Twisting, Binding, and Probing Matter Waves in a
Rubidium Cavity QED System
byChengyi Luo
B.S., Sun Yat-sen University, 2017

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
2024

Committee Members:
James K. Thompson, Chair
Ana Maria Rey
Joshua Combes
Adam Kaufman
Jun Yeii
Luo, Chengyi (Ph.D., Physics)
Twisting, Binding, and Probing Matter Waves in a Rubidium Cavity QED System
Thesis directed by Prof. James K. Thompson

In this thesis work, I have explored a novel platform for quantum metrology and many-body
physics by realizing matter-wave interferometric controls in a high finesse cavity. By correlating
the internal states of the atoms to the external degrees of freedom, we demonstrated direct entanglement generation on the momentum states of the atoms with two distinct approaches, quantum
non-demolition measurement and one-axis twisting dynamics. After injecting the squeezed momentum states into a matter-wave interferometer, we realized an entanglement-enhanced matter-wave
interferometer for the first time.
Decoupling the momentum states from the internal states with all atoms in the same atomic
spin state, we realized a novel cavity-mediated collective momentum-exchange interaction in whichpairs of atoms swap their momenta by exchanging photons through the cavity. The momentumexchange interaction leads to an observed all-to-all Ising-like interaction in a matter-wave interferometer, which is useful for entanglement generation. A many-body energy gap also emerges, effectively binding interferometer matter-wave packets together to suppress Doppler dephasing withanalogies to Mössbauer spectroscopy. In the same system, by adding new laser frequency control
for driving pair creation/annihilation processes, we realized Hamiltonian engineering of collective
XYZ spin models between two momentum states and the first demonstration of the long-sought
two-axis counter-twisting dynamics.
The entanglement-enhanced matter-wave interferometer experiment shed new light on improving future atom interferometers by reducing the fundamental quantum source of imprecision.
The momentum-exchange interaction provides new options for interacting momentum states enabled by the cavity. The Hamiltonian engineering realized here not only enables new dynamics
for entanglement generation but also offers new possibilities for quantum simulation with atomic

iii
momentum states. All these opportunities arise from coupling the atoms to a high-finesse cavity,
known as cavity quantum electrodynamics systems. Combining the matter-wave interferometriccontrol and cavity QED, our system provides a new platform for the study of quantum metrology,
quantum simulation and many-body physics with qubits based on atomic momentum states.

iv

Acknowledgements

My time in graduate school has been fun, mainly owing to James. He is for sure a fantastic(if not the best) advisor and my role model for research. I have greatly benefited from his patience.
I still remember, during my first month in the lab, he would even give hands-on advise on how toturn the knobs. Later in the Ph.D., he was also very understanding when I got distracted fromthe original plan or when I felt like more time for dealing with technical issues. Apart from giving
helpful suggestions, he would allow me with a lot of freedom by saying “Do whatever makes sense
to you.” I always enjoyed the moment when he came in the lab, dropped down his laptop and coffee,
and said “Hey, I have an idea/question.” Some of our late night discussions are now turning into
actual experimental results.
Though one year younger, Dylan Young joined the group only a few months later than I
did. He is undoubtedly the kindest person in the lab, and I’m very glad to have overlapped with
him the most. Unfortunately, we worked on different sides of the room, it would have been socool to directly collaborate with him. On the Rb experiment, Baochen Wu and Graham Greve are
the ones who put me on track at the beginning. Vanessa Koh provided significant help with themomentum-exchange project and preparing the new lab space. I wish her all the best with her
new career. Chitose Maruko joined our experiment during my last year. She is by far the brightest
first-year student I have ever seen, and I regret not having more time to work with her. I’m also indebt to her for bravely taking over recovering the experiment during my unforeseen sudden absence,right after the big lab move. I believe she will be able to bring the experiment to a new level.
One unique aspect of JILA is that you get to work with brilliant theorists like Ana Mariav
Rey, who has deep understanding of both theory and experiment. I’m grateful to have worked
with Haoqing Zhang, Anjun Chu from Rey group and John Wilson from Murray Holland’s group.
In particular, I spent a lot of time discussing with Haoqing through the second half of my Ph.D..
There are even days he actually came down to the lab, watched me taking new data, readjusted
his simulation and fed me new ideas. You just can’t beat this bandwidth! The support from JILA
machine shop is also fantastic. I would like to thank James and Calvin for such a great job withthe move. To quantify how successful the move was, the lasers are still well coupled to the cavity
after we turned them back on.
Before joining graduate school, I was very lucky to work with many great scientists as well.
Prof. Jiahao Huang and Prof. Chaohong Lee introduced me to quantum metrology during myundergraduate studies. Jiahao was a senior Ph.D. student at the time, but he was also patientenough to give very detailed guidance on derivation and paper writing to a second-year undergraduate student. Prof. Wes Campbell offered me my first opportunity to work in an AMO lab, andhis passion has influenced me to choose experiment for graduate school. During my last year incollege, I had the privilege of working with Dr. Tao Shi. Although I have forgotten most of thetheoretical techniques he taught me, his childlike curiosity about physics continues to inspire me.
I would like to thank my parents for their unconditional support for me to pursue research.
I still remember one of the defining moments for me to choose physics as college major is when
my dad handed me the public information about the 2012 Nobel prize in physics. I was mostly
brought up by my grandfather who passed away a few years ago. Hopefully, I have inherited someof his qualities. They might be best described by two Cantonese catchphrases “是但啦” and “顶
硬上”, which roughly correspond to “be less opinionated” towards trivial things but “can bite thebullet” when facing challenges. Lastly, I would like to thank my partner Wenqin for always being
encouraging, and for accompanying me through all the ups and downs of our Ph.D. times. I lookforward to exploring more in life with her in the future.

Contents

Chapter
1 Motivation and overview

1

1.1

Quantum metrology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.2

Matter waves and atom interferometry . . . . . . . . . . . . . . . . . . . . . . . . . .

6

1.3

Cavity quantum electrodynamics with cold atoms in an optical cavity . . . . . . . .

7

1.4

Outline of thesis work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

1.4.1

Entanglement-enhanced matter-wave interferometer . . . . . . . . . . . . . .

8

1.4.2

Cavity-mediated momentum-exchange interactions . . . . . . . . . . . . . . .

9

1.4.3

Hamiltonian engineering of atomic momentum states . . . . . . . . . . . . . .

92 Introduction and theory

10

2.1

87 Rb atoms coupled to an optical cavity

2.2

Quantum metrology and spin squeezing . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3

Matter-wave interferometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

. . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3.1

Interferometric sequence on a Bloch sphere . . . . . . . . . . . . . . . . . . . 19

2.3.2

Effective spin 1/2 system with momentum states . . . . . . . . . . . . . . . . 20

2.3.3

Momentum states coupled to cavity . . . . . . . . . . . . . . . . . . . . . . . 22

3 Apparatus

24

3.1

Experimental sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2

Vacuum glass cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

vii
3.3

Science cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.4

Laser system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.5

3.6

3.7

3.4.1

Overview of all lasers and the locking chain . . . . . . . . . . . . . . . . . . . 30

3.4.2

REF, MOT and REP lasers for laser cooling . . . . . . . . . . . . . . . . . . 31

3.4.3

813 nm laser for red-detuned lattice . . . . . . . . . . . . . . . . . . . . . . . 31

3.4.4

760 nm laser for blue-detuned dipole trap . . . . . . . . . . . . . . . . . . . . 32

3.4.5

Narrowing laser linedwith with optical feedback . . . . . . . . . . . . . . . . . 36

3.4.6

Cavity Probe: tracking bare cavity resonance . . . . . . . . . . . . . . . . . . 42

3.4.7

Atomic probe for homodyne detection . . . . . . . . . . . . . . . . . . . . . . 44

3.4.8

Raman laser for driving two-photon transition . . . . . . . . . . . . . . . . . 46

Microwave and RF system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.5.1

Generation of low-noise 6.8 GHz signal . . . . . . . . . . . . . . . . . . . . . . 49

3.5.2

Coherent rotations for all three transitions . . . . . . . . . . . . . . . . . . . . 50

Degenarate Raman side-band cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.6.1

Polarization twisting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.6.2

Two-dimensional lattice formed by three beams . . . . . . . . . . . . . . . . . 52

3.6.3

Cooling sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.6.4

Radial temperature measurement with dispersive shifts . . . . . . . . . . . . 54

Relevant parameters for the experiment . . . . . . . . . . . . . . . . . . . . . . . . . 56

4 Entanglement enhanced matter-wave interferometer

57

4.1

Experimental setup and overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.2

Manipulating atomic momentum states. . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.3

Squeezing on momentum states. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.4

Wineland criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.5

Entangled matter-wave interferometry. . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.6

Vibration noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

viii
4.7

Error budget for standard quantum limit. . . . . . . . . . . . . . . . . . . . . . . . 76

4.8

Conclusion and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5 Cavity mediated momentum-exchange interaction

79

5.1

Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.2

Experiment setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.3

Modulation sidebands created by atomic density grating. . . . . . . . . . . . . . . . . 83

5.4

Different interpretations of modulation sidebands . . . . . . . . . . . . . . . . . . . . 85

5.5

Effective Hamiltonian of momentum-exchange interaction. . . . . . . . . . . . . . . . 86

5.6

One-axis twisting dynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.7

Gap protection: binding wave packets together. . . . . . . . . . . . . . . . . . . . . . 91

5.8

Collective recoil and Suppression of Doppler Dephasing . . . . . . . . . . . . . . . . 96

5.9

Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

6 Hamiltonian engineering of momentum states

103

6.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

6.2

Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

6.3

Four-photon spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

6.4

Mean-field dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

6.5

Dynamics on the Bloch sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6.6

Two-axis counter-twisting with unstable points at north and south poles . . . . . . . 118

6.7

Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

7 Conclusion and outlook

122

7.1

Direct entanglement generation between momentum states . . . . . . . . . . . . . . . 122

7.2

Squeezing with balanced dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

7.3

Continuous quantum phase measurement . . . . . . . . . . . . . . . . . . . . . . . . 125

7.4

QND measurement of the atomic density grating . . . . . . . . . . . . . . . . . . . . 126

ix
7.5

Going beyond two-body interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

7.6

Simulating BEC-BCS cross-over with momentum qubits . . . . . . . . . . . . . . . . 128

7.7

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

Bibliography129

Tables

Table
3.1

Table for cavity and atomic parameters. . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.1

Table for different sources of error. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

Figures

Figure1.1

Controlling matter waves with an optical cavity. Adapted from original graphic in
the public information for 2012 Nobel prize in physics [1]. In this thesis, we explore
how to generate interactions between matter waves by coupling them to a high-finesse
cavity formed by two mirrors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2

3

The quantum state or a two-level system represented by a Bloch vector on a Bloch
sphere. The azimuthal angle on the Bloch sphere corresponds to the relative phasebetween the two levels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.3

4

A matter-wave interferometer is realized by splitting and recombining the wave packets. Here, the blurry blobs represent the wave packets for different momentum states.
The goal is to measure the accumulated phase ϕ between different trajectories which
depends on gravity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

2.1

Rb atoms in high-finesse cavity.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2

Homodyne detection of dressed cavity resonance.

2.3

Ramsey sequence on Bloch spheres. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.4

Ramsey sequence on Bloch spheres. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.5

Mach-Zehnder matter-wave interferometer sequence. . . . . . . . . . . . . . . . . . . 18

2.6

Bloch spheres defined with different two levels. . . . . . . . . . . . . . . . . . . . . . 20

2.7

Bloch sphere formed by momentum states.

. . . . . . . . . . . . . . . . . . . 12

. . . . . . . . . . . . . . . . . . . . . . . 21

xii
3.1

A picture of the main system after the move. . . . . . . . . . . . . . . . . . . . . . . 24

3.2

Picture and drawing for the glass cell . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.3

A rendered image of the science cavity.. . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.4

Locking chain for stabilizing all laser frequency.

3.5

Optics for 813 nm laser.

3.6

CAD drawing for 760 nm laser. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.7Interference between LG(1,0) mode and a Gaussian mode.

3.8

LG(1,0) mode is a superposition of HG(1,0) and HG(0,1) modes.

3.9

All optics for 760 nm laser. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

. . . . . . . . . . . . . . . . . . . . 30

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

. . . . . . . . . . . . . . 34
. . . . . . . . . . 34

3.10 Atomic probe optical feedback narrowing setup. . . . . . . . . . . . . . . . . . . . . . 36
3.11 PZT-actuated mirror for reflecting the picked off light.

. . . . . . . . . . . . . . . . 37

3.12 Transfer function for the PZT-actuated mirror. . . . . . . . . . . . . . . . . . . . . . 37
3.13 Beatnote between two different narrowed lasers.

. . . . . . . . . . . . . . . . . . . . 38

3.14 Setup for active stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.15 Error signal for active stabilization.

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

3.16 Measured laser linewdith when locking to different offset. . . . . . . . . . . . . . . . 40
3.17 Laser linewidth response to change in current/temperature with/without active feedback.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.18 All optics for cavity probe laser.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.19 Frequencies of all three laser relative to atomic transition and cavity resonances. The
6.45 MHz is the FM modulation frequency for creating PDH sidebands on cavity
probe laser.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.20 Atomic probe to cavity probe beatnote setup.
3.21 All optics for atomic probe laser.

. . . . . . . . . . . . . . . . . . . . . 44

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.22 Path length stabilization for atomic probe.

. . . . . . . . . . . . . . . . . . . . . . . 46

3.23 Beam path for the Raman laser. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.24 Microwave, Raman and Bragg transitions connecting three different energy levels

. 48

xiii
3.25 Microwave chain.

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

3.26 Microwave chain for coherent rotation for microwave, Raman and Bragg transiions.
3.27 Relevant energy levels for 87 Rb.

49. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.28 Polarization twisting for creating Raman coupling. . . . . . . . . . . . . . . . . . . . 51
3.29 Polarization twisting for Blue-detuned beams . . . . . . . . . . . . . . . . . . . . . . 52
3.30 Three beams for Raman sideband cooling (RSBC.). . . . . . . . . . . . . . . . . . 53

3.31 Pictures of three beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.32 Timing sequence for RSBC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.33 Dispersive shift decay as expansion time increases. . . . . . . . . . . . . . . . . . . . 55
4.1

Artistic conceptualization of entanglement-enhanced matter-wave interferometer. Atoms(blurry blobs) are entangled to each other. Each atom can be spitted into two paths(colored as red and blue) and get recombined later. Credit: Steven Burrow . . . . . 57

4.2

Laser cooled rubidium atoms are trapped inside a high-finesse cavity. . . . . . . . . . 59

4.3

Space-time diagram and the corresponding Bloch sphere depictions for the entangled
matter-wave interferometer.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.4

Simplified energy-level diagram for 87 Rb.

4.5

Driving Raman transition with two EOM sidebands. . . . . . . . . . . . . . . . . . . 61

4.6

Velocimetry results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.7

Velocimetry after selection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.8

Large momentum transfer with sequential Bragg pulses. . . . . . . . . . . . . . . . . 63

4.9

Left: frequency diagram for QND and OAT. Right: Homodyne signal when scanning. . . . . . . . . . . . . . . . . . . . . . . . 60

atomic probe laser across cavity resonance with or without atom in the cavity . . . . 66
4.10 (A) Contrast lost (top) and spectroscopic enhancement W (bottom) as a function of
the photon number inside the cavity. (B) State tomography of coherent spin state(CSS) and squeezed states generated by QND and OAT.

. . . . . . . . . . . . . . . 67

4.11 Entangled matter-wave interferometer. . . . . . . . . . . . . . . . . . . . . . . . . . . 73

xiv
4.12 Interferometer contrast fringes with Tevol = 0.112 ms shown for no squeezing Jc(black) and with squeezing Js (red). . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.13 Phase sensitivity with different evolution time. . . . . . . . . . . . . . . . . . . . . . 74
4.14 Measuring vibration noise with an optical Mach-Zehnder interferometer.

. . . . . . 75

4.15 Power spectral density of the vibration noise as measured by the optical MachZehnder interferomter and a commercial vibrometer.

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

5.1

Experimental setup for observing momentum-exchange interaction. . . . . . . . . . . 80

5.2

Modulation sidebands created by atomic density grating . . . . . . . . . . . . . . . . 82

5.3

Frequency diagram of the optical atomic transition frequency . . . . . . . . . . . . . 83

5.4

Matter-wave superradiance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.5

The measured interferometer phase shift scales linearly with the initial spin projection Jz = ⟨Jˆz ⟩, while now holding ∆d fixed. The orange data points and fitted line
is for χ/2π = +2.1 Hz, and green for χ/2π = −2.5 Hz). . . . . . . . . . . . . . . . . 87

5.6

Matter-wave interferometer sequence and space-time diagram for observing all-toall Ising or One-Axis Twisting dynamics. The interferometer fringe amplitude andphase shift ∆ϕ are measured by scanning the phase of the final rotation ϕ.

5.7. . . . . 88

The observed phase shift ∆ϕ of the interferometer fringe versus the dressing laser’s
detuning from the dressed cavity resonance, displaying the predicted (line) functional
form of χ from Eq. 5.2. The insets illustrate the relative alignment of the modulation
sideband to the cavity resonance for three characteristic detunings.

5.8. . . . . . . . . 89

Visualizations of the phase shift ∆ϕ in both the pseudo-spin picture (Bloch spheres)and in the atomic density grating picture.

. . . . . . . . . . . . . . . . . . . . . . . 90

xv
5.9

Using the sequence in the top panel, the bottom panel shows the contrasts of theinterferometer fringe measured at the end of the exchange interaction period with χ =0 (black), χ/2π = 6 Hz (red). The ratio between the two (inset) displays significant
gap protection of coherence due to the momentum-exchange’s Jˆ2 contribution. The
simulated results (solid lines) show good agreement with the data. . . . . . . . . . . 91

5.10 We run a similar sequence to that of Fig. 6.8C except with an additional 40 µs
delay after wave packet overlap before application of the dressing laser for a variable
time tx . We see that as the interaction strength is increased relative to the rms
inhomogeneous broadening σin = 2π × 2 kHz, there is a transition in the dynamics
for N χ/σin > 0.9. Strikingly, there are also clear oscillations that were only hinted
at in Fig. 6.8C. The lines are theory predictions. . . . . . . . . . . . . . . . . . . . . 92
5.11 (A) The theory prediction with residual superradiance on (solid) and turned off(dashed) are shown with three example points in the oscillations labeled, and withno interactions (grey). (B) The total length of the pseudo-spin Bloch vector J⃗
oscillates in time because the individual Bloch vectors oscillate as shown for J⃗p>0
and J⃗p<0 in green and orange respectively, with χN /σin = 2.8. . . . . . . . . . . . . 93
5.12 (left) In a co-moving frame, the wave packets oscillate in time about their average
position in space (blue and red wave packets and centers solid lines, non-interacting
system dashed lines.) The momentum-space wave packets (right) also oscillate butwith a π/2 phase shift in time relative to the position space wave packets, as would
be the case for a harmonic oscillator.

. . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.13 The interferometer contrast as a function of imbalance in the time from nominal
perfect reoverlap of the wave packets with: no momentum-exchange (red data andfit), momentum-exchange applied right after first π/2 at intermediate power (bluedata and theory) and high power (brown data and theory), and momentum-exchange
applied when wave packets are separated (green data and fit).

. . . . . . . . . . . . 95

5.14 Simulated evolution of the wave packets. . . . . . . . . . . . . . . . . . . . . . . . . . 99

xvi
5.15 Simulation of collective recoil. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.1

Experimental overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6.2

Frequency diagram.

6.3

Pair creation.

6.4

Four-photon spectroscopy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

6.5

Initial states for four-photon spectroscopy. . . . . . . . . . . . . . . . . . . . . . . . . 110

6.6

Four-photon spectroscopy with initial states on the equator. (A) The initial Bloch. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

vectors. (B) Interaction induced change in Jz as a function of the four-photon
detuning. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.7

TACT dynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

6.8

Evolution under different Hamiltonians . . . . . . . . . . . . . . . . . . . . . . . . . . 113

6.9

Dynamics near saddle points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

6.10 Parrallel projection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.11 Perpendicular projection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.12 Two-axis counter-twisting with unstable points at north/south poles. . . . . . . . . . 119
6.13 Frequency diagram and projection of dynamics. . . . . . . . . . . . . . . . . . . . . . 119
6.14 Deformation of initial state distributions. . . . . . . . . . . . . . . . . . . . . . . . . 120
7.1

Excess noise in the measured spin projection Jz induced by dressing laser. The two
black lines correspond to quantum projection noise (QPN) for the amount of atoms
used in the experiments, and the combined noise of QPN and photon-shot noise(PSN) related to the detection processes. . . . . . . . . . . . . . . . . . . . . . . . . . 123

7.2

Generating squeezing with dissipation. . . . . . . . . . . . . . . . . . . . . . . . . . . 125

7.3

Three-body interactions with six-photon processes. . . . . . . . . . . . . . . . . . . . 127

Chapter 1
Motivation and overview

Wave-particle duality, the concept that matter can exhibit both particle-like or wave-like
properties, plays a central role in quantum mechanics since its development. Originated from thetheory for describing electrons by Louis de Broglie [2], this concept has been generalized to andverified in many systems, including cold atoms as studied in this thesis. Going beyond the studyof fundamental quantum theory, matter waves are now being applied in quantum sensing [3, 4],which holds great promise for providing new insights into nature, and for searching for undiscovered
physics [5, 6].
To understand how matter waves help improve measurements, it is helpful to take a moment
and think about how do we measure lengths in daily life. We measure the lengths of any objects
with a ruler or measuring tape by counting the ticks on the ruler. Ignoring the details about
how one position the ruler and the objects, the precision of the measurements is limited by how
finely and evenly spaced the ticks are. The finest spacing is typically 1 mm1 . The need of better
length measurements motivate the invention of optical interferometer [7, 8]. Visible lights havecharacteristic wavelengths of λ ≈ 1µm. By separating the light into two paths and recombining
them, the interference pattern changes between bright and dark when the differential path lengthchanges by λ/2. With this new “ruler”, the “ticks” are now separated by about 1µm, which isabout three orders of magnitude better than almost any rulers one can buy.
With matter waves, the wavelength of a particle (de Broglie wavelength) is determined by1

Or 1/16 of an inch if you live in the U.S.

2h
λ = mv, here h is the Planck constant, m and v are the mass and velocity of the particle. Takean atom as an example, for a rubidium atom moving at 5 m/s, the de Broglie wavelength is about
λRb ≈ 1 nm. This is another factor of 1000 better than the optical interferometers, which indicatesthe potential of matter-wave interferometer for performing measurements with high precision.
Staring at the de Broglie wavelength formula, one might ask: “Wait a minute. If I throw a
baseball (roughly 0.1 kg) at 10 m/s, doesn’t that give me an even smaller wavelength?” The caveat
is that the baseball is hot! As compared to the absolute zero, room temperature (roughly 300 ◦ K)
is hot enough such that different particles inside the baseball move at different velocities (let alone
the difference in mass.) Or in other words, there are many tiny “rulers” with different tick spacing,which thus blur out the ticks and make it impossible to read. Going back to the case with atoms,
in order to keep the “atomic ruler” visible, we need keep all the atoms moving at the same speed.
To reduce the random motions of the atoms, we need to cool the particles down. Only after being
cooled to a certain degree2 , typically < 1 µK, can the atoms start exhibiting quantum feature.
Going into the quantum regime, an extra bonus for precision measurement is the new resource
called entanglement, or “spooky action at a distance” according to Einstein. This is another central
concept in quantum mechanics describing correlation between objects. To give an intuition forunderstanding the benefit, we can go back to the “atomic ruler” example. Assuming there are N
atoms in an ensemble, every atom can serve as an individual ruler. When the length does not alignwith any tick, every individual measurement with single atom has an error of 1 mm (tick spacing.)
When performing N different measurements, the uncorrelated error for individual measurement√
can be reduced by N times. Establishing entanglement between atoms can lead to correlation
between measurements and reduce the error even more. For example, before every measurement,
the atoms “know” the error in the former measurement and compensate it by changing its reading,and therefore can lead to even better precision.
Apart from precision measurements, the quantum mechanical features can also enable thestudy of quantum simulation. By designing certain properties and interactions, matter waves can
2

Pun intended.

3
mimic the behaviors of quantum magnets, superconductors [9] and even black holes [10]. For bothquantum metrology and simulation, we need to understand how to manipulate matter waves, how
to establish interactions between different matter waves and how to perform measurements on
matter waves.

Figure 1.1: Controlling matter waves with an optical cavity. Adapted from original graphic in the
public information for 2012 Nobel prize in physics [1]. In this thesis, we explore how to generateinteractions between matter waves by coupling them to a high-finesse cavity formed by two mirrors.

In this thesis, I will present our approach for solving these questions with a platform in whichmatter waves are coupled to a high-finesse cavity. In this system, we are able to use the cavity forentangling momentum states [11][G] 3 , applying one/two-axis (counter) twisting dynamics between
momentum state and binding matter-wave packets together to suppress the Doppler dephasing witha novel collective recoil mechanism [12, 13][G] . As the motivations for these experiments, in thischapter, I will start by highlighting a few concepts including quantum metrology, cavity quantum
electrodynamics and matter waves by answering what they are and why they are interesting. In
the end, I will give an outline of the whole thesis.

1.1

Quantum metrology
Recent developments in quantum science have enabled precision measurements assisted byquantum mechanics [14]. A classic paradigm is sensing with two-level systems. The key to the3[G]: papers from the group (Thompson lab.)

4
precision is the encoding of information in the phase ϕ that appears in the superposition of thetwo levels. The two levels could be two different atomic internal states such as in atomic clocks
or two momentum states in matter-wave interferometers. This phase must be estimated fromquantum measurements to extract the desired information. For N atoms, the phase estimation
is fundamentally limited by the independent quantum collapse of each atom to an rms angular√
uncertainty ∆θSQL = 1/ N rad known as the standard quantum limit (SQL) [15].

Figure 1.2: The quantum state or a two-level system represented by a Bloch vector on a Bloch
sphere. The azimuthal angle on the Bloch sphere corresponds to the relative phase between the
two levels.

Reducing this fundamental quantum source of imprecision would provide a new resource thatcan be exploited to directly enhance measurement precision, bandwidth, and accuracy or operate
at reduced size. The only way to surpass the standard quantum limit set by the quantum collapseof independent atoms is to correlate atoms to each other, also known as quantum entanglement. In
this thesis, I’ll focus on one of the most metrologically useful entangled states, the spin squeezed
state [16, 17]. Specifically, the squeezing generation between matter waves and the applicationin enhancing the precision of matter-wave interferometer. In Chapter 2 and Chapter 4, I will
present two different approaches for generating squeezed state: one-axis twisting (OAT) dynamics
and quantum non-demolition (QND) measurement. The more traditional approach for squeezing

5
generation is to induce a certain many-body interaction such as OAT, and let the system evolve
into a entanglement state. In contrast, QND measurement takes advantage of the other aspect ofquantum mechanics, by performing certain measurements on the system, one can project an unentangled quantum state into a squeezed state. I will here give a minimal example for understanding
this process.
Consider an isolated quantum system with two atoms labeled as 1 and 2, where each has two
possible spin states |↑⟩ and |↓⟩. By preparing both atoms in equal superpositions of the two states√|ψ⟩1,2 = |↑⟩+|↓⟩, the initial state of the system is thus2|ψ⟩i = |ψ⟩1

O|ψ⟩2 =|↑1 ↑1 ⟩ + |↑1 ↓2 ⟩ + |↓1 ↑2 ⟩ + |↓2 ↓2 ⟩
,
2

(1.1)

which is a superposition of all four possible measurement outputs. Let’s assume one can prepare
many copies of the same quantum system and measure the atom number N↑ in |↑⟩. On average,
there is only one atom in |↑⟩. However, each individual measurement could give N↑ =0, 1 and 2,
resulting from the uncorrelated random collapses of the two atoms into the measurement operator
projects each atoms into either |↑⟩ or |↓⟩.
Instead of directly measuring the individual state of the two particles, if one performs a
pre-measurement on the system of the total atom number in |↑⟩ and gets a result of 1, the initialquantum state will then be projected to|ψ⟩s =|↑1 ↓2 ⟩ + |↓1 ↑2 ⟩
√
.
2

(1.2)

This is an entangled state with the two atoms always in different spin states. The second or final
measurements on the system will always give N↑ = 1 with no fluctuation.
Apart from showing how does entanglement reduce the measurement noise, this example also
highlights that, squeezing with QND measurement relies on asking the right question (how many
atoms in |↑⟩) without collapsing the whole system (which atom is in |↑⟩.) Later, we will show how
to satisfy both requirements by using an optical cavity.

