collective dispersive interaction of atoms and light in a high finesse
TRANSCRIPT
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
ii
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
2. DIPOLE TRAPPING AND ALL-OPTICAL BOSE-EINSTEINCONDENSATION
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
2. DIPOLE TRAPPING AND ALL-OPTICAL BOSE-EINSTEINCONDENSATION
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. DIPOLE TRAPPING AND ALL-OPTICAL BOSE-EINSTEINCONDENSATION
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
2. DIPOLE TRAPPING AND ALL-OPTICAL BOSE-EINSTEINCONDENSATION
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
2. DIPOLE TRAPPING AND ALL-OPTICAL BOSE-EINSTEINCONDENSATION
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
2.3 Discussion of crossed beam traps
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. DIPOLE TRAPPING AND ALL-OPTICAL BOSE-EINSTEINCONDENSATION
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
2.3 Discussion of crossed beam traps
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
2. DIPOLE TRAPPING AND ALL-OPTICAL BOSE-EINSTEINCONDENSATION
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
2.3 Discussion of crossed beam traps
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
2. DIPOLE TRAPPING AND ALL-OPTICAL BOSE-EINSTEINCONDENSATION
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
2.3 Discussion of crossed beam traps
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
2. DIPOLE TRAPPING AND ALL-OPTICAL BOSE-EINSTEINCONDENSATION
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
2.3 Discussion of crossed beam traps
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
2. DIPOLE TRAPPING AND ALL-OPTICAL BOSE-EINSTEINCONDENSATION
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.
2.4 Experimental setup
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
2. DIPOLE TRAPPING AND ALL-OPTICAL BOSE-EINSTEINCONDENSATION
Figure 2.7: Schematic of master laser optics.
Figure 2.8: Schematic of slave laser optics.
28
2.4 Experimental setup
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
2. DIPOLE TRAPPING AND ALL-OPTICAL BOSE-EINSTEINCONDENSATION
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
2.4 Experimental setup
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
2. DIPOLE TRAPPING AND ALL-OPTICAL BOSE-EINSTEINCONDENSATION
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
2.4 Experimental setup
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
2. DIPOLE TRAPPING AND ALL-OPTICAL BOSE-EINSTEINCONDENSATION
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
2.4 Experimental setup
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
2. DIPOLE TRAPPING AND ALL-OPTICAL BOSE-EINSTEINCONDENSATION
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
2.4 Experimental setup
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
2. DIPOLE TRAPPING AND ALL-OPTICAL BOSE-EINSTEINCONDENSATION
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
2.5 Bose-Einstein condensation experiment
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
2. DIPOLE TRAPPING AND ALL-OPTICAL BOSE-EINSTEINCONDENSATION
Figure 2.19: Free evaporation at constant trap depth (∼ 2.2 mK).
40
2.5 Bose-Einstein condensation experiment
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
2. DIPOLE TRAPPING AND ALL-OPTICAL BOSE-EINSTEINCONDENSATION
Figure 2.20: Forced evaporation in the primary trap only.
42
2.5 Bose-Einstein condensation experiment
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
2. DIPOLE TRAPPING AND ALL-OPTICAL BOSE-EINSTEINCONDENSATION
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
2.5 Bose-Einstein condensation experiment
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
2. DIPOLE TRAPPING AND ALL-OPTICAL BOSE-EINSTEINCONDENSATION
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
2. DIPOLE TRAPPING AND ALL-OPTICAL BOSE-EINSTEINCONDENSATION
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
3.2 Cavity quantum electrodynamics
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
3. COLLECTIVE CAVITY QUANTUM ELECTRODYNAMICS WITHMULTIPLE ATOM LEVELS
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
3.2 Cavity quantum electrodynamics
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
3. COLLECTIVE CAVITY QUANTUM ELECTRODYNAMICS WITHMULTIPLE ATOM LEVELS
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
3.2 Cavity quantum electrodynamics
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
3. COLLECTIVE CAVITY QUANTUM ELECTRODYNAMICS WITHMULTIPLE ATOM LEVELS
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
3.2 Cavity quantum electrodynamics
(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
3. COLLECTIVE CAVITY QUANTUM ELECTRODYNAMICS WITHMULTIPLE ATOM LEVELS
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
3.2 Cavity quantum electrodynamics
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
3. COLLECTIVE CAVITY QUANTUM ELECTRODYNAMICS WITHMULTIPLE ATOM LEVELS
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
3.2 Cavity quantum electrodynamics
|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
3. COLLECTIVE CAVITY QUANTUM ELECTRODYNAMICS WITHMULTIPLE ATOM LEVELS
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
3.2 Cavity quantum electrodynamics
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
3. COLLECTIVE CAVITY QUANTUM ELECTRODYNAMICS WITHMULTIPLE ATOM LEVELS
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
3.2 Cavity quantum electrodynamics
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
3. COLLECTIVE CAVITY QUANTUM ELECTRODYNAMICS WITHMULTIPLE ATOM LEVELS
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.
