Collective Dispersive Interaction of Atoms and Light in a High Finesse

COLLECTIVE DISPERSIVE

INTERACTION OF ATOMS AND

LIGHT IN A HIGH FINESSE CAVITY

KYLE JOSEPH ARNOLDB.S. Eng. Physics, University of Illinois Urbana-ChampaignB.S. Mathematics, University of Illinois Urbana-Champaign

A THESIS SUBMITTED FOR THE DEGREE

OF DOCTOR OF PHILOSOPHY

CENTRE FOR QUANTUM TECHNOLOGIES

NATIONAL UNIVERSITY OF SINGAPORE

2012

mailto:[email protected]

http://www.quantumlah.org/

http://www.nus.edu.sg

Declaration

I herewith declare that the thesis is my original work and

it has been written by me in its entirety. I have duly

acknowledged all the sources of information which have

been used in the thesis.

The thesis has also not been submitted for any degree in

any university previously.

Kyle Joseph Arnold

1 December 2012

Acknowledgements

First and foremost, I would like to thank my supervisor, Dr. Murray Bar-

rett. We have worked closely over the years and without the wealth of

atomic physics, optics, and electronics knowledge I have received from him,

the work in the thesis would not have been possible.

Next I would like to thank Markus Baden, my partner on the cavity

experiments. In particular, for many fruitful physics discussions and taking

the time to proof read my thesis. Also, for introducing me to python, my

go-to tool for scientific computing and source of many quality plots in this

thesis.

Many thanks to my other fellow PhD students, Arpan Roy, Chuah Boon-

Leng, and Nick Lewty, who, though not directly involved in my experiments,

have all contributed to our common efforts in developing the lab. Thanks

also to the many RAs who have helped out in the lab, in particular Andrew

Bah who produced the 3D-rendered experiment schematics for my thesis.

I’m grateful for work of our CQT support staff, especially our procurement

officer, Chin Pei Pei, our electronics support staff, Joven Kwek and Gan Eng

Swee, and our machinists, Bob and Teo, who have made numerous parts

for me on short order.

Finally, I would like to thank my wife, Vicky, for her continuous love

and support during these years, and my parents who having always been

supportive of my chosen path even though it has taken me to distant lands

far from home.

Contents

List of Tables ix

List of Figures xi

1 Introduction 1

1.1 Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Dipole Trapping and All-Optical Bose-Einstein Condensation 5

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.1 Dipole traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.2 Laser cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.3 Evaporative cooling . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.4 Scaling laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.5 Atom losses due to inelastic collisions . . . . . . . . . . . . . . . 15

2.3 Discussion of crossed beam traps . . . . . . . . . . . . . . . . . . . . . . 16

2.3.1 General thermal distribution of a trapped gas . . . . . . . . . . . 17

2.3.2 Crossed beam distribution: numeric solution . . . . . . . . . . . 18

2.3.3 Crossed beam distribution: approximate analytic solution . . . . 18

2.3.4 Thermalization in crossed beam traps . . . . . . . . . . . . . . . 21

2.3.5 Analysis of a recent cross-beam result . . . . . . . . . . . . . . . 23

2.3.6 Elliptical beams . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3.7 Basics of Bose-Einstein condensates . . . . . . . . . . . . . . . . 25

2.4 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.4.1 Cooling lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.4.2 Imaging diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . 30

iii

CONTENTS

2.4.3 MOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.4.4 Dipole trap loading . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.4.5 Trap lifetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.4.6 Hyperfine changing collisions . . . . . . . . . . . . . . . . . . . . 35

2.4.7 Measuring trap frequencies . . . . . . . . . . . . . . . . . . . . . 36

2.4.8 Thermal lensing . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.5 Bose-Einstein condensation experiment . . . . . . . . . . . . . . . . . . . 38

2.5.1 Primary beam geometry . . . . . . . . . . . . . . . . . . . . . . . 38

2.5.2 Primary beam free evaporation . . . . . . . . . . . . . . . . . . . 39

2.5.3 Primary beam forced evaporation . . . . . . . . . . . . . . . . . . 39

2.5.4 Secondary beam geometry . . . . . . . . . . . . . . . . . . . . . . 41

2.5.5 Cross-beam compression . . . . . . . . . . . . . . . . . . . . . . . 41

2.5.6 Observation of a condensate . . . . . . . . . . . . . . . . . . . . . 43

2.5.7 Comments of observing a bi-modal distribution . . . . . . . . . . 46

2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3 Collective Cavity Quantum Electrodynamics with Multiple Atom Lev-

els 49

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.2 Cavity quantum electrodynamics . . . . . . . . . . . . . . . . . . . . . . 50

3.2.1 Jaynes-Cummings model . . . . . . . . . . . . . . . . . . . . . . . 50

3.2.2 Real systems: dissipation . . . . . . . . . . . . . . . . . . . . . . 55

3.2.3 Cavity QED for N multi-level atoms . . . . . . . . . . . . . . . . 58

3.2.4 Semi-classical model for multi-level atoms . . . . . . . . . . . . . 63

3.3 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.3.1 High finesse cavity . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.3.2 Cavity laser system . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.3.3 Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.3.4 Optical lattice transport . . . . . . . . . . . . . . . . . . . . . . . 71

3.3.5 808 nm intra-cavity FORT . . . . . . . . . . . . . . . . . . . . . 72

3.3.6 Optical pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.4 Experimental results: cavity transmission spectra . . . . . . . . . . . . . 74

3.4.1 Experiment procedure . . . . . . . . . . . . . . . . . . . . . . . . 75

iv

CONTENTS

3.4.2 Two-level atoms: the cycling transition . . . . . . . . . . . . . . 76

3.4.3 Multi-level atoms: π-probing . . . . . . . . . . . . . . . . . . . . 77

3.4.4 Driving both cavity modes . . . . . . . . . . . . . . . . . . . . . 78

3.4.5 Optical pumping by the cavity field . . . . . . . . . . . . . . . . 79

3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4 Self-Organization of Thermal Atoms Coupled to a Cavity 83

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.2 Derivation of the threshold equations . . . . . . . . . . . . . . . . . . . . 87

4.2.1 Lattice geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.2.2 Traveling wave geometry . . . . . . . . . . . . . . . . . . . . . . . 94

4.3 Experimental set-up and methods . . . . . . . . . . . . . . . . . . . . . . 95

4.3.1 Dual-wavelength high finesse cavity . . . . . . . . . . . . . . . . . 95

4.3.2 Cavity laser system . . . . . . . . . . . . . . . . . . . . . . . . . . 97

4.3.3 Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

4.3.4 Atom transport: translation of the dipole trap . . . . . . . . . . 101

4.3.5 1560 nm intra-cavity FORT . . . . . . . . . . . . . . . . . . . . 103

4.4 Experimental results: self-organization threshold scaling . . . . . . . . . 106

4.4.1 Experimental procedure . . . . . . . . . . . . . . . . . . . . . . . 107

4.4.2 Comparison to the threshold equations . . . . . . . . . . . . . . . 107

4.4.3 Lattice geometry threshold results . . . . . . . . . . . . . . . . . 109

4.4.4 Traveling wave geometry threshold results . . . . . . . . . . . . . 110

4.4.5 Discussion of threshold scaling . . . . . . . . . . . . . . . . . . . 111

4.5 Experimental results: dynamics of self-organization . . . . . . . . . . . . 113

4.5.1 Lattice geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

4.5.2 Traveling wave geometry . . . . . . . . . . . . . . . . . . . . . . . 120

4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

5 Bragg Scattering, Cavity Cooling, and Future Directions 123

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

5.2 Bragg scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

5.2.1 Future Bragg scattering related experiments . . . . . . . . . . . . 127

5.3 Cavity cooling of atomic ensembles . . . . . . . . . . . . . . . . . . . . . 128

5.3.1 Cavity cooling via the collective mode . . . . . . . . . . . . . . . 129

v

CONTENTS

5.3.2 Cavity cooling via self-organization . . . . . . . . . . . . . . . . . 134

5.3.3 Conclusions and future experimental directions for cavity cooling 138

5.4 Future directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

A High finesse cavities: technical details 141

A.1 ATF mirrors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

A.1.1 Brief History of low-loss mirrors . . . . . . . . . . . . . . . . . . 141

A.1.2 Mirrors from ATF: 2008-2011 . . . . . . . . . . . . . . . . . . . . 142

A.1.3 Mirror handling and cleaning . . . . . . . . . . . . . . . . . . . . 144

A.2 Contamination of mirrors by Rb . . . . . . . . . . . . . . . . . . . . . . 145

A.3 Cavity construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

B Self-organization threshold equations 149

B.1 Lattice geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

B.2 Traveling wave geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

C Self-organization: temperature, entropy and phase space density 153

References 163

vi

Summary

This thesis details experimental investigations into the interaction of an

ultracold atomic ensemble with a single mode high finesse optical cavity.

To this end, simple and efficient experimental methods are developed to

cool and transport atoms. These include the all-optical production of a

Bose-Einstein condensate in a 1 µm wavelength crossed beam dipole trap

and direct mechanical translation of cold atoms into a high finesse cavity

over ∼ 1 cm.

First, we study the cavity transmission spectra for weak driving of a single

mode cavity coupled to a cold ensemble of rubidium atoms. The multi-level

structure of the atoms together with the collective coupling to the cavity

mode leads to complex spectra which depend on atom number and probe

polarization. We model the linear response of the system as collective spin

with multiple levels coupled to a single mode of the cavity. The observed

spectra are in good agreement with this reduced model.

Second, we study transverse pumping of a thermal ensemble of atoms cou-

pled to a cavity which results in self-organization. The differences between

probing with a traveling wave and a retro-reflected lattice are investigated.

We derive threshold conditions for self-organization in both scenarios and

verify a threshold scaling consistent with the mean field prediction over a

range of atom numbers and cavity detunings.

Most recently, a 2D lattice potential is used to organize the atoms into a

Bragg crystal, and coherent scattering into the cavity is observed without

threshold. This configuration is ideal for future investigations into either

cavity sideband cooling of the collective motion or simulation of Dicke model

via the collective spin.

CONTENTS

viii

List of Tables

2.1 Scaling for forced evaporation in optical traps (η = 10). . . . . . . . . . 14

2.2 Example initial conditions for a 1.06 µm cross beam trap (η = 8). . . . . 22

3.1 High finesse cavity parameters . . . . . . . . . . . . . . . . . . . . . . . 67

4.1 Dual-wavelength high finesse cavity parameters . . . . . . . . . . . . . . 96

A.1 First coating run . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

A.2 Second coating run . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

ix

LIST OF TABLES

x

List of Figures

2.1 Schematic and photograph of the experimental apparatus. . . . . . . . . 7

2.2 Density distribution in cross beam trap. . . . . . . . . . . . . . . . . . . 19

2.3 Cross beam potential profile. . . . . . . . . . . . . . . . . . . . . . . . . 19

2.4 Fraction of atoms in wings vs η . . . . . . . . . . . . . . . . . . . . . . . 20

2.5 Migration of atoms to wings during evaporation. . . . . . . . . . . . . . 22

2.6 Trap volume for elliptical beams. . . . . . . . . . . . . . . . . . . . . . . 25

2.7 Schematic of master laser optics. . . . . . . . . . . . . . . . . . . . . . . 28

2.8 Schematic of slave laser optics. . . . . . . . . . . . . . . . . . . . . . . . 28

2.9 Frequency tuning scheme for cooling lasers. . . . . . . . . . . . . . . . . 29

2.10 Schematic of imaging system. . . . . . . . . . . . . . . . . . . . . . . . . 30

2.11 Loading sequence for dipole traps. . . . . . . . . . . . . . . . . . . . . . 33

2.12 Dipole trap loading vs repump intensity. The peak corresponds to a

repump intensity of ≈ 5 µW/cm2 . . . . . . . . . . . . . . . . . . . . . . 33

2.13 Atoms loaded vs FORT power. . . . . . . . . . . . . . . . . . . . . . . . 34

2.14 Atoms loaded into FORT vs MOT number. . . . . . . . . . . . . . . . . 34

2.15 Trap lifetime for several Rb source currents. . . . . . . . . . . . . . . . . 35

2.16 Hyperfine changing collisional losses. . . . . . . . . . . . . . . . . . . . . 36

2.17 Measuring trap frequencies by parametric excitation. . . . . . . . . . . . 37

2.18 Measured radial trap frequencies of the primary beam. . . . . . . . . . . 38

2.19 Free evaporation in the primary trap . . . . . . . . . . . . . . . . . . . . 40

2.20 Forced evaporation in the primary trap only. . . . . . . . . . . . . . . . 42

2.21 Composite trap evaporation cycle. . . . . . . . . . . . . . . . . . . . . . 43

2.22 BEC Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.23 Condensate fraction below the critical temperature. . . . . . . . . . . . . 45

xi

LIST OF FIGURES

3.1 Cavity experiment schematic and photo . . . . . . . . . . . . . . . . . . 50

3.2 Jaynes-Cummings ‘ladder’ of dressed states. . . . . . . . . . . . . . . . . 52

3.3 Vacuum-Rabi splitting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.4 Eigenenergies for N-atoms. . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.5 Transmission spectrum including dissipation . . . . . . . . . . . . . . . . 56

3.6 Schematic of experiment configuration . . . . . . . . . . . . . . . . . . . 58

3.7 Collective enhancement example . . . . . . . . . . . . . . . . . . . . . . 61

3.8 Collective coupling to multiple hyperfine transitions. . . . . . . . . . . . 63

3.9 Analytic eigenspectrum for N -alkali atoms . . . . . . . . . . . . . . . . . 64

3.10 Transmission spectrum for dispersion theory . . . . . . . . . . . . . . . . 65

3.11 Vibration isolation stack. . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.12 Experiment cavity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.13 PZT circuit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.14 Schematic of experiment laser system. . . . . . . . . . . . . . . . . . . . 70

3.15 Intra-cavity FORT lifetimes . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.16 Optical pumping scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . 75

3.17 Lifetime after optical pumping. . . . . . . . . . . . . . . . . . . . . . . . 75

3.18 Experiment configuration for an effective two-level system. . . . . . . . . 76

3.19 Transmission spectrum on cycling transition . . . . . . . . . . . . . . . . 76

3.20 Transmission spectra probing π-transitions . . . . . . . . . . . . . . . . 77

3.21 Transmission spectra probing both cavity modes . . . . . . . . . . . . . 78

3.22 Optical pumping by cavity probe field . . . . . . . . . . . . . . . . . . . 80

4.1 Schematic and picture of experiment apparatus . . . . . . . . . . . . . . 84

4.2 Schematic of self-organization . . . . . . . . . . . . . . . . . . . . . . . . 85

4.3 Self-organization potentials . . . . . . . . . . . . . . . . . . . . . . . . . 92

4.4 Picture of dual-wavelength cavity . . . . . . . . . . . . . . . . . . . . . . 97

4.5 Cavity birefringence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.6 780/1560 nm laser setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.7 Beatnote between our 1560 nm laser and a frequency comb phase-locked

to a narrow reference laser. . . . . . . . . . . . . . . . . . . . . . . . . . 101

4.8 Cavity coupling and detection set-up. . . . . . . . . . . . . . . . . . . . 102

4.9 Lowest electronic states of 87Rb. . . . . . . . . . . . . . . . . . . . . . . 104

xii

LIST OF FIGURES

4.10 Cavity mode overlap for 780/1560 nm lattices . . . . . . . . . . . . . . . 105

4.11 Measured cavity coupling at nodes and anti-nodes . . . . . . . . . . . . 106

4.12 Heating rate due the scattering of probe beam . . . . . . . . . . . . . . 109

4.13 Heating due to adiabatic compression by the probe beam . . . . . . . . 109

4.14 Results of threshold measurements in the lattice geometry . . . . . . . . 110

4.15 Results of threshold measurements in the traveling wave geometry . . . 111

4.16 Self-organization traces: lattice probe, very large dispersive shift, ∆c < −κ115

4.17 Self-organization traces: lattice probe, very large dispersive shift, ∆c = −κ116

4.18 Self-organization traces: lattice probe, large dispersive shift . . . . . . . 117

4.19 Self-organization traces: lattice probe, µ→ 1 . . . . . . . . . . . . . . . 118

4.20 Heating due to non-adiabatic dynamics . . . . . . . . . . . . . . . . . . 120

4.21 Self-organization traces: traveling wave probe, unstable configuration . . 121

4.22 Self-organization traces: traveling wave probe, stable configuration . . . 121

5.1 Bragg scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

5.2 Suppression of Bragg scattering . . . . . . . . . . . . . . . . . . . . . . . 126

5.3 Entropy phase diagram for self-organization. . . . . . . . . . . . . . . . . 136

5.4 Entropy and phase space density gain. . . . . . . . . . . . . . . . . . . . 136

5.5 Realization of Dicke model . . . . . . . . . . . . . . . . . . . . . . . . . . 139

A.1 Various cavities designs. . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

A.2 Cavity construction set-up inside laminar flow hood. . . . . . . . . . . . 146

C.1 Entropy and phase space density across threshold . . . . . . . . . . . . . 160

xiii

LIST OF FIGURES

xiv

Chapter 1

Introduction

Since the discovery of laser cooling over two decades ago [1, 2, 3, 4, 5], the use of laser

light to precisely manipulate both internal and external atomic degrees of freedom has

been an essential tool for atom physics research. Near to an atomic resonance, scat-

tering of laser light via spontaneous emission is the dominant process. Laser cooling

harnesses this scattering to cool the motion of atoms down to micro-Kelvin temper-

atures. Additionally, optical pumping [6] methods utilize spontaneous emission to

prepare atoms in a specific internal state. For laser light far detuned from an atomic

resonance, spontaneous emission is suppressed and the dominant processes are coherent

scattering and the dispersive interaction with the light field. In this regime, even more

precise control is possible; by using Raman processes [7] to coherently manipulate the

internal electronic state or the motional state of the atom, and by using the dipole

force which arises from the dispersive interaction of the induced dipole with the light

field. For sufficiently large detuning, the optical potential resulting from the dipole

force is effectively conservative [8]. Far-off resonance optical traps (FORT) are a ver-

satile tool for the trapping and manipulation of cold atoms with which a variety of

trapping geometries can be realized, such as optical lattice potentials [9] and crossed

beam traps [10]. Furthermore, with evaporative cooling, the FORT offers an all-optical

route to producing Bose-Einstein condensate (BEC) [11] as a practical alternative to

the standard magnetic trapping methods [5, 12].

Inside a high finesse optical resonator, the normally weak dipole interaction is en-

hanced by multiple passes of the light such that a single photon may strongly couple

to a single atom. Although the physics of cavity quantum electrodynamics (QED) was

1

1. INTRODUCTION

first described nearly 50 years ago by the Jaynes-Cummings model (JCM) [13], only in

the last 20 years has mirror technology advanced to the point where the strong coupling

regime has been experimentally accessible. This has opened the way for experimental

demonstration of phenomena which predicted by the JCM, such as the photon blockade

effect [14] and the vacuum-Rabi splitting due to a single atom [15, 16]. For a sufficiently

good cavity, an atom can even be trapped by the dispersive backaction from the field of

a single photon [17]. Unlike a free-space FORT, the intra-cavity optical potential in a

driven cavity depends on the position of the atom [18] and thus strongly couples to the

atomic motion. This coupling can result in dissipative forces [19] which have been used

to cool a single atom [20]. Cavity cooling via coherent scattering is of particular interest

because it can, in principle, be applied to any polarizable particles [21, 22]. Extend-

ing cavity cooling schemes to ensembles of particles in a cavity remains a tantalizing

possibility and area of active theoretical [23, 24] and experimental research [25, 26].

For an ensemble of atoms trapped inside a cavity driven by a laser field, the atom-

field coupling increases due to the interaction of N atoms with the single cavity mode.

This system is described quantum mechanically by the Tavis-Cummings model [27] in

which the atoms couple collectively to the field with an effective rate√N greater than

the single atom rate. The√N enhancement of the coupling is readily observed, for

example, in the normal mode splitting [28, 29]. The dynamical effects of backaction on

the atomic motion due to the dispersive interaction are complicated by the fact that

the total dispersive coupling depends on the positions of all of the atoms. As a result,

the motion of one atom couples to all the others via long-range cavity-mediated light

forces [30]. Theoretical work has shown that consequently the cooling rates for some

cavity cooling mechanisms do not scale favorably with N [31, 32]. Recently however,

cavity cooling of a single collective phonon mode at a collectively enhanced rate has been

experimentally demonstrated [25]. Whether this can be used to efficiently cool all modes

of an ensemble [25, 26], and whether this result holds for transverse pumping [33, 34, 35]

are still under investigation.

When driving an atomic ensemble with a laser field transverse to the cavity, the

behavior of the system is significantly different and interesting phenomena emerge. Co-

herent scattering into the cavity mode is enhanced by constructive interference only if

the atoms are spatially ordered to match the phase of the driving and cavity fields.

For uniformly distributed atoms, scattering is suppressed by destructive interference.

2

1.1 Outline of the Thesis

In a standing wave cavity, the backaction on the atoms from the scattered field gives

rise to a phase transition to a spatially organized array for sufficiently strong pump-

ing [36]. The first experiments to observe self-organization [37, 38] transversely pumped

a thermal ensemble falling through a cavity. In addition to collectively enhanced co-

herent scattering, these experiments reported cooling and a deceleration of the center

of mass motion resulting from resonator induced light forces. Later experiments with

a BEC in a standing wave cavity explored the transition from a superfluid to the self-

organized phase [39, 40], which was mapped to the Dicke Hamiltonian and associated

quantum phase transitions [41, 42]. However, there are still open questions related to

self-organization, specifically concerning the effective threshold scaling [34, 43] and the

extent to which self-organization can be used for cavity cooling of a thermal ensemble

[36, 44].

1.1 Outline of the Thesis

In the following chapters of this thesis, I describe work spanning five years and three

generations of experimental apparatus touching on many of these topics. The focus

of our research has evolved over time, together with the capability of the apparatus.

The structure of this thesis follows the evolution of the experimental apparatus with a

chapter devoted to each. The common thread throughout is the dispersive interaction

between light and atomic ensembles. First, in free space, where optical potentials

are used for the trapping and manipulation of atoms. Second, in a cavity, where the

collective dispersive interaction with the cavity gives rise to the dispersive shift and

dynamic optical potentials.

Chapter 2 In our first experiment, we all-optically produce a BEC in crossed beam

dipole trap of 1 µm wavelength. Our experimental setup is relatively simple

compared to earlier 1 µm all-optical BEC experiments and thus could be easily

integrated into more complex setups, such as a cavity QED experiment. We

discuss the effects of beam geometry on the thermal distribution and highlight the

obstacles to condensing all-optically in short wavelength traps. The techniques

for cooling, trapping, and manipulating atoms described in this chapter form the

foundation for experiments described in subsequent chapters.

3

1. INTRODUCTION

Chapter 3 In our second experiment, we integrate trapped ultracold rubidium atoms

with a high finesse optical cavity in the strong coupling regime. For a weakly

driven cavity, we study the transmission spectra of a single mode cavity strongly

coupled to an ensemble of multi-level atoms. The linear response of this system

is that of a collective spin coupled to multiple levels for which the correspond-

ing effective couplings to the cavity mode are enhanced by√N . We develop a

reduced quantum mechanical model for the system which is in good agreement

with the experimental results. We also interpret the results from a semi-classical

perspective using linear dispersion theory.

Chapter 4 Presently, in our third experiment, we employ a dual wavelength high

finesse cavity which allows for trapping of the atoms at every alternate anti-node

of the cavity mode. This configuration is ideal for studying coherent scattering

into the cavity from a transverse pumping field. In this chapter, we detail our

systematic study of self-organization of thermal atoms for two probing geometries:

a retro-reflected lattice [37, 38, 43] and a traveling wave. Self-organization in a

linear cavity with a traveling wave probe has not been previously reported. We

derive threshold conditions for both probe configurations in the mean field limit.

We experimentally measure the scaling of the self-organization threshold over a

wide range of parameters and characterize the behavior of the induced dynamical

potentials.

Chapter 5 By using an additional transverse trapping lattice with our third exper-

iment apparatus, we are are able to organize the atom into a Bragg crystal by

static FORT potentials alone. We report our observation of threshold-free coher-

ent scattering into the cavity and discuss potential future research directions with

such a system. Demonstrating cavity cooling of atomic ensembles has been one

of our long standing research objectives, but has thus far produced null results.

We end on a discussion of the practical and fundamental difficulties of cooling

trapped atomic ensembles coupled to a single mode cavity.

4

Chapter 2

Dipole Trapping and All-Optical

Bose-Einstein Condensation

This chapter covers the first generation experimental apparatus, pictured in Fig. 2.1, and is

largely based on ’All-optical Bose-Einstein condensation in a 1.06 µm dipole trap’ K. J. Arnold

and M. D. Barrett Optics Communications 284, 3288 (2011).

2.1 Introduction

Since the first observation of a Bose-Einstein condensate (BEC) in 1995 [45, 46], there

has been a tremendous volume of experimental and theoretical work which continues to

this day. So many labs around the world produce BEC that it can now be considered

routine, though still by no means trivial. The most common method is to use radio

frequency (rf) induced forced evaporation of atoms in a magnetic trap. An alternative

method, first demonstrated in 2001 [11], uses only optical trapping. Since then, several

atomic species have been successfully condensed via this method. For instance, using

a 10 µm wavelength CO2 laser, 87Rb [11, 47], 133Cs [48], and 23Na [49] have all been

condensed. Using a near 1 µm wavelength fiber laser, 87Rb [50, 51, 52], 133Cs [53], and

52Cr [54] have been condensed.

The advantages of the all-optical approach over magnetic trapping include a rela-

tively simple experimental setup, comparatively high repetition rates, and the ability

to trap arbitrary spin states. The primary difficulty is associated with relaxing the

trapping potential to induce evaporation. Relaxation of the trap leads to a continual

5

2. DIPOLE TRAPPING AND ALL-OPTICAL BOSE-EINSTEINCONDENSATION

decrease in the collision rate as the evaporation progresses. The decreasing collision

rate can stagnate the evaporation process before degeneracy is achieved. In the earliest

all-optical BEC experiments [11, 47], researchers were able to achieve sufficiently high

initial phase space and spatial densities by directly loading a 10.6 µm-wavelength trap

that BEC was reached before the evaporation process stagnated.

The original goal for the experiment described in this chapter was to produce BEC

in exactly the same simple crossed-beam optical trap geometry used in [11], except

with a 1.06 µm wavelength trap rather than a 10.6 µm wavelength trap. The 1.06 µm

wavelength is practically advantageous over the 10.6 µm wavelength because special

optics and vacuum windows are not required. This is important for integration into

more complex experiments which have limited optical access, such as our subsequent

cavity QED experiments. Although previous experiments had produced BEC in near

1 µm wavelength traps, they generally started from worse initial conditions and instead

used methods of varying complexity to circumvent a stagnating evaporation rate. In

Ref. [50] a mobile lens system dynamically compressed the atoms to offset the decreasing

density. In other experiments evaporation was forced without relaxing the optical trap

using either a strong magnetic field gradient [53], or a large displaced auxiliary beam

[52].

Our original goal proved elusive, however, as the low beam divergence due to the

1 µm wavelength was, unexpectedly, a significant complicating factor for the crossed

beam trap geometry. Later in this chapter, we will discuss the nature of the thermal

distribution in crossed beam traps in order to understand the source of our early diffi-

culties. Although focused primarily on cross beam traps, the considerations discussed

would equally apply to any trapping geometry composed of disparate volumes.

Ultimately, we were able to produce a BEC all-optically in a 1 µm cross-beam trap.

We used a method similar in spirit to the dynamic compression experiment [50], but

with the same minimal complexity of the crossed beam geometry. Fig. 2.1 shows a

schematic representation of our experiment next to a photograph of the experiment

chamber. A single tightly focused dipole trap is loaded directly from a magneto-optical

trap (MOT) and only later in the evaporation is a second transverse beam used to the

compress the atoms. Using this approach, we are able to a achieve a gain in phase

space density of 105, higher than is typically possible in all-optical traps. We reach

6

2.2 Background

Figure 2.1: Schematic representation and photograph of the experiment. Not all optics

used in the experiment are present in the photograph.

degeneracy after 3 seconds of evaporation resulting in a mostly pure condensate of

3.5× 104 atoms.

2.2 Background

Throughout the work presented in this thesis, optical traps in a variety of wavelengths

and geometries are used to confine atoms. In this section we review the basic physics

of dipole traps, the loading of dipole traps by laser cooling, and evaporative cooling in

optical potentials.

2.2.1 Dipole traps

For a comprehensive review of optical dipole force traps, the reader may refer to Grimm,

Weidemuller, and Ochinnikov [8], on which the following discussion is based.

2.2.1.1 Two-level atom

The optical dipole force results from the interaction of an induced atomic dipole moment

with the intensity gradient of a light field. For an atom placed into laser light with

an electric field E oscillating at frequency ω, the amplitude p of the induced dipole

moment is related to the electric field amplitude E by

p = α(ω)E (2.1)

7


where α(ω) is the frequency dependent polarizability. The interaction potential is then

given by

Udip = −1

2〈pE〉 = − 1

2ε0cRe(α)I, (2.2)

where the field intensity is I = 2ε0c|E|2, and the factor of 1/2 is because the dipole

is an induced, not a permanent one. The dipole force arises from the gradient of the

interaction potential and is thus a conservative force proportional to the gradient of

the laser intensity.

While the real part of the polarizability governs to dispersive interaction which gives

rise to the dipole force, the imaginary part governs the absorption. Power absorbed by

the oscillating dipole is re-radiated via spontaneous emission processes at a scattering

rate

Γsc =1

~ε0cIm(α)I. (2.3)

For a two level atom, the frequency dependent polarizability can be found semi-

classically by treating the atom as a classical driven oscillator with resonance frequency

ω0, but with a damping rate determined by the dipole matrix element between the

excited and ground state,

Γ =ω3

0

3πε0~c3|〈e|µ|g〉|2. (2.4)

The polarizability can then be found by integration of the equation of motion with the

result

α(ω) = 6πε0c3 Γ/ω2

0

ω20 − ω2 − i(ω3/ω2

0)Γ. (2.5)

Substituting this equation for polarizability into Eq. 2.2 and Eq. 2.3 we arrive at:

Udip = −3πc2

2ω30

(Γ

ω0 − ω+

Γ

ω0 + ω

)I, and (2.6)

Γsc =3πc2

2~ω30

(ω

ω0

)3( Γ

ω0 − ω+

Γ

ω0 + ω

)2

I. (2.7)

These equations are valid for any detuning ∆ ≡ ω−ω0 greater than the natural atomic

linewidth Γ. Typically |∆| ω0, in which case the so-called counter rotating term

can be neglected and also ω/ω0 ≈ 1. In the ‘rotating wave approximation’ (RWA), the

equations reduce to simply,

Udip =3πc2

2ω30

(Γ

∆

)I (2.8)

Γsc =3πc2

2~ω30

(Γ

∆

)2

I. (2.9)

8

2.2 Background

From these equations two key aspects of far-off resonant traps (FORT) are readily

observed. First, for ∆ < 0 (known as ”red detuning”) the potential is negative and

thus atoms are attracted to the light field. Second, while the potential scales as I/∆,

the scattering rate scales as I/∆2. Therefore by going to large detuning and high

intensity, the scattering rate can be made negligibly small while maintaining the same

potential depth.

2.2.1.2 Multi-level alkali atoms

Here we consider the results specifically for the ground state of alkali atoms, where

spin-orbit coupling leads to the D-line doublet of S1/2 → (P1/2, P3/2). Coupling to the

nuclear spin then produces the hyperfine structure of the ground and excited states. In

this case, the potential and spontaneous scattering rate for an atom in a ground state

with total angular momentum F and magnetic quantum number mF are given by [8],

Udip(r) =πc2Γ

2ω30

(2 + P gF mF

∆2,F+

1− P gF mF

∆1,F

)I(r) (2.10)

Γsc(r) =πc2Γ2

2~ω30

(2

∆22,F

+1

∆21,F

)I(r) (2.11)

where P = (1, 0,−1) for transitions (σ+, π, σ−) respectively, ∆1,F is the detuning of

the FORT light from the P1/2 state, ∆2,F is the detuning of the FORT light from the

P3/2 state, (F,mF ) are the hyperfine and magnetic sub-levels, and gF is the Lande

g-factor. These equations are valid as long as the detuning from both D-lines is large

with respect to the excited state hyperfine splitting. Usually linear polarization is used

so that the potential is mF -state independent.

Note that the RWA has been made in the above equations. For the experiments in

subsequent chapters, 87Rb will be trapped in both 1064 nm and 1560 nm wavelength

FORTs where the counter-rotating terms should not be neglected. Keeping the counter

rotating terms results in a correction factor to the potential depth (Eq. 2.10) of 1.15

for a 1064 nm FORT, and 1.33 for a 1560 nm FORT.

2.2.1.3 Rayleigh vs Raman scattering

The total scattering rate given by Eq. 2.11 is composed of two parts: coherent Rayleigh

scattering does not change the internal state, and incoherent Raman scattering for

9


which the atom changes hyperfine or Zeeman level in the S1/2 ground state manifold.

The ratio of these two rate is given by

ΓRaman

ΓRayleigh= 2

(∆2,F −∆1,F

∆2,F + 2∆1,F

)2

. (2.12)

For detunings larger than the fine structure splitting, spontaneous Raman processes

are suppressed by quantum interference of the scattering amplitudes via the D1 and

D2 lines [55]. For example, Raman scattering is suppressed by roughly a factor of 1000

relative to Rayleigh scattering for 87Rb trapped in a 1064 nm wavelength FORT.

2.2.1.4 Single focused beam FORT

The simplest possible trapping geometry is a single Gaussian laser beam of wavelength

λ focused to a waist w0, and thus with Rayleigh range zR = πw20/λ. The FORT

potential is determined from the intensity profile of the Gaussian beam, i.e.

