Development of a new Dy quantum gas experimentbyWilliam Lunden
B.S., Stony Brook University (2013)
Submitted to the Department of Physics
in partial fulfillment of the requirements for the degree ofDoctor of Philosophy
at theMASSACHUSETTS INSTITUTE OF TECHNOLOGY
February 2020
c Massachusetts Institute of Technology 2020. All rights reserved.
○

Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Department of Physics
January 17, 2020

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

Accepted by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Nergis Mavalvala
Associate Department Head

2

Development of a new Dy quantum gas experimentbyWilliam Lunden
Submitted to the Department of Physics
on January 17, 2020, in partial fulfillment of therequirements for the degree ofDoctor of Philosophy

Abstract
Since the first realization of Bose-Einstein condensation in an atomic gas at the end of
the twentieth century, ultracold atomic gases have become a widely adopted platform
for the study of various quantum phenomena. In recent years, attention has increasingly turned to species with large magnetic dipole moments due to the much stronger
long-range interactions that these species exhibit in comparison with the more commonly studied alkalis. Dysprosium, with a magnetic moment of about 10𝜇𝐵 , is themost magnetic atomic species and therefore has become an attractive platform forstudying systems in which the long-range (dipole-dipole) interactions compete withor dominate over the contact interactions.
In this thesis I describe the design and optimization of a new dysprosium quantumgas machine. Apart from giving a detailed description of the components of the apparatus and their performance, I describe in detail the characterization and optimizationof the "angled slowing" technique which is used to enhance the loading rate of ourmagneto-optical trap (MOT). I also describe in detail the production and detectionof the first Bose-Einstein condensates (BECs) produced using the apparatus.
This thesis also contains a detailed description of the development of new control
hardware and software which are used in the dysprosium experiment, but can be(and have been) used with other quantum gas experiments. On the hardware side,
I discuss the design of high-performance analog voltage control channels which offer
advantages over commercially available alternatives. On the software side, I discuss alaboratory control and logging database system which I designed, which both expands
the capabilities of our control software and simplifies the storage of and accessibilityof lab data.
Thesis Supervisor: Wolfgang Ketterle
Title: John D. MacArthur Professor of Physics

3

4

Acknowledgments
I have a lot of people to thank for a variety of things over the last 6.5 years. I owe myadvisor Wolfgang more thanks than I can describe here. From my first weeks until
my last week he has helped me to develop my physical intuition, hone my sense ofprioritization and how to be goal-oriented in the lab, and learn to assess and carefully
consider my arguments before making them. He also trusted me to work in not justone, but two of his labs and on a variety of different projects. I’ve learned so muchthrough both his direct mentorship and the projects I’ve worked on under him, andI want to thank him for giving me so many opportunities to do so.
I also want to thank Professor Martin Zwierlein for all of the extremely helpfuladvice he’s given me over the years. Aside from all of his help with navigating
course selections and doctoral exam preparations when I was a younger student, healso helped me to figure out that it was a good idea for me to switch from my firstlab (working on lithium) to the dysprosium experiment. I also want to thank him
for putting me in touch with Jamil from Vector Atomic, whom I’m excited to startworking for when I leave MIT.
Alan Jamison has been a great friend and mentor over the years. He joined theKetterle group as a postdoc only a year after I arrived, and in my earlier years he
was a great person to ask all sorts of random physics questions. When I switched tothe dysprosium experiment his leadership and advice were invaluable in helping me
to understand this new platform, and how to best go about making the experimentwork.
I spent my first 4 years of grad school working in BEC 5 with Jesse, Niki, andIvana, finishing the construction of and doing some first experiments with a bosoniclithium machine. All three of them helped me learn about BEC experiments superquickly, and gave me the opportunity to make meaningful contributions even early on,
for which I’m very grateful. They’ve also been great friends, with whom I’ve gotten
the chance to travel to both Italy (for the Varenna summer school) and to Bulgaria(for Ivana’s wedding), plus have countless beers with.
5

Li, Michael, and Pierre have been awesome labmates in my 2.5 years in Dypole (thedysprosium lab). It’s been awesome seeing how quickly all three of them have become
experts in various aspects of physics and lab work. And it’s been a pleasure to learn
about dysprosium alongside such talented people. I’m excited to see what happens
next in the lab with such a strong team: Li’s work ethic is extremely impressive, andthis is something I was very grateful for during the long nights/overnights of debugging
and data taking as we were optimizing our MOT and eventually evaporating to BEC.
Michael’s thorough experience with circuit design and broad problem solving skills
are going to be invaluable as the needs of the lab continue to evolve. And Pierre’s
deep understanding of atomic physics and software savvy will likewise enormously
benefit the lab for as long as he is there.
I also want to extend thanks to some of the friends I met in Boston outside of grad
school, and with whom I had the chance to work on a variety of musical endeavors.
The senses of accomplishment and teamwork that came out of these musical projects
made the last several years extremely enjoyable, and also had a cross-pollinating
effect, motivating me and giving me new perspective that helped with lab work.
Since 2014 I’ve been involved in a death metal band called The Beast of Nod. Paul,
Brendan, and Nate have become my best friends over the years, probably because of
the sheer number of things we’ve done together: we wrote and recorded three CDs ina variety of studios around the state, toured a large chunk of the U.S., filmed music
videos and performance videos, shared the stage with internationally acclaimed metal
artists, and developed a genuine online fanbase. So I’m grateful to have met theseguys and have had the chance to have all of these great experiences (and more tocome!).
In 2016 I joined a progressive metal project called Episodes that was formed at
Berklee college of music. I owe a lot of my musical development and understanding oflive performance technology to Amaury, OG, Billy, and David, who are all insanely
talented and were a genuine pleasure to perform with. I wish we’d been able to tour
more together, but it was great fun while it lasted, and I’m excited for the future
incarnation of Episodes that’s in the works.
6

In 2018 I started a new project called Mimesis, with three super talented musicians
I’d met in the New England music scene: Jen, Tony, and Kilian. We pulled off analbum which combined aspects of modern rock and metal in a way that none of us had
heard before, and the reception has been extremely positive. I’m extremely gratefulto all three of them for working on this project with me, and for everything I’ve
learned from them about multimedia, promotion, the music industry.
I also want to separately thank Jen for being such a supportive partner and positive
influence on my life. Her disciplined approach to life is inspiring when it seems likethere is an overwhelming number of things to do. Certain periods of the last year ofgrad school were particularly draining, and I’m super grateful for her endless patience
when I was semi-sleep deprived for weeks at a time.
Finally, I want to thank my family for their endless support, both during grad
school and in all the years before it. My mom and step father Lewis gave me a lotof encouragement in the right directions all through high school and college, helping
me to get on a good path without forcing me onto it, and I’m very grateful for this.
Moving to MA meant being able to see my father more often too, and I’m gratefulfor all of the life advice and support he gave me during my grad school years. My
grandparents have also always been extremely supportive of my academic endeavors,
and I want to thank them for that.

7

8

Contents
1 Introduction
1.1

13Bose-Einstein Condensation . . . . . . . . . . . . . . . . . . . . . . .

14

1.1.1

The Dipolar Quantum Gas . . . . . . . . . . . . . . . . . . . .

16

1.2

Why Dysprosium? . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

1.3

Scientific Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

1.3.1

Spin-Orbit Coupling Schemes . . . . . . . . . . . . . . . . . .

20

1.3.2

Suppressing Dipolar Relaxation . . . . . . . . . . . . . . . . .

21This Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

1.4

2 Properties of Dysprosium

25

2.1

Isotopes of Dysprosium . . . . . . . . . . . . . . . . . . . . . . . . . .

25

2.2

Electronic Structure of 162 Dy . . . . . . . . . . . . . . . . . . . . . . .

26

2.3

Angular Momentum and Magnetic Moment . . . . . . . . . . . . . .

27

2.4

Scattering: Dipolar Interactions and Feshbach Resonances . . . . . .

32

2.4.1

Cold Collisions and Interaction Potentials . . . . . . . . . . .

32

2.4.2

Partial Waves . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

2.4.3

Feshbach Resonances . . . . . . . . . . . . . . . . . . . . . . .

34

2.4.4

Dipolar Relaxation . . . . . . . . . . . . . . . . . . . . . . . .

36

Properties of the Useful Electronic Transitions . . . . . . . . . . . . .

37

2.5.1

421 nm Transition . . . . . . . . . . . . . . . . . . . . . . . . .

37

2.5.2

626 nm Transition . . . . . . . . . . . . . . . . . . . . . . . . .

38

2.6

741 nm Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

2.7

Raman Transitions in Dysprosium . . . . . . . . . . . . . . . . . . . .

39

2.5

9

2.7.1

Spin-Orbit Coupling Using Raman Transitions . . . . . . . . .

3 Experimental Apparatus

40
47

3.1

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47

3.2

Vacuum System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47

3.2.1

Oven . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

48

3.2.2

Ion Pumps and UHV . . . . . . . . . . . . . . . . . . . . . . .

49

3.2.3

Zeeman Slower . . . . . . . . . . . . . . . . . . . . . . . . . .

50

3.2.4

Main Chamber . . . . . . . . . . . . . . . . . . . . . . . . . .

52

Laser Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

3.3.1

421 nm Light . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

3.3.2

626 nm Light . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

3.3.3

1064 nm Light . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

3.4

Layout of Optical Beams . . . . . . . . . . . . . . . . . . . . . . . . .

60

3.5

Magnetic Field Coils . . . . . . . . . . . . . . . . . . . . . . . . . . .

63

3.6

Electronics and Control . . . . . . . . . . . . . . . . . . . . . . . . . .

64

3.6.1

Control and Power Electronics . . . . . . . . . . . . . . . . . .

64

3.6.2

Cameras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65

3.6.3

Cicero Word Generator . . . . . . . . . . . . . . . . . . . . . .

65

3.3

4 Cooling and Trapping of Dysprosium

67

4.1

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67

4.2

Loading a Magneto-Optical Trap . . . . . . . . . . . . . . . . . . . .

68

4.2.1

Dysprosium Oven . . . . . . . . . . . . . . . . . . . . . . . . .

68

4.2.2

Zeeman Slowing . . . . . . . . . . . . . . . . . . . . . . . . . .

70

4.2.3

Transverse Doppler Cooling . . . . . . . . . . . . . . . . . . .

73

4.2.4

Angled Slowing . . . . . . . . . . . . . . . . . . . . . . . . . .

76

4.2.5

Magneto-Optical Trapping . . . . . . . . . . . . . . . . . . . .

81

Compression of the MOT . . . . . . . . . . . . . . . . . . . . . . . . .

87

4.3.1

Compression Approach 1 . . . . . . . . . . . . . . . . . . . . .

87

4.3.2

Compression Approach 2 . . . . . . . . . . . . . . . . . . . . .

88

4.3

10

4.4

Loading of Shallow-Angle Crossed Dipole Trap . . . . . . . . . . . . .

89

4.4.1

Stark Shift of a Gaussian Beam . . . . . . . . . . . . . . . . .

89

4.4.2

Crossed Optical Dipole Traps . . . . . . . . . . . . . . . . . .

91

4.4.3

Loading the XODT . . . . . . . . . . . . . . . . . . . . . . . .

92

4.4.4

Calibration of Trap Depth via Modulation . . . . . . . . . . .

95

4.4.5

Scattering in the Trap . . . . . . . . . . . . . . . . . . . . . .

96

4.5

Evaporation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

97

4.6

Bose-Einstein Condensation . . . . . . . . . . . . . . . . . . . . . . . 100

4.7

4.6.1

Detection of a Condensate . . . . . . . . . . . . . . . . . . . . 101

4.6.2

Observation of Bimodality . . . . . . . . . . . . . . . . . . . . 102

4.6.3

Phase Space Density of the Measured Cloud . . . . . . . . . . 104

Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5 Low-Noise Analog Control Channels
5.1

5.2

107

Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.1.1

Ground Loops . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

5.1.2

Noise Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5.1.3

Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

Analog Channel Design . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.2.1

Schematic of System . . . . . . . . . . . . . . . . . . . . . . . 112

5.2.2

FPGA/SOC and Network Interface . . . . . . . . . . . . . . . 114

5.2.3

20-bit DAC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

5.2.4

LTZ1000A Voltage Reference . . . . . . . . . . . . . . . . . . 117

5.2.5

Supercapacitors . . . . . . . . . . . . . . . . . . . . . . . . . . 118

5.2.6

Poseidon Server . . . . . . . . . . . . . . . . . . . . . . . . . . 121

5.3

Description of Operation . . . . . . . . . . . . . . . . . . . . . . . . . 122

5.4

Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.4.1

The Low-Cost Variant . . . . . . . . . . . . . . . . . . . . . . 124

5.4.2

Noise Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . 125

5.4.3

Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
11

6 Experimental Control and Logging Database

127

6.1

Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

6.2

Summary of Developments . . . . . . . . . . . . . . . . . . . . . . . . 128

6.3

The Database . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6.3.1

6.4

Structure of the Database . . . . . . . . . . . . . . . . . . . . 130

The Zeus Server . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
6.4.1

Data Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

6.4.2

External Hardware Monitoring. . . . . . . . . . . . . . . . . 137

6.5

Genetic Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

6.6

The Future: Lab Monitoring with a Mesh Network . . . . . . . . . . 141

7 Outlook

145

7.1

741 nm and Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

7.2

Toward Spin-Orbit Coupling . . . . . . . . . . . . . . . . . . . . . . . 146

7.3

Technological Improvements and Automation . . . . . . . . . . . . . 146

A C# API for Image Database

149

A.1 Structure of the API . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
A.1.1 The Constructor . . . . . . . . . . . . . . . . . . . . . . . . . 150
A.1.2 The Data Types . . . . . . . . . . . . . . . . . . . . . . . . . . 150
A.1.3 The Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

12Chapter 1
Introduction
As time goes on, mankind’s ability to meaningfully describe the world around us –that is, to accurately predict the behavior of real physical systems – is increasingly
limited not by the validity of our physical models but by the tractability of the necessary calculations. This is especially true when it comes to describing the quantum
mechanical behavior of systems of many particles, where the required number of states
to keep track of grows exponentially with the system size. Sometimes, we luck out andcan apply approximations such a mean-field approach, where we essentially solve aself-consistent single particle problem in place of the true manybody problem. A classic example of this is the description of 𝑁 identical, contact-interacting bosons in an
external potential. Formally, one must solve a problem with a potentially enormous
Hamiltonian which acts on the N-body, symmetrized wavefunction [57]:∏︀Ψ(⃗𝑟1 , ..., ⃗𝑟𝑁 ) = 𝑁
𝑟𝑖 )𝑖=1 𝜓𝑖 (⃗∑︀ −ℎ̄2 ∇2
∑︀
∑︀
2 2
𝐻 = 𝑁 2𝑚 𝑁 + 𝑁 𝑉 (𝑟⃗𝑁 ) + 𝑖<𝑗 4𝜋ℎ̄𝑚 𝑎𝑠 𝛿(⃗𝑟𝑖 − ⃗𝑟𝑗 )

(1.1)
(1.2)

Fortunately, much can be learned about the system by instead solving the singleparticle ”non-linear Schrodinger equation,” also known as the Gross-Pitaevski equation [57, 22]:˙ 𝑟) =𝑖ℎ̄𝜓(⃗(︂)︂−ℎ̄2 ∇24𝜋ℎ̄2 𝑎2𝑠 2+ 𝑉 (⃗𝑟) +|𝜓 (⃗𝑟)| 𝜓(⃗𝑟)
2𝑚
𝑚
13

(1.3)

which is a much more tractable problem.
It is not always possible to construct a meaningful mean-field description of a problem, and so alternative approaches to finding a solution must be considered. Naive
numerical methods, such as diagonalization of a manybody Hamiltonian, are usually
only computationally tractable for small numbers of particles. More sophisticated approaches such as density-matrix renormalization group (DMRG) have been successful
primarily in the case of 1D systems, but most 2D and 3D manybody systems remaincomputationally intractable [60].
An alternative approach to studying quantum manybody problems is to develop a"quantum simulation" of the problem of interest [19, 24, 26, 1]: a different quantum
system which is easy to manipulate and measure is used to implement the Hamiltonian, and the measured behavior of this system can be used to understand the Hamiltonian of interest. Quantum degenerate gases – dilute ensembles of atoms which havebeen cooled to the point that the effects of quantum statistics dominate the behaviorof the gas – have become an increasingly popular platform for developing quantum
simulations of a wide variety of Hamiltonians. Examples include simulation of gauge
fields [20], realization of spin-orbit coupling [8], and the observation of phase transitions in lattice systems [22]. In addition to allowing us to simulate Hamiltonians
related to real-world problems of interest, the tunability of quantum gas systems also
opens up the consideration of totally novel Hamiltonians. This enables the study ofthe physics of systems which have no known analog in nature, such as the spin-orbit
coupling of high-spin bosons [73].

1.1

Bose-Einstein Condensation

Quantum degenerate atomic gases of bosons1 exhibit a striking behavior called BoseEinstein condensation. Bose-Einstein condensation is a phenomenon that occurs whenan ensemble of particles obeying bosonic statistics is cooled down to the point that1

Individual atoms have a total composite spin (consisting of the summed spin angular momentaof all nucleons and all electrons) which is either integer or half-integer, meaning that the atoms arerespectively composite bosons or composite fermions.

14

entropy is maximized when the absolute ground state is macroscopically occupied[46]. The resulting state of matter – the BEC – is essentially made of 103 − 106 copiesof the ground state wavefunction (for the case of the ideal noninteracting BEC) on
top of one another. The large particle number allows for the techniques of atomic
physics, including various imaging techniques, to be used to manipulate and probe
the behavior of the condensate, allowing for an unprecedented level of control overan inherently quantum system [46, 57].
The temperature (and corresponding density) at which condensation occurs can beunderstood with a simple physical picture. Let us consider the atoms in the ensemble
to be individual wavepackets, with a spatial extent given by their thermal deBroglie
wavlengths [46]:
√︃
𝜆dB =

2𝜋ℎ̄2
𝑚𝑘𝐵 𝑇

(1.4)

where m is the atomic mass, 𝑘𝐵 is the Boltzmann constant, and 𝑇 is the temperatureof the ensemble. The interparticle spacing is given by the inverse of the atomic number
density 𝑛. If the temperature is lowered, eventually, the deBroglie wavelength will beon the order of the interparticle spacing:
𝑛𝜆3dB ≈ 1

(1.5)

and the particles will "overlap" with one another. This corresponds to the notion of
the particles losing their distinguishability, and beginning to macroscopically occupy
the same state. A more rigorous calculation yields that the critical value of 𝑛𝜆3dB ,the so-called phase space density of the ensemble, for Bose-Einstein condensation is
actually [57, 46]
𝑛𝜆3dB ≈ 2.612(1.6)

The Bose-Einstein condensation of dilute atomic gases, first achieved in 1995 [5, 16],
opened up a new field of quantum research. The seminal work on achieving BEC
soon led to the preparation of quantum-degenerate Fermi gases [47], the loadingof ultracold (i.e., quantum degenerate) atoms into optical lattices [22], and a wide15

variety of other milestones. Initially a topic of study in its own right due to its
relationship to e.g. superfluid He and superconductivity, the BEC has increasingly
become a general platform for the study of quantum mechanical phenomena. This isdue to the unprecedented level of control over a quantum system that a BEC offers toexperimenters: highly tunable "designer" single-particle and manybody Hamiltonians
can be implemented by addressing the BEC with the tools of atomic physics such as
Raman transitions, periodic optical potentials, and Feshbach resonances.

1.1.1

The Dipolar Quantum Gas

Many real quantum systems involve interactions long-ranged interactions. The realization of quantum gases with long-range, anisotropic interactions opens the door
to studying a variety of effects which cannot be studied with the purely contactinteracting alkali species that were first Bose-condensed [34]. The effects range fromroton instabilities in anisotropic trap geometries [68] to interacting lattice gases which
don’t require superexchange [6]. Dipolar quantum gases – both BECs and degenerate
Fermi gases – have thus increasingly become a focus of attention within the atomic
physics community, with experimental progress being made on producing and studying gases of both ultracold polar molecules and of dipolar atoms.
The interaction potential between two dipoles oriented along a guiding magnetic
field can be written as [34]𝑈𝑑𝑑 =𝜇0 𝜇2 1 − 3 cos2 𝜃
4𝜋
𝑟3

(1.7)

where 𝜇 is the magnetic moment of a dipole, 𝜃 is the angle between between the quantization axis and the internuclear separation, and 𝑟 is the magnitude of the internuclear
separation. This interaction is repulsive in the case of side-by-side orientations of thedipoles, attractive in the case of head-to-tail orientations, and continuously variable
as a function of 𝜃 between these two cases. The effect of this interaction potential
on the behavior of the quantum gas can be quite striking; Figure 1-1 shows someexamples of phenomena which have been observed so far in dipolar atomic gases.
16

Figure 1-1: Some recently observed phenomena in ultracold dipolar gases. Top left:
the roton-like dispersion of a dipolar gas in an anisotropic trap [14]. Top right: theobservation of supersolid-like properties in a dipolar BEC [13]. Bottom: droplet
formation as reported in [30].

17

1.2

Why Dysprosium?

The path to quantum degeneracy in all of the stable alkalis (Li, Na, K, Rb, Cs)
has been paved for decades, and an enormous number of systems have been realizedusing these species, ranging from BCS pairing of fermions [70] to lattice-based spinmodels [52]. With alkalis alone it is possible to create bosonic or fermionic systems(or mixtures thereof) in highly controllable potentials and with tunable interactions[57]. Alkalis are typically cooled, trapped, and manipulated entirely using light which
addresses the lowest-lying S-to-P transition of the single valence electron (further
split into the D1 and D2 lines by fine structure and into further manifolds by thehyperfine interaction, of course). The properties of the transition are similar acrossall of alkalis, namely that they are all optically accessible and all have linewidths thatare suitable for creating magneto-optical traps (MOTs). Although there are virtually
endless systems that can be realized with alkalis, these species have their limitations.
Namely,1. The interactions between the atoms are completely dominated by short-range
interactions. At low temperatures, the atoms interact with an effective isotropic
contact potential. This precludes the study of, e.g., lattice models with significant next-nearest neighbor interactions.
2. Unlike alkaline earth and transition metal species, they possess no narrowtransitions (loosely speaking, less than 1 MHz), which can be used for lowertemperature Doppler cooling, population shelving, and optical clocks.
3. The optical Raman transitions between hyperfine sublevels, which are often
leveraged for simulating gauge fields, effectively require that the single-photon
detuning be the order of the fine structure splitting. This means that theresultant heating due to Rayleigh scattering from each beam limits the lifetimeof the cold Raman-coupled system [74]. The ratio of Raman coupling strength
to heating rate can be made arbitrarily large by using larger detuning and morelaser power in atomic species in which there is already spin-orbit coupling in18

the electronic ground state (see Chapter 2 for more details).
Atomic dysprosium offers a platform for quantum gas experiment which suffers fromnone of these limitations. As will be described in Chapter 2, Dysprosium has alarge magnetic moment of 𝜇 ≈ 10𝜇𝐵 in its ground state, approximately ten times
higher than the corresponding moment of alkali atoms and in fact the highest of any
atomic species [49]. As a result, the dipole-dipole interaction between two dysprosium
atoms is up to 100 times stronger than the corresponding interaction between alkalis;
combined with the tunability of the contact interaction strength by way of Feshbachresonanes (also described in Chapter 2), this means that dysprosium can be studied
in regimes where either the contact interaction dominates, the dipolar interaction
dominates, or the two interactions are comparable.
Dysprosium also has a wealth of optically accessible, closed (or mostly closed)transitions with different linewidths, allowing for a high level of control over slowing,cooling, and trapping using different lasers.
Finally, the high angular momentum of the Dy ground state means that Raman
transition amplitudes among ground state sublevels scale favorably compared to theheating rate as large detunings are used; this alleviates a practical limitation of implementing Raman-coupling based synthetic gauge field schemes in alkali species [74].

