Dipolar Shielding and Subwavelength Bilayers
in a Quantum Gas of DysprosiumbyMichael Alan Cantara
Submitted to the Department of Physics
in partial fulfillment of the requirements for the degree ofDoctor of Philosophy
at theMASSACHUSETTS INSTITUTE OF TECHNOLOGY
February 2023
© Massachusetts Institute of Technology 2023. All rights reserved.

Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Department of Physics
December 20, 2022

Certified by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Wolfgang Ketterle
John D. MacArthur Professor of Physics
Thesis Supervisor

Accepted by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Lindley Winslow
Associate Department Head of Physics

2

Dipolar Shielding and Subwavelength Bilayers
in a Quantum Gas of DysprosiumbyMichael Alan Cantara
Submitted to the Department of Physics
on December 20, 2022, in partial fulfillment of therequirements for the degree ofDoctor of Philosophy

Abstract
In our paper Can the Dipolar Interaction Suppress Dipolar Relaxation?, by using
magnetic dysprosium atoms and an optical lattice we engineered the dipolar suppression of dipolar relaxation in an ultra-cold gas prepared in an excited Zeeman level.
The atoms were confined to ultra-thin layers, with their magnetic moments aligned
perpendicularly to the trap such that the dipolar interaction was purely repulsive.
In such a configuration we observed an order of magnitude extension of lifetime inhigher-lying states, opening up new possibilities for quantum simulation with multiple spin-species of highly magnetic atoms. Theoretical efforts applying Fermi’s golden
rule numerically corroborated the observed suppression factors, including the fascinating dependence of dipolar relaxation on both the external magnetic field and theoptical trap confinement.
In our paper Atomic physics on a 50 nm scale: Realization of a bilayer systemof dipolar atoms, we utilize dysprosium’s disparate Clebsch-Gordan coefficients to
produce two independent optical lattices capable of tuning the interlayer separation
to 50 nm. The head-to-head orientation of the layers comprising the bilayer provide a new avenue of research for dipolar physics. Since the dipole-dipole interaction
scales as 1/𝑟3 , the reduction in interlayer separation leads to an approximately 500x
enhancement of the dipole-dipole interaction. With our new subwavelength tool we
explored dipolar coupling across the layers by observing both the transfer of thermal
excitation, as well as the transfer of a dipole oscillation in the harmonic trap, fromone layer to another. The Born approximation, applied to a dipolar interaction potential, yields excellent agreement with the observed interlayer thermalization rate.
The observed interlayer transfer of a dipole oscillation in the harmonic trap results in
an in-phase oscillation rather than the mean-field predicted out-of-phase oscillation,and can be partly ascribed to the friction force observed in the interlayer thermalization experiment, but may benefit from further exploration (e.g., correlated interlayer
density fluctuations). The super-resolution provided by our bilayer is amazingly flexible, limited only by the width of the layer, such that the interlayer separation canbe tuned arbitrarily small if sufficient localization of the layers is achieved. Such3

a tool will hopefully prove highly useful for the exploration of dipolar physics on apreviously unattainable length scale.
Thesis Supervisor: Wolfgang Ketterle
Title: John D. MacArthur Professor of Physics

4

To my family

5

6

Contents
1 Introduction

17

1.1

Demagnetization Cooling . . . . . . . . . . . . . . . . . . . . . . . . .

19

1.2

Overview of Dipolar Many-body Physics . . . . . . . . . . . . . . . .

21

1.3

Brief Summary of Publications . . . . . . . . . . . . . . . . . . . . . .

23

2 The Machine

27

2.1

The Vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

2.2

The Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

2.3

The Zeeman Slower . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

2.4

The Magneto-Optical Trap . . . . . . . . . . . . . . . . . . . . . . . .

51

2.5

The Optical Dipole Trap . . . . . . . . . . . . . . . . . . . . . . . . .

52

2.6

The Optical Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

3 Magnetic Dipole Moments

55

3.1

Landé g-factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

3.2

Hund’s Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57

3.3

Illustration of Hund’s Rules in the p-orbital . . . . . . . . . . . . . .

59

3.4

Magnetic Moments and g-factors of the Lanthanides . . . . . . . . . .

61

3.5

Electronic Configurations in the f -orbital . . . . . . . . . . . . . . . .

63

3.6

Magnetic Dipole Moment of Dysprosium . . . . . . . . . . . . . . . .

70

4 Dipolar Suppression of Dipolar Relaxation
4.1

Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7

71
72

4.2

3D Dipolar Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . .

74

4.3

Dipolar Collisions with Harmonic Confinement . . . . . . . . . . . . .

79

4.4

Pure 2D and Quasi-2D Dipolar Collisions . . . . . . . . . . . . . . . .

80

4.4.1

Double Spin Flip in the Pure 2D System . . . . . . . . . . . .

82

4.4.2

Introduction to the Quasi-2D System . . . . . . . . . . . . . .

87

4.4.3

Double Spin Flip in the Quasi-2D System . . . . . . . . . . .

89

4.4.4

Single Spin Flip in the Quasi-2D System . . . . . . . . . . . .

92

4.4.5

Theoretical Results . . . . . . . . . . . . . . . . . . . . . . . .

94

4.5

The Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

97

4.6

Extracting Dipolar Relaxation Rate Coefficient From Atomic Decay
Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

4.7

Zeroing The Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . 104

4.8

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5 A Tuneable Subwavelength Bilayer

109

5.1

The Tensor AC Stark Shift . . . . . . . . . . . . . . . . . . . . . . . . 110

5.2

Controlling Separations in the Bilayer . . . . . . . . . . . . . . . . . . 115

5.3

Calibrating Separations in the Bilayer . . . . . . . . . . . . . . . . . . 117

5.4

Spin-resolved Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . 119

5.5

Creating Spatially Isolated Layers . . . . . . . . . . . . . . . . . . . . 120

5.6

Thermal Transfer Across Layers . . . . . . . . . . . . . . . . . . . . . 123

5.7Interlayer Transfer of a Dipole Oscillation in the Harmonic Trap . . . 125

6 Outlook

129

A An Acousto-optic Modulator Driving Circuit with Intensity Stabilization

131

A.1 The Faceplate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
A.2 The Driver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
A.3 The Feedback in Action . . . . . . . . . . . . . . . . . . . . . . . . . 149
A.4 The RF Amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
8

A.5 The Housing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

9

10

List of Figures
1-1 A schematic diagram of the steps to perform demagnetization cooling.

20

2-1 Cross-sectional view of a Gamma Vacuum ion pump. . . . . . . . . .

29

2-2 An ion gauge comprising a filament, grid and collector. . . . . . . . .

30

2-3 A circuit representation of the bias voltages and current measurement
used in an ion gauge. . . . . . . . . . . . . . . . . . . . . . . . . . . .

30

2-4 A schematic diagram of a turbomolecular pump. . . . . . . . . . . . .

33

2-5 A schematic diagram of an axial compressor used in a Rolls Royce jetengine. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

2-6 A turbomolecular pump. . . . . . . . . . . . . . . . . . . . . . . . . .

34

2-7 A schematic diagram of the operation of a scroll pump. . . . . . . . .

35

2-8 A legend for the optical elements shown in the figures. . . . . . . . .

38

2-9 A schematic diagram showing the optical path of the 1064 nm outputof Amplifier 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

2-10 A schematic diagram showing the optical path of the 1064 nm outputof Amplifier 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

2-11 A schematic diagram showing the optical path of the 421 nm laser. .

412-12 A schematic diagram showing the optical path of the angled slowing.

42

2-13 A schematic diagram showing the optical path of the 626 nm laser. .

43

2-14 A schematic diagram showing the optical paths of the Zeeman slowing,
angled slowing, and horizontal MOT beams in the chamber. . . . . .

43

2-15 A schematic diagram showing the optical paths of numerous beams in
the horizontal plane of the lab.

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

44

2-16 A schematic diagram showing the optical path of the 741 nm light used
to create two optical lattices, as well as fiber couplings to frequency
metrology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

2-17 Zeeman slower design and magnetic field profile . . . . . . . . . . . .

47

2-18 Dependence of the MOT population on the detuning and power of theangled slowing beams. . . . . . . . . . . . . . . . . . . . . . . . . . .

49

2-19 Atomic population in the MOT after loading. . . . . . . . . . . . . .

50

3-1 Possible one-electron configurations for maximizing 𝐿 . . . . . . . . .

59

3-2 Possible four-electron configurations for maximizing 𝑆 and 𝐿 . . . . .

60

3-3 Magnetic Moments of the Lanthanides . . . . . . . . . . . . . . . . .

61

3-4 𝑔-factors of the Lanthanides . . . . . . . . . . . . . . . . . . . . . . .

61

3-5 One-electron ground state in the f-orbital . . . . . . . . . . . . . . . .

63

3-6 Two-electron ground state in the f-orbital . . . . . . . . . . . . . . . .

63

3-7 Three-electron ground state in the f-orbital . . . . . . . . . . . . . . .

64

3-8 Four-electron ground state in the f-orbital . . . . . . . . . . . . . . .

64

3-9 Five-electron ground state in the f-orbital . . . . . . . . . . . . . . . .

65

3-10 Six-electron ground state in the f-orbital . . . . . . . . . . . . . . . .

65

3-11 Seven-electron ground state in the f-orbital . . . . . . . . . . . . . . .

66

3-12 Eight-electron ground state in the f-orbital . . . . . . . . . . . . . . .

66

3-13 Nine-electron ground state in the f-orbital . . . . . . . . . . . . . . .

67

3-14 Ten-electron ground state in the f-orbital . . . . . . . . . . . . . . . .

67

3-15 Eleven-electron ground state in the f-orbital . . . . . . . . . . . . . .

68

3-16 Twelve-electron ground state in the f-orbital . . . . . . . . . . . . . .

68

3-17 Thirteen-electron ground state in the f-orbital . . . . . . . . . . . . .

69

3-18 Fourteen-electron ground state in the f-orbital . . . . . . . . . . . . .

69

4-1 Effective potentials and wave functions for different axial confinements

95

4-2 Channel-by-channel decomposition of the dipolar relaxation rates . .

96

4-3 Experimental results of the dipolar relaxation rates . . . . . . . . . . 100
4-4 Zeroing the magnetic field with imaging light . . . . . . . . . . . . . . 106
12

5-1 The AC Stark shift of each magnetic sublevel in dysprosium for 𝜋, 𝜎+ ,and 𝜎− transitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5-2 Creating and loading a subwavelength potential using tensor and vector
polarizabilities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5-3 Creating the subwavelength bilayer array. . . . . . . . . . . . . . . . . 116
5-4 Calibrating bilayer separations via Kapitza-Dirac. . . . . . . . . . . . 117
5-5 The Kapitza-Dirac signal as a function of the relative detuning. . . . 118
5-6 Spin-resolved imaging of the bilayer. . . . . . . . . . . . . . . . . . . 119
5-7 A spin-resolved absorption image showing the atoms loaded into the𝜎− and 𝜎+ lattices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5-8 Measuring the atom loss to determine the overlap of layers in the bilayer.121
5-9 Observation of interlayer thermalization. . . . . . . . . . . . . . . . . 124
5-10 Observation of momentum transfer between layers in the bilayer. . . . 126
A-1 A high-level schematic of the RF amplitude and frequency control in
the AOM Driver Box. . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
A-2 A schematic diagram of the Faceplate board showing the PCI-36 cardslot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
A-3 A schematic diagram of the Faceplate board showing the SMA inputs
and outputs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
A-4 A schematic diagram of the Faceplate board showing the switches. . . 137
A-5 A schematic diagram of the Faceplate board showing the potentiometers.137
A-6 A schematic diagram of the Faceplate board showing a solid-stateswitch, the output of which is then sent to one of two pins of a bicolor LED. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
A-7 A schematic diagram of the Faceplate board showing the PCB layout. 138
A-8 A schematic diagram of the Driver board showing the PCI-36 card-edgeof the PCB, which slots into the PCI-36 card-slot of the Faceplate board.140
A-9 A schematic diagram of the Driver board showing two voltage regulators.14013A-10 A schematic diagram of the Driver board showing an instrumentation
amplifier providing ground isolation for the External Amplitude Setpoint input, as well as a solid-state switch for selecting external orinternal amplitude setpoint control. . . . . . . . . . . . . . . . . . . . 141
A-11 A schematic diagram of the Driver board showing a dual-package opamp that 1) sums the offset voltage and the amplitude setpoint voltage
and 2) buffers the offset voltage. . . . . . . . . . . . . . . . . . . . . . 141
A-12 A schematic diagram of the Driver board showing an instrumentation
amplifier providing ground isolation for the External Amplitude Measure input. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
A-13 A schematic diagram of the Driver board showing a dual-package opamp that 1) provides proportional gain and 2) provides integral gain. A
solid-state switch is placed in the feedback of the integrator to prevent
wind-up (‘railing’) in the feedback capacitor. . . . . . . . . . . . . . . 142
A-14 A schematic diagram of the Driver board showing a dual-package opamp that 1) sums the proportional and integral gain signals and 2)
generates the error signal via summing. . . . . . . . . . . . . . . . . . 143
A-15 A schematic diagram of the Driver board showing a solid-state switch
for selecting feedback or feedthrough control of the amplitude setpoint.
Two Zener diodes are arranged to limit the feedback output to positive
voltages. Additionally, a dual-package op-amp 1) buffers the signal
applied to the IF port of the RF mixer and 2) provides 14x gain to theerror signal, which is then sent to an absolute value circuit implemented
with an instrumentation amplifier. The output of the absolute value
circuit is then sent to the Faceplate board to act as logic for the solidstate switch that powers the bi-color LED. . . . . . . . . . . . . . . . 143
A-16 A schematic diagram of the Driver board showing an instrumentation
amplifier providing ground isolation for the External Frequency Tune
input, as well as a solid-state switch for selecting external or internalfrequency control. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
14

A-17 A schematic diagram of the Driver board showing a fast variant of an
instrumentation amplifier providing ground isolation for the External
TTL input, as well as a solid-state switch for selecting external orinternal TTL control. . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
A-18 A schematic diagram of the Driver board showing an RF coupler used
to monitor the RF frequency, an RF mixer for adjusting the power ofthe RF signal, and an RF Switch for rapidly switching the RF output
on and off. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
A-19 A schematic diagram of the Driver board showing the PCB layout.

. 145

A-20 The voltage from a photodiode (cyan) and the amplitude setpoint voltage (yellow), as a function of time, in feedthrough operation. . . . . . 150
A-21 A Bode plot from a photodiode (cyan) compared to unity gain (yellow),in feedthrough operation. . . . . . . . . . . . . . . . . . . . . . . . . . 150
A-22 The voltage from a photodiode (cyan) and the amplitude setpoint voltage (yellow), as a function of time, in feedback operation. . . . . . . . 151
A-23 A Bode plot from a photodiode (cyan) compared to unity gain (yellow),in feedback operation. . . . . . . . . . . . . . . . . . . . . . . . . . . 151
A-24 A schematic diagram of the RF Amplifier board showing the two PCB
mounted power supplies and a voltage regulator. . . . . . . . . . . . . 152
A-25 A schematic diagram of the RF Amplifier board showing the RF amplification. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
A-26 A schematic diagram of the RF Amplifier board showing the PCB layout.153
A-27 The front panel, partition panel, and back panel.

. . . . . . . . . . . 156

A-28 3D printed insert for electrical insulation of the SMAs. . . . . . . . . 156
A-29 An assembled AOM Driver Box. . . . . . . . . . . . . . . . . . . . . . 157

15

16

Chapter 1
Introduction
In this thesis I will detail the construction of a new lab to study the dipolar interaction
between highly magnetic dysprosium atoms cooled to ultracold temperatures.
Dysprosium has a few key properties that truly differentiate it from alkali andalkaline earth atomic species. As the most magnetic element, dysprosium acts as
a powerful tool in the investigation of the dipolar interaction between two atoms,
which scales as the square of the magnetic moment and is responsible for wonderfulphenomena such as the Rosensweig instability in dipolar quantum droplets [1], demagnetization cooling [2], and dipolar supersolids [3]. As a result of its large magnetic
moment (10𝜇𝐵 ), the dipolar interaction between two dysprosium atoms is 100x larger
than alkali atoms.
In addition to its strong magnetic interactions, dysprosium has transitions ofhighly differing linewidths, allowing for powerful Zeeman slowing and imaging (Γ421 =2𝜋 × 32.2 MHz), a cold magneto-optical trap (Γ626 = 2𝜋 × 136 kHz, 𝑇Doppler = 3 𝜇K),

and an easily tunable optical lattice (Γ741 = 2𝜋 ×1.78 kHz). Furthermore, dysprosium
possesses large angular momentum in its ground state, an important feature that is
explored below.
Raman coupling between ground states can be used to engineer various synthetic
gauge field schemes, though alkali and alkaline earth species suffer from a fundamental
limitation when compared to dysprosium [4]. Due to the inability of electric dipole
transitions to couple with electronic spin, Raman processes with a spin flip (or between
17

two different spin states) in alkali and alkaline earth atoms rely on excited state spinorbit coupling to provide a pathway to changes of electronic spin in the ground state.
In order to utilize the spin-orbit coupling in the excited state for such purposes, theRaman laser must be able to resolve the excited state fine-structure splitting. If
the Raman laser detuning is larger than the fine-structure splitting, the contributions
from each line cancel one another, resulting in an unfavorable ratio of Raman coupling
to light scattering. On the other hand, it is desirable to achieve large detunings
for Raman coupling schemes such that light scattering is minimized, therefore sucha limitation is problematic. With spin-orbit coupling in the ground state, Raman
transititons for dysprosium do not rely upon the spin-orbit coupling in the excitedstate, thus allowing for large detunings that further diminish light scattering while
maintaining strong Raman coupling.
Nature has also favored dysprosium with a high natural abundance (≈ 20%) for
two bosonic species (162 Dy and 164 Dy) as well as two fermionic species(161 Dy and 163 Dy) [5]. While the bosonic species have no nuclear spin (𝐼 = 0),
the fermions have nuclear spin (𝐼 = 5/2) and corresponding hyperfine structure.
Furthermore, the differences in background scattering length and Feshbach spectraof each isotope may enable additional experimental opportunities.
Having outlined in broad strokes why dysprosium provides an interesting platform for ultracold atomic research, the rest of this chapter will be used to discuss aninteresting application of magnetic atoms in the setting of demagnetization cooling,
as well as an overview of dipolar many-body physics and a summary of the results
achieved in our lab during my tenure. Since a main result of this thesis is the suppression of dipolar relaxation, it is interesting to understand the below example ofdemagnetization cooling, which utilizes dipolar relaxation for a positive effect – thecooling of an atomic cloud.

18

1.1

Demagnetization Cooling

Due to the anisotropy of the dipole-dipole interaction, which leads to the couplingof partial waves of different angular momenta with one another, no purely magnetic
trapping of dense dysprosium clouds is possible. Although the sum of 𝑚𝐽 and 𝑚𝐽 ′of the colliding atoms is not conserved, the total angular momentum is conserved,
and one or both spins can be flipped in a process known as dipolar relaxation. The
probability for single spin flips scale as ∼ 𝐽 3 , and so dysprosium (𝐽 = 8) has arelaxation rate that is ∼ 4000x faster than alkali atoms (𝐽 = 1/2).

Purely magnetic trapping requires the atoms to be in a low-field seeking magnetic
sublevel due to Earnshaw’s theorem, which forbids magnetic field maxima, but notminima. However, these states will quickly undergo dipolar relaxation, heating theatoms as they release internal Zeeman energy from the spin flipping process andcausing loss. Due to this heating and loss, an optical dipole trap is used to produce a
Bose-Einstein condensate of dysprosium, spin-polarized in its lowest magnetic subleveland placed in a suitable magnetic field to adequately separate the Zeeman energies
relative to the thermal energy of the atoms.
Demagnetization cooling is a clever method that uses the typically unfavorable
dipolar relaxation process to enhance the phase-space density of an already cold (𝜇K)sample by an order of magnitude, with minimal loss of atom number [2]. With a spinpolarized sample of highly magnetic atoms in the state |𝑚𝐽 = −𝐽⟩, one can decrease

the bias magnetic field from a value large compared to the thermal energy of thesample to one that separates the magnetic sublevels by an energy comparable to thethermal energy, as shown in the top panel of Figure 1-1. In such a configuration
dipolar relaxation results in spin-flips that transfer population from the |𝑚𝐽 = −𝐽⟩
state to |𝑚𝐽 = −𝐽 + 1⟩ state. In order to do so, the atoms convert their externalmotional energy into internal Zeeman energy. Correspondingly, the temperature of
the atoms is reduced. In order to reset the cycle and allow for further cooling, thebottom panel of Figure 1-1 shows an optical pumping beam that is used to shuffle
atoms back into the stretched state while the emitted photon carries away the Zeeman
19energy.

Figure 1-1: A schematic diagram of the steps to perform demagnetization cooling.
Reproduced from [2].
This process is analogous to evaporative cooling, where energy is removed fromthe sample and it is allowed to thermalize before further cooling. A magnetic fieldramp, rather than discrete jumps, combined with a continuous optical pumping beam,
allows for demagnetization cooling to be run continuously. The optical pumping beam
should operate far-detuned from resonance (2–40Γ) and in the low saturation limit(𝑠 ≈ 10−5 ) to prevent heating. Ultimately, the recoil heating due to optical pumping

is the fundamental limitation that prevents further cooling. In general, the figure of
merit for a cooling process is given by𝜒=−ln(𝜌𝑓 /𝜌𝑖 )ln(𝑁𝑓 /𝑁𝑖 )where 𝜌𝑖,𝑓 is the phase-space density and 𝑁𝑖,𝑓 is the atom number before and after thecooling process. Evaporative cooling in ideal conditions can provide 𝜒evaporative = 4,20

whereas demagnetization cooling highly magnetic chromium atoms from 20 𝜇K to
10 𝜇K achieved 𝜒demagnetization = 11 [2]. According to calculations accounting forcooling limitation resulting from recoil energy, reabsorption of the scattered photons,
collisions with background gas, and 3-body collisions, a final temperature of 1 𝜇K
should be possible. The unexpected limitation is thought to result from imperfectcontrol of the magnetic field at the mG level, which results in a less than perfect 𝜎−polarization of the optical pumping beam.
Demagnetization cooling can also be applied to highly magnetic fermions andmolecules, leading to interesting possibilities of cooling fermions without a bosonicbuffer gas. Evaporative cooling of fermions in a single spin state is also possible
with dipolar interactions because the collision cross section does not vanish at zerotemperature due to the mixing of different angular momenta partial waves.

1.2

Overview of Dipolar Many-body Physics

For dipolar Bose-Einstein condensates (BECs) in the weakly-interacting regime, populated in the ground state with polarization axis 𝑧, the Gross-Pitaevskii equation is
given by [3]:
[︂
]︂
𝜕
~2 2
2
𝑖~ 𝜓 = −∇ + 𝑉trap (𝑟) + 𝑔|𝜓| + Φdd (𝑟, 𝑡) 𝜓,
𝜕𝑡
2𝑚

(1.1)

2

where 𝑔 = 4𝜋~
𝑎 is the coupling constant for contact interactions, characterized by𝑚𝑠-wave scattering length 𝑎, 𝑉trap (𝑟) is the trapping potential, and the mean field term
associated with the dipole-dipole interaction is given by:Φdd (𝑟, 𝑡) =∫︁𝑑𝑟′ |𝜓(⃗𝑟 ′ , 𝑡)|2 𝑈dd (⃗𝑟 − ⃗𝑟 ′ ),𝑈dd (⃗𝑟) = 𝐷

1 − 3 cos2 (𝜃)
,
|𝑟|3
21

𝜇0
where 𝐷 = 4𝜋(𝜇𝐵 𝑔𝐽 𝐽)2 and 𝜃 is the angle between the polarization axis (𝑧) and ⃗𝑟.

Following [3], we can define a dipolar length
𝑎dd =

𝐷𝑚
3~2

which differs from the dipolar length used in chapter 4 by only numerical prefactors.
In order to quantify the strength of the dipole-dipole interaction relative to 𝑠-wave
contact interactions, we can use the dimensionless parameter:
𝜖dd =

𝑎dd
.
𝑎

Although the validity of the dipolar Gross-Pitaevskii equation of Equation 1.1
requires further investigation, in the weakly dipolar regime (𝜖dd < 1), it has acted asa useful experimental and theoretical tool for the analysis of fundamentally different
properties derived from the anisotropic and nonlocal character found in dipolar BECs.
For example, an untrapped (𝑉trap = 0) uniform 3D cloud has a dispersion relationthat is anisotropic due to the dipole-dipole interaction. Breaking the spatial symmetry(e.g., via tight anisotropic confinement along the dipoles’ orientation), in conjunction
with the long-range anisitropic nature of the dipole-dipole interaction, can affect thetransverse excitation spectrum. Such is the case for the predicted and observed roton
excitation modes found in dipolar BECs confined to a pancake geometry [6, 7].
For 𝜖𝑑𝑑 ≥ 1, the attractive component of the dipole-dipole interaction allows low-energy elementary excitations of phonons in the plane perpendicular to the dipoles,

possibly causing mechanical collapse even for positive values of the 𝑠-wave scattering
length. The competition of the contact interaction and the dipole-dipole interaction
can lead to fascinating phenomena, including an ordered liquid-like arrangement ofdroplets [1], analogous to the Rosensweig instability found in a classical ferrofluid,with an observed stability that ultimately suggests beyond mean field effects resulting
from quantum fluctuations.

22

The dipole-dipole interaction can also lead to new physics when systems with confinement are explored. In 1D, for instance, the dipole-dipole interaction can break the
integrability found in systems with purely contact interactions and produce chaotic
dynamics [8].
Lastly, the dipole-dipole interaction, in the context of a lattice arrangement, leads
to the extended Hubbard model, adding anisotropic on-site and off-site interactions
and a competition of energy scales that is absent in systems with only the contact
interaction.

1.3

Brief Summary of Publications

In our paper Enhancing the capture velocity of a Dy magneto-optical trap with twostage slowing, we developed the means of enhancing our atom number in the magnetooptical trap (MOT) by a factor of 20 through angled slowing. In the experiment,
atoms escape from a hole in our oven and are collimated by a second hole to form an
atomic beam. This beam is slowed by a Zeeman slower through the combination ofmagnetic fields and a laser. The MOT is situated at the end of the atomic beam’s
path in order to capture the resulting slowed atoms. The main issue we encountered
was the relatively large distance the atoms must traverse from the end of the Zeeman
slower to the MOT’s location. In order for the MOT to capture the atoms, theirvelocity must be very slow. The combination of a large distance and slow velocity
along the direction of propagation unfortunately lends itself to transverse bloomingof the atomic beam, directly translating to a smaller flux of atoms through the MOT
capture region. Our paper successfully explored a workaround to such an issue byletting the atoms leave the Zeeman slower at a velocity higher than the MOT cancapture. Positioned just before the MOT region are two additional beams that crossand provide a final kick to the atoms, thereby negating the necessity of fully slowing
the atoms at the end of the Zeeman slower. By allowing the atoms to maintain a higher
velocity at the end of the Zeeman slower, the transverse blooming is mitigated andthe number of atoms reaching the MOT position is greatly enhanced. By boosting the23

number of cold atoms in our MOT we best prepare ourselves for evaporative coolingto a Bose-Einstein condensate.
In our paper Can the Dipolar Interaction Suppress Dipolar Relaxation?, by using
magnetic dysprosium atoms and an optical lattice we engineered the dipolar suppression of dipolar relaxation in an ultra-cold gas prepared in an excited Zeeman level.
The atoms were confined to ultra-thin layers, with their magnetic moments aligned
perpendicularly to the trap such that the dipolar interaction was purely repulsive.
In such a configuration we observed an order of magnitude extension of lifetime inhigher-lying states, opening up new possibilities for quantum simulation with multiple spin-species of highly magnetic atoms. Theoretical efforts applying Fermi’s golden
rule numerically corroborated the observed suppression factors, including the fascinating dependence of dipolar relaxation on both the external magnetic field and theoptical trap confinement.
In our paper Atomic physics on a 50 nm scale: Realization of a bilayer systemof dipolar atoms, we utilize dysprosium’s disparate Clebsch-Gordan coefficients to
produce two independent optical lattices capable of tuning the interlayer separation
to 50 nm. The head-to-head orientation of the layers comprising the bilayer provide a new avenue of research for dipolar physics. Since the dipole-dipole interaction
scales as 1/𝑟3 , the reduction in interlayer separation leads to an approximately 500x
enhancement of the dipole-dipole interaction. With our new subwavelength tool we
explored dipolar coupling across the layers by observing both the transfer of thermal
excitation, as well as the transfer of a dipole oscillation in the harmonic trap, fromone layer to another. The Born approximation, applied to a dipolar interaction potential, yields excellent agreement with the observed interlayer thermalization rate.
The observed interlayer transfer of a dipole oscillation in the harmonic trap results in
an in-phase oscillation rather than the mean-field predicted out-of-phase oscillation,and can be partly ascribed to the friction force observed in the interlayer thermalization experiment, but may benefit from further exploration (e.g., correlated interlayer
density fluctuations). The super-resolution provided by our bilayer is amazingly flexible, limited only by the width of the layer, such that the interlayer separation can
24be tuned arbitrarily small if sufficient localization of the layers is achieved. Sucha tool will hopefully prove highly useful for the exploration of dipolar physics on apreviously unattainable length scale.