U(r, z) = −

(U0

1 + ( zzR

)2

)exp

−2r2

w20

(1 + ( z

zR)2) (2.13)

where U0 is the depth of the trap. In the harmonic approximation, the radial and axial

trapping frequencies are given by

ωr =

√4U0

mw20

, (2.14)

and

ωax =

√2U0

mz2R

=

√2U0λ2

πmw40

. (2.15)

To get a general idea of typical FORT parameters, 10 W from a 1064 nm wavelength

laser focused to a 40 µm waist has a depth of U0/kb = 600 µK, radial trapping fre-

quencies 2π × 1.9 kHz, and an axial tapping frequency 2π × 12 Hz. The scattering

rate is only Γsc = 2π × 0.35 Hz. Since recoil temperature for an elastically scattered

1064 nm photon is Trec ≈ 190 nK, the heating rate due to off-resonant scattering is

T ≈ TrecΓsc ≈ 63 nK/s, which is very low relative to the trap depth.

10

2.2 Background

2.2.2 Laser cooling

Because the FORT is a conservative potential, the atoms must be cooled into the trap.

This is done using well-established laser cooling techniques. We will not cover details

of the laser cooling or magneto-optical traps (MOT) here as many excellent sources

are available, in particular the book “Laser Cooling and Trapping” by Metcalf and van

Straten [56]. The basis for the magneto-optical trap is Doppler cooling, whereby high

velocity atoms are Doppler shifted into a resonance with a cooling laser which slows

the atoms by elastic scattering of photons. Resolution of the Doppler shift is limited

to natural linewidth Γ of the atomic transition and results in a minimum temperature

of the atoms Tdoppler = ~Γ/2kb [57]. For 87Rb, this Doppler limit is 146 µK [58].

Even lower temperatures can be reached by sub-Doppler cooling methods [3, 59]. Sub-

Doppler cooling methods are still reliant on photon scattering processes and so are

limited by the recoil energy of the spontaneously emitted photons. In practice, sub-

Doppler cooling is limited to ∼ 10Trec. For the 780 nm cooling transition of 87Rb,

Trec ≈ 360 nK, and thus temperatures on the order of a few µK can be reached [60].

For creating a Bose-Einstein condensate, the quantity of interest is the phase space

density ρ = nΛ3, where Λ =√

2π~2/mkbT is the thermal de-Broglie wavelength and n

is the particle density. Thus we want to not only cool the atoms via laser cooling, but

also achieve high densities. In a MOT, the typical particle density is ∼ 1010 cm−3 which

is limited by radiation pressure from reabsorption of light scattered by the trapped

atoms and collisions with excited state atoms [56]. By increasing the magnetic field

gradients as well as increasing the detuning of the cooling lasers, a MOT can be com-

pressed to higher densities as a result of the reduced rate of excitation. Densities as

high as 5 × 1011 cm−3 are reported by this method [61]. Alternatively by creating a

“dark spot” in the center of MOT where there is no repump intensity and thus a much

lower rate of excitation, densities close to 1012 cm−3 have been reported [62]. Using

these laser-cooling methods, the highest reported phase-space densities in a MOT are

in range 10−5 to 10−4 [63, 64]. In order to cool atoms to higher phase space densities,

methods which do not require excitation of the atom must be employed. The only

successful methods have been forced evaporation in either a magnetic trap or FORT.

11


2.2.3 Evaporative cooling

Evaporation is the natural process by which the highest energy atoms escape the con-

fining potential, thereby lowering the average thermal energy of the remaining trapped

atoms. Collisions continually redistribute energy between atoms to bring the total

distribution towards the Maxwell-Boltzmann distribution. By this thermalization pro-

cess, atoms in the high energy tail of the distribution, which have sufficient energy

to escape the trap, are replenished. The process eventually stagnates for decreasing

temperature as the tail of distribution above of the trap depth becomes exponentially

small. The evaporation must then be forced by lowering the trap depth, truncating the

Maxwell-Boltzmann distribution at ever lower energies.

In magnetic traps, this is accomplished by driving rf-transitions to remove the higher

energy atoms. Because atoms higher in the trapping potential have a larger Zeeman

shift, they can be selectively addressed, flipping their spin, and thus ejecting them

from the trap as they experience an anti-trapping potential. Ramping down the rf

frequency, the effective trap depth decreases as lower energy atoms are removed. Since

the trapping frequencies remain the same, both the density and collision rate increase

as the evaporation progresses. This is known as ‘runaway’ evaporation because the

rate at which the trap depth can be efficiently lowered accelerates. By this method the

first BEC was created, which required nearly a 107 gain in phase space density from

forced evaporation [45]. A general review of evaporative cooling methods, including

rf-induced evaporation, can be found in the report of Ketterle and Druten [65].

In optical traps, the evaporation is usually forced by lowering the laser beam in-

tensity. Although this method has the advantage of simplicity, a major limitation

is that the thermalization rate slows as evaporation progress. Because the trapping

frequencies decrease along with the trap depth, the density and collision rate simulta-

neously decrease even while the phase space density increases. We investigate this more

quantitatively by considering the scalings laws for evaporation in optical potentials.

2.2.4 Scaling laws

In order to get a rough idea of how the atom number N , phase space density ρ, and

elastic collision rate γ, vary as the potential depth U is lowered, it is useful to consider

the simple scaling laws derived in [66]. These scaling laws provide simple analytic

12

2.2 Background

expressions which are in good agreement with a Boltzmann equation model. We briefly

outline the derivation in [66] here.

An important parameter for evaporation is the ratio of the trap depth to tempera-

ture, η ≡ U0/kbT . As an initial condition, we assume free evaporation processes have

already stagnated such that η is large. Furthermore, we assume that the evaporation

forced by lowering the optical potential proceeds sufficiently slowly that η remains con-

stant. For large η, the atoms are well confined near the bottom of the potential which

is, to good approximation, a harmonic oscillator potential. Thus the energy is given by

E = 3NkbT , from which one finds the rate equation

E = 3NkbT + 3NkbT . (2.16)

From this equation, we see there are two contributions to the energy loss: the first

is associated with the evaporative loss of atoms, and the second with a decreasing

temperature due to adiabatic lowering of the potential. The average energy taken by

single atom which escapes from the trap is (η + α)kbT , where 0 ≤ α ≤ 1 [67]. The

factor α depends on the trap geometry, but for any potential which is harmonic near

the bottom, α = (η − 5)/(η − 4) [66]. The energy loss rate due to evaporation is thus

Eevap = η′NkbT , where η′ = η + α. To find the rate of energy loss due to changing

potential, we note that the phase space density, given by ρ = N(hν)3/(kbT )3 for a

harmonic oscillator in the classical regime, is adiabatically invariant at fixed N . Thus

the quantity ν/T is constant, where ν is the geometric mean trap frequency. Given

E ∝ T , at fixed N , and ν ∝√U , we have d

dt(√U/E) = 0 from which we find the

rate due to the changing potential Epot = (3NkbT/2)(U/U). Equating Eevap + Epot to

Eq. 2.16 we have

η′NkbT +3NkbT

2

U

U= 3NkbT + 3NkbT . (2.17)

Substituting U = ηkbT , by definition, and U = ηkbT , assuming constant η, we find

N

N=

3

2

1

η′ − 3

U

U. (2.18)

Direct integration yields the scaling law for atom number

Nf

Ni=

(Uf

Ui

) 32

1η′−3

. (2.19)

13


We are particularly interested in the scaling of phase space density, for which we see

ρ ∝ NU−3/2 and thus

ρf

ρi=

(Ui

Uf

) 32η′−4η′−3

. (2.20)

Finally, we find the scaling for the elastic collision rate. For a Bose gas in harmonic

potential, the elastic collision rate is given by γ(sec−1) = 4πNmσν3/(kbT ) [66], where

m is the particle mass, σ is the elastic scattering cross section, and ν is the mean

trapping frequency in Hz. Thus γ ∝ NU1/2 and

γf

γi=

(Uf

Ui

) 12

η′η′−3

. (2.21)

Table 2.1: Scaling for forced evaporation in optical traps (η = 10).

phase space density atom number trap depth collision rate

ρf/ρi Ni/Nf Ui/Uf γi/γf

103 2.8 200 37

104 3.9 1200 133

105 5.4 6600 440

Let us consider the numbers listed in Table 2.1, which are calculated for a typical

η = 10. We can understand the importance of the initial conditions because gaining

significantly more than 103 in phase space density is impractical. This is for two reasons.

First, while controllably lowering the trap depth to 1/100 of the maximum depth is

straight forward, going to 1/1000 and beyond becomes experimentally problematic.

Second, the collision rate sets the time scale for how fast the potential depth can be

lowered while maintaining thermalization. Eventually the loss of phase space density

from non-evaporative atom losses, which were neglected in the scaling laws, overtakes

gain of phase space density from evaporation.

Experiments which have produced BEC by directly loading crossed beam traps

and simply lowering the potential mostly report starting phase spaced densities >

10−3 [11, 47, 49]. In the experiment reported in Ref. [68], they start at a phase space

density 2 × 10−4 and are unable to achieve condensation, stagnating at ρf ≈ 0.2 after

14

2.2 Background

decreasing the potential by a factor of 100. These results are all consistent with the

scaling laws and demonstrate the necessity of ρi ∼ 10−3 as a starting condition in the

absence of more sophisticated methods.

One outlying example is Ref. [47] which reports production of a BEC in a single

beam trap starting from ρi = 1.2 × 10−4 while only decreasing the beam power by a

factor of 140. In this case, it is most likely that gravity is playing a significant roll

in truncating the trap near the end of the evaporation. The ratio of their reported

initial temperature and critical temperature suggests Ui/Uf ≈ 800, assuming constant

η, which is much closer to the expected scaling.

2.2.5 Atom losses due to inelastic collisions

The scaling laws in the previous section neglect the effects of atom losses through

inelastic collisions. Different collisional processes contribute to one-, two-, and three-

body loss rates, labeled α, β, and L respectively. As a result, the density of the atomic

cloud decreases at the rate

dn

dt= −αn− βn2 − Ln3. (2.22)

One-body losses

The one-body loss rate α results from collisions between atoms in the trap and the

background gas at 300 K. While the exact loss rate depends on the composition of the

background gas (see Ref. [69] for relevant collisional cross-sections), it is on the order

of α/nbg ∼ 5 × 10−9 cm−3/sec, where nbg is the particle density of the background

gas. Experiments which load a MOT from Rb vapor pressure, such as ours, typically

operate at a vacuum in the range of 10−10 to 10−9 Torr, which corresponds to a trap

lifetime on the order of 10 seconds. We report the lifetime due to background collisions

for our experiment in Sec. 2.4.5. Other more complex experiments which employ cold

atom sources such a Zeeman slower [70] or 2D MOT [71] are able to reach a vacuum

of ∼ 10−12 Torr and thus background limited lifetimes exceeding 10 minutes.

Two-body losses

There are two principle mechanisms for two-body losses [72]: light assisted collisions

and hyper-fine ground state changing collisions. Light-assisted collisions occur when

15


near-resonant light excites a pair of nearby atoms to a molecular potential. The atoms

are accelerated together and gain sufficient energy before relaxing to the ground state,

via radiative emission, such that both atoms are ejected from the trap [73, 74]. This

type of collision can be avoided by ensuring all near-resonant light, such as from the

MOT and repump lasers, is completely blocked (e.g. by mechanical shutters) after

loading the FORT.

Hyperfine ground state changing collisions can occur when at least one of the col-

liding atoms is in the upper hyperfine ground state, |F = 2〉 for 87Rb. During the

collision, the atom undergoes a spin-flip from the |F = 2〉 to the |F = 1〉 ground state,

resulting in a release of energy corresponding to the ground state splitting of 6.8 GHz.

This is over 100 times the typical depth of a dipole trap, and so both atoms are lost

after the collision. These collision can be avoided by pumping all the atoms into the

|F = 1〉 ground state manifold. For a measurement of the two-body loss rate β due to

hyperfine changing collisions in our experiment, see Sec. 2.4.6.

Three-body losses

Three-body recombination results in two atoms forming a molecule with the third atom

carrying away resulting energy, and thus all three atoms are lost from the trap. The

rate has been measured for 87Rb to be L ∼ 10−29cm6s−1 [75, 76]. Thus three-body

losses only become an issue for peak densities > 1014 cm−3, which can be the case in

crossed beam dipole traps [11]. For the experiments in this thesis, however, three-body

losses do not play a significant role.

2.3 Discussion of crossed beam traps

Due to the unfavorable scaling discussed in Sec. 2.2.4, the key to reaching degeneracy in

dipole traps is good initial conditions. The crossed beam configuration is particularly

advantageous for facilitating high initial densities and phase space densities. A simple

picture is that the comparatively large volume of the individual beams allows effective

loading of a large number of atoms from the MOT. During free-evaporation immediately

after loading, the atoms redistribute towards a Boltzmann distribution with most of

remaining atoms settling in the much smaller volume of the crossed beam region. In

this way densities ni > 1014 cm−3 and phase space densities ρi > 10−3, roughly two

16


orders magnitude higher than in the compressed MOT, are quickly obtained [11, 47].

From these starting conditions, evaporating to BEC is straightforward.

But this simple picture is not everything. In our initial experiments, we expected

to load a 1.06 µm wavelength crossed beam trap with similar results to those earlier ex-

periments using the 10.6 µm wavelength CO2 laser. Much like the experiment reported

in Ref. [68], which was virtually identical to ours, we had difficultly getting such favor-

able starting conditions. In this section, we present a detailed analysis of the thermal

distribution of crossed beam traps to understand how the difference in wavelength has

such a significant impact.

2.3.1 General thermal distribution of a trapped gas

For a thermal gas in a infinitely deep potential, the phase-space distribution is given

by [67]

f0(r,p) = n0Λ3 exp[−(U(r) + p2/2m)/kbT ], (2.23)

where n0 is the peak density located at the minimum of the potential, Λ is the thermal

de-Broglie wavelength. By integrating over momentum states we find the familiar

thermal density distribution

n(r) = n0 exp[−U(r)/kbT ]. (2.24)

For physical trap of finite depth εt, a truncated Boltzmann energy distribution leads

instead to the phase-space distribution [67]

f(r,p) = f0(r,p)Θ(εt − U(r)− p2/2m), (2.25)

where Θ(x) is the Heaviside step function. Integrating over momentum states yields

the density distribution

n(r) = n0 exp(−U(r)/kbT )P (3/2, εt − U(r)/kbT ) , (2.26)

where P (3/2, x) is the incomplete gamma function and n0P (3/2, εt/kbT ) ≈ n0 is the

peak density. The normalization n0 can be found by integrating the spatial density

over all space,

N = n0

∫V

exp(−U(r)/kbT )P (3/2, εt − U(r)/kbT ) d3r. (2.27)

17


2.3.2 Crossed beam distribution: numeric solution

Here we consider the potential formed by two orthogonal dipole traps both focused to

a waist w0 and each with potential depth U0. Following methods described in Ref. [77],

the integral of Eq. 2.27 can be simplified by introducing the function V (U), which is

the volume of space enclosed by the equipotential surface U . This reduces Eq. 2.27 to

the one dimension form

N = n0

∫ εt

0exp(U/kbT )P (3/2, (εt − U)/kbT )

dV

dUdU. (2.28)

It is convenient to express this equation in terms of scaled parameters,

N = n0

∫ β

0exp(−ηu)P (3/2, η(β − u))

dV

dudu (2.29)

where η = U0/kbT , β = εt/kbT , and u = U/U0. Note that u ∈ [0, 2] for the crossed

beam potential. We do not have an analytic expression for dVdu , but can calculate it

numerically. As discussed in Ref. [77], it is necessary to take β < 2 truncating the

trap depth due to the divergence in the density of states at the top of the trap. For

relevant values of η, the divergence only becomes numerically significant for β & 1.95

and, in practice, the trap depth is typically less than this due to external effects such

as gravity. For 1.85 < β < 1.95 the integrals only vary by 10% and smaller values of β

are only relevant when the potential is significantly modified by an external influence.

In Fig. 2.2, we compare the numerically calculated density profiles of 10.6 µm and

1.06 µm wavelength crossed beam traps for w0 = 40 µm and η = 8. The difference in

density profiles for the two wavelengths is easily interpreted by considering the geometry

of the two potentials, which are plotted in Fig. 2.3. Although the potentials are identical

in the cross region (µ < 1), the shallow divergence of the 1.06 µm wavelength trap

results in much weaker confinement outside the cross region (µ > 1). The comparatively

large increase in the density of states outside the cross region in the 1.06 µm wavelength

trap results in a significant fraction of the atoms remaining outside the center region

even for a thermalized distribution. This is after free evaporation has already stagnated,

given that η = 8, and the truncation of the distribution is negligible.

2.3.3 Crossed beam distribution: approximate analytic solution

We can find approximate analytic expressions for the number of atoms in the center

region, Nc, and number of atoms in the wings, Nw, by considering the integral in

18


Figure 2.2: Density profiles found by nu-

meric integration for w0 = 40 µm, η = 8,

and β = 1.9.

Figure 2.3: Optical potential depth along

the axis of one beam in a crossed beam

trap. For both wavelengths the beam

waists are w0 = 40 µm.

Eq. 2.29 separately for the two regions µ < 1 and µ > 1. In the center region (µ < 1),

the potential is, to a good approximation, independent of the divergence of the beams

as can be seen in Fig. 2.3. Since the density is weighted towards lower values of

u, we can make a harmonic approximation to the potential and neglect the effects

of truncation. The volume of the center region is given by V (u) = π3w

30u

3/2 in the

harmonic approximation. Thus we have

Nc = n0

∫ 1

0exp (−ηu)P (3/2, η(β − u))

dV

dud u (2.30)

≈ πn0w30

2

∫ ∞0

exp (−ηu)u1/2d u =n0w

30

4

(π

η

)3/2

. (2.31)

For the wings (µ > 1) we have,

Nw = n0

∫ β

1exp (−ηu)P (3/2, η(β − u))

dV

dud u (2.32)

= n0e−η∫ β−1

0exp

(−ηu′

)P(3/2, η(β − 1− u′)

) dVdu′

d u′ (2.33)

where we have made the substitution u′ = u−1. Note in Eq. 2.33 the factor e−η out the

front can be interpreted as the reduction factor of the density in the wings relative to the

19


Figure 2.4: Comparison of the fraction of atoms in the wings outside the dimple (blue

lines) and the peak density (red lines) as a function of η for 1.06 µm and 10.6 µm traps.

The red and blue dashed lines are the analytic approximations from equations (2.35) and

(2.36) respectively. The solid lines result from numerical integration of the exact equations

(2.30) and (2.32) with β = 1.9. All calculations assume an atom number N = 2× 106 and

beam waists w0 = 40 µm.

trap center. For u > 1 the function V (u) is, to a good approximation, independent of

the beam intersection and can be approximated by the sum of two independent single

focus beams. As before, they can be approximated by two infinitely deep harmonic

oscillator potentials, each with a volume V (u) = 2π3 w

20zRu

3/2. Integration of Eq. 2.33

yields

Nw = n0w20zR

(π

η

)3/2

e−η. (2.34)

Since the total atom number N = Nc + Nw, the fraction of atoms in the wings of the

cross beam trap, NwN , and the peak density at the center, n0, are then easily found and

we haveNw

N≈

4πw0λ e−η

1 + 4πw0λ e−η

(2.35)

and

n0 ≈4N

w30

(ηπ

)3/2 1

1 + 4πw0λ e−η

. (2.36)

These expressions clearly highlight the influence of the wavelength λ. As πw0/λ

increases, a larger fraction of the atoms appear in the wings of the potential and the

peak density decreases accordingly. Figure 2.4 specifically illustrates the differences

20


between a CO2 laser with λ = 10.6 µm and a fiber laser with λ = 1.06 µm. We see

that agreement of the analytic expressions (dashed lines) to the numeric result (solid

lines) is reasonably good considering the crude approximations made. Under typical

experimental conditions, the initial free evaporation proceeds quickly, due to the very

large densities and corresponding collision rates, until stagnating near η ∼ 8. For both

wavelengths there is a relatively small fraction of atom in the wings for this value of η.

However, during forced evaporation, the collision rate drops which can result in a slight

decrease in η. Although this drop in η is inconsequential for a CO2 laser trap, it can

have a substantial effect for a 1.06 µm laser trap with large number of atoms migrating

to the wings. While this migration of atoms to the wings does result in a further

decrease in the density, the main impact is on the thermalization and evaporation rates

of the overall distribution.

2.3.4 Thermalization in crossed beam traps

The collision rate sets the time scale for how fast the evaporation can be forced without

a significant increase in η. The problem with the atoms in the wings is their vastly

reduced collision rate as compared to atoms in the center region. To illustrate the

problem we consider the experiment of two 8 W 1.06 µm beams with waist w0 = 40 µm

and the initial conditions shown in Table 2.2. These would normally be ideal conditions

to start evaporation as discussed in Sec. 2.2.4. The collision rate in the center region

is sufficiently high to maintain continuous thermalization as evaporation is forced. For

atoms in the wings however, the density is suppressed the factor e−η, and typically

the collision rate will be less than the single beam axial trapping frequency. Thus it is

reasonable to assume that the atoms move independently in the wings until they reach

the center of the trap where they have a high probability of undergoing a collision. The

thermalization rate will therefore be roughly determined by the axial frequency of the

individual beams.

When we attempting forced evaporation with the same geometry and initial con-

ditions of this example, we observe a migration of the atoms into the wings, which is

shown in Fig. 2.5. When the power is not ramped down, nearly all the atoms are in the

cross region of the trap after 2 seconds, as expected from Eq. 2.35 for large η. As the

power is lowered, atoms ‘evaporate’ from the center of the trap into the wings where

there is insufficient density to thermalize with other atoms in the wings and evaporate

21


Table 2.2: Example initial conditions for a 1.06 µm cross beam trap (η = 8).

trap depth U0 490 µK

temperature T 61 µK

mean trap frequency (dimple) ω 1.9 2π kHz

axial trap frequency ωax 10 2π Hz

atom number N 1 106

peak density n0 2.4 1014 cm−3

peak collision rate γ 11 kHz

peak phase space density ρ 3.5 10−3

Figure 2.5: Fraction of atoms in the wings after linearly ramping the optical power from

an initial power of 12 W (6 W per beam) to a final power over 2 seconds.

from the trap entirely, at least on a reasonable timescale. The wings thus act as a

reservoir for atoms of higher energy than those in the dimple. These atoms primarily

thermalize only with the colder atoms in the dimple at a rate determined by the axial

frequency. This rate is too slow to maintain thermaliztion during the 2 second ramp

and, as a result, η increases and essentially all the atoms end up in the wings.

Although we abandoned starting from a crossed beam trap because of this compli-

cation, there are several ways one might circumvent atoms accumulating in the wings.

The issue is that we want to take advantage of the trapping volume of the individual

beams to load a large number of atoms, but do not want atoms to be trapped in the

wings during evaporation. To this end, one could turn on an additional potential gra-

dient, from either a magnetic or optical field, after loading to truncate the trap depth

22


so that atoms are not trapped in the wings. This would function like the ‘RF knife’

but in an optical trap. Both magnetic [53] and optical [52] methods have already been

employed in this way to achieve ‘run-away’ evaporation in optical traps. A simpler

method would be to put one of the crossed beams at an angle so that gravity truncates

the trap depth in the wings. However, this is disadvantageous because fewer atoms will

be loaded into the dimple during the initial free evaporation. Finally, one could con-

sider decreasing the waist in order to increase the axial frequency which scales as λ/w30.

However, this is also disadvantageous because the peak density would increase by the

same order of magnitude. As illustrated in Fig. 2.4, peak densities at w0 = 40 µm al-

ready exceed 1014 cm−3 where three-body losses become significant [75]. Increasing the

density further would simply result in a substantial loss of atoms by this mechanism.

Rather than any of these methods, which either complicates the experiment or

have other disadvantages, we used a cross beam trap with a modified geometry and

evaporation cycle to reach BEC. Briefly, a single tightly focused beam is loaded from

the MOT and only later in the evaporation is a secondary cross beam used to compress

the atoms. The experimental setup is the same as for loading directly into a cross

beam trap, and thus quite simple. However, this method allows us to (1) load a large

number of atom from the MOT given the relatively large trapping volume of the single

beam, (2) avoid accumulation of atoms in the wings which disrupt thermalization, (3)

circumvent the unfavorable scaling of optical evaporation by boosting the collision rate

through recompression, and (4) avoid high densities and thus three-body losses. The

experiment is described in further detail in Sec. 2.5.

2.3.5 Analysis of a recent cross-beam result

Some time after our BEC results, Lauber et al. were able to produce a BEC in a

cross-beam 1070-nm wavelength trap loaded directly from a MOT [78]. It is interesting

to consider their results in the context of the discussion thus far. Their crossed beams

are focused to an average waist of 43 µm with a total beam power of 12 W. They state

that atoms initially in the ‘wings’ are rapidly spilled at the beginning of evaporation

cycle. This could be due to several factors such as astigmatism in beam (see Sec. 2.4.8)

or gravity if either beam is at a modest angle. Regardless of the reason, if the atoms in

the wings are quickly spilled then they are not being efficiently loaded into the center

region. This is consistent with their unfavorable starting conditions of Ni = 3 × 105,

23


ni = 1.1×1013, and ρi = 2×10−5. These conditions are roughly what one would expect

by loading the center region only from a compressed MOT and complete loss of atoms

in the wings. As discussed in Sec. 2.2.4, it is very difficult to reach BEC from these

conditions due to the adverse scaling laws. Indeed, Lauber et al. employ logarithmic

photo diodes to actively stabilize their beam power over three order of magnitude and

require a long 12 second evaporation cycle. Their MOT is loaded from a Zeeman slower

which results in a low background pressure and allows for long evaporation times. In

all of our experiments the dipole trap in loaded from a vapor pressure MOT, and the

one-body loss rate precludes such a slow evaporation (see Sec. 2.4.5). Their end result

of a BEC with 15× 103 atoms after evaporation over Ti/Tcrit = 3700 is consistent with

the scaling laws.

In summary, while Ref. [78] has shown it is clearly possible to evaporate to BEC

from directly loading of ∼ 1 µm crossed dipole trap, this is not inconsistent with our

arguments that the ∼ 1 µm wavelength is geometrically unfavorable, in particular as

compared to the ∼ 10 µm wavelength. Although the problems with thermalization were

avoided by dumping the atoms in the wings, the bi-partite nature of the cross beam trap

was not utilized to produce the high initial phase space densities from which a larger

condensate can be easily produced in the much shorter time of a few seconds [11, 47].

2.3.6 Elliptical beams

We briefly discuss how focusing a dipole trap to an elliptical waist can be a useful

method for engineering better trap geometries with increased harmonic confinement

but still large capture volume. Let us consider the potential due to a single Gaussian

beam focused to confocal waists wx and wy,

U(r, z) = U0

1√1 + ( z

zRx)2

1√1 + ( z

zRy)2

exp

−2x2

w2x

(1 + ( z

zRx)2) +

−2y2

w2y

(1 + ( z

zRy)2) .

We define an ellipticity parameter ξ ≡ wx/wy, and let wx =√ξw0 and wy = w0/

√ξ

so that the trap depth is constant as we vary ξ. In the harmonic approximation, the

trapping frequencies are then

ωx = ξ

√4U0

mw20

ωy =1

ξ

√4U0

mw20

ωz =

√U0

mz2R

ξ2(1 +1

ξ4). (2.37)

24


Figure 2.6: Volume enclosed up to the equipotential u as a function of beam ellipticity

ξ. Calculated for w0 = 40 µm.

For ξ > 1, the axial trapping frequency thus scales approximately as wz ∝ ξ and the

geometric mean trapping frequency as ω = (ωxωyωz)1/3 ∝ ξ1/3. Thus at constant trap

depth, increasing the beam ellipticity can help to compensate the typically weak axial

confinement. The other important consideration is trap volume, which determines the

number of atoms loaded from the MOT. We find the trap volume, V (u, ξ), by numeric

integration as in Sec. 2.3.2 with the result shown in Fig. 2.6. The volume is scaled by

the circular case (ξ = 1). We can see the volume near the bottom of the trap (u < 0.5)

decreases significantly faster than the volume at the top (u = 0.9 − 0.95). Thus by

tightening the harmonic confinement at the bottom of the trap without significantly

compromising on trap volume, beam ellipticity can be used to tailor more favorable

single beam, as well as crossed beam, trap geometries.

2.3.7 Basics of Bose-Einstein condensates

Before moving on to the experiment, we briefly review the basic properties of a BEC.

Here we focus only on the details necessary for detection of a condensate in the exper-

iment. A more comprehensive review of Bose-Einstein condensation can be found in

25


the papers [12, 79, 80] and books [81, 82].

Below a critical temperature, an N particle Bose gas condenses to a phase where the

ground state is macroscopically occupied. This critical temperature, for a 3-dimensional

harmonic potential with the geometric mean trapping frequency ω, is given by

kbTc =~ωN1/3

[ζ(3)]1/3' 0.94~ωN1/3, (2.38)

where ζ(s) is the Riemann zeta function. Below this critical temperature, the occupa-

tion of the ground state grows as

N0

N= 1−

(T

Tc

)3

. (2.39)

These exact results are specific for the harmonic oscillator potential, and in Ref. [83]

the corresponding expressions for a variety of confining potentials are found.

When the s-wave scattering length as is much smaller than the average distance

between atoms, the gas is well described by the Gross-Pitaevskii (GP) equation

i~∂

∂tΦ(r, t) =

(− ~

2m∇2 + U(r) +

4π~asm|Φ(r, t)|2

)Φ(r, t), (2.40)

where Φ(r, t) is the condensate wave function in the mean field approximation. The time

dependence can be separated out as Φ(r, t) = φ(r)e−iµt using the chemical potential

µ. For large atomic clouds, the kinetic energy term can be neglected in the so-called

Thomas-Fermi (TF) approximation. The ground state can then be found by solving

the GP equation reduced to the time independent form

[U(r) + gφ2(r)− µ

]φ(r) = 0, (2.41)

where g = 4π~as/m. This is trivially solved to find the density

n(r) = φ2(r) =µ− U(r)

g, (2.42)

which is non-zero only in the region where µ − U(r) ≥ 0. The condensate is thus an

inverse paraboloid with a radius along each axis

Ri =

√2µ

mω2i

, (2.43)

26

2.4 Experimental setup

at which the density goes to zero. The chemical potential is then determined from the

normalization condition∫d3rφ2(r) = N0 ' N to be

µ =~ω2

(15Nasaho

)2/5

, (2.44)

where aho =√

~/mω is the characteristic length of the harmonic oscillator.

Finally, we would like to know how the condensate expands when released from

the confining potential so that it can be distinguished via absorption imaging. After

sudden release from the trap, the condensate retains its parabolic shape and the radii

vary in time as Ri(t) = Ri(0)λi(t), where the parameters λi must satisfy the ordinary

differential equation [84]

λi =ω2i (0)

λiλ1λ2λ3. (2.45)

Solving Eq. 2.45 for the initial conditions λi(0) = 1 and λi(0) = 0, one finds the

time varying density profile of the condensation for free expansion. An important

feature to note is that for anisotropic trapping frequencies, the condensate expands

anisotropically in contrast to a thermal cloud. Anisotropic expansion is easily detected

in the experiment and is a key signature of a Bose-Einstein condensate.


In this section we turn to the technical details of experiment apparatus. We detail the

cooling laser setup, imaging system, MOT, and dipole trap loading. These systems

form the starting point not only for the BEC experiment, but for all the experiments

reported in subsequent chapters.

2.4.1 Cooling lasers

Laser cooling of Rubidium is greatly simplified by the wide availability of inexpensive

100 mW single mode laser diodes at 780 nm1. Frequency stabilization of diode lasers

to less than a MHz for laser cooling is, by now, relatively straight forward. Using many

of the methods described in these references [85, 86, 87], we have developed our own

laser housing and the accompanying electronics. Our cooling laser setup of consists of

five such diode lasers: one master, three slaves, and one repumping laser.

1For example, the Sharp GH0781RA2C diode.

27


Figure 2.7: Schematic of master laser optics.

Figure 2.8: Schematic of slave laser optics.

28


Figure 2.9: AOMs are used to tune the cooling lasers anywhere from the |F = 2〉 →|F ′ = 3〉 transition (center) to the |F = 2〉 → |F ′ = 2〉 transition (right).

The master laser is an extended cavity diode laser (ECDL) in which a diffraction

grating placed a few centimeters from the diode forms the external cavity. Feedback

from the grating allows for mode-hop free tuning of laser wavelength as well as narrow-

ing of the linewidth to ∼ 200 kHz. Fig. 2.7 shows the optics layout for the master laser.

An electro-optic modulator (EOM) provides 25 MHz phase modulation for fm saturated

absorption spectroscopy [88] of the 87Rb S1/2 to P3/2 transition. The laser frequency

is offset by the acousto-optic modulator (AOM) #1 in a double pass configuration and

stabilized to the F = 2 to F ′ = 1 : 3 crossover transition. The double-passed AOM #2

provides 200 MHz of frequency tuning range. By coupling the laser through an optical

fiber and actively stabilizing the power on the output, residual beam steering effects

and variation of the AOM’s diffraction efficiency with RF frequency are eliminated.

Three free-running slave lasers, see Fig. 2.8, are optically injection locked [89] to the

master laser to increase the optical power available for the MOT. The slave lasers each

have an AOM for fast switching of the intensity, and a mechanical shutter for blocking

the residual leakage light. The combined frequency shifts from the AOMs #1− 3 allow

for tuning the light sent to the experiment anywhere from the |F = 2〉 → |F ′ = 3〉cycling transition to the |F = 2〉 → |F ′ = 2〉 transition, see Fig. 2.9.

Finally, the repumper laser is a free-running diode locked via the diode injection

29


Figure 2.10: Schematic of imaging system.

current only. This laser is frequency stabilized, again to a Rb vapor cell using fm

spectroscopy, near to the |F = 1〉 → |F ′ = 2〉 transition and brought into resonance

with another AOM. We measured the laser diodes (Sharp GH0781RA2C) to have a the

free-running linewidth of ≈ 1.2 MHz, easily less than the natural atomic linewidth of

≈ 6 MHz. Without an external cavity, this laser is impervious to external perturbations,

short of a power outage, and thus remains locked indefinitely for all practical purposes.

2.4.2 Imaging diagnostics

Our primary diagnostic tool for probing cold atom clouds, in the MOT or a dipole trap,

is imaging by either fluorescence or absorption imaging methods. Both the number and

temperature of the atoms can be simply extracted from an image using either method.

A detailed discussion of both techniques, as well other non-destructive methods, can

be found in [12].

