Experimental realization of atomtronic circuit elements in
non-equilibrium ultracold atomic systems
bySeth C. Caliga
B.A., University of Colorado, 2009
M.S., University of Colorado, 2013

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillmentof the requirements for the degree ofDoctor of Philosophy
Department of Physics
2016This thesis entitled:
Experimental realization of atomtronic circuit elements in non-equilibrium ultracold atomic
systems
written by Seth C. Caliga
has been approved for the Department of Physics

Dana Z. Anderson

Murray J. Holland

Date

The final copy of this thesis has been examined by the signatories, and we find that boththe content and the form meet acceptable presentation standards of scholarly work in theabove mentioned discipline.

iii
Caliga, Seth C. (Ph.D., Physics)
Experimental realization of atomtronic circuit elements in non-equilibrium ultracold atomic
systems
Thesis directed by Dana Z. Anderson

Research in the field of atomtronics aims to develop a new paradigm for the use of ultracold atomic systems in a manner that mimics the functionality of electronic circuits anddevices. Given the ubiquity of the electronic transistor and its application to a vast array ofsignal processing tasks, the development of its atomtronic counterpart is of significant interest. This dissertation presents the experimental studies of two atomtronic circuit elements: abattery and transistor. Experiments are conducted in an atom-chip-based apparatus utilizing hybrid magnetic and optical trapping techniques that enable one to “pattern” atomtronic
circuit elements. An atomtronic battery is realized in a double-well trapping potential inwhich a finite-temperature Bose-Einstein condensate is prepared in a non-equilibrium state
to generate thermodynamic gradients that drive atom current flow. Powered by the atomtronic battery, a triple-well atomtronic transistor is demonstrated, and quasi-steady-statebehavior of the device is characterized. Results are found to be in agreement with a semiclassical model of the transistor that is also used to study the active properties of the device,including current gain. Based on these results, future directions regarding signal processing
operations are proposed.

Dedication

To my wife, Kristen, and parents, David and Tina.
In loving memory of Grandma Margaret and Grandpa Charles

v

Acknowledgements

It goes without saying that the work contained in a Ph.D. dissertation is not the resultof a singular effort. First among those who deserve recognition is my wife, Kristen, who
has stood by my side since the very beginning. Her unending encouragement significantly
eased the stresses of graduate school. I cannot thank her enough for the sacrifices that she
made along the way to support the two of us. She adds balance to my life, keeping me
focused on my goals while also providing enough distraction to keep me sane. Next are myparents, David and Tina, who raised me in an environment that encouraged the pursuit of
any activity that intrigued me. Their love and guidance have been invaluable. A big thankyou to my sister, Sasha, and her beautiful family, my awesome in-laws, Dave and Pattie,
along with all of my extended family in both TX and CO for their continued support. I
would also like to acknowledge my best friend, Kyle Thatcher. After living together for many
years (and a few emergency room visits) he has become a brother to me.
Thank you to my advisor, Dana Anderson, who has overseen my development as anexperimental physicist for nearly nine years. I am grateful that he recognized my passion for
physics during his undergraduate optics lab course and invested his time to help me pursue
my dream. He has afforded me a great deal of freedom in the lab, allowing me to steer the
research path while ensuring that I didn’t stray too far off course. He also taught me theimportance of both written and verbal scientific communication, and impressed upon me his
knack for conjuring intuitive descriptions of complex physics concepts.
While working in Dana’s lab, I had the pleasure of working alongside many fantastic
individuals, some of whom have moved on to bigger and better things. My first mentor,

viStephen Segal, exhibited unyielding patience as I cut my teeth in the research world. I
consider myself lucky to have added his methodical approach to apparatus design and construction to my own toolbox. Evan Salim’s intuition for all things mechanical never ceases toamaze me. Even more so does his range of talents outside of the lab. My first postdoc, Dan
Farkas, shared just a sliver of his knowledge with me during the time we overlapped at JILA,
including the ins and outs of laser cooling, the importance of proper electrical shielding andcable construction, and a healthy appreciation for hip hop. Leslie Czaia, with her steadyhands and immaculate attention to detail, helped create the numerous vacuum chambers
used throughout my research. In addition to her technical skills, I greatly appreciate Leslie’s
outlook on life and the many interesting conversations that we shared. Lastly, Kai Hudek, a
man after my own heart. The knowledge that Kai passed on to me is difficult to summarize.
He continues to be an inspiring role model and a true friend.
In the latter half of my time at JILA, I had the pleasure of working with several talented
graduate students. I could not have found two better labmates and friends than Cameron
Straatsma and Carrie Weidner. The significant contributions of Cameron are reflected inmuch of the work presented in this dissertation. As an optical engineer, Cameron’s skill set
was originally quite different from my own. In the end, I hope to have taught him as much
physics as he taught me about programming and optics. I wish Cameron the best of luckas he continues on the atomtronics experiment. While I did not work with Carrie directly,
her humor and passion for physics (and the Chicago Cubs) were contagious. I will miss thehours of ice time that the three of us spent together on our IM hockey team (Champions,
Spring 2016!). Thank you also to my second postdoc, Hoon Yu, fellow graduate students
Brad Dinardo, Doug Bopp, Ava “Queso” Ashby, Phoebe Tengdin, and Evolene Premillieu,
and undergraduate, Taylor Froelich, for keeping life in the lab interesting.
Outside of Dana’s lab, JILA houses a myriad of talented individuals who helped mealong the way. Members of the instrument shop Hans Green, Blaine Horner, Todd Asnicar,
Tracy Keep, Kim Hagen, Ariel Paul, and of course Kyle Thatcher, have tolerated innumerable

vii
questions regarding machining and mechanical design for which I am most grateful. The
same appreciation goes to the electronics shop staff. Thanks to Terry Brown, Carl Sauer,
and James Fung-A-Fat, for always pointed me in the right direction when a circuit was
misbehaving.
I also had the pleasure of collaborating with a handful of excellent people. Alex Zozulya
helped develop a great deal of the theory presented in this dissertation. Imaging guru Jay
Margolis of Infinity Photo-Optical was instrumental in developing the optical system usedin my research. David Van Wie of Boulder Labs developed the LiveAtom software used tomodel our atom chip traps. Also, Marine Pigneur whose visit to JILA may have been short
but resulted in a lasting friendship.
Finally, I would like to thank Wes Mangney, my first mentor in physics, who inspired
many young minds and was taken from us too soon. Both he and Kevin LaBella helped me
to discover my passion for physics at Air Academy High School.

viiiContents

Chapter
1 Introduction1

1.1

Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

Atomtronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.3

The pursuit of an atomtronic transistor . . . . . . . . . . . . . . . . . . . . .

5

1.4

Fundamental atomtronic circuit elements . . . . . . . . . . . . . . . . . . . .

6

1.5

Dissertation outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

82 Theory

10

2.1

Theoretical description of Bose-Einstein condensates . . . . . . . . . . . . . .

10

2.1.1

Non-interacting Bose-Einstein condensates . . . . . . . . . . . . . . .

11

2.1.2

Interacting Bose-Einstein condensates at zero-temperature . . . . . .

15

2.1.3

Corrections due to finite temperature, size, and interatomic interactions 17

2.2

Theoretical study of a triple-well atomtronic transistor . . . . . . . . . . . .

20

2.2.1

The theoretical system . . . . . . . . . . . . . . . . . . . . . . . . . .

22

2.2.2

Current flow mechanism . . . . . . . . . . . . . . . . . . . . . . . . .

22

2.2.3

Source and gate well ensembles in steady-state . . . . . . . . . . . . .

25

2.2.4

Current gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30

2.2.5

Transistor power output and transconductance . . . . . . . . . . . . .

34

2.2.6

Connection to electronic transistor functionality . . . . . . . . . . . .

37

ix
3 Atom chip development

39

3.1

Magnetic trapping of neutral atoms . . . . . . . . . . . . . . . . . . . . . . .

39

3.2

Designing an atom chip for atomtronic devices . . . . . . . . . . . . . . . . .

40

3.2.1

Side-guide trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

3.2.2

Split-guide trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

3.2.3

H-wire trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47

3.2.4

T-wire trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

3.3

3.4

Atom chip designs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

3.3.1

Atom chip standardization . . . . . . . . . . . . . . . . . . . . . . . .

53

3.3.2

The V2 window chip for in-trap imaging . . . . . . . . . . . . . . . .

53

3.3.3

The V3 window chip, a double split-guide and other design upgrades

57

Technical considerations for atom chip traps . . . . . . . . . . . . . . . . . .

62

3.4.1

Ensemble heating in the split-guide trap . . . . . . . . . . . . . . . .

63

3.4.2

Wire heating comparison in window and non-window chips . . . . . .

66

4 High-resolution imaging & optical control of ultracold atoms
4.1

Atom-light interactions in neutral atoms . . . . . . . . . . . . . . . . . . . .
4.1.1

4.2

4.3

4.4

73
73

Refractive index, absorption, and scattering rate of atoms in a light field 75

Imaging methods in the atomtronics experiment . . . . . . . . . . . . . . . .

77

4.2.1

Weak absorption imaging of atomic ensembles in free expansion . . .

78

4.2.2

Strongly saturated absorption imaging of trapped atomic ensembles .

80

Optical trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

4.3.1

Heating in optical dipole traps . . . . . . . . . . . . . . . . . . . . . .

82

4.3.2

Blue-detuned optical barriers . . . . . . . . . . . . . . . . . . . . . .

83

A high-resolution microscope system for imaging trapped atoms and projecting optical potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85

4.4.1

85

Through-chip imaging . . . . . . . . . . . . . . . . . . . . . . . . . .

x
4.4.2

Optical potential projection system . . . . . . . . . . . . . . . . . . .

5 Experimental demonstration of atomtronic devices

93
98

5.1

BEC production and preparation for atomtronics experiments . . . . . . . .

5.2

The atomtronic battery system . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.2.1

5.3

5.5

Charging the battery . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

Impedance matching in atomtronic systems . . . . . . . . . . . . . . . . . . . 102
5.3.1

5.4

An atomtronic terminator . . . . . . . . . . . . . . . . . . . . . . . . 105

Characterizing the atomtronic battery . . . . . . . . . . . . . . . . . . . . . 107
5.4.1

Modeling the battery . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5.4.2

Battery performance . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

An atomtronic transistor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.5.1

The experimental triple-well system . . . . . . . . . . . . . . . . . . . 120

5.5.2

Transport dynamics in the atomtronic transistor . . . . . . . . . . . . 122

5.5.3

Source and gate wells in steady-state . . . . . . . . . . . . . . . . . . 124

5.5.4

Further discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

6 Outlook
6.1

98

132

Current controlled atomtronic transistors . . . . . . . . . . . . . . . . . . . . 132
6.1.1

More complex atomtronic devices . . . . . . . . . . . . . . . . . . . . 133

6.2

Coherent behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

6.3

Tailored barrier transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

Bibliography

139

Appendix
A Derivation of the normalized temperature and chemical potential drops

150

xi

FiguresFigure1.1

Original thermionic valve and point contact transistor . . . . . . . . . . . . .

2

1.2

Double-MOT vacuum chamber used to conduct atomtronics experiments . .

4

1.3

In situ images of the atomtronic battery and transistor during operation . .

7

2.1

Thermal cloud density at temperatures T > Tc , T = Tc , and T < Tc . . . . .

18

2.2

Node biasing in two analogous circuits, the atomtronic transistor and common
source FET amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

2.3

Schematic of the steady-state transistor model parameters . . . . . . . . . .

26

2.4

Fractional gate-drain current versus feedback parameter . . . . . . . . . . . .

28

2.5

Temperature drop across the source-gate barrier in steady-state . . . . . . .

29

2.6

Normalized chemical potential drop across the source-gate barrier in steadystate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

2.7

Differential current gain in the atomtronic transistor. . . . . . . . . . . . .

33

2.8

Transconductance at various above-threshold feedback parameters . . . . . .

35

2.9

Peak theoretical power dissipation at the transistor output . . . . . . . . . .

36

3.1

Magnetic field created by a side-guide chip trap . . . . . . . . . . . . . . . .

42

3.2

Quadrupole versus harmonic side-guide traps . . . . . . . . . . . . . . . . . .

43

3.3

Split-guide magnetic field configuration . . . . . . . . . . . . . . . . . . . . .

46

3.4

Split-guide chip wire limited numerical aperture . . . . . . . . . . . . . . . .

48

xii
3.5

Example H-wire trapping potential . . . . . . . . . . . . . . . . . . . . . . .

50

3.6

T-wire trapping potential used during chip loading . . . . . . . . . . . . . .

52

3.7

V2 window chip pictures . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

54

3.8

V2 window chip H- and T-wire traps . . . . . . . . . . . . . . . . . . . . . .

56

3.9

Ambient, vacuum, and x-ray view of the V3 window chip . . . . . . . . . . .

58

3.10 Finite element analysis of the current density at the intersection of the chip
T-wires . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

60

3.11 Comparison of the V2 and V3 longitudinal split-guide chip trap fields . . . .

61

3.12 Ensemble heating rate in the split-guide trap . . . . . . . . . . . . . . . . . .

65

3.13 Test setup used to determine the thermal properties of atom chips during
current flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67

3.14 Wire heating behavior of window versus non-window chips . . . . . . . . . .

70

3.15 Increase in the relative chip wire resistance versus current magnitude . . . .

70

3.16 Burned out wires on the V1 window chip and direct evaporation chip . . . .

72

3.17 Relative increase in the non-window chip wire resistance at different experiment duty cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

72

4.1

Complete imaging and projection system used in the atomtronics experiments 86

4.2

Fluorescence and absorption imaging beam paths . . . . . . . . . . . . . . .

4.3

In situ images from fluorescence imaging resolution test with ∼ 2.5 µm fringes 90

4.4

Fourier transforms of images used to test fluorescence imaging resolution . .

91

4.5

Comparison of fluorescence and absorption imaging techniques . . . . . . . .

93

4.6

Static and dynamic projection systems used in atomtronics experiments . . .

95

4.7

First demonstration of optical control over trapped atoms . . . . . . . . . . .

96

5.1

Schematic of the atomtronic battery potential . . . . . . . . . . . . . . . . . 101

5.2

Schematic showing the method used to charge the atomtronic battery . . . . 103

5.3

Percent error in the Thomas Fermi approximation in a half-harmonic well . . 104

87

xiii
5.4

Diagram of an atomtronic terminator . . . . . . . . . . . . . . . . . . . . . . 106

5.5

Reservoir ensemble evolution during battery discharge . . . . . . . . . . . . . 108

5.6

Experimental measurement of κ, the excess energy parameter

5.7

Atomtronic battery Thévenin equivalent circuit . . . . . . . . . . . . . . . . 114

5.8Internal resistance of the atomtronic battery for various barrier heights . . . 116

5.9

Terminal voltage of the atomtronic battery for various barrier heights . . . . 116

5.10 Atomtronic battery current capacity. . . . . . . . 112. . . . . . . . . . . . . . . . . . . . . . 118

5.11 Battery performance characterized by the energy capacity and peak power
output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
5.12 Schematic of the atomtronic transistor potential and in-trap atomic density . 121
5.13 Gate well population measured with a saturated probe (Ip  Isat ) . . . . . . 123
5.14 Net source-gate atom current during transistor operation . . . . . . . . . . . 123
5.15 Time series of the atomtronic transistor operation . . . . . . . . . . . . . . . 125
5.16 Gate well approach to equilibrium population . . . . . . . . . . . . . . . . . 127
5.17 Quasi-steady-state gate well population versus feedback parameter . . . . . . 127
5.18 Quasi-steady-state temperature drop versus feedback parameter . . . . . . . 128
5.19 Quasi-steady-state chemical potential drop versus feedback parameter . . . . 130
6.1

Schematic for the proposed current controlled atomtronic transistor . . . . . 133

6.2

2D current controlled atomtronic transistor and a multi-transistor circuit . . 134

6.3

Model system used to study coherent transistor output current . . . . . . . . 136

Chapter 1
Introduction

1.1

Motivation
The historical development of devices for the purpose of generating and manipulating

electronic signals arguably began following Thomas A. Edison’s invention of the lightbulb,
at the end of the 19th century [1]. Shortly after its invention, more complicated lightbulbs enclosing multiple electrically isolated filaments were constructed, leading Owen W.
Richardson to discover thermionic emission from hot bodies, previously known as the Edison
effect [2]. By 1904, John A. Fleming invented the first vacuum tube specifically for the
purpose of measuring AC signals and signal rectification (see Figure 1.1(a)), marking the
beginning of electronic signal processing. Fleming’s thermionic valve gave way to manysuccessive iterations of vacuum tube amplifier designs, including triodes and pentodes thatcan still be found in the amplifiers of today’s audio enthusiast. Signal processing technology
took its next big leap in 1948, with the invention of the semiconductor transistor [3], at
Bell Laboratories (see Figure 1.1(b)), which ushered in the era of integrated circuitry andcontemporary electronics.
Moore’s law observes that the number of transistors on an integrated circuit doubles
approximately every two years. Quite apart from a comment about the remarkable evolutionof electronics and fabrication techniques, Moore’s law highlights the cardinal role of thetransistor in modern technology. What is it that distinguishes the transistor from otherelectronic elements? The canonical transistor is a three-terminal device that derives its

2

a)

b)

Figure 1.1: a) Original thermionic valve invented by J. A. Fleming [4]. When heated byan alternating electric current (AC), thermally excited electrons are ejected from the coiled
filament in a process known as thermionic emission [5, 6]. The second, straight filamentin the tube acts as a sink to the liberated electrons when charged positively, resulting ina measurable current between the two wires. b) Original point contact transistor invented
at Bell Laboratory by J. Bardeen, W. H. Brattain, and W. Shockley [3]. This somewhat
haphazard looking device was constructed with a PN-doped slab of Germanium (Ge) resting
atop a gold sheet, which serves as the base contact of the transistor [7]. Two gold additional
gold contacts, named the emitter and collector, are supported by a plastic wedge and seenpressed against the Ge slab by a vertical metal spring-like structure.

3
utility by enabling a small electric current or voltage to control a large current. In particular,transistors are active devices that provide electrical gain—gain that can be used to amplify asignal, sustain coherent oscillation, perform switching, provide storage, and implement logic.
Such are a few of the many functions that reflect the ubiquity of the transistor in modern
electronics.
Along with the aforementioned statements about transistors, research thrusts in thefield of quantum signal and information processing (see References [8, 9]) are ongoing. These
endeavors motivate research towards the realization of a device capable of performing theclassical signal processing duties of electronic transistors in the realm of quantum mechanicalsignals. Therefore, the road towards developing quantum signal processing technology brings
us back to vacuum tubes, or should I say ultra high vacuum (UHV) chambers, in which weseek to realize an atomtronic transistor. At the heart of the Anderson lab atomtronics apparatus is the double-MOT vacuum chamber shown in Figure 1.2. The atom chip that forms
the top wall of the vacuum chamber is designed specifically for the experiments describedin this dissertation. In particular, it includes a transparent window that, in conjunction
with external optics, enables the projection of optical potentials with micron length-scalefeatures onto the magnetic potential generated by the atom chip. Just as the layout ofelectronic circuit dictates its functionality, this optical potential determines the flow anddynamics of ultracold atoms as they propagate through the atomtronic system. The work
described throughout this dissertation represent one vein of research in the growing field of
atomtronics.
1.2

Atomtronics
The field of atomtronics has emerged in tandem with the development of experimental

techniques in the field of atomic, molecular, and optical physics (AMO), that allow one tocreate and manipulate the state of ultracold atoms. Utilizing said atoms in place of electrons,
atomtronics lives in a challenging arena of physics involving many-body, interacting, and

4

Atom
Chip
3D MOT
Chamber
2 l/s Ion
Pump
CopperPinch-Off

2 cm

2D MOT &Source Tube
Figure 1.2: Double-MOT vacuum chamber used to conduct atomtronics experiments. The
inset shows the atom chip, which forms the upper wall of the vacuum chamber and is used
to form the confining magnetic potentials in the atomtronics experiments. Once constructed
and baked-out to pressures < 10−10 torr, the vacuum chamber is pinched off from a larger
station, after which the UHV pressure required for experiments is maintained solely by a
2 l/s ion pump and other passive gettering elements. The atom source is located in thelower of the two glass cells, labeled “2D MOT & Source Tube”, in which initial laser coolingtechniques produce a stream of pre-cooled atoms upwards, into the 3D MOT chamber. The
remainder of the experiment occurs in this upper chamber, which appears dark in the figure
due to an anti-reflection coating. Additional information regarding the construction andfunctionality of the Double-MOT vacuum chamber can be found elsewhere [10, 11, 12, 13].

5open quantum systems in non-thermal equilibrium. It is likely not possible to develop
complete descriptions of any but the simplest atomtronic circuits. Indeed, most theoreticaland experimental works thus far involve fully quantum atomtronic circuits, i.e., operating asa closed system that is described by unitary evolution [14, 15, 16, 17, 18, 19, 20]. Transistorlike gain, on the other hand, requires an open system approach [21, 22, 23, 24, 25, 26].
Moreover, accepting the connection between information processing and entropy changesuggests that dissipation and heat generation are fundamental aspects to be taken into
account in atomtronic circuit design [27, 28, 29, 30, 31]. In classical circuits, electrons
are strongly coupled to the thermal environment of the medium in which they propagate.
Ultracold atomic systems, by contrast, are isolated from the surrounding environment suchthat dissipation and heat generation contribute to the state of the system. To make headway
toward the understanding of atomtronic circuits one can either treat such dissipation as aperturbation to an otherwise quantum system, or introduce quantum effects to an otherwiseclassical system. Both can be instructive.
1.3

The pursuit of an atomtronic transistor
Driving an interest in atomtronics is the possibility of developing a paradigm for ad-

dressing problems in quantum signal and information processing that parallels the power ofelectronics in the classical realm. Thus, an atomtronic version of transistor action has been
a central theme of research in atomtronic devices and circuits. Transistor-like behavior has
been studied in both cold and ultracold atomic systems revealing a variety of approaches
for realizing an atomtronic transistor. The mechanisms responsible for transistor action explored thus far can be distinguished into three categories: (1) interatomic interactions ofa single species in a single state, A, i.e., Ain controls Aout [21, 22, 23, 24]; (2) Interatomic
interactions of two species or states, A and B, i.e., Bin controls Aout [25, 32]; and, (3) anexternal coupling field, C, that drives internal state transitions of atom A, i.e., Cin controls
Aout [33, 34]. The atomtronic transistor described in this work falls into the first category

6
and follows from the analogy with electronic transistors, which operate based on the flow orbuildup of a single carrier species, namely electrons. The first two categories, both of which
derive functionality from interatomic interactions, are differentiated solely by experimental
implementation. By introducing a second species or state, B, to control the flow of A,
the transistor system becomes more complex, requiring the superposition of two species- orstate-dependent trapping potentials. By contrast, the third category utilizes a coupling, C,
other than atoms to control the flow of A. In such systems the coupling strength of field C
is converted into the flow of atoms A, a process akin to transduction. As in electronics, thedevelopment of various types of transistors for specific applications emphasizes their utility.
1.4

Fundamental atomtronic circuit elements
The experimental work presented in this dissertation characterizes the behavior of

two atomtronic devices and represents a step towards the development of devices that canbe readily applied to the challenges of quantum signal processing. The first device, often
overlooked due to its integral role in nearly every circuit, is a power supply. In the context
of the experiments presented here, a finite-temperature Bose-Einstein condensate (BEC)of Rubidium 87 atoms (87 Rb) is stored in a double-well potential. This devices is more
aptly referred to as a battery, providing both particles and energy to the connected circuitcomponents in order to maintain circuit functionality, until the stored energy of the BEC isdepleted. The atomtronic battery, when considered as part of the entire circuit, embodies asystem that is far from thermodynamic equilibrium, with many atoms stored in a reservoir,
waiting to be doled out to the rest of the circuit as required. Figure 1.3(a) shows an in situ
image of the atomtronic battery as it outputs a current of ultracold atoms. In the end, all
but the most mundane circuits, electronic or otherwise, operate in a state of non-equilibrium,
and it is the roll of the power supply to provide and sustain the potential gradients that drive
circuit operation. The second device is, of course, an atomtronic transistor, which can beseen in Figure 1.3(b).

7

a)
5 μm

b)
5 μm
Figure 1.3: a) In situ image of the atomtronic battery illustrating the current output fromthe reservoir well due to the non-equilibrium distribution of atoms in a double-well potential(indicated by the red outline of a representative energy contour) that is separated by arepulsive barrier with finite height. b) A similar image, this time showing the triple-welltransistor, illustrates the transport dynamics between the three wells, separated by barriersof differing height, result in transistor-like behavior.

8
The name “transistor,” an amalgam of “transfer” and “resistor,” refers to the effective
resistance, or transresistance of the device when treated as a two-port, input/output device.
It is precisely that this resistance can be negative that corresponds to the gain that transistors
are known for [35, 36]. More than just a conceptual convenience, the negative resistance ofa transistor is a thermodynamic reality: the transresistance corresponds to negative power
dissipation during current flow [37]. Here, the concepts of negative transresistance and gain
are applied to the atomic system to investigate the principles of an atomtronic transistor
where transistor-like behavior arises due to the interplay of atom currents between the threewells.
1.5

Dissertation outline
The remainder of this dissertation is organized as follows: Chapter 2 presents the the-oretical treatment of Bose-Einstein condensates, as it pertains to the experimental system.
Additionally, a semiclassical treatment of the triple-well atomtronic transistor is presented
that analyzes the system in steady-state and calculates relevant transistor-like properties,
such as current gain and negative transconductance. Chapter 3 provides an overview of thetheory and development of a key component in the experimental apparatus, the atom chip.
As executed, the atomtronics experiment presents experimental challenges as far as atom
chip development is concerned. These difficulties were overcome by including an opticallytransparent window, which enables the simultaneous use of optical potentials in conjunction with the magnetic potential generated by the atom chip. Chapter 4 then describes theprocesses of both imaging atoms trapped in the magnetic potential as well as the projection of dynamic optical potentials that define the functionality of our atomtronic devices.
Finally, Chapter 5 presents the experimental realization of an atomtronic battery, which ischaracterized according to a Thévenin equivalent circuit. The battery is subsequently used
to power the atomtronic transistor in order to study the transport dynamics of atoms between the three wells of the potential and the resulting quasi-steady-state behavior of the9system. Chapter 6 provides some concluding remarks, including the future direction of theatomtronics experiment.

