Theory of Steady-State Superradianceby
Minghui Xu
B.S., Electronic Engineering, Peking University, 2008
M.S., Physics, Peking University, 2011

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
2016This thesis entitled:
Theory of Steady-State Superradiance
written by Minghui Xu
has been approved for the Department of Physics

Prof. Murray J. Holland

Prof. James K. Thompson

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.

iiiXu, Minghui (Ph.D., Physics)
Theory of Steady-State Superradiance
Thesis directed by Prof. Murray J. Holland

In this thesis, I describe the theoretical development of the superradiant laser, or laser in theextreme bad-cavity regime. In this regime, the cavity decay rate is much greater than the atomic
dynamics. The atoms emit photons into the cavity mode superradiantly in steady state. We
develop group-theoretic methods that enable us to exactly solve mesoscopic systems with hundredsof atoms. We demonstrate the synchronization of atomic dipoles in steady-state superradiance.
With this synchronized system, we propose conditional Ramsey spectroscopy which allows us toobserve Ramsey fringes indefinitely, even in the presence of atomic decoherence. Furthermore,
we explore manifestations of synchronization in the quantum realm with two superradiant atomic
ensembles. We show that two such ensembles exhibit a dynamical phase transition from two
disparate oscillators to quantum phase-locked dynamics. Finally, we study the mechanical effect of
the light-atom interaction in the steady-state superradiance. We find efficient many-body coolingof atoms. The work described in this thesis lays the theoretical foundation for the superradiantlaser and for a potential future of active optical frequency standards.

Dedication

To Mom, Dad, and Tianxiao

Acknowledgements

I owe a tremendous thank you to my advisor, Prof. Murray Holland. Murray allowed me tobe part of his group in the summer of 2012 when my first-year advisor Chris left JILA. Since thenMurray has financially supported me. During the fall semester in 2013, I was stuck in China due
to my F1 visa denial. In that semester, Murray did all he could to eventually get me back to the
US. He worked through an exhaustive sequence of events, including working closely with lawyers
and officials of the University. Under the intervention by Congressman Jared Polis, the situation
was eventually reviewed and finally the determination was made that the original decision by the
US Embassy was actually made in error. I am now typing the acknowledgements, full of gratitude,
also because Murray has been an excellent scientific mentor. Murray lead me to the fruitful projectof steady-state superradiance, in which he has keen physical insights and a wealth of creative ideas.
He is always available for discussion and providing potential solutions and guidance. On top of allthese things, Murray has also been a good friend.
Special thanks to John (a.k.a. Jinx) Cooper. Jinx is a quantum optics guru and has beenenthusiastic about steady-state superradiance. Among numerous things that he has taught me, I
feel that these two are most important for research. As he always says, “Once you get the physics
right, math is simple”, and “Do not trust anything unless you derive it by yourself”. There is a good
example for the latter. Jinx was acknowledged by J. D. Jackson in the Classical Electrodynamics (astandard graduate textbook) because he found quite a few improvements when he taught that class.
I would like to thank the faculty here at the University of Colorado. I am especially gratefulto Chris Greene, Ana Maria Rey, Juan Restrepo, Victor Gurarie, Lijun Chen, Oliver Dewolfe,viMichael Hermele, James Thompson, and Jun Ye. I benefited a lot from interactions with them.
Here are even more people who helped me understand physics: David Tieri, Effie Fine, Stefan
Schütz, Giovanna Morigi, Dominic Meiser, Tony Lee, Ching-Kit Chan, Jose D’Incao, Bihui Zhu,
Victor Colussi, Peiru He, Athreya Shankar, Zhexuan Gong, Alexey Gorshkov, Michael Foss-Feig,
Johannes Schachenmayer, and Kaden Hazzard. Special thanks to David. He has been a super niceofficemate and extremely helpful.

viiContents

Chapter
1 Introduction

1

2 The superradiant laser

5

2.1

Dicke superradiance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

2.2

Laser theory in a nutshell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

2.3

Steady state superradiance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3 Group-theoretic methods for solving the laser equation

22

3.1

Lindbladian in terms of generators of the SU(4) group . . . . . . . . . . . . . . . . . 22

3.2

Generalized Dicke states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.3

Calculating observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.4

Transform to the angular-momentum basis representation . . . . . . . . . . . . . . . 29

3.5

Further simplifying: U(1) symmetry and quantum jump . . . . . . . . . . . . . . . . 32

4 A new perspective on steady state superradiance

35

4.1

Synchronization of atomic dipoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.2

Probing the synchronization with Ramsey spectroscopy . . . . . . . . . . . . . . . . 37

4.3

Robustness of the synchronization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.4

Conditional Ramsey spectroscopy with synchronized atoms . . . . . . . . . . . . . . 42

viii
5 Quantum synchronization of two ensembles of superradiant atoms

47

5.1

Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.2

Mean-field picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.3

Quantum signatures of the synchronization . . . . . . . . . . . . . . . . . . . . . . . 50

6 Supercooling of atoms based on steady-state superradiance

59

6.1

Cavity cooling in the strong coupling regime . . . . . . . . . . . . . . . . . . . . . . . 59

6.2

Supercooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

7 Summary and outlook

78

Bibliography

79

Appendix
A Adiabatic elimination

89

B SU(4) generators and λ-matrices

91

C |S, M ihS, M 0 | representation

93

ix

TablesTable
3.1

The action of the SU(4) superoperators on the elements of the fundamental representation u, d, s, c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

x

FiguresFigure2.1

Source: Ref. [42]. Comparison between the general characteristics of ordinary fluorescence and superradiance experiment. (a) Ordinary spontaneous emission is essentially isotropic with exponential decaying intensity (time constant τsp ). (b) Superradiance is anisotropic with an emission occurring in a short burst of duration∼ τsp /N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2

Dicke states of N two-level atoms. S denotes the symmetrizer with respect to atomic
labels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.3

6

7

Schematic for implementing the incoherent pumping in a three-level configuration.
The coherent driving laser is indicated by the green-dashed arrow and spontaneous
decay into the excited state |ei is indicated by the orange-dotted arrow. . . . . . . . 10

2.4

Numerical simulation (blue dashed line) of a bad cavity laser. Left panel: Atomatom correlation as a function of time. Right panel: Spectrum of the laser light (redsolid line is the fit to Lorentzian). The parameters are N = 200, C = 0.2, w = 18γ,
T2−1 = 2γ, κ = 2500γ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.5

Source: Ref. [93]. Power as a function of pump rate w and atom number N . The
rapid buildup of power above threshold w ∼ γ can be seen as well as the decrease ofemitted power for too strong a pump. The dashed line shows the boundary of theregion of collective emission. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

xi
2.6

Source: Ref. [93]. (a) Linewidth vs w and N . The white dashed lines indicate (fromleft to right) the spontaneous decay rate γ, the inhomogeneous relaxation rate T2 ,and the maximum pump rate w = N Cγ. (b) is a cut through (a) for N = 106 atoms. 21

3.1(a) Young tableau for determining the irreducible representations contained in the
direct product representation of the SU(2) group for N = 4. The dimension for
S = 2, 1, 0 is 5, 3, 1 respectively, and ns = 1, 3, 2, so the total Hilbert space dimension
is 5 × 1 + 3 × 3 + 1 × 2 = 24 as expected. (b) “Pascal’s triangle” to evaluate nS in an
iterative way for any N (the case considered in (a) is the fourth row down from thetop). (c) Pyramid representation of the density operator in the |S, M i representation
for N = 4. Each layer of the pyramid represents the block matrix for each S. . . . . 31

4.1(a) Ramsey spectroscopy where atoms are coupled collectively to a cavity and pumped
individually with incoherent rate w. (b) Ramsey sequence showing preparation instate |g (pseudospins pointing down to the south pole of the Bloch sphere), the π/2
y-axis rotation from the south pole to the equator, precession around the equator,and second π/2 x-axis rotation, after which the z-axis projection carries information
about the cosine of the accumulated phase. . . . . . . . . . . . . . . . . . . . . . . . 38

4.2

Calculations of Eq. (4.1) using SU(4) theory in Chapter 3 with N = 250, ΓC =0.2/T1 , T2 = T1 , and w = N ΓC /2. (a) Ramsey fringes with synchronized atoms (redsolid line) versus T . Conventional Ramsey fringes (blue dashed line) for the same T1
and T2 . (b) During the interrogation time, the atomic inversion hσ̂jz i (blue dashedline), spin-spin correlation hσ̂j+ σ̂k− i (red solid line), hσ̂j+ σ̂k− i- hσ̂j+ ihσ̂k− i (red dotdashedline) and hσ̂j+ σ̂kz i/(hσ̂j+ ihσ̂kz i) (green dotted line). . . . . . . . . . . . . . . . . . . . . 39

4.3

The decay rate λ of the visibility of Ramsey fringes at ΓC = 0.2/T1 and T2 = T1 as
a function of repumping for N = 200 and as a function of N for w = N ΓC /2 (Inset). The dots are numerical solutions of Eq. (4.1), and the solid blue line is thesemiclassical approximation for comparison. . . . . . . . . . . . . . . . . . . . . . . . 40

xii
4.4

(a) The decay rate of the visibility of the Ramsey fringes as a function of standard
deviation of the g factor at ΓC = 0.2/T1 , T2 = T1 , and w = N ΓC /5 for N = 250
atoms. (b) The decay rate of the visibility of the Ramsey fringes as a function of
number of atoms at ΓC = 0.2/T1 , T2 = T1 , w = N ΓC /5, and ∆g/ḡ = 0.2. ΓCin both (a) and (b) is defined using the mean of the g distribution, ḡ. Statistics
are accumulated by sampling g from a Gaussian distribution. The dots are averagevalues for 100 runs, and error bars represent the standard deviations. . . . . . . . . . 43

4.5

Quantum state diffusion calculations of conditional Ramsey fringes subject to continuous homodyne measurement of the cavity output field for N = 10 and w = N ΓC /2.
The blue dashed lines are the ensemble average for reference. (a) A typical Ramsey
fringe for a single experimental trial (red solid line). (b) Histograms are the statisticsof the positions of zero crossings of each fringe for 4000 trials. The blue solid lines
are fitted Gaussian distributions with variance of T ΓC centered on the zero crossingof the ensemble average. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.1

Two ensembles of driven two-level atoms coupled to a single-mode cavity field. The
atoms in ensemble A are detuned above the cavity resonance (dashed line). Ensemble
B contains atoms detuned below the cavity resonance by an equivalent amount. . . . 48

5.2

Steady state value of population inversion hσ̂1z i, spin-spin coherence within each en+ −σ̂B1 i as a function of detunsemble hσ̂1+ σ̂2− i and correlation between ensembles hσ̂A1ing. Red dots are exact numerical calculation by applying the SU (4) group theory.
Blue lines are semiclassical results based on solutions to Eq. (5.7) to (5.9).
5.3. . . . . 52

Steady-state relative phase precession for two ensembles as a function of detuning at
w = N γC /2 for N = 100 (blue dashed line), N = 500 (purple dot dashed line) andN = 106 (red solid line). The straight dotted line is δ = ∆. . . . . . . . . . . . . . . 53

xiii
5.4

(a) Nonequilibrium phase diagram of the quantum synchronization represented byΓ (in units of γC ) on the w-δ parameter plane, where the dissipative coupling
N γC (N = 104 ) is fixed. An abrupt peak is observed at the boundary between
the synchronized and unsynchronized phases. (b) As for (a) but on the w-N γCparameter plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.5

Finite size scaling behavior of the quantum criticality for δ = N γC /2. For N → ∞,the critical pump rate is wc = δ. The red dots show the offset between the criticalpump rate wN for finite N and wc . The blue squares show Γ (in units of γC ) at wN .
Both exhibit linear scalings on the log-log plot. . . . . . . . . . . . . . . . . . . . . . 56

5.6Classical correlation and quantum discord between one atom in ensemble A and one
atom in ensemble B. The blue dashed line is based on semiclassical approximation
described in the paper. The red dots are exact numerical calculations. There are
30 atoms in each ensemble. The comparison between these two lines indicates the“quantumness” of correlations present in our system. . . . . . . . . . . . . . . . . . . 57

6.1

An atom with velocity v is moving freely along the cavity axis. The atom coherently
scatters the photons of the transverse laser into the cavity mode. The frequency ofthe laser ωL is red detuned from the cavity resonance frequency ωc by ∆c . Due toDoppler effect, the atom will scatter more photons into the direction along which it
moves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

xiv
6.2

Atoms with ultranarrow transition g ↔ e are confined to the axis of a standingwave mode of an optical cavity. Different implementations of pumping may beconsidered [12, 93]. In the simplest scenario shown, a transition is driven from theground state |gi to an auxiliary state |ai that rapidly decays to the excited state |ei.
In this way |ai can be adiabatically eliminated and a two-state pseudospin description
in the {|gi, |ei} subspace used, with repumping corresponding to an effective rate
w from |gi to |ei. If the repumping laser is directed normal to the cavity axis, theabsorption does not modify the momentum. Momentum recoil is induced by the onaxis component of the wavevector k~0 of the dipole radiation pattern for the |ai ↔ |eitransition.

6.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

Time evolution of the average momentum square (red dots) evaluated from 4000 trajectories simulated by integrating Eqs. (6.28) and (6.32) for 1 atom (a), 20 atoms (b),and 60 atoms (c). The blue solid line is a fit to an exponential decay. The parametersare ∆ = κ/2 = 100, γC = 0.1, and ωr = 0.25. The repumping rates are chosen suchthat the average atomic population inversion in all cases is the same [w = 0.15 (a),0.28 (b), 1.3 (c)]. Insets show the momentum statistics. The blue solid line is a fit
to a Gaussian distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

6.4(a) Cooling rate (in units of the single atom cooling rate RS ) as a function of atompnumber. (b) Final momentum width (∆p = hp2 i, blue squares) and spin-spincorrelation (red dots) as a function of atom number. The parameters are the sameas those in Fig. 6.3.

6.5

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71(a) Final momentum width as a function of γC for 40 atoms. The parameters are∆ = κ/2 = 200, w = N γC /4, and ωr = 0.25. (b) Final momentum width as afunction of repumping strength for 40 atoms without (k 0 = 0, blue squares) and withrecoil associated with repumping (k 0 = k, red dots). The parameters are ∆ = κ/2 =200, γC = 0.5, and ωr = 0.25. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

Chapter 1

Introduction

Our era has seen quantum physics beginning to enable new technologies with unprecedented
capabilities, e.g., optical lattice clocks with stability at the 10−18 level [9, 49], small quantum
processors implementing Shor’s factoring algorithm [82, 144], and quantum teleportation over 100
kilometers [84, 145]. The light-matter interaction has been a valuable resource for probing andcontrolling quantum systems, and for synthesizing new quantum states. It is therefore the basis
for many quantum technologies such as quantum computation, communication, simulation, sensing
and metrology.
Cavity quantum electrodynamics (QED) represents a particularly important system for applications in quantum physics. First, cavity output photons are information carriers. It was shownthat cavity fields could be used to probe the Mott insulator to superfluid phase transition [94] andto detect topological order and statistics of anyonic excitations [58]. Second, cavity dissipation provides the source to dissipatively control quantum systems. Based on the idea of quantum reservoir
engineering [29], it was shown that an atomic ensemble coupled to a decaying cavity mode could
be driven dissipatively to the targeted spin-squeezed state [25]. The cavity dissipation can also be
used for the feedback control of quantum systems [138] and for cooling the objects coupled to thecavity [108, 141]. Third, cavity mediated interactions give rise to interesting quantum many-bodystates, e.g., crystallinity and frustration with Bose-Einstein condensates in a multimode cavity [38],frustration and glassiness with cavity-confined spin systems [39], and to the Dicke quantum spinglass of atoms and photons with atomic qubits coupled to discrete cavity modes [119]. In this thesis,

2
it will be shown that the cavity-mediated dissipative coupling leads to synchronization states ofatomic dipoles [140, 142]. Last, optical cavities are ideal for conversion between the stationary gate
qubits and flying qubits (photons), which is essential for quantum networks. Quantum gates can be
implemented in optical cavities with high fidelities [15]. An elementary quantum network of single
atoms in optical cavities has been demonstrated [109]. Therefore, an important theme of current
research in quantum physics is to explore how engineered photon fields interacting with quantum
systems could be harnessed to probe and control the system, synthesize new quantum states, andprocess quantum information. Over the past decade, cavity QED has been expanded to new physical systems, such as atoms coupled to chiral bosonic waveguides [102], nitrogen-vacancy (NV)centers in diamond coupled with photonic structures [21], quantum dots in photonic crystals [124],and superconducting qubits coupled with an inductance/capacitance resonator [146]. This will not
only advance quantum technologies, but also allow for better understanding of quantum many-body
physics and fundamental questions in quantum mechanics.
The laser is a typical cavity QED system. Since its first demonstration in 1960 [85], the laser
has had a profound impact on fundamental science research and has found widespread applications
in society in general. Lasers typically operate in the good cavity regime of cavity QED where thelinewidth of the cavity is much narrower than the bandwidth of the gain medium [95]. The atoms
generate a coherent electromagnetic field in the cavity by means of stimulated emission [113].
An important application of lasers is as a stable local oscillator for optical atomic clocks [83].
The precision currently achievable by atomic clocks is remarkable; for example, the accuracy andinstability of state-of-the-art optical lattice clocks lies in the realm of 10−18 [9, 49]. The pursuit ofeven more stability is motivated by the potential benefit to a wide range of fields in the physical andnatural sciences, facilitating progress in diverse areas such as; redefinition of the system of physicalunits in terms of time [14], clock-based geodesy [20], gravitational wave detection [40], and tests offundamental physics and cosmology [6, 26]. Atomic clock developments have also enabled spin-off
applications, including precision measurements [111], quantum state control [77], and investigationsof quantum many-body physics [88, 106].