25

26

Chapter 2
The Machine
Creating and manipulating quantum gases requires a simultaneous orchestration of
every component in the lab to take hot atoms from an oven down to the coldest
temperatures in the universe over the span of a few seconds. A network of digital
and analog voltages facilitates communication between a computer and our machine’s
power supplies, acousto-optic modulators, magnetic field coils and cameras, while ahost of software developed in the Ketterle group (e.g., Cicero, Poseidon, Zeus) provides the precisely tailored control sequences required to accomplish this challenging
task [9]. Gaining such control is critical when considering the sheer number of elements acting harmoniously in each stage of the experiment.
This chapter will cover the various vacuum and laser systems we employ in the
dysprosium machine. A detailed description of an AOM driving circuit with intensity
stabilization can be found in Appendix A. Further discussion on analog voltages andcomputer software can be found in chapters 5 and 6, respectively, of [4].

27

2.1

The Vacuum

Ultracold samples are maintained by avoiding collisions with atoms or molecules athigher temperatures. In order to prevent dysprosium atoms from colliding with room
temperature air, we create an ultra-high vacuum (UHV) environment for the atoms
to reside in. The construction of the machine has involved bolting together various
vacuum pumps, pressure gauges, valves and viewports, followed by wrapping themachine in heating tape and slowly increasing the temperature so as to bake out thegases embedded in the walls of the system and expedite their removal via vacuum
pumps.
UHV is maintained by titanium sublimation pumps and ion pumps. Titanium
sublimation pumps operate by sublimating titanium, converting it from a solid toa gas, such that the gaseous titanium solidifies and coats the nearby walls of thechamber. Sublimation is performed using a filament which is heated ("fired") bycirculating ∼ 50 A of current through it. The titanium coating is highly reactive, suchthat air molecules colliding with the coating are transformed into a stable product.
The titanium sublimation pump can be fired periodically, typically several times after
breaking vacuum, in order to replenish the titanium coating that has been consumed
via reactions with air molecules.
Ion pumps, such as the one shown in Figure 2-1, can be operated continuously
and contain an anode and a cathode with a voltage of several kV across them. The
anode attracts free electrons, which are then trapped in the form of cyclotron motion
due to a ∼ 1000 G magnetic field parallel to the electric field. The swirling plasma

has enough energy to ionize air molecules that enter its interaction region, with theresulting positively charged ions attracted to the titanium coated cathode. Upon
colliding with the cathode, several processes may occur: the ions can chemicallyreact with titanium, the titanium coating can be sputtered onto nearby surfaces andact as a getter, and the ions can become embedded in the titanium surface throughphysisorption.
28

Figure 2-1: Cross-sectional view of a Gamma Vacuum ion pump.
While titanium sublimation pumps and ion pumps are used to maintain UHV, theion gauge is a triode device that is capable of measuring ultra-low pressures (∼ 10−11torr) inside of the vacuum chamber. A filament acts as a hot cathode and is heated
via electrical current such that it emits electrons due to thermionic emission. The
emitted electrons are then attracted to the fine wire grid, which is DC biased above
the voltage of the filament. The emitted electrons do not immediately enter thewire grid, and during their motion prior to entering the grid, the energetic electrons(∼ 150 eV) can collide with surrounding air molecules, ionizing them in the process.
The ionized air molecules are then attracted to the grounded collector. Since thenumber of ionization events depends on the number of air molecules, the collector
current then serves as a measurement of the pressure in the region of the ion gauge.
Figure 2-2 shows a typical ion gauge design and Figure 2-3 shows the corresponding29

bias voltages and current measurement in each component.

Figure 2-2: An ion gauge comprising a filament, grid and collector. Reproduced from
[10].

Figure 2-3: A circuit representation of the bias voltages and current measurement
used in an ion gauge. Reproduced from [10].

30In order to achieve UHV, we must first traverse through regimes of higher pressure.
Vacuum pumps only function properly when they are operating within a range ofmanufacturer specified pressures. The order in which various pumps are utilized toachieve UHV is as follows:
1. Roughing pump (e.g., a dry scroll pump)
2. Turbomolecular pump
3. Titanium sublimation pump and ion pump
The roughing pump reduces the atmospheric pressure in the chamber down to ∼ 10−3

torr, at which point the turbomolecular pump is activated. If the chamber has been
exposed to atmosphere, then baking the chamber will be required. Baking is performed by using numerous heating tapes, each with a respective variac (variable ACtransformer) that controls the amount of heat produced by the fiberglass tape. Thermocouples are used on surfaces of the chamber close to the heating tape in order tomeasure the temperature. With the roughing pump and turbomolecular pump running, the chamber should be slowly heated (typically over the course of a day or two)to ∼ 180 ∘ C. The goal is to evaporate residual gases from the walls of the chambersuch that the pumps are able to remove the air molecules that would normally act asa leak in steady-state operation. Baking the chamber is a purposefully slow process,such that the thermal expansion of various vacuum components are able to equilibriate before further heating. Typically the angle valve is the most sensitive to changes
in temperature, such that following its manufacturer specified heating ramp should be
followed for the entire system. Furthermore, due to the high temperature of the ovenchamber, we used silver plated copper gaskets near the oven to prevent the melting
and consequential binding of flanges. Silver plated screws were also used everywhere
to prevent melting during baking. After a steady-state pressure is reached at highbaking temperatures, the temperature should be lowered to ∼ 80 ∘ C. The higher

pressure gauge used in the earlier stages of the pump down procedure is typicallyineffective below 1 mtorr, at which point the ion gauge is activated. The turbomolecular pump, together with the roughing pump backing it, further reduces the pressure31

in the chamber to ∼ 10−7 torr. With the turbomolecular pump and roughing pumpstill running, the titanium sublimation pump and ion pump are then activated. After

steady-state pressure is reached, the temperature should be slowly ramped back toroom temperature. As the pressure in the chamber approaches ∼ 10−9 torr the turbomolecular pump and roughing pump act as a leak and are turned off, in addition

to closing the gate valve which separates the chamber from the turbomolecular pumpand roughing pump. Patience and several firings of the titanium sublimation pumpshould lead to UHV at ∼ 10−11 torr or lower. If the chamber is struggling to achieve
UHV, leak checking should be performed. Using a bottle of methanol or isopropanol,squirt the alcohol into the small gap designed in the flange which provides easy accessto the gasket. If the flange is not properly tightened, a portion of the alcohol will
evaporate into the chamber, causing a large spike on the ion gauge reading. Tightening the flange should resolve the leak. Additionally, the dysprosium oven may also be
ramped to higher temperatures when the pressure is less than 10−7 torr, which bakes
the oven and the chamber surrounding it.
As discussed above, the turbomolecular pump and roughing pump play a criticalrole in achieving UHV. A turbomolecular pump is a multistage axial compressor
composed of numerous compression stages, each comprising a rotor and stator pair,with successively higher and higher pressures after each stage. The turbomolecularpump operates under the same simple principle as a room fan: spinning rotor blades
impart momentum to air molecules, thus directing them away from the chamber being
pumped. The rotor blades impart angular velocity to the air, forming a current known
as swirl. Stator blades are stationary blades that act as diffusers, randomizing theswirl into the other velocity components of the air and thus leading to an increase inpressure.

32

Figure 2-4: A schematic diagram of a turbomolecular pump. Reproduced from [11].
Without a stator to diffuse the swirl generated by a rotor, the pressure differential after
each stage would be minimal. Furthermore, successive rotors would be minimiallyeffective since the air would have a minimal velocity relative to the subsequent stage’s
rotor.

Figure 2-5: A schematic diagram of an axial compressor used in a Rolls Royce jetengine. The numerous rotor and stator stages lead to an oscillation in the air velocity
which is converted into an increase in pressure after each stage. Reproduced from
[12].

33

The angle between the rotor blades and the stator blades in each stage is chosensuch that air molecules accelerated by the rotor blades are pushed further down in thepump, towards the next compression stage. The outlet of the turbomolecular pumpis connected to the roughing pump, which is able to remove the higher pressure air
resulting from the numerous compression stages.

Figure 2-6: A turbomolecular pump. Reproduced from [11].

34Roughing pumps can be used to pump atmospheric pressure to low vacuum, typically 10−3 torr. We use a dry scroll pump, which is dry in the sense that it does notuse a lubricating oil which is exposed to vacuum. A scroll pump consists of a fixedspiral and an orbiting spiral which eccentrically orbits about the fixed spiral. Air is
continuously compressed to smaller volumes and brought towards the center of thetwo spirals, where it is ultimately ejected out of an exhaust port.

Figure 2-7: A schematic diagram of the operation of a scroll pump. Gas is pumped by
two spirals compressing the gas into progressively smaller volumes, eventually being
released via the exhaust port in the center. Reproduced from [13].
Having presented the broader aspects of creating an ultra-high vacuum environment, it is useful to consider the specific features of our machine. The dysprosiummachine was designed to mitigate unnecessary exposures of vacuum to atmosphereby using two gate valves to divide the machine into three sections. Each section has
an ion pump, a titanium sublimation pump, an angle valve, and an ion gauge, thus
allowing for each section to be isolated from the rest of the machine. The redundancyof pumps reduces the risk of failure for the machine as a whole, which proved useful when dysprosium flakes formed on the walls of the vacuum chamber in front of
the oven and preceded to fall into the ion pump situated underneath. The metallic
dysprosium flakes electrically shorted the cathode and anode, rendering the pumpinoperable. The machine as whole is still operable, however, because of the generous
overhead on pumping capabilities built into the design.
35

Many cold atom machines transmit Zeeman slowing light through a viewport thatis directly in the path of the atomic beam. As atomic layers build up on the viewport,
the transmitted optical intensity will diminish. To solve this issue, heating is applied
to the viewport to remove the atomic layers via sublimation.
A unique design feature of the dysprosium machine is due to the reflectivity ofdysprosium itself. I constructed an in-vacuum mirror mount in that is placed in
the atomic beam path to reflect Zeeman slowing light from a viewport that is notin the atomic beam path, leading to the buildup of dysprosium layers on the invacuum mirror rather than the viewport. Since dysprosium is reflective, as layers ofdysprosium build up, the mirror continues to reflect. The in-vacuum mirror mount isscrewed into a flange and suspended upside down into the vacuum chamber. Vented
screws were used to allow air to escape after the screwing process.
The highly magnetic nature of dysprosium led to design decisions that minimized
the potential for undesired magnetic fields near the main chamber. Accordingly,
the main chamber was custom-made out of titanium and is thus non-magnetic, andstandard optical mounts were substituted with non-magnetic variants. A surprisingsource of magnetism was found in large beam splitter cube mounts, which often
contain magnets for holding the two constituent components together. We were able
to remove the magnets without adverse effects.
In an effort to provide substantial differential pumping between the oven andthe remainder of the machine, a homemade solution was devised. A copper metal
gasket forming a solid disk was cut in the center, with a copper tube passed throughthe opening and held in place by friction. Due to better than expected vacuum
performance in the oven section, we believe the differential pumping to be superfluous.
Lastly, a rotatable atomic beam shutter is located in the atomic beam path, before
the Zeeman slower. Since atoms emitted from the oven can have a small but nonnegligible effect on a cold atom cloud in the main chamber, it is useful to havethe ability to eliminate atomic flux in the main chamber. We have found that forexperiments requiring long hold times, such as the measurement of 3-body losses,
this is a critical feature to employ. A mirror is also embedded in the atomic beam36

shutter, allowing for the Zeeman slower beam to be reflected out through a viewportfor alignment purposes. Using just visual inspection the intensity of the fluorescenceof the atomic beam at the oven, resulting from Zeeman slowing light, can also be used
for alignment.

2.2

The Lasers

The unique level structure of dysprosium ([Xe] 4𝑓 10 6𝑠2 ) provides a plethora of diverse
atomic transitions for our lasers to address. The valence electrons form a filled 𝑠-shell,allowing for helium-like transitions that may be used for cycling transitions in laser
cooling and imaging. The submerged and partially occupied 𝑓 -shell produces thestrong magnetic dipole moment (10 𝜇𝐵 ) while also providing transitions that differ
from helium-like elements.
We utilize the broad 421 nm transition for Zeeman slowing and imaging, while
the narrow 626 nm transition is used for forming a magneto-optical trap (MOT).
Lastly, the ultra-narrow 741 nm dipole-forbidden transition, which excites an 𝑓 -shellelectron, is used to create optical lattices. The narrowness of the 741 nm transition
allows for tunability between a red and a blue detuned optical dipole trap or opticallattice while also enabling tight lattices with trap frequencies on the order of hundredsof kHz or more.
There are lasers for each of the aforementioned transitions, as well as high-power
far-detuned 1064 nm light for the creation of optical dipole traps (ODTs). Below
are schematic diagrams showing the optical paths of the various lasers, while later
sections in this chapter detail how the lasers are utilized in the experiment.
37

Acousto-optic
deflector
Acousto-optic
modulator
Fiber
coupling
MirrorConvex lensConcave lens
Cylindrical lensHalf waveplate
Quarter waveplate
Polarizing
beamsplitter
Beam shutter
Beam block

Figure 2-8: A legend for the optical elements shown in the figures.

38

XODT 1
Vertical
ODT+200 MHz-200 MHz1064 nm
Azurlight
Amplifier 1 (45W)

Figure 2-9: A schematic diagram showing the optical path of the 1064 nm output of
Amplifier 1.

XODT 2+80 MHz1064 nm
Azurlight
Amplifier (45W)

Figure 2-10: A schematic diagram showing the optical path of the 1064 nm outputof Amplifier 2.
39

Figure 2-9 and Figure 2-10 show the three 1064 nm ODT beams used in the experiment. The acousto-optic modulator (AOM) frequencies are chosen to be sufficiently
different such that any overlap in the polarizations of the beams results in a rapid
traveling wave whose effect on the atoms is rapidly averaged out. The AOMs also
provide intensity stabilization and control. Crossed ODT 1 (XODT 1) and Crossed
ODT 2 (XODT 2) are crossed at a small angle relative to one another in the horizontal plane of the lab. The Vertical ODT travels along the vertical axis of the lab andintersects the two crossed ODTS. The various polarizing beam splitters (PBSs) canbe used to diminish the optical power going to the optical fibers, allowing for safer
fiber coupling with less risk of thermal damage. Due to thermal lensing in the AOM,
it is generally best practice to keep the optical power traversing the AOM constantduring fiber and AOM coupling, instead diminishing the power after the AOM to prevent fiber damage. If the optical power traversing the AOM is not constant, then the
couplings may have to be adjusted to correct for steady-state differences in thermal
lensing.

40

Horizontal
Imaging

VerticalImaging
AngledSlowing

ULE
CavityZeemanSlower+200 MHz

Wavemeter+335 MHz421 nmM2
Doubling Cavity

842 nmM2
Ti:Sapphire

532 nmSproutDiode

Figure 2-11: A schematic diagram showing the optical path of the 421 nm laser.

Figure 2-11 shows the the main optical breadboard used for 421 nm light. We use a
532 nm laser to pump an 842 nm Ti:Sapphire laser, which is then frequency doubled
to produce the desired 421 nm blue light for the strong transition of dysprosium. The
Zeeman slower requires high power to function optimally, thus we tune the Ti:Sapphirefrequency such that the output of the doubling cavity can be used for Zeeman slowing
without further frequency shifting. In place of an AOM, a shutter is used as an opticalswitch for the light in the Zeeman slower path. The input of the Angled Slowing fiber
is sent to a separate breadboard, shown in Figure 2-12. A portion of the 421 nm lightis double-passed by a +335 MHz AOM, with part of the output sent to be further
double passed through a +200 MHz AOM for horizontal and vertical imaging andthe remainder sent to the wavemeter and the ultra-low expansion (ULE) cavity forfrequency metrology. A shutter is used in the wavemeter path since the wavemeter41

can also be used for frequency stabilization or measurement of other lasers in the lab.

AngledSlowing

Angled Slowing
Arm 1+500 MHz

Angled Slowing
Arm 2

Figure 2-12: A schematic diagram showing the optical path of the angled slowing.
Figure 2-12 shows the angled slowing optical breadboard. A double passed +500
MHz AOM shifts the fiber optic input light, the output of which is divided between
two angled slowing fibers with output beams that are crossed just before the MOT
capture region, as shown in Figure 2-14.

42

ULE
Cavity626 nmToptica
Diode-200 MHz
MOT

Wavemeter

Figure 2-13: A schematic diagram showing the optical path of the 626 nm laser.
The MOT beams are derived from a 626 nm Toptica diode laser, as shown in Figure 213. An additional optical breadboard splits the light from the MOT fiber into three
retro-reflected beams, two of which are shown in Figure 2-14 and Figure 2-15.

Figure 2-14: A schematic diagram showing the optical paths of the Zeeman slowing,
angled slowing, and horizontal MOT beams in the chamber. Image provided courtesyof Pierre Barral.
Figure 2-14 shows the optical path of angled slowing in relation to the center of the
MOT.
43

CameraDichroic

In-vacuum
MirrorAtomic beam

XODT 1
Dichroic

XODT 2

Cleaning
Beam

MOT 2

MOT 1

Horizontal
Imaging

ZeemanSlowerFigure 2-15: A schematic diagram showing the optical paths of numerous beams in
the horizontal plane of the lab.
Using a lens, XODT 1 and XODT 2 are crossed at a small angle relative to one
another in the horizontal plane of the lab, as shown in Figure 2-15. The beams of
MOT 1 and MOT 2 are allowed to directly expand to their final diameter by using a2" collimating lens such that no telescoping is necessary. Additional 2" optics (e.g.,
waveplates, beam splitters, mirrors) are also used. The Cleaning Beam is used toquickly remove atoms from the imaging region and is derived from the zeroth orderof the imaging beam in Figure 2-11, which is subsequently shifted -500 MHz by an
AOM. The Cleaning Beam is useful for kinetics mode imaging (e.g., with an Andorcamera) such that images with and without atoms can be taken in rapid succession
in an effort to minimize the effects of time-varying fringes.

44532 nmSproutDiode842 nmM2
Ti:Sapphire

Lattice 1-80 MHz+200 MHz

Lattice 2
ULE
CavityWavemeter

Figure 2-16: A schematic diagram showing the optical path of the 741 nm light used
to create two optical lattices, as well as fiber couplings to frequency metrology.
Figure 2-16 shows the 741 nm optical breadboard used for the creation of two
optical lattices. An acousto-optic deflector (AOD) is used to provide broad frequency
tunability while maintaining high diffraction efficiency. The AOD frequency bandwidth of ≈ ±40 MHz is used to change the relative separation of the two opticallattices which form the bi-layer. With this bandwidth the AOD is able to maximallyseparate the lattice sites of the atoms ("interlaced") as well as fully merge the twolattice sites on top of one another. The AOM and AOD frequencies are chosen to be
sufficiently different such that any overlap in the polarizations of the beams results
in a rapid traveling wave whose effect on the atoms is rapidly averaged out.

45

2.3

The Zeeman Slower

The Zeeman slower is designed to operate at the broad Γ = 2𝜋 × 32.2 MHz transition

found at 𝜆 = 421 nm [5]. The increasing magnetic field profile produced by a textbook
Zeeman slower is given by:
(︂
)︂
√︂
𝑧
𝐵(𝑧) = 𝐵0 1 − 1 −+ 𝐵bias
𝐿0
where
𝐵0 =

ℎ𝑣0
𝜆𝜇𝐵

𝐿0 =

𝑣02
𝑎max

𝑎max =~𝑘 Γ.
𝑀2

To produce the Zeeman slower used in our experiment we generated a similar but
different magnetic field profile, as shown in Figure 2-17, that prioritized homogeneous
deceleration of the atomic beam.
For our experiment, we operate the oven at 1250°C, which leads to a most probablevelocity of 𝑣0 = 483 m/s for the modified Maxwell Boltzmann distribution that thedysprosium atoms follow after effusing from the oven. The maximum deceleration
the atoms undergo, 𝑎max , occurs when the intensity of the slowing light approaches
infinity. From these values, we determine that
𝐵0 = 82 mT

𝐿0 = 42.5

46𝑎max = 550, 000 m/s

Figure 2-17: Top panel: a schematic diagram of the magnetic field coils of the Zeeman
slower. Bottom panel: designed (solid blue) and measured (red dots) magnetic fieldvalues as a function of position within the Zeeman slower. Reproduced from [4].
The decision to construct an increasing field Zeeman slower has its origins in thefollowing reasoning. A Zeeman slower operates through the interplay of the atom’s
detuning from the atomic resonance resulting from the Zeeman shift, the Doppler
shift and the laser’s detuning. By designing an increasing field slower, our magnetic
field at the start of the slower is at its minimum, 𝐵(𝑧 = 0) = 𝐵bias .
Correspondingly, the laser’s detuning must offset the lack of a Zeeman shift suchthat it addresses the maximum velocity class that is to be cooled. The result isthat the laser is further red-detuned from resonance than in the decreasing-field case.
This provides two advantages when utilizing such a broad slowing transition: 1) itallows the laser to off-resonantly slow atoms traveling faster than the planned capture
velocity as they travel from the oven to the start of the Zeeman slower and 2) as the47magneto-optical trap (MOT) is loaded, the Zeeman slowing light pushes the MOT
cloud less because it is further detuned from resonance at zero magnetic field.
If a nonzero 𝐵bias is used, it must be compensated by further detuning the laser
frequency. A large bias is useful to insure that the Zeeman beam is far-off resonancefrom the atoms loaded into the MOT, however, if too-large of a bias is applied thenthe atoms will scatter less slowing light between the oven and the entrance to the
Zeeman slower coils.
Since the magnetic field is maximal at the end of the Zeeman slower, a large
counter-wound magnetic coil is used to cancel the magnetic field at the location of
the atoms. The sharp reduction in magnetic field quickly removes the atoms fromresonance, which leads to an experimental challenge. The non-negligible distancefrom the end of the Zeeman slower to the center of the MOT leads to production of
low velocity atoms that have poor flux into the MOT, or to higher velocity atoms
that cannot be caught by the MOT.
We successfully turned this challenge into an advantage through the addition of
two 421 nm beams which intersect in the path of the atomic beam, just before the
MOT center. The two beams perform angled slowing by providing a last second
kick to the atoms emerging from the Zeeman slower, allowing them to fall into the
MOT. With angled slowing the Zeeman slower is made to produce atoms which arefast enough to provide high atomic flux to the MOT, but slow enough to be loaded
into the MOT when provided the final bit of scattering from the two beams. The
angled slowing is directly launched into the chamber from two optical fibers with
attached lenses and quarter waveplates, all situated atop two standalone pedestals,
see Figure 2-14.

48

The angled slowing beams are red-detuned from resonance, where the optimal
frequency for loading the MOT can be found for a given intensity, as shown in
Figure 2-18.

Figure 2-18: Dependence of the MOT population on the detuning and power of theangled slowing beams. Reproduced from [4].

49With the optimal frequency and intensity of the angled slowing beams selected,
we were able to observe a powerful atomic population enhancement of ≈ 20x whencompared to loading without angled slowing.

Figure 2-19: Atomic population in the MOT after loading. Note the (x10) factor
applied for No Angled Slowing or Transverse Cooling.
We have previously used two 421 nm injection-locked laser diodes to create a 2D
optical molasses before the atoms enter the Zeeman slower, providing transverse (tothe atomic beam) cooling. The diodes inherently have a broad emission spectrum, but
by sending in a tiny sample of our narrow fundamental laser, the diode’s output snaps
to the same emission spectrum as that of the seed light. As shown in Figure 2-19, thetransverse cooling provided modest gains to atomic population, but was ultimately
removed from the setup due to its complexity and lack of utility.

50

2.4

The Magneto-Optical Trap

We generate ground state spin-polarized dysprosium atoms using the 626 nm transition and magnetic field coils in an anti-Helmholtz configuration, together forming amagneto-optical trap (MOT). The final atomic population after MOT loading benefits from high power 626 nm light, thus we use three retro-reflected beams rather
than 6 independent beams.
Spin-polarization to the ground state is critical for dysprosium since rapid heating
resulting from dipolar relaxation will rapidly destroy the sample. However, dipolar
relaxation is energetically forbidden when the two participating atoms are in thelowest energy state. Utilizing dysprosium’s large mass and detuning the 626 nm
MOT beams further from resonance, we are able to directly spin-polarize dysprosium
to its ground state and prevent dipolar relaxation [14]. The spin-polarization occurs
through a beautiful interplay between gravitational and radiative forces. Consider
only the vertical axis of the MOT. In order to trap dysprosium, a 𝜎− beam sent frombelow must counteract the gravitational force acting on the atom. The radiative forceof the 𝜎− beam depends on the both the intensity and the detuning of the beam withrespect to the atom’s resonance. However, if the intensity of the beam is reduced,
then the atomic cloud will sag to a finite magnetic field below the center of the MOT.
When the atomic cloud is near the center of the MOT, the magnetic field is near zeroand the splitting between excited state Zeeman levels is small, leading to negligible
differences in detuning for different polarizations of light. As the atomic cloud sags
to a finite magnetic field, the Zeeman levels split further. The splitting, combinedwith further detuned MOT beams, causes 𝜋 and 𝜎+ transitions to be detuned fromresonance.
It has been shown that, at the equilibrium position between gravitational andradiative forces, the detuning of the 𝜎− beam with respect to the transition only
depends on the intensity of the beam, and not on the bare detuning of the laser[14]. For example, the detuning of the 𝜎− beam increases with increasing intensity ofthe beam, since a farther detuned brighter beam is able to provide the same gravity51

canceling radiative force as a less detuned weaker beam. Essentially, the atomic cloudsags to a finite magnetic field value that compensates for the bare detuning of thelaser. The detuning of the 𝜎− beam from the transition is relatively constant asa result of this equilibrium and is thus preferentially absorbed over the 𝜎+ and 𝜋transitions, which possess detunings that are not compensated by the magnetic field.
The horizontal MOT beams project their circular components onto the 𝑧^ bias field of
the atoms resulting from the finite magnetic field, and since only the 𝜎− light is close
to resonance, the horizontal beams also provide optical pumping to the ground state.
In order to enhance the capture velocity of the MOT, the MOT beams are dithered
via an AOM with a frequency of modulation at 120 kHz and a depth of ditheringthat broadens the laser linewidth to ≈ 2.6 MHz. The myriad resulting 120 kHz

spaced sidebands provide an artificially broadened laser for the atoms to absorb.
After loading the MOT we perform compression by simultaneously lowering boththe magnetic field gradient and the intensity of the MOT beams, turning off thefrequency dithering to achieve a lower Doppler limit, and tuning the frequency of the
MOT beams closer to resonance. The compressed MOT thus provides a several 𝜇K
atomic cloud that can be loaded into the optical dipole trap.

