Homogeneous Atom-cavity Coupling for Quantum
MetrologybyBaochen Wu
B.S., Nanjing University, 2014
M.S., University of Colorado Boulder, 2018

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillmentof the requirements for the degree ofDoctor of Philosophy
Department of Physics
2021
Committee Members:
James K. Thompson, Chair
Murray J. Holland
Konrad W. Lehnert
Jun YeShu-Wei Huang

ii
Wu, Baochen (Ph.D., Physics)
Homogeneous Atom-cavity Coupling for Quantum Metrology
Thesis directed by Prof. James K. Thompson

Interactions between cold atoms and optical cavities have been proved successful in generatinglarge amounts of spin-squeezing. However, the inhomogeneous coupling between the atoms andoptical cavity, due to the standing wave nature of the cavity modes and the incommensuratenessbetween the trapping and probing light, can limit the applications of spin-squeezed states. In
this thesis, I will summarize our efforts towards creating homogeneous atom-cavity coupling, andattempts at utilizing spin-squeezed states in which the entanglement is more evenly distributed
among all the atoms for quantum metrology.
I demonstrate a method to obtain more homogeneous atom-cavity coupling in the Lamb-Dicke
regime by initially loading atoms into thousands of lattice sites and then using a spectroscopic
method to select and keep only those atoms at lattice sites with near maximal coupling to thedesired cavity mode. The degree of residual inhomogeneity is expected to reduce quadraticallywith the number of the retained atoms. We are able to select 4% of the atoms and get an average
coupling strength of more than 91% the peak coupling strength.
I present an ongoing project to build an intracavity, guided atom interferometer with spinsqueezed states. Spin-squeezing homogeneously distributed between all the atoms are required,
which will be realized by time-averaged probing of falling atoms, in the non-Lamb-Dicke regime. I
will give a detailed account about how all the ingredients for the squeezed atom interferometry are
realized and and a brief summary of our recent progress.
I will also talk about a novel cooling method utilizing adiabatic transfer on Raman transitions,and introduce a proposal for continuous real-time tracking of a quantum phase and the creation of
spin-squeezed states in the quantum phase quadrature.

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Acknowledgements

Attending graduate school at JILA is one of the best decisions I have ever made. JILA is aplace full of great physicists and wonderful engineers, where cool ideas come true.
I would like to thank my advisor James for his patience, guidance, tolerance, support andencouragement, as a mentor with great communication skills, a physicist with excellent intuition
and a researcher full of passion. I have also worked with great labmates. Thank you to Kevin,Matt, Graham, Julia, Juan, Chengyi, Dylan, Matthew and Denton for all the help and discussions.
I would also like to thank all the engineers providing technical support for us. Thank you
to Hans, Todd, Adam, Calvin, James, Kim and Tracy for helping us with machining problems.
Thank you to Terry, Carl and other folks for helping us with electronics. Thank you to Dave andMark for maintaining the clean room and making customized pieces.
At JILA, experimentalists can easily find theorists to discuss crazy ideas and ask questions.
Thank you to Peiru, Haonan, Athreya and Murray.
There are fantastic visiting scholars at JILA. Thank you to Prof. Nicola Poli and Prof. Barak
Dayan for discussing interesting ideas.
Life in Boulder has been pretty enjoyable. Thank you to my friends for playing computer
games, playing Texas hold’em and working out together. Thank you to Jian for all the help andadvice both on my life in Boulder and my career as a physicist.
I also want to acknowledge my high school physics teacher, Feng Yan. Thank you for motivating me to major in physics.
Finally, I would like to thank my parents for their unconditional love and support.

iv

Contents

Chapter
1 Introduction

1

1.1

Quantum Metrology and Spin Squeezing . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.2

Inhomogeneous Atom-cavity Coupling . . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.3

Atom Interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.4

Doppler Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

1.5

Outline of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Experimental Apparatus
2.1

11

Laser Cooling and Trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.1

3D MOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.1.2

PGC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.1.3

Λ-enhanced Gray Molasses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.1.4

Raman Sideband Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2

The Optical Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3

Glass Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4

Atomic Source: 2D MOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.5

Frequency Stabilization of Lasers and Cavities . . . . . . . . . . . . . . . . . . . . . . 22

3 Site-dependent Selection of Atoms for Homogeneous Atom-cavity Coupling
3.1

24

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

v
3.1.1

Optomechanically-induced Oscillations in QND Measurement . . . . . . . . . 25

3.1.2

Alternatives for Achieving Homogeneous Atom-Cavity Coupling . . . . . . . 29

3.2

Experimental Setup and Working Principle . . . . . . . . . . . . . . . . . . . . . . . 31

3.3

Theory Plots for Scaling in Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.4

Microwave Spectrums and Fluorescence Measurements . . . . . . . . . . . . . . . . . 35

3.5

Mitigation of Optomechanically-induced Oscillations . . . . . . . . . . . . . . . . . . 38

3.6

Conclusion

3.7

Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.7.1

Simulations in Sec. 3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.7.2

Effects from Finite Atomic Radial Extent . . . . . . . . . . . . . . . . . . . . 42

3.7.3

Factors Affecting the Microwave Spectrums . . . . . . . . . . . . . . . . . . . 42

3.7.4

Simulation of the Contribution of the Radial Oscillations to Cavity-ResonanceFrequency in Fig. 3.8a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.7.5

Calculating r for a Certain Distribution of Atoms . . . . . . . . . . . . . . . . 49

3.7.6

Electronic Block Diagram for Probing the Optomechanically-induced Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4 Squeezed Atom Interferometry: Scheme and Progress

52

4.1

Introduction to Squeezed Atom Interferometry . . . . . . . . . . . . . . . . . . . . . 52

4.2

Ingredients for Intracavity, Guided Squeezed Atom Interferometry . . . . . . . . . . 53
4.2.1

Two-photon Transitions along the Cavity Axis . . . . . . . . . . . . . . . . . 53

4.2.2

Guided Falling of the Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.3

Progress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.4

Sequence for Squeezed Atom Interferometry . . . . . . . . . . . . . . . . . . . . . . . 59

4.5

Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.5.1

Two-photon Rabi Frequency for Atom Interferometry . . . . . . . . . . . . . 60

4.5.2

Frequency Sweep of the Two Tones for Two-photon Transitions . . . . . . . . 61

vi
4.5.3

Amount of Squeezing Available . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.5.4

Change in Effective Gravitational Acceleration g . . . . . . . . . . . . . . . . 62

4.5.5

Relative Phase Shift of the Two Tones Induced by Frequency Jittering of theRaman Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.5.6

Cavity Induced Phase to Amplitude Noise Conversion . . . . . . . . . . . . . 65

5 Laser Cooling with Adiabatic Transfer on Raman Transitions

68

5.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.2

Laser Cooling 87 Rb with Adiabatic Transfers . . . . . . . . . . . . . . . . . . . . . . 70

5.3

Adiabatic Transfer in the Presence of Scattering . . . . . . . . . . . . . . . . . . . . 74

5.4

Conclusion

5.5

Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.5.1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

Optical Bloch Equations and Adiabatic Elimination for a Three-Level Raman
System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.5.2

Including Internal Momentum States . . . . . . . . . . . . . . . . . . . . . . . 80

5.5.3

Probabilitistic Model for Raman SWAP Cooling . . . . . . . . . . . . . . . . 82

6 Continuous Real-time Tracking of a Quantum Phase below the Standard Quantum Limit

86

6.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

6.2

Overview of Theoretical Proposal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

6.3

Mean-field Picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

6.4

Implementation in 87 Rb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

6.5

6.4.1

Quantum Phase in Spin-1 Systems . . . . . . . . . . . . . . . . . . . . . . . . 94

6.4.2

Experimental Scheme and Numerical Results . . . . . . . . . . . . . . . . . . 98

6.4.3

Technical Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

7 Conclusion and Outlooks

105

vii
7.1

Creation of Dicke States by Quenching Superradiance . . . . . . . . . . . . . . . . . 105

7.2

Understanding Limitations on the Creation of Spin Squeezing . . . . . . . . . . . . . 106

7.3

Closing Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

Bibliography

109

viii

TablesTable
2.1

Parameters of the optical cavity at 780 nm . . . . . . . . . . . . . . . . . . . . . . . 15

2.2

List of lasers

5.1

First-order possibilities for atom state trajectories . . . . . . . . . . . . . . . . . . . 83

6.1

Proposed parameters for continuous phase tracking experiment . . . . . . . . . . . . 98

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

ix

FiguresFigure1.1Working principles of quantum metrology . . . . . . . . . . . . . . . . . . . . . . . .

3

1.2

Coherent spin state and spin squeezed state . . . . . . . . . . . . . . . . . . . . . . .

4

1.3

Inhomogeneous atom-cavity coupling . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

1.4

Mach-Zehnder optical interferometer . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

1.5

Two-photon Raman transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

1.6

Mach-Zehnder type atom interferometer . . . . . . . . . . . . . . . . . . . . . . . . .

8

2.1

Lens tubes for collimating and tuning the polarizations of the MOT beams . . . . . 12

2.2

Level diagram for 87 Rb D2 line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3

SolidWorks design of the optical cavity spacer . . . . . . . . . . . . . . . . . . . . . . 16

2.4

Jigs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.5

SolidWorks drawing of the vacuum system . . . . . . . . . . . . . . . . . . . . . . . . 17

2.6

2D MOT cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.7

Optics layout drawing for the 2D MOT . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.8

2D MOT Image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.9

Loading curve of the 3D MOT from 2D MOT . . . . . . . . . . . . . . . . . . . . . . 21

3.1

Experimental diagram for spin squeezing . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2

Cause of optomechanically-induced oscillations . . . . . . . . . . . . . . . . . . . . . 28

3.3

Optomechanically-induced oscillations of the dressed cavity resonance . . . . . . . . 29

x
3.4

Alternatives for achieving homogeneous atom-cavity coupling . . . . . . . . . . . . . 30

3.5

Experimental setup and working principle . . . . . . . . . . . . . . . . . . . . . . . . 31

3.6

Simulation with minimal model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.7

Microwave spectrums and fluorescence measurements . . . . . . . . . . . . . . . . . . 36

3.8

Dressed cavity equilibrium resonance change due to optomechanics . . . . . . . . . . 39

3.9

Comparison between measured microwave spectrum and that predicted by theory . . 43

3.10 Effect of imperfect π-pulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.11 Effect of varying Ωm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.12 Effect of varying δs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.13 Effect of non AC Stark shift induced broadening of the microwave transition
3.14 Effect of inhomogeneous broadening of the microwave transition. . . . 47

. . . . . . . . . . . 48

3.15 Electronic block diagram for probing the optomechanically-induced oscillations . . . 50
4.1

Squeezed Mach-Zehnder type atom interferometer . . . . . . . . . . . . . . . . . . . 52

4.2

Two-photon transitions in free space . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.3

Standing waves of cavity modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.4

Degeneracy of counter-propagating beams lifted . . . . . . . . . . . . . . . . . . . . . 55

4.5

Creation of the dipole trap

4.6

Intensity pattern of LG01 mode and phase plate pattern for generating it

4.7

Velocimetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.8

Two-photon Rabi flopping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.9

Transition frequency between adjacent momentum states . . . . . . . . . . . . . . . . 61

5.1

Comparison between Single-photon and Two-photon SWAP Cooling . . . . . . . . . 70

5.2

Experimental Setup and Cooling Procedure . . . . . . . . . . . . . . . . . . . . . . . 71

5.3

Performance of cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.4

Characterization of adiabatic transfer and discussion of the quanlity parameter Q . . 75

5.5

Simulated time dynamics of adiabatic transfer on a three-level system . . . . . . . . 77

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
. . . . . . 57

xi
5.6

Simulation of populations including momentum labels during adiabatic transfer . . . 81

5.7

Optimization of experimental parameters with Q . . . . . . . . . . . . . . . . . . . . 85

6.1

Schematic and working principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

6.2

Cavity-assisted Raman superradiance . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

6.3

Driving two paths of superraidance at the same time . . . . . . . . . . . . . . . . . . 92

6.4

Level diagram for 87 Rb D2 line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

6.5

Raman superradiance in spin-1 systems . . . . . . . . . . . . . . . . . . . . . . . . . 95

6.6

Population evolution of spin-1 systems . . . . . . . . . . . . . . . . . . . . . . . . . . 97

6.7

Evolution of the three spin components . . . . . . . . . . . . . . . . . . . . . . . . . 97

6.8

Schematic diagram of continuous trakcing of a quantum phase with spin-1 systems . 99

7.1

Raman superradiance visualized on Bloch spheres . . . . . . . . . . . . . . . . . . . . 107

Chapter 1

Introduction

The development of metrology has played a crucial role in human activity. More and more
precise measurement of time, distance, weight, temperature, etc., has not only made human life
more convenient, but also boosted the development of science and technology.
Let’s take the metrology of time as an example. In ancient days, people kept track of timeby monitoring the position of the sun, with hour-level uncertainty. The inventions of sundial,
clepsydra and sandglass improved the uncertainty to minute level. Mechanical clocks and quartz
clocks further advanced the imprecision to sub-second level. Quantum metrology was developed
as physicists gained better understanding of quantum mechanics. Quantum metrology measuresa physics quantity by probing the interactions between the quantity and quantum systems withdiscrete energy levels, whose precision is limited by the principles of quantum mechanics. With
its help, atomic clocks based on masers at room temperature were able to give an uncertainty of
10−9 in fractional frequency unit in the 1950s [36], and have reached 10−16 in the 2010s [47] withthe development of atomic fountains. Optical clocks have even reached 10−19 level of uncertainty[16, 70]. The current US time standard is based on caesium fountain clocks. Scientists from NIST
and JILA have proposed a new time standard based on optical clocks [74].
Precise time-keeping has brought about great convenience to people’s everyday life. For
aeroplanes moving at 1000 miles/hour, the navigation can easily be off by miles, if the imprecisionin time is at seconds level. It is also common that good deals will be fully claimed just several
seconds after they start during Black Fridays! Even though quartz clocks are usually precise enough2
for the two examples above, people may start to get an idea of how important precise time-keeping
is. There are actually situations that we have to turn to quantum metrology with atomic clocks,
for example, autonomous driving and navigation of missiles enabled by atomic clocks on satellites.
With the development of optical clocks, tests of fundamental theory are enabled, say, variations offine structure constant [90] and searches for dark matter [59].
Quantum metrology also opens up other possibilities in addition to precise time-keeping, forexample, measurement of gravitational acceleration variation for mining. I will move on to discuss
how quantum metrology works, what limits its precision and how we could break the limit.

1.1

Quantum Metrology and Spin Squeezing
Quantum metrology has become a powerful tool for precision measurement of physical quan-tities. It works by encoding the information of the quantity of interest into the quantum phase √φ of the superposition state of a two-level system |ψi = |↓i + eiφ |↑i / 2. The information of
the quantity of interest will be written into φ if it modifies the energy difference between |↑i and|↓i since the rate that φ changes is set by the energy difference φ̇ = (E↑ − E↓ )/~, where ~ is the
reduced Planck constant. We then infer φ by applying couplings between |↑i and |↓i to map φ into
a population difference between |↑i and |↓i which we can then measure to infer φ. This process isnamed the Ramsey sequence. We will consider an electron in a magnetic field as an example. The
spin of the electron aligned or anti-aligned to the magnetic field is the state we call |↑i or |↓i. If
we initialize the spin perpendicular to the magnetic field, the spin will precess at a rate set by themagnetic field, and the azimuthal angle the spin precesses through is the phase φ. We are able tobuild a magnetometer by tracking φ.
We often visualize the Ramsey sequence on Bloch spheres as shown in Fig. 1.1. The Bloch
sphere is a geometrical representation of the quantum state of two-level systems. With |↓i represented by a vector pointing at the south pole with coordinate (0, 0, − 21 ) and |↑i represented
by a vector pointing at the north pole with coordinate (0, 0, 12 ), a superposition state |ψi =
sin (θ/2) |↓i + cos (θ/2) eiφ |↑i can be represented on the Bloch sphere by the vector (cos (φ) sin (θ) ,3sin (φ) sin (θ) , cos (θ))/2, with θ and φ now being interpreted as the polar coordinates of a vectorof length 1/2 for a spin-1/2 system. When we have N spin-1/2 systems which are in a productstate of the wavefunction above, their individual Bloch vectors will connect tip-to-tail to form acollective Bloch vector, with length N/2. We show collective Bloch vectors in Fig. 1.1.

Figure 1.1: Working principles of quantum metrology. We initialize the collective Bloch vector
representing the quantum state of the two-level systems onto the equator in the preparation stage.
The Bloch vector will precess during the interaction stage under the effect of the quantity to bemeasured, for instance, the flow of time, accelerations, or a magnetic or electric field. Finally, we
convert the quantum phase φ into atom number for readout. This is accomplished by once again
applying a coupling between |↑i and |↓i with the correct phase to cause a rotation about the green
axis through an angle π/2. By measuring the number of atoms in |↑i and in |↓i, we can estimate
the phase φ accrued during the interaction period.

Population measurement usually results in independent collapse of individual wavefunctions
into either |↓i or |↑i, which is subject to random-walk statistics. This sets a limit on the rms
phase resolution the measurement can achieve, which we call the standard quantum limit (SQL),√∆φSQL = 1/ N radians, as indicated by the pink blobs in Fig. 1.1. This is the fundamental limit
on quantum metrology with unentangled atoms, and we call such a state a coherent spin state(CSS). As can be seen from the scaling, larger atom number will help to improve the precision of
the measurement.
However, the number of atoms cannot be arbitrarily large, because of both technical con-

4
straints and potential atom-atom interactions that can lead to systematic errors and decoherence.
We will eventually need to turn to quantum entanglement to break this limit and approach thefundamental limit set by quantum mechanics, which we call the Heisenberg limit, ∆φHL = 1/N[39]. By engineering entanglement between the atoms, we could get a spin squeezed state (SSS),
where atoms will conspire during the population measurement to cancel out the noise in the spin
projection corresponding to the population basis, as indicated by the pink blobs in Fig. 1.2. We
will refer to the noise in population basis as in atom number quadrature, and the noise in quantum
phase (azimuthal angle on the Bloch sphere) as in phase quadrature throughout this thesis.

Figure 1.2: Coherent spin state and spin squeezed state

There have been many efforts to create spin squeezing through spin dynamics [14, 65, 97, 13],
direct mapping from squeezed light [69] and quantum non-demolition measurement [51, 27, 20, 11].
We are able to create about 18 dB of squeezing, or 60 times improvement in the variance of atom
number measurement, by cavity-based quantum non-demolition measurement in the Dicke statebasis. More details of our method can be found in Sec. 3.1.1 [27, 26].
We move on in two directions. One is to make the squeezing better. We developed a sitedependent selection technique to achieve homogeneous atom-cavity coupling. The other one is to
apply the squeezed states to quantum sensors. Here I will discuss the application of spin-squeezing

5
to an atom interferometer. I will define the inhomogeneous atom-cavity coupling problem in cavity
QED experiments in Sec. 1.2, and give a brief introduction to atom interferometer in Sec. 1.3.

1.2

Inhomogeneous Atom-cavity Coupling
We typically send a lattice laser and an atomic probe laser into a two-mirror optical cavity

for trapping and probing 87 Rb atoms, with wavelengths of 813 nm and 780 nm respectively. There
will be inhomogeneity in the coupling of individual atoms to the cavity because of the standing
wave nature of the cavity modes [28]. Atoms are always trapped at the maxima of the opticallattice, but their coupling to the cavity can range between zero coupling and a maximum coupling
depending on where they are trapped relative to the standing wave of the cavity mode of interest.
For example in Fig. 1.3, atom A is trapped near an anti-node of the blue cavity mode of interest andso has maximal coupling, while atom D is trapped near a node of the cavity mode and so has very
little coupling to the mode. The inhomogeneous coupling reduces the effective atom number [52],creates dephasing in driven systems [75], and brings about optomechanically-induced oscillations[27].
In Chapter 3, I will discuss these effects in detail and present our solution to this inhomogeneous coupling problem with site-dependent selections.

1.3

Atom Interferometry
An atom interferometer is the matterwave analog of an optical interferometer, where we takeadvantage of the wave nature of atoms. To understand how optical interferometers work, we firstconsider a Mach-Zehnder type optical interferometer in Fig. 1.4. One light beam is split into twopaths first with a beam splitter. The two beams will gain differential phase ∆Φ along different
paths before they are recombined and interfered with mirrors and a second beam splitter. The
intensities out of the two ports are proportional to cos2 (∆Φ/2) and sin2 (∆Φ/2) respectively.
The matterwave counterpart of beam splitters and mirrors are realized with two-photon
Raman transitions, as shown in Fig. 1.5. For a cloud of free-falling atoms, a π/2-pulse could impart

6

Figure 1.3: Inhomogeneous atom-cavity coupling. The lattice laser (red) is for trapping the atoms.
The atomic probe (blue) is for probing the atoms. Atoms are always trapped at the maxima of
the optical lattice, but not necessarily the maxima of the atomic probe, due to the difference inwavelengths between the lattice laser and the atomic probe laser, typically here 813 and 780 nm
respectively.