3
Such lasers rely on stabilization against reference cavities. The most advanced such laser
realized to date has been achieved at JILA and reaches linewidths about 40mHz corresponding toquality factors of Q > 1015 [22]. The principal limiting factor in the way of further improvementsof these local oscillators is thermal vibrations of the dielectric coatings on the cavity mirrors [65].
To overcome this technical challenge, researchers have proposed an alternate approach using anactive system based on steady state superradiance on a clock transition to create an even morestable light source [18, 93].
As shown in Ref. [93], a steady-state superradiant laser is a possible system to consider for
generating millihertz linewidth light. It was demonstrated in a recent proof of principle experimentusing a two-photon Raman transition [12]. In the regime of steady-state superradiance, the cavity
decay is much faster than all other processes. Therefore, the cavity mode plays the role of adissipative collective coupling for the atoms that leads to the synchronization of atomic dipoles [140,142]. The emergence of a macroscopic collective dipole induces an extremely narrow linewidth forthe generated light [93, 140]. The optimal parameters are in the weak-coupling regime of cavity
QED [95] [single-atom cooperativity parameter C < 1, see definition in Eq. (2.50)], that is opposite
to the strong-coupling situation usually considered. Narrow-linewidth superradiant lasers require
weak-dipole atoms (e.g. using intercombination lines or other forbidden transitions) confined in ahigh-finesse optical cavity.
In this thesis, I provide comprehensive studies of the steady-state superradiance. The thesis
is structured as follows:• Chapter 2 introduces Dicke superradiance, laser theory, and features of steady state superradiance. We emphasize that the superradiant laser is indeed collective spontaneousemission in steady state.
• Chapter 3 describes novel group-theoretic methods to efficiently solve the laser equation.
The method may be applied to a wide variety of applications for simulating open quantum
systems with large system size.

4• Chapter 4 presents our new perspective on steady state superradiance: the synchronizationof atomic dipoles. This offers an alternative understanding of the narrow linewidth of thesuperradiant laser. By demonstrating the synchronization effect, we propose conditionalRamsey spectroscopy with synchronized atoms. This could have important consequences
for atomic clocks.
• Chapter 5 studies quantum synchronization with two ensembles of superradiant atoms. We
show quantum synchronization as a dynamical phase transition. Quantum fluctuations andquantum correlations are found to be important, shedding light on current understandingof the quantum aspects of synchronization.
• Chapter 6 proposes to utilize steady-state superradiance for the efficient cooling of atoms.
The temperature limit is determined by a modified atomic linewidth rather than the cavity
linewidth as in normal cavity assisted cooling. The cooling rate is enhanced by the superradiance. The new cooling scheme is dubbed “supercooling”. It represents a first example
of many-body laser cooling.
• Chapter 7 concludes the thesis and discusses the potential future of active optical standards.

Chapter 2

The superradiant laser

2.1

Dicke superradiance
Superradiance is a fundamental quantum optical phenomenon in the photon emission process

by many emitters. It was first predicted by Robert H. Dicke [28] in the context of spontaneousemission. Consider a gas of atoms (or molecules) initially prepared in the excited state decaying
by spontaneous emission to a lower state. In a dilute atomic system, the photon emission byeach atom can be considered as independent. As shown in Fig. 2.1(a), the emission obeys an
exponential decay law (decay time constant τsp ) and the radiation pattern is essentially isotropic.
These features are notably changed when the atomic ensemble becomes dense enough. As shownin Fig. 2.1(b), the atomic ensemble starts to radiate photons much faster and stronger than theemission of independent atoms, and that radiation occurs in a well defined direction depending on
the geometry of the atoms [42].
This phenomenon was termed “superradiance” by Dicke in 1954. Essentially, it is due to theindiscernibility of atoms with respect to the photon emission. This causes constructive interferencein the photon emission by the ensemble. The intensity becomes proportional to the square of thenumber of atoms N . In the above case [Fig. 2.1(b)], when the wavelength of light is much greaterthan the separation of atoms, the atoms interact with the light in a collective and coherent way.
As the emission begins, the quantum fluctuation of the vacuum field acts on independent atoms,
and an order appears in the atomic ensemble which can be defined as phase-locking of the atomic
dipoles. As a result, the atoms emit light as a high intensity pulse, with rate ∝ N 2 and short

to a process in which the atoms (or molecules) of the sample can be considered as interacting
independently from each other with the radiation field. The emission, obeying an exponential law, takesa characteristic time r~,equal to the reciprocal of the radiative decay rate F of the initially excited level
and the radiation pattern of the atomic sample is essentially isotropic (see fig. la). These features are6
D~ectortime.a.
Detectorb.

time

Fig. 1. Comparison between the general characteristics of ordinary fluorescence and superradiance experiments. (a) Ordinary spontaneous emission
Figure isotropic
2.1: with
Source:Ref. [42].
Comparison
betweenthe general characteristics of ordinary fluois essentiallyan exponentiallydecaying
intensity (time constant;~,).(b) Superradiance is anisotropic with an emission occurring in a short
rescence
superradiance experiment. (a) Ordinary spontaneous emission is essentially isotropic
burstof durationand~~rw/N.

with exponential decaying intensity (time constant τsp ). (b) Superradiance is anisotropic with anemission occurring in a short burst of duration ∼ τsp /N .

duration of the order of τsp /N .
Since Dicke’s original work [28], superradiance has been extensively studied and demonstrated
in a wide variety of physical systems [42]. More recently, it was demonstrated in quantum dots [114],
NV centers [62], and atoms trapped along a photonic crystal waveguide [37]. In spite of thecomplications of different physical situations, the essential mechanism of superradiance is pretty
simple.
To illustrate the key aspects of superradiance, it is easiest to introduce the permutation
invariant atomic states. Consider an ensemble of N two level atoms, identified by the indices1, 2, ..., j, ...N . The upper level and lower level of atom j are represented as |eij and |gij respectively.
All the atoms are prepared in the upper level at time 0 so that|ψ(0)i = |ei1 |ei2 ...|eiN .

(2.1)

We make the basic assumption that the atoms are indiscernible with respect to the photon emission
or absorption process in the subsequent evolution. Therefore, the system must remain in a Hilbert
subspace invariant to atomic permutations. Note that the two-level atom can be described as a

7fictitious pseudospin-1/2 system with |ei and |gi being the spin up and down state respectively.
So the N -atom permutation invariant state is isomorphous to a symmetrical superposition of N
spin-1/2 states [42], which are the familiar angular momentum eigenstates |J, M i with J = N/2
and M = −N/2, −N/2 + 1, ..., N/2 − 1, N/2. There are N + 1 such states, see Fig. 2.2. Due to theirapplication in superradiance, these states are also referred to as Dicke states. Note that J + M
is the number of atoms in the excited state. So the state in which all atoms are excited may be
written as |ψ(0)i = |N/2, N/2i.

Symmetrical state of N two-level identical atomsȁ‫ܬ‬ǡ ‫ ܯ‬ൌ ‫ۄܬ‬‫ ؠ‬ȁ݁ǡ ݁ǡ Ǥ Ǥ Ǥ ǡ ݁‫ۄ‬ȁ‫ܬ‬ǡ ‫ ܯ‬ൌ ‫ ܬ‬െ ͳ‫ ܵ ؠ ۄ‬ȁ݃ǡ ݁ǡ Ǥ Ǥ Ǥ ǡ ݁‫ۄ‬ȁ‫ܬ‬ǡ ‫ ܯ‬ൌ ‫ ܬ‬െ ʹ‫ ܵ ؠ ۄ‬ȁ݃ǡ ݃ǡ ݁ǡ Ǥ Ǥ Ǥ ǡ ݁‫ۄ‬J , M = 0 º S[| g , g ,..., g , e,...
,...,., e, e >]
,.
N /2

N /2

ȁ‫ܬ‬ǡ ‫ ܯ‬ൌ െ‫ ܬ‬൅ ͳ‫ ܵ ؠ ۄ‬ȁ݃ǡ ݃ǡ ݃ǡ Ǥ Ǥ Ǥ ǡ ݁‫ۄ‬ȁ‫ܬ‬ǡ ‫ ܯ‬ൌ െ‫ۄܬ‬‫ ؠ‬ȁ݃ǡ ݃ǡ ݃ǡ Ǥ Ǥ Ǥ ǡ ݃‫ۄ‬

Figure 2.2: Dicke states of N two-level atoms. S denotes the symmetrizer with respect to atomic
labels.

To define the electric dipole of j-th atom, we introduce the atomic raising and lowering
operators
σ̂j+ = |eij hg|j ,σ̂j− = |gij he|j ,(2.2)

and the atomic population inversion operatorsσ̂jz = |eij he|j − |gij hg|j .

(2.3)

These are Pauli matrices and obey the well-known commutation relations[σ̂i+ , σ̂j− ] = δij σ̂jz ,[σ̂iz , σ̂j± ] = ±2δij σ̂j± .

(2.4)

The dipole operator for the j-th atom is
Dj = (σ̂j+ + σ̂j− )Dˆ
,

(2.5)

8
where D is the electric dipole matrix element of the |ei ↔ |gi transition and ˆ is the normalized
vector along the polarization direction.
Under the condition of Dicke superradiance, individual dipoles may phase-lock. It is natural
to assume that N atoms behave like a macroscopic dipole resulting from the sum of all N individual
dipolesD̂ =NX

D̂j = (Jˆ+ + Jˆ− )Dˆ
,

(2.6)

j=1

where we have introduce the collective operators
Jˆ± =

N
X

σ̂j± .

(2.7)

j=1

Recalling that the rate of photon emission by a single atom is proportional to hσ̂j+ σ̂j− i, we naturally
generalize this to get the superradiant photon emission rate [42],WS ∝ hJˆ+ Jˆ− i .

(2.8)

For the Dicke state |J, M i, we immediately get
WS ∝ (J + M )(J − M + 1) .

(2.9)

We have used
Jˆ± |J, M i =p(J ∓ M )(J ± M + 1)|J, M ± 1i .

(2.10)

From Eq. (2.9), we can see that the photon emission rate increases from N for M = N/2 (fully
excited) to 12 N ( 21 N + 1) for M = 0 (half excited). Therefore, the photon emission rate strongly
increases when the system cascades down the ladder of the |J, M i states as shown in Fig. 2.2. The
maximum rate of emission is proportional to N 2 for M = 0.
The N 2 scaling of the photon emission rate is directly related to the build up of correlations
between atomic dipoles, which is the very essence of Dicke superradiance. By using Eq. (2.7) andpermutation symmetry, we can rewrite Eq. (2.8) as
*N
+
N
X
X
hJˆ+ Jˆ− i =σ̂ +σ̂ − = N hσ̂ + σ̂ − i + N (N − 1)hσ̂ + σ̂ − i .
i

i=1

j

j=1

i

i

i

j

(2.11)

9
There are N terms hσ̂i+ σ̂i− i representing the population in the excited state and N (N − 1) crossterms hσ̂i+ σ̂j− i that represent the atomic correlations. These correlations are reminiscent of the spinspin correlations appearing in a ferromagnetic system. In order for the superradiance to manifest,
the atom-atom correlation, measured by hσ̂i+ σ̂j− i, should be nonzero.
It is easy to get
hJ, M |σ̂i+ σ̂j− |J, M i =J2 − M2.
N (N − 1)(2.12)

Therefore, the atom-atom correlation increases from 0 to a maximum value of 1/4 when M goesfrom N/2 to zero and decreases to 0 when M = −N/2.

2.2

Laser theory in a nutshell
As noted by Serge Haroche in Ref. [42],“One of the interests in superradiance study lies in its close connection with thephysics of laser emission. A superradiant medium is indeed nothing but a mirrorlesslaser pumped in a percussional way at time t = 0. The superradiant phase-lockingof the dipoles belonging to different atoms thus exhibits a strong relationship withthe ordering process by which the atoms in a laser amplifier acquire a common
phase and start emitting coherently.”
In this section, I provide a simple theory of the laser. The basis of laser action is stimulated

emission of radiation. If a net population inversion of atoms is maintained, the electromagnetic
field will be amplified. As lasing takes place, atoms lose their inversion. Some mechanism must be
provided to pump them back up again so as to maintain the population inversion. Note that by coherent drive, the atom will never achieve population inversion regardless of the pump strength [36].
This is because coherent drive involves both atomic excitation and deexcitation. Therefore, a laser
requires the incoherent repumping of the atoms (see below for a possible implementation).
I use the simplest possible laser model as the basis for my analysis. The well-known model [44]accounts for a single-mode electromagnetic field in a cavity interacting with N identical two-levelatoms in terms of the Jaynes-Cummings Hamiltonian [95], and for the irreversible photon escape (atomic and cavity decay) and atomic decoherence in terms of Liouville operators. The above

10
model is treated in terms of the following quantum master equation for the atom plus field density
operator ρ̂:
N

X
d
1
1
L[σ̂jz ]ρ̂ + wL[σ̂j+ ]ρ̂ρ̂ = [Ĥ, ρ̂] + κL[â]ρ̂ +γL[σ̂j− ]ρ̂ +
dt
i~
2T2

(2.13)

j=1

where L[Ô]ρ̂ = (2Ôρ̂Ô† − Ô† Ôρ̂ − ρ̂Ô† Ô)/2 is the Lindbladian superoperator [80] describing theincoherent processes. The Hamiltonian Ĥ is
Ĥ = ~ωa

N
X
j=1

N

σ̂j+ σ̂j− + ~ωc â† â +

~g X  † −â σ̂j + σ̂j+ â ,2

(2.14)

j=1

where ωa and ωc are the atomic transition frequency and the frequency of the cavity mode respectively, and g is the single atom-cavity coupling strength. Here, â and â† are the annihilation andcreation operators for cavity photons. Note that the Hamiltonian has been specified in the dipole
and rotating wave approximations. The cavity dissipation with power decay rate κ is described byκL[â]ρ̂. The atomic spontaneous emission with rate γ is described by γL[σ̂j− ]ρ̂. The incoherent
repumping takes the atoms from the ground state to the excited state incoherently and is modeled as inverse spontaneous emission with rate w (term wL[σ̂j+ ]ρ̂). The additional dephasing of
the atomic dipole is modeled phenomenologically as a T2 process and is described by 2T1 2 L[σ̂jz ]ρ̂.
Note that the Liouvillian is derived in the usual Born-Markov approximation [81], assuming weaksystem-reservoir coupling and delta-correlation of the reservoir.

