Single Atom Delivery into a Bottle Beam Trap Using an
Optical Conveyor Belt and Quantum Coherent Gain in a
Matterwave Transistor
byBrad A. Dinardo
B.S., Physics and Mathematics, Juniata College, 2011
M.S., Physics, University of Colorado, Boulder, 2014

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
2018

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This thesis entitled:
Single Atom Delivery into a Bottle Beam Trap Using an Optical Conveyor Belt and Quantum
Coherent Gain in a Matterwave Transistor
written by Brad A. Dinardo
has been approved for the Department of Physics

Prof. Dana Z. Anderson

Prof. Murray Holland

Date

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

iii
Dinardo, Brad A. (Ph.D., Physics)
Single Atom Delivery into a Bottle Beam Trap Using an Optical Conveyor Belt and Quantum
Coherent Gain in a Matterwave Transistor
Thesis directed by Prof. Dana Z. Anderson

The work of this dissertation falls into two broad categories. In the first part, I describe
loading a single atom from a reservoir into a blue-detuned crossed vortex bottle beam trap using a
dynamic 1D optical lattice. The lattice beams are frequency chirped using acousto-optic modulators, which causes the lattice to move along its axial direction and behave like an optical conveyor
belt. A stationary lattice is initially loaded with approximately 6000 atoms from a reservoir, andthe conveyor belt transports them 1.1 mm from the reservoir to a bottle beam trap, where a single
atom is loaded via light-assisted collisions. Photon counting data confirm that an atom can bedelivered and loaded into the bottle beam trap 13.1 % of the time.
In part II, I describe a theory and experiment in the field of atomtronics that displays acoherent gain mechanism for a triple-well matterwave transistor oscillator. I start with a wellestablished semi-classical description of an atomtronic transistor but model the system using a
many-body formalism. The quantum model predicts interesting physics when the atoms flowing
through the transistor have sufficiently low enough temperatures such that the motional state of adipole oscillating BEC, placed in the transistor itself, couples atom transitions between high lying
transistor energy eigenstates. In this regime, the coupling gives rise to a new gain mechanismthat increases the flux of matterwaves flowing out of the transistor system, compared to when the
coupling is absent. Our experiments suggest that the gain mechanism is coherent and increases the
spread of matterwave energy that flows out of the transistor.

Dedication

To my parents, Karen and Joe Dinardo, whose unconditional love and support throughout
my life made me the person that I am today.

v

Acknowledgements

There are so many people that I must thank for helping me throughout my academic career.
First, I thank my advisor, Dana Z. Anderson, for his constant support, advice, and help throughout
my time here at JILA. Dana has made me a better scientist and person and I will forever be gratefulfor his absolute dedication to my graduate career. Second, I must absolutely thank Leslie Czaia
for her enormous help throughout my time at JILA. Leslie assisted with so many crucial aspects ofmy experiments behind the scenes, and saved me weeks (or months) with her expertise. Moreover,I will forever be indebted to the graduate students of the Anderson lab. An old saying goes “If you
want to go fast, go alone. If you want to go far, go together.” With that I must thank my fellow
graduate students: Kai Hudek, Seth Caliga, Cameron Straatsma, Carrier Weidner, and Curtis Rao.
Without you, I would never have made it through this wonderful, yet challenging experience.
I also need to thank former Anderson Lab post docs Hoon Yu and Marko Zgonik. I mustalso credit the JILA machine and instrument shop, as they did so many incredible things behindthe scenes to help me with my research. For that I thank Hans Green, Blaine Horner, Tracy Keep,and Kim Hagen. I cannot tell you how many times I went to the JILA electronic shop with a friedcircuit and was given great advice on how to fix the problem. Thus, I must thank Terry Brown, Carl
Sauer, and James Fung-A-Fat. Additionally, I need to extend a very big thanks to the Universityof Wisconsin, Madison neutral atom quantum computing team, including Mark Saffman, Marty
Lichtman, and Michal Piotrowicz. I also extend a great thanks to Evan Salim and Steve Hughesfrom ColdQuanta as well as Sterling McBride and Joey Michalchuk from SRI Sarnoff for their
help. I was also very fortunate to have very memorable professors here at CU Boulder. I thankvi
Chris Greene, K.T. Mahanthappa, Shanta de Alwis, Oliver DeWolfe, James Thompson, and Victor
Gurarie for very memorable classes.
I would never be here without the incredibly talented professors from Juniata College. I
thank Jamie White, Jim Borgardt, Norm Siems, Mary Atchley, Mark Pearson, Ben Sunderland,
Henry Escuadro, John Bukowski, Catherine Stenson, Kim Roth, Jerry Kruse, and Paul Schettler.
I remember on my very first day at Juniata College meeting Dr. Siems in his office. He told me“When you think you’ve worked hard, just work a little harder.” I never forgot that advice and I
credit it to helping me get through college and my graduate studies. Additionally, I must thanksome of the great teachers I had growing up in Altoona, PA: From Logan Elementary, I thankBill Killian and Randy Bickel. From D.S. Keith Jr. High School, I thank John Banyas, Stephen
Lightner and Frank Swalga. Finally, from Altoona Area High School, I had the great pleasure oflearning from Joe Falger, Eric Stoudnour, Jim Krug, Mike Ergler, David Borst, Roger Menard,
Marie Suter, and E. Preston Rice.
Finally, I want to thank my parents, Karen and Joe Dinardo as well as my brother Derek.
Without them, I would never be here. For as long as I remember, they encouraged me to pursue
my passions and did whatever they could to support me through life. Thank you all.

vii

Contents

Chapter
1 Part I: Introduction and Motivation
1.1

1.2

1.3

AQuA Project Summary: Neutral Atom Quantum Computing . . . . . . . . . . . .

1

1.1.1

Scaling a Neutral Atom Quantum Computer to Many Qubits . . . . . . . . .

2

1.1.2

Drawback of a Neutral Atom Qubit Array: Atom Loss . . . . . . . . . . . . .

4

1.1.3

Solution to Atom Loss: Transporting Atoms to Reload Distant Unoccupied
Array Optical Traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

Ultracold Matterwave Transistor Oscillator . . . . . . . . . . . . . . . . . . . . . . .

7

1.2.1

Atomtronics: Ultracold Atomic Circuits . . . . . . . . . . . . . . . . . . . . .

7

1.2.2

Analogue between Electronic and Matterwave Transistors . . . . . . . . . . .

9

1.2.3

Emergence of a Gain Mechanism for a Matterwave Transistor . . . . . . . . .

9Dissertation Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Laser Cooling and Trapping 133 Cs and 87 Rb Atoms
2.1

1

12

Atom-Photon Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.1

Hamiltonian and Energy Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1.2

Optical Dipole Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.1.3

Radiation Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.1.4

Magneto-Optical Trapping of Atoms . . . . . . . . . . . . . . . . . . . . . . . 20

2.1.5

Sub Doppler Cooling with σ+ /σ− Polarization Gradients . . . . . . . . . . . 23

viii
2.2

2.3

Cooling Further to Degeneracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2.1

Magnetic Trapping Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.2.2

Radio Frequency Evaporation . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.2.3

Reaching Bose-Einstein Condensation . . . . . . . . . . . . . . . . . . . . . . 28

Collisions Between Pairs of Low Energy Atoms in a BEC . . . . . . . . . . . . . . . 31
2.3.1

BEC Many-Bodied Hamiltonian and Gross-Pitaevskii Equation . . . . . . . . 32

2.3.2

BEC Characteristics in the Thomas-Fermi Approximation . . . . . . . . . . . 33

3 Compact Ultra High Vacuum Systems for the Optical Conveyor Belt and Transistor Oscillator Experiments
3.1

Atomic Qubit Array Cell

35
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.1.1

Cesium Source Chamber, 2D and 3D MOT Chambers . . . . . . . . . . . . . 38

3.1.2

Hex Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.1.3

Anodically Bonded Ion Pumps . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.2

Atomtronics Double MOT Vacuum Cell . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.3

Atom Chip: Making Magnetic Traps for Ultra Cold Atoms . . . . . . . . . . . . . . 45

3.4

3.3.1

Producing 3D Harmonic Magnetic Potentials . . . . . . . . . . . . . . . . . . 45

3.3.2

Double Guide Wire IP Trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

Conclusion to Part I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4 Part II: Individual Atom Delivery of Neutral Cs Atoms into Bottle Beam Traps Using a
Dynamic 1D Optical Lattice

54

4.1

Motivation for Single Atom Delivery into Bottle Beam Traps . . . . . . . . . . . . . 54

4.2

Laser System and Beam Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.2.1

Layout of 852 nm Cooling, Repump, and Probe Lasers . . . . . . . . . . . . . 57

4.2.2

1064 nm Transport Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.2.3

780 nm Bottle Beam Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.2.4

Layout of Laser Beams in the AQuA Hex Cell

. . . . . . . . . . . . . . . . . 65

ix
4.3

4.4

4.5

4.6

Producing a Reservoir of Cold Atoms. . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.3.1

2D+ MOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.3.2

3D MOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.3.3

Polarization Gradient Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.3.4

3D MOT and PGC Pagoda Coil System . . . . . . . . . . . . . . . . . . . . . 72

Atom Transport System: Optical Conveyor Belt

. . . . . . . . . . . . . . . . . . . . 74

4.4.1

1D Dynamic Lattice Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.4.2

Properly Accelerating the Dynamic 1D Lattice . . . . . . . . . . . . . . . . . 76

4.4.3

Demonstration of Atom Transport with an Optical Conveyor Belt . . . . . . 78

4.4.4

Laser Beam Overlap and Alignment Considerations. . . . . . . . . . . . . . 82

Trapping Single Atoms in a Crossed Vortex Bottle Beam Trap . . . . . . . . . . . . 84
4.5.1

Overview of Optical Bottle Beams . . . . . . . . . . . . . . . . . . . . . . . . 84

4.5.2

Generating a Crossed Vortex Bottle Beam Trap . . . . . . . . . . . . . . . . . 85

4.5.3

Detecting a Loaded Single Atom . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.5.4

Loading a Bottle Beam Trap from a Background MOT. . . . . . . . . . . . 89

Demonstration of Single Atom Delivery and Loading into a Crossed Vortex Bottle
Beam Trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.6.1

Atom Transport and Reloading Operation . . . . . . . . . . . . . . . . . . . . 91

4.6.2

Further Discussion for Continuous, Uninterrupted Loading

4.6.3

Demands to Upscale Transport and Loading System . . . . . . . . . . . . . . 97

. . . . . . . . . . 96

5 Steering the Optical Convyeor Belt: Addressing an Array of Bottle Beam Traps

98

5.1

Ineffectiveness of an Array of Crossed Vortex BoB Traps . . . . . . . . . . . . . . . . 98

5.2

Atom Trapping in a Gaussian Beam Array of BoBs . . . . . . . . . . . . . . . . . . . 99
5.2.1

Gaussian Beam Array (GBA) . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.2.2

Analysis of Trapping Potential of Gaussian Beam Array . . . . . . . . . . . . 100

5.2.3

Atom Trapping in the Focused Array. . . . . . . . . . . . . . . . . . . . . . 105

x
5.3

Steering the Optical Conveyor Belt . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.3.1

Single BoB Site Addressing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

5.3.2

Atom Transport and Delivery Sequence with Steerable Conveyor Belt . . . . 112

6 Conclusion to Part II

117

6.1

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

6.2

Future Work and Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6.2.1

Route to Continuous Atom Reloading: Vertical then Horizontal Transport . . 118

6.2.2

Using the Optical Conveyor Belt to Address a 2x3 BoB Array . . . . . . . . . 120

6.2.3

Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

7 Part III: An Ultracold Gain Mechanism for a Matterwave Transistor Oscillator

123

7.1

Introduction to Atomtronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

7.2

Background and Motivation for an Atomtronic Transistor Oscillator . . . . . . . . . 125
7.2.1

Analogue to Electronic Transistors . . . . . . . . . . . . . . . . . . . . . . . . 125

7.2.2

Extending the Semiclassical Descriptions of an Atomtronic Transitor . . . . . 127

7.3

Matterwave Transistor Oscillator Principle of Operation . . . . . . . . . . . . . . . . 129

7.4

Modeling Matterwave Transmission Through 1D Transistor Potential . . . . . . . . . 132

7.5

7.4.1

Source, Gate, and Drain Well Setup and Formalism . . . . . . . . . . . . . . 132

7.4.2

Coupling Gate Well Energy Eigenstates With a Dipole Oscillating BEC . . . 133

7.4.3

Reduced Representation of the Quantized Gate Well . . . . . . . . . . . . . . 137

Reduced Gate Well Many-Bodied Hamiltonian . . . . . . . . . . . . . . . . . . . . . 139
7.5.1

Single-Particle Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

7.5.2

Atom-BEC Collision Contributions . . . . . . . . . . . . . . . . . . . . . . . . 140

7.5.3

Expressing Gate-Well Hamiltonian in the Basis of Normal Modes and Normal
Mode Basis Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

7.5.4
7.6

Eigenstates and Eigenvalues of the Interaction Hamiltonian . . . . . . . . . . 145

Matterwave Current Flow From Source to Gate to Drain . . . . . . . . . . . . . . . . 147

xi
7.6.1

Empty Gate-Well with No BEC Present. . . . . . . . . . . . . . . . . . . . 148

7.6.2

Effect of Interaction Potential on Matterwave Current and Gain . . . . . . . 150

7.6.3

Gain Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

7.6.4

Characterizing the Matterwave Gain Mechanism . . . . . . . . . . . . . . . . 154

8 Experimental Observations to Verify Transistor Gain Mechanism

156

8.1

Criteria for Observing Matterwave Transistor Gain Mechanism . . . . . . . . . . . . 156

8.2

Production of a 50 nK Atom Ensemble . . . . . . . . . . . . . . . . . . . . . . . . . . 158

8.3

8.2.1

Laser System and Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

8.2.2

Loading Cold Atoms into the Atom Chip Trap . . . . . . . . . . . . . . . . . 162

8.2.3

Transferring pre-Cooled Atoms to the Atom Chip . . . . . . . . . . . . . . . . 163

8.2.4

Reaching Bose-Einstein Condensation with Forced RF Evaporation . . . . . . 163

Forming the Transistor Oscillator Model Triple-Well Potential . . . . . . . . . . . . . 167
8.3.1

8.4

Projecting SG and GD Barriers onto the Magnetic Potential . . . . . . . . . 167

Matterwave Transistor Oscillator Experiments: Flow of an Ultracold Atom Current
Through Transistor Oscillator Potential . . . . . . . . . . . . . . . . . . . . . . . . . 171

8.5

8.4.1

Loading Ultracold Atoms into the Source-Well . . . . . . . . . . . . . . . . . 171

8.4.2

Powering the Source-Well Ensemble to Commence Atom Currents . . . . . . 173

8.4.3

Atom Current Flow Through the Transistor Oscillator . . . . . . . . . . . . . 175

8.4.4

Using a Colder Source-Well . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

Discussion of Results: Verifying Transistor Gain and Coherence . . . . . . . . . . . . 178
8.5.1

Observation of Transistor Gain Mechanism . . . . . . . . . . . . . . . . . . . 180

8.5.2

Additional Evidence of Gain Mechanism: Observing Canonically Conjugate
Relationship Between Gate-well Position and Momentum Wavefunctions

8.5.3

Coherence in the Matterwave Gain Mechanism . . . . . . . . . . . . . . . . . 185

9 Conclusion to Part III
9.1. . 183

189

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

xii
9.2

9.3

Future Work and Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
9.2.1

Measuring Differential Matterwave Gain . . . . . . . . . . . . . . . . . . . . . 191

9.2.2

Observing Transmission of Gate-Well Normal Modes . . . . . . . . . . . . . . 192

Outlook: Integrated Atomtronic Transistors . . . . . . . . . . . . . . . . . . . . . . . 198

Bibliography

199

Figures

Figure1.1a) Concept of the neutral atom qubit array. Single qubits are located in individual
optical traps, each separated by distance d. b) 8x8 site qubit array made with individual optical traps where each is capable of confining a single atom. c) Fluorescence
image of single atoms occupying each optical trap of the array. These images aretaken from references 1 and 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2

3a) Two-body interaction strength for two ions (due to the Coulomb interaction), twoground state rubidium atoms (due to the van der Waals and magnetic dipole-dipole
interaction), and two Rydberg states excited to the 100s level. b) CNOT truth tableobtained with two Rydberg state Rb atoms excited to n = 97 with a separation of
10 µm. Images are taken from [1] [2] . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.3

4

Dissertation Part II: Concept of reloading an empty qubit array site with a new
atom from some distant cold atom reservoir. A single atom can be delivered using a
dynamic one-dimensional optical lattice without running into adjacent array sites. .

6

xiv
1.4

Dissertation Part III: Setup and principle of operation of the matterwave transistor
oscillator. a) A 1D potential energy diagram of the triple well matterwave transistor
where an ensemble of atoms at chemical potential µ and temperature T occupy thesource well. The source well and drain wells are separated by the gate well, whichis created with two repulsive Gaussian barriers called the Source-Gate (SG) and
Gate-Drain (GD) barriers, which have heights VSG and VGD , respectively. Imagesb) and c) convey the principle of operation of the transistor, showing that a greater
matterwave flux Φ0 flows through the gate when a dipole oscillating BEC occupies
the gate well, compared to the matterwave flux Φ when a BEC is absent from thegate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1

Optical Dipole Potential: Illustration showing how a laser beam red-detuned (δ < 0)or blue-detuned (δ > 0) from resonance can trap or repel atoms, respectively. For
both cases, the function drawn is the potential energy curve Vdip as a result of the realcomponent of the atom-photon interaction −d̂ · E. For the red-detuned laser beam,
the Vdip is a potential energy well and can trap sufficient cooled atoms, while for theblue-detuned case, Vdip is a potential energy barrier which can repel atoms. Bothread and blue-detuned optical dipole potentials are relied on heavily throughout thisdissertation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

xv
2.2

a) 1D configuration for optical molasses featuring two counter-propagating laser
beams of equal intensity overlapping a the atom. b) Plot showing the radiation force
Frad from equation 2.15 that is imparted on an atom moving with velocity v in the+x̂ from a pair of laser beams detuned from resonance by δ. The plot shows the
maximum frictional force imparted to the atom occurs at a frequency detuning to
the red (δ > 0) of the atomic resonance. c) Solution to the equation of motion for an
atom subjected to the radiation force from a pair of counter-propagating lasers whenplaced in an inhomogeneous magnetic field (equation 2.17) as described in section
2.1.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3a) Schematic showing the setup of a 1D MOT. Current running in opposite directions
through a pair of coils produces an anti-Helmholtz field with an approximate linear
gradient B 0 . A pair of counter-propagating lasers with orthogonal, circular polarization overlaps the zero point of the B-field. b) Energy level diagram showing how
the spatially varying B-field Zeeman shifts the m = ∓1 sublevel closer to resonancewith the σ∓ laser when in the ±x̂ region of the B-field. Thus, the atom is always
experiencing a net force directed towards the zero point of the B-field. . . . . . . . . 22

2.4a) Digram showing the 1D σ− /σ+ PGC mechanism setup. b) Atoms moving againstthe σ− laser feature a greater probability of absorbing the σ− beam and are preferentially pumped to the m = −2 Zeeman sublevel over the m = +2 sublevel (thispopulation imbalance of optical pumping is illustrated by thick, red lines for thedominant absorption path and dim grey lines for the less common absorption path).
c) The reverse outcome occurs when atoms move against the σ+ laser beam. The
population imbalance of the Zeeman sublevels creates an unbalanced radiation pressure and subsequently establishes a net frictional force directed against the atom’s
motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

xvi
2.5

a) Illustration of the RF evaporation process. An ensemble of atoms is initiallypumped to the |mF = +2i trapping state and confined in a magnetic potential. The
RF photon frequency is the swept from infinity down to ωflip . All atoms with energy
above the energy corresponding to the ωflip RF photons (pictured above the thickdotted line) undergo spin flips and are pumped to the |mF = −2i anti-trapping stateand expelled from the trap. By continuing the RF sweep down to just above thetrap bottom ωb , only the coldest of the cold atoms will remain in the trap and canundergo transition into a Bose-Einstein condensate. b) Illustration showing howeach RF knife sweep removes the outer tails of the Maxwell-Boltzmann distributionof the trapped atom ensemble. By continuously cutting out the hottest atoms whileallowing rethermalization time, the RF evaporation process leaves a large enoughdensity of atoms at low temperature to result in a high enough phase space density
for Bose-Einstein condensation to occur.

2.6. . . . . . . . . . . . . . . . . . . . . . . . 27

Illustration depicting the overlap of de Broglie waves as atoms are cooled from above
the critical temperature Tc to well below it. As T << Tc , the de Broglie waves
overlap so much that all atoms are completely indistinguishable from one another. . 28

2.7

Plot in “energy space” showing the energy of an atom in the anisotropic harmonic
oscillator as a function of occupying the nx , ny , and nz energy levels. Note that for
an atom occupying some arbitrary energy state (nx , ny , nz ), the total atom energy
is a plane  = x + y + z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.8

Density profile (and solution to the Gross-Pitaevskii equation) of the BEC in theThomas-Fermi approximation when confined to a harmonic potential. . . . . . . . . 34

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3.1

a) Schematic of the AQuA cell vacuum chamber, showing the locations of the threemain regions of the device: the Cs source and 2D MOT region, the 3D MOT region,and the Hex cell. The location of the pinhole that isolates the 2D and 3D MOT
chambers is shown. Additionally, the positions of the ion pumps and non-evaporable
getters are also shown. The cell manifold serves as a mechanical foundation for the
AQuA cell. The manifold is a 1 cm thick, machined Pyrex block that is polished on all
faces. Channels are machined throughout the interior of the manifold to connect the
3D MOT chamber to the Hex cell and also to connect each ion pump to its respective
pumping load. The long lever-arm aspect of the 2D and 3D MOT chamber leaves thecell very susceptible to vibrations. Therefore, two Pyrex support rods are fixed tothe bottom of the 2D MOT chamber and are connected to the AQuA cell manifold.
b) The AQuA cell prior to vacuum processing and bakeout being held with hands forscale. The glass to metal is used to connect the cell to a bakeout station. The final
vacuum seal is completed by anodically bonding a Pyrex window in-vacuo (whilestill connected to the bakeout station) onto the cell manifold. The glass to metal iscompletely removed after pumping the AQuA cell down to UHV. . . . . . . . . . . . 37

3.2a) Anodically bonded source tube, 2D MOT, and 3D MOT chambers. The bottomof the source tube, which contains the conductive feedthroughs, is not yet anodicallybonded to the chamber. b) Close up view of the AQuA Cell source tube showing theglowing Cs dispenser and the non-evaporable getter. The dispenser is glowing due torunning 3.5 A through it in order to expel a cesium gas via thermal emission. c) View
looking up vertically through the bottom of the source tube through the 2D MOT
chamber where the bright dot in the center is fluorescence from a 2D+ MOT that
centered on the 2D-3D pinhole. d) 2D and 3D MOT chamber with the permanent
MOT magnets attached. The silver contacts that connect to the Cs dispenser are
also shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

xviii
3.3

AQuA Hex cell, which serves as the location for the single atom transport andreloading experiments, as well as the location of the qubit array. . . . . . . . . . . . 40

3.4

Left: Schematic showing the construction of the AQuA cell ion pumps. The body ismade from a machined Pyrex block with tubular bore holes running horizontally andvertically through the Pyrex. A titanium cylinder is placed in the center. A metal
rod is welded to the titanium cylinder and is connected to the silicon anode by aspring. The spring pushes on both the the rod and the andode and forms the anodefeedthrough. The cathode is formed in a similar manner by using a bowed titanium
disc to act as a spring to ensure a constant connection between the titanium cylinderand metalized silicon. The titanium disc forms the cathode feedthrough and forms
an electrical connection between the anode and cathode. The opening on the rightside of the Pyrex body is then anodically bonded to the AQuA cell manifold. Right:
Completed AQuA cell ion pumps. Note that the ion pumps in this image are rotated
90 degrees from the given schematic. The metalized silicon forming the anode andcathode is clearly visible. The bottom of the pump is then anodically bonded to the
AQuA cell manifold. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.5

Image of the atomtronics vacuum cell. The cell is still attached to the bakeout station
via the copper pinch off tube and has not yet been pinched-off. . . . . . . . . . . . . 44

3.6a) Vacuum cell with atom chip serving as the top of the cell. b) Underside of theatom chip that is inside the vacuum. The guide wires are shown at the center ofthe chip. c) 2D cross section of the magnetic field lines produced by running current
through one guide wire. d) Uniform bias field oriented purely in the y-direction. e)
Summation of the guide wire field and the y-bias field. The result gives a magnetic
quadrupole field featuring a local minimum a distance d below the chip guide wire.
This plot is a 2D cross-section that is symmetric in and out of the page and thus
the field minimum runs below the entire length of the wire, giving a region where
magnetic trapping of atoms is permitted. . . . . . . . . . . . . . . . . . . . . . . . . 47

xix
3.7

Plots of the guide-wire magnetic field in the ẑ and x̂ directions. The solid and dashedlines represent the magnetic fields with and without the additional x-bias field. . . . 48

3.8a) Top view of the ambient side of the atom chip. The vertically running pairs ofwires are the H-wires and are centered about the chip window. b) Plot of the H-wiremagnetic field BH (equation 3.7) produced by running current of equal direction
and magnitude through two pairs of H-wires centered about the chip window. c)
Combined magnetic potential forming an Ioffe-Pritchard trap from the guide-wirefield (equation 3.6) and the H-wire field (equation 3.7) . . . . . . . . . . . . . . . . . 49

3.9a) 2D cross section of the magnetic field produced by running currents Ix and −Ixthrough the pair of guide-wires separated by distance l (equation 3.8). b) Uniform
bias field oriented purely in the z-direction. c) Summation of the split guide-wirefield (equation 3.8) and the z-bias field. The result gives a magnetic quadrupole fieldfeaturing a local minimum a distance D below the center of the atom chip window.
This plot is a 2D cross-section that is symmetric in and out of the page and thus
the field minimum runs below the entire chip window, giving a region where bothmagnetic trapping of atoms and optical access is permitted. . . . . . . . . . . . . . . 50

4.1

(Color online) a) Schematic of a qubit array constructed with a 2D grid of equally
spaced optical bottle beam traps. b) Concept of using pairs of entangled Rydberg
state atoms to perform quantum logic gates and operations involving multiple qubits
simultaneously. c) A stray background gas atom collides with a trapped qubit andexpels it from the BoB trap. d) A new pre-cooled cesium atom is transported with
an optical conveyor belt to the empty BoB site. e) Exactly one atom at most is thentransferred into the BoB trap, reloading the empty site and keeping the array fully
occupied. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

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4.2

(Color online) a) Setup for the 852 nm cooling and repump lasers. b) Setup for the
852 nm probe lasers. In both schematics, mechanical shutters are used to turn on
and off the laser beams. Additionally, all fibers are 852 nm, polarization maintaining
fibers. c) Laser locking scheme for the Cs D2 transitions used in the single atomtransport and loading work. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.3

(Color online) a) Setup of the 1064 nm atom transport laser system. b) Schematicof the 780 nm blue-detuned bottle beam laser system. . . . . . . . . . . . . . . . . . 60

4.4

(Color online) a) Drawing illustrating a SPP transforming a TEM00 mode Gaussianlaser beam into an LG01 mode Laguerre-Gaussian beam. The phase profile of the
SPP used in this work, showing how one full azimuthal rotation retards the TEM00
phase by 2π, is shown below. b) The LG01 mode created in this experiment bysending the TEM00 mode through the SPP. . . . . . . . . . . . . . . . . . . . . . . . 63

4.5

(Color online) This set of images shows the formation of the Gaussian beam arrayat various locations of the optical setup. a) Intensity plot and the direction of linear
polarization of the 4 identical Gaussian beams taken after a single Gaussian beam
passes through the diffractive beam splitter. b) Intensity and polarization diagramsfor the array after the 4 identical Gaussians pass through the calcite and are focusedby the 200:60 demagnifying telescope. c) Image showing the array after focusingwith the 500:23.125 demagnifying telescope. As shown in section 4.9, this intensity
profile in c) is what is focused into the Hex cell to make an array of bottle beam traps. 64

4.6Matterwave transistor oscillator concept . . . . . . . . . . . . . . . . . . . . . . . . . 65

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4.7

(Color online) a) Optical setup to form a 3D MOT in the AQuA Hex cell. The inset
shows the orientation of the 6 3D cooling beams, the vertically oriented push beam,
MOT coils, and shim coils. A weak probe beam intersects the MOT location andis focused onto a Basler CCD for absorption imaging. b) Optical setup to form a
2D+ MOT in the AQuA 2D chamber. Inset 1 shows the orientation of the 4 coolingbeams. Inset 2 shows the location of the vertically oriented push beam. The pushbeam enters at the base of the 2D chamber and propagates vertically through theentire AQuA cell, exiting through the top surface of the Hex cell. Since it overlaps
the 2D MOT, it “pushes” atoms into the Hex cell that are used to seed the 3D MOT. 67

4.8(Color online) Fluorescence image of A 3D MOT of approximately 107 cesium atoms
and a diameter of 3 mm created in the AQuA Hex cell. The atoms that source the3D MOT are initially cooled in a 2D+ MOT located in a chamber below the Hex
cell. The 2D+ MOT atoms are subsequently pushed up into the Hex cell by use ofa push beam. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.9

(Color online) a) Time-of-flight (TOF) absorption images of the 3D MOT after
performing a CMOT stage. From top to bottom the TOF expansion times are 5
ms, 10 ms, 15 ms, 20 ms and 25 ms. b) and c) show the expansion of the Gaussian
width σx (t) and σy (t) of the 3D MOT as a function of time. The TOF expansion
data corresponds to atom temperatures of Tx = 80.1 µK and Ty = 67.7 µK along
the x̂ and ŷ directions, respectively, giving an average temperature of 73.9 µK. . . . 70

4.10 (Color online) a) and b) show Gaussian width σx and σy time of flight expansions of
the 3D MOT atom cloud after performing 10 ms of polarization gradient cooling. Expansion of the atom cloud along the x̂ and ŷ directions gives respective temperature
measurements Tx = 22.3 µK and Ty = 23.4 µK and average atom cloud temperatureof 22.9 µK. The inset of b) shows an absorption image of a cesium 3D MOT withatom density of 1010 atoms/cm3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

xxii
4.11 (Color online) Pagoda coil mount: a) 3D rendering of the pagoda coil mount. b)
Rendering showing the pagoda mount position with respect to the Hex cell, the 3D
MOT beams, as well as any high numerical aperture lenses. Note that optical accessof all Hex cell windows is still permitted. c) Photograph of the pagoda coil system.
d) Plot showing the direction of the magnetic field gradients from the 3 pairs of x/yshim coils as well as the z-bias coils. . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.12 (Color online) Schematic of the optics and RF electronics used to create and transport the optical conveyor belt. An RF spectrum analyzer measures the velocity of
the conveyor belt by measuring ∆ω of the lattice beams. . . . . . . . . . . . . . . . . 74
4.13 (Color online) Plot showing the Moving Standing Wave Potential for positive acceleration (a > 0), negative acceleration (a < 0), and no acceleration (a = 0).

. . . . . 77

4.14 (Color online) Plot of the adiabatic parameter ξ as a function of the lattice chirpingparameters d∆ω/dt. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.15 Absorption images showing the sequence of transporting cold atoms with an opticalconveyor belt: 1) Atoms are loaded into the conveyor belt by overlapping the 1D
lattice and the 3D MOT. 2) The MOT is turned off, leaving behind a reservoir ofcold atoms in the lattice. 3) The frequency of one lattice beam is linearly frequency
chirped and the optical conveyor belt accelerates to a maximum velocity. After
this acceleration, the frequency of this lattice beam is linearly chirped again but
with the opposite sign in order to slow the atoms down to a halt. The atoms
are transported back to their original starting location by repeating the process inreverse. 4) Atoms are then held in place at the desired, final location. The inset
shows a plot displaying the relative detuning ∆ω of the lattice beams as a functionof time during the conveyor belt transport process. . . . . . . . . . . . . . . . . . . . 80

xxiii4.16 Change in atom density (atoms/cm3 ) in the moving standing wave dipole trap after
transport. The final density ρf is compared to the initial atom density ρ0 for various
AOM detunings and frequency sweep times for the moving standing wave dipole
trap. The numbers written inside the plot show the transportation distance for each
set of parameters. Note the black points indicate atom survival in the stationary
lattice.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.17 (Color online) Beat note spectrum measured by overlapping the two 1064 nm opticalconveyor belt laser beams. The frequency spectrum peaks at 400 kHz and has a fullwidth at half-max of 1.33 kHz as measured at the -3dB point. This signal is used toquantify the lattice beam overlap, relative polarization, and phase jitter. . . . . . . . 82
4.18 (Color online) Various Bottle Beam Intensity Profiles: a) Gaussian interference BoB
made from the destructive interference of two TEM00 Gaussian modes with different
waists. b) Crossed Vortex BoB made from a Laguerre-Gaussian LG01 mode. c)Gaussian Beam Array BoB made from four TEM00 Gaussian modes.

. . . . . . . . 84

4.19 (Color online) Top: Optical setup for generating and imaging the crossed vortex
bottle beam trap. The added optical molasses light is used for fluorescence imaging
on an atom trapped in the BoB. A fraction of atom fluorescence is collected with the
0.4 NA lens and propagated into a counter to perform photon counting statistics.
Bottom: Intensity profile at of the crossed vortex BoB trap forming as the two LG01
mode lasers are focused at the focal plane of the 0.4 NA lens. Images a), b), c), andd) show the trap at axial lengths z = 34 µm, z = 26 µm, z = 12 µm, and z = 0,
respectively from the focus of the 0.4 NA lens. . . . . . . . . . . . . . . . . . . . . . 86
4.20 Crossed Vortex Bottle Beam Trapping Potential . . . . . . . . . . . . . . . . . . . . 87
4.21 Numerically calculated a) radial and b) longitudinal trap depths produced by thecrossed vortex BoB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

xxiv
4.22 (Color online) Histogram of photon counting data recorded during fluorescence imaging of the BoB trap. Two Poisson distributions are present with means of 125 counts
and 215 counts, corresponding to 0 or 1 atom present in the BoB trap, respectively.
The histogram is fitted with a compound Poisson distribution over atom and photon
number. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.23 (Color online) a) Density profiles of 133 Cs atoms at different time-of-flight (TOF)
expansion times after horizontal transport with the optical conveyor belt. b): Graphshowing the size of the atom clouds at different TOF expansion times when extra
PGC is applied (blue dots) compared to no PGC (purple dots). The TOF expansion
data shown here corresponds to an atom cloud temperature of 14.9 µK when PGC
is applied to the transported atoms. Without the extra PGC, the transported atom
temperature is considerably higher at 72.2 µK. . . . . . . . . . . . . . . . . . . . . . 93
4.24 (Color online) Summary of the Atom Transport and Loading Experiment: Left: Timing scheme for different stages of the atom transport and delivery process involving
the 780 nm BoB lasers, 852 nm MOT lasers, and the 1064 nm conveyor belt lasers. A
graph shows the mutual detuning ∆ω between the 1D lattice beams as a function oftime during the conveyor belt transport process. Right: Absorption images showing
atoms transported 1.1 mm with the optical conveyor belt at various times during
travel. Images a), b), c), d), e) show the loaded conveyor belt at 0, 5, 10, 15, and20 ms, respectively. Furthermore, images a) and e) show the conveyor belt at theinitial and final transport locations. Images b) and d) show the conveyor belt when
it reaches the maximum velocity of 212 mm/s. Image c) shows the conveyor belt
at the turn around point. Approximately 3,000 atoms remain in the conveyor belt
after transport. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

xxv
4.25 (Color online) Histogram of photon counting data recorded during fluorescence imaging of the BoB trap after delivering the atom via an optical conveyor belt. Two
Poisson distributions are present with means of 300 counts and 570 counts. The
peaks correspond to loading 0 or 1 atom, respectively, in the BoB trap with theoptical conveyor belt. The histogram shows that a single atom is delivered into the
BoB trap 13.1% of the time. The histogram data is fitted to a compound Poisson
distribution over atom and photon number. . . . . . . . . . . . . . . . . . . . . . . . 95
5.1

(Color online) Bottle Beam Array Concept: A combination of a calcite beam displacer and a diffractive beam splitter is used to form and array of crossed vortex
BoB traps. However, polarization fluctuations between interfering tightly overlapping LG01 beams at the focal plane of the 0.4 NA lens render this concept uselessfor quantum computation purposes. . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.2

(Color online) Top: Optical setup to generate a GBA that produces two bottle beam
trapping sites. Bottom: Intensity images of the GBA as one sweeps through the focusof the array. As one moves away from the focus in the positive and negative axial
direction, each Gaussian beam in the array rapidly diverges and interferes to cap
off the bottle beam trap and produce a barrier in the z-direction. This permits theintensity pattern to function as a 3D trapping potential. The locations of the focusof the two BoB traps are indicated with a red arrow. . . . . . . . . . . . . . . . . . . 101

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5.3

(Color online) Optical intensity image showing a central, dark region of minimum
light intensity surrounded by four intense Gaussian beams. Each beam has an equal
waist of w0 and is separated from it’s nearest neighboring beam by distance d. Adding
the intensities of all four beams creates bright saddle points with intensity Isp thatsurrounds the central, dark region with intensity Idark . Because the laser light used tocreate this pattern is blue-detuned from resonance, the gradient in optical intensity
Isp − Idark creates a potential energy well that can confine atoms in the central, dark
region.

5.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102(Color online) a) Plots of the normalized intensity of the bright saddle points Isp /I0(solid line) and the central dark region Idark /I0 (dashed line) as a function of thearray aspect ratio s. b) Plot of the normalized trapping intensity Itrap /I¯ = (Isp −Idark )/I¯ (solid line) between the bright saddle-point and the central dark region of
one trapping site in the GBA as a function of the aspect ratio s. The dashed line
shows the intensity of the dark, central region that is present at the same aspect ratios, thus indicating the non-zero bias to the trap bottom present in the setup. From
this plot, it is determined that the deepest trapping depth occurs at s = d/w0 = 2.2. 104

5.5(Color online) Intensity image of the Gaussian beam array at the focus of the 0.4
NA f = 23.125 mm lens. Each beam in the array has a beam waist of w0 = 2.21
µm and is separated from its nearest neighbor by d = 4.95 µm. This gives an aspectratio of s = d/w = 2.23. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.6(Color online) Optical trapping potential produced at the bottle beam locations of
the 2x1 array in the radial (x,y) and axial (z) directions. Using 750 mW of λ = 780
nm laser power, a trapping barrier of depth 547 µK is created in the x,y radial
direction and a trap depth of 578 µK is created in the z, or axial direction. The
trapping frequencies for this trap geometry are 8.35 kHz in the radial direction and2.58 kHz in the axial direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

xxvii
5.7(Color online) Demonstration of atom trapping in the 2x1 array of BoB traps. The
fluorescence images in a) and b) show a sum of 10 individual images. The schematic
for fluorescence imaging the array is shown in c). By changing the final imaging
lens from f = 500 mm to f = 1000 mm, the resolution was doubled which is thedifference between images a) and b). . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

5.8(Color online) The optical conveyor belt setup is modified by adding two additional
AOMs along the beam paths of 1D lattice lasers in order to execute angular deflections of the beams along the x and y dimensions. As in the optical conveyor belt forthe crossed vortex BoB setup (section 4.4.4) lattice beam overlap is monitored bymeasuring their relative beat frequencies with an RF spectrum analyzer. . . . . . . . 108

5.9(Color online) Schematic of the beam steering system for the optical conveyor belt.
The 1064 nm beams are split into two beam paths. Each beam passes through an
x-AOM, which alters the angle of the beam in the x dimension. A 1:1 telescope
relay images the center of the x-AOM on the center of the y-AOM. This y-AOM
changes the beam angle in the y dimension. A second 1:1 telescope relays the centerof the y-AOM onto the center of a 5x telescope, which magnifies the beam to 1 cm indiameter. The beam is then focused with a 0.4 NA custom lens onto the qubit arrayin the AQuA Hex cell. At the qubit array plane, the x and y positions of the beams
can be laterally translated by δx and δy by varying the x and y angular deflectionsof the AOMs.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

xxviii
5.10 (Color online) Demonstration that the beam steering system is capable of addressing
both BoB trapping sites in the GBA. These images were taken by only changing the1064 nm beam deflecting x-AOM and y-AOM driving frequencies with no manual
realignment of any beams. a) Focused GBA located inside the Hex cell. b) The focusof the optical conveyor belt spatially overlaps the focus of the GBA. The yellow arrow
points to light from the focused conveyor belt. Using the lattice deflected x-AOMs
and y-AOMs, the optical conveyor belt is repositioned to overlap the first BoB trap,
shown in image c), and the second BoB trap, shown in image d).

. . . . . . . . . . 111

5.11 (Color online) a) Setup for fluorescence imaging of the 1D optical lattice. An opticalmolasses is applied to atoms trapped in the 1D lattice and a fraction of the induced
fluorescence is collected with the 0.4 NA custom lens and focused onto an Andor
iXon EMCCD. b) Histogram of photon counting data recorded during fluorescenceimaging of the 1D lattice. The data shows multiple Poisson distributions that emerge
due to different amounts of atoms loaded in the 1D lattice. Each number N shows
the Poisson distribution corresponding to loading N atoms in the lattice. The distribution is fitted to a compound binomial-quasi Poisson function, which accounts
for the random and discrete atom loading along with the sub-Poissonian spread ofobserved photon counts. The statistics show that the most probable loading number
is 5 atoms.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

5.12 (Color online) Fluorescence images, all showing the same area of interest, of approximately 5 atoms after being transported 100 µm from a 3D MOT to an empty array
BoB trap using the optical conveyor belt. Images a) and b) shows atoms transported
to BoB2 and BoB1, respectively. Image c) is a combination of a) and b) to show
their spatial separation of 5 µm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

xxix
6.1

a) MM MOT setup in the lower 3D MOT Chamber that can propel atoms up into
the Hex Cell. Diagram taken from the PhD thesis of Kai Hudek [3]. b) Sequence of
105 atoms traveling vertically from the 3D MOT chamber into the Hex cell chamber.
A 12W, 1064 nm dipole trapping beam helps guide the atoms upwards. Once theatoms reach a maximum height in the Hex cell, they will be trapped in a crossed
dipole trap which serves as a “secondary cold atom reservoir.”

6.2. . . . . . . . . . . . 119a) Diagram showing the location of the “secondary cold atom reservoir” in relation to
the Hex cell lasers and the BoBs. Atoms confined to this reservoir can be transported
and loaded into a BoB using the methods from chapter 4 and 5 of this dissertation. b)
Atoms from the MM MOT that have traveled vertically into the uppermost chamber
above the 3D chamber and have reached their highest trajectory. The inset showsa small crossed dipole trap as a result of intersecting this vertically traveling MOTwith a horizontal dipole trap.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

6.3

Put in description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

6.4a) Image of the 2x3 BoB trapping array at the focus of the 0.4 NA, f = 23.125 mm
lens. The optical configuration creates 6 BoB traps separated by d = 4.67 µm. b)
Demonstration of the beam steering system addressing all 6 BoB traps in the array.
The yellow arrow points to the optical conveyor belt light. . . . . . . . . . . . . . . . 121

6.5

Chart taken from reference [1] conveying that single atom transport and reloading
is but one element of a successfully operating neutral atom quantum computer. . . . 122

7.1

Concept of atomtronic transistor gain mechansim: . . . . . . . . . . . . . . . . . . . 126

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7.2

Summary of the semiclassical atomtronic transistor: 1D potential energy diagramof the triple-well transistor potential where an source-well ensemble of atoms at achemical potential µS and temperature TS flow with current ISD into the drain wellby classically traversing the barriers separating the centrally located gate from thesource and drain. As the current flows, some atoms rapidly form a BEC in the gate
well with a well-defined, steady state chemical potential µG and TG . Previous experimental work shows the occupancy of this BEC can increase or decrease the current
flowing from to source to drain, which provides an atom current gain mechanism.