2.4.2.1 Imaging system

For both fluorescence and absorption imaging, the imaging system shown in Fig. 2.10

is used. Standard two inch diameter achromatic lenses image the atom cloud onto the

CCD array. The iris at the intermediate focal plane acts to mask the CCD array so

that only a part of the array is exposed. This is necessary for using the ‘kinetics’ mode

of camera where the exposed section of the array is shifted to a masked section of the

array. This mode allows multiple images to be taken in rapid succession after which to

whole array is read out slowly, which is necessary for low read-out noise.

The imaging system has evolved through several modifications over the course of the

experiments in this thesis. However, in all experiments the size of the vacuum chamber

and optical access for other laser beams required that R ≈ 25 cm. Given D ≈ 5 cm,

the imaging resolution should be ∼ 7 µm if diffraction-limited. However at the time of

30


the BEC experiment, the resolution of the entire imaging system, as measured with a

resolution target, was only ∼ 18 µm.

2.4.2.2 Fluorescence imaging

For fluorescence imaging, the atoms are excited by a short (< 200 µs) pulse of light

tuned to the |F = 2〉 → |F ′ = 3〉 resonance. The atoms are probed well above the

saturation intensity so that each atom can be assumed to scatter at a rate Γ/2 ≈2π × 3.0 MHz. The total number of counts, n, detected on the CCD is then given by

n = NΓ

2τ

(D

4R

)2

η, (2.46)

where N is the number of atoms, τ is the pulse duration, D and R are shown in

Fig. 2.10, and η is the combined efficiency of the optics and camera. Although the

quantum efficiency of the most recent generation of cameras (including the PIXIS 1024

BR) is extraordinarily high (∼ 98%) at 780 nm, a small numerical aperture limits the

usefulness of fluorescence imaging. In our case(D4R

)2 ≈ 0.3%, and florescence imaging

is useful only for N & 105 atoms.

2.4.2.3 Absorption imaging

While fluorescence imaging detects only the photons scattered into the small solid angle

subtended by the imaging system, absorption imaging measures the photons scattered

into the full solid angle as a reduction of the probe beam intensity. For this reason,

absorption imaging is roughly 100 times more sensitive than fluorescence imaging. For

absorption imaging a weak probe (I < Isat) is used to ensure linear absorption as

the beam passes through the cloud. To accurately measure the transmission requires

three images: one of the beam with the shadow of the atoms Sa(x, y), one of the beam

without the atoms So(x, y), and a background image without the beam Sb(x, y). The

normalized and background-subtracted transmission of probe is then given by

T (x, y) =Sa(x, y)− Sb(x, y)

So(x, y)− Sb(x, y). (2.47)

The two images Sa and S0 are taken as close as together as possible so that inference

fringes intensity profile of probe beam do not shift between images. We typically

separate the images by 10 ms, just long enough for the atoms to fall away so they are

31


not present in the So image. The background image is taken earlier and reused over

many experiments.

The column density, n(x, y), of the atom cloud is related to the transmission by

n(x, y) =−ln(T (x, y))

σ0, (2.48)

where σ0 is the scattering cross section. The total number of atoms is simply determined

from integration, N =∫n(x, y)dxdy.

2.4.2.4 Temperature

The temperature of atoms can determined from the rms radius of the cloud after

ballistic expansion for a fixed time. After sudden release from the trap, the rms spread

of an initially Gaussian density profile expands as [12]

σ(t)2 = σ(0)2 +kbT

mt2. (2.49)

In almost all cases, the atoms are dropped sufficiently long that σ(t) σ(0) along

at least one dimension of the atom cloud. Thus, the initial size of the cloud can be

neglected by fitting the integrated density profile along that direction.

2.4.3 MOT

The physics behind the MOT can be found in the references in Sec. 2.2.2. Here we

outline only the details of our experimental setup. The three slave lasers are each

coupled to non-polarization maintaining (PM) fibers and delivered to the experiment

chamber. As long as the fibers are mechanically secured, the polarization is observed to

be more stable than with PM fibers and rarely requires adjustment. The polarization at

the output of each fiber is cleaned up with a polarizing beam splitter and the intensity

of each beam actively is stabilized to 25 mW. The intensity stabilization was added

to improve the repeatability of loading the dipole trap. A 15 mW beam from the

repumper is spatially overlapped with of the cooling lasers. The three MOT beams are

expanded to a waist of 9.5 mm clipped at a diameter of 23 mm, resulting in an intensity

of ∼ 16 mW/ cm2 = 10 Isat. The cooling lasers are typically detuned by ≈ −22 MHz

from the |F = 2〉 → |F ′ = 3〉 resonance to yield the largest MOT. Water-cooled anti-

Helmholtz coils generate a quadrupole magnetic field with a gradient of 10 G/ cm along

32


Figure 2.11: Loading sequence for dipole

traps.

Figure 2.12: Dipole trap loading vs re-

pump intensity. The peak corresponds to a

repump intensity of ≈ 5 µW/cm2

the strongest axis. Rubidium is dispensed from a SEAS getter inside the experiment

chamber. The MOT typically traps 50-200 million atoms, depending on the flux from

the Rb dispenser.

2.4.4 Dipole trap loading

A detailed analysis of the loading dynamics of FORTs is reported in Ref. [90]. Essen-

tially, loading depends on two primary considerations, the flux of atoms into the FORT

region and volume of the trapping region. In order to maximize the density of atoms

at the FORT, we use a loading sequence similar to that used in [61] for compressing

a MOT to peak densities > 1011 cm−3. Our loading sequence is shown in Fig. 2.11

and has been manually optimized starting from the loading sequence described in [11].

The critical components are a slow (> 30 ms) ramp of detuning and reduction of the

repump intensity during loading. For example, Fig. 2.12 shows the number of atoms

loaded in a dipole trap for different values of repump intensity. The maximum occurs at

a repump intensity of ≈ 5 µW/cm2. The loading sequence shown in Fig. 2.11 appears

to be nearly optimal independent of FORT waist and depth.

The volume of the trapping region is roughly that enclosed by the equipotential on

the order of the temperature of the compressed MOT atoms, ∼ 30 µK. Not surprisingly,

deeper and larger traps capture more atoms. Fig. 2.13 shows the number of atoms

loaded into a 1064 nm wavelength trap with waists [wx, wy] = [110, 30] µm as the

33


Figure 2.13: Atoms loaded vs FORT

power.

Figure 2.14: Atoms loaded into FORT vs

MOT number.

trap depth is varied. Because the compressed MOT is density limited by laser cooling

processes, increasing the number of atoms in the MOT only increases its size. For a

trap volume smaller than the MOT, one would expect the number of atoms loaded to

be independent of the number of atom in the MOT. For the 1064 nm FORT, although

the radial dimensions are much smaller than the MOT, the extent along the beam axis

is typically greater. Increasing the size of the MOT thus also increases the geometric

overlap and the number atoms loaded, as seen in Fig. 2.14. Needless to say, the spatial

overlap of the FORT and MOT is crucial for optimal loading.

2.4.5 Trap lifetime

The lifetime of atoms in the FORT is principally determined by the background collision

rate. The background pressure is dependent on the current run through the Rb source

in the experiment chamber. Fig. 2.15 shows the lifetime of atoms in the dipole trap

for several values of the Rb dispenser current. The lifetime is determined from a fit

to the data after 5 seconds where free evaporation has stagnated and the temperature

constant (lower right). While both the size of MOT and number of atoms loaded into

in the dipole trap are increased at higher source currents, this comes at the expense of

a shorter lifetime. The optimal Rb source current is thus a compromise which depends

on the duration of the experiment. Considering our longest experiment lasted 3 second

after loading the trap, the shortened lifetime is not a critical limitation.

It is worth noting from the temperature plot in Fig. 2.15 that, after free evaporation,

34


Figure 2.15: Trap lifetime for several Rb source currents.

there is negligible temperature increase over the next 10 seconds. Fluctuations in

beam power can result in technical heating [8, 91], and [92] has suggested this puts

stringent requirements on the laser power stability. Nevertheless, with a commercial

fiber laser1 and no active power stabilization, we do not observe technical heating.

Another potential source of heating is off-resonant scattering from the FORT itself.

However, as estimated in Sec. 2.2.1.4, this is expected to be negligible for this far-

detuning, at least on the time scale of several seconds.

2.4.6 Hyperfine changing collisions

In order to avoid the hyperfine changing collisions described in Sec. 2.2.5, it is necessary

to optically pump all the atoms into the |F = 1〉 ground state manifold at the end of

the trap loading sequence. In Fig. 2.16, the lifetime for the atoms prepared in |F = 1〉manifold is compared with the lifetime of atoms prepared in the |F = 2〉 manifold. The

atoms in |F = 2〉 have enhanced losses which are observed to be density dependent.

From the |F = 2〉 data, we infer a two-body loss rate of ∼ 2 × 10−12 cm3s−1. This

is comparable to the values on the order of ∼ 5 × 10−12 cm3s−1 reported by [93] for

87Rb atoms in a mixture of the magnetic sub-states in the |F = 2〉 manifold.

1IPG model YLR-25-1064-LP-SF

35


Figure 2.16: Trap lifetime for atoms pumped into either the |F = 2〉 or |F = 1〉 manifold.

2.4.7 Measuring trap frequencies

Estimation of the density and phase space density requires knowledge of the geometric

mean trap frequency in addition to the directly measured atom number and tempera-

ture. Although the trap frequencies can be inferred from the beam waist and power, it

is a useful diagnostic check to measure the trap frequencies directly. This is done via

sinusoidal modulation of the beam power to drive parametric excitation [94]. It is well

known that for a harmonic oscillator if the trap frequency is parametrically modulated

at twice the natural frequency, ωmod = 2ωo, the energy of the system grows exponen-

tially. Additional parametric resonances occur at the subharmonics ωmod = 2ωo/n,

where n is a positive integer [95]. In an optical trap, the parametric resonance is

marked by the heating and loss of atoms as seen in Fig. 2.17. This data was taken

by modulation of the trap depth by five percent for 500 ms in a FORT with 13 W

of optical power and waists [wx, wy] = [110, 30] µm, as measured on a CCD camera.

Parametric resonances are thus expected to occur at [2ωx, 2ωy]/(2π) = [1.0, 4.0] kHz,

in good agreement with the observed heating peaks in Fig. 2.17. The atoms losses are

asymmetric and skewed to lower frequencies due to anharmonicity of the potential [96].

The peak in temperature is observed when atoms are excited primarily at the bottom of

the potential but not completely out of the trap. We thus use the peak in temperature

as the more accurate indicator of the harmonic resonance frequency.

36


Figure 2.17: Measuring trap frequencies by parametric excitation.

2.4.8 Thermal lensing

Thermal lensing is an issue with high power laser beams whereby heating of the bulk

material in optical elements results in spatial variation of their refractive properties.

Other groups, for example [78], using high power 1 µm dipole traps report limiting

the power per beam to ∼ 10 W because of thermal effects. We found the acousto-

optic modulator (AOM) to be the main source of the problem. Using an AOM1 with a

1.2 mm active aperture, the spatial mode became severely distorted in a doughnut shape

above 8-10 W optical power, rendering it unusable for optical trapping. This problem

was mitigated by using a 3 mm active aperture AOM2 which produced no significant

laser mode distortion until > 15 W. Even before distortion occurs, asymmetric thermal

lensing in the AOM results in an astigmatism which had to be corrected with cylindrical

lenses after the AOM. It is particularly important to correct astigmatism because it

can severally reduce the axial confinement of a single beam FORT.

Thermal lensing from other cylindrically symmetric optical elements, such as lenses,

only shifts the focal position. Using fused silica optics rather than BK7 or another

standard lens glass helped to reduce the thermal lensing. These lensing effects are

manageable in the experiment, but a considerable nuisance.

1Crystal Technologies 3110-1972Isomet M1099-T80L

37


Figure 2.18: Measured radial trap frequencies of the primary beam.

2.5 Bose-Einstein condensation experiment

In this section we describe the experiment in which a BEC was successfully produced.

Having tried various crossed beam geometries unsuccessfully, we opted for a slightly

modified beam configuration and evaporation cycle. By loading first into a tightly

focused single beam trap and using a secondary crossed beam only later in the evap-

oration cycle, the problems discussed in Sec. 2.3 are avoided. The single beam traps

a large number of atoms and the tight focus results in a sufficiently high densities for

rapid initial evaporation. The single beam also forms part of the cross beam geometry

where the tight focus ensures the atoms are well localized to the intersection region.

Compression of the atoms into the crossed beam region compensates for the decreas-

ing collision rate, allowing for further forced evaporation. In this way we are able to

produce a BEC with 3.5× 104 atoms after 3 seconds of total evaporation time every 10

seconds.

2.5.1 Primary beam geometry

The configuration of the beams and labels for the coordinate axes were introduced at

the start of the chapter in Fig. 2.1. Most of the available optical power is diverted

to a horizontal primary beam which has maximum of 15 W. This single-beam trap is

focused to a waist of ≈ 25 µm. While a larger waist would allow more atoms to be

captured from the MOT, a smaller waist has the advantage of higher densities and

38


collision rates. From previous attempts using a few different waists, 25 µm appeared

a good compromise. After correcting for astigmatism in the beam, the resulting focus

is slightly elliptical. We measure the radial trap frequencies at several beam powers by

parametric excitation, with the results shown in Fig. 2.18. The black lines are a fit to

the measure values with the beam waists as free parameters. The waists implied by

the fit are [wx, wz] = [22, 28] µm, close to the values measured by a CCD camera beam

profiler outside the chamber.

The axial trap frequency can also be parametrically excited, but typically requires

stronger modulation of the trap depth (30 − 50%) and is broadened as result. Still,

the measured values are still consistent within error to the calculated value from the

implied beam waists.

It is difficult to make accurate frequency measurements at a low beam powers due

to the limited atom numbers. Thus, we assume that the ∝√P scaling holds and

extrapolate from the fit to determine the trap frequencies at lower powers.

2.5.2 Primary beam free evaporation

By holding the dipole trap at a constant 15 W power after loading, we observe free

evaporation and are able estimate our initial conditions for forced evaporation. Fig. 2.19

shows the evolution in time of several key quantities after loading. The free evaporation

stagnates after 2 seconds as η converges towards ≈ 10. The phase space density peaks

at 2 × 10−5 with 1.5 × 106 atoms remaining in the trap. As discussed in Sec. 2.2.4,

these are not favorable starting conditions and this can be explicitly demonstrated by

attempting forced evaporation in the single beam trap only.

2.5.3 Primary beam forced evaporation

Starting from 15 W, the beam power is lowered on a exponential trajectory with a time

constant of 300 ms. Fig. 2.20 displays the results for the quantities of interest. The

optimal time constant is a compromise between more efficient evaporation at a high η

and the non-evaporative atom losses. Shorter time constants result in excess atom loss

due to a decreasing eta, and longer time constants lead to excess loss due to background

collisions. In this case, the time constant was chosen to be as fast as possible such the

trap depth to temperature ratio η remains roughly constant. Time constants of 300-400

ms result in the highest final phase space density for our background limited lifetime.

39


Figure 2.19: Free evaporation at constant trap depth (∼ 2.2 mK).

40


We see in Fig. 2.20 that η ≈ 7 throughout the evaporation until the anomalous increase

near the end. This is due to the effect of gravity truncating the potential depth and

eventually tipping the atoms out of the trap near the end of the evaporation.

For each data point the phase space density ρ and collision rate γ are estimated from

the measured temperature, measured atom number, and the mean trap frequency we

previously determined as a function of beam power. It is interesting to compare these

results directly to the scalings laws. The black lines shown in Fig. 2.20 are calculated

from scaling laws for phase space density (Eq. 2.20) and collision rate (Eq. 2.21) with

the conditions ρi = 2 × 10−5, γi = 1000 Hz, and η = 6. The agreement with the

scaling laws is quite good with the only deviation at low trap depth, again due to

the unaccounted for effects of gravity. As expected, decreasing the beam power by a

factor of ∼ 700 we are able to gain in phase space density by only slightly more than

three orders of magnitude. With collision rates approaching ∼ 1 Hz, we would have to

considerably slow down the rate of forcing evaporation near the end in order to push

for further gains in phase space density. However, this is simply not possible due to our

modest vacuum pressure which limits the trap lifetime. Instead, we use a secondary

beam to compress the atoms and boost the collision rate.

2.5.4 Secondary beam geometry

The secondary beam has a maximum optical power of 1 W and is focused to an elliptical

spot with waists [wx, wy] = [20, 80] µm. The beam waists are inferred from a fit to trap

frequencies measured at several powers, just as for the primary beam. The secondary

beam axis is vertical and the 80 µm waist is aligned to the primary beam axis. Due

to thermal lensing, the focal position of the primary beam shifts by as much as one

millimeter during the course of evaporation. The secondary beam therefore has to be

aligned to cross the primary beam at this displaced position. Since the atom cloud has

a spatial radius along the primary beam axis of σy = zR/√

2η ≈ 0.6 mm, the exact

alignment is not crucial but the lensing must be taken into account. More crucial is

the transverse alignment (x-axis) of the crossed beams.

2.5.5 Cross-beam compression

Fig. 2.21 shows the trajectory of the trap depths and trap frequencies in the combined

evaporation cycle. The secondary beam power is increased linearly over 500 ms such

41


Figure 2.20: Forced evaporation in the primary trap only.

42


Figure 2.21: Composite trap evaporation cycle.

that when the beam reaches its maximum power of 1 W the trap depth of the primary

and secondary beam are roughly equal. From that point, both beams are lowered

exponentially with a time constant of 300 ms. The exponential ramp of each beam ends

at a non-zero power which is manually optimized for producing the largest condensate.

The final beam powers are approximately 3 mW and 20 mW for the primary and

secondary beams respectively. In the final stages of evaporation, the atoms are barely

supported against gravity. In fact, gravity plays a useful role in truncating the trap

depth and forcing the last bit of evaporation without significantly impacting the trap

frequencies.

The trap frequencies shown in Fig. 2.21 are calculated for the harmonic approxi-

mation of the combined potential due to the two beams. While this not an accurate

approximation during the compression phase, the figure illustrates how the mean trap-

ping frequency is boosted in particular by shoring up the weak trapping along the

primary beam axis, ωy. From 0.5 to 1.5 seconds in the evaporation cycle, the atomic

density distribution is split between the primary beam and the dimple region. During

this period it is difficult to extract useful data on the number of atoms in the cross and

the temperature from time of flight images.

2.5.6 Observation of a condensate

Following the evaporation shown in Fig. 2.21, we reach the critical temperature at

240 nK after 2.4 seconds with approximately 1.9× 105 atoms remaining. At 3 seconds,

43


Figure 2.22: Absorption images and density line profiles at and below the critical tem-

perature. From left to right, at 2.4, 2.75, and 3 seconds into the evaporation cycle.

we have a mostly pure condensate with 5.5×104 atoms remaining. Fig. 2.22 shows three

absorption images after 20 ms time of flight from 2.4 to 3 seconds in the evaporation

cycle.

The absorption images are fit by the column density function

n(z, y) = nth(0, 0)e−(z2+y2)

σ2th + nc(0, 0)max

(1− z2

σ2c,z

− y2

σ2c,y

, 0

)3/2

, (2.50)

where the first term is a Gaussian for the thermal component and the second term is

the Thomas-Fermi profile for the condensed component. The fit is relatively good, as

seen from the line density profiles shown below in Fig. 2.22, except very near criticality

where a Bose-enhanced distribution would provide a better fit to the central region [12].

The softening at the edge of the condensed component is due to the 18 µm resolution

of the imaging system.

The condensate fraction, r, is extracted from the fit and the temperature is deter-

mined from the rms radius of the thermal component in the usual way. Fig. 2.23 shows

the measured condensate fraction versus temperature as compared to the Eq. 2.39

44


Figure 2.23: Condensate fraction below the critical temperature.

(black line). In this plot, the measured temperatures are scaled by the critical temper-

ature calculated from Eq. 2.38 using the measured atom number and the ω implied by

the FORT beam powers. Fig. 2.23 suggests that the critical temperature is being sys-

tematically overestimated by ∼ 30%. Principle sources of the systematic discrepancy

are likely the m-state distribution and uncertainty in ω near the end of the evaporation.

There is some uncertainty in ω for the cross beam trap because it is inferred from

the single beam frequencies measured at higher powers and assumes optimal alignment

of the crossed beams. Additionally, thermal effects are known to affect the shape of

beams as well as the calibration for the AOM controlling the beam power. These

thermal effects are observed to depend on the exact power ramp used and even vary

over several iterations of the experiment, making them difficult to account for.

The other consideration is the magnetic sub-state distribution of the atoms. So far

all quantities have been calculated assuming a one component Bose gas. However, the

linearly polarized FORT traps all magnetic sub-states equally. The loading sequence

ensures the atoms are pumped into the |F = 1〉 manifold but does nothing to pump

them to a particular Zeeman sub-level. The distribution between sub-levels can be

measured directly in a Stern-Gerlach type experiment where a magnetic field gradient

is switched on (using the MOT coils) to separate the mF components of the gas after

the atoms are released from the trap. In order to resolve the component atoms clouds,

the atoms must be cold enough that the ballistic expansion of the clouds during time

of flight is less than the separation resulting from the magnetic field gradient. Near the

45


critical point in our experiment, this measurement showed the atoms divided between

the mF = −1, 0, 1 states in approximately a 5:2:1 ratio. Because this measurement can

only be done near the end of the evaporation, we cannot be sure whether the skewed

distribution is due to unintended optical pumping during the loading phase or a stray

magnetic gradient tipping atoms out of the shallow trap near the end in an mF selective

manner. Nonetheless, the fact that only ≈ 60% of atoms are in the predominate m-state

explains most of the discrepancy in the observed critical temperature.

2.5.7 Comments of observing a bi-modal distribution

The secondary beam in this experiment has been deliberately made highly elliptical to

ensure an anisotropic trap so that the bi-modal distribution could be clearly seen. In a

previous beam configuration, the trap frequencies were isotropic and we were unable to

distinguish the Thomas-Fermi profile from the thermal component. Absorption imaging

requires the optical depth be on the order of one. For a much lower optical depth the

image signal to noise is poor and for much higher optical depth the probe beam is

completely extinguished, resulting in an unreliable measurement of transmission. This

places a constraint on the allowable ballistic expansion time.

For an isotropically expanding condensate, it may happen for reasonable parameters

that the condensate radius is similar the rms radius of the thermal distribution during

this window of time for imaging. In this case the joint density profile is practically

indistinguishable from a Gaussian.

2.6 Summary

In summary, we have demonstrated an all-optical method for BEC production using a

1 µm wavelength FORT. This has produced a BEC equivalent in size and in the same

short evaporation time as earlier 10 µm-wavelength experiments [11]. The compar-

atively simple single chamber setup allows for straight-forward integration into more

complex apparatuses utilizing BEC.

It should be acknowledged that this experiment, as implemented, was not very

robust. In particular, the primary beam power had to be lowered from an initial 15

W to 3 mW; a factor of 5000. Given the beam power was set simply by an voltage-

controlled AOM driver and a calibrated look-up table, the stability of the beam power

46

2.6 Summary

near the end of evaporation was questionable. One way to improve stability over several

orders of magnitude would be to use logarithmic photodetectors and actively stabilize

the beam power, as done by [78] and others. Alternatively, one could avoid going to

such low powers by using another method to force the final stages of evaporation. The

least complicated method would be to modify the potential using magnetic gradients in

order to controllably force the final stage of evaporation, as demonstrated in Ref. [53].

The important considerations for success in this experiment were the scaling laws

for evaporation in optical traps and the effects of the trapping geometry on the thermal

distribution. In our comparison of the 1 µm and 10 µm wavelength optical traps, we

have explained why for geometric reasons the high initial phase space densities observed

in 10 µm cross beam traps are difficult to achieve in 1 µm traps. Finally, although

we discussed the cross-beam geometry specifically, the same analysis could easily be

applied for optimizing evaporative cooling in other geometries which combine disparate

trapping volumes.

47


48

Chapter 3

Collective Cavity Quantum

Electrodynamics with Multiple

Atom Levels

This chapter covers the second generation experimental apparatus, pictured in Fig. 3.1, and is

largely based on ’Collective cavity quantum electrodynamics with multiple atomic levels’ K. J.

Arnold, M. P. Baden, and M. D. Barrett Physical Review A 84, 033843 (2011).

3.1 Introduction

The experimental apparatus, which is pictured in Fig. 3.1, was designed to be a versatile

tool for studying cavity quantum electrodynamics (QED) in the strong coupling regime

over a wide range of atoms numbers. Using the method of the previous chapter to

produce a BEC, the BEC could in principle be transfered into an optical lattice and

transported to the cavity [29]. Alternatively, only single atom [97] or a few atoms

could be transported allowing for investigation of quantum information protocols such

as entanglement of a single atom and photon [98], or entanglement of two atoms via

the cavity interaction [99].

In our first foray into the field of cavity QED, we sought to characterize the system

with many atoms coupled to the cavity. When N atoms are coupled to a single mode of

a cavity, the atom-cavity interaction is collectively enhanced [27]. In our system, this

collectively enhanced coupling was comparable to the hyperfine splitting of the atom

49

3. COLLECTIVE CAVITY QUANTUM ELECTRODYNAMICS WITHMULTIPLE ATOM LEVELS

Figure 3.1: Schematic representation of the experiment and photograph of the apparatus.

such that the atom’s multi-level structure could not be neglected in the interaction with

cavity [100]. In this chapter, we experimentally investigate the collective interaction of

N multi-level 87Rb atoms with the mode of a high finesse cavity by studying the cavity

transmission spectra. The transmission spectra have a complex structure which can

be understood through the underlying atom-field interaction. We develop a reduced

model for the system, an extension from the Tavis-Cummings model [27], which is in

good agreement with our experimental observations.

3.2 Cavity quantum electrodynamics

In this section we review the basics of cavity quantum electrodynamics and then gener-

alize to our more complex physical system. We begin with a two-level atom coupled to a

single mode of the electromagnetic field inside a cavity, known as the Jaynes-Cummings

model [13, 101]. This basic model already captures the essence of the atom-cavity in-

teraction. We then consider the implications of N two-level atoms coupled to the single

cavity mode, which is described by the Tavis-Cummings model [27]. Finally, we extend

the model to fully describe our physical system: N multi-level alkali atoms coupled to

a cavity with two orthogonal polarization modes.

3.2.1 Jaynes-Cummings model

Let us first consider the simplest cavity QED system: a single two-level atom at fixed

position coupled to a single mode of an ideal cavity. The Hamiltonian of this system

50


consists of three terms,

H = Hc +Ha +Hint, (3.1)

where Hc describes the cavity mode subsystem, Ha describes the atom subsystem, and

Hint the interaction between the two. The quantized electromagnetic mode of the cavity

with resonant frequency ωc is simply a quantum harmonic oscillator for which

Hc = ~ωca†a. (3.2)

Here a† (a) is the creation (annihilation) operator for a single excitation and the zero

point energy term, ~ωc/2, is omitted. For a two-level atom with a ground state |g〉 and

excited state |e〉, the atom subsystem is described by

Ha = Eg|g〉〈g|+ Ee|e〉〈e| = ~ω0|e〉〈e| = ~ω0σ†σ, (3.3)

where the ground state energy, Eg, has been set to zero, ω0 = (Ee − Eg)/~ is the

frequency of the atomic transition, and the atomic state transition operators are defined

as σ† = |e〉〈g| and σ = |g〉〈e|.Assuming the cavity field is nearly uniform over the extent of the atom, the interac-

tion between atom and the cavity field can be approximated by the dipole interaction.

This interaction is given by

Hint = µ ·E , (3.4)

where µ = −e · r is the dipole operator and E the electric field operator. The dipole

operator can be written in terms of the atomic state transition operators as

µ =∑

i,j∈g,e

|i〉〈i|µ|j〉〈j| = 〈g|µ|e〉(|g〉〈e|+ |e〉〈g|

)= µ0(σ† + σ), (3.5)

where µ0 is the dipole matrix element of the transition. It is useful to write the electric

field amplitude as E(r) = E0f(r), where f(r) is the spatial mode function defined by

the cavity geometry. Thus the cavity mode volume can be defined as Vm ≡∫f2(r)dr,

and the electric field due a single photon in the cavity is then E0 =√

~ωc2ε0Vm

. Assuming

that the atom is located at the position of the maximum field strength, the electric field

operator becomes E = Eo(a + a†). Introducing the conventional atom-cavity coupling

parameter g0 ≡ µ0E0/~, we write the complete Hamiltonian

H = Hc +Ha +Hint

= ~ωca†a+ ~ω0σ†σ + ~g0(σ† + σ)(a+ a†). (3.6)

51


Figure 3.2: Jaynes-Cummings ‘ladder’ of

dressed states.

Figure 3.3: Vacuum-Rabi splitting.

Although it was recently shown this Hamiltonian is in fact solvable [102], analytic treat-

ment is greatly simplified by the rotating wave approximation (RWA). This approxi-

mation is most easily understood by transforming to the interaction picture (H(t) =

H0 +H1(t)) choosing H0 = Hc +Ha. In this rotating frame, the time dependent piece

of the Hamiltonian is

H1(t) = ~g0

(σ†ae−i(ω0−ωc)t + σ†a†ei(ω0+ωc)t + σae−i(ω0+ωc)t + σa†e−i(ω0−ωc)t

). (3.7)

In the RWA, the ‘counter-rotating’ terms oscillating at the frequency ω0+ωc are dropped

under the assumption they average to zero on the time scale of relevant dynamics. This

is assumption is valid when the cavity frequency is near to the atomic resonance such

that |ω0 − ωc| (ω0 + ωc), which is well satisfied in all of our experiments.

Neglecting the counter-rotating terms and returning to the Schrodinger picture, we

arrive at the Jaynes-Cummings model (JCM):

H = ~ωca†a+ ~ω0σ†σ + ~g0(σ†a+ σa†). (3.8)

3.2.1.1 Dressed states and rabi-splitting

In the resonant case ω = ωc = ω0, it is easily verified that the eigenstates and energy

eigenvalues of this Hamiltonian are

|±〉n =1√2

(|n, g〉 ± |n− 1, e〉

)(3.9)

E±,n = n~ω ±√n~g0 (3.10)

52


for n excitations in the system. These eigenstates form a ‘ladder’ as shown in Fig. 3.2.

The anharmonicity of the level splittings for n > 1 excitations is a distinctly quantum

feature of this system which differs from the semiclassical prediction of E±,n = n~ω ±

n~g0. A consequence of this non-linearity is the ‘photon blockade’ effect, whereby

after one resonant excitation of the system, additional excitation is suppressed [14], as

illustrated in Fig. 3.2.

In the case of weak probing, we are interested only in the first excitation manifold.

For an arbitrary detuning of the cavity from the atomic resonance, ∆ = ωc − ω0, the

eigenstates and energies for the first excitation manifold are

|+〉 =1√2

(sin(θ)|1, g〉+ cos(θ)|0, e〉

)(3.11)

|−〉 =1√2

(cos(θ)|1, g〉 − sin(θ)|0, e〉

)(3.12)

E± = ~ω0 +~∆

2± ~

2

√∆2 + 4g2

0, (3.13)

where θ is given by

tan(2θ) = −2g0

∆. (3.14)

The eigenenergies, which are plotted in Fig. 3.3, demonstrate a characteristic avoided

crossing of the normal modes known as the vacuum-Rabi splitting [103]. On resonance

the frequency separation of the eigenstates is minimum and given by 2g0, the single-

photon Rabi frequency. A weakly driven cavity will transmit light when the probe

frequency matches the an energy of an eigenstate ~ω = E±. The amount of transmission

will be proportional to how ‘cavity-like’ the eigenstate is, or more precisely, 〈±|a†a|±〉

for the eigenstates |±〉, indicated by the color scale in Fig. 3.3. In the experiment, we

probe the eigenspectrum of our system in essentially this way.

3.2.1.2 JCM for N atoms

Let us now consider many two-level atoms coupled to a single mode of an ideal cavity.

For N atoms, the JCM of Eq. 3.8 can be written as

H = ~ωca†a+N∑i=1

~ω0σ†iσi +

N∑i=1

~gi(σ†i a+ σia†), (3.15)

53


Figure 3.4: Eigenenergies for N-atoms.

which is also known as the Tavis-Cummings model [27]. Here we assume that the

atoms are at a fixed position but with non-identical couplings, |gi| ≤ |g0|, depend-

ing on their spatial position in the cavity. Tavis and Cummings realized that this

Hamiltonian is analytically solvable in the basis of Dicke states [41], in which the ex-

citations are collectively shared among the atoms. Given the N -atom ground state

|gN 〉 ≡ |g1g2 . . . gi . . . gN 〉, we define the Dicke state for a single excitation as

|eN 〉 ≡1√N

N∑i=1

|g1g2 . . . ei . . . gN 〉. (3.16)

On resonance, ω = ωc = ω0, the following are readily observed to be eigenstates of

Eq. 3.15,

|±N 〉 =1√2

(|1, gN 〉 ± |0, eN 〉

),with (3.17)

E± = ~ω ±√N~g, (3.18)

where g =√∑

i |gi|2/N is the averaged atom-cavity coupling. In the first excitation

manifold the system behaves like the single-atom JCM except with an effective coupling,

geff =√Ng, which is collectively enhanced by the factor

√N . For higher excitation,

the behavior differs because the atomic-subsystem can store N -excitations rather than

just a single one. Additional excitations lead to further splitting of degeneracy in the

eigenspectum seen in Fig. 3.4. The Tavis-Cummings model is analytically solvable for

54


higher excitations, though it is algebraically more complex [41, 104] than the JCM case.

Note that for N 1 but n N , the normal mode splitting converges to n2geff , that

of classical coupled oscillators. In this limit the excitation spectrum is harmonic as

illustrated in Fig. 3.4.

3.2.2 Real systems: dissipation

So far we have assumed an ideal cavity and neglected the effects of dissipation. Real

cavity systems have dissipation through both spontaneous emission of the atoms and

decay of the cavity field resulting from both transmission and scattering losses at the

mirrors. By convention, these are characterized by the dipole decay rate γ, which is

half the natural atomic linewidth γ = Γ0/2, and the cavity field decay rate κ, which is

half the cavity linewidth κ = ∆νc/2. For the normal mode splitting of the JCM to be

observed in a real system, both rates of dissipation must be less than the cavity-atom

coupling rate g0. This is known as the ‘strong-coupling’ regime, where g0 > (γ, κ).

