HIGH-PRECISION MEASUREMENTS IN
ATOMIC CESIUM SUPPORTING A
LOW-ENERGY TEST OF THE STANDARD
MODEL
by

STEPHEN CARROLL BENNETT
B. A., Middlebury College, 1992
M. S., University of Colorado, 1995

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial ful llmentof the requirements for the degree ofDoctor of Philosophy
Department of Physics
1998

This thesis entitled:
High-Precision Measurements in Atomic Cesium Supporting a Low-Energy Test of
the Standard Model
written by Stephen Carroll Bennett
has been approved for the Department of Physics
byCarl E. Wieman

Eric A. Cornell

Date

The nal copy of this thesis has been examined by the signatories, and we nd thatboth the content and the form meet acceptable presentation standards of scholarly
work in the above mentioned discipline.

iii
Bennett, Stephen Carroll (Ph. D., Physics)
High-Precision Measurements in Atomic Cesium Supporting a Low-Energy Test of
the Standard Model
Thesis directed by Professor Carl E. Wieman
The measurement of parity nonconservation (PNC) in atomic cesium provides the most precise low-energy test of the standard model of electroweak interactions. However, the test is limited by the uncertainty in the ab initio calculations thatare required to interpret the measurement. This thesis describes one measurementthat suggests that the accuracy of the theory may be better than its authors claim,
and a second measurement that may be used in place of a less accurate calculation.
A 0.11% measurement of the dc Stark shift of the 6S ! 7S transition in cesium using high-precision laser spectroscopy removes the largest discrepancy between
experiment and theory. With this new measurement, and several recently improved
measurements by other groups, the uncertainty of the theory is re-evaluated. We
nd that the standard deviation of the di erences between experiment and theory is
0.40%. This standard deviation suggests that the quoted uncertainty of the theory
can be reduced from 1% to 0.40%.
A 0.16% measurement of Mhf = using similar methods is also presented.
The quantity Mhf is the o -diagonal hyper ne-interaction-induced magnetic dipole
amplitude, and is the tensor transition polarizability; both are for the 6S ! 7S
transition in cesium. This ratio is combined with a 0.25% semi-empirical determination of Mhf from another group to determine the value of with a precision of
0.30%. Previously, the value of used in the test of the standard model was calculated using the ab initio theory, thus increasing the uncertainty due to theory in
the nal test. Using the new measured value of , the current values of the theory
with improved precision, and the previous measurement of PNC in cesium we makea 0.61% test of the standard model.
The precision of this test is likely to motivate further work in this eld.
Therefore, two experiments are discussed that may be useful in improving the signalto-noise ratio on a future PNC measurement: an experiment that phase modulates
the dye laser used to drive the 6S ! 7S transition to eliminate the spatial intensity
variation inside a Fabry-Perot etalon, and an experiment that transversely cools theatomic cesium beam. Finally, a di erent{but related{PNC measurement is discussed.

For Rebecca.

ACKNOWLEDGEMENTS
This document has but a single author; every other document upon which
my name has been placed has at least two, and usually more. The simple fact ofthe matter is that in all of these works, the number of people who contributed is
far larger than the author list would suggest. I could not have produced this thesis
without the help of a very long list of people both inside JILA and out, where they
helped me along my professional and personal journeys. The list is, of course, toolong to remember to thank everyone, but here is where I shall try.
First and foremost I must thank my advisor Dr. Carl Wieman. Somehow
Carl saw something in me worth growing when I was a rst-year graduate student.
Although he was uncertain, at times, whether I would nish graduate school witha Ph. D., he never doubted that I could. He has taught me a great deal abouthigh-precision experiments and how to do them right. I probably taught him athousand ways he never dreamed about to do them wrong. I am a much betterscientist because of his tutelage, and for that he has my gratitude.
There are two colleagues who worked with me daily until they went theirseparate ways: Jake Roberts and Chris Wood. Chris, being the senior graduate
student, taught me lots about the ins and outs of the experimental details. Jake,
despite being the \junior" graduate student, was an excellent source of ideas, andwas always a valuable resource for when I did not completely understand the theory,or when I needed someone to gure out how to analyze the data I just took while I
slaved away taking more. Thanks to you both.
I am convinced that JILA is a such a great place to work because of the
support sta .
In no particular order, Hans Greene, Hans Rohner, Blaine Horner, Dave
Alchenberger, and Seth Wieman, were always around in the shop to help me buildthat part I needed yesterday, or to solve what I usually thought was an unsolvableproblem. Terry Brown, Paul Buckingham, James Fung-A-Fat, and Mike Whitmore
were invaluable for their help with electronics. Whether it was trying to gureout what was wrong with my high-voltage ampli er, or trying to get my favorite
oscilloscope xed, they were always willing to help.
There is a group of people in JILA who some would call \secretaries" or\assistants." I call them saviors. Many, many thanks to Karen Melcher, Fran Haas,
Leslie Haas, Lisa Perry, LaReina Romero, Agnes Polanowski, and Susannah Jarvis,
who take care of all the details, make the co ee, and make JILA a more pleasant
place to live. . . um, I mean work.

vi
The rst time I ordered something from the Supply Oce, I think I got
on Ed's bad side. I quickly corrected that mistake and stayed on his good side eversince (I think). Everyone should give Ed Holliness a big thanks, because his ocemakes buying equipment painless, and without him we'd have to buy our own stu !
I did not nd out how valuable the Scienti c Reports Oce is until I started
writing real papers. Many thanks must go to Marilee DeGoede and Laurie Kovalenko
for taking care of a lot of the little details I never realized existed.
On a more personal note, I have had many friends who have helped me out
along the way. Thanks to Andy Steinbach for studying with me for comps. Thanks
to Thomas Crawford for commiserating with me during the bad times. Thanks
to Ruth Rhodes for all the trips up to Dairy Queen for Blizzards (and where are
you now?). Thanks to Michelle{then Pierce, now Ruprecht{and Pete Ruprecht for
all the card playing (do you still owe me a bomber?). Thanks to everyone at the\Physics House" for all our Thanksgivings and Christmases together. Thanks to
Matt Wingate for being a great roommate until Rebecca came along. Thanks to
Mrs. Sama for making me hate English. Thanks to Rich Wolfson and Je Dunhamfor helpful advice when making big decisions. Thanks to Mom, Dad, and Kristy forbeing proud.
One friend and colleague deserves special mention: Kristan Corwin. I am
not sure how it worked out this way, but Kristan always seemed to be around when I
wanted to talk to someone because I was bored, because I had an exciting new result
to share, because I had a problem I did not understand, because I needed someone
to cheer me up{whatever. My graduate school experience was much more pleasant,and I have a di erent outlook because of her. Thank you very much, Kristan.
Finally, there are no words to express my gratitude to my dear wife Rebecca.
This is the second thesis she has had to live through, and I promise there will not bea third. Rebecca has been with me through thick and thin and has supported myevery decision, even when she felt that she did not. It is no understatement to saythat without her support I would not still be in physics. Thank you, Rebecca. Now
it is your turn.

CONTENTS

CHAPTER
1 INTRODUCTION
1.1 Search for Understanding . . . . . . . . . . . . . . . . . . . . . . . .
1.2 The Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Parity Nonconservation in Atoms . . . . . . . . . . . . . . . . . . . .
1.4 Parity Nonconservation Interference Measurements in Cesium . . . .
1.5 Motivation for the Present Work . . . . . . . . . . . . . . . . . . . .

1
1
3
4
5
6

2 THEORY
2.1 Weak Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Cesium Energy Level Structure. . . . . . . . . . . . . . . . . . . . . .
2.3 Electric Dipole Amplitudes . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Parity Nonconserving Amplitude . . . . . . . . . . . . . . . .
2.3.2 Stark-Induced Amplitude . . . . . . . . . . . . . . . . . . . .
2.4 Magnetic Dipole Amplitudes . . . . . . . . . . . . . . . . . . . . . .
2.5 Electric Quadrupole Interaction . . . . . . . . . . . . . . . . . . . . .
2.6 E ect of Misaligned B~ . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7 Measured Transition Rates . . . . . . . . . . . . . . . . . . . . . . .
2.8 Zeeman E ect in Cesium . . . . . . . . . . . . . . . . . . . . . . . .
2.8.1 E ect on Eigenstates . . . . . . . . . . . . . . . . . . . . . . .
2.8.2 E ect on Spectrum . . . . . . . . . . . . . . . . . . . . . . . .

8
8
10
12
12
13
15
16
16
18
20
20
20

viii
3 APPARATUS
3.1 The Cesium Oven . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Optical Pumping Region . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Interaction Region . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Electric and Magnetic Field Production . . . . . . . . . . . .
3.3.2 Dye Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.3 Fast Frequency Stabilization . . . . . . . . . . . . . . . . . .
3.3.4 Power Build-up Cavity . . . . . . . . . . . . . . . . . . . . . .
3.3.5 Slow Frequency Stabilization . . . . . . . . . . . . . . . . . .
3.4 Detection Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5 Data Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25
27
29
31
31
31
33
33
35
36
37

4 MEASURING THE STARK SHIFT OF THE 6S ! 7S TRANSITION
4.1 Theory of the dc Stark E ect . . . . . . . . . . . . . . . . . . . . . .
4.2 Details of the Experiment . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Data Scans and Frequency Drift . . . . . . . . . . . . . . . .
4.2.2 Coupling Light into the Reference Cavity . . . . . . . . . . .
4.2.3 Production and Measurement of the Electric Field . . . . . .
4.3 Determination of the dc Stark Shift . . . . . . . . . . . . . . . . . . .
4.3.1 Determination of Line Centers . . . . . . . . . . . . . . . . .
4.3.2 Determination of Frequency Separation of two Electric Fields
4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40
41
43
43
44
48
49
49
51
51

5 MEASURING THE TENSOR TRANSITION POLARIZABILITY
5.1 Experimental Concept . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Experimental Considerations . . . . . . . . . . . . . . . . . . . . . .
5.2.1 Measuring the Areas Correctly . . . . . . . . . . . . . . . . .
5.2.2 Comparing the Areas Correctly . . . . . . . . . . . . . . . . .
5.2.3 Photoionization Correction . . . . . . . . . . . . . . . . . . .

58
59
60
60
62
64

ix
5.2.4 Non-Bz Magnetic Field Systematic Error . . . . . . . . . . .
5.3 Preliminary Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4 Accounting for the Electric Quadrupole Amplitude . . . . . . . . . .

65
68
71

6 CONCLUSIONS AND IMPLICATIONS
6.1 Re-evaluation of Atomic Theory . . . . . . . . . . . . . . . . . . . .
6.2 Final Result and Implications . . . . . . . . . . . . . . . . . . . . . .

74
74
77

7 CONSIDERATIONS FOR FUTURE EXPERIMENTS
7.1 Cavity Sidebands . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.1.1 Experimental Idea . . . . . . . . . . . . . . . . . . . . . . . .
7.1.2 Experimental Implementation . . . . . . . . . . . . . . . . . .
7.1.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.1.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Improving the Signal-to-Noise Ratio . . . . . . . . . . . . . . . . . .
7.2.1 Increasing Beam Flux . . . . . . . . . . . . . . . . . . . . . .
7.2.2 Transversely Cooling the Atomic Beam . . . . . . . . . . . .
7.2.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3 Prospects for Further Parity Nonconservation Measurements . . . .
7.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

80
81
81
84
86
86
87
87
88
94
95
97

BIBLIOGRAPHY

115

APPENDIXF 0 M 0 COEFFICIENTS
A CFM

121

B EFFECT OF THE ELECTRIC QUADRUPOLE AMPLITUDE

123

C DERIVATION OF INTRACAVITY INTENSITY WITH SIDEBANDS

132

FIGURES

Figure
1.1 Picture Showing Beta Decay . . . . . . . . . . . . . . . . . . . . . . .
1.2 Feynman Diagrams of Electromagnetic and Weak Interactions . . . .

2
4

2.1
2.2
2.3
2.4
2.5

Lowest Energy Levels of the Cesium Atom . . . . . . . . . . . . . . .
Misaligned B~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Four 6S ! 7S Hyper ne Transitions . . . . . . . . . . . . . . . . . .
Zeeman Spectrum With A Small Magnetic Field . . . . . . . . . . .
Scans of the E 1 Transition With B = 0 and B = 6:6G . . . . . . . .

11
17
21
22
24

3.1
3.2
3.3
3.4
3.5
3.6
3.7

Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Detail of the Atomic Beam Path . . . . . . . . . . . . . . . . . . . .
The Cesium Oven . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Dye Laser Frequency Locking Schematic . . . . . . . . . . . . . . . .
Mechanical Design of the Power Build-up Cavity . . . . . . . . . . .
Schematic of Data Acquisition System . . . . . . . . . . . . . . . . .
Typical Data for Measurement of the dc Stark Shift . . . . . . . . .

26
27
28
32
34
38
39

4.1
4.2
4.3
4.4

Sample Scans at Various Electric Fields . . . . . . . . . . . . . . . .
A Double-Passed AOM . . . . . . . . . . . . . . . . . . . . . . . . .
Three Modes of a Reference Cavity . . . . . . . . . . . . . . . . . . .
Shifted Lock Points Due to Changing Mode Coupling . . . . . . . .

43
45
47
48

xi
4.5 Voltage Divider Measurement of E . . . . . . . . . . . . . . . . . . .
4.6 The 6S ! 7S Transition With and Without Saturation . . . . . . .
4.7 Typical Drift in Line Centers . . . . . . . . . . . . . . . . . . . . . .
4.8 Determination of k from Pairs of Electric Field Values . . . . . . . .
4.9 Determination of k from a Fit to Stark = kE 2 . . . . . . . . . . . .
4.10 Comparison of dc Stark Shift Values . . . . . . . . . . . . . . . . . .

49
53
54
55
56
57

Hypothetical 6S ! 7S Scan . . . . . . . . . . . . . . . . . . . . . . .
Comparison of E 1 and M 1 Line Shapes . . . . . . . . . . . . . . . .
Data and Theory of Misaligned Bx . . . . . . . . . . . . . . . . . . .
Measured Ratios Mhf = at Di erent Values of Bz . . . . . . . . . . .
Plot of (M , Mhf )= vs. Laser Intensity . . . . . . . . . . . . . . . .
An Historical Comparison of Values of . . . . . . . . . . . . . . . .

60
64
67
68
70
73

6.1 Comparison of Values of QW . . . . . . . . . . . . . . . . . . . . . . .

79

5.1
5.2
5.3
5.4
5.5
5.6

7.1 Phase Modulation Sidebands . . . . . . . . . . . . . . . . . . . . . . 82
7.2 Intensity Inside PBC With Sidebands . . . . . . . . . . . . . . . . . 84
7.3 Modi ed Mounting Scheme of the PBC Input Mirror . . . . . . . . . 85
7.4 Line Shapes With and Without Sidebands . . . . . . . . . . . . . . 98
7.5 Atomic Beam Size vs. Oven Temperature. . . . . . . . . . . . . . . . 99
7.6 Spin Polarization vs. Beam Size. . . . . . . . . . . . . . . . . . . . . 100
7.7 Dressed States of an Atom. . . . . . . . . . . . . . . . . . . . . . . . 101
7.8 Transverse Cooling by Stimulated Emission . . . . . . . . . . . . . . 102
7.9 Doppler Pro les of the Atomic Beam at Various Collimator Positions 103
7.10 Transverse Velocity Center at 0.6 MHz . . . . . . . . . . . . . . . . . 104
7.11 Transverse Velocity Center at 3.1 MHz . . . . . . . . . . . . . . . . . 105
7.12 Transverse Velocity Center at 5.5 MHz . . . . . . . . . . . . . . . . . 106
7.13 Transverse Velocity Center at 7.9 MHz . . . . . . . . . . . . . . . . . 107

xii
7.14 Transverse Velocity Center at 10.3 MHz . . . . . . . . . . . . . . . . 108
7.15 Transverse Velocity Center at 12.3 MHz . . . . . . . . . . . . . . . . 109
7.16 Convolution of a Population Distribution Ansatz with the Natural
Line Shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
7.17 Velocity Distribution of the Atomic Beam with No Collimation . . . 111
7.18 Fractional E ect of Cooling as a Function of Doppler Shift . . . . . . 112
7.19 Population Distribution of the Atomic Beam With and Without Transverse Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
7.20 Atomic Beam and 6S ! 7S Line Shapes With and Without Cooling 114

xiii

TABLES

Table
5.1 Population Distributions After Hyper ne Pumping . . . . . . . . . .
5.2 Raw Data for (M  Mhf )= . . . . . . . . . . . . . . . . . . . . . . .
5.3 Final Results for the Mhf = Experiment . . . . . . . . . . . . . . . .

66
69
73

6.1 Comparison of Theory with Experiment . . . . . . . . . . . . . . . .

77

B.1 E2 Transition Rates With Uniform Hyper ne Populations for the F =3 to F 0 = 4 Transition . . . . . . . . . . . . . . . . . . . . . . . . . .
B.2 E2 Transition Rates With Uniform Hyper ne Populations for the F =4 to F 0 = 3 Transition . . . . . . . . . . . . . . . . . . . . . . . . . .
B.3 E2 Transition Rates With Nonuniform Hyper ne Populations for the
F = 3 to F 0 = 4 Transition . . . . . . . . . . . . . . . . . . . . . . .
B.4 E2 Transition Rates With Nonuniform Hyper ne Populations for the
F = 4 to F 0 = 3 Transition . . . . . . . . . . . . . . . . . . . . . . .

128
129
130
131

CHAPTER 1
INTRODUCTION
1.1 Search for Understanding
At some very basic level, physicists are driven by the desire to understandthe world around them. We perform experiments and invent theories that probe anddescribe how things work. Good experiments are sometimes described as \beautiful" and simple theories are sometimes called \elegant." It seems as if there is anunderlying desire to have nature, its behavior, and descriptions of its behavior be
beautiful, elegant, and simple.
When formulating theories one would like things to be symmetric in someway, just like nature. For example, the physics of a particle in motion is the same
regardless of whether time is running forward or backward. Indeed, symmetries
are so fundamental that the de nite relationship between invariance or symmetry
properties of a system and its conserved quantities is described in the mathematical
Noether's Theorem [1]. So, when a symmetry is broken, physicists take notice.
Such was the case when parity nonconservation (PNC) was rst observedin nuclear beta decay by Wu and collaborators [2]. In this experiment, the scientists
studied the decay 27 Co60 !28 Ni60 + e +  by measuring the direction of the emitted
electrons relative to the magnetic moment of the 27 Co60 nuclei. They found that thedirection of emission was not symmetric with respect to the plane perpendicular to
the magnetic moment. Instead, there is a preferred direction of emission that is \left-

2
handed" with respect to the direction of an imaginary current that would produce
the same magnetic moment of the 27 Co60 nucleus. The physics of the mirror image (aparity reversal) of this reaction is not the same, as shown in Fig. 1.1. In the mirror
image, the electron emission is right handed. Thus, the decay does not conserve
parity.

Figure 1.1: A picture of beta decay in the 27 Co60 nucleus. Electrons are emitted
in a preferred direction opposite the magnetic moment of the nucleus. In a parity
reversal, the direction of the magnetic moment changes, but the direction of electron
emission does not. Thus, beta decay process does not conserve parity.

3

1.2 The Standard Model
The force that is responsible for beta decay is the weak force, which has
been uni ed with the electromagnetic force by a theory developed by Glashow [3],Salam [4], and Weinberg [5] in the 1960's. The Glashow-Salam-Weinberg model
uni es the two forces into the electroweak force and, when combined with quantum
chromodynamics, is known as the standard model. Part of this model describes theelectroweak interaction between six quarks (the up, down, strange, charm, bottom,
and top quarks) and six leptons (the electron, muon, tauon, and their three neutrinos). The electroweak force is mediated by four particles: the neutral photon andZ 0 boson, and the charged W  bosons.
Although the standard model predicts the electroweak interactions between
particles, it is silent as to the masses of those particles. There are also three additional parameters, the ne-structure constant, , the Fermi constant GF , and theWeinberg angle, W , whose values must be determined from experiment. In addition, the mechanism for the quarks and leptons to acquire mass is the so-called Higgs
mechanism, which is mediated by the Higgs boson. The Higgs boson has never been
seen, and the Higgs mechanism is poorly understood.
For the above reasons, physicists have been devoting massive amounts oftime and e ort to test the standard model. The high energy physics community
has achieved great success in their experiments [6] (although the funding for theSuperconducting Super Collider, which may have found the Higgs boson, was terminated), and many in the atomic physics community have devoted their e orts tounderstanding PNC in atoms to test the standard model, as suggested by Bouchiat
and Bouchiat [7, 8]. Both communities' experiments have now reached the level ofprecision where they can test the radiative corrections and search for new physics
beyond the standard model.

4

1.3 Parity Nonconservation in Atoms
Parity nonconservation in atoms comes from the the exchange of a Z 0
boson between an electron and a quark inside the nucleus. This interaction is to
the weak force as the exchange of a photon between an electron and a proton is tothe electromagnetic force. Feynman diagrams comparing these two interactions areshown in Fig. 1.2. The exchange of a Z 0 boson gives rise to a Hamiltonian that isFigure 1.2: Two Feynman diagrams showing (a) electromagnetic interaction via an
exchange of a photon and (b) the weak interaction via the exchange of a Z 0 boson.
Here N is a nucleon.
proportional to the Dirac matrix 5 and the weak charge QW . The weak charge isa neutral analog to the electric charge e, and wherever there is an electromagnetic
interaction via the exchange of a photon there is also a neutral weak interaction.
In atoms, the neutral weak interaction mixes eigenstates with opposite parity, andthus provides a mechanism for electric dipole transitions that do not conserve parity.
Experiments that measure the PNC electric dipole transition give access to the valueof QW , which can then be compared to the standard model prediction.

5
There are two types of experiments that are typically used to measure thee ect of PNC mixing: optical rotation experiments and Stark interference experiments. In optical rotation experiments [9, 10, 11, 12], linear laser light is directed
through an atomic vapor. The PNC mixing induces birefringence in the vapor, andcauses a rotation of the polarization of the incident light. The angle of rotation ismeasured and can be used to determine QW . Stark interference experiments measure
the interference between the small PNC-induced amplitude and a larger electric- eld
induced electric dipole amplitude.