2~k momentum kicks to transfer the atoms into a superposition state |ψi = √12 (|↓, 0~ki + |↑, 2~ki),with ~ here the reduced Planck constant, and k the wavevector of the laser. The momentum
transferred to the atoms causes the superposition state to move apart spatially, just as a beam
splitter causes an incident field to travel along two distinct paths. A π-pulse could reverse therelative movement direction of the two atomic clouds, so that the spatially separated superposition
state could re-overlap in space at some later time, just as a mirror changes the propagation directionof the incident field. The differential phase ∆Φ is obtained from the physical quantity we wantto measure, for example inertial effects such as rotations and accelerations. Here, we will consider
an interferometer that is sensitive to accelerations such as gravity, as shown in the top panel ofFig. 1.6. We start with a cloud of atoms in free space. The first separation π/2-pulse transfers theatoms into a superposition state and results in spatial separation of the superposition state. After

7

Figure 1.4: Mach-Zehnder optical interferometer

Figure 1.5: Two-photon Raman transitions

8

Z

0

1

T

2

3

2T

4

t

5

Figure 1.6: Mach-Zehnder type atom interferometer. The top panel shows how the position of theatoms evolve as a function of time t. The bottom panel maps the process to the quantum metrology
paradigm discussed in Sec. 1.1. The first π/2-pulse initializes the Bloch vector on the equator. The
rotation axis is indicated by the green line. Then the Bloch vector will precess through a quantum
phase φA in time T. The π-pulse rotates the Bloch vector 180◦ with reference to its initial positionon the equator, as indicated by the purple rotation axis. The Bloch vector will precess throughanother phase φB in time T afterwards. The final π/2-pulse will interfere the atoms, which converts
information from the phase quadrature to the atom number quadrature for readout. The rotation
axis is indicated by the green line. Here φB − φA =∆Φ.

an evolution time of T, a mirror π-pulse is applied, which reverses the relative movement directionof the two atomic clouds so that they could recombine in space after another evolution time of

9
T. We use a final π/2-pulse to interfere them. The differential phase ∆Φ arises from the different
kinetic energy difference between the two clouds of atoms in the two evolution windows, because
the kinetic energy is proportional to the square of the momentum. The atom numbers in |↓i and |↑iafter interference are proportional to sin2 (∆Φ/2) and cos2 (∆Φ/2) respectively, with ∆Φ = 4gkT 2 .
In this way, we are able to build a Mach-Zehnder type atom interferometer. It is straightforward toconnect the process back to the quantum metrology paradigm discussed in Sec. 1.1 after relabelingthe north and south pole of the Bloch spheres with |↑, 2~ki and |↓, 0~ki, as shown in the bottom
panel of Fig. 1.6.
In Chapter 4, I will discuss our scheme to introduce squeezed states into atom interferometers
and our progress towards building squeezed atom interferometers.

1.4

Doppler Cooling
Colder atomic temperature is usually favorable for quantum metrology. Doppler cooling isa widely applied cooling technique, which works by allowing the cooling laser to interact morestrongly with atoms moving opposite to its propagation direction.
There are several restrictions on the Doppler cooling. First, it requires well-defined cyclingtransitions to work. Second, the cooling rate and equilibrium temperature is set by the linewidthof the excited state Γ. The maximum rate at which momentum can be removed from the atom is
set by the maximum photon scattering rate for a saturated atom Rmax = Γ/2. The equilibrium
temperature is Teq = ~Γ/2kB , with kB the Boltzmann constant.
In Chapter 5, I will disccuss how these restrictions can be relaxed by adiabatic transfer andintroduce a sub-Doppler cooling technique via adiabatic transfer on Raman transitions, where weachieved a final 1D temperature in 87 Rb 25 times lower than the Doppler cooling limit, and theeffective excited state decay rate can be modified in time, presenting another degree of freedom
during the cooling process.

10

1.5

Outline of Thesis
I will devote Chapter 2 to describing the experimental apparatus, with a focus on changes

from the previous generation of the experiment. In Chapter 3, I will talk about the principle
and the performance of the site-dependent selection technique. I will describe our scheme forbuilding a squeezed atom interferometer with current progress in Chapter 4. I will review a 1D
sub-Doppler cooling technique via adiabatic transfer on Raman transitions in Chapter 5. I willsummarize a theoretical proposal for countinuous tracking of a quantum phase, which could function
as an alternative to the Ramsey sequence, focusing on a proposed implementation and technicalrequirements needed to implement the scheme in Chapter 6. Finally, I will conclude the thesis anddiscuss future directions in Chapter 7.

Chapter 2

Experimental Apparatus

The current generation of the experiment was reconstructed starting in late 2017. The general
structure and many of the details of our current experimental apparatus are similar to the former
generations and can be found from the previous theses of the group [19, 26, 9, 106]. The changes
mainly happen in the parameters of the optical cavity, the way we load atoms, and the coolingprocedure. I will summarize the changes and provide only a brief overview of the key experimental
tools and techniques described in other theses.

2.1

Laser Cooling and Trapping
Our experiment always starts by cooling and trapping 200-300 thousand atoms into the

TEM00 mode of the science cavity. It usually takes three steps. First, we create a 3D MOT ofseveral million atoms by transferring atoms from a 2D MOT. Second, we cool the atoms withpolarization gradient cooling (PGC) to 10 − 20 µK and transfer 200-300 thousand atoms into an
intracavity red lattice, which is created by driving a TEM00 mode of the science cavity with 813
nm laser light. Third, we further cool the atoms with Λ-enhanced gray molasses and 2D Raman
sideband cooling in the plane transverse to the cavity axis to achieve a transverse temperature ofabout 1.5 µK.

12
2.1.1

3D MOT

The magneto-optical trap (MOT) is created with three pairs of counter propagating beams
and a quadrupole magnetic field. The polarization and the frequency of the beams are tuned suchthat the atoms tend to get momentum kicks from the beams that are opposing its motion or its
position from the zero point of the quadrupole field. As a result, the atoms will be cooled andtrapped near the zero magnetic field point. I will refer the readers to ref. [71] for a detailed accountof the physics of the MOT.
The magnetic field gradient is around 10 G/cm, with a driving current of 2.2 A. We typicallyhave 8 mW per beam in the MOT laser and 6 mW total in the repumper laser. The waist size of thebeams is about 0.5 cm. For compactness, we use lens tubes to collimate and tune the polarizationsof the beams as shown in Fig. 2.1. The collimator and lens used have focal lengths of 1.1 cm and15 cm respectively.

Figure 2.1: Lens tubes for collimating and tuning the polarizations of the MOT beams

2.1.2

PGC

Polarization gradient cooling (PGC), also known as Sisyphus cooling, works by creating alight potential in which atoms always have to climb potential hills. Every time they reach the top

13of a potential hill, they are preferentially optically pumped to another ground zeeman state forwhich the potential now appears as the bottom of a valley with a new hill to climb, thus losing
kinetic energy. I will also refer the readers to ref. [71] for a detailed account of the physics.

Figure 2.2: Level diagram for 87 Rb D2 line

After we turn off the MOT coils and zero the magnetic field with bias coils, we could do
PGC directly with our MOT beams. The detuning we use for the MOT beams is 30-40 MHz to theblue of F=2 to F’=2 transition of the 87 Rb D2 line. The level diagram is shown in Fig. 2.2. The
power of the MOT beams is maximized, while the power of the repumper beams is decreased bya factor of 2 to 3. We typically cool and load atoms into the optical lattice for about 30 ms. The
atomic temperature after PGC is 10-20 µK with the higher temperatures for deep optical lattices
with Ttrap = 200 µK or more.

14
2.1.3

Λ-enhanced Gray Molasses

Λ-enhanced Gray molasses employs the principles from PGC and dark state, which is asuperposition of the ground states whose excitation probability to resonant lasers is canceled out
due to interference [42, 92]. We split and phase modulate the MOT laser to get a tone that is near
detuned to 1-2’ transition of the 87 Rb D2 line. The modulation frequency is about the hyperfine
splitting between F=1 and F=2 manifolds. The power of the tone from phase modulation is about
70 µW per beam. This tone and the MOT beam can be used to apply gray molasses because they
are coherent. We are able to cool the atoms to about 7 µK with about 5 ms of Λ-enhance molasses.

2.1.4

Raman Sideband Cooling

Raman sideband cooling starts by optically pumping the atoms into a stretched state. If we
could induce Raman transitions between the stretched state and another Zeeman substate whose
vibrational quantum number in the optical lattice is one quanta lower, then we could keep decreasing
the vibrational quantum number of the atoms by tranferring them into that substate and opticallypumping them back, until the vibrational quantum number reaches 0, after which there is no longer
any lower vibrational levels to transfer into and no more Raman transitions will occur [60].
Our final cooling stage is a 2D degenerate Raman sideband cooling [60] transverse to thecavity axis, i.e., in the horizontal plane. Three beams derived from the same laser subtends 128◦ ,116◦ and 116◦ to their neighbors. The polarization of the three beams is each 10◦ clockwisely
rotated from vertical if viewed along the wavevector of each beam. The power in the three beams
is about 55 mW, 45 mW and 45 mW respectively. The detuning of the sideband cooling laser isabout 35 GHz to the red of 1-2’ transition of the 87 Rb D2 line. The polarization of the repumping
light is a combination of σ+ and π. One cooling cycle is defined as ramping up the 2D lattice,cooling and ramping down the 2D lattice, which takes about 2 ms. We are able to cool the atoms
to about 1.5 µK in the transverse direction with two cooling cycles.

15

2.2

The Optical Cavity
The core part of our apparatus is a high-finesse optical cavity mounted in high vacuum. The

SolidWorks design of the optical cavity spacer is shown in Fig. 2.3.
The cavity spacer was CNC machined from macor in the JILA machine shop. The twoceramic screws allow tuning of the position of the top mirror, and thus the free spectral range(FSR) of the cavity. The desired FSR is 10-47 MHz either red or blue detuned from the ground
hyperfine spliting of 87 Rb and what we currently have is 42 MHz red detuned. The reason forrequiring such a detuning range can be found in Sec. 4.2.1. Our tuning procedure of the FSR ischanging the position of the top mirror in the JILA clean room and then measuring the FSR in
the lab, and iterating. The cavity mirrors (from FiveNine Optics) and the piezos (Piezomechanik
HPCh 150/12-6/2) were glued together in the JILA clean room with Torrseal (KJLC 9530001 and
9530002). We machined an aluminum jig, as shown in Fig. 2.4 Left, to ensure that the mirrors
and the piezos are concentric. The piezos and the cavity spacer were then joined with additional
drops of Torrseal, assisted by a concentric-shape Delrin jig, see Fig. 2.4 Right. Kapton-dippedwires (KJLC FTAK01420) were glued to the electrodes of the piezo with silver epoxy (EPO-TEK
H21D), for fine control of the cavity resonance frequency. The cavity parameters at 780 nm aresummarized in Table. 2.1. By specs, the cavity linewidth at 760 nm and 813 nm are 154 kHz
and 170 kHz respectively. The main difference of the new cavity is its finesse at 780 nm has been
increased from about 2.5 × 103 to 1.3 × 105 .
Parameter
Free spectral range
Linewidth κ
Mirror radius of curvature
FinesseCavity length
TEM00 mode waist
Single-atom vaccum Rabi splitting g2
87 Rb D2 transition
Cooperative C= 4g
κΓ forValue
6792(1) GHz
52.1(1) kHz
5.00(1) cm
1.3 × 105
2.21 cm
71.7 µm
0.403 MHz
2.06

Table 2.1: Parameters of the optical cavity at 780 nm

16Figure 2.3: SolidWorks design of the optical cavity spacer. The spacer is designed to be made frommacor. The mirrors and the piezos are already glued on, indicated by the small cylinders. The
length of the cavity is tuned with the two ceramic screws at the top.

Figure 2.4: Left: SolidWorks design of jig for joining the mirrors and the piezos. Right: Picture of
the jig for joining the piezos and the cavity spacer.

The cavity is mounted inside a vacuum chamber, as shown in the SolidWorks drawing Fig. 2.5,
pumped with an ion pump and a titanium-sublimation pump connected to the 6-way cube at the17bottom right corner in the figure. The pressure is about 1.0 × 10−8 torr, which limits the atomic
lifetime to about 400 ms due to background collisions [53]. The vacuum pressure is limited bytechnical reasons from the construction of the glass cell, which I will elaborate in Sec. 2.3.

Figure 2.5: SolidWorks drawing of the vacuum system. The 2D MOT on the left-hand side isconnected to the glass cell through a protrusion. The cavity is mounted inside the glass cell, whichis connected to a 6-way cube at the bottom.

2.3

Glass Cell
The glass cell shown at the top right corner in Fig. 2.5 was made by the JILA machine shopwith silicate bonding [1]. The top surface is an AR coated 2” × 2” quartz plate, and the sides
are four AR coated 2” × 5” quartz plates. The reflection coefficient for the AR coated surfaces
are 0.25% at 0 degree of incidence angle and 0.6% at 45 degrees for 780 nm. All quartz plates
were purchased from Specialty Glass Products. Some corner support pieces, which were nominallyright-angle wedges, were also added, which were custom parts from Esco Optics. The protrusion

18
is connected to a 2D MOT as the atomic source. The details of the 2D MOT will be discussed inSec. 2.4.
The limitation to the vacuum pressure came from unsuccessful applications of the silicate
bonding. We eventually decided to seal the leakage with torr seal (KJLC 9530001 and 9530002).
However, a wrong type of torr seal was applied unfortunately, which was not able to tolerate abaking temperature of 40◦ C.

2.4

Atomic Source: 2D MOT
In the previous generation of the experiment, atoms were loaded into a 3D MOT from abackground rubidium vapor produced by a heated ampoule of rubidium attached to the vacuum
chamber. We decided to switch to a 2D MOT as the atomic source in the current generation toprotect the high finesse optical cavity, whose finesse has been increased by more than 50 fold.
The sideview of the SolidWorks drawing of the 2D MOT could be found in Fig. 2.6a. Fig. 2.6bshows the real parts made and assembled by the JILA machine shop. We could disassemble the2D MOT setup with the two-way crossing as a reference point, which is shown in Fig. 2.6c. The
righthand side of the two-way crossing in Fig. 2.6a was a stainless steel to pyrex adaptor (Larson
SP-150-F2) fused with a pyrex cuvette (Fireflysci, CM1484B, dimensions 25 mm×25 mm×75 mm).
The lefthand side was a zero length 2-3/4” to 1-1/3” CF reducer with the bore sealed and welded
to a stainless steel tube (McMaster-Carr 8457k55), whose OD was 3/8”. The other side of the tube
was welded to a polished round copper disk, which would work as a mirror to allow axial coolingfor the 2D MOT. A tiny hole of about 0.8 mm diameter was drilled at the center of the copper
disk to allow atoms from the 2D MOT to be transferred to the 3D MOT, and provide differential
pumping. The top 1-1/3” port was connected to a feedthrough (KJLC IFTAG065102) to drive
the getter and dispensers mounted on a macor disk as shown in Fig. 2.6d. The dispenser is therubidium source, and the getter is like a mini pump. The macor disk was held fixed on the stainlesssteel tube with retaining rings (McMaster-Carr 98410A665). The bottom port was later connected
to a Rb chunk (ESPI, Rb 1 gram breakseal ampoule, 3N+ purity) as a backup atomic source, as

19
(a)

(b)

(c)
(d)

Figure 2.6: 2D MOT cell (a) Sideview of the SolidWorks drawing of the 2D MOT cell. From rightto left are pyrex cuvette, pyrex to 2-3/4” CF adapter, two-way crossing, and zero-length 2-3/4” to1-1/3” CF reducer. The stainless steel tube which we can see through the pyrex adapter is weldedto the zero-length reducer. The shiny copper disk in (b) is on the other side of the tube. The 2D
MOT will be formed in the pyrex cuvette. The atoms from the 2D MOT will be pushed througha tiny hole on the copper disk, the stainless steel tube and the protrusion shown in Fig. 2.5, andform a 3D MOT in the glass cell. (b) 2D MOT cell made and assembled by the JILA machine
shop. (c)Two-way crossing from Kimball physics (KPI 53-650010). (d) Two Rb dispensers (SAES
5G0125) and one getter (SAES S5K0088) mounted on a macor disk.

shown in Fig. 2.5.
The 2D MOT coils were made with 3d-printed frames as shown in Fig. 2.5. The typicalmagnetic field gradient applied was 15-20 Gauss/cm at a current of about 1.2 A. Our optics layout
drawing is shown in Fig. 2.7. The typical power out of the fiber couplers for MOT and Repumper
beams is 100 mW and 5 mW respectively.
An image of the string-like 2D MOT is shown in Fig. 2.8. With 2.9 A of dispenser current,

20

y
z

Repumper

x
MOT

z

M9

y

x

QWP4
QWP3

QWP1

Cuvette
M6
QWP2

FC2

FC1

M8

M7

PBS2

QWP3
Cuvette

L2
L1

M5

PBS2

HWP2
HWP1

HWP3

M6 QWP1

M1

CL1

PBS1

M2

M4
CL2

M3

Figure 2.7: Optics layout drawing for the 2D MOT. FC1 and FC2 are fiber couplers. L1 and L2
are convex lenses with focal length 20 cm. The beams have a collimated diameter of 1.5 cm after
theses lenses. HWP1-HWP3 are half waveplates. QWP1-QWP4 are quarter waveplates. M1-M8
are mirrors. PBS1 and PBS2 are polarized beam splitters. The beam reflected by PBS2 goes inthe z-direction. CL1 and CL2 are cylindrical lenses, which work together to magnify one dimentionof the beam to 3.5 cm in diameter. The cuvette is where the 2D MOT is located. The 3D MOT isin the far y-direction, which is not shown in the layout drawing.

and 2.3 A of getter current, we were able to load a 3D MOT of about 1.3 × 107 atoms with 1/eload time of 1.22 s, as shown in Fig. 2.9, from which we could estimate an atomic flux of about

21

Figure 2.8: 2D MOT image. The shiny string is the 2D MOT formed inside the pyrex cuvette. The
disk on the right-hand side is the copper disk. The atoms in the 2D MOT will be pushed throughthe center hole and eventually sent to the glass cell to form a 3D MOT.

1.07 × 107 atoms/s. Our result is comparable to a typical 3D MOT size of about 2 × 107 atoms[91].

Figure 2.9: Loading curve of the 3D MOT from 2D MOT. The horizontal axis is time in second,
and the vertical axis is fluorescence counts of the 3D MOT. We start with a 3D MOT in steadystate. The coils are turned off at about 10 s, and back on at about 25 s to reload the 3D MOT.
From the loading curve, we fit a 1/e loading time of about 1.24 seconds. At about 42 s, the 2D
MOT beams are blocked. They are unblocked at about 60 s to reload the 3D MOT. The 1/3 loading
time is fitted to be abobut 1.2 seconds. We report a typical loading time of 1.22 seconds which isthe average of the two measured values of the loading time.

22

2.5

Frequency Stabilization of Lasers and Cavities
The experiment relies on stabilizing the frequencies of various lasers and optical cavities.

Here, I give a short summary of how this is accomplished. In all, there are two cavities and ninelasers.
One cavity is a passive reference cavity which has a typical resonance frequency drift rate ofabout 50 − 100 MHz/hour. The other cavity is the high finesse science cavity.
The nine lasers are listed in Table. 2.2
Name
Reference laser
MOT laser
Repumper laser
Atomic probe
Cavity probe
Raman laser
Red lattice laser
Blue lattice laser
Sideband cooling laser

Wavelength
780 nm
780 nm
780 nm
780 nm
780 nm
780 nm813 nm
760 nm
780 nmFunction
Frequency reference, locked to 85 Rb 3-4’ transition
Create the 2d and 3d MOT, PGC and optical pumping
Create the 2d and 3d MOT, PGC and optical pumping
Probe the cavity resonanceCommon mode cancellation of the cavity frequency noise
Drive intracavity Raman transitions
Create intracavity red lattice
Create intracavity blue optical dipole trap [28]Drive Raman sideband cooling in the transverse plane

Table 2.2: List of lasers

The science cavity is Pound-Drever-Hall (PDH) [7] locked to the blue lattice laser, whose 1storder sideband, created by an EOM modulated at 5-9 GHz, is PDH locked to the passive reference
cavity.
The red lattice and the cavity probe are both PDH locked to the science cavity.
The reference laser is locked to a Rb vapor cell. Both MOT laser and repumper laser are
beatnote (BN) locked to the reference laser with phase-frequency detectors (PFDs).
The Raman laser is BN locked to the 7th sideband of the cavity probe, and is about 95 GHz
detuned from the cavity probe. The sidebands of the cavity probe is created by phase modulatingwith modulation frequency of 13.6 GHz.
The atomic probe is either locked to the 12th sideband of the cavity probe or to the sciencecavity, controlled by a high-speed switch (Minicircuits ZASWA-2-50DR+). When locking atomic

23probe to the sideband of the cavity probe, their detuning is more than 100 GHz.
The sideband cooling laser is free running, since we just need to keep it 10s of GHz red
detuned to F=1 to F’=2 transition of the 87 Rb D2 line.

Chapter 3

Site-dependent Selection of Atoms for Homogeneous Atom-cavity Coupling

In this chapter, I will talk about a selection technique to achieve homogeneous atom-cavity
coupling, and how the technique helps suppress optomechanically-induced oscillations. I will also
discuss how different factors could affect our experiment and present some of the technical details.

3.1

Introduction
Atomic ensembles in optical cavities have become a versatile and powerful platform for creat-

ing atomic entanglement [51, 27, 44], generating superradiant lasers [80, 10], synthesizing quantum
matter [23], interacting many-body pseudo-spin and related systems [79, 31] and precision measurement [76, 82]. As I explained previously in Sec. 1.2, the inhomogeneity in the coupling of individual
atoms to the cavity is brought about by the standing wave nature of the cavity modes and theincommensurateness between the atomic probe and the lattice laser. The inhogeneous coupling
reduces the effective atom number [52], creates dephasing in driven systems [75], and brings about
optomechanically-induced oscillations [27]. What motivates us to develop the selection techniqueis that the optomechnically-induced oscillations were believed to be the primary limitation to thedegree of spin squeezing we could create. It will potentially benefit the whole cavity-qed community
as well.
I will review how we created a spin squeezed state (SSS) with quantum non-demolition (QND)
measurement, how the optomechanical effects were excited, and how it could set a limit on spinsqueezing. I will also briefly introduce some alternative solutions to the inhomogeneous coupling25
problem.

3.1.1
3.1.1.1

Optomechanically-induced Oscillations in QND Measurement
Creating Spin Squeeing with QND Measurement

As discussed in Sec. 1.1, a two-level system is like a spin-1/2 particle. The quantum statesof N two-level systems can be represented in the Dicke basis |J, mJ i, with J = N/2 the total spin,
and mJ the spin projection along z. A coherent spin state (CSS) parameterized with azimuthal
angle φ and polar angle θ can be represented in the Dicke basis, as shown in Eq. (3.1) [43],|θ, φi =JX

cm (θ)e−i(J+m)φ |J, mi ,

(3.1)

m=−J

with cm (θ) subjected to a binomial distribution peaked around θ, described by Eq. (3.2),s
cm (θ) =(2J)!
cosJ−m (θ/2) sinJ+m (θ/2).
(J + m)!(J − m)!