3.2.2.1 Master equation

The JCM can be extended to include dissipation using a master equation approach for

the evolution of the density operator ρ. For a single atom in a cavity driven by probe

field with amplitude E and frequency ω, the master equation is given by

ρ = − i~

[H, ρ] + γ(2σρσ† − σ†σρ− ρσ†σ) + κ(2aρa† − a†aρ− ρa†a), (3.19)

with the Hamiltonian

H = ~∆ca†a+ ~∆aσ

†σ + ~g0(σ†a+ σa†) + E(a+ a†). (3.20)

Here ∆c = ω − ωc is the cavity detuning and ∆a = ω − ω0 is the atomic detuning.

The master equation is valid in all regimes of our experiments and can be used to

numerically simulate both the dynamic and steady-state behavior of the system. For

example, Fig. 3.5 shows the steady-state transmission of a weakly driven cavity with the

parameters (g0, κ, γ) = 2π×(9.2, 3.7, 3) MHz, as in our experiment. The transmission is

normalized to the intra-cavity photon number for an empty cavity driven on resonance.

Our cavity is in the strong-coupling limit and so the vacuum-Rabi splitting is resolved

even for a single atom.

55


Figure 3.5: Cavity transmission spectrum for weak probing including dissipation with

the parameters (g0, κ, γ) = 2π × (9.2, 3.7, 3) MHz. A single atom maximally coupled (left)

and 10 atoms maximally coupled (right) which act as a single collective spin with effective

coupling g =√

10g0. The gray dashed lines are the eigenenergies of the Jaynes-Cummings

model.

While the full quantum description using the maser equation is possible for small

systems, it quickly becomes computationally impractical when including many exci-

tations, atoms, internal atomic states, or external motional states. For our system

in which N 1, we are primarily going to study the cavity transmission with weak

probing. Since we are in the strong-coupling regime, the eigenspectum resulting from

a Jaynes-Cummings-like model will be sufficient to describe the system. As seen in

Fig. 3.5, the vacuum-Rabi splitting is already resolved for a single atom with maximum

transmission at the eigenenergies (dashed lines). With additional collective enhance-

ment,√Ng (κ, γ) and the effects of dissipation on the structure of the transmission

spectra are negligible.

3.2.2.2 Semi-classical approach

It is useful to compare semi-classical results to quantum mechanical models discussed

so far. As introduced in Sec. 2.2.1 on the FORT, in the semi-classical formulation the

atoms are treated as radiating dipoles with a damping rate (Eq. 2.4) determined by

the dipole matrix element from quantum mechanics. Using the dynamic polarizability

56


(Eq. 2.5) for a classical driven dipole, the mean intra-cavity field for a pumped atom-

cavity system can found by linear dispersion theory [105, 106]. For N atoms in a cavity,

the input electric field, E, is related intra-cavity field, α = 〈a〉, by the equation [107]

E = αq

1 +NC

1 + δ2a + 4g2

γ2 α2

+ i

δc +NCδa

1 + δ2a + 4g2

γ2 α2

, (3.21)

where q is the transmission coefficient for the mirrors1, δc = ∆c/κ is the scaled cavity

detuning, δa = ∆a/γ is the scaled atomic detuning, and C ≡ g2/(κγ) is the average

cooperativity per atom. Above the intra-cavity photon number α2 =⟨a†a⟩≈ γ2/(4g2),

also known as the critical photon number n0, this equation is non-linear due to the sat-

uration of the atomic transition. The non-linearity of Eq. 3.21 leads to the phenomenon

of optical bistability [108]. Note that in the strong-coupling regime, by definition C > 1

and the critical photon number is less than one. Thus in the regime of many atoms,

N 1, and strong single-atom coupling, n0 < 1, the effects of optical bistability can be

observed by pumping the cavity mode with less than one excitation on average [109].

Well below saturation (α2 n0), the cavity field depends linearly on the input field

and we find the cavity transmission from Eq. 3.21 to be

Pout

Pin=α2q2

E2=

[(1 +

NC

1 + δ2a

)2

+

(δc +

NCδa1 + δ2

a

)2]−1

. (3.22)

This equation correctly describes the transmission for either a single atom or many

atoms as long as the weak probing condition is satisfied. The steady state results of

the master equation shown in Fig. 3.5 are identical to the prediction of Eq. 3.22. The

semiclassical model fails above the linear regime if the excitation spectrum is anhar-

monic, as predicted and observed [14] for a single atom in the strong coupling regime.

In Ref. [110], the authors performed spectroscopy on a small number of atoms (N . 3)

in the strong coupling regime but had difficulty experimentally resolving anharmonic-

ity from quantum effects because the system response asymptotically approaches the

semi-classical prediction for N > 1. In the limit N 1, as in our experiment, the

semi-classical result is applicable in both probing regimes; Eq. 3.22 in the linear regime

and Eq. 3.21 in the non-linear regime.

1Here we assume identical mirrors with coefficients for transmission q and reflection r for which

|q|2 + |r|2 = T + R = 1.

57


Figure 3.6: Schematic of experiment configuration

3.2.3 Cavity QED for N multi-level atoms

In this section we develop a JCM-like reduced Hamilton for the physical system realized

by our experiment, summarized in Fig. 3.6. We consider N 87Rb atoms trapped in an

intra-cavity 808 nm FORT where the cavity is weakly probed with linearly polarized

light. The cavity supports two degenerate and orthogonal polarization modes. A

magnetic field transverse to the cavity defines the quantization axis such these modes

correspond to π and ⊥= (σ+ + σ−)/√

2 transitions of the atom. The atoms are in a

mixture of mF states in the |F = 2〉 ground state manifold and probed on the D2-

transition at 780 nm. The collectively enhanced coupling is on the order of the excited

state hyperfine splitting and so the multi-level structure of the atom must be taken

into account. Our model Hamiltonian describing this system is

H = ~ωc(a†a+ b†b) + ~

N∑i=1

3∑F ′=1

ωF′ |F ′〉〈F ′|i +

~N∑i=1

3∑F ′=1

(gia†D0

i (2, F′) + gib

†D⊥i (2, F ′) + H.c.). (3.23)

Here, a and b are the annihilation operators for the cavity modes corresponding to π

and ⊥ transitions of the atom, ωc is the resonance frequency of the cavity, |F ′〉〈F ′|i is

the operator projecting the i-th atom onto the excited state |F ′〉, ωF′ is the frequency

of the transition from the |F = 2〉 ground state to the |F ′〉 excited state, gi is the atom-

cavity coupling constant for the i-th atom, and ‘H.c.’ denotes the Hermitian conjugate.

58


D0i (2, F

′) and D⊥i (2, F ′) are the atomic dipole transition operators for the i-th atom

coupling the |F ′〉 excited state to the |F = 2〉 ground state by π and ⊥ transitions

respectively.

Following the notation of [100], the dipole transition operators for an atom inter-

acting with different polarizations of the light field are defined as

Dq(F, F ′) =∑mF

|F,mF 〉⟨F,mF |µq|F ′,mF + q

⟩〈F ′,mF + q|, (3.24)

where q = −1, 0, 1 and µq is the dipole operator for σ−, π, σ+-polarization, nor-

malized such that 〈µ〉 = 1 for the cycling transition from the |F = 2,mF = 2〉 to the

|F ′ = 3,mF ′ = 3〉 state. For ⊥-polarized light we identify D⊥i = (D+1i +D−1

i )/√

2.

Here we consider the specific case of weakly driving only the π-polarized cavity mode

and generalize to other couplings later when discussing the experimental results. We

now detail how Eq. 3.23 is reduced to a simple effective Hamiltonian by (1) averaging

over the atomic ensemble to find an effective collective coupling for the driven mode,

and (2) dropping the coupling terms for undriven mode because they are not collectively

enhanced.

3.2.3.1 Effective coupling for the driven mode

First we consider the interaction terms of Hamiltonian Eq. 3.23 corresponding to the

driven mode,

HI,π = ~N∑i=1

3∑F ′=1

(gia†D0

i (2, F′) + H.c.

). (3.25)

We first simplify the dipole transition operator,

D0i (2, F

′) =∑mF

|2,mF 〉i⟨2,mF |µ0|F ′,mF

⟩i〈F ′,mF |i , (3.26)

by considering its action on a product of state of N atoms, each in a particular mF

state. The sum over mF states in the operator D0i (2, F

′) will pick out only one non-zero

term for every atom. The operator then becomes an effective lowering operator taking

the i-th atom from the |F ′〉 to the |F = 2〉 state, i.e.

D0i (F, F

′) =⟨F,mF |µ0|F ′,mF

⟩i|F = 2〉〈F ′|i, (3.27)

with a coupling strength scaled by the appropriate Clebsch-Gordon coefficient,

〈F,mF |µ0|F ′,mF 〉i.

59


Generalizing from the previous discussion of the Tavis-Cummings model for two-

level atoms, we introduce the uncoupled basis states

|π, gN 〉 = |π〉N∏i=1

|F,mF 〉i , and (3.28)

|0, eN 〉 = |0〉

1√N

N∑i=1

|F ′,mF ′〉i∏j 6=i|F,mF 〉j

, (3.29)

where the state |π, gN 〉 represents a single excitation in the π-cavity mode with all

atoms in the ground state, and |0, eN 〉 represents a Dicke state in which one atom is

excited and the cavity is empty. As before in the Tavis-Cummings model, the coupling

between these states is collectively enhanced, i.e.,

〈0, eN |HI,π|π, gN 〉 =√NgF ′ , (3.30)

where gF ′ is the average coupling to the |F ′〉 manifold. Specifically,

gF ′ =

√√√√ 1

N

N∑i=1

| 〈F,mF |µ0|F ′,mF 〉i gi|2, (3.31)

is the coupling averaged over the position dependent constants gi including themF -state

dependence of coupling strength via the Clebsch-Gordon coefficients. If we assume each

mF state has a population p(mF ) which is sufficiently large that the position dependent

coupling for the atoms in that state independently averages to g =√∑

i |gi|2/N , then

gF ′ can be written

gF ′ ≈ g√∑

mF

p(mF )| 〈F,mF |µ0|F ′,mF 〉 |2. (3.32)

The interaction Hamiltonian of Eq. 3.34 thus reduces to the form

HI,π = ~3∑

F ′=1

(√NgF ′a

†|F = 2〉〈F ′|+ H.c.). (3.33)

3.2.3.2 The undriven mode

Next we consider the interaction with undriven mode,

HI,⊥ = ~N∑i=1

3∑F ′=1

(gib†D⊥i (2, F ′) + H.c.

). (3.34)

60


|φ′g〉 = 1√N1

N1∑i=1|g1 · · · g(i)

2 · · · g1, g2 · · · g2,⊥〉

|φg〉 = |g1 · · · g1, g2 · · · g2, π〉

|φe〉 = 1√N1

N1∑i=1|g1 · · · e(i) · · · g1, g2 · · · g2, 0〉

= ~√N1g

〈φe|HI,π|φg〉

= ~ g〈φ′g|HI,⊥|φe〉

Figure 3.7: Example demonstrating collective enhancement of the driven mode but not

the undriven mode.

The undriven mode of the cavity is only populated by a transition from an excited

state |0, eN 〉 into the state where the cavity holds a ⊥-photon and the atoms are in a

superposition of one of them having changed mF states, i.e.,

| ⊥, g′N 〉 = | ⊥〉

1√2N

∑q=±1

|F,mF + q〉i∏j 6=i|F,mF 〉j

. (3.35)

The rate at which this process occurs is not collectively enhanced [111, 112], i.e.,⟨⊥, g′N |HI,⊥|0, eN

⟩= gF ′ , (3.36)

even if all mF states are macroscopically occupied [113].

This is more easily understood from a minimum working example, summarized in

Fig. 3.7. Let us consider only three atomic states: two ground states |g1〉 and |g2〉, and

one excited state |e〉. The states |g1〉 and |e〉 are only coupled through π transitions,

while states |g2〉 and |e〉 are only coupled though ⊥ transitions. If we assume N1 atoms

are in |g1〉, N2 atoms in |g2〉, and the cavity is empty, we write our initial state as

|φ0〉 = |g1 . . . g(N1)1 , g2 . . . g

(N2)2 , 0〉. (3.37)

By weakly probing the cavity with π-polarized light, the cavity mode is excited, bringing

the system into the state

|φg〉 = |g1 . . . g(N1)1 , g2 . . . g

(N2)2 , π〉. (3.38)

61


The cavity interaction, HI,π, couples the state |φg〉 to the Dicke state

|φe〉 =1√N1

N1∑i=1

|g1 . . . e(i) . . . g

(N1)1 , g2 . . . g

(N2)2 , 0〉 (3.39)

at the collectively enhanced rate 〈φe|HI,π|φg〉 = ~√N1g. As we have seen, the coupling

results in eigenstates of the system which are a superpositions of the bare states |φg〉and |φe〉. For the undriven ⊥-mode of the cavity to become populated, an atom in the

state |e〉 must transition to the state |g2〉 via the interaction HI,π. This can only occur

through the interaction HI,π coupling the state |φe〉 to a state

|φ′g〉 =1√N1

N1∑i=1

|g1 . . . g(i)2 . . . g

(N1)1 , g2 . . . g

(N2)2 ,⊥〉, (3.40)

where one atom originally in |g1〉 has transfered to |g2〉 generating one | ⊥〉 excitation in

the cavity. However, the rate of this process is not collectively enhanced, 〈φ′g|HI,⊥|φe〉 =

~ g. By writing the atom states as separable, we have assumed there are no coherences

between the atoms. In this case, it is clear that population of atoms originally in the

|g2〉 state play no role populating the | ⊥〉 mode because they have no interaction with

the state |φe〉. Since HI,⊥ couples only a single excited atom, decay into |φ′g〉 occurs at

the single-atom rate g. In our experiments N 1 and thus we will neglect emission

into the undriven mode.

3.2.3.3 Effective Hamiltonian

Omitting the terms associated with the undriven mode and substituting in the inter-

action Hamiltonian of Eq. 3.33, we arrive at the effective Hamiltonian

H = ~ωca†a+ ~

3∑F ′=1

ωF′ |F ′〉〈F ′|+ ~3∑

F ′=1

(√NgF ′a

†|F = 2〉〈F ′|+ H.c.). (3.41)

This Hamiltonian describes the ensemble as single collective spin which is coupled

to three excited states with the effective couplings geff,F ′ =√NgF ′ , summarized in

Fig. 3.8. As in the JCM, this Hamiltonian would predict a strong non-linearity for ex-

citations greater that one. This is of course not the case for our system which will have

the linear excitation spectrum of a harmonic oscillator so long as n N . Nevertheless,

since both quantum and semi-classical models agree in the linear regime, the Hamilto-

nian of Eq. 3.41 correctly predicts the eigenspectrum for the weak probing. This model

62


Figure 3.8: Collective coupling to multiple hyperfine transitions.

will also be useful for interpreting the spectra in other probing configurations, to which

it can be easily generalized.

The eigensystem of the reduced Hamiltonian Eq. 3.41 is easily solved numerically

and the resulting eigenspectra are shown in Fig. 3.9 for several values of N . We assume

that the mF magnetic sub-states of |F = 2〉 are equally populated. Taking into account

the Clebsch-Gordon coefficients [58], this constrains the effective couplings g1, g2, g3

to the ratio 0.18, 0.41, 0.68g. If the collectively enhanced couplings√NgF are small

relative to the excited state hyperfine structure, we expect three resolved splittings

with a size 2√NgF at the respective ωF ′ resonances. For larger atom numbers, the

splittings become dispersively shifted and eventually merge into a single splitting, as

observed in Fig. 3.9.

3.2.4 Semi-classical model for multi-level atoms

By including the multi-level structure of the atoms into the dynamic polarizability, the

semi-classical model using linear dispersion theory can also be extended to our system.

Keeping with π-probing scenario discussed in the previous section, the complex dynamic

63


Figure 3.9: Theoretical transmission spectra for weak driving. Three avoided crossings

between the energy of the bare cavity (solid diagonal line) and the energies of the bare atom

(solid horizontal lines) contribute to the transmission spectrum. Transmission through the

cavity is expected at the eigenenergies of the system. The amount of transmission is propor-

tional to⟨a†a⟩

of the corresponding eigenstate, indicated by the color scale. Calculations

were performed for an equal distributions of mF states and a g of 6.5 MHz. All detunings

are with respect to the |F = 2〉 to |F ′ = 3〉 transition.

64


Figure 3.10: Transmission spectra for N 87Rb atoms from classical linear dispersion

theory.

polarizability of an atom in state |F = 2,mF 〉 is given by1

α(ω,mF ) =3∑

F ′=1

|〈F ′,mF |µ0|2,mF 〉|2

~

(1

ωF ′ − ω+ i

γ

(ωF ′ − ω)2

), (3.42)

where µ0 is the dipole operator for π-polarization as before. In Eq. 3.42 the RWA

approximation has been made and the linear regime assumed. By averaging over themF

populations as before, we find an average polarizability, α(ω) =∑

mFp(mF )α(ω,mF ).

Following the same procedure detailed in [106] for a two-level atom, it is straightforward

to show the cavity transmission for N atoms with average polarizability α(ω) is given

by

Pout

Pin=

(1 +

3∑F ′=1

Ng2F ′

κγ

1

1 + δ2F ′

)2

+

(δc +

3∑F ′=1

Ng2F ′

κγ

δF ′

1 + δ2F ′

)2−1

, (3.43)

where δF ′ = (ω − ωF ′)/γ are the scaled detunings of the probe from the respective

resonances.

Fig. 3.10 shows the cavity transmission spectra predicted by Eq. 3.43 for N = 500

and N = 3000 with a g of 6.5 MHz and a uniform population of mF states. The

spectra of Fig. 3.10 are equivalent to the spectra in Fig. 3.9 obtained from the model

1From Eq. 2.5 after the RWA and generalization to multiple levels by superposition.

65


Hamiltonian, but with the transmission features having a width determined by the

cavity linewidth. That the two results are identical in the linear regime is not surprising

as we have already seen that the vacuum-Rabi splitting of the Jaynes-Cummings model

can be understood purely from linear dispersion [105].

The semi-classical model can also be extended beyond the linear regime by writing a

bistability equation, analogous to Eq. 3.21, including the multi-level structure of atom.

In this regime the features of the transmission spectra are power broadened due to

saturation of the atomic transition and optical bistability is observed. Because we are

in the strong-coupling regime, saturation and the effects of bistability occur even for

low excitation number. We observed optical bistability in the experiment when first

characterizing the cavity at high probe strength, though we did not investigate this in

detail.


In this section we cover the technical details of our methods and the experimental

apparatus itself, pictured in Fig. 3.1. Many of the systems which were discussed in

the previous chapter, such as the MOT and 1064 nm FORT, are also used in this

experiment. Here will discuss the new additions to the apparatus: a high finesse cavity,

optical lattice transport, a 780 nm cavity probe laser and a 808 nm FORT laser.

3.3.1 High finesse cavity

The cavity used in this experiment is characterized by the parameters listed in Table 3.1.

The mirror substrates are ground down to a 4 mm major diameter which tapers to a

2 mm diameter at the high reflective surface, as seen in the photograph of the cavity

(Fig. 3.12). This allows for the cavity mirrors to be placed much closer together while

still having optical access from the side. The cavity had a finesse of 120, 000 when first

constructed, but after baking the vacuum chamber and out-gassing of the 87Rb source,

the finesse degraded by roughly a factor a 2. Even so, the cavity is still in the strong-

coupling regime for a single atom1. Additional technical information on the high finesse

mirrors is reported in Appendix A.

1The single-atom coupling constant is determined from the dipole coupling for the |F = 2,mF = 2〉to |F ′ = 3,mF = 3〉 cycling transition.

66


Table 3.1: High finesse cavity parameters at λ = 780.24 nm, the D2-line of 87Rb.

cavity length lc 500 µm

free spectral range ∆νFSR 300 GHz

FWHM linewidth ∆νc 5.3 MHz

finesse F 57× 103

mirror transmission T 17 ppm

mirror absorptive losses A 38 ppm

mirror radius of curvature Rc 2.5 cm

mode waist (TEM00) w0 25 µm

atomic dipole decay rate γ 2π × 3.0 MHz

cavity field decay rate κ 2π × 2.6 MHz

atom-cavity coupling constant g0 2π × 9.2 MHz

single atom cooperativity (g20/κγ) C 11

One point of note is that this cavity had no detectable birefringence. Birefringence

is a common problem in high finesse cavities whereby stress induced asymmetry in

the mirrors causes a frequency separation of the two linear polarization modes of the

cavity. Because our cavity had no birefringence, the polarization modes are degenerate

and the cavity can support circularly polarized light. This is necessary for coupling

the cavity to σ± transitions of the atom and an obstacle for many other high finesse

cavity experiments. We did not realize until later that we were quite lucky to build

a birefringence-free experiment cavity. Subsequent cavities we constructed developed

birefringence, as we will see for the cavity used in next chapter (Sec. 4.3.1).

3.3.1.1 Cavity length stabilization

High finesse cavities demand extremely precise control of cavity length. With our cavity

parameters, a length change of δlc = λ/(2F) ≈ 7 pm shifts the cavity resonance by one

cavity linewidth. We aim to stabilize the cavity to within ∼ 1% of this value, which

is on order of femtometers, or, to give a sense of scale, a typical nuclear radius. This

requires both careful passive and active stabilization.

The cavity is made passively stable with the isolation stack shown in Fig. 3.11. The

cavity mount rests on top of four-tiers of stainless steel separated by Viton rubber spac-

ers which isolate the cavity from acoustic noise coupled through the vacuum chamber.

67


Figure 3.11: Vibration isolation

stack.

Figure 3.12: Experiment cavity.

Figure 3.13: PZT circuit.

A hole runs through the center to allow optical access for probe lasers in the vertical

direction. The isolation stack performs so well that we see no fluctuations of the cavity

length attributable to acoustic noise in the lab.

The cavity length is actively stabilized with a PZT. As pictured in Fig. 3.12, the

mirrors are attached with Torr seal to a 2 mm thick square PZT plate1 which tunes the

cavity by one FSR (390 nm) per 600 volts. In terms of PZT voltage noise, the ∼ 1% of

the linewidth stability criterion corresponds to 100 µV of voltage. Our home built 1000

volt PZT driver boards have ≈ 10 mV residual noise, so we use the circuit shown in

Fig. 3.13. The high voltage is heavily filtered to the required noise level and applied to

one side of the PZT, while the other side is controlled by standard low voltage precision

op-amps. We servo the cavity length using the low voltage side with a bandwidth of

100 Hz, which is limited by the first mechanical resonance of the PZT at ∼ 1 kHz.

The largest remaining source of length instability is pick up of electronic noise on

the PZT from external sources, principally the ion pump. This jitter is on the order

of a few percent of the cavity linewidth and so is acceptable for our experiments. If

necessary in the future, the pick-up could be reduced by better wiring inside the vacuum

chamber with twisted wire pairs and grounded shielding.

1Channel Industries C5500 Navy II PZT material

68


3.3.2 Cavity laser system

Two additional lasers were required for the cavity experiment, a 780 nm laser for weakly

probing the cavity near the atomic resonance, and an 808 nm laser for stabilizing the

length of the cavity as well as providing an intra-cavity FORT to trap the atoms.

Fig. 3.14 shows the layout of these laser in a simplified schematic. Both lasers are locked

by the Pound-Drever-Hall (PDH) method to a passively stable transfer cavity, which

serves as a frequency reference for the lasers and to narrow their linewidth. In order to

eliminate the slow drift of the transfer cavity, which is approximately ±10 MHz over

the course of a day, a double pass AOM actively steers the 780 laser to a Rb transition.

For the 808 laser, cavity drift is compensated by feeding forward the same oscillator

from the 780 AOM to the 808 AOM, although there is a 3.5% tracking error due to the

difference in the wavelengths.

Each laser then goes though a fiber based electro-optic modulator1 (EOM) which

can put sidebands at any frequency from DC to 10 GHz. The two lasers are then

combined into the same fiber and sent to the experiment cavity. By adjusting the

EOM frequencies2, the sidebands can be tuned such that a 780 nm sideband is resonant

with the atomic transition while at the same time both 780 nm and 808 nm sidebands

are resonant with the experiment cavity. The PZT on the experiment cavity actively

stabilizes the length of the cavity to remain resonant with the 808 nm sideband.

In order for both lasers to have clean transmission through the experiment cav-

ity, their linewidths must be much narrower the experiment cavity linewidth. This is

particularly an issue for the 808 nm laser which serves as an intra-cavity FORT. The

cavity converts laser frequency jitter into intensity jitter which results in parametric

heating and loss of atoms. To avoid this problem, we set out to narrow the linewidth

of both lasers to . 30 kHz, approximately 1% of the cavity linewidth. This was accom-

plished through two improvements to our laser system: a redesigned laser housing in

which the feedback grating can be placed up to 20 cm from the diode, and redesigned

electronics which increased the lock bandwidth to ∼ 3 MHz. With a longer extended

cavity, the ‘free-running’ linewidth of ECDL is reduced, facilitating further narrowing

1EOspace PM-0K5-10-PFA-PFA-780-UL2The rf drive for the EOspace EOMs is synthesized by a Phase Matrix QuickSyn FSW-0010 which

has 10 GHz frequency range in addition to 100 µs switching speed for fast ramps of the rf frequency

during an experiment.

69


Figure 3.14: Schematic of 780 nm and 808 nm laser system for locking and probing the

cavity. All cubes are polarizing beam splitters except the ones explicitly labeled as 50:50

which are non-polarizing beam splitters. See text for description.

70


of the linewidth by high bandwidth electronic feedback. By locking the lasers down

to at least a 10−3 fractional stability with respect to the transfer cavity linewidth, we

estimate a linewidth of ∼ 20 kHz for both lasers.

3.3.3 Detection

Transmission of the 780 nm probe laser is detected by fiber coupling the cavity output

to a single photon counting module1 (SPCM). A narrow band interference filter2 blocks

the unwanted 808 nm light which is also transmitted by the cavity. The 780 nm light

emitted from the cavity is detected with a total efficiency of

η = ηSPCM ηfc ηopt = (0.65)(0.65)(0.9) = 38%, (3.44)

where ηSPCM is the quantum efficiency of the SPCM, ηfc is the fiber coupling efficiency,

and ηopt is the efficiency of all the other optics including lenses, windows, and the

spectral filter. From the SPCM count rate, r, we infer the mean intra-cavity photon

number, n, from the relation

r =nηT

τcav= nηT

c

2lc, (3.45)

where τcav = c/2lc is the cavity round trip time and T is the transmission per mir-

ror. The combined rate of dark counts and background counts on the SPCM is 150

counts/sec. Given that n = 1 corresponds to a count rate of 2.3 × 106 counts/sec, we

are able to easily detect n 1 with good signal to noise.

3.3.4 Optical lattice transport

In order to transport the atoms from the location of the MOT to the cavity 1 cm

below, we employ the method of optical lattice transport [114, 115]. The optical lattice

is formed by two counter propagating 1064 nm wavelength beams, each with 1 W of

power, focused to a waist of 50 µm inside the cavity. Both beams are coupled through

optical fibers to ensure good mode quality and to aid in the alignment. Coupling each

beam back into the fiber of the opposing beam guarantees optimal spatial overlap of

the two beams.

1Perkin Elmer SPCM-AQRH-132Semrock LL01-780-12.5

71


The atoms are loaded into the lattice using a sequence similar to that of the all-

optical BEC experiment. Atoms from the MOT are first loaded into a dipole trap

formed by a 12 W beam focused to a 25 µm waist, which has a depth of 1.6 mK. Over

a 500 ms evaporation cycle, the dipole trap is ramped down while the optical lattice is

simultaneously ramped up. At the position of the dipole trap, the optical lattice has

a waist of 80 µm with a corresponding depth of 50 µK. Even though > 106 atoms

are loaded into the dipole trap from the MOT, we only transfer a maximum of only

1.5× 104 atoms into the lattice because of the shallow depth. The atoms in the dipole

trap have an rms radius of ≈ 4 µm along the lattice axis prior to transfer, and so we

assume the atoms in transport lattice are distributed over ≈ 15 lattice sites.

By changing the frequency difference, δν, between the counter-propagating beams,

the initially stationary lattice becomes a moving lattice with velocity v = λ δν/2.

We control the frequency difference using AOMs driven by home-built direct digital

synthesis (DDS) boards1. The atoms are transported a distance D in a time interval

T following the polynomial function [115]

v(t) =

DT (−1760

9 ( tT )4 + 3203 ( tT )3) for 0 < t ≤ T/4

DT (800

9 ( tT )4 − 16009 ( tT )3 + 320

3 ( tT )2 − 1609 ( tT ) + 10

9 ) for T/4 < t ≤ 3T/4

DT 2 (−1760

9 ( tT )4 + 60809 ( tT )3 − 2560

3 ( tT )2 + 41609

tT −

8009 ) for 3T/4 < t ≤ T

This piece-wise smooth function has zero initial acceleration, v(0) = 0, and jerk, v(0) =

0, to ensure smooth transport. In our experiments the transport distance is fixed at

9.2 mm and we vary the transport time T for optimal transport efficiency. For transport

times longer than ∼ 800 ms we observe no additional heating of the atoms and assume

the transport is fully adiabatic. We transfer up to 6000 atoms into the cavity, which

corresponds to a transfer efficiency of 50% from the starting position. This transfer

efficiency includes atoms losses due to background collisions and evaporation during

transport to the lattice focus.

3.3.5 808 nm intra-cavity FORT

Inside the cavity, the atoms are trapped by a combination of the transport lattice and an

intra-cavity lattice formed by the 808 nm light. The optical power of the 808 nm laser is

1In first iteration of the transport, a voltage controlled oscillator source controlled the AOM. Phase

noise from this source lead to excess losses in the transport lattice. A DDS source eliminated all phase

noise problems. DDS boards were designed by Christian Kurtseifer.

72


Figure 3.15: Lifetime of atoms trapped in the 808 nm intra-cavity FORT. Each data

point is an averaged over 10 runs. Exponential fit to data after the first 500 ms yields a

lifetime of 2.9 seconds.

actively stabilized to 27 mW of circulating optical power in the cavity. Given the cavity

mode waist of 25 µm, this corresponds to a lattice potential with a depth of 150 µK,

the same as the depth of the transport lattice inside the cavity. The 808 nm FORT is

linearly polarized so that all magnetic substates are trapped equally. The scattering

rate due to the 808 nm FORT is Γsc/2π ≈ 2.5 Hz. While this is still not problematic as

a heating rate, T ≈ 0.8 µK/s, it should be noted that the detuning is only somewhat

larger than the fine structure splitting and consequently 10% of scattering events are

Raman scattering (Sec. 2.2.1).

Other groups [20] have reported short trap lifetimes in intra-cavity FORTs, on the

order of 10 ms, which they attribute to parametric heating. By trapping the atoms

in the intra-cavity FORT only, we measure the lifetime in the 808 nm trap to be 2.9

seconds, as shown in Fig. 3.15. We can only assume the comparatively longer lifetime

is due to better length stability of the cavity and that our FORT laser linewidth is

much narrower than the cavity linewidth.

Along the cavity axis, the atom cloud has an rms radius of ≈ 8 µm in the transport

lattice prior to entering the cavity mode. We assume several 808 nm lattice sites are

occupied with populations weighted by the 8 µm rms Gaussian profile. Due to the

difference in wavelength between the 808 nm FORT lattice and 780 nm probe lattice,

the two intra-cavity lattices completely dephase over a distance of 6 µm. Because

73


FORT lattice sites are occupied over a length longer than the dephasing length, the

average atom-cavity coupling to the 780 nm mode will be roughly equivalent to that of

a uniform distribution of atoms along the cavity axis. The spatially averaged coupling

for a uniform distribution in the axial direction is g = g0/√

2 ≈ 2π × 6.5 MHz.

In the transverse dimension, the 808 nm confines the atoms well within the mode

field diameter and the reduction in coupling due to the transverse spread is only 8%.

We will neglect this reduction in coupling due to the transverse dimension because it

is small compared to other systematic uncertainties in the experiment.

3.3.6 Optical pumping

For some experiments, we want to prepare the ensemble in a single magnetic substate.

This is done with the optical pumping scheme shown in Fig. 3.16. As discussed in

Sec. 2.4.1, the master cooling laser can be tuned via AOMs to the |F = 2〉 → |F ′ = 2〉transition so that the same beam we used for resonant imaging can also be used for

optical pumping. After the atoms have been loaded from the MOT into the dipole trap,

a 100 µs pulse from the optical pumping laser and repumping laser is applied. Normally

the lifetime of |F = 2〉 atoms is reduced by hyperfine-changing collisions as we observed

in Sec. 2.4.6. However, if the atoms are fully pumped into the |F = 2,mF = 2〉 state,

hyperfine changing collisions are forbidden by conservation of total angular momentum.

In Fig. 3.17 we compare the lifetime with and without optical pumping. We take the

unaltered lifetime after the optical pumping pulse, in contrast to the shortened lifetime

after repumping only, as verification that the ensemble has been successfully pumped

to the |F = 2,mF = 2〉 state.

3.4 Experimental results: cavity transmission spectra

In this section we analyze the experimental cavity transmission spectra in a variety

of probe configurations. Depending on the specific combination of probe polarization,

magnetic field direction, and atomic state preparation, the spectra exhibit unique char-

acteristics. The structure of the transmission spectra can are explained using the model

discussed in Sec. 3.2.3. We also discuss observed dynamic behavior resulting from state

changing processes.

74


Figure 3.16: Optical pumping scheme.

Optical pumping laser (red) and repump

laser (blue).

Figure 3.17: Lifetime after optical

pumping.

3.4.1 Experiment procedure

We briefly review the full procedure of a single experiment cycle, which takes a total

of 10 seconds. Prior to each experiment, the detuning of the cavity relative to the

atomic resonance is set via the frequency of the 808 nm sideband, to which the cavity

is locked. Starting from a MOT, the atoms are loaded into a single beam 1064 nm

dipole trap. Immediately after loading the dipole trap, the atoms are optionally either

pumped into the |F = 2,mF = 2〉 state or left in the |F = 1〉 manifold. Over 500

ms, the dipole trap depth is adiabatically lowered while the optical transport lattice is

raised, transferring the atoms into the transport lattice. Over the subsequent 800 ms,

the optical lattice transports the atoms 9.2 mm down to the cavity mode. During this

time the magnetic field1 is rotated to set the quantization axis to the desired direction

for probing, either transverse to the cavity axis or along the cavity axis. The rotation

of the field is adiabatic relative to the Larmor precession frequency such that the spin

polarization of the ensemble tracks the magnetic field direction.