1.3

Scientific Goals

The scientific goals of our experiment involve the exploration of dipolar-interacting
systems which involve more than one spin state (i.e., more than one ground state
𝑚𝑗 projection). Specifically, we are interested in studying the spin-orbit coupling ofdipolar-interacting atoms, in two and possibly three dimensions. Because the validity of mean-field approaches to the problem of an interacting 2D spin-orbit coupledsystem of bosons is questionable, it is an interesting goal in its own right to realize
and study the properties of such a system’s ground state [73]. 2D spin-orbit couplingof dipolar-interacting bosons is also predicted to give rise to exotic quasicrystalline
ground states [21], and so realizing such a system would constitute a first detection of19

these states. By tuning the dipolar interactions with a tilted or rotating [66] guiding
magnetic field, we can also study the effect of dipolar interactions on 1D spin-orbit
coupled bosons in both the "classic" Raman-coupled scheme [8, 36, 12] and in topologically interesting lattice-based spin-orbit coupling schemes.
Unfortunately, the rapid dipolar relaxation of spin mixtures of bosons precludes
the study of systems with more than one spin state populated unless measures tosuppress this relaxation are taken [8]. The first goal of our experiment is thus toattempt to suppress dipolar relaxation by means of geometrical constraints imposedby optical lattice potentials.
In the section below, spin-orbit coupling and the physical mechanisms that canlead to the suppression of dipolar relaxation are discussed in detail.

1.3.1

Spin-Orbit Coupling Schemes

Spin-orbit coupling is a phenomenon that arises naturally in the description of electrons in many real systems and materials. The motion of an electron through anelectric potential, such as the crystalline potential of a solid material, causes it to see
an effective magnetic field, which in turn gives rise to a spin-dependent energy shift.
From special relativity, we know that a particle moving with velocity ⃗𝑣 through an⃗ sees an effective magnetic field 𝐵⃗ given by [51]electric field 𝐸⃗ × ⃗𝑣⃗ = 1𝐸𝐵𝑐(1.8)

For an electron with spin ⃗𝜎 = ℎ̄2 𝜎
ˆ , there is then a motional- and spin- dependent
energy shift given byℋ𝑆𝑂𝐶 = −𝑔𝑒 𝜇𝐵⃗⃗𝜎 · 𝐵ℎ̄(1.9)

If we associate the electric field with a potential 𝑉 and denote the electron’s momentum as 𝑝⃗, we generically end up with a term in the Hamiltonian of the form
ℋ𝑆𝑂𝐶 ∝ ⃗𝜎 · (∇𝑉 × 𝑝⃗) ∝ ∇𝑉 · (⃗𝜎 × 𝑝⃗)
20(1.10)

In real systems, this phenomenon has a number of effects:∙ Spin-orbit coupling contributes to the fine structure splitting in atomic transitions.
∙ It creates the spin galvanic effect in solid materials: because spin and momentum
are locked in the band structure of the material, preparing a spin density inducesa charge current to flow.
∙ It also creates the inverse spin galvanic effect: inducing a charge current gives
rise to a transverse spin density.
∙ It is the origin of the spin hall effect in solid materials: spins accumulate on theoutsides of a current-carrying piece of material, analogous to charge accumulation in Hall effect∙ It gives rise to topological states like 2D and 3D topological insulators

1.3.2

Suppressing Dipolar Relaxation

A consequence of dysprosium’s large dipole-dipole interaction strength is that dipolar
relaxation – inelastic two-body collisions which couple internal angular momentum toexternal angular momentum, converting Zeeman energy into kinetic energy – occurs
at a much higher rate than in alkali species. Pairs of atoms in the absolute lowestenergy state (𝑚𝑗 = −8 in the case of 162 Dy) cannot undergo dipolar relaxation because
there is no lower-energy state to flip into, and so clouds which are spin-polarized into
this state are collisionally stable.
Spin-orbit coupling requires that at least the next-highest spin state is populated.
As will be discussed in Chapter 2, the rate of dipolar relaxation of fermions is lower
than that of bosons, and can be suppressed by increasing the magnitude of the guiding√
magnetic field. While the suppression of dipolar relaxation in fermions scales as 𝐵[25], the enhancement of dipolar relaxation in bosons scales by the same factor. In
the one experimental implementation of spin-orbit coupling with highly dipolar atoms
21

so far, a 1D Raman-coupling scheme, the loss rate of the bosonic spin mixture wasprohibitively high and a fermionic spin mixture was utilized instead [8, 9].
A short (< 10ms) lifetime was observed for Bosons even at low magnetic fields[8], suggesting that suppressing dipolar relaxation by working at small controlled
fields2 , would not be a feasible approach. We thus are interested in implementing
new approaches to suppressing dipolar relaxation by confining the geometry of oursystem to one or two dimensions.
One approach, which has been demonstrated in Cr atoms [56], is to load theatoms into 1D tubes (i.e., freeze out motion along two directions using a 2D opticallattice) and point the guiding field for the atoms along the unconfining direction. To
conserve angular momentum, dipolar relaxation of a spin mixture in this configurationnecessitates the creation of transverse relative motion of the particles; however, if thereleased Zeeman energy falls into the bandgap of the transverse directions, the thenthe dipolar relaxation process is suppressed. This is illustrated in Figure 1-2. For
the lattice depths we can achieve, this approach would require very careful control
of a small magnetic field – a technical challenge. If successful, it could allow forinvestigation of 1D spin-orbit coupled Hamiltonians along the unconfined direction.

1.4

This Thesis

The work described in this thesis constitutes the development of a new experiment toinvestigate the physics of dipolar Bose gases. A new apparatus for creating BoseEinstein condensates of dysprosium is described in detail, including the array ofcooling and trapping techniques used to create a degenerate gas. This description
includes a detailed report of "angled slowing," a new technique for which we have
performed the first rigorous characterization and which we believe to be of general
2This is a technical challenge in its own right. At large magnetic fields, stray transverse components of the magnetic field are "projected out", and magnetic field control is essentially equivalent
to current control in the coils generating the guiding field; at low magnetic fields, precise controlof the magnetic field necessitates measurement and stabilization against stray fields. The scale ofdistinction between "high" and "low" is approximately a few hundred mG – the magnitude of theearth’s magnetic field.

22

Figure 1-2: Suppression of dipolar relaxation in 1D tubes as reported in [56]. Left:the dependence of the temperature of the confined atoms after a fixed hold time on
the angle between the guiding magnetic field and the long axis of the tubes. Center:the population of the first excited band of the lattice as a function of the Zeeman
energy of the atoms in the guiding field. Right: the temperature of the atoms as afunction of the Zeeman energy after a fixed hold time for two different lattice depths(the shaded regions correspond to the widths of the first excited band in each case).
interest to experiments employing narrow transitions for magneto-optical trapping[45]. In addition to the description of the apparatus, the detailed description of two
technology development projects is given. The first is the development of low-noise,high-precision analog control voltage hardware; initially developed to improve upon
the limitations of commercially available analog channels for precision work on another experiment, this newly developed hardware forms an isolated, low-noise control
backbone of the dysprosium apparatus. The second is the development of a databasebackend and associated applications for our laboratory control system, which has
extended the interoperability, flexibility, and automation capabilities of the existing Cicero control system; this new software has been implemented in several labs,
including the dysprosium experiment.

23

24

Chapter 2
Properties of Dysprosium
A fairly unremarkable element outside the field of atomic physics, dysprosium possesses a number of properties which make it a very attractive platform for studying
physics [15, 42, 7, 10, 29, 48, 49, 44, 67, 11] in the ultracold regime. Most of theseproperties arise from its [Xe]4𝑓 10 6𝑠2 electronic structure: a large (𝜇 ≈ 10𝜇𝐵 ) magnetic moment, several closed (or nearly closed) optical transitions with a variety oflinewidths, and spin-orbit coupling in the ground state which enables long-lived Raman coupling schemes.

2.1

Isotopes of Dysprosium

The four most abundant isotopes of dysprosium are 161 Dy (18.9% abundance), 162 Dy(25.5%), 163 Dy(24.9%), and 164 Dy (28.3%).

162

Dy and 164 Dy are bosons with nonuclear spin (and hence no hyperfine structure), while 161 Dy and 163 Dy are fermions
with nuclear spin 𝐼 = 25 [49, 41]. The work described in this thesis was done using
the 162 Dy isotope (see Figure 2-1).
This isotope was chosen from between the two abundant bosons out of conveniencebased on available AOMs and the distance between the dysprosium 626 nm resonanceand the available spectroscopic lines in our iodine reference cells. In the future thework may be extended to other isotopes. As has been demonstrated by other groups[40, 48] there is no need for consideration of additional repumping schemes when25

Figure 2-1: Single-beam absorption spectroscopy at 421 nm of the atomic beam, measured transverse to the direction of beam propagation, showing the presence of bothabundant bosons and a variety of hyperfine states of the abundant fermions in theatomic beam. The isotopes have been identified by comparison to the spectra reported
in [49].
switching isotopes, and it is a straightforward matter of adjusting the frequencies of
the cooling lasers to form a MOT of a different isotope.

2.2

Electronic Structure of 162Dy

The electronic structure of dysprosium is rich, giving rise to many transitions that are
optically accessible and which have a useful range of linewidths. The full electronic
structure of the ground state is(𝑙𝑠2 2𝑠2 2𝑝6 3𝑠2 3𝑝6 3𝑑10 4𝑠2 4𝑝6 4𝑑10 5𝑠2 5𝑝6 )4𝑓 10 6𝑠2(2.1)

A cartoon depiction of the electronic structure is shown in Figure 2-2. Several of thestrong optical transitions only involve the excitation of one of the two 6𝑠 electrons
to a 6𝑝 state; in this sense, Dy’s spectrum has many similarities to the spectra ofalkaline earth species. Transitions involving the 4𝑓 electrons give rise to additional
26

Figure 2-2: Cartoon depiction of 162 Dy, showing the electrons which are relevantto the transitions and properties used in quantum gas experiments. The green 4𝑓
shell has been unconventionally placed outside of the 6𝑠 shell to emphasize the larger
spatial extent of the radial wavefunctions of these high-𝑙 orbitals.
lines which can be very narrow; for example, the 4𝑓 to 5𝑑 transition at 1001 nm wasrecently measured to have a linewidth of around 2𝜋 × 10 Hz [64].
Although seemingly intractably complicated (see Figure 2-4), the spectrum contains a few optically allowed transitions which have been well characterized and possess useful properties for laser cooling. The three transitions of interest to us (421 nm,
626 nm, and 741 nm), along with their electronic configurations [33], are shown in Figure 2-3. As shown in the figure, the 421 nm and 626 nm transitions involve the same
orbital configuration in the excited state, but different electron spin orientations of
the outer (n=6) electrons. The singlet transition is the usual, allowed transition andhence has a larger matrix element and broader linewidth than the triplet transition,
which is only allowed as a result of spin-orbit coupling in the atom.

2.3

Angular Momentum and Magnetic Moment

The ground state of Dysprosium has a large magnetic moment of approximately
10𝜇𝐵 . We can understand the physical origin of this magnetic moment intuitively by27

Figure 2-3: The three transitions relevant to our experiment. The 421 nm and 626 nm
transitions involve the same electron orbital configurations in the excited states, butcorrespond to different orientations of the electron spins of the outer 6𝑠 electrons:
the broad 421 nm transition is a spin singlet-to-singlet transition while the narrower
626 nm transition is an intercombination line (i.e., a spin singlet-to-triplet transition).
separately considering the orbital and spin angular momentum of all of the valence
electrons. In reality, the two angular momentum components couple and the total
atomic angular momentum is best described in terms of composite angular momentum
𝐽 = 𝐿 + 𝑆.
First we consider the contribution of the orbital angular momentum of the outerelectrons. The 6𝑠 electrons possess no angular momentum of course, but the 4𝑓
electrons each carry up to 3ℎ̄ of angular momentum, depending on their 𝑚𝑙 quantum
number. Only the angular momenta of unpaired electrons contribute to the total
atomic angular momentum, and so we must identify which electrons are unpaired. By
applying Hund’s rules [49], we can conclude that the arrangement of the 10 electrons
in the 4𝑓 shell should leave the electrons with 𝑚𝑙 = {3, 2, 1, 0} unoccupied. This is
depicted in Figure 2-5.
Thus the sum of the orbital angular momentum projections of the 4𝑓 electronsis 𝐿𝑧 = 0ℎ̄ + 1ℎ̄ + 2ℎ̄ + 3ℎ̄ = 6ℎ̄, implying a total angular momentum of 𝐿 = 6.
Considering now the contribution of the spin angular momentum, we again note that
only unpaired spins contribute to the total atomic spin angular momentum. Thus
only the 4 unpaired 4𝑓 electrons contribute their spin angular momentum, with each28

Figure 2-4: A portion of the energy level spectrum of Dy from the NIST database[33], showing the energy in units of wavenumbers for states of various total angularmomentum. The ground state is shown at zero energy with J = 8. Orange lines
represent states of even parity, and blue lines represent states of odd parity.

29

Figure 2-5: Visualization of the arrangement of the electrons among the orbital angular momentum 𝑚𝑙 levels in the 4𝑓 shell in dysprosium. The four unpaired electrons
have orbital angular momenta which align with both each other and with the spin
angular momenta, giving rise to the large ground state angular momentum of 𝐽 = 8.

contributing 12 ℎ̄.
We can now consider the magnetic moment that results from all of these contributions to the total atomic angular momentum. Again appealing to Hund’s rules,
we conclude that because the 4𝑓 orbital is more than half-full, that the state withthe largest possible total angular momentum 𝐽 corresponds to the ground state;
this is equivalent to the physical picture that all of the constituent orbits and spins
contribute angular momentum in the same direction. Noting that the g-factor forelectron orbital angular momentum is exactly 1 and that the g-factor for electron
spin is approximately 2, we can calculate the total magnetic moment as

𝜇=−

3
∑︁
𝑚𝑙 =0

𝑔𝑙 𝑚𝑙 𝜇𝐵 −3
∑︁1
𝑔𝑠 𝜇𝐵 ≈ −10𝜇𝐵
2
𝑖=0

(2.2)

The proper description of the angular momentum and hence magnetic moment of
the atom is given by considering the total angular momentum 𝐽 = 8, with magnetic
moment30

Figure 2-6: The angular momentum structure of the 𝐽 = 8 ground state and any of
the 𝐽 = 9 excited states in dysprosium. The three arrows indicate transitions fromthe 𝑚𝑗 = −8 state, with the shading of the arrows indicating the relative strengthsof the three transitions.

𝜇 = −𝑔𝑗 𝑚𝑗 𝜇𝐵(2.3)

The experimentally measured value of 𝑔𝑗 for bosonic dysprosium is 𝑔𝑗 = 1.2415867[49], yielding a moment for the polarized |𝑚𝑗 | = 8 states of 9.932𝜇𝐵 .
For 162 Dy, the ground state has a multiplicity of 17, corresponding to integerspaced projections 𝑚𝐽 of the total angular momentum 𝐽 onto the quantization axis
running from -8 to +8. Since the nuclear spin of both bosons is 0, there is no hyperfine
structure in the ground state for 162 Dy or 164 Dy. All of the excited states of thetransitions we address in our experiment have 𝐽 = 9, which has a multiplicity of 19.
Since (for reasons discussed in the following chapters) we are usually addressing atoms
that are in the polarized state 𝑚𝑗 = −8, it is worth noting how this large angularmomentum structure gives rise to unusually different Clebsch-Gordon coefficients fortransitions to different excited state 𝑚𝑗 levels.
Specifically, if we consider the case of electric dipole transitions from the 𝑚𝑗 = −8
state to 𝑚𝑗 = {−7, −8, −9} by respectively 𝜎 − , 𝜋, 𝜎 + light, then we find that the ratioof the corresponding Clebsch-Gordon coefficients is 1: 31 : 3√117 , corresponding to a ratioof transition strengths of 1:0.11:0.0065. The full ground and excited state sublevelstructure, as well as arrows indicating the transitions from the 𝑚𝑗 = −8 state andtheir relative transition strengths, are shown in Figure 2-6.
31

2.4

Scattering: Dipolar Interactions and Feshbach
Resonances

Dysprosium possesses an interesting array of scattering properties. The large magnetic moment of dysprosium means that the strength of two-body dipole-dipole interactions is much higher than is the case in alkali species, leading to both the persistenceof higher partial wave scattering at low temperatures and enhanced rates of (inelastic) dipolar relaxation. The large number of possible electronic configurations also
gives rise to a dense spectrum of Feshbach resonances [49, 7], resulting in a sensitive
dependence of both elastic and inelastic scattering on the magnetic field experienced
by the atoms.

2.4.1

Cold Collisions and Interaction Potentials

Many properties of the interactions between dysprosium atoms in an ultracold gas
can be described using a combination of short range interactions and dipolar interactions. At short ranges, the potential is complicated, containing contributions fromthe van der Waals (𝐶6 ) interaction, as well higher-order dispersion terms. The shortrange potential is usually approximated by a pseudopotential which gives a usefuldescription at the low collisional energies that occur at ultracold temperatures; mostcommonly, a contact pseudopotential is used:
𝑉 (𝑟) = 𝑔𝛿(𝑟)

(2.4)

4𝜋ℎ̄2
𝑔=
𝑎
𝑚

(2.5)

where 𝛿(𝑟) is a delta function andwhere 𝑚 is the mass of a single atom and 𝑎 is the s-wave scattering length [57].
The scattering length is a notion that arises from the consideration of the scatteredwavefunction in a two-body scattering problem; it is the value of the relative radial32

coordinate at which the scattered wavefunction amplitude crosses zero. The scattering
length characterizes the behavior of the wavefunction away from the complicatedoscillations it exhibits at short ranges (where the scattering potential is relatively
complicated). This is specifically the s-wave scattering length because, as will be
discussed below, at low temperatures only spherically symmetric s-wave scattering
occurs for the short-range interaction.
The dipole-dipole interaction is a longer-ranged interaction which is also responsible for many of the collisional properties of dysprosium. For two atoms whose magnetic moments are aligned along a guiding magnetic field, the interaction potential is[34, 57]
𝑈𝑑𝑑 =𝜇0 𝜇2 1 − 3 cos2 𝜃
4𝜋
𝑟3

(2.6)

where 𝜇 is the dipole moment of an atom, 𝑟 is the separation, and 𝜃 is the angle
between the guiding field and the separation vector. The moment of a polarized
dysprosium atom is approximately 10 times as large as the moment of a polarizedalkali (whose magnetic moment arises almost entirely from the spin of the single 𝑠shell valence electron), meaning that the dipolar interaction between two dysprosium
atoms is 100 times stronger than alkali-alkali dipolar interactions.

2.4.2

Partial Waves

For low-energy scattering we often use the method of a partial wave expansion todetermine the scattering cross section. This gives us the usual results that for very
low energies, identical bosons only interact through s-wave scattering and thus havescattering properties that are well-described by the notion of a scattering length,
and identical fermions do not interact because they do not have sufficient relative
momentum to exceed the centrifugal barrier for p-wave interactions (and are forbiddenfrom interacting through s-wave scattering) [57].
The fact that higher-l partial waves become negligible in the limit of low energy can beseen by considering how the scattering phases associated with each partial wave scalewith the momentum k. Borrowing a result from general scattering theory (reference33

37 in [34]), we know that for a scattering potential of the form 𝑈 ∝ 𝑟1𝑛 ,lim 𝛿𝑙 (𝑘) =

𝑘→0

⎧
⎪
⎨𝑘 2𝑙+1 for 𝑙 < (𝑛−3)
2

(2.7)

⎪
⎩𝑘 𝑛−2 otherwise

For the van der Waals interaction potential 𝑈 ∝ 𝑟16 which constitutes the dominant
contribution to short-range interactions discussed above, we see that 𝛿𝑠 ∝ 𝑘, 𝛿𝑝 ∝ 𝑘 3 ,and 𝛿𝑑,𝑓,... ∝ 𝑘 4 . Thus as claimed, the relative contributions of 𝑙 > 0 partial waves
vanish as k goes to zero.
For the dipole-dipole interaction 𝑈 ∝ 𝑟13 , we instead see that all phase shifts scalewith the first power of 𝑘. Thus, we must include all partial waves even at very low
energies, invalidating the treatment of the scattering problem through truncation of
the partial wave expansion to only one or two terms. Treatment of scattering due tothe dipole-dipole interaction can be carried out in the first Born approximation [25],leading to such results as the scaling of the inelastic collision rate with the magnetic
field mentioned in Chapter 1.

2.4.3

Feshbach Resonances

The s-wave scattering length, which characterizes the short range interactions of an
ultracold gas, can be experimentally tuned by taking advantage of a phenomenon
called Feshbach resonance.
Consider the interaction potential between two free atoms with some specified
quantum numbers. The set of quantum numbers is referred to as a channel. The
scattering properties between the atoms in the original "entrance" channel dependon the presence of the bound state in the other "closed" channel as long as somecoupling between the channels exists. Even a small coupling can give rise to strong
modification of the scattering in the entrance channel if the energy of the bound statein the closed channel is tuned into the vicinity of the energy of the scattering particles.
This is illustrated in Figure 2-7. In quantum gas experiments, this tunability comesfrom the fact that the different channels correspond to atoms in different combinations
34

Figure 2-7: Schematic depiction of a Feshbach resonance. Two free particles with anenergy indicated by the orange arrow interact with the orange potential curve denoted"Entrance Channel." The energy of the free particle pair is nearly-resonant with abound state in another interaction potential, drawn in blue and denoted "Closed
Channel." A coupling between these potentials means that the closed channel boundstate gives rise to a Feshbach resonance.

of 𝑚𝑗 or 𝑚𝑓 sublevels, which have different total magnetic moments. As a result, theenergy of the closed channel can be shifted up and down relative to the energy of theentrance channel by applying a uniform magnetic field and tuning its strength. The
scattering length of a colliding pair then exhibits a dependence on the magnetic field,
allowing for the contact interactions to be tuned in value, and made either attractive
or repulsive [57, 49].

Compared to alkali species, dysprosium has a large number of possible channels(e.g., 17 magnetic sublevels in the ground state of 162 Dy), and so has many more
Feshbach resonances. Many of these resonances are very narrow, and are essentiallyonly detectable by the additional inelastic losses they give rise to. Fortunately, someresonances are sufficiently broad to allow for useful control of the scattering length
[49].
35

2.4.4

Dipolar Relaxation

To consider the types of processes, both elastic and inelastic, which occur as a resultof the dipolar interaction, we can write the potential in terms of in terms of spin
operators [25, 9] 𝐹⃗1,2 as
𝑈𝐷𝐷𝐼 = 𝜇0 (𝑔𝐹 𝜇𝐵 )2𝐹⃗1 · 𝐹⃗2 − 3(𝐹⃗1 · 𝑟ˆ)(𝐹⃗2 · 𝑟ˆ)
4𝜋𝑟3

(2.8)

Expanding the numerator in terms of angular momentum projection and ladder operators, we see that the the interaction potential admits three types of processes [9]:𝐹⃗1 · 𝐹⃗2 − 3(𝐹⃗1 · 𝑟ˆ)(𝐹⃗2 · 𝑟ˆ) = 𝐹1𝑍 𝐹2𝑍
1+ (𝐹1+ 𝐹2− + 𝐹1− 𝐹2+ )2
3
𝑧 𝐹1𝑍 + 𝑟˜− 𝐹1+ + 𝑟˜+ 𝐹1− ) · (2˜𝑧 𝐹2𝑍 + 𝑟˜− 𝐹2+ + 𝑟˜+ 𝐹2− )− (2˜4where 𝑧ˆ = 𝑧𝑟 , 𝑟ˆ+ = 𝑥+𝑖𝑦
, and 𝑟ˆ− = 𝑥−𝑖𝑦[25]. The first line includes terms which𝑟𝑟
give rise to elastic scattering, the second line includes spin-exchanging terms, and thethird line includes terms which couple orbital motion and the spin states (giving riseto inelastic scattering). In general we see that the spin projection of either particle
can change by 1 or 0, and the orbital angular momentum projection can change by 0
or 2 [9].
The inelastic terms in the third line are the ones that give rise to dipolar relaxation.
Because the strength of the dipolar interaction is approximately 100 times higher
than in the alkalis, losses due to dipolar relaxation can have a much more significant
impact on experiments. For example, evaporation in a magnetic trap, which requires
atoms to be in the highest-Zeeman energy (weak field seeking) state is infeasible fordysprosium because of the high rate of dipolar relaxation a gas prepared in such astate would experience.
Since there can be no inelastic dipolar relaxation collisions among atoms spinpolarized in the absolute lowest-energy (strong field seeking) state, the most feasible
approach to performing a spin mixture experiment is to use a mixture of the lowest
36and next-lowest energy sublevels. As mentioned in Chapter 1, spin mixtures of Dy
bosons (|𝑚𝑗 = −8⟩,|𝑚𝑗 = −7⟩) have a much higher rate of dipolar relaxation thanthe equivalent spin mixture of Dy fermions (|𝑚𝑗 = −21/2⟩,|𝑚𝑗 = −19/2⟩); this has
been explored in detail in [8] and [9].