(3.2)

In the limit 2J, 2J sin(θ/2) and 2J cos(θ/2) are large, the binomial distribution can beapproximated by the Gaussian distribution, as shown in Eq. (3.3),22(J+m−2J cos (θ/2))
1
2×(2J sin2 (θ/2) cos2 (θ/2)) .
e|cm (θ)|2 ≈ p(3.3)
2π(2J sin2 (θ/2) cos2 (θ/2))√
The standard deviation of the Gaussian distribution in Eq. (3.3) is N /2, for J = N/2 and√θ = 90◦ , so the corresponding angular uncertainty in θ is ∆θ = 1/ N radians. For a CSS, we also

have ∆φ = ∆θ.
If we could project the CSS onto the Dicke basis with a measurement, the standard deviationof the distribution obtained after the measurement will be determined by the measurement noise.
√
Only if the measurement noise is smaller than N /2 in atom number quadrature, we are able tocreate a SSS. We call this measurement a QND measurement because we preserved the coherence
and thus the quantum nature of the state. I will explain how we perform the measurement in the
Dicke basis below.

26
Consider the hyperfine F=1 and F=2 states in the 52 S1/2 manifold of 87 Rb, where F = I + J,
with I the quantum number for nuclear spin, and J for the total angular momentum. The twostates are labeled as |↓i and |↑i and they are separated by ωHF ≈ 2π × 6.8 GHz. The transition
wavelength from 52 S1/2 to 52 P3/2 , which is labeled as |ei, is about 780 nm, as shown in Fig. 3.1
Left.

Figure 3.1: Left: Simplified 87 Rb level diagram. Right: Simplified experimental diagram. The
cavity resonance is measured by homodyne detection.

If we tune the cavity near resonance to |↑i → |ei transition, the photons in the cavity areallowed to exchange between the cavity mode and the atoms in |↑i. ωc in Fig. 3.1 indicates thecavity resonance, and δc is typically 400-500 MHz.
The coupling between the atoms in |↑i and cavity can be treated as two coupled harmonic
oscillators in the low atomic excitation limit, with Holstein-Primakoff approximation. The Hamiltonian is [21]pĤ = ~δc ĉ† ĉ + ~ N↑ g(âĉ† + â† ĉ),

(3.4)

27
where â† and â are creation and annihilation operators for the atoms with â† creating one atomin the excited state, and â destroying one atom in the excited state, ĉ† and ĉ are creation andannihilation operators for the cavity photons, N↑ is the number of atoms in |↑i, and 2g is the single
atom vacuum Rabi frequency.
The eigenfrequencies of the Hamiltonian in Eq. (3.4) are

ω± =

q
p
δc ± δc2 + (2g N↑ )2
2

.

(3.5)

In the large δc limit, the two eigenfrequencies correspond to one cavity-like mode and one atom-likemode. The cavity-like mode is shifted by the interaction with atoms in |↑i by an amountg 2 N↑
δc . By

probing the cavity resonance change with homodyne detection, as indicated by Fig. 3.1 Right, weare able to learn how many atoms are in |↑i, without collapsing individual atomic wavefunctions.
This is exactly performing measurements in the Dicke basis.
Lastly, the coupling strength of the i-th atom in |↑i is gi due to inhomogeneous coupling
between the atoms and the cavity. The measured N↑ and the coupling strength g for calculating
the dressed cavity resonance shift are effective quantities, which are related to the total number ofatoms in |↑i N↑,t and the peak coupling strength for atoms sitting at the antinodes of the atomic
probe g0 as N↑ ≈ 23 N↑,t and g 2 ≈ 34 g02 . These relations are chosen to preserve the average couplingstrength and the projection noise [27, 52, 20, 11, 66].

3.1.1.2

Optomechanically-induced Oscillations

Fig. 3.2 explains the cause of the optomechanically-induced oscillations. All atoms are initiallytrapped at the maxima of the optical lattice. Another potential is developed when we turn on theatomic probe. The gradient of the potential is force, so the equilibrium position of the atoms in
the optical lattice will be displaced by an amount that depends on the force induced by the atomic
probe, if they are not trapped near the extrema of the atomic probe. The displacement will change
the dressed cavity resonance because the atom-cavity coupling is changed.

28

Figure 3.2: Cause of optomechanically-induced oscillations

We could expect an oscillation of the dressed cavity resonance at the axial trapping frequencyof the optical lattice whenever the power in the atomic probe is changed. This oscillation will reduce
the precision of the QND measurement of the cavity resonance.
Shown in Fig. 3.3 was the atomic probe induced oscillation of the dressed cavity resonance(red trace) in the previous spin squeezing generation experiment of our lab [27]. The oscillation
could be partially canceled with a staggered turn-on sequence. An initial half-power turn-on of theatomic probe induced the oscillation, while the second full-power turn-on 2.5 µs (1/4 of an axial
oscillation period) later coherently canceled out the initial axial oscillation. This staggered turn-on
sequence suppressed the oscillation significantly, but only improved the optimal squeezing by an
estimated 0.6 dB.
The optomechanical effect was still the primary limitation of the amount of squeezing available. Mitigation of this effect would be crucial for creating more squeezing.

29

Figure 3.3: Atomic probe induced oscillations partially cancelled by a staggered turn-on sequence.
ωc and ωc0 are the original and dressed cavity resonances. The oscillations are fully present withno kick (red, 43 traces averaged) during a 40 µs measurement, but greatly reduced by a half-power
2.5 µs kick (blue, 30 traces averaged). The 2.5 µs kick length corresponds to a quarter of the axial
trap oscillation period. There is an 80 MHz offset subtracted from the vertical axis. Reprinted
from ref. [27].

3.1.2

Alternatives for Achieving Homogeneous Atom-Cavity Coupling

There have been previous efforts to circumvent the issue of inhomogeneous coupling. One way
is to utilize a commensurate trapping wavelength, such as λl = 2λp [51, 64], with λl the trapping
wavelength, and λp the probing wavelength, as illustrated in Fig. 3.4a. However, the wavelengthof the trapping laser could be a degree of freedom we want to reserve for tuning, for example, toengineer state-dependent traps and magic-wavelength traps [57, 68, 48]. Tightly confining magnetic
traps have also allowed atoms to be loaded into single lattice sites, requiring a high atomic density
at fixed total atom number. If localizing the atoms is undesired, one can also allow the atoms tomove along the cavity axis to time-average away the standing-wave inhomogeneity [28], as shownin Fig. 3.4c. The flat axial potential is engineered by phase modulating the lattice laser withmodulation frequency equals the FSR of the cavity, injecting the ±1-order sideband of the latticelaser into the cavity along with the carrier, and tuning the relative intensity. Finally, one can use
ring-cavity geometries with running waves [95], for example, Fig. 3.4b, but this adds complexity,
results in larger mode volumes, and breaks the degeneracy between polarization modes of the cavity.

30

Figure 3.4: Alternatives for achieving homogeneous atom-cavity coupling. (a) Commensurate
trapping wavelength. (b) Ring cavity. (c) Effective intracavity dipole trap to allow the atoms tomove and time-average their coupling.

Here we demonstrate a method to obtain more homogeneous atom-cavity coupling by initiallyloading atoms into thousands of lattice sites and then using a spectroscopic method to select andkeep only those atoms at lattice sites with near maximal coupling to the probe cavity mode. While
this approach involves discarding atoms, it has favorable scaling in that the degree of homogeneity
is expected to scale quadratically with the number of retained atoms.

87

Rb

Y

Stark Shift BeamZ

Atomic Probe

Intensity

Lattice Laser

A

B

C

e

D

Stark Shift Beam

1 mm

(b)

Atomic Probe

X

Blow-away Beam

Cavity Mirror

31

(a)

↑

Z

0

↓

(c)

Microwave

Energy Level

(d)
A

B

C

D

↑

↓
Z

Figure 3.5: Experimental setup and working principle. (a) Three lasers are sent into the cavity fortrapping (lattice laser), probing (atomic probe) and dressing (Stark shift beam) the atoms. The
atoms are always trapped at the maxima of the optical lattice. The intensity of the atomic probe
and the Stark shift beam the atoms see are site dependent and opposite near the center of thecavity. (b) Level diagram of 87 Rb, and the relationship between the lasers and energy levels. (c)Atomic probability distribution P (η) as a function of η = g 2 /g02 for atoms trapped in the opticallattice without selection. TheRgray region indicates the atoms we want to select. The cumulative
1
distribution function C (η) = 0 P (η)dη is also presented to confirm that P (η) is normalized. (d)Principles of the selection. The dashed lines are the original energy levels without the presence of
the Stark shift beam. The solid lines are the Stark-shifted energy levels. Atoms with unchanged
microwave transition frequencies are maximally coupled to the atomic probe, and can be selectedwith microwave π-pulses at well tuned frequencies.

3.2

Experimental Setup and Working Principle
To further explain the problem and our approach, let us consider our experimental system

in which initially N ≈ 105 87 Rb atoms are trapped at the center of the cavity using an intracavity
optical lattice at λL = 813 nm. The atoms are trapped at antinodes of the lattice light as shownin Fig. 3.5a red. Two additional lasers, the atomic probe and Stark shift beams, can be injected32
into the cavity at or near resonance with other longitudinal modes of the cavity. Their functionsin the spectroscopic selection process will be returned to shortly.
A simplified level diagram is shown in Fig. 3.5b. The two primarily-relevant hyperfine
ground states here are the magnetic-field insensitive states |↑i ≡ |F = 2, mF = 0i and |↓i ≡|F = 1, mF = 0i with a splitting of 6.834 GHz. Only the optical excited state |F 0 = 3, mF = 0i
denoted by |ei is shown for simplicity with transition wavelengths λa ≈ 780 nm.
Here, we wish to enhance the homogeneity of the coupling of the atoms to the atomic probe.
The atomic probe and an associated longitudinual cavity mode is typically detuned by ∆p /2π =700 MHz from resonance with the |↑i to |ei transition. The atomic-probe can then be used to
2measure the shift in the frequency of the cavity mode by an amount δc ≈ g02 /∆p ΣN
i cos (2πzi /λp )[21] where 2g0 is the single-particle vacuum rabi frequency, atom i’s position along cavity axis
relative to the cavity mid-plane is zi , and only atoms in |↑i are implicitly included in the sum. Thissame frequency shift has been used previously to realize entanglement generation via collective
measurements, cavity-optomechanics, and cavity-mediated spin-spin interactions [51, 27, 79, 15].
The goal is then to select atoms at antinodes of the probe mode (i.e. cos2 (2πzi /λp ) ≈ 1), andremove all other atoms from the cavity mode. To reduce notational complexity, we define thecoupling gi of atom i relative to that of a maximally coupled atom as ηi ≡ (gi /g0 )2 .
In conceptual analogy to what is done in NMR imaging by applying magnetic field gradients[85, 8], here we engineer a spatially-dependent shift of the |↓i to |↑i transition frequency ω↑↓ →ω↑↓ + δ (z). The frequency shift is generated by an AC Stark shift δ (zi ) = δs sin2 (2πzi λs ) that is
induced by injecting light into a cavity mode one free spectral range away from that of the probe
mode. Adjacent longitudinal modes have the desired opposite symmetry. The free spectral rangeof the cavity 6.791 GHz is close to the hyperfine splitting ω↑↓ , so that the Stark shift mode is
detuned by approximately ∆s /2π ≈ 900 MHz from the |↓i to |ei transition, leading primarily to a
Stark shift of the state |↓i. Because of the small frequency difference between the probe mode andthe Stark shift mode, the standing-waves are to good approximation locally out of phase near thecenter of the cavity where the atoms are located.

33
To select atoms with peak coupling to the probe mode, we begin by optically pumping atoms
into |↑i, applying the spatially-dependent Stark shift beam, and then applying microwaves fortm ≈ 200 to 250 µs at frequency ωm = ω↑↓ + δm to perform a π-pulse for atoms whose transition
frequency is resonant with the microwaves. The bandwidth of transition frequencies that undergo
the spin flip is set by the Rabi frequency Ωm of the applied microwaves. Atoms that are not flippedto |↓i are then removed from the trap using a radiation pressure force from the blow away beamthat is tuned to resonance with the |↑i to |ei transition and applied for 100 µs. If the microwaves
are resonant with the unshifted transition frequency, i.e., δm = 0, then atoms at nodes of the
Stark beam will remain in the trap. We depicted the selection process in Fig. 3.5d, where atom
A will survive after the blow-away stage, while atom B, C and D will be killed. We also plot theprobability density of atoms as a function of η in Fig. 3.5c. The grey region indicates the selection
bandwidth set by Ωm .
The probability that an atom with a given coupling is selected is simply given by the Rabi
spin flip probability as q
2 + ((η − η)δ )2
Ω
c
s
π
m
,
F (ηc , η, Ωm , δs ) =sin2 
2ΩmΩ2m + ((ηc − η) δs )2
Ω2m

(3.6)

where the microwave detuning is parameterized as ηc = δm /δs . In Fig. 3.6c we show the spin flip
probability for a given coupling ηi for different ratios of Ωm /δs at ηc = 0.

3.3

Theory Plots for Scaling in Selection
In Fig. 3.6a, we show the theoretical trade off between the fraction of of atoms retained

after selection Ns /N and the ensemble averaged coupling η̄ ≡ hηi i for one, two, three and four
selections. In Fig. 3.6b, we show the trade off versus the fractional standard deviation of thepcoupling about the mean ∆η/η̄ ≡ hη 2 i − η̄ 2 /η̄. In this model, the microwave detuning is δm = 0
and the atomic ensemble’s axial spatial extent is much larger than the differential wavelength
−1
−1
λ−1
d = λl − λp . From numerical simulations such as in Fig. 3.6a and Fig. 3.6b, we find in

(a)

(b)

(c)

(d)

34

Figure 3.6: Simulation with minimal model. (a) Curves for 1 − η̄ vs fraction of atoms selected, forone, two and three selections, coded in black solid, red solid and purple dashed. We also plotted
the same relationship for two selections √with varying Rabi frequencies, with the Rabi frequencyof the second selection pulse Ωm,2 = Ωm 2, which is coded in green dashed. The curve for threeselections is hard to identify because it is pretty much buried under the red and green curves. The
simulation is done with δm = 0, and the change in the fraction of atoms is achieved by changing theratio Ωm /δs . (b) Curves for ∆η/η̄ vs fraction of atoms selected, for one, two and three selections,
coded in the same way as in (a). The curve for two selections with varying Rabi frequencies isalso presented. (c) Transfer probability vs η for one selection (solid curves) and two selections(dashed curves). Different colors correspond to different Ωm /δs and Ns /N . (d) Curves for ∆η/η̄ vsfraction of atoms selected, for one and two selections, color coded in black and red. These curves
are obtained with δm = 12 δs .

the region Ns /N < 0.1 that 1 − η̄ = A (Ns /N )α , where α = 2.00 and A = 1.83, 1.04, 1.00, 1.00
for 1, 2, 3 and 4 selection pulses with the same Rabi frequency respectively. The fitting result
for two selections with varying Rabi frequencies is α = 2.00 and A = 1.01. Similarly, we find∆η/η̄ = A (Ns /N )α , where α = 1.47, 2.00, 2.00, 2.00 and A = 1.75, 1.41, 1.12, 1.09 for 1, 2, 3, and4 selection pulses with the same Rabi frequency respectively. The fitting result for two selections

35
with varying Rabi frequencies is α = 2.00 and A = 1.27. For two selections or greater, the qudratic
scaling of the inhomogeneity is highly favorable. We also note that the inhomogeneity is only
marginally improved for greater than two selection pulses. Applying two selection pulses withvarying Rabi frequencies can improve the averaged coupling and homogeneity compared to twoselection pulses with the same Rabi frequency, but it cannot beat three selection pulses.
For comparison, we show the trade off versus the fractional standard deviation of the coupling
about the mean ∆η/η̄ when δm = 21 δs in Fig. 3.6d. We could select atoms located at the maximum
slope of the atomic probe intensity profile in this way, which are potentially useful in certain
circumstances, for example, ponderomotive squeezing [15]. Because of symmetry, η̄ = 0.5 holds all
the time, as noted in the figure. Likewise, we find ∆η/η̄ = A (Ns /N )α , where α = 0.48, 0.99, 1.00
and A = 0.93, 1.70, 1.30 for 1, 2 and 3 selection pulses with the same Rabi frequency respectively.
Only linear scaling can be obtained with two or more selection pulses.

3.4

Microwave Spectrums and Fluorescence Measurements
How the Stark shift beam modifies the microwave spectrum is shown in Fig. 3.7a. The

dashed trace is the original microwave spectrum without the presence of the Stark shfit beamwith long-time Rabi flopping. The solid trace is taken with microwave π-pulses of Rabi freqeucnyΩm0 = 2π × 2.04 kHz, when the Stark shift beam is on. The horizontal axis can also be converted
into η = g 2 /g02 , which the microwave transition frequency change is proportional to, as shown in
Fig. 3.7b. η is physically meaningful in the range from 0 to 1, and the corresponding microwave
frequency range suggested that the peak AC Stark shift δs0 induced by Stark shift beam is about
2π × 32.7 kHz. The finite cavity frequency shift in the grey regions comes from non-zero Rabi
frequency. The cavity frequency shift δωc after a microwave π-pulse takes the form in Eq. (3.7),Z 1
δωc = Ntot δωc0

ηP (η)F (ηc , η, Ωp , δs )dη,

(3.7)

0

where Ntot is the total number of atoms, δωc0 is the cavity frequency shift induced by a single atom

36
(a)

(b)

(c)

(d)

Figure 3.7: Microwave spectrums and fluorescence measurements. (a) Microwave spectrums foratoms with (solid line) and without (dashed line) the Stark shift beam. For reference, a single
atom sitting at the anti-nodes of the atomic probe gives a cavity frequency shift of about 150 Hz.
(b) Microwave spectrum for atoms with the Stark shift beam on in the cavity frequency shift vsmicrowave frequency basis can be mapped onto approximate η 2 P (η) vs η basis, with P (η) theatomic probability distribution. Physically unavailable regions are shaded in gray. (c) Microwave
spectrums for atoms after one (black) and two (red) selections are mapped in the same way as the
spectrum in (b). The insets are in P (η) vs η basis obtained after approximating window functions
as delta functions. The grey line in the background is the original probability density function foratoms without any selections. (d) Spectrums in fluorescence counts vs cavity frequency shift basis
for atoms after one selection (red) and without any selections (black).

sitting at the anti-node of the atomic probe, P (η) is the probability density, and F (ηc , η, Ωp , δs ) is
as defined in Eq. (3.6), with Ωp the Rabi frequency for spectroscopy π-pulse.
F (ηc ) will approach a Dirac delta function δ(ηc ) in the limit Ωp → 0. As a result, the verticalaxis of Fig. 3.7b can only be approximated as ηP (η) due to finite Ωp , represented by the tilde on
the axis label. It is worth mentioning that we used Ωp = Ωm0 for the spectrum in Fig. 3.7b, and1Ωp ≈ 12Ωm0 for the spectrums in Fig. 3.7c.

37
The black and red traces in Fig. 3.7c are microwave spectrums for atoms after one selection
pulse and two selection pulses. The spectrums are taken with microwave π-pulses of Rabi frequencyΩp = 2π × 0.17 kHz, and δm = 2π × −2.7 kHz. Likewise, the horizontal and vertical axes can bemapped to η, and ηP (η). If we anyway approximate F (ηc ) as Dirac delta function δ(ηc ), we
get the discrete probability density distribution Pd (η), normalized to the total area under the
curve, truncated in the range 0.4 < η < 1, as shown in the inset of Fig. 3.7c. We do not include
η < 0.4 in calculating the area because the data points in that range are too noisy, since the cavity
frequency shift is comparable to our measurement noise floor there. The results are summarized asfollows. For one selection, we measure η̄ = 0.88(3), ∆η/η̄ = 0.14(3), Ns /N = 0.11(1), and predictη̄ = 0.90, ∆η/η̄ = 0.12, Ns /N = 0.18. For two selections, we measure η̄ = 0.92(4), ∆η/η̄ = 0.1(1),Ns /N = 0.083(8), and predict η̄ = 0.92, ∆η/η̄ = 0.04, Ns /N = 0.12.
Another way to measure η̄ is by comparing the measured cavity frequency shift to the observed
fluorescence from the atoms with and without selections. Atoms contribute differently to the cavity
frequency shift depending on their coupling to the cavity, but contribute equally to fluorescenceimaging. Fig. 3.7d is a plot of fluorescence counts versus cavity frequency shift. The black andred data points are for atoms without selections and after one selection, whose slopes are fitted to
be au = 1.60(1) and as = 0.92(3) Counts/kHz respectively. From the ratio of the slopes, we inferthat η̄ = au /2as = 0.87(3), in agreement with the numerical prediction η̄ = 0.90 and the previous
estimate from microwave spectrum η̄ = 0.88(1).
In principle, one would expect that η̄ could go even closer to 1, if we select atoms withδm = 0 kHz microwave center frequency. We decided to take the data with detuned microwave
pulses for signal to noise reasons.
One important goal we want to achieve is to suppress the optomechanical oscillations withmore homogeneous atom-cavity coupling. We will move on to actively probe and characterize thiseffect.