Excited state

Ground state

Figure 2.3: Schematic for implementing the incoherent pumping in a three-level configuration.
The coherent driving laser is indicated by the green-dashed arrow and spontaneous decay into theexcited state |ei is indicated by the orange-dotted arrow.

Since incoherent pumping generates population inversion in a two-level system, it is inducedby the system coupling to a negative-temperature reservoir and therefore can only be implemented

11
using auxiliary atomic levels. A typical approach is coherently driving a transition from the groundstate |gi to a third state that rapidly decays to the excited state |ei, as shown in Fig. 2.3. In thelimit of large γs , this will lead to an effective incoherent pumping rate w = |Ω|2 /γs .
It turns out that simple mean-field theory of Eq. (2.13) can reveal a great deal about theproperties of a laser. The complete set of quantum Langevin equations according to Eq. (2.13) inthe reference frame rotating at frequency ω [36]d1iN g −â = − (κ + 2iωc − 2iω)â −Ŝ + F̂ a ,
dt
2
2
d −
1
ig
Ŝ
= − (Γ + 2iωa − 2iω) Ŝ − + âŜ z + F̂ − ,
dt
2
2




d z
Ŝ = −(w + γ) Ŝ z − d0 + ig â† Ŝ − − âŜ + + F̂ z ,dt(2.15)(2.16)(2.17)

where Γ ≡ w + γ + 2/T2 is the generalized single-atom decoherence and d0 = (w − γ)/(w + γ)
characterizes the atomic inversion that would be obtained for a single-atom in the absence of thecavity. We have defined the collective operators,
N

1 X ±
Ŝ =
σ̂k ,
N
±

k=1

N

1 X z
Ŝ =
σ̂k .
N
z

(2.18)

k=1

The quantum noise operators F̂ µ have zero mean and second-order correlations given by
hF̂ µ (t)F̂ ν (t0 )i = 2Dµν δ(t − t0 ) .

(2.19)The diffusion matrix elements Dµν can be obtained [36]


1
z
2D
= κ, 2Dw + (1 + hŜ i)
T2


1
1
2w
2D−+ =
γ + (1 − hŜ z i) , 2D+z = − hŜ + i
N
T2
N
2γ +
2γ −
2Dz+ =hŜ i, 2D−z =
hŜ i
N
N
2w
2γ
2w
2Dz− = − hŜ − i, 2Dzz =(1 + hŜ z i) +(1 − hŜ z i).
N
N
N
aa†

+−

1
=
N

(2.20)√
From Eq. (2.20), we can see that the noise is 1/ N smaller than the deterministic terms in
the quantum Langevin equation. To zeroth order, we can discard the noise terms and arrive at the12mean-field equations of motiond1
iN g −
a0 = − (κ + 2iωc − 2iω)a0 −
S ,
dt
2
2 0
d −
1
ig
S0 = − (Γ + 2iωa − 2iω) S0− + a0 S0z ,
dt
2
2

d z
S0 = −(w + γ) (S0z − d0 ) + ig a∗0 S0− − a0 S0+ ,dt(2.21)(2.22)(2.23)

where the 0 subscript denotes the mean value, e.g. hâi = a0 .
A closed-form solution of Eqs. (2.21)–(2.23) can be obtained in steady-state by setting theleft hand sides to zero. We find from Eq. (2.22) and Eq. (2.23)
S0z =

d0
g 2 |a0 |22Γ
1 + w+γ Γ2 +4(ω2a −ω)

(2.24)

for the steady-state population inversion. Plugging it to Eq. (2.21), we have


2
Ng
d0
 = 0.
a0 1 −
(κ + 2iωc − 2iω)(Γ + 2iωa − 2iω) 1 + g2 |a0 |2 2 2Γ
2

(2.25)

w+γ Γ +4(ωa −ω)

There are two solutions for the average photon number |a0 |2 ,a0 = 0,(2.26)

andw + γ Γ2 + 4(ωa − ω)2|a0 | =
g22Γ2

d0 N g 2−1(κ + 2iωc − 2iω)(Γ + 2iωa − 2iω).

(2.27)In Eq. (2.27), for |a0 |2 to be real, we need to haveω=κωa + Γωc,
κ+Γ(2.28)

that is the laser frequency ω is the weighted average of the cavity frequency and atomic frequency.
Plugging Eq. (2.28),into Eq. (2.27), we can get a simple expression when |ωc − ωa |  κ + Γ(w + γ)Γ
|a0 | ≈
2g 2
2

d0 N g 2−1
κΓ= n0 (G − 1),(2.29)

2g
where n0 = (w+γ)Γ
is called the saturation photon number and G = NκΓd0 is called the generalized
2g 2

cooperativity parameter. Clearly, the second solution is only possible if G > 1. G can be varied

13
by changing the repumping w. The behavior of the system is completely different depending onwhether G is greater than or less than 1, and there is a threshold at G = 1. Below the threshold,G < 1, a0 = 0. There is no mean coherent field established. Above the threshold, when G > 1,a mean coherent photon field arises. More importantly, we emphasize that a nonzero collective
atomic dipole S0− develops, which is a signature of atomic dipole phase locking.
As introduced in Chapter 1, one of the most important properties of the laser is its spectral
linewidth. Indeed, one impetus to develop a useful form of quantum optical theory came from theneed to understand the properties of laser light. Here, I follow Haken’s phase diffusion method [43,44] to provide a closed-form solution to obtain the spectral linewidth of the laser light.
Differentiating Eq. (2.15) with respect to time and substituting Eqs. (2.15) – (2.16) we obtain2¨ = − 1 (κ + Γ)â˙ − κΓ â + N g âŜ z + F̂ ,
â
2
4
4

(2.30)




2
d †
z
†
† a
a†
(w + γ) + F̂ −(â â) + κâ â − â F̂ − F̂ â,
N dt(2.31)

where
z

Z t

Ŝ =

0 −(w+γ)(t−t0 )

dt e
0

and
F̂ =

Γ a iN g −˙F̂ −F̂ + F̂ a .
2
2(2.32)

The annihilation operator â is decomposed according toâ = (a0 + r̂)eiφ̂ .

(2.33)

Above threshold, amplitude fluctuations are small so that r̂ can be neglected. We then obtain for
the two time correlation function of the field amplitudehâ† (t)â(0)i = a20 hei[φ̂(t)−φ̂(0)] i .

(2.34)

After substituting Eq. (2.33) into Eq. (2.30), we take the imaginary part to first order inproducts of operators, and find,11
˙¨φ̂ = − (κ + Γ)φ̂ + Im[F̂ ] ,2
a0(2.35)

14
where a factor of e−iφ has been absorbed into F̂ . Equation (2.35) is then integrated, assuming that(κ + Γ) is large, to arrive at2
φ̂(t) − φ̂(0) =a0 (κ + Γ)Z t



Γ a iN g −
dt Im
F̂ −
F̂
2
2
0
0


.

(2.36)

Since F̂ a and F̂ − are Gaussian, we can use
12

hei[φ̂(t)−φ̂(0)] i = e− 2 h[φ̂(t)−φ̂(0)] i .

(2.37)

Therefore, we use Eq. (2.36), along with Eqs. (2.20) to find2h(φ̂(t) − φ̂(0)) i =Γg2κ
(G/d0 + 1)
t,2(G − 1) (w + γ) (κ + Γ)2(2.38)

so that the linewidth ∆ν in the unit of Hz is given by
∆ν =

Γ
g2κ
(G/d0 + 1).
2(G − 1) (w + γ) (κ + Γ)2(2.39)

Lasers typically operate in the good cavity regime of cavity QED where the linewidth of thecavity is much narrower than the bandwidth of the gain medium, i.e., κ  Γ. In this case,
∆ν ≈
∝

κ
G/d0 + 1 κ
1 S0z + 1 κ
G/d0 + 1==
2(G − 1) (w + γ)Γ/g 2
4|a0 |2
4 S0z |a0 |2
κ
.
|a0 |2(2.40)

This is the famous Schawlow-Townes linewidth for laser [113]. In a good-cavity laser, the macroscopic phase information that is associated with the coherence of the generated radiation is encodedin the light field.

2.3

Steady state superradiance
Fritz Haake made an important connection between Dicke superradiance and bad-cavity laser.

As he wrote in Ref. [13]As has already been pointed out by Dicke the “coherence brightening” occurs
practically exclusively in the axial direction for such a pencil-shaped arrangement.
Emission of radiation does take place in non axial directions, too, but to a much

15
lesser extent and essentially unfavored by cooperative effects. It may therefore belooked upon as an incoherent loss mechanism for the atoms. The latter artificegreatly reduces the complexity of the problem, for it leaves us with the axial modes
only as dynamical field variables. As a device which further simplifies the problem,
we quantize the electromagnetic field with respect to the volume of the cavity. · · ·The escape of photons through the non mirrored end face of the pencil into theradiation field is taken care of as a loss mechanism for the field mode in the cavity.
What we have just described as our model of a superradiant system is nothing
but the well-known simplest possible laser model. · · · However, we have to solve
the laser master equation in a limit that, for good reasons, has hardly ever been
considered in laser theory. We are here concerned with a low-quality cavity. · · ·In contrast to the laser, the superradiant device must not be designed to lock thephotons in the cavity but rather to release them as fast as they can escape according
to the velocity of light. Moreover, again as opposed to what is typical for a laser,
the incoherent atomic decay process must be so slow that the individual atomic
dipoles do not get out of phase with each other before they can involve themselves
cooperatively in the interaction with the field mode.
Haake’s statement can be made quantitative by a “superradiance master equation” that was first
derived in Ref. [13]. It describes the cooperative decay of atomic excitations without allowing for
any feedback of the field on the atoms.
Let us consider the laser equation Eq. (2.13) in the bad cavity regime, or the superradiantlimit. In this limit, the cavity decay rate κ is much faster than any other time scale. Therefore, it
allows for an expansion in powers of κ−1 and the elimination of the field variable (see Appendix
A). To this end, we start with the quantum Langevin equation for the cavity field according to thequantum master equation Eq. (2.13) in the rotating frame of the cavity frequency,√dκgâ = − â − i∆â − i Jˆ− + κξˆκ (t) ,dt
22(2.41)

where ∆ = ωc − ωa and ξˆκ is the quantum white noise originating from the cavity output withhξˆκ (t)ξˆκ† (t0 )i = δ(t − t0 ). The formal solution to Eq. (2.41) isâ(t) = ewhere F̂(t) =−(κ/2+i∆)∆tgâ(t0 ) − i2Z ∆tdse−(κ/2+i∆)s Jˆ− (t − s) + F̂(t) ,(2.42)

0√ R ∆tκ 0 dse−(κ/2+i∆)s ξˆκ (t − s) is the noise term and ∆t = t − t0 . Under the approx-

imation of coarse graining, the first term on the right-hand side of Eq. (2.42) vanishes, and it is16
shown in Appendix A thathF̂(t)Fˆ† (t0 )i ≈√κδ(t − t0 ) .

(2.43).

(2.44)

κ2 /4 + ∆2γˆIt would be convenient to choose F̂(t) = −i g/2C ξ(t),
with
γC =

g 2 κ/4
κ2 /4 + ∆2

Furthermore, the integral in Eq. (2.42) can be expanded in powers of 1/(κ/2 + i∆) (see Appendix
A). As a result we obtainâ(t) ≈

−i g2 Jˆ−+ F̂(t) + O[(κ/2 + i∆)−2 ] .
κ/2 + i∆(2.45)

Because we are interested in the deep bad-cavity regime as motivated above, the expansion can betruncated in lowest order. This yields the “superradiance master equation”.
NX
1 ~γ∆ ˆ+ ˆ−
1dρ̂ = [−J J , ρ̂] + γC L[Jˆ− ]ρ̂ +γL[σ̂j− ]ρ̂ +L[σ̂jz ]ρ̂ + wL[σ̂j+ ]ρ̂ ,
dt
i~
2
2T2

(2.46)

j=1

where
γ∆ =

g 2 ∆/2.
κ2 /4 + ∆2

(2.47)

Before discussing the physics of Eq. (2.46), a proof is given of the superradiance masterequation by comparing the equations of motion for spins (i.e. σ̂j− , σ̂jz ) derived from Eq. (2.46) andEq. (2.13) in the large κ limit. The complete set of equations of motion for spins according to
Eq. (2.46) is
d −
w
1
γ∆
γC z
σ̂ = −( + )σ̂j− − i σ̂jz cos(kx̂j )Jˆ− +σ̂ cos(kx̂j )Jˆ−
dt j
2
T2
2
2 j
r
√ z †
2 ˆz† ˆz −√zˆˆ+ γC σ̂j cos(kx̂j )ξ − wσ̂j ξj +(ξ − ξj )σ̂j ,
T2 j
h
i
d z
+ ˆ−
z
σ̂ = w(1 − σ̂j ) + 2Re (iγ∆ − γC )σ̂j J cos(kx̂j )dt j√√− 2 γC (σ̂j+ ξˆ + ξˆ† σ̂j− ) cos(kx̂j ) + 2 w(σ̂j+ ξˆj† + ξˆj σ̂j− ),(2.48)

where ξˆj and ξˆjz are the independent repumping noise and T2 dephasing noise for each atom,
respectively. While the same set of equations according to Eq. (2.13) is
r
√ z †
d −
w
1 −
g z
2 ˆz† ˆz −σ̂ = −( + )σ̂j + i σ̂j â cos(kx̂j ) − wσ̂j ξˆj +(ξ − ξj )σ̂j ,
dt j
2
T2
2
T2 j
√
d z
σ̂j = w(1 − σ̂jz ) − igσ̂j+ â cos(kx̂j ) + igâ† σ̂j− cos(kx̂j ) + 2 w(σ̂j+ ξˆj† + ξˆj σ̂j− ) .
dt(2.49)

17
Plugging Eq. (2.45) into Eq. (2.49) results in a set of equations identical to Eq. (2.48). Consequently,
the superradiance master equation describes the same spin dynamics of atoms as Eq. (2.13) in thelarge κ limit.
From the superradiance master equation Eq. (2.46), we see two new terms coming from theelimination of the photon field:• γC L[Jˆ− ]ρ̂ termThis is a Lindbladian term with decay operator being the collective operator Jˆ− . Therefore,
this term gives rise to the collective photon emission by the atoms into the cavity mode.
We can expand this term as
γC L[Jˆ− ]ρ̂ = γCN
X

L[σ̂j− ]ρ̂ + γC

j=1

X1
i6=j

2

(2σ̂i− ρ̂σ̂j+ − ρ̂σ̂j+ σ̂i− − σ̂j+ σ̂i− ρ̂) .

