by rockson chang€¦ · secondary material which had to be cut due to time-constraints....

Exploring matter-wave dynamics with a Bose-Einstein condensate

by

Rockson Chang

A thesis submitted in conformity with the requirementsfor the degree of Doctor of Philosophy

Graduate Department of PhysicsUniversity of Toronto

c© Copyright 2013 by Rockson Chang

Abstract

Exploring matter-wave dynamics with a Bose-Einstein condensate

Rockson Chang

Doctor of Philosophy

Graduate Department of Physics

University of Toronto

2013

Bose-Einstein condensates of dilute gases provide a rich and versatile platform to study both

single-particle and many-body quantum phenomena. This thesis describes several experiments

using a Bose-Einstein condensate of 87Rb as a model system to study novel matter-wave effects

that traditionally arise in vastly different systems, yet are difficult to access. We study the

scattering of a particle from a repulsive potential barrier in the non-asymptotic regime, for

which the collision dynamics are on-going. Using a Bose-Einstein condensate interacting with a

sharp repulsive potential, two distinct transient scattering effects are observed: one due to the

momentary deceleration of particles atop the barrier, and one due to the abrupt discontinuity

in phase written on the wavepacket in position-space, akin to quantum reflection. Both effects

lead to a redistribution of momenta, resulting in a rich interference pattern that may be used

to reconstruct the single-particle wavefunction. In a second experiment, we study the response

of a particle in a periodic potential to an applied force. By abruptly applying an external force

to a Bose-Einstein condensate in a one-dimensional optical lattice, we show that the initial

response of a particle in a periodic potential is in fact characterized by the bare mass, and

only over timescales long compared to that of interband dynamics is the usual effective mass

an appropriate description. This breakdown of the effective mass description on fast timescales

is difficult to observe in traditional solid state systems due to their large bandgaps and fast

timescale of interband dynamics. Both these experiments make use of the condensate’s long

coherence length, and the ability to shape and modulate the external potential on timescales

fast compared to the particle dynamics, allowing for observation of novel matter-wave effects.

ii

Preface

My graduate school experience has taken me on an unexpectedly winding road through

the realm of experimental physics. Eight years is a long time to devote to a single optical

table, even one filled to the brim with optical elements, bursting at the seams with electronics,

and constantly humming with the sounds of vacuum pumps and cooling fans. The time spent

has not always been pleasant. There were many a dark day when the experiment refused to

cooperate and I questioned the direction of my work. But then there were those beautiful

moments when the world of the quantum was made visible. In that regard I am fortunate to

have chosen ultracold atoms - for me, a single absorption image can contain more wonder than

a text book written to describe it.

I am grateful to my thesis supervisor Professor Aephraim Steinberg for supporting me

throughout my degree. His ability to think through a problem and apply his unique under-

standing of quantum mechanics never ceases to amaze me, and I hope that some small fraction

of that skill has rubbed off on me. I have greatly benefited from my interaction with the multi-

tude of undergraduates, fellow graduate students, and postdoctoral fellows who have overlapped

with my time in the lab. The great diversity of interests and expertise represented in our re-

search group - including but not limited to quantum information, condensed matter, ultracold

atoms, and foundational concepts in quantum mechanics - has been a boon to my graduate

career. I owe thanks to a number of my peers who have contributed to the work described in

this thesis. Chris Ellenor and Mirco Siercke built the foundation upon which the work described

in this thesis is based. In particular, much of the apparatus described in Chapter 3, and the

early theory and experimental effort on the momentum interference experiment of Chapter 4. I

spent my early years on the experiment under their guidance, learning the do’s and don’ts1 of

ultracold atoms. Our local electronics wizard Alan Stummer has been invaluable to my work

and my education. He has had a hand in nearly all the electronics developed for this experiment.

Julian Schmidt developed and tested the push coil drivers used in the waveguide and later the

effective mass experiments. Shreyas Potnis worked with me during the past 3 years, which have

been by far the most scientifically productive years of my Ph.D. He has been a constant foil

upon which I can test out and exchange ideas, while sharing duties on the long data runs that

sometimes stretched over multiple days. He is also responsible for the design and construction

of the 2nd generation barrier described in Chapter 5. Alex Hayat’s manic and infectious energy

has been a surprising but welcome addition to my graduate experience. I have undoubtedly

learned a great deal from listening to him and Aephraim argue about physics, semantics, and

the smell of the wavefunction. Furthermore, the exciton-polariton condensation experiments he

has developed in our group has helped me realize that the physics in strange new systems and

regimes may be different, yet still familiar. It has been a pleasure working with Professor J.

E. Sipe and Federico Duque-Gomez on the effective mass experiment described in Chapter 6.

1Do try and spice up your presentations with Labview aids and polar bears. Don’t forget to turn on the watercooling before running 30 Amps of current and dumping 450 Watts through your coils.

iii

They provided the theory which is described at the beginning of that chapter, and appears in

the supplementary material of the associated paper.

The scientific output of this thesis is represented by two papers, currently in review at the

time of submission of this thesis. Their pre-prints may be found online:

• Observation of transient momentum-space interference during scattering of a condensate

from an optical barrier

R. Chang, S. Potnis, C. Ellenor, M. Siercke, A. Hayat, and A. M. Steinberg

arXiv:1303.1137

• Observing the onset of effective mass

R. Chang, S. Potnis, R. Ramos, C. Zhuang, M. Hallaji, A. Hayat, F. Duque-Gomez, J.

E. Sipe and A. M. Steinberg

arXiv:1303.1139

I have found that trying to summarize my many years of work in Toronto, in a single doc-

ument and in a timely manner, has been a challenging task. Perhaps it is my own fastidious

nature, but I feel like I could easily have spent an additional 6 months combing through the

numbers and analyses to ensure accuracy, logical consistency, and style, as well as including

secondary material which had to be cut due to time-constraints. Nonetheless, I hope that this

thesis will be useful to those seeking to develop a stable Bose-Einstein condensation apparatus,

and perhaps interesting to those still amazed by the wonders of quantum mechanics.

Rockson Chang

July 5, 2013

iv

http://arxiv.org/abs/1303.1139

http://arxiv.org/abs/1303.1139

Contents

1 Introduction 1

2 Theoretical concepts 4

2.1 Quantum statistics and Bose-Einstein condensation . . . . . . . . . . . . . . . . . 4

2.2 Weakly interacting Bose gases at zero temperature . . . . . . . . . . . . . . . . . 6

2.2.1 Free-space expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 1D lattice theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Bose-Einstein condensation apparatus 13

3.1 Vacuum system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.2 Cooling and diagnostic laser light . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2.1 Light generation and control . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2.2 MOT beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2.3 Push beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2.4 Optical pumping beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.3 Absorption probe and imaging system . . . . . . . . . . . . . . . . . . . . . . . . 22

3.4 Magnetic potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.4.1 Quadrupole trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.4.2 Bias coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.4.3 Time-orbiting potential trap . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.5 Radio-frequency evaporation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.6 Optical potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.7 Computer control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.8 Ramp to Bose-Einstein condensation . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.9 Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4 Transient interference during scattering 50

4.1 Scattering in the non-asymptotic limit . . . . . . . . . . . . . . . . . . . . . . . . 50

4.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.2.1 Transmitted-incident interference . . . . . . . . . . . . . . . . . . . . . . . 51

4.2.2 Chirped wavepackets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

v

4.2.3 Pushed-incident interference . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.3 Experimental scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.3.1 Optical dipole force barrier . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.3.2 Measuring the momentum distribution . . . . . . . . . . . . . . . . . . . . 59

4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.5 Discrepancy with initial expectations . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.5.1 Simulation of interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.5.2 Imaging simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.6 Phase profile reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5 Waveguide experiments:

Towards measurement of tunneling times of a BEC 75

5.1 Tunneling times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.1.1 A brief overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.1.2 The Buttiker-Landauer time . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.2 Experimental scheme and apparatus . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.2.1 2nd generation barrier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.2.2 Waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.2.3 Magnetic push coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.3 Momentum filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.4 First attempts: discovery of a lattice . . . . . . . . . . . . . . . . . . . . . . . . . 92

5.5 Potential solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.6 Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

6 Effective mass dynamics of a BEC in an optical lattice 106

6.1 Dynamics in a 1D lattice theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

6.1.1 Tilted lattice: modified Bloch states . . . . . . . . . . . . . . . . . . . . . 108

6.1.2 Bloch oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

6.1.3 Effective mass dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

6.2 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6.2.1 Optical lattice implementation . . . . . . . . . . . . . . . . . . . . . . . . 112

6.2.2 Generating an external force . . . . . . . . . . . . . . . . . . . . . . . . . 114

6.2.3 Observation of Bloch oscillations . . . . . . . . . . . . . . . . . . . . . . . 115

6.2.4 Landau-Zener tunneling and scattering effects . . . . . . . . . . . . . . . . 117

6.2.5 Observation of effective mass dynamics . . . . . . . . . . . . . . . . . . . 121

6.2.6 Estimating experimental parameters . . . . . . . . . . . . . . . . . . . . . 122

6.3 Analysis of data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

vi

7 Summary and outlook 129

Bibliography 132

vii

List of Figures

2.1 Scattering and indistinguishability in quantum mechanics . . . . . . . . . . . . . 5

2.2 Classical and quantum particle statistics . . . . . . . . . . . . . . . . . . . . . . . 5

2.3 Scattering from a periodic potential . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.4 Dispersion relation for a sinusoidal lattice potential . . . . . . . . . . . . . . . . . 10

3.1 Schematic of vacuum system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2 87Rb D2 lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.3 Schematic of laser system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.4 Schematic of injection-lock setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.5 Schematic of doppler-free polarization rotation spectroscopy setup . . . . . . . . 19

3.6 Polarization rotation spectroscopy signals . . . . . . . . . . . . . . . . . . . . . . 20

3.7 Upper MOT layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.8 Lower MOT optics module and layout . . . . . . . . . . . . . . . . . . . . . . . . 23

3.9 Lab temperature from April 28th to May 7th, 2010 . . . . . . . . . . . . . . . . . 24

3.10 Push beam and optical pumping layout . . . . . . . . . . . . . . . . . . . . . . . 24

3.11 Imaging system layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.12 Imaging system calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.13 Absorption image fringe reduction . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.14 Lower chamber magnetic coil layout . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.15 Quadrupole coil mechanical drawing . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.16 Quadrupole coil temperature during experiment operation. . . . . . . . . . . . . 32

3.17 Calculated Feshbach magnetic field and scattering length . . . . . . . . . . . . . 33

3.18 Cross-dipole trap layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.19 Spectroscopy of trap shape by amplitude modulation of trap laser . . . . . . . . 38

3.20 TOP trap ramp to BEC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.21 Hybrid optical and magnetic trap ramp to BEC. . . . . . . . . . . . . . . . . . . 44

3.22 Hybrid potential during transfer and tilt evaporation . . . . . . . . . . . . . . . . 45

3.23 Hybrid ramp results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.24 Bose-Einstein condensation in hybrid trap . . . . . . . . . . . . . . . . . . . . . . 47

3.25 Tilt evaporation in hybrid trap to BEC . . . . . . . . . . . . . . . . . . . . . . . 48

3.26 Condensate fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

viii

4.1 Transient momentum interference during scattering . . . . . . . . . . . . . . . . . 52

4.2 Momentum-space interference for a chirped wavepacket . . . . . . . . . . . . . . 53

4.3 Momentum-space interference with a finite width barrier . . . . . . . . . . . . . . 54

4.4 Illustration of barrier push effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.5 Barrier push effect for typical experimental parameters . . . . . . . . . . . . . . . 56

4.6 Extracting the momentum-space phase profile from interferograms . . . . . . . . 56

4.7 Transient momentum-space interference experimental sequence . . . . . . . . . . 58

4.8 780 nm barrier optical layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.9 780 nm barrier profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.10 Focusing and calibration of 780 nm barrier with atoms . . . . . . . . . . . . . . . 61

4.11 Mis-match between position and momentum distributions . . . . . . . . . . . . . 61

4.12 Effect of imaging system limitations on expected profiles . . . . . . . . . . . . . . 62

4.13 Transient momentum-space interference during a collision: images . . . . . . . . 63

4.14 Transient momentum-space interference during a collision: 1D profiles . . . . . . 63

4.15 Evidence for transmitted-incident momentum-space interference . . . . . . . . . . 64

4.16 Anomalous features in data not seen in simulations . . . . . . . . . . . . . . . . . 66

4.17 Comparison of simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.18 Simulation of imaging of a collided cloud of atoms . . . . . . . . . . . . . . . . . 69

4.19 Off cloud centre imaging 1D cloud profiles . . . . . . . . . . . . . . . . . . . . . . 70

4.20 Extracted phase curvature from off cloud centre imaging simulation . . . . . . . 71

4.21 Comparison of profiles with and without anomalous features . . . . . . . . . . . . 71

4.22 Extracting the phase profile from experimental data . . . . . . . . . . . . . . . . 73

5.1 Buttiker-Landauer tunneling time . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.2 Buttiker-Landauer tunneling time observables . . . . . . . . . . . . . . . . . . . . 79

5.3 Scheme for studying tunneling with a BEC in a waveguide . . . . . . . . . . . . . 81

5.4 404 nm barrier objective design in OSLO . . . . . . . . . . . . . . . . . . . . . . 83

5.5 404 nm barrier optical layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.6 404 nm barrier beam profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.7 Calibration of 2nd generation barrier . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.8 Calculated 2nd generation barrier transmission function . . . . . . . . . . . . . . 86

5.9 Waveguide design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.10 Push circuit timescales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.11 Transmission spectra for various matter-wave Bragg structures . . . . . . . . . . 91

5.12 Illustration of momentum filtering by far-field shuttering . . . . . . . . . . . . . . 93

5.13 Expansion of a thermal cloud in the waveguide . . . . . . . . . . . . . . . . . . . 94

5.14 Schematic illustration of formation of accidental lattice . . . . . . . . . . . . . . . 95

5.15 Group velocity dispersion for a shallow lattice . . . . . . . . . . . . . . . . . . . . 96

5.16 Structured velocity distributions in lattice due to group velocity dispersion . . . 96

5.17 Angle dependence of bi-modal expansion. . . . . . . . . . . . . . . . . . . . . . . 97

ix

5.18 Angle dependence of diffusely scattered light . . . . . . . . . . . . . . . . . . . . 98

5.19 Observation of matter-wave scattering in the Kapitza-Dirac regime . . . . . . . . 99

5.20 Resonant Bragg scattering from lattice due to diffusely scattered light . . . . . . 100

5.21 Estimating lattice depth from Bragg scattering signal . . . . . . . . . . . . . . . 100

5.22 Phase modulation sideband intensity for even and odd orders . . . . . . . . . . . 102

5.23 Condensate propagation in gravity-compensate waveguide . . . . . . . . . . . . . 103

5.24 Condensate collision with barrier in gravity-compensate waveguide . . . . . . . . 104

6.1 Illustration of effective mass dynamics . . . . . . . . . . . . . . . . . . . . . . . . 107

6.2 Microscopic view of effective mass dynamics . . . . . . . . . . . . . . . . . . . . . 112

6.3 Schematic of optical lattice layout . . . . . . . . . . . . . . . . . . . . . . . . . . 113

6.4 Simulated dependence of effective mass dynamics on force rise time . . . . . . . . 115

6.5 Upgraded push coil current driver . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

6.6 Experimental sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

6.7 Observation of Bloch oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

6.8 Analysis of Bloch oscillation data . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

6.9 Landau-Zener tunneling and inter-particle scattering . . . . . . . . . . . . . . . . 119

6.10 s-wave scattering halos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

6.11 Raw effective mass data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

6.12 Fitted effective mass dynamics data . . . . . . . . . . . . . . . . . . . . . . . . . 122

6.13 Extracted average velocity showing the effective mass dynamics . . . . . . . . . . 122

6.14 Estimate of lattice depth from first-order sideband amplitude . . . . . . . . . . . 123

6.15 Sample fits of average velocity data . . . . . . . . . . . . . . . . . . . . . . . . . . 124

6.16 Timescales of effective mass dynamics . . . . . . . . . . . . . . . . . . . . . . . . 125

6.17 Effective mass and dynamical mass results . . . . . . . . . . . . . . . . . . . . . . 126

6.18 Failure of 4 peak fits for large lattice depths . . . . . . . . . . . . . . . . . . . . . 127

6.19 Failure of 4 peak fits: effect on amplitude of Bloch oscillation . . . . . . . . . . . 128

x

List of Acronyms

1D one-dimensional

3D three-dimensional

AOM acousto-optic modulator

AOD acousto-optic deflector

BEC Bose-Einstein condensate

BS beam-splitter

CCD charged-coupled device

FDM finite-difference method

FET field effect transistor

FPGA field programmable gate array

HWP half-wave plate

LMOT lower magneto-optical trap

MOT magneto-optical trap

NA numerical aperture

OD optical density

ODT optical-dipole trap

PBS polarizing beam splitter

P-I pushed-incident

PI proportional and integral (gain)

PID proportional, integral, and differential (gain)

PM polarization maintaining

QWP quarter-wave plate

RF radio frequency

RGA residual gas analyzer

RMS root-mean-square

ROI region-of-interest

SS split-step

TA tapered amplifier

T-I transmitted-incident

TOF time-of-flight

TOP time-orbiting-potential

UHV ultra-high vacuum

UMOT upper magneto-optical trap

xi

Chapter 1

Introduction

In the early 1980’s, a handful of groups around the world were working on a fringe topic trying

to trap and cool a dilute cloud of neutral atoms to temperatures surpassing that possible by

cryogenics. At first there was a lot that was not known, such as the theory of laser cooling,

the collisional properties of the various candidate atoms, and the optimal cooling strategies.

The technology of narrow-band lasers was just beginning to be realized as a powerful tool for

experimental physics. The field advanced slowly at first but rapidly picked up steam, and

these pioneering efforts led to the development of powerful techniques of laser cooling [1, 2] and

evaporative cooling [3]. In 1995, their work lead to Bose-Einstein condensation of a dilute gas

[4, 5], a phase transition from the classical to the quantum regime. This new state of matter,

called a Bose-Einstein condensate, exhibits macroscopic quantum coherence of such purity never

before observed [6, 7]. In the years following their achievement, the field of ultracold atoms

has absolutely exploded. Today, there are over a hundred groups around the world [8] working

with a wide variety of atoms, both bosonic and fermionic, alkali and even lanthanide.

The level of control demonstrated over these systems has been unprecedented. Neutral

atoms interact with both electric and magnetic fields, allowing for manipulation of their internal

and external degrees of freedom. The dispersive interaction with laser light has allowed for

the creation of near arbitrary time-dependent external potentials [9, 10]. Condensates have

spatial coherence on the 10 µm length scale, and the typical timescales of their dynamics is

as slow as milliseconds. On the other hand, optical potentials can be made with sub-micron

spatial features, limited only by the wavelength of light used to induce them, and can be

modulated on the microsecond timescale. This contrast makes quantum coherent phenomena

readily accessible. Optical potentials have been used for atom traps [11], atomic waveguides

[12], optical lattices [13], ring traps [14], and to create random scattering potentials [15]. In

addition, the weakly interacting nature of ultracold systems of dilute gases have allowed for the

observation of a range of matter-wave phenomena such as condensate interference [6], coherent

scattering from a periodic potential, and Anderson localization[15]. Furthermore, the strength

of inter-particle interactions can also be tuned through the use of Feshbach resonances [16],

allowing for studies in the strongly interacting regime. These tools and techniques make it

1

Chapter 1. Introduction 2

possible to engineer the Hamiltonian, making ultracold atoms one of the most interesting and

flexible systems to work in. For example, atoms in optical lattices have become an excellent

testbed for ideas from solid state physics. Typical semiconductor systems have lattice spacings

d ≈ 5 nm and bandgap frequencies ∆ ≈ 200 THz. In contrast, optical lattices have d ≈ 500

nm, and ∆ ≈ 10 kHz. The vastly different regime of ultracold atoms provides much more

direct access to the system for manipulation and probing of the dynamics. Perhaps more

importantly, optical lattices are essentially defect-free, allowing for clear observation of long

range coherent phenomena such as Bloch oscillation [17] and the associated Wannier-Stark

ladder [18]. The recent demonstration of high numerical aperture imaging of ultracold atoms has

allowed for single-site resolution, realizing a powerful tool capable of simultaneously studying

the microscopic and mesoscopic regimes [19].

This thesis describes my own modest contribution to the field of ultracold atoms. I describe

several experiments conducted at the University of Toronto, using a Bose-Einstein condensate

of 87Rb as a model system to explore matter-wave phenomena that are difficult to access in their

native systems. The long term goal of the Bose-Einstein condensation apparatus is to study

tunneling and tunneling times of massive particles. Tunneling is one of the most well known

and dramatically quantum effects. The question of how long a particle spends in the classically

forbidden region has a long and storied history, complicated by the absence of a quantum

mechanical operator for time. In place of a well defined quantum mechanical observable, a

host of operational definitions have arisen [20, 21, 22]. Many of these timescales have never

actually been measured for massive particles. Ultracold systems, with their low energies and

long coherence lengths, seem like a natural system to explore these timescales. Beyond the

single-particle physics, the many-body regime is ripe for exploration. Despite the advantages

of ultracold atoms, these experiments still pose a significant technical challenge. I will describe

the progress made towards this goal, and the science explored along the way. In particular, this

thesis presents two completed projects.

The first project studied the scattering of a particle from a repulsive potential barrier in the

non-asymptotic regime, for which the collision dynamics are on-going. Scattering is traditionally

described asymptotically, connecting the initial and final states while bypassing the dynamics

during the collision itself. Using a Bose-Einstein condensate colliding with a sharp repulsive

potential, two distinct scattering effects are observed: one due to the momentary deceleration

of particles atop the barrier, and one due to the abrupt discontinuity in phase written on

the wavepacket in position-space, akin to quantum reflection [23]. These effects are transient,

appearing only during the collision and vanishing in the asymptotic limits, and result in a

rich momentum-space interference pattern that may be used to reconstruct the single-particle

wavefunction.

The second project studied the response of a particle in a periodic potential to an applied

force. By abruptly applying a force to a Bose-Einstein condensate in an optical lattice and

observing the resulting motion, we show that the initial response is in fact characterized by

Chapter 1. Introduction 3

the bare mass, and only over timescales long compared to the interband dynamics is the usual

effective mass an appropriate description. This breakdown of the effective mass description on

fast timescales was first predicted in 1954 [24], and is difficult to observe in traditional solid

state systems due to the large bandgap and fast timescale of interband dynamics. This work

provides a more complete picture of the effective mass, a concept widely used in both science

and engineering.

These experiments represent the first science on the Bose-Einstein condensation apparatus,

and were made possible by many years of work in upgrading the apparatus to improve its overall

stability while providing new capabilities. The content of this thesis is divided into chapters as

follows:

• Chapter 2 provides a basic overview of the phenomenon of Bose-Einstein condensation,

and the weakly interacting Bose gas at zero temperature. It concludes with a review of

the quantum treatment of a particle in a periodic potential.

• Chapter 3 describes the apparatus for Bose-Einstein condensation as it currently stands.

Two different approaches to quantum degeneracy are presented. The time-orbiting poten-

tial based ramp was the approach first utilized to reach condensation. Recently, a hybrid

optical and magnetic approach has been implemented that has allowed for the production

of larger, more robust condensates at a higher rate.

• Chapter 4 describes the first experiment on the apparatus - a study of particle scattering

from a sharp repulsive potential. By abruptly removing the barrier, we are able to ob-

serve the dynamics mid-collision and observe the transient scattering phenomena that are

only visible during the scattering event. The applications of this effect to wavefunction

tomography are discussed.

• Chapter 5 describes our progress towards the tunneling experiments, and the challenges

encountered along the way. It presents an in-depth description of the Buttiker-Landauer

semiclassical time, a main candidate for tunneling time measurement, and its potential

implementation with ultracold atoms. The tools developed to perform this experiment

are discussed in detail. During testing, we observed the presence of a weak accidental

optical lattice that prevents further progress on the tunneling experiments as designed,

yet motivated a new direction for the apparatus.

• Chapter 6 describes the effective mass experiment. The theoretical description of a particle

in a titled lattice and the dynamical behaviour of the effective mass is presented. The

implementation with a condensate in an optical lattice, and the results of the experiment

to observe the effective mass dynamics are presented.

• Chapter 7 concludes the thesis, summarizing the scientific contributions, and providing

some final comments on the future directions of the apparatus.

Chapter 2

Theoretical concepts

This chapter provides a basic overview of the concepts of Bose-Einstein condensation and Bose-

Einstein condensates, with a focus on their properties in a harmonic potential. The expansion

of thermal and condensed gases in free-space is discussed, as this is the basis upon which most

measurements are made. The theoretical treatment of a particle in a one-dimensional lattice is

also reviewed, which will be useful for understanding many of the lattice phenomena presented

in Chapter 5, and will form the basis for understanding the effective mass dynamics studied in

Chapter 6.

2.1 Quantum statistics and Bose-Einstein condensation

The quantum mechanical description of particles is dramatically different from the classical

description. Classically, every particle is distinct. Quantum mechanically however, particles

are fundamentally indistinguishable, leading to coherent many-particle interference. A system

of bosons, integer-spin particles, must have a symmetric wavefunction under exchange of any

two particles. A system of fermions, half-integer spin particles, must be anti-symmetric under

particle exchange. This symmetry implies that bosons have increased amplitude for scattering

into the same quantum state while fermions will avoid doing so, and gives rise to Bose-Einstein

and Fermi-Dirac statistics.

A collection of non-interacting bosons in thermal equilibrium is described by the Bose-

Einstein distribution

n(Eν) =1

e(Eν−µ)/kBT − 1, (2.1)

where µ is the chemical potential, kB is Boltzmann’s constant, T is the temperature, and Eν is

the energy of the single-particle state labelled ν. At high temperatures, the chemical potential

is large and negative, and the average occupation of any state is much less than one. In this

limit, the distribution yields the Boltzmann distribution, and particles behave like classical,

distinguishable particles. However at low temperatures, when the particle’s thermal deBroglie

wavelength becomes comparable to the inter-particle spacing, λdb ∼ ρ−1/3, quantum statistics

4

Chapter 2. Theoretical concepts 5

+k'

-k'

-k+k -k+k

+k'

-k'

(a) (b)

Figure 2.1: For indistinguishable particles incident at +k and −k, and scattered to +k′ and−k′, the two scattering events (a) and (b) are indistinguishable, and thus interfere. For bosonsthis interference is constructive leading to enhanced probability for scattering. For fermionsthe interference is destructive leading to suppressed probability for scattering.

kBT>>hω Bosons

T=0FermionsT=0

(a) (b) (c)

Figure 2.2: Illustration of particle distributions in a harmonic oscillator for (a) a thermal state(kBT hω), and (b) bosons and (c) fermions at zero temperature.

become important. Here, ρ is the particle density, and the deBroglie wavelength is

λdb =

√2πh2/mkBT (2.2)

where m is the particle’s mass, and h is the reduced Planck constant. At this point, the single-

particle wavefunctions begin to overlap, and quantum mechanics can no longer be ignored. As

the temperature approaches zero, the chemical potential approaches the ground state energy

and the average occupation of the ground state exceeds unity. This is the phenomenon of

Bose-Einstein condensation.

A Bose-Einstein condensate (BEC) is a macroscopic occupation in the ground state of the

system while in equilibrium. This occurs when the phase space density is D ∼ 1, where the

phase space density is defined as

D = ρλ3db (2.3)

The transition point is commonly expressed as a critical temperature. Consider a three-


dimensional (3D) harmonic oscillator potential

V (x, y, z) =1

2mω2

xx2 +

1

2mω2

yy2 +

1

2mω2

zz2, (2.4)

where ωi is the trap frequency along direction i = x, y, z. For a gas of N particles, the critical

temperature is [25]

kBTC =1

ζ(3)hωN1/3 ≈ 0.94hωN1/3. (2.5)

ω = (ωxωyωz)1/3 is the geometric average of trap frequencies, and ζ(n) is the Riemann Zeta

function. We typically reach the threshold for condensation with ω ∼ 2π·100 Hz andN ∼ 2×106

atoms, yielding a transition temperature of Tc = 570 nK. Below this temperature exists a

mixture of a thermal cloud and condensate, where the condensate fraction is

N0

N= 1−

(T

Tc

)3

. (2.6)

Note that the transition temperature depends on the atom number. In typical cooling schemes,

the atom number is not held constant and the transition temperature usually falls as cooling

proceeds. Thus, the job of a Bose-Einstein condensation apparatus is to cool and compress a

gas as efficiently as possible to create the largest and most pure condensate possible.

2.2 Weakly interacting Bose gases at zero temperature

An ultracold gas is one in which the deBroglie wavelength is larger than the characteristic size

of the particle. Under this condition, the interactions are predominantly s-wave in nature, and

are characterized by a scattering length as. In the ultracold limit, the two-body interactions can

be modelled as a contact interaction of form gδ(~r− ~r′), where g = 4πh2as/m is the interaction

strength. A weakly interacting gas is one in which the inter-particle spacing is large compared to

the length scale for inter-particle interaction, asρ−1/3 1. For typical dilute gas condensates,

the deBroglie wavelength is λdB ∼ 1µm, the density is ρ ∼ 1013 cm−3, and the scattering length

for 87Rb is as = 5 nm, yielding as/λdB ∼ 0.02 1 and asρ−1/3 ∼ 0.01 1. A condensate in the

weakly interacting limit can be accurately described by a meanfield approach. The N -particle

ground state is described by a symmetrized product of single-particle states

Ψ(~r1, ~r2..., ~rN ) =N∏i

φ(~ri), (2.7)

where the φ(~r) are the normalized single particle wavefunctions. The wavefunction for the

condensed state may be represented by

ψ(~r) =1√Nφ(~r). (2.8)


The weak interparticle interactions are treated in a meanfield approach, which averages over

the microscopic interaction between particles. The equilibrium state of the system is described

by the Gross-Pitaevskii equation (GPE)[− h

2∇2

2m+ V (~r) + g|ψ(~r)|2

]ψ(~r) = µψ(~r) (2.9)

where µ is the chemical potential, and V (~r) is the external potential. The GPE has the form of

a non-linear Schroodinger equation for the many-body wavefunction ψ(~r), where the energy per

particle E has been replaced by the chemical potential µ. Thus a particle in the condensate can

be described a single-particle moving within the meanfield potential created by the surrounding

atoms. For the experiments described in this thesis, we work in the low-density limit, where

the meanfield interactions are either negligible or may be considered as a perturbation to the

single-particle behaviour.

The Thomas-Fermi approximation

An approximation to the ground state solution can be obtained by recognizing that the mean-

field energy typically dominates over the kinetic energy. By neglecting the kinetic term in the

GPE, known as the Thomas-Fermi approximation, we can obtain a simple analytic expression

for the ground state. For a harmonic trap, this solution is

ψTF (~r) =µ

g

[1−

∑i

r2i

R2i

]1/2

, for∑i

r2i

R2i

≤ 1 (2.10)

Outside the ellipsoid described by∑

ir2i

R2i

= 1, ψTF = 0. Ri is the the Thomas-Fermi radius in

direction i = x, y, z,

Ri =

√2µ

mω2i

(2.11)

and ωi is the trap frequency in direction i. The wavefunction is normalized such that the

integrated probability density yields the particle number N . Within the Thomas-Fermi ap-

proximation, the density distribution, ρ(~r) = ψ∗(~r)ψ(~r), takes on the form of the potential. For

a harmonic trap, the Thomas-Fermi cloud has a parabolic shape.

2.2.1 Free-space expansion

The most common measurement of cold and ultracold gases is to image the cloud during time-of-

flight (TOF) expansion, after the cloud has been abruptly released from the potential. These

images integrate over the line-of-sight, providing two-dimensional information on the cloud

column density and distribution. The size and rate of expansion is used to infer thermodynamic

properties such as the cloud temperature and density.


Thermal cloud expansion

A thermal cloud is described by the Boltzmann distribution. In a 1D harmonic potential, the

density distribution is

ρ(~r) = ρ0 exp

(−∑i

r2i

2σ2i

)(2.12)

where ρ0 is the peak density, and σi =√kBT/mω2

i is the root-mean-squared (RMS) width.

