Many-body physics in long-range interacting quantum
systemsbyBihui Zhu
B.S., Physics, Peking University
M.S., Physics, Chinese Academy of Sciences

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillmentof the requirements for the degree ofDoctor of Philosophy
Department of Physics
2017This thesis entitled:
Many-body physics in long-range interacting quantum systems
written by Bihui Zhu
has been approved for the Department of Physics

Prof. Ana Maria Rey

Prof. Murray J. Holland

Date

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

iii
Zhu, Bihui (Ph.D., Physics )Many-body physics in long-range interacting quantum systems
Thesis directed by Prof. Ana Maria Rey

Ultracold atomic and molecular systems provide a useful platform for understanding quantum
many-body physics. Recent progresses in AMO experiments enable access to systems exhibiting
long-range interactions, opening a window for exploring the interplay between long-range interactions and dissipation. In this thesis, I develop theoretical approaches to study non-equilibrium
dynamics in systems where such interplay is crucial.
I first focus on a system of KRb molecules, where dipolar interactions and fast chemical reactions coexist. Using a classical kinetic theory and Monte Carlo methods, I study the evaporative
cooling in a quasi-two-dimensional trap, and develop a protocol to reach quantum degeneracy. I
also study the case where molecules are loaded into an optical lattice, and show that the strong
dissipation induces a quantum Zeno effect, which suppresses the molecule loss. The analysis requires including multiple bands to explain recent experimental measurements, and can be used todetermine the molecular filling fraction.
I also investigate a system of radiating atoms, which experience long-range elastic and dissipative interactions. I explore the collective behavior of atoms and the role of atomic motion. The
model is validated by comparison with a recent light scattering experiment using Sr atoms. I also
show that incoherently pumped dipoles can undergo a dynamical phase transition to synchronization, and study its signature in the quantum regime.

Note: This work was jointly supervised by Prof. Murray Holland.

Dedication

To my Mom.

Acknowledgements

First and foremost, I would like to thank my advisors Prof. Ana Maria Rey and Prof. Murray
Holland, for giving me the invaluable opportunity to work with them. Without them the completionof this thesis would have been impossible. I am very grateful for their generous financial supportand helpful guidance. Through their encouragement, ingenious ideas and enthusiasm for physics,
Ana Maria and Murray have been not only constant sources of inspiration, but also incredible
mentors and collaborators to me in the past six years of my life. I feel very fortunate to have hadthem as my advisors.
I would also like to acknowlege other members of my thesis committee. The courses I tookwith Prof. Victor Gurarie were my favorites in graduate school. They stimulated my interest inquantum many-body physics. I also benefited from discussions with Prof. James Thompson andProf. Juan Restrepo, who led me to think from a different angle on the same topic and helped me
to expand my knowledge.
I owe special gratitude to Prof. Jun Ye. Jun is one of the most brilliant scientists I havemet, with inspiring perspective and ingenious ideas. He also possesses such a great personality,and everyone around him respects and likes him from the bottom of their hearts. It has been mytremendous pleasure to collaborate with him and join his group meetings.
I benefited from interactions with many people, at or outside of JILA. Prof. Deborah Jin
was an inspiring and kind collaborator and teacher to me. Prof. Mikhail Lukin, Prof. Susanne
Yelin, Prof. Robin Kaiser and Prof. John Cooper helped me to gain a better understanding oflight-matter interactions. I would like to thank Goulven Quéméner, Sergey Syzranov, Michael Wall,vi
and Johannes Schachenmayer for the helpful discussions we had.
My years at JILA and University of Colorado have left me with a lot of memories. I would liketo thank those who have influenced me in one way or another during these years, although due tothe limited space I cannot list all of them. To name just some of them: Bo Yan, Xibo Zhang, Kaden
Hazzard, Michael Foss-Feig, Steven Moses, Jake Covey, Sarah Bromley, Bryce Gadway, Norman
Yao, Prof. John Bohn, Prof. Matthew Davis, Prof. Darrick Chang, Prof. Alexey Gorshkov,
Prof. Leo Radzihovsky, Prof. Juha Javaneinen, Arghavan Safavi-Nani, Leonid Isaev, Tony Lee,
Marco Manzoni, Salvatore Manmana, Dong-Ling Deng, Zhe-Xuan Gong, Xiao-Peng Li, Minghui
Xu, David Tieri, Dominic Meiser, Yiheng Lin, Shuming Li, Hao Wu, Hao Song and Andrew Koller.
Finally, I am deeply indebted to my parents, for their love and support.

Contents

Chapter
1 Introduction

1

1.1

Quantum many-body physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

Long-range interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.3

Dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.4

Outline of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42 Radiating dipoles8

2.1

Dipolar interactions: origin and characteristics . . . . . . . . . . . . . . . . . . . . .

2.2

Near-field and far-field interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3

Superradiance and subradiance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3 Evaporative cooling of reactive polar molecules confined in a two-dimensional geometry

8

16

3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2

Evaporative cooling with anisotropic collisions . . . . . . . . . . . . . . . . . . . . . . 18
3.2.1

Anisotropic elastic collisions between polar molecules . . . . . . . . . . . . . . 18

3.2.2

Thermalization rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2.3

Evaporation with anisotropic collisions . . . . . . . . . . . . . . . . . . . . . . 22

3.3

Evaporative cooling in 2D traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.4

Evaporative cooling of KRb molecules . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.5

Anti-evaporation in quasi-2D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

viii
3.6

Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4 Suppressing the loss of ultracold molecules via the continuous quantum Zeno effect

34

4.1

Experiment with KRb molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.2

Theoretical description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.3

4.2.1

Single-band rate equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.2.2

Multiband rate equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.2.3

Mean-field and DMRG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

Comparison with experiment measurement . . . . . . . . . . . . . . . . . . . . . . . 41
4.3.1

4.4

Mean-field vs. experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5 Synchronization of Interacting Quantum Dipoles

44

5.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5.2

Dipole-dipole interaction and master equation

5.3

Mean-field treatment and connection to the Kuramoto model . . . . . . . . . . . . . 48

5.4

Quantum synchronization for the collective system . . . . . . . . . . . . . . . . . . . 52

. . . . . . . . . . . . . . . . . . . . . 47

5.4.1

Phase Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.4.2

Quantum correlations and entanglement. . . . . . . . . . . . . . . . . . . . 53

5.5

Synchronization with finite-range interactions . . . . . . . . . . . . . . . . . . . . . . 55

5.6

Synchronization of dipoles with elastic interactions . . . . . . . . . . . . . . . . . . . 57

5.7

Experimental implementation

5.8

Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

6 Light scattering from dense cold atomic media

63

6.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

6.2

Coherent dipole model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
6.2.1

Equations of motion for coherent dipoles . . . . . . . . . . . . . . . . . . . . . 66

ix
6.2.2

Collective effects in fluorescence . . . . . . . . . . . . . . . . . . . . . . . . . . 67

6.2.3

Anisotropic features of scattered light . . . . . . . . . . . . . . . . . . . . . . 73

6.3

Random walk model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

6.4

Role of atomic motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
6.4.1

Modified frozen dipole model . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

6.4.2

Semi-classical approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

6.5

Comparison with experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

6.6

Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

7 Emergent Weyl excitations in systems of polar particles

90

7.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

7.2

Phenomenological argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

7.3

Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

7.4

Atoms in a deep lattice

7.5

Effects of quenched disorder and dissipation

7.6

Proposal for experimental observation . . . . . . . . . . . . . . . . . . . . . . . . . . 97

7.7

Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
. . . . . . . . . . . . . . . . . . . . . . 96

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

Bibliography

100

Appendix
A Theoretical approaches for evaporative cooling with long-range interactions

119

A.1 Monte Carlo simulation for evaporative cooling . . . . . . . . . . . . . . . . . . . . . 119
A.2 Kinetics of evaporative cooling in 2D . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
A.3 Master equation approach for losses in a quasi-2D trap . . . . . . . . . . . . . . . . . 122

x
B Modeling dissipative dynamics

124

B.1 Mean-field treatment (MF) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
B.2 Rate equation (RE) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
C Phase coherence and correlations in interacting dipoles

127

C.1 Mean-field approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
C.2 Cumulant expansion approach and two-time correlation between dipoles . . . . . . . 128
C.3 Conditional evolution, entanglement and quantum correlations . . . . . . . . . . . . 129
D Combining dipole-dipole interaction and motion

131

D.1 Long-range dipole-dipole interaction, force and diffusion . . . . . . . . . . . . . . . . 131
D.2 Effect of motion on single-atom fluorescence . . . . . . . . . . . . . . . . . . . . . . . 134
E Calculation of Weyl dispersion

138

E.1 Dispersion near Weyl nodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
E.2 Excitation dispersion in a deep lattice . . . . . . . . . . . . . . . . . . . . . . . . . . 139
E.3 Details of disorder averaging. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

Tables

Table
3.1

Derived parameters from the fit of Eq. (3.1) for the scattering energies shown in
Fig. 3.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Figures

Figure2.1

Spatial dependence of dipole-dipole interactions. The diagonal parts of the imaginary(f zz ) and real (g zz ) dipolar interactions between two dipoles are plotted, for threedifferent interparticle spacings. While both types of interactions depend on geometry,
they are nonidentical, and the elastic interactions are highly sensitive to the spatial
separation and orientation of the particles. The maximum value for imaginary andreal interactions for fixed |rab | is denoted as fmax and gmax , respectively. The cone√
illustrates the magic angle, θm = arccos(1/ 3), along which the elastic interactions
vanish. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2

Typical transitions relevant for dipolar interactions in several AMO systems. a is the
optical lattice spacing estimated from experimental realizations [260, 187, 249, 250].
Different choices of internal states allow us to access different dipolar regimes . . . . 13

2.3

Dicke states for three atoms. |ei and |gi denote the excited and ground state in theindividual atom basis. Also indicated are the decay rates under the dissipator Jˆ+ Jˆ− .
States in red colors decays at least at rate Ne Γ, while the decay rate for states inblue (or black) color is < Ne Γ (or =0), where Ne denotes the number of excitationsin the system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

xiii
3.1

(a) Two-dimensional differential scattering cross-section for elastic collisions as afunction of the scattering angle. Four collision energies are shown; Ec = 1 nK (bluedisk), 10 nK (green diamond), 100 nK (red triangle), and 1 µK (black star). The
induced dipole moment is d = 0.2 debye and the harmonic frequency of the onedimensional confinement potential is ν = 23 kHz, The solid lines show the empiricalformula Eq. (3.1). (b) Total scattering cross section for elastic collisions (solid line)
and reactive processes (dashed line) as a function of the collision energy for an
induced dipole moment of d = 0.2 debye and a one dimensional confinement ofν = 23 kHz. The ratio of elastic to reactive is indicated at four different collisionenergies; Ec =1 nK (blue point), 10 nK (green point), 100 nK (red point), and 1 µK
(black point). (c) Number of collisions κ required for thermalization for different
differential scattering cross-sections. The magenta solid line shows Eq. (3.10). The
purple circles are results from the MC simulations, and are shown with their small
statistical error bars. The four symbols (blue disk, green diamond, red triangle,
black star) correspond to the same energies as (a). . . . . . . . . . . . . . . . . . . . 19

3.2

Increase in phase space density vs truncation parameter η for different dλ(φ)
dφ . Here Ω0is the initial phase-space density, and the final phase space density, Ωf , was calculated
when the ratio of molecule numbers was Nf /N0 = 0.1. Blue circles: dλ(φ)
dφ =const;dλ(φ)
red crosses: ( dλ(φ)
dφ )Ec =1µK ; green squares: ( dφ )Ec =100 nK ; black dotted line: criticalphase space density Ωc for quantum degeneracy.
3.3. . . . . . . . . . . . . . . . . . . . 22dλ(φ)
Trajectories of evaporation for (a): ( dλ(φ)
dφ )Ec =1µK at η ∼ 3.4 and (b):dφ =constat η ∼ 4.3, under the same initial conditions as in FIG. 3.2. In comparison with(a), it is apparent that in (b) the temperature T drops faster and the phase-spacedensity Ω increases much more rapidly. . . . . . . . . . . . . . . . . . . . . . . . . . . 23

xiv
3.4

Comparison of the efficiency of evaporative cooling in two-dimensional (2D) andthree-dimensional (3D) harmonic traps. ln(Ωf /Ω0 )’s are calculated at Nf /N0 = 0.1,
and ζ = 1/200. Red (blue) line: 3D (2D) harmonic trap from method in Appendix
A.2; red squares: 3D harmonic trap from MC simulation; blue disks: 2D harmonic
trap from MC simulation; blue (red) dotted line: Ωc for a 2D (3D) trap. . . . . . . . 26

3.5

Evaporative cooling trajectories for KRb molecules inside a 2D trap of 2π × 20 Hz,
with scattering cross-sections as computed in FIG. 3.1(b). The initial temperaturewas ∼ 200 nK, with an initial-phase space density Ω0 ∼ 0.1Ωc , and η = 3.8. . . . . . 27

3.6

Heating due to two-body losses in a two-dimensional harmonic trap. Here (a) and(b) show the MC result in a two-dimensional harmonic trap where all molecules arein the lowest band of the lattice potential. The results for (c) and (d) are obtainedwith the master equation approach. The blue line assumes only 1 band along thelattice direction is populated. The red dashed line (green dots) assumes 2 (3) bands
are populated. N0 is the initial total molecule number. The initial temperature ischosen to be around 800 nK, so that when two bands are included, there are ∼25% of
molecules on the second band, and when three bands are included, there are ∼19%of molecules on the second band and ∼6% on the third band. . . . . . . . . . . . . . 29

3.7

Collisions between molecules in a quasi-2D trap. (a) intra-band collisions, occuring
at rate Γ̃(1) . (b) inter-band collisions, occuring at rate Γ̃(2) (see Appendix A.3).

4.1. . 30(a) A 50:50 mixture of fermionic KRb molecules in two rotational states, |0, 0i (red)and |1, −1i (blue), is prepared in a deep 3D lattice, which is suddenly made shallow
along one dimension (y). Along y, molecules tunnel with a rate J/~ and have alarge on-site loss rate Γ0 because of chemical reactions. (b) In the Zeno regime,~Γ0  J, doubly occupied sites are only virtually populated, and the loss occurs
at a significantly slower rate Γeff  Γ0 for molecules on adjacent sites. For KRb, amultiband analysis of this process is required for all experimental lattice parameters. 35

xv
4.2

(a) Measured number loss of |↓i molecules for an axial (transverse) lattice depth of
Vy = 5 ER (V⊥ = 25 ER ) (circles) and best fit using a rate equation (RE), Eq. (4.1)(black dashed line). (b) Number loss rate, κ, as a function of Γ0 (fixing J ≈ 570 Hz
and varying the bare on-site rate via V⊥ ). (c) Number loss rate, κ, versus J for
fixed Γ0 ≈ 87 kHz (varying Vy and adjusting V⊥ accordingly). Vy (V⊥ ) was varied
from 5 to 16 ER (20 to 40 ER ). Black circles are experimental measurements (error
bars represent one standard error). Green short-dashed lines show solutions of the
RE Eq. (4.4) using an effective loss rate Γeff (single-band approximation). The blue
long-dashed line shows the multiband RE using Γ̃eff in Eq. (4.4). The multiband
and single-band RE results were obtained by fixing the filling fraction to be 6%, and
25% respectively. Panels (b) and (c) directly manifest the continuous quantum Zeno
effect: in (b) the measured loss rate κ decreases with increasing on-site Γ0 ; in (c) a
fit to the experimental data supports κ ∝ J 2 , with a χ2 (sum of the squared fitting
errors) several times smaller than for a linear fit. . . . . . . . . . . . . . . . . . . . . 37

4.3

Comparison of experimental loss dynamics for the deepest considered lattice to MF
and t-DMRG calculations. (a) Molecule loss vs. time for V⊥ = 80 ER and Vy = 5 ER
[Identical conventions/conditions to Fig. 4.2(b)]. The MF matches the experimental
data better than the RE (experimental fit). (b) Comparison of t-DMRG simulations(χMPS = 128, 2000 trajectories) to MF, for two different cases: i) an identical trap
for the two spin states with trap frequency ω↓ = ω↑ = 2π × 38 Hz; and ii) slightly
different trap frequencies ω↓ = 2π × 38 Hz, ω↑ = 2π × 34.2 Hz. Shaded areas indicate
the standard error of the mean. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

xvi
4.4

(a) Number loss rate, κ, as a function of Γ0 (same data as Fig. 2(b)). (b) Number
loss rate, κ, versus J for fixed Γ0 (same data as Fig. 2(c)). Black circles are experimental measurements. Blue long-dashed and solid red lines show RE and MF
solutions, respectively, using Γ̃eff (multiband model). The shaded area accounts for±2% variations around the MF estimate of f ∼ 9% arising from the uncertainty inthe initial molecule distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5.1

Arrays of quantum dipoles spontaneously emit and absorb photons at rate Γ. The
photons mediate dipolar interactions between dipoles separated by a distance rab
with both dissipative, f (rab ), and elastic, g(rab ), components. A repumping source
at a rate W provides energy to maintain the oscillations and can be implemented
using additional internal states that are not shown. A possible implementation can

5.2

be realized with cold atoms in an optical lattice. . . . . . . . . . . . . . . . . . . . . 46√(a) Mean-field phase diagram calculated from the order parameter 0 ≤ Z ≤ 1/ 8.
The insets show snapshots of the tips of the Bloch vectors (red points) for dipoles
prepared with random initial phases and then evolved to steady-state in both regimes.
√(b) Quantum phase diagram calculated from 0 ≤ ZQ ≤ 1/ 8. (c) The time evolutionof the conditional QFI exhibits entanglement (dashed line: single trajectory, solidline: mean value of a few trajectories). Panel (d) shows the steady state QFI vs W/Γ
after averaging over many trajectories. The solid line corresponds to the conditionalcase, and indicates entanglement over the repumping range where synchronization
exists. Upon computing the ensemble average one recovers the reduced density
matrix which leads to a calculated QFI below the entanglement witness threshold(dashed line). (c) and (d) are shown for feff = 15Γ. For all panels, δa = g(rab ) = 0
and f (rab ) = feff /N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

xvii
5.3

Quantum correlations and total correlations. The total and quantum correlations inthe steady state are quantified by the mutual information 0 ≤ I ≤ 2 and quantum
discord 0 ≤ D ≤ 2. (a) In the synchronized phase, there are nonzero quantum
correlations and classical correlations (I − D), and both show a dependence on W
that qualitatively agree with ZQ . (b) Even in the thermodynamic limit, quantum
correlations remain a significant fraction of the total correlations. For both panels,
δa = g(rab ) = 0 and f (rab ) = feff /N . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.4

d , in linear clusters containing d dipoles at optimal(a) Spin-spin correlations, ZQ αwith lattice spacing a. We
repumping W for power-law couplings f (rab ) = Γ4 raabset δa = g(rab ) = 0 and consider N = 900 dipoles arranged in both linear (D = 1) andsquare lattice (D = 2) geometries. For α . D global synchronization is observed andthe order parameter is independent on cluster size d. For D . α the order parameterd = 0.14) provides
starts to clearly decay with increasing d. The magenta line (ZQ

an indicative scale of the boundary between global and local synchronization. The
white contour lines provide an indication of the decrease of the synchronized domains
with increasing α. (b) Pair wise two-time correlation functions in the steady-state areparametrized by Za,b (τ ) = A cos(ντ ) exp(−γτ ) where a is chosen as the central dipoleof a linear chain of N = 200 dipoles. The dipoles are assigned random detunings
δa distributed uniformly in [−Γ/2, Γ/2]. The dark blue, red, and light blue symbols
correspond to α = 0, 0.65 and 2 respectively. The histogram of frequencies ν exhibits
similar synchronization regimes than those seen in (a).

. . . . . . . . . . . . . . . . 57

xviii5.5(a) Synchronization in dipole arrays is demonstrated for N = 12 dipoles on a line
when subjected to incoherent pumping (optimal rate). In this geometry, regardlessof the strong angular variation of g with the lattice spacing a (see contours) the order√max = 1/ 8) exhibits a weak dependence on θ andparameter, ZQ , (normalized by ZQ
a and reaches a maximum at θ = θm . (b) The order parameter is computed forP
N = 16 dipoles on a line with θ = θm and feff = a,b6=a f (rab )/(N − 1) (symbols),
and for a system with constant f (rab ) = feff /N and g(rab ) = 0 (dashed line)s. Similar
dependence on W is found for these two different systems. Here the order parameterfor dipoles is always smaller in the presence of elastic interactions. . . . . . . . . . . 58

6.1

Scheme. (a) Microscopic models: random-walk model (left) and coherent dipole
model (right). In the random-walk model, a photon is randomly scattered by theatoms. Scattering events are characterized by the incident and outgoing wavevectorof the photons and their corresponding polarization. In the coherent dipole model,
atoms are coupled by dipole-dipole interactions (Gab ) and all atoms contribute to
the fluorescence. (b) Experimental setup for measuring the fluorescence from a cloudof atoms. An incident laser drives an atomic transition with a spontaneous emission
rate Γ. Atoms absorb and emit light. The detectors collect scattered photons at anangle θ measured from the incident beam direction.

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

xix
6.2

CD model: on-resonance intensity. (a) Angular distribution: due to the constructive
interference along the forward direction (θ = 0), the intensity is drastically enhanced
within a small angular region. (b)-(c) The intensities for three different directions(θ = 0, π/10, π/2) are shown as a function of atom number, each normalized to
the value at N = 50. All are calculated for a spherical cloud of fixed size. The
right vertical axes label the intensity for θ = π/2, π/10. In (b) the OD and thedensity are relatively low (when N = 500, τ =2,n0 k0−3 =0.0015). The transverse
intensity increases ∼ N , while the forward intensity increases ∼ N 2 , showing acollective enhancement for small window of θ around zero. Outside this narrow
angular window the enhancement disappears and the intensity becomes almost θ
independent as indicated by the identical behavior observed for two different angles,θ = π/10 and π/2. In (c) the OD and density are relatively large (when N = 500,
τ =10, n0 k0−3 =0.017). The rate at which the intensity increases with N slows down
in both the forward and transverse direction: with I ∼ N 1 for θ = 0 and I ∼ N 0.5
for θ = π/2, respectively. (d) On-resonance intensity as a function of OD: it is highly
suppresed at large OD. Here, the intensity is normalized to the corresponding valueat τ = 25 for each direction.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

xx
6.3

CD model: collective broadening. (a) FWHM linewidth as a function of OD. At small
OD, the FWHM increases linearly with OD, but as OD increases density effects set
in, multiple scattering events become relevant and the linewidth dependence on OD
is no longer linear. Two different atom numbers are used for the blue (N=1000) andblack lines (N=200), and the OD is varied by changing the cloud size. For the blueline, n0 k0−3 = 0.14 at τ =25, 0.003 at τ =2. For the black line, the density at the same
OD is doubled. With smaller density, the linewidth increases to a larger value inthe large OD regime. (b) The line shape at small OD values (τ =2, n0 k0−3 = 0.002)
for different angles θ (θ is defined in Fig. 6.1(a)) is mainly Lorentzian. Here theintensity is normalized to the on-resonance intensity for each θ. (c) At large OD(τ =20, n0 k0−3 = 0.06), the fluorescence line shape significantly broadens and stops
being Lorentzian. The brown dots for θ = 0 show the Lorentzian fit, which fails to
describe the line shape. At intermediate θ, a double-peak structure shows up. For
all panels, the cloud shape is spherical.

6.4. . . . . . . . . . . . . . . . . . . . . . . . . 70

CD model: frequency shift. For small density, the shift calculated from the coherent
dipole model (blue line) increases linearly with density as predicted by the meanfield theory. However, when density is large, there is a significant deviation from themean-field result. When motional effects are taken into account (red line, Doppler
width = 5Γ, see Sec. 6.4.1), the non linear suppression of the frequency shift withdensity is less severe.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

xxi
6.5

CD model: finite size scaling. (a) The linewidth is calculated for different numberof atoms at the same OD (τ =4 for N =3000) by varying the density (blue line withcircles). From N = 1000 to 5000 the linewidth is not obviously changed. In contrast,by keeping the same density (n0 k0−3 = 0.0037 for N =3000) while varying the OD(magenta line with triangles), the linewidth keeps changing. (b) The frequency shift
is calculated for different number of atoms. By keeping constant density (magenta
line with triangles), the frequency shift remains almost constant, while for constantOD (blue line with circles), there is a significant variation of the frequency shift with
N . Here the cloud aspect ratio is Rx : Ry : Rz = 2 : 2 : 1 [33]. . . . . . . . . . . . . . 72

6.6

CD model: effect of laser polarization. (a) Intensity distribution in the plane perpendicular to the laser propagation direction, i.e., θ = π/2 for all φ’s (inset). φ = 0 is
the z-direction. Here, the incident laser is polarized along z. The intensity detectedalong the laser polarization is suppressed compared to other directions. (b) and (c)show the line shape and linewidth of light detected at θ = π/2, φ = 0. (b) At small
OD, the FWHM linewidth is below the natural linewidth. (c) As OD increases, thelinewidth is collectively broadened due to multiple scattering processes (Fig. 6.3(a)).
For all panels, degeneracy in excited levels has been assumed.

6.7. . . . . . . . . . . . 73

RW: transformation of Stokes vectors. In the random walk model, a scattering event
is determined by two consecutive transformations of local coordinates: {e3 , er , el } →{e03 , e0r , e0l } via rotation φ, and {e03 , e0r , e0l } → {e003 , e00r , e00l } via rotation θ [15]. . . . . . 75

xxii
6.8

RW Model. (a) Distribution of on-resonance intensity in the plane θ = π/2. Alongthe direction where single scattering events are forbidden (φ = 0), the light intensity is suppressed. (b) The FWHM linewdith increases linearly with τp in a dilute
medium. The linewidth along the direction parallel to the laser polarization candrop below Γ for small (purple squares). (c) Lin eshapes for different τp s and angles
of observation. The line shape is Lorentzian for low τp but gets distorted and develops a double-peak structure as the τp increases. Here, for all angles, the signal iscollected within a small angular region δθ = 5◦ . The intensity is normalized to thevalue at zero detuning for θ = 0 and τp =0.3, and the width of laser beam is 5 times
the Gaussian width of the atomic cloud. Here all τp are labeled as the value at ∆ = 0. 77

6.9

Modified frozen dipole model: peak intensity. (a) Collective forward enhancement:
forward intensity normalized by the transverse intensity as a function of Doppler
width. (b) Intensity detected along different directions at Doppler width ∆D = 6Γ asa function of atom number for fixed cloud size (when N = 500, τ =2,n0 k0−3 =0.0015).
I ∼ N 1.6 for θ = 0, and I ∼ N 0.7 for θ = π/2. The right vertical axis labels theintensity for θ = π/2, π/10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

6.10 Semi-classical model: single-atom line shape. The excitation line shape is calculated
for three different driving strength, each normalized by Ω2 . The atomic motion isallowed to be one-dimensional and parallel to the laser propagating direction. Here
we used ωr = 0.6Γ (the value of the 1 S0 → 3 P1 transition of Sr [108]). . . . . . . . . 81
6.11 Semi-classical model: effect of motion on frequency shift. Two atoms are driven bya pair of counter-propagating lasers, with Ω = 0.1Γ, ωr = 0.6Γ. The red dashed line
shows the result of semi-classical model where motion is allowed in three dimensions.
The blue line shows the result from the modified CD. The center of excitation line
shape is calculated at t = 5/Γ for different Doppler width, for both models. . . . . . 82
6.12 The short caption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6.13 The short caption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

xxiii6.14 Forward Scattering. (a) Comparison of forward scattering intensity versus angle
using a red and blue probe beam. The inset shows the setup used to block theprobe beam. After interacting with the atoms the probe beam is focused using alens, which also collimates the fluorescence from the atoms. We block the probe
beam using a beam stopper, which we translate perpendicular to the probe beam
to change the angular range of fluorescence collected by the detector, characterized
by the angle (θ) between x̂ and the edge of the beam stopper (see Ref. [33]). The
measured intensity, Ix,0 (θ), for each probe beam is normalized to the intensity atθmax = 7.5 mRad. The dephasing caused by motion reduces the forward intensity
peak for the red transition. (b) Comparison of intensity in the forward direction,
Ix , versus intensity in the transverse direction, Iz . Both are varied by changing N .
All measurements are made at θ = 2 mRad (arrow in (a)) and normalized to theintensity, Ix,0 , for the atom number used in (a). (c) Linewidth broadening in theforward direction measured by scanning the blue probe beam detuning, ∆, acrossresonance. Example lineshapes for different optical depths (ODs) are shown in theinset. Two different atom numbers are used, N = 1.7(2) × 107 (blue squares) andN/4 (cyan triangles). The dashed line represents Γ for reference. All solid curves
are based on the full theory of coupled dipoles and the band in (c) is for a ±20%atom number uncertainty. All errorbars are for statistical uncertainties. . . . . . . . 85

xxiv
6.15 Transverse Scattering. (a) Linewidth broadening for the blue transition in the transverse direction for ŷ-polarization (open squares) and ẑ-polarization (open triangles).
(b) Intensity ratio, Iypol /Izpol , of ŷ-polarization to ẑ-polarization measured in thetransverse direction when a blue probe beam is used. For low optical depths single
particle scattering is dominant, and for single particle scattering almost zero intensity
is predicted for ẑ-polarized fluorescence as this polarization points directly into thedetector. (c) Linewidth broadening for the red transition in the transverse direction
for ŷ-polarized light showing a similar trend to the blue transition. This transition
is more sensitive to magnetic fields so a large magnetic field is applied to probe only
the m = 0 to m’ = 0 transition. All solid curves are based on the full theory of
coupled dipoles and the band in (c) is for a ±20% atom number uncertainty. Allerrorbars are for statistical uncertainties.
7.1. . . . . . . . . . . . . . . . . . . . . . . . 87

Weyl quasiparticles in 3D dipolar arrays. (a) Schematics of the 3D lattice potentialthat traps an array of dipolar particles. The lattice is deep enough to pin theparticles, most of which are prepared in the J = 0 ground state (blue spheres). Onlya few particles are excited to the J = 1 states. Dipolar interactions between the
J = 0 and J = 1 states give raise to Weyl excitations. (b) Schematics of dipole
mediated interactions: An excited J = 1 state of one particle can be transferred
to another particle in the J = 0 state by dipole-dipole interactions (virtual photon
exchange is shown with a yellow wiggly line). Three types of allowed processes
include |00ii |1σij ↔ |1σii |00ij , |00ii |10ij ↔ |1, ±1ii |00ij , and |00ii |1, ±1ij ↔
|1, ∓1ii |00ij . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

xxv
7.2

Weyl quasi-particle dispersion and eigenstates. (a) Brillouin zone for the simple cubic
lattice. (b) Dispersion along high-symmetry lines in the absence of magnetic field(all energies are measured in units of (3/4)γ0 /(k0 a)3 ). (c) Dispersion in the presenceof magnetic field B = γ0 /(k0 a)3 demonstrating the existence of Weyl nodes (redpoints) with linear quasiparticle dispersion near them. Each node is characterisedby the monopole charge ±1. In agreement with the fermion doubling theorem [181](the Nielsen-Ninomiya no-go theorem), there is an even number (six) of Weyl points
in the first Brillouin zone. (d) Dispersion in the horizontal (kx − ky ) plane (shown by
blue color in panel (a)) containing the Weyl node near the R point. Quasiparticles inthis plane are similar to quasiparticles in graphene and are characterised by a nontrivial Berry phase of π. The arrows show the pseudospin hσ̂i (the Pauli matricesσ̂i act in the space of the angular-momentum projections Jz = 0 and Jz = −1).
(e) Dispersion along the (yellow in panel (a)) plane consisting of vectors k = (π +√√kd / 2, π + ky , 0.71π + kd / 2) containing the Weyl point. Color shows the weight of
the |1 − 1i state in the quasiparticle eigenstate, and arrows represent the pseudospin
hσ̂i. (f) Dispersion along the (green in panel (a)) vertical plane (ky − kz ) containing
the Weyl point near the R point. For each momentum k the color represents theweight of the |1 − 1i state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

xxvi
7.3

Effects of disorder and dissipation on Weyl quasiparticles. (a) Real part of the quasiparticle dispersion in the presence of dissipation (including spontaneous emission andcollective radiative decay). The parameters of the J = 1 → J = 0 transition correspond to those of the electronic 3 P0 −3 D1 levels of bosonic 88,84 Sr atoms, trapped ina magic optical lattice potential with a = 206.4nm [187] at unit filling. Momentum
k is measured in units π/a. (b) The upper bound on the inelastic scattering rateestimated from simulating the full quasiparticle spectra including all allowed elastic
and inelastic dipolar processes. The dissipation is significantly suppressed near the
Weyl node (blue line). In striking contrast, the dissipation is enhanced close to theΓ point (red line) due to the enhanced collective emission (superradiance). (c) Disordered case: Quasiparticle dispersion for a lattice with filling fraction f = 0.93 in
the limit of small dissipation k0 a  1 (the energy is measured in units γ0 /(k0 a)3 ).
The line thickness shows the inelastic scattering rate. . . . . . . . . . . . . . . . . . . 97

7.4

Observation of Weyl quasiparticles. (a) Momentum-selective Ramsey spectroscopy:
interfering Raman beams create an excitation with the angular momentum projection
Jz = −1 and with translational momentum k near the Weyl node (see text). Aftera waiting time t the second pulse is applied to measure the fraction of particles in
the Jz = 0 angular-momentum state. (b) Six quasimomenta near the Weyl node.
(c) The fraction of particles in the Jz = 0 state oscillates as a function of time twith the frequency (Ek1 − Ek2 )/(2π), where Ek1 − Ek2 is the energy splitting between
the two branches of the quasiparticle dispersion. (d) The oscillations in the presenceof dissipation for k0 a ∼ 0.5. (e) The oscillations in a 99% randomly filled lattice.
(f) The oscillations in a 93% filled lattice. For (c) and (d) a 3D cubic lattice of
100 × 100 × 100 sites was used and we took advantage of the translational symmetry.
For (e) and (f) a 3D cubic lattice of 10 × 10 × 10 was used. . . . . . . . . . . . . . . 98

xxviiC.1 Pair-wise two-time correlation functions in the steady-state parametrized by Z1,2 (τ ) =Aexp(−γτ ). The correlations are calculated for a pair of dipoles in an ensemble of
N = 70 dipoles identically coupled with f (rab ) = feff /N . The cumulant expansion
solution agrees with the full calculation except at W  Γ where subradiant behavior
dominates (purple region) [166]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

Chapter 1

Introduction

1.1

Quantum many-body physics
Atoms and molecules are the building blocks of the objects surrounding us. However, it issomehow amazing that collections of them exhibit behaviors not only drastically different from asingle atom or molecule, but also very different between each other. Such many-body behaviors
include a kaleidoscope of phenomena, ranging from superconductivity, which promises high performance electronic devices, to light harvesting, which is essential for photosynthesis in green plants.
It is then natural to ask, how do these rich phenomena emerge from microscopic rules of quantum
mechanics? And perhaps more interestingly, how to engineer materials with certain properties andhow to harness the benefits of quantum physics? The understanding of these many-body behaviors
has been a long-sought-after goal of physics. Yet with so many degrees of freedom and myriads ofingredients in different systems, it is a challenging task and many open questions remain.
Thanks to impressive experimental advances in the last two decades, systems of ultracold
atoms and molecules provide a new avenue for addressing these questions. Experiments nowadays
can access and selectively control internal levels of various types of bosonic and fermionic species,
trap them in flexible choices of confinement geometries and tune their interaction strength, evenwith single-atom resolution [26, 143, 74, 121, 190]. In addition, experiments can control the degrees
of complexity in the system and separate the target objectives we want to study from the undesired
ones.
The power of these tools has been demonstrated in numerous studies that have, for instance,

2
observed the crossover from a molecular BEC to a BCS superfluid of Cooper pairs [286, 204], theKibble-Zurek mechanism through a phase transition [138, 177], realized the paradigmatic BoseHubbard model in optical lattices and observed the superfluid to Mott insulator quantum phasetransition [93, 83]. In those studies, the interaction between atoms is mediated by contact collisions,which are intrinsically short-ranged. Only recently, experimental advances have enabled us toaccess a variety of atomic, molecular and optical (AMO) systems with long-range interactions, thus
opening a window for the exploration of long-range interacting systems, which are ubiquitous innature but poorly understood. The use of AMO systems to gain a better understanding of longrange interacting physics can not only boost our knowledge of fundamental quantum sciences but
also lead to important advances in quantum metrology and technology. This thesis aims to provide
a step forward towards reaching this goal. In the remainder of this chapter, the key subjects thisthesis revolves around will be briefly highlighted.