2.5Properties of the Useful Electronic Transitions

The three transitions we use (or plan to use) in our experiment span four orders ofmagnitude in linewidth, and as a result have different properties in the context oflaser cooling and trapping.

2.5.1

421 nm Transition

The 6s-to-6p 1 𝑃1 transition from the dysprosium ground state has a linewidth ofΓ = 2𝜋 × 32.2 MHz [49], making it the broadest optical transition from the groundstate. Since the photon scattering rate of an atom is limited to Γ2 in the limit oflarge saturation parameter, we have chosen to use this transition for Zeeman slowing
and angled slowing. Specifically, the maximal optical force we can generate from thistransition is𝐹max =ℎ̄𝑘Γ= 1.6 × 10−19 N
2

(2.9)

corresponding to an acceleration of over 60,000 times gravity.
The broad linewidth also means that the cross section of the atoms is relatively
insensitive to variations in the laser frequency and/or magnetic field, making it aconvenient choice of frequency for absorption imaging.
The Doppler temperature of the transition is given by
𝑇𝐷 =

ℎ̄Γ
= 772 𝜇K
2𝑘𝐵

(2.10)

which, compared to the Doppler temperature of the 626 nm transition discussed in
the following subsection, is very high and thus not the transition we chose for our
MOT light.
37

2.5.2626 nm Transition

The 6s-to-6p 3 𝑃1 transition at 626 nm has a much narrower linewidth than the 421 nm
transition, with Γ = 2𝜋 × 136 kHz [49]. This makes it useful for Doppler cooling,with a low Doppler temperature of
𝑇𝐷 =

ℎ̄Γ
= 3.3 𝜇K
2𝑘𝐵

(2.11)

which is similar to the temperatures reached by sub-Doppler cooling in alakli species.
A drawback of utilizing this transition to form a MOT is the correspondingly low
capture velocity of the transition: since the width of this transition is approximately
50 times smaller than the width of typical alkali D-line transitions, the maximum√
speed that can be slowed to a stop by a 626 nm MOT beam is approximately 50 ≈ 7times slower than the corresponding maximum speed for an alkali MOT.
As will be described in Chapter 4, we dither the frequency of this transitionin order to increase the capture velocity. The angled slowing technique, described in
Chapters 3 and 4, can also be thought of as a way of effectively increasing the capture
velocity of this narrow-line MOT.

2.6

741 nm Transition

A transition of a 4𝑓 electron to the 5𝑑 shell gives rise to an exceptionally narrowtransition at 741 nm, with a linewidth of Γ = 2𝜋 × 1.78 kHz. This transition has a
Doppler temperature of
𝑇𝐷 =

ℎ̄Γ
= 43 nK
2𝑘𝐵(2.12)

which is actually lower than the transition’s recoil temperature of𝑇𝑟 =(ℎ̄𝑘)2= 215 nK
𝑚𝑘𝐵

(2.13)

This transition has been utilized to form a MOT out of atoms initially captured in a
MOT formed on the 421 nm tranisition [43], but the final temperature of a compressed38741 nm MOT is not lower than the temperatures later achieved using 626 nm MOTs
[48].
The narrow linewidth of this transition (and resulting large tunability with respect to the natural linewidth) makes it an ideal candidate for implementing Raman
couplings with low heating rates, as will be described in the following section.

2.7

Raman Transitions in Dysprosium

When Raman transitions are induced using pairs of laser beams, there is inevitably
some heating of the atoms due to Rayleigh scattering. The photon scattering rate ofatoms being excited on a transition of width Γ by a laser of Rabi frequency Ω at a
detuning of ∆ scales as [74, 62]
𝑅s ∝

Ω2 Γ
∆2

(2.14)

likewise, the two-photon Rabi frequency of a Raman transition induced by two lasersof the same Rabi frequency with a single-photon detuning ∆ scales as
Ω2
Ω𝑅 ∝
∆

(2.15)

An ideal Raman-coupling scheme is realized when the ratio of scattering to Raman
coupling strength is small; the ratio scales as
𝑅scat
Γ
∝
Ω𝑅
∆

(2.16)

Thus as power is increased and detuning is correspondingly increased while maintaining a fixed Raman coupling strength, the heating rate due to scattering becomes
more neglibile. Since this ratio scales as the inverse detuning in units of linewidths,
it is technically easier to use AOMs to reach the desired detuning if the linewidth isnarrow; this makes the 741 nm transition an ideal line to use for Raman coupling in
Dy.
This ratio can already be made favorable given the typical laser power budgets
39

2and linewidths in alkalis; however, Raman transition amplitudes do not scale as ΩΔ

indefinitely in alkali atoms. The physical reason for this is simple: Raman transitions
among ground state hyperfine sublevels in an alkali are transitions between states
which only differ in the orientation of the electron spin. Because electric dipole
transitions do not couple to the electron spin, the spin-orbit coupling of the atom
must be used in order to use lasers to move population from one sublevel to another.
For detunings much larger than the spin-orbit coupling strength of the atom – i.e.,larger than the fine structure splitting ∆FS of the D-line excited state – the Raman2Ω
coupling strength instead scales as ΔFS[74]. This caps the ratio of scattering to
Δ2

Raman coupling at
Γ
𝑅scat
∝
Ω𝑅
∆FS

(2.17)

This result can also be found by explicitly summing up the transition dipole moments of the excited states in the D1 and D2 lines, and finding that they contribute
oppositely to the two-photon transition amplitude when the detuning is larger inmagnitude than the fine structure splitting.
In dysprosium, the ground state sublevels represent projections of the coupledorbital and spin angular momenta (and nuclear spin angular momenta in the case of
the fermionic isotopes). Since there is already spin-orbit coupling in the ground state,
arbitrarily large detunings may be used for Raman transitions without abandoning thefavorable scaling of the scattering rate with respect to the Raman coupling strength.

2.7.1

Spin-Orbit Coupling Using Raman Transitions

The physics of spin-orbit coupling, discussed in Chapter 1, can be studied in quantum
gases by utilizing Raman transitions. In some schemes, one takes advantage of thefact that Raman transitions among ground state sublevels change both the internal
state and the momentum of the atoms. Alternatively, in other schemes it is purely
the state-selectivity of Raman transitions which is leveraged, creating state-dependent
tunneling terms in optical lattice potentials. The following subsection describes eachof these types of schemes in detail.
40

|𝑒⟩∆1 , ∆2 (∆1 ≈ ∆2 )

Ω𝑒−𝑖𝑘𝑅 𝑥Ω𝑒𝑖𝑘𝑅 𝑥

| ↓⟩

| ↑⟩

Figure 2-8: Three-level Raman coupling scheme used in the seminal work on 1D SOC.
The pseudospin- up and down states are two hyperfine sublevels, typically isolated
from the the rest of the hyperfine manifold by a quadratic Zeeman shift.

Raman-Induced Spin-Orbit Coupling

By using hyperfine (or in the case of dysprosium, 𝑚𝑗 ) states as pseudospins and engineering light-matter couplings that introduce a spin-momentum interaction, SOC
can be explored using cold atoms. The natural way to implement a spin-momentum
interaction among atomic spin states is to use Raman transitions, allowing for spin
flips to be associated with momentum exchanges of up to 2𝑘𝑅 , where 𝑘𝑅 is the recoil
momentum of the Raman laser.
The scheme works as follows [62, 74, 37]. Consider a generic three-level atom withground states | ↓⟩ and | ↑⟩ and excited state |𝑒⟩ coupled to two counterpropagating
lasers along the 𝑥ˆ direction as shown in Figure 2-8. The detunings of the two transitions are given by ∆1,2 respectively. To avoid spontaneous scattering from the excitedstate, we want to consider two-photon transitions between the two ground states
while only minimally populating the excited state. This is accomplished by using
sufficiently large detunings. If we assume that ∆1,2 are much larger than the excitedstate decay rate Γ, then we can arrive at the following rotating-frame, rotating-wave
41

approximation Hamiltonian⎛ℋ=ℎ̄2 𝑘2+ ℎ̄∆1⎝ 2𝑚ℎ̄Ω𝑅 −2𝑖𝑘𝑅 𝑥𝑒2⎞ℎ̄Ω𝑅 2𝑖𝑘𝑅 𝑥𝑒2⎠ℎ̄2 𝑘2+ ℎ̄∆22𝑚(2.18)2Ω
2
where Ω𝑅 ≡ 2Δ
, and ∆ ≡ Δ1 +Δ. If the coupling strengths of the two lasers were2

different, we would additionally have Stark shift terms on the diagonals; for the caseof equal coupling strengths, we can shift the energy origin of the system and ignore
them. Defining 𝛿 ≡ ∆1 − ∆2 and shifting the energy origin again, we have⎛ℋ=ℎ̄2 𝑘2+ ℎ̄ 2𝛿⎝ 2𝑚ℎ̄Ω𝑅 −2𝑖𝑘𝑅 𝑥𝑒2⎞ℎ̄Ω𝑅 2𝑖𝑘𝑅 𝑥𝑒2⎠ℎ̄2 𝑘2− ℎ̄ 2𝛿2𝑚(2.19)In terms of Pauli matrices,
ℋ=

𝛿
Ω𝑅 ℎ̄
ℎ̄2 𝑘 2
1 + 𝜎𝑧 +(𝜎𝑥 + 𝜎𝑦 )2𝑚22(2.20)

To make this look more like the traditional SOC Hamiltonians from solid state physics,
we will perform two unitary transformations. First,⎛
𝑈 =⎝−𝑖𝑘𝑅 𝑥

0

0

𝑖𝑘𝑅 𝑥

𝑒

𝑒

⎞
⎠

(2.21)

giving:𝐻=ℎ̄Ω𝑅ℎ̄2 (𝑘𝑥 + 𝑘𝑅 𝜎𝑧 )2 ℎ̄𝛿+ 𝜎𝑧 +𝜎𝑥2𝑚2
2(2.22)

Often a rotation of coordinate system 𝜎𝑧 → 𝜎𝑦 , 𝜎𝑦 → 𝜎𝑥 , 𝜎𝑥 → 𝜎𝑧 is then performed to
put the Hamiltonian into a form more directly comparable to the SOC Hamiltonians
considered in condensed matter systems.

Optical Raman Lattices
Realizing SOC in more than one dimension requires schemes that go beyond thesimplicity of the three-level, two-photon Raman transition approach which has been42

used to generate 1D SOC. Many proposals have described schemes involving tripod/multipod level structures and additional state-selective Raman processes, butthese are relatively complex and require phase stability of many beams [74].
An alternative approach has been developed recently - the optical Raman lattice. The
scheme, which generates a spin-orbit coupled band structure in 1D or 2D (and which
may be possible to extend to 3D [74]), was proposed a few years ago by Liu and Zhang[39, 74]. The scheme involves a combination of a blue-detuned1 , retroreflected opticallattice and additional laser beams (1 in 1D, 2 in 2D) which, together with the lattice
beams, induce Raman transitions between a pair of ground states in a simple lambda
configuration. The combination of spin-independent and spin-dependent tunnelingcreates spin-orbit coupled bands, and additionally gives rise to topological effects.
Let us consider the 1D optical Raman lattice. Ignoring the spatial profiles ofthe beams, the total electric field due to the lattice laser, its retroreflection, and theRaman coupling laser is
𝐸 = 𝐸𝐿 cos(𝑘𝑥 − 𝜔𝐿 𝑡) + 𝐸𝐿 cos(𝑘𝑥 + 𝜔𝐿 𝑡) + 𝐸𝑅 cos(𝑘𝑦 − 𝜔𝑅 𝑡 + 𝛽)(2.23)

The absolute phase of the Raman beam is irrelevant and so 𝛽 will be suppressed fromhere on. We will assume that the 𝐸𝐿 fields are orthogonal to the 𝐸𝑅 field; we thenget:
1. An optical lattice along the x direction from the first 2 terms
2. A Raman coupling which varies periodically along the x direction, with a period
twice that of the optical lattice, from the combination of all three terms
3. A DC potential offset which we can ignore
1

The 𝑀0 term (see equation 2.24) gives rise to a spin-flip tunneling term. The magnitude ofthis tunneling term will depend on the spatial overlap of the Wannier functions of atoms sitting
on adjacent sites, multiplied by the spatial profile of the Raman coupling. Figure 2-9 shows therelative arrangement of the optical lattice and the Raman coupling. If the optical lattice were red
detuned, atoms would sit at the regions of highest lattice intensity, and thus the regions of highest
Raman coupling would sit inside the lattice wells - away from the regions of high spatial overlap
between neighboring Wannier functions. For a blue detuned lattice, as depicted in the figure theatoms occupy the regions of lowest light intensity. Thus, the Raman coupling strength is highest in
the regions where neighboring Wannier functions overlap.

43

Figure 2-9: The 1D optical Raman lattice scheme. Figures reproduced from [74].
Top left: the relative arrangement of the lattice beam and the additional Raman
transition-inducing beam. Top right: The 3-level system considered in the optical
Raman lattice scheme. Bottom: The relative arrangements of the optical potential
and Raman coupling, illustrating the fixed phase relationship.
Our total Hamiltonian in the basis of the ground states (spin states) of the Ramancoupled system thus becomes𝐻=−ℎ̄2 𝑘𝑥2
𝛿+ 𝑉0 cos2 (𝑘𝑥) + 𝑀0 cos(𝑘𝑥)𝜎𝑥 + 𝜎𝑧2𝑚2(2.24)

Band Structure
To understand how this Hamiltonian gives rise to spin-orbit coupling, let’s recast it
in terms of a tight-binding Hamiltonian [74]:
𝐻=

∑︁

†
𝑡𝑖,𝑗
0 𝑐𝑖𝜎 𝑐𝑗𝜎 +⟨𝑖,𝑗⟩,𝜎∑︁ 𝛿2(𝑛𝑖↑ − 𝑛𝑖↓ ) +∑︁ 𝑖,𝑗 †(𝑡𝑆𝑂 𝑐𝑗↑ 𝑐𝑗+1↓ + ℎ.𝑐.)(2.25)⟨𝑖,𝑗⟩

Given the periodicity of the lattice, we can express the spin-conserving tunnelingmatrix element as𝑡𝑖𝑗
0 = −𝑡0 = −∫︁𝑑𝑥𝜑(0)
𝑠 (𝑥)

[︂

]︂
ℎ̄2 𝜕
2
+ 𝑉0 cos (𝑘𝑥) 𝜑(0)𝑠 (𝑥 − 𝑎)2𝑚 𝜕𝑥244(2.26)

(𝑗)where 𝜑𝑠 (𝑥) is the Wannier function of an atom with spin 𝑠 on site 𝑗. We assume
the Wannier functions to be spin-independent, but this assumption can be relaxed
if the optical lattice potential is inherently spin-dependent. Noting that the sign of
the Raman coupling flips every other lattice site, we can express the spin-flipping
tunneling matrix element as𝑡𝑗,𝑗±1= ±(−1)𝑗 𝑡𝑆𝑂
𝑆𝑂(2.27)

where∫︁𝑡𝑆𝑂 = 𝑀0(0)𝑑𝑥𝜑(0)
𝑠 (𝑥) cos(𝑘𝑥)𝜑𝑠 (𝑥 − 𝑎)(2.28)

Let’s absorb the sign-changing into a redefinition of the spin-down annihilation operator:𝑐↓ → −𝑒𝑖𝜋𝑥𝑗 /𝑎 𝑐𝑗↓(2.29)

Bringing this all together our Hamiltonian is now

𝐻=−

∑︁

𝑡0 (𝑐†𝑖↑ 𝑐𝑗↑ − 𝑐†𝑖↓ 𝑐𝑗↓ ) +

∑︁ 𝛿
𝑗

⟨𝑖,𝑗⟩

2

[︃
(𝑛𝑗↑ − 𝑛𝑗↓ ) +]︃∑︁𝑡𝑆𝑂 (𝑐†𝑗↑ 𝑐𝑗+1↓ − 𝑐†𝑗↑ 𝑐𝑗−1↓ ) + ℎ.𝑐.

𝑗(2.30)
So, we see that the sign of the tunneling for one of the two spin states is effectively
reversed. We can calculate the band structure for this Hamiltonian by the usualapproach (write all terms as Fourier series, truncate the resulting reciprocal space
matrix, determine the quasimomentum-dependent eigenvalues), but the work for thelowest band has been done for us already in [74]:ℋ(𝑘) = (𝑚𝑧 − 2𝑡0 cos(𝑘𝑎))𝜎𝑧 + 2𝑡𝑆𝑂 sin(𝑘𝑎)𝜎𝑦(2.31)

where 𝑚𝑧 = 2𝛿 .
Figure 2-10 is from [74] and illustrates the origin of the SOC band structure. In
the absence of an SOC term we have two bands, each of which corresponds to one spinstate. A finite SOC term mixes the bands, giving rise to a spin texture throughout
the first Brillouin Zone.
This scheme, as well as its two-dimensional extension, has been demonstrated in45

Figure 2-10: Illustration of the band structure for the 1D optical Raman lattice.
Figures reproduced from [74]. The first panel shows the lowest band of a bare opticallattice. After our gauge transformation, the ground band is split into two spindependent bands. For nonzero SOC strength, a gap opens between the bands, andthere is now a quasimomentum-dependent spin texture.
experiments with both Rb (bosons in 2D) [72, 65] and Yb (fermions in 1D) [61] atoms.

46

Chapter 3
Experimental Apparatus
3.1

Overview

In order to produce a quantum degenerate gas of dysprosium, we must begin bycreating a (hot) thermal vapor of dysprosium and then employ a variety of cooling andtrapping techniques. Carrying out these techniques requires an array of mechanicalcomponents, electronics, and lasers. The physics behind and characterization of theseries of steps which brings us from the 103 K regime to the 10−8 K regime is describedin detail in Chapter 4. In the present chapter we will describe the details of theapparatus itself, namely the vacuum system, magnetic field generation coils, laser
systems, and control electronics.

3.2

Vacuum System

The preparation and probing of a quantum gas requires an environment that is largely
free of background gases, necessitating that we obtain ultra-high vacuum (UHV) in
our apparatus – pressures of 10−9 Torr or less. This obtained through first pumping with traditional pumps (i.e., scroll pumps and turbo pumps) into the 10−7 Torr
regime, and then baking out all of the components of our assembled vacuum system
in order to remove water and other contaminants from the steel.
Once the bakeout has been completed, we can lower the pressure into the UHV
47

Figure 3-1: The vacuum system for our experiment.
regime by firing titanium sublimation pumps and switching on ion pumps. The ion
pumps then maintain vacuum against any residual outgassing.
Figure 3-1 depicts the layout of the UHV apparatus we constructed for this experiment. Each ion pump is responsible for maintaining vacuum in one section of
the apparatus, allowing for individual sections to be closed off with gate valves (notshown in the figure) for vacuum maintenance without affecting the vacuum quality
in the rest of the apparatus. In the following subsections, the individual componentsof the vacuum system are described in detail.

3.2.1

Oven

Compared to the alkali species most commonly employed in quantum gas experiments,
dysprosium must be heated to significantly higher temperatures to create sufficient
atomic flux to load a MOT. In order to reach the temperatures in excess of 1000𝑜 C
that are required to generate the flux required to load our MOT, we purchased acommercial molecular beam epitaxy oven from SVT Associates (model SVTA-DF20-450). We obtain about a year of running time from each filling of the oven (with48around 30-50g of Dy) before the dysprosium has run out.
Due to the construction of the oven the end from which the atomic vapor is emitted
sags downward by a few degrees. To prevent this from reducing the flux reaching ourmain chamber, we attached the oven to our apparatus using a conflat bellows, whichallowed us to adjust the angle of the oven until maximum population of our MOT
was reached.

3.2.2

Ion Pumps and UHV

Our apparatus consists of three regions which are separated by low-conductance channels, allowing for differential pumping among the different regions. The three regions
are the oven chamber (connected to the rest of apparatus through a 7 mm apertureand a 10 cm long by 7 mm diameter differential pumping tube made of copper), anintermediate differential pumping region, and the main chamber region (separated
from the differential pumping region by the Zeeman slower, which itself forms aneffective differential pumping tube. The pressures we reach in each region are:1. Oven: 1.3 × 10−9 Torr
2. Differential: 1.4 × 10−10 Torr
3. Main Chamber: 6.5 × 10−11 Torr
Each has a 45L/s ion pump made by Gamma Vacuum dedicated to pumping that
region, although the ion pump in the oven region was damaged and is no longer
operated. Despite the lack of dedicated pumping in the oven chamber, we are able topump it through the differential pumping path down to this pressure.
Due to the heat delivered to the vacuum apparatus by the high-temperature oven,the pressure in the oven chamber actually reaches the 10−8 Torr regime without
additional cooling of the steel. We obtain about an order of magnitude reduction in
the pressure in the oven chamber by running cooling water through a braized coolingtube which winds around the nipple and cross which surround the oven.
49

3.2.3

Zeeman Slower

Preparation of an atomic vapor requires high temperatures, which correspond totemperatures which are much larger than the capture velocity of our MOT (on theorder of a few m/s without angled slowing beams). In order to slow the atomic beam
to within the capture velocity of the MOT, we use Zeeman slowing. The idea is asfollows:
The scattering force due to a resonant laser addressing a transition of linewidthΓ is [62]
𝐹 =

ℎ̄𝑘Γ𝑠
2(𝑠 + 1)

(3.1)

where 𝑘 = 2𝜋
is the wavenumber of the laser and 𝑠 is the saturation parameter. As
𝜆an atom decelerates due to this scattering force, it will become Doppler shifted outof resonance, and eventually no longer scatter light from the fixed-frequency laser.
We can compensate for the Doppler shift by adding a spatially varying magnetic field
profile and leveraging the Zeeman effect; optimally,𝑘𝑣(𝑧) = 𝜇𝐵(𝑧)(3.2)

where 𝑣(𝑧) is the velocity as a function of longitudinal coordinate 𝑧 along the Zeeman
slower, 𝜇 is the magnetic moment of the atom, and 𝐵(𝑧) is the magnitude of themagnetic field as a function of 𝑧. If the field is well-designed, such that the atoms
always scatter resonantly and hence decelerate at a fixed rate 𝑎, then from elementary
kinematics we can relate velocity and position as√︁𝑣(𝑧) = 𝑣𝑖2 + 2𝑎𝑧(3.3)

and hence the magnetic field should follow a square-root profile:𝑘𝐵(𝑧) =𝜇√︁
𝑣𝑖2 + 2𝑎𝑧

(3.4)

Our Zeeman slower consists of three segments of coils: a primary set of coils which
50

Figure 3-2: Increasing-field Zeeman slower. The field increases from left (oven side) toright (main chamber side) due to field generated by the windings of the "primary" coil.
The "counterwind" on the right side consists of several windings act to counteract
the fringing field of the primary coils which reach the position of the MOT (around70cm). The "bias" coil provides a homogeneous offset field along the entire length of
the Zeeman slower.

follows an approximate square-root profile; an oppositely-wound set of coils at the endof the slower which cancels out the stray fields from the primary coils inside the mainchamber; and a uniformly wound bias coil which provides an offset to the magnetic
field inside the slower. The design of this coil configuration, shown in Figure 3-2, wasperformed by colleague Li Du and will be described in detail in a future thesis.
In order to reduce the largest magnitude of the magnetic field required for Zeeman
slowing, one can use a so-called "spin-flip" Zeeman slower, in which the magnetic fieldundergoes a zero crossing somewhere along the length of the slower. The disadvantageof this design is that, since the slowing laser must now be resonant with a lower veloc51

Figure 3-3: The AR coatings of the eight viewports on the main chamber. The
532 nm/1064 nm coating also transmits well at 421 nm.

ity than in the case of a strictly increasing field slower, the off-resonant scattering fromthe beam can become a limitation in the steady-state population of the MOT. Given
the broad linewidth of our slowing transition, and reports from other dysprosium experiments about the impact of slower beam scattering on their MOT populations, we
opted for the increasing-field design (necessitating a larger counterwound segment).