1.4 Parity Nonconservation Interference Measurements in Cesium
The most precise atomic PNC measurement to date [13] measures the interference between the PNC amplitude and the Stark-induced amplitude between
the 6S and 7S states of cesium. The PNC amplitude is given by

EPNC  h7S j D~ j 6S i = ( QNW )kPNC ;(1.1)

where the bars over the bra and the ket indicate that they have small amounts of
opposite parity states mixed into the pure parity eigenstates. Here, e is the electron
charge, a0 is the Bohr radius, N is the number of neutrons in the atom, and kPNC
is the value of a combination of the relevant parity conserving and PNC matrix
elements. It is calculated using the ab initio theory of Blundell et al. [14] and of
Dzuba et al. [15]. The quantity measured in Ref. [13] is the ratio of two amplitudes,
Im(EPNC )= , where is the tensor transition polarizability that characterizes thestrength of the Stark-induced transition. The test of the standard model is throughthe equation
Im(EPNC ) = ,i QW k :
(1.2)
N PNC
This equation contains the motivation for all of the work presented in this thesis.

6

1.5 Motivation for the Present WorkEquation (1.2) connects the measurement of Im(EPNC )= and the calculation kPNC , allowing the extraction of QW . This is atomic physics' link to thestandard model.
The left side of the equation is the 0.35% measurement of Ref. [13]. The
amplitude EPNC is too small to measure by itself, so the experiment measures it
relative to . The right side contains QW , which is the quantity we wish to know,
but it also contains two other parameters needed to interpret the experiment: kPNC
and .
The constant kPNC contains matrix elements of the Dirac matrix 5 thatcan only be calculated. Therefore, prior to this work, the uncertainty of kPNC waslimited by the 1% uncertainty in the atomic theory calculations. Further, while
has been determined semi-empirically, the best value was again from a 1% atomic
theory calculation [14].
Because the goal of PNC measurements in cesium is a high-precision testof the standard model, as a rst objective we would like to reduce the uncertainty
due to the calculations. The work presented in this thesis achieves this objective.
The main stumbling block in the way of reducing the uncertainty in kPNC
is a 2% di erence between the measurement and the calculation of the dc Stark
shift of the 6S ! 7S transition in cesium. Because the level of agreement between
experiment and theory is one indicator of the accuracy of the theory, this di erence
was of great concern and prevented a reduction in the uncertainty below 1%. In
order to resolve this problem, we have measured the dc Stark shift, and our new
measurement agrees with the predictions to 0.3%.
Our previous result for QW [13] used the calculated value of from Ref. [14].
We have performed a new measurement that allows the determination of to 0.3%,
which can then be used instead of the calculation. This reduces the number of7quantities that need to be calculated to determine QW from two to one. In addition,
our result for can be used as an additional test of the atomic theory.
This thesis is arranged as follows. Chapters 2 and 3 discuss the theory
and apparatus used in the experiments described in subsequent chapters, Chapter 4
discusses the measurement of the dc Stark shift, Chapter 5 covers the determinationof , and Chapter 6 discusses the implications of the two experiments on tests ofthe standard model. Finally, Chapter 7 discusses the possibility of additional improvements or new measurements that may be useful for future tests of the standard
model.

CHAPTER 2
THEORY

Using atomic PNC to test the standard model brings together two disparate
elds of physics: high-energy physics and atomic physics. It is not the purpose ofthis chapter to write at length about the underlying theories. Instead, this chapter
presents enough theory so that the atomic physics measurements can be understood,
and it also illustrates the way in which the measurements provide a test of the standard model. The theory of the weak interaction is covered in detail in the literature,and the reader is referred there for technical details and for general information on
atomic theory [16, 17, 18, 19]. Here only an overview will be provided.
This chapter covers the theory necessary to understand the experiments discussed and the implications of those experiments. In addition to the general theory,
this chapter also covers the e ects of external magnetic elds and their alignment
with respect to the experimental apparatus, and it details the exact transition rates
the experiments measure. Much of the content of this chapter closely follows thetheses of Gilbert [20], Noecker [21], Masterson [22], and Wood [23].

2.1 Weak Interaction
The weak interaction can be mediated by the exchange of a Z 0 boson
between an electron and a quark. This exchange, shown in Fig. 1.2(b), is analogous
to the exchange of a photon for the electromagnetic interaction, shown in Fig. 1.2(a).
In the present form of the standard model, the weak interaction is described by a

9Hamiltonian given by

HPNC = p1 GF QW 5 N (r);

(2.1)

2 2

where GF is the Fermi constant, N (r) is the nuclear density, and 5 is the Diracmatrix, which is responsible for PNC. The weak charge of the nucleus, QW , is analogous to the electric charge for the electromagnetic interaction. For an atom withatomic number Z and N neutrons, the value of QW is given by

QW = 2[(2Z + N )C1u + (Z + 2N )C1d ];(2.2)

where C1u and C1d are electron-quark coupling constants for the up and down quarks,
respectively, which are described in Chapter 1. Thus, QW is atomic physics' \window" into the standard model. At tree level QW is given by

QW = ,(N , Z + 4 sin2 W );(2.3)

where sin2W is the Weinberg angle. Any deviation from the tree level value of
QW indicates the need for higher order corrections (such as the so-called \radiative
corrections") or signals a breakdown in the standard model predictions.
The weak interaction does not conserve parity, so HPNC mixes states of
opposite parity as in

j i =j + i +Xi,j ,i i hi Ej H,PNCE j

i

+i

(2.4)

where the \+" and \," indicate opposite parity states. The matrix elements h,i j HPNC j + i
are essentially matrix elements of 5 . The operator 5 does not appear in any
atomic observable, so the values of these matrix elements must be calculated. The


Tree level is essentially the lowest order of the theory.

10
Bouchiats [8] have shown that in the non-relativistic approximationn` (r)
hn0`0 j HPNC j n`i / Rn0`0 (r) @R@r

r=0;

(2.5)

where n is the principal quantum number, ` is the orbital angular momentum quantum number, and Rn` (r) is the radial wave function of the j n`i state. BecauseRn`(r)  r`Z `+ 21 for small r, HPNC mixes only S and P states, and to a good
approximation it only mixes states with J = 1=2.

2.2 Cesium Energy Level Structure.
Atomic cesium has a single valence electron with a Xenon-like core. We
use the stable isotope 133 Cs, which has 55 protons and 78 neutrons. It has nuclear
spin I = 7=2, electronic spin S = 1=2, orbital angular momentum L = 0; 1; : : :, andtotal electronic angular momentum J~ = L~ + S~ . Its total angular momentum is given
by F~ = I~ + J~, so with J = 1=2 there are two hyper ne levels: F = 3 and 4. WithJ = 3=2 there are four hyper ne levels: F = 2; 3; 4; and 5. The projection of F
onto the quantization axis is given by mF = ,F; ,F + 1; : : : ; F , 1; F . The mostimportant of these energy levels are shown in Fig. 2.1.
Perhaps the most familiar spectral features of atomic cesium are the socalled \D1" and \D2" lines, which are electric dipole transitions from the 6S1=2
ground state to the 6P1=2 and 6P3=2 states, respectively. The work in this thesis uses
the D2 line for optical pumping and for detection, and it uses the normally forbidden
transition between the 6S1=2 and 7S1=2 states to study the interesting science.

11

Figure 2.1: The lowest energy levels of the cesium atom. The mechanisms giving riseto the 6S ! 7S transition are shown, as well as hyper ne structure of all the states
and the Zeeman structure of the 6S and 7S states. The 6P1=2 state is not shown
because of its relative unimportance for the work presented in this thesis. Energy
splittings are not shown to scale in this gure.

12

2.3 Electric Dipole Amplitudes
2.3.1 Parity Nonconserving Amplitude
In cesium, the mixing of S and P states by the weak interaction discussed
in Section 2.1 is given by

j nSFmF i =j nSFmF i +Xn00
j nSFmF i :j n0P i hn P jEHPNC
,E 0
nS

nP

(2.6)

In the presence of an oscillating laser eld with electric eld polarization ~, there is
an E 1 amplitude between the 6S and 7S states of cesium. This amplitude is givenby(E 1PNC )FFmmFF = h7SF 0 m0F j , e~  ~rj 6SFmF i
X  h7SF 0 m0F j HPNC j nP i=
hnP j D~ j 6SFm i00F
E7S , EnP

h
nP
j
H
j
6
SFm
i
PNC
F
0
0
~
+ h7SF mF j D j nP i
E6S , EnP
00
= iIm(EPNC )~  hF mF j ~ j FmF i;(2.7)
nwhere the unprimed quantum numbers are for the ground state and the primed quantum numbers are for the excited state, and the operator ~ is the Pauli spin matrix.
The constant EPNC contains all the radial information as well as the connection to
the standard model:

EPNC =

X h7S j HPNC j nP ihnP j D~ j 6S i+
E7S , EnPn!
h7S j D~ j nP ihnP j HPNC j 6S i :E6S , EnP

(2.8)

(2.9)

the geometrical matrix element hF 0 m0F j ~ j FmF i can be written in terms

13
F mF
of the CFmcoecients:
F00

F 0mF +q CFmF

 1 jqj
p hF 0 mF + q j 1q j FmF i;
2(2.10)

where q = 0; 1. These coecients are simply combinations of Clebsch-Gordancoecients, and they are tabulated in Appendix A. The PNC amplitude can thenbe written as(E 1PNC )FFmmFF = iIm(EPNC ) [(x + iy )00

F 0mF 1 
F 0m
CFm
mF m0F 1 + z CFmFF mm0 ];F(2.11)

where we have evaluated the dot product ~  ~ in Eq. (2.7). Note that the operators
x and y drive mF = 1 transitions and z drives mF = 0 transitions.

2.3.2 Stark-Induced Amplitude
An applied external dc electric eld E~ polarizes the cesium atom and also
mixes states in a similar way the weak interaction does in Eq. (2.4) with HPNC
replaced with eE~  ~r. An oscillating laser eld then drives the Stark-induced 6S ! 7S
amplitude given by
0 0(E 1Stark )FFmmFF

X ( h7SF 0 m0F j eE~  ~r j nPJ F 00m00F ihnPJ F 00 m00F j e~  ~r j 6SFmF i=
E7S , EnPJ
n;J;F 00;m00F
0 m0 j e~  ~r j nPJ F 00 m00 ihnPJ F 00 m00 j eE~  ~r j 6SFmF i )
h
7
SF
F
F
F
+
: (2.12)
E7S , EnPJThe Bouchiats [8] showed that this complicated expression can be written in a simple
form using an e ective dipole operator:(E 1Stark )FFmmFF = ~  h7SF 0 m0F j ~re ective j 6SFmF i
0

0

=
=

E~  ~ F;F 0 mF m0F + i ~  hF 0 m0F j ~  E~ j Fmi
E~  ~ F;F 0 mF m0 + i (E~  ~)  hF 0 m0F j ~ j Fmi:
F

(2.13)

14
Here and are the scalar and tensor transition polarizabilities, respectively. They
are given by [8, 24]

"

2X
= e9
h7S j r j nP1=2 ihnP1=2 j r j 6S i E ,1E+ E ,1E
7S
nP1=2
6S
nP1=2
n

1

1

+ 2 h7S j r j nP3=2 ihnP3=2 j r j 6S i E , E+
7S
nP3=2 E6S , EnP3=2!#!

; (2.14)

and

"

2X
h7S j r j nP1=2 ihnP1=2 j r j 6S i E ,1E, E ,1E
= e9
7S
nP1=2
6S
nP1=2
n

1

1

, h7S j r j nP3=2 ihnP3=2 j r j 6S i E , E,
7S
nP3=2 E6S , EnP3=2!#!

: (2.15)The F = 0 transitions are dominated by the term because = = ,9:905(11) [25].
Here hnS j r j nP i are e ective radial integrals which are related to reduced matrix
elements by [24]hnS jj r jj n0P

1=2

i = hn0 P

1=2 jj r jj nS i =r2

0

3 hnS j r j n P1=2 i(2.16)

0(2.17)andhnS jj r jj n0P

3=2

i = ,hn0P

3=2 jj r jj nS i =r4

3 hnS j r j n P3=2 i:

We use these forms of Eqs. (2.14) and (2.15) to emphasize the fact that is thesum of similar sized terms and is the di erence of similar sized terms. Thus, thefractional uncertainty of is comparable to or better than the fractional uncertainty
of its individual terms, while the fractional uncertainty of is much worse than thatof its individual terms.
If we evaluate the dot product as before, the Stark-induced amplitude can

15
be written as(E 1Stark )FFmmFF =00+

F 0mF 0E~  ~ FF 0 mF m0F + i (E~  ~)z CFm
F mF mF
F 0mF 1 
[i(E~  ~)x , (E~  ~)y ]CFmmF m0 1 :F

(2.18)F

The Stark-induced and PNC electric dipole amplitudes di er signi cantly
in size. If the D1 amplitude has a strength on the order of unity, E 1Stark has astrength of 3  10,5 when E = 500 V/cm, and E 1PNC has a strength of 10,11 .

2.4 Magnetic Dipole Amplitudes
Although to rst order magnetic dipole (M 1) amplitudes vanish between
states with di erent n, relativistic e ects [26] and the o -diagonal hyper ne interaction [8] both contribute to a small amplitude between the 6S and 7S states. The
Hamiltonian for this interaction connecting S states is given by

HM 1 = ~  B~ ac = hB (L^ + 2S^)  B~ ac= B~  B~ac

(2.19)(2.20)^ h is the Pauli spin operator, and if ~k is thewhere B is the Bohr magneton, ~ = 2S=
propagation vector of the laser, B~ ac = ~k  ~ is the oscillating magnetic eld of thelaser. L^ and S^ are the electron angular momentum and spin operators, respectively.
The amplitude for the M 1 transition is then given by
0 0
M 1FFmmFF = h7SF 0 m0F j HM 1 j 6SFmF i = M 1(~k  ~)  hF 0m0F j ~ j FmF i; (2.21)

where the radial integrals have been incorporated into the constant M 1. The radial
part can be written as
M 1 = M  Mhf F;F 01 ;(2.22)

16
where M is from relativistic e ects, and Mhf is from the o -diagonal hyper ne interaction. The M 1 amplitude has a size of about 2  10,6 compared with the D1line. As before, Eq. (2.21) can be written as

M 1FFmmFF = M 1f[ (~k  ~)x + i(~k  ~)y ]F 0mF 1 
F 0m
CFm
mF m0F 1 + (~k  ~)z CFmFF mF m0F g:
F
00(2.23)

2.5 Electric Quadrupole Interaction
Another transition that is driven by a laser eld is the electric quadrupoletransition (E 2), which arises from mixing of hn0 Di states with hnSi states by the o diagonal hyper ne interaction. This amplitude is very small compared to the allowed
E 1 transitions and even the M 1 transition (E 2=M  0:01). However, in order totest the standard model at the few tenths of a percent level, we must account forthe small e ects of the E 2 amplitude, as discussed in Section 5.4 and Appendix B.
In our case, the E 2 amplitude requires us to correct our data by only  0:08%;
in other experiments the necessary correction is as large as 3% [27]. Because thee ect of E 2 on our measurements is so small, we can determine the ratio E 2=Mhf
by comparing our result for Mhf =M with other results that were a ected by E 2.
See, for example, Ref. [28] and Section 5.4. A phenomenological discussion of E 2
following the treatment of Ref. [29] will be presented in Section 5.4.

2.6 E ect of Misaligned B~
In the foregoing discussion it has been assumed that the quantization axis
is along z^. It is important to ask what happens if the magnetic eld is slightly
misaligned from the z^ axis.
The e ect of a misaligned magnetic eld is to change the de nition of theoperator ~. The Pauli spin matrix is de ned assuming a quantization axis alongz^, so we must rotate ~0 , which is de ned with B~ = B z^0, into the experimental

17

Figure 2.2: A picture showing magnetic eld misaligned into the x^ direction.
coordinate system where B~ = Bx x^ + By y^ + Bz z^. Using Fig. 2.2 it is easy to see thatz^ = cos z^0 , sin x^0 and x^ = sin z^0 +cos x^0 , where cos  = Bz =B and sin  = Bx=B .
A similar rotation can be performed if y^ replaces x^ in the formulas. The componentsof the Pauli spin matrix are then given byx = BBz x0 + BBx z0y = BBz y0 + BBy z0z = BBz z0 , BBx x0 , BBy y0 :(2.24)

F mF
Simple algebra then shows that the CFmcoecients are transformed intoF00

F 0 m0F 1
F 0 m0 1
0 1 ! CFm F
CFmmF m0F 1 
m
m
F F
F
F

 Bx

 0
B
y
F mF
B  i B CFmF mF m0F :(2.25)

This means that a misaligned magnetic eld introduces mF = 0 transitions where
F 0mF 0 is also
there should only be mF = 1 transitions. The form of CFm
F mF mF
changed, but it is not important for our purposes and so it is omitted.

18

2.7 Measured Transition Rates
As we have seen, in the presence of an external dc electric eld and a cw laser
eld there are ve di erent amplitudes connecting the 6S and 7S states of cesium:the PNC and Stark-induced E 1 amplitudes, the M 1 amplitudes from relativistic
e ects and o -diagonal hyper ne interactions, and the tiny E 2 amplitude. There
are four hyper ne components to the 6S ! 7S transition: the F = 0 F = 3 toF 0 = 3 and F = 4 to F 0 = 4, and the F = 1 F = 3 to F 0 = 4 and F = 4 toF 0 = 3. All the experiments discussed in this thesis use E~ = E x^, B~ = B z^ , and~k = y^, and a general normalized polarization ~ = z z^ + (x + ii)^x.
The most recent PNC measurement performed by our group measured theinterference between the Stark-induced and PNC electric dipole amplitudes. The
geometry in that experiment used E ' 500 V/cm, B ' 6:6 G, i =z ' 1 and 2,and x = 0. The nal quantity measured was the ratio of the two amplitudes forF = 1 transitions given byP f d C F 0mF 12
Im(EPNC ) i mF mF mF FmF
F mF
RFm
=
2
F
E z P f d C F 0mF 1 2 ;0

0

mF mF mF

FmF

(2.26)

where fmF is the fractional population in a Zeeman sublevel mF , dmF is a detuning
factor that accounts for the non-degeneracy of di erent Zeeman transitions (seeSection 2.8). The value Im(EPNC )= can then be extracted if all the other quantities
are measured.
In order to measure the dc Stark shift (see Chapter 2.18) we use the F = 3
to F 0 = 3 transition. We use electric elds from 1 kV/cm to 10 kV/cm, with E~ = E x^,
In this section we ignore the problem of misaligned magnetic elds. That will be addressed in
Chapter 5
 Actually, the ratios measured on the F = 1 transitions are slightly di erent because of
nuclear PNC. This di erence will be ignored in this thesis.
19

B = 0, and ~ = xx^ + z z^. The transition rate from Eq. (2.12) is simply
2 2 2
2 2 2 FmF 2
E1
F 2
j (E 1Stark )Fm
FmF j ' [ E x mF m0F + E z (CFmF ) mF m0F 1 ] (I;  ); (2.27)
00where E 1 (I;  ) is a line shape factor that depends on the intensity I and frequency
 of the excitation laser (see Ref. [30]), and the small M 1 and E 2 rates have been
ignored.
To measure the tensor transition polarizability we use the F = 1
transitions. We also have E = 700 and 0 V/cm, B = 4 G, z = 1, and x = i = 0.
With E = 0 V/cm the transition rates from Eq. (2.23) are
F 1mF 2
j (M 1)FFm1FmF j2' (M  Mhf )2 M 1(I;  )(CFm
):
F
0

0

(2.28)

With E ' 700 V/cm, [using Eq. (2.18)] they are
F 1mF 2
j (E 1Stark )FFm1FmF j2 ' 2E 2 E1(I;  )(CFm
F ) :
0

0

(2.29)

Here again, (I;  ) is a line shape factor. The total rate for a given transition isfound by summing over mF and integrating over all  . If we take the ratio of thetotal of these two rates, we haveR 

 M  Mhf 2E:(2.30)

These two ratios can be combined with a semi-empirical value of Mhf [27] to determine .

20

2.8 Zeeman E ect in Cesium
2.8.1 E ect on Eigenstates
The presence of a dc magnetic eld mixes the hyper ne states and shifts
the energies of the 2F + 1 Zeeman sublevels. For a small magnetic eld, the e ect
can be treated as a perturbation, and the Hamiltonian is given by~
HZ = hB L^ + gS S^  B~ = hB gS S^  B;

(2.31)

for L = 0 states. In large magnetic elds, the di erent F states can be mixed into
one another. However, for the elds used in this thesis and in Ref. [13, 23] the mixing
is on the order of or less than 10,3 and is negligible. The energy shift for a given
state is"nSFmF = hnSFmF j HZ j nSFmF i(2.32)= B gF mF B;
where gF is the Lande g-factor. For the F = 3 and 4 states j gF j= 4, so the magnitudeof the frequency shift of each sublevel is given by
mF = (0:35 MHz=G) mF B:(2.33)

2.8.2 E ect on Spectrum
While the mixing of states due to an external magnetic eld is negligible,
the energy shifts due to the Zeeman e ect have profound implications for PNC measurements and for the other measurements presented in this thesis. It is this shift inenergy that made the 1988 PNC measurement [31] possible by completely resolving
di erent 6SFmF ! 7SF 0 m0F transitions in a large eld. In the 1997 PNC measurement [13], the magnetic eld was smaller but the atomic beam was spin polarized.

21
Therefore, the Zeeman splitting allowed a correction for unwanted transitions thatwould not be possible without energy shifts. For the experiments described in thisthesis, no discrimination is made between di erent Zeeman sublevels, so the energy
shifts serve only to broaden the spectral features observed. In the absence of a magnetic eld, there are only the four hyper ne transitions described in Section 2.7.
These are shown in Fig. 2.3. The widths of these transitions will depend on, for

Figure 2.3: The four 6S ! 7S Stark-induced electric dipole hyper ne transitions inatomic cesium in the absence of an external magnetic eld.
example, the Doppler width of the atomic beam or the intracavity intensity of theexcitation laser.
In the presence of a magnetic eld, the spectrum becomes much more complicated. As the Zeeman sublevels lose their degeneracy, each 6S ! 7S hyper netransition splits into its underlying Zeeman transitions. For F = 0 transitions,22
there are three spectral lines, one each for the mF = 0; 1 transitions, as shownin Fig. 2.4(a). However, for F = 1 transitions, the Lande g-factors for di erent
F levels have opposite signs. The result is the very complicated spectrum shown in
Fig. 2.4(b).

Figure 2.4: The Zeeman spectrum for 6S ! 7S transitions in a small magnetic eld.
(a) The spectrum for the F = 3 to F = 3 transition with the mF = 0 peak greatly
reduced. (b) The spectrum for the F = 3 to F = 4 transition. The Stark-induced
rates are shown in solid lines and the Stark-PNC interference terms are shown indashed lines and o set for clarity.