1.2Long-range interactions
In systems such as trapped ions, Rydberg atoms, polar molecules and magnetic atoms, inter-action decays with interparticle spacing in a power-law form V ∼ 1/rα , with smaller α indicating alonger-ranged interaction.

While there have been important developments on the understandingof the nonequilibrium dynamics and equilibrium phases in systems with short-range interactions,long-range interacting systems can exhibit richer but much less understood behaviors. For example,
Lieb and Robinson showed that in short-range interacting systems, information can only propagate
with a finite speed, and thus quantum correlations only exist within a linear “light-cone” [147].
This behavior has been experimentally confirmed in a one-dimensional array of ultracold bosonicatoms [46]. However, as observed in an experiment with trapped ions that interact with longrange Ising or XY spin-spin interactions, such a bound is violated [206]. A follow-up question isthen, if long-range interactions can facilitate entanglement generation and be useful for quantum
computation.
For isolated systems with short-range interactions, sufficient disorder can drive the system into

3
a many-body localized phase, where thermal relaxation is prohibited. It has been conjectured thatsuch localization can not exist when the decay rate of interaction α is smaller than the dimensionof the system [176]. However, much is still unclear about the transport properties of systems withlong-range interactions.
The observation of many-body effects requires interactions to be stronger than other energy
scales, such as loss and heating. This is a particular aspect where long range interactions can bring
important opportunities. Let us consider for example the case of two particles sitting at adjacentsites of a lattice. If they interact via short-range interactions, to see each other one of them needs
to first tunnel, then interact locally with the other and finally tunnel back. This type of tunnelingmediated interactions, known as superexchange interactions, can be very weak ∼nK. Their low
energy scales have been an important obstacle to experimentally observing them in the laboratory.
On the contrary, if they can interact at long distance via dipolar interactions, no motion is required.
This has facilitated their experimental observation as reported in recent experiments using KRb
molecules or magnetic atoms such as Cr and Er [278, 10, 61]. The strong dipolar interactions are also
experienced by Rydberg atoms, which are becoming an attractive platform for quantum information
processing. Dipolar interactions between polar molecules are anisotropic, and consequently can giverise to geometrically frustrated interactions. Those are the underlying mechanism behind exotic
states of matter such as spin liquids, which are highly entangled and topologically nontrivial [11].
Moreover, dipolar interactions naturally give rise to spin-orbit coupling (SOC)—a key element for avariety of topological phenomena, such as Majorana fermions and topological insulators [168, 279].
Spin-orbit coupling emerges in dipolar particles because dipolar interactions can couple externalangular momentum and internal states.
The fact that dipolar interactions can directly take place between frozen particles also meansthat they could be a source of undesirable systematic effects in atomic clocks as the clocks become
sensitive enough to detect them [153, 45]. Further improving the accuracy of atomic clocks thus
demands a thorough knowledge of the underlying many-body dipolar effects. What is more, if
frozen dipolar arrays can exhibit rich physics due to the long-range and collective character of

4dipolar interactions, then one can anticipate the rich and complex physics that emerges when the
dipoles are allowed to move. The study of the interplay between motional and dipolar interactions
also constitutes a major part of this thesis.

1.3

Dissipation
To utilize many-body quantum mechanical systems for technological benefits, such as building

a powerful quantum computer, it is important to overcome processes that can smear out quantum
coherences. Those are in general unavoidable due to the coupling with environment. In AMO
systems, the main decoherence mechanisms typically come from two-body or three-body collisions
that induce number loss, spontaneous emission that changes internal level of an atom, and heatingfrom light scattering processes. Usually these effects are considered as detrimental since theyoften result in an exponential decay of coherence. However, it is now appreciated that dissipative
processes can also give rise to nontrivial effects and even be used to generate entanglement between
particles [56, 148]. In fact, the interplay between dissipation and interactions is far from trivial,
especially when the interactions are nonlocal and collective. Superradiance and subrradiance are
two iconic examples. Superradiance, as will be shown in chapter 5, can drive the system into a robust
phase coherent state. Subradiance can maintain long-lived coherence and help to protect interesting
quantum states [248, 195, 20]. Understanding how robust phases can emerge as a consequence ofsuch interplay is highly desirable. However, dissipative many-body interacting quantum systems
are generically hard to model and understand. In this thesis we present some of the methods
developed to deal with several cases relevant to recent experiments.

1.4

Outline of this thesis
In this thesis, comprehensive studies are conducted on non-equilibrium many-body behaviors

in long-range interacting systems, especially where the interplay with dissipation is relevant.
Chapter 2 introduces the origin and form of dipole-dipole interactions and their remarkablefeatures, and provides the necessary background for the physics discussed in the following chapters.

5
The investigation of dipolar physics starts in Chapter 3, which discusses thermalization processes of dipolar molecules, explicitly KRb, confined in a quasi-two-dimensional trap. The main
focus of the chapter is to determine possible ways to achieve evaporative cooling of the moleculargas. For efficient evaporation it is important that inelastic collision processes, for KRb molecules
mainly coming from chemical reactions, become subdominant. We discuss the competition between
elastic and inelastic events and find regimes where cooling to quantum degeneracy could be possible. The calculations presented in this chapter can guide current experiments in their effort tocreate high-phase-space-density gases of polar molecules.
Chapter 4 studies a different geometry, where molecules are confined in an optical lattice andtunneling along one dimension is allowed. We develop theoretical models to describe the tunnelingdynamics in the presence of fast on-site losses. Our calculations reveal the important role played bymultiple lattice bands and are capable of reproducing the experimental observations. They explain
the suppression of molecule losses as a result of the quantum Zeno effect. Although in this chapter
we focused on KRb molecules, the theory presented is expected to be applicable to a variety ofother highly dissipative systems.
Alkaline earth atoms have important applications in quantum many-body physics and quantum metrology. They also represent a system where the interplay between long-range dispersiveand dissipative interactions is relevant. All the following chapters of this thesis are devoted to
explore the collective phenomena that can result from these interactions.
Chapter 5 discusses a possible way to realize a quantum analogue of the phenomenon ofsynchronization of classical oscillators in an array of incoherently pumped atoms. In this system, macroscopic phase coherence is generated by long-range interactions. Quantum correlations,entanglement, and the spectral properties of atoms in the coherent and incoherent phases are investigated in detail, providing an understanding of the open questions regarding the quantum aspectof synchronization.
Chapter 6 develops theoretical models to study the fluorescence properties of an ensembleof atoms driven by an external laser. Dipole-dipole interaction is shown to collectively affect the6line center and linewidth of the fluorescence spectrum. It is also found that atomic motion cansignificantly modify the spectral lineshape. We further show that, the theories presented can beapplied to successfully model a recent experiment done in 88 Sr atoms.
In chapter 7, we show that the same model used to describe the fluorescence in Sr atoms
in chapter 6 predicts the emergence of Weyl quasiparticles in a dense array of frozen atoms in athree-dimensional cubic lattice. Weyl excitations possess fascinating topological properties, andremarkably, they are found in this implementation to be robust against dissipation, due to themechanism of subradiance.
The theoretical approaches developed here are applicable to general long-range interacting
systems, such as trapped ions, atoms trapped in photonic crystals, an ensemble of nitrogen vacancy
spin impurities in diamond, systems of Rydberg atoms, among others.
Publications summarized in this thesis:
Chapter 3:
Evaporative cooling of reactive polar molecules confined in a two-dimensional geometry
B. Zhu, G. Quémenér, A. M. Rey and M. J. Holland, Phys. Rev. A, 88, 063405, 2013.
Chapter 4:
Suppressing the loss of ultracold molecules via the continuous quantum Zeno effect
B. Zhu, B. Gadway, M. Foss-Feig, J. Schachenmayer, M. Wall, K. R. A. Hazzard, B. Yan, S. A.
Moses, J. P. Covey, D. S. Jin, J. Ye, M. Holland and A. M. Rey, Phys. Rev. Lett., 112, 070404,
2014.
Chapter 5:
Synchronization of Interacting Quantum Dipoles
B. Zhu, J. Schachenmayer, M. Xu, F. Herrera, J. G. Restrepo, M. J. Holland and A. M. Rey, New
J. Phys., 17, 083063, 2015.

7
Chapter 6:
Light scattering from dense cold atomic media
B. Zhu, J. Cooper, Jun Ye, and A. M. Rey, Phys. Rev. A, 94, 023612, 2016.
Collective atomic scattering and motional effects in a dense coherent medium
S. Bromley, B. Zhu, M. Bishof, X. Zhang, T. Bothwell, J. Schachenmayer, T. L. Nicholson, R.
Kaiser, S. F. Yelin, M. D. Lukin, A. M. Rey and J. Ye, Nature Communications, 7, 11039, 2016.
Chapter 7:
Emergent Weyl quasiparticles in three-dimensional dipolar arrays
S. V. Syzranov, M. L. Wall, B. Zhu, V. Gurarie and A. M. Rey, Nature Communications, 7, 13543,
2016.

Chapter 2

Radiating dipoles

2.1

Dipolar interactions: origin and characteristics
The origin of electric dipole-dipole interactions can be formally derived by considering acollection of atoms exposed to an electromagnetic field. Specifically, we consider an ensemble ofatoms exhibiting a J = 0 → J 0 = 1 transition driven by an electromagnetic field. The Hamiltonian
governing this system is [94, 45, 110, 187]Ĥ = Ĥ0 + V̂ ,Ĥ0 = ~ω0X α†Xb̂j b̂αj + ~ωk â†k âk ,
j,α

V̂
D̂j

= −
=

X

X

(2.1)
(2.2)

k,

D̂j · Ê(rj ),

(2.3)

α
êα d(b̂α†j + b̂j ),(2.4)

gk k (eik·r âk + H.c.),

(2.5)

j

α

Ê(rj ) =Xk

where the notation |αi is used to denote the three different excited levels, and j to label the atoms.
For convenience, the Cartesian basis is chosen, e = (x, y, z), |αi = |xi , |yi or |zi. The latter are√√
related to the angular momentum basis by |xi = −(|+1i − |−1i)/ 2, |yi = −i(|+1i + |−1i)/ 2
and |zi = |0i. b̂α†j = |αj i hgj | is the raising operator from the ground state to the state |αi of thej th atom, and d is the atomic dipole moment. ω0 and ωk denote the atomic angular frequency andthe photon angular frequencies, respectively. k () is the momentum (polarization) of the photon.
ε0 is the vacuum permittivity, and V is the photon quantization volume.

9The evolution of the composite system of atoms and photons is described by
dρ̂af
dt

i
= − [V̂ 0 (t), ρ̂af ],
~

(2.6)

V̂ 0 (t) = eiĤ0 t/~ V̂ (t)e−iĤ0 t/~

(2.7)

=

XX
j,α k,

† −ik·rj +i(ω0 +ωk )tik·rj +i(ω0 −ωk )t
gk d(êα · ˆ)[b̂α†+ b̂α†
j âk ej âk e† −ik·rj −i(ω0 −ωk )t+b̂αj âk eik·rj −i(ω0 +ωk )t + b̂α†].
j âk e(2.8)

Given the large number of photon modes, the above equation is not very useful. Instead, we look
for an equation describing the system of atoms only. To this end, we can formally integrate Eq.
(2.6), and obtain
dρ̂af
dt

= −

i
~

Z t

dτ [V̂ 0 (t − τ ), ρ̂af (t − τ )].

(2.9)

dτ [V̂ 0 (t), [V̂ 0 (t − τ ), ρ̂af (t − τ )]].

(2.10)0

Substituting this back to Eq. (2.6) leads to
dρ̂af
dt

1
= − 2
~

Z t
0

This integro-differential equation gives the density matrix of the composite system via its dependence on all its past history. To further simplify it, we adopt the Born-Markov approximation,
which assumes that the correlation between photon modes can be neglected, and that the atomfield correlation time is short compared to the characteristic evolution time of the atomic system.
These assumptions are reasonable for many practical applications, considering that the large number of degrees of freedom of the photon field and the fact that the atom-field coupling is too weak tosignificantly modify the field. In this thesis, we are only interested in cases where these assumptions
are valid. Thus we can factorize the density matrix as ρ̂af (t − τ ) = ρ̂(t) ⊗ |0i h0|, where we also let
the photon field remain in the vacuum state |0i, following from the above assumptions. ρ̂ is the
reduced density matrix of the atomic system, which evolves as

dρ
dt

1
= − 2 Trf
~

Z ∞
0


dτ [V̂ (t), [V̂ (t − τ ), ρ̂(t) ⊗ |0i h0|]] ,00(2.11)

10
where Trf is the trace over the photonic degrees of freedom. Substituting the expression for V 0 (t)into the above, we obtain
dρ
dt

1
= − 2
~

Z ∞

dτ

0

XX

gk2 d2j,l k,
0Xα,α0(êα · ˆ)(êα0 · ˆ)0α† ik·rjl +i(ω0 −ωk τ )αα× {(b̂α†j b̂l ρ̂ − b̂l ρ̂b̂j )e
0

0

α† α
α
−ik·rjl +i(ω0 +ωk τ )+(ρ̂b̂α†j b̂l − b̂j ρ̂b̂l )e00+(b̂αj b̂αl † ρ̂ − b̂αl † ρ̂b̂αj )eik·rjl −i(ω0 +ωk τ )00+(ρ̂b̂αl † b̂αj − b̂αj ρ̂b̂αl † )e−ik·rjl −i(ω0 −ωk τ ) },(2.12)

where rjl = rj − rl .
V
The summation over k can be replaced by an integral (2π)
3R
the integral dΩ can be explicitly implementedZdΩ±ik·rXX α,α0(êα · ˆ)(êα0 · ˆ) = 4π

X
α,α0

R

k 2 dkdΩ. The angular part of[A(k)(êα · ˆ)(êα0 · ˆ) + B(k)êα · êα0 ],sin(kr)cos(kr)sin(kr)
+3
−3
,
3
kr
(kr)
(kr)2
sin(kr) cos(kr) sin(kr)
−
+
.
kr
(kr)3
(kr)2

(2.13)

A(k) = −(2.14)

B(k) =(2.15)

These seemingly convolved forms come from the fact that photons have two different polarizations.
The integral over time can be performed using the following relations
Z ∞
0
Z ∞

dτ ei(ω0 ±ω)τdτ e−i(ω0 ±ω)τ01,
ωk ± ω01= πδ(ωk ± ω0 ) ∓ iP.V.
.
ωk ± ω0= πδ(ωk ± ω0 ) ± iP.V.

(2.16)(2.17)

where P.V. denotes the principal value. Carrying out the correspondent integrals result in
dρ̂
dt

i
= − [Ĥdd , ρ̂] + L[ρ̂].
~(2.18)In the above expression, the dissipator L comes from the integration over the Dirac Delta
function, which enforces ω = ω0 . The Hamiltonian part comes from the principal value integral,

11which involves ω 6= ω0 , accounting for the processes involving virtual photons. Explicitly, we haveĤdd = ~XX00αα α† αgjlb̂j b̂l ,(2.19)

j,l α,α0L[ρ̂] =

XXj,l α,α0

0

0

0

α† α
fjlαα [2b̂αj ρ̂b̂α†i − {b̂i b̂j , ρ̂}],(2.20)

with dipole-dipole interactions
αα
gjl

0

= Re[Gααjl ],0= Im[Gαα
jl ],

fjlαα

0

0

(2.21)(2.22)

where
Gαα
jl

0

=

a(r) =

3Γ
[a(rjl )(êα · r̂jl )(êα0 · r̂jl ) + b(rjl )(êα · êα0 )],4
eik0 reik0 reik0 r+ 3i 2 2 − 3 3 3 ,
k0 r
k0 r
k0 r

b(r) = −eik0 reik0 reik0 r−i 2 2 + 3 3.
k0 r
k0 r
k0 r

(2.23)(2.24)(2.25)

k 3 d20is the spontaneous emission rate, k0 = ω0 /c is the wavevector
In the above equations, Γ = 3π~ε0of the dipole transition, and êα is the dipole polarization direction corresponding to the transition|gi → |αi. Note that although the above equations were derived for the case of a J = 0 → J 0 = 1
transition, the procedure can be generalized straightforwardly to more general dipole transitions.

2.2

Near-field and far-field interactions
In this section we take a closer examination of the dipole-dipole interactions. In addition to

the dispersive interactions indicated by the real part of G, the coupling to radiation field results
in an imaginary part that represents the dissipative interactions. An illustration of these interactions is shown in Fig. 2.1. The structures and spatial dependence of the dissipative and dispersiveinteractions are generally very different. For dipole-dipole interactions, Γ sets the dominant energy scale, which depends on the dipole moment, and 1/k0 sets the relevant length scale in theseinteractions. When k0 r  1, namely, when the interparticle spacing is much smaller than the00αα diverges as ∼ Γ/(k r)3 , while f αα approaches a constant Γ, thus becomes
photon wavelength, gjl
0
jl

12
negligible. Then in this limit the dipolar interactions reduce to the conventional electric dipolarinteraction, namely

Figure 2.1: Spatial dependence of dipole-dipole interactions. The diagonal parts of the imaginary(f zz ) and real (g zz ) dipolar interactions between two dipoles are plotted, for three different interparticle spacings. While both types of interactions depend on geometry, they are nonidentical, andthe elastic interactions are highly sensitive to the spatial separation and orientation of the particles.
The maximum value for imaginary and real interactions for fixed |rab |√is denoted as fmax and gmax ,respectively. The cone illustrates the magic angle, θm = arccos(1/ 3), along which the elastic
interactions vanish.

V̂dd (r) =d2α0[(êα · êα0 ) − 3(êα · r̂)(êα0 · r̂)]b̂α†j b̂l .
3
4πε0 r0X
αα(2.26)An exemplary system of this regime is ultracold KRb molecules prepared in the lowest two rotational
states.
ik r

In the opposite limit, k0 r  1, the dominant term in the dipole-dipole interactions is ∝ e r0 ,which shares the same behavior as the Green’s function of a photon propagating in free space. Thisis the far-field regime. In fact, it is also possible to modify the form of the interactions by selecting
the photon mode to which atoms are coupled. By loading atoms inside a single-mode Fabry–Perotcavity, a collective interaction that does not decay with r can be generated. Tunable power-law
long-range interactionscan can also be engineered by coupling atoms to photonic crystals [69].
In the case where k0 r ∼ 1, the near-field and far-field terms ∝ 1/r3 , 1/r2 , 1/r are of similar
magnitude. Neither the dissipative and dispersive part of the Hamiltonian, nor the near-field andfar-field components commute with each other, and thus this case corresponds to an interesting

13
regime where all terms are relevant. Fig. 2.2 lists the properties of dipole transitions in several experimentally relevant systems. For alkaline earth atoms, such as Sr atoms, it is possible to probe theintermediate regime where the system is affected by the complex interplay between all components
of dipole-dipole interactions. While this complexity imposes a challenge to theoretical treatment, italso enables extra control knobs and allows for new phenomena. In a recent experiment, Sr atoms
have been loaded into a three-dimensional lattice, and the clock transition (1 S0 →3 P0 ) has beenprobed [38]. Even though the transition matrix element of the transition is ultranarrow (Γ ∼ mHz),
dipolar interactions might play an important role in the future performance of these 3D opticallattice clocks.
Polar molecules+-

Alkaline earth atoms

+

Sr

-

Rotational levels
Vibrational levels
N=0	
  →	
  N=1 (KRb) v=0 →	
  v=1 (KRb)

Electronic orbitals
1S →	
  3P (87Sr)
00Electronic orbitals
3P →	
  3D (87Sr)
01

k0 a ⇠ 2 ⇥ 10 5

k0 a ⇠ 2 ⇥ 10 2

k0 a ⇠ 3.6

k0 a ⇠ 0.5

d ⇠ 0.5Debyed ⇠ 0.004Debyed ⇠ 0.001Debyed ⇠ 4Debye

Figure 2.2: Typical transitions relevant for dipolar interactions in several AMO systems. a isthe optical lattice spacing estimated from experimental realizations [260, 187, 249, 250]. Different
choices of internal states allow us to access different dipolar regimes .

In general, dipole-dipole interactions depend on the orientation of atomic dipoles. The offdiagonal elements in Eq. (2.23), i.e., α 6= α0 , couple states with different angular momentum
projections. This coupling is a quantum analog of the Einstein-de Haas effect [73], and can be also
seen as an effective spin-orbit coupling. As will be discussed in Chapter 7, it gives rise to Weyl
quasiparticles in a three-dimensional array of dipoles [248].

14

2.3

Superradiance and subradiance
To illustrate some of the interesting phenomena arising from long-range dipole-dipole inter-actions, we first consider a simplified case. We include only two levels in the transition, |gi and|zi, thus we can define σ̂j− = b̂zj = |gi hz|. We also consider the case where dipoles are very close toeach other k0 r  1 and thus fijzz ∼ f . We focus on the dissipative part of the interactions. Then
only the second part of Eq. (2.18) is relavant, and the dissipator becomesL[ρ̂] = f (2Jˆ− ρ̂Jˆ+ − Jˆ+ Jˆ− ρ̂ − ρ̂Jˆ+ Jˆ− ),where Jˆ± =(2.27)±j σ̂j are collective angular momentum operators. Since the dissipator commutes withP

the total angular momentum J, it is convenient to use a representation in the angular momentum
basis |J, M, βi, where J = N/2, N/2 − 1, ...(N mod2)/2 is the total angular momentum, M =−J, −J + 1, ...J is its projection on the quantization axis, and β is an additional quantum number
to label the states with the same J and M . Overall, the total number of states equals 2N , thus forgeneral J < N/2, there are multiple states with the same J and M . The manifold with J = N/2
is the symmetrized Dicke states. The above dissipator induces decay only within the same J
manifold. An example of three atoms is given in Fig. 2.3, where the decay rates can be explicitly
calculated for all states. For states in the symmetrized Dicke manifold, the decay rate can exceed
the independent atom spontaneous emission rate. This phenomenon, known as superradiance, wasfirst theoretically predicted by Dicke in 1954 [67]. In fact, the decay rate of symmetrized Dicke
states is Γ(J + M )(J − M + 1), which is maximal for the state |N/2, 0i [67]. Physically this can beunderstood from the fact that there are N indistinguishable decay channels, so the decay rate isenhanced by ∼ N . For states in other manifolds, the individual atomic states are not symmetric,
and the decay rates can be even smaller than the rate of independent atoms. These are the so-called
subradiant states, which are less affected by the dissipative environment.
Outside the near-field limit, dipole-dipole interactions are not collective, but superradiant andsubradiant states still exist, though consisting of a mixture of the above Dicke states and requiring
more sophisticated theoretical approaches. Superradiance, which was originally conceived for a

3
J=
2
M=

3
2

3

1
2

4

M=

1
J= ,
2

1
J= ,
2

=1

15

=2

|eeei

M=

1
2

M=

3
2

3

|geei + |egei + |eegi
p3

1p (2|eegi
6|egei|geei)1
p (|geei
2|egei)|ggei + |gegi + |eggi
p3

1
p (2|ggei
6|gegi|eggi)1p (|eggi
2|gegi)|gggi

Figure 2.3: Dicke states for three atoms. |ei and |gi denote the excited and ground state in theindividual atom basis. Also indicated are the decay rates under the dissipator Jˆ+ Jˆ− . States in red
colors decays at least at rate Ne Γ, while the decay rate for states in blue (or black) color is < Ne Γ(or =0), where Ne denotes the number of excitations in the system.

small dense sample, can be observed experimentally in large and dilute ensembles. A demonstrationof this phenomenon is performed in a recent experiment with bosonic Sr atoms. We show inchapter 6 that using a microscopic theory derived from Eqs. (2.19) and (2.20), we are able toquantitatively describe the experiment. Collective emission has also been observed in various
experiments performed using photonic crystal waveguides and in cavity systems [86, 29, 182].
In contrast, subradiant states are usually difficult to prepare or probe in experiments withmany atoms. In chapter 7, we propose a protocol to realize subradiant states in a dipolar array,
without requiring complicated control of external fields. We also show there, that subradiancestabilizes Weyl quasiparticles.

Chapter 3

Evaporative cooling of reactive polar molecules confined in a two-dimensional
geometry

3.1

Introduction
Polar molecules that exhibit strong dipole-dipole interactions have tremendous applications,

ranging from quantum many-body physics [253, 14, 136] and quantum information processing [66]to precision measurements [282] and ultracold chemistry [41]. The parameter regime of interestis generally ultralow temperature and high phase-space density, where novel quantum features
can emerge. In practice, it turns out to be difficult to cool molecules into the desired regime
using standard methods due to their complicated internal level structure. In the past few years,
significant experimental progress has been made towards this goal through the demonstration of amethod for preparing a gas of fermionic KRb molecules in the lowest electronic, vibrational, androtational quantum state, with a temperature T at the verge of quantum degeneracy (i.e. T /TF ∼ 1
where TF is the Fermi temperature) [179, 189, 180]. From this starting point, one would like tofurther increase the phase-space density by implementing evaporative cooling—demonstrated to be
one of the most useful cooling methods for quantum gases [32, 58, 65].
A fundamental limitation to the effectiveness of evaporative cooling for polar molecules istheir fast losses. For KRb molecules, the losses arise mainly from exothermic chemical reactions,i.e. KRb + KRb→ K2 + Rb2 . In such reactions, molecules prepared in different internal states
can undergo barrierless collisions, with a lifetime of only ∼10 ms [180, 201]. In contrast, identicalfermionic KRb molecules at ultra-low temperature are protected by the p-wave barrier, which

17
potentially results in a much slower reaction rate [180]. However, due to the anisotropic nature of
the dipole-dipole interaction, molecules of the correct orientation can be attracted towards eachother by experiencing “head-to-tail” collisions. As a result, the p-wave barrier can be greatly
suppressed. Such a loss mechanism is further facilitated by the application of a strong external
electric field that increases the dipole strength.
Significantly, these obstacles can be overcome by confining the molecules into quasi-twodimensional traps, which can be generated by a one-dimensional optical lattice. In this case, theadverse collisions can be greatly suppressed and the reaction barrier effectively raised [60, 202,200]. Furthermore, both the elastic and inelastic collision rates can be tuned by controlling theapplied external electric field and the trapping potential. Thus, while most previous experiments
implementing evaporative cooling were performed in three-dimensional geometries [123], here weare explicitly interested in focussing on evaporative cooling in two-dimensions, with the anisotropic
collisions that arise from the dipolar interaction. A detailed understanding of this situation would
be beneficial for future experimental realizations.
Given this motivation, we theoretically investigate evaporative cooling of molecules. We use
both Monte Carlo (MC) simulations and models developed on the basis of kinetic theory to study
the efficiency of evaporative cooling with parameters applicable to current state-of-the-art KRb
experiments. This chapter is structured as follows. In Sec. 3.2 we explore the effect of anisotropic
collisions on evaporative cooling. In Sec. 3.3 we consider the effect of reduced trap dimension,i.e. two-dimensional rather than three-dimensional traps. In Sec. 3.4 we apply the MC method todetermine the optimum evaporative cooling trajectory for KRb molecules. In Sec. 3.5 we discuss
the potential anti-evaporation mechanism arising from the energy dependence of the inelastic loss.

18

3.2

Evaporative cooling with anisotropic collisions

3.2.1

Anisotropic elastic collisions between polar molecules

Evaporative cooling relies on removing particles with above-average energy and redistributing
the residual energy among the remaining particles by elastic collisions so that the temperature falls.
For polar molecules, the characteristic parameters that encapsulate the elastic scattering process
can be dramatically modified by the application of external fields. In the presence of an external
electric field, the dipole-dipole interaction between polar molecules can mix different partial waves
and give rise to highly-anisotropic scattering. Moreover, for identical fermions, the lowest total
angular momentum partial wave that has the correct symmetry is p-wave, so the elastic collisions
are anisotropic at ultralow temperature even in the absence of applied fields [87].
Considering current experimental conditions [60], we adopt the differential and total scattering cross-section computed for KRb molecules in two-dimensions at an induced dipole momentd = 0.2 debye as a function of the collision energy Ec .

FIG. 3.1(a) shows the dependence ofdifferential cross-section on the scattering angle, φ, which is the precession angle of the relative
momentum during the collision. Scattering is mainly forward (0 or 2π) and backward (π). The
peak is more pronounced as the collision energy increases due to the larger contribution of higher
partial waves. This differential cross-section is well parametrized by the empirical function


dλ(φ)
dφ





0
= λ a(cosφ)2α + a0 (cosφ)2α ,

(3.1)

Ec

with a, a0 , α and α0 constants that are real and positive, and with the total cross-section given by
Z 2π
λ=

dφ
0

dλ(φ)
.
dφ

(3.2)

The dependence of λ on collision energy is shown in FIG. 3.1(b). Our best parametrization forvarious scattering energies is given in Table 3.1. Note that at Ec ∼ 1 nK, the angular dependenceof the differential cross-section is well approximated by that of a p-wave collision, since a  a0 andα ≈ 1.

19
HaL

3.0
ò

2.0

÷

÷

Ec=1ΜKò

Ec=100nKì

Ec=10nKò æ

Ec=1nK

ò
ò÷ò

ò

ò

ò
÷ ÷

ò

ò

HbL

ò
ò÷

12
ò

0.5

Κ
ì ÷ ò
ì

ò ÷
ò
ò ÷ ì
ì ì÷
ì
ì
ì
ò
òì
÷
÷
÷
ìò
ò
òì
ò
ì ÷
÷ ì
÷
÷
ò
ò
ò
ò
ì
ì
÷ ò
÷ ò
ò ÷
ò ÷
ì
÷ ìò
÷ ì
ò ÷÷
ò ÷÷
ì
÷ ÷ òòæ æ æ æ÷ ÷ ì
æ
æ
æ
æ
ì
÷
÷
æ æ æ
ò
ò
ò
æ æ æ
÷ æ æ
÷ ÷÷
ì
ì
æ÷æ
òì
ò÷æ
òòò
÷÷
÷ ÷æ æ
æ
æ
ì
òò
÷÷÷
÷÷÷
æ
æòæ
ò
÷ì

0.0
0.0

÷

10

÷
ò

1.5
1.0

HcL

14

Λ

dΛdΦ H10-6cmradL

ò

2.5 ÷

8
6

ò

ò

0.5

1.0
Φ HΠL

1.5

æì

4
2
0

2.0

0

1

2

3
Α

4

5

6

Figure 3.1: (a) Two-dimensional differential scattering cross-section for elastic collisions as afunction of the scattering angle. Four collision energies are shown; Ec = 1 nK (blue disk), 10 nK(green diamond), 100 nK (red triangle), and 1 µK (black star). The induced dipole moment
is d = 0.2 debye and the harmonic frequency of the one-dimensional confinement potential isν = 23 kHz, The solid lines show the empirical formula Eq. (3.1). (b) Total scattering crosssection for elastic collisions (solid line) and reactive processes (dashed line) as a function of thecollision energy for an induced dipole moment of d = 0.2 debye and a one dimensional confinement ofν = 23 kHz. The ratio of elastic to reactive is indicated at four different collision energies; Ec =1 nK(blue point), 10 nK (green point), 100 nK (red point), and 1 µK (black point). (c) Numberof collisions κ required for thermalization for different differential scattering cross-sections. The
magenta solid line shows Eq. (3.10). The purple circles are results from the MC simulations, andare shown with their small statistical error bars. The four symbols (blue disk, green diamond, red
triangle, black star) correspond to the same energies as (a).
Ec (nK)
1
10
100
1000

a
0.31
0.27
0.24
0.34

a0
0.0005
0.06
0.21
0.40

α
1.00
1.00
1.19
2.47

α0
2.19
2.09
7.03
26.40

λ(10−6 cm)
0.38
3.67
5.99
3.42

Table 3.1: Derived parameters from the fit of Eq. (3.1) for the scattering energies shown in Fig. 3.1.