7.3

. 128

Setup and principle of operation of the matterwave transistor oscillator. a) A 1D
potential energy diagram of the triple well matterwave transistor potential wherean ensemble of atoms, at chemical potential µ and temperature T , occupies thesource-well. The source-well and drain-well are separated by the gate-well, which iscreated with two repulsive Gaussian barriers called the Source-Gate (SG) and GateDrain (GD) barriers, which have potential energy heights VSG and VGD , respectively.
Images b) and c) convey the principle of operation of the transistor, showing that agreater matterwave flux Φ0 flows into the drain-well when a dipole oscillating BEC
occupies the gate-well, compared to the matterwave flux Φ when a BEC is absent
from the gate-well.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

xxxi
7.4

Setup and principle of operation of the matterwave transistor: a) Plot showing thesource-gate (SG) and gate-drain (GD) barriers at potential energy heights VSG = 30
kHz and VGD = 33 kHz, respectively. The barriers separate the transistor into thethree source, gate, and drain-well regions. In this model, the source and drain-wells
are treated as flat potentials VS (x) = VD (x) = 0 and the gate-well is a slightly
anharmonic oscillator. b) Plot showing the difference in energy ∆E between sequential energy eigenstates of the transistor gate-well. The horizontal axis lists the nthgate-well energy level and the vertical axis is the corresponding energy difference∆E = En − En−1 . For example, the left-most data point shows the energy difference
between the first excited state and the ground state: E1 − E0 . The next data pointshows the energy difference E2 − E1 , and so on. From this plot, it is clear thatfor our model with barrier parameters VSG = 30 kHz, VGD = 33 kHz, w0 = 2 µm
and d = 4.85 µm, a degeneracy exists between the ground and first excited stateof the gate-well and the highest lying pair of gate-well energy eigenstates. That is,
E1 − E0 = E23 − E22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

7.5

Plots showing the energy value of each gate-well eigenstate as well as the energy
difference ∆E between adjacent energy eigenstates for various SG and GD barrier
separations for the gate-well potential when VSG = 30 kHz, VGD = 33 kHz, andbarrier widths are held constant. The solid red line indicates two pairs of adjacenteigenstates with the same ∆E value. Thus, altering the barrier separation d canchange what transitions are degenerate with each other. . . . . . . . . . . . . . . . . 135

7.6

Plots showing the energy value of each gate-well eigenstate as well as the energy
difference ∆E between adjacent energy eigenstates for various SG and GD barrier
widths for the gate-well potential when VSG = 30 kHz, VGD = 33 kHz, and barrier
separation is held constant. The solid red line indicates two pairs of adjacent eigenstates with the same ∆E value. Thus, altering the barrier separation d can change
what transitions are degenerate with each other. . . . . . . . . . . . . . . . . . . . . 136

xxxii
7.7

Reduced representation of the transistor gate well . . . . . . . . . . . . . . . . . . . . 138

7.8

Plots showing a time lapse of the matterwave superposition of normal modes ψ+ +ψ−occupying states ψ2 and ψ3 and oscillating with a phase φζ as well as the excitedcondensate occupying state ψ1 and oscillating with a phase φα . The phase ϕ used for
determining the eigenvalue of the interaction Hamiltonian is the difference of thesephases, given as ϕ = φζ − φα .

7.9

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146a) Transmission probability of the transistor gate-well as a function of incident matterwave energy when no BEC occupies the gate-well, as calculated by the general
impedance method. Resonances in transmission occur when incident matterwaves
have energy coinciding with a gate-well energy eigenstate. b) This inset shows thetransmission resonances near energies coinciding with the highest lying pair boundenergy eigenstates (which correspond to reduced gate-well energy eigenstates |2i and|3i). If incident matterwaves have a slightly different energy, the probability of transmission through the gate-well and into the drain-well decreases significantly and canbe reduced by more than 10 orders of magnitude. Essentially, transmission throughthe “empty” gate-well occurs only when incident matterwaves have energy coinciding
with a gate-well eigenstate. Thus, in the reduced gate-well, incident matterwaves
must have energy equal to E2 or E3 for transmission into the drain-well to occur.
Otherwise, the matterwaves are reflected. . . . . . . . . . . . . . . . . . . . . . . . . 149

xxxiii
7.10 a) Diagram showing how the presence of a dipole oscillating BEC in the transistor
gate-well that specifically couples the highest lying pair of bound gate-well energy
eigenstates broadens the gate-well resonant tunneling bandwidths by introducing theinteraction Hamiltonian, Hint (equation 7.30). The resonant tunneling bandwidths
are broadened from a very narrow transmission resonance (as shown in figure 7.9),
to a wider band Λ2 and Λ3 (given in equation 7.40a and equation 7.40b). b) As
a result of the interaction Hamiltonian, in the example given in section 7.6.2, as asource-well matterwave ψs , initially off-resonance from a gate-well eigenstate, enters
the gate-well, the interaction Hamiltonian permits the atom to exchange energy withthe dipole oscillating BEC in order to match gate-well resonant boundary conditions
while conserving energy. Consequently, the symmetric mode ψs+ couples into thegate-well with subsequent flow into the drain while the antisymmetric mode ψs−is reflected with probability unity. The net effect is the interaction potential has
permitted a matterwave to resonantly tunnel through the gate-well and into thedrain with considerably greater probability than if the interaction potential was
missing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
7.11 a) Plot showing the gate-well resonant tunneling transmission probability for thecase, described above, without the dipole oscillating BEC and interaction potential(dashed line) and when a dipole oscillating BEC occupies the gate-well with Hint 6= 0(solid line). b) Ratio of matterwave resonant tunneling transmission curves whenthe BEC is present to when the BEC is absent. This quantifies that gain for thisgiven example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
8.1

Transitions for the 780 nm cooling, repump, optical pumping, and probe lasers.

. . 160

8.2

Complete laser setup for the atomtronics transistor oscillator experiment. . . . . . . 161

xxxiv
8.3

Diagram a) and b) show the orientation of the 3D MOT lasers with respect to theatomtronics vacuum cell. In a) the pair of vertically oriented MOT beams intersecting in an X shape 15 mm below the bottom surface of the atom chip. In b) the third
set of MOT beams oriented normal to the X shape beams. Note that the top andbottom diagrams are rotated 90 degrees with respect to each other. This diagram istaken from the thesis of Evan Salim in reference [4], as it was his design that is still
being used today. Image c) shows the orientation of the MOT coils and magnetictransport coils with respect to the 3D MOT chamber. Not visible in this image are
the y-bias and z-bias coils. d) Image of the RF antenna that is located approximately
3 mm above the top layer of the atom chip. e) Imaging showing Cooner wire bonded
to individual conductive pads of the atom chip using a conductive epoxy. . . . . . . 164

8.4

Time-of-flight absorption images showing the momentum distribution of the atomic
cloud after a 22 ms TOF. From left to right the images show the atom cloud momentum just before the condensate, at the appearance of the condensate, and at nearlya pure condensate.The transition to a BEC occurs during the forced RF evaporation
stage for RF frequencies of approximately 2.66 MHz. At this stage, the condensate
temperature was measured to be T = 50 nK. The appearance of an inverted parabola
profile shown in the central and right image is characteristic of the onset of a BEC
as derived in equation 2.46 and shown in figure 2.8.

8.5

. . . . . . . . . . . . . . . . . . 166a) Optical setup for producing dynamically controllable SG and GD barriers. b)
Setup to project the SG and GD barriers through atom chip window and onto theharmonic magnetic potential. The same lens that focuses the barriers also serves tofor in-situ absorption imaging of atoms. c) Optical intensity image of the focused
SG and GD barriers with beam waists of w0 = 2 µm and a separation of d = 4.8µm. d) An in-situ absorption image taken through the atom chip window showing
ultracold atoms trapped in the triple-well transistor potential.

. . . . . . . . . . . . 169

xxxv
8.6

Image showing the experimental setup of the SG and GD barrier projection system
as well as the in-situ imaging system relative to the 3D MOT cell and the atom chip.
The inset shows the 0.6 NA Zeiss objective located approximately 3 mm above the
top surface of the atom chip. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

8.7

The following are in-situ absorption images of ultracold atoms at various stages ofpreparing the source-well ensemble. a) A Bose-Einstein condensate with approximately 6,000 atoms at a temperature of 50 nK loaded into the harmonic magnetic
potential. b) By sweeping the SG barrier from the longitudinal edge of the magnetic
potential, nearly all atoms are confined into the transistor source-well. c) The SG
barrier is lowered to its desired height of 30 kHz and the GD barrier is raised to 31
kHz. d) Source-well atom ensemble after raising the source-well chemical potential(described in the upcoming section 8.4.2). By allowing source-well atoms to flow
into the gate-well, we can directly measure the size of the gate-well to verify the SG
and GD barriers are separated by d = 4.8 µm.

8.8. . . . . . . . . . . . . . . . . . . . . 172a) Voltage driving the AOD that controls the x̂ (or longitudinal) position of the SG
barrier as a function of time. By using this sweeping function for the SG barrier,the time averaged optical potential for the SG barrier is shown in b). Image b) is anintensity image of the SG and GD barriers after the above sweeping function. This
serves to raise the source-well chemical potential with the corresponding transistor
potential shown in c). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

8.9

in-situ absorption images of the transistor oscillator at various flow times as an atom
current flows from the source-well, through the gate-well and into the drain-well. The
spatial locations of the SG and GD barriers are indicated with the dashed lines.

. . 176

8.10 Plot showing the temperature Ts of the source-well atom ensemble as a function of
atom current flow time. The source-well ensemble temperature in each data point
was measured using time-of-flight imaging after each subsequent current flow time.
Error bars are propagated from the standard error of the mean for Ts .

. . . . . . . 177

xxxvi8.11 in-situ absorption images of the transistor oscillator at various flow times as an atom
current flows from the source-well, through the gate-well and into the drain-well.
The spatial locations of the SG and GD barriers are indicated with the dashed lines.
Insets show the longitudinal density profiles of the source-well ensemble immediately
before the flow of an atom current (at ∆t = 10) and after an atom current flow timeof 25 ms. Both images show that the source-well ensemble is a BEC as the system
begins to evolve and remains a BEC as current flows though the gate-well and into
the drain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
8.12 Plot conveying how one observes the spread of energy ∆E of the drain-well flux.
The longitudinal position of the flux of atoms in the drain-well is indicative of theirenergy and the longitudinal spread of the atoms is an indication of the spread ∆Eof the atom flux energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

xxxvii
8.13 a) in-situ absorption images showing the flow of ultracold atoms through the transistor potential with VSG = 30 kHz, VGD = 31 kHz, and source well chemical potentialµ = 25 kHz after 25 ms of flow time. The images show a comparison of the atom
current for cases of a 78 nK and 57 nK source-well ensemble. From the images,
it is immediately apparent that a wider range of matterwave energy flows into thedrain-well when the current is sourced with the 57 nK BEC, rather than 78 nK
thermal source. This important result indicates that gate-well coupling strength fora matterwave current increases as the source-well ensemble temperature approaches
the 50 nK energy spacing of the eigenstates of the gate-well (as explained in section
8.1). The correspondence between the increased spread in matterwave energy flowing
into the drain-well as gate-well coupling strength increases suggests the presence of amatterwave gain mechanism. b) in-situ absorption images showing the flow of ultracold atoms through the transistor potential with VSG = 15 kHz, VGD = 16 kHz, andsource well chemical potential µ = 12 kHz. In the 78 nK images, the atom current
has flowed for 22.5 ms, 25 ms, and 27.5 ms respectively (from top to bottom). The
57 nK image shows the atom current after 25 ms of flow time. Similar to figure a),we still observe a wider range of matterwave energy flows into the drain-well when
the current is sourced with the 57 nK BEC, rather than 78 nK thermal source. . . . 182

xxxviii
8.14 a) Envelope of the total position space wavefunction, Ψ(x), of the gate-well, formedby summing symmetric modes ψ+ over a broad range of initial energies that couple
into the gate-well. b) Envelope of the corresponding momentum space wavefunction, Φ(p). Because momentum and position are canonically conjugate variables, by
taking the Fourier transform of the gate-well position space wavefunction Ψ(x), we
immediately arrive at the gate-well momentum space wavefunction Φ(p). This resultshows the effects of the gain mechanism: as more atoms couple into the gate-well, thecorresponding momentum spread in Φ(p) also increases. This is directly observed infigure 8.13, as the spread in momenta (and energy) of the drain-well flux increases
as source-well temperature decreases and gate-well coupling increases. . . . . . . . . 184
8.15 in-situ absorption images after 20 ms of flow time showing the observed differenceof a drain-well matterwave current of coherent and incoherent matterwaves. The
white dashed lines show the locations of the SG and GD barriers. Top: With thegate-well and the gain mechanism present, a drain-well atom flux only appears at
the longitudinal turning point of the drain-well, with no drain-well atoms located atlongitudinal positions from the GD barrier to the turning point. This is attributed todestructive interference occurs within the matterwave superposition at longitudinal
positions between the GD barrier and the drain-well turning point. Moreover, thematterwaves of different frequencies interfere constructively at the drain-well turning
point to form a localized atom flux. This behavior suggests that coherence exists
between drain-well matterwaves. We can test that the interference effects are dueto the gain mechanism by removing it entirely. Bottom: The gate-well is removed,
which elimates the gain mechanism. With a now incoherent matterwave current,no such interferences occur and atoms appear at all longitudinal locations of thedrain-well. From this set of experiments, we conclude that the interaction potential(equation 7.30) establishes a coherent gain mechanism in the atomtronic transistor.

187

xxxix9.1

Plot a) is some function f (x) composed of sine waves with various frequencies. The
plot in c) is the Fourier transform showing what frequencies are present in f (x). Plotb) is a function ψ(x) that is comprised of a linear combination of harmonic oscillator
functions (equation 7.4). The plot in d) is the Hermite-Gaussian transform of ψ(x)which shows what harmonic modes are present in the function ψ(x).

9.2. . . . . . . . 194

Images a) and b) show a 2D plot of |ψ+ (x)|2 for the symmetric mode ψ+ (x) =√1 [ψ22 (x) + ψ23 (x)] (which realize corresponds to ψ+ = √1 [ψ2 (x) + ψ3 (x)] in the22

reduced gate-well model, section 7.4.3) located in the transistor drain-well and withtwo different oscillation phases. Images c) and d) are the respective probability amplitudes |ψ+ (x)|2 . Finally, image e) is the Hermite-Gaussian transform of |ψ+ (x)|2 ,showing that the transmitted symmetric mode is comprised of Harmonic oscillator
modes ψ22 (x) and ψ23 (x), as described in equations 9.7 and 9.8.

. . . . . . . . . . . 197

Chapter 1

Part I: Introduction and Motivation

The work of this dissertation falls into two broad categories: transporting and loading single
neutral cesium atoms into optical bottle beam traps in order to facilitate a continuously operating
quantum computer and developing a purely quantum thermodynamic model and experiment fora matterwave transistor oscillator that demonstrates the existence of a matterwave gain mechanism. The former is part of a five year collaboration between institutions working on the Atomic
Qubit Array (AQuA) project, which itself is part of the IARPA Multi-Qubit Coherence Operations(MQCO) program. The latter is in the field of atomtronics and aims to demonstrate the existenceof a matterwave gain mechanism in an atomtronic transistor oscillator for the eventual use in ultracold atomic experiments. This chapter briefly summarizes both research areas as well as introduces
motivation for each subject. The fundamental and basic theoretical methods that are essential tounderstanding both experiments, as well as the ultra high vacuum systems necessary to performthe work, are described in part I of this dissertation.

1.1

AQuA Project Summary: Neutral Atom Quantum Computing
Part II1 of this dissertation describes work occurring within the field of neutral atom quantum

computing. First proposed by Richard Feynman at Caltech in 1959 [6], quantum computing is a
general, all-encompassing term that describes storing information in a two-state quantum system
in order to perform algorithms analogous to how classical computers perform algorithms using
1

The fundamental and basic theoretical methods that are essential to understanding both experiments as well as
the ultra high vacuum systems necessary to perform the work are described in part I of this dissertation.

2
binary transistor states, or bits. The two-state quantum system, called a qubit, could be the spinof an atomic ground state, a ground state and Rydberg state atom pair, electronic states of anion, a Josephson junction, or a SQUID to name a few examples. Irrespective of the type of qubit,
the ambition of quantum computing is to perform algorithms exponentially faster than classicalcomputers. A major milestone in the field was achieved in 1995, when quantum computing was
first demonstrated by using a single trapped ion to serve as a qubit to perform quantum logic[7]. Since then, multi-qubit quantum logic gate operations have been performed experimentallywith multiple trapped ions [8, 9, 10], numerous superconductors [11, 12, 13], collections of photonic
qubits [14, 15], and neutral Rydberg atom pairs [2, 16, 1, 17, 18]. Just like a classical computer, thenumber of qubits in a quantum computer directly influences its productivity. Most recently, using
superconducting qubits, IBM has steadily increased their commercially available quantum computerfrom 2 qubits in 2016, to 16 qubits in 2017, and finally to using a 20 qubit register in 2018. It isclear that the current state-of-the-art in quantum computing has progressed to the point where itis necessary to increase the number of qubits. Large scale quantum computers that would computewith tens to hundreds of qubits would exceed the capabilities of classical computers by being able
to solve integer factorization, such as using Shor’s algorithm [19], or inverting functions by using
Grover’s algorithm [20], faster than any classical computer. While experiments with trapped ions[21, 22, 23] and superconducting qubits [24, 25] have already succeeded in realizing high fidelity
many-qubit quantum gates, using an array of neutral atom qubits also remains an attractive andviable solution to scaling a qubit register to an arbitrarily large amount.

1.1.1

Scaling a Neutral Atom Quantum Computer to Many Qubits

The Atomic Qubit Array-49 (AQuA) project aims at developing and demonstrating a continuously operating atomic qubit array of 49 individual qubits. The qubit array concept, shown infigure 1.1, features a 2D array of neutral atom qubits all with equal separation between each neighboring qubit. The AQuA-49 project operates with qubit gates based on exciting neutral atoms to
Rydberg states. Using neutral atoms, one can theoretically scale the amount of qubits N to as high3

Figure 1.1: a) Concept of the neutral atom qubit array. Single qubits are located in individual
optical traps, each separated by distance d. b) 8x8 site qubit array made with individual opticaltraps where each is capable of confining a single atom. c) Fluorescence image of single atoms
occupying each optical trap of the array. These images are taken from references 1 and 2.

as one desires with very minimal crosstalk between the remaining N − 1 qubits. This is because
neutral atoms differ from ions with respect to their state dependent interaction properties. This isevident in figure 1.2, which shows the two particle interaction strength as a function of separation
distance r for two ions, two neutral ground state rubidium atoms, and two rubidium atoms excited
to Rydberg states.
First, a pair of ions separated by a distance r interact predominantly with the 1/r Coulomb
potential. Because the Coulomb potential only decreases with a single power of r, it remains
strongly interacting, even at long distances, which are great attributes for qubits. The drawback
is the Coulomb interaction is always on, making it difficult to create an arbitrarily large registerof ion qubits. Next, the interaction between two ground state neutral atoms is dominated by the
1/r6 van der Waals potential at short distances and by the 1/r3 magnetic dipole-dipole interaction
at large distances2 . With a separation of 10 µm, this 1/r6 ground state interaction is very weak at
approximately U/~ ≈ 10−5 Hz in frequency units. However, at this same separation, two Rydberg
state atoms have a very strong r−3 electric dipole-dipole interaction of approximately 107 Hz. This
result gives the core principle of operation of the neutral atom qubit array: selectively turning on
the Rydberg interaction (by exciting an atom to such a state) allows one to selectively increase
an atom-atom interaction by 12 orders of magnitude [1]. The ability to control the interaction
2Here, long distances refers to any atom separation r ≈ 30nm.

4

Figure 1.2: a) Two-body interaction strength for two ions (due to the Coulomb interaction), twoground state rubidium atoms (due to the van der Waals and magnetic dipole-dipole interaction),and two Rydberg states excited to the 100s level. b) CNOT truth table obtained with two Rydberg
state Rb atoms excited to n = 97 with a separation of 10 µm. Images are taken from [1] [2]strength by a factor of 1012 over a wide spatial range appears to be unique to the Rydberg system
and makes an array of Rydberg-mediated quantum gates a viable solution to a many-qubit quantum
computer. In fact, recently, a controlled-NOT (CNOT) gate has been demonstrated between two
Rydberg state qubits [2] and it’s measured truth table is shown in figure 1.2.

1.1.2

Drawback of a Neutral Atom Qubit Array: Atom Loss

A drawback of using neutral atom qubits, as opposed to ion qubits and superconducting
qubits, is that neutral atoms must be held in optical traps enclosed inside ultra high vacuum
chambers. This results in unavoidable atom losses due to finite lifetimes on all of the neutral
atom qubits trapped in the array shown in figure 1.1. Atom loss poses a serious problem sincefinite trapping lifetimes endangers the ability of using an array of neutral atom qubits to perform
arbitrarily long quantum algorithms. The solution to this problem, which is the subject of part IIof this dissertation, is to reload any array traps that lose a qubit (due to background gas collisions)with a new, pre-cooled atom from some distant cold atom reservoir. Providing or transporting a precooled atom from some reservoir has been accomplished in past experiments [26, 27, 28, 29, 30, 31].
However, common to each of these experiments is that they take place in large vacuum systems

5
where optical isolation and stray field isolation throughout the chamber occurs easier than in avery compact vacuum cell (such as the vacuum cell used in this dissertation). Moreoever, theseexperiments all involve loading single atoms into a physically visible object (such as a cavity) as
opposed to an infrared optical trap, whose location cannot be assumed a priori.

1.1.3

Solution to Atom Loss: Transporting Atoms to Reload Distant Unoccupied
Array Optical Traps

Site selective atom reloading of the qubit array can be accomplished by using a dynamic,
one-dimensional optical lattice (also called an optical conveyor belt) to deliver a single atom to anunoccupied array site [32]. The concept of such an atom reloading scheme is shown in figure 1.3.
Moreover, the atom transport system must be scalable such that it is capable of addressing anddelivering to an array of sites and not just a single site. Accordingly, the atom transport system
must be “steerable” in that it could deliver an atom to one site and then be repositioned anddeliver an atom to another array site. Finally, the qubit array in the AQuA-49 project is located ina miniaturized and portable vacuum chamber which makes optical isolation an intricate problem:
Any stray light from the atom transportation lasers can destroy the qubit array coherence andwill disrupt any ongoing quantum computations. Thus, to transport atoms from a reservoir to an
empty qubit array site without disrupting any occupied array sites, the issue of optical isolation
presents a compelling, but otherwise difficult problem.

6

Figure 1.3: Dissertation Part II: Concept of reloading an empty qubit array site with a new atomfrom some distant cold atom reservoir. A single atom can be delivered using a dynamic onedimensional optical lattice without running into adjacent array sites.

7

1.2

Ultracold Matterwave Transistor Oscillator
Part III of this dissertation involves a drastically different theory and ongoing experimentin the field of atomtronics that introduces a new gain mechanism for a matterwave transistor
oscillator. We start with a well-established, semiclassical description and experiment [33][5][34]of an atomtronic transistor and model the system with a purely quantum mechanical formalism.
This quantum model differs from the semiclassical results only when the atoms flowing throughthe transistor have sufficiently low enough temperatures. In this “low temperature regime”, themotional state of a dipole oscillating BEC, placed in the transistor itself, couples atom transitions
between high lying transistor energy eigenstates as an atom current flows through the transistor.
In this case, the transistor oscillator displays interesting physics that was not observed in thesemiclassical theory or experiments, namely the presence of coherent gain mechanism that broadens
the range of atom energy (and hence momentum) that may flow through the transistor.

1.2.1

Atomtronics: Ultracold Atomic Circuits

Atomtronics [35][36] is an emerging field of ultracold atomic physics that focuses on quantum
circuits that operate as atom analogues to electrical circuits, where an atom current takes theplace of an electron current and the chemical potential takes the place of a voltage. In their design,
atomtronic circuits are drastically different from their electronic circuit counterparts, as they involve
manipulating ensembles of ultracold atoms to flow through a variety of potential energy landscapes,
such as• Optical potentials produced from laser beams (in particular, the experiments in [37][38])• Harmonic magnetic potentials created with atom chips (as demonstrated in references[39][40][41][42][43])• A hybrid combination of optical and magnetic potentials, such as the transistor potential
used throughout chapters 7 and 8 of this dissertation.

8
At the heart of all atomtronic circuits is a Bose-Einstein condensate (BEC) [44], which supplies asource of coherent, ultracold atoms that are controlled to flow throughout a network of quantum potentials. Some recently demonstrated atomtronic circuits of note include BECs flowing throughout
double-well [33] and triple-well potentials [5], as well as BECs flowing through 1D optical lattices[45][46][47] and optical ring lattices [48][49][50]. Irrespective of the circuit potential geometry, theobjective of atomtronics remains the same: manipulate a current of ultracold atoms in some tailored potential in order to study fundamental physics or to create purely quantum mechanicaltechnologies [51].
Accompanied with such high aspirations of atomtronics comes a difficult, innate challenge
ingrained in the field: such ultracold atom circuits are inherently a many-bodied, open quantum
system, since atoms and BECs interact and dissipate as they flow throughout the circuit from somesource to a sink or drain. While this immensely complicates the theoretical description of atomtronic
circuits, complete descriptions are usually simplified by invoking various approximations, such as• Neglecting dissipation and treating the atomtronic circuit as a closed system (as demonstrated in references [52][53][54][55])• Reducing the many-bodied system from a very large number to a three or four-body problem• Ignoring the BEC mean-field interaction effects (as described in reference [34]).
Despite the the necessity of using approximations just to solve for something, atomtronics has
made unwavering progress in ultracold atom technologies such as realizing quantum simulators[56][57][58][59], improving precision measurements [60][61], creating an atomtronic analogue to a
SQUID (called an AQUID, or atomtronic quantum interference device [62][63]), and realizing thefirst-ever atom battery [33]. Moreover, if used in conjunction with well-established matterwave
interferometry experiments, atomtronics can help enhance inertial sensing and gravimetry sensitivities by up to 10 orders of magnitude as compared to their light wave interferometry counterparts[64]. Finally, atomtronics can conceive new types of quantum technologies by combining various

9
atomtronic circuit elements into an integrated ultracold atom circuit. One such integrated ultracold
circuit, which is the subject of part III of this dissertation, is a matterwave transistor oscillator.

1.2.2

Analogue between Electronic and Matterwave Transistors

Fundamental to many technological developments since its demonstration by Bardeen, Shockley, and Brattain at Bell Labs in 1948, the transistor is a three-terminal semiconductor electronic
device that uses an applied voltage or current on one terminal to amplify or switch electrical signals
on another terminal [65]. An atomtronic transistor would behave similarly to its semiconductor
counterpart by switching or amplifying atom, or matterwave, currents rather than electrical currents. Moreover, in place of three electrical terminals (commonly known as the source, gate, anddrain terminals) found on semiconductor transistors, an atomtronic transistor features three quantum potential energy wells that are capable of confining ultracold atoms. In similar form andfunction to the semiconductor transistor, these three potential energy wells, which are labeled thesource-well, gate-well, and the drain-well, together behave as an atomtronic transistor source (tosource an ultracold atom ensemble to flow through the system) a gate (confining an additional
ultracold ensemble to provide a gain mechanism), and a drain (which functions as the output portof the transistor).

1.2.3

Emergence of a Gain Mechanism for a Matterwave Transistor

Most importantly, for such source-gate-drain system to be called an atomtronic transistor,
it must display a gain mechanism for a matterwave or atom current. In this atomtronic transistor
oscillator, gain occurs due to the emergence of a new atom-BEC interaction that occurs whenatom currents collide with a dipole oscillating BEC confined to the transistor gate-well. This
new interaction, which specifically arises due to the motional state of an oscillating gate-well BEC
coupling high lying gate-well energy eigenstates, is very similar to the Rabi model interaction
between two-level atoms and a photon field. The interaction serves to increase the atom current
flowing through the transistor by ultimately amplifying the matterwave probability amplitude in

10the transistor drain-well. This manifests itself in an increase of the atom flux Φ, or atoms per unit
time flowing into the drain-well, as well as broadening the range of energies (and momenta) carriedby the atom current flowing through the potential (figure 1.4).

1.3

Dissertation Outline
The remainder of this dissertation is organized in the following way: Chapter 2 provides

some fundamentals and the basic theory for the methods that are essential to understanding thework contained within this dissertation. The concepts of laser cooling, optical trapping, magnetic
trapping, evaporation, and Bose-Einstein condensation as they pertain to the experiments in thisdissertation are summarized. Chapter 3 describes in detail the ultra high vacuum systems and theatom chip system that together, provide the experimental bedrock for this thesis. Next, we present
single atom transport experiments to reload optical traps for quantum computing (as motivated in
section 1.1.3): In chapter 4, we describe an atom transport system for loading a single atom from areservoir into a blue-detuned crossed vortex bottle beam trap using a dynamic 1D lattice. Chapter
5, details scaling the atom transport and reloading system to deterministically load any arbitrarysite of an array of bottle beam traps. Chapter 6 concludes the atom transport and reloading
work and gives an outlook for the field. Finally, we present the atomtronic transistor oscillator
theory and experiments (as motived in section 1.2): Chapter 7 details a model for the matterwave
transistor oscillator and explicitly derives the many-bodied Hamiltonian for the system, showing
the emergence of a gain mechanism. Chapter 8 describes the ongoing experimental efforts andresults with the matterwave transistor oscillator and presents data suggesting the presence of acoherent gain mechanism as well as coherence in the ultracold current flowing through the system.
Chapter 9 concludes by looking ahead at the future of the atomtronic transistor experiments aswell as providing interesting applications for such a matterwave transistor device.

11

Figure 1.4: Dissertation Part III: Setup and principle of operation of the matterwave transistor
oscillator. a) A 1D potential energy diagram of the triple well matterwave transistor where an
ensemble of atoms at chemical potential µ and temperature T occupy the source well. The sourcewell and drain wells are separated by the gate well, which is created with two repulsive Gaussian
barriers called the Source-Gate (SG) and Gate-Drain (GD) barriers, which have heights VSG and
VGD , respectively. Images b) and c) convey the principle of operation of the transistor, showingthat a greater matterwave flux Φ0 flows through the gate when a dipole oscillating BEC occupies
the gate well, compared to the matterwave flux Φ when a BEC is absent from the gate.

Chapter 2
Laser Cooling and Trapping 133 Cs and 87 Rb Atoms

The work and experiments covered in this dissertation involves manipulating and imaging
cold and ultracold atoms. For the AQuA-49 single atom reloading experiment, we must be able
to cool 133 Cs atoms to a temperature of approximately 15 µK and confine them in optical dipole
traps. For the atomtronics transistor oscillator experiments, we must be able to trap even colder
atoms of 87 Rb in magnetic potentials and rapidly condense them to a Bose-Einstein condensate
with a temperature of about 50 nK. Remarkably, the atoms used in our experiments begin at aconsiderably higher temperature of 300 K, meaning the atoms for our experiment must be cooledby as much as twelve orders of magnitude. Thus, producing any type of successful results relies on
our ability to efficiently cool and trap atoms in a compact vacuum apparatus. Therefore, a brief
overview on the fundamental principles involved in our laser cooling and trapping techniques usedthroughout both experiments is given. This chapter is outlined as follows:
Starting from the Hamiltonian of a two-level atom interacting with a laser field (section
2.1.1) we quickly arrive at forces that give rise to optical dipole trapping (2.1.2), magneto-opticaltrapping (2.1.3), and polarization gradient cooling (2.2.1); all of which are absolutely essential tothe experiments in this dissertation and can cool an initially room temperature ensemble of 109
atoms to about 10 − 20 µK in about one second. We then show how these pre-cooled atoms aresubsequently trapped in magnetic potentials (2.4.1) that are produced on an atom chip, whichallows further cooling with forced radio-frequency evaporation (2.5.1) to produce a Bose-Einsteincondensate (2.6) with a temperature as low as 50 nK.

13

2.1

Atom-Photon Interactions

2.1.1

Hamiltonian and Energy Shift

Consider a two-level atom with a ground state |gi and an excited state |ei, with energies Eg
and Ee , respectively. By placing the atom in an oscillating electric field E produced by a laser beamwith frequency ω, the atoms will gain an induced electric dipole moment d̂. In such a configuration,
the atom experiences an energy shift [66, 67] given by the interaction Hamiltonian
H 0 = −d̂ · E.

(2.1)

Because H 0 is a perturbation to the energy of the atom, the associated change in the atom’s groundstate energy Eg is given by standard perturbation theory [68, 69, 70] as∆Eg =X | he| H 0 |gi |2
e6=g

Ee − Eg= hg| d̂ · E |ei

11
he| d̂ · E |gi + hg| d̂ · E |ei
he| d̂ · E |gi
Ee − Eg + ~ωEe − Eg − ~ω
|
|
{z
}
{z
}
absorption(2.2)

emissionwhere the first term is the energy shift due to the atom absorbing a photon of energy ~ω fromthe laser and the second term represents the energy change due to the atom emitting a photon ofenergy ~ω. In both the single atom loading and transistor oscillator experiments described in thisdissertation, the cooling laser frequency ω is very close to resonance. That is Ee − Eg ≈ ~ω. Thus,
the absorption contribution to the ground state energy shift (equation 2.2) is negligible comparedto the emission contribution. Using this approximation, the ground state energy shift for an atomin the laser field can be approximated as∆Eg ≈ hg| d̂ · E |ei

1| hg| d̂ · E |ei |2| hg| d̂ · ˆ |ei |2 2
he| d̂ · E |gi ==E
Ee − Eg − ~ωEe − Eg − ~ωEe − Eg − ~ω

(2.3)

where ˆ is a unit vector indicating the direction of the electric field with magnitude E produced by
the laser.
The ground state energy shift (equation 2.3) can also be examined using classical electrodynamics. Using that the induced dipole moment is proportional to the electric field by means of the14atom polarizability [67], the shift to the ground state energy of the atom in the laser beam can also
be expressed as
Z∆Eg = −d̂ZdE = −α(ω)

E · dE(2.4)

1= − α(ω)E 2
2

where the relation d̂ = α(ω)E was used with the atomic polarizability α(ω) defined as a functionof the frequency ω of the laser electric field E. By equating equations 2.3 and 2.4, one finds thatthe atomic polarizability for a two-level atom in the laser electric field isα(ω) ≈| hg| d̂ · ˆ |ei |2.
Ee − Eg − ~ω

(2.5)

Before continuing on, a correction factor must be applied to the atomic polarizability result of equation 2.5. In its current state, the result of equation 2.5 intrinsically assumes an infinite lifetime1
for an atom in the excited state |ei. This, of course, is false since an excited state decays exponentially with time constant τ = 1/Γe , where Γe is the linewidth of the transition |ei → |gi. The
polarizability in equation 2.5 can be corrected to reflect this finite lifetime by including the complex
energy shift of i~Γe /2 due to transitions to the ground state2 . That is Ee → Ee − i~Γe /2 = ∆Ee .
Applying this energy shift in equation 2.5, the atomic polarizability becomes| hg| d̂ · ˆ |ei |2| hg| d̂ · ˆ |ei |2α(ω) ==
Ee − i~Γe /2 − Eg − ~ω~(δ 2 + Γ2e /4)



Γe
δ+i
2

(2.6)

where we have introduced the laser detuning δ = ωe − ωg − ω with ωe = Ee /~ and ωg = Eg /~ being
the energy eigenvalues of the |ei and |gi states as expressed in frequency units.
This argument is also extended to the ground state |gi where in a very general case, the1

The infinite lifetime arises because the energy Ee of the excited state |ei is purely real. Thus, the probability offinding the atom in the excited state is a stationary state and will never decay, since




Ee
Ee
|Ψ|2 = |A|2 exp i t exp −i t = |A|2 = constant.
~~2

By including the imaginary energy shift −i~Γe /2 to the excited state energy, the probability amplitude correctlyreflects a decay with time constant τ = 1/Γe due to the finite lifetime 1/Γe of the excited state:
 
 
 
 
Ee − i~Γe /2
Ee + i~Γe /2|Ψ|2 = |A|2 exp it exp −it = |A|2 e−Γe t .
~~15ground state energy shift ∆Eg in a laser beam is also written as a complex quantity [68]:∆Eg = Vg − i~Γg /2.

(2.7)

The real component Vg represents the energy shift of the atom ground state due to the laser electricfield E and the imaginary component −i~Γg accounts for the finite lifetime τ = 1/Γg of |gi due
to transitions |gi → |ei that are induced by the atom absorbing the laser radiation field. We canthen insert equation 2.6 into equation 2.4 and equate the result to equation 2.7 to formulate an
expression representing the ground state energy shift of an atom due to a laser field:
1
Vg − i~Γg /2 = − α(ω)E 22
1 | hg| d̂ · ˆ |ei |2
Γe
=−
δ
+
i
E2
2 ~(δ 2 + Γ2e /4)
2


~
Ω2
Γe
=− 2
δ+i
2 δ + Γ2e /4
2

(2.8)

where we have introduced the Rabi frequency Ω = | hg| d̂ · E |ei |/~, which is merely the magnitudeof the original perturbation H 0 = −d · E (equation 2.1) expressed in frequency units [71]. Equation
2.8 is a very important result for the laser cooling and trapping used throughout this dissertation:
By equating the purely real components, equation 2.8 gives the atom energy shift due to scattering
photons from the laser electric field and directly leads to the optical dipole force that is used totrap and repel atoms. Additionally, by equating the purely imaginary components of equation
2.8, one arrives at the atom energy shift due to absorbing photons from the laser field and leads
immediately to the radiation force, which permits the cooling of atoms in an optical molasses andleads to more effective cooling by means of a magneto-optical trap.

2.1.2

Optical Dipole ForceBy equating the purely real components of both sides of equation 2.8, we arrive at an expression for the effective potential Vg :Vg = −~δΩ2.
2 δ 2 + Γ2e /4

(2.9)

16
This effective potential Vg is called the optical dipole potential [72] and while not obvious in thisform, Vg permits atom trapping at the maxima of an intense laser beam. To see this, some simple
algebra is used to rewrite equation 2.9 in a form that is more enlightening to the experimentalist:

Vg = −~δΩ2
δ
| hg| d̂ · ˆ |ei |2 2~E=−
2 δ 2 + Γ2e /4
2 δ 2 + Γ2e /4
~2
!


2
ˆ
δ
3 πc2
|
hg|
d̂
·

|ei
|
=−
0 cE 2ω03
2 ω03 δ 2 + Γ2e /4
3~c3 π0(2.10)

where c is the velocity of light, ω0 = ωe −ωg , and 0 is the permittivity of free space. Some identities
can be used to further simplify this result. We first use that the intensity I(x) of an electromagnetic
wave with amplitude E can be expressed as I = c0 E 2 [67]. Additionally, we use that linewidth ofthe excited state |ei can be alternatively expressed as Γe = ω03 | hg| d̂ · ˆ |ei |2 /3~c3 π0 [72]. Finally,
the lasers used for dipole trapping in the experiments described in this dissertation (section 4.5.1
and 8.2.1) are detuned far3 from resonance and hence we can state that δ 2 >> Γ2e . Using theseidentities and the approximation δ 2 >> Γ2e , we can write that the optical dipole potential imparted
onto an atom by a far-detuned laser beam is
3 πc2
Vdip ≈ −
2 ω03



Γe
δ


I(x).

(2.11)

Equation 2.11 shows that the optical dipole potential can be attractive (and trap atoms) or repulsive(and repel atoms) depending on the sign of the laser detuning (figure 2.1). It immediately follows
that if the laser features a spatially varying intensity (∇I(x) 6= 0), then the atom will experiencean attractive or repulsive force (depending on the sign of the laser detuning δ) directed towards oraway from the laser intensity maximum given as
3 πc2
Fdip =
2 ω03



Γe
δ


∇I(x).

(2.12)The result Fdip of equation 2.12 is called the optical dipole force and is a very useful tool to trap
(or repel) neutral atoms that have been previously cooled to milliKelvin temperatures or lower
3

As described in chapters 4 and 5, we use a 1064 nm and 760 nm laser for dipole trapping and repelling of 133 Cs
and Rb atoms, respectively. That corresponds to detunings of order 1012 Hz for 133 Cs and 109 Hz for 87 Rb. For
both types of atoms, the linewidths are of order 106 Hz, thus validating δ 2 >> Γ2e .
87

17Figure 2.1: Optical Dipole Potential: Illustration showing how a laser beam red-detuned (δ < 0)or blue-detuned (δ > 0) from resonance can trap or repel atoms, respectively. For both cases, thefunction drawn is the potential energy curve Vdip as a result of the real component of the atomphoton interaction −d̂ · E. For the red-detuned laser beam, the Vdip is a potential energy well andcan trap sufficient cooled atoms, while for the blue-detuned case, Vdip is a potential energy barrier
which can repel atoms. Both read and blue-detuned optical dipole potentials are relied on heavily
throughout this dissertation.

18[73]. The experimental consequences of equation 2.12 are frequently used throughout both experiments presented in this dissertation, most notably facilitating the optical trapping and transportof single neutral 133 Cs atoms (section 4.4 and 4.5) and for constructing and preparing atoms in thematterwave transistor oscillator potential (section 8.2 and 8.3).

2.1.3

Radiation Force

Returning to equation 2.8, if we instead equate the purely imaginary components, we immediately arrive at an expression for Γg , or the rate that atoms absorb a photon from the laser beam:

Γg =

Γe
Ω2
2 δ 2 + Γ2e /4(2.13)

where again, δ is the detuning of the laser beam from resonance. Now, for simplicity, consider atwo-level atom placed in a one-dimensional light field produced by a single pair of counter propagating laser beams along the x̂ direction (figure 2.2a). Since the atom receives ~k momentum after
absorbing a photon from the beam, the rate of change of momentum, or force, due to absorbing aquanta of light from each laser beam is thenFrad = ~kΓg = ~kΓeΩ222 δ + Γ2e /4(2.14)

which is called the radiation force. Equation 2.14 is only valid when the atom in the laser field is
at rest. In actual experiments in the lab, atoms are always moving with some non-zero velocity.
An atom moving along the +x̂ direction with velocity v will observe4 the frequency of each laser
beam (with wavenumber k) to be Doppler shifted [74] by an amount k · v [75]. Thus, the radiation
force5 on an atom moving at velocity v due to a pair of counter-propagating laser beams is


Γe
Ω2
Ω2
Frad = F−x̂ + F+x̂ = ~k−.
2 (δ + kv)2 + Γ2e /4 (δ − kv)2 + Γ2e /4(2.15)For a red-detuned laser beam (δ > 0), equation 2.15 imparts a velocity dependent, frictional force
on the atoms [76] with the dependence on δ shown in figure 2.2b.
4
5This Doppler shift of the laser beams is observed by the atom in it’s rest frame.
The subscripts ±x̂ denote the force from each laser beam propagating in the ±x̂ direction.

19

Figure 2.2: a) 1D configuration for optical molasses featuring two counter-propagating laser beams
of equal intensity overlapping a the atom. b) Plot showing the radiation force Frad from equation
2.15 that is imparted on an atom moving with velocity v in the +x̂ from a pair of laser beams
detuned from resonance by δ. The plot shows the maximum frictional force imparted to the atom
occurs at a frequency detuning to the red (δ > 0) of the atomic resonance. c) Solution to theequation of motion for an atom subjected to the radiation force from a pair of counter-propagating
lasers when placed in an inhomogeneous magnetic field (equation 2.17) as described in section 2.1.4.

20
Physically, the radiation force occurs due to the Doppler effect: the atom absorbs (andscatters) more photons from the laser that propagates against the atom’s motion because thislight, if red-detuned in the lab frame by some amount δ, is subsequently blue-shifted in the atom
frame back onto (or near) resonance with the |gi ↔ |ei transition. Consequently, the atoms absorb
far less light from the other laser beam (which propagates in the same direction of the atom) sincein the atom frame this laser is Doppler shifted out of resonance. By red-detuning the lasers to the
optimal δ (figure 2.2b), the radiation pressure from the laser light imparts a maximum frictional
force, or optical molasses, that removes kinetic energy from the atoms, cooling them in the process.
This method of laser cooling is called Doppler cooling, since it relies on the Doppler effect tosystematically remove energy from atoms that move against a counter-propagating laser beam.
While the radiation force given in equation 2.15 has the capabilities of cooling atoms to a few
hundred microKelvin, atom trapping with it is unstable as the radiation force Frad is divergenceless.6

Because ∇ · Frad = 0, the radiation force cannot possibly be center-seeking at all points

in space and consequently fails as a stable means to trap atoms [77]. The force given in equation
2.15 can be made stable by introducing a positionally dependent magnetic force in addition to theradiation force by placing the entire atom-radiation field system into an inhomogeneous magnetic
field (figure 2.3a).

2.1.4

Magneto-Optical Trapping of Atoms

Consider a one-dimensional case where an atom is placed into a magnetic quadrupole field
generated by anti-Helmholtz coils (figure 2.3a). The magnetic field B varies linearly along the x̂
direction with a gradient of magnitude B 0 = ∂B/∂x. Due to the Zeeman effect [78], when the atom
6
Because the electric field intensity (encapsulated in the Rabi frequency Ω) from the laser beam is Gaussian, theradiation force can be described more generally in cylindrical coordinates as

Frad ∝

22ẑ −r2 /z2re+ r̂ 3 e−r /z2zz

and taking the divergence yields∇ · Frad =1 ∂ r −r2 /z2∂ 1 −r2 /z2
e
+
e
= 0.
3
r ∂r z∂z z 2

21
is located at position x in the magnetic field B, the atomic energy levels change according to∆E = gµb mB 0 x(2.16)

where g is the Lande g factor, µb is the Bohr magneton, and m is the magnetic quantum numberof the atom. The important result of equation 2.16 is that magnetic field establishes a positionallydependent energy shift for the atom. Combining this with the discussion in section 2.1.3, given an
atom moving with velocity v in an inhomogeneous magnetic field, the atomic transition |gi → |ei
experiences an effective Doppler shift (in frequency units) of ∆νdop = ±kv as well as an additional
Zeeman shift (in frequency units) of ∆νzee = gµb xB 0 /~. Therefore, when inserting values for the
total detuning δ for the radiation force Frad given in equation 2.15, one must consider the effects
of both the Doppler shift of the laser light and the Zeeman shift of the atomic transition. In thiscase, the total radiation force on an atom becomes
Frad = ~k

Γe 

2

= −2~kΓeΩ20δ + kv + gµ~b B x

2
+ Γ4e

!