23
As mentioned earlier, this separation of di erent transitions made the 1988
PNC measurement possible. This is because the interference terms on opposite sidesof the spectrum have opposite signs, as is shown in Fig. 2.4(b). Without the Zeeman
splitting, and assuming an approximately uniform distribution among the Zeeman
sublevels, all the interference terms sum to zero. If optical pumping is used to spinpolarize the atomic beam as in the 1997 PNC experiment, the Zeeman splitting
reduces the level of precision needed when measuring the degree of spin polarization.
This point will be discussed in more detail in Section 7.3.
For measuring the dc Stark shift (Chapter 4), the Zeeman e ect is unimportant because we use the F = 3 to F 0 = 3 transition, where all mF = q transitions
that have the same value of q are degenerate. Here q = 0 or 1. For measuring(Chapter 5), the Zeeman e ect broadens the spectral feature but does not a ect
the total transition amplitudes we measure. Sample scans over the E 1 F = +1
transition with small and large B~ are shown in Fig. 2.5.
In our measurement of we would like to use as small a magnetic eld as
possible. This is because it is dicult to make scans over very large frequency ranges,
and the smaller the magnetic eld is, the narrower the transition will be. However,there is a systematic error that is proportional to the ratio of Bi =Bz , where i = x ory, so we compromise and use an intermediate magnetic eld of 4 G.

24

Figure 2.5: Sample scans of the E 1 transition in (a) small and (b) large magnetic
eld showing broadening due to the Zeeman e ect. The total area of each transition
does not change because the Zeeman e ect does not change the total transition rate.

CHAPTER 3
APPARATUS

Much of the apparatus used in the experiments described in this thesis wasinherited from the previous PNC measurement. Intricate details of the apparatus
will not be repeated where they have been presented in the literature or in previous
theses, such as for the Pound-Drever-Hall locking scheme that stabilizes the dye laser
to the power build-up cavity, or the details of the external cavity diode lasers usedin the optical pumping and detection.
The entire apparatus ts onto a large optical table that is isolated frombuilding vibrations by nitrogen- lled legs. A vacuum chamber, which is a large
aluminum box measuring about 30  56  66 cm is bolted to one end of the optical
table. Figure 3.1 is a schematic of the apparatus showing the vacuum chamber on
the right and the other assorted elements above and to the left. Cesium comesout of an oven in an atomic beam and enters the vacuum chamber. The atomic
beam then enters the optical pumping region where the atoms are placed in a single
hyper ne state by the hyper ne pumping laser. The atoms then enter the interaction
region where there are mutually orthogonal electric, magnetic, and laser elds. The
laser eld is an intense standing wave with a wavelength of 540 nm. The standing
wave is created in a Fabry-Perot etalon with a nesse of 105 . This etalon increases
the power available to drive the 6S ! 7S transition in the cesium atoms and it iscalled the power build-up cavity (PBC). The 540 nm light is provided by a tunable
dye laser that is locked to the PBC using the Pound-Drever-Hall method [32, 33]

26

Figure 3.1: The apparatus used in the experiments described in this thesis. SMOF isa single-mode optical ber, RefCav is the invar reference cavity, AOM is an acoustooptic modulator, EOM is an elecro-optic modulator, T is a mode-matching telescope,
HFP is the hyper ne-pumping diode laser, OP is the optical pumping region, OpIs
is an optical isolator, =2 is a half-wave plate, P is a polarizer, PBC is the powerbuild-up cavity, D is the detection region, and LAPD is a large area photodiode.
PD1, PD2, PD3 are photodiodes.
and the PBC is then locked to a stable reference cavity. The atomic beam nallyenters the detection region where atoms that repopulated the depleted hyper nestate after making the 6S ! 7S transition scatter photons from the probe laser. The
scattered photons are collected on a large area photodiode, and the photocurrent is
proportional to the number of atoms that made the 6S ! 7S transition. A schematic
view of just the atomic beam path is shown in Fig. 3.2.

27

Figure 3.2: Detail of the atomic beam path showing its travel from the liquid nitrogen(LN2 ) cooled plate, through the optical pumping region, through the interaction
region, and into the detection region.

3.1 The Cesium Oven
The cesium oven used as the source of the atomic beam in these experiments
was designed by Carl Wieman and built by Blaine Horner in the JILA instrument
shop. It replaces the oven that had been used since the rst PNC experimentswith cesium were performed by Sarah Gilbert at the University of Michigan [20].
A diagram of the oven is shown in Fig. 3.3. The oven has two sections, the main
section in the back of the oven where ampoules of cesium are placed and heated, andthe nozzle in the front where the atoms receive their initial collimation.
To load the oven, its pieces are completely assembled except for the rearange. Then, in a glove box back- lled with argon gas so that the cesium does
not react with air, two 10 g cesium ampoules are broken and placed in a small glass\boat." The boat is then placed in the main section of the oven and the rear ange is
attached. The oven is then removed from the glove box and attached to the vacuum

28

Figure 3.3: A schematic of the cesium oven. Shown are the back section where thecesium ampoules are loaded, the nozzle where the atoms rethermalized to 250 C,
and the assorted heaters, capillary array, etc.
chamber. When the vacuum chamber has been evacuated, the oven gate [34] canbe opened. The rear section of the oven is heated by two ceramic heaters fastened
around the oven. The entire oven is then wrapped in insulation and temperature

29
stabilized at around 180 C.
The hot cesium atoms travel from the back of the oven through a small tube
to the nozzle which is kept at 250 C to dissociate cesium dimers. A glass plate blocks
the direct line-of-sight from the rear of the oven so that the atoms must rethermalize
to the nozzle temperature. Once rethermalized, the atoms can leave the nozzle
through a glass capillary array that is purchased from Galileo Electo-optics. The
arrays are made from blocks of many glass tubes 10 m in diameter, close packed,
and sliced into 0.5 mm thick, 2.54 cm square wafers. Each wafer is then cut into
0.852.54 cm rectangles by dragging the corner of a razor blade across its surfaces
until enough glass has been scraped away for the array to break cleanly along thescratch. The array covers an opening in the nozzle that is 0.8 cm high and 2.5 cm
wide, which gives the initial de nition to the atomic beam shape. After leaving theoven, the atomic beam passes through an aperture 0.5 cm high  2.5 cm long in alarge copper plate, which is cooled to liquid nitrogen (LN2 ) temperature, and thenthe beam passes through a vertical vane collimator with vane separation of 1 mm.
After the vane collimator the beam has a divergence of 30 rad.
It is important to note that capillary arrays are also sold as \microchannel
plates", which are made of leaded glass. These plates are often intentionally cut ona bias. That is, the slices are not cut normal to the long direction of the glass tubes.
The arrays we received from our supplier varied in the angle at which they were cut
from zero to ve degrees, despite our speci cation of zero bias angle. To ensure ahigh ux in the atomic beam, we measured the bias angle of every array and rejectedany arrays that had a bias angle of greater than 0.5 . We were able to use 65% of
the arrays delivered to us.

3.2 Optical Pumping Region
Once collimated by the capillary array and the vane collimator, the atomic
beam enters the optical pumping (OP) region, which is 10 cm from the oven. Here

30
the atoms interact with the hyper ne pumping laser (HFP), which is tuned to the
6S1=2 to 6P3=2 transition. We use the F = 3 to F 0 = 4 transition to deplete theF = 3 state when we want to drive transitions from the 6S1=2 F = 4 hyper nestate and we use the F = 4 to F 0 = 3 transition to deplete the F = 4 when wewant to drive transitions from the 6S1=2 F = 3 hyper ne state. These transitions
were chosen in previous experiments [22] to minimize the amount of light scattereddown the beam that could \undo" the initial optical pumping and to minimize thenumber of atoms left in the depleted state. The requirements remain the same forthe present experiments, so the hyper ne pumping scheme has not been changed.
The hyper ne pumping takes place in a 2.5 G magnetic eld, and thepolarization of the light is linear and perpendicular the the magnetic eld. The
light is therefore equal parts + and , so that atoms should not be preferentiallypumped to mF < 0 or mF > 0 states.
The HFP is an external cavity diode laser constructed using an SDL 5401G1 semiconductor laser locked to a saturated absorption spectrometer as describedin Ref. [35]. This laser provides approximately 20 mW/cm2 of light, which is well
above the 1.1 mW/cm2 saturation intensity of cesium.
In previous experiments, the atomic beam was spin polarized as well as
pumped into a single hyper ne state. That \Zeeman pumping", which placed all
the atoms into an extreme Zeeman sublevel, is not needed or used for the present
experiments.
The result of this mechanical and optical manipulation is a well collimatedbeam of atoms predominantly (99.95%) in one hyper ne level of the ground state.
The horizontal distribution of the atoms is fairly uniform with 64% of the atoms in
the center 50% of the beam. The vertical distribution is roughly Gaussian with aheight of  0:8 cm. Typical uxes were 1013 atoms/sec, estimated from the shotnoise on the signal. (See Section 3.5.3 of Ref. [23] for a discussion of this estimation.)

31

3.3 Interaction Region
3.3.1 Electric and Magnetic Field Production
In the interaction region the atoms encounter a region of mutually orthogonal electric, magnetic, and laser elds. The electric eld is produced by two plates,
made either of molybdenum evaporated on glass or of solid molybdenum (dependingon the experiment). The plates are centered on both the atomic beam and on thelaser eld. The magnetic eld is produced mainly by a pair of Helmholtz coils, whichis also centered on the atomic beam and the laser beam. The laser eld is produced
by a dye laser coupled into a high- nesse Fabry-Perot etalon as discussed below.

3.3.2 Dye Laser
The 6S ! 7S transition is excited by a 540 nm laser eld. For the previous
PNC experiment and for the present experiments, the frequency of the laser driving
the 6S ! 7S transition must be extremely stable. To achieve the necessary stability,
we use the locking scheme shown in Fig. 3.4. The dye laser is a heavily modi ed
Spectra-Physics 380 pumped by a Coherent Innova-90 argon ion laser. Typical ion
laser powers range from 3 to 5 W, depending on the age of the dye, and give 200500 mW of tunable, single-frequency 540 nm light. We use a dye solution made of
1 g/` of Pyromethene 556 dye dissolved in ethylene glycol.
Coarse tuning of the laser is accomplished using an intracavity birefringent tuner, while thick and thin etalons select the longitudinal lasing mode. A
Faraday rotator prevents bidirectional lasing. Fine tuning of the laser frequency is
performed using three additional intracavity elements. Two galvonometer-mounted
glass plates with a range of about 20 GHz control the frequency on a slow time scale(f <2 Hz). A piezoelectric transducer behind one of the cavity mirrors has a range
of approximately 300 MHz and controls the frequency at an intermediate time scale(2 Hz< f <2 kHz). Finally, an electro-optic modulator controls the frequency at

32

Figure 3.4: A schematic of the dye laser frequency stabilization scheme. The dyelaser passes through the EOM where it is phase modulated at 7 MHz and throughan AOM before entering the PBC. The light re ected o the PBC is directed to a
photodiode (PD1). The signal from PD1 is mixed with the 7 MHz driving signal
at A to generate an error signal. The error signal is sent to the Pound-Drever-Hall
electronics and then to the dye laser transducers at B. The dye laser is also sent
through a second AOM to the reference cavity. The input piezo-electric transducer(PZT) is dithered at 300 Hz. The modulated transmission of the reference cavity
is collected on PD2 and sent to the dither lock electronics where an error signal isgenerated and sent to the PBC PZT at C. The transmission of the PBC collected
on PD3 is used to stabilize the intensity of the dye laser.
fast time scales (1.3 MHz> f >2 kHz).
The gross location of the dye laser frequency is referenced to a group ofmolecular iodine lines. The proper mode of the laser can be found by scanning thefrequency of the laser and observing the uorescence in an iodine cell. If the laser
is far away from the correct frequency it is sometimes necessary to use a portable33
scanning double monochromator to set the frequency of the laser near 540 nm.

3.3.3 Fast Frequency Stabilization
The light from the dye laser passes through an electro-optic modulator thatputs 7 MHz sidebands on the light. The light then passes through an acousto-optic
modulator used to stabilize the laser intensity. The light passes through a telescopefor proper mode matching into the etalon. An optical isolator protects the laserfrom optical feed back, and a half-wave plate and a polarizer are used to control thepolarization. The laser light is then incident on the Fabry-Perot etalon.
The frequency of the dye laser is locked to the etalon using the PoundDrever-Hall method [32]. This method works by looking at the 7 MHz sidebands
re ected o the input mirror of the etalon after they are demodulated on a photodiode. When the laser frequency is directly on the resonance of the etalon, thesidebands are phase shifted by equal and opposite amounts. The two sidebands sum
together and the demodulated signal size is zero. When the dye laser frequency isslightly detuned from resonance, the sidebands are not phase shifted equally andthe demodulated signal is nonzero. Thus, an error signal can be derived from thedemodulated sum of the sidebands and used to feed back to the galvos, the piezo
and the EOM in the dye laser to keep the frequency of the dye laser on the etalon
resonance. When locked to the etalon the dye laser has a line width much less than
the 5 kHz resonance width of the etalon.

3.3.4 Power Build-up Cavity
The Fabry-Perot etalon is constructed of two high re ectivity multilayer
dielectric mirrors 2.5 cm in diameter. The input mirror has a radius of curvature of
10 m and the output mirror has a radius of curvature of 6 m. The two mirrors are
separated by 27 cm, which gives the etalon a free spectral range of '550 MHz and a
relatively large beam radius of !0 = 0:41 mm. Since the etalon geometry is almost

34\ at- at" there is minimal focusing of the beam. In fact, the beam waist only varies
by 1% over the length of the etalon.
The two mirrors are of very high quality with the transmittance of each
expressed in parts per million rather (ppm) than per cent. The input mirror has
T = 40 ppm and the output mirror has T = 13 ppm. The geometry and there ectivity combine to give the etalon a nesse of 105 , and the power inside theetalon is approximately 30,000 times the incident power. For this reason, the etalon
is referred to as the power build-up cavity (PBC), and it is one of the main reasonsthe experiments described in this thesis and the PNC experiment are possible atall. The power density inside the cavity is typically 800 kW/cm2 , which makes
observation of the very weak 6S ! 7S transitions easy.
The mechanical design of the PBC is shown in Fig. 3.5. The two mirrors
are mounted on aluminum brackets using silicone rubber to minimize stress-induced
birefringence. The brackets are then mounted in optical mounts. The input mirror
is also mounted to a tube piezo-electric transducer (piezo) to allow for tuning of the

Figure 3.5: Detailed mechanical design of the PBC.

35
resonant frequency of the PBC. The two mounts are then bolted to a large granite
block, which is very rigid and provides a large inertial mass. The granite block issuspended in the plane of its center of mass by four beryllium-copper springs, which
are damped by a small amount of sorbothane rubber. This arrangement is to minimize tilts between the cavity and the optical table and to retain a gentle suspension
to isolate the cavity from vibration and shock. Since temperature variations willcause frequency shifts, both mirror mounts are temperature stabilized.

3.3.5 Slow Frequency Stabilization
While the line width of the dye laser is less than 5 kHz on a short timescale when locked to just the PBC, the PBC does bounce slightly, and the piezo on
the input mirror can creep. Therefore the longer term stability can be unacceptably
poor. In order to stabilize the frequency of the dye laser on long time scales, the PBC
is locked to a stable reference cavity. This reference cavity was locked to the peakof the 6S ! 7S transition for the PNC experiment, and was only used to stabilize
the PBC on a long time scale. However, for the experiments described in this thesis
the cavity must serve as a frequency reference by itself.
To improve the stability of the reference cavity, which is already constructedof Invar, the nearly confocal cavity was hermetically sealed, and a second stage oftemperature stabilization was added. In addition, light is coupled into a single
mode optical ber before going into the reference cavity because motion of the beamincident upon the reference cavity can cause a change in the point at which the PBC
locks to the reference cavity. (See Chapter 4.)
One of the mirrors of the reference cavity is mounted on a piezo. The
voltage on this piezo is dithered at 300 Hz, and the transmission through the cavity
is monitored on a lock-in ampli er. The signal from the lock-in ampli er provides
an error signal that is fed back to the PBC piezo, and thus the PBC is kept locked
to the reference cavity.

36

3.4 Detection Region
After the atoms interact with the frequency-stabilized dye laser light andare excited to the 7S state they relax back to the 6S state via transitions to the 6P1=2
and 6P3=2 states and then to the ground state. The atoms decay to both hyper nestates of the ground state with 75% of the atoms decaying to the state with the same
quantum number F to which they were excited That is, of the 7S1=2 F 0 = 4 atoms,
75% decay to the 6S1=2 F = 4 and 25% decay to the 6S1=2 F = 3. We then detect
the number of atoms that made the 6S ! 7S transition by \counting" the numberof atoms that are in the initially depleted hyper ne state.
To detect atoms that relax to the F = 4 hyper ne state we lock anotherexternal cavity diode laser{the probe laser [23, 35]{to the 6S1=2 F = 4 to 6P3=2
F 0 = 5 transition. Since the dipole selection rule requires F = 0; 1, atoms driven
to the 6P3=2 F 0 = 5 state can only relax back to the 6S1=2 F = 4 (since there are no
F = 5 or 6 states in the ground state) where they are available to be excited again.
We collect roughly 200 scattered photons per atom on this \cycling" transition ona large area photodiode located directly beneath the region where the atomic beamand probe beam intersect. We place a gold mirror above the atomic beam to re ect
photons emitted upward back into the photodiode. In order to maximize the numberof photons an individual atom can scatter we use two cylindrical telescopes to widen
the probe laser beam to about 2.5 cm in width and 1 cm in height.
The situation is a little di erent for atoms that are excited out of the F = 4
hyper ne state. The appropriate cycling transition here is the 6S1=2 F = 3 to 6P3=2
F 0 = 2. This transition su ers from the problem that it is similar to a \lambda"
transition where there are fewer excited state levels than there are ground state levels.
Atoms that are excited can eventually evolve into a superposition of ground states
are no longer resonant with the excitation recitation; they \go dark" [36, 37, 38].
In order to prevent the atoms from \going dark", the probe beam is linearly37polarized and re ected back upon itself, passing twice through a quarter wave plate.
The result is that the polarization of the re ected light is rotated by 90 forming
a con guration known as \lin-perp-lin" [39], which has a spatial variation of thepolarization. At one point in space the light is linearly polarized, then =8 away thelight is + polarized, then another =8 away the light is linearly polarized, then , ,and so on. In addition, a magnetic eld gradient is applied in the detection region
so that as an atom moves down the atomic beam it is in a di erent magnetic eldeach time it is excited to the 6P3=2 state. In this way the atoms never encounter theright conditions to \go dark." The scheme is not perfect, however, and the numberof photons scattered on the F = 3 to F 0 = 2 transition is only 45% of the number
scattered on the F = 4 to F 0 = 5 transition

3.5 Data Acquisition
All of the work presented in this thesis is basic spectroscopy. That is,
the frequency of the dye laser is scanned across the 6S ! 7S transition, and thetransition rate as a function of laser frequency is recorded. A computer controls thedata acquisition in the following manner, schematically illustrated in Fig. 3.6.
The computer sends a signal to a digital-to-analog converter that provides
a control voltage to change the frequency of the dye laser. For low-precision measurements of the line shape (Section 7.1) the computer changes the voltage on thereference cavity piezo, thus changing the reference frequency. For the dc Stark shiftmeasurement (Chapter 4) and for the tensor transition polarizability measurement(Chapter 5) the computer controls the frequency of an AOM that shifts the frequencyof the light coupled into the reference cavity. The frequency of the dye laser mustchange to account for the shift.
After a xed amount of time after the dye laser control voltage changes(to let transients settle, etc.), the computer triggers two gated integrators. These
integrators integrate the signal from the probe photodiode and the voltage controlling38

Figure 3.6: A schematic of the data acquisition system. The computer controls thelaser frequency through a digital-to-analog converter (DAC). As the laser scans over
the transition, the scattered light is collected on the probe photodiode (PPD) andampli ed. The voltage to the frequency control and the signal from the PPD are
integrated in 16.667 ms intervals, converted on the analog-to-digital converter (ADC)and stored on disk.
the laser frequency. The length of the integration interval, which is also controlled
by the computer, is 16.667 ms. This particular interval is chosen to average away any
60 Hz noise. The two voltages are then converted to a digital signal by an analogto-digital converter and are stored on disk. As the control voltage is scanned, we
generate a transition rate versus voltage plot that, with suitable calibration, may be
converted to a transition rate versus frequency plot similar to that shown in Fig. 3.7.
Plots like this make up the data for all of the experiments described in this thesis.

39

Figure 3.7: A sample of the data taken for the measurements in this thesis. The
x-axis is the frequency of the AOM that shifts the frequency of the dye laser before
it is incident on the reference cavity, and the y-axis is the signal from the probe
photodiode.

CHAPTER 4
MEASURING THE STARK SHIFT OF THE 6S ! 7S
TRANSITION

As discussed in Chapter 1, a test of the standard model using measurementsof PNC in atomic cesium requires an accurate and precise calculation of atomic
structure using atomic theory. The most objective way to test the accuracy of thetheory is to compare its predictions with measurements of various atomic properties.
There are many measurements of the properties of cesium, including the
lifetimes of the 6P states [40], and the hyper ne structure constants of the 6S ,7S [41], 6P1=2 , and 7P1=2 [42] states. All of these measurements agree with thepredictions of Blundell et al. [14, 43, 44, 45] and Dzuba et al. [46, 15, 47, 48] to better
than 1%. There was a notable exception to this good agreement in the measurementof the 6S ! 7S dc Stark shift, where the di erence between the theory and theexperiment [49] was 2%.
This chapter presents the theory of the dc Stark e ect along with the details of our experiment that re-measures the dc Stark shift of the 6S ! 7S transitionand eliminates the previous discrepancy. The details of the experiment include discussions of the requirement of a stable reference frequency and its implications forour data taking method and frequency stabilization schemes, the production andmeasurement of the electric eld causing the Stark e ect, and the determination of
the dc Stark shift from our data. A presentation of our results concludes the chapter.