3.2.2

Thermalization rate

The thermalization rate characterizes the timescale needed for a system to redistribute energy
after an evaporative cut. To quantitatively investigate the thermalization under anisotropic collisions, we adopt the typical experimental procedure of cross-dimensional thermalization [173, 64].
In order to isolate the effect of anisotropic elastic collisions, we will neglect inelastic collisions
completely for these thermalization calculations.
Consider a gas of N molecules in a two-dimensional harmonic trap with a slight initial20
imbalance of temperatures along each axis. This initial condition can be prepared in experimentby parametric heating of an equilibrium gas [173]. We define an effective temperature Ti in theith direction in terms of the total energy Ei by kB Ti = Ei /N = p2i /(2m) + mωi2 x2i /2, where ωiis the trapping frequency, m is the molecule mass, and p2i /(2m) and mωi2 x2i denote the averagekinetic energy and potential energy. We assume that the system is well described by the Boltzmann
distribution
 X p2f0 (x, p) = n0 exp −( i + mωi2 x2i )/(2kB Ti ) ,
m

(3.3)

i

where n0 guarantees normalization, i.e.
1(2π~)2Z

d2 xd2 p f0 (x, p) = N .

(3.4)

Without loss of generality, we assume Ty = (1 − ξ)Tx with ξ > 0. Elastic collisions lead to an
exchange of energy between the x and y directions and reduce the relative temperature difference.
The rate of such change is [205]
dEx
dt

dTx
1
= N kB=dt
2m(2π~)4Z

d2 xd2 p1 d2 p2 dφ0

× f0 (x, p1 )f0 (x, p2 )|p1 − p2 |dλ(φ0 )
∆Ex .
dφ0

(3.5)

This involves the energy change per collision given by∆Ex =1|p1 − p2 |2 (cos2 φ0 − cos2 φ),
4m

(3.6)

where φ and φ0 are the angles between the relative momentum and the total momentum before andafter the collision respectively. The period of time it takes to thermalize, τ , can be defined as the
1/e decay time of the temperature difference, so that
dTx
=
dt

1
− (Tx − T0 ),
τ

(3.7)

where T0 = (Tx + Ty )/2 is the temperature at equilibrium. From Eq. (3.5) and Eq. (3.7), we obtain
the average number of elastic collisions required for the system to thermalize, κ, in the following
way. We begin by defining the collision rate γZλγ=
d2 xd2 p1 d2 p2 f0 (x, p1 )f0 (x, p2 )|p1 − p2 |,m(2π~)4(3.8)

21
so that κ ≡ limξ→0 (τ γ), independent of the initial temperature and trapping frequencies [64, 207].
Combining Eq. (3.3)–Eq. (3.8) with the form of the differential cross-sections given in Eq. (3.1)
leads to the following result:
κ=

8
√
15 π



a Γ (α + 1/2)
2α + 2 Γ(α + 1)0
Γ (α0 + 1/2) −1a.
+ 0
2α + 2 Γ(α0 + 1)

(3.9)

When a0 = 0 and α0 = 0, this reduces to the particularly simple expressionκ=

16(2α + 2),15

(3.10)

which corresponds to the differential cross-sectiondλ(φ)= a cos2α φ.
dφ(3.11)In order to interpret these results, for each scattering energy Ec , we calculate κ using Eq. (3.9), andplot the result on the (α, κ) line, as prescribed by Eq. (3.10), with markers (star, square, triangle,
diamond, and disk), as shown in FIG. 3.1(c). The use of the linear relationship between α and κ
is supported by results obtained from MC simulations, as described in Appendix A.1, utilizing
the differential cross-section form of Eq. (3.11) directly (purple circles). The exception is at large
α, where the assumption adopted in deriving Eq. (3.9)—namely that of a Boltzman distribution
parametrized by an effective temperature—becomes poor when the thermalization is too slow.
The calculations demonstrate a strikingly strong dependence on the anisotropy of the collisions for rethermalization. For α = 0 the collisions are isotropic (s-wave like) and κ ≈ 2.1. This
compares with κ ≈ 4.3 at α = 1, which describes p-wave collisions. For Ec = 1 µK we find κ ≈ 9.7,
which is many times that for isotropic collisions, implying a significant increase in the numberof collisions required for rethermalization. One may have anticipated this since the differential
cross-section is sharply peaked in the forward and backward directions and energy is not efficiently
redistributed. Also note that the outcome for the differential cross-section of KRb with Ec =1 nK isvery close to that from Eq. (3.10) with α = 1, indicating p-wave elastic collisions between identicalfermions at ultra-low temperature.

22
3.2.3

Evaporation with anisotropic collisions

In evaporative cooling experiments, inelastic collisions compete with elastic collisions, andthus a slow thermalization rate gives rise to a reduction in cooling efficiency. In this section, we
use MC simulations to quantify the efficiency of evaporative cooling in two ways: by the achieved
increase in phase-space density Ωf /Ω0 at the expense of losing a certain portion of the molecules inthe trap (see FIG. 3.2), and by the time required for the temperature to go down and Ω to increaseby a given amount (see FIG. 3.3).
6

lnHWf W0 L

4

2

0

isotropic
Ec =100nK
Ec =1ΜK
critical

-2
3

4

5

6

7

Η

Figure 3.2: Increase in phase space density vs truncation parameter η for different dλ(φ)
dφ . Here Ω0is the initial phase-space density, and the final phase space density, Ωf , was calculated when thedλ(φ)
ratio of molecule numbers was Nf /N0 = 0.1. Blue circles: dλ(φ)
dφ =const; red crosses: ( dφ )Ec =1µK ;green squares: ( dλ(φ)
dφ )Ec =100 nK ; black dotted line: critical phase space density Ωc for quantum
degeneracy.
.
In these simulations, we assume an instantaneous removal of all particles with energy greaterthan a cut-off energy. The cut-off energy, t , evolves during the evaporation trajectory with theconstraint that a truncation parameter, η = t /E, is kept constant, where E is the average energyof the nonequilibrium distribution. The comparison between isotropic and anisotropic collisions ismade by calculating the evaporation trajectory for a variety of differential cross-sections, startingwith the same initial molecule number. To simplify the comparison, for both elastic and inelastic
collisions, we keep the total cross-sections constant while varying the differential cross-section for

HaL
NN0

1

23

0.5
0
0.3
0.2
0.1
0
60
40
20
0
0

WW0

0
0.3
0.2
0.1
0
5
4
3
2
1
0
0

T HΜKL

0.5

WW0

T HΜKL

NN0

1

HbL

1

2
3
time HsL

4

1

2
3
time HsL

4dλ(φ)
Figure 3.3: Trajectories of evaporation for (a): ( dλ(φ)
dφ )Ec =1µK at η ∼ 3.4 and (b): dφ =const atη ∼ 4.3, under the same initial conditions as in FIG. 3.2. In comparison with (a), it is apparent
that in (b) the temperature T drops faster and the phase-space density Ω increases much more
rapidly.

elastic scattering. We also assume the cross-sections to be energy independent.
The molecules are initially simulated from a truncated Boltzmann distribution utilizing the
cut-off energy [155] t :f (x, p) =
p2+ mω 2 x2 )/(2kB T )mp2
1
×Θ t − (+ mω 2 x2 ) ,2m 2

n00 exp −((3.12)

where Θ is the Heaviside function, n00 is the normalization, and T is the initial temperature, held
fixed for different simulations. Although the energy distribution of the molecules is intrinsically
non-equilibrium here, we may assign it a phase-space density, Ω, as that of the corresponding
equilibrium distribution that has the same molecule number and average molecule energy.
In FIG. 3.2, Ω is plotted for the case of evaporation trajectories in which 90% of the initialmolecules are lost. For anisotropic collisions, we calculate evaporation trajectories for the differential cross-sections for Ec = 100 nK and Ec = 1µK given in the subsection 3.2.2. The resultspresented in FIG. 3.2 show that the maximum efficiency achievable by varying η is much lower

24
for anisotropic collisions than that achievable with isotropic collisions. In addition, the maximum
efficiency is reached at smaller η. This suggests, for example, that if t is set by the trap depth
in an experiment, a shallower trap is desirable for evaporating a gas with anisotropic collisions,compared with evaporating a gas with isotropic collisions, at the same temperature and collisionrate.
For these calculations, the inelastic to elastic collision rate ratio was held fixed at ζ = 1/200.
This was chosen based on typical experimental conditions for KRb molecules (see Sec. 3.4). The
value of ζ does not qualitatively change the above comparisons; it only changes the shape of eachcurve [123], and as one would expect, the maximum efficiency and the corresponding η decreases
as ζ increases.
In FIG. 3.3, we plot the time-dependent trajectories of evaporative cooling for isotropic
collisions with η = 4.3, and for anisotropic collisions for Ec = 1 µK with η = 3.4. These valuesof η were chosen such that Ωf /Ω0 is maximum in FIG. 3.2, and thus most efficient. It is evident
from this plot that to reach the same increase in Ω and decrease in T , it takes much shorter timewith isotropic collisions (FIG. 3.3(b)). This indicates a lower efficiency with anisotropic collisions,consistent with the results of FIG. 3.2.

3.3

Evaporative cooling in 2D traps
Physics in two-dimensional geometries can often be quite different from that in three-dimensions.

Unlike many previous evaporative cooling experiments, KRb molecules need, in general, to be confined in traps with reduced dimensionality to be stable. Such a configuration was also used toevaporate Cs atoms [105]. Here, we study the effect of reduced dimensionality on evaporative
cooling by comparing the efficiency in two-dimensional and three-dimensional traps.
The previous section showed that the existence of highly-anisotropic collisions slows down
thermalization and decreases the efficiency of evaporative cooling. A calculation analogous to Sec.

25
3.2.2 but for a three-dimensional harmonic trap gives5κ = (2α + 3),
6(3.13)

which implies κ = 2.5 when α = 0 (isotropic collisions). This result may suggest a potentiallyfaster thermalization rate and more efficient evaporation in a two-dimensional harmonic trap whencompared with a three-dimensional harmonic trap with everything else held fixed. However, sucha conclusion based soley on Eq. (3.13) would be premature, since the density of states depends
on the dimensionality. Specifically, it was found for the comparison between quadratic potentials
and linear potentials in three dimensions that changing the density of states can also change theevaporative cooling efficiency [123].
To incorporate such differences, we use MC simulation and a truncated-Boltzmann (TB)
method based on kinetic theory (see Appendix A.2) to calculate the evaporation trajectories. Here,we consider harmonic traps and isotropic energy-independent collisions in order to isolate the effectof dimensionality. Following a similar procedure to that of the previous section, we calculateln(Ωf /Ω0 ) at fixed Nf /N0 , under equivalent initial conditions for both types of traps. As shownin FIG. 3.4, both the MC and TB approaches show a lower achievable increase in the phase-spacedensity for a two-dimensional harmonic trap. This suggests that evaporative cooling is intrinsicallyless efficient in two-dimensional traps. The discrepancy between MC and TB, which increases forsmaller η, is caused by the discrepancy between the form of the energy distributions, which for TB
is constrained. At smaller η, the trucated Boltzmann distribution deviates significantly from theactual distribution of molecules, as calculated in MC.

3.4

Evaporative cooling of KRb molecules
In a quasi-two-dimensional trap, the elastic and reactive collisions between KRb molecules

both depend on both the strength of the external electric field and the confinement induced by thelattice. Under current trapping conditions in experiments, a favorable ratio between the elastic
and reactive processes can be reached at a moderate eletric field [60, 201]. We show in FIG. 3.1(b)

26
6
5

lnHWf W0 L

4
3
æ

æ æ

æ

æ

æ æ

à à
æ æ æ æ æàæà à à à à à

à

à

æ
æ

à
à

æ

æ

2

2D MC

3D MC

1

2D TB
3D TB

2D critical
3D critical

0
3

4

5

à

6

à
à

æ
æ
æ
æ

7

8

Η

Figure 3.4: Comparison of the efficiency of evaporative cooling in two-dimensional (2D) and threedimensional (3D) harmonic traps. ln(Ωf /Ω0 )’s are calculated at Nf /N0 = 0.1, and ζ = 1/200. Red(blue) line: 3D (2D) harmonic trap from method in Appendix A.2; red squares: 3D harmonic trapfrom MC simulation; blue disks: 2D harmonic trap from MC simulation; blue (red) dotted line: Ωcfor a 2D (3D) trap.

the total scattering cross-section for the elastic process and reactive process of KRb molecules at
d = 0.2 debye with the confinement along the lattice direction given by ν = 23 kHz. In the ultracold
3/2

1/2

regime, the elastic cross-section scales as Ec , while the reactive cross-section scales as Ec[146].

Furthermore, elastic processes are found to be generally faster than reactive processes, supporting
the potential for successful evaporative cooling [200] while the large elastic cross-section favors fast
rethermalization.
The quantitative knowledge of the elastic and reactive collisions between KRb molecules
allows us to apply MC simulation with realistic experimental parameters. In FIG. 3.5, we show
evaporation trajectories in a 2π × 20 Hz two-dimensional gaussian trap. For this simulation, we
have assumed the instantaneous removal of energetic molecules above the cut, as in Sec. 3.2.3
and Sec. 3.3. We also assume that the inelastic losses are only due to the reactive collisions thatare discussed above. The initial temperature was chosen to be within the accessible regime ofexperiments. The results show that a considerable increase in phase-space density can be achieved
via evaporative cooling. We note that there are further experimental effects that can in principle
reduce the achieved phase-space density increase from that calculated here, such as the finite rate

27
of removing energetic molecules and other sources of loss and heating. On the other hand, progress
towards realizing deeper lattices ( i.e., ν > 23 kHz), can potentially reduce ζ and thereby enhance
the cooling efficiency.

0.5

0
0.20
0.15
0.10
0.05
0.00
2.0
1.5
1.0
0.5
0.0
0.0

WWc

T HΜKL

NN0

1.

0.5

1.0
1.5
time HsL

2.0

Figure 3.5: Evaporative cooling trajectories for KRb molecules inside a 2D trap of 2π × 20 Hz,
with scattering cross-sections as computed in FIG. 3.1(b). The initial temperature was ∼ 200 nK,
with an initial-phase space density Ω0 ∼ 0.1Ωc , and η = 3.8.
.

3.5

Anti-evaporation in quasi-2D
Fast inelastic collisions are disadvantageous for evaporative cooling, since they not only reducethe molecule number, but can also cause heating known as “anti-evaporation” [180]. Since theability to identify the relative importance of different heating mechanisms is important in thedesign of experiments, in this section we quantitatively investigate heating due to two-body inelastic
collisions.
Anti-evaporation arises from the fact that inelastic collisions are more frequent in the highdensity region of the trap (density-selection). Molecules there have on average total energies thatare less than the ensemble average. Thus one may anticipate evaporative heating rather thanevaporative cooling to occur. In competition with this, however, is the effect that the inelastic

28
collision cross-section is dependent on the collision energy, i.e., molecules with energies higher than
the average are more likely to engage in a reactive collision and be removed (velocity-selection). In
three-dimensional harmonic traps, the density-selection mechanism wins over the velocity-selection
mechanism and consequently there is net heating caused by p-wave two-body losses [180]. In twodimensional trapping potentials, however, this is not necessarily true as we now show.
We consider a two-dimensional harmonic trap and assume a Boltzmann distribution as given
in Eq. (3.3), for which the number loss rate is
dN
=2
dt

Z

d2 xd2 p1 d2 p2 λ(Erel )vrel f0 (x, p1 )f0 (x, p2 ),(3.14)and the corresponding energy loss is
dE
=
dt

Z

d2 xd2 p1 d2 p2 λ(Erel )vrel f0 (x, p1 )f0 (x, p2 )(E1 + E2 ),(3.15)

where Erel and vrel are the relative energy and velocity between the two coliding molecules, andE1 + E2 is the total energy. From Eq. (3.14) and Eq. (3.15), with λ(Erel ) ∝ E 1/2 for p-wave
collisions (Sec. 3.4), which dominates for ultracold identical fermionic KRb molecules [60], one canderive the energy loss per molecule, which is given bydE= 2kB T.
dN

(3.16)

This is precisely equal to the average energy per molecule. Thus we have a remarkable result that
there is no net heating or cooling in a pure two-dimensional harmonic trap due to reactive loss. In
contrast, a similar argument leads to heating for three-dimensional traps, as shown in Ref. [180](heating rate there ∼100 nK/s).
However, we emphasize that Eq. (3.14) to Eq. (3.16) only takes into account the two-bodyinelastic collsions and assumes a Boltzmann distribution, which implies sufficiently fast rethermalization. Note that at zero electric field, the elastic collisions that lead to thermalization are
infrequent and can generally be neglected in experiments with KRb molecules [180], thus a Boltzmann distribution is not guaranteed.

29
To study possible effects caused by the deviation from a Boltzmann distribution during theevolution, we use MC simulations to calculate the molecule loss and temperature change in a
two-dimensional harmonic trap, assuming only two-body inelastic processes between molecules.
The results are plotted in FIGs. 3.6(a) and (b) respectively. The reactive collision rate is takenfrom FIG. 3.1 (b). The heating rate is found to be very small, illustrating a drastically different
quantitative behavior from that seen in three-dimensional traps.

HaL

1

NN0

NN0

1

0.5

0

0.40
0.36
0

0.5

time HsL
1.0

1.5

T HΜKLT HΜKL0.88

0.44

1 band2 bands
3 bands

0.5

00.48 HbL

HcL

0.84

2.0

0.76
0

HdL

0.80
0.1

time HsL

0.2

0.3

0.4

0.5

Figure 3.6: Heating due to two-body losses in a two-dimensional harmonic trap. Here (a) and (b)show the MC result in a two-dimensional harmonic trap where all molecules are in the lowest bandof the lattice potential. The results for (c) and (d) are obtained with the master equation approach.
The blue line assumes only 1 band along the lattice direction is populated. The red dashed line
(green dots) assumes 2 (3) bands are populated. N0 is the initial total molecule number. The initial
temperature is chosen to be around 800 nK, so that when two bands are included, there are ∼25%of molecules on the second band, and when three bands are included, there are ∼19% of moleculeson the second band and ∼6% on the third band.
.
The periodic array of two-dimensional pancake traps for confining KRb molecules is generated
by a one-dimensional optical lattice [60]. This forms energy bands separated by ∼ ~ωz , whereωz = 2πν and ν is the frequency of the harmonic approximation of the lattice potential. When thetemperature is sufficiently low, the longitudinal degree-of-freedom along the lattice axis is effectively
frozen out and one may consider solely the lowest band. However, as the temperature increases,

30
higher bands can be populated. For example, at T ∼ 800 nK in a lattice of ν = 23 kHz, about 25%
of molecules occupy higher bands.
With multiband population, collisions between molecules can occur as inter-band or intraband, as shown in FIG. 3.7. The inter-band collisions involve additional degrees-of-freedom along
the longitudinal direction (see FIG. 3.7(b)), and thus do not satisfy the assumptions leading to
Eq. (3.16) and can give rise to anti-evaporation heading [180].

Figure 3.7: Collisions between molecules in a quasi-2D trap. (a) intra-band collisions, occuring at
rate Γ̃(1) . (b) inter-band collisions, occuring at rate Γ̃(2) (see Appendix A.3).
.
We adopt a many-body Hamiltonian to describe these p-wave inelastic collisions [119, 243,
162]
Ĥp =

3πb3p ~2mZ

d3 r[(∇Ψ̂† (r))Ψ̂† (r) − Ψ̂† (r)(∇Ψ̂† (r))]·[(∇Ψ̂(r))Ψ̂(r) − Ψ̂(r)(∇Ψ̂(r))],(3.17)

where Ψ̂(r) is a fermionic field operator that annihilates a molecule at position r, and b3p is theinelastic p-wave scattering volume. The value of bp for KRb molecules can be found from thereactive cross-section calculated in the previous section. In a trap with transverse confinement ωrand longitudinal confinement ωz , the field operator can be expanded in the basis of non-interacting
harmonic oscillator eigenstates φrn and φzn as:Ψ̂(r) =

Xnĉn φrnx (x)φrny (y)φznz (z),(3.18)

31
where n = (nx , ny , nz ) enumerates the mode number along each dimension, and ĉn annihilates afermion in mode n.
Considering elastic collisions to be negligible, we assume there is no change of modes during
the losses, i.e., a molecule initially in mode n remains in mode n. With such an assumption,
each molecule can be labeled with its mode indices, and for identical fermions, there are no twomolecules with the same indices. The two-body losses can then be accounted for by jump operatorspÂn,m = Γn,m ĉn ĉm , which remove two molecules in mode n and m at a rate Γn,m determined by
the corresponding p-wave two-body inelastic collisions Eq. (3.17) (see Appendix A.3 for details).
When nz = mz , these correspond to the intra-band collisions (FIG. 3.7(a)); when nz 6= mz , these
correspond to the inter-band collisions (FIG. 3.7(b)).
The dynamics can then be described by the quantum master equation
1X
dρ̂
=(2Ân,m ρ̂Â†n,m − Â†n,m Ân,m ρ̂ − ρ̂Â†n,m Ân,m ).
dt
2 n,m

(3.19)

Since we have assumed that the molecules do not change modes during evolution, in order to solve
Q P
Eq.( 6.1), we can use ρ̂ = i n ρin,n δ(n − ni )|nihn|, where i = 1, 2, 3...N represents the ithmolecule that initially is in mode ni , N is the initial total number of molecules. Since the quantum
jumps here correspond to the reactive loss of pairs of molecules, the off-diagonal coherence terms
have been dropped. The relevant equation of motion under these assumptions isdρini ,nidtX

= −ρini ,niΓni ,mj ρjmj ,mj .

(3.20)

j6=i

The total molecule number at time t is thus
N (t) =

X

ρini ,ni (t),(3.21)iand the average temperature can be found fromT (t) =

X1
~ωr (nix + niy )ρini ,ni (t).
N (t)kB(3.22)i

Furthermore, the initial population in different bands can be determined byNαe−(1/2+α~ωz )/kB T= P −(1/2+α~ω )/k T ,
z
B
N
αe

(3.23)

32
where α = 0, 1, ... is the band index. For the initial condition, the density matrix is simulated
such that N α molecules are assigned to band index niz = α, and the transverse modes nix , niy are
randomly generated from a Boltzmann distribution with given initial temperature T . The final
result is averaged over many different simulations. In order to show the effect of including different
number of bands, we assum a cut-off in the highest band index. In our calculation, if a total of nmax
bands are taken into account, the population of the lower nmax − 2 bands is determined according
to Eq. (3.23). All of the reminder molecules are assigned to the highest populated band, nmax − 1.
In FIG. 3.6(c) and (d), we show the result with temperature T = 800 nK and lattice frequency ν = 23 kHz, which are typical for KRb experiments. While the small amount of multiband
population does not lead to a significant change in the loss rate for molecules (FIG. 3.6(c)), there
can nevertheless be significant heating. In consequence, it is necessary to greatly suppress the
population outside the lowest band for evaporative cooling experiments to be effectively performedin quasi-two-dimensional geometries when reactive loss of the form considered here is present.

3.6

Conclusion
Motivated by recent experiments of KRb molecules, we have applied MC simulation meth-ods and semianalytical approaches to perform a detailed study of evaporative cooling properties
in two-dimensional traps with anisotropic collisions. We have quantitatively analyzed the dependence of the thermalization rate on the anisotropy of the elastic collisions. Specifically, for our
calculations of the differential and total scattering cross-section for KRb molecules confined in aquasi-two-dimensional trap, we were able to investigate the efficiency of evaporative cooling for thepractical parameter regime of recent experiments. The dipole-dipole interactions resulted in highly
anisotropic elastic collisions which were disadvantageous for evaporative cooling when comparedwith isotropic collisions.
The reduced dimension of the trapping potential further decreased the efficiency of evaporative cooling when compared with conventional evaporation procedures for three-dimensional traps.
Nevertheless, we showed that the phase space density of KRb molecules can potentially be increased

33
using evaporative cooling with parameters that are accessible under current experimental conditions. Although this could be limited by further complicated real-world experimental details that
we have not considered, it is also the case that future experimental progress in increasing the ratio
between the elastic and inelastic collision rates could lead to more efficient evaporative cooling of
KRb molecules.
We also developed theoretical models to investigate the anti-evaporation induced by twobody losses. Our results highlighted the distinctions between the evaporation processes in twodimensional and three-dimensional geometries through the role of multiband excitations in a quasitwo-dimensional trap. We point out that the importance of multiband physics has also beenaddressed in recent literature [233, 161, 267].

Chapter 4

Suppressing the loss of ultracold molecules via the continuous quantum Zeno
effectFast inelastic two-body losses, induced for example by the exothermic chemical reactions
discussed in the previous chapter for KRb molecules, impose a severe limit on molecule lifetimes [188, 180, 59]. For the use of ultracold molecules as a quantum resource for the explorationof rich physical phenomena, it is crucial to find appropriate means to overcome losses. Recent
experiments with KRb molecules [278] have reported an inhibition of losses when the molecules
are confined in an array of one-dimensional tubes with a superimposed axial optical lattice along
the tubes (Fig. 4.1). Similar loss suppression by strong dissipation was previously observed inbosonic Feshbach molecules [244]. Extending the molecules’ lifetime over timescales much longer
than those determined by tunneling opens a path for the exploration of itinerant magnetism andother many-body phenomena arising from the interplay between dipolar interactions and motion,even in these in highly reactive systems.
In Chapter 3, we have shown that free KRb molecules react rapidly. In a lattice, the two-bodyinelastic collision rates are larger than all other lattice energy scales, including the band separationenergy. Consequently, this system is an example of a strongly correlated system that defies simple
treatment in terms of single-particle physics. As such, description of the loss suppression basedon the assumption that inelastic interactions do not affect the single-particle wave-functions isincorrect [161]. Evidence of this issue was reported in Ref. [278] where a heuristic “single-band”
treatment of the losses was found to significantly overestimate the molecule filling fraction f .

35

Figure 4.1: (a) A 50:50 mixture of fermionic KRb molecules in two rotational states, |0, 0i (red)and |1, −1i (blue), is prepared in a deep 3D lattice, which is suddenly made shallow along one
dimension (y). Along y, molecules tunnel with a rate J/~ and have a large on-site loss rate Γ0
because of chemical reactions. (b) In the Zeno regime, ~Γ0  J, doubly occupied sites are only
virtually populated, and the loss occurs at a significantly slower rate Γeff  Γ0 for molecules on
adjacent sites. For KRb, a multiband analysis of this process is required for all experimental lattice
parameters.

Here, we develop a theoretical description of the dissipative dynamics that non-perturbatively
includes three dimensional multi-band effects. Our analysis allows us to attribute the observed loss
suppression to the continuous quantum Zeno effect [170, 109, 76, 99, 82] – a suppression of coherent
transitions due to strong dissipation – and to generalize previous single-band treatments [82, 16]
to the strongly dissipative regime. We perform systematic measurements of the KRb lifetime
under controlled and reproducible lattice conditions that allow us to validate the calculations. The
observed dependence of the loss rate on lattice parameters is consistent with Ref. [278] and isfully reproduced by the multiband theory. Moreover, the inclusion of multiple bands reduces thedetermined filling f by a factor of ∼5, giving results consistent with the filling predicted by Ramsey
spectroscopy measurements of molecules pinned in a 3D lattice and prepared under similar initialconditions [278, 100].
The multiband calculations are first applied to derive a simple rate equation (RE) for twobody losses, which assumes instantaneous redistribution of molecules between collision events. We
show that the RE can describe the measured dissipative dynamics fairly well over a broad rangeof lattice parameters, but fails for the deepest lattice configurations. We develop a simple andunified theory capable of describing the loss dynamics in all parameter regimes based on a mean-

36
field (MF) approximation of the many-body master equation. We validate the MF formulation bycomparing it to a numerically exact time-dependent density matrix renormalization group method(t-DMRG) [259, 55, 266], which we combine with a quantum trajectory technique [39, 172, 72].
The MF, t-DMRG, and experimentally observed loss dynamics quantitatively agree.

4.1

Experiment with KRb molecules
The experiment begins by loading ∼104 fermionic KRb ro-vibrational ground-state molecules,|N = 0, mN = 0i, into the lowest band of a deep 3D cubic optical lattice with lattice constanta = 532 nm. Here, N is the principal rotational quantum number and mN is the projection onto
the quantization axis, which in our case is determined by an external magnetic field angled 45◦
between the x and y lattice directions. We next apply a π/2 microwave pulse to rotationallyexcite half of the molecules to |N = 1, mN = −1i. We consider |0, 0i and |1, −1i as |↓i and |↑icomponents of a pseudo-spin 1/2 system. We choose the lattice polarizations so that the tensor
AC polarizabilities of |0, 0i and |1, −1i are similar [178]. However, a residual differential AC Stark
shift introduces single-particle dephasing that results in a spin-coherence time for the entire sampleof ∼1 ms. This dephasing allows us to prepare an incoherent 50:50 spin mixture of |↓i and |↑i byholding the molecules in the deep lattice for 50 ms. Losses are then initiated by quickly ramping
down the lattice depth in the y direction (within 1 ms) to allow tunneling. We measure the numberof remaining molecules |↓i, i.e., N↓ (t), as a function of the subsequent holding time in the lattice.
We experimentally determine the initial loss rate κ by fitting N↓ (t) to the solution of a
two-body loss RE of the formdN↓κ=−[N↓ (t)]2 ,dtN↓ (0)(4.1)

with N↓ (0) the initial number of |↓i molecules. A typical experimental fit is shown in Fig. 4.2(a).
To avoid the saturation of the losses that originates from the finite number of molecules per tube(∼6 per tube on average), which cannot be captured by the RE, we fit only up to times when
∼ 25% of the molecules are lost.

10

6.0
5.5
5.0

20

20

37
60

HcL

80

single-band RE, f~25%
multiband RE, f~6%
experiment

10

loss rate @1sD

6.5

HbL

experimentexperimental fit

loss rate @1sD

molecule number H103L

7.0 HaLV¦ @ER D
40

15

single-band RE, f~25%
multiband RE, f~6%
experiment

10
5

4.5
4.0

0

0.01

0.02
time @sD

0.03

0.04

0
0.3´105

6´104

9´104
G0 @1sD

12´104

0
0

100

200

300 400
JÑ @1sD

500

600

Figure 4.2: (a) Measured number loss of |↓i molecules for an axial (transverse) lattice depth of
Vy = 5 ER (V⊥ = 25 ER ) (circles) and best fit using a rate equation (RE), Eq. (4.1) (black dashedline). (b) Number loss rate, κ, as a function of Γ0 (fixing J ≈ 570 Hz and varying the bare on-siterate via V⊥ ). (c) Number loss rate, κ, versus J for fixed Γ0 ≈ 87 kHz (varying Vy and adjustingV⊥ accordingly). Vy (V⊥ ) was varied from 5 to 16 ER (20 to 40 ER ). Black circles are experimental
measurements (error bars represent one standard error). Green short-dashed lines show solutionsof the RE Eq. (4.4) using an effective loss rate Γeff (single-band approximation). The blue longdashed line shows the multiband RE using Γ̃eff in Eq. (4.4). The multiband and single-band RE
results were obtained by fixing the filling fraction to be 6%, and 25% respectively. Panels (b) and(c) directly manifest the continuous quantum Zeno effect: in (b) the measured loss rate κ decreases
with increasing on-site Γ0 ; in (c) a fit to the experimental data supports κ ∝ J 2 , with a χ2 (sumof the squared fitting errors) several times smaller than for a linear fit.

The loss rate κ in general depends on the tunneling rate J/~ and the on-site “bare” loss rate
Γ0 . If a single-band approximation is used, the on-site bare loss rate Γ0 is given by [82]
Γ0 = β (3D)R

|W (x)|4 d3 x,(4.2)

where W (x) is the lowest-band single-particle 3D Wannier orbital. The two-body loss rate coefficient for molecules in |0, 0i and |1, −1i, β (3D) , was measured to be β (3D) = 9.0(4)×10−10 cm3 s−1 [278].
To experimentally extract the dependence of κ on Γ0 and the single-particle hopping energy
J, we perform similar measurements to those reported in Ref. [278]. However, here we ensure
reproducibility of the initial conditions to fix f for all lattice conditions. To measure the Γ0 dependence we set Vy = 5 ER , which fixes J, and then tune Γ0 by modifying V⊥ [Fig. 4.2(b)]. Here,ER = ~2 π 2 /2ma2 is the recoil energy and m is the KRb mass. To study the J-dependence, we vary
Vy while simultaneously adjusting V⊥ to keep Γ0 fixed [Fig. 4.2(c)]. The loss rate κ is found todepend quadratically on J for fixed Γ0 and to decrease with increasing Γ0 for fixed J. This scaling

38
is consistent with the continuous quantum Zeno effect, as we now explain.

4.2

Theoretical description

4.2.1

Single-band rate equation

A simple way to understand the loss suppression is to consider two opposite spin particles ina double well, |↑, ↓i. Left and right sides in this notation represent left and right wells [Fig. 4.1(b)].
When two molecules occupy the same site the singlet component decays with rate Γ0 , while thedecay of the triplet component is suppressed by the centrifugal barrier in odd partial-wave chan√nels [59]. Consequently, the loss rate is determined by Js = 2J, which is the tunneling computed√
after projecting the initial wavefunction into the singlet state |si = (|↑, ↓i − |↓, ↑i)/ 2.
When ~Γ0  Js , second-order perturbation theory can be applied and gives a net |↓i loss
2

rate of 4Γeff with Γeff = 2(J/~)Γ0 . This loss rate can be connected to number loss dynamics with a
RE,

dnj↓
dt = −4qΓeff nj+1↑ nj↓ , where nj↓ is the number of |↓i molecules at site j and q is the numberof nearest neighbor lattice sites (q = 2 for tunneling along the tube direction) [16]. Assuming auniform distribution, the 50:50 mixture implies nj+1↑ = nj↓ = n↓ and
dn↓
dt
dN↓
dt

= −8Γeff [n↓ (t)]2
= −

or

κSB
[N↓ (t)]2 ,N↓ (0)

(4.3)
(4.4)

where κSB = 8Γeff n↓ (0). All parameters are known except the filling fraction f = 2n↓ (0). The RE
assumes that the loss rate depends only on the average density. This assumption is valid when
the redistribution of density after a loss process occurs faster than the typical time between losses(J  Γeff ) 1 .
This simplified single-band model qualitatively reproduces the measured dependence of κ on
lattice parameters [green-short-dashed line in Figs. 4.2(b)-(c)]. However, since Γ0 is larger than theband gap (e.g., 4 times larger for a Vy = 5 ER and V⊥ = 40 ER lattice), this single-band theory is
1

Note: Although a transition dipole matrix element exists between |↑i and |↓i, we neglect dipolar interactions
because of the lack of spin coherence in our 50:50 mixture

39
known to be inadequate. Moreover, in order to fit the experiment, the single-band theory requiresf ≈ 25%, which is known to be inconsistent with estimates of the filling f . 10% from Ramsey
spectroscopy procedures [100, 49]. Resolution of this discrepancy requires including multiple singleparticle bands, which are admixed by strong two-body losses.