3.3 Experimental setup
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
3.3 Experimental setup
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
3. COLLECTIVE CAVITY QUANTUM ELECTRODYNAMICS WITHMULTIPLE ATOM LEVELS
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 Experimental setup
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
3. COLLECTIVE CAVITY QUANTUM ELECTRODYNAMICS WITHMULTIPLE ATOM LEVELS
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
3.3 Experimental setup
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
3. COLLECTIVE CAVITY QUANTUM ELECTRODYNAMICS WITHMULTIPLE ATOM LEVELS
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
3.3 Experimental setup
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
3. COLLECTIVE CAVITY QUANTUM ELECTRODYNAMICS WITHMULTIPLE ATOM LEVELS
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
3.4 Experimental results: cavity transmission spectra
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
3. COLLECTIVE CAVITY QUANTUM ELECTRODYNAMICS WITHMULTIPLE ATOM LEVELS
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
3.4 Experimental results: cavity transmission spectra
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
3. COLLECTIVE CAVITY QUANTUM ELECTRODYNAMICS WITHMULTIPLE ATOM LEVELS
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
3.4 Experimental results: cavity transmission spectra
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
3. COLLECTIVE CAVITY QUANTUM ELECTRODYNAMICS WITHMULTIPLE ATOM LEVELS
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
3. COLLECTIVE CAVITY QUANTUM ELECTRODYNAMICS WITHMULTIPLE ATOM LEVELS
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
4. SELF-ORGANIZATION OF THERMAL ATOMS COUPLED TO ACAVITY
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.
4.2 Derivation of the threshold equations
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
4. SELF-ORGANIZATION OF THERMAL ATOMS COUPLED TO ACAVITY
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
4.2 Derivation of the threshold equations
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
4. SELF-ORGANIZATION OF THERMAL ATOMS COUPLED TO ACAVITY
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
4.2 Derivation of the threshold equations
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
4. SELF-ORGANIZATION OF THERMAL ATOMS COUPLED TO ACAVITY
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
4.2 Derivation of the threshold equations
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
4. SELF-ORGANIZATION OF THERMAL ATOMS COUPLED TO ACAVITY
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.
4.3 Experimental set-up and methods
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
4. SELF-ORGANIZATION OF THERMAL ATOMS COUPLED TO ACAVITY
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
4.3 Experimental set-up and methods
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
4. SELF-ORGANIZATION OF THERMAL ATOMS COUPLED TO ACAVITY
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
4.3 Experimental set-up and methods
Figure 4.6: Layout for 780 nm and 1560 nm lasers which probe and lock the experiment
cavity. See text for description.
99
4. SELF-ORGANIZATION OF THERMAL ATOMS COUPLED TO ACAVITY
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
4.3 Experimental set-up and methods
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
4. SELF-ORGANIZATION OF THERMAL ATOMS COUPLED TO ACAVITY
Figure 4.8: Cavity coupling and detection set-up.
102
4.3 Experimental set-up and methods
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
4. SELF-ORGANIZATION OF THERMAL ATOMS COUPLED TO ACAVITY
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
4.3 Experimental set-up and methods
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
4. SELF-ORGANIZATION OF THERMAL ATOMS COUPLED TO ACAVITY
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
4. SELF-ORGANIZATION OF THERMAL ATOMS COUPLED TO ACAVITY
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
4.4 Experimental results: self-organization threshold scaling
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
4. SELF-ORGANIZATION OF THERMAL ATOMS COUPLED TO ACAVITY
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
4.4 Experimental results: self-organization threshold scaling
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
4. SELF-ORGANIZATION OF THERMAL ATOMS COUPLED TO ACAVITY
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.
4.5 Experimental results: dynamics of self-organization
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
4. SELF-ORGANIZATION OF THERMAL ATOMS COUPLED TO ACAVITY
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
4.5 Experimental results: dynamics of self-organization
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
4. SELF-ORGANIZATION OF THERMAL ATOMS COUPLED TO ACAVITY
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
4.5 Experimental results: dynamics of self-organization
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
4. SELF-ORGANIZATION OF THERMAL ATOMS COUPLED TO ACAVITY
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
4.5 Experimental results: dynamics of self-organization
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
4. SELF-ORGANIZATION OF THERMAL ATOMS COUPLED TO ACAVITY
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
4.5 Experimental results: dynamics of self-organization
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
4. SELF-ORGANIZATION OF THERMAL ATOMS COUPLED TO ACAVITY
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
5.2 Bragg scattering
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
5. BRAGG SCATTERING, CAVITY COOLING, AND FUTUREDIRECTIONS
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
5.2 Bragg scattering
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
5. BRAGG SCATTERING, CAVITY COOLING, AND FUTUREDIRECTIONS
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 Cavity cooling of atomic ensembles
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
5. BRAGG SCATTERING, CAVITY COOLING, AND FUTUREDIRECTIONS
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
5.3 Cavity cooling of atomic ensembles
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
5. BRAGG SCATTERING, CAVITY COOLING, AND FUTUREDIRECTIONS
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
5.3 Cavity cooling of atomic ensembles
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
5. BRAGG SCATTERING, CAVITY COOLING, AND FUTUREDIRECTIONS
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
5.3 Cavity cooling of atomic ensembles
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
5. BRAGG SCATTERING, CAVITY COOLING, AND FUTUREDIRECTIONS
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
5.3 Cavity cooling of atomic ensembles
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
5. BRAGG SCATTERING, CAVITY COOLING, AND FUTUREDIRECTIONS
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
5. BRAGG SCATTERING, CAVITY COOLING, AND FUTUREDIRECTIONS
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
A. HIGH FINESSE CAVITIES: TECHNICAL DETAILS
Table A.1: First coating run
Parameters at 780 nm
F 160,000
T 3 ppm
A 17 ppm
Parameters at 1560 nm
F 450,000
T 0.5 ppm
A 6.5 ppm
Table A.2: Second coating run
Parameters at 780 nm
F 160,000
T 9 ppm
A 11 ppm
Parameters at 1560 nm
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
A. HIGH FINESSE CAVITIES: TECHNICAL DETAILS
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
A. HIGH FINESSE CAVITIES: TECHNICAL DETAILS
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
C. SELF-ORGANIZATION: TEMPERATURE, ENTROPY AND PHASESPACE DENSITY
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
C. SELF-ORGANIZATION: TEMPERATURE, ENTROPY AND PHASESPACE DENSITY
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
C. SELF-ORGANIZATION: TEMPERATURE, ENTROPY AND PHASESPACE DENSITY
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
C. SELF-ORGANIZATION: TEMPERATURE, ENTROPY AND PHASESPACE DENSITY
162
References
[1] T.W. Hansch and A.L. Schawlow. Cooling of gases by laser radiation. Optics
Communications, 13(1):68 – 69, 1975.