3.2.4

Main Chamber

Our main experiment chamber is an octagonal chamber made by Kimble physics.
Due to the large magnetic moment of dysprosium and its dense spectrum of Feshbachresonances, we chose a titanium chamber in order to eliminate systematic uncertainties in our magnetic fields due to magnetization of the apparatus. The optical accessto the trapping region is provided by a total of 8 viewports, AR coated as shown in
Figure 3-3.
The 6 side viewports have a view diameter of 1.4" and the top and bottom viewports have a view diameter of 2.69".
52

3.3

Laser Systems

We use three different wavelengths in order to perform all cooling, trapping, and detection of the atoms in our experiment: 421 nm, 626 nm, and 1064 nm. The following
subsections discuss the locking schemes, power distribution, and frequency shifting
schemes employed for each of these wavelengths.

3.3.1

421 nm Light

Our 421 nm light is generated by an M-Squared Lasers SolsTiS Ti:Sapphire laser
which lases at 842nm, which is then frequency doubled by an M-Squared ECD-X
fixed-frequency doubling cavity. The Ti:Sapphire is pumped with a 15W 532 nm
Sprout laser made by Lighthouse Photonics.
When the alignment of the pump light and the fundamental are optimal, and theTi:Sapphire crystal has been recently cleaned, the system delivers 1.6 W of 421 nm
light. Typical day-to-day operation yields at least 1.2 W.
Laser System Layout
Figure 3-4 depicts the frequencies and power splittings of our 421 nm laser setup.
The laser itself and all frequency shifting doublepassed AOMs (with the exception of
the angled slowing doublepass) are mounted on a honeycomb breadboard made byThorlabs, which is in turn isolated from the rest of the optical table by Sorbethane
squares. This isolation helps prevent unlocking of the doubling cavity due to vibrations or disturbances elsewhere on the table.
Thermal lensing of the 421 nm beam inside the TeO2 AOMs became a limiting
factor in our ability to fiber couple any frequency shifted beams for distribution to
the experiment. In order to mitigate the effect of the thermal lensing, we lock thefrequency of the laser to the frequency needed by the highest power beam used inour experiment (the Zeeman slower beam) so that this beam can be directly split off
from the output of the laser.
The choice to frequency shift the angled slowing light independently of the remain53

Figure 3-4: Layout of the 421 nm laser system. Because it requires the most power andwould thermally lens the most in an AOM, the light used for the Zeeman slower beam
does not pass through an AOM. The frequencies shown for each AOM represent thecenter of the tunable range of each AOM; the AOM frequencies have been fine-tuned
about these values.
ing beams (i.e., imaging, locking, pumping, transverse cooling seed) was also made to
avoid thermal lensing. Specifically, if we had used the shared 350MHz AOM and anan additional 200MHz AOM to perform the frequency shift, then the total amount of
power in the 350MHz AOM would have exceeded the threshold for significant thermal
lensing, affecting the fiber coupling and coupling stability of all AOMs downstreamof it. It is for this reason that we use a separate 500MHz AOM to shift the angled
slowing light (while we do experience some lensing in this 500MHz doublepass, theimpact is limited to only the angled slowing fibers, and can be minimized by ensuring
the power sent to this doublepass is the same day-to-day).

Frequency Stabilization of 421 nm
Frequency stabilization of 421 nm lasers for dysprosium experiments is typically accomplished by using either a hollow-cathode lamp to obtain an atomic dysprosium
reference, or by locking to a mode of an ultra-low expansion (ULE) cavity. The typ54

ical approach used in alkali experiments of using a vapor cell of the atomic species
being cooled is intractable for dysprosium due to the high temperatures required forappreciable vapor pressures.
We briefly explored the approach of performing modulation transfer spectroscopyon the hot atomic beam in one of the vacuum crosses preceding the Zeeman slower.
The signal-to-noise we could obtain with this method was prohibitively small, however, with somewhere around one linewidth of uncertainty in the lockpoint arisingfrom electronic noise and polarization drifts.
Our currently implemented scheme relies on the short-term passive stability ofa WS-7 HighFinesse wavelength meter. To stabilize the 421 nm laser frequency, we
run a Python script which polls the wavemeter several times per second. If thereported frequency deviates from the desired lock frequency by at least ∆, then the
Python script sends a command to a connected Arduino Due board to increment ordecrement its DAC by the correct amount. The ∆ parameter is set by the resolutionof the Arduino’s DAC.
This scheme maintains the frequency to within about 1 MHz as reported by the
wavemeter. We have found by referencing the reported wavelengths to atomic sources
at various frequencies that the wavemeter drift is on the order of a few MHz over a
period of several days. Due to the broad 32 MHz linewidth of the 421 nm transition,
we are effectively insensitive to MHz-scale drifts of the lockpoint. We adjust for any
drifts periodically by scanning the frequency of our absorption imaging light andfinding the frequency which yields the peak OD.

3.3.2

626 nm Light

Our 626 nm light is generated by a Toptica DL-TA-SHG system (an ECDL followed
by a tapered amplifier, followed by an SHG cavity) with a dedicated digital control
module. The system produces up to 1100 mW of 626 nm light through frequency
doubling of a 1252 nm fundamental. This light is used to generate the beams for
our MOT, as well as for temporary setups such as a Doppler-sensitive fluorescence
excitation beam.
55

Figure 3-5: Layout of the 626 nm laser system. The frequencies shown for the AOMs
are the center frequencies of the tunable range for each AOM. The frequencies havebeen fine-tuned about these values.
Laser System Layout
Figure 3-5 shows the frequency shifts and power splittings for the 626 nm laser. Since
the frequency shifted MOT light is split into three separate beams to form the retroreflected beams using polarizing beamsplitters (on the output end of the fiber depictedin the figure), a quarter-wave plate is used to correct the polarization of the MOT
light before it enters the polarization-maintaining fiber.

Iodine Locking Setup
Frequency stabilization of our 626 nm laser is accomplished by performing Dopplerfree spectroscopy on a vapor cell of iodine. Our setup is depicted in Figure 3-6.
In order to avoid intensity- and polarization-dependent drifts in the baseline of
our locking signal, we use a modulation transfer spectroscopy (MTS) scheme [53].
56Figure 3-6: Iodine setup for locking the 626nm laser. A frequency modulated pumpbeam produces an amplitude modulation in the detected probe beam. Two iodine
cells are used in sequence to increase the signal-to-noise ratio. A fiber is used to clean
up the probe light before detection in order to prevent backreflected pump light fromreaching the photodiode.
In MTS, the power is split roughly evenly between the traditionally named "pump"and "probe" beams. An error signal is generated in the detected probe beam whena frequency-modulated pump beam mixes non-linearly with the unmodulated probe
beam inside the vapor cell, generating sidebands on the probe beam which then beatwith the center frequency of the probe on the photodector. Demodulation of thedetected light yields an error signal which is centered about zero, and whose zerocrossing is independent of the amount of light in each beam. This makes the error
signal an ideal feature to stabilize the laser to. The error signal generated by our
MTS setup, as well as the Doppler-free peaks generated by unmodulated saturated
absorption spectroscopy, are shown in Figure 3-7.

Iodine Lock Performance
The performance of our lock to the iodine cell is limited by the small absorption for theline we are addressing in the vapor cells, as well as by backreflections of the modulated
probe beam off various surfaces in the spectrocopy setup and onto the photodetector,
where they can beat with the probe beam and cause offsets in the resultant locking
signal. The slope to which we lock spans about 5 MHz as measured by our wavemeter,
57

Figure 3-7: Spectroscopy of iodine. The top curve is an example trace of a measured
error signal generated by our modulation transfer spectroscopy setup. The bottom
curve is a corresponding (unmodulated) saturated absorption spectroscopy trace.

58

so in order to precisely address the 626 nm transition (Γ = 2𝜋 × 136 kHz) we want
to stabilize the laser to around the percent level with respect to this error signal.
Backreflections of the pump beam, even with all optical surfaces slightly tilted tominimize the effect, give rise to 10% or more shifts in the baseline of the error signal
which vary interferometrically as the optics vibrate. The backreflections were almost
completely eliminated by using an optical fiber to mode-filter the detected probe
beam and hence reject most of the backreflected pump light, as shown in the figure.
The width of the error signal when the laser is locked, together with the uncertainty in the peak OD vs. frequency that we measure when taking absorption images
with the 626 nm laser, indicate that we obtain about 2 to 3 Γ of precision with ourlock. Improving the SNR of the locking signal with additional light or a longer path
length may help push this to the sub-Γ level.

3.3.31064 nm Light

Our 1064 nm light is generated by a pair of 45 W Azurlight Systems amplifiers which
are seeded by the same Coherent Mephisto laser. The output of each amplifier goes
into an AOM in order to both provide intensity control and to frequency-shift thetwo amplifiers away from one another. The diffracted order of each AOM is coupled
into a photonic crystal fiber for delivery to the main chamber. In order to improvethe coupling into the optical fiber, a slightly defocused 3-to-4 telescope is placed in
between the amplifier and the fiber for each arm. A backside-polished mirror on the
output end of each fiber allows for monitoring of the output power and stabilizationof the delivered intensity by servoing the RF amplitude of each AOM.

Laser System Layout
Figure 3-8 shows the layout of the two amplitude control/fiber coupling setups forour 1064 nm light.
59

Figure 3-8: The fiber coupling and amplitude control setups for the two ALS amplifiers. The modes emerging from the ALS outcoupling heads are slightly converging,with a waist of 1.0 mm. The telescopes shown in the figure are slightly defocused inorder to correct for the slightly focused beam and to compensate for thermal lensingthat occurs inside the AOM.

3.4

Layout of Optical Beams

All beams are delivered by optical fibers from their respective laser sources to one ofof the optical breadboards surrounding the experimental chamber. Figure 3-9 shows
the geometry of all of the cooling, trapping, and imaging beams which pass into or
through the main chamber as viewed from the top of the machine.
The 421 nm Zeeman slower beam has a power of 400 mW and enters the apparatus
through a viewport followed by a 45-degree in-vacuum mirror which has line-of-sightto the oven. This approach was chosen to prevent the entrance window for the Zeeman
slower beam from becoming opaque due to accumulation of Dy from the atomic beam.
The beam is blown up with a 30-to-250 telescope which has been adjusted so thatthe beam comes to a focus at the oven; this roughly matches the diverging profile of
the atomic beam.
The MOT light consists of 3 retroreflected beams of 626 nm light, each with apower of up to 60 mW. In order to eliminate the need for beam-expanding telescopes
and save space on the optical breadboards, the beams are shaped by placing 100 mm
60

Not shown are the vertical MOT beam and vetical imaging path. These beams are
identical to the ones for the two horizontal arms. They originate on the bottom andare combined on a dichroic mirror for passage through the vertical viewports, before
being split on another dichroic on a small breadboard suspended above theexperiment. This breadboard holds the MOT retroreflection mirror, the imaging
lenses, and a camera for vertical imaging.
Figure 3-9: The layout of all beams in our experiment which enter the main chamber
through a horizontal viewport. The atomic beam arrives from the top of the image,and is represented by the thick grey arrow. Note that the angled slowing beams crossthe atomic beam upstream of the position where the MOT, ODT, and imaging beams
cross.
Not shown are the vertical MOT beam and vetical imaging path. These beams are
identical to the ones for the two horizontal arms. They originate on the bottom
and are combined on a dichroic mirror for passage through the vertical viewports,
before being split on another dichroic on a small breadboard suspended above theexperiment. This breadboard holds the MOT retroreflection mirror, the imaging
lenses, and a camera for vertical imaging.
61

aspheric lenses directly after the output fibers. 2-inch mirrors and polarizing beamsplitting cubes then clean up the polarization and steer the beams into the chamber.
The 421 nm angled slowing beams are delivered through the same windows as we
use for MOT light, and as a result require a compact beam delivery setup. Eachbeam is launched from an optical fiber with a 50mm asphere (giving a beam waistof 4.5 mm). Since we assume that atoms are pumped into the 𝑚𝑗 = −8 state by
the Zeeman slower, and since the polarization out of the PM optical fiber is roughly
linear, we use a single quarter-wave plate on each arm to set the polarization of theangled slowing beams for maximum enhancement of the MOT population.
The 1064 nm optical dipole trap beams enter through the same viewport as the
421 nm imaging light. The beams start with a waist of 1.6 mm, and propagate inparallel to the focusing lens with a separation of a few cm. The 200 mm focusinglens both crosses and focuses the beams, creating beams with waists of about 40𝜇m
which cross at about 5 degrees. The maximal angle that we could cross our beams
at, given the size of the viewport, is about 11 degrees, but the current arrangementof MOT and imaging optics constrains us to use a smaller initial separation of the
ODT beams and hence obtain a smaller crossing angle.
The 421 nm horizontal imaging beam has a diameter of about 1.8cm, and has its
polarization set to correspond to 𝜎 − when the imaging guide magnetic field is on.
To work with a magnification of 1, we use a 2 lens, 4𝑓 configuration consistingof one 200 mm achromatic doublet positioned 𝑓 away from the atoms, 400 mm of
spacing between the lenses, and another 200 mm achromat positioned 𝑓 away fromthe camera sensor.
To instead work with a magnification of 4, we insert a telescope consisting of a
300 mm and 75 mm lens in the region between the two 200 mm achromats.
The 421 nm vertical imaging beam also has a diameter of about 1.8cm. We obtain amagnification of 3 with a 1-to-1 4𝑓 configuration consisting of two 200mm achromats,followed by a 1-to-3 4𝑓 configuration consisting of a 100 mm and 300 mm achromat
which "relays" the image to the camera with magnification.
The 421 nm transverse cooling light is generated by injection-locking two Nichia
62

Main Coils(Along Z)
Gauss/Amp in
Helmoltz
Configuation
Gauss/cm/Amp in
Anti-Helmholtz
Configuration

X Bias Coil Y Bias Coil Z Bias Coil

3.47

0.67

0.67

0.59

0.45

N/A

N/A

N/A

laser diodes (one per axis). The diodes are held in Thorlabs LDM56F laser diodeheads, and output 90-95 mW (power measured before an optical isolator). The beams
are expanded to approximately 1cm in size and then shaped with a f = -50 mm,
f=+150 mm cylindrical telescope in order to match the roughly match the light distribution spatial to the profile of the atomic beam.

3.5

Magnetic Field Coils

We need to generate inhomogeneous magnetic fields for Zeeman slowing, magnetooptical trapping, evaporation, and levitation. We also need to generate homogeneous magnetic fields for background/earth’s field compensation, steering of forcesby magnetic field gradients during and after evaporation, and utilization of Feshbachresonances.
The magnetic fields around the chamber are generated by four sets of coils. The
largest set of coils is oriented along the vertical ("z") axis of the main chamber, andis connected to an H-bridge so that it can either be used to generate magnetic field
gradients (such as for the MOT) or large homogeneous fields along the z direction.
Three sets of lower-current coils allow us to produce roughly homogeneous fields
along the x, y, and z directions in the main chamber, allowing for cancellation ofstray magnetic fields or the creation of bias fields in any direction up to a few Gaussin magnitude.
The table above summarizes the properties of all of the field-generation coils
around the main chamber. The magnetic field profile of the Zeeman slowing coils
is detailed in Chapter 4.
63

3.6

Electronics and Control

Experimental control is accomplished through a combination of commercial andhomebuilt instrumentation.

3.6.1

Control and Power Electronics

We use an array of commercial power supplies to generate all of the currents for ourmagnetic field coils. Our compensation coils are driven by Agilent E3614A supplies
with up to 6 A of output current each. Our main gradient/Feshbach field coils are
driven by a Delta Elektronika SM18-50 supply which can supply up to 50 A of current.
The primary, counterwind, and bias coils of the Zeeman slower are respectively driven
by a Delta SM35-45, a Delta SM15-100, and an Agilent N7971A.
Most of our AOMs are driven by a full-featured driver circuit designed by Colin
Kennedy and Michael Cantara. This will be detailed in future theses and/or publications.
To toggle devices on and off during the experimental sequence, we use a National
Instruments PXIe-1073 chassis with a PXIe-6535 digital output card to generate TTLcompatible digital control signals. The clock for these digital outputs is derived froman Opal Kelly FPGA which is configured to work with our in-house control software
suite (described below); the FPGA provides us with a variable timebase, allowing for
fewer buffer samples to be generated during portions of the experimental sequencewhen control values are static or change slowly, and more samples to be generated
when devices must be controlled more quickly.
Analog control voltages are used to vary VCO frequencies, laser powers, andmagnetic fields during the experiment. These voltages are generated by a newly
developed low-noise system using high-precision DACs and floating power supplies
which has been fully integrated into the Cicero word generator suite. The details of
this system can be found in Chapter 5.
64

3.6.2

CamerasWe use two ZWO ASI183MM CMOS cameras to perform absorption imaging andreal-time fluorescence detection of the atoms in our experiment. These cameras have
2.4 𝜇m pixels with a total resolution of 5496×3672 pixels.
Because the cameras are designed for astronomy applications, they do not have
hardware triggering capabilities. We must trigger their acquisition through softwarecommands, which introduces some tens of milliseconds of timing jitter of the camera’s
exposure window with respect to the rest of the experiment.

3.6.3

Cicero Word Generator

We control our experimental sequence using the Cicero word generator [32] and associated applications, which were developed at MIT and have since been adopted at
institutions around the world. Cicero breaks the experimental sequence into "steps,"
each of which has a specified duration and can involve the toggling of digital controls,
the arbitrary variation of analog control voltages, and the transmission of messages
to devices over RS-232 or GPIB connections.
Some extensions to the functionality of Cicero, as well as new applications, havebeen developed for our experiments. The details of these new developments aredescribed in Chapters 5 and 6.

65

66

Chapter 4
Cooling and Trapping of Dysprosium
4.1

Overview

The process of creating and detecting a Bose-Einstein condensate of dysprosium inour vacuum chamber consists of many steps which occur sequentially over about
40 seconds. The experimental sequence can be broken down into a few conceptual
phases:1. Loading of a magneto-optical trap (MOT)
2. Compression of the MOT
3. Transfer into a crossed optical dipole trap (XODT)
4. Evaporation
5. Imaging
Figure 4-1 shows a complete experimental sequence as it appears in our lab control
software, with the regions of the sequence corresponding to the five aforementioned
phases highlighted in different colors.
Each of these phases requires several experimental technologies to work together
in parallel and/or in sequence. In this chapter, we will discuss each of the components of each of the five phases of the experimental sequence, including the physicalmechanisms, characterization, and optimization of each component.
67

Figure 4-1: The five phases of our experimental sequence as viewed in the Cicero
control software. Each column represents a timestep of fixed duration. The white
traces in the middle of the pane represent the value of analog control channels during
the sequence, and the colored boxes at the bottom of the pane represent the valuesof digital control channels.

4.2

Loading a Magneto-Optical Trap

Loading a MOT with dysprosium atoms is the first step in creating our BEC. While
MOT loading corresponds to a single, static set of control values in our experiment, itis the step that involves the greatest number of separate technologies and techniques
working together.
An unusual aspect of our MOT loading approach is the inclusion of two "angled
slowing" beams. As will be described in the subsection below, these angled slowing
beams increase the population of our MOT by more than an order of magnitude.
While in our specific case the enhancement due to angled slowing works to offset
some limitations specific to our apparatus (giving us MOT populations similar to
those reported by other Dy experiments), we believe that our demonstration of its
effectiveness will also be of general interest to other experiments which use narrowtransitions to load MOTs.

4.2.1

Dysprosium Oven

Formation of the hot atomic beam, which is subsequently slowed and then cooled
and trapped, occurs in the oven. In order to achieve the atom flux necessary to load68

a MOT, we need to heat solid dysprosium to high enough temperatures to create asufficient vapor pressure. Calculating the vapor pressure from the Antoine equation(using the coefficients found in [49]), we can calculate the flux as a function of velocity
which emitted from the oven as [49](︂𝑓 (𝑣, 𝑇 ) =

𝜋𝐷
2

)︂2

𝑝(𝑇 )
𝑘𝐵 𝑇

(︂

𝑚
2𝜋𝑘𝐵 𝑇

)︂3/2

− 𝑚𝑣

2

𝑣 3 𝑒 2𝑘𝐵 𝑇 sin2 𝜃(4.1)

where p(T) is the saturated vapor pressure, v is the longitudinal velocity, T is thetemperature, D is the diameter of the aperture of the oven, m is the average massof Dy, and 𝜃 is the angle into which emission is being considered. The selection of
the oven temperature affects not only the flux out of the oven, but also the velocity
distribution of atoms in the atomic beam and the lifetime of the dysprosium reservoir
loaded into the oven. In practice, selection of the temperature of the oven is acompromise between maximizing flux and minimizing how often the oven needs tobe reloaded (a procedure which requires breaking vacuum). The parameters of the
Zeeman slower can be adjusted to match the velocity distribution of the atoms theresults from the selected temperature.
We operate our oven, which has an output aperture diameter of about 2mm,
at 1250 ∘ C, which is similar to the temperatures used by other groups. This yields
roughly double the flux obtained from running the oven at 1200 ∘ C, and gives rise
to a velocity distribution which is depicted in Figure 4-2. A complete filling of theoven, consisting of about 30 g of Dy, yields about one year of running time (the oventemperature is turned down by several hundred degrees when the experiment is notrunning in order to conserve Dy).
The way in which oven is constructed naturally causes it to sag by a few degrees.
As a result, the maximal flux out of the oven was not directed along the axis definedby the apertures and Zeeman slower. We corrected for this by mounting the ovento the experiment with a 4.5-inch conflat bellows, which was then wedged at a small
angle.
69

Figure 4-2: Velocity distribution out of the oven at 1250 ∘ C. The most probablevelocity is 483 m/s. The scale of the vertical axis corresponds to the flux for a
specific value of 𝜃.

4.2.2

Zeeman Slowing

As described in Chapter 3, we use a Zeeman slower to reduce the longitudinal velocityof the atomic beam in preparation for MOT loading. To optimize the performanceof the Zeeman slower, we analyzed the velocity distribution via Doppler-sensitive
spectroscopy. For this, we shined a 626 nm laser beam at small angle from the normal to the atomic beam inside the main chamber. The frequency of this beam was
swept across a range of about 100 MHz, and the resulting fluorescence of the atomic
beam was measured using a photomultiplier tube. The detected fluorscence vs. laser
frequency reveals the range of Doppler shifts of the atoms, and thus their velocity
distribution.
Initially, we turned off the Zeeman slower coils and adjusted the frequency of the
421 nm slowing beam until we observed a feature consisting of a hole and a pileup in the velocity distribution near the most-probable velocity. The detuning fromresonance that corresponds to the most-probable velocity (483 m/s) is about 1.1 GHz.
An example of this hole burning is shown in Figure 4-3.
We then turned on the Zeeman slower coils to varying percentages of the fullcurrents we calculated to slow atoms from 480 m/s to a few m/s. We kept the primary70

Figure 4-3: "Hole-burning" in the velocity distribution of the atomic beam by the
Zeeman slowing laser beam with the slower coils off.

and counterwind coils in a fixed ratio which was calculated to give rise to zero field at
the position where the MOT would form (see the future thesis of colleague Li Du for
details of the coil design and calculations). After turning on the magnetic fields, thepointing and polarization of the Zeeman slower beam was adjusted until pile-up of
population at lower velocity groups started to appear in the Doppler-sensitive spectra.
We incrementally increased the currents of the primary and counterwind coils to movethe peak corresponding to the Zeeman-slowed atoms to lower velocities. Figure 4-4
shows an example spectrum where flux has been piled up into a lower velocity classby the action of the Zeeman slower.
For reasons that will be discussed in detail in the subsection on angled slowing,
we were unable to move this peak to within the estimated capture velocity of our
MOT; as the Zeeman slower field was increased to larger values, the observed peakwould move to smaller velocities but decrease in height. Without the addition angled
slowing beams, only the tail of the slowed distribution could be captured by the MOT.
71

Figure 4-4: Zeeman slowing. Atoms have piled up into a lower velocity class due tothe action of the Zeeman slower.