6

1.2

Matter waves and atom interferometry
One central concept of quantum mechanics are superposition and coherence, which leads toanother spooky feature of quantum mechanics: delocalization. As described earlier, a single atom
can be in more than one place at the same time, and different parts of the wave function caninterfere with each other. A classic example of quantum sensing with delocalized atoms is theatom interferometer [18]. By exploiting the wave nature of particles, we are now able to use theinterference of de Broglie waves to perform state-of-the-art measurements of gravity [19], rotation[4],fine structure constant [20, 21] and even dark matter and dark energy [22, 6].
The space-time trajectories of Mach-Zehnder matter-wave interferometers are shown in Fig. 1.3.
A single wave packets can be spitted into two, which later spatially separate from each other while
free falling under gravity. The relative phase ϕ between the two matter waves is determined by the
kinetic energies or the difference in couplings to the environment, see Chapter 2 and 4 for details.
The accumulated phase shift can be measured by the population difference between the two wave

Position

packets after applying additional laser pulse when the two wave packets overlap.

Time

Figure 1.3: A matter-wave interferometer is realized by splitting and recombining the wave packets.
Here, the blurry blobs represent the wave packets for different momentum states. The goal is tomeasure the accumulated phase ϕ between different trajectories which depends on gravity.

In this thesis, I will present the first entanglement enhanced matter-wave interferometer bygenerating squeezing between matter-wave packets.

7

1.3

Cavity quantum electrodynamics with cold atoms in an optical cavity
Before getting into the details, I will start by motivating with discussions on what is cavity

quantum electrodynamics (QED) and why is it interesting. Cavity QED is a broad concept forthe study of the interaction between matter and quantized electromagnetic fields. Such systems
generally describe an atom that emits light (or in our case many atoms) coupled to a resonator.
This concept can be generalized to include ions trapped in an optical cavity, superconducting qubits
couple by electrical resonators [23], rare-earth ion doped in photonic crystal [24], etc.
The squeezing between matter wave packets is realized in a cavity quantum electrodynamics
platform with 87 Rb trapped inside a high-finesse Fabry-Perot cavity formed by two macroscopic
mirrors. Such system havs succeeded in generating the most amounts of directly observed entanglement [25, 26] of any experimental platform, including degenerate gases [27, 28, 29], trapped ions[30, 31, 32] to superconducting qubits [23], etc.
Apart from entanglement generation, with the atoms strongly coupled to the photons inside
the cavity, cavity QED systems allow one to exploit all possible tools offered by quantum mechanics: unitary evolution, quantum measurement, and dissipation. Unitary dynamics or many-bodyinteractions between atoms can be mediated by the common cavity mode that couples all the atoms.
Measurements of the atomic states can be achieved by measuring the photonic state, since all atoms
are coupled to the cavity. Dissipation can be induced by engineering cavity field (bath), such thatthe atomic information is lost due to photon leaking out of the cavity. All these capabilities also
make cavity QED an appealing platform for quantum simulation.

1.4

Outline of thesis work
This thesis is organized as follows: In Chapter 2 I will first introduce the theory for under-

standing the atom-cavity coupled system and the basics of spin squeezing and atom interferometry.
In Chapter 3, I will review the design and construction of our apparatus. In Chapter 4, I will discuss an experiment for generating entanglement between atomic momentum states for realizing an8entanglement-enhanced matter-wave interferometer. In Chapter 5, I will discuss a novel many-bodyinteraction directly between two atomic momentum states and a collective recoil mechanism thatarises from this interaction. In Chapter 6, I will discuss an experiment that generalize the interactions between momentum states and realizes Hamiltonian engineering of atomic momentum states.
With the tunable all-to-all Heisenberg XYZ model, the long-sought two-axis counter-twisting dynamics are witnessed at the mean-field level for the first time. In Chapter 7, I will conclude byidentifying a few promising future directions, as well as challenges and possible solutions.
The following are abstracts for the three experiments that highlight our efforts to better
understand and control interactions between matter waves. Going from establishing many-bodyinteractions using the atomic spin state, we can now directly induce interactions between momentum
states with the same spin, and even design arbitrary interactions with quadratic Hamiltonians torealize more interesting and useful dynamics.

1.4.1

Entanglement-enhanced matter-wave interferometer

By tagging the momentum state to atomic internal states, we realize cavity-QED entanglement of external degrees of freedom to realize a matter-wave interferometer of 700 87 Rb atoms
in which each individual atom falls freely under gravity and simultaneously traverses two paths
through space while also entangled with the other atoms [11][G] . We demonstrate both quantum non-demolition measurements and cavity-mediated spin interactions for generating squeezed+0.6
momentum states with directly observed metrological gain 3.4+1.1−0.9 dB and 2.5−0.6 dB below thestandard quantum limit respectively. An entangled state is for the first time successfully injected
into a Mach-Zehnder light-pulse interferometer with 1.7+0.5−0.5 dB of directly observed metrologicalenhancement. Reducing the fundamental quantum source of imprecision provides a new resourcethat can be exploited to directly enhance measurement precision, bandwidth, and accuracy oroperate at reduced size.

9
1.4.2

Cavity-mediated momentum-exchange interactions

Without the atomic internal degrees of freedom, we can also induce exchange interactions
between momentum states through the interplay between the atomic density grating and the cavity.
In this experiment, we realize for the first time momentum-exchange interactions in which atoms
exchange their momentum states via collective emission and absorption of photons from a common
cavity mode [12][G] . The momentum-exchange interaction leads to an observed all-to-all Ising-likeinteraction in a matter-wave interferometer, which is useful for entanglement generation. A manybody energy gap also emerges, effectively binding interferometer matter-wave packets together tosuppress Doppler dephasing, akin to Mössbauer spectroscopy. The tunable momentum-exchangeinteraction provides a new capability for quantum interaction-enhanced matter-wave interferometry
and for realizing exotic behaviors including simulations of superconductors and dynamical gauge
fields.

1.4.3

Hamiltonian engineering of atomic momentum states

In Chapter 6, we show that we can create more complex cavity-mediated interactions, andeventually realize Hamiltonian engineering of XYZ models between momentum states [13][G] . Thisis achieved by adding a second dressing laser tone and carefully controlling it amplitude and phaserelative to the first dressing laser tone. We demonstrate the tunability of the Hamiltonian by
explicitly mapping out the evolution of Bloch vectors on the Bloch sphere at the mean-field level
and witness the two-axis counter-twisting dynamics for the first time. Our work paves the way
towards quantum simulation of more complex systems and fast robust entanglement generation
with momentum states for future quantum enhanced metrology.

Chapter 2
Introduction and theory

This thesis is focused on a system with matter waves coupled to a high-finesse cavity. In
order to establish a basic theory framework for the experiments, this chapter aims to provides introduction on atom-cavity coupling, spin squeezing, and atom interfrometry. I will first introduce
the language for understanding atom-cavity coupling. Secondly, I will discuss quantum sensing
with two level systems, the standard quantum limit, and the concept of spin squeezing. Finally, I
will introduce matter-wave interferometry, and present the mapping from the momentum states toeffective spin states and an intuitive picture for understanding matter-wave interferometers usinga Bloch vector representation of the two paths that the matter-wave packets take as they traverse
the interferometer. Parameters relevant for the experiment will be presented during the discussion
with a table given in the end for summarizing them.

2.1

87

Rb atoms coupled to an optical cavity

As mentioned before, cavity QED systems can allow for unitary dynamics, quantum measurements and dissipations. Here, I will use our experimental system with 87 Rb atoms trappedin a two-mirror Fabry-Perot cavity (as shown in Fig. 2.1) as an example, to give a unified picture
for understanding the cavity-mediated interactions [33] and non-demolition measurements [34, 35]
realized in cavity QED systems. Dissipation is not a core focus of this thesis, but will be discussed
in Chapter 5.

11

Figure 2.1: Rb atoms in a two-mirror high-finesse cavity vertically oriented along Ẑ.

A simplified energy diagram for 87 Rb is shown in Fig. 2.2A on the right (detailed level
diagrams [36] will be given in later chapters when relevant.) The optical excited state is labeled as|e⟩ ≡ 52 P3/2 , F = 3 . The two ground states |↑⟩ ≡ 52 S1/2 , F = 2 and |↓⟩ ≡ 52 S1/2 , F = 1 areseparated by a hyperfine spiting ωHF ≈ 2π × 6.834 GHz. The frequency for the optical transition
connecting the ground state |↑⟩ and the excited state |e⟩ is labeled as ωa with a natural linewidthof Γ = 2π × 6.06 MHz[36] determined by the decay rate of the excited state. The corresponding
wavelength is λ = 780 nm.
For the high-finesse cavity studied in this thesis (Fig. 2.2A, left), the two mirrors are separatedby Lcav = 2.02 cm, giving a free-spectral range (FSR) ωF SR = 2π × 6.788 GHz. The FSR is chosen
to roughly match the hyperfine splitting for driving two-photon transitions (see Chapter. 4 fordetails.) For the atomic transition wavelength λ = 780 nm, the transmission coeﬀicients of the
top and bottom mirror are roughly T1 ≈ 2 ppm and T2 ≈ 40 ppm, which leads to a finesse of
F = 130, 000 after accounting for additional scattering loss (about 2 ppm per mirror.) The cavity
linewidth is κ = ωF SR /F ≈ 2π × 52 kHz. The cavity mode waist is w0 = 72 µm. For an atom
at the anti-node of the cavity mode, the maximal atom-cavity coupling is g0 = 0.48 MHz (for thecycling transition |F = 2, mF = 2⟩ → |F ′ = 3, m′F = 3⟩), which leads to a single atom cooperativity

12

Figure 2.2: Homodyne detection of dressed cavity resonance. (A) Homodyne detection is performedby scanning the frequency of the incident photon in to the cavity and overlap with an local oscillator
beam to measure the cavity induced phase shift. (B) Dressed cavity resonance (green) is shifted byan amount proportional to the number of atoms in the spin up N↑ inside the cavity. Here in thisplot, there are 800 atoms all prepared in |↑⟩ in the cavity. Bare cavity resonance is shown by graylines.

2gof C = κΓof order 1, depending on the specific atomic transitions of interest. For the cyclingtransition |F = 2, mF = 2⟩ → |F ′ = 3, m′F = 3⟩ with the highest Clebsch–Gordan coeﬀicient, thecooperativity is C0 = 0.73, assuming all atoms are at the anti-nodes of the cavity mode. A cavity
mode frequency ωc is detuned from |↑⟩ → |e⟩ transition ωa by ∆a = ωa − ωc .
For the experiments presented in this thesis, the atom-cavity detuning is typically set to be∆a < 2π × 500 MHz, much smaller as compared to the hyperfine spplitting ωHF . Up to smallenergy shift on |↓⟩, we can ignore the coupling between |↓⟩ and |e⟩ for simplicity. With N atoms
in the cavity, the system is described by the Tavis-Cummings Hamiltonian†ĤT C = h̄ωc â â +

N
X
i=1

ωe |e⟩i ⟨e|i + h̄g0

N 
X


â |e⟩i ⟨↑|i + â† |↑⟩i ⟨e|i ,

(2.1)

i=1

where the three terms correspond to quantized cavity field, atomic state, and the coupling between
the two. Here, â† and â are the creation and annihilation operators for the cavity mode, i is theindex for the ith atom.
For simplicity, all atoms are assumed to have the same coupling coeﬀicient g0 with the cavity.

13
Other than trapping atoms in a commensurate lattice, this can also be achieved by performing side
selection on the atoms [37][G] . The spatial dependence of the atom-cavity coupling can modify theeffective atom number, atom-cavity coupling and induce opto-mechanical effect. Here, the optomechanical effects (cavity field changing atomic motion) are unwanted in many experiment, but incontrast, it is crucial for the momentum-exchange interaction and will be discussed in Chapter 5.
We rewrite the Hamiltonian in the rotating frame of the cavity field at ωc , with the HamilP
tonian by Ĥr = h̄ωc â† â + N
|e⟩
⟨e|
i
i
i . In this rotating frame, under the transformation givenÛ †by the unitary Û = exp iĤr t , the Hamiltonian Ĥ ′ = Û † ĤT C Û + ih̄ ∂∂t
Û takes the form:′

Ĥ = h̄∆aNX|e⟩i ⟨e|i + h̄g

i=1

N 
X


â |e⟩i ⟨↑|i + â† |↑⟩i ⟨e|i .

(2.2)

i=1

We can then drive the equation of motion for the operator |e⟩i ⟨↑|i
h

i


d (|e⟩i ⟨↑|i )= ih̄ |e⟩i ⟨↑|i , H ′ = ih̄ −∆a |e⟩i ⟨↑|i + g â |e⟩i ⟨e|i − â† |↑⟩i ⟨↑|i .
dt

(2.3)

In the large detuned limit with ∆a ≫ Γ, κ, g0 , the excited states are not populated and thus
could be adiabatically eliminated with ⟨|e⟩i ⟨e|i ⟩ ≈ 0. Solving for the steady state response withd(|e⟩i ⟨↑|i )dt= 0, we have |e⟩i ⟨↑|i =gâ† |↑⟩i ⟨↑|i.
∆a

Substituting this back to the original Hamiltonian

Eq. 2.2 gives an effective HamiltonianĤef f = h̄χQN D

N
X|↑⟩i ⟨↑|i â† â = h̄χQN D N̂↑ â† â,

(2.4)

i

P
with χQN D = g 2 /∆a . Here, a constant single particle term proportional to N
i |e⟩i ⟨e|i has been
PN
neglected. The collective population operator is defined as N̂↑ = i |↑⟩i ⟨↑|i , which measures thetotal atom number in |↑⟩.
This effective Hamiltonian enables the quantum nondemolition (QND) measurements byintroducing an interaction between the sub-system that needs to be measured (atomic state) and themeasurement apparatus (photonic state of the optical cavity.) With finite population N↑ = ⟨N̂↑ ⟩ inthe cavity, the effective Hamiltonian can be intuitively understood as the cavity resonant frequency
being shift by χN↑ , which is exactly proportional to the population in |↑⟩ as counted by the operator

14N↑ , with χQN D being the cavity frequency shift induced by a single atom. Therefore, by measuring
the cavity frequency shift, we can then measure the collective atomic population. For typicalnumbers in this thesis, the single atom frequency shift is χQN D ≈ 2π × 175 Hz.
Having this picture in mind, we can now design an experiment to perform atom number
measurements by probing the cavity resonances. As shown in Fig. 2.2,A the cavity resonance isprobed using homodyne detection. By scanning the frequency of the photon incident at the cavity,
collecting the retro-reflected photon and measuring the relative phase shift against a local oscillator (LO), an example experimental homodyne signal is shown in Fig. 2.2 with the Q-quadrature
response [38][G] . With no atoms in the cavity, the cavity resonance is centered at zero (gray line).
The cavity frequency is then shifted by about 300 kHz with 800 atoms (green line.) For this experimental data, atoms are prepared in |F = 2, mF = 0⟩ state, the cavity is detuned by 175 MHz
from the atomic transition and is probed by light with σ + polarization.
The cavity-mediate interaction between atoms can be understood with the same Hamiltonian.
We again consider the same experiment, but now apply a pump laser tone with a fixed detuningδpump (∆a ≫ δpump > κ/2) from the dressed cavity resonance instead of scanning the frequencypacross the resonance. With the pump laser amplitude αpump (in unit of photons/second), thedriven cavity Hamiltonian now become∗
e−iδpump t â ,Ĥpump = h̄χQN D N̂↑ â† â + h̄ αpump eiδpump t â† + αpump
with the jump operator L̂ =

√

(2.5)

κâ describing the photon loss of the cavity.

In an empty cavity with no atom inside (⟨N̂↑ ⟩ = 0), we can solve for the steady state intracavity amplitude αcav =

αpump
δ

1+i pump
κ/2

by going into the rotating frame of the pump photon frequency.

With N atoms inside the cavity prepared in equal superpositions of |↑⟩ and |↓⟩, the steady-state
intra-cavity amplitude then becomes
αcav =

αpump
δpump +χQN D (N /2+Jˆz )
1+i
κ/2

,

(2.6)



where Jˆz = N̂↑ − N̂↓ /2 is the population difference operator. With equal superpositons between

15
the two spin states, we can then perform an expansion of the amplitude around ⟨Jˆz ⟩ = 0 and get⟨â† â⟩ ≈|αpump |2(
1+

δpump +χQN D N /2
κ2 /4

)

2

+ χOAT Jˆz .

In the presence of atoms inside the cavity, the laser-cavity detuning thus depends on the atom
number, which then changes the intra-cavity photon number. In other words, the photon number
established inside the cavity now depends on the atomic population difference â† â ∝ Jˆz (up to a
constant term.) After substituting back into Eq. 2.4 and ignoring the constant term, the effective
Hamiltonian becomesĤOAT = h̄χOAT Jˆz2 ,(2.7)

which describes a many-body interaction such that the relative phase between the two spin state
accumulates at a rate proportional to the population difference ⟨Jˆz ⟩ at the mean-field level. Thisis exactly the quadratic Hamiltonian gives rise to the one-axis twisting dynamics [16] for squeezing
generation. See Chapter 4 for details.

2.2

Quantum metrology and spin squeezing
Having discussed the basics of cavity-QED, I will now turn to introduce one of its applications(specifically, entanglement generation) in quantum metrology, starting with quantum sensing andthe definition of standard quantum limit.
The basic principle for quantum sensing with two-level systems is to encode an informationof interest (time, magnetic field, gravity, etc.) into the relative phase between the two levels andmeasure this phase. As an example, the two hyperfine ground state of rubidium atoms can beutilized for sensing time because finite energy difference (h̄ωHF ) between them. For pseudo-spin
systems formed by two levels, it is helpful to understand the representation of quantum stateson Bloch spheres. As an example, the north and south poles of the Bloch sphere can be definedby the two hyperfine ground states |↑⟩ and |↓⟩. Quantum states (for a single atom or many
atoms) are represented by a Bloch vector J⃗ = ⟨Jˆx ⟩x̂ + ⟨Jˆx ⟩ŷ + ⟨Jˆx ⟩ẑ in a fictitious coordinate
space. Here, the collective spin projection operators are defined as Jˆz = N̂↑ − N̂↓ /2, Jˆx =16PN

ˆ = PN (|↑⟩ ⟨↓| − |↓⟩ ⟨↑| ) / (2i). In this picture, a specific
i=1
i
i
i
i

i=1 (|↑⟩i ⟨↓|i + |↓⟩i ⟨↑|i ) /2 and Jy

quantum state |ψ⟩ = cos (θ/2) |↑⟩ + sin (θ/2) eiϕ |↓⟩ is represented by a Bloch vector with a polar
angle θ (opening angle from the north pole) and an azimuthal angle ϕ.
zxyFigure 2.3: Ramsey sequence on Bloch spheres. Starting with an initial state on the equator, theenergy difference between the two levels changes the relative phase. The accumulated phase shift
is turned into population difference with a π/2-pulse for readout.

A classic method for phase measurement is the Ramsey sequence. Let’s first start with all
atoms in the |↓⟩. Applying a π/2-pulse (with microwave or two-photon transition), one can prepare√
an initial state into |ψ⟩ = (|↑⟩ + |↓⟩) / 2, represented by a Bloch vector pointing along x-axis on
the equator. Due to the energy difference, the relative phase between the two state change by
ϕ = ωHF T after certain evolution time T . By applying another π/2-pulse along y-axis, the phase
information is now turned into population difference. By measuring the population difference, we
can then extra relative phase and thus estimated the evolution time T .
In Fig. 2.3, the blurry blob around the Bloch vector represent the quantum noise distribution
due to the random collapse of individual atoms when one makes a measurement of the populations
to determine the spin projection Jz . For example, with a Bloch vector prepared on the equator,
the spin projection Jz measurements give an average of zero. But, the individual measurements√
vary shot-to-shot by N /2. In the large number limit N ≫ 1, the resulting angular uncertainty for√
estimating the polar angle θ is then ∆θ = ∆Jz /J = 1/ N , which defines the standard quantum
limit.
With QND measurements or OAT dynamics, we can create spin squeezed states with reduced

17
noise along one of the quadratures, represented by the elliptical distribution as shown in Fig. 2.4. By
preparing the initial state of Ramsey sequence to be a squeezed state along the phase quadrature,
we can explicitly see how squeezing reduces the noise in the readout process, and thus reduce theuncertainty in estimating the accumulated phase.
zxyFigure 2.4: Ramsey sequence with squeezed state along phase quadrature. Starting with theazimuthal phase-squeezed state, the noise for the final population measure measurement is reduced.

A detailed comparison between QND and OAT can be found in [39][G] . As a result, for any
3, QND is fundamentally better
cavity-QED systems with properly define quantum eﬀiciency η > 16

than OAT for squeezing generation. However, we note that which approach is best to use truly
depends strongly on additional technical constraints and imperfections in an experiment.

2.3

Matter-wave interferometer
Apart from two-level systems based on atomic internal states, one can also prepare quantum

superposition with motional states or external degrees of freedom and realize a matter-wave interferometer. An atom interferometer is a technique that really exploit the wave nature of particles.
Just like optical interferometer, atoms can also be spitted, recombined and interfered. Because
atoms are massive, they are sensitive to inertial quantities unlike photons. They can perform someof the most precise measurements thanks to the ultra-short de Broglie wavelengths.
A simplified diagram for understanding the matter-wave interferometer is shown in Fig.2.5.
Atoms are all in a single momentum state to begin with. A laser pulse prepares the atoms ina quantum superposition of two different momentum states, corresponding to having kicked by18
photons or not. Due to the finite spread in momentum distributions, the two wave packets start toseparate from each other. At the same time, a relative phase starts to accumulate. An extra pulseof laser is applied to flip the momenta of the two wave packets, such that they start to recombine.

T

T

Position

By the time they re-overlap, a final laser pulse is applied for reading out the interference.

Time

Figure 2.5: Mach-Zehnder matter-wave interferometer sequence. Horizontal axis is time, verticalaxis is spatial position. The slopes represent different momentum states. Different colors label thetwo paths of wave packets.

There are different (however equivalent) interpretations of the accumulated phases. In one
picture, one can consider the first laser pulse imprints an optical phase to the atomic density grating.
When the two wave packets recombine at the different location, the interference then serves as aphase memory for comparing to the local laser phase when the readout pulse is applied [40][G] . In
other words, the atomic wave function works as a ruler for measuring the optical phase of the laser
field when traveling through space.
An alternative way for understanding the phase is by tacking the kinetic energy difference
between the two wave packets. Let’s consider the initial momentum state of the wave packet is|p0 − h̄k⟩. The two-photon transition prepare equal superposition between p0 ± h̄k. However, both
wave packet are free-falling under gravity, which results in time-varying momentum p0 ± h̄k + mgt.
Here, g is the gravitation acceleration, m is the mass of the atom and t is the time. The difference

19
in kinetic energy at time t is thus∆E (t) =(p0 + h̄k + mgt)2 (p0 − h̄k + mgt)2−= 2h̄k (v0 + gt) ,
2m
2m

(2.8)

where v0 = p0 /m is the average initial velocity of the two wave packet. This differential kinetic
energy can also be understood as the Doppler shift due to the falling atoms. In order to resonantly drive the two-photon transition connecting the two momentum states |p0 ± h̄k + mgt⟩,
the frequency separation between the two photons needs to be chirped at a rate α = 2kg ≈=2π × 25.1 kHz/ms. Assuming the two lasers tone are separated by ω0 at t = 0, this separation
become ω0 + αt at time t.
For the Mach-Zehnder interferometer sequence shown in Fig. 2.5, the duration T of phaseaccumulated is the same before and after applying the π-pulse for refocusing the wave packets. At
any instantaneous time T ′ through the sequence, the relative phase between the two laser tones fordriving the two-photon transition is
ϕ T′ =Z T′(ω0 + αt) dt

(2.9)

0

The total phase shift between the two wave packets is
ϕ = ϕ (0) − 2ϕ (T ) + ϕ (2T ) = 2kgT 2 ,(2.10)

which scales quadratically with the accumulation period T .
In the experiment, we scan the phase of the final π/2-pulse when the two wave packets
overlap with each other and measure the population in the two momentum states. The populationdifference varis depending on the readout phase and give rise to the fringe of the interferometer.

2.3.1

Interferometric sequence on a Bloch sphere

The Bloch sphere representation of quantum sensing is a powerful way to understand thequantum sensors, and for understanding how squeezed states can improve the sensitivity of suchinterferometers. There exists a one-to-one mapping between the rotations on the Bloch sphereand the laser pulses in matter-wave interferometer sequence. The matter-wave Mach Zehnder20interferometer can be understood as a π/2 − π − π/2 spin echo sequence [41] with pseudo-spinstates.
The initial π/2-pulse that prepares equal superposition of two momentum states corresponds
to the π/2 rotation that prepares a Bloch vector on the equator. The relative phase accumulated
between the wave packets is exactly the change in azimuthal phase on the Bloch sphere. The πpulse that recombines the wave packets together can be understood as a echo π-pulse on the Bloch
sphere that cancel the inhomogeneity that arises from the finite spread in momentum.

2.3.2

Effective spin 1/2 system with momentum states

In this thesis, different Bloch spheres are defined with different choices of two levels. The
specific definitions will be explained in each chapter. Here in Fig. 2.6, we summarized three different
interferometers with different two levels.

Microwave
interferometer

Raman
interferometer

Bragginterferometer

Figure 2.6: Bloch spheres defined with different two levels. A microwave interferometercan be realized with atomic internal two levels, such as the hyperfine states |↑⟩ and |↓⟩. A Raman
matter-wave interferometer is based on spin-momentum states |↑, +h̄k⟩ and |↓, −h̄k⟩. A Bragg
matter interferometer involves only the atomic momentum states |+h̄k⟩ and |−h̄k⟩.

In a Bragg interferomter where atoms are in the same internal states, the interference of the
wave packets (when overlapped) also leads to a spatially varying atomic density distribution. Here,I’ll discuss the connection between the pointing of the Bloch vector and the atomic density grating.

For the works in Chapter 5 and 6 on momentum-exchange interactions and tunable XYZ

21Figure 2.7: Bloch sphere formed by momentum states. The corresponding wave functions
are given for different Bloch vectors on the equator.

models, the atoms are prepared in the same internal states. The effective spin-1/2 system is thendefined by two momentum states |±h̄k⟩ (ignoring the average momentum), which correspond to thenorth and south poles of the Bloch sphere. The polar angle is still determined by the populationdifference between the two momentum states. The azimuthal angle again corresponds to the relative
phase between the two momentum states, which also defines the spatial phase of the atomic wave
function in the context of a Bragg interferomter. As shown in Fig. 2.7, the north/south poles
are |↑⟩ ≡ |ψ+ ⟩ ≡ |+h̄k⟩ and |↓⟩ ≡ |ψ− ⟩ ≡ |−h̄k⟩. The corresponding atomic wave functions are⟨Z|ψ± ⟩ = e±ikZ . Here, |Z⟩ is the position operator, ⟨Z|ψ± ⟩ is the wave function in position space,
and Z labels the position along the cavity axis.
 √For a Bloch vector on the equator, the quantum state is |ψϕ ⟩ = eiϕ/2 |↑⟩ + e−iϕ/2 |↓⟩ / 2
with the corresponding wave function ⟨Z|ψϕ ⟩ = eiϕ/2 cos (kZ + ϕ/2). Therefore, the azimuthal
phase of the Bloch vector also determines the spatial phase of the atomic density grating. This is
an important intuition for understanding the momentum-exchange interaction experiment.