Ω2 δ
δ 2 + Γ4e

2

2

kv +

−

gµb B 0Ω20
δ − kv − gµ~b B x

2

2
+ Γ4e



(2.17)

x .

~With the result of equation 2.17, the radiation force Frad is now proportional to both the velocity
and position of the atom. In this instance, the divergence of Frad is non-zero and in cases wherek 2 >> gµb B 0 /~ (which is common in our cold atom experiments) the motion of an atom subjected
this force is that of an overdamped harmonic oscillator (figure 2.2c). Thus, the radiation force inequation 2.17 permits the trapping of atoms in the minimum of the magnetic quadrupole field.
In order to ensure that atoms remain trapped, the radiation force must always be directed
towards the B-field minimum (located at x = 0). This can occur by giving the counterpropagating
lasers orthogonal, circular polarizations σ+ and σ− (figure 2.2a). The physics behind this trapping
scheme is illustrated in figure 2.3b and is explained as follows: The atom in the +x̂ region of themagnetic field will absorb the greatest amount of light from the σ− beam due to the magnetic field
Zeeman shifting the m = −1 magnetic sublevel much closer to resonance with the σ− laser while22
subsequently shifting the m = +1 magnetic sublevel further out of resonance with the σ+ laser.
Thus, when the atom is in the +x̂ region, it experiences a net force opposing its motion. The
opposite effect occurs when the atom is in the −x̂ region of the magnetic field, as the atom absorbs
a greater amount of the σ+ laser and experiences a net force directed in the +x̂ direction. Thus,
the total net force is always directed towards the origin x = 0. This atom trapping configuration iscalled a magneto-optical trap (MOT) [79] and with it, we can cool atoms to the Doppler temperature
limit, which corresponds to a temperature of 145.57 µK for 87 Rb and 125 µK for 133 Cs. The MOT
is a ubiquitous tool in cold atom science and serves as the starting point for both experiments
presented in this dissertation.

Figure 2.3: a) Schematic showing the setup of a 1D MOT. Current running in opposite directions
through a pair of coils produces an anti-Helmholtz field with an approximate linear gradient B 0 . A
pair of counter-propagating lasers with orthogonal, circular polarization overlaps the zero point of
the B-field. b) Energy level diagram showing how the spatially varying B-field Zeeman shifts them = ∓1 sublevel closer to resonance with the σ∓ laser when in the ±x̂ region of the B-field. Thus,
the atom is always experiencing a net force directed towards the zero point of the B-field.

23
2.1.5

Sub Doppler Cooling with σ+ /σ− Polarization Gradients

Using the laser cooling methods described in section 2.1.4, we can create a MOT of approximately 109 133 Cs or 87 Rb atoms but only cool them to their respective Doppler temperature limitsof 125 µK and 145.57 µK [80]. These temperatures are too hot to achieve any of the experimental
goals of this dissertation. In order to further cool our atoms to between 10-20 µK, we continuously
remove more kinetic energy from the atoms by utilizing polarization gradient cooling (PGC) [81].
To understand the key physics of PGC that leads to cooler temperatures than the Doppler
limit, we must abandon the simple two-level atom model and consider that real atoms have multiple
ground states. As a simple example, consider an atom with a triplet ground state (Jg = 1) coupled
to a quintuplet excited state (Je = 2) in a region with no external magnetic field7 . Since thecooling lasers are red-detuned from resonance, atoms absorb more light from the laser beam they
are propagating against (as described in section 2.1.3). Thus, in the one-dimensional setup withno external magnetic field (figure 2.4a), an atom moving with positive velocity v > 0 absorbs
considerably more light from the σ− laser than the σ+ laser.
As a result, there is a considerably greater population of atoms optically pumped to the|m = −2i magnetic sublevel than the |m = +2i sublevel (figure 2.4b), creating an unbalanced
radiation pressure and subsequently a net frictional force directed in the propagation directionof the σ− laser. The opposite effect occurs when the atoms move with negative velocity v < 0(figure 2.4c). Therefore, the σ− /σ+ polarization gradient creates a net damping force that would(theoretically) continue until the atoms absorb both σ− and σ+ beams with equal efficiency. Thisadditional damping mechanism is capable of removing energy well below the Doppler limit because
the actual atom cooling in σ− /σ+ PGC is a direct result of a population imbalance of magnetic
sublevels and not purely from a Doppler shift [83] (such as cooling in section 2.1.3). The lowest
obtainable temperature is limited by the recoil energy atoms gain when emitting a photon during
the cooling process. This temperature limit, called the recoil limit, corresponds to T = 198 nK and
T = 180 nK for 133 Cs and 87 Rb atoms, respectively [84].
7

It should be noted that this method of PGC is suppressed when atoms are in an external magnetic field [82].

24

Figure 2.4: a) Digram showing the 1D σ− /σ+ PGC mechanism setup. b) Atoms moving againstthe σ− laser feature a greater probability of absorbing the σ− beam and are preferentially pumped
to the m = −2 Zeeman sublevel over the m = +2 sublevel (this population imbalance of opticalpumping is illustrated by thick, red lines for the dominant absorption path and dim grey lines forthe less common absorption path). c) The reverse outcome occurs when atoms move against theσ+ laser beam. The population imbalance of the Zeeman sublevels creates an unbalanced radiationpressure and subsequently establishes a net frictional force directed against the atom’s motion.

25

2.2

Cooling Further to Degeneracy
The laser cooling and trapping methods outlined in section 2.1 are used in this dissertation

to cool ensembles of both 133 Cs and 87 Rb to about 10-20 µK (see sections 4.3.3 and 8.2.2). Thistemperature is sufficiently cold enough for the IARPA single atom transport and loading experiments but is still too warm by orders of magnitude for the ultracold matterwave transistor. In thatexperiment, ensembles of 87 Rb atoms must be cooled to approximately 50 nK, which is beyond thecapabilities of the laser cooling methods outlined in this dissertation. Furthermore, the phase spacedensity of the atoms must be high enough such that the 87 Rb atoms transition into a Bose-Einsteincondensate (BEC). In order to reach BEC, pre-cooled atoms (by means of PGC described above)
are confined to a harmonic magnetic potential produced on an atom chip8 and then cooled to degeneracy by performing forced radio-frequency (RF) evaporative cooling. The remaining sectionsof chapter 2 outline the basis for understanding the final stages of producing ultracold 87 Rb atoms
on an atom chip for the matterwave transistor oscillator experiments (chapters 7 and 8).

2.2.1

Magnetic Trapping Interaction

Consider a neutral atom placed in a magnetic field. The potential9 energy between theatom’s magnetic moment µ̂ and the external magnetic field B is given by the Zeeman effect as
U = −µ̂ · B = mf gF µB B

(2.18)

where mf is the z-component of the atomic angular momentum (or magnetic quantum number),gF is the Landé-g factor, and µB is the Bohr magneton10 . If the magnetic field is spatially varying,then the atom experiences a force given by the gradient of equation 2.18:
F = −∇U = ∇(µ̂ · B) = −mf gf µB ∇B
8(2.19)

Specifics of the atom chip used for the experiments in this dissertation is covered in-depth in section 3.3
While this interaction is very weak, magnetically trapping atoms provides a very efficient method to compress
and manipulate previously cooled atoms into a Bose-Einstein condensate [85].
10Where the Bohr magneton is µB = e~/2me where e and me are the charge and mass of the electron.
9

26
The magnetic force can be attractive (thus strong-field seeking) or repulsive (weak-field seeking)depending on the sign of mf gF . This is very important for carrying out magnetic trapping experiments as all magnetic fields are divergenceless (∇ · B = 0), which means they do not feature a
local maximum. This renders strong-field seeking magnetic potentials useless as a means to magnetically trap neutral atoms. Thus, the potential in equation 2.18 must be made to be a repulsive,
weak-field seeking potential. We can satisfy this requirement experimentally (section 8.2.2) by
optically pumping the atoms into a magnetic sublevel where mf gF < 0. With U = −µ̂ · B < 0
fulfilled, atoms can now be magnetically trapped but only if the external magnetic field B featuresa local minimum. Magnetic fields featuring such localized minima suitable for atom trapping in
the transistor oscillator experiments (as well as how the fields are experimentally created with anatom chip) are detailed in sections 3.3.1 and 3.3.2.

2.2.2

Radio Frequency Evaporation

Using the methods of section 2.1.5 and 2.2.1, an ensemble of atoms at a temperature of
approximately 10-20 µK can be prepared in a harmonic magnetic trap. To further cool the atoms
such that their phase space density is high enough to form a BEC, we use a process called radio
frequency (RF) evaporation to continuously remove the warmest atoms from the magnetic trap,
leaving only the “coldest of the cold” to remain in the magnetic potential. Consider an ensembleof 87 Rb atoms optically pumped to a magnetic trapping sublevel, such as |F = 3, mF = 2i (figure2.5). To lower the temperature of this trapped atom ensemble, RF radiation with frequency ωRFirradiates the atoms. Since the energy difference between adjacent Zeeman sublevels in the magnetic
potential is approximately11 ∆E = gF µB B, RF photons that are tuned to the frequency
ωRF = gF µB B/~(2.20)

induce RF spin flips where atoms undergo multi-photon transitions from the |mF = +2i trappedstate to the |mF = −2i anti-trapped state and are subsequently scattered from the trap (figure
11This relation only holds for cases where the magnetic field strengths are small. An analysis of RF evaporative
cooling in large magnetic fields is covered in reference 21.

272.5a). Consequetly, by sweeping the RF frequency ωRF from infinity down to some arbitraryωflip , atoms with energy above ~ωflip will be pumped from the |mF = +2i → |mF = −2i magnetic
sublevel and subsequently expelled from the trap (figure 2.5a). This process permits selectively
removing warmer atoms above a certain energy in the magnetic potential without changing thetightness of the trap, allowing for efficient rethermalization. The expelled atoms carry away energy,
leaving the atom ensemble at a reduced temperature. By continuously ramping ωRF lower and lower,
forced RF evaporation can maintain a high cooling rate by regularly removing the hottest atoms
in the trap as long as the RF sweeping speed is slow enough to allow the remaining cooler atoms
time12 to rethermalize. RF evaporation and rethermalization is repeated until the atom phasespace density is high enough for a BEC. Shown in section 8.2.4, we use RF evaporation to cool 20µK atoms to energies 10-20 kHz above the trap bottom, which then condense into a 50 nK BEC.
12

The phenomenon of runaway evaporation, which is the maximum efficiency of RF evaporative cooling, occurs
when the time per collision, τcol , decreases as the atom ensemble cools. Runaway evaporation emerges when the trap
tightness and RF sweeping rate are such that τtrap /τth ' 100 where τtrap is the 1/e lifetime of the magnetic trap.

Figure 2.5: a) Illustration of the RF evaporation process. An ensemble of atoms is initially pumped
to the |mF = +2i trapping state and confined in a magnetic potential. The RF photon frequency
is the swept from infinity down to ωflip . All atoms with energy above the energy corresponding
to the ωflip RF photons (pictured above the thick dotted line) undergo spin flips and are pumped
to the |mF = −2i anti-trapping state and expelled from the trap. By continuing the RF sweep
down to just above the trap bottom ωb , only the coldest of the cold atoms will remain in the trap
and can undergo transition into a Bose-Einstein condensate. b) Illustration showing how each RF
knife sweep removes the outer tails of the Maxwell-Boltzmann distribution of the trapped atom
ensemble. By continuously cutting out the hottest atoms while allowing rethermalization time, theRF evaporation process leaves a large enough density of atoms at low temperature to result in ahigh enough phase space density for Bose-Einstein condensation to occur.

28
2.2.3

Reaching Bose-Einstein Condensation

For the experiments in this dissertation, RF evaporation is the final stage to condensing
atoms into a BEC. For the transition from a gas to a BEC to occur during evaporation, the de
Broglie wavelength
λ=

h
h
.
=√
p
3mkT

(2.21)

of each atom with momentum p and temperature T must become larger than the inter-atomic
spacing of the gas. That is, the phase space density, PSD must be greater than unity
PSD ≡ λ3 n ≥ 1(2.22)

where n is the number of atoms per unit volume. Equation 2.21 reveals that as the gas is cooled,
the de Broglie wavelength of each atom increases, eventually reaching a critical temperature Tc
whereby the de Broglie waves of each atom begin to overlap. By cooling the gas even further to theultracold regime (T << Tc ), the de Broglie wavelength of each atom becomes so large that all of theatom de Broglie waves completely overlap, rendering each atom of the gas indistinguishable (figure2.6). At this temperature, nearly the entire gas has condensed into a BEC that must be treatedquantum mechanically and is described en masse according to the Bose-Einstein13 distribution.
13
For temperatures where the de Broglie wavelengths of each atom are much smaller than the inter-atomic spacing
d between each atom, the particles are distinguishable from each other and hence the gas behaves classically and is
described with Maxwell-Boltzmann statistics.

Figure 2.6: Illustration depicting the overlap of de Broglie waves as atoms are cooled from above
the critical temperature Tc to well below it. As T << Tc , the de Broglie waves overlap so muchthat all atoms are completely indistinguishable from one another.

29
In order to gain a fundamental understanding of the onset of a BEC from trapped ultracold
atoms (with respect to the transistor oscillator experiment) we examine the phase transition withthe constraint that the atoms are not interacting with each other. After an RF evaporation andrethermalization sequence (section 2.2.2), consider an ensemble of atoms in a state of thermal
equilibrium at temperature T and chemical potential µ in a magnetic potential. The averageoccupancy n of a state with energy  is described by the Bose-Einstein distribution:n() =1e(−µ)/kT − 1.

(2.23)

Moreover, in the transistor oscillator experiment described in chapters 7 and 8 of this dissertation,
the cold atom gas is confined to an anisotropic harmonic oscillator potential given as
1
V (x, y, z) = m ωx2 x2 + ωy2 y 2 + ωz2 z 2
2(2.24)

where m is the mass of each atom and the ωi represents the atom oscillation frequencies along thei = x̂, ŷ, ẑ axis of the potential. The corresponding energy levels of the anisotropic potential are






1
1
1
(nx , ny , nz ) = nx +~ωx + ny +~ωy + nz +
~ωz
2
2
2
|
{z
} |
{z
} |
{z
}
x

y

(2.25)

zwhere nx , ny and nz are integers. Each atom thus has an energy x , y and z due to oscillation
along the respective axis. The energy of this system (equation 2.25) is plotted in figure 2.7 and thenumber of available energy states of this system when the atom has total energy  = x + y + z
is readily given by the integral141
η() = 3~ ωx ωy ωzZ Z −x −yZ −x
dx

0

dy
0

dz =
0

3
.
6~3 ωx ωy ωz(2.26)

It immediately follows that the density of states, or the number of available energy states between
 and  + d isg() =
14

dη()
2
= 3
.
d
2~ ωx ωy ωz(2.27)The integral η(), and hence the number of available energy states in the anisotropic potential, is merely thevolume of first octant of figure 3 bounded by the plane  = x + y + z .

30

Figure 2.7: Plot in “energy space” showing the energy of an atom in the anisotropic harmonic
oscillator as a function of occupying the nx , ny , and nz energy levels. Note that for an atom
occupying some arbitrary energy state (nx , ny , nz ), the total atom energy is a plane  = x +y +z .

Knowing the density of states (equation 2.27), as well as the Bose-Einstein distribution (equation
2.23), one can readily determine the experimental conditions on atom temperature T and trapped
atom number N where all atoms of the gas are in an excited state (meaning the ensemble temperature is exactly at criticality Tc ). For this simplified case of a non-interacting gas, the number ofparticles in the excited state Nex reaches a maximum when the chemical potential µ = 0. That is
Z ∞g()n()dNex (µ = 0, Tc ) =
0
Z ∞
1
2
=
d
(2.28)
2~ωx ωy ωz 0 e/kTc − 1
k 3 Tc3=Γ(3)ζ(3)2~ωx ωy ωzwhere Γ(n) and ζ(n) are the Gamma and Riemann zeta functions, respectively. Rearranging
equation 2.28 and solving for Tc gives the critical temperature
1/3
1/3 ωx ωy ωz≈ 0.94~ωN 1/3
Tc = ~kN
Γ(3)ζ(3)(2.29)

where for T > Tc all atoms in the gas are in an excited state. It immediately follows that if theatom ensemble is cooled lower than Tc , atoms begin to abruptly occupy the anisotropic harmonic
potential ground state, which is the onset of Bose-Einstein condensation. By cooling the atom
temperature T << Tc , one achieves a macroscopic occupation of the harmonic oscillator groundstate and can produce a pure BEC.

31

2.3

Collisions Between Pairs of Low Energy Atoms in a BEC
While section 2.2.3 gives a very good description of the key characteristics of the BEC, it

does not address any interactions or collisions between atoms in the ultracold ensemble. In thetheory and experimental progress of the ultracold transistor oscillator (chapter 7 and 8), interactions
between low energy, ultracold atoms are not only omnipresent, but they drive the matterwave gain
mechanism and must be accounted for. Specifically, the interactions that we must address arebinary elastic scattering events between pairs of atoms with very small total energy. By small, wemean that the scattering process is dominated by the s-wave contribution to the wavefunction15 .
For s-wave scattering of indentical bosons, the scattering cross section is σ = 8πa2 where a is thescattering length given in the Born approximation [70] as
mr
a=
2π~2

Z

0

ei(k −k)·r V (r)d3 r(2.30)

where mr is the reduced mass of the two atoms, V (r) is the effective two-body interaction andwhere k and k0 are the wave vectors of the atom before scattering and after scattering, respectively.
Because the interactions here are due to s-wave scattering, the momentum transfer ~k0 − ~k due
to the scattering process is approximately zero. Thus, equation 2.30 can be approximated as
mr
a=
2π~2

Z
V (r)e

m
d r≈
4π~2

i(k0 −k)·r 3

ZV (r)d3 r.

(2.31)The effective two-body interaction V (r, r0 ) between identical atoms at points r0 and r in equation
2.31 is treated as a discrete contact interaction
V (r0 , r) = gδ(r0 − r)

(2.32)

where g denotes the energy of the interaction (defined shortly) and δ(r0 − r) is the Dirac delta
function.
15

That is, we only consider components of each atom’s wavefunction with orbital angular momentum l = 0. As aresult, the total scattering cross section σ drastically simplifies to
∞

σ=

4π X
4π
(2l + 1) sin2 δl → 2 sin2 δ0
k2
k
l=0

where δl is the phase shift experienced by the l = 0 partial wave due to the scattering event.

32
Inserting equation 2.32 into equation 2.31, one arrives at the scattering length between two
atoms in the BEC as
m
a≈g
4π~2

Z

δ(r0 − r)d3 r = g

m
.
4π~2

(2.33)

Using the result of equation 2.33, we can write the very important result that the effective interaction between two colliding atoms at low energy in the BEC is constant and is given byg=

4π~2 a.
m(2.34)

Equation 2.34 is essential to the transistor oscillator theory and its implications will be used frequently throughout the remainder of chapter 2 as well as chapters 7 and 8 of this dissertation.

2.3.1

BEC Many-Bodied Hamiltonian and Gross-Pitaevskii Equation

Given that there are N interacting atoms in the BEC, the BEC is described with a manybodied Hamiltonian

N 
X
X
~2 2
H=
−
∇i + V (ri ) + g
δ (ri − rj ) .
2m
i=1(2.35)

i<jUsing this many-bodied Hamiltonian in the Schrödinger equation yields the Gross-Pitaevskii equation [86]:


~2 2
2
−
∇ + V (r) + g|ψ(r)| ψ(r) = µψ(r)2m(2.36)

where the eigenvalue µ = ∂E/∂N is the BEC chemical potential and ψ(r) is the total wavefunction
for a system of N indistinguishable bosons making up the BEC, which is given as
ψ(r) = ψ1 (r1 )ψ2 (r2 )...ψN (rN )(2.37)

where ri refers to the position of the ith boson in the BEC. The Gross-Pitaevskii equation is aform of the Schrödinger equation that consists of a linear contribution from the external potential
V (r) confining the atoms and a non-linear contribution g|ψ(r)|2 modeling the averaged, or mean
field, interaction with all of the other atoms in the BEC. An important approximation can be33made to equation 2.36 when the kinetic energy of the atoms is much less than the mean-field16 . In
such cases, the kinetic energy operators −~2 ∇2i /2m are ignored, which reduces the Gross-Pitaevskii
equation to the much simpler
V (r) + g|ψ(r)|2 ψ(r) = µψ(r).

(2.38)

This result is called the Thomas-Fermi approximation and equation 2.38 has the solution|ψ(r)|2 =µ − V (r).
g

(2.39)

Equation 2.39 is very important as it gives the atom density profile of a BEC. It states that in
the Thomas-Fermi approximation, atoms condensed into a BEC fill the external trapping potential
V (r) uniformly up to the condensate chemical potential µ [87].

2.3.2

BEC Characteristics in the Thomas-Fermi Approximation

In the transistor oscillator theory and experiments described in chapters 7 and 8 this thesis,
the external trapping potential is harmonic along the x̂, ŷ, and ẑ axes. From equation 2.39, theatomic density n of the BEC is an inverted parabola (figure 2.8). In the transistor oscillator
experiments (chapter 8) observing this characteristic inverted parabola profile for the atom density
establishes that the atoms have reached Bose-Einstein condensation. The radius R (in position
space) of the BEC, or Thomas-Fermi radius, is defined as the distance from the center of thecondensate to the position r that satisfies µ = V (r). Atoms located within this radius are Bosecondensed while atoms outside the radius remain as thermal atoms. For a harmonic potential
V (r) = mω 2 r2 /2, the Thomas-Fermi radius, R is readily calculated as
r
R=

2µ
.
mω

(2.40)

16
At equilibrium, the mean-field pseudopotential and the external potential are both proportional to R2 , while thekinetic energy is proportional to 1/R2 . Additionally, the mean-field increases linearly with atom number N . Thus
for atom numbers of N > 104 (which is very common in the ultracold atom experiments performed in chapters 7
and 8 of this dissertation) we can accurately state that the kinetic energy of atoms in the BEC is indeed negligible
compared to V and g.

34
The number of atoms in the condensate can be found by inserting the harmonic potential into thesolution to the Gross-Pitaevskii equation (equation 2.39) givingmµ1 mω 2 r2
2|ψ(r)| =1−
4π~2 a
2 µ


r2
mµ
1− 2 .
=
4π~2 aR(2.41)

Integration over the radial coordinate from the origin to the Thomas-Fermi radius R gives the total
atom number in the BEC as a function of the chemical potential µ:

Z
Z R
mµ
r2
2
N = |ψ(r)| dr =1 − 2 4πr2 dr
4π~2 a 0Rmµ 8π 3=R
4π~2 a 15
8π
2µ 3/2 µ=
15 mω 2g

(2.42)

where in the final line of equation 2.42, the definitions of the Thomas-Fermi radius (equation 2.40)
and the scattering length (equation 2.33) are used. Rearranging this result and solving for µ yields
the Thomas-Fermi approximation for the BEC chemical potential:~ωµ=2
where a0 =



15N a
a0

2/5
(2.43)

p~/mω. Equation 2.43 is used throughout the remainder of this dissertation whenreferring to the BEC chemical potential.

Figure 2.8: Density profile (and solution to the Gross-Pitaevskii equation) of the BEC in theThomas-Fermi approximation when confined to a harmonic potential.

Chapter 3

Compact Ultra High Vacuum Systems for the Optical Conveyor Belt andTransistor Oscillator Experiments

The single atom transport and reloading experiments (chapter 4 and chapter 5) as well as thematterwave transistor oscillator experiments (chapter 8) are conducted in an ultra-high vacuum(UHV) environment, which corresponds to pressures below 10−10 Torr. While such laboratoryenvironments have been available to scientists since the 1970s, the UHV systems custom designed
specifically for the experiments in this dissertation are unique in that each can produce an UHV
environment suitable for MOTs and BECs, all the while remaining small enough to carry on acommercial airplane. As an added bonus, the vacuum chamber for the atom transport experiments
has the interesting design of containing no metal1 nuts, bolts, or screws and is completely held
together with optical and anodic bonds.
The goal of this chapter is to describe the structure and layout of two UHV systems constructed particularly for the work in this dissertation: one for the optical conveyor belt experiments(section 3.1) and the other for the transistor oscillator experiment (section 3.2). In both sections,
we show how the UHV design assists in completing the scientific goals of each experiment. Components that play pivotal roles in achieving experimental results, such as the 2D and 3D MOT
chambers, the Hex Cell (section 3.1.2, specific the atom transport system), and the atom chip (section 3.3, specific to the transistor oscillator system) are further described in detail and are referred
to many times in subsequent chapters.
1

This design constraint was implemented in order to eliminate any and all unwanted magnetic fields that would
introduce noise into the Rydberg atom qubit gate operations performed by the UW Madison team (see section 1.2)

36

3.1

Atomic Qubit Array Cell
The first set of experiments in this dissertation, the transport and delivery of single atoms

into optical bottle beam traps (chapters 4 and 5), is conducted in the Atomic Qubit Array (AQuA)
Cell, and is shown in figure 3.1. This system is designed to provide a small footprint, portable,
UHV environment with 6 windows of optical access in order to permit:• Single atom transport and reloading into vacant optical bottle beam sites of the qubit arraywith an optical conveyor belt during simultaneously occurring qubit logic gate operations.
• Providing a continuous source of pre-cooled cesium atoms in an adjacent chamber for use
in the atom transport and reloading procedure.
• Isolation between the atom production and dispenser region (a region of higher pressure)and the MOT, bottle beam, and atom transport regions (regions of UHV pressures).
• Multiple ion pumps assigned to different chambers of the vacuum system in order to permit
differential pumping [88] between higher and lower pressure regions• Apertures between each adjacent chamber such that atoms can be deterministically transported from the lower 2D and 3D MOT chambers into the hex cell using a movable opticalmolasses.
In order to achieve the above listed performances, the AQuA cell design was based off of a double
MOT vacuum configuration [89] featuring a 133 Cs source region, a 2D MOT region, a 3D MOT
region, and a hexagonal chamber featuring six windows with anti-reflection (AR) coating on bothambient and vacuum faces. The AQuA cell was fabricated at the Sarnoff Corporation where theentire vacuum system was constructed with no metal nuts, bolts, or screws and is entirely held
together by over 50 optical contact bonds [90][91] and anodic bonds [92][93][94]. Even the ion pumps
themselves are anodically bonded to the cell. Using a much larger vacuum pump in conjunction
with a bakeout process, the AQuA cell was pumped down to UHV pressures at ColdQuanta. Detailsof the principle components of the AQuA cell are described briefly in the following sections.

37

Figure 3.1: a) Schematic of the AQuA cell vacuum chamber, showing the locations of the threemain regions of the device: the Cs source and 2D MOT region, the 3D MOT region, and the Hex
cell. The location of the pinhole that isolates the 2D and 3D MOT chambers is shown. Additionally,
the positions of the ion pumps and non-evaporable getters are also shown. The cell manifold serves
as a mechanical foundation for the AQuA cell. The manifold is a 1 cm thick, machined Pyrex blockthat is polished on all faces. Channels are machined throughout the interior of the manifold toconnect the 3D MOT chamber to the Hex cell and also to connect each ion pump to its respective
pumping load. The long lever-arm aspect of the 2D and 3D MOT chamber leaves the cell very
susceptible to vibrations. Therefore, two Pyrex support rods are fixed to the bottom of the 2D
MOT chamber and are connected to the AQuA cell manifold. b) The AQuA cell prior to vacuum
processing and bakeout being held with hands for scale. The glass to metal is used to connect
the cell to a bakeout station. The final vacuum seal is completed by anodically bonding a Pyrex
window in-vacuo (while still connected to the bakeout station) onto the cell manifold. The glass
to metal is completely removed after pumping the AQuA cell down to UHV.

38
3.1.1

Cesium Source Chamber, 2D and 3D MOT Chambers

Shown in figure 3.2a, at the very bottom of the AQuA cell is the cesium source, which contains
an SAES alkali metal cesium dispenser (figure 3.2b) that evaporates a nearly pure gas of 133 Cs into
the 2D MOT chamber when the dispenser is heated above a critical temperature. The dispenser
temperature is held constant by running an electrical current of 3.5 A through it at all times during
the experiments. Additionally, a SAES non-evaporable getter (NEG) is also included inside thecesium source region for improved vacuum quality. Current is supplied to the dispenser and NEG
via electrical feedthroughs connected to conductive silicon pads located on the bottom, ambient
side of the source chamber (figure 3.2a and d).
Anodically bonded directly above the cesium source is the 2D MOT chamber, which is a 1
cm x 1 cm x 4.5 cm Pyrex vacuum chamber designed to support a 2D+ MOT from the background
133 Cs gas produced in the source.

Because this chamber is filled with warm, cesium vapor, thepressure in this chamber is higher than UHV at approximately 10-100 nanotorr. The 2D MOT
chamber is capped with a 500 µm thick silicon wafer that features a 750 µm diameter pinhole
drilled in the center. As described in section 4.3.1, a 2D+ MOT loaded from this background vapor
provides a transversely cooled beam of atoms that can be vertically transported to the 3D MOT(and also Hex cell) chamber by passing through the 2D-3D pinhole (figure 3.2c). Also separating
the 2D and 3D MOT chambers is a Pyrex cylinder, which is bonded to the 2D MOT pinhole andcontains an additional getter for improved vacuum quality. Anodically bonded to the top of thiscylinder is the 3D MOT chamber, which has the same dimensions as the 2D MOT chamber andis also made of Pyrex. In order to have MOT lifetimes of 10-20 seconds, the 3D chamber must be
kept at a lower pressure of approximately 0.1 nanotorr, which is orders of magnitude lower than theadjacent connecting 2D chamber pressure. Sufficient isolation between the high and low pressure
MOT chambers is provided by the 2D-3D pinhole, as the 0.024 mL/sec conductance of the pinhole
is negligible compared to the 0.4 L/sec pumping rate of the ion pumps (described in section 3.1.2).
Because of the very low background pressure of 0.1 nanotorr in the 3D chamber, a 3D MOT cannot

39
be loaded from a background gas. Instead, a stream of pre-cooled 2D+ MOT atoms are pushed
through the 2D-3D pinhole and into the 3D MOT chamber, where they are used to source the 3D
MOT. This design of using a 2D MOT to directly source a 3D MOT in an entirely separate vacuum
chamber is called a double-MOT cell and it improves the 3D MOT lifetime by decoupling the atom
lifetime (which is set by the 3D chamber vapor pressure) from the MOT load time (which is set
by the 2D chamber vapor pressure) [89]. Similar to the 2D MOT chamber, the 3D chamber is also
capped with a silicon disc but features a larger pinhole with a 3.18 mm diameter. This pinhole islarger so that the 3D MOT itself can be transported vertically into the Hex cell (figure 6.1), locatedabove the 3D MOT chamber. This entire source-2D-3D MOT vacuum ensemble is then anodicallybonded to the AQuA cell manifold. Finally, we note that the magnetic fields that are required
to make the 2D+ and 3D MOTs are produced with permanent magnets that are mounted on 3D
printed braces that firmly mate to the outside of the MOT chambers (figure 3.2d).

Figure 3.2: a) Anodically bonded source tube, 2D MOT, and 3D MOT chambers. The bottom of
the source tube, which contains the conductive feedthroughs, is not yet anodically bonded to thechamber. b) Close up view of the AQuA Cell source tube showing the glowing Cs dispenser andthe non-evaporable getter. The dispenser is glowing due to running 3.5 A through it in order to
expel a cesium gas via thermal emission. c) View looking up vertically through the bottom of thesource tube through the 2D MOT chamber where the bright dot in the center is fluorescence from a
2D+ MOT that centered on the 2D-3D pinhole. d) 2D and 3D MOT chamber with the permanent
MOT magnets attached. The silver contacts that connect to the Cs dispenser are also shown.

40
3.1.2

Hex Cell

Located above the 3D MOT chamber is the most visually prominent feature of the AQuA
cell: the six sided Hex Cell chamber (figure 3.3), which provides an UHV environment of less than
0.1 nanotorr with six faces of optical access and serves as the location for a majority of the single
atom transport and reloading experiments as well as the qubit array and bottle beam trappingsites. The Hex cell walls are AR coated on all inside and outside faces for the 532, 780, 852, 1038,
and 1064 nm wavelength lasers used in the qubit logic gate (from UW Madison) and single atom
reloading experiments. The Hex cell is optically contact bonded to the top of the Pyrex manifold(figure 3.1) and is pumped by its own individual ion pump, separate from the 2D-3D ion pump.
At the base of the Hex cell exists an elliptically shaped aperture that is aligned with the 2D-3D
pinhole to allow the 2D+ MOT atoms to be transported directly into the Hex cell, bypassing the
3D chamber entirely, such that a small 3D MOT can be created in the Hex cell to serve as a cold
atom reservoir for the single atom transport and reloading experiments (figure 4.7).

Figure 3.3: AQuA Hex cell, which serves as the location for the single atom transport and reloading
experiments, as well as the location of the qubit array.

41
3.1.3

Anodically Bonded Ion Pumps

The AQuA cell features two 0.4 L/sec ion pumps that are constructed by anodically bonding
each component together rather than using large nuts, bolts, screws, or epoxies. A schematic of
the ion pumps is shown in figure 3.4a. The body of each ion pump is a machined Pyrex block thatfeatures two drilled out bore holes that are oriented at right angles to each other: one positioned
horizontally and the other vertically. The principle components built around this Pyrex block are:
• The anode is formed by attaching a titanium cylinder to the inside of the Pyrex blockthat runs along the vertically oriented bore hole. A conductive rod is placed inside thehorizontally oriented hole and is welded to the titanium cylinder. This rod extends all theway to the very left-most side of the ion pump which is capped with metalized conductive
silicon. Electrical contact between the silicon and the rod is made with a metal springwhich forms the anode feedthrough.
• The cathode is formed by capping the top and bottom of the pump with metalized conductive silicon. A small Pyrex ring is anodically bonded to each cathode. Placed in between
this Pyrex ring and the cathode is a titanium disc that is slightly bowed, giving the discspring-like properties and forming the cathode feedthrough.
• Connectivity between the anode and cathode is provided by the spring force present in thetitanium disc. Due to its slight radius of curvature, the disc is constantly pushing on boththe cathode and the titanium cylinder, which maintains constant electrical connectivity.
• Two neodymium magnets are mounted on the top and bottom cathodes to produce themagnetic field required for the ion pump to function. A custom Mu Metal box placed overthe ion pumps provides magnetic shielding.
This entire ion pump apparatus is then anodically bonded directly to the Pyrex manifold foundationof the AQuA cell. The 1 cm thick Pyrex manifold features channels machined throughout its interior
to connect the ion pumps to their respective vacuum load.

42

Figure 3.4: Left: Schematic showing the construction of the AQuA cell ion pumps. The body ismade from a machined Pyrex block with tubular bore holes running horizontally and verticallythrough the Pyrex. A titanium cylinder is placed in the center. A metal rod is welded to thetitanium cylinder and is connected to the silicon anode by a spring. The spring pushes on boththe the rod and the andode and forms the anode feedthrough. The cathode is formed in a similarmanner by using a bowed titanium disc to act as a spring to ensure a constant connection between
the titanium cylinder and metalized silicon. The titanium disc forms the cathode feedthrough andforms an electrical connection between the anode and cathode. The opening on the right side of the
Pyrex body is then anodically bonded to the AQuA cell manifold. Right: Completed AQuA cellion pumps. Note that the ion pumps in this image are rotated 90 degrees from the given schematic.
The metalized silicon forming the anode and cathode is clearly visible. The bottom of the pump isthen anodically bonded to the AQuA cell manifold.

43

3.2

Atomtronics Double MOT Vacuum Cell
The UHV system, called the “atomtronics cell,” used for the ultracold transistor oscillator

experiments is also a double-MOT cell, although it is a different design. An image of the UHV
cell and its principle components is shown in figure 3.5. The fundamental source-2D-3D chamber
design described in section 3.1.1 for the AQuA cell is also used for the atomtronics cell.
• Similar to the AQuA cell system, the 87 Rb source for this cell is also an SAES alkali metal
dispenser located at the base of the 2D MOT chamber. An NEG is included adjacent to
the dispenser to improve vacuum quality.
• The 2D and 3D MOT chambers, made with Pyrex walls with dimensions of 2 cm x 2 cm x
4 cm, are larger than those of the AQuA cell. The ambient and vacuum side walls of the
3D MOT chamber are AR coated for 780 nm laser light.
• A major difference between the AQuA cell and the atomtronics cell is that an atom chip
(section 3.3) is anodically bonded to the top of the 3D chamber and forms the uppersurface of the cell. The atom chip, which described in section 3.3, is a major component inproducing BECs with the atomtronics cell.
• Unlike the AQuA cell, the base of the 2D and 3D MOT chambers are attached to ConFlat
flanges via glass-to-metal anodic bonds. The atomtronics cell is then held together bybolting each MOT chamber onto a spherical cube.
• Isolation between the 2D and 3D MOT chambers is provided by a 750 µm pinhole, which
is drilled into a silicon wafer placed at the top of the 2D MOT chamber (see figure 3.5).
The pinhole permits differential pumping between the 2D and 3D chambers, resulting in a
background pressure 0.1 nanotorr in the 3D chamber.
• An Agilent 2 L/sec ion pump is also bolted to the spherical cube to maintain UHV.

44
When bolting each chamber or ion pump onto the spherical cube, a copper gasket is placed between
the base of the ConFlat and the spherical cube. A vacuum seal is established when knife edges,
found at the base of the ConFlat, as well as on the spherical cube, are tightly pressed into thecopper gaskets, creating a knife-edge seal. The atomtronics cell was initially pumped down to
UHV by bolting the entire vacuum cell to a much larger vacuum station via a copper pinch off
tube and following a standard bakeout procedure at approximately 300◦ C. Following bakeout, theatomtronics cell was separated from the bakeout station by “pinching” the pinch off tube with aset of hydraulic jaws. After removal, the system is soft baked for 5-7 days at 120◦ C to remove any
water moisture that erroneously entered the cell during pinch-off.

Figure 3.5: Image of the atomtronics vacuum cell. The cell is still attached to the bakeout station
via the copper pinch off tube and has not yet been pinched-off.

45

3.3

Atom Chip: Making Magnetic Traps for Ultra Cold Atoms
The top surface of the atomtronic cell 3D chamber is an atom chip [95, 96, 97] (figure 3.6a).

Found on both the ambient and vacuum sides of the chip are small gold wires, ranging in width from
100-500 µm (figure 3.6b). Running electrical currents through various atom chip wires produces
the magnetic fields necessary to setup a harmonic magnetic potential in order to perform magnetic
trapping, evaporative cooling, and ultimately Bose-Einstein condensation of an ensemble of 87 Rb
atoms. The atom chip is also used to form the harmonic magnetic potential for the matterwave
transistor oscillator (figure 1.4). A 2 mm wide Pyrex window built into the center of the chip (figure3.6b) allows for optical access of atoms trapped in the magnetic potentials as well as in-situ imagingof the BEC confined to a magnetic potential (section 8.3.1). In this final section of chapter 3, wegive a brief overview of the magnetostatics of the atom chip relevant to the transistor oscillator
theory and experiments covered in chapters 7 and 8.

3.3.1

Producing 3D Harmonic Magnetic Potentials

While there many wires on the atom chip, to form harmonic potentials for the transistor
oscillator experiment, we only need to consider a small set of wires positioned at the centrally
located chip window, specifically:• A single pair of parallel oriented “guide-wires.” The guide-wires are shown in figure 3.6b
as the set of parallel running wires extending over the atom chip window.
• Two pairs of parallel oriented “H-wires.” The H-wires can also be found in figure 3.6b as
the yellow colored wires running along the outer edge of the chip window and oriented
perpendicular to the guide-wires.
Consider running a current Ix through one of the chip guide-wires. Compared to the size ofthe atoms as well as the dimensions of a BEC2 , the guide-wire is considered to be infinitely long.
2

In the transistor oscillator experiments, typical BEC dimensions are 50-100 µm in the longitudinal direction.

46
From Ampere’s law [98], the current Ix running through the guide-wire produces a magnetic field(called the “guide-wire field”) given by
B=

µ0 Ix
φ̂
2π r

(3.1)

with corresponding field gradient B0 and curvature B00 of
µ0 Ix
φ̂
2π r2
µ0 I x
B00 =
φ̂
π r3
B0 = −(3.2)

where µ0 is the permeability of free space and r is the distance from the wire (figure 3.6c). To
produce a local minimum in this field (to permit magnetic trapping), we apply a bias field in theŷ direction with strength βy ŷ (figure 3.6d). As shown in figure 3.6e, the vector summation of
the y-bias field with the guide-wire field (equation 3.1) creates a quadrupole field by canceling theguide-wire field below the wire at a depth d given by
d=

µ0 Ix
.
2π βy

(3.3)

Since the magnetic field features symmetry along the axis of the guide-wire (in and out of thepage with figure 3.6e), a local minimum exists a distance d below the entire length of the guide-wire.
This forms a hollow, tube-like region of zero magnetic field surrounded by high magnetic field, which
permits loose confinement along the direction of the guide-wire (i.e. the longitudinal direction) andstrong confinement perpendicular to the guide-wire (i.e. the radial direction). While the guide-wiremagnetic field, gradient, and curvature features symmetry in r̂, φ̂, ẑ cylindrical coordinates, it isuseful for the transistor oscillator experiment to express all fields and their derivatives in rectilinear
coordinates3 . Accordingly, the total guide-wire magnetic field from equation 3.1 (with the y-bias
field included) in rectilinear coordinates isµ0 I x
B=
2π
3





z
y
− βy ŷ + 2ẑ .
y2 + z2
y + z2(3.4)

When performing the actual experimental work in the lab, we will be applying many additional bias fields purely
in the x̂, ŷ, and ẑ directions (section 5.2), making rectilinear units easier to visualize in the lab.

47

Figure 3.6: a) Vacuum cell with atom chip serving as the top of the cell. b) Underside of the atom
chip that is inside the vacuum. The guide wires are shown at the center of the chip. c) 2D crosssection of the magnetic field lines produced by running current through one guide wire. d) Uniform
bias field oriented purely in the y-direction. e) Summation of the guide wire field and the y-bias
field. The result gives a magnetic quadrupole field featuring a local minimum a distance d below
the chip guide wire. This plot is a 2D cross-section that is symmetric in and out of the page andthus the field minimum runs below the entire length of the wire, giving a region where magnetic
trapping of atoms is permitted.

48
By choosing to orient the guide-wire along the x̂ axis, the local minima of the quadrupole
field is found by simply taking the limit of equation 3.4 as y → 0 and z → d. In this limit, we findapproximations for the guide-wire fields:µ0 I x 1− βy
2π z2
µ0 I xyBz ≈.
2π
y 2 + d2

By ≈(3.5a)(3.5b)

With equations 3.5a and 3.5b, one of the requirements for magnetic trapping is satisfied: a localized
minimum for a magnetic field has been produced. In the lab, however, the quadrupole fields fromequations 3.5a and 3.5b produced by the atom chip are very problematic for atom trapping andreaching BEC. Plotting the fields By and Bz (figure 3.7), we immediately observe that a cusp exists
at the zero of both fields. This cusp not only perturbs the harmonic potential, but it also resultsin atom losses [99] due to heating4 via Majorana spin flips [100]. The cusp in fields By and Bz
can be removed by adding another bias field βx x̂, along the direction of the guide-wire (figure 3.7).
With the addition of this x-bias field, the total field of the guide-wire is approximately
B = βx x̂ +
µ0 Ix 1y
µ0 Ix
− βy ŷ +ẑ.
2π z 22π y 2 + d2

(3.6)

Figure 3.7: Plots of the guide-wire magnetic field in the ẑ and x̂ directions. The solid and dashedlines represent the magnetic fields with and without the additional x-bias field.
4

The cusp in the magnetic field creates a discontinuity where the atoms cannot adiabatically follow the field. At
the zero crossing, the field gradient is undefined, which can induce spin flips [100] (called Majorana spin flips) into astrong-field seeking state (that is mf gF > 0) where the atom can no longer be trapped [99].