41

4.1 Theory of the dc Stark E ect
The theory of the dc Stark e ect, which gives rise to the dc Stark shift, isdiscussed in detail in Ref. [50]. The full theory will not be repeated here. Instead,an outline of the calculation following that in Ref. [51] is presented. The interactionof an atom with a dc electric eld E~ can be described by the Hamiltonian

HStark = ,eXi~r~i  E;

(4.1)

where e is the electric charge, and r~i is the vector from the nucleus to the ith electron.
In Ref. [50] it is shown that this Hamiltonian can be broken into two components:
HStark = Hscalar + Htensor . The term Hscalar depends on only the magnitude of theelectric eld, while the term Htensor depends on the eld's direction as well as its
magnitude.
The Stark Hamiltonian shifts energy levels as well as gives rise to the mixing
discussed in Section 2.3.2, which permits the Stark-induced electric dipole transition.
If the hyper ne structure is neglected, the energy shift is given by!

2 , J (J + 1) 3E 2 , E 2z;"(n; J; mJ ) = , 12 0 E 2 , 21 2 3mJJ (2J , 1)
2

where 0 is the scalar polarization due to Hscalar anddue to Htensor . These two polarizations are given by

2 is the tensor polarization

2 X j hnJ j p j n0 J 0 i j2=,0
3 (2J + 1)(E , E 0 0 )n0 J 0

and

nJ

(4.2)

nJ

(4.3)

42
2


1=2
J (2J , 1)
= 2 3(2J +103)(
J + 1)(2J + 1)

8
9
< J J0 1 >=X j hnJ j p j n0J 0 i j2 J +J 0 +1 >
(
,
1)
;

>
:1 2 J>;
n0 J 0 EnJ , En0 J 0(4.4)

where EnJ is the energy of the j nJ i state.
In this approximation it is clear that the tensor polarizability vanishes
for J = 1=2. However, when higher orders of the theory are considered there
is a contribution to 2 from the combination of the o -diagonal matrix elementof Htensor between J = 1=2 and J = 3=2 states and the matrix element of thehyper ne structure operator between the same states. For the L = 0 states weare considering, this mechanism vanishes, but there are still tiny contributions dueto the spin-dipolar part of the magnetic hyper ne structure and the quadrupole
interaction [50]. In cesium, the tensor polarizability has been measured to be,2
2 = ,0:1372(79)  10,7 Hz(V=cm) [52]. When compared with the scalar polarizability 0  0:7 Hz (V/cm),2 it is clear that the tensor polarizability is negligible.
Therefore, we are only concerned with the scalar part of the dc Stark shift, which is
independent of hyper ne level for energy shifts that are small compared to the ne
structure splitting [53].
The energy shift due to the dc Stark shift of an j nS1=2 i state is given simply
by(4.5)
"nS = , 21 nS E 2 ;where nS is the scalar polarizability. The dc Stark shift of the 6S ! 7S transitionis given by
Stark = , 7S 4, 6S E 2 = kE 2 :(4.6)
To determine the constant k we scan across the 6S ! 7S transition at high and low
electric elds, measure the line centers of the two scans, and calculate the frequency
separation, Stark , of the two centers. Sample scans are shown in Fig. 4.1. The

43

Figure 4.1: Four di erent scans of the 6S!7S transition with dc electric elds of(a) 1 kV/cm, (b) 6 kV/cm, (c) 8 kV/cm, and (d) 10 kV/cm.
value of k is then given by

k = E 2Stark, E2 :
high

low

(4.7)

Because the value of k is independent of hyper ne level, we use the transition withthe highest signal-to-noise ratio: the F = 3 to F 0 = 3 transition.

4.2 Details of the Experiment4.2.1 Data Scans and Frequency Drift
The major challenge in this experiment is determining the frequency separation of two scans to 0.1%. In order to make such a determination we must have astable reference frequency. We use use the reference cavity described in Section 3.3.5,
which is the same cavity discussed in section 5.1.3 of Ref. [23]. In order to make44
the cavity stable enough for the present experiment we have hermetically sealed thecavity and added a second stage of temperature stabilization. In addition, we use
a simple peak locking scheme rather than the Pound-Drever-Hall locking scheme
because of the latter's suceptibility to electronic o sets. The nal frequency drift
rate of the reference cavity averaged 0.5 MHz/min with a maximum observed driftof 0.8 MHz/min.
As described in Section 3.5, a computer controls the data acquisition. For
this experiment the computer uses GPIB commands to control a Hewlett-Packard
frequency synthesizer. The output from the synthesizer drives an acousto-optic modulator (AOM), which shifts the frequency of the dye laser light incident on the reference cavity. Since the dye laser is locked to the PBC, and the PBC is locked to
the reference cavity, changing the AOM frequency causes the dye laser frequency tochange by twice that amount. (Twice because the AOM is double passed. See thenext section.)
The scans over the 6S ! 7S transition are 75 MHz wide and take approximately 8 seconds. The frequency separation between a scan at E =1 kV/cm and ascan at E =10 kV/cm is 70 MHz. Therefore, the maximum allowable drift rate of
the reference cavity is 0.5 MHz/min. Even smaller drift rates are required for lower
electric elds. For all the data used in the nal analysis we measured the drift before
and after the measurement to ensure the rate was at an acceptable level.

4.2.2 Coupling Light into the Reference Cavity
The AOM is used to shift the frequency of the dye laser light incident on
the reference cavity. Because the angle through which the light is di racted by theAOM changes as the frequency of the AOM is changed, the AOM is \double-passed"
as shown in Fig. 4.2. Double passing the light in the AOM works in the following
way.
First we examine the horizontal motion by looking at the top view in45

Figure 4.2: Top and side views of light \double-passed" through an AOM. Double
passing the light eliminates most of the angular motion associated with a change in
AOM frequency.
Fig. 4.2. The light to be passed through the AOM is centered horizontally on alens that has a focal length f . The center of the AOM transducer is placed at the
focal point of the lens, and the action of the AOM di racts the light at some angle .
The light then travels to another lens placed a distance f away from the AOM. Thissecond lens also has a focal length f . The second lens bends the light back parallelto the initial direction of the laser beam. The beam is then retro-re ected o a prismand retraces its path back through the lenses and the AOM. As the frequency of theAOM changes, so does the angle . However, because of the positioning of the second
lens, the laser beam is always normal to the prism and always gets retro-re ected.
Therefore, the horizontal motion of the beam has been nearly eliminated.
A consequence of removing the horizontal motion of the laser beam is thatthe re ected beam travels back upon itself. This makes it dicult to use the lightthat has passed through the AOM. To describe the solution to this problem, we use

46
the side view in Fig. 4.2. The light incident on the lens-AOM system comes in above
the center of the rst lens. The rst lens bends the light down through the AOM,
causing the light to hit the second lens below its center. The second lens then brings
the light back parallel to the initial direction and the prism re ects the light backwith a vertical shift, and the light passes back through the second lens above its
center. The second lens repeats the action of the rst by bending the light down
through the AOM and then low on the rst lens. The beam is then bent back toparallel by the rst lens. The result of the whole system is that the beam coming outof the lens-AOM system is lower than the one going in, but the horizontal motion
does not change with changing AOM frequency. The lower beam is easily directedto the reference cavity.
The double-passed AOM system does not eliminate all the motion; theangular displacement with a change in frequency of 40 MHz is 0.3 mrad, comparedwith the usual 260 mrad. This is sucient to keep the laser beam from moving o
the reference cavity. It is not sucient for measuring 70 MHz separations to 0.1%,
however. The reason is as follows.
The reference cavity was constructed to be a confocal cavity. However, thelimitations of experimental science preclude us from building a perfectly confocal
cavity, so the cavity is only nearly confocal. As a result, adjacent modes are not
perfectly degenerate. The transmission through the reference cavity of three suchmodes, (a), (b), and (c), into which the light might couple is shown in Fig. 4.3. As
the light incident on the reference cavity moves, the coupling into each mode changes.
Figure 4.4(d) shows the results when 60% of the light is coupled into mode (a), 30%
is coupled into mode (b), and 10% is coupled into mode (c). Figure 4.4(e) shows
the results when the fractions coupled into mode (a) and mode (c) are reversed.
Because we use a locking scheme that locks the laser frequency to the frequencyof the maximum transmission of the reference cavity, the lock point of the laser
frequency will change as the coupling changes. To eliminate this problem from beam

47

Figure 4.3: Transmission versus frequency of three nearly degenerate modes of thereference cavity.
motion on the reference cavity, we couple the light from the double-passed AOM
into one end of a 0.3 m length of single mode optical ber. The other end of theber is xed to an optical mount. By using this scheme, the intensity of the lightincident on the reference cavity changes by up to 20% because the light from theAOM moves on the face of the ber, which changes the coupling eciency into theber. However, the output end of the ber remains xed, and thus the light into thereference cavity remains motionless. The variations in intensity transmitted throughthe ber do not a ect the lock point of the dye laser.

48

Figure 4.4: Total transmission through the reference cavity versus frequency. The
transmission when 60% of the light is coupled into mode (a) (from Fig. 4.3), 30% of
the light is coupled into mode (b), and 10% of the light is coupled into mode (c).
The transmission shown in (e) is when the fraction of light coupled into modes (a)and (c) has been reversed.

4.2.3 Production and Measurement of the Electric Field
The electric eld plates used in this experiment are di erent from those usedin the PNC measurement and in the measurement of . The plates are 2  5  0:5 cm
blocks of molybdenum mounted in a plexiglas bracket. The plates are separated by
0.48994(25) cm. They were made smaller than the plates used in the PNC experimentto keep them as far away from other conducting surfaces as possible in order to handle
safely the 5 kV applied across the plates. In addition, all the corners and sharp edges
on the plates are rounded or smoothed to prevent arcing.
The voltage on the plates is measured with a high-voltage probe that was

49

Figure 4.5: The voltage divider measurement of E at point A for the dc Stark shiftmeasurement. Vprecision is the high-voltage source, RCL is a current-limiting resistor,
and the two resistors and the dashed box are the high-voltage probe. The experimentis always run with the probe in place because the probe and RCL in series make asecond voltage divider that changes the voltage at A.
calibrated with a NIST-traceable voltage source in the JILA electronics shop. The
probe divides the voltage applied to the plates by 901.52(5). The method for measuring the voltage is shown in Fig. 4.5. It is important to note that when thehigh-voltage probe is in place it makes a voltage divider with the current-limiting
resistor RCL . This divider causes a voltage drop across RCL . If the experiment isperformed with the probe removed the voltage on the plates will be di erent thanthe voltage that was measured. Because of this di erence, the probe is left connected
at all times.

4.3 Determination of the dc Stark Shift
4.3.1 Determination of Line Centers
To obtain the transition rate versus frequency plots for this experiment,
we lock the laser and the cavities with the AOM frequency set in the middle of its

50
range. We then change the voltage on the reference cavity piezo to tune the dyelaser frequency to the peak of the 6S ! 7S transition. The computer then quickly
sweeps the voltage controlling the AOM frequency to a point below the 6S ! 7S
transition and then sweeps the voltage across the transition in small steps, taking a
16.667 ms long data point at each step.
An accurate determination of the center of each scan is critical to the successof this measurement. Because the ac Stark e ect can cause broadening of the line
shape and large frequency shifts of its e ective center [21 MHz/(MW/cm2 )] [30] wekeep the intensity inside the PBC as low as possible. The intensity in this experimentis 200 W compared with the intensity of 4 kW typical for the PNC experiment. Our
intensity reduces the ac Stark shift to 0.84 MHz. This is much larger than theacceptable uncertainty on our measurement of high- and low-E peak separations.
However, the ac Stark shift does not depend on the dc electric eld. Therefore, thissmall ac Stark shift does not a ect the accuracy of our measurement.
Because of the asymmetry introduced by the ac Stark shift, it is impossible
to t our line shapes to a simple function. Therefore it is not straight forward
to determine the line center. The line shape does not change signi cantly with
E , however, so it is possible to determine an e ective line center for each scan.
The \center" of a line is found by taking the two frequencies at which the probe
signal is n=10 times its peak value and averaging the pair of frequencies for eachn = 1; 2; : : : ; 9. The e ective center is then the average of these nine values. We
test the reproducibility of this method by changing slightly the value of the probe
signal at which we determine the two frequencies. For example, instead of taking thevalues at n=10, we can take the values at 1.01n=10. The reproducibility of nding
line centers in this manner is 0.02 MHz.
We nd that saturation of the 6S ! 7S transition can cause small shifts in Of course, we only adjust the reference cavity before the initial scan. For successive scans thecavity serves as the frequency reference.

51
the e ective line center. Scans over the transition with and without saturation are
shown in Fig. 4.6. As can be seen in Eq. (2.27), the 6S ! 7S \ " transition rate
has two terms. To avoid an error from saturation we rotate the linear light untilx is roughly 10% smaller than its smallest value at which we can measure a shiftof the line center. Therefore, there is no e ect on our determinations of the line
centers from saturation. The small terms are larger as a result, but the relative
size of the \ " rate to the \ " rate is ( = )2 = 98:1, so the small terms are still
unimportant. In the end we determine the line center to approximately 2 parts in
1000.

4.3.2 Determination of Frequency Separation of two Electric Fields
If the reference cavity were perfectly stable, determining the frequency separation between a low-E scan and a high-E scan would be trivial. However, thereference cavity has a nonzero drift rate as previously discussed. As can be seen in
Fig. 4.7, the drift can be nonlinear, even reversing direction from time to time. Taking several scans at high and low electric elds in succession is sucient to eliminate
a linear drift. To account for the nonlinear drift we use a standard least squares
technique [54] to t the centers of 16 scans, which alternate between high and low
E , to a third order polynomial plus a constant. The extracted constant is the dc
Stark shift. Our nal fractional uncertainty in Stark , which is dominated by theuncertainty in the determination of the line centers, is 0.04% for each ten scan set.

4.4 Results
Once we can determine the separation between two scans taken with different electric elds we can, in principle, determine the value of k. However, a more
precise determination can be made if we extract k from the t to data taken at
Even if the \ " terms do become appreciable, their contribution to the line shape scales thesame way the \ " contribution does. Therefore, the frequency shift between e ective line centerswill be the same as in the absence of the \ " transition.
52
several di erent values of the high eld. Determinations of k from pairs of elds
(eg. 1 kV/cm and 10 kV/cm) are shown in Fig. 4.8. The weighted average of the 14scans is 0.72618(19) Hz (V/cm),2 . The t of the data to Stark = kE 2 , along witha plot of the residuals is shown in Fig. 4.9. The slope of the line tting the data is
k = 0:72620(13) Hz (V/cm),2 . The reduced 2 's for the t and the weighted averageboth indicate a 60% probability that the data are from a random distribution. The
contributions to the uncertainty in our nal result for k are 0.1% from the measurement of the eld plate separation , 0.04% from the determinations of Stark , and
0.01% from the measurement of the applied voltage. This yields a total fractional
uncertainty of 0.11%. Our nal result is

k = 0:7262(8) Hz(V=cm),2;(4.8)

which disagrees with the previous result of 0.7103(24) Hz (V/cm),2 [49], but is inexcellent agreement with theoretical prediction k = 0:7257 Hz (V/cm),2 of Blundell,2 of Dzubaet al. [14], and is within 0.3% of the prediction k = 0:7237 Hz (V/cm)
et al. [55]. These data are shown in Fig. 4.10. The di erence in polarizabilities of
the two states, 7S , 6S can be extracted from these numbers. In atomic units,
our result is 7S , 6S = 5837(6) compared with the predictions of Blundell et al.
( 7S , 6S = 5833) and Dzuba et al. ( 7S , 6S = 5817). We have attempted
to determine the source of the disagreement between the present work and that ofWatts et al. [49]. However, Watts, who carried out the primary data analysis, isdeceased and the records of his analysis are no longer available.

We measure the separation of the plates by using a high-precision height-micrometer and adial indicator to measure the di erence in each plate's height above a at granite surface. A more
precise measurement could be made optically, but the constraints of the experiment prevent us
from constructing the plates out of materials that would allow such a measurement. (See Ref. [13],Section 5.2.)53

Figure 4.6: Scans over the 6S ! 7S transition (a) with and (b) without saturation.
In (a) the electric eld is 9 kV/cm and in (b) the electric eld is 1 kV/cm. Notethat in (b) the signal has been increased by a factor of 100.

54

Figure 4.7: Typical data from a data \set" showing the drift of the reference cavity.
The separation between the data points is the dc Stark shift.

55

Figure 4.8: Determinations of k from paris of electric elds. The high eld value
is shown on the x^ axis. The dashed horizontal line is the weighted average k =0:72618(19) Hz (V/cm),2 and the solid lines show the one  error bars.

56

Figure 4.9: (a) Measured values and (b)residuals of Stark plotted as a function of
E 2 . In (a) The error bars are smaller than the data points and values from scanswith the same values of E have been averaged together. The solid line is a t to thedata assuming Stark / E 2 . The result is k = 0:72620(13) Hz (V/cm),2 .

57

Figure 4.10: A comparison of determinations of the dc Stark shift of the 6S ! 7S
transition. The number of Watts et al. is from Ref. [49], the number of Dzuba et al.
is from a private communication, and the number of Blundell et al. is from Ref. [14].

CHAPTER 5
MEASURING THE TENSOR TRANSITION POLARIZABILITY

As discussed in Chapter 1, the standard model can be tested by extracting
the value of the weak charge QW from the equation
Im(EPNC ) = ,i QW k

N PNC ;

(5.1)

where the value of Im(EPNC )= has been measured to 0.35% [13]. Also discussed
was the fact that atomic theory calculations are needed to determine the values of
kPNC and  . Because the uncertainty in the theory is the limiting factor in the testof the standard model, it is desirable to reduce the dependence on theory by making
an independent measurement of .
While the measurements of the dc Stark shift and are both accomplishedby scanning a dye laser across the 6S ! 7S transition and measuring the transition
rate as a function of frequency, the latter experiment is susceptible to many more
problems. Measuring the dc Stark shift requires only the accurate measurement of
the frequency separation of two spectral lines. The line shapes, contributions frombackground signals, magnetic eld perturbations, and contamination of the signal
from unwanted amplitudes are all examples of e ects that do not a ect the dc Stark
shift measurement but are important for measuring .
This chapter details our measurement of . It begins with a description
has also been determined semi-empirically, but the calculation from the atomic theory has
higher precision.
59of the basic idea behind the measurement and continues with a detailed discussionof the experimental complications that must be addressed to make an accurate andprecise measurement. The raw data are then presented, and the chapter concludes bycovering the issue of the presence of an electric quadrupole transition. The electric
quadrupole amplitude has a ected previous experiments, and its e ects must be
considered carefully in the present measurement. A full mathematical treatment ofthese e ects is given in Appendix B.

5.1 Experimental Concept
As discussed in Section 2.7, in order to determine , we need to rst measure
the E 1 and M 1 amplitudes on the F = 1 transitions. We do this by scanning thedye laser across the appropriate 6S ! 7S transitions and integrating the total uorescence collected on the probe photodiode. These integrated areas are proportional
to the transition rates. If we then take the ratio of the M 1 area (measured with
no applied E ) and the E 1 area (measured with E = 700 V/cm), all the unknown
parameters (such as beam ux, probe detection eciency, and intensity inside thePBC) cancel, and to a good approximation we are left with the ratios

2R  M EMhf ;(5.2)

where R correspond to the F = 1 transitions. From these ratios we can extract
the value of Mhf = . [See Eq. (5.6) and Eq. (5.7) for minor corrections.] Then we
can use the value of Mhf determined in Ref. [27] to extract . The amplitude Mhf
can be determined very precisely because it is related to the well-known hyper nesplittings of cesium. The goal is to measure Mhf = with a precision of 0.1%. The
uncertainty in the nal value of is then dominated by the 0.25% determination of
Mhf [27].

60

5.2 Experimental Considerations
5.2.1 Measuring the Areas Correctly
Determining a transition amplitude by measuring the area underneath aspectral feature is straight forward if the line shape ts a well known function, forexample a Voigt function. Our line shapes do not t simple functions, so we mustnd another way to extract the amplitude.
Consider the sample line shape shown in Fig. 5.1. The signal detectedon the photodiode as a function of laser frequency is f ( ), which is the sum of
two quantities: the signal from atoms making the 6S ! 7S transition, representedby h( ), and the signal from all other contributions, represented by BG( ). The

Figure 5.1: A hypothetical scan across the 6S ! 7S transitions. The total signal isf (i ) and the background is BG(i). See text for discussion of the symbols in the
gure.

61
quantity we want to measure is given by

ATotal =Z1,1h( )d:(5.3)

Of course, we make measurements of the signal size at discrete frequencies, andwe can only scan the frequency of the laser within its tunable frequency range. In
addition, we cannot measure h( ) directly. Rather, we must calculate it from thedi erence between f ( ) and BG( ). We then have

ATotal =
=

1
X

X
i<a

[f (i ) , BG(i)]i +X[f (i ) , BG(i)]ii>b
Alow + Ameasured + Ahigh:+=[f (i ) , BG(i )]i

i=,1

(5.4)
b
X
i=a

[f (i ) , BG(i )]i

Clearly we must choose a and b such that Alow and Ahigh are negligible. Then we areleft with Ameasured . We are still left with two problems: the noise on the backgroundand the fact that we can only measure f (i ), not h(i ) and BG(i ).
If the background is a linear function of frequency then we can measure
its value at a low reference point and a high reference point where the contribution
from h(i ) is negligible. Then, if these two reference points are centered aroundP
the peak of the transition, we can take their average and use [f (i ) , BG(i )] =P[f ( ) , BG ] to subtract o the contribution from the background. If theiAVGbackground is nonlinear, this method does not give the right answer.
The contributions to the background level from known sources are all essentially linear functions of the dye laser frequency or independent of it. These sources
are atoms left in the wrong hyper ne state, scattered light from the PBC mirrors,
and scattered light from the probe and hyper ne pumping lasers. The only contri-

62bution to the background that might be nonlinear is from molecules interacting withthe dye laser and being detected by the probe laser in a frequency sensitive manner.
However, we measure the background with varying oven nozzle temperatures (whichvary the fraction of Cs2 molecules in the beam) and nd no measurable molecular background. Therefore, the procedure in the previous paragraph does give thecorrect answer.
Another complication is the fact that there is noise on the background, andthis 1=f noise is our limiting source of noise. Thus, if we measure the backgroundat the high and low reference points and subsequently scan across the transition,
the background will be slightly di erent at each measurement of f (i ) than it waswhen we rst measured it. To solve this problem, we take advantage of our 25 Hz
data taking frequency: before and after each measurement of f (i ) we measure thebackground at the high and low reference points. In this way we get the averagebackground at the time of the measurement of f (i ). We also alternate whichreference comes rst to eliminate problems that might arise due to hysteresis.
Finally, we must know the frequency separation i between each measurement of the transition rate. This is because we are essentially determining thearea of the line shape using the trapezoidal rule. We must know both the height andthe width of each \rectangle" we use to determine the total area. If the rectangles
are not of uniform width and we assume they are, we will get an incorrect result.
Therefore, we calibrate the control voltage from the computer to obtain sucientaccuracy in our knowledge of i .

