynands. Physics - the atomic wrist-watch. Nature, 429(6991):509–510, 2004.
[57] J. Yu, M. Schell, M. Schulze, and D. Bimberg. Fourier-limited 1.6-ps pulses with
variable repetition rate from 1 to 26 ghz by passive mode-locking of a semiconductorlaser in an external-cavity. 7(5):467–469, 1995.
[58] X. Zeng, Z. Wu, T. Call, E. Miron, D. Schreiber, and W. Happer. Experimentaldetermination of the rate constants for spin exchange between optically pumped k, rb,
and cs atoms and 129xe nuclei in alkali-metal noble-gas vanderwaals molecules. Phys.
Rev. A, 31(1):260–278, 1985.
144
[59] X. Z. Zeng, C. J. Wu, M. X. Zhao, S. L. Li, L. Y. Li, X. T. Zhang, Z. D. Liu, and
W. Y. Liu. Laser-enhanced low-pressure gas nmr signal from xe-129. Chem. Phys.
Lett., 182(6):538–540, 1991.