2.5

The Optical Dipole Trap

After MOT compression we load the dysprosium cloud into a crossed optical-dipole
trap (XODT). The XODT consists of three 1064 nm laser beams: two beams with
40 𝜇m beam waists crossed at 8∘ in the (horizontal) 𝑥^ − 𝑦^ plane, and a beam witha 64 𝜇m waist propagating along the (vertical) 𝑧^ direction. An interesting featureof dysprosium is that it possesses disparate Clebsch-Gordan coefficients for 1064 nm
light, which must be accounted for when determining the trapping potential that isformed. For instance, if the atomic sample is biased with a magnetic field along 𝑧^,
then the trapping potential can vary by a factor of two depending on whether a 𝜎−polarization or a 𝜎+ polarization is propagated along the vertical axis.
52

2.6

The Optical Lattices

After reaching ultra-cold temperatures we are then able to load the dysprosium cloud
into one or two optical lattices, depending on the experiment.
With one lattice, an array of quasi-2D crêpes can be created from the atomic
sample by retro-reflecting a 741 nm laser beam along 𝑧^. The lattice beam frequency
is easily adjustable about the narrow 1.8-kHz 741 nm transition, thus enabling neardetuned operation that provides tight axial confinement while still avoiding excessive
light scattering.
As will be discussed in [the bi-layer section], disparate Clebsch-Gordan coefficientsallow for two 741 nm optical lattices to be simultaneously employed, with negligible
cross-interference, along the vertical 𝑧^ direction. The two lattices are of opposite circular handedness and are detuned by hundreds of MHz so as to prevent low-frequency
beat notes.

53

54

Chapter 3
Magnetic Dipole Moments
The magnetic moment of an atom can determined with a simple application of Hund’srules to determine the ground state configuration, along with a calculation of the
Landé 𝑔-factor corresponding to the ground state. The long-range and anisotropic
dipole-dipole interaction provides a beautiful counterpoint to the traditional settingof ultracold bosonic collisions which are characterized by a spherically symmetriccontact interaction.

3.1

Landé g -factor

The Landé 𝑔-factor is related to the classical quantity known as the gyromagnetic
ratio. The gyrommagnetic ratio (𝛾) is a multiplicative prefactor that linearly scales
the magnetic field (𝐵) a magnetic dipole moment sits in to the precession frequency(𝜔) of that magnetic dipole moment about the axis of the applied field: 𝜔 = 𝛾𝐵. The
Landé 𝑔-factor for an electronic spin (𝑔𝑆 ≈ 2) is the dimensionless prefactor of thecorresponding dimensional gyromagnetic ratio 𝛾𝑒 = 𝑔𝑆 𝜇~𝐵 .

Classically, the gyromagnetic ratio scales the precession frequency of the magnetic
dipole in a magnetic field, while quantum mechanically the Landé 𝑔-factor scales themagnetic Zeeman energy sublevels of an atom in a magnetic field. To determine
the Landé 𝑔-factor of an atom requires a quantum mechanical summation of thevarious angular momenta that contribute to the magnetic dipole moment of the atom.
55

Contributions to the magnetic moment derived from electronic angular (𝐿) and spin(𝑆) momenta contain the Bohr magneton (𝜇𝐵 ) as a prefactor while contributions from
nuclear spin (𝐼) contain the nuclear magneton (𝜇𝑁 ) as a prefactor.
The Bohr magneton can be derived by considering a single electron creating aring current about a proton. A ring current (𝐼current ) with an area (𝐴) has a magnetic
moment 𝜇 = 𝐼current 𝐴. The spatial extent of this hydrogen atom is given by the Bohrradius 𝑎0 , while current is defined as the the rate at which charge flows through asurface. In this example there is one electron of charge 𝑒 which flows through a given
0
area of the ring once every period 𝑇 = 2𝜋𝑎, where 𝑣 is the velocity of the electron.
𝑣

𝑒𝑣
Thus 𝜇𝐵 = 2𝜋𝑎(𝜋𝑎20 ) = 𝑎02𝑒𝑣 . The Heisenberg uncertainty principle tells us that the0minimum orbital, of size 𝑎0 , must have a corresponding momentum uncertainty such𝑒~that 𝑎0 Δ𝑝 = ~. Therefore 𝑎0 𝑚𝑒 𝑣 = ~ and we arrive at 𝜇𝐵 = 2𝑚.
𝑒

In order to determine the nuclear magneton the same calculation can be performed
for the proton, with the ring current instead formed by the proton orbiting around the𝑒~electron and ultimately yielding 𝜇𝑁 = 2𝑚
. Due to the differences in mass between𝑝the electron and the proton, the Bohr magneton is ≈ 1836x greater than the nuclear

magneton. Consequentially, we may neglect the nuclear contributions to the magnetic
dipole moment for the remainder of this thesis.
The magnetic moment of an isolated spin is given by 𝜇𝑆 = 𝑔𝑆 𝑆𝜇𝐵 and the magnetic
moment of an isolated orbital angular momentum is given by 𝜇𝐿 = 𝑔𝐿 𝐿𝜇𝐵 , where𝑔𝐿 ≈ 1 is the Landé 𝑔-factor for the electronic orbital angular momentum. The⃗ + 𝑆)⃗ is then given bymagnetic moment of the total angular momentum (𝐽⃗ = 𝐿⃗ 𝐵 = 𝜇𝜇
⃗ 𝐽 = 𝑔𝐽 𝐽𝜇⃗𝐿 + 𝜇⃗ 𝑆 . Previously, the magnetic moments have been defined as
scalar quantities since they arose from only one angular momentum vector. Since
the total angular momentum is now a sum of orbital and spin angular momenta, the⃗ and 𝑆,⃗
vector 𝐽⃗ is necessary to account for the possible relative angles between 𝐿
which leads to allowed total angular momentum values |𝐿 − 𝑆| ≤ 𝐽 ≤ 𝐿 + 𝑆. For

the case of dysprosium we will find the ground state configuration to be a stretched⃗ and 𝑆⃗ can be simply added together to findstate (𝐽 = 𝐿 + 𝑆), such that the 𝐿

𝜇Dy = 𝜇𝐿 + 𝜇𝑆 .
56

In general we can determine the Landé 𝑔-factor for the total angular momentum
via the following derivation:⃗ + 𝑔𝑆 𝑆⃗≈𝐿⃗ + 2𝑆⃗𝑔𝐽 𝐽⃗ = 𝑔𝐿 𝐿
⃗ · 𝐽⃗ + 2𝑆
⃗ · 𝐽⃗
𝑔𝐽 𝐽⃗ · 𝐽⃗ = 𝐿
Therefore,⃗ ·𝑆⃗
𝑔𝐽 𝐽(𝐽 + 1) = 𝐿2 + 𝑆 2 + 3𝐿
⃗ + 𝑆,⃗ therefore 𝐽⃗ · 𝐽⃗ = 𝐽(𝐽 + 1) = 𝐿2 + 𝑆 2 + 2𝐿
⃗ · 𝑆.
⃗ Substitution and
But 𝐽⃗ = 𝐿
simplification leads to the Landé 𝑔-factor for the total angular momentum𝑔𝐽 ≈3 𝑆(𝑆 + 1) − 𝐿(𝐿 + 1)+
2
2𝐽(𝐽 + 1)

with corresponding magnetic moment 𝜇𝐽 = 𝑔𝐽 𝐽.

3.2

Hund’s Rules

Armed with the understanding of how to determine the magnetic dipole moment fora given 𝐿, 𝑆, and 𝐽 we now seek to determine these quantum numbers by applying
Hund’s rules to find the ground state for a given atom. Hund’s rules provide aconstrained optimization for the ground state based on the hierarchy of the energies
involved. Electrons in singly occupied orbitals are less shielded than their doubly
occupied counterparts, thus leading to a tighter binding energy and a lower energy
configuration (Rule 1). In a similar vein, electrons orbiting in the same direction
encounter each other less frequently compared to the case of electrons orbiting in
opposite directions (Rule 2). Lastly, the spin-orbit coupling must be minimized tofind the ground state (Rule 3). It is important to note that Rule 2 does not apply
for the case of electronic shells with only one electron since the different 𝑚𝑙 orbitals
are degenerate in such a system. Proceeding to Rule 3 then breaks the degeneracy,
ironically producing a maximal L state that Rule 2 could have produced.
57

To summarize, Hund’s rules are:
1. Maximize 𝑆
2. Maximize 𝐿
3. Account for spin-orbit coupling:1ℋ𝑆𝑂 ∝ 𝐿 · 𝑆 = 𝜁(𝐿, 𝑆)(𝐽 2 − 𝐿2 − 𝑆 2 )2where 𝜁(𝐿, 𝑆) = +1 for less than half-filled shells and 𝜁(𝐿, 𝑆) = −1 for morethan half-filled shells. Therefore the ground state will minimize (maximize) 𝐽
for less than (more than) half-filled shells.
Rule 3 is often taught in a convoluted manner, but as will be shown in the following
examples, it is quite trivial to implement if one only focuses on minimizing the spinorbit coupling and forgoes the complicated wording and equations.

58

3.3

Illustration of Hund’s Rules in the p-orbital

In order to maximize 𝐿 we first maximize the projection of 𝐿 onto the quantization
axis, 𝑚𝑙 , which then dictates the maximum possible 𝐿. The four options for where
we could place the first electron in the 𝑝-shell are illustrated below:

Figure 3-1: Possible one-electron configurations for maximizing 𝐿
To determine which of these four configurations is the ground state we must lookat the spin-orbit term and minimize it: 𝐿 · 𝑆 < 0 when 𝐿 and 𝑆 are anti-aligned,
therefore only configurations i) and iv) are correct. When 𝐿 and 𝑆 are anti-aligned,

𝐽 is minimized: 𝐽 = |𝐿 − 𝑆|. Thus for the case of one electron in a 𝑝-orbital, theground state has 𝐿 = 1, 𝑆 = 1/2, and 𝐽 = 1/2.

The fact that two equivalent ground states were found is a result of the fact that
it is only the relative orientation of 𝐿 and 𝑆 that matters since their orientations in
the lab are arbitrary.

59

We now understand the first half of Hund’s Rule 3, so let’s look at the case when
the shell is more than half-filled. Again, we have four possible configurations due tomaximizing 𝑆 and 𝐿 (Hund’s rules 1 and 2):

Figure 3-2: Possible four-electron configurations for maximizing 𝑆 and 𝐿
We notice now that when the shell is more than half-filled, the electrons we add
have the opposite sign compared to the first half of the shell. By looking at the four
configurations we can see that for the fourth electron, 𝐿 · 𝑆 < 0 only for ii) and iv),
where the 𝐿 and 𝑆 of the newly added electron are anti-aligned. The newly added

electron, however, is placed in an 𝑚𝑙 that is aligned with the remaining unpaired
spins! Thus we see that for more than half-filled shells, we maximize 𝐽 to find theground state: 𝐽 = 𝐿+𝑆. Thus for the case of four electrons in a 𝑝-orbital, the groundstate has 𝐿 = 1, 𝑆 = 1, and 𝐽 = 2.
It is now straightforward to apply Hund’s rules to the lanthanide elements whichcontain electrons in the 𝑓 -orbital. Atomic states can be succinctly defined via theirterm symbol 2𝑆+1 𝐿𝐽 , where 2𝑆 + 1 is the multiplicity of the spin states.

60

3.4

Magnetic Moments and g -factors of the Lanthanides

μ
10
9
8
7
6
5
4
3
2
1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

13

14

electrons

Figure 3-3: Magnetic Moments of the Lanthanides
g - factor
2

1.5

1

0.5

1

2

3

4

5

6

7

8

9

10

11

Figure 3-4: 𝑔-factors of the Lanthanides

61

12electrons

There are several interesting features worth noting in Figure 3-3 and Figure 3-4.
As electrons are added to the empty 𝑓 -shell the 𝑔-factor initially decreases due to 𝐿
and 𝑆 being anti-aligned and 𝐿 increasing less rapidly than 𝑆, such that the larger 𝑔𝑆
is able to dominate over the weaker 𝑔𝐿 . In the case of an 𝑓 -shell with six electrons,
𝐿 and 𝑆 cancel to produce 𝐽 = 0, thus producing no magnetic moment in the atom.
The 𝑔𝐽 is chosen to be zero in this case, although any value of 𝑔𝐽 coupled with 𝐽 = 0
would produce no magnetic moment.
With seven electrons in the shell, 𝐿 = 0 and 𝑆 = 7/2 such that 𝑔𝐽 = 𝑔𝑆 = 2 since
there is no contribution from the orbital angular momentum. Although 𝑔𝐽 decreases
as more than seven electrons occupy the shell, the magnetic moment increases to a
maximum as a result of increasing 𝐽, thus making terbium (9e− ) and dysprosium(10e− ) the most magnetic elements. In the case of a full 𝑓 -shell both 𝐿 = 0 and
𝑆 = 0, again producing no magnetic moment in the atom. Similar to the six electron
case, 𝑔𝐽 is thus chosen to be zero.
For pedagogical purposes, the next section will apply Hund’s rules to all of thelanthanide elements in an explicitly clear fashion. Recall that the Landé 𝑔-factor forthe total angular momentum is given by𝑔𝐽 ≈3 𝑆(𝑆 + 1) − 𝐿(𝐿 + 1)+
2
2𝐽(𝐽 + 1)

and its corresponding magnetic moment by 𝜇𝐽 = 𝑔𝐽 𝐽.

62

3.5

Electronic Configurations in the f -orbital

1e−Figure 3-5: One-electron ground state in the f-orbital

𝐿=3
𝑆 = 1/2
𝐽 = 3 − 1/2 = 5/2

Term Symbol: 2 F5/2
𝑔𝐽 = 6/7
𝜇 = ( 67 )( 52 ) = 15/7
2e−Figure 3-6: Two-electron ground state in the f-orbital

𝐿=3+2=5
𝑆=1
𝐽 =5−1=4

Term Symbol: 3 H4
𝑔𝐽 = 4/5
𝜇 = ( 45 )(4) = 16/5
63

3e−

Figure 3-7: Three-electron ground state in the f-orbital

𝐿=3+2+1=6
𝑆 = 3/2
𝐽 = 6 − 3/2 = 9/2

Term Symbol: 4 I9/2
𝑔𝐽 = 8/11
8
𝜇 = ( 11)( 92 ) = 36/11

4e−

Figure 3-8: Four-electron ground state in the f-orbital

𝐿=3+2+1+0=6
𝑆=2
𝐽 =6−2=4

Term Symbol: 5 I4
𝑔𝐽 = 3/5
𝜇 = ( 35 )(4) = 12/5

64

5e−

Figure 3-9: Five-electron ground state in the f-orbital

𝐿=3+2+1+0−1=5
𝑆 = 5/2
𝐽 = 5 − 5/2 = 5/2

Term Symbol: 6 H5/2
𝑔𝐽 = 2/7
𝜇 = ( 27 )( 52 ) = 5/7
6e−Figure 3-10: Six-electron ground state in the f-orbital

𝐿=3+2+1+0−1−2=3
𝑆=3
𝐽 =3−3=0

Term Symbol: 7 F0
𝑔𝐽 = 0
𝜇=0

65

7e−

Figure 3-11: Seven-electron ground state in the f-orbital

𝐿=3+2+1+0−1−2−3=0
𝑆 = 7/2
𝐽 = |0 − 7/2| = 7/2
Term Symbol: 8 S7/2
𝑔𝐽 = 2𝜇 = (2)( 72 ) = 7
8e−

Figure 3-12: Eight-electron ground state in the f-orbital

𝐿=3
𝑆=3
𝐽 =3+3=6
Term Symbol: 7 F6
𝑔𝐽 = 3/2
𝜇 = ( 32 )(6) = 9

66

9e−

Figure 3-13: Nine-electron ground state in the f-orbital

𝐿=3+2=5
𝑆 = 5/2
𝐽 = 5 + 5/2 = 15/2
Term Symbol: 6 H15/2
𝑔𝐽 = 4/3
𝜇 = ( 43 )( 15
2 ) = 1010e−Figure 3-14: Ten-electron ground state in the f-orbital

𝐿=3+2+1=6
𝑆=2
𝐽 =6+2=8
Term Symbol: 5 I8
𝑔𝐽 = 5/4
𝜇 = ( 54 )(8) = 10

67

11e−

Figure 3-15: Eleven-electron ground state in the f-orbital

𝐿=3+2+1+0=6
𝑆 = 3/2
𝐽 = 6 + 3/2 = 15/2
Term Symbol: 4 I15/2
𝑔𝐽 = 6/5
𝜇 = ( 65 )( 152) = 912e−

Figure 3-16: Twelve-electron ground state in the f-orbital

𝐿=3+2+1+0−1=5
𝑆=1
𝐽 =5+1=6
Term Symbol: 3 H6
𝑔𝐽 = 7/6
𝜇 = ( 76 )(6) = 7

68

13e−

Figure 3-17: Thirteen-electron ground state in the f-orbital

𝐿=3+2+1+0−1−2=3
𝑆 = 1/2
𝐽 = 3 + 1/2 = 7/2
Term Symbol: 2 F7/2
𝑔𝐽 = 8/7
𝜇 = ( 87 )( 72 ) = 4
14e−

Figure 3-18: Fourteen-electron ground state in the f-orbital

𝐿=3+2+1+0−1−2−3=0
𝑆=0
𝐽 =0Term Symbol: 1 S0
𝑔𝐽 = 0
𝜇=0
69

3.6

Magnetic Dipole Moment of Dysprosium

A dysprosium atom contains ten f-shell electrons that produce its strong magnetic
moment. We have determined that its ground state configuration has 𝐿 = 6, 𝑆 = 2,
and 𝐽 = 8. Since 𝐽 = 𝐿 + 𝑆, dysprosium’s total angular momentum is in a stretchedstate and the individual contributions to the magnetic moment from the spin andthe orbital angular momentum can be directly summed to determine its magnetic
moment 𝜇𝐷𝑦 = 𝜇𝑆 + 𝜇𝐿 = 10𝜇𝐵 .
In general, for all electronic shells with greater than half-filling, Hund’s third rulemaximizes 𝐽 such that the total angular momentum is in a stretched state and similar
direct summations of the individual spin and angular momentum contributions canbe performed.

70

Chapter 4
Dipolar Suppression of Dipolar
Relaxation
Just as a cavity can modify the rate of spontaneous emission for an atom, an opticallattice can modify the rate of inelastic collisions between two atoms. Here we describe
the suppression of dipolar relaxation using dysprosium atoms confined in ultra-thinlayers, with their magnetic moments aligned perpendicularly to the layers such thatthe dipolar interaction is purely repulsive. In such a configuration we observed anorder of magnitude extension of the lifetime, and expect another factor of ten is withinreach based on the models which we have validated with our experimental study. The
loss suppression opens up many new possibilities for quantum simulations with spin
mixtures of highly magnetic atoms.
This chapter will begin with the dipolar relaxation process and why it is beneficial
to suppress it. Next, we will apply Fermi’s golden rule to calculate the modificationof dipolar relaxation in different regimes of optical confinement. Theoretical andexperimental results will be shown, followed by an overview of the experiment itself.
A detailed calculation for converting atom loss into a dipolar relaxation rate will
be performed, and the chapter will conclude with a description of a magnetic fieldzeroing technique with prophetic implications for the realization of a subwavelengthbilayer (see chapter 5).
71

4.1

Background

Cold atoms with a spin degree of freedom provide a promising platform for the studyof quantum materials, having enabled studies ranging from spin-orbit coupled Bose–
Einstein condensates [15] to Heisenberg spin models [16, 17]. State-dependent potentials can be used in lattices to create spin-dependent tunneling and interaction terms[18], or in the bulk to engineer artificial gauge potentials which can be used to study
non-trivial phases of matter, such as vortex lattices [19] and the fractional quantum
Hall effect [20].
A parallel direction of research with molecules [21], Rydberg atoms [22], and atoms
with a strong magnetic moment [3] has added dipolar interactions to the toolkit ofquantum simulation. Systems with strong dipolar interactions may be used to engineer phases of quantum magnets by implementing spin interactions directly throughthe long-range interaction [23] rather than second-order tunneling [24]. Many lanthanides (e.g., erbium and dysprosium) also possess orbital angular momentum intheir ground state, making them attractive candidates for engineering artificial gauge
potentials without the large scattering rates that alkali species suffer from. Additionally, dysprosium’s large contrast of Clebsch-Gordan coefficients allows for highly
spin-dependent atom-light couplings at large detunings.
Experiments with ultracold atoms or molecules, however, are often limited byunfavorable inelastic collision rates. Several methods have been developed to control inelastic collisions, such as isolating atoms in deep optical lattices [25], reducing
collisional channels via confinement [26], or by mitigating their effects through theenhancement of elastic collisions via Feshbach resonances [27]. Polar molecules, inparticular, have been shielded from chemical reactions at short range by using repulsive interactions between electric dipoles, either in two dimensions or via microwave
dressing [28, 29].
Here we explore how dipolar shielding can be realized in dysprosium, a highly
magnetic atom for which the dipolar interaction is two orders of magnitude smaller
than for polar molecules. Compared to the rovibrational and chemical complexity72

of molecules, magnetic atoms can achieve lower temperatures while providing a controlled, tunable, and relatively simple framework for exploring novel forms of matterwith long-range forces [20, 30, 31, 32, 33]. Dysprosium, with a magnetic moment of
10𝜇𝐵 , has a magnetic dipole-dipole interaction that is 100 times larger than that ofalkali atoms.
In the presence of an external magnetic field, however, the dipolar interaction inlanthanides can cause an undesirable coupling between the spin and orbital motion of
two atoms. As a result, the sum of the projections of the angular momentum operators of the colliding atoms, 𝑚𝑗 + 𝑚𝑗 ′ , is not conserved. Correspondingly, one or both
spins can be flipped, with spin angular momentum converted into orbital angularmomentum and conserving only the total angular momentum. Known as dipolar relaxation, this mechanism converts Zeeman energy into motional energy incoherently,
resulting in heating or an escape from the trap [34]. If the participating atoms arein the lowest energy spin state, such as 𝑚𝐽 = −8 for the case of bosonic dyspro-sium, then the relaxation process does not occur. Due to the unfavorable scaling ofdipolar relaxation with magnetic moment (∼ 𝜇4 ), highly magnetic atoms populated
in excited Zeeman levels suffer from significantly shorter lifetimes compared to theiralkali counterparts. The dominance of dipolar relaxation has, thus far, precluded theexperimental realization of many proposed new phenomena in spin mixtures of highly
magnetic atoms [35, 36].
In order to gain a quantitative understanding of the dipolar relaxation rate Γ, andpotential suppression mechanisms, we can apply Fermi’s golden rule:~Γ = 2𝜋 |⟨Ψout |𝑉𝑑𝑑 |Ψin ⟩|2 𝜌(𝐸).

(4.1)

Here, Ψin,out are the incoming and outgoing collisional wave functions, 𝑉𝑑𝑑 is the
dipole-dipole operator, and 𝜌(𝐸) is the density of final states at the outgoing kinetic
energy of the pair of atoms.
Previous work with chromium [26] modified the density of final states by reducing
the magnetic field and by confining the atoms in a one-dimensional lattice, such that73

the released kinetic energy remains below the first excited state of the trap. Although
the confinement succeeded in reducing the dipolar relaxation rate by a factor of three
for low magnetic fields, this method constrains the accessible fields, since the density ofstates rapidly approaches the three-dimensional limit when new vibrational channels
open. Due to the weak dipole-dipole interaction of chromium, approximately threetimes smaller than that of dysprosium, the incoming wave function is essentially freeof dipolar repulsion in the range of trap frequencies used.
Another experiment with dysprosium [37] utilized the anti-symmetric nature offermionic atoms in a stretched state, such that the incoming wave function is given⃗⃗𝑘⟩√ 𝑖 . At high magnetic fields, and therefore high outgoing momentum, theby |𝑘𝑖 ⟩−|−
2

incoming momentum has only a small effect on the outgoing wave functions. Thus
the two opposite and nearly equal matrix elements cancel out, leading to a much
reduced relaxation rate.
In this chapter we implement a new dipolar suppression scheme for a wide rangeof magnetic fields. By applying tight 1D optical lattice confinement, we reduce thedensity of final states and directly suppress the incoming wave function amplitude atshort-range via the strongly repulsive dipole-dipole interaction [28, 38, 39]. Previous
suppression schemes have been unable to eliminate losses to the level required formany experiments, making the dipolar shielding presented in this chapter all themore valuable due to its ability to be combined with other suppression mechanisms.

4.2

3D Dipolar Collisions

To understand collisions between two particles it is necessary to account for their kinematics and their interactions. The relevant kinematics in dysprosium are the kinetic
energy and centrifugal potential, while the relevant interactions are the magnetic
dipole-dipole interaction and the van der Waals interaction. The magnetic dipoledipole interaction rapidly dominates the induced electric dipole interaction (van der74

Waals) for 𝑟 > 𝑅𝑣𝑑𝑊 , where 𝑅𝑣𝑑𝑊 is the van der Waals length and is determined by
63
𝐶6 /𝑅𝑣𝑑𝑊= 𝐷/𝑅𝑣𝑑𝑊.

In the case of dysprosium, 𝐶6 = 2273 (atomic units) [40] and 𝑅𝑣𝑑𝑊 = 2.7 nm. Thus,
for 𝑟 > 𝑅𝑣𝑑𝑊 , the dipole-dipole interaction between two dysprosium atoms is givenby:2

𝑉𝑑𝑑 (𝜌, 𝑧) = 𝐷

1 − 3 𝑧2𝑧+𝜌2(𝑧 2 + 𝜌2 )3/2

,

(4.2)

𝜇0
𝜇0
𝜇0
with 𝐷 = 4𝜋(𝜇𝐵 𝑔𝐽 𝐽)2 = 4𝜋
(𝜇Dy )2 = 4𝜋
(10𝜇𝐵 )2 being the standard pre-factor for the

dipole-dipole interaction. For dysprosium, 𝑔𝐽 = 5/4 and 𝐽 = 8.