Chapter 2TheoryThe atomtronic devices studied throughout this dissertation rely on the non-linear
behavior of interacting Bose gases at finite-temperature. This chapter covers the salient
concepts of trapped BEC systems and their role in transistor-like behavior. To start, atheoretical description of Bose-Einstein condensates is provided, beginning from the idealizedcase of a non-interacting BEC. Building upon this simplified description, the influencesof both intraparticle interactions as well as finite-temperature effects are included to thedescription of the trapped gas. Next, a semiclassical description of an atomtronic transistor
is presented. Within this theory, we begin by determining the steady-state behavior of
finite-temperature ensembles trapped within a triple-well potential. Based on the steadystate characteristics, transistor-like current gain and power dissipation are explored. Finally,
the theoretical basis for generating the hybrid magnetic and optical triple-well potential isprovided.
2.1

Theoretical description of Bose-Einstein condensates
Following the experimental realization of BEC, the field of ultracold atomic physics

has burgeoned, experiencing a thrust towards the study of quantum phenomena in manybody systems, both in the pursuit of fundamental physics research [38], as well as real-world
applications [39, 40]. This chapter begins by introducing the salient equations that describe
trapped, non-interacting BECs that can be found in many statistical mechanics texts [41]11and others dedicated solely to BEC physics [42, 43]. Corrections to these equations arethen discussed, which account for finite size and temperature effects as well as interatomic
interactions [43, 44]. The expressions provided in this section are then used in Section 2.2
to derive analytic relations that describe steady-state behavior in a triple-well transistor-like
potential.
2.1.1

Non-interacting Bose-Einstein condensates

While the atomic systems studied throughout this dissertation rely heavily on theinteractions between atoms, it is informative to start from the simplest scenario in whichinteractions are not considered in order to develop a notion for the behavior of trapped Bose
gases. At temperatures, T , that are well above the onset of BEC, the distribution of particle
energies, , within the gas is approximately described by the Boltzmann distribution,fM B () '1e(−µ)/kB T

,

(2.1)

where the chemical potential µ   and kB is the Boltzmann constant. As the temperature of
the gas is reduced further, quantum statistics become non-negligible and the bosonic natureof the atoms begins to play a dominant role. In this ultracold regime, the distribution ofparticle energies is described by the Bose-Einstein (BE) distribution,fBE () =1e(−µ)/kB T − 1

,

(2.2)

which accounts for macroscopic occupation of the ground state of the system and the formation of a BEC.
When the gas is confined in an externally applied potential, the density of states of
the trap modifies the distribution and requires the inclusion of the energy level degeneracy,g(). Our atomtronics experiments are conducted in an anisotropic 3-dimensional potentialthat can be described by
V (r) =

m 2 2
ωx x + ωy2 y 2 + ωz2 z 2 ,
2

(2.3)

12
where xi and ωi are the three spatial coordinates and corresponding harmonic trapping
frequencies, respectively, and m is the atomic mass. The length scales of the potential arepthen defined by the harmonic oscillator lengths, ai = ~/mωi , and the overall confinement
is characterized by the geometric mean trap frequency ω̄ = (ωx ωy ωz )1/3 . This potential hasa degeneracy factor
g () =

2,2 (~ω̄)3(2.4)

where ~ is the reduced Planck constant [43]. Given the degeneracy factor, the total particle
number is obtained by integrating over the distribution of particles,
Z ∞
g ()
N=
d (−µ)/k T
.
B
e
−1
0

(2.5)

The transition point between a gas that behaves classically and the onset of quantum
statistics, i.e., macroscopic occupation of the ground state, is determined by the criticaltemperature, Tc . In the absence of interatomic interactions, T = Tc when µ = 0 and
N = Nth . Solving Equation (2.5) under these conditions, one finds
Z ∞
g ()
N = Nth =d /k Tc,
e B −103
Γ(3)ζ(3) kB Tc
=
,
2
~ω̄

(2.6)
(2.7)

where Nth is the number of thermally excited atoms at the threshold of condensation, Γ isthe gamma function, and ζ is the Riemann zeta function. The integral in Equation (2.6) isevaluated according to Appendix D in Reference [41]. Solving Equation (2.7) for the criticaltemperature yields
s
~ω̄
Tc =
kB

3

N
.
ζ(3)

(2.8)

From Equation (2.7) it is clear that the total number of atoms scales relative to the ratioT̃ ≡ T /Tc and is related to the number of thermally excited atoms by Nth = N T̃ 3 , whichleads to the expression for the condensate fraction,Nc= 1 − T̃ 3 .
N

(2.9)

13
The total energy of the trapped atomic ensemble can be calculated in a similar mannerto the total atom number by including the energy of each state in the integrand,
Z ∞
E=
0

 g ().
d (−µ)/k T
Be−1(2.10)

Solving Equation (2.10) with µ = 0 yields the energy of the condensed phase,ζ(4) 3T̃ .
ζ(3)Ec = 3N kB T

(2.11)

From E, the heat capacity can be calculated according to the relation C = ∂E/∂T . Therefore, the heat capacity of the condensed phase is
Cc = 12N kB

ζ(4) 3T̃ .
ζ(3)(2.12)

To determine the energy of the normal phase Equation (2.10) is solved given the approximation fBE () ' exp[−( − µ)/kB T ] + exp[−2( − µ)/kB T ] [43], which yields
ζ(3)Eth = 3N kB T 1 −,16T̃ 3(2.13)and the corresponding heat capacity,ζ(3)Cth = 3N kB 1 +.
8T̃ 3(2.14)

Equations (2.12) and (2.14) behave differently as T → Tc , which causes the heat capacity topeak at the critical temperature, indicating the transition between normal and condensed
phases. As the temperature becomes large, T  Tc , one recovers the classical limits, E =3N kB T and C = 3N kB .
In addition to thermodynamic variables of the gas, the in-trap spatial distribution of
the atoms plays a role in the behavior of our atomtronic systems. Furthermore, when imaging atoms in situ, the spatial distribution of the atoms can be used to extract quantitative
information about the atomic ensemble. As one might expect, the condensed and thermal
components of the gas exhibit different spatial distributions. Whereas the thermal gas is

14best described as particle-like, the condensate behavior is wave-like, since the intraparticle
spacing is on the order of the de Broglie wavelength. The thermal cloud density is determined by Boltzmann statistics, with exp[/kB T ] → exp[V (r)/kB T ], and exhibits Gaussiancharacteristics,
nth =

N

2

π 3/2 σx σy σz

2

2

2

2

2

e−x /2σx −y /2σy −z /2σz .

(2.15)

Thus, the spatial extent along each direction of the trap is dictated by the characteristicpwidth, σi = 2kB T /mωi2 , which relates the thermal energy of the atoms to the strength of
the harmonic confinement. The momentum distribution is also Gaussian but isotropic anddetermined by the mean thermal energy,2pth ∝ ep /2mkB T .

(2.16)

Both the spatial and momentum distributions of trapped thermal gases are used extensively
in the simulations discussed in Section 5.4.1. The spatial distribution of the condensate isaddressed in the next section under the Thomas-Fermi (TF) approximation.
Near the critical temperature and below, the gas can be described by the semiclassicaldistribution function,f (r, p) =1e(p2 /2m+V (r)−µ)/k

BT

−1

,

(2.17)

in which atoms occupy a phase space drdp. The density of thermally excited atoms is thengiven byZ

dp
f (r, p) ,(2π~)3
g3/2 (z(r))=,(2.18)
λthpwhere z = exp [−(V (r) − µ)/kB T ] is the fugacity, λth = 2π~2 /mkB T is the de Broglienth (r) =wavelength, and g3/2 is the polylogarithm function for atoms in a 3D harmonic oscillator potential. Whereas the Boltzmann distribution is often trivial to integrate, the same operations
on the Bose-Einstein distribution produce polylogarithm functions, which are described in15depth in Appendix D of Reference [41] and Appendix B.1 in Reference [45]. Equation (2.18)
provides an intuition to the behavior of Bose gases near the transition temperature where
the effects of Bose enhancement manifest as a peak in the central density of the cloud.

2.1.2

Interacting Bose-Einstein condensates at zero-temperature

While the expressions introduced in Section 2.1.1 are sufficient in describing a noninteracting gas, the theoretical work described in Section 2.2 and experiments described inchapter 5 utilize 87 Rb atoms. In its electronic ground state, 87 Rb is a neutral species, devoidof long-range interaction mechanisms. However, as a massive boson, short range interactions
via collisions are sufficient to alter the characteristics described in the previous section andprovide non-linear behavior that is fundamental to our atomtronic systems.
The collision process can be understood via scattering theory, which is described thoroughly in chapter 5 of Reference [43]. At the low temperatures of the atomic ensembles weconsider, the scattering cross-section is solely s-wave and given by
σs = 8πa2s ,(2.19)

where as = 95.47a0 ≈ 5 nm is the s-wave scattering length of 87 Rb atoms in the |F =2, mF = 2i state, and a0 is the Bohr radius. The scattering process can be viewed as aneffective contact interaction between two atoms and is well characterized by the LippmannSchwinger (LS) equation. In the Born approximation, the LS equation predicts an effective
pseudopotential,1 Uef f = gδ(r − r0 ), with an interaction strengthg=4π~2 as,m(2.20)

which is equivalent to the scattering matrix at zero energy [43].
1

The effective interaction due to collisions is typically treated according to the regularized zero-range
pseudopotential, Vef f ψ = (4π~2 as /m)δ(r)∂(rψ)/∂r, which corrects for the divergent behavior of the Diracdelta function as r → 0 [43, 46]. Treating the collision process in the Born approximation, which neglects
the scattered wave, simplifies the effective interaction to Uef f = (4π~2 as /m)δ(r), since the role of ∂(rψ)/∂ris to eliminate the contribution of the scattered wave to the potential.

16
Given this scattering interaction, one can accurately describe the condensed phaseusing the effective many-body Hamiltonian,

N 
X
X
~2 ∇2iH=−+ V (ri ) + g
δ(ri − rj ).
2m
i=1
i<j

(2.21)

Constraining the solution of the wave function by the relation ∂E/∂N = µ ensures consistency between E and N and yields the time independent Gross-Pitaevskii equation,
 2 2

~∇
2
−
(2.22)+ V (r) + g|ψ(r)| ψ(r) = µψ(r),2mwhere the eigenvalue for the non-linear condensate eigenstate ψ(r) is the chemical potential [45]. The GPE is a powerful tool that can be used to model the behavior of zerotemperature gases, and simulation techniques are well documented [47]. If the contribution
due to the kinetic energy is small compared to the potential or collision pseudopotential,
then Equation (2.22) can be reduced toµψ(r) = V (r) + g|ψ(r)|2 ψ(r),(2.23)

under the well-known Thomas-Fermi (TF) approximation. The TF approximation holds
ptrue for trapped clouds that satisfy N as /ā  1, where ā = ~/mω̄ is the mean harmonic
oscillator length [43]. Since the atoms are confined within a harmonic potential, the TF
approximation leads to an inverted parabola density profile of the condensate,nc (r) =with TF widths Ri =µ − V (r),g(2.24)

p2µ/mωi . The total atom number in the condensate can then be

determined by integrating nc across Ri , yielding

3/2
8π µ
2µ
Nc =
.
15 g mω̄ 2Solving for µ, one obtains the TF approximation for the chemical potential,2/5~ω̄ 15Nc as
µ=
,
2
ā
which is used throughout the remainder of this dissertation.

(2.25)(2.26)

17
2.1.3

Corrections due to finite temperature, size, and interatomic interactions

Corrections to the thermodynamic quantities introduced in Section 2.1.1 arise due tofinite number of particles, intraparticle interactions, and finite temperature. Throughout thework covered in this dissertation, the atomic systems that are considered have temperatures
below the critical temperature yet higher than the effective temperature associated withinteratomic interactions, T0 = 2µ/7N kB . In this regime, Tc & T > T0 , the two componentsof the cloud can be treated separately under the Hartree-Fock (HF) approximation [43, 45].
Using this description of the system, the presence of a condensate produces an effective
potential known as the mean field,
UHF (r) = Vtrap (r) + 2g [nc (r) + nth (r)] ,(2.27)

which is experienced by the thermal cloud. In typical situations the thermal density of thetrapped gas can be neglected, owing to the fact that nth (r)  nc (r) [45].
One can now consider the in-trap density distribution of an interacting Bose gas at
finite temperature using Equations (2.18) and (2.24). The thermal cloud density depends
on the fugacity,zHF (r) = e−gnc (r)/kB T ,(2.28)

that now includes the repulsive, positive-valued, TF chemical potential within nc (r), thezero-temperature condensate density from Equation (2.24). Figure 2.1 shows a plot of thethermal cloud density for various temperatures that exemplify the behavior of the finitetemperature, interacting gas. The three curves illustrate the stark contrast in behavior at
T > Tc , T = Tc , and T < Tc . At temperatures above the critical temperature, the thermal
cloud density has a Gaussian distribution. However, as the cloud is cooled to the criticaltemperature, Bose enhancement is noticeable as the ground state begins to acquire higher
occupancy. Well below the critical temperature, where the chemical potential is positive, themean field potential results in the repulsion of thermal atoms by the condensate and reveals

183

ζ(3/2)/λ3th

μ = kBT/2μ=02nth λ3th

μ = -kBT/2

1

0

-2

-1

0
r/σ

1

3

Figure 2.1: Thermal cloud density for clouds at temperatures T > Tc , T = Tc , and T < Tc .
The density is determined from the fugacity with chemical potentials equal to µ = −kB T /2,
0, and kB T /2, respectively. To compare the three cases, the thermal
density is scaled by λ3thp
and the position vector by the Gaussian thermal width, σ = 2kB T /mω 2 .

19
an inverted-parabola shaped void. Despite the repulsive interaction with the condensed
atoms, the peak density of the thermal atoms remains well approximated by nth evaluatedat r = 0, or nth,0 = ζ(3/2)/λ3th ; however, the position of the peak density is displaced by thecondensate.
The presence of interactions and thermal excitations results in a depletion of the central
density of the cloud. Subsequently, the critical density required for the cloud to begin tocondense is achieved at a lower temperature. This reduction of the critical temperature
is determined by expanding  as  − 0 in Equation (2.5), where 0 = 3~ωm /2 + 2nc g andωm = (ωx + ωy + ωz )/3 is the arithmetic mean trap frequency. The leading first ordercorrections to Tc include terms for ∂N/∂T , ∂N/∂µ, and ∂nc /∂µ, yield the finite-temperatureinteracting gas expression for the critical temperature,


as √
ωm 1
6
N ,
Tc = Tc,0 1 − 0.73
√ − 1.33
ω̄ 3 Nā(2.29)

where Tc,0 is the non-interacting gas critical temperature from Equation (2.8). The firstcorrective term depends on the ratio of arithmetic and geometric mean trap frequencies andindicates that the reduction in the critical temperature is dependent on the anisotropy of the√potential, with ωm /ω̄ = (2 + λT )/3 λT and λT = ωk /ω⊥ [43]. Therefore, the first term is
minimized by a homogeneous trap with large atom number. The second correction accounts√
for the strength of the repulsive interactions on the thermal atoms, which scale as 6 N as /ā.
Next, the finite temperature corrections for the condensate fraction can be addressed.
Expanding Equation (2.5) to first order in the chemical potential, one finds that, in responseto the reduction of the critical temperature, Nth at finite-temperature is larger than thezero-temperature value. The corrected expression for Nth is∂NthNth (T, µ) ≈ Nth,0 + µ,∂µζ(2) µ3= N T̃ 1 +,ζ(3) kB T(2.30)

which shows that at a given temperature the number of thermal particles increases according

20
to the ratio of interaction and thermal energies. As before, the condensate fraction for aninteracting, finite-temperature Bose gas is given by Nc /N = (N − Nth )/N .
Finally, corrections to the energy and specific heat of the cloud can be addressed. The
contribution to the total energy by the condensate in the TF approximation is given by
Ec = 5µ(T )Nc (T )/7 and includes the temperature dependence shown in Equation (2.30).
The energy of the thermal cloud is again calculated from Equation (2.10), but now the effects
due to interactions are considered by determining ∂E/∂µ and including terms to first orderin µ/kB T . Therefore, the total energy of the cloud is given by"#
ζ(4) 4 5 + 16T̃ 3 µ(T )T̃ +.
E = N kB Tc 3
ζ(3)
7
kB Tc

(2.31)As before, the specific heat can then determined via the relation, C = ∂E/∂T . Equations
for Nth , E, and C can be made to depend solely upon N , T , and Tc by inserting the noninteracting gas expression for the condensed atom number Nc = N (1 − T̃ ). The equations
are as follows:
"

#
2/5
1/6
N
a
s
T̃ 2 (1 − T̃ 3 )2/5 N,
Nth ≈ T̃ 3 + 2.15
ā"#
 1/6 2/5Nas
E = 2.7T̃ 4 + 1.12
(1 + 3.2T̃ 3 )(1 − T̃ 3 )2/5 N kB Tc ,ā"
 1/6 2/5
N as
3
C = 10.8T̃ + 10.752
T̃ 2 (1 − T̃ 3 )2/5
ā#
 1/6 2/5
N asT̃ 2 (1 + 3.2T̃ 3 )/(1 − T̃ 3 )3/5 N kB .
− 1.344ā(2.32)The set of Equations (2.32) is used extensively in the numerical calculations of theatomtronic battery and transistor described in chapter 5.
2.2

Theoretical study of a triple-well atomtronic transistor
We now look to apply the characteristics of interacting Bose gases at finite temperature

to the field of atomtronics. In this section, concepts from analog electronics are drawn upon

21
to theoretically study the behavior of a trapped, ultracold atomic system in the context of athree terminal atomtronic device, namely a transistor.2 A semiclassical analysis is presented
that studies the flow of ultracold atoms having finite temperature, T , and chemical potential,µ, through a triple-well potential designed to mimic the behavior of an electronic transistor.
The theoretical triple-well system that we consider is sectioned by two repulsive potential barriers, each of which is analogous to the built-in potential in the space charge region of
a semiconductor junction or the work function of the cathode in a vacuum tube [49, 50, 51].
Much like the thermionic emission of electrons across the energy barriers in these electronics
examples, atom currents in our system flow due to those atoms that are energetic enoughto traverse the repulsive barrier. Thus, atom currents can be described using a formalism
common to evaporative cooling. Building upon the work of References [52] and [53], relations
for atomic currents driven by thermal and chemical potentials are defined in Section 2.2.2.
Similar non-equilibrium dynamics have been studied in double-well systems containing cold
and ultracold atoms. These works exemplify the role of gradients in temperature and chemical potential in driving the system towards thermal equilibrium [54, 55, 56, 57, 58, 59]. By
considering one of the three wells open to the surrounding environment, we introduce atom
loss that prohibits the system from equilibrating. The resulting quasi-steady-state chemical
potentials and temperatures of the wells are determined in Section 2.2.3 and analyzed in thecontext of an electronic circuit with potential biases between nodes.
Transistor-like functionality of the device is established in Sections 2.2.4 and 2.2.5,
where current gain and maximum power output are calculated given an externally applied
input current to the gate well. Here, transistor action arises due to the interplay between
the input current and the steady-state thermodynamic properties of the gate. Dependingon the Q-point, the current into the drain is either attenuated or amplified in response to
the applied gate current, demonstrating a current gain greater than unity. Additionally, thetransresistance is found to be negative indicating the active nature of the transistor in that2

The contents of this section are published in Reference [48].

22
it draws power from a reservoir to control power output to a load. The maximum power toan impedance matched load is then shown for a range of Q-points.

2.2.1

The theoretical system

Our semiclassical treatment of an atomtronic transistor begins by borrowing the nomenclature of the semiconductor field-effect transistor. The three regions of the triple-well potential, shown in Figure 2.2(a), are labeled the “Source,” “Gate,” and “Drain” wells. Inlabeling the regions of the triple-well potential, field-effect transistor nomenclature is used
as it describes the role of each well in an atomtronic transistor: Atoms flow from the source,
through a gate region, and into a drain well where they are removed from the system. This
naming scheme is meant to convey an intuition for the role of each region, rather than imply
the behavior of the triple-well system is more analogous to the behavior of a field-effecttransistor than a bipolar junction transistor. Emulating the experimental system describedin chapter 5 we take the trapping potential to be cigar shaped with corresponding harmonic
trap frequencies ωk  ω⊥ . This potential is sectioned in the longitudinal direction by two repulsive Gaussian barriers having peak heights VGS and VGD , respectively, to form the source,
gate, and drain wells. The longitudinal profile and separation of the barriers determines thelongitudinal gate well trap frequency. Furthermore, the longitudinal trap axis of the sourcewell is assumed to be half harmonic for simplicity. The degree of overlap of the two barriers
contributes to a potential bias in the gate, VG,0 , with respect to the source and drain. Finally,
the drain well is modeled as a reflectionless output port that would feed into a subsequentcircuit element.

2.2.2

Current flow mechanism

Previous works have modeled the flow of atoms across a potential barrier in order tostudy both the selective removal of atoms involved in evaporative cooling as well as thereverse process in trap loading [52, 53, 60]. Here, the expressions in Ref. [60] are modified

23

a)

SourceGate

VGD Drain

VGS

μsg +

μg

μs

Load

μd
+

μsd
Vsg

b)

+

Vs

G

Vg
Vd

S

D

+

Vsd

Figure 2.2: (a) Triple-well potential with source, gate, and drain wells. Chemical potential
levels in each well are depicted in the context of potential biases. The current output into
the drain can then be coupled to the desired ‘load.’ In the model presented here, the flowof atoms through the potential is controlled by adjusting the source-gate chemical potential
difference via an external current input to the gate. (b) A simplified schematic of an electronic
common-source amplifier circuit with analogous functionality to its atom based counterpart.

24
to include the chemical potential due to the presence of a BEC in the source and gate wells.
Under the HF theory, the mean field of the condensate shifts the energies of the thermal atom
distribution up by µ and is conceptually analogous to an applied bias voltage in an electronic
circuit. In the absence of a BEC, the chemical potential can be calculated according to selfconsistent mean field methods [61]. However, in this model the minimum single particle
energy is Vi,0 and thus negative µi are set to zero. Therefore, the atom currents in the
triple-well system are defined by the set of equations:
Isg = γs Nth,s exp[−(VGS − µs )/kB Ts ],Igs = γg Nth,g exp[−(VGS − VG,0 − µg )/kB Tg ],(2.33)

Igd = γg Nth,g exp[−(VGD − VG,0 − µg )/kB Tg ],where the product γi Nth,i is the effective collision rate of thermally excited atoms in thei-th well, and the exponential factor reflects the thermodynamic probability that an atom
possesses sufficient energy to traverse the barrier. Note that the subscript order indicatesthe direction of current flow, e.g. Isg describes the atom flux from the source to the gate well,and capitalized subscripts indicate model parameters. Building upon the collision processpdescribed in Section 2.1.2, the equilibrium collision rate (γ = 2/πnth σ0 ∆v) then depends
only on the peak density of the thermal component, nth = ζ(3/2)/λ3th , the s-wave collisioncross-section σs , and the mean thermal velocity, ∆v [43]. Thus, the collision rate for the i-thwell is given by
γi = 32π 2 ζ(3/2)m(as kB Ti )2 /h3 ,(2.34)and the expression for Nth,i is given by Equation (2.30).
As alluded to previously, the currents in Equation (2.33) are reminiscent of the current
in thermionic devices, J = AT 2 exp[−(V0 − Vb )/kB T ], where A is the Richardson constant,which depends on the density of states and other material properties [49]. Devices that derive
their functionality from thermionic emission convert thermal energy into electric current
between a heated filament and an adjacent, charged electrode [62]. The magnitude of the25bias voltage, Vb , applied to the electrode, relative to the built-in potential, V0 , then tunes
the current flow. In both electronic and atomtronic systems, Bose-Einstein and FermiDirac statistics are approximated by the Maxwell-Boltzmann distribution, given that thetruncation parameter η ≡ V /T is sufficiently large, i.e., η & 3 [49, 52, 53]. Factors prior to
the exponential term in Equations (2.33) as well as the expression for thermionic emission
describe the abundance of particles which may contribute to the current.

2.2.3

Source and gate well ensembles in steady-state

Steady-state circuit operation is analyzed by enforcing particle number and energy
conservation, expressed using analogues of Kirchhoff’s current and voltage laws:
Isg = Igs + Igd ,Isg (VGS + κGS kB Ts ) = Igs (VGS + κGS kB Tg ) + Igd (VGD + κGD kB Tg ),(2.35)(2.36)

where the κ’s, which indicate the average excess energy of atoms traversing the barriers, areof order unity [60]. It is supposed that some external reservoir supplies atoms to the source
well, maintaining a fixed chemical potential, µs , and temperature, Ts . For barrier heightsthat are large compared to the chemical potential and thermal energies, the three wells are
weakly coupled by thermal atoms that have sufficient energy to traverse the barriers. Thus,
as current flows from source to gate and subsequently from the gate to the drain, we assume
that the gate acquires a well-defined chemical potential, µg , and temperature, Tg , once a
steady-state is reached. Additionally, we impose that current into the drain is removed fromthe system such that no current flows from the drain back towards the gate. An illustrationof the system is shown in Figure 2.3 that elucidates important model parameters.
With the source well ensemble held constant, we seek steady-state values of the gatechemical potential and temperature in terms of µs and Ts . It proves useful to define a tem-

26

Isg

Igs

κGSkTsμs

κGDkTgVGDΔVκGSkTg

VGS

Ts

Igd

ΔT
Tg
μg
μsg

VG,0

μd

Figure 2.3: Schematic of atom and energy flow across the gate barriers. The atom current
carries with it an energy in excess of the barrier height by the factor κkT relative to eachbarrier. Light blue shading indicates the offset in energy of the thermal component, shownin red, due to the condensate chemical potential. An example steady-state for source andgate ensembles is shown, in which the temperature drop, τ , is positive and the chemical
potential drop, µsg , is negative.