Once the atoms are inside the cavity, a repump beam is turned on to keep the atoms

in the |F = 2〉 manifold for the remaining duration of the experiment. If the atoms had

1The bias magnetic field is set by three Helmholtz coil pairs which provide ≈ 1 G/A. These are

driven by home-built 3-amp voltage-controlled bipolar current sources.

75


Figure 3.18: Experiment configu-

ration for an effective two-level sys-

tem.

Figure 3.19: Transmission spectrum for N =

590(70) atoms (g = 6.5 MHz).

been left in the |F = 1〉 manifold, they are, presumably, pumped into a distribution

across all mF states in the |F = 2〉 manifold. If the atom had been prepared in the

|F = 2,mF = 2〉 state, the repump beam does nothing other than repump atoms that

later fall to |F = 1〉 during cavity probing. The detuning of a weak probe beam coupled

to the cavity is swept ±500 MHz relative to the cycling transition over 200 ms while

monitoring the cavity output with the SPCM. The intensity of the probe beam is set

such that, on resonance with the empty cavity, the intra-cavity photon number is less

than one. After probing, the atoms are released from the trap and the atom number

is measured by absorption imaging. The experiment cycle is repeated over a range of

cavity detunings to obtain a full transmission spectrum.

3.4.2 Two-level atoms: the cycling transition

First we consider the probe configuration summarized in Fig. 3.18. The atoms are

optically pumped into the |F = 2,mF = 2〉 state, the magnetic field is aligned along

the cavity, and the probe is circularly polarized such that it couples to σ+-transitions.

In this configuration each atom is in effect a two-level system because only the cycling

transition is coupled. Fig. 3.19 shows an example transmission spectrum for N =

590(70) atoms. As predicted by the Tavis-Cummings model (white dashed lines), there

is a single splitting of the size 2√Ng = 315 MHz.

76


Figure 3.20: Cavity transmission spectra coupling to π-transitions. All detunings are with

respect to the |F = 2〉 to |F ′ = 3〉 transition. The average probe intensity is nempty ≈ 0.3.

3.4.3 Multi-level atoms: π-probing

Next we consider the probe configuration which we discussed at length in Sec. 3.2.3. The

atoms are pumped into a distribution of mF -states in the |F = 2〉 manifold which we

assume to be uniform. The probe polarization is linear and parallel to a magnetic field

transverse to the cavity axis (see Fig. 3.6). The magnetic field has an amplitude of 2.6

G which results in differential Zeeman shifts of 600 kHz. We neglect the Zeeman shifts

because they are small compared to the cavity linewidth of 5.3 MHz. From the model

presented in Sec. 3.2.3, we expect to see three splittings with effective couplings gF ′ ,

which are calculated by averaging over variations in coupling due the spatial distribution

of the atoms and the mF state distribution. We find (g1, g2, g3) = (0.18, 0.41, 0.68)g =

(1.2, 2.7, 4.4) MHz for a uniform population of the mF -states. By varying the number of

atoms loaded into cavity, we go from a regime where the collectively enhanced couplings

are small compared to the separation between atomic levels to a regime where both

are of comparable size. The transmission spectra are shown in Fig. 3.20, plotted on a

logarithmic scale to highlight the weaker features. The gray lines overlying the data

77


Figure 3.21: Cavity transmission spectra probing both σ+ and σ− modes. In left frame,

the atoms are optically pumped into the |F = 2,mF = 2〉 state prior to probing. In the

right frame, the atoms are prepared in a mixture of mF states prior to probing. The

gray dashed lines indicate the upper eigenenergies (N = 3000) of the σ+ and σ− modes

considering only |F ′ = 3〉 coupling for either a perfectly pumped |F = 2,mF = 2〉 ensemble

(left) or a uniform distribution (right).

are the eigenenergies of Eq. 3.41 calculated without any free parameters using the gF ′

above and the average atom number measured by absorption imaging.

3.4.4 Driving both cavity modes

In the previous two examples, the probe polarization and magnetic field were configured

to drive only one mode of the cavity. In this next configuration we drive both cavity

modes. The probe is linearly polarized and the magnetic field is aligned to the cavity

axis. Interaction with the atoms splits the degeneracy of the cavity polarization modes

into the the modes corresponding to σ+ and σ− as defined by the atoms. The linear

input polarization decomposes into (|σ+〉+ |σ+〉)/√

2 and thus couples to both modes.

In Fig. 3.21 we see the resulting spectra both with and without initially optical pumping.

The inset of each plot illustrates the respective couplings, considering only transitions

from |F = 2〉 to |F ′ = 3〉 for simplicity.

On the top half of the left plot, we see a separation of the eigenstates correspond

78


to the two polarization modes. By polarization spectroscopy on the cavity output, it

was verified the lower line corresponds to left hand circularly polarized light, and the

upper line right hand circularly polarized light, transmitted through the cavity. The

separation of the eigenstates can be understood from the difference in coupling strength

between the σ+ and σ− transitions. For a perfectly pumped gas as in the left inset of

Fig. 3.21, the relative coupling strengths are gσ− =√

1/15gσ+ as determined from the

appropriate Clebsch-Gordon coefficients. The dashed lines indicate the eigenenergies

for the the two polarization modes for a maximally polarized gas with N = 3000.

For the right plot, the mF distribution is assumed to be symmetric and therefore

unpolarized. In this case there is no difference in coupling between σ+ and σ− and

the eigenenergies of the two modes are degenerate (dashed line, right). In the left plot,

the separation of the two modes in the data is less than expected for a fully polarized

gas (dashed line), which suggests the ensemble is only partially spin polarized. This is

believed to be a result of optical depumping by the cavity probe beam itself during the

sweep.

3.4.5 Optical pumping by the cavity field

When the system is probed on resonance, the atoms scatter from the intra-cavity

field into free space, potentially changing their mF state. Redistribution of the mF

populations changes the effective couplings and can move the system out of resonance

with the probe. Consequently we are not necessarily observing steady-state behavior

with the transmission spectra. We demonstrate this by changing the direction of the

probe frequency sweep and observing a different spectra.

In Fig. 3.22(left) we repeat the same experiment as Fig. 3.21(left) except sweeping

the probe only over positive detunings and increasing the probe strength to nempty ≈ 1.2

to increase the optical pumping rate. By sweeping the probe frequency up, the σ− mode

comes into resonance first. Scattering of σ− light pumps to mF distribution towards

|F = 2,mF = −2〉, increasing the effective coupling to the σ− mode. Thus the probe

beam pushes the σ− mode resonance higher as the sweep proceeds, until the gas is

depolarized and the σ+ mode is simultaneously resonant.

In contrast, we have Fig. 3.22(right) for which the probe frequency is swept down.

Here, the σ+ mode is resonant first and scattering maintains the initial mF distribution.

When the σ− mode does become resonant, optical pumping quickly moves the system

79


Figure 3.22: Optical pumping by the probe field. For both spectra the atoms are prepared

in the |F = 2,mF = 2〉 and probed with both σ+ and σ−. In the left spectra the probe

frequency is sweep from zero detuning to +600 MHz, and in the right spectra in the

opposite direction. Both insets show the trace from the single experiment run marked by

the vertical gray line. The probe strength was increased to nempty ≈ 1.2 to exaggerate the

pumping effect.

80

3.5 Summary

out of resonance in opposite direction of the sweep. This is evidenced by the narrow

σ− feature in the trace shown in Fig. 3.22(right-inset).

For the π-probing configuration, on which we primarily focused in the theoretical

analysis, optical pumping is less of an issue. Scattering of π-light is symmetric in its

pumping effect and so an initially unpolarized ensemble will remain so. The effective

couplings will tend not to deviate far from that of a uniform distribution. For the

π-probing configuration we observe the same spectra independent of sweep direction or

probe strength.

3.5 Summary

In summary, we experimentally measured the transmission spectra for N 87Rb atoms

coupled to a high finesse cavity in a variety of probing configurations. The system

was modeled as a collective spin with an effective coupling found by averaging over

the ensemble. The experimental results are in good agreement with the predictions

of this model. By considering the appropriate dipole interaction, the structure of

the transmission spectra depending on probe polarization, magnetic field, and atomic

distribution have been explained. Finally, we demonstrated the dynamic effects of

mF -state changing processes while probing the cavity.

81


82

Chapter 4

Self-Organization of Thermal

Atoms Coupled to a Cavity

This chapter covers the third generation experimental apparatus, pictured in Fig. 4.1, and is

largely based on ‘Self-Organization Threshold Scaling for Thermal Atoms Coupled to a Cavity’

K. J. Arnold, M. P. Baden, and M. D. Barrett Physical Review Letters 109, 153002 (2012).

4.1 Introduction

Design concept of the experiment

After the experiments of previous chapter, we constructed a new dual-wavelength

780/1560 nm high finesse experiment cavity1. By using the 1560 nm wavelength for the

intra-cavity FORT, the atoms can be trapped exactly at every second anti-node of the

probe mode such that each trap site is identically and maximally coupled. This config-

uration is particularly well suited for studying collectively-enhanced coherent scattering

from an ensemble into a cavity when probed from the side. We had several specific

research directions for this system, all of which revolve around probing an ensemble of

atoms from the side and scattering into the cavity. These include, but are not limited

to: self-organization, Bragg scattering, cavity-assisted ensemble cooling for transverse

pumping, and simulation of the Dicke model. This chapter details our investigation of

self-organization.

1Due to a technical problem of the Rb source being depleted, we were unable to proceed the previous

cavity and had to reopen the vacuum at this point in time anyway.

83

4. SELF-ORGANIZATION OF THERMAL ATOMS COUPLED TO ACAVITY

Figure 4.1: Schematic and picture of third generation experiment apparatus.

Self-Organization

In nature, many open systems driven away from thermal equilibrium exhibit phase

transitions into an organized state [116, 117]; a few examples from physics are the

laser [118, 119] and Rayleigh-Benard convection cells [120]. Often these systems exhibit

spontaneous symmetry breaking above a critical point, behavior familiar from second

order equilibrium phase transitions. Additionally, like second order phase transitions,

there exists an order parameter which is zero below the critical point and converges

continuously to ±1 above a critical point. Self-organization of polarizable particles

coupled to the standing wave mode of a cavity is another such non-equilibrium phase

transition which demonstrates these qualities.

The polarizable particles will organize for sufficiently strong transverse pumping

due to the backaction of the collectively scattered light field on the atomic motion [36].

Essentially, light scattered into the cavity results in a potential which acts to localize

the atoms into a lattice configuration more favorable for scattering. This further en-

hances the localization of the atoms. Thus above a critical pump intensity, an initially

uniform distribution of atoms undergoes a phase transition, spontaneously breaking

symmetry [121] and organizing into one of two possible lattice configurations, as seen

in the top half of Fig. 4.2.

An experiment at Stanford [37] was the first to observe collectively enhanced coher-

ent scattering into a standing wave cavity when probing thermal atoms with an optical

lattice transverse to the cavity. Self-organization was suggested [36] as an explanation

84

4.1 Introduction

Figure 4.2: Schematic representation of self-organization in the lattice (top) and traveling

wave (bottom) geometry. (a, c) Below threshold, the atoms (black) are confined by the

intra-cavity 1560 nm FORT (yellow). (b, d) Above threshold, the atoms organize into a

λ-spaced lattice trapped by the probe and scattered fields (red). (b) For a lattice probe,

the atoms can form one of two possible λ-spaced lattices (filled or open circles). (d) For a

traveling wave probe, interference between probe and scattered fields results in a λ-spaced

transverse lattice out of phase with the atoms by θ.

85


of this phenomenon, and a subsequent experiment [38] confirmed the atoms were self-

organizing into a Bragg crystal which collectively enhanced emission into the cavity

mode. Later experiments at ETH Zurich [39, 121] explored self-organization with a

Bose-Einstein condensate, which was mapped to a quantum phase transition (i.e the

Dicke model) [39, 42, 121].

In the first self-organization experiments [37, 38], cooling and deceleration of the

falling atomic ensemble by a velocity dependent force, which was enhanced by the

collective scattering, were observed. Self-organization is of particular interest as a

platform for cooling as it can be applied to all polarizable particles, including molecules

[21, 22], which do not have a closed transition. In addition, theoretical studies have

suggested a cooling rate which is independent of particle number N [36, 44] as result of

the collective effects. This is in contrast to other stochastic ensemble cooling schemes

for which the cooling rate decreases linearly with N [32, 122]. However, numerical

simulations have suggested that if statistical fluctuations are required to trigger the self-

organization, the effective threshold may scale as N−12 instead of N−1 [43]. For large

ensembles, an N−12 threshold scaling places severe constraints on the required probe

power, greatly impacting the practicality of self-organization as a cooling method [34].

Although several theoretical papers on cavity cooling related to self-organization [21,

22, 33, 36, 44] followed the experimental result of [37, 38], there have been no further

experimental studies to validate the theory associated with self-organization of thermal

atoms or investigate to what extent an ensemble of atoms can be cooled.

Outline

In this chapter we detail our experimental study of self-organization of thermal 87Rb atoms

coupled to a high finesse cavity. We directly measure the threshold behavior over a

wide range of experimental parameters for two transverse probing configurations: a

retro-reflected lattice and a traveling wave. Fig. 4.2 shows a schematic representation

of both probing configurations above and below threshold. Previously, only the lattice

probe configuration has been considered in the literature. Unlike earlier studies, in our

system the atoms are trapped intra-cavity by a 1560 nm FORT which locates the atoms

at every second anti-node of the 780 nm cavity mode. Accounting for both the external

FORT potential and the additional traveling wave probe configuration requires mod-

ification to the original threshold equation given in Ref. [43]. However, the resulting

86

4.2 Derivation of the threshold equations

equations still maintain a N−1 scaling and, in the appropriate limit, exactly agree with

the mean-field result of Ref. [43]. Our measurements of the threshold clearly demon-

strate a N−1 scaling of the threshold and are in good agreement with the threshold

equations we derive.

We also consider the potentials above threshold and discuss the possibility of sta-

ble defect atoms, expanding on the discussion in Ref. [43] to include both axial and

transverse dimensions. Above threshold, the strong non-linear coupling between the

spatial configuration and induced potential leads to complex dynamics driven by the

opto-mechanical forces. We characterize the experimentally observed cavity output by

parameter regimes and qualitatively interpret the dynamics by considering the mu-

tual inter-dependence of the system parameters. However, in regards to using self-

organiztion as a means of cooling trapped ensembles, we see little prospect of useful

cooling.


In this section we derive a threshold condition for self-organization in both the retro-

reflected lattice and traveling wave probe configurations. The original equations de-

termining the threshold intensity given in Ref. [43] are specific to the lattice probe

configuration. They are also restricted to a particular probe detuning relative to the

cavity, and assume the transverse extent of the atomic distribution can be neglected.

We extend the previous theoretical treatment of Ref. [43] to include an external trap-

ping potential and the spatial extent of the atomic distribution transverse to the cavity.

If we assume the atoms are in thermal equilibrium, the spatial density distribution

in a confining potential V (x) is given by

ρa(x) =1

Zexp

(−V (x)

kBT

), (4.1)

where Z =∫

exp (−V (x))/(kBT )) d3x. For our case the confining potential consists of

two components

V (x) = VT(x) + VS(x), (4.2)

where VT(x) is the potential associated with the 1560 nm FORT and VS(x) is the

potential associated with the probe beam itself and light scattered from the probe

beam into the cavity.

87


Two spatial averages quantify scattering into the cavity, and thus VS(x). Namely,

α =

∫f2(r, z)ρa(x) d3x (4.3)

and

β =

∫f(r, z)h(x)ρa(x) d3x, (4.4)

where f(r, z) = cos(kz) exp(−r2/w2) is the mode function of the cavity and h(x) the

mode function of the probe beam. The factor α accounts for the average dispersive

coupling to cavity due to the spatial overlap of the atomic density distribution and the

cavity mode. The factor β accounts for the scattering into the cavity by coherently av-

eraging the scattering amplitude over the atomic distribution. For unorganized atoms,

β ≈ 0 due destructive interference and thus scattering of light into the cavity mode is

suppressed. For atoms organized in a Bragg crystal, β = ±1 and scattering into the

cavity is enhanced by constructive interference.

Self-organization arises from the fact that the potential VS(x) and hence ρa(x) de-

pend on α and β. As the atoms scatter more light into the cavity, a potential is produced

which localizes the atoms in a more favorable configuration for scattering. This fur-

ther enhances the localization of the atoms, and, under sufficient driving strength, an

initially uniform distribution of atoms will undergo a phase transition spontaneously

reorganizing into one of two possible lattice configurations. The phase transition is

characterized by the parameter β, which serves as the order parameter. It has a value

near zero prior to the phase transition and it rapidly approaches ±1 as the probe

coupling, Ω, rises above a threshold value.

The threshold condition can be found by self-consistently solving Eq. 4.1 with

Eq. 4.3 and 4.4. First, the potential VS(x) is determined by looking at the steady

state solutions describing the system for arbitrary α and β. Since we will operate at a

large atomic detuning and well below atomic saturation, we can adiabatically eliminate

the atomic excited states and the resulting Hamiltonian is

H = (−∆c +NU0α)a†a+N(ΩRβ)∗a+N(ΩRβ)a† (4.5)

with the cavity decay described by the Liouvillian

Lρ = (κ+NΓ0α)(2aρa† − a†aρ− ρa†a). (4.6)

88


In these expressions, ∆c = ωp−ωc is the cavity detuning with respect to the probe, U0

is the single atom dispersive shift, ΩR is the per atom pump rate of the cavity due to

scattering of the probe field, and Γ0 is a correction to the cavity decay associated with

spontaneous emission. In terms of the atom-cavity coupling, g, atom-probe coupling,

Ω, the detuning ∆ of the probe from the atomic resonance, and the atomic linewidth

γ, the parameters U0, Γ0, and ΩR may be written

U0 =g2

∆, Γ0 = γ

g2

∆2, ΩR =

Ωg

∆.

In our system NΓ0 κ and so we neglect this term in all that follows.

The steady solution of Eq. 4.5 and Eq. 4.6 for the cavity mode excitation is a

coherent state with amplitude

λ =NΩRβ

∆c + iκ, (4.7)

where ∆c = ∆c − NU0α is the probe detuning from the dispersively shifted cavity

resonance.

The total potential due to the scattered and probe fields, VS(x), is then given by

VS(x) =~∆|Ωh(x) + λgf(x)|2

=~Ω2

∆

[|h(x)|2 + 2εf(r, z)Re

(β∗e−iθh(x)

)+ ε2|β|2f2(r, z)

](4.8)

where

ε = − NU0√∆2c + κ2

and eiθ =−∆c + iκ√

∆2c + κ2

. (4.9)

For the specific case of a 1560 nm intra-cavity lattice that coincides with every

second antinode of the cavity, the external potential is given by

VT = −VT0 exp

(−2 r2

w2T

)cos2(kT z). (4.10)

Using the resulting potential V (x) = VT(x) +VS(x) to determine the density distri-

bution from Eq. 4.1 leads to self-consistency equations for α and β via their definitions,

Eq. 4.3 and 4.4. Near to the critical point the order parameter β ≈ 0, and so the

self-consistency equations can be expanded to first order in β. From Eq. 4.4 this ex-

pansion yields an equation for the threshold probe coupling given the associated value

of α from the first order expansion of Eq. 4.3. The details of this derivation are given

89


in Appendix B for two probing geometries: h(x) = cos(kx − φ) and h(x) = eikx, cor-

responding to a lattice or traveling wave probe respectively. In both cases we assume

that the transverse extent of the ensemble is much larger than the wavelength of the

probe. The two geometries give qualitatively different results and so we discuss each

separately.

4.2.1 Lattice geometry

The threshold condition for the probe coupling strength is conveniently expressed in

terms of the dimensionless quantity µ = −~Ω2/(∆kbT ) which is the probe trap depth

relative to the atoms’ thermal energy, kbT . In the lattice probe geometry the threshold

value, µ∗, is then determined by the equation(1 +

I1(µ∗/2)

I0(µ∗/2)

)µ∗ =

1

NU0α

∆2c + κ2

∆c

(4.11)

where

α =1

2

1 + e−4/η

1 + 2/η, (4.12)

Ik(x) are modified Bessel functions of the first kind, and η = VT0/(kbT ) is the ratio of

the FORT depth to the atoms’ thermal energy. The factor α given by Eq. 4.12 accounts

for the spatial averaging of the atom-cavity coupling given the thermal distribution in

the external potential. Thus the term NU0α is exactly the dispersive shift of the

cavity resonance prior to self-organization. In the experiment, we will be able to non-

destructively measure NU0α and fully determine the right hand side of Eq. 4.11. We

note that the probe beam is red detuned from the atomic resonance (∆ < 0) and so

µ is a positive quantity and NU0α is negative. Consequently this requires ∆c < 0 for

self-organization to occur. Also note the minimum threshold occurs at the effective

cavity detuning ∆c = −κ for a given dispersive shift.

4.2.1.1 Comparison to previous threshold condition result

The threshold condition given by Eq. 4.11 is derived under the assumption that the

atoms are trapped at every second antinode of the cavity and have a transverse spatial

extent that is large relative to the wavelength. The result given in Ref. [43] on the other

hand neglects the transverse dimension and assumes a uniform distribution along the

transverse z-axis. It is therefore of interest to consider how these differences impact

90


on the threshold behavior of the system. Firstly, it should be noted that the FORT

restricting the position of the atoms along the cavity axis to every second antinode

does not significantly alter the threshold condition. It merely alters the parameter α

and the net effect is then a slight rescaling of the threshold value. The most significant

difference lies in the consideration of the transverse dimension. Consideration of this

dimension gives rise to the ratio I1(µ∗/2)/I0(µ∗/2). This ratio should be set to ≈ 1 if

the atoms are confined to length scales much smaller than the wavelength along this

dimension. We can recover the exact expression [Eq (19)] reported in Ref. [43] from

Eq. 4.11 by setting α = 1/2 to account for the initially uniform distribution, setting

I1(µ∗/2)/I0(µ∗/2) = 1 to account for neglecting the transverse dimension, and using

their chosen detuning of ∆c = −κ. Our threshold equation is therefore a generalization

of the result given in [43] to account for an arbitrary detuning of the cavity and for the

transverse extent of the atomic distribution.

4.2.1.2 Stable defect atoms

Although there is no significant change to the threshold conditions by including the

transverse dimension, there is a significant difference concerning the possibility of defect

atoms, or atoms that are stably trapped at the minority sites. Neglecting the Gaussian

dependence of Vs, we have the potential

Vs(x, z) =~Ω2

∆

[cos2(kx) + 2εβ cos(kz) cos(kx) cos(θ) + ε2β2 cos2(kz)

](4.13)

where ε and cos(θ) are given by Eq. 4.9.

In Ref. [43], the authors discuss the possibility of stable defect atoms trapped at

minority sites. They consider only the 1D case where atoms are restricted to the cavity

axis. For the potential given by Eq. 4.13, this is equivalent to setting cos(kx) = 1. For

this 1D axial potential, stable minima at the minority sites only appear when the term

proportional to β2 begins to dominate. If, without loss of generality, we assume β ≥ 0,

the condition for stable minima to appear and their associated depth, Vms, are given

by

β >∆c

NU0, Vms =

~Ω2

∆

(NU0β − ∆c)2

∆2c + κ2

. (4.14)

It is clear that minima at the minority sites cannot occur if |NU0| < |∆c|. When

|NU0| > |∆c| stable minority sites only appear for sufficiently large β and their depth

91


Figure 4.3: One dimensional potentials before (β = 0) and after (β = 1) self-organization.

The potentials are calculated for the probe intensity at threshold (µ = µ∗) for α = 1 and

∆c = −κ. Both the regimes of the µ∗ < 1 (left column) and µ∗ > 1 (right column) are

shown for comparison.

continues to increase as β rises. This leads to a strong possibility of defect atoms

arising. In the limit |∆c| → −κ, where the threshold is lowest, these two regimes

correspond to either a small dispersive shift and |µ∗| < 1 or a large dispersive shift and

|µ∗| > 1. The top row of Fig. 4.3 shows the potential along the cavity axis in both

regimes, illustrating that minority sites occur only when |µ∗| < 1. Note also that the

depth of minority sites is not limited by the depth of the probe potential because the

scattered field which builds up in the cavity can be larger than the probe field.

For our system, the atoms are pinned to every second antinode site along the cavity

axis by the external FORT potential. Thus we set cos(kz) = 1 in Eq. 4.13 and instead

consider the 1D potential in the transverse dimension. From Eq. 4.13 it is straight

92


forward to show the condition for minority sites to exist and their depth are given by

β < µLT, Ums =~Ω2

∆

(1− β

µLT

), (4.15)

where

µLT ≡1

NU0

∆2c + κ2

∆c

=

(1 +

I1(µ∗/2)

I0(µ∗/2)

)αµ∗. (4.16)

Note that the factor(

1 + I1(µ/2)I0(µ/2)

)∈ [1, 2] and typically α ∈ [1/2, 1]. Thus, up to at

most a factor of 2, the parameter µLT is the threshold.

In contrast to the axial dimension, here the minority sites exist before the self-

organization begins. At the onset of organization, where β ≈ 0, the only potential is

the probe lattice potential and we see from Eq. 4.15 that the depth of the minority sites

is simply determined by that of the probe. Above threshold, the behavior again differs

for the two regimes |µLT| < 1 and |µLT| > 1, as illustrated in the bottom row of Fig. 4.3.

For |µLT| < 1, the depth of probe lattice at threshold is less than the thermal energy

of the atoms and provides little impediment to motion in the transverse direction. As

the organization process proceeds, β increases and the depth of the minority sites is

reduced until they no longer exist. Thus organization is expected to proceed rapidly

and completely. In contrast, for |µLT| > 1 the probe depth at threshold is greater than

the thermal energy of the atoms and thus the probe lattice has a strong influence on

the motion of the atoms in the transverse direction. For a sufficiently large |µ|, we

would expect the atoms to be locked into position by the probe lattice before threshold

is reached and self organization would not occur. Moreover, in the regime |µLT| > 1

the minority sites provide stable minima even for a fully organized distribution (β = 1)

and so stable defect atoms can be expected.

Note that the conditions for stable defects given by Eq. 4.15 and Eq. 4.14 are in

direct contrast to each other. In the 1D transverse case defect atoms are only likely to

occur if the threshold parameter |µLT| ≈ |µ∗| & 1, and in the axial case defect atoms

can only appear if |µ∗| . 1. One is therefore inclined to ask what happens if the atoms

are free to move in both directions. In this case, consideration of the full 2D potential

given by Eq. 4.13 reveals that condition for local potential minima to exist is

∆2c

∆2c + κ2

|µLT| < β < |µLT|. (4.17)

93


In the fully organized phase (β ≈ 1), we see that minority sites can only exist in a narrow

window of threshold values bounded by 1 < |µLT| < (1 + κ2/∆2c). For large effective

detuning (|∆c| 1), this window becomes vanishingly small. Note that Eq. 4.17 is

a necessary condition for the existence of local minima at minority sites but not a

sufficient condition for the existence of stable defect atoms. A more detailed analysis

of the thermal distribution in the potential would be required to establish the stability

region for trapped defect atoms. Nonetheless, outside the small range of parameters

defined by Eq. 4.17, we can definitively say that stable defect atoms cannot occur unless

the system is influenced by an external potential.

4.2.2 Traveling wave geometry

For the traveling wave probe configuration, the threshold condition, derived in Ap-

pendix B, is given by

µ∗ =

√∆2c + κ2

−NU0αe−iθ (4.18)

where α is given by Eq. 4.12 and e−iθ is given by Eq. 4.9. The phase term e−iθ at face

value would imply that no stable organization pattern can actually form. Physically,

this is because the phase shift arising from the cavity detuning results in an interference

pattern that has a minimum at a location displaced from the atoms position by the

phase θ = tan−1(−κ/∆c) (see Fig. 4.2). Thus, if the atoms form an organized lattice,

the resulting scattered field will cause the lattice to move transversally. Hence it would,

in principle, be possible for the atoms to form a quasi-stable moving lattice in a manner

similar to that of a collective atomic recoil lasing (CARL) [123]. In the case of the

CARL, it has been shown that additional forces can stabilize the lattice [124]. In

our case this is also possible due to the Gaussian dependence of the cavity mode in

the transverse dimension. We can expect the atoms to be displaced off center from

the cavity axis in such a way that the transverse force moves the atoms away from

the lattice minimum in order to effectively cancel the phase term. The phase shift

does not appear to significantly affect the onset of self-organization and we use the

threshold condition given by Eq. 4.18 with the phase term ignored in the analysis of

the experimental data.

Note that the threshold behavior in this case is very different from the lattice case.

In the lattice case we required ∆c < 0 in order for a solution to Eq 4.11 to exist. In the

94

4.3 Experimental set-up and methods

case of the traveling wave probe, organization can occur for any ∆c. The only difference

between positive and negative cavity detuning is the sign of θ, and hence to which side

of the cavity mode the lattice is displaced. Also note that for large detuning, |∆c| κ,

the phase θ → 0 and the organized configuration becomes inherently more stable.


In this section we describe the two major upgrades from the previous experimental

apparatus: the dual-wavelength high finesse cavity, and a new method of transporting

atoms to the cavity. In addition, we detail experimental methods which have not yet

been introduced.

4.3.1 Dual-wavelength high finesse cavity

The dual-wavelength experiment cavity is characterized by the parameters listed in

Table 4.1. This cavity was made considerably longer than the previous cavity in order

to have narrow linewidth. Our goal was for both the cooperativity to be greater than

one and the cavity linewidth to be less than the anticipated axial trapping frequency of

the intra cavity FORT. This would allow us to investigate cavity cooling of an atomic

ensemble in the resolved side-band regime. Further discussion of this topic is postponed

until the final chapter. For studying self-organization, the exact cavity parameters are

not very crucial and the important characteristic of this cavity is the FORT lattice

which has exactly twice the wavelength of the probe.

4.3.1.1 Cavity mount design: preventing Rb contamination

The experiment cavity is pictured in Fig. 4.4, fully assembled on the same vibration

isolation stack described in Sec. 3.3.1. Prior to the design which can be seen in Fig. 4.4,

the experiment apparatus had been fully constructed with two different cavity designs.

However, after operation of the Rb source to form a MOT, the experiment cavity rapidly

degraded in finesse. This is described in more detail in Appendix A. Consequently, the

cavity mount shown in Fig. 4.4 was designed to protect the cavity mirrors from Rb

as much as possible. The block is made of stainless steel with as few holes for optical

access as possible. The cavity mode passes through a 1 mm hole inside the block, which

required precise alignment of the optic axis. The cavity construction and alignment

95


Table 4.1: Dual-wavelength high finesse cavity parameters at 780 nm and 1560 nm

wavelengths.

780 nm 1560 nm

cavity length lc 9.6 mm

free spectral range ∆νFSR 15.6 GHz

FWHM linewidth ∆νc 140 kHz 100 kHz

finesse F 110× 103 160× 103

mirror transmission T 11 ppm 7 ppm

mirror absorptive losses A 17 ppm 13 ppm

mirror radius of curvature Rc 2.5 cm

mode waist (TEM00) w0 50 µm 70 µm

atomic dipole decay rate γ 2π × 3.0 MHz

cavity field decay rate κ 2π × 0.07 MHz

atom-cavity coupling constant g0 2π × 1.06 MHz

single atom cooperativity (g20/κγ) C 5

methods are also described in Appendix A. Two 1000 volt ring PZTs are used in order

to have a range of travel sufficient to span a full FSR at 1560 nm.

Our previous experiments had all used a Rb getter which sprayed Rb throughout

the experiment chamber when forming a MOT. As an additional precaution to protect

the cavity, a directional Rb oven was designed. The oven is the white object made of

Macor ceramic which is can be seen in Fig. 4.4. A hole in the oven funnels the Rb only

to the MOT region. After the bakeout of the vacuum system, the absorptive losses

at 780 nm increased from 12 ppm to 17 ppm. However, no further degradation has

occurred since then, over nearly one year of operating the Rb source.

4.3.1.2 Cavity birefringence

When first constructed, this cavity had a small birefringence of 70 kHz, as seen in

Fig. 4.5 (left). After baking the vacuum system at 150 C, the birefringence significantly

increased to 365 kHz, as seen in Fig. 4.5 (right). This birefringence is almost certainly

due to stress induced on the mirrors by the Torr seal which attaches them to the

PZT [18]. Unlike the experiment cavity from the previous chapter (Fig. 3.1) for which

the mirrors were attached by Torr seal at only one point, the mirrors of this cavity were

96


Figure 4.4: Picture of dual-wavelength cavity

necessarily attached at multiple points. Presumably this exerted asymmetric stress on

the mirror surface as the Torr seal hardened, which worsened after the thermal cycling

from the bake out. Possibly baking at a lower temperature would have been better

since 150 C is the absolute limit specified for Torr seal.

For self-organization, birefringence is not a significant problem since we will be

scattering linearly polarized light. Unfortunately the polarization axes of the cavity

are rotated by 21 with respect to the vertical and horizontal axes and our experiment

geometry restricts us to the vertical direction when probing transverse to the cavity.

For a vertical probe beam linearly polarized orthogonal to the cavity axis, scattering

into the nearest cavity polarization mode is therefore reduced to cos2(21) = 87% of

the maximum possible rate.

4.3.2 Cavity laser system

The cavity laser setup is similar to one described in Sec. 3.3.2 of the previous chapter,

except with a 1560 nm ECDL replacing the 808 nm laser. The new setup is shown

schematically in Fig. 4.6. Both 780 nm and 1560 nm ECDLs are again locked to a

transfer cavity to narrow their linewidths and fix their relative frequencies. Since we

will be operating at large atomic detuning for these experiments (|∆| > 100 GHz),

the exact frequency of the 780 nm laser is not important. Thus we don’t require the

wideband fiber EOM and simply use a double-pass AOM for tuning the 780 laser with

respect to the experiment cavity. A ±10 MHz tuning range is more than sufficient

because of the narrow linewidth of the experiment cavity. So that we may measure

97


Figure 4.5: Cavity transmission showing birefringence before and after baking the vacuum

system. The polarization is set such that both polarization modes of the cavity are equally

coupled. The red line is a fit for two Lorentzians of equal height and width.

the threshold over a wider range of parameters, the optical power of the probe laser is

boosted by injection locking a free-running slave laser. The 780 nm probe light is then

diverted into three optical paths: the local oscillator used in heterodyne detection, a

probe beam coupled directly to the cavity, and the probe beam transverse to the cavity

which drives the self-organization. The transverse probe has most of the optical power,

up to 30 mW in the chamber.