Upon release from the trap, the cloud shape remains gaussian, but grows in time as

σi(t) = σi(0)√

1 + ω2i t

2 (2.13)

In the limit of t 1/ω, the cloud width grows linearly with a velocity vrms =√kBT/m,

independent of trap frequency. Thus in the far-field, the cloud aspect ratio approaches unity.

By studying the expansion of the cloud, the trap cloud size σi and temperature T can be

extracted.

Condensate expansion

The interparticle interactions in a Bose-Einstein condensate play a dominant role in the expan-

sion of the gas in free-space. Upon release from the confining potential, the meanfield potential

is no longer compensated by the potential energy of the trap, resulting in rapid expansion. A

particle locally feels a force given by the density profile of the atoms around it. For a cloud

initially in the ground state of a harmonic potential, this means a force for a particle at position

x of F (x) = −∇gρ(x) ∝ x. This spatially dependent force leads to self-similar expansion of the

condensate [26], with a Thomas-Fermi radius that grows in time as

Ri(t) = λi(t)Ri(0) (2.14)

where the scaling parameters λi(t) obey a set of coupled differential equations

λi =ω2i (0)

λiλxλyλz− ω2

i λi (2.15)

with initial conditions λi(0) = 1 and λi(0) = 0.

The timescale for interaction driven expansion in direction i = x, y, z is characterized by

the trap frequency ωi. In the far-field limit, t 1/ωi, the meanfield energy has dissipated and

the particles behave like free particles. In contrast to the isotropic expansion of thermal clouds,

condensate expansion is anisotropic, leading to inversion of the aspect-ratio. Although not a

probe of 2nd order coherence (g(2) = 1 is a defining property of the condensate), this inversion

is commonly used as one of the signatures of the condensed state. Confusingly, the inversion

of the aspect ratio is also expected for a non-interacting gas in the ground state of a potential.

However the rate of expansion is significantly higher for interacting clouds as the interactions


(a) (b)

Figure 2.3: Scattering from a periodic structure for (a) a classical particle, and (b) a quantumparticle.

dominate over the transform limited momentum width.

2.3 1D lattice theory

The properties of a particle in a periodic potential is markedly different in quantum mechanics

than in classical mechanics. Consider, for example, the two-dimensional (2D) scattering from

a periodic structure illustrated in Fig. 2.3. For a classical particle, the behaviour is determined

entirely by the local potential, and the scattering is insensitive to the long range order. How-

ever, for a quantum particle (a wavepacket) de-localized over a length scale long compared

to the lattice spacing, the long range periodicity influences the subsequent behaviour of the

state. The coherent reflection from the hills and valleys of the periodic potential can interfere

constructively, creating a superposition of classical scattering trajectories. Thus the periodic

structure dramatically modifies the momentum distribution and energy spectrum of a quantum

particle, affecting, for example, its motion and response to external forces.

An atom in a one-dimensional (1D) lattice V (z), with periodicity d and reciprocal lattice

vector k = 2π/d, is described by the Hamiltonian

H0 =h2

2m

d2

dz2+ V (z), (2.16)

This system exhibits discrete translational symmetry and has eigenstates described by Bloch’s

theorem [27]:

ψn(q, z) =1√2πun(q, z)eiqz, (2.17)

where n is the band index, and hq the quasimomentum. These states are de-localized plane

waves modulated by the amplitude function un(q, x), which has the periodicity of the lattice.

The quasimomentum wavevector is a modular quantity with periodicity of the reciprocal lattice

vector, ψn(q, z) = ψn(q + k, z).


0

2

4

6

8

10

12

14

16

s = 0

quasimomentum

−kr

0 +kr

E(q

) [E

r]

0

2

4

6

8

10

12

14

16

s = 4

quasimomentum

−kr

0 +kr

E(q

) [E

r]

0

2

4

6

8

10

12

14

16

s = 8

quasimomentum

−kr

0 +kr

E(q

) [E

r]

Figure 2.4: Example lattice dispersion relation for a sinusoidal lattice potential with depths = 0, 4, and 8.

The lattice potentials used in the experiments described in this thesis are generated by a

pair of counter-propagating laser beams of wavelength λ, and wavevector kr = 2π/λ. This

creates a sinusoidal potential of periodicity d = λ/2, and lattice wavevector k = 2kr. The

recoil wavevector sets a natural momentum scale hkr, and energy scale Er = h2k2r/2m. These

quantities describe a particle’s momentum and energy change due to absorption of a photon,

respectively. The lattice depth is commonly expressed in terms of the dimensionless number

s = V0/Er. We will use these units henceforth in this thesis.

Shown in Fig. 2.4 is the typical dispersion relation for a sinusoidal lattice potential, displayed

for increasing lattice depth and shown in the reduced Brillouin zone (defined from q = −kr to

+kr). The periodicity of the lattice couples free-particle states separated in momentum by

hk, resulting in avoided crossings and the formation of a band structure. The strength of the

coupling is reduced at higher orders, thus the spectrum at high energies approaches that of a

free-particle parabolic dispersion. In the limit of deep lattices, the bands flatten and the band

structure approaches that of a harmonic oscillator energy spectrum.

The Bloch states ψn(q, z) form an ortho-normal basis, and can be used to describe a gener-

alized wavepacket state

Ψ(z, t) =∑n

∫dq cn(q, t)ψn(q, z), (2.18)

where cn(q, t) are time-dependent, complex coefficients. For example, a particle in the ground

band of the lattice (n = 0) is described by the wavepacket

Ψ0(z, t) =

∫dq c0(q, t)ψ0(q, z). (2.19)

In the real momentum basis, a Bloch state decomposes into a superposition of real-momentum


states separated by p = 2hkr. Thus a quantum particle scattering from a lattice potential

couples momentum states in discrete units of the lattice recoil momentum 2hkr, as is illustrated

in Fig 2.3b.

The modification of the dispersion relation by the periodic potential has profound effects on

the dynamics of a particle moving in the lattice. For example, the group velocity of a particle,

which is proportional to the slope of the dispersion relation

vg(n, q) =1

h

d

dqEn(q), (2.20)

oscillates with quasimomentum. When an external force F is applied, the quasimomentum of

a Bloch state increases linearly with time1

hq = F. (2.21)

Note the similarity of the response of a particle in a lattice with quasimomentum hq, to that of

a particle in free space with real momentum p = hk. Thus the motion of a particle in a periodic

potential and under an applied force, is predicted to be periodic in time. This phenomenon is

known as Bloch oscillation [27], and will be discussed further in Chapter 6.

The band theory has proved tremendously useful in understanding and describing the dy-

namics of particles in periodic systems. In particular the concept of the effective mass became

widely used in the study of electronic conduction in semiconductors. Under the external force

F , a particle in band n of a periodic potential will respond as if it had an effective mass m∗n,

distinct from its bare mass m,

〈a〉 =F

m∗n, (2.22)

where 〈a〉 is the particle’s acceleration. The effective mass is inversely related to the curvature

of the band,1

m∗n≡ 1

h2

d2

dq2En(q). (2.23)

The concept of the effective mass, somewhat miraculously, buries the details of the interaction

of the particle with its surrounding potential, allowing for a semi-classical treatment of the

particles dynamics. The effective mass can take on values quite different from the particle’s

bare mass, and can be both positive and negative. For example, one of the main reasons why

Silicon is the basis of semiconductor technology is its small effective mass, resulting in a high

drift velocity.

Note that under the external force, the discrete translational symmetry of H0 is broken,

and the Bloch states ψn(q, z) are no longer the eigenstates of the system. In fact, the force

unavoidably couples Bloch states from different bands, and thus the band index used in Equa-

tions 2.22 and 2.23 does not refer to the bands associated with the un-perturbed Hamiltonian.

1There is some subtlety in showing how this behaviour comes about. This will be discussed in Section 6.1.1.


Clarifying this subtle point will be the starting point of discussion in Chapter 6, leading towards

the theory and experimental observation of effective mass dynamics.

Chapter 3

Bose-Einstein condensation

apparatus

The apparatus for production of Bose-Einstein condensates is a complicated piece of machinery.

From a purely technical perspective, the construction and maintenance require simultaneous

expertise with classical optics, laser systems, ultra-high vacuum (UHV), electronics, and elec-

tromagnetics. A number of graduate students have contributed to the decade long development

of the 87Rb Bose-Einstein condensation apparatus. This work is documented in detail in their

respective theses. Ana Jofre began the project, building the vacuum system and early laser

systems [28]. Mirco Siercke continued the work on the laser systems, and developed the mag-

netic trapping potentials [29]. With the assistance of Chris Ellenor, the first Bose-Einstein

condensate on the apparatus was produced in 2005. Although condensation had been achieved,

the overall stability of the experiment remained an issue. Both shot-to-shot fluctuations and

long term drifts in the condensate atom number were observed. Chris Ellenor led a strong

effort to address these stability issues, in particular rebuilding the laser systems to improve the

frequency, power, and alignment stability [30].

When I became project leader on the BEC apparatus in 2010, some progress had been made

in improving the stability of the apparatus, yet it was still not at the point where scientific ex-

periments could be reliably performed. Firstly, there were significant shot-to-shot fluctuations

in the condensate centre-of-mass momentum upon release from the magnetic trap, making time-

of-flight measurement unreliable. To remedy this issue, an optical dipole trap was developed

to confine the atoms, removing the need to abruptly switch the large magnetic fields, avoiding

the associated instability [31]. Secondly, the apparatus still suffered from long-term condensate

atom number stability issues. This is suspected to be caused by the low overall efficiency of the

evaporative cooling process, placing it just on the borderline of runaway evaporation. Minor

fluctuations in the magneto-optical trap (MOT) size and temperature would have significant

impact upon the size of the final condensate. The ramp was re-designed, making use of a

recently acquired fibre laser system to implement a hybrid optical and magnetic trap scheme.

13

Chapter 3. Bose-Einstein condensation apparatus 14

This hybrid approach resulted in condensates 3 times larger than previously obtained on the

apparatus, and with a shorter overall duty cycle. The science described in Chapters 4, 5, and

6 of this thesis were made possible by these improvements to the apparatus.

The production of a BEC is broken into three steps:

• The first step is collection and pre-cooling of a large number of 87Rb atoms in the upper

MOT (UMOT) from a hot vapour. A high collection efficiency requires a large Rubidium

vapour pressure, incidentally resulting in a poor trap lifetime due to collisions with un-

trapped atoms.

• In the second step, the UMOT atoms are transferred to the lower UHV chamber, and

re-collected in the lower MOT (LMOT). A brief optical molasses phase further cools the

atoms. In net, laser cooling reduces the centre-of-mass motion from room temperature

down to 80 µK. Though an impressive reduction in temperature, laser cooled atoms are

typically still 8 orders of magnitude in phase space density (D) away from condensation.

• The final step is to then transfer the cloud into a near-conservative trapping potential

(TOP trap, hybrid optical and magnetic trap) for evaporative cooling, which selectively

ejects the hottest atoms from the trap, reducing the average temperature of the remaining

atoms after re-thermalization. Despite losing 99.99% of the 3×109 MOT atoms during

evaporation, we finally produce a pure condensate of 3×105 atoms, more than sufficient

for the experiments at hand.

This chapter summarizes the apparatus - firstly providing a detailed technical description of

the components of the apparatus (Sections 3.1 to 3.7), and secondly describing how they fit

together to produce a Bose-Einstein condensate (Section 3.8). Given the vast literature available

describing the now standard techniques of laser cooling, evaporative cooling, trapping, and

imaging (see [3, 25, 32, 33, 34] and references therein), I will forgo discussion of these topics,

opting instead to focus on the technical details and results of our specific implementation.

3.1 Vacuum system

One of the first limiting factors in the preparation of ultracold atoms is the loss of trapped

atoms due to collision with background gases [3]. The room temperature background gases are

travelling at the speed of sound, approximately 3 orders of magnitude faster than the typical

velocity of laser cooled atoms. The collision with these background particles ejects particles

from the trap, limiting the lifetime of the trapped atoms. The lifetime increases dramatically

with the quality of the vacuum. The upper chamber, used for initial atom collection, has a

pressure of 10−9 Torr, providing a collisional lifetime of approximately 1 second [28]. The lower

chamber, used for production and study of Bose-Einstein condensates, has a pressure below

10−11 Torr providing a collisional lifetime of 140 seconds [30].


The vacuum system is illustrated schematically in Fig. 3.1. The system is divided into

two chambers, respectively for the upper and lower MOTs. The UMOT chamber is stainless

steel with quartz windows. There are 6 windows for MOT beams, 1 for the push beam and top

camera access, and 1 for side camera access. The main chamber is connected through 4-way split

to an ion gauge and the ‘Rubidium arm’. The Rubidium arm is a half-nipple chamber where

an ampule of Rubidium (ESPI metals, 99.95% purity) is stored and heated to approximately

40C, flooding the upper chamber with Rubidium vapour. A block valve and feedthrough are

installed to facilitate changing of the ampule without breaking the upper chamber vacuum.

The upper chamber is maintained at a pressure of 10−9 Torr by a Varian Turbo V-70LP pump

with pumping speed 70 L/s, backed by a Varian SD-40 mechanical pump. The LMOT chamber

is stainless steel and feeds into a glass cell through a metal-to-glass interface. The cell is made

of Corning 7980 glass, and is 2×2 cm in cross-section, with 2 mm thick walls. It is pumped

on by a Varian VacIon Plus 150 ion pump with a pumping speed of 150L/s. Built into the

ion pump is a Titanium Sublimation pump with a pumping speed of 500L/s. The Titanium

sublimation pump is typically fired once a month, and the overall UHV system provides lower

MOT pressures below 10−11 Torr. An ion gauge and a residual gas analyzer (SRS RGA 100)

are installed to provide diagnostics on the vacuum.

The upper and lower chambers are connected through a 5” long, 0.5” wide feedthrough.

A pneumatic gate valve (Kurt J. Lesker SG0063-PCCF) with expected lifetime of 105 cycles,

is used to seal the lower chamber and maintain the UHV. Despite the low conductance of the

narrow feedthrough, it is necessary to close the gate valve every experimental cycle, following

the LMOT loading stage. The repeated cycling of the valve led to its failure in late 2006, forcing

us to break UHV and replace the valve. Clearly this is not an ideal setup. Unfortunately it was

decided at the time that the situation could not be improved without a substantial re-design of

the vacuum system. The installed valve is the same model, and as of submission of this thesis,

6 years after installation, it has likely exceeded its expected lifetime.

3.2 Cooling and diagnostic laser light

3.2.1 Light generation and control

We generate narrow-band light for laser cooling, repumping, optical pumping, and imaging.

The relevant 87Rb level structure is shown in Fig. 3.2. The laser system to produce this light is

illustrated schematically in Fig. 3.3. It consists of two groups of lines: those based around the

F = 2→ F ′ transitions (cooling line), and those around the F = 1→ F ′ transitions (repumping

line). Each line begins with a commercial grating-stabilized laser diode (New Focus Vortex I

6013 for the repumping, and Vortex II TLB-6913 for the cooling) with an output power of 40

mW, a mode-hop free tuning range of 140 GHz, and linewidth below 300 kHz.

The cooling line is locked to the F = 2 → F ′ = 3 cycling transition using doppler-free

polarization rotation spectroscopy (see Fig. 3.5 for setup and Fig. 3.6 for typical signals) [36].


1.5” quartz window

Pneumatic gate valve

Varian Turbo V-70LP

4-way connection toUMOT ion gauge and Rb arm

to LMOT chamber

to RGA LMOT ion gauge

11” nipple extension toVarian VacIon Plus 150 ion pump(backside)

to UMOT chamber

2 cm X 2 cmQuartz cell

Upper MOT chamber

Lower MOT chamber

Figure 3.1: Schematic of vacuum system. The dual MOT system is divided into chambersfor the upper MOT at pressure 10−9 Torr, and the lower MOT at pressure below 10−11 Torr.The upper chamber is consists of a steel body with 8 quartz windows. It is connected througha 4-way split to an ion gauge and the Rubidium arm, and pumped on by a Turbo pump. Itis connected to the lower chamber through a pneumatic gate valve. The lower chamber isconnected to an ion gauge and a RGA, and is pumped on by an ion pump. This chamber feedsinto a glass cell which allows for large optical access. The glass cell is at an approximate angleof +18 relative to the optical table in the x−z plane, the body of the lower chamber is alignedto the optical table, and the upper chamber is at -18.


52P3/2

52S1/2

F' = 3

F' = 2

F' = 1

F' = 0

F = 2

F = 1

267 MHz

157 MHz

72 MHz

6.8 GHz

780.24 nm

cool

, ab

sorp

tion

prob

e

repu

mp

optic

al p

ump

Figure 3.2: 87Rb D2 lines. Based on data from reference [35]. Specific hyperfine transitionsused are indicated by coloured arrows (detunings are not shown).

The spectroscopy signal feeds a proportional and integral gain (PI) controller, which in turn

modulates the Vortex lasers. Fast modulation of the laser frequency is provided by control over

the diode current, and slow modulation by controlling the grating piezoelectric voltage. The

full system thus provides a stable frequency light source, locked to an absolute reference. This

light is shifted in frequency by -110 MHz by double passing through acousto-optic modulator 1

(AOM1). AOM1-6 are Neos TeO2 acousto-optic modulators driven by Intraaction drivers with

a centre frequency of 80 MHz. After frequency shifting and spectroscopy, there is insufficient

power remaining for amplification. The remaining power is used to injection-lock a free-running

laser diode (Sanyo DL-7140-201W, 782 ± 2 nm) providing up to 60 mW of power see (Fig. 3.4).

Each free-running laser diode in the setup uses a Wavelength Electronics LFI-4502 200 mA cur-

rent driver, and is temperature stabilized using a home-built copper mount with thermoelectric

cooler controlled by a Wavelength Electronics HTC3000 temperature controller. The cooling

light is shifted by AOM2 such that the net detuning from the F = 2 → F ′ cooling transition

is -15 MHz. AOM1 provides frequency control over the cooling light, while AOM2 provides

power control. The repumper line has a similar setup, and after passing through AOM3, is

slightly off-resonance from the F = 1 → F ′ = 1 transition. Note that the ideal transition to

be repumping on is actually F = 1 → F ′ = 2. Repumping on the F = 1 → F ′ = 1 transition

requires about 3 times as many scattering events to pump an atom out of the F = 1 ground

state, than repumping on the F = 1 → F ′ = 2 transition. Strangely, we have found that we


Spectroscopy lockF=2 F'=3

AOM1 -55 MHzDouble pass

Spectroscopy lockF=1 F'=0

AOM3 +76 MHzSingle pass

AOM2 +95 MHzSingle pass

Tapered amplifier20 dB gain



AOM4 +55 MHzDouble pass Absorption probe

Optical pump

Push beam

Upper MOT

Lower MOTaxial beams

Lower MOTnon-axial beams

Coo

ling

line

Rep

umpi

ng li

ne

HWP

Injection lock

Laser diode

Vortex laser

Optical isolator

AOM

Spectroscopy

Figure 3.3: Schematic of laser system. Shown is the production, frequency locking, manipula-tion, amplification, and distribution of the several beams required for cooling and probing. Allbeams are eventually fibre coupled for delivery to the atoms.


Polarizer

Faraday rotator

HWP

Seed ~ 2 mW

Output ~ 60 mW

Slave laser diode

Optical isolator

Figure 3.4: Schematic of injection-lock setup. A small quantity of light from a master lasersource is fed into a slave laser diode, locking its output to the master. The optical isolatorprotects the slave from feedback due to back reflections that can destabilize the laser.

Rb vapour cell 50/50 BS PBS

PBS

90/10 BS probe

pump

QWP

PD2

PD1

Figure 3.5: Schematic of doppler-free polarization rotation spectroscopy setup. Light is splitinto a weak, linearly polarized probe and a counter-propagating, strong, circularly polarizedpump. The pump induces an optical anisotropy for the probe. The polarization rotation signalis measured by biased photodiodes. PD2-PD1 provides a doppler and background-free signalused for locking.

obtain significantly larger MOTs at the same temperature when tuned to the F = 1→ F ′ = 1

transition. This oddity remains unexplained.

At this point, approximately 25 mW of power remain from the cooling and repumper slave

diodes. These beams are combined on a polarizing beam-splitter (PBS) as orthogonal polar-

izations, and the combined beam is amplified using an Eagleyard tapered amplifier (TA) chip,

producing 1 W of total power. This chip is driven at 2 Amps by a Wavelength electronics

LDTC2/2, and resides in a mount designed and constructed at Laboratoire Charles Fabry de

l’Institut d’Optique [37]. The TA gain is strongly polarization dependent. A half-wave plate

(HWP) is used after beam combination to control the relative power in the output between the

cooling and repumping frequency modes. Typically the repumper is set to 1/10 of the total

power, checked by measuring the TA output spectrum using a Burleigh Fabry-Perot interfer-

ometer (free-spectral range 150 MHz, linewidth 1.5 MHz). The output of the TA thus contains


−100 0 100 200∆ [MHz]

Repumper Spectroscopy (F=1→ F′)

0 0−1 1 0−2 1−2 2

(a)

−300 −200 −100 0 100 200∆ [MHz]

Trap Spectroscopy (F=2→ F′)

32−31−321−2

(b)

Figure 3.6: 87Rb 52S1/2 → 52P3/2 polarization rotation spectroscopy signals. Vertical linesindicate resonance and cross-over features, and the thick vertical line indicates the locking point.(a) F = 1→ F ′ repumper signal. The horizontal arrow indicates the final frequency after AOMfrequency shifting. (b) F = 2 → F ′ trap signal. The F ′ = 1 feature is outside the frequencyscan range shown here.

a mixture of cooling and repumper light, both in the same spatial mode, making for perfect

overlap in the MOT. The power from this beam is divided up for the UMOT, LMOT and push

beam.

The light for the optical pumping and absorption probe beams are derived from a laser

diode, injection-locked by the cooling line Vortex. These beams are used at separate times in

our experiment. To make maximum use of the available power, we use a mechanical switch

to shift the power generated by the diode between these two beams. Each line double passes

through an AOM, and providing frequency and power control. The absorption probe is shifted

up in frequency such that it is back on resonance with the F = 2 → F ′ = 3 transition. The

optical pumping beam is shifted down in frequency such that it addresses the F = 2→ F ′ = 2

transition.

All beams are eventually coupled into polarization maintaining (PM) fibre, providing a

clean single spatial-mode upon delivery to the atoms. More importantly, this setup allows

us to modularize the experiment, separating the light generation stage from the light-matter

interaction stage. This approach has essentially removed the instability due to alignment drifts

which were observed to occur on a timescale as short as hours. However, due to the multi-spatial

mode of the TA output, the fibre coupling efficiency is typically limited to 50%. In addition,

there is a 50% loss due to the fibre splitters. At output of the fibres and after beam shaping,

approximately 80 mW are available for the UMOT, 100 mW for the LMOT, 10 mW for the

push beam, and a few milliwatts for absorption probe and optical pumping. Due to the low

optical power efficiency, this system sacrifices some control over the MOT light to increase the

available optical power, necessary to achieve large atom collection efficiency. For example, the

frequency of the cooling line is shifted by changing the frequency of AOM1. We limit this shift

to ±20 MHz to ensure the subsequent slave laser diode remains injection-locked. Ideally, all

frequency shifting and power control would occur after locking and light generation. The poor


QWP

QWP

QWP

σ-

σ-

σ+

PD

PD

QWP

Mirror

QWP

Mirror

σ-

σ-

σ-

σ-

Axial UMOT beam into page and retro-reflected

σ+- polarization

Axial beam

Non-axial beam 2

Non-axial beam 1

Non-axial beam 1

Non-axial beam 2

(a) (b)

Magnetic quadrupole

coils

Figure 3.7: Schematic of UMOT layout. (a) Free-space splitting and shaping of UMOT beams.The telescopes have magnification of about 30, and the beams are clipped by the 1” apertureof the 2nd telescope lens. This produces fairly flat beam profiles, ideal for laser cooling. Thebeam power is monitored at two points by photodiodes positioned to measure the clipped partof the beams. (b) Setup of retro-reflection MOT. The non-axial beams are incident from below.They pass through the chamber and are retro-reflected by a mirror on the far side. A QWPensures the polarization is unchanged with respect to the wavevector. Not shown is the axialbeam, aligned to the quadrupole coil axis into/out of the page. All polarizations are indicatedwith respect to the direction of beam propagation.

power efficiency is currently one of the remaining limitations of this system. A higher power

tapered amplifier may help relieve this issue, allowing more control over the lasers without

sacrificing the overall stability of the experiment.

3.2.2 MOT beams

The upper MOT light is split into three beams in free space as illustrated in Fig. 3.7 and used

in a retro-reflection scheme. The beams are first expanded up to 1.5” and apertured to 1” to

provide large, flat beam profiles. These beams are directed incident on the MOT cloud through

the UMOT vacuum chamber windows. Their polarizations are set to be circularly polarized

as indicated, and for the quadrupole magnetic field polarity described in Equation 3.4. The

transmitted light is reflected, and rotated in polarization by a QWP such that the reflected light

is again the same polarization of the incident light but with respect to its new direction. This

retro-reflection MOT is technically simpler to implement, requiring significantly less optics.

However, due to the light absorption on the first pass through the atoms, the intensity profile

of the reflected beam will be slightly distorted. This effect can create imbalances in the UMOT

and limit the final temperature reached by doppler cooling. Since the UMOT is just the first

stage of cooling, the retro-reflection scheme is sufficient.

On the other hand, the size and stability of our final condensate depends sensitively on the

lower MOT. The LMOT is formed from 3 pairs of counter-propagating beams. The LMOT


light is split using PM splitters from OZ Optics. A 1:2 splitter is used for the axial beams and a

1:4 splitter for the non-axial beams. To minimize the instability due to drifts in alignment, we

deliver the light as close as possible to the atoms. This is done by using custom built modules

that shape the beam profile and set the polarization (see Fig. 3.8). A Thorlabs FDS100 Silicon

photo-diode is mounted on the end of the module, picking up a small fraction of the light, and

is used to track drifts in power. The full assembly has a length of less than 10 cm, and are

positioned typically within 20 cm of the atoms. This design has proved extremely effective

in minimizing the alignment instabilities, in particular due to temperature fluctuations in the

lab (see Fig. 3.9). Unless installing new optics, we have not found it necessary to adjust the

LMOT beam alignment over a timescale of many months, a welcome improvement after the

daily tweaks required by the original free-space design.

3.2.3 Push beam

The push beam facilitates transfer of atoms from the UMOT to the LMOT. The beam is

focused to a spot size of 25 µm, positioned approximately one cloud width above the UMOT,

and is designed such that the atoms predominantly scatter photons only when in the UMOT.

By providing just enough force to allow a fraction of atoms to escape, we create a continuous

flux of cold atoms from the upper chamber to the lower chamber, which are then recollected in

the LMOT. The LMOT loading rate is optimal when the push beam is tuned in between the

F = 2 → F ′ = 2 and F = 2 → F ′ = 3 transitions [29] with a beam power of about 12 mW.

When everything is well aligned, the lower MOT loading rate is approximately 2×108 atoms

per second.

3.2.4 Optical pumping beam

The optical pumping beam is tuned to the F = 2 → F ′ = 2 transition, circularly polarized

and directed vertically upwards. In the presence of a vertical bias field, this beam drives σ+

transitions, eventually pumping atoms to the |F = 2,mF = +2〉 dark state. This state is weak-

field seeking, ideal for magnetic trapping. During the optical pumping stage, the repumper

light is also present to avoid depumping into the F = 1 manifold.

3.3 Absorption probe and imaging system

All information regarding the state of our cold atoms is gathered from images of the cloud.

Quick qualitative information on the MOT can be obtained from imaging the fluorescence.

This signal is gathered from multiple angles (along x and z for the LMOT), providing full 3D

information on the shape and size of the cloud. Quantitative information, such as atom number,

size, and temperature are inferred from absorption images taken from the main imaging line

along x. The images are taken with a National Instruments DALSA CA-D1 12-bit CCD camera


Axial LMOT beam Into/out of pageσ+- polarization

Magnetic quadrupole coils

σ-

σ-

σ-

σ-

Magnetic bias coils

f = 75 mm doublet

Polarizer

QWP

QWP

PD

780 nm PM fibreNA=0.12

σ+/-

(a)

(b)

x

y

Figure 3.8: (a) Schematic of LMOT optics module. The output of a fibre (NA = 0.12)is collimated using a f = 75 mm lens, creating a gaussian beam of width w0 = 9 mm. Apolarizer and quarter-wave plate are locked at a relative angle of 45 such that the outputlight is circularly polarized. These optical elements can be rotated together relative to theaxis of the PM fibre to provide individual power control over each beam, which is monitoredby a photodiode mounted at the edge of the assembly. (b) Schematic of LMOT layout. TheLMOT modules are positioned within 20 cm of the atoms. The axial beams directed into/outof the page are not shown. All polarizations are indicated with respect to the direction of beampropagation.


0 50 100 150 20022

24

26

28

30

time [hrs]

Tem

pera

ture

[ °C

]

Lab temperature − April 28th

to May 7th, 2010

Figure 3.9: Lab temperature from April 28th to May 7th, 2010. In addition to the 24 hourperiod oscillation, there is a large variation in temperature over several days that is typical forthe spring and fall seasons.

f = 150 mmf = 25 mm

f = 50 mm

QWPUMOT

LMOT

(a) (b)

PDPBS

UMOT top view

y

x

y

Figure 3.10: (a) Push beam layout. The push beam enters through the top window of theupper vacuum chamber and is focused above the UMOT centre. When required, the top windowalso allows access for imaging the UMOT from above providing x−z information on the UMOTshape and size. The push beam is blocked and a 50/50 beamsplitter is slid in place, collectingthe UMOT fluorescence, which is then imaged by a Pulnix TM-7AS CCD camera. (b) Opticalpumping beam layout. The output of the PM fibre is collimated producing a w0 = 6 mm beam,much larger than the LMOT cloud size. Polarization optics ensure a circular polarization atthe output. A small fraction of the light is monitored on a photodiode.


with 255×256 pixels, 16×16 µm in size. This data is acquired by a National Instruments PCI-

1424 frame grabber [38]. In absorption imaging, resonant light is shone on the atoms resulting

in an absorption profile that varies spatially with the optical density profile of the atom cloud.

The remaining light is imaged onto a CCD.

I(y, z) = I0(y, z)e−OD(y,z) (3.1)

where the optical density, OD(y, z) = σ∫ρ(x, y, z) dx, depends on the absorption cross section

σ, and column density along the line of sight. The incident probe intensity, I0 generally has

spatial dependence that must be accounted for to obtain accurate estimates of the optical

density. This is done by additionally taking a reference image in the absence of atoms. By

studying the shape, size, and depth of the imaged shadow, we can infer the number of atoms

that were present [32]. Furthermore, by imaging after time-of-flight (TOF) expansion, we can

infer the temperature and momentum spread of the atom cloud [26].

The size of the cold atom cloud varies quite dramatically over the course of the experimental

cycle (see Tables 3.2 and 3.3). During the MOT stage, the cloud diameter is roughly of 3 mm,

while the final trapped condensate is less than 10 µm in size. Our imaging system thus has

two modes of operation, a zoomed-out and a zoomed-in mode with pixel sizes 50 µm and 3.0

µm, respectively. The absorption probe beam also changes from a diameter of 12.75 mm to

0.77 mm as we switch from out to in. Both imaging modes use a 1:1 telescope to transport the

image of the atoms to a more accessible point 30 cm from the atoms. Zoomed-out mode then

images that field onto the CCD using a commercial objective with net magnification Mout = 3,

resulting in pixel size σout = 50 µm, and a field of view of 12.75×12.75 mm. Zoomed-in mode

uses a single f = 5 cm achromatic doublet lens to image onto the CCD, resulting in Min = 1/5,

pixel size σin = 3.0 µm, and a field of view of 750×750 µm. The f = 7.5 cm, 1” telescope lenses

limit the numerical aperture (NA) of the imaging system to 0.12, giving a diffraction limited

performance of 1.8 µm. Our measured imaging resolution is 3.0 ± 0.1 µm, limited by optical

aberrations. Using the ray tracing software OSLO we have designed a new system which should

allow for diffraction limited performance, however we have yet to implement it.