[2] D. J. Wineland and Wayne M. Itano. Laser cooling of atoms. Phys. Rev. A,
20:1521–1540, Oct 1979.
[3] J. Dalibard and C. Cohen-Tannoudji. Laser cooling below the Doppler limit
by polarization gradients: simple theoretical models. J. Opt. Soc. Am. B,
6(11):2023–2045, Nov 1989.
[4] Steven Chu. Nobel Lecture: The manipulation of neutral particles. Rev. Mod.
Phys., 70:685–706, Jul 1998.
[5] William D. Phillips. Nobel Lecture: Laser cooling and trapping of neutral
atoms. Rev. Mod. Phys., 70:721–741, Jul 1998.
[6] William Happer. Optical Pumping. Rev. Mod. Phys., 44:169–249, Apr 1972.
[7] N.V. Vitanov, M. Fleischhauer, B.W. Shore, and K. Bergmann. Coherent
manipulation of atoms and molecules by sequential laser pulses. 46:55 – 190,
2001.
[8] Rudolf Grimm, Matthias Weidemller, and Yurii B. Ovchinnikov. Optical
Dipole Traps for Neutral Atoms. 42:95 – 170, 2000.
[9] I. Bloch. Ultracold quantum gases in optical lattices. Nature Physics, 1(1):23–30,
2005.
[10] C.S. Adams, H.J. Lee, N. Davidson, M. Kasevich, and S. Chu. Evaporative
cooling in a crossed dipole trap. Physical review letters, 74(18):3577–3580, 1995.
[11] M. D. Barrett, J. A. Sauer, and M. S. Chapman. All-Optical Formation of an
Atomic Bose-Einstein Condensate. Phys. Rev. Lett., 87:010404, Jun 2001.
[12] W. Ketterle, D. S. Durfee, and D. M. Stamper-Kurn. Making, probing, and
understanding bose-einstein condensates. 1998.
163
REFERENCES
[13] E.T. Jaynes and F.W. Cummings. Comparison of quantum and semiclassical
radiation theories with application to the beam maser. Proceedings of the IEEE,
51(1):89–109, 1963.
[14] K.M. Birnbaum, A. Boca, R. Miller, A.D. Boozer, T.E. Northup, and H.J.
Kimble. Photon blockade in an optical cavity with one trapped atom. Nature,
436(7047):87–90, 2005.
[15] R. J. Thompson, G. Rempe, and H. J. Kimble. Observation of normal-mode
splitting for an atom in an optical cavity. Phys. Rev. Lett., 68:1132–1135, Feb 1992.
[16] A. Boca, R. Miller, K. M. Birnbaum, A. D. Boozer, J. McKeever, and H. J.
Kimble. Observation of the Vacuum Rabi Spectrum for One Trapped Atom.
Phys. Rev. Lett., 93:233603, Dec 2004.
[17] PWH Pinkse, T. Fischer, P. Maunz, and G. Rempe. Trapping an atom with
single photons. Nature, 404(6776):365–368, 2000.
[18] C.J. Hood. Real-time measurement and trapping of single atoms by single photons. PhD
thesis, California Institute of Technology, 2000.
[19] Vladan Vuletic and Steven Chu. Laser Cooling of Atoms, Ions, or Molecules
by Coherent Scattering. Phys. Rev. Lett., 84:3787–3790, Apr 2000.
[20] P. Maunz, T. Puppe, I. Schuster, N. Syassen, P.W.H. Pinkse, and G. Rempe.
Cavity cooling of a single atom. Nature, 428(6978):50–52, 2004.
[21] Weiping Lu, Yongkai Zhao, and P. F. Barker. Cooling molecules in optical
cavities. Phys. Rev. A, 76:013417, Jul 2007.
[22] Thomas Salzburger and Helmut Ritsch. Collective transverse cavity cooling
of a dense molecular beam. New Journal of Physics, 11(5):055025, 2009.
[23] A Vukics, J Janszky, and P Domokos. Cavity cooling of atoms: a quantum
statistical treatment. Journal of Physics B: Atomic, Molecular and Optical Physics,
38(10):1453, 2005.
[24] Almut Beige, Peter L Knight, and Giuseppe Vitiello. Cooling many particles
at once. New J. Phys., 7(1):96, 2005.
[25] Monika H. Schleier-Smith, Ian D. Leroux, Hao Zhang, Mackenzie A.
Van Camp, and Vladan Vuletic. Optomechanical Cavity Cooling of an Atomic
Ensemble. Phys. Rev. Lett., 107:143005, Sep 2011.
[26] M. Wolke, J. Klinner, H. Keßler, and A. Hemmerich. Cavity Cooling Below
the Recoil Limit. Science, 337(6090):75–78, 2012.