For the 1560 nm laser, the light is sent through a fiber EOM and coupled directly

to the cavity. The experiment cavity is locked via the PZTs to a frequency sideband

of the 1560 nm laser (up to ±10 GHz from the carrier). The transmission through the

cavity at 1560 nm is used to actively stabilize the intra-cavity intensity and ensure a

constant FORT depth. This is important for repeatability of the experiment because

the cavity coupling drifts throughout the day due to thermal effects.

The narrow linewidth of this experiment cavity places even more stringent require-

ments on the frequequncy stability of the lasers than the previous cavity. To stabilize

the laser linewidths to within ∼ 1% of the cavity requires linewidths on the order of

1 kHz. We were able to narrow the laser linewidths further using the same extended

laser housings and electronics but locking to a better transfer cavity. The new transfer

cavity uses the same dual-wavelength high finesse mirrors as the experiment cavity and

98


Figure 4.6: Layout for 780 nm and 1560 nm lasers which probe and lock the experiment

cavity. See text for description.

99


is estimated to have a linewidth of 100 kHz at 780 nm and 60 kHz at 1560 nm. Locked

to the transfer cavity, the estimated linewidths of both lasers are less than 1 kHz.

4.3.2.1 Measurement of 1560 nm laser linewidth

We were able to independently verify the linewidth of the 1560 nm laser by performing

a beat note measurement with the frequency comb in the lab down the hall. The

frequency comb is stabilized on short time scales by phase locking to an ECDL laser

which itself is locked to a reference cavity (linewidth = 320 kHz). The comb reference

laser was previously measured to have a linewidth of 600 Hz. The results of a beatnote

measurement between our 1560 nm laser and the frequency comb are shown in Fig. 4.7.

In the primary plot we observe a beat note with 400 Hz linewidth at 1 Hz resolution

bandwidth for a ‘slow’ sweep time of 400 ms for the full span shown. The inset show a

‘fast’ sweep where the beatnote is resolution bandwidth limited by spectrum analyzer.

This implies for that times scales of ≈ 1 ms, the two lasers in different labs have

better than 20 Hz relative frequency stability! Considering the beatnote is more narrow

than the previous linewidth measurement for the comb reference laser, it is likely the

beat linewidth is limited by the comb reference laser and our laser is in fact even

narrower that this measurement. Given our reference cavity linewidth is a factor of 5

narrower than the comb reference cavity, one would expect a better linewidth assuming

both lasers are locked with a comparable fractional stability relative to their respective

reference cavities.

4.3.3 Detection

Fig. 4.8 shows the cavity coupling and detection setup. The cavity output signal is

fiber coupled, with 85% efficiency, and split with a 50:50 spliced fiber into two detection

setups: the SPCM, as before, and a heterodyne setup. The SPCM allows us to detect

weak signals on the order of tens of femto-Watts, but is easily saturated at a few

pico-Watts. The heterodyne detection can also be used to detect signals as low as a

pico-Watt, depending on bandwidth, but has a much larger dynamic range. Backscatter

from the heterodyne local oscillator is enough to saturate the SPCM and so only one

detection method can be used at a time. A mechanical shutter allows us to block the

local oscillator and switch between detection methods mid experiment. Approximately

100


Figure 4.7: Beatnote between our 1560 nm laser and a frequency comb phase-locked to

a narrow reference laser.

25% of the total power emitted from the cavity goes to the SPCM and another 25%

goes to the heterodyne detection.

The local oscillator and cavity probe beams are separated by 190 MHz. The hetero-

dyne beat note is generated using a Thorlabs PDB430A-AC differential photodetector

and the amplitude is measured with an Aglient E4405B spectrum analyzer in zero-span

mode. The voltage amplitude of the 190 MHz beat is given by V = C√ISILO, where

IS is the optical intensity of signal, ILO the intensity of the local oscillator, and C is

a constant. Since IS ILO, we use the DC voltage on one of the photo detectors to

actively stabilize ILO. We calibrate C by measuring V over a range of known IS and

fitting to a x1/2 power law.

4.3.4 Atom transport: translation of the dipole trap

Each experiment starts by loading atoms from a MOT into a 1064 nm dipole trap 15

mm above the cavity. The dipole trap has waists [wx, wy] = [110, 30] µm and 16 W of

optical power. We load a maximum of 15 million atoms into this trap from the MOT.

Previously, we evaporatively cooled the atoms from the dipole trap into an optical

101


Figure 4.8: Cavity coupling and detection set-up.

102


lattice which transported the atoms into the cavity. This transfer was very inefficient

and limited the number of atoms we could load into the cavity.

For this experiment apparatus, we instead use a mechanical stage1 to translate the

dipole trap directly into the cavity over one second. Once in the cavity mode, the beam

power is lowered over 300 ms as the atoms are transfered from the 1064 nm dipole trap

into the 1560 nm intracavity FORT. With this method we are able to load up to 7×105

atoms into the cavity. Compared to the lattice transport, it is both simpler and more

effective, loading roughly 100 times more atoms.

4.3.5 1560 nm intra-cavity FORT

For all the experiments reported in this chapter, the optical power at 1560 nm trans-

mitted through the cavity is actively stabilized to 33 µW. This corresponds to 4.7 W

of circulating power in the high finesse cavity. For a waist of 70 µm, the trap depth

is calculated to be 230 µK with trapping frequencies [ωax, ωr] = 2π × [133, 0.7] kHz.

We measure a trap lifetime in the 1560 nm FORT of a few seconds, comparable to the

lifetime in the 1064 nm dipole trap.

4.3.5.1 Differential AC Stark shift

The wavelength 1560 nm happens to be near to the 5P3/2 → 4D3/2 transition in

87Rb (see Fig. 4.9). This results in large AC Stark shift of the 5P3/2 excited states

such that for the D2 transition the excited states are shifted by approximately 42 times

the ground state light-shift [125]. For atoms at the bottom of our 230 µK trap, the

differential light shift is thus about 200 MHz. For the self-organization experiments

which are performed at very large detuning, this light shift is inconsequential. However

when probing on resonance, i.e. for optical pumping, the light shift must be taken

into account. Even accounting for light-shifted transition frequency, we are unable to

optically pump the atoms in the 1560 nm dipole trap although we are able to the

808 nm and 1064 nm traps. This is most likely due to non-uniform light shifts of the

mF states in the excited manifold. Currently, we are planing to optically pump instead

on the D1 line, for which the differential light shift is lower in magnitude by a factor

of 2.6.

1Newport UTS50CC

103


Figure 4.9: Lowest electronic states of 87Rb.

4.3.5.2 Positioning of atoms in the cavity

By selecting the appropriate cavity length, we are able to position the 1560 FORT

anti-nodes to coincide with either nodes or anti-nodes of the probe field. Given a probe

wavelength λ, for the cavity to be resonant it must have length lc = nλ2 where n is

a positive integer. If the FORT wavelength λ′ is to be simultaneously resonant, the

wavelength must be chosen such that λ′ = 2lcm for an integer m ≈ n/2. When n is even,

the atoms will be trapped at nodes of the probe field near the middle of the cavity.

While for n odd, they will be trapped at anti-nodes. This is illustrated in Fig. 4.10

which shows the coupling g at each trap site for n odd (left side) and n even (right

side). Several resonant λ′ are shown for the values of m nearest n/2. Note that the

atom cloud has an rms spread of ≈ 20 µm along the cavity axis, much less than the

length scale of the cavity shown in Fig. 4.10. Thus we can safety assume that all trap

sites have identical coupling.

Experimentally, we do not know a priori which cavity length corresponds to an odd

or even integer number of half wavelengths. By measuring both the dispersive shift and

atom number, we can infer whether we are at the nodes or anti-nodes. The dispersive

shift of the cavity is given by ∆d = Ng2α/∆. When the trap sites are positioned on

the anti-nodes of the probe mode, the factor α is given by Eq. 4.12. When the trap

104


Figure 4.10: Atom-cavity coupling at the position of the FORT antinodes along a cavity

of length lc = nλ/2 = 9.6 mm.

sites are centered on the nodes of the probe mode, α is instead given by

α =1

2

1− e−4/η

1 + 2/η. (4.19)

We can easily measure the dispersive shift by sweeping the frequency of a weak probe

beam coupled to the cavity and observing cavity transmission. Since we have > 105

atoms trapped in the cavity, the number of atoms and temperature can be accurately

measured by absorption imaging after ballistic expansion. Using these measured quan-

tities, we calculate the parameter α and compare the result directly to Eq. 4.12 and

Eq. 4.19. Fig. 4.11 shows several measurements over a range of η. The temperature

of the atoms was varied by displacing the transport beam and transferring the atoms

into the cavity FORT off-axis. The agreement with Eq. 4.12 and Eq. 4.19 (black lines)

verifies we are in fact trapping the atoms at either the nodes or anti-nodes. We can

easily switch between the two configurations by simply changing the cavity length by

one FSR at the probe wavelength. We note it is also possible to position the traps

sites at an arbitrary phase between the nodes and anitonodes of the probe lattice by

displacing the z position of the atom cloud from the center of the cavity.

105


Figure 4.11: The parameter α as a function of trap depth to temperature ratio η. For

each data point, the parameter α is calculated using the measured atom number, measured

dispersive shift, ∆ = −265 GHz, and g =√

23g0 = 2π × 0.87 MHz. The black lines are

calculated from Eq. 4.12 and Eq. 4.19.

4.3.5.3 Transverse probe beam

The transverse probe beam is focused to a waist of 120 µm at the position of the atoms;

large enough to ensure a nearly uniform intensity across the atom cloud. The power is

actively stabilized with a maximum of 25 mW available. The probe polarization is linear

and aligned orthogonal to the cavity axis. A magnetic field defining the quantization

axis is aligned transverse to the cavity and parallel the probe beam polarization. We

note again that the polarization of the cavity mode is misaligned by 21 with respect

probe as result of the cavity birefringence. This reduces the scattering rate into the

cavity by 13%.

4.4 Experimental results: self-organization threshold scal-

ing

This section details the experimental procedures for measuring the threshold and

presents the resulting measurements. The threshold is measured for both the lat-

tice and traveling wave probe configurations over a range of atoms numbers at two

detunings from the atomic resonance, -110 GHz and -265 GHz.

106

4.4 Experimental results: self-organization threshold scaling

4.4.1 Experimental procedure

Starting from a MOT, we load atoms into a 1064 nm dipole trap which is translated

15 mm over 1 second into the cavity mode. Once in the cavity, the 1064 nm FORT is

adiabatically ramped off in 300 ms, transferring the atoms into the intra-cavity 1560 nm

FORT. By varying the MOT atom number, we control the number of atoms delivered

to the cavity FORT, up to a maximum of 7× 105.

Once the atoms are trapped inside the cavity, the frequency of a weak probe beam

(n < 10) coupled to the cavity is swept over the dispersively shifted cavity resonance in

10 ms. The cavity transmission is detected on the SPCM and fit with a Lorentzian to

extract the dispersive shift. Measurement of the dispersive shift has virtually no effect

on the atoms for the large detunings at which we are operating1.

Next we measure the threshold by linearly ramping up the power of the transverse

probe beam over 10 ms at a known probe detuning ∆c and monitoring the cavity out-

put. This is done with either the SPCM or heterodyne detection. For measuring the

threshold, we use the SPCM because of the higher sensitivity and nearly zero back-

ground. However, above threshold the SPCM quickly saturates and so the heterodyne

detection must be used to accurately measure the cavity output.

After the ramp of the transverse probe, the atom number and temperature are

measured by absorption imaging. This information is not used for inferring the mea-

sured threshold, but is useful for interpreting the above threshold dynamics of the

self-organization process.

4.4.2 Comparison to the threshold equations

The threshold condition, given by Eq. 4.11 for the lattice case and Eq. 4.18 for the

traveling wave case, depends on three parameters: the threshold parameter µ, the

dispersive shift NU0α, and the effective cavity detuning ∆c. Having directly measured

the dispersive shift, we can infer ∆c since the probe detuning from the empty cavity,

∆c, is known. The threshold parameter is obtained from the ratio of the probe depth

at which light begins to scattering into the cavity to the temperature of the atoms at

that threshold. Without probing the atoms, they have a temperature of 31 ± 2 µK

1For example, at a detuning ∆ = −265 GHz and intra-cavity photon number n = 10 on resonance

with the cavity, the scattering rate per atom is only Γsc/2π ≈ 3× 10−4 Hz.

107


corresponding to η ≈ 7 in the 1560 nm FORT. However since the probe beam itself

can heat the atoms prior to organization, we must account for this effect in order to

accurately determine the temperature at the threshold.

4.4.2.1 Determining temperature at threshold

The probe heats the atoms prior to self-organization in two-ways: scattering into free

space and adiabatic compression. Heating effects due to scattering are reduced at large

detuning because the scattering scales as ∝ ∆−2 but the dispersive interaction, and

hence threshold, scales as ∝ ∆−1. Fig. 4.12 shows the heating rate for our system at

∆ = −265 GHz for a 10 mW probe beam in both the traveling wave and retro-reflected

lattice configurations. From the slope we infer a heating rate of ≈ 0.06 µK/ ms for

the traveling wave and ≈ 0.2 µK/ ms for the lattice. This is consistent with heating

due to off resonance scattering at the rate determined by Eq. 2.11. By appropriate

scaling of this heating rate for detuning and optical power, we can infer the net heating

during the intensity ramp up to the observed threshold in each experiment. Since the

intensity ramp is executed in 10 ms, the heating due to scattering is usually negligible.

The dominant heating process is adiabatic compression; this effect is already seen in

Fig. 4.12 where compression by the lattice probe is responsible for the jump in the

initial temperature (red).

We calibrate the temperature increase due to compression by measuring the tem-

perature throughout a ramp of the probe intensity with the cavity far detuned so that

self-organization does not occur. Fig. 4.13 shows an example of the temperature mea-

sured at several points during a 10 ms ramp of the optical power up to 20 mW in the

lattice configuration. At this atomic detuning, 20 mW corresponds to a 100 µK deep

lattice. Using such calibrations, we infer the temperature of the atoms at the threshold

power for which organization is observed. Note that, in the traveling wave configura-

tion, the probe does not significantly alter the confinement prior to organization and

so this compression heating effect is not observed in this case.

We also correct for the change in the dispersive shift which depends on the tem-

perature via the parameter α. By comparing the temperature at threshold to the

temperature at which the dispersive was measured, 31 ± 2 µK, we can calculate the

change in α via Eq. 4.12 and thus infer the correct dispersive shift at the observed

threshold.

108


Figure 4.12: Heating rate due the scat-

tering from the probe beam at 10 mW con-

stant power for both the traveling wave

(blue) and retro-reflected lattice (red) con-

figurations. The green data shows the base-

line temperature without probing.

Figure 4.13: Heating due to adiabatic

compression by the probe beam in the

lattice configuration. The temperature is

measured at several points throughout a

10 ms linear ramp of the optical power.

4.4.3 Lattice geometry threshold results

The left plot of Fig. 4.14 shows the results of several threshold measurements at fixed

probe detuning, ∆c, with respect to the empty cavity. As the atom number fluctuates

from shot to shot of the experiment, we sample a range of effective detunings ∆c. By

coarsely controlling the atom number via the initial MOT size, we fill out the curve

shown in the left side of Fig. 4.14. Several data sets were taken at fixed ∆c ranging

from −65κ to −5κ. The black line shows the theoretical threshold calculated ab initio

from Eq. 4.11.

On the right side of Fig. 4.14 are plotted a subset of points from all the data sets for

which |∆c + κ| ≤ κ/2; or in other words, the points near to the minimum value of the

threshold for a given ∆c. This is compared to the minimum threshold calculated from

Eq. 4.11 for ∆c = −κ (black line). The results are in reasonable agreement with Eq. 4.11

and a N−1 scaling is clearly observed. We note there is a deviation of the measured

threshold as µ → 1. This is due to the rate of self-organization slowing dramatically

near µ = 1 such that the measured threshold is systematically overestimated. The cause

109


Figure 4.14: Threshold measurements in the lattice probe geometry. (left) One set

measurements at constant ∆c. Black line is an ab initio calculation of the threshold

from Eq. 4.11. (right) All measurements from the data sets at various ∆c for which

|∆c + κ| ≤ κ/2. Black line is the calculated threshold from Eq. 4.11 for ∆c = −κ. Red

circles are data taken at the atomic detuning −265 GHz, and blue circles at −110 GHz.

of this is discussed further in the following sections in which we consider the dynamics of

self-organization. Essentially this is a consequence of the external potential restricting

organization to the transverse dimension in our system.

4.4.4 Traveling wave geometry threshold results

The threshold in the traveling wave geometry is measured in essentially the same way

as in the lattice case. The left plot of Fig. 4.15 shows the results of several threshold

measurements at fixed probe detuning, ∆c. The black line shows the theoretical thresh-

old calculated from Eq. 4.18. In contrast to the lattice geometry, here self-organization

occurs for both signs of the cavity detuning with a minimum at ∆c = 0. The right

side of Fig. 4.15 plots the subset of all points for which |∆c| ≤ κ/2, i.e. the points

within ±κ/2 of the minimum threshold. The black line is the threshold calculated from

Eq. 4.18 for ∆c = 0. Again the threshold measurements are in good agreement with

the expected threshold and a N−1 scaling.

110


Figure 4.15: Threshold measurements in the traveling wave probe geometry. (left) One

set of measurements at fixed ∆c. The black line is an ab initio calculation of the threshold

from Eq. 4.18. (right) All measurements from the data sets at various ∆c for which

|∆c| ≤ κ/2. The black line is the calculated threshold from Eq. 4.18 for ∆c = 0. Red

circles are data taken at the atomic detuning −265 GHz, and blue circles at −110 GHz.

4.4.5 Discussion of threshold scaling

In this section we briefly discuss our threshold scaling results in the context of Ref. [43].

In Ref. [43], the authors derive a threshold equation [Eq.19] using mean-field methods

for self-organization in the lattice configuration which has a N−1 scaling . This equation

is derived assuming a tight transverse confinement and a uniform initial distribution.

Our threshold equation Eq. 4.11 reduces to [Eq.19] of Ref. [43] in the appropriate

limits: I1(µ/2)I0(µ/2) = 1 for tight transverse confinement and α = 1/2 for a uniform atomic

distribution. Incidentally, the parameter we define as µLT is exactly the threshold

condition of Ref. [43].

In Ref. [43], the authors report a multi-particle simulation of self-organization where

the pump power is linearly ramped through threshold over 4 ms, essentially the same

as the experiments we performed. In this simulation the atoms are assumed a priori

to have a thermal distribution and only couple via interaction with the collectively

scattered field (i.e. collisions are not considered). They observe that as the particle

number is increased, the rate of organization slows such that the effective threshold

deviates from the mean field prediction (Fig.6 in [43]). This can be understood as a

consequence of their model assumptions, namely that the atoms can only thermalize via

111


coupling to the scattered field. Prior to organization, β = 0 on average and scattering

into the cavity only results from statistical fluctuations away from initial distribution.

Presumably since fluctuations are suppressed for increasing N , the atom-atom coupling

via the scattered field is suppressed and so ‘thermalization’ to organized phase becomes

slow. Consequently the authors of Ref. [43] propose an alternative threshold condition

which requires that the scattered field due the statistical fluctuations have a depth on

the order ∼ kbT , sufficient to organize the atoms “instantly”. This alternative threshold

condition naturally has a scaling of N−1/2.

We would argue that, provided there are sufficiently many atoms and a physical

mechanism by which the atoms thermalize on the timescale of interest, the mean field

approach should yield accurate results for the threshold. In our experiments, the densi-

ties are such that collision times are ∼ 100 µs with particle numbers of ∼ 104 per site of

the 1560 nm lattice potential. In this case we can expect a thermodynamic description

to be valid, and, indeed, our threshold measurements show an N−1 scaling, in good

agreement with the mean field description. However, one can imagine cases in which

a thermodynamic description may not apply, such as for fermions or low densities of

molecules where the collision rate is negligible. In such cases a sub-N−1 scaling of the

effective threshold, as observed in the simulation of Ref. [43], may still be applicable.

We briefly mention a final point concerning the ‘hysteresis’ of threshold reported in

the simulations of Ref. [43] (Fig.6). In Ref. [43], they also simulate what happens if the

atoms are already fully organized (β = 1) and the probe intensity is ramped from above

to below threshold. In their simulation the atoms revert to the unorganized phase at

the mean field threshold independent of atom number. In Ref. [43] there is a factor

of 2 discrepancy between ‘ramp down’ threshold and the mean field prediction. This

is likely because they have implicitly assumed α = 1/2 in their mean field threshold

[Eq.19]. When starting from an organized phase, as in their ‘down’ simulation, α =

1 and the correct mean field threshold condition should be a factor of 2 lower (i.e.

µ∗(α=1) = µLT /2). If this is the case, then the ‘ramp down’ simulation in fact agrees

exactly with the expected mean field threshold. In our experiment when we ramp

the power down starting from above threshold, the heating and atom loss due to the

non-adiabatic dynamics of organizing always results in a higher threshold than if we

ramp up. Nevertheless, we would expect whether ramping up or down, as long as the

112

4.5 Experimental results: dynamics of self-organization

thermodynamic description of the system is valid, the threshold will be determined by

Eq. 4.11.


In this section we analyze the cavity output traces to characterize the dynamics of self-

organization. Due to the mutual inter-dependence of T , α, ∆c, and µ, these dynamics

are complex and non-linear. Generally, the degree of non-linearity in this system de-

pends on the magnitude of the dispersive shift. When the dispersive shift is large, a

small change in the spatial configuration of the atoms results in a significant change

in ∆c and thus the field inside the cavity. In the following sections we discuss the two

probing configurations broken down into the relevant parameter regimes.

4.5.1 Lattice geometry

In our system the 1560 nm FORT tightly confines the atoms in the axial direction but

only weakly in the transverse direction. In order to organize, the atoms must rearrange

themselves in the transverse direction and thus we are primarily concerned with the

potential along this axis. For the following discussion we will consider the 1D transverse

potential which, neglecting the Gaussian dependence, can be conveniently expressed in

the form

−Vs(x, z = 0)

kbT= µ cos2(x) + 2

(µ

µLT

)β cos(x), (4.20)

where

µLT =1

NU0

∆2c + κ2

∆c

(4.21)

Again we note the term µLT is approximately equal to the exact threshold condition µ∗

to within a factor of 2. In Eq. 4.20, the first term results from the probe lattice itself

and the second term results from inference between the probe and cavity fields. The

threshold condition, µ = µ∗ ≈ µLT, thus corresponds to the cross term having a depth

of ∼ kbT as it drives the system to the self-organized phase, β → ±1. Previously in

Sec. 4.2.1.2 we analyzed the potential with regards to appearance of local minima at

minority sites and observed two distinct regimes, |µ∗| < 1 and |µ∗| > 1. Perhaps not

surprisingly, the above threshold dynamics also differ depending on which these two

regimes one is in.

113


Regime of |µ∗| < 1

First we briefly summarize the nature of the potential in this regime. At threshold

the depth of the probe lattice itself is less than the thermal energy of the atoms by

definition, and thus the atoms are able to move freely in the transverse direction. Above

threshold (µ > µ∗), the cross term dominates as β → ±1. Since there are no stable

minority sites we expect the organization to proceed rapidly and completely. Note that

since the probe depth is < kbT , the depth of the 780 cavity mode must be > kbT for

the cross term to be ∼ kbT .

In this regime the cavity dispersive shift, ∆d = NU0α, must be large relative to

the cavity linewidth, as can been seen from the threshold Eq. 4.11. Naively, one might

expect the dispersive shift not to change very much after organization because the 1560

FORT already traps all the atoms at the anti-nodes of the cavity field. However, as we

saw in Fig. 4.11 for atoms trapped in the 1560 FORT with an η = 7, the dispersive shift

is reduced by the factor α ≈ 0.6 due to the thermal spread in the 1560 lattice sites. If

instead the atoms are confined deeply in the 780 cavity mode itself, the tighter axial

and transverse confinement can significantly increase α. Since the regime of |µ∗| < 1

is exactly where the potential depth of the cavity mode is greater than the ensemble

temperature above threshold, during organization the dispersive shift will increase due

to tighter confinement.

The left side of Fig. 4.16 shows a heterodyne trace of the cavity output while ramping

the transverse power linearly to drive self-organization. The shape of this trace, a single

sharp pulse, is typical when crossing threshold in the regime of very large dispersive shift

and ∆c < −κ. This is likely because above threshold, compression leads to an increase

in the dispersive shift and thus brings ∆c closer to −κ. The threshold parameter µLT

has a minimum at ∆c = −κ and will decrease as organization proceeds. Thus for the

initial condition ∆c < −κ, just above threshold both coefficients determining the depth

of the organizing potential, µ/µLT and β, are increasing. The observed cavity output

suggests that this results in the potential coming up too fast (i.e. non-adiabatically).

In the middle frames of Fig. 4.16, we see the phase and amplitude of the cavity

output1 for an experiment in the same regime as the left frame of Fig. 4.16. From the

1These traces were capture by a fast oscilloscope using IQ-demodulation of the heterodyne beatnote

to observe the fast dynamics. All other heterodyne traces are from the spectrum analyzer which is

limited to a 100 kHz sampling rate in zero-span mode.

114


Figure 4.16: (left) Trace of cavity output power (heterodyne measurement) while ramping

the depth of the transverse probe lattice over 10 ms. The ∆d and ∆c measured prior to

the trace are given in the plot, with the implied threshold µ∗ from Eq. 4.11. The atomic

detuning is ∆ = −265 GHz and here 50 nW of power output from the cavity corresponds

to an intra-cavity lattice with a depth 165 µK. (middle) Intensity and phase of the cavity

output pulse for an experiment in the same regime as the left frame. (right) Absorption

image ≈4 ms after self-organization event. Image is clipped at an optical depth of 2 to

highlight the jets of atoms escaping the trap.

cavity output power (middle top), we see the atoms both self-organize and dissipate

on the time scale 5 µs, which is nearly κ−1, as fast as the cavity field can respond.

From the phase of the output light (middle bottom), we can infer information about

the organization and transverse motion of the atoms. Initially the atoms are unorga-

nized (β ≈ 0) and we define phase here as zero. As the cavity field builds, the phase

approaches ±π/2 corresponding to β = ±1, the two possible lattice configurations. As

the pulse amplitude decays, we see the phase reverses and shifts by ≈ π, meaning β has

flipped sign. One interpretation is that the atoms are accelerated towards the majority

positions as they rapidly organize but then overshoot to the minority positions because

the potential has come up too fast. This conclusion is supported by absorption images

taken a few milliseconds after such a pulse occurs which show jets of atoms ejected from

the either side of the cavity along the probe axis (Fig. 4.16 right). From absorption

images taken well after the single pulse at threshold has occurred, we observe that a

significant fraction of the atoms are lost from the trap (for example in the left trace of

Fig. 4.16, we measured 22% of the atoms were lost due to organization event).

115


Figure 4.17: Heterodyne trace in the regime of very large dispersive shift for an effective

detuning ∆c ≈ −κ. The atomic detuning is ∆ = −265 GHz.

Next we consider the trace in Fig. 4.17 where the dispersive shift is similarly large,

but for which ∆c ≈ −κ. The organization still occurs rapidly, but the output is

sustained for several milliseconds before suddenly switching off. Since ∆c ≈ −κ, µLT is

already at a minimum, and thus an increase in the dispersive shift due to compression

above threshold will increase the threshold parameter. So in contrast to the previous

case, here an increase in α results in a decrease of (µ/µLT ). It is possible this dynamic

of (µ/µLT ) decreasing while β is increasing results in a less violent self-organization

where a stable equilibrium is reached. As for the sudden reversion to the unorganized

phase far above threshold, we suspect heating and atom loss due to the intense intra-

cavity field results in the system falling below threshold. For the trace in Fig. 4.17,

13% of the atoms were lost and the final temperature was 50 µK. This is compared to

2% of atoms lost and a final temperature of 33 µK if the probe beam was swept but

no organization occurred.

Fig. 4.18 shows two additional traces both still with large dispersive shift but less so

than the previous examples. We see the qualitative shape of the traces is the same as the

previous examples depending on whether ∆c < −κ or ∆c ≈ −κ. As the dispersive shift

becomes smaller two trends are observed which are illustrated by these traces: first,

after the sudden reversion towards the unorganized phase there is still a significant

amount of scattering into the cavity, and second, the initial build up of the cavity

field above threshold slows. The continued scattering after the sudden fall in cavity

116


Figure 4.18: Heterodyne traces in the regime of large dispersive shift plotted on a log

scale. The same traces are plotted on a linear scale in the insets for comparison to the

previous traces. The atomic detuning is ∆ = −265 GHz.

output indicates that the subsequent distribution of atoms in the probe lattice still

has a residual population imbalance between majority and minority trap sites. For the

traces shown in Fig. 4.18, the probe lattice is ramped up to a depth nearly 3 times the

initial temperature of the atoms. At the point where system reverts to below threshold,

the probe lattice depth alone is comparable to or exceeds the temperature of the atoms

and thus it could be expected that a residual majority of atoms remains trapped in the

organized configuration.

Regarding the slowing rate of self-organization, we believe this is a consequence of

the probe lattice acting as a barrier to self-organization. In the limit that µ∗ 1, this

barrier is greater than the thermal energy of the atoms and will prevent them from

rearranging in the transverse direction; thus self-organization can not occur. However,

it is somewhat surprising that we see a strong effect on the organization due to probe

lattice even for µ∗ approaching unity. We suspect the organization is initially limited by

the time required for the scattered potential to reduce the potential barrier of the probe

lattice and facilitate runaway self-organization. This is supported by the observation

that organization again becomes rapid if we probe further above threshold. This is

demonstrated by the traces shown in Fig. 4.19. In these experiments, the probe lattice

was ramped up to some depth in 200 µs and held at constant power. For the left trace,

the probe lattice was held at depth of 1.5 times the implied threshold condition and we

117


Figure 4.19: Heterodyne traces showing the rate of self-organization for µ∗ ≈ 0.35. In

these experiments, the probe lattice is ramped up to a depth above threshold in 200 µs

and held at constant power.

see the self-organization starts very slowly, building up over tens of milliseconds. If we

probe slightly harder at 2 times the implied threshold condition, as in the right trace,

the self-organization again becomes rapid. This is slightly counter-intuitive because in

the latter case the probe lattice potential is deeper and thus a greater barrier to the

atoms rearranging to the organized phase. On the other hand, the increased scattering

rate results in a larger organizing potential for a given β. The rate of thermalization

into the organized phase will be determined from the combination of both effects. As

the threshold increases, both the effects work against us to dramatically slow down the

rate of organization at threshold.

Regime of µ∗ > 1

In this regime either the dispersive shift is small, or the effective detuning ∆c from the

cavity is large. In either case, we are unable to explore self-organization in this regime

as a consequence of our external potential geometry for the reasons just discussed.

Nevertheless, we can consider what would happen in this regime if the atoms were

free to move in the axial direction. As discussed in Sec. 4.2.1.2, µ∗ ' 1 is exactly the

regime where there are no stable minima at defect sites for the axial potential. With

no potential barrier prior to organization, one would expect the atoms to self-organize

rapidly at threshold, like we observe for |µ∗| < 1 in the transverse dimension.

118


However, there is one complication we observe in our experiments which would

apply to the axial case as well: that the probe lattice itself necessarily increases the

temperature of the atoms by adiabatic compression. Although we confirmed that the

threshold parameter scales linearly as N−1, the optical intensity which was required to

reach threshold did not scale linearly in our experiment because the temperature was

not constant. This is only a problem in the regime µ∗ ≥ 1, where the heating due to

adiabatic compression in significant. In the limit µ∗ 1, the atoms must be deeply

trapped in the probe lattice itself. In this case if we assume the atoms are adiabatically

compressed as the probe lattice is ramped up to threshold, the temperature will scale

with the beam intensity ∝ I1/2. Thus in practice the intensity required to satisfy the

threshold condition will scale quadratically for large µ∗.

4.5.1.1 Comments on temperature

One of our objectives in studying self-organization was to experimentally investigate

cavity cooling. By loading such a large number of atoms in the cavity, we are able

to directly measure the ensemble temperature via ballistic expansion. As a general

observation, self-organization always resulted in an increased temperature of the atoms.

For example, Fig. 4.20 shows the temperatures measured by ballistic expansion after

the probe lattice has been ramped on ( in 100 µs) to a value above threshold and held

at constant power for 0.8 ms. We observe the atoms have a temperature proportional to

the depth of the induced intra-cavity optical potential as one would expect. However,

the hysteresis in the temperature as the scattered potential rises and falls indicates

that the temperature increase is not simply due to adiabatic compression. It suggests

the presence of a strong heating mechanism possibly due to non-adiabatic dynamics

resulting from the rapidly changing potentials.

In regimes where the cavity output indicates the atoms have settled in a quasi-

stable configuration, we observe a slow decay of the cavity output over at most tens of

milliseconds. This is roughly consistent with the heating and atom loss expected from

trapping atoms in a relatively near-resonant FORT. For our experimental observations,

we conclude it is unlikely that self-organization can be a practically useful cooling

method for a trapped ensembles of atoms. We further justify this claim in the closing

chapter in which we discuss cavity cooling schemes for ensembles in further detail.

119


Figure 4.20: (left) Measured temperature after self-organization by probing above thresh-

old at constant power for 0.8 ms plotted against the depth of the scattered field in the cavity

as determined by the cavity output at the time of measurement. (right) Two example traces

of the cavity output scaled the peak output corresponding to data points in from the left

plot. For red line (circles) the cavity output was above 50% of its peak at the measurement

time, whereas for blue line (circles) the transmission was already below 50% and falling.

For reference, an intra-cavity photon number of 4 × 106 corresponds to a 530 µK deep

intra-cavity optical lattice potential (∆ = −265 GHz).