Final adjustment of the Zeeman slower fields was done by observing the fluorescenceof a loaded MOT in steady-state, and maximizing the fluorescence (and hence thepopulation) with respect to the slower fields.
Varying the curent in the bias coil is almost entirely equivalent to varying thedetuning of the Zeeman slower beam. In principle, the bias field can be set such that
the Zeeman slower beam is resonant with atoms far past the most-probably velocity
in the region before the bias coil begins. The bias can then shift the resonances of
the atoms by up to several hundred MHz as soon as they enter the slower, allowing
for the capture of a larger portion of the distribution out of the oven. In practice, it
is much easier to vary the bias field than to vary the Zeeman slower laser detuning
due to the arrangement of our laser systems, and so we have empirically optimized
the slowing by setting the laser to 1.1 GHz of detuning and varying the bias current
to maximize MOT population.
72

4.2.3

Transverse Doppler Cooling

While we usually consider the longitudinal velocity distribution of the atomic beam,
there is also a small transverse velocity distribution. The flux delivered to the MOT
in the main chamber can be reduced due to spreading of the atomic beam (andsubsequent loss of atoms) before the Zeeman slower, within the Zeeman slower, or
between the Zeeman slower and the MOT. Applying a stage of Doppler laser coolingto the atomic beam before Zeeman slowing allows us to reduce the losses which occur
before the Zeeman slower.1
Doppler cooling works by the following mechanism [62]. Let us consider the caseof cooling and motion along a single dimension. The scattering force due to a laser
close to resonance with a single transition can be written as𝐹 =𝑠Γ ℎ̄𝑘2
2 1 + 4ΔΓ2

(4.2)

where 𝑠 = 𝐼/𝐼sat is the saturation parameter, Γ is the linewidth of the transition,𝑘 = 2𝜋is the wavenumber of the laser, and ∆ is the detuning from resonance.
𝜆The total force due to two couterpropagating lasers is simply𝑠Γ𝐹sum =
2

(︃

ℎ̄𝑘
4Δ2

−

1 + Γ21

ℎ̄𝑘

)︃
(4.3)

4Δ2

1 + Γ22

If an atom is moving with velocity 𝑣, then it will see the right-moving (left-moving)laser light as lower (higher) in frequency due to the relativistic Doppler effect. For
small velocities, the Doppler shift is given by 𝑘𝑣 where 𝑘 is the laser wavevector. For
two counterpropagating lasers of equal frequency addressing an atom moving at 𝑣,
we then have a total force𝑠Γ
𝐹sum =
2

(︃

ℎ̄𝑘
2

1 + 4(Δ−𝑘𝑣)
Γ2

1

−

)︃

ℎ̄𝑘
2

1 + 4(Δ+𝑘𝑣)
Γ2

(4.4)

Our simulations have actually shown that while the transverse Doppler distribution is not significantly narrowed by the Doppler cooling we employ here, the amount of flux which makes it
through to the Zeeman slower is increased. These simulations will be discussed in the future thesisof colleague Michael Cantara.

73

Figure 4-5: Optical molasses. The sum of the two Lorentzians in the expression for
the total force, offset from one another in velocity space by the Doppler shift, gives
rise to a feature with a roughly linear force as a function of velocity near the origin.
For negative detunings, this force is a restoring force that moves atoms toward 0
velocity.
For red detunings, the sum of these forces gives rise to what is called an opticalmolasses, depicted in Figure 4-5.
For a range of velocities centered about 𝑣 = 0, atoms experience a restoring force
toward 𝑣 = 0. An ensemble of atoms with some thermal distribution of velocities willthus be cooled by the molasses. In the absence of technical limitations, the lowest
temperature that can be reached by Doppler cooling is a result of the fluctuationsin the force that arise from the discrete nature of the scattering process (i.e., momentum is imparted in discrete amounts as individual photons are scattered). Thistemperature is:𝑇 =ℎ̄Γ 1 + (2∆/Γ)2
4𝑘𝐵 (−2∆/Γ)

(4.5)

ℎ̄Γ
2𝑘𝐵

(4.6)

which is minimal at ∆ = − Γ2 :𝑇𝐷 =This limit 𝑇𝐷 is called the Doppler temperature.
74

As described in Chapter 3 and illustrated in the figure therein, our transverse
cooling stage consists of two retroreflected beams which are shaped with cylindricallenses to roughly evenly distribute the cooling light over the atomic beam. To optimize
the performance of our transverse cooling, we set the detuning of the injection lock
seed light to − Γ2 and aligned one axis at a time. We used the integrated 626 nm MOT
light fluorescence, as detected by our camera running in a continuous acquisitionmode, to obtain a real-time measurement of the MOT population as we aligned thetransverse cooling beams.
We saw a maximal total gain of about 40% in the population of the MOT, as
measured with absorption imaging, after optimizing the alignment and detuning of
the transverse cooling beams [48]. This is a smaller gain in population than has
been reported by similar experiments which employ transverse Doppler cooling. We
attribute this to two factors.
First, we do not have enough optical power to see the maximal population gain
which is possible with transverse cooling. When running the Zeeman slower andloading a MOT, we are only able to provide about 85 mW of cooling light per axis,
limited by the output power of our injection-locked laser diodes and subsequent losses
through optical isolators. We investigated the gain in atom flux reaching the mainchamber vs. transverse cooling power more thoroughly by diverting the majority ofthe power of the 421 nm laser directly to the transverse cooling beam paths, andmeasuring the gain in flux using 626 nm fluorescence and a PMT (as described in theabove subsection on Zeeman slower optimization). We found that the measured flux
continued to increase until about 200-250 mW of cooling light was used on each axis,
indicating that we do not generate enough power with our injection locked diodes to
see the maximum flux enhancement from the transverse cooling stage.
Second, our experiment is particularly susceptible to losses due to transverse
spread of the atomic beam during the free-flight of the atoms. Although the transverse velocity distribution is narrowed by transverse cooling prior to the slower, it canbe re-broadened due to recoil heating along the transverse directions by the Zeeman
75slower laser beam 2 . This, combined with the unusually long free-flight time (see thefollowing subsection for detailed discussion of this), reduces the enhancement in MOT
loading we should expect to see from the transverse cooling stage.

4.2.4

Angled Slowing

Atoms that have been slowed by a Zeeman slower must travel some nonzero distanceat 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 thebeam 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 and the narrow linewidth of the MOT transition creates asituation in which the transverse spread plays a significant role. To clarify this point,
we compare Dy to the more common alkali MOTs.
The capture velocity of a MOT can be estimated by calculating the largest velocity
that can possibly be slowed to a stop within the profile of the MOT beams. Assumingthat the atoms scatter photons at the maximum possible rate Γ2 across an entire beam
diameter 𝐷, an expression for the capture velocity is given by√︂𝑣cap =

2

ℎ̄𝑘Γ
𝐷
2𝑚

(4.7)

where 𝑚 is the atomic mass and 𝑘 = 2𝜋
is the wavenumber of the MOT light.
𝜆The spatial spread 𝜎 of the atomic beam can be estimated as𝜎 ≈ 2𝑑𝑣trans
𝑣long

(4.8)

where 𝑑 is the free-flight distance, 𝑣trans is the RMS transverse speed, and 𝑣long is theaverage longitudinal speed. Collimation of the atomic beam by one or more apertures
2

To slow an atom using the 421 nm transition from the most probable velocity of 483 m/s to about
10 m/s would require about
𝑁 = 80, 000 photons. The transverse spread due to these scattered√photons is thus about 𝑁 ℎ̄𝑘𝑚 = 1.6 m/s.

76

typically leads to a transverse velocity distribution with an RMS speed around 1% of
the average (unslowed) longitudinal velocity 3 .
Let us compare the case of a 87 Rb MOT to a 162 Dy MOT, taking typical valuesof 𝐷 = 2 cm for the MOT beam diameters. For Rb, 𝜆 = 780 nm, 𝑚 = 87 amu,
and Γ = 2𝜋 × 6 MHz. This corresponds to a capture velocity of 67 m/s according toEquation 4.7, but at velocities this high atoms are Doppler shifted far out of resonancewith the MOT beams. A more realistic estimate of the Rb MOT capture velocity
might be half this, or 33 m/s (similar to the value reported in in [63]). A typicalinitial most-probable velocity for atoms effusing from a Rb oven is about 330 m/s,
and so 3.3 m/s is a reasonable estimate of the RMS transverse speed of the atoms.
If we consider an atomic beam slowed to the capture velocity value and an example
free-flight distance of 10 cm we can estimate the spread of the atomic beam to be
𝜎Rb ≈ 2 cm

(4.9)

which is still comparable to the size of the MOT beams.
For a Dy MOT, 𝜆 = 626 nm,, 𝑚 = 162 amu, and Γ626 = 2𝜋 × 136 kHz. This gives
a capture velocity of 8 m/s. The most probable velocity of the Dy atoms effusingfrom our oven is about 480 m/s, so 4.8 m/s is a reasonable estimate of the averagetransverse speed. For a free-flight distance of 10 cm, we estimate the spread of theatomic beam to be𝜎Dy ≈ 12 cm(4.10)

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 spreading of
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 the larger Zeeman slower laser detuning, further exacerbates
the transverse spreading problem. One reason is that the larger detuning reduces
3

When transverse Doppler cooling of the atomic beam is employed prior to Zeeman slowing, thetransverse velocity distribution can become much narrower; however, the light scattering during
Zeeman slowing causes the transverse velocity distribution to re-broaden.

77the amount of off-resonant slowing that occurs during the free-flight distance. The
more off-resonant slowing that occurs during the free-flight, the larger the initial exit
velocity from the Zeeman slowing region can be. We can estimate the typical effect ofoff-resonant slowing in a Dy experiment: a typical Zeeman slower beam detuning ina spin-flip slower is around −18Γ421 , with (resonant) saturation parameters of 𝑠0 ≈ 1[48]. If we assume that the slowed atoms scatter at a (detuned) saturation parameter𝑠0−4of 𝑠 = 1+4Δ
over a 10 cm free-flight distance, then we can estimate
2 /Γ2 ≈ 7.7 × 10

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 distance due to the need for fieldcancelling coils near the MOT. In an increasing-field Zeeman slower, the largest numbers of windings are closest to the MOT. As a result, it is 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 the position of the MOT is close
to zero. Slowed atoms must thus travel an extra distance of several cm comparedto the 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 byallowing 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 ofΓ421 intersect the atomic beam to provide a net longitudinal slowing force, slowing theatomic beam to within the capture velocity of the MOT. The transverse componentsof the two beams’ scattering forces are oppositely oriented and thus cancel. In effect,
the addition of the angled slowing beams increases the capture velocity of the MOT.
The advantage of using a pair of angled beams over a single beam colinear withthe 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 than the recently reported core-shell MOTs foralkaline-earth-like atoms [35].
Without employing angled slowing, optimization of our Zeeman slowing parame78

Figure 4-6: Population of the MOT as a function of loading time. The blue curve(circles) shows the population when both transverse cooling and angled slowing withthe optimal parameters of 𝛿 = −50 MHz and 7 mW per beam are employed. The
purple curve (diamonds) shows the population when angled slowing with the same
optimal parameters is employed, but the transverse cooling beams are turned off. Theorange curve (squares) shows the population when both the transverse cooling andangled slowing beams are turned off, multiplied by a factor of ten for visual clarity.

79ters led to a steady-state MOT population of about 107 atoms. We observe more thana factor of 20 gain in the final population of our MOT when using angled slowing.
As described in the next section, we found optimal 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 4-6 shows the population with and without angled slowing as a function of
MOT loading time. As shown in the figure, the improvement in MOT population
we see due to transverse Doppler cooling is approximately a factor of 1.4, limited bylaser power. This factor is independent of whether or not angled slowing is employed.

Optimization of Angled Slowing
We determined the optimal alignment of our angled slowing beams by maximizing
the steady-state MOT population as measured by the integrated 626 nm fluorescencescattered by the MOT. While 𝜎 − light is used to pump 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 the polarization 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 tothis 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 beam in
our angled slowing light.
Given this power constraint, we looked for an optimal combination of power anddetuning for the angled slowing beams. Figure 4-7 shows the population after a fixedload time and fixed compression sequence (see the following section) as a function of80

both detuning (always red) and power per beam. The uncertainty in our beam power
was at most ±15%, and the uncertainty in our frequency was about ±2 MHz, withthe latter uncertainty arising from drifts in our wavemeter.
The general trends are explained by a simple physical picture: 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 nearlyzero-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 turn theatoms 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 field in the slower, which isequivalent 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 thestep size we explored (steps of 1 A ≈ Γ421
of effective Zeeman slower detuning). We
2
also found that the same bias field was optimal when loading a MOT without angled
slowing. Together, these observations suggest that the final velocity distribution of
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.

4.2.5

Magneto-Optical Trapping

Once the atomic beam has been slowed by the Zeeman slower and the angled slowing
beams, we use a magneto-optical trap (MOT) to capture as many of the slowed
atoms as possible, and further cool them down. A MOT provides a restoring force
in both position space and in velocity space, and is formed by a combination of an
inhomogeneous magnetic field and one or more pairs of counterpropagating lasers
81Figure 4-7: Optimizing angled slowing. Dependence of the MOT population on
the detuning and power of the angled slowing beams. A fixed loading time andcompression sequence were employed for all of the data shown. The powers in thetwo beams were balanced to within 15%. The detuning from the 421 nm resonancewas known to within ±2 MHz.

82

with opposite4 circular polarizations. For a 3D MOT, as we have in our experiment,
there are three pairs of counterpropagating beams.
The confinement in position space comes from the interplay of the spatially varying Zeeman shift due to the magnetic field and the laser detuning. This works asfollows (see Figure 4-8 for an illustration for the simple case of 𝐽 = 0, 𝐽 ′ = 1. The
case of higher-𝐽 is not qualitatively different, since optical pumping moves the population into the stretched state.). A pair of magnetic field coils, in anti-Helmholtz
configuration, gives rise to a quadrupole field profile near the region in the center ofthe coil arrangement. The field is zero at the center of the coil arrangement, and eachcomponent of the magnetic field varies linearly near the zero.
For simplicity, we will restrict the discussion to one dimension for now. The
linearly varying magnetic field gives rise to a linearly varying Zeeman shift as:∆𝐸 = −𝑚𝑗 𝑔𝑗 𝜇𝐵 (∇𝐵)𝑥(4.11)

Both lasers have the same circular polarization orientation with respect to their propagation direction; because the magnetic field flips from positive to negative at theorigin, this means that on each side, one laser has 𝜎 + polarization and one laser has
𝜎 − polarization. At zero field, both lasers are red-detuned. As atoms move to one sideor another, the 𝑚𝑗 → 𝑚′𝑗 = 𝑚𝑗 − 1 transition gets shifted down into resonance withthe laser with 𝜎 − polarization. As long as the directions of the circular polarizationsof the beams have been chosen appropriately, the beam which provides a force towardmagnetic field zero will always be the 𝜎 − beam.
In addition to providing spatial confinement, a MOT also cools through the
Doppler mechanism described in the previous subsection; each pair of beams forms
an optical molasses.
Our MOT is formed by three retroreflected 626 nm beams. Each beam has a 1/𝑒2
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
4

The circular polarizations are opposite with respect to the magnetic field at any point inside the
MOT, but are the same with respect to the lasers’ propagation directions.

83

Figure 4-8: Operation of a MOT in 1D. Both beams are right-hand circularly polarized. On the left side, where the magnetic field points left, Beam 1 is 𝜎 − and thus isbrought into resonance by the Zeeman shift. The atoms are thus kicked more strongly
by Beam 1 than by Beam 2 (which is 𝜎 − on this side of the origin), and experiencea net force toward the center. On the right side, the roles of the beams are reversed,
and the atoms once again experience a net force toward the center.

84

Figure 4-9: Dependence of the MOT population on the laser detuning. The horizontal
axis is in units of control volts applied to a VCO which determines the MOT frequency(10 mV ≈ 100kHz.)
direction5 is approximately 2.5 G/cm.
To improve the capture velocity of the MOT we dither the frequency of the MOT
light using a double-passed AOM. The dithering occurs at a frequency of 120 kHz
and broadens the laser linewidth to 2.6 MHz (30Γ626 ).
To optimize the number of atoms captured by our MOT, we optimized the laser
detuning and magnetic field gradient. For a fixed gradient, we scanned the detuningof the laser light in steps of about 100 kHz and chose the value which gave the largest
population. Population was measured by compressing the MOT to increase the signalto-noise of our imaging procedure; compression and imaging are described in detailin later sections of this chapter. An example scan of population vs MOT detuning isshown in Figure 4-9
Although the minimal temperature of the MOT occurs at ∆ = − Γ2 , more atoms
can be loaded into the MOT if the detuning is initially larger (this gives a wider
capture range in velocity space for the molasses and a larger cloud diameter). The
minimal temperature can then be reached later during the compression phase, when
the frequency is brought closer to resonance.
The combination of a large atomic mass (162 amu in our particular case) and anarrow MOT transition gives rise to an interesting effect. The ratio of the maximal
5

By symmetry, the magnetic field gradient is the same in any direction normal to the axis con𝜕𝐵𝑦
𝜕𝐵𝑧𝑥necting the two coils. If we call this axis 𝑧, then by ∇ · 𝐵 = 0 we know that 𝜕𝐵𝜕𝑥 = 𝜕𝑦 = −2 𝜕𝑧85force (for 𝑠 >> 1) on resonance that a 162 Dy atom can experience from the optimally
detuned 626 nm light during to the force of gravity on the atom isℎ̄𝑘Γ/2
𝐹scat
=
≈ 170
𝐹grav
𝑚𝑔

(4.12)For comparison, this is approximately 1000 times smaller than the corresponding ratio
for lithium. For non-zero detunings, such as the ∆ ≈ 30Γ detuning at which we load
the MOT, the force of gravity can become even more comparable to the scattering
force from the MOT beams, causing the trapped cloud to sag until the force ofgravity is compensated by the scattering force. Specifically, the MOT position moves
downward (by more than the initial cloud size) until the Zeeman shift due to thegradient has brought the 𝜎 − transition back close to resonance. This phenomenon is
discussed in detail in [17].
In a typical alkali MOT, atoms are pumped back and forth across the manifold ofground state sublevels as they explore the trapping potential, since their orientationin the lab frame must flip whenever crossing through the field zero. The sagging
effect in the Dy MOT causes the cloud’s equilibrium position to sit on one side of
the quadrupole field zero, so the steady-state MOT is always spin-polarized in thelowest-energy state (i.e., in 𝑚𝑗 = −8) [17].
For the large detunings at which we load our MOT, we expect nearly all of theatoms to occupy the 𝑚𝑗 = −8 state [17]. For smaller detunings, the equilibrium
spin distribution is expected to change and spread out among several of the lowestenergy states. Although we have not observed multiple spin components in attempts
to perform Stern-Gerlach type measurements, this may be due to limitations in theramping rate of the power supply that we use to generate magnetic field gradients. We
have verified that we have a high degree of spin polarization by comparing the opticaldepth of the compressed MOT during absorption imaging for oppositely oriented
circular polarization; we find that the optical depth varies by about an order ofmagnitude.
86

4.3

Compression of the MOT

Once a MOT has been loaded, we can begin a compression sequence in order to drastically increase the phase space density of the trapped cloud. Compression consists ofreducing the physical size of the MOT while simultaneously bringing the MOT lasers
closer to resonance. Care must be taken to avoid light-assisted collisions during thisprocess, so the laser intensities must be lowered as the cloud is compressed.
Since bringing the lasers close to resonance causes the cloud to become muchsmaller already, the magnetic field gradient can actually be decreased during thecompression sequence. As will be discussed in the following subsections, we have
implemented two different compression schemes: one in which we relaxed the gradientand optimized phase space density of our compressed MOT for the first time, andone in which we increased the gradient at the expense of atom loss in order to aid
in loading of our crossed optical dipole trap. Since past published work utilized thefirst compression approach, it is detailed here, but work since then has utilized thesecond approach.

4.3.1

Compression Approach 1

In the first compression approach, we relax the magnetic field gradient. 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, andramping the frequency from the initial detuning to within about a few linewidthsof resonance. To minimize losses due to light-assisted collisions, and to reach thelowest 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 order to further minimize losses. At the end of the compression,
the MOT is approximately 400 𝜇m × 800 𝜇m × 800 𝜇m (dimensions of the opticallydense region). We lose up to half of our atoms during the 300 ms of post-compression
equilibration, but obtain a net gain in phase space density due to the simultaneous
reduction in temperature.
87

Figure 4-10: The compressed MOT. Gravity points into the top-left corner of thisimage, which was taken using a camera mounted at an angle. The two strips correspond to projections of two shells, which are oriented above and below the quadrupole
center (i.e., along the direction where the gradient is strongest). We interpret the twoshells as two regions where the atoms in the 𝑚𝑗 = −8 state are Zeeman shifted suchthat the magnetic gradient force is balanced by the MOT beam scattering force.

4.3.2

Compression Approach 2

Loading of the crossed optical dipole trap is sensitive to the position of the compressed
MOT, which is in turn sensitive to variations in the 626 nm laser frequency. Due
to noise picked up in our locking setup for this laser, the frequency at the end ofcompression varies shot-to-shot, and so the position of the MOT fluctuates. In orderto prevent this fluctuation from causing large fluctuations in the number of atoms
loaded into the dipole trap, we implemented an alternative compression approach: thegradient is ramped up to about 14 G/cm, and the detuning is set to approximately−10Γ. This forms a MOT with a double-shell shape (shown in Figure 4-10). Although
the higher density of this compressed MOT leads to more rapid losses, due to thehigher gradient, the position of the MOT is less sensitive to fluctuations in laser
detuning and has been found to be favorable for loading repeatable populations into
the dipole trap.
88

4.4

Loading of Shallow-Angle Crossed Dipole Trap

In order to increase the phase-space density beyond the 10−5 level, we transfer afraction of the atoms in the compressed MOT into an optical dipole trap, allowing us
to perform evaporatively cooling until degeneracy is reached. An optical dipole trap
works via the AC stark shift: the ground state energy of the atoms varies spatiallywith the intensity of one or more red-detuned laser beams, giving rise to a conservative
potential.

4.4.1

Stark Shift of a Gaussian Beam

We consider a trap formed by the AC Stark shift of an off-resonant laser beam withsome spatial profile. In the rotating wave approximation, a simple expression for the
AC Stark shift for a two-level atom is [22, 62]
𝑈𝑑𝑖𝑝 (⃗𝑟) =𝑑2𝑔𝑒 𝐸(⃗𝑟)2ℎ̄Ω𝑅 (⃗𝑟)
=
4∆
4ℎ̄∆

(4.13)

where 𝐸 is the spatially dependent electric field strength of the laser, 𝑑𝑔𝑒 is the
dipole matrix element for the atomic transition in the model two-level system, and∆ = 𝜔 − 𝜔0 is the detuning of the laser from the transition. It is convenient toexpress this shift in terms of known parameters: the laser intensity and the transition
linewidth:
𝜔03Γ=𝑑2 ⇒ 𝑑𝑔𝑒 =3𝜋𝜖0 ℎ̄𝑐3 𝑔𝑒√︃1
𝐼 = 𝜖0 𝑐𝐸 2 ⇒ 𝐸 =2√︂3𝜋𝜖0 ℎ̄𝑐3 Γ
𝜔03(4.14)

and
2𝐼
𝜖0 𝑐

(4.15)

Finally, for a Gaussian beam of waist 𝑤0 , we can relate the spatial intensity profile
to the total power 𝑃 carried by the beam as𝐼(𝑟, 𝑧) =

2
2𝑃
− 2𝑟
𝑤2 (𝑧)𝑒𝜋𝑤2 (𝑧)89(4.16)

where 𝑧 is the coordinate along the propagation direction, 𝑟2 = 𝑥2 + 𝑦 2 , 𝑤(𝑧) =√︁𝜋𝑤2
𝑤0 1 + ( 𝑧𝑧𝑅 )2 , and 𝑧𝑅 = 𝜆 0 . So an expression for the AC stark shift due to a single
Gaussian beam in the rotating wave approximation is
𝑈𝑑𝑖𝑝 (⃗𝑟) =2
2
3𝑐2 𝑃 Γ − 𝑤2𝑟2 (𝑧)− 2𝑟
𝑤2 (𝑧)𝑒≡𝑈𝑒
0
𝑤2 (𝑧)𝜔03 ∆(4.17)

Alternatively, we may relax the rotating wave approximation and use the more general
expression
3𝑐2 𝑃
𝑈𝑑𝑖𝑝 (⃗𝑟) = 2
𝑤 (𝑧)𝜔03(︂ΓΓ−∆ 2𝜔0 + ∆

)︂

− 2𝑟

2

𝑒 𝑤2 (𝑧)(4.18)

which includes the counter-rotating term.