38

3.5

Mitigation of Optomechanically-induced Oscillations
I have explained in Sec. 3.1.1.2 the cause of optomechanically-induced oscillations. In prin-ciple, oscillations of the dressed cavity resonance at the axial trapping frequency are expected
whenever the atomic probe power is changed. However, we were only able to observe the equilibrium dressed cavity resonance change as we toggled the power of the atomic probe, because thelinewidth of our cavity was about 50 kHz, much smaller than the axial trapping frequency of the
optical lattice, which is about 205 kHz.
To probe the optomechanical effect, we took a sequence as follows, after two selection pulses,
we pumped atoms to |F = 2, mF = 2i, actively locked the atomic probe frequency to the dressedcavity resonance, and monitored the dressed cavity resonance as we toggled the atomic probe power
up and down with windows of 100 µs long. The polarization of the atomic probe was σ+ , so that
optical pumping to darker states due to free space scattering can be ignored on the |F = 2, mF = 2i
to |F = 30 , mF = 3i cycling transition. The high and low power levels were different by a factorof 3.3. The results are presented in Fig. 3.8a. The insets are zoom-ins of the grey regions, which
include two toggling cycles each. The transients of the locks are removed by subtracting off the
trace obtained with an empty cavity.
As a figure of merit of the degree of inhomogeneity, we consider the degree to which the
optomechanical oscillations are suppressed by the selection process via the ratio r = as /a0 of
the fractional amplitudes of oscillations of the cavity resonance frequency with as and without
a0 selection. We find a measured ratio r = 0.32 and fraction of retained atoms Ns /N = 0.04(1)with N = 8 × 104 after a double selection with δm = 0 and Ωm /δs = 0.08 in Fig. 3.8a. For
comparison, the theoretical predictions are r = 0.08, Ns /N = 0.14 and η̄ = 0.96. To get a senseof the sensitivity of the prediction to system parameters, if we allow for a small shift in detuningof the applied microwave frequency of 2 kHz with the same sign as the applied AC stark shift,
the simulation predicts r = 0.32, however this also predicts Ns /N = 0.14, and η̄ = 0.91 and thesource of a possible 2 kHz shift is unknown. Other effects that have been considered to explain

39

(a)
L

L

H

(b)

L

L
H

H

(c)
2

2

3

3

Cavity Frequency
Shift [kHz]

1

1

H

Figure 3.8: Dressed cavity equilibrium resonance change due to optomechanics. (a) Dressed cavity
resonant frequency shift as a function of time when the power of the atomic probe is toggled
between high and low power level. The insets are zooming-in of the grey regions. The black andred traces are for atoms without any selections and with two selections. (b) The ratio of fractional
amplitudes of oscillations of the cavity resonance frequency with and without selections r, as afunction of the distance from the MOT center to the cavity center. (c) Bottom panel shows the
dressed cavity resonant frequency shift as a function of time with a constant atomic probe power,
where the grey curve is an exponential fit. The inset is the simulated contribution of the radial
oscillations to cavity-resonance frequency in Fig. Fig. 3.8b as we toggle the power in the atomic
probe. The residue of the exponential fit is presented in the middle panel. The top panel sketches
the relationship between the trap depth and the distribution of the atomic ensemble in position
space, before, at the moment, and after the atomic probe is turned on, as indicated by the greyarrows.

40
the discrepancy but are not sufficiently large enough include: finite extent of axial wavefunctions,
non-AC Stark shift induced inhomogeneous broadening of the microwave transition, finite radial
temperature, imperfect π-pulses, errors in the ratio of the Rabi frequency to peak AC Stark shiftΩm /δs , finite optical pumping to F = 1, and potential heating during blow away step.
We also observe that the toggling of the atomic probe power drove radial oscillations of theatomic ensemble, which makes a considerable contribution to the observed oscillation amplitudeof the dressed cavity resonance. The atomic probe not only provides an axial force on the atoms,
but also changes the radial confining potential. As a result, the finite temperature of the atomic
ensemble and a non-adiabatic change in the radial confinement combine to drive a breathing of theradial extent of the atomic ensemble, with a period that is half of the radial oscillation period Tr
as illustrated in the top panel of Fig. 3.8c. Fig. 3.8c middle and bottom panels show the observed
oscillations of the cavity resonance frequency due to the radial breathing. The oscillations are
induced here by turning on the atomic probe and keeping it at a constant power level.
The inset of Fig. 3.8c shows the simulated contribution of the radial oscillations to thecavity-resonance frequency in Fig. 3.8a. Applying a correction for the amplitude ratio r, we finda corrected r = 0.25 closer but still off the predicted value of r = 0.08. This does not, however,affect the discrepancy in the number of selected atoms.
The quality of the selection also depends on the local orthogonality of the atomic probe andStark shift standing waves, i.e., they are out of phase. We measure r as a function of the position of
the atomic cloud along the cavity axis as shown in Fig. 3.8b, and it is well described by a quadratic
fit with a minimum value r = 0.30(2). All preceding data presented in this paper was taken at theposition indicated by the red point, which was 0.30(4) mm off the cavity center, not sufficiently far
enough away to account for any residual discrepancies in r and Ns /N .

3.6

Conclusion
In conclusion, we have demonstrated a method to obtain homogeneous atom-cavity coupling.

We select atoms that are near-maximally coupled to the cavity with the help of a Stark shift

41
beam that is one FSR away from the cavity mode used to probe the atoms. In this way, we
could always keep the atoms trapped in the Lamb-Dicke regime and retain the optical lattice
wavelength as a tunable degree of freedom, which opens the route for state-dependent trapping,magic wavelength trapping and other wavelength-dependent tricks. We have also shown that thismethod helps suppress the optomechanical oscillations, which was a main obstacle for achievinga large amount of squeezing for inhomogenous atom-cavity coupling systems [27]. We believe themethod demonstrated in this thesis could benefit experiments based on atom-cavity interactions ina broad sense.

3.7

Discussions

3.7.1

Simulations in Sec. 3.3

The simulations in Sec. 3.3 are done with the Rabi spin flip probability function Eq. (3.6),
where free space scattering is ignored.
I will take one selection pulse with δm = 0 as an example. The probability density function
1 √of the atoms in terms of η is P (η) = π√1−η. We could calculate Ns /N , η̄ and ∆η with P (η).
η

Ns
=
N

Z 1
P (η)F (0, η, Ωm , δs )dη

(3.8)

0

R1
η̄ =

sR
∆η =

0 P (η)ηF (0, η, Ωm , δs )dη

(3.9)

N s/N
1
2
0 P (η)η F (0, η, Ωm , δs )dη

N s/N− η̄ 2(3.10)All of theses quantities can be numerically calculated with a well-defined ratio of Ωm /δs . We
are then able to get the curves in Fig. 3.6. Results for two or more selection pulses can be obtained
similarly.

42
3.7.2

Effects from Finite Atomic Radial Extent

The distributions of the atoms in x and y directions are both Gaussian distributions withstandard deviations σx = σy =8 µm. As an atom goes off the cavity axis, its coupling to the cavity
will be decreased because of the Gaussian profile of the cavity mode. To take into account theeffects from finite atomic radial extent, we just need to integrate over the distribution in x and y
in Eq. (3.8), Eq. (3.9) and Eq. (3.10).
For Ωm /δs = 0.08, η̄ is estimated to decrease by about 2%.

3.7.3

Factors Affecting the Microwave Spectrums

The simulation done as discussed in Sec. 3.9 shows that the peak of the microwave spectrumwith the Stark shift beam on is located at about η = 0.97, as shown by the blue curve in Fig. 3.10,
while the peak of the spectrum we measured as shown in Fig. 3.7a and also in Fig. 3.9 was locatedat about η = 0.92. I will discuss possible factors that could lead to such a discrepancy.

3.7.3.1

Imperfect π-pulses

The effects of imperfect π-pulses can be found in Fig. 3.10. If we use 0.9 π-pulse, the peakcan be shifted slightly to about η = 0.96. 0.5 π-pulse is able to displace the peak to about η = 0.93,
and the corresponding lineshape actually matches the experimental data better. However, it was
impossible to have such a big discrepancy in the calibration of the π-time, which was measured tobe 0.245(3) ms with Rabi flopping.

3.7.3.2

Error in Ωm

The effects of error in Ωm is presented in Fig. 3.11. If I make Ωm twice as large, the theory
predictions are more close to the spectrum we got experimentally. However, we measured Ωm =2π × 2.04(2) kHz, so the error in Ωm is minor, and such a small error is not enough to explain thedeviations of the data from the prediction. I just present this discussion here for completeness.

43

Figure 3.9: Comparison between measured microwave spectrum (black) and that predicted bytheory (blue). The vertical axis of the theory plot is in a.u., and is rescaled for comparison withthe experimental data. Both are with the Stark shift beam on.

3.7.3.3

Error in δs , the Peak AC Stark Shift

The effects of error in δs is presented in Fig. 3.12. The x-axis of Fig. 3.12 is in units of
microwave frequency rather than η because η is dependent on δs as discussed in Sec. 3.4. Changing
δs only changes the peak transfer probability, instead of the position where the peak is located.

3.7.3.4

Non AC Stark shift Induced Homogeneous Brodening of the Microwave
Transition

Empirically, the non AC Stark shift induced broadening of the microwave transition, if wedenote its HWHM as γnAC , will modify the Rabi spin flip probability Eq. (3.6) into

44

Figure 3.10: Effect of imperfect π-pulse. The blue trace is the result with a perfect π-pulse. The
brown, red and green traces are the results with 0.5 π-pulse, 0.9 π-pulse and 1.1 π-pulse.

 q
2 + ((η − η)δ )2
Ω
c
s
m
2 π
.
Fb (ηc , η, Ωm , δs ) =
2
2 sin
2
2
Ωm
Ωm + ((ηc − η) δs ) + (γnAC )Ω2m(3.11)We measured γnAC =1.1(2) kHz, by measuring the long-time Rabi flopping spectrum with
the Rabi frequency as small as possible. As a reference, the Rabi frequency used is estimated to besmaller than 400 Hz.
Fig. 3.13 shows the effects of non AC Stark shift induced broadening of the microwave transition. A larger broadening will decrease the transition probability, but only shift the position of
the peak slightly. With 5.5 kHz of broadening, the peak of the spectrum is displaced to η ≈ 0.95,
which is not enough to explain the discrepancy.

45

Figure 3.11: Effect of varying Ωm . The blue trace is the result with Ωm,0 =2.04 kHz, which we used
for all the analysis in this chapter. The red and green traces are the results with 0.5 Ωm,0 and 2Ωm,0 .

3.7.3.5

Inhomogeneous Broadening of the Microwave Transition

Consider some type of inhomogeneous broadening δin that is subject to a Gaussian distribu2−δin1
tion √2πσ
e 2σ2 , with σ the standard deviation of the distribution. The Rabi spin flip probability

Eq. (3.6) will be modified into q
2 + ((η − η)δ − δ )2
Ω
c
s
in
π
m
.
Fb (ηc , η, Ωm , δs ) =sin2 
2ΩmΩ2m + ((ηc − η) δs − δin )2Ω2m(3.12)

To calculate the expected spectrums, the integral over δin weighted by the Gaussian distribution is required.
Fig. 3.14 shows the effects of the inhomogeneous broadening of the microwave transition.
With σ =3.5 kHz, the peak of the spectrum is displaced to η ≈ 0.92, which is adequate to explain
the discrepancy. The next question is what mechanism could induce such a broadening, which I

46

Figure 3.12: Effect of varying δs . The blue trace is the result with δs,0 =32.6 kHz, which we used
for all the analysis in this chapter. The red and green traces are the results with 0.8 δs0 and 1.2
δs,0 .

will discuss below in Sec. 3.7.3.6.

3.7.3.6

Finite Extent of the Axial Wavefunction

The inhomogeneous broadening of the microwave transition required in Sec. 3.7.3.5 seems
large, so we are motivated to think about possible broadening mechanisms from the Stark shift
beam. I will estimate how large the broadening induced by the finite extent of the axial wavefunctionis.
The wavelength of our lattice is 813 nm. The trap depth is 150 µK. We could estimate thesquare of Lamb-Dicke parameter for the ground state, LD2 ≈0.018. The ground state broadening
approximates LD2 δs , which is about 0.6 kHz.
The axial temperature of the atomic ensemble is about 1/10 of the trap depth, from which
we could estimate the average occupation number in the harmonic trap, n̄ ≈2. The square of the47

Figure 3.13: Effect of non AC Stark shift induced broadening of the microwave transition. The
blue trace is the result without any broadening. The green trace is the result with broadening
γnAC =1.1 kHz. The red and brown traces are the results with 2.5γnAC and 5γnAC .

Lamb-Dicke parameter is related to the energy level n in the harmonic trap as LD2 (n) ∝ 2n + 1.
For n=2, the broadening is about 3 kHz. This broadening is close to the expected value to explain
the discrepancy in the microwave spectrum we saw on its own.

3.7.3.7

Summary

To summarize, we have considered the effects on the microwave spectrums from imperfect πpulses, errors in Ωm and δs , non AC Stark shift induced homogeneous broadening of the microwave
transition and inhomogeneous broadening of the microwave transition. Finite extent of the axial
wavefunction induces an inhomogeneous broadening with σ ≈3 kHz, which is close to the value
needed (about 3.5 kHz) for explaining the discrepancy. It is also possible that the discrepancy resultsfrom a combination of contributions, or some other sources that have not yet been identified.

48

Figure 3.14: Effect of inhomogeneous broadening of the microwave transition. The blue trace isthe result without any broadening. The green trace is the result with the standard deviation of thebroadening σ =1 kHz. The red and brown traces are the results with σ =3.5 kHz and σ =5 kHz.

3.7.4

Simulation of the Contribution of the Radial Oscillations to Cavity-ResonanceFrequency in Fig. 3.8a

For an atom with initial position and momentum x0 and p0 in a harmonic trap, its position
and momentum will evolve following Eq. (3.13) and Eq. (3.14)p0
sin(ωt)mω

x(t)= x0 cos(ωt) +p(t)= −x0 mω sin(ωt) + p0 cos(ωt),

where ω is the trapping frequency of the harmonic trap.
If we define pn = p/mω, the equations of motion will be modified into

(3.13)(3.14)

49x(t)= x0 cos(ωt) + pn0 sin(ωt)(3.15)

pn (t)= −x0 sin(ωt) + pn0 cos(ωt).

(3.16)

When we have an ensemble of atoms in a harmonic trap, the standard deviation in x and pn
will evolve following Eq. (3.17) and Eq. (3.18)

∆x(t)

=

p

(∆x0 cos(ωt))2 + (∆pn0 sin(ωt))2(3.17)∆pn (t)=p(∆x0 sin(ωt))2 + (∆pn0 cos(ωt))2 .

(3.18)

If the trap depth changes abruptly, we just need to rescale pn and the following evolution
will be governed by the same set of equations, i.e., Eq. (3.17) and Eq. (3.18).

3.7.5

Calculating r for a Certain Distribution of Atoms

We have talked about how to extract the distribution of atoms in Sec. 3.7.1. For a certain
distribution of atoms, r can be estimated as follows.
The gradient force from the lattice is
Fg = −d
UL cos(kL z)2 = UL kL sin(2kL z)dz(3.19)

where UL and kL = 2π/λL are the trap depth and the wavevector of the lattice.
We could then calculate the displacement zd of the i-th atom with the following equation of
motion,mz̈ + UL kL sin(2kL zd ) + UA kA sin(2kA (zd + zi )) = 0(3.20)

where UA and kA = 2π/λA are the equivalent trap depth and the wavevector of the atomic probe,
zi is the initial position of the i-th atom, and m is the mass of an atom.

50
The change of the effective coupling of the i-th atom to the cavity can be calculated after
we know its displacement, which simply changes from cos(kA (zi ))2 to cos(kA (zd + zi ))2 . The ratior as we toggle the power of atomic probe can be extracted from the effective total atom number
change induced by the displacement.

3.7.6

Electronic Block Diagram for Probing the Optomechanically-induced OscilHold probe
frequency whenprobe is off

lations½fs=40.55MHz,φs

Homodyne
Detector

REF

RF

From
Fig. S2

-

DC

τ=1s

Path length stabilization

VCO

PFD

Lstab

-

26 dB
Attenuator

81.1 MHz

DAQ
SampleHold

Remove DC offset

Sample/Hold
Cavity
Probe
Laser

-

13.6 GHz

EOM

Homodyne
AOM

SHA

SHA

Vref
Function
generator forvoltage sweep

Phase-lock atomic probe
to cavity probe

Atomic
Probe
Laser

La

PFD

REF

fca

LSHA

15 dB
Attenuator

VCO

I

R

L

90-110 MHz

sweep

DDS

Freq. Offset
30 MHzFigure 3.15: Electronic block diagram for probing the optomechanically-induced oscillations

Fig. 3.15 shows the electronic block diagram for probing the optomechanically-induced oscillations. It is pretty similar to what we used before [27]. The changes are mainly about how we look
for the resonance of the cavity and hold the output voltage of the loop filter, and how to make thelock to the held DC voltage and the lock of the atomic probe to the cavity resonance compatible.
We look for the resonance of the cavity by actively scanning the input voltage of the VCOwith a function generator. Vref is a tunable reference voltage. When the cavity resonance is found,51
we will hold the VCO input voltage with the help of the sample/hold. The held voltage is what
the DC lock is locked to, so that we could jump the output frequency of the VCO, and thus thefrequency of the atomic probe to the cavity resonance when needed. The lock of the atomic probe
to the cavity resonance is used to probe the cavity resonance with high precision. We want to makethese two locks compatible so that the lock of the atomic probe to the cavity resonance is close
to its lock point, when we switch the feedback path. The DC lock requires two integrator stages,
however, our cavity linewidth is about 50 kHz, which works like a built-in integrator in terms of
the shape of its transfer function. We thus add a low-pass filter in the DC lock path, and turnon only one integrator stage of the loop filter. The differential gains required by the two locks arebridged with the 26-dB attenuator. The 15-dB attenuator is meant for improving the precision of
the measurement of the control voltage of the VCO for cavity resonance frequency readout.

Chapter 4

Squeezed Atom Interferometry: Scheme and Progress

In this chapter, I will explain what squeezed atom interferometry is, introduce our scheme
for building an intracavity, guided squeezed atom interferometer and report our current progress.
I have explained what the Mach-Zehnder type atom interferometer is in Sec. 1.3. I will keep taking
the Mach-Zehnder type squeezed atom interferometer as an example here. The spin-squeezing
generated for an atom interferometer needs to be homogeneously distributed between all the atoms,
which is guaranteed by time-averaged probing of falling atoms in this context [28].

4.1

Introduction to Squeezed Atom Interferometry
In Sec. 1.3, I mapped the process of the Mach-Zehnder type atom interferometer onto Bloch

spheres, as shown in Fig. 1.6. If we replace the coherent spin state with a squeezed state, we get asqueezed Mach-Zehnder type atom interferometer, as shown in Fig. 4.1.

ࢥA

ࢥ

B

Figure 4.1: Squeezed Mach-Zehnder type atom interferometer. The only difference from a normal
atom interferometer is that the coherent spin state is replaced with a squeezed state.

53
The squeezed state in the initialization step is a phase-squeezed state, meaning that thenoise in the phase quadrature is reduced, while that in the atom number quadrature is increased.
It can be transformed from the number-squeezed state introduced in Sec. 1.1 with a π/2-pulse
whose rotation axis is aligned with the Bloch vector. As discussed in Sec. 1.3, the atom numbersin |↑i and |↓i after interference are proportional to cos2 (∆Φ/2) and sin2 (∆Φ/2) respectively. The
definition of ∆Φ can be found in Sec. 1.3. It is clear that the noise in the final readout depends on
the precision of the quantum phase, and that is why we have to use a phase squeezed state.

4.2

Ingredients for Intracavity, Guided Squeezed Atom Interferometry
We mainly need three ingredients, that is, generating squeezed states, driving two-photontransitions along the cavity axis and guiding the falling of the atoms. I have described how generation of squeezed states with quantum nondemolition measurement works in Sec. 3.1.1.1, and wewill keep using the same technique. I will explain the other two ingredients in this section.

4.2.1

Two-photon Transitions along the Cavity Axis

Consider a two-photon transition in free space between |↓, 0~ki and |↑, 2~ki driven by two
counter-propagating beams at frequencies ω1 and ω2 , as shown in Fig. 4.2. The atoms will betransferred from |↓i to |↑i, and get 2~k momentum kicks along the z-direction, if they start in |↓i andthe two beams are propagating as shown. Here k is the wavevector of the lasers, k = ω1 /c ≈ ω2 /c,with c the speed of light. We could make the momentum kicks the atoms received −2~k by switching
the propagation direction of the two beams or pumping atoms into |↑i initially.
However, the modes supported by the cavity are standing waves, that is, if we inject two tones
at ω1 and ω2 into the cavity, we will get both upwards-propagating and downwards-propagating
beams at ω1 and ω2 , as shown in Fig. 4.3. This is bad because the direction of the momentum
kicks imparted becomes ambiguous. It is necessary to lift the degeneracy between the upward anddownward beams.
Our method to lift the degeneracy is to allow the atoms to fall to gain an initial velocity

54

X
Z

Figure 4.2: Two-photon transitions in free space. The transitions are driven by two counterpropagating beams at frequencies ω1 and ω2 . Atoms are pumped into |↓i initially.

Figure 4.3: Standing waves of cavity modes

55v, and take advantage of the Doppler shift, as shown in Fig. 4.4. Beams co-propagating with orcounter-propagating against the atoms will be shifted to the red or to the blue by kv. We plan totake |↓i ≡ |F = 1, mF = 0i and |↑i ≡ |F = 2, mF = 0i of 87 Rb, so the detuning between |↓i and |↑i
is about ωHF , the hyperfine splitting between F=1 and F=2 manifolds. If we choose to make thepair with frequencies ω1 + kv and ω2 − kv on resonance with the two-photon transition, the otherpair will be detuned by 4kv. Momentum kicks in deterministic directions are now enabled.

Figure 4.4: Degeneracy of counter-propagating beams lifted. Left figure shows the frequenciesof the counter-propagating beams in the atom’s rest frame. Right figure explains how we could
deterministically apply two-photon momentum kicks in certain directions.

In Sec. 2.2, we mentioned that the desired FSR of the science cavity is 10-47 MHz detuned
from the hyperfine spliting of 87 Rb, ωHF . The reason is that we want the FSR to be close enough toωHF , so that the two beams with frequencies ω1 and ω2 can be injected into adjacent cavity modes,
while we also want the FSR to be far enough from ωHF , so that the vacuum fields of adjacent cavity
modes will not make significant contributions to the two-photon transitions, and to avoid strong

56
conversion of laser frequency noise into intensity noise which can then drive Bragg transitions toother momentum states.
In the experiment, we generate the two tones by phase modulating the Raman laser withmodulation frequency of about ωHF /2. The carrier of the Raman laser is locked to the cavity probe.
The frequency of the Raman laser is thus indirectly stabilized to the science cavity since the cavity
probe is locked to the science cavity. The arrangement is that the ±1 sidebands of the Raman
laser are detuned to their nearest cavity modes by about ±∆r =±(ωHF − F SR)/2=±21 MHz,
respectively.