27
perature drop τ and barrier height difference υ, both normalized to the source temperature:τ ≡ (Ts − Tg ) /Ts ≡ ∆T /Ts ,(2.37)υ ≡ (VGD − VGS )/kB Ts ≡ ∆V /kB Ts .

(2.38)

Using the currents defined in Equation (2.33), along with Equations (2.35) and (2.36), it ispossible to derive relations for both the temperature drop,τ = e−υ/(1−τ )υ + (κGD − κGS ),κGS + κGD e−υ/(1−τ )(2.39)and the normalized chemical potential drop,
h

4



υ
− 1−τ

i



− τ V̂GS − µ̂sµ̂sg = (1 − τ ) ln (1 − τ ) 1 + e



ζ(2)
µ̂s − V̂G,0 − µ̂sg
1 − τ + ζ(3),
+ (1 − τ ) ln ζ(2)1 + ζ(3) µ̂s

(2.40)

where µsg ≡ µs − µg and the hat (ˆ) indicates quantities normalized to the source temperature. Within these equations, e−υ/(1−τ ) represents the ratio of forward current into thedrain relative to current fed back into the source, Igd /Igs . Given its effect on the balance of
currents into and out of the gate well, the value υ is referred to as the feedback parameter.
The impact of the feedback parameter on the gate-drain, normalized to the source-gate current, can be seen in Figure 2.4. The ratio of currents is independent of η, but the absolutemagnitude of the Igd depends strongly on µs , Ts , and η due to the exponential nature of
the currents in Equations (2.33). In deriving Equation (2.40) we have used the fact that thethermal and chemical potential drops are zero at υ = ∞ along with the relations introducedin Section 2.1.3 to determine the ratio of γg Nth,g /γs Nth,s that satisfies Equation (2.35). Complete derivations of Equations (2.39) and (2.40) are provided in Appendix A. In the absenceof an external gate input, Equation (2.40) shows that the source-gate junction is self-biased
to some Q-point characterized by µsg . The magnitude of this bias is primarily dependent
on the feedback parameter, both directly and through the temperature drop. This self-bias

28

Igd/Isg

1

1/2

10-1

10-2
-1

0

1

2
υ

3

4

5

Figure 2.4: Log plot of the net drain current relative to the source-gate current, which reflects
the role of the feedback parameter.

is illustrated in Figure 2.2(a) within the atomic system and Figure 2.2(b) for an equivalent,
simplified electronic circuit. Equations (2.39) and (2.40) form the basis of our model andcan be solved self-consistently in order to characterize the thermodynamic variables of thetriple-well system in steady-state.
To calculate the values of τ and µ̂sg , the κ parameters of the trapping potential must
be determined. Trap geometry has a strong effect on the average energy removed by anatom leaving via a controlled trajectory. For atoms escaping isotropically from a potential
well, as in typical evaporative cooling schemes, κ ' 1 [52, 53]. However, limiting the escapetrajectory to purely 1D, for instance along the loose axis of a cigar-shaped trap, κ ' 2.9
for truncation parameters η ≡ V /kB T = 4 − 7 [60]. The difference in κ factors is a direct
result of allowed escape trajectories. The direct simulation Monte-Carlo (DSMC) method
is used to confirm κ ' 2.9 for the geometry and feedback parameters modeled here. Moreinformation regarding these simulations are given in Section 5.4.1.
The temperature and chemical potential drops are shown as functions of the feedback
parameter in Figure 2.5, with κGS = κGD = 2.9. The behavior shown by the temperaturedrop is somewhat non-intuitive in that for values of positive feedback (VGD > VGS ), the

29

0.2

μs

0.1

-

Isg,net+μgτNegative Resistance0μs Isg,net μg
+

- 0.1
- 0.2
-1

Positive Resistance

0

1

2
υ

3

4

5

Figure 2.5: Plot of the temperature drop vs. feedback parameter. Insets illustrate power
dissipation due to the static source-gate resistance. Negative υ results in a negative temperature drop in the direction of net current across the source-gate barrier, indicating heattransfer into the gate and Tg > Ts in steady-state. Conversely, positive υ results in net
cooling (negative power dissipated) and τ increases until µsg peaks. As υ → ∞, Igd → 0 andequilibrium between the gate and source wells is reached.

30
temperature of the gate is actually lower than the source, despite current from the source
into the gate having an average energy VGS + κGS kB Ts . Furthermore, Figure 2.6 shows that
there will generally exist a range of feedback parameters for which the chemical potential
drop is negative, meaning the source-gate junction is reverse-biased. Also shown in Figure 2.6
are two threshold feedback parameters. The first, υBEC , indicates the formation of a BEC
in the gate well. For υ ≥ υBEC the balance of particle and energy currents leads to a steadystate gate well ensemble with sufficiently high phase-space density to condense. The second
threshold, υTC , indicates the feedback parameter above which negative transconductanceoccurs for the steady-state parameters. Section 2.2.5 provides an in-depth discussion of theυTC threshold.
To better understand the steady-state results, one can consider the power dissipated2
Rsg . Here Isg,net = Isg − Igs > 0 is the total
within the gate well of the transistor, Psg = Isg,net

atom current flowing into the gate. Note that the power dissipated within the gate well does
not include the gate to drain current, as power output to the drain is available to do workon a connected load. The static source-gate resistance in steady-state is
Rsg ≡ µsg /Isg,net .

(2.41)

Given the sign of the chemical potential drop, Rsg can be negative indicating an ohmic coolingsynonymous with the evaporative cooling process, which leads to positive τ . Negative power
dissipation within the gate well peaks at υ ≈ 1, which is consistent with the maximum valueof τ .

2.2.4

Current gain

Within the steady-state model presented, current gain is the most readily studied
quantity since voltage and power gain require more explicit knowledge of the connected‘load.’ To study gain, we use a method similar to determining the two-port admittanceparameters of an electronic circuit, which is commonly used to describe transistor action in

31

1.0

η=3

η=4

3

4

η=5

- 0.5
-1

υTC

0

υBEC

^
μ
sg

0.5

0

1

2
υ

5

Figure 2.6: Chemical potential drop between the source and gate wells, normalized to thesource temperature, versus the feedback parameter. Red, blue, and black curves show thesteady-state value of µ̂sg for truncation parameters of 3, 4, and 5, respectively. Two verticaldashed lines indicate threshold behavior linked to the feedback parameter. The first, υBEC ,above which a BEC is predicted to form in the gate well and a second, υT C , where thetransconductance becomes negative.

32
electronic transistors [7]. The coupled system of equations for the triple-well atomic system
is given by










 dIN,g   a11 a12   dµg 

=

,
dIQ,g
a21 a22dTg(2.42)

where IN,g and IQ,g are the particle and heat currents into the gate well. The ‘a’ parameters relate the currents to either a change in chemical potential or temperature assuming
the other is constant, e.g. a11 = (∂IN,g /∂µg )dTg =0 . Equation (2.42) is akin to the Onsager relations [29, 63] for particle and heat diffusion that have been utilized to describe
non-equilibrium transport dynamics in atomic systems [56, 57]. Here, they are used to describe the particle and heat currents applied to the gate well in order to elicit the changesµg → µg + dµg and Tg → Tg + dTg . The particle and heat currents on the left-hand sideof Equation (2.42) are given by the differential of the sum of terms in Equations (2.35)
and (2.36), respectively, with current flow into (out of) the gate taken to be positive (negative). In steady-state, dIQ,g = 0; thus, the temperature response to a change in chemical
potential is given by ∂Tg /∂µg = −a21 /a22 . It is not surprising that this quantity is negative,
as the chemical potential varies inversely with temperature at constant atom number, as seenfrom Equations (2.9) and (2.26). For small deviations from the steady-state, this relation isused in conjunction with Equation (2.42) to determine the input current required to change
the gate chemical potential by some dµg :


a21
,
dIg = dµg a11 − a12a22(2.43)

where the quantity in the parentheses is equivalent to the gate well input admittance. The
differential current gain about the Q-point is then defined as the ratio of the resulting change
in gate-drain current and the external gate well input current:
AI ≡

dIgd
,
dIg

(2.44)

0where dIgd = Igd− Igd and the primed current is evaluated at µ0g = µg + dµg and Tg0 =Tg + (∂Tg /∂µg )dµg .

33

3

_

_

ωg= ωc____ωg= 2ωc ωg= 3ωc

2

AI

1
0
-1
-2
-1

0

1

2
υ

3

4

5

Figure 2.7: Plot of the differential current gain at the drain terminal of the atomtronictransistor. Three curves are shown that illustrate the increase in current gain magnitudewith increasing mean trap frequency of the gate well. Horizontal dashed lines indicate unity
gain.

Figure 2.7 shows the current gain for the range of feedback parameters shown in Figure 2.5, illustrating the current gain at various Q-points. Two complementary operating
regimes arise in which the transistor provides either positive or negative differential current
gain. For feedback parameters near zero, AI is positive, indicating that a positive (negative)
dIg leads to an increase (decrease) in Igd . For increasingly negative feedback parameters, i.e.,υ < υBEC , the current gain becomes nonphysical as dµg becomes ill-defined in the absence ofa condensate in the gate well. At υ ∼ 0.5, the sign of AI flips where the changes in Igd due
to dµg and dTg become equal. Finally, at υ ∼ 1.6, the gain reaches a maximum, negative
amplitude.
Positive and negative differential current gain regimes arise due to the interplay betweenµg and Tg . To better understand each regime, consider the scaling of the chemical potential
2/5

with respect to the number of condensed atoms in the Thomas-Fermi limit, µ ∝ Nc . In
the negative differential current gain regime, steady-state µg is larger than in the positive
gain regime. Therefore, a larger number of atoms at T < Tg must be injected into the gate

34
well to elicit a positive dµg . The resulting decrease in Tg causes a reduction of Igd that ismore substantial than the increase of Igd due to the additional chemical potential. Thus,dµg is positive, but the net change in the gate-drain current is negative. The opposite effectis responsible for positive differential current gain.
With the sign of the current gain understood, the magnitude of the output current isdetermined. Multiple traces are shown in Figure 2.7 that indicate the gain for different gate
well geometric trap frequencies, ω̄g , relative to the trap frequencies of the external well that
2 1/3
injects the control current, ω̄c , where ω̄ = (ωk ω⊥) . The enhancement in the magnitudeof the current gain arises due to the scaling of the specific heat, C ∝ 1/ω̄ 3 , and chemical
potential, µ ∝ ω̄, of a tight well [43]. More specifically, as ω̄ increases, the scaling in µ andC reduce the number of atoms needed to alter the chemical potential and temperature of
the steady-state gate well population. In the case that the trap frequencies are equal, one
recovers the expected result that AI = 1/2 at υ = 0, as there are two equally probable output
channels from the gate. This behavior indicates that the ratio of trap frequencies is a key
design parameter for achieving greater than unity gain.

2.2.5

Transistor power output and transconductance

Despite a lack of explicit knowledge regarding the load circuit connected to the drain
well, it is possible to determine the maximum power delivered to an impedance matched load.
Stated generally, impedance matching is the process of selecting the input impedance of theload circuit to be equal to the output impedance of the device such that reflection of theoutput signal is minimized. The reflection of particles with a given energy impinging upona potential landscape can be calculated using a number of methods (e.g. [64, 65]) and has
been experimentally studied using cold [66] and ultracold atoms [67]. For a matched load,
we borrow the electronics definition for the maximum power at the transistor output [6],Pmax =

gm dµ2g,4(2.45)

35

-0.25

^
μ
sg

10%

-0.35
50%

-0.45

100%

0

0.2

0.4
0.6
I gd /I ref

0.8

1

Figure 2.8: Plot of the transconductance of the atomtronic transistor for η = 5. Three curves
are shown for feedback parameters 10%, 50%, and 100% above threshold. The horizontal
dashed line indicates the chemical potential drop at threshold.

where dµg is the amplitude of the gate chemical potential modulation and gm is the transconductance, given by
gm

dIgd
=
≡
dµsg



∂Ts
∂µsg



∂Igd
∂Ts


,

(2.46)

where the partial derivatives are evaluated at constant µs . Transconductance is useful indescribing operation of the devices as an active element, i.e., one that supplies power. From
Equation (2.45), it can be seen that if gm is negative, the power dissipated at the transistor
output is negative. Effectively, the transistor converts power supplied by the source wellinto power output from the drain controlled by dµg . As alluded to in Figure 2.6, there is a
threshold feedback parameter, υTC , above which the system exhibits negative transconductance in steady-state. This threshold is determined from the inflection point of gm . Both thesign and magnitude of gm are determined from the derivative of the transfer characteristic
curves. Examples of these curves are shown in Figure 2.8 for υ = 10%, 50%, and 100%
above υTC .
The maximum power dissipated at the output of the transistor is shown in Figure 2.9 asa function of the fraction above threshold, (υ − υTC )/υTC . The values of Pmax are calculated

360

P max/| P max( υ T C )|

-1
-2
-3
-4
-5
-6
-7

0

0.5
1
(υ-υ TC) / υ TC

1.5

Figure 2.9: Maximum power dissipated when connected to an impedance matched load,
normalized to the magnitude of the threshold value, |Pmax (υTC )|, as a function of the feedback parameter. Above threshold, the transistor exhibits negative transconductance. The
triangle, square, and circle represent the peak power for feedback parameters that are 10%,50%, and 100% above the threshold value υT C , respectively. Negative dissipated power issynonymous with power supplied to the load circuit.

37
using the same fractional value of dµg with respect to the steady-state µg for all υ. The power
output corresponding to the three transfer characteristic curves in Figure 2.8 are includedin Figure 2.9.
The absolute magnitude of the power depends on the value of dµg ; therefore, to better
illustrate the behavior of Pmax it is scaled by Pmax (υ = υTC ) in Figure 2.9. The magnitude of
the maximum power dissipated is seen to peak at υ ≈ 1.75 × υTC ≈ 2.6. As was the case forthe current gain, the power delivered by the transistor is ultimately limited by the gate-draincurrent, which decreases exponentially with increasing υ. However, the power is deliveredin the form of higher energy thermal atoms. As a result, the maximum power dissipation
peaks at a higher value than the current gain.
Negative power dissipation, a familiar concept introduced early in the literature onelectronic oscillators, originates due to the negative transconductance or transresistance ofa device [36]. When coupled to a circuit containing frequency dependent elements, an active
device that exhibits negative transresistance cancels the resistive power loss in the load,
resulting in the buildup of a resonant, oscillatory signal [35, 36]. The oscillator concept is
attractive in the field of atomtronics, given the historical impact of RF and higher frequency
signal generation within the field of analog electronics.

2.2.6

Connection to electronic transistor functionality

Within the semiclassical approach presented in this section, atom currents driven bychemical potentials and thermal energy are used to describe the particle and energy transport
throughout the system. By varying the height of the barriers that separate the three wells,
it is shown that the steady-state chemical potential and temperature differences between thesource and gate wells can be controlled. This process is analogous to biasing an electronictransistor to the desired quiescent point. Returning to the analogy with electronics, transistor
functionality largely depends on the bias or quiescent operating point set by the voltages
at each of its three terminals. This steady-state theory indicates that regimes of positive

38
and negative potential drops between the source and gate wells exist, which are analogous
to forward and reverse-bias modes in electronic transistors, respectively.
Transistor-like behavior of the triple-well atomic system was investigated by calculating
the current gain given an external current input to the gate well. The device exhibits regionsof both negative and positive differential current gain, depending on the feedback parameter,that can exceed unity. The magnitude of the gain was shown to depend on the mean trap
frequency of the gate well, a characteristic that arises due to the nonlinear dependence of thechemical potential and specific heat of the gas on the trap frequency. Therefore, the current
gain is a widely tunable parameter, controlled by both υ and ω̄g .
In addition to transistor-like gain, the maximum power delivered to an impedance
matched load was calculated. For feedback parameters above the threshold value, υT C , thetransistor exhibits negative power dissipation at its output. In other words, the triple-wellpotential supplies power to the load circuit. As in analog electronics, active devices, i.e., thosewhich exhibit negative transresistance behavior, can be coupled with frequency dependent
loads to create an oscillatory output signal. Therefore, if coupled to the appropriate load, thetransistor-like triple-well system described here could be used to generate oscillating atomic
currents.

Chapter 3
Atom chip development

Applications for atom chips in AMO experiments are numerous, including rapid production of BEC [68, 69, 70, 71] as well as much of the work within the field of atom optics [72, 73, 74, 75, 76, 77]. In order to realize the atomtronic battery and transistor, pictured
in Figure 1.3, one must first synthesize the multi-well trapping potential. This chapter begins with a brief theoretical discussion of magnetic trapping, followed by an overview of thevarious atom chip wire patterns that are utilized in the atomtronics experiment. Finally,other important issues such as heating of the atomic ensemble due to technical noise, andresistive heating in the atom chip wires due to current flow are analyzed.
3.1

Magnetic trapping of neutral atoms
Despite the charge neutral nature of 87 Rb atoms, they possess a finite magnetic dipole

moment, µmag , generated by their total atomic angular momentum. As a result, in thepresence of a magnetic field, B, the atoms experience an interaction energy,
Umag = −µmag · B.

(3.1)

If the magnetic field is inhomogeneous, the atom will experience a spatially dependent force,
Fmag = −∇Umag = ∇ (µmag · B) .

(3.2)

Due to the behavior of magnetic fields in regions free of charges and currents, it is only
practical to consider fields that force atoms towards a field minimum. More explicitly, local

40
maxima of a quasi-static magnetic field occur only at the source of the field, e.g. a current
carrying wire, (see Reference [78] for a rigorous proof). While it is possible to confine
atoms to a Kepler-like orbiting trajectory around a current carrying wire, as was shown inReference [73], the resulting trap is not localized and is impractical for many atom trapping
applications, including atomtronics. Therefore, neutral atoms are typically trapped in a
weak-field-seeking atomic state, i.e., µmag > 0, and the magnetic trap is designed about aminima of the magnetic field.
In the context of magnetic trapping the state of 87 Rb atoms is defined by the hyperfine
structure quantum number F , the sum of the nuclear spin angular momentum, I, and total
electron angular momentum, J. The 5S1/2 ground state of 87 Rb has I = 3/2 and J = 1/2,
which result in either F = 1 or 2. In the F = 1 ground state manifold, Zeeman split sublevels range from mF = −1, 0, 1 with Landé g-factor gF = −1/2. Similarly, the F = 2 groundstate manifold has sub-levels mF = −2, −1, 0, 1, 2 and gF = 1/2. Therefore, the magnetic
dipole moment of these atoms is given by µ = mF gF µB , where µB is the Bohr magneton.
In order to maximize the magnetic trapping force experienced by the atoms, they should be
prepared in the F = 2 ground state manifold, in the mF = 2 Zeeman sub-level, which has
twice the dipole moment of other trappable states.
3.2

Designing an atom chip for atomtronic devices
This section covers the basics of magnetic trapping on an atom chip along with thechip conductor patterns used for atomtronics experiments. Supplemental derivations of themagnetic fields generated by these chip patterns can be found elsewhere [10, 11, 12, 72, 76,79]. Special considerations for high-resolution, through-chip, in situ imaging are describedalong with two atom chip design iterations used to complete the work in this dissertation
are discussed.

41
3.2.1

Side-guide trap

To begin, consider the most simple configuration used to trap atoms on a chip, theside-guide. While it does not form a complete, three dimensional trapping potential the side
guide addresses the basic concepts used in all static chip trap designs. A waveguide potential
is formed by the combination of an infinitely long and thin wire carrying a current Ix and a
uniform magnetic bias field oriented perpendicular to the flow of current, By . The magnetic
field generated by the line current is given by
Bwire =

µ0 Ix
,
2πr

(3.3)

where µ0 is the vacuum permeability, r2 = y 2 + z 2 , and the field circulates according to theright-hand rule. When combined with the external bias field, a minimum in the magnetic
field forms a distance dz away from the wire, where the magnitudes of Bwire and By are
equal. In order to determine the properties of the trap, one must consider the total magnetic
field:zµ0 Ix,
22π (y + z 2 )µ0 Ix −y+ By ,=
2π (y 2 + z 2 )Bŷ,side =(3.4)

Bẑ,side(3.5)

where field components are decomposed along the ŷ and ẑ directions. Figure 3.1 illustrates
magnitude and direction of the magnetic field around the waveguide and current carrying
wire. The distance dz is determined by solving Equations (3.4) and (3.5) for z where thetotal field magnitude is zero,
dz =

µ0 Ix
.
2π By

(3.6)

At the position y = 0, z = dz the potential is a linear quadrupole field with a gradient
determined by
B 0 |dz =2π By2
By
=
,
µ0 Ix
dz

(3.7)

where the relation for dz from Equation (3.6) has been used. The dotted lines in Figure 3.2(a)and (b) show the magnetic field magnitude along both the ŷ− and ẑ−directions, respectfully.

42

Bias Field

a)

c)

-z
y

Wire Field

b)
-z

dz
Ix

-z

By

x
y

y

Figure 3.1: Illustration of the magnetic field lines in the side-guide configuration, which isformed by the superposition of a uniform bias field along −ŷ, shown in (a), and the magnetic
field generated by a current carrying wire, shown in (b). c) The resulting waveguide is locateda distance dz below the conductor. Colors indicate the magnetic field strength in the y-z
plane, increasing from a local minimum (purple) to the maximum field at the conductor(red).

a)

By|Bz,side||By,side|

By

43

b)

Bx

y

wire

{

Bx
dz

z

Figure 3.2: Addition of a bias field along the wire current in the side-guide trap converts thefield gradient near the minimum from quadrupole (dotted) to harmonic (solid). (a) and (b)show the magnetic field magnitude along the ŷ− and ẑ−directions, respectively.

Equation (3.7) reveals one of the attractive characteristics of atom chips. By enabling atoms
to be trapped in close proximity to a current carrying wire (typically tens to hundreds ofmicrons) magnetic traps with large field gradients (> 103 G/cm) can be realized with currentsof only a few amperes.
The quadrupole side-guide is an acceptable trap configuration for cold atoms (T &1µK); however, as the temperature of the trapped atoms reaches lower temperatures (e.g.
near the BEC phase transition) they become more susceptible to Majorana spin flips. At thezero of the magnetic field, where the Larmor precession frequency of the atoms is lowest, theatoms are no longer able to adiabatically follow the change in the field gradient and are likely
to undergo a spin-flip transition to an untrapped, strong-field seeking state [80]. However,this problem can be ameliorated by adding a second bias field, Bx , along the direction ofcurrent flow in the wire. In doing so, the magnetic field minimum is lifted from zero and thefield gradient near the minimum becomes approximately harmonic, forming a Ioffe-Pritchard(IP) trap [81]. So long as the Larmor frequency, ωLarmor = µBy /~, is much larger than theharmonic IP trap frequency, ωIP , losses due to Majorana transitions can be considered tobe negligible [76]. Solid lines in Figure 3.2 show the magnetic field of a side-guide IP trap
relative to the same field where Bx is absent.

44
Two key figures of merit for IP traps are the trap frequency and depth. The trap depth
is determined by the field magnitude infinitely far away from the wire, which is set by thebias field perpendicular to the current carrying wire, By . In order to determine the trap
frequency we again look at the field near the trap minimum, this time taking the second
spatial derivative,00

B |dz =

2π
µ0

2

By4
By2
=
.
Bx Ix2Bx d2z(3.8)

Assuming the potential to be harmonic (i.e., Uz,mag = mω 2 z 2 /2), Equation (3.8) in conjunction with Equation (3.1) to yield the side-guide IP trap frequency,rωIP,sideµmag 00
By
B |dz =
=
m
dz
2
By
√ .
∝
Ix Bx

r

µmag 1
,
m Bx
(3.9)

When designing an atom chip trap, Equations (3.6) and (3.9) are instrumental in understanding the interplay between the trap position with respect to the chip surface and thetrap frequency, given control parameters Ix , Bx , and By .
3.2.2

Split-guide trap

The side-guide IP trap is sufficient for many atom-chip-based experiments. However,
the atomtronics experiments described in Chapter 5 require unobstructed optical access tothe atoms in order to project optical potentials down onto the trapped atoms and also image
them in situ. Thanks to the atom chips discussed in Section 3.3, which are constructed fromcompound substrates that incorporate co-planar regions of silicon and glass, through-chip
optical access is not an issue. Instead, one is restricted by the fact that the side-guide IP
trap confines the atoms directly beneath one of the chip wires. While the wire is an excellent
conductor, this also means it is completely opaque at the optical wavelengths relevant to ourexperiment. To side step this issue, the chip design is modified as follows: a second chip wireis added that runs parallel to the original side-guide wire and carries the same magnitude

45
current but in the opposite direction. Additionally, the external bias field is rotated to point
perpendicular to the plane of the two wires. The resulting magnetic field minimum is shiftedout from under the single wire to a position half-way between the two wires. The totalmagnetic field is now described by


µ0 Ix
z
z
Bŷ,split =,−2π(y − dw /2)2 + z 2 (y + dw /2)2 + z 2


µ0 Ix
y + dw /2
y − dw /2
Bẑ,split =− Bz ,−
222π(x + dw /2) + z(y − dw /2)2 + z 2(3.10)(3.11)

where dw is the distance between the two wires. The total split-guide trap configuration isshown in Figure 3.3.
To determine the distance between the trap and the plane of the two wires, we follow
the same procedure used for the side-guide and solve dz = z where the total field magnitudeis zero. With currents of equal magnitude in both wires this is simplified further, as the trap
is centered along the plane y = 0. The distance dz for the split-guide trap is then given bysµ0 Ix dw d2w− .
(3.12)dz =2π Bz
4Upon examination, the expression for dz reveals a subtle issue. If the magnitude of thecurrent flowing in the wires is not sufficiently large (Ix < πdw Bz /2µ0 ), the solution becomes
imaginary. This occurs when the field from the wires is no longer large enough to cancel thebias field at dz > 0. Therefore, the trap is located in plane of the two wires, and the single
trap minimum splits, creating two side-guide traps between the wires.
Assuming the chip wire current is large enough to form a single, split-guide trap, thegradient of the magnetic field at z = dz is then given by the first spatial derivative,
B 0 |dz =4π Bz2 dz.
µ0 Ix dw

(3.13)

As with the side-guide, the split-guide trap can be made harmonic, i.e., suitable for trapping
ultracold atoms, by adding Bx , and the trap depth is now determined by the magnitude ofBz . Following the procedure from Section 3.2.1, the trap frequency is calculated using the

46

Bias Field

a)

c)

-z

-Bz
y

Wire Field

b)

-Ix

dz

-z
x

dw

-z
y

Ix

y

Figure 3.3: Illustration of the magnetic field lines in the split-guide configuration, which
consists of a uniform bias field along ẑ, shown in (a), and the magnetic field generated bycounter-propagating currents in a pair of parallel wires separated by a distance dw , shownin (b). c) The resulting waveguide is located a distance dz from the plane of the conductors.
Colors indicate the magnetic field strength in the y-z plane, increasing from a local minimum(purple) to the maximum field at the two conductors (red).