4.2.2

Multiband rate equation

A single-band model overestimates Γ0 , predicting it to be larger than the band gap. Incorporating higher bands decreases Γ0 and hence decreases the f estimated from experiment (since theeffective loss rate is inversely proportional to Γ0 ). We extract a renormalized effective loss rate bynumerically computing the loss of two molecules trapped in a double well along y. We expand thenon-Hermitian Hamiltonian Ĥ = Ĥ0 − i~β (3D) δreg (r) /2, where δreg (r) = δ (r) (∂/∂r) r is a regularized pseudopotential [104] and Ĥ0 the single-particle Hamiltonian, in the 3D Wannier function
basis. This model accounts for interaction-mediated band excitations in all three dimensions. We
initialize the system with two molecules in the singlet |si and infer the effective loss rate by fitting
the norm decay to exp(−4Γ̃eff t). Convergence is achieved with 6 bands in each dimension.
Surprisingly, as shown in Figs. 4.2(b–c), both effective loss rates Γeff and Γ̃eff scale similarly
with Γ0 and J. This similarity explains why qualitative experimental signatures of Zeno suppression
expected from a single-band model survive even though such a model is invalid. However, themultiband Γ̃eff is ∼5 times larger than Γeff . Once these effective loss rates are calculated, the only
free parameter to fit the experimental measurements is the filling f , which was fixed to be the samefor all data shown in Figs. 4.2(b–c). The ∼ 5 times faster loss rate from the multiband model leads
to a ∼ 5 times smaller filling fraction of f = 6% [Figs. 4.2(b–c), blue-long-dashed line] compared to
the grossly overestimated 25% extracted using Γeff [Figs. 4.2(b–c) green-short-dashed lines]. The
inadequacy of the single-band model to extract the correct filling fraction, and the success of themultiband model, are key results of this work.

40
4.2.3

Mean-field and DMRGThe RE, with parameters extracted from the multiband model, describes the experimental
observations fairly well at intermediate V⊥ , but deviates from them for the largest V⊥ . We attributethese deviations to the suppression of tunneling at the cloud’s edges due to the energy mismatch
between adjacent sites in the harmonic potential generated by the lattice beams. By inhibitingtransport, this effect invalidates the assumption that molecules are redistributed rapidly between
loss events, and therefore the losses are not determined exclusively by the average density but
depend on the detailed dynamical redistribution of molecules.
Although this redistribution is absent from the RE, it can be accounted for by solving a masterequation with a density matrix, ρ̂, projected into the states with at most one molecule per site
after adiabatic elimination of doubly occupied states. We keep terms up to order Γeff [82], and we
simultaneously account for multiband effects by replacing the single-band Γeff by the renormalized
loss rate extracted from the multiband double well solution, obtainingdiρ̂ = − [Ĥ0 , ρ̂] + Lρ̂.
dt~
Here Ĥ0 = −J†j,σ (ĉjσ ĉj+1σ + h.c) +

P

1
σ †
j,σ Vj ĉjσ ĉjσ , Lρ̂ = 2

P

(4.5)
P h
j

i
2Âj ρ̂Â†j − ρ̂Â†j Âj − Â†j Âj ρ̂ 2

, and Vjσ = 21 mωσ2 j 2 a2 is the parabolic trapping potential felt by molecules in state σ at site j.
The average trap frequency (ω↑ + ω↓ )/2 varies between ≈ 2π × (15 − 40) Hz for the experimental
range of V⊥ . The σ-dependence is due to residual differential AC Stark shifts between the tworotational states. L is a Lindblad superoperator that accounts for losses, and the jump operatorship
are Âj = 2Γ̃eff (ĉj↑ ĉj+1↓ + ĉj↑ ĉj−1↓ ) − (ĉj↓ ĉj+1↑ + ĉj↓ ĉj−1↑ ) . We have checked the validity of therenormalized single-band model by confirming that it reproduces the dynamics of the multiband
problem for the case of two molecules in four wells.
To solve Eq. (B.1) we map the hardcore fermions onto hardcore spin-1/2 bosons [255], andQPthen use a mean-field ansatz ρ̂ = j ρ̃j with ρ̃j ≡ α,β={↑,↓,0} ρα,βj |αihβ|. Here, ρ̃j is the reduced
projected density matrix at site j, and ↑, ↓, 0 label the three possible local states of spin up, down,
2

Note: The sums run over all lattice sites, operators acting outside the bounds are implicitly ignored.

41
and the vacuum, respectively. This ansatz leads to closed equations of motion for ρα,β[see Appendix
j0

6=σB]. Due to the rapid dephasing of spin coherence resulting from ω↑ 6= ω↓ , we set ρσ,σ
= 0, whichjsimplifies the equations further. Although the MF treatment predicts no coherent tunneling for apure Fock state, we initiate it by assuming non-zero particle/hole coherence |ρσ,0j | = 1/2.
Fig. 4.3(a) shows the dynamics for the largest V⊥ , where the coherent tunneling is strongly
suppressed by the large parabolic potential ω↑/↓ . We see that the dynamics is poorly describedby the RE, and the MF solution better describes the data. Admittedly, the MF assumption is anextreme approximation precluding entanglement between parts of the system. In order to test its
validity, we also solve Eq. (B.1) numerically by combining t-DMRG algorithms [259, 55, 266] with astochastic sampling over quantum trajectories [39, 172, 72, 220]. The results, shown in Fig. 4.3(b),are converged in the matrix product state dimension χMPS and are therefore numerically exact. The
differential stark shift for the lattice parameters of Fig. 4.3 (ω↑ /ω↓ ∼ 0.9) gives rise to an effective
spatially dependent magnetic field that disrupts spin correlations generated during the dynamics.
In this case the data, t-DMRG, and MF (which explicitly ignores spin-correlations) agree up to thetimes used to extract loss rates from the data, when the contrast has decayed by ∼ 20%. However,in the absence of a differential Stark shift (ω↓ = ω↑ ), we note that the density calculated from
t-DMRG saturates at a higher value than predicted by the MF theory, which we attribute to thegrowth of spin correlations in the absence of dephasing [77].

4.3

Comparison with experiment measurement

4.3.1

Mean-field vs. experimentWith the validity of the MF established, we use it to model the experiment. In the MF
calculation, we assume that molecules are initially uniformly distributed within a shell with inner(outer) radius of 20 (50) lattice sites. The shell distribution is expected because molecules are
created from a Mott insulator of Rb and a band insulator of K. Assuming only sites with one Rb
and one K can yield molecules during STIRAP [188, 49], sites in the trap center initially doublymolecule number H103L

HaL

8

HbL

42

7
6
5
4

Ω­Ω¯=1,t-DMRGΩ­Ω¯=0.9,t-DMRG
mean-field

experimentexperimental fit
mean-field0

0.04
time @sD

0.08

0

0.04
time @sD

0.08

Figure 4.3: Comparison of experimental loss dynamics for the deepest considered lattice to MF
and t-DMRG calculations. (a) Molecule loss vs. time for V⊥ = 80 ER and Vy = 5 ER [Identicalconventions/conditions to Fig. 4.2(b)]. The MF matches the experimental data better than the
RE (experimental fit). (b) Comparison of t-DMRG simulations (χMPS = 128, 2000 trajectories)
to MF, for two different cases: i) an identical trap for the two spin states with trap frequencyω↓ = ω↑ = 2π × 38 Hz; and ii) slightly different trap frequencies ω↓ = 2π × 38 Hz, ω↑ = 2π × 34.2 Hz.
Shaded areas indicate the standard error of the mean.

occupied by Rb atoms are lost [188]. We then average over random initial configurations, since
the experiments measure an ensemble of 1D tubes. Figures 4.4(a-b) show the MF results (redline), where we used f = 9% to match the experiment. This is slightly larger than that from themultiband RE (dashed blue line in Figures 4.4), f = 6%, since the RE overestimates the loss rateby assuming instantaneous redistribution of the molecules.
Since the molecule distribution in the experiment is known only approximately, we vary theshell width and find that the estimated MF filling fraction that best fits the experimental loss hasa range f ∼ 9 ± 2%. The MF accounts better for the dependence of the loss rate on V⊥ . Remaining
deviations between MF and experiment are seen only for the shallowest V⊥ , where the transverse
tunneling rate is only three times smaller than the axial one, which may indicate the breakdown of
1D dynamics.

4.4

Conclusion
The understanding of the underlying physical mechanism responsible for the loss suppression

in KRb opens the path for laboratory explorations of iconic models of quantum magnetism com-

10

loss rate @1sD

HaL
20

V¦ @ER D
20
40

43
60

80

multiband RE, f~6%
mean-field, f~9%
experimentHbL
15

multiband RE, f~6%
mean-field, f~9%
experiment

10
10

0
3´104

5
6´104 9´104 12´104
G0 @1sD00

100 200 300 400 500 600
JÑ @1sD

Figure 4.4: (a) Number loss rate, κ, as a function of Γ0 (same data as Fig. 2(b)). (b) Number loss
rate, κ, versus J for fixed Γ0 (same data as Fig. 2(c)). Black circles are experimental measurements.
Blue long-dashed and solid red lines show RE and MF solutions, respectively, using Γ̃eff (multiband
model). The shaded area accounts for ±2% variations around the MF estimate of f ∼ 9% arisingfrom the uncertainty in the initial molecule distribution.

bining motional and spin degrees of freedom, previously believed to be inaccessible due to losses.
These include the extended t-J model, predicted to exhibit itinerant ferromagnetism, d-wave superfluidity [90, 129], and topological phases [136, 13]. Our findings also extend to other dissipative
systems, such as alkaline earth atoms [152, 23, 252, 149] and other chemically reactive molecularspecies [285].

Chapter 5

Synchronization of Interacting Quantum Dipoles

5.1

Introduction
Arrays of synchronized oscillators [197] are ubiquitous in biological [277, 236], physical [52]and engineering [210] systems and are a resource for technological advances [145]. Although
there has been significant progress in the study of synchronization in classical systems [8], theunderstanding of the same phenomena in the quantum realm remains limited. A major obstacle
so far is the general problem of the exponential scaling of the Hilbert space with system size whichmakes calculations dealing with quantum arrays very challenging. In fact, current investigations
have been limited to the exact treatment of arrays of a small number of coupled quantum oscillators[262, 226, 283, 163, 85, 84, 199, 3, 106, 261, 159], and large ensembles at the mean field level orby including quantum corrections perturbatively [154, 140, 141]. Highly symmetric situations withcollective coupling mediated, for example, by a cavity mode [9, 273, 160], have also been studied.
Ensembles of radiating dipoles are a natural platform to study quantum synchronization,
where coherence can be generated from an incoherent source. One might regard laser systems,
where radiation is amplified by the stimulated emission of photons, as a prototypical example.
However, lasing is fundamentally a distinct phenomenon from quantum synchronization. This canbe seen from the fact that lasing is possible even in the absence of coupling between the atomic
dipoles, as is clear in the single atom laser [164], or in atomic beam lasers where only one atom ispresent in the cavity at any given time. A more relevant situation is the quantum synchronizationthat takes place in the context of superradiance [94, 158]. It has recently been understood that,45
in contrast to lasers, steady-state superradiance can produce spectrally pure light [94, 158, 29,
166, 96, 272, 241, 97] without stimulated emission. So far this has been demonstrated using acavity mode as a communication channel that spatially selects an optical mode and enhances thecoupling (through the cavity finesse). A more generic and relevant scenario, with great potential andapplicability, is the emergence of spontaneous macroscopic quantum synchronization in radiating
dipole arrays without a cavity but naturally coupled by the intrinsic anisotropic and long-range
dipolar interactions. This is the situation considered in this chapter.
Here we demonstrate that in the presence of an incoherent repumping source, dipole induced
cooperative emission can dominate over spatial inhomogeneities and quantum fluctuations and leadto a resilient steady-state that exhibits macroscopic quantum phase coherence and intrinsic quantum correlations. An iconic example of a macroscopic coherent state is a Bose-Einstein condensate,achieved in ultra-cold gases at thermal equilibrium. In our case, however, the macroscopic order isreached in the steady-state of an interacting and driven-dissipative system. Moreover, the cooperative behavior can be detected by measuring the spectral purity of the emitted radiation. We notethat in clear distinction to previous studies [106, 154, 262, 3, 261, 141], our proposal does not relyon an external coherent source or externally generated nonlinearities to seed the collective phase.
In our model synchronization emerges as a spontaneously broken symmetry driven by incoherent
processes in naturally coupled dipole arrays. As we show, and somewhat counterintuitively, an
incoherent drive is sufficient to generate phase coherence in these systems.
Specifically, the systems we consider are dense arrays of frozen quantum dipoles modeled
as quantized two-level systems. By dense arrays of frozen dipoles we mean arrays separated bya distance much closer than the wavelength of the emitted photons and with motional degrees of
freedom evolving at a much slower rate than their internal dynamics (Fig. 5.1). These conditions
can be readily satisfied in a variety of quantum systems found in atomic, molecular and opticalphysics (e.g., Rydberg gases [257, 1, 5], alkali vapors [122], alkaline-earth atoms [187], and polar
molecules [278]), chemistry (e.g., J-aggregates of dye molecules [28, 192, 234]), and biology (e.g.,light-harvesting complexes [193, 222]). In cold vapors, one possible way to freeze the motion and46
tightly trap the particles is via an optical lattice potential (Fig. 5.1). In this case a sub-opticalwavelength transition must be used in order to reach the tight-packing regime [278, 187].

Figure 5.1: Arrays of quantum dipoles spontaneously emit and absorb photons at rate Γ. The
photons mediate dipolar interactions between dipoles separated by a distance rab with both dissipative, f (rab ), and elastic, g(rab ), components. A repumping source at a rate W provides energy
to maintain the oscillations and can be implemented using additional internal states that are notshown. A possible implementation can be realized with cold atoms in an optical lattice.

To fully understand synchronization in the complex dipolar system, we analyze each of theingredients that compete and affect synchronization in a step-by-step procedure: the interplay
between repumping and collective emission, inhomogeneity in the coupling constants, quantum
correlations, and the competition between elastic and inelastic interactions. This chapter is organized as follows: In Sec. 5.2 we introduce the system in consideration and the master equation we
use to describe the dynamics. In Sec. 5.3 we first provide a simple mean-field description and discuss
connections to the classical Kuramoto model—the iconic model used to describe synchronizationin non-linear coupled oscillators. In Sec. 5.4 we discuss the phase diagram for the quantum system
assuming collective (all-to-all) coupling in the absence of inhomogeneity and elastic interactions,and compare it with the mean-field solution. For this exactly solvable case we are able to explicitly
quantify the entanglement and correlations present in the steady state. In Sec. 5.6 we study how

47inhomogeneity in the inelastic couplings affects synchronization and focus on the case of power-law
decaying interactions. In Sec. 5.5 we study the emergence of quantum synchronization in radiating
dipoles taking the full long-range and anisotropic dipolar interactions into account. Specifically,
we consider a one dimensional geometry, where both the elastic and inelastic interactions can varysignificantly accross the dipoles and can be easily adjusted. In Sec. 5.7 we discuss experimental
implementations of our model, and in Sec. 5.8 we provide a conclusion and an outlook.

5.2

Dipole-dipole interaction and master equation
In this work we consider arrays of quantum dipoles with two accessible levels, which wedenote as |↓i and |↑i. The interactions between two dipoles a and b, can be explicitly written as3Γ n 2 cos ζabcos ζab sin ζab osin θ+(3 cos2 θ−1)[
+
] ,
2
ζab
(ζab )3 (ζab )2
3Γ n 2 sin ζabsin ζab cos ζab of (rab ) =sin θ+(3 cos2 θ−1)[
−
].
2
ζab
(ζab )3 (ζab )2g(rab ) = −In the above, ζab = k0 |rab |. Both g(rab ) and f (rab ) depend on the dipoles’ separation, |rab |, and theangle θ between the mean dipole moment and the vector joining the dipoles. As noted in Chapter
2, when ζab  1, the elastic g interactions with a strong angular variation are dominant except√
close to the magic angle θm = arccos(1/ 3), at which they are greatly suppressed 1 . In contrast,
f (rab ) is almost isotropic in the near-field regime.
The spatially uniform behavior of f (rab ) at short distance is what gives rise to cooperative
effects and superradiant emission [94]. Under generic conditions, however, superradiance is a transient effect that substantially limits the lifetime of dipole excitations. To compensate for the fast
decay here we add an incoherent repumping driving term at a rate W . This term is needed to generate a synchronized steady state where long-lasting coherence persists. An incoherent repumping
drive is commonly used in laser systems to maintain population inversion. It can be implemented
1
For generic cases, the pathological divergence of g(r) at r = 0 can be removed by introducing an additional term
4π
into the expression of g(rab ), − 3k
3 δ(rab ) [175, 43]. This gives rise to the so called Lorentz-Lorentz shift [80]. For thediscrete arrays we consider in this work, there is always a cut-off distance and thus no divergence. For example, inthe suggested implementation using Sr atoms that is discussed in Sec. 5.7, the cut-off distance is determined by the
optical lattice spacing.

48
by coherently driving, at a rate Ω, the | ↓i state to an excited level that spontaneously decays,
at a rate γ  Ω, to the state |↑i [224]. Due to the fast depletion of the excited state, it can be
adiabatically eliminated and thus the net process is just an incoherent transfer of population from|↓i to |↑i at a rate W = Ω2 /γ.
The evolution of N dipoles is modeled by a quantum master equation for the reduced density
matrix ρ̂ of the dipoles [94]:
dρ̂
dt
Ĥ0

i
= − [Ĥ0 , ρ̂] + Lf [ρ̂] + LW [ρ̂],
~
N
N
i
X
~ Xh
=
δa σ̂az +g(rab )σ̂a+ σ̂b− ,
2
a=1

(5.1)
(5.2)

b=1,b6=aLf [ρ̂] =1X
f (rab )(2σ̂b− ρ̂σ̂a+ − σ̂a+ σ̂b− ρ̂ − ρ̂σ̂a+ σ̂b− ),
2

(5.3)

LW [ρ̂] =W X + −
(2σ̂a ρ̂σ̂a − σ̂a− σ̂a+ ρ̂ − ρ̂σ̂a− σ̂a+ ).
2 a

(5.4)

a,b

(+,−,z)The Hamiltonian Ĥ0 generates the coherent evolution of the dipole array where σ̂aare thePauli spin operators for dipole a, δa denotes its oscillation frequency in the rotating frame definedby the mean frequency of the dipole ensemble, and ~ is the reduced Planck constant. The Lindblad operator functionals, Lf,W , describe the inelastic photon emission and incoherent repumping
processes, respectively.

5.3

Mean-field treatment and connection to the Kuramoto model
To obtain a qualitative picture of how synchronization can happen among the dipoles, we first

perform a mean-field treatment and show the close connection between our quantum model andthe prototype models for classical synchronization. The mean-field approach assumes uncorrelatedNP
00
dipoles, i.e., ρ̂ = a ρ̂a , where each ρ̂a = σ,σ0 =↑,↓ ρσ,σa |σihσ | is a 2 × 2 matrix in the pseudospin
1/2 basis {|↑i, |↓i}. The components of the single-dipole density matrix, ρ̂a , can be visualized as
↓↑
↑↓
↓↑
↑↑
↓↓
a Bloch vector {Sa⊥ (t) cos φa (t), Sa⊥ (t) sin φa (t), Saz (t)} = (1/2){ρ↑↓a + ρa , −i(ρa − ρa ), ρa − ρa }(Fig. 5.2). The mean-field solution yields a system of coupled nonlinear differential equations for0ρσ,σa . For each a = 1, 2, . . . , N the parameters evolve as

49

0.08

0.08

0.04

0.04

synchronized

0.04
0.1
0 0.00unsynchronized
20
40
60

0

W êG

(c) N=15

3

F Q(ρc)/N

3

2

-

2

1

1

-

F Q(ρc)/N

4

0 0
0 02

24

46
Γt Γt

68

N=70

0.08
0.04

0.12
0.08
0.04

40

W êG

60

(d)

conditional
conditional
unconditional
unconditional

N=15

0.00

80

0.00
100

0.12
0.08
0.04

quantum discord

4

0.12

0.00
0.00
80 0 100 20

W/Γ=0.5
W/Γ=0.5
0.12
5 W/Γ=9
5 W/Γ=9
W/Γ=60W/Γ=60mutual information

-6
-4
-2
0246

entangled

mutual information

0.08
0.2

(b)

quantum discord

0.3

N=70
0.12
S?

quantum discord

0.12

Sz

mutual information

(a)

0.00

0 0.5
20 10
8 0.5
10
2040 2030 603040 80
40 100W êG
W/ΓW/Γ

√Figure 5.2: (a) Mean-field phase diagram calculated from the order parameter 0 ≤ Z ≤ 1/ 8.
The insets show snapshots of the tips of the Bloch vectors (red points) for dipoles prepared withrandom initial phases and then evolved√ to steady-state in both regimes. (b) Quantum phase
diagram calculated from 0 ≤ ZQ ≤ 1/ 8. (c) The time evolution of the conditional QFI exhibits
entanglement (dashed line: single trajectory, solid line: mean value of a few trajectories). Panel(d) shows the steady state QFI vs W/Γ after averaging over many trajectories. The solid line
corresponds to the conditional case, and indicates entanglement over the repumping range wheresynchronization exists. Upon computing the ensemble average one recovers the reduced density
matrix which leads to a calculated QFI below the entanglement witness threshold (dashed line).
(c) and (d) are shown for feff = 15Γ. For all panels, δa = g(rab ) = 0 and f (rab ) = feff /N .

dSaz (t)dt= −Sa⊥ (t)

X
b6=a

h
i
Sb⊥ (t) f (rab ) cos[δφba ] − g(rab ) sin[δφba ]

dSa⊥ (t)
dt
dφa
dt




1
1
z
z
−Γ
+ Sa (t) + W− Sa (t) ,
2
2
h
i Γ+W
X
= −Saz (t)Sb⊥ (t) g(rab ) sin[δφba ]− f (rab ) cos[δφba ] −Sa⊥ (t),
2

(5.5)
(5.6)

b6=a

= δa +

NX

Saz

b=1,b6=a

i
Sb⊥ h
g(r
)
cos[δφ]+f(r)
sin[δφ
]
,
ab
ba
ab
ba
Sa⊥

(5.7)

50
where δφba (t) = φb (t) − φa (t). The term proportional to f (rab ) in Eq. (5.7) that contains the sinefunction can be identified with a similar term in the Kuramoto model (KM) [130]:NX
dφa= δa + K
sin[δφba ],
dt

(5.8)

b=1

where K, the coupling strength per oscillator, must be large enough and positive for synchronization to occur. The term proportional to g(rab ) that contains the cosine function appears in the
Sakaguchi-Kuramoto model [216]—a more general but similar synchronization model to the KM.
Compared to the basic KM, the situation here is more complex. This is due to the fact that in
Eq. (5.7) the coupling constants are nonuniform and effectively time-dependent, since Sa⊥ (t) andSaz (t) are dynamic variables.
To investigate whether the mean-field model admits spontaneous synchronization we consider
first the simplified case where δa = 0 for all dipoles, impose g(rab ) = 0 for all pairs, and assume
a constant collective decay rate N f (rab ) ≡ feff . We define a global order parameter Z as ZeiΦ =1 P⊥ iφa and look for a solution in which Z is time-independent and synchronized oscillators
a Sa eNpossess a collective frequency ω, and thus a macroscopic phase Φ = ωt. These conditions lead totwo equations for the order parameter Z and the collective frequency ω (see C.1):
ω = 0,pfeff (W − Γ) − (W + Γ)2
√
Z =
.
2feff

(5.9)
(5.10)The solution is shown in Fig. 5.2(a). The insets show the phase distribution in the steadystate for an array of oscillators initially prepared with a random distribution of phases for two
different values of the repump rate W . For a slow repumping rate (bottom inset), the systemremains unsynchronized. As the repumping rate is increased beyond a threshold value, the systementers a synchronized state, as can be seen by the appearance of phase locking and the resulting
narrow phase spread (top inset). This can be explained by the fact that one necessary condition forsynchronization in the KM is K > 0, which translates to the requirement Saz > 0 on average in ourmodel and thus the need to have sufficiently large repump rate. In the limit feff  Γ (e.g., for large

51N), maximum synchronization is achieved at Wopt = feff /2, where the order parameter Z reachespa maximum value Zmax ≈ 1/8. For this optimal condition for synchronization the quantum
dipoles are ordered with the same phase and radiate with atomic inversion Saz ≈ 1/4. Note thatthe maximum order parameter is smaller than 1/2 even when fully synchronized because of thisrequired finite value of the atomic inversion. One intriguing aspect is that repumping, which is theprocess that builds up synchronization, is itself an incoherent process. It is crucial that repumping
does not preserve the norm of the collective Bloch vector, allowing it to extend or contract. For
large W > Wopt , Z decreases again reflecting a suppression of synchronization. In this limit thedipoles are repumped so fast that they are all driven to the |↑i state (Saz → 1/2 and Sa⊥ → 0) andphase coherence between them cannot build up.
The cases of a heterogeneous distribution of δa ’s can be treated at the mean-field level in a
simple way. The results, summarized in the C.1, are qualitatively similar. In general, the inclusionof a finite spread ∆ in δa decreases the value of the order parameter Z. For instance, if δa is sampledfrom a Lorentzian distribution p(δa ) = ∆/[π(∆2 + δa2 )],qpfeff P − Q2 + 2∆2 − 2∆ ∆2 + feff P
√
Z=
,
2feff

(5.11)

where Q = Γ + W and P = W − Γ. Optimal synchronization is obtained at a smaller repumping√
rate, Wopt ≈ feff /2 − ∆/ 2. We note that for given feff and W , synchronization is destroyed (thatis, Z = 0) at ∆c = (Q2 − feff P )/(2Q).
For the case with finite g(rab ) 6= 0, Eq. (5.7) predicts dephasing caused by different g(rab ) 6= 0,
and a global rotation that does not affect synchronization if all g(rab ) are identical [see C.1].
However, the mean-field ansatz is not appropriate for treating the case where elastic interactions
are present since it neglects quantum entanglement between dipoles. The case with finite g(rab )will be discussed in Sec. 5.6 by exactly solving the many-body problem.

52

5.4

Quantum synchronization for the collective system
In the simplified case where δa = 0 for all dipoles, g(rab ) = 0 for all pairs, and a constantcollective decay rate N f (rab ) ≡ feff , it is possible to exactly solve Eq. (5.1), i.e., the full quantum
dynamics, even for many particles, allowing us to benchmark the validity of the mean-field solution.
This is due to the invariance of the master equation under individual dipole permutations that
reduces the scaling of the Liouville space from exponential, 4N , to polynomial, of order N 3 [275].

5.4.1

Phase Diagram

Quantum fluctuations can lead to phase diffusion and to decay of single particle coherencesin the steady-state (it is possible for hσ̂a+ i → 0 even in a synchronized state), so Z cannot be
used as a measure of synchronization in a beyond mean-field treatment. However, phase locking
in quantum mechanics can be quantified by the degree of spin-spin correlations ZQ , defined by
2 ≡ hσ̂ + σ̂ − i, where the bar indicates an average over all pairs of different dipoles a and b. For anZQa b√max = 1/ 8 [166, 29].
unsynchronized state ZQ is 0 and for a completely synchronized state ZQ is ZQ

The corresponding phase diagram, shown in Fig. 5.2(b), closely resembles the mean-field one.
To demonstrate that ZQ can be used to quantify the emergence of quantum synchronization,
regardless of the inherent non-equilibrium and dissipative character of our system, we have also
computed pairwise two-time correlation functions (see Appendix C.2). The decay rate of these
correlations encodes information about the spectral coherence of the emitted radiation. The rangeof W/Γ values where the emitted light is maximally coherent agrees with the regime where thesystem is optimally synchronized according to ZQ . Moreover, we have also confirmed the moderate importance of higher order correlations in the synchronized steady-state by comparing theexact solution with a cumulant expansion calculation (which includes lowest order corrections to
the mean-field result). We find the cumulant expansion agrees well with the exact solution (seeAppendix C.2). The only limit where there are important deviations is at very weak pumping
W  Γ where the system favors subradiant emission arising from strong atom-atom correlations

53[indicated by the purple region in Fig. 5.2(b)] [275].

5.4.2

Quantum correlations and entanglement

The robust macroscopic quantum coherence exhibited by the synchronized state leads to thenatural question of whether or not entanglement can be present in the steady-state even in thisdissipative environment. Most previous studies that attempted to address this question have been
limited to small systems [84, 141, 159, 160] and focused on the entanglement between a pair ofsynchronized oscillators. Here, to determine the non-separability of the many-body steady-state,
we compute the average of the quantum Fisher information (QFI) and use it as an entanglement
witness [237, 107]. Any N particle state with (N 2 + 2N )/3 ≥ F̄Q (ρ̂) > 2N/3 is entangled (nonseparable) and a quantum resource for phase estimation (see Appendix C.3 for details).
Due to dissipation, the density matrix of the system is reduced to a mixed state, as obtained
from Eq. (5.1), which describes the dynamics of the system after a statistical average over many
experimental trials. However, the evolution of the system for an individual experimental realization
can be quite different. We consider a Gedanken experiment in which one monitors the systemevolution and keeps a measurement record of the emitted photons. The evolution of the systemis then conditioned on the measurement record [40, 198]. This type of conditional evolution has
been widely studied in quantum optics and utilized together with quantum feedback control inexamples such as the optimal generation of spin squeezed states (See Refs. [251, 98]). It shouldbe emphasized that the conditional evolution based on the measurement record gives a quantum
trajectory that should not be regarded simply as a numerical tool to allow the efficient assemblyof ensemble averages. Each quantum trajectory is a potentially realizable physical outcome (evenif hard to perform in practice) as allowed by the quantum dynamics of the open quantum system
under consideration.
We calculate F̄Q (ρ̂c ) (with the c in ρ̂c meaning conditional) for each conditional trajectoryand in Fig. 5.2(d) we show its average over a sufficiently large set of trajectories at steady state (seeAppendix C.3). For this conditional evolution we observe entanglement in a parameter regime that54
correlates with ZQ > 0 [see Fig. 5.2(c) and (d)]. On the other hand, if we discard the information
present in the measurement record, by using the ensemble averaged ρ obtained from directly solving
Eq. (5.1), and then computing FQ (ρ̂), the QFI falls below the entanglement witness threshold [seeFig. 5.2(d)].
To differentiate quantum effects from classical ones, we further calculate the quantum discord
D [see Appendix C], which can be considered as a measure of quantum correlation more general thanentanglement and more robust in a dissipative environment [186, 54]. Separable states with nonzero
D are intrinsically nonclassical, since local measurements performed on a subsystem inevitably
disturb the whole system [186, 171]. We measure classical correlations of the steady state by thedifference between the mutual information I [see Appendix C] and D. We find the mixed steadystate contains nonzero quantum correlations in the synchronized regime Fig. 5.3(a). Moreover weobserved that although both D and ZQ exhibit a similar dependence with pumping rate W , they
do not exactly peak at the same value [3].
(a)

(b)

Figure 5.3: Quantum correlations and total correlations. The total and quantum correlationsin the steady state are quantified by the mutual information 0 ≤ I ≤ 2 and quantum discord
0 ≤ D ≤ 2. (a) In the synchronized phase, there are nonzero quantum correlations and classicalcorrelations (I − D), and both show a dependence on W that qualitatively agree with ZQ . (b)Even in the thermodynamic limit, quantum correlations remain a significant fraction of the total
correlations. For both panels, δa = g(rab ) = 0 and f (rab ) = feff /N .

Although the existence of a nonzero D has been reported to exist in several quantum synchronization studies [159, 171], we want to emphasize that those have been always limited to small
systems. To our knowledge our calculations are the first to consider D in macroscopic samples. In

55
Fig. 5.3(b) we show the dependence of D with system size. Our calculation shows that quantum
correlations remain a significant fraction of I even in the thermodynamic limit.