Therefore, this collective decay term introduces spontaneous decay of atoms into the cavity
mode with rate γC and dissipative coupling between atoms. The dissipative coupling isessential for the build up of atom-atom correlations hσ̂i+ σ̂j− i and thus the superradiantemission.
1• i~[− ~γ2∆ Jˆ+ Jˆ− , ρ̂] termThis is a Hamiltonian term with the Hamiltonian − ~γ2∆ Jˆ+ Jˆ− . This term introduces thecoherent coupling between atoms and the energy shift of atom j by −~γ∆ /2. The underlying
mechanism of this energy shift is formally similar to that of the Lamb shift [81], that is,
the interaction of atoms with off-resonant light modes through emission and absorption ofvirtual photons. When ∆ = 0, the energy shift disappears. An intuitive way to understandthis energy shift in the dispersive limit (∆  κ) is to view it as ac Stark shift due tovacuum fluctuations of the cavity field.
It is the collective decay term that gives rise to the superradiance. To focus on the superradiantbehavior of atoms in the bad-cavity laser, we consider the case ∆ = 0 in this section. As a result,

18
γC = g 2 /κ. In this case, we rename γC to ΓC = Cγ. Here,
C=

g2
κγ

(2.50)

is the so-called single-atom cooperativity parameter of the cavity [64]. It is a dimensionless parameter that describes the ratio of atomic photon emission rate into the cavity mode as enhanced by
the cavity finesse to atomic photon emission rate into the free space.
A remarkable feature of the superradiant limit of the laser is its linewidth property. According
to the linewidth formula for laser Eq. (2.39), in the bad cavity limit, i.e., κ  Γ,∆ν ≈G/d0 + 1 ΓΓC .
2(G − 1) w + γ(2.51)In the limit that w  γ, 1/T2 , d0 ≈ 1 and Γ ≈ w + γ, this can be further simplified to
∆ν ≈

G+1
1 + S0z
ΓC ≈Cγ.
2(G − 1)2(1 − S0z )(2.52)The linewidth formula in the superradiant limit Eq. (2.52) differs qualitatively from the SchawlowTownes linewidth for good-cavity laser [113]. As the cavity linewidth κ drops out of the linewidthformula Eq. (2.52), the coherence property of the bad cavity laser does not depend on the cavity
linewidth. By comparing Eq. (2.40) and Eq. (2.52), we reach the conclusion that the coherence isstored in the photons for the good cavity laser but is stored in the atoms for the bad-cavity laser.
To validate the above discussion about the superradiance emission by atoms in the badcavity laser, we employ the methods described in Chapter 3 to numerically solve the laser masterequation Eq. (2.13) in the bad-cavity limit. As shown in Fig. 2.4, we find the build-up of atomatom correlations hσ̂i+ σ̂j− i in steady state, which is a signature of superradiance. Furthermore,
the linewidth of the laser light shown in Fig. 2.4 is Lorentzian and proportional to Cγ, which isconsistent with the prediction of Eq. (2.52).
Therefore, the superradiant decay of an ensemble of atoms in steady state can be described
realistically in terms of the simplest possible laser model Eq. (2.13) in the bad cavity limit. In
2009, Meiser et. al. proposed to use a clock transition of strontium atoms in an optical cavity

Arbitrary unit

19

(Hz)

Figure 2.4: Numerical simulation (blue dashed line) of a bad cavity laser. Left panel: Atom-atomcorrelation as a function of time. Right panel: Spectrum of the laser light (red solid line is the fit
to Lorentzian). The parameters are N = 200, C = 0.2, w = 18γ, T2−1 = 2γ, κ = 2500γ.

as a possible realization of such model [93]. The motivation here was to generate an ultranarrow
linewidth laser source by using atoms with extremely small spontaneous emission rate γ and abad cavity with C  1. In this system, the weak radiative power due to small γ of the clocktransition (for example, for 106 fully inverted 87 Sr atoms the power of the spontaneous emitted
light is of the order 10−16 W) is not of major concern. This is due to the fact that the photon
emission rate is boosted by the superradiance and scales favorably with atom number. Below I will
give detailed discussion of the proposal.
106

105
10-15
4

10

10-18
wmax

Γ

P @WD

10-12
N

10-21

103
10-3 10-2 10-1 1 101 102 103
w @s -1 D

Figure 2.5: Source: Ref. [93]. Power as a function of pump rate w and atom number N . The
rapid buildup of power above threshold w ∼ γ can be seen as well as the decrease of emitted power
for too strong a pump. The dashed line shows the boundary of the region of collective emission.

20
Consider N ultracold two-level 87 Sr atoms confined in an optical cavity. The lasing transitionis the ultranarrow clock transition from 3 P0 to 1 S0 with extremely weak dipole matrix element of
the intercombination transition of order 10−5 ea0 . Here e is the electron charge and a0 is the Bohrradius. The linewidth γ is about 0.01 s−1 . The atoms are located in a high-Q optical cavity withcavity decay rate κ = 1 MHz. The atoms are confined along the axis of the cavity mode in a
1D optical lattice at antinodes of the field, so that the atom-cavity coupling g is uniform to good
approximation. For a bad cavity, the single-atom cooperativity parameter C ≈ 0.15 and g ≈ 37 s−1 .
The T2 dephasing time can be pushed to T2 ∼ 1 s in modern optical lattice clock experiment [9].
The repumping rate w can be widely tuned from 0 to the values of order 104 s−1 .
According to the parameters, the proposed system is thus an extreme case of a bad cavity
laser. It has astounding potential linewidth of ∼ Cγ = 1 mHz according to Eq. (2.52). For reference,
the most stable conventional laser based on stabilization against carefully designed passive cavities
achieve linewidth ∼ 40 mHz [22].
In such a laser, the atoms emit light collectively. As shown in Fig. 2.5, the resulting power
is of order 10−12 W for 106 atoms, which would be large enough to be technological relevant. We
can also identify two lasing thresholds from Fig. 2.5. In the first threshold w ∼ γ, the repumping islarge enough so that the atoms get population inverted and start lasing. For the second threshold,the repumping is so large that the atoms get fully inverted and the coherence between atoms is
destroyed. This threshold is given by w = N Cγ. The maximum power is achieved at w = N Cγ/2,
where hσ̂jz i = 1/2. A minimum number of atoms is necessary to have superradiance emission.
Assuming T2 γ  1, it is found that Ncritical = 2/CγT2 . Physically, this means that there must be
enough atoms for the system to be in the collective strong coupling regime.
As has been discussed, the most striking feature of this laser is its ultranarrow linewidth.
Fig. 2.6 shows the linewidth as a function of w and N . The leftmost dashed line in Fig. 2.6(a) is γ
corresponding to the threshold for collective behavior. When the pump strength w passes throughthat threshold the linewidth gets rapidly smaller with increasing w. When w reaches T2−1 , indicatedby the second dashed line, essentially all atoms are phase locked together. From that point on the

5

N
104

Γ

T2-1

wmax

HaL
10

3

10-3

10-1
101
-1
w @s D

103

T2-1
103

DΝ @s -1 D
103
102
101
1
10-1
10-2
10-3

101

DΝ @s -1 D

Γ
10

21

105

106

10-1

HbL
10-3
10-3 10-1 101 103 105
w @s -1 D

Figure 2.6: Source: Ref. [93]. (a) Linewidth vs w and N . The white dashed lines indicate (from left
to right) the spontaneous decay rate γ, the inhomogeneous relaxation rate T2 , and the maximum
pump rate w = N Cγ. (b) is a cut through (a) for N = 106 atoms.

pump noise due to w grows in proportion to the size of the collective spin vector. Therefore thelinewidth is approximately constant and is given by ∼ Cγ. The actual numerical factor in frontof Cγ does not change much as a function of w. When w increases beyond N Cγ, indicated by
the third dashed line, the collective dipole is destroyed and the linewidth increases rapidly until it
is eventually given by w. It is important to note that the parameters for achieving the maximum
output power and the small linewidth are compatible with each other.

Chapter 3Group-theoretic methods for solving the laser equation

The laser equation Eq. (2.13) is a quantum master equation in the Lindblad form [80] underthe Born and Markov approximations [81]. It describes the time evolution of a paradigmaticquantum optical system, i.e., an open quantum system of N two-level atoms (analogous to pseudospin-1/2 systems or qubits) symmetrically coupled to a single-mode photon field. In this chapter,
we show that the laser equation can be efficiently simulated by applying group theoretic methods
to the quantum master equation. This is important since many examples in quantum optics fall
into this class, such as cavity QED, atoms coupled to chiral bosonic waveguides [102], nitrogenvacancy (NV) centers in diamond coupled with photonic structures [21], quantum dots in photonic
crystal [124], and superconducting qubits coupled with inductance/capacitance resonator [146].
The method described in this chapter will find numerous applications for simulating open quantum
systems with large system size.

3.1

Lindbladian in terms of generators of the SU(4) group
In general, for all but the smallest system sizes, exact analytic solutions to Eq. (2.13) are

intractable. Various approximation methods have been introduced, e.g. perturbation theories [61],mean-field approaches [30, 98], cummulant expansions [91, 92, 93], linear response theories [25] andc-number Langevin equations [7, 69]. However, it is often necessary to benchmark approximate
methods with exact numerical solutions. Existing numerical simulation approaches, such as thequantum Monte Carlo method [24, 33, 104], scale exponentially with the underlying dimensionality

23of the Hilbert space. Therefore, treating any appreciable system size is extremely difficult. It
should be emphasized that the fully-symmetrical Dicke states [28] do not suffice as a reduced basis
since basis vectors with mixed symmetry are required.
It has been realized that because of the permutation symmetry with respect to the exchange ofspins in Eq. (2.13), the number of independent elements of the atomic density matrix is (N +1)(N +2)(N +3)/6 [74]. This leads to a tremendous reduction of the number of required basis states needed
to provide an exact solution of the master equation (from exponential to N 3 ). The permutation
symmetry also lead to Hartmann’s development of generalized Dicke states [46], which are fully
symmetrical states in the Liouvillian space. Furthermore, Hartmann found that the Lindbladianin the laser equation can be expressed in terms of the generators of the SU(4) group. For thispurpose, 18 superoperators O+ , O− and O3 where O ∈ {Q, Σ, M, N , U, V} are definedQ± ρ :=NX

N

σj± ρσj∓ ,
1X 3
σj ρ + ρσj34Q3 ρ :=j=1

Σ± ρ :=

N
X

j=1
N

σj± ρσj± ,Σ3 ρ :=

j=1

j=1

M± ρ :=NX

1 + σj3
σj± ρ
2

j=1

N± ρ :=

N
X

σj± ρ

j=1

U± ρ :=

V± ρ :=

1 − σj3
2NX
1 + σj3
j=1

2

NX
1 − σj3
j=1

2


1X 3
σj ρ − ρσj3
4
N,

1 X 3 1 + σj3
σj ρ
2
2

M3 ρ :=

j=1

N,N3 ρ :=

1 X 3 1 − σj3
σj ρ
2
2
j=1
N

ρσj∓ ,

U3 ρ :=

1 X 1 + σj3 3
ρσj
2
2
j=1
N

ρσj∓ ,

V3 ρ :=

1 X 1 − σj3 3
ρσj .
2
2

(3.1)

j=1

Although this list, Eq. (3.1), contains 18 operator definitions, only 15 of them are independent(it is possible to write N3 , U3 , V3 in terms of the others). One can also demonstrate that the 15
remaining superoperators are linear combinations of the familiar Gell-Mann matrices that are thegenerators of the SU(4) group, λ1 , ..., λ15 (see Appendix B).
In order to see how the superoperators [Eq. (3.1)] are related to generators of the SU(4)group (Gell-Mann matrices), consider first the fundamental one atom case. We interpret the 2 × 2

24
density matrix as a 4 × 1 vector in the representing vector space (i.e. Liouville space).
 
a


 
 
a
c
c




→
 .


d b
d
 
b

(3.2)

The relations are then given by [143]1
Q± → (λ9 ± iλ10 ) ,2
1
Σ± → (λ6 ± iλ7 )21
M± → (λ4 ± iλ5 )21
N± → (λ11 ± iλ12 )2
1
U± → (λ1 ± iλ2 )21V± → (λ13 ± iλ14 )
2

,
,
,
,
,

r
1
1
1
Q3 → λ3 + √ λ8 +
λ15
4
6
4 3
√
1
3
Σ3 → − λ 3 +
λ8
4
√4
1
3
M3 → λ 3 +
λ8
4
4
r
1
1
1
N3 → − λ3 + √ λ8 +
λ15
4
6
4 3
1
U3 → λ32
r
1
1
V3 → − √ λ8 +
λ15 .
6
2 3

(3.3)

The commutation relations of the superoperators are given in both Ref. [46] and [101]. The SU(4)group has three Casimir operators, one of which is quadratic in the generators, and the others areof higher order. The quadratic Casimir operator C1 can be expressed in terms of superoperators
C1 =

X
O

1
1
(O− O+ + O3 ) + U32 + (U3 + 2Σ3 )2 + (3Q3 − 2U3 − Σ3 )2 .
3
6

(3.4)

We can also identify six SU(2) subalgebras,[O+ , O− ] = 2O3 , [O3 , O± ] = ±O± ,(3.5)

so that it is useful to define six corresponding quadratic superoperators O2 = O− O+ + O32 + O3 ,which commute with O3 .
The Lindblad operators in the laser equation Eq. (2.13) can be written compactly with

25
superoperators O+,−,3N

1X ± ∓
N(2σ̂j ρσ̂j − ρσ̂j∓ σ̂j± − σ̂j∓ σ̂j± ρ) = − ± Q3 + Q± ,
2
2
j=1

N

1X z z(2σ̂j ρσ̂j − 2ρ) = 4M3 − 2Q3 − 2Σ3 − N .
2

(3.6)

j=1

The completeness of O+,−,3 implies that an arbitrary full-symmetrical Hamiltonian can be expressedby them. Therefore, the Hamiltonian term in Eq. (2.13) can be rewritten as
1[H, ρ] =
i~

3.2

i
1
gh
[~ωc â† â, ρ] − 2iωa Σ3 ρ − i â(M+ + N+ )ρ + â† (M− + N− )ρ
i~
2
i
gh
+ i (U+ + V+ )ρâ† + (U− + V− )ρâ .
2

(3.7)

Generalized Dicke states
The generalized Dicke states are the fully symmetric representation of the SU(4) group.

Transcribing notation from the four-flavor quark model, a model with the same symmetry structure,
the fundamental representation is given by u = |1ih1|, d = |0ih0|, s = |1ih0|, and c = |0ih1| (up,down, strange, and charm). For the fully symmetric case, the normalized basis is:
Pα,β,γ,δ =α!β!γ!δ! X h α β γ δ i
P u d s c ,
N!

(3.8)

P

where P denotes all possible permutations and α + β + γ + δ = N . Therefore, the number of thebasis is (N + 1)(N + 2)(N + 3)/6. Note that only basis states with γ = δ = 0 have non-vanishing
trace of unity.
Although it is perfectly fine to use any three of α, β, γ, δ to label the generalized Dicke states,
Hartmann uses three quantum numbers [46] associated with SU(2) subalgebras [cf. Eq. (3.5)]. To
specify such quantum numbers, we derive the following relations using the definition of the basis
states in Eq. (3.8) and the action of the superoperators on the fundamental representation shown

26
Q+
Q−
Q3
Σ+
Σ−
Σ3

u
0
d
1
2u
0
0
0

d
u
0
- 12 d
0
0
0

s
0
0
0
0
c
1
2s

c
0
0
0
s
0
- 12 c

M+
M−
M3
N+
N−
N3

u
0
c
1
2u
0
0
0

d
0
0
0
s
0
- 12 d

s
0
0
0
0
d
1
2s

c
u
0
- 12 c
0
0
0

U+
U−
U3
V+
V−
V3

u
0
s
1
2u
0
0
0

d
0
0
0
c
0
- 12 dsu0- 12 s
0
0
0

c
0
0
0
0
d
1
2c

Table 3.1: The action of the SU(4) superoperators on the elements of the fundamental representation u, d, s, cin Table 3.1
Q− Q+ Pα,β,γ,δ = (α + 1)βPα,β,γ,δ ,Σ− Σ+ Pα,β,γ,δ = (γ + 1)δPα,β,γ,δ ,M− M+ Pα,β,γ,δ = (α + 1)δPα,β,γ,δ ,N− N+ Pα,β,γ,δ = (γ + 1)βPα,β,γ,δ ,U− U+ Pα,β,γ,δ = (α + 1)γPα,β,γ,δ ,V− V+ Pα,β,γ,δ = (δ + 1)βPα,β,γ,δ ,1Q3 Pα,β,γ,δ = (α − β)Pα,β,γ,δ
21
Σ3 Pα,β,γ,δ = (γ − δ)Pα,β,γ,δ2
1M3 Pα,β,γ,δ = (α − δ)Pα,β,γ,δ2
1N3 Pα,β,γ,δ = (γ − β)Pα,β,γ,δ
2
1U3 Pα,β,γ,δ = (α − γ)Pα,β,γ,δ2
1V3 Pα,β,γ,δ = (δ − β)Pα,β,γ,δ .
2

(3.9)

We can now define the quantum number associated with the six SU(2) subalgebras. By
requesting, as usual, that O2 Pα,β,γ,δ = o(o + 1)Pα,β,γ,δ and O3 Pα,β,γ,δ = o3 Pα,β,γ,δ , we obtain1
q = (α + β)21
σ = (γ + δ)2
1
m = (α + δ)
21
n = (β + γ)
21
u = (α + γ)
21
v = (β + δ)
2
Note that q + σ = m + n = u + v = N/2.