164
REFERENCES
[27] Michael Tavis and Frederick W. Cummings. Exact Solution for an N-
Molecule¯Radiation-Field Hamiltonian. Phys. Rev., 170:379–384, Jun 1968.
[28] A. K. Tuchman, R. Long, G. Vrijsen, J. Boudet, J. Lee, and M. A. Kase-
vich. Normal-mode splitting with large collective cooperativity. Phys. Rev. A,
74:053821, Nov 2006.
[29] F. Brennecke, T. Donner, S. Ritter, T. Bourdel, M. Kohl, and T. Esslinger.
Cavity QED with a Bose–Einstein condensate. Nature, 450(7167):268–271, 2007.
[30] P. Munstermann, T. Fischer, P. Maunz, P. W. H. Pinkse, and G. Rempe.
Observation of Cavity-Mediated Long-Range Light Forces between Strongly
Coupled Atoms. Phys. Rev. Lett., 84:4068–4071, May 2000.
[31] Markus Gangl and Helmut Ritsch. Collective dynamical cooling of neutral
particles in a high-Q optical cavity. Phys. Rev. A, 61:011402, Dec 1999.
[32] Peter Horak and Helmut Ritsch. Scaling properties of cavity-enhanced atom
cooling. Phys. Rev. A, 64:033422, Aug 2001.
[33] Peter Domokos and Helmut Ritsch. Mechanical effects of light in optical
resonators. J. Opt. Soc. Am. B, 20(5):1098–1130, May 2003.
[34] Benjamin L. Lev, Andras Vukics, Eric R. Hudson, Brian C. Sawyer, Peter
Domokos, Helmut Ritsch, and Jun Ye. Prospects for the cavity-assisted laser
cooling of molecules. Phys. Rev. A, 77:023402, Feb 2008.
[35] T. Grieer, H. Ritsch, M. Hemmerling, and G. R.M. Robb. A Vlasov approach
to bunching and selfordering of particles in optical resonators. The European
Physical Journal D, 58:349–368, 2010.
[36] Peter Domokos and Helmut Ritsch. Collective Cooling and Self-Organization
of Atoms in a Cavity. Phys. Rev. Lett., 89:253003, Dec 2002.
[37] Hilton W. Chan, Adam T. Black, and Vladan Vuletic. Observation of
Collective-Emission-Induced Cooling of Atoms in an Optical Cavity. Phys.
Rev. Lett., 90:063003, Feb 2003.
[38] Adam T. Black, Hilton W. Chan, and Vladan Vuletic. Observation of Collec-
tive Friction Forces due to Spatial Self-Organization of Atoms: From Rayleigh
to Bragg Scattering. Phys. Rev. Lett., 91:203001, Nov 2003.
[39] Kristian Baumann, Christine Guerlin, Ferdinand Brennecke, and Tilman
Esslinger. Dicke quantum phase transition with a superfluid gas in an optical
cavity. Nature, 464(7293):1301–1306, 04 2010.
165
REFERENCES
[40] D. Nagy, G. Szirmai, and P. Domokos. Self-organization of a Bose-Einstein
condensate in an optical cavity. Eur. Phys. J. D, 48:127–137, 2008.
[41] R. H. Dicke. Coherence in Spontaneous Radiation Processes. Phys. Rev., 93:99–
110, Jan 1954.
[42] D. Nagy, G. Konya, G. Szirmai, and P. Domokos. Dicke-Model Phase Tran-
sition in the Quantum Motion of a Bose-Einstein Condensate in an Optical
Cavity. Phys. Rev. Lett., 104:130401, Apr 2010.
[43] J. K. Asboth, P. Domokos, H. Ritsch, and A. Vukics. Self-organization of
atoms in a cavity field: Threshold, bistability, and scaling laws. Phys. Rev. A,
72:053417, Nov 2005.
[44] W. Niedenzu, T. Grieer, and H. Ritsch. Kinetic theory of cavity cooling and
self-organisation of a cold gas. EPL (Europhysics Letters), 96(4):43001, 2011.
[45] M.H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman, and E.A. Cor-
nell. Observation of Bose-Einstein condensation in a dilute atomic vapor.
science, 269(5221):198–201, 1995.
[46] KB Davis, M.O. Mewes, M.R. Andrews, NJ Van Druten, DS Durfee,
DM Kurn, and W. Ketterle. Bose-Einstein condensation in a gas of sodium
atoms. Physical Review Letters, 75(22):3969–3973, 1995.
[47] G. Cennini, G. Ritt, C. Geckeler, and M. Weitz. Bose–Einstein condensation
in a CO2-laser optical dipole trap. Applied Physics B: Lasers and Optics, 77(8):773–
779, 2003.
[48] T. Weber, J. Herbig, M. Mark, H.C. Nagerl, and R. Grimm. Bose-Einstein
condensation of cesium. Science, 299(5604):232–235, 2003.
[49] R. Dumke, M. Johanning, E. Gomez, JD Weinstein, KM Jones, and PD Lett.
All-optical generation and photoassociative probing of sodium Bose–Einstein
condensates. New Journal of Physics, 8(5):64, 2006.
[50] T. Kinoshita, T. Wenger, and D.S. Weiss. All-optical Bose-Einstein conden-
sation using a compressible crossed dipole trap. Physical Review A, 71(1):011602,
2005.