49
Using the magnetic field from equation 3.6, we can form a magnetic potential that can confine atoms along the radial direction but cannot trap atoms along the longitudinal direction, since∇x B = 0. In order to confine atoms along the x̂, or longitudinal direction, we run a current IH
through a pair of “H-wires” centered about the atom chip window (figure 3.8a) that run perpendicular to the guide wire (figure 3.6b). The H-wires are fabricated to be a rectangular wire witha width w. They are wider than the guide-wires because they carry a considerably higher current
load. Each pair of H-wires and its conjugate pair placed symmetrically on the opposite side of thechip window is separated by distance l. Therefore, running a current IH through the H-wire pair
produces a magnetic field given by Ampere’s law as
 



µ0 I Hw
w
BH =ln 1 +− ln 1 +x̂.
2πw
x + l/2x − l/2(3.7)

By combining the magnetic fields from the guide-wire (equation 3.6) and the H-wire (equation 3.7),
the total field forms a 3D harmonic magnetic potential capable of trapping atoms in a local field
minimum. This configuration is called an Ioffe-Pritchard trap and is shown in figure 3.8c).

Figure 3.8: a) Top view of the ambient side of the atom chip. The vertically running pairs ofwires are the H-wires and are centered about the chip window. b) Plot of the H-wire magnetic
field BH (equation 3.7) produced by running current of equal direction and magnitude throughtwo pairs of H-wires centered about the chip window. c) Combined magnetic potential forming an
Ioffe-Pritchard trap from the guide-wire field (equation 3.6) and the H-wire field (equation 3.7)

50
3.3.2

Double Guide Wire IP Trap

The magnetic fields used to radially confine ultracold atoms (equation 3.6) was formed byrunning current Ix through a single guide-wire. This places the magnetic trap a distance d (equation
3.3) below the wire, which blocks optical access. In order to gain optical access to ultracold atoms
trapped in the magnetic potential, we must move the trap out from under the guide wire. Thiscan be accomplished by adiabatically ramping up an oppositely flowing current through the second
guide-wire (running parallel to the first, separated by distance l; figure 3.6b) from 0 to −Ix . Under
this new set of conditions, the magnetic field (and hence the potential) is shifted directly under thechip window, forming what is called the “split guide-wire trap.” This is readily shown by applying
Ampere’s law to the case of running currents of equal magnitude Ix but with opposite directions
through both guide wires separated by distance l (figure 3.6b). The resulting magnetic field is
B=

µ0 Ix
2π



z
z
−
(y + l/2)2 + z 2 (y − l/2)2 + z 2




ŷ +

y + l/2y − l/2+(y + l/2)2 + z 2 (y − l/2)2 + z 2

 
ẑ
(3.8)

and is plotted in figure 3.9a.

Figure 3.9: a) 2D cross section of the magnetic field produced by running currents Ix and −Ixthrough the pair of guide-wires separated by distance l (equation 3.8). b) Uniform bias field oriented
purely in the z-direction. c) Summation of the split guide-wire field (equation 3.8) and the z-bias
field. The result gives a magnetic quadrupole field featuring a local minimum a distance D below
the center of the atom chip window. This plot is a 2D cross-section that is symmetric in and outof the page and thus the field minimum runs below the entire chip window, giving a region where
both magnetic trapping of atoms and optical access is permitted.

51
Similar to the single guide-wire trap, a local minimum in equation 3.8 can be formed byapplying a bias field, this time oriented in the ẑ direction with magnitude βz (figure 3.9b). By
adding the bias field to equation 3.8, a quadrupole field oriented along the x-axis is formed at a
depth D below the chip window wheresD=µ0 Ix l l2
− .
2π βz
4

(3.9)

Physically, the quantity D represents the z value where the z-bias field, βz , cancels the ẑ componentof the split-guide wire field (equation 3.8) along the entire line y = 0 (figure 3.9c). As in the
single-guide wire case, Majorana spin flipping is avoided by transforming the quadrupole trap into
an Ioffe-Pritchard trap by adding an x-bias field βx along the direction of the split guide-wires.
Therefore, the total field that gives rise to the split guide-wire Ioffe-Pritchard trap iszzµ0 I x−ŷB=
2π (y + l/2)2 + z 2 (y − l/2)2 + z 2
y + l/2y − l/2µ0 I x++ βz ẑ + βx x̂.
+
2π(y + l/2)2 + z 2 (y − l/2)2 + z 2(3.10)The local minima of the total magnetic field that forms the split guide-wire Ioffe-Pritchard trap isfound by taking the limit of equation 3.10 as y → 0 and z → d, giving
Bx = βx(3.11a)


µ0 Ix
D
D
−
2
2
2π (y + l/2) + D(y − l/2)2 + D2


µ0 Ix
l
Bz =
− βz .
2π l2 /4 + z 2



By =

(3.11b)(3.11c)From equations 3.11b and 3.11c, the magnitude of the total5 field at the location z = D isq
|B| = Bx2 + By2

=

µ0 Ix
2π



D
D
−
2
2
(y + l/2) + D(y − l/2)2 + D2

2

(3.12)

!1/2
+ βx2

.

While this expression gives the total B-field located a distance D under the chip window, from anexperimentalist point of view, it is more useful to also only consider the field at small y values,
5

Recall from equation 3.9 and the sentence that follows it, that Bz vanishes when z = D. Therefore, equation
3.11a is not used in the magnitude of the total field at the location z = D.

52since after all, atoms are localized to y ≈ 0 when trapped in the field minimum. Therefore, thefield is expanded in a Maclaurin series|B| = βx +8π 2 D2 βz4 2
y + O(4).
µ20 Ix2 l2 βx(3.13)The trapping forces on an atom in this spatially varying magnetic field are readily calculated
by evaluating the first and second spatial derivatives of equation 3.13, giving corresponding field
gradient B 0 and curvature B 00 of|B 0 | =16π 2 D2 βz4yµ20 Ix2 l2 βx(3.14a)|B 00 | =8π βz3µ0 I x l β x(3.14b)where the definition of D given in equation 3.9 is used in solving |B 00 |. Using equations 3.14a and3.14b, we immediately arrive at the radial trapping force imparted onto atoms in the split guidewire trap:
Frad = −∇ (µ̂ · B) ≈ −µb B 0 = −16µb π 2 D2 βz4yµ20 Ix2 l2 βxwhich features a trapping frequency in the radial direction ofsrµ 00
8π µ βz3
ωrad =B =.
mµ0 Ix ml βx(3.15)(3.16)The trapping potential along the longitudinal direction is still provided by the same H-wire fieldgiven in equation 3.7. Thus the trapping force and corresponding trapping frequencies in thelongitudinal direction are found by evaluating the first and second spatial derivatives of equation
3.7 and are
4µ0 IH
Flong = −∇x (µ̂ · BH ) = −
πw
r
ωH =

3.4

µ 00
B ≈
m H

r



x
l2 − 4x2µµ0 2IH.
πw ml
(3.17)(3.18)

Conclusion to Part I
With the laser cooling and trapping theory outlined in chapter 2 and the vacuum cells andatom chip systems outlined in chapter 3, we have all of the universal tools and methods required

53
to perform the atom transport and atom transistor experiments. Any additional methods that are
specific to either experiment are described within their respective chapters.

Chapter 4

Part II: Individual Atom Delivery of Neutral Cs Atoms into Bottle Beam
Traps Using a Dynamic 1D Optical Lattice

4.1

Motivation for Single Atom Delivery into Bottle Beam Traps
Advances in neutral atom quantum computing have led to the recent development anddemonstration of a 49 element atomic qubit array, where quantum information is stored in the
ground states of neutral cesium atoms and two-qubit gates implemented with Rydberg states[101, 102, 1, 103, 104, 105]. The qubit array is comprised of a 7x7 grid of individual opticalbottle beam (BoB) traps (figure 4.1a) whereby each BoB confines exactly one cesium atom. The
ambitious goal of the IARPA AQuA-49 project, as described in section 1.1, is to use such a qubit
array, located inside an UHV chamber, to perform quantum logic gates involving multiple qubits foran arbitrarily long length of time (figure 4.1b). Due to collisions with thermal background atoms inthe surrounding vacuum chamber (figure 4.1c), trap lifetimes of qubits are on the order of τ = 100
sec, [3, 1, 103, 105, 106, 107] and thus an array of N = 49 cesium qubits will lose one qubit every
τ /N ≈ 2.0 seconds, on average. Even with extreme high vacuum of 10−4 nanotorr, trapped atom
lifetimes would be about 1 day. [108] For a quantum computer to run arbitrarily long algorithms,
atom loss would be an issue. Therefore, to sustain a functional neutral atom qubit array for anindefinite period of time, a new atom must be reloaded into the vacant BoB array site in a timely
manner. Past work has made advances in loading single atoms into BoB traps by overlapping the
BoB with a magneto-optical trap (MOT)[109, 110]. This solution will certainly reload the newly
vacant BoB site but will also disturb other loaded, neighboring sites due to collisions with other55
MOT atoms. A more suitable solution involves reloading a lost atom by delivering a new atomfrom a distant cold atom reservoir by means of a movable, dynamic 1D lattice (figure 4.1d) similar
to the atom transport experiments covered in references [111, 112, 113]. The latter solution is
the subject of this chapter, which covers our work in solving the problem of atom reloading anddescribes experiments that demonstrate transporting and subsequent loading of a single neutral
cesium atom into a distant, unoccupied optical BoB trap.

Figure 4.1: (Color online) a) Schematic of a qubit array constructed with a 2D grid of equally
spaced optical bottle beam traps. b) Concept of using pairs of entangled Rydberg state atoms toperform quantum logic gates and operations involving multiple qubits simultaneously. c) A straybackground gas atom collides with a trapped qubit and expels it from the BoB trap. d) A new
pre-cooled cesium atom is transported with an optical conveyor belt to the empty BoB site. e)
Exactly one atom at most is then transferred into the BoB trap, reloading the empty site andkeeping the array fully occupied.

56
The experiment functions by initially loading a few thousand atoms into a red-detuned 1D
optical lattice from a background cold atom reservoir. Precise frequency control over the lattice
beams allows us to move the lattice back-and-forth along its axial direction, making it function as
an “optical conveyor belt.” With this setup, we demonstrate transporting the small atom ensemble
approximately 1.1 mm to an empty optical BoB trap followed by transferring exactly one atom atmost into the BoB trap. An added challenge for this experiment arises due to the lack of an a-priori
physical reference point for the BoB trap. The remainder of this chapter is organized as follows:
Section 4.2 describes the laser system and beam layout for the 852 nm cooling, repump, and probe
lasers, as well as the 780 nm bottle beam laser system and the 1064 nm optical conveyor belt
lasers. The layout of each beam with respect to each other in the AQuA Hex cell is also provided.
Section 4.3 details the laser cooling processes to produce a reservoir of sufficiently cold atoms inthe Hex cell for use in the transport and reloading experiments. Section 4.4 describes generating
and implementing the optical conveyor belt. Section 4.5 outlines the crossed vortex BoB trap aswell as the system for detecting single atoms confined to such an optical potential. Finally, section
4.6 demonstrates the single atom delivery into a BoB trap using the optical conveyor belt.

4.2

Laser System and Beam Layout
The single atom transport and reloading experiment requires the use of three sets of lasers

with various detunings from the 133 Cs D2 line (λ = 852.347 nm, ν = 351.725 THz) :• Three 852 nm lasers that are on resonance with the 133 Cs D2 line.
• One 780 nm laser for BoB traps that is blue-detuned from resonance.
• One 1064 nm laser for atom transport that is red-detuned from resonance.
The 852 nm resonant laser light was produced using Vescent Photonics D2-100-DBR-852 distributed
Bragg reflector (DBR) diode lasers. The transport and reloading experiment requires three individual DBRs: one to provide cooling light, one to provide repump light, and one to act as a probe57
laser to perform absorption imaging. In addition, a New Focus VAMP TA-7616 tapered amplifier(TA) was used to amplify the light emitted from the cooling DBR from 70 mW to 500 mW of
total usable laser power. Blue detuned light at 780 nm, used to form optical bottle beam traps
and bottle beam arrays, is produced with a Sanyo DL 7140-201 S laser diode and is amplified bya Toptica BoosTA Pro to 3 W of laser power. Finally, the red detuned 1064 nm laser that is used
for optical dipole trapping is a 30 W, IPG Photonics YLR-30-1064-LP-SF laser. A block diagramof the 852 nm, 780 nm, and 1064 nm laser systems are shown in figure 4.2 and figure 4.3.

4.2.1

Layout of 852 nm Cooling, Repump, and Probe Lasers

The 852 nm cooling, repump, and probe laser setup as well as each laser’s corresponding
atomic transition used within the 133 Cs D2 manifold is shown in figure 4.2 and described here. The
frequency locking scheme for the cooling laser1 involves picking off 5 mW of the cooling light fromthe main beam and blue shifting it by 145 MHz with an acousto-optic modulator (AOM). Thisblue-shifted light was then sent into a Vescent Photonics saturated absorption spectroscopy (SAS)module and locked to the |F = 4i → |F 0 = 5i D2 cycling transition. Using this locking scheme,
realize that the main cooling light that was not picked off is subsequently left red-detuned from thecycling transition by 145 MHz. This red-detuned light was then used to seed the New Focus TA,
which then outputs 500 mW of laser light. The output of the TA supplies cooling light for boththe 2D and 3D MOTs. Acousto-optic modulators are used to shift the frequency of the 2D and
3D MOT light such that the 2D beam is detuned −2.9Γ ≈ 95 MHz and the 3D beam is detuned−2.3Γ ≈ 76 MHz from the |F = 4i → |F 0 = 5i cycling transition. It is important to note thatwith respect to the 3D MOT cooling light path, the AOMs are setup in the cat’s eye, double passconfiguration [114], which allows one to shift the 3D MOT frequency without imparting any spatial
deflections to the beam. After passing through the AOMs, a total of 100 mW of 2D MOT lightand 60 mW of 3D MOT light is then coupled into a polarization maintaining fiber.
1

It should be noted that since the lasers are DBR lasers, they are extremely sensitive to back-reflected light.
Thus, the cooling and repump laser outputs initially pass through two optical isolators providing 60 dB of isolation.

58

Figure 4.2: (Color online) a) Setup for the 852 nm cooling and repump lasers. b) Setup for the
852 nm probe lasers. In both schematics, mechanical shutters are used to turn on and off the laser
beams. Additionally, all fibers are 852 nm, polarization maintaining fibers. c) Laser locking scheme
for the Cs D2 transitions used in the single atom transport and loading work.

59
In order to keep atoms contained within the cycling transition, the repump laser is locked
resonant to the |F = 3i → |F 0 = 4i transition (figure 4.2c). Repumping light is then split such thatit can be supplied to both 2D and 3D MOT setups. Variable attenuation of the 3D repumping lightis provided by the combination of an electo-optic modulator (EOM) and a polarizing beam splitter(PBS) on the input and output of the EOM. As seen in the laser schematic shown in figure 4.2a,the 2D MOT repumping light is coupled into its own polarization maintaining optical fiber whilethe 3D MOT repump light first spatially overlaps the 3D cooling light and is coupled into the 3D
MOT polarization maintaining fiber2 . This arrangement results in 10 mW of 2D repump and 5
mW of 3D repump light coupled into the respective fibers. Finally, the probe laser, which is used
for absorption imaging (described in chapter 4.2), is locked to the |F = 4i → |F 0 = 3i / |F 0 = 5i
crossover transition. The probe light is blue shifted onto resonance with the |F = 4i → |F 0 = 5i
cycling transition by using an AOM driven at 125.5 MHz. Additionally, a small amount of probe
light is picked off from the main beam line to act as a resonant beam3 .

4.2.2

1064 nm Transport Lasers

The setup for the 1064 nm atom transport laser and the 780 nm bottle beam laser is shownin figure 4.3.
The red-detuned, 1064 nm fiber laser supplies 30 W of laser power that is split with a PBS
to send 18 W to a vertical dipole trap setup4 and 12 W to form the optical conveyor belt. Afterthe PBS, a 2:1 telescope de-magnifies the beam waist from 3 mm to the desired waist of 1.5 mm.
The light is then equally split with another PBS, where each beam subsequently passes through adouble pass AOM setup. Each AOM is driven by its own individual Agilent 33250 80 MHz arbitrary
waveform generator (AWG) that shifts the frequency of one beam by ∆ν1 = 2 × 78 = 156 MHz
2
Note that while the 3D MOT cooling and 3D repumping light are coupled into the same polarization maintaining
fiber, the repump light polarization is oriented perpendicular to the polarization maintaining axis.
3
While not actually used in the single atom transport and loading experiment, the resonant beam is a very useful
tool for the experimentalist as it can be overlapped with other non-resonant beams (such as the 780 nm and 1064
nm light used in the experiment) and used as a nifty alignment tool.
4
During the development of this experiment, work involving vertical transportation of atoms from a MOT to theoptical BoB traps was implemented in a moving molasses MOT. While this is not the subject of this chapter, it opens
some interesting extensions to the atom transport work and thus is discussed further in appendix A.

60

Figure 4.3: (Color online) a) Setup of the 1064 nm atom transport laser system. b) Schematic of
the 780 nm blue-detuned bottle beam laser system.

61
and the other by a variable amount of ∆ν2 = 2 × (78 + δ) = 156 + 2δ MHz, where the amount δ istunable. To minimize phase jitter between the two beams, both AWGs are frequency locked andphase synchronized via their internal 10 MHz clocks. The frequency shifted light emerging from the
double pass AOMs is then combined and spatially overlapped at a final PBS. This setup produces
two co-propagating 1064 nm beams with orthogonal linear polarizations with a mutual frequency
detuning of ∆ν = 2δ. The pair of beams is then sent to the AQuA cell (section 4.2.4) to form theoptical conveyor belt for atom transport and delivery (section 4.4.1).

4.2.3

780 nm Bottle Beam Lasers

Shown in figure 4.3b, for the blue detuned, 780 nm laser, 30 mW of light is sent directly toseed the Toptica BoosTA, which outputs a total of 3 W of light. Output from the TA splits at
a PBS where it is either sent to a crossed vortex bottle beam setup or the Gaussian beam arraybottle beam setup (used in chapter 5). For both paths, the light is first spatially filtered by passing
through an anti-reflection coated, single mode fiber and then propagated through a system of beam
shaping lenses to set the correct beam waist of w0 = 0.8 mm for the crossed vortex BoB setup and
w0 = 1.2 mm for the BoB array setup.

4.2.3.1

Crossed Vortex BoB Setup

The objective of the crossed vortex BoB setup is to generate two co-propagating LaguerreGaussian (LG01) beams with equal intensity and orthogonal polarizations. Shown in figure 4.4,
the LG01 beams are created by propagating a TEM00 mode, Gaussian laser beam with initial
beam waist of 0.8 mm, through an RPC Photonics spiral phase plate (SPP). The SPP imparts an
azimuthal dependent phase delay of the form e±ilφ on the incident TEM00 mode by slowing thephase of the Gaussian beam as a function of it’s azimuthal angle, where l is an integer called the
topological charge5 of the outgoing LG mode.
5

Here, l indicates how many times the phase of the beam is shifted by 2π for every full azimuthal rotation of the
TEM00 beam mode. The ± sign of the topological charge indicates the sign of the helicity of the outgoing LG beam.

62
For every complete 2π rotation of the azimuthal angle φ of the TEM00 mode, the total phasedelay ∆φ imparted on the light with wavelength λ passing through the SPP is∆φ = 2πl =2π(n − 1) d
λ(4.1)

where d and n are the thickness and refractive index of the SPP, respectively [115]. As a result of
the phase delay, the SPP transforms the laser electric field from a Gaussian mode into a LaguerreGaussian mode LGp,l described by√ !|l|






2p!
1
r 2
2r2
r2
r2
|l|
Lp
exp −ikE(r, φ, z) =exp − 2π (p + |l|)! w(z) w(z)w (z)w2 (z)
2R(z)

 
z
× e−ilφ e−ikz exp i(|l| + 2p) tan−1
zR
s

(4.2)

where p is the radial index, l is the topological charge, Llp (x) are the associated Laguerre polyno2 /z. The phase profile
mials, and R(z) is the radius of curvature of the beam, given as R(z) = z + zRof the SPP used in this experiment is shown in figure 4.4a and it was carefully manufactured to
impart a topological charge of exactly l = +1 on the incoming TEM00 mode of λ = 780 nm light,
which transforms it into an LG01 mode beam in the far-field (figure 4.4b) with intensity profile

 
2
2r2
2r2
1
exp − 2L0w (z)w2 (z)


 2 
2r2
4P
r
exp − 2=2
2w (z)πw0 w (z)2P
I(r, z) =
πw02



2r2
w2 (z)

(4.3)

where P is the total laser power in the LG01 mode [116]. The LG01 beam is then split into two
co-propagating beams separated by 2.5 mm with equal intensity and orthogonal polarizations bypassing the beam through a calcite beam displacer. Upon exit from the calcite, the two LG01
beams pass through a 2x telescope designed to increase both the beam separation to 5 mm andthe beam waist to 1.61 mm. This configuration of two co-propagating “vortex” LG01 beams withorthogonal polarization is then sent to the AQuA Hex cell (section 4.2.4), where tightly overlappingthem forms a crossed vortex BoB trap (section 4.5.2, figure 4.19).

63

Figure 4.4: (Color online) a) Drawing illustrating a SPP transforming a TEM00 mode Gaussianlaser beam into an LG01 mode Laguerre-Gaussian beam. The phase profile of the SPP used in thiswork, showing how one full azimuthal rotation retards the TEM00 phase by 2π, is shown below.
b) The LG01 mode created in this experiment by sending the TEM00 mode through the SPP.

4.2.3.2

Gaussian Array BoB Setup

Returning to the 780 nm laser system shown in figure 4.3b, blue-detuned light can also be
directed to the Gaussian beam array setup to create an array of bottle beam traps. After emerging
from the AR coated, single-mode fiber, the beam is focused through a Holo/Or MS-248-X-Y-A
diffractive beam splitter which diffracts a single TEM00 mode beam into a 2x2 array of identical
Gaussian beams, all with with parallel linear polarizations (figure 4.5a). A set of telescopes andadditional calcite beam displacers serves to de-magnify the array to the desired dimensions while
increasing the number of beams in the array. First, a 66:35 telescope images the 2x2 array through a
calcite beam displacer cut to a thickness of 353 µm which replicates the 2x2 array into an array of 8
identical Gaussian beams separated from their nearest neighbor by 353 µm (figure 4.5b). Since thecalcite is a birefringent material, the 4 replicated Gaussian beams have an orthogonal polarization
to the 4 original beams. Additionally, in order to have equal spacing of each array beam (fromtheir nearest neighbor), the fast axis of the calcite must be rotated by 45 degrees with respect tothe diffractive beam splitter.

64
After passing through the calcite, the array of 8 Gaussian beams is further de-magnified by a
200:60 telescope and then demagnified again with a 500:23.125 telescope (figure 4.5c) and imaged
into the center of the AQuA Hex cell (section 4.2.4) for use in the experiments. At the focus of the
0.4 NA, 23.125 mm lens, each Gaussian beam has a focused beam waist of w0 = 2.21 µm and isseparated from its nearest neighboring beam by a distance d = 4.95 µm. The aspect ratio6 s, which
is defined as the ratio of the focused beam spacing to focused beam waist, is s = d/w = 2.23. After
passing through all of the optics required to produce the array, the total optical power remaining
in the array is 750 mW, which gives a uniform distribution of 47 mW per beam in the array. As
described in section 5.1.1, the optical intensity profile of the array at the focus of the 23.125 mm
lens (figure 4.5c) creates a two BoB traps.

Figure 4.5: (Color online) This set of images shows the formation of the Gaussian beam array at
various locations of the optical setup. a) Intensity plot and the direction of linear polarization of
the 4 identical Gaussian beams taken after a single Gaussian beam passes through the diffractive
beam splitter. b) Intensity and polarization diagrams for the array after the 4 identical Gaussians
pass through the calcite and are focused by the 200:60 demagnifying telescope. c) Image showing
the array after focusing with the 500:23.125 demagnifying telescope. As shown in section 4.9, thisintensity profile in c) is what is focused into the Hex cell to make an array of bottle beam traps.

6
While not explained until section 4.9, the section devoted to atom trapping in the array, it should be notedthat the aspect ratio s = d/w is not merely some spurious detail but rather is a very critical parameter for actuallytrapping atoms in the Gaussian beam array.

65
4.2.4

Layout of Laser Beams in the AQuA Hex Cell

The 852 nm cooling and MOT lasers from section 4.2.1 as well as the 1064 nm and 780 nm
transport and BoB lasers from section 4.2.2 and 4.2.3 all converge at the center of the AQuA Hex
cell. Figure 4.6 shows the placement of each beam inside the vacuum chamber. The BoB traps
are formed at the focus of the 780 nm BoB lasers and are offset from the 1064 nm atom transportlasers (functioning as an optical conveyor belt) by an angle of Θ = 60◦ . Additionally, the coolinglasers that form a 3D MOT intersect at an angle of Θ = 60◦ (as opposed to the traditional 90◦
angle from most MOT setups).

Figure 4.6: (Color online) Transport, loading, and imaging lasers entering the vacuum cell. The
780 nm beams generate the BoB trap at the center of the cell, while 1064 nm lasers are used togenerate the dynamic 1D optical lattice for atom transport. Finally, laser cooling, optical molasses,
and fluorescence imaging are performed with the 852 nm beams.

66

4.3

Producing a Reservoir of Cold Atoms
A 3D magneto-optical trap [117][118][119][120] of cesium atoms located in the AQuA Hex cellprovides a continuous reservoir of cold atoms for the experiment. As described in section 3.1, atoms
that source the 3D MOT are initially trapped and cooled in a 2D+ MOT located in a chamber
below the Hex cell and then sent upwards into the 3D MOT chamber via a “push beam.” Here,the experimental process of creating such a cold atom reservoir in the AQuA cell is described.

4.3.1

2D+ MOT

Shown in figure 4.7, the 2D+ MOT is created by overlapping two pairs of retroreflecting
50 mW, cylindrically shaped beams with orthogonal σ+ /σ− circular polarizations in the lower, 2D
MOT chamber. Both beams consist of a combination of 2D cooling and repump light that enters thesystem through individual fibers, is collimated with an f = 50 mm aspherical lens, and is spatiallyoverlapped7 at a polarizing beam splitter (PBS). A telescope made from a pair of cylindrical lenses
then enlarges the beam and gives it an elliptical profile with a beam waist of w = 0.5 mm along
the semi-minor axis and a waist of w = 1.0 mm along the semi-major axis. Using a PBS, the lightis then split into two beams with equal intensity. A pair of λ/4 waveplates for each beam controls
the circular polarization such that all beams overlap at an angle of 90◦ at the center of the 2D
chamber (inset 1 of figure 4.7b) with orthogonal σ+ /σ− circular polarizations.
Additionally, four permanent magnets located outside the 2D chamber creates a 2D quadrupole
field with gradient 15 G/cm. A 2D+ MOT is then created by red-detuning the 2D cooling light byδ = 2.9Γ. The resulting MOT force provides strong confinement of atoms in the x̂ and ŷ directions
but no confinement in the ẑ direction. A vertically oriented push beam overlaps the 2D MOT along
the loose ẑ axis (inset 2 of figure 4.7b) and provides a momentum kick to transfer cooled atoms up
through the 2D and 3D pinholes to provide a flux of approximately 109 cooled atoms per second
into the AQuA Hex cell to subsequently source a 3D MOT.
7

Prior to overlapping the cooling and repump, approximately 3 mW of cooling light is picked off and sent verticallythrough the cell to function as the push beam.

67

Figure 4.7: (Color online) a) Optical setup to form a 3D MOT in the AQuA Hex cell. The inset
shows the orientation of the 6 3D cooling beams, the vertically oriented push beam, MOT coils, andshim coils. A weak probe beam intersects the MOT location and is focused onto a Basler CCD for
absorption imaging. b) Optical setup to form a 2D+ MOT in the AQuA 2D chamber. Inset 1 shows
the orientation of the 4 cooling beams. Inset 2 shows the location of the vertically oriented pushbeam. The push beam enters at the base of the 2D chamber and propagates vertically through theentire AQuA cell, exiting through the top surface of the Hex cell. Since it overlaps the 2D MOT,
it “pushes” atoms into the Hex cell that are used to seed the 3D MOT.

68
4.3.2

3D MOT

Figure 4.7a diagrams the 3D MOT setup. The 3D MOT is created by retroreflecting threepairs of λ = 852 nm laser beams that are red-detuned δ = −2.9Γ from the |F = 4i → |F 0 = 5i
cycling transition and overlapped with repumping light resonant with the |F = 3i → |F 0 = 4itransition at the center of the AQuA Hex cell. Both cooling and repump beams enter the systemthrough the same fiber. A total of 30 mW of 3D cooling light and 5 mW of 3D repump lightis collimated from the fiber launcher by using an f = 50 mm aspherical lens. Immediately after
collimation, a 90:35 telescope demagnifies the beams to a collimated beam waist of w0 = 1.5 mm. A
series of polarizing beam splitters then separates the laser light into three beams of equal intensity.
Lastly, before entering the AQuA Hex cell, each beam passes through a λ/2 waveplate followed by a
λ/4 waveplate combination to give the MOT beams a σ+ (or σ− ) circular polarization. The opticsof the 3D MOT setup are completed by retroreflecting each beam to spatially overlap itself inside
the Hex cell. An additional λ/4 waveplate located in front of each retroreflecting mirror ensuresthe circular polarization of the retroreflected light is orthogonal to the initial light beam. Thus,
shown in the inset of figure 4.7a, each pair of 3D MOT beams has the σ+ /σ− circular polarizationnecessary for cooling (as described in figure 2.2a).
The final orientation of the 3D MOT beams as they appear in the actual experiment in thelab is shown in the inset of figure 4.7a. It is important to note that the three pairs of σ+ /σ− MOT
beams do not actually overlap at 90◦ angles along three orthogonal directions (which is the “modus
operandi” for most MOT setups). Rather, two pairs of σ+ /σ− MOT beams oriented on the samevertically oriented plane intersect in an “X” shape which crosses at an angle of θ1,2 = 40◦ . The
third pair of σ+ /σ− beams intersects the “X” pair at an angle of θ3 = 120◦ (see inset of figure4.7a). While this optical setup differs from the conventional MOT setup of three pairs of orthogonal
beams, 3D atom confinement is still achieved (although it is weaker than the conventional setup)since components of each beam are projected along the orthogonal x̂, ŷ, and ẑ directions.
With the laser setup completed, a pair of coils, where each is made with 100 turns of 20

69
gauge wire, is oriented along the ẑ direction of the AQuA Hex cell (inset of figure 4.7a). Running
1 A of current through the coils when the coil pair is oriented in the anti-Helmholtz configuration
generates a positionally dependent magnetic field B(x) with gradient magnitude of |B 0 | = 16 G/cm.
A flux of atoms is sent into the AQuA Hex cell from the 2D+ MOT formed in a chamber below
the Hex cell and sources the 3D MOT [121]. Using this described 2D+/3D double MOT setup, a
3D MOT of approximately 107 atoms is created within 1 second of loading time (figure 4.8). This
small 3D MOT functions as the reservoir of pre-cooled 133 Cs atoms for use in the atom transportand loading experiments described in sections 4.4, 4.5 and 4.6.

Figure 4.8: (Color online) Fluorescence image of A 3D MOT of approximately 107 cesium atoms
and a diameter of 3 mm created in the AQuA Hex cell. The atoms that source the 3D MOT are
initially cooled in a 2D+ MOT located in a chamber below the Hex cell. The 2D+ MOT atoms
are subsequently pushed up into the Hex cell by use of a push beam.

After loading the 3D MOT, the cooling beams were further detuned to −20Γ which serves toact as an effective compressed MOT (CMOT) stage, since only atoms at the most intense region of
the MOT beams will consistently absorb the laser light [3]. After the CMOT stage, time-of-flightabsorption imaging (described in reference [122]) was performed to measure the temperature of theatoms to be Tx = 80.1 µK and Ty = 67.7 µK along the x̂ and ŷ directions, respectively, giving an
average 3D MOT temperature of T = 73.9 µK (figure 4.9).

70

Figure 4.9:
(Color online) a) Time-of-flight (TOF) absorption images of the 3D MOT after
performing a CMOT stage. From top to bottom the TOF expansion times are 5 ms, 10 ms, 15
ms, 20 ms and 25 ms. b) and c) show the expansion of the Gaussian width σx (t) and σy (t) of
the 3D MOT as a function of time. The TOF expansion data corresponds to atom temperaturesof Tx = 80.1 µK and Ty = 67.7 µK along the x̂ and ŷ directions, respectively, giving an averagetemperature of 73.9 µK.

71
4.3.3

Polarization Gradient Cooling

In order to obtain maximum single atom loading probability and transport efficiency in the
BoB trap and optical conveyor belt, we implement a polarization gradient cooling (PGC) stage
after forming the 3D MOT to decrease the MOT temperature and increase the MOT atom density[123, 124, 125]. To perform PGC, the MOT coils are switched off while shim coils are turnedon in order to eliminate or reduce stray magnetic fields to milligauss levels at the center of theHex cell such that effective PGC can be applied to the atoms. During PGC, the 3D MOT lasers
are left on but detuned -50 MHz or approximately −9.6Γ. After 10 ms of PGC, time-of-flightabsorption imaging was performed to measure the temperature and density of the 3D MOT atoms
to be approximately T = 22.9 µK and 1010 atoms/cm3 , respectively (figure 4.10). Polarization
gradient cooling leaves the atoms with an energy that is a factor of about 100 smaller than the
crossed vortex bottle beam trapping potential and a factor of about 25 smaller than the Gaussian
beam array bottle beam trapping potentials. Most importantly, as discussed in sections 4.5.5 and4.6.1, the high atom density remaining after PGC permits the average number of atoms that canbe loaded into the crossed vortex BoB traps to be of order unity.

Figure 4.10: (Color online) a) and b) show Gaussian width σx and σy time of flight expansionsof the 3D MOT atom cloud after performing 10 ms of polarization gradient cooling. Expansion of
the atom cloud along the x̂ and ŷ directions gives respective temperature measurements Tx = 22.3
µK and Ty = 23.4 µK and average atom cloud temperature of 22.9 µK. The inset of b) shows anabsorption image of a cesium 3D MOT with atom density of 1010 atoms/cm3 .

72
4.3.4

3D MOT and PGC Pagoda Coil System

Before concluding this section on generating a sufficiently cold 133 Cs atom reservoir, it isimperative to briefly outline the coil system for generating the larger magnetic quadrupole field
necessary for the 3D MOT and the considerably smaller shim fields needed for PGC. We required
the coil system to be capable of providing shim coils for all six AQuA Hex cell windows in order toproduce bias fields along the optical axis of all cooling beams.
Moreover, the system must also provide shim coils for the ẑ direction as well as providing
a set of anti-Helmholtz coils for producing the 3D MOT magnetic fields. Most importantly, themounting scheme must fully permit optical access to all windows of the AQuA Hex cell. With theassistance from professor Mark Saffman and Marty Lichtman from the University of Wisconsin,
Madison, we were able to meet all of the above described design constraints by constructing a“pagoda” style coil mount8 (figure 4.11a).
The pagoda coil mount is 3D printed from ABS plastic and features a heat deflection temperature of 98◦ C at 0.46 MPa and 88◦ C at 1.8 MPa. A recessed lip is found on all six vertical faces aswell as the top and bottom faces of the pagoda mount in order to wrap the shim and 3D MOT coils.
Once each coil was wrapped on its respective face of the pagoda mount, each coil was permanently
fastened to the mount using Araldite epoxy. The finished coil system was then mounted over the
Hex cell (figure 4.11b). When fixed around the Hex cell, there exists 5 mm of clearance between
the outer surface of the Hex cell windows and the inner surface of the pagoda mount. The finishedproduct (figure 4.11c) features 3 pairs of x-y shim coils for magnetic bias fields oriented normal to
the Hex cell windows with gradients of 0.624 G/A, 0.639 G/A, and 1.54 G/A. The 1.54 G/A shim
is higher to help cancel residual fields from the 0.624 G/A and 0.639 G/A shims (figure 4.11d). The
shim coils oriented in the vertical direction create a magnetic bias field in the ẑ direction with agradient of 1.73 G/A. Finally, the set of anti-Helmholtz coils used to create the 3D MOT magnetic
quadrupole field has a magnetic field gradient of 16 G/A.
8
The “pagoda” mount aptly gets its name due to its slight resemblance of the religious pagoda structures foundin eastern as well as southern Asia.

73

Figure 4.11: (Color online) Pagoda coil mount: a) 3D rendering of the pagoda coil mount. b)
Rendering showing the pagoda mount position with respect to the Hex cell, the 3D MOT beams,
as well as any high numerical aperture lenses. Note that optical access of all Hex cell windows
is still permitted. c) Photograph of the pagoda coil system. d) Plot showing the direction of themagnetic field gradients from the 3 pairs of x/y shim coils as well as the z-bias coils.

74

4.4

Atom Transport System: Optical Conveyor Belt

4.4.1

1D Dynamic Lattice Potential

Atoms are transported from the 3D MOT to a single crossed vortex BoB trap by utiliing an
optical conveyor belt. The optical conveyor belt is a 1D optical lattice potential that can moveatoms back and forth along the axial direction of the lattice [126, 127, 128, 129]. Our opticalconveyor belt is created inside the AQuA Hex cell by overlapping two P = 5 W, λ = 1064 nm,
counter-propagating Gaussian laser beams with frequencies ω1 and ω2 and equal intensities of
I = 3.75 × 108 mW/cm2 (figure 4.12). The resulting standing wave potential is [126, 128]


w02
2r2
U (r, z, t) = U0
exp −cos2 (π∆ωt − kz)
w(z)2
w(z)2

(4.4)

where w0 is the laser beam waist size, w(z) is the Gaussian beam waist, k = 2π/λ is the wavenumber,and ∆ω = ω1 − ω2 is the detuning between the two 1064 nm beams.

Figure 4.12: (Color online) Schematic of the optics and RF electronics used to create and transport
the optical conveyor belt. An RF spectrum analyzer measures the velocity of the conveyor belt bymeasuring ∆ω of the lattice beams.

75
The quantity U0 is the maximum dipole potential and is defined as [72]U0 =~Γ P Γ
,2 πw02 I0 ∆(4.5)

where ~ = 1.05 × 10−34 J·s is the reduced Planck’s constant, P and I0 are the total laser power andsaturation intensity at the atoms, ∆ = 70 THz is the detuning between the 1064 nm beam and thecesium D2 transition frequency, and Γ = 2π × 5.22 MHz is the natural linewidth of the cesium D2transition. In the atom transport experiments, both λ = 1064 nm lasers have an equal power of
P = 5 and intensity of I = 3.75 × 108 mW/cm2 . From equation 4.6, focusing the beams to a waistof w0 = 30µm at the center of the Hex cell produces a 1D lattice with with a maximum trapping
depth of U0 /kB = −1.88 mK. where kB = 1.38 × 10−23 J/K is the Boltzmann constant. The axial
and radial harmonic oscillation frequencies Ωz and Ωrad of atoms trapped at the focus of the 1D
lattice arerΩz = 2πsΩrad =

2U0= 2π × 454 kHz
mλ2

(4.6)

4U0
= 2π × 3.6 kHz
mw02

(4.7)

where the numerical values are computed using the values for U0 and w0 listed above.
The velocity v of the lattice potential U (r, z, t) along the axial direction of the lattice iscontrolled by changing the lattice beam detuning ∆ω. The velocity9 of the 1D lattice as a functionof laser detuning is
v=

λ∆ω
.
4π

(4.8)

The value of ∆ω, and hence v is set by passing each lattice beam through an acousto-optic modulator(AOM) initially driven at 78 MHz in a double-pass configuration (full schematic shown in figure
4.3 and concisely summarized in figure 7.8). The double-pass geometry allows for changes in laser
frequency while avoiding changes in beam pointing, which is critical for maintaining the lattice9
To understand why detuning the lattice beams induces motion of the lattice, realize that there exists a reference
frame moving along the axial direction of the lattice at a velocity v where both lattice beams are Doppler shiftedto the same frequency, that is ∆ω = 0 in this frame. From the lab frame of reference, this is consistent with thelattice propagating along the axial direction with the same velocity v. Consequently, atoms that are trapped in this1D lattice potential U (r, z, t) are also translated with the same velocity v.

76beam overlap throughout the experiment. To minimize phase jitter in the lattice, both AOMs are
driven with independent RF drivers that are frequency locked and phase synchronized via the RF
drivers’ internal 10 MHz clocks.

4.4.2

Properly Accelerating the Dynamic 1D Lattice

It is important to note that in our atom transport experiments, the lattice detuning ∆ω,and hence conveyor belt velocity v, cannot just be “snapped” on instantaneously, as the resulting
acceleration will heat the atoms out of the lattice. Rather, motion of the lattice is achieved byadiabatically frequency chirping one lattice beam by ∆ω. Frequency chirping the beams at an
adiabatic rate is important since the total potential experienced by atoms trapped in the opticalconveyor belt becomes a tilted lattice potential. In the atom reference frame, the complete opticalconveyor belt potential along the direction of the lattice acceleration becomes
Utot (r = 0, z, t) = U0

w02
cos2 (kz) + maz
w(z)2

(4.9)

2

≈ U0 cos (kz) + maz
where a is the acceleration of the atoms. Shown in figure 4.13, the total potential is tilted upward
or downward when the respective conveyor belt acceleration is positive or negative. During any
atom transport sequence, as long as the 1D lattice potential is greater than the maz contributions
from atom accelerations, then atoms will remain trapped in the optical conveyor belt. To solve forthe maximum permitted acceleration in the tilted, moving potential, the local minimum of equation
4.9 as a function of acceleration a is computed by equating the spatial derivative of Utot to zero:dU= −2U0 k cos(kz) sin(kz) + ma = 0
dz(4.10)and solving for aa = 2U0kcos(kz) sin(kz).
m(4.11)

77
Since the maximum value of cos(kz) sin(kz) = 0.5, it immediately follows that the maximum ofequation 4.11, and hence maximum optical conveyor belt acceleration amax , isamax = U0k≈ 5 × 105 m/s2
m

(4.12)

where the numerical value is computed using the values for U0 , k, and m listed in section 4.4.1.
With the result in equation 4.12, we have an absolute speed limit for how fast we can transport
the atoms with the optical conveyor belt. However, this does not tell us how to accelerate theatoms. During transport, a minimal amount of heating will occur when frequency chirping one of
the lattice beams in a fashion that minimizes the rate of change of acceleration of the atoms.
Using equation 4.8, the velocity of the optical conveyor belt as a function of the mutual latticebeam detuning is
v=

c ∆ω
2 Ωz

(4.13)

where ω = 2πc/λ is the initial, un-shifted frequency of the λ = 1064 nm transport laser. It
immediately follows that the time rate of change of acceleration of the optical conveyor belt is

Figure 4.13: (Color online) Plot showing the Moving Standing Wave Potential for positive acceleration (a > 0), negative acceleration (a < 0), and no acceleration (a = 0).

78
expressed as
da
d2 v
= 2 =
dt
dt



1 d2 ∆ω1 d∆ω ∆ω
− 2
+ 3
2
Ωz dt
Ωz dt
Ωz


.

(4.14)

In our experiments, the lattice chirping ∆ω will be a linear ramp. Thus, the d2 ∆ω/dt2 term
vanishes. Additionally, as shown in upcoming section 4.4.3, the lattice axial trap frequency Ωz ∆ω, and thus we can approximate that ∆ω/Ω3z ≈ 0. Using these approximations, the magnitudeof the time rate of change of the conveyor belt acceleration is
da
1 d∆ω
= 2
.
dt
Ωz dt

(4.15)In order to minimize this function, the lattice chirping must satisfy
Ω2z 

d∆ω
.
dt

(4.16)

Equation 4.16 can be used to define a dimensionless adiabatic chirping parameterξ=Ω2zd∆ω/dt(4.17)

whereby ξ  1 must be satisfied to transport atoms with minimal heating imparted from changes
in acceleration of the optical conveyor belt. In our experiments, the desired adiabaticity parameteris ξ ≈ 10−3 . Figure 4.14 displays a plot of the adiabaticity parameter ξ as a function of thefrequency acceleration d∆/dt and shows that ξ = 10−3 is possible for d∆ω/dt ≈ 2 × 108 Hz/sec,
which corresponds to a frequency chirp of 500 kHz over 5 ms. Thus, any slower acceleration will
always satisfy the desired ξ parameter. With proper acceleration schemes understood, we now
demonstrate transporting atoms with the optical conveyor belt.