,
,
,
,
,
,

1
q3 = (α − β)21
σ3 = (γ − δ)21
m3 = (α − δ)21n3 = (γ − β)21u3 = (α − γ)21v3 = (δ − β) .
2(3.10)

27
Hartmann chooses q, q3 , σ3 to characterize the generalized Dicke states [46].

Note thatα, β, γ, δ can be expressed in terms of q, q3 , σ3 ,α = q + q3,

β = q − q3

γ = σ + σ3,

δ = σ − σ3 ,(3.11)

where σ = N/2 − q. From q = (α + β)/2, we conclude that q ranges from 0 to N/2 in steps of 1/2
unit. From the SU(2) subalgebras, we conclude that q3 ranges from −q to q in steps of 1 and σ3
ranges from −σ to σ in steps of 1. Therefore, we denote Pα,β,γ,δ by Pq,q3 ,σ3 with
• q: Q2 Pq,q3 ,σ3 = q(q + 1)Pq,q3 ,σ3 , q = 0, 1/2, ... , N/2• q3 : Q3 Pq,q3 ,σ3 = q3 Pq,q3 ,σ3 , q3 = −q, −q + 1, ... , q• σ3 : Σ3 Pq,q3 ,σ3 = σ3 Pq,q3 ,σ3 , σ3 = −σ, −σ + 1, ... , σ (σ = N/2 − q).
It is straightforward to determine actions of all the raising and lowering superoperators on Pq,q3 ,σ3 ,Q± Pq,q3 ,σ3= (q ∓ q3 )Pq,q3 ±1,σ3 ,Σ± Pq,q3 ,σ3= (σ ∓ σ3 )Pq,q3 ,σ3 ±1 ,M± Pq,q3 ,σ3= (m ∓ m3 )Pq±1/2,q3 ±1/2,σ3 ±1/2 ,N± Pq,q3 ,σ3= (n ∓ n3 )Pq∓1/2,q3 ±1/2,σ3 ±1/2 ,U± Pq,q3 ,σ3= (u ∓ u3 )Pq±1/2,q3 ±1/2,σ3 ∓1/2 ,V± Pq,q3 ,σ3= (v ∓ v3 )Pq∓1/2,q3 ±1/2,σ3 ∓1/2 .

(3.12)

We note that the fully symmetrical basis are also eigenstates of the quadratic Casimir operator C1
with common eigenvalue 3N (N + 4)/8.
In order to solve the laser equation Eq. (2.13), we expand the density matrix as
ρ=

X

m,n
Cq,q
P
m n ,
3 ,σ3 q,q3 ,σ3(3.13)

q,q3 ,σ3 ,m,nm,nwhere Cq,q
3 ,σ3 are complex coefficients, and |ni is the photon Fock state. Combining Eqs. (3.6) and(3.7) with the action rules of the SU(4) superoperators on the basis states Eq. (3.12) gives a closed

28
solution of the laser equation Eq. (2.13). In general, this can be solved analytically or numericallywith standard methods. With generalized Dicke states, the numerical complexity with respect tothe atom number reduces from exponential to N 3 .

3.3

Calculating observables
Having established the procedure for determining the time evolution of ρ, it is now important

to describe how to calculate physical observables. We begin with the trace given by:Tr[ρ] =

X

m,m
CN/2,q3,0= 1,(3.14)m,q3

which is an invariant during evolution to represent probability conservation. Average values a
and a† a are found analogously. For the spin-operators, we provide the following examples up toquadratic order:hσjz i = 2Tr[Q3 ρ]/N,
hσjz σkz i = (4Tr[(Q23 − Σ23 )ρ] − N )/[N (N − 1)],

(3.15)

hσj± i = Tr[(M± + N± )ρ]/N,
hσj+ σk− i = Tr[V− (M− + N− )ρ − Q− ρ]/[N (N − 1)],
where j 6= k.
For coherence properties it is necessary to calculate products of operators evaluated at different times. Of particular interest are the first-order and second-order correlations, which can befound by applying the quantum regression theorem [95]:hihÔ1 (t + τ )Ô2 (t)i = Tr Ô1 eLτ [Ô2 ρ(t)] ,hihÔ1 (t)Ô1 (t+τ )Ô2 (t+τ )Ô2 (t)i = Tr Ô2 eLτ [Ô2 ρ(t)Ô1 ]Ô1 ,(3.16)

where eLτ [ρ] is the time propagation from Eq. (2.13) starting with the initial density matrix ρ. For
example, in order to obtain the first-order correlation of Ô1 and Ô2 , one takes Ô2 ρ(t) as an initialcondition, time evolves it for τ according to Eq. (2.13), applies Ô1 , and computes the trace. A

29similar procedure follows for the second-order correlation. In this way, field quantities, ha† (t+τ )a(t)iand ha† (t)a† (t + τ )a(t + τ )a(t)i are directly calculated. For spin-coherence, the required expressions
are:
N
X



hσj+ (t + τ )σk− (t)i = Tr (M+ + N+ )eLτ [(M− + N− )ρ(t)] ,

j,k=1
N
X

hσj+ (t)σj+0 (t + τ )σk− (t + τ )σk−0 (t)i =j,j 0 ,k,k0 =1
Tr V− (M− + N− )eLτ [V− (M− + N− )ρ(t)] .

3.4(3.17)

Transform to the angular-momentum basis representation
Although at this point we have provided a theoretical framework that is complete and pro-vides exact and efficient solutions to the general quantum master equation, it is often inconvenient
to work in the Pq,q3 ,σ3 representation of the density operator. For example, it can be a nontrivial procedure to characterize the many-body spin-state in this representation by quantifying thedegree of entanglement, which is derived from a functional (i.e. Tr[ρ log(ρ)]). For this reason, we
illustrate now the procedure for efficiently projecting the density operator from the SU(4) basis
representation onto the usual representation of density matrices formed from the Hilbert space basis vectors [143]. These Hilbert space basis vectors are specified by the angular momentum eigenket|S, M i, where S = N/2, N/2 − 1, ..., (1/2 or 0) is the total spin and M = −S, −S +1, . . . , S is the
spin-projection. Note that S also labels the symmetry of the states, e.g. S = N/2 corresponds tothe fully symmetrical Dicke states.
In order to illustrate how this projection is done, it is instructive for us to first examine
explicitly the N = 2 case where the Hilbert space is 4 dimensional. Two spins form a symmetric
triplet state and an antisymmetric singlet state, corresponding to total spin S = 1 and S = 0

30
respectively. In this case, the complete density matrix from Eq. (3.13) for given m, n is
h1, 1|
m,n
C1,1,0|1, 1i  C m,n
√|1, 0i  1/2,1/2,−1/2
2


m,n
|1, −1i  C0,0,−1
|0, 0i0h1, 0|m,n
C1/2,1/2,1/2
√
2
m,n
m,n
+C0,0,0
C1,0,02

h1, −1|m,nC0,0,1m,nC1/2,−1/2,1/2

√

2

m,n
C1/2,−1/2,−1/2√2m,nC1,−1,000h0, 0|

0



0

.


0


m,n
m,n

C1,0,0 −C0,0,02

Notice that the resulting matrix is block diagonal in the S = 1 and S = 0 subspaces (a 3 × 3 blockand a 1 × 1 block). In addition, the complex coefficients contributing to the matrix element for
Pm,n=|S, M ihS, M 0 | all satisfy q3 +σ3 = M and q3 −σ3 = M 0 . Finally, the trace is simply 1q3 =−1 C1,q3,01.
These results can be systematically extended to higher N . For any N , the density matrix isblock diagonal in S, with each block given by
=
ρm,n
S

X

m,n
0
DS,M,M
0 |S, M ihS, M |,(3.18)

M,M 0

where DS,M,M 0 are density matrix elements for the symmetry type S. There are nS ways for N spinsP
to construct the basis for each S, so that S (2S + 1)nS = 2N , i.e. the Hilbert space dimension [4].
To find nS , we note that |S, M i forms a basis of the (2S + 1)-dimensional irreducible representationof the SU(2) group. Determining nS is accomplished with the help of the Young tableau of theSU(2) group, where one can obtain the number of equivalent representations iteratively. Fig. 3.1(a)shows the Young tableau for the N = 4 case. A corresponding tabular method for evaluating nS
for any N is shown in Fig. 3.1(b), which contains about one half of Pascal’s triangle.
With this in mind, one can now derive a systematic algorithm for obtaining density matrixm,nm,nelements DS,M,M
0 given SU(4) expansion coefficients Cq,q3 ,σ3 . The procedure is outlined as follows.

For each layer of the pyramid [cf. Fig. 3.1(c)], one may start with a corner element (M and M 0
maximal) and fill out the matrix by successive application of the angular momentum loweringP−ˆˆ
operator Jˆ− = N
j=1 σj (noting that ρJ− = (U+ + V+ )ρ) to recursively fill out each row, and J− ρ(or Hermiticity of ρ) to fill out each column. The layers are filled upwards from the base, starting

31

Figure 3.1: (a) Young tableau for determining the irreducible representations contained in the
direct product representation of the SU(2) group for N = 4. The dimension for S = 2, 1, 0 is 5, 3, 1
respectively, and ns = 1, 3, 2, so the total Hilbert space dimension is 5 × 1 + 3 × 3 + 1 × 2 = 24 as
expected. (b) “Pascal’s triangle” to evaluate nS in an iterative way for any N (the case consideredin (a) is the fourth row down from the top). (c) Pyramid representation of the density operator
in the |S, M i representation for N = 4. Each layer of the pyramid represents the block matrix foreach S.

m,nm,nwith DN/2,N/2,N/2= CN/2,N/2,0
as the corner element of the lowest layer, and finding the corner

element of higher layers by Gaussian elimination from the trace constraint Eq. (3.14). In Appendix
C, we demonstrate explicit application to 3 atoms, with extrapolation to higher N straightforward.
Being able to express the density operator in the |S, M i representation makes easy the calculation of functionals, such as the purity Tr[ρ2 ], or the von Neumann entropy− Tr(ρ ln ρ) = −X

λj ln λj ,(3.19)jwhere λj are eigenvalues of ρ. The point is that, because the density matrix is block diagonal in
the |S, M i representation, we do not need to diagonalize the whole density matrix, which would
be a daunting task. Instead, we only need to diagonalize a series of bN/2c + 1 blocks of dimension
2S + 1.
Therefore, we have developed powerful methods to transform the density operator in theSU(4) basis representation to the |S, M i representation. This has enabled us to efficiently diagonalize the whole density matrix and thus provided complete information about the system, including
state information and functional properties of the density operator.

32

3.5

Further simplifying: U(1) symmetry and quantum jump
We have formulated and applied a SU(4) theory to numerically solve the quantum masterequation, which has reduced the exponential scaling of the problem to cubic in N . It turns outthat it is possible to further simplify the problem by applying U(1) symmetry and by using thequantum jump method.
Let us first discuss applying the U(1) symmetry. By inspecting the laser equation Eq. (2.13),
we find that it has global U(1) symmetry. By performing the following transformationâ → âe−iϕ,â† → â† eiϕ ,σ̂j− → σ̂j− e−iϕ,σ̂j+ → σ̂j+ eiϕ ,(3.20)

where ϕ is an arbitrary phase, Eq. (2.13) is invariant. As a result, if the initial state of the systemis phase invariant, the time evolution of the system will remain phase invariant. This will simplify
the expansion of the density matrix in Eq. (3.13), where five quantum numbers q, q3 , σ3 , m, n areneeded. In the case of phase-invariant states, the following condition must be satisfied
2σ3 = n − m .

(3.21)

Therefore, only four quantum numbers are needed. This reduces the numerical complexity withrespect to the atom number from N 3 to N 2 . This tremendous reduction should be comparedwith the full dimensionality of the Liouville space given by 4N . Therefore, it enables us to obtain
numerical solutions up to a few hundred pseudospins and tens of photons.
Now let us extend the SU (4) method by using quantum jump [24, 33, 104]—a standard
method in quantum optics. As will be shown, this allows us to simulate systems with a moderate
number of photons and atoms. The unusual feature here is that the unraveling is performed in
Liouville space rather than in Hilbert space, because it is in Liouville space that the SU (4) method
operates.
According to Eq. (3.13), the dimension of the density matrix grows as the square of thephoton number. This would impose great difficulties in numerical simulations of the laser region

33
due to the large number of photons. To overcome this difficulty, we unravel the master equation
into Monte-Carlo trajectories in Liouville space enabling us to eliminate the photon basis from thesimulation. The essential idea behind the method is that we are able to deduce the photon stateby keeping track of the total number of quanta Nq in the system.
The quantum Monte Carlo method decomposes the density operator evolution into a setof quantum trajectories where, between applications of random jumps into random channels, thesystem evolves under an effective Hamiltonian [24, 33, 104]. The random jumps are chosen withprobabilities such that the correct density operator evolution is obtained when an average is takenover trajectories. To construct a single trajectory, we first need to identify the jump operators.
In our problem, there are four decay channels: repumping, spontaneous emission, dephasing, andcavity decay. The corresponding jump operators Ji are
J1 ρ̂ = w

NX
(σj+ ρ̂σj− ) = wQ+ ρ̂ ,j=1

J2 ρ̂ = γNX(σj− ρ̂σj+ ) = γQ− ρ̂ ,j=1
NJ3 ρ̂ =

1 X z z(σj ρ̂σj )
2T2
j=1

=

1
(4M3 − 2Q3 − 2Σ3 )ρ̂ ,2T2

J4 ρ̂ = κaρ̂a† .

(3.22)

When a repumping quantum jump occurs, Nq increases by one. When a spontaneous emission ora cavity-decay quantum jump happens, Nq decreases by one. The dephasing quantum jumps leave
Nq unchanged. Therefore, during the evolution of a single trajectory, Nq is uniquely determined atevery time step by keeping track of the numbers of jumps of the different types. With knowledge of
Nq , the photon number does not need to be treated as an independent variable but is determinedfrom the number of excited atoms. In Appendix C we have shown that(s)(s)
Jˆz Pq,q= (q3 + σ3 )Pq,q
,
3 ,σ3
3 ,σ3
(s)
(s)
Pq,q
Jˆ = (q3 − σ3 )Pq,q,
3 ,σ3 z3 ,σ3(3.23)

34
where Jˆz =PNz
j=1 σj /2 is the collective spin operator. Therefore, the atomic state for a particular

fully symmetrical atomic basis state in terms of the number of excited atoms is |q3 + σ3 + N/2ihq3 −σ3 + N/2|. And thus the corresponding photon state ism n = Nq − (q3 + σ3 + N/2) Nq − (q3 − σ3 + N/2) .

(3.24)The simulation of jump times and decay channels is completely analogous to the wave-function
Monte Carlo method. The effective evolution of the system is governed by the master equation
excluding the above jump operators. As a result, under the effective evolution, the trace of thedensity operator is no longer conserved, but decreases as a function of time. This is analogous to
the decay of the norm of the wavefunction in the wave function Monte Carlo method. A jumpoccurs when the trace of the density operator is less than a random number uniformly distributedin the interval [0, 1]. When a decay occurs we stochastically determine the channel i into which thesystem decays according to the probability distribution,
Tr[Ji ρ̂]
Pijump = P4.
k=1 Tr[Jk ρ̂](3.25)

Finally, in order to get the density operator at each time step, an ensemble average of many
quantum trajectories is required. Then, various observables can be calculated according to Section
3.3. It is also worth noting that if one is only interested in the steady state density operator, atime average in the steady state can be applied instead of the ensemble average.

Chapter 4

A new perspective on steady state superradianceIn this chapter, we demonstrate a new perspective on the superradiant laser. We show thatthe cavity-mediated dissipative coupling between atoms in a superradiant laser acts to synchronize
their phases. With this new perspective, we investigate Ramsey spectroscopy performed on asynchronized ensemble of two-level atoms. We show that, in principle, with this synchronizedsystem it is possible to observe Ramsey fringes indefinitely, even in the presence of spontaneousemission and other sources of individual-atom dephasing. This could have important consequences
for atomic clocks and a wide range of precision metrology applications.