[51] A. Couvert, M. Jeppesen, T. Kawalec, G. Reinaudi, R. Mathevet, and
D. Gury-Odelin. A quasi-monomode guided atom laser from an all-optical
Bose-Einstein condensate. EPL (Europhysics Letters), 83(5):50001, 2008.
[52] J.F. Clement, J.P. Brantut, M. Robert-de Saint-Vincent, R.A. Nyman,
A. Aspect, T. Bourdel, and P. Bouyer. All-optical runaway evaporation to
Bose-Einstein condensation. Physical Review A, 79(6):061406, 2009.
166
REFERENCES
[53] C.L. Hung, X. Zhang, N. Gemelke, and C. Chin. Accelerating evaporative
cooling of atoms into Bose-Einstein condensation in optical traps. Physical
Review A, 78(1):011604, 2008.
[54] Q. Beaufils, R. Chicireanu, T. Zanon, B. Laburthe-Tolra, E. Marechal,
L. Vernac, J.-C. Keller, and O. Gorceix. All-optical production of chromium
Bose-Einstein condensates. Phys. Rev. A, 77:061601, Jun 2008.
[55] R. A. Cline, J. D. Miller, M. R. Matthews, and D. J. Heinzen. Spin relaxation
of optically trapped atoms by light scattering. Opt. Lett., 19(3):207–209, Feb 1994.
[56] H. J. Metcalf and P. Sraten. Laser Cooling and Trapping. Springer, 1999.
[57] Y. Castin, H. Wallis, and J. Dalibard. Limit of Doppler cooling. J. Opt. Soc.
Am. B, 6(11):2046–2057, Nov 1989.
[58] Daniel A Steck. Rubidium 87 D Line Data, 2010. Ver. 2.1.4.
[59] G. Nienhuis, P. van der Straten, and S-Q. Shang. Operator description of
laser cooling below the Doppler limit. Phys. Rev. A, 44:462–474, Jul 1991.
[60] C. D. Wallace, T. P. Dinneen, K. Y. N. Tan, A. Kumarakrishnan, P. L. Gould,
and J. Javanainen. Measurements of temperature and spring constant in a
magneto-optical trap. J. Opt. Soc. Am. B, 11(5):703–711, May 1994.
[61] Wolfgang Petrich, Michael H. Anderson, Jason R. Ensher, and Eric A.
Cornell. Behavior of atoms in a compressed magneto-optical trap. J. Opt.
Soc. Am. B, 11(8):1332–1335, Aug 1994.
[62] Wolfgang Ketterle, Kendall B. Davis, Michael A. Joffe, Alex Martin, and
David E. Pritchard. High densities of cold atoms in a dark spontaneous-force
optical trap. Phys. Rev. Lett., 70:2253–2256, Apr 1993.
[63] M. Drewsen, Ph. Laurent, A. Nadir, G. Santarelli, A. Clairon, Y. Castin,
D. Grison, and C. Salomon. Investigation of sub-Doppler cooling effects in a
cesium magneto-optical trap. Applied Physics B, 59:283–298, 1994.
[64] C. G. Townsend, N. H. Edwards, C. J. Cooper, K. P. Zetie, C. J. Foot, A. M.
Steane, P. Szriftgiser, H. Perrin, and J. Dalibard. Phase-space density in
the magneto-optical trap. Phys. Rev. A, 52:1423–1440, Aug 1995.
[65] Wolfgang Ketterle and N.J. Van Druten. Evaporative Cooling of Trapped
Atoms. 37:181 – 236, 1996.
[66] K. M. O’Hara, M. E. Gehm, S. R. Granade, and J. E. Thomas. Scaling laws for
evaporative cooling in time-dependent optical traps. Phys. Rev. A, 64:051403,
Oct 2001.
167
REFERENCES
[67] O. J. Luiten, M. W. Reynolds, and J. T. M. Walraven. Kinetic theory of the
evaporative cooling of a trapped gas. Phys. Rev. A, 53:381–389, Jan 1996.
[68] T Mther, J Nes, A-L Gehrmann, M Volk, W Ertmer, G Birkl, M Gruber,
and J Jahns. Atomic quantum systems in optical micro-structures. 19, page 97,
2005.
[69] S. Bali, K. M. O’Hara, M. E. Gehm, S. R. Granade, and J. E. Thomas.
Quantum-diffractive background gas collisions in atom-trap heating and loss.
Phys. Rev. A, 60:R29–R32, Jul 1999.
[70] Michael A. Joffe, Wolfgang Ketterle, Alex Martin, and David E.
Pritchard. Transverse cooling and deflection of an atomic beam inside a
Zeeman slower. J. Opt. Soc. Am. B, 10(12):2257–2262, Dec 1993.
[71] K. Dieckmann, R. J. C. Spreeuw, M. Weidemuller, and J. T. M. Walraven.
Two-dimensional magneto-optical trap as a source of slow atoms. Phys. Rev.
A, 58:3891–3895, Nov 1998.
[72] D. Sesko, T. Walker, C. Monroe, A. Gallagher, and C. Wieman. Collisional
losses from a light-force atom trap. Phys. Rev. Lett., 63:961–964, Aug 1989.
[73] Alan Gallagher and David E. Pritchard. Exoergic collisions of cold Na∗-Na.
Phys. Rev. Lett., 63:957–960, Aug 1989.
[74] A. Fuhrmanek, R. Bourgain, Y. R. P. Sortais, and A. Browaeys. Light-
assisted collisions between a few cold atoms in a microscopic dipole trap.
Phys. Rev. A, 85:062708, Jun 2012.