Validity of the Two-Level and Rotating Wave Approximations
We first consider a crossed dipole trap made up of two 1064 nm beams. We will
perform the following calculations of trap depth and scattering rate under a twolevel approximation, considering only excitation via the 421 nm transition. For larger
detuning this immediately seems reasonable because the transition strength (dipole
matrix element squared) is approximately 72 times larger for this transition than forthe 626 nm transition, which is the second strongest transition from the ground state:𝑑2421
Γ421 𝜆3421
=
≈ 72
𝑑2626
Γ626 𝜆3626(4.19)

More formally, we can calculate the ratio of the trapping potentials that 1064nm lightwould induce due to each transition:
Γ421 𝜆3421 ∆626
𝑈0421
=
≈ 33
𝑈0626
Γ626 𝜆3626 ∆421

(4.20)

So we can probably obtain a meaningful estimate by neglecting transitions aside fromthe 421 nm transition.
Let us now consider the relative magnitudes of the co- and counter-rotating terms in90

our expressions for the trap depth and scattering rate. We see that1
= 3.7 × 10−16 seconds
𝜔421 − 𝜔1064(4.21)

1= 1.6 × 10−16 seconds
𝜔421 + 𝜔1064(4.22)

and

These two terms are quite comparable, so we will not employ the rotating wave
approximation in the following calculations.

Comparison to Measured Polarizability at 1064 nm
We can recast expression 4.18 in terms of the polarizability of the atoms at the
specified wavelength, which is a quantity that can be measured in the lab. This
yields
2
𝛼
2𝑃
− 2𝑟
2 (𝑧)
𝑤
𝑈𝑑𝑖𝑝 (⃗𝑟) = −𝑒
2𝜖0 𝑐 𝜋𝑤2 (𝑧)(4.23)

Calculating the polarizability from 4.18 yields a value of 161 in atomic units. The(scalar) polarizability of the ground state at 1064nm has been measured to be 136 ±15 in atomic units [49]; this indicates that the two-level approximation is too simplistic
to accurately predict the depth of an optical trap. The measured value will be used
for the calculations that follow.

4.4.2

Crossed Optical Dipole Traps

Because the confinement along the propagation direction is much weaker than along
the transverse directions for a focused Gaussian beam, we want to use an overlapping
pair of Gaussian beams which cross at an angle. Due to optical access constraints,
we are limited to small crossing angles of less than about 10 degrees for our ODTbeams. The total optical potential is then given by the sum of two intensity profiles:222((𝑥 cos 𝜃−𝑧 sin 𝜃) +𝑦 )−2𝑃1
2 ((𝑧 cos 𝜃+𝑥 sin 𝜃))
𝑤1
𝐼1 =
𝑒
𝜋𝑤12 ((𝑧 cos 𝜃 + 𝑥 sin 𝜃))

91(4.24)

and

2

2

)
2𝑃2 − 2(𝑥𝑤2+𝑦
2 (𝑧)
𝑒
𝐼2 =
𝜋𝑤22 (𝑧)(4.25)

where 𝜃 is the crossing angle of the beams. The coordinate transformation in the firstexpression is just the result of a rotation in the x-z plane by 𝜃. In our experiment,
the crossing angle is 5 degrees and the beam waists are 40 𝜇m, corresponding to aRayleigh range 𝑧𝑅 of 4.7 mm. Using the measured polarizability, we can calculate
the full 3D optical potential. We stabilize the powers in our trapping beams to a
maximum of 11.8 W in one ODT beam and 7.8 W in the other, corresponding to apeak trap depth of 240 𝜇K. The potential along x, y, and z for our crossed ODT(XODT) at full power is shown in Figure 4-11.
If we approximate the region near the center of the trap as a 3D harmonic oscillator, we can associate trapping frequencies with the potential:111
𝑉 (𝑥, 𝑦, 𝑧) ≈ 𝑚𝜔𝑥2 𝑥2 + 𝑚𝜔𝑦2 𝑦 2 + 𝜔𝑧2 𝑧 222
2(4.26)

We can calculate the trapping frequencies along x, y, and z by Taylor-expanding thefull potential about the origin. At full power, these are:𝜔𝑥 ≈ 𝜔𝑦 = 2𝜋 × 880Hz𝜔𝑧 = 2𝜋 × 49Hz

(4.27)The effect of the force of gravity can become significant as the depth of the trap
is lowered. Figure 4-12 shows the profile along the vertical coordinate 𝑦 at boththe initial power (full power) and at the final power at the end of our evaporation
sequence, with the force due to gravity included. As will be described in the following
section, we use a magnetic field gradient to partially cancel the force of gravity inorder to maintain confinement.

4.4.3

Loading the XODT

Because the volume of the XODT is very small compared to that of the MOT (bothbefore and after compression), we turn on the ODT beams during the MOT loading
92

Figure 4-11: The optical potential of the XODT at full power in temperature units.
The higher-power beam is the beam which propagates in the z direction. Top: theprofile along z for 𝑥, 𝑦 = 0, showing the ehancement to the confinement from thesecond (crossed) beam. Middle: the profile along x. Bottom: the profile along y.

93

Figure 4-12: The effect of gravity on the optical potential. Top: the effect of gravity
along the 𝑦 (vertical) direction when the beams are at full power. Bottom: theeffect of gravity when the beams are at the final powers used in our evaporation
sequence. Gravity significantly affects the ability of the trap to confine along thevertical direction at low ODT powers.

94

Figure 4-13: Crossed optical dipole trap. The atoms initially populate both thecrossed region and the uncrossed "wings" of the XODT, but these higher-energy
atoms leave the trap when the evaporation ramp begins. The wings of one beam are
populated more than the wings of the other because one beam has about 50% more
power than the other during the initial loading phase.
stage and see no adverse effects on the MOT population. With the beams on during
the MOT compression stage, we find that the maximal population is loaded into the
XODT if we hold the compressed MOT for about 40 ms after the end of compression.
We load about 3 × 106 atoms into the ODT at a temperature of between 30 𝜇K and
50 𝜇K6 . Prior to the implementation of the new compression sequence which mitigates
position fluctuations of the MOT, we were able to load roughly twice this number
into the XODT; this implies that further improvements to the loading stage can be
made.
An absorption image of the loaded XODT, as viewed from the top, is shown in
Figure 4-13.

4.4.4

Calibration of Trap Depth via Modulation

To measure the trap frequencies of the XODT in our experiment, we can use amplitude
modulation spectroscopy [23]. By modulating the depth of one of the beams at twice
the trap frequency, we can parametrically drive the trapped atoms, heating them andeventually causing them to leave trap.
6

Although the temperature of the MOT at the end of compression is lower than this, we attributethe increase in temperature to the acceleration (followed by collisional thermalization) of the atoms
as they "fall" into the XODT potential from the MOT.

95

Figure 4-14: Amplitude modulaton in the XODT. Atom number after 500 ms of 10%
amplitude modulation of one of the two beams at full power vs. modulation frequency.
The broad peak is centered around 1200 Hz, corresponding a trap frequency of 2𝜋 ×600Hz. The trapping frequency along the z-direction was not observed.
Although we have not yet worked extensively to experimentally determine ourtrapping frequencies, preliminary data suggests that our x and y trapping frequencies
may be about 600 Hz instead of the calculated 880 Hz. This would correspond toan actual trap depth that is 0.46 times the calculated one, likely arising due topower losses on the beam delivery optics and/or the beams having larger waists thancalculated. An example preliminary spectrum is shown in Figure 4-14.

4.4.5

Scattering in the Trap

In addition to forming a conservative potential via the AC Stark shift, the trapping
laser will also cause atoms to scatter photons and thus gain kinetic energy, heating
the cloud. We can roughly estimate the scattering rate for our optical trap when it isat maximum depth by considering the two-level, RWA expression from the beginningof this section. The scattering rate in the rotating wave approximation for a two-level
atom is𝑅=Γ

𝐸 2 (⃗𝑟)𝑑2𝑔𝑒ΓΩ2𝑅 (⃗𝑟)=Γ𝑟)
2 2 = 𝑈𝑑𝑖𝑝 (⃗
2
4∆
ℎ̄∆
4ℎ̄ ∆

(4.28)

This corresponds to a photon scattering rate of less than 1 Hz. Each scatteredphoton heats the atoms due to the added photon recoil energy, which is much lessthan the depth of the trap at full power:ℎ̄2 𝑘22𝑚

𝑘𝐵 𝑉 (𝑟 = 0)96

≈ 10−4(4.29)

Even at the lowest power, the trap depth is at least 100 recoil deep (not counting theeffect of gravity). The effect of photon scattering is thus insignificant in terms of lossof atoms from the trap. The lifetime of the BEC eventually formed in our trap islimited by the scattering rate (lower than 1 Hz at the final trap depth), since a single
photon can knock an atom out of the condensate.

4.5

Evaporation

Evaporative cooling allows us to cool our atoms from the 10−5 K regime to the transition temperature to form a Bose-Einstein condensate, around 100 nK. The idea behindevaporative cooling is as follows [54, 46]. Above the transition temperature, atoms
held in the XODT exhibit a thermal distribution of energies. If the trap depth issuddenly lowered, then the atoms in the high-energy tail of this distribution can haveenergies that exceed the trap depth, allowing them to escape. The remaining atoms
are then described by a non-equilibrium distribution, and subsequent rethermalization of the atoms leads to a new distribution with a lower temperature. By turning
down the trap depth as a function of time, sufficiently slowly that the collisions between the trapped atoms allow them to rethermalize as the trap is ramped down, thetemperature can be lowered by several orders of magnitude. Combined with the resulting increasing in density, this process of evaporative cooling can increase the phasespace density of the cloud by at least 5 orders of magnitude, eventually resulting incondensation.
We perform our evaporation in 6 steps:
1. Hold the atoms at full power for 500ms
2. 4 steps of linearly ramping the power of the ODT beams
3. Hold the atoms at the final power for 6000ms
Immediately after loading the XODT, holding the atoms in the trap leads to rapid
loss of atoms. Over two seconds of holding time, we observe the number of atoms to97

decrease by roughly an order of magnitude, while the phase space density increases
by approximately a factor of 2-3. We attribute the losses to a combination of dipolar
relaxation due to imperfect spin polarization of the loaded atoms and evaporation of
the initially loaded cloud. Dipolar relaxation at this stage of the experiment can be
mitigated by optically pumping the cloud; this will be investigated in the future. The
temperature of the initially loaded cloud is measured to be about 30 𝜇K, implying aratio 𝜂 of trap depth to the thermal energy 𝑘𝐵 𝑇 of between 𝜂 = 4 (for the preliminarily
measured trap depth) and 𝜂 = 8 (for the calculated trap depth).
The duration of the initial hold step, as well as the 4 subsequent ramping steps
and the final hold step, were optimized in sequence. The optimization was performedby measuring the relative increase in the phase space density as a function of theduration of each step. A single-shot metric that allowed us to optimize the phasespace density was constructed as follows:
The de Broglie wavelength scales as √1𝑇 , and for a fixed time-of-flight, the temperature of the cloud and the width 𝜎 we measure are related as 𝑇 ∝ 𝑣 2 ∝ 𝜎 2 . Thus thede Broglie wavelength scales as 𝜎1 . The number density of the atoms in the trap also
depends on the temperature. In the harmonic approximation, the size of the cloud√in the trap in each direction depends on the temperature as 𝑚𝜔 2 𝑥2 𝑘𝐵 𝑇 ⇒ 𝑥 ∝ 𝑇 .
And so the in-trap volume of the cloud scales as 𝜎 3 . Thus, for a fixed trap depth andfixed time-of-flight, we can use the measured atom number and TOF width of thecloud to construct a metric that is proportional to the phase space density:
PSD = 𝑛𝜆3dB ∝𝑁
𝜎6(4.30)An example of a PSD-optimizing duration scan is shown in Figure 4-15.
The final powers of each step were initially chosen such that each ramp reduced
the power in each beam by a factor of two, but then were changed in order to optimize
PSD. The final hold time was chosen to maximize the observed bimodality in the BEC
that was produced.
Figure 4-16 shows the laser powers as a function of time during our evaporation.
98

Figure 4-15: Example optimization of phase space density using the single-shot metric. The metric is plotted vs. the duration of one of the linear ramp segments of ourevaporation. The larger variation at longer times is due to the limited signal-to-noiseof the atom counting when the number gets too low.

Figure 4-16: Laser powers vs. time during the evaporation. The two curves represent
the powers each of the two ODT beams during the 16-second evaporation sequence.

99

Figure 4-17: Compensating for gravity with a magnetic field gradient. The orange
trace shows the profile of the XODT potential along the vertical direction at the endof the evaporation sequence including the effect of gravity. The blue trace shows thesame profile when gravity is partially compensated using a magnetic field gradient.

As mentioned in the previous subsection, the effect of gravity on the verticalconfinement becomes more significant as the powers are ramped down. In order tomitigate this, we apply a magnetic field gradient during the entire evaporation which
results in a force on the atoms that partially compensates for the force of gravity.
Specifically, we apply a gradient of 1.8 G/cm, which yields a force of about 63% theforce of gravity. Figure 4-17 shows the (calculated) potential at the end of evaporation
with and without the gradient applied.
In addition to the magnetic field gradient, we apply a bias field along the verticaldirection of about 3 G. The value of this field was varied experimentally to minimize
losses due to any Feshbach resonances at low field.

4.6

Bose-Einstein Condensation

Our experiment currently produces Bose-Einstein condensates of 162 Dy with around
5000-10000 atoms. In this section the detection of our first BECs is described.
100

4.6.1

Detection of a Condensate

As described in Chapter 1, Bose-Einstein condensation occurs when the phase spacedensity of the trapped atom cloud becomes on the order of unity. Optimization of ourevaporative cooling sequence as described above leads to temperatures and numbers
which, combined with the estimated in-situ size of the cloud based on the calculated
XODT potential, approach and exceed unity. It is at this point that we can begin
looking for signatures of a partially (or fully) condensed cloud.
There are a few "smoking gun" signatures to look for when trying to determine
whether evaporative cooling has resulted in a BEC [57, 46]. One is the inversion of
the aspect ratio due to mean-field expansion. The expansion of a BEC in time-offlight is predominantly due to the interactions between the particles. The conversionof mean-field interaction energy into kinetic energy will result in more kinetic energy
along the directions of the BEC that have the steepest density gradient. This has
the consequence that the in-trap aspect ratio inverts after sufficient time-of-flight.
By comparison, the aspect ratio of a thermal cloud in equilibrium approaches 1 intime-of-flight, regardless of the initial in-trap distribution. The aspect ratio inversion
is thus a tell-tale sign of the phase transition to BEC.
Another smoking gun of condensation is the bimodality of the expansion of thecloud in time-of-flight. When the cloud is partially condensed, some atoms remain
described by a thermal distribution around the transition temperature while otheratoms condense into the BEC. The initial shape of the BEC in the trap can beunderstood by considering the Gross-Pitaevski equation [57]:
𝑖ℎ̄

𝑑𝜓
ℎ̄2 2
=−
∇ 𝜓 + 𝑈 (𝑟)𝜓 + 𝑔|𝜓|2 𝜓
𝑑𝑡
2𝑚(4.31)

Here we are ignoring the contribution of the nonlocal dipolar interactions, for thesake of simplicity; in most cases, this will not change the qualitative behavior. Fora typical BEC, the interactions are strong enough that we can consider the ThomasFermi approximation, and neglect the kinetic energy term (the approximation is thatthe interaction energy and potential energy are much larger than the kinetic energy).
101

We can thus write a simplified time-independent GPE:𝜇𝜓 = 𝑈 (𝑟)𝜓 + 𝑔|𝜓|2 𝜓(4.32)

where 𝜇 is the chemical potential. This can be solved to yield the Thomas-Fermi
profile:𝜓∝√︀𝜇 − 𝑈 (𝑟)(4.33)

which shows that the shape of the condensate in the trap follows the shape of thetrapping potential 𝑈 (𝑟).
The thermal atoms have much more kinetic energy than the condensate, and so
expand more quickly than the condensate and with a gaussian velocity distribution.
The condensate’s shape remains defined by a Thomas-Fermi distribution which only
expands due to mean-field repulsion in time-of-flight. After sufficient time-of-flight,
the thermal atoms have spread out enough compared to the condensate that a clear
Thomas Fermi profile-plus-gaussian profile appears.
While the aspect ratio inversion can be detected whether the cloud is partially or
fully condensed, the bimodality is only detectable if there is a measurable thermal
fraction. In our experiment, we image our condensate along the longitudinal ("𝑧")axis of the XODT in order to maximize the signal-to-noise of our imaging. Even
though our trap – and resulting condensate – has a large initial aspect ratio, our
imaging integrates along this axis and so we are unable to observe the anisotropic
expandion. We are, however, able to observe the bimodality of the expansion in TOF.

4.6.2

Observation of Bimodality

At the end of the optimized evaporation sequence, we look for bimodal expansion of
the cloud in time-of-flight. In order to be able to observe the cloud for TOF valuesof 10-15ms or more without it falling out of the sensor area (or away from the highSNR region of the composite image), we increase the magnetic field gradient usedduring evaporation from 1.8 G/cm to 2.86 G/cm. This field gradient nearly perfectlycompensates for gravity.
102

Figure 4-18: One of the first BECs detected with our experiment. Although thesignal-to-noise ratio is low due to the low atom number and fringing, bimodality of
the cloud in TOF is evident when the horizontal coordinate is integrated over.

Figure 4-19: Bimodality in the composite TOF image. The darker region in the centerof the image corresponds to the condensate

Starting after about 8-10ms of TOF, we observe the separation of the distribution
into a wide Gaussian component and a narrow, roughly parabolic component. Becauseof the small number of atoms left after our evaporation sequence (roughly 5000-6000
for most runs), the images are typically too noisy to observe clear bimodality byinspection of the composite image. The bimodality becomes clear when either rows
or columns are summed over, producing a 1D integrated lineout of the composite
image. An example of this is shown in Figure 4-18.
Figure 4-19 shows one example of a cloud where bimodality is visible by inspectionof the composite image.
103

4.6.3Phase Space Density of the Measured Cloud

We can separately verify that the observed cloud corresponds to a BEC by measuringthe "temperature" of the condensate and calculating the phase space density. Thatis, if we assume the expansion of the dense central part of the cloud to be due tothermal kinetic energy, then we can calculate what the corresponding phase space
space density of the cloud would be. The size of a thermal cloud in the trap canbe determined by noting that a thermal cloud "fills" the bottom of a trap up to
the thermal energy 𝑘𝐵 𝑇 . By using the full 3D calculation of the potential discussed
above, we can use the measured temperature of the cloud to calculate the in-situ size,and hence the density, of the cloud.
The expansion of the BEC was measured for a range of TOF times ranging from
3 ms to 15 ms, and a line was fit to extract the velocity (on the order of 0.1 mm/s).
Because of the effects of fringing in the composite image, there was uncertainty inthe widths at large TOF values, and hence in the expansion velocity. Associating
this expansion velocity with a temperature gives a range of possible "temperatures"
between 5 nK and 20 nK.
To calculate the PSD this corresponds to, the intersection of 𝑘𝐵 𝑇 with the modeled
XODT potential (including the effects of gravity and the gradient) was calculated
along each dimension to determine the in-situ size in each direction. The PSD wasthen evaluated as
𝑁
𝜆3dB
𝜋𝑥
𝑦
𝑧
0
0
0
3

𝑛𝜆3dB = 4(4.34)

where 𝑥0 , 𝑦0 , 𝑧0 are the numerically determined sizes of the cloud along each dimension, and an ellipsoidal shape for the cloud is assumed. For 5 nK, this corresponds toa calculated PSD of 4500; for 20 nK, the calculated PSD is 135. The fact that both ofthese numbers are much larger than 2.612 indicates that the cloud cannot be a in athermal state. These PSDs were calculated using the predicted optical potential, but
as mentioned in Chapter 4 there is preliminary evidence that our optical potential isroughly 0.46 the depth that we predicted. Multiplying the optical potential by thisfactor and recalculating the PSD for 20 nK yields a "worst case" PSD of 58, which is104

still more than an order of magnitude above 2.612.
According to the numerical calculation, the transition temperature for this atom
number is approximately 60 to 100nK.

4.7

Imaging

We take absorption images of our atoms in order to measure the number, momenta,
and spatial profile of our atoms during various stages of the experiment. Absorption
imaging consists of the collection of three images for each run of the experimental
sequence, and is performed as follows:
1. The trap which is confining the atoms (i.e., MOT or XODT) is suddenly
switched off, allowing the atoms to freely expand.
2. A guiding magnetic field of a few G is turned on along the direction of propagation of our imaging beam while all other fields are turned off in order to rotate
the spins of the atoms to point along this direction.
3. After a specified time-of-flight (TOF) a pulse of 𝜎 − polarized 421 nm light isshone through the atom cloud and detected on a camera, forming the "atom"
image. This pulse is typically 100 𝜇s in duration and has an intensity of approximately 1 mW.
4. After the image has been acquired and the atoms have been given sufficient
time to spread out/fall out of the field of view of the camera, a second identicalpulse of imaging light is used to take a reference "light" image.
5. A third image is then taken without any pulse of laser light, allowing us tocreate a background "dark" image. This image allows us to subtract out theeffects of both sensor dark counts and stray light.
We then create a composite image out of the three images as [46]:Composite =Atoms − Dark
Light − Dark

105

(4.35)

We then have an array of values corresponding to the attenuation of the imaging
light by the atoms along each "column" of the cloud that corresponds to a single pixel.
In order to extract the number of atoms from this composite image, we perform thefollowing computation:(∆𝑥)2 ∑︁𝑁=
ROI
𝜎0 𝑀 Pixels(4.36)

where ∆𝑥 is the width of a single pixel, 𝜎0 is the resonant cross-section for photon
scattering by an atom, 𝑀 is the magnification ratio, and "ROI" refers to the subsetof the "Composite" image containing the atom cloud. In principle, the effective 𝜎0of the cloud is a complicated function of the polarization of the imaging light, thedistribution of the population among different spin states, and any optical pumpingthat occurs during the imaging pulse. The situation is much simpler if the cloud isspin-polarized in the 𝑚𝑗 = −8 states and 𝜎 − light is used; in this case, a cyclingtransition between the 𝑚𝑗 = −8 and 𝑚′𝑗 = −9 states is induced, creating an effective
3 2
two-level system and meaning that the full resonant cross-section 𝜎0 = 2𝜋
𝜆 can beused. Since we have not yet measured what our spin state distributions are during
various parts of the experimental sequence, we may be underestimating our measured
atom number by using the full cross-section.

106

Chapter 5
Low-Noise Analog Control Channels
To control our experiment, we use three types of control signals: digital outputs,
analog outputs, and serial communication buses. Digital outputs are used to toggle devices, such as high-current relays on magnetic field coils, on and off. Analog
outputs are used to provide continuously variable control voltages to devices, often
acting as the setpoint of a feedback system or the tuning voltage on a VCO. Serial communication, which lacks the precise hardware-clocked timing of the digital
and analog controls, is usually used to communicate with specialty devices such as
high-bandwidth function generators.
Commercially available digital output hardware fulfils all of the needs of our experiment "out of the box," with the exception of providing isolation between the groundsof the channels and the control computer. Fortunately, digital isolation at the 1 MHz
level (the fastest timescale associated with our experimental control system) is easy to
implement with low-cost optoisolators. Likewise, our needs for serial communication
hardware (such as USB-to-RS-232 adapters) are easily met by commercially available
solutions.
In contrast, commercially available analog voltage hardware can become a limiting
factor in certain aspects of our experimental control. This is due in part to the noiselevel and limited resolution of the DAC circuitry used in these devices, and in partto the ground loops that the deployment of this hardware can create. This chapter
discusses the development and implementation of new analog control voltage hardware
107which avoids the limitations of the commercially available hardware.

5.1

Motivation

The development of new analog control channels was motivated by limitations in theperformance of the standard National Instruments analog output cards which are typically used in labs in our research center. The primary limitation was the multitude ofground loops created when setting up a control system using this hardware: all of theoutput channels on one of the NI cards share a common ground, which is grounded
to the lab’s mains ground through the host PC. It is possible to use various typesof analog isolation circuitry to eliminate these ground loops, but these inevitably
increase the noise level of the output channel. Attenuating this noise with filtering
stages reduces the bandwidth of the channel. For specific applications, such as controlled ramps of the intensity of PID-stabilized optical lattice beams, it is desirable
to have the lowest noise possible in the control signal while also maintaining enoughbandwidth – perhaps 10kHz – to rapidly vary the lattice intensity during portions of
the ramp.
In the absence of ground loops, the inherent output noise level of the NI cards still
exceeds that of standalone DACs which can be purchased from electronics suppliers.
A secondary limitation of the NI hardware was the limited precision of the output
voltages, due to the limited resolution of the DACs.