47
second spatial derivative of the total magnetic field, given by 24π
Bz4 d2z00
B |dz =,µ0
Bx Ix2 d2w(3.14)

to yieldrωIP,split =µ 00
B |dz =
m



4π
µ0



Bz2 dz
Ix dw

r

µ 1
,
m Bx

3/2

Bz
∝ √
.
Ix Bx

(3.15)

Interestingly, ωIP,split = ωIP,side for dw = 2dz , which is satisfied when Ix = 2πdz Bz /µ0 .
The design process for a split-guide atom chip trap is the same as for the side-guide,
except one must consider the distance between the wires. Since the purpose of the splitguide trap is to gain unobstructed optical access to the trapped atoms, dw and dz must
accommodate a solid angle larger than that of the primary objective in the optical system.
Assuming the window in the chip substrate has a large enough clear aperture, the chip wire
limited numerical aperture is given by
NAwiredw,= sin arctan
2dz
1/d˜z
= q
,
4 + 1/d˜2z(3.16)

where d˜z = dz /dw . A schematic illustrating the through-chip optical access is shown in
Figure 3.4(a). To provide context to the optical system described in Chapter 5, Figure 3.4(b)shows NAwire versus d˜z , which indicates that the maximum ratio of dz to dw for a 0.6 NA
primary objective is ∼ 2/3.
3.2.3

H-wire trap

Thus far, only waveguides that confine atoms along two dimensions have been considered. Both the side- and split-guide potentials can be modified to form full, three-dimensional
trapping potentials by superimposing another magnetic field with spatially varying magnitude along the x̂-direction. This is accomplished by running current through a pair of parallel

48b) 1.0θNAdzdw

Chip Wire Limited NA

a)

0.8
0.6
0.4
0.2
0

2/3
0

0.5

1.0
~
dz

1.5

2.0

Figure 3.4: a) Illustration of the maximum thru-chip numerical aperture: The maximum
ray angle depends on the chip wire separation, dw , and trap-to-chip distance, dz . b) Plot of
the chip wire limited NA versus d˜z . Dotted lines indicate the maximum d˜z = 2/3 that still
accommodates the imaging and projection system described in Chapter 4, which has an NAof 0.6.

49
conductors oriented perpendicular to the main guide wire(s) to form an ‘H’-shape that gives
the H-wire trap configuration its name. The contribution to the total magnetic field alongx̂ is treated in the same manner as a side-guide. Figure 3.5 shows an example potential
generated by the H-wire configuration with a single side-guide wire. Here, the H-wires are
separated by a distance dH = 2.2 mm, both the H- and guide-wire currents are IH,G = 2 A,
and bias fields along (x, y, z) = (1, −30, 0) G. The resulting trap is ∼ 130 µm below the chipwith trap frequencies ω(x, y, z) ' (25, 2100, 2100) Hz and a depth of ∼ 30 G.
Design parameters for the H-wire trap are dictated primarily by the ratio of currents,
IH /IG , and the ratio of distances, dz /dH . The potential along the x̂−direction is seen to
be quite anharmonic; however, through careful selection of IH,G and dH,z the anharmonic
contributions can be reduced [11]. Additionally, there is a noticeable twist to the potential
in which the longitudinal trap axis is rotated away from the x̂-axis by θH . This twist arises
due to the perpendicular contribution of the H-wire field and is accentuated at higher fieldmagnitudes. Near the trap minimum, the twist θH ∝ (dz /dH )2 , which indicates that thetwist is minimized by increasing the distance between the H-wires or moving the trap closer
to the chip.

3.2.4

T-wire trap

The primary utility of the H-wire trap is to provide loose confinement along the untapped axis of the guide traps. If instead a potential with tight confinement along x̂ isdesired, as is required for rapid rethermalization during the RF evaporation process used toform a BEC, a single conductor oriented perpendicular to the guide wire(s) can be added tothe chip wire pattern. By flowing current in the opposite direction of the H-wire current, a
dimple is produced in the longitudinal potential profile. The wire pattern can form either a‘+’ or ‘>’ relative to the horizontal guide wire, where the former is referred to as a dimple
trap and the latter as a T-wire trap. This section considers the T-wire trap, as the vertical
dimple wire would obscure optical access to the atoms. More information on dimple traps

50

Bx

IH

H-WireH-Wire

30

IH

20

IG

10

|Bx| (G)

By

20

12

36

30

20 10
|By| (G)

0 -2

12

20

4

20

12
28

-1

0.4
0.2
0
-0.2
-0.4

36

36

4

Y-Position (mm)028

0X-Position (mm)

36

4

28

1

2

Figure 3.5: H-wire trap configuration: The inset at the upper left illustrates the conductor
scheme along with the wire current and bias field directions. The central energy contour plotshows the highly elongated trap shape. Each contour line is labeled with the field magnitudein Gauss (G). The dotted red and blue lines through the center of the trap indicate theposition of slices through the trap along the x̂− and ŷ−directions that are shown above andto the left, respectively.

51
can be found in the theses of Salim and Hudek [82, 13]. Figure 3.6 illustrates the T-wirepotential, again with H-wires separated by a distance dH = 2.2 mm, but with wire currents
IH = 1.75 A, IG = 2.5 A, IT = 0.75 A, and bias fields along (x, y, z) = (6.7, −30, 0) G. The
resulting trap is also located ∼ 130 µm below the chip with a depth of ∼ 30 G, but now has
trap frequencies ω(x, y, z) ' 2π × (350, 2100, 2100) Hz.
Design parameters for the T-wire trap are almost identical to those of the H-wire trap
and are conceptually quite similar. The T-wire current can be setup to either run parallelor antiparallel with respect to IG in the main guide wire, resulting in increased or decreased
total current through the shared section of the guide wire, respectively. Both configurations
can be used to generate T-traps, and the radial trap parameters can be tuned to match theH-wire trap by decreasing or increasing the magnitude of IG , leaving By unchanged. To
account for the magnetic field contribution from the T-wire current that runs antiparallel tothe H-wires, the magnitude of Bx is increased, allowing IH to be decreased. In doing so, thetrap bottom and depth can be tuned to match that of the H-wire trap. Typical T-wire trap
parameters do not require a large current in the T-wire. As Figure 3.6 shows, less than 1
A in the T-wire generates a dimple in the longitudinal trapping potential that is ∼ 5 Gauss(350 µK) deep and increases the mean trap frequency to ∼ 2.5× that of the H-wire trap.
3.3

Atom chip designs
The previous section covered the basic concepts for both side- and split-guide traps

along with H- and T-wire traps. Each of these trap geometries is used at different stagesof the atomtronics experimental cycle, described further in Section 5.1. The sections that
follow give an overview of the conductor patterns that were designed for and implemented
in the experimental work of this dissertation.

52

Bx

IH

T-WireH-WireH-Wire
30

IH

20

IG

IT

10

|Bx| (G)

By

12

12

30

20 10
|By| (G)

0 -2

20

20

4

20

28

36

-1

0.4
0.2
0
-0.2
-0.4

36

36

28

Y-Position (mm)00X-Position (mm)

12

28

36

1

2

Figure 3.6: T-wire trap configuration: The inset at the upper left illustrates the conductor
scheme along with the wire current and bias field directions. The central energy contour
plot shows the elongated trap shape similar to the H-wire trap with an additional dimple
generated by the T-wire at trap center. Each contour line is labeled with the field magnitudein Gauss (G). The dashed red and blue lines through the center of the trap indicate theposition of slices through the trap along the x̂− and ŷ−directions that are shown above andto the left, respectively.

53
3.3.1

Atom chip standardization

Atom chip fabrication in the Anderson lab is largely standardized. The historicalevolution of our atom chip construction can be found in the theses of Du [83], Squires [10],Segal [11], Salim [82], and Hudek [13]. At the heart of our chip technology is the compoundsubstrate on which the chip conductors are lithographically patterned. Constructed of coplanar regions of high conductivity silicon and optically transparent Pyrex, the substrateprovides hermetically sealed electrical and optical feedthrough [82, 84]. This co-planar design
enables conductors to be patterned directly atop transparent regions of the chip, and in turnallows atoms to be trapped below an optical surface. When bonded to one of our double-MOT
cells, the chip then serves as one of the walls of the vacuum chamber, allowing high-numericalaperture optics to be placed . 1 mm from the atoms.
3.3.2

The V2 window chip for in-trap imaging

The V2 window chip is the second generation of window chip used to produce BECs
and image them through the chip window (the first being the V1 window chip describedin the thesis of Evan Salim [82]). This chip design, shown in Figure 3.7, incorporates thepreviously discussed split-guide design patterned directly over the central, 2 mm diameterchip window. The chip design includes wires for both H- and T-wire trap configurations. As
a design constraint, the infinite wire approximation only remains valid so long as the widthof the wires, ww , is much less than the distance from the wire to the atoms. To combat thehigh resistance of thin wires, the width of individual conductors on the vacuum side of theatom chip is ww = 100 µm.
One caveat to this conductor pattern is the inevitable junction between the guide, T-,
and the H-wires, if all conductors are on the same side of the chip. Multiple intersecting (i.e.,
shorting) wires require independent, floating current drivers for each separate current path.
This is feasible but becomes cumbersome and undesirable for more than two intersecting

54

a)

b)

c)

d)

IG
IM,T,G

IM
IH

IT

IH

Figure 3.7: a) Bare compound substrate prior to metallization process. b) Vacuum side of
the V2 window chip with copper conductors. c) Ambient side of V2 window chip with gold
conductors. d) X-ray schematic view of the central region of the chip with the chip current
paths used in the V2 window chip. As shown, the lower T-wire is used in the side-guide,
T-wire, and split-guide traps. The current direction in the lower of the two horizontal wires
are labeled IM , IT , and IG , respectively.

55
current paths. By running multiple floating current supplies, one runs the risk of developing
high voltages between supplies and therefore across the chip, which can cause breakdown
in the dielectric layer. Therefore, the V2 window chip conductor design places the H-wires
on the ambient side of the chip substrate while the T- and guide-wires are on the vacuum
side. The dielectric coating on the chip provides electrical isolation between the vacuum andambient sides of the substrate, so the H- and guide-wires are driven by current supplies witha common ground reference and the T-wire driver floats. The downside to this configurationis that the H-wires are now displaced by 420 µm (the thickness of the chip) from the guidewires. This results in a decrease in the longitudinal confinement that goes as ∼ 1/r. To
account for this decrease, the H-wire width is increased to 150 µm to accommodate larger
currents and doubled up, with two conductors running immediately parallel to one another.
By moving the H-wires to the ambient side of the chip, the overall trapping potential
generated by both the H-and T-wire traps discussed in Sections 3.2.3 and 3.2.4 are modified.
Figure 3.8 illustrates these changes. Here, the conditions are identical except the H-wirecurrent is doubled and a second pair of H-wires separated by 2.4 mm is added, reproducing
the V2 window chip design. The increased distance between the H-wires and the trap
location, due to the wires being on the ambient side of the chip, causes the magnetic field
generated by the H-wires to appear more like an additional x-bias field, which smooths
out the longitudinal trapping potential and reduces both the trap depth and mean trap
frequency. The trap profiles indicated by the solid lines in Figure 3.8 show the trap in which
the majority of the evaporation is completed during typical experimental cycles.
Once a BEC is created in the T-wire trap, the atoms must be transferred out from under
the wire and into a split-guide potential to gain optical access to the atoms. The vacuum side
conductor pattern is mirrored across the center of the chip, which also provides a redundantT-wire. This can be seen in Figure 3.7(b), in which the top and bottom halves of the vacuumside metallization are mirror images separated by dw = 340 µm. Given this separation and
ww = 100 µm, the split-guide wires satisfy the condition that the N Awire > 0.6 for values of56|B| (G)

a)

30
20
10
-2

-1

0
Position (mm)

1

2

-2

-1

0
Position (mm)

1

2

b)
|B| (G)

30
20
10Figure 3.8: a) Comparison of H-wire traps in which the H-wires are on the same side (dashed)
and opposing sides (solid) of the chip substrate as the guide wire. b) Comparison of T-wiretraps in which the H-wires are on the same side (dashed) and opposing sides (solid) of thechip substrate as the guide and T-wires. In both (a) and (b) the longitudinal and radial trap
profiles are shown in red and blue, respectively.

57
dz . 160 µm.
The imaging trap is generated by running equal currents, IG , antiparallel through thetwo guide wires and a vertical bias field, as described in Section 3.2.2. By this point in
the experimental cycle, the atomic ensemble is ultracold and the trap depth, even at low
wire currents, is much greater than the energy scales of the ensemble. Typical experimental
parameters are IG = 2 A, IH = 1 A, and bias fields along (x, y, z) = (1, 0, −30) G. Finetuning of the radial and longitudinal trap frequencies is accomplished through small changes
to Bx and IH . These two parameters vary the magnitude of the magnetic field at the trap
bottom, which changes the field curvature without adjusting the position of the trap. In
this way, the trap frequencies are widely tunable from hundreds of Hz to a few kHz radiallyand from tens of Hz to hundreds of Hz longitudinally. This chip design is utilized for the
majority of the experimental work described in this dissertation.

3.3.3

The V3 window chip, a double split-guide and other design upgrades

Despite the great success of the V2 window chip, several aspects of the design required
minor adjustments. Modifications include the addition of a second set of H-wires withincreased spacing along with a second pair of guide wires for the split guide trap. The
central window in the chip substrate is also anti-reflection (AR) coated to improve optical
performance. These changes look to improve the utility of the chip without drastic changes
to its overall functionality. The V3 chip design, including the current path schematic, isshown in Figure 3.9.
Thus far, chip design has been discussed under the approximation of infinitely thinchip wires. As mentioned previously, this assumption holds when the trap is far from theconductor dz  ww . Even then, a trap located dz . ww from a straight conductor will only
suffer from a reduced gradient due to the finite spatial extent of the current density in the
conductor, which can be accounted for analytically [13]. However, this issue is accentuatedin the case that atoms are in close proximity to a wire that bends, widens, or intersects

58

VacuumAmbient

X Ray

Current Schematic
IG,1

IG,2
IM,T,U

IM,U
IH

IU

IT

IU

IH

Figure 3.9: View of the ambient and vacuum sides of the V3 window chip that incorporates
a second pair of guide wires. An X-ray view of the chip is also shown along with the current
direction for various chip trap configurations including H- and T- traps, the split-guide trap,
and the U-trap.

59
another wire. The current density within wires at locations such as these is non-trivial andin order to model the trap correctly, finite element analysis (FEA) is necessary. Modelingthe traps generated by our atom chips is done using FEA software that was developed inconjunction with Boulder Labs Inc. [85].
The first addition to the V3 window chip ameliorates unwanted kinks in the magnetic
field that occur at the T-wire junctions of the split guide wires. As current flows through eachguide wire, two issues arise. First, a region of reduced current density manifests within theguide wire as electrons pass through a region of increased cross-section at the wire junction.
Figure 3.10 shows an example of the current density in the central region of the mirroredT-wires. Second, as a result of the modified current density, a non-zero projection of themagnetic field is generated perpendicular to the intended direction. These factors result in a
distortion of the magnetic field near the trap minimum. The longitudinal profile of the splitguide trap, shown in Figure 3.11 as the solid line, is impacted the most and exhibits a ‘kink’in the potential energy with magnetic field artifacts on the order of ∼ 1/4 G (∼ 20 µK).
The typical temperature of the atomic ensemble in the split-guide trap is . 1 µK. Therefore,
the atoms become trapped in the new minima and the longitudinal trap frequency is nolonger dictated by the H-wires. Instead, the longitudinal trap curvature is determined by
the distance between the atoms and the T-wire junction as well as the relative decrease incurrent density within the conductor. The simple solution to this problem is the additionof a second pair of parallel guide wires that serve as the new split-guide trap. These wires
are colored light blue in the current schematic shown in Figure 3.9. By removing the Twire junction from the split-guide wires the longitudinal profile, shown as the dashed line in
Figure 3.11, is once again harmonic and tunable by IH .
The second addition to the V3 window chip is an additional set of H-wires, withtwice the original spacing and a width of ww = 200 µm. The addition of more H-wires
provides flexibility to chip loading schemes as well as the longitudinal confinement of thechip traps. Use of the wider spaced H-wires enables lower longitudinal trap frequencies (∼ 10

60

0

25
50
Current Density (A/mm2)

Figure 3.10: FEA of the current density at the intersection of the chip T-wires. Current
density increases from red to green. The brief excursion of the guide wire current into thejunction leads to a depletion in the current density of ∼ 20%.

61

b)

9

16

8.6
8.2

8
V2

0
-2.4

V3

-1.6

V3-w

-0.8
0
0.8
1.6
Longitudinal Position (mm)

-0.4

-0.2

0

0.2

Position (mm)B-Field (G)Chip Trap B-Field (G)

a) 24

7.8
0.4

2.4

Figure 3.11: a) Comparison of the split guide trap fields for V2 (solid) and V3 (dashed) window chips. Elimination of the T-wire intersection in the V3 chip enables looser longitudinal
confinement controlled by the H-wire current. Furthermore, the V3 chip includes a second,wider spaced set of H-wires. The longitudinal profile with this new set of H-wires (V3-w) isshown by the dot-dashed line for the same chip current and bias field conditions. b) Close up
around the field minimum of the V2 and V3 traps indicated in (a) by the dotted rectangle.
The kink in the longitudinal profile is eliminated when guide wires without a T-wire junction
are used for the split-guide.

62Hz) without also greatly reducing the trap depth. Figure 3.11 includes a comparison of thelongitudinal confinement in the split-guide trap using the inner or outer pairs of H-wires,with equal IH .
Another important consideration for chip design is the method of initially loading
atoms from the previous, external magnetic trap into the chip trap. To achieve optimal chiploading efficiency, it is necessary to mode match the quadrupole trap created by externalcoils to the magnetic field generated by the chip [71]. The chip trap configurations discussed
thus far do not provide optimal mode matching; however, by running current in a U-shapeon the chip with a single bias field it is possible to generate a trap that mimics the external
quadrupole. This current path is shown in Figure 3.9, labeled IU , and utilizes the inner
H-wires along with one of the main guide wires in conjunction with a bias field oriented
along the same direction as in the side-guide. Systems that load the chip trap directly froma mirror-MOT commonly use this loading scheme [83, 86].
Lastly, in addition to the updated conductor pattern, the chip window is coated forseveral wavelengths (< 0.6% reflectivity from 760 − 790 nm and 99.92% reflectivity at 1064
nm at normal incidence). The high-R coating was utilized in conjunction with steering optics
anodically bonded to the chip to demonstrate an on-chip optical lattice by Straatsma andcolleagues at ColdQuanta [87]. In that work, optics were bonded to the atom chip in regions
where the dielectric coating was stripped during chip fabrication.
3.4

Technical considerations for atom chip traps
Once the atom chip design is determined to adequately meet the requirements of theexperimental goals, one must consider other, more technical aspects. These include thepower dissipation within the chip during the experiment cycle and the effect of technicalnoise in the wire currents on the atoms trapped in the magnetic potential. While they
impact different parts of the system, both heating related issues require careful attention
when designing and conducting experiments in an atom-chip-based apparatus. The analysis

63
provided in the following sections is useful in understanding both topics.

3.4.1

Ensemble heating in the split-guide trap

When implementing any magnetic trap design to produce and study ultracold atomic
systems, one must consider the heating rate of the trapped ensemble due to various mechanisms. Heating in magnetic traps can be caused by fluctuating currents in the chip wires,due to either technical noise or thermal currents, as well as collisional processes such as
dipolar relaxation [76, 88]. This section addresses the role of technical noise in the heating
rate of atoms confined in the split-guide potential and follows the analysis reported on in
References [76, 89, 90].
Under the harmonic oscillator approximation of the atom chip magnetic trap, the rate
at which atoms are excited is given by
Γex =

a2
SF (ωt ),4~2(3.17)

where a is the harmonic oscillator length and SF (ωt ) is the noise spectrum of the trapping
force at the oscillator frequency, ωt . From this, the rate of energy transfer to the excited
atom is given by dE = ~ωt Γex . A more accessible parameter than the force exerted by themagnetic trap is the position of the trap minimum, which are related by F = mωt2 z where z
is the position of the trap relative to the chip. The rate of energy transfer due to fluctuationsin the trap position is
dE =

mωt4
Sz (ωt ),4(3.18)and depends strongly on the trap frequency, scaling as dE ∝ ωt4 .
These equations can be applied to the split-guide trap by looking at the trap position
fluctuation spectrum caused by noisy current supplies. The position fluctuation spectrum
is determined from Equation (3.12) for z = dz , in conjunction with measurements of the64current driver noise spectra. Recalling the form of the trap position in ẑ relative to the chip,sµ0 Ix dw d2wdz =− ,(3.19)
2π Bz
4one sees that fluctuations in dz depend on both the chip currents as well as the external bias
field. The magnetic field generated by the square z-bias Helmholtz coil pair is given by
8µ0 Nt Ib
Bz = √,
125W(3.20)

where Nt is the number of turns in each coil, Ib is the current in the coil, W is the halfwidth of the coil, and the coil pair is separated by 2W . Combining the current noise spectraof both the coil and chip currents yields the total position fluctuation spectrum, Sz (ωt ) =(∂dz /∂Ix )SIx (ωt ) + (∂dz /∂Ib )SIb (ωt ). The current noise spectra for the chip and coil current
drivers, both of which are designed and built in our lab, were measured using a Stanford
Research Systems SR780 spectrum analyzer in a frequency band from 1−2 kHz. Across these√√frequencies, the noise spectra are fairly flat with SIx ≈ 35 nA/ Hz and SIb ≈ 120 nA/ Hz.
The heating rate in the split-guide trap was determined by measuring the temperatureof the atomic ensemble after hold times of various duration, using the temperature fitting
method described in Section 4.2.1. Measurements were made for radial trap frequencies
ranging from ∼ 200−3400 Hz, with an ensemble of ∼ 104 atoms at a temperature T ≈ 1.5Tc .
Results of this measurement are compared to the theoretical heating rate in Figure 3.12.
Disagreement in the measured and calculated heating rates at low trap frequencies is in partdue to the excitation of a slosh mode while loosening the trap and higher values of 1/fnoise of the current drivers. Unfortunately, technical noise limits the trap frequencies usedin the atomtronics experiments to below ∼ 2 kHz. At higher trap frequencies, the resulting
decrease in condensate lifetime is too severe, dropping below ∼ 100 ms for ωt /2π > 2 kHz.
In order to reach higher trap frequencies without sacrificing the condensate lifetime, thecurrent noise of the coil driver must be decreased, as it contributes more significantly to theobserved heating than the chip current driver.

Heating Rate (nK/s)

65

10

4

10

3

10

2

0

1

2

3

4

5

6

Radial Trap Frequency (kHz)
Figure 3.12: Comparison of the theoretical and measured heating rate in the split-guidetrap. Red points indicate the measured heating rate. The fit to the data as well as the 95%
confidence interval for the measured heating rate are shown by the black dotted line andgray band, respectively. At low radial trap frequencies the fit to the heating rate is higher
than the theoretical rate, which is calculated using the measured current driver noise spectraand shown by the blue dashed line, but the two agree at higher frequencies.

66
3.4.2

Wire heating comparison in window and non-window chips

The previous sections have discussed chip conductor pattern designs that achieve thedesired magnetic trap functionality in our atomtronics experiments. However, one of the
primary concerns in chip design is heat management. Compared to other atom chip groups,
whose chips are mounted to large copper blocks [91] and sometimes temperature stabilized [92, 93] to facilitate heat transfer out of the chip substrate, our atom chips have limited
thermal mass and rely on convective cooling. In the event that thermal conduction to thechip substrate is inadequate, the results can be catastrophic, often resulting in the evaporation (burn-out) of the chip wire [94]. Yet, with proper consideration for the thermal
conductivity of the chip substrate, conductor properties, and insulating dielectric layers,
even complex, multilayered chips can be implemented successfully [95]. In this section thethermal performance of our two-layer, compound substrate atom chip is assessed using models for fast and slow heating. Chip designs with both 20 µm and 100 µm wide conductors arecompared along with designs that include conductors patterned over the transparent chip
window. Additional measurements provide information relevant to safe operating limits.
Wire heating test setup
Two atom chip designs were tested: A direct evaporation chip used in the thesis ofK. Hudek [13] was chosen to represent the non-window, silicon substrate chips (labeled S),
and the V1 window chip used in the thesis of E. Salim [82] represented the window chip
substrate (labeled W). Each chip was bonded to a standard 3D MOT glass chamber andpumped down to ∼ 10−6 torr in order to reduce vacuum side convective cooling and emulateconditions in typical experiments. Electrical connection to the chip was made using pogopins rated to 6A continuous current in air. Electrical properties of the chip were monitored
by measuring the resistance of the chip wire in a 4-point probe configuration. Voltage andcurrent data were recorded by a LabVIEW script and analog input card (NI cRIO-9215).