Compared to typical 𝑠-wave collisions at ultracold temperatures, dipolar collisions
are often universal [41]. For bosonic ultracold collisions dominated by contact interactions, the scattering length 𝑎 fully characterizes 𝑠-wave collisions and must normally
be determined through experimental measurement due to the difficulty in calculating accurate interatomic potentials. In contrast, dipolar collisions are characterized
as universal, since their collisional cross section can be determined directly by the
dipole-dipole interaction and its resulting dipolar length, 𝑅𝐷 , thus providing exact
collisional wave functions. Just as the van der Waals length is found by determining the regime where, for 𝑟 < 𝑅𝑣𝑑𝑊 , the van der Waals interaction dominates overthe magnetic dipole-dipole interaction, the dipolar length is found by determining theregime where, for 𝑟 < 𝑅𝐷 , the dipole-dipole interaction dominates over the centrifugal
potential:

𝐷
~2
=
3
2
𝑅𝐷
𝜇𝑅𝐷

where 𝜇 = 𝑚/2 is the reduced mass. For Dy this yields:
𝑅𝐷 =𝜇𝐷= 10.43 nm~275

with a corresponding dipolar energy:
𝐸𝐷 =~2= 𝑘𝐵 (27.42 𝜇K).
2
2𝜇𝑅𝐷

Another commonly used definition of the dipolar length, 𝑎𝑑𝑑 = 23 𝑅𝐷 , is often usedin the context of many-body physics with dipolar quantum gases [3]. However, 𝑅𝐷 is
the natural length scale in the context of two-body collisions, which will be the focusof this chapter.
There are, however, a few instances under which the universality of dipolar collisions may be ruined. Let 𝑅0 denote a hard wall potential, typically on the order ofthe van der Waals length 𝑅𝑣𝑑𝑊 , behind which the short-range physics occurs. Firstly,
if 𝑅0 lies beyond the dipolar length 𝑅𝐷 , then at low energies where it is found the2cross section scales as 𝜎 ∼ 𝑅𝐷
, we would instead find 𝜎 ∼ 𝑅02 . Secondly, if the𝑠-wave scattering length 𝑎 would be larger than 𝑅𝐷 , then the universality would be

modified by 𝑎 at low energy. Lastly, at high energy the universal cross section behaves as 𝜎 ∼ 𝑅𝐷 /𝑘, where 𝑘 is the relative wavenumber of the colliding atoms. As
energy is increased, the universal cross section will diminish until the short-range

cross section 𝑅0 dominates: 𝑅02 > 𝑅𝐷 /𝑘. The cross-over between the low energy
cross section and high-energy scaling roughly occurs at the natural energy scale ofthe dipolar interaction, 𝐸𝐷 . Above this energy the scattering becomes semi-classicaland many partial-waves contribute, while below this threshold the effects of non-zeropartial waves is merely perturbative. The atoms in our experiment have temperatures of several 𝜇K, well below the dipolar interaction energy, thus the effects ofhigher partial-waves are negligible.
With a firm understanding of the unique properties of elastic dipolar collisions, weare now ready to explore inelastic dipolar collisions in the form of dipolar relaxation.
To do so, we can write the dipole-dipole interaction in terms of the angular momentum
operators (F) of the two interacting particles:
𝑉𝑑𝑑 =

𝜇0
(F1 · F2 ) − 3(F1 · ^r)(F2 · ^r)
(𝑔𝐹 𝜇𝐵 )2
3
4𝜋
𝑟
76

where r = r1 − r2 is the interatomic separation.
Narrowing our focus to the angular momentum operators, we can define:𝑉𝑑𝑑, F = (F1 · F2 ) − 3(F1 · ^r)(F2 · ^
r)

(4.3)
(4.4)

= 𝐹1𝑧 𝐹2𝑧
1
(𝐹1+ 𝐹2− + 𝐹1− 𝐹2+ )
2
3
− (2¯
𝑧 𝐹1𝑧 + 𝑟¯− 𝐹1+ + 𝑟¯+ 𝐹1− )(2¯𝑧 𝐹2𝑧 + 𝑟¯− 𝐹2+ + 𝑟¯+ 𝐹2− )
4
+

(4.5)
(4.6)

with
𝑧
𝑥 ± 𝑖𝑦𝑧¯ = , 𝑟¯± =, 𝐹± = 𝐹𝑥 ± 𝑖𝐹𝑦 .
𝑟𝑟
Equation 4.4, together with the term −3¯
𝑧 2 𝐹1𝑧 𝐹2𝑧 from Equation 4.6, show thediagonal terms of the dipole-dipole interaction that are responsible for elastic scattering: 𝐹1𝑧 𝐹2𝑧 (1 − 3¯𝑧 2 ). Additionally, off-diagonal terms allow colliding spins to swap
one unit of spin in a process known as spin exchange, as shown in Equation 4.5.
Other off-diagonal terms such as 𝑟¯− 𝐹1+ 𝑟¯− 𝐹2+ (such as seen in demagnetization cooling, as in section 1.1), 2¯𝑧 𝐹1𝑧 𝑟¯+ 𝐹2− (single spin flip), and 𝑟¯+ 𝐹1− 𝑟¯+ 𝐹2− (double spinflip) in Equation 4.6 are all forms of dipolar relaxation. For each dipolar relaxation
process there is a corresponding ladder operator, 𝑟¯± , which modifies the relative orbital angular momentum of the collisional pair such that total angular momentum isconserved.
In the case of two bosonic dysprosium atoms (𝐼 = 0, 𝐹 = 𝐽 = 8) prepared in thespin state:|𝜑in ⟩ = |𝐽 = 8, 𝑚𝐽 = +8; 𝐽 = 8, 𝑚𝐽 = +8⟩ ≡ | + 8; +8⟩,
the allowed symmetrized output spin states for dipolar relaxation are:1
|𝜑out, 1 ⟩ = √ [ | + 7; +8⟩ + | + 8; +7⟩ ] ,2|𝜑out, 2 ⟩ = | + 7; + 7⟩,
77

representing single spin flip and double spin flip processes, respectively.
In three dimensions it has been shown, through the application of Fermi’s golden
rule [26], that dipolar relaxation can be interpreted as occurring at a point:
𝑅𝐷𝑅 =

16
(𝑗)

3𝜋𝑘𝑓

(4.7)

(𝑗)

where 𝑘𝑓 is the outgoing wavevector of the collisional pair and the superscript represents a single (𝑗 = 1) or double (𝑗 = 2) spin flip event.
Conservation of energy dictates:(𝑗)~2 (𝑘𝑓 )2~2 𝑘𝑖2=+ Δ𝐸 (𝑗)2𝜇2𝜇and assuming 𝑘𝑖 → 0, we find that
(𝑗)
𝑘𝑓 =

√︀
2𝜇Δ𝐸 (𝑗)~where 𝑘𝑖 is the incoming wavevector of the collisional pair andΔ𝐸 (1) = 𝑔𝐽 𝜇𝐵 𝐵
Δ𝐸 (2) = 2𝑔𝐽 𝜇𝐵 𝐵
are the Zeeman energies released in single and double spin flip processes, respectively.
As a result, the dipolar relaxation rate is proportional to the probability of thetwo colliding particles being at a distance of 𝑅𝐷𝑅 from one another. Within a few
factors on the order of unity, this distance is the same as the distance 𝑅cross where
the input and output molecular channels cross:√︂𝑅cross = ~√𝑙(𝑙 + 1)∝1/
𝐵.
𝑚Δ𝐸 (𝑗)In a semi-classical picture, the Franck-Condon principle predicts spin flips to occur78

at the classical turning point of the outgoing wave function [26], i.e. when the released
Zeeman energy equals the energy of the centrifugal barrier. Thus, as the external
magnetic field increases, the interatomic separation at which dipolar relaxation occurs
decreases.
At magnetic fields of 10 mG, 100 mG, and 300 mG and assuming 𝑘𝑖 = 0, therespective 3D dipolar relaxation points found from Equation 4.7 are equal to 72 nm,
23 nm, and 13 nm for double spin flip processes. For single spin flips processes,√
each distance should be multiplied by 2 to account for the difference in released
Zeeman energy. As will be shown in the following sections, the dipolar barrier weform via confinement can have a spatial extent of several 𝑅𝐷 , translating to tensof nanometers in the case of dysprosium. Thus, the dipolar shielding can strongly
modify the probability of two particles approaching the relaxation point.

4.3

Dipolar Collisions with Harmonic Confinement

Due to the anisotropy of the dipole-dipole interaction, the confinement of dipolar
quantum gases can lead to fascinating changes in the collisional properties. Since
dipolar relaxation can be characterized as occurring at a single location in 3D, we
can attempt to prevent the colliding atoms from approaching the dipolar relaxation
point by using their dipole-dipole interaction to repel one another. Such intuition willneed modification in lower dimensions, but is useful nevertheless, as will be shown.
Applying a 1D optical lattice along 𝑧^ constrains the dipoles to a plane of relatively
free motion (^𝑥 − 𝑦^). The dipole-dipole interaction is attractive in the case of a tip-

to-tail orientation and repulsive for the side-by-side orientation. For dipoles aligned
perpendicular to the plane of weak confinement, the addition of axial (^𝑧 ) harmonic
confinement leads to a total interaction energy given by [42]:
[︂ 3
]︂
3
𝑧 2 𝑙⊥𝑧2
1 2 2 𝐷 𝑙⊥
𝑉 (⃗𝑟) = 𝑉𝑑𝑑 (𝜌, 𝑧) + 𝜇𝜔𝑧 𝑧 = 3−3 5 + 2
2
𝑙⊥ 𝑟3
𝑟
2𝑙⊥
79

where 𝑙⊥ = (𝐷𝑚/8~2 𝑎𝑧 )1/5 𝑎𝑧 and the two-body axial harmonic length is given by√︁
𝑎𝑧 = 𝜇𝜔~ 𝑧 . As expected, a dipolar barrier is found by constraining the interacting

dipoles to a largely side-by-side orientation. The saddle point of the potential is thenfound to occur at
𝑅𝑠 ≈ 1.28𝑙⊥

𝑧𝑠 ≈ ±0.64𝑙⊥

with a potential barrier height𝐷
𝑉 (𝑅𝑠 , 𝑧𝑠 ) ≈ 0.34 3 .
𝑙⊥Assuming the collisional energy is less than the the potential barrier height, a semiclassical approximation estimates an exponentially suppressed tunneling probability
𝑃tunneling ≈ 𝑒−5.86(𝑅𝐷 /𝑎𝑧 ) [42]. To maximally inhibit the approach of two colliding
2

atoms, it is therefore beneficial to maximize both the dipolar interaction and theaxial trap frequency.

4.4

Pure 2D and Quasi-2D Dipolar Collisions

We will now compute the dipolar relaxation rate of an atomic gas in a tightly confining
1D lattice. There are two scenarios that will be considered in the calculation: thepure 2D regime (𝑎𝑧 = 0) and the quasi-2D regime (𝑅𝐷 > 0.34 𝑎𝑧 > 0) [38]. In our
experiment we observe the pure 2D regime to be out of reach with our current latticeconfinement, while a quasi-2D calculation yields results that match our experimental
findings. The calculations will be carried out using Fermi’s golden rule, which gives
the collision rate for a pair of particles interacting through a potential 𝑉𝑗 to be:
2𝜋
Γ𝑗 =
~

[︂∫︁

Ψin 𝑉𝑗 Ψout 𝑑𝑉

]︂2

𝜌1𝐷

where the 1D density of final states for the two-body collision process is 𝜌1𝐷 = 𝜋~𝜇𝐿
2𝑘 ,𝑓
𝐿 is the size of the system (which will be normalized out later in the calculation), and𝑘𝑓 is the outgoing wavevector for the pair of particles. The incoming and outgoing80

collisional wave functions are given by Ψin,out .

In the case of a collision between two particles in the highest energy Zeeman level(𝑚𝐽 = +𝐽), Equation 4.6 shows that dipolar relaxation can occur through either asingle or a double spin flip collision, with corresponding interaction potentials:
3 2 (𝑥 + 𝑖𝑦)2𝑉2 = 𝐽𝑑2
𝑟5(𝑥 + 𝑖𝑦)𝑧
𝑉1 = 3𝐽 3/2 𝑑2
𝑟5

(4.8)

𝜇0
where 𝑑2 = 4𝜋
(𝑔𝐽 𝜇𝐵 )2 , which differs from the definition of 𝐷 in order to factorize out

the role of the total orbital angular momentum (𝐽).

Now we will look at the matrix element involving transitions between the harmonic
oscillator eigenstates, assuming the incoming pair of particles is in the ground state𝑣𝑖 = 0. Let’s find which vibrational levels have non-zero matrix elements with thisstate.

For single spin flip transitions between two atoms occupying the 𝑚𝐽 = +𝐽 state,
𝑉1 ∝

𝑧
𝑧
= 2
,
5
𝑟
(𝑧 + 𝜌2 )5/2

an odd function of 𝑧. Here, 𝜌 is the radial distance in the cylindrical coordinate
basis. Meanwhile, for double spin flip transitions between two atoms occupying the𝑚𝐽 = +𝐽 state,
𝑉2 ∝

1
1
= 2
,
5
𝑟
(𝑧 + 𝜌2 )5/2

an even function of 𝑧. Correspondingly, single spin flips only occur between harmonic
oscillator states of the opposite parity, while double spin flips only occur between
states of the same parity.
814.4.1

Double Spin Flip in the Pure 2D System

In the pure 2D case of two particles interacting through only the dipole-dipole interaction, the exact incoming radial wave function at 𝑘𝑖 = 0 is given by [43]:𝐹in, dipolar (𝜌) = 𝛼in 𝐾0

(︃√︃

8𝑅𝐷
𝜌

)︃

where 𝛼in is the amplitude, 𝐾0 is the modified Bessel function of the second kind,and 𝑅𝐷 is the dipolar length.
We wish to patch this wave function together with a free, finite-𝑘 incoming wave
function, so that we may simultaneously capture the modification to the short-range
wave function in the strongly dipolar regime while using the finite-𝑘 free solution at
long-range. Thus we have:𝐹in (𝜌) =⎧⎪⎨𝐹in, dipolar (𝜌) 𝜌 ≤ 𝜌in⎪⎩𝐹in, free (𝜌)𝜌 > 𝜌inAll calculations are done in a disk of radius 𝐿, though the problem will eliminate
all 𝐿-dependence in the final calculation of the dipolar relaxation rate. After proper∫︀ 𝐿
normalization of the radial wave function ( 0 |𝐹in (𝜌)|2 𝜌 𝑑𝜌 = 1) the free solution is
given by [44]:𝐹in, free (𝜌) =√︂𝜋𝑘𝑖[𝑎𝐽0 (𝑘𝑖 𝜌) + 𝑏𝑌0 (𝑘𝑖 𝜌)] = 𝑎
𝐿

√︂

𝜋𝑘𝑖
[𝐽0 (𝑘𝑖 𝜌) − tan(𝛿in )𝑌0 (𝑘𝑖 𝜌)]𝐿where 𝑎2 + 𝑏2 = 1 due to normalization and tan 𝛿in ≡ −𝑏/𝑎. The derivatives of Bessel
functions of the first and second kind are given by:𝑑𝑘
𝐽𝑚 (𝑘𝜌) = [𝐽𝑚−1 (𝑘𝜌) − 𝐽𝑚+1 (𝑘𝜌)]
𝑑𝜌
2

𝑑
𝑘
𝑌𝑚 (𝑘𝜌) = [𝑌𝑚−1 (𝑘𝜌) − 𝑌𝑚+1 (𝑘𝜌)] .
𝑑𝜌282Then we can find:𝑑𝐹in, free (𝜌) = 𝑎
𝑑𝜌

√︂

(︁
)︁]︁
𝜋𝑘𝑖 𝑘𝑖 [︁𝐽−1 (𝑘𝑖 𝜌) − 𝐽1 (𝑘𝑖 𝜌) − tan(𝛿in ) 𝑌−1 (𝑘𝑖 𝜌) − 𝑌1 (𝑘𝑖 𝜌) .
𝐿 2

To perform the patching of wave functions, we need to set the dipolar solution
amplitude (𝛼in ) and the free solution phase (𝛿in ) so that the dipolar and free solutions
have the same amplitude and derivative at a point. We start by taking the derivativeof the dipolar wave function and find:𝑑𝐹in, dipolar (𝜌) = 𝛼in
𝑑𝜌

√︃

2𝑅𝐷
𝐾1
𝜌3

(︃√︃

8𝑅𝐷
𝜌

)︃

.

Let’s perform the wave function patching at a point 𝜌in :𝑑𝐹(𝜌 )𝑑𝜌 in, free in

𝐹in, free (𝜌in )

=
=

𝑘
2

[︁

(︁
)︁]︁
𝐽−1 (𝑘𝑖 𝜌in ) − 𝐽1 (𝑘𝑖 𝜌in ) − tan(𝛿in ) 𝑌−1 (𝑘𝑖 𝜌in ) − 𝑌1 (𝑘𝑖 𝜌in )

𝑑
𝐹
(𝜌 )
𝑑𝜌 in, dipolar in

𝐽0 − tan(𝛿in )𝑌0 (𝑘𝑖 𝜌in )𝐹in, dipolar (𝜌in )≡ 𝑅in .
Then we find that
2𝑅in
𝐽0 (𝑘𝑖 𝜌in ) − 𝐽−1 (𝑘𝑖 𝜌in ) + 𝐽1 (𝑘𝑖 𝜌in )𝑏tan(𝛿in ) = − = 2𝑅𝑘𝑖in𝑎
𝑌0 (𝑘𝑖 𝜌in ) − 𝑌−1 (𝑘𝑖 𝜌in ) + 𝑌1 (𝑘𝑖 𝜌in )
𝑘
𝑖

and

𝛼in =

where 𝑎 =

√︁

𝑎

√︁

𝜋𝑘𝑖
[𝐽0 (𝑘𝑖 𝜌in ) − tan(𝛿in )𝑌0 (𝑘𝑖 𝜌in )]
𝐿

𝐾0

(︁√︁

8𝑅𝐷
𝜌in

)︁

1
.
1+tan2 (𝛿in )Let us now determine the matching point where the dipolar and free solutions arejoined. The free solution becomes viable when the kinetic energy becomes comparable83

to the dipolar energy:
𝑘2 ≈

2𝑅𝐷
.
𝜌3

Thus the matching points of the dipolar wave functions to the free wave functions
for the incoming and outgoing solutions are given by:

𝜌in =

(︃

2𝑅𝐷
𝑘𝑖2

)︃1/3

𝜌out =

(︃

2𝑅𝐷
𝑘𝑓2

)︃1/3

.

We can now apply a similar strategy to the outgoing wave function. The exact
outgoing radial wave function of two dipolar particles at 𝑘𝑓 = 0 is given by:

𝐹out, dipolar (𝜌) = 𝛼out 𝐾4

(︃√︃

8𝑅𝐷
𝜌

)︃

where 𝛼out is the amplitude. After proper normalization of the radial wave function∫︀ 𝐿( 0 |𝐹out (𝜌)|2 𝜌 𝑑𝜌 = 1) the outgoing free solution is given by:
𝐹out, free (𝜌) =√︂𝜋𝑘𝑓[𝑐𝐽2 (𝑘𝑓 𝜌) + 𝑑𝑌2 (𝑘𝑓 𝜌)] = 𝑐
𝐿

√︂

𝜋𝑘𝑓
[𝐽2 (𝑘𝑓 𝜌) − tan(𝛿out )𝑌2 (𝑘𝑓 𝜌)]𝐿where 𝑐2 + 𝑑2 = 1 due to normalization and tan 𝛿out ≡ −𝑑/𝑐. Then we find:𝑑𝐹out, free (𝜌) = 𝑐𝑑𝜌

where 𝑘𝑓 =

√︂

(︁
)︁]︁
𝜋𝑘𝑓 𝑘𝑓 [︁𝐽1 (𝑘𝑓 𝜌) − 𝐽3 (𝑘𝑓 𝜌) − tan(𝛿in ) 𝑌1 (𝑘𝑓 𝜌) − 𝑌3 (𝑘𝑓 𝜌)
𝐿 2√︀
𝑘𝑖2 + 2𝑔𝐽 𝜇0 𝐵𝑚/~2 . We once again take derivative of the outgoing

dipolar radial wave function and find:𝑑𝐹out, dipolar (𝜌) = 𝛼out𝑑𝜌√︃[︃ (︃√︃
)︃
(︃√︃
)︃]︃
𝑅𝐷
8𝑅𝐷
8𝑅𝐷
𝐾3
+ 𝐾5
.
2𝜌3
𝜌
𝜌

Thus we patch together the dipolar and free solutions by matching the outgoing
dipolar radial wave function to the outgoing free solution via 𝛼out and 𝛿out . Let’s
84perform the wave function patching at a point 𝜌out :𝑑𝐹(𝜌 )𝑑𝜌 out, free out

𝐹out, free (𝜌out )

=
=

𝑘𝑓
2

[︁
(︁
)︁]︁
𝐽1 (𝑘𝑓 𝜌out ) − 𝐽3 (𝑘𝑓 𝜌out ) − tan(𝛿out ) 𝑌1 (𝑘𝑓 𝜌out ) − 𝑌3 (𝑘𝑓 𝜌out )𝐽2 (𝑘𝑓 𝜌out ) − tan(𝛿out )𝑌2 (𝑘𝑓 𝜌out )𝑑𝐹(𝜌 )𝑑𝜌 out, dipolar out
𝐹out, dipolar (𝜌out )≡ 𝑅out
Then we find that:
2𝑅out
𝐽2 (𝑘𝑓 𝜌out ) − 𝐽1 (𝑘𝑓 𝜌out ) + 𝐽3 (𝑘𝑓 𝜌out )𝑑𝑘𝑓tan(𝛿out ) = − = 2𝑅out𝑐
𝑌2 (𝑘𝑓 𝜌out ) − 𝑌1 (𝑘𝑓 𝜌out ) + 𝑌3 (𝑘𝑓 𝜌out )
𝑘
𝑓

and

𝛼out =

where 𝑐 =

√︁

𝑐

√︁

𝜋𝑘𝑓
[𝐽2 (𝑘𝑓 𝜌out ) − tan(𝛿out )𝑌2 (𝑘𝑓 𝜌out )]
𝐿

𝐾4

(︁√︁

8𝑅𝐷
𝜌out

)︁

1
.
1+tan2 (𝛿out )Thus, the outgoing radial wave function is given by:

𝐹out (𝜌) =⎧⎪⎨𝐹out, dipolar (𝜌) 𝜌 ≤ 𝜌out⎪
⎩𝐹out, free (𝜌)𝜌 > 𝜌out

The last ingredients are the double spin flip dipolar potential term and the collisional
wavefunctions, which give3
1 (𝑥 + 𝑖𝑦)2
3 2 1 2𝑖𝜑𝑉2 (𝜌, 𝜑) = 𝐽𝑑2 3=
𝐽𝑑 3 𝑒2𝜌𝜌22𝜌1Ψin (𝜌, 𝜑) = Φin (𝜑)𝐹in (𝜌) = √ 𝐹in (𝜌)
2𝜋
𝑒2𝑖𝜑
√
Ψout (𝜌, 𝜑) = Φout (𝜑)𝐹out (𝜌) =𝐹out (𝜌).
2𝜋
We can perform the angular integral to remove the 𝜑 dependence from the prob85

lem:

∫︁ 2𝜋

Φ*out (𝜑) 𝑒2𝑖𝜑 Φin (𝜑) 𝑑𝜑 =0such that⟨𝑉2 ⟩𝜑 ≡

∫︁ 2𝜋
0

1
2𝜋

∫︁ 2𝜋

𝑒−2𝑖𝜑 𝑒2𝑖𝜑 𝑑𝜑 = 1

03
1
Φout (𝜑) 𝑉2 (𝜌, 𝜑) Φin (𝜑) 𝑑𝜑 = 𝐽𝑑2 32
𝜌Fermi’s golden rule then tells us the rate of a dipolar relaxation event occurring,assuming cylindrical waves, to be:2𝜋
Γ2𝐷,𝑐 =~[︂∫︁Ψin (𝜌, 𝜑)𝑉2 (𝜌, 𝜑)Ψout (𝜌, 𝜑) 𝑑𝜑 𝜌 𝑑𝜌]︂2[︂∫︁ ∞
2𝜋
𝐹in (𝜌)⟨𝑉2 ⟩𝜑 𝐹out (𝜌)𝜌 𝑑𝜌 𝜌1𝐷
=
~
0

]︂2

𝜌1𝐷

Symmetrization and conversion of cylindrical waves to plane waves results in:4
Γ2𝐷, 𝑐𝜋𝑘𝑖 𝐿]︂2[︂∫︁ ∞
8𝜇=𝐹in (𝜌)⟨𝑉2 ⟩𝜑 𝐹out (𝜌) 𝜌 𝑑𝜌
𝜋~3 𝑘𝑖 𝑘𝑓 0
]︂2[︂∫︁ ∞
18𝜇
1
2 4
=
𝐽 𝑑
𝐹in (𝜌)𝐹out (𝜌) 2 𝑑𝜌 .
𝜋~3 𝑘𝑖 𝑘𝑓
𝜌
0

Γ2𝐷 =

Then we find:
∫︁

𝑑𝑛
𝑑𝑁
𝑑𝐴 =
= −Γ2𝐷 𝑁 2 = −𝛽2𝐷
𝑑𝑡
𝑑𝑡

∫︁

𝑛2 𝑑𝐴 = −𝛽2𝐷 𝑁 2.
𝐴

and therefore:
𝛽2𝐷 = 𝐴Γ2𝐷 = 𝜋𝐿2 Γ2𝐷 .
The above provides us the pure 2D dipolar relaxation rate parameter. To relate thisto our experiment, we must consider the third dimension (^𝑧 ) of the trap, which resultsfrom the lattice confinement. We assume that the atoms are in the vibrational groundstate,
𝜒0 (𝑧) =

(︂

1
𝜋𝑎2𝐻𝑂

)︂1/4

86

2

2

𝑒−𝑧 /2𝑎𝐻𝑂

where we have used the one-body axial harmonic length, 𝑎𝐻𝑂 =√︁~, since we are𝑚𝜔𝑧looking at the decay rate across the entire cloud. We then have the relation:√︃𝑛3𝐷 (𝑟) = 𝑛2𝐷 (𝜌)|𝜒0 (𝑧)|2 = 𝑛2𝐷 (𝜌)1 −𝑧2 /𝑎2𝐻𝑂
𝑒
.
𝜋𝑎2𝐻𝑂

In order to have consistency, for all 𝑟 we must have 𝛽3𝐷 𝑛3𝐷 (𝑟) = 𝛽2𝐷 𝑛2𝐷 (𝜌), thus at√
𝑟 = 𝜌 = 0 we find 𝛽3𝐷 𝑛3𝐷 (0) = 𝛽2𝐷 𝑛2𝐷 (0), ultimately leading to 𝛽3𝐷 = 𝜋𝑎𝐻𝑂 𝛽2𝐷 =√︀𝜋/2 𝑎𝑧 𝛽2𝐷 . Finally, we find the 3D dipolar rate coefficient to then be:
𝛽3𝐷 =

√︂

[︂ ∫︁ ∞
]︂2
1
𝜋 18𝜇
2 4
𝐹in (𝜌)𝐹out (𝜌) 2 𝑑𝜌 .
𝑎𝑧 𝐽 𝑑 𝐿
2 ~3 𝑘𝑖 𝑘𝑓
𝜌0√
The normalization of 𝐹in, out (𝜌) produces a 1/ 𝐿 coefficient for each radial wavefunction, thus cancelling the 𝐿 dependence.
An interesting and analytically simple result can be found in the case of a purely
2D system without dipolar interactions. In such a system the radial wavefunctions
are given by
𝐹in, free (𝜌) =𝐹out, free (𝜌) =

√︂

√︂

𝜋𝑘𝑖
[𝑎𝐽0 (𝑘𝑖 𝜌) + 𝑏𝑌0 (𝑘𝑖 𝜌)]𝐿

𝜋𝑘𝑓[𝑐𝐽2 (𝑘𝑓 𝜌) + 𝑑𝑌2 (𝑘𝑓 𝜌)] .
𝐿

Since lim𝑥→0 𝑌0 (𝑥) and lim𝑥→0 𝑌2 (𝑥) diverge, 𝑏 = 𝑑 = 0 and 𝑎 = 𝑐 = 1. Additionally,
since lim𝑘𝑖 →0 𝐽0 (𝑘𝑖 𝜌) ≈ 1, we find the free (unshielded) 2D dipolar rate coefficient to

be:

[︂
]︂2
∫︁ ∞
18𝜇 2 4 √︀𝐽2 (𝑘𝑓 𝜌)
𝐽 𝑑 𝜋 𝑘𝑖 𝑘𝑓𝑑𝜌~3 𝑘𝑖 𝑘𝑓
𝜌202𝜇𝜋 2 2 4 2= 3 𝐽 𝑑 𝑘𝑓 .
~

free
=
𝛽2𝐷

4.4.2

(4.9)
(4.10)

Introduction to the Quasi-2D System

Now that we have considered the pure 2D system in the plane, we will consider thethird dimension of our system that is defined by tight harmonic confinement, leading87

to a quasi-2D system. Averaging the elastic term of the dipolar potential yields a one
dimensional radial Schrödinger equation given by:
[︂

~2
2𝜇

(︂
)︂
]︂
𝑑2
~2 𝑘𝑖2𝑚2 − 1/4
elastic
− 2+
+
⟨𝜒
|𝑉
|𝜒
⟩
𝜑
=
𝜑.
0
0
𝑑𝑑
𝑑𝜌
𝜌2
2𝜇

(4.11)

In contrast to 3D, the centrifugal potential in 2D can be attractive (𝑚 = 0), as is
the case for the incoming collisional channel of our experiment.
Dipolar relaxation turns internal Zeeman energy into motional energy, thus it ispossible that when the dipolar relaxation occurs in a high enough magnetic field,
the collisional pair will be excited to a higher vibrational level of the lattice. Fora collisional pair of atoms, the 𝑣 = 0 and 𝑣 = 1 two-body vibrational states of thelattice are given by:
𝜒0 (𝑧) =where 𝑎𝑧 =

√︁

~
.
𝜇𝜔𝑧

(︂

1
𝜋𝑎2𝑧

)︂1/4

−𝑧 2 /2𝑎2𝑧𝑒𝜒1 (𝑧) =

(︂

4
𝜋𝑎6𝑧

)︂1/4

2

2

𝑧𝑒−𝑧 /2𝑎𝑧

By performing integration over 𝑧, one can easily see that when

the collisional pair starts in the ground-band, 𝜒𝑜 (𝑧), a selection rule will emerge forallowable final states for a given interaction potential. The 𝑣 = 0 state is even, while
the single spin flip (double spin flip) interaction potential is odd (even). Thus, the final
vibrational state must be odd (even). This result precludes a single spin flip collisionfrom occurring for low enough magnetic fields, since the final vibrational state must be
odd. Thus, at low magnetic fields, the double spin flip collision will be the only sourceof dipolar relaxation, with the final vibrational state 𝑣𝑓 = 0. Additionally, the double
spin flip requires a final vibrational state that is even, precluding 𝑣𝑓 = 1. Let us now
determine exactly when higher vibrational exit channels will become energeticallyallowed. Recall that the energy released for a single and double spin flip is given by:
Δ𝐸 (1) = 𝑔𝐽 𝜇𝐵 𝐵
Δ𝐸 (2) = 2𝑔𝐽 𝜇𝐵 𝐵
88

We have determined that higher vibrational channels will be excited when 𝑣𝑓 = 1
for single spin flip collisions and 𝑣𝑓 = 2 for double spin flip collisions. The energy
released by dipolar relaxation must therefore match or exceed ~𝜔𝑧 (single spin flip)and 2~𝜔𝑧 (double spin flip). Since the energy released from single and double spin flip
collisions differ by a factor of 2, both vibrationally excited exit channels open up at
precisely the same moment: ~𝜔𝑧 = 𝑔𝐽 𝜇𝐵 𝐵. In our experiment the gas of dysprosium
atoms is loaded into the ground-band of the lattice, 𝑣 = 0. To perform the simplest
calculation we will first look at double spin flip collisions in the low magnetic field
limit, such that we have only one exit channel: 𝑣𝑓 = 0.