5.5

Synchronization with finite-range interactions
Up to this point we have only considered all-to-all interactions; now we consider the effectof finite range interactions on synchronization. In the dipole array both f (rab ) and g(rab ) arenontrivial functions of rab and contain terms decaying as a power-law with distance, ∝ 1/|rab |αwith α = 1, 2, 3. Instead of dealing with all these terms together, to gain insight on how spatial
inhomogeneities affect quantum synchronization, we first study a simpler case assuming a powerlaw cooperative decay f (rab ) ∝ |rab |−α , with the exponent α as a variable parameter and set bothg(rab ) = 0 and δa = 0.
In the classical regime [53, 238, 50, 256], analytical calculations and numerical simulations
considering arrays of oscillators interacting via power law interactions on a one-dimensional lattice
had identified αc = 3/2 as the critical value of the power law exponent below which long-range phaseorder is possible [50]. For α < αc , a transition to a state in which a finite fraction of the oscillators is
entrained takes place for a sufficiently strong but finite coupling strength in the large system limit.
Generalizations of these results to oscillators in D dimensions [50] have also identified three different
regimes for synchronization: perfect phase ordering for α ≤ D, entrainment with long-range phaseorder for α < 3D/2 and a crossover to exponential decay of correlations at α = (3D + 1)/2.
Reference [256] has also suggested that in the regime α > D global synchronization is absent but
local synchronization persists for arbitrary weak coupling with a slowly decaying order parameter.
To quantify the effect of finite-range interactions on synchronization in the quantum regimed )2 ≡ hσ̂ + σ̂ − i ,we compute spin-spin correlations within linear clusters that contain d dipoles, (ZQa b dusing a cumulant expansion method as described in Appendix B. Here the bar followed by asubscript d indicates an average over the pairs of different dipoles a and b contained in a linear
cluster of size d. The linear clusters start at the central spin as shown in Fig. 5.4. We have confirmedthat the cumulant expansion method reproduces well the correlation functions by performing direct

56
d as
comparisons with the exact solution (see Appendix B). Fig. 5.4(a) shows the behavior of ZQ

a function of cluster size d and power-law decay exponent α in arrays of dimension D = 1 andd independent of d is observed for
2. Clear global synchronization with an order parameter ZQ
d is almost independent of α and d and
α . D. For 0 < α . D/2, the local order parameter ZQ

the system behaves almost like the all-to-all system. For D/2 . α . D synchronization remains
global and almost independent of d, but the order parameter slowly decreases with α. For α & D,synchronization becomes local and correlations quickly decrease with cluster size. The magenta
contour provides an indicative scale of the boundary between global and local synchronization.
The white contour lines also provide information about the decrease of the order parameter withincreasing α and d. We observe that, as in the classical case, α ∼ D roughly marks the transition
between global and local synchronization, although a more quantitative comparison would require
far larger systems.
An alternative way to characterize domain formation and the fact that it can persist evenwhen there is a variation in the local detunings, δa 6= 0, is to examine pairwise two-time correlation
functions, Za,b (τ ) ≡ limt→∞ h(σ̂a+ (t + τ ) + σ̂b+ (t + τ ))(σ̂a− (t) + σ̂b− (t))i, which can be related to theemission spectrum of the pair of atoms [40]. The oscillations in Za,b (τ ) encode information about
the relative precession rate between different dipoles. By parameterizing Za,b (τ ) as Za,b (τ ) =A cos(νab τ ) exp(−γτ ) we can extract the relative precession frequency νab between dipoles a and b,
where entrainment of dipoles a and b corresponds to νab = 0. To explore the entrainment of dipole
pairs in our system, we assign random detunings distributed uniformly in [−Γ/2, Γ/2] to a linear
chain of N = 200 dipoles and calculate νab for b = 1, 2, . . . , 100 with a = 101 corresponding to
the central dipole in the chain. Synchronization regimes similar to those shown in Fig. 5.4(a) are
observed for this D = 1 system, which we illustrate in Fig. 5.4(b) by plotting a histogram (top panel)and the distribution of frequencies ν (bottom panel) for three values of α. For global coupling,
α = 0 (dark blue symbols), all the dipoles become entrained with each other (ν = 0), indicatingcomplete synchronization; for α = 0.65 (red symbols) dipoles split into entrained (ν = 0) anddrifting (ν 6= 0) groups. While not all dipoles are entrained, the entrained dipoles are distributed

57
along the whole array, and thus synchronization is still global; and for α = 2 (light blue symbols)
the majority of dipoles are not entrained. These observations of relative precession frequencies
between pairs of oscillators are consistent with the regimes obtained from the order parameterplotted in Fig. 5.4(a).

d , in linear clusters containing d dipoles at optimal(a) Spin-spin correlations, ZQ αrepumping W for power-law couplings f (rab ) = Γ4 raabwith lattice spacing a. We set δa =g(rab ) = 0 and consider N = 900 dipoles arranged in both linear (D = 1) and square lattice (D = 2)
geometries. For α . D global synchronization is observed and the order parameter is independent
on cluster size d. For D . α the order parameter starts to clearly decay with increasing d. The
d = 0.14) provides an indicative scale of the boundary between global and local
magenta line (ZQ
synchronization. The white contour lines provide an indication of the decrease of the synchronizeddomains with increasing α. (b) Pair wise two-time correlation functions in the steady-state areparametrized by Za,b (τ ) = A cos(ντ ) exp(−γτ ) where a is chosen as the central dipole of a linear
chain of N = 200 dipoles. The dipoles are assigned random detunings δa distributed uniformly in[−Γ/2, Γ/2]. The dark blue, red, and light blue symbols correspond to α = 0, 0.65 and 2 respectively.
The histogram of frequencies ν exhibits similar synchronization regimes than those seen in (a).

Figure 5.4:

5.6

Synchronization of dipoles with elastic interactions
We now treat the full problem of radiating quantum dipoles incorporating elastic interactions

g(rab ) and the intricate competition of spatially-dependent and anisotropic couplings [both g(rab )and f (rab ) have terms with power law dependence α = 1, 2, 3 on distance] (Fig. 2.1). We solve thefull master equation without any approximation [40] for systems of up to twenty dipoles in a chain
using the actual spatial dependence of both f (rab ) and g(rab ), and set δa = 0. We observe a robust

58
synchronized state that exists in a wide parameter space. As long as feff ≡ N f (rab ) is large enough,we find that synchronization takes place and is only weakly affected by substantial differences ing(rab ), e.g., variations in the dipole array that modify g(rab ) by two orders of magnitude only
decrease the order parameter by a factor of two (Fig. 5.5) in the steady state. This is in striking

Figure 5.5: (a) Synchronization in dipole arrays is demonstrated for N = 12 dipoles on a line whensubjected to incoherent pumping (optimal rate). In this geometry, regardless of the strong angularvariation of g √with the lattice spacing a (see contours) the order parameter, ZQ , (normalized
max
by ZQ = 1/ 8) exhibits a weak dependence on θ and a and reaches a maximum at θ = θm .
(b)
P The order parameter is computed for N = 16 dipoles on a line with θ = θm and feff =a,b6=a f (rab )/(N − 1) (symbols), and for a system with constant f (rab ) = feff /N and g(rab ) = 0(dashed line)s. Similar dependence on W is found for these two different systems. Here the orderparameter for dipoles is always smaller in the presence of elastic interactions.

contrast to the situation in a system without dissipation, where the elastic interaction is known
to generate entanglement between spins and to cause a decay of the order parameter during time
evolution [157].
√
For the orientation θm = arccos(1/ 3), the order parameter reaches a significant fraction
max , indicating the emergence of macroscopic spontaneous synchronization of the radiating
of ZQ

59
quantum dipole array (Fig. 5.5). To further emphasize the relevant role played by the inelastic
term, in Fig. 5.5(b) we compare a solution of the master equation [Eq. (5.1)] for two cases: asystem of coupled dipoles arranged in a 1D chain and oriented at the magic angle (symbols) and anarray of identically coupled dipoles with the same feff but experiencing only inelastic interactions[g(rab ) = 0, dashed lines]. The calculated order parameters agree well for the two different cases.
The similar behavior demonstrates that in spite of the complex geometry of the dipolar interactions,the capability of the dipole system to synchronize can be characterized to great extent by thequantity feff .

5.7

Experimental implementation
Our calculations above demonstrate the potential for synchronization in a dense array of

dipoles. The flexible and precise control exhibited by ultracold atomic systems make them ideal
platforms to experimentally investigate the synchronization phenomenon predicted here. Atomic
systems operate with a large number of quantum oscillators and also allow for the tunability of theinteraction parameters over a broad range.
One possible set-up to observe synchronization consists of arrays of ultracold 87 Sr atoms
prepared in two electronic internal states that form the two-level system. The | ↓i could thencorrespond to the long-lived 5s5p 3P0 state, with an intercombination line narrower than 10−3 s−1. This is the state used to operate the most precise atomic clocks [27]. The |↑i could correspondto the 5s4d 3D1 state with a natural linewith Γ = 290 × 103 s−1 . Both states can be trapped in
an optical lattice at the magic wavelength a = 0.2 µm [187], that generates the same trapping
potential for both states minimizing Stark shifts and inhomogeneities in the coupling constants.
The dipole-dipole interactions are mediated by photons at the wavelength λ = 2.6 µm and thus,
as shown in Fig. 5.5, the ratio a/λ < 0.08 falls in the parameter regime where dipoles can besynchronized.
By changing the angle between the laser beams used to form the lattice potential, the latticespacing can be varied allowing tunability of the interaction strength between dipoles. The incoher-

60
ent pumping can be realized by coherently transferring the 5s5p 3P0 population to one or several
appropriate intermediate states that decay rapidly to the 5s4d 3D1 state [167]. An example would
be the 4d5p 3P1 state [217].
The polarization of the dipoles can be oriented in an arbitrary direction by an electromagnetic
field. Although all the dipoles cannot be oriented at the magic angle in a 3D geometry, one may still
suppress the elastic interactions by dynamical decoupling techniques adopted from NMR [71]. Those
have been already demonstrated in ultracold polar molecule systems [278]. Another possibility is
to use a spatial configuration of external fields that induces an ‘averaging out’ of the dominantelastic interactions [184]. Moreover, by slightly departing from the magic-wavelength condition,
the dipoles can be subjected to onsite inhomogeneities that generate different detunings δa .
The phase synchronization can be probed by measuring ZQ , which experimentally can bedirectly obtained from the fluorescence intensity. As suggested in Sec. 5.5, phase locking can also
be extracted from two-point correlations which can be determined by analyzing the fluorescence
spectrum [167].
An intriguing but also more speculative and less controllable realization of our quantum
dipole model is the case of fluorescent organic molecules. A possible two-level configuration in
those systems consists of a vibrational level of the ground electronic potential chosen as |↓i and thelowest vibronic level of the first excited potential chosen as |↑i. Incoherent pumping can be realized
by driving an optical transition to a higher excited vibronic level |φi in the first excited potential,
which decays on picosecond timescales to the state |↑i via non-radiative transitions [127]. Typicalvalues of the fluorescence wavelength λf and lifetime τf for organic chromophores under a varietyof environmental conditions put these systems in a regime of near optimal synchronization. For
instance, pseudoisocyanine chloride (PIC) and merocyanine derivatives commonly used in organic
light-emitting diodes (LED) [68, 17, 120, 6] typically form low-dimensional molecular aggregates inliquid solution with a ≈ 0.5 − 2.0 Å, and ratios a/λf on the order of 10−3 . The typical fluoresence
decay rate for these organic chromophores is Γ ∼ 0.1 − 1 GHz [127]. In order to achieve W/Γ = 1
and enter the synchronized phase, the required pumping laser intensity is IW ∼ 1 − 10 kW/cm2 ,61which is lower than the theoretical lasing threshold intensities Ith ∼ 0.1 − 1 MW/cm2 of dye lasers[127]. Therefore, it should be feasible to achieve steady state synchronization of organic dipoles via
incoherent optical driving.

5.8

Conclusion
We have demonstrated that a system of radiating quantum dipoles can be synchronized in

the presence of repumping. Our analytic mean-field approach provides a direct analogy between
synchronization of quantum dipoles and synchronization of classical phase oscillators. Using exact
solutions of the master equation and a cumulant expansion approach, we determined the necessary
conditions for synchronization, and the entanglement properties in the steady state of macroscopic
ensembles under different measurement protocols. We also analyzed the effect of finite-range interactions in large arrays. To our knowledge those have been previously explored only in the classicalregime. These calculations, although restricted to g(rab ) = 0, emphasize how the interplay between
incoherent repumping and cooperative dipole decay can give rise to robust quantum synchronization. For treating the general case of dense packed dipoles, we numerically solved the masterequation exactly for up to twenty dipoles, and studied the effect of elastic interactions. The one
dimensional geometry chosen allows us to tune the dipolar interactions accross a wide range ofparameters by simply adjusting the spacing and polarization angle, and investigate the competingroles of the elastic and inelastic interactions. One dimensional systems are amenable for theoretical investigation since they help us to minimize finite size effects. They can be implemented incurrent experiments using ultra-cold gases and are relevant for the suggested implementation usingorganic molecules. There are several organic species such as PIC molecules [17] that are known toform quasi-one dimensional arrays both as single crystals and in liquid solution. These systems are
typically modelled as 1D arrays of two-level dipoles with energetic disorder to match spectroscopic
data [75]. For this configuration, we found synchronization in the presence of a large variation ofelastic interactions. We expect that in more general geometries, similar degree of synchronization
could be achieved provided the collective dissipative coupling feff dominates over the elastic pair-62
wise couplings, g(rab ). These conditions might be easier to be satisfied in higher dimensional dense
packed dipolar arrays given the anisotropic character of the elastic terms which fluctuate both insign and magnitude across the array. In higher dimensions, geometric effect becomes more important. Nevertheless, whether or not spontaneous synchronization in dipole arrays can be observed
under more generic high dimensional geometries remains an open question.
Our results show that the intrinsic macroscopic coherence of the superradiant steady state isinherently resilient to single particle decoherence, spatial inhomogeneities, and noisy environmental effects. This observation could have relevant application to the development of low-thresholdorganic lasers, highly efficient solar cells, materials with enhanced chemical reactivity, as well as
ultra-precise quantum devices, where these effects are anticipated to play an important role. Moreover, since quantum synchronization is imprinted in the spectral purity of the emitted radiation[166], the generated light may potentially serve as a direct diagnostic tool of quantum coherencesin generic systems beyond cold gases such as organic molecules.

Chapter 6

Light scattering from dense cold atomic media

6.1

Introduction
Light-matter interactions are fundamental for the control and manipulation of quantum sys-tems. Thoroughly understanding them can lead to significant advancements in quantum technologies, quantum simulations, quantum information processing and precision measurements [124, 51,
62, 69, 24, 118, 25]. Over the past decades, cold atom experiments have provided a clean andtunable platform for studying light-matter interactions in microscopic systems where rich quantum
effects emerge, such as superradiance and subradiance, electromagnetically induced transparency,
and non-classical states of light [30, 95, 214, 191, 34]. However, in spite of intensive theoreticaland experimental efforts over the years, long-standing open questions still remain regarding the
propagation of light through a coherent medium, especially when it consists of large and dense ensembles of scatterers [110, 142, 213, 113, 223, 242, 22, 122, 232, 112, 231, 185, 115, 111, 114, 7, 31].
In fact, by studying small systems where analytical solutions are obtainable, it has been realizedthat atom-atom interactions can significantly modify the spectral characteristics of the emitted
light. These effects yet need to be understood in large systems [187, 158, 281, 284, 19] where finite
size effects and boundary conditions become irrelevant. The situation is even more complicatedwhen the coupling with atomic motion is non-negligible [18, 144]. It is timely to develop theories
capable of addressing these questions, given the rapid developments on cold atom experiments andnanophotonic systems. The experiments are entering strongly coupled regimes, where atom-atom
and atom-photon interactions need to be treated simultaneously and sometimes fully microscopi-

64cally [187, 116, 88].
A widely adopted approach to describe light scattering consists of integrating out the atomic
degrees of freedom and treating the atoms just as random scatterers with prescribed polarizability[63, 258, 134, 36]. While this approach can successfully capture some classical properties of thescattered light, it does not fully treat the roles of atom-atom interactions and atomic motion [131,133, 227, 132]. An alternative route consists of tracing over the photonic degrees of freedom. In thiscase the virtual exchange of photons induces dispersive and dissipative dipole-dipole interactions
between atoms, which can be accounted for by a master equation formulation [110, 142]. Thisapproach has been used to study systems of tightly localized atoms where the dynamics only takes
place in the atomic internal degrees of freedom. It has been shown to successfully capture quantum
effects in light scattering [242, 48, 280, 22, 21, 240]. However, due to the computational complexity,
it has been often restricted to weak excitation and small samples [169, 165, 281, 284], and a directcomparison with experiments containing a large number of atoms has been accomplished only
recently [33, 212]. In general, most theories have not properly accounted for atomic motional
effects and atomic interactions on the same footing and many open questions in light scattering
processes remain.
Here, we present a unifying theoretical framework based on a coherent dipole (CD) model(see Fig. 6.1(a)) to study the light scattering from cold atoms with possible residual motion. In
the low intensity and slow motion regime, we use the CD to investigate the collective effects in thelight scattered by a large cloud and show the interplay of optical depth (OD) and density. These
results are compared with the random walk (RW) model (see Fig. 6.1(a)) that only accounts forincoherent scattering and thus ignores coherent dipolar interaction effects. To address the role ofatomic motion, we perform different levels of generalization of the CD model. With these modified
models, we show that atomic motion not only reduces phase coherence and collective effects, but also
impacts the line shape and line center of the spectral emission lines via photon recoils. Motivatedby a recent experiment at JILA (see Fig. 6.1(b)) [33], we focus our discussions on a J = 0 → J 0 = 1
transition, but the methods presented here can be extended to more complicated level structures

65
without much difficulty.

(a)

~✏out
~✏in ~kin

~kout

a

(b)

Gab

b

detector~✏✓⌦eik0 xFigure 6.1: Scheme. (a) Microscopic models: random-walk model (left) and coherent dipole model(right). In the random-walk model, a photon is randomly scattered by the atoms. Scattering eventsare characterized by the incident and outgoing wavevector of the photons and their corresponding
polarization. In the coherent dipole model, atoms are coupled by dipole-dipole interactions (Gab )and all atoms contribute to the fluorescence. (b) Experimental setup for measuring the fluorescencefrom a cloud of atoms. An incident laser drives an atomic transition with a spontaneous emission
rate Γ. Atoms absorb and emit light. The detectors collect scattered photons at an angle θmeasured from the incident beam direction.

This chapter is organized as follows. In Sec. 6.2, we provide the mathematical description of
the CD model, which treats atoms as coupled, spatially fixed dipoles sharing a single excitation.
Its predictions on the collective properties of the emitted light such as the light polarization anddensity dependence of the line shape and peak intensity are discussed. In Sec. 6.3, we introduce
the RW model and compare its predictions on the linewidth and peak intensity of the scattered
light to the ones obtained from the CD model. Those comparisons allow us to explore the role ofphase coherence in the atom-light interaction. In Sec. 6.4, we present extended models to study
motional effects on atomic emission. We first include motion in the CD model by assuming thatits leading contributions comes from Doppler shifts. Those are accounted for in the frozen model
approximation when we introduce local random detunings for atoms sampled from a MaxwellBoltzmann distribution. Then we go beyond the frozen model approximation and explicitly include

66
atomic motion by means of a semi-classical approach. This treatment also allow us to go beyond thelow excitation regime. In Sec. 6.5, we apply the CD model to a recent experiment measuring thefluorescence from an ensemble of 88 Sr atoms. We finish in Sec. 6.6 with conclusions and outlook.

6.2

Coherent dipole model

6.2.1

Equations of motion for coherent dipoles

As derived in chapter 2, for an ensemble of N atoms with internal dipole transition J = 0 →J 0 = 1, the master equation is
i

dρ̂
dt

= −
+

X

α
∆α [b̂α†i b̂i , ρ̂] +i,αXΩα [(eik0 ·ri b̂α†i + H.c.), ρ̂]i,ααα0 α† α0gi,j[b̂i b̂j , ρ̂]X
i6=j,α,α0+X
i,j,α,α0

0

0

0

α† α
αα
fi,j
(2b̂αj ρ̂b̂α†i − {b̂i b̂j , ρ̂}),(6.1)
where we have added the term describing the effect of an external driving laser with polarizationα, wavevector k0 , and Rabi frequency Ωα . The Hamiltonian is written in the rotating frame of thelaser, with ∆α denoting the detuning between the laser and the transition |0i → |αi. The dipoledipole interactions include both the far-field (1/r) and near-field (1/r2 , 1/r3 ) contributions. The
0αα encapsulate collective dissipative process responsible for the superradiantterms described by fi,j0αα account for elastic interactions between
emission in a dense sample. The terms described by gi,jatoms which can give rise to coherent dynamical evolution. These elastic interactions compete withand can even destroy the superradiant emission [79, 284].
When the atoms’ thermal velocity v satisfies k0 v  Γ, atoms can be assumed to be frozen
during the radiation process. Moreover, in the weak driving regime, Ω  Γ, to an excellent approximation, the master equation dynamics can be captured by the 3N linear equations describing
the atomic coherences bαj = hb̂αj i of an excitation propagating through the ground state atomic

67
medium. The corresponding steady state solution can be found from:
0

bαj

=

X
Gαα
Ωα δα,γ eik0 ·rj /2
0
jn
+
bαn ,
α
α
∆ + iΓ/2∆ + iΓ/2
0(6.2)

n6=j,αwhere we have specified the polarization of the driving laser to be along γ. For obtaining Eq. (6.2),
we have assumed there is only one excitation in the system, and ignored multi-atom coherences(see Appendix D.1). This assumption is justified in the weak driving field regime [21, 234]. The
fluorescence intensity measured at the position rs in the far-field can be obtained by the summation
[110]
0

0

For dilute samples the dipolar interactions are weak, G ≡P

I(rs ) ∝X

e−iks ·rjn

X
α,α0

jn

(δα,α0 − r̂αs r̂αs )bαj bα∗
n ,

(6.3)

α
where bα∗n is the complex conjugate of bn , and rjn = rj − rn .

6.2.2

Collective effects in fluorescence
αα0
i6=j,α,α0 |Gij |/(N Γ)  1, and Eq.

α,1α,2(6.2) can be solved perturbatively using G as an the expansion parameter, bαj = bα,0j +bj +bj +...,(bα,n∝ G n ) which results in
j
bα,n
j

=

αn−1 αn γ
α1 α2
ik0 ·rln
1
X Gαα
jl1 Gl1 l2 ...Gln−1 ln Ω δαn ,γ ein (∆α + i Γ2 )(∆α1 + i Γ2 )...(∆αn + i Γ2 )l1 6=jl2 6=l1...ln 6=ln−1
α1 ,α2 ...αn

.

(6.4)

In the expansion, terms of order n account for nth order scattering events. For simplicity, in thefollowing we assume the atomic sample has a spherical shape, with density distribution n(r) =22n0 e−r /2R , unless otherwise specified. However, the conclusions can be generalized to other geometries. Here, n0 = 2π√N2πR3 is the peak density.
To the zeroth order, the atomic response is driven by the external field only and is not
modified by the scattered light:
bα,0
=
j

Ωα δα,γ eik0 ·rj.
∆α + iΓ/2

(6.5)

68
Substituting it into Eq. (6.3), the intensity of scattered light is22

I ∝ N + N 2 e−|ks −k0 | R .

(6.6)

There are two contributions to the intensity, the first term ∝ N represents the incoherent contribution, and the second term ∝ N 2 is the collective emission resulting from coherent scattering
processes[223, 33]. The phase coherence is restricted to a narrow angular region around the incident
laser direction, with δθ ∼ 1/k0 R. The enhanced emission arises from the constructive interferenceof the radiation from N dipoles [21]. Along other directions, the random distribution of atom positions randomizes the phases of the emitted light, smearing out the phase coherence after averaging
over the whole sample [33].
Including first order corrections, the intensity of the scattered light is given byI(rs ) ∝N Ω2,(∆ − Γ<[G])2 + (Γ + 2Γ=[G])2 /4(6.7)

for transverse directions, where we have denoted Ω = Ωγ and ∆ = ∆γ , andG =X−ik0 ·rij/(N Γ).
Gγγ
ij e

(6.8)

i6=j

For the forward direction, the intensity has the same form, except that the factor N is replaced by
N 2 , due to the phase coherence. Therefore, the line shape of the scattered light is Lorentzian, with
its line center frequency shifted by the elastic interactions [211, 211], and the linewidth broadenedby the radiative interactions. If we temporarily neglect the effect of polarization, when the atomatom separation is large, the dipole-dipole interaction is dominated by far-field terms, i.e., Gγγ
ij ∼ik0 rijγ γ e− 3Γ
4 (1 − r̂ij r̂ij ) k0 ri j . In this limit analytical expressions for the linewidth broadening Γ, anddensity shift, ∆, can be obtained: Γ = 2Γ=[G] = 3N Γ/(8k02 R2 ) = τ4 Γ, with τ = 2k3N
2 2 being the0R√
OD of the sample, and ∆ = Γ<[G] = −Γn0 k0−3 /4 2π. Therefore, while the collectively broadened
linewidth depends on the OD of the atomic cloud, the frequency shift depends on the density.
In a dense medium dipolar interactions are strong, G & 1, higher order scattering events
become important, and the interplay between the radiative interactions and elastic interactions
becomes non-negligible. As a consequence the above perturbative analysis is no longer applicable.

69
In Fig. 6.2 and Fig. 6.3, we show the numerical solution of Eq. (6.2), which takes into accountall the scattering orders. As shown in Fig. 6.2(a),���������
[����units)�����]
Intensity (arb.
-0.3
-0.4

Intensity (norm.)

◇
◇

◇◇
=0
-0.4
�� ΔD =0◇Δ◇D◇
�◇ ΔD =40kHz-0.5
� Δ◇D =40kHz
-0.5
◇
MF
-0.6
� MF
�
�
���
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���
���
���
���
-0.6
0.00
0.05
0.10
0.15
0.20
0.25
�
-3
0.00
0.05
0.10
0.20
0.25
n0.15
k
0
-3

���������
Intensity(arb.[����]units)

Intensity(arb.
units)
���������
[����]

Intensity (norm.)-0.2-0.1 -0.1
��
�
◇
◇◇◇
���
◇
-0.2
◇
-0.2 -0.2
◇◇
��
�
◇◇◇
◇◇◇
-0.3
◇◇◇
-0.3
◇
◇
-0.3
◇◇
��
�
◇◇◇
��
-0.4ΔD =0ΔD =0 ◇◇◇◇◇
ΔD =0
-0.4
-0.4
◇◇
◇
◇◇
�� ΔD =40kHz
�Δ
ΔD =40kHz-0.5◇◇D =40kHz
-0.5
◇◇
◇
-0.5
◇ MF
�
MF
◇
-0.6
� MF
�
-0.6
�
���
���
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���
-���
-���
���
���
��� -0.6
0.00
0.05
0.10
0.15
0.20
0.25
0.00
0.05
0.10
0.15
0.20
0.25
θ/π -30.00
0.05
0.10 � 0.15
0.20
0.25
-3
n0 k
n0 k
(d)
n0k -3
��
��
0.0
0.0
�
θ=�
θ=�
��
-0.1
θ=π/��
θ=π /�
-0.1
��◇◇◇◇◇◇◇◇◇◇◇ θ=π/�
�
◇◇◇
��
-0.2
◇◇◇◇◇
-0.2
◇
◇
◇◇◇◇◇
◇◇◇
◇
��
-0.3
��
◇
-0.3
�
-0.1

Intensity���������(θ>0)(θ>�) (arb.
[����] units)

-0.1

��

θ=�
θ=π/��
θ=π/����������
(θ>�)(arb.
[����]
Intensity(θ>0)
units)

(c)

0.0

(b)
0.0 ���
0.0

0.0
����

��������� (θ=�)[����]
Intensity(arb. units)(a)most of the scattered light is distributed

-0.4
��
-0.5
-0.6
�
�
0.00

ΔD =0
ΔD =40kHz
MF
�

��

0.05

��

0.10
τ
n0 k

��

0.15

��
0.20

0.25

-3

k
Figure 6.2: CD model:n0on-resonanceintensity. (a) Angular distribution: due to the constructive
interference along the forward direction (θ = 0), the intensity is drastically enhanced within a small
angular region. (b)-(c) The intensities for three different directions (θ = 0, π/10, π/2) are shown as afunction of atom number, each normalized to the value at N = 50. All are calculated for a spherical
cloud of fixed size. The right vertical axes label the intensity for θ = π/2, π/10. In (b) the OD
and the density are relatively low (when N = 500, τ =2,n0 k0−3 =0.0015). The transverse intensity
increases ∼ N , while the forward intensity increases ∼ N 2 , showing a collective enhancement forsmall window of θ around zero. Outside this narrow angular window the enhancement disappears
and the intensity becomes almost θ independent as indicated by the identical behavior observed
for two different angles, θ = π/10 and π/2. In (c) the OD and density are relatively large (whenN = 500, τ =10, n0 k0−3 =0.017). The rate at which the intensity increases with N slows down inboth the forward and transverse direction: with I ∼ N 1 for θ = 0 and I ∼ N 0.5 for θ = π/2,
respectively. (d) On-resonance intensity as a function of OD: it is highly suppresed at large OD.
Here, the intensity is normalized to the corresponding value at τ = 25 for each direction.

within a narrow peak around the forward direction (laser direction). Outside this narrow region
fluorescence is almost uniformly distributed among other directions 1 . The forward emission is1

Note: Around the θ = π direction there is a small enhancement ∼ 2 due to the coherent backward scattering[134].

�

�
�

�

�

��

��
τ

�

�

Intensity (norm.)���������/Γ

�

�

�

�

�

��

��

(b)

0.0

-�

��
-�

�

-�

-0.2
-0.3

(c) 0.00.00
�-�(c) 0.00.00

-0.1

ΔD =0

�

ΔD =40kHz�MF��������/Γ
�0.05
0.10

n0k -3

-�

0.15

-�
0.20

0.25
-�

Intensity (θ>0) (arb. units)

τ

��

-0.6

-�

-0.1

-0.6

-0.5

��
-�

-0.5

-0.4

-�

0.0

-0.4

-0.3

��

Intensity (θ>0) (arb. units)

�

-0.2
ΔD =0
ΔD =40kHz
MF
��������/Γ
�0.05
0.10

n0 k

-0.1

70

0.15

0.20

0.25

0.15

0.20

0.25

-3

-0.2
-0.3
-0.4
-0.5

ΔD =0
ΔD =40kHz
MF

-0.6
�0.00

��������/Γ
0.10
� 0.05Intensity (θ>0) (arb. units)

(b)

(a) �

Intensity (norm.)�Intensity (norm.)���������/Γ

-0.1

13

�

ntensity (θ>0) (arb. units)�

n0k -3

Intensity (θ>0) (arb. units)

Intensity (norm.)

Intensity (norm.)

Intensity (norm.)

Intensity (θ>0) (arb. units)-�
Figure 2.3: CD Model: collective
broadening. (a) FWHM linewidth as a function of OD. At s
-0.2
Figure -0.1
6.3: CD model: collective broadening.
FWHM
linewidth
as aOD,
functionof OD.
At smallOD, the(a)FWHM
increases
linearly with
but as OD
increases
density e↵ects set in, mul
-0.3
-0.2
scattering
eventsbecomerelevant density
and the linewidth
dependence
on OD is no longer linear.
ΔD =0
OD, theFWHM
increases
linearly
with
OD,
but
as
OD
increases
effects
setin,
multiple-0.4-0.3
di↵erent
ΔD =40kHz atom numbers are used for the blue (N=1000) and black lines (N=200), and the O�
ΔD =0scattering-0.5 the linewidth dependence on OD is no longer 3linear. Two-0.4 events become relevant andvariedby changing the cloud size. For the blue line, n0 k0 = 0.14 at ⌧ =25, 0.003 at ⌧ =2. ForMF�
ΔD =40kHz�-0.6-0.5 atom numbers are used for
different
the blackblueline,(N=1000)
lines(N=200),the
OD theis linewidth increases
the densityand
at theblacksame OD
is doubled.
With andsmaller
density,�MF��������/Γ
0.10
0.15 −3
0.20
0.25
-�
-�
�0.00 larger
� 0.05 value
-0.6
in
the
large
OD
regime.
(b)
The
line
shape
at
small
OD
valuesvaried by changing
For the blue line,
n0 k0 = 0.14 at τ =25, 0.003 at τ =2. For the(⌧ =2, n0 k0 3 = 0.002��������/Γthe cloud size.
-�n0k -3
-�
-�
�
�0.05
0.00.00
0.10
0.15
0.20
0.25
di↵erent angles✓ (✓smaller
is defineddensity,
in Fig. 2.1(a))
is mainly Lorentzian.
Here
(c)
black
OD is doubled.
With
the linewidth
increasesto thea intensity is norma-� line, the density at-3 the same-�
n0kModel:-0.1 Figure 2.3: CDcollective
broadening.
FWHM linewidtha functionof OD.
At smallto the (a)on-resonanceintensityasfor
each ✓. (c)At large
OD(⌧ =20, n0 k0 3 = 0.06), the fluoresc−3-�larger
value
inthe
the
large increases
OD regime.
The
small
OD andvalues=2,
n0 k0 = 0.002)
for dots for ✓ = 0 show
-0.2 OD,
FWHM
linearly(b)withOD,linebut shapeas
OD atincreases
densitye↵ectsset
in, multiple
line
shape
significantly
broadens
stops(τbeing
Lorentzian.
The brown-0.3angles
scattering
eventsbecome relevantthelinewidthis no thelongerlinear.
different
θ (θis definedin Fig.and
6.1(a))
is mainly
Lorentzian.
Herethenormalized
Lorentzian
fit,dependence
which
failsonto OD
describe
lineintensity
shape. Two
Atisintermediate✓, a double-peak struc
ΔD =0−3 isandblue(N=1000)
black
lines
(N=200),the ODis fluorescence-0.4 di↵erent atom numbers are used for theshows
up. large
For alland
panels,
the
cloud
shape
spherical.
=
0.06),
theto theon-resonanceintensity
for
each
θ.
(c)
At
OD
(τ
=20,
n
k
0
ΔD =40kHz�
03-0.5 varied by changing the cloud size. For the blue line, n0 k0 = 0.14 at ⌧ =25, 0.003 at ⌧ =2. For theMFline� shapesignificantly
broadens
and
stops
being
Lorentzian.
The
brown
dots
for to
θ=
black
line,
the
density
at
the
same
OD
is
doubled.
Withsmaller
density,thelinewidth
increasesa 0 show the-0.6
3��������/Γ
Lorentzian
fit,
which
fails
describetheshape.
At intermediate
θ,0 ka0 double-peakstructurelargervalue
in thelarge
ODregime.
Thelineline shapeat smallOD values (⌧ =2, n= 0.002) for
0.10
0.15 to0.20
0.25 (b)
-�
-�
�0.00
� 0.05-�(✓ isthedefinedin Fig.
2.1(a))
mainly Lorentzian. Here the intensity is normalizedn0✓k -3
shows
up.di↵erent
For allangles
panels,cloud
shape
is isspherical.
to the on-resonance✓. (c) of
AtOD.
large
(⌧ =20, n0 k0 3 = 0.06), the fluorescence-�e 2.3: CD Model: collective
broadening.
(a) FWHMintensity
linewidthforaseacha function
AtOD
small
line
shape
significantly
broadens
and
stops
being
Lorentzian.
the FWHM increases linearly with OD, but as OD increases density e↵ects set in, multiple The brown dots for ✓ = 0 show theLorentzian
fit, whichfails toondescribeline shape.
intermediate ✓, a double-peak structure
ering events become relevant andthe linewidth
dependence
OD is theno longer
linear.AtTwoup.(N=1000)
For all panels,
the cloudis spherical.
ent atom numbers are used for shows
the blueand blacklines shape(N=200),and the OD is-�d by changing the cloud size. For the blue line, n0 k0 3 = 0.14 at ⌧ =25, 0.003 at ⌧ =2. For2thecollectively enhanced. For low OD it increases as ∼ N (Fig. 6.2 (b)) while the transverse intensity
line, the density at the same OD is doubled. With smaller density, the linewidth increases to avalue in the large OD regime. (b) The line shape at small OD values (⌧ =2, n0 k0 3 = 0.002) for
increasesas ∼ Nent angles ✓ (✓ is definedin Fig. 2.1(a))
is .mainly Lorentzian. Here the intensity is normalized
e on-resonance intensity for each ✓. (c) At large OD (⌧ =20, n0 k0 3 = 0.06), the fluorescencehape significantly broadens Dipolarand stops interactions
being Lorentzian.
The
dots for0 showthetend
tobrown
suppressthe✓ =
rate
at which
the intensity grows with N (Fig. 6.2(c)).
tzian fit, which fails to describe the line shape. At intermediate ✓, a double-peak structures up. For all panels, This
the cloudis spherical.
canshape
be qualitatively
understood from Eq. (6.7), which predicts that the intensity is reduced as

OD increases. Despite the fact that the perturbative analysis is only valid in the weak interaction
limit, this tendency remains and becomes more pronounced in the large OD regime as shown by
the numerical solution presented in Fig. 6.2(d). Broadly speaking, multiple scattering events tend
to suppress collective behavior [112, 135]. Similar physics is also observed in the behavior of thelinewidth. At small OD, the FWHM linewidth increases linearly with OD (Fig. 6.3(a)) and theline shape is well described by a Lorentzian (Fig. 6.3(b)), as expected from Eq. (6.7). However,when OD is large and the density is high, in addition to a significant broadening, the line shape
becomes non-Lorentzian (Fig. 6.3(c)) and the FWHM increases slowly with OD (Fig. 6.3(a)). To
further illustrate this, in Fig. 6.3(a) we plot the FWHM for the same OD but with smaller density(by using a larger atom number). The figure shows that the linewidth indeed keeps increasing

71
until saturation at a larger value of OD. Another interesting feature is the double-peak structure
observed only at intermediate angles θ, which arises from the interplay between the stimulated
photon emission driven by the probe field and the scattered photons emitted via dipolar exchageprocesses (Fig. 6.3(c)).
The drastic modifications of the perturbative expectations from multiple scattering are alsopresent in the frequency shift of the scattered light. From a mean-field point of view, the linecenterof scattered light is shifted according to the Lorentz-Lorenz shift πn0 k0−3 Γ [80]. As shown by
the numerical calculation in Fig. 6.4 (blue line), at small density, the frequency shift is indeed
linear with n0 k0−3 , but as density increases, the shift is quickly suppressed [112]. For atom density∼ 5 × 1013 cm−3 (reached for example in cold 87 Rb atom experiments [12]), the calculated density
shift is approximately a factor of two lower than the mean-field prediction.
0.0

Frequency shift/Γ

-0.1
-0.2
-0.3
-0.4
-0.5

frozen atoms
moving atoms
mean-field

-0.6
0.00

0.05

0.10
0.15
n0k0-30.20

0.25

Figure 6.4: CD model: frequency shift. For small density, the shift calculated from the coherent
dipole model (blue line) increases linearly with density as predicted by the mean-field theory.
However, when density is large, there is a significant deviation from the mean-field result. Whenmotional effects are taken into account (red line, Doppler width = 5Γ, see Sec. 6.4.1), the nonlinear suppression of the frequency shift with density is less severe.