4.2.2

Guided Falling of the Atoms

We need to create a spin-squeezed state that is homogeneously shared by all the atoms, which
is obtained by time-averaged probing of falling atoms. The guided falling of the atoms is achieved
with a dipole trap, whose axial trapping potential becomes flat but radial trapping potential isstill maintained [28]. The creation of the dipole trap is illustrated in Fig. 4.5. The adjacent cavity
modes have different symmetries near the center of the cavity, i.e., if the intensity pattern of one
mode is described by cos2 (kz), with k the wavevector of the laser and z the position from thecavity center, the intensity patterns of its two adjacent modes will be approximately sin2 (kz). I
take the wavevector of all three modes to be k, because FSR of the cavity is much smaller than
the frequency of the laser. By injecting the lattice laser and its ±1 sidebands into the cavity andtuning the relative intensities right, we get a flat axial potential near the center of the cavity because
cos2 (kz) + sin2 (kz) = 1.
We were creating the dipole trap with Hermite-Gaussian HG00 mode of the cavity. To further
reduce the bumpiness along the cavity axis, we decide to use Laguerre-Gaussian LG01 mode, i.e.,the donut beam, as shown in Fig. 4.6 Left. The LG01 mode is generated by diffracting a Gaussian
beam (the 760 nm blue lattice laser) off a phase plate, whose pattern is shown in Fig. 4.6 Right.
The diffraction efficiency is about 10%. The phase plate was fabricated by the JILA Keck lab bychromium deposition.

57

Figure 4.5: Creation of the dipole trap. What is shown in the figure is the axial intensity patternof the intracavity optical trap.

Figure 4.6: Intensity pattern of LG01 mode and phase plate pattern for generating it. Left is theintensity pattern of LG01 mode measured with a camera. Right is the pattern of the phase platefor generating LG01 mode, which we learnt how to make and is reprinted from ref. [101].

4.3

Progress
In Sec. 4.2.1, we explained how to make the two beams with frequencies ω1 +kv and ω2 −kv on

resonance with the two-photon transition for atoms moving at velocity v. It is natural to think that
if we change the detuning between ω1 and ω2 , we could interact with atoms with different velocities,
and thus map out the velocity distribution of the atoms. We call such a spectrum velocimetry. The

58
sequence for velocimetry is as follows, we first allow atoms to fall along the cavity axis to gain some
initial velocity and pump all the atoms into |↓i, then we apply a two-photon π-pulse to transfer
atoms into |↑i as we vary the detuning between ω1 and ω2 , and we finally measure the atom numberin |↑i. One example velocimetry is shown in Fig. 4.7. The huge peak around 0 kHz corresponds to
the Doppler-free transition, which is driven by either ω1 + kv and ω2 + kv, or ω1 − kv and ω2 − kv.
No momentum kicks are imparted on the Doppler-free transition.

Figure 4.7: Velocimetry. The horizontal axis is the two-photon detuning (ω1 − ω2 )/2π in kHz. The
vertical axis is the measured atom number in arbitrary unit (a.u.).

We could measure that the HWHM of the distribution of the Doppler shift is about 100 kHz
from the velocimetry. We need velocity selections to select a group of atoms with a much narrower
velocity distrbution. Velocity selections are done simply by two-photon π-pulses with a small Rabi
frequency, followed by a blow-away pulse to remove those atoms that are not selected. We are able
to select atoms with 1 kHz HWHM out after two selection π-pulses with 1.4 kHz of Rabi frequency.
With the atoms after velocity selections, we are able to drive the two-photon Rabi flopping,as shown in Fig. 4.8. The decoherence is dominated by the ratio of the Rabi frequency for the Rabi
flopping to the HWHM of the corresponding Doppler shift of the velocity distribution in the shorttime limit, and the radial temperature, or radial spread of the atomic cloud in other words, in the59long time limit.
This summarizes the progress made as of Dec. 2020. We expect preliminary results bysummer 2021.

Figure 4.8: Two-photon Rabi flopping. The horizontal axis is time in ms. The vertical axis is themeasured atom number in a.u.

4.4

Sequence for Squeezed Atom Interferometry
We plan to try a sequence as follows. First, we will load atoms into the red lattice, cool

the atoms to about 7 µK axially and about 1.5 µK radially. Then atoms will be transferred into
the blue dipole trap, and allowed to fall for about 15 ms. Next, velocity selections will be appliedand all the selected atoms will be optically pumped into |↓i. Afterwards, we will apply a twophoton π/2-pulse to transfer the atoms into a superposition state |ψi = √12 (|↓, 0~ki + |↑, 2~ki).
The following quantum nondemolition measurement will create a spin squeezed state in the atom
number quadrature, and we can apply an aligned π/2-pulse to transform the number squeezed state
into phase squeezed state. Now we arrive at the first state shown in Fig. 4.1. All steps left will
follow what is presented in Fig. 4.1.

60

4.5

Discussions

4.5.1

Two-photon Rabi Frequency for Atom Interferometry

I mentioned that the short-time decoherence of the Rabi flopping shown in Fig. 4.8 wasdominated by the ratio of the Rabi frequency for the Rabi flopping to the HWHM of the corresponding Doppler shift of the velocity distribution. I will explain why the Rabi frequency for the
Rabi flopping cannot be made arbitrarily large.
For an atom in free space, the transition frequency between two states equals the frequencychange in the internal states plus that in the motional states. Consider an atom in |↓i moving
downwards at velocity v, whose state we label as |↓, 0~ki, as shown in Fig. 4.2. The transition
frequency δω20 between |↓, 0~ki and |↑, 2~ki in the lab frame takes the formδω20 = δω↑↓ +(mv + 2~k)2 1− mv 2
2m
2

!
/~,

(4.1)

δω↑↓ is the transition frequency between the two internal states, |↓i to |↑i, and m is the mass of
the atom. After simplification of Eq. (4.1), we getδω20 = δω↑↓ +
2~k 2+ 2kv .
m

(4.2)

Likewise, the transition frequency between |↓, 0~ki and |↑, −2~ki isδω−20 = δω↑↓ +
−2~k 2+ 2kv ,
m

(4.3)

and the transition frequency between |↓, 4~ki and |↑, 2~ki isδω24 = δω↑↓ +
6~k 2+ 2kv .
m

(4.4)

2

For 87 Rb, we have δω24 − δω20 = δω20 − δω−20 = 4~km ≈ 2π × 30.0 kHz. That is to say, if thetwo-photon transition driven by the two lasers is on resonance with the transition between |↓, 0~ki
and |↑, 2~ki, it will be off resonance from the transition between |↓, 0~ki and |↑, −2~ki and the61transition between |↓, 4~ki and |↑, 2~ki by 30 kHz, as shown in Fig. 4.9, where we set δω↑↓ = 0 for
simplicity.

Figure 4.9: Transition frequency between adjacent momentum states. If a two-photon transitionis on resonance with the transition between |↓, 0~ki and |↑, 2~ki, it will be off resonance from thetransition between |↓, 0~ki and |↑, −2~ki and the transition between |↓, 4~ki and |↑, 2~ki.

It is clear now that if the square of the Rabi frequency for the Rabi flopping becomes comparable to (30 kHz)2 , the probability of the transitions into |↓, 4~ki and |↑, −2~ki become significant and
we no long have well-defined two-level systems. We typically use about 9.5 kHz of Rabi frequency.

4.5.2

Frequency Sweep of the Two Tones for Two-photon Transitions

We need to sweep the frequencies of the two tones to compensate for the change of the
Doppler shift from the free-falling of the atoms. As mentioned previously, the two tones are
generated by phase modulating the Raman laser. The sweep rate of the modulation frequency is
f˙m = g/λr = 12.56 kHz/ms, in which λr is the wavelength of the Raman laser.

62
4.5.3

Amount of Squeezing Available

The maximum amount of squeezing available ξm on clock transitions scales as ξm =p
8/(qN C),

where q is the total quantum efficiency, N is the total atom number and C=2.06 is the single atom
cooperativity [21]. Definition of C can be found in Table. 2.1.
We plan to use N=1000 for the interferometry. The total quantum efficiency is about 10%.
The maximum amount of squeezing available is estimated to be 7.1 dB.
√
The standard quantum limit with 1000 atoms is ∆φSQL = 1/ 1000 ≈ 3.2 × 10−2 . With
maximum amount of squeezing available, the phase resolution is improved into ∆φM ≈ 1.4 × 10−2 .
This is the number that we should compare all the potential noise sources with in order to see ifthey will limit our ability to observe that we are generating a squeezed state that can be observed
to pass through the interferometer.

4.5.4

Change in Effective Gravitational Acceleration g

The change in effective gravitational acceleration g can arise from the tilt of the cavity, andthe axial potential from the blue optical dipole trap.

4.5.4.1

Tilt of the Cavity

We measured the tilt of the cavity with a dial gauge and were able to bound the tilt of thecavity to less than φt =3 mrad.
The effective acceleration g 0 = g cos(φt ) ≈ g 1 − (φt )2 /2 . This will induce a 10−5 fractional
error on g and thus give a small correction to the sweep rate of the modulation frequency which
was discussed in Sec. 4.5.2.

4.5.4.2

Axial Potential from the Blue Optical Dipole Trap

In a dipole trap, the beam diverges so that the intensity changes. Just as in previous sections
discussing optomechanical forces, the gradient of this intensity change leads to a force along the axisof the cavity. For atoms at zero radial temperature, the blue guiding potential has zero intensity

63
on axis so there is no force. However, the atoms are at finite temperature and so here we estimate
the size of the acceleration due to the finite axial gradient in the intensity of the blue optical dipole
trap for the finite temperature atoms. The radial mode pattern of the LG01 mode in the plane of
the mode waist is2

2r2 − 2r2
I(r) = I0 2 e w0 .
w0

(4.5)

The trap depth of the blue optical dipole trap is more than 50 µK, while the radial temperature of the atoms is Ta ≈1.5 µK, so the standard deviation of the radial distribution of the atoms
is at most σr = 5.2 µm.
I will try to estimate the effective change in g for the atoms sitting σr off the cavity axis.
The axial potential these atoms see takes the form

V (z) =kB Ta,
1 + (z/zR )2(4.6)

where z is the displacement from the cavity center, kB is the Boltzmann constant, and zR ≈ 1.97 cm
is the Rayleigh range of the blue optical dipole trap.
The acceleration induced by the gradient of the axial potential is|a(z)| =dkB Ta,
dz 1 + (z/zR )2(4.7)

where we take the absolute value of a(z) for simplicity. The maximum of |a(z)| is estimated to be
am = 3.6 × 10−3 m/s2 . We have am /g = 3.7 × 10−4 , which is significant and we should be careful
if we want to have accuracy in measurement of g.

4.5.5

Relative Phase Shift of the Two Tones Induced by Frequency Jittering of
the Raman Laser

The phase of the two photon coupling is set by the difference phase between the two tones thatdrive the two-photon Raman transition. Anything that changes the relative phase in an uncontrolled
way during an interferometer sequence will add phase noise to the atom interferometer (kind of

64
like vibrating mirrors in an optical interferometer) and obscure our ability to see that a squeezed
state has passed through the interferometer. Here, we consider how laser frequency noise might beconverted into a phase difference between the two tones. Ideally, the two tones are ±∆r ≈ ±21 MHz
detuned from their nearest cavity modes respectively. Consider a small frequency shift δr of theRaman laser, the detunings of the two tones from their nearest cavity modes becomes ∆r + δr and−∆r + δr .
To calculate the relative phase shift of the two tones, we need the transfer function of thecavity. For an incident field Einc (t) = Einc e−iωt , and a circulating field inside the cavity Ecav (t) =Ecav e−iωt , the following relation always hold,
Einc t1 + Ecav r2 r1 ei2Lω/c = Ecav ,(4.8)

where t1 and r1 , and t2 and r2 are the transmission and reflection coefficients of the input andouput mirror, L is the length of the cavity, and c is speed of light.
c
In the limit |ω − ωc |  2L, the transfer function is

√
Ecav
F T1
1
,
T (ω) ==cEinc
π 1 − i ω−ω
κc /2

(4.9)

in which ωc is the cavity resonant frequency, F is the finesse of the cavity, T1 = |t1 |2 , and κc is the
linewidth of the cavity. We then have arg(T (ω)) = tan−1 ( κc∆/2 ), where ∆ = ω − ωc is the detuning.
Let’s now estimate the change in the difference phase if during the interferometer sequence,the central laser frequency used to create the two laser tones shifts by δr . The change in the
difference phase isδφ = (tan−1 (∆r−∆r∆r + δr−∆r + δr
) − tan−1 ()) − (tan−1 () − tan−1 (
)),
κ/2
κ/2
κ/2
κ/2

(4.10)

where κ =52.1 kHz is the linewidth of the science cavity. In the limit ∆r  δr , κ, Eq. (4.10) can2r
be approximated as δφ ≈ − κδ.
∆3r

δr can be suppressed below κ easily, so we have δφ < 6.2 × 10−6  ∆φM . So, by operating

65
at large detuning, the laser frequency noise will not hamper our ability to see the squeezed statepass through the interferometer.

4.5.6

Cavity Induced Phase to Amplitude Noise Conversion

When the frequency of the laser for driving two-photon transitions jitters, the coupling of thedriving tones to the cavity will change, so there will be amplitude noise on the two photon Rabi
frequency Ω2ph due to the laser frequency jittering. Luckily, there will be first order cancellation of
the amplitude noise induced by frequency jittering, because the two-photon Rabi frequency Ω2phr
1
1
, where δ∆ denotes the frequency jittering. In the limit
scales as Ω2ph ∝(∆ +δ∆)2(∆r −δ∆)2
1+ r
1+(κ/2)2(κ/2)2


2
δ∆2
1
−
. We will focus on the amplitude noise converted∆r  δ∆, κ, we have Ω2ph ∝ (κ/2)
∆2
∆2
r

r

from the phase noise with the help of the cavity here.
The phase noise of a laser is presented as the finite linewidth of the laser. With the help of thecavity, the phase noise can be converted into amplitude noise, because the transfer function of thecavity is frequency-dependent. We will derive a formalism for estimating the converted amplitude
noise and bound the amplitude noise in our interferometry experiment.
Consider a laser whose phase noise is modeled with a time-dependent phase fluctuation φ(t),Ei (t) = Ei,0 cos(ω0 t + φ(t)).

(4.11)

Apply Fourier transform to φ(t), and we getZ ∞Ei (t) = Ei,0 cos ω0 t +φ̃(ω) cos(ωt)dω ,(4.12)

0where we have assumed the lineshape of the laser is symmetric. If we discretize Eq. (4.12) andfocus on a single frequency component ω = ωs , we haveEs (t) = Ei,0 cos ω0 t + φ̃(ωs ) cos(ωs t) ,where φ̃(ωs ) =(4.13)p
Sφ df , with Sφ the power spectral density of phase fluctuation and df an infinites-∆ν
imal frequency range. For a Lorentzian lineshape, Sφ (f ) = πf
2 , with ∆ν the FWHM of the laser,

66
and f the frequency component of the power spectral density. Eq. (4.12) is a standard form of theq2π∆νphase modulation, whose modulation index is β = φ̃(ωs ) =dω, with dω = 2πdf . In theπω 2slimit β is small, we haveEs (t) = Ei,0 [cos(ω0 ) +ββcos(ω0 + ωs ) − cos(ω0 − ωs )].
22(4.14)We have derived the transfer function of the cavity as shown in Eq. (4.9). I will rewrite it as
√
c)
1
F T1
i tan−1 ( ω−ωκc /2 .
qe
T (ω) =π
1 + ( ω−ωc )2(4.15)

κc /2

The carrier of the field described by Eq. (4.14) after injected into the cavity becomesω −ω
1i tan−1 ( κ0 /2c )cEc,0 = q
Ei,0 eiω0 t .
eω0 −ωc 2
1 + ( κc /2 )(4.16)The two sidebands becomes

Ec,+

=

ω +ωs −ωcβ1i tan−1 ( 0 κ /2)cqe
Ei,0 ei(ω0 +ωs )t
2 1 + ( ω0 +ωs −ωc )2(4.17)

κc /2

ω −ωs −ωc1βi tan−1 ( 0 κ /2)ce
Ei,0 ei(ω0 −ωs )t .
=− q
2 1 + ( ω0 −ωs −ωc )2
κc /2

Ec,−

(4.18)In the limit ω0 − ωc  κc /2 and ω0 − ωc  ωs , we get

Ec,0
Ec,+
Ec,−
E

/2
κc /2 i( π2 − ωκc−ω
)
iω0 t
c E
0
e
i,0 e
ω0 − ωc
κc /2
β
κc /2
i( π −)=
e 2 ω0 +ωs −ωc Ei,0 ei(ω0 +ωs )t2 ω0 + ωs − ωc
κc /2
β
κc /2
i( π −)=−
e 2 ω0 −ωs −ωc Ei,0 ei(ω0 −ωs )t .
2 ω0 − ωs − ωc=E

We could then express Ec,+and Ec,−as follows
c,0c,0(4.19)(4.20)(4.21)

67

Ec,+
Ec,0
Ec,−
Ec,0
E

ωs κ/2βωsi)e (ω0 −ωc )2 eiωs t
ω0 − ωc2
ωs κ/2ωsβ−i= −(1 +)e (ω0 −ωc )2 e−iωs t .
ω0 − ωc2= (1 −(4.22)(4.23)

E

If we decompose Ec,+
+ Ec,−
into amplitude modulation and phase modulation basis, with the formc,0c,0Ec,+ Ec,−+= (P ei(ωs t+ψ) − P e−i(ωs t+ψ) ) + (Aei(ωs t+η) + Ae−i(ωs t+η) ),(4.24)
Ec,0
Ec,0
q
1
2π∆νdω
s
we find that 2A = −β ω0ω−ω=−π
ω0 −ωc . In the limit 2A is small, the time-dependent formcof the amplitude modulation component of the cavity field can be written as

Ea (t) = Ei,0 (1 + Σs 2A cos(ωs t + φs )),(4.25)

where we sum over noise frequency ωs which goes from 0 to ∞ with step size dω.
For a two-photon transition driven by two tones generated from phase modulations, theinstantaneous two-photon Rabi frequency Ωtp (t) follows the relationship shown in Eq. (4.26),Ωtp (t) ∝ (1 + Σs 2A cos(ωs t + φs ))(1 − Σs 2A cos(ωs t + φs )).

(4.26)

If we denote ensemble average with hi, time average with¯and assume that the frequency ofthe phase modulated carrier is perfectly centered between two adjacent cavity modes, we get theform of the amplitude noiseqΩ̄2tp − Ω̄tp
Ω̄tp

q

2

=

sT )
Σs (A)4 1−cos(2ω(T ωs )2

1 − 2A2,(4.27)

where T is the π-time of the two-photon Rabi flopping, which is the period that the time-average
is done over.
Notice that only the term containing A4 ∝ (dω)2 survives, so the amplitude noise converted
from phase noise is also subject to first order cancellation.

Chapter 5

Laser Cooling with Adiabatic Transfer on Raman Transitions

In this chapter, I will review a 1D sub-Doppler cooling technique via adiabatic transfer on
Raman transitions. This technique was inspired by Sawtooth Wave Adiabatic Passage (SWAP)
laser cooling demonstrated using a narrow-linewidth single-photon optical transition in 88 Sr [81, 3],
and in 163 Dy [83]. Here we demonstrate that two-photon Raman transitions with arbitrarily-tunable
linewidths of 87 Rb can be used to achieve 1D SWAP cooling without significantly populating theintermediate excited state. Unlike SWAP cooling on a narrow transition, Raman SWAP coolingallows for a final 1D temperature well below the Doppler cooling limit (here, 25 times lower), andthe effective excited state decay rate can be modified in time, presenting another degree of freedom
during the cooling process. Here, I reproduce the work published in ref. [41] with little modification.

5.1

Introduction
Advances in laser cooling techniques have opened new scientific vistas for neutral atoms, ions,and mechanical resonators [22, 32, 18]. Doppler cooling techniques have been widely applied to
produce atoms at mK to sub-µK temperatures and high phase-space density [100, 34, 54, 67, 103, 5,
104, 58, 6]. However, standard Doppler cooling is limited in both final temperature and maximum
force by the linewidth Γ and wavelength λ ≡ 2π/k of the available optical transitions – properties
that are provided by nature and not under control of the experimentalist. Understanding methods
and limitations for removing entropy from a system is of fundamental interest and continues to be
widely explored [102, 18, 50, 25, 88, 84, 35].

69
Recently, a new mechanism called Sawtooth Wave Adiabatic Passage (SWAP) cooling wasobserved using a narrow-linewidth optical transition in 88 Sr atoms [81], and a related deflection
force was previously reported in He atoms [73]. The forces that give rise to SWAP cooling rely on
adiabatic transitions back and forth between a ground state |ai and a long-lived optically excitedstate |bi with lifetime τ = 1/Γ. This allows for many photon-recoils worth of momentum reduction
before spontaneous emission occurs, but so far, SWAP cooling has been limited to single-photontransitions.
In this chapter, we present proof-of-principle experiments in 87 Rb to demonstrate that the
SWAP cooling mechanism does not require a closed, single-photon transition. Rather, it is amenable
to atoms and molecules with at least two long-lived ground states. The core idea is to dress theseground states (here labeled |ai and |bi) using externally applied lasers tuned off-resonance from anintermediate optically excited state |ii. This permits us to engineer effective optically excited stateswith tunable lifetimes or, equivalently, linewidths instead of relying on the properties of the opticaltransitions |a, bi ↔ |ii (Fig. 5.1(a)). Unlike the previous report of SWAP cooling with 88 Sr, we~Γachieve an equilibrium temperature in 1D that is 25 times lower than the usual TD ≈ 2k
= 146 µK
B

cooling limit for standard Doppler cooling in 87 Rb.
In SWAP cooling using a single-photon transition, the magnetic field gradients, laser directions, and polarizations are essentially identical to those of standard Doppler cooling except thatthe pair of counter-propagating laser beams are ramped in frequency in a sawtooth pattern frombelow to above the transition frequency ωa (Fig. 5.1(c and e)). The relative Doppler shift of the twolaser beams causes the beam counter-propagating to the atom’s motion to pass through resonancebefore the co-propagating beam. As a result, the counter-propagating beam drives an adiabatic
transition from |ai to |bi along with a momentum kick due to photon absorption that opposes theatomic motion. The co-propagating beam then drives an adiabatic transition back from |bi to |aialong with a momentum kick due to stimulated emission that, again, slows the atom. In net, each
sweep ideally removes 2~k of momentum, reducing the atom’s speed regardless of the direction it
is moving.

a)

b)

c)

70

d)
e)

Figure 5.1: Comparison between Single-photon and Two-photon SWAP Cooling. (a) and (b)SWAP cooling level diagram for a single-photon narrow-linewidth transition and a two-photon
Raman transition. In the Raman scheme, the large detuning ∆ from the intermediate state ensuresthe effective lifetime of the excited state |bi is effectively infinite except when it is useful to induce
decay back to |ai following each sweep. (c) In the single-photon case, the ω2 laser (red) is counterpropagating and incident upon an atom (green). (d) The two-photon case is identical except asecond ω1 laser (blue) serves to dress the |bi state. (e) The frequency of ω2 is sawtooth-swept
through the atomic transition frequency.