4.4.3

Double Spin Flip in the Quasi-2D System

In order to calculate the transition rate, we first perform an integration over 𝑧 to
reduce our problem to the remaining dimensions of 𝜑 and 𝜌.
Let us now evaluate the matrix element for a double spin flip collision that occurs
in a quasi-2D setting. Recall that the double spin flip potential is given by:
3(𝑥 + 𝑖𝑦)2𝑉2 = 𝐽𝑑2.
2
𝑟5
In the pure 2D system we accounted for the third dimension only at the end of
the calculation, to convert 𝛽2𝐷 into 𝛽3𝐷 . We would now like to consider how theconfinement along 𝑧^ modifies the dipolar interaction and dipolar relaxation. The
procedure is simple: we determine a 𝑧-averaged interaction potential, ⟨𝑉2 ⟩𝑧 , and thenapply Fermi’s golden rule as before, but using the averaged potential in place of the89

original 𝑉2 potential:∫︁ ∞3
𝜒*0 (𝑧)𝑉2 (𝜌, 𝜑, 𝑧)𝜒0 (𝑧) 𝑑𝑧 = 𝐽𝑑2
2
−∞
3
= 𝐽𝑑2
2

√︃
√︃

1
(𝑥 + 𝑖𝑦)2
𝜋𝑎2𝑧

∫︁ ∞

−∞

1
(𝜌2 + 𝑧 2 )5/2

22𝑒−𝑧 /𝑎𝑧 𝑑𝑧

1(𝑥 + 𝑖𝑦)2
𝜋𝑎2𝑧(︂ 2 )︂
(︂ 2 )︂]︂
22 [︂
2𝑒𝜌 /2𝑎𝑧 2
𝜌
𝜌
2
2
×
𝜌 𝐾0
+ (𝑎𝑧 − 𝜌 )𝐾1
4
2
2
3𝑎𝑧 𝜌
2𝑎𝑧
2𝑎2𝑧
√︃
1 2𝑖𝜑
= 𝐽𝑑2
𝑒
𝜋𝑎2𝑧
(︂ 2 )︂
(︂ 2 )︂]︂
2
2 [︂𝑒𝜌 /2𝑎𝑧 2
𝜌
𝜌
2
2
−
𝜌
)𝐾
×
𝜌
𝐾
+
(𝑎
.
0
1
𝑧
𝑎4𝑧
2𝑎2𝑧
2𝑎2𝑧
We are left with a two dimensional problem, and we can push forward with our
calculation by integrating over the angular component. First, let’s look at the fullincoming and outgoing wave functions for the double spin flip collision:1
Ψin (𝜌, 𝜑, 𝑧) = 𝜒0 (𝑧)Φin (𝜑)𝐹in (𝜌) = 𝜒0 (𝑧) √ 𝐹in (𝜌)2𝜋𝑒2𝑖𝜑Ψout (𝜌, 𝜑, 𝑧) = 𝜒0 (𝑧)Φout (𝜑)𝐹out (𝜌) = 𝜒0 (𝑧) √ 𝐹out (𝜌).
2𝜋

We can perform the angular integral to remove the 𝜑 dependence from the problem:∫︁ 2𝜋

Φ*out (𝜑) 𝑒2𝑖𝜑 Φin (𝜑) 𝑑𝜑 =

0

1
2𝜋

∫︁ 2𝜋

𝑒−2𝑖𝜑 𝑒2𝑖𝜑 𝑑𝜑 = 10such that:∫︁𝜒*0 (𝑧)Φ*out (𝜑) 𝑉2 (𝜌, 𝜑, 𝑧) 𝜒0 (𝑧)Φin (𝜑) 𝑑𝜑 𝑑𝑧(︂ 2 )︂]︂[︂(︂ 2 )︂
𝜌2 /2𝑎2𝑧
𝜌
𝜌
2 𝑒
2
2
2
= 𝐽𝑑 √ 5 𝜌 𝐾0
+ (𝑎𝑧 − 𝜌 )𝐾1
.
2
2𝑎𝑧
2𝑎2𝑧
𝜋𝑎𝑧

⟨𝑉2 ⟩𝜑,𝑧 ≡The remaining dependence on 𝜌 is thus confined to the bracketed term.
90

Fermi’s golden rule then gives:
2𝜋(2)Γ2𝐷,𝑐 =[︂∫︁Ψin (𝜌, 𝜑)𝑉2 (𝜌, 𝜑, 𝑧)Ψout (𝜌, 𝜑) 𝑑𝜑 𝜌 𝑑𝜌]︂2[︂∫︁ ∞
2𝜋
𝐹in (𝜌)⟨𝑉2 ⟩𝜑,𝑧 𝐹out (𝜌)𝜌 𝑑𝜌 𝜌1𝐷 .
=
~
0
~

]︂2

𝜌1𝐷

Symmetrization and converting cylindrical waves to plane waves results in:
(2)

4 (2)
Γ
𝜋𝑘𝑖 𝐿 2𝐷, 𝑐[︂∫︁ ∞]︂2
8𝜇=𝐹in (𝜌)⟨𝑉2 ⟩𝜑,𝑧 𝐹out (𝜌) 𝜌 𝑑𝜌
𝜋~3 𝑘𝑖 𝑘𝑓 0
1
8𝜇𝐽 2 𝑑4 10= 2 3
𝜋 ~ 𝑘𝑖 𝑘𝑓
𝑎𝑧[︂∫︁ ∞(︂(︂ 2 )︂
]︂2(︂ 2 )︂)︂
𝜌𝜌
𝜌2 /2𝑎2𝑧2
2
2
𝐹in (𝜌)𝐹out (𝜌) 𝑒×𝜌 𝐾0+ (𝑎𝑧 − 𝜌 )𝐾1
𝜌 𝑑𝜌
2𝑎2𝑧
2𝑎2𝑧
0

Γ2𝐷 =

Once again, 𝛽2𝐷 = 𝜋𝐿2 Γ2𝐷 so that𝛽2𝐷 =8𝜇𝐿2 2 4 1
𝐽 𝑑 10𝜋~3 𝑘𝑖 𝑘𝑓
𝑎𝑧[︂∫︁ ∞(︂(︂ 2 )︂
]︂2(︂ 2 )︂)︂
𝜌
𝜌
22
𝜌2 /2𝑎2𝑧
2
𝐹in (𝜌)𝐹out (𝜌) 𝑒×𝜌 𝐾0+ (𝑎𝑧 − 𝜌 )𝐾1
𝜌 𝑑𝜌
2𝑎2𝑧
2𝑎2𝑧
0

and 𝛽3𝐷 =
𝛽3𝐷 =

√︂

√︀
𝜋/2 𝑎𝑧 𝛽2𝐷 , thus

𝜋 8𝜇1𝐽 2 𝑑4 93
2 𝜋~ 𝑘𝑖 𝑘𝑓𝑎𝑧[︂ ∫︁ ∞
(︂(︂ 2 )︂
(︂ 2 )︂)︂
]︂2𝜌
𝜌
2
𝜌2 /2𝑎2𝑧
22× 𝐿
𝐹in (𝜌)𝐹out (𝜌) 𝑒𝜌 𝐾0+ (𝑎𝑧 − 𝜌 )𝐾1
𝜌 𝑑𝜌 .
2𝑎2𝑧
2𝑎2𝑧
0Note that, as before, the normalization of the radial wavefunctions remove dependence on 𝐿 from the relaxation rate.
91

4.4.4Single Spin Flip in the Quasi-2D System

Let us now evaluate the matrix element for a single spin flip collision that occurs ina quasi-2D setting. It is assumed that the magnetic field is large enough to allow
excitation to the 𝑣 = 1 vibrational state. The full incoming and outgoing wave
functions for the single spin flip collision are given by:1
Ψin (𝜌, 𝜑, 𝑧) = 𝜒0 (𝑧)Φin (𝜑)𝐹in (𝜌) = 𝜒0 (𝑧) √ 𝐹in (𝜌)2𝜋𝑒𝑖𝜑√𝐹out (𝜌).
Ψout (𝜌, 𝜑, 𝑧) = 𝜒1 (𝑧)Φout (𝜑)𝐹out (𝜌) = 𝜒1 (𝑧)2𝜋Recall that the single spin flip potential is given by:
𝑉1 = 3𝐽 3/2 𝑑2(𝑥 + 𝑖𝑦)𝑧.
𝑟5

Similar to before, we can determine a 𝑧-averaged interaction potential, ⟨𝑉1 ⟩𝑧 . In thefollowing, 𝑈 (𝑎, 𝑏, 𝑥) is the confluent hypergeometric function of the second kind:
∫︁ ∞

√︃

∫︁ ∞

𝑧2
2
2
𝑒−𝑧 /𝑎𝑧 𝑑𝑧
22
5/2−∞ (𝜌 + 𝑧 )
√︃
[︃ √
(︃
(︂ )︂2 )︃]︃2
𝜋3
𝜌(𝑥 + 𝑖𝑦)𝑈, 0,= 3𝐽 3/2 𝑑2
4
2
𝜋𝑎𝑧
2𝜌
2
𝑎𝑧
[︃
(︃
(︂ )︂2 )︃]︃3 3/2 2 1 (𝑥 + 𝑖𝑦) 1
3𝜌=√ 𝐽 𝑑 2
𝑈
, 0,
𝑎𝑧
𝜌
𝜌
2
𝑎𝑧
2
[︃
(︃
)︃]︃
(︂ )︂2
3 3/2 2 1 𝑖𝜑 13𝜌= √ 𝐽 𝑑 2𝑒
𝑈
, 0,
.
𝑎𝑧
𝜌
2
𝑎𝑧
2

𝜒*0 (𝑧)𝑉1 (𝜌, 𝜑, 𝑧)𝜒1 (𝑧) 𝑑𝑧 = 3𝐽 3/2 𝑑2−∞

2(𝑥 + 𝑖𝑦)
𝜋𝑎4𝑧

We can perform the angular integral to remove the 𝜑 dependence from the problem:∫︁ 2𝜋

Φ*out (𝜑) 𝑒𝑖𝜑 Φin (𝜑) 𝑑𝜑 =

0

92

1
2𝜋

∫︁ 2𝜋
0

𝑒−𝑖𝜑 𝑒𝑖𝜑 𝑑𝜑 = 1

such that:∫︁𝜒*1 (𝑧)Φ*out (𝜑) 𝑉1 (𝜌, 𝜑, 𝑧) 𝜒0 (𝑧)Φin (𝜑) 𝑑𝜑 𝑑𝑧[︃(︃(︂ )︂2 )︃]︃3 3/2 2 1 1
3𝜌=√ 𝐽 𝑑 2
𝑈
, 0,
.
𝑎𝑧 𝜌
2
𝑎𝑧
2

⟨𝑉1 ⟩𝜑,𝑧 ≡The remaining dependence on 𝜌 is thus confined to the bracketed term.

Fermi’s golden rule then gives:
2𝜋(2)Γ2𝐷,𝑐 =[︂∫︁Ψin (𝜌, 𝜑)𝑉1 (𝜌, 𝜑, 𝑧)Ψout (𝜌, 𝜑) 𝑑𝜑 𝜌 𝑑𝜌]︂2[︂∫︁ ∞
2𝜋=
𝐹in (𝜌)⟨𝑉1 ⟩𝜑,𝑧 𝐹out (𝜌)𝜌 𝑑𝜌 𝜌1𝐷 .
~
0
~

]︂2

𝜌1𝐷

Symmetrization and converting cylindrical waves to plane waves results in:
(2)

4 (2)
Γ
𝜋𝑘𝑖 𝐿 2𝐷, 𝑐]︂2[︂∫︁ ∞
8𝜇=𝐹in (𝜌)⟨𝑉1 ⟩𝜑,𝑧 𝐹out (𝜌) 𝜌 𝑑𝜌
𝜋~3 𝑘𝑖 𝑘𝑓 0[︃∫︁(︃(︂ )︂2 )︃ ]︃2
∞
36𝜇
𝜌
3
3 4 1
=
𝐹in (𝜌)𝐹out (𝜌) 𝑈
, 0,
𝐽 𝑑 4
𝑑𝜌 .
3
𝜋~ 𝑘𝑖 𝑘𝑓
𝑎𝑧 0
2
𝑎𝑧

Γ2𝐷 =

Once again, 𝛽2𝐷 = 𝜋𝐿2 Γ2𝐷 so that
36𝜇𝐿2 3 4 1
𝐽 𝑑 4
𝛽2𝐷 = 3~ 𝑘𝑖 𝑘𝑓𝑎𝑧
and 𝛽3𝐷 =[︃∫︁∞𝐹in (𝜌)𝐹out (𝜌) 𝑈0√︀
𝜋/2 𝑎𝑧 𝛽2𝐷 , thus:

𝛽3𝐷 =

√︂

(︃

3
, 0,
2

(︂

𝜌
𝑎𝑧

)︂2 )︃

]︃2

𝑑𝜌

[︃ ∫︁
(︃
(︂ )︂2 )︃ ]︃2∞
𝜋 36𝜇 3 4 1
3
𝜌
𝐽 𝑑 3 𝐿𝐹in (𝜌)𝐹out (𝜌) 𝑈
, 0,
𝑑𝜌 .
3
2 ~ 𝑘𝑖 𝑘𝑓
𝑎𝑧
2
𝑎𝑧
0
93

4.4.5

Theoretical Results

To illustrate the suppression mechanism, we perform an average of the elastic termin dipolar interaction over the harmonic wave function induced by the lattice confinement, which together with the attractive (𝑚𝑖 = 0) centrifugal potential (see Equation 4.11) produces the quasi-2D curve shown in Fig. 4-1a. The dipolar barrier pushes
the maximum of the incoming wave function to larger 𝜌 values (Fig. 4-1b), leading toa reduction in the overlap of the incoming wave function and the dipolar interaction(Fig. 4-1c), as well as a corresponding reduction in transition rates as calculated byFermi’s golden rule.
Three parameters determine the loss rate in quasi-2D: the ratio 𝑎𝑑𝑑 /𝑎𝑧 set by theconfinement, the temperature 𝑇 , and the magnetic field 𝐵. The potential barrier
increases with increasing confinement, ultimately reaching the pure 2D limit. As thetemperature decreases, the wave function of an incoming pair is suppressed by thebarrier over a longer range, thereby decreasing the chance of two atoms reaching close
range (Figure 4-2d). Since the majority contribution to loss is derived from the first
oscillating lobe of the outgoing wavefunction (≈ 1/𝑘𝑓 in extent), the range of dipolar
relaxation decreases as the outgoing momentum increases. Correspondingly, higher
magnetic fields cause spin flips to occur at shorter range, behind the barrier felt by
the incoming atoms, and are therefore more strongly suppressed by dipolar repulsion(Figure 4-2c). Shielding qualitatively changes the magnetic field dependence of thedipolar relaxation rate. In both the 3D and unshielded 2D geometries, the relaxation
rate for bosons increases with magnetic field. When accounting for the barrier in
2D, however, the signature of dipolar shielding appears: the relaxation rate decreases
with magnetic field (Figure 4-2a-b).

94

2D
Ei

0.0

Quasi
2D
3D

0.5

Integrand(a.u.)

Wave function(a.u.)

Energy (Edd)

0.5 a

b
0
Outgoing Free
3D Free
Quasi 2D Shielded
Pure 2D Shielded

c
0
0

5

(add)

10

15

Figure 4-1: Effective potentials and wave functions for different axial confinements.
(a) Effective radial potential from Equation 4.11 for: no confinement (light blue,3D), quasi-2D with 𝜔𝑧 /2𝜋 = 475 kHz (steel blue) and pure-2D (dark blue). The
incoming energy 𝐸𝑖 = ~2 𝑘𝑖2 /2𝜇 is 𝑘𝐵 × (1 𝜇K). The unit is dipolar energy definedas 𝐸𝑑𝑑 = ~2 /2𝜇𝑎2𝑑𝑑 . (b) Wave function solutions of Equation 4.11 with 𝑚𝑖 = 0 for
the three confinement strengths described above, and in red the 𝑚𝑓 = 2 outgoingwave function with 𝑘𝑓2 > 𝑘𝑖2 . (c) Integrand of Fermi’s golden rule (Equation 4.1).
Each curve is the product of the respective wave function in (b), the outgoing freewave function, and the double spin-flip operator from Equation 4.8 integrated withthe harmonic oscillator in the 𝑧-direction for a Δ𝑣 = 0 transition. The shielding we
implement here corresponds to the difference between the light blue and steel blue
curves. The minimum attainable decay rate corresponds to the dark blue curve.

95

b
n=3

1.0
0.5
0.0

j = 1, mi = 0
j = 2, mi = 0
All j, n, m

n=1
n=0

0

500
B (mG)0

B (mG)

1000

Suppression factor

3D (10 11 cm3/s)

a

1.5

103

1
2

pure2D
Analytical

c

d

1
2

pure2D
Analytical

102
101
100

0 50

T1 = 10 nK

B1 = 170 mG

B1 200B (mG)

T1 102 103
Temperature (nK)

Figure 4-2: Channel-by-channel decomposition of the dipolar relaxation rates of the𝑚𝑖 = 0 incoming state in a 300 kHz trap, both for free wave functions (a) andshielded ones (b). The blue and red curves correspond to single and double spin
flips, respectively. The different shades correspond to different harmonic oscillator
states as they open up with increasing magnetic field. The black-dashed curve isthe contribution of all the different channels, including 𝑚𝑖 ̸= 0, and only shows adeviation at magnetic fields comparable to the incoming energy, which is on the orderof a few mG. Also shown is the suppression factor, as defined by the ratio of 𝛽3𝐷
obtained from shielded and free wave functions, as a function of magnetic field (c)and temperature (d). For each of the graphs we present curves for 𝜔1 /2𝜋 = 300 kHz(dotted), 𝜔2 /2𝜋 = 1.8 MHz (dashed dotted), the pure-2D case (solid line) as wellas the analytical approximation (grey dotted). The theoretical work performed by
Pierre Barral extended the efforts of this chapter to higher vibrational channels, theresults of which were used to create the theory curves.

96

4.5

The Experiment

In three dimensions, the rate of dipolar relaxation is fully characterized by the wave
function amplitude at the interparticle separation where the molecular potentials of
the input and output channel intersect [26]. In contrast, Fermi’s golden rule applied
to the two-dimensional case shows that a significant contribution to the dipolar relaxation rate occurs for all interparticle separations less than ∼ 1/𝑘𝑓 . To suppress

the incoming wave function in this region, we load 162 Dy atoms into the ground bandof an optical lattice applied along the magnetic field quantization axis (^𝑧 ).
To begin, we prepare samples of ∼ 8 × 104 162 Dy atoms at 150 nK which are spinpolarized in the |𝐽 = 8, 𝑚𝐽 = −8⟩ state. The samples are obtained after evaporative

cooling in a crossed optical-dipole trap (XODT) (section 2.5) which is loaded froma narrow-line magneto-optical trap (MOT) (section 2.4). The XODT consists ofthree 1064 nm laser beams: two beams with 40 𝜇m beam waists crossed at 8∘ inthe (horizontal) 𝑥^ − 𝑦^ plane, and a beam with a 64 𝜇m waist propagating alongthe (vertical) 𝑧^ direction. We work with a thermal gas in order to avoid a dipolar

relaxation rate which depends on BEC fraction. Following evaporation, atoms in thelowest Zeeman sublevel |𝑚𝐽 = −8⟩ are transferred to the highest Zeeman sublevel|𝑚𝐽 = +8⟩ via adiabatic rapid passage (ARP) using an RF sweep in a magnetic

field of 3.5 G along the 𝑧^ direction. The transfer efficiency of the ARP is greater than
95%, which is verified by a Stern-Gerlach measurement, and by lifetime measurements
after transferring the atoms back to the |𝑚𝐽 = −8⟩ absolute ground state with an

RF sweep in the opposite direction.

An array of quasi-2D crêpes is created from the thermal sample using a 1D opticallattice formed by retro-reflecting a 741 nm laser beam along 𝑧^. The lattice beam isblue-detuned from the 1.8-kHz 741 nm transition by either 2.25 GHz or 14.25 GHz,
thus providing tight axial confinement while avoiding the excessive adiabatic heating
a comparable red lattice would inflict.
At low magnetic fields, linear lattice light induces Raman couplings between |𝑚𝐽 =8⟩ and other even 𝑚𝐽 states. Perturbatively, some |𝑚𝐽 = +6⟩ character will be
97admixed into the |𝑚𝐽 = +8⟩ state, thereby opening additional dipolar relaxation
channels that conserve total spin while releasing Zeeman energy (Equation 4.5). Spinexchange removes a unit of spin angular momentum from one of the colliding atoms,
transferring it to its collisional partner. While this form of spin exchange dominates
the observed lifetimes at low magnetic fields, as the magnetic field is increased, thetwo-photon detuning increases sufficiently to suppress the Raman coupling. After
mitigating undesired Raman couplings by ensuring the lattice polarization is highly
circular, we observe that magnetic fields of approximately 100 mG are sufficient for
dipolar relaxation to dominate.
A coaxial vertical ODT is used to provide confinement to compensate the transverse repulsion resulting from the blue-detuned lattice, as shown in Figure 4-3a. The
maximum achievable axial trap frequencies were limited to 𝜔𝑧 = 2𝜋 × 260 kHz due

to the maximum laser power of the compensation beam. Due to the combinationof circular lattice polarization and disparate Clebsch-Gordan coefficients, we prepare
the unstable |𝑚𝐽 = +8⟩ state prior to loading the lattice so as to prevent the atoms

from seeing a changing lattice potential during the ARP. We then turn up the latticeand ODT beams using 50 ms exponential ramps to adiabatically load the atoms into

the lowest vibrational level of the lattice. During the first 40 ms of lattice ramp, wequickly reduce the magnetic field to 40 mG within 100 𝜇s to minimize the dipolar
relaxation losses. The magnetic field is then ramped up to its final value during thelast 10 ms of the lattice loading ramp, after which we preform a lifetime measurement.
The experimental data in Figure 4-3(b-c) was collected by varying the externally applied magnetic field (for 𝜔𝑧 /2𝜋 = 185 kHz) and the axial trap frequency (for 𝐵 = 200mG), respectively.
Our experimental and theoretical results are shown in Figs. 4-3(b-c). Compared
to data obtained in the ODT, the lattice confinement leads to more than an order ofmagnitude suppression for most magnetic fields, with larger confinement leading to
a decrease in the dipolar relaxation rate. In contrast to the 3D loss rate (red), which√
increases with 𝐵, we observe the signature of shielding: a much weaker dependence
on magnetic fields (blue). Our experiment operates in the quasi-2D regime, with axial
98confinement frequencies up to 300 kHz, which differs from the 2D regime in several
aspects. Compared to pure 2D, the finite axial extent of the quasi-2D geometry softens the radial barrier, such that our temperatures are comparable to the height ofthe barrier. Furthermore, for Zeeman energies that are larger than the axial trapping
frequency, new collisional channels open, with a portion of the energy converted into
axial excitation and the remainder into radial motion. As a result, the relaxation forthese processes is shifted to larger distances, thereby weakening the shielding. The
first channel opening is clearly visible on the Figure 4-3b at 106 mG. The aforementioned factors lead to a relaxation rate that does not decrease with magnetic field, as
it would in the pure 2D case, but still lead it to increase less compared to the casewithout a dipolar barrier (dashed blue). Figure 4-3(c) shows the loss rate coefficient
as a function of axial confinement. The loss rate decreases with confinement due
to enhanced shielding by the dipolar repulsion, and by closing collision channels toaxially excited states.

99

1064
741
red beam blue lattice

Figure 4-3: (a) Trap geometry. A blue-detuned 741 nm retroreflected beam repels
the atoms to create a 1D lattice. The finite contrast of the lattice and the zero-point
motion of the atoms in the ground state creates a repulsive transverse potential. A
1064 nm red-detuned beam compensates for that and creates an adjustable transverse
harmonic confinement. (b and c) Experimentally measured 𝛽3𝐷 in an ODT (red) andin the thin layer (blue). The lines are theory curves. The red one corresponds to theBorn approximation treatment in three dimensions as explained in [34, 37], the dashedblue one corresponds to Fermi’s golden rule for non-shielded atoms in a lattice. The
solid blue line is Fermi’s golden rule, taking into account the shielding induced by theelastic dipole-dipole interaction. (b) Measurement of 𝛽3𝐷 as a function of magnetic
field in a fixed geometry: the ODT and a 𝜔𝑧 /2𝜋 = 185 kHz lattice which corresponds
to 𝑎𝑧 = 26 nm. (c) Measurement of 𝛽3𝐷 in a constant magnetic field of 200 mG whilevarying the trap frequency. The uncertainties are set by the atom number stability,
cloud temperature measurement and trap frequency measurements.