22
2.3.3

Momentum states coupled to cavity

The spatial dependence of the atom-cavity coupling was ignored previously, but we will now
revisit this feature after introducing atomic momentum states and the idea that superpositionsof momentum states give rise to atomic density gratings. Depending on the positions, the atomcoupling coupling coeﬀicient gi of an specific atom i is gi = g0 cos (kxi ), where g is the maximal
coupling at the anti-node, k is the wave vector of the cavity field and xi is the position of the ith
atom along the cavity axis. Allowing for the spatially varying atom-cavity coupling, the effective
Hamiltonian Eq. 2.4 thus becomesĤef f = h̄NX
g 2 cos (kxi )
0

i

∆a

†

|↑⟩i ⟨↑|i â â = h̄NX
g02 e2ikxi + e−2ikxi + 1
i

2∆a


|↑⟩i ⟨↑|i â† â.

(2.11)The Hamiltonian now depends on the spatial distribution of the atoms. In the far detuned limit,
P g02 (e2ikxi +e−2ikxi +1) †all atoms are in the ground state, and therefore Ĥef f = h̄ Nâ â.
i2∆a
In the pseudo-spin system defined by the two momentum states |p0 ± h̄k⟩, the real space
wave functions of the two spin states are ⟨x|p0 ± h̄k⟩ = ei(p0 /h̄±k)xi for the ith atom. The e±2ikx̂iare exactly the momentum displacement operators that change the atomic momentum by ±2h̄k.
Specifically, e±2ikx̂i |p0 ∓ h̄k⟩ = |p0 ± h̄k⟩. Equivalently, they manifest as the raising and lower
operators e±2ikx̂i ≡ ĵi,± for the pseudo-spin system. Collective operators can be defined Jˆ± =PN ±2ikxi ˆPNPˆˆˆ +Jˆ−+ikxi − exp−ikxi . By applying, Jx = J+ +2 J− , Jˆy = J+2i
and Jˆz = Ni ei ĵi,± =i exp
a Bragg π/2-pulse, one can prepare all atoms in superpositions of the two momentum states with⃗ = ⟨Jˆx ⟩, ⟨Jˆy ⟩, ⟨Jˆz ⟩ = N /2.
the same relative phase, such that the length of the Bloch vector |J|
Substituting the collective operators back to the effective Hamiltonian, we have
g02 Jˆ+ + Jˆ− + 1â† â.
Ĥef f = h̄2∆a(2.12)

Here, the cross terms between collective raising and lowering operators give rise to different cavitymediated interaction between momentum states. Details will be discussed later. Specifically, in
Chapter 5, we will focus on the momentum-exchange interaction (described by Jˆ+ Jˆ− ) where pairsof atoms exchange their momenta by exchanging photons. In Chapter 6, we will incorporate the23pair creation Jˆ+ Jˆ+ and pair annihilation processes Jˆ− Jˆ− to realize tunable XYZ models.

Chapter 3
Apparatus

This chapter aims to introduce the apparatus for realizing matter-wave inerferometry in ahigh-finesse cavity. I took part in building the whole setup with two other graduate students
Baochen Wu and Graham Greve. Additional details of the setup can be found in their theses[42, 43][G] . We started building the apparatus in 2018, and the major parts were completed in
2021. The whole setup was moved from JILA X1B21 to X1B30 in early 2024.

Figure 3.1: A picture of the main system after the move.

The whole experimental apparatus can be roughly divided into mainly three sub-systems
to accomplish different requirements for studying science. The first one is the ultra-high-vacuum

25
system. We need to keep the atoms in a low-pressure environment for reducing the collisions or
chemical reactions between the atoms of interest (rubidium in our experiment) and the backgroundgases of molecules, such that the atoms can have long enough lifetime to be trapped and manipulated. The second key subsystem are the various lasers of different wavelengths and spectral
properties that we use to trap/cool the atoms, manipulate the atomic internal/external states andto perform measurements on the cavity and the atoms. The third key sub-system is the electronicsystem for controlling frequencies and powers of all the lasers, and for direct manipulation of atomic
state with microwaves. In this chapter, we will begin by high lighting the typical experimental sequence and then provide detailed descriptions of the three key sub-systems.

3.1

Experimental sequence
I will start by giving an overview of the experimental sequence for preparing the atomic

states before the matter-wave interferometer, which will highlight different requirements for andcapabilities of the apparatus.
For every run of the experiment, we start by cooling and trapping atoms in a two-dimensional
magneto-optical trap (MOT) [44] in one of the glass cells (source cell.) At the same time, theatoms are also continuously pushed by a laser beam (on resonance with 87 Rb |F = 2⟩ → |F ′ = 3⟩transition) from the source cell into the main science glass cell. The atoms are then recaptured,cooled and trapped in a three-dimensional MOT which is positioned at the center of the high-finesse
cavity. The source glass cell and the science glass cell are connected using a long glass tube witha tiny pinhole at the center. With this configuration, we are able to reduce the rubidium vapor
pressure in the science glass cell to prevent coating the high-finesse mirror with rubidium atoms.
Given the small mode waist (w0 = 71 µm) of the cavity, the majority of the atoms in the
3D-MOT are not well coupled to the cavity. We apply polarization gradient cooling (PGC) by
jumping the frequencies of two cooling laser by few hundreds of MHz and to load around 105
atoms into a 1D red-detuned lattice. The lattice is formed by exciting a cavity longitudinal modewith an 813 nm laser. After lowering the red-detuned lattice depth from 1 mK down to 100 µK,

26
we apply gray molasses cooling to further lower the atomic temperature down to 5 µK. An extra
stage of two-dimensional degenerate Raman sideband cooling is applied to reduce the atomic radial
temperature down to 1 µK1 . We then adiabatically ramp off the red-detuned lattice to transfer
the atoms into a blue-detuned 1D dipole trap (trapped depth 5̃0µK) formed by a 760 nm laser.
This dipole trap is always on during the sequence for radially confining the atoms while allowing
them to free-fall.
After free-falling for 20 ms, the average atomic momentum is given by p0 = mv0 with anaveraged velocity v0 =20 cm/s. We then optically pump the atoms into |F = 2, mF = 0⟩ state andperform velocity selection by driving velocity-dependent two-photon Raman transitions connecting|F = 2, mF = 0, p0 + h̄k⟩ → |F = 1, mF = 0, p0 − h̄k⟩. Here, h̄ is the reduced Planck constant andk is the wave number of the Raman laser used to drive this transition. By adjusting the duration
and Rabi frequency of the pulse, a small group of atoms (typically around 1000 atoms) with narrow
momentum distribution (RMS momentum σp < 0.05h̄k) are transferred to |F = 1⟩. The un-selected
atoms are blown away out of the cavity mode with a laser beam (resonant with |F = 2⟩ → |F ′ = 3⟩transition) applied from the side of the cavity.
At this point, all atoms are prepared in the same atomic internal and external state, while
homogeneously coupled to the cavity. The specific sequences for the Raman or Bragg interferometer
may vary between different experiments, and will be discussed in Chapter 4, 5 and 6.

3.2

Vacuum glass cell
As explained before, atoms needs to be prepared in ultra-high vacuum for longer lifetime.

The complete vacuum setup includes an atomic source glass cell (where we form a two-dimensional
magneto-optical trap), a science glass cell which houses the high-finesse cavity (science cavity), a
Titanium sublimation pump, an ion pump and some connection parts. Here I’ll only focus on thescience glass cell.
1

There is no additional cooling for axial temperature. We rely on velocity selection to reduce the momentum
spread along axial direction.

27
Compared to conventional metal chamber, glass cells have large optical access and low electrical conductance (for reducing Eddy current when changing magnetic fields.) It’s however challenging to have anti-reflection (AR) coated on both sides of the cell and to have good optical quality
at the surface with using traditional glass blowing technique. For these reasons, we decided to givethe optical contact technique a shot.
We started with some two-side AR coated quartz plates purchased from Specialty Glass
Products. For the side walls, we have “Fused Quartz Rectangle; 5” ± 0.01” × 2.165” ± 0.01” ×
0.120” ± 0.01” thickness; top and bottom surfaces polished finish and AR coated centered at 780nm
for < 0.2% reflectance at 0 degree AOI and < 0.6% reflectance at 45 degree AOI; 5” long sides
polished finish; 2” long sides ground/machined finish; 3mm max rail/shadow marks due to coating”.
The detailed spec is the same for the top piece but with different dimensions (2.29” ± 0.01” square×0.120” ± 0.01” thick). We then asked the machine shop to make a box out of it. The opticalcontact trick indeed worked to bond two sidewalls to each other, but failed to complete the wholebox. Hans then tried to use silicate bonding to “glue” all edges together. The picture and thedrawing are shown in Fig. 3.2.

Figure 3.2: Picture and drawing for the glass cell

It was working at the beginning but then started to leak. Thin layers of epoxy (Hysol 1C-LV,

28
not recommended due to the low glass-to-metal transition temperature) are applied around all edges
to seal the leaks. We successfully pumped down the system and baked it up to 100 Celsius degree(which is unfortunately around the glass-to-metal transition temperature of this specific epoxy).
When cooling down from the bake, leaks started to develop. We had to use the same epoxy (toavoid reaction or any incompatibility with other epoxy) to seal the leaks again (Nov. 2018.) Aftersealing the leaks, the pressure became acceptable (1.0-2.0×10−8 Torr) to perform experiments thatlast typically less than one second. We then moved on with other construction and eventually theexperiments.
But four years later (Jun. 2022), the pressure rose to 8.5×10−8 Torr and the MOT size
reduced by a lot. Luckily, we were able to fix it again by repeating the same leak checking andsealing process. The predominant leak point is found to be around the connection between the slim
glass tube and the corner of the glass cell. The pressure went back to normal and even survived
the moving of the apparatus from the X1B21 lab to the X1B30 lab across the hall.
Though expensive, there are new companies that can make customized optical contact glass
cells. But since we have verified it ourselves that epoxy-glued 2 glass cell can still work, new parts
are already purchased for making a new cell to prepare for any future failure.

2

Make sure you use Torr Seal!

29

3.3

Science cavity
As shown in Fig. 3.3, the science cavity is form by two mirrors with 5 cm radius of curvature

separated by L =2.02 cm. The free spectral range of the cavity is measured to be 6.787893(1) GHz.
The linewidth of the cavity is measured to be 52(2) kHz, which corresponds to a cavity finesse ofF ≈ 130000 at 780 nm.

Figure 3.3: A rendered image of the science cavity..

The mode waist of the cavity is calculated to be w0 = 71.8 µm, with the associated Rayleighrange of the cavity field zR = 2.07 cm for λ = 780 nm. The mode column of the cavity is
8.934 × 10−11 m3 , which gives the peak atom-cavity coupling g0 = 0.485 MHz. For an atoms at theanti-node of the cavity, the maximal atom-cavity Rabi coupling is 2g0 = 0.970 MHz.

3.4

Laser system
In the experiments, we use different lasers to cool/trap atoms, to perform matter-wave inter-

ferometric control and to probe the atoms and cavity. To address these needs, we require in totalnine lasers including two distributed Bragg reflector lasers (DBR) at 780 nm, three grating external
cavity diode lasers (ECDL) lasers at 813 and 780 nm, one interference filter ECDL at 760 nm, andthree DBR lasers with linewidth narrowed by optical feedback. This is a fairly complex system.
For comparison, the number of lasers is comparable to what are considered to be laser-intense
experiments on molecules.

30
3.4.1

Overview of all lasers and the locking chain

Starting with a grating ECDL laser locked to atomic transition, we stabilized two DBR
lasers to REF laser with offset phase lock for different laser cooling techniques. An interference
filter ECDL at 760 nm is locked to a reference cavity with the Pound-Drever-Hall (PDH) technique[45]. The science cavity is locked to the 760 nm laser such that an intra-cavity dipole trap can beformed for confining the atoms radially while allowing them to under-go free fall along the cavity
axis. An 813 nm Littrow-style grating ECDL is PDH-locked to the science cavity for creating a
one-dimensional red detuned optical lattice for trapping the atom. A narrowed DBR laser named
cavity probe is PDH locked to the science cavity to serve as a tracer beam. Another two narrowed
DBR lasers (atomic probe and Raman lasers) are phase locked to the cavity to have a well-defined
detuning with the cavity for probing the atomic population and driving Raman transitions inside
the cavity. Apart from the lasers described above, there are two extra tampered amplifiers (TA) for
amplifying for the MOT and Raman sideband cooling laser to achieve high enough optical power
for trapping and cooling.
We frequency stabilize the lasers using the chain below, and we will explain each of theseelements in the following sections. A complete drawing of the locking chain is shown in Fig. 3.4.

Figure 3.4: Locking chain for stabilizing all laser frequency.

31
3.4.2

REF, MOT and REP lasers for laser cooling

The REF laser is frequency stabilized to the 85 Rb |F = 3⟩ → |F ′ = 4⟩ transition with modulation transfer spectroscopy [46]. The laser is a grating ECDL to which we apply both current andpiezo tuning. We then stabilize MOT and REP lasers to the REF laser with frequency offset locks.
For the error signal of the frequency offset lock, we first overlap two lasers on a fast photodiode(Hamamatsu G4176 3 ) to measure the beatnote, and send the beatnote into a phase-frequency
detector (PFD) with a RF frequency reference generated by direct digital synthesizer (DDS.) The
PFD (TJ-005)is made by JILA electronic shop, based on HMC-440.
One of the lasers (MOT laser) is mainly for driving the cooling transition connecting |F = 2⟩
ground state and the excited states. The other (REP laser) is mainly connecting |F = 1⟩ to excitedstates for re-pumping. By using offset frequency locks, we can flexibly tune the laser frequencies
by changing the reference RF frequency sent into the PFD. For reference, the laser frequencies canjump by 500 MHz and the lock can recapture within 50 µs. With this agile frequency tuning ability,
we can tune the two laser frequencies for various purposes (Doppler cooling, polarization gradientcooling, optical pumping, blowing atoms away, etc.)

3.4.3

813 nm laser for red-detuned lattice

We use a red-detuned 813 nm 1D lattice to load the atoms from the MOT into the lattice,such that all the atoms in the lattice are maximally coupled to the cavity mode. We directly locka 813 nm laser to the science to create this 1D red lattice.
The 813 nm laser is a grating-style ECDL. In order not to heat up the atoms after releasing
the lattice, we need to adiabatically ramp down the lattice depth, which is realized by a intensity
servo as shown in Fig. 3.5. We use the transmission photodiode signal as the error signal for theintensity lock and feedback on an AOM that controls the power going into the cavity to ramp up
or down the lattice depth.
3This product is now obsolete. We are now using a EO-4000 photodiode for the new phase lock setup.

32

Figure 3.5: Optics for 813 nm laser.

3.4.4

760 nm laser for blue-detuned dipole trap

After releasing the atoms from the red-detuned lattice, we use a 760 nm laser to create a 1D
dipole for trapping the atoms radially while allowing them to free fall. With a blue-detuned laser,
the radial confinement is achieved by exciting the Laguerre-Gauss (1,0) mode [47], which looks likea donut with atoms trapped at center where there is no light. The “dipole trap” is realized byexciting multiple cavity longitudinal modes (separated by FSR), which cancel the standing wave
when only exciting single cavity resonance. The same laser also serves as the frequency reference
for the science cavity. The 760 nm laser is first stabilized to a passive reference cavity with PDH
technique. The science cavity is then frequency locked to this 760 nm laser, such that the cavity
resonances are stabilized.
We need only a few mW of optical power for establishing reasonable trap depths and performing PDH lock. However, this wavelength is not a popular option for commercial lasers, so
we decided to try out a new ECDL design based on interference filter [48] and build it ourselves.
This was an exciting project for me as a new graduate student who had very little experimental

33
experience. The diode was purchased from Eagleyard (RWE-0760-02010-1500-SOT12-0000) and acustom frequency filter (centered at 760 nm with 0.3 nm bandwidth) is purchased from Iridian.
Here is the CAD drawing for the design:

Figure 3.6: CAD drawing for 760 nm laser. The laser diode is installed in the diode holder on theleft. The output light then pass through the interference filter mounted in the half-inch Thorlabs
kinematic mount in the middle. This kinematic mount is installed on a rotational mount foradjusting the angle. The light is focused on an output-coupler (a mirror with 70% transmission)
which then get collimated again, which improves the robustness against beam misalignment. Thebase is about 4-inches long for reference.

As the first interference filter ECDL built in JILA, the performance is satisfying and it has
served till today. The mode-hop-free tuning range can go up to 23 GHz with the PZT driven
by a high-voltage amplifier. Practically, we directly drive the PZT with the “PZT output” fromthe JILA loop filter, and can still get 7 GHz of mode-hop-free tuning range. I did a rough phasenoise measurement on this laser. This is done by coupling and parking the laser on the side of
the transmission peak of a 10 MHz linewidth cavity. The frequency (phase) noise of the laser
is converted into the amplitude noise of the transmitted power. We can then analyze the power
spectral density of the transmission signal to extract the frequency noise of the laser. With thisapproach, I was able to extract a Lorentzian linewidth of 40 kHz.
For exciting the donut-shape trap, we first modify the spatial mode of the laser with a forkpatterned phase plate before coupling into the cavity. The two diffracted beams out of the phase
plate are both LG(1,0) mode. To verify the generation of the LG(1,0) mode, we overlap the LG

34beam with a fundamental Gaussian mode colinearly , the interference pattern is shown in Fig. 3.7
with a spiral pattern resulting from the radial phase distribution of the LG beam.

Figure 3.7: Interference between LG(1,0) mode and a Gaussian mode.

In reality, the spatial eigen mode of the cavity are Hermite–Gaussian (HG) modes insteadof the Laguerre-Gauss modes, which originates from the deformation of the mirrors likely due tostress in how the mirrors are mounted.. In a perfect cylindrically symmetric cavity, LG(1,0) mode
is a superposition of HG(1,0) and HG(0,1) mode [49] which have the same frequency.

=
LG(1,0) mode+
HG(1,0) mode

HG(0,1) mode

Figure 3.8: LG(1,0) mode is a superposition of HG(1,0) and HG(0,1) modes.

In our science cavity, the two HG modes are separated by about 350 kHz. In order tostill create the donut-trap, we couple both diffracted beam from the phase plate to the cavity to35
individually excite both the HG(1,0) and (0,1) mode. To compensate for the 350 kHz frequency
splitting, one of the beams passes through two different AOMs. One AOM shifts the frequency
up by 80 MHz, the other shifts the frequency back down by 79.65 MHz to introduce a 350 kHz
frequency difference from the other beam. To create the PDH error signal, the other beam passthrough an EOM to create sidebands, and the polarization is rotated by 90◦ . The two beams
with orthogonal polarizations are combined with a polarizing beam splitter (PBS) before being
coupled into the science cavity. For canceling the standing wave, we use a fiber EOM to create
sidebands with frequency separation matching the cavity free-spectral range (FSR). By exciting
multiple cavity longitudinal mode [50][G] , we are able to cancel the axial variation of the trap depth
around the center of the cavity to allow the atoms free-fall while being guided at all time. All
associated optics are shown below.

Figure 3.9: All optics for 760 nm laser.

36
3.4.5

Narrowing laser linedwith with optical feedback

The linewidth of our high-finesse cavity is about 50 kHz which is narrower than most of
the lasers either commercially or home made. Over the years, we have learned our lessons that
injecting a broader laser into a narrow cavity is always a bad idea. It can cause problems fromdriving parametric heating to low-eﬀiciency probing and exciting unwanted two-photon transition,etc. Eventually, we identify three different lasers that we really need to narrow down the linewidth.
They are atomic probe, cavity probe and the Raman lasers.
The specific use of these lasers will be described later. I will first focus on the technique we
use for narrowing the linewidth. This trick is also called self-injection-locking which has been rediscovered by many people overtime. We first learned it from Vuletić group [51, 52]. The main idea
is picking off some of the optical power and then sending it back to the laser diode. In this setup,
one can intuitively (although not rigorously) understand it as building an extra cavity for further
selecting (narrowing) the lasing mode. The laser frequency is then dominated by the external cavity
length (with higher pulling coeﬀicient) but not the laser current, and therefore we need to feedbackon the optical feedback path length for stabilizing the laser frequency.

Figure 3.10: Atomic probe optical feedback narrowing setup. Blue lines label the output path, redlines label the feedback path.

37
To achieve wide dynamic range while maintaining fast actuation, we decide to use a piezoactuated mirror for slow feedback and a free-space EOM for fast feedback. An example setup is
shown in Fig. 3.10. The PZT-actuated mirror is formed by gluing a tiny hand-cut mirror 4 to afast PZT (Thorlabs: PA3CKW) and a lead-filled copper mount [53] to increase the bandwidth, asshown in Fig. 3.11.

Figure 3.11: PZT-actuated mirror for reflecting the picked off light.

To verify the performance, the transfer function (i.e. bandwidth and detrimental mechanicalresonances) can be easily measured by a Michelson interferometer.

Figure 3.12: Transfer function for the PZT-actuated mirror with the amplitude response on theleft and the phase response on the right. Different colors correspond to different choices of piezo
and bonding technique with CBond stands for crystal bond. Gluing PA3CKW piezo to the mountwith the Hardman epoxy gives the best performance with no obvious resonance up to 50 kHz.

The most tricky part is to accommodate the fast feedback on the EOM and the slow feedbackon the PZT-actuated mirror. For that, I need to individually estimate the effective gain (frequencychange given feedback voltage) for both the EOM and PZT-mirror, and tune the relative ratio
4

A new batch of tiny mirrors (1 mm by 1 mm by 1 mm in size) are cut by the clean room with a dicing machine.

38
between the two with a homemade voltage divider and frequency filter to eventually have a stable
feedback loop.
For characterizing the linewidth of the narrowed lasers, we first form a beanote between
two narrowed lasers and send the beannote into a microwave delay line interferomter to turn thefrequency noise on the input signal into output voltage noise and measure the voltage power spectral
density. The Lorentzian linewidth is measured to be about 900 Hz 5 , which roughly agrees withdirect linewidth measurement fitted from the beatnote spectrum.

Figure 3.13: Beatnote between two different narrowed lasers.

3.4.5.1

Active stabilization for improving stability

Though having narrow linewidth, this optical feedback setup is not very stable. The reason isdue to the free-running laser diode. Because the laser current/temperature is not being stabilized,
the gain profile of the laser might drift overtime. If the drift is big enough as compared to the FSRof the “feedback cavity”, different lasing mode will start to compete with each other and eventually
drop the lock. Practically, the three narrowed lasers can stay locked for about 30 minutes after
re-locking.
One way to improve the stability is to feedback/feedforward on the laser current such thatthe laser gain profile can follow the lasing mode selected by the “feedback cavity”. Inspired by the5
The relative linewidth between the two lasers are measured to be 1800 Hz. The two lasers are assumed toperforms the same.

39
PDH technique, we came up with a novel scheme for this active stabilization of the laser current.
The setup is shown in Fig. 3.14. The rough idea is that, one can modulate the laser current athigh frequency (70 MHz in our case, 150 MHz is the FSR of “feedback cavity”), pick off some of
the light reflected by the feedback mirror, detect the signal on the photodiode and demodulate itwith a local oscillator to get the error signal.

Figure 3.14: Setup for active stabilization

By scanning the laser current, an error signal (Fig. 3.15) can be found for active stablization of thethe DBR free running laser frequency (via tuning the current) to the resonance frequency of theexternal feedback path.
For centering the gain profile to the resonance of the “feedback cavity”, we need to lock thelaser current to the zero-crossing of the error signal. One might also ask, what happens if one does
not lock to the zero of the error signal? To answer this question, we lock to different offset voltage
and realize the laser linewidth will increase when not locking to the center.
Apparently, the laser linewidth change with the alignment between the gain profile andthe “feedback caity resonances”. Another test for showing improved stability is to perturb the

40

Figure 3.15: Error signal for active stabilization.

Figure 3.16: Measured laser linewdith when locking to different offset.

current/temperature of the laser and see the response of the laser with and without the active
stabilization. As shown in Fig. 3.17, the robustness against temperature change is improved by 30
times. Without the stabilization, the laser will drop lock under a change of 0.016 degree Celsius.
This number is improved to 0.5 degree. About the same amount of improvement can also be seenby the change of laser current. Now the laser can stay locked over 12 mA change of laser current
as compared to only 0.3 mA before.

41
1500

Lorentzian linewidth [Hz]Raman laser linewidth [Hz]

1500

1000
500
No Active feedback
With active feedback

0

-2

-1

0

1

Current drift [mA]2

No active feedback
With active feedback

1000

500

0

-0.2

-0.1

0.0

0.1

0.2

Temperature change [°C]

Figure 3.17: Laser linewidth response to change in current/temperature with/without active feedback.

This extra active stabilization can keep the narrowed laser locked almost indefinitely while
maintained being narrowed. However, this technique is not being applied at the moment limitedby the space and resources. In the future, one could integrate a low-frequency loop filter circuit
and a VCO (frequency source) into a small box to further simplify the setup.

42
3.4.6

Cavity Probe: tracking bare cavity resonanceWe need to send multiple laser tones that are close to atomic resonance into the science cavity
for manipulating or probing the atomic states. These laser tones are only applied for short amountsof time (25 µs to few ms), otherwise they will cause unwanted scattering or AC Stark shift on theatomic states. However, they need to be frequency stabilized to the cavity throughout the whole
sequence. In our experiment, we solve this problem by locking a laser (cavity probe) to a cavity
mode that is far from atomic transition, and then stabilize the other close-detuned lasers to thecavity probe.
The complete optics setup for cavity probe laser is shown below in Fig. 3.18.

Figure 3.18: All optics for cavity probe laser.

The cavity probe wavelength is chosen to be 779.915 nm (160 GHz away from atomic tran-

43
sition) to reduce the perturbation on the atoms. The frequencies of cavity probe, atomic probe
and Raman laser are summarized in Fig. 3.19. For bridging the huge frequency gap (>100GHz)
between cavity probe and atomic probe or Raman lasers, we first send the cavity probe laser intoa fiber-coupled EOM. The EOM is driven by 13.6 GHz with high modulation index, which creates
reasonable amounts of power at the 7th and 12th order sidebands. These two sidebands are closer
to the atomic probe and Raman laser frequency. We then overlap the fiber EOM output withatomic probe and Raman laser to form beatnotes for the frequency offset locks.

Figure 3.19: Frequencies of all three laser relative to atomic transition and cavity resonances. The
6.45 MHz is the FM modulation frequency for creating PDH sidebands on cavity probe laser.