The imaging system is focused by obtaining a small cloud, either by compression or cooling,

and imaging the cloud for various imaging lens positions. Provided the probe beam is on

resonance, the imaging plane that brings the atom cloud into focus is the one that observes

the smallest cloud. Figure 3.12 shows a typical set of data. Note that based on this data, the

accuracy with which we can find the focal plane is ± 60 µm. The pixel size is calibrated by

tracking the centre-of-mass of a free-falling atom cloud under the influence of gravity.

Absorption signal fringe reduction

The absorption imaging signal is extracted by comparing the absorption image to the reference

image taken without atoms, where the data is respectively encoded into 2D matrices A and R.


(b)

Imaging objective

(a)

1:1 telescopef = 7.5 cm

f = 50 mm

Barrier optical path

Barrier objective f = 50 mm

f = 50 mmimaging lens

1:1 telescopef = 7.5 cm

f = 45 mm

Barrier objective f = 50 mm

f = 200 mm

CCD

CCD

Zoomed-out

Zoomed-in

x

z

x

z

Dichroic mirror T 405, R 780


NA = 0.13

f = 45 mmNA = 0.13

Figure 3.11: Schematic illustration of imaging system and absorption probe. (a) Zoomed-out:Probe φ = 12.75 mm, Mout = 3, and σout = 50 µm. (b) Zoomed in: Probe φ = 0.77 mm, Min =1/5, and σin = 3.0 µm.

9.5 10 10.5 115

5.5

6

6.5

7

7.5

8

8.5

9

Clo

ud R

MS

siz

e [pix

els

]

Imaging lens axial position [mm]

(a) H width

V width

0 5 10 15 20100

110

120

130

140

150

Clo

ud V

centr

e [pix

els

]

Time−of−flight [ms]

(b)

Figure 3.12: (a) Focusing the imaging system by imaging a condensate. This representativedata set gives a focus of 10.24±0.06 µm. (b) Pixel size measurement by tracking the centre-of-mass of a free-falling cloud. Representative data shown in zoomed-out imaging mode, gives apixel size of 50 ± 1 µm/pixel. Plot error bars represent uncertainty in fitting to clouds.


Ideally, the only difference between these images is the shadow cast by an atom cloud. However

if the absorption probe shifts between these images, this will result in a false absorption signal.

The absorption probe unavoidably contains fringes due to interference between the incident

beam and the multiple reflections, notably off the walls of the glass cell. This small scale

structure means that the absorption signal is highly sensitive to the small shifts in the probe

beam which are caused by vibrations in the optical surfaces. The resulting absorption signal

exhibits fringes that can have false optical densities as high as 0.3, severely inhibiting our ability

to reliably observe small signals and fine structure.

This problem is solved by recognizing that the shifts in absorption probe fringes arise due

to mechanical vibrations, and obey an underlying pattern which can be accurately captured by

a finite number of reference images [39]. It is thus possible to construct a linearly-independent

set of basis images Bi, each of which contains unique information on the background fringes.

From this basis, we can generate any possible reference image R =∑

i biBi, where bi is the

amplitude of each basis image. We construct such a basis using Gram-Schmidt decomposition

with a pool of reference images. From this basis set, we then construct an ideal reference image

to match the fringe structure in the absorption images

R =∑i

(A ·Bi) Bi, (3.2)

thus suppressing the fringe noise in the absorption signal. Ideally, the region in A chosen for

this projection should not include any contribution from the atoms. However, in practice, we

have found that as long as the cloud is small compared to the full image, the error incurred is

small. Figure 3.13 shows an example of the image cleaning algorithm at work on a noisy, low

OD image. Using a pool of as few as 8 reference images, we observe dramatic suppression of

the background fringes.

3.4 Magnetic potentials

The interaction between an atom with magnetic dipole moment ~µ and a magnetic field ~B is

described by the Zeeman shift. The linear Zeeman shift, valid in the limit where the field is

weak such that energy shifts are small compared to the hyperfine structure, is

V = −~µ · ~B = gmFµBBz (3.3)

where gF is the Lande g-factor for hyperfine state with total spin F , mF the projection of

the spin along the quantization axis z, and Bz the magnetic field projection along z. For 87Rb

atoms in the |F = 2,mF = 2〉 ground state, this corresponds to a level shift of 1.4 MHz/G. Thus

for reasonably accessible magnetic fields, large energy shifts can be created to manipulate the

internal and external degrees of freedom of the atoms. These fields are created using magnetic

field coils.


Original

(a)

0 200 400 600−0.2

0

0.2

0.4

y [µm]

OD

3 basis images

(b)

0 200 400 600−0.2

0

0.2

0.4

y [µm]O

D

8 basis images

(c)

0 200 400 600−0.2

0

0.2

0.4

y [µm]

OD

Figure 3.13: Absorption signal fringe reduction, images and 1D line profiles. (a) Originalimage. (b) Image using reference based on 3 basis images. (c) Image using reference based on8 basis images.

Coil Quadrupole TOP TOP x-bias y-bias z-bias

Axis z z x x y z

Size [cm] φ8.0 φ5.0 1.85× 8.35 12.6× 12.6 φ16.5 φ11.0

Spacing [cm] 2.6 2.6 2.6 8.2 7.0 5.5

Turns 64 14 20 12 30 12

Wire [AWG] hollow core 16 16 20 20 20

Expected field per amp. 4.15 G/cm 3.88 G 3.1 G 1.37 G 3.54 G 1.90 G

Measured field per amp. 3.87 G/cm 3.92 G 2.78 G 1.49 G 3.71 G 1.87 G

Table 3.1: Parameters for science chamber magnetic coils. Spacing refers to the inner face-to-face coil spacing. Expected field calculation based on [40, 41]. See [30] for details of the fieldcalibration.

A number of magnetic field coils are used in this experiment. The layout of the science

chamber coils are illustrated schematically in Fig. 3.14. The characteristics of these coils are

summarized in Table 3.1. The main set of coils are the large quadrupole coils wound with hollow

core wire, and oriented along z. These coils are used throughout much of the experimental

cycle: during the MOT, quadrupole trap, old time-orbiting potential (TOP) trap, and the

current hybrid trap stages of the experiment. The TOP trap fields are produced by two sets

of Helmholtz coils oriented along x and z. Lastly there are 3 sets of Helmholtz coils, providing

the ability to apply magnetic bias fields along x, y, and z.


x-biasz-bias

y-bias

TOP-x

Quadrupole,TOP-z

z

y

x

z

x

y

z-view x-view

y-viewisometric viewx

z

y

Figure 3.14: Lower chamber magnetic coil layout. Shown are the bias coils (yellow), quadrupolecoils (cyan), and TOP coils (grey). The z-TOP coils are embedded within the quadrupole coils,see Fig. 3.15. All coil dimensions are given in Table 3.1. The isometric view (upper left)additionally shows all mounting brackets (dark grey), UHV cell (blue), and MOT beams (red).


3.4.1 Quadrupole trap

A quadrupole magnetic field is generated by a pair of coils in which the current is running in

opposite directions. The field near the centre of the coils is

~B = B′(

1

2xx+

1

2yy − zz

), (3.4)

where B′ is the field gradient along z. This field creates a potential for the atoms

V (x, y, z) = µB′√x2

4+y2

4+ z2. (3.5)

Near the centre of the trap the potential has linear slope in all directions, forming a cone-shaped

potential for which atoms with magnetic dipoles aligned to the field (weak-field seeking) can be

trapped. Classically, the magnetic dipoles precess around the local orientation of the magnetic

field at the Larmor frequency ωL = µB/h, adiabatically following as they move through the

potential. At the trap centre lies a magnetic field zero, across which there is an abrupt change

in field direction. A particle may pass smoothly from +z to −z on a 3D trajectory that avoids

the zero, with its spin adiabatically following the local field. However a particle that crosses this

zero may have its spin flipped, going from weak-field to strong-field seeking, and subsequently

be ejected from the trap. The rate at which these Majorana spin flips occurs depends on the

size of the trapped cloud, kBT/µB′. At high temperatures, atoms spend little time near the

field zero and the trap lifetime is limited by collisions with background gases. However for lower

temperatures and tight confinement, Majorana spin flips are the main limitation [42].

The lower chamber quadrupole coils are designed to generate large field gradients while being

low in inductance and thermally and mechanically stable. To this end, each coil is composed

of 64 turns of hollow core wire (4 mm × 4 mm square cross-section with a 2 mm diameter

circular core). This wire was manufactured by Wolverine Fabricated Products Inc., covered

with a double polyester glass insulation applied by S.&W. Wire Co [43], and purchased from

Joseph Thywissen’s lab. The hollow-core wire was chosen for its ability sustain large currents

while efficiently dissipating the heat generated by running cooling water through the core.

Figure 3.15 shows the coil design and dimensions. Each coil is composed of an inner section

(2 layers deep) and an outer section (6 layers deep). Between these sections is a z-axis TOP

coil with 20 turns of 16 AWG wire with radius 26.25 mm. This coil has a depth of 14 mm,

with the remaining space occupied by a Delrin spacer with grooves cut to allow for connection

from inner coil to outer coil, and for a temperature sensor (NTC thermistor 0.5 kΩ). The coils

were hand wound and glued together with a generous application of the epoxy Wet-Bond with

TekFlex, able to sustain temperatures up to 93C. After winding, the inside surface of the coil

was flattened by applying epoxy putty J-B Weld to fill in the gaps, making a clean and flat

surface.

The coils in anti-helmholtz configuration produce field gradients B′ = 3.87 G/cm/A. The


26.25 mm

54.50 mm

15.55 mm

40.05 mm

Inside face Outside face

Inner to outer coil connection

Cutaway

Temperature sensor slot

TOP coil slot

Inner coil

Delrin spacer

Outer coil

14.00 mm

Figure 3.15: Quadrupole coil mechanical drawing, highlighting the different sections. Thecoils are constructed in an inner and outer set of layers, separated by a gap. The z-TOP coils,temperature sensor, and mid-coil electrical connection fit into this gap.

coils are connected in series. Including all control electronics, they have a total resistance of 77

mΩ, and an estimated inductance of 340 µH. We use building water (approximately 22C in

temperature) for cooling at a flow rate of 11 mL/s. During our typical experimental cycle the

coils do not rise by more than 8C. The coil is driven by an Agilent 6682A in constant current

mode, which can deliver up to 240 A at 21 V.

Accessing the Feshbach resonance

Feshbach resonances are 2-particle scattering resonances between free particle states and a

molecular bound states [16]. These states may be tuned into resonance using an external

magnetic field. They have found significant use in ultracold atom systems for studies of quantum

chemistry [44], and for tuning inter-particle interactions [45]. In our immediate experiments

it is desirable to be able to tune the interactions to zero in order to minimize the momentum

width of a condensate expanding in an optically-induced waveguide. This momentum width is

dominated by the repulsive interparticle interactions and turns out to be one of main limiting

factors in the planned tunneling experiments (see Chapter 5). 87Rb has a number of fairly

narrow Feshbach resonances [46, 47]. Most of these resonances are prohibitively narrow to use.

The broadest resonance is between atoms in the |F = 1,mF = 1〉 state, and is still technically

challenging, sitting at B = 1007.40 G with a width of ∆B = 0.20 G. To access this field, we

require field control and stability on the 10 mG level.

The lower chamber coils were constructed in two sections to facilitate the generation of

a spatially flat Helmholtz field which may be used to access the Feshbach resonance. Given

the turn distribution and coil spacing, the full coils will have residual field curvature of 0.8

G/mm2/A. Due to the ultra-narrow width of the resonance, this represents a significant increase

in the scattering length to 15% of the field-free value, 200 µm from the centre. However, the

outer sections alone produce a field which only increases to 4% of the field-free value over the

same span (see Fig. 3.17). To allow access to the outer turns only, the insulation on the section


15 15.5 16 16.523

24

25

26

27

28

29

30

Coil

tem

pera

ture

[°C

]

Time [hrs]

Figure 3.16: Measured quadrupole coil temperature during typical TOP ramp operating con-ditions. Each spike in coil temperature coincides with an experimental cycle in which the coilswere driven at 120 A for 27 seconds (see Fig. 3.20). A condensate is produced approximatelyevery 60 seconds here. Upon coil turn-off, the water cooling rapidly cools the coils in prepara-tion for the next cycle. The slow drift on the hour timescale is due to the variation in watercooling temperature, supplied by the building.

of hollow core wire that feeds from the inner section to the outer section has been removed and

a copper sleeve has been fitted on top. This serves as an electrical contact point through which

the current can be controlled through the various sections. To operate in this configuration, a

current of approximately 125 A is expected to be required to generate 1007.40 G.

The quadrupole trap and the Feshbach field are clearly mutually exclusive experimental

tools. The final stage of our condensate preparation transfers to an optical trap, freeing up

the quadrupole coils for use. With the current fully off, a series of low-resistance mechani-

cal switches (Tyco V23132) allows us to change the direction and path of current flow. The

Agilent is run in voltage mode, and the current stabilized by a home-built PID circuit. This

circuit was designed and constructed by Alan Stummer. It uses a stable Thaler VRE3025A

voltage reference, which is compared to a feedback signal provided by a Danfysik Ultrastab

867-200I current transducer which sample the Feshbach current. The full system should allow

for stable currents up to 200 A with a stability of ±1 mA, corresponding to an 8 mG field pre-

cision. For further details, see Alan Stummer’s documentation of this circuit on his website at

http://www.physics.utoronto.ca/~astummer. Although the circuitry has been assembled, it

has yet to be installed in the experiment. The large currents and precision required to reliably

make use of the Feshbach resonance to control inter-particle interactions make this a technically

challenging goal.

Further details on the design, construction, and testing of the quadrupole coils and circuitry

may be found in references [30, 48, 49].

http://www.physics.utoronto.ca/~astummer


−300 −200 −100 0 100 200 3001007.58

1007.6

1007.62

1007.64

1007.66

1007.68

1007.7

Bia

s f

ield

[G

]

z [µ m]

All 8 layers

6 layers

5 layers

4 layers

−300 −200 −100 0 100 200 300−5

0

5

10

15

20

25

30

a/a

BG

[%

]

z [µ m]

All 8 layers

6 layers

5 layers

4 layers

Figure 3.17: Calculated spatial variation in Feshbach magnetic field along z and resultingvariation in scattering length. The scattering length is expressed as a fraction of the field-freescattering length aBG. The field has been set to be 1007.60 G, where the scattering length isas = 0. Due to the spatial variation in the field, the scattering length also varies. Calculationsare for different configurations of the quadrupole coils, only using the outer layer of turns inan effort to reduce the field curvature. The coils have been constructed to allow access to the6 outer layers.

3.4.2 Bias coils

In addition to the main quadrupole coils, pairs of bias coils are installed along all 3 orthogonal

axes. Their dimensions and field gradients are listed in Table 3.1. The vertical bias coil along

y is used to cancel the Earth’s magnetic field (mainly along y in Toronto), which is important

in reaching the lowest temperatures possible in the optical molasses. They are later used to

provide a strong quantization axis during optical pumping in preparation for loading into the

quadrupole trap, and during the loading itself to ensure the cloud centre and trap centre are

matched. The vertical field is driven by a Kepco BOP 36-12M. The x and z coils are driven by

home-built drivers which can deliver up to 4 A of current, but are less commonly used. The z

coils were later co-opted to generate field gradients for exerting forces directed along z on the

ultracold atoms (see Chapters 5 and 6).

3.4.3 Time-orbiting potential trap

At low temperatures, evaporative cooling in the quadrupole magnetic trap becomes inefficient

due to the loss from Majorana spin flips. One way to circumvent this and continue cooling is to

switch to a time-orbiting potential trap [42]. The TOP trap uses two sets of of Helmholtz coils

in the x−z plane to create a rotating bias field, displacing the magnetic zero of the quadrupole

trap in a circular path around the atoms with a frequency ωT . The time-dependent field is

~B =

[1

2B′x+B0 sin(ωT t)

]x+

1

2B′yy +

[−B′z +B0 cos(ωT t)

]z (3.6)

where B0 is the amplitude of the TOP bias field. The field rotation must be slow compared to

the Larmor frequency of the magnetic dipoles in the field ωL = µB0/h ωT , while being fast


compared to the motion of the atoms ωT ωtrap. Under these conditions, the resulting time-

averaged potential observed by the atoms is approximately harmonic with trapping frequencies

ωx =

√µB′

4mB0(3.7)

ωy =

√µB′

2mB0(3.8)

ωz =

√µB′

mB0(3.9)

More importantly, the field at the centre of the trap is now raised to B0. The magnetic field

zero has been shifted-off centre, and sweeps out an elliptical path of size xCOD = B0/B′ and

zCOD = B0/2B′, called the circle-of-death1. Any atom that wanders out to the circle-of-death

may be spin-flipped and ejected from the trap, effectively setting a trap depth of V0 = µB0/4.

The field along z is generated by the round TOP coils embedded in the quadrupole coils (see

Fig. 3.15), and the field along x is generated by the square TOP coils. The placement of these

coils is shown in Fig. 3.14, and the parameters are listed in Table 3.1. These coils are driven by

a home-built system that uses an FPGA to generate two 10 kHz sine waves with a relative π/2-

phase. The FPGA signal is amplified using a variable gain amplifier. A 10 mΩ sense resistor is

used to provide feedback through a PI controller. This circuit is able to deliver up to 20 Ap of

current to the coils, generating bias fields of up to 56 G and 78 G, along x and z respectively.

The FPGA sets the relative phase between channels, and is programmed with a ramp that

allows for smoothly modulation of the amplitude during TOP compression. See Alan Stummer’s

documentation of this circuit on his website at http://www.physics.utoronto.ca/~astummer.

3.5 Radio-frequency evaporation

Forced evaporative cooling is performed in the quadrupole and TOP magnetic traps by applying

a radio-frequency (RF) field introduced by a single turn of AWG 27 wire wrapped around the

cuvette, approximately 2 cm above the centre of the trap. This field couples adjacent spin states

which have been split by the magnetic field. Due to the spatial variation in the magnetic field,

this results in an evaporation surface which defines the trap depth. The RF field is generated

by an Agilent 33250A 80 MHz waveform generator amplified by a Mini circuits ZHL-1-2W

amplifier.

3.6 Optical potentials

The oscillating electric field of a laser ~E can induce an atomic dipole moment ~p = α~E, where

α is the polarizability. This induced dipole moment in turn can interact with the electric field

1Although technically incorrect, creative liberties have been allowed in the naming.



resulting in an interaction potential [9]

V = −1

2〈~p · ~E〉 = −1

2Re (α) | ~E|2. (3.10)

The interaction potential depends on the beam intensity I = 2ε0c|E|2, and the real part of

the complex, frequency dependent polarizability. The latter term characterizes the dispersive

interaction of the light field with the atom. The angular brackets indicate time average over

the rapidly oscillating 2ω terms. The full potential for a two-level atom is

V (~r) = −3πc2

2ω30

(Γ

ω0 − ω+

Γ

ω0 + ω

)I(~r), (3.11)

where ω0 and Γ are the frequency and linewidth of the atomic transition, ω and I(~r) are the

frequency and spatial intensity profile of the light. For multi-level atoms, the contributions

from the multiple transitions sum. This can lead to spin-dependent potentials which can have

interesting applications such as for studying many-body dynamics in a spin-dependent lattice.

In our case, the detunings are typically much larger than the 6.8 GHz groundstate fine structure,

and so the optical potential is spin-independent. The optical dipole interaction can be used

to create both attractive and repulsive potentials by controlling the detuning of the laser field

relative to atomic resonance, ∆ ≡ ω − ω0. In particular, we create potential barriers using

blue-detuned light, and traps using red-detuned light.

While the real part of the polarizability corresponds to a dispersive interaction, the imag-

inary part describes light absorption. The photon scattering rate scales as Γsc ∝ I/∆2. The

scattering rate falls much more rapidly with detuning than the trap depth. Thus by detuning

the addressing laser from atomic resonance and increasing the beam intensity, the scattering

rate can be suppressed while maintaining a desired potential strength. Due to the large 200

nm detunings we use for trapping, photon scattering is usually not considered a limitation. For

example, our trapped condensate is held in a potential of depth of 2.9 µK, and has an expected

photon scattering lifetime of 39 seconds.

The optical dipole potential offers several distinct advantages over magnetic potentials.

The induced potential is proportional to the intensity profile of the beam, and thus a variety

of potentials with optical wavelength sized spatial features can be implemented, limited only

by the ability to shape and control the optics. These potentials can be modulated on the

microsecond time-scale, significantly faster than magnetic fields and without the need to worry

about inductive coupling to surrounding conductors. This ability to rapidly modulate the

potentials opens up greater flexibility in creating time-dependent, or time-averaged potentials

[10]. In this thesis we use the optical dipole interaction to create 3D traps, waveguides, barriers,

and 1D periodic potentials.


Diode laser based optical trap

The first generation of optical traps were based on a pair of 980 nm laser diodes (Thorlabs

L980P300J), each with 300 mW output power. The power of these beams was controlled using

a single pass AOM (Intraaction AOM-402 AF3 modulator with ME-45 modulator driver). Each

diode was devoted to one beam, fibre coupled, and focused on the atoms. After all optics, up to

110 mW is available in each beam. For a single beam, the intensity profile is gaussian, resulting

in the trapping potential

V (r, z) = −V0(z) exp

(− 2r2

w2(z)

). (3.12)

where V0(z) is the trap depth at axial position z

V0(z) =3πc2Γ

2ω30∆

2P

πw2(z), (3.13)

w(z) = w0

√1− (z/zR)2 is the beam size, and zR = πw2

0/λ is the beam Rayleigh range. w0 is the

1/e2 focused beam size. For deep traps, the potential near the minimum is well approximated

by a harmonic potential with transverse and axial frequencies

ωr =

√4V0(0)

mw20

(3.14)

ωz =

√2V0(0)

mz2R

(3.15)

Single beam traps are typically strongly anisotropic, with tight transverse confinement and weak

longitudinal confinement. This makes them ideal for our 1D waveguide experiments described

in Chapter 5. For initial confinement, after cooling in the magnetic trap, we create a 3D trap

by crossing a pair of optical dipole traps (see Fig. 3.18) . Each beam is generated by a separate

diode and focused to about w0 = 12 µm. ODT-z propagates along the z quadrupole coil axis

and acts as our primary beam, providing x, y confinement. ODT-x propagates at a 22 angle

below the x-axis, and is almost certainly not focused on the position of the atoms. This beam

serves to provide z confinement.

To characterize this trap we measure the trap frequencies by modulation of the beam power

[50, 51]. The time-dependent, 1D Hamiltonian is

H =p2

2m+

1

2mω2 (1 + ε(t))x2 (3.16)

where ε(t) = ε0 sin(ωmt) is the fractional harmonic fluctuation in the spring constant with

amplitude ε0 and frequency ωm. When modulated on parametric resonance, ωm = 2ω, the x2


x

z

x

y

Absorption probe, barrier line

Axial MOT beam

ODT-zODT-x

(a) (b)

1:1 telescopefor MOT beam

f = 20cm f = 2.54 cm


f = 15 cmf = 5 cmf = 10 cm

Axial MOT beam

Beam dump

Figure 3.18: Cross-dipole trap layout. ODT-z propagates in the x − z plane, and is directedalong the quadrupole coil axis z. ODT-x is oriented at angle of 22 below the x axis. Thebeams cross at the position of the atoms, resulting in a 3D potential. (a) Top-down view, along+y. (b) Side view along −z.

perturbation drives transitions between vibrational states separated by two modulation quanta,

〈n+ 2|x2|n〉 =h

2mω

√(n+ 1)(n+ 2) (3.17)

〈n− 2|x2|n〉 =h

2mω

√n(n− 1), (3.18)

where n is the vibrational state quantum number. This parametric excitation results in heating

and an exponentially growing energy with modulation time. By varying the frequency of mod-

ulation and measuring this heating, we map out excitation of spectrum of quadrupole modes.

This spectrum exhibits resonances at 2ωr,z. Sub-harmonics should also exist at 2ω/N , where

N is an integer [52]. However these resonances have widths that scale as εN0 ω, and for the small

modulation depths typically used they are not resolved. A sample spectrum and mapping of

ωr versus beam power is shown in Fig. 3.19. Evaluating the trap shape through parametric

excitation serves as an alternative to the more common method of inducing dipole oscillations

and tracking the cloud centre-of-mass motion. One advantage of this approach is the ability to

obtain information of the trap shape along the line-of-sight. This is seen in our spectra, where

the two distinct y, z modes arise due to a slight beam shape ellipticity and the additional ODT-

x confinement. The estimated beam sizes from these measurements are wx = 14.7± 1.1µm and

wy = 12.7± 1.2µm.


2 4 62

3

4

5

6

7x 10

4

Ato

m n

um

be

r

frequency [kHz]

(a)

2 4 626

28

30

32

34

36

38

x−

wid

th [

µm

]

frequency [kHz]

(b)

0 50 100 1500

1

2

3

4

5

fre

qu

en

cy [

kH

z]

Pz [mW]

(c)

Figure 3.19: Spectroscopy of trap shape by amplitude modulation of ODT-z trap laser. (a,b)Sample spectra for Px,z = 110 mW, 2.6% modulation depth, and 500 ms modulation duration.Lines are a 2-gaussian fit to the data. (a) Atom number remaining in trap and (b) x-widthafter modulation. The two observed resonances correspond to the y and z quadrupole modes ofthe trap. (c) Measured parametric resonances (2ω) versus beam power. Lines are a square-rootfit to the data, yielding ODT-z spot-size estimates wx = 14.7± 1.1µm and wx = 12.7± 1.2µm.Uncertainty on atom number is estimated to be 10%. Error bars on centre-frequencies areencompassed by the marker, and are typically less than 10%.

This crossed optical trap potential was used as the final stage of trapping in the TOP trap

based ramp (see Section 3.8). After cooling close to the transition in the TOP trap, we begin to

ramp the quadrupole field down while ramping up the optical power, transferring approximately

50% of the atoms into the optical trap. With the atoms loaded in the optical trap, the magnetic

fields are gently switched off. The ability to avoid abrupt changes in the magnetic fields avoids

the previous limitation of inductive coupling during shut-down which resulted in significant

jitter in the free-expansion velocity [30]. Next we perform forced evaporative cooling in the

optical trap by lowering the trap depth. This type of evaporation is generally inefficient since

the trap frequency also falls with trap power, thus reducing the confinement and lowering the

collision rate making it impossible to reach runaway evaporation [34]. In our setup however,

we do most of the cooling relatively efficiently in the TOP trap and only minimal cooling is

required in the optical trap. In this way, we typically produced fairly pure condensates of 8×104

atoms.

Fibre laser based optical trap

Following the conclusion of the transient momentum interference experiment (Chapter 4), the

980 nm diode lasers were replaced by a 1064 nm fibre laser. This system consists of an NP

Photonics ROCK Fibre Laser seed source (5 kHz linewidth), and a Nufern 15 W fibre amplifier.

This provided significantly more power allowing for greater flexibility in the trap design and

usage. In particular, the increased available power meant we could create larger traps with

weaker transverse confinement, more suited for our waveguide experiments (Chapter 5), and

yet be able to achieve larger trap depths for trapping much hotter atoms. This allowed us to

implement a new hybrid optical magnetic trapping scheme which is technically simpler than

the TOP trap, yet has been shown to be capable of producing significantly larger condensates


[53]. The hybrid trap implementation and results are described in Section 3.8.

The optical component of this trap consists of a single beam optical trap. This single

beam also acts as our waveguide for future experiments. The final design had a focused spot

size w0 = 75 µm. The beam design was firstly motivated by the stringent requirements of

the waveguide experiments, and secondly by the hybrid trap. The details of the design and

experiments performed with it are presented in Chapter 5. The layout is very similar to ODT-z

shown in Fig. 3.18, with appropriately chosen lenses. The optics to realize this required some

modification from that used for the diode based trap, primarily due to the high powers involved.

Waveplates and lenses were purchased from CVI, and high power end-capped fibres from OZ

Optics. Unfortunately, despite their specification, the fibres were damaged in use at 5 Watts of

input power2. The beam was thus manipulated in free-space. This is not ideal since the beam

focus should be positioned 170 µm below the quadrupole trap centre, within 50 µm. Given the

temperature fluctuations in our lab and the path length involved, a drift in the beam position

of this order, is possible over a timescale of several days. Furthermore, the free-space approach

poses more of a laser safety concern than the fibre-based approach.

3.7 Computer control

A series of National Instruments PCI boards interface the experiment to a control computer.

A pair of PCI-6713 boards provide analog output channels with voltage range of ±10 V and

a resolution of 5 mV. Each channel runs through a home-built buffer circuit that protects the

card against electronic feedback, and increases the current output to up to 30 mA, sufficient

to drive 1 V through a 50 Ω load. A PCI-6229 board provides 4 analog output channels, and

32 digital channels. The digital channels are useful for synchronizing the various parts of the

experiment. In total there are 20 analog output and 32 digital channels. The entire system is

clocked by a 80 MHz crystal oscillator divided down to 250 kHz, giving a timing resolution of

4 µs. The clock electronics are built into the TOP driver circuitry.

3.8 Ramp to Bose-Einstein condensation

The previous sections detailed the various elements of the apparatus separately. This section

describes the results of putting these elements together to create a ramp for production of

Bose-Einstein condensates. The original ramp was based on the TOP trap. The momentum

interference experiment (Chapter 4) and accidental lattice experiments (Chapter 5) used this

ramp. In the Spring of 2012 a new ramp based on a hybrid magnetic and optical trap was

implemented. The waveguide experiments (end of Chapter 5) and the effective mass experiment

2Interestingly there seems to be a non-linear dependence on how the uncoupled light was dissipated with fibrecoupling efficiency. At efficiencies of 50%, the fibre collimation package was noticeably warm to the touch forinput powers as low as 1 W. At coupling efficiency of 70% up to 5 W were coupled before noticing a rise inhousing temperature.


(Chapter 6) used the hybrid trap ramp.

TOP trap ramp

To reach quantum degeneracy, the original ramp to BEC made use of a time-orbiting potential

trap. The TOP trap ramp sequence is illustrated in Fig. 3.20, and the parameters and results

are summarized in Table 3.2.

We begin by loading about 3×109 atoms into the lower MOT in 20 seconds by continuous

transfer from the upper MOT. We then seal the gate valve, restoring UHV, and perform a brief

optical molasses stage, cooling the atoms to 80 µK. This relatively high molasses temperature

is attributed to the high optical density of the atom cloud, leading to absorption of the cooling

light and low cooling efficiency in the central region of the cloud. In addition, the limited

detuning in the cooling light we can currently achieve, as discussed in Section 3.2.1, is likely to

contribute to the relatively high temperature. The molasses cooled cloud is optically pumped

to the |F = 2,mF = +2〉 state and the quadrupole field is abruptly turned on to a gradient of

B′ = 120 G/cm. This potential captures about two-thirds of the atoms, and significantly heats

the atoms up to 420 µK, due to the large size of the molasses cloud. We adiabatically compress

the quadrupole trap, raising the gradient to B′ = 470 G/cm, which raises the collision rate

while preserving the phase space density. In this compressed trap we perform RF evaporation,

sweeping the frequency from 60 to 17 MHz over 8.3 seconds. The resulting temperature is 220

µK in the compressed trap, and we find that Majorana loss begins to play a role. We then

transfer the atoms to the TOP trap by abruptly turning the fields on to B0 = 40 G, and resume

evaporative cooling. This is done by lowering the TOP field while sweeping the frequency from

84 to 80 MHz in 15 seconds, which simultaneously lowers the trap depth while also compressing

the trap. The final TOP field has an amplitude of B0 = 4 G. At this point, only 2.5×106 atoms

remain, but are at a temperature of 20 µK and a phase space density of D ∼ 10−3. The RF is

further ramped to a final frequency of about 3 MHz in 2 seconds, driving the atom cloud from

a thermal cloud to a fairly pure condensate of typically 1.5× 105 atoms. Further details on the

TOP ramp to BEC can be found in references [29, 30].

Depending on the experiment, we may finally transfer the atoms from the TOP trap to the

crossed-ODT before condensation by shortening the final RF ramp, cooling just to the threshold

for condensation, and ramping down the quadrupole field while ramping up the optical trap

power over 1 second. Finally, all magnetic fields are switched off. The transfer efficiency is

typically 50%. We perform forced evaporation in the optical trap, leaving us with about 8×104

atoms in a nearly pure BEC.