4.4.3

Demonstration of Atom Transport with an Optical Conveyor Belt

Approximately 6,000 atoms are loaded into the conveyor belt by overlapping the 1D lattice
and the 3D MOT for 530 ms [130]. After loading the lattice, the MOT is turned off, leaving
behind a reservoir of cold atoms in the lattice. To transport atoms using this optical conveyor
belt, one lattice beam is linearly frequency chirped by 200 kHz over 5 ms. Because the AOMs are

79

Figure 4.14: (Color online) Plot of the adiabatic parameter ξ as a function of the lattice chirpingparameters d∆ω/dt.

in the double pass configuration, this causes a lattice beam detuning of 400 kHz. With 400 kHz
detuning, the optical conveyor belt accelerates to a maximum velocity of 212 mm/s. After thisacceleration, the lattice beam is linearly chirped down -400 kHz over 5 ms to give zero detuning
for the lattice. This process displaces any loaded atoms by 0.53 mm in 10 ms. The atoms aretransported back to their original starting location by repeating the process in reverse. The entire
procedure transports the atoms a total distance of 1.1 mm in 20 ms. A composite time-lapse imageof this entire transport process is shown in figure 4.15.
As the optical conveyor belt motion abruptly starts and stops, large forces are imparted to
atoms trapped in the 1D lattice. Consequently, cold atoms in the conveyor belt can be heatedand even ejected from the lattice. As described in section 4.4.2 and equation 4.16, minimizing thisheating effect is achieved by chirping the moving lattice at a much lower rate than the square of thelattice axial oscillation frequency. For motion of our optical conveyor belt the inequality holds asΩ2z ∼ 1012 s−2 and d∆ω/dt ∼ 108 s−2 . Additionally, we found that this transport process preserves
approximately 87 percent of the atoms trapped in the optical conveyor belt. This atom survival
probability is approximately equal to the atom survival in the stationary lattice over the same time

80

Figure 4.15: Absorption images showing the sequence of transporting cold atoms with an opticalconveyor belt: 1) Atoms are loaded into the conveyor belt by overlapping the 1D lattice and the3D MOT. 2) The MOT is turned off, leaving behind a reservoir of cold atoms in the lattice. 3)
The frequency of one lattice beam is linearly frequency chirped and the optical conveyor belt
accelerates to a maximum velocity. After this acceleration, the frequency of this lattice beam islinearly chirped again but with the opposite sign in order to slow the atoms down to a halt. The
atoms are transported back to their original starting location by repeating the process in reverse.
4) Atoms are then held in place at the desired, final location. The inset shows a plot displaying therelative detuning ∆ω of the lattice beams as a function of time during the conveyor belt transport
process.

81
period (figure 4.16). Because the process of transferring a single atom from the conveyor belt andloading it into a BoB trap relies heavily on the atom density in the lattice, we chose to use the
400 kHz frequency chirping, as it permitted the farthest and fastest transport with the highest
remaining atom density after transport. Also shown in figure 4.16, we tested transporting atoms
at a much faster velocity and greater transport distances by using higher frequency chirps andlonger transportation times. While we achieved greater transport distances of up to 5.32 mm, themore intense accelerations resulted in the atom survival rate rapidly declining to below 50 percent.
More importantly, for these more intense accelerations, the remaining atom density after transport
drops far below the remaining atom density of the stationary lattice over the same time durations,
showing that heating due to these greater accelerations is outweighing the natural heating rate of
the lattice.

Figure 4.16: Change in atom density (atoms/cm3 ) in the moving standing wave dipole trap
after transport. The final density ρf is compared to the initial atom density ρ0 for various AOM
detunings and frequency sweep times for the moving standing wave dipole trap. The numbers
written inside the plot show the transportation distance for each set of parameters. Note the black
points indicate atom survival in the stationary lattice.

82
4.4.4

Laser Beam Overlap and Alignment Considerations

Successful atom transport with the optical conveyor belt depends critically on the degree ofoverlap between the two 1064 nm lattice beams and their relative polarization. The amplitude of
the beat note ∆ω of the 1D lattice beams is a maximum when the beams are entirely overlappedwith parallel polarization. [131] Moreover, the line width of the beat note is used to quantify thephase jitter, and hence the axial position fluctuations of the conveyor belt. [131, 132, 133] Thebeat note is measured by focusing the overlapped 1D lattice beams onto a photodetector that isconnected to an RF spectrum analyzer (Fig. 4.12). The lattice beams are focused using a lens withthe same focal length used to focus the lattice in the vacuum cell.

Figure 4.17: (Color online) Beat note spectrum measured by overlapping the two 1064 nm opticalconveyor belt laser beams. The frequency spectrum peaks at 400 kHz and has a full-width at
half-max of 1.33 kHz as measured at the -3dB point. This signal is used to quantify the latticebeam overlap, relative polarization, and phase jitter.

Additionally, to avoid beam distortions on the 780 nm BoB when passing through the 7.3 mm
thick cell windows, the front face of the 0.4 NA lens must be aligned normal to the cell walls. [134]
Otherwise, the BoB will never completely close in the axial direction, giving atoms an opportunistic

83
path to escape the trap. In order to have precise control over the position of the 0.4 NA lens, thelens is mounted with a compact lens positioner that provides adjustments along the x, y, and z
axes in addition to control over θx and θy .
To deliver and load an atom into the BoB trap, the foci of the 780 nm and 1064 nm beams
must overlap. As shown in Fig. 4.6, the optical axis of the 1064 nm conveyor belt and 780 nm BoB
lie on the same plane but are at a 60 degree angle with respect to each other. To overlap their foci,an 852 nm resonant beam is sent backwards though the photon collecting fiber shown in Fig. 4.19
and is focused through the BoB trap. After transporting atoms to the BoB site, the conveyor belt
is held in place while the resonant beam is snapped on. The resonant beam causes any trapped
atoms in its path to be ejected from the conveyor belt. Performing absorption imaging provides
visual proof of atoms ejecting from the trap and approximates the degree of overlap between theconveyor belt and BoB trap. Course adjustments of the conveyor belt position were made until
only atoms at the focus of the conveyor belt were observed to be ejected. Fine alignment of theconveyor belt position was made by turning on the 3D MOT lasers after transporting atoms to
the BoB site and counting the photons emitted from the atoms while still held in the lattice using
the methods from section 4.5.3. Micron-scale adjustments of the conveyor belt position were madeto optimize the fluorescence signal. With the photon counting signal at a maximum, there exists
adequate overlap between the foci of the BoB trap, photon detection optics, and conveyor belt
lasers. However, this degree of overlap contains the largest source of error for this technique, due
to the lack of a distinct, physical reference point for the microscopic BoB trap.
With the optical conveyor belt system and the atom transport process described, we now
introduce the optical bottle beam traps that we will deliver a single transported atom into.

84

4.5

Trapping Single Atoms in a Crossed Vortex Bottle Beam Trap

4.5.1

Overview of Optical Bottle Beams

An optical bottle beam (BoB) is an electric field distribution that features a region of low
optical intensity completely surrounded by high optical intensity. Various BoB intensity profiles
taken from the PhD thesis of Siyuan Zhang10 are shown in figure 4.18. In this dissertation, thelaser light used to form the BoB intensity profiles has wavelength λ = 780 nm and is blue-detuned
from resonance. From equations 2.11 and 2.12, by using blue-detuned light, the regions of the
BoB with high intensity produce a repulsive potential, forcing the atoms to seek regions of low
optical intensity. Thus, atoms can become trapped in the central, dark region of the BoB. While a
red-detuned dipole trap is considerably easier to implement for trapping atoms, the upside of theblue-detuned BoB trapping scheme is that it is very useful for confining single cesium atoms foruse in quantum computing (which is the intended application of this work). Atoms in the dark,
blue-detuned traps subsequently scatter less photons than if they were trapped in a red-detuned
dipole trap of comparable depth. Since the qubit decoherence time is proportional to the photon
scattering rate of the trapped qubit, trapping in blue-detuned BoBs can result in coherence times
300 times longer than those observed in red-detuned dipole traps with similar trapping depths [135].

Figure 4.18: (Color online) Various Bottle Beam Intensity Profiles: a) Gaussian interference BoB
made from the destructive interference of two TEM00 Gaussian modes with different waists. b)
Crossed Vortex BoB made from a Laguerre-Gaussian LG01 mode. c) Gaussian Beam Array BoB
made from four TEM00 Gaussian modes.
10

From Mark Saffman’s group at the University of Wisconsin, Madison.

85
4.5.2

Generating a Crossed Vortex Bottle Beam Trap

The intensity profiles of the BoBs shown in figure 4.18a,b would generate radial confinement(along x̂ and ŷ) but not axial confinement along the direction of beam propagation (ẑ, i.e. in andout of the page). To provide confinement along the axial direction, additional blue-detuned lightmust close the ends of the BoB (or “cap” the bottle beam). A BoB trapping potential is formed forthe experiment in this dissertation by using a high numerical aperture (0.4 NA) lens to tightly cross
two co-propagating Laguerre-Gaussian LG01 beams with orthogonal linear polarizations at a tightfocus. Because the LG01 beams are sometimes called “vortex beams” this optical configuration
gives the trap the appropriate name of “Crossed Vortex Bottle Beam Trap.”
The complete, detailed schematic for generating the co-propagating pair of orthogonallypolarized LG01 beams with identical beam waists of w = 1.61 mm and separated by distance d = 5
mm is shown back in section 4.2.2, figure 4.3b. With the LG01 beams created, figure 4.19 shows
how we form the BoB trap by using a high quality, diffraction limited 0.4 NA lens with focal lengthf = 23.125 mm to tightly focus and overlap the LG01 beams at the center of the AQuA Hex cell.
As the beams are focused with the lens, they are rotated by dθ = tan−1= 6.17◦ .
2f(4.18)

Because the rotation is small (θ < 10◦ ) the changes in polarization to each LG01 beam can beneglected. The resulting crossed vortex intensity profile ICV is also shown in figure 4.19 and isgiven as the sum of both linearly polarized LG01 beams:
ICV (r, z) = Il,p (r+ , z+ ) + Il,p (r− , z− ) = I1,0 (r+ , z+ ) + I1,0 (r− , z− )(4.19)

where the intensity of each LG beam, Il,p (r, z) is defined in equation 4.3. For the intensities inequation 4.19, the radial coordinates r± are defined [102] aspy 2 + (x cos θ ± z sin θ)2(4.20)z± = z cos θ ∓ x sin θ.

(4.21)r± =and the axial coordinates z± are

86

Figure 4.19: (Color online) Top: Optical setup for generating and imaging the crossed vortex bottle
beam trap. The added optical molasses light is used for fluorescence imaging on an atom trappedin the BoB. A fraction of atom fluorescence is collected with the 0.4 NA lens and propagated into
a counter to perform photon counting statistics. Bottom: Intensity profile at of the crossed vortex
BoB trap forming as the two LG01 mode lasers are focused at the focal plane of the 0.4 NA lens.
Images a), b), c), and d) show the trap at axial lengths z = 34 µm, z = 26 µm, z = 12 µm, andz = 0, respectively from the focus of the 0.4 NA lens.

87

Figure 4.20: (Color online) Crossed Vortex Bottle Beam trapping potentials along the xz, yz, andxy planes at the focus of the 0.4 NA lens.

where the ± subscripts refer to the π+ and π− linearly polarized LG01 beams, respectively.
The total optical trapping potential of the crossed vortex BoB trap is then numerically calculated
by inserting equation 4.19 into equation 2.11. The calculated trapping profiles along the orthogonalxz, yz, and xy planes at the focus of the crossed vortex BoB are plotted in figure 4.20. With respect
to the parameters of this experiment, at the focus of the 0.4 NA, f = 23.125mm lens, the crossed
vortex BoB trap has a width of 5.5 µm in the radial direction and an axial length of 52 µm. Using
300 mW of 780 nm light produces a calculated BoB trap depth of U/kB = 1.65 mK, with harmonic
oscillation trap frequencies of Ωrad = 2π × 51 kHz in the radial direction and Ωz = 2π × 5.3 kHz in
the axial direction (figure4.21).

Figure 4.21: Numerically calculated a) radial and b) longitudinal trap depths produced by thecrossed vortex BoB.

88
Finally, in order to get an intuitive understanding of how the final size of the crossed vortex
BoB affects the trapping depth, the equations for the trapping potential can be vastly simplified
by choosing to perform a Taylor series expansion of the intensity (equation 4.19) about the origin.
Such expansion of the trap depth about (x = 0, y = 0, z = 0) gives [102]2αP cos2 θ 2xπ0 cw4
2αP 2
U (y) ≈ −yπ0 cw4
2αP sin2 θ 2
U (z) ≈ −z .
π0 cw4

U (x) ≈ −(4.22a)(4.22b)(4.22c)Equations 4.22a, 4.22b, and 4.22c convey a very important result that the crossed vortex BoB
trap potential scales as w−4 , meaning (for the experimentalist) that meticulous care must be madewhen focusing the LG01 beams though the thick Hex cell walls to ensure that as little distortions
as possible occur such that the final beam waist of w0 = 5.5 µm is correct.

4.5.3

Detecting a Loaded Single Atom

Fluorescence imaging is used to detect the presence of a single atom loaded into the crossed
vortex BoB trap. When a single atom is loaded, a 3D optical molasses is applied to the trapped
atom by turning on the 3D MOT lasers and detuning them -30 MHz. Repump light is also turned
on. Additionally, the MOT coils are turned off and shim coils are used to eliminate any residual
magnetic fields. This induces atom fluorescence while continuously cooling the trapped single atom.
As shown in figure 4.19, the fluorescence is collected and collimated with the same 0.4 NA, f=23.125
mm lens used to focus the BoB trap. Photons collected from the trapped atom are separated fromthe 780 nm trapping light by an 800 nm short-pass dichroic mirror (also shown on figure 4.19). The
collimated fluorescence is then focused by an f=60 mm achromatic doublet lens into a 10 µm core
diameter single mode optical fiber that is coupled into a single photon counting module (SPCM).
The 10 µm fiber functions as a pinhole for spatial filtering of the fluorescence. Since the opticalmagnification from the BoB trap to the pinhole is 2.6, a window 3.85 µm in diameter is thus selected
at the BoB trap focus. Two 852 nm laser line filters are used to eliminate stray 780 nm BoB light.

89
The SPCM works alongside an avalanche photodiode to count individual photons emitted
from the trapped atom over an arbitrary time interval. When exposed to optical molasses, thetrapped atom scatters on the order of 105 photons over a 50 ms interrogation time. Since thefluorescence collecting lens has an effective focal length of 23.125 mm and an aperture of 18 mm,
the collection angle is Ω/4π = 4.67%. Total transmission of the photon collecting optics is approximately 50% and the quantum efficiency of the SPCM at 852 nm is 54%. Thus, the maximum photon
collection efficiency is approximately 1.1%. Additionally, noise of approximately 100 counts/sec of
780 nm light and approximately 25 dark counts/sec is recorded using the photon counter.

4.5.4

Loading a Bottle Beam Trap from a Background MOT

To verify that our BoB is capable of trapping a single atom, we first attempted to load
the BoB trap directly from a 3D MOT. The MOT was loaded for 1 second followed by PGC (asdescribed in section 4.3.3) for 10 ms. During the final 5 ms of PGC, the BoB is snapped on in 300µs and overlaps the central and most dense region of the 3D MOT. If multiple atoms are initiallyloaded into the BoB trap, a useful process between pairs of atoms called light assisted collisions11
will reduce the trap occupation to zero or one atom [136]. After loading the BoB trap and allowing
time for light assisted collisions, all MOT lasers are turned off for 20 ms to let any untrapped atoms
fall away due to gravity. After this fall time, the MOT beams are turned back on and fluorescenceimaging on the trapped single atom is performed (described in section 4.4.3). A histogram of thephoton counting data recorded by the SPCM is shown in figure 4.22.
The photon counting statistics for 1002 independent atom loading attempts show two Poisson
distributions corresponding to either 0 or 1 atom confined in the BoB trap after any given loading
trial. When 0 atoms are present in the BoB trap, a mean of nbg = 125 background photons arecounted over 50 ms (figure 4.22). This signal arises from stray 852 nm molasses light and 780 nm
11
As atoms confined in the BoB approach each other in the presence of the MOT laser light, there exists a criticalinter-atomic separation where the laser light excites a long-range, attractive dipole-dipole interaction between pairs of
atoms, which results in the atom pair gaining considerable kinetic energy leading to collisions that expel both atoms
from the BoB trap. Consequently, if an even number of atoms occupy the BoB trap, they will all pair-collide until noatoms remain in the trap, whereas an odd number of atoms will pair-collide until only one atom remains in the trap.

90
trapping light being collected by the detection optics. During trials when one atom is loaded in the
BoB trap, the number of photons detected over 50 ms increases to a mean of n1 = 215 counts (Fig.
4.22) due to the added fluorescence from the atom trapped in the BoB. Photon counting statistics
show that a single atom is loaded into the BoB trap in 45.2% of all loading trials. The near complete
absence of multi-atom loading events is due to light assisted collisions and the collisional blockade
effect. [136, 137, 138]

Figure 4.22:(Color online) Histogram of photon counting data recorded during fluorescenceimaging of the BoB trap. Two Poisson distributions are present with means of 125 counts and 215
counts, corresponding to 0 or 1 atom present in the BoB trap, respectively. The histogram is fittedwith a compound Poisson distribution over atom and photon number.

The photon counting data is fitted to a compound Poisson fit"#
2X(n − (nbg + kn1 ))2
N k e−N1pP(n) = Aexp −k!
2 (nbg + kn1 )2π (nbg + kn1 )(4.23)

k=0

For random loading following a Poisson distribution
P (k) =

N k e−Nk!

(4.24)

the probability of loading a single atom (assuming that on average we load one atom) has an
upper bound of P (1) = e−1 = 0.368. Thus, following random loading, P (1) ≤ 0.368. However,91
from figure 4.22, a single atom is loaded 45.2% of all loading trials, corresponding to a variance of(∆N )2 = 0.452N which is smaller than the variance of the Poisson distribution, (∆N )2 = 0.368N ,which shows that the atom loading is sub-Poissonian in nature.

4.6

Demonstration of Single Atom Delivery and Loading into a Crossed
Vortex Bottle Beam Trap
This section chronicles the experimental results of delivering and loading a single atom into acrossed vortex BoB trap. A discussion on applying the results to qubit array systems (as discussed
in section 1.3) as well as motivating scaling the experiment to address an array of BoB traps (asdiscussed in chapter 5) is included.

4.6.1

Atom Transport and Reloading Operation

Each run of the experiment begins with using the 3D MOT system described in section 4.3.2
to create a reservoir of 106 atoms at T ≈ 75 µK within 1 second. During the last 480 ms of the 3D
MOT loading time, the 1064 nm optical conveyor belt lasers with no mutual detuning (∆ω = 0)
are un-shuttered and overlap the MOT. Approximately 5000 to 6000 of the 3D MOT atoms areloaded into the stationary 1D lattice. Polarization gradient cooling, as described in section 4.3.3 isapplied throughout the final 10 ms of the conveyor belt loading time, cooling the loaded atoms toT ≈ 25 µK and increasing the atom density to ρ ≈ 1010 atoms/cm3 .
It is essential to mention that the probability of loading a single atom into the BoB trap
is directly proportional to the atom density trapped in the optical conveyor belt. With the BoB
approximated as an ellipsoid, an upper bound for the BoB trap volume is
4 2
V = πrradrz
3(4.25)

where rrad is the radius along the radial direction and rz is the axial radius. Fig. 4.19 showsthat rrad = 2.75 µm and rz = 26 µm, which gives a total BoB trapping volume of approximately
8.23 × 10−10 cm3 . After the entire atom transport sequence (Fig. 4.15), approximately 3,000 atoms

92
remain in the conveyor belt and the atom density is approximately 1.4 × 109 atoms/cm3 . Thus,
an upper bound to the expected number of atoms that can be overlapped with the BoB trap is
hN i = VBoB × ρ [139] which corresponds to overlapping an average of 1 atom per trial. Therefore,
maintaining an atom density of 109 atoms/cm3 in the conveyor belt after transport is crucial.
After loading the optimal atom density into the optical conveyor belt, atoms are transported
1.1 mm to the BoB trap site by following the frequency chirping sequence outlined in section 4.4.3
where one lattice beam is linearly frequency chirped by 400 kHz over a 5 ms time interval andaccelerated to a maximum velocity of 212 mm/s. The lattice beam is then linearly chirped down
-400 kHz over another 5 ms period to instantaneously stop the conveyor belt before repeating thechirping sequence in reverse to bring the atoms back to the starting point. This series of latticebeam frequency chirps first transports atoms a distance of 0.530 mm along the axial direction
followed by an additional 0.530 mm in the reverse direction for a total transport distance of 1.1
mm in 20 ms. Time-of-flight temperature measurements performed on the transported atoms showthat atom temperatures rise to 70 µK after transport in the conveyor belt. Ballistic expansion in theradial direction of the conveyor belt was chosen for time-of-flight temperature measurements since
it expands faster and therefore shows greater temperature dependence than in the axial direction[140]. To load a single atom in the BoB trap from the conveyor belt and also to successfully image
the loaded atom, the transported atom temperature should optimally be a factor of 50 colder thanthe BoB trap depth [106]. To cool the transported atoms to such an optimal temperature, anadditional 10 ms of PGC is applied to the atoms immediately after transport while still held in
the 1D optical lattice. The intense light shift imparted by the lattice (equation 2.11) detunes theatoms an additional −8Γ, giving a total detuning of −14.6Γ. When applying PGC after transport
the atoms have an average temperature of 15 µK as compared to a temperature of 70 µK when no
extra PGC is applied (figure 4.23).
During the final 5 ms of the additional PGC, the BoB trap is snapped on and overlaps thefocus of the optical conveyor belt. This permits an atom to load from the transported 1D lattice
into the BoB trap. After loading, the optical conveyor belt is turned off while still holding the BoB

93

Figure 4.23: (Color online) a) Density profiles of 133 Cs atoms at different time-of-flight (TOF)
expansion times after horizontal transport with the optical conveyor belt. b): Graph showing thesize of the atom clouds at different TOF expansion times when extra PGC is applied (blue dots)
compared to no PGC (purple dots). The TOF expansion data shown here corresponds to an atom
cloud temperature of 14.9 µK when PGC is applied to the transported atoms. Without the extra
PGC, the transported atom temperature is considerably higher at 72.2 µK.

94
trap to allow all untrapped atoms to fall away. At this point in the experiment, the BoB is loadedwith either zero or one atom. A complete synopsis of the timing scheme for single atom loading
via an optical conveyor belt is presented in figure 4.24.

Figure 4.24: (Color online) Summary of the Atom Transport and Loading Experiment: Left:
Timing scheme for different stages of the atom transport and delivery process involving the 780
nm BoB lasers, 852 nm MOT lasers, and the 1064 nm conveyor belt lasers. A graph shows themutual detuning ∆ω between the 1D lattice beams as a function of time during the conveyor belttransport process. Right: Absorption images showing atoms transported 1.1 mm with the opticalconveyor belt at various times during travel. Images a), b), c), d), e) show the loaded conveyor belt
at 0, 5, 10, 15, and 20 ms, respectively. Furthermore, images a) and e) show the conveyor belt at
the initial and final transport locations. Images b) and d) show the conveyor belt when it reaches
the maximum velocity of 212 mm/s. Image c) shows the conveyor belt at the turn around point.
Approximately 3,000 atoms remain in the conveyor belt after transport.

95
Fluorescence imaging, as described in sections 4.5.3 and 4.5.4, is performed on the BoB trap
for 100 ms to observe if zero or a single atom is successfully delivered and loaded into the trap.
A histogram of the photon counting data is shown in figure 4.25. During fluorescence imaging, anaverage of 300 photons are detected when no atom is present in the BoB trap. When an atom isloaded into the BoB trap, the average number of counted photons increases to 570 counts. The
photon counting statistics for 843 atom delivery and loading attempts confirm that an atom canbe delivered and loaded into the BoB trap 13.1% of the time, with no multi-atom loading events
ever recorded.

Figure 4.25:(Color online) Histogram of photon counting data recorded during fluorescenceimaging of the BoB trap after delivering the atom via an optical conveyor belt. Two Poisson
distributions are present with means of 300 counts and 570 counts. The peaks correspond toloading 0 or 1 atom, respectively, in the BoB trap with the optical conveyor belt. The histogramshows that a single atom is delivered into the BoB trap 13.1% of the time. The histogram data is
fitted to a compound Poisson distribution over atom and photon number.

96
4.6.2

Further Discussion for Continuous, Uninterrupted LoadingThis technique provides a method for correcting atom loss errors for single atom BoB traps
used in neutral atom quantum computing systems. In order to provide an optimal vacuum environment for trapped qubits and qubit gate operations, the 3D MOT that supplies the source ofcold atoms for atom reloading should be located in a separate vacuum chamber positioned adjacentto the chamber that houses the BoB traps. This also permits the 3D MOT to be on continuously,as the MOT footprint will not disturb any previously trapped atoms. Using an uninterrupted 3D
MOT, the presented technique for single atom delivery and loading takes a total of 680 ms, as
the procedure (fig. 4.24) involves loading the conveyor belt (530 ms), transporting and loading
the atom into a BoB trap (50 ms), and then finally imaging to verify the presence or absence ofa single atom (100 ms). The total time for delivering and loading an atom can be reduced to alittle as 55 ms total, as far-off resonance dipole traps, and hence the optical conveyor belt, can beloaded in as little as 10 ms, depending on the trap detuning, intensity, and the vacuum backgroundpressure,[130] while single atom signals can be observed using fluorescence imaging with exposure
times as little as 5-10 ms [1, 141]. Additionally, the BoB hold time discussed in both section 4.6.1
and figure 4.24 can be decreased to 10 ms. Even shorter hold times are possible but at the expenseof introducing fluorescence noise from untrapped atoms that may be collected during imaging.
With a 13.1% probability of successful single atom delivery and loading, 33 loading trials willneed to run in order to have single atom loading with 99% confidence. Working in the lower limitof a 55 ms loading time, the procedure will require 1.82 seconds to reload an atom if all 33 trials areneeded, which is within the necessary interval of time of 2 seconds to keep a 49 element qubit array
continuously occupied (discussed in sections 1.2 and 1.3). Additionally, with a 13.1% probability ofsuccessful atom reloading, 50 loading trials are required to reload an atom with 99.9% confidence,
which requires 2.75 seconds if all trials are needed, which is slightly greater than the necessary time
to reload a site.
Finally, the single atom loading efficiency of this technique may be improved by incorporating

97
methods that suppress the inherit stochastic characteristics present with using red-detuned, lightassisted collisions to prepare a single atom in the BoB trap. One such method uses a feed-backmechanism to rapidly prepare a 3D MOT with a single atom,[142] while another method uses
additional blue-detuned lasers, rather than MOT lasers, to induce repulsive light-assisted collisions
to prepare single atoms in dipole traps [143, 144, 145, 137]. These methods have the capabilities toprepare single atoms directly from a MOT within 20-200 ms with average efficiencies ranging from
80% to 90%. If implemented after atoms are delivered to the BoB trap via transport as discussed
in section 4.4, they may increase the single atom loading efficiency.

4.6.3

Demands to Upscale Transport and Loading System

We have shown a method to transport and load a blue-detuned BoB trap with a single atomby using a dynamic 1D optical lattice. The transporting lattice, called an optical conveyor belt, isinitially loaded with a few thousand atoms. Using the optical conveyor belt, atoms are transported
a distance of 1.1 mm to a crossed vortex BoB trap. After transport, a single atom is then loaded
into the BoB trap via light assisted collisions. Photon counting data confirms that an atom canbe delivered and loaded into the bottle beam trap 13.1% of the time. Implementing this work
to reload a functional qubit array[3, 1, 103, 105, 106, 107] requires deterministically loading any
arbitrary site of an array of BoB traps. An additional pair of acousto-optic modulators for each of
the 1D lattice beams can reposition the optical conveyor belt to permit loading of any BoB arraysite at an arbitrary location [146, 147]. Modifications were made to the atom transport and loading
apparatus in order to “steer” the optical conveyor belt with the eventual goal of loading any site ofan arbitrarily large array of bottle beam traps such as those shown in figure 1.1. By “steering” theoptical conveyor belt, we mean giving the atom transport system the capabilities of delivering an
atom to one BoB trap and the repositioning the conveyor belt and delivering an atom to anotherarray BoB trap. Scaling up the atom delivery system to accommodate an array of BoB traps is
described in detail in chapter 5.

Chapter 5

Steering the Optical Convyeor Belt: Addressing an Array of Bottle Beam Traps

In order to transport and load atoms into an array of multiple BoB traps, the optical conveyor
belt must be “steerable.” As mentioned in section 4.6.3, the conveyor belt can be transported not
only along its axial direction, but also along the x̂ and ŷ directions by utilizing an additional pair of
acousto-optic modulators for each lattice beam. This allows one to reposition the optical conveyor
belt to permit loading of a BoB trap at an arbitrary (x, y, z) location. To properly demonstrate asteerable optical conveyor belt, an array of BoB traps, rather than one single trap, must be made.

5.1

Ineffectiveness of an Array of Crossed Vortex BoB Traps
The process of making an array of crossed vortex BoB traps involves taking a pair of LG01

beams and using a diffractive optic element (DOE) to replicate or “copy” the beams into an array ofLG01 beams on a conjugate plane of the image plane of the final 0.4 NA lens used to focus the BoBs
into the Hex cell (figure 5.1). While this setup concept is relatively simple to construct, it creates
extensive problems with the resulting array of crossed vortex BoBs. The calcite beam displacerand DOE result in the tightly crossing BoBs at the image plane having the same polarization,rather than orthogonal polarizations. As a result, considerable phase dependent interference effects
commonly arise, making it very difficult to implement as a qubit array capable of performing
sensitive quantum computations. Accordingly, the geometry of the BoB array was modified1 from1

Our collaborators at UW-Madison, who actually perform the quantum algorithms with the qubit array, firmly
insisted on making this change. Since our work at CU-Boulder was to be able to reload the UW-Madison array, wenaturally changed our geometry as well.

99
an array of crossed vortex BoBs to a “Gaussian Beam Array” (GBA), that subsequently forms anarray of BoB traps. This final chapter for the atom transport and reloading project shows the GBA
system (5.1), the steerable conveyor belt (5.2), and the corresponding atom transport results (5.3).

Figure 5.1: (Color online) Bottle Beam Array Concept: A combination of a calcite beam displacer
and a diffractive beam splitter is used to form and array of crossed vortex BoB traps. However,polarization fluctuations between interfering tightly overlapping LG01 beams at the focal plane of
the 0.4 NA lens render this concept useless for quantum computation purposes.

5.2

Atom Trapping in a Gaussian Beam Array of BoBs

5.2.1

Gaussian Beam Array (GBA)

The optical system for creating the Gaussian beam array (GBA) with two BoB trappingsites is shown formally in figure 4.3b and summarized in figure 5.2a. A Toptica BoosTA Pro sends
3W of 780 nm light into a polarization maintaining, anti-reflection coated, fiber with 65% coupling
efficiency. After emerging from the fiber, the Gaussian beam is collimated and sent through apolarizing beam splitter (PBS). This PBS serves to keep the polarization of the outgoing beam
constant by mapping minor polarization drifts from the TA and fiber into minor power drifts in thebeam. This single Gaussian beam is then focused through a Holo/Or MS-248-X-Y-A diffractive
beam splitter (DBS), which diffracts the single beam into a 2x2 array of identical Gaussian beams
which are all collimated with a f = 13.86 mm aspherical lens. A 66:35 telescope then images
the 2x2 array pattern through a 353 µm thick calcite crystal to replicate the 2x2 array into an

100arrangement of 8 identical Gaussian beams separated from their nearest neighbor by d = 353 µm.
In order to have equal spacing of each of the 8 array beams, the fast axis of the calcite is rotatedby 45◦ with respect to the diffractive beam splitter axis. After passing through the calcite, thearray of 8 Gaussian beams is demagnified by a 200:60 telescope and then further demagnified andimaged through the AQuA Hex cell walls (figure 4.6) with a 500:23.125 telescope2 . At the focusof the 23.125 mm lens, each Gaussian beam has a focused beam waist of w0 = 2.21 µm and isseparated from its nearest neighboring beam by a distance d = 4.95 µm. The aspect ratio s, which
is defined as the ratio of the focused beam spacing to focused beam waist is s = d/w = 2.23. After
passing through all of the optics required to produce the array, the total optical power remaining
in the array is 750 mW, which gives a uniform distribution of 47 mW per beam in the array setup.
An image showing the optical intensity of the GBA as it is swept through the focus of the 200:60
telescope is shown in figure 5.2b. As this 8 beam array is focused, there exists two regions of low
optical intensity surrounded by high optical intensity (similar to the figure 4.18c) that creates two
distinct BoB traps capable of confining atoms (figure 5.2b).

5.2.2

Analysis of Trapping Potential of Gaussian Beam Array

To understand how the array of Gaussian beams in figure 5.2b creates two BoB traps, let us
examine the array at the plane of the array focus. Specifically, we examine one the two identicaldark regions of minimum light intensity surrounded by four intense Gaussian beams that is outlined
clearly in figure 5.3. Each of the four Gaussian beams in this arrangement has intensity [148]22

I(r) = I0 e−2r /w0(5.1)

where w0 is the focused beam waist and I0 is the maximum intensity that occurs at r = 0. As
shown in figure 4.5b, each of the four beams has a polarization that is orthogonal to its nearest
neighboring beam. It immediately follows that the total optical intensity exactly in-between two
2This f = 23.125 mm lens is the same 0.4 NA, high quality lens used to focus the crossed vortex BoB traps insection 4.5.2.

101

Figure 5.2: (Color online) Top: Optical setup to generate a GBA that produces two bottle beam
trapping sites. Bottom: Intensity images of the GBA as one sweeps through the focus of the array.
As one moves away from the focus in the positive and negative axial direction, each Gaussian beamin the array rapidly diverges and interferes to cap off the bottle beam trap and produce a barrierin the z-direction. This permits the intensity pattern to function as a 3D trapping potential. The
locations of the focus of the two BoB traps are indicated with a red arrow.

102
nearest-neighboring beams, which corresponds to an array saddle point Isp , isIsp = I(d/2) + I(d/2)
2

2

2

2

= I0 e−2(d/2) /w0 + I0 e−2(d/2) /w0
2

(5.2)

2

= 2I0 e−d /2w0

where d is the separation between neighboring Gaussian beams. The lowest optical intensity in

Figure 5.3: (Color online) Optical intensity image showing a central, dark region of minimum lightintensity surrounded by four intense Gaussian beams. Each beam has an equal waist of w0 andis separated from it’s nearest neighboring beam by distance d. Adding the intensities of all four
beams creates bright saddle points with intensity Isp that surrounds the central, dark region withintensity Idark . Because the laser light used to create this pattern is blue-detuned from resonance,
the gradient in optical intensity Isp − Idark creates a potential energy well that can confine atoms
in the central, dark region.
√
the central, dark region ID , occurs at a distance r = d/ 2 from the middle of each focused beam,√
and is equal to the sum of intensities from all four Gaussian beams at this central point r = d/ 2:



q
√
√
√
√
ID = 2 × I(d/ 2) + I(d/ 2) + 2 I(d/ 2)I(d/ 2) cos φ


q
√
√
√
√
2 /w 2 )2 /w 2 )−2((d/ 2)2 /w02 )−2((d/ 2)2 /w02 )−2((d/2)−2((d/
2)
0 I0 e0= 2 × I0 e
+ I0 e+ 2 I0 e(5.3)
h= 2 × I0 e2−d2 /w02+ I0 e−d2 /w02+ 2I0 e−d2 /w02i2= 8I0 e−d /w0 .
Comparing equations 5.2 and 5.3, there exists a non-zero, optical intensity gradient between thebright saddlepoints Isp and the dark central region ID . Since the laser light used to create these103
intensity profiles is blue-detuned from resonance, equations 2.11 and 2.12 state that a potential
energy well is created that permits 2D atom confinement at the dark regions; that is the four
beams together form a single 2D BoB trap. Furthermore, from equation 2.11, the total trapping
depth is proportional to the difference of bright optical intensity (equation 5.2) and dark opticalintensity (equation 5.3)
Itrap = IS − ID
2

2

2

2

= 2I0 e−d /2w0 − 8I0 e−d /w0−d2 /2w02−d2 /2w02= 2I0 e1 − 4e22= 2I0 e−s /2 1 − 4e−s /2(5.4)

where we use the definition s = d/w0 , which was defined in section 5.2.1 as the aspect ratio of thearray.
It is useful to the experimentalist to express the results of equation 5.4 in terms of the average
optical intensity of the entire 2D trap (shown in the inset of figure 5.3). Since each BoB trap isconstructed from four Gaussian beams separated by distance d, the average optical intensity is
πw02 I0πI0P= 2I¯ ≈ 2 =
d
2d2
2s

(5.5)

where P is the total power of all four Gaussian beams combined.
Thus, we can use equation 5.5 to express I0 in terms of the average intensity [106] with
I0 ≈

2s2 ¯
I.
π

(5.6)

Inserting equation 5.6 into equation 5.4 gives the total 2D trapping intensity ass2 ¯ −s2 /2 
−s2 /2Itrap = 4 Ie
1 − 4e.
π(5.7)

By plotting equation 5.7 (figure 5.4), it is immediately evident that the largest intensity difference,and hence the deepest BoB trap, will occur when the set of four focused Gaussian beams has an
aspect ratio of s = d/w = 2.2. This is a very important result, as it dictates what lenses and (moreimportantly) what calcite thicknesses to use when forming the GBA in the Hex cell, as shown figure
5.2a.

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Figure 5.4: (Color online) a) Plots of the normalized intensity of the bright saddle points Isp /I0(solid line) and the central dark region Idark /I0 (dashed line) as a function of the array aspect ratios. b) Plot of the normalized trapping intensity Itrap /I¯ = (Isp − Idark )/I¯ (solid line) between thebright saddle-point and the central dark region of one trapping site in the GBA as a function of
the aspect ratio s. The dashed line shows the intensity of the dark, central region that is present at
the same aspect ratio s, thus indicating the non-zero bias to the trap bottom present in the setup.
From this plot, it is determined that the deepest trapping depth occurs at s = d/w0 = 2.2.

The total optical intensity in equation 5.7 results in 2D atom confinement in the radial
direction but provides no confinement in the loose, axial direction of the array. In order for thearray BoBs to trap atoms in three dimensions, a repulsive optical potential in the axial direction
must also be formed. Similar to how tightly overlapping two LG01 beams “caps” off the crossed
vortex BoB trap with high light intensity (section 4.5.2), the overlap of the Gaussian beams as
they rapidly diverge when propagating out of the GBA focus (figure 5.2b) “caps” the array BoBs
with high optical intensity and provides a repulsive potential barrier to give axial confinement inaddition to radial confinement.

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5.2.3

Atom Trapping in the Focused Array

When finally focusing the array into the AQuA Hex cell to its desired size (using with thef = 23.125 mm lens), the GBA has focused beam waists of w0 = 2.21 µm and beam separations of
d = 4.95 µm (figure 5.5). This configuration gives a final aspect ratio ofs=d= 2.23w(5.8)

which is very close to optimal aspect ratio of s = 2.20. Note that the focal plane of the array iscarefully aligned to overlap with the focus of the optical conveyor belt (figure 4.6). As explained
in section 4.5.2, scrupulous care must be taken when focusing the GBA though the thick Hex cell
walls to ensure that as little distortions as possible occur to the array beams such that the aspectratio s = 2.23 is preserved. Inspection of figure 5.5 shows that this focused GBA pattern contains
two low intensity dark regions that are each surrounded by four bright saddle points, thus forming
two 3D BoB traps.

Figure 5.5: (Color online) Intensity image of the Gaussian beam array at the focus of the 0.4 NA
f = 23.125 mm lens. Each beam in the array has a beam waist of w0 = 2.21 µm and is separated
from its nearest neighbor by d = 4.95 µm. This gives an aspect ratio of s = d/w = 2.23.

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To get an explicit calculation for the trap depth as a function of the atom position in each
BoB, we add the intensities [102] from each diagonally opposite pairs of Gaussian beams:c0 |E(x − d/2, y − d/2, z) + E(x + d/2, y + d/2, z)|2
2

c0 
|E(x + d/2, y − d/2, z) + E(x − d/2, y + d/2, z)|2+2

I(x, y, z) =(5.9)

where c is the velocity of light, 0 is permittivity of free space, and E(x, y, z) is the electric field ofa Gaussian laser beam given asE(x, y, z) = E0

w0
w(z)



  2

x + y2exp −.
w2 (z)(5.10)We plug equation 5.9 into equation 2.11 to numerically calculate a dipole trapping depth of 547µK along the radial (x, y) directions and a trap depth of 578 µK along the axial z direction when
the total optical power in the entire array is 750 mW. The corresponding trapping frequencies areΩrad = 2π × 8.35 kHz along the radial direction and Ωz = 2π × 2.58 kHz along the axial direction.
The radial and axial trapping potentials for each BoB trap in the array are plotted in figure 5.6.

Figure 5.6: (Color online) Optical trapping potential produced at the bottle beam locations of the
2x1 array in the radial (x,y) and axial (z) directions. Using 750 mW of λ = 780 nm laser power,a trapping barrier of depth 547 µK is created in the x,y radial direction and a trap depth of 578µK is created in the z, or axial direction. The trapping frequencies for this trap geometry are 8.35
kHz in the radial direction and 2.58 kHz in the axial direction.

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Finally, to demonstrate the BoB array was capable trapping atoms, the array was overlappedwith the center of a 3D MOT for 500 ms. After loading the array from a background MOT,
polarization gradient cooling was performed on the atoms for 5 ms and then all MOT beams
were extinguished for an additional 25 ms to allow un-trapped atoms to fall away. Fluorescenceimaging (as described in section 4.5.3) was then performed on any trapped atoms by shining athree-dimensional optical molasses detuned -25 MHz from resonance. Repump light was also used
to sustain the continuous fluorescence of each trapped atom. Each fluorescence image was takenwith an Andor iXon EMCCD with an EM gain of 200, vertical pixel shift speed of 0.5 µs, and ahorizontal pixel readout rate of 5 MHz. Results of the fluorescence imaging are shown in figure5.7. Each trapping image shows the sum of 10 individual fluorescence imaging results of atoms
confined in the two BoB traps. By improving the resolution of the florescence imaging system, theseparation of the two BoB traps was observed to be 4.5 µm, which is slightly less than the expected
separation of 4.95 µm that was observed from the imaged intensity profile in figure 5.53 .

5.3

Steering the Optical Conveyor Belt
To steer the optical conveyor belt, we modify the dynamic 1D lattice setup shown in figure

4.12 by placing two additional 4.2 mm x 4.2 mm aperture, 80 MHz acousto-optic modulators(AOMs) along the beam paths of both counter-propagating 1D lattice beams (figure 5.8). The
AOMs, labeled the x-AOM and y-AOM for each beam path, execute angular deflections θx andθy of each lattice beam in the respective x̂ and ŷ directions by diffracting4 each laser into the±1 order. An optical relay system shown in figure 5.9 was designed to precisely map the angulardeflections θx and θy from the x-AOMs and y-AOMs into lateral displacements δx and δy at the3

It is important to mention that these results do not demonstrate that only a single atom is present in the BoB
trap but rather that the array of BoBs can trap atoms. At this point in the IARPA MQCO experiment, things
started become very rushed due to funding coming to an end. This data was taken in April 2016 and the entire
project funding was scheduled to end in July 2016. As a result, I decided to move very quickly. The data showedthat our array was capable of trapping atoms, whether it was one or more. Knowing that we had to still incorporate
the optical conveyor belt, I decided to immediately sprint to that milestone.
4
Because the beam steering system requires each lattice beam to be diffracted via an AOM, power loss starts toreally play a factor. Thus, painstaking care was taken to optimize the AOM diffraction efficiency such that 90% of
the beam power was diffracted into the +1 order.

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Figure 5.7: (Color online) Demonstration of atom trapping in the 2x1 array of BoB traps. The
fluorescence images in a) and b) show a sum of 10 individual images. The schematic for fluorescenceimaging the array is shown in c). By changing the final imaging lens from f = 500 mm to f = 1000
mm, the resolution was doubled which is the difference between images a) and b).

Figure 5.8: (Color online) The optical conveyor belt setup is modified by adding two additional
AOMs along the beam paths of 1D lattice lasers in order to execute angular deflections of thebeams along the x and y dimensions. As in the optical conveyor belt for the crossed vortex BoB
setup (section 4.4.4) lattice beam overlap is monitored by measuring their relative beat frequencies
with an RF spectrum analyzer.