67

a)

Vacuum stationb)
Pogo pins

Ruby Bearings

d)

c)

e)

f)

Figure 3.13: Pictures of the test setup used to determine the thermal properties of atom
chips during current flow. a) Top side of direct evaporation chip bonded to a 3D MOT
chamber and connected to the vacuum station. Ruby ball bearings epoxied to the cell walls
serve to locate the prototype pogo pin chip connector. b) Underside of chip connector shows
two pogo pins lined up with the gold pads on the top of the chip. c) Electrical breakout
board for the chip connector. d) Complete prototype pogo pin chip connector attached to thecell and secured with a tap stand. e) Direct evaporation chip used to test non-window chipproperties. Image credit [13]. f) V1 window chip used to test wires over Pyrex substrate.
Image credit [82]. Insets of (e) and (f) show the wires used in the heating tests. Green
arrows indicate current directions in the tests.

68
Models for fast and slow chip heating
Heating of the atom chip wires and substrate is modeled following the work of Groth etal [96]. Both fast and slow time scales are considered. Fast heating is determined primarily
by the resistivity of the wire and the surface conductance, k, of the chip substrate. During
the fabrication process, prior to metallization, a thin dielectric layer (SiO2 ) is grown on thewafer in order to electrically insulate the high conductivity silicon of the chip substrate fromthe wires. Therefore, the surface conductance is determined by properties of the Cu-SiO2 -Si
or Cu-SiO2 -Pyrex interface. The relative thermal conductivity of the substrate materials are
kSiO2 ≈ kPyrex ≈ kSi /100 [97]. The change in conductor temperature just after a current, I,
begins to flow is given by∆Tf ast (t) =hρj 2
1 − e−t/τf ast ,2k − hj αρ

(3.21)

where j = I/(hw) is the current density, h and w are the height and width of the conductor,
respectively, and ρ is resistivity of the copper, which has a linearly approximated temperature
coefficient α. The time constant associated with the heating is determined by the heat
capacity of the wire, Cw , and is given by τf ast = Cw h/(k − hj 2 αρ). The slow heatingprocess depends more strongly on the bulk thermal conductivity, λ, of the substrate, which
determines how quickly the heat disperses through the substrate. After the initial fast
temperature increase, the change in temperature is approximated by
 2 
ρIj
4π λt
∆Tslow (t) ≈ln,
2πλCs w 2(3.22)

where Cs is the substrate heat capacity. Since the substrate material varies across the lengthof the conductor, Cs is left as a fit parameter when analyzing the chip heating data.
Chip heating comparison
Chip heating in 100 µm wide wires on both the window and non-window chips iscompared for current densities ranging from j = 1 − 2 × 103 A/cm2 . The increase in the

69
chip wire temperature is determined from the relative change in the conductor resistance,
R/R0 = 1 + α(T − T0 ), where R0 and T0 are the initial chip wire resistance and temperature,
respectively. Figure 3.14 shows four representative data sets in which the calculated temperature is shown as the black solid line and fits to the data using Equations (3.21) and (3.22)
are shown as dashed lines. Since the fast heating time constant is dominated by the physical properties of the conductor rather than the substrate, the initial heating behavior ofboth the direct evaporation and window chips are similar, with τf ast = 41.0(2.6) µs. Fits
to the data also enable extraction of the effective values of k and λ for the two chips. Fits
to the fast temperature increase yield the ratio kS /kW = 8.71(0.73), indicating that heattransport from the wire, across the SiO2 insulating layer, and into the substrate is nearly
an order of magnitude lower for the window chip. This results in a much more substantial
initial increase in the measured wire temperature within the first few milliseconds. While
the ratio of surface conductivity is nearly flat across the range of current density, the ratioof bulk heat conductivity shows a strong dependance on the current density, j. For low j,
λU /λW ≈ 3.75, but the ratio increases exponentially as a function of the current density.
Doubling j leads to λU /λW ≈ 14, as λW decreases and λU remains nearly constant. Sucha pronounced decrease in the window chip substrate conductivity reflects saturation of theheat capacity of the chip substrate. Ultimately, the reduction of the heat capacity sets theupper limit on the duration for which current can be run through the wire.
An array of additional heating measurements were performed, encompassing the typicaloperating parameters of our atom chips during normal operation in experiments. Whileneither of the chips failed under typical experimental conditions, it is valuable to determine
the maximum current supported by the chip wires. Measurements of the relative change inthe wire resistance were performed for increasing currents, each run for 3 seconds, i.e., thetypical length of an BEC production cycle, until the conductor failed. The results are shownin Figure 3.15. In the case of the window chip, 4 A flowing in the 100 µm trace lead to
failure after only 1.39 s, at which point the wire temperature reached 240◦ C followed by a

70

200

Temperature ( C)

W4A
100

W3A
S4A
S3A
20
0

0.2

0.4

0.6

0.8

1

Time (s)Figure 3.14: Comparison of the rate at which 100 µm wide chip traces heat up under current
flow for two chips: one with wires patterned only across silicon (S) and the other where partof the wire is patterned over a 3 mm diameter Pyrex window (W). Green and red dashedtraces indicate fits to the data using the fast and slow heating model equations, respectively,
which are used to extract the surface heat conductance, k, and bulk heat conductivity, λ ofthe two chips. It is clear that the presence of the pyrex window reduces the rate at whichheat is removed from the conductor.

1.5

∆R/R0

30 s
pulse

2s
pulse

1
1.74 s
pulse

0.5

0

1

2

3

4
5
Current (A)67Figure 3.15: Relative increase in the chip wire resistance during a 3 second current pulse.
Data sets for a 100 µm wide wire on the window chip (red square), along with 100 µm (bluetriangle) and 20 µm (green circle) wide wires on the non-window chip are shown. Chip wireburn-outs for each data set are indicated by stars along with the respective time until failure.

71
rapid increase to > 500◦ C. As expected, the wire burned out directly over the chip window(see Figure 3.16(a)). By comparison, the heat conductivity of the 100 µm trace on the
non-window test chip is sufficient to survive 4 A for > 30 seconds and 7A for > 3 seconds.
Additional tests show that smaller, 20 µm traces on the non-window test chip fail after 30
seconds at 5 A and 2 seconds at 7 A, both near ∆R/R0 ≈ 1.25 − 1.4. The failure locationsin these tests are shown in Figure 3.16(b).
Knowledge of the chip wire resistance under repetitive use is of equal importance tothe single shot maximum current. Given insufficient time to cool after an experimental
cycle, the chip wire will reach higher peak temperatures during successive current pulses.
The change in chip wire resistance at various duty cycles was measured for the non-window
chip, and the results are shown in Figure 3.17. During a typical experimental run in theatomtronics apparatus an average of 2.5 − 3 A is run through the main wire for a durationof 2 − 3 seconds. Given the lower heat conductivity of the window chip and accounting forthe currents in the second guide wire along with the H- and T-wires, the experimental cycle
is conservatively limited to a duty cycle of 0.05, or once per minute.

72

a)

b)

BurnedRegions

BurnedRegions

BurnedRegions

Figure 3.16: a) Picture of the burned out 100 µm wire that runs across the Pyrex window
on the V1 window chip after 4 A of current was run through the wire for 1.39 s. The wire ispartially lifted away from the window with dislocations in the wire at the points indicated.
b) Picture of the central region of the direct evaporation chip showing two regions of theburned out 20 µm wire. The two traces failed after 7 A of current run for 2 s in the rightangle wire, and 5 A of current for 30 s in the straight wire. Note that the failures occurredin the center of the narrowest region.

0.5

6A

5A

4A

3A

0.4
∆ R/R0

2.5 A

0.3
0.2
2A

0.1
0

0

0.2

0.4
0.6
Duty Cycle (ton/ttotal)

0.8

1

Figure 3.17: Increase in the 100 µm chip wire resistance on non-window chip under various
current loads and a range of duty cycles. Points indicate measured values, while the lines
are guides to the eye. Low currents can safely be run at a near unity duty cycle, as seen for
2A and 2.5 A. In conjunction with the data from Figure 3.15, the maximum safe duty cycle
for a fixed chip current is given by the region to the left of each line.

Chapter 4
High-resolution imaging & optical control of ultracold atoms

Research over the past two decades has ushered in a new paradigm of quantum simulation that utilizes trapped, ultracold gases to study the behavior of complex condensed mattersystems [98]. This research thrust has led to the development of high-resolution imaging systems to study both collective and microscopic phenomena in ultracold quantum gases. Suchsystems include both bosonic [99, 100] and fermionic [101, 102] quantum gas microscopesthat exhibit single atom sensitivity and enable studies of quantum magnetism [103], atomic
correlations [104], and other exotic states [105]. In this chapter, a similar high-resolution
system is presented that enables the study of atomtronic circuit elements.
The organization of this chapter is as follows: First, the atom-light interaction thatenables one to image and optically manipulate atoms is derived in Section 4.1. Building uponthese results, Section 4.2 discusses absorption imaging, the primary technique for observing
our atomtronic systems. Optical trapping and its application to the atomtronic transistor
system are then described in Section 4.3. Finally, Section 4.4 introduces the high-resolution
microscope system, and the remaining sections characterize the microscope system’s ability
to simultaneous project optical patterns onto and image the trapped atoms.
4.1

Atom-light interactions in neutral atoms
The interaction between atoms, e.g. 87 Rb, and an applied light field can be understoodaccording to the classical treatment of a two level atom with a single resonant frequency, ω0 .

74
In the presence of a light field, the electron distribution of the atom is driven harmonically
about its nucleus leading to a charge displacement governed by the equation of motion,me ẍ + me γ ẋ + me ω02 x = −eE,

(4.1)

where E is the electric field of the light wave, γ is the classical damping rate, and me and e
are the mass and charge of the electron, respectively [106]. For an electric field of the form
E(t) = E0 Re{exp(iωl t − φl )}, solutions to Equation (4.1) are given byx(t) = XRe{exp(iωl t − φe )}.

(4.2)

Here, X and φe are the amplitude and phase of the charge displacement, whereas E and φl
are the amplitude and phase of the light field, with frequency ωl . Solving for X, one finds
X=−

eE
1
,
2
me ω0 − ωl2 − iγωl(4.3)

which can then be used to determine the induced polarization of the atom, p = −eX. The
polarization can also be expressed as p = αE, where α is the complex atomic polarizability,
α=

e2
1
.
2
me ω0 − ωl2 − iγωl

(4.4)

Within X and α, the classical damping rate describes the radiative energy loss as the charge
distribution oscillates,γ=

e2 ω 2,6π0 me c3(4.5)

where 0 is the permittivity of free space and c is the speed of light in vacuum [107]. Replacing 6π0 c3 γ/ωl2 = e2 /me and inserting the on-resonance damping rate Γ = (ω0 /ωl )2 γ,
the complex atomic polarizability becomes
6π0 c3
Γ
α=
.
2
2
2
ω0 ω0 − ωl − i(ωl3 /ω03 )Γ

(4.6)

In the case of the two-level atom, Γ is equivalent to the natural linewidth of the transition
and is inversely related to the lifetime of the excited state. The following sections show

75
that the complex polarizability dictates both the refractive index and absorption coefficientof a cloud of atoms. These two properties of the atom cloud are integral to experimental
techniques such as imaging and optical trapping.

4.1.1

Refractive index, absorption, and scattering rate of atoms in a lightfield

Building upon the classical treatment for the response of a two level atom to an opticalfield, the refractive index of an atomic ensemble with density n is determined by nref =√1 + 4πnα [108]. By re-expressing the complex polarizability as
30 λ2l
α=
(2π)2



i

δ
−
2
1 + δ + s0 1 + δ 2 + s0


,

(4.7)

it is possible to separate the real and imaginary parts of α corresponding to the phase shift
and absorption of light by the atom, respectively [109]. Here, λl = 2πc/ωl is the wavelengthof the optical field. The polarizability is seen to depend on two parameters of the field: thesaturation parameter, s0 = I/Isat , where Isat = ~ω03 Γ/12πc2 , is the saturation intensity, andthe normalized detuning, δ = ∆/(Γ/2), where the detuning, ∆ = ωl − ω0 , is a measure of
the frequency difference between the light field and the resonant frequency of the two level
atom [108]. Therefore, both δ and s0 dictate the strength of the atom-light interaction. It
follows that the refractive index of the atoms is
σ0 nλlnref = 1 +
4π



i

δ
−
2
1 + δ + s0 1 + δ 2 + s0


,

(4.8)

so long as nref − 1  1 [108]. As the light field propagates through the cloud of atoms,
the real and imaginary parts of the refractive index cause the field to accumulate additional
optical phase and lose amplitude due to photon absorption, respectively. The resulting phaseshift is calculated by integrating the contribution to the spatially varying refractive index

76
due to the atoms along the propagation direction of the optical field (ẑ),
Z
2π ∞φ(x, y) =dz (nref (x, y, z) − 1),
λl −∞
σ0 ñ
δ
= −
,
2 1 + δ 2 + s0

(4.9)

where σ0 = 3λ2l /(2π) is the on-resonance scattering cross-section and ñ is the column density
along the path of the optical beam [108]. The transmission of the light field through thecloud is calculated in a similar manner, by considering the attenuation of the beam as itpropagates through the cloud, under the modified Beer-Lambert law,dI/dz = −nσ0Ii ,1 + δ 2 + s0

(4.10)

where Ii is the intensity of the incident optical field. The transmission coefficient,


Z ∞
q
nIi
t(x, y) =If /Ii = exp −σ0dz,
1 + δ 2 + s0
−∞


1
σ0 ñ
,
= exp −
2 1 + δ 2 + s0

(4.11)

then relates the transmitted intensity, If , to the intensity incident on the cloud. Therefore, the phase shift scales linearly with the column density, while the transmission decays
exponentially [108].
The modifications to the light field described by Equations (4.9) and (4.11) treat only
the transmitted field. However, as alluded to previously, atoms absorb photons from thelight field and subsequently re-emit them. These photons are emitted isotropically into 4π
steradians. The rate of emission, or scattering rate,
Γsc =

Γ
s0
,
2 1 + δ 2 + s0

(4.12)

has a Lorentzian line shape and depends on both the intensity and detuning of the light field
as well as the natural line width of the atomic transition [110].

77
4.2

Imaging methods in the atomtronics experiment
Perhaps one of the most important aspects of cold and ultracold atom experimen-

tation is the extraction of information regarding the state of the atomic system. Common optical imaging techniques include phase-contrast, absorption, and fluorescence imaging. The governing relations of these methods were derived in the previous section (Equations (4.9), (4.11), and (4.12), respectively). The goal of this section is to provide an in-depth
treatment of the absorption imaging technique used throughout the atomtronics experiments.
Within this treatment, the use and utility of weak and strongly saturating probe beams are
discussed.
Absorption imaging of 87 Rb atoms is performed by illuminating the atomic ensemble of interest with light resonant to the 5S1/2 → 5P3/2 cycling transition, between the|F = 2i → |F 0 = 3i hyperfine states. Typically, a resonant probe beam with Ip  Isat isused to maximize the atom’s sensitivity to the probe light. The transmitted light is thenrecorded using a lens system that images the shadow of the atoms onto the sensor of a camera. However, if sufficiently optically thick, the cloud may reduce the light transmission tolevels below the dynamic range of the camera’s sensor, resulting in the loss of information.
Inspection of Equation (4.11) reveals two avenues to side-step this issue, without the useof more exotic, non-optical imaging techniques, such as scanning electron microscopy [111].
The solution lies in either detuning the probe field from resonance, thus increasing δ, or
saturating the optical transition with an intense probe field to increase s0 .
By detuning the beam from resonance, the atoms behave like a spatially varying refractive index, see Equation (4.9), which results in lensing of the probe field. The effective
refraction angle is ∼ 2λl φ0 /πd for a cloud with diameter d and peak phase shift φ0 [108].
If the refractive angle is greater than the NA of the imaging system, the refracted portionof the probe field is lost, resulting in a false absorption signal [112, 113]. In order to avoid
false signals, one can instead saturate the optical transition with an intense probe field. This

78
method is discussed further in Section 4.2.2 and was used to image trapped atoms in theexperimental work presented in Chapter 5.

4.2.1

Weak absorption imaging of atomic ensembles in free expansion

Absorption imaging techniques are widely used throughout the AMO community toimage and extract quantitative information regarding atomic ensembles. As a result, theabsorption imaging methods are well documented [42, 108, 114, 115, 116]. When imaging
with a weak probe, Ip  Isat , it is necessary to release the atoms from the trap and allow
them to expand freely for a period of time, tT OF , a method often referred to as time-offlight (TOF) imaging. The typical absorption imaging sequence consists of three images:
the first captures the probe beam and shadow due to the atoms (Ip,a ), the second contains
only the probe beam (Ip ), and a final background shot (Id ) after the probe beam has been
extinguished. While this last ‘dark’ image is not strictly necessary, it captures dark counts
inherent to the camera sensor and residual scattered light in the absence of the probe beam.
Since the beam propagates along the ẑ-direction, the intensity patterns are 2D images, e.g.
Ip (x, y). The optical depth is then calculated according toOD(x, y) = ln

Ip − Id
Ip,a − Id,(4.13)

which reflects the exponential relationship between the transmission of the probe beam andthe column density (ñσ0 = OD). From the measured OD, the atom number can then be
calculated using
N=

Apix X X
OD(x, y),σ0 x y(4.14)

where Apix is the area per pixel and σ0 the on-resonance scattering cross section. The
on-resonance scattering cross section depends on the polarization of the probe field and isgiven by σ0,π = 1.938 × 10−9 cm2 or σ0,σ = 2.907 × 10−9 cm2 for π- and σ-polarized light,
respectively [110].

79
The beauty of imaging after a period of free expansion is that the momentum distribution of the cloud is translated into the spatial degrees of freedom similar to a far-field Fouriertransform, assuming tT OF is sufficiently long. In fact, one of the most stark signatures of
the BEC phase transition in TOF images, the emergence of a bimodal spatial distribution inwhich the narrow momentum spread of the condensate sprouts from the surrounding thermal
distribution of the non-condensate atoms. Once the two components are distinguishable, it
is possible to extract the thermodynamic variables of the atomic ensemble from the spatial
variation in OD(x, y).
Using knowledge of the harmonic trap frequencies of the final confining potential, ωi ,the corresponding temperature along trap axis i is given by
σi2 ωi2m,
Ti =2kB 1 + ωi2 t2T OF(4.15)and can be extracted from the spatial distribution of thermal atoms by fitting to the Gaussian
distribution given by Equation (2.15) [108]. If a BEC is present, the thermal atoms only
occupy the ‘wings’ of the distribution, and it becomes necessary to only fit the part of theatomic distribution in which the condensate is absent [108, 115]. When imaging atoms thatwere released from a highly asymmetric trap (ω⊥  ωk ), it is not uncommon for temperature
measurements along the two axes to disagree. In this case, the corrected temperature can becalculated according to T ≈ 2τx2 /(1 + 3τx2 )Ty + (1 + τx2 )/(1 + 3τx2 )Tx , where τi = ωi tT OF [117].
At temperatures below Tc , under the TF approximation the chemical potentialmRi2 ωi2µi =,(4.16)
2 1 + ωi2 t2T OF
is extracted by fitting an inverted parabola, given by Equation (2.24) with half width Ri , tothe portion of the OD profile that contains the condensate [108]. Fits to the cloud can be
improved by using a Bose-enhanced Gaussian fit that accounts for effects due to the Bosonicnature of the atoms [118]. These methods were used extensively to analyze the evolution of
the atomic ensemble trapped in both the battery reservoir and transistor source well, which
are discussed in Chapter 5.

80
4.2.2

Strongly saturated absorption imaging of trapped atomic ensembles

Whereas imaging after a sufficiently long tT OF ensures the density of the cloud is within
the dynamic range of the sensor, the density of trapped atoms can be ∼ 10 − 100× greater.
At such large densities it is necessary to operate with probe powers Ip  Is , in order toreduce the absorption enough to completely probe the trapped ensemble,. When the probetransition is saturated, Equation (4.13) is no longer accurate. Following the work of G.
Reinaudi et al. in Reference [114], the corrected OD profile is given by!
I˜p − I˜p,a
I˜p+,
OD(x, y)sat = ln
Isat
I˜p,a

(4.17)and the (˜) indicates that Id has been subtracted from a respective image. Equation (4.17)
accounts for the saturating of the atomic transition and the reduction in the on-resonancescattering cross section. The effective scattering cross section,
σef f =

σ0,1 + s0(4.18)

indicates that the saturated absorption imaging technique enables one to probe clouds withdensities that are a factor of (1 + s0 ) larger than with a weak probe field.
The method for extracting thermodynamic quantities from in-trap atomic distributions
is similar to that of the weak probe. Given knowledge of the trapping potential, both thetemperature and chemical potential can be extracted using the relations for the spatial
density profiles of the thermal and condensed components, respectively, that were derived in
Chapter 2. In the instance that the in-trap spatial profile allows the differentiation of nc andnth , the chemical potential is determined by fitting to Equation (2.24), and subsequently thetemperature is extracted by fitting to Equation (2.18) using the fugacity calculated with thefit value for the TF chemical potential. Data extraction from images of trapped atoms in
atomtronic systems is discussed further in Chapter 5.

81
4.3

Optical trapping
The treatment of the atom-light interaction from Section 4.1 can also be applied to-wards optical trapping techniques. This section presents the optical dipole potential experienced by an atom within a light beam that is detuned from resonance. In conjunction withthe magnetic potential generated by the atom chip, optical potentials are implemented in
the atomtronics experiments to realize repulsive barriers according to the transistor system
studied in Section 2.2
Like magnetic trapping, optical trapping techniques have been studied extensively andare well understood [107]. Whereas the magnetic potential originates from the magnetic
dipole moment, the optical potential is the result of the induced polarization, p = αE, thatwas derived in Section 4.1. Therefore, the atom can be treated as a polarized medium in anelectric field and is subject to the interaction potential ∝ hpEi,
Udip =−Re(α)
I.
20 c(4.19)

Here, the intensity I = 20 c|E|2 is obtained by time averaging the electric field, and α isthe familiar value from Equation (4.6). As discussed previously, the imaginary part of thepolarizability is responsible for the absorption of photons. In the context of optical trapping,Im(α) results in heating of the atoms and is discussed in the next section. By re-expressing
the real part of the polarizability as
3π0 c3Re(α) =
ω03



Γ
Γ
+
ω0 − ωl ω0 + ωl,

(4.20)

and assuming that the detuning of the drive field is small compared to ω0 , the second termin the parenthesis can be eliminated by taking the rotating wave approximation [119]. In
doing so, the optical dipole potential experienced by ground state atoms simplifies to
Udip =

3πc2 Γ
I,
2ω03 ∆(4.21)and has spatial dependence dictated by the intensity pattern, I = I(x, y, z). In the semiclassical treatment of the optical dipole potential, the energy shift is referred to as the AC

82
Stark shift, and atoms in the ground and excited states experience shifts of the opposite
magnitude [107].
In the case of multi-level atoms, the two-state approximation is no longer accurate.
Due to the fine-structure splitting of the electronic p-orbital, one must account for boththe D1 and D2 transitions of 87 Rb atoms between 5S1/2 → 5P1/2 and 5P3/2 states at
ω1 = 2πc/(795 nm) and ω2 = 2πc/(780 nm), respectively. In the case that the detuning
is comparable to the difference in D1 and D2 transition frequencies, the relative detunings
from both transitions must be considered, and the dipole potential becomesπc2
ΓUdip =
2ω03



2
1
+
∆2 ∆1


I,

(4.22)

where ∆1,2 are the respective detunings. Within the limit ∆  ∆1,2 , Equation (4.21)
becomes a valid approximation.
From the relation for the dipole potential, one can see that the sign of the detuning
determines whether the potential is attractive or repulsive, i.e., blue-detuned light is repulsive
while red-detuned is attractive. Therefore, when designing an optical potential, ∆ is a crucial
parameter. Section 4.3.2 provides further discussion regarding the application of the optical
dipole potential towards the generation of repulsive barriers.