5.1.1

Ground Loops

The term "ground loop," in many circles, carries an undue air of mystery. A few
different context-dependent definitions exist for the term [71]. A comprehensive definition of a ground loop is, perhaps, an unwanted electrical potential difference formed
between two points of a circuit - that ideally should be at the same potential - arisingfrom multiple current return paths. The basic mechanism is easy to understand: Anycurrents (DC or AC) that a device is sinking into its connection to ground will always
give rise to a small Ohmic voltage drop across the finite impedance of the ground108

Figure 5-1: Ground loops. In the top circuit there is no ground loop present, whilein the bottom circuit there is a ground loop.
connection. If two devices share a path to ground, then the ohmic voltage drop fromone device will appear as a voltage source between the second device and ground.
This is illustrated in Figure 5-1.
Let’s consider the two circuits shown in the figure. In the top circuit, there are
two load resistors connected to the same voltage source, each with its own return pathwith a resistance of 1 m𝜔. There is no ground loop in this example, but the effectof the non-zero impedance of the return path is still measurable: if we were to placea voltmeter across the terminals of Resistor A, we would measure the power supply
voltage, less the small voltage drop across the finite impedance in the return path.
Resistor A and the return path impedance simply form a voltage divider:

𝑉𝑅𝐴 = 𝑉Out𝑅𝐴𝑅𝐴 + 1 𝑚Ω

109

(5.1)

If Resistor A is 1000 Ω, then this is a 1 part-per-million effect.
In the bottom circuit the two load resistors’ "ground" terminals are connected ina second location - forming a ground loop. Let’s consider the effect of the current
flowing through Resistor B on the voltage we would measure across the terminals ofResistor A:
Let the voltage source be a 120 V RMS AC waveform (we could then imagine that
𝑅𝐴 and 𝑅𝐵 represent the current draw of two devices powered off of the same outlet).
Without a ground loop, we’d expect to measure 120 V RMS across the terminalsof RA (well, actually 119.99988 V, due to the milliohm of return-path impedance).
When we form the ground loop, the 120 A RMS of current flowing through 𝑅𝐵 is now
split between the two possible return paths - one of which is shared with 𝑅𝐴 . Since
we’ve assumed that both return paths have the same impedance, the current will beevenly split between the two paths, leading to an additional 60A of current flowing
through 𝑅𝐴 ’s return path! From Ohm’s Law, we know that there must now be an
additional 60 mV drop across this impedance.
Thus, a measurement of the voltage across RA would now yield 119.99988V 60 mV = 119.4 V, a half part-per-thousand effect. If we imagine, for example, thatRA represents the output of a DAC, the effect of the ground loop is to create an AC
waveform in the measured output at the 11-bit level; that is, even if we have a 16-bit
DAC, roughly 5 bits of resolution will be "lost" in the AC waveform arising from theground loop. If the current flowing through 𝑅𝐴 ’s return path contains a broad rangeof frequencies due, for example, to switching power supplies in the device represented
by 𝑅𝐵 , then the effect of the ground loop is the introduction of some complicatednoise to the measured voltage across 𝑅𝐴 .

5.1.2

Noise Spectrum

A commonly used National Instruments analog output card, the PXI-6733 [55], is
specified to have 80 𝜇𝑉RMS of noise in the band between DC and 1 MHz; this cor√
responds to an average noise spectral density of 80 nV/ Hz. With the addition of
analog isolators or instrumentation amplifiers to the system to suppress ground loops,
110this noise level typically will increase. The noise spectral density of a PXI-6733 card
channel running through an analog isolator was measured in one of the labs in our re√search group to exceed 150 nV/ Hz; a variety of peaks in the noise spectrum brought
the total noise level far above this at specific frequencies.
Commercially available standalone DACs have noise spectral densities that are
far below this level; as will be discussed below, the DACs used in our analog control√
channels have a NSD of 7.5 nV/ Hz [3].

5.1.3

Resolution

The maximum resolution of the analog channel cards produced by National Instruments is 16 bits. This corresponds to 1.5 parts in 105 , which is more than enoughprecision for many applications (tuning voltages on VCOs rarely need more than a
few mV of resolution for example). For certain applications, more precise voltages are
required. One example is the measurement of Feshbach resonances of lithium atoms
using a technique called "interaction spectroscopy." Lithium atoms are loaded into
an optical lattice, and ramped into a Mott insulator giving rise to both singly anddoubly occupied sites. The RF frequency required to spin-flip an atom on a singly
occupied site from one hyperfine state to another varies as the differential Zeeman
shift of the two states, roughly 33 kHz/G. On doubly occupied sites, the difference ininteraction energy between the initial state (two atoms in the same hyperfine state)
and the final state (two atoms in different hyperfine states) gives rise to an additional
shift in this transition frequency due to the presence of Feshbach resonances between
700G and 900G. Figure 5-2 illustrates how this works with figures pulled from [2].
The precision with which the transition frequency can be measured at each magnetic field value is limited (due to the relative Zeeman shift) to the 100 Hz level dueto the limited precision with which the magnetic field can be stabilized (around one
part in 105 ).
Another example within our research center is the precise control of the magnetic
field required for sweeping Li and Na atoms into weakly bound Feshbach molecules.
The sweep is performed across a 1 mG-wide Feshbach resonance at 754 G, which is111necessitates nearly 1 part-per-million control of the magnetic field.
Smaller voltage steps can be reached by using a voltage divider on the output of
the DAC, but this reduces the dynamic range of the DAC and can quickly create aninconvenient scenario where various voltages need to be summed in order to meet theneeds of the experiment. More conveniently, low-noise DACs which meet the timing
requirements of our experiment are available with up to 20-bit resolution, and so a
20-bit DAC was chosen for the newly developed analog output channels.

5.2

Analog Channel Design

The analog voltage channels which were developed consist of a few key components:
1. A network-interfaced 32-bit ARM processor running a lightweight Linux which
is able to receive and store voltage buffers sent by a control computer
2. An FPGA for communicating the buffer data to the DAC using its specified
serial protocol.
3. A low-noise, high-resolution digital-to-analog converter (DAC)
4. A stable voltage reference circuit
5. Isolated, low-noise power rails to prevent ground loops
The requirements and specifications of each of these components are detailed in thefollowing section.

5.2.1

Schematic of System

Figure 5-3 shows a schematic overview of one of the analog control channels.
The system consists of a combined FPGA/ARM processor IC called a system ona chip, or SOC; a a low-noise, high-resolution DAC; a ppm-stable voltage reference; a
PCB containing isolated banks of supercapacitors and low-noise regulators for powering the DAC; and an optical isolator for isolating the incoming clock signal whichis shared with all other channels in the lab.
112

Figure 5-2: Interaction spectroscopy of 7 Li. Top: the differences in interaction energies between pairs of atoms in different hyperfine states gives rise to shifts in thetransition frequencies due to the presence of Feshbach resonances. Bottom: the measured interaction-induced shift in the transition frequency for atoms initially in thesame hyperfine state vs magnetic field, relative to the transition frequency for atoms
on singly occupied sites.

113

Figure 5-3: Schematic of the SOC system. The Opal Kelly clock block represents
the FPGA which generates the variable-timebase clock for all digital and analog
control channels in our lab. The blue blocks are all contained on the DE0-Nano
SOC evaluation board. The red blocks representing the supercapacitor banks are allcontained on one PCB.

5.2.2

FPGA/SOC and Network Interface

Interfacing the DACs in the analog voltage channels with our laboratory control
system requires specialized digital hardware. Specifically, hardware is required which
can communicate with our control system, store large arrays of output words forthe DAC (buffer samples), and update the output word when clock edges from ourlaboratory clock arrive.
Storage and hardware-timed output of the buffers is a job well-suited to a fieldprogrammable gate array (FPGA). An FPGA is an IC which is made of thousandsof discrete units individual "logic elements" (LEs), each of which contains a small
register and basic logical arithmetic capabilities. FPGAs can be configured to perform an enormous variety of digital tasks, including many-input combinatorial logic,
counting/buffering, and even emulation of an entire microprocessor, which can in turnrun custom software. FPGAs are programmed using a hardware description language
such as VHDL or Verilog.
Various methods of transmitting the buffers from the control system to the FPGAs
and DACs were considered, but communication over ethernet was the clear winner
in terms of speed and reliability over 10+ meter distances. Communication with our114lab control system thus requires at least a very basic processor. In early prototypesof the analog control hardware, this role was filled by a simple 8-bit microcontroller,
which communicated with the lab network over a 25 MHz SPI bus to an ethernetto-SPI adapter. The limited memory of the microcontroller necessitated a different
solution, and so in the next iteration a NIOS II soft-processor was instantiated on
the same FPGA which handled the buffer storage and hardware-timed updating of
the DAC. The NIOS II solution, combined with an external RAM chip on which the
NIOS could store large buffers before sending them to the DAC, was fully functional.
The 25MHz SPI bus used to communicate between the lab network and the NIOS
became a bottleneck, however, leading to tens of seconds of buffer transmission timefor every run of the experimental cycle.
To speed up the communication of the buffers over the ethernet connection, communication had to be established with the processor with dedicated ethernet hardware. A type of IC well-suited to this problem was Altera’s line of "systems on a chip"(SOCs), which incorporate both a 32-bit ARM processor and a high-performance
FPGA into the same physical IC [69, 28]. The ARM processor is capable of interfacing with gigabit ethernet physical-layer hardware (and comes with this hardware
when purchased on an evaluation board), and has a variety of high-speed connections
to the FPGA portion inside the IC. The final version of the analog control channels
thus use Intel DE0-Nano-SOC1 evaluation boards, based on a Cyclone V SOC, toperform all of the network communication and buffer sample manipulation required
to interface the lab control system with the DACs. Figure 5-4 shows the schematiclayout of the SOC evaluation board.
The hard processor system (HPS) component of the SOC runs a lightweight Linux
operating system, using an SD card as a hard drive. The software which is used toestablish communication with the control system is compiled using a toolchain whichtargets the ARM processor. The executables are then copied over to the SD card,1

A note about names. Intel recently acquired Altera, and so many devices formerly branded
as Altera are now branded as Intel. The DE0-Nano-SOC is also sold under another name, theAtlas SOC, which is an identical piece of hardware which comes with a differently configured Linux
operating system on its SD card.

115

Figure 5-4: Schematic of the DE0-Nano SOC evaluation board. Reproduced from[69]. The FPGA and hard processor system (HPS) are on the same physical chip,enabling high-bandwidth communication between the two components without any
external components. The general-purpose in/out (GPIO) headers connected to the
FPGA are used to send signals – including the serial communication line for the DAC– to external peripherals. The HPS is connected to both a gigabit ethernet physicallayer chip and a 1 GB external RAM chip, allowing for the rapid receipt and storageof large buffers from the experimental control system.

116

where they are activated remotely by control application, as will be described laterin this chapter.

5.2.3

20-bit DAC

The DAC chosen for this system is the AD5791, a 20-bit DAC with a noise spectral√
density of 7.5 nV/ Hz [3]. 20 bits of resolution corresponds to greater than 1 partper-million of resolution; specifically, with ±10 V rails, the smallest voltage step is
19 𝜇𝑉 .
Communication with this DAC occurs over an SPI bus with a maximum clock
rate of 35 MHz; since 24 bits must be transmitted to communicate an output word to
the DAC, this in principle allows for a greater than 1MHz update rate; however, thesettling time for the DAC (the time for the output voltage to reach within 0.02% of its
final value after the output word has been changed) is up to 1𝜇s. Since the absolute
maximum update rate of our lab’s control system is 500 kHz, the timing capabilitiesof the DAC are well-matched to our experimental control needs.
The DAC is available as a standalone IC and on evaluation boards from the
manufacturer. Although a custom PCB for the IC was designed and successfully
tested, it was found that the overall cost (both monetary and time/labor) for thesmall number of channels needed by our experiment was lower if a pre-made evaluation
board was purchased for each channel. Figure 5-5 shows the layout of the evaluation
board [4] used for all of the channels in our lab.
The evaluation board has an onboard high-speed output buffer amplifier, which
prevents errors due to loading of the DAC and decreases the settling time. The buffer
also adds a small amount of noise, and so can be bypassed (and an external buffer
can be supplied) if desired.

5.2.4

LTZ1000A Voltage Reference

The reference voltage which a DAC uses to convert a digital word into an output
voltage must be provided from an external source. The high precision of a DAC is a117

Figure 5-5: The evaluation board for the AD5791 DAC. The optional onboard power
supplies can provide all of the necessary voltages using a single 3.3 V input, which
can float with respect to the rest of our experiment. This revision of the evaluation
board includes an onboard voltage reference which, although inferior to the reference discussed in the following subsection, meets the needs of many simple control
applications in the lab.
useless specification if the voltage reference used with the DAC isn’t stable to at least
the level of precision of the DAC; A 20-bit DAC such as the AD5791 thus requires
a roughly part-per-million stable voltage reference in order to be used in precisionapplications [18].
A high-performance voltage reference exists in the form of the LTZ1000A reference
made by Linear Technology [38]. These buried Zener references come in rigid TO99 packages, shielding the voltage reference circuitry from perturbations due to air
currents or PCB strain. With the addition of external circuitry, the LTZ1000A is
also temperature stabilized, preventing changes in the voltage due to changes in theambient temperature.
The PCB which was designed to work with the LTZ1000A reference included
low-TCR resistors, is shown in Figure 5-6.

5.2.5

Supercapacitors

In order for each analog output channel to float with respect to the rest of the equipment in the lab, some sort of isolated power supplies must be used. The most cost118

Figure 5-6: The PCB designed for the LTZ1000A voltage reference. The reference istemperature-stabilized above room temperature. The pins on the bottom of the PCBinterface with one of the supercapacitor banks.
effective types of isolated supplies, switching converters, create noise at their switchingfrequencies in the power rails. While this noise can be filtered out to a large degree,
the switching frequency is also broadcast from the magnetics of the switching converter, and can be picked up by sensitive portions of the circuit such as the voltage
reference or DAC output traces.
Linear power supplies, although much easier to filter down to low noise levels after voltage regulation, are extremely bulky and expensive, requiring a sizeable transformer and set of windings for each isolated voltage rail that needs to be produced.
Batteries offer a simple, quiet source of isolated power and were thus an appealing
option. A cheap and simple substitute for batteries, which doesn’t require any detailed
consideration of the charging cycles of the batteries (and which also doesn’t require
the battery voltage to be stepped up to meet the requirements of the power rails forthe DAC), is a bank of supercapacitors connected in series. Supercapacitors, withcapacitances of 1 − 10F, fill a role somewhere between that of a battery and that ofa capacitor. They are able to be charged to the desired voltage in a few seconds, andsupply tens of milliamps of current for on the order of a minute, which is sufficientfor the needs of the analog output channel2 .
2An exception is the SOC evaluation board, which requires several watts of power. This is
powered using a separate isolated power supply which is included with the evaluation board. While119Figure 5-7: The supercapacitor bank PCB. Left: the layout of the PCB showing
the 4 individual supercapacitor banks, relays and current-limiting resistors used forcharging the banks, and the ultralow-noise voltage regulators on the output. Right:
Three assembled PCBs, with the supercapacitor banks populated.

A board with 4 banks of supercapacitors, able to be charged by a single linear
supply and then isolated from one another using a set of relays, was developed to
power the output channels. The 4 banks correspond to the positive and negative
15 V rails for powering the DAC and associated amplifiers (voltage reference scaling
amplifiers and output buffer) on the DAC evaluation board, a 10 V rail for poweringthe LTZ1000A voltage reference, and a 5V rail to act as a digital power line for the
DAC. The 5 V rail is also used as the supply on the output end of a circuit which
optically isolates the laboratory clock, incorporated onto the supercapacitor board. At
output of each bank of supercapacitors is an ultralow-noise voltage regulator (Texas
Instruments TPS7A47 or TPS7A33) which specifies the final output voltage of thebank without adding significant extra noise.
The layout of the supercapacitor bank PCB, as well as a photograph of the assembled system, is shown in Figure 5-7.

this supply is not particularly low-noise, the separation of the digital and analog grounds on the
DAC evaluation board prevents this noise from coupling into the output of the DAC.

120

Figure 5-8: The Poseidon server, showing several connected SOCs. Additional SOCs
can be added using the SOCList parameter on the left.

5.2.6

Poseidon Server

The Cicero word generator application specifies the functional forms and output sample rates of all analog channels during each step of the experimental sequence, but
this information needs to be processed to create arrays of voltages (in floating pointform) and then turned into arrays of integer words for the DACs that correspond tothese voltages. These arrays of output words are usually referred to as buffers.
The job of creating these buffers and transmitting them to the SOCs associatedwith each output channel is handled by an application called Poseidon. A screenshot
of Poseidon is shown in Figure 5-8.
Every analog channel in the lab has an SOC associated with it, each of which
has a unique IP address on the lab’s network. Poseidon stores a list of all SOCs
that output has been specified for, along with their IP addresses, and establishes asocket connection to each one prior to the start of any experimental sequences. In
this socket connection, each SOC runs an application which acts as a simple socket
server, and Poseidon connects to each SOC as a client. At the beginning of eachsequence, Poseidon receives all of the information about the experimental sequence121

and generates buffers based on code which was originally written for an application
called Atticus – the hardware control server that was originally developed in tandemwith Cicero. The buffers are then transmitted over the socket connection to the SOC,
which stores them in memory.

5.3

Description of Operation

This section describes the operation of the analog voltage channels in detail. The
purpose of including this detailed description is both to serve as a resource for those
who are looking to extend the capabilities of the existing analog channels and toprovide ideas and insight for those who may be developing a similar system from theground up.
First, if any channels are using supercapacitor banks, the supercapacitors must beallowed to charge. This is accomplished by sending a logical high to the supercapacitor
charging boards, activating the relays which connect all of the supercapacitor banks
to the charging supply rails. When it is time for the channel to be used, the control
signal is set back to logical low, isolating the banks from the charging supply andfrom one another.
When the sequence begins, Cicero sends a command to Poseidon to generatebuffers from the sequence data. When the buffers have been generated, they are sent
to the SOCs over the individual socket connections. At the end of buffer transmission,
the socket confirms that all data has been received, and Poseidon performs no further
tasks until the end of the sequence.
After data has been received by an SOC, the application running on the SOC
forks into two processes: one process listens for the eventual abort command from
Poseidon, while the other moves buffer samples onto the FPGA. The latter process
works by polling whether a 4096-word FIFO buffer on the FPGA is full or not. This
haardware buffer, which stores the next 4096 words which will be outputted to the
DAC, is used to ensure that variations or lags in the timing of the HPS never leadto a scenario where a new output sample is required but the HPS has not loaded it122

from RAM yet.
The DAC is capable of storing one word in its output buffer (i.e., the word thatis used to generate the DAC’s output voltage) and one word in a separate input shift
register. The purpose of this is to allow for a new word to be outputted immediately
when a logical signal (i.e., clock edge) arrives, without waiting for it to be transmitted
over the SPI bus from the FPGA. There is then sufficient time before the next clockedge for a new word to be transmitted over the SPI bus into the input shift registerof the DAC.
When a clock edge from the lab control clock arrives, the clock signal is passedalong to the "LDAC" ("Load DAC") pin on the DAC which immediately moves theword on the input shift register to the output register. The arrival of this clock edge
also triggers the transmission of the next word from the FIFO buffer to the input
shift register over the SPI line. As successive edges arrive, this process repeats, withthe application running on the HPS ensuring that the FIFO is kept full (or as close
to full as possible, if the clock edges are arriving at a high rate) with new samples
from the buffer it received from Poseidon.
When the experimental sequence has completed, the FIFO buffer is flushed andthe DAC output is returned to its dwell output value, specified by Cicero, until thenext sequence begins.

5.4

PerformanceThis section details the characterization of the analog control channels, including
their noise levels and voltage stability.
Two variants have been realized in the lab: a low-cost variant with higher noise
which can be assembled purely from evaluation boards, and a more costly highperformance variant which uses all of the components described in the previous section. Both variants outperform the National Instruments PXI-6733 in terms of noise,resolution, and stability, and the low-cost variant has a lower cost per output channel
than the National Instruments unit.
123Figure 5-9: The low-cost variant. The two evaluation boards are connected by jumper
wires to enable serial communication between the FPGA and the DAC. An external(isolated) 3.3V power source and the (isolated) power adapter provided with the SOC
evaluation board provide power for the entire system.

5.4.1

The Low-Cost Variant

For lab applications which do not require the lowest noise and tightest stability,
isolated control channels can be assembled using only a pair of evaluation boards: anevaluation board for the AD5791 which has onboard switched-mode power supplies
and an inexpensive voltage reference, and the SOC evaluation board. This is shownin Figure 5-9.
The cost of assembling one of these channels amounts to the cost of the two
evaluation boards, an optoisolator for the lab clock signal, and a 3.3V power supply
capable of supplying about 0.3A. This amounts to approximately $300 per channel.
By comparison, a PXI-6733 has a cost per channel of about $345. By comparison, theadditional cost of the LTZ1000A and associated PCB, plus the supercapacitor bank124

PCB and charging power supply, add at least $300 of additional cost per channel.

5.4.2

Noise Spectrum

The noise spectrum of even the low-cost variant shows a marked improvement overthat of a National Instruments analog output channel. The noise spectrum of a lowcost variant channel was measured by using a 500MS/s digital oscilloscope made byPicoscope to take an FFT spectrum. The measured spectrum yields the total noise
power in each frequency bin of the spectrum in units of dBV (decibels referenced to1V). To convert the measured noise spectrum in dBV to a noise spectral density in√
V/ Hz is simple [31]:𝑁 𝑆𝐷 = 10

Measurement in dBV
20√/ Bin Width in Hz(5.2)

The measured noise spectrum of the output of the DAC with and without the
onboard output buffer amplifier, with a 1V specified DC level, is shown in Figure5-10. Also shown is the noise spectrum of a National Instruments output channel forcomparison.

5.4.3

Stability

In the absence of slowly drifting DC offsets due to ground currents, the precision of
the DAC output voltage is limited by the DAC’s resolution and the precision of thevoltage reference.
The stability of the LTZ1000A + AD5791 system was verified by using a Keysight
34465A 6.5 digit multimeter to measure the change in DAC output voltage vs time
for a fixed output word. In two ten-minute tests, a standard deviation of 8 𝜇V abouta mean of 7.158771 V was measured (test 1) and a standard deviation of 7 𝜇V abouta mean of 7.161054V was measured (test 2).

125

Figure 5-10: Noise spectra of analog output channels outputting 1V DC. Top: The
measured noise spectrum of a National Instruments analog output channel from√DCto about 800 kHz. The power spectral density of the noise is about 130 nV/ Hz
across the entire frequency range, with additional peaks on top of this. Bottom left:
The measured noise spectrum of the unbuffered output from√ a DAC integrated intoa low-cost variant
outputchannel.
The
PSD
is
about
25nV/
Hz at 100 kHz and falls√
to about 11nV/ Hz by 700 kHz. Bottom right: The measured noise spectrumof the√buffered
output
of
the
same
DAC.
The
PSD
ranges
from
about
28
nV/
Hz
to
about
√
30nV/ Hz over the entire frequency range.

126

Chapter 6
Experimental Control and Logging
Database
This chapter describes the implementation of a relational database management system (RDBMS) and associated databases into the experimental control and datalogging flow of our experiment. Changes to existing applications in the Cicero word
generator software suite and new applications that were developed to interface withthe database are also described in detail.