[75] E. A. Burt, R. W. Ghrist, C. J. Myatt, M. J. Holland, E. A. Cornell, and
C. E. Wieman. Coherence, Correlations, and Collisions: What One Learns
about Bose-Einstein Condensates from Their Decay. Phys. Rev. Lett., 79:337–
340, Jul 1997.
[76] J. Sding, D. Gury-Odelin, P. Desbiolles, F. Chevy, H. Inamori, and J. Dal-
ibard. Three-body decay of a rubidium BoseEinstein condensate. Applied
Physics B, 69:257–261, 1999.
[77] M. D. Barrett. PhD thesis, Georgia Institute of Techology, 2002.
[78] T. Lauber, J. Kuber, O. Wille, and G. Birkl. Optimized Bose-Einstein-
condensate production in a dipole trap based on a 1070-nm multifrequency
laser: Influence of enhanced two-body loss on the evaporation process. Phys.
Rev. A, 84:043641, Oct 2011.
168
REFERENCES
[79] Franco Dalfovo, Stefano Giorgini, Lev P. Pitaevskii, and Sandro Stringari.
Theory of Bose-Einstein condensation in trapped gases. Rev. Mod. Phys., 71:463–
512, Apr 1999.
[80] Anthony J. Leggett. Bose-Einstein condensation in the alkali gases: Some
fundamental concepts. Rev. Mod. Phys., 73:307–356, Apr 2001.
[81] L. Pitaevskii and S. Stringari. Bose-Einstein Condensation. International Series of
Monographs on Physics. Oxford University Press, USA, 2003.
[82] C.J. Pethick and H. Smith. Bose-Einstein Condensation in Dilute Gases. Cambridge
University Press, 2001.
[83] Vanderlei Bagnato, David E. Pritchard, and Daniel Kleppner. Bose-Einstein
condensation in an external potential. Phys. Rev. A, 35:4354–4358, May 1987.
[84] Y. Castin and R. Dum. Bose-Einstein Condensates in Time Dependent Traps.
Phys. Rev. Lett., 77:5315–5319, Dec 1996.
[85] C.E. Wieman and L. Hollberg. Using diode lasers for atomic physics. Review
of Scientific Instruments, 62(1):1–20, 1991.
[86] KB MacAdam, A. Steinbach, and C. Wieman. A narrow-band tunable diode
laser system with grating feedback, and a saturated absorption spectrometer
for Cs and Rb. American Journal of Physics, 60:1098, 1992.
[87] R. A. Boyd, J. L. Bliss, and K. G. Libbrecht. Teaching physics with 670-
nm diode lasers—experiments with Fabry–Perot cavities. American Journal of
Physics, 64(9):1109–1116, 1996.
[88] G.C. Bjorklund, M.D. Levenson, W. Lenth, and C. Ortiz. Frequency modu-
lation (FM) spectroscopy. Applied Physics B, 32:145–152, 1983.
[89] G. Hadley. Injection locking of diode lasers. Quantum Electronics, IEEE Journal
of, 22(3):419–426, 1986.
[90] S. J. M. Kuppens, K. L. Corwin, K. W. Miller, T. E. Chupp, and C. E. Wieman.
Loading an optical dipole trap. Phys. Rev. A, 62:013406, Jun 2000.
[91] M. E. Gehm, K. M. O’Hara, T. A. Savard, and J. E. Thomas. Dynamics of
noise-induced heating in atom traps. Phys. Rev. A, 58:3914–3921, Nov 1998.
[92] T. A. Savard, K. M. O’Hara, and J. E. Thomas. Laser-noise-induced heating
in far-off resonance optical traps. Phys. Rev. A, 56:R1095–R1098, Aug 1997.
169
REFERENCES
[93] S. D. Gensemer, P. L. Gould, P. J. Leo, E. Tiesinga, and C. J. Williams.
Ultracold 87Rb ground-state hyperfine-changing collisions in the presence and
absence of laser light. Phys. Rev. A, 62:030702, Aug 2000.
[94] S. Friebel, C. D’Andrea, J. Walz, M. Weitz, and T. W. Hansch. CO2-laser
optical lattice with cold rubidium atoms. Phys. Rev. A, 57:R20–R23, Jan 1998.
[95] L.D. Landau and E.M. Lifshitz. Mechanics. Course of theoretical physics. Pergamon
Press, 1976.
[96] R. Jauregui, N. Poli, G. Roati, and G. Modugno. Anharmonic parametric
excitation in optical lattices. Phys. Rev. A, 64:033403, Aug 2001.
[97] J. A. Sauer, K. M. Fortier, M. S. Chang, C. D. Hamley, and M. S. Chapman.
Cavity QED with optically transported atoms. Phys. Rev. A, 69:051804, May
2004.
[98] T. Wilk, S.C. Webster, A. Kuhn, and G. Rempe. Single-atom single-photon
quantum interface. Science, 317(5837):488–490, 2007.
[99] Almut Beige, Hugo Cable, and Peter L. Knight. Dissipation-assisted quan-
tum computation in atom-cavity systems. pages 370–381, 2003.
[100] K. M. Birnbaum, A. S. Parkins, and H. J. Kimble. Cavity QED with multiple
hyperfine levels. Phys. Rev. A, 74:063802, Dec 2006.
[101] Bruce W. Shore and Peter L. Knight. The Jaynes-Cummings Model. Journal
of Modern Optics, 40(7):1195–1238, 1993.
[102] D. Braak. Integrability of the Rabi Model. Phys. Rev. Lett., 107:100401, Aug
2011.