The interplay between the imaginary and real part of the dipole-dipole interaction has tobe carefully accounted for to compare numerical simulations with experimental measurements by
doing finite size scaling. For typical computation resources the numerical solution of Eq. (6.2) islimited to ∼ 104 atoms. On the other hand experiments usually operate with ensembles of tens of
millions of atoms. To theoretically model these large systems a proper rescaling in the cloud size is72necessary. Equation (6.7) implies that in order to characterize the effect of radiative interactions(a) ���scales as ∼ N151/2 . On the other hand, to
one should aim to match the experimental OD, which△△△ △△△ ○○ ○○○
○○
○ △
○ △
○○ △

△

△ △△

���������/Γ○○○○○○○○○○○the dimensionless number○○ match○ ○toproperly reproduce the effect of elastic interactions it is���better
△
○○
△ △

△△△
△△△
△△△
△
△

○

○ ○○

n0 k0−3 , which scales as N 1/3 . Therefore, there exist two different
ways of rescaling
the cloud size,△○○ �� ����
���

△

△

△
△
△
△

������� ����either by keeping the same OD or the same density. In Fig. 6.5 we show the effect of finite N for���
a moderate range of atom numbers. Indeed, except fromsmall deviations seen at very low atom
�

����

����

����

����

����

�

���������/Γ��� ○○○
○

���

○ ○

△

△

○○ ○○○○○○○○○○

△△

△△△△
△△△
△△△

△

△ △△
△ △△△
△ △
○○ ○○○ ○ ○○
○○
○ △
○ △
○○ △
△△

△
△
△
△

○

○

�� ����

△������� ����

����

����

���
�

����

����

����

(b) ���� △△△△ △ △ △△ △△△△△△ △△△
○○
○ △△
○○○ △
○○ △△△
○○ △△
△△
○ △△
○ △
○○ △
○○○○○○○○○○

��������� �����/Γ

���

(a)

○

-����

△△△ △ △ △△ △△△△△△ △△△
○○
○ △△
○○○ △
○○ △△△
○○ △△
△△
○ △△
○ △
○○ △
○○○○○○○○○○

○
△

○○

○
○

-���� ○
-���� ○
-����

�

����

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○

�� ����

△������� ����

����

����

����

�

�

(b) ���� △○○△��������� �����/Γ

FigureCD Model:size scaling.
(a) The linewidthis calculated
Figure 6.5: CD model: finite
size2.5:
scaling.
(a) Thefinite
linewidth
is calculated
for different
number forof di↵erent numbe-���� ○
atoms at the same OD (⌧ =4 for N =3000) by varying the density (blue line with circles). F
atoms at the○ same OD (τ =4 for N =3000) by varying the density (blue line with circles). FromN = 1000 to 5000 the linewidth is not obviously changed. In contrast, by keeping the same den
N = 1000
to○5000 the linewidth is3 not obviously changed. In contrast, by keeping the same density-����(n0 k○0 ��= 0.0037
for N =3000) while varying the OD (magenta line with triangles), the linewi
����
while
varying
the OD (magenta line with triangles), the linewidth(n0 k0−3 = 0.0037 for N =3000)
keeps
changing.
(b)△ ������� ���� The frequency shift is calculated for di↵erent number of atoms. By keep
keeps changing.
(b) The frequency
shift
iscalculateddifferent
numberof atoms.
keeping-���� ○
constant density (magenta linefor
withtriangles),the frequency
shiftByremains
almost constant, w
constant density (magenta linewithtriangles),thefrequency
shift
remains
almostconstant,
while
for constant OD (blue line with circles), there is a significant variation of
the frequency shift w-���� OD (blue line withfor constantcircles),there
is a significant
shift with
Here
the
cloud
aspectratio is Rx :variation
Ry : Rz =of2 :the2 : 1frequency
[24].
�
����
���� N . ����
����
����
○

○

○○

N . Here the cloud aspect ratio� is Rx : Ry : Rz = 2 : 2 : 1 [33].

gure 2.5: CD Model: finite size scaling. (a) The linewidth is calculated for di↵erent number ofoms at the same number
OD (⌧ =4(<
for 1000),
N =3000)
varying thedensity (bluethebylinewidth
broadening
canlinebe withwell circles).
capturedFrom
by keeping the OD constant,= 1000 to 5000 the linewidth is not obviously changed. In contrast, by keeping the same density
3
N =3000)
while varying
theisOD
linebywithtriangles),the linewidth0 k0 = 0.0037 forwhile
the frequencyshiftwell(magenta
described
using
a constantdensity. In contrast, if a constanteps changing. (b) The frequency shift is calculated for di↵erent number of atoms. By keeping
nstant density (magenta
line withtriangles),thethefrequency
frequency shift
almostconstant,
while
OD is used
to compute
shift,remains
the result
wouldconsiderablly
overestimate the shift,constant OD (blue line with circles), there is a significant variation of the frequency shift with
Here the cloud aspectratio
is Rx of
: R10
Rz = 2the: 2N: 1 in[24].
y : whene.g., bya factor
numerical simulation is 1/100 the atom number in experiment.

The interplay of multiple scattering and density effects is more prominent for larger N values. To
deal with the OD vs density scaling issues in comparing with experiments the most appropriate
rescaling procedure that we found is the following: when computing the linewidth or peak intensity,
the theory is rescaled accordingly with the experimental OD. However, the actual OD value is notexactly matched to the experimental one but to a slightly modified value, τe = ητ to account for73
density effects [33]. For a moderate window of τ values, for example achieved experimentally byletting the cloud expand for different times, η should be kept fixed. For the frequency shift thetheory should be rescaled according to density.

6.2.3

Anisotropic features of scattered light

For independent atoms radiation along the polarization of the driving laser is forbidden.
However, dipole-dipole interactions can generate polarization components different to the driven
ones if the atoms exhibit internal level structure, e.g., degenerate Zeeman levels in the excitedstate. This is the case of a J = 0 → J 0 = 1 transition, where, as shown in Fig. 6.6, the fluorescenceemitted along the laser polarization direction (z-direction, θ = π/2 ) is nonzero. It is, however,

Figure 6.6: CD model: effect of laser polarization. (a) Intensity distribution in the plane perpendicular to the laser propagation direction, i.e., θ = π/2 for all φ’s (inset). φ = 0 is the z-direction.
Here, the incident laser is polarized along z. The intensity detected along the laser polarization issuppressed compared to other directions. (b) and (c) show the line shape and linewidth of lightdetected at θ = π/2, φ = 0. (b) At small OD, the FWHM linewidth is below the natural linewidth.
(c) As OD increases, the linewidth is collectively broadened due to multiple scattering processes(Fig. 6.3(a)). For all panels, degeneracy in excited levels has been assumed.

much weaker than the intensity emitted along other directions. On the contrary, for two-leveltransitions the polarization of the scattered light is conserved and thus the emission parallel tothe laser polarization is completely suppressed. The strong dependence of the scattered light onpolarization and atomic internal structure is most relevant along the transverse direction. Alongthe forward direction those effects are irrelevant, as verified by our numerical simulations. From Eq.
(6.4), the lowest order contribution to the intensity detected along the laser polarization direction
comes from the first order scattering processes, thus I ∝ (∆2 +Γ1 2 /4)2 , which lead to a “subradiant”

74
line shape (i.e. the full width at half maximum (FWHM), ΓFWHM =p√2 − 1Γ < Γ, is smaller

than the one for independent particles). The analytic result agrees perfectly with the numericalsimulation at low OD (Fig. 6.6(b)). As the OD increases and interactions become stronger, higher
order scattering contributions lead to a collective broadening (linewidth larger than Γ) even along
this “single dipole forbidden” direction, as shown in Fig. (6.6 (c)).

6.3

Random walk model
In this section we use the random walk model to investigate the role of incoherent scattering

processes in collective emission. We focus on the low-intensity regime. Classically, light transport ina disordered medium can be described by a sequence of random scattering events experienced by aphoton (see Fig. 6.1(a)) [134, 133, 135]. The expected number of scattering events is roughly givenby (τp )2 , where τp is the peak optical depth, which depends on detuning ∆ as 3N/[k02 R2 (1+4∆2 /Γ2 )]. For simplicity here we also assume a spherical cloud. The transmission of light is given by e−τp[44]2 . For the J = 0 → J 0 = 1 transition (degenerated J 0 = 1 states), the differential scatteringcross section that defines a scattering event is given by [134, 135]dσ(kin , in → kout , out ) =
dΩ

3σ0 ∗
| · out |2 ,
8π in

(6.9)

where kin,out is the incident/scattered wavevector, in,out is the polarization of the incident/scatteredphoton [134, 135, 203] and σ0 = 3λ2 /[2π(1 + 4∆2 /Γ2 )], with λ the wavelength of the driving laser.
To simulate the polarization dependent scattering events as dictated by Eq. (6.9), it is convenient to use the Stokes-Mueller formalism [103]. A photon in a given state of polarization can be
2
Note: For a cloud with a Gaussian distribution, the size of the incident laser beam needs to be taken into
account, since a broad beam experiences fewer scattering events at the edge of the cloud and the actual optical depth
depends on the transverse coordinates. The finite beamsize is not taken into account in the CD presented here. In
RW we choose the beamsize to be 5 times the Gaussian width of the cloud.

75
described by a Stokes vector [35]|El |2 + |Er |2 S0  

 


 


 S1   |El |2 − |Er |2 
=
,
S = 
 


 S2   E ∗ Er + E E ∗ 
l
r
 


l
 


∗
∗
S3
i(El Er − El Er )(6.10)

where El , Er are the electric field components projected onto the two orthorgonal axis êl andêr in the plane perpendicular to the wavevector k. For example, S = (1, 1, 0, 0) represents a
photon linearly polarized along the reference axis êl . A scattering event kin , in → kout , out can be
determined by two angles: θ and φ [see Fig. 6.7]. The change of polarization can be obtained fromthe transformation S00 = M (θ)R(φ)Sin , where Sin is the incident Stokes vector, S00 is defined withrespect to the axis ê00l , ê00r and ê003 , and then Sout = R(ψ)S00 transforming back to the original frameêl , êr and ê3 [15], with Sout the scattered Stokes vectors. The scattering matrix that we use, M , isthe scattering matrix that describes Rayleigh scattering [103, 135]. It is given byer

ele003 = kout

e00l

e0l

✓

e3 = kin

e0r

e03

e00r

Figure 6.7: RW: transformation of Stokes vectors. In the random walk model, a scattering event isdetermined by two consecutive transformations of local coordinates: {e3 , er , el } → {e03 , e0r , e0l } via
rotation φ, and {e03 , e0r , e0l } → {e003 , e00r , e00l } via rotation θ [15].

M (θ) =
2 θ + 1 cos2 θ − 1
cos
0
0




 2

2
cos
θ
−
1
cos
θ
+
1
0
0

3

.


4
0
0
2cosθ
0 




0
0
0
2cosθ

(6.11)The matrix, R(φ), is the rotation matrix that rotates the incident axis êr to ê0r (perpendicular tothe scattering plane),

76




1
0
0
0






0
cos(2φ)sin(2φ)0R(φ) = 0 −sin(2φ) cos(2φ) 0




0
0
0
1

(6.12)

and R(ψ) is the rotation matrix that transforms the coordinate system ê00l , ê00r and ê003 back to theoriginal coordinate system êl ,êr and ê3 , and can be found from θ and φ. The polarization of thephotons is encoded in the Stokes vector. The probability of a scattering event can be directly
calculated from S000 /S0in . Complete trajectories of the photons can be found from Monte Carlosampling of scattering events. As the phase information of photon is not kept in this approach, it ismore suitable for describing classical media or hot atoms where phase coherence is not important.
The polarization of the incident photon is randomized after multiple scattering events. Since
the intensity detected along the polarization direction of the incident photon requires at least two
scattering events, it is suppressed compared to other directions but nonzero (Fig. 6.8(a)) and thelinewidth at small τp also drops below the natural linewidth (Fig. 6.8(c)) along this direction. For
the other directions, the intensity distribution is roughly homogeneous and does not exhibit thecollective enhancement along the forward direction observed in the coherent dipole model. The
FWHM linewidth for different directions increases linearly with OD up to a moderate value ofOD (see Fig. 6.8(b)), and displays a polarization dependence similar to the prediction of CD.
Under this classical treatment more scattering processes are expected to occur with increasing τpand those processes tend to inhibit the transmission of light. As the scattering becomes more
frequent, forward scattering decreases and more light is scattered backwards [133]. Since τp is
maximum at resonance, ∆ = 0, the linewidth develops a “double-peak” profile as the medium
becomes denser (Fig. 6.8(c)). Before the distortion in line shape develops, the FWHM linewidthalso linearly increases with τp . We note that a similar “double-peak” structure also appears in thecoherent dipole model. However, there, it only happens at specific small angles (see Fig. 6.3(c)), andnever happens along the forward direction, where the stimulated photon emission dominates or the

77
(b)

-0.3
���
-0.4

θ=�
θ=π /�
■ θ=π /�

���

���
-0.1
���
-0.2

���������/Γ

Intensity(arb.
units)
���������
[�����](a) ���
0.0

���

ΔD =0

ΔD =40kHz-0.5 �-���������

���
-0.6

MF
-���
-���
0.00
0.05

���
���
���

���
���
0.10
0.15
ϕ/π
n0k -3���
0.20■ ■ ■ ■ ■■ ■ ■ ■ ■■ ■ ■ ■ ■■ ■ ■ ■ ■

���
�
0.25

0.0
�
-0.1
-0.2
�
-0.3
-0.4
�
-0.5
�
-0.6
-�
0.00

�
τ�

�

�

θ=π/��τ� =���
θ=π/��τ� =���

θ=π/��τ� =���ΔD =0
ΔD =40kHz

θ=��τ� =���

MF
-�
0.05

�

Intensity (θ>0) (arb. units)��������� [���� �]
Intensity (norm.)

(c)

���

θ=��τ� =���
�0.10
0.15
�
�������� /Γ -3
n0 k

0.20
�

0.25
θ=��τ
� =���

Figure 6.8: RW Model. (a) Distribution of on-resonance intensity in the plane θ = π/2. Alongthe direction where single scattering events are forbidden (φ = 0), the light intensity is suppressed.
(b) The FWHM linewdith increases linearly with τp in a dilute medium. The linewidth along thedirection parallel to the laser polarization can drop below Γ for small (purple squares). (c) Lin
eshapes for different τp s and angles of observation. The line shape is Lorentzian for low τp but gets
distorted and develops a double-peak structure as the τp increases. Here, for all angles, the signal
is collected within a small angular region δθ = 5◦ . The intensity is normalized to the value at zerodetuning for θ = 0 and τp =0.3, and the width of laser beam is 5 times the Gaussian width of theatomic cloud. Here all τp are labeled as the value at ∆ = 0.

transverse direction, where the photons scattered via dipolar exchange dominate. We attribute theappearance of the “double-peak” structure to the interplay between these two types of emission. Insummary, despite of the fact that the RW model does not include coherent emission mechanisms,
it is able to reproduce the collective broadening observed with increasing optical depth and the
sub-natural linewidth present in the direction parallel to the laser polarization at relatively small
optical depths. The RW model on the other hand ignores coherent elastic dipolar interactions andthus does not predict a density shift.

78

6.4

Role of atomic motion
An assumption made in Sec. 6.2 is that the position of the atoms remains fixed. This

assumption is only valid when atoms move at a rate slower than the radiative decay rate. Whenmotion is significantly faster, e.g. hot atomic clouds, the coherence during radiation is smearedout and classical approaches, such as the RW model, are usually satisfactory [258, 36] to describe
collective light emission. However, many experiments operate in an intermediate regime where
coherences cannot be totally disregarded and it is not a good approximation to treat atoms as
frozen. For example, for the 1 S0 → 3 P1 transition of 88 Sr atoms, the Doppler boadening at ∼ 1 µK,∆D ≈ 6Γ, and recoil frequency ωr = ~k02 /2m ≈ 0.6Γ, both comparable to the natural linewidth[108, 254]. Consequently for a proper description of light scattering one needs to account for bothphoton coherences and atomic motion on an equal footing. Below we present two approximate
ways to accomplish this:

6.4.1

Modified frozen dipole model

In this section, we discuss a simple way to include the effects of atomic motion on lightscattering via a modified CD model. For a single atom, to leading order, a major modificationfrom motion in the emitted light intensity is the Doppler shift, a velocity dependent modification of
the effective laser detuning experienced by an atom (see Appendix D.2). We include this effect inthe many-body system by introducing random detunings for each atom, i.e., ∆ → ∆ + δν, and by


2
1
δν
sampling them according to a Maxwell-Boltzmann distribution P (δν) = √ e exp − e 2 that
2π ∆D
2∆D
√
e
accounts for the Doppler shifts [112]. Here ∆D = ∆D / 8ln2. We denote this approximation as themodified frozen dipole model. The random detunings modify atomic coherences as (for simplicity
we assume a single beam illumination):bj=Ωeik0 ·rj.
(∆ − δνj ) + iΓ/2

(6.13)

79
Using Eq. (6.13), the intensity along a generic direction becomes
Iincoh =

√

1

Z

eD
2π ∆

2

e2

dδνj |bj |2 e−δνj /2∆D .

(6.14)On the other hand, the coherent scattering in the forward direction which takes into accountpairwise atomic contributions, becomes
Icoh =

1
e2
2π ∆

Z

2

e2
e 2 −δν 20 /2∆Djdδνj dδνj 0 bj b∗j 0 e−δνj /2∆D e.

(6.15)D(a) ����

The on-resonance enhancement factor is thus

����

Icoh
Iincoh2e=1e 2 /Γ2
8∆
D -����

�����������
�����������

pπ

���
-����

erfc( √ e1)
2 2∆D /Γ
���
,
-����
e D /Γ
2∆
���
-����

(6.16)-������� induces an exponential suppression of thewhere erfc is the complementary error function. This
�

�

�

�

�
Δ� /Γ27e D /Γ [33], as shownforward interference that depends on ∆in Fig.
�� 6.9(a).
��

�

�

�

��

��

Δ� /Γ

(b)
0.0���
0.0

���
-����

��
-0.1-0.1
Intensity (norm.)

-����
���
-����
���
-����
-����
���

�

�

�

���������
[����]
Intensity(arb.
units)

����

�

�

�

�
�
Δ� /Γ
��
Δ� /Γ

�

��

�

��

(b)
0.0�������

θ=�
θ=π/��
θ=π/�

�
◇
◇◇◇
◇
-0.2
◇
-0.2 ��
�
◇◇◇
◇◇◇
◇◇◇
-0.3
◇◇◇
◇
◇
-0.3 ��
◇◇◇�
◇◇◇-0.4 ΔDΔ=0
◇◇◇
D =0
◇
◇
-0.4
◇
◇
◇
��
�
=40kHzΔDΔ◇=40kHz
◇◇
D
-0.5
◇
-0.5
◇◇ MF
◇
-0.6
� MF
�
-0.6
�
��� 0.20
������
0.00 ���
0.05 ���0.10��� 0.15
0.25
0.00
0.05
0.10 � -3
0.15
0.20
0.25
n0 k
n0k -3��������� (θ>�) [����]Intensity (θ>0) (arb. units)�����������
�����������

(a) ����

���������
[����]
Intensity(arb.
units)

Intensity (norm.)��������� (θ>�) [����]Intensity (θ>0) (arb. units)

Figure2.9: model:
Modifiedfrozendipole model:
peak intensity.
Collective forward
θ=�frozen
Figure 6.9:0.0Modified
dipolepeakintensity.
(a) Collective
forward(a)enhancement:
for- enhancement:θ=π/�� ward intensity normalized-0.1
��� by the transverse intensity as a function of Doppler width. (b) Inten-0.1
ward intensity normalized
by the transverse intensity as a function of Doppler width. (b) Intensity◇ θ=π/�
◇◇◇ directions at Doppler widthalongdi↵erent= 6 as a function of atom number◇◇◇-0.2-0.2
detectedalong
different detected
directions
atDoppler���width ∆D = 6Γ as a functionD of atom number for
◇◇◇
◇
◇
3
1.6 for ✓ = 0, and I ⇠ N 0.7 for ✓ = ⇡
◇◇
◇
◇
fixed
cloud
size
(when
N
=
500,
⌧
=2,n
k
=0.0015).
I
⇠
N
−3
◇
0 0for θ = 0, and I ∼ N 0.7◇◇◇τ =2,n0 k
fixed cloudsize (when N = ◇500,
I ∼ N 1.6
for θ = π/2.
-0.3-0.3
0 =0.0015).
◇◇◇
��
�
◇
right vertical axis labels the intensity for ✓ = ⇡/2, ⇡/10.
ΔDΔ=0
◇◇
◇The
-0.4
◇
D =0
The right
verticalaxislabelstheintensity
for
θ
=
π/2,
π/10.
◇
◇
-0.4
◇◇

◇
��
�
=40kHzΔDΔ◇=40kHz
◇◇
D
-0.5
◇
-0.5
◇◇ MF
◇
-0.6
� MF
�
-0.6
�
���
���
���
���
���
���
0.00
0.05
0.10
0.15
0.20
0.25
The0.00
reducedlight0.10
scatteringon the other hand, competes with dephasing since it� 0.15 probability,
0.05
0.20
0.25
n0k -3���-3n0 k
Ω=���Γ
suppresses multiple scattering,
and as a consequence promotes collective enhancement.
Effectively,
Ω=���Γ
��� enhancement: forure 2.9: Modified frozen dipole model: peak intensity. (a) Collective forward

Ω=Γ����������

d intensity normalized by the transverse intensity as a function of Doppler���width. (b) Intensity
it brings the system closer to the small OD regime (Fig. 6.2(b)). The motion induced suppressionected along di↵erent directions at Doppler width D = 6 as a function of atom number for���
3
d cloud size (when= 500, ⌧ =2,n0 kprocesses
I ⇠ signaled
N 1.6 for ✓in=the0, andI ⇠ N 0.7
for ✓(Fig.
= ⇡/2.
0 =0.0015).
of N
multi-scatteringis alsofrequency
shift6.4) [112, 114], which keeps
���e right vertical axis labels the intensity for ✓ = ⇡/2, ⇡/10.

increasing until a larger value of density in the presence��� of motion.
-��

���
���

-�

�
��������/Γ���Figure 2.10: Semi-classical model: single-atom line shape. The excitation line shape is calcula
Ω=���Γ
for three di↵erent
driving strength, each normalized by ⌦2 . The atomic motion is allowed toΩ=���Γ

80
6.4.2

Semi-classical approach

Laser light mediated forces on atoms are a fundamental concept in atomic physics and lay thefoundations of laser trapping and cooling [196, 44]. They can be accounted for at the semiclassicallevel by explicitly including the position ri and the momentum pi degrees of freedom of the atoms,
and solving for their dynamics while feeding those back into the quantum dynamics of the internal
degrees of freedom. An explicit description of this procedure is presented below.
For simplicity we will assume a two-level transition. This condition is achievable in experiments, for example, by applying a large magnetic field to split apart (|∆α − ∆γ |  GΓ) the
excited state levels and thus energetically suppressing population of the ones not directly driven by
the laser. For driving the atoms we will consider the case of two counter-propagating lasers withwavevector ±k0 , propagating along x, and Rabi frequency Ω. The internal atomic variables evolve
according to (see Appendix D.1)
dbj
dt

= iΩ cos(k0 ·rj )sj −(XΓ+ i∆)bj +i
sj bl (gjl −ifjl ),2
l6=j(6.17)
dsj
dt= 2iΩ cos(k0 ·rj )(bj − H.c.) − Γsj
−2i

X
l6=j

(gjl − ifjl )b∗j bl + 2i

X(gjl + ifjl )b∗l bj ,l6=j(6.18)
where ∆ = ∆α , and the last terms describe the effect of dipole-dipole interactions g(r) and f (r)(see Appendix D.1). In general the position of atoms can also change with time as
drj
dt

=

pj
,
m

(6.19)

so the response to the local driving field and dipole-dipole interactions in Eq. (6.17) and Eq. (6.18)
change accordingly. The momentum changes not only due to the force exerted by the driving
laser, but also the long-range forces gjl = −∇gjl and fjl = −∇fjl , which originate from thedispersive and dissipative dipole-dipole interaction, respectively (see Appendix D.1), and depend

81on the instantaneous atomic coherences and positions:dpj
dt= −2~k0 Ωsin(k0 · ri )<{bi (t)}
+

X
l6=j

[gjl (bj b∗l + H.c.) − ifjl (bj b∗l − H.c.)].

(6.20)

The atomic coherence and external motion are now coupled together, so Eq. (6.17)∼Eq. (6.20)
need to be solved simultaneously to obtain the dynamics. In deriving Eq. (6.20), we have ignored
the role of momentum diffusion (see Appendix D.1). We have verified in our numerical simulations
that it can be safely ignored for the parameters of interest presented in this work.
From previous sections, it has been shown that except for a narrow cone around the forwardP
direction, the fluorescence intensity is I(rs ) ∝i si (also see Appendix D.2). Since we expect
motion to further reduce the effect of coherence, we will focus on the transverse scattering, andP
compute i si . Fig. 6.10 shows the line shape of a single atom for different Ω calculated from thesemi-classical approach.
���Ω=���Γ
Ω=���Γ
Ω=Γ����������

���
���
���
���
���
-��

-�

�
��������/Γ���Figure 6.10: Semi-classical model: single-atom line shape. The excitation line shape is calculated
for three different driving strength, each normalized by Ω2 . The atomic motion is allowed to be
one-dimensional and parallel to the laser propagating direction. Here we used ωr = 0.6Γ (the valueof the 1 S0 → 3 P1 transition of Sr [108]).

As a result of atomic motion, the laser-atom system is not in a stationary state. We show theresults for driving time t = 5/Γ. When Ω  Γ, the line shape is a Voigt profile with the Doppler
width determined by the velocity. When Ω ∼ Γ, there is a distortion in the line shape, with more
intensity at ∆ < 0. This is because for ∆ < 0 the laser force decelerates the atom, while for ∆ > 0

82
the atom is heated up. With reduced/increased velocity, the atom is on average more/less excited,resulting in a distorted line shape. The center of the line is therefore shifted to the red.
Motion can significantly modify the interactions between atoms [57]. We study the effectof motion on the frequency shift in Fig. 6.11. Dipolar induced frequency shifts were previously
discussed in Sec. 6.2. Since we focus on low driving fields, again we neglect momentum diffusionin these calculations. The calculations show that unless ∆D is very small, atomic motion leads toa fast suppression of the frequency shift. Only when ∆D  Γ, the frequency can be increased bymotion and this is the regime where the modified frozen dipole model is qualitatively valid; recall,it predicts always an increase of density shift with Doppler broadening. We note that at ∆D → 0
the frequency shift obtained using the modified frozen dipole model, is slightly smaller than that
one obtained from the semi-classical approach. This is a consequence of the distortion caused bylaser cooling/heating which additionally shifts the spectral line. For ∆D & Γ motion needs to beproperly accounted for and the modified frozen dipole model is not reliable.

��������� �����/Γ

����
-����
-����
-����
-����
-����
�

�

�

�
Δ� /Γ

�

�

�

Figure 6.11: Semi-classical model: effect of motion on frequency shift. Two atoms are driven by apair of counter-propagating lasers, with Ω = 0.1Γ, ωr = 0.6Γ. The red dashed line shows the resultof semi-classical model where motion is allowed in three dimensions. The blue line shows the result
from the modified CD. The center of excitation line shape is calculated at t = 5/Γ for different
Doppler width, for both models.

83

6.5

Comparison with experiment
In this section, we describe an experiment that measures the collective emission from coher-

ently driven ultracold 88 Sr atoms.

88 Sr atoms have both a strong 1 S −1 P blue transition (λ = 46101nm) and a spin-forbidden weak 1 S0 −3 P1 red transition (λ = 689 nm), with a strict 4-level geometry(Fig. 6.12(a)). When the atoms are cooled to a temperature of ∼1 µK, Doppler broadening at 461
nm is about 55 kHz, which is almost three orders of magnitude smaller than the blue transition
natural linewidth, Γ =32 MHz. To an excellent approximation atomic motion is negligible foratomic coherence prepared by the 461 nm light. To the contrary, the red transition with a naturallinewidth Γ = 7.5 kHz is strongly affected by atomic motion. By comparing the behaviors of thesame atomic ensemble observed at these two different wavelengths (Fig. 6.12(b)) we can thus collectclear signatures of motional effects on coherent scattering and dipolar coupling[112, 132].

Figure 6.12: (a) The experimental scheme and concept. We weakly excite the strontium atoms
with a linearly polarized probe beam and measure the fluorescence with two detectors, one in
the forward direction, x̂, and the other almost in the perpendicular direction, ẑ. We probe twodifferent J = 0 to J’ = 1 transitions. The first transition is a 1 S0 −1 P1 blue transition with anatural linewidth of Γ = 32 MHz and the second is a 1 S0 −3 P1 red transition with Γ = 7.5 kHz.
(b) In the coherent dipole model photons are shared between atoms. When the Doppler broadened
linewidth becomes comparable to the natural linewidth, dephasing must be considered. At our∼ 1µK temperatures the Doppler broadening is ≈ 40 kHz meaning motional effects are important
only for the red transition.

84
We use the experimental scheme shown in Fig. 6.12(a) to perform a comprehensive set ofmeasurements of fluorescence intensity emitted by a dense sample of 88 Sr atoms. The sample isreleased from the trap and then illuminated with a weak probe laser. We vary the atomic density,
cloud geometry, observation direction, and polarization state of the laser field, and we report
the system characteristics using three key parameters: the peak scattered intensity, the linewidth
broadening, and the line center shift. For example, along the forward and transverse directions weobserve different values of intensity and linewidth broadening, as well as their dependence on lightpolarization (see Fig. 6.13). We also observe motional effects on the red transition in contrast to
the same measurements on the blue transition.

Figure 6.13: The 3D intensity distribution predicted for a blue probe beam. The coupled-dipole
model predicts a strong 103 enhancement of the forward intensity compared to other directions
and a finite fluorescence along a direction parallel to the incident polarization. The speckled
pattern is due to randomly positioned atoms and can be removed by averaging over multiple atom
configurations.