In SWAP cooling using a two-photon transition, the frequency of the laser dressing the
ground state |bi is held fixed (figure. 5.1(d)). The laser dressing the ground state |ai is swept in
an asymmetric sawtooth pattern through a two-photon resonance, driving adiabatic two-photon
Raman transitions between the states |ai and |bi. At the end of the sweep, optical pumping isbriefly applied to transfer atoms erroneously remaining in |bi back to |ai. In comparison to thework in strontium [81], this is equivalent to being able to set Γ ≈ 0 during the frequency sweep,
but then setting Γ up to the lifetime of the intermediate state for a very brief period of time inbetween sweeps, potentially offering a different degree of freedom for optimizing cooling.

5.2

Laser Cooling 87 Rb with Adiabatic Transfers
Fig. 5.2 shows the experimental setup for demonstrating 1D Raman SWAP cooling in 87 Rb.

The quantization axis is established by a uniform magnetic field applied along the propagation axisof the cooling beams. The two ground states are the hyperfine Zeeman states |ai ≡ |F = 2, mF = 0i
and |bi ≡ |F = 1, mF = 0i. These states are coupled by lasers at frequencies ω1 and ω2 , both far-

a)

71

c)

OP

sweep

into

OP

sweep

into

1
Frequency in atom
rest frame [MHz]

0

-1

b)

AOM

d)

1
Atomicpopulation
0

Magnetic field /quantization axis
Atom, velocity

Polarizatione)

Averagemomentum0

1time [ms]Figure 5.2: Experimental Setup and Cooling Procedure. (a) 87 Rb level diagram. Two-photontransitions between |ai and |bi are induced by ω1 (blue) and ω2 (red). (b) Experimental layout. A
moving atom (green) interacts with two pairs of counter-propagating laser beams (blue, red) with
ω2 varied in a saw-tooth manner. (c) The four unique laser frequencies observed in the rest frameof the atom after including the Doppler shifts that separate counter-propagating lasers in frequency
by 2kv. Offsets have been subtracted so that points marked with circles correspond to allowed twophoton resonances which involve pairs of orthogonally-polarized, counter-propagating laser beams.
(d) Two-photon Landau-Zener transitions transfer an atom from |ai to |bi and back to |ai each
sweep. (e) The counter-propagating lasers and sweep direction ideally remove 4~k momentum percooling sweep.

detuned from the intermediate state |ii by an amount ∆ ≈ 2π × 2 GHz. The allowed two-photontransitions involve absorption and stimulated emission of pairs of orthogonal linearly polarized
photons differing in frequency by the ground state hyperfine splitting of ωHF = 6.834 GHz. The
cooling beams incident from the left are vertically polarized and the beams incident from the rightare horizontally polarized. Dipole selection rules disallow two-photon transitions with pairs ofphotons of the same linear polarization, i.e., the only allowed two-photon Raman transitions arethose that also impart a net photon recoil momentum 2~k as the internal state changes.
Both beam directions have two distinct frequency components ω1 and ω2 (Fig. 5.2(b)) createdby combining the output of two phase-locked lasers. We choose to hold the frequency ω1 fixed
while the frequency component ω2 is swept linearly in time downward in frequency through the72
two-photon atomic resonance at ω1 − ω2 ≈ ωHF .
Accounting for Doppler shifts of the laser frequencies as seen by an atom moving at speed v,
there are two two-photon resonances that occur when δ(t) ≡ ω1 − ω2 − ωHF = ±2kv (Fig. 5.2(c)).
Ideal adiabatic passage through each resonance imparts a net momentum kick of 2~k per transition,i.e., 4~k per sweep. The direction of the frequency sweep is chosen such that the time ordering of
the passage through the two resonance frequencies leads to a reduction of the atom’s speed in thelaboratory frame.
As the atom approaches zero speed and the Doppler shift is of the same order as Ωab , the time1 Ω2ordering of the adiabatic transfers detailed above is invalid. Here, Ωab ≈ Ω2∆is the two-photon

Rabi frequency and Ω1 ≈ Ω2 are the single photon Rabi frequencies of each frequency component.
With the inevitable failure of a transfer, either due to broken time-ordering or imperfect adiabatic
transfer, an atom can have considerable probability to be in |bi following a sweep so that the next
iteration results in heating. We apply π-polarized optical pumping light for 100 µs to return any
atoms remaining in |bi to |ai before the next frequency sweep. Optical pumping light is applied
from a direction orthogonal to the cooling beams, and the duration could have been decreased.
In our experiment, around 107 atoms are loaded into a magneto-optical trap (MOT) andpre-cooled to about 42 µK with polarization gradient cooling (PGC) to narrow the initial velocity
distribution and lessen the requirements on the sweep’s effective capture range. The atoms are
optically pumped into |ai. Frequency sweeps are then applied with typical sweep times of 1 ms
and sweep range 1 MHz as shown in Fig. 5.2(c). The ideal internal state populations and change
in momentum are shown in Fig. 5.2(d and e). This sequence of optical pumping and sweeping isrepeated several times. The final 1D temperature of the atoms is measured by velocimetry. All
atoms are optically pumped back to |ai and then velocity-selective Raman transitions drive atoms
within a small velocity range into |bi [17]. The population in |bi is determined using fluorescenceand the resulting Voigt profiles are fit to extract the temperature (Fig. 5.3(a)).
We observe cooling from an initial temperature of 42(3) µK to 10 µK after the applicationof 9 sweeps with fixed two-photon Rabi frequency Ωab = 2π × 22 kHz and sweep range of ∆swp ≡

73

0
1
11 μK
(4 sweeps)

01

Temperature [μK]42 μK
(0 sweeps)

0
-0.3

0

0.3

Detuning of velocity-selective
Raman beams [MHz]

c)

40
1.0

30
20

0.5
10
0

6 μK
(9 sweeps)

b)

1.5

0

2

4

6

8

0

2

Number of sweeps

d)

4

6

Relative phase spacedensity [ρ/ρ0]50

Velocity [mm/s]

Atoms [arbitrary units]

a) 1

0

8

Number of sweeps

50
0
-50
0

2

4

6

8

10

Number of Sweeps

Figure 5.3: Performance of cooling. (a) The reduction in temperature is determined by measuring
the initial velocity distribution (red) and the distribution after four and nine sweeps (purple andblue). The velocity distributions shown here are determined by driving Raman transitions thatare only resonant for atoms in a narrow velocity-class whose center is set by the detuning of theRaman beams. (b) The atomic ensemble is cooled as low as 5.9 µK in one dimension over thecourse of several sweeps. For the purple points, all sweeps used a two-photon Rabi frequencyΩab ≈ 2π × 22 kHz and sweep range ∆swp = 2π × 0.8 MHz. For the green points, Ωab and ∆swp
were successively reduced between sweeps as described in the text. (c) The relative one-dimensional
phase-space density for the corresponding purple or green points in (b). The phase-space density
begins to decrease after four sweeps despite the decreasing temperature because atoms begin toleave the velocimetry beams. (d) To cool into a moving reference frame (blue points), we apply afrequency offset δAOM = 2π × 150 kHz in the lab reference frame between the beams from opposite
directions. By sweeping ω2 downwards, the atoms cool and equilibrate into a moving reference
frame which has a velocity vF ≡ δAOM /2k (dashed orange line). If the sweep direction is reversed(pink points), the atoms accelerate in the opposite direction without being bound by |vF |.

2π × 0.8 MHz (purple points in Fig. 5.3(b)). The sweep range is chosen to allow over 95% of theatoms to pass through two-photon resonance during the sweep when accounting for the Doppler
shifts of the initial velocity distribution of the atoms.
The temperature can be reduced by progressively decreasing the two-photon Rabi frequency
and the sweep range, as is done for the green points in Fig. 5.3(b). After the first three sweeps√(purple points), the Rabi frequency is reduced to 0.5 × Ωab and the the sweep range is reduced to
0.5 × ∆swp such that the Landau-Zener adiabaticity parameter ξ (discussed below) is unchanged.

74
The sweep range can be reduced because the velocity distribution was reduced by the initial sweeps.
√
After two sweeps, the remaining sweeps are performed with two-photon Rabi frequency 0.4 × Ωaband sweep range 0.4 × ∆swp . After 9 total sweeps, the measured temperature reaches 5.9(3) µK.
An increase in phase-space density demonstrates a reduction of entropy and not merely aselective loss of atoms or a redistribution of the density in phase space. In Fig. 5.3(c), we see thatthe the relative 1D phase space density ρ/ρ0 = ∆x0 ∆v0 /(∆x∆v) is increased, where ∆x and ∆v(∆x0 and ∆v0 ) are the measured cloud size and velocity spread after (before) cooling.
Although the polarization scheme is reminiscent of polarization gradient cooling [30], here
there is a large magnetic field present that breaks the degeneracy of the ground states that is
typically required for polarization gradient cooling to such a low temperature. In addition, when
the sweep direction was reversed, we observed heating as expected in the SWAP model of cooling.
To further emphasize the critical role of the sweep direction in the present work, we apply afixed relative offset frequency δAOM between the counter-propagating beams such that one would
expect that atoms are cooled into a moving reference frame with velocity vF = δAOM /2k. If the laser
frequency is swept downward as was done for the cooling experiments above, we observe that theatoms are accelerated into and equilibrate into the moving frame (blue points in Fig. 5.3(d)). Theorange dashed line indicates the velocity of the predicted moving frame for the applied frequency
offset δAOM = 2π × 150 kHz. In contrast, if we simply reverse the frequency sweep direction, we
observe that the atoms are accelerated in the opposite direction despite the direction of the moving
reference frame remaining unchanged (pink points of Fig. 5.3(d)). The atoms are accelerated to
speeds larger than the calculated |vF | as expected.

5.3

Adiabatic Transfer in the Presence of Scattering
It is important to understand the possible limitations of using Raman transitions associatedwith the sweep rate α ≡ dω2 /dt being too fast or too slow. To achieve high quality adiabatic
transfer, one would like to sweep slowly such that the ideal Landau-Zener diabatic transition probability is small, Pd = e−ξ  1 where ξ = π2 Ω2ab /α. To avoid off-resonant spontaneous scattering of

75
light from the applied cooling laser beams, one would like the probability to have not scattered aphoton during the total sweep time to be close to unity, Psc = e−Rsc ∆swp /α ≈ 1. Rsc is the total
spontaneous scattering rate from the far-from-resonance intermediate state(s).
In Fig. 5.4(a) (circles), we measure the probability to successfully transfer from |ai to |bi
after a single adiabatic transfer using σ + polarized beams from a single direction and such thatthe two-photon Rabi frequency is the same as for the data in Fig. 5.3(b). The data shows that
there is an optimum sweep rate α that maximizes the transfer efficiency as desired for efficientcooling. For comparison, the red dashed line is the predicted transfer efficiency 1 − Pd ignoring freespace scattering. The measurements qualitatively match predictions from numerically integrating
optical-Bloch equations including spontaneous emission (Fig. 5.4(a) orange).

0.8
0.6
0.4
0.2
0
0.1

1

10

Laser frequency sweep rate

100[MHz/ms]

Maximum Δp per sweep
for SWAP cooling [ħk]Population in after
one adiabatic sweep

b)

1.0

4

80

3

60

2

40

1

20

0

1

10

Quality parameter Q

100

0

Δp|1/e : momentum transferred
when 1/e molecules remain [ħk]a)Figure 5.4: Characterization of adiabatic transfer and discussion of quanlity parameter Q. (a)Population in |bi after one adiabatic transfer attempt, measured using fluorescence detection (black
points). Here, transitions were driven using co-propagating σ + -polarized light with frequencies ω1and ω2 . ω1 was constant and ω2 was swept through resonance at different sweep rates α = dω2 /dt.
Transfers were performed with (Ωab , Γ, ∆, ∆swp ) = 2π × (20 kHz, 6 MHz, 2 GHz, 0.8 MHz). The
familiar Landau-Zener prediction (red dashed) fails at low sweep rate due to off-resonant scatteringfrom the intermediate state |ii. A numerical simulation using the experimental parameters andincluding off-resonant scattering shows qualitative agreement (orange) with the data. (b, left) The
achievable momentum transfer per SWAP cooling sweep (blue dashed) approaches the ideal 4~k2
at large quality parameter Q ≡ π2 Γ∆Ωswp . (b, right) For SWAP cooling of molecules, we consider aworst-case scenario in which a molecule is lost if it undergoes a single spontaneous emission event.
The predicted achievable momentum transfer when 1/e molecules remain (green) is shown versus
the quality parameter Q.

To generalize predictions for how scattering will limit SWAP cooling, we consider the threelevel system shown in Fig. 5.1(b). We will assume that the intermediate state |ii decays with equal

76
rates of Γ/2 into both |bi and |ai. We also assume Ω1 = Ω2 = Ω, and that the large detuning limit2Ω∆  Γ, Ω is satisfied. The two photon Rabi frequency is then Ωab = 2∆
, and the total scattering
2rate is Rsc = ΓΩ. More details about the following treatment are provided in the Sec. 5.5.
4∆2
We use this simplified system to find the optimum momentum transferred ∆p during a coolingsweep after optimizing for sweep rate α. In Fig. 5.4(b, blue), we plot ∆p against a dimensionless2

quality parameter Q ≡ ln Pd / ln Psc ≈ π2 Γ∆Ωswp . When Q  1, the momentum transfer per sweep
saturates to the ideal value 4~k, and the effective force is 4~k/tswp , where the time to complete2∆each sweep is tswp . At the optimized sweep rate, tswp = 8 ln(2Q)∆ . In terms of experimentallyπΩ4 swpcontrollable parameters, the quality factor scales with laser intensity I and wavelength as Q ∝ Iλ3 ,but does not depend on the dipole matrix element M between the states |bi, |ai and the intermediate2

state |ii. The sweep time scales as roughly tswp ∝ M∆4 I 2 ∆swp .
For cooling molecules, where avoiding spontaneous emission is of chief importance, one must
understand how much momentum can be removed before the molecule is lost. We take the worst
case scenario, where every molecule that spontaneously emits a photon is completely lost. In
Fig. 5.4(b, green), we plot ∆p|1/e , the average momentum transfer when 1/e molecules remain,
again numerically optimizing the sweep rate. For Q  1, the numerical result is well approximated
2.1Q
by ∆p|1/e ≈ ln(4(Q+14))~k. By engineering systems with high quality parameter Q, one can remove

many photon recoils of momentum from a molecule before it is likely to be lost. The optimized
2∆
sweep time is nicely approximated by tswp ≈ ln(4(14+Q))∆ .
3.2Ω4 swp5.4

Conclusion
We have demonstrated that Raman transitions may be employed for SWAP cooling atoms

without a single-photon narrow transition, achieving final temperatures well below the Doppler
cooling limit. The technique is straightforward to implement, is amenable towards working in thepresence of a large magnetic field, is robust against small changes in atomic transition frequency,
and might prove useful for cooling molecules. Future work may look towards using the techniqueto cool in more dimensions or to manipulate ensembles via accelerations and decelerations. More

77
complex waveforms for the laser intensity and detuning could potentially decrease the required
sweep range, increasing the effective cooling rate [4]. We have also identified a quality parameterQ that provides guidance as to what experimental systems are needed for Raman SWAP coolingto work efficiently.

5.5

Discussions
As discussed in the main text, adiabatic transfer in the presence of scattering is optimized

by a balance between the need to sweep slowly to preserve adiabaticity and the need to sweep fastenough to avoid significant scattering. Numerical simulations were used to better understand the
dynamics and limitations of adiabatic passage. Here we consider a slightly simplified model: thisatom (Fig. 5.5(a)) has stable ground states |ai and |bi and an optically excited state |ii that decays
into them with equal probability at total rate Γ. The energy of these states are ~ωa , ~ωb , and ~ωi .
A laser at frequency ω1 (t) couples |ai ↔ |ii with single-photon Rabi frequency Ω1 , and a laser at
frequency ω2 (t) couples |bi ↔ |ii with Rabi frequency Ω2 . The average detuning ∆ of these lasers
from the excited state is large compared to the two-photon detuning δ(t).
b)
Population

a)

1.0

0.5

0.0

δ [∆swp]

0.5
0
-0.5
0.0

0.25

0.50

0.75

1.0

Time [∆swp ⁄ α]

Figure 5.5: Simulated time dynamics of adiabatic transfer on a three-level system. (a) Diagram
for simplified three-level system. An excited state |ii spontaneously decays equally to |ai and |bi
at total decay rate Γ. Lasers connect |ai and |bi to a state detuned from |ii by ∆. (b) Simulated
time dynamics of adiabatic transfer, ignoring momentum states. In units of the scattering rate, thesweep range is ∆swp = Γ/3, ∆ = 300Γ, Ωab = Γ/150. The scattering rate Γ is responsible for thecoherence decay.

78
5.5.1

Optical Bloch Equations and Adiabatic Elimination for a Three-Level Raman
SystemThe density matrix describing this three-level system is ρii ρia ρib ρ = ρai ρaa ρab ρbi ρba ρbb
and the Hamiltonian for this system can be written H = HA +V1 +V2 , with the interaction between
an atom and a field provided by
Z t
V1 = −d · E1 cos

0

0



ω1 (t )dt
0

 Rt

 Rt

R
R
~Ω1 
−i 0 ω1 (t0 ) dt0
i 0t ω1 (t0 ) dt0−i 0 ω1 (t0 ) dt0
i 0t ω1 (t0 ) dt0|ii ha| e+e+ |ai hi| e
+e
=
2

Rt
Rt
~Ω1 
0
0
0
0
≈
|ii ha| ei 0 ω1 (t ) dt + |gi hi| e−i 0 ω1 (t ) dt
2

Rt
Rt
~Ω2 
0
0
0
0
V2 ≈
|ii hb| ei 0 ω2 (t ) dt + |bi hi| e−i 0 ω2 (t ) dt
2

andafter making the rotating wave approximation, with ~Ω1 ≡ − hg | d · E1 | ii and ~Ω2 ≡ − he | d · E2 | ii
where
δ(t)
2
δ(t)
ω2 (t) = ωi − ωb + ∆(t) −2
ω1 (t) = ωi − ωa + ∆(t) +such that ∆(t) is the average detuning of the lasers from their respective transitions. In our
experiment and simulations, δ(t) = αt −∆swp
is swept linearly in time.
2

From the Liouville equation, ρ̇ = ~i [ρ, H] − γρ, with the dissipation term γ representing population
relaxation, the equations of motion for the coherences are
Rt
Rt


0
0
0
0
Γ
iΩ1 e−i 0 ω1 (t ) dtiΩ2 e−i 0 ω2 (t ) dt
ρ̇ia = − i(ωi − ωa ) +ρia +(ρii − ρaa ) −
ρba ,
2
2
2
Rt
Rt


0
0
0
0
Γ
iΩ2 e−i 0 ω2 (t ) dt
iΩ1 e−i 0 ω1 (t ) dt
ρ̇ib = − i(ωi − ωb ) +ρib +(ρii − ρbb ) −
ρab ,
2
2
2
Rt

0

0

Rt

0

(5.1)
(5.2)

0

iΩ1 e−i 0 ω1 (t ) dtiΩ2 ei 0 ω2 (t ) dt
ρ̇ba = −i (ωb − ωa ) ρba +
ρbi −
ρia ,
2
2

(5.3)

79
and for the populations, because the branching ratios from the intermediate state are balanced,
ρ̇ii =

 iΩ  R t

Rt
RtiΩ1  i R t ω1 (t0 ) dt0
0
0
0
0
0
0
2
ρia − e−i 0 ω1 (t ) dt ρai +e 0
ei 0 ω2 (t ) dt ρib − e−i 0 ω2 (t ) dt ρbi − Γρii ,
2
2
(5.4)

 Γ
Rt
iΩ1  −i R t ω1 (t0 ) dt000
e 0
ρai − ei 0 ω1 (t ) dt ρia + ρii ,
2
2
 Γ
Rt
iΩ2  −i R t ω2 (t0 ) dt000ρ̇bb =e 0
ρbi − ei 0 ω2 (t ) dt ρib + ρii .
22ρ̇aa =

(5.5)
(5.6)

To enter the rotating (natural) frame, the transformation to “slow” variables is used:
Rt

0

0

Rt

0

0

ρia = ρ̃ia e−i 0 ω1 (t ) dt ,
ρib = ρ̃ib e−i 0 ω2 (t ) dt ,
Rt

0

0

0

ρba = ρ̃ba e−i( 0 ω1 (t )−ω2 (t ) dt )Substituting these into Eqns. 5.1-5.3, the coherences in this frame become ˙ρ̃ia = i ∆(t) + δ(t) − Γ ρ̃ia + iΩ1 (ρii − ρaa ) − iΩ2 ρ̃ba ,
2
2
2
2


 
Γ
iΩ2
iΩ1
δ(t)
−
ρ̃ib +(ρii − ρbb ) −ρ̃ab ,
ρ̃˙ ib = i ∆(t) −
2
2
2
2
iΩ1
iΩ2
ρ̃˙ ba =ρ̃bi −
ρ̃ia + iδ(t)ρ̃ba .
2
2
The populations of Eqns. 5.4-5.6 can be rewritteniΩ1iΩ2(ρ̃ia − ρ̃ai ) +(ρ̃ib − ρ̃bi ) − Γρii ,
2
2
iΩ1
Γ
ρ̇aa =(ρ̃ai − ρ̃ia ) + ρii ,
2
2
iΩ2
Γ
ρ̇bb =(ρ̃bi − ρ̃ib ) + ρii .
22ρ̇ii =

(5.7)
(5.8)
(5.9)

We now reduce the equations of motion to an effective two-level system by adiabaticallyeliminating ρia and ρib . This is justified when the population of the intermediate state is small,which is valid when ∆  Γ, δ. The time derivatives ρ̃˙ ia , ρ̃˙ ii , and ρ̃˙ ib are set to zero, and henceiΩ1iΩ2(ρii − ρaa ) −ρ̃ba ,
Γ + 2i∆(t) − iδ(t)Γ + 2i∆(t) − iδ(t)Ω2Ω1ρ̃ib =(ρii − ρbb ) −ρ̃ab .
Γ + 2i∆(t) + iδ(t)Γ + 2i∆(t) + iδ(t)ρ̃ia =

80
These coherences are substituted into Eqns. 5.7-5.9, and after transforming back into theoriginal coordinates, we arrive at the final set of Bloch equations


Γρii
ρ̃ba
ΓΩ21 (ρii − ρaa )
Ω1 Ω2
ρ̃ab
ρ̇aa =,+ 2−+2
Γ + (2∆(t) + δ(t))22
Γ − 2i∆(t) − iδ(t) Γ + 2i∆(t) + iδ(t)


Γρii
Ω1 Ω2
ρ̃ba
ΓΩ22 (ρii − ρbb )ρ̃abρ̇bb =−,+ 2+2
Γ + (2∆(t) − δ(t))22
Γ + 2i∆(t) − iδ(t) Γ − 2i∆(t) + iδ(t)


Ω21
Ω22
˙ρ̃ba = iδ(t)ρ̃ba − ρ̃ba+2
Γ + 2i∆(t) − iδ(t) Γ − 2i∆(t) − iδ(t)ρbb − ρii
Ω1 Ω2
ρaa − ρii−+2
Γ + 2i∆(t) − iδ(t) Γ − 2i∆(t) − iδ(t)
with
ρii ≈

Ω21 ρaa + Ω22 ρbb + Ω1 Ω2 ρ̃ba + Ω1 Ω2 ρ̃ab.
Γ2 + 4∆(t)2 + Ω21 + Ω22(5.10)

Equation 5.10 has been greatly simplified by the assumption that δ(t)  ∆(t), Ω1 , Ω2 .