[103] J. J. Sanchez-Mondragon, N. B. Narozhny, and J. H. Eberly. Theory of
Spontaneous-Emission Line Shape in an Ideal Cavity. Phys. Rev. Lett., 51:550–
553, Aug 1983.
[104] I. P. Vadeiko, G. P. Miroshnichenko, A. V. Rybin, and J. Timonen. Algebraic
approach to the Tavis-Cummings problem. Phys. Rev. A, 67:053808, May 2003.
[105] Yifu Zhu, Daniel J. Gauthier, S. E. Morin, Qilin Wu, H. J. Carmichael,
and T. W. Mossberg. Vacuum Rabi splitting as a feature of linear-dispersion
theory: Analysis and experimental observations. Phys. Rev. Lett., 64:2499–2502,
May 1990.
[106] Haruka Tanji-Suzuki, Ian D. Leroux, Monika H. Schleier-Smith, Marko
Cetina, Andrew T. Grier, Jonathan Simon, and Vladan Vuletic. Chapter 4
170
REFERENCES
- Interaction between Atomic Ensembles and Optical Resonators: Classical
Description. 60:201 – 237, 2011.
[107] A. T. Rosenberger, L. A. Orozco, H. J. Kimble, and P. D. Drummond. Ab-
sorptive optical bistability in two-state atoms. Phys. Rev. A, 43:6284–6302, Jun
1991.
[108] Hyatt M. Gibbs. Optical bistability : controlling light with light / Hyatt M. Gibbs.
Academic Press, Orlando :, 1985.
[109] Subhadeep Gupta, Kevin L. Moore, Kater W. Murch, and Dan M. Stamper-
Kurn. Cavity Nonlinear Optics at Low Photon Numbers from Collective
Atomic Motion. Phys. Rev. Lett., 99:213601, Nov 2007.
[110] R. J. Thompson, Q. A. Turchette, O. Carnal, and H. J. Kimble. Nonlin-
ear spectroscopy in the strong-coupling regime of cavity QED. Phys. Rev. A,
57:3084–3104, Apr 1998.
[111] M. L. Terraciano, R. Olson Knell, D. L. Freimund, L. A. Orozco, J. P.
Clemens, and P. R. Rice. Enhanced spontaneous emission into the mode of a
cavity QED system. Opt. Lett., 32(8):982–984, Apr 2007.
[112] James P. Clemens. Probe spectrum of multilevel atoms in a damped, weakly
driven two-mode cavity. Phys. Rev. A, 81:063818, Jun 2010.
[113] C.F. Brennecke. Collective interaction between a Bose-Einstein condensate and a co-
herent few-photon field. PhD thesis, ETH Zurich, 2009.
[114] Stefan Kuhr, Wolfgang Alt, Dominik Schrader, Martin Mller, Victor
Gomer, and Dieter Meschede. Deterministic Delivery of a Single Atom.
293(5528):278–280, 2001.
[115] Stefan Schmid, Gregor Thalhammer, Klaus Winkler, Florian Lang, and
Johannes Hecker Denschlag. Long distance transport of ultracold atoms
using a 1D optical lattice. New Journal of Physics, 8(8):159, 2006.
[116] H. Haken. Synergetics. Naturwissenschaften, 67(3):121–128, 1980.
[117] H. Hermann. Synergetics. An Introduction. Nonequilibrium Phase Transitions
and Self-Organization in Physics, Chemistry and Biology, 1978.
[118] V. DeGiorgio and Marlan O. Scully. Analogy between the Laser Threshold
Region and a Second-Order Phase Transition. Phys. Rev. A, 2:1170–1177, Oct
1970.
171
REFERENCES
[119] R. Graham and H. Haken. Laserlight first example of a second-order phase
transition far away from thermal equilibrium. Zeitschrift fr Physik, 237:31–46,
1970.
[120] S. Ciliberto and P. Bigazzi. Spatiotemporal Intermittency in Rayleigh-
Benard Convection. Phys. Rev. Lett., 60:286–289, Jan 1988.
[121] K. Baumann, R. Mottl, F. Brennecke, and T. Esslinger. Exploring Symmetry
Breaking at the Dicke Quantum Phase Transition. Phys. Rev. Lett., 107:140402,
Sep 2011.
[122] M. G. Raizen, J. Koga, B. Sundaram, Y. Kishimoto, H. Takuma, and T. Tajima.
Stochastic cooling of atoms using lasers. Phys. Rev. A, 58:4757–4760, Dec 1998.
[123] R. Bonifacio, L. De Salvo, L. M. Narducci, and E. J. D’Angelo. Exponential
gain and self-bunching in a collective atomic recoil laser. Phys. Rev. A, 50:1716–
1724, Aug 1994.
[124] D. Kruse, C. von Cube, C. Zimmermann, and Ph. W. Courteille. Observation
of Lasing Mediated by Collective Atomic Recoil. Phys. Rev. Lett., 91:183601, Oct
2003.
[125] J. P. Brantut, J. F. Clement, M. Robert de Saint Vincent, G. Varoquaux,
R. A. Nyman, A. Aspect, T. Bourdel, and P. Bouyer. Light-shift tomography
in an optical-dipole trap for neutral atoms. Phys. Rev. A, 78:031401, Sep 2008.
[126] Vladan Vuletic, Hilton W. Chan, and Adam T. Black. Three-dimensional
cavity Doppler cooling and cavity sideband cooling by coherent scattering.