4.3.1

Heating in optical dipole traps

Whereas the magnetic potential from Section 3.1 is strictly conservative, the atomlight interaction includes an imaginary component that results in a finite scattering rate
(see Equation (4.12)). Due to the inherent momentum transfer in this scattering process,the atom gains an energy proportional to the recoil temperature, Trec. = (~kl )2 /2m, wherekl = 2π/λl is the wavenumber of the laser light. In the context of optical trapping ina detuned light field, the scattering rate is determined in a similar fashion to the dipole

83
potential in Equation (4.22), yielding−Im(α)
I,
~0 c


1
πc2 2 2
I.
Γ
+
=
2~ω03
∆22 ∆21

Γsc =(4.23)In terms of scaling, it is useful to note that the scattering rate is related to the dipole
potential by Γsc = Udip × Γ/~∆ [107].
In order to determine the heating rate of a trapped atomic ensemble, first consider the
total mean energy of the atoms, which can be expressed as Ē = (3/2)kB T (1 + β). Here, thefactor β ≡ Ēpot /Ēkin dictates the penetration of the atomic wavefunction into the opticalfield. For example, a slowly varying harmonic potential has β = 1, while a steep walled
box potential has β = 0. Using the relations for Ē, Trec , and Γsc , the heating rate is givenby [107]Ṫ =2/3
Trec Γ̄sc .
1+βHere, Γ̄sc is the mean scattering rate experienced by the atoms,Γ
3βΓ̄sc =
kB T ,|Udip,0 | +~∆2(4.24)(4.25)and depends on the maximum potential (height or depth), Udip,0 , as well as the temperatureof the ensemble. Equations (4.24) and (4.25) can then be used to determine the heating rate
in both red- and blue-detuned traps:
2/3
Γ Udip,0
Trec
,
1+β
~ ∆
β
Γ
=
Trec
kB T.
1+β
~∆

Ṫred =

(4.26)

Ṫblue(4.27)

From this set of equations, it is clear that steep, box-like potentials result in larger (smaller)
heating rates for red- (blue-)detuned traps.
4.3.2

Blue-detuned optical barriers

The semiclassical treatment of the atomtronic transistor presented in Section 2.2 does
not specify or depend on the shape of the repulsive barriers, so long as they provide a potential

84barrier with heights VGS or VGD and impede atom currents due to quantum tunneling. The
most straight forward barriers to realize experimentally are anisotropic Gaussian beams witha spatially dependent intensity,
I = I(x, y, z) =2P22e−2(x/wx (z)) −2(y/wy (z)) ,πwx (z)wy (z)(4.28)

where wx,y (z) are the beam waists along the x̂- and ŷ-directions, and the beam propagatesalong the ẑ-direction. Highly asymmetric optical fields that are thin along the loose, longitudinal magnetic trap axis and wide along the tight, radial trap axis are used to generate thebarriers for the triple-well transistor structure. This ensures that atoms cannot skirt aroundthe barriers, and that atom currents obey Equations (2.33). The waist defines how quickly
the beam diverges as a function of the distance along z from the focus, given by
s

2
z
,
wi (z) = w0,i 1 +zR,i(4.29)

2/λl is the Rayleigh range along
where w0,i are the barrier waists at the focus, and zR,i = πw0,ithe i-th direction [120]. Therefore, the tighter the focus, the faster the beam diverges as itpropagates away from the focal plane. For the beam to serve as a constant height potential
across the radial width of the magnetic trap, the decrease in beam intensity due to its divergence must occur over longer length scales than the increase in the magnetic trap magnitude.
One final point of interest relates back to the heating rate given by Equation (4.27). If the
optical barrier waist along the longitudinal trap axis is sufficiently small, i.e., the intensity
increases rapidly compared to the thermal width of the atoms in the magnetic trap, thenthe barrier appears to be a step function and the heating due to light scattering becomes
negligible, as β → 0.

85
4.4

A high-resolution microscope system for imaging trapped atoms andprojecting optical potentials
The microscope system used for all atomtronics experiments, is schematically illus-

trated in Figure 4.1.1 Consisting entirely of commercially available components, due to thetransparent regions of the atom chip, the imaging and projection system achieves high numerical aperture performance without the need for custom in vacuo optics. The use of an
infinity-corrected objective lens and a beam splitter allow for simultaneous in-trap absorption
imaging and the projection of optical potentials. Detailed descriptions of individual components are provided in sections 4.4.1 and 4.4.2. The entire microscope system, including thecamera and optics, are mounted on a precision XYZ translation state (Z Stage: Newport
MVN80, XY Stage: Optosigma 123-0400) that is used to position the microscope relative tothe atoms in the vacuum chamber.
At the heart of the microscope system is the primary objective. The NA of the primaryobjective dictates the feature sizes resolved by both imaging and projection beam paths.
Throughout the experimental work contained in this dissertation, a 0.6 NA Zeiss LD Plan
Neofluar 40× objective was used. This off-the-shelf biological microscope objective wasselected for its high-resolution performance and integrated adjustment collar that corrects forvariable thickness coverslips. This second feature is crucial, as it allows for the compensationof spherical aberration introduced by the finite thickness and refractive index of the chip
window.

4.4.1

Through-chip imaging

Atoms confined in the magnetic potential generated by the atom chip are imaged
using either fluorescence or absorption imaging techniques. The probe beam path for these
two methods are illustrated in Figure 4.2. The fluorescence imaging probe is incident on
1

The work presented in the following sections is partially published in Reference [84] and represents thefirst demonstration of through-chip imaging and projection of optical potentials.

f = 75 mm

0th-Order
Block

Fourier
Plane

Model KCInFocus

InfiniProbe TS-160

NA = 0.6
40x

Probe

NA = 0.6
40x

Atom
Chip
VacuumChamber

BS

BPF

Figure 4.1: Simplified schematic of the optical system used to conduct atomtronics experiments. Imaging and projection beampaths are distinguished by red and blue marginal rays, respectively. The expanded region around the atom chip shows the
projected optical potential and in-trap absorption imaging probe light as they pass through the plane of the atoms (dark red
spatial distribution). System components and their respective functionality are described in the text.

AOD

AWG + RF
Amplifier

to
Andor
iXon

86

a)

c)

87

To AndorTo
Basler1 cmb)
To Andor

θM

1 cm
1 cm

Figure 4.2: a) Schematic of the fluorescence imaging probe beam path (also used for TOF
absorption imaging, with a Basler scA1400 - 17fm CCD camera). b) An expanded viewof the cell, chip, and microscope objective during the fluorescence imaging process that
illustrates the collection of scattered photons. Assuming proper atom chip design, discussed
in Section 3.2.2, the marginal ray half-angle (θM ) of the imaging system is dictated by the
NA of the primary objective. c) Schematic of the in-trap absorption imaging probe beam
path in the double MOT cell. The beam is focused through the pinhole separating the 2D
and 3D MOT chambers.

88
the atoms from the side, parallel to the surface of the atom chip, such that the scatteredphotons are imaged by the microscope system rather than the probe light. By contrast, theabsorption probe beam is oriented vertically in order to image the shadow of the atoms. In
both configurations, light is collected by the Zeiss objective and passes through one port ofan AR coated, 50:50 plate beam splitter (Edmund Optics; 45-853) that connects the imaging
and projection beam paths. The transmitted light is then imaged onto an Andor iXon 897E#BV electron multiplying charge-coupled device (EMCCD) by an Infinity Photo-Optical
Model KC InFocus lens system. The resolution limit is given by the Rayleigh criterion,
dmin =

0.61λl,NA(4.30)

where λ is the probe light wavelength. Therefore, the 0.6 NA lens has a minimum resolvabledistance between features of 793 nm. Through-chip optical resolution was tested using a1951 USAF hi-resolution target (Edmund Optics; 58-198) that includes features as small as
780 nm. With the correction collar set to 0.42 µm, i.e., the thickness of the chip, diffraction
limited optical performance is achieved. Coupled with the Andor, which has square pixels,
16 µm on a side, the 40× magnification Zeiss objective followed by the unity magnification
Model KC InFocus lens system results in an object-space pixel size of 0.4 µm.
Fluorescence collection efficiency
Early in the development of the microscope system, fluorescence imaging was used toimage atoms in the chip trap. To acquire a fluorescence image, a resonant probe beam is
flashed on at a set intensity for a time τp , during which photons are scattered isotropically at
a rate according to Equation (4.12). The ratio of the solid angle subtended by the primaryobjective, Ω, to the photon emission solid angle, Ωmax = 4π steradian, is
ΩR ≡

Ω
Ωmax

=

1 − cos(θM ),2(4.31)

where θM is the marginal ray angle of the primary objective. For the 0.6 NA objective,θM = arcsin(0.6), and ΩR = 0.1; thus, the Zeiss objective collects only 10% of the total

89
scattered light. The collected light is then subject to losses in the imaging system itself.
These include 85% transmission through the Zeiss objective at 780 nm, 50% transmission
through the beam splitter, the 99% transmission of the Model KC lens tube, and 80% quantum efficiency of the Andor. Overall, the fluorescence imaging process is ∼ 3.4% efficient. To
improve the signal to noise of the images, the Andor EMCCD is cooled to −70◦ C to reducedark counts, and the electron multiplying gain setting is set such that several photoelectrons
are generated per photon incident on the sensor [121].
Assuming the net efficiency of the optical system is fixed, the next best way to increase
the signal is to scatter more photons by increasing the probe pulse duration. However, despite
the tight radial confinement of the magnetic chip trap, the recoil imparted to the atoms byeach photon results in a random walk and a root-mean-square (RMS) displacement,rNP
vrec ∆t,(4.32)
rrms =
3
where NP is the number of photons scattered, vrec ≈ 5.88 mm/s is the recoil velocity, and∆t is the probe duration [108, 110]. This effect is referred to as recoil blurring and poses a
problem when the spatial position of atoms within the magnetic trap is of interest.
Fluorescence imaging resolution tests using trapped atoms
In order to determine the minimum resolvable feature size in the trapped atom cloud,fluorescence imaging was performed by exposing the atoms to an interfering probe beam.
Two s-polarized beams were overlapped at the location of the atoms with an included halfangle of θI . This configuration creates a probe field with a standing wave pattern withsinusoidally varying intensity with a spatial period of of ΛI = λl /2sin(θI ) along the loose
axis of the magnetic trap. Due to the high in-trap atomic density, the probe field was
detuned by several linewidths of the optical transition, Γ ≈ 6 MHz, in order to probe theentire spatial extent of the trapped ensemble. Since fluorescence imaging detects only thescattered photons, lensing of the probe beam is not a concern.

90

Radial Pos. (µm)

a)
20

0

-20
-100

-80

-60

-40

0

20

40

60

80

0

Longitudinal Position (µm)b)
Photon Counts

-20

100

350

60
40
20
0
-100

-80

-60

-40

-20

0

20

40

60

80

100

Longitudinal Position (µm)

Figure 4.3: a) Superposition of 10 in situ fluorescence images exposed by a standing wave
probe field with ΛI ≈ 2.5 µm. b) An averaged longitudinal slice of the image shown in (a)illustrates the fringe contrast.

Figure 4.3(a) shows an accumulation of 10 fluorescence images taken with a detuningof ∆ = 4Γ and θI ≈ 18◦ , resulting in a fringe spacing of ≈ 2.5 µm. The fringes are clearly
evident, as seen in Figure 4.3(b), which shows the longitudinal profile of the cloud obtainedby averaging the 2D image along the radial axis. Fourier analysis of the spatial frequency
content in Figure 4.3(a) reveals the periodicity of the probe beam imprinted on the atoms
and the resulting spatial variation in the number of scattered photons. Figure 4.4 shows both
1D and 2D Fourier transforms of the accumulated image. The 1D Fourier transform along
the longitudinal trap axis reveals spatial frequency content at ±0.38(1) µm−1 , corresponding
to a spatial period of Λ = 2.63(9) µm. Repeating the measurement using interfering probe
fields with a standing wave period nearer the resolution limit of the imaging system did notprovide measurable spatial frequency content.
When imaging an extended object, the limiting factor is not the transverse, but the-0.6-0.4 -0.20
0.2 0.4
Spatial Frequency (µm-1)

b)

0.6

Spatial Frequency (µm-1)

Magnitude (arb)

a)

0.4

0.6

91

0.2
0
-0.2
-0.4
-0.6
-0.6 -0.4 -0.20
0.2 0.4
Spatial Frequency (µm-1)

0.6Figure 4.4: Fourier spectra of the image in Figure 4.3(a). a) Averaged 1D Fourier transforms
along the longitudinal (solid, blue) and radial (dotted, red) trap dimensions. Peaks centeredat ±0.38 µm−1 indicate a fringe spacing of ∼ 2.63(9) µm. b) Corresponding 2D Fouriertransform of the average image.

longitudinal resolution, i.e., the depth of field. An objective operating at a given NA has adepth of field,
dDOF =λl.
NA2(4.33)For the 0.6 NA Zeiss objective, dDOF = 2.17 µm. The effect of finite depth of field is shownin Figure 4.3(b), where the fringe pattern contrast is lowest in the region where the radialextent of the cloud is largest (∼ 15 µm). In this region, the contrast is . 15%, which isapproximately the ratio of the depth of field to the cloud thickness.
In order to improve the resolving power in the cloud using fluorescence imaging techniques, it is necessary to either lower the NA of the primary objective or reduce the cloud
thickness along the imaging direction. Lowering the NA of the objective is unfavorable, asboth the collection efficiency and transverse resolving power suffer. The finite extent of thecloud in the image is the product of two factors: trap frequency, which determines the initialradial extent of the cloud, and recoil blurring. The first factor can be improved by increasing

92
the radial frequency of the magnetic trap; however, this leads to increased trap heating. The
second factor can be improved by reducing the average distance an atom is displaced as aresult of scattering multiple photons. Ultimately, fluorescence imaging was abandoned infavor of absorption imaging. The two methods are compared in the following section.
Comparison of in situ absorption and fluorescence imaging techniques
Absorption imaging is attractive due to the large number of photons collected by
the detector relative to fluorescence imaging since the probe beam is imaged directly, as
shown in Figure 4.2(c). In conjunction with a saturating probe intensity, τp can be madesignificantly shorter, without suffering a loss of signal. By comparison to the fluorescenceimaging sequence, absorption imaging is more complex. In order to ensure Ip and Ip,a containthe same probe beam, i.e., displacement in structure on the probe beam is minimized, theAndor is operated in fast kinetics mode and only a portion of the EMCCD exposed to theprobe light. Andor iXon 897 cameras come stock with an EMCCD array that is 512 × 1024
pixels, with an opaque mask shielding half of the array for storage purposes. The camera
reads out pixels from the active array at 10 MHz with 0.9 µs shift time per row. Therefore,
in order to readout the entire 512 × 512 active array into the storage array, the minimum
time between exposures is ∼ 27 ms. This is far too long of a time between exposures andresults in unwanted fringing across the absorption image due to shifting interference patterns
caused by vibrations within the system. To reduce the time between shots, two thirds ofthe active detector region are masked off and the Andor is run in fast kinetics mode. The
minimum time between shots is ultimately limited by the vertical shift speed, which for 170
rows is ∼ 170 µs. With this time delay, interference effects are reduced below the level ofother noise sources. As in fluorescence imaging, the Andor EMCCD is cooled to −70◦ C to
reduce dark counts, but the electron multiplying gain is reduced such that a single photon
generates a single photoelectron, where the sensor has the highest signal to noise, as theprobe beam contains many more photons.

93

Radial Pos. ( µm)

a)
10
0
-10
-60

-40

0

20

Longitudinal Position (µm)b)

Radial Pos. ( µm)

-20

40
0

60

40
0

60

450

10
0
-10
-60

-40

-20

0

20

Longitudinal Position (µm)

3Figure 4.5: a) Accumulation of 20 fluorescence images acquired using a 20 µs pulse of 4Γ
detuned probe light. b) Accumulation of three absorption images with 6 µs pulses andresonant probe light. Atomic ensembles were prepared under identical circumstances inboth (a) and (b), and were imaged with probe light below the saturation intensity.

To combat the issue of recoil blur, the AOM that is used to quickly shutter the probe
beam is triggered using a circuit that generates a train of TTL pulses. This enables probe
pulses shorter than the minimum TTL pulse of 20 µs of the control system DAC. Pulses
as short as 1.4 µs are possible, but little cloud size variation is detected below the typicalprobe pulse time of 4 − 6 µs. Figure 4.5 illustrates the substantial difference in the recorded
atomic distribution between fluorescence and absorption imaging techniques. The ability toimage the in-trap spatial distribution of atoms is an integral part of data collection in theatomtronics experiments described in Chapter 5.

4.4.2

Optical potential projection system

The projection system can be broken into two decoupled sub systems, one that generates the desired optical pattern and another that images the optical pattern and projects the
optical field onto the atoms with a given magnification. Figure 4.6 illustrates the two systems used to generate repulsive optical potentials. The InfiniProbe TS-160 lens system, also

94
from Infinity Photo-Optical, is integrated into the microscope system. The remaining optics
are mounted on a precision XYZ translation stage identical to the one that positions the
microscope system. Optical systems used to generate static and dynamic optical potentials
are discussed in the sections that follow.
Initial alignment of the optical potential is straightforward, owing to the integrationof the imaging and projection optical system. First, the microscope is positioned to image
one of the wires on the vacuum side of the atom chip near the center of the window. With
the wire in focus, the projection system XYZ stage is used to center and focus the projected
optical potential in the field of view, using the video mode of the Andor. The magnificationof the projected optical pattern is controlled via the adjustment collar on the InfiniProbe
TS-160 and its working distance, which is set by the position of the 75 mm achromat.
Information regarding the magnification of the InfiniProbe is available in its data sheet.
Finally, the projection and microscope system translation stages are both displaced thenecessary distance to move the microscope system to the plane of the trapped atoms.
Static optical potential generation
The first projection experiments conducted in the apparatus consisted of the generationof an array of asymmetric beams, like those described in Section 4.3.2, to split a trapped
BEC into multiple wells. Upwards of 100 mW of blue detuned, 760 nm light is derived from a
Topical DLX110 external cavity diode laser. The light is spatially filtered by an optical fiberand coupled into the optical system shown in Figure 4.6(a), which was used to generate five
repulsive optical barriers spaced evenly along the longitudinal axis of the atom chip trap. A
5× cylindrical telescope elongates the initial 1.5 mm diameter beam along the radial extent
of the atoms, while a transmission grating creates multiple diffracted orders. The diffraction
pattern is shown in the inset of Figure 4.6(a).
A 75 mm achromatic doublet lens Fourier transforms the output of the transmission
grating to generate the desired optical potential pattern at the front focal plane of an Infinity

M

M

M

HWP

Terminator
Input

f = 15.3 mm
AsphereSingle
Mode
Fiber

M

AOD

90:10 BS

Dump
Port toServo PD

Transmission
Grating

5x Cylindrical f = 8 mm
AsphereTelescope

SwitchSwitchAmplifier2W

VCO orAWG

VariableA�enuator

InfiniProbe TS-160

InfiniProbe TS-160

VCO orAWG

FourierPlane

Fourier
Plane

VariableA�enuator

0th-Order
Block

Amplifier2W

f = 75 mm
Achromatf = 75 mm
AchromatSingleMode
Fiber

Figure 4.6: a) Diagram of the optical system used to create multiple static potential barriers. b) Optical system used in atomtronic
experiments that enables dynamic control of multiple optical barriers. Mirror (M), half wave plate (HWP), photo-diode (PD).

b)

a)

95

96

Radial Pos. (µm)

a)
20

0

-20
-100

-80

-60

-40

-20

0

20

40

Radial Pos. (µm)

800Longitudinal Position (µm)

b)

60

100

1

20

0

-20
-100

-80

-60

-40

-20

0

20

Longitudinal Position (µm)

40

60

0

80

100

300

Figure 4.7: a) An array of repulsive barriers generated by a highly asymmetric Gaussian
beam and a 1D transmission grating. The intensity pattern is recorded by imaging thepartial reflection from the vacuum side of the atom chip window. The false color scale isnormalized to the maximum intensity. b) Fluorescence image of a BEC containing ∼ 30×103
atoms trapped in the atom chip trap and exposed to the projected optical pattern in (1).
Areas of reduced atomic density indicate the position of the repulsive barriers.

Photo-Optical InfiniProbe TS-160 lens system. The InfiniProbe images the optical patternof the barriers through the reflected port of the 50:50 beam splitter and subsequently thebarriers are projected onto the atoms by the Zeiss objective. This setup was used to demonstrate the first optical control of atoms through an atom chip [84]. Figure 4.7 shows theresulting optical potential and a fluorescence image of a BEC split into multiple clouds.
Dynamic optical potential generation
Our atomtronics experiments described in the next chapter require dynamic control of
the optical potential in order to prepare the initial state of the system. Figure 4.6(b) showsa diagram of the optical system used in said experiments. As in the static potential, 760
nm light from the Topic DLX110 is spatially filtered and coupled into the optical system by

97
a single mode optical fiber. The shape and position of the optical barriers is controlled by
the RF waveforms that drive a 2D acousto-optic deflector (AOD) from IntraAction Corp.
(DTD-6010RH29). This AOD has a center frequency of 60 MHz and deflection bandwidthof 32 MHz. With an input beam diameter of 3.6 mm, the time bandwidth product of thedeflector yields ≈ 170 resolvable spots with an access time of 5.4 µs, i.e., pattern update
frequency of ≈ 185 kHz. Power diffracted into the +1, +1 order of the AOD forms thedesired optical potential. The RF drive signals are generated by either a voltage controlled
oscillator (VCO) or a two-channel arbitrary waveform generator (AWG), (Lecroy ArbStudio:
125 MHz maximum frequency, 16 bit resolution, 1 GS/sec sample rate). To generate theradial extent of the optical barrier, the AWG RF frequency is rastered about the center
frequency of the AOD. The raster frequency is set to 20 kHz, which is considerably larger
than the radial magnetic trap frequency such that the atoms experience the time averaged
intensity of the rastered field. The longitudinal position of each barrier is then set by thefrequency of two VCO outputs combined using a resistive splitter-combiner.

Chapter 5
Experimental demonstration of atomtronic devices

This chapter presents the results of two experiments that explore the behavior of atomtronic analogues to electronic circuit elements. The first is a power supply that sources anatom current to a connected load and is driven by the thermodynamic potentials of a finitetemperature BEC, namely a battery. The second experiment utilizes the battery to drive
an atom current through a triple-well potential to study the steady-state behavior of theatomtronic transistor described in Section 2.2. To begin, an overview of the experimental
procedure used to prepare the finite-temperature BEC that powers these atomtronic devices
is provided. Section 5.2 provides a detailed description of the atomtronic battery, which isrealized in a double-well potential and analyzed according to an electronic Thévenin equivalent circuit. Important experimental and numerical techniques relevant to both the battery
and transistor are also introduced. Finally, experimental results are presented that validate the steady-state behavior of the triple-well atomtronic transistor system presented in
Section 2.2.3.
5.1

BEC production and preparation for atomtronics experiments
Due to the standardization of vacuum chamber and atom chip technologies, BEC pro-duction is also standardized throughout the experiments in our lab and described thoroughly
in the theses of previous students [10, 12, 13]. A detailed description of the early stages of
the BEC production cycle that lead up to capturing atoms in the chip trap, along with

99
schematics for the optical, electrical, and control systems of the atomtronics apparatus canbe found in the thesis of E. A. Salim [12]. The final stages including the initial process of
capturing atoms on the chip, trap compression, and forced RF evaporation in the V2 window
chip (see Figure 3.7) are described here.
After laser cooling the atoms to ∼ 10 µK, they are optically pumped into the |F =2, mF = 2i state, and captured in a magnetic quadrupole trap generated by external coils.
The atoms are then transferred to a location just below the atom chip window using aquadrupole ladder scheme. The initial side-guide IP trap is loaded directly from the coilgenerated quadrupole field by quickly shutting off the current in the external coils (. 600 µs)
and subsequently ramping on the chip wire currents to IM = 3 A and IH = 4.75 A alongwith the necessary bias fields, B(x, y, z) = (14, −24, 0) G, over 10 ms. The initial chip trap
is located 230 µm below the chip and has frequencies ωload (x, y, z) ' 2π × (80, 270, 270)
Hz. This method captures ∼ 20 − 30 × 106 atoms in the initial chip trap at a temperatureof ∼ 50 µK.
The chip trap is then compressed in order to increase the collision rate of atoms in thetrap and improve the efficiency of forced RF evaporation. After a short, 50 ms RF sweep from40 → 28 MHz in the initial chip trap, the wire currents are ramped to IM = 2.5 A, IT = 0.6 A
and IH = 2.5 A while simultaneously ramping the bias fields to B(x, y, z) = (3.6, −31, 0) G.
The resulting trap is located 130 µm below the chip and has frequencies ωevap (x, y, z) '2π × (280, 1500, 1500) Hz. This compression stage occurs over 400 ms, during which the
RF knife is ramped from 28 → 19 MHz. The remainder of the evaporation cycle is completed
in four stages that are ∼ 300 ms long by linearly decreasing the RF knife frequency as follows:
19 → 11 → 7 → 5 → 3.8 MHz. This scheme results in a nearly exponential decay of the
RF knife frequency that allows for optimal rethermalization [122, 123]. Prior to the final RF
evaporation sweep, the T-wire current is ramped to zero in order to match the longitudinal
confinement of the split-guide trap (described below) and avoid excitations of the cloud. At
this point in the sequence, a finite-temperature BEC has been produced with a temperature

100
and chemical potential that are controlled via the final RF frequency.
The procedure for loading the finite-temperature BEC into the final split-guide chip
trap is much more efficient than the initial chip trap loading stage, since the side- and splitguides are well mode matched. By linearly ramping the chip wire and bias coil currents
from the final evaporation trap parameters to those that generate the split-guide trap, themagnetic field is smoothly deformed into the configuration desired for the atomtronics experiments. Parameters for the split-guide trap include chip wire currents IG = 2 A and
IH = 0.5 A along with bias fields B(x, y, z) = (3.2, 0, −21) G. The resulting cigar shaped
magnetic trap is located 130 µm below the chip and has trap frequencies ωexpt (x, y, z) '2π × (67, 1500, 1500) Hz.1 Transfer to the split-guide trap is accomplished with near-unity
efficiency and little to no heating of the atoms. An image of an initial ultracold atomic
ensemble was shown in Figure 4.5(b). This point in the experimental cycle serves as theinitial condition for the experiments described in the remainder of this chapter.
5.2

The atomtronic battery system
Circuits, both electronic and atomtronic alike, fundamentally require a source of power

to operate. Atomtronic batteries have been proposed in both open quantum systems constructed of reservoirs with different chemical potentials connected by a lattice potential [21,24], as well as mesoscopic systems containing a finite-temperature Bose-Einstein condensate(BEC) that sources condensate atoms to the connected load [124]. The work presented
in this section describes the realization and characterization of an atomtronic battery that
stores energy in the form of a finite-temperature BEC and sources an atom current to asimulated load element. A schematic of the experimental system is shown in Figure 5.1.
1

The trap frequencies of the split-guide trap used during the atomtronics experiment were measured
by exciting the center of mass mode along the longitudinal or radial trap axes. The radial slosh mode isexcited by modulating the z-bias coil current with a low amplitude sinusoid and measuring the temperatureof atoms in the trap. The induced heating near resonance can be understood under the discussion of chipheating in Section 3.4.1. The longitudinal mode is excited by displacing the atoms with an optical barrier
and measuring the harmonic center of mass motion after the barrier is removed.