The large detuning (160 GHz) of the cavity probe laser is chosen to reduce the differential
AC Stark shift between the two hyperfine levels, and to avoid unwanted Bragg scattering off thesmall intracavity field. Furthermore, in order to lower the intra-cavity power of the cavity probe,
we use an avalanche photodiode for detecting the retro-reflected beam and lock to the sideband
created by the EOM. In conventional PDH lock setup, one locks the carrier of the laser to the cavity
resonance. However, the slope of the PDH error signal at the sideband frequency is only a factor of
2 lower as compared to carrier regardless of the modulation index. By locking to the sideband, we
can afford much lower intra-cavity optical power while allowing the carrier to act as a heterodyne
reference that amplifies the fundamental photon vacuum noise above the technical noise floor of
the photodiode. Specifically, in our setup, we send only 200 pW power into the cavity (1 µW for44carrier and 200 pW in the sideband) while having the relative frequency fluctuation between thecavity probe and the science cavity to achieve a QND readout noise floor of 1 kHz.
Here is the optical setup forming a beatnote between cavity probe and atomic probe before
moving to X1B30. After moving the experiment, we decide to rebuild the beatnote setup with fiber
splitter (Thorlabs, PN780R5A1) and fiber coupled photodiode (Thorlabs, DX12DF), which should
in principle be much more stable.

Figure 3.20: Atomic probe to cavity probe beatnote setup.

3.4.7

Atomic probe for homodyne detection

Atomic probe is the most “busy” laser in our experiment. The frequency is set to be close
to 87 Rb |F = 2, mF = 2⟩ → |F ′ = 3, m′F = 3⟩ transition with the wavelength around 780.24 nm.
We primarily use it for probing the dressed cavity resonance with a homodyne detection setup.
For the momentum-exchange interaction experiment, we also use the same laser for driving Bragg
rotation and the cavity-mediated interaction. The frequency of this laser is tuned with AOM, EOMor jumping frequency offset lock in different cases. The complete optical setup can be found below

45
in Fig. 3.21.

Figure 3.21: All optics for atomic probe laser.

For homodyne detection, we first split the atomic probe laser into two paths. One goes tothe cavity with the retro-reflected beam overlapped with the other beam (local oscillator, LO) toform the homodyne signal at the detector. The LO beam is shifted by 80 MHz from the original
atomic probe laser. For the beam going into the cavity, it first passed through a fiber-EOM driven
by 80 MHz. The 80 MHz EOM sideband then forms the homodyne signal with the LO beam. The
80 MHz beatnote between the carrier tone and the LO beam is then compared with a reference
80 MHz signal, to detect the relative path length fluctuation between the two paths. The path
length drift is then corrected by changing the VCO frequency (around 80 MHz) that drives the
AOM for the LO beam, to keep the relative phase between the LO and atomic probe beam stable.

46

Figure 3.22: Path length stabilization for atomic probe.

3.4.8

Raman laser for driving two-photon transition

We have a dedicated laser (Raman laser) for driving the velocity-selective two-photon Raman
transition connecting |F = 1, mF = 0, p0 − h̄k⟩ and |F = 2, mF = 0, p0 + h̄k⟩. The wavelength of
the Raman laser is set to be 780.08 nm, which is 80 GHz from the atomic transition. This large
detuning reduces the amount of free-space scattering when driving the two-photon transitions.
Similar to the atomic probe laser, the Raman laser is also frequency stabilized to the cavity probe
laser, such that the laser frequency is stable to within 1 kHz relative to the science cavity.
Starting from the Raman laser output, the laser light is frequency modulated by a fiber EOM
at around 3.417 GHz to create two sidebands with separation matching the hyperfine splitting(6.834 GHz) plus the change in kinetic energy (about 500 kHz) between the two momentum states.
With the cavity FSR=6.788 GHz close to the atomic hyperfine splitting, the two Raman laser
sidebands are symmetrically detuned from two neighboring cavity longitudinal modes by ±23 MHz
to reduce FM to AM conversion, which could cause variation in the two-photon Rabi frequency

47
and drive unwanted Bragg transitions to other momentum states. With 5 mW of optical powerincident on the cavity, we are able to achieve up to 15 kHz of Rabi frequency for the two-photon
Raman transition.

Figure 3.23: Beam path for the Raman laser.

3.5

Microwave and RF system
Other than the generic microwave signal which could be generated by direct digital synthesizer(DDS) and arbitrary waveform generator (AWG), there are three microwave signals that require
special engineering. One is the low-noise microwave signal at around 6.834 GHz for driving thehyperfine transitions. The other two are around 3.417 GHz and 500 kHz for driving Raman and
Bragg transition, which require agile phase/frequency control and the ability of frequency chirping.
Finally, we would also like to drive microwave, Raman and Bragg transitions in a phase coherent
way with respect to each other.
Driving Bragg rotations between |p0 − h̄k⟩ and |p0 + h̄k⟩ requires two different laser tones
separated by the two-photon Doppler shift ωz = 2kv0 where v0 = p0 /m. These two tones are

48

Figure 3.24: Microwave, Raman and Bragg transitions connecting three different energy levels

generated by first red shifting the atomic probe frequency by 75 MHz with one acousto-optic
modulator (AOM) and then blue shifting it back with another AOM driven with two radio frequency(RF) tones at ωRF1 = 75 MHz and ωRF2 = 2π × 75 − ωz MHz. The phase of the Bragg rotation
ϕB = ϕRF2 − ϕRF1 is defined by the relative RF phases of ωRF2 and ωRF1 . To compensate thechanging Doppler shift due to the free-falling, the frequency separation between the two tones are
chirped by 25.11 kHz/ms with a DDS (AD9959) in linear frequency ramp mode.

Figure 3.25: Microwave chain.

For driving the interactions, we need two tones offset by 2ωz with the separation chirped by
50.22 kHz/ms. This is again realized by first red shifting the atomic probe by 75 MHz with one

49
AOM and then blue shifting it back with another AOM. In this case, the second AOM is driven by
two RF tones at ωRF1 and ωRF3 = 2π × 75 − 2ωz MHz. Here, ωRF3 = 2ωRF2 − ωRF1 is generated
by first frequency doubling ωRF2 then mixing with ωRF1 to shift the frequency back down after
RF1
proper frequency filtering. By doing so, we maintain the differential phase ϕint = ϕRF3 −ϕ− ϕB
2between the Bragg rotations and the interactions stabilized. This differential phase can also be
rapidly controlled by the RF phase of ωRF2 .
3.5.1

Generation of low-noise 6.8 GHz signal

A detailed description can be found in [54][G] , the main idea is to start with a 100 MHz
signal generated by a low-noise crystal oscillator (Wenzel ULN series) and then perform frequency
multiplication and frequency filtering to eventually reach 6.8 GHz. The low-offset-frequency noise
and the long-term stability is improved by locking the crystal oscillator to a 10 MHz Rb frequency
standard (SRS FS725).

Figure 3.26: Microwave chain for coherent rotation for microwave, Raman and Bragg transiions.

50
3.5.2

Coherent rotations for all three transitions

In order to drive microwave, Raman and Bragg transitions all coherently with each other,
we constructed a microwave frequency generation (shown in Fig. 3.26) chain to maintain the phasecoherence between the three microwave signal.

3.6

Degenarate Raman side-band cooling
For cavity QED experiments, the atomic temperature is usually not a big concern since

the equivalent energy is much smaller than the energy gap between the spin states. For atominterferometers, the effective axial temperature or the axial momentum distribution indeed mattersa lot for the performance, which we will fix by performing velocity selection along the cavity axis.
The radial temperature also turns out to be an important factor. It will affect the radial distributionof the atomic cloud and lead to a reduced atom-cavity coupling due to the spatial spread. The
inhomogeneous atom-cavity coupling will then lead to an inhomogeneous Rabi frequency for driving
the Raman transition along the cavity axis and causes a reduction of the interferometer contrast.
In our experiment, apart from the standard Doppler cooling, polarization gradient coolingand the Λ-enhanced gray molasses, we applied an an extra step of two-dimensional degenerate
Raman sideband cooling to further reduce the atomic radial temperature to about 1 µK.

Figure 3.27: Relevant energy levels for 87 Rb.

51
Sideband cooling is a popular technique for trapped ion and optical tweezers experiments. In
those systems, the strong trapping potential naturally create a harmonic trap for the atoms. The
rough idea of sideband cooling is that one can use motional quanta to induce spin-flips and thenuse the optical pumping to serve as the dissipation process for resetting the spin state of the atom.
Degenerate sideband cooling (RSBC) [55] is a version of this where the magnetic field difference
between the spin states is degenerate with the trapping frequency. In our implementation, thetrapping is also generated by the very same laser beams. Specifically, in our experiment, the atoms
start in |F = 1, mF = 1⟩ and the cooling is done by driving a two-photon transition connecting|F = 1, mF = 1, n⟩ → |F = 1, mF = 0, n − 1⟩. A optical pumping beam is always on for pumping
the atoms back to |F = 1, mF = 1⟩. All possible path for the two-photon transitions are shown in
Fig. 3.27 with the numbers labeling the Clebsch-Gordan coeﬀicient for each transition.

3.6.1

Polarization twisting

Optical lattices are usually formed by interfering two counter-propagating laser beams withthe same polarization. However, in this configuration, the atoms are trapped at the node or antinode of the standing wave where the electric field gradient goes to zero, which fails to drive thesideband transition that changes the parity of the motional atomic wave function.

Figure 3.28: Polarization twisting for creating Raman coupling. The polarizations of the twocounter propagating beams are rotated by a small angle θ (along the propagation axis) from x̂. ŷ
is perpendicular to x̂ and ŷ, which goes out of the paper. θ is set to be 10◦ in our experiment.

52
To fix this issue, we need to introduce polarization twisting to the two counter-propagating
beams. Here is a 1D example of two counter-propagating beams with orthogonal polarization. Theinterference of the two beams will create two sets of lattices, one in plane and one out of the plane.
For atoms trapped at the anti-node of the black standing wave, they also experience the strong
electric field gradient coming from the other lattice.

Figure 3.29: Polarization twisting for Blue-detuned beams

The example above is for red-detuning beams. Since we already have a background radialtrapping from the 760 nm dipole trap, we decided to use blue-detuned beams (about 80 GHz fromatomic transition) to further reduce the free-space scattering with the atoms trapped at the nodeof the lattice. The polarization twisting can still work for blue-detuning.

3.6.2

Two-dimensional lattice formed by three beams

In our experiment, in order to cool the atoms on both radial axes, we form a hexagonal
2D lattice with three different laser beams separated by about 120 degrees from each other, as
shown in Fig. 3.30. Due to the constraint of space, we could not achieve exactly 120 degrees as
the separations between beams, which could lead to a non-degenerate trapping frequency along thetwo axis x̂ and ŷ. We purposefully introduce a power imbalance between the beams to balance thetrapping frequencies.

53

Figure 3.30: Three beams for Raman sideband cooling (RSBC.)

The pictures of the actual beam launchers (circled red) as shown in Fig. 3.31.

Figure 3.31: Pictures of three beams

3.6.3

Cooling sequence

Different from the free-space RSBC experiment, the atoms are also subjected to a background
dipole trap in our case. After releasing the RSBC lattice, the atoms will start to accelerate towards
the center of the cavity and heat up again. In other words, not only do we need to remove thekinetic energy of the atom, we also need to remove the potential energy.
As shown in Fig. 3.32 (horizontal axis is for time and vertical axis is for trap depth), we
design a effective “delta-kick” sequence for increasing the on-axis density of the atom. In eachcooling cycles,(1) Raman beams create a strong lattice (compared to the radial confinement of the red lattice)
to hold and cool the atoms to ground state of the lattice.

54

Figure 3.32: Timing sequence for RSBC.

(2) Releasing the Raman lattice adiabatically, atoms will regain kinetic energy due to the initialradial distribution.
(3) Wait a period of time that is approximately 1/4 of the radial trapping oscillation period.
(4) Adiabatically turn on the Raman lattice and sideband cool.
By repeating the above processes, we could eventually reduce both kinetic and potential energy toachieve a radial temperature of 0.7 µK which allows for better radial confinement.

3.6.4

Radial temperature measurement with dispersive shifts

The temperature measurements are usually done by time-of-flight imaging. Without relying
on cameras, the high-finesse cavity can actually serve as a lens for “imaging” the radial distributionof the atom. In our experiment, we use the dispersive shift of the cavity for characterizing theradial temperature of the atoms.
Starting with all atoms trapped in the blue-detuned dipole trap, we turn off the trap andlet the atoms start expanding radially. We then measured the cavity frequency shift after various
delay times. Based on how fast the cavity frequency shift changes, we can estimate the radial
temperature.
To derive the expression of the decaying dispersive shift, we start with the atom phase space

55

Figure 3.33: Dispersive shift decay as expansion time increases.

distribution (x, y, px and py are the positions and momentum along x, y axes)
P (x, y, px , py ) = Px (x)Py (y)Ppx (px )Ppy (py )!
− 12 mωr2 x2= norm exp
exp
kB T

− 12 mωr2 y 2
kB T

!


exp

−p2x /2m
kB T


exp

−p2y /2m
kB T

(3.1)
!
,
(3.2)

where norm is the normalization factor, ωr is the radial trapping frequency, m is the mass of
rubidium atom and kB is the Boltzman constant. After time-propagating the distribution, theprobability for finding an atom at position (x, y) at time t after the radial dipole trap has been
turned off isZ
P (x, y, t) =


Px

mωr 2

py 
e 2kB T (1+ω) mω 2
px  
x − t Py y − t Ppx (px )Ppy (py )dpx dpy =.
mm
2πkB T (1 + ωt2 )
−

(3.3)

After integrating the atomic spatial distribution with the cavity mode profile, we havefdisp (t) =mw02 ω 2,
4kB T (1 + ω 2 t2 ) + mw02 ω 2(3.4)

where fdisp is the dispersive shift at time t, w0 is the mode waist of the cavity, ω is the radialtrapping frequency and T is the radial temperature of the atoms.

56

3.7

Relevant parameters for the experiment
The relevant parameters are summarized in the table below
Cavity Parameters
Maximal single-atom vacuum Rabi splitting g
Bottom mirror transmission T1
Top mirror transmission T2
Estimated loss per mirror L
Linewidth κFinesse (780 nm)
Free spectral range
TEM00 waist size w0
Cavity length
Mirror radius of curvature
Atomic Parameters
Excited state decay rate Γ
Ground state hyperfine splitting ωHF
Axial temperature after cooling Taxial
Radial temperature before interferometer sequence Tradial
Thermal RMS cloud radius rrms
RMS momentum spread after velocity selection σp2π × 0.48(3) MHz
40 ppm2 ppm
2 ppm
2π × 52(3) kHz
130,000
6.788GHz
72(1) µm
2.02 cm
4.999(5) cm
2π × 6.06 MHz
2π × 6.834 GHz
5 µK
1.5 µK
3.0(5) µm
0.05 h̄k

Table 3.1: Table for cavity and atomic parameters.

Chapter 4
Entanglement enhanced matter-wave interferometer

Figure 4.1: Artistic conceptualization of entanglement-enhanced matter-wave interferometer.
Atoms (blurry blobs) are entangled to each other. Each atom can be spitted into two paths(colored as red and blue) and get recombined later. Credit: Steven Burrow

Light-pulse matter-wave interferometers exploit the quantized momentum kick given to atoms
during absorption and emission of light in order to split atomic wavepackets so that they traverse
distinct spatial paths at the same time. Additional momentum kicks then return the atoms to thesame point in space to interfere the two matter-wave packets. Such matter-wave interferometers
are exquisitely precise and accurate quantum sensors for a vast range of applications including
inertial sensing and navigation [56, 19, 57], searching for dark matter[6, 5] and dark energy[22, 58],
determination of fundamental constants and the most precise test of a physical theory to date [20,5821], geodesy, geophysics and mineral and hydrocarbon exploration[59], exploring general relativity
with quantum probes [60, 61, 62], and detecting gravitational waves [63, 40, 64].
Quantum entanglement between the atoms allows the atoms to conspire together to reduce
their total quantum noise relative to their total signal during the phase estimation process [16, 17].
Such entanglement has been generated between atoms and ions using direct collisional [27, 28, 65,
66, 67, 68, 69] or Coulomb [70, 32] interactions, with applications to entangled spatial interferometryin trapped geometries [27, 69] and mapping of internal entanglement onto momentum states[71].
We demonstrate for the first time, the realization of cavity-QED entanglement generation
between the external momentum states of different atoms using two distinct approaches that both
rely on the strong collective coupling between the atoms and an optical cavity. In the first approach,
we realize cavity-enhanced quantum non-demolition measurements [25, 26, 72, 34] to essentiallymeasure and subtract out the quantum noise. In the second approach, we utilize the cavity tomediate unitary interactions between the atoms to realize so-called one-axis twisting [16, 73, 74,
75, 76] or an all-to-all Ising interaction. Both approaches have been realized for generating as
much as 18.5 dB of entanglement [26, 25], but only between internal states of atoms and with
only the realization of directly observed enhancements in entangled microwave clocks [77, 78] andmagnetometers [79].

4.1

Experimental setup and overview
Strong collective coupling to the cavity N C ≫ 1 is the key requirement for both approaches

to generate entanglement, where C is the single particle cooperativity parameter [38, 74, 80].
Previously, an interferometer was operated in a low finesse cavity [81, 82], to provide power build-up,
spatial mode filtering, and precise beam alignment. Here, we achieve matter-wave interferometriccontrol [3, 83] simultaneously with strong collective coupling N C ≈ 500 by operating inside a highcavity finesse F = 1.3 × 105 with small mode waist w0 = 72µm.
Our two-mirror cavity is vertically-oriented along Ẑ (Fig. 4.2). The cavity has a power
decay rate κ = 2π × 56(3) kHz, mirror separation L = 2.2 cm, and free spectral range ωFSR =59

Squeezing
light

Raman
tones

Homodyne
readout

Entangled anddelocalized atoms

Figure 4.2: Laser cooled rubidium atoms are trapped inside a high-finesse cavity. The
atomic wavepackets are split and recombined by driving two-photon Raman transitions to provide
quantized momentum kicks to the atoms. (inset) Intracavity atomic probe light generates entanglement between the atoms via either one-axis twisting dynamics or quantum non-demolition
measurements made by detecting the reflected atomic probe field with a homodyne detector.

2π × 6.7879 GHz. Rubidium atoms are loaded into a red-detuned 813 nm standing-wave intracavity
lattice and laser-cooled to a radial temperature of 0.7(3)µK. The lattice is adiabatically reduced
to allow the atoms to accelerate under gravity for a duration Tfall , guided tightly along the cavity
axis by a hollow (Laguerre-Gauss LG01 -like) blue-detuned optical dipole guide[50] with thermal
rms cloud radius of rrms = 3.0(5) µm ≪ w0 .
The generation and injection of the entanglement into a Mach-Zehnder matter-wave interferometer are shown by the space-time and Bloch sphere depictions in Fig. 4.3. Squeezing is first
generated in the population basis, and then a Raman beam splitter pulse orients the squeezing forenhanced interferometer phase sensitivity. The two paths (red and blue) accrue a relative phase ϕover time Tevol , the mirror pulse serves to reoverlap the wavepackets, and the readout beam splitter
pulse creates interference that is read out as a population difference with sub-standard quantum
limit sensitivity. Representative noise distributions are depicted on the Bloch sphere for various

60
Momentum squeezing
via QND or OAT
Generate
entanglement

Matter-wave interferometer
BeamsplitterMirrorpulse

Homodyne
readout

Readout
beam
splitterFigure 4.3: Space-time diagram and the corresponding Bloch sphere depictions for the entangled
matter-wave interferometer.

points in the interferometer.

4.2

Manipulating atomic momentum states.
We manipulate matter-wave wavepackets using velocity-sensitive two-photon transitions withwavelength λ = 780 nm. The combined absorption and stimulated emission of photons imparts
2h̄k momentum kicks along the cavity axis, where k = 2π/λ and h̄ is the reduced Planck constant.

Figure 4.4: The empty-cavity resonance (green) is detuned by δc from the |↑⟩ → |e⟩ transition ωa .
The Raman tones (blue) injected into the cavity drive a spin-changing |↑⟩ ↔ |↓⟩ transition with
two-photon detuning δ defined in a falling reference frame.

61
For Raman transitions in which both momentum and spin states are changed, we utilize themagnetically-insensitive 87 Rb clock states, |↓⟩ ≡ |F = 1, mF = 0⟩ and |↑⟩ ≡ |F = 2, mF = 0⟩, separated by the hyperfine transition frequency ωHF ≈ 2π × 6.835 GHz. The driving laser’s frequency isstabilized between two TEM00 longitudinal modes approximately ∆ = 2π × 85 GHz blue-detunedof |↑⟩ → |e⟩ ≡ 52 P3/2 , F = 3 . The cavity free spectral range is tuned such that two sidebands at±ωR are approximately ±23 MHz from resonance with the closest TEM00 mode when 2ωR = ωHF .
This configuration allows enough light to nonresonantly enter the cavity for a two-photon Rabi
frequency Ω = 2π × 10 kHz. By injecting the Raman tones non-resonantly and with opposite detunings, we greatly suppress laser frequency noise from being converted into phase and amplitude
noise inside the cavity. Such noise manifests as noise in the Raman rotations and undesired Bragg
scattering to other momentum states.

Raman tones

Frequency Figure 4.5: The Raman tones are derived from a laser locked between two adjacent TEM00 modes(grey) and modulated at ωR ∼ ωHF /2, leaving them detuned from the cavity resonances by ±23
MHz, close enough to resonance to allow light to enter the cavity while also suﬀiciently detuned to
avoid strong laser FM to intracavity AM and PM conversion that would interfere with manipulating
the matter-waves with a precision below the standard quantum limit.

As the atoms fall under gravity inside the cavity, the relative Doppler shift for light propagating upwards versus downwards chirps linearly in time. We compensate this effect by linearly ramping the instantaneous frequency of the sidebands as 2ωR = ωHF + δ − b(t − tvs ) with62
b = 2kg = 2π × 25 kHz/ms with the local acceleration due to gravity g = 9.8m/s2 , δ the two-photon
detuning in the falling frame of reference, and tvs the time at which we will apply the first π pulse
for velocity selection described below (see Fig. 4.5).

Resonant atoms[atoms/kHz]

600

800 atoms,
Δp=0.1 ħk

400
200
0
-600

-500

-400

-300

-200

-100

0

Two-photon detuning [kHz]Figure 4.6: Atoms are prepared in |↓⟩ and allowed to fall for a duration Tfall = 7.5 ms (orange)or 15 ms (blue). The Raman coupling is applied at a fixed detuning δ, after which the number ofatoms in |↑⟩ is measured, revealing the axial velocity distribution. The full-width half-maximum ofboth distributions corresponds to a momentum spread of 5ℏk, too broad for interferometry. During
velocity selection, a group of about 800 atoms with rms momentum spread ∆p = 0.1ℏk (red) are
kept from the latter distribution while the rest are removed with transverse radiation pressure.

In Fig. 4.6, we show the initial axial velocity spectrum of the atoms as mapped out by
inducing velocity-dependent spin flips. We use this same process to select atoms within a narrow
range of initial velocities for coherent manipulation of matter-waves in the remainder of this work.
After falling for Tfall = 15 ms, atoms are optically-pumped to |↑⟩, and the two-photon detuning
is set to δvs = −400 kHz to transfer a group of atoms to |↓⟩ from the center of the axial velocity
distribution [84]. Atoms in |↑⟩ are removed by a transverse radiation pressure force. The velocityselected atoms are returned to |↑⟩ with a Raman π pulse and the selection process is repeated,
resulting in approximately N0 = 800 − 1200 atoms in |↓⟩ with rms momentum spread ∆p = 0.1h̄k
set by choice of the two-photon Rabi frequency Ω = 2π × 1.4 kHz.
In Fig. 4.7 we demonstrate the quantized nature of the momentum kicks imparted by the
intracavity Raman transitions. After velocity selection, a π/2 pulse is followed by a second Raman

EnergyResonant atoms[atoms/kHz, arb.]

63

-30

0

30

60

Two-photon detuning90

0 2 4 6 8 10

[kHz]

Momentum []Figure 4.7: After velocity selection, a pair of Raman transitions can be used to place atoms intoa superposition of |0ℏk, ↓⟩ and |4ℏk, ↓⟩. Raman spectroscopy is used to verify the discrete velocity
distribution.

π pulse to place the atoms into a superposition of |0h̄k, ↓⟩ and |4h̄k, ↓⟩ in the falling frame ofreference. We observe this as two distinct peaks separated in the subsequent velocity spectrum.
Though not leveraged here, future interferometers might evolve in such superpositions so as tominimize systematic errors and dephasing due to environmental couplings to the spin degree ofEnergyResonant atoms[atoms/kHz, arb.]

freedom.

-30

0

30

60

Two-photon detuning

90

120

150

0 2 4 6 8 10

[kHz]

Momentum []Figure 4.8: Large momentum transfer with sequential Bragg pulses. Bragg transitions canbe driven by adding amplitude modulation to the Raman tones. Here, a Bragg π/2 pulse splits
the wavepacket, and consecutive π pulses transfer additional momentum to create a superposition|0ℏk, ↓⟩ and |2nℏk, ↓⟩ with the momentum difference as large as 10ℏk shown here.

Complementary to hyperfine spin-state changing Raman transitions, we also demonstrate

64
intracavity Bragg transitions in this high finesse and high cooperativity cavity. The Bragg coupling
connects states |nh̄k⟩ ↔ |(n + 2)h̄k⟩ with no change in the spin degree of freedom, as shown in
Fig. 4.8. The Bragg transitions are driven by two laser tones derived from the same laser withdifference frequency ωB = δvs − b (t − tvs ). After velocity selection, the wavepacket is coherently
split by a Bragg π/2 pulse, followed by successive π pulses to transfer momentum to one of the
wavepacket components for a momentum difference of up to 10 h̄k. Access to Bragg transitions
opens the door to both large momentum transfer operations for greater sensitivity and to improved
coherence times in future work.

4.3

Squeezing on momentum states.
With the ability to manipulate matter-waves in our cavity, we turn our attention to creat-ing entanglement between atoms that includes this external degree of freedom. We describe thecollective state of our matter-wave interferometer using a Bloch sphere with average Bloch vectorJ⃗ = ⟨Jˆx x̂ + Jˆy ŷ + Jˆz ẑ⟩ of length J ≡ J⃗ ≤ N0 /2 in a fictitious coordinate space[85]. The collec
tive pseudospin projection operators are defined as Jˆz ≡ 21 N̂↑ − N̂↓ with collective population
P N0
P 0
projection operators N̂↑ = N
i |b⟩i i ⟨b|, and similarly for other pseudospin
i |a⟩i i ⟨a| and N̂↓ =projections. In this work, |a⟩i = |2ℏk, ↑⟩i and |b⟩i = |0ℏk, ↓⟩i for the ith atom.
We use a Raman π/2 pulse to nominally prepare all atoms in an unentangled coherent spin
1
0√state |ψ⟩ = ΠN(|a⟩i + |b⟩i ) described by the Bloch vector J⃗ = J x̂. The quantum noise thati2will appear in a measurement manifests in the non-zero variance of the spin projection operators(∆Jz )2 = ⟨Jˆz2 ⟩ − ⟨Jˆz ⟩2 ̸= 0, etc. and is visualized on the Bloch sphere as a quasi probability
distribution of the orientation of the Bloch vector from trial to trial. It is the ratio of rms spin
projection noise amplitude to the length of the Bloch vector that sets the standard quantum limit
in the quantum phase estimation of the polar and azimuthal angles.
The Wineland parameter characterizes the phase enhancement of a squeezed state with phase

65
uncertainty ∆θ that is certified to arise from entanglement between the atoms [17]

2
∆θ
W =
.
∆θSQL

(4.1)

Physically, it is the reduction in the angular noise variance of the phase estimation relative to√the standard quantum limit ∆θSQL = 1/ N one would have for a pure state with a Bloch vector
length Jc = N /2 equal to that of the actually prepared mixed or partially decohered state absent
the squeezing operation.
We prepare squeezed momentum states through two separate cavity-based interactions: quantum non-demolition (QND) measurements [38, 25, 26] and one-axis twisting (OAT) [16, 73, 74]. In
both cases, quantum noise is reduced in one spin-momentum projection at the expense of increased
quantum noise along the orthogonal projection.
Population readout is achieved through collective or QND measurements of the free falling
atomic samples that ideally give information about the fraction of the atoms in different spinmomentum states without revealing single-particle information [38, 50][G] . The two momentum
states interact differently with the optical cavity because they carry distinct spin labels. We tunea TEM00 cavity mode with resonance frequency ωc to the blue of the |↑⟩ → |e⟩ transition ωa byδc = ωc − ωa . After adiabatically eliminating the excited state |e⟩ and ignoring mean-field lightshifts that will be spin-echoed away, the effective Hamiltonian[74] describing the atom-cavity QND
interaction can be expressed in a rotating frame at the atomic transition frequency as
ĤQND = δc + χQND N̂↑ ĉ† ĉ

(4.2)

where the cavity field is described by creation and annihilation operators ĉ† and ĉ. The QND
interaction generates entanglement between the cavity field and the collective population N↑ in |↑⟩so that measuring the phase of the field that emerges from the cavity allows us to determine thepopulation N↑ . Alternatively, atoms in |↑⟩ create a round trip phase shift of the intracavity lightthat causes the cavity resonance to shift by an amount χQND = 2π × 336(2) Hz per atom in |↑⟩ at
a detuning δc = 2π × 175 MHz and accounting for interaction of the cavity mode with all excitedstate hyperfine levels.