The evaporation efficiency is characterized by the logarithmic ratio of gain in D to loss in

atom number N [3],

γ = −ln(Df/Di)ln(Nf/Ni)

. (3.19)

The goal of evaporative cooling is to maximize the overall γ such that the largest condensates


Quad.

load

ing (1

sec)

Quad.

Ram

p (0

.5 se

c)

Quad.

RF (8

.3 se

c)

TOP tran

sfer (

1 se

c)

TOP com

pres

sion+

RF (15

sec)

TOP RF (2

sec)

ODT tran

sfer (

1 se

c)

ODTOpt

ical p

umpin

g (3

ms)

Mola

sses

(25

ms)

Quadrupole field

TOP field

RF frequency

Optical dipole trap

Ramp time

470 G/cm

120 G/cm

84 MHz

3 MHz

100 mW

50 G

4 G

10 MHz60 to 17 MHz

Figure 3.20: TOP trap ramp to BEC.

Stage N T [µK] σz,y [µm] D B′ [G/cm]

Molasses (20 sec MOT load) 3B 80 1270, 1440 - -

Load quad. 2B 420 742, 1424 1.4·10−8 120

Quad. compressed 2B 1000 662, 1200 1.7·10−8 470

End quad. RF 280M 220 232, 387 10−6 470

Load TOP 220M 110 382, 547 10−6 470

End TOP comp. 25M 20 122, 130 6.8·10−4 470

End TOP RF 150k < 0.03 12, 9 > 1 470

Load ODT 80k < 0.03 14, 11 > 1 470

Table 3.2: Parameters and results for TOP trap ramp to BEC. Average efficiency is γ ≈ 2.1.


can be produced. The overall efficiency of the TOP trap ramp is γ ≈ 2.1. The higher the

collision rate and the lower the loss rate, the higher the evaporation efficiency can be. This

is one of the advantages of chip traps, with their tight confinement and high collision rates,

resulting in high evaporation efficiencies [54, 55]. If the evaporation efficiency is high enough,

it enters the regime of runaway evaporation in which the overall collision rate increases despite

the lower temperature and falling atom number. For a fixed harmonic trap, the minimum

evaporation efficiency to be in the runaway regime is γmin = 2 [3]. Runaway evaporation is

fairly essential to the production of large condensates. Perhaps more importantly, being deep

in the runaway regime makes the final condensate size more robust against fluctuations in

the initial MOT conditions. Near the threshold for runaway evaporation, any slight variation

in atom number or temperature may be enough to push the evaporation above or below this

threshold. Over the fixed cooling cycle, this may rapidly compound leading to large fluctuations

in condensate number. The borderline evaporation efficiency is believed to be a major source

of instability in the TOP trap ramp.

Hybrid optical and magnetic trap ramp

In the Spring of 2012, we began using a hybrid optical and magnetic trap in place of the TOP

trap. The implementation of the hybrid trap was a significant departure from the familiar

method of the TOP trap used in our lab for nearly a decade. However it was clear that

the atom number stability of the apparatus was an issue that could not be solved by further

‘software’ optimization, and a hardware change was required. The hybrid approach makes

use of the complementary advantages of the quadrupole magnetic trap and the optical dipole

trap, while avoiding their disadvantages [53]. The magnetic trap has a very large trap depth,

allowing for collection of a large number atoms. The linear potential also has high evaporative

cooling efficiency [3], effectively providing continual compression as the temperature is lowered.

However the Majorana spin flips set a limit to the phase space density that can be efficiently

reached. In contrast, the optical trap is small in volume requiring similarly sized clouds for

efficient transfer into the trap. This small volume however also implies tight confinement, ideal

for efficient evaporative cooling. Thus the hybrid trap uses the magnetic trap for the early

trapping and cooling stages, and the optical trap for the final stages. The hybrid trap ramp

sequence is illustrated in Fig. 3.21, and the parameters and results is summarized in Table 3.3.

The hybrid potential is the sum of contributions from the quadrupole magnetic trap (Equa-

tion 3.5), the optical dipole trap (Equation 3.12), and the gravitational potential.

V (x, y, z) = µB′√x2

4+y2

4+ z2 − V0 exp

[−2

x2 + (y − y0)2

w20

]+mgy (3.20)

where g is the acceleration due to gravity, y0 is the vertical displacement of the ODT below

the quadrupole centre, and V0 is the ODT trap depth. At high temperatures, the optical trap


plays a minimal role and the potential can be approximated as

V (x, y, z) ≈ µB′√x2

4+y2

4+ z2 +mgy. (3.21)

At low temperatures, the optical trap dominates and the total potential is approximated as a

harmonic

V (x, y, z) ≈ 1

2m(ω2xx

2 + ω2yy

2 + ω2zz

2)

(3.22)

where the trap frequencies are

ωx,y =

√4V0

mw20

(3.23)

ωz =

√2µB′

my0(3.24)

The above assumes the trap is deep relative to the potential tilt. This is a fair approximation

until the final tilt evaporation stage of the ramp, which weakens the confinement along y. The

x, y confinement are provided primarily by the ODT, while the z confinement is provided by the

quadrupole trap, which is displaced from ODT centre. A field gradient of B′ = mg/2µ = 30.8

G/cm balances against the gravitational potential.

The initial stages, from laser cooling to quadrupole trap loading, are similar to that of the

TOP ramp. While in the B′ = 102 G/cm quadrupole trap, the optical trap is ramped up over

1 second to 5 W. At this stage the trap has a depth of 65 µK, and thus has negligible effect on

the 420 µK cloud. We perform radio-frequency evaporation in the quadrupole trap, ramping

the RF from 20 to 3 MHz in 8 seconds. This efficiently cools the atoms to 32 µK. At this point

the Majorana loss becomes a limiting factor and we begin transfer to the 65 µK deep optical

trap. The optical trap is focused to a spot size of w0 = 75 µm, and positioned approximately

two beam waists below the magnetic field zero. This ensures that as the atoms are cooled in the

combined potential, they do not pile up on top of the the magnetic field zero. To transfer into

the optical trap, the quadrupole trap is decompressed while further evaporating from 3 to 2

MHz in 1 second, cooling into the optical trap. The transfer is completed by decompressing the

quadrupole trap to 23.8 G/cm (see Fig. 3.22(a)). This residual gradient provides some balance

against gravity as well as weak longitudinal confinement.

Figure 3.23 plots the results of the hybrid ramp, cooling just to the edge of Bose-Einstein

condensation. Data is shown for the quadrupole and ODT evaporation stages. The cloud

shape during transfer is typically bi-modal, making fits unreliable. The cloud temperature is

observed to follow the trap depth with an average truncation parameter [3] of η ≈ 8. The

collision rate is observed to grow rapidly during quadrupole evaporation indicating efficient

runaway evaporation. However evaporation in the ODT by reduction of the beam power also

results in decompression, and the collision rate drops. Nonetheless, the high initial collision


Quad.

load

ing (1

sec)

Quad.

Ram

p (0

.5 se

c)

Quad.

RF (8

.3 se

c)

TOP tran

sfer (

1 se

c)

TOP com

pres

sion+

RF (15

sec)

TOP RF (2

sec)

ODT tran

sfer (

1 se

c)

ODTOpt

ical p

umpin

g (3

ms)

Mola

sses

(25

ms)

Quadrupole field

TOP field

RF frequency

Optical dipole trap

Ramp time

470 G/cm

120 G/cm

84 MHz

3 MHz

100 mW

50 G

4 G

10 MHz60 to 17 MHz

Figure 3.21: Hybrid optical and magnetic trap ramp to BEC.

Stage N T [µK] σz,y [µm] D B′ [G/cm] ODT [W]

Molasses (20 sec MOT load) 3B 90 1270, 1440 - - -

Load quad. 2B 420 742, 1424 1.4 · 10−8 102 5

End quad. RF 72M 32 80, 160 2.5 · 10−4 102 5

Transfer - - - - 78.2 5

Start ODT evap. 20M 10 82, 13 5 · 10−2 24 5

Start ODT tilt evap. 2M 0.3 21, 10 ∼1 16 0.22

Hybrid final 300k < 0.03 10, 7 > 1 24 0.22

Table 3.3: Parameters and results for hybrid optical and magnetic trap ramp to BEC. Averageefficiency is γ= 2.68 ± 0.04.


−1000 −500 0 500 10000

100

200

300

y−position [µm]

U−

U(0

) [µ

K]

Quadruope trap to ODT transfer

(a)

102 G/cm

70 G/cm

30.8 G/cm

−400 −300 −200 −100 0 100 200 300 400

−2

0

2

4

6

8

y−position [µm]

U−

U(0

) [µ

K]

Tilt evaporation in ODT

(b)

30.8 G/cm

25 G/cm

20 G/cm

15 G/cm

Figure 3.22: Hybrid potential along y. (a) During transfer from quadrupole trap to opticaltrap. (b) During tilt evaporation. Perfect gravity compensation occurs for B′ =30.8 G/cm.

rate and tight ODT confinement allow for efficient evaporation, even if it is not sustainable.

Some groups using a similar approach have reported observing significant gains in phase space

density during the transfer. This is called the ‘dimple effect’, and occurs due to the changing

of the trap shape leading to a dramatic change in density of states [53, 56]. Although we

observe a significant increase in phase space density, we also lose atoms (see Fig. 3.23(d)). It

is not clear if these two observations are related, as in evaporative cooling, or unrelated with

the dimple effect compensating for the low transfer efficiency. At the end of the the ODT

evaporation, the phase space density is of order 1 and the cloud is just below the threshold for

condensation. We reach the transition with approximately 2×106 atoms. In net, we observe

an overall evaporation efficiency γ = 2.68± 0.04, a significant gain over the previous TOP trap

ramp efficiency of γ ≈ 2.1.

With the falling collision rate, ODT evaporation becomes increasingly inefficient. At this

point we fix the ODT power and begin tilt evaporation by lowering the magnetic field gradient.

By tilting the potential, we effectively lower the trap depth while only weakly perturbing the

confinement (see Fig. 3.22(b)). This technique has been used to demonstrate runaway evapora-

tion in an optical trap [34]. By lowering the gradient from the initial 23.8 G/cm to 14 G/cm over

1 second, we have produced pure condensates of atom number as large as 4 × 105 (Fig. 3.24).

Unfortunately due to the large optical densities, there are likely image saturation issues at the

central peak, causing us to mis-estimate the condensate atom number. Keeping this in mind,

we fit the 1D line profiles to the sum of a gaussian and integrated Thomas-Fermi function.

Fig. 3.25 plots the condensate size and background thermal cloud temperature for various final

field gradients. The transition temperature for a non-interacting Bose gas in a harmonic trap is


0 5 10 1510

−7

10−6

10−5

10−4

10−3

Te

mp

era

ture

[K

]

Time [sec]

(I) (II) (III)

(a)

0 5 10 1510

6

107

108

109

1010

Ato

m n

um

ber

Time [sec]

(I) (II) (III)

(b)

0 5 10 1510

0

101

102

103

Avg

. C

olli

sio

n r

ate

[H

z]

Time [sec]

(I) (II) (III)

(c)

107

108

109

10−8

10−6

10−4

10−2

100

102

Atom number

Ph

ase−

space d

ensity

(I)(II)(III)

(d)

Figure 3.23: Hybrid ramp results. The ramp is broken up into 3 segments: quadrupole trap RFevaporation (I), transfer (II), optical trap evaporation (III). (a) Temperature during the ramp.The dashed blue curve indicates the calculated trap depth during (I) set by the RF knife. Theblue solid line indicates the ODT trap depth. (b) Atom number. (c) Estimated average collisionrate. (d) Phase space density and atom number on log scale, showing an overall evaporationefficiency γ= 2.68 ± 0.04.


B′ = 23.8 G/cm

−200 0 2000

2

4

6

z−position [µm]

OD

B′ = 20.4 G/cm

−200 0 2000

2

4

6

z−position [µm]

OD

B′ = 14.3 G/cm

−200 0 2000

2

4

6

z−position [µm]

OD

Figure 3.24: Tilt evaporation in hybrid trap to BEC. From left to right: At B′ = 23.8 G/cmthe cloud is thermal, exhibiting a Maxwell-Boltzmann distribution of velocities. At B′ = 20.4G/cm a dense central peak is observed on top of the thermal background. Finally at B′ = 14.3G/cm we are left with a nearly pure BEC which exhibits a characteristic Thomas-Fermi profileand inversion of the aspect ratio. Due to the large optical densities, there are likely imagesaturation issues at the central peak.

kBTc ≈ 0.94hωN1/3 (Equation 2.5) [25]. The estimated trap depth and transition temperature

are both qualitatively consistent with the observed behaviour of the condensate and thermal

cloud, showing the onset of condensation at a final field gradient of 24 G/cm. Figure 3.26 plots

the condensate fraction N0/N as a function of the estimated fractional temperature T/Tc. The

condensate fraction for a harmonic trap is expected to follow N/N0 = 1−(T/Tc)3 [25]. Although

we observe the same qualitative behaviour, our estimated transition temperature disagrees with

the prediction by a factor of 0.67.

The finite-size and repulsive meanfield interactions of the system are known to lower the

transition temperature [57]. The expression for Tc, Equation 2.5, assumes a smooth density of

states with zero ground state energy. The finite-size correction accounts for the ground state

energy, shifting the transition temperature. For the relatively weak ω = 2π × (62, 50, 40) Hz

final confinement, this correction is less than 1%. The meanfield correction arises due to the

lowered particle density in the centre of the cloud for repulsively interacting particles. This

correction is estimated to be

Tc − T 0c

T 0c

≈ −1.326as

√mω

hN1/6 (3.25)

where T 0c is the non-interacting transition temperature. For our parameters, this amounts to

a -5% correction. Despite the correction, the uncertainty in our trapping frequencies (10%),


14 16 18 20 220

5

10

15

x 105

B′ [G/cm]

Ato

m n

um

ber

Condensate

Thermal cloud

14 16 18 20 22

10−1

100

B′ [G/cm]

Te

mpe

ratu

re [µ

K]

Temperature

Trap depth

Transition temp.

Figure 3.25: Tilt evaporation to BEC. Measured condensate atom number, and thermal cloudatom number and temperature for various final field gradients. The trap depth and transitiontemperature (solid lines) are estimated from measured field gradient, and ODT powers.

0 0.2 0.4 0.6 0.8 1 1.20

0.2

0.4

0.6

0.8

1

T/Tc

0

Condensate

fra

ction, N

/N0

Data

Theory

Figure 3.26: Condensate fraction versus T/T 0c , where T 0

c is the transition temperature fora non-interacting Bose gas. The transition temperature is estimated based on measured trapparameters, and assumes a harmonic trapping potential. Red circles are data, and blue solidline is the expected behaviour.


atom number (10%)3, and temperature (15%) do not seem to be enough to account for the

disagreement. The image saturation also cannot account for the disagreement since this does

not become an issue until a large condensate is formed, well below Tc. It may be possible this

discrepancy arises due to the strong trap anharmonicity for large tilts, resulting in a higher

density of states and a lower transition temperature.

3.9 Summary and outlook

The hybrid optical and magnetic ramp to Bose-Einstein condensation reliably produces pure

condensates of 3 × 105 atoms, every 40 seconds. This is a major upgrade to the previous

TOP trap ramp, producing significantly larger condensates in a shorter amount of time. The

experiments described in Chapters 5 and 6 may not have been possible without this upgrade.

In particular, the effective mass experiment required continuous data runs of over 6 hours per

data set. Thanks to the reliability of the condensate production, we were able to take the full

set of data for that experiment in just over a month, spread over a half-dozen data runs.

Nonetheless, it is always possible to further improve upon the apparatus - whether to im-

prove the long term stability or to provide new capabilities. Some of the planned upgrades are

listed below:

• Upgrading the imaging system (see Section 3.3). The current imaging system is aberration

limited to a resolution of 3 µm. A new system has already been designed and the optics

ready, but still requires some testing before installation. The Buttiker-Landauer tunneling

time experiment, proposed in Chapter 5, would benefit from enhanced resolution.

• Modularizing the upper MOT optics (see Section 3.2.2). The modular design of the

lower MOT optics have proven quite successful in minimizing the effects of temperature

fluctuation on MOT beam alignment. The upper MOT beams, currently distributed in

free-space, remain susceptible to this instability. An upgrade would be a minor investment

for a significant benefit.

• Upgrading the laser system with a 2 W tapered amplifier (see Section 3.2.1). To maintain

a reliable minimum level of available optical power for laser cooling, some control over

the laser system has been sacrificed. While this provides large atom collection efficiency,

this limits our ability to cool them. An upgraded tapered amplifier chip would go a long

way to improving the control and flexibility.

• Installation of Feshbach circuitry (see Section 3.4.1). The design work and construction

of the hardware for accessing the 87Rb Feshbach resonance is already finished, and the

circuitry should be ready for installation. It remains to be seen if the full system can

realize the technically challenging goal of stably generating a 1000 G field with 10 mG

stability.

3This is the estimated atom number uncertainty for unsaturated images.

Chapter 4

Transient momentum-space

interference during scattering

This chapter describes the first experiment performed on the Bose-Einstein condensation appa-

ratus. We study the scattering of a BEC from repulsive potential barrier, observing a transient

interference effect that arises only during the collision, and vanishes in the asymptotic limits.

This experiment was originally conceived as a test of our experimental capabilities, in particu-

lar our barrier system, before pursuing the long-term goal of the study of tunneling times (see

Chapter 5). The experiment was initially plagued by an instability in the initial momentum of

the condensate in time-of-flight, suspected to be due to the abrupt turn-off of the large magnetic

fields used in the TOP trap. This resulted in a seemingly random velocity kick for the initial

state of δv ∼ 1 µm/ms, making averaging of the data impossible. The optical dipole trap was

developed to avoid this issue, successfully reducing these fluctuations to less than 0.1 µm/ms.

In the end, this experiment required significantly more work than we first expected, requiring

a better understanding of the underlying physics, extensive simulation, and the development of

new tools and methods for characterizing the system. Yet it was also more interesting, with two

distinct scattering effects occurring that had to be disentangled to make sense of the results.

4.1 Scattering in the non-asymptotic limit

Scattering is traditionally described asymptotically, detailing the initial and final states, while

ignoring the dynamics during the collision itself [58]. In typical particle scattering experiments,

measurements are made long after the collision and thus the asymptotic solutions provide

an accurate description of the observations. Furthermore, the collision dynamics are often

far too fast to directly probe. Yet, there are circumstances in which the scattering cannot

be described asymptotically and knowledge of the full wavefunction is required, for example in

understanding the quantum kinetics of moderately dense gases [59, 60]. Brouard and Muga [61,

23] theoretically studied the one-dimensional scattering of a wavepacket from a delta-function

50

Chapter 4. Transient interference during scattering 51

potential. They showed that during the collision with a repulsive potential, the momentum-

space wavefunction could vary dramatically, for instance exhibiting the generation of high-

momentum components with classically forbidden probabilities. This non-classical momentum

enhancement is a consequence of wave-particle duality, in the spirit of quantum reflection [62,

63, 64] - an effect that occurs when the potential changes abruptly on the scale of the deBroglie

wavelength irrespective of the sign of the change. However the effect predicted by Brouard and

Muga is distinctly transient, manifesting itself only during the scattering event, and vanishing

in the asymptotic limits.

Over the past few decades, impressive advances in experimental techniques have granted ac-

cess to ever faster timescales and lower energies, allowing for direct probes of the non-asymptotic

regime. For example, ultrafast laser pulses can now be used to probe sub-femtosecond timescales,

providing time-resolved probes of electron dynamics [65, 66, 67]. In parallel, neutral atom cool-

ing techniques [2, 3] have matured, and now routinely produce samples in the nanokelvin regime

[4, 5, 32]. The scattering dynamics of these ultracold systems occur on an easily accessible mi-

crosecond timescale, making it an ideal system for investigation of non-asymptotic scattering.

Bose-Einstein condensates of dilute gases have been used extensively to observe matter-wave

phenomena [6, 68, 64]. Due to the relatively low densities, the inter-particle interactions are

weak, allowing for clear observation of single-particle quantum effects. Furthermore, the dy-

namic Stark effect allows for creation of nearly arbitrary potentials with spatial features limited

only by the wavelength of light used to induce them [9], and can be modulated much faster

than the dynamics of the condensate [10].

The experiment described here makes use of these advantages, using a Bose-Einstein con-

densate as a matter wavepacket interacting with a thin, optically-induced potential barrier, as

a model system to study scattering in the non-asymptotic limit.

4.2 Theory

4.2.1 Transmitted-incident interference

Brouard and Muga [61] studied the momentum distribution of a transform-limited wavepacket

impinging upon a delta-function barrier potential. They predicted that during the collision there

would be a dramatic reshaping of the momentum distribution, resulting in a transient enhance-

ment of high momenta. At first glance this might be interpreted as a spatial compression of the

wavepacket as it impinges upon the barrier, resulting in a broadened momentum distribution.

Indeed, they showed that the enhancement increases with the height of the barrier, saturating

in the opaque barrier limit [23]. However Perez, Brouard and Muga [69] later showed that

a much more significant momentum enhancement unexpectedly occurs for wavepackets with

kinetic energy well above the barrier height. They interpreted this effect as an interference be-

tween incident and transmitted portions of the wavepacket. In the limit of a nearly transparent

barrier, the primary effect on the wavepacket is to write a phase shift. During the collision, half


−30 −20 −10 0 10 20 300

0.2

0.4

0.6

0.8

1

k [a.u.]|Ψ

(k)|

2

0

0.5

1

y

|Ψ(y)|2 φ [π]

Figure 4.1: Transient enhancement of momentum during scattering from a delta-functionbarrier. Shown is the simulated momentum distribution, in the wavepacket centre-of-massframe, long before/after the collision (red) and during the collision (blue). Inset: Position-space distribution during scattering (blue), and phase-profile (red). The barrier height hasbeen chosen such that it results in a π-phase shift across the wavepacket in position-space.

of the wavepacket has yet to reach the barrier and half has been transmitted. Thus there ex-

ists a phase discontinuity dividing the incident and transmitted components of the wavepacket

(Fig. 4.1 inset). This sharp spatial feature in the phase results in destructive interference of the

central momentum component and constructive interference in the tails [70]. The net result

is a symmetrically broadened momentum distribution which exhibits non-classical momentum

enhancement, and tails that fall off as 1/k2 (see Fig. 4.1). The momentum-space wavepacket

simulations used throughout this chapter to illustrate the scattering phenomena are performed

using the 3-step, split-operator method described in Section 4.5.2.

To quantify the momentum enhancement, Brouard and Muga defined a quantity, G(p, t),

that measures the accumulated probability to have enhanced momentum during the collision

at time t. For an ensemble of classical particles, characterized by a momentum distribution

P (p, t), scattering from a repulsive potential (V (y) ≥ 0 for all y), this quantity can never be

positive:

Gcl(p, t) ≡∫ ∞p

[P (p′, t)− P (p′, 0)

]dp′ ≤ 0. (4.1)

However for a quantum mechanical particle, described as a momentum-space wavepacket Ψ(p, t),

the quantity

Gq(p, t) ≡∫ ∞p

[|Ψ(p′, t)|2 − |Ψ(p′, 0)|2

]dp′ (4.2)

has no such restriction, and can in general be greater than zero. Thus Gq(p, t) may be used as

a quantitative measure of this non-classical effect.

4.2.2 Chirped wavepackets

For a wavepacket with linear chirp, as would occur during free-expansion, the momentum distri-

bution during scattering contains a much richer structure compared that of a transform-limited


−15 −10 −5 0 5 10 15

0

0.2

0.4

0.6

0.8

1

y [µm]

|Ψ(y

)|2

(a)

−10 −5 0 5 10

0

0.5

1

1.5

2

k [1/µm]

|Ψ (

k)|

2

(b)

Figure 4.2: Momentum-space interference for a chirped wavepacket. (a) Position distribution(black line), illustrating decomposition into transform-limited gaussians, each associated with adistinct central momentum. The π-phase discontinuity (red dashed line) only affects the centralmomentum component. (b) Resulting momentum distribution in the wavepacket centre-of-massframe, long before/after the collision (red) and during the collision (blue).

wavepacket. A chirped wavepacket can be thought of as a train of transform-limited gaussians,

each representing a component of the momentum distribution. The slow components which have

yet to enter the barrier acquire no phase, while the fast components which have fully traversed

the barrier acquire an overall phase. Only the central momentum component, predominantly

located near the centre of the spatial wavepacket, acquires the phase discontinuity, and the

associated long momentum tails (Fig. 4.2(a)). The resulting momentum distribution exhibits

rapid oscillations due to interference between the redistributed central momentum component,

and components in the tails of the distribution (Fig. 4.2(b)).

4.2.3 Pushed-incident interference

For finite-width barriers, the phase profile written is no longer an abrupt step. For a gaussian

shaped barrier with width σy and height V0, the phase acquired by a particle travelling from

the left with velocity v and currently at position y is

φ(y) =V0σyvh

√π

2

[erf

(y√2σy

)+ 1

]. (4.3)

Due to the finite-width of the phase profile, the predicted interference effect is significantly

reduced. This reduction depends exponentially on the ratio of the barrier width to the initial

wavepacket width ∆y (equivalently, it depends on the product σy · ∆k, where ∆k = 1/∆y

is the momentum width of the wavepacket). In addition to the reduction of the momentum

enhancement, there is a secondary effect which is that a centre-of-mass momentum is imparted

(see Fig. 4.3). Components of the wavepacket in mid-collision receive a net deceleration. These

components are effectively pushed to lower momenta, resulting in interference between incident

and pushed components of the wavepacket. This breaks the symmetry of the momentum distri-


0

1

2

3

|Ψ (

k)|

2

∆φ = 1π

σy = 0.5 µm

0

1

2

3

|Ψ (

k)|

2

∆φ = 2π

σy = 1.0 µm

0

1

2

3

|Ψ (

k)|

2∆φ = 1π

σy = 0.0 µm(a)

0

1

2

3

|Ψ (

k)|

2

∆φ = 1π

σy = 1.0 µm(b)

−10 −5 0 5 100

1

2

3

|Ψ (

k)|

2

∆φ = 1π

σy = 1.0 µm

k [1/µm]−10 −5 0 5 100

1

2

3

|Ψ (

k)|

2

∆φ = 3π

σy = 1.0 µm

k [1/µm]

Figure 4.3: Momentum-space interference during scattering for a linearly chirped wavepacketwith a finite width barrier. Simulated momentum distributions, in the centre-of-mass frameof the incident wavepacket, before/after the collision (red dashed line) and during the collision(blue solid line). (a) The left column shows the dependence on barrier width. For a delta-function barrier, T-I interference is dominant, creating a symmetric distribution. For a finite-width barrier, P-I interference becomes dominant, breaking the symmetry. (b) The right columnshows the effect of increasing the barrier height.

bution and is distinct from the interference between the transmitted and incident components.

Figure 4.4 illustrates the momentum redistribution, showing the mapping of an initial mo-

mentum wavevector ki to final wavevector kf , and the effect it has on the momentum-space

phase profile. Due to the correlation of position and momentum in the chirped wavepacket, in

mid-collision the barrier changes the momenta of the central momentum components, break-

ing the 1:1 relationship between initial and final states. Figure 4.5 shows the resulting phase

profile for typical experimental parameters. Due to the momentum redistribution, there are

components of the wavepacket that began with different initial momenta, but end up at the

same final momentum. The resulting interference pattern depends on the relative phase ∆φ

accumulated along the different pathways. In the limit of a sharp barrier potential, the expected

momentum-space phase profile for a non-interacting wavepacket is

∆φ =1

h

∫ tcol

t=0dt

(h2k2

i

2m−h2k2

f

2m

), (4.4)


−5 −4 −3 −2 −1 0 1 2 3 4 5−5

0

5

kfinal

kin

itia

l

Final momentum as a function of initial momentum

(a)

No barrier

With gaussian barrier (∆ K=1,2,3)

−5 −4 −3 −2 −1 0 1 2 3 4 50

0.5

1

1.5

2

kfinal

φ [

π]

Momentum space phase profile

(b) No barrier

With gaussian barrier (∆ K=1,2,3)

Figure 4.4: Illustration of barrier push effect. (a) Mapping of initial momenta to final momentawith and without the barrier. The effect of the barrier on the chirped wavepacket breaks the1:1 mapping of initial and final states. (b) Effect on momentum-space phase profile. In theabsence of the barrier, the phase profile is quadratic. The redistribution of momenta distortsthis profile leading to pair-wise interference between distinct paths that reach the same finalmomentum. The interference depends on the relative phase accumulated along these paths.

where tcol is the time from wavepacket release to barrier collision. This quadratic relative

phase breaks down at small momenta for interaction with a finite width barrier, as shown in

Fig. 4.5. The finite width means there is a continuous range of momenta imparted. For large

momenta however, the quadratic phase is still a good approximation. For our experimental

implementation with an initially interacting matter wavepacket (discussed in the following

section), this is still a fair approximation to the expected phase profile, despite neglecting

the dynamics during the non-linear expansion. While accounting for the non-linear expansion

is important in describing the pre-collision momentum distribution [26], it provides only a minor

contribution to the phase profile when compared to the contribution from free expansion since

tcol 1/ωy. An additional consequence of the finite-width scattering potential is that the strict

2π-phase periodicity in transmitted-incident interference, that would occur for a delta-function

potential, is washed out by the continuous range of phases written on the wavepacket. By

studying the momentum-space interferogram, it is possible to reconstruct the phase profile as

shown in Fig. 4.6. This phase extraction technique will be further explored in Section 4.6.

For our typical experimental parameters, σy ·∆k ≈ 4, the transmitted-incident (T-I) inter-

ference effect is strongly suppressed, and the scattering is dominated by pushed-incident (P-I)

interference. However, since the latter only changes the distribution of slow-moving components

of the wavepacket, it is still be possible to unambiguously observe the T-I interference effect

on the fast-moving, transmitted components. Both T-I and P-I interference phenomena are a


−4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 00

1

2

3

4

5

k [1/um]

φ [

π]


(a) Full solution

Quadratic phase α k2

−4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 00

1

2

3

4

5

k [1/um]

∆φ [

π]

Phase difference between pushed and incident wavepacket components

(b) ∆ φ un−pushed − pushed (low k)

∆ φ un−pushed − pushed (high k)

∆ φ pushed (low k) − pushed (high k)

Quadratic phase

Figure 4.5: Barrier push effect for typical experimental parameters: tcol = 3.5 ms, ttot = 30ms, σy = 2.2 µm, ∆φ = 5π. (a) Momentum-space phase profile. There are up to 3 branches ofthe profile, representing different paths to reach the same final momentum state. (b) Relativephase between different branches of phase profile.

−8 −6 −4 −2 0 2 4 6 80

0.5

1

1.5

Momentum space distribution

k [1/um]

|Ψ(k

)|2

−8 −6 −4 −2 0 2 4 6 80

2

4

6

8

k [1/um]

∆φ [

2π]


Quadratic phase

∆ φ un−pushed − pushed (low k)

∆ φ un−pushed − pushed (high k)

Interference max/min

Figure 4.6: Extracting the momentum-space phase profile from interferograms. Each interfer-ence peak/valley is marked and a relative π-phase assigned. The extracted phase profile (redcircles) closely resembles the expected quadratic phase described by Equation 4.4


consequence of the wave nature of atoms. It is important to emphasize the transient nature

of these effects. For our experimental implementation, the barrier height is typically 8 times

smaller than the average kinetic energy of the wavepacket; there is no classical reflection and

a probability of quantum reflection less than 10−13. Yet even in this near-transparent barrier

limit, there is a dramatic modification of the momentum distribution during interaction with

the barrier, which afterwards vanishes, returning to the original distribution. In contrast, the

only classical effect would be a temporary decrease in the momentum of the small fraction of

atoms in mid-collision at any given moment

4.3 Experimental scheme

The experimental sequence is illustrated in Fig. 6.6. We prepare a Bose-Einstein condensate

in the ground state of an optical dipole trap formed by the intersection of two focused 980

nm beams oriented in the horizontal x-z plane (see Section 3.6). The trap is cylindrical, with

approximate radial and axial trapping frequencies ωx = ωy = 2π × 300 Hz and ωz = 2π × 100

Hz, respectively. In this trap, we prepare nearly pure condensates of about 8 × 105 atoms.