4.1

Synchronization of atomic dipoles
Synchronization is an emergent phenomenon that describes coupled objects spontaneously

phase-locking to a common frequency in spite of differences in their natural frequencies [121]. It
was famously observed by Huygens, the seventeenth century clock maker, in the antiphase synchronization of two maritime pendulum clocks [59]. Dynamical synchronization is now recognized
as ubiquitous behavior occurring in a broad range of physical, chemical, biological, and mechanicalengineering systems [16, 103, 121].
Theoretical treatments of this phenomenon are often based on the study of phase models [1,71], and as such have been applied to an abundant variety of classical systems, including thecollective blinking of fireflies, the beating of heart cells, and audience clapping [16, 103, 121]. The
concept can be readily extended to systems with an intrinsic quantum mechanical origin such as

36nanomechanical resonators [23, 50], optomechanical arrays [47], Josephson junctions [57, 130], andviscous collective atomic recoil lasing [110, 123]. When the number of coupled oscillators is large, it
has been demonstrated that the onset of classical synchronization is analogous to a thermodynamic
phase transition [135] and exhibits similar scaling behavior [68].
It turns out that in the superradiant laser regime, an effective synchronization model for thelaser equation Eq. (2.13) can be derived under the mean-field approximation [140]. According to
the model, we find that bad-cavity mediated coupling between atoms acts to synchronize the phaseof atomic dipoles.
First of all, in an extreme regime of bad-cavity, the cavity field is slaved to the atomic field√and can be adiabatically eliminated [13]. This requires κ  N g, w, T1−1 , T2−1 . In a recent 87 Sr
lattice clock experiment at JILA [9] (T1 ≈ 160s, T2 ∼ 1s), this condition can be satisfied very well.
Consider typical values of κ = 107 s−1 and C = 0.16, then g = 102 s−1 and ΓC = C/T1 = 10−3 s−1 .
√Even for N = 106 atoms, N g = 105 s−1 , and w < N ΓC = 103 s−1 . The role of the cavity fieldthen is to simply provide a source for a dissipative collective coupling for the atoms. The effective
evolution is given by a quantum master equation containing only atoms:
N
N 

X
X
dρ
i
1
z
−
ˆ
= − ∆ν
[σ̂j , ρ] + ΓC L[J ]ρ +L[σ̂jz ] ρ,wL[σ̂j+ ] + γL[σ̂j− ] +
dt
2
2T2
j=1

(4.1)

j=1

where ∆ν is the difference between the cavity frequency (in resonant with atomic frequency) andan external frequency (e.g. the frequency of a local oscillator). The collective decay rate can betaken to be small, i.e. ΓC  ΓS , where ΓS = (T1−1 + 2T2−1 )/2 is the single-atom decoherence rate.
This is because C is a dimensionless geometric cavity parameter that for real systems is typicallymuch less than 1. For a current generation 87 Sr optical clock experiment [9], ΓS ∼ 1s−1 , while ΓC
can be as small as 10−3 s−1 for C ≈ 0.16.
To derive the effective synchronization model, we first make the mean-field ansatz that theQdensity matrix in Eq. (4.1) is a product of density matrices for each atom, i.e., ρ = j ρj . We havechecked that this ansatz is accurate to O(1/N ). Plugging this ansatz into Eq. (3) in the paper, we

37
obtain the equation of motion for j-th atom by tracing out all other atoms:
dρj
dt

=

N 

X
1 ∆ν z1
1
L[σ̂jz ] ρj[σ̂j , ρj ] +wL[σ̂j+ ] + ( + ΓC )L[σ̂j− ] +
i 2
T1
4T2
j=1

ΓC −
ΓC
(σ̂j ρj − ρj σ̂j− )O +(ρj σ̂j+ − σ̂j+ ρj )O∗ ,+22
where O =(4.2)+
m6=j hσm i. Eq. (4.2) is self-consistent since the effect of all the other atoms is approx-

P

imated by a mean field O. O acts as an order parameter for the synchronization phase transition:in the absence of synchronization, or phase correlation between atoms, |O| = 0, while |O| > 0 in
the synchronized phase, breaking the U (1) symmetry of Eq. (4.1).
There are two factors at work in Eq. (4.2), the interaction with the mean field [resulting fromthe dissipative coupling in Eq. (4.1)] and quantum noises on individual atoms. We can see thisfrom the quantum Langevin equation for σ̂j+ ;d +1/T1 + 1/T2 + w + ΓC + ΓCσ̂ = i∆ν σ̂j+ −σ̂j +Oσ̂jz + F(t),
dt j
2
2

(4.3)

where F(t) is the quantum noise contributed by spontaneous emission, inhomogeneous dephasing,
repumping and collective decay. The quantum noises randomize the phase of individual atoms, andthus inhibit phase locking between atoms. To find the effect of the dissipative coupling between
atoms, we parameterize hσ̂j+ i as αj e−iφj and derive the equation of motion for φj ,dΓC hσ̂jz i X
φj = −∆ν +αm sin(φm − φj ).
dt
2 αj m(4.4)

Eq. (4.4) is equivalent to the well-known Kuramoto model [70, 120] for describing the phase synchronization. In the case of hσ̂jz i > 0, the coupling gives rise to phase attraction between atoms.

4.2

Probing the synchronization with Ramsey spectroscopy
Ramsey spectroscopy offers an unambiguous way to show the synchronization dynamics ofatoms in a superradiant laser. As shown in Fig. 4.1(b), Ramsey spectroscopy consists of three steps:(i) initial preparation of a coherent superposition between two quantum states, (ii) accumulationof a phase difference between the atoms and a local oscillator reference over an interrogation time

38
(a)

ߨ
ʹ

ȁ ‫ۄ‬
ȁ‫ۄ‬Interrogation time T

(b)
ߨ
ʹ

Figure 4.1: (a) Ramsey spectroscopy where atoms are coupled collectively to a cavity and pumped
individually with incoherent rate w. (b) Ramsey sequence showing preparation in state |g (pseudospins pointing down to the south pole of the Bloch sphere), the π/2 y-axis rotation from thesouth pole to the equator, precession around the equator, and second π/2 x-axis rotation, after
which the z-axis projection carries information about the cosine of the accumulated phase.

T , and (iii) mapping of the phase difference to a population readout. The population measurement
gives the well-known Ramsey fringes [105]. Due to the atomic decoherence, the visibility of Ramsey
fringes decay as a function of T .
It is worth to point out that conventional Ramsey spectroscopy is based on independentatom physics, with the role of a large number of atoms entering only through improving the signal
by statistical averaging. Here, however, Ramsey spectroscopy is performed over atoms that arecoupled collectively to a cavity. Conventional Ramsey spectroscopy is recovered by setting ΓC = 0
and w = 0 in Eq. (4.1), with the result that the Ramsey fringe visibility then decays exponentiallywith the single-atom decoherence rate ΓS [see Fig. 4.2(a)].
According to Eq. (4.4), the dissipative coupling between atoms will phase lock the atomic
phases. Thus, we expect that the decay of Ramsey fringe visibility will be slower than that ofconventional Ramsey spectroscopy. Indeed, Fig. 4.2(a) shows that the fringe visibility decays muchslower than that of conventional Ramsey spectroscopy under the same T1 and T2 decoherences,demonstrating the robustness to individual-atom decoherence. When compared to conventionalRamsey spectroscopy with independent atoms, the principal difference here is that strong spin-spincorrelations between atoms hσ̂j+ σ̂k− i (j 6= k) develop due to the dissipative coupling, as shown in
Fig. 4.2(b). This feature is a characteristic of phase-locking [72, 93]. After a brief initial transient

Population difference

39

1
(a)

0.5
0
-0.5
-1
0

1

2

3

4

5

6

Dynamical quantities

Time H2ΠDΝL
1.0
(b)

0.8
0.6
0.4
0.2
0.0

0

1

2

3

4

5

6

Time H2ΠDΝL

Figure 4.2: Calculations of Eq. (4.1) using SU(4) theory in Chapter 3 with N = 250, ΓC = 0.2/T1 ,T2 = T1 , and w = N ΓC /2. (a) Ramsey fringes with synchronized atoms (red solid line) versus T .
Conventional Ramsey fringes (blue dashed line) for the same T1 and T2 . (b) During the interrogation
time, the atomic inversion hσ̂jz i (blue dashed line), spin-spin correlation hσ̂j+ σ̂k− i (red solid line),hσ̂j+ σ̂k− i- hσ̂j+ ihσ̂k− i (red dotdashed line) and hσ̂j+ σ̂kz i/(hσ̂j+ ihσ̂kz i) (green dotted line).

evolution, the fringe fits well to an exponentially decaying sine function, i.e., Ae−λt sin ∆νt, where
λ is the decay rate of the fringe visibility and A is an amplitude (we derive this behavior later.)Intuitively, one may expect that in order to effectively phase-lock the atoms, it should benecessary for the dissipative coupling that provides rephasing to dominate over the ‘random-walk’
due to quantum noises that destroy phase correlations. Because of the all-to-all nature of theinteraction of atoms through the cavity mode, the dissipative coupling strength scales with N andis given by N ΓC /2. We show the effect of this in the inset of Fig. 4.3. For small atom number,
the individual quantum noises dominate over the rephasing, and the fringe envelope decays more
rapidly than in conventional Ramsey spectroscopy, i.e. λ > ΓS . As N increases, the dissipative
coupling increases, and we reach the regime λ < ΓS . For large atom number, we find λ approaches
ΓC . The ΓC limit arises from quantum fluctuations associated with the collective pseudospin decay

40Λ

through the cavity.

GS

Λ

GC
0

50 100 150 200 250

GS

N

GC
GS

0.2

0.4

0.6

0.8

1.0

w HNGC L

Figure 4.3: The decay rate λ of the visibility of Ramsey fringes at ΓC = 0.2/T1 and T2 = T1 as
a function of repumping for N = 200 and as a function of N for w = N ΓC /2 (Inset). The dots
are numerical solutions of Eq. (4.1), and the solid blue line is the semiclassical approximation forcomparison.

Therefore, due to the synchronization effect, the coherence time of atoms in steady-statesuperradiance is significantly longer than the single-atom coherence time and is essentially independent of single atom decoherences. A valid question to consider is: Why does the large incoherent
repumping rate w not destroy the synchronization? One may have thought that the repumping
would simply give rise to additional decoherence channels, on top of the usual T1 and T2 processes,
and cause the Ramsey fringe visibility to decay more rapidly. Somewhat paradoxically, repumping
is crucial for building up phase correlations among atoms. In Fig. 4.3, we show the effect of w on
the decay rates of the Ramsey fringe visibility λ. When the repumping rate is too small or too large
we find λ > ΓS , so that the system performs worse than conventional Ramsey spectroscopy. Thiscan be understood since the effective Kuramoto model Eq. (4.4) shows that population inversionof the pseudospins is a necessary condition for phase synchronization. The repumping strengthmust be large enough that there is more probability for the atoms to be in the excited state thanin the ground state. However, if the repumping rate is too large, the associated quantum noise
destroys the phase correlations before they can develop. As has also been seen in the case of thesuperradiant laser [93], the most coherent system is realized at an intermediate pump strength.
We emphasize that there are three timescales here. At short times, quantum correlations

41
develop as the atoms synchronize. This can be seen in the initial transient part of the evolution of
the observables shown in Fig. 4.2(b), and is characterized by the timescale w−1 . This phase-locking
time should be less than the atomic coherence time Γ−1
S in order to observe high-visibility fringes.
There is also a long timescale provided by the collective decay time Γ−1
C . It is important to operate
in the parameter regime in which w  ΓS  ΓC .
It is possible to develop an accurate semiclassical approximation to get analytical expressionof λ. The approximation is valid in the case of large numbers of atoms. Taking advantage of the factthat all expectation values are symmetric with respect to atom exchange, we find from Eq. (4.1),
d +
Γt
ΓC
hσ̂j i = i∆νhσ̂j+ i − hσ̂j+ i +(N − 1)hσ̂j+ σ̂kz i,
dt
2
2

(4.5)

where j 6= k and Γt = 2ΓS + w + ΓC is the total decay rate of the atomic coherence. We firstpoint out that instead of calculating the population difference measured at the end of the Ramsey
sequence, it is equivalent to calculate 2Im[hσ̂j+ i] just before the second π/2 pulse. The decay rate ofhσ̂j+ i during the interrogation time T is therefore the same as that of the Ramsey fringe visibility.
As seen in Fig. 4.2(b), the quantities α(t) = hσ̂j+ σ̂kz i/(hσ̂j+ ihσ̂kz i) and hσ̂jz (t)i rapidly approach steadystate on the short timescale of the phase-locking, w−1 . We therefore substitute the steady-statevalues αss and hσ̂jz iss into Eq. (4.5). This produces the exponentially decaying sine function solution
noted earlier with decay constantλ=1Γt − (N − 1)ΓC αss hσ̂jz iss .
2

(4.6)

Furthermore αss ≈ 1, see Fig. 4.2(b). At the level of mean-field, hσ̂jz iss ≈ Γt /(N − 1)Γc giving thetrivial result λ = 0. It is therefore necessary to develop a semiclassical expression for hσ̂jz iss that
goes beyond mean-field.
To find hσ̂jz iss , we employ the cumulant approximation method [91, 92, 93]. Expectation
values of the atoms are expanded in terms of hσ̂jz i and hσ̂j+ σ̂k− i. Their equation of motion can then42
be found from Eq. (4.1) in the paper,


d z
1
hσ̂j i = −(ΓC + ) hσ̂jz i + 1 − w hσ̂jz i − 1 − 2ΓC (N − 1)hσ̂j+ σ̂k− i
dt
T1

d + −
ΓC z
hσ̂j σ̂k i ≈ −Γt hσ̂j+ σ̂k− i +hσ̂j i 1 + hσ̂jz i + ΓC (N − 2)hσ̂j+ σ̂k− ihσ̂jz i.
dt2
where we have factorized hσ̂jz σ̂kz i ≈ hσ̂jz i2 and hσ̂j+ σ̂k− σ̂lz i ≈ hσ̂j+ σ̂k− ihσ̂jz i. hσ̂jz iss can then be foundby setting the time derivatives to zero, and the resulting algebraic equations form a close set andcan be solved exactly.
Fig. 4.3 compares λ from the semiclassical expression with the quantum master equation
solution, showing good agreement over the full range of pumping rates.

4.3

Robustness of the synchronization
In the above discussion, the atom-cavity coupling strength g is assumed to be identical for

all atoms. In principle, this could be achieved by trapping the atoms at the antinodes of the cavity
mode by an optical lattice in the Lamb-Dicke regime. Even so, there is still weak inhomogeneity ofg. In this section, we show that the synchronization of atoms is robust against spatial variationsof the atom-photon coupling strength, g.
We calculate the decay rate of the visibility of Ramsey fringes, λ, under an inhomogeneous
g factor. As shown in Fig. 4.4(a), λ will increase slowly when the spread in g (∆g) increases.
However, this can be interpreted as a reduced effective atom number. Fig. 4.4(b) shows that given
the inhomogeneity of g, λ will approach ΓC simply by increasing the number of atoms. This isprecisely due to the fact that the collective dissipative coupling strength is given by N ΓC /2. Aslong as the dissipative coupling can be made to dominate the individual-atom quantum noises,
which can be done by increasing N , the spread in g does not play an important role.