QND

-100

OAT

0

100

200

300

bare cavity resonance [kHz]400

Homodyne
readout [
]

66
1
0
-1
-300

0

300

600

bare cavity resonance [kHz]Figure 4.9: Left: frequency diagram for QND and OAT. Right: Homodyne signal when scanningatomic probe laser across cavity resonance with or without atom in the cavity

The cavity frequency shift is estimated by detecting the Q quadrature of probe light reflectedfrom the cavity input mirror as the laser frequency is swept across resonance (Fig. 4.9). A typicalmeasurement lasts 150µs. The population N↓ of atoms in the momentum state with spin label |↓⟩ ismeasured with the same technique after transferring the atoms to |↑⟩ using a Raman π pulse. The
Raman π pulse serves the additional functions of re-overlapping the wavepackets and cancelling the
average light shift of the probe.
Collective QND measurements are used in creating conditional spin squeezing. The spinmomentum projection in the population basis is measured once with the pre-measurement outcome
Jzp = 12 (N↑ − N↓ ) |pre . The same projection is then measured a second time with the final measurement outcome labeled Jzf = 21 (N↑ − N↓ ) |fin . Each final population measurement is made after
first optically pumping atoms in |↑⟩ to |F = 2, mF = 2⟩ to achieve lower readout noise (estimatedat more than 15 dB below the projection noise level) by using the optical cycling transition to|F = 3, mF = 3⟩.
If suﬀiciently precise, the pre-measurement localizes the state to below the initial coherent
spin state level, producing a squeezed state. It can be considered a measurement of the quantum
fluctuation of the orientation of the state on a given trial, and the measurement outcome canthen be used to partially subtract the quantum fluctuation from the final measurement outcome
by considering the difference Jzd = Jzf − Jzp . The quantum fluctuation is common to the two

67
measurements, but any rotation of the state (i.e. the signal) that occurs in between the two measurements appears only in the final measurement outcome so that one can estimate the angulardisplacement as sin(θ) ≡ Jzd /Js . The length of the Bloch vector Js after the pre-measurement
has prepared a squeezed state is measured in a separate set of experiments in which a π/2 pulse
about azimuthal angle ϕ is inserted between the pre- and final measurements. The length of the
Bloch vector is estimated from the fringe amplitude of Jzf versus ϕ as it is varied between 0 to 2π.
The initial length of the Bloch vector Jc needed for estimating the spectroscopic enhancement isestimated in the same manner, but without the pre-measurement applied.

(A)

(B)

1

QND spectroscopicenhancement

0

CSS

3

Standard
quantum
limit

2

1

8
6
5
4
3

QND
2

3

4

5 6 7 89

10³Interacting photons

2

3

OAT

Figure 4.10: (A) Contrast lost (top) and spectroscopic enhancement W (bottom) as a functionof the photon number inside the cavity. (B) State tomography of coherent spin state (CSS) andsqueezed states generated by QND and OAT.

Fig. 4.10(A) shows the spectroscopic enhancement W versus the strength of the QND interaction as parameterized by Mi , the average number of incident photons that enter the cavity
during each population pre-measurement window. At low Mi , the probe’s vacuum noise limits the
spectroscopic enhancement, while at high Mi , the spectroscopic enhancement is limited by freespace scattering of the probe light that leads to a reduction in Js and transitions to other groundstates that decorrelate the pre- and final measurements. Near Mi = 600, N = 1170(30) atoms, and+1.1
δc = 2π × 175 MHz, we achieve W = 0.46(11) or 3.4−0.9
dB of directly observed squeezing in the68
momentum-spin basis.
We also realize momentum-spin entanglement via cavity-mediated interactions [73, 74, 80].
The one-axis twisting (OAT) Hamiltonian [16]ĤOAT = χOAT Jˆz2

(4.3)

is generated by applying a fixed frequency drive tone offset from the average dressed cavity resonanceby δp ≳ κ/2. Briefly, the populations in each momentum-spin state tune the cavity closer to or
further from resonance with the fixed frequency drive tone, allowing more or less light into thecavity. After adiabatic elimination of the dressed cavity mode and neglecting terms that will be
spin-echoed away, one finds that to first approximation, ĉ† ĉ ∝ N̂↑ . As a result, the spin-dependent
portion of the QND Hamiltonian is transformed after a spin-echo pulse into Eq. 4.3, a pure spinspin Hamiltonian proportional to N̂↑2 + N̂↓2 = 4Jˆz2 with the cavity mode acting as a mediator of theinteraction. The unitary interactions drive shearing of the atomic quantum noise distribution witha resulting squeezed state minimum noise projection oriented at a small angle α0 from ẑ (Fig. 4.11).
To suppress free-space scattering, it is ideal to operate at δp = κ/2, however, we work atlarger detunings. First, this reduces deleterious QND interactions (or equivalently, photon shotnoise from the applied drive tone) that were neglected in the above description of the emergenceof the unitary dynamics [86]. Secondly, this allows operation in a linearized regime even in thepresence of shot-to-shot total atom number fluctuations. We empirically find an optimum detuningof δp = 2.7 × κ/2 with χOAT ≈ 2π × 10 Hz.
After the OAT interaction is applied, the state is rotated using the Raman beams to couple
the momentum-spin states so that the minimum noise projection is along ẑ. The momentumspin populations are destructively readout out as described previously with measurement outcome
labeled Jzf . The Bloch vector lengths Js (Jc ) with (without) OAT squeezing are measured in
the same manner as was done for the QND measurements by measuring the fringe amplitudeof Jzf versus the azimuthal phase of a π/2 rotation prior to the final readout. We achieve a
directly observed spectroscopic enhancement from OAT of W = 0.56(8) or 2.5+0.6−0.6 dB. The optimal

69
configuration was realized with Mi ≈ 700 photons, δc = 2π × 350 MHz, and N = 730(10) atoms.

4.4

Wineland criterion
Here in this section, we will justified the definition Eq. 4.1 presented before. The Wineland

criterion is often presented in the following form [26]W =(∆Jz )2 Ci,
2∆Jz,SQL
Cf2(4.4)

where the contrasts are related to Bloch vector lengths here by Ci ≡ 2JC /N0 and Cf ≡ 2Js /N0
for total atom number N0 . By rearranging terms, it can also be expressed in a more physicallyz
meaningful form as the ratio W = (∆θ/∆θSQL )2 between the observed angular resolution ∆θ = ∆J
Js√√with entanglement and the standard quantum limit ∆θSQL = 1/ N ≡ 1/ 2JC for a pure statewith the same Bloch vector length JC as that of the actual mixed state when entanglement is notcreated.
We now establish the connection between the spin operators and actual experimental measurements. We define the cavity frequency shifts induced by a single atom in |F = 2, mF = 2⟩,|F = 2, mF = 0⟩ and |F = 1, mF = 0⟩ as χ2 , χ0 ≡ χQND and χ↓ respectively.
For OAT squeezing, we estimate the angular resolution ∆θ after the squeezing generationor the full squeezed interferometer sequence as follows. To measure the final spin projection Jzf ,we optically pump the atoms in |↑⟩ to |F = 2, mF = 2⟩, measure the cavity frequency shift withoutcome labeled ω1f , blow away atoms in |F = 2⟩, apply a Raman π pulse, optically pump the atomsin |↑⟩ to |F = 2, mF = 2⟩, and measure a second cavity frequency shift with outcome labeled ω2f .
We estimate the final spin projection Jzf from the difference between the two cavity frequency shifts
Jzf =ω1f −ω2f
χ /2− χϵ2 ω2f , where ϵ = χ↓ 2 . To convert the spin projection Jzf into an estimate of the2χ2

Bloch vector polar angle θf , we measure the length of the Bloch vector Js by scanning the azimuthal
phase ϕ of the readout π/2 pulse. In the case of the squeezed interferometer, this is the final π/2
pulse of the interferometer and just prior to the measurement Jzf . In the case of OAT squeezed state
generation, this is an added π/2 pulse after the squeezing and just prior to the measurement Jzf . We

70
fit the resulting differential cavity frequency shifts (ω1f − ω2f ) |ϕ to the function y0 +Af sin (ϕ − ϕ0 ).
A

f
The Bloch vector length is then estimated by Js = 2χ2 −χ
. The Bloch vector polar angle θf from↓J

the final measurement is thus estimated by θf = Jzfs =resolution ∆θ is approximated as ∆θ = ∆θf ≈ω1f −ω2f
ω +ωω
− ϵ 1fAf 2f + 2ϵ2 A2ff . The angularAf∆(ω1f −ω2f ), where we note the scale factors χ2Afetc. are canceled at the order of ϵ0 . With a typical |ϵ|< 1/50 and the fractional total numberω +ωfluctuation ∆ 1fAf 2f being less than 0.03, the corrections of order ϵ1 would need to be included
for squeezing 30 dB below the SQL.
For the QND measurements, we perform pre-measurements to localize the quantum state
and use the final measurements to verify the squeezing generated by the pre-measurements as
described before. The phase resolution is defined as the phase fluctuation between the pre- andfinal measurements ∆θ = ∆ (θp − θf ). The Bloch vector polar angle of the final measurements
θf is estimated as in the OAT measurement with the atomic population optically pumped to|F = 2, mF = 2⟩. For the pre-measurements, we measure pairs of cavity frequency shifts ω1p andω2p separated by π pulses but without the optical pumping so the atomic population is in |↑⟩during the cavity frequency shift measurements. The spin projection Jzp in the pre-measurements is
ω

−ω

2p
estimated from the differential frequency shift Jzp = 2 1p
. The length of the Bloch vector Js just(χ0 −χ↓ )after the pre-measurement is measured by adding a π/2 pulse just after the pre-measurement andscanning its azimuthal phase ϕ, after which we perform a single cavity frequency shift measurementwith outcome labeled ω1f |ϕ . We then fit the resulting fringe to the function y0 + Ap sin (ϕ − ϕ0 )Apand estimate the Bloch vector length Js = χ0 −χ
. The Bloch vector polar angle θp is evaluated↓Jθp = Jzp=pω1p −ω2p
2Ap . As before, the angular phase resolution is suﬀiciently approximated by keeping

only to the order of ϵ0 as ∆θ = ∆ (θp − θf ) ≈ ∆ω −ω
ω1p −ω2p− 1fAf 2f
2Apwith no dependence on scale

factors χ2 , χ0 or χ↓ .
For estimating the standard quantum limit ∆θSQL , we measure the length of the Bloch vector
Jc = Js |Mi =0 =Ap |Mi =0χ0 −χ↓

using the same sequence for measuring Js in the QND pre-measurements

described just above but setting the photon number Mi to zero during the pre-measurements
or squeezing for QND measurement or OAT respectively. To estimate the standard quantum

71√√
limit ∆θSQL = 1/ N = 1/ 2Jc we therefore need to know accurate values of χ0 and χ↓ . To
B1
2
suﬀicient approximation χ0 = g 2 Bδc3 + δcB+δ+δc +δ1 with atom-cavity coupling g discussed below,
26
hyperfine splittings δ2 = 2π × 266.7 MHz, δ1 = 2π × 423.6 MHz and branching ratios B3 = 15
, B2 =
1= 60of the excited states |F ′ = 3, 2, 1, mF = 1⟩ to the ground state |↑⟩ transition that


B
B1,↓
interact with the probe light. To suﬀicient approximation χ↓ = g 2 δc +δ22,↓
with
+
−ωHF
δc +δ1 −ωHF
3
12 , B1

3
5
branching ratios B2,↓ = 12
, B1,↓ = 12of the |F ′ = 2, 1, mF = 1⟩ to the ground state |↓⟩ transition.

Though not used in the calculation, the cavity frequency shift from a single atom in |F = 2, mF = 2⟩2

is approximated by χ2 = gδc for the cycling transition between the excited state |F = 3, mF = 3⟩
and ground state |F = 2, mF = 2⟩.
The maximum single-atom vacuum Rabi splitting 2g0 = 2

q

2D 2 ωc= 2×2π×0.4853(5) MHz
πLw02 ϵ0 h̄[34] with fractional uncertainty dominated by the fractional uncertainty (1.1 × 10−3 ) on the dipole
matrix element D for the |F = 2, mF = 2⟩ to |F = 3, mF = 3⟩ transition, and ϵ0 the vacuum permeability. The cavity length L and mode waist w0 are determined very precisely by measuring the free spectral range and transverse mode frequency splitting. Since the atoms traverse
many standing waves of the cavity during the measurement windows, we can coarse grain over theg0 √estanding waves to arrive at a time-averaged spatially dependent coupling gt (r, z) = √2

2−r 2 /w0

1+

(

z
Zr

)2 ,

where Zr = 2.1 cm is the Rayleigh range of the cavity [50]. The effective single atom-cavity coupling frequency is given by the ensemble averaged moments of the spatially dependent gt (r, z) asr⟨gt (r,z)4 |gt (r,z)4 ⟩g0
g=(1 − fcor ) = 2π × 0.341(2) MHz [34]. The final fractional uncertainty= √2⟨gt (r,z)2 |gt (r,z)2 ⟩(6 × 10−3 ) on g is dominated by the uncertainty on the correction factor fcor ≈2
z02 +σz2
rrms+, where2
2Zr
w02

z0 = 1 (2) mm is the axial position of the cloud relative to the cavity center, σz = 0.5(3) mm is
the RMS axial spread of the cloud, and rrms is the RMS cloud radius of the atoms. The fractional
uncertainty on g contributed from z0 , σz and rrms are 5 × 10−3 , 4 × 10−3 and 2 × 10−3 respectively.
The uncertainties on the cavity detuning δc = 175(2) MHz or 350(2) MHz lead to fractional
uncertainties ≤ 0.01 on χQND − χ↓ . Because the atoms move along the cavity axis, the probe
light is Doppler shifted by of order δvs /2; however, here δc ≫ δvs so that there is only a negligible

72
fractional correction to χQND of order (δvs /2δc )2 ≲ 10−6 . The effect of spread in momentum states
is even more negligible.
Combining uncertainties from g and δc , the fractional uncertainty on χQND − χ↓



is ≤

1.4 × 10−2 . This uncertainty combined with the fractional uncertainty on the fitted fringe amplitude Ap of 9 × 10−3 yields a total fractional uncertainty on the standard quantum limit variance2∆θSQL of 1.7 × 10−2 . To estimate the angular resolutions (∆θ)2 , we typically use 100 to 200
experimental trials, which leads to a typical statistical fractional uncertainty on (∆θ)2 of 0.1 to 0.2.
The final reported uncertainties on the Wineland parameters are thus dominated by the statisticaluncertainties on the phase resolution (∆θ)2 .
Without the QND pre-measurements or one-axis twisting, the mixed state actually performs
worse than the standard quantum limit, conceptually due to the spin noise from the dephased or
decohered fraction of the atoms that contribute noise but no signal. This is why the observed
improvement in the interferometer sensitivity is larger than the Wineland parameter; however, theWineland parameter captures what fraction of the improvement can be certified to arise due toentanglement between the atoms and not due to just cancellation of spin noise alone.

4.5

Entangled matter-wave interferometry.
We now turn to injecting the prepared entangled states into a matter-wave interferometer.

We use a slightly modified Mach-Zehnder Raman interferometer with (π/2 - π - π/2) pulse sequence
to coherently separate, undo the separation, and interfere the atomic wavepackets. In analogy toan optical Mach-Zehnder interferometer, the π/2 pulses play the role of 50/50 beam splitters at theentrance and exit of the interferometer and the π pulse plays the role of the mirrors that redirect
the beams from the input beam splitter to the output beam splitter.
After preparing a squeezed state with OAT, a Raman beam splitter rotation orients thesqueezing along ŷ. The spin projection Jy will change if a small signal phase ϕ is applied. The
orienting of the squeezing is accomplished via a (π/2 + α0 ) pulse aligned to the atomic Bloch vector
along x̂. A relative phase accumulates between the wavepackets during a free evolution time Tevol ,

73

Time

OAT

OAT

Readout

Spectroscopicenhancement

10²

10¹

Standard quantum limit

10

10 - 2

10 -1

10 0

Rotated angle α [rad]Figure 4.11: Entangled matter-wave interferometer. Top: The squeezed interferometer sequence, including entanglement generation (purple), the interferometer (blue), and state readout(green). Each Raman transition (white rectangle) is labeled with magnitude (within) and axis ofrotation (above). Bottom: The spectroscopic enhancement W is compared for three configurations:a Mach-Zehnder interferometer with OAT (red circles, sequence above), an unentangled interferometer without OAT (black squares), and OAT-squeezed states without the interferometer (bluecircles). The duration of a π/2 + α rotation is scanned to minimize the projected spin noise at
α0 . An ellipse is fit with 68% confidence bands to the OAT-squeezed interferometer data, giving aminimum variance of W = 0.68(8) or 1.7+0.5−0.5 dB. The interferometer here had Tevol = 0.112 ms.

a Raman π “mirror” pulse is applied, followed by another free evolution time Tevol . Finally, areadout π/2 pulse transfers the signal ϕ and the squeezing into a displacement in the momentumspin population basis ẑ with a measurement outcome Jzf . The Bloch vector lengths Js and Jc
are measured in separate experiments with and without OAT applied by scanning the azimuthal
phase of the final π/2 pulse of the interferometer and measuring the fringe amplitude as before (seeFig. 4.12.
We achieve a directly observed spectroscopic enhancement as measured by the Wineland

74[atoms]

600

0
-600
0

π

2π

3π

4π

Readout phase [rad]Figure 4.12: Interferometer contrast fringes with Tevol = 0.112 ms shown for no squeezing Jc (black)and with squeezing Js (red).

parameter of 1.7+0.5−0.5 dB beyond the standard quantum limit with N = 660(15) atoms as shownin Fig. 4.11. We also note that the actual phase variance of the interferometer is enhanced by
3.4+0.9−1.2 dB compared to with no one axis twisting, consistent with expectations from Jc .

4

Spectroscopicenhancement

3
2

1.0

Standard quantum limit

8
6
4

1.0

0.1

Evolution time[ms]

Figure 4.13: Phase sensitivity is maintained below the SQL for the squeezed interferometer (redcircles, left Bloch sphere) up to Tevol = 0.7 ms. By contrast, if the squeezed spin projection isoriented along the population basis (blue circles, right Bloch sphere), spectroscopic enhancement
was seen to persist beyond Tevol = 1 ms because this orientation is insensitive to phases accrued
during the evolution time.

Phase sensitivity beyond the SQL was limited to evolution times Tevol < 0.7 ms (Fig. 4.13).
Evidence from purely microwave interferometers with no momentum states involved suggests thisloss of observable squeezing was primarily due to magnetic field fluctuations in the lab that lead75
to added fluctuations in the azimuthal phase accrued during the interferometer evolution times.
The matter-wave interferometer is sensitive to vibration noise, but the measured accelerations are
not suﬀicient to explain the loss of directly observed entanglement. Single-particle decoherence isalso inadequate as an explanation because Jc decreased by less than 5% over these evolution times.
We also observe that if the squeezed spin projection is left in the population basis Jz during theinterferometer, then the squeezing persists for several milliseconds. From this, we conclude thatthe entangled state persists for longer than we can directly confirm because the interferometeris detecting an undesired magnetic field signal that masks the quantum noise reduction as Tevol
increases.

4.6

Vibration noise
Mechanical vibrations of the cavity mirrors are equivalent to a fluctuating phase reference forthe atoms. We measure the vibration noise with two different approach. First, we form an optical
Mach-Zehnder interferomter (setup is shown in Fig. 4.14) with the science cavity mirror and an
extra fixed mirror on the optical table. The measured vibration noise is dominated by the verticalvibration of the optical table.

Figure 4.14: Measuring vibration noise with an optical Mach-Zehnder interferometer.

Secondly, a commercial vibrometer (Herzan VA-2) was used to measure the noise at a locationon the optical table close to the portion that supports the vacuum chamber. The measurement
results (power spectral density of vibration noise) are shown in Fig 4.15. In the limit of zero-

76
duration pulses, the transfer function for a matter-wave Mach-Zehnder interferometer |T (ω)|2 =
4
R∞
64k2
sin ωT2evo converts accelerations to an integrated phase noise ϕ2 = 0 |T (ω)|2 Sa (ω) dω. Forω4a sequence with Tevo =0.3 ms, we estimate the phase noise caused by vibrations is 20 dB lower
than the phase resolution set by the SQL of 1000 atoms.

Figure 4.15: Power spectral density of the vibration noise as measured by the optical Mach-Zehnderinterferomter and a commercial vibrometer.

4.7

Error budget for standard quantum limit
We estimated the observed squeezing by comparing the measured phase uncertainty ∆θ tothe calculated standard quantum limit ∆θSQL . The SQL is estimated based on the atom number
measurements. The atom number is measured by the cavity frequency shift divided by the single
atom frequency shift χQN D , which relies on the details of the atom-cavity coupling.
The maximal atom-cavity coupling g0 is calculated based on the atomic transition dipole

77
matrix element, cavity parameters and fundamental constants. The cavity parameters are estimatedbased on frequency measurements. Tough trapped inside the cavity, the finite radial temperatureand finite axial extend could also affect the atom-cavity coupling. To account for different sourcesof error, different parameters and the resulting fractional error in terms of estimating the SQL aresummarized in Table 4.1.
Parameters

Value

Error

Unit

Dipole matrix element
for cycling transition
Free spectral range
Mode splitting between
TEM00 and TEM10
mode
Atomic radial temperature
Radial trap depth of
donut mode
Initial position of latticefrom center
Detuning of probe beamfrom atomic transition
Cavity frequency shift
with all atoms in |↑⟩2.534 × 10−29

0.003 × 10−29

C*m

Fractional uncertainty on ∆θSQL
0.223 %

6.78792.11450.00010.0003GHz
GHz

0.002%
0.016%

1.4

0.5

µK

0.307%

58

5µK

0.069%

0

0.002

m

0.903%

175

1

MHz

0.497%

238

3

kHz

1.266 %

Table 4.1: Table for different sources of error.

4.8

Conclusion and outlook
In the future, the magnetic field noise can be reduced and looking further ahead, the combi-

nation of Raman and Bragg techniques demonstrated here would enable the most delicate portionof the interferometer to be operated fully with the two portions of the superposition possessing thesame spin label. To further improve interferometer sensitivity, the entanglement can be combinedwith large momentum transfer sequences, one could inject the squeezed state into a lattice interferometer to hold the atoms longer [82], or prepare the entanglement in the cavity and allow theatoms to undergo free fall and subsequent fluorescence measurement readout [77].

78
The amount of momentum squeezing could be improved with larger collective cooperativity
N C. The need for velocity selection limits our final number of atoms, so higher atom density
in momentum space through improved axial cooling or the use of a Bose-Einstein condensate
could lead to significant improvements [87, 88, 28, 89]. It will also be possible to perform theentanglement generation utilizing optical cycling transitions in rubidium, strontium, and ytterbium[38, 90, 26, 50, 76] for which the fundamental scaling of the achievable Wineland parameter would√
improve to W ∝ 1/N C from the current scaling W ∝ 1/ N C.
This proof-of-principle light-pulse matter-wave interferometer paves the way for utilizing
cavity-generated entanglement as a quantum resource, enabling the next generation of interferometers with higher precision, enhanced measurement bandwidth, higher accuracy, and smaller size.
Such devices will advance the frontiers of both practical applications and discoveries in fundamental
science, from particles and fields to gravitational wave detection [5, 22, 58, 20, 21, 60, 61, 63, 40,
64, 19, 56, 4, 81, 87, 88, 89, 82, 59, 91], and build a bridge to a future where cavity-QED-based
quantum many-body simulations will move beyond mean-field physics to probe and manipulatequantum fluctuations in large ensembles of atoms [92, 93, 94, 95, 96].

Chapter 5
Cavity mediated momentum-exchange interaction

Large ensembles of laser-cooled atoms interacting via infinite-range photon-mediated interactions are powerful platforms for quantum simulation and sensing. Here we realize for the firsttime momentum-exchange interactions in which pairs of atoms exchange their momentum states
via collective emission and absorption of photons from a common cavity mode, a process equivalent
to a spin-exchange or XX collective Heisenberg interaction. The momentum-exchange interaction
leads to an observed all-to-all Ising-like interaction in a matter-wave interferometer, which is usefulfor entanglement generation. A many-body energy gap also emerges, effectively binding interferometer matter-wave packets together to suppress Doppler dephasing with analogies to Mössbauer
spectroscopy. The tunable momentum-exchange interaction provides a new capability for quantum interaction-enhanced matter-wave interferometry and for realizing exotic behaviors including
simulations of superconductors and dynamical gauge fields.

5.1

BackgroundMany-body quantum states of laser-cooled atoms can be exquisitely controlled, making them

powerful platforms for quantum simulation, metrology, and computing. In particular, quantum
sensing and metrology rely on understanding how to realize new forms of interactions between theatoms to achieve the next generation of ultra-precise quantum sensors and to emulate both complex
quantum phases of matter as well as non-equilibrium systems that are diﬀicult to access in realmaterials.

80
Optical cavities can be used to enhance the interaction of atoms with light in quantum manybody systems in which either the atomic internal [33, 97, 98, 99, 100], motional [101, 102, 103, 104,105], or both [106, 11] degrees of freedom are coupled between different atoms. In addition, thestrong light-atom interaction has enabled the largest directly observed entanglement generation todate in any system [107, 25], with applications in quantum sensing with matter-wave interferometers[11] and clocks [76, 108, 78, 25].

A

B
Intermediate state

Atom jEnergyAtom iMomentum

Atom iAtom jInitial state

Final state

Momentumexchange χFigure 5.1: Experimental setup for observing momentum-exchange interaction. (A) A momentumexchange interaction is realized between atoms in different momentum states (p0 ± h̄k) Ẑ by exchange of photons via a standing-wave optical cavity, illustrated for two particular atoms in red
and blue. The dressing laser (light blue arrow) is injected into the cavity. (B) The energy versus
momentum diagrams illustrate the steps of the momentum-exchange between the example red andblue atoms. Eliminating the intermediate states leads to an effective momentum-exchange Hamiltonian involving only the atoms.