5.2.2 Comparing the Areas Correctly
As mentioned in Chapter 4, the line shapes of the 6S ! 7S transition areasymmetric because of the ac Stark e ect [30]. The ac Stark e ect causes anotherproblem: it shifts and distorts the M 1 transitions di erently from the E 1 transitions.
This is because the E 1 transition is driven by the oscillating electric eld (ac ) of

63
the laser and the M 1 transitions are driven by the oscillating magnetic eld (bac ) of
the laser, while the ac Stark e ect causes the largest shift where ac is large. Thus,in the regions of large ac there is a large ac Stark shift and a high probability ofdriving an E 1 transition. In regions of small ac (and a correspondingly large bac )there is a small ac Stark shift and a large probability of driving an M 1 transition.
This e ect is shown dramatically in Fig. 5.2.
If the line shapes for the two transitions had simple Voigt pro les with thesame widths, for example, and we missed 1% of each area in the wings, the ratio will
still be correct. That is M 1= = (0:99  M 1)=(0:99  ). However, since the twoline shapes do not have the same width and they are o set, it would be easy to miss
1% of and only miss 0.1% of M 1. We avoid this problem by making the scans
across the transitions wide enough to measure all the area for both transitions. We
also con rm that the scan width is wide enough by reanalyzing the data by omitting
increasingly large amounts of data from the beginning and end of the scans. We do
not nd a signi cant variation in the ratios when we do the reanalysis.
The discussion so far has implied that all we need to do is scan across theM 1 transition and then scan across the E 1 transition and we have the two values weneed to determine the ratio. However, when we scan across the E 1 transition, theM 1 transition is not absent, it is just very small. The ratio we actually measure is

Mhf )2 :Rmeasured= 2 E(2M+ (M  M )2
hf

(5.5)

The ratio we actually want is given by2R = (M 2 EM2hf ) = Rmeasured:
1 , Rmeasured(5.6)

The terms \high" and \low" are only used in the relative sense. For example, there may be a\high" probability of making a Stark-induced E 1 transition, but that transition amplitude is overseven orders of magnitude smaller than the D1 electric dipole transition.
64

Figure 5.2: Plot comparing the E 1 and M 1 line shapes. The asymmetries in the line
shapes and the di erences in widths and center frequencies are due to the ac Starke ect as discussed in Ref. [30].

5.2.3 Photoionization Correction
Just as the ac Stark e ect's dependence on ac combines with the dependence of E 1 on ac and M 1 on bac to give di erent line shapes, photoionization also
causes a di erence in detection eciency between E 1 and M 1 because it is also
maximum where ac is maximum. This is explained in the following manner.
Atoms that are in a region with a large ac have a high probability of making
an E 1 transition. The same atoms also have a high probability of being photoionized.
Atoms in a region with a large bac have a high probability of making an M 1 transition.
However, these atoms have a small probability of being photoionized. Any atoms
that are photoionized are no longer available to be detected downstream in thedetection region. Thus, a smaller fraction of atoms that made the E 1 transition isavailable for detection relative to the fraction of atoms that made the M 1 transition.

65
The result is that as the intensity inside the PBC gets higher, the ratio M  Mhf =gets correspondingly larger. With this e ect, the ratios on the two transitions are
given by
RI  ( M EMhf )(1 + I ):(5.7)
The intensity inside the PBC is proportional to the power that is transmitted throughthe output mirror, which we measure on a photodiode. I is this voltage, and  is aparameter that describes the di erence in photoionization probabilities. In order tocorrect for this e ect, we must take data at several intensities and extrapolate theresult to zero intensity. The data taken for this correction are shown in Section 5.3.

5.2.4 Non-Bz Magnetic Field Systematic Error
As mentioned in Section 2.6, a misaligned magnetic eld can change thede nition of the operator ~ . This, in turn, changes the rates we measure. Using
Eq. (2.28) and Eq. (2.25) we nd that the ratios we measure becomeR= Bx 2 F 1m0F 2
P  Bz 2 (C F 1m0F )2  0
0
+
)

(
C
mF mF 1
mF mF
FmF
FmF
B
B
0
P
F 1mF 2
2E2
0 (C
)

(M  Mhf )2 m0F

mF

FmF

(5.8)

for a small rotation of B~ into the x^ axis, andP

F 1mF 2(M  Mhf )2 m0F (CFm
)
F


R=




0
F 1m0F 2
By 2
Bz 2 (C F 1mF )2  0
2 E 2 Pm0
mF mF 1 + B (CFmF ) m0F mF
FmF
B
F
0

(5.9)

for a small rotation of B~ into the y^ axis. This would not be a problem if the
populations of the mF sublevels were uniform. This is becauseXmF

F 1mF )2 = 2
(CFm
F

X
mF

F 1mF 1 )2 :(CFmF

Therefore, the ratios are constant, regardless of the value of Bx or By .

(5.10)

66
However, we know that pumping the atoms into a single hyper ne state
redistributes the atoms in a nonuniform way. The populations from a uniform distribution, from a simple Monte Carlo rate equation simulation, and from a fullquantum mechanical treatment are shown in Table 5.1. When we pump atoms into
the F = 3 state, more atoms tend to fall into the Zeeman sublevels near mF = 0,
and when we pump atoms into the F = 4 state, more atoms tend to fall into the
Zeeman sublevels near j mF j= 4.
F 0m0FThe CFm
F coecients have a nonuniform dependence on the initial mF
state that is similar to the dependence of the Zeeman sublevel population distribution. The mF = 0 transitions are stronger for transitions starting from near
mF = 0, and the mF = 1 transitions are weaker for the same transitions. The
result is that on the F = 3 to F 0 = 4 (F = 4 to F 0 = 3) transitions the numerator
gets larger (smaller) as B~ rotates into x^ and weights the mF = 0 transitions moreheavily. Similarly the denominator gets larger (smaller) as B~ rotates into y^. Thisbehavior has been veri ed experimentally. The results of the theory are compared
Table 5.1: The fractional population in a given Zeeman sublevel after hyper ne
pumping has been performed. The uniform distribution assumes that no redistribution of the populations takes place. \Bennett" refers to the populations found using
a simple Monte Carlo simulation of hyper ne pumping. \Marte" refers to the resultsof a full quantum mechanical calculation performed by Peter Marte [56].

F mF
3

3
2
1
0
-1
-2
-3

MethodUniform Bennett
0.1429 0.1030
0.1429 0.1570
0.1429 0.1598
0.1429 0.1598
0.1429 0.1598
0.1429 0.1570
0.1429 0.1030

Marte F mF
0.1163 440.1442
30.1632
2
0.1666
1
0.159300.1369-1
0.1134-2-3-4

MethodUniform Bennett
0.1111 0.1430
0.1111 0.1308
0.1111 0.1067
0.1111 0.0831
0.1111 0.0749
0.1111 0.0831
0.1111 0.1067
0.1111 0.1308
0.1111 0.1430

Marte
0.12300.10610.10380.10400.11090.10280.10680.11040.1321

67
with data in Fig. 5.3. The results shown are for the F = 4 to F 0 = 3 transition
with the magnetic eld rotated into the x^ direction. It is clear that the full quantum
mechanical treatment gives reasonable qualitative agreement.

Figure 5.3: Data and theory showing the e ect of misaligned B~ on the ratio M +Mhf = on the F = 4 to F 0 = 3 transition. Here the magnetic eld is rotated into
the x^ direction.
This error due to misaligned magnetic elds is the most serious systematic
error we need to address. Using the data we collected (not the theoretical prediction),
we nd that in order to have the e ect of a misaligned magnetic eld contribute lessthan 0.02% we must keep Bx and By less than 220 mG when the main eld is set at
Bz = 4 G. This is an easy requirement to meet since we routinely kept these elds
less than 1 mG for the PNC measurement. To con rm the absence of any e ect
we also take data with main magnetic elds smaller than 4 G. If there is a residual
misaligned component, the fractional error in the ratio will get larger as the main68
eld gets smaller. These data are shown in Fig. 5.4. The fractional di erence between

Figure 5.4: Measured ratios Mhf = as a function of Bz . Any deviation as the valueof Bz is scaled indicates the presence of a misaligned Bx or By .
the measurements at 1 G and 4 G is 0:1(1:0)  10,3 . The misaligned magnetic eldthat is required to give this change is 9.595 mG. If we assume the actual stray eld
is 100 mG, then the eld is a factor of two smaller than the maximum misaligned
eld we can tolerate. Thus, the error due to a misaligned eld is negligible in our
measurements.

5.3 Preliminary Data
The nal data taken to measure the ratio Mhf = were taken over a periodof one week and the conditions of the experiment were changed twice during that
period to ensure the measured ratio did not depend on things that we thought itshould not, for example collimator angle (which changes the direction of the atomic

69beam), alignment of the dye laser into the PBC, and beam density. The raw datawith no corrections are shown in Table 5.2. A least squares t of these data toTable 5.2: Table of raw data for the ratios RI = (M  Mhf = E )(1 + I ) with nocorrections and with E = 707:63(68).

I

2.300
2.000
1.500
1.000
0.800

R+ (10,3 ) R, (10,3 )2.5112(06) 1.1602(08)2.5047(12) 1.1573(16)2.4926(10) 1.1522(24)2.4793(29),2.4775(34),Eq. (5.7) with the correction shown in Eq. (5.6) gives

RI+ = 2:4636(24)  10,3 [1 + 0:009522(13)I ]:(5.11)The data plotted against this line are shown in Fig. 5.5. The reduced 2 for the t is
0.20 indicating a probability of 90% that the data come from a random distribution.
The correction for photoionization is dominated by the fraction of atoms
that make the E 1 transition. Therefore, the fact that M 13!4 = M , Mhf comparedto M 14!3 = M + Mhf has no e ect of the correction (1 + I ) and we expect thatthe correction factor  is the same for both ratios. If we use the slope from the R+
t to nd the least squares t to the R, data we nd that

RI, = 1:1357  10,3 [1 + 0:009522I ]:(5.12)The reduced 2 for this t indicates a 90% probability that the data come from arandom distribution. This means the data are consistent with our assumption thatboth ratios scale the same way with intensity. However, we do not use this t todetermine Mhf = because the poor signal-to-noise ratio on the F = 4 to F 0 = 3
transition at low intensities does not allow sucient precision in the determination

70

Figure 5.5: Plot of M , Mhf = vs. laser intensity showing the e ect of photoionization.
of R0, . Instead, we use the ratio RI+ =RI, using data at higher intensities where thesignal-to-noise ratio is high. This does allow sucient precision, and it eliminates
the need to determine R0, .
The equationq
Mhf = 1 , qRI+ =RI,
(5.13)
M 1 + RI+ =RI,
gives the value of Mhf =M without the need to make a correction for photoionizatione ects. The weighted average of RI+ =RI, for the three largest intensities shown in
Table 5.2 corrected according to Eq. (5.6) is RI+ =RI, = 2:1673(14): This gives apreliminary value of Mhf =M = ,0:1910(2).

71The equation

0

v
u

1

RI, A EMhf = qR+ @1 , u
t
0
RI+ 2

(5.14)

gives the value of Mhf = . Here it is only necessary to use the determination of R0+ ,and we can use the more precise values of R, at higher intensities. The preliminary value is Mhf = = ,5:6325(85) V/cm. We can combine this value with Mhf =,151:86(38) a30 V=cm [27] to arrive at the value = 26:962(41)expt (67)theory a30. However, these values need to be corrected for E 2.

5.4 Accounting for the Electric Quadrupole Amplitude
The data given in the previous section is very nearly the nal data, with thee ects of magnetic eld misalignments, photoionization, o set line centers, asymmetric line shapes, and nonzero background taken into consideration. However, we have
not yet considered the e ects of the electric quadrupole amplitude (E 2) in the measurements. If we had a uniform population distribution among the mF states there
would be no correction for the E 2 amplitude because its e ects cancel exactly. (SeeAppendix B.) We do not have a uniform distribution, as discussed in Section 2.6, so
we need to correct the data.
Electromagnetic amplitudes beyond the usual electric and magnetic dipole
amplitudes are commonly ignored because of their relatively small size compared toallowed E 1 and M 1 amplitudes. However, when dealing with amplitudes such as the
PNC E 1 amplitude and the rst-order forbidden M 1 amplitude, the E 2 amplitude
must be considered to reach accuracies of 0.1%.
Bouchiat and Guena [27] consider the E 2 amplitude in their reanalysis ofseveral experiments that determine the ratio Mhf =M for the 6S ! 7S transition incesium. In particular, they say that the work of Ref. [28] does not measure Mhf =M ,72
but instead measures the ratioE2  :
R = MMhf 1 , 34 M

(5.15)

hf

Clearly, if the ratio E 2=Mhf is small enough, there is no error. However, the resultsof Ref. [27] indicate that E 2=Mhf = 42(13)  10,3 and so Mhf =M = ,0:1886(17)rather than the value Mhf =M = ,0:1830(4) found in Ref. [28]. Indeed, if we compare
this with our result of Mhf =M = ,0:1910(2), there is clearly a discrepancy.
As shown in Appendix B, if we have the population distributions calculated
by Peter Marte, then the measurements of the ratio R+ need a ,0:094% correction
and the measurements of the ratio R, need a +0:079% correction. This is a small
but signi cant correction to our data. The revised numbers that follow assume that
there is a 50% uncertainty in these corrections, mainly from the uncertainty in thedetermination of the population distribution.
To arrive at the fractional corrections in the previous paragraph, we begin
by assuming that the E 2 amplitude does not a ect our measurements. The di erence
between the values of Mhf =M determined by the present experiment and in Ref. [28]then gives us the value of E 2=Mhf using the equation

M 
hf

M

present = MMhfRef : [28]E 2 ,1 :
1 , 34 M
hf(5.16)

Then we use this value of E 2=Mhf to correct our data for the E 2 contributions. Our
new value of Mhf =M can again be compared with that of Ref. [28] to determine an
improved value of E 2=Mhf , and so on. This procedure is iterated until we arrive at
an internally consistent value of E 2=Mhf . Our nal value is E 2=Mhf = 53(2)  10,3 ,which gives the corrections mentioned above. The nal results for this experimentare given in Table 5.3. Our new value of is compared to previous results in Fig. 5.6.

73

Table 5.3: Final results for the Mhf = experiment. The uncertainties shown are one
sigma values. In the rst three values, the uncertainty is purely experimental, andin the nal value the uncertainties have been separated into their contributions fromexperiment and theory.
Quantity
E 2=Mhf

Without E 2 Correction,

Mhf =M,0:1910(2)Mhf = (V/cm),5:6325(85)(a30 )
26:962(41)expt (67)theory

With E 2 Correction
0.053(3),0:1906(3),5:6195(91)27:024(43)expt (67)theory

Figure 5.6: An historical comparison of the determinations of . Gilbert is from
Ref. [57], Bouch1 is from Ref. [58], Bouch2 is from Ref. [59], Dzuba is from Ref. [24],and the theory is from Ref. [14].

CHAPTER 6
CONCLUSIONS AND IMPLICATIONS

We have now presented the two main results of this thesis. We nd thatthe 6S ! 7S dc Stark shift is k = 0:7262(8) Hz(V=cm),2 , and we nd that the
6S ! 7S tensor transition polarizability is = 27:024(43)expt (67)theory a30 . We will
now use the measurement of the dc Stark shift to re-evaluate the uncertainty of theatomic theory calculations, and then use our new measurement of , along with ourprevious measurement of Im(EPNC )= , to obtain a value for the weak charge QW .
This value of QW will be compared with the prediction of the standard model. The
comparison provides a test of the standard model.

6.1 Re-evaluation of Atomic Theory
The test of the standard model using a measurement of PNC in atomic
cesium requires calculations of the cesium matrix elements. If the uncertainty of theexperiment is signi cantly smaller than the uncertainty of the calculations, then thetest will be limited by the theory.
There are two groups that have made a considerable e ort to develop thetheory to determine the structure of cesium: Blundell, et al. [43, 14, 45] at the
University of Notre Dame and Dzuba, et al. [15, 47, 48] now at the University of
New South Wales. The most important result of their calculations is the value of75

kPNC , which is given by

X h7S j D~ j nP ihnP j HPNC j 6S i
kPNC = QN
E6S , EnP
W n
~ j 6S i !
h
7
S
j
H
j
nP
ih
nP
j
D
PNC
+
:
E7S , EnP

(6.1)

The quantity kPNC is made up of two types of matrix elements: allowed
electric dipole matrix elements like h6P j D~ j 6S i, and unmeasurable matrix elements
like h6P j HPNC j 6S i. Because HPNC is proportional to the Dirac matrix 5 , theselatter matrix elements are essentially the matrix elements of 5 .
Of equal importance to the calculated value of kPNC is the accuracy of thecalculation; if the calculation is only good to 2%, then the subsequent test of thestandard model is limited to 2%. Therefore, the two groups have spent considerable
time evaluating the accuracy of their calculations. Because the behavior of the two
sorts of matrix elements is so di erent|the matrix elements of D~ are sensitive tothe value of the wave function at large ~r and the matrix elements of 5 are sensitive
to values of small ~r|one must be careful to test the predictions of the theory using
quantities that behave in a similar fashion to both of these matrix elements. The mostobjective way to estimate the accuracy of the theory is to compare its predictions
with the results of experiments.
In order to provide these comparisons, many groups have performed highprecision measurements of the properties of cesium. Among the most precise of
these are measurements of the hyper ne structure constants. These measurements
are particularly relevant because of the similarity of their behavior to the behavior of
the matrix elements of 5 . Other precise measurements are of the lifetimes of excitedstates such as the 6P 12 ; 32 states. The lifetime measurements are important because
they involve the same radial matrix elements as kPNC , such as h6P j D~ j 7S i.
To quote from Ref. [14]: \If all known properties of cesium could be reproduced by some ab initio calculation scheme to within 0.1%, we feel it would be rea-

76
sonable to trust a PNC calculation carried out in the same fashion to this same level,
subject to a scatter analysis along the lines presented above." The \scatter analysis"
mentioned is one where the measured values of EnS , EnP , and hnS j D~ j nP i areused in di erent combinations with ab initio calculations to estimate how much thevalue of kPNC would change with small changes in the calculated parameters .
In Blundell et al. [14], the authors quote a variation of 0.78%, but thestandard deviation of their calculated numbers is more accurate:  = 0:40%. Thus,
if we can show that the theory's predictions agree with experiments at the 0.4% level,
then it should be reasonable to trust the calculation of kPNC to that same level.
All of the measurements made prior to the publication of Ref. [14] show only
small di erences from the theory (less than 1%) with the exception of the measurement of the dc Stark shift [49], where there was a 2% di erence between the theory
and experiment (Chapter 4). This di erence is now 0.3%. In addition, there are newexperiments that reveal errors in earlier lifetime measurements of sodium [60, 61] andlithium [62]. These new experiments eliminate what had appeared to be troubling
1% errors in calculations for these atoms that are equivalent to the calculations incesium. Finally, the agreement is very good for the newly-measured 7S ! 9S interval in 210 Fr [63, 64]. Now is an opportune time to revisit the comparison between
the experiments and the predictions and re-evaluate the uncertainty that should be
attributed to the theory.
In Table 6.1 we have collected the most precise measurements of several
quantities in cesium that provide tests of the theory, along with the \rescaled" predictions of Dzuba, et al. and the ab initio predictions of Blundell et al.
Particularly notable are the top three lines of the table, which show thatthe agreement has dramatically improved from the 1-2% agreements of the older
 This scatter analysis is performed by Blundell et al. to estimate their uncertainty. They do notchange the numbers they report, however. When the ab initio calculation of Dzuba et al. did not\reproduce the energy level, (they) multiplied the correlation correction by some numerical factor
to t the energy." [47] That is, they use the experimental energies to \correct" their calculations.

77
Table 6.1: Fractional di erences (103 ) between measured and calculated values ofquantities relevant for testing PNC calculations in atomic cesium. We only list themost precise experiments. The second column lists the most relevant aspects of the
wave functions that are being tested. h1=r3 inP is the average of 1=r3 over the wave
function of the electronic state nP . Where the experiment has improved or changed
signi cantly since the publication of Ref. [14], the di erence from the old experimentis listed in brackets.
Quantity
Calculation
Di erence (103 )measuredtestedDzubaa Blundell b Expt
6S!7S dc Stark shift c
h7P jj D~ jj 7S i
,3.4[19] ,0.7[22] 1.0[4]d~6P1=2 lifetime
h6S jj D jj 6P1=2 i,4.2[,8]4.3[1]
1.0[43]
6P3=2 lifetime dh6S jj D~ jj 6P3=2 i ,2.6[,41] 7.9[,31] 2.3[22]eh7S jj D~ jj 6P1=2 i, andh7S jj D~ jj 6P3=2 i
,
,1.4
3.2
f
same as,,0.8
3.0
6S HFS g
(
r
=
0)
1.8
,
3.1
,
6S
h
7S HFS
,6.0
,3.4
0.2
7S (r = 0)
6P1=2 HFS ih1=r3 i6P
,6.1
2.6
0.2
j
3
7P1=2 HFS
h1=r i7P
,7.1
,1.5
0.5
experiments .
The standard deviation of the fractional di erences between theory andexperiment in Table 6.1 is 4:0  10,3 . We believe this to be the most objective
number to use to represent the 68% con dence level for the atomic theory, and thus
for the uncertainty on kPNC . Using the average of kPNC = 0:905  10,11 iea0 [14]and kPNC = 0:908  10,11 iea0 [15], we nd the value

kPNC = 0:9065(36)  10,11 iea0 :(6.2)

6.2 Final Result and Implications
We now have all the numbers|with less than 0.5% uncertainty|to determine the value of the weak charge and to test the standard model. When we
The footnotes in Table 6.1 are as follows. a Refs. [15, 47, 48]; b [14, 43, 44, 45]; c This work;
Ref. [40]; e Using the present work and = from Ref. [25]; f This work; g De ned; h Ref. [41];iRef. [65], i Ref. [42].


d

78
combine the 0.40% value of kPNC from Eq. (6.2) with our new 0.30% value for andthe 0.35% measurement Im(EPNC )= from Ref. [13], we nd

QW = ,72:06(28)expt (34)theory :(6.3)

This value is virtually the same as our previous result QW = ,72:11(27)expt (89)theory [13]but more precise. It would be extremely satisfying to use this number to say thatthe standard model is either right or wrong, but a quick look at a recent 105 page
review paper on tests of the standard model [66] shows that it is not an easy task to
draw such a simple conclusion.
The value of QW given by Eq. (2.2) depends on the two coupling constants
C1u and C1d . These constants depend on the de nition of sin2 W , and they depend
on which radiative corrections are included in the calculation. Using the analysisof [67, 68], we nd that the standard model predicts QW = ,73:20(13). This value
di ers from our measured value by 2.5. A comparison of our values of QW and thestandard model prediction is shown in Fig. 6.1.
Assuming that the di erence from the standard model is not due to an
experimental error or a statistical uctuation, it suggests several possibilities. The
rst is that there is some very perverse feature of the calculated electronic wave
functions and the 5 operator that causes errors in the calculation of this matrix
element to be much larger than for directly measurable matrix elements. In thiscase, our assumption that the standard deviation of the values in Table 6.1 represents the accuracy of kPNC is wrong. The second is that there are contributions orcorrections to atomic PNC within the standard model that have been overlooked.
We see no justi cation for either of these possibilities, but they clearly need to be
explored further. The rst o ers a formidable but not overwhelming challenge toboth theoretical and experimental atomic physicists.
The nal possibility is that the discrepancy is indicative of the presence

79

Figure 6.1: A comparison of values from Ref. [13], this work, and the standard model.
The di erence between this work and the standard model is 2.5.
of some new physics not contained in the standard model. Physics that would becharacterized by the S parameter is not a likely candidate because the size of thecontribution needed (S = ,1:44(35)expt (46)theory ) would be in con ict with otherdata [6]. However, there are other types of new physics, such as an additional Z
boson, that would be consistent all other current data. For example, there is atheory that predicts the existence of an extra Z boson [67] with a mass given by2mWNQW (55+
55 Cs) ' 0:4(2N + 55) m2 :
Z

(6.4)

Using our value of QW this theory predicts mz = 692(29) GeV/c2 . The possibility
that this theory|or other theories like it [69, 70, 71, 72, 73]|is correct also needs
to be explored further.