4.6

Extracting Dipolar Relaxation Rate Coefficient
From Atomic Decay Measurement

We wish to analyze an experiment that measures the number of atoms in a cloud asa function of time. We expect to be dominated by dipolar relaxation, a two-body= −𝛽𝐷𝑅 𝑛2 . Since the experiment will only measure
decay process described by 𝑑𝑛
𝑑𝑡

atom loss, we must be careful to calculate the correct volume of the gas so that we
may accurately determine 𝛽𝐷𝑅 . We will take a quick detour to develop the tools toanalyze the above expression for 𝑑𝑛. The density and atom number of a gas trapped𝑑𝑡100

in a potential 𝑈 (r) is given by:−𝑈 (r)/𝑘𝐵 𝑇𝑛(r) = 𝑛0 𝑒

𝑁 = 𝑛0∫︁𝑒−𝑈 (r)/𝑘𝐵 𝑇 𝑑r

where 𝑛0 and 𝑇 are the peak density and temperature of the cloud, respectively. We
can then define the effective volume of the trapped gas to be:
𝑉𝑒 ≡ 𝑁/𝑛0 =∫︁𝑒−𝑈 (r)/𝑘𝐵 𝑇 𝑑r.

Lastly, we can now define the mean collisional volume of the trapped gas to be:𝑉¯ ≡ 𝑁/⟨𝑛(r)⟩⟨𝑛(r)⟩ =∫︁

𝑛(r)𝑝(r)𝑑r =

∫︁

𝑛(r)

𝑛(r)
𝑑r.
𝑁

We can now apply these relations to the case of a harmonically trapped gas, as wehave in our experiment. Along a harmonically confined dimension, we have:1
𝑈 (r) = 𝑚𝜔 2 𝑟2 .
2√
We define 𝜎 as the 1/ 𝑒 width of the cloud such that22𝑛(r) = 𝑛0 𝑒−r /2𝜎 = 𝑛0 𝑒−𝑈 (r)/𝑘𝐵 𝑇 → 𝜎 2 =

𝑘𝐵 𝑇
.
𝑚𝜔 2

The effective volume of an isotropic harmonically trapped gas is then given by:
𝑉𝑒 =

∫︁ ∞

2

2

𝑒−𝑟 /2𝜎 4𝜋𝑟2 𝑑𝑟 = (2𝜋)3/2 𝜎 3 .

0Let’s now look at anisotropic harmonic trapping, where a similar calculation gives:
𝑉𝑒 =

∫︁ ∞

2

2

2

2

2

2

𝑒−𝑥 /2𝜎𝑥 𝑒−𝑦 /2𝜎𝑦 𝑒−𝑧 /2𝜎𝑧 𝑑𝑥 𝑑𝑦 𝑑𝑧 = (2𝜋)3/2 𝜎𝑥 𝜎𝑦 𝜎𝑧 .

−∞

101

We populate the lowest state of the harmonic oscillator along the tight confinement~= 12 𝑎2𝐻𝑂 , such that:
direction (^𝑧 ) of our experiment, and correspondingly 𝜎𝑧2 = 𝑚𝜔𝑧

𝑉𝑒 = (2𝜋)

3/2

𝜎𝑥 𝜎𝑦

(︂

1
√ 𝑎𝐻𝑂
2

)︂

𝑘𝐵 𝑇
where 𝜎𝑖2 = 𝑚𝜔
2 for 𝑖 ∈ (𝑥, 𝑦).
𝑖

The mean density of the harmonically trapped gas can be found by
⟨𝑛⟩ =

∫︁

𝑛(r)𝑝(r)𝑑r =

∫︁

𝑛(r)

𝑛0
𝑛(r)
𝑑r = √ .
𝑁
8

The above result is independent of trap frequency due to the magic of Gaussian
integrals and the cancellation that occurs, leaving only numerical factors from theintegration.
The mean collisional volume of the harmonically trapped gas is given by:𝑉¯ ≡ 𝑁/⟨𝑛⟩ =√√√8𝑁/𝑛0 = 8𝑉𝑒 = 8(2𝜋)3/2 𝜎𝑥 𝜎𝑦 𝜎𝑧 .

With the above definitions, we can continue with the analysis of 𝑑𝑛= −𝛽𝐷𝑅 𝑛2 by𝑑𝑡

integrating over the volume of a harmonically trapped atomic cloud [45]:𝑑𝑁= −𝛽𝐷𝑅
𝑑𝑡
Therefore we have∫︁𝑛2 𝑑𝑉 = −𝛽𝐷𝑅 ⟨𝑛⟩𝑁 = −𝛽𝐷𝑅 𝑉¯ −1 𝑁 2 .

∫︁ 𝑁 (𝑡)𝑁0𝑑𝑁=−
𝛽𝐷𝑅 𝑉¯ −1 𝑁 2

∫︁ 𝑡

𝑑𝑡.

0

Solving this for 𝑁 (𝑡) yields
𝑁 (𝑡) =

𝑁0.
1 + 𝛽𝐷𝑅 𝑉¯ −1 𝑁0 𝑡

Let’s now apply this analysis to the situation created by a lattice. For the 𝑖𝑡ℎ pancake102

in the lattice we have a loss rate governed by:
𝑑𝑁𝑖−1= −𝛽𝐷𝑅 𝑉¯𝑖 𝑁𝑖2 .
𝑑𝑡
𝑉¯𝑖 is the same for all the pancakes:
𝑉¯𝑖 =where 𝑉¯𝑂𝐷𝑇 ≡

√

8(2𝜋)

3/2

𝜎𝑥 𝜎𝑦

(︂

1
√ 𝑎𝐻𝑂
2

)︂

𝑎𝐻𝑂
= 𝑉¯𝑂𝐷𝑇 √2𝜎𝑧√8(2𝜋)3/2 𝜎𝑥 𝜎𝑦 𝜎𝑧 . Next, we perform a sum over all pancakes:∑︁ 𝑑𝑁𝑖
𝑖

𝑑𝑡

=

∑︁
𝑑𝑁
−1
𝑁𝑖2 .
= −𝛽𝐷𝑅 𝑉¯𝑖𝑑𝑡𝑖

Now let’s use the fact that the lattice is slicing the gas while it is in a thermal ODTwith a harmonic trapping potential along 𝑧^, such that:1
𝑁𝑖 = 𝑁 √︀
2𝜋𝜎𝑧2∫︁ 𝑧𝑖+1

2

− 𝑧2

𝑒

2𝜎𝑧

𝑧𝑖

2

𝑧
1
− 𝑖
𝑑𝑧 ≈ 𝑁 √︀𝑒 2𝜎𝑧2
2𝜋𝜎𝑧2

(︂ )︂
𝜆
.
2

Next we can replace the sum over pancakes by using the helpful relation:
∑︁
𝑖

→

∫︁

𝑑𝑧𝑖
𝜆
2

.

Thus allowing us to write:∑︁
𝑖

1
𝑁𝑖2 ≈ 𝑁 2
2𝜋𝜎𝑧2

(︃ )︃
(︂ )︂ ∫︁ ∞ 𝑧2
(︂ )︂
𝜆
2
𝑖
√
𝜆
1
𝜆
𝑁
−
2
√
𝑒 𝜎𝑧2 𝑑𝑧𝑖 = 𝑁 2
𝜋𝜎
=
.
𝑧
2
2
2𝜋𝜎
2
𝜎
2
𝜋
𝑧
−∞
𝑧

Therefore, when loaded into a ground-band of a 1-D lattice, our atom number equation
becomes:
𝑁 (𝑡) =

with 𝑉¯lattice ≡𝑁0𝑁0)︁ (︁ 𝜆 )︁ =(︁
−1
1 + 𝛽𝐷𝑅 𝑉¯lattice
𝑁0 𝑡2
1 + 𝛽𝐷𝑅 𝑉¯𝑖−1 𝑁0 2√1 𝜋𝑡𝜎𝑧√ (︁ 𝑎 )︁
2𝜋 𝐻𝑂
𝑉¯𝑂𝐷𝑇 .
𝜆2103

(4.12)

This can be intuitively understood by considering atoms in the ODT being slicedby the lattice in segments of 𝜆2 and confining the atoms from each segment to a harmonic length 𝑎𝑧 along the lattice axis. Equipped now with the expression for atom
loss 𝑁 (𝑡) in a lattice undergoing dipolar relaxation, we may fit our experimental decay
curves to determine 𝑁0 and 𝛽𝐷𝑅 𝑉¯lattice , and by calculating 𝑉¯lattice from our experimentally known parameters (trap frequencies, wavelength of light, and temperature) we
may directly determine 𝛽𝐷𝑅 .
The treatment can easily be extend to the case where losses result from bothone-body and two-body decay:𝑑𝑁= −𝛾𝑁 − 𝛽𝐷𝑅 𝑉¯ −1 𝑁 2 .
𝑑𝑡
Therefore we find that∫︁ 𝑁 (𝑡)𝑁0𝑑𝑁=−
𝛾𝑁 + 𝛽𝐷𝑅 𝑉¯ −1 𝑁 2

∫︁ 𝑡

𝑑𝑡.

0

Solving this for N(t) yields
𝛾𝑒−𝛾𝑡
𝑁 (𝑡) =𝑁0 .
𝛾 + 𝛽𝐷𝑅 𝑉¯ −1 𝑁0 (1 − 𝑒−𝛾𝑡 )We measure the losses as a function of the magnetic field and transverse confinement, and fit the experimental decay curves using Equation 4.12. Calculating 𝑉¯latticeusing experimentally known trap parameters (i.e. the trap frequencies, the lattice
wavelength, and the temperature), we may directly determine 𝛽.

4.7

Zeroing The Magnetic Field

The rate of dipolar relaxation depends on the external magnetic field, which modifies
both the allowed outgoing vibrational channels as well as the outgoing collisional
wavefunction. For our study, it was crucial to control small magnetic fields.
We have devised a method to zero the magnetic field that relies on the large dis104

parity of Clebsch-Gordan coefficients for different hyperfine states, which will also
play a prominent role in the following chapter on the creation of a subwavelengthbilayer. At a high level, a magnetic dipole moment oriented parallel to the propagation of an imaging beam will scatter the circularly polarized light in vastly different
amounts than a magnetic dipole moment oriented anti-parallel to the propagationof the imaging beam. By using absorption imaging for various external magnetic
fields, as shown in Fig. 4-4, it is then possible to observe when the dipole moment
has flipped, thus setting the zero of the external magnetic field.
More specifically, in a spin-polarized (𝑚𝐽 = −8) sample of bosonic dysprosium,

the square of the Clebsch-Gordan coefficients for 𝜎− , 𝜋, and 𝜎+ transitions are given
by:
𝜎− :

|⟨8, −8; 1, 1| 9, −9⟩|2=1(4.13)𝜋:|⟨8, −8; 1, 0| 9, −8⟩|2= 1/9(4.14)𝜎+ :|⟨8, −8; 1, −1| 9, −7⟩|2 = 1/153.

(4.15)

We perform imaging of a spin-polarized sample with left-circularly polarized (𝜎𝐿 )light along the magnetic field quantization axis (𝑧). At large positive magnetic fieldbias, the atoms see 𝜎 − light with a corresponding Clebsch-Gordan coefficient of 1.
The imaging therefore produces a strong atom count. At large negative magnetic fieldbias, the atoms see 𝜎 + light with a corresponding Clebsch-Gordan coefficient of 1/153.
The imaging therefore produces a low atom count due to the lack of absorption. We
then lower the magnetic field bias, and as it becomes comparable to the magnetic
field along 𝜌, the 𝜎𝐿 light is seen as a mixture of 𝜎 − , 𝜋 and 𝜎 + light, resulting in an
atom count between that found in the large positive and large negative bias scenarios.
Thus, scanning the magnetic field bias and imaging with 𝜎𝐿 light results in an
atom number that smoothly transitions from high to low values due to the disparity
in Clebsch-Gordan coefficients. The steepness of the transition is maximized when themagnetic field along 𝜌 is minimized, since then the large positive and large negative
magnetic field regimes are valid for the largest region of bias magnetic fields. With
105

Imaging signal (103 a.u.)

120
100

B<0

80

σL

60

B>0
σL

40
20
0
−50

0
B (mG)

50

Figure 4-4: Determining the zero of the magnetic field with spin-polarized 𝑚𝐽 = −8
dysprosium atoms and left-circularly polarized imaging light. As the axial magnetic
field traverses through zero, the drastic difference in Clebsch-Gordan coefficients for𝜎+ and 𝜎− transitions produce a step-like change in imaging signal.
the magnetic field along 𝜌 set by finding the maximal steepness, we then set the zeroof the bias field along 𝑧 as the magnetic field value at the nearly step-like transition
between 𝜎 − and 𝜎 + imaging.

106

4.8

Conclusions

By applying a moderate magnetic field bias and tight optical lattice confinement,
dipolar relaxation can be suppressed by an order of magnitude compared to thethree dimensional scenario. Our theoretical results agree with the measured dipolar
relaxation rate. Our calculations suggest that a further increase in the suppression
factor may be obtained by implementing even tighter trap frequencies in the bluedetuned optical lattice, by lowering the temperature of the cloud, or by enhancing thepolarization purity of the lattice. In principle, arbitrarily low loss rates and infinite
shielding factors are possible at very low temperatures or at very high magnetic fields.
The techniques described in this chapter enable the creation of dense dysprosium
clouds consisting of multiple long-lived spin-species, thus opening new possibilities
for quantum simulation experiments.

107

108

Chapter 5
A Tuneable Subwavelength Bilayer
We have thus far considered the lowest magnetic sublevel, 𝑚𝐽 = −8 in dysprosium,

to be the ground state. In this chapter, we will find that by using light of opposite circular polarizations, we gain the ability to create strong AC Stark shifts inboth the 𝑚𝐽 = −8 and the 𝑚𝐽 = +8 states. For small detunings and low magnetic
fields, these AC Stark shifts can dominate over the Zeeman shifts, thus generating

two allowed ground states of opposite orientations. Furthermore, we will show thatthese two ground states can coexist while being simultaneously addressed by both 𝜎+
and 𝜎− beams. Through this individual addressability, we are able to break throughthe wavelength-limited separations of an individual optical lattice, instead generating two optical lattices capable of high trap frequencies and arbitrarily small spatial
separations from one another. The resulting tunable subwavelength bilayers are characterized via three-body losses, and are subsequently used to study the transfer ofthermal energy and the transfer of a dipole oscillation in the harmonic trap across
50 nm separated layers through the long-range dipole-dipole interaction.

109

5.1

The Tensor AC Stark Shift

Optical lattices formed by two counter-propagating laser beams utilize the AC Stark
shift of one or more atomic transitions to create a periodic trapping potential foratoms. By utilizing the spin dependence of atomic transitions, it is therefore possible to craft spin-dependent lattices [46]. Highly magnetic atoms present a unique
opportunity for the creation of spin-dependent atom-light couplings. Since the magnetic moment of an atom is proportional to its total angular momentum, 𝐽, highly
magnetic atoms correspondingly possess large total angular momentum. However, afurther consequence of large 𝐽 values can be found in the disparity of Clebsch-Gordan
coefficients, which characterize the strength of the atom-light coupling. As discussed
in section 4.7, the strength of the atom-light interaction scales with the square of
the Clebsch-Gordan coefficient of a given transition. In the case of a spin-polarized
𝑚𝐽 = −8 dysprosium atom, this results in a strength of the atom-light interactionthat is 153x larger for resonant 𝜎− light compared to 𝜎+ light. Similarily, due tothe symmetry of the Clebsch-Gordan coefficients about the 𝑚𝐽 = 0 state, a spinpolarized 𝑚𝐽 = +8 dysprosium atom has a strength of the atom-light interactionthat is 153x larger for resonant 𝜎+ light compared to 𝜎− light. Atomic transitions
from a stretched state (𝑚𝐽 = ±𝐽) possess the highest contrast of Clebsch-Gordan coefficients for 𝜎+ and 𝜎− polarizations of light, with larger 𝐽 values producing higher
contrast. Dysprosium and other magnetic atoms are often prepared in their spinpolarized ground state, a stretched state, in order to minimize losses resulting from
dipolar relaxation. Since the Clebsch-Gordan coefficient is maximal for transitions
between two stretched states, we find that the maximal AC Stark shift occurs forthe states |𝐽 = 8, 𝑚𝐽 = +8⟩ and |𝐽 = 8, 𝑚𝐽 = −8⟩, which each act as a groundstate when ignoring Zeeman shifts. It has been shown that a vector AC Stark shift,which scales linearly with 𝑚𝐽 , is mathematically equivalent to an effective magnetic

field [47]. Although such a picture is not exact for dysprosium due to its tensor AC
Stark shift in addition to vector and scalar components, it is helpful to think of theground state |𝐽 = 8, 𝑚𝐽 = +8⟩ instead as the state |𝐽 = 8, 𝑚𝐽 = −8⟩ in an effective
110magnetic field that is oriented anti-parallel to the original quantization axis. In sucha picture, 𝜎− and 𝜎+ beams create anti-aligned effective magnetic fields that trap
atoms in ground states which are also anti-aligned with respect to one another. Figure 5-1 extends theses results, showing the AC Stark shift of each magnetic sublevelfor different polarizations of light.

Figure 5-1: The AC Stark shift of each magnetic sublevel in dysprosium for 𝜋, 𝜎+ ,and 𝜎− transitions.
The full AC Stark shift an atom experiences in a light field can be decomposed intoa scalar, vector, and tensor component. The scalar component provides an equal shift
for all 𝑚𝐽 sublevels, while the vector component is proportional to 𝑚𝐽 , analogous to
the linear Zeeman effect, thus allowing the vector AC Stark shift to be viewed as aneffective magnetic field. Lastly, the tensor component contains a dependence on 𝑚2𝐽 .
The 𝜎± transitions in Figure 5-1 have contributions from all three components, whilethe 𝜋 transition contains only scalar and tensor components [48].
A more systematic illustration of how the bilayer array is formed in such an atomlight coupled system is shown in Figure 5-2 (a-f), where the adiabatic potentials of all
17 𝑚𝐽 states are plotted with different polarizability components taken into account.
The scalar polarizability 𝛼𝑠 leads to an AC Stark shift that is degenerate for all 17
𝑚𝐽 states (Figure 5-2 (a, d)). The light shift caused by the vector polarizability 𝛼𝑣 islinear in 𝑚𝐽 (Figure 5-2 (b, e)), therefore mathematically it is equivalent to a Zeeman
shift caused by a fictitious sinusoidal magnetic field Blat along the z direction. The
111

presence of 𝛼𝑣 lifts the degeneracy except for one point in the unit cell where thefictitious magnetic fields from the 𝜎+ and the 𝜎− lattices cancel. By further adding in
the tensor polarizability 𝛼𝑡 , the remaining 17-fold degenerate point is lifted, resulting
in a potential structure with the 𝑚𝐽 = +8 state being the local ground state of the𝜎+ potential minima, and the 𝑚𝐽 = −8 state being the local ground state of the 𝜎−potential minima (Figure 5-2 (c, f)). The removal of the 17-fold degenerate pointby the tensor shift protects the system against spin-flip processes caused by weaktransverse magnetic fields.

112

only

b

+

+

d

e

+

+

Adding v

Adding t

c
+

f
+

+

+=

a

s

g

|Bext | |Blat |h|Bext | |Blat |i|Bext | |Blat |jlFigure 5-2: Creating and loading a subwavelength potential using tensor and vector
polarizabilities. (a-f) Illustration of how the different polarizability components makeup the potential through two independently controllable, red-detuned standing waves
of 𝜎+ and 𝜎− light. Solid lines denote potential curves for different 𝑚𝐽 states (−8 ≤𝑚𝐽 ≤ 8 – blue to red), and the shaded regions refer to the intensities of the lattice
for a particular layer separation. (a, d) The scalar polarizability (𝛼𝑠 ) alone creates an
𝑚𝐽 -independent potential. (b, e) The vector polarizability (𝛼𝑣 ) lifts the degeneracy
between states, and acts as an effective magnetic field. At the maximum of the 𝜎+ (𝜎− )potential, the ground state is 𝑚𝐽 = 8 (-8). There is an exact crossing in the middle,
and small (stray) fields could cause spin flips there. (c, f) The tensor polarizability (𝛼𝑡 )lifts this degeneracy and detunes the spin flip process. (a-c) When using lattice beams
with degenerate frequencies (Δ+ = Δ− ) the crossings turn into avoided crossings,
resulting in spin flips. (d-f ) Raman-induced spin flips are suppressed by using different
frequencies for both lattices (Δ+ ̸= Δ− ). (g-j) Loading the bilayer lattice; during
the loading procedure the atoms continuously follow the (effective) magnetic field(black arrows). (g) Starting with the lattices in a fully interlaced configuration in
the presence of a strong transverse field, the atoms do not see a potential gradient,
as the transverse field dominates over the artificial field generated by the lattice. (h)As the transverse field is ramped down, the artificial field from the lattice becomesapparent. (i) By completely switching off the transverse field we complete the loading
process. (j) Finally, the layer spacing is reduced by changing the frequency of one of
the lattices.

113

An optical field possessing both 𝜎− and 𝜎+ polarization components not only
causes 𝑚𝐽 - dependent AC Stark shifts, but can also induce two-photon Raman couplings among different 𝑚𝐽 states, as shown in Figure 5-2 (a-c) and Figure 5-2 (d-f).
To preserve the spin polarization, a two-photon detuning |Δ+ − Δ− | of more than

300 MHz is introduced by offsetting the frequencies of the 𝜎− and 𝜎+ optical lattices.

Under such an optical configuration and zero magnetic field, the ground-state of thelight-atom coupled system consists of isolated pancake-shaped atomic clouds with
opposite spin orientations (𝑚𝐽 = −8 and 𝑚𝐽 = +8) locally trapped in the potential

minima of the 𝜎+ and 𝜎− lattices, hence forming a bilayer array with repulsive interlayer dipole-dipole interactions. We note that the bilayer array cannot be derived
from a lin-𝜗-lin polarization gradient such as used in Refs. [46, 49, 50]. In such an
optical configuration both polarization components have the same frequency, and so
the aforementioned Raman couplings would invariably lead to spin flips.
The ground-state of the bilayer is loaded using an adiabatic transfer method, as
depicted in Figure 5-2(g)-(j). We start by preparing the 𝑚𝐽 = −8 atoms in a magnetic

field with a transverse component 𝐵𝑥 = 200 mG and an axial component 𝐵𝑧 around
10 mG. We then ramp up the 𝜎+ and the 𝜎− optical lattices in interlaced configurationin 100 ms, and subsequently ramp down the transverse magnetic component 𝐵𝑥 in
15 ms to form a bilayer with head-to-head dipoles. By varying the final power of the
optical lattices and the axial magnetic field 𝐵𝑧 , we can offset the 𝜎+ and 𝜎− potential
minima. A balanced loading occurs when the energy offset Δ = 𝑈+ − 𝑈− + 𝐸𝑍 iszero. This is set by the AC Stark shifts in the layers 𝑈± , and the differential Zeeman

energy 𝐸𝑍 = 16𝑔𝐽 𝜇𝐵 𝐵𝑧 . However, it is crucial that the atoms stay in the ground statethroughout the experiment to prevent losses and heating due to dipolar relaxation. An
implication of this is the fictitious magnetic field |Blat | from the optical lattices must

dominate over the external magnetic field 𝐵𝑧 , otherwise one of the spin components
would end up being an excited state. This precludes the use of an external axial field
to guide the spins such as can be done in spin-dependent lattices using alkali’s [50].

114After loading a balanced bilayer array, we ramp up the power of the two lattice
beams such that the two layers share the same trap frequencies of (𝜔𝑥 , 𝜔𝑦 , 𝜔𝑧 ) =2𝜋 × (0.7, 0.7, 144) kHz. The strong axial confinement results in a harmonic oscillator√︀
length 𝑎𝐻𝑂 = ~/𝑚𝜔𝑧 of 21 nm. We then adjust the bilayer array from interlaced

configuration to different interlayer distances in 0.5 ms by changing the frequencyof the 𝜎− optical lattice. The interlayer distance 𝑙 is calibrated by Kapitza-Diracmeasurements (see section 5.3). At the end of each experimental sequence, we release the atoms from the bilayer array within 1 𝜇s and image them in time of flight.
With the small axial magnetic field 𝐵𝑧 serving as a guiding field, atoms in two layers
are projected onto the 𝑚𝐽 = −8 and 𝑚𝐽 = +8 Zeeman sublevels along the z direction, which are then imaged by our spin-resolved absorption imaging system (seesection 5.4). This imaging setup allows us to perform a spin-resolved measurementof the population and momentum distribution of both layers in a single experimental
iteration.

5.2

Controlling Separations in the Bilayer

To control the distance between two layers, we shift the frequency of the 𝜎− lattice,leading to a variable layer displacement due to the accumulated phase shift of the
optical lattice at the position of the atoms that is distance 𝐿 away from the retro
mirror.
With 𝛿𝑓 = 80 MHz tuning range of the 𝜎− lattice frequency and 1.9 m of retropath length, we are able to shift the 𝜎− lattice sites by a distance 𝐿 × 𝛿𝑓 /𝑓 = 375 nmwith respect to the 𝜎+ lattice sites, where 𝑓 is the frequency of the 741 nm laser.

115

zσ- polarizationxy

σ+ polarization

s
(II)

(I)

Figure 5-3: Two overlapping laser beams with opposite circular polarizations 𝜎+ and𝜎− are retro-reflected by mirror (I) to form two optical standing waves. The twostanding waves are shifted at the position of the atoms (II) by a distance 𝑠 controlled
by the frequency offset between the two laser beams. Dy atoms in this configuration
form an array of pancake-shaped bilayers of head-to-head dipoles with adjustableinterlayer distance 𝑠.

116

5.3

Calibrating Separations in the Bilayer

To verify this tuning capability, we perform a Kapitza-Dirac experiment [51] with a
Bose-Einstein condensate in the 𝑚𝐽 = −8 Zeeman state polarized along the transverse

direction 𝑥^ in a magnetic field of 2 G. After the preparation of the condensate, we
simultaneously pulse on the two lattices with the same intensity for 𝜏 = 5 𝜇s, suchthat most of the population is in the the first order of the Kapitza-Dirac diffraction
pattern when the lattices are overlapped.

Figure 5-4: A Kapitza-Dirac pulse is performed by simultaneously applying the vertical 𝜎𝐿 and 𝜎𝑅 lattice beams, represented in yellow, to dysprosium atoms in the𝑚𝐽 = −8 state, which are polarized along the horizontal axis by a magnetic field of
2 G.

Since the AC Stark shift experienced by the atoms polarized along the transverse direction is composed of 1/2 𝜋 component, 1/4 𝜎− component, and 1/4 𝜎+component in both 𝜎+ and 𝜎− lattices taking the atomic spin as the quantization
axis, the light-atom interaction Hamiltonian for the 𝑚𝐽 = −8 state can be expressedas the superposition of two phase-shifted sinusoidal potentials of the same depths
𝑉0 (𝑥, 𝑦)[sin2 (𝑘𝑧) + sin2 (𝑘𝑧 + 𝜑)]. Figure 5-5 shows the typical 1st-order Kapitza-Diracsignals in the short pulse limit 𝑉0 (𝑥, 𝑦)𝜏 /ℎ ≪ 1.
117

(I)

(III)

(II)

(IV)

Δ Δ

Figure 5-5: The Kapitza-Dirac signal as a function of the relative detuning.