Here, we realize a unitary cavity-mediated momentum-exchange interaction in a many-bodysystem in which pairs of atoms exchange their momentum states, as shown in Fig. 5.1. The
momentum exchange interaction arises from an atomic density grating creating sideband tones
on an applied dressing laser, similar to as occurs in cavity opto-mechanical systems [109, 110,

81
111, 112, 113, 114, 115], as illustrated in Fig. 5.2. The momentum-exchange can be modeled as
an all-to-all pseudo-spin-exchange interaction, analogous to that observed for internal spin states[98, 97, 116, 99, 100]. While previous theoretical proposals have considered the generation ofsuch momentum-exchange in a ring cavity, as well as extensions to two-mode squeezing involving
additional spin degrees of freedom [117, 118], here we experimentally realize a momentum-exchangeinteraction in a standing wave cavity by exploiting the Doppler shift of the falling atoms.
The observed momentum-exchange interaction allows for the realization of the collective XXHeisenberg model, an iconic model in quantum magnetism and superconductivity [119, 120, 121],
now generated in a momentum-only basis of states with no internal atomic degrees of freedom
involved, as compared to previous [98, 97, 116, 99] and contemporaneous work [122]. The exchangeinteraction manifests firstly as a magnetization-dependent global spin precession of the collective
pseudo-spin Bloch vector, referred to as one-axis twisting (OAT). Secondly, it generates a manybody energy gap that realizes a collective recoil mechanism that suppresses dephasing due toDoppler broadening (i.e. single-particle dispersion), analogous to, but distinct from Mössbauer andLamb-Dicke spectroscopy, which have been keys to realizing state-of-the-art quantum metrology
and searches for new physics [123, 124].

5.2

Experiment setup
In the experiment, 87 Rb atoms are laser-cooled inside a two-mirror standing wave cavity thatis vertically-oriented along Ẑ, see Fig. 5.1 and [11, 125]. The atoms are allowed to fall along thecavity axis, guided by a blue detuned intracavity optical dipole trap. A pair of laser beams withdifferent frequencies are injected non-resonantly into the cavity in order to drive velocity-sensitive
two-photon Raman transitions between ground hyperfine states (for state preparation and readout)or Bragg transitions that only change momentum states (for manipulating the superposition of
momentum states.)
The atoms are prepared in the ground hyperfine state |F = 2, mF = 2⟩ with momentum along
the cavity axis p0 − h̄k and rms momentum spread σp < 0.1h̄k after Raman velocity selection from

82

A

B
time

maximalalignmentintermediatealignment

minimalalignment

Densitygrating

Figure 5.2: Modulation sidebands created by atomic density grating (A) Space-time diagram of the matter-wave interferometer. Bragg pulses are applied to manipulate atoms in superpositions of momentum states, causing the wave packets to separate in time with subsequent pulses
reoverlapping the wave packets. When the wave packets overlap with each other, their interference
forms a density grating along Ẑ. (B) As the atomic density grating moves, its spatial overlap withthe standing-wave cavity mode (light blue on the left) varies, with three snapshots in time (purple,
green and orange) shown on the right.

the laser-cooled cloud [11], where h̄ is the reduced Planck constant, the wavenumber is k = 2π/λ,
and the wavelength is λ = 780 nm. As shown in Fig. 5.2A, the Bragg lasers are then applied toplace the atoms in a superposition of two wave packets with momenta centered on p0 ± h̄k withaverage momentum p0 and separated by two photon recoil momenta 2h̄k. The average momentum
p0 continues to increase due to gravity, but we compensate this by appropriate chirping of appliedlaser frequencies [11, 12], such that one can consider p0 to be constant in the following discussion forsimplicity. Inserting additional Bragg pulses, we can realize a matter-wave interferometer, in which
the atomic wave packets move apart and then reoverlap at later time. Just after the wave packet
splitting and just before reoverlapping, the two portions of the wave packets interfere, leading toa spatially varying atomic density grating with periodicity λ/2 matching the periodicity of thestanding-wave cavity mode.
As shown in Fig. 5.3, a cavity mode’s frequency is detuned by about 500 MHz to the blue of
the D2 cycling transition |F = 2, mF = 2⟩ → |F ′ = 3, mF ′ = 3⟩. A dressing laser with photon flux
|αd |2 (in unit of photons per second) drives the cavity at frequency ωd that is typically within a
MHz of the cavity resonance frequency. The input coupling of the cavity κ1 is determined by thetransmission of the input mirror. The detuning of the dressing laser from the atomic transition,

83

freq

Figure 5.3: Frequency diagram of the optical atomic transition frequency ωa (black solid line),bare cavity frequency ωc with no atoms in the cavity (black solid Lorentzian), and the atomdressed cavity resonance frequency (dashed purple, green, orange Lorentzians) for the corresponding
snapshots in time from (C). The average dressing laser detuning ∆d is shown. The cavity isfrequency modulated at ωz , leading to sidebands on the dressing laser at ±ωz (lower sideband
shownas wiggly black line) that with the dressing laser couple the momentum states p0 ± h̄k torealize the momentum-exchange.

∆a is large compared to all other relevant frequency scales including the excited state decay rate
Γ = 2π × 6 MHz and the cavity power decay rate κ = 2π × 56(3) kHz. In this far-detuned limit, an
g2

atom at position Z shifts the cavity resonance by ∆0a cos2 (kZ), where g0 = 2π × 0.48 MHz is the
maximal Jaynes-Cummings atom-cavity coupling at a cavity anti-node [38].

5.3

Modulation sidebands created by atomic density grating.
As the atomic density grating moves along the cavity axis at velocity v0 = p0 /m, with m themass of 87 Rb, the density grating goes from being aligned to misaligned with the cavity standing
wave shown in Fig. 5.2 from left to right. This leads to a modulation of the cavity resonancefrequency at the two-photon Doppler frequency ωz = 2kv0 as shown in Fig. 5.3. The modulation of
the cavity resonance frequency leads to optical modulation sidebands on the dressing laser inside
the cavity at frequencies ωd ± ωz , with the closer to resonance sideband shown in Fig. 5.3 (black
wiggly line), in a direct analogy to cavity opto-mechanical systems [115, 114, 113]. The modulation
sidebands can also be understood as the Doppler-shifted reflection of the dressing laser from themoving matter-wave grating.
We directly observe that a modulation sideband combined with the dressing laser form a
Bragg coupling that drives collective population transfer from p0 − h̄k to p0 + h̄k as shown by

Momentum state populations

A

84

B

1.0

Spatial phase matching

0.8
freq

0.6
0.4

+

+

0.2
0
0

10

20

30

Dressing photons |αd|² tx

3

40 x10Figure 5.4: Matter-wave superradiance. (A) When a modulation sideband generated by themoving atomic density grating is tuned to resonance with the cavity (top inset), the light escapes
the cavity and population is collectively or superradiantly transferred (bottom inset) between thetwo momentum states at p0 ± h̄k (solid points and lines.) (B) The system is well phase matchedwhen the wave numbers of the Bragg laser (that generates the density grating) and the dressing
laser (that drives the momentum-exchange interaction) closely match each other (on the left). A
difference in wave number (kBragg ̸= kd ) will lead to a spatially varying phase that eliminates thesuperradiance (on the right).

the solid points and lines in Fig. 5.4(A). This occurs when we tune a modulation sideband to be
nominally on resonance with the dressed cavity by setting the dressing laser detuning from theaverage cavity resonance frequency (see Fig. 5.3 to ∆d = ωz . In this regime, the sideband light canescape from the cavity before being re-absorbed by the atoms, such that the collective populationtransfer can also be understood as a superradiant decay between momentum states [109, 111, 112].
To confirm the collective nature of the decay, in a separate experiment, we prepare the initialsuperposition of states using an initial Bragg π/2-pulse 85 GHz detuned from the dressing laser.
The difference in wave numbers of the dressing laser kd and the Bragg laser kBragg causes a slip
in the spatial alignment of the cavity standing wave and the atomic density grating by a phase
2 |kd − kBragg | Lcloud = 3.5 radians across the axial extent Lcloud = 1 mm of the atomic cloud(Fig. 5.4B). In this case, we observe no superradiant transfer of population in Fig. 5.4(A) (opencircles and dashed lines).

85
We now realize the momentum-exchange interaction by tuning the dressing laser so that themodulation sidebands are far from resonance with the cavity, i.e. |∆d ± ωz | ≫ κ/2. In this limit,
photons emitted at the sideband frequencies are more likely to be re-absorbed by the atoms thanto escape from the cavity. This process of emitting and absorbing sideband photons leads to a
momentum-exchange as illustrated in Fig. 5.1A and B.

5.4

Different interpretations of modulation sidebands
In the former section, we have presented the creation of the frequency sidebands. We will now

elaborate on different interpretations of the origin of the sideband creation to give more physicalintuition. There are mainly three ways to understand this process.
• Atoms inside the cavity induce a phase shift on the light inside the cavity as discussed in
Chapter 2. Therefore, classically, atoms can be consider as dielectric material with index
of refraction proportional to the atomic density. Inside the cavity, the density grating canthen be considered as stacks od dielectric material similar to a Bragg grating such as is used
to make dielectric mirrors.. The cavity photons are reflected by this “grating” from both
direction. But because the “grating” is moving along the cavity axis, the reflected light is
Doppler shifted. The Doppler shifted light correspond to the blue and red sidebands.
• The microscopic picture of the red/blue sideband photons can be consider as the stimulated
emission of the atoms. When atoms are in higher momentum state, absorbing a dressing
laser photon and emitting a higher frequency photon leads to the transfer into a lower
momentum state, and vice versa.
• With the same periodicity between the cavity mode and the atomic density grating, one canconsider the cavity resonance being phase modulated, however, because the two sideband
tones are not the same distance from cavity resonance, they are not equally well supportedby the cavity, leading to an additional AM quality to the generated tones.

86

5.5

Effective Hamiltonian of momentum-exchange interaction.
To model the momentum exchange process, we begin by defining ψ̂ † (p) and ψ̂ (p) as creation

and annihilation field operators of an atom with momentum p which are related to creation andRannihilation operators in position space by ψ̂(Z) = ψ̂(p)eipZ/h̄ dp. Because the wave packets
centered at p0 ± h̄k have a narrow momentum spread h̄k ≫ σp , we define ψ̂↑ (p) = ψ̂(p + p0 + h̄k),ψ̂↓ (p) = ψ̂(p+p0 −h̄k) operators that annihilate atoms at momentum p+p0 ±h̄k within a momentum
range p ∈ [−h̄k, +h̄k]. Doing this will support understanding in terms of both wave packets andan effective pseudo-spin language.
We divide the differential kinetic energy between the two momentum states p+p0 ±h̄k into twohiterms: a homogeneous or common kinetic energy difference Ĥz (p) = h̄ω2 z ψ̂↑† (p)ψ̂↑ (p) − ψ̂↓† (p)ψ̂↓ (p)ihand an inhomogeneous contribution Ĥin (p) = h̄ωin2 (p) ψ̂↑† (p)ψ̂↑ (p) − ψ̂↓† (p)ψ̂↓ (p) with ωin (p) =2kp/m.
We can adiabatically eliminate the cavity fields using second order perturbation theory (see
√
√
g02
α κ1
Fig. 5.1B), and in the perturbative limit |∆d ± ωz | ≫ N ∆dd+iκ/2
4∆a , we obtain an effective
atomic-only momentum-exchange Hamiltonian
Z Z h̄k hĤmx =−h̄kih̄χ+ ψ̂↑† (p)ψ̂↓ (p)ψ̂↓† (q)ψ̂↑ (q) + h̄χ− ψ̂↓† (p)ψ̂↑ (p)ψ̂↑† (q)ψ̂↓ (q) dp dq

(5.1)

R h̄k
R h̄k
with the total Hamiltonian Ĥ = Ĥmx + −h̄k Ĥin (p) dp + −h̄k Ĥz (p) dp. The momentum exchange
couplings are given by

χ± =

g02
4∆a

2

|αd |2 κ1∆d ± ωz,22∆d + κ /4 (∆d ± ωz )2 + κ2 /4(5.2)

where we have included finite cavity damping via appropriate Lindblad operators (see supplement.)
As discussed in Chap. 2 To map this to a pseudo-spin model, we define ladder operators
h
i
†
†
1
ĵ+ (p) = ψ̂↑ (p)ψ̂↓ (p), ĵ− (p) = ψ̂↓ (p)ψ̂↑ (p) and spin projection operators ĵx (p) = 2 ĵ+ (p) + ĵ− (p) ,
i
h
i
h
†
†
1
1
ĵy (p) = 2i ĵ+ (p) − ĵ− (p) and ĵz (p) = 2 ψ̂↑ (p)ψ̂↑ (p) − ψ̂↓ (p)ψ̂↓ (p) . Integrating over all momenR h̄ktum states, we can then define collective operators Jˆα = −h̄k ĵα (p) dp where α ∈ [x, y, z, +, −]. The
momentum-exchange Hamiltonian Ĥmx is then equivalent to an effective spin-exchange Hamilto-

87nian Ĥsx = χ+ Jˆ+ Jˆ− + χ− Jˆ− Jˆ+ . This can be viewed as a collective XX-Heisenberg or RichardsonGaudin integrable model where the non-local spin-spin couplings χ compete with an inhomogeneousaxial field—a model often used in quantum magnetism and superconductivity via the spin Anderson mapping [119, 120, 121]. We also note that the standing-wave cavity mode’s spatial intensity
2 and J 2 that we can neglect due to the same perturvariation cos2 kZ produces additional terms J+−

bative limit because these terms do not conserve energy between the initial and final states shownin Fig. 5.1B (see supplement and [126].)

5.6

One-axis twisting dynamics.

Interferometer fringe
phase shift [rad]0.6χ/2π = +2.1 Hz
χ/2π = -2.5 Hz

①

0.3
0.0

②

③

-0.3
-0.6
-1

-0.5

0.0

0.5

Initial spin projection (2Jz /N)

1

Figure 5.5: The measured interferometer phase shift scales linearly with the initial spin projection
Jz = ⟨Jˆz ⟩, while now holding ∆d fixed. The orange data points and fitted line is for χ/2π = +2.1 Hz,
and green for χ/2π = −2.5 Hz).
The exchange Hamiltonian can be re-written as Ĥsx ≈ χ Jˆ2 − Jˆz2 with χ = χ+ +χ− , ignoring
single-particle terms. At the mean field level, the one-axis twisting Hamiltonian χJˆz2 ≈ 2χ⟨Jˆz ⟩Jˆz
induces a rotation of the collective Bloch vector about the z direction at a constant frequency,
2χ⟨Jˆz ⟩, that depends on the initial momentum population difference ⟨Jˆz ⟩, which is conserved by
the Hamiltonian Ĥsx . In the equivalent matter-wave picture, the azimuthal phase, ∆ϕ = 2χ⟨Jˆz ⟩tx
accumulated when the exchange interaction is applied for a time tx , appears as a shift of the spatial

88
interference fringe between the two wave packets, see Fig. 5.5.

Time

Figure 5.6: Matter-wave interferometer sequence and space-time diagram for observing all-to-all
Ising or One-Axis Twisting dynamics. The interferometer fringe amplitude and phase shift ∆ϕ aremeasured by scanning the phase of the final rotation ϕ.

To observe this phase shift, we run a matter-wave interferometer sequence (see Fig. 5.6)
beginning with a Bragg π/4-pulse lasting 15 µs that prepares the atoms with population difference⟨Jˆz ⟩
N /2≈ −0.7. After waiting a delay time td = 25 µs, we apply the dressing laser to create theexchange interaction for tx = 25 µs. To re-overlap the wave packets or equivalently undo theinhomogeneity from Ĥin , we then apply a Bragg π-pulse, and apply the dressing laser again before
applying a final Bragg π/2-pulse with various phase ϕ. The final π/2-pulse maps the phase shift∆ϕ into a change in ⟨Jˆz ⟩. We measure the population in each momentum state by using velocitysensitive Raman π-pulses and cavity-assisted quantum non-demolition measurements (see [11][G]and supplement.) We repeat the experiment while scanning the phase of the final π/2-pulse. The
D E
phase shift ∆ϕ is then determined from the phase of the observed fringe Jˆz versus ϕ.
The momentum-exchange coupling of Eq.(5.2) predicts a triple-dispersive structure as thedetuning of the dressing laser ∆d varies. We observe this predicted structure in Fig. 5.7 by measur-89Interferometer fringe
phase shift [rad]

0.6
0.3
0.0
-0.3
-0.6
-0.8

-0.4

0.0

0.4

0.8

Δd /2π dressing laser detuning fromdressed cavity resonance [MHz]Figure 5.7: The observed phase shift ∆ϕ of the interferometer fringe versus the dressing laser’s
detuning from the dressed cavity resonance, displaying the predicted (line) functional form of χfrom Eq. 5.2. The insets illustrate the relative alignment of the modulation sideband to the cavity
resonance for three characteristic detunings.

ing the induced phase shift ∆ϕ as we vary the dressing laser detuning ∆d from the dressed cavity
resonance. In this data, the incident dressing laser power (350 photons/µs) is held fixed. The twoouter dispersive features arise as the two sideband frequencies at ±ωz pass through resonance withthe cavity as shown in the insets. The dispersive feature near ∆d = 0 arises from the carrier passing
through resonance with the cavity. At ∆d = 0, the exchange interaction parameters are χ+ ≈ −χ−
leading to a cancellation of the total exchange interaction (χ ≈ 0).
D E
The phase shift ∆ϕ is expected to scale linearly with Jˆz . We observe this by replacing
the initial π/4-pulse with variable-length pulses to vary ⟨Jˆz ⟩ while holding ∆d fixed instead. For
the orange data in Fig. 5.5(A), the frequency of the relevant sideband is higher than the cavity
resonance frequency leading to a measured χ/2π = +2.1 Hz. For the green data, we retune thedetuning ∆d so that the relevant sideband frequency is lower than the cavity resonance frequency

90
leading to a measured χ/2π = −2.5 Hz. We observe a linear phase shift ∆ϕ for χ > 0 and χ < 0with opposite slopes as expected.

①

②

③

Figure 5.8: Visualizations of the phase shift ∆ϕ in both the pseudo-spin picture (Bloch spheres)and in the atomic density grating picture.

We observe that the size of the phase shift ∆ϕ decreases if the wave packets are allowed
to separate for a time td before applying the dressing laser for time tx = 25 µs to induce themomentum-exchange interaction. Fig. 5.6 (top) shows the pulse sequence used to measure thisdecay of the phase shift (bottom) for both positive and negative χ (orange and green points). For
comparison, the solid lines indicate the predicted phase shifts for the full momentum-exchange
Hamiltonian while the dashed lines indicate the predicted phase shift for an OAT Hamiltonian−χJˆz2 . The wave packet separation or equivalently the inhomogeneity Ĥin would not affect a pure
OAT Hamiltonian as was the case in [11][G] , whereas the phase shift is decreased by dephasing forexchange interaction as was observed in a spin system [98]. The wave packet separation leads todephasing or shortening of the Bloch vector as visualized in Fig. 5.8. As the wave packets separate,
the corresponding collective Bloch vectors are shortened while the projection ⟨Jˆz ⟩ is conserved.

91

5.7

Gap protection: binding wave packets together.
The additional non-linear term Jˆ2 in the momentum-exchange Hamiltonian gives rise to a

many-body energy gap between states of higher symmetry (large J) and lower symmetry (smallerJ) [98, 116]. To explore how matter-wave coherence is protected by the gap, we run a Mach-Zehnderinterferometer as shown in Fig. 5.9 (top) in which we apply the dressing laser for a time tx startingat the point of maximum reoverlap of the wave packets (with T = 70 µs).

1.0
Enhancement

Contrast after
momentum exchange

maximally
reoverlapped

0.8
0.6
0.4

12
10
8
6
4
2
0

0

200

400

tx [µs]

600

0.2
0.0
0

200

400

600

Interaction duration tx [µs]Figure 5.9: Using the sequence in the top panel, the bottom panel shows the contrasts of theinterferometer fringe measured at the end of the exchange interaction period with χ = 0 (black),
χ/2π = 6 Hz (red). The ratio between the two (inset) displays significant gap protection ofcoherence due to the momentum-exchange’s Jˆ2 contribution. The simulated results (solid lines)
show good agreement with the data.

The coherence at the end of the dressing laser application is estimated from the amplitude of
the interferometer fringe using an appropriately timed π-pulse and a final π/2-pulse shown (with92
T ∗ = 70 µs.) To account for the atomic loss resulting from free-space scattering and superradianceinto higher momentum states, the contrast is calculated by normalizing the fitted fringe amplitude
to the residual population in the two momentum states p0 ±h̄k. The actual coherence of the system
is higher due to the finite possibility of under-estimating the number of atoms that underwent freespace scattering. In Fig. 5.9, the experiment is performed with the dressing laser off (χ = 0, black
points and fitted black curve) and the dressing laser on (χ/2π = 6 Hz, red points and theory curve),
for which we observe appreciable fringe contrast survives out to 600 µs. In Fig. 5.9 inset, one seesthat the momentum-exchange enhances the contrast by as much as a factor of 10(2). In Fig. 5.9,
the coherence undergoes a slight rise before ultimately falling. This behavior can be accentuatedby allowing the wave packets to undergo a small amount of separation for 40 µs before applying
the momentum-exchange interactions for a duration tx .

0

/σ in 0.9
N
χ

1.7
2.1
2.8

Contrast

1.0
0.5
0.0
0

Inte 200
rac 400
tion
600
tx [µ durati 800
ons]

Figure 5.10: We run a similar sequence to that of Fig. 6.8C except with an additional 40 µs
delay after wave packet overlap before application of the dressing laser for a variable time tx . We
see that as the interaction strength is increased relative to the rms inhomogeneous broadening
σin = 2π × 2 kHz, there is a transition in the dynamics for N χ/σin > 0.9. Strikingly, there are also
clear oscillations that were only hinted at in Fig. 6.8C. The lines are theory predictions.

The observed interferometer contrast versus tx (see Fig. 5.10) is measured at different dressing
laser powers to obtain different ratios of χN to the rms inhomogeneity from Ĥin expressed as a

93frequency σin . We observe a sharp transition in the dynamical behavior between χN /σin = 0.9 and1.7 with the emergence of oscillations of the contrast that extend to long times as χN increases.
The oscillations become faster and have larger amplitudes at shorter times as χN increases. Thisbehavior is reasonably consistent with the overlayed theory simulations (colored traces in Fig. 5.10)
that include finite superradiance and where only σin is fit from the data with χN = 0.

B

1.0

Contrast

A

0.5

0

0

100 200 300 400

tx [µs]Figure 5.11: (A) The theory prediction with residual superradiance on (solid) and turned off(dashed) are shown with three example points in the oscillations labeled, and with no interactions(grey). (B) The total length of the pseudo-spin Bloch vector J⃗ oscillates in time because theindividual Bloch vectors oscillate as shown for J⃗p>0 and J⃗p<0 in green and orange respectively, with
χN /σin = 2.8.

The extension of coherence to longer times and the observed oscillations can be understood as
the momentum-exchange interaction causing the wave packets to become bound to each other suchthat they no longer freely separate. In Fig. 5.11A, we show the simulated variation of the contrast
versus time without superradiance, high lighting three example points that we explain using thesimulated trajectories in Fig. 5.11B for the collective pseudo-spin Bloch vectors evaluated for p > 0
D
E
R h̄k
R0
or p < 0 with J⃗p>0 = 0 ⃗j(p) dp and J⃗p<0 = −h̄k ⃗j(p) dp where ⃗j(p) = ĵx (p)x̂ + ĵy (p)ŷ + ĵz (p)ẑ .
In Fig. 5.12, we also show the simulated results without superradiance for the individual
wave packets in both momentum and position space. In the pseudo-spin picture, the momentumexchange causes the displayed vectors J⃗p>0 and J⃗p<0 to undergo orbits that oscillate symmetricallyabove and below the equator such that the total Bloch vector length oscillates in time. In the wave
packet picture, with no interactions, the wave packet centers would follow the diverging dashedtx 0 µs|ψ↑(p)|²

60 µs|ψ↓(p)|²

120 µs|ψ↓(Z)|² , |ψ↑(Z)|²180 µs

94

-2.5

0

Z/λ

2.5 -0.3 0 0.3 -0.3 0 0.3

p/ħk

p/ħk

Figure 5.12: (left) In a co-moving frame, the wave packets oscillate in time about their average
position in space (blue and red wave packets and centers solid lines, non-interacting system dashedlines.) The momentum-space wave packets (right) also oscillate but with a π/2 phase shift in time
relative to the position space wave packets, as would be the case for a harmonic oscillator.

lines. With interactions, the wave packets oscillate in position with respect to each other, while
also oscillating in their momentum p, as though the wave packets are now connected by a spring withcharacteristic frequency set by the exchange interaction strength χN in the limit that χN ≫ σinand for small wave packet separation. If the wave packets are allowed to initially separate before
the spring-like coupling is turned on, then the amplitude of the oscillations of the wave packet
separations (and hence the contrast) will be larger as was observed in Fig. 5.10.
To further explore this idea of wave packets becoming bound to each other, we run a MachZehnder matter-wave interferometer with the sequence shown in Fig. 5.13 (top). If the exchangeinteraction is not applied (Fig. 5.13 red points and fit), then the fringe contrast is maximized whenthe echo time difference is ∆T = 0 since this is when the wave packets have maximal reoverlap.
If the momentum exchange interaction is applied just after the first splitting pulse, we see thatthe point of maximum contrast is shifted to ∆T = −55 µs (blue points and simulation), andInterferometer fringe contrast

95

1.0
0.8
0.6
0.4
0.2
0
-400

-200

0

200

400

Re-overlap imbalance ΔT [µs]Figure 5.13: The interferometer contrast as a function of imbalance in the time from nominal
perfect reoverlap of the wave packets with: no momentum-exchange (red data and fit), momentumexchange applied right after first π/2 at intermediate power (blue data and theory) and high power(brown data and theory), and momentum-exchange applied when wave packets are separated (green
data and fit).

becomes non-Gaussian (brown points and simulation) at even higher dressing laser power. We
rule out single-particle effects that might also shift the maximal reoverlap time by repeating theexperiment, but with the dressing laser applied 1 ms after the first splitting pulse when the wave
packets are not overlapped (green points and fit), which also suppresses collective superradiance,
leading to the higher observed contrast. The fact that the delay is modified by 55 µs rather thantx = 25 µs (as one might naively expect should one think of the wave packet separation as being
frozen in place during the exchange interaction) arises from a 25 µs delay between the end of the
π/2-pulse and the beginning of the interaction, and the fact that the nature of the coupling of the
wave packets is harmonic-oscillator-like.

96
In order to emphasize the extreme unusualness of the binding of the wave packets, consider
a gedanken experiment in which a single photon combined with a coherent state of light drives
the two-photon Bragg transition with total momentum transfer to the atomic cloud 2h̄k. Given
that one can not tell which atom underwent the two-photon transition, the initial state should besymmetrized with respect to which atom absorbed the single photon, analogous to a Dicke or W
state (see supplement). Without exchange interactions, at long times one would observe a single
atom eventually emerge from the cloud with velocity vrec = (2h̄k)/m while all other atoms remain
at their initial momentum. In contrast, with the momentum-exchange interaction, one would neverobserve a single atom to emerge with velocity vrec . Instead the whole cloud of N atoms would
collectively recoil with velocity vrec /N (see supplement).
This collective recoil is analogous to Mössbauer spectroscopy (or Lamb-Dicke spectroscopy) in
which atoms embedded in a crystal cause the whole crystal to recoil when the atoms absorb light. In
our case the collective recoil mechanism enabled by strong exchange interactions suppresses Doppler
dephasing or line broadening. However, unlike Mössbauer spectroscopy, it does not suppress thephoton recoil shift which here defines the transition frequency ωz between the two momentum
states.