101

Battery Reservoir

Load

V (kHz)40
Terminator

20
0

-60

-40

-20
0
20
Longitudinal Position (µm)

40

0

60

1

2

Figure 5.1: Schematic of the double-well atomtronic battery potential: The top panel shows
the longitudinal potential energy landscape of the hybrid magnetic and optical potential. A
resonant “terminator” beam removes atoms that flow into the load well from the system.
The bottom panel shows a false color, in situ absorption image of atoms occupying bothwells of the potential.

102
5.2.1

Charging the battery

The first step in realizing an atomtronic battery is loading the atomic ensemble into
the reservoir well, i.e., charging the battery. To do so, the cigar shaped magnetic trap
is transformed into a double-well potential. In the process, the finite-temperature BEC
is compressed into one of the two wells, labeled “Battery Reservoir” in Figure 5.1. Thisis accomplished by ramping on a repulsive barrier, with a height VS = 100 kHz  kB T ,at the edge of the longitudinal spatial extent of the cloud and sweeping it adiabaticallyto the center of the magnetic trap, as shown in Figure 5.2(a)-(c). When converting fromtemperature units to Hz, note that 1 kHz ≈ 50 nK. Dynamic control over the position andheight of the barrier are achieved by adjusting the power and frequency of the RF signal
driving the AOD described in Section 4.4.2. In this way all of the atoms are loaded into
the reservoir well and the battery is considered charged. The stored energy of the battery isthen characterized by the thermal and chemical potentials of the trapped atomic ensemble.
With the barrier present the reservoir well is approximated as a half-harmonic wellwith ωx,batt ≈ 2 × ωx,expt = 2π × 134 Hz and a density of states that is reduced by half
relative to the harmonic magnetic potential. The half-harmonic well approximation is used
to determine the chemical potential in the TF limit. Under this approximation, the reservoir
well chemical potential differs by < 5% from the corresponding values determined from 3D
simulations of the Gross-Pitaevskii equation for a BEC in the ground state of the hybrid
magnetic and optical potential. A plot of the relative error in the chemical potential versus
the number of condensed atoms in the reservoir well is shown in Figure 5.3.
5.3

Impedance matching in atomtronic systems
Impedance matching is a fundamental concept in electrical system design, especiallyin power transmission applications. The same can be said for atomtronic systems that
supply an atom current to a connected load. Some fraction of the current will be reflected

103

a)
b)
VS

Sweep

c)
d)

Lower
VB

e)

-60

-40

-20
0
20
Longitudinal Position (µm)

40
0

1

60
2

Figure 5.2: Battery charging procedure: a) Atoms begin in the split-guide magnetic trap.
b) An optical barrier is turned on at the far-right extent of the atoms and swept towards
the center of the magnetic trap. c) Atoms trapped in one half of the magnetic trap after thebarrier is swept. d) The barrier height is the reduced to connect the battery to the load. e)
Battery output after 30 ms of discharge with VB = 30 kHz.

104

5

% error in μ

0
-5
- 10
- 15
- 20

0

1000

4000
3000
2000
Condensed Atom Number

5000

Figure 5.3: Error in the TF chemical potential in a half-harmonic well, relative to resultsfrom GPE simulations. Here the % error in µ is defined as (1 − µT F /µGP E ) × 100. The grayshaded region indicates the range of condensed atoms in the battery reservoir under typicalexperimental conditions.

105
back towards the current supply, depending on the characteristics of the load. Therefore,
the input impedance of the load circuit must be matched to the output impedance of thecircuit element supplying the current in order to achieve maximum power transfer. Whereas
electrical impedance, Z(ω), depends on the signal frequency, the transmission spectrumof massive particles through a given potential is energy dependent, i.e., Z(E) [65, 125].
Examples of Bragg-matching to a periodic potential have been experimentally studied [66,67]. The impedance of non-trivial potentials such as the Gaussian barriers used in the battery
and transistor experiments can be calculated numerically using the method of generalized
impedances [65, 126].

5.3.1

An atomtronic terminator

In order to study the behavior of the atomtronic battery rather than the reflection
properties of the connected potential, it is necessary to simulate an impedance matchedload. Therefore, during the evolution of the atomtronic battery, the load well is illuminatedwith laser light (labeled “terminator” in Figure 5.1) that is tuned to atomic resonance suchthat any atoms reaching the load well are removed from the trap. This is accomplished by
optically pumping atoms from the magnetically trapped |F = 2, mF = 2i state to any of
the untrapped |F = 2, mF ≤ 0i states, as shown in Figure 5.4. To illustrate the effectof the terminator beam, the harmonic potential in the load well has been flattened out in
Figure 5.1, indicating that atoms entering the load region are effectively coupled to vacuum
modes and escape the system. Therefore, the terminator beam serves an analogous purposeto termination in RF electronics, where proper termination impedance matches the device
output to the load circuit in order to eliminate signal reflection and possible interference.
Furthermore, the terminator beam introduces localized dissipation that enables the study of
non-equilibrium dynamics in atomtronic systems as they evolve.
In the experimental apparatus, the terminator beam is projected along the same beam
path as the optical barriers, as shown in Figure 4.6, and has a typical power of ∼ 10−12

106

a)
Terminator BeammF

Battery Reservoir
VB
urr
mC

T, μ

Ato

ent

mF

=2

=1

mF = 0m

F=

Energy

-1

=
mF
-2

x-Positionb)

5 P3/2, F' = 2
m'F g'F μB BTRAP

-1

0

1

2

Δν

m'F = -2

-2

-1

0

mF = 2mF gF μB BTRAP

5 S1/2, F = 2

Figure 5.4: a) Schematic illustrating the role of the terminator beam. Atoms that flow into
the right-hand load well of the battery system are optically pumped by the terminator beam
into states that experience zero or repulsive magnetic forces and subsequently escape thetrap. b) Energy level diagram illustrating the optical pumping scheme. The terminatorbeam is red-detuned from the |F = 2i → |F 0 = 2i optical transition by ∆ν, the average
Zeeman splitting of mF states.

107
W. The quantization axis for the optical transition is determined by the bias field along
the x̂-direction; therefore, the terminator beam must be aligned to have a non-zero fieldprojection along the longitudinal trap axis. Displacing the terminator beam, which has afull width at 1/e of 16 µm, 32 µm into the load well ensures that only atoms in the load well
are out-coupled.
5.4

Characterizing the atomtronic battery
The battery is characterized via observations of the atomic ensemble in the reservoir

well and the emitted atom current while the battery supplies current to the load. The
discharge behavior was recorded for several emission rates dictated by the height of thebarrier, which is varied from VB = 20 − 70 kHz, in 10 kHz increments between data sets.
Throughout the following analysis the unit convention of Ref. [124] is followed, and both
energies and particle currents are reported in units of Hz. Evolution of the battery reservoir
is measured using both TOF and in situ absorption imaging techniques. Thermal, Nth ,and condensed, Nc , atom numbers as well as the ensemble temperature are extracted fromtwo-dimensional bimodal fits to the spatial density profile of atoms in TOF images (seeSection 4.2.1).
One such dataset is presented in Figure 5.5, which shows the time evolution of the reservoir ensemble during battery discharge with VB = 50 kHz. In this data set, the reservoir has
an initial population of N = 19.03(97) × 103 atoms, of which Nc = 1170(84) are condensed,
corresponding to chemical and thermal potentials of µ = 2.86(8) kHz and T = 678(24) nK,
respectively. These thermodynamic potentials provide a gradient between the reservoir andload wells that drives an initial atom current of I = 334(22) kHz. As the battery reservoir
discharges, it cools. After ∼ 30 ms a nearly constant chemical potential of µ = 3.86(16)kHz is reached and sustained for the remaining ∼ 50 ms. After 80 ms the thermal potential
energy has decreased to T = 440(11) nK, as shown in the inset of Figure 5.5, and the current
to I = 60.8(7.7) kHz. Battery discharge with other barrier heights exhibits similar behavior,Temperature (nK)108

Atom Number (x10 3)

20
15

N

10

Nth

5
0

675
575
475
375

0

20

40

60

80

Nc
0

20

40

60

80

Time (ms)Figure 5.5: Time evolution of the total (black squares), thermal (red circles), and condensed(blue triangles) atom numbers in the reservoir for VB = 50 kHz. The temperature of thereservoir ensemble is shown in the inset. Error bars represent the standard error of the meanfor five experimental realizations. Solid lines show the results from numerical calculation of
the reservoir population described in Section 5.4.1.

109with lower values of VB resulting in a more rapid depletion of both the reservoir atom number
and temperature.
In-trap images are used to probe the spatial distribution of the atoms emitted into
the load well and also to calibrate the height of the repulsive barrier. By extinguishing theterminator beam ∼ 4 ms prior to imaging (∼ 1/4 × 2π/ωx,expt ), atoms emitted into theload well are allowed to interact with the magnetic potential for a time equal to roughly
one quarter of the longitudinal trap period. As a result, the images serve as a rudimentary
spectrum analyzer, translating the displacement of atoms at their classical turning pointin the load well into a measurement of their longitudinal energy. An example is shown in
Figure 5.2(e). The longitudinal position and spatial distribution yield information regarding
the height of the barrier and the momentum spread of atoms entering the drain well, respectively. The longitudinal momentum spread of the battery output current is determinedby fitting a Gaussian to the longitudinal spatial distribution that peaks at ∼ 40 µm, whichcorresponds to a spread in kinetic energy, ∆E. We find that ∆E ≈ kB T , indicating that thecurrent emitted into the load has an energy spread on the order of the temperature of theensemble in the battery reservoir.

5.4.1

Modeling the battery

Presuming that the contribution to the total atom current due to tunneling is negligible,
only atoms in the high energy tail of the distribution can contribute to the output current.
Therefore, the system relies on thermalizing collisions to repopulate energy levels from whichatoms are emitted. Given the tight radial confinement of the magnetic potential, the reservoir
well is in the hydrodynamic regime along the longitudinal dimension of the trap when σ  l,where σ is the thermal width of the reservoir ensemble from Equation (2.15), and l = 1/nσs
is the mean free path between collisions [43, 127]. Under typical experimental conditions
σ/l ∼ 100, and the collision rate is sufficiently high for the reservoir ensemble to remain in astate of dynamical quasiequilibrium [128]. Therefore, the atomic ensemble can be described110
in terms of equilibrium thermodynamic variables.
The discharge behavior of the atomtronic battery is modeled using rate equations forboth the atom number and energy stored in the reservoir well:dN= −γNth f (η, µ, T ) ,
dt
dE
dN
=
(η+κ) kB T + βC,
dt
dt

(5.1)
(5.2)

which are derived from Equations (2.33) and (2.36), respectively. The magnitude of theparticle current is governed by the probability that an atom has sufficient energy to traverse the barrier, f (η, µ, T ) =1/exp(η − µ/kB T ), where η = VB /kB T is referred to as thetruncation parameter. This probability is scaled by the quantity γNth , which encapsulates
the rate at which the reservoir atoms encounter the barrier. Here, the collision rate, γ, isgiven by Equation (2.34). The energy current on the other hand is given by the product ofthe particle current and the average energy removed per particle given by (η+κ) kB T . The
κ-parameter indicates the average energy removed from the reservoir in excess of the barrierheight energy, by each particle traversing the barrier. The value of κ is on the order ofunity and depends on the dimensionality of the ensemble’s momentum distribution relative
to the allowed escape trajectories [53, 60]. For the potential studied here, κ ∼ 3, whereas
in isotropic evaporation, e.g., forced-RF evaporation, κ ∼ 1 [122]. Determination of theκ−parameter is discussed further in the following section.
To account for additional heating or cooling effects external to the flow of atoms, therate βC is included. This term does not affect the total atom number, but transfers energy
to or from the reservoir ensemble according to the heat capacity, C, (see Equation (2.32))
and the total external heating or cooling rate, β. In the context of this demonstration, only
heating due to technical trap noise2 is included; however, this term could also be used to incorporate heat generation associated with irreversible processes during circuit operation [27].
2
The heating rate due to technical noise, measured with VB = VS , increases the temperature of thereservoir ensemble by ≈ 80 nK over an 80 ms period.

111
Determining the κ-parameter
The excess energy parameter, κ, is determined from battery discharge data across a
range of truncation parameters according to the relation

κ=


3
+ δ3D (α + 1) − η,
2

(5.3)

where δ3D = 3/2 for a three-dimensional harmonic oscillator [122]. Here, α represents theenergy transfer efficiency,
α=

Ṫ /T
.
Ṅ /N

(5.4)

The time derivatives of the temperature and number, Ṫ and Ṅ , respectively, are extracted
from the time evolution of the battery reservoir, shown in Figure 5.5. Values of κ forvarious truncation parameters are shown in Figure 5.6. Also shown are results from DSMC
simulations3 for the experimental conditions. The DSMC method is well-documented [129],and has been used to study collisions and evaporative processes in cold and ultracold atomic
systems [128, 130, 131]. The horizontal and vertical positions of each point indicate η̄ andκ̄, the mean values of η and κ across the measured discharge time, respectively. Similarly,
error bars indicate the range of η and κ values during discharge and incorporate statisticaluncertainty from five experimental realizations. The horizontal dashed line at κ = 2.9 reflects
the result of molecular-dynamics simulations reported in Reference [60]. As the truncation
parameter is decreased, the κ-parameter determined from experimental data decreases to∼ 2.34 at η̄ = 1.77, following the trend of the DSMC results. The drop in κ at lower
η reflects the increased coupling between the transverse modes of the reservoir well andthe load, allowing less energetic atoms to traverse the barrier. Therefore, decreasing thebarrier height increases the magnitude of the current, but each emitted particle possesses
less energy in excess of VB . The interplay between the magnitude of the current and κ playsa determining role in the power output of the battery, which is described in Section 5.4.2.
3

Simulations of the κ-parameter were performed with code developed following References [129] and [130].

112

3

κ

2.8
2.6
2.4
2.2
1

2

3

4

5

6

η
Figure 5.6: Comparison of the κ-parameter from experiment (red circles) and DSMC simulation results (blue circle). Horizontal and vertical error bars on the red points represent
the standard deviation of η and κ during the time evolution of each experimental data set,
respectively, while the points indicate the mean values, η̄ and κ̄. Results from DSMC simulations are compiled from 50 ms of evolution time during which κ is determined for eachtest particle that traverses the barrier. The data points and error bars at η = 2, 3, and 4
indicate the mean value and standard deviation of κ obtained by averaging over all emitted
test particles. The horizontal, dashed blue line indicates the results of molecular-dynamics
simulations reported in Reference [60].

113
Testing the validity of the rate equation model
To compare the battery rate equation model to experimental data, Equations (5.1)
and (5.2) are integrated numerically using the fourth-order Runge-Kutta (RK4) method (seeReference [132]) in which parameters of the ensemble (e.g. N , E, and C) are updated at eachtime step using expressions derived in Section 2.1.3. Experimental parameters including κ̄and η̄ from Figure 5.6 along with initial reservoir numbers and temperatures are used toinitialize the calculation. Representative results of the numerical calculation for N , Nth ,Nc , and T of the battery reservoir during discharge with VB = 50 kHz are shown as solidlines in Figure 5.5 overlaying the experimental data. Results of the numerical calculation
show agreement with the evolution of the reservoir ensemble to within the uncertainty of theexperimental data. The numerically calculated state of the reservoir ensemble and output
current are utilized in the analysis of the battery presented in the next section.
Thévenin equivalent circuit
The atomtronic battery is analyzed using the Thévenin equivalent circuit of a non-ideal
electronic battery, shown in Figure 5.7, where equivalent elements of the two systems arelabeled. Of particular interest is the role of dissipation in the atomtronic system. Dissipationin electronic devices manifests in the form of heat generated during the flow of current, i.e.,
Ohmic heating. Electronic batteries store energy, ε, and necessarily contain a finite internal
resistance, Rint . When connected to a load with resistance RL , current IL = ε/ (Rint + RL )flows through the combined battery/load circuit. Due to the non-zero internal resistance,
power is dissipated within the battery equivalent to Pint = IL2 Rint . These relations are used
to study dissipation in the atomtronic system, where µ replaces ε and dissipation in thereservoir is represented by an effective internal resistance, Rint .
The power dissipated within the atomtronic battery is calculated using the numericalresults for the evolution of the reservoir ensemble during battery discharge. Here, power

114

a)

IL

ε, Rint
b)

VT

RL

IL

ε

Rint

VT

RL

Figure 5.7: a) Diagram of the atomtronic battery potential showing elements that are conceptually equivalent to those of a non-ideal electronic battery. b) The Thévenin equivalent
electronic circuit. The reservoir well contains the thermodynamic energy that is analogous
to the electromotive force, ε, stored in the battery. Dissipation in the reservoir during discharge is represented by an effective internal resistance Rint . Current, IL , emitted from thereservoir into the load well carries with it an average energy that is likened to the terminal
voltage, VT , of the electronic battery.

115
dissipated in the reservoir is equivalent to the change in internal energy of the ensemble
stored in the reservoir well, given by Pint = Ė − µṄ . Therefore, the internal resistance isĖ − µṄIL2µ − (η + κ) kB T',
IL

Rint =(5.5)

which has units of [Hz/Hz]. In simplifying the expression for Rint to obtain Equation (5.5),
Ṅ and Ė have been replaced by Equations (5.1) and (5.2), respectively, and IL = −Ṅ .
Additionally, an additive factor of βC/IL2 has been omitted, as it is small compared to othercontributions to the internal resistance. We find that the internal resistance is negative,
synonymous with the observed cooling of the reservoir, and its magnitude increases in time.
Figure 5.8 shows the evolution of the internal resistance as the atomtronic battery discharges
for a subset of the experimental data, denoted by their mean truncation parameter, η̄ =
1.77, 3.18, and 4.57. The magnitude of the internal resistance, |Rint |, increases more rapidly
in time with smaller η̄, but the initial value of |Rint | is smaller. The observed increase in|Rint | during battery discharge is indicative of an increased impedance to current flow, andis consistent with the behavior of electronic batteries.
The terminal voltage of the Thévenin equivalent circuit is given by
VT = µ − IL Rint
' (η + κ)kB T,(5.6)

where Equation (5.5) has been inserted for Rint . Therefore, VT is the average energy carriedby each atom as it flows into the load, as in Equation (5.2). This result is intuitively satisfying
as VT represents the potential of the emitted atoms to do work on the connected load. The
magnitude of VT of the fully charged atomtronic battery depends on both the height ofthe barrier and the reservoir ensemble temperature, and to first order the terminal voltage
evolves according to V̇T ∝ κkB Ṫ . The time-evolution of VT is shown in Figure 5.9 for thesame subset of η̄ shown in Figure 5.8. Depletion of stored energy, accompanied by increasing

116
0

R int (Hz/Hz)

-0.5
-1
-1.5
-2
η = 1.77
-2.5
-3

3.18
4.57

0

20

40

60

80

Time (ms)Figure 5.8: Internal resistance of the atomtronic battery during discharge calculated using
Equation (5.5) for three barrier heights: η̄ = 1.77 (solid, red), 3.18 (dashed, black), and 4.57
(dash-dot, blue). Curves indicate results from the numerical model, while the data points
show the corresponding results from experimental data for N, µ, T, and η̄ along with thecalculated values of κ̄ and IL . Error bars are propagated from the standard error of themean for N , µ, and T .

100

V T (kHz)

80
60
40
20

η = 1.77
0

4.57

3.18

20

40

60

80

Time (ms)Figure 5.9: Effective terminal voltage, VT , of the atomtronic battery during discharge forthree barrier heights with mean truncation parameters: η̄ = 1.77 (solid, red), 3.18 (dashed,black), and 4.57 (dash-dot, blue). Data points show the corresponding results from theexperimental data, calculated using Equation (5.6). Error bars are propagated from thestandard error of the mean for N , µ, and T .

117
Rint , leads to a decrease in the voltage supplied by the battery. Again, this behavior isconsistent with that of electronic batteries.

5.4.2

Battery performance

Performance of the atomtronic battery depends on η̄, and is characterized by boththe capacity of the battery and the peak power delivered to the impedance matched load.4
Electronic battery capacity is characterized in multiple ways; however, two common metrics
are the total current and total energy delivered to the load during the time it takes theterminal voltage to drop below a set depletion voltage [133]. In the atomtronic battery,the current capacity, Cc , is determined by integrating the current output from the time thebarrier is lowered (from VS → VB ) until the discharge time, td , when VT falls below 90% of
its initial value. The calculated current capacity, which is plotted in Figure 5.10 for data setswith η̄ ≈ 1.77 − 5.28, indicates that increasing the discharge time by an order of magnitude
results in a 1.6(2)× increase in Cc of the atomtronic battery. During the discharge time,shown as an inset in Figure 5.10, nearly half of the initial ≈ 20 × 103 atoms are emitted into
the load. Current capacity in electronic batteries is often characterized in terms of Peukert’slaw, which states that battery current capacity obeys the relationship Cc = ILρ td , where ρis the Peukert constant [134]. An ideal battery has ρ = 1 while typical lead-acid electronic
batteries have ρ = 1.1 − 1.5, and ρ > 1 indicates internal power loss [135]. Therefore, byincreasing td the current capacity of a battery can be increased. Using measured values
for the current and discharge time, along with the calculated current capacity, the Peukert
constant of the atomtronic battery is found to be ρ = log(IL )/log(Cc td ) = 2.1(3). This large
value of ρ reflects the dissipation within the atomtronic battery reservoir.
The energy capacity, CE , of the atomtronic battery is an extension of Cc , describing
the total energy supplied by the battery, rather than the number of atoms. Figure 5.11
4

Impedance matching implies that PL = VT2 /RL , where RL = Rint . Using the calculated values of Rint
and VT , the load resistance is calculated using RL = Rint /(µ/VT − 1). For all data sets we find |RL | = |Rint |,confirming that the load well is impedance matched by the terminator beam.

118

10

100
80

6

td (ms)Cc (kHz x s)

8

4

40
20

2
0

60

0

0

20

40

td (ms)

2

3

η

60

4

5

80

100

Rt
Figure 5.10: Current capacity, Cc = 0 d IL dt, shown here in units of kHz × s, is found toincrease monotonically with the discharge time, td , which is shown in the inset. The discharge
time increases exponentially with the mean truncation parameter, which is consistent with
Equation (5.2).

CE (x105 kHz)

5

5

4

4.5
4

3

3.5
2
1

PL,peak (x104 kHz2)

5.5

3
2

3

η

4

5

2.5

Figure 5.11: Atomtronic
battery performance versus truncation parameter. Energy caR td
pacity, CE = 0 IL VT dt, (filled blue circles, left axis) increases linearly with increasing
barrier height. Peak power output, PL,peak , (open red circles, right axis) decreases monotonically with increasing barrier height. Values are determined from experimental data for
N, µ, T, and η̄ along with the calculated values of κ̄ and IL . Error bars are propagatedfrom the standard error of the mean for N , µ, and T .

119shows CE , which is calculated in a similar manner to the current capacity, by integrating thepower output to the load during the discharge time. Like the current capacity, the energy
capacity shows an increasing trend with respect to the truncation parameter, in part due tothe linearly increasing terminal voltage. Ultimately, the energy capacity is limited by the
total energy of the ensemble initially stored in the reservoir, Etot = 6.5(9) × 105 kHz.
The peak power output, PL,peak , of the atomtronic battery is determined from experimental data using the relation PL = IL VT , and is shown in Figure 5.11. Output power of theatomtronic battery varies during the discharge time, providing a peak power immediately
after reducing the barrier height to VB . The battery is capable of supplying a maximum
peak power of PL ≈ 5 × 104 kHz2 for η̄ . 3. At higher truncation parameters the peak power
output declines due to the exponentially decreasing IL , despite increasing VT . Figure 5.11
encapsulate the performance of the atomtronic battery and illustrates that the battery canbe operated in regimes of either large peak output power or large energy capacity.
5.5

An atomtronic transistor
With its behavior understood, the atomtronic battery can now be used to power otheratomtronic circuit elements. Owing to the importance and ubiquity of its electronic counterpart, the device we seek to emulate is a three-terminal transistor. This is accomplishedby adding a second optical barrier to the atomtronic battery system. In this configuration,
the role of the battery is to maintain the chemical potential and temperature of the source
well, allowing the triple-well system to be treated according to Section 2.2. In the following
sections, the steady-state behavior of the atomtronic transistor5 is explored by varying therelative height of the two barriers, which sets the feedback parameter, υ ≡ (VGD −VGS )/kB Ts .
Measurements of the normalized temperature drop, τ = (Ts − Tg ) /Ts , and chemical potential drop, µ̂sg = (µs − µg ) /kB Ts , are compared to the analytic results of Section 2.2.3 and arefound to confirm the ability to control the steady-state behavior of the atomtronic transistor
5

The results discussed here are published in Reference [136].

120
via the feedback parameter.

5.5.1

The experimental triple-well system

Keeping with the nomenclature of field-effect transistors adopted in Section 2.2.1, thethree regions of the experimental triple-well potential are labeled “source,” “gate,” and“drain.” These regions are indicated in the longitudinal profile of the hybrid magnetic andoptical trapping potential that is shown in the top of Figure 5.12. The two barriers are
separated by 4.8 µm resulting in a gate well longitudinal trap frequency of ωx,G ≈ 2π ×
850 Hz.6 In conjunction with the Gaussian profile of the barriers, this spacing minimizes theanharmonicity of the central well. Note that the degree of overlap between the two opticalbarriers contributes to an offset in the potential energy of the gate well. The shift in theminimum energy of each well due to the Gaussian shape of the barriers is denoted by Vi,0 . For
the data presented here, the barrier separation of 4.8 µm results in VS,0 ≈ VG,0 ≈ VD,0 ≈ 0.
As in the experimental characterization of the battery, the terminator beam is used to outcouple atoms that enter the drain well during transistor operation in order to observe thetime dynamics of the source and gate wells.
The initial state of the transistor is prepared in a similar manner to the atomtronic
battery (see Figure 5.2). By sweeping the source-gate barrier, with a height of VS ≈ 100 kHz,all of the atoms are compressed into the source well before turning on the gate-drain barrier,leaving the gate and drain wells initially empty. Each experimental realization begins witha total source well atom number of Ns = 20.0(2) × 103 atoms, at a temperature Ts =15.0(5) kHz, which corresponds to a TF chemical potential of µs = 3.0(2) kHz in the halfharmonic trap. At time t = 0 the barrier heights are set to the desired energies, VGS and
VGD , and the system is allowed to evolve for a time ∆t. Measurements on the state of the
6

The longitudinal gate frequency is measured using the same method that was used to determine theradial trap frequency of the split-guide trap. By displacing the barriers slightly, and in unison, the first order
longitudinal mode of the gate well is excited and the resulting heating is used to map out the excitation
spectrum.