The result presents an oscillatory behavior as we vary the detuning Δ2 . When
two optical lattices are in an interlaced configuration (where 𝜑 = 𝜋/2), the atoms
experience a constant potential 𝑉0 (𝑥, 𝑦), hence the first-order Kapitza-Dirac signal
vanishes. Whereas when two optical lattices are in an overlapped configuration where(𝜑 = 0), the amplitude of the sinusoidal potential is maximized, corresponding to thestrongest Kapitza-Dirac signal. The resulting oscillation period of the Kapitza-Diracsignal indicates our tuning capability of the interlayer separation to be one lattice
site, approximately 370 nm.
118

5.4

Spin-resolved Imaging

Our spin-resolved absorption imaging system operates in the weak saturation limitof the dysprosium 421 nm transition. It utilizes the large contrast of photon scattering rates of atoms in the stretched 𝑚𝐽 = ±8 Zeeman states for two opposite

circular polarizations of light. The resonant imaging light addressing the 421 nm
transition is linearly polarized along x, and propagates along the lattice axis z. Taking z as the quantization axis, the imaging light contains an equal amount of 𝜎−
and 𝜎+ polarization components. Due to the stark difference between the ClebschGordan coefficients for the |𝐽 = 8, 𝑚𝐽 = −8⟩ → |𝐽 ′ = 9, 𝑚′𝐽 = −9⟩ and the|𝐽 = 8, 𝑚𝐽 = −8⟩ → |𝐽 ′ = 9, 𝑚′𝐽 = −7⟩ electric dipole transitions, the 𝜎+ photons
are predominantly scattered by the atoms in the 𝜎+ layer, whereas the 𝜎− photons
are predominantly scattered by the atoms in the 𝜎− layer in the weak saturation
limit. The two polarization components are then spatially separated by a 1∘ angle via a quarter-wave plate and a Wollaston prism (see Figure 5-6), leading to twonearly-independent imaging channels for the 𝜎+ and the 𝜎− layers on the camera.
The duration of the imaging pulse is adjusted to reduce the optical pumping effect,
such that the crosstalk between the two imaging channels are negligible.

Figure 5-6: The spin-resolved absorption imaging scheme. The atoms are exposed
to linearly polarized light that is resonant with the 421 nm transition. Subsequently,
this is projected onto the circular basis using a quarter-wave plate and a Wollaston
prism. The latter creates a spatial separation between clouds which is recorded by a
CMOS camera (the lenses of the imaging system are not shown).
119Figure 5-7: A spin-resolved absorption image showing the atoms loaded into the 𝜎−
and 𝜎+ lattices.

5.5

Creating Spatially Isolated Layers

The oscillator length, 𝑎𝐻𝑂 , constrains the shortest interlayer distance we can achieve
without having contact interaction. To experimentally map out this contact-free
region, we measured the population loss of the bilayer dipolar gases at different interlayer distances.
The results shown in Figure 5-8 present a Gaussian-shaped loss rate feature: when
two layers are completely overlapping, the loss rate is maximized; as the separation
between the two layers exceeds the oscillator length, the loss rate exponentially diminishes with increasing interlayer distance.

120

a

b
σ+ layerσ- layer

Figure 5-8: (a) Evolution of the population in 𝜎+ layers at two different interlayer
distances 𝑠 = 185 nm and 0 nm. The loss is much faster when the layers are overlapped. Initial loss rates Γ are obtained from fits to the decay curves. (b) Fits of
the initial loss rates Γ to the interlayer distances 𝑠 according to Eq. (5.1) (solid lines)
provide a value of 𝜎𝑧 = 26.33(4) nm for the layer thickness.

121This behavior is well-described by a rate equation containing loss terms arisingfrom the intralayer and interlayer contact interactions2𝑙Γ = Γintra + Γinter 𝑒− 3 ( 𝜎𝑧 )

2

(5.1)

where Γ = 𝑁˙ tot /𝑁tot is the total loss rate of a layer, which contains an intralayer2

𝑙2

loss term Γintra and an interlayer loss term Γinter 𝑒− 3 ( 𝜎𝑧 ) . Since only the groundvibrational level is significantly populated along the axial direction of the bilayers,
the interlayer loss term, given by the overlap integral of the densities of the twolayers, possesses a Gaussian dependence on the interlayer distance 𝑙. Fitting the datawith Eq. (5.1), we obtain Γinter /Γintra ≈ 50, indicating the population loss to be 50

times faster when two layers are overlapped. This makes the loss measurement ahighly sensitive probe for the contact-free region. The obtained thickness of eachlayer 𝜎𝑧 = 26.33(4) nm ≈ 1.3 × aHO , and we regard the interlayer contact interaction

when the two layers are separated by more than 50 nm to be negligible due to theloss of overlap of wavefunctions.
With the ability to bring two layers of magnetic dipoles to as close as 50 nm without putting them in contact, we enhance the dipole-dipole interaction between twolayers by a factor of 1200 compared to magnetic dipoles trapped at a typical distancescale of 532 nm provided by optical lattices of infrared lasers at 1064 nm wavelength.
The peak interaction strength between dysprosium atoms with a magnetic momentof 10𝜇𝐵 trapped in two such layers is 𝑈dd /ℎ = 12 kHz. To demonstrate such a strong
interlayer interaction, we performed two out-of-equilibrium experiments presented in
the following sections.

122

5.6

Thermal Transfer Across Layers

Having shown and characterized co-trapping for atomic layers of opposite spin (i.e. ahead-to-head orientation) of Dy with miniscule separations of 50 nm, we now explore
the dipolar physics enabled by such a configuration.
In the first experiment, we heated the 𝜎+ layer via a parametric drive, and measured how the system relaxes back to thermal equilibrium. This was achieved bymodulating the depth of the 𝜎+ layer at twice the transverse trap frequency for
30 ms, followed by a 5 ms hold to ensure any residual breathing motion is damped
out. This results in an increase of the temperature of the 𝜎+ layer up to 8.5 𝜇K,
which is 3 𝜇K hotter than the 𝜎− layer. We then change the distance between the
two layers in 0.5 ms and measure the evolution of the temperatures of both layers.
The temperature evolution shows a strong dependence on the interlayer distance 𝑙(insets of Figure 5-9). Simulations show that the residual overlap between the two
clouds cannot explain that thermalization via 𝑠-wave collisions, and we find excellent
agreement with the expected collision rate from dipolar interaction. Using the Born
approximation with a dipolar potential we find that the collisional cross-section for
infinitely thin pancakes separated by a distance 𝑠 is:𝜎2𝐷 = 𝑎2𝑑𝑑 𝑘𝑖𝜋(𝑘𝑖 𝑠)3(5.2)

Here, 𝑎𝑑𝑑 is the dipolar length and 𝑘𝑖 the thermal momentum. For our temperatureand 𝑠 = 60 nm, the cross section is ≈ 0.5 nm. We fit the temperature difference
between two layers with an exponential function Δ𝑇 = Δ𝑇0 exp (−Γ𝑡) to obtain theinterlayer thermalization rate Γ. Data in Figure 5-9 shows that significant thermalization occurs when the interlayer distance is between 50 nm and 100 nm. This isa clear indication that it is the strong long-range dipole-dipole interaction between
the two layers which causes the system to thermalize, an effect that would otherwise
exponentially vanish as the interlayer distance extends outside the 50 nm range wherecontact interaction due to wavefunction overlap is absent.

123

Pure 2D
Pure 2D - limit
Quasi 2D100

10T ( K)

10 2
60
50
40
30
20
10
0

Thermalization Rate (s 1)

Scattering crosssection (nm)

a

102

8
6
0

50

Dipolars-Wave050100Interlayer Distance (nm)150

Figure 5-9: Thermalization rate between lattices as a function of interlayer separation.
Idnsets: temperature as measured in time of flight for both layers at small separation(63 𝜇m, left) and large separation (100 nm, right). Solid lines are exponential decay
curves from which the relaxation rate is extracted. Blue disks and red diamonds
correspond to the stationary and the driven layer, respectively.

124

5.7

Interlayer Transfer of a Dipole Oscillation in the
Harmonic Trap

In the second experiment, we excited a transverse dipole oscillation in the 𝜎+ layer
and observed how the excitation is transferred to the 𝜎− layer over time due tothe interlayer dipole-dipole interaction. After a balanced bilayer array was loaded,
we adjusted the interlayer distance to a designated value in 0.5 ms, and adiabatically displaced the 𝜎+ layer along the transverse direction 𝑦^ in 10 ms by imposing a
misaligned circularly-polarized dipole beam that is 458 MHz blue-detuned from thedysprosium 626nm transition. Because of the large anisotropy in polarizability, it
only affects atoms in the 𝜎+ lattice, and leaves atoms in the the 𝜎− one untouched.
By quenching the dipole beam off in 1 𝜇s, a dipole oscillation of the 𝜎+ layer was
excited at the transverse trap frequency of 700 Hz with an adjustable amplitude from0 to 8 𝜇m depending on the final power of the dipole beam. We then measured thein-trap velocity evolution of each layer from time-of-flight images.
Figure 5-10 shows the interlayer transfer of a dipole oscillation in the harmonic
trap from the 𝜎+ layer to the 𝜎− layer, at interlayer distances of 50 nm and 70 nm,
in terms of the measured time evolution of the velocity of each layer.

125aσ+ layerσ- layerb

Figure 5-10: (a) Center-of-mass oscillation of the 𝜎+ layer excited by suddenly switching off a displacement force. The 𝜎− layer oscillates due to dipolar coupling.
(b) Comparison of the spatial amplitudes of oscillations for the two layers. Amplitudes were determined during the first 2 ms. The vertical line is at the radial cloud
diameter.

126

The harmonic oscillation of the 𝜎+ layer shows some damping which is fasterfor larger amplitudes, probably caused by anharmonicities of the trapping potential.
The dependence of the amplitude 𝑥− of the 𝜎− layer on the amplitude 𝑥+ of the 𝜎+layer is shown in Figure 5-10(b). We first discuss a simple model for the momentum
transfer 𝑘− during the first pass (i.e. during half an oscillation period 𝑇 ). For small
amplitudes, the number of interlayer collisions during the time 𝑇 /2 is independent of
the amplitude, and we expect a constant fraction of the momentum to be transferred.
This fraction is identical to the collisional density, since in half a period each atom
moves on average by one cloud diameter. When the displacement of the 𝜎+ layer
becomes larger than the radial diameter of the cloud, the fraction of momentum
transferred is also given by the collisional density. However, the collisional crosssection (Equation 5.2) decreases since the center-of-mass momentum 𝑘+ is now larger
22),than the thermal momentum 𝑘𝑡ℎ . With a collision energy proportional to 1/(𝑘𝑡ℎ+𝑘+22
the momentum transfer is proportional to 𝑘+ /(𝑘𝑡ℎ+ 𝑘+) which has a maximum for𝑘+ = 𝑘𝑡ℎ , i.e. when layer 1 is displace by approximately one radial diameter 𝐷.
Expressed in terms of spatial amplitudes, the model predicts 𝑥− ∝ 𝑥+ /(𝐷2 + 𝑥2+ ), in
semi-quantitative agreement with observations (Figure 5-10(b)).

However, after the first half period, the 𝜎− layer oscillates at almost constant
amplitude, perfectly in phase with the 𝜎+ layer, instead of being constantly driven
by the 𝜎+ layer. Any simple model of coupled harmonic oscillators would lead to fulltransfer of momentum (after half a beat period). If the coupling were frictional innature a model would lead to irreversible transfer of half the momentum. Here weobserve a maximum momentum transfer of only 10%. In some cases, especially forlarge displacements of the 𝜎+ layer, we observed a small increase in the amplitude ofthe 𝜎− layer over two or three cycles, but in all of our observations, the friction force
diminished greatly after the clouds had passed once. This result needs further study.
Part of the explanation may be given by the fact that excitation of the kicked layer
is not only a center-of-mass motion, but also results in strong quadrupole motion, orby the presence of spatial correlations between the two layers. Since the maximum
dipolar energy between two atoms at 50 nm separation is 20 kHz or 1 𝜇K in temper127

ature units, we don’t expect strong correlation effects at our temperature of 5 𝜇K,
but future experiments at lower temperature should answer this question.

128

Chapter 6
Outlook
The techniques described in chapter 4 enable the creation of dense dysprosium clouds
consisting of multiple long-lived spin mixtures, thus opening new possibilities forquantum simulation experiments such as long-lived spin-orbit coupling schemes andthe creation of artificial gauge potentials.
With the ability to create dipolar bilayers at 50 nm separations, as discussed inchapter 5, we can ask what new phenomena we expect from such a system. Interestingly, the mean-field contribution to the dipolar interaction between two layers goes tozero as the system size goes to infinity, such that a mean-field approach only accounts
for edge effects. By performing mean-field calculations with trial wave functions determined by particle statistics, it is predicted that an out-of-phase oscillation will be
induced in a second layer if a center of mass motion is created in a first layer [52, 53].
Therefore, our observation of an in-phase oscillation in chapter 5, resulting in partfrom a frictional force, emphasizes the experimental and theoretical opportunities
awaiting this new regime of ultra-strong dipolar coupling.
These results open up many new directions for research. Regarding the bilayer,
lower temperatures should lead to strong correlations between the layers. Adding
transverse optical lattices to the layers could realize arrangements where the dipolarinteraction is either repulsive or attractive [54, 55, 56, 57]. It is also possible to projectseparate arbitrary potentials into the 𝜎+ and 𝜎− layers which would be a generalization of twisted bilayer potentials [58]. We are currently applying the super-resolution
129technique to optical tweezers to place two atoms at 50 nm scale separation. This willallow the study of superradiance and radiative shifts at separations much smaller thanthe optical wavelength, and the study of magnetic interactions and spin exchange between two isolated atoms, as done recently with polar molecules [59, 60, 61]. The
tweezer setup can be generalized to a linear array of atoms alternating in spin-up
and spin-down states. Moving the spin-up atoms back and forth would provide fullconnectivity along the chain, and realize a spin chain with strong magnetic coupling
between nearest neighbors. These ideas can be generalized to higher dimensions. Using higher laser power and tighter confinement, even 10 nm separations should bepossible. Furthermore, dipolar quantum gates may be realized in such tweezer arrays, where the strong dipolar interaction, combined with a high configurability ofcouplings, could provide a robust platform for quantum operations.
Lastly, the combination of strong dipolar interactions and a head-to-head configuration additionally shows great promise for the exploration of new quantum phases
in bilayer systems, such as pair supersolids and pair superfluids in bilayer systems[55], superfluid pairing in a fermionic bilayer [62], as well as pair superfluidity in
two-component systems [63] which can be mapped onto the bilayer, with the bilayer
providing the added benefit of an interaction that is controllable through the separation.

130

Appendix A
An Acousto-optic Modulator Driving
Circuit with Intensity Stabilization
Acousto-optic modulators (AOMs) are important devices that allow a lab to modify
both the frequency and intensity of an input laser through the absorption or emissionof phonons. AOMs require radio frequencies (RF) at powers of approximately 1W inorder for the internal transducer to create acoustic waves in the crystal through whichlaser light traverses. The frequency of the RF signal (10 MHz to over 1 GHz) sets
the frequency spacing between diffraction orders, while the Bragg condition and thepower of the RF signal determine the amount of light diffracted into different orders.
Due to their high diffraction efficiency, the +1 or −1 orders of the light diffracted froman AOM are typically used. A standard technique to obtain larger frequency shifts
is to double-pass an AOM, wherein a diffracted order is returned through the AOM
for a second pass to double the frequency shift. By varying the applied RF power,
AOMs can enable exquisite control of the laser intensity, such as performing ramps forevaporative cooling, rapidly toggling imaging beams on and off, creating sidebands
in the MOT beams through modulation, or stabilizing laser intensity. Stability inthe intensity of trapping lasers is of great utility for optical dipole traps and opticallattices, since intensity fluctuations result in heating of the atoms.
The AOM Driver Box I have designed consists of three separate printed circuitboards (PCBs) which provide a range of features that a cold atom experiment de131

mands of its AOMs while having several advantages over commercially available options. First, at approximately 1/4 the cost of AOM drivers without feedback control,
the price per box is significantly cheaper. Additionally, commercial amplifiers areoften bulky and radiate large amounts of heat into the lab through their heat sink.
In contrast, the heat load of the AOM Driver Box is minimal. When consideringthat a typical AMO lab requires ten or more of these amplifiers, the heat production
becomes a concern that can be eliminated by using the home-built option. Lastly, thedesign was created to be used in a compact box (EFP206A97), with a power supply
mounted directly to the PCB and direct integration with the other PCBs required
to drive an AOM. In the current market there are extremely limited options for devices combining RF generation, amplification, and intensity stabilization, with prices
more than 10x the cost of the AOM Driver Box. In terms of functionality, the AOM
Driver Box provides numerous inputs and outputs that can be absent in commercial
alternatives. For instance, the amplitude and frequency can be set either internally(via potentiometer) or with an externally applied voltage. An RF switch, controlled
again by either internal or external signals, can be used to turn off diffraction of theAOM. Furthermore, an output SMA carrying the error signal can be used to tune
the feedback loop, while a frequency monitor port provides an easily accessible, lowpower output to measure the RF frequency. With many electrical grounds sharedacross the lab, ground-loops can create myriad hard-to-diagnose issues across a variety of seemingly unconnected circuits. By incorporating instrumentation amplifiersin each AOM Driver Box, the pain of ground-loops is avoided. Additionally, the AOM
Driver Box utilizes two linear power supplies, thus avoiding the 60 Hz noise present
in switching power supplies.
The ultimate function of the PCBs is to produce an RF signal from a voltage
controlled oscillator (VCO), amplify the signal to ≈ 1 W, and provide the capabilityof stabilizing the intensity of the light diffracted by the AOM through the use of

feedback circuitry. The amplitude of the RF signal is adjusted by using an RF mixer
to mix the RF signal with a DC signal provided by either feedback or feedthrough.
The feedback bandwidth of the AOM Driver Box is roughly 80 kHz, with unity132

feedback gain for signals under 10 kHz. Due to delays from the speed of sound ina crystal, AOMs are fundamentally limited to hundreds of kHz feedback. For the
purposes of performing ramps on the scale of a millisecond, we have observed thetens of kHz feedback to be ample.
Figure A-1 shows a high-level schematic of the RF amplitude and frequency control
in the AOM Driver Box. Amplitude Setpoint, Frequency Tune and TTL may be
set internally or externally, as explained in section A.1. The three PCBs and theenclosure housing them, as well as representative examples of intensity stabilization,
are discussed in detail in the following sections.

133

Proportional Gain
Integral Gain
OffsetAmplitude
MeasureFeedback
Circuitry

Amplitude
Setpoint

Amplitude
Error

Feedback
SignalFeedback

FeedthroughSolid-state
SwitchFrequency
MonitorFrequency
Tune

VCO

RF Mixer

TTL

RF
SwitchRF Output
Signal

RF Amplifier
Board

Figure A-1: A high-level schematic of the RF amplitude and frequency control in the
AOM Driver Box.
134

A.1

The Faceplate

The Faceplate board serves as an interface between the user and the Driver board. To
minimize clutter from SMAs, a PCI-36 card-slot is used to transfer signals between
the Driver board and the Faceplate board. The Faceplate board is inserted directly
into the front panel of the enclosure (see section A.5), and provides the following
inputs, outputs, and controls.
SMA Inputs: External TTL, External Frequency Tune, External Amplitude
Setpoint, External Amplitude Measure.
SMA Outputs: Amplitude Error (= Setpoint - Measure + Offset), Frequency
Monitor, RF Output Signal.
Potentiometers: Proportional Gain, Integral Gain, Offset, Internal Frequency
Tune, Internal Amplitude Setpoint.
Switches:
TTL: (1) Internal On (2) Internal Off (3) ExternalFrequency Tune: (1) Internal (2) External
Amplitude Setpoint: (1) Internal (2) External
RF Power Control: (1) Feedthrough (2) Feedback
In the above list each switch is separated by a comma, and the position of eachswitch is denoted by the number inside of the parentheses (e.g., (1) represents thefirst position of the switch).
Lastly, there is an LED which outputs green when the Amplitude Error is small,approximately <100 mV, and outputs red otherwise. A successfully locked box should
therefore continuously output green.

135

Figure A-2: A schematic diagram of the Faceplate board showing the PCI-36 cardslot.

Figure A-3: A schematic diagram of the Faceplate board showing the SMA inputs
and outputs.

136

Figure A-4: A schematic diagram of the Faceplate board showing the switches.

Figure A-5: A schematic diagram of the Faceplate board showing the potentiometers.

8/3/2022 9:24 PM C:\Users\Cantara\Documents\Studies\MIT\Eagle\projects\AOMDriverBox\AOMDriverFaceplateMkV.sch (Sheet: 3/5)

FigureA-6: A schematic diagram of the Faceplate board showing a solid-state switch,
8/3/2022 9:24
PM C:\Users\Cantara\Documents\Studies\MIT\Eagle\projects\AOMDriverBox\AOMDriverFaceplateMkV.sch(Sheet: 4/5)
the output of which is then sent to one of two pins of a bi-color LED.
137

Figure A-7: A schematic diagram of the Faceplate board showing the PCB layout.

138

The above figures show the electrical connections of the Faceplate board. For
clarity, the schematic is divided into multiple figures, with electrical connections acrossthe figures determined by matching names.
External control can be provided by one or more external voltage sources via the
SMAs shown in Figure A-3. The switches in Figure A-4 allow the user to switch
between internal and external control for the amplitude and frequency setpoints, aswell as between feedback or feedthrough control of the light intensity. Additionally,
the RF output of the board can be toggled on and off by a logic voltage (TTL)
provided externally, or by internal means, dependening on the position of the switch
in Figure A-4.
The potentiometers in Figure A-5 allow the user to internally tune the frequency
and amplitude of the RF directly, as well as the ability to tune the feedback loop for
intensity stabilization.
Figure A-6 shows a solid-state switch (ADG1419BRMZ) with a comparator signal
derived from the Driver board as input. Depending on whether the comparator signal
is logic low or high, the drain terminal of the switch is routed to one of the two anodesof the LED, which produces either red or green emission by the LED.
Figure A-7 shows a schematic diagram of the PCB layout for the Faceplate board.
The proportional, integral, and offset potentiometers (CT6EP104) are recessed insideof the housing and are accessible with a screwdriver, thus minimizing the risk of accidental adjustments. Conversely, the internal frequency and amplitude setpoints aretunable with potentiometers (P260T-D1BS3CB100K) that protrude from the housing.

A.2

The Driver

The Driver board is a circuit that, in conjunction with the tightly integrated Faceplateboard, produces a small RF signal (< 10 dBm) that is later amplified by the RF
Amplifier board. All user interactions with the Driver board are performed throughthe Faceplate board.
139

Figure A-8: A schematic diagram of the Driver board showing the PCI-36 card-edgeof the PCB, which slots into the PCI-36 card-slot of the Faceplate board.

Figure A-9: A schematic diagram of the Driver board showing two voltage regulators.

140

Figure A-10: A schematic diagram of the Driver board showing an instrumentation
amplifier providing ground isolation for the External Amplitude Setpoint input, as
well as a solid-state switch for selecting external or internal amplitude setpoint control.

Figure A-11: A schematic diagram of the Driver board showing a dual-package opamp that 1) sums the offset voltage and the amplitude setpoint voltage and 2) buffers
the offset voltage.

141

Figure A-12: A schematic diagram of the Driver board showing an instrumentation
amplifier providing ground isolation for the External Amplitude Measure input.

Figure A-13: A schematic diagram of the Driver board showing a dual-package opamp that 1) provides proportional gain and 2) provides integral gain. A solid-stateswitch is placed in the feedback of the integrator to prevent wind-up (‘railing’) in the
feedback capacitor.

142

Figure A-14: A schematic diagram of the Driver board showing a dual-package opamp that 1) sums the proportional and integral gain signals and 2) generates theerror signal via summing.

Figure A-15: A schematic diagram of the Driver board showing a solid-state switch
for selecting feedback or feedthrough control of the amplitude setpoint. Two Zener
diodes are arranged to limit the feedback output to positive voltages. Additionally,a dual-package op-amp 1) buffers the signal applied to the IF port of the RF mixer
and 2) provides 14x gain to the error signal, which is then sent to an absolute value
circuit implemented with an instrumentation amplifier. The output of the absolutevalue circuit is then sent to the Faceplate board to act as logic for the solid-stateswitch that powers the bi-color LED.

143

Figure A-16: A schematic diagram of the Driver board showing an instrumentation
amplifier providing ground isolation for the External Frequency Tune input, as well
as a solid-state switch for selecting external or internal frequency control.

Figure A-17: A schematic diagram of the Driver board showing a fast variant of an
instrumentation amplifier providing ground isolation for the External TTL input, as
well as a solid-state switch for selecting external or internal TTL control.

Figure A-18: A schematic diagram of the Driver board showing an RF coupler used
to monitor the RF frequency, an RF mixer for adjusting the power of the RF signal,
and an RF Switch for rapidly switching the RF output on and off.
144

145

Figure A-19: A schematic diagram of the Driver board showing the PCB layout.
The above figures show the electrical connections of the Driver board. For clarity,
the schematic is divided into multiple figures, with electrical connections across thefigures determined by matching names.
In order to prevent the formation of ground loops, instrumentation amplifiers(INA114BU and AD8251ARMZ) are used to provide ground isolation. Figure A-9,
Figure A-12, Figure A-16 show the INA114BU providing such ground isolation for theexternal amplitude and frequency setpoints and for the measured voltage provided
by the photodiode. The AD8251ARMZ in Figure A-18 was chosen for its large slew
rate of 20 V/𝜇s, minimizing the switch on/off time of the RF output.
The inputs to each instrumentation amplifier are connected to the board groundby a 20 kΩ resistor, preventing the common-mode of the inputs from floating beyond
the range allowed by the instrumentation amplifiers. If it is preferred to directly use
floating (no connections to ground) voltage sources for external control, the inputs ofeach INA114BU have a jumper (PRPC002SAAN-RC) to connect each input groundto board ground, as well as an available connection for a 0 Ω resistor to allow each
input signal to bypass the INA114BU and connect to its output. In such use, theINA114BU should not be soldered to the Driver board.
In feed-through mode, a voltage is sent directly to the RF mixer with no consideration of the actual intensity of light diffracted by the AOM. By using feedback we gain
the ability to stabilize the light level and to have the light intensity closely track the
input waveforms, such as fast sinusoidal modulations or exponential ramps, withoutworrying about the nonlinear response of the RF mixer. The corner frequency of the
feedback loop is around 10 kHz.
At high RF power (> 1 W) the AOM can enter the overdriven regime, wherein
the diffraction efficiency decreases with increasing RF power. The feedback circuitry
is configured to work only in the regime where increasing RF power leads to increasing diffraction efficiency. Accordingly, the maximum amplitude setpoint is limited
by an adjustable voltage divider formed with a fixed 10 kΩ resistor and a 100 kΩ
potentiometer (CT6EP104), as shown in Figure A-15. Furthermore, the voltage sent146

to the IF port is limited to only positive values by the arrangement of Zener diodes.
In order to monitor the RF frequency, an RF directional coupler (ADC-10-4+)
picks off a small amount of the signal which is then sent to the Faceplate board via
an SMA, as shown in Figure A-18. A voltage provided from feedback or feedthroughcontrol is sent to the IF port of the RF mixer (ADEX-10H), thus modifying theamplitude of the RF signal provided from the VCO. Lastly, an RF Switch (KSWH20-1+) is used to rapidly turn on or off the RF signal before it is sent to the RF
Amplifier board.
Operationally, feedback is straightforward: a photodiode measures a small sampleof the light, typically picked off from the beam path with a beam sampler, and sends
its signal to the External Amplitude Measure input of the Faceplate board. The
Driver board then compares this measured signal with the amplitude setpoint, which
can be set internally via a potentiometer or externally with a voltage applied toExternal Amplitude Setpoint input, and then sends an appropriate voltage to the RF
mixer in order for the measured signal to match the amplitude setpoint. An offset
voltage can be adjusted internally using a potentiometer, shown in Figure A-5, and issummed with the amplitude septoint such that the summed signal is initially matchedwith the measured signal.
It is easiest to configure the feedback parameters using a wonderful and rather
inexpensive device, the Analog Discovery 2. It has the ability to generate manydifferent waveforms, read two different oscilloscope channels, and serve as a network
analyzer to generate Bode plots, all through interfacing with a computer.
In order for feedback to work correctly, follow the procedure below:
1. Using internal amplitude setpoint control and feedthrough control, turn theInternal Amplitude Setpoint potentiometer to increasing values until you noticeno change in the diffracted light level.
2. On an oscilloscope, which could be the Analog Discovery, observe the photodiode signal and adjust the photodiode gain until you have the largest signal thatis under 5V.
147

3. Use a coaxial T adapter to also send the photodiode signal into the External
Amplitude Measure input of the Faceplate board.
4. Turn the proportional and integral gains fully counter-clockwise. This minimizes
proportional gain and maximizes the DC gain of the integrator.
5. Use the waveform generator of the Analog Discovery to generate an External
Amplitude Setpoint, and put this signal on the oscilloscope with the photodiodesignal. Make sure the switch is set to External Amplitude Setpoint.
6. Switch to feedback control and adjust the offset potentiometer until the measureand setpoint signals are well overlapped on the oscilloscope.
7. Now use the waveform generator to make a 1 kHz sine wave that spans theentire dynamic range of the photodiode values. Ensure that the photodiodehas a lens tube to reduce ambient light pickup, thereby allowing for the lowest
possible diffraction amplitudes to be feedback stabilized.
8. Increase proportional gain. You should see the measured signal more closely
resemble the setpoint and then get worse as you turn the potentiometer further.
Turn the potentiometer back a bit at this point until there are no oscillations.
9. Now adjust the integral gain until it most closely matches the setpoint.
10. You will probably have to iterate between steps 8 and 9 a few times while
adjusting the offset small amounts until the optimal stability is found.
It is recommended to test several different setpoint signal frequencies and amplitudes and make sure the AOM Driver Box is properly locked at all settings. A
variety of plots generated from such tests are shown in section A.3 and demonstrate
the stability of the lock over the full dynamic range of the AOM Driver.