4.5.2 Traveling wave geometry

For the traveling wave probe geometry, the dynamics above threshold are complicated

by the fact that the induced organizing potential is out of phase with atoms’ positions

by θ = tan−1(−κ/∆c), as discussed in Sec. 4.2.2. When |∆c| . κ, this results in

the organization switching off as the atoms are pushed away from the axial center

of the cavity mode, decreasing the coupling to cavity and increasing the threshold

condition. Presumably as the atoms move back into the cavity mode, self-organization

is re-initiated, resulting in the observed pulsing output seen in Fig. 4.21. The left side

shows the cavity output for a 10 ms sweep of the probe intensity through threshold. The

right side shows the typical pulsing output captured on a high bandwidth oscilloscope,

where we see the features are only a few microseconds long. Note that the cavity output

is much lower than in the lattice case because the configuration is not stable and the

atoms never become fully organized.

The organized phase does approach stability in the limit that |∆c| κ. Here the

phase θ ≈ 0 and a small displacement in the transverse Gaussian potential of the FORT

120


Figure 4.21: (left) Trace of cavity output power (heterodyne measurement) while ramping

the depth of the traveling wave over 10 ms. The ∆d and ∆c which were measured prior

to the trace are given in the plot, with the implied threshold µ from Eq. 4.18. The atomic

detuning is ∆ = −265 GHz. (right) Pulsing cavity output cavity as seen on a fast timescale.

Figure 4.22: SPCM traces of the cavity output for a traveling wave probe. Both traces

are for a 10 ms sweep time and at the atomic detuning ∆ = −110 GHz.

121


leads to a stable configuration where the atoms are located at the potential minima. In

Fig. 4.22 (left) we observe stable output from the cavity for several milliseconds. Note

that unlike the previous heterodyne traces, these traces are from the SPCM because

of the weak signals involved. The SPCM saturates at roughly an intra-cavity photon

number of 800 and thus the cavity output may not be as stable as the left trace suggests.

As |∆c| decreases, we observe the cavity output again becomes unstable, as seen in right

side of Fig. 4.22.

4.6 Summary

In summary, we have systematically measured the threshold scaling for self-organization

and analyzed the dynamics above threshold for two probing configurations over a wide

range of system parameters. We have shown in both cases the measured threshold has

an N−1 scaling and agrees well with a model based on simple mean field considerations.

Above threshold, the cavity output is indicative of complex and non-linear dynamics

resulting from the optomechanical forces driving self-organization. We have attempted

to qualitatively explain these dynamics by considering the inter-dependence of the

relevant parameters in different regimes.

122

Chapter 5

Bragg Scattering, Cavity

Cooling, and Future Directions

In the closing chapter, we present preliminary data of Bragg scattering into a cavity

mode, discuss the practicality of cavity cooling atom ensembles, and consider possible

future directions with the current experimental apparatus.

5.1 Introduction

By adding an additional 1560 nm FORT lattice transverse to the cavity along the

probe direction, the combination of the intra-cavity and transverse 1560 nm FORTs

organizes the atoms into a Bragg crystal. In this configuration, coherent scattering into

the cavity mode from a transverse probe field is enhanced by constructive interference

for any probe strength. We verify the power scattered into the cavity scales with the

number of scatterers as N2, a signature of superradiance.

This experimental configuration is particularly suited for investigating cavity cool-

ing via coherent scattering from a transverse probe. By avoiding the complex dynamics

of driving the system to self-organization, we can study coupling to the collective mo-

tion in a controlled way. In particular, our cavity is designed to be in the side-band

resolved regime where the cavity linewidth is less than the typical axial trap frequency

(2κ < ωz). Additionally, the single atom cooperativity is greater than unity (C > 1),

potentially enabling ground state cooling of the collective motion [126, 127]. A recent

experiment [25] at MIT has demonstrated cavity cooling of a collective mechanical

123

5. BRAGG SCATTERING, CAVITY COOLING, AND FUTUREDIRECTIONS

mode nearly to the theoretical limit when driving the cavity in the regime 2κ ≈ ωz and

C < 1. Our configuration should allow us to investigate ground state cooling of the

collective mode for either a driven cavity as in Ref. [25] or transverse pumping.

Since we have sufficiently many atoms to directly measure the ensemble temperature

by ballistic expansion, we hoped to demonstrate cavity cooling of a thermal atomic

ensemble in the dispersive regime, as suggested in Ref. [25]. This would be an important

proof of principle step towards the realization of practically useful cooling for a more

general class of polarizable particles [34]. Unfortunately we have thus far been unable

to observe any cooling, or even reduced recoil heating, in either the driven cavity or

transverse configurations. In this chapter we will discuss cavity cooling in a variety of

configurations and conclude that, in fact, the cooling rate can not be expected to scale

better than N−1. Thus for large ensembles (N > 105), for which we can measure the

temperature, we believe impractically long cooling times can not be avoided and do

not expect to observe cavity cooling of the ensemble temperature.

The chapter is ended with a brief overview of other potential research directions

with the current experimental apparatus.

5.2 Bragg scattering

Instead of driving the system to self-organize, we trap the atom in a Bragg crystal

configuration using two 1560 nm FORT lattices, one intra-cavity and one transverse to

the cavity along the probe direction. The lattice transverse to cavity is generated using

a 10 W fiber amplifier1 seeded by the same 1560 nm laser to which the cavity is locked.

The high power 1560 nm beam is focused to an elliptical spot at the position of atoms

in the cavity with waists of [50, 110] µm, where the 110 µm waist is aligned to the cavity

axis, and retroreflected to form an optical lattice. Due to technical problems with the

amplifier, the beam has only 1.2 W of optical power at the experiment chamber instead

of the 6-7 W we anticipated. Consequently, the transverse lattice only has a depth of

38 µK, as compared to the depth of the intra-cavity FORT lattice, 240 µK, and the

temperature of the atoms, 33 µK. Although the transverse lattice is not deep enough

to well localize the atoms, it does modulate the density distribution in the transverse

dimension which is sufficient for enhancing Bragg scattering into the cavity.

1NuFern PSFA-1550-01-10W-2-2

124


Figure 5.1: (left) Cavity transmission traces scaled to intracavity photon number for

a frequency sweep of a transverse probe beam (∆ = −265 GHz, Ω = 2π × 210 MHz,

g = 0.8 MHz, α = 0.65). Four selected traces that different numbers of are shown.

(right) Peak intracavity field when probing on the dispersively shifted resonance versus the

dispersive shift. Black line is calculated from Eq. 5.1 for β = 0.07 and ∆c = NU0α.

For transverse pumping with a traveling wave probe, we see from Eq. 4.7 that the

steady-state intracavity photon number n, is given by

n = |λ|2 = N2

(Ωgβ

∆

)2 1

(∆c −NU0α)2 + κ2. (5.1)

The 2D 1560 nm lattice ensures that β is non-zero and thus there is scattering into

the cavity even for probe intensities much less than required for self-organization. As

long as the probe and scattered field intensities are sufficiently weak that back action

on the density distribution is negligible, we can assume β is constant and the cavity

transmission will depend linearly on the probe intensity. From Eq. 5.1, we see that

as a function of probe-cavity detuning the scattering into the cavity is maximal at

the dispersively shifted cavity resonance (∆c = NU0α) and has a Lorentzian lineshape

given by the cavity. This is can be observed on the left side of Fig. 5.1, which shows

the intracavity photon number inferred from the cavity output as the frequency of the

transverse probe is swept over the cavity resonance. Here the probe intensity is far

below the threshold for self-organization and we can assume the back-action on the

density distribution due to the probe and scattered fields to be negligible. Sample

traces for several values of the dispersive shift are shown.

125


Figure 5.2: Steady steady intra-cavity photon number for at various fixed ∆c as func-

tion of dispersive shift (atom number). Lines are calculated from Eq. 5.1 for β = 0.07.

Data points are extracted from traces as shown in Fig. 5.1(left). The black line is the

measurement noise floor.

On the right side of Fig. 5.1, the peak intra-cavity photon number from each trace

is plotted against the dispersive shift. The black line is a fit to Eq. 5.1 for ∆c = NU0α

and β as a free parameter. The fit yields β = 0.07, which is quite a bit lower than unity

because the atoms are not fully localized by the transverse lattice1. Nevertheless, the

N2 scaling consistent with superradiant scattering is clearly observed.

In Ref. [128], the authors describe a phenomenon referred to as ‘suppression of Bragg

scattering’ which is readily observed in our data. The authors of Ref. [128] theoretically

consider atoms pinned into position for Bragg enhanced scattering into a cavity from a

transverse pump; exactly the configuration we have effectively realized experimentally.

For a fixed ∆c less than the dispersive shift and fixed probe strength, they point out

that as the particle number N is increased, the scattered field tends to a constant

independent of N . Specifically, in the limit |NU0α| |∆c|, κ, the intensity of the intra-

cavity field (from Eq. 5.1) is given by(

Ωgβα

)2, which is independent of N . This can be

observed in Fig. 5.2 where the intra-cavity photon number given by Eq. 5.1 is plotted

for several value of fixed ∆c as function of dispersive shift (atom number). Intuitively

we can interpret these curves in terms of the dispersive shift moving the cavity into or

away from resonance with the probe. To illustrate, let us consider the power scatter for

1ab intio calculation of β for the thermal distribution (η = 7) for our trap parameters yields β ≈ 0.2.

We are not yet sure what is the cause this discrepancy with the observed value of β ≈ 0.07.

126


fixed ∆c = −κ as we add atoms to the cavity. Initially the scattering increases as the

cavity is dispersively shifted in resonance with the probe. As more atoms are added,

the cavity is dispersively shifted out of resonance, which, in the tail of of the Lorentzian,

suppresses scattering as ∝ N−2. However, since the height of the Lorentzian increases

as ∝ N2 due to superradiance, the net effect is scattering independent of N in the

regime |NU0α| |∆c|, κ. This can also be observed in Fig. 5.1(left) where the tails of

the Lorentzian profiles overlap independent of dispersive shift.

5.2.1 Future Bragg scattering related experiments

Once the 1560 nm fiber amplifier has been repaired to produce the specified 10 W, we

will be able to localize the atoms tightly in a 2D lattice, increasing the Bragg enhanced

coupling by an order of magnitude (β = 0.07 → 0.7). In addition to better data than

shown in Fig. 5.2 demonstrating the ‘suppression of Bragg scattering’ effect, it would be

interesting to show the associated suppression of free space scattering [128]. While it is

probably not feasible to measure this directly, it could be inferred indirectly through the

rate of spontaneous Raman processes pumping the hyperfine ground state populations,

which can be easily measured.

In the previous chapter on self-organization, we saw that the transverse lattice

potential for a traveling wave probe, which results from interference between the cavity

and probe fields, is out of phase with the position of atoms by θ = tan−1(−κ/∆c). This

was the most important factor governing the dynamics and resulted in a transverse

force on the atoms which is maximal for ∆c = 0. For the Bragg scattering data just

presented, the ‘self-organization’ potential was small relative to the transverse FORT

and would have resulted in only a small displacement in the FORT lattice. In principle,

the phase of the cavity output field relative to the probe provides a precise measure of

the transverse position of the atoms such that even a small displacement of the Bragg

lattice of atoms could be measured. By using the FORT as a variable stabilizing force

and the phase as a measure of the displacement, we will be able to more quantitatively

study the transverse forces due to the phase shifted ‘self-organization’ potential in the

traveling wave probe geometry.

Lastly, with the atoms tightly confined to the 2D lattice and trap sites positioned

at nodes of the cavity mode, we can investigate resolved-sideband cavity cooling for

127


transverse pumping. This brings us to our discussion of cavity cooling of atomic en-

sembles.

5.3 Cavity cooling of atomic ensembles

For a recent review of both theoretical and experimental research on cavity cooling

of single particles and ensembles the reader is referred to Ref. [129]. Cavity cooling

of single particles is well established [33] and has been demonstrated both with blue-

detuned light [20] and red-detuned light [130]. This cooling is a consequence of frictional

forces resulting from the time-delayed action of the electric dipole force on the atoms in

the cavity. A fundamental result is that the steady-state temperature is limited only by

the cavity linewidth, kbT ≈ ~κ [131], even to below the recoil temperature, kbT ≈ ~ωR,

unlike standard free-space laser cooling.

For many atoms in the cavity, the situation is complicated by the fact that the

potential depends on the position of all the atoms, and thus the friction force felt by

one atom is correlated to the motion of other atoms. Ref [132] showed, in a detailed

study for two atoms, that these correlations result in a reduced efficiency of the cavity

cooling. Thus, one might expect, for large ensembles, that cavity cooling effects decrease

further in efficiency or even become washed out. However, it has been argued [32] that,

in the regime of small dispersive shift, NU0 < κ, cooling is possible since, for an

individual atom, changes in the cavity field are correlated only to its own motion while

the influence of the other atoms averages to zero. The simulations of cavity cooling in

this regime [32] confirm atoms are cooled independently to the ~κ limit and the rate

of total kinetic energy dissipation is independent of atom number. Unfortunately this

implies that the rate of kinetic energy dissipation per atom scales as N−1, and thus

impractically long cooling times would be required for large ensembles. The cooling

rate is thus the most significant bottleneck to realization of cavity cooling for atomic

ensembles.

A current focus of research is to utilize collective enhancement of the coherent

scattering to achieve a better scaling and more efficient cooling. Here we discuss two

paradigms for cavity cooling which employ collective scattering: sideband cooling of a

single collective phonon mode in the tightly confined regime [25], and cooling associated

with self-organization [36, 38, 43].

128


5.3.1 Cavity cooling via the collective mode

In this section we establish that, in the tight confinement regime, the cavity couples at

an enhanced rate only to a single collective mode of the atomic ensemble. We discuss

the scaling for cooling the collective mode, and, sympathetically, all modes of an atomic

ensemble.

5.3.1.1 Coupling to the collection motion for transverse probing

We first establish that, for our experimental configuration, the cavity couples only

to the common collective vibrational mode with a rate enhanced by the factor√N .

We consider N atoms coupled to a cavity mode with the position dependent rate

g(z) = g0 sin(kz), where k = 2π/780 nm. The atoms are trapped in a 2D FORT lattice

with wavenumber kT = k/2 = 2π/1560 nm and trap sites located at the nodes of the

cavity mode. The atoms are harmonically trapped with the frequencies ωz and ωx,

which we assume to be equal. The atoms are probed transverse to the cavity by a

classical field Ω(x) = Ω0eikx. For the purposes of this discussion, we will consider only

the motion of the atoms along the cavity axis. At large atomic detunings, ∆, we can

adiabatically eliminate the excited state and write the Hamiltonian for this system as

H = ∆ca†a+

N∑j=1

ωzb†jbj +

N∑j=1

(g0 sin kzj)(Ω0eikxj )

∆(a+ a†), (5.2)

where bj and b†j are the ladder operators for the vibrational excitation of the jth atom.

The confinement by the external potential is characterized by the Lamb-Dicke param-

eter, η = k√~/2mωz. We assume the Lamb-Dicke regime η 1, where the trap

vibrational energy ~ωz exceeds the recoil energy Erec = (~k)2. Coupling to the atomic

motion results from the momentum transfer of scattering of light into the cavity. In

the Lamb-Dicke regime the scattering spectrum consists of three components to first

order [2]: carrier transitions which do not change the vibrational excitation number,

and sideband transitions which change the excitation number by ±1. Since every trap

site is positioned at the nodes of the cavity, the coupling to the z-motion, to first order,

is sin kzj ≈ kzj = η(bj + b†j). Note that, due to the symmetry of g(z), carrier tran-

sitions are not allowed and only sideband transitions are coupled. In the transverse

dimension, the trap sites are coupled with identical phase to the classical field due to

129


the periodicity of our FORT lattice and therefore Ω0eikxj ≈ Ω0. Thus the Hamiltonian

reduces to

H = ∆ca†a+

N∑j=1

ωzb†jbj +

N∑j=1

ηg0Ω0

∆(bj + b†j)(a+ a†). (5.3)

Instead of describing the motion as N harmonic oscillators (b†j , bj), associated with

the individual atoms, we can equivalently describe the system in term of N collective

phonon modes with operators (c†j , cj). It is well established that, in the Lamb-Dicke

regime, the cavity couples only to a single collective mode of the atomic motion [25,

133, 134]. In our system, it is clear from Eq. 5.3 that this collective mode is exactly

the common center of mass mode, ccom ≡ c1 = 1√N

∑Nj=1 bj .

1 We can thus write the

Hamiltonian in terms of the single collective mode

H = ∆ca†a+ ωzc

†comccom + η

√Ng0Ω0

∆(ccom + c†com)(a+ a†), (5.4)

where we have dropped the N−1 terms ωzc†jcj representing the energy in the other col-

lective modes since they are decoupled from the cavity. This Hamiltonian is that simply

two harmonic oscillators, the collective phonon mode and the cavity field, coupled at a

rate collectively enhanced by√N .

5.3.1.2 Sideband cooling of the collective mode

Cooling arises from the differential rates of scattering on the heating and cooling side-

bands when cavity is detuned from the pump field. Given that the Hamiltonian of

Eq. 5.4 is effectively that of a single oscillator coupled to the cavity, we can draw on

existing results from similar Hamiltonians [127, 133]. In particular, the Refs. [25, 133]

consider sideband cooling an atomic ensemble in a driven cavity. For a driven cavity,

the Hamiltonian is essentially the same as Eq. 5.4 except the coupling comes in as g2(z)

instead of g(z)Ω(x). The Refs [25, 133] consider atoms tightly trapped in a lattice with

wavelength incommensurate with cavity mode. It is shown that in this case the cavity

also only couples to single collective mode, although it is not the center of mass mode.

Our 1560 lattice allows us to position the trap sites such that the coupled collective

mode is exactly the center of mass mode. For a driven cavity, this corresponds to

positioning the atoms half-way between the nodes and anti-nodes of the cavity mode.

1The 1/√N factor is for normalization and preserves the commutation relation [cj , c

†j ] = 1.

130


In either case, transverse pumping or a driven cavity, the physics of cavity sideband

cooling the collective mode is essentially the same and thus we start our discussion

from the result of Ref [133].

Using methods developed in Ref. [135], the authors of Ref [133] derive a rate equa-

tion for heating/cooling of the collective mode. For a cavity detuned to the cooling

sideband, ∆c = −ωz, the cooling rate of the collective mode is given by [133]1

d

dtEcom = Nnκ

(g2

0

∆κ

)22η2

1 + ( κ2ωz

)2

[(κ

2ωz

)2

~ωz − Ecom

], (5.5)

where Ecom = ~ωz⟨c†comccom

⟩is energy in the collective mode. Thus the cooling rate

of the collective mode is

Rcom = Nnκ

(g2

0

∆κ

)22η2

1 + ( κ2ωz

)2. (5.6)

In the experiment reported in Ref. [25], the cooling rate of the collective mode was

verified to scale linearly with N for atom numbers in the range 103 − 104.

5.3.1.3 Sympathetic cooling of all modes

We are more interested in cooling the ensemble, and thus all N modes2, not just the

collective mode. Other modes are only cooled sympathetically via mixing with the

collective mode. For now, we will assume the best case scenario of ‘strong mixing’,

meaning that all modes thermalize to the same temperature, 〈b†b〉 ≈ 〈c†comccom〉, on a

fast timescale compared to the cooling rate of the collective mode. In this case, we

can say that average thermal energy per atom, ET = ~ωz〈b†b〉, is cooled at the rate

Rcom/N . In order to find the fundamental cooling limit, we must include heating which

results from scattering into free space. The free space scattering per atom is given by

the rate [133]

Rfs = nκ

(g2

0

∆κ

)22Erec

C0, (5.7)

1This is Eq.19 from the supplemental material of Ref. [133] for ∆c = −ωz.2For purposes of this discussion, we consider only the N oscillator modes in the axial dimension.

Of course there are 3N modes if one includes the transverse dimensions. This does not significantly

impact our discussion or conclusions.

131


where C0 = g20κ/γ is the single atom cooperativity1. Adding Rfs to Eq. 5.5, we find

the rate equation for the energy per atom ET:

d

dtET =

Rcom

N

[(κ

2ωz

)2

~ωz +~ωzC0

(1 +

(κ

2ωz

)2)− ET

]. (5.8)

From Eq. 5.8, the fundamental limit of cavity sideband cooling for the average excitation

per atom is found to be

〈a†a〉∞ =1

~ωzlimt→∞

ET =

(κ

2ωz

)2

+1

C0

(1 +

(κ

2ωz

)2). (5.9)

Note that Eq. 5.9 is consistent with the single atom cooling limit found in both Ref. [126]

and Ref. [127]. From Eq. 5.9, it is apparent that two conditions are required for ground

state cooling. First, the cavity must resolve the vibrational sidebands, κ < ωz. Second,

the cavity must be ‘good’ (C0 1) in order to suppress the heating contribution from

free space scattering. Using typical parameters for our experiment (κ/2π = 70 kHz,

ωz/2π = 140 kHz, C0 = 5 × 23 = 3.3), one finds the limit 〈a†a〉∞ ≈ 0.3 and thus near

ground state cooling is in principle possible with our experimental parameters.

5.3.1.4 Scaling of cooling rates

In order to determine the timescale for cooling large ensembles, we must consider the

scaling of the per atom cooling rate, given by

RT =Rcom

N= κ

(Ng2

0

∆κ

)2n

N2

2η2

1 + ( κ2ωz

)2. (5.10)

Ostensibly, this cooling rate is independent of particle number N , but we have expressed

the rate in this form to highlight a few key points. The quantity(Ng2

0∆κ

)is the ratio of

the dispersive shift to the cavity linewidth. In practice, it is difficult to operate in the

regime of large dispersive shift because fluctuations in atom number leads to significant

changes in the effective detuning from the dispersively shifted cavity. Thus for scaling

purposes, we assume(Ng2

0∆κ

)is held constant by increasing the atomic detuning to kept

the dispersive shift manageable, Ng20/∆ ≈ κ. Consequently, to maintain a cooling

rate independent of N , the pump rate must scale as n ∝ N2 ∝ ∆2. Keeping (n/∆2)

1We have assumed the atoms are trapped between the nodes and anti-nodes, i.e. g = g0/√

2, for

consistency with Refs. [25, 133] and thus included a factor of two in Eq. 5.7 to account for the actual

cooperativity being C = C0/2.

132


constant is equivalent to a constant excited state population, and thus constant recoil

heating rate per atom. The final term in Eq. 5.10 can simply be viewed as the modified

mechanical coupling. The argument is that cooling of one collective mode at the rate

enhanced by N leads to sympathetic cooling of all N modes, such that the per atom

cooling rate RT is independent of N . Thus the cooling efficiency for a thermal ensemble

approaches that of a single atom which is limited only by the quality of the cavity, C0,

and the resolution of the vibrational sideband, κ/ωz.

However, this picture of N independent scaling breaks down for two reasons (1)

the cooling rate of the collective mode has a maximum limit of Rcom . ωz, and (2)

the ‘strong mixing’ requirement further constrains Rcom. Just as for conventional side-

band cooling, the Raman rate must be less than ωz in order to resolve the vibrational

sideband. This implies that even for ideal mode mixing, the per atom cooling rate is

limited to RT . ωz/N . For typical parameters of our experiment (ωz/2π = 140 kHz,

N = 3 × 105), this implies the best possible cooling rate is ≈ 0.5 Hz, which is far too

slow.

The other consideration is the mode mixing rate. By conservation of momentum,

elastic collisions between individual particles alone do not change the center of mass

motion of the ensemble and thus do not mix the common mode with other collective

modes1. Mode mixing results from atoms dephasing with the common mode when

exploring the anharmonicity of the potential. We can estimate the mixing rate from

the time required for the thermal atoms to become out of phase with the common

mode frequency by 180. For an intra-cavity lattice potential U(z) = U0 cos2(kT z) with

harmonic trap frequency ωz, one finds the period of one oscillation is approximately

Tosc ≈2π

ωz

(1 +

(ktzm)2

4

)(5.11)

for a displacement amplitude zm from a potential minimum. For atoms with trap

depth to temperature ratio η′ = U0/kbT , the rms radius of the thermal distribution is

z = kT /√

2η′. Taking this as the average displacement amplitude, we find

ωzTosc

2π= 1 +

1

8η′. (5.12)

1This is true even for atoms trapped in an incommensurate lattice as in the experiment of Refs. [25,

133]. Although the coupled collective mode is not the center of mass mode for whole ensemble, within

each trap site only the center of mass mode of atoms in that site contributes to the collective coupling.

133


The average thermal atom becomes completely out of phase with the common mode in

4η′ cycles and thus we define a mode mixing rate γm = ωz/4η′. Note that the mixing

rate decreases for large η′ because the bottom of the potential is more harmonic. For

a perfectly harmonic potential, there would be no mixing of the common mode with

other modes.

We thus expect efficient sympathetic cooling of all modes requires Rcom . γm,

which places a further limitation on the per atom cooling rate, RT . ωz/(4Nη′).

We can conclude that, in practice, the scaling for cavity sideband cooling of a large

thermal ensemble is ∝ N−1. One can view this as a fundamental limitation of cooling

N oscillators via coupling to a single oscillator mode.

5.3.2 Cavity cooling via self-organization

The first theoretical paper on self-organization in a linear cavity by Ritsch and Domokos [36]

suggested the collective effects may provide a fast and efficient means of cooling large

ensembles. In 2002, Vuletic et al. reported cooling and center-of-mass deceleration

of falling atomic ensemble (N ≈ 106) resulting from self-organization and collective

scattering into a cavity [37, 38, 136]. Several theoretical papers [34, 35, 43, 137] have

further investigated cooling associated with self-organization since then. In this section

we discuss the possibility of cooling in three regimes: in the organized phase, crossing

the phase transition, and below threshold.

5.3.2.1 The organized phase

First we consider the coupling to motion when the atoms are already self-organized

and thus well-localized to less than a wavelength, i.e. the Lamb-Dicke regime. Since

the atoms are located at the anti-nodes, we expand the coupling to the motion as

cos(zj) ≈ 1 + 12(kzj)

2 = 1 + η2

2 (bj + b†j)2, noting there is no first order coupling to

the motion. For this case the interaction part of the Hamiltonian, after adiabatic

elimination of the excited states, can be written

HI =

N∑j=1

g0Ω0

∆(1 +

η2

2(bj + b†j)

2)(a+ a†) (5.13)

= Ng0Ω0

∆(a+ a†) +

η2g0Ω0

2∆

N∑j=1

(b2j + (b†j)2 + bjb

†j + 1)(a+ a†), (5.14)

134


where, as before, we have only considered the coupling to the motion along cavity axis.

From Eq. 5.14, we see the coupling to the motion occurs only through second order

processes. Although all modes are coupled to second order, the rate is not collectively

enhanced and additionally suppressed by the factor η2. By far, the dynamics are

dominated by carrier transitions which couple at a rate enhanced by N . This results in

the superradiant scattering which sustains the intra-cavity field organizing the atoms

and, as carrier transitions, neither heat nor cool the atoms. Thus in the self-organized

phase, we expect to observe only residual recoil heating from scattering into the free

space. This is consistent with observations in our experiment, although in practice

it is difficult to isolate recoil heating from the heating associated with the onset of

self-organization.

5.3.2.2 Crossing the phase transition

In Ref. [43], the authors simulate self-organization for a small number of atoms and

report an increase in phase space density crossing the phase transition. Because the

temperature necessarily increases due to compression when crossing the phase transi-

tion, in Ref. [43] phase space density is used as a measure for the cooling. Thus the

increase in phase space density, which is observed to be independent of N , is suggestive

of scalable cavity cooling [43].

As an alternative interpretation, we show that the increase in phase space density

can be attributed to the change in entropy associated with crossing the phase transition.

First, we note that phase space density can increase even for adiabatically varying

potentials if the density of states changes. This is beautifully demonstrated in the

experiment of Ketterle et. al. [138] which reversibly crossed the BEC phase transition

simply by altering the form of the confining potential. Adiabatic change is defined by

constant entropy S, and is the natural quantity to consider here, in particular since a

phase transition is involved.

We would describe the self-organization process by the phase diagram shown in

Fig. 5.3. Prior to organization, the atoms are trapped in a external potential which

is weakly confining and approximately harmonic. After organization, the atoms are

well localized to lattice sites which, individually, are also approximately harmonic.

Thus both the initial and final configurations of the system are harmonic for which the

entropy is functionally related to the phase space density, S = SHO(ρ). Because there

135


Figure 5.3: Entropy phase diagram for

self-organization.

Figure 5.4: Entropy change and phase

space density increase as a function of the

threshold parameter µLT.

is a decrease in entropy associated with the phase transition, it follows that the phase

space density must have increased. By calculating the change in entropy at the phase

transition, we can quantify the increase in phase space density.

We can calculate the change in entropy explicitly for the one dimensional transverse

case. We assume the atoms are tightly confined to anti-nodes along the cavity axis so

that we can neglect this dimension. This simplifies the problem since the dispersive

shift is constant. By expanding the self-consistency equation in orders of β, we find the

change in entropy at the phase transition by taking the limit µ → µ∗ from above and

below threshold. The details of the calculation can be found in Appendix C. For this

1D transverse case, we find that the change in entropy at the phase transition is given

by

∆S

kb=

−3(

(µ∗)2

µLT+ ( µ∗

µLT)2 − µ∗

2

)3 µ∗

µLT− 2

(µ∗

µLT

)2 (1− 1

µ∗ + 1µLT

) . (5.15)

In limits µLT 1 and µLT 1, Eq. 5.15 is approximately

∆S

kb≈

−(2 + µLT

6 ) if µLT 1

−1.5 if µLT 1. (5.16)

Far above threshold (µ → ∞, β → 1), one finds the resulting increase in phase space

136


density ρ is simply given by (see Appendix C)

ρf

ρi= exp

[−∆S

kb

]. (5.17)

Thus we expect the phase space density to increase by a finite amount, less than one

order a magnitude depending on the exact value of µLT (see Fig. 5.4), independent of

particle number. This is consistent with the simulation [Fig.4] of Ref. [43] for which

the phase density decreased by the same factor for both atom numbers simulated. The

fact that the trajectory in phase space was the same, only with less fluctuations for

the larger atom number, is consistent with our interpretation of the increase in phase

space density as a mean field phenomena. If our interpretation is correct, the practical

utility of a small finite gain in phase space density as a cooling mechanism is doubtful.

One might imagine doing an experiment similar to that of Ketterle’s [138] whereby a

cold gas is prepared near to the critical temperature for Bose-Eisenstein condensation

but self-organization is used for the last bit of ‘cooling’ to cross the threshold. Such an

experiment may be complicated in practice because, as we have seen, there is typically

excess heating due to non-adiabatic dynamics during self-organization.

In summary, for systems where a thermodynamic description is valid, the mean field

approach describes the essential characteristics of the self-organization phase transition:

both the threshold and the increase in phase space density. The fixed increase in

phase space density represents the best possible outcome assuming adiabatic dynamics.

However, observing this phase space density gain in real systems will be difficult, since

above threshold a thermodynamic description is often not valid as a result of rapidly

changing potentials. Although quite interesting, the drop in entropy associated with

the phase transition does not appear to be a means for scalable cooling.

5.3.2.3 Below threshold, dissipative cooling

Recently, dissipative cooling below the self-organization threshold in the regime of

very small dispersive shift (NU0 κ) has been proposed [44]. Below threshold, the

atoms are unorganized and coherent scattering into the cavity is suppressed. Thus it is

somewhat counterintuitive to expect efficient cooling in a regime where both scattering

is low relative to the pump strength and coupling to the cavity is weak (NU0 κ).

Nevertheless, it is argued in Ref. [44] that efficient cooling is possible in this regime.

Assuming their result is correct given the model assumptions, we note that the reported

137


optimal cooling time still scales linearly with N . Also, from a practical perspective, we

note that the parameter regime considered is extreme. The parameters simulated in

Ref. [44] are a dispersive shift of NU0 = κ/100 at the optimal detuning ∆c = −κ. From

the threshold Eq. 4.11, this implies µ∗ ≈ 100, that probe lattice have depth 100 times

the temperature of the atoms. Considering the heating which results from adiabatic

compression as discussed in Sec. 4.5.1, reaching the regime µ∗ ≈ 100 will generally

require extremely high laser power where recoil heating can not be neglected.

5.3.3 Conclusions and future experimental directions for cavity cool-

ing

In conclusion, self-organization is unlikely to be an effective means of cooling a trapped

ensemble in a cavity because (1) the self-organization process is not well controlled, (2)

in the best case scenario one can only expect a finite gain in phase space density from

the phase transition, and (3) once localized to the antinodes, the motion of the atoms is

no longer coupled to first order. This is not inconsistent with the results of the earlier

experiments [37, 136]. There the MOT atoms were falling through the cavity and, as

result of self-organization, ordered into a Bragg grating. As the grating fell through the

cavity nodes/anti-nodes, the center-of-mass mode alone is strongly coupled and thus

decelerated by scattering at the collectively enhanced rate. The earliest experiment [37]

also reported a decrease in the temperature of the decelerated atoms. This cooling was

later understood as result of Raman processes [136] and was only observed for near-

detunings. In a later experiment further in the dispersive regime, only coupling to the

center of mass mode was observed [38].

For collective cooling of trapped ensembles, the optimal configuration would appear

to be the one realized by our experiment: atoms trapped in a 2D λ-lattice and posi-

tioned at the cavity nodes. Here the phase transition to the self-organization phase

is unnecessary, Bragg scattering (∝ N2) on the carrier is avoided, and the enhanced

scattering (∝ N) on the vibrational sideband is the dominate process. However, since

the collective enhancement goes hand in hand with coupling to a single collective mode,

this cooling method has limited scalability for cooling all modes in a large ensemble.

Given the large number of atoms (N > 105) in our experiment, it is not surprising

we have been unable to observe any cooling of the ensemble temperature as measured

by time of flight in our experiments. Nevertheless, our experimental configuration is

138

5.4 Future directions

|0〉

|1〉

|r〉|s〉

gs

gr

Ωr

Ωs

ωHF

∆s

∆r

Figure 5.5: Proposed realization of the Dicke model [139]. Classical laser fields (Ωr, Ωs)

and interaction with the cavity mode (gr,gs) forms Raman couplings between the hyperfine

ground states separated by ωHF/2π = 6.8 GHz.

ideal for cooling the center of mass mode, in principle, to the ground state given our

system parameters. In order to investigate this, we plan to measure the excitation of

the collective mode directly using the methods of Ref. [25]. Excitation of the collective

motion leads to a fluctuation of the cavity dispersive shift which can be detected via

the cavity transmission. The method of Ref. [25] is essentially to perform spectral

analysis on the cavity output to infer the excitation number of the collective mode.