5.8

Collective recoil and Suppression of Doppler Dephasing
To further elaborate on the unusualness of the momentum-exchange interaction and thegedanken experiment. Here, we expand and analyze this observation from three different perspectives. We first focus on the case with only one atom undergoing the two-photon transition andelucidate the collective recoil with an analytical derivation. We then generalize it to allow for more
initial recoiling atoms using mean-field numerical simulations of the wave packet dynamics. The
simulations explicitly show the the collective recoil of the total atomic wave packet which resultsin the enhanced coherence observed in the Ramsey-type experiment in the main text.
In Mössbauer spectroscopy, Doppler broadening of a Rabi spectrum is suppressed because thewhole crystal collective recoils rather than just a single photon-absorbing atom embedded in the

97
crystal. We present numerical simulations that show that Doppler broadening of a Rabi spectrum
is also suppressed by the gap protection term Jˆ2 .
We first consider initializing the system with N atoms having the same momentum distributions centered at p0 −h̄k. The quantum state of the system can be expressed in second quantization
R
N
+h̄k
as |ψini ⟩ = √1N ! −h̄k ϕ(p)ψ̂ † (p)dp|vac⟩ with |vac⟩ the vacuum state in which there are no atom
present and ϕ(p) ∝ exp −p2 /4σp2 the probability amplitude distribution with rms spread in probability σp . We then introduce a different basis set |Na , Nb ⟩ in which Na denote the number ofatoms with momentum ∈ (p0 − 2h̄k, p0 ) and Nb atoms the number of atoms with momentum∈ (p0 , p0 + 2h̄k). In this new basis, the initial state can be expressed as |ψini ⟩ = |N, 0⟩.
The gedanken experiment described in the main text considers a single photon combinedwith a coherent state of light driving the two-photon Bragg transition. One can not tell which atom
absorbed the single photon and underwent the two-photon transition, such that the state should besymmetrized with respect to which atom experiences the 2h̄k momentum transfer. The resulting
state can be expressed asP|ψ0 ⟩ = |N − 1, 1⟩ =†p ψ̂↑ (p)ψ̂↓ (p)√N|N, 0⟩ ,(5.3)

with the sum implicitly over discrete states between ±h̄k. This initial state then evolves under the
Hamiltonian Ĥ = Ĥkin + Ĥint :Ĥkin =X (p0 − h̄k + p)22mpĤint = χXψ̂↓† (p)ψ̂↓ (p) +X (p0 + h̄k + p)2p2mψ̂↑† (p)ψ̂↑ (p)ψ̂↑† (p)ψ̂↓ (p)ψ̂↓† (q)ψ̂↑ (q)

(5.4)
(5.5)

p,q

The first term is the kinetic energy and second the exchange-interaction.
Without applying the interaction (χ = 0), after time evolution the final state becomes|ψf ree (t)⟩ = Û0 |ψ0 ⟩ ,hiwith Û0 = exp −iĤkin t/h̄ . We can then calculate the spatial probability distribution:

(5.6)

98
 

⟨ψf ree (t)| ĉ†Z ĉZ |ψf ree (t)⟩ ∝ (N − 1) exp −Z − p0 −h̄km t2σZ2 (t)

2 

 



 + exp −Z − p0 +h̄km t
2σZ2 (t)

2 

,

(5.7)

P †
†
−i(p0 −h̄k+p)Z/h̄ +−i(p0 +h̄k+p)Z/h̄ is the operator for creating anp ψ̂↓ (p)ep ψ̂↑ (p)er
 2 22σ t
atom at position Z, and σZ (t) = 2σh̄p 1 + mh̄p
is the rms spread of the wave packets in position
where the ĉ†Z ∝P

space. The result above corresponds to two Gaussian distributions moving at different velocities
with heights proportional to the number of atoms in each momentum distribution centered at
p0 ± h̄k.
Now we turn to the case with interaction applied. The initial Dicke-like state |ψ0 ⟩ of Eq. 5.3
is an eigenstate of Ĥint with eigenvalue χN as discussed in before. On the other hand, Ĥkincouples the symmetrized Dicke-like state to other eigenstates of the interaction Hamiltonian thatare separated by an energy gap χN[98]. In the limit χN ≫ σin , we can consider Ĥkin as aperturbation of Ĥint , with the time evolution being governed by the projection of Ĥkin onto theinitial Dicke-like state with operator P̂ = |N − 1, 1⟩ ⟨N − 1, 1|.
|ψint (t)⟩ = Û0′ |ψ0 ⟩ ,

(5.8)

i
h
with Û0′ = exp −iP̂ Ĥkin P̂ t/h̄ . We can then calculate the spatial probability distribution:"#p̄t 2Z−m⟨ψint (t)| ĉ†Z ĉZ |ψint (t)⟩ ∝ exp −,2σZ2 (t)(5.9)

which corresponds to a single Gaussian wave packet moving at the velocity p̄/m = NN−1 p0 −h̄k+m
1 p0 +h̄k
N m .

Relative to the co-moving frame of the atoms before absorbing the single photon, thiscorresponds to a velocity of N1 p0 +h̄k
m . Thus the momentum-exchange interaction leads to collective
recoil of the atomic ensemble at a velocity reduced by 1/N .
We now consider the case where N atoms starts with a momentum distribution centered at
p0 − h̄k and a coherent two-photon Bragg coupling is applied to place the atoms in a superpositionof momentum states such that on average M̄ atoms undergo the two-photon transition. In this
scenario, the Bragg pulse transfers M̄ 2h̄k of total momentum to the atomic ensemble. In Fig. 5.14,

99
we perform mean-field numerical simulations of the wave-packet spatial probability distribution
without interactions and with a Heisenberg interaction Jˆ2 provided by the momentum-exchangeinteractions but neglecting the Ising or OAT contribution Jˆz2 . We consider three initial Bragg pulses
areas (π/4, π/2 and 3π/4) to achieve M̄ /N ≈ 0.15, 0.5 and 0.85 and simulate the evolution of thespatial probability distributions of the atoms as a function of time of flight period TT OF after the
initial Bragg pulse.
500

TTOF [μs]Non-interacting

A

B

M/N ≈ 0.15

C

M/N = 0.5

M/N ≈ 0.85

|ψ|2

250

0
500

TTOF [μs]Interacting

D

1.0

E

0.5

F

250

0
-5

0

Z0/λ

5

-5

0

Z0/λ

5

-5

0

5

0.0

Z0/λ

Figure 5.14: Simulated evolution of the wave packets (credit: Haoqing Zhang.) Evolutionof the spatial probability distributions in the moving frame defined by the p0 state with Z0 = Z− pm0 t.
With all atoms initialized in p0 − h̄k momentum state, π/4, π/2 and 3π/4 Bragg pulses are applied
to achieve different fractions of the initial recoiling atoms. With no interaction applied (χ=0), theinterference fringes start to disappear as the wave packets separate into two for M̄ /N =0.25 (A),
0.5 (B) and 0.75 (C). As a comparison, with the interaction applied (χ± N ≫ σin ), the wave packets
do not separate but instead move at an averaged velocity determined by M̄ /N as shown in (D),(E) and (F).

As shown in Fig. 5.14 (A) to (C), without the interaction (χN = 0), the wave packets
separate into two with increasing TT OF , which also leads to the vanishing interference fringes. The
disappearance of the interference fringes caused by the two wave packets separating is precisely
Doppler dephasing in this wave-packet picture. With the interaction applied (χN ≫ σin ), as

100
shown in Fig. 5.14 (D) to (F), the wave packets no longer separate but instead move at velocity
proportional to M̄ /N . Note that the interference fringes now persist because the atoms collectively
recoil as a single wave packet. This is why the collective recoil mechanism leads to a suppressionof the Doppler dephasing.
One also notices that the fringes are stationary in the reference frame of Fig 5.14, independentof M̄ /N . This implies that the fringes always move at a velocity p0 /m in the lab frame. If
one considers an acceleration aZ (such as due to gravity), the momentum p0 now goes to p0 →p0 +m aZ t. The changing velocity of the interference fringes provides the interferometric sensitivity
to accelerations, even in the presence of the Heisenberg interaction Jˆ2 .

A

Non-interacting

0.5
0.0
-5

0

5

Z-/λ

10

15

Interactingthwi Ns M/ 1.0
om er
at nsf 0.75
f
o tra
n
io um 0.5t
ac tn
Fr em 0.25
o
m 0.0
1.0

|ψ(Z’)|²|ψ(Z’)|²thwi Ns M/ 1.0
om er
at nsf 0.75
f
o tra
n
io um 0.5t
ac nt
Fr me 0.25
o
m 0.0
1.0

B

0.5
0.0
-5

0

5

Z-/λ

10

15

Figure 5.15: Simulation of collective recoil (credit: Haoqing Zhang.) Simulated spatial
probability distributions of the wave functions after evolution in the moving frame defined by thep0 − h̄k state with Z− = Z − p0 −h̄km t. (A) Without the interaction applied, the wave packets split
into two regardless of the fraction of initial recoiled atoms. (B)After turning on the interaction,
the wave packets no longer separate but instead recoil together with an averaged momentum acrossthe whole ensemble.

By fixing the time of flight period TT OF = 500 µs, we can also compare the results with different fractions of initial recoiling atoms M̄ /N . As shown in Fig. 5.15, when there is no interaction
applied, as one expects, the atoms are only found in two spatially distinct regions corresponding
to either having not absorbed two-photon recoil or having absorbed two-photon recoil and gaining
a velocity vrec = 2h̄k/m. Varying the number of two-photon momentum kicks M̄ only varies the101fraction of atoms found in one spatial region or the other, but not the centers of the two distinct
spatial regions.
However, when the momentum-exchange interaction is much larger than the Doppler broadening |χN | ≫ σin , one sees in Fig. 5.15B that now the atoms recoil as a single wave packet withwave packet velocity v = M̄N vrec that depends on the fraction of atoms M̄ /N that underwent twophoton absorption. For simplicity, the simulations above included only the gap protection term Jˆ2
and neglected the Ising term Jˆz2 as this latter term only shifts the phase of the interference fringes
within the wavepacket, but does not alter the collective wavepacket recoil velocity.

5.9

Summary.
We have realized a cavity-mediated momentum-exchange interaction between different mo-mentum states for the first time. By measuring the phase shift induced by this momentum exchangeinteraction, we observe the collective OAT dynamics, which paves the way for entanglement generation between momentum states [117, 118] and the study of beyond mean-field physics. We also
directly observe an extension of the coherence time of the system which we identify with a collective
recoil mechanism. The collective recoil mechanism still allows for the sensing of accelerations aZsuch as due to gravity (see supplement), since the phase difference that accrues between the two
momentum states still depends on ωz which chirps as ωz → 2k (p0 /m + aZ t). Equivalently, theatoms still act as a phase memory of the optical Bragg pulses with which they interact [40][G] . This
opens interesting new paths for enhancing quantum memory lifetime [127], for Doppler-broadeningfree spectroscopy, and for matter-wave interferometers that do not rely on spin-echo like sequences
and therefore would also allow measurements of velocities rather than accelerations.
Finally, we note that the momentum-exchange Hamiltonian here is equivalent to the model
Hamiltonian often used to describe BCS s-wave superconductors. From this perspective, the observed oscillations can be identified with Higgs oscillations following a quench of the exchangeinteraction strength [119, 128, 129]. This would enable quantum simulation of BCS superfluidity
and also sets the stage for quantum simulation that goes beyond two-level systems by encoding

102
degrees of freedom in the larger ladder of momentum states [130, 131] as well as internal states [132,133, 134, 135, 136, 137]. The large number of synthetic dimensions in combination with the longrange cavity mediated interactions, open unique opportunities for the emulation of self-generated
spin-orbit coupling [138, 139], pair production [97, 140, 141, 142, 143, 144, 145, 146], long sought
but never seen topological superfluids [147], and dynamical gauge fields [148, 149, 150, 151, 152].
Lastly, the generation of sideband tones may open a path to transduce excitations between matterwaves and mesoscopic opto-mechanical systems [115, 114] or back-action evading measurements ofmatter waves as proposed for spins [153].

Chapter 6
Hamiltonian engineering of momentum states

Quantum control of momentum states is important for precision measurements. However,limited by the lack of controllable many-body interaction, momentum qubits are not ideal for
quantum simulation and entanglement generation. Recent progress demonstrated an exchangeinteraction between two momentum states assisted by a high-finesse cavity [12]. Here, we generalize
such approach to achieve more four-photon processes and realize an all-to-all interaction with
arbitrary quadratic Hamiltonian or effectively an infinite range tunable Heisenberg XYZ model
between two atomic momentum states. We demonstrate the tunability of the Hamiltonian by
explicitly mapping out the evolution of Bloch vectors on the Bloch sphere at the mean-field level
and witness the two-axis counter-twisting dynamics for the first time. Our work paves the way
towards quantum simulation of more complex systems and fast robust entanglement generation
with momentum states for future quantum enhanced metrology.

6.1

Introduction
The ability to create and control different many-body interactions is key for entanglement

generation, optimization, quantum sensing, and quantum simulation. Long-range interacting systems have recently shown much promise for the engineering of Hamiltonians that generate interesting correlations that propagate across the system. Several experimental platforms are making
rapid progress, including Rydberg atoms [154, 155, 156], polar molecules [157, 158], trapped ions
[159, 160], cavity QED systems [138] and defect centers in solids [161, 162]. Short range contact

104interactions in ultra-cold atomic systems are another promising approach [163, 164, 165], however,it remains an open challenge to reach the suﬀiciently low-temperatures.
So far, most efforts to engineer Hamiltonians have been limited to XXZ spin models or models
that feature both exchange and Ising interactions. Common to these models is the fact that the totalmagnetization of the spin ensemble is preserved. However, limited progress has been achieved inengineering more general spin models, such as XYZ models, which can break both SU(2), and U(1)symmetries and lead to more general ground-state and out-of-equilibrium many-body behaviors. A
few exceptions include experiments in Rydberg atoms using two-color dressing [166] with dynamics
limited to pairs of atoms, or Floquet engineering in disordered arrays [155].
XYZ Hamiltonians can in fact provide important gains in the context of quantum sensing andmetrology applications. For instance, the so called two-axis counter-twisting (TACT) [16] model is
one type of XYZ model well known for its capability to generate entanglement exponentially fast
all the way to the Heisenberg bound. It was proposed more than 30 years ago [167, 16] together
with a simpler version, the so-called one-axis twisting (OAT) featuring only Ising interactions.
So far, only the OAT has been realized in experiments including trapped ions [30, 168], BoseEinstein Condensates [28, 29], atomic cavity-QED [33, 78, 98, 116], superconducting qubits [23]and optical interferometers [169]. Experiments have approximated TACT dynamics locally on
the Bloch sphere by combining OAT with a transverse drive (the so-called Lipkin-Meshkov-Glick(LMG) model) [170, 171, 172] or equivalent approximations in spin-nematic dynamics in higher spin
systems [66, 67]. However, there has been no demonstration of genuine TACT in any platforms
due to the challenge of how to realize this more complex non-linear Hamiltonian.
Here we experimentally show that photon-mediated interactions between atoms inside an
optical cavity can be tuned to realize a tunable all-to-all Heisenberg XYZ Hamiltonian, including
the TACT model, with the added benefit of not requiring us to prepare extremely low entropy
quantum degenerate gases. While there have been prior theory proposals to realize TACT models
in atom-cavity systems using four laser tones [173, 174], here we experimentally realize the modelusing only two dressing lasers. We demonstrate the tunability of the XYZ Hamiltonian by explicitly

105
mapping out the evolution of Bloch vectors on the Bloch sphere at the mean-field level and witness
the TACT dynamics for the first time.
The pseudo-spin system here is novel, consisting of two momentum states of atoms freelyfalling inside the cavity, making our results of great interest for Bragg matter-wave interferometers[175] that are important for both inertial navigation and fundamental science such as searches
for dark matter and dark energy, detection of gravitational waves, and determination of the finestructure constant [20, 21]. This approach can also be straightforwardly applied to systems withadditional internal levels, making it ideal for developing next-generation quantum-enhanced sensors
for technology and exploring a broad range of science from atomic clocks [76, 108] and magnetometers [176, 177, 178] to geodesy [179].

6.2

Experimental setup
In the experiment, 87 Rb atoms are laser-cooled inside a vertically-oriented two-mirror stand-ing wave cavity, see Fig. 6.1 and [12, 11][G] . A repulsive intra-cavity doughnut dipole trap confines
the atoms radially, but allows the atoms to fall along the cavity axis. To prepare atoms in awell-defined momentum state, a pair of laser beams are injected into the cavity to drive velocitydependent two-photon Raman transitions between ground hyperfine states |F = 1, mF = 0⟩ and|F = 2, mF = 0⟩. After removing the unselected atoms with a resonant laser push beam, successive
microwave pulses are applied to prepare the internal states of the selected atoms in the ground
hyperfine state |F = 2, mF = 2⟩ (see [12][G] ).
With about 700 atoms centered at momentum p0 − h̄k, another pair of laser beams is injectedalong the cavity axis to drive two-photon Bragg transitions connecting the two momentum states|p0 ± h̄k⟩ which defines a two-level spin-1/2 system [11, 12][G] . Here the average momentum is
p0 , h̄ is the reduced Plank constant, and k = 2π/λ where λ is the wavelength of the Bragg laser
beams. Due to finite momentum spread, one can consider momentum wave packets centered atthese two momentum states as considered in detail previously [12][G] . Ignoring the finite momentum†
spread of the selected momentum states, we define ψ̂↑,↓and ψ̂↑,↓ as the operators for creating andAtom jAtom i

106

Figure 6.1: Experimental overview. Illustration of the microscopic momentum pair raisingprocess described by the Hamiltonian Jˆ+ Jˆ+ , using two momentum states p0 ± h̄k as a pseudo-spin
1/2 degree of freedom. Initially, two atoms i and j are in the same momentum state along thecavity axis. Dressing lasers are applied to the cavity (red and blue arrows) that allow atom i toabsorb a dressing laser photon and emit a photon (squiggly line) into the cavity such that the net
photon recoil flips its momentum state by 2h̄k. The emitted photon is absorbed by atom j, also
flipping its momentum state by 2h̄k. There also exists the separate momentum exchange process
Jˆ+ Jˆ− , where atoms initially in opposite momentum states flip their momentum states by emitting
and absorbing photons.

annihilating an atom in momentum states |↑⟩ ≡ |p0 + h̄k⟩ and |↓⟩ ≡ |p0 − h̄k⟩. For mapping to a
pseudo-spin model, we define ladder operators Jˆ+ = ψ̂↑† ψ̂↓ , Jˆ− = ψ̂↓† ψ̂↑ and spin projection operators






1
Jˆx = 21 Jˆ+ + Jˆ− , Jˆy = 2iJˆ+ − Jˆ− and Jˆz = 21 ψ̂↑† ψ̂↑ − ψ̂↓† ψ̂↓ .
As shown in Fig. 6.2, the cavity frequency ωc is detuned from the atomic cycling transition|F = 2, mF = 2⟩ → |F ′ = 3, mF ′ = 3⟩ by ∆a = ωc − ωa = 2π × 500 MHz, which is much larger thanthe excited state decay rate Γ = 2π × 6 MHz and the cavity power decay rate κ = 2π × 56(3) kHz.
A series of Bragg pulses can be applied to realize a Mach-Zehnder matter-wave interferometer(i.e. π/2 − π − π/2), in which the wave packets first separate in position and then re-overlap. Whenthe two wave packets are overlapped, the interference between them forms an atomic density gratingwith period λ/2, which matches the standing wave of a cavity mode.

107

freq

Figure 6.2: Frequency diagram of the applied dressing lasers with frequencies ω1,2 and coherent
state amplitudes α1,2 . The emitted photons (squiggly line) are Doppler shifted by ωz from the
dressing laser frequencies and detuned by ∆c from the cavity resonance frequency ωc . The cavity
is far detuned from the atomic transition frequency ωa .

As the atoms move along the cavity axis, the density grating is periodically aligned to thestanding-wave of the cavity mode, leading to modulation of the cavity resonance frequency at thetwo-photon Doppler frequency ωz = 2kp0 /m ≈ 2π × 500 kHz, where m is the mass of 87 Rb. To
have this modulation to mediate an effective atom-atom interaction, we typically apply two σ +polarized dressing laser tones (see Fig. 6.1) at frequencies ω1,2 within a few MHz of the cavity
resonance frequency (see Fig. 6.2), with complex amplitudes α1,2 corresponding to the field that√would be established inside the cavity were no atoms in the cavity and implicit units photons.
The atom–induced cavity frequency modulation leads to the generation of modulation sideband
tones at frequencies ω1,2 ± ωz . In the following simplifications, we will assume ω2 > ω1 .
Intermediate state

Atom i

Em

Initial state

Energyits

Atom jAbs

orbs

Final state
Momentum

Figure 6.3: Representation of the emission/absorption processes described in (A) but depicted interms of the atomic energy versus momentum for atoms i and j. The reverse process is also allowed,
giving rise to a collective lowering operator described by Jˆ− Jˆ− .

108The key insight is that different combinations of the dressing lasers and their atom-induced
sideband tones will induce different virtual four-photon processes which will manifest as all-toall exchange interactions Jˆ+ Jˆ− [12] and pair-raising Jˆ+ Jˆ+ (lowering Jˆ− Jˆ− ) processes as shown in
Fig. 6.1 and 6.3. After adiabatically eliminating the cavity fields using second order perturbation
theory (see [12][G] ), we obtain an effective time-dependent atom-only Hamiltonian in an appropriate
frame rotating at ωzĤ = χe Jˆ+ Jˆ− + χp eiδt Jˆ+ Jˆ+ + χ∗p e−iδt Jˆ− Jˆ− ,(6.1)

with the exchange and pair-raising/lowering couplings given by|α1 |2|α2 |2χe =+,∆c̄ + δ/2 ∆c̄ − δ/2 2 2|α1 α2 |eiϕint
1
1
g0
+
.
χp =
4∆a
2
∆c̄ + δ/2 ∆c̄ − δ/2
g02
4∆a

2 

(6.2)

Here, ∆c̄ = (ω2 + ω1 ) /2−ωc is the average detuning of the two dressing lasers from cavity resonance,
typically set to be less than 1 MHz. δ = (ω2 − ω1 )−2ωz is the detuning from four-photon resonance,
wg0 = 2π ×0.96 MHz is the maximal atom-cavity Rabi coupling at an anti-node of the cavity mode,and ϕint = arg (α2 α1∗ ) − ϕB is the differential phase between the two dressing laser tones relative tothe phase of Bragg coupling ϕB which forms initial density grating. Collective cavity dissipation isdealt with separately.
We will focus on the resonant case δ = 0 and ϕint = 0, though we show example data in
Fig. 6.4 that clearly exhibits a resonance in the interaction-induced dynamics at δ = 0 as thepair raising/lowering processes are tuned into and out of resonance by tuning the dressing laser
frequency difference. In the resonant case, the Hamiltonian of Eq. (6.1) reduces to:Ĥ = (χe + χp ) Jˆx2 + (χe − χp ) Jˆy2= χe Ĵ · Ĵ + χp Jˆx2 − Jˆy2 − χe Jˆz2 ,(6.3)

where we defined the collective angular momentum operator, Ĵ = {Jˆx , Jˆy , Jˆz } and introduced thecollective Heisenberg interaction Ĵ·Ĵ. The latter acts as a constant for any eigenstate of the collective
angular momentum operator Ĵ · Ĵ, such as the collective states with eigenvalue N /2(N /2 + 1). In

Change in Bloch vector
azimuthal angle φ [rad]109

Detuning from four-photon
resonance δ/2π [kHz]Figure 6.4: Observation of the four-photon resonance that generates pair raising and lowering
processes. We scan the dressing laser frequency difference to vary the detuning δ. With equal
dressing laser amplitudes corresponding to realizing the Jˆx2 Hamiltonian when δ = 0, we see clear
resonances in the observed change of the Bloch vector’s azimuthal angle ϕ for a Bloch vector
prepared near the south pole (θ = π/4, red data points) and near the north pole (θ = 3π/4, blue
data points). All error bars reported are 1σ uncertainties. Simulation results are shown in solidlines with shaded area allowing for 5% uncertainty in interaction strength.

the presence of single particle inhomogeneities, this term opens a many-body gap that help promote
spin locking which we explored before for the protection of coherences against dephasing [98, 12][G] .
Here we instead focus on the fully collective dynamics and therefore without loss of generality wecan add a generic χz Ĵ · Ĵ term without affecting the dynamics. As such, in our system we are able
to engineer dynamics governed by an XYZ Hamiltonian Ĥ = χx Jˆx2 + χy Jˆy2 + χz Jˆz2 with interaction
strengths χx = (χe + χp + χz ) and χy = (χe − χp + χz ). The XYZ Hamiltonian is highly tunable
by simply adjusting the relative power in the two applied dressing lasers since χe scales as |α2 |2 ,|α1 |2 and χp scales as |α2 α1 |.

6.3

Four-photon spectroscopy
In Fig. 6.4, we have shown a spectroscopic result of the four-photon resonance. For thisexperiment, we prepare two different initial Bloch vectors to be π/4 above or below the equatorwith projections nominally along ŷ, as shown in Fig. 6.5.

110

z
x

y

Figure 6.5: Initial states for four-photon spectroscopy.

When on resonance with δ = 0, the ratio of the two dressing laser amplitudes is balanced toachieve an effective Hamiltonian in the form of Ĥ = χJˆx2 , which preserve the azimuthal angle.
The four-photon detuning δ can be intuitively understood as inducing a time-dependent one2
where the twisting axis rotates as Jˆϕ(t) = cos (δt/2) Jˆx +axis twisting interaction Ĥ = χJˆϕ(t)sin (δt/2) Jˆy . With the four-photon detuning δ ≫ χN , the effective Hamiltonian recovers to the
exchange interaction Ĥ = χ Ĵ · Ĵ − Jˆz2 which induces finite changes in the azimuthal angle of theinitial Bloch vectors with the signs depending on the initial projection along ẑ.

(A)

(B)

z

x

y

Figure 6.6: Four-photon spectroscopy with initial states on the equator. (A) The initial Blochvectors. (B) Interaction induced change in Jz as a function of the four-photon detuning.

With a different set of initial states, the four-photon resonance can also be witnessed by thechange in polar angle or equivalently spin projection Jz . In this experiment, we prepare initial

111
Bloch vectors on the equator but with different azimuthal angle ϕ = −π/4, 0 and +π/4 as shownin Fig. 6.6. When on resonance with δ = 0, the effective Hamiltonian Ĥ = χJˆx2 causes a positive/negative change in Jz for initial states with ϕ = ±π/4. While for ϕ = 0, the initial Blochvector is along Jˆx which commutes with the Hamiltonian and thus experience no change in Jz .
With δ ≫ χN , Jz in conserved under the effective exchange interaction Ĥ = χ Ĵ · Ĵ − Jˆz for all
three initials states.

6.4

Mean-field dynamics
At the mean-field level, we can define the Bloch vector J ≡ (Jx , Jy , Jz ) = (⟨Jˆx ⟩, ⟨Jˆy ⟩, ⟨Jˆz ⟩)and approximate the Hamiltonian as Ĥ = B(J) · Ĵ. In this way, the collective dynamics are driven
by a self-generated effective magnetic field B(J) = (2χx Jx , 2χy Jy , 2χz Jz ) which depends on the
instantaneous collective spin projections. We can derive the equations of motion of the collective
Bloch vector, from Eq. (6.3), which simplify to a nonlinear torque equation dJ/dt = B(J)×J ≡ T(J).
One can identify the fixed points Jfix as the points where T(Jfix ) = 0.

x

z

z

y

y

z
x

Figure 6.7: Dynamics induced by two-axis counter twisting in the form of a Hamiltonian Jˆx2 − Jˆz2
is represented on a Bloch sphere with north and south poles defined by |p0 ± h̄k⟩. Top right shows
the local circular flows around the stable point at −x̂. The local flows for unstable saddle points
at ŷ with exponential squeezing and anti-squeezing along x̂ ± ẑ are shown in bottom right.