CHAPTER 7CONSIDERATIONS FOR FUTURE EXPERIMENTS

The implications of atomic parity nonconservation for the standard model
depend on both experiments and theory. The present work lessens the dependenceof the test of the standard model on the theory, but does not eliminate it. If there-evaluation of the uncertainty in the theory stands up to scrutiny, and if the calculated value of the PNC matrix elements does not change signi cantly with further
re nement of the theory, then atomic PNC reveals a need for extensions to thestandard model.
It is worthwhile to consider improvements in the experiments. Therefore,
this chapter explores ways that the measurement presented in Refs. [13, 23] might
be improved. There are three approaches I have studied: reduce (or eliminate) thee ect of the largest systematic errors, increase the signal-to-noise ratio to reduce theamount of time it takes to reach a given precision, or make another measurement
using the same technique on the so-called \ " transition. Section 7.1 describes amethod that can be used to make the 6S ! 7S line shape symmetric and thus eliminate or reduce many of the systematic errors that are enhanced by an asymmetricline shape. Section 7.2 details an experiment on transversely cooling the atomic
beam to increase the signal-to-noise ratio. Section 7.3 discusses repeating the PNC
measurement using the \ " transition.

81

7.1 Cavity Sidebands
7.1.1 Experimental Idea
Without the high- nesse Fabry-Perot etalon, the PNC measurement presented in Refs. [13, 23] would not have been possible. The increase in laser power
inside the cavity makes it possible to drive very weak transitions with high signal-tonoise ratios. The standing-wave nature of the laser eld inside the etalon does have
one large drawback: a spatially-varying ac Stark shift due to the sinusoidal variationof laser intensity.
Because the intensity inside the cavity is proportional to sin ky, atoms
crossing through the laser beam at di erent positions su er from di erent size ac
Stark shifts. Furthermore, atoms that have a nonzero transverse velocity see achanging intensity and thus experience a varying ac Stark shift. The spatially varying
ac Stark shift causes the asymmetry and broadening in the spectral lines of the 6S !
7S transition as discussed in Ref. [30]. The asymmetry couples with other e ects
such as E 1-M 1 interference to produce PNC systematic errors. If the asymmetry
can be eliminated, some of the most troublesome systematic errors could also be
eliminated.
Because the asymmetry comes directly from the spatially varying laser intensity, if we could make the intracavity power uniform, the asymmetry would be
eliminated. This can be accomplished by phase modulating the laser light incident
upon the PBC at the free spectral range of the PBC. The following discussion willshow how this is possible.
A phase modulated laser eld of the form  = A cos[(!t ,  sin !m t)] canalso be written in terms of Bessel functions as [74]

 = A[J0 () cos !t + J1 () cos(! , !m )t , J1 () cos(! + !m )t+ J2 () cos(! , 2!m )t + J1 () cos(! + 2!m )t];

(7.1)

82
where terms involving Bessel functions higher than order two are neglected. Presented in this form it is easy to see that the phase modulation at !m puts sidebandson the laser that are separated from the main frequency by integer multiples of!m. The size of the sidebands relative to the carrier with a modulation index of = 1:2045 is shown in Fig. 7.1.

Figure 7.1: Relative sizes of phase modulation sidebands with  = 1:2045. The
height of each line indicates the amount of power in each sideband relative to thecarrier [J02 ()]. A negative height indicates a 180 phase shift with respect to theother sidebands.
If the laser eld is incident on a perfectly stable Fabry-Perot etalon with afree spectral range !FSR = 4`=c, where ` is the cavity length, and if !m = !FSR ,then Appendix C shows that the standing wave inside the etalon is given by83

 = A[2J0 () sin ky sin !t + 2J1 ()(sin k+ y sin !+t , cos k,y sin !,t)(7.2)

+ 2J2 ()(sin k,, y sin !,, t + sin k++ y sin !++t)];where !  !  !m , !  !  2!m and so on. If the natural lifetime of an atomp
is long compared to both ! and !m , and A = 1= 2, then the atom \sees" a time
averaged intensity given by

I = sin2 ky[J02 () + 2J12 () cos 2km y + 2J22 () cos 4km y](7.3)+ 2[J12 () sin2 km y + J22 () sin2 2km y]:
When  = 0, there is no phase modulation and the intensity is simply
I = sin2 kz. This is exactly the situation discussed in Ref. [30] that gives rise to a
spatially varying ac Stark shift and asymmetric line shapes. When  6= 0, there is
still the sin2 kz term. Now, however, that term is multiplied by a much more slowly
spatially varying term that depends on km . In addition, there is a term that does nothave the fast sin2 kz dependence. The result is a rapidly spatially varying intensity
superimposed on a more uniform pro le. The envelope of the rapidly varying partis shown in Fig. 7.2. The question now is what e ect will this have on the ac Stark
shift and on the 6S ! 7S transition rate for each atom?
The size of the ac Stark shift depends on the detuning of the laser lightfrom allowed transitions such as the 6S1=2 ! 6P3=2 transition. The detuning of thecarrier from this transition is approximately 2  108 MHz, and the rst side bandis only di erent by 500 MHz, which is 2:5  10,6 of the carrier detuning. For the
purposes of the ac Stark shift, the sideband and the carrier are indistinguishable.
Thus, the ac Stark shift depends on the time average of the sidebands and the carrier
combined.
Conversely, the strength of the 6S ! 7S transition depends on the laser
detuning from the 6S ! 7S resonance. Therefore, the carrier is on resonance, but

84

Figure 7.2: A plot showing the intensity of the laser light inside the PBC as afunction of position in the cavity when the incident light is phase modulated at thefree spectral range of the PBC. The intensity is the sum of two terms: one that varies
rapidly with position (sin2 ky) and one that varies slowly (trigonometric functionsof km y.) Close to either end of the cavity the intensity varies rapidly, but near thecenter where the atomic beam intersects the laser beam, the intensity is very close
to uniform.
the sidebands are well detuned ( 100,) from resonance. Thus, the strength of the
6S ! 7S transition depends on the time average of only the carrier. The result isthat we can eliminate the spatial dependence of the ac Stark shift at the expense ofa factor of two in signal size.

7.1.2 Experimental Implementation
In order to put sidebands on the light inside the PBC we borrowed a homebuilt electro-optic modulator (EOM) from Dr. John Hall's laboratories. We drive
the EOM using a tunable frequency synthesizer. The EOM has a resonant frequencyof 503 MHz and is tunable from 489 MHz to 518 MHz. The resonant frequency is

85very sensitive to temperature, so any attempt to make permanent use of the EOMmust include temperature stabilization. For the purpose of demonstrating the e ectof the sidebands, temperature stabilization is not needed. In its usual con guration,
the free spectral range of the PBC is slightly larger than 518 MHz, so during thisexperiment the input mirror of the PBC is mounted on a modi ed optical mount asshown in Fig. 7.3.

Figure 7.3: Under normal conditions, the length of the PBC is 27 cm. In order tochange the free spectral range of the PBC to match the resonant frequency of theEOM used to phase modulate the dye laser the input mirror has to be moved in
front of its mount to make the cavity length 30 cm.
To monitor the sidebands on the dye laser we place a scanning Fabry-Perot
etalon after the PBC and use it as a spectrum analyzer. The amount of power in the
sidebands is easily monitored using the etalon. The dye laser with sidebands wasthen simply locked to the PBC using the usual Pound-Drever-Hall scheme.

86
The asymmetry of the 6S ! 7S line shape is emphasized if the photoionization current on the electric eld plates is monitored. Therefore, we connect anelectrometer to the eld plates during this experiment so we can look at both theprobe scattered light and the photoionization line shapes. We use the 6S1=2 F = 3
to 7S1=2 F 0 = 4 transition for these measurements.

7.1.3 Results
Sample scans over the \ " 6S ! 7S E 1 transition with and without sidebands are shown in Fig. 7.4. The dramatic reduction of the asymmetry in the line
shapes is evident by comparing the pairs of scans. However, a signi cant reductionin peak signal also occurs. Scans were also taken (unintentionally at rst) with themodulation index slightly o the optimum  = 1:0245. In that case, the peaks of thescans are very sharp and the asymmetry is prominent. It only takes a change in of about 1% to begin to see the asymmetry again.

7.1.4 Conclusion
The addition of the sidebands on the dye laser to remove the asymmetry
in the line shape of the 6S ! 7S transition can be considered a success, but it is not
at all clear that this is a viable solution to the problem of line shape asymmetry for
two reasons.
First, the price of having the sidebands to produce a uniform intensity isa factor of two loss in the amount of laser light that is resonant with the 6S ! 7S
transition. Losing a factor of two in signal means we need to take twice as muchdata to get the same uncertainty as before. The whole purpose of the sidebands wasto eliminate a major source of systematic error so that we would not have to take asmuch data. Reducing the resonant intensity by a factor of two does not achieve thisgoal. If the line shape asymmetry were the limiting systematic the sidebands might
be an excellent solution, but that is not the case with the PNC measurement.

87
Second, putting sidebands on the dye laser is a complicated task. The additional temperature stabilization that would be required, as well as nding some way
to determine when the modulation index is correct, would add many complications
to an almost hopelessly complicated experiment. So, while the sidebands do makethe line shape symmetric, in the end it seems that phase modulating the dye laser is
not a modi cation that we would like to make to a next-generation PNC experiment.

7.2 Improving the Signal-to-Noise Ratio
7.2.1 Increasing Beam Flux
The second way to improve upon the last PNC measurement would be to
improve the signal-to-noise ratio. Increasing the signal-to-noise ratio by a factor ofn also reduces by a factor of n the amount of time required to reach a given level ofprecision, since the PNC measurement is shot noise limited.
We investigated the e ect of turning up the oven temperature, which should
increase the number of atoms in the beam. As shown in Fig. 7.5, the beam size asmeasured by the probe laser does increase with temperature initially, but at higher
temperatures the number of atoms reaching the detection region begins to plateau.
A plausible explanation for this e ect is that at high temperatures there are so many
atoms emerging from the oven nozzle that not all of the atoms pass through the rst
beam-de ning aperture. The result is that a cloud of atoms forms just in front of
the oven nozzle and becomes the e ective source of the atomic beam. A hotter oven
only means a bigger cloud, not a denser beam. Details of such a process are given
in Ref. [75].
At approximately 225 C the oven repeatably exhibits behavior we call\crashing": the size of the beam increases by 30 to 50% and then settles to a value
10 to 20% lower than when the temperature was changed. This decreased level thenslowly decreases with time. If the oven is turned o , allowed to cool, and turned back88
on again, the signal size at a given temperature is lower than before but stable. We
cannot explain this behavior, but it gives us an upper limit for the oven temperature.
Another drawback of a very dense atomic beam is that the eciency withwhich the optical pumping lasers spin-polarize the atomic beam decreases as thedensity of the atomic beam increases. Data showing this e ect are shown in Fig. 7.6.
At low beam ux, the spin-polarization is near 98%, but as the beam size is increased,
the spin-polarization drops o signi cantly. Future PNC experiments using thisapparatus will rely on highly spin-polarized samples of atoms, so an increase in thedensity of the atomic beam that causes a reduction in the degree of spin-polarizationis undesirable.

7.2.2 Transversely Cooling the Atomic Beam
7.2.2.1 Experimental Concept
An alternative to simply increasing the number of atoms in the beam isto transversely cool the atomic beam to decrease its divergence and to increase
the number of atoms available for making the 6S ! 7S transition. However, thetechniques normally used to cool atoms do so by making the hot atoms scattermany photons and thus dissipate energy. If we were to use the same techniquesin the PNC experiment, a signi cant portion of those scattered photons would bescattered back along the beam where they can be absorbed by atoms that havealready been spin-polarized. The result is a subsequent depolarization of the beam,
which is unacceptable.
In 1986 Dallibard and Cohen-Tannoudji [76] published a paper describing
the so-called \Dressed-Atom" approach that describes the interaction of atoms withhigh-intensity standing-wave laser eld. In the regime of high intensity, simple pictures of the light-atom interaction (as used in describing Doppler cooling) break
down. The dressed atom approach uses the basis states j i; ni, where i is either 1 or
2, and n indicates the number of photons in the electromagnetic eld. At low laser

89
intensities these \dressed" states reduce to the normal basis states j s; ni, where s
represents the ground (g) or excited (e) state of the atom. At high intensities, thestates j 1; ni and j 2; ni are di erent linear combinations of the j g; ni and j e; ni states.
A comparison of the two sets of basis states is shown in Fig. 7.7.
Where the laser intensity is small, the states reduce to the normal j s; niset. Where the laser intensity is high, the j i; ni basis states must be used, and theenergies of these states depend on the intensity. Therefore, in any laser beam (where
the intensity pro le is, for example, Gaussian) there is a maximum energy shift at
the center of the beam. The energy shift gets smaller as the atom moves away fromthe center. If the atom is in a standing wave, the intensity varies sinusoidally in thelongitudinal direction as well as varying as a Gaussian in the transverse direction.
This longitudinal variation is shown in Fig. 7.8. In the Dressed-atom picture, atoms
moving along a standing wave can be cooled via stimulated emission. Using Fig. 7.8
as a guide, we now describe the idea behind the cooling.
Consider an atom in the j 1; n + 1i state moving along the standing wave.
The probability of this atom making a transition to another state is small in thenodes and large in the anti-nodes. As the atom moves from node to anti-node it
loses energy. If it makes a transition at the peak of the anti-node (point A), thenthat energy loss is permanent. The atom, now in the j 2; ni state (point B), again
moves along the standing wave. Now, however, the probability of making a transitionis large at the nodes and small at the anti-nodes. If the atom again makes a transition(at point C), the energy loss from anti-node to node is again permanent. As the atom
moves along the standing wave the process repeats itself (points D and E) and theatom is cooled. More important, since the force due to stimulated emission cooling ismuch greater than that due to Doppler cooling, many fewer photons are required to
do the same amount of cooling and the problem of depolarizing the beam is minimal.
There are two demonstrations of this cooling e ect on a cesium atomic
beam where the standing wave is detuned from the 6S1=2 ! 6P3=2 transition. One

90experiment [77] uses a high power ring dye laser, and another experiment [78] uses
a low power diode laser coupled into a power build-up cavity. In both cases, coolingof atoms with transverse velocities of less than 4 m/s was achieved. However, it was
unclear whether the same mechanism could cool atoms with transverse velocities of
up to 17 m/s. If this is possible, it may lead to an increased signal-to-noise ratio for
the PNC experiment.

7.2.2.2 Horizontal Cooling
In order to utilize transverse cooling of the atomic beam we need a stable,
high-power laser and a power build-up cavity into which we can inject the power.
We use a Ti:sapphire laser that provides 300 mW of 852 nm light and a cavity
constructed of two at, 1 in diameter mirrors separated by 5 cm. The nesse of
the cavity is roughly 40, and the ultimate power density inside is 3.4 W/cm2 . The
cavity is placed in the optical pumping region with the optical pumping magnetic
eld coils removed. Once aligned, the cavity does not creep out of alignment; allthat is needed to peak up the power inside the cavity is minor realignment of the
input laser beam.
All horizontal collimating elements after the cavity are removed (a verticalvane collimator and two beam de ning apertures) so as many atoms as possible
actually reach the detection region. In the two previous references, detection of thevelocity distribution is performed using a hot wire detector. The transverse velocity
of a group of atoms is inferred from its position in space. In our experiment we detect
the atoms by collecting the light scattered by the atomic beam as we sweep the probe
laser across the 6S1=2 ! 6P3=2 transition. The transverse velocity is inferred fromthe Doppler shift of a particular group of atoms and we use a saturated absorption
spectrometer as a reference. In the rest of the chapter the units of velocity are MHz.
These may be converted to m/s using the formula v =  , where  = 852 nm and is the Doppler shift.

91
Our detection method limits our sensitivity to the amount of cooling. Imagine an atomic beam with a Gaussian distribution transverse velocities. If the widthof the distribution is 20 MHz FWHM, then we would measure a line width very close
to 20 MHz. Now imagine we can cool all the atoms in the beam to zero transverse
velocity. A hot wire detector would measure a large signal right near zero velocity;
we would measure a line width of about 10 MHz since we are limited by the naturalline width of the 6S1=2 ! 6P3=2 transition.
In order to overcome this limitation, we leave in the rst vertical-vane
collimator. This allows us to \steer" the beam. Thus, we can move the beam sothat the transverse velocities are centered at various Doppler shifts, not necessarilynear zero shift. Then, if we cool all the atoms to zero transverse velocity we will see
the entire line shape move to zero Doppler shift. Sample uncooled line shapes areshown in Fig. 7.9.
The previous experiments used carefully chosen detunings to optimize their
cooling; once we saw evidence of cooling we selected the detuning that appeared togive the \best" e ect. The detuning for all the data shown here is  = 48 MHz = 9,,
where , is the natural line width of the 6S1=2 ! 6P3=2 transition. It is possible tostudy the e ects of detuning extensively, but the discussion that follows shows thatthe exact detuning does not ultimately a ect the increase in signal that is possible,
so we do not perform this study.
Taking the data for this experiment is almost trivial. Interpreting thedata, however, is not. Figures. 7.10 through 7.15 show scans with and without
cooling for a variety of collimator positions. In all cases, the e ect of the coolingis evident, whether it is a narrowing of the line width (Fig. 7.10) or a signi cantshift of the atoms toward zero Doppler shift (Fig. 7.15). Also shown in each gure is
the di erence between the cooled and the uncooled lines showing clearly that atoms
are removed from areas with high transverse velocity and placed in areas of lowtransverse velocity.

92
In order to quantify the e ects of the cooling we would like to be able tolook at the number of atoms having a given transverse velocity. That is, we wantto know the population distribution (or equivalently, the velocity distribution) of
the beam. Since the uorescence we measure is the convolution of the population
distribution with the natural line shape, we should, in principle, be able to deconvolve
the population distribution from the uorescence if we know the natural line shape.
In practice this is very dicult because the solution to the deconvolution is notunique.
As an alternative to a deconvolution, we can make an ansatz about the
population distribution and then convolute that with the natural line width of the
6P3=2 ! 6S1=2 transition. We can then use the di erence between the line shapeprediction of the ansatz and the measured line shape to modify our initial ansatz.
Several iterations give good agreement with measured line shape. An example of an
initial ansatz and the nal population distribution for the data shown in Fig. 7.16.
The asymmetry is not surprising because of the probable bias angle in the capillaryarray as mentioned in the caption of Fig. 7.9. The velocity distribution of the wholeatomic beam with no collimation is shown in Fig. 7.17.
Ultimately, we want to know the nal velocity distribution of the atomic
beam after cooling. To do this, we need to compare the velocity distributions in
the beam before and after cooling. So, we perform the iteration discussed for eachcollimator position. This gives us the velocity distribution for each position before
and after cooling. We then ask the question: \What fraction of atoms in a given
velocity class is transferred to near the zero transverse velocity class?" The answer
gives us the e ect of cooling as a function of transverse velocity. The maximum,
average, and minimum e ects of cooling are shown in Fig. 7.18.
Now that we know what fraction of the atoms with a given transverse velocity will be cooled to near zero transverse velocity, we can determine the population
distribution of the whole atomic beam with and without cooling. These data are93
shown in Fig. 7.19 The number of atoms with zero transverse velocity increases by
about a factor of eight. Clearly the cooling technique is successful at slowing atoms
with transverse velocities between 10 MHz and 4 MHz to less than 4 MHz.
The net e ect of the cooling on the atomic beam is not as dramatic as
Fig. 7.19 would suggest, however. The line shapes we would expect if we measured
the beam and the 6S ! 7S transition are shown in Fig. 7.20. The severe reductionin the improvement of the signal is due to the fact that in the PNC experiment
we must always convolve the population distribution with the natural line width ofthe detection. In addition, the ac Stark broadening [30] will broaden the 6S ! 7S
transition regardless of cooling, and this e ect is not shown in the gures. The
variation in intensity in the radial direction of the dye laser is unavoidable, andthe variation of the intensity along the laser due to the standing wave can only beeliminated if we sacri ce a factor of two in power, as discussed in Section 7.1. We
estimate the reduction in the bene t of cooling by the ac Stark shift is 50%, which
limits the improvement in the 6S ! 7S signal size to a factor of 1.5.