In the experiment, up to 20 million 88 Sr atoms are cooled to ∼1 µK in a two-stage magnetooptical trap (MOT), the first based on the blue transition and the second on the red transition.
The atomic cloud is then released from the MOT and allowed to expand for a variable time of flight(TOF), which allows us to control its optical depth and density. They are subsequently illuminatedfor 50(100) µs with a large size probe beam resonant with the blue (red) transition (Fig. 6.14(a)).
The resulting scattered light is measured with two detectors far away from the cloud (see Fig. 6.12(a)). One detector is along the forward direction x̂ (detector DF ), and the other transverse direction

85ẑ (detector DT , offset by ∼10◦ ). For a short TOF, the atomic cloud is anisotropic and has an
approximately Gaussian distribution with an aspect ratio of Rx : Ry : Rz = 2 : 2 : 1 , where
R{x,y,z} are r.m.s radii.
The coherent effect manifests itself most clearly in the forward direction (Fig. 6.14). To

Figure 6.14: Forward Scattering. (a) Comparison of forward scattering intensity versus angle using a
red and blue probe beam. The inset shows the setup used to block the probe beam. After interactingwith the atoms the probe beam is focused using a lens, which also collimates the fluorescence fromthe atoms. We block the probe beam using a beam stopper, which we translate perpendicular to theprobe beam to change the angular range of fluorescence collected by the detector, characterized by
the angle (θ) between x̂ and the edge of the beam stopper (see Ref. [33]). The measured intensity,
Ix,0 (θ), for each probe beam is normalized to the intensity at θmax = 7.5 mRad. The dephasing
caused by motion reduces the forward intensity peak for the red transition. (b) Comparison of
intensity in the forward direction, Ix , versus intensity in the transverse direction, Iz . Both arevaried by changing N . All measurements are made at θ = 2 mRad (arrow in (a)) and normalized
to the intensity, Ix,0 , for the atom number used in (a). (c) Linewidth broadening in the forward
direction measured by scanning the blue probe beam detuning, ∆, across resonance. Examplelineshapes for different optical depths (ODs) are shown in the inset. Two different atom numbers
are used, N = 1.7(2) × 107 (blue squares) and N/4 (cyan triangles). The dashed line represents Γfor reference. All solid curves are based on the full theory of coupled dipoles and the band in (c)is for a ±20% atom number uncertainty. All errorbars are for statistical uncertainties.
separate the forward fluorescence from the probe beam, we focus the probe with a lens (L1 ) after it
has passed through the atomic cloud and then block it with a beam stopping blade, which can betranslated perpendicular to the probe beam (Fig. 6.14(a) inset). The same lens (L1 ) also collimates
the atomic fluorescence so that it can be imaged onto DF . The position of the beam stopper canbe used to vary the angular range of collected fluorescence, characterized by the angle (θ) betweenx̂ and the edge of the beam stopper (See Ref. [33]). Using the maximum atom number available in

86
the experiment, the measured intensity Ix,0 (θ) is normalized to that collected at θmax = 7.5 mRad.
Both the blue (square) and red (triangle) transition results are displayed in Fig. 6.14(a). For theblue transition a thousand-fold enhancement of the normalized intensity is observed for θ < 0.5
mRad. We note that this enhancement as a function of θ far exceeds the diffraction effect by thebeam stopper. The latter effect becomes significant only when θ . 0.1 mRad, and is also suppressedin our differential measurement scheme which compares measurements with and without atoms.
The enhancement is also present for the red transition, but it is reduced by nearly two-orders ofmagnitude at small θ due to the motional effect. On the other hand, the wider angular area ofenhancement is attributed to the longer wavelength of the red transition. The forward intensity
strongly depends on the atom number. Fig. 6.14(b) shows measurements of the forward intensity Ix
versus the transverse intensity Iz at a fixed θ = 2 mRad for different atom numbers. The intensities
are normalized to those obtained at the peak atom number as used in Fig. 6.14(a). To the firstorder approximation, the transverse fluorescence intensity scales linearly with the atom number.
Hence, the forward intensity of both the blue and red transitions scales approximately with theatom number squared. These observations are consistent with the conclusions obtained in Sec. 6.2.
The solid lines in Figs. 6.14(a) and (b) represent the full solution of the CD model, which agreewith experiment for both transitions.
In the forward direction we have also investigated the linewidth broadening of the bluetransition as a function of the atomic OD. By scanning the probe frequency across resonance we
extract the fluorescence linewidth, which is found to be determined primarily by the OD of thecloud (open squares in Fig. 6.14(c)). For the range of 0 < OD < 20 the lineshape is Lorentzian (seeinsets); however, the observed lineshape starts to flatten at the center for OD>20. We have also
varied the atom number by a factor of four, and to an excellent approximation the linewidth data
is observed to collapse to the same curve when plotted as a function of OD (open triangles). It isalso evident that the scaling of the linewidth vs. OD turns nonlinear at large values of OD. The
experimental data falls within the shaded area in Fig. 6.14(c), which represents the full solution of
CD model with a 20% uncertainty in the experimental atom number.

87
For independent emitters the forward fluorescence should have no dependence on the probe
beam polarization; however, the transverse fluorescence (along ẑ) should be highly sensitive to theprobe polarization, and it is even classically forbidden if the probe is ẑ-polarized. However, multiple
scattering processes with dipolar interactions can completely modify this picture by redistributing
the atomic population in the three excited magnetic states and thus scrambling the polarizationof the emitted fluorescence, with an intensity that increases with increasing OD. Consequently,
even for a ẑ-polarized probe there should be a finite emission along ẑ. This is consistent with thetheoretical prediction of the CD model, as shown in Fig. 6.13. The experimental investigation of thefluorescence properties along the transverse direction is summarized in Fig. 6.15. Under the same

Figure 6.15: Transverse Scattering. (a) Linewidth broadening for the blue transition in the transverse direction for ŷ-polarization (open squares) and ẑ-polarization (open triangles). (b) Intensity
ratio, Iypol /Izpol , of ŷ-polarization to ẑ-polarization measured in the transverse direction when ablue probe beam is used. For low optical depths single particle scattering is dominant, and for single
particle scattering almost zero intensity is predicted for ẑ-polarized fluorescence as this polarization
points directly into the detector. (c) Linewidth broadening for the red transition in the transverse
direction for ŷ-polarized light showing a similar trend to the blue transition. This transition ismore sensitive to magnetic fields so a large magnetic field is applied to probe only the m = 0 to m’= 0 transition. All solid curves are based on the full theory of coupled dipoles and the band in (c)is for a ±20% atom number uncertainty. All errorbars are for statistical uncertainties.
OD along ẑ, the ŷ-polarized probe beam (square) gives rise to a much more broadened lineshapefor the blue transition than the ẑ-polarized probe beam does (triangle), as shown in Fig. 6.15(a).
Meanwhile, the peak intensity ratio of Iypol /Izpol decreases significantly with an increasing OD,
indicating the rapidly rising fluorescence with a ẑ-polarized probe when OD increases (Fig. 6.15(b)).

88
For both intensity and linewidth broadening observed in the transverse direction, under either ŷ orẑ probe polarization, the full solution of the CD model (shown as shaded areas in both Fig. 6.15(a)and (b)) reproduces well the experimental results on 1 S0 −1 P1 .
For the red transition, the existence of Doppler broadening requires the lineshape data to be
fitted to a Voigt profile. With the Doppler linewidth ∆D fixed from the thermal velocity measured
in free expansion, the Voigt profile determines the line center as well as the Lorentzian linewidthwith the Gaussian linewidth determined by the temperature. Fig. 6.15(c) displays the Lorentzian
linewidth obtained with a ŷ-polarized red probe showing a strong increase of the linewidth with OD.
Taking into account motional dephasing (see Sec. 6.4.1), the transverse broadening for 1 S0 −3 P1 isalso well reproduced as shown in Fig. 6.15(c).

6.6Conclusion
We theoretically studied the propagation of light through a cold atomic medium. We pre-sented two different microscopic models, the “coherent dipole model” and the “random walk model”,
and analyzed how the light polarization, optical depth and density affect the linewidth broadening,
intensity and line center shift of the emitted light. We showed that the random walk model, which
neglects photonic phase coherence, can fairly capture the collective broadening (narrowing) of theemission linewidth but on the other hand does not predict a density shift. Due to the limitation ofcomputation capacity, the numerical simulation of CD is usually restricted to ∼ 104 atoms, which
is much smaller than that in some cold atom experiments [95, 12]. Nevertheless, the understandingof the underlying physics allowed us to perform an appropriate rescaling in the cloud size to compare with experiments. We further developed generalized models that explicitly take into accountmotional effects. We showed that atomic motion can lead to drastic dephasing and reduction in
the collective effects, together with a distortion in the line shape. While the modified frozen dipole
model predicts a monotonic increase of the density shift with increasing motion, the semiclassicalmodel, which properly accounts for recoil effects, predicts that this behavior only holds at slow
motion ∆D  Γ. Instead, as atoms move faster, motional effects start to become dominant, the89
cloud expands and the frequency shift decreases. We show, as an explicit example, an experiment
done with 88 Sr atoms [33], and apply the CD model to explain all the experiment observations forthe strong transition, where motional effects are weak. The modified frozen dipole model is also
shown to capture the fluorescence intensity and linewidth broadening measured for the weak transition, where motional effects are significant. None of the presented theoretical models, however,can explain the large density shift measured in the 1 S0 → 3 P1 transition of 88 Sr atoms [108]. It
will be intriguing to determine what are the actual physical processes that cause this large shift.

Chapter 7

Emergent Weyl excitations in systems of polar particles

7.1

Introduction
Recent predictions [263, 37] and experimental observations [276, 156, 151] of Weyl semimet-

als in solid state systems have instigated intensive studies of their properties, such as non-local
electrodynamics and chiral anomaly [194], topologically protected Fermi arcs on the surfaces [263,
276, 156], non-Anderson disorder-driven transitions[78, 91, 246, 245, 219, 174, 125] and unusual
dependencies of transport properties on doping and temperature [218, 228, 209]. In parallel, enormous research efforts are now directed at finding Weyl excitations in new systems. A promising platform for exploring Weyl physics is tunable and fully controllable ultracold atomic gases[139, 117, 4, 70, 81, 150]. However, henceforth proposed cold-atom realisations of Weyl quasiparticles have focussed mostly on non-interacting systems, and all have required implementations ofexternally imposed spin-orbit coupling through laser assisted tunneling schemes [139, 117, 4, 70, 81],other optical means [271, 89, 225], or external rotating fields [150].
In this chapter we demonstrate that excitations with Weyl dispersion generically emerge inthree-dimensional (3D) arrays of dipolar particles in the presence of a weak magnetic field, as a resultof the dipole-interactions-induced transitions between their internal angular-momentum J = 0 and
J = 1 states. These excitations exhibit the same single-particle physics as Weyl fermions [265] but,similarly to other non-fermionic Weyl excitations [151], their many-particle properties are expected
to be different, opening alternative research directions, new functionalities and applications beyond
those accessible with solid-state systems [276, 156].

91
We also show that, experimentally such excitations can be observed, for instance, in trappedalkaline-earth atoms (AEAs) in a 3D optical lattice with lattice spacings smaller than the wavelengthof the electronic J = 0 → J = 1 transition. The simple and unique internal structure of these atoms
has already lead to record levels of precision and accuracy in atomic clocks [27]. Taking advantageof the well developed spectroscopic tools to interrogate and manipulate AEAs, we propose toprobe the Weyl quasiparticle dispersion and non-trivial chirality by means of momentum-resolvedRamsey spectroscopy. Our proposal opens a path for a feasible experimental realisation of Weyl
quasiparticles in clean and controllable atomic systems. Moreover, it lays the groundwork for the yet
unexplored regime of topologically protected sensing, owing to the topological robustness of Weyl
quasiparticles that could be used to push the stability and accuracy of optical-lattice AEA-based
clocks.

7.2

Phenomenological argument
We assume that the system has long-lived excitations (quasiparticles) with (integer) angularmomentum J. Due to the translational invariance, the (quasi)momentum k is a good quantum
number. In the long-wave limit the effective quasiparticle Hamiltonian is insensitive to the detailsof the potential of the periodic lattice that the particles may be placed in. To preserve rotation andinversion symmetries in the absence of magnetic field the Hamiltonian has to be an even functionof (k · Ĵ) and a function of |k| and Ĵ2 . In the presence of a sufficiently weak uniform magnetic field,
ω, that creates a perturbation −ω · Ĵ independent of k in the limit k → 0, the most generic formof the quasiparticle Hamiltonian is given byhiĥ(k, J) = F |k|, (k · Ĵ)2 , Ĵ2 − ω · Ĵ,(7.1)

where F is an arbitrary function of three arguments.
The small quasimomentum k can be measured from any high-symmetry point in the Brillouinzone characterised by an isotropic dispersion ξk = ξ(|k|) of non-interacting particles in the limit
k → 0.

92
For the particular case of J = 1, the Hamiltonian (7.1) has nodes at momenta K k ω, suchthat F (|K|, |K|2 , 2) ± ω = F (|K|, 0, 2), corresponding to two intersecting branches with angularmomentum projections Jz = 0 and Jz = 1 or Jz = −1 on the direction of magnetic field. We
note that such nodes always exist for weak magnetic fields and Hamiltonians that are regular as afunction of k.
The excitation Hamiltonian near a node is obtained by expanding the function F in small
momentum p = k − K. For a 3D system, it has Weyl dispersion of the form (see Appendix E)ĥeff (p) = ζ(p) + v⊥ σ̂x px + v⊥ σ̂y py + vk σ̂z pz(7.2)

with Pauli matrices σ̂i acting in the space of the respective two angular-momentum projections.

7.3

Model
In what follows we confirm the above phenomenological argument by microscopic calculations

for a 3D system of dipolar particles described by the HamiltonianĤ =X

Ĥ0 (ri ) +

i



1X
Ĥdip d̂i , d̂j , ri − rj ,
2

(7.3)

i,j

where d̂i is the dipole moment operator of the i-th particle, andĤ0 (ri ) = −ˆ2∇i+ U (ri ) + BJ Ĵ2i − Ĵi · B
2m

(7.4)
ˆ2

i
is the single-particle Hamiltonian that includes the particle kinetic energy − ∇2m (hereinafter ~ = 1),

the periodic potential U (ri ) of the lattice that the system may be placed in, the energy BJ Ĵ2i ofinternal levels with Ĵi being the angular momentum of the i-th particle, and the interaction −Ĵi · B
with magnetic field (measured in units of the gyromagnetic ratio) that splits the J = 1 levels.
The most generic form of the dipole-dipole interaction, which accounts for retardation effects,
is given by 1
Ĥdip (d̂i , d̂j , r) = a(r)(r̂ · d̂i )(r̂ · d̂j ) + b(r)d̂i · d̂j

(7.5)

P
σ† σ†
σ
σ σ
See chapter 2. d̂ = σ êσ d(bσ†i + bi ). Terms like bi bj and bi bj do not play an role, since they are enegeticallysuppressed. See Appendix E.2 for further explanation.
1

93
0
0
where r̂ = r/r; a(r) = 3γ[y (k r) − ij2 (k0 r)] and b(r) = 3γ
4d2 2 04d2

P1

nn=0 (−1) [yn (k0 r) − ijn (k0 r)]/(k0 r)

for r 6= 0, with yn and jn being the n-th-order spherical Bessel functions of the second and first kindrespectively and k0 – the wavevector of the J = 0 to J = 1 transition. The terms proportional to yn
describe elastic interactions between dipoles a distance r apart, while the terms with jn account for
the inelastic collective photon emission (radiation). In this chapter we use γ0 to denote the naturallinewidth of the transition to distinguish it from the high symmetry point Γ. If the dipoles are muchcloser to one another than the wavelength of the dipole transition, k0 r  1, retardation effects canbe ignored, and one recovers the more familiar form of the dipolar interactions, a(r) ∝ −3/r3 ,b(r) ∝ 1/r3 , common for NMR solid-state systems [47], polar molecules [278] and Rydberg atoms[264, 208].
We note that the above phenomenological derivation of the dispersion of Weyl-type quasiparticles carries over straightforwardly to other dimensions. For example, a 2D system of dipolar
particles with an in-plane magnetic field hosts 2D Dirac excitations with the dispersion of monolayer graphene [183]. We emphasise that such 2D excitations are distinct from the 2D “chiron”
excitations [247] that exist in a perpendicular magnetic field and resemble electrons in bilayer
graphene.

7.4

Atoms in a deep lattice
While the above phenomenological argument demonstrates the existence of Weyl quasiparti-

cles in a generic 3D system of dipolar particles in magnetic field, below we focus on the experimentally important case of particles pinned in a deep unit-filled cubic lattice (Fig. 7.1 (a)) with smalllattice spacing a; ak0  1.
We assume that all particles are initially prepared in the Ji = 0 state and that the energy BJof internal levels significantly exceeds the interaction strength (usually in dipolar gases [278, 187]
|Ĥdip |/BJ . 10−6 ), leading to the conservation of the number of sites excited to the J = 1 state toa good accuracy (cf. Appendix E).
If an excitation with the angular momentum J = 1 is created on site i, the dipole-dipole

94

a)
Ji

b)
i
σ = +1 σ = 0 σ = −1

Ji = 1j

σ = +1σ=0

σ = −1

Jj = 1Jj
Ji = 0Jj = 0Figure 7.1: Weyl quasiparticles in 3D dipolar arrays. (a) Schematics of the 3D lattice potentialthat traps an array of dipolar particles. The lattice is deep enough to pin the particles, most ofwhich are prepared in the J = 0 ground state (blue spheres). Only a few particles are excited to
the J = 1 states. Dipolar interactions between the J = 0 and J = 1 states give raise to Weyl
excitations. (b) Schematics of dipole mediated interactions: An excited J = 1 state of one particle
can be transferred to another particle in the J = 0 state by dipole-dipole interactions (virtualphoton exchange is shown with a yellow wiggly line). Three types of allowed processes include|00ii |1σij ↔ |1σii |00ij , |00ii |10ij ↔ |1, ±1ii |00ij , and |00ii |1, ±1ij ↔ |1, ∓1ii |00ij .

interaction can transfer it to another site j, possibly changing the projection of the angular momentum on the direction of the magnetic field; |1σii → |1σ 0 ij . The quasiparticles in the system arethus hard-core bosons corresponding to the angular momentum-degrees of freedom that hop fromsite to site as described by the effective Hamiltonian (see Appendix E for a detailed derivation)Ĥlat =X

0

Mijσσ b̂†iσ b̂jσ0 =i,j,σ,σ 0X

0Mkσσ b̂†σ,k b̂σ0 ,k ,(7.6)

σσ 0 k0Mijσσ = h1σ|i h00|j |Ĥdip d̂i , d̂j , ri − rj |00ii |1σ 0 ij
− σδσσ0 δij B.

(7.7)

Due to the translational invariance, the single-excitation Hamiltonian can be diagonalised in thebasis of momentum states k, with the results shown in Figs. 7.2a,c. In accordance with the above
general phenomenological argument, for B 6= 0 the dispersion has Weyl nodes (six in the first
Brillouin zone, Figs. 7.2a,c).
To demonstrate the chiral nature of Weyl quasiparticles we show in Figs. 7.2d,e,f the pseudospins hσ̂i (with the Pauli matrices σ̂i acting in the space of the Jz = −1 and Jz = 0 angular-

95
Weyl points

10

(a)

(b)

10

R

10

(c)

10

B 6= 0Weyl points

B=0

M

y
x

Γ

0

1

2

0

5

5

-5

(d)

1.2

kd
0.1

M

a/⇡ 0

k

1

2

3

4

X
(e)

.1

1.
1.0

M

k

R

(f)

0.0

0.8

Energy

0.2

0.75

-5

R

hb̂† 1,k b̂ 1,k i

k xa/⇡

4

5

X

X

0.8

3

Energyz

Energy

5

0.4

0.2

1.2
0.9

0.8

ky a/⇡

1.2

0
0.0

0.9

ky a/⇡

0.8

ky a/⇡

EnergyEnergy0.6

1.2 0.55

1.8

kz a/⇡

0.8

1.1

Figure 7.2: Weyl quasi-particle dispersion and eigenstates. (a) Brillouin zone for the simple cubic
lattice. (b) Dispersion along high-symmetry lines in the absence of magnetic field (all energies aremeasured in units of (3/4)γ0 /(k0 a)3 ). (c) Dispersion in the presence of magnetic field B = γ0 /(k0 a)3
demonstrating the existence of Weyl nodes (red points) with linear quasiparticle dispersion nearthem. Each node is characterised by the monopole charge ±1. In agreement with the fermion
doubling theorem [181] (the Nielsen-Ninomiya no-go theorem), there is an even number (six) of
Weyl points in the first Brillouin zone. (d) Dispersion in the horizontal (kx − ky ) plane (shown by
blue color in panel (a)) containing the Weyl node near the R point. Quasiparticles in this plane aresimilar to quasiparticles in graphene and are characterised by a non-trivial Berry phase of π. The
arrows show the pseudospin hσ̂i (the Pauli matrices σ̂i act in the space of the angular-momentum
projections Jz = 0 andalong the (yellow in panel (a)) plane consisting of√ Jz = −1). (e) Dispersion√
vectors k = (π + kd / 2, π + ky , 0.71π + kd / 2) containing the Weyl point. Color shows the weightof the |1 − 1i state in the quasiparticle eigenstate, and arrows represent the pseudospin hσ̂i. (f)Dispersion along the (green in panel (a)) vertical plane (ky − kz ) containing the Weyl point near
the R point. For each momentum k the color represents the weight of the |1 − 1i state .

momentum projections) for the eigenstates with momenta k in the horizontal (pz = 0), tilted(pz = px ), and vertical (px = 0) planes (Fig. 7.2 (a)) that contain a Weyl node. Excitations in theseplanes are equivalent to quasiparticles in graphene, the 2D counterpart of a Weyl semimetal, andare characterised by the non-trivial Berry phase π. Figs. 7.2d,e,f demonstrate that the pseudospinsσ̂ of these states are linked to their momenta p, measured from the Weyl node, in agreement withthe effective Hamiltonian (7.2).

96

7.5

Effects of quenched disorder and dissipation
In general, quasiparticles in interacting systems have finite lifetimes due to elastic and inelastic

scattering processes. Indeed, deep optical lattices under consideration are ususlly not completely
filled by particles and thus inherently disordered due to the randomness of the particle disptribution.
Also, spontaneous and dipolar collective emission from the internal J = 1 levels to the ground state
can lead to the decay of the excitations.
To analyse the effects of dissipation in a unit-filled lattice we compute numerically the quasiparticle dispersion for retarded dipolar interactions, Eq. (7.5), with parameters of the J = 1 to
J = 0 transition of the electronic 3 P0 − 3 D1 levels of bosonic 88,84 Sr atoms trapped in a magic
optical lattice with a = 206.4nm considered in Ref. [187]. The wavelength and the dipole moment
for this transition are 2.6µm and d = 4.03D, leading to the linewidth γ0 = 290 × 103 s−1 and thedissipation parameter ak0 ∼ 0.5. Albeit quasiparticle damping in this regime is rather strong, it
is significantly suppressed (by more than three orders of magnitude) near the Weyl nodes, as our
simulations show, Fig. 7.3 (a) (b). Our results indicate that the quasiparticle scattering in sucha system would be dominated by quenched disorder rather than by collective radiative decay orspontaneous emission.
To account for the effects of disorder we evaluate numerically the quasiparticle dispersion fora lattice filling of 93%. This filling fraction could be achieved by preparing a cold bosonic Mott
insulator using moderate atom numbers that allow one to suppress doubly occupied states at thetrap centre. Mott insulators have already been realised with bosonic AEAs in in the ground 1 S0state [239, 235]. These atoms can be excited to the desired 3 P0 state by laser pulses [2].
As our simulations demonstrate, the characteristic energy scales of Weyl excitations significantly exceed the elastic scattering rate, demonstrating that the excitations could be conveniently
observed in current experiments.

97
10

max[|Im[E(k)]|]/γ 0(b)(c) 10

5

8200

0

6100

-5 (1,1,0.7)(0.15,0,0)R(0.65,0,0)

1000
Dissipation near the BZ center100
101Dissipation near the Weyl point
0.100 (1,1,0.7)
R
0.010
0.001
k

E(k)/γE(k)Re[E(k)]/γ0

(a)

4

0

2

-100
0

X

-200

-2

-4 Γ

X

M

R

k

Figure 7.3:
Effects of disorder and dissipation on Weyl quasiparticles. (a) Real part of thequasiparticle dispersion in the presence of dissipation (including spontaneous emission and collective
radiative decay). The parameters of the J = 1 → J = 0 transition correspond to those of theelectronic 3 P0 −3 D1 levels of bosonic 88,84 Sr atoms, trapped in a magic optical lattice potential
with a = 206.4nm [187] at unit filling. Momentum k is measured in units π/a. (b) The upper boundon the inelastic scattering rate estimated from simulating the full quasiparticle spectra including
all allowed elastic and inelastic dipolar processes. The dissipation is significantly suppressed near
the Weyl node (blue line). In striking contrast, the dissipation is enhanced close to the Γ point (redline) due to the enhanced collective emission (superradiance). (c) Disordered case: Quasiparticle
dispersion for a lattice with filling fraction f = 0.93 in the limit of small dissipation k0 a  1 (theenergy is measured in units γ0 /(k0 a)3 ). The line thickness shows the inelastic scattering rate.

7.6

Proposal for experimental observation
For probing the Weyl character of the excitations we propose a Ramsey protocol illustrated

in Fig. 7.4 (a). After preparing a Mott insulator of particles in the J = 0 state, a pulse of interfering
Raman beams is used to create excitations in the |1, −1i angular-momentum state with translational
momentum k. Here we consider the case when k is set to be close to the Weyl point with intersecting
Jz = 0 and Jz = −1 branches. For the proposed 3 P0 − 3 D1 electronic levels in Sr, two intermediate
states |ei, |e0 i could be used to create the Raman pulses, imparting a net momentum to the atoms
proportional to k = k1 + k2 + k3 (see Fig. 7.4 (a)). A possible excitation level scheme consists on
using 5s6s 3 S1 and 5s6p 3 P1 as the intermediate |ei and |e0 i levels respectively. After a waiting timet, another pulse is applied to measure the fraction of particles in the Jz = 0 angular-momentum
state. Because of the interference of the two branches, this fraction oscillates with the frequency

98

(a)

|+i

|ei

⌦2 eik2 ·r|1i⌦1 eik1 ·r
|gi

ik·r

×

⌦

|gi

(d)

0.10
0.08
0.08

0.06

0.06

0.04
0.04
0.02
0.02

t

0.10
0.08
0.06
0.04
0.02
0.00

0.00

0

0.10 0

5

10

10

15

20

0 tγt 20

0.12

25

30

(f)

γ0 t

0.08

0.10

0.08
0.06
0.06

0.04

0.04

0.02
0.02
0.00
0.0
0.00

0

0

40
Jz =0 population

Jz =0
Jz =0population
population

Weyl point

⌦

| i

0

Jz =0 population=0 population
population
JJzz=-1

⌦3 e
| 1i

| 1i

0.00

(e)

ky

ik3 ·r

0.10

(c)

kz

|e0 i

|0i

⌦e

(b)

10

20

0.10

0γt0 t

30

40

2.0

2.5

0.08
0.06
0.04
0.02
0.00

0.5

1.0

10

1.5
0γt0 t

2.0

20

2.5

30

3.0

0.0

40

0.5

1.0

1.5
γ0tt
0

3.0

γ0 t
Figure 7.4: Observation of Weyl quasiparticles.
(a) Momentum-selective Ramsey spectroscopy:
interfering Raman beams create an excitation with the angular momentum projection Jz = −1 andwith translational momentum k near the Weyl node (see text). After a waiting time t the second
pulse is applied to measure the fraction of particles in the Jz = 0 angular-momentum state. (b)Six quasimomenta near the Weyl node. (c) The fraction of particles in the Jz = 0 state oscillates
as a function of time t with the frequency (Ek1 − Ek2 )/(2π), where Ek1 − Ek2 is the energy splitting
between the two branches of the quasiparticle dispersion. (d) The oscillations in the presence ofdissipation for k0 a ∼ 0.5. (e) The oscillations in a 99% randomly filled lattice. (f) The oscillations
in a 93% filled lattice. For (c) and (d) a 3D cubic lattice of 100 × 100 × 100 sites was used and we
took advantage of the translational symmetry. For (e) and (f) a 3D cubic lattice of 10 × 10 × 10
was used.

(Ek1 − Ek2 )/(2π), where the energy splitting Ek1 − Ek2 between the two branches is linear in k andvanishes near the Weyl node.
Another signature of the Weyl node is the strong dependency of the amplitude of such
oscillations on momentum k near the node, as the amplitude is determined by the projection of

99
the Bloch vector on k̂z (the magnetic field direction).
In Fig. 7.4 (c-f) we show the fraction of particles in the Jz = 0 state as a function of time at
the end of the above described Ramsey protocol, for the six different quasi-momenta in the kz − kyplane near the Weyl point indicated in Fig. 7.4 (b). Panel (c) shows the dynamics for an ideal unit
filled lattice in the dissipationless limit k0 a  1. Panel (d) shows the dynamics in the presenceof dissipation for the experimentally relevant scenario discussed above. The population dynamics
in disordered systems is shown in panels (e) and (f) for 99% and 93% filled lattices respectively.
Quasiparticles scattering on empty sites in a disordered system leads to the decay of the oscillations.

7.7

Conclusion
We demonstrated that Weyl quasiparticles generically emerge in 3D systems of polar par-ticles in magnetic field. This opens intriguing prospects of observing chiral anomaly, non-local
electrodynamics, non-Anderson disorder-driven transitions, and other fascinating phenomena in
the realm of fully controllable atomic systems. We showed that observing Weyl excitations is currently possible in arrays of AEA in 3D lattices, in particular, using the 3 P0 − 3 D1 levels of bosonic
Sr atoms. Other experimentally convenient schemes, that deserve further exploration, include using
metastable levels of Sr or Yb atoms that can be trapped in magic lattices with spacings smaller
than the wavelength [215] or arrays of polar molecules with the rotational levels dressed to avoid thesplitting of J = 1 levels in the presence of hyperfine interactions [278]. The long lifetimes and the
topological character of Weyl excitations in interacting dipolar systems also open new possibilities
for implementing optical-lattice clocks with sensing capabilities beyond those of non-interacting
systems.

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Appendix A

Theoretical approaches for evaporative cooling with long-range interactions

A.1

Monte Carlo simulation for evaporative coolingWith its stochastic nature, the Monte Carlo method is capable of simulating the individual

collisions that are intrinsic to evaporative cooling phenomena [270, 64, 269], and this leads to a flexible algorithim in which it is easy to incorporate a variety of different conditions in a straightforward
way. The detailed simulation algorithm consists mainly of the following steps:
1. Preparation of an ensemble of particles with coordinates and velocities generated from agiven probability distribution, e.g. an initial equilibrium Boltzman distribution.
2. Evolution of the particles between collisions that follows the classical Hamilton’s equationsof motion, i.e.
dx
dt
dv
dt

= v,
= −∇V (x),(A.1)

where V (x) is the external potential. The time step is chosen to be both small enough to guarantee
numerical convergence for the computed trajectory, as well as much less than the time between
adjacent collision events.
3. Checking every pair of particles to decide if there is a collision. Since we are interested
in two-body collisions, the collision events are determined by the distance between particles. Thisdistance is characterized by the scattering length as . In three-dimensional collisions, there is acollision cross-section with units of area, σ = πa2s . In two-dimensional collisions, there is an120
analogous cross-section with units of length, λ = 2as .
4. Changing the state of the pair of paticles if they collide. For an elastic collision event, thevelocities after collisions are determined from the conservation of total momentum and energy. The
scattering angle is simulated from a probability distribution determined by the differential crosssection. For instance, in two-dimensional s-wave scattering, the outgoing angles φ are uniformly
distributed in the interval [0, 2π), while for p-wave scattering, the distribution of the outgoingangles follows the distribution cos2 φ. In three-dimensional isotropic scattering, there are two
random angles: an azimuthal angle φ uniformly distributed in the interval [0, 2π) and a polar angle
θ from uniformly distributed according to cos θ. The non-uniform distribution of the scattering
angles does not affect the collision rate that is determined by the total cross-section, but the rateof redistributing energies does vary. For an inelastic collision event corresponding to two-bodyreactive collisions, the two particles are lost from the trap after the collision, and thus deleted fromthe simulation when such processes occur.
5. Averaging over many initial samples and trajectories. The physical quantities such as thetotal number of particles, temperature, collision rate and phase space density can be computed
statistically from the simulation ensemble.

A.2

Kinetics of evaporative cooling in 2D
An alternative algorithm which is partially analytic can be derived by assuming that in theprocess of evaporative cooling, the energetic particles can be efficiently removed from the trap. The
system is assumed to follow a truncated Boltzmann distribution f (x, p) (Eq. 3.12), with a cut-offenergy t [155, 102], meaning that there is no particle with energy E > t . Then similar to thedescription of an equilibrim ensemble, the measured quantities of the system can be expressed byaveraging over this distribution function. For example, the collision rate is
γ =

λΛ2
m

Z

d2 xd2 p1 d2 p2

1 ,2× f (x, p1 )f (x, p2 )|p1 − p2 |,

(A.2)

121
in which Λ = 1/(2π~)2 , 1,2 = p21,2 /2m + V (x1,2 ) are the energies of the incident particles. For
simplicity, here we have assumed isotropic energy-independant elastic and inelastic collisions, so λin Eq. (A.2) is constant. We also assume the two-body inelastic collisions happen rate ζγ. Duringforced evaporative cooling, t decreases with time, so does the temperature T , and the distributionfunction f = f (x, p). As a result, the evolution of the system from t to t0 can be modeled via
three steps: the change due to f when t decreases to 0t (dN1 , dE1 ), the change due to evaporation(dN2 , dE2 ), and the change due to the inelastic losses (dN3 , dE3 ). These can be represented by thefollowing equations:Z td2 xd2 pf,(A.3)

d2 xd2 pf,

(A.4)

dN1 =

Λ

dE1 =

Λ

dN2
=
dt
dE2
=
dt
dN3
=
dt
dE3
=
dt

Z
λΛ2
d2 xd2 p1 d2 p2 dφ0 f1 f2 |p1 − p2 |,
m Σ
Z
λΛ2
d2 xd2 p1 d2 p2 dφ0 sf1 f2 |p1 − p2 |4 ,
m Σ
Z
λΛ2
ζ
d2 xd2 p1 d2 p2 f1 f2 |p1 − p2 |,
m Σ0
Z
λΛ2
ζ
d2 xd2 p1 d2 p2 (1 + 2 )f1 f2 |p1 − p2 |,
m Σ0

=0
Z t t
=0t

(A.5)
(A.6)
(A.7)

(A.8)
where f1 , f2 are the distribution function for each of the two colliding particles, φ0 specifies thescattering angle,  = (x, p) is the energy of a particle and is a function of the coordinate andmomentum of the particles, 3 and 4 are energies after collision, which are determined once p1 , p2
and φ0 are known, and Σ = {1 , 2 , 3 < 0t , 4 > 0t } and Σ0 = {1 , 2 < 0t } specify the integration
region. These contributions add up to give the total changes: N (t0 ) = N (t) − dN1 − dN2 − dN3
and E(t0 ) = E(t) − dE1 − dE2 − dE3 . The new temperature T (t0 ) is found from0N (t ) =

Λ

2

Z 0t

d2 xd2 pf (x, p),(A.9)

d2 xd2 pf (x, p).