148

A.3

The Feedback in Action

Below are experimentally measured oscilloscope traces and Bode plots derived froman AOM driven by the AOM Driver Box. In the following figures, the amplitude
setpoint is externally driven using a voltage signal with a 500 mV offset, 300 mV
peak-to-peak amplitude, and 1 kHz frequency (oscilloscope traces) or 50 Hz - 1 MHz
frequencies (Bode plots).
Figure A-20 and Figure A-21 show the AOM Driver Box in feedthrough (openloop) operation. Figure A-20 directly shows the poor matching between the photodiode signal and the amplitude setpoint, while Figure A-21 shows the underlying
non-unity gain, in the form of a Bode plot, over a wide range of frequencies.
Figure A-22 and Figure A-23 show the AOM Driver Box in feedback (closed-loop)
operation. Figure A-22 directly shows the strong matching between the photodiodesignal and the amplitude setpoint, while Figure A-23 shows the underlying unity gain,in the form of a Bode plot, over a wide range of frequencies. Optimal performance of
feedback control can be achieved for frequencies less than 10 kHz, due to the roll off
from unity gain at higher frequencies that is shown in the upper plot of Figure A-23.

149

Figure A-20: The voltage from a photodiode (cyan) and the amplitude setpoint voltage (yellow), as a function of time, in feedthrough operation.

Figure A-21: A Bode plot from a photodiode (cyan) compared to unity gain (yellow),in feedthrough operation.

150

Figure A-22: The voltage from a photodiode (cyan) and the amplitude setpoint voltage (yellow), as a function of time, in feedback operation.

Figure A-23: A Bode plot from a photodiode (cyan) compared to unity gain (yellow),in feedback operation.

151

A.4

The RF Amplifier

The RF Amplifier board is a circuit that takes a small RF input signal (10 MHz - 1GHz), such as that produced by a VCO, and amplifies it to a maximum of approximately 32 dBm. Recall that dBm is an absolute unit of power, where 0 dBm = 1
mW and 30 dBm = 1 W.

Figure A-24: A schematic diagram of the RF Amplifier board showing the two PCB
mounted power supplies and a voltage regulator.

Figure A-25: A schematic diagram of the RF Amplifier board showing the RF amplification.

152

Figure A-26: A schematic diagram of the RF Amplifier board showing the PCB
layout.

153

The above figures show the electrical connections of the RF Amplifier board.
For clarity, the schematic is divided into multiple figures, with electrical connections
across the figures determined by matching names.
In Figure A-24, the mounted Acopian power supplies require AC wall current tobe provided through the screw terminal labeled AC. The screw terminal labeled 10
V is used to power a fan that cools the amplifier chips. The third screw terminal,
labeled ±15 V, is sent to the Driver board. A voltage regulator, used for stability,

minimally steps the 10 V input down to 9.3 V.

In Figure A-25, the RF enters through an SMA input and then passes througha medium-gain amplifier (PHA-13HLN+), after which it is split into two paths by
an RF power splitter/combiner (ADP-2-10+). Each of these paths is again amplified
by the PHA-13HLN+, and the two paths are merged into one via the ADP-2-10+.
Limitations in the amount of power each amplifier can provide, combined with the factthat each amplifier has the largest gain for lower power inputs, made this seemingly
convoluted approach optimal. The signal is then sent off the board to the AOM via
an SMA.
Figure A-26 shows the PCB layout of the RF Amplifier board. Numerous vias
surrounding the RF transmission line are employed as a via fence in order to mitigate parasitic coupling of the transmission line to other signal lines. Where feasible,
straight and rounded transmission lines are used and right angle turns avoided, suchthat impedance mismatch within the transmission line is minimized.

A.5

The Housing

A 4U subrack (CMA15-20/90) is used to hold up to six AOM Driver modules, each ofwhich are housed in compact metal enclosures (EFP206A97). Both the 4U subrackand the enclosure are provided by Vector Electronics. The EFP206A97 has several
slots running the length of the enclosure to allow for PCBs to be conveniently slotted
into place. The standard front and back panels provided with the EFP206A97 aresubstituted with custom-built panels. The custom-built front panel is designed such154

that the SMAs, potentiometers, switches, and LED of the Faceplate board are accessible to the user. Horizontal vents are also incorporated to provide airflow into theenclosure. The custom-built back panel is designed to hold an AC power module totransmit wall power into the enclosure, as well as numerous smaller holes for airflow
out of the enclosure. In order to minimize heat transfer from the RF Amplifier board
to the control electronics of the Faceplate and Driver boards, a partition panel creates
two compartments inside of the EFP206A97. The two rectangular protrusions at the
top and bottom of the partition allow the partition to be slotted into the PCB slots ofthe enclosure. The partition has a square cutout near the bottom for the RF outputof the Driver board to be transmitted to the RF Amplifier board via SMA. The cutout
on the upper right side allows for smaller electrical cables to be connected between
the Driver board and the RF Amplifier board, and for the amplified RF signal to
be sent to the front panel via SMA. Lastly, a large circular cutout with four smallerholes allows for a cooling fan to be attached, preventing the RF Amplifier board fromoverheating while directing warm air out the back panel.
The front panel, partition, and back panel were machined using the water jet
cutter at the MIT Hobby Shop and are shown in Figure A-27. The front panel labels
were produced using black laser marking spray, which is sprayed to cover the front
panel entirely and reacts with laser light from a laser cutter to produce the desired
markings. After laser cutting, the portions of the spray that do not interact withthe laser can be washed off with water. To prevent the SMA connections mounted
to the Faceplate board from contacting the metal of the front panel, a 3D printed
insert was created to cover the circular cutouts through which the SMAs protrude(see Figure A-28). The fully assembled AOM Driver Box is shown in Figure A-29.

155

Figure A-27: The front panel, partition panel, and back panel.

Figure A-28: 3D printed insert for electrical insulation of the SMAs.
156

Figure A-29: An assembled AOM Driver Box.

157

Enhancing the capture velocity of a Dy magneto-optical trap with two-stage slowing
William Lunden, Li Du, Michael Cantara, Pierre Barral, Alan O. Jamison, and Wolfgang Ketterle
Research Laboratory of Electronics, MIT-Harvard Center for Ultracold Atoms, Department of Physics,
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA(Dated: March 15, 2022)
Magneto-optical traps (MOTs) based on the 626 nm, 136 kHz-wide intercombination line of Dy,
which has an attractively low Doppler temperature of 3.3 µK, have been implemented in a growing
number of experiments over the last several years. A challenge in loading these MOTs comes fromtheir low capture velocities. Slowed atomic beams can spread out significantly during free-flightfrom the Zeeman slower to the MOT position, reducing the fraction of the beam captured by the
MOT. Here we apply, for the first time in a Dy experiment, a scheme for enhancing the loading rateof the MOT wherein atoms are Zeeman-slowed to a final velocity larger than the MOT’s capture
velocity, and then undergo a final stage of slowing by a pair of near-detuned beams addressing the421 nm transition directly in front of the MOT. By reducing the free-flight time of the Zeemanslowed atomic beam, we greatly enhance the slowed flux delivered to the MOT, leading to morethan an order of magnitude enhancement in the final MOT population.
I.

INTRODUCTION

Dysprosium, which possesses the largest magnetic moment (µ ≈ 10µB ) of any atomic species, has grown in
popularity in the ultracold quantum gas community overthe last decade [1–11] The large magnetic moment, as
well as several other useful properties, arise from its[Xe]4f 10 6s2 electronic configuration. The two 6s electrons give rise to a helium-like excitation spectrum, including a strong transition at 421 nm and a weak, intercombination transition at 626 nm. The unfilled 4f shell
gives rise to narrow clock-like transitions. It also leads
to spin-orbit coupling in the ground state, which is useful for many quantum simulations, including simulating
gauge fields [11, 12].
The narrow linewidth of the 626 nm transition in Dy
corresponds to a low Doppler temperature of 3.3 µK,
making it an attractive option for magneto-optical trapping. The downside of using a narrow transition is thatthe capture velocity of the MOT is lower than for abroader transition. Slowing an atomic beam to withina low capture velocity can lead to a situation where theslowed beam transversely spreads out so much that manyslowed atoms miss the MOT. Cooling the transverse degrees of freedom of the atomic beam and increasing thecapture velocity of the MOT by frequency-dithering the
MOT light are two measures which are typically employed to mitigate this limitation [2–6, 13], but their effectiveness can be limited.
In the present work, we add a new approach which
we refer to as “angled slowing,” which applies a second
stage of slowing to the atomic beam with a pair of lowpower beams that intersect directly in front of the MOT.
This allows us to choose a sufficiently large final velocity
for the first stage of slowing (i.e., Zeeman slowing) that
atoms do not spread out appreciably before reaching the
MOT. This approach was introduced in an Yb experiment, where it gave a small enhancement to the MOT
loading rate [14]. In our experiment, angled slowing enhances the MOT population by more than a factor of 20.

Compared to other methods which have been employed
to increase the capture velocity of narrow-line MOTs —such as the two-stage MOT [1] and the core-shell MOT[15]— the angled slowing approach requires fewer beams
and less laser power.
In section II we briefly describe the aspects of our experiment that are similar to those previously reported byother experiments. In section III, the idea behind angled
slowing and how it is particularly applicable to experiments with narrow-line MOTs is discussed. In section
IV, we describe how we optimized the performance of angled slowing with respect to beam pointing, laser power,
and frequency. In section V, the compression and detection sequence that follows the loading of our MOT is
described, and the temperature and phase space densityof the compressed MOT are reported.

II.

EXPERIMENTAL SETUP

Due to the recent explosion in popularity of dysprosium, several groups have developed similar cooling andtrapping protocols in parallel [2–6, 13]. Here we briefly
describe our approach, and give references to more detailed explanations of similar systems.
Our atomic beam of Dy is generated by a commercial
molecular beam epitaxy oven [16] heated to 1250◦ C. The
dysprosium vapor is collimated into an atomic beam by a
7 mm diameter nozzle, which is 90 mm from the openingof the oven, followed by a 10 cm long, 7 mm diameter differential pumping tube that starts 19 cm from the nozzle.
We use 421 nm laser light for Zeeman slowing, transverse cooling, and absorption imaging. This light isgenerated using an M-Squared Ti:Sapphire laser and
ECD-X fixed-frequency second-harmonic generation cavity (1.6 W total output), as well as two injection-locked
laser diodes (90 mW total output each). The frequencyof the Ti:Sapphire laser is stabilized by measuring the frequency of the doubled light with a HighFinesse WLM-7
wavemeter, and feeding back on a piezo-actuated mirror

2

FIG. 1. Geometry of our Zeeman slowing, MOT, and angled slowing beams. The vertical MOT beam pair is not shown in thistop view of the machine. The angled slowing beams enter through the same viewports as the MOT beams, and are aligned
such that they intersect the atomic beam without hitting the MOT (indicated by the dark circle in the center of the chamber).

in the laser cavity using an Arduino Due. We periodically adjust for drifts in the calibration of the wavemeter(which are typically a few MHz per day) by checking theresonance frequency of the 421 nm transition via absorption imaging. A Toptica TA-SHG system (700 mW total
output) generates the 626 nm light for the MOT. For
frequency stabilization of this laser, we shift the light by
about +1 GHz and employ a modulation transfer spectroscopy scheme to lock the laser frequency to a transition in a room-temperature iodine cell.
In the present work we slow the bosonic isotope 162 Dy,
which has 25.5% natural abundance [17]. Our Zeeman
slowing light consists of 300mW of light addressing the421 nm transition (Γ421 = 32.2 MHz), which comes toa focus at the position of the oven. Light enters thevacuum chamber with a beam diameter of about 2 cm,
bouncing off of a 45-degree in-vacuum mirror as shown in
Figure 1. This scheme was implemented so the entrance
window for the slowing light does not get coated by the
Dy atomic beam.
To minimize the effect of the Zeeman slower light on
atoms trapped in our MOT, we use an increasing-field
Zeeman slower design. This allows us to employ a larger
detuning in our slowing beam, which reduces the losses

due to scattering in the MOT. A counter-wound segment of coils at the end of the slower cancels the fringing
magnetic field from the slowing coils at the position of
the MOT. We use light detuned about 1.1 GHz fromthe zero-velocity transition, which resonantly addresses
atoms moving at 480 m/s (close to the most probable velocity of the atoms emitted from the oven). We have anadditional, uniformly wound bias coil running the lengthof the Zeeman slower, which creates a constant offset
magnetic field inside the slowing region. This allows adjustment of the effective detuning of the Zeeman slower
beam by up to several hundred MHz without needing toemploy an accousto-optic modulator (AOM).
Our MOT is formed by three retroreflected 626 nm
beams. Each beam has a diameter of 2.3 cm and a total power of 42 mW (±5%), corresponding to a (peak)saturation parameter of s ≈ 280. The quadrupole
field’s gradient along the strong direction is approximately 2.5 G/cm. To improve the capture velocity of the
MOT we dither the frequency of the MOT light using adouble-passed AOM. The dithering occurs at a frequencyof 120 kHz and broadens the laser linewidth to 2.6 MHz(30Γ626 ). Three pairs of rectangular coils in Helmholtz
configuration allow us to cancel background magnetic

3fields, and will also allow us to employ feedback- andfeedforward-based magnetic field stabilization schemes
during future experiments.
III.

ANGLED SLOWING

Atoms that have been slowed by a Zeeman slower musttravel some nonzero distance at their final, slowed velocity from the end of the Zeeman slower to the position of
the MOT. During this period of free flight, the transverse
velocity distribution of the beam causes the atomic beam
to spread out. If the free flight time is sufficiently long,then the atoms can spread out far enough that they are
not captured by the MOT.
While this is not typically a limiting factor in experiments, the combination of an increasing-field slower andthe narrow linewidth of the MOT transition creates a situation in which the transverse spread plays a significant
role. To clarify this point, we compare Dy to the morecommon alkali MOTs.
The capture velocity of a MOT can be estimatedby calculating the largest velocity that can possibly beslowed to a stop within the profile of the MOT beams.
Assuming that the atoms scatter photons at the maximum possible rate Γ2 across an entire beam diameter D,
an expression for the capture velocity is given byr~kΓ
vcap = 2D(1)2m
where m is the atomic mass and k = 2π
λ is the wavenumber of the MOT light.
The spatial spread σ of the atomic beam can be estimated as
vtransσ ≈ 2d(2)
vlongwhere d is the free-flight distance, vtrans is the RMS transverse speed, and vlong is the average longitudinal speed.
Collimation of the atomic beam by one or more apertures typically leads to a transverse velocity distribution
with an RMS speed around 1% of the average (unslowed)
longitudinal velocity [18].
Let us compare the case of a 87 Rb MOT to a 162 Dy
MOT, taking typical values of D = 2 cm for the MOT
beam diameters. For Rb, λ = 780 nm, m = 87 amu, andΓ = 2π × 6 MHz. This corresponds to a capture velocityof 67 m/s. A typical initial most-probable velocity foratoms effusing from a Rb oven is about 330 m/s, andso 3.3 m/s is a reasonable estimate of the RMS transverse speed of the atoms. If we consider an atomic beamslowed to the capture velocity value and an example freeflight distance of 10 cm we can estimate the spread of theatomic beam to be
σRb ≈ 1 cm
which is smaller than the size of the MOT beams.

(3)

For a Dy MOT, λ = 626 , m162 amu, and Γ626 =
2π×136 kHz. This gives a capture velocity of 8 m/s. The
most probable velocity of the Dy atoms effusing from ouroven is about 480 m/s, so 4.8 m/s is a reasonable estimateof the average transverse speed. For a free-flight distanceof 10 cm, we estimate the spread of the atomic beam tobeσDy ≈ 12 cm(4)which is much larger than the size of the MOT beams.
We thus see that the narrow linewidth of the 626 nm
transition already lead to a significantly larger spreadingof the atomic beam than in a typical alkali MOT.
Employing an increasing-field slower, while effective
in reducing scattering losses in the MOT due to thelarger Zeeman slower laser detuning, further exacerbates
the transverse spreading problem. One reason is thatthe larger detuning reduces the amount of off-resonantslowing that occurs during the free-flight distance. The
more off-resonant slowing that occurs during the freeflight, the larger the initial exit velocity from the Zeeman slowing region can be. We can estimate the typical effect of off-resonant slowing in a Dy experiment:a typical Zeeman slower beam detuning in a spin-flip
slower is around −18Γ421 , with (resonant) saturation parameters of s0 ≈ 1 [2]. If we assume that the slowed
atoms scatter at a (detuned) saturation parameter ofs0−4
s = 1+4∆
over a 10 cm free-flight dis2 /Γ2 ≈ 7.7 × 10
tance, then we can estimate that atoms with exit velocities as high as 13 m/s will be decelerated to within thecapture velocity of the MOT.
A second reason is the increased free-flight distancedue to the need for field-cancelling coils near the MOT.
In an increasing-field Zeeman slower, the largest numbers of windings are closest to the MOT. As a result, itis necessary to compensate for the large fringing fields
with an oppositely-wound compensation coil, so that the
total residual magnetic field and field curvature at theposition of the MOT is close to zero. Slowed atoms must
thus travel an extra distance of several cm compared tothe travel distance in spin-flip Zeeman slowers. In our
experiment, the free-flight distance is 16 cm.
The purpose of the angled slowing scheme is to reduce
the free-flight time by allowing atoms to exit the Zeeman slower at velocities well above the MOT’s capture
velocity. A few cm before the MOT, two beams with
red detuning on the order of Γ intersect the atomic beam
to provide a net longitudinal slowing force, slowing theatomic beam to within the capture velocity of the MOT.
The transverse components of the two beams’ scattering
forces are oppositely oriented and thus cancel. In effect,
the addition of the angled slowing beams increases thecapture velocity of the MOT.
The advantage of using a pair of angled beams over a
single beam colinear with the main Zeeman slower light,
or adding a near-resonant sideband to the Zeeman slower,
is that scattering losses in the MOT are avoided. Angled
slowing also requires fewer beams and less laser power

4

FIG. 2. Population of the MOT as a function of loading time.
The blue curve (open circles) shows the population when bothtransverse cooling and angled slowing with the optimal parameters of δ = −50 MHz and 7 mW per beam are employed.
The orange curve (filled squares) shows the population whenboth the transverse cooling and angled slowing beams wereturned off, multiplied by a factor of ten for visual clarity.

than the recently reported core-shell MOTs for alkalineearth-like atoms [15]. The setup for the angled slowing
beams is depicted in Figure 1.
Without employing angled slowing, optimization of our
Zeeman slowing parameters led to a MOT population of
about 107 atoms. We observe more than a factor of 20
gain in the final population of our MOT when using angled slowing. As described in the next section, we foundoptimal angled slowing performance with only 7 mW per
beam, and a detuning of −50 MHz (−1.6Γ421 ). The
beam diameters are about 5 mm, putting us far below
the saturation regime (Isat = 56 mW/cm2 ). Figure 2
shows the population with and without angled slowing
as a function of MOT loading time.

IV.

OPTIMIZATION OF ANGLED SLOWINGWe determined the optimal alignment of our angled
slowing beams by maximizing the steady-state MOT
population as measured by the integrated 626 nm fluorescence scattered by the MOT. While σ − light is used topump and cycle atoms that are being slowed by our Zeeman slower, the magnetic field magnitude and direction
at the position where the angled slowing beams intersect
the atomic beam are not easily known, so we varied thepolarization of the angled slowing beams to maximize the
MOT population after the pointing had been optimized.
The angled slowing light is prepared by frequency shifting light from our 421 nm master laser with a 500 MHz
AOM in a double-pass configuration, and then splitting

the shifted light into two separate fibers. With more than
200 mW of input power to this frequency shifting setup,thermal lensing in the AOM causes sensitively powerdependent variations in the spatial mode of the beams
reaching the fibers, greatly reducing the fiber coupling
efficiency. To avoid thermal lensing (and allow for more
controlled variation of the angled slowing power via the
RF power), we keep the power going to this AOM low,
resulting in a maximum power of about 10 mW per beamin our angled slowing light.
Given this power constraint, we looked for an optimal
combination of power and detuning for the angled slowing
beams. Figure 3 shows the population after a fixed loadtime and fixed compression sequence (see the following
section) as a function of both detuning (always red) andpower per beam. The uncertainty in our beam power was
at most ±15%, and the uncertainty in our frequency was
about ±2 MHz, with the latter uncertainty arising fromdrifts in our wavemeter.
The general trends are explained by a simple physicalpicture: At small detunings, a small amount of power
kicks some of the slowed flux to below the capture velocity of the MOT, but increasing power causes significant
additional scattering in the nearly-zero-velocity atoms
and causes them to turn around. At intermediate detunings, more flux is kicked out of the broad, slowed distribution to velocities below the capture velocity. Eventually,
with enough power, off-resonant slowing begins to turnthe atoms around again. At large detunings, the majority of the slowed flux is only addressed off-resonantly by
the angled slowing beams. Eventually, atoms will also be
turned around off-resonantly and thus there should be an
optimal power for any given detuning.
At each detuning, we scanned the constant offset fieldin the slower, which is equivalent to scanning the Zeeman
slower laser frequency and hence the final velocity. We
found that the optimal bias field was the same for all
detunings to within the step size we explored (steps of
1 A ≈ Γ421
of effective Zeeman slower detuning). We
2
also found that the same bias field was optimal whenloading a MOT without angled slowing. Together, theseobservations suggest that the final velocity distributionof the Zeeman-slowed atoms is broad compared to Γ421
If the final velocity distribution were narrow, we would
expect the optimum to vary with the choice of angled
slowing detuning.

V.

COMPRESSION AND DETECTION

We load about 3 × 108 atoms in 2 seconds with our
optimized angled slowing parameters. To prepare thecaptured atoms to be loaded into an optical dipole trap
(ODT) for evaporation, we compress the cloud over 50ms
and let the compressed cloud equilibrate for at least
300 ms. Compression consists of switching off the dithering of the MOT light frequency, and ramping the frequency from the initial detuning to within about a few

5mean speed of the cloud and hence the temperature. We
observe faster expansion along the vertical direction thanalong the horizontal direction, corresponding to a “vertical temperature” of 6 µK and a “transverse temperature”
of 13 µK.
To obtain the optimal phase space density, nλ3T , wevaried the MOT frequency, detuning, and gradient during
the compression sequence. We used the size in large TOF(20−25 ms) as a proxy for velocity (and therefore temperature), which in combination with the measured number
allowed for single-shot estimation of the phase space density. We optimized the phase space density both throughmanual parameter scans and by automating the searchusing a genetic algorithm, which converged after about 4
generations. We obtained similar results from both approaches, and measured an optimal phase space densityof 10−5 after 10 seconds of loading.
FIG. 3. Optimizing angled slowing. Dependence of the MOT
population on the detuning and power of the angled slowing
beams. A fixed loading time and compression sequence were
employed for all of the data shown. The powers in the two
beams were balanced to within 15%. The detuning from the421 nm resonance was known to within ±2 MHz.

linewidths of resonance. To minimize losses due to lightassisted collisions, and to reach the lowest final temperature of the cloud, the MOT beams are ramped down to a
final power of 22 µW per arm. We also reduce the magnetic field gradient from 2.5 G/cm to 1.75 G/cm in orderto further minimize losses. At the end of the compression,
the MOT is approximately 400 µm × 400 µm × 800 µm.
We lose up to half of our atoms during the 300 ms of
post-compression equilibration, but obtain a net gain inphase space density due to the simultaneous reduction intemperature.
To detect the number of atoms captured in our trap,
we perform absorption imaging using light resonant withthe 421 nm transition. We expect a high degree of spinpolarization in the mJ = −8 spin state as a result of
the force of gravity on our narrow-line MOT as discussed
in [4], and so we image using σ − light to address themJ = −8 → m0J = −9 transition, which has a ClebschGordon coefficient of nearly unity [19]. We let the cloud
expand freely for between 10ms and 30ms before shining
a 100 µs imaging light pulse. We have verified that wehave a high degree of spin polarization by using σ + lightinstead of σ − light, and observing that the optical depth
was reduced by more than an order of magnitude.
We measure the temperature of our cloud after compression by loading successive MOTs with identical parameters and varying the time-of-flight (TOF) after turning off the MOT beams and quadrupole. By fitting thecloud size as a function of the TOF, we can observe the[1] M. Lu, N. Q. Burdick, S. H. Youn, and B. L.
Lev,
Strongly
dipolarbose-einsteinconden-VI.

CONCLUSION

In conclusion, we used the described angled slowing
technique to reduce the effect of transverse atomic beam
spreading on our MOT loading, effectively increasing thecapture velocity of our narrow-line MOT. We observe
more than an order of magnitude increase in the numberof atoms captured in the MOT when the angled slowing
is operated with optimal parameters, allowing us to load
MOTs in the 108 regime in a few seconds. In our experiment, the combination of a narrow cooling transition,
long free-flight distance, and reduced off-resonant slowing means that the free-flight time is particularly long;
we believe that angled slowing can be of use in similarly
designed experiments using species with narrow coolingtransitions (such as Dy, Er, or Yb). Even in experiments
where transverse spread can be avoided by employing
other techniques, such as transverse Doppler cooling or
the core-shell MOT configuration, the low power requirements and simple geometry of the angled slowing scheme
may make it a comparatively attractive option.

VII.

ACKNOWLEDGEMENTS

We would like to thank Benjamin L. Lev, Tilman
Pfau, and Isaac L. Chuang for fruitful discussions during the early construction stages of our experiment aswell as Rudy Pei, Rokas Veitas, and Cody Winkleblackfor experimental assistance. We acknowledge support
from ONR (Grant No.N00014-17-1-2253), and from a
Vannevar-Bush Faculty Fellowship. M.C. acknowledges
support by the National Science Foundation Graduate
Research Fellowship under Grant No. 1122374.

sate

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