4.4

Conditional Ramsey spectroscopy with synchronized atoms
The performance of conventional Ramsey spectroscopy is limited by the atomic coherencetime, which causes decay of the fringe visibility as a function of T . Due to this decay, an optimal

43
2.0

3.5

(a)

1.8

2.5

1.4

Λ HGC L

Λ HGC L

1.6
1.2

2.0

1.0

1.5

0.8

1.0

0.6
0.0

(b)

3.0

0.1

0.2

0.3

0.4

Dg  g

0.5

0.5

100

150

200

250

N

Figure 4.4: (a) The decay rate of the visibility of the Ramsey fringes as a function of standard
deviation of the g factor at ΓC = 0.2/T1 , T2 = T1 , and w = N ΓC /5 for N = 250 atoms. (b) The
decay rate of the visibility of the Ramsey fringes as a function of number of atoms at ΓC = 0.2/T1 ,T2 = T1 , w = N ΓC /5, and ∆g/ḡ = 0.2. ΓC in both (a) and (b) is defined using the mean of the gdistribution, ḡ. Statistics are accumulated by sampling g from a Gaussian distribution. The dots
are average values for 100 runs, and error bars represent the standard deviations.

strategy is typically used that involves setting T to be of the order of the coherence time, and fillingup the total measurement interval τ by repeated Ramsey cycles [54]. This gives an uncertainty in√the frequency difference between the atoms and local oscillator that scales as 1/( N τ ), with the√
N coming from the quantum projection noise at each readout. This scaling τ −1/2 is much worse
than the fundamental Fourier limit τ −1 .
There are two paths to improving on the standard limit for Ramsey spectroscopy, apartfrom simply increasing N . Firstly, the projection noise can be reduced by preparing spin-squeezed
states [66, 132, 133]. Pursuing this direction, there have been numerous efforts to produce spinsqueezing in various physical situations [3, 19, 34, 41, 76, 107, 115]. It is worth pointing outthat entangled states are often fragile and sensitive to decoherence processes, which may limit
their potential for providing significant improvements to the sensitivity [2, 55]. Secondly, one
can increase the coherence time of atoms. One approach has been to increase the dephasingtime of magnetically and optically trapped atomic ensembles by spin self-rephasing induced by theexchange interaction between two identical particles [27, 67]. In recent lattice clock experiments [9],the atomic dephasing time T2 has been pushed to ∼1s. Even if further technical improvements are44
made, there is a fundamental upper limit to the atomic coherence time provided by the lifetime, T1 ,of the long-lived excited clock state (∼160s for 87 Sr) [100].
In this section, we propose that Ramsey Spectroscopy using synchronized atoms is potentially more robust against decoherence [140]. We show that synchronized atoms undergo only acollective quantum phase diffusion. However, the collective phase can be continuously monitored byobserving the cavity output field. Consequently, this system provides a kind of conditional Ramsey
spectroscopy, conditioned on the cavity output, where fringes of high visibility may be observed
indefinitely.
We have shown that with synchronized atoms the Ramsey fringe decays extremely slowly
with rate ΓC . This results from a statistical average of independent trials. The decay of the fringe
visibility is really due to the averaging itself, as we will now see. In each trial, the quantum phaseis diffusing as a function of interrogation time. This means that as time goes on, different trials
begin to add out of phase, and so the fringe visibility decays.
This motivates us to consider the properties of a single experimental run, where the behavior
is qualitatively different. Although in a single run, the fringe undergoes a quantum phase diffusion, it does so with non-decaying visibility. This quantum phase diffusion has a simple physicalinterpretation in terms of quantum measurements. Since the cavity field follows the atomic coherence through adiabatic elimination, measuring the phase of the cavity output field, for example byhomodyne measurement, is equivalent to a continuous non-destructive measurement on which information is gathered about the evolving collective atomic phase. The back-action of this measurement
introduces fluctuations that cause the collective atomic phase to undergo a random-walk [11].
We demonstrate this in Fig. 4.5(a), where we show a typical Ramsey fringe for a single
experimental trial by using the method of quantum state diffusion [137, 136] to yield conditional
evolution of the system subject to continuous measurements of the cavity field. The phase diffusionof the synchronized atoms is evident from the phase fluctuation of the Ramsey fringe. To findthe phase diffusion coefficient, Fig. 4.5(b) shows the statistics of the positions of the zero crossingsof the fringe for 4000 trials. They fit well to Gaussian distributions with variance given by T ΓC ,

Population difference

45

1

(a)

0.5
0
-0.5
-1
0

1

2

3

4

5

6

7

Time H2ΠDΝL
Statistics

(b)

0

0.5

1

1.5

2

Time H2ΠDΝL
Figure 4.5: Quantum state diffusion calculations of conditional Ramsey fringes subject to continuous homodyne measurement of the cavity output field for N = 10 and w = N ΓC /2. The blue
dashed lines are the ensemble average for reference. (a) A typical Ramsey fringe for a single experimental trial (red solid line). (b) Histograms are the statistics of the positions of zero crossings ofeach fringe for 4000 trials. The blue solid lines are fitted Gaussian distributions with variance ofT ΓC centered on the zero crossing of the ensemble average.

clearly demonstrating that it is a diffusion process and that the diffusion coefficient is√ΓC . Notethat this is the same mechanism that also sets the quantum limited linewidth in a superradiantlaser to be ΓC [93], observed here in the time rather than frequency domain.
We should emphasize that the quantum phase diffusion does not itself provide a fundamental
limit to the performance of conditional Ramsey spectroscopy, since the collective atomic phase canbe tracked by measuring the light output from the cavity. This opens up the exciting possibilityof observing conditional Ramsey fringes (meaning an experimental trial conditioned on the measurement record of the output field) of near maximum fringe visibility for as long as the atoms
can be stored, even in the presence of T1 and T2 processes. Of course a practical limit is also set
by the length of time for which the local oscillator can remain phase coherent. In principle, if46experimentally achieved, this work could lead to dramatic advances in the sensitivity of Ramsey
spectroscopy, since the entire measurement interval could then be used to determine frequency at
the Fourier limit.

Chapter 5

Quantum synchronization of two ensembles of superradiant atoms

Recently, there has been increasing interest in exploring manifestations of synchronizationin the quantum realm [35, 48, 53, 56, 72, 73, 75, 86, 87, 127, 128, 147]. Connections between
quantum entanglement and synchronization have been revealed in different systems [72, 73, 87, 147].
Based on Heisenberg uncertainty principle, quantum synchronization measures have been applied
to coupled optomechanical cells [87]. The effect of quantum noise has been shown to reduce thesynchronization region of a driven self-sustained oscillator [127]. Since the phenomenon is inherently
non-equilibrium, all of these systems share the common property of competition between coherent
and incoherent driving and dissipative forces.
In this chapter, we propose a modern-day realization of the original Huygens experiment [59].
We consider the synchronization of two active atomic clocks coupled to a common single-mode optical cavity. It has been predicted that in the regime of steady-state superradiance [10, 12, 91, 92, 93]a neutral atom lattice clock could produce an ultracoherent optical field with a quality factor (ratioof frequency to linewidth) that approaches 1018 . We show that two such clocks may exhibit adynamical phase transition [30, 52, 61, 122] from two disparate oscillators to quantum phase-locked
dynamics. The onset of synchronization at a critical pump strength is signified by an abruptly
increased relative phase diffusion that diverges in the thermodynamic limit. Besides being of fundamental importance in nonequilibrium quantum many-body physics, this work could have broad
implications for many practical applications of ultrastable lasers and precision measurements [93].

48

5.1

Setup

Figure 5.1: Two ensembles of driven two-level atoms coupled to a single-mode cavity field. The
atoms in ensemble A are detuned above the cavity resonance (dashed line). Ensemble B contains
atoms detuned below the cavity resonance by an equivalent amount.

The general setup is shown schematically in Fig. 5.1. Two ensembles, each containing N twolevel atoms with excited state |ei and ground state |gi, are collectively coupled to a high-quality
optical cavity. The transition frequencies of the atoms in ensembles A and B are detuned fromthe cavity resonance by δ/2 and −δ/2 respectively. This could be achieved by spatially separating
the ensembles and applying an inhomogeneous magnetic field to induce a differential Zeeman shift.
The atoms in both ensembles are pumped incoherently to the excited state as in the steady-statesuperradiance [10, 12].
This system is described by the Hamiltonian in the rotating frame of the cavity field:~δ ˆz~g(JA − JˆBz ) + (â† JˆA− + JˆA+ â + â† JˆB− + JˆB+ â) ,
(5.1)
2
2
PN
P
−
z
z
ˆ−
where g is the atom-cavity coupling. Here JˆA,B= 21 Nj=1 σ̂(A,B)j are thej=1 σ̂(A,B)j and JA,B =Ĥ =collective atomic spin operators, written in terms of the Pauli operators for the two-level system−+zσ̂(A,B)jand σ̂(A,B)j= (σ̂(A,B)j)† .

In addition to the coherent atom-cavity coupling, incoherent processes are critical and include:
the cavity intensity decay at rate κ, the pump rate w, the free-space spontaneous emission rateγ, and a background dephasing of the |ei–|gi transition at rate T2−1 . The total system is thendescribed using a master equation for the reduced density operator ρ:
N 

X X
dρ
1
1
= [Ĥ, ρ] + κL[â] ρ +γL[σ̂T−j ] + wL[σ̂T+j ] +L[σ̂Tz j ] ρ .
dt
i~
2T2
T =A,B j=1

(5.2)

49
Again, the regime of steady-state superradiance is defined by the cavity decay being much
faster than all other incoherent processes [10, 12, 91, 93]. In this regime, the cavity can be adiabatically eliminated, resulting in a field that is slaved to the collective atomic dipole of the twoensembles of atoms:â ' −

ig ˆ−
ig ˆ−
JA −
J .
κ + iδ
κ − iδ B(5.3)

For small detuning on the scale of the cavity linewidth, δ  κ, Eq. (5.3) reduces to â ' −ig Jˆ− /κ,where Jˆ− = JˆA− + JˆB− is the total collective spin-lowering operator. In this limit, the net effect of
the cavity is to provide a collective decay channel for the atoms, with rate γC = Cγ. This collective
decay should be dominant over other atomic decay processes, i.e., N γC  γ, T2−1 , so that the time
evolution is effectively given by a superradiance master equation containing only atoms:
N

X
δ
dρ
+
+
]) ρ.
] + L[σ̂Bj=[JAz − JBz , ρ] + γC L[Jˆ− ] ρ + w(L[σ̂Aj
dt
2i~

(5.4)

j=1

With this system we naturally provide the three necessary ingredients for quantum synchronization; a controllable difference between the oscillation frequencies of two mesoscopic ensembles, adissipative coupling generated by the emission of photons into the same cavity mode, and a driving
force produced by optical pumping.

5.2

Mean-field picture
It is possible to understand a great deal about the synchronization of the proposed systemwith a simple mean-field treatment. Starting from Eq. (5.4), we make the mean-field ansatz that theNdensity matrix is a product of density matrices for each atom, i.e., ρ = j ρAj ⊗ ρBj . Substituting
this ansatz into the master equation, we obtain the equation of motion for the j-th atom in eachensemble by tracing out all other atoms, i.e.
d
ρAj
dt
d
ρBj
dt

δ z
γC −
γC
+
−
+
+
= −i[ σ̂Aj
, ρAj ] + wL[σ̂Aj]ρAj +(σ̂Aj ρAj − ρAj σ̂Aj
)O +(ρAj σ̂Aj− σ̂AjρAj )O∗ ,
2
2
2
δ z
γC −
γC
+
−
+
+
= −i[− σ̂Bj
, ρBj ] + wL[σ̂Bj]ρBj +(σ̂Bj ρBj − ρBj σ̂Bj
)O +(ρBj σ̂Bj− σ̂BjρBj )O∗ ,222

where O =++j hσ̂Aj i + hσ̂Bj i is the mean-field.

P50
The above mean-field equation enables us to obtain an analogous two-component phase modelto the well-known Kuramoto model for studying synchronization. To this end, we first derive theequation of motion for hJˆA− i and hJˆB− i,





 

γC ˆz
γC ˆz
−
−
−
w
δ
ˆ
ˆ
ˆ
d hJA i
2 hJA ihJA i −i 2 − 2 + 2 hJA i hJA i

=M

=
.
dt
γC ˆz
γC ˆz
−
−
−
w
δ
ˆ
ˆ
ˆ
hJB i
i 2 − 2 + 2 hJB i
hJB i
hJB i
2 hJB i(5.5)

By parameterizing hJˆA− i as α exp(iθA ) and hJˆB− i as β exp(iθB ), we havedδ γC β ˆzθA = − +hJ i sin(θB − θA ),
dt
2
2 α A

d
δ γC α ˆzθB = +
hJ i sin(θA − θB ).
dt
2
2 β B

(5.6)

From Eq. (5.6), we find a phase attraction between the two ensembles of atoms that exists as longz i is positive. It is theas the atoms are repumped so that the atomic population inversion hJˆA,B

phase attraction resulting from the dissipative coupling that gives rise to the synchronization.
We may identify the synchronization transition within the mean-field theory. In the synchronized phase, the frequencies of the atoms are locked to the cavity frequency, leading to a
non-oscillatory steady state in Eq. (4.2). In this case, it is easy to calculate the steady state valuez i to be (w 2 + δ 2 )/(2wγ ). However, as δ > w, the matrix M in Eq. (5.5) acquires complex
of hJˆA,B
C(non-real) eigenvalues, indicating oscillations and therefore a precession of the relative phase. We
refer to this as the unsynchronized parameter region.
The photons emitted by the cavity provide directly measurable observables. Synchronization
is evident in the properties of the photon spectra. In the case of two independent ensembles in
the unsynchronized phase, each ensemble radiates photons at its own distinct transition frequency.
This leads to two Lorentzian peaks that are typically well-separated. In the synchronized phase,
all of the atoms radiate at a common central frequency resulting in a single peak.

5.3

Quantum signatures of the synchronization
Although the mean-field treatment correctly predicts the phase transition, the associatedquantum noises in the open quantum system and quantum correlations between the pseudospins

51
are absent. These are the aspects that distinguish quantum synchronization from classical synchronization.
To solve this problem quantum mechanically, we use a semiclassical approximation thatis applicable to large atom numbers. Cumulants for the expectation values of system opera±z
tors {σ̂(A,B)j, σ̂(A,B)j} are expanded to second order [91, 93]. All expectation values are symmetric+ −+ −with respect to exchange of atoms within each ensemble, i.e. hσ̂Bi
σ̂Bj i = hσ̂B1
σ̂B2 i, for all i 6= j.
±
Due to the U (1) symmetry, hσ̂(A,B)ji = 0. Therefore, all nonzero observables can be expressed in
+
−
z
z
z
terms of hσ̂(A,B)ji, hσ̂(A,B)iσ̂(A,B)ji, and hσ̂(A,B)iσ̂(A,B)ji. Expectation values involving only one en-semble are the same for both ensembles and for these cases we omit the superfluous A,B subscripts.
The equations of motion can then be found from Eq. (5.4):
d z
hσ̂ i
dt 1

= −γC (hσ̂1z i + 1) − w (hσ̂1z i − 1)+ −σ̂B1 i + c.c ,− 2γC (N − 1)hσ̂1+ σ̂2− i − γC N hσ̂A1

d + −hσ̂ σ̂ i
dt 1 2= −(w + γC )hσ̂1+ σ̂2− i +

(5.7)

γC
(hσ̂1z σ̂2z i + hσ̂1z i)
2+ γC (N − 2)hσ̂1z ihσ̂1+ σ̂2− iγC+ −σ̂B1 i + c.c ,N hσ̂1z i hσ̂A1
2
γC
+ −
z
z
σ̂B1 i += −(w + γC − iδ)hσ̂A1(hσ̂A1σ̂B1i + hσ̂1z i)
2+ −σ̂B1 i + hσ̂1+ σ̂2− i ,+ γC (N − 1)hσ̂1z i hσ̂A1+

d + −hσ̂ σ̂ i
dt A1 B1

(5.8)

(5.9)

describing population inversion, spin-spin coherence within each ensemble, and correlation between
ensembles, respectively. In deriving Eq. (5.8) and (5.9), we have dropped third order cumulants.
zz
We also factorize hσ̂(A,B)iσ̂(A,B)ji ≈ hσ̂1z i2 , which we find to be valid outside the regime of very

weak pumping where a non-factorizable subradiant dark state plays an important role [91]. Aftermaking these approximations, Eq. (5.7) to (5.9) form a closed set of equations. The steady stateis found by setting the time derivatives to zero and the resulting algebraic equations can be solved
exactly.
We have validated Eq. (5.7) to (5.9) by comparison with exact numerical solutions of the52quantum master equation based on applying the SU (4) group theory (see Chapter 3). As shownin Fig. 5.2, the comparison is very good even for small system size of 30 atoms in each ensemble.
Due to the presence of multiple ensembles it is difficult to implement exact calculations for morethan about 50 atoms.