6.1

Motivation

Most quantum gas experiments generically run in a cycle: atoms are trapped andcooled to degeneracy, some experimental parameter is varied compared to the previous run, and an image or series of images are taken with one or more scientificcameras. Saved images can then be imported into scientific programming suites foranalysis. Building off of this, various labs have written various custom applications
which perform lab-specific tasks to meet their data acquisition needs; for example, aroutine might poll a folder for new image files and automatically load any new files
for analysis as they are saved, or an application may save the value of some experimental parameter that is being varied into a running text file. Often, there has been
redundant/parallel development of solutions for different experiments, and many of127the custom solutions need to be heavily modified or replaced when the needs of thelab change.
The Cicero software suite is also limited in its automated running capabilities:
in its original form, experimental parameters could only be varied in predetermined
lists. As quantum gas experiments have become more complex, there has been agrowing interest in the utilization of search and optimization algorithms (such as
genetic algorithms) to minimize the cycle time of experiments, maximize some signalof interest, or even to determine information about the phase diagram of a system[58, 59].
The goal of developing a database backend to the Cicero software suite - and somenew applications to interface with it - was thus manyfold:∙ Automating the storage of all experimental (and eventually, environmental)
parameters during each run of the experiment to allow for searching and sortingin high-level ways.
∙ Creating a system where the same set of simple platform-independent commands (i.e., SQL commands) could be used to access laboratory data from any
computer or application.
∙ Enabling the existing experimental control software to be able to run search andoptimization algorithms by allowing Cicero to interface generically with outside
applications∙ Developing new protections to enable reliable automated operation of Cicero,including rigorous synchronization between the experimental parameters andimages and the ability to interlock on external conditions

6.2

Summary of Developments

The software development related to the implementation of the database consisted ofseveral key parts:
128∙ The definition of a database schema which does not contain redundant or interdependent columns, and which is capable of being easily modified as the needs
and scope of the experiment change∙ The development of a database-interfacing application called Zeus which serves
as a "traffic director" for the control of the experiment.
∙ The development of a USB-interfaced laboratory hardware monitor, which allows for Zeus to determine whether the experimental sequence should run or bepaused based on external conditions such as measured laser powers.
∙ The development of a C# API which allows for Window-based applications toquickly access all of the images, processed data, and experimental parameters
stored in the database, without requiring detailed knowledge of the internal
structure of the database.
In addition to these primary developments, a few other pieces of hardware and software have been developed which demonstrate the expanded capabilities of the experimental control system. These include two genetic algorithm-based optimization
applications which can optimize experimentally measured quantities (e.g., total atom
count or condensate fraction), and a variety of wirelessly interfaced sensors which
open the door to both the idea of "big data" logging of environmental variables andflexible software-based interlocking schemes to protect the experiment during long
stretched of automated running.

6.3

The Database

Because the Cicero suite of laboratory control applications was written for the Windowsspecific .NET framework, and because many pieces of hardware used in the laboratory(such as cameras) only have Windows-compatible drivers, all of our laboratory control
computers run versions of the Windows operating system.
In contrast, RDBMS software is arguably the most straightforward to set up, backup, and maintain when it is running in a Linux environment. We thus set up a central
129

data logging server running an Ubuntu distribution of Linux, and set up the MariaDB
RDBMS on it. MariaDB is a fork of the more popular MySQL software, sharing allof its essential characteristics.
Figure 6-1 shows all of the computers used in running the dysprosium experiment,
which communicate with one another through the gigabit ethernet-compatible LAN.

6.3.1

Structure of the Database

The database is divided into several tables, each of which has a column-row structure,with columns corresponding to data fields and rows corresponding to individual saved
entries in the table. One column of each table serves as a unique integer, called thetable’s primary key, which is used to give a unique identifier to each row.
The tables used are the CiceroOut, Images, Cameras, InputValues, Updates, Sequence, and Analysis tables. The layout of the database is shown in Figure 6-2.
The CiceroOut table stores the values of all of the variables defined within the
Cicero word generator suite for every run of the experiment; these variables define
the laser powers, laser detunings, and power supply currents during each run, as
well as durations that are varied. The timestamp of the run and any user-enterednotes about the run are also stored in the database. Because the variables values are
stored as numerical values and can thus be used in the construction of comparison or
logical SQL commands, this allows for streamlined searching through the databasefor experimental runs meeting certain criteria. For example, one can easily construct
a command which allows them to search for the ID numbers of all runs for which thedetuning of an imaging laser was negative:
SELECT runID FROM CiceroOut where ImagingDetuning < 0;
The Images table stores the raw binary data of each image acquired by a camera
during the experiment. In each row there are up to three images: Atoms, Light, andDark. For a typical absorption image, all three images are taken, and are then saved
to be later used to construct the composite image from which the total atom number
and fits of the cloud profile can be extracted. Along with each entry in this table
130Figure 6-1: The network-enabled devices used in our experiment. Most control software runs on a single computer. The Poseidon server discussed in Chapter 5 passes
buffer information to each of the SOCs over the network connection as previously
described. The Zeus server, memcached daemon, and genetic algorithm applications
noted in the diagram will be discussed later in this chapter.

131

Figure 6-2: The tables and fields of the experimental control/logging database. The
large text indicates the name of a table and the small text below it indicates thenames of the fields belonging to that table. The two tables in the dashed box always
contain exactly one row of data, which is updated by various applications during theexperimental sequence. The lines indicate relationships between primary keys andforeign keys, as described in the text.

132

is stored the CameraID of the camera which acquired the images (see the following
paragraph), the runID of the associated experimental sequence run by Cicero, anda unique ImageID primary key field. A database call which can pull up all images
associated with specific experimental criteria is easy to construct:
SELECT runID FROM CiceroOut where ImagingDetuning < 0;
SELECT imageID FROM Images where runID_fk = $result;where the "$result" variable represents the result of the first line. The abbreviation"fk" here refers to the fact that in the images table, the runID refers to a field in
another table (CiceroOut) which is the primary key for that table; the runID_fk fieldin Images is thus a "foreign key" while the runID field in Cicero is a primary key.
The Cameras table stores the specifications of each camera (or special mode of useof each camera) in the lab. These specifications include the sensor dimensions andthe physical pixel size, and can be called to correctly reconstruct the images stored
in the Images table and to obtain the correct measured atom number from them;
images must stored in the Images table as 1D arrays, and so must be reshaped using
the saved values of the camera’s sensor dimensions. For example, the commands todetermine the sensor dimensions for a particular image might look like this:SELECT cameraID_fk from Images where ImageID = 16234;
SELECT width,height from Cameras where cameraID = $result
The InputValues table allows for Cicero variables to be bound to values which arecontrolled by external applications which have access to the database. Prior to theimplementation of this database table, the values of experimental control variables
in Cicero could only be specified prior to a batch of runs commencing, precluding
the implementation of search or optimization algorithms. The InputValues tableconsists of a fixed number (twenty in our case) of rows – although more can be added
easily – each of which stores the value to be associated with a variable, and a fieldindicating whether the value has been updated for the next run of the experimental
sequence. Using this table consists of binding a variable in Cicero to one of the rowsof this table, and then having an external application (such as a search/optimization133

algorithm application) update all of the bound fields each run. Since it may takeseveral seconds to e.g. fit the detected cloud shape after an image is taken, the fieldindicating whether the bound fields have been updated can be used by the Zeus server
to halt the execution of the next experimental sequence until after the fields have been
updated.
The Updates table is used to store some simple binary variables which are used as
flags to maintain synchronization between applications interfacing with the database.
The "newRun" field is set to true whenever a new experimental sequence begins,alerting any data acquisition applications that the data they are about to collect isassociated with the most recent runID value that can be found in the CiceroOut table.
The "newImage" field alerts Zeus that new images have been collected. As will be
described in the following section on Zeus, images are initially saved into a RAM-only
cache on the Linux server to facilitate immediate viewing and/or analysis, but only
permanently saved if the user has opted to save all images to the database. The
newImage field tells Zeus that new image data is ready to be saved to the database
if this has been enabled.
The Sequence table stores the name of and short description of each experimental
sequence, as specified by the user within the Cicero control software. This allows forconvenient string-based searching of the database for specific sequenceIDs, which arethen associated through the Images table with any saved data and the experimental
parameters in the CiceroOut table.
Oftentimes, image analysis is set up to be performed immediately as images are
acquired, but then images are later loaded and reanalyzed more carefully later. The
results of the real-time analysis are stored in a table called Analysis, which associates
the measured atom number, parameters of functional fits, and composite quantities
derived from these values with a specific runID (as a foreign key in the table). The
reason that the values are associated with a runID and not an imageID is that one
can elect to save hard drive space by not saving any of the image data during a longbatch run, but only saving the real-time analysis results; in this case, there are no
imageIDs in the database to associate with the analysis results.
134

6.4

The Zeus Server

The additional functionality made possible by the implementation of the databasebackend, as well as the inevitable increase in complexity, warranted the developmentof an all-in-one application to streamline the usage of the database. Zeus is anapplication which performs a variety of auxiliary tasks involving communication withand through the database. The tasks performed by Zeus are
1. Saving the experimental variable values and timestamp to the CiceroOut table
during each run of the experiment
2. Ensuring that acquired images have been associated with the correct runIDby only allowing a new run to begin after all images have been acquired. This
especially prevents hang-ups or lag in the data acquisition software from causing
images to be erroneously associated with the next run’s entry in the CiceroOut
table.
3. Monitoring user-specified hardware and software conditions to determine whether
the experimental sequence should run or be paused
4. Saving captured image data, initially stored in a temporary RAM cache for fastaccess, into the Images table if the user has specified this
5. Ensuring that if Cicero’s variables are bound to fields in the InputValues table,
Cicero does not run until these fields have been updated by external applications
The connection between Cicero and Zeus follows the same model as the connection
between Cicero and the hardware control server Atticus; the connection is made using.NET remoting, wherein Cicero (the client) passes data to Zeus (the server) andremotely executes functions defined on Zeus. Most of Zeus’s functionality is thus
controlled by Cicero, with checkboxes and input fields in the main Windows form
primarily allowing users to specify options.
135

6.4.1

Data Flow

A complete run of the experiment, using the database and the Zeus application,
proceeds as follows.
When an experimental sequence begins, Cicero first runs a function called "CheckIfCiceroCanRun" on Zeus.1 Zeus then checks for various conditions to be met, basedon the user-specified options regarding software and hardware conditions, and whether
the "manual override" checkbox has been clicked to prevent the sequence from running. Examples of these conditions include whether a specific piece of software is stillrunning (monitored by ensuring that Zeus is receiving a regular "heartbeat" fromthat application, i.e. a new random number every second or so) or whether a digital
voltage line in the lab is reading logical high.
Once all conditions have been met and the manual override box has been unchecked,
Zeus checks to see whether any variables which have been bound to the database arestill waiting for updated values. This may be the case if the analysis of the previous
image has not yet completed, or if an optimization application is still computing what
the next set of variable values to try should be. As soon as each bound variable hasreceived an updated value, the variable values are moved into Cicero’s application
memory and the database fields are reset to their non-updated states in preparation
for the next run.
Once any bound variables have been updated, Cicero passes all of the information
about the experimental sequence to Zeus, and Zeus saves all of the variable values,
a timestamp, and the user-entered description of the sequence to the CiceroOut andSequence tables. If new variables have been added since the last time data was saved
to the CiceroOut table, Zeus modifies the structure of the table and associates a newcolumn with the new variable.
The experimental sequence then runs as specified by Cicero and carried out by
1

Technically, all functions called over .NET remoting by Cicero are run on all connected servers,
including the hardware control servers Atticus and Poseidon. All of these servers must have adefinition for all functions that Cicero will call, but the functions can simply be set to return "true"
if that server is not relevant to that function call. For example, the CheckIfCiceroCanRun function
is defined in Atticus and Poseidon, but simply returns "true" when called.

136

hardware servers Atticus and Zeus. At the end of the sequence, data acquisition
software (typically camera control software) is triggered and one or more images
is acquired. These images are then saved immediately into a RAM-only memory
cache on the Linux server. The motivation behind this step is to move the image
data out of program memory and into a place where other applications in the lab
can read it immediately, but prevent hard drive reading/writing times from adding
additional time to the experimental sequence. For example, the routine determinationof the resonance frequency of a transition involves taking many absorption images andcomparing the optical depths at different frequencies; there is no need to save any ofthese images as soon as the OD is extracted, however, and so it is sufficient to havethe images be analyzed in real time as they are moved into the cache.
Once image data has been acquired and stored in the cache, Zeus will move theimage data from the cache to the database for permanent storage if the user has
elected for this. In this case Zeus will also halt the execution of the next experimental
sequence until the image data has been saved, in order to ensure that the image
acquisition software associated the image data with the correct runID, and to preventthe sequence from repeating more quickly than image data can be saved (in the caseof very short sequences of around a second in duration).

6.4.2

External Hardware Monitoring

A well-built quantum gas experiment can perform hundreds or thousands of experimental sequences in a row without any human attention. Inevitably, certain things
will degrade over the course of a day or night, such as the fiber coupling efficiency of
trapping lasers or the alignment of a sensitive beam. While hardware interlocks willusually be set up in the lab to prevent degradations like these from causing damage tothe equipment, there was no functionality in place to prevent the continued executionof the experimental sequences (and acquisition of meaningless data) if some hardware
condition or conditions were not met.
To address this limitation, a small microcontroller-based circuit which can measure several digital and analog voltages simultaneously, and communicate their values137

Figure 6-3: A USB port provides both power for the board and serial communication
with the host computer. A series of screw terminals break out the digital and analog
pins of the ATMEGA328 microcontroller for easy use in the lab. The Zener diodes D8
and D7 allow the 5V-compatible microcontroller to communicate over USB without
risking exceeding the common-mode voltage of the USB chip on the PC. The 16MHz
onboard crystal stabilizes the timing of the microcontroller sufficiently to communicate over the USB port.

to Zeus many times per second, was developed. The microcontroller, an 8-bit AVR
microcontroller without any native USB capabilities, uses the VUSB library to communicate as a USB human input device (HID) over the PC’s USB port using its own
native USART port. Figure 6-3 shows the layout of this peripheral.

The advantage to using a custom microcontroller circuit over a commercial ADC/digital
I/O peripheral is that the firmware can be highly customized. In addition to performing simple acquisition of signals, the microcontroller can be programmed to perform
other tasks such as on-board computation and interlocking, low-bandwidth feedback,and control of laboratory devices. For example, one peripheral was programmed tosend a series of logical pulses to a hardware interlock in the lab to reset the interlock
state under certain conditions.
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6.5

Genetic Algorithm

The database backend has allowed us to control our experiment (i.e., control the
Cicero sequence) using external applications which run optimization routines. A
genetic algorithm-based application has been developed which allows for optimizationof any metric constructed out of the atom count, fitted widths, and fitted position of
an imaged cloud.
The idea behind a genetic algorithm is to optimize the value of this metric (the"fitness function") by a heuristic approach that mimics the "survival of the fittest"
notion in evolutionary biology [50]. A trial solution to the optimization problem isreferred to as an "organism," each of which contains a set of "genes" which represent discrete parts of the trial solution. For our genetic algorithm application, theorganisms are a set of variable values that specify e.g., magnetic field values, VCO
frequencies, and timestep durations. The value of each variable (up to 20 can beused) constitutes a gene. The algorithm is as follows:
1. Create an initial population of N organisms.
2. Evaluate the fitness function of each organism by running the experimental
sequence with its parameters
3. Create N new organisms (a new "generation") by selecting 2 parents for eachnew organism and "breeding" them. The probability of an organism being
selected as a parent is proportional to the value of its fitness function. Breeding
is performed by beginning by copying a gene from the first selected parent;
for each successive gene, the probability of swapping to the other parent (andtaking that gene from them instead) is given by the "cross-over rate."
4. With a probability given by the "mutation rate," genes in the new set of N organisms are mutated: they are assigned new random values in order to introduce
variability into the population.
5. Steps 2-4 are repeated until a consistent solution with a consistent fitness function has emerged. This is referred to as "convergence" of the algorithm.
139

Figure 6-4: Genetic algorithm application. The variables on the left specify eachof the "genes" that is being evolved by the application. The scatter plot shows the
distribution of fitness functions of all organisms for each generation.

The application which performs the genetic algorithm optimization is shown in Figure
6-4.
This application interfaces with both the main database described in the previous
sections, and a secondary database. The secondary database stores all of the information about all of the organisms as the algorithm is running. This allows the genesof the optimal organisms to be looked up by the user after running the genetic algorithm, and also allows the application to resume from a previously, partially evolved
population. The structure of this secondary database is shown in Figure 6-5.
The analysisResults table consists of a single row of data which is updated bysome analysis script (or manually by a user, if desired).
The genetic algorithm has been used to optimize the phase space density of acompressed MOT in the dysprosium experiment, and the evaporation of lithium atoms
into a BEC in a separate experiment.
140

Figure 6-5: The genetic algorithm application database. The line indicates the primary key-foreign key relationship between the two fields. The en/name/max/min
columns in the experimentSettings table specify the name, max value, and min valueof the 20 possible variables being used by the algorithm. The en field indicates whether
or not that variable is being used in the specified experiment. The organisms tablesaves every organism in every generation of the algorithm, so that the optimal organisms’ parameters can be looked up even if the algorithm does not converge to a
globally optimal solution.

6.6

The Future: Lab Monitoring with a Mesh Network

In addition to logging experimental parameters during each sequence, the databasecan be extended to log measurements (such as laser powers or measured magnetic
fields) or environmental variables (such as temperature or humidity) every time thesequence is run. These values could also be integrated into Zeus, allowing for interlocking or alerts based on measured values from the lab; for example, Zeus may
advise against running a sequence if a thermocouple on a magnetic field coil indicatesthat it has become too hot.
Inexpensive wireless hardware to perform these measurements has been developed
using the Zigbee protocol, a short-ranged radio protocol that is ideal for low-frequency
measurement of a large number of values around a building or room. The Zigbee protocol establishes a self-healing mesh network among distributed radio transceivers.
The idea behind a mesh network is that rather than having many devices communicating with a central "coordinator" (e.g., the wireless router in a typical home or lab
141

Figure 6-6: A Zigbee mesh network. Reproduced from [27]. A single coordinator
establishes the network, and is often the destination of data measured by distributed
sensors on the network. Routers are full-featured nodes, able to both transmit theirown messages onto the network and relay messages to and from other nodes on the
network. Limited-functionality end devices can only transmit into the network; often
used in low-duty cycle sleep modes to save power in between periodic measurements,
end devices can run for months or years on single battery charges, but are not "awake"enough to participate in the relaying of messages.

WiFi setup), devices can pass messages to their final destinations by relaying themthrough other devices on the network. This allows the network to reach beyond thereceiving range of a single coordinator. The self-healing aspect means that a newroute to the final destination will automatically be found (if possible) if a device
on the previously established route fails. For distributed arrays of sensors potentiallyrunning on battery power, this minimizes the impact of devices running out of batteryon the rest of the network.
The structure of an example Zigbee network is shown in Figure 6-6.
Compact circuits containing a Zigbee transceiver, an 8-bit microcontroller, and a
few peripherals have been developed. The ADCs on the microcontroller can be used
to measure the outputs of sensors such as photodiodes, humidity sensors, etc. The
results of the measurements can then be sent to the Zigbee module over a serial bus,
142

Figure 6-7: Zigbee device schematics. Left: the low-power variant designed to be
run using batteries. The four SMA connectors are 4 measurement inputs. Right: thehigher-power variant with onboard switching regulators and instrumentation amplifiers with selectable gain.
and then sent to the network coordinator through the mesh of other Zigbee devices.
The coordinator is connected to the lab control computer, and communicates with a
C application to make the data available to the lab control software.
Two variants have been designed: one which uses only the onboard ADC hardwareon the microcontroller, and which uses very little power if the microcontroller andzigbee are put into sleep mode periodically; and and one which uses more power inorder to implement instrumentation amplifiers (and onboard switching regulators topower them off of a 5V input) for a wider range of measurement applications. The
former are well-suited to act as battery-powered end devices, while the latter require
an external power supply and so are well-suited to act as routers.
The schematic layouts of both boards are shown in Figure 6-7. Photographs ofthe devices are shown in Figure 6-8. While the devices function and communication
has been established between a C application and the end devices/routers, there isadditional programming work to be done to fully implement the sensors into Zeus or
to log their values into the database.

143

Figure 6-8: Zigbee hardware. A lower-power variant assembled and successfully tested
in the lab.

144

Chapter 7
Outlook
While the story of this thesis ends with the production of BECs of dysprosium, thestory of this experiment is just beginning. The following sections discuss the shortterm and longer-term possibilities for the experiment now that it produces quantum
degenerate gases dysprosium.

7.1

741 nm and Lattices

The immediate next step for the experiment is to address the BEC with 741 nm
light. Eventually, the 741 nm light can be used to implement the spin-orbit coupling
schemes that were described earlier; prior to this, it will be a useful transition for
creating optical lattice potentials which can be made very deep (at the expense of
scattering lifetime, by tuning closer to resonance) if desired. This will be useful for
exploring how dipolar relaxation can be suppressed by geometrical effects, such as
the effect described in [56].
A 741 nm source is already set up in the laboratory, and so it should not be long
until the investigations of dipolar relaxation and spin-orbit coupling can begin.
145

7.2

Toward Spin-Orbit Coupling

The spin-orbit coupling schemes discussed earlier will only work for bosonic dysprosium if the dipolar relaxation rate can be suppressed in some way. If the relaxation
can be suppressed, then the door is open to studying long-range interacting 1D and2D spin-orbit coupled bosons for the first time.
If the relaxation cannot be suppressed, then investigations of spin-orbit coupling
can still be performed using the fermionic isotopes, as was demonstrated in 1D by
the Lev group. Converting the experiment to cool and trap one of the two abundantfermionic isotopes should require only adjustment of the locking frequencies of the
626 nm and 421 nm lasers. Realizing a long-range interacting 2D spin-orbit coupledsystem of fermions would be a novel achievement.

7.3

Technological Improvements and Automation

A few technology upgrades which are in the works will also improve the cycle time ofthe experiment as well as its performance. Several seconds of the current experimental
sequence are currently dedicated to the triggering and acquisition time for the three
separate exposures that form the absorption image. Replacing the software-triggered
camera with a hardware-triggered scientific camera will eliminate the majority of thisextra time from the sequence, and will also allow for lower expsoure times to be used,decreasing the effects of stray light and fringing.
Switching from the iodine-based locking setup for the 626 nm laser to a ULE
cavity is expected to improve the signal-to-noise of the locking signal, yielding better
control of the MOT laser. The lower power requirements of the ULE will also free
more power to be used for the MOT beams, which we expect will slightly improve
the MOT population.
The groundwork has been laid for the implementation of higher level- and remotecontrol capability in the lab through the database system and Zeus application. Perhaps the operation of the experiment in the future will further take advantage of this146

to functionality in order to employ algorithms for optimization of experimental cycle
time, or even to explore the phase diagrams of newly realized quantum systems as
has been reported in [58].

147

148

Appendix A
C# API for Image Database
To make it simpler to integrate a new application with the SQL database described in
Chapter 7, a C# API was developed. The API exists in the form of a .NET assembly– specifically, a DLL – called the "Database Helper." By calling functions from thisDLL during runtime, an application can perform all of the most common queries on
the database without requiring the user to construct the correct SQL commands.

A.1

Structure of the API

The API contains the following components:
1. A constructor which establishes the connection to the database using userspecified parameters
2. A set of specialized data types which are used to store the results of methods
3. A set of methods which carry out specified queries on the database
Minor modifications to the methods in this API – for example, to change the getAnalysisResults() method if new fields are added to the Analysis table – are straightforward
to make and allow for high level of flexibility.
149

A.1.1The Constructor

To use the API, the constructor must be passed the connection parameters for the
database when the DLL is instantiated. The constructor takes as inputs the IP address of the computer running the memcached server, the IP address of the computerrunning the MariaDB server (usually the same computer), the username and password to use to log into the database, and the name of the database to be accessed(e.g., "DysprosiumLabDatabase").

A.1.2

The Data Types

There are several new data types defined within the API∙ imageStruct: Used for the results of methods which call images from thedatabase. Contains arrays for Atoms, Light, and Dark images which are used
to create a composite absorption image; the timestamp; the height and widthof the image so that the returned 1D arrays can be reshaped into 2D images;
the pixel size of the camera used; and a list of all of the saved Cicero variables
associated with that image, in form of an array of variable types.
∙ variable: Stores the name and value of a Cicero variable that is read from thedatabase∙ description: Can be used to store the sequence name and sequence descriptionof a stored entry from the sequences table

A.1.3

The Methods

The most commonly used/most useful methods in the API are described below
1. readImage() or readImage(int imageID): Overloaded method which returns an imageStruct. The data returned corresponds to the most recent image
if no argument is passed, or to the specified imageID if one is passed to themethod.
150

2. getAnalysisResults(int runID): Returns the saved results of the analysisof image "imageID" as an array of floating point numbers. This can include
the atom number, peak widths, peak positions, and any other metrics that are
saved to the database.
3. readImageFromCache(): Returns on imageStruct, populated by the data
currently held in the memcached cache
4. readVariableValues or readVariableValues(int runID): Returns a structcontaining an array of variables datatypes, populated by the list of variables
and their values from either the most recent runID of the specified runID.
5. getLastRunID(): Returns the most recent runID
6. checkForRunUpdate(): Returns a 1 (0) if the "new run" field has (hasn’t)
been set to true in the Updates table in the database

151

152

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