5.5.2

Including Internal Momentum States

We now include the momentum states of an atom, using a second label such as |a, 0~ki. The
basis of states is truncated to only include 0~k, 1~k, 2~k, . . . , (m − 1)~k momentum states, so thedensity matrix becomes 2m × 2m. Terms that are non-physical, involving a momentum transfer
with no change of state, can also be negated:



..

 .


. . .
. . .

ρ = . . .
. . .
. . .



..
.

..
.

..

.

..
.

..
.

ρaa00

0

0

. . . ρab00

ρab01

ρab02

0

ρaa11

0

ρab10

ρab11

ρab12

0
..
.

0

ρaa22 . . . ρab20
..
..
..
.
.
.

ρab21

ρab22
..
.

0

0

ρba00 ρba01 ρba02 . . . ρbb00
ρba10 ρba11 ρba12

0

ρbb11

0

ρba20 ρba21 ρba22 . . .
..
..
..
.
.
.

0
..
.

0

ρbb22
..
.





. . .. . .


.. 
.



. . .. . .

.. 
.

81
Normal adiabatic passage from |a, 0~ki through |i, 1~ki to |b, 2~ki is represented in the matrix
elementsΓρii00ρ̃ba20ΓΩ21 (ρii11 − ρaa00 )Ω1 Ω2
ρ̃ab02ρ̇aa00 =
,
+ 2
−
+
2
Γ + (2∆(t) + δ(t))22
Γ − 2i∆(t) − iδ(t) Γ + 2i∆(t) + iδ(t)


Ω1 Ω2
ρ̃ba20Γρii22ΓΩ2 (ρii11 − ρbb22 )ρ̃ab02−,
ρ̇bb22 =
+ 2 2
+
2
2
Γ + (2∆(t) − δ(t))
2
Γ + 2i∆(t) − iδ(t) Γ − 2i∆(t) + iδ(t)


Ω21
Ω22
˙ρ̃ba20 = ρ̃ba20 2iδ(t) −−2
Γ + 2i∆(t) − iδ(t) Γ − 2i∆(t) − iδ(t)


Ω1 Ω2
ρbb22 − ρii11
ρaa00 − ρii11−+2
Γ + 2i∆(t) − iδ(t) Γ − 2i∆(t) − iδ(t)withρii11 ≈

Ω21 ρaa00 + Ω22 ρbb22 + Ω1 Ω2 ρ̃ba20 + Ω1 Ω2 ρ̃ab02.
Γ2 + 4∆(t)2 + Ω21 + Ω22

Other momentum states are coupled in due to the scattering terms. For example, |b, 2~kimay absorb a photon, losing ~k of momenta, and enter |a, 1~ki. This state may in turn decay
incoherently to |a, 1~ki or |b, 1~ki.
Because the momentum states are now distinguishable within the density matrix, we canseparate atoms that undergo adiabatic passage without scattering from other possibilities (Fig. 5.6).
We can also construct expectation values for the momentum change.
1.0

Population

0.8
0.6
0.4
0.2
0.0
0.0

0.25

0.50

0.75

1.0

Time [∆swp ⁄ α]

Figure 5.6: Simulation of populations including momentum labels during adiabatic transfer. The
parameters used match those of Fig. 5.5(b). The expected change in momentum at the end of theadiabatic transfer is 1.71~k.

A Runge-Kutta fourth-order numerical integration method is used to simulate adiabatic
transfer (Fig. 5.5(b)). In practice, we find a 10 × 10 density matrix is sufficient, and population

82
starts in |a, 1~ki (ρaa11 ). When determining final populations, oscillations in the simulation canbecome significant, so we take the average of during the last 1/30th of time steps. This leads to aslight error due to the average slope during this time. To find the expected change of momentum,
we compute"h∆p(Γ, ∆, Ω, ∆swp )AT i =#Xn ~k (ρbbnn + ρaann ) − 1~k.

nFinally, this model ignores effects such as laser noise and atomic dephasing which can set
substantial limitations on transfer fidelity [63, 78]. In Fig. 5.4(a), a separate two-level simulation
was used which allowed us to enter this dephasing by hand. However, these details are beyond thescope of this model.

5.5.3

Probabilitistic Model for Raman SWAP Cooling

Adiabatic transfer from |ai to |bi adds 2~k momentum from an atom, and transfer from |bi
to |ai removes 2~k momentum. If we assume that during a scattering process, absorption from|ai adds one photon’s worth of momentum, and absorption from |bi removes one photon’s worth
momentum, but decay into either state causes no coherent momentum change, then to first-order,
the possible change in momentum is detailed in Table. 5.1. The first-order expected change inmomentum, from summing the tabulated momentum changes multiplied by the probability of that
trajectory, is found to beh∆pAT i = 1 + x − 2x1+Q × ~kwith x ≡ e−Rsc

∆swp
α

2

2

2

ΓΩ
ΓΩ
π Ω
, Rsc = Γ2 +4∆
2 +2Ω2 ≈ 4∆2 , and Q ≡ 2 Γ∆swp . Numerical simulations of theprevious section serve to validate this model exceptionally well.
The momentum transfer during a full SWAP cooling sweep comes from cooling during thefirst half of the sweep, and cooling or heating during the second half depending on if an atom
successfully transferred from |ai to |bi. The expected momentum transfer during a full cooling

Row
1
2
3
4
5
6
7
8
9
10
11
12

Before T /2
Scatter to |bi
Scatter to |bi
Scatter to |bi
Scatter to |bi
Scatter to |ai
Scatter to |ai
Scatter to |ai
Scatter to |ai
No scatter
No scatter
No scatter
No scatter

At T /2
Transfer to |ai
Transfer to |ai
No transfer
No transfer
Transfer to |bi
Transfer to |bi
No transfer
No transfer
Transfer to |bi
Transfer to |bi
No transfer
No transfer

After T /2
Scatter
No scatter
Scatter
No scatter
Scatter
No scatter
Scatter
No scatter
Scatter
No scatter
Scatter
No scatter

Probability
1
2 SAS
1
2 SA(1 − S)1
2 S(1 − A)S
1S(1− A)(1 − S)
2
1
2 SAS
1
SA(1
− S)
2
1
S(1
−
A)S
2
1
S(1
−
A)(1
− S)
2
(1 − S)AS(1 − S)A(1 − S)
(1 − S)(1 − A)S(1 − S)(1 − A)(1 − S)

∆p
0 ~k
−1 ~k
0 ~k
1 ~k
2 ~k
3 ~k
2 ~k
1 ~k
1 ~k
2 ~k
1 ~k
0 ~k

83

Table 5.1: First-order possibilities for atom state trajectories. An atom starts in |ai and may
scatter before time T /2. At time T /2, adiabatic transfer may occur. From T /2
 to T ,the atom
Ω2

may scatter again. The probability to undergo adiabatic transfer is A ≡ 1 − exp − π2 αab , and the


∆swp
ΓΩ2
ΓΩ2
probability to scatter is S ≡ 1 − exp −Rsc 2αwith scattering rate Rsc = Γ2 +4∆
2 +2Ω2 ≈ 4∆2 .

sweep is then h∆pSWAP i = h∆pAT i (1 + (P (e) − P (g))). The fraction of atoms cooled vs. heatedin the second half of the sweep is represented by (P (e) − P (g)), where P (a) is the probability to
be in state |ai at the conclusion of the first half of the sweep. We findh∆pSWAP i = h∆pAT i 1 + x(1 − 2xQ ) .

(5.11)

To evaluate the benefits of SWAP cooling over Doppler cooling, we would like to know how
much momentum could be removed before an atom (or molecule) detrimentally scatters a photon.
Although we assumed equal branching ratios up until now, suppose that every scattering event
causes an atom to be lost. We assume each atom scatters one photon from optical repumping if
it finishes the cooling sweep in |bi. The expected change in momentum per sweep comes entirely
from the events of Table. 5.1 row 10 and 12:∆p0AT = (1 − S)A(1 − S) × 2~k = x(1 − xQ ) × 2~k.
To be consistent, we need to normalize this to the number of atoms remaining in the system:

84h∆pAT i=h∆p0AT i(1 − S)A(1 − S) + (1 − S)(1 − A)(1 − S)

= (1 − xQ ) × 2~kh∆pSWAP i= h∆pAT i × [1 + ((1 − S)A(1 − S) − (1 − S)(1 − A)(1 − S))]= 1 + x − xQ − 3x1+Q + 2x1+2Q × 2~kNext we choose to find the average momentum transferred per atom at a time when 1/e
atoms remain, with the assumption that any scattering event will cause the atom to be lost. The
two ways an atom does not scatter a photon (during the sweep or from optical repumping) comeagain from rows 10 and 12:

P (no scatter) = [(1 − S)A(1 − S)]2 + [(1 − S)(1 − A)(1 − S)]2 .
Let the probability that an atom survives n sweeps be 1/e; then the momentum change for
an atom at the point when it has probability 1/e not to be lost isn h∆pSWAP ih∆pSWAP i− ln (P (no scatter))(1 − xQ )(x(2xQ − 1) − 1)=
× 2~k.
ln (x2 + 2xQ+2 (xQ − 1))=(5.12)

This function is plotted in Fig. 5.7(b) for Q = 25 and maximized with respect to sweep rate
in Fig. 5.4(b).
The optimal sweep rate for transferring momentum in the model where scattering is unimportant is found by taking the derivative of Eq. (5.11) with respect to α and equating it to zero:

αopt ≈

1π Ω4.
2
2 4∆ ln [2 (1 + Q)](5.13)

However, the optimal sweep rate for transferring momentum before atoms are lost to recoiled
photons Eq. (5.12) could only be found numerically (Fig. 5.7(a)). The scaling at significant Q

85remains the same as in Eq. (5.13) – the optimal sweep rate can be increased as roughly Ω4 /∆2(though Q also changes as Ω2 ).
b)

10

1
1

10

Quality parameter Q

100

Momentum transfer before
photon is scattered [ħk]Sweep rate αopt for maximal ∆p
per scattered photon [Rsc ∆swp]

a)

15

10

5

0
0.1

x = exp(-Rsc Δswp / α)1.0

Figure 5.7: Optimization of experimental parameters with Q. (a) The momentum transferred during
a single simulated SWAP cooling sweep was found and maximized with respect to the sweep rate α.
When the quality parameter Q is very small, the ideal strategy is to sweep very slowly in order tocoherently scatter once. When Q is large, efficient adiabatic sweeps are possible. (b) The maximum
momentum transferable before a photon scatters due to Raman cooling beams or optical pumping
beams.

Chapter 6

Continuous Real-time Tracking of a Quantum Phase below the Standard
Quantum Limit

In this chapter, I will present a theoretical proposal for countinuous tracking of a quantum
phase, which could function as an alternative to the Ramsey sequence. Quantum nondemolition
measurements of a lossy cavity mode interacting with an atomic ensemble are used to directly probe
the quantum phase of the collective atomic spin without converting it into a population difference.
Spin-squeezed states in phase quadrature develop naturally with long enough measurement time[99]. This work is in collaboration with Dr. Athreya Shankar and Prof. Murray Holland. I will
reproduce the text of ref. [99] closely in Sec. 6.1 to introduce the background and summarize
the main conclusions, but then go on to discuss a mean-field picture to provide some additional
intuition. I will focus on a proposed implementation in 87 Rb and technical requirements needed to
implement the scheme later.

6.1

Introduction
Quantum systems have become robust platforms for metrology and tests of fundamental

physics. Many applications rely on the dynamics of pseudospin-1/2 systems with two long-lived
quantum states, |↑i and |↓i. After preparing an equal superposition of these two states, a physicalinteraction is studied by investigating its effect on the relative phase φ(t), with the state of each √spin evolving in time as |ψ(t)i = |↓i + eiφ(t) |↑i / 2. We propose a novel scheme that enables
continuous tracking of this relative phase. Our scheme continuously and directly measures the87real-time phase φ(t) unlike the widely used Ramsey sequence [86, 77, 49, 56, 2, 89, 98, 105, 87, 45,
40, 37], which indirectly measures the net accumulated phase φ(T ) during an interrogation time
T . The typically destructive readout in a Ramsey sequence requires multiple state resets, rotations
and repetitions of the sequence to infer the phase at different times from a population difference.
In contrast, a single run of our protocol yields a continuous time series of phase measurements.
Therefore, our scheme enables real-time tracking of time-varying signals that are not reproducible.
As an added benefit, our scheme yields continuous phase estimates with precision well beyond√
the standard quantum limit (SQL) of ∆φSQL = 1/ N radians that limits readout precision with N
unentangled spins. In comparison to several proposals and experiments [61, 65, 96, 62, 11, 27, 51]
that have demonstrated squeezed states with precision beyond the SQL, our scheme enjoys theadvantage that the squeezing is produced, the phase accumulated, and the readout performed, allin the same spin quadrature.
Recent experiments have demonstrated phase tracking of a spin using quantum nondemolition(QND) measurements via a Faraday rotation angle [24]. In contrast, our proposal is based oninterfering Raman transitions in a cavity and enables an intuitive interpretation of phase tracking
in terms of elementary atom-cavity interactions that nearly balance one another. Our scheme
directly reveals a phasor precessing in the equatorial plane of a Bloch sphere, in the spirit of the“hand on a clock” analogy at the core of quantum metrology.

6.2

Overview of Theoretical Proposal
We represent the collective angular momentum of N atomic spins by a classical Bloch vectorof length N/2 with components Jx , Jy , Jz (Fig. 6.1(d, left)). With all spins initially in the sameequal superposition state, the Bloch vector lies in the equatorial plane along a direction that
we define as the y-axis. As the phase evolves, the Bloch vector acquires a small x-component,
Jx = N2 sin φ(t) ≈ N2 φ(t), for small deflections, and we propose a straightforward extension tolarge deflections in the conclusion. We arrange atom-cavity interactions wherein a cavity field
quadrature is sourced by Jx . Continuous homodyne detection of this quadrature amounts to real-

88

Figure 6.1: Schematic and working principle. (a) Two lasers drive a collection of atoms to interact
with a cavity mode. The relative phase φ(t) can be continuously tracked by homodyne detectionof the field leaking out. (b) Cavity-assisted Raman transitions: The red (blue) pathway leads tothe emission of a cavity photon accompanied by a spin flip |↓i → |↑i (|↑i → |↓i). (c) Hierarchyof frequencies. (d) Classical Bloch vector picture: The red and blue pathways set up balanced,
opposing superradiance pathways that lead to a coherent cancellation of the intracavity field whenthe Bloch vector (green) is along the y-axis (φ = 0). When the Bloch vector has a small xcomponent (φ 6= 0), the intracavity field from the two pathways add constructively, giving rise tonon-zero output field.

time, continuous, QND measurement of φ(t).
We consider N atoms trapped at the antinodes of a cavity with resonance frequency ωc and89
decay rate κ, as shown in Fig. 6.1(a). The states |↓i and |↑i have an energy separation ~ω0  ~κand form a pseudospin-1/2 system described by the Pauli spin operators σ̂i , i = x, y, z, withraising (lowering) operators σ̂+ (σ̂− ). The N atoms form a collective spin with total angularPjmomentum components Jˆx , Jˆy , Jˆz , with Jˆi = N
j=1 σ̂i /2. We assume the dipole-allowed transitions|↓i ↔ |ei and |↑i ↔ |ei with frequencies ω↓e and ω↑e to be respectively driven using lasers withfrequencies ω1 and ω2 in a far-detuned regime with detuning ∆  ω0 , κ, allowing for the adiabatic
elimination of |ei [55]. The two drive lasers differ by a frequency 2ω0 (Fig. 6.1(c)) and do not bythemselves drive |↓i ↔ |↑i Raman transitions; however, they are symmetrically detuned by ω0 fromωc and participate in cavity-assisted Raman transitions as illustrated in Fig. 6.1(b). When the Rabi
frequencies of the two drive lasers are balanced, i.e. Ω1 = Ω2 = Ω0 , the atom-cavity Hamiltonian,
to leading order in 1/∆, is simply the sum of a Jaynes-Cummings and an anti-Jaynes-Cummings
interaction and is given by
ĤQND =~ΩQND ˆX̂ Jx .
2

(6.1)√
Here X̂ = (â + â† )/ 2 is the amplitude quadrature, with â, â† the annihilation and creation√
operators for the cavity mode, and Ŷ = (â − â† )/ 2i is the conjugate phase quadrature such that√
[X̂, Ŷ ] = i. The atom-cavity interaction strength is ΩQND = 2Ω0 g0 /∆ with g0 the single atomcavity vacuum Rabi frequency. If the two drive lasers have initial phases ψ1 and ψ2 , the cavity
quadrature â† ei(ψ1 +ψ2 )/2 + H.c. is coupled to the spin component Jˆ+ ei(ψ1 −ψ2 )/2 + H.c. , whereJˆ+ = Jˆx + iJˆy . Here we assume ψ1 = ψ2 = 0 without loss of generality.
Classically, the intracavity fields established by the two balanced drives exactly cancel when
D E
D E
Jx = 0 (Fig. 6.1(d)). However, even with Jˆx = 0, Jˆx2 6= 0, i.e. quantum fluctuations source
the Y quadrature of the cavity field. In the regime κ2  N Ω2QND , Ŷ is slaved to Jˆx as
Ŷ (t) ≈ −ΩQND ˆJx (t) + F̂(t),
κ

(6.2)

where the noise operator F̂(t) arises from coupling of the cavity mode to external modes throughthe lossy mirror (Fig. 6.1(a)) [107, 72]. The field leaking out is to be monitored via balanced
homodyne detection using a local oscillator at frequency (ω1 + ω2 )/2 with phase tuned to detect

90
the output field quadrature that is sourced by the intracavity Y quadrature. The photocurrent
thus recorded is a measurement of the Y quadrature which, from Eq. (6.2), amounts to measuringJx .
The precision of the phase estimate in a window is determined by the window duration. A
characteristic time, T0 = (ηCγsc )−1 /(N/4), is the time required to average down the measurement
2
noise of Jx to the standard quantum limit ∆Jx,SQL= N/4, with η the quantum efficiency of thedetection system. In principle, we could improve the precision with longer measurement window
duration, until free space scattering becomes significant and degrades the precision by inducing
random rotation about the z-axis, Raman spin flips and atom loss.

6.3

Mean-field Picture
We try to understand the mechanism in a mean-field picture. Consider a cavity-assisted

superradiance process on a Raman transition as shown in Fig. 6.2 [10]. After loading atoms into|↑i, we could expect cavity-assisted superradiance1 if the frequency of the cavity is tuned such thatthe cavity vacuum field and the dressing laser (indicated by the straight line) is on resonance witha two-photon Raman transition from |↑i to |↓i.
The Hamiltonian of the Raman superradiance process is [12]Ĥ = ~ωc ĉ† ĉ + ~ωe N̂e + ~ω↑ N̂↑ + ~ω↓ N̂↓Ω  ˆ −iω1 t−ψ1+~Je↑ e
+ H.C. + ~g ĉ† Jˆ↓e + H.C. ,
2

(6.3)

where ĉ and ĉ† are creation and annihilation operators of the cavity field, ~ωe , ~ω↑ and ~ω↓ are
the energy of |ei, |↑i and |↓i, N̂e , N̂↑ and N̂↓ are the number operators of |ei, |↑i and |↓i, Ω isthe Rabi frequency of the driving laser, 2g is the cavity vacuum Rabi frequency, Jˆe↑ = |ei h↑| andJˆ↓e = |↓i he|.
1

The collective emission rate must also be larger than atomic decoherence rates.

91

Figure 6.2: Cavity-assisted Raman superradiance. The red dots represent atoms. The dressing laser(Ωeiψ1 ) and the emission light (ESR1 ) are indicated by the straight and wavy lines respectively. The
cavity resonance is tuned near the frequency of the emission light. The frequency of the dressing
laser is ω1 . The cavity resonance is ωc . The cavity vacuum Rabi frequency is 2g. The energy
splitting between |↓i and |↑i is ~ω0 .

After adiabatic elimination, with the help of Heisenberg equations of motion and assumingthat atoms in |↑i don’t interact with the emission cavity mode, we get

ĉ˙ =



κ
− −i
2



g2
N↓ + ωc∆1ĉ − igΩ ˆ −iω1 t−ψ1
J− e
,
2∆1

(6.4)

where Jˆ− = |↓i h↑| and κ is the linewidth of the cavity.
When the cavity field and the driving laser are on resonance with the two-photon Raman
transition, we havegΩ/2∆1 ˆ −iψ1Cˆ = −iJ− e,κ/2where Cˆ = ĉe−iωc t and Jˆ− = Jˆ− e−iωc t , in steady state C˙ = 0.