112
To understand the dynamics near the fixed points, it is useful to follow a standard stability
analysis by diagonalizing the Jacobian matrix M (J) = ∂T/∂J|J=Jfix . The local motion near these
fixed points is illustrated in Fig. 6.7 (right). We use red circles for stable points with purely
imaginary eigenvalues. The Bloch vector evolves on stable closed orbits indicated by the blue
circular traces. The red squares are used to denote unstable saddle points with real eigenvalues
with opposite signs. The unstable saddle points are labelled by the red squares. The eigenvalues of
the Jacobian matrix at the saddle points are real but with opposite signs. In Fig. 6.7(bottom right),
the dynamics show exponential divergence from the origin (indicated by outward blue arrows) alongx̂ + ẑ, corresponding to the positive eigenvalue [180]. The negative eigenvalues are indicated by theconvergence towards the origin (inward blue arrows) along x̂ − ẑ.

6.5

Dynamics on the Bloch sphere
In the experiment, we probe the local dynamics induced by the above Hamiltonian withvarious values of χe and χp . To do this, we vary the phase and duration of the Bragg pulse toprepare an initial pseudo-spin coherent state Ji . Before the atomic wave packets separate, wethen apply the interaction for a short time ∆t satisfying χN ∆t ≪ 1 and measure the change inazimuthal angle dϕ and polar angle dθ to obtain the final Bloch vector Jf after the interaction.
This is achieved by repeating the experiment and applying additional appropriate rotations before
measuring populations in the two momentum states. The local flow vector is then determined by
the torque T(Ji ) ≈ ∆J/∆t = (Jf − Ji ) /∆t.
In Fig. 6.8(A), we show the predicted flow vectors T(Ji ) on the Bloch sphere for three example
Hamiltonians of interest. Different Hamiltonians are obtained by changing the ratio of the dressing
laser amplitudes |α2 /α1 | = 0, 0.17, and 1.0.
We show the measured flow vector T(Ji ) in Fig. 6.8(B) with each row aligned to the corresponding example presented in Fig. 6.8(A). In each case, the flow vector start at Ji and end at Jf .
The left and right panels are polar plots (radial coordinate linear in polar angle) of the dynamics on
the south/north hemisphere looking from the north poles of the Bloch sphere. The middle panels

(A)(B) South hemisphere

Equirectangular projection
onto plane (ϕ,θ)

113

1

0.17

0

1

π/2

θ [rad]0π2North hemisphere

(C)

0

0.5π

π

ϕ [rad]

1.5π

2π

1

π

1

2

3

3

1

2

π/2

2
1

2

1

3

2

2
4

0

2

θ [rad]

4

1

0

0.5π

2

π

ϕ [rad]

1.5π

2π

2

π/2

θ [rad]

1

2

0

2

1

π

(D)

0

1

0.5π

π

ϕ [rad]

2

1.5π

2π

Figure 6.8: Evolution under different Hamiltonians. (A) Setting different amplitude ratios
between the two dressing tones |α2 /α1 | gives rise to different XYZ Hamiltonians, with specificexamples shown here including one-axis twisting Ĵ · Ĵ − Jˆz2 , two-axis counter twisting Ĵ · Ĵ + Jˆx2 − Jˆz2
and one-axis twisting Jˆx2 . The experimentally observed dynamics are shown in corresponding rowsin (B, C, D), where the tail of each vector indicates the initial position of the Bloch vector Ji
on the Bloch sphere, and the arrow indicates the displacement T after a brief period of evolution
under the corresponding Hamiltonians. The left (right) panels are for initial Bloch vectors on thesouth (north) hemisphere. The polar angle θ of the initial Bloch vector linearly increase from π/2
at the rim to π in the middle for the left plot, whereas θ decrease from π/2 at the rim to 0 in the
middle for the right plot. The middle panels are equirectangular projections. In each case, thequalitatively observed stable fixed points are marked with numbered red circles, and the unstable
fixed points are marked with numbered red squares. The blue lines are shown to indicate directionsof observed flow. The red lines indicate lines where the dynamics are zero and separate regions of
opposite flows on the Bloch sphere.

are equirectangular projections to show dynamics near the equator. From these vector maps, we
can make qualitative comparisons based on the geometry of the flow. For both theoretical andexperimental results, the stable fixed points and unstable saddle points labeled as numbered red
circles and squares respectively [180, 181].

114
In the first row of Fig. 6.8, we consider the simplest case with χp = 0. This is achieved byturning off one of the dressing lasers (|α2 /α1 | = 0), leading to the Hamiltonian Ĥ = χe Ĵ · Ĵ − Jˆz2 .
This Hamiltonian, referred to as the OAT Hamiltonian [16], maintains U(1) symmetry, thereby
conserving Jz . At the mean-field level, it features a constant effective magnetic field along the ẑ
direction, which results in the rotation of the collective Bloch vector about the ẑ-axis at a uniform
constant angular frequency,−2χe Jz . As expected, we observe two stable fixed points (red 1 and2 circles), and a reversal of the circulation across the equator (i.e. Jz = 0) where there are no
dynamics (red line). This Jz dependent circulation leads to shearing of the quantum noise in
the orientation of a Bloch vector on the Bloch sphere, a semiclassical explanation for how OAT
dynamics generate spin-squeezed states [16, 11]. We note that while the term Ĵ · Ĵ is trivial for ourcurrent observations at short times, at longer times, when inhomogeneities in our system manifest,
it can lead to important dynamical effects as shown in Ref. [12][G] .
Next, we consider the last row in Fig. 6.8 with χp = χe . This is achieved by using equal
dressing laser amplitudes |α2 /α1 | = 1. Here, the Hamiltonian is Ĥ = 2χe Jˆx2 , leading to OAT
dynamics along the x̂ direction. The corresponding dynamics are induced at the mean-field levelby a magnetic field along x̂ that preserves Jx and induces a rotation about x̂ with constant angularfrequency 4χe Jx . It is noteworthy that the interaction strength here is twice that of the Jˆz2 case,attributed to the use of two dressing laser tones. The data qualitatively shows two stable fixed
points along x̂ labeled by red circles 1 and 2, and a reversal of the sign of circulation across Jx
highlighted by the red lines.
Finally, we come to the case that achieves TACT, as shown in the middle row of Fig. 6.8. In√ √this case, the ratio of dressing lasers amplitudes is set to be |α2 /α1 | =2 − 1 / 2 + 1 ≈ 0.17,
which produces χe = 3χp . Starting with a single dressing tone α1 , by introducing a second dressingtone with tiny amount of power (|α2 /α1 |2 ≈ 3%), the Hamiltonian become Ĥ = 2χp 2Jˆx2 + Jˆy2 =2χp Ĵ · Ĵ + Jˆx2 − Jˆz2 .
The corresponding mean-field magnetic field is now B(J) = 4χp (Jx , 0, −Jz ), where we haveagain ignored the Heisenberg term since we will only consider dynamics at constant Bloch vector

115
length. The theoretical flow line for the dynamics is depicted in Fig. 6.8(A) (middle), showing four
stable fixed points along ±x̂ and ±ẑ, as well as two unstable fixed points along ±ŷ connected bygreat circles inclined at ±π/4 to the equatorial plane (red circles). In comparison to OAT, there
exist two twisting fields thus only the two points intersected by two blue circles with Jx = Jz = 0,
exhibit the maximum shearing dynamics, which serve as the unstable saddle points.
We experimentally explore the flow lines. In the left and right panels, the observed stable
fixed points are labeled with red circles numbered 1 to 4. The circulations of the flow lines areopposite for stable fixed points on opposite sides of the Bloch sphere. In the middle panel, weobserve two unstable points labeled with red squared 1 and 2 in Fig. 6.8(C). The two unstable
fixed points have flow lines that either diverge from them (blue arrows outward) or converge towards
them (blue arrows inward), with the two flows orthogonal to each other (see the discussion below).
These findings constitute the first direct observation of genuine TACT dynamics. We note thatfor a particular set of parameters of the LMG Hamiltonian, for example when Ĥ = χJˆz2 + δ Jˆywith δ ∼ χN , and when the Bloch vector initially points along the ŷ direction [180, 181], the flowlines can resemble the ones close to a saddle point of the TACT model. However, the instability is
restricted to this single point, in contrast to the full TACT which features two independent unstable
points.
For a quantitative comparison, we examine the dynamics near the saddle point Jsad along theŷ axis on the Bloch sphere. In Fig. 6.9, we map the displacement T (Ji ) as a function of the initial
Bloch vector orientation Ji = Jsad + dJi . We scan the initial Bloch vector angles θi and ϕi over arange ±π/12 centered about π/2 (i.e. about ŷ) with discrete points sampled using a detailed 11×11√
grid. The mean-field equations of motion for the two orthogonal directions n̂± = (x̂ ± ẑ) / 2 are:d(Jx + Jz ) = 4χp Jy (Jx + Jz )dt
d(Jx − Jz ) = −4χp Jy (Jx − Jz ).
dt

(6.4)

For the small range of angles around the ŷ axis sampled in these measurements, one can assume
Jy ≈ N /2, and therefore find out that the displacement’s time derivative increases linearly with

-π/12-π/240

θ-π/2 [rad]

π/24

π/12

116

-π/12

0

-π/24

π/24

ϕ-π/2 [rad]π/12Figure 6.9: Dynamics near saddle points. Measured local flow vector map around the saddle
point of the TACT dynamics.

the displacement itself, indicating dynamics that change exponentially over time.

0.1
0
-0.1
-0.2

Projection of flow

0.2

Parallel projection

-0.4

-0.2

0

0.2

Initial displacement from
saddle point

0.4

Figure 6.10: Blue and red points are parallel projections of the local flow vector onto the n̂± axis
∆J · n̂± / (N /2) as a function of initial displacement dJi · n̂± / (N /2) from the saddle point along theblue and red lines in Fig. 6.9. Blue and red solid lines are simulation results. The linear dependenceshows exponentially growing / shrinking dynamics along the squeezing / anti-squeezing axes.

We first focus on the data points along the two directions n̂± (depicted by blue and red

117
lines in Fig. 6.9) and compute the parallel projection of the flow ∆J · n̂± / (N /2) as a function of
initial displacement from the saddle point dJi · n̂± / (N /2), as shown in Fig. 6.10. The observed
linear relationship between the projection of the flow and initial displacement matches well withthe prediction from the simulation results (solid lines) which go beyond the linear approximation
by solving the non-linear equations (Eq. (6.4)). The small differences observed between the ∆J ·n̂± / (N /2) slopes stem from the finite duration of the interaction, which extends the dynamics
beyond the linear response regime.
The unstable dynamics explain why an initial circular distribution centered around the saddle
point shears as a function of time by squeezing (anti-squeezing) along the n̂− (n̂+ ) direction at a
rate exponentially faster than the linear growth seen in OAT. This combined with the global (overthe full Bloch sphere) dynamical behavior allows the TACT to directly approach the fundamental
Heisenberg limit on phase estimation [16, 180].

0.1
0
-0.1
-0.2

Projection of flow

0.2

Perpendicular projection

-0.3

-0.15

0

0.15

0.3

Initial displacement from
saddle point
Figure 6.11: Green and orange points are the perpendicular projection of the local flow vector as afunction of initial angular displacement from the saddle point along the green and orange lines in(A) (n̂ = x̂ and ẑ) onto n̂ × ŷ = ẑ and −x̂ axis (∆Jz and −∆Jx ). Orange and green solid lines are
simulation results.

118
In Fig. 6.11, we analyze the orthogonal projection of interaction-induced flows, especially
focus on the data points that initially displaced from the saddle point along n̂ = x̂ and ẑ axes
(depicted by green and orange lines in Fig. 6.9), and calculate their projections along the n̂ × ŷ = ẑand −x̂ axes (∆Jz and −∆Jx ), respectively. The dynamics can be explained by noticing that whenthe Bloch vector is initially prepared in the y-z plane, the effective mean-field magnetic field isalong the ẑ axis with a magnitude of −4χp Jz . Conversely, when prepared in the y-x plane, thefield is along the x̂ axis, with an amplitude of 4χp Jx . Thus one expects these two perpendiculardisplacements to grow linearly in magnitude with initial Bloch vector displacement, as we observedin Fig. 6.11.

6.6

Two-axis counter-twisting with unstable points at north and south poles
The original TACT Hamiltonian, as proposed by Kitagawa and Ueda [16], is defined as




2
2
Ĥ ′ = χ Jˆ++ Jˆ−= 2χ Jˆx2 − Jˆy2 ,(6.5)

with the theoretical flow lines shown in Fig. 6.12. This Hamiltonian bears resemblance to thepreviously discussed TACT, with an additional π/2 rotation around the y-axis. It is characterized
by unstable points at the north and south poles of the Bloch sphere, along with four stable points
located on the equator at ±x̂ and ±ŷ.
In our setup, it seems to be impossible to choose the parameters canceling χe in Eq. (6.2) at
first glance. However, this can be achieved by placing the two detuned dressing lasers tones on thesame side of the cavity resonance as shown in Fig. 6.12. In this configuration, the previously ignored
lower modulation sideband of the red dressing laser becomes significant. This sideband introduces
an exchange interaction with the opposite sign of the other generated exchange interactions. By
carefully selecting the detuning and the amplitude ratio of the two dressing laser tones, we canachieve a configuration where χe = 0, effectively only leaving the χp term. Ĥ ′ offers the advantage ofpossibly being less sensitive to the presence of superradiance or collective decay for future squeezing
generation in our system.

119

1
3
4

2
1

2

1

π/2

(B)
θ [rad](A)πFigure 6.12: Two-axis counter-twisting with unstable points at north/south poles. Dynamics shown as flows on the Bloch sphere with three characteristic circles (purple: θi = 0.9π,
black: θi = 0.5π and green: θi = 0.1π) highlighted.

2

3

4

1

0

freq
0

π

ϕ [rad]2πFigure 6.13: (A)Frequency diagram with both dressing lasers positioned on the same side relative
to the cavity resonance. (B)The equirectangular projection of the resulting dynamics (bottom).

In Fig. 6.13(B), we show the measured flow lines for Ĥ ′ in the equirectangular projection.
One can clearly identify four stable fixed points on the equator as expected.
To have a better intuition of the dynamics, instead of focusing on the dynamics on the whole
Bloch sphere, we take a few cuts with initial Bloch vectors Ji prepared with θi = 0.1π, 0.5π and
0.9π and study the dynamics separately. For the initial Bloch vectors prepared on the two circles
with θi = 0.1π and 0.9π (green and purple in Fig. 6.12 and Fig. 6.14 (A) and (B) near thenorth/south poles, the distribution of the states after the interaction Jf is plotted in Fig. 6.14 (A)

120

(A)

(B)

1

2

(C)
3
2
4
1

Figure 6.14: Deformation of initial state distributions. (A), (B) and (C) are the distributionsof the states after the interaction with the prepared initial state θi = 0.1π (green circle), 0.9π
(purple circle) and 0.5π (black circle). The deviations from the initial state distribution (graydashed circles) are shown by the colored arrows.

and (B)(solid lines with fitted curves). The elliptical distributions with major axis orthogonal toeach other explicitly show the squeezing and anti-squeezing axis near the north and south polesin this small displacement limit. In Fig. 6.14 (C), the initial Bloch vector is prepared on theequator with different azimuthal angles (grey dashed line) and we plot Jf (black dot). The four
zero crossings correspond to the four stable points on the equator. Between the stable points, theobserved final states are deflected alternately above or below the equator as expected.

6.7

Conclusion
Here we have demonstrated the flexibility of our optical cavity simulator to engineer tunable

XYZ Hamiltonians using two selected momentum states, without the need for Floquet engineering
which in some cases might be challenging to allow access beyond mean-field dynamics in large many-

121
body systems due to amplitude and phase noise on the Floquet control fields [182][G] . The pairraising/lowering processes are verified to be present for the first time by the resonant spectroscopic
signal when the detuning δ is scanned as well as the direct cancellation of the exchange interactions
2 + Jˆ2 . By combining the correct relativethat yielded dynamics of the Hamiltonian Ĥ ′ = χ Jˆ+−balance of exchange and pair-raising/lowering contributions, we observed TACT dynamics for the
first time.
While so far our focus has been on the characterization of the Hamiltonian by probing theshort-time dynamics, extension to longer times can open untapped opportunities for quantum
simulation and metrology. For example, by using an implementation where two dressing lasers areused to generate one-axis twisting dynamics, one should be able to balance the collective dissipation
generated by the absorbed and emitted photons facilitating the emergence of beyond mean-field
effects and entanglement generation.
Furthermore, while here we focused on only two momentum states as a first step, by combining
them with actual internal states of the atoms or by adding more selected momentum states anddressing tones one should be able to engineer a toolbox for quantum state engineering, similar towhat has been done with momentum states in Bose-Einstein condensate [130, 131, 183]. In our casehowever, in addition to the internal and external level control that tunes the synthetic dimensions,
we can further enjoy the rich opportunities offered by the tunable cavity-mediated interactions toengineer phenomena ranging from superfluidity and supersolidity [102, 184, 185, 103] to dynamical
gauge fields and non-trivial topological behaviors [186, 93, 151, 138, 150].

Chapter 7
Conclusion and outlook

In this thesis, I have discussed how a high finesse cavity can be used to measure, bind together
and introduce interaction between atoms in superpositions of momentum states. Specifically, thematter-wave-cavity coupled system has demonstrated an entangled matter-wave interferometer,a novel collective recoil mechanism and a new way for simulating XYZ model. Not only can weentangle different momentum state using different spins, we can now also directly introduce tunableinteractions between momentum states with the same spin label.
In terms of the performance of the matter-wave interferometer, we are able to achieve aresolution of ∆g/g = 10−5 − 10−6 with about 1000 atoms and a separation period of few ms.
Although not comparable with state-of-the-art matter wave interferometer, it’s an exciting platform
with many possible prospects. In this chapter, I will give a brief overview on some future directions,the associating technical challenges and what might be the possible solutions.

7.1

Direct entanglement generation between momentum states
One missing opportunity for my PhD work is the direct entanglement generation between

the momentum states. With the momentum-exchange interaction, there is a clear path towards
squeezing using the one-axis-twisting dynamics. However, we are limited to above SQL due to a
few technical issues.
(1) Finite laser linewidth as discussed in Chapter 5 and Chapter 6. The kHz linewidth laser
we use to drive interactions is also inducing noisy rotations between the two momentum

123
states and thus spoils the squeezing. In a simple experiment, we prepare the initial stateon the equator of the Bloch sphere, scan the dressing laser power applied at the cavity,and measure the excess noise on the population difference induced by the dressing laser.
The extra noise quickly increases above few times of SQL, faster as compared to squeezing
dynamics. This is one of the key reasons why did we not observe entanglement. To suppress
this effect, we need to narrow the dressing laser more, with the critical reduction of phasenoise required for phase noise components at offsets from carrier ωz = 2π × 500 kHz. In
collaboration with Blumenthal’s group from UCSB, we believe that we can significantly
reduce the phase noise at 500 kHz offset from carrier by one to two orders of magnitude[187]. Given the relatively large offset from carrier, we also believe that we can improve oursystem using more traditional filter cavity to reduce the relevant phase noise components.

sigma/SQL

6

4

2

QPN + PSN
0

QPN
0

10

20

30

40

Dressing laser incident photons

50

3

60 x10Figure 7.1: Excess noise in the measured spin projection Jz induced by dressing laser. The two
black lines correspond to quantum projection noise (QPN) for the amount of atoms used in theexperiments, and the combined noise of QPN and photon-shot noise (PSN) related to the detection
processes.

(2) Secondly, the strong free-space scattering due to the finite detuning of the dressing/Bragg
laser reduce the interferometer contrast significantly. In order to satisfy the phase matchingcondition, the same laser is used for driving Bragg rotation and interactions. At the moment
it is only 500 MHz detuned from the atomic transition, such that every Bragg π-pulse causes

124
10% loss in contrast. The atomic loss from the scattering leads to a reduced contrast of theinterferometer and washes out the squeezing if any. This can be fixed by simply detuning
the cavity and dressing laser further away from the atomic transition, however, at the costof increased sensitivity to the laser phase noise, which in terms also requires a better laser
with lower phase noise at ≈ 500 kHz offset frequency.
(3) Lastly, finite eﬀiciency for transferring atomic population for the readout also reduces theobserve metrological gain. Equivalently, the atomic loss due to the state transfer leads toa reduced size in signal and could therefore wash out the squeezing if any. In the experiment, for enhancing the interaction between momentum states with larger Clebsh-Gordon
coeﬀicients, the atomic internal state is first transferred to |F = 2, mF = 2⟩ to utilize the cycling transition |F = 2, mF = 2⟩ → |F ′ = 3, m′F = 3⟩. However the eﬀiciency of the current
readout process is limited by the complex combination of microwave rotations and Raman
rotations, leading to an extra loss of atoms. Microwave rapid adiabatic passage may be employed in the future to improve the eﬀiciency. It may also be possible to simply perform thevelocity-selective readout and measurement using the F=1, mF=1 to F=2,mF=2 if inhomogeneous broadening and time dependent variation of the micrwave transition frequency
can be made smaller than 5 kHz.

7.2

Squeezing with balanced dissipation
Squeezing with unitary evolution (OAT) and quantum measurement (QND) have been re-alized in many systems. There are recent proposals for squeezing generation with engineered
dissipation[188, 189, 190]. One example is to drive balanced superradiance processes (J− and J+ ).
Specifically for our experiment, one can apply two laser tones at ±ωz away from the dressed cavity resonance to purposefully induce balanced superradiance between the two momentum states,
as shown in Fig. 7.2. Theory has predicted that the steady state shows appreciable amount ofsqueezing (> 7 dB) even for low atom number (700, as in our current setup.)

A

B

C

125

freq

Figure 7.2: Generating squeezing with dissipation. (A) Applying two dressing laser tones at±ωz from the cavity resonance can induce the balanced dissipation processes Jˆ− and Jˆ+ as shownin (B). As a result, the steady state is a squeezed state with squeezing on the phase quadrature asshown in (C).

The challenge here is how well balanced can we achieve for the two superradiance processes.
Aside from the classical noise for the laser power at the two frequencies, a key problem here is
the fluctuating dressed cavity resonance. In other words, the atom number fluctuation could leads
to the dressed cavity resonance fluctuate from shot to shot, such that it is hard to keep the two
dressing laser tones equally detuned from the cavity resonance. One way to circumvent this issue isto detune the cavity resonance much further away from the atomic transition to reduce the cavity
frequency shift induced by the presence of the atoms. This will again require that we work tonarrow the dressing lasers to avoid making the noise of the first issue greater.

7.3

Continuous quantum phase measurement
Quantum sensing relies on measuring the accumulated phase between the two levels. The

classic paradigm for phase measurement is the Ramsey sequence, because population measurement
is more viable as compared to phase measurement. However, the finite pulse lengths and phaseaccumulation time reduce the measurement bandwidth. It would be a big win to directly measure
the quantum phase.
In collaboration with our lab, Athreya from Holland’s group proposed a new approach for
performing phase measurement[153]. This idea is directly related to the balanced dissipation processes as described above. In this experiment, in stead of only applying photons at the cavity, we
also need to collect the photons (black squiggly line in Fig. 7.2) coming out of the cavity. A simple

126
intuitive way to understand this measurement is that the photon created by the two superradiance
processes carry the atomic information ⟨â⟩ ∝ ⟨Jˆ+ ⟩ and ⟨Jˆ− ⟩. Because photons coming from thetwo processes are indistinguishable, the total output is determined by the sum ⟨Jˆ+ ⟩ + ⟨Jˆ− ⟩ = ⟨Jˆx ⟩,which carries the phase information.
With the complex level structure, it might be challenging to do this experiment with atomic
hyperfine states. But with two momentum states, we could isolate a pure two-level system. The
relative phase between momentum states is directly related to the phase of atomic density grating
as discussed below.

7.4

QND measurement of the atomic density grating
OAT and QND essentially share the same Hamiltonian as discussed in Chapter. 2. Since we

can induce OAT between two momentum states, one should be able to perform QND measurementon the momentum states as well. There are two different steps.
The first one is the atom number counting in a specific momentum state. For QND measurement of the spin state (rubidium hyperfine states), one need to tune the cavity resonance close
to the transition between the excited state and one of the spin state. The QND measurement on
momentum states is only proposed for the narrow linewidth transition in strontium[191]. Thisis however diﬀicult for rubidium, due to the small energy difference between momentum state ascompared to the excited state lifetime. A possible approach is to use an extra laser tones to dress an
auxiliary spin state to realize an effective excited state with narrow and tunable linewidth, whichallows one can probe one leg of the two-photon transition between the two momentum states. Or
in other words, one can perform QND measurement of the two-photon transition to selectively
measure the atoms number in a specific momentum state.
A step further will be the direct phase measurement between two momentum states. This isconnected to the balanced superradiance idea or the XYZ interaction experiment. Again, one canapply two dressing laser at the cavity driving two superradiance processes, the photons coming outof the cavity carry phase information between the two momentum states. Direct phase measurement

127
can be performed by collecting and detecting the photon exiting the cavity. Combined with thewave packet binding mechanism, a continuous gravity measurement can be performed, akin to
dropping a ruler and continuously reading the position while falling.

7.5

Going beyond two-body interactions
The momentum-exchange interaction and the XYZ models realized in this thesis are both

two-body interactions involving pair of atoms. In other words, the associated Hamiltonian are
quadratic in terms of pseudo-spin operators, including the one-axis twisting and two-axis countertwisting dynamics. There are recent proposals on fast entanglement generations with three-bodyinteraction[192], or more complex squeezing protocol for phonon states in trapped ion systems[193].
Engineering controllable all-to-all three-body interactions is challenging, however, this might beachievable for cavity-QED systems with six-photon processes as drawn below:

A

B
freq

Figure 7.3: Three-body interactions with six-photon processes. (A) Frequency diagramwith two dressing lasers separated by 3ωz . (B) Six-photon process that raises three atoms fromlower momentum state |↓⟩ to the higher momentum state|↑⟩.

By separating two dressing laser tones for (3ωz ), there exist a resonant six-photon processthat promote three atoms into higher momentum states at once, and thus leads to a Hamiltonian
3 + Jˆ3 . Bigger frequency separation might lead to even higher order processes.
Ĥ6ph = χ6ph Jˆ+−However, detailed calculation and analysis is still needed to verify the interaction strength is strong
enough as compared to other dissipative processes (superradiance, free-space scattering, etc.) inorder to study these complex dynamics.

128

7.6

Simulating BEC-BCS cross-over with momentum qubits
In a recent work from the Sr experiment in our lab [9], dynamical phases of the BCS modelare observed with the Higgs oscillation due to the spin-exchange interaction. With the momentumexchange interaction, the dephasing process is more controlled and predictable such that one cansimulate the BEC-BCS crossover by preparing initial state with different decoherence and observing
the coherence oscillation under the momentum-exchange interaction.
Simulations have shown an obvious change in the time constants for the decay of oscillation
with different initial states. The challenge here is to suppress superradiance process to allow for
longer interaction time. One should be able to solve it by carefully balancing the dressing laser
powers above and below the cavity resonances.

7.7

Summary
Apart from the future directions listed above, Zhijing Niu is now having rapid progress on

suppressing Doppler shift on a narrow optical transition with momentum-exchange interactions
realized between strontium atoms in a ring cavity. All these fun physics are realized by coupling
matter waves to a high-finesse cavity. With this new approach for introducing interactions between
wave packets, momentum states could potentially become an interesting qubit system with many
new opportunities for quantum metrology and simulations.

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