Once we are able to observe the excitation of the collective mode, we will be able to

experimentally investigate the limits of sideband cooling of the collective motion. For

smaller ensembles (N < 103), it may also be possible to cool all modes sympathetically

on a reasonable timescale.

5.4 Future directions

We conclude with a brief summary of other potential research directions for current

experimental apparatus. Especially promising is the proposed realization of the Dicke-

model via cavity-mediated Raman processes [139], shown schematically in Fig. 5.5.

This system realizes an effective Dicke Hamiltonian [41] for which the parameters of

the Hamiltonian can be controlled experimentally via the relative detuning of the lasers

fields and cavity [139]. This proposal take advantage of the collectively enhanced in-

teraction of an atomic ensemble with a single cavity mode to ensure the coupling rates

139


of the effective Dicke Hamiltonian far exceed the dissipation rate due to spontaneous

emission and cavity decay. With our experiment it will be possible to satisfy these

requirements in a practically reasonable parameter regime since we are able to load

such a large number of atoms into our strongly coupled cavity. Realization of the Dicke

model in this way is a potentially rich test bed for investigation of first and second

order quantum phase transitions [140, 141], quantum chaos [142], entanglement [143],

and the dynamical Casimir effect [144].

Additionally, we are currently laying the groundwork for investigation of cavity

QED with a Rydberg-blocked atomic ensemble [145]. As a result of the Rydberg block-

ade phenomena [146, 147], a small ensemble can become an effective two-level system

coupled to the cavity. This is system is described by an effective Jaynes-Cummings

model, and, for strong effective coupling, has a non-linear excitation spectrum (see

Sec. 3.2.1.1). While it is experimentally difficult to achieve sufficiently strong coupling

to detect this non-linearity for a single particle in an optical cavity [14], it should

be readily observable with a Rydberg-blocked ensemble for reasonable experimental

parameters [145].

140

Appendix A

High finesse cavities: technical

details

This appendix provides technical details from our experience building high finesse cav-

ities using mirrors manufactured by Advanced Thin Films (ATF). This information

may be of practical use to other experimentalists.

A.1 ATF mirrors

The finesse of a cavity with identical mirrors is given by F = πL

where L is the total

losses per mirror. These losses are a combination of transmissive loss T, and absorptive

loss A, such that L = T +A. The transmissive loss is determined by the number layers

in the reflective dielectric coating and can be made extremely low, values of ∼0.1 ppm

have been reported [18]. Absorptive losses are the limiting factor to the finesse. The

absorptive loss term includes two components, diffractive losses due surface roughness

of the mirrors and absorptive losses due to point defects in the coating and on surface

of the mirrors.

A.1.1 Brief History of low-loss mirrors

Here we give a brief history of low-loss mirrors to provide context for our experience

with the high finesse mirrors commercially available from ATF. The development of

ultra-low loss mirrors for cavity QED research was spearheaded by Ramin Lalezari at

the company Research Electro-Optics (REO) in collaboration with researchers at the

141

A. HIGH FINESSE CAVITIES: TECHNICAL DETAILS

California Institute of Technology (Caltech). In 1992, they reported an optical cavity

with a finesse of 2× 106 (at 850 nm), the highest quality optical cavity recorded to this

day [148]. The losses of this cavity broke down into T = 0.5 ppm and A = 1.1 ppm.

Throughout the 1990’s REO provided low-loss mirrors for numerous research groups

throughout the world, which were able to build cavities with finesses from 300, 000 to

1× 106 for use in atomic physics experiments.

Around 2002, Ramin Lalezari left REO and co-founded his own company Advanced

Thin Films (ATF). According to the thesis of Tracy Northup (Caltech group) [149],

from 2005-2007 they requested a new batch of high finesse mirrors from both REO and

ATF. Both companies were able to produce mirrors with L ∼ 1.6 ppm at this time.

In this case of ATF, this required several coating runs and active involvement of the

Caltech researchers themselves [149]. It was in 2008 that we placed our first order for

780 nm high finesse mirrors.

A.1.2 Mirrors from ATF: 2008-2011

In 2008, REO informed us they were no longer providing small scale coating runs for

research grade mirrors. This left ATF as the only company which could provides low

loss mirrors. To our knowledge, this is still the case.

2008, 780 nm coating run: We ordered this first coating run with a specification

of T ≈ 15 ppm and absorptive losses on a best effort basis. We ordered several 7.75

mm super-polished substrates with radii of curvature of 2.5 cm, 5 cm, and 10 cm. Half

of these substrates were machined down to 2mm/4mm diameter tapered cone shape

seen in Fig. A.1(a). We understand that ATF outsources this machining process to a

third party after the substrates have been coated. During machining, the coating is

protected by a varnish which is cleaned off afterwards.

We inspected the mirrors under a microscope1 and found that the unmachined

substrates where quite clean with sparse surface contamination. A test cavity built with

these mirrors had a finesse of 150,000 with the loss breakdown of T/A ≈ 17/5 ppm.

The machined mirrors, however, were covered in a residue, presumably left behind by

the spin cleaning process used by ATF after the machining. This residue was easily

removed by cleaning (see A.1.3). After cleaning, an experiment cavity built with the

machined mirrors also had a finesse of nearly 150,000. This suggests the machining

1It is easier see defects on the mirror surface if dark field imaging is used.

142

A.1 ATF mirrors

process in and of itself does not degrade the mirrors, an observation also reported in

Ref. [149]. Of the ≈ 5 ppm absorptive losses, it was not possible to determine whether

these are limited by defects in the coating itself or our ability the clean the surface

contaminates.

2008, broad-band coated utility mirrors: Along with the high finesse mir-

rors, we also ordered broadband mirrors with a reflectivity of 99.7 − 99.9% over the

wavelength range 650− 1064 nm. These mirrors proved extremely useful for reference

and transfer cavities. For these mirrors, A/T 1 and so the total power transmitted

by a cavity is nearly 100%. With careful mode matching, we have measured ∼ 99%

overall transmission though cavities constructed with these mirrors. For this reason

these mirrors also make excellent narrowband filter cavities, which can, for example,

be used to spectrally isolate the frequency sideband of a phase modulated laser.

2011, first coating run for 780/1560 nm mirrors: In 2011 we ordered a dual-

wavelength coating in order to construct an experiment cavity with high finesse at

both 1560 nm and 780 nm. Due to a miscommunication with ATF, they coated the

substrates to as low transmission as possible instead of the T ≈ 10 ppm at 780 nm we

requested. According to their spectrophotometer data, these mirrors have T780 = 5 ppm

and T1560 = 0.3 ppm. Unfortunately, upon inspection we found these mirrors to be

severely contaminated. There were numerous point defect on all the mirrors which

could not be cleaned off with solvents. This suggest the defects are embedded in the

coating. Selecting the cleanest mirrors, an experiment cavity was constructed anyway.

Measurements of this cavity are summarized in Table A.1. Even though the finesse is

relatively high, at both wavelength the losses are predominately absorptive and so the

cavity has very low overall transmission.

2011, second coating run for 780/1560 nm mirrors: ATF reported to us

they had relocated their coating facilities prior to the first 2011 coating run which had

possibly lead to the contamination we observed. They agreed to repeat the coating run,

this time with a transmission specification of T780 = 10 ppm. Upon inspection, these

mirrors also had many visible point deflects, though less than the previous run. Yet

again, most defects could not be removed by cleaning. Typical parameters for a cavity

constructed with these mirrors are given in Table A.2. Even though the finesse is the

same as the previous run, the ratio of A and T is much improved at both wavelengths.

143


Table A.1: First coating run

Parameters at 780 nm

F 160,000

T 3 ppm

A 17 ppm


F 450,000

T 0.5 ppm

A 6.5 ppm

Table A.2: Second coating run


F 160,000

T 9 ppm

A 11 ppm


F 400,000

T 3 ppm

A 5 ppm

Summary

ATF’s capability to produce low loss mirrors appears to have degraded over time. The

defects in the 2011 coatings runs seem particularly serious because they cannot be

cleaned, which suggests the contamination occurred in the coating chamber prior to

the annealing of the coating. We can only speculate on the source of the problems

and have not gotten a clear explanation from ATF. From our perspective, ATF can

no longer reliably provide mirrors better than A ∼ 10 ppm. Possibly with more active

involvement in the coating process, such as going to ATF to provide real-time testing

of the mirrors and immediate feedback, one might get a better outcome.

A.1.3 Mirror handling and cleaning

We use cleaning methods similar to those detailed in Ref. [18]. Under a laminar flow

hood, the mirror is inspected with a microscope using dark field imaging. The mirror

surface is wiped once with a lens tissue wetted with clean ethanol and then re-inspected.

This typically take several attempt until the central ∼ 400 µm region is free of debris

and solvent streaks. Care must taken to not to pull large debris, such as glass particles

from the machined edge of the substrate, across the mirror as this can scratch the

surface.

For the coating runs in 2011, most of the visible defects could not be cleaned, and

generally attempts to clean the mirrors only made them worse. Instead we simply

inspected all the mirrors under the microscope and selected the mirrors with the fewest

defects in the central region for use in the experiment cavity.

144

A.2 Contamination of mirrors by Rb

Figure A.1: Various cavities designs.

A.2 Contamination of mirrors by Rb

This section briefly describes the various cavity designs, seen in Fig. A.1, we have used

and how they were affected by Rb contamination in situ. For the first experiment cavity

Fig. A.1(a), we observed a drop in the cavity finesse which occurred either during bake-

out of the vacuum chamber or setting-up the experiment. We did not perform regular

linewidth measurements on this cavity and so can not be sure if the finesse degraded

at one time or slowly over months. This loss in finesse corresponded to a change in

the absorptive losses per mirror A = 5 ppm → 15 ppm. At the time, we assumed the

degradation was a one off occurrence due to the bake-out.

Our next experiment cavity, using the dual-coated mirrors, again used an open de-

sign which look similar to Fig. A.1(b). This cavity was longer than the first and used

the larger unmachined substrates. For this cavity, we measured the linewidth immedi-

ately before and after baking out the vacuum system and observed no change. However,

a few weeks later after running the Rb source to set-up the MOT and atom transport,

we remeasured the linewidth and there had been a drastic degradation from 130 kHz to

2 MHz. Clearly this was a direct result of Rb contaminating the mirrors. This cavity

was made using the first run of dual coat mirrors and was going to be replaced anyway

because the A/T ratio was not very good even before the contamination.

For the replacement cavity, using the better mirrors from the second dual coat run,

we added a shield as seen in Fig. A.1(c). This was intended to protect the cavity mirrors

from rubidium by eliminating direct line of sight to the source. Considering the first

145


Figure A.2: Cavity construction set-up inside laminar flow hood.

cavity survived prolonged exposure to background Rb vapor, we thought this would

be sufficient. It was not, and this cavity also degraded severely after a week or two of

operating the Rb source.

For the next cavity we switched to the a completely enclosed design shown in

Fig. A.1(d). The steel mount has a 1 mm through hole for the cavity mode. For

this design, any trajectory of Rb atoms to the cavity mirrors would require multiple

surface collisions. Since Rb has a high sticking coefficient, it is unlikely to reach the

mirror surfaces. For this cavity we observed an increase in the cavity linewidth from

110 kHz to 140 kHz after bake-out. Also, the birefringence worsened significantly

after bakeout. This was likely a result of stress on the mirrors from to the Torr seal

epoxy, applied somewhat over zealously on this particular cavity. More importantly,

we regularly measured the linewidth of this cavity and have not observed any further

degradation despite prolonged operation of a vapor pressure MOT above the cavity.

A.3 Cavity construction

For construction of the experiment cavities, we use the optical set-up pictured in

Fig. A.2. The construction is performed inside a laminar flow hood to keep dust

146

A.3 Cavity construction

off the mirrors. An unisolated free-running diode laser is fiber coupled and used as a

tracer beam for alignment of the cavity axis. A custom tweezer holds the substrate and

mechanical stages allow for 5-axes of motion control in positioning the mirrors. The

alignment procedure is as follows:

1. Position the back mirror first, centering the substrate on the beam by imaging

the back of the mirror with a CCD camera.

2. Adjust the back mirror to retro-reflect to tracer beam back into the fiber coupler.

3. Glue the back mirror down to the PZT with a small amount of TorrSeal and leave

to harden overnight.

4. Lift the tracer beam by a fixed amount; center front substrate on tracer beam

using the imaging system and back reflect the beam into the fiber.

5. Return the tracer beam to its original position and lower the front mirror into

position

6. Feedback from the high finesse cavity causes the unisolated laser to latch onto

the most strongly coupled mode of the cavity [150]. The order of the cavity mode

can be determined by viewing the the cavity output with a CCD camera.

7. With minor adjustments to front mirror, the coupling to the fundamental mode

can be maximized.

8. Glue the front mirror with a minimal amount TorrSeal.

Using this method the optic axis of the cavity can be precisely aligned to a tracer

beam and both mirrors well centered. This was particularly crucial in the construction

of the experiment cavity in Fig. A.1(d), for which the optic axis had to be aligned

through 1 mm holes in the mount.

147


148

Appendix B

Self-organization threshold

equations

In this appendix we give a detailed derivation of the threshold equations, Eq. 4.11

and Eq. 4.18, for the lattice and traveling wave respectively. We start by using the

full potential V (x) = VT(x) + VS(x) to determine the density distribution via Eq. 4.1.

From there we find self-consistent equations for α and β via their definitions in Eq. 4.3

and Eq. 4.4, which yields the equations

α =1

Z

∫e−(VT(x)+VS(x))/(kBT )f2(r, z) d3x (B.1)

and

β =1

Z

∫e−(VT(x)+VS(x))/(kBT )f(r, z)h(x) d3x (B.2)

with

Z =

∫e−(VT(x)+VS(x))/(kBT ) d3x. (B.3)

These equations are then expanded to lowest non-zero order in β from which we can

obtain the desired threshold conditions.

From this point we consider the specific external potential VT used in our experi-

ment: an intra-cavity FORT which has a wavelength twice that of the probe and traps

the atoms at every alternate antinode of the cavity. In this case, integration along the

z direction (cavity axis) can be carried out by integrating over single trapping sites of

the FORT lattice potential and weighting each site in accordance with the probability

that the site is occupied. Since the integrands have the same periodicity as the lattice

149

B. SELF-ORGANIZATION THRESHOLD EQUATIONS

potential, the integrations over each site are equal and we need only consider a single

site. This is equivalent to considering all the atoms being located in one site. Fur-

thermore, we assume the atoms to be sufficiently well localized by the FORT potential

that the trapping site can be well represented by its harmonic approximation. Thus we

replace the potential VT by its harmonic approximation

VT =1

2mω2

rr2 +

1

2mω2

zz2

and extend the z integration out to infinity.

B.1 Lattice geometry

For a lattice probe the mode function is h(x) = cos(kx − φ) and the potential Vs is

given by

Vs =~Ω2

∆

[cos2(kx− φ) + 2εβf(r, z) cos(kx− φ) cos(θ)

](B.4)

where ε and θ are determined by Eq. 4.9. Note we have dropped a term proportional

to β2 as we will only be interested in terms up to first order. When we expand Eq. B.1

and B.2 to lowest non-zero order in β we encounter integrals of the form

I =

∫ ∞−∞

e−x2/(2σ2)W (cos(kx− φ)) d3x

for some function W . These integrals can be greatly simplified when kσ 1, which

means the spatial extent of the atomic distribution is larger than the wavelength. In

this case the integrand can be averaged over the wavelength. Denoting the averaging

by 〈〉 we then have

I ≈ 〈W (cos(kx− φ))〉∫ ∞−∞

e−x2/(2σ2) d3x

= 〈W (cos(kx))〉σ√

2π (B.5)

Thus, within this approximation, the phase φ of the lattice potential is irrelevant and

we subsequently set it to zero. The expansions of Eq. B.1 and B.2 can now be readily

evaluated. The zeroth order expansion of Eq. B.1 gives

α =

∫e−VT /(kBT )eµ cos2(kx)f2(r, z) d3x∫

e−VT /(kBT )eµ cos2(kx) d3x

=

∫e−VT /(kBT )f2(r, z) d3x∫

e−VT /(kBT ) d3x(B.6)

150

B.2 Traveling wave geometry

Within the harmonic approximation for VT(x), this expression is readily evaluated to

give

α =1

2

1 + e−4/η

1 + 2/η, (B.7)

where η = VT0/(kbT ) and VT0 is the depth of the FORT potential. Note we have

utilized the fact that the waist of the FORT beam is√

2 larger than that of the cavity

mode associated with the probe wavelength.

For the expansion of Eq. B.2 the zeroth order term is proportional to an integral of

the form ∫ ∞−∞

e−x2/(2σ2)eµ cos2(kx) cos(kx) dx. (B.8)

Within the approximation kσ 1 this term is zero, as one would expect. If the

approximation kσ 1 is not satisfied then the zeroth order term would be non-zero

and β would be non-zero for any value of the probe coupling. This is simply because

the atoms already have sufficient localization to provide some level of scattering into

the cavity for any probe intensity and thus no threshold would exist.

The expansion of Eq. B.2 to first order then gives

β = 2εµβ cos(θ)

∫e−VT /(kBT )eµ cos2(kx) cos2(kx)f2(r, z) d3x∫

e−VT /(kBT )eµ cos2(kx) d3x

= 2εµβ cos(θ)α

⟨eµ cos2(kx) cos2(kx)

⟩⟨eµ cos2(kx)

⟩=

[εµ cos(θ)α

(1 +

I1(µ/2)

I0(µ/2)

)]β (B.9)

A nontrivial solution for β requires the expression within the square parentheses to be

unity and thus gives the required threshold condition, Eq. 4.11.

B.2 Traveling wave geometry

For a traveling wave probe, the mode function is h(x) = e−ikx and the potential Vs

given by

Vs =~Ω2

∆

[2εf(r, z)Re

(β∗ei(kx−θ)

)]=

~Ω2

∆

[2εf(r, z)|β| cos(kx− θ)

](B.10)

151

B. SELF-ORGANIZATION THRESHOLD EQUATIONS

where θ = θ+ arg(β). Note we have dropped the constant term ~Ω2/∆ and, as before,

we have neglected the term proportional to β2 as we are only interested in terms up to

first order.

Taking the zeroth order expansion of Eq. B.1, we get exactly the same expression

for α as for the lattice case. This is not surprising as α represents the averaging of the

dispersive shift due to the spatial distribution of the atoms. Since we assume the spatial

extent in the transverse direction is much larger than the wavelength, this averaging is

unaffected even if a lattice potential is invoked along this dimension.

For the same reason as for the lattice case, the zeroth order term in the expansion

of Eq. B.2 vanishes and to first order we have

β = 2εµ|β|∫e−VT /(kBT )f2(r, z)eikx cos(kx− θ) d3x∫

e−VT /(kBT ) d3x

= 2εµ|β|α⟨eikx cos(kx− θ)

⟩= εµ|β|αeiθ

=(εµαeiθ

)β. (B.11)

Equating the term in parentheses to unity then yields the threshold condition, Eq. 4.18.

152

Appendix C

Self-organization: temperature,

entropy and phase space density

In this appendix we consider self-organization in the one dimensional traverse case

and calculate the temperature T , entropy S, and phase space density ρ, as the probe

lattice intensity is ramped from zero to well above threshold. We assume the atoms

are tightly confined to the anti-nodes along the cavity axis such that the the dispersive

shift does not change as the system organizes (α = 1). This simplifies the treatment of

the problem since we only need consider the self-consistency equation for β.

We consider atoms trapped in the external potential VT = 12mω

2x2, for which the

rms radius of the thermal cloud is σ =√

kbTmω2 . The potential due to the scattered and

probe fields is given by

VSkbT

= −µ(

cos2(kx) +2β

µLTcos(kx)

), (C.1)

where all variables are defined the same as in Sec. 4.2. The partition function is then

given by

Z =1

~

∫ ∞−∞

∫ ∞−∞

e−p2

2mkbT e−x2

2σ2 eµ(

cos2(kx)+ 2βµLT

cos(kx))dxdp (C.2)

=

√2πmkbT

~

∫ ∞−∞

e−x2

2σ2 eµ(

cos2(kx)+ 2βµLT

cos(kx))dx (C.3)

and the self-consistency equation for β (from Eq. B.2) is

β =

∫∞−∞ cos(kx)e

−x2

2σ2 eµ(cos2(x)+ 2β

µLTcos(kx))

dx∫∞−∞ e

−x2

2σ2 eµ(cos2(x)+ 2β

µLTcos(kx))

dx. (C.4)

153

C. SELF-ORGANIZATION: TEMPERATURE, ENTROPY AND PHASESPACE DENSITY

Reformulation of the integrals

First we reformulate the integrals into a more useful working form. With the substitu-

tions u = x/σ and λ = kσ, Eq. C.4 becomes

β =

∫∞−∞ cos(λu)e

−u2

2 eµ(cos2(λu)+

βµLT

cos(λu))du∫∞

−∞ e−u2

2 eµ(cos2(λu)+

βµLT

cos(λu))du

. (C.5)

As before when we derived the threshold equations (Appendix B), we assume the radius

of the atom cloud is much greater than the wavelength σk = λ 1. In this case the

integrands are oscillating rapidly and we can simplify the integrals by replacing the

integrands with their average over a single cycle:

β ≈√

2π 12π

∫ 2π0 cos(v)e

µ(cos2(v)+βµLT

cos(v))dv

√2π 1

2π

∫ 2π0 e

µ(cos2(v)+βµLT

cos(v))dv

. (C.6)

With the substitution w = cos(v), we find

β =

∫ 1−1

w√1−w2

eµ(w2+

βµLT

w)dw∫ 1

−11√

1−w2eµ(w2+

βµLT

w)dw

(C.7)

=

∫ 10

w√1−w2

eµ(w2+

βµLT

w)dw +

∫ 0−1

w√1−w2

eµ(w2+

βµLT

w)dw∫ 1

01√

1−w2eµ(w2+

βµLT

w)dw +

∫ 0−1

1√1−w2

eµ(w2+

βµLT

w)dw

(C.8)

=

∫ 10

w√1−w2

eµ(w2+

βµLT

w)dw +

∫ 10

w√1−w2

eµ(w2− β

µLTw)dw∫ 1

01√

1−w2eµ(w2+

βµLT

w)dw −

∫ 10

1√1−w2

eµ(w2− β

µLTw)dw

(C.9)

=2∫ 1

0w√

1−w2eµw

2sinh(2µ β

µLTw)dw

2∫ 1

01√

1−w2eµw2 cosh(2µ β

µLTw)dw

. (C.10)

The integrals can be compactly written in term of the functions

bn(µ, βµLT

) =

2π

∫ 10

wn√1−w2

eµw2

cosh(2µ βµLT

w)dw, n even2π

∫ 10

wn√1−w2

eµw2

sinh(2µ βµLT

w)dw, n odd. (C.11)

Where we will make extensive use of the relationships,

∂bn∂β

=2µ

µLTbn+1 (C.12)

∂bn∂µ

= bn+2 +2β

µLTbn+1, (C.13)

154

and the Taylor expansions in β,

bn(µ, βµLT

) =∞∑k=0

1

k!

dkbndβk

∣∣∣∣β=0

βk (C.14)

=

∞∑k=0

1

k!

(2µ

µLT

)kbn+k(µ, 0)βk. (C.15)

We can now write Eq. C.2 and Eq. C.4 in terms of the functions bn as

Z =

√2πmkbT

~

(√σ2

2πb0(µ, β

µLT)

)(C.16)

=kbT

~ωb0(µ, β

µLT) (C.17)

and

β =b1(µ, β

µLT)

b0(µ, βµLT

). (C.18)

Entropy and phase space density below threshold

The entropy in a canonical ensemble is given by

S = −∂F∂T

=∂

∂T(kbT lnZ). (C.19)

We calculate the entropy below threshold, S<, by evaluating b0 for β = 0 to find

S<

kb=

∂

∂(kbT )

(kbT ln

(kbT

~ωeµ/2I0(µ/2)

))(C.20)

= ln

(kbT

~ω

)+ 1 + ln(I0(µ/2))− µ

2

I1(µ/2)

I0(µ/2). (C.21)

Assuming the initial conditions T = T0 and µ = 0, we can find the temperature as the

probe lattice depth is adiabatically increased by requiring that the entropy be constant.

Thus,

S<0 = S<f (C.22)

ln

(kbT0

~ω

)+ 1 = ln

(kbTf

~ω

)+ 1 + ln(I0(µ/2))− µ

2

I1(µ/2)

I0(µ/2)(C.23)

Tf

T0=

eµ2I1(µ/2)I0(µ/2)

I0(µ/2). (C.24)

155


The partition function Z is exactly the normalization for the phase-space distribution

(i.e. see Eq. 2.23), and thus the normalized phase space density is given by

ρ(x) =e−(VT (x)+VS(x))

kbT

Z, (C.25)

where the peak phase space density (x = 0) below threshold is

ρ< =eµ

Z. (C.26)

The dependence of ρ< and Tf on the probe lattice depth below threshold is simply a

result of the change in the density of states when going from a pure harmonic potential

to a harmonic plus lattice potential.

Threshold condition

As in Appendix B, we find the threshold condition by Taylor expansion of the self-

consistency equation (Eq. C.18) to first order,

β =b1(µ, β

µLT)

b0(µ, βµLT

)≈ 2µ

µLT

b2(µ, 0)

b0(µ, 0)β. (C.27)

Requiring a non-trivial solution yields

1 =2µ∗

µLT

b2(µ∗, 0)

b0(µ∗, 0)=

µ∗

µLT

(1 +

I1(µ∗/2)

I0(µ∗/2)

), (C.28)

and thus we have recovered exactly the lattice threshold Eq. 4.11 for α = 1.

Temperature and phase space density at threshold

Substituting the threshold condition into Eq. C.24, we find the temperature at threshold

is

T ∗

T0=

eµLT−µ

∗2

I0(µ∗/2). (C.29)

From which we find the partition function at threshold

Z∗ =kbT0

~ωeµLT , (C.30)

and the peak phase space density at threshold,

ρ∗ =eµ∗

Z∗=

~ωkbT0

eµ∗−µLT

2 . (C.31)

156

Note for µ∗ 1, the probe lattice up to threshold is only a small perturbation on the

initial distribution and thus ρ∗ ≈ ~ωkbT0

= ρ0 as expected. For µ∗ 1, we find1 that

ρ∗ ≈ ρ0e1/2, and thus by simply deforming the potential from a harmonic well to a

deep lattice, the peak phase space density increases by the factor e1/2.

Change in entropy at threshold

Above threshold, the entropy is still given by Eq. C.19, but evaluated for non-zero β.

Thus,

S>

kb=

∂

∂(kbT )

(kbT ln

(kbT

~ωb0

))(C.32)

= ln

(kbT

~ω

)+ 1 + ln(b0)− µb2

b0−(

2µ

µLT

b1b0β +

2µ2

µLT

b1b0

dβ

dµ

)(C.33)

We calculate the jump in entropy at the phase transition by taking the limit from above

and below threshold,

∆S∗

kb=

limµ+→µ∗

S> − limµ−→µ∗

S<

kb(C.34)

= limµ+→µ∗

(− 2µ

µLTβ2 − 2µ2

µLTβdβ

dµ

). (C.35)

Of the two remaining terms, the first term does not contribute since β → 0 as µ+ → µ∗.

However since the factor dβdµ →∞ as µ+ → µ∗, we must carefully evaluate the limit of

the second term,∆S∗

kb= lim

µ+→µ∗

(− µ2

µLT

dβ2

dµ

), (C.36)

to find the change in entropy. In order to find this limit, we will express dβdµ in terms of

the functions bn and then Taylor expand for small β near to threshold. From there we

find dβ2

dµ in terms of the expansion coefficients and take the limit µ→ µ∗.

Starting from dβdµ = d

dµb1b0

, we have

dβ

dµ=

1

b20

(∂b1∂µ

b0 − b1∂b0∂µ

)+

1

b20

(∂b1∂β

b0 − b1∂b0∂β

)dβ

dµ(C.37)

(b20 −

2µ

µLTb2b0 +

2µ

µLTb21

)dβ

dµ=

(b3b0 +

2β

µLTb2b0 − b2b1 −

2β

µLTb21

)(C.38)

1Note I1(µ∗/2)I0(µ∗/2)

≈ 1− 34µ∗

1+ 14µ∗ ≈ 1− 1

µ∗ .

157


and hence

dβ

dµ=

2βµLT

b2b0− b1

b0b2b0

+ b3b0− 2β

µLT( b1b0 )2

1− 2µµLT

b2b0

+ 2µµLT

( b1b0 )2. (C.39)

For small β, we have the following Taylor expansions:

b3b0≈ 2µ

µLT

b4

b0β (C.40)

b2b0≈ b2

b0+ 2

(µ

µLT

)2(b4

b0−(b2

b0

)2)β2 (C.41)

b1b0≈ 2µ

µLT

b2

b0β, (C.42)

where bn = bn(µ, 0) denotes evaluation at β = 0. Keeping only the terms up to second

order in β, Eq. C.39 becomes

dβ

dµ≈

2βµLT

b2b0− 2µ

µLT

(b2b0

)2β + 2β

µLT

b4b0

1− 2µµLT

b2b0− 4

(µµLT

)3 (b4b0−(b2b0

)2)β2 + 8

(µµLT

)3 (b2b0

)2β2

. (C.43)

Hence dβ2

dµ is given by

dβ2

dµ= 2β

dβ

dµ=

2

[1µ

(2µµLT

b2b0

)− µLT

2µ

(2µµLT

b2b0

)2− 2µ′

µLT

b4b0

]β2

[1− 2µ

µLT

b2b0

]+

[3µµLT

(2µµLT

b2b0

)2− 4

(µµLT

)3b4b0

]β2

. (C.44)

In order to correctly find the limit of dβ2

dµ , which has the form

limµ→µ∗

dβ2

dµ= lim

µ→µ∗2Aβ2

C +Bβ2, (C.45)

we must apply l’Hopital’s rule,

limµ→µ∗

dβ2

dµ= lim

µ→µ∗

2dAdµβ2 + 2Adβ2

dµ

dCdµ + dB

dµ β2 +B dβ2

dµ

. (C.46)

158

It can be shown that limµ→µ∗

dC

dµ= −A, from which

limµ→µ∗

dβ2

dµ= lim

µ→µ∗

2Adβ2

dµ

−A+B dβ2

dµ

(C.47)

limµ→µ∗

dβ2

dµ= lim

µ→µ∗3A

B(C.48)

limµ→µ∗

dβ2

dµ= lim

µ→µ∗

3

[1µ

(2µµLT

b2b0

)− µLT

2µ

(2µµLT

b2b0

)2− 2µ

µLT

b4b0

]3µµLT

(2µµLT

b2b0

)2− 4

(µµLT

)3b4b0

. (C.49)

We note that 2µµLT

b2b0

= 1 is exactly the threshold condition from Eq. C.27, which together

with the limit

limµ→µ∗

2µ

µLT

b4

b0= 1− 1

µ∗+

1

µLT, (C.50)

yields the result

limµ→µ∗

dβ2

dµ=

3(

1 + 1µLT− µLT

2µ∗

)3µ∗

µLT− 2

(µ∗

µLT

)2 (1− 1

µ∗ + 1µLT

) . (C.51)

Plugging this result into Eq. C.36, we find the change in entropy at the phase transition,

∆S∗

kb=

−3(

(µ∗)2

µLT+ ( µ∗

µLT)2 − µ∗

2

)3 µ∗

µLT− 2

(µ∗

µLT

)2 (1− 1

µ∗ + 1µLT

) . (C.52)

Temperature and phase space density above threshold

To find the final temperature above threshold, Tf , we again consider the initial and

final entropy,

S<0 + ∆S∗ = S>

ln

(kbT0

~ω

)+ 1 +

∆S∗

kb= ln

(kbTf

~ω

)+ ln b0 + 1− µb2

b0−(

2µ

µLTβ2 +

µ2

µLT

dβ2

dµ

)Tf

T0=

exp[µb2b0

]b0

exp

[2µ

µLTβ2 +

µ2

µLT

dβ2

dµ+

∆S∗

kb

]. (C.53)

Here the first exponential determines the temperature rise prior to organization, and

the second exponential, what happens after self-organization. Thus by setting β = 0,

∆S∗ = 0, and dβ2

dµ = 0, we recover Eq. C.24, the temperature below threshold.

159


Figure C.1: Entropy and phase space density change across threshold. The blue line

is entropy change calculated from Eq. C.52. The red line is phase space density gain

calculated from Eq. C.59. The dotted red line is the contribution from of the entropy term

only (Eq. C.60).

Given the partition function above threshold,

Z> =kbTf

~ωb0 (C.54)

=kbT0

~ωexp

[µ

(b2b0

+2β2

µLT

)+

µ2

µLT

dβ2

dµ+

∆S∗

kb

](C.55)

we can determine the peak phase space density again from Eq. C.25,

ρ> =eµ(

1+ 2βµLT

)Z>

(C.56)

=~ωkbT0

exp

[µ

(1− b2

b0+

2β

µLT− 2β2

µLT

)− µ2

µLT

dβ2

dµ− ∆S∗

kb

]. (C.57)

If we consider the phase density far above threshold (µ → ∞), there β → 1, b2b0→ 1,

and dβ2

dµ → 0. We find this limit, verified numerically, to be

ρ∞ =~ωkbT0

exp

[µLT

2(1 + µLT)− ∆S∗

kb

]. (C.58)

Thus the gain in phase space density from just below threshold, ρ∗, to far above

threshold isρ∞

ρ∗= exp

[µLT

2− µ∗ +

µLT

2(1 + µLT)− ∆S∗

kb

]. (C.59)

In both limiting cases of µ∗ 1 and µ∗ 1, this reduces to simply

ρ∞

ρ∗= e−∆S∗

kb . (C.60)

160

Both Eq. C.59 and Eq. C.60 are plotted in Fig. C.1 for comparison, note the additional

terms in Eq. C.59 contribute only a small correction in the intermediate regime.

161


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