7.2.2.3 Vertical Cooling
It is important to remember here that the previous discussion is relevant
only for cooling in the horizontal direction. For the vertical direction, the naturalline widths are not important. What is important is the total number of atoms
interacting with the dye laser that is driving the 6S1=2 ! 7S1=2 transition. For thisanalysis we assume that we can cool the beam perfectly in the vertical direction.
First, the relative number of atoms emerging from a long thin tube at anangle  is given approximately by [75]N () = 1:00978 , 0:85898 , 0:094892 + 0:151633 :(7.4)

The capillary array in the oven nozzle can be thought of as many in nitely thin tubes

94
stacked one upon another. The total number of atoms coming out of a vertical slicein the array is then given by

NTotal = 2

Z =2
0

N ()d  5 mm: = 6:35:(7.5)

This is simply twice the sum of all the atoms with trajectory angles from 0 to 90
times the height of the array. The total number of atoms in an uncooled beamintersecting the dye laser beam is given by
Hot
NIntersect=2

Z 2:5mm ,2:5mm

N Sin,1 (jxj=260mm) dx  1mm = 0:5:

(7.6)

This is twice the contribution from each of the in nitely thin tubes in one half of thearray that are a distance x from the middle of the array that reach the laser beam
26 cm away. If we can perfectly collimate the beam, then all the NTotal atoms will
be compressed into the height of the array. The number of atoms intersecting thebeam will be
1mm  6:35 = 1:27
Cooled = height of laser  N=
NIntersect
Total
height of array
5mm

(7.7)

In this case, we gain the ratio of
CooledNIntersect= 2:54:Hot
NIntersect(7.8)

7.2.3 Conclusion
The best improvement we can get by cooling is approximately a factor of
1.5 in the horizontal direction and a factor of 2.54 in the vertical direction givinga factor of 3.8 total improvement. This assumes that the ac Stark shift [30] only
reduces the improvement in horizontal cooling by 50% and that we can cool the beam
perfectly in the vertical direction. This seems to be an overly optimistic situation.

95
Further, attempts to increase the beam ux by increasing the oven temperature arecounter productive.
The bottom line is that, while we get approximately eight times more atoms
within a 4 MHz Doppler width, the largest improvement in signal on the 6S ! 7S
transition will be about a factor of three. With only this marginal improvement,a second measurement of PNC using the same technique as Refs. [13, 23] is not
attractive.

7.3 Prospects for Further Parity Nonconservation Measurements
Equation (2.26) shows the quantity measured in the recent PNC measurement [13, 23]. In that experiment we measure the interference between a PNCelectric dipole amplitude and the \ " Stark-induced electric dipole transition. This
experiment provides the value Im(EPNC )= . A similar experiment that uses the \ "Stark-induced amplitude from Eq. (2.13) can provide the value Im(EPNC )= . This
would be a useful quantity to measure because it would provide another determination of the quantity Im(EPNC ) as well as providing a consistency check of the pastmeasurements of Im(EPNC )= and = [25]. Perhaps the most important reason to
make a measurement of Im(EPNC )= is that we could check the value of the nuclear
anapole moment determined in Ref. [13]. However, there are several problems thatbring the feasibility of such an experiment into question.
The measurement of Im(EPNC )= is performed on both the F = 3 to F 0 = 4
and F = 4 to F 0 = 3 6S ! 7S transitions. When using the F = +1 (F = ,1)
transition, the F = 4 (F = 3) hyper ne state is depleted. Atoms making the
6S ! 7S transition repopulate the F = 4 (F = 3) state where they are detectedusing the 6S1=2 F = 4 to 6P3=2 F 0 = 5 ( 6S1=2 F = 3 to 6P3=2 F 0 = 2) cycling transition.
These detection schemes are favorable because 75% (67%) of the atoms making the
6S ! 7S transition repopulate the proper detection state.
However, the measurement of Im(EPNC )= is performed on the F = 0

96transitions. Here only 33% (24%) of the atoms excited to the F 0 = 3 (F 0 = 4) state
end up in the proper F = 4 (F = 3) state where they can be detected. So, attempting
a measurement of Im(EPNC )= starts with a signal-to-noise ratio more than a factorpof 2 smaller than that of the already-dicult measurement of Im(EPNC )= .
Another consideration is the problem of fractional populations alluded toin the discussion of Eq. (2.26). The previous measurement of PNC in cesium used aspin-polarized atomic beam with roughly 98% of the atoms in a single extreme mF
sublevel. Equation (2.26) included detuning factors dmF , which indicate the relative
importance of transitions that are detuned from the main transition because of the
Zeeman e ect. The largest of the detuning factors is 0.845, while the smallest is0.127. The e ect of these factors is to reduce the precision with which we need toknow the populations of the mF levels.
On the F = 0 transitions the Zeeman e ect does not resolve transitions
from di erent mF sublevels because the g-factors for hyper ne levels with the same F
are equal. (See Fig. 2.1.) The result is that all the detuning factors in the \ " analog
of Eq. (2.26) are unity. For the Im(EPNC )= measurement we only needed to measure
the populations to 1% to have an uncertainty of 0.1% in the nal measurement; for ameasurement of Im(EPNC )= that requirement becomes an order of magnitude more
severe. We would need to measure the populations to 0.1% to have 0.1% in the nal
measurement. At present, our ability to measure the populations is 0.1 to 0.2%. In
order to be comfortably below the uncertainty required, a much better modeling of
the optical pumping processes that we use to spin polarize the beam would have tobe accomplished.
Further, while one would not expect a measurement of Im(EPNC )= to takethe more than 15 years that the measurement of Im(EPNC )= took, a considerable
amount of time (perhaps two graduate student lifetimes) would be needed to nd andeliminate all the systematic errors that will inevitably plague the new measurement.
These three considerations make a measurement of Im(EPNC )= using a

97spin-polarized atomic beam an unlikely next step.

7.4 Summary
The previous 0.35% measurement of PNC in atomic cesium is a notableachievement that, when combined with the 0.3% measurement of and the reevaluation of the ab initio theory, provides a 0.6% test of the standard model. As
with any measurement with this importance, a con rming experiment is always
comforting. However, it does not seem feasible to make another measurement of
PNC using the present methods, whether it is a measurement of Im(EPNC )= orIm(EPNC )= , mainly because of signal-to-noise concerns.
Therefore, the work on parity nonconservation using an atomic beam hascome to a conclusion after more than 15 years. Future work may involve cold,
trapped atoms or measurements on a chain of isotopes to eliminate the need for theatomic theory calculations.

98

Figure 7.4: Sample scans over the \ " 6S ! 7S E 1 transition with and without sidebands using (a) photoionization detection and (b) probe detection. The asymmetry
disappears with sidebands, but the peak signal also drops signi cantly

99

Figure 7.5: This plot shows the size of the atomic beam as a function of the oventemperature. A \saturation" e ect at higher temperatures can be seen, as well as
evidence of an oven \crash" at about 225 C.

100

Figure 7.6: This plot shows the spin polarization as a function of the size of theatomic beam. Signi cant deterioration of the spin-polarization at large beam sizes
is obvious.

101

Figure 7.7: The bare and dressed states of an atom in an intense standing-wave laser
eld. Here  is the detuning of the laser from the jgi to jei transition, g and e are
ground and excited states, n is the number of photons in the laser eld, !` is thefrequency of the laser, 1 and 2 are linear combinations of the g and e states, andis the generalized Rabi frequency.

102

Figure 7.8: The energy levels of an atom in an intense standing-wave eld. The
atoms starts in j 1; n + 1i and ends in j 1; n , 1i and is cooled by losing energy as it
moves from troughs to peaks and then moves to a di erent state.

103

Figure 7.9: Doppler pro les of the atomic beam with no cooling at various collimator
positions. The fact that the line with the largest signal is slightly o set from zerodetuning indicates that the capillary array in the oven nozzle is slightly tilted.

104

1.0

Signal Size (arb. units)

0.8

w/o coolingw/ cooling

0.6

0.4

0.2

0.0

Difference (with cooling - without cooling)

0.20

0.15

0.10

0.05

0.00

-0.05

-0.10
-20

-10

0

10

20

Doppler Shift (MHz)Figure 7.10: Data showing the atomic beam without cooling with transverse velocity
centered at 0.6 MHz. Also shown is the cooled beam and the di erence between thescans with without cooling.

105

1.0

Signal Size (arb. units)

0.8

w/o coolingw/ cooling

0.6

0.4

0.2

0.0

Difference (with cooling - without cooling)

0.3

0.2

0.1

0.0

-0.1

-0.2
-20

-10

0

10

20

Doppler Shift (MHz)Figure 7.11: Data showing the atomic beam without cooling with transverse velocity
centered at 3.1 MHz. Also shown is the cooled beam and the di erence between thescans with and without cooling.

106

0.8
w/o coolingw/ coolingSignal Size (arb. units)

0.6

0.4

0.2

0.0

Difference (with cooling - without cooling)

0.3

0.2

0.1

0.0

-0.1

-0.2
-20

-10

0

10

20

30

Doppler Shift (MHz)Figure 7.12: Data showing the atomic beam without cooling with transverse velocity
centered at 5.5 MHz. Also shown is the cooled beam and the di erence between thescans with and without cooling.

107

0.6

Signal Size (arb. units)

0.5

w/o coolingw/ cooling

0.4

0.3

0.2

0.1

0.0

Difference (with cooling - without cooling)

0.25
0.20
0.15
0.10
0.05
0.00
-0.05
-0.10
-0.15
-20

-10

0

10

20

30

Doppler Shift (MHz)Figure 7.13: Data showing the atomic beam without cooling with transverse velocity
centered at 7.9 MHz. Also shown is the cooled beam and the di erence between thescans with and without cooling.

108

0.30
w/o coolingw/ coolingSignal Size (arb. units)

0.25

0.20

0.15

0.10

0.05

0.00

Difference (with cooling - without cooling)

0.15

0.10

0.05

0.00

-0.05

-0.10
-40

-30

-20

-10

0

10

20

30

Doppler Shift (MHz)Figure 7.14: Data showing the atomic beam without cooling with transverse velocity
centered at 10.3 MHz. Also shown is the cooled beam and the di erence between
the scans with and without cooling.

109

0.25

Signal Size (arb. units)

0.20

w/o coolingw/ cooling

0.15

0.10

0.05

0.00

Difference (with cooling - without cooling)

0.12
0.10
0.08
0.06
0.04
0.02
0.00
-0.02
-0.04
-0.06
-20

-10

0

10

20

30

Doppler Shift (MHz)Figure 7.15: Data showing the atomic beam without cooling with transverse velocity
centered at 12.3 MHz. Also shown is the cooled beam and the di erence between
the scans with and without cooling.

110

0.30

(a)

initial guess
final distribution

0.25

0.20

0.15

number of atoms (arb. units)

0.10

0.05

0.00
0.20

(b)

data
from initial ansatz
after interation

0.15

0.10

0.05

0.00
0

50

100

150

200

Doppler Shift (arb. units)

Figure 7.16: Determination of the velocity distribution within the atomic beam using
the convolution of an ansatz with a natural line width. Plot (a) shows the initial andnal estimates of the velocity distribution, while plot (b) compares the predicted line
shape of these two estimates with the measured line shape. The data shown used
a total of 14 iterations to arrive at the nal ansatz. This number of iterations wasused in all the analyses.

111

Figure 7.17: The velocity distribution of the atomic beam determined using theiterative procedure discussed above.

112

Figure 7.18: The minimum, average, and maximum fractional cooling e ect as afunction of Doppler shift. We t to the maximum e ect because that is the most
likely e ect we can achieve with optimization of detuning, laser power, etc.

113

Figure 7.19: Population distribution of the atomic beam with and without cooling. The distribution without cooling was obtained using the iterative convolution
technique discussed above. The distribution with cooling was obtained by using the
distribution without cooling and the fractional e ect of cooling derived from the data
in Figs 7.10 through 7.15

114

Figure 7.20: The line shapes for the (a) atomic beam and (b) the 6S ! 7S transition.
The improvement from transverse cooling in signal size is not as dramatic as for theseline shapes as it was for the velocity distribution. This is because the limitations of
the natural line widths plays an important role.

BIBLIOGRAPHY[1] E. Noether, Nachrichten Gesell. Wissenschaft. Gottingen 2, 235 (1918).
[2] C. S. Wu et al., Phys. Rev. 105, 1413 (1957).
[3] S. L. Glashow, Nucl. Phys. 22, 579 (1961).
[4] A. Salam, in Elementary Particle Theory: Relativistic Groups and Analyticity (8th Nobel Symp.), N. Svartholm, ed., (Almquist and Wicksell,
Stockholm, 1968).
[5] S. Weinberg, Phys. Rev. Lett. 19, 1264 (1967).
[6] M. E. Peskin, \Electroweak reconciliation," Science 281, 1153 (1998).
[7] M. A. Bouchiat and C. Bouchiat, \Parity violation induced by weak neutral
currents in atomic physics. Part I," J. Phys. (Paris) 35, 899 (1974).
[8] M. A. Bouchiat and C. Bouchiat, \Parity violation induced by weak neutral
currents in atomic physics. Part II," J. Phys. (Paris) 36, 493 (1975).
[9] D. M. Meekhof, P. Vetter, P. K. Majumder, S. K. Lamoureaux, and E. N.
Fortson, \High-precision measurement of parity nonconserving optical rotation
in atomic lead," Phys. Rev. Lett. 71, 3442 (1993).
[10] R. B. Warrington, C. D. Thompson, and D. N. Stacey, \A new measurement ofparity nonconserving optical rotation at 648 nm in atomic bismuth," Europhys.
Lett. 24, 641 (1993).
[11] N. H. Edwards, S. J. Phipp, P. E. G. Baird, and S. Nakayama, \Precise measurement of parity nonconserving optical rotation in atomic thallium," Phys.
Rev. Lett. 74, 2654 (1995).
[12] P. A. Vetter, D. M. Meekhof, P. K. Majumder, S. K. Lamoreaux, and E. N.
Fortson, \Precise test of electroweak theory from a new measurement of parity
nonconservation in atomic thallium," Phys. Rev. Lett. 74, 2658 (1995).
[13] C. S. Wood, S. C. Bennett, D. Cho, B. P. Masterson, J. L. Roberts, C. E.
Tanner, and C. E. Wieman, \Measurement of parity nonconservation and ananapole Moment in Cesium," Science 275, 1759 (1997).

116[14] S. A. Blundell, J. Sapirstein, and W. R. Johnson, \High-accuracy calculation ofparity nonconservation in cesium and implications for particle physics," Phys.
Rev. D 45, 1602 (1992).
[15] V. A. Dzuba, V. V. Flambaum, and O. P. Sushkov, \Summation of the highorders of perturbation theory for the parity nonconserving E1-amplitude of the
6S-7S Transition in the Caesium Atom," Phys. Lett. A 141, 147 (1989).
[16] G. Kane, Modern Elementary Particle Physics (Addison Wesley, New
York, 1993).
[17] C. Caso et al., \1998 Review of particle physics," Euro. Phys. J. C3, 1 (1998).
[18] G. K. Woodgate, Elementary Atomic Structure (Oxford University Press,
New York, 1980).
[19] C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics (John Wileyand Sons, New York, 1977).
[20] S. L. Gilbert, Parity Violation in Atomic Cesium and Associated Experiments, Ph.D. thesis, University of Michigan, 1984.
[21] M. C. Noecker, Precision Measurement of Parity Nonconservation in
Atomic Cesium, Ph.D. thesis, University of Michigan, 1988.
[22] B. P. Masterson, Prospects for Measuring Parity Nonconservation in
an Optically-Pumped Cesium Beam, Ph.D. thesis, University of Colorado,1991.
[23] C. S. Wood, High Precision Atomic Parity Nonconservation Measure-ment Using a Spin-Polarized Cesium Beam and the Nuclear Anapole
Moment of 133 Cs, Ph.D. thesis, University of Colorado, 1996.

[24] V. A. Dzuba, V. V. Flambaum, and O. P. Sushkov, \polarizabilities and parity
nonconservation in the Cs atom and limits on the deviation from the standard
electroweak model," Phys. Rev. A 56, 4357 (1997).
[25] D. Cho, C. S. Wood, S. C. Bennett, J. L. Roberts, and C. E. Wieman, \Precision
measurement of the ratio of scalar to tensor transition polarizabilities for thecesium 6S-7S transition," Phys. Rev. A 55, 1007 (1997).
[26] V. V. Flambaum, I. B. Khriplovich, and O. P. Sushkov, \g-Factor anomalies
and strongly forbidden M1 transitions in heavy atoms," Phys. Lett. 67A, 177(1978).
[27] M. A. Bouchiat and J. Guena, \The E2 6S-7S amplitude in cesium and its
importance in a precise calibration of E1PV ," J. Phys. (Paris) 49, 2037 (1988).
[28] S. L. Gilbert, B. P. Masterson, M. C. Noecker, and C. E. Wieman, \Precision
measuement of the o -diagonal hyper ne interaction," Phys. Rev. A 34, 3509
(1986).

117[29] C. Bouchiat and C. A. Piketty, \Magnetic dipole and electric quadrupole amplitudes induced by the hyper ne interaction in the cesium 6S-7S transition andthe parity violation calibration," J. Phys. (Paris) 49, 1851 (1988).
[30] C. E. Wieman, M. C. Noecker, B. P. Masterson, and J. Cooper, \Asymmetricline shapes for weak transitions in strong standing-wave elds," Phys. Rev. Lett.
58, 1738 (1987).
[31] M. C. Noecker, B. P. Masterson, and C. E. Wieman, \Precision measurement ofparity nonconservation in atomic cesium: A low-energy test of the electroweak
theory," Phys. Rev. Lett. 61, 310 (1988).
[32] R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J.
Munley, and H. Ward, \Laser phase and frequency stabilization using an opticalresonator," Appl. Phys. B 31, 97 (1983).
[33] N. M. Sampas, A Ring Laser Gyroscope with Optical Subtraction, Ph.D.
thesis, University of Colorado, 1990.
[34] S. L. Gilbert and C. E. Wieman, \Easily constructed high-vacuum valve," Rev.
Sci. Instrum. 53, 1627 (1982).
[35] K. B. MacAdam, A. Steinbach, and C. Wieman, \A narrow-band tunable diodelaser system with grating feedback, and a saturated absorption spectrometer for
Cs and Rb," Am. J. Phys. 60, 1098 (1992).
[36] R. G. Brewer and E. L. Hahn, \Coherent two-photon processes: Transient andsteady-state cases," Phys. Rev. A 11, 1641 (1975).
[37] P. M. Radmore and P. L. Knight, \Population trapping and dispersion in athree-level system," J. Phys. B 15 (1982).
[38] A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji,\Laser cooling below the one-photon recoil energy by velocity selective coherent
population trapping: theoretical analysis," J. Opt. Soc. Am. B 6, 2112 (1989).
[39] J. Dalibard and C. Cohen-Tannoudji, \Laser cooling below the Doppler limit
by polarization gradients: simple theoretical models," J. Opt. Soc. Am. B 6,2023 (1989).
[40] R. J. Rafac, C. E. Tanner, A. E. Livingston, and H. G. Berry, \Fast-beam laser
liftime measurements of the cesium 6p2 P 12 ; 23 states," to be published (1998).
[41] S. L. Gilbert, R. N. Watts, and C. E. Wieman, \Hyper ne structure measurement of the 7S state of cesium," Phys. Rev. A 27, 581 (1983).
[42] E. Arimondo, M. Inguscio, and P. Violino, \Experimental determinations fo thehyper ne structure in the alkali atoms.," Rev Mod. Phys. 49, 31 (1977).

118[43] S. A. Blundell, W. R. Johnson, and J. Sapirstein, \High-accuracy calculationof the 6S1=2 -7S1=2 parity-nonconserving transition in atomic cesium and implications for the standard model," Phys. Rev. Lett. 65, 1411 (1990).
[44] S. A. Blundell, W. R. Johnson, and J. Sapirstein, \Third-order many-body
perturbation theory calculations of the ground-state energies of cesium andthallium," Phys. Rev. A 42, 3751 (1990).
[45] S. A. Blundell, W. R. Johnson, and J. Sapirstein, \Relativistic all-order calculations of energies and matrix elements in cesium," Phys. Rev. A 43, 3407
(1991).
[46] V. A. Dzuba, V. V. Flambaum, P. G. Silvestrov, and O. P. Sushkov, \Relativistic
many-body calculations in atoms and parity Violation in Caesium," J. Phys. B
18, 597 (1984).
[47] V. A. Dzuba, V. V. Flambaum, and O. P. Sushkov, \Summation of the highorders of perturbation theory in the correlation correction to the hyper ne structure and to the amplitudes of E1-transitions in the caesium atom," Phys. Lett.
A 142, 373 (1989).
[48] V. A. Dzuba, V. V. Flambaum, and O. P. Sushkov, \Summation of the perturbation theory high order contributions to the correlation correction for theenergy levels of the caesium atom," Phys. Lett. A 140, 493 (1989).
[49] R. N. Watts, S. L. Gilbert, and C. E. Wieman, \Precision measurement of the
Stark shift of the 6S-7S transition in atomic cesium," Phys. Rev. A 27, 2769
(1983).
[50] J. R. P. Angel and P. G. H. Sandars, \The hyper ne structure Stark e ect: I.
Theory," Proc. Roy. Soc. London A 305, 125 (1968).
[51] L. R. Hunter, D. K. Jr., S. Murthy, and T. W. Sung, \Precision measurementof the Stark shift of the cesium D lines," Phys. Rev. A 37, 3283 (1988).
[52] H. Gould, E. Lipworth, and M. C. Weisskopf, \Quadratic Stark shift between
Zeeman substates in Cs133 , Rb87 , Rb85 , K39 , and Na23 ," Phys. Rev. 188, 24(1969).
[53] A. Khadjavi and A. Lurio, \Stark e ect in the excited states of Rb, Cs, Cd, andHg ," Phys. Rev. 167, 128 (1968).
[54] W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipies FORTRAN Version (Cambridge Unviersity Press, New York,1989).
[55] V. A. Dzuba, private communication.
[56] P. Marte (unpublished).