After preparation, we abruptly turn off the optical trap, dropping the atoms under gravity

(y-direction) onto an optically-induced barrier potential. The barrier is positioned beneath our

atoms such that the collision typically occurs after tcol ∼ 4 ms of free-fall, during which the

wavepacket acquires a linear chirp. The acquired chirp arises from the non-linear, self-similar

expansion of the condensate [26]. In a timescale of 1/ωy, the repulsive inter-particle interac-

tions are converted into kinetic energy, determining the wavepacket momentum distribution.

This wavepacket then collides with the barrier. To study the momentum distribution during

the collision, we abruptly turn-off the optical barrier within 20 µs, freezing out the dynamics

of the collision. We then perform a long time-of-flight expansion for time tprop, followed by

absorption imaging after a total time ttot = tcol + tprop = 30 ms. This time-of-flight maps the

momentum of the particles during the collision to their final position, such that the imaged

position distribution is representative of the momentum distribution of the atoms during the

collision.

Note that for an interacting wavepacket, the described phase-writing would result in soliton

formation [71, 72]. The effect studied here is a single-particle interference effect. The repulsive

inter-particle interactions in the system are only relevant during the first millisecond of the

experiment, and are negligible during the collision itself. This has been confirmed by compar-

ing an interacting 3D Gross-Pitaevskii equation simulation to a non-interacting Schrodinger

equation simulation of the collision with matched pre-collision momentum distributions. The

details of these simulations are provided in Section 4.5.2.


−15

−10

−5

0

5

10

y

−15

−10

−5

0

5

10

−15

−10

−5

0

5

10

−15

−10

−5

0

5

10

−15

−10

−5

0

5

10

−15

−10

−5

0

5

10

Figure 4.7: Experimental sequence. From left to right: A BEC (red circle) is formed in theground-state of an ODT (black parabola). The trap is abruptly turned off, and the atomsundergo expansion while falling under gravity. The atoms drop onto a weakly repulsive barrierpotential (blue line). In the middle of the scattering process at time tcol ∼ 4 ms, the barrier isabruptly turned off, freezing the dynamics. The momentum distribution is then measured byTOF expansion followed by absorption imaging at time ttot = 30 ms.


4.3.1 Optical dipole force barrier

The optical barrier potential is generated by a 780 nm, beam blue-detuned from the 87Rb D2

transition by 150 GHz. This beam is approximately gaussian in shape with an aspect ratio of

10, and propagates along the x-axis. The 1/e2 barrier width along the direction of wavepacket

propagation is σy = 2.2 ± 0.1 µm. Transverse to the wavepacket propagation, the beam has a

size of σz ∼ 20 µm, and has a Rayleigh range of xR ∼ 20 µm. σz is comparable to the size of

the expanding cloud at the time of collision. We smooth out the potential along z by rapidly

scanning the position of the barrier using an acouto-optic deflector (Neos 23080-1), extending

the length of the barrier to ±60 µm. This scan occurs at 100 kHz and is much faster than

the motion of the atoms, resulting in a time-averaged potential [10]. Furthermore, by imaging

the barrier beam, we tailor the AOD scanning waveform to generate a time-averaged potential

flat to within 1% of the barrier height. This ensures that the collision dynamics we observe

are truly one-dimensional. For further details on the scanning, see reference [30]. The barrier

height, calibrated by reflection of the atomic wavepacket, is typically around kB× 1 µK, where

kB is Boltzmann’s constant. The height is chosen such that the effect of the barrier is to write a

phase shift, typically around 3π, while being small compared to the typical wavepacket average

kinetic energy of kB× 8 µK acquired in free-fall.

The layout of the barrier beam is shown in Fig. 4.8. The beam shaping consists of a cylin-

drical telescope, which sets the beam aspect ratio. The beam is focused using a Thorlabs f = 5

cm asphere, which is specified by the manufacturer to provide diffraction limited performance.

The diffraction limited width is expected to be 0.7 µm. We measure the barrier profile outside

the vacuum chamber by knife edge scanning. The edge used was a USAF test target with edge

sharpness specified to 0.5 µm. The scanning was performed using a Melles-Griot NanoMover

with 50 nm position resolution. Sample beam profiles near focus are shown in Fig. 4.9. The

beam profile exhibits clear signatures of optical aberration, particularly off-focus. On-focus

however, the beam resembles that of a gaussian with width σy = 2.2 ± 0.1 µm. All calcu-

lations and simulations assume this form of potential. Simulations show that the resulting

interferogram is not strongly dependent on small deviations in the shape of the barrier profile.

By positioning the barrier below the trap centre, we can choose the velocity with which

the particles impinge upon the barrier. The placement of the barrier focus is determined by

dropping atoms on the barrier and maximizing the reflection signal. Similarly, the barrier height

is calibrated by reflection, and assuming the barrier width measured outside the vacuum system.

See Fig. 4.10. By measuring the reflection point for several incident wavepacket velocities, the

barrier width can be independently verified (see calibration of 2nd generation 405 nm barrier

described in Section 5.2.1).

4.3.2 Measuring the momentum distribution

The momentum distribution described in section 4.2 is revealed by a long TOF expansion. In

order for the position distribution to be a fair representation of the momentum distribution,


M=10 cylindrical telescope

Thorlabsf = 50 mmasphere

x

z

PBS

Absorption probeoptical path

AOD

Barrier PD

Figure 4.8: 780 nm barrier optical layout. The optics produces a beam of 1/e2 width σy =2.2 ± 0.1 µm, σz ∼ 20 µm. Scanning of the AOD at 100 kHz generates a time-averaged, flatpotential, extending the range along z to ±60 µm.

0 10 20 30

0

0.2

0.4

0.6

0.8

1

11.74 mm

0 10 20 30

0

0.2

0.4

0.6

0.8

1

Knife transverse position [um]

Inte

gra

ted

PD

sig

na

l /

diffe

ren

tia

l P

D s

ign

al

11.69 mm

11.7 11.72 11.74 11.76 11.781

1.1

1.2

1.3

1.4

1.5

1.6

1.7

Knife longitudinal position [mm]

Fitte

d g

au

ssia

n r

ms r

ad

ius [

um

]

Figure 4.9: Barrier profile measured by knife edge scanning. The scan has 100 nm resolution,and the knife edge has 0.5 µm edge sharpness. Shown are two sample profiles: on-focus atlongitudinal position 11.74 mm, and off-focus at 11.69 mm. The integrated photodiode signalis plotted in blue, and the differential signal in red. The fitted gaussian widths are plottedon the right showing a focused spot size of w0 = 2.2 ± 0.1 µm. The double peaked structureoff-focus, and the asymmetry along the beam longitudinal direction are characteristic of opticalaberration.


Barrier height

y−

po

sitio

n

(a)

2.6 2.65 2.7 2.75 2.8 2.85

0.5

0.6

0.7

0.8

0.9

1

Barrier focus [µm]

Tra

nsm

issio

n

(b)

0.2 0.4 0.6 0.8

0

0.2

0.4

0.6

0.8

1

Barrier height [mW]

Re

fle

ctio

n/T

ran

sm

issio

n

(c)

Figure 4.10: Focusing and calibration of 780 nm barrier with atoms. (a) Composite of sampleabsorption images after collision with barrier. Atom cloud falls under the influence of gravitybouncing off the barrier oriented in the horizontal plane. Barrier height is increasing from leftto right. (b) Focusing barrier on atoms based on transmission signal. Line is a gaussian fit tothe data, used to identify the optimal focal point. (c) Calibration of barrier height by reflection(blue) and transmission (red) signal. Lines are error function fits, used to identify the 50%reflection point.

−200 −100 0 100 2000

0.5

1

1.5

Momentum and position space profile comparison

|Ψ|2

y [µm]

tcol

= 3.5 ms, ttot

= 30 ms

σy = 1.1 µm, ∆φ = 5π

y−distribution

Scaled k−distribution

Figure 4.11: Mis-match between position |Ψ(x)|2 distribution, and scaled momentum distri-

bution |Ψ(htpropm k)|2.

the cloud must expand by several times its initial size. The momentum-space resolution is

δk ≈ mh

∆xtprop

. The mapping from momentum to position is only exact in the limit of tprop →∞.

For our typical experimental parameters, there is a small but observable discrepancy from being

truly far-field. This is illustrated in Fig. 4.11, where the simulated position distribution |Ψ(x)|2

after 30 ms TOF is compared to a scaled momentum distribution |Ψ(htpropm k)|2. This scaling

acts on the wavefunction to mimic a perfect mapping from momentum to position. Given this

discrepancy, when analyzing our data we must compare to the position-space distribution. The

position-space wavepacket simulations used for comparison to experiment are performed using

the quasi-interacting 3-step, split-operator method described in Section 4.5.2.

In addition to the limitations of the TOF method, the imaging system has a finite resolution.

As discussed in Section 3, the imaging system has resolution of 3 µm, and pixel size of 3 µm.


−200 −100 0 100 2000

0.5

1

1.5

Simulation of imaging system limitations

|Ψ(y

)|2

y [um]

tcol

= 3.5 ms, ttot

= 30 ms

σy = 1.1 µm, ∆φ = 3π

Original distribution

With imaging filters

Figure 4.12: Simulated cloud profiles for a perfect imaging system (red) and for our experi-mental setup (blue). The imaging system has a resolution of 3 µm, CCD pixel size 3 µm, andabsorption probe pulse duration 24 µs.

Furthermore, the cloud is moving during our imaging pulse. If the pulse duration is long

compared to the time it takes for an interference fringe to move by one fringe spacing, the

fringes may be washed out. A short pulse means fewer photons reaching the CCD and less

signal, and so we typically limit the duration of our imaging pulse to 20 µs. For a cloud falling

under gravity for ttot = 30 ms, and an imaging pulse of duration 20 µs, the cloud moves 5.9

µm. These three imaging effects limit the spatial resolution of fringes, and markedly reduce the

visibility of fringes near the edges of the cloud. Fortunately, these effects are easily modelled,

and are applied to the simulated distributions when comparing to experiment (see Fig. 4.12).

4.4 Results

We measure the condensate momentum distribution during scattering for a variety of scatter-

ing parameters. Figure 4.13 shows a representative set of absorption images for pre-collision

expansion time tcol = 3.3 ms and increasing barrier height (corresponding integrated 1D pro-

files are shown in Fig. 4.14). The barrier height is expressed in terms of the estimated phase

shift imparted to the fully transmitted atoms. As the barrier height is increased we observe

the development of a rich interference pattern, consistent with P-I interference. For higher

barriers, the central momentum component is more strongly decelerated, thus increasing the

range over which interference is observed. The fringe pattern that develops is reflective of the

momentum-space phase profile of the condensate prior to collision. For condensate expansion,

we expect a quadratic phase profile [68], and a fringe spacing that decreases linearly from cloud

centre.

Given the finite barrier width, T-I interference is strongly suppressed and the scattering

is dominated by P-I interference. However the latter effect is asymmetric, only affecting the

low-momentum side of the distribution, thus T-I interference can still be observed on the high-


y [

µm

]

0.0 π−250

−200

−150

−100

−50

0

50

100

150

200

250

1.3 π 2.4 π 3.6 π 5.5 π 7.0 π

Figure 4.13: Transient momentum-space interference during a collision. Representative ab-sorption images of a BEC after collision with a barrier at tcol = 3.3 ms and total TOF ttot =30 ms. The barrier height, expressed in terms of the phase shift imparted to fully transmittedatoms, is increasing from left to right. The cloud is moving downwards (+y) under the influenceof gravity. The images have been low-pass filtered in the transverse z direction for presentation.

0

0.1

0.2

∆φ = 0.0 π

|Ψ(y

)|2

0

0.1

0.2

∆φ = 1.3 π

|Ψ(y

)|2

−200 −100 0 100 200

0

0.1

0.2

∆φ = 2.4 π

|Ψ(y

)|2

y [µm]

0

0.1

0.2

∆φ = 3.6 π

|Ψ(y

)|2

0

0.1

0.2

∆φ = 5.5 π

|Ψ(y

)|2

−200 −100 0 100 200

0

0.1

0.2

∆φ = 7.0 π

|Ψ(y

)|2

y [µm]

Figure 4.14: Transient momentum-space interference during a collision with a barrier at tcol =3.3 ms and total time-of-flight expansion ttot = 30 ms. Integrated experimental 1D densityprofiles corresponding to images in Fig. 4.13.


0

5

10

x 10−3 Experiment

|Ψ(y

)|2

(a)

−2

0

2

4

6

8

∆ (

× 1

000)

−200 −100 0 100 200

−0.1

−0.05

0

G

y [um]

0

5

10

x 10−3 Simulation

|Ψ(y

)|2

(b)

−2

0

2

4

6

8

∆ (

× 1

000)

−200 −100 0 100 200

−0.1

−0.05

0

G

y [um]

Figure 4.15: Evidence for transmitted-incident momentum-space interference for tcol = 6.3ms, ttot = 30 ms, and an estimated barrier phase shift of 3.1π. (a) Sample experimental1D profile (average of 20 images), and (b) simulation (including image resolution effects; seetext). Shown are the density profiles for un-collided (red) and collided clouds (blue), and thedifference between these profiles, ∆. The pushed-incident interference is clearly visible on thelow-momentum side (negative y), while the transmitted-incident interference is visible on thehigh-momentum side (positive y) around y = 50 µm.

momentum side. Since the range of the P-I interference is limited to near the centre of the

momentum distribution, the effect is most visible for phase curvatures which are large such that

at least a full fringe is observed, yet small such that the fringes are resolved by our imaging

system. In our parameter range, this is achieved with long pre-collision times. Figure 4.15 shows

a representative 1D profile (averaged over 20 images) for tcol = 6.3 ms and an estimated barrier

phase shift of 3.1π. The experimental profiles show excellent agreement with our simulation,

which accounts for the 3 µm imaging resolution and camera pixel size. Some residual jitter

in the shot-to-shot position of the cloud may contribute to the reduced fringe visibility in the

averaged experimental profiles when compared to simulation. At this pre-collision time, we

begin to see T-I interference fringes on the high-momentum side (positive y; oscillations around

50 µm).

In principle, this type of interference also results in momentum enhancement, which may

be quantified by calculating Gq(p, t) (Equation 4.2) during the collision. This is done by nu-

merically integrating ∆(y) from y to yend, the end of the imaging window. Since the cloud is

fully contained within the region-of-interest, this is the practical equivalent to taking the limit


y →∞,

G(y) =

∫ yend

y∆(y′) dy′. (4.5)

G(y) is calculated and plotted in Fig. 4.15 for the tcol = 6.3 ms sample profile. Although

there is nice qualitative agreement, the size of the momentum enhancement is buried within the

experimental noise. Due to the integration in G(y), this signal is very sensitive to low-frequency

noise that may not average away within the integration region. For example, a small mis-

match between the collided and un-collided cloud positions or shapes can result in a dramatic

background in G(y). While the optical trap has greatly improved the experiment stability,

yet small fluctuations remain. Although the T-I interference fringes are clearly observed, the

expected momentum enhancement signal is buried within the experimental noise.

4.5 Discrepancy with initial expectations

At first glance there is good agreement between the observed experimental profiles and the

simulation. However a closer look reveals a number of anomalous features present in the data

never seen in the single-particle simulation:

• Larger fringe spacing than expected

• Fringes which extend out further from cloud centre than expected

• A split central peak

• Lower amplitude for central peak than expected

These features are highlighted in Fig. 4.16. Furthermore, certain data sets exhibit these features,

while others do not. For example, the data set shown in Fig. 4.15 is free of these features.

These discrepancies were no small source of frustration. A host of theories were explored to

explain the discrepancies, primarily through simulation. One theory was that the shape of the

barrier inside the vacuum system was not well known. As described in Section 4.3.1, the 780

nm optical barrier was aberration limited. Beam profiling outside the vacuum system showed

that at the beam focus, the profile was well approximated by a gaussian. However off-focus, the

beam shape changes and can exhibit multiple peaks. Motivated thus, several different barrier

profiles were tested however they were never able to reproduce the anomalous features. Another

theory was that the finite Rayleigh range was playing a role in the observed absorption images,

however this too was ruled out by simulation. In the end, a substantial amount of effort was

expended testing various theories. In particular, two theories proved insightful and are discussed

in the following sub-sections.

4.5.1 Simulation of interactions

In the experiment, inter-particle interactions had always been assumed to be negligible. This

is because during free-expansion, the interactions are expected to dissipate on a timescale of


−60 −40 −20 0 20 40 60

0

0.1

0.2

0.3

0.4

Sample experimental profile (bc20)

tcol

= 5 ms

y [um]

A,B

C,D

−60 −40 −20 0 20 40 60

0

0.1

0.2

0.3

0.4

Simulated profile

tcol

= 5 ms

∆φ=2.5π

y [um]

Figure 4.16: Anomalous features in data not seen in simulations. (A) Split central peak, (B)lower amplitude for the central peak than expected, (C) larger fringe spacing than expected,and (D) fringes which extend out further from centre than expected.


1/ωy tcol. Based on such order-of-magnitude calculations, we did not expect interactions to

be important, yet could never really convince ourselves that they were un-important. Including

inter-particle interactions in the simulation is a computationally expensive task, since the in-

teractions they can couple the longitudinal (y) and transverse degrees-of-freedom (x, z). Thus

a full 3D calculation is required. A variety of simulations were implemented with increasing

complexity to model this system.

1) 1D non-interacting 3-step split-operator. We began with a simple 1D single-particle

simulation using a 3-step split-operator method. The three steps involve k-space propaga-

tion to barrier, x-space collision with barrier, and k-space propagation to imaging. With the

single-particle simulations, we assume that the repulsive interactions only affect the initial mo-

mentum distribution, resulting in a broadening. To account for this we simply choose the initial

wavepacket size such that we match the momentum distribution at the barrier to the observed

distribution.

2) 1D quasi-interacting 3-step split-operator. Instead of artificially setting the initial

cloud size to obtain the correct momentum distribution, we can use the Thomas-Fermi scaling

solution [26] to find the wavefunction at the barrier. Assuming the inter-particle interactions

have dissipated at this point, it is fair to continue the propagation using the non-interacting

Schrodinger equation.

3) 1D interacting split-operator and finite difference method simulations. We sim-

ulated the Gross-Pitaevskii equation in 1D, which incorporates the effect of inter-particle in-

teractions through a repulsive meanfield. If interactions are present, a 1D model isn’t really

appropriate since interactions can couple the longitudinal and transverse modes of the con-

densate. However this simulation should show if interactions have any affect on the fringe

spacing.

4) 3D interacting split-operator and finite difference method simulations. A full 3D

simulation was performed assuming cylindrical symmetry, thus reducing the 3D equation to

2D. These simulations required a significant amount of time to perform and only a few test

cases were done.

The results of the simulation are summarized in Fig 4.17. Although there does seem to be some

variation from method to method around the centre of the distribution, for the most part the

extracted phase curvature is consistent with the simple quadratic phase profile described by

Equation 4.4, indicating that interactions do not play a role.

4.5.2 Imaging simulations

Based on a suggestion received at DAMOP in 2011, we began looking at simulating the imag-

ing process. We simulated a 2D plane-wave electric field propagating through our (simulated)

atomic cloud distribution and the subsequent propagation to the imaging plane. The simu-


−100 −50 0 50 1000

0.5

1

−100 −50 0 50 1000

0.5

1

−100 −50 0 50 1000

0.5

1

−100 −50 0 50 1000

0.5

1

y position [um]0 20 40 60 80

0

2

4

6

8

10

12

∆ y [um]

∆φ [

π]

1D 3SS

1D FDM Ndz=8192 interacting

3D SS Ndz=2048 interacting

3D FDM Ndz=2048 interacting

Expected phase profile

Figure 4.17: Comparison of several different simulations, both interacting and non-interacting,and the extracted phase profiles for tcol = 5 ms, ttot = 30 ms, and ∆φ = 3π. Shown are theresults for the 1D quasi-interacting 3-step split-operator (blue), 1D interacting finite-differencemethod (green), 3D interacting split-operator (red), and 3D interacting finite-difference method(black). The extract phase profiles are compared to the expected curvature α (magenta).


imaging−plane, x [um]

y [um

]

−500 0 500

−150

−100

−50

0

50

100

150

Figure 4.18: Simulation of imaging of a collided cloud of atoms (on-resonance, tcol = 5 ms).Each vertical slice is the 1D cloud profile along the direction of gravity (y). This profile isshown for object plane at various distance from the cloud centre along x.

lation allowed us to look at the effect of probe detuning and imaging different object planes.

Figure 4.18 shows an example of this simulation. Plotted is the inferred 1D optical density

profile along the cloud propagation direction (y) for the imaging system focused at various

object-plane distances (x). What is immediately clear from this calculation is that the fringe

spacing increases as one moves the object plane away from the centre of the cloud. This is

analogous to imaging a slit pattern off-focus, the resulting image is a diffraction pattern with

fringe spacing that increases the further from the slit the object plane is.

Figure 4.19 shows a few sample profiles for imaging increasingly off cloud centre. The

profiles ∼ 300 µm off-centre exhibit features consistent with experimental observations: split

central peak, reduced height of central peak, larger fringe spacing than expected, and fringe

that extend out further from centre than expected. To make this quantitative, Fig. 4.20 plots

the extracted phase curvature for off-centre imaging. Clearly the extracted phase curvature

drops rapidly when the object plane is moved off cloud centre. Lastly, Fig. 4.21 compares

two sample experimental profiles: The tcol = 4.7 ms profile (A) shows the anomalous features

associated with imaging off cloud centre and has a phase curvature that deviates significantly

from expected. On the other hand, the tcol = 6.3 ms profile (B) does not exhibit these features,

and has a phase curvature close to expected. Also see the extracted phase curvatures and


−150 −100 −50 0 50 100 150

0

0.5

1

OD

xslice

= 0 µm

−150 −100 −50 0 50 100 150

0

0.5

1

OD

xslice

= 160 µm

−150 −100 −50 0 50 100 150

0

0.5

1

OD

xslice

= 320 µm

y [um]

Figure 4.19: Sample 1D cloud profiles corresponding to full simulation data shown in Fig. 4.18.Shown are profiles imaged off cloud-centre (blue) and the on centre profile (green). The anoma-lous features observed in the data shown in Fig. 4.16 are also observed in the simulation forobject plane off cloud centre

comparison to theory in Fig. 4.22 .

The imaging simulations are able to explain all of the anomalous features observed in the

data, assuming the imaging is approximately 300 µm off cloud centre. There are several possible

explanations for how this could occur:

• The imaging system is focused by imaging a small cloud shortly after the optical trap

turn-off. By translating the imaging lens along the imaging axis, the object plane is

scanned. The lens position which gives the minimal cloud size is chosen. Given this

procedure and the typical data obtained, the focus can only be located to an accuracy of

∼ 50 µm. This also assumes that the probe is on resonance, otherwise lensing effects can

skew the inferred focal position.

• Our images are taken after 30 ms time-of-flight, during which the atoms fall under gravity

by a distance of 4.4 mm. This requires that we translate our imaging system vertically by

the same amount. Any slight angle between our translation and the direction of gravity

will result in a shift in object plane. An angle of 4 degrees would result in a displacement

of 300 µm. In our setup, the imaging system is mounted on an optical table extension,

forming a diving board which can bend and sag under its own weight [30]. Furthermore,

there is no guarantee that the table itself is level, especially with the majority of the

weight on one side of the table. A rough measurement estimates the angle to be 1.

• As noted, the atoms fall 4.4 mm. However the imaging system was typically translated


−500 0 500

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

Ph

ase

cu

rva

ture

, α

[π/u

m2*1

e3

]

Imaging plane, x [um]

0 2 4 60

1

2 x = 0 um

α

0 2 4 60

1

2 x = 160 um

α

0 2 4 60

1

2 x = 320 um

α

tcol

[ms]

Figure 4.20: Extracted phase curvature from off cloud centre imaging simulation. Left: Cur-vature is plotted for various collision times from 2 ms to 7 ms, and compared to theory (dashedline). The curvature drops rapidly when imaging away from the cloud centre. Right: Changein extracted curvature versus collision time for imaging at various object planes (red line). Thisis compared to the simple theory described by equation 3 (blue line).

−50 0 50

0

0.5

1

1.5

2

2.5bc20 − t

col = 4.7 ms

y [µm]

(a)

−50 0 50

0

0.5

1

1.5

2

2.5mg01 − t

col = 6.3 ms

y [µm]

(b)

Figure 4.21: Comparison of profiles with and without anomalous features. (a) tcol = 4.7 msprofile exhibits the anomalous features, and a phase curvature 33% from expected. (b) tcol =6.3 ms profile does not exhibit these features, and has a phase curvature consistent withinexperimental uncertainty with the expected curvature. See extracted phase curvatures andcomparison to theory in Fig. 4.22.


only 3.0 mm, and the probe was angled in such a way that the it passed through the atoms

and onto the CCD to compensate. This was necessitated by the fact that the absorption

probe beam must also pass through the barrier objective. It is not entirely clear how

this could affect the imaging. A simple model is to assume we are simply imaging off

the axis of the imaging system. Away from the imaging axis, the object plane (for fixed

image plane position) shifts towards the imaging lens. This shift follows a radius called

the Petzvel radius. For our imaging system, this provides a shift of 40 µm.

Given the consistency of the results of the imaging simulations with our data, and the arguments

provided above, it does not seem unreasonable to say that our imaging is the source of the

discrepancy between data and theory. Unfortunately these simulations were performed after

the apparatus had long been re-purposed for the waveguide experiments.

To summarize, off cloud-centre imaging of the interferogram results in distinct diffraction

features, and a reduction in extracted experimental fringe curvature from expected. These

imaging artifacts are well understood and entirely reproduced by our imaging simulations.

Thus we compare our extracted fringe curvature to that predicted from our simulations for a

range of imaging planes from on-centre to 300 µm off-centre. Note the tcol = 6.3 ms data set,

where the T-I interference effect is most visible, is consistent with imaging on-focus.

4.6 Phase profile reconstruction

The observed momentum distributions during the collision exhibit a rich structure, containing

both amplitude and phase information about the single-particle momentum-space wavefunction

prior to collision. Both T-I and P-I effects result in interference between momentum components

initially at ki and scattered by the barrier at tcol to kf , and un-scattered components which

have momentum kf . The interference depends on the relative phase accumulated along the

respective trajectories as described by Equation 4.4. This expression neglects the dynamics

during the non-linear expansion. While accounting for the non-linear expansion is important

for describing the pre-collision momentum distribution, it provides only a minor contribution to

the phase profile when compared to the contribution from free expansion since tcol 1/ωy. By

studying the fringe pattern we can extract the phase profile of the single-particle wavefunction.

To illustrate, we estimate the phase profile of our expanding condensate, as shown in Fig. 4.22.

To do this, we mark the maxima and minima of each P-I interference feature in the observed

profiles and assign a relative π-phase shift between adjacent fringes. For condensate expansion,

a quadratic phase profile with a curvature that increases with the pre-collision time is expected

for both momentum- and position-space distributions [26]. The position-space phase profile

after TOF is φ(z) = αz2, where the phase curvature is

α ' m

2h

(1

ttot − tcol− 1

tcol

). (4.6)


−200 −100 0 100 200

0

0.05

0.1

0.15

0.2

y [um]

|Ψ|2

(a)

0 20 40 60 80 100 1200

2

4

6

8

∆y [um]

∆φ [

π]

(b)

0 1 2 3 4 5 6 70

0.5

1

1.5

2

2.5x 10

−3

tcol

[ms]

Ph

ase

cu

rva

ture

[π/u

m2]

(c) α

Sim. − on−centre

Sim. − 150 um off−centre

Sim. − 300 um off−centre

Figure 4.22: Extracting the phase profile from experimental data. (a) Sample profile for tcol =3.3 ms, where each peak/trough is marked and a relative π phase is assigned. (b) The phaseprofile is fitted to a quadratic function. (c) The fitted phase curvatures are plotted againstpre-collision time. Thick black line indicates expected free-expansion curvature, α. Red linesindicate curvature extracted from simulated data for imaging on cloud centre, and increasinglyoff-centre (top to bottom; see text). The colour and symbols of fringe data in (b) match thatof the fitted curvatures in (c).

We find close agreement between α and the phase curvature extracted from simulated data

using the above technique, as shown in Fig. 4.6.

The technique used here to reconstruct the expanding condensate phase profile can be used

for measurement of arbitrary single-particle phase profiles. This interferometric approach bears

some similarities to existing condensate phase profile reconstruction techniques [73, 68]. In

contrast to these approaches, our technique, in principle, allows for single-shot reconstruction,

provided the scattering potential is sharp enough. For sharp scattering potentials, T-I interfer-

ence dominates, symmetrically scattering the central momentum component across the cloud.

Thus the central momentum component acts like a local oscillator, beating against the other

momentum components of the condensate simultaneously. The sharper the barrier, the less P-I

interference there is, and the cleaner the signal. On the other hand, for broad scattering po-

tentials the P-I interference dominates. Assuming a symmetric phase profile, it is still possible

to extract the full phase information, as we have shown for condensate expansion. When using


P-I interference, detailed knowledge of the scattering potential and its effect on the wavepacket

is required to accurately recover the relative phase between momentum components. Lastly,

it is interesting to note that since the fringe visibility is linked to the off-diagonal elements of

the density matrix, this technique may be extended to perform full tomography on the single-

particle state. This could be used as a measure of the onset of many body-correlations in

ultracold systems, for example during the superfluid to Mott insulator transition [74].

4.7 Summary

This chapter described our first foray into studying the interaction of a Bose-Einstein condensate

with a sharp potential barrier. An experiment we initially believed to be quite straight-forward

somehow turned into a 7 year effort, spread over several iterations of the apparatus. The

effort however has not been for nothing. We have observed the non-asymptotic scattering of

a Bose-Einstein condensate from a sharp repulsive potential. Two distinct momentum-space

interference effects arise: one due to the momentary deceleration of particles atop the barrier,

and one due to the abrupt discontinuity in phase written on the wavepacket in position space,

akin to quantum reflection. These effects are transient, manifesting only during the collision and

vanishing in the asymptotic limits, and result in a rich interference pattern that may be used to

reconstruct the single-particle wavefunction. Although we have studied the scattering from a 1D

model potential, these scattering phenomena are general features of non-asymptotic scattering

from near-transparent potentials and should be present in higher dimensions. Furthermore,

the presence of the effects demonstrated do not depend on the specific shape of the scattering

potential, only the spatial size of features present and potential height, and are transient,

occurring only during the scattering event and disappearing in the asymptotic limits. Lastly,

beyond the science explored, this first experiment forced us to work out many of the bugs in the

machine and led to the development of new tools, such as the optical dipole trap and barrier,

which will prove useful for future experiments.

Chapter 5

Waveguide experiments:Towards measurement of tunneling times of a

Bose-Einstein condensate

The study of tunneling and tunneling times has always been one of the main goals of the

Bose-Einstein condensation apparatus. The question of how long a particle takes to tunnel

through a potential barrier has a long history, and has seen its share of controversy. The root

of the problem lies in the absence of a Hermitian time operator in quantum mechanics. As

a result, a number of proposals for tunneling times have emerged based on different physical

scenarios. Over time, some of these proposals have been rejected, and others have found to be

linked. Presently, a handful of tunneling times remain in discussion. Tunneling times have been

measured in systems exhibiting frustrated total internal reflection [75, 76], microwave guides

beyond cutoff [77, 78, 79], and in photonic band-gap materials [80, 81]. These measurements

have confirmed the superluminal behaviour of the group delay and the Larmor time. In contrast,

there have been very few experiments with massive particles. This is largely due to the differing

energy scales involved in the problem. The tunneling of electrons has seen some attention in

solid state systems [82], however their electric charge and the typically high densities render

it difficult to extract a single particle measurement. Recently, interest has been re-ignited by

experiments in the Keller group at ETH Zurich, studying tunnel-ionization [83].

This chapter describes the work to date towards realizing a measurement of tunneling times

using an ultracold cloud of atoms. I begin with a brief discussion of candidate tunneling times,

with a particular focus on the Buttiker-Landauer traversal time and its potential implemen-

tation. The tools developed to realize this experiment are described. The final sections will

describe the initial experimental results using the developed apparatus, and the complications

which arose.

75

Chapter 5. Waveguide experiments 76

5.1 Tunneling times

5.1.1 A brief overview

MacColl (1932) was one of the first to address the tunneling time problem, introducing the

concept of the group delay [84]. He defined the group delay as the time interval between the

arrival and emergence of a wavepacket peak from the front and back interfaces of a barrier.