109
focus of the 0.4 NA, f = 23.125 mm lens. For each lattice beam, the relay system (shown in figure5.9) works by imaging the center of the x-AOM crystal onto the center of the y-AOM crystal using
a 1:1 telescope. The “center” of the AOM crystal refers to the distance half-way through the crystal
and the center of the face of the crystal. Next, a second 1:1 telescope images the center of they-AOM crystal onto the center of the first lens of a 5x enlarging telescope. Each 1:1 telescope also
serves to help minimize any beam walk-off, as each beam is imaged onto the center of the next
subsequent optical element. The 5x telescope enlarges the lattice beam to 1 cm in diameter before
being focused onto the plane of the BoB array with the 0.4 NA, f = 23.125 mm lens (figure 4.6)
to a 1/e beam waist of approximately 1.7 µm. The reduced beam waist of the optical conveyor
belt (as compared to the original setup in section 4.4.1) is to demonstrate atom delivery to single
BoB traps in the GBA without overlapping adjacent BoB sites that are separated by 4.8 µm. With
the smaller beam waist of the conveyor belt beams, we only need to use P = 500 mW of total
power in the beams to still obtain the same trapping depths U0 = −1.88 mK (equation 4.5) andaxial trapping frequency Ωz = 2π × 454 kHz (equation 4.6) as in section 4.4. However, the radialtrapping frequency (equation 4.7) has increased to Ωrad = 2π × 75 kHz.
As a result of this relay system, by driving the beam deflecting AOMs, we can laterally
displace each 1D lattice beam by δx and δy at a resolution 1.06 µm/MHz. Since we can control
the drivers to a resolution of δν = ±0.1 MHz, the beam steering setup can laterally displace theoptical conveyor belt by some distance at a resolution of 0.1 µm. It is important to note that withthis setup, in order for both lattice beams to remain overlapped after repositioning, the x-AOM
drivers are set to the same driving frequencies but opposite sign (that is x-AOM1=+δνx and xAOM2=−δνx ) while the y-AOM drivers are set to the same frequency (y-AOM1=y-AOM2=δνy )in the steerable conveyor belt setup5 . Moreover, while each AOM is driven with an independent
RF driver, they are all phase locked to the same 10 MHz clock reference signal in order to minimize
phase jitter of the lattice beams.
5This can cause a non-zero mutual detuning δ 6= 0 between the transport beams, thus giving it an unwanted
velocity. To correct this error, one must simply adjust the detuning δ in the original frequency chirping AOMs fromfigure 4.12

110

Figure 5.9: (Color online) Schematic of the beam steering system for the optical conveyor belt.
The 1064 nm beams are split into two beam paths. Each beam passes through an x-AOM, whichalters the angle of the beam in the x dimension. A 1:1 telescope relay images the center of thex-AOM on the center of the y-AOM. This y-AOM changes the beam angle in the y dimension.
A second 1:1 telescope relays the center of the y-AOM onto the center of a 5x telescope, which
magnifies the beam to 1 cm in diameter. The beam is then focused with a 0.4 NA custom lens
onto the qubit array in the AQuA Hex cell. At the qubit array plane, the x and y positions of thebeams can be laterally translated by δx and δy by varying the x and y angular deflections of the
AOMs.

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5.3.1

Single BoB Site Addressing

To demonstrate the single BoB site addressing capabilities with the steerable optical conveyor
belt, the focused BoB array was imaged onto an Andor iXon EMCCD (figure 5.10a) using the same6
imaging system used in figure 5.7. On this image, light from the focused optical conveyor belt
can also be seen. By only changing the driving frequencies of the lattice deflecting x-AOMs and yAOMs, the focus of the optical conveyor belt is repositioned such that it overlaps the center of eitherof the two BoB traps in the GBA (figure 5.10b, figure 5.10c, figure 5.10d). The beam deflecting
AOM driving frequencies can be adjusted electronically within approximately 10 microseconds,
allowing for very fast optical conveyor belt reposition times.
6
Although it must be slightly modified as the optical conveyor belt is not drawn on this figure (simply notethat both counter-propagating λ = 1064 nm beams spatially overlaps the 780 nm BoB array beams). After passing
through the Hex cell, the 1064 nm and 780 nm beams are separated by simply placing 1000 nm low pass dichroic
mirror directly after the f = 23.125 mm lens. However, 0.1% of the optical conveyor belt light still passes through thedichroic mirror and is then focused onto the Andor iXon EMCCD with an f = 500 mm lens. This forms a 23.125:500
confocal microscope that images the focus of the array-conveyor belt combination.

Figure 5.10: (Color online) Demonstration that the beam steering system is capable of addressing
both BoB trapping sites in the GBA. These images were taken by only changing the 1064 nm beam
deflecting x-AOM and y-AOM driving frequencies with no manual realignment of any beams. a)
Focused GBA located inside the Hex cell. b) The focus of the optical conveyor belt spatially overlaps
the focus of the GBA. The yellow arrow points to light from the focused conveyor belt. Using thelattice deflected x-AOMs and y-AOMs, the optical conveyor belt is repositioned to overlap the first
BoB trap, shown in image c), and the second BoB trap, shown in image d).

112
5.3.2

Atom Transport and Delivery Sequence with Steerable Conveyor Belt

The sequence of delivering atoms to either of the array BoB traps begins by first choosing a
BoB trap to deliver atoms to. With the spatial coordinates of both BoB traps known, the driving
frequencies of the beam deflecting AOMs was set to the proper value such that the focus of the
optical conveyor belt is repositioned to overlap the center of the target BoB trap. Specifically, inthis setup shown in figure 5.9:
• Setting the x-AOMs to ±79.83 MHz and setting both y-AOMs to +79.69 MHz positions
the focus of the optical conveyor belt at the center of BoB-1 (figure 5.10c)• Setting the x-AOMs to ±84.45 MHz and setting both y-AOMs to +79.69 MHz repositions
the conveyor belt to overlap BoB-2 (figure 5.10d).
After the optical conveyor belt is repositioned, the degree of overlap for the lattice beams isalways verified by using the same method described in section 4.4.4, where an RF spectrum analyzer
measures the amplitude of the relative beat note between the both lattice beams. The procedure fortransporting atoms from the 3D MOT to the programmed BoB trap follows the sequence describedin section 4.6.1 and figure 4.24, although (as described in section 5.3.2.3) a different frequency chirp
is applied to the lattice beams and, as described next, a considerably smaller number of atoms isloaded into the conveyor belt.

5.3.2.1

Loading the Optical Conveyor Belt With a Small Atom Number

Once a BoB destination is chosen, a 3D MOT is loaded into the AQuA Hex cell for 1 secondusing the methods described in section 4.3.2. During the final 530 ms of MOT loading, the 1064
nm transport beams are un-shuttered allowing the optical conveyor belt to overlap the MOT where
5 atoms on average7 are loaded into the optical conveyor belt. After loading the conveyor belt, the7Compared to the atom transport experiments in section 4.5.3, considerably fewer atoms (by about a factor of 103 )are loaded into the optical conveyor belt. This is explained by the appearance of the collisional blockade effect due
to reducing the lattice beam waists from 35 µm (as in section 4.5.3) to w0 = 1.7 µm [149]. The collisional blockade,combined with any light assisted collisions from the cooling light prevents more than a single atom occupying in eachpotential well of the 1D lattice potential [149].

113
MOT was extinguished by shuttering the 852 nm cooling and repump beams as well as turning off
the 3D MOT magnetic fields. The conveyor belt was held in place for 20 ms to allow the remaining
untrapped atoms to fall away due to gravity.

5.3.2.2

Measuring Conveyor Belt Occupancy

The occupancy of the conveyor belt was measured by illuminating it with a 3D opticalmolasses by turning on the 3D MOT lasers and detuning them -30 MHz. Repump light is also
turned on. Additionally, shim coils are used to eliminate any residual magnetic fields. As a result,the 3D molasses light induces atom fluorescence from any atoms loaded into the optical conveyor
belt. The 852 nm scattered fluorescence light is collimated with one of the 0.4 NA, f = 23.125
mm lenses used to focus the conveyor belt and separated from the 1064 nm trapping light witha 900 nm short pass dichroic mirror (figure 5.11a). Using an f = 500 mm achromatic lens, thefluorescence light is then focused onto an Andor iXon EMCCD with single photon sensitivity where
photon counting is performed over a 10 ms time interval. The overall imaging system features amagnification of 21.6 and results in a 0.740 µm/pixel ratio for the EMCCD. A set of three 852 nm
laser line filters are used to eliminate any stray 1064 nm lattice light. A histogram showing thephoton counting results of 500 individual loading sequences is shown in figure 5.11b and was fitted
to a compound binomial-Poisson function:
"
#
 
n
2
k e−hN i
X
(n
−
(n
+
kn
))
n
hN
i
1
1
bg
p
f (x) = A
(1 − p)n−k
.
exp −
k
k!
2 (nbg + kn1 )2π (nbg + kn1 )k=0(5.11)

Physically, the appearance of this compound probability distribution is due to the combination of• The random, discrete number of atoms loading into the optical conveyor belt
• The Poissonian spread of observed photon counts from each of the loaded atoms.
The fitting result is p = 0.5, N = 5, n = 10, nbg = 120, and n1 = 120. The appearance of multiple
peaks in the photon counting histogram (as compared to 2 peaks in figure 4.22) is due to loading
more than one atom in the conveyor belt. From the photon counting statistics the most probableloading number is 5 atoms (figure 5.11b).

114

Figure 5.11: (Color online) a) Setup for fluorescence imaging of the 1D optical lattice. An opticalmolasses is applied to atoms trapped in the 1D lattice and a fraction of the induced fluorescenceis collected with the 0.4 NA custom lens and focused onto an Andor iXon EMCCD. b) Histogramof photon counting data recorded during fluorescence imaging of the 1D lattice. The data shows
multiple Poisson distributions that emerge due to different amounts of atoms loaded in the 1D
lattice. Each number N shows the Poisson distribution corresponding to loading N atoms in thelattice. The distribution is fitted to a compound binomial-quasi Poisson function, which accounts
for the random and discrete atom loading along with the sub-Poissonian spread of observed photon
counts. The statistics show that the most probable loading number is 5 atoms.

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5.3.2.3

Transporting Small Atom Number to BoB Site

Unlike the crossed vortex BoB trap loading (section 4.6), the BoB traps formed by the GBA
are never turned off during the atom transport sequence. Turning off the GBA in order to reload
a new atom to an empty BoB trap would ruin any previously loaded traps and ongoing quantum
computations would also be ruined. Because the GBA and this optical conveyor belt are focusedthrough the same 0.4 NA, f = 23.125 mm lens, the conveyor belt will pierce atoms through theweaker, axial confining potential of the array BoBs. During this process, atoms confined to theconveyor belt will not be expelled, as the axial trapping frequency of the conveyor belt, Ωz = 2π×450
kHz is considerably greater than the axial potential of the array BoB trap Ωz = 2π × 2.58 kHz.
To transport the small number of loaded atoms to the desired array BoB trap, one latticebeam is linearly frequency chirped by δ = 20 kHz over 5 ms. Because the frequency chirping AOMs
(figure 5.8) are in the double pass configuration, this causes a total lattice beam detuning of 40
kHz, With 40 kHz detuning, the optical conveyor belt accelerates to a maximum velocity of 21mm/s. After this frequency chirp, the lattice beam is linearly chirped down -40 kHz over 5 ms tobring the conveyor belt to a halt. The entire procedure transports the atoms a total distance ofapproximately 100 µm in 10 ms8 Additionally, the adiabatic criterion for accelerating the opticalconveyor belt (equation 4.17) is satisfied, as we are still chirping the moving lattice at a much
lower rate than the square of the lattice axial oscillation frequency. That is Ω2z >> d∆ω/dt, whereΩ2z ≈ 1012 s−2 and d∆ω/dt ≈ 107 s−2 . Fluorescence imaging, as described in section 5.3.2.2, isperformed for 10 ms to observe atoms transported into either BoB trap (figure 5.12).
Note that this only shows atom transport to each array BoB trap and does not demonstratereloading the empty traps. The next experimental plans were to optimize the atom transportsequence in order to load a single atom deterministically into either array BoB trap using thesteerable conveyor belt (similar to section 4.6.1). Unfortunately, at this point in the atom transportand reloading project, funding was discontinued and the experiment reached its premature ending.
8This 100 µm transport distance is considerably smaller than the 1.1 mm transport distance in section 4.6. Thisis due to the steerable optical conveyor belt having a focused beam waist of w0 = 1.7 µm, giving it a Rayleigh range
approximately 225 times smaller than the optical conveyor belt in section 4.6.

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Figure 5.12: (Color online) Fluorescence images, all showing the same area of interest, of approximately 5 atoms after being transported 100 µm from a 3D MOT to an empty array BoB trap
using the optical conveyor belt. Images a) and b) shows atoms transported to BoB2 and BoB1,
respectively. Image c) is a combination of a) and b) to show their spatial separation of 5 µm.

Chapter 6

Conclusion to Part II

6.1

Summary
Part II of this dissertation has investigated individual atom delivery into optical bottle beamtraps using an optical conveyor belt. In chapter 4, we describe a system for loading a single atomfrom a reservoir into a blue-detuned crossed vortex bottle beam trap using a dynamic 1D lattice.
The lattice beams are frequency chirped using acousto-optic modulators, which causes the lattice
to move along its axial direction and behave like an optical conveyor belt. A stationary lattice isinitially loaded with approximately 6000 atoms from a reservoir, and the conveyor belt transports
them 1.1 mm from the reservoir to a bottle beam trap, where a single atom is loaded into the bottle
beam trap 13.1% of the time.
In chapter 5, we showed how this atom transport and reloading system can be scaled-up
to deterministically transport approximately 5 atoms on average to an arbitrary site of an arrayof BoB traps. Accordingly, the atom transport system is “steerable” in that it could transport
atoms to one BoB site and then be repositioned in order to deliver an atom to another array site.
The beam steering of the optical conveyor belt was accomplished by using an additional pair of
acousto-optic modulators for each of the 1D lattice beams making up the conveyor belt in orderto reposition the optical conveyor belt to permit loading of any BoB array site at an arbitrary
location [146, 147]. Modifications could be made to the single atom loading apparatus to reload
an arbitrarily large array of BoB traps, such as figure 1.1. Unfortunately, funding for the IARPA
AQuA-49 project ended and the reloading of an array of BoB traps was not completed. However,

118
with the results described in chapters 4 and 5, there are still interesting future experiments that
could be performed to improve and extend this research.

6.2

Future Work and Experiments

6.2.1

Route to Continuous Atom Reloading: Vertical then Horizontal Transport

Demands for maintaining a satisfactory vacuum environment for trapped qubits and qubit
gate operations suggest that the 3D MOT which supplies the source of cold atoms for atom reloading
be located in a separate vacuum chamber positioned adjacent to the cell that houses the BoB array.
This also permits the 3D MOT to be on continuously, as the MOT footprint will not disturb any
previously trapped atoms. Subsequently, one practical atom reloading sequence would transport
atoms vertically from a 3D MOT, using what is called a Moving Molasses MOT (figure 6.1b), to a
position in the Hex cell that is off-axis from the BoB array (figure 6.2).
By making a 6-Beam MOT1 with 3 pairs of counter-propagating cooling beams in a 2D
magnetic field, we propel the entire MOT vertically, in what is called a Moving Molasses MOT[150], by mutually detuning the vertically oriented MOT beams by δω (figure 6.1a). Similar to thephysics behind the optical conveyor belt, this detuning creates a reference frame where where bothpairs of cooling beams are Doppler shifted to the same frequency. This frame moves vertically (as√an atom fountain) with velocity v = 2 2λδω. A 12 W, 1064 nm dipole trapping beam helps toguide the atoms upwards. Once the atoms reach a maximum height in the Hex cell, they will be
trapped in a crossed dipole trap which serves as a “secondary cold atom reservoir.” The atoms fromthis secondary reservoir could then be horizontally transported to a BoB site using the methods
described in chapters 4 and 5 of this dissertation. Using this transport scheme, while considerably
more difficult, will not only maintain a better vacuum in the Hex cell, but would solve isolation
issues by keeping the cold atom reservoir from disrupting any concurrently operating qubit gatestaking place in the array, which allows for a continuous atom reloading.
1
For more information on this Moving Molasses MOT and a complete diagram of the system, please refer to the
PhD thesis of Kai Hudek.

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Figure 6.1: a) MM MOT setup in the lower 3D MOT Chamber that can propel atoms up into theHex Cell. Diagram taken from the PhD thesis of Kai Hudek [3]. b) Sequence of 105 atoms travelingvertically from the 3D MOT chamber into the Hex cell chamber. A 12W, 1064 nm dipole trapping
beam helps guide the atoms upwards. Once the atoms reach a maximum height in the Hex cell,they will be trapped in a crossed dipole trap which serves as a “secondary cold atom reservoir.”Figure 6.2: a) Diagram showing the location of the “secondary cold atom reservoir” in relation
to the Hex cell lasers and the BoBs. Atoms confined to this reservoir can be transported andloaded into a BoB using the methods from chapter 4 and 5 of this dissertation. b) Atoms from the
MM MOT that have traveled vertically into the uppermost chamber above the 3D chamber andhave reached their highest trajectory. The inset shows a small crossed dipole trap as a result ofintersecting this vertically traveling MOT with a horizontal dipole trap.

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6.2.2

Using the Optical Conveyor Belt to Address a 2x3 BoB Array

The range of the steerable optical conveyor belt can be increased to address six BoB traps
rather than just two. Such demonstration would show a fully capable atom reloading system fora qubit array without disturbing neighboring, previously loaded BoB traps. The optical systemfor creating the larger Gaussian beam array with 6 bottle beam trapping sites is shown in figure
6.3a and is just an extension of the GBA setup in section 5.2.2 and figure 5.2. A 2x2 array ofidentical Gaussian beams is created by a Holo/Or MS-248-X-Y-A diffractive beam splitter. The
100:35 telescope images the arrays in between two calcites. The first calcite replicates the 2x2
array into an array of 8 identical Gaussian beams separated by 353 µm, while the second calcite
duplicates the 8 beams into a full 16 beam array with each beam separated by 250 µm. In order tohave equal spacing of each array site, the fast axis of each of the calcites is rotated by 45 degrees
with respect to each other. After passing through both calcites, the 2x3 array of Gaussian beams
is demagnified by a 200:60 telescope to give each beam a waist of 34m and beam separation of 74µm. The array is then demagnified one final time with a 500:23.125 telescope to image the arraythrough the Hex cell. At the focus of the 0.4 NA, f = 23.125 mm lens, each beam has a separationof d = 4.67 µm and each beam waist is w = 2.22 µm, giving an aspect ratio of d/w = 2.10 (figure6.4a). The steerable conveyor belt system in chapter 5 could then be used to address all 6 BoB
traps (figure 6.4b).

6.2.3

Outlook

The work in part II of this dissertation demonstrates single atom reloading on a small scale,but does not demonstrate loading in a manner that leaves neighboring BoB sites undisturbed.
Moreover, the atom reloading process is inherently stochastic and thus does not provide a 100%
reliable reloading system for every reloading attempt. Since the publication of the small scalereloading system [32], methods to reload arrays of up to 50 atoms within 400 ms have been demonstrated [151][152]. As neutral atom qubit arrays begin to grow to larger sizes of up to 100 sites [1]121

Figure 6.3: a) Optical setup to increase the BoB array from two traps to a 2x3 array of BoB traps.
b) Intensity image of the 2x3 array as one sweeps through the focus of the array.

Figure 6.4: a) Image of the 2x3 BoB trapping array at the focus of the 0.4 NA, f = 23.125 mm
lens. The optical configuration creates 6 BoB traps separated by d = 4.67 µm. b) Demonstrationof the beam steering system addressing all 6 BoB traps in the array. The yellow arrow points tothe optical conveyor belt light.

122
or even 3D array geometries [16], one conclusion remains perfectly clear: the process of reloading
atoms due to finite trapping lifetimes presents a serious challenge to continuously operating neutral
atom quantum computers. The atom transport and reloading system is but one cog in the neutral
atom quantum computer (figure 6.5). Just like the rest of the quantum computing architectures,
atom reloading must be sufficiently solved in order for neutral atoms to become a viable approachfor quantum computation.

Figure 6.5: Chart taken from reference [1] conveying that single atom transport and reloading is
but one element of a successfully operating neutral atom quantum computer.

Chapter 7

Part III: An Ultracold Gain Mechanism for a Matterwave Transistor Oscillator

Part III of this dissertation involves a drastically different theory and ongoing experimentin the field of atomtronics that introduces a new gain mechanism for a matterwave transistor
oscillator. We start with a well-established, semiclassical description and experiment [33][5][34] ofan atomtronic transistor and model the system with a purely quantum mechanical formalism. The
quantum model predicts dynamics differently from the semi-classical results only when the atoms
flowing through the transistor oscillator have sufficiently low enough temperatures such that the
motional state of a dipole oscillating BEC, placed in the transistor itself, couples atom transitions
between high lying transistor energy eigenstates. In this regime, the transistor oscillator displays
interesting physics that was not observed in the semiclassical theory or experiments, namely theexistence of a matterwave gain mechanism.
The remaining sections of this dissertation are organized as follows: Chapter 7 first gives abrief overview of atomtronics in relation to this work (section 7.1) and motivates the matterwave
transistor oscillator (section 7.2). We then present the quantum model (sections 7.3 and 7.4) of thematterwave transistor oscillator and formally derive the gain mechanism whose physics resembles
the Rabi model of a two-level atom interacting with a photon field (section 7.5). We then show the
manifestation of matterwave gain in the model by comparing a matterwave current flowing throughthe transistor with and without the gain mechanism (section 7.6). Chapter 8 then presents theongoing work to demonstrate the matterwave gain mechanism. Chapter 9 concludes.

124

7.1

Introduction to Atomtronics
Atomtronics [35][36] is an emerging field of ultracold atomic physics that focuses on quantum

circuits that operate as atom analogues to electrical circuits, where an atom current takes theplace of an electron current and the chemical potential takes the place of a voltage. In their design,
atomtronic circuits are drastically different from their electronic circuit counterparts, as they involve
manipulating ensembles of ultracold atoms to flow through a variety of potential energy landscapes,
such as• Optical potentials produced from laser beams (in particular, the experiments in [37][38])• Harmonic magnetic potentials created with atom chips (as demonstrated in references[39][40][41][42][43])• A hybrid combination of optical and magnetic potentials, such as the transistor potential
used throughout chapters 7 and 8 of this dissertation.
At the heart of all atomtronic circuits is a Bose-Einstein condensate (BEC) [44], which supplies asource of coherent, ultracold atoms that are controlled to flow throughout a network of quantum potentials. Some recently demonstrated atomtronic circuits of note include BECs flowing throughout
double-well [33] and triple-well potentials [5], as well as BECs flowing through 1D optical lattices[45][46][47] and optical ring lattices [48][49][50]. Irrespective of the circuit potential geometry, theobjective of atomtronics remains the same: manipulate a current of ultracold atoms in some tailored potential in order to study fundamental physics or to create purely quantum mechanicaltechnologies [51].
Accompanied with such high aspirations of atomtronics comes a difficult, innate challenge
ingrained in the field: such ultracold atom circuits are inherently a many-bodied, open quantum
system, since atoms and BECs interact and dissipate as they flow throughout the circuit from somesource to a sink or drain. While this immensely complicates the theoretical description of atomtronic
circuits, complete descriptions are usually simplified by invoking various approximations, such as

125• Neglecting dissipation and treating the atomtronic circuit as a closed system (as demonstrated in references [52][53][54][55])• Reducing the many-bodied system from a very large number to a three or four-body problem• Ignoring the BEC mean-field interaction effects (as described in reference [34]).
The transistor oscillator model described in this dissertation uses all three of the above bulleted
approximations and justifies the validity of each when applied to this transistor oscillator model insections 7.4.1, 7.4.3, and 7.5.2, respectively.
Despite the the necessity of using approximations just to solve for something, atomtronics
has made unwavering progress in ultracold atom technologies such as realizing quantum simulators[56][57][58][59], improving precision measurements [60][61], creating an atomtronic analogue to a
SQUID (called an AQUID, or atomtronic quantum interference device [62][63]), and realizing thefirst-ever atom battery [33]. Moreover, if used in conjunction with well-established matterwave
interferometry experiments, atomtronics can help enhance inertial sensing and gravimetry sensitivities by up to 10 orders of magnitude as compared to their light wave interferometry counterparts[64]. Finally, atomtronics can conceive new types of quantum technologies by combining various
atomtronic circuit elements into an integrated ultracold atom circuit. One such integrated ultracold
circuit, which is the subject of part III of this dissertation, is a matterwave transistor oscillator.

7.2

Background and Motivation for an Atomtronic Transistor Oscillator

7.2.1

Analogue to Electronic Transistors

Fundamental to many technological developments since its demonstration by Bardeen, Shockley, and Brattain at Bell Labs in 1948, the transistor is a three-terminal semiconductor electronic
device that uses an applied voltage or current on one terminal to amplify or switch electrical signals
on another terminal [65]. An atomtronic transistor would behave similarly to its semiconductor
counterpart by switching or amplifying atom, or matterwave, currents rather than electrical currents. Moreover, in place of three electrical terminals (commonly known as the source, gate and126drain terminals) found on semiconductor transistors, an atomtronic transistor features three quantum potential energy wells that are capable of confining ultracold atoms, as shown in one of theoriginal atomtronic transistor design concepts in figure 7.1. In similar form and function to thesemiconductor transistor, these three potential energy wells, which are labeled the source-well, gatewell, and the drain-well, together behave as an atomtronic transistor source (to source an ultracold
atom ensemble to flow through the system) a gate (confining an additional ultracold ensemble toprovide a gain mechanism), and a drain (which functions as the output port of the transistor).
Most importantly, for such source-gate-drain system to be called an atomtronic transistor,
it must display a gain mechanism for a matterwave or atom current. The overall concept of an
atom current gain mechanism in an atomtronic transistor is shown in figure 7.1. As matterwaves
flow from source, to gate, to drain via tunneling between each potential energy well, the gain
mechanism emerges due to a change in the gate-well chemical potential (as a result of an ultracold
atom population building up in gate-well) resulting in an increase of tunneling probability to flow
through the gate-well and into the drain-well. This manifests itself in an increase of the atom flux
Φ, or atoms per unit time, flowing through the potential [35].

Figure 7.1: Concept of atomtronic transistor gain mechansim: a) With no atoms in the gate, no
atom current flows from source (S) to gate (G) to drain (D), giving an atom flux Φ = 0. b) As
the gate occupancy increases, so does the gate-well chemical potential and a flux of atoms (Φ > 0)
flows into the drain. c) As the gate occupancy increases, so does the magnitude of the atom current
and flux continues to increase (Φ >> 0).

127
7.2.2

Extending the Semiclassical Descriptions of an Atomtronic Transitor

Before introducing our quantum model, we note that it begins with the setup and results ofreferences [33][5][34], which investigates transistor-like behavior of ultracold atoms flowing throughnetworks of quantum potential energy wells in a semiclassical formalism by creating matterwave
analogues of Kirchoff’s current and voltage laws for atoms flowing across a double Gaussian barrier
potential. Compared to the quantum model, the semiclassical picture is defined as the set ofexperimental conditions where the thermal energy of the atoms flowing through the transistor
potential is much larger than the energy level spacing of the transistor gate-well. In the semiclassicalmodel summarized in figure 7.2, atom currents are driven by collisions and their flow is describedusing classical Boltzmann statistics [34]. Atoms that have enough thermal energy to traverse
the Gaussian barriers flow into the drain and establish an atom current ISD . A non-intuitive
result of the semiclassical model is that given an initially empty gate-well, an atom population
quickly grows in the gate and rapidly condenses into a Bose-Einstein condensate. Not only was
this remarkable effect experimentally observed [153], but the BEC steady state population was also
shown experimentally to influence the atom current ISD flowing from source to drain [34], which(as described in section 7.2.1) is characteristic of a gain mechanism.
For the remainder of this chapter, we start with the results of the semiclassical model andexamine the transistor behavior and gain mechanism in a purely quantum description. As describedin subsequent sections of chapter 7, there are major differences between the semiclassical andquantum transistor models. Namely, in the quantum model:• The thermal energy of the source-well atom ensemble is comparable (ideally equal) to thegate-well energy level spacing• Atoms flowing through the transistor potential are described as matterwaves.
• The gate-well condensate is treated as a coherent state.
• The flow of atoms through the transistor is a result of resonant tunneling from the source-

128
well, through the gate-well, and into the drain-well (as opposed to the collisions primarily
driving the semiclassical atom current).
• When the atom current reaches low enough energies, a new interaction, not observed in
the semiclassical model or experiment, arises in the gate-well between the gate-well BEC
and the atom current.
This new interaction, which arises due to the motional state of an oscillating gate-well BEC couplinghigh lying gate-well energy eigenstates, is similar to the Rabi model interaction between two-levelatoms and a photon field. The interaction serves to increase the atom current by ultimately
amplifying the matterwave probability amplitude in the transistor drain-well. This introduces anadditional gain mechanism that only appears when the atom current is at sufficiently low enoughtemperatures for the BEC coupling (described above) to occur. We now describe this quantum
model and principle of operation of our matterwave transistor oscillator.

Figure 7.2: Summary of the semiclassical atomtronic transistor: 1D potential energy diagram of thetriple-well transistor potential where an source-well ensemble of atoms at a chemical potential µS
and temperature TS flow with current ISD into the drain well by classically traversing the barriers
separating the centrally located gate from the source and drain. As the current flows, some atoms
rapidly form a BEC in the gate well with a well-defined, steady state chemical potential µG and
TG . Previous experimental work shows the occupancy of this BEC can increase or decrease thecurrent flowing from to source to drain, which provides an atom current gain mechanism.

129

7.3

Matterwave Transistor Oscillator Principle of Operation
Following references [33][5][34], our matterwave transistor is constructed by overlapping aharmonic, magnetic potential with two repulsive Gaussian barriers. This setup forms three potential
energy wells which are labeled the source well, gate well, and drain well (figure 7.3a). Because thebarriers separate the source-well from the gate-well and the gate-well from the drain-well, they are
fittingly labeled the “source-gate” (SG) and “gate-drain” (GD) barriers.
Given a thermal atom ensemble originating in the source-well, a flux Φ of atoms per unit timeflows through the transistor potential and into the drain-well by resonantly tunneling [154][155][156]through the transistor gate-well. Our model operates with the assumption that only the phenomenon of tunneling facilitates the flow of atoms throughout the transistor potential and does notallow atoms to classically traverse the SG or GD barriers. Thus, the atoms are hereafter describedas matterwaves. With this definition, we arrive at the first principle of our matterwave transistor
model:• Only the source matterwaves satisfying the resonant tunneling boundary conditions of thegate-well will successfully flow with flux Φ into the drain-well (figure 7.3b).
This means that the incident source matterwaves must match gate-well boundary conditions (i.e.
a gate-well energy eigenstate) to experience transmission into the drain. Otherwise the matterwave will be reflected off of the GD barrier with probability unity. It should be noted that thephysics of a matterwave current flowing through our transistor via resonant tunneling is synonymous to the resonance phenomena observed in already well-understood systems, such as resonant
tunneling diodes [157][158][159][160], optical Fabry-Perot resonators [161][162][163][164] and coupled transmission line resonators. Additionally, numerous references have examined matterwaves
resonantly tunneling through double barrier potentials [165][166][167][168]. All of these examples oftransmission phenomena given here, as well as our transistor model, share them same fundamental
characteristic: only the incident waves matching boundary conditions of a resonator in the system
experience transmission while all other fields are reflected.

130

Figure 7.3: Setup and principle of operation of the matterwave transistor oscillator. a) A 1D
potential energy diagram of the triple well matterwave transistor potential where an ensemble ofatoms, at chemical potential µ and temperature T , occupies the source-well. The source-well anddrain-well are separated by the gate-well, which is created with two repulsive Gaussian barriers
called the Source-Gate (SG) and Gate-Drain (GD) barriers, which have potential energy heights
VSG and VGD , respectively. Images b) and c) convey the principle of operation of the transistor,
showing that a greater matterwave flux Φ0 flows into the drain-well when a dipole oscillating BEC
occupies the gate-well, compared to the matterwave flux Φ when a BEC is absent from the gate-well.

131This leads to the second principle of our matterwave transistor model:• By placing a dipole oscillating BEC in the gate-well, the matterwave current flowing into
the drain-well with flux Φ0 is amplified due to the BEC broadening the resonant tunnelingboundary conditions (figure 7.3c).
The mechanism responsible for this increase of matterwave current is the emergence of an atom-BEC
interaction that occurs only when the motional state of a gate-well BEC couples high lying gate-wellenergy eigenstates. An interesting consequence is the atom-BEC interaction allows matterwaves
that do not initially match a resonance condition of the gate-well to absorb energy from or deposit
energy into the interaction, allowing off-resonant matterwaves to shift their energy and satisfy theresonant boundary conditions of the gate-well. The now-resonant matterwaves are then permitted
to resonate and build up inside the gate-well followed by subsequent flow out into the drain.
Combining principles 1 and 2 gives the basis of our matterwave transistor model: When a dipole
oscillating BEC occupies the transistor gate well, the matterwave current flowing into the drainwith flux Φ0 is greater than the flux Φ when the BEC is absent. This characteristic shows thepresence of a gain mechanism, where gain G is defined as
G = Φ0 /Φ > 1.

(7.1)

The remainder of this chapter proceeds as follows: In section 7.4, we introduce the source,
gate, and drain-well formalism in the quantum theory as well as present a reduced representation of
the gate-well that is necessary to formulate a solvable Hamiltonian. Section 7.5 derives the manybodied Hamiltonian for our transistor and explicitly shows the emergence of a gain mechanism
when atom temperatures are sufficiently cold and an oscillating BEC is present in the gate-well.
Section 7.6 directly shows that the gain mechanism amplifies an atom current flowing through thetransistor by comparing the flux of matterwaves flowing from source-well to drain-well when thegain mechanism is absent to when it is present. Thus, we now introduce the formalities of thequantum model.

132

7.4

Modeling Matterwave Transmission Through 1D Transistor Potential

7.4.1

Source, Gate, and Drain Well Setup and Formalism

We begin the model by simplifying the problem by considering the flow of matterwaves
through the transistor potential in one dimension only. In this 1D system, both the source anddrain wells are treated as constant, flat potentials VS (x) and VD (x) (figure 7.4a) withVS (x) = VD (x) = 0(7.2)

where matterwaves are described for free particles and the source and drain wavefunctions, ΨS (x, t)and ΨD (x, t), are given as plane wave solutions
ΨS (x, t) = AS e±i(kS x−ωS t)(7.3)
ΨD (x, t) = AD e
where wavenumber ki =√±i(kD x−ωD t)

2mEi /~ for kinetic energy Ei (where i = S, D for source and drain) andmass m. The centrally located gate-well is separated from the source-well and drain-well by the
SG and GD repulsive Gaussian barriers featuring equal 1/e waists of w0 = 2 µm and asymmetricheights1 of VSG /~ = 30 kHz and VGD /~ = 33 kHz. The asymmetry in the barrier heights ischosen such that the ratio υ = (VSG − VGD ) /Ts , called the “feedback parameter” of the transistor
oscillator [34], is equal to υ = 1. As derived in reference [34], when barrier heights VSG , VGD andsource-well ensemble temperature Ts give a feedback parameter υ = 1, there will be an optimalbalance of particle and energy currents flowing into and out of the gate-well to yield a stable atom
ensemble in the gate well with high enough phase-space density to condense2 in steady state. Thus,
the barrier height asymmetry in the model allows for a steady-state BEC to collect in the gate-well
as source atoms flow through the gate and into the drain-well.
By separating the SG and GD barriers by a distance d = 4.85 µm, the gate-well becomesa slightly anharmonic, asymmetric oscillator potential capable of confining ultracold atoms. An
1

We shall assume throughout the rest of this dissertation that potential energy is expressed in frequency units.
This remarkable behavior is equivalent to a resistor with negative resistance since the source atoms at temperature
Ts that enter the gate and condense into a BEC are cooled to a lower temperature. If one were to flip the barrierheights so that VSG = 33 kHz and VGD = 30 kHz, the atom current would actually heat up upon entering the gate
and no BEC would form. This latter case is equivalent to a resistor with positive resistance.
2

133
important result (that is critical to the emergence of gain in the model) of the anharmonicity3is that it establishes a degeneracy amongst transitions between the gate-well ground and firstexcited states and the highest lying pair of bound energy eigenstates (figure 7.4b). Additionally,
this degeneracy (with difference ∆E) is unique to this particular pair of transitions and will notexist between any other pairs of states. By varying the 1/e widths w0 and separation d of the SG
and GD barriers, the degeneracy between the lowest lying and highest lying energy eigenstates canbe transferred between various pairs of eigenstates or even removed entirely (figure 7.5 and 7.6).
Finally, despite the slight anharmonicity of the gate-well, we approximate the gate-well wavefuctions ψn (x, t) as solutions to the quantum harmonic oscillator with1  mω 1/4
ψn (x) = √Hn
2n n! π~

r


mω
2
x e−mωx /2~
~

(7.4)

where n corresponds to the nth energy eigenstate of the gate well and Hn (x) is the nth Hermite
polynomial. With the gate-well eigenfunctions assumed to be harmonic oscillator solutions, as well
as noting the gate-well potential used in this model features 24 bound energy eigenstates (figure7.9b), the degeneracy described above exists between gate-well energy eigenstates ψ0 (x, t) andψ1 (x, t) with ψ22 (x, t) and ψ23 (x, t).

7.4.2

Coupling Gate Well Energy Eigenstates With a Dipole Oscillating BEC

With the degeneracy between lowest and highest lying pairs of eigenstates present, we canwrite that
E1 − E0 = E23 − E22 = ~ωp(7.5)

where each En is the energy eigenvalue for the ψn gate well eigenstate and ωp is the fundamental
oscillation frequency of the gate well potential defined as E0 = ~ωp /2. As a consequence of thisdegeneracy (equation 7.5), given a gate-well BEC occupying ψ0 with an excited portion occupying
ψ1 and undergoing dipole oscillation at frequency ωp , the BEC will couple the highest lying pair3
The anharmonicity of the gate-well potential can also be observed in the non-linear increase of the gate-wellenergy eigenvalues. As shown in figure 7.9b, the gate-well energy eigenvalues do not increase linearly but rather
follow a (very slight) s-curve function.

134

Figure 7.4: Setup and principle of operation of the matterwave transistor: a) Plot showing thesource-gate (SG) and gate-drain (GD) barriers at potential energy heights VSG = 30 kHz and
VGD = 33 kHz, respectively. The barriers separate the transistor into the three source, gate,
and drain-well regions. In this model, the source and drain-wells are treated as flat potentials
VS (x) = VD (x) = 0 and the gate-well is a slightly anharmonic oscillator. b) Plot showing thedifference in energy ∆E between sequential energy eigenstates of the transistor gate-well. The
horizontal axis lists the nth gate-well energy level and the vertical axis is the corresponding energy
difference ∆E = En − En−1 . For example, the left-most data point shows the energy difference
between the first excited state and the ground state: E1 − E0 . The next data point shows theenergy difference E2 − E1 , and so on. From this plot, it is clear that for our model with barrierparameters VSG = 30 kHz, VGD = 33 kHz, w0 = 2 µm and d = 4.85 µm, a degeneracy exists
between the ground and first excited state of the gate-well and the highest lying pair of gate-wellenergy eigenstates. That is, E1 − E0 = E23 − E22 .

of energy eigenstates, ψ22 and ψ23 . Therefore, as atoms occupying ψ22 and ψ23 collide with theoscillating BEC, stimulated emission and absorption of phonons with energy ~ωp results in continuous Rabi flopping between the upper two energy eigenstates. It is from this continuous atom-BEC
interaction (as a result of the BEC coupling the highest lying pair of states) that leads to a gain
mechanism in the transistor oscillator model. The presence and effects of such gain mechanism
emerges directly from the gate-well many-bodied Hamiltonian (section 7.5).

135

Figure 7.5: Plots showing the energy value of each gate-well eigenstate as well as the energy
difference ∆E between adjacent energy eigenstates for various SG and GD barrier separations forthe gate-well potential when VSG = 30 kHz, VGD = 33 kHz, and barrier widths are held constant.
The solid red line indicates two pairs of adjacent eigenstates with the same ∆E value. Thus,
altering the barrier separation d can change what transitions are degenerate with each other.

136

Figure 7.6: Plots showing the energy value of each gate-well eigenstate as well as the energy
difference ∆E between adjacent energy eigenstates for various SG and GD barrier widths for thegate-well potential when VSG = 30 kHz, VGD = 33 kHz, and barrier separation is held constant.
The solid red line indicates two pairs of adjacent eigenstates with the same ∆E value. Thus,
altering the barrier separation d can change what transitions are degenerate with each other.

137
7.4.3

Reduced Representation of the Quantized Gate Well

Prior to formulating the gate-well many-bodied Hamiltonian that includes the atom-BEC
interaction, our model simplifies the quantized energy structure of the gate-well by ignoring4 any
intermediate energy eigenstates located in-between the lowest and highest lying pairs of states.
That is, the model only considers states ψ0 , ψ1 , ψ22 and ψ23 . Using this approximation, we reduce
the gate well from 24 states to a four-level system and relabel the aforementioned four meaningfulenergy eigenstates to |0i, |1i, |2i and |3i, where the first two states (|0i and |1i) are the ground andfirst excited states and the latter two states (|2i and |3i) pertain to the highest lying pair of boundenergy eigenstates in the gate well (figure 7.7). The respective corresponding energy eigenvalues
are E0 , E1 , E2 and E3 , where
E1 − E0 = E3 − E2 = ~ωp .

(7.6)

In this reduced representation, as atoms transition between upper states |2i and |3i by colliding
with the BEC, the excited condensate occupying |1i consequently gains (or loses) a phonon ofenergy ~ωp by absorbing (or depositing) a quanta of energy ~ωp from (or into) |0i. Our reduced
gate well picture is completely analogous to the Rabi model for two-level atoms interacting with aquantized electromagnetic field [169]. Where the Rabi model features transitions between excitedand ground electronic states |ei ↔ |gi by absorbing (and emitting) photons from an oscillating
electric field, our gate model features similar transitions between upper harmonic oscillator states|2i ↔ |3i by absorbing (and emitting) phonons from an oscillating BEC.
Finally, it is important to comment about the validity of reducing the gate-well down from24 states to a four-level system. We may neglect the intermediate eigenstates of the gate-well sinceomitting such states will not change the final results of this model. Regardless of how many energy
eigenstates make up the transistor oscillator gate-well, the model only requires a dipole oscillating
BEC in the ground and first excited states to couple the highest lying pair of bound eigenstates
and no other states. Whether the gate well has 24 states, 100 states, or 4 states, as long as this4

We make this assertion our model because as shown in sections 7.4.1 and 7.4.2. the dipole oscillating BEC
couples only the highest lying pair of energy eigenstates and is decoupled from any other transition.

138
constraint is fulfilled, the gate-well many-bodied Hamiltonian, as well as the results of the model,
will remain unchanged. In this regard, the reduced gate-well Hamiltonian is presented.

Figure 7.7: Diagram showing the reduced representation of the transistor gate well where thesystem has been reduced from a 24 state to a 4 state system. Stimulated emission and absorption(purple arrows) between |2i or |3i occurs due to phonon exchange from atom-BEC interactions
with the dipole oscillating BEC (blue arrows)

139

7.5

Reduced Gate Well Many-Bodied Hamiltonian
To explicitly show the emergence of a transistor oscillator gain mechanism in this model, themany-bodied Hamiltonian of the reduced gate-well confining a BEC undergoing dipole oscillation
with frequency ωp must be formulated. Treating the reduced gate-well shown in figure 7.7 as aclosed system with four energy eigenstates |0i, |1i, |2i, and |3i, the many-bodied Hamiltoniantakes the form5
H=

3
X

g
hi| ~ωi |li a†i al +

2

il=0

3
X

hij| V |kli a†i a†j ak al .

(7.7)

i,j,k,l=0

where ~ωi is the single particle energy of gate well eigenstate |ii and where a†i and ai are theeigenstate’s corresponding creation and annihilation operators. Since this model uses harmonic
oscillator eigenfunctions, the operators are defined in the number state basis as
ai |Ni i =p
Ni |Ni − 1ipa†i |Ni i = Ni + 1 |Ni + 1ihiai , a†i = 1hiai , a†j = 0.

(7.8)

The factor g in equation 7.7, which is derived in section 2.6.2 (equation 2.41), is the interactionenergy between two low energy, long wavelength atoms with equal masses m, given as
g=

4π~2
a
m

(7.9)

where a is the s-wave scattering length of the colliding atoms. Finally, the inner product hij| V |kli
is the overlap integral over four harmonic oscillator eigenfunctions (equation 7.4)
Z
hij| V |kli = Vijkl =ψi (x)ψj∗ (x)ψk (x)ψl∗ (x)dx(7.10)

giving the relative amplitude of a transition a†i a†j ak al between the normalized reduced gate welleigenstates |ii, |ji, |ki, and |li [170]. Using the definitions in equations 7.8, 7.9, and 7.10, the firstsummation of equation 7.7 physically represents all energy contributions from individual gate atoms
5

The factor of 1/2 in equation 7.7 prevents double counting of atom-BEC interactions.