119[57] S. L. Gilbert and C. E. Wieman, \Atomic-beam measurement of parity nonconservation in cesium," Phys. Rev. A 34, 792 (1986).
[58] C. Bouchiat and C. A. Piketty, \A new method for the evaluation of the parityviolating electric dipole 6S to 7S amplitude in atomic cesium," Europhys. Lett.
2, 511 (1986).
[59] M. A. Bouchiat and J. Guena, \The E2 6S! 7s amplitude in cesium and its
importance in a precise calibration of Epv
1 ," J. Phys. (Paris) 49, 2037 (1988).
[60] C. W. Oates, K. R. Vogel, and J. L. Hall, \High-precision linewidth measurement of laser-cooled atoms: resolution of the Na 3p2 P3=2 lifetime discrepancy,"
Phys. Rev. Lett. 76, 2866 (1996).
[61] U. Volz, M. Majerus, H. Liebel, A. Schmitt, and H. Schmoranzer, \Precision
lifetime measurements on Na I 3p2 P1=2 and 3p2 P3=2 by beam-gas-laser spectroscopy," Phys. Rev. Lett. 76, 2862 (1996).
[62] W. I. McAlexander, E. R. I. Abraham, and R. G. Hulet, \Radiative lifetime of
the 2P state of lithium," Phys. Rev. A 54, R5 (1996).
[63] V. A. Dzuba, V. V. Flambaum, and O. P. Sushkov, \Calculation of energy
levels, E1 transition amplitudes and parity violation in francium," Phys. Rev.
A 51, 3454 (1995).
[64] J. E. Simsarian, W. Shi, L. A. Orozco, G. D. Sprouse, and W. Z. Zhao, \7S1=2
to 9S1=2 two-photon spectroscopy of trapped francium," Opt. Lett. 21, 1939
(1996).
[65] R. J. Rafac and C. E. Tanner, \Measurement of the 133 Cs 6p2 P3=2 state hyper ne
structure," Phys. Rev. A 56, 1027 (1997).
[66] P. Langacker, M. Luo, and K. Mann, \High-precision electroweak experiments:
A global search for new physics beyond the standard model," Rev. Mod. Phys.
64, 87 (1992).
[67] W. J. Marciano and J. L. Rosner, \Atomic parity violation as a probe of new
physics," Phys. Rev. Lett. 65, 2963 (1990).
[68] W. J. Marciano and J. L. Rosner, \Atomic parity violation as a probe of new
physics, eratta," Phys. Rev. Lett. 68, 898 (1992).
[69] D. London and J. L. Rosner, \Extra gauge bosons in E6 ," Phys. Rev. D 34,
1530 (1986).
[70] B. W. Lynn and P. G. H. Sandars, \Electroweak radiative corrections to atomic
parity nonconservation," J. Phys. B 27, 1469 (1994).
[71] J. L. Rosner, \Radiative corrections and electroweak observables," Phys. Rev.
D 42, 3107 (1990).

120[72] K. T. Mahanthappa and P. K. Mohapatra, \Limits on mixing angle and massof Z 0 using  and atomic parity violation," Phys. Rev. D 43, 3093 (1991).
[73] P. Langacker and M. Luo, \Constraints on additional Z bosons," Phys. Rev. D
45, 278 (1992).
[74] A. Yariv, Quantum Electronics (John Wiley and Sons, New York, 1989).
[75] N. F. Ramsey, Molecular Beams (Oxford University Press, Oxford, 1956).
[76] J. Dalibard and C. Cohen-Tannoudji, \Dressed-atom approach to atomic motion
in laser light: the dipole force revisited," J. Opt. Soc. Am. B 2, 1707 (1985).
[77] A. Aspect, J. Dalibard, A. Heidmann, C. Salomon, and C.Cohen-Tannoudji,\Cooling atoms with stimulated emission," Phys. Rev. Lett. 57, 1688 (1986).
[78] C. E. Tanner, B. P. Masterson, and C. E. Wieman, \Atomic beam collimation
using a laser diode with a self-13, 357 (1988).
[79] R. N. Zare, Angular Momentum (John Wiley and Sons, New York, 1988).

APPENDIX A
F 0 M 0 COEFFICIENTS
CFM

F 0m0 are de ned by the relation
The coecients CFm jqjF 0 m+q  p1
CFm

2

hF 0 m + q j 1q j Fmi(A.1)

where q = 0; 1, and 1q are the components of the Pauli spin operator:

11 = , p1 (x + iy )1 ,1 = p1 (x , iy )

2

and

2

10 = zy = pi (11 + 1 ,1 ):x = , p1 (11 , 1 ,1 )
2

2

(A.2)

(A.3)

With the rotations given in Section 2.6, we have

 Bx By 1p
1 1  2 B  i B 10 :1  1 =  0

(A.4)

F mF
The values of the CFm
are given by
F
00

C44mm1 =  18qp

16 , m2 ;4(A.5)(5  m) (4  m);(A.6)

C34mm = C43mm =

122

q

C33mm1 =  18 (4  m) (3  m);q

C43mm1 = , 18 (4  m) (5  m);

q

(A.7)
(A.8)

C34mm1 = 81 (3  m) (4  m)

(A.9)

C33mm = , m4 ; and(A.10)

C44mm = m4 :(A.11)

APPENDIX B
EFFECT OF THE ELECTRIC QUADRUPOLE AMPLITUDE

This appendix gives a detailed treatment of the e ects of the electric dipole
operator (E 2) discussed by Bouchiat and Guena in Ref. [27]. The angular momentum
text by Zare [79] is referred to extensively in the discussion, and the notation \Zare(2.45)" means \equation 45 in Chapter 2 from Ref. [79]."
In Ref. [27] Bouchiat and Guena introduce the phenomenological transition
operator

T (nS ,n0S ) = , E~  ~ , 2i S~  E~  ~ + a1S~  ~  ~k(B.1)
+ ia2 (S~  I~)  (~  ~k ) + ia3 [(S~  ~)(I~  ~k ) + (S~  ~k)(I~  ~)];where a1 = ,2M , a2 = ,Mhf =2, and a3 = ,E 2=2. The rst two terms are the
Stark-induced amplitudes from Eq. (2.13), the terms proportional to a1 and a2 are
the magnetic dipole amplitudes from Eq. (2.21), and the term proportional to a3
is a new amplitude|the electric quadrupole amplitude|caused by the o -diagonal
hyper ne mixing of j nDi states into the j nP i states.
In Ref [27] Bouchiat and Guena re-analyze several di erent experimentsthat determine the value of Mhf =M and show that the contribution from the E 2
amplitude is non-negligible in some cases. Speci cally, they examine the result ofRef. [28]. In the geometry of that experiment the relevant transition rates R(F 0 mF ; FmF )124
are given by
and


a
3
R(4  4; 3  3) = E (a1 + 4a2 ) 1 , 3 a + 4a
1
2

(B.2)



R(3  3; 4  4) = E (a1 , 4a2 ) 1 + 3 a ,a34a

(B.3)



1

2

Taking the ratio of these two transition rates and inserting the values of the ai 's
gives
M + Mhf , 3E 2=4 :
(B.4)
R= M
, M + 3E 2=4
hf

Then

R , 1 = Mhf 1 , 3 E 2  :
R+1 M
4 Mhf

(B.5)

Clearly, if E 2=Mhf is large enough the result is not Mhf =M . If the contribution fromE 2 to the present measurement of Mhf = is similarly non-negligible, then we must
make a similar correction.
We now consider E 2 with the geometry in the present experiment, whichis the same as in Ref. [28]: E~ = E x^, B~ = B z^, and ~ = z^.
The transition operator of concern is

TE2 = ia3 [(S~  ~)(I~  ~k) + (S~  ~k)(I~  ~)](B.6)

= ia3 (Sz Iy + Sy Iz ):An angular momentum spherical tensor operator of rank one is de ned as

J1;1 =  p1 (Jx  iJy ) and J1;0 = Jz :
2

(B.7)

Equation B.6 can then be written as

TE 2 = , pa3 [S0(I+1 + I,1 ) + (S+1 + S,1 )I0]:
2

(B.8)

125
The product of two spherical tensor operators is another spherical tensor operator
as de ned in Zare (5.36):

X (k; q) =

X
q1 q2

hk1 q1k2q2 j kqiA(k1 ; q1)B (k2 ; q2 ):(B.9)

We can then write down the following components of X (k; q).

X (1; 1) = p1 (S1 I0  S0I1 ):

2
1
X (2; 1) = p (S1 I0 + S0 I1 ):2(B.10)(B.11)

Then, the products of two spherical tensor operators may be written as

1 [X (1; 1)  X (2; 1)]; andS0 I1 = p

2
1S1 I0 = p [X (1; 1)  X (2; 1)]:
2(B.12)(B.13)

Thus, we need to nd the matrix elements of S0 I1 + S1 I0 :h 0 j S0I1 + S1I0 j i(B.14)= p1 h 0 j [X (1; 1)  X (2; 1)]  [X (1; 1)  X (2; 1)] j i21= p h 0 j X (1; 1)  X (1; 1) + X (2; 1) + X (2; 1) j i
p2 0= 2 h j X (2; 1) j i:

Thus, we haveh 0 j TE2 j i = ,a3h 0 j X (2; 1) + X (2; ,1) j i:

(B.15)

This matrix element can be evaluated using the formulas Zare(5.64):

126

h j1 j2 jm j X (k; q) j 0 j10 j20 j 0 m0i 0(B.16)
1j k j0 C= (,1)j ,m B@
A h j1 j2 j jj X k jj 0j10 j20 j 0 i;
0,m q mand Zare (5.68):

h j1 j2 j jj X k j j 0j10 j20 j 0 i = [(2j + 1)(2j 0 + 1)(2k + 1)]1=2(B.17)

8
9>0>>
j1 j1 k1 >
>
>
<
=X
0
 > j2 j2 k2 > h j1 jj Ak1 jj 00j10 ih 00 j2 jj B k2 jj 0 j20 i:00>: j j0 k >;For our situation these give

h7SIF 0 m0F j X (k; q) j 6SIFmF i = (,1)F 0 ,m0F

(B.18)

0
1
0
F k F C[(2F 0 + 1)(2F + 1)(2k + 1)]1=2 B@ 0
A,mF q m0F
8
9>>
S S 1>>><=X
 > I I 1 > h 7sS jj S 1 jj 00 S ih 00 I jj I 1 jj 6sI i:>> 00>: F0 F k >;Note that in the 3-j symbol we must satisfy the triangle condition with m0F , mF = q
with q = 1. That means we either need the matrix elements of X (2; +1) or of
X (2; ,1) but never both.
The reduced matrix elements in Eq. (B.18) can be evaluated using the
Wigner-Eckart Theorem. The results areqph1=2 jj S (1) jj 1=2i = 3=2 and h7=2 jj I (1) jj 7=2i = 3 14:(B.19)

127
We can now use Eq. (B.18) to calculate the relative size of the E 2 amplitude
compared with the M and Mhf amplitudes as well as calculate the size of the E 2-M
and E 2-Mhf interference terms. Tables B.1 and B.2 show the relevant calculations
if we assume the population in the j 6SF i state is uniform across the mF sublevels.
In that case, the interference terms between the M 1 amplitudes and the E 2 amplitude cancel and the pure E 2 rate is tiny. Therefore, with uniform populations, thecontributions from the E 2 amplitude can be ignored.
The population across the hyper ne sublevels is not uniform, however, as
we discussed in Chapter 5. Tables B.3 and B.4 show the e ect of the nonuniform
population distribution calculated by Peter Marte. (See Chapter 5.) In this case, theR+ we measure is 0.084% too large, and R, is 0.068% too small. If we are conservative and assume a 50% uncertainty in the E 2 correction because of uncertainties
in the size of E 2 and the exact distribution of the population, then the correction
makes our result for j Mhf = j smaller by 0.2% and increases our uncertainty from
0.15% to 0.16%.
These results can also be con rmed by rewriting the basis states j FmF iin terms of the electron and nucleus spin quantum numbers and Clebsch-Gordancoecients using

j FmF i =X
mS mI

j SmS ImI ihSmS ImI j FmF i:

Using that formalism gives identical results.

(B.20)

128

Table B.1: Table showing coecients for F = 3 to F 0 = 4 magnetic dipole andelectric quadrupole amplitudes for various transitions. The transition rates in thetable are the contributions to the total transition rate relative to the M amplitude assuming uniform populations in the hyper ne levels, E 2=Mhf = 0:053, and
Mhf =M = ,0:1906. The sum of all the hyper ne transitions is shown at the bottom
and the fractional correction needed to account for the presence of E 2 is shown inthe \Correction" row.
F = 3 to F 0 = 4 Uniform PopulationCoecients
Transition Rates (100=M 2 )0
0
F mF
mF m0F CFm
E2
Pop
2ME 2 2Mhf E 2 (E 2)2
F
3
4 0.93541 ,0.70151 0.1429 ,0.1486 0.0280 0.0004
3
2 0.17678 ,0.39772 0.1429 0.0159 ,0.0030 0.0001
2
3 0.81009 ,0.20251 0.1429 ,0.0371 0.0070 0.0000
2
1 0.30619 ,0.53578 0.1429 0.0371 ,0.0070 0.0003
1
2 0.68465 0.17115 0.1429 0.0265 ,0.0050 0.0000
1
0 0.43301 ,0.54122 0.1429 0.0531 ,0.0100 0.0003
0
1 0.55902 0.41923 0.1429 0.0531 ,0.0100 0.0002
0 ,1 0.55902 ,0.41923 0.1429 0.0531 ,0.0100 0.0002,1 0 0.43301 0.54122 0.1429 0.0531 ,0.0100 0.0003,1 ,2 0.68465 ,0.17115 0.1429 0.0265 ,0.0050 0.0000,2 ,1 0.30619 0.53578 0.1429 0.0371 ,0.0070 0.0003,2 ,3 0.81009 0.20251 0.1429 ,0.0371 0.0070 0.0000,3 ,2 0.17678 0.39772 0.1429 0.0159 ,0.0030 0.0001,3 ,4 0.93541 0.70151 0.1429 ,0.1486 0.0280 0.0004
Sum(100)000.0027
Correction to R+,0.002%

129

Table B.2: Table showing coecients for F = 4 to F 0 = 3 magnetic dipole andelectric quadrupole amplitudes for various hyper ne transitions. The transition rates
in the table are the contributions to the total transition rate relative to the M
amplitude assuming uniform populations in the hyper ne levels, E 2=Mhf = 0:053,
and Mhf =M = ,0:1906. The sum of all the hyper ne transitions is shown at thebottom and the fractional correction needed to account for the presence of E 2 isshown in the \Correction" row.
F = 4 to F 0 = 3 Uniform PopulationCoecients
Transition Rates (100=M 2 )0
0
F mF
mF m0F CFm
E2
Pop
2ME 2 2Mhf E 2 (E 2)2F
4
3 0.93541 0.70151 0.1111 ,0.1155 ,0.0218 0.0003
3
2 0.81009 0.20251 0.1111 ,0.0289 ,0.0054 0.0000
2
3 0.17678 0.39772 0.1111 0.0124 0.0023 0.0001
2
1 0.68465 ,0.17115 0.1111 0.0206 0.0039 0.0000
1
2 0.30619 0.53578 0.1111 0.0289 0.0054 0.0002
1
0 0.55902 ,0.41923 0.1111 0.0412 0.0078 0.0001
0
1 0.43301 0.54122 0.1111 0.0412 0.0078 0.0002
0 ,1 0.43301 ,0.54122 0.1111 0.0412 0.0078 0.0002,1 0 0.55902 0.41923 0.1111 0.0412 0.0078 0.0001,1 ,2 0.30619 ,0.53578 0.1111 0.0289 0.0054 0.0002,2 ,1 0.68465 0.17115 0.1111 0.0206 0.0039 0.0000,2 ,3 0.17678 ,0.39772 0.1111 0.0124 0.0023 0.0001,3 ,2 0.81009 ,0.20251 0.1111 ,0.0289 ,0.0054 0.0000,4 ,3 0.93541 ,0.70151 0.1111 ,0.1155 ,0.0218 0.0003
Sum(100)000.0021
Correction to R,
0.002%

130

Table B.3: Table showing coecients for F = 3 to F 0 = 4 magnetic dipole andelectric quadrupole amplitudes for various transitions. The transition rates in thetable are the contributions to the total transition rate relative to the M amplitude
assuming nonuniform populations in the hyper ne levels, E 2=Mhf = 0:053, and
Mhf =M = ,0:1906. The sum of all the hyper ne transitions is shown at the bottom
and the fractional correction needed to account for the presence of E 2 is shown inthe \Correction" row.
F = 3 to F 0 = 4 Nonuniform PopulationCoecients
Transition Rates (100=M 2 )0
0
F mF
mF m0F CFm
E2
Pop
2ME 2 2Mhf E 2 (E 2)2
F
3
4 0.93541 ,0.70151 0.1163 ,0.1548 0.0295 0.0006
3
2 0.17678 ,0.39772 0.1163 0.0166 ,0.0032 0.0002
2
3 0.81009 ,0.20251 0.1442 ,0.0480 0.0091 0.0001
2
1 0.30619 ,0.53578 0.1442 0.0480 ,0.0091 0.0004
1
2 0.68465 0.17115 0.1632 0.0388 ,0.0074 0.0000
1
0 0.43301 ,0.54122 0.1632 0.0776 ,0.0148 0.0005
0
1 0.55902 0.41923 0.1666 0.0792 ,0.0151 0.0003
0 ,1 0.55902 ,0.41923 0.1666 0.0792 ,0.0151 0.0003,1 0 0.43301 0.54122 0.1593 0.0757 ,0.0144 0.0005,1 ,2 0.68465 ,0.17115 0.1593 0.0379 ,0.0072 0.0000,2 ,1 0.30619 0.53578 0.1369 0.0455 ,0.0087 0.0004,2 ,3 0.81009 0.20251 0.1369 ,0.0455 0.0087 0.0001,3 ,2 0.17678 0.39772 0.1134 0.0162 ,0.0031 0.0002,3 ,4 0.93541 0.70151 0.1134 ,0.1509 0.0288 0.0006
Sum(100)0.1150 ,0.0220 0.0042
Correction to R+,0.094%

131

Table B.4: Table showing coecients for F = 4 to F 0 = 3 magnetic dipole andelectric quadrupole amplitudes for various hyper ne transitions. The transition rates
in the table are the contributions to the total transition rate relative to the M
amplitude assuming uniform populations in the hyper ne levels, E 2=Mhf = 0:053,
and Mhf =M = ,0:1906. The sum of all the hyper ne transitions is shown at thebottom and the fractional correction needed to account for the presence of E 2 isshown in the \Correction" row.
F = 3 to F 0 = 4 Nonuniform PopulationCoecients
Transition Rates (100=M 2 )0
0
F mF
mF m0F CFm
E2
Pop
2ME 2 2Mhf E 2 (E 2)2F
4
3 0.93541 0.70151 0.123 ,0.1637 ,0.0312 0.0006
3
2 0.81009 0.20251 0.1061 ,0.0353 ,0.0067 0.0000
2
3 0.17678 0.39772 0.1038 0.0148 0.0028 0.0002
2
1 0.68465 ,0.17115 0.1038 0.0247 0.0047 0.0000
1
2 0.30619 0.53578 0.104 0.0346 0.0066 0.0003
1
0 0.55902 ,0.41923 0.104 0.0494 0.0094 0.0002
0
1 0.43301 0.54122 0.1109 0.0527 0.0100 0.0003
0 ,1 0.43301 ,0.54122 0.1109 0.0527 0.0100 0.0003,1 0 0.55902 0.41923 0.1028 0.0489 0.0093 0.0002,1 ,2 0.30619 ,0.53578 0.1028 0.0342 0.0065 0.0003,2 ,1 0.68465 0.17115 0.1068 0.0254 0.0048 0.0000,2 ,3 0.17678 ,0.39772 0.1068 0.0152 0.0029 0.0002,3 ,2 0.81009 ,0.20251 0.1104 ,0.0367 ,0.0070 0.0000,4 ,3 0.93541 ,0.70151 0.1321 ,0.1758 ,0.0335 0.0007
Sum(100),0:0589 ,0:0112 0.0034
Correction to R,
0.079%

APPENDIX CDERIVATION OF INTRACAVITY INTENSITY WITHSIDEBANDS

This appendix derives the intensity inside a Fabry-Perot etalon when theincident light is phase modulated at the free spectral range of the etalon. Thisintensity is given in Eq. (7.2).
An electric eld of the form

 = A cos(!t ,  sin !m t)(C.1)

can be written in terms of Bessel functions as [74]

 = A[J0 () cos !t + 2J1 () sin !t sin !m t

(C.2)

+ 2J2 () cos !t cos 2!m t + : : :]where the series extends including higher order Bessel functions and higher multiples
of !m , and  is the phase modulation index. If the eld is part of an electromagnetic
wave, it must have spatial dependence as well. If we replace !t with !t , ky andreplace !m t with !m t , km y, and use the identitiescos  cos  = 12 [cos( , ) + cos( + )]
sin  sin  = 12 [cos( , ) , cos( + )]

(C.3)
(C.4)

133
we get a wave traveling in the y^ direction with the form

 = AfJ0 () cos(!t , ky) + J1 () [cos(!, t , k, y) , cos(!+ t , k+ y)](C.5)+ J2 () [cos(!,, t , k,, y) + cos(!++ t , k++ y)]g
where k = k  km and ! = !  !m . If this wave is injected into a stable FabryPerot etalon with a free spectral range of FSR = 2!m , then the standing wave
formed inside the etalon is given by = 2A[J0 sin !t sin ky + J1 (sin !,t sin k,y , sin !+ t sin k+ y)(C.6)+ J2 (sin !++ sin k++ y + sin !,, sin k,, y)];where the label  for the phase modulation index has been dropped. Using theidentities sin(  ) = sin  cos   cos  sin , the electric eld can be written as = 2A[J0 sin !t sin ky(C.7), 2J1 (cos !t sin !mt sin ky cos km y + sin !t cos !mt cos ky sin km y)+ 2J2 (cos !t sin 2!m t cos ky sin 2km y + sin !t cos 2!m t sin ky cos 2km y)]:We square the eld to nd the intensity, and since the atoms do not react
to changes in the intensity on time scales near 2=!, or even 2=!m , we will takethe time average of the intensity. Therefore we will drop terms containing odd
trigonometric functions or products of orthogonal trigonometric functions. The time
averaged intensity inside the cavity is given byh2it = 2A2 [ J02 sin2 ky + 2J12 sin2 ky cos2 km y + cos2 ky sin2 km y
(C.8)


2
2
2
2
2
+ 2J2 sin ky cos 2km y + cos ky sin 2km y ] :Using the identities cos2  = 1 , sin2  and cos2  , sin2  = cos 2, and setting

134

p

A = 1= 2, the time averaged intensity inside the cavity can nally be written ash2it = sin2 ky[J02 () + 2J12 () cos 2km y + 2J22 () cos 4km y]+ 2[J12 () sin2 km y + J22 () sin2 2km y]:(C.9)