Einsenbud (1948), Bohm (1951), and Wigner (1955) later generalized the result, showing how

it could be calculated from the transmission phase shift [85, 86, 87]. Hartman (1962) would

highlight the difficulty with the group delay, showing explicitly that the delay time becomes

independent of barrier thickness beyond the evanescent decay length [88]. Some have taken this

to argue that tunneling can be superluminal and violate causality [78, 77], an interpretation

which others strongly reject [21, 89].

Later work moved beyond the group delay, introducing new timescales. Smith (1960) pro-

posed the dwell time, defined as the probability to find a particle in the barrier divided by

the incident flux [90]. While the dwell time seems intuitively appealing, it fails to distinguish

between transmitted and reflected particles. Baz and Rybachenko’s Larmor time (1967) uses

the interaction of an an electron with a magnetic field localized within the barrier as a clock

that ticks only during the tunneling process [91, 92], thus encoding the timing information in

the spin degree-of-freedom of the electron. Buttiker and Landauer’s semiclassical time (1982),

on the other hand, uses the barrier itself as a probe of the tunneling time [93]. By modulating

the barrier height and studying the transmission as a function of the modulation frequency,

the semiclassical time can be inferred. A more recent proposal from Sokolovski (1995) takes

a Feynman path integral approach, summing over the time durations of each classical path

weighted by the amplitude for each path [94]. Due to the relative phase accumulated along

different paths, this time is in general a complex value.

Many of the proposed tunneling times are connected in some way. The real part of

Sokolvski’s complex time is found to be equal to the in-plane component of the Larmor time

and the dwell time; and the imaginary component of the complex time is equal to the out-of-

plane component of the Larmor time [95], and the semiclassical time. This seems to suggest

that these varied approaches to estimating the tunneling time are really characterizing different

aspects of the tunneling process. Steinberg (1990) used the weak measurement formalism to

show how these real and imaginary components were linked to measurement and measurement

back-action [96, 97].

The timescales described in the sources cited above represent some of the major contribu-

tions, however it is by no means an exhaustive listing. See [20, 21, 22] and references therein

for a more complete review of tunneling time proposals. It is also worth noting that there

have been some efforts to define a quantum mechanical time operator [22, 98]. Nonetheless,

it is obvious that a host of timescales have emerged where classically a single timescale would

suffice.


E+ΔE

E-ΔE

E

ω

Figure 5.1: Buttiker-Landauer tunneling time. Particles tunnel through a barrier with heightthat oscillates at frequency ω. The oscillations write energy sidebands on the carrier. Thebeating between these components gives rise to fringes in position space. As the modulationfrequency increases, a sideband asymmetry develops in the transmission which marks the criticalfrequency ω = 1/τBL.

5.1.2 The Buttiker-Landauer time

Buttiker and Landauer’s semiclassical time is of particular interest to us, since its physical

implementation is relatively straight forward in our experimental setup. Here I describe the

timescale in detail and its potential implementation.

Buttiker and Landauer proposed a method to measure the tunneling time by using the

barrier itself as a probe [93]. They imagined applying a small modulation on the barrier height

at frequency ω and studying the particle transmission. The time-dependent potential is

V (x, t) = V0(x) + V1(x) cos(ωt), (5.1)

where V0(x) is the barrier height, and V1(x) the modulation amplitude. For modulation periods

slow compared to the timescale of the motion of the atoms through the barrier, ω 1/τ , each

atom sees only a small part of the barrier cycle and thus interacts with an effectively static

barrier. In the limit of rapid modulation, ω 1/τ , each atom sees many barrier cycles and

the transmission acquires energy sidebands due to the absorption and emission of modulation

quanta. Buttiker and Landauer argue then that there is a critical frequency which separates

these two regimes, and this frequency sets a timescale which is natural to call a traversal time

τBL ∼ 1/ωc. In the limit of weak modulation and opaque barriers, the traversal time for a

particle with kinetic energy E is

τBL =

∫ x2

x1

=m

hκ(x)dx, (5.2)

where κ(x) =√

2m(V (x)− E)/h2 is the under-barrier wavevector, and x1 and x2 are the

classical turning points. For a rectangular barrier with height V0, τBL is the time it would take

for a classical particle with kinetic energy V0 − E to travel a distance d = x2 − x1. For this


reason, the Buttiker-Landauer time is also referred to as the semi-classical time.

For an incident particle of energy E and momentum hk, the barrier modulation generates

sidebands with energy E± = E ± hω and momentum hk± =√

2mE±. To first order in the

modulation depth, the transmitted wavefunction is

ψ(x, t) ≈ Deikx−iEt/h +D+eik+x−iE+t/h +D−e

ik−x−iE−t/h, (5.3)

where D and D± are the transmission amplitudes for the carrier and first-order sidebands,

respectively. For a rectangular barrier of width d, the sideband amplitudes are related to the

carrier amplitude,

D± = −DJ±(V1

hω

)e∓i

mωd2hk(e±ωτBL − 1

), (5.4)

where J±(x) is the Bessel function of the first kind of order ±1. Equation 5.4 is valid in

the limit hω E and hω V0 − E, such that k± ≈ k ± mω/hk and κ± ≈ κ ∓ mω/hκ.

The Buttiker-Landauer timescale becomes evident when one looks at the first order sideband

asymmetryT+ − T−T+ + T−

= tanh(ωτBL), (5.5)

where T± = D∗±D± is the sideband transmission probability. The sideband asymmetry is

plotted in Fig. 5.2(a).

In our experimental scheme, directly measuring the sideband asymmetry is difficult. Al-

though time-of-flight allows us to measure the momentum distribution, the momentum reso-

lution required is unrealistic. The sidebands typically travel at velocity δv = ω/k relative to

the carrier, and for our parameter regime this is on the order of 0.1 mm/sec. It would require

100 ms of TOF in order to separate these components by just 10 µm. Another possibility is to

use velocity-selective Bragg scattering to map out the momentum distribution. This approach

would require multiple measurements to reconstruct the full spectrum.

A potentially more straight-forward observable is the fringe visibility in the transmitted

distribution. These fringes arise due to interference between the sidebands and the carrier. In

the limit of hω E0, the transmitted probability density is

|ψ(x, t)2| ≈ |D|2 + |D+|2 + |D−|2 − 2(|D||D+|+ |D||D−|) cos

(mω

hk

[x− d

2

]− ωt

). (5.6)

The interference between the positive and negative sidebands that lead to oscillations at 2ω has

been neglected here. The 1ω fringe visibility is

V ≡ 2(|D||D+|+ |D||D−|)|D|2 + |D+|2 + |D−|2

(5.7)

V ≈ 2V1

hωsinh(ωτBL) (5.8)


0 100 200 300 400 500 6000

0.2

0.4

0.6

0.8

1

Sideband asymmetry

ω/2π [Hz]

(a)

0 100 200 300 400 500 6000

0.1

0.2

1ω fringe visibility

ω/2π [Hz]

(b)

Figure 5.2: Buttiker-Landauer tunneling time observables. (a) Sideband asymmetry and (b) 1ωfringe visibility versus modulation frequency for a rectangular tunnel barrier with V0 = kB×50nK, V1 = 0.01 × V0, E = kB × 30 nK, and barrier width d = 1.0, 1.5, and 2.0 µm (red,blue, green). Plotted is the numerical calculation (solid line) and the analytical approximation(dashed line). Solid dots mark the critical frequency characterizing transition from low to highfrequency behaviour.


Thus, the Buttiker-Landauer timescale can also be extracted by monitoring the fringe visibility

as a function of modulation frequency. The 1ω fringe visibility is plotted in Fig. 5.2(b).

The increasing visibility of fringes with modulation frequency arises due to the preferential

transmission of the upper sideband. For small modulation frequencies, hω E, the sidebands

have similar energies to the carrier. As the modulation frequency increases, the energy dif-

ference from the carrier increases. The transmission probability of the upper sideband grows

exponentially, while the lower falls. Thus the upper sideband is preferentially transmitted, and

the ratio of upper sideband to carrier increases, as described by Equation 5.5.

The characterization of ωBL as the frequency for which the transmitted wavefunction is

perturbed suggests that the correct interpretation of the Buttiker-Landauer timescale is in fact

that of a measurement back-action. The barrier modulation modifies the transmitted state

wavefunction, preferentially transmitting the upper sideband with momentum hk+, analogous

to how the magnetic field in the Larmor time causes preferential transmission of the +z spin

state, resulting in a out-of-plane component of the Larmor time [95]. Thus it is not surprising

that the semiclassical time is equivalent to the imaginary part of the complex time. This leads

one to question whether there is an analogous ‘real’ component of the semiclassical time. Fol-

lowing the Larmor time, it seems that the relative phase between sidebands should correspond

to the real part of the semiclassical time, and the true measurement result. These ideas are spec-

ulative, and need further investigation to check their validity. However, if a real component of

the semiclassical time could indeed be found within Buttiker and Landauer’s proposal, it would

be conceptually very satisfying, further connecting it to the other tunneling time proposals and

fitting in with the general framework of quantum measurement.

5.2 Experimental scheme and apparatus

The goal of the experiment is to study tunneling and tunneling times in a one-dimensional scat-

tering configuration. This configuration is readily generated using optically induced potentials

for our Bose-Einstein condensate.

The proposed experimental sequence is as follows: We begin by loading a BEC in a waveg-

uide, generated by a single red-detuned gaussian beam (see Section 3.6). The waveguide poten-

tial has tight transverse confinement and weak longitudinal confinement, thus all the relevant

dynamics occur along the longitudinal direction and ideally the dynamics can be considered

one-dimensional. The prepared matter wavepacket is then briefly accelerated along the waveg-

uide by pulsing on a magnetic field gradient, setting the incident kinetic energy. The wavepacket

impinges upon a potential barrier, generated by a tightly focused blue-detuned sheet of light

intersecting the waveguide. Great care is taken in designing the optics for shaping the barrier

beam to achieve the tightest focus possible, and thus obtain a detectable signal of tunneling.

In this setup a number of the proposed tunneling times can in principal be measured. For

example, the Buttiker-Landauer time can be measured by modulating the barrier height and


t1

t2

t3

zx

z

Figure 5.3: Scheme for studying tunneling with a Bose-Einstein condensate in an opticallyinduced waveguide. A wavepacket is prepared to the left of the barrier. The initial momentumis controlled by brief application of a magnetic field gradient. The wavepacket impinges uponthe tunnel barrier, partially reflecting and partially transmitting.

observing the fringe visibility in the transmitted particles, and the Larmor time can be mea-

sured by introducing Raman coupling beams within the barrier region and measuring the spin

rotation of the transmitted particles.

5.2.1 2nd generation barrier

Optical design

Our 1st generation barrier (see Section 4.3.1), used for the scattering experiment, was generated

from a 780 nm beam and had an 1/e2 radius of 2.2 ± 0.1 µm. This barrier thickness is too

large to resolve the tunneling signal within our experimental regime. The width of the beam

is set by the geometry (characterized by the numerical aperture), the quality and design of

the optics (characterized by the amount of optical abberations), and ultimately the diffraction

limit (determined by the wavelength). Our system has NA≈ 0.2, and is already near maximal.

However the other factors can be improved upon. A 2nd generation barrier was designed, based

on 404 nm light that addresses the 4S → 5P 420 nm transition [99]. This beam provides

nearly a factor of two reduction in the diffraction limit. This light is generated by a Mitsubishi

ML320G2-11 laser diode with 120 mW output. The design of the optics was developed with the


assistance of the ray-tracing software OSLO, resulting in a near-diffraction limited design using

standard off-the-shelf optics available from Thorlabs. Although the design is near-diffraction

limited, the performance of the constructed objective is limited by the tolerances of the optics,

and the spacing of the optical elements. The latter in particular is believed to be the largest

source of deviation from the designed performance, which is sensitive to 100 µm relative shifts

in the centration and axial position of the optical elements. An alternative to building our own

objective would have been to purchase a professionally produced custom objective. This would

have been significantly more expensive, and given the tight physical space available, finding a

manufacturer able to provide us with what we needed at a reasonable price was unrealistic.

The design and construction were carried out by Shreyas Potnis. The final OSLO objective

design is shown in Fig. 5.4. The optical layout is similar to that of the 780 nm barrier, and

is shown in Fig. 5.5. The first step is to pass the light through an optical fibre which acts

primarily as a single spatial mode filter. The beam is then passed through and acousto-optic

deflector (Intraaction AOD-70), from which the 1st order diffraction is taken. This provides

both power control, and the ability to dynamically translate the beam at the focus. Next,

the beam is expanded using a cylindrical telescope with aspect ratio 20. Lastly, the custom

objective focuses the beam into a tightly focused, highly elliptical shape. The beam is integrated

into the existing apparatus by passing through the absorption probe line. Since the absorption

probe is only required to be a relatively large collimated beam, the optics are relatively easily

modified to accommodate the blue barrier. Like the 780 nm barrier beam, the 404 nm beam

profile is measured outside the vacuum system using a scanning knife-edge technique. The test

setup includes a single uncoated glass slide, used to mimic the walls of the vacuum system

cuvette. It was found that the presence of this slide did not noticeably affect the beam shape.

The results of the beam profiling are shown in Fig. 5.6. The profiling reveals clear signatures of

optical aberration. This is most notable off-focus. On-focus however, the profile is reasonably

approximated by a gaussian with 1/e2 width w0 = 1.1 ± 0.1 µm. This is a factor of two

improvement over the original 780 nm barrier beam, and should put us in a regime where

tunneling is visible.

Barrier calibration

For the 404 nm beam, the closest transition is the 4S → 5P transition at 420 nm [99]. However

the strength of this transition is small, with a narrow Γ = 2π· 0.29 MHz linewidth. Surprisingly

it turns out that the dominant interaction is with the familiar 780 nm 5S → 6P transition.

Similar to the 780 nm barrier, the 404 nm barrier was calibrated by reflection of a BEC falling

under the influence of gravity. In particular, we are interested in the beam waist at the focus.

By measuring the 50% transmission points for several different incident wavepacket energies,

and comparing to calculation, the beam intensity can be extracted. The intensity is roughly

I ≈ 2P/√

2πw0L, where P is the power in the beam and L is the effective transverse length

of the barrier generated by the AOD scanning. The beam power in the vacuum chamber is


Figure 5.4: 2nd generation barrier objective design in OSLO. The design is based on λ = 404nm light, and is near diffraction limited with a focused spot size w0 = 0.7 µm.

M=20 cylindrical telescope

f = 45 mmcustom objective

x

z


Absorption probeoptical path

AOD

Barrier PD

Figure 5.5: 2nd generation barrier optical layout. The optics produces a beam of 1/e2 widthσy = 1.1 ± 0.1 µm, σz ∼ 14 µm. The AOD may be scanned at 100 kHz to generate atime-averaged potential which can be used to flatten the potential along y, or to generate amulti-barrier structure along z.


18

20

22

24

26

4050607080

z [µm]

x [

µm

]

40 50 60 70 801

2

3

4

z [µm]

w0 [

µm

]

(b)(a)

18 20 22 24 261.5

2

2.5

3

3.5

x [µm]

I

(c)

18 20 22 24 260

0.5

1

1.5

x [µm]

dI/

dx

(d)

Figure 5.6: 2nd generation barrier beam profile, measured by scanning knife method with100 nm resolution. (a) 2D map of beam profile near focus. (b) Fitted 1/e2 beam waist. (c)Integrated intensity profile at focus. (d) Derivative of integrated intensity profile, taken to bethe beam profile. Red line is a gaussian fit yielding a minimum beam width w0 =1.1 ±0.1 µm.

carefully estimated by using the test setup, including the glass slide to account for losses at the

interface of the of vacuum cell. We expect this gives us the power inside the vacuum chamber

to better than 5%. The effective beam length is estimated by imaging the beam directly onto

the imaging system CCD. This profile is approximately trapezoidal, from which we extract an

effective length which corresponds to a flat profile with the same area and height. Putting this

all together, and comparing to the reflection data (example data shown in Fig. 5.7), we estimate

the barrier width to be w0 = 1.19± 0.04 µm, consistent with the beam profiling

Assuming a gaussian barrier profile of width w0 = 1.1 µm, we calculate the transmission

function. This is done numerically using a transfer matrix approach [100]. This approach breaks

each section of the spatial potential V (x) into a flat regions located at xi and with height V (xi).

In these regions, the form of the wavefunction is know to be ψ(x) = A exp ikx+B exp −ikx,where k =

√2m(E − V )/h2 is the local wavevector. The boundary conditions between sections

is imposed, to produce a system of linear equations which may be solved by specifying the initial

conditions. In this way, the transmission function can be evaluated for arbitrary V (x). The

results are plotted Fig. 5.8.

5.2.2 Waveguide

The waveguide beam is generated by a single 1064 nm beam which primarily addresses the

5S → 6P transitions at 780 and 795 nm. This light comes from the 15 W fibre laser system


0

0.5

1

t = 1.92 ms

0

0.5

1

Fra

ctio

nal re

flectio

n a

nd tra

nsm

issio

n

t = 2.40 ms

5 10 15 20 25

0

0.5

1

t = 2.90 ms

Barrier power [mW]0 1 2 3

0

5

10

15

20

25

30

Time to barrier [ms]

50%

reflectio

n p

ow

er

[mW

]

Figure 5.7: Calibration of 2nd generation barrier by reflection of a free-falling condensate.Left: Sample transmission (red) and reflection (blue) curves, fitted to an error function. Right:The 50% reflection points are plotted against the BEC free-fall time to reach the barrier. Theexperimental data is compared to theory, giving an estimated barrier width of w0 = 1.19 ±0.04 µm.


0 20 40 60 80 1000

0.5

1V

0 = 50 nK

2 3 4 50

0.5

1v

0 = 3.1 um/ms

50 100 1500

0.5

1V

0 = 100 nK

3 4 5 60

0.5

1v

0 = 4.4 um/ms

100 120 140 160 180 2000

0.5

1V

0 = 150 nK

Incident energy [nK]4 5 6 7

0

0.5

1v

0 = 5.4 um/ms

Incident velocity [um/ms]

Figure 5.8: Calculated 2nd generation barrier transmission function, numerically calculated bythe transfer matrix method. Barrier profile is assumed to be gaussian of width w0 = 1.1 µm.Plotted are the transmission curves for barrier heights V0 = kB× 50, 100, and 150 nK (red,blue, green solid lines) versus incident energy and the corresponding velocity. The dashed linesindicate the corresponding classical transmission.

(see Section 3.6), and is the same beam used for optical trapping and cooling stages of the

hybrid trap ramp. The design of this beam thus needs to balance the needs of the condensate

preparation stage with the experiment stage. In the former, a tight confinement is desired

to maintain a high collision rate necessary for efficient evaporative cooling; during the latter,

relatively weak transverse confinement is desired to reduce the effect of interparticle interactions.

For a single gaussian beam trap, the principal design parameter is the 1/e2 beam waist, w0. As

discussed in Section 3.6, the total potential in absence of the quadrupole field is

V (x, y, z) = −V0(z) exp

(−2

x2 + y2

w2(z)

)+mgy, (5.9)

where V0(z) is the optical trap depth along the beam axis, and w(z) is the beam transverse

waist. For deep traps, the potential is approximated to be harmonic near the trap minimum,

giving trap frequencies

ωx,y =

√4V0

mw20

(5.10)

ωz =

√2V0

mz2R

(5.11)


The potential exhibits tight confinement in the transverse directions, and weak longitudinal

confinement due to the beam divergence. For shallow traps, the gravitational potential begins

to play a role, causing a sag in the y-direction. The trap depth is defined as the potential

difference between the local minimum and the local maximum caused by the gravitational tilt,

and there exists a minimum beam power in order for a local minimum to be present and for a

trap to exist.

After satisfying the basic condition of the existence of a local minimum in the potential, the

next concern is the momentum width of the matter wavepacket. Ideally, the particles incident

on the barrier would be monochromatic. This would allow us to approach arbitrarily close to the

threshold for classical transmission without exceeding it, to observe significant tunneling. For

example, for the kB×100 nK barrier plotted in Fig. 5.8, an particle incident with energy kB×90

nK would have a tunneling probability of 20%. A finite momentum width will blur out the

transmission as a function of incident energy, and makes it difficult to unambiguously observe

tunneling. For example, for the sample calibration data shown in Fig. 5.7, the transmission

curves qualitatively resemble that of the calculated monochromatic transmission function of

Fig. 5.8. Any claims of having observed tunneling then require careful quantitative comparison

between the classical and quantum transmission for a wavepacket. In other words, there may

not be a clear and distinct signature we can use to verify the observation of tunneling.

To avoid this situation, we would like to be in a regime where there is negligible classical

transmission. Thus a finite momentum width sets a minimum energy scale for the tunneling

problem, barrier heights V0 must be chosen to be greater than the sum of the incident energy

and the width. Since the tunneling probability falls exponentially with the energy difference

V0 − E and linearly with the energy ratio E/V01, the most important design consideration

on the waveguide for the experimental stage is to minimize the momentum width. In this

regard, the repulsive interparticle interactions play a prominent role. Upon release into the

waveguide, this meanfield energy is rapidly converted into kinetic energy, determining the final

wavepacket momentum width as it impinges upon the barrier. For times long compared to

1/ωz, the interparticle interactions are negligible and the condensate wavefunction is composed

of a superposition of non-interacting plane waves. In 1D the wavefunction is

ψ(z, t) =1√2π

∫dkψ(k, t)eikz. (5.12)

The momentum distribution is found using the Thomas-Fermi scaling solution [26], and is [101]

P (k) = |ψ(k, t)|2 ≈ 3N

4kmax

(1− k2ξ2

init

)Θ(1− kξinit) (5.13)

where Θ(k) is the heaviside step function. The momentum distribution exhibits a high-

1True of square tunnel barriers, T ≈ 16 EV0

(1 − E

V0

)exp

−2√

2m(V0 − E)d/h

.


0 0.5 10

100

200

300

400

P [W]

fx [Hz]

0 0.5 10

100

200

300

400

P [W]

fy [Hz]

0 0.5 10

0.5

1

1.5

2

2.5

3

P [W]

fz [Hz]

0 0.5 10

2

4

6

8

10

P [W]

Tdepth

[uK]

0 0.5 10

1

2

3

4

5

6

7

P [W]

vTF

[um/ms]

0 0.5 10

20

40

60

80

100

P [W]

U0,int

/kB [nK]

Figure 5.9: Waveguide properties for focused beam spot sizes w0 = 50, 60, 70, 80, 90, and100 µm (plotted in colours cyan, magenta, black, green, blue, red). Plotted are the waveguidetrapping frequencies (fx, fy, fz), trap depth, and the condensate peak Thomas-Fermi velocityand peak meanfield energy V0,int assuming an initial 10 Hz longitudinal confinement and N =100 k atoms.

momentum cut-off, given by the inverse of the initial healing length,

kmax =1

ξinit=

√4mµ

h(5.14)

This momentum width is significantly greater than that of the single-particle groundstate.

Figure 5.9 plots the numerically calculated properties of the waveguide versus beam power

for several focused spot sizes, ranging from 50 to 100 µm. The effect of the gravitational

potential can be seen for low beam powers, distorting the trap, and imposing a minimum

power to maintain a potential minimum. The velocity width vTF = hkmax/m and the peak

meanfield energy V0,int = gn0 are also plotted for a condensate of N = 105 atoms and an initial

longitudinal confinement of 10 Hz. These numbers suggest that the barrier height must be at

least on the order of kB× 100 nK to avoid significant classical transmission.

A choice of w0 = 75 µm was made as a compromise between the weak confinement required

for the experiment stage and reasonable trap depths for the preparation stage. However, it

is clear that the transverse confinement is still significant enough to produce longitudinal mo-

mentum widths comparable to the desired barrier heights. The role of interactions and the

minimum confinement required by the gravitational tilt, significantly affects our ability to per-

form the experiments as planned. Figure 5.8 shows that in our parameter regime, there is only a


window of about 0.5 µm/ms in incident velocity over which we can observe tunneling. Without

additional tricks to reduce the momentum width, it seems that the experiment as it stands is

not possible. One possibility is to make use of a Feshbach resonance to control the strength

of inter-particle interactions. The details of the apparatus developed to access the 87Rb Fesh-

bach resonance, and a discussion of the associated experimental challenges, have already been

presented in Section 3.4.1. Another possibility is to create a potential that balances against

gravity, thereby allowing for significantly weaker transverse confinement. Section 5.4 describes

our efforts to provide gravity compensation by using the existing quadrupole coils. A third

possible solution is to filter the incident momenta prior to interaction with the tunnel barrier.

This proposal is discussed in Section 5.3.

5.2.3 Magnetic push coils

To set the wavepacket in motion, we pulse on an external force using a magnetic field gradient.

~F = −∇V = gmFµB∇ ~B (5.15)

This gradient is generated by coils oriented along z and wound around the quadrupole coils

which were originally used as bias coils (see Section 3.4). These coils generate a magnetic field

gradient at the position of the atoms ∇ ~B = B′pz, and a force on the atoms ~F = gµBmFB′pz.

Depending on the desired range and direction of the force, one coil alone or two coils in anti-

helmholtz configuration can be used. A circuit was developed to generate field gradients,

with the main design considerations being current pulses of up to 10 A, and pulse-to-pulse

repeatability of better than 1%. To generate the large currents necessary, a bank of 8 super-

capacitors are used, each with capacitance of 1 F. These capacitors are slow to charge, but still

fast compared to the experiment cycle time of 40 seconds. The discharge of the capacitors is

controlled by a high current FET, which is gated by an operational amplifier with a negative

feedback signal from a shunt resistor sampling the current generated. See Alan Stummer’s

documentation of this circuit on his website at http://www.physics.utoronto.ca/~astummer.

During testing, some quirks were found in the initial design. For large drives, the control

circuit would begin oscillating introducing noise to the applied field. This was compensated

for by slowing the response of the operational amplifier using capacitors. The result was a

relatively slow rise time of typically 200 µs. Furthermore, the rise and delay of the current

pulse was strongly dependent on the desired amplitude of the current pulse. However, since

our primary concern is the integrated acceleration leading to a final velocity, this rise time was

deemed satisfactory for this experiment. The repeatability of the pulses was tested, and verified

to be better than 1 part in 1000. The field gradient is calibrated by observing the centre-of-mass

velocity of a freely falling condensate after a field pulse.



2 4 6 8 1050

100

150

200

250

300

Vctrl

[V]

Ris

e t

ime

[u

s]

(a)

2 4 6 8 10500

1000

1500

2000

Vctrl

[V]

De

lay [

us]

(b)

Figure 5.10: Push circuit timescales. (a) Rise time, and (b) delay versus control voltage. A10 V control signal delivers 10 A of current. Solid lines are a guide to the eye.

5.3 Momentum filtering

In this section, I discuss some proposals to reduce the momentum width of the incident

wavepacket. These proposals assume that the interparticle interactions have dissipated and

can be neglected during filtering.

A matter-wave Fabry-Perot

The ability to rapidly scan the position of the barrier faster than the motion of the atoms has

the potential to create near-arbitrary 1D time-averaged potentials for the atoms to interact

with. This was first demonstrated in the previously described scattering experiment to extend

and flatten the time-averaged barrier potential. We can apply this here to generate a Bragg

structure for matter-waves. Using the acousto-optic deflector, the position of the barrier can

be rapidly scanned at megahertz rates, creating a time-averaged potential. A ramp can be

designed such that the barrier sits at discrete positions, separated by δz, for brief periods

of time. As long as the modulation cycle is fast compared to the motion of the atoms, the

potential appears as a static multi-barrier structure. The simplest case is a double-barrier

Fabry-Perot structure [102]. In analogy to the optical Fabry-Perot, the transmission spectrum

will exhibit a series of resonances that depends on the spacing between barriers, and with a

linewidth that depends on the barrier reflectivities. The resonance energies can be shifted by

changing the spacing between the barriers, and the width of the resonances tuned by changing

the barrier heights. This approach offers a highly tunable way to select out different momentum

components, without the need for adjusting the optics.

The transmission spectrum of a few example structures is numerically calculated using the

transfer matrix method, and shown in Fig. 5.11. One thing that is clear is that the width of the

resonances are quite narrow, around 0.2 µm/ms. Though such a narrow velocity distributions

would be ideal for the tunneling experiments, given the expected velocity widths of greater

than 2 µm/ms, a very small fraction of the wavepacket would be transmitted. Slightly broader

widths up to 0.5 µm/ms would be preferred. The sharpness of these resonances can be tuned by


0

0.5

1

Tra

nsm

issio

n

0

0.5

1

Tra

nsm

issio

n

0

0.5

1

Tra

nsm

issio

n

0

0.5

1

Tra

nsm

issio

n

0

0.5

1

Tra

nsm

issio

n

0

0.5

1

Tra

nsm

issio

n

20 40 60 800

0.5

1

Tra

nsm

issio

n

Incident energy [nK]2.5 3 3.5 4

0

0.5

1

Tra

nsm

issio

n

Incident velocity [um/ms]

Ba

rrie

r p

rofile V

0 = 50 nK, δz = 4 µm, 2 barriers

Ba

rrie

r p

rofile V


Ba

rrie

r p

rofile V


−30 −20 −10 0 10 20 30

Ba

rrie

r p

rofile

z−position [µm]

V0 = 50 nK, δz = 8 µm, 5 barriers

Figure 5.11: Transmission spectra for various matter-wave Bragg structures. Two barrierand five barrier structures for spacings δz = 4 and 8 µm and barrier height V0 = 50 nK areshown (first column). The transmission spectra versus incident kinetic energy (middle column)and the corresponding incident velocity (third column) are shown in blue. The single barriertransmission is shown in red for contrast.


increasing the number of barriers. Unfortunately, the width of these resonance windows can not

be tuned below a minimum set by the finite width of the barrier δz ≈ w0. Thus, although an

attractive proposal for a tunable matter-wave filter, this does not seem like a realistic approach

for our parameter regime.

Far-field shuttering

A less elegant, but potentially more effective filtering method is to create a time-dependent

shutter. Assuming the interaction energy has been fully converted into kinetic energy, the con-

densate wavefunction is composed of non-interacting plane waves as described by Equation 5.12.

In the far field, the momentum components are strongly correlated to their position. By using

a time-dependent barrier to abruptly switch between transmission and reflection, a specific ve-

locity class can be selected. By choosing the timing of the shuttering relative to the wavepacket

propagation, the central velocity and the width of the transmitted distribution can be selected.

An example shuttering sequence is illustrated in Fig. 5.12, in which the central momentum has

been selectively transmitted. Provided the wavepacket is in the far-field in which the position

and momenta are well correlated, the approach should be more flexible than the matter-wave

Fabry-Perot proposal. It falls to experiment to demonstrate the effectiveness of this approach.

5.4 First attempts: discovery of a lattice

With much of the experimental apparatus in place, we proceeded to attempt to demonstrate

a number of basic capabilities: wavepacket propagation, velocity control, and reflection from

the optical barrier while in the waveguide. The elements of the apparatus were assembled, and

tested separately. Conceptually, the implementation was clear. However, being experimental-

ists, we are aware that design and reality often do not quite match up. Indeed, we stumbled

upon an un-anticipated problem almost immediately.

The first test was to observe the free-expansion of the wavepacket in the waveguide. This

test would provide some data on the wavepacket momentum width, excitation of transverse

modes, and residual longitudinal confinement. We prepared a thermal cloud in the hybrid trap.

The quadrupole field was then adiabatically relaxed such that the longitudinal confinement

was about 10 Hz, defining the initial size of the wavepacket. Finally, the quadrupole field was

abruptly switched off, allowing for propagation in the waveguide. For condensate expansion

from a harmonic trap, the expansion should be self-similar. The shape of the cloud at all times

is expected to be an inverted parabola given by the Thomas-Fermi spatial distribution, and

the momentum distribution is described by Equation 5.13. What we observed was significantly

different. The data, shown in Fig. 5.13, shows a symmetric double peaked structure developing.