140
occupying eigenstates |ii , |ji , |ki, and |li, while the second summation is energy contributions fromatom-BEC collisions that result in transitions a†i a†j ak al between gate-well eigenstates.
7.5.1

Single-Particle Contributions

Since the gate-well eigenstates are orthogonal, expanding the single-particle contributions toequation 7.7 (which we will label H0 ) yields
H0 =

3
X

hi| ~ωi |li a†i al =

il

3
X

~ωi a†i al δij = ~ω0 a†0 a0 + ~ω1 a†1 a1 + ~ω2 a†2 a2 + ~ω3 a†3 a3(7.11)

il

where δij is the Kronecker delta. The model treats the BEC occupying state |0i of the gate-well
as a macroscopic occupation of the gate-well ground state and accordingly uses the Bogoliubov
approximation6 to state
a0 ≈

p
N0

(7.12)

where N0 is the number of atoms in the ground state BEC [172][173]. Using this approximation,
equation 7.11 can be expressed as
H0 = ~ω0 N0 + ~ω1 a†1 a1 + ~ω2 a†2 a2 + ~ω3 a†3 a3 .
7.5.2

(7.13)

Atom-BEC Collision Contributions

From equation 7.7, the summation for energy contributions due to atom-BEC collisions, given
as
g
H=
2

3
X

hij| V |kli a†i a†j ak al .

(7.14)

i,j,k,l=0

is considerably larger than the summation for single particle energies (equation 7.11) and contains
256 terms when evaluated exactly. Thankfully, some helpful approximations can eliminate all but
four of the terms without altering the results of the model. Justification for excluding certain
collective sets of terms in the summation in equation 7.14 is provided as follows:√6
Equation
7.8 shows the ground state annihilation and creation operators are a0 |N0 i = N0 |N0 − 1i and√†a0 |N0 i = N0 + 1 |N0 + 1i. However,√ if the BEC is contains a large number of atoms, then N0 ≈ N0 − 1 and we canreasonably approximate a0 |N0 i ≈ N0 |N0 i. Therefore,
the result of this Bogoliubov approximation is to replace√
the ground state operators a0 and a†0 with the real N0 [171].

141• Mean-field energy shifts are neglected. That is, terms in the summation containing factorsof a†i a†i ai ai or a†i a†i ak al such as a†0 a†0 a0 a0 , a†0 a†0 ak al , or a†1 a†1 ak al are excluded from the model.
Since mean-field shifts increase all gate-well energy levels equally, the shifts do not change
the overall eigenstate layout of the gate-well and, more importantly, preserve the coupling
between the highest pair of bound eigenstates via the oscillating condensate. Whether themean-field energy shifts are included or not, the physics of the model remains unchanged.
Accordingly, the model ignores the mean-field terms in equation 7.14.
• Nonsensical transitions with the form a†i a†i ai aj are neglected from the model. These represent meaningless transitions that are either unphysical such as a†3 a†3 a3 a0 or transitions thatleave the system unchanged, such as a†3 a3 a3 a†2 or a†1 a1 a1 a†0 .
• The rotating wave approximation is used to eliminate the terms a†0 a†1 a2 a3 and a0 a1 a†2 a†3 .
These contributions in equation 7.14 oscillate rapidly compared to the other terms and thus
average approximately to zero.
• The Bogoliubov approximation (equation 7.12) is once again used since the number ofatoms occupying the ground state BEC is considered to be large.
Using the first two bullet points in the above described set of approximations, the Hamiltonian for
atom-BEC collisions, which is now labeled the interaction Hamiltonian Hint , is reduced to
3gX
Hint =
hij| V |kli a†i a†j ak al
2
ijkl

1 h
= V a†0 a†1 a2 a3 + a†0 a†1 a3 a2 + a†1 a†0 a2 a3 + a†1 a†0 a3 a22+ a†0 a†2 a1 a3 + a†0 a†2 a3 a1 + a†2 a†0 a1 a3 + a†2 a†0 a3 a1+ a†0 a†3 a2 a1 + a†0 a†3 a1 a2 + a†3 a†0 a1 a2 + a†3 a†0 a2 a1+ a†1 a†2 a3 a0 + a†1 a†2 a0 a3 + a†2 a†1 a3 a0 + a†2 a†1 a0 a3+ a†1 a†3 a0 a2 + a†1 a†3 a2 a0 + a†3 a†1 a2 a0 + a†3 a†1 a0 a2i+a†2 a†3 a0 a1 + a†2 a†3 a1 a0 + a†3 a†2 a0 a1 + a†3 a†2 a1 a0 .

(7.15)

142
Using the commutation relations for creation and annihilation operators [a†i , a†j ] = [ai , aj ] = 0 and[ai , a†j ] = δij , equation 7.15 is further reduced to only six unique terms and is expressed as
3

Hint =

gX
hij| V |kli a†i a†j ak al
2(7.16)ijklhi= V a†0 a†1 a2 a3 + a†0 a1 a†2 a3 + a†0 a1 a†3 a2 + a†1 a0 a†2 a3 + a†1 a0 a†3 a2 + a0 a1 a†2 a†3 .
Finally, using the third and fourth bullet point in the list of approximations, the rotating wave and
Bogoliubov approximations reduce the interaction Hamiltonian to its most condensed form in theharmonic oscillator basis:
3

gX
hij| V |kli a†i a†j ak al
Hint =
2
ijkl

i
h
 h
i
N0 a1 a†2 a3 + a†3 a2 + a†1 a†2 a3 + a†3 a2
p = gV N0 a1 + a†1 a†2 a3 + a†3 a2 .

= gV

7.5.3

p

(7.17)

Expressing Gate-Well Hamiltonian in the Basis of Normal Modes and Normal
Mode Basis Definitions

The Hamiltonians H0 (equation 7.13) and Hint (equation 7.17) for single-particle and atomBEC contributions to the gate well many-bodied Hamiltonian are expressed in the harmonic oscillator (or number) basis. Considering that the dipole oscillating BEC couples upper states |2i and|3i, the reduced gate-well (shown in figure 7.7) is a coupled harmonic oscillator and it proves to bemuch more illuminating to express the many-bodied Hamiltonian in a basis where the symmetricand antisymmetric normal modes of the reduced gate-well are the energy eigenstates. This new basis, which for the remainder of this dissertation is labeled as the normal mode basis, has orthogonalbasis states1|+i = √ (|2i + |3i)2i|−i = √ (|2i − |3i)2

(7.18a)(7.18b)

143
where |+i and |−i are the symmetric and antisymmetric normal modes7 , respectively. Additionally,in the normal mode basis, the oscillating, excited portion of the gate-well condensate that occupies|1i is described as a coherent state |αi with magnitude |α|.
When using matrix notation, the symmetric and antisymmetric modes given in equations
7.18a and 7.18b are defined as

 
 
1
 
0
|+i =   , |−i =   .
01(7.19)

To convert H0 (equation 7.13) and Hint (equation 7.17) into the normal mode basis, the following
symmetric and antisymmetric operator transformations are introduced:
1
a+ ≡ √ (a3 + a2 )2ia− ≡ √ (a3 − a2 )2a ≡ a1 .

(7.20a)(7.20b)(7.20c)Applying the operator transformations 7.20a, 7.20b, and 7.20c to equation 7.13 outputs
H0 =

3
X

hi| ~ωi |li a†i al(7.21)

il

~ω3 †
~ω2 †
(a+ − ia†− ) (a+ + ia− ) +(a+ + ia†− ) (a+ − ia− )= ~ω0 N0 + ~ω1 a† a +2
2
and applying the operator transforms to equation 7.17 produces
3

gX
Hint =
hij| V |kli a†i a†j ak al
2
ijkl

o
p n †
N0 a[a+ a+ − a†− a− ] + a† [a†+ a+ − a†− a− ]op n= gV N0 (a + a† )(a†+ a+ − a†− a− ) .

= gV(7.22)

Combining equations 7.21 and 7.22 gives the many-bodied Hamiltonian for the reduced gate-well
7

Realize that the normal mode basis, with eigenstates |+i and |−i, is simply a 2D vector space rotated Θ = π/4from the original |2i and |3i basis states.

144
in the normal mode basis:
H = H0 + Hint
~ω2 †
~ω3 †
= ~ω0 N0 + ~ω1 a† a +(a+ − ia†− ) (a+ + ia− ) +(a+ + ia†− ) (a+ − ia− )
2
2
o
p n
+ gV N0 (a + a† )(a†+ a+ − a†− a− ) .

(7.23)

Equation 7.23 can be simplified further by using the following identities:a†+ a+ = |+i h+|(7.24a)a†− a− = |−i h−|(7.24b)(a†+ ± ia†+ ) (a+ ∓ ia− ) = 1 ± σ2(7.24c)(a + a† )(a†+ a+ − a†− a− ) = (a + a† )σ3(7.24d)where 1 is the identity matrix, σ2 is the Pauli y-matrix, and σ3 is the Pauli z-matrix defined as






0
1 0 
0 −i
1
1=
(7.25)
 , σ2 =  , σ3 = 
.
0 1
i 0
0 −1
Plugging equations 7.24a-7.24d into equation 7.23 and simplifying gives the many-bodied Hamiltonian for the reduced gate-well in the normal mode basis as
H = H0 + Hint
(7.26)p~~= ~ω0 N0 + ~ω1 a a + (ω2 + ω3 )1 − (ω3 − ω2 )σ2 + gV N0 (a + a† )σ3 .
22†Treating8 matterwaves occupying the reduced, coupled gate-well as a linear combination of normalmodes
1
ΨG (x) = √ [A+ |+i + A− |−i]2(7.27)

where A± are the probability amplitudes for the |±i modes, there exists matterwave interference
between the constituent (and simultaneously occupying) |+i and |−i modes. As a result, the total
wavefunction in the gate-well has a beat note that oscillates with a relative phase9 φζ (see figure
8

The assumption can be made because the normal modes |+i and |−i are orthogonal and complete.
This phase described here is a relative phase defined with respect to some reference in the gate well, such as theground state BEC occupying |0i. An example of the superposition of gate well normal modes, as well as the relative
phase φζ , is shown in figure 7.8.
9

145
7.8 for a depiction of φζ ). Therefore, the amplitude of the atom-BEC interaction Hamiltonian,√gV N0 , gains a phase dependence and can be expressed as a complex valued function ζ as
ζ = gVp
N0 eiφζ .

(7.28)

Using equation 7.28 in equation 7.26, the entire gate-well many-bodied Hamiltonian can be expressed in the normal mode basis as a sum of a free space term H0 , a condensate term Hc , and theatom-BEC interaction term Hint as
HG = H0 + Hc + Hint1
= ~ω − ~ωp σ2 + ~ω0 N0 + ~ωp a† a + (ζ ∗ a + ζa† )σ3
{z
}
2
{z
} |
|
{z
} |
interaction(7.29)

condensate

free space

where ω = (ω2 + ω3 )/2 and ωp = ω3 − ω2 is the phonon frequency.
Other than a change of basis and the fixed condensate energy, equation 7.29 is identical to the
Rabi Hamiltonian10 for a two-level atom interacting with an electromagnetic field. Where the Rabi
Hamiltonian has electronic states interacting with photons, the gate-well Hamiltonian HG involves
center-of-mass states (meaning the symmetric and antisymmetric normal modes) interacting withphonons.

7.5.4

Eigenstates and Eigenvalues of the Interaction Hamiltonian

As shown in equation 7.29, the atom-BEC interaction Hamiltonian expressed in the normal
mode basis isHint = (ζ ∗ a + ζa† )σ3 .

(7.30)

Eigenstates of the interaction Hamiltonian are |α, ±i = |αi ⊗ |±i where |αi is a coherent state withmagnitude |α|, and |±i are the symmetric and antisymmetric normal modes of the gate-well. The
10

Similar to the Rabi Hamiltonian, in the gate-well free space Hamiltonian component H0 derived in equation
7.29, the σ2 term represents spontaneous emission:Hse =~(ω3 − ω2 ) σ22

and couples states of opposite symmetric or anti-symmetric parity.

146
corresponding eigenvalues Eint of the interaction Hamiltonian are
Eint = hα, ±| Hint |α, ±i = ± (ζ ∗ α + ζα∗ )= ±2|ζ||α| cos (φζ − φα )(7.31)= ±2|ζα| cos ϕwhere ϕ is the relative phase difference between φζ , the phase of the matterwave interference ofthe gate-well normal modes, and φα , the phase of the gate-well dipole oscillating BEC. Therefore,
the physical interpretation of ϕ = φζ − φα is that it represents the phase of a difference of phases.
The oscillating condensate and matterwave interference are shown relative to each other in figure
7.8 in order to conceptualize the phases φζ , φα , and ϕ. It immediately follows from the domainof equation 7.31 that the eigenvalues of the interaction Hamiltonian take on a range of values
depending on the relative phase ϕ:−2|ζα| ≤ Eint ≤ +2|ζα|.

(7.32)The interaction energy eigenvalue in equation 7.31 can be written in a more complete form bytaking the expressions for g (equation 2.41) and V (equation 7.10) and plugging them into theexpression for ζ (equation 7.28), givingZ ∞p  4π~2 a iφζ = N0eψ0∗ (x)ψ1 (x)ψ2∗ (x)ψ3 (x)dx.
m υ−∞(7.33)

Figure 7.8: Plots showing a time lapse of the matterwave superposition of normal modes ψ+ + ψ−
occupying states ψ2 and ψ3 and oscillating with a phase φζ as well as the excited condensate
occupying state ψ1 and oscillating with a phase φα . The phase ϕ used for determining the eigenvalueof the interaction Hamiltonian is the difference of these phases, given as ϕ = φζ − φα .

147
Additionally, since the dipole oscillating BEC occupying |1i is described as a coherent state |αi, we
use the definition of the coherent state magnitude to write
|α| =

p
N1 .

(7.34)Taking the results of equation 7.33 and 7.34 and inserting them into equation 7.31 yields a complete
expression for the eigenvalue of the interaction Hamiltonian Hint in the normal mode basis:
hα, ±| Hint |α, ±i = ±2|ζα| cos ϕ Z ∞
p
4π~2 a∗∗= ±2 N0 N1
ψ0 (x)ψ1 (x)ψ2 (x)ψ3 (x)dx cos ϕ.
m υ−∞(7.35)

With the eigenvalue of the interaction Hamiltonian fully defined in the normal mode basis, thefollowing section of this dissertation demonstrates how the interaction Hamiltonian (equation 7.30)
acts as a gain mechanism.

7.6

Matterwave Current Flow From Source to Gate to Drain
The gain mechanism of the matterwave transistor oscillator is understood by comparing theflow of a matterwave current resonantly tunneling from the source-well, through the gate-well, andinto the drain-well in two different cases:• When no dipole oscillating BEC occupies the transistor gate-well; that is when the gate-wellis “empty” and Hint = 0• When a dipole oscillating BEC occupies the gate-well and couples high lying gate-welleigenstates; that is when Hint 6= 0.
In this section, we show that a greater flux of source-well matterwaves flow via resonant tunnelingthrough the gate-well and into the drain-well when a dipole oscillating BEC occupies the gate (whenHint 6= 0) as opposed to when the gate has no such BEC (when Hint = 0). As a result, a greater
matterwave probability amplitude exists in the transistor drain-well when the interaction potential
Hint is present compared to when it is missing from the system, thus establishing the interaction
potential Hint as a matterwave gain mechanism.

148
7.6.1

Empty Gate-Well with No BEC Present

Consider the triple-well transistor potential with no oscillating BEC present in the gate-well(which is labeled as an “empty gate”). As a source-well matterwave flows through the gate-well, it’s
probability amplitude in the drain-well is a function of the transmission coefficients of the gate-well.
Such transmission coefficients are calculated for our specific gate-well potential (figure 7.4) by utilizing a numerical technique called the general impedance method [174][175][176][177][178][179], which
cleverly formulates an equivalence between matterwaves tunneling through the gate-well (whose solutions are governed by the 1D time-independent Schrodinger equation) and electromagnetic waves
propagating through transmission line junctions (whose solutions are governed by transmissionline theory from classical electrodynamics). Relying on the philosophy that “similar problems have
similar solutions”, the general impedance method is capable of calculating transmission coefficientsfor arbitrary potential barriers in quantum systems, making it useful for determining transmission
properties of the asymmetric, double Gaussian gate-well for the atomtronic transistor. The actual
calculation of applying the general impedance method to our gate-well potential to solve for thetransmission coefficients is covered in the Master’s thesis of Ava Ashby [180].
Examination of the transmission spectrum (figure 7.9) reveals that sharp resonances in thegate-well tunneling probability occur when the energy of incident source-well matterwave coincides
with an energy eigenstate of the gate-well11 . Using the reduced gate-well nomenclature, whenincident source-well matterwaves match a gate-well resonance boundary condition by having energy
E2 or E3 , the matterwaves couple into the gate-well and resonate in the gate-well, followed bysubsequent tunneling into the drain-well with probability τ2 ≈ 10−4 or τ3 ≈ 10−4 , respectively.
Slight deviations from these transmission resonances (by having an energy E just off-resonancefrom E2 or E3 ) results in a substantial decrease in matterwave probability amplitudes in the drainwell. Consequently, for the case of no BEC in the gate well: the resulting observable effect isthat an appreciable atom flux Φ flows into the drain-well only when source-well atoms are initially11
As mentioned in section 7.3, this transmission behavior is synonymous with Fabry-Perot resonators, resonant
tunneling diodes, and other systems that feature maximum transition when incident fields match resonance boundary
conditions.

149

Figure 7.9: a) Transmission probability of the transistor gate-well as a function of incident matterwave energy when no BEC occupies the gate-well, as calculated by the general impedance method.
Resonances in transmission occur when incident matterwaves have energy coinciding with a gatewell energy eigenstate. b) This inset shows the transmission resonances near energies coinciding
with the highest lying pair bound energy eigenstates (which correspond to reduced gate-well energy
eigenstates |2i and |3i). If incident matterwaves have a slightly different energy, the probabilityof transmission through the gate-well and into the drain-well decreases significantly and can be
reduced by more than 10 orders of magnitude. Essentially, transmission through the “empty”
gate-well occurs only when incident matterwaves have energy coinciding with a gate-well eigenstate. Thus, in the reduced gate-well, incident matterwaves must have energy equal to E2 or E3
for transmission into the drain-well to occur. Otherwise, the matterwaves are reflected.

150
prepared to match a gate-well eigenstate.

7.6.2

Effect of Interaction Potential on Matterwave Current and Gain

We now examine matterwave current flow from source to drain when a dipole oscillating BEC
occupies the gate-well and couples upper states |2i and |3i. With the presence of the BEC, theatom-BEC interaction Hamiltonian Hint (equation 7.30) must be included in the gate-well manybodied Hamiltonian. As we will now show, by including Hint , the matterwave current (an atomflux) flowing from source to drain increases. To understand why the current is increased, consider
a single matterwave ψs (x, t) with frequency ωs originating in the source-well and whose energy does
not coincide with a gate-well eigenstate; that is ωs 6= ω2 6= ω3 . Using the definitions from section
7.4.1, the model treats the source-well matterwaves as plane wave solutions (equation 7.3). We also
operate in the normal mode basis and additionally, treat the source-well matterwave as a linear
combination of the gate-well symmetric ψs+ (x, t) and antisymmetric ψs− (x, t) normal modes:ψs (x, t) = A+ ψs+ (x, t) + A− ψs− (x, t)(7.36)

with amplitudes A+ and A− (and s denotes a source-well quantity). Using the definitions for thesymmetric and anti-symmetric modes (equation 7.18a and 7.18b) produces expressions for ψs+ (x, t)and ψs− (x, t) as1
ψs+ (x, t) = √ [ψs3 (x, t) + ψs2 (x, t)]2ωp= As+ (x, t)ei(ks x−ωs t) cost+ϕ21ψs− (x, t) = √ [ψs3 (x, t) − ψs2 (x, t)]2ωp= As− (x, t)ei(ks x−ωs t) sin
t+ϕ2(7.37a)(7.37b)with wavenumber ks and frequency ωs . Since the symmetric (equation 7.37a) and antisymmetric(equation 7.37b) components of ψs (x, t) are sinusoids that are offset by π/2 radians, they are characterized as the in-phase and quadrature components of the source-well matterwave, respectively.

151
Using the above source-well matterwave definitions, when ψs (x, t) (equation 7.3) penetrates
the SG barrier and enters the gate-well, the atom-BEC interaction, described by Hint , occurs. In
order for the incoming wave ψs (x, t) to simultaneously match the gate-well boundary conditions
and conserve energy, the wave must deposit energy into, or absorb energy from the atom-BEC
interaction energy. The energy shift ∆E imparted by the atom-BEC interaction was calculated inequation 7.31 as ∆E = ±2|ζα| cos ϕ where the sign of the energy shift is positive for the symmetriccomponent ψs+ (x, t) and negative for the antisymmetric component ψs− (x, t). It immediately
follows that for an initially off-resonant matterwave ψs (x, t) with frequency ωs to match a gate-welleigenstate with frequency ωg and satisfy gate-well resonance boundary conditions, the atom-BEC
interaction must change the incoming wave energy by ∆E = ~ωs − ~ωg . Using equation 7.31, thisenergy shift defines an in-phase12 condition
ϕ = − cosωg − ωs2|ζα|
(7.38)

whereby if satisfied, the in-phase, symmetric component ψs+ (x, t) (equation 7.37a) is shifted onto agate-well eigenstate and couples into the gate-well, while the quadrature, antisymmetric component
ψs− (x, t) (equation 7.37b) is completely reflected since ψs− (x, t) is shifted even further from thegate-well resonance.
The domain of equation 7.38 reveals that the range of source-well matterwave frequencies ωsthat may be shifted onto a gate-well resonance with frequency ωg via the atom-BEC interaction is
given byωg − 2

|ζα|
~




≤ ωs ≤ ωg + 2

|ζα|
~


(7.39)

which indicates the interaction Hamiltonian broadens the gate-well resonant tunneling bandwidths
12

One could, without changing the model results, define instead a quadrature condition ofωg − ωsπωg − ωsϕ = − cos
+
= sin
2|ζα|
2
2|ζα|

whereby if satisfied, the antisymmetric component ψs− (x, t) couples into the gate with the in-phase, symmetriccomponent being reflected.

152
Λ from the very narrow peaks shown in figure 7.9a to the wider bands



|ζα|
Λ2 = ω2 ± 2
~


|ζα|
Λ3 = ω3 ± 2~(7.40a)(7.40b)where subscripts 2 and 3 correspond to the resonant frequencies ω2 or ω3 of gate-well eigenstates|2i and |3i, respectively. The result of equations 7.40a and 7.40b are shown pictorially in (figure7.10). From these equations, it is immediately apparent that the resonant tunneling bandwidth iscontrolled by the magnitude |ζα|, which is an experimentally tunableparameter in the sense thatthe larger the gate-well condensate, the greater the magnitude of |α|, and hence a larger interaction
potential Hint , which results in broader resonant tunneling bandwidths Λ2 and Λ3 .

153

Figure 7.10: a) Diagram showing how the presence of a dipole oscillating BEC in the transistor gatewell that specifically couples the highest lying pair of bound gate-well energy eigenstates broadens
the gate-well resonant tunneling bandwidths by introducing the interaction Hamiltonian, Hint
(equation 7.30). The resonant tunneling bandwidths are broadened from a very narrow transmission
resonance (as shown in figure 7.9), to a wider band Λ2 and Λ3 (given in equation 7.40a and equation
7.40b). b) As a result of the interaction Hamiltonian, in the example given in section 7.6.2, as asource-well matterwave ψs , initially off-resonance from a gate-well eigenstate, enters the gate-well,
the interaction Hamiltonian permits the atom to exchange energy with the dipole oscillating BEC
in order to match gate-well resonant boundary conditions while conserving energy. Consequently,
the symmetric mode ψs+ couples into the gate-well with subsequent flow into the drain while
the antisymmetric mode ψs− is reflected with probability unity. The net effect is the interaction
potential has permitted a matterwave to resonantly tunnel through the gate-well and into the drainwith considerably greater probability than if the interaction potential was missing.

154
7.6.3

Gain Mechanism

By comparing the analysis of section 7.6.2 (which treats matterwave transition with the gatewell BEC) and section 7.6.1 (which examines transmission without the gate-well BEC) we arrive at
the matterwave gain mechanism of our transistor oscillator: compared to when no BEC is present
in the gate-well, the atom-BEC interaction Hint increases the range of matterwaves that may couple
into the gate-well with subsequent resonant tunneling out to the drain-well. The gain mechanism
thus amplifies the matterwave probability density in the drain-well, which results in the observableeffect of intensifying the atom flux tunneling into the drain-well when a gate well BEC is present
compared to when it is absent.

7.6.4

Characterizing the Matterwave Gain Mechanism

The equations derived in section 7.5.3 communicate a significant result of the model: Since
each of the symmetric and antisymmetric components ψ+ (x, t) and ψ− (x, t) constitute fractions|A+ |2 and |A− |2 of the total wavefunction ψs (x, t), we thus find that given a gate-well, dipole
oscillating BEC described by |αi with magnitude |α|2 that couples gate-well eigenstates |2i and|3i, if a source-well matterwave with frequency ωs is within either frequency bands Λ2 or Λ3 , thenthe symmetric component ψs+ (x, t) is transmitted through the gate-well 100 × |A+ |2 percent of thetime with probability τ2 or τ3 , respectively. The remaining fraction is reflected with certainty.
The physics in the transistor oscillator model draws very similar comparisons to the cold atom
micromaser [181][182], where a slow, excited-state atom approaching a microwave cavity will berandomly reflected from or transmitted through with 50% probability. This behavior occurs due toa coherent interaction between a quantized field of microwave photons and two-level atoms, which
creates a potential barrier that reflects one component of the atom wavefunction with certainty andtransmits the other component with probability close to unity. In the micromaser case, since eachof the components constitutes half of the wavefunction, about half of the atoms are transmitted
through the maser cavity with the other half reflected. The atomtronic transistor oscillator is155
analogous, except that it additionally demands resonant tunneling for transmission.
Finally, as an example to characterize the gain mechanism, the general impedance method
is used to numerically calculate the gate-well transmission spectrum when a dipole oscillating
BEC (described as a coherent state |αi) with N0 = 20, 000 atoms occupies the ground state |0i andN1 = |α|2 = 5 atoms occupies the first excited state |1i of the gate-well. The transmission spectrum
for this specific case is shown in figure 7.11a and is compared to the transmission spectrum for theexact same gate well, but without a BEC present. With the dipole oscillating BEC coupling upper
states |2i and |3i, the atom-BEC interaction potential described by Hint (equation 7.30) broadens
the resonant transmission peaks by Λ2 = Λ3 = 150 Hz (equations 7.40a,b). To approximate the gainof the transistor oscillator in this example, the ratio of matterwave transmission curves when the
BEC is present to when the BEC is absent is plotted in figure 7.11b. The gain plot shows that withthe dipole oscillating BEC coupling upper states |2i and |3i, matterwaves have their transmission
probabilities increased by the interaction potential from as low as τ0 ≈ 10−12 (in the empty-gatewell case) to τ ≈ 10−3 (when the gate-well dipole oscillating BEC is present), which subsequently
increases matterwave current flowing from source to drain, giving a gain G > 1 (equation 7.1).

Figure 7.11: a) Plot showing the gate-well resonant tunneling transmission probability for thecase, described above, without the dipole oscillating BEC and interaction potential (dashed line)
and when a dipole oscillating BEC occupies the gate-well with Hint 6= 0 (solid line). b) Ratio ofmatterwave resonant tunneling transmission curves when the BEC is present to when the BEC isabsent. This quantifies that gain for this given example.

Chapter 8

Experimental Observations to Verify Transistor Gain Mechanism

This chapter presents the current experimental progress to observe the gain mechanism of thematterwave transistor oscillator. In section 8.1, we describe the experimental criteria that must be
satisfied in order to observe effects of the interaction Hamiltonian (equation 7.30) and subsequent
matterwave gain of the transistor oscillator. In section 8.2, we give an overview of the experimental
apparatus as well as the procedure for preparing a source-well atom ensemble at a temperature low
enough for the predicted transistor behavior to occur. Section 8.3 provides a detailed description ofexperimentally forming the proper transistor oscillator triple-well potential that was introduced in
section 7.4.1. Section 8.4 details the matterwave transistor oscillator experiments aimed at observing the dynamics of an ultracold atom current flowing through the transistor. In-situ absorption
images showing the flow of an ultracold atom current through the transistor are presented. Section
8.5 provides a discussion of the results that verifies some of the principles predicted by the model,
namely the existence of a coherent matterwave gain mechanism.

8.1

Criteria for Observing Matterwave Transistor Gain Mechanism
Following the transistor oscillator model derived in chapter 7, there exist several criteriathat must be satisfied in order to experimentally demonstrate the matterwave gain mechanism.
First, for a gate-well BEC to couple the highest lying gate-well energy eigenstates, we must prepare
the source-well atom ensemble at a temperature comparable (and ideally equal) to the gate-wellenergy level spacing (corresponding to E3 − E2 in the reduced gate-well in figure 7.7). In the157
theory presented in chapter 7, the model triple well potential has sequential energy levels spaced
by approximately 1 kHz, which corresponds to a temperature of Ts ≈ 50 nK. Thus, when sourcing
the atom current, the source-well ensemble thermal energy must be as close as possible to Ts = 50
nK.
Second, for the gain mechanism (equation 7.30) to be present in the transistor, we need toarrange the gate-well potential to sustain a steady-state BEC as atom current flows from source,
to gate, to drain. As shown in section 7.4.1, the optimal balance of particle and energy flow that
leads to a high enough phase-space density to maintain a stead-state gate-well BEC occurs whenthe “feedback parameter” υ = 1.1, where recall υ is defined asυ=(VGD − VSG )
Ts

(8.1)

where Ts is the temperature of the source-well ensemble. As described above, we restrict the valueof kTs = 1 kHz. It follows from equation 8.1 that the difference in SG and GD barrier heights
must be 1.1 in order to maintain an optimal steady-state BEC in the gate-well. Consequently, weconstrain1 the SG and GD barrier heights at VSG =30 kHz and VGD =31 kHz.
Third, we need the atoms to be prepared in the source-well at an energy near the the highest
lying pair of gate-well eigenstates (to within eigenvalue of the interaction Hamiltonian given byequation 7.31). Otherwise, the atoms will first, never tunnel2 into the gate-well to begin withmeaning atom currents never leave the souce-well, and second, none of the atoms will ever couple toeither of the highest gate-well eigenstates, making the gate-well many-bodied Hamiltonian (equation
7.29) not applicable. As described above, for this experiment the source-well ensemble thermalenergy is constrained at Ts = 50 nK, meaning source-well atoms are considerably below theseenergy values, and will never flow through the transistor. Physically, a loaded source-well functions
as an atomtronic battery, making this particular scenario equivalent to sourcing an electrical circuitwith a “poorly charged” battery. Since the stored energy in the source-well is characterized by the1

This moderate deviation from the model barrier heights (shown in figure 7.4) will not change the ∆E sequential
energy level spacing of the gate-well (and hence will not change the constraint on Ts ) as ∆E ≈ 1kHz remains true.
2
This is because, in energy, the atoms are at the wide base of the SG barrier, where the tunneling probability isessentially non-existent.

158
ensemble thermal energy and its chemical potential, we must raise the source-well chemical potentialµs from 0 (as in the model in figure 7.4) to µs ≈ 27 kHz such that the ensemble with thermal energyof 1 kHz has energy very near to the highest gate-well eigenstates.
Finally, with all of the above described criteria satisfied, we must verify the presence of amatterwave gain mechanism. The remainder of this chapter details the experiment and results
aiming to verify the matterwave transistor oscillator model presented in chapter 7.

8.2

Production of a 50 nK Atom Ensemble
A finite temperature 87 Rb Bose-Einstein condensate of approximately 7 × 103 atoms at atemperature of 50 nK serves as the atom ensemble used to source the transistor oscillator system.
The BEC is formed using an atom chip in the Atomtronics Double MOT UHV cell outlined in
sections 3.2 and 3.3, as well as figure 3.5. The setup and procedure for making such a 50 nK atom
ensemble is described in the following sections.

8.2.1

Laser System and Layout

A summary of all laser cooling, repump, optical pumping, and probe transitions are shownin figure 8.1 and the specific 780 nm laser setups used to produce a BEC is shown in figure 8.2.
Note that all rubidium transitions described in this section refer to the D2 line. With respect tothe cooling, repump, and optical pumping setups, each features a JILA made3 master external
cavity diode laser (ECDL) that is used to injection lock a slave laser.
The 780 nm cooling master ECDL is locked to the 85 Rb |F = 3i → |F 0 = 2i / |F 0 = 4i
crossover transition by picking off a small amount of light from the output of the ECDL using
Doppler free saturated absorption spectroscopy. An additional 5 mW of cooling light is picked offfrom the main ECDL output to be used as a frequency reference which is stabilized using a phaselocked loop [4]. Using a frequency offset locking setup, the light is blue-shifted 1.2 GHz such that3

The ECDL master lasers were designed by Scott Papp, built in the JILA electronics shop, and are detailed in
his PhD thesis [183]. Additionally, each of the JILA made lasers are powered by JILA made current drivers. Whilea stark contrast to the commercially manufactured lasers systems used in the Cs experiments in chapter 4, they stillwork masterfully over 15 years later.

159
the laser is red-detuned by 2Γ from the 87 Rb |F = 2i → |F 0 = 3i cooling transition. Additionally,
40 mW of cooling light is used to seed a Toptica BoosTA tapered amplifier, which then outputs 1000mW of 780 nm laser light. Using a polarizing beam splitter, the TA output is split into two paths,
where 500 mW of 2D MOT cooling light and 70 mW of 3D MOT cooling light are coupled with
50 % efficiency into their individual polarization maintaining fibers and sent to the experimental
apparatus.
For repumping light, a 780 nm repump master ECDL is locked to the 87 Rb |F = 1i →|F 0 = 1i / |F 0 = 2i crossover transition by picking off a small amount of light from the output of
the ECDL and using Doppler free saturated absorption spectroscopy. Approximately 1 mW ofrepump light is picked off from the main beam line and used to seed a repump slave laser. The
output of both the master and slave lasers are each sent through a AOM driven at 87 MHz toblue-shift the beams onto resonance with the 87 Rb |F = 1i → |F 0 = 2i transition. Each beam isthen coupled into a polarization maintaining fiber, where the master light is sent to the 2D MOT
and the slave light is sent to the 3D MOT. With the described configuration, the master repump
setup provides 1 mW of 3D repump light and the slave provides 12 mW of 2D repump light.
The final 780 nm laser system used for producing a BEC is the optical pumping and probe
setup. The pump-probe master ECDL laser is locked to the 87 Rb |F = 2i → |F 0 = 2i / |F 0 = 3i
crossover transition by picking off a small amount of the main ECDL light and using Doppler
free saturated absorption spectroscopy. The main beam line is then split into two paths using apolarizing beam splitter, with one beam used for optical pumping and the other for a resonant probebeam. With respect to the optical pumping light, an AOM red-shifts the beam by 121 MHz suchthat the pump light is blue-detuned 2Γ from the 87 Rb |F = 2i → |F 0 = 2i transition. Moreover,the probe light is blue-shifted by 133.5 MHz onto resonance with the |F = 2i → |F 0 = 3i coolingtransition. After passing through their respective AOMs, a total of 3 mW of optical pumping lightand 3 mW of resonant probe light are each coupled into their own polarization maintaining fiber,where the light is then sent to the experimental apparatus. Using the above described laser system,
we can now create a BEC using the atom chip located at the top of the atomtronics UHV cell.

160

Figure 8.1: Transitions for the 780 nm cooling, repump, optical pumping, and probe lasers.

161

Figure 8.2: Complete laser setup for the atomtronics transistor oscillator experiment.

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8.2.2

Loading Cold Atoms into the Atom Chip Trap

The production of a BEC in the Atomtronics Double MOT UHV cell follows similarly to themethod developed by Evan Salim in [4] and begins by generating a 3D MOT of 87 Rb inside the 3D
chamber of the UHV cell (figure 3.5). The 3D MOT is created by retroreflecting three pairs of 10
mW, 780 nm cooling laser beams with orthogonal σ+ /σ− circular polarization and a beam waistof w0 = 5 mm. The MOT beams are spatially overlapped with 3D repump light and all three pairsof lasers intersect at an angle of 90◦ with respect to each other approximately 15 mm below thebottom surface of the atom chip (figure 8.3a,b). This configuration provides 3D cooling along theorthogonal x̂, ŷ, and ẑ directions. Anti-Helmholtz coils oriented around the 3D chamber (figure8.3b) provide a position dependent 8 G/cm magnetic field gradient necessary for creating the 3D
MOT. Similar to the Cs MOT described in section 4.3.1, atoms are sourced into the 3D MOT froma 2D+ MOT formed in the adjacent 2D MOT vacuum chamber located below the 3D chamber(figure 3.5). During BEC production, a typical 3D MOT contains approximately 109 atoms and isloaded within 40 seconds.
After loading the 3D MOT, the 2D MOT lasers are shuttered and the 3D MOT is compressed into a CMOT by ramping the anti-Helmholtz field gradient up an additional 10 G/cm
while simultaneously detuning the cooling lasers by an additional 40 MHz and attenuating the 3D
repump to 50 µW of optical power. Typically, the CMOT stage lasts 15 ms and the MOT atom
temperature after compression is approximately T ≈ 130 to 160 µK. After CMOT, sub-Doppler
cooling is performed. The MOT anti-Helmoltz coils are switched off while x-bias, y-bias, and z-bias
coils are turned on in order to eliminate any stray magnetic fields at the location of the atoms
such that a PGC stage can properly be performed. After eliminating stray magnetic fields at theatoms, we red-detune the 3D cooling lasers an additional 105 MHz for 3 ms, which cools the atoms
to temperatures of approximately T ≈ 20 µK. After PGC, all cooling light is extinguished while
optical pumping light illuminates the atoms for 800 µs to pump as many atoms as possible to the|F = 2, mF = 2i Zeeman magnetic sublevel. As shown in equations 2.18 and 2.19, since mF > 0,163
this permits the optically pumped atoms to be trapped in a minimum of a spatially varying magnetic field. At the conclusion of the optical pumping stage, all lasers are extinguished and no morelaser cooling stages are applied.

8.2.3

Transferring pre-Cooled Atoms to the Atom Chip

Given that the atoms are prepared in the |F = 2, mF = 2i Zeeman sublevel, if we linearlyramp current through the 3D MOT coils in the Helmholtz configuration over a span of 35 ms,
the optically pumped atoms are then captured in a magnetic quadrupole field with a 60 G/cm
field gradient. To vertically transport the atoms from the location of the 3D MOT up to theatom chip, current is linearly run through a pair of magnetic transport coils, also in the Helmholtz
configuration (figure 8.3c), to produce a transport field gradient of 110 G/cm while the quadrupole
field is linearly ramped to zero over the same time period. This operation vertically shifts thezero point of the quadrupole field that the atoms are trapped in. As long as the movement isdone adiabatically, the atoms remain trapped in the magnetic quadrupole field as it is transported
vertically. Upon reaching the height of the atom chip, the transported atoms are transferred intoa single-guide wire Ioffe-Pritchard trap produced on the atom chip (described in section 3.3.1).
To transfer atoms into this chip trap, the transport field is first compressed over a time spanof 65 ms such that the magnetic quadrupole field gradient matches the (anticipated) chip trap
gradients. After such compression, the chip guide-wire and H-wire currents, as well as the y-bias
current are linearly ramped on over a span of 10 ms and a magnetic potential, as shown in section
3.3.1, is formed. Approximately 20 × 106 atoms at a temperature of 17 to 20 µK remain in the
Ioffe-Pritchard trap after this transfer sequence and are located approximately 150 µm below thebottom surface of the atom chip.

8.2.4

Reaching Bose-Einstein Condensation with Forced RF Evaporation

To condense the atoms into a BEC, a forced RF evaporation (section 2.5.1) stage is applied tothe trapped atom ensemble to further cool the atoms below the BEC critical temperature (equation

164

Figure 8.3: Diagram a) and b) show the orientation of the 3D MOT lasers with respect to theatomtronics vacuum cell. In a) the pair of vertically oriented MOT beams intersecting in an X
shape 15 mm below the bottom surface of the atom chip. In b) the third set of MOT beams
oriented normal to the X shape beams. Note that the top and bottom diagrams are rotated 90
degrees with respect to each other. This diagram is taken from the thesis of Evan Salim in reference[4], as it was his design that is still being used today. Image c) shows the orientation of the MOT
coils and magnetic transport coils with respect to the 3D MOT chamber. Not visible in thisimage are the y-bias and z-bias coils. d) Image of the RF antenna that is located approximately 3
mm above the top layer of the atom chip. e) Imaging showing Cooner wire bonded to individual
conductive pads of the atom chip using a conductive epoxy.

1652.29) necessary for condensation to occur. In order to perform efficient RF evaporation, the chipmagnetic trap is compressed along the radial and longitudinal directions in 500 ms in order toincrease the atom collision and re-thermalization rate during the evaporative cooling stage. Using
an RF antenna located 3 mm above the top surface of the atom chip (figure 8.3c), RF radiation
with a frequency of νRF = 40 MHz is illuminated onto the atoms. The RF frequency is then linearly
swept from νRF = 40 MHz to 26 MHz in 150 ms, which expels the absolute warmest atoms fromthe chip trap. Afterwards, the RF frequency is swept over a span of 1800 ms fromνRF = 26 MHz → 11 MHz → 6.5 MHz → 4.8 MHz → 2.62 MHz(8.2)

where the final RF value is approximately 100-200 kHz above the trap bottom, leaving on thecoldest of the cold atoms remaining in the magnetic trap.
After the above described RF evaporation sequence, a Bose-Einstein condensate of about
7 × 103 atoms is produced in the chip guide-wire trap. As explained in section 3.3.1, this BEC isproduced under the chip guide wire and thus blocks optical access to the condensate. To move
the BEC out from under the wire and align it with the center of the atom chip window (as shownin figure 3.6b), we slowly ramp a current of equal magnitude but opposite direction through thesecond chip guide-wire (as detailed in section 3.3.2) in 80 ms. This adiabatically deforms the single
guide-wire magnetic trap into the split guide-wire magnetic trap located 150 µm below the centerof the atom chip window. At this final location, a Bose-Einstein condensate of 7 × 103 is confined
to a cigar shaped magnetic potential with trapping frequencies along the x̂, ŷ, and ẑ directions of
ω(x, y, z) = 2π × (67, 1500, 1500) Hz.

(8.3)

Absorption images showing the momentum distribution of the atoms (figure 8.4) taken at final RF
evaporation frequencies of 2.68 MHz, 2.66 MHz, and 2.62 MHz show the appearance of an inverted
parabola profile, which (as derived in equation 2.39) is characteristic of atoms that have undergonea phase transition to a Bose-Einstein condensate. The atom density profile of the condensate
shown in 8.4 was fitted to a binomial distribution, which is the sum of a Gaussian distribution

166(representing the thermal atoms in the ensemble) and an inverted parabola (for the condensed
atoms in the ensemble). The final temperature of the condensate was measured by performing
time-of-flight imaging on the expansion of the Gaussian tails of the BEC density profile. Using a
final RF evaporation frequency of 2.62 MHz, the BEC temperature was measured at approximately
T = 50 nK.

Figure 8.4: Time-of-flight absorption images showing the momentum distribution of the atomic
cloud after a 22 ms TOF. From left to right the images show the atom cloud momentum just before
the condensate, at the appearance of the condensate, and at nearly a pure condensate.The transition
to a BEC occurs during the forced RF evaporation stage for RF frequencies of approximately 2.66
MHz. At this stage, the condensate temperature was measured to be T = 50 nK. The appearanceof an inverted parabola profile shown in the central and right image is characteristic of the onsetof a BEC as derived in equation 2.46 and shown in figure 2.8.

167

8.3

Forming the Transistor Oscillator Model Triple-Well Potential
The triple-well transistor oscillator potential (figure 7.3) is formed by overlapping a harmonicmagnetic potential with two repulsive Gaussian barriers. Here, we detail the creation of bothelements. First, the harmonic magnetic potential is the same cigar shaped Ioffe-Pritchard trapwith radial and longitudinal trapping frequencies of ωrad = 2π × 67 Hz and ωrad = 2π × 1600
Hz, respectively that confines the 50 nK BEC formed in section 8.2.3. The magnetic potential
is located approximately 150 µm below the bottom surface of the atom chip window. Next, therepulsive source-gate (SG) and gate-drain (GD) barriers are formed by focusing two blue-detuned,
760 nm Gaussian laser beams with equal beam waists of w0 = 2.0 µm and a separation of d = 4.8µm onto the center of this harmonic magnetic potential.