121

V (kHz)Source

Gate

40
VGS

Drain

VGD

20
Terminator

0

-60

-40

-20
0 4.8
20
Longitudinal Position (μm)

40
0

1

60
2

Figure 5.12: Top panel: Longitudinal potential energy profile of the hybrid magnetic andoptical trap. Center panel: Calculated 2D energy contour. Bottom panel: In situ absorption
image of approximately 4.5 × 104 87 Rb atoms at T ≈ Tc , revealing the three distinct regionsof the triple-well potential

122system are performed using absorption imaging techniques, either in TOF or in situ.

5.5.2

Transport dynamics in the atomtronic transistor

Quantitative data regarding the thermodynamic evolution of the source ensemble isextracted using TOF imaging. Figure 5.13 shows the total, Ns , thermal, Nth,s , and condensed,
Nc,s , atom numbers determined from bimodal fits to the momentum distribution of the cloudin free expansion (see Section 4.2.1), for a representative data set with VGS = 30 kHz and
VGD = 33 kHz. The decay of the total atom number in the source well reflects the flux ofatoms emitted into the gate, and can be used to quantify the source-gate atom current, whichis shown in Figure 5.14. As one might expect, the evolution of the source well mimics thatof the battery reservoir. However, the magnitude of the net source-gate current is reduced
due to the reverse current, Igs , that flows from the gate, back into the source due to thepresence of the gate-drain barrier.
Transport dynamics in the triple-well system are calculated according to the same RK4
numerical integration method that was used to model the battery evolution. The calculation is expanded to suit the transistor potential by including the three atom currents from
Equation (2.33) and imposing particle and energy conservation given by Equations (2.35)
and (2.36), respectfully. The steady-state solution for source and gate ensemble properties
obtained via the RK4 calculation reproduces the analytical predictions for the temperatureand chemical potential drops shown in Figures 2.5 and 2.6. Numerical calculations of thetime evolution of the source well atom number and temperature are overlaid with the experimental data in Figure 5.13. The net source-gate current is also calculated and shown in
Figure 5.14. After an initial transient period of ∼ 20 ms, the numerically calculated atom
numbers, temperature, and net source-gate current agree with the experimentally measured
values to within the uncertainty of the data. It is suspected that the disagreement between
the model and data at early evolution times is due to the initial non-equilibrium state of thegate well.

Source Atom Number (x103)

20
15
Ns

Temperature (nK)

123
700
600
500
400

0

10

10

20

30

40

50

Nth,s

5
0

Nc,s

0

10

20
30
40
Evolution Time (ms)50

Figure 5.13: Gate well population as a function of the feedback parameter. The populationis measured using a saturated probe beam intensity and undercounts the true number. Error
bars represent the standard error on the mean from five experimental realizations at each
feedback parameter.

Isg,net (x103 s-1)

750

500

250

0

0

10

20
30
40
Evolution Time (ms)50

Figure 5.14: Net source-gate atom current determined from the first time derivative of thesource well total atom number. Error bars indicate the error propagated from data in
Figure 5.13.

124
Transport dynamics are further observed by probing the system in situ. Figure 5.15
presents a series of images showing the in-trap atomic distribution that includes the initialstate, with all of the atoms contained in the source well, followed by the evolution of thesystem for ∆t = 0 − 50 ms, during which VGS = 30 kHz and VGD = 33 kHz. This series ofimages was recorded using a weak probe beam (Ip . Isat ) in order to observe atoms thathave reached the drain. As can be seen in the third frame of the series, atoms becometrapped in the gate well by ∆t = 10 ms. If atoms were not trapped in the gate, but rather
only traversing it, one would expect there to be at most twice the density seen in the drain
well, just to the right of the gate-drain barrier. We note that at early evolution times atoms
that enter the gate well experience a region of substantial population inversion since energy
states lying below the height of the barriers are initially unpopulated. While the routeto steady-state is an analytically difficult problem, the role of interatomic interactions andmanifestation of dissipation are made evident by the population growth in the gate well.

5.5.3

Source and gate wells in steady-state

In order to probe the steady-state behavior of the atomtronic transistor, the system
must first evolve for a sufficiently long time such that the source and gate ensembles reach aquasi-steady-state. The required equilibration time is determined by measuring the gate well
population as a function of time. In order to extract quantitative data from measured intrap atomic density, the probe transition is strongly saturated, as described in Section 4.2.2.
In the saturated probe regime one gains access to the high density source and gate wellensembles, but loses information regarding the atom current flowing in the drain. The
evolution of the total gate well atom number, Ng is shown in Figure 5.16, for υ = 0.6. Here,the atom number is calculated according to Equation (4.17). The data shows a rapid rise in
the gate population, increasing to Ng = 1380(55) atoms after 20 ms, after which it settles toa value that agrees with the steady-state gate population, shown by the blue dashed line in
Figure 5.16, that is calculated using the previously described RK4 method. In conjunction

125

2

OD

0 ms

1
10 ms

0

20 ms
30 ms

Radial Pos. (μm)

40 ms
15
10
5
0

50 ms
-60

-40

0
-20
20
Longitudinal Position (μm)

4060Figure 5.15: Time series of the atomtronic transistor as current flows from the source,
through the gate, and into the drain. Vertical dashed lines indicated the location of the two
repulsive, Gaussian barriers, here with heights of VGS = 30 kHz and VGD = 33 kHz.

126
with the agreement between the measured and calculated net source-gate current after ∼ 20
ms, equilibration of the gate well population verifies that the system reaches a quasi-steadystate by ∆t = 30 ms.
The tunability of the steady-state behavior of the atomtronic transistor is explored bymeasuring the in-trap atomic distribution with different feedback parameters, after an evolution time of ∆t = 30 ms. Measurements are made with a gate-source barrier height fixed at
VGS = 30 kHz, while the gate-drain barrier height is varied between data sets. The measured
gate population is plotted in Figure 5.17 as a function of the feedback parameter. The datashows a rapid increase in the quasi-steady-state population of the gate well beginning at
the feedback parameter υ ≈ −0.25. Population growth in the gate well when VGD < VGS
provides further evidence that dissipation occurs as atoms traverse the gate region. The
source of dissipation is thought to arise due to collisions between particles entering the gate
well. This hypothesis is supported by other works that study the filling rate of an initiallyempty well [59] and collision induced decoherence [137].
The temperatures of the source and gate well ensembles are also extracted from insitu images. Equation (2.15) is fit7 to the thermal tails of the in-trap atomic density profiles√
along the radial trap direction to yield the characteristic widths that scale as σi ∝ Ti . The
temperature drop, τ ≡ (Ts − Tg )/Ts , is calculated from the measured values of Ts and Tg andis plotted in Figure 5.18. The data shows that τ > 0 for υ & 0, indicating that atoms in thegate do in fact become colder than those in the source, just as was predicted in Section 2.2.3.
Experimental measurements of the steady-state gate atom number are compared toresults of the RK4 numerical calculation performed for a range of feedback parameters thatspan those of the experimental data. The numerical results are indicated by the solid line
in Figure 5.17. Deviation between the model prediction for the gate population and the7

Ideally one would fit the thermal distribution using Equation (2.18), which includes g3/2 (z(x)), wherez(x) is given by Equation (2.28). Fits using this function were unstable and required care to ensure selfconsistency of µi and Ni . Instead, the temperature is fit using Equation (2.15), which neglects the HF
mean-field but provides a more robust, less hands-on fitting procedure.

127

Gate Atom Number

1500

Ng,eq

1000

500

0

0

10

20
30
40
Evolution time (ms)50

Figure 5.16: Time evolution of the measured gate well atom number versus time, relative tothe calculated steady-state population, Ng,eq , for υ = 0.6. Error bars represent the standard
error on the mean of three experimental realizations at each time.

2000

Ng

1500
1000

500
0
−1

−0.5

0
0.5
υ ≡ (VGD−VGS)/Ts

1

Figure 5.17: Chemical potential drop between the source and gate wells of the atomtronictransistor after an evolution time of 30 ms. Error bars represent the standard error on themean for five experimental realizations at each feedback parameter.

128

0.2

τ ≡ (Ts−Tg)/Ts

0.1
0
−0.1
−0.2
−0.3
−1

−0.5

0
0.5
υ ≡ (VGD−VGS)/Ts

1

Figure 5.18: Temperature drop between the source and gate wells of the atomtronic transistor
after an evolution time of 30 ms. Error bars represent the 95% confidence interval of theGaussian fit to wings of the in-trap density profile for five experimental realizations at each
feedback parameter.

129
experimental data at large feedback parameters is a result of the increasing atom density
in the gate. Large densities lessen the reduction of the scattering cross section because thecloud is optically thicker.8 As a result, the experimental data for the gate atom number
at large positive feedback parameters is less than the numerically calculated population.
The temperature drop data is also compared to the numerical calculation. As alluded topreviously, results of the RK4 calculation accurately reproduce τ as a function of υ. The
numerical results are indicated by the solid line in Figure 5.18. We find good qualitative
agreement to within the error bars of the temperature drop data, which represent the 95%
confidence interval of the fit to the in-trap density profile.
Using the gate well atom number from Figure 5.17 and temperature data from Figure 5.18, the chemical potential in the gate well is calculated in the TF limit in a harmonic
well. The chemical potential drop, µ̂sg ≡ (µs − µg )/kB Ts , characterizes the chemical potential gradient between the source and gate wells, and is shown in Figure 5.19. For values of
υ > 0.2 the chemical potential drop is negative, revealing that the chemical potential in thegate well exceeds that of the source well. Furthermore, we determine the critical feedback
parameter, υBEC = −0.24, above which a condensate forms in the gate well in steady-state.
Below this threshold (υ < υBEC ) the chemical potential drop reflects the positive chemical
potential of the source well relative to VG,0 . The same behavior was found in the analytic
model for the steady-state behavior in the triple-well system shown in Figure 2.6 where thecurves are dashed in the absence of a BEC.

5.5.4

Further discussion

It is worth elaborating further on the presence of a negative chemical potential gradient
between the source and gate wells. At first glance it may appear that in order to sustain
8

The saturation parameter, s0 = Ip /Isat , used in the experiment results in a reduction of the effective
cross section (see Equation (4.18)) that is determined to be C = (σ/σef f ) = 9.3(1). Steady-state gate
numbers at higher υ can be determined by further saturating the optical transition, but this also leads to adecrease in sensitivity to the population at υ . 0.

130

0.6

µsg ≡ (µ s −µ g )/Ts

0.4
0.2
0

−1

υ

−0.4

BEC

−0.2

−0.5

0
0.5
υ ≡ (VGD −VGS )/T s

1

Figure 5.19: Chemical potential drop between the source and gate wells of the atomtronictransistor after an evolution time of 30 ms. Error bars represent the standard error on themean propagated from the uncertainty in Ngate and Tgate .

131
a positive net source-gate current in this operating regime, work must constantly be doneon the system. However, current flows due to gradients in both chemical potential andtemperature. Thus, although the chemical potential gradient is negative, there is a positive
temperature gradient that sustains current flow from source to gate. Additionally, a negative
gradient in the chemical potential represents the conversion of thermal energy in the source
to chemical potential energy in the gate. This process decreases the local entropy of thesystem; however, the presence of the terminator beam makes this an open system such that
globally the entropy of the system increases and the second law of thermodynamics is upheld.
The data in Figures 5.18 and 5.19 confirm that by tuning the feedback parameter it ispossible to achieve quasi-steady-state ensembles in the source and gate that are related bytemperature and chemical potential drops. Returning to the analogy of electronic transistor
operation, the resulting potential drops are a demonstration of the self-biasing behavior thatwere predicted in Section 2.2.3. Electronic transistor functionality largely depends on thebias or quiescent operating point set by the voltages at each of its three terminals. The
experimental results demonstrated here show that the atomtronic transistor can be tuned toregions of either positive or negative chemical potential drops between the source and gate
wells, indicating operation in forward and reverse-bias modes, respectively.

Chapter 6
OutlookThis dissertation has described the experimental realization of two fundamental atomtronic circuit elements: a battery and transistor. In addition to the demonstration of theseatomtronic devices, a semiclassical analytical model was used to explore the transistor-likebehavior of the triple-well system, explicitly its ability to exhibit current gain. Furthermore,
the atom-chip-based apparatus that was constructed to demonstrate these atomtronic devices provides the experimenter the ability to reconfigure the trapping potential in orderto prototype future atomtronic circuit elements. Building upon this body of work, threeavenues of continued research are outlined in the following sections.
6.1

Current controlled atomtronic transistors
The first research avenue is a direct continuation of experimental work presented in

Chapter 5 towards the goal of demonstrating a current controlled transistor, using the current
gain predictions of the semiclassical model (Figure 2.7) as a guide. Ultimately, the utilityof a transistor lies in the ability to control a large current or voltage at the output of thedevice via the application of a smaller control current or voltage. Such an ultracold atom
amplifier would be immediately applicable to the AMO community.
In order to realize a current controlled atomtronic transistor, it is necessary to create asecond reservoir to serve as the source of the control current. This is theoretically possible inthe current experimental apparatus owing to the flexibility of the 2D AOD, though system

133

Source VGS
μs,TsIc
ControlGate VGD
μg
Tg

Drain

I gd +dI gd

VGC

Figure 6.1: Possible experimental potential to study a current controlled atomtronic transistor. A contour of the total potential (solid black curve) is shown that has three wells. The
bare magnetic potential is also shown (red dashed curve). A blue-detuned optical barrier
bisects the original source well to create a control well. The three wells are then separated
by independently controlled barriers, VGS , VGC , and VGD . The initial state would be prepared according to the previously described procedure, followed by ramping up a barrier toseparate the source and control wells.

initialization and actuation must be carefully considered. Figure 6.1 shows an illustration of
one possible design that could be used to study current gain. An alternative approach would
be to utilize the trapping techniques described in References [20, 138] or [139], in which an
all optical potential allows atom currents to flow along two dimensions.

6.1.1

More complex atomtronic devices

Once a current controlled transistor has been realized, it becomes possible to again
draw on the analogy with electronics to construct more complex atomtronic devices. In
two dimensions, the current controlled transistor potential could be realized as is shown in
Figure 6.2(a), which leaves the source, gate, and drain wells relatively unchanged from theimplementation used to conduct the experiments of Chapter 5. One can imagine interconnecting multiple transistors to realize atomtronic analogues to logic gates and more complex
transistor based circuits. As an example, a possible implementation of a flip-flop is shownin Figure 6.2(b). Since the current gain described in Section 2.2.4 is differential, the characteristic behavior of the electronic counterparts to these devices would then appear as adeviation from the steady-state output of each of the individual atomtronic transistors.

134

a)

Source

VGS

VGD

Gate

Drain
I gd +dI gd

VGC

Control
Input signal (Ic)

b)

S1

D1

G1

Vset

δ

C1

D2

C2

δ

G2

VresetS2

Figure 6.2: a) A possible two dimensional potential for a current controlled atomtronictransistor generated using all optical potentials. This geometry also allows the input current
to originate from a separate potential from which the signal to be amplified originates.
The behavior of the current controlled transistor is described in Section 2.2.4. b) One
possible realization of a flip-flop circuit built from two transistor potentials shown in (a).
This configuration would exhibit flip-flop-like behavior about the steady-state operating
conditions determined by the feedback parameter of each transistor. For example, assuminga small coupling ratio, δ, between the drain current in D1 and the connected control C2 , the0drain current in D1 would decrease to Igd,1= (1−δ)Igd,1 , while the drain current in D2 would0
increase to Igd,2= (1 + δAI )Igd,2 . The same behavior would occur in the other transistor,
and the drain output currents of the two transistors would alternate with one higher and theother lower than the steady-state Igd , and vice versa, depending on the sign of AI set by υ.
As in an electronic flip-flop, set and reset elements are included, here in the form of barriers.
In analogy to grounding the gate input, a large V(re)set shuts off current flow into the control
gate of the respective transistor potential, i.e., δ → 0, returning the drain current to thesteady-state value.

135
6.2

Coherent behavior
The work in this dissertation has approached the problem of realizing an atomtronictransistor from a semiclassical perspective in which the bulk of the quantum behavior ishidden within interactions that give rise to the positive valued chemical potential. However,an attractive concept in atomtronics is the ability to influence the coherence of an atomic
signal. In the context of the triple well system studied here, coherence can refer to phaselocking between condensates in the source- and gate-wells, which is studied theoretically in
Reference [26], or the coherence of the drain current. The remainder of this section discusses
ongoing attempts to control the degree of coherence of the atom current emitted into thedrain.
Conceptually, it should be possible to couple the motional state of the condensate
trapped in the gate well to higher lying energy states, thus establishing coherence between
energy levels that contribute to the gate-drain current. Figure 6.3 illustrates the simplified
model of the gate well that is used to discuss the generation of coherence in the gate-draincurrent. Notice that states below the barrier height are considered, indicating the presenceof quantum mechanical tunneling. Also, consider that the properties of the gate barriers
including their widths, heights, and separation have been designed to provide preferential
transmission of atoms in levels |gi and |ei. Methods for designing such a potential are
discussed in Section 6.3. The energy level spacing of |gi and |ei is given by E/~ = Ω,and is equivalent to the oscillation frequency of the center of mass dipole mode of the gate
well condensate. Intraparticle interactions, e.g. mean-field effects, lead to the conversion ofenergy stored in the collective motion of the condensate to the higher lying states, a processsimilar to Landau-damping [43]. In this manner, it is expected that the output modes of thegate, i.e., γg and γe , develop a coherent relationship to both the condensate in the gate welland one another. Ongoing theoretical treatment of this model system draws upon methods
from semiclassical laser theory [140] with possible extension to the quantum master equation.

136

Pe
|e

Pg

Ω

|g

γe
γg

Ω
Figure 6.3: Schematic of the model system for realizing coherent transistor output. Gate
well populations in energy levels |gi and |ei are pumped by the source-gate current, Pg and
Pe , respectively, and decay by emitting current into the drain, at γg and γe . The gate wellcondensate prepared in the dipole mode (center of mass oscillation) then couples the twostates, generating a coherent output current into the drain.

137
Coherence properties of atoms can be probed using several methods including spatial
interference [141] and measurements of correlation functions [142, 143, 144]. Images ofthe spatial distribution of trapped atoms have been used to show that the correlations
persist within the trap, but require Fourier analysis and well-characterized, high-resolution
imaging [145]. Future implementation of such techniques may prove useful in measuring
spatial correlations in the drain current of the atomtronic transistor.
6.3

Tailored barrier transmission
As alluded to in previous chapters, the ability to design the potential landscape of an

atomtronic circuit element to realize specific behavior of the atomic current and thermodynamic potentials is an attractive and powerful concept. Ultimately, the relevant characteristics of the potential are the energy dependent transmission and reflection of matter waves.
While this problem can be approached analytically, e.g. using the WKB approximation, solutions for all but the simplest of potentials are non-trivial and instead require numericalcalculation. Drawing upon the analogy to the propagation of an electromagnetic wave in atransmission line, the generalized impedance (GI) concept provides an easily implemented
numerical approach to determine the wave transmission and reflection of an arbitrary potential [65].
Originally developed to treat currents due to quantum mechanical tunneling in semiconductor electronics [146], the GI concept has been extended to treat wave transmission
through arbitrary potentials [126]. Calculations using this method accurately reproduce theresults of recent experiments that measure energy dependent reflection from a periodic potential synonymous with a Bragg reflector [67]. Synthesized potentials of this type could
be applied towards the realization of resonant atomtronic circuit elements as well as othersystems in the field of atom optics, such as a matter wave Fabry-Pérot interferometer [147].
Theoretical application of the GI concept to the atomtronic transistor system is ongoing andshows promise towards the experimental realization of previously discussed coherent output

138
currents and other interesting behavior that arises due to quantum mechanical tunneling.

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Appendix A
Derivation of the normalized temperature and chemical potential drops

This section provides the derivations of the key relations in the analytic steady-statemodel of the atomtronic transistor, the normalized temperature and chemical potential drops
provided in equations (2.39) and (2.40), respectfully. Given that the source well is maintained
at a finite µs & Ts , and there are no external sources of atoms into the system, the potential
drops are derived in a form that depends solely on µs and Ts . The first step in both derivations
is to re-cast the current conservation relation from equation (2.35),
Isg = Igs + Igd ,in terms of the feedback parameter, υ, which controls the ratio of Igd /Igs . Beginning fromthe expression for Igd from equation (2.33),
Igd = γg Nth,g exp[−(VGD − VG,0 − µg )/kB Tg ]and inserting unity as 1 = exp [(VGS − VGS )/kB Tg ], one finds
Igd = γg Nth,g exp [−(VGD − VGS )/kB Tg ] exp [−(VGS − VG,0 − µg )/kB Tg ] .

(A.1)

Next, inserting the barrier height difference, ∆V = VGD − VGS , and pulling out a factor ofIgs simplifies equation (A.1) to
Igd = Igs e−∆V /kB Tg ,

(A.2)

151
which shows that the ratio Igd /Igs = exp [−∆V /kB Tg ]. Plugging equation (A.2) into theexpression for current conservation and inserting the feedback parameter, υ ≡ (VGD −VGS )/kB Ts ≡ ∆V /Ts , along with Tg /Ts = (1 − τ ) simplifies equation (A.1) toIsg = Igs 1 + e−υ/(1−τ ) .

(A.3)

Equation (A.3) is the modified current conservation relation and it is used in derivations of
the transcendental equations for both τ and µ̂sg .
Temperature drop
In the derivation of the transcendental relation for the temperature drop temperature drop, τ ≡ (Ts − Tg ) /Ts , one begins from the energy conservation equation, originallypresented in equation (2.36):
Isg (VGS + κGS kB Ts ) = Igs (VGS + κGS kB Tg ) + Igd (VGD + κGD kB Tg ).
Dividing through by Igs , one then uses equation (A.3) to replace the ratio Isg /Igs and equation (A.2) to replace to replace the ratio Igd /Igs , yielding(VGS + κGS kB Ts ) 1 + e−υ/(1−τ )= (VGS + κGS kB Tg )+ (VGD + κGD kB Tg ) e−υ/(1−τ ) .

(A.4)

The remaining steps include algebraic manipulation of equation (A.4),
κGS (kB Ts + kB Tg ) = e−υ/(1−τ ) (∆V + κGD kB Tg − κGS kB Ts ) ,followed by dividing through by kB Ts and inserting τ where appropriate,κGS τ = e−υ/(1−τ ) (υ + κGD (1 − τ ) − κGS ) .
Finally, one obtains the expression for the temperature drop,τ = e−υ/(1−τ )υ + (κGD − κGS ).
κGS + κGD e−υ/(1−τ )

(A.5)

152
Chemical potential drop
The transcendental relation for the normalized chemical potential drop is derived ina similar manner to the temperature drop. Beginning again from the modified current
conservation relation, equation (A.1), this time insert the expressions for currents fromequation (2.33):
VGS − µs
γs Nth,s exp −
kB Ts





VGS − VG,0 − µg
= γg Nth,g exp −
kB Tg



1 + e−υ/(1−τ )



(A.6)

To begin, the previous equation is rearranged to group like-terms in the exponentials,





µs
µg + VG,0γg Nth,g
VGS
VGS
exp
−
=
exp
−
1 + e−υ/(1−τ ) .
(A.7)
kB Ts
kB Tg
γs Nth,skB Ts kB Tg
Next, take the natural log of both sides and use the identity: ln[A × B] = ln[A] + ln[B],

 



µg + VG,0γg Nth,g
VGS
VGS
µs
−
= ln
+
−
+ ln 1 + e−υ/(1−τ ) .
(A.8)
kB Ts
kB Tg
γs Nth,skB Ts kB Tg
Inserting the definition for the feedback parameter and the temperature drop further simplifies the expression toµ̂g + V̂G,0= −V̂GS
µ̂s −
1−τ



τ
1−τ








γg Nth,g+ ln+ ln 1 + e−υ/(1−τ ) ,γs Nth,s(A.9)

where the (ˆ) indicates the quantity is normalized to the source temperature. Using theexpressions for the number of thermally excited atoms in the ensemble, equation (2.30), thecritical temperature, equation (2.8), and the thermal atom collision rate, equation (2.34),
the ratio γNth factors simplifies toγg Nth,g=γs Nth,s
=



Tg
Ts

5 1 + ζ(2) µg

ζ(3) kB Tg,µs
1 + ζ(2)ζ(3) kB Ts
1 − τ + ζ(2)µ̂
ζ(3) g4(1 − τ ).
1 + ζ(2)µ̂ζ(3) sFinally, one obtains the expression for the chemical potential drop,
h

i


υ
µ̂sg = (1 − τ ) ln (1 − τ )4 1 + e− 1−τ − τ V̂GS − µ̂s



ζ(2)
1 − τ + ζ(3)µ̂s − V̂G,0 − µ̂sg,+ (1 − τ ) ln ζ(2)1 + ζ(3)µ̂s(A.10)

153
after replacing the normalized gate chemical potential by µ̂g = µ̂s − V̂G,0 − µ̂sg .