The distribution seemed to be composed of a dense central core, and a rapidly expanding

background. By observing the cloud in TOF, we confirmed there was no condensate present.


time

Shutter height

t1

t2

t3

(c)

t1

k

z

t2

k

z

t3

k

z

(a)k

k-projectionNo shuttering

(b) k

k-projectionShuttering

k

z

k

z

k

z

t1

t2

t3

x0

x0

x0

k0

k0

k0

k0

Figure 5.12: Illustration of momentum filtering by far-field shuttering. Illustrated arethe phase-space distributions at various times during the wavepacket evolution for (a) free-propagation, and (b) shuttering. (c) Shutter timing. The initial state at t1 has centre momen-tum hk0. As the wavepacket evolves, its centre-of-mass moves in the positive z direction, andthe distribution acquires a linear chirp. For shuttering, the barrier, located at x0, is initiallyhigh such that all incident wavepacket components are reflected. At t2 the fast tail of the distri-bution is reflected from +k to −k (negative momenta not shown). The barrier is turned-off fora brief period of time, allowing a fraction of the chirped wavepacket to pass, before turning backon and reflecting the remaining components. The final transmitted distribution at t3 exhibitsa much narrower spectrum than before, having selectively transmitted only those componentswithin the desired velocity range. This final momentum distribution is illustrated in the k-spaceprojection.


z−position [µm]

tim

e [m

s]

−400 −200 0 200 400

0

5

10

15

20

25

30

Figure 5.13: Expansion of a thermal cloud in the waveguide. The cloud temperature isestimated to be 130 nK. Each horizontal slice is an absorption image of a separate preparationof the cloud, and for variable expansion time in the waveguide.

As the atoms were cooled to lower temperatures, the dense core got larger while the background

got smaller. A condensate was observed to contain no broad, fast expanding tails. Next we tried

applying a force on the atoms, to see how each structure responded. To our utter confusion,

the dense central core did not move whatsoever. The broad background, on the other hand,

seemed to respond as expected.

After some consideration of the observations in sum, it was suggested that there was an

optical lattice present which was modifying the dispersion relation of the expanding particles.

This lattice was speculated to arise due to interference between the incident optical trap beam

and light reflected from the glass-to-air interfaces of the vacuum system. The beam geometry

is illustrated in Fig. 5.14. The following section presents the results of experiments used to test

this claim.

Lattice dispersion

We speculated that a bi-modal looking structure could arise in the presence of a lattice, which

modifies the dispersion relation of the expanding cloud. The measured temperature of the

atoms in the lattice was 130 nK. Assuming the cloud is thermal, the state can be described as a

statistical mixture of Bloch states completely filling the first Brillouin zone, and spilling over into

the 2nd zone. For reference, the expected photon recoil velocity is vr = hkr/m = 4.3µm/ms,

where kr is the recoil wavevector for the λ = 1064 nm waveguide light. The characteristic


2θ

δH

L

2δθ

vacuumair

z

x

Figure 5.14: Schematic illustration of formation of accidental lattice. The incident opticaltrap beam (red arrow) is partially reflected from the glass cuvette. The incident and reflectedbeam overlap at the position of the atoms, creating a standing wave interference pattern. Fora beam incident at angle θ, the specular reflection occurs at angle 2θ from incidence, and thuscan be avoided by angling the beam such that δH > w0. However a diffuse reflection (light redarrow) with angular spread δθ around the specular reflection remains. Given the small size ofthe cuvette (L = 2 cm) and the optical access (θmax ∼ 8), it is not possible to avoid the diffusereflection.


−2 −1 0 1 2

0

1

2

3

4

En

erg

y [

Er]

Momentum [kr]

(a)

−2 −1 0 1 2−10

−5

0

5

10

vg [

µm

/ms]

Momentum [kr]

(b)

Figure 5.15: (a) Dispersion relation and (b) corresponding group velocity for a free particle(red line) and for a shallow lattice of depth 2Er (black line).

2.5 sec RF, T = 130 nK

−200 0 2000

0.5

1

1.5

Position [µm]

(a)

2.0 sec RF, T = 340 nK

−200 0 2000

0.5

1

1.5

z−position [µm]

(b)

Figure 5.16: Structured velocity distributions in lattice due to group velocity dispersion. Clouddistribution after 19.3 ms of expansion in waveguide for cloud temperatures (a) 130 nK, and (b)340 nK. Arrows are a guide to the eye, marking the locations of dips in density correspondingto wavevectors k = ±kr and k = ±2kr

recoil energy is Er = h2k2r/2m = kB × 97 nK. For shallow lattices, the dispersion relation

resembles that of a free-particle except near the Brillouin zone boundaries where the lattice

coherently couples momentum states separated by the lattice recoil 2hkr. The group velocity

of any particle is given by the slope of the dispersion relation (Equation 2.20). The calculated

dispersion relation and group velocity for a shallow 2Er lattice are plotted in Fig. 5.15. The

resulting far-field distribution reflects the distribution of group velocities. The density of states

with group velocity vg = 0 is higher in the presence of a lattice than for a free-particle, resulting

in a build-up of particle density near the wavepacket centre. The central core consists of atoms

near k = 0 and k = ±kr, while the broad background consists of states with dispersion nearly

that of a free-particle. Over time, the zero group velocity points at k = ±kr are distinguished

as an absence of atoms in the far-field distribution. These notches in the atomic density after

a 19.3 ms expansion are shown in Fig. 5.16 for thermal clouds with temperatures 130 nK and

340 nK, corresponding to populations up to the 2nd and 3rd Brillouin zones respectively.


z−position [µm]

δH

[m

m]

−400 −200 0 200 400

−0.6

−0.4

−0.2

0

0.2

0.4

0.6−0.5 0 0.50

0.2

0.4

0.6

0.8

1

Re

lative

ato

m n

um

be

r

δH [mm]

Figure 5.17: Angle dependence of bi-modal expansion. The angle is converted into an expectedreflected beam displacement δH. The 1D profile is fit to the sum of two concentric gaussians,and the relative populations are plotted.

Angle dependence

A second test was to see if the effect could be suppressed by simply mis-aligning the incident

and reflected beams. This was done by angling the incident beam such that the reflection was

at least a beam waist away from the incident at the position of the atoms, δH > w0. A clear

variation in the relative amplitudes of the central core and broad background was observed with

incident angle (see Fig. 5.17). When aligned parallel to the cuvette face, the lattice depth is

highest, and the group velocity dispersion is expected to be flattest, hence the higher amplitude

of the central core. The angular width of the amplitude dependence was consistent with our

beam geometry and beam spot size w0 = 75 µm (see Fig. 5.14).

However beyond this angular width, the effect was observed to persist, seemingly inde-

pendent of incident angle. Based on the geometry of the setup, this cannot be accounted for

by specular reflection. This suggests that there is diffuse scattering which is giving rise to a

residual lattice. Diffuse scattering can occur if there is any roughness on the surface. This

too should have an angular dependence, depending on the level of roughness, and an intensity

which should fall off as approximately 1/r2 from the point of reflection. However, since our

cuvette is particularly small, there is only 0.8 cm between the atoms and the first reflecting

surface (inside face of cuvette), resulting in non-negligible diffuse scattering.

An attempt to measure the angular spread of the diffuse reflection was made by shining light

onto the surface of a cuvette, and imaging the scattered light for increasing numerical aperture.

The surface of the cuvette shows a seemingly random distribution of bright scattering points.

This structure may be due to many years of Rubidium build-up. Focusing on one of these

scattering points, and measuring the integrated intensity as a function of numerical aperture,

it should be possible to obtain an estimated of the angular spread δθ of the scattering from this

specific point. The results are shown in Fig. 5.18. These measurements were limited by the NA

of our imaging system, and we can only conclude that the δθ is at least 8. Unfortunately, the


0 2 4 6 840

60

80

100

120

Max acceptance angle [deg.]

Mean integrated intensity

(a)

0 2 4 6 80

5

10

15

Angle [deg.]

Mean intensity

(b)

Figure 5.18: Angle dependence of diffusely scattered light. Measurements are limited by thenumerical aperture of the imaging system to 8. (a) Integrated intensity as a function of NA.(b) Numerical derivative of the integrated intensity

optical trap beam is limited by the same optical access, and it seems that the diffuse reflection

can not be avoided by changing the incident angle. A brief attempt was made to use light-

induced atom desorption [103] to remove the structure with 404 nm. The light source had an

estimated intensity of 10 mW/cm2, and was exposed to the glass cell for a period of 2 hours.

No clear change in the diffuse scattering was observed.

Matter-wave scattering from a periodic potential

Thus far we have studied how broad momentum wavepackets behave in the waveguide poten-

tial. A more dramatic verification of the lattice should be possible with a narrow momentum

wavepacket. We prepare a Bose-Einstein condensate, release it from its trap, pulse on the

waveguide beam for 60 µs, and look for matter-wave diffraction. The initial observations are

displayed in Fig. 5.19, where a clear diffraction patter is observed. By abruptly pulsing on the

lattice, we project the atoms from the initial free-particle states, to Bloch states, and back.

This process is diabatic because our optical pulse duration, τ = 60 µs, is short compared

to the inverse of the expected lattice bandgap (1/∆ ∼ 1 ms). The resulting scattering is of

the Kapitza-Dirac type [104] and couples our transverse E = 0, k = 0 state to E = ±2nEr,

k = ±2nkr states, where n an is the integer number of lattice recoil quanta exchanged. In this

fast pulse regime, the bandwidth of the energy exchange is broadened by δE ∼ h/τ = kB × 130

nK, allowing for inelastic scattering.

The angular dependence of the scattering signal is found to be consistent with the bi-modal

expansion measurements. However in this case, no signal is observed from the diffusely scattered

light. Since Kaptiza-Dirac scattering is an off-resonant scattering mechanism, and the lattice

depth due to diffuse scattering is expected to be small, it is not surprising that no scattering

is observed. A much more sensitive scattering signal can be obtained by using resonant Bragg

scattering. The Bragg condition is satisfied when our matter wavepacket has a velocity along

the lattice direction of vr = hkr/m [105]. To achieve this, we briefly pulse on the magnetic field


velocity [vr]

δ H

[m

m]

−5 −4 −3 −2 −1 0 1 2 3 4 5

−0.35

−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Figure 5.19: Observation of matter-wave scattering in the Kapitza-Diarc regime for 60 µslattice pulses. Dependence on beam angle, expressed as the displacement of incident andspecularly reflected beams δH.

gradient, accelerating the atoms up to a desired velocity before interacting with the lattice. To

avoid the Kapitza-Dirac regime, the lattice is slowly ramped up and ramped back down. The

results are shown in Fig. 5.20.

The observed scattering unequivocally shows that avoiding specular reflection is not suffi-

cient to avoid creating of an optical lattice. Using the Bragg scattering signal, it is possible

to obtain an experimental measure of the lattice depth VL. Bragg scattering resonantly scat-

ters velocity states of −vr to +vr by the scattering two lattice photons, one from each beam.

Assuming no off-resonant effects, this forms a closed two-level system, with Rabi frequency Ω

which depends on the lattice depth [105],

Ω =VL2h

(5.16)

Θ =

∫Ω(t)dt (5.17)

P2,1 =1

2(1± cos Θ) (5.18)

P1,2 are the fractional populations in the unscattered and scattered states, and Θ is the inte-

grated time-dependent Rabi frequency. By comparing the population transferred for a given

Bragg pulse area, we can estimate the lattice depth VL. Figure 5.21 shows the results of our

analysis. The analysis accounts for the Bragg pulse shape and the motion of the condensate


vfinal

[vr]

vin

itia

l [v

r]

(a)

−10 −8 −6 −4 −2 0 2 4 6 8 10

−1.5

−1

−0.5

0

0.5

1

1.5

0 200 400 6000

0.2

0.4

0.6

0.8

1

Pulse area [a.u.]

Sca

tte

red

po

pu

latio

n

(b)

Figure 5.20: Resonant Bragg scattering from lattice due to diffusely scattered light. (a)Composite of absorption images. Each row is for a separate initial velocity. (b) Fractionalscattered population versus pulse area (proportional to Θ) for vinitial = vrec.

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

time [ms]

Norm

aliz

ed inte

raction

(a) ODT ramp

Atom motion

Normalized interaction

0 10 20 30 40 500

0.2

0.4

0.6

0.8

1

Lattice depth [nK]

Popula

tion

(b) Unscattered

Scattered

Figure 5.21: Estimating lattice depth from Bragg scattering signal. (a) The normalizedinteraction strength (green) is calculated from the Bragg pulse shape (red) and accounting forthe motion of the condensate through the Bragg pulse (blue). (b) The expected populationtransfer is plotted for this interaction profile, scaled by the lattice depth VL, as described bythe optical Bloch equations.

as it free-falls through the lattice potential under the influence of gravity. Comparing the scat-

tered population data (Fig. 5.20(b)) to the calculation (Fig. 5.21(b)), we estimate our lattice

depth for this experiment is approximately kB × 12 nK. This estimate should be regarded as

an order of magnitude estimate as it only represents one set of measurements, and there may

be effects unaccounted for in the analysis, for example the effect of interparticle interactions

[106]. However it is clear that the residual lattice depth is a fraction of Er = kB × 100 nK.

When compared to the typical optical dipole trap depth of 6.5 µK (in the absence of gravity),

this seems like a small effect. Indeed it corresponds to a reflected intensity at the 10−6 level.

However the relevant comparison isn’t to the trap depth but to the kB × 100 nK energy scale

of the tunneling experiments we wish to conduct.


5.5 Potential solutions

The presence of the residual lattice due to diffuse scattering poses a significant problem to the

tunneling experiments. This is because the tunneling experiments require that we work in the

low energy regime around kB × 100 nK, and so even a lattice on the scale of 0.5 Er is already

significant. There are a number of potential solutions which I will describe here:

Amplified spontaneous emission source

The most obvious solution is purchase a broadband seed laser for our fibre laser amplifier.

The current seed is a NP Photonics ROCK fibre laser with 5 kHz linewidth. To wash out

the interference, the laser bandwidth must be greater than c/2L = 7.5 GHz, where L = 2

cm is the path length difference of interfering fields. Typical diode lasers may specify such

a bandwidth, however this really represents an envelope of sharp features, where the sharp

features are defined by the cavity linewidth of several megahertz. Such a spectrum will also

result in a static interference pattern. Generally speaking, any light source with a decent cavity

will produce features too sharp for our desired spectrum. Thus, the only option is to replace

our fibre laser seed with an amplified spontaneous emission source, which has a truly broad and

flat spectrum that can be nanometers wide. Currently, we are searching for such a source that

will satisfy our desired specifications.

Laser phase modulation

Another possible solution is apply a modulation to attempt to suppress the lattice. For example,

in phase-modulation, sidebands in the frequency spectrum are generated at ±n∆f of the carrier,

where n is an integer. The intensity pattern created from the interference of this field and

its reflection results in a set of static and moving lattices. The moving lattices arise due to

interference between field components of different frequencies, and the static lattices are due to

interference between field components of the same frequency. The moving lattices move much

faster than the atoms, and thus time-average to a spatially flat contribution. However the

static lattices, with slightly different periodicity, will sum together in intensity. By modulating

at c/2L = 7.5 GHz, the 1st order sideband lattices are out of phase with the carrier lattice at

the position of the atoms, thus resulting in a lowered depth. In general, all the even orders will

increase the lattice depth, while all the odd orders will lower it. By controlling the modulation

depth of the phase-modulation, it is possible in principal to completely suppress the lattice

potential at the position of the atoms (see Fig. 5.22). Of course, this extinction is only perfect

at one point in space, with the lattices instead summing in intensity at the point of reflection.

However the expected region of interest of 1 mm is small compared to the distance to the

common node for all orders at L/2 = 1 cm.


0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

Phase modulation depth [π]S

ide

ba

nd

in

ten

sitiy

odd orders

even orders

Figure 5.22: Phase modulation sideband intensity for even and odd orders. These contributionsare exactly equal at specific values of the modulation depth, resulting in perfect extinction ofthe lattice at a distance c/2L from the reflecting surface.

Gravity cancellation

To hold our atoms against gravity, a minimum optical trap power is required (see Fig. 5.9). This

minimum power also means a minimum lattice depth. A magnetic field gradient can be used to

levitate the atoms, reducing or completely cancelling the gravitational potential. For example,

without need to install additional hardware, this can be done using the quadrupole magnetic

coils used in the hybrid trap. However, such an approach also introduces additional field cur-

vatures in the other directions, notably along the longitudinal z-direction (see Equation 3.24).

This residual confinement sets a timescale for which experiments must be performed. To be

able to approximate the waveguide as being flat along z, our experiments must be completed

in a time much less than this period. Thus there is a trade-off between the supression of the

residual lattice and the longitudinal z confinement.

Using this configuration, a few tests of wavepacket propagation in the waveguide were per-

formed. In these experiments, a condensate was prepared in the hybrid trap with longitudinal

confinement 10 Hz provided by the quadrupole field. The bias field along y was abruptly jumped

to its maximum value of 25 G, shifting the quadrupole field centre up and suddenly reducing the

confinement. Subsequently, the magnetic field gradient is pulsed on along z, accelerating the

atoms to a final velocity of 20 µm/ms. The condensate is imaged 2 ms after all the potentials

have been switched off.

Figure 5.23 shows the time evolution of the wavepacket in this gravity balanced potential.

Two cases are shown: the case where the incident and specular reflection beams are aligned

(δH = 0 mm), and where they are mis-aligned (δH = 0.6 mm). The external force causes the

atoms to accelerate, gaining momentum. In the presence of a lattice, when the atoms reach

the Brillouin zone boundary, they coherently Bragg scatter, reflecting from +k to −k. If the

lattice is weak however, the bandgap is small and Landau-Zener tunneling can be important,

causing coupling to the higher bands. At higher momenta, the bandgaps become smaller and the


δ H = 0.0 mm

z−position [µm]

tim

e [m

s]

(a)

0 200 400 600

0

2

4

6

8

10

12

14

16

18

20

δ H = 0.6 mm

z−position [µm]

tim

e [m

s]

(b)

0 200 400 600

0

2

4

6

8

10

12

14

16

18

20

Figure 5.23: Condensate propagation in gravity-compensate waveguide for (a) δH = 0 (inci-dent and specular reflection aligned), and (b) δH = 0.6 mm (incident and specular reflectionmis-aligned).

particles behave more like free-particles. Thus the combination of the Landau-Zener tunneling

and Bragg scattering lead to the observed distortion of the wavepacket. In contrast, when the

specular reflection is avoided, the presence of the lattice can no longer be seen. In reality, there

is certainly still a weak lattice present. However, in this case, the Landau-Zener tunneling is

dominant, accelerating the entire wavepacket out of the first Brillouin zone. Although the lattice

effects have indeed been suppressed, the trade-off residual z-confinement results in deceleration

of the wavepacket as it moves away from the trap centre.

Though the waveguide potential is not ideal, we pushed forward to look for interaction with

the tunnel barrier. The above sequence is repeated, but now with the barrier in the path of

the atoms. The results are displayed in Fig. 5.24. In these images, the barrier is located at

approximately z = 100 µm. An incident wavepacket impinges upon it, and the subsequent

dynamics are observed after 2 ms TOF. In this data, interaction with the barrier is observed,

resulting in partial transmission and reflection of the wavepacket. It also shows the residual

effects of the lattice leading to wavepacket distortion, and the z-curvature introduced by the

gravity compensation. This data represents the first results of the long efforts to develop and

combine the many tools required to one day study tunneling and tunneling times. Although

beautiful in their own way, they also highlight the technical challenges which must be overcome

before reaching our stated goal.


First collision

z−position [µm]

tim

e [

ms]

(a)

0 200 400 600

5

15

25

35

45

55

65

10.5 µm/ms collision

z−position [µm]

tim

e [

ms]

(b)

0 200 400 600

5

15

25

35

45

55

65

5.2 µm/ms collision

z−position [µm]

tim

e [

ms]

(c)

0 200 400 600

5

15

25

35

45

55

65

Figure 5.24: BEC collision with 404 nm barrier in gravity-compensate waveguide. (a) Firstcollision with barrier in which the barrier was centered 100 µm below waveguide (along y).The resulting collision couples the longitudinal to the transverse waveguide mode. (b) Collisionwith properly aligned barrier with an expected wavepacket velocity of 10.5 µm/ms. Barrierheight has been chosen to be partially transmissive. (c) Collision for same barrier height andwavepacket velocity 5.1 µm/ms. The effects of both the residual lattice and the residual z-confinement are observable here, particularly in (b).


5.6 Summary and outlook

In this chapter I described the progress towards observation of tunneling in a scattering con-

figuration and measurement of tunneling times. This involves the design and construction of

a repulsive barrier thin enough to observe tunneling, preparation of a matter wavepacket in-

side a quasi-1D optical waveguide, and the ability to manipulate the centre-of-mass of this

wavepacket. Although there has been significant progress towards the stated goals, a number

of serious obstacles have also arisen. The first obstacle is the broad momentum widths that

result from the repulsive interparticle interacitons. This width lowers our signal-to-noise ratio,

and blurs out the tunneling signal. Potential solutions were dicussed, such as use of a Feshbach

resonance to control interactions or the application of a filter to obtain a narrow momentum

distribution for our studies. The second obstacle is the observation of a residual optical lattice

due to the diffuse scattering from the walls of our glass cuvette. This modifies the dispersion

relation of our matter wavepacket. Both these issues are fairly critical given the low energy

scales of our planned experiments, and must be solved before any further progress can be made.

Chapter 6

Effective mass dynamics of a

Bose-Einstein condensate in an

optical lattice

The un-intentional creation of an optical lattice led to observation of a bevy of interesting

matter-wave effects. Motivated by these observation, and a recent theoretical proposal from

Duque-Gomez and Sipe [107] on the behaviour of the lattice effective mass on fast timescales,

we decided to further explore this rich vein of ultracold physics.

The concept of the effective mass is ubiquitous in the modern description of electronic

conduction, providing a simple and intuitive way to interpret and characterize the motional

dynamics of charge carriers in periodic structures. However, the simplicity of the statement of

the effective mass (Equation 2.23) masks a complex interplay between interband and intraband

lattice dynamics. In 1954, Pfirsch and Spenke [24] studied the initial response of a particle to

a force. Curiously, they predicted that the initial response is in fact described by the bare mass

and not the effective mass, seemingly contrary to the effective mass theorem. Subsequently,

the particle response undergoes rapid oscillations, centered around the response described by

the effective mass. This is illustrated schematically in Fig. 6.1. In the presence of interband

dephasing, these rapid oscillations damp out, and the response eventually settles to that de-

scribed by the usual effective mass. In typical semiconductor systems, the fast timescales of

interband dynamics have thus far prohibited study of these effective mass dynamics.

This chapter summarizes our experimental work to study the response of an ultracold cloud

in an optical lattice to an abruptly applied force. Our observations are the first experimental

demonstration that the initial response of a particle to an abruptly applied force is in fact

characterized by the bare mass, and only over timescales long compared to that of the interband

dynamics is the usual effective mass an appropriate description.

106

Chapter 6. Effective mass dynamics 107

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−1

−0.5

0

0.5

1

Time [τB]

mo<

a>

/F

t1

t2

t3

(b)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−1

−0.5

0

0.5

1

quasimomentum−k

r0 +k

r

E(q

)

t1

(a)

quasimomentum−k

r0 +k

r

t2

quasimomentum−k

r0 +k

r

t3

Figure 6.1: Illustration of effective mass dynamics. (a) Lattice dispersion relation showingthe time evolution of an initially single-band wavepacket under an abruptly applied force. (b)The corresponding acceleration 〈a〉, normalized by the applied force F . The force results in aninteraction energy with both intraband and interband contributions. The response accordingto the intraband contribution alone is characterized by the lattice effective mass m∗ and leadsto Bloch oscillations (dashed black line). The interband contribution coherently couples theground band to the excited bands. This modifies the dynamics such that the total acceleration(black solid line) is initially that of a free particle with bare mass m0 (bare mass responseshown by green dash-dotted line), and then rapidly oscillates around the usual effective massbehaviour.


6.1 Dynamics in a 1D lattice theory

This section builds upon the lattice theory introduced in Section 2.3. An atom in a one-

dimensional periodic potential V (z), with periodicity d and reciprocal lattice vector k = 2π/d,

is described by the Hamiltonian

H0 =h2

2m0

d2

dz2+ V (z). (6.1)

Here m0 is the particle’s bare mass1. The eigenstates of this system are the Bloch states

ψn(q, z), described by Equation 2.17.

6.1.1 Tilted lattice: modified Bloch states

In the presence of an external force F , the Hamiltonian becomes

H =h2

2m0

d2

dz2+ V (z)− Fz. (6.2)

The discrete translational symmetry of H0 is broken, and the Bloch states ψn(q, z) are no

longer the eigenstates of the system. The application of the force unavoidably introduces

coupling between states of differing band index and quasimomentum. In the limit of a weak

force, Wannier showed that it is possible to compose new states, formed by the superposition

of Bloch states, such that the bands remain decoupled [108]. To first order in F , these modified

Bloch states are

φn(q, z) ≈ ψn(q, z) +∑n′ 6=n

ψn′(q, z)Fξn′,n(q)

En′,n(q), (6.3)

where En′,n(q) ≡ En′(q)−En(q) is the band energy difference, and ξn′,n(q) is the Lax connection

relating the periodic parts of the Bloch functions [109]. The modified Bloch state φn(q, z)

is composed primarily of the Bloch state in the corresponding band ψn(q, z), but with small

contributions from adjacent bands. This description is valid only when Landau-Zener tunneling

is negligible. A single-band wavepacket built from the modified Bloch states evolving under the

Hamiltonian H can be described at all times as

Φn(z, t) =

∫dq bn(q, t)φn(q, z). (6.4)

A wavepacket prepared in modified band n has the property that it remains in that band at all

later times. The amplitudes bn(q, t) to first order in F are

bn(q, t) ≈ bn(κ)e−iγn(κ,t) (6.5)

1Note the change in notation for the bare mass, used throughout this chapter.


where κ ≡ q − Ft/h, and γn(κ, t) is an energy dependent phase factor. This wavepacket has

the property that it obeys the effective mass theorem (Equation 2.22)

d

dt〈Φ(t)|v|Φ(t)〉 =

(∫dq

1

m∗n(q)|bn(q, t)|2

)F, (6.6)

where m∗n(q) is the effective mass of particle with quasimomentum q and in band n. Thus under

an applied force, a particle, described by a wavepacket of form Φ(z, t) with narrowly distributed

quasimomentum centered at q, will respond like classical particle with an acceleration 〈a(q)〉 =

F/m∗n(q).

6.1.2 Bloch oscillations

The modified Bloch state wavepacket Φn(z, t) described by Equations 6.4 and 6.5 represents a

state with quasimomentum increasing linearly in time, hq = F . Under the applied force, the

wavepacket will be uniformly accelerated across the Brillouin zone. In the reduced zone picture,

when the wavepacket reaches the zone boundary it coherently reflects from q = +kr to −kr,before continuing its acceleration. Thus, in the limit that interband Landau-Zener tunneling

can be neglected, the motion of the wavepacket, as described by the group velocity (Equation

2.20), is periodic in time with period

τB =h

Fd. (6.7)

This behaviour is known as Bloch oscillation and is illustrated in Fig. 6.1(b). The prediction of

Bloch oscillation is contrary to our everyday experience, since it describes periodic motion of the

charge carriers in a semiconductor when a potential difference is applied, and no net conduction.

The resolution to this mystery requires the inclusion of imperfection in the system. Scattering

from lattice defects effectively erases the phase information carried by a particle propagating

in a lattice, inhibiting the build-up of coherence necessary to observe Bloch oscillation. Thus,

in typical solids, conduction occurs through a diffusive process. Under the applied force, a

particle accelerates for some time, reaching a velocity v, then incoherently scatters losing its

velocity. This process repeats, and the average motion is characterized by a drift velocity which

is inversely proportional to the effective mass.

Despite being an incomplete description of conduction in typical solid state systems, it

remains of fundamental interest to see if Bloch oscillation could occur in a defect-free system.

However, for even the cleanest semiconductor systems, the timescales of the Bloch oscillation

are much longer than the typical scattering times, and this phenomenon went unobserved for

over 60 years. In 1992, artificial structures of layered semiconductors, called superlattices,

were used to observe Bloch oscillation for the first time [110]. This was made possible by the

relatively large lattice spacing of 100 angstroms, compared to the typical 5 angstroms for natural

semiconductors, allowing for significantly shorter Bloch periods. In 1996, Bloch oscillations

were observed for cold atoms in an optical lattice [17]. The optical lattice periodicity is several


hunderd nanometers, leading to Bloch periods on the millisecond timescale. Bloch oscillations

have also been observed with Bose-Einstein condensates in optical lattices [106].

It is important to emphasize that Bloch oscillations, and the effective mass treatment, arise

for wavepackets constructed from modified Bloch states of a single band and evolving in the

modified band. It is in this sense that Bloch oscillation is commonly considered a single-band

phenomenon. In the un-perturbed Bloch basis, the surrounding bands play a small but essential

role in realizing this effective mass behaviour.

6.1.3 Effective mass dynamics

Suppose now that the force is abruptly applied at time t = 0. A naive application of the effective

mass theorem would suggest that the particle should respond with an acceleration proportional

to its effective mass. However, for a particle initially in the ground state of the lattice potential

(described by the state ΨN (z), Equation 2.19), a simple application of Ehrenfest’s theorem

predicts a response described by the bare mass at all times,

〈a(t)〉 ≡ d

dt〈Ψ(z)|v|Ψ(z)〉 =

1

ih〈[v, H

]〉 =

F

m0, (6.8)

seemingly contrary to the effective mass theorem. However, we have already shown in Sec-

tion 6.1.1 that the effective mass response only occurs for the special case of a wavepacket

composed of the modified Bloch states of a single band Φn(z, t). Furthermore, the calculation

described in Equation 6.8 is valid only at t = 0, since the initially single-band state inevitably

couples to neighbouring bands under the applied force. Thus it is predicted that the response

of a wavepacket in a single band is initially described by the bare mass m0, and not the effective

mass m∗n [24].

To describe the subsequent dynamics, it is natural to study this problem in the modified

Bloch basis where the bands are decoupled. In this basis, the initial Bloch state in band N is

a superposition of states in multiple bands

Φ(z, t) =∑n

∫dq bn(q, t)φn(q, z), (6.9)

where the amplitudes to first order in the force are

bn(k, t) ≈

cN (κ)e−iγn(κ,t) , if n = N

−cN (κ)(FxnN (k)EnN (k)

)e−iγn(κ,t) , if n 6= N

(6.10)

In the modified basis, the bands remain decoupled, and the components in each band indepen-

dently experience Bloch oscillation. The net response of this wavepacket contains contributions


from multiple bands, with an acceleration given by

〈a(t)〉 ≈ F

m0

∫dk|cN (κ)|2

(m0

m∗N (k)+∑n6=N

2

m0

EnN (k)

(EnN (κ))2Re[pNn(k)pnN (κ)eiγNn(κ,t)

])(6.11)

where pnN are the momentum matrix elements between bands n and N . The contributions

from neighbouring bands depends on the relative phase between these states γNn(κ, t). At

t = 0, these contributions are in phase, and the initial acceleration is described by the bare

mass

〈a(t = 0)〉 ≈ F

m0

∫dq|cN (q)|2

(m0

m∗N (q)+∑n6=N

2

m0

p2Nn(q)

EnN (q)

), (6.12)

as expected from Ehrenfest’s theorem (Equation 6.8), and can be seen in Equation 6.12 by

applying the effective mass sum rule [111]

∑n6=N

2

m0

p2nN (q)

EnN (q)= 1− m0

m∗N. (6.13)

It is instructive to consider the acceleration for narrow momentum width wavepacket initially

in the ground band N = 1. The interband coupling is primarily to the first excited band n = 2.

For this two-band case, and at times shortly after the force is applied, the acceleration is

[107, 111]

〈a(t)〉 ≈ F

m0

(m0

m∗1(q)+

2

m0

p221(q)

E21(q)cos

[E21(q)

ht

]). (6.14)

The particle response is predicted to oscillate with a frequency corresponding to the bandgap

energy E21(q) = E2(q)−E1(q). These effective mass oscillations occur around the steady-state

response F/m∗1. Thus the full dynamics involve a slow Bloch oscillation associated with the in-

traband dynamics, and a rapid oscillation associated with the interband dynamics, respectively

the first and second terms in Equation 6.14. Note that the the frequency and amplitude of

the effective mass oscillations change periodically in time as the particle traverses the Brillouin

zone while undergoing Bloch oscillation. This effect becomes less prominent at large lattice

depths since the bands flatten and the variation in bandgap is relatively small. In the Bloch

basis, the initial wavepacket state evolves into a superposition of states in different bands with

time-dependent amplitudes. These dynamics are illustrated in Fig. 6.1. In the presence of

interband dephasing, the phase relationship between components in different bands disappears,

and the acceleration is again described by the effective mass theorem2.