(A.10)

00

E(t ) =

Λ

2

Z 0t
0

122
Then the trajectory of evaporative cooling in two-dimensional traps can be solved from the above
equations. Similar calculations also apply to three-dimensional harmonic traps, and the results areequivalent to solving the rate equations following the method developed by Walraven et al., aslong as the truncated Boltzmann distribution is a good approximation [155].

A.3

Master equation approach for losses in a quasi-2D trap
Represented in the basis of harmonic oscillator eigenstates, the rate coefficients Γn,m definedin the jump operators in Sec. 3.5 can be written explicitly for intraband collisions (nz = mz ,FIG. 3.7(a)) [162]Γn,m ==

Γ̃(1)√√3 2πb3p ωr ωz
Is(nz , mz , nz , mz )(arho )3X×
Is(nσ , mσ , nσ , mσ )Ip(nσ0 , mσ0 , nσ0 , mσ0 ),σ6=σ 0(A.11)
and for interband collisions (nz 6= mz , FIG. 3.7(b))Γn,m ==

Γ̃(2)√√3 2πb3p ωr ωz(arho )3X
σ1 6=σ2 6=σ3

ωσ2
Is(nσ1 , mσ1 , nσ1 , mσ1 )ωr×Ip(nσ2 , mσ2 , nσ2 , mσ2 )Is(nσ3 , mσ3 , nσ3 , mσ3 ),(A.12)

123
where σ, σ 0 = {x, y}, σ1 , σ2 , σ3 = {x, y, z}, arho =ZIs(n, m, p, q) =p~/mωr ,
2e−2u Hn (u)Hm (u)Hp (u)Hq (u)p,du
π 2n+m+p+q n!m!p!q!
(A.13)Z
Ip(n, m, p, q) =

2

e−2u
du p
π 2n+m+p+q n!m!p!q!

×[∂Hn (u)Hm (u) − Hn (u)∂Hm (u)]×[∂Hp (u)Hq (u) − Hp (u)∂Hq (u)],(A.14)and Hn , Hm , Hp , Hq are Hermite polynomials.

Appendix B

Modeling dissipative dynamics

We model the dissipative dynamics by means of a master equation for the density operatorρ̂:diρ̂ = − [Ĥ0 , ρ̂] + Lρ̂,
(B.1)
dt
~
P
with Ĥ0 describing the unitary evolution and Lρ̂ = 21 j (ρ̂Ĉj† Ĉj + Ĉj† Ĉj ρ̂ − 2Ĉj ρ̂Ĉj† ) accountingP
for dissipation. In a 1D tube with a superimposed lattice potential, Ĥ0 = −J j,σ (ĉ†jσ ĉj+1σ +h.c.) + V̂trap , where J represents the hopping of molecules between sites, and V̂trap represents theparabolic trapping potential. The two-body losses that occur between KRb molecules are described√by jump operators Ĉj = Γ0 ĉj↑ ĉj↓ , which destroy a pair of molecules in different rotational states(|↑i or |↓i) at the same site j, and Γ0 is the on-site loss rate. In the regime Γ0  J/~, thedoubly occupied states can be adiabatically eliminated [82], projecting the Hilbert space to stateswith at most one molecules per site. As a result, the jump operators Ĉj are replaced by Âj =hi√2Γeff (ĉj↑ ĉj+1↓ + ĉj↑ ĉj−1↓ ) − (ĉj↓ ĉj+1↑ + ĉj↓ ĉj−1↑ ) , and Γeff = 2(J/~)2 /Γ0 . These new jump
operators Âj effectively remove spin singlet states formed in neighoring sites, and account forsecond order tunneling processes. Even within the reduced space, the above master equationis generally difficult to solve for experimentally relevant lattice sizes. In 1D, however, various
numerical techniques can be applied. In the following we provide the details of the numericalmethods utilized in this paper for solving the above master equation.

125

B.1

Mean-field treatment (MF)The above master equation can be significantly simplified by adopting a MF ansatz thatQ
allows for efficient numerical treatment of realistic system sizes. Under this ansatz, ρ̂ = j ρ̃jPwith ρ̃j ≡ α,β={↑,↓,0} ρα,βj |αihβ|. ↑, ↓, 0 label the three possible local states of spin up, spin down,
and the vacuum, respectively. Solving the master equation reduces to solving a set of coupled andnon-linear differential equations for ρα,β
j :
dρσσ
j
dt
dρσσ
j
dt

=

X 0
X 0 0
i X σ0 0σ0
0 σ0 σσ0(ρl ρj − ρ0σJ(ρσl σ ρσσ+ ρσσ
ρσl σ ρσσjj ,l ρj ) + 2Γeffl ρj ) − 4Γeff
~
l

l

0

=

i
i
0
(Ωσ − Ωσ0 )j 2 ρσσ
+ J
j
~
~

X
l

(B.2)

l

XX
0 αα
0σ 0
αα σσ 00σ 0 σ0(ρσσ(ρσ0l ρj − ρl ρj ),l ρj − ρl ρj ) + 2Γeff
l

α

(B.3)
dρ0σ
j
dt

=

X 0
i
i X X 0α ασ
0σ 0 σ 0 0σ00ρj ),(ρσl σ ρ0σΩσ j 2 ρ0σ(ρl ρj − ρ0σ
j − ρlj + J
l ρj ) + 2Γeff
~
~
α

(B.4)

dρσ0
j
dt

X
i
i X σ0 00 X α0 σα0 σ0 0σ 0 σ 0 σ0
(ρσσ
ρj ),
ρl ρj ) + 2Γeff= − Ωσ j 2 ρσ0(ρl ρj −j + Jl ρj − ρl
~
~
α

(B.5)

dρ00
j
dt

l

l

l

l

=

i
J
~

XX
l

α

α0
α0 0α
(ρ0α
l ρj − ρl ρj ) + 4Γeff

XX X
l

0 0

αβ
,
(−1)δαβ ραβ
l ρj

(B.6)

α6=α0 β6=β 0where σ, α, α0 , β, β 0 ∈ {↑, ↓}, σ 6= σ 0 , the summation of l is over the nearest neighbors of j, and2 a2 with a the lattice spacing. In deriving the above Eqs. (B.2)-(B.6), we haveΩ↑(↓) = 12 mω↑(↓)

neglected terms such as n̂jσ ĉ†j+1σ0 ĉj−1σ0 ρ̂ and ĉ†j+1σ ĉjσ ĉ†jσ0 ĉj−1σ0 ρ̂ in Lρ̂, which correspond to thecorrelated hopping processes involving three sites. We have performed calculations to confirm thecontributions from these terms are small for the systems we treat in this paper.

B.2

Rate equation (RE)A further simplification can be obtained by neglecting the dynamical redistribution of molecules

and the effect of inhomogeneity, as well as coherences. This is equivalent to dropping the unitaryevolution terms in Eqs. (B.2)-(B.6), and setting all off-diagonal terms of the density matrix ρ̃j to↓↓
0. With these approximations, for a 50:50 mixture of molecules, ρ↑↑
j = n↑ = ρj+1 = n↓ , Eq. (B.2)

126
simplifies todn↓= −8Γeff [n↓ (t)]2 ,
dt

(B.7)

which is the same as the RE Eq. (3) in the main text. The RE gives a fairly good understanding of
the loss dynamics at short times, as discussed in the main text. However, a quantitative estimateof the filling fraction requires going beyond these approximations. This can be accomplished byusing the MF treatment.

Appendix C

Phase coherence and correlations in interacting dipoles

C.1

Mean-field approach
The mean-field ansatz, ρ̂ =Na ρ̂a , reduces the dynamics to 3N coupled nonlinear differential

equations presented in the main text. In the most generic case we define local order parameters totake into account the effect of the inhomogeneous couplings:
Xa eiΦ =X

f (rab )Sb⊥ eiφb , Ya eiΦ =

b6=a

X

g(rab )Sb⊥ eiφb .

b6=aIf the local order parameters vary slowly over the system size, and can be approximated to be thesame for all dipoles one can define Xa ≈ feff Z, Ya ≈ geff Z, where the global order parameter Z isP
P P
defined as ZeiΦ = N1 a Sa⊥ eiφa and the effective couplings are given by feff = a b6=a f (rab )/(N −P P
1) and geff = a b6=a g(rab )/(N − 1).
The steady-state solution Ż = 0, Φ = ω̄t leads to two self-consistent equations for the orderparameter Z and the collective frequency ω
ZP [feff Q+2geff (δa +ω)]a N Q[4(δa +ω)2 +(2f 2 Z 2 +2g 2 Z 2 +Q2 ] ,
eff
eff

(C.1)

ZP [geff Q−2feff (δa +ω)]a N Q[4(δa +ω)2 +(2f 2 Z 2 +2g 2 Z 2 +Q2 ] ,
eff
eff

(C.2)

Z=

PN

0=

PN

which can be evaluated in the N → ∞ limit as integrals when the detunnings δa have a known
distribution.
We note that in this treatment, when g(rab ) 6= 0, the elastic couplings simply induce a global
frequency shift ω = geff Q/(2feff ) that can be eliminated by moving to a rotating frame. However, as

128
discussed in the main text the role of elastic interactions is not just to introduce a simple rotation.
In fact the elastic terms tend to generate entanglement and correlations which can not be capture
by a simple mean-field treatment. Therefore the applicability of the mean-field solution is restricted
to regimes where elastic interactions are highly suppressed.

C.2

Cumulant expansion approach and two-time correlation between dipoles
The cumulant expansion method is a useful theoretical tool for including correlation effects

beyond the mean-field approximation [166, 274, 101]. We keep two-point correlations such ashσ̂a+,−,z σ̂b+,−,z i, but factorize three-point correlations and higher [128]:hσ̂aα σ̂bβ σ̂cγ i = hσ̂aα σ̂bβ ihσ̂cγ i + hσ̂aα ihσ̂bβ σ̂cγ i + hσ̂aα σ̂cγ ihσ̂bβ i−2hσ̂aα ihσ̂bβ ihσ̂cγ i.

(C.3)

This factorization closes the set of dynamical equations of motion for all single particle observableshσ̂a+,−,z i and equal time two-point correlations. Two-time correlation functions can be computedby solving [40]:dhσ̂a+ (t + τ )σ̂b− (t)i=dτhΓ+Wi +− iδa +hσ̂a (t + τ )σ̂b− (t)i2
1
+ fab hσ̂az (t)ihσ̂b+ (t + τ )σ̂b− (t)i2
1 X+
faj hσ̂az (t)ihσ̂j+ (t + τ )σ̂b− (t)i,2
j6=a,bwhere we have introduced the approximation hσ̂az (t + τ )σ̂j+ (t + τ )σ̂b− (t)i ≈ hσ̂az (t)ihσ̂j+ (t + τ )σ̂b− (t)i.
Comparisons with exact numerical solutions show that the cumulant expansion captures well thesteady-state behavior for inhomogeneous couplings f (rab ), provided the elastic couplings g(rab ) aresufficiently small. In Fig. C.1 we compute the pair-wise two-time correlation function, Za,b (τ ) ≡limt→∞ h(σ̂a+ (t + τ ) + σ̂b+ (t + τ ))(σ̂a− (t) + σ̂b− (t))i, using both the cumulant expansion and the exact
solution. The decay rate of these correlations, Za,b (τ ) = Ae−τ γ , encodes information about the
spectral coherence of the emitted radiation (note that here ν = 0). The result shows that Γ/γexhibits the same dependence on W/Γ as ZQ .

0.5 f0.000.000

0.12129
exact
cumulant

ÊÊ
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
ÊÊÊÊÊÊÊÊÊÊÊÊ
Ê
ÊÊÊÊ
Ê
ÊÊÊ
Ê
ÊÊÊ
Ê
Ê
Ê
ÊÊÊ
ÊÊÊ
Ê
ÊÊÊ
Ê
ÊÊÊÊ
ÊÊ
ÊÊÊ
Ê
ÊÊ
ÊÊ
ÊÊÊ
e↵
e↵
ÊÊÊÊÊÊÊ
ÊÊÊÊÊÊÊÊÊÊÊ
e↵

1.5

0.04

1.0

0.08
0.04

0.08
0.04

0.5
/ = 30 30
0.00 f0.00
0.00
f / =
0.0
2010 40 20 60 30 80 40 100
40 20 60 30 80 400100
0
W
êG
W
êG
WêG
WêG

/ = 70

2010

2.0

Gêg

1.5

0.041.0

mutual informationGêg

mutual information

0.082.0

exact
2.5
cumulant
0.08

0.12

quantum discord

2.5

0.12

quantum discord

0.12

Figure C.1: Pair-wise two-time correlation functions in the steady-state parametrized by Z1,2 (τ ) =Aexp(−γτ ). The correlations are calculated for a pair of dipoles in an ensemble of N = 70 dipoles
identically coupled with f (rab ) = feff /N . The cumulant expansion solution agrees with the fullcalculation except at W  Γ where subradiant behavior dominates (purple region) [166].

C.3

Conditional evolution, entanglement and quantum correlations
An individual experimental realization can be considered as a single trajectory, whose evolu-tion can be quite different from the ensemble averaged solution of the master equation. Tracking
the evolution of an individual trajectory is equivalent to performing continuous measurements thatcollect the record of the emitted photons, for example homodyne measurements. The conditional
evolution of the system subject to continuous measurements can be modeled by the method of quantum state diffusion [40, 268]. For a single run the state of the system remains pure, ρ̂c = |ψi hψ|,but the average over many trials reduces the system into a mixed state and recovers the density
matrix obtained from the master equation.
To probe the entanglement of the dipoles, in Fig. 2 we calculate the average quantum Fisher
information for each individual trajectory [107, 237]1
F Q (ρ̂c ) = (FQ (ρ̂c ; Jˆx ) + FQ (ρ̂c ; Jˆy ) + FQ (ρ̂c ; Jˆz )),3
where FQ (ρˆc ; Ĥ) = 4(hψ| Ĥ2 |ψi−hψ| Ĥ |ψi2 ), and Jˆx,y,z are collective angular momentum operators.
States with zero entanglement can still be nonclassical. Two systems are correlated if theyshare information with each other. The total amount of correlation can be quantified by the quan-

130
tum mutual information I = SA +SB −SAB , where Si is the von Neumann entropy of the subsystemi ∈ {A, B, AB} (AB is the total system spanned by A and B together) , Si = −Tr[ρ̂i log2 ρ̂i ], with
ρ̂i the reduced density matrix of the subsystem i. A value varying between 0 and 2 is obtained
when A and B are pure states or maximally correlated respectively. The mutual information can beseparated into a classical and a quantum part. The classical part is JB|A = max{SB − SB|A }. Here
SB|A is the von Neumann entropy of subsystem B conditioned on the measurement performed on
A and max represents maximum value obtainable over all local measurements on A. The quantum
part, known as the quantum discord, DB|A = I − JB|A , measures the amount of correlations thatexceed the classical part and characterizes the “quantumness” of the system [171]. A state with
nonzero quantum discord behaves in a way intrinsically non-classical, since a local measurement
performed on one of its subsystems can disturb the whole system. In order to calculate JB|A , we1
consider a set of von Neumann measurements Π̂Ank · ~σ A ) with |~nk |2 = 1, made on the
k=1,2 = 2 (1 ± ~Psubsystem A and minimize the corresponding conditional entropy SB|A = 2k=1 Tr[pk S(ρ̂B|ΠA )],k

A
where pk = Tr[Π̂Ak ρ̂], ρ̂B|ΠA = TrA [Π̂k ρ̂]/pk [171]. In Fig. 2(e) we calculate the mutual informationkfrom I and the quantum discord from D using as subsystems A and B a pair of dipoles, a and b
respectively.

Appendix D

Combining dipole-dipole interaction and motion

D.1

Long-range dipole-dipole interaction, force and diffusion
In Chapter 2, we presented a derivation of dipole-dipole interactionsbased on a direct fac-torization of the atomic and photonic density matrices. Here we adopt an alternative approach,and derive the form of dipole-dipole interactions and the corresponding forces for two-level moving
atoms, with ωa = k0 c the frequency of transition and d the dipole moment. We include motionin the dipolar coupling after eliminating the electromagnetic vacuum modes. The Hamiltonian,
including the atoms and free-space electromagnetic field is
H = ~ωa

X †
XX
X
gk
b̂j b̂j +ωk â†k âk −
j

k,

ik·rj

×(d · ˆk )[(ejk,âk + H.c.)(b̂†j + H.c.)].

(D.1)

The atomic dipoles and the field modes evolve according to
dâk
dt

= −iωk âk +iX
gk (dj · k )[e−ik·rj (b̂j + H.c.)],
~
j

(D.2)
db̂j
dt

= −iωa b̂j + i

X gk
k,

~

(dj · ˆk )[eik·rj âk ŝj + H.c.],
(D.3)

dŝj
dt

= −i

X gk
k,

~

(dj · ˆk )[eik·rj âk b̂†j − e−ik·rj â†k b̂j−eik·rj âk b̂j + e−ik·rj â†k b̂†j ],

(D.4)

132
where ŝj = b̂†j b̂j − b̂j b̂†j , and sj = hŝj i gives the inversion of the j th atom. We have assumed
that internal operators commute with external operators, and neglected the diffusion of the atomic
wavepacket. Eq. (D.2) can be formally integrated to obtainâk (t) = âk (0) − i

X gk
j

~

(dj · ˆk )Z0

dt0 eik·rj +iωk (t −t)×(b̂j + H.c.).

(D.5)

Assuming the external motion is much slower than the internal dynamics relevant inside the integral,so that rj (t0 ) ≈ rj (t), and the interaction between atoms and the field modes is weak so that0b̂j (t0 ) ≈ b̂j (t)e−iωa (t −t) . Substituting Eq. (D.5) into Eq. (D.3), we obtain the equation for thequantum averaged quantitiesdb̂jdt= −iωa b̂j + ŝj

X X g2
k

l

k,

~2

(dj · ˆk )(dl · ˆk )×{eik·rjl [b̂†l (πδ(ωk + ωa ) − iP+b̂l (πδ(ωk − ωa ) − iP1)]ωk − ωa−e−ik·rjl [b̂†l (πδ(ωk − ωa ) + iP+b̂l (πδ(ωk + ωa ) + iP1)
ωk + ωa1)
ωk − ωa1)]},
ωk + ωa(D.6)

where we have utilized the fact that hâk (0)i = 0, and assumed that atomic motion is classicalR
PV[229, 230]. Changing k = (2π)
dΩ dk k 2 , and applying rotating wave approximation, we have
3
db̂j
dt

= −iωa b̂j + iX
l6=jŝj b̂l (gjl − ifjl ) −

Γ
b̂j ,
2

(D.7)

with f (0) = Γ, and



3Γ
cosk0 rcosk0 r sink0 r
g(r) = −z1 (θ)+ z2 (θ)
+ 2 2
,
4
k0 r
k03 r3
k0 r(D.8)
f (r) =

3Γ
sink0 rsink0 r cosk0 r
z1 (θ)+ z2 (θ)
− 2 2
,
4
k0 r
k03 r3
k0 r
(D.9)

133
where z1 (θ) = sin2 θ, and z2 (θ) = (3cos2 θ − 1). These expressions can also be obtained from Eq.
(2.23)∼ Eq. (2.22) by keeping only a single α component. We further assume that multi-atomcorrelations can be factorized, hŝj b̂l i ≈ hŝj ihb̂l i, so that the atomic coherence evolves asdbj
dt= −iωa bj + iX
l6=jsj bl (gjl − ifjl ) −

Γ
bj ,
2

(D.10)

where O = hÔi for any atomic operator Ô. It is coupled to sj , the equation of which can be derived
in a similar way and is given bydsj
dt= −Γsj − 2iXX(gjl − ifjl )b∗j bl + 2i
(gjl + ifjl )b∗l bj .
l6=j

l6=j

(D.11)
If we consider very low external driving field and sj ≈ −1, Eq. (D.10) is decoupled from sj andcan be reduced to the form of Eq. (6.2). For the momentum,dp̂jdt= −∇Ĥ= −Xk,

gk (dj · k )(ikeik·rj âk b̂†j − ike−ik·rj â†k b̂j+ikeik·rj âk b̂j − ike−ik·rj â†k b̂†j ).

(D.12)

After substituting Eq. (D.5), taking the quantum average and performing a similar integration
procedure as above, we obtain
dpj
dt

=

X
l6=j

[gjl (bj b∗l + H.c.) − ifjl (bj b∗l − H.c.)],(D.13)

with gjl = −∇gjl and fjl = −∇fjl . As the dispersive force gjl is a steep function of rjl , itdominates at short distance, and atoms are drastically accelerated/decelerated. Both the dispersiveand dissipative forces are anisotropic and couple motion along different directions.
Due to the presence of spontaneous emission and radiative interactions, the atomic momentum also diffuses over time, which can be described by including classical noise dξiα in the equationof motion for pαi . The components of these noises are correlated, and characterized by the diffusion

134
matrixE[dξiα (t)dξiβ (t0 )] = δα,β2 − δα,z 2 2~ k0 Γ(si + 1)δ(t − t0 ),10(D.14)

E[dξiα (t)dξjβ (t0 )] = −2~2 k02 ∇α ∇β f (rij )<[b∗i bj ]δ(t − t0 ),(D.15)
where E[·] denotes the expectation value.
The momentum diffusion matrix can be found fromDjl ==dhp̂j p̂l idhpl i dhpj i− hpj i
−
hpl i
dt
dt
dt
X X
gk1 gk2 (dj · 1 )(dl · 2 )(ik1 eik1 ·rj âk1 1 b̂†j

(D.16)

k1 ,1 k2 ,2−ik1 e−ik1 ·rj â†k1 1 b̂j+ik1 eik1 ·rj âk1 1 b̂j − ik1 e−ik1 ·rj â†k1 1 b̂†j )×(ik2 eik2 ·rl âk2 2 b̂†l − ik2 e−ik2 ·rl â†k2 2 b̂l+ik2 eik2 ·rl âk2 2 b̂l − ik2 e−ik2 ·rl â†k2 2 b̂†l )= −2~2 k02 ∇∇fij <[b∗j bl ].

(D.17)In dense clouds momentum diffusion from radiative interactions can give rise at long times
to significant heating. This heating was reported to be one of the main limiting mechanisms inlaser cooling [42, 230]. At short times, tΓ ∼ 1, with low densities and weak probes, Ω < Γ, themomentum diffusion is not prominent, and since this is the regime we are interested in this work,
we will ignore momentum diffusion in our calculations presented in the main text.

D.2

Effect of motion on single-atom fluorescence
Here we derive the fluorescence intensity emitted by a weakly driven atom, using a fullquantum approach with motional effect included. We consider the states including at most one
excitation, and label the relevant quantum states by |pα, 0i , |pg, k i, with α = {e, g}, p the135
momentum of the atom, and k() being the momentum (polarization) of the photon in vacuum(|pα, 0i stands for no photon). For generality we assume here that two counter propagating lasers
are used to drive the atoms, carrying momentum k0 and −k0 respectivelly. The Hamiltonian is 1
Ĥ =

p̂2
+ ~ωa b̂† b̂ + ~Ω[cos(k0 · r)(e−iωL t b̂† + H.c.)]2mXX+~ωk â†k âk, − ~(d · ˆk )gk [eik·r âk b̂† + H.c.].
k,

k,

(D.18)The state vector of the system is
|ψi =

XZ
α

+

dp |pα, 0i Aα0 (p, t)e−i(Eα +Ep )t/~XZk,dp |pg, k i Bgk (p, t)e−i(Eg +Ep +~ωk )t/~ ,(D.19)2p, |Aα0 (p, t)|2 represents the population in the state α possessing
where Eα = ωa δα,e , Ep = 2m

momentum p, and Bgk (p, t) is the amplitude of having a photon k with polarization . The state1

Note: The counterrotating terms such as âk b̂ have been dropped because the relevant states are restricted toincluding at most one excitation.

136
of the system evolves according toidAg0 (p, t)
dt= ΩAe0 (p + k, t)ei(ωL −ωa )t e−iEp+k0 ,p t/~+ΩAe0 (p − k, t)ei(ωL −ωa )t e−iEp−k0 ,p t/~ ,(D.20)idAe0 (p, t)
dt

= −

X
k,

(d · ˆk )gk Bgk (p − k, t)×e−i(ωk −ωa )t e−iEp−k,p t/~+ΩAg0 (p − k, t)e−i(ωL −ωa )t e−iEp−k0 ,p t/~+ΩAg0 (p + k, t)e−i(ωL −ωa )t e−iEp+k0 ,p t/~ ,(D.21)
idBgk (p, t)
dt= Bgk (p, t) − (d · ˆk )gk Ae0 (p + k, t)×ei(ωk −ωa )t e−iEp+k,p t/~ ,where Ep1 ,p2 =(D.22)

p21 −p22
2m . The first term in Eq. (D.21) describes the effect of vacuum photons, whichaccording to Wigner-Weisskopf approach leads to the spontaneous decay with rate Γ, and can be
rewritten as [137]idAe0 (p, t)dtΓ= −i Ae0 (p, t) + ΩAg0 (p − k, t)2×e−i(ωL −ωa )t e−iEp−k0 ,p t/~+ΩAg0 (p + k, t)e−i(ωL −ωa )t e−iEp+k0 ,p t/~ .
(D.23)

Consider the initial condition Bgk (p, 0) = 0, Ag0 = δ(p − p0 ), where p0 is the initial momentumof the atom. The steady state solution is
Ae0 (p, ∞) =Ω[δ(p0 − p + k0 ) + δ(p0 − p − k0 )],∆L + Ep0 ,p + i Γ2
(D.24)

137
with ∆L = ωL − ωa . Thus the atomic excitation Ae =R

dp|Ae0 (p, ∞)|2 indicates two Lorentzianwith FWHM=Γ and centered at ωr ± k0m·p0 . The photon emission rate along a given direction ks is
Iks

=

V ~c
(2π)3

Z
dk k

3

XZ

|Bgk (p, t)|2,t→∞t

dp lim(D.25)
with k = k k̂s . We consider the transverse intensity case where, k·k0 = 0, and for atomic transitions
ωr , p0 · k0 /m  ωL , thenIks≈

ωL4 d2 Ω2
1
1
[
2 +
2 ].
3
2
k
·p
Γ
0
0
2
c 8π ε0 (∆ − ωr +) +(∆ − ωr − k0 ·p0 )2 + Γm

4m(D.26)

4

The emitted light intensity exhibits the same profile as the atomic excitation Ae .
As indicated by the above expression, motion modifies the emitted light intensity by adding
two natural corrections: a Doppler shift ∝ k0m·p0 , modifying the effective laser detuning, and avelocity independent recoil shift, ωr , which physically accounts for the fact that to compensate
for the energy imparted to the atom via photon recoil, the incident laser needs to have a higher
frequency to be resonant with the atomic transition.

Appendix E

Calculation of Weyl dispersion

E.1

Dispersion near Weyl nodes
We define the quasiparticle dispersion as the poles of the retarded Green’s function averaged

with respect to quenched disorder.
While long-wave quasiparticles (k → 0) are insensitive to the details of the lattice potential,
their effective Hamiltonian preserves rotation and inversion symmetries, and in the absence ofmagnetic field– time-reversal symmetry, with the generic form of the Hamiltonian given by Eq. (7.1)
and with the vector ω parallel to the magnetic field.
We assume the existence of excitations with momentum J = 1 and focus on the respective
manifold of states in what follows. The dispersion of such excitations has three branches for each
momentum k.
For momenta k parallel to ω the respective excitations have momentum projections Jz = 0
and Jz = ±1 on the direction ω. The branch with Jz = 0 intersects the branch with Jz = ±1 atmomenta K k ω such that
F (|K|, K 2 , 2) ± ω = F (|K|, 0, 2),(E.1)

where we used that Jˆ2 = J(J + 1) = 2 for the states under consideration.
The quasiparticle dispersion near the nodes can be found by expanding the Hamiltonianin small momenta p = k − K. Momentum deviation from a node along the z axis leads to the
splitting F K + pz , (K + pz )2 , 2 ± ω − F [K + pz , 0, 2] between the intersecting branches. Using

139
2

2

that [(K + p) · Ĵ]2 ≈ K2 + K2 σ̂z + Kpz + Kpz σ̂z ± √K2 (px σ̂x + py σ̂y ), with σ̂i being the Pauli matrices
in the space of momentum projections Jz = +1 (Jz = −1) and Jz = 0, we obtain the quasiparticle
Hamiltonian (7.2) with
 1 

1 
ζ(p) = F K + pz , (K + pz )2 , 2 − F K, K 2 , 2
22
1
1+ F (K + pz , 0, 2) − F (K, 0, 2),221v⊥ = ± √ KF (2) (K, K 2 , 2)
2

(E.2)
(E.3)

vk = KF (2) (K, K 2 , 2)
1
+ [F (1) (K, K 2 , 2) − F (1) (K, 0, 2)]
2

(E.4)

where the upper (lower) sign in Eq. (E.3) applies to the intersection of the Jz = 0 branch with
Jz = +1 (Jz = −1), and F (1) and F (2) are the derivatives of the function F with respect to thefirst and the second argument.

E.2

Excitation dispersion in a deep lattice
As particles cannot move from site to site in a deep optical lattice, the quasiparticles are

represented by the angular-momentum degrees of freedom that propagate through the system.
ˆσAssuming there is one particle per site and introducing bosonic operators dˆσ†i and di for creating
and annihilating the particle state |1σii on site i with angular momentum J = 1 and projection σ
and the operators â†i and âi for creating and annihilating the momentum state J = 0 on site i, the140system Hamiltonian can be rewritten asĤ =X
i,j,σ,σ 0



ˆσ0 †
h1σ|i h00|j Ĥdip d̂i d̂j , rij |00ii |1σ 0 ij dˆσ†i dj âj âi



X h
ˆσ0 †
+
h1σ|i h1σ 0 |j Ĥdip d̂i d̂j , rij |00ii |00ij dˆσ†i dj âj âii,j,σ,σ 0i0+h00|i h00|j Ĥdip d̂i d̂j , rij |1σii |1σ 0 ij â†j â†i dˆσi dˆσj
−B

X σ†
X
σ dˆi dˆσi âi â†i + BJ
dˆi σ † dˆσi
iσ

iσ




i
X h σ†  σ†
ˆσ â† âi .
+Udˆi dˆσi dˆi dˆσi − 1 + â†i âi â†i âi − 1 + dˆσ†
d
i i
i

(E.5)

iσ

The first term in the Hamiltonian (E.5) is responsible for moving angular-momentum excitations
from site to site; the angular-momentum state |1σii can be transferred by the dipole-dipole interactions from site i to another state |1σ 0 ij on site j that initially was in the J = 0 state. The terms
in the second sum in Eq. (E.5) change pairs of sites i and j from the J = 0 to J = 1 angularmomentum states or vice versa. The term ∝ B is the Zeeman energy. The term ∝ BJ accounts forthe internal rotation (internal levels) of the particles. The terms ∝ U = ∞ in Eq. (E.5) enforce the†hard-core constraints for the bosons created by the operators dˆσ†i and âi , taking into account that

there is one particle on each site.
In chapter 7 we consider excitations on top of the ground state with all sites (particles) in
the J = 0 state. Exciting the internal degree of freedom of a particle on a site costs the rotationenergy BJ that significantly exceeds all the other energy scales, except U = ∞, including thematrix elements ∼ |Ĥdip | of hopping of such angular-momentum degrees of freedom between sites(for instance, for dipolar molecules and alkaline-earth atoms |Ĥdip |/BJ . 10−6 ). As a result, thenumber of sites excited to the J = 1 states is conserved to a good approximation, and the second
sum in Eq. (E.5), that creates or annihilates pairs of J = 1 excitations, can be neglected when
considering the angular-momentum dynamics.
Therefore, the quasiparticles in the system are hard-core bosons that carry angular-momentum(J = 1) degrees of freedom and hop from site to site as described by the effective Hamiltonian (7.6)(7.7) with b̂iσ = dˆσi â†i .

141

E.3

Details of disorder averaging.
Realistic systems of particles pinned in deep optical lattices are inherently disordered due tothe randomness of the spatial distribution of the particles. Each lattice site hosts either a particle
with probability f or a vacancy with probability 1 − f .
For a small concentration of vacancies, excitations in the system are delocalised and theirdispersion is close to that in the disorder-free system but acquires a small finite imaginary part
ImEk due to the scattering on the vacancies.
In order to numerically obtain the quasiparticle spectra in such a disordered system wediagonalise the Hamiltonian Ĥ = Ĥ0 + V̂ , where Ĥ0 is the excitation Hamiltonian in the clean caseand the operator V̂ models vacancies as sites with infinite potential V (ri ) = ∞. We compute theretarded Green’s function
G(r1 σ1 , r2 σ2 , E) =∗ (r )ψ
X ψασ
1 ασ2 (r2 )1α

E − Eα + iη

(E.6)

for multiple disorder realisations, where ψα and Eα are the eigenfunctions and eigenenergies fora particular disorder realisation, σ1 and σ2 label projections of the angular momentum J = 1,and η is a small positive number introduced to ensure that the disorder-averaged Green’s functionhG(r1 σ1 , r2 σ2 , E)idis is a smooth function of its arguments for a given number of disorder realisations. At the same time, η has to be chosen sufficiently small to not affect the results for thequasiparticle dispersion. The energy E has to be chosen close to the energies of the quasiparticles
of interest.
Disorder averaging restores translational invariance, yielding an averaged Green’s functionthat depends only on the coordinate difference r2 − r1 . Computing the Fourier transform of thefunction hG(r1 σ1 , r2 σ2 , E)idis with respect to r2 − r1 and diagonalising it in the angular-momentum
space gives 1/(E − Ekn ), where n = 1, 2, 3 labels the dispersion branch for a given k, ReEkn is thequasiparticle dispersion and −2ImEkn – the scattering rate.
In chapter 7 we perform averaging over 100 disorder realisations on a 10 × 10 × 10 cubic
lattice with periodic boundary conditions for the filling fraction f = 0.93, close to that in the142
recent experiments [126, 221, 92]. The results for the quasiparticle dispersion and scattering rates
are shown in Fig. 7.3c.