8.3.1

Projecting SG and GD Barriers onto the Magnetic Potential

The 760 nm laser setup for producing the SG and GD barriers is shown in figure 8.2 andthe schematic for projecting them onto the magnetic potential is shown in figures 8.5a and 8.5b.
A total of 35 mW of 760 nm laser light enters the barrier projection system from a polarization
maintaining, single mode fiber and is collimated to a beam waist of w = 4 mm with an f=15mm
aspherical lens. The 760 nm light then propagates into an IntraAction Corp DTD-6010RH29 2D
acousto-optic deflector4 (AOD), which can dynamically control the shape and positions of the SG
and GD barriers. To generate the longitudinal (or x̂) position of the barriers, two RF tones, one foreach barrier, drive the AOD and diffracts each beam into its +1 order along the x̂ direction. The
RF tone for the GD barrier is generated by a VCO while the tone for the SG barrier is produced
by a Lecroy ArbStudio Arbitrary Waveform Generator (AWG). Because the RF driving voltages
are easily adjustable, this scheme provides a mechanism to dynamically change the longitudinal
positions of the SG and GD barriers during the experiment (which, as shown in section 8.4, is anabsolute necessity for the experiment). The radial (or ŷ) position of the both SG and GD barriers
4

The 0 order beams are blocked leaving only the +1, +1 diffracted order beams for the experiment.

168
is generated by using a second channel of the AWG to continuously raster the SG and GD barriers
back-and-forth in the ŷ direction at a frequency of 20 kHz.
Using an Infinity Probe TS-160 objective in conjunction with a Zeiss LD Plan Neofluar 0.6
NA, 40x magnification microscope objective, the optical barrier potential is projected onto theharmonic magnetic potential (figure 8.5b). Note that the barriers are focused through the 2mm
wide, 0.420 µm thick atom chip window (as shown in figure 3.6b). The Zeiss objective features anadjustable correction collar to eliminate aberration introduced when the light focuses through the
0.42 µm thick window. Additionally, because the 20 kHz raster frequency of the barriers in theradial direction is considerably faster than the 1.6 kHz radial trapping frequencies of the magnetic
potential, atoms confined in the magnetic potential encounter the constant time-averaged opticalintensity of the laser beams in the ŷ direction. Thus, the above description produces two barriers
with optical intensity profiles that are Gaussian beams (with width w = 2 µm) in the longitudinal
direction and a “top hat” potential (with length 8 µm) in the radial direction (figure 8.5c). By
overlapping an atom ensemble in the magnetic potential, the SG and GD barriers can section off
the potential into the source-well, the gate-well, and the drain-well (figure 8.5d).
This barrier projection system also doubles as an imaging system to observe the flow ofultracold atoms through the triple-well transistor potential. To observe atoms in the transistor
potential, in-situ absorption imaging is performed (as opposed to time-of-flight imaging) while theatoms are still confined to the triple well potential. A probe beam resonant with the 87 Rb D2|F = 2i → |F 0 = 3i transition is propagated vertically through the vacuum cell (entering throughthe base of the 2D MOT chamber and exiting though the atom chip window, see figure 8.5b)
where it intersects ultracold atoms trapped in the hybrid magnetic and optical transistor potential.
Positioned 3 mm above the outside surface of the atom chip, the same Zeiss 0.6 NA microscope
objective that focuses the barriers is used to collect and collimate the absorption image signal.
An Infinity PhotoOptical KC Lens then focuses the probe light onto an Andor iXon EMCCD for
imaging (figure 8.6). An absorption image of approximately 20,000 ultracold atoms trapped in the
triple-well transistor potential is shown in figure 8.5d.

169

Figure 8.5: a) Optical setup for producing dynamically controllable SG and GD barriers. b) Setup
to project the SG and GD barriers through atom chip window and onto the harmonic magnetic
potential. The same lens that focuses the barriers also serves to for in-situ absorption imaging ofatoms. c) Optical intensity image of the focused SG and GD barriers with beam waists of w0 = 2µm and a separation of d = 4.8 µm. d) An in-situ absorption image taken through the atom chip
window showing ultracold atoms trapped in the triple-well transistor potential.

170

Figure 8.6: Image showing the experimental setup of the SG and GD barrier projection system aswell as the in-situ imaging system relative to the 3D MOT cell and the atom chip. The inset shows
the 0.6 NA Zeiss objective located approximately 3 mm above the top surface of the atom chip.

171

8.4

Matterwave Transistor Oscillator Experiments: Flow of an Ultracold
Atom Current Through Transistor Oscillator Potential
Experiments to observe transistor behavior begin with following the procedure of section 8.2

and loading a Bose-Einstein condensate of approximately 7 × 103 atoms with a temperature of
T = 50 nK (figure 8.4) into a harmonic magnetic potential located 150 µm below the center of theatom chip window (figure 8.7a).

8.4.1

Loading Ultracold Atoms into the Source-Well

Ultracold atoms are then loaded into the transistor source-well by turning on the SG barrier
and adiabatically sweeping it from the right-most longitudinal boundary of the harmonic magnetic
potential to near the center of the magnetic potential in 40 ms (figure 8.7b). As the SG barrieris swept across the magnetic potential, the barrier height is set at VSG = 100 kHz, which isconsiderably higher than the corresponding temperature of the trapped atoms (T = 50 nK ≈1kHz) and as a result, nearly all of the atoms are compressed into the transistor source well by therepulsive SG barrier (figure 8.7b). Any atoms that happen to remain in the adjacent drain well arequickly removed by optically pumping them to the |mF = −2i anti-trapping state by illuminatinga laser beam, called the drain terminator, that is locked to the 87 Rb |F = 2i → |F 0 = 2i transition.
The drain terminator, which is a 10−13 W, 780 nm beam focused to a waist of 16 µm, is projected
onto the center of the transistor drain-well using the same optical projection system for the SG
and GD barriers (section 8.3) and can remove all drain-well atoms within 5 ms without disturbing
any atoms confined to the source or gate well. After clearing any residual atoms not swept into
the source-well, the SG barrier height is ramped down to its desired height of VSG = 30 kHz while
simultaneously, the GD barrier is ramped on to a height of VGD = 31 kHz and is displaced from theSG barrier by a distance d = 4.8 µm. After this sequence, the matterwave transistor is preparedin a state shown in figure 8.7c.

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Figure 8.7: The following are in-situ absorption images of ultracold atoms at various stages ofpreparing the source-well ensemble. a) A Bose-Einstein condensate with approximately 6,000 atoms
at a temperature of 50 nK loaded into the harmonic magnetic potential. b) By sweeping the SG
barrier from the longitudinal edge of the magnetic potential, nearly all atoms are confined into thetransistor source-well. c) The SG barrier is lowered to its desired height of 30 kHz and the GD
barrier is raised to 31 kHz. d) Source-well atom ensemble after raising the source-well chemical
potential (described in the upcoming section 8.4.2). By allowing source-well atoms to flow into thegate-well, we can directly measure the size of the gate-well to verify the SG and GD barriers are
separated by d = 4.8 µm.

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8.4.2

Powering the Source-Well Ensemble to Commence Atom Currents

With their current energy of approximately 1 kHz, the source-well atom ensemble will not flow
through the transistor-oscillator potential. The energy of the atoms places the source-well ensembleat the base of the SG barrier (figure 8.7c) making subsequent flow into the gate and drain-wells
highly unlikely. In this condition, the source-well in the atomtronic transistor is equivalent to anelectronic battery with insufficient energy to power an electrical circuit. To raise the energy of thesource-well atom ensemble to high enough energies such that an atom current flows, the source-wellchemical potential is raised from 0 to approximately µS ≈ 27 kHz (figure 8.7d).
This is accomplished by rapidly sweeping the SG barrier back-and-forth at a frequency of 19
kHz, extending over a majority of the source well, while ramping up the optical power of laser beam
forming the SG barrier. Specifically, an AWG is used to ramp the driving voltage of the AOD that
controls the longitudinal position of the SG barrier. The sweeping sequence begins by holding the
SG barrier at its initial position (as shown in figure 8.7b) for 4 µs. Then the AOD driving voltage
is ramped up by 1 V over the span of 24 µs. This serves to sweep the SG barrier from its initial
position to the left-most attainable longitudinal position such that the barrier sweeps over as muchlength of the source-well as possible. The AOD voltage is then ramped down over 22 µs, bringing
the SG barrier back to its starting point, where the sweeping sequence is repeated for the remainderof the experiment. Note that the longer the barrier is held at its starting point, the higher the SG
barrier will be.
The total sweeping frequency of the SG barrier is 19 kHz, which is considerably larger thanboth the 1.6 kHz and 125 Hz radial and longitudinal trapping frequencies of the transistor sourcewell potential. As a result, by continuously repeating this waveform, the atoms experience the time
averaged intensity of the sweeping optical field, which is shown in figure 8.8b. As the SG barrieris swept back-and-forth, the optical power of the laser beam forming the SG barrier is linearlyramped from its initial value of 1.7 mW to 2.5 mW over a span 10 ms. This process serves to raise
the bottom of the source-well potential from 0 to 27 kHz and places the source-well ensemble at a174much higher chemical potential of µs = 27 kHz (figure 8.8). Finally, by supplying 2.5 mW and 0.8
mW of optical power to the laser beams that form the SG and GD barriers respectively, the final
time averaged optical intensity of the barriers still maintains a gate-well with SG and GD barrierheights of VSG = 30 kHz and VGD = 31 kHz in addition to the raised source-well with chemical
potential µs ≈ 27 kHz (figure 8.8c). The source-well is now prepared in a state shown in figure 8.7c
where an ultracold atom current may flow into the transistor gate-well and subsequently into thedrain.
Time-of-flight imaging performed on the atom ensemble in the raised source-well shows thatthe atom temperature has increased to approximately 78 nK and the ensemble is no longer condensed into a BEC. The heating is attributed to technical noise from position fluctuations of the

Figure 8.8: a) Voltage driving the AOD that controls the x̂ (or longitudinal) position of the SG
barrier as a function of time. By using this sweeping function for the SG barrier, the time averaged
optical potential for the SG barrier is shown in b). Image b) is an intensity image of the SG and GD
barriers after the above sweeping function. This serves to raise the source-well chemical potential
with the corresponding transistor potential shown in c).

175√
magnetic potential due to current noise of 35 nA/ Hz in the current drivers that supply the atom√
chip currents Ix and 120 nA/ Hz in the current drivers that form the z-bias field βz were measured in reference [153] to account for a 1 nK/ms heating rate. We did not observe any measurable
heating from sweeping the SG barrier back and forth.

8.4.3

Atom Current Flow Through the Transistor Oscillator

With an ultracold atom ensemble prepared in the raised source-well, the flow of an atom
current through the transistor oscillator is observed by probing the system in-situ. Each run of theexperiment begins with the raised source-well loaded with approximately N = 7 × 103 atoms at
a temperature of approximately Ts = 78 nK and a chemical potential of µs = 27 kHz. The atom
current begins to flow at time t = 0, which is defined as the time when the source-well chemical
potential is raised to µs = 27 kHz and the SG and GD barriers are set to their desired heights of
VSG = 30 kHz and VGD = 31 kHz.
The transistor oscillator system is allowed to evolve5 for some time ∆t, during which an atom
current flows throughout the transistor potential landscape. After an ultracold atom current flows
for some time ∆t, the occupancy of the source, gate, and drain wells of the transistor oscillator isobserved by performing in-situ absorption imaging. Figure 8.9 shows a series of absorption images
demonstrating the flow of an atom current from source-well, through the gate-well, and into thedrain-well at various flow times. In figure 8.9, the first image shows the source-well atom ensemble
at a chemical potential of µs = 27 kHz and temperature of T = 78 nK immediately before the startof an atom current at t = 0. Within 10 ms, a very large population of atoms becomes trapped in
the gate-well. However, it takes approximately another 5 ms for the current flow into the drain-welland collect at the drain classical turning point. Over the next 7.5 ms, the atom current flowing
into the drain-well steadily increases, with a corresponding decrease in the gate-well occupation.
5

As explained in great detail in reference [153], as the system evolves, the drain-well is constantly illuminatedwith the drain terminator beam (section 8.4.1) to remove any atoms that flow into the drain-well. The terminatorultimately prevents the transistor oscillator system from reaching an equilibrium. Note however, that the terminatorbeam is extinguished approximately 6 ms prior to taking an absorption image in order to take an instantaneous
snapshot of the atom current accumulating in the drain-well. The 6 ms corresponds to approximately 1/4νx whereνx is the longitudinal trapping frequency of the harmonic magnetic potential.

176
At ∆t = 22.5 ms, the atom flux flowing into the drain-well reaches a maximum, and subsequently
starts to decay, decreasing over the next 6.5 ms until the current stops flowing into the drain-well.
After a flow time of ∆t = 30 ms, no atom current was observed to flow into the drain-well.

Figure 8.9: in-situ absorption images of the transistor oscillator at various flow times as an atom
current flows from the source-well, through the gate-well and into the drain-well. The spatial
locations of the SG and GD barriers are indicated with the dashed lines.

177
Time-of-flight measurements show that during the first 10 ms of the atom current flow time,
the source-well ensemble temperature increases from 78 nK to approximately 109 nK. However,once the current begins to flow into the drain-well, the source-well ensemble continuously cools(figure 8.10). This is consistent with the semi-classical atomtronic transistor experiments [5][34]and shows that as the atom current flows into the gate and drain wells, it carries away energy fromthe source-well. This source-well heating and cooling behavior should be expected since the in-situ
absorption images show that an atom current isn’t flowing during the first 10 ms, so we shouldn’t
expect any flow of energy out of the source well. Once current begins to flow into the gate anddrain wells we see a noticeable drop in source-well temperature.

Figure 8.10: Plot showing the temperature Ts of the source-well atom ensemble as a function of
atom current flow time. The source-well ensemble temperature in each data point was measured
using time-of-flight imaging after each subsequent current flow time. Error bars are propagatedfrom the standard error of the mean for Ts .

178
8.4.4

Using a Colder Source-Well

The ultracold atom current flowing through the transistor oscillator in figure 8.9 begins at atemperature of T = 78 nK, but is not condensed into a BEC. At the cost of a smaller source-wellatom number, we performing a more intense RF evaporation stage when preparing the source-wellensemble, by using a higher intensity of RF radiation. With the more intense RF evaporation,
the atom ensemble in the raised source-well is at a cooler temperature of T ≈ 54 nK compared
to section 8.4.3. More importantly, the ensemble remains condensed in a BEC. We then repeatedthe methods of section 8.4.3 to observe the atom current flow when the source-well ensemble is a
BEC. As in the previous section, the flow of an atom current through the transistor oscillator isobserved by probing the system in-situ and is shown in figure 8.11a. Additionally, time-of-flight(TOF) absorption images of the source-well ensemble following a 22 ms TOF show the presenceof a source-well condensate at times immediately before the flow of atom currents (t = 0, figure
8.11b) and after an atom current flow time of 25 ms (figure 8.11c).

8.5

Discussion of Results: Verifying Transistor Gain and Coherence
Verifying the existence of a coherent gain mechanism in the atomtronic transistor oscillator

can be accomplished by examining in-situ absorption images of ultracold atom currents flowing
through the transistor triple-well potential, such as the images shown in figures 8.9 and 8.11. We
now confirm the transistor model formulated in chapter 7 by explicitly showing the presence ofa matterwave gain mechanism. We directly observe an increase in the spread of the energy ofthe drain-well atom flux as the source-well temperature is reduced (and coupling strength to thegate-well subsequently increases). This behavior is consistent with the presence of the matterwave
gain mechanism that was derived in chapter 7 and further described in sections 7.6.2 and 7.6.3.
Finally, we conclude this chapter by further verifying the transistor model by showing experimental
evidence suggesting coherence between the matterwaves flowing into the drain-well and accordingly
establishes the interaction potential (equation 7.30) as a coherent gain mechanism.

179

Figure 8.11: in-situ absorption images of the transistor oscillator at various flow times as an
atom current flows from the source-well, through the gate-well and into the drain-well. The spatial
locations of the SG and GD barriers are indicated with the dashed lines. Insets show the longitudinaldensity profiles of the source-well ensemble immediately before the flow of an atom current (at∆t = 10) and after an atom current flow time of 25 ms. Both images show that the source-wellensemble is a BEC as the system begins to evolve and remains a BEC as current flows though thegate-well and into the drain.

180
8.5.1

Observation of Transistor Gain Mechanism

Before discussing the atom current results that lead to conclusions showing the presence ofa matterwave gain mechanism in our transistor oscillator, it is important to first establish what
the in-situ absorption images convey about the energy of a flux of matterwaves flowing into thedrain well. When examining in-situ absorption images of a matterwave current flowing though
the transistor, it is important to note that the longitudinal position of the atom flux flowing into
the drain-well is indicative of drain-well ensemble’s energy. That is, the greater total energy thedrain-well atoms have, the farther the longitudinal distance they will flow into the drain-well until
reaching their turning point in the potential (figure 8.12). It then follows that the longitudinal
width of the drain-well ensemble is a measurement of the distribution, or spread, of the energyof the drain-well flux. For example, if every single atom entering the drain-well had the sameenergy, then every single atom would travel to the same longitudinal turning point in the drainwell. However (as in our atom currents shown in sections 8.4.3 and 8.4.4), if the drain-well current
contains atoms with a spread of energy ∆E, then there will be a corresponding spread of the drainwell ensemble’s absorption image that is centered at the location of the longitudinal turning point
corresponding to the average energy in the drain-well flux. (figure 8.12).

Figure 8.12: Plot conveying how one observes the spread of energy ∆E of the drain-well flux.
The longitudinal position of the flux of atoms in the drain-well is indicative of their energy and thelongitudinal spread of the atoms is an indication of the spread ∆E of the atom flux energy.

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Depending on whether the prepared6 source-well atom ensemble was a purely thermal ensemble with temperature T = 78 nK (similar to the methods described in section 8.4.3), or a condensate
with temperature T = 54 nK (as in section 8.4.4), we observed different spreads of energy ∆E in
the drain-well current (see figure 8.13). Specifically, the drain-well flux sourced with the colder,
T = 54 nK source-well ensemble, contains a broader spread of energy as compared to drain-well flux
sourced with the warmer T = 78 nK source-well ensemble. This important result, shown in figure
8.13a, indicates that the gate-well coupling strength for a matterwave current has a dependence
on the source-well ensemble temperature. The correspondence between the increased spread inmatterwave energy flowing into the drain-well as the gate-well coupling strength increases agrees
with the quantum, many-body model of the transistor presented in chapter 7 and thus suggests
the presence of a matterwave gain mechanism in the gate-well. This behavior was also observed for
an atom current flowing through a modified triple-well potential with parameters VSG = 15 kHz,VGD = 16 kHz, and source chemical potential µ = 12 with the results shown in figure 8.13b.
By comparing all of the drain-well ensembles in figure 8.13, it is clear that the drain-wellatom flux sourced with the T = 57 nK ensemble covers a wider longitudinal range in position spacethan the drain-well atom flux sourced with the T = 78 nK ensemble. This result demonstrates
that the source-well atoms that couple into the gate-well, with subsequent flow into the drain, forthe 57 nK case contains a broader range of energy, as compared to the 78 nK case. Not only dothese results agree with the gain mechanism of the model derived in chapter 7, but the results also
demonstrate a canonically conjugate relationship between the total position space wavefunction
Ψ(x) and momentum space wavefunction Φ(p) in the transistor gate-well. As we now explain, thisconjugate relationship is also consistent with the existence of a matterwave gain mechanism in
the gate well and cannot be explained using the semi-classical models from [33][5][34], but rather
can only be interpreted by describing the flow of the atom currents with a quantum mechanical,
matterwave interference, description.
6

Recall from section 8.4.2, the atom current is flowing though the triple-well transistor potential with SG and
GD barrier heights of VSG = 30 kHz and VGD = 31 kHz and with a source-well chemical potential µ = 25 kHz (figure
8.8).

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Figure 8.13: a) in-situ absorption images showing the flow of ultracold atoms through the transistor
potential with VSG = 30 kHz, VGD = 31 kHz, and source well chemical potential µ = 25 kHz after
25 ms of flow time. The images show a comparison of the atom current for cases of a 78 nK and
57 nK source-well ensemble. From the images, it is immediately apparent that a wider range ofmatterwave energy flows into the drain-well when the current is sourced with the 57 nK BEC,
rather than 78 nK thermal source. This important result indicates that gate-well coupling strength
for a matterwave current increases as the source-well ensemble temperature approaches the 50 nK
energy spacing of the eigenstates of the gate-well (as explained in section 8.1). The correspondence
between the increased spread in matterwave energy flowing into the drain-well as gate-well couplingstrength increases suggests the presence of a matterwave gain mechanism. b) in-situ absorption
images showing the flow of ultracold atoms through the transistor potential with VSG = 15 kHz,VGD = 16 kHz, and source well chemical potential µ = 12 kHz. In the 78 nK images, the atom
current has flowed for 22.5 ms, 25 ms, and 27.5 ms respectively (from top to bottom). The 57 nK
image shows the atom current after 25 ms of flow time. Similar to figure a), we still observe a wider
range of matterwave energy flows into the drain-well when the current is sourced with the 57 nK
BEC, rather than 78 nK thermal source.

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8.5.2

Additional Evidence of Gain Mechanism: Observing Canonically Conjugate
Relationship Between Gate-well Position and Momentum Wavefunctions

While the model formulated in chapter 7 is derived considering a single matterwave flowing
through the transistor potential, the experimental results shown in sections 8.4.3, 8.4.4, and 8.5.1
are for a source-well containing an ensemble of many atoms. Thus, we aim to describe the observed
matterwave current dynamics by considering a superposition of many source-well matterwaves,spanning a range of energies, just as in the above experiments. As described in section 7.6.2,
the gain mechanism permits only the symmetric (or antisymmetric) mode ψ+ (or ψ− ) to couple
into the gate-well with the mode of opposite parity, which in this case is the antisymmetric (orsymmetric) mode ψ− (or ψ− ), always reflected. By summing7 all of the symmetric modes ψ+ (orantisymmetric modes ψ− ) that couple into the gate-well over a broad range of initial energies, theresulting envelope of the matterwave superposition (figure 8.14a) gives a sinc function:
Ψ(x) = sinc(x).

(8.4)

Because position and momentum space wavefunctions are conjugates of each other, we can immediately surmise the total momentum space wavefunction Φ(p) that couples into the gate-well(figure 8.14b) by performing the Fourier transform of equation 8.4. That is, the envelope of thetotal momentum space wavefunction Φ(p) coupling into the gate-well is a “box” function
0 p ≤ pminΦ(p) =
P pmin ≤ p ≤ pmax





 0 p ≥ pmax

(8.5)

where P is constant and the width of Φ(p) is equal to the range of momenta ∆p = pmax − pmin of
the matterwaves that couple into the gate-well. Because Ψ(x) and Φ(p) are canonically conjugate
parameters, as the gate-well coupling increases, the Ψ(x) = sinc(x) position wavefunction becomes
7
In this summation, we assume that all symmetric modes contribute with equal amplitudes. Note that a symmetricmodes ψ+ with different initial energies will have a different phase φξ relative to each other in order to remain coupled
into the gate-well. This occurs because the phase condition (equation 7.38), unique to each matterwave energy must
always remain satisfied for each symmetric mode ψ+ to couple to the gate-well, otherwise the wave would have beenreflected. Thus the superposition in the gate-well is a summation of symmetric modes ψ+ with different phases.

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tighter while the width of the Φ(p) box momentum wavefunction becomes wider. Therefore, due
to the gain mechanism, as more source-well atoms couple into the gate-well, the momentum spreadof the resulting superposition increases. As the gate-well superposition subsequently flows into
the drain-well, the observed atom flux will be distributed over a wider longitudinal position in
the drain-well, since the matterwave flux contains a wider distribution of momentum (and hence
energy).
This exact behavior was observed8 in our experiments: as the temperature of the sourcewell ensemble becomes closer to the desired T = 50 nK (where coupling is max), we observe an
increase in the spread of energy (and hence momentum) in the drain-well flux (figure 8.13). As
described in section 8.5.1, this indicates that coupling to the gate-well increases as the source-wellensemble temperature decreases from 78 nK to 54 nK, which is consistent with the existence of thematterwave gain mechanism derived in chapter 7. Our observations in figure 8.13 would not occur
8

The absorption images, such as those shown in figures 8.9 and 8.11, are also measurements of the atom current
probability amplitudes at all longitudinal positions of the transistor oscillator.

Figure 8.14: a) Envelope of the total position space wavefunction, Ψ(x), of the gate-well, formedby summing symmetric modes ψ+ over a broad range of initial energies that couple into the gatewell. b) Envelope of the corresponding momentum space wavefunction, Φ(p). Because momentum
and position are canonically conjugate variables, by taking the Fourier transform of the gatewell position space wavefunction Ψ(x), we immediately arrive at the gate-well momentum space
wavefunction Φ(p). This result shows the effects of the gain mechanism: as more atoms couple into
the gate-well, the corresponding momentum spread in Φ(p) also increases. This is directly observedin figure 8.13, as the spread in momenta (and energy) of the drain-well flux increases as source-welltemperature decreases and gate-well coupling increases.

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without this gain mechanism, as there cannot be matterwaves possessing a continuous spread ofenergy that couple into the gate-well without it (shown in figure 7.10). Consequently, if there were
no gain-mechanism, one would not observe an increase in matterwave momentum (and energy)
flowing into the drain-well as source-well temperature decreases.
With the presence of a matterwave gain mechanism experimentally verified, we conclude thischapter by describing how the ultracold atom current results shown in sections 8.4.3, 8.4.4, 8.5.1,
and 8.5.2 suggest that the matterwave gain mechanism is also coherent.

8.5.3

Coherence in the Matterwave Gain Mechanism

The model presented in chapter 7 suggests that coherence exists between matterwaves coupling into the gate-well and flowing into the drain-well. This coherence occurs due to the following:• In order for source-well matterwaves to couple into the gate-well and flow into the drain,each matterwave must satisfy gate-well resonant boundary conditions.
• The gain mechanism (i.e. the interaction potential given in equation 7.30) transmits only
one gate-well normal mode: either the symmetric ψ+ or antisymmetric ψ− mode. The
mode of opposite parity is always reflected.
• For matterwaves ψ+ (or ψ− ) to match gate-well boundary conditions and couple to thegate-well, the precise phase condition (equation 7.38) must always be satisfied, otherwise
the wave is reflected from the gate.
• Axiomatically, the relative phase difference between each normal mode ψ+ (or ψ− ) that
couples into the gate-well cannot change and remains constant (if it does change then themode will not couple to the gate, is reflected, and is not part of the gate-well superposition).
Therefore, the superposition that couples into the gate-well features a sum of purely symmetric (orantisymmetric) modes ψ+ (or ψ− ) whose phases relative to each other do not change; giving rise
to coherence between each matterwave in the gate-well superposition.

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Furthermore, the results of the matterwave current from sections 8.4.3 and 8.4.4 show evidence suggesting the coherence between matterwaves is preserved as the superposition flows into
the drain-well. Shown in figure 8.15a, in-situ absorption images demonstrate that a drain-well atom
flux only appears at the longitudinal turning point9 of the drain-well. Nearly the entire remaining
regions of the drain-well contain no atom current. This result is attributed to:• Destructive interference occurring within the matterwave superposition at longitudinal positions between the gate-drain barrier and the drain-well turning point.
• The matterwaves of different frequencies in the drain-well interfering constructively at thedrain-well turning point to form a localized atom flux.
The results seen in the drain-well current (summarized in figure 8.15) and behaviors described in
the above bullet ponts suggests that coherence exists between the drain-well matterwaves.
We can verify that coherence is in fact imparted by the gain mechanism by intentionallymaking the matterwave current incoherent by removing the gate-drain barrier from the triple-welltransistor potential (thus only having the source-well and drain-well). Removal of the gate-welleliminates the matterwave gain mechanism. In this case, the matterwave superposition flowing
into the drain-well contains many matterwaves with different phases that are not fixed, and thus
the current is incoherent. In-situ absorption images of the atom current in this regime shows atoms
appearing in all locations throughout the entire drain-well (figure 8.15b) and shows that destructive
interference is not taking place without the presence of the matterwave gain mechanism, verifyingthat the atom current here is incoherent. Additionally, previous semi-classical transistor experiments described in references [153] examine a considerably warmer atom current at temperature
T = 850 nK flowing through the gate-well. In this case, even with the inclusion of the gate-well,no matterwave interference effects are observed in the drain-well and a very large population of
atoms is found at all longitudinal locations in the drain. This is attributed to the current being
9
Although a small population of atoms appears in between the GD barrier and the drain well at ∆t=20 ms infigure 8.9.

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Figure 8.15: in-situ absorption images after 20 ms of flow time showing the observed difference ofa drain-well matterwave current of coherent and incoherent matterwaves. The white dashed lines
show the locations of the SG and GD barriers. Top: With the gate-well and the gain mechanism
present, a drain-well atom flux only appears at the longitudinal turning point of the drain-well,with no drain-well atoms located at longitudinal positions from the GD barrier to the turning
point. This is attributed to destructive interference occurs within the matterwave superposition
at longitudinal positions between the GD barrier and the drain-well turning point. Moreover, thematterwaves of different frequencies interfere constructively at the drain-well turning point to form
a localized atom flux. This behavior suggests that coherence exists between drain-well matterwaves.
We can test that the interference effects are due to the gain mechanism by removing it entirely.
Bottom: The gate-well is removed, which elimates the gain mechanism. With a now incoherent
matterwave current, no such interferences occur and atoms appear at all longitudinal locations of
the drain-well. From this set of experiments, we conclude that the interaction potential (equation
7.30) establishes a coherent gain mechanism in the atomtronic transistor.

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far too hot10 for the gain mechanism to take place. Thus, the matterwaves in this case are also
incoherent. We conclude that coherence between matterwaves in the drain-well occurs only when
the gate-well is present and the atom current is cold enough for the gain mechanism (equation 7.30)
to materialize. Finally, the empirical observation of the canonically conjugate relationship between
the envelopes of the gate-well position space wavefunction Ψ(x) and the drain-well momentum
space wavefunction Φ(p) is also suggestive that the matterwaves flowing into the drain-well havecoherence. Without coherence, this canonically conjugate relationship would not be observed in
the transistor oscillator results.
From the set of experiments described in this chapter (summarized in figure 8.13 and figure 8.15), we conclude that the interaction potential (equation 7.30) establishes a coherent gain
mechanism in the atomtronic transistor.

10

Recall from section 8.1 that for the matterwave gain mechanism to be relevant, the atom current must havea temperature on the order of the spacing between adjacent gate-well eigenstates, which for this experiment isapproximately 50 nK.

Chapter 9

Conclusion to Part III

9.1

Summary
In part III of this dissertation, I described a theory and experiment in the field of atomtron-

ics that displays an ultracold gain mechanism for a matterwave transistor oscillator. The theory
presented in chapter 7 starts with a well-established semi-classical model of an atomtronic transistor but models the system with a purely quantum mechanical formalism. The quantum model
predicts dynamics different from the semi-classical results only when the atoms flowing throughthe transistor oscillator have sufficiently low enough temperatures such that the motional state of adipole oscillating BEC, placed in the transistor itself, couples atom transitions between high lying
transistor energy eigenstates. In this ultracold temperature regime, interesting physics that were
not observed in the semi-classical theory or experiments occurs, specifically the manifestation ofa coherent gain mechanism that broadens the range of energy (and by extension, momenta) of amatterwave current flowing from the source-well, through the gate-well, and into the drain-well.
In chapter 8, I reported on the current experiment aimed at verifying the quantum transistor
oscillator model presented in chapter 7. In this experiment, an ultracold atom ensemble is loaded
into the source-well at a temperatures of 57 nK and 78 nK. By raising the chemical potential of
the source-well, I showed that an atom current flows throughout the transistor by first collecting in
the gate-well followed by flowing into the drain. Time-of-flight temperature measurements of thetransistor source-well ensemble confirm that as an atomic current flows through the transistor, thecurrent constantly carries energy away from the source-well ensemble, which provides continuous

190
cooling to the source-well atoms. Interestingly, our experimental results showed that a wider energy spread in the atom flux flowing into the drain-well is detected when the source-well ensemble
contains a condensate at cooler temperature of T = 57 nK, compared to when the source-well ensemble is a purely thermal collection of atoms at a temperature of T = 78 nK. The correspondence
between the increased spread in matterwave energy flowing into the drain-well as gate-well couplingstrength increases (as source-well temperature approaches the 50 nK energy spacing between adjacent gate-well eigenstates) agrees with the quantum, many-body model of the transistor presented
in chapter 7 and thus suggests the presence of a matterwave gain mechanism in the gate-well.
Additionally, this result confirms a canonically conjugate relationship between the spatial
wavefunction of the gate-well and the momentum wavefunction of the drain-well, which suggests
coherence in the transistor gain mechanism and the drain-well atom current. Evidence of a coherent
drain-well atom current can also be observed in the complete absence of an atom flux at all positionsin the drain-well with the exception of an atom flux located at the longitudinal turning point. We
conclude that this observation is due to a superposition of coherent matterwaves where destructive
interference occurs throughout the entire drain-well except at the longitudinal turning point, where
the matterwave superposition constructively interferes. From the set of experiments described inchapter 8 (summarized in figure 8.13 and figure 8.15), we conclude that the interaction potential(equation 7.30) establishes a coherent gain mechanism in the atomtronic transistor.
The results given in chapter 8 help to verify principles of the quantum model derived inchapter 7. Our experiment suggests the presence of a coherent, matterwave gain mechanism in
the transistor gate-well. However, further experimentation must be carried out in order to draw acomplete conclusion. Specifically, the drain-well atom flux must be measured for many more initialconditions of the source-well. It will be interesting to observe the position-momentum conjugate
relationship as the source-well ensemble is cooled to lower and lower temperatures. If the model
presented in chapter 7 is correct, absorption images of the drain-well flux will show an atom ensemble
that continuously broadens along the longitudinal direction until the atom current is so cold that it
no longer couples to the gate-well. Additionally, there are still many more experiments that pique

191
my interest, with differing degrees of veracity1 , to investigate the quantum model further and toverify some of it’s very fine details.

9.2

Future Work and Experiments
Here, I briefly outline two interesting routes the experiments in chapter 8 could take in orderto further verify more principles of our matterwave transistor oscillator model. I first describe
measuring the differential gain of the transistor followed by a “possible” method for explicitly
detecting the transmission of the gate-well normal modes ψ+ or ψ− (equations 7.37a and 7.37b)
into the drain-well.

9.2.1

Measuring Differential Matterwave Gain

In order to measure the differential matterwave gain due to the atom-BEC interaction (equation 7.30) we must measure the change in atom flux flowing into the drain-well as a function of thechange of occupancy of the gate-well BEC. If we are working in an experimental regime where amatterwave current only flows via tunneling through the SG and GD barriers, this measurement
could provide a more robust method of quantifying matterwave gain, as it removes any impact ofstray source-well atoms that may randomly enter the gate and drain-wells (as noise) by classicallytraversing the SG and GD barriers. This is because we would be measuring the changes in drainwell flux as a function of the change in gate-well occupancy, rather than strictly2 measuring thedrain-well flux as a function of gate-well atom number. This method would essentially “ignore”
noise in the atom flux.
A challenge of measuring differential gain is changing the occupancy of the gate-well BEC
without altering the gate-well potential. From the semiclassical models [5] and experiments [34],1
These experiments may be very hard to do. I am not suggesting they are even possible in the current iterationof this experiment, but rather they are brief overviews (or brainstormings, for a lack of a better term) of experiments
I’ve been thinking about for some distant future iteration of the experiment.
2
In this case, any stray atoms that enter each well by classically traversing each barrier will pollute the measurement with noise. If we instead measure change in atom flux flowing into the drain-well as a function of the change ofoccupancy of the gate-well BEC, we can get a measurement of the gain of the transistor that may be less influenced
by noise from atoms that do not flow via tunneling.

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the size of the gate-well BEC is controlled by the ratio of SG and GD barrier heights, VSG and VGD .
While we can readily control the BEC occupancy by adjusting VSG and VGD , the quantum modelshows (section 7.4.2, figures 7.5 and 7.6) that doing so changes the degeneracy (and subsequent
coupling) between the dipole oscillating BEC and the highest lying pair of gate-well eigenstates.
This degeneracy and coupling, being an essential precondition to matterwave gain, cannot bealtered. This renders the methods developed in references [5] and [34] ineffective for increasing(and decreasing) the size of the gate-well BEC.
Instead, we can adjust the flux of source-well atoms that enter the gate-well by increasingor decreasing the source-well chemical potential, µs . Extending the semiclassical formalism fromreference [5], an atom current flowing from the source-well to the gate-well ISG is proportional to
the fugacity factor
ISG ∝ exp

 µ 
s

kT s

(9.1)

where k is the Boltzmann constant, Ts is the source-well temperature, and µs is the source-wellchemical potential. Since the source-well chemical potential can be freely adjusted (section 8.4.2),
the parameter µs becomes a tunable characteristic of the transistor without impacting the SG and
GD barrier heights or the gate-well eigenstates. From this, we have an experimentally adjustable“knob” that we may be able to use to control the amount of atoms that enter the gate-well, allowing
us to measure the transistor differential gain.

9.2.2

Observing Transmission of Gate-Well Normal Modes

Explicitly measuring the transmission of the gate-well symmetric or antisymmetric modes
ψ+ (x) and ψ− (x) (equations 7.37a and 7.37b) “may” be accomplished by performing a HermiteGaussian transform of the drain-well atom flux in-situ absorption images. A Hermite-Gaussiantransform is a mathematical operation that would decompose the drain-well probability amplitude
signal (which note is just the drain-well absorption image), such as in figure 8.9) which is a functionof position, into the Hermite-Gaussian functions that make it up. That is, the Hermite-Gaussiantransform could decompose a drain-well absorption image signal into the harmonic oscillator func-

193
tions (and thus gate-well modes) that make it up. To further investigate this idea, we define
the Hermite-Gaussian transform: Given some Hermite function Hm (x) of order m, the HermiteGaussian transform H {Hm (x)} is defined as
Z ∞H {Hm (x)} =2

e−x Hn (x)Hm (x)dx =√π2n n!δ(n − m).

(9.2)

−∞

Realize that this operation is completely analogous to the Fourier transform which decomposes afunction of time into the frequencies that make it up (figure 9.1). To give an example comparing
the Fourier and Hermite-Gaussian transforms, consider some sine wave function, such as f (x) =sin(2x) + sin(5x). The Fourier transform of f (x) is expressed in the frequency basis and wouldfeature delta functions F (ω) = δ(ω ± 2) + δ(ω ± 5) corresponding to the frequencies ω = 2 andω = 5 present in the original function f (x). Now, given some harmonic oscillator wavefunction,such as ψ(x) = ψ2 (x) + ψ5 (x), the Hermite-Gaussian transform of ψ(x) decomposes the function
to the harmonic oscillator basis and would feature delta functions H(n) = δ(n − 2) + δ(n − 5)corresponding to the Hermite-Gaussian modes n = 2 and n = 5 present in ψ(x).
Next, in the transistor-oscillator experiment in chapter 8, the observable signal (i.e. the drainwell in-situ absorption image) is comprised of the square of Hermite-Gaussian functions (since theabsorption image really represents the matterwave probability amplitude squared). Thus, we cannot
take the Hermite-Gaussian transform of the drain-well wavefunction, but rather we must take thetransform of the square of the wavefunction.
The corresponding Hermite-Gaussian transform for this signal can be found by taking thedefinition of the Hermite-Gaussian transform (equation 9.2) and applying the convolution theorem
to arrive at
H {Hm (x)Hl (x)} = H {Hm (x)} ∗ H {Hl (x)} .

(9.3)

It then follows that the Hermite-Gaussian transform of a product of Hermite-Gaussian functions is

H {Hm (x)Hl (x)} =√ √ l 
π2n n!
π2 l! [δ(n − m) ∗ δ(n − l)]
(9.4)

= π2

n+p

n!l! [δ(n − m − l))]

194

Figure 9.1: Plot a) is some function f (x) composed of sine waves with various frequencies. The
plot in c) is the Fourier transform showing what frequencies are present in f (x). Plot b) is afunction ψ(x) that is comprised of a linear combination of harmonic oscillator functions (equation
7.4). The plot in d) is the Hermite-Gaussian transform of ψ(x) which shows what harmonic modes
are present in the function ψ(x).

195
where the “∗” operation denotes the convolution. We must now apply this formalism to our casein order to determine the expected Hermite-Gaussian transform of the drain-well signal. Given thesymmetric mode that transmits into the drain-well is
1
ψ+ (x) = √ [ψ22 (x) + ψ23 (x)]
2

(9.5)

the observable signal will be∗|ψ+(x)ψ+ (x)| =
1|ψ22 (x)|2 + |ψ23 (x)|2 + 2|ψ22 (x)ψ23 (x)| .
2

(9.6)

Applying equation 9.3 to take the Hermite-Gaussian transform of each product, we find the transform would be proportional to:
1
H |ψ+ (x)|2 = [H {|ψ22 ψ22 (x)|} + H {|ψ23 ψ23 (x)|} + H {|ψ22 ψ23 (x)|}]
2

(9.7)

∝ δ(n − 44) + δ(n − 45) + δ(n − 46)
Therefore, we find that the Hermite-Gaussian transform of the observed symmetric modetransmitted into the drain-well takes the form of three delta functions in the harmonic oscillator
basis:• one at twice the mode number corresponding to the highest lying gate-well eigenstate(corresponding to |3i in the reduced model)• one at twice the the mode number corresponding to the second highest lying gate-welleigenstate (corresponding to |2i in the reduced model)• one at the sum of both gate-well eigenstate modes making up the symmetric normal mode.
A plot showing this result of a Hermite-Gaussian transform applied to a hypothetical transmitted
symmetric mode is shown in figure 9.2. This could be a useful tool to explicitly measure whatmode has transmitted into the transistor gate-well, which also tells us what modes are having
their probability amplified as a result the gate-well gain mechanism. As a generalized extensionof equation 9.6 and 9.7, if the constructed gate-well potential has N bound energy eigenstates,

196
then the Hermite-Gaussian transform of the drain-well signal in the presence of the ultracold gain
mechanism is
H |ψ+ (x)|2 ∝ δ(n − (2N − 2)) + δ(n − (2N − 1) + δ(n − 2N )(9.8)

Finally, recall from equation 7.31 that the atom-BEC interaction responsible for transmitting
symmetric (or antisymmetric) modes into the drain well has a phase dependence ϕ. Consequently,
the phase of the transmitted symmetric mode that enters the drain-well may not be the same foreach successive run of the experiment. This complicates the problem of averaging drain-well images(such as averaging in-situ absorption images to filter random noise) due to the symmetric mode
having a different phase between shot-to-shot. Because the Hermite-Gaussian transform of thedrain-well signal will produce the same result irrespective of the phase of the transmitted symmetricmode, one must perform many data runs of the experiment and average all of the Hermite-Gaussiantransforms of the absorption images rather than the raw absorption image themselves in order tocollect reliable data to verify the presence of a specific symmetric of antisymmetric normal mode
flowing into the drain-well. Such a measurement will very difficult to make, however, if successful,
would be truly fascinating.

197

Figure 9.2: Images a) and b) show a 2D plot of |ψ+ (x)|2 for the symmetric mode ψ+ (x) =√1 [ψ22 (x) + ψ23 (x)] (which realize corresponds to ψ+ = √1 [ψ2 (x) + ψ3 (x)] in the reduced gate2
2
well model, section 7.4.3) located in the transistor drain-well and with two different oscillation
phases. Images c) and d) are the respective probability amplitudes |ψ+ (x)|2 . Finally, image e)is the Hermite-Gaussian transform of |ψ+ (x)|2 , showing that the transmitted symmetric mode iscomprised of Harmonic oscillator modes ψ22 (x) and ψ23 (x), as described in equations 9.7 and 9.8.

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9.3

Outlook: Integrated Atomtronic Transistors
The work in part III of this dissertation demonstrates results that suggest the existence of acoherent matterwave gain mechanism. If such an atom transistor is realized (and perfected), more
complex, integrated atomtronic circuits with multiple transistors could theoretically be developed.
This would permit atomtronic equivalences of logic gates. Theorists have recently proposed an
atomtronic controlled NOT (CNOT) gate that would be implemented with two independently
controlled atom transistors [184]. Since the CNOT gate is a “universal” gate, any other logic gate
can be constructed from sets of CNOT gates. The realization of an atomtronic CNOT gate would
be extremely influential in computing and eventually realizing an “atomtronic computer.”
Finally, the work in this dissertation could act as another route for building a continuous,
coherent matterwave source otherwise known as a “practical atom laser.” While nothing in thisdissertation addresses testing the coherent output (such as performing interferometry experiments),
such tests could certainly be a future milestone that evolves from this work. Within the previous
year, various experiments in AMO physics have provided excellent starting points for a continuous
atom laser, such as continuously cooling a BEC such that it could (in theory) exist for long periodsof time and provide a constant source of coherent atoms [185]. If achieved, the continuous coherent
matterwaves could be used to source integrated matterwave circuits. These integrated circuits
examine the physics of coherent matterwaves launched into waveguides where the matterwave
current can switched, divided, and recombined as the matterwaves flow throughout the system[186]. It would undoubtedly be fascinating if the these circuits could be sourced with a continuous,
coherent matterwave source rather than sourced with a ”pulse” of coherent atoms. The resultspresented in part III of this dissertation can provide an additional method of realizing such apractical atom laser.

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