(6.5)

92
Now we consider the case when we drive two paths of superradiance from |↑i to |↓i and from|↓i to |↑i at the same time as shown in Fig. 6.3.

Figure 6.3: Driving two paths of superraidance at the same time. Left is the superradiance from|↓i to |↑i. Right is the superradiance from |↑i to |↓i.

2 ˆ−iψ2 .
For the superradiance from |↓i to |↑i, we have Cˆ† = −i gΩ/2∆
κ/2 J+ e

In the limit ∆1 ≈ ∆2 ≈ ∆  ω0 , the total emission field Etot = ESR1 + ESR2 takes the form

Etot = −iψ1 −ψ2gΩ/2∆ −i ψ1 +ψ2 i ψ1 −ψ2 ˆ2(e 2 J+ + e−i 2 Jˆ− ),
e
κ/2

(6.6)

which is the same as the general form discussed in the paragraph below Eq. (6.1). The field wewant to monitor for continuous phase tracking results from the interference of the two superradiancepaths from |↑i to |↓i and from |↓i to |↑i in the mean-field picture. Remember that Jˆ+ = Jˆx + iJˆy
and Jˆ− = Jˆx − iJˆy , Eq. (6.7) can be rewritten as





 
gΩ/2∆ −i ψ1 +ψ2
ψ1 − ψ2 ˆ
ψ1 − ψ2 ˆ
2Etot = −ie
2 cos
Jx − sin
Jy ,
κ/2
2
2
which says that Jˆx and Jˆy couple to different quadratures of the emission field.

(6.7)

93

6.4

Implementation in 87 Rb

Figure 6.4: Level diagram for 87 Rb D2 line

The level diagram of 87 Rb D2 line is shown in Fig. 6.4. At first sight, one may expect
to implement the phase tracking experiment with |F = 1, mF = −1i and |F = 1, mF = 1i as |↓iand |↑i. However this choice is difficult because the two-photon transition between the two states
is forbidden in the limit the detuning from the excited state ∆  ωHF,e , the excited hyperfine
splitting of 87 Rb. Instead, we propose an experiment based on spin-1 systems realized with the
F=1 manifold.
I will first explain how to understand the quantum phase in spin-1 systems. I will then present
our scheme, and give some numerical results with proposed parameters. Finally, I will discuss sometechnical requirements needed for the implementation.

94
6.4.1

Quantum Phase in Spin-1 Systems

Here we provide two ways to understand how the quantum phase φ appears in a spin 1
system.

6.4.1.1

Understanding Quantum Phase with Wavefunctions

The wavefunction of a spin-1/2 system in an equal superposition state in the Dicke basis|J, mJ i is1|θ = 90◦ , φi = √ (|1/2, −1/2i + e−iφ |1/2, 1/2i),
2

(6.8)

according to Eq. (3.1).
Likewise, in a spin-1 system, we have√1|θ = 90◦ , φi = (|1, −1i + 2e−iφ |1, 0i + e−i2φ |1, 1i),
2

(6.9)

so the quantum phase in spin-1 systems is just the phase difference between adjacent energy levels
φ as shown in Eq. (6.9).
We can also reproduce the same sets of equations for spin-1 systems (see Fig. 6.5) as thosefor spin-1/2 system, from Eq. (6.3) to Eq. (6.5).
If the vacuum Rabi frequency for |g1 i to |e1 i transition is 2g1 and that for |g2 i to |e2 i transition
is 2g2 , the Hamiltonian for spin-1 Raman superradiance can be written as

Ĥ = ~ωc ĉ† ĉ + ~ωe1 N̂e1 + ~ωe2 N̂e2 + ~ωg0 N̂g0 + ~ωg1 N̂g1 + ~ωg2 N̂g2
Ω1  ˆ
+~Je1 g0 e−iω1 t−φ1 + H.C. + ~g1 ĉ† Jˆg1 e1 + H.C.
2Ω2  ˆ
+~Je2 g1 e−iω1 t−φ2 + H.C. + ~g2 ĉ† Jˆg2 e2 + H.C. .
2(6.10)

After adiabatic elimination and assuming that atoms in |g0 i don’t interact with the emission cavity
mode, we get

95

Figure 6.5: Raman superradiance in spin-1 systems

ĉ˙ =


 2

κ
g1
g2 Ω2 ˆg22
g1 Ω1 ˆ− −i
Ng1 +
Ng2 + ωc
Jg0g1 e−iω1 t−φ1 − i
Jg1g2 e−iω1 t−φ2 , (6.11)ĉ − i
2
∆1
∆1
2∆1
2∆1

where Jg0g1 = |g0i hg1| and Jg1g2 = |g1i hg2|. Notice that both Jg0g1 and Jg1g2 can be denoted withJ− .
If g1 Ω1 = g2 Ω2 and φ1 = φ2 , we haveg1 Ω1 /2∆1 ˆ −iφ1Cˆ = −2i
J− e
,
κ/2

(6.12)

where Cˆ = ĉe−iωc t and Jˆ− = Jˆ− e−iωc t , in steady state C˙ = 0, when the two-photon transition isdriven resonantly.
Eq. (6.12) takes the same form as Eq. (6.5) up to numerical prefactors, so we should expect
the same phase tracking performance for spin-1 systems as that in spin-1/2 systems.

96
6.4.1.2

Understanding Quantum Phase with Spin Components

D E
D E
In spin-1/2 systems, the quantum phase φ can be defined with Jˆx and Jˆy as

φ = tan−1

DE D E
Jˆx / Jˆy ,(6.13)

where hi denotes expectation values, because the dynamics of the three spin components Jˆx , Jˆyand Jˆz follow rigid body rotations on the Bloch sphere during Rabi flopping. I will show that thethree spin components still follow rigid body rotations for spin-1 systems.
If we denote the three Zeeman sublevels of the spin-1 system in 87 Rb as |1i, |2i and |3i, andrepresent a quantum state |φi = a |1i + b |2i + c |3i with a vector (a, b, c), we could represent thethree spin components of spin-1 systems L̂x , L̂y and L̂z in matrix forms as shown below,

0 1 0 
1 L̂x = √ 1 0 1 ,

2


0 1 0



(6.14)

0 −1 0 i L̂y = √ 1 0 −1 ,

2


0 1
0



(6.15)

1 0 0 L̂z = 0 0 0  .

0 0 −1(6.16)The Hamiltonian describing the on-resonance Rabi flopping of a spin-1 system reads

0 1 0 

~Ω 
~Ω


Ĥ = − L̂x = − √ 1 0 1 .


2
2 2

0 1 0
From Eq. (6.17), we could get the the population evolution (Fig. 6.6).

(6.17)

97

Figure 6.6: Population evolution of spin-1 systems. Horizontal axis is time. Vertical axis is population.

D E2 D E2
D E2
We could further get the evolution of the three spin components L̂x , L̂y and L̂z ,as shown in Fig. 6.7. The rigid body rotation of the three spin components still holds for spin-1
systems, so we can still define the quantum phase as φ = tan−1 (hLx i / hLy i).

Figure 6.7: Evolution of the three spin components. Horizontal axis is time. Vertical axis is thelength of the corresponding spin components.

98
6.4.2

Experimental Scheme and Numerical Results

The schematic diagram of continuous trakcing of a quantum phase with spin-1 systems implemented on 87 Rb D2 line can be found in Fig. 6.8. We have a single ended cavity, whose transimission
at the open port is more than 20 times that of the closed port. We plan to drive the Raman transitions through the open port at both ω1 and ω2 , and detect the emitted field at the cavity resonanceωc through the open port with homodyne detection (see Fig. 6.8c).
The quantization axis and the laser polarizations are shown in Fig. 6.8d. There will be offresonant emissions from the σ− component of the blue driving tone and the σ+ component of theorange driving tone. These off-resonance emissions, which are detuned by ±2ω0 from the cavity
resonance, are suppressed by (2ω0 / (κ/2))2 from the resonant emission. The typical suppression ismore than 5000 times, so we simply ignore it.
The parameters we propose to try are listed in Table. 6.1.
Parameter
∆
ω0
P1
P2

Description
Detuning of the dressing laser from F’ (see Eq. (6.7))
Energy difference between adjacent Zeeman sublevels
Power in the blue driving tone at frequency ω1Power in the orange driving tone at frequency ω2

Value
63 GHz
1 MHz
10 µW
10 µW

Table 6.1: Proposed parameters for continuous phase tracking experiment

We could calculate the 4 two-photon Rabi frequencies from the proposed parameters, as
shown in Eq. (6.18) to Eq. (6.21), the AC Stark shifts the three Zeeman sublevels get, as shown in
Eq. (6.22) to Eq. (6.24), and the free space scattering rate, as shown in Eq. (6.25).

99

(a)

(c)

(b)

(d)

Figure 6.8: Schematic diagram of continuous trakcing of a quantum phase with spin-1 systems. (a)Relevant energy levels of 87 Rb and two dressing lasers are presented. The emission field is indicatedwith wavy lines. (b) Hierarchy of frequencies. (c) Homodyne detection implementation is shown.
The dressing tones have frequency components at ω1 and ω2 . The frequency of the LO is at ωc ,which is the average of the two frequencies in the dressing tones. (d) Quantization axis and laser
polarizations are shown. The quantization axis is indicated by ~q. The polarizations of the two
beams are both perpendicular to the paper.

100Ω2ph, mF =−1, mF =0, σ+= 2π × 119.2 Hz(6.18)Ω2ph, mF =0, mF =1, σ+= 2π × 119.1 Hz(6.19)Ω2ph, mF =1, mF =0, σ−= 2π × 119.2 Hz(6.20)Ω2ph, mF =0, mF =−1, σ−= 2π × 119.1 Hz.

(6.21)

δAC, mF =−1= 2π × 8392.0 Hz(6.22)

δAC, mF =0= 2π × 8391.4 Hz(6.23)

δAC, mF =1= 2π × 8392.0 Hz.

(6.24)RF SS, mF =−1 = RF SS, mF =0 = RF SS, mF =1 = 2π × 2.15 Hz.

(6.25)The typical detection quantum efficiency is 10%, so the time for resolving the SQL is estimatedto be T0 = 330.0 µs. The time for creating 3 dB squeezing is thus 2T0 . The decoherence from freespace scattering during 2T0 is less than 3%, so we should be able to observe direct phase squeezing.

6.4.3
6.4.3.1

Technical Requirements
Compensating Quadratic Zeeman Shift

The splitting between Zeeman sublevels is created with a bias magnetic field. However, it will
also induce quadratic Zeeman shift which is proportional to m2F . As a result, the frequency splitting
between adjacent Zeeman sublevels will not be the same anymore, which excites the nematic spincomponents and the rigid body rotation of L̂x , L̂y and L̂z no longer holds. The rotation for spin-1
system is described by SU(3) group, which has 8 infinitesimal generators. The spin components
other than L̂x , L̂y and L̂z are called the nematic spin components.
The quadratic Zeeman shift between |F = 1, mF = 0i and |F = 1, mF = ±1i is estimated to
be 64 Hz, if ω0 = 2π × 1 MHz, comparable to the two-photon Rabi frequency which is about

101
120 Hz. We could take advantage of the differential AC Stark shift between |F = 1, mF = 0i and|F = 1, mF = ±1i from off-resonant π-polarized microwaves to compensate the quadratic Zeeman
shift, which has been demontrated to be good to below 1 Hz level [38, 109].

6.4.3.2

Phase Noise from the Driving Tones

Fig. 6.8b shows the hierarchy of frequencies. With perfect balance of the two superradiancepaths, we don’t expect any emission at ωc , so the output of the homodyne detector comes from thebeating between the 1/2 vacuum photon at ωc and the local oscillator (LO). However, because thefinite linewidth of the driving tones and the LO, there is actually additional noise contributions
to the homodyne detection from the beating between the driving tones and the LO, which is thesource of noise of our detection. The number of vacuum noise photons is 1/2, so the quantum
1/2
efficiency of the detection is QE= 1/2+N, with Nn the number of noise photons from the drivingntones. We want a small Nn to maintain a large quantum efficiency.
To understand why the beating between the driving tones and the LO matters, we willconsider the case where we replace the cavity in Fig. 6.8c with a mirror first. As discussed inSec. 4.5.6 (see Eq. (4.12)), the phase noise of a laser can be modeled as phase modulations of
the carrier with different modulation frequencies. The modulation index β depends on the power
spectral density of instantaneous phase fluctuations. The electric field of the blue driving tone,
the orange driving tone and the LO, including the carrier and the noise components at modulation
frequency ω0 only, can be written asEb= Eb0 cos(ωc t + φc0 + ω0 t + φ0 + β cos(ω0 t + φp ))(6.26)

Eo= −Eo0 cos(ωc t + φc0 − ω0 t − φ0 + β cos(ω0 t + φp ))(6.27)= ELO0 cos(ωc t + φc0 + φh + β cos(ω0 t + φp )),(6.28)

ELO

in which we assume that all three tones are derived from the same master laser, and that the blue
and orange driving tones are generated by phase modulating a laser so that Eb0 = Eo0 . ωc t + φc0

102
is the phase of the master laser. ω0 t + φ0 is the phase of microwave tone for the phase modulation.
φh is the additional phase gained by the LO. φp is the phase of the noise components.
In the limit β is small, Eq. (6.26) can be rewritten as Eb = Ebc + Eb+ + Eb− . The expressions
for Ebc , Eb+ and Eb− areEbc= Eb0 cos(ωc t + φc0 + ω0 t + φ0 )β
Eb0 cos(ωc t + φc0 + 2ω0 t + φ0 + φp )2β
= − Eb0 cos(ωc t + φc0 + φ0 − φp ).
2

Eb+

=

Eb−

(6.29)(6.30)(6.31)

Ebc , Eb+ and Eb− are at ωc + ω0 , ωc + 2ω0 and ωc respectively.
Likewise, we could write Eo = Eoc + Eo+ + Eo− , with Eoc , Eo+ and Eo− are at ωc − ω0 , ωcand ωc − 2ω0 respectively, and ELO = ELOc + ELO+ + ELO− , with ELOc , ELO+ and ELO− are at
ωc , ωc + ω0 and ωc − ω0 respectively.
The beating between the components at the same frequency will potentially contribute tothe output of the homodyne detector, that is, the beating between ELO+ and Ebc , between ELO−
and Eoc , between ELOc and Eb −, and between ELOc and Eo+ .
The output voltage vh of the homodyne detector from the beating of the three electric fieldcomponents is

1
vh ∝
2π

Z 2π
ωc

(Eb− + Eo+ ) × ELOc + Eoc ELO− + Ebc ELO+ dt,(6.32)

0

We in fact have vh = 0 here, because the transfer function of a mirror is not frequency-dependent.
If we switch back to the configuration shown in Fig. 6.8c, where the two driving tones reflect
off a cavity, the electric field of the blue driving tone, the orange driving tone and the LO, will be
modified as

103

Eb

= Ebc + Eb+ − Eb−(6.33)

Eo= Eoc − Eo+ + Eo− ,(6.34)

Notice that the Eb− and Eo+ gain a minus sign, because these two tones are on resonancewith the cavity and will gain an additional phase shift of π when reflecting off the cavity.
Now we get vh ∝ 2βEb0 ELO0 cos(φh ) sin(φp − φ0 ). Notice that φh = 0 here, because according2
to the discussions below Eq. (6.1), φc0 + φh = ψ1 +ψ+ π2 with ψ1 = φc0 + φ0 and ψ2 = φc0 − φ0 − π.
2

As a result, cos(φh ) = 1 holds.
If we generate the blue and orange driving tones with amplitude modulation, we then havevh ∝ 2βEb0 ELO0 sin(φh ) sin(φp − φ0 ). With the same argument above, we now have φh = π/2, sosin(φh ) = 1 still holds.
Next, I will estimate how large the noise component is in unit of photon numbers. For a laser
with linewidth κl , the number of noise photons is

Nn =

2
πκl

1+

1


ω0
κl /2

2

P1 + P2
P F,
Eph(6.35)

where Eph is the single photon energy, and PF is a polarization filtering factor since the LO and2
the driving tones have orthogonal polarizations. Notice that in Eq. (6.35), πκl 1 2ω0
1+ κ /2is thel

Lorentzian profile of a laser, so Eq. (6.35) can be intuitively viewed as the number of photons in
the mode ω0 away from the carrier with total laser power P1 + P2 . The typical value of PF is 10−4 ,which we will use in the following discussions.
The typical laser linewidth of an ECDL is 200 kHz. The number of noise photons is estimatedto be 246 according to Eq. (6.35), so QE=0.002, which is awful.
If we could narrow down the linewidth to 1 kHz, the number of noise photons will be de1/2
creased to 1.2. The quantum efficiency can be calculated as QE= 1/2+1.2=0.29, which starts to be

acceptable. Building lasers with 1 kHz intrinsic Lorentzian linewidth is challenging, but DBR lasers
with external feedback could be a potential way to achieve that linewidth [93, 108]. Of course, there

104
could be other solutions to beat down the number of noise photons, for example, suppressing laser
phase noise in a certain frequency range with a notch filter, but I will not extend the discussions
here.

6.5

Conclusion
I have introduced a theoretical proposal for continuous tracking of a quantum phase and direct

creation of phase squeezed states, and presented and analyzed a potential experimental implementation in 87 Rb. The implementation is demanding on the laser linewidth, which is challenging, but
started to be within reach.

Chapter 7

Conclusion and Outlooks

In this thesis, I have presented an experiment for creating homogeneous atom-cavity couplingwith site-dependent selections, where we achieved an average coupling strength of more than 91%
the peak coupling strength with 4% remaining atoms. I have described an ongoing project, where
we apply homogeneous spin-squeezed states, created by time-averaged probing of falling atoms, tobuild a spin-squeezed atom interferometer. These efforts are projected to make atom-cavity systems
more powerful and improve the precision of quantum metrology. I have demonstrated a new coolingscheme based on adiabatic transfer on artificial Raman transitions, where we achieved 1D equilibrium temperature 25 times below the Doppler cooling limit in 87 Rb. This cooling method could
potentially be helpful for cooling quantum systems without cycling transitions, say, molecules. I
have also reviewed a continuous phase tracking scheme and discussed a possible experimental implementation. The scheme is promising and could especially find its applications in the measurementof non-replicable processes, for example, searches for magnetic monopoles. The implementation ischallenging in terms of its technical requirements, but the development of narrow linewidth DBR
lasers with external feedback starts to catch up. In this chapter, I will talk about some future
directions we could pursue.

7.1

Creation of Dicke States by Quenching SuperradianceThis idea first came from a discussion with Prof. Nicola Poli from University of Florence,

Italy. The definition of Dicke states was discussed in Sec. 3.1.1.1. Consider a cavity-assisted

106
superradiance process on a Raman transition (see Fig. 6.2) [10], which we have discussed in Sec. 6.3.
The superradiance process can be visualized on Bloch spheres as shown in Fig. 7.1. If we quench
the superradiance process by turning off the Raman dressing laser and count the number of photons
emitted, we will be able to learn about how many atoms are left in |↓i and how many atoms aretransferred into |↑i, without collapsing individual atomic wavefunctions. In the meantime, theBloch vector is actually in a superposition state of all possible azimuthal angles, since no phase
information is collected. A Dicke state is thus created. If single photon resolution is not achievable
due to noise, a collection of Dicke states will be created, and I will discuss this case in detail in thenext paragraph.
The detection of the emitted photons is subject to a binomial distribution. Let’s assume
that the probability that a photon is detected is Q, and that the probability that a photon is
missed by the detector is 1 − Q. The spin projection of the created state along z is mJ = (J −p
Nd /Q)± Nd (1 − Q)/Q if we initialize the atomic ensemble in |J, Ji, and detect Nd photons before
quenching the superradiance. If we take Q = 0.9, J = N/2 and Nd /0.9 = N/2 (assume N is an even√number) as an example, the projection along z is mJ = 0 ± 0.47∆NSQL , where ∆NSQL = N /2 isthe standard quantum limit in mJ for a coherent spin state. The cavity cooperativity parameter
C or the atom number should be large enough to ensure that the collective emission rate is larger
than the atomic decoherence rates. Single particle decoherence will shorten the length of the Blochvector, and increases the standard deviation of the polar angle distribution of the state.
To confirm a Dicke state or a collection of Dicke states has been created, we could measure
the number of atoms in |↑i following π/2-pulses with different phases on resonance with |↑i to |↓itransition.

7.2

Understanding Limitations on the Creation of Spin Squeezing
In the 18 dB spin squeezing experiment [27], we actually had a technical noise floor of 25 dB

below the quantum projection noise. We believed that the failure in saturating the technical noise
floor should be attributed to the optomechanically-induced oscillations of the cavity resonance.

107

Figure 7.1: Raman superradiance visualized on Bloch spheres. The Bloch vector starts at the northpole (|↓i). Quantum fluctuations disturb the system from its unstable equilibrium transition, andeventually ends at the south pole. Reprinted from ref. [80]. To describe Raman superradiant decay
in the Dicke basis, we should replace all the Bloch vectors with a superposition of Bloch vectors
with all possible azimuthal angles and the same polar angle.

With the site-dependent selection technique, we expect to suppress the amplitude of the
optomechanically-induced oscillations by more than a factor of 4 in the short time limit. We are
now able to understand the limitations on the creation of spin squeezing better.

7.3

Closing Remarks
The early cQED experiments were done with Rydberg atoms flying through microwave cav-

ities [46]. Nowadays, cQED experiments can be done with optical cavities, neutral atoms trappedin the Lamb-Dicke regime, and engineered homogeneous atom-cavity coupling, and thus enable
explorations in spin-wave multiplexing [29], creation of atomic entanglement [51, 27, 44], cavitymediated spin-spin interactions [79, 31], dynamical phase transitions [75] and synthetic quantum
matter [23], to name a few.
Personally, the most fascinating application of cQED systems is still to precision measurements. Imagine that all you need to do to improve the precision of your quantum sensor is to108
enclose it with two mirrors. That is how squeezed atomic clocks, squeezed atom interferometers,squeezed magnetometers, etc. can be built.
There are still more advanced ingredients that can be added to cQED systems, for example,
high-order cavity modes [94] and atom assembly with optical-tweezer-arrays [33], to make the cQED
systems more powerful. More exciting applications are yet to come.

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