Phys. Rev. A, 64:033405, Aug 2001.
[127] Stefano Zippilli and Giovanna Morigi. Cooling Trapped Atoms in Optical
Resonators. Phys. Rev. Lett., 95:143001, Sep 2005.
[128] Stefano Zippilli, Giovanna Morigi, and Helmut Ritsch. Suppression of Bragg
Scattering by Collective Interference of Spatially Ordered Atoms with a High-
Q Cavity Mode. Phys. Rev. Lett., 93:123002, Sep 2004.
[129] H. Ritsch, P. Domokos, F. Brennecke, and T. Esslinger. Cold atoms in
cavity-generated dynamical optical potentials. arXiv preprint arXiv:1210.0013,
2012.
[130] S. Nußmann, K. Murr, M. Hijlkema, B. Weber, A. Kuhn, and G. Rempe.
Vacuum-stimulated cooling of single atoms in three dimensions. Nature Physics,
1(2):122–125, 2005.
172
REFERENCES
[131] Peter Domokos, Peter Horak, and Helmut Ritsch. Semiclassical theory of
cavity-assisted atom cooling. Journal of Physics B: Atomic, Molecular and Optical
Physics, 34(2):187, 2001.
[132] Janos K. Asboth, Peter Domokos, and Helmut Ritsch. Correlated motion of
two atoms trapped in a single-mode cavity field. Phys. Rev. A, 70:013414, Jul
2004.
[133] K.W. Murch, K.L. Moore, S. Gupta, and D.M. Stamper-Kurn. Observation of
quantum-measurement backaction with an ultracold atomic gas. Nature Physics,
4(7):561–564, 2008.
[134] Dvid Nagy, JnosK. Asbth, and Pter Domokos. Collective cooling of atoms in
a ring cavity. Acta Physica Hungarica A) Heavy Ion Physics, 26:141–148, 2006.
[135] Florian Marquardt, Joe P. Chen, A. A. Clerk, and S. M. Girvin. Quantum
Theory of Cavity-Assisted Sideband Cooling of Mechanical Motion. Phys. Rev.
Lett., 99:093902, Aug 2007.
[136] Adam T Black, James K Thompson, and Vladan Vuleti? Collective light
forces on atoms in resonators. Journal of Physics B: Atomic, Molecular and Optical
Physics, 38(9):S605, 2005.
[137] Yongkai Zhao, Weiping Lu, P. F. Barker, and Guangjiong Dong. Self-
organisation and cooling of a large ensemble of particles in optical cavities.
Faraday Discuss., 142:311–318, 2009.
[138] D. M. Stamper-Kurn, H.-J. Miesner, A. P. Chikkatur, S. Inouye, J. Stenger,
and W. Ketterle. Reversible Formation of a Bose-Einstein Condensate. Phys.
Rev. Lett., 81:2194–2197, Sep 1998.
[139] F. Dimer, B. Estienne, A. S. Parkins, and H. J. Carmichael. Proposed real-
ization of the Dicke-model quantum phase transition in an optical cavity QED
system. Phys. Rev. A, 75:013804, Jan 2007.
[140] S. Morrison and A. S. Parkins. Dynamical Quantum Phase Transitions in
the Dissipative Lipkin-Meshkov-Glick Model with Proposed Realization in
Optical Cavity QED. Phys. Rev. Lett., 100:040403, Jan 2008.
[141] S. Morrison and A. S. Parkins. Collective spin systems in dispersive opti-
cal cavity QED: Quantum phase transitions and entanglement. Phys. Rev. A,
77:043810, Apr 2008.
[142] Alexander Altland and Fritz Haake. Quantum Chaos and Effective Ther-
malization. Phys. Rev. Lett., 108:073601, Feb 2012.
173
REFERENCES
[143] Neill Lambert, Clive Emary, and Tobias Brandes. Entanglement and the
Phase Transition in Single-Mode Superradiance. Phys. Rev. Lett., 92:073602, Feb
2004.
[144] G. Vacanti, S. Pugnetti, N. Didier, M. Paternostro, G. M. Palma, R. Fazio,
and V. Vedral. Photon Production from the Vacuum Close to the Superra-
diant Transition: Linking the Dynamical Casimir Effect to the Kibble-Zurek
Mechanism. Phys. Rev. Lett., 108:093603, Feb 2012.
[145] Christine Guerlin, Etienne Brion, Tilman Esslinger, and Klaus Mølmer.
Cavity quantum electrodynamics with a Rydberg-blocked atomic ensemble.
Phys. Rev. A, 82:053832, Nov 2010.
[146] D. Jaksch, J. I. Cirac, P. Zoller, S. L. Rolston, R. Cote, and M. D. Lukin.
Fast Quantum Gates for Neutral Atoms. Phys. Rev. Lett., 85:2208–2211, Sep 2000.
[147] M. Saffman, T. G. Walker, and K. Mølmer. Quantum information with Ry-
dberg atoms. Rev. Mod. Phys., 82:2313–2363, Aug 2010.
[148] G. Rempe, R. J. Thompson, H. J. Kimble, and R. Lalezari. Measurement of
ultralow losses in an optical interferometer. Opt. Lett., 17(5):363–365, Mar 1992.
[149] T.E. Northup. Coherent control in cavity QED. PhD thesis, California Institute of
Technology, 2008.
[150] B. Dahmani, L. Hollberg, and R. Drullinger. Frequency stabilization of semi-
conductor lasers by resonant optical feedback. Opt. Lett., 12(11):876–878, Nov
1987.
174