The effective mass dynamics have an intuitive microscopic interpretation, see Fig. 6.2. In

position space, the wavepacket can be thought of as a superposition of smaller wavepackets, each

localized at the minima of the periodic potential. Since these small wavepackets are centered

around the potential minima, they feel no net gradient in the potential. When an abrupt force

2This is only approximately true since there will be an incoherent population in the excited modified Blochbands.


−1.5 −1 −0.5 0 0.5 1 1.5

(a)

−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4

(b)

−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4

z [d]

(c)

Figure 6.2: Microscopic view of effective mass dynamics. (a) The de-localized Bloch statecan be thought of as a superposition of localized wavepackets. (b) Each wavepacket is initiallycentered at the local minimum of the potential. (c) The abruptly applied force tilts the localpotential, inducing an initial response at the bare mass, and subsequent rapid dipole oscillationsfor each wavepacket.

is applied, the potential shifts, and the small wavepackets are now centered around a potential

with slope dV/dx = F . The instantaneous response of the small wavepacket is thus described

by the bare mass, F/m0. Only at later times, when the wavepacket begins to move, does the

presence of the lattice potential begin to affect the dynamics of the particle’s motion, giving

rise to the effective mass behaviour.

6.2 Experiment

6.2.1 Optical lattice implementation

The layout of the optical lattice optics is shown schematically in Fig. 6.3. An optical lattice

was created by splitting off a portion of the λ = 1064 nm optical trap beam in free space, and

redirecting it such that it was counter-propagating to the original beam at the position of the

atoms, forming a standing-wave. This re-directed beam was first passed through an acousto-

optic modular (Intraaction ATM-802), collecting the 0th order. This provided some control

over the lattice depth, while ensuring the two beams forming the lattice remain at the same

frequency. The full 1D potential is

Vtot = V0 + V1 + 2√V0V1 cos(2krz) +

1

2mω2

zz2, (6.15)


x

z

Axial MOT beam

1:1 telescopefor MOT beam

f = 20cm f = 2.54 cm


f = 10 cm

Axial MOT beam


f = 20 cm

Figure 6.3: Schematic of optical lattice layout. The fibre laser 1064 nm beam is split into a+y and −y beam which overlap at the position of the atoms. The +y beam is the same beamused for the waveguide. The −y beam is used only for the lattice. The 0th order of the AOMis used to provide some control over the lattice depth.

where V0 and V1 are the potential depths corresponding to each beam acting on the atoms

alone, and ωz is hybrid trap frequency along z provided by the quadrupole magnetic field (see

Section 3.8).

The construction of the lattice in this way does not allow for the lattice to be fully extin-

guished without also extinguishing the optical trap used to produce the condensate. Ideally, the

condensate preparation and transfer to the optical lattice would occur in two separate steps.

While the AOM is able to provide up to 75% extinction in the beam power, this only corresponds

to a 50% reduction in the lattice depth. Thus in this configuration, there is an optical lattice

present at all optical trapping stages. The lattice modifies the evaporative cooling process.

At relatively high temperatures, the lattice is inconsequential. However, as the temperature

is lowered, the presence of the lattice inevitably becomes an important factor, since the final

state is now the ground state of the total potential. As a consequence, motion along z be-

comes inhibited at low temperatures. In this intermediate phase where kBTc < kBT < V0, the


system can be thought of as a one-dimensional array of pancake-shaped atom clouds, each act-

ing independently of the others. While thermalization within each lattice site occurs rapidly,

thermalization across the entire sample is slower. Thus the net effect is a reduction in the

evaporation efficiency. In the presence of the lattice, condensates are smaller, typically con-

taining around 1 × 105 atoms. This construction of the lattice was used for the simplicity of

its implementation. An ideal construction would require independent control of both power

and frequency for each lattice beam. This can be achieved by passing each beam separately

through an AOM, requiring additional optics and electronics and adding complexity to the sys-

tem. Though not ideal, the approach settled on was able to produce condensates of sufficient

size. Furthermore, for the current experiment, many-body effects are undesirable, and thus

smaller condensates are favourable.

The quadrupole magnetic field is used to balance against the gravitational potential. Under

this condition, the residual z confinement results in a trapping frequency of approximately

ωz = 2π × 20 Hz. This corresponds to a timescale much longer than the millisecond Bloch

periods studied here, the slowest lattice timescale of interest. Thus we ignore the effect of the

harmonic confinement for the present experiments.

6.2.2 Generating an external force

To apply a force on the atoms, an external bias field Bz was applied by the pair of bias coils in

Helmholtz configuration, oriented along z (see Section 3.4). These coils were also used for the

waveguide experiments described in the previous chapter. They were driven by a custom built

driver, adapted from the driver used for the previous experiment. The driver was upgraded to

deliver up to 50 A of current in 10 ms pulses. This bias field abruptly shifts the centre of the

quadrupole magnetic field by a distance zshift = Bz/B′, resulting in a potential gradient at

z = 0. The total potential around z = 0 can be written as

Vtot ' VL cos2(2kz)− F (t)z, (6.16)

where VL = 4√V0V1 is the lattice depth, and F (t) = mω2

zzshift is the external force.

To observe the effective mass dynamics predicted by Pfirsch and Spenke, the force must be

applied abruptly compared to the timescale for interband dynamics 1/∆ ≈ 100 µs. Figure 6.4

plots the particle response for several different rise times of the force. For rise times comparable

to the timescale for interband dynamics, the rapid oscillations begin to wash out, leaving the

usual Bloch oscillations. This occurs because the slowly applied force adiabatically changes the

state from the Bloch basis to the modified Bloch basis. The original design of the push coil driver

had a slow rise time of τrise ≈ 250 µs, much slower than the typical bandgap frequencies. Under

these conditions, we observe Bloch oscillations. The results of this experiment are described

in the next section. Following this experiment, the push coil driver was upgraded to improve

the switching speed, resulting in rise times of τrise ≈ 20 µs, much faster than the timescale for


0 100 200 300 400−1

−0.5

0

0.5

1

time [µs]

<v>

[µ

m/m

s]

τ × ∆ = 0.0

τ × ∆ = 0.1

τ × ∆ = 0.9

τ × ∆ = 5.3

Figure 6.4: Simulation of effective mass dynamics for various force rise times. Calculationsare for 85Rb in a 7Er, d = 390 nm lattice. The external force has amplitude F/m0 = 2.5g,and is modelled as an error function with temporal width τ . This rise time is compared to thebandgap frequency ∆. Calculation courtesy of Federico Duque-Gomez [107].

0 50 100 150 200 250

0

5

10

15

20

Time [us]

Cu

rre

nt

[A]

(a)

0 0.5 1 1.5 20

20

40

60

80

Vctrl

[V]

Tim

e [

us]

(b) Rise time

Delay time

Figure 6.5: Upgraded push coil current driver. (a) Current pulses of various amplitude as setby the control voltage Vctrl. (b) Each trace is fit to an error function, and the delay and risetime are plotted.

interband dynamics. Under these conditions, we are able to observe the effective mass dynamics

predicted by Pfirsch and Spenke. The results of this experiment are described in Section 6.2.5.

Both the original and upgraded push coil circuit exhibit a rise time and delay that depended

on the strength of the desired field. These timescales are plotted in Fig. 6.5 for the upgraded

current driver.

6.2.3 Observation of Bloch oscillations

The experimental sequence is illustrated in Fig. 6.6. We prepare a Bose-Einstein condensate

in the hybrid optical and magnetic trap, with the lattice suppressed using the AOM. Next we

ramp up the optical lattice over 100 ms, adiabatically deforming the potential such that the

lattice becomes dominant. After sitting in this potential for another 50 ms, we apply a magnetic


(a) (b) (c)

Figure 6.6: Experimental sequence for observation of Bloch oscillations and effective massdynamics. (a) A condensate is prepared in the hybrid trap with weak lattice. (b) The lattice isramped up, adiabatically loading the condensate into the ground state of the lattice potential.(c) An external force is applied, tilting the potential.

field gradient with rise time τrise ≈ 250 µs. This switch-on is slow compared to the bandgap

frequency such that the initial ground state wavepacket Ψ0(z, t) adiabatically follows into the

corresponding single-band modified Bloch state wavepacket Φ0(z, t). The atoms are allowed to

evolve for a variable time, before the entire potential - trap, lattice, and external force - are

abruptly switched off. We perform a 20 ms time-of-flight expansion, followed by absorption

imaging. The abrupt switch-off is fast compared to the dynamics of the atoms in the lattice,

and thus what is observed after time-of-flight is representative of the momentum distribution

of the atoms in the lattice.

Shown in Fig. 6.7 is a representative set of raw absorption images. Each horizontal slice

is an image from a separate preparation of the condensate with increasing evolution time in

100 µs steps. The free-space momentum distribution is a diffraction pattern consisting of

components separated by the lattice recoil momentum 2hk (the recoil velocity is vr = hk/m =

4.3 µm/ms). For weak potential gradients and deep lattices, we can neglect interband Landau-

Zener tunneling [112]. For each diffracted order, the momentum grows linearly in time while

changing in amplitude as the particle travels across the Brillouin zone. As one component at

−k rises, a component at +k falls, resulting in a periodic modulation in the average velocity.

The data show a clear oscillation in the average velocity on a millisecond timescale, consistent

with Bloch oscillations.

The 1D profiles generated from these images is fitted to 4 equally-spaced, equal-width

gaussians. The normalized amplitude Ai and velocity vi = v0 +2nvr of each peak are extracted,

and the average velocity 〈v〉 of the particles in the lattice is then reconstructed from a weighted

sum of these peaks,

〈v〉 =∑i=1..4

Ai(v0 + 2nvr) (6.17)

Figure 6.8 plots the results of the fit. The velocity of each peak is observed to evolve linearly

in time, while the amplitude rises and falls. The net result is an average velocity that evolves

periodically in time, with a frequency that scales with the applied force.

The delay, rise time, and drive dependence of the applied force are clearly visible in the first

millisecond of data. For example, in Fig. 6.7(a), for the first 700 µs, the state does not change.

Over the next 500 µs, the acceleration begins to increase until it reaches a constant around


Figure 6.7: Bloch oscillation raw data. (a) F/m0 = 4.8 µm/ms2, (b) F/m0 = 7.6 µm/ms2.The slow rise time and delay of the applied force are clearly visible

t = 1200µs.

6.2.4 Landau-Zener tunneling and scattering effects

It is useful to pause here to highlight some effects we seek to avoid when studying Bloch

oscillations and the effective mass dynamics. Figure 6.9 shows an early attempt at studying

Bloch oscillations with a relatively dense condensate in a weak lattice. In fact this experiment

was performed using the lattice created from specular reflection, estimated to have a depth of

about 2Er.

In this data set, the shallow lattice resulted in prominent Landau-Zener tunneling from the

ground band to the first excited band. Landau-Zener tunneling is important when the applied

acceleration a = F/m0 is comparable to the critical value ac ≈ d∆2 [112]. This is typically most

relevant at the edge of the Brillouin zone at momentum ±hkr, where the bandgap is smallest.

In contrast to Fig. 6.7 the cloud splits at −vrec (for example, this first occurs at 3.8 ms) with

a portion remaining in the ground band and a portion tunneling into the first excited band.

Each component then continues to evolve under the applied force, Bloch oscillating in their

respective bands.

A second effect observed is that in this parameter regime, the residual harmonic confinement

becomes important. The applied force is relatively weak, leading to long Bloch oscillation times.

Given the timescale of the harmonic confinement 1/ωz ≈ 10 ms, over a few oscillation cycles the

confinement begins to play a role, and the quasimomentum no longer grows linearly in time.


0 1 2 3 4−20

−10

0

10

20Zbias −0.1V

Peak v

elo

city [um

/ms](a)

0 1 2 3 40

0.2

0.4

0.6

0.8

Peak a

mplit

ude

0 1 2 3 4−0.6

−0.4

−0.2

0

0.2

0.4

<v>

[um

/ms]

time [ms]

0 1 2 3 4−20

−10

0

10

20Zbias −0.2V

Peak v

elo

city [um

/ms](b)

0 1 2 3 40

0.2

0.4

0.6

0.8

Peak a

mplit

ude

0 1 2 3 4

−0.4

−0.2

0

0.2

0.4

time [ms]

<v>

[um

/ms]

Figure 6.8: Analysis of Bloch oscillation data. (a) F/m0 = 4.8 µm/ms2, (b) F/m0 = 7.6µm/ms2. The 1D profile of each image is fit to a sum of gaussians. The velocity and amplitude ofeach peak are extracted and plotted in the first two rows. The average velocity is reconstructedfrom the sum of velocity components, weighted by their amplitude.


Figure 6.9: An early attempt to observe Bloch oscillations using a large condensate in a shallow2Er lattice, and external force F/m0 = 2.8 µm/ms2. Landau-Zener tunneling, inter-particlescattering, and the role of the residual z-confinement are all visible in this data.


Figure 6.10: s-wave scattering halos, observed for high condensate densities and at the firstBrillouin zone boundary where there is a superposition of +hkrec and −hkrec momentum states.At low densities (A), scattering is not observed. The two velocity components pass througheach other largely undisturbed. As the density is increased (B,C) a scattering halo is observed.

Lastly, the effect of particle-particle scattering is also evident in this data set. Bloch oscil-

lation and the effective mass dynamics are both single-particle effects. However, it has been

shown that Bloch oscillations in the presence of weak interparticle interactions can persist [106].

The effect of weak repulsive interactions is accounted for by assuming a lattice with depth low-

ered by the meanfield, valid for peak particle densities below 3×1013 atoms/cm3. For higher

densities, the non-linearity acts to dephase and smear out the Bloch oscillations [106, 113].

Particle-particle scattering is most prominently observed near the Brillouin zone edge, when

there exists a superposition of states with distinct momenta +hkrec and −hkrec (for example,

this first occurs at 3.8 ms). The scattering at these energies is predominantly s-wave [58],

which has a distinctive scattering halo observable in the far-field. Figure 6.10 shows represen-

tative data after release from a lattice at the Brillouin zone boundary and 12 ms TOF. s-wave

scattering is isotropic in momentum space. Here, the imaging integrates over the line-of-sight

dimension, and what is observed are ring-like structures which become increasingly prominent

as the particle density is increased. The distinctive scattering pattern can be used as an in-

dicator of many-body effects. For the effective mass experiment, we operate at peak densities

below 3×1013 atoms/cm3, for which we do not see particle-particle scattering.


Figure 6.11: Raw effective mass data for s = 9.3, F/m0 = 11.8 m/s2. Data taken at a timeresolution of 4 µs. Composite image is constructed from the central slice of each image. Sampleimages, 1D profiles, and fits are shown.

6.2.5 Observation of effective mass dynamics

We next set out to observe the effective mass dynamics predicted by Pfirsch and Spenke. To

observe these dynamics, we use the upgraded coils with 20 µs rise time, and follow the time

evolution of the momentum distribution at a much higher time resolution of 4 µs.

Plotted in Fig. 6.11 is a composite of absorption images from a sample data run (s = 9.3,

a = 11.8 µm/ms2). From the raw images, the long timescale Bloch oscillation is again readily

apparent. The effective mass dynamics are an interband phenomenon, which we expect to

result in a modulation in the relative amplitude of the diffracted orders. To observe this, we

extract the amplitude and velocity of each peak (Fig. 6.12). The velocity of each peak again

grows linearly in time. The amplitudes grow and fall with the Bloch period. In addition, there

appears to be a small oscillation in the amplitude. The average velocity is then reconstructed

from the sum of diffraction peak velocities, weighted by their amplitude (Fig. 6.13). In addition


0 200 400 600 800−30

−20

−10

0

10

20

30

Ve

locity [

um

/ms]

Time [us]0 200 400 600 800

0

0.2

0.4

0.6

0.8

1

Am

plit

ud

e

Time [us]

Figure 6.12: Fitted effective mass dynamics data. The 1D profiles are fit to the sum of 4gaussians, and the diffraction peak velocity and normalized amplitude are extracted.

0 100 200 300 400 500 600 700 800−1

−0.5

0

0.5

1

<v>

[u

m/m

s]

Time [us]

Figure 6.13: The average velocity is reconstructed by the sum of the velocity componentsweighted by their amplitudes. The solid black curve is the result of low-pass filtering this data,and serves as a guide to the eye.

to the slow Bloch oscillation in the average velocity, we observe a much faster oscillation with

periodicity consistent with the timescale of the bandgap frequency.

6.2.6 Estimating experimental parameters

Experimentally, the effective mass dynamics are determined by the lattice depth s ≡ VL/Er and

the applied force F . The applied force is estimated from the acceleration of the diffraction peaks

(Fig. 6.12). The ±1 sideband amplitude of the ground state (q = 0) is used to estimate the

depth of the lattice. The observed amplitudes are compared to a numerical calculation including

meanfield interactions. Comparing to a non-interacting calculation, the lattice dynamics for a

system with weakly repulsive meanfield interaction occur as if they were in a lattice of depth

Veff < VL [106].


0 0.05 0.1 0.15 0.2 0.250

2

4

6

8

10

12

14

16

18

La

ttic

e d

ep

th,

s

±1 sideband amplitude

Non−interacting

Interacting

Figure 6.14: Estimate of lattice depth from first-order sideband amplitude. Plotted are thenumerically calculated ±1 sideband amplitude for the q = 0 ground state, for a non-interactingsystem and a weakly interacting system.

6.3 Analysis of data

To quantitatively study the dynamics, we fit the the average velocity to the sum of two sinusoids,

explicitly separating the Bloch oscillation from the effective mass dynamics:

〈v(t)〉 = Ad sin(ωdt+ φd) +AB sin(ωBt+ φB) (6.18)

where A is the amplitude of the oscillation, ω the frequency, φ the phase, and the subscripts d

and B indicate fitting parameters for the effective mass and Bloch oscillation, respectively. Due

to the variation of the bandgap as the particle traverses the Brillouin zone, the fitting function

used is not strictly correct at times comparable to the Bloch period. Thus when extracting

the effective mass oscillation we fit only the first 300 µs of data (roughly 3 periods of the fast

oscillation). This approach estimates the effective mass dynamics near the centre of the band

at q = 0.

Figure 6.16 plots the dependence of these timescales on the applied force F and lattice depth

s. A small correction to the lattice depth is made to account for the mean-field interaction (less

than 10%, see Fig. 6.14) . The frequency of the slow oscillation is consistent within measurement

uncertainty with a linear increase with the applied force (Fig. 6.16(a), lower), as expected for a

Bloch oscillation. Figure 6.16(b) upper, plots this frequency, scaled by the applied force, against

lattice depth, showing that the frequency is independent of lattice depth. The fast oscillation

increases in frequency with lattice depth (Fig. 6.16(b), lower) and thus the bandgap, while

being independent of applied force (Fig. 6.16(a), upper). The fitted frequency is compared to

the numerically calculated bandgap at q = 0 and q = kr, representing the range of frequencies

the particle samples as it undergoes a complete Bloch oscillation. A more direct comparison


50 100 150 200 250 300

−0.2

0

0.2

0.4

0.6

0.8

Time [us]

<v>

[um

/ms] s=5.9, a=10.8 um/ms

2

50 100 150 200 250 300

−0.2

0

0.2

0.4

0.6

0.8

Time [us]

<v>

[um

/ms] s=8.2, a=12.1 um/ms

2

50 100 150 200 250 300

−0.2

0

0.2

0.4

0.6

0.8

Time [us]

<v>

[um

/ms] s=13.0, a=11.8 um/ms

2

Figure 6.15: Sample fits of average velocity data, for increasing lattice depth and roughlyconstant applied force. The rapid oscillation frequency can be seen to increase with latticedepth, and Bloch oscillation amplitude decrease.


8 10 120.5

1

1.5

2

F/mo [um/ms

2]

f B [

kH

z]

0.5

1

1.5

f d /

∆

(a)

5 10 150

5

10

15

s, lattice depth [Er]

f d [

kH

z]

100

150

f B /

(F

/mo)

(b)

Figure 6.16: Timescales of effective mass dynamics data. Effective mass oscillation frequencyfd, and Bloch oscillation frequency fB. Blue lines are the expected behaviour of these timescales.(a) Dependence of timescales on applied force. (b) Dependence on lattice depth. Lower: bluelines are bandgap frequencies at q = 0 and q = kr, dashed black line is a fit to numericallysimulated data.

to the data is made by fitting Equation 6.18 to the first 300 µs of data generated from a

Gross-Pitaevskii equation simulation, in the same way as we fit to the experimental data.

From the average velocity fits we extract the initial response of the particle in the N = 1,

q = 0 state to an applied force. The external force is applied by turning on a magnetic bias

field, shifting the quadrupole trap centre within 20 µs after a 20 µs delay. This delay time is

accounted for by the phase in the sinusoids of our fitting function. This phase is negligible for

the Bloch oscillation but not for the effective mass dynamics. The initial response is evaluated

at the point in time t0 = φm/ωm when the phase of the effective mass oscillation is zero. From

the fit to the average velocity we extract the initial acceleration

〈a(t = t0)〉 = Adωd +ABωB cos(ωBt0 + φB) (6.19)

We characterize the initial response in terms of a dynamical mass, md ≡ F/〈a(t = t0)〉. The

lattice effective mass is estimated by considering the Bloch dynamics only

〈aB(t = t0)〉 = ABωB cos(ωBt0 + φB) (6.20)

The extracted masses are plotted in Fig. 6.17. The effective mass theorem (Equation 2.22)

describes the response of a particle to a force over timescales long compared to the interband

dynamics; thus we estimate this effective mass from the Bloch oscillation alone. Since the

band curvature decreases with increasing lattice depth, the effective mass increases. However,


4 6 8 10 12 140

1

2

3

4

5

6

7

8

s, lattice depth [Er]

m*/

mo

Effective mass

Dynamical mass

Figure 6.17: Effective mass and dynamical mass results. The steady-state effective mass atthe centre of the ground band (N = 1, q = 0) associated with the Bloch oscillation is plottedas green squares. As the lattice depth increases, the band flattens, and the effective mass isexpected to increase. The dashed black line is the theoretical effective mass. The initial (t = t0)response, characterized by the dynamical mass, is extracted from the full dynamics (red circles),and is expected to be independent of lattice depth. The solid blue line is the expected baremass response m∗/m0 = 1, independent of lattice depth.

the full response contains contributions from both the ‘single-band’ Bloch oscillation and the

‘multi-band’ effective mass dynamics. The dynamical mass is estimated from the sum of these

contributions, and at t = t0 is observed to be consistent with the bare mass, independent of

lattice depth.

In deep lattices (s >10), the deduced effective mass begins to deviate from the predictions.

This is due to the growth of high-order diffraction peaks in the momentum distribution which

lie beyond the imaging window. Neglecting these peaks causes us to over-estimate the ampli-

tude of the Bloch oscillation, and thus underestimate the effective mass. This is illustrated

in Figures 6.18 and 6.19). Using simulated data, Figure 6.18 plots the reconstructed average

velocity 〈v〉 using the 4 largest peaks in the momentum-space distribution, compared to the

actual velocity. The reconstructed average velocity depends sensitively to the cancellation of

momentum components at ±vrec. With a limited number of peaks accounted for there are

inevitably situations where a contribution is un-accounted for, breaking this careful balance.

In our case, with a 4-peak fit, this occurs at q = 0, as illustrated in Fig. 6.18. Although the

amplitude of the un-accounted for peak is small, its velocity is high. The significance of this


−6 −4 −2 0 2 4 60

0.5

1q = -1 hkrec

−6 −4 −2 0 2 4 60

0.5

1q = -0.5 hkrec

−6 −4 −2 0 2 4 60

0.5

1q = 0 hkrec

−6 −4 −2 0 2 4 60

0.5

1q = +0.5 hkrec

−6 −4 −2 0 2 4 60

0.5

1q = +1 hkrec

Momentum−space diffraction peaks

Velocity [vrec

]

−1 −0.5 0 0.5 1

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

Reconstructed average velocity, s=12

<v(q

)> [um

/ms]

Quasimomentum, q [krec

]

All peaks

4 peaks

0 0.2 0.4 0.6 0.8 1

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

<v(T

)>

Time [τB]

All peaks

4 peaks

Figure 6.18: Failure of 4 peak fits for large lattice depths. Left: reconstructed average velocityusing all peaks versus 4 peaks. The choice of the 4 peaks is indicated by the blue squareson the right. The limited number of peaks forces an asymmetry at some point in the Blochoscillation. In our case of 4 peaks, this occurs at q = 0. This seemingly small contribution ofthe un-accounted for peak can have a large impact on the reconstructed velocity.

effect grows dramatically above s = 10, as illustrated in Fig. 6.19. Note that this has a minimal

impact on the estimate of the dynamical mass since the dominant contribution in the deep

lattice regime is from the effective mass dynamics.

The error bars in Figures 6.16 and 6.17 are given by the fit uncertainties, where the main

contribution in the effective mass uncertainty comes from fitting the long timescale Bloch os-

cillation. In the sample data shown in Fig. 6.11, at 550 µs and 650 µs there are abrupt changes

in the peak width. The average velocity depends only on the relative amplitude of the diffrac-

tion peaks, and their amplitude. The width of the diffraction peaks plays no role. This likely

occurred when the cooling lasers were improperly re-locked. The full data set shown here rep-

resents a continuous 6 hour data run, and is a testament to the improvements to the overall

stability of the experiment made over the past few years.

6.4 Summary

This chapter described our work to study the timescales of the effective mass description. The

concept of the effective mass allows for a simple and intuitive description of the response of


4 6 8 10 12 140

1

2

3

4

5

6

Lattice depth [Erec

]

P ±2 sideband @ q=0 (x 1000)

(a)

4 6 8 10 12 140

0.5

1

1.5

Lattice depth [Erec

]

Bloch osc. amplitude [um/ms]

(b) All peaks

4 peaks

4 6 8 10 12 141

1.2

1.4

1.6

1.8

Lattice depth [Erec

]

Fractional deviation

(c)

Figure 6.19: Effect of 4 peak fits on the amplitude for Bloch oscillation. (a) Amplitude of ±2order sidebands. (b) Amplitude of Bloch oscillation estimated by accounting for 4-peaks, andall peaks. (c) Fractional deviation in Bloch amplitude from expected.

a particle in a periodic potential to an external force. However, the simplicity of its state-

ment is deceptive, masking a complex interplay of interband and intraband dynamics, and its

application makes some subtle assumptions. Our work shows that the initial response is in

fact characterized by the bare mass, and only over timescales long compared to the interband

dynamics is the effective mass an appropriate description. To our knowledge, the effective mass

dynamics explored in this work have never before been observed, despite its initial prediction

nearly 50 years ago. In the presence of excited band decay and interband dephasing, the ef-

fective mass dynamics are expected to reduce to the behaviour described by the usual effective

mass. In our system, this can occur due to inter-particle scattering [113, 114]. However, within

the parameter range and timescales probed, we do not expect to see dephasing of the effective

mass oscillation. The study of dephasing effects may be an interesting future direction for this

project.

Chapter 7

Summary and outlook

This thesis describes experiments in which a Bose-Einstien condensate of 87Rb was used as

model system for exploring matter-wave phenomena which are difficult to access in their native

systems. The inherent length and time scales of ultracold atoms allows for clear observation

of single-particle quantum effects. In our experiments we make use of these advantages, in

particular the ability to create sharp spatial features and to modulate the external potential

much faster than the dynamics of the atoms. These experiments represent the first science

performed on the Bose-Einstein condensation apparatus after many years and significant effort

to improve its stability and upgrade its capabilities. In this final chapter I briefly review the

technical and scientific work detailed in this thesis.

Hybrid optical and magnetic trap for Bose-Einstein condensation

When I inherited the experimental apparatus from my predecessor, it exhibited a number of

stability problems. One immediate concern was the reliance on large magnetic fields to confine

the atoms. Upon switch-off, these fields would induce eddy currents in the surrounding metallic

elements resulting in an instability in the time-of-flight measurement. This instability meant

that the far-field atomic distributions were unreliable, and averaging impossible. These issues

were resolved by the development of an optical dipole trap to confine the atoms, in place of

the magnetic trap. It was this upgrade that allowed for reliable measurements of the collisional

transient momentum-space interference described in Chapter 4. A more general concern was

the overall instability in condensate number and purity, which would fluctuate on a timescale as

short as days. The sensitivity to the external environment was a serious concern, and thus the

experiment was modularized as much as possible to ensure that a small fluctuation at step A

of the experiment would not propagate to step C. The experiment cycle was largely revamped,

dispensing with the high fields of the TOP trap and implementing a hybrid optical and mag-

netic trap that allowed for faster and more robust production of condensates. As a result of

these upgrades we are now able to produce pure condensates of 3×105 atoms every 40 seconds.

The success of these efforts is evidenced by the ability to perform the continuous 6 hour data

runs required for the effective mass experiment described in Chapter 6.

129

Chapter 7. Summary and outlook 130

Transient momentum-space interference during scattering

The upgrades to the experiment allowed us to move beyond the technical issues that plagued us

for many years. The first science conducted on this apparatus was the observation of transient

momentum-space interference of a Bose-Einstein condensate during scattering with a repulsive

potential. During particle scattering, the momentum-space wavefunction can exhibit dramati-

cally different features including a non-classical enhancement of high momenta. These features

are present only during the scattering event, disappearing in the asymptotic limits. Using a

Bose-Einstein condensate colliding with a optically induced barrier, we observed these transient

interference effects, providing the first direct verification for a phenomenon first predicted in

1998. Furthermore, we showed that this phenomenon can be used to reconstruct the conden-

sate momentum-space phase profile. The key tool that allowed for this observation was the thin

optically-induced barrier, which could be abruptly switched-off on a timescale much faster than

the motion of the atoms, thereby halting the dynamics of the collision. This ability to probe

fast collisional dynamics can be thought of as the ultracold atom analog to the use of ultrafast

laser pulses to probe electron dynamics.

Waveguide experiments and the accidental lattice

The transient interference experiment made use of the optical dipole trap and 1D barrier, tools

developed to pursue our long term of goals of studying tunneling times. As we made progress

towards these experiments, new experimental challenges emerged. The un-intentional creation

of an optical lattice lead to observation of a number of lattice effects, including matter-wave

scattering in the Bragg and Kapitza-Dirac regimes, dipole oscillations of a thermal-condensate

mixture, s-wave scattering halos, Bloch oscillation, and Landau-Zener tunneling. Dazzled by

the bright lights of the lattice physics we were observing, we decided to momentarily change

direction, and work on a lattice project proposed by our U of T colleagues.

Effective mass dynamics

The response of a particle in a periodic potential is commonly characterized by an effective

mass, which accounts for the complex interaction of the particle with its surrounding poten-

tial, allowing for a simple and intuitive description of the particle’s dynamics. The concept of

the effective mass is widely used, for example in characterizing the dynamics of electrons in

semiconductor materials. Using a Bose-Einstein condensate in a 1D optical lattice, we studied

the response of a particle to an abruptly applied force. We show that the initial response of

a particle is in fact characterized by its bare mass, and only over timescales long compared

to that of the interband dynamics is the effective mass an appropriate description. This is

the first observation of an effect first predicted in 1954, and is made possible by the relatively

small bandgaps of optical lattice potentials. To observe this effect for an electron in a typical

semiconductor system, with their 1 eV bandgaps, would require excitation at speeds of 100’s of

Chapter 7. Summary and outlook 131

terahertz.

Outlook

With the conclusion of the effective mass experiment, the apparatus is back on track towards

the tunneling experiments. Significant challenges remain. The immediate challenges are to

suppress the accidental lattice and to reduce the momentum width. Longer term challenges

are to further investigate the role of interactions, the coupling to the transverse waveguide

modes, and whether the signal-to-noise ratio will be good enough. The time-averaged barrier

system is a particularly interesting and flexible tool, potentially allowing for creation of tunnel

barriers, matter-wave Fabry-Perots, and box potentials. This may one day allow for the study

of non-linear tunneling out of a quasi-bound state and bi-stability in a matter-wave cavity.

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