Figure 5.2:
Steady state value of population inversion hσ̂1z i, spin-spin coherence within each+ −+ −ensemble hσ̂1 σ̂2 i and correlation between ensembles hσ̂A1
σ̂B1 i as a function of detuning. Red dots
are exact numerical calculation by applying the SU (4) group theory. Blue lines are semiclassicalresults based on solutions to Eq. (5.7) to (5.9).

In order to calculate the photon spectrum, we employ the quantum regression theorem [95]to obtain the two-time correlation function of the light field, hâ† (τ )â(0)i, where time 0 denotes
an arbitrary time-origin in steady-state. In the limit δ  κ, according to Eq. (5.3), the phasediffusion of the atoms and light are the same, i.e. hâ† (τ )â(0)i ∼ hJˆ+ (τ )Jˆ− (0)i. We begin by+−+−
deriving equations of motion for hσ̂A1(τ )σ̂B1(0)i and hσ̂B1(τ )σ̂B2
(0)i:







+
−
+
−
hσ̂
(τ
)σ̂
(0)i
X
Y
hσ̂
(τ
)σ̂
(0)i
d  A1
B1
B1
 1
  A1


= 

,
dτ
2
+
−
+
−
hσ̂B1
(τ )σ̂B2
(0)i
Y X∗
hσ̂B1
(τ )σ̂B2(0)i(5.10)

53
where X = γC (N − 1)hσ̂1z (0)i − γC − w + iδ , Y = γC N hσ̂1z (0)i . We have systematically factorized:+−+−
hσ̂1z (τ )σ̂A1(τ )σ̂B1(0)i ≈ hσ̂1z (0)ihσ̂A1(τ )σ̂B1(0)i ,+−+−
hσ̂1z (τ )σ̂B1(τ )σ̂B2(0)i ≈ hσ̂1z (0)ihσ̂B1(τ )σ̂B2(0)i .

(5.11)+−+−
Similarly, one finds that hσ̂A1(τ )σ̂A2(0)i and hσ̂B1(τ )σ̂A1(0)i satisfy the same equation of motion as+−
Eq. (5.10). The solution of this coupled set is straightforward and shows that both hσ̂A1(τ )σ̂B1(0)i

and thus also hâ† (τ )â(0)i evolve in proportion to the exponential:

 

q
1
exp −
w + γC − (N − 1)γC hσ̂1z i − (N γC hσ̂1z i)2 − δ 2 τ ,2(5.12)

which we parametrize by exp [−(Γ + i∆)τ /2], where Γ represents the decay of the first-order correlation and ∆ the modulation frequency. Laplace transformation yields the photon spectrum which
consists of Lorentzians of halfwidth Γ/2 centered at frequencies ±∆/2.

DNΓc

1.5
1
0.5
0
0

0.5

1

1.5

∆NΓc
Figure 5.3: Steady-state relative phase precession for two ensembles as a function of detuning at
w = N γC /2 for N = 100 (blue dashed line), N = 500 (purple dot dashed line) and N = 106 (redsolid line). The straight dotted line is δ = ∆.

The importance of the two-time correlation function is that it provides direct access to thecorrelated phase dynamics of the two ensembles. The parameter ∆ physically represents the precession frequency of the phase of the collective mesoscopic dipoles with respect to one another. In
Fig. 5.3, we show ∆ as a function of δ at w = N γC /2 for several values of N . For large detuning,∆ approaches δ, indicating that the dipoles precess independently at their uncoupled frequency.
Below a critical δ, we find ∆ to be zero, indicating synchronization and phase locking.

54
There is not yet a generally accepted meaning of the term ‘quantum synchronization’. In
general, we believe that in the quantum synchronization case, quantum mechanics introduces two
effects. The first is the quantum noise, which is due to the oscillator gaining or losing individual
quanta. This can be equivalently viewed as the fundamental fluctuations due to the interactionof the system with its environment. The second effect is that the oscillators can be quantum
mechanically correlated.

Figure 5.4: (a) Nonequilibrium phase diagram of the quantum synchronization represented by Γ (inunits of γC ) on the w-δ parameter plane, where the dissipative coupling N γC (N = 104 ) is fixed.
An abrupt peak is observed at the boundary between the synchronized and unsynchronized phases.
(b) As for (a) but on the w-N γC parameter plane.

We first discuss the quantum noise aspect. The fact that this system undergoes a synchro-55
nization transition that is fundamentally quantum mechanical is evident in the manifestation ofquantum noises associated with the repumping process and the cavity decay. It is shown that quantum noises result in the phase diffusion of the collective atomic dipoles (see Chapter 4), setting theultimate quantum limit for the linewidth of the superradiant laser to be γC . They also give riseto the relative quantum phase diffusion between the two ensembles, as quantified by the linewidth
Γ of the Lorentzian peak(s). Therefore, Γ/γC can be a dimensionless measure of the degree ofthe synchronization between the two ensembles as normalized by the phase diffusion within eachensemble.
This system has three independent control variables; the detuning δ, the dissipative coupling
N γC and the pumping w, so we show Γ/γC on the w-δ parameter plane in Fig. 5.4(a) and on the
w-N γC parameter plane in Fig. 5.4(b). In the region of no quantum correlation, the quantum noisedue to pumping destroys the coherences between spins faster than the collective coupling inducedby the cavity field can reestablish them. Therefore the mesoscopic dipole is destroyed and theobserved spectra are broad. In both the synchronized and unsynchronized regions, spins withineach ensemble are well-correlated so that the corresponding Lorentzian peaks have ultranarrow
linewidth (∼ γC ). As is apparent in Fig. 5.4(a), the two ensembles cannot be synchronized when
N γC < δ since then the coherent coupling is not sufficient to overcome the relative precession thatarises from the detuning.
For strong coupling, N γC > δ, the synchronization transition occurs as the pump rate passes
a critical value. The two phases on either side of the critical region are abruptly separated. As
one approaches the synchronized phase from the unsynchronized one by variation of either δ or w,
the linewidth increases rapidly, showing amplification of the effect of quantum noise in vicinity of
the critical point. After passage of the critical region, the linewidth drops rapidly, leading to rigid
phase locking between the two collective dipoles.
We emphasize that the synchronization dynamics shown in Fig. 5.3 and 5.4 is a dynamicalphase transition [30, 52, 61, 122] that is reminiscent of a second-order quantum phase transition.
To capture features of the quantum criticality, we numerically study the finite size scaling behavior.

56

Figure 5.5: Finite size scaling behavior of the quantum criticality for δ = N γC /2. For N → ∞, thecritical pump rate is wc = δ. The red dots show the offset between the critical pump rate wN for
finite N and wc . The blue squares show Γ (in units of γC ) at wN . Both exhibit linear scalings on
the log-log plot.

Fig. 5.5 shows both the critical pump rate wN for finite N and the corresponding Γ at wN . The
scaling laws of (wN − wc )/wc ' N −0.34 and Γ/γC ' N 0.66 can be identified.
In Hamiltonian systems, a quantum phase transition results from the competition between
two noncommuting Hamiltonian components with different symmetries on changing their relative
weight. The transition between the two distinct quantum phases can be identified from the nonanalytical behavior of an order parameter, and the scaling behavior of certain correlation functions
that diverge at the critical point. By analogy, the synchronization phase transition is caused by
the competition between unitary dynamics that is parametrized by δ and enters asymmetrically forthe two ensembles, and driven-dissipative dynamics parametrized by γC that is symmetric. The
order parameter ∆ is zero in the synchronized phase and non-zero in the unsynchronized phase.
The critical behavior is encapsulated by the divergence of the relative quantum phase diffusion.
It should be emphasized that the treatment given here is quite different to the typical analysis
since the transition is embodied by the characteristic features of the two-time correlation functions,rather than the behavior of an energy gap or correlation length.
In the thermodynamic limit, simple expressions for hσ̂1z i to leading order in 1/N can be57
obtained:hσ̂1z i =









w
2N γC ,

if δ = 0

w2 +δ 2

if 0 < δ < w ,2wN γC ,





w

N γC ,(5.13)

if δ ≥ w

where w should be such that hσ̂1z i < 1. A critical point at wc = δ can be found by substituting
Eq. (5.13) into Eq. (5.12). In particular, ∆ = (δ 2 −w2 )1/2 in the unsynchronized phase, which showsan analogous critical exponent to that of a second-order quantum phase transition, i.e., β = 1/2.
As a second point, we provide in-depth detail to quantify the effect of correlations. We do
this by a calculation of what is known as the quantum discord. It is worth to point out that theconnection between synchronization phenomena and quantum correlations are subtle and subjectof current research. Quantum discord can generally be thought of as a measure of the degree of
nonclassical correlations between two subsystems. Fig. 5.6 shows the degree of “quantumness” of
the correlation presented in our system. We also see that the cumulant expansion provides a good
description of the important quantum correlations in the system.

Figure 5.6: Classical correlation and quantum discord between one atom in ensemble A and one
atom in ensemble B. The blue dashed line is based on semiclassical approximation described in
the paper. The red dots are exact numerical calculations. There are 30 atoms in each ensemble.
The comparison between these two lines indicates the “quantumness” of correlations present in oursystem.

We have presented a system that exhibits quantum synchronization as a modern analogue ofthe Huygens experiment but is implemented using state-of-the-art neutral atom lattice clocks of the58highest precision. It sheds lights on the ongoing efforts of understanding the quantum aspects ofsynchronization. It will be intriguing in future work to study the many possible extensions that are
inspired by these results, such as the effect of an atom number imbalance on the synchronization
dynamics, and the sensitivity of the phase-locking to external perturbation.

Chapter 6

Supercooling of atoms based on steady-state superradianceIn the previous discussion of steady-state superradiance, the external motions of atoms wereassumed to be frozen. Therefore, the mechanical effects of cavity-photon atom interaction was
neglected. In this chapter, we investigate the motional effects on atoms in steady-state superradiance. We focus on how the novel features of steady-state superradiance can be used for efficientcooling of the atoms. We demonstrate that when atomic dipoles are synchronized in the regimeof steady-state superradiance, the motion of the atoms may be subject to a giant frictional force
leading to potentially very low temperatures. The ultimate temperature limits are determined bya modified atomic linewidth, which can be orders of magnitude smaller than the cavity linewidth.
The cooling rate is enhanced by the superradiant emission into the cavity mode allowing reasonablecooling rates even for dipolar transitions with ultranarrow linewidth.

6.1

Cavity cooling in the strong coupling regime
The discovery of laser cooling [129] has enabled new areas of quantum gas physics and quan-tum state engineering [8]. Laser cooling is an essential technology in many fields, including precision
measurements, quantum optics, and quantum information processing [83, 126, 131]. Doppler laser
cooling [45, 134] relies on repeated cycles of electronic excitation by lasers followed by spontaneous
relaxation, reaching temperature limits determined by the atomic linewidth. Only specific atomic
species can be Doppler cooled because they should possess an internal level structure that allows
for closed cycling transitions.

60
Cavity-assisted laser cooling [32, 108] utilizes the decay of an optical resonator instead of
atomic spontaneous emission for energy dissipation. It is based on the preferential coherent scattering of laser photons into an optical cavity [51, 125]. Temperatures that can be achieved inthis way are limited by the cavity linewidth. Since the particle properties enter only through thecoherent scattering amplitude, cavity-assisted cooling promises to be applicable to any polarizable
object [63, 78, 79, 89, 90, 96, 97, 116, 139], including molecules [79, 97] and even mesoscopic systems
such as nanoparticles [63, 96]. The requirement of cavity cooling [51, 125] is that the single-atomcooperativity parameter C should be much greater than 1, since we want the atom to scatter lightmostly into the cavity rather than the free space.
In this section, I provide a simple model to illustrate the underlying mechanism of cavity
cooling. Consider that an atom is confined within a standing-wave resonator and is illuminatedby a transverse laser (see Fig. 6.1). A dipolar transition of the atom couples far-detuned with thelaser fields and scatters photons from the laser into the cavity. Therefore, the cavity is effectively
‫ݒ‬ൌ

‫݌‬
݉

ߢ
߱௅

Lorentzian lineshape

߂

߱௅ ߱௖

ߢ Τʹ ߨ݂ ଶ ൅ ߢ Τʹ ଶ݂Figure 6.1: An atom with velocity v is moving freely along the cavity axis. The atom coherently
scatters the photons of the transverse laser into the cavity mode. The frequency of the laser ωL is
red detuned from the cavity resonance frequency ωc by ∆c . Due to Doppler effect, the atom will
scatter more photons into the direction along which it moves.

61
pumped by laser photons. The effective Hamiltonian of the cavity-atom system is [117]
Ĥ =

p̂2
Sd
+ ~ (â + â† ) cos(kx̂) + ~∆c â† â,
2m
2

(6.1)

where Sd is the effective pumping rate of the cavity and ∆c is the detuning between the cavity
frequency and laser frequency. Here, the motion of the atom is restricted to the cavity axis (xaxis). The position and canonically conjugated momentum of the atom are given by x̂ and p̂ with[x̂, p̂] = i~. Note that the shift of the cavity frequency due to the interaction with the atom isneglected in Eq. (6.1) since it is much smaller than ∆c [117].
In the regime C  1, spontaneous emission of the atom can be neglected [117]. Therefore,
the only decay channel of the system is the cavity decay and the system can be described by thequantum master equation for the density matrix ρ,d1
ρ = [Ĥ, ρ] + κL[â]ρ .
dt
i~

(6.2)

Let us focus on the condition under which the motion of the atom can be treated semiclassically.
This requires that~k  ∆p,(6.3)

where ~k is the photon momentum and ∆p is the momentum width of the atom.
In the parameter regime of interest, the cavity linewidth is much larger than the atomic
motion,k∆p |κ + i∆c |.
m

(6.4)

Note that when the recoil frequency ωr = ~k 2 /2m is much less than κ, this condition is consistent
with Eq. (6.3). Therefore, the cavity field can be adiabatically eliminated. In order to correctly
encapsulate the cavity cooling mechanism, the adiabatic elimination of the cavity field has to be
expanded beyond the leading order. Specifically, the retardation effects between the cavity field
and atomic variables should be included.
The Langevin equation of motion for the cavity field is√dκ
Sdâ = − â − i∆c â − i cos(kx) − κξˆκ (t),
dt
2
2

(6.5)

62
where x is now the semiclassical position and ξˆκ is the quantum white noise originating from thecavity output with hξˆκ (t)ξˆκ† (t0 )i = δ(t − t0 ). Following the procedure of doing adiabatic elimination (see Appendix A), we have


Z t
√
Sd
−(κ/2+i∆c )s
ˆ
−i cos(kx) − κξκ
dse
â ≈
2
0

√ ˆ
Sd ∆c
d
d
−i S4d κ cos(kx)cos(kx) i S2d (κ2 /4 − ∆2c ) dt
cos(kx)κ∆c S2d dtκξκ
2 cos(kx)
≈
.
− 2
+
+
−
2
2
2
2
2
2
2
2
2
κ /4 + ∆cκ /4 + ∆c(κ /4 + ∆c )(κ /4 + ∆c )
κ/2 + i∆c
(6.6)

We can see from Eq. (6.6) that the retardation between the cavity field and the atomic motion has
been included to the second order in 1/κ.
The total force on the atom is given by the derivative of the Hamiltonianf = −∇H = ~k sin(kx)Sd(â + â† )
2

(6.7)

By plugging Eq. (6.6) into Eq. (6.7), we find the conservative force to be
Sd Sd ∆ccos(kx),2 κ2 /4 + ∆2c

(6.8)

d
cos(kx)κSd2 /4
Sd κ∆c Sd dt
2∆c k/m
2
=
−~k
sin
(kx)
p,
2
2
2
2
2
2 (κ /4 + ∆c )
κ /4 + ∆c κ2 /4 + ∆2c(6.9)

f c = −~k sin(kx)the frictional force to be
f f = ~k sin(kx)and the diffusion force has the correlation matrix
hdpdpi = ~2 k 2

κSd2 /4
sin2 (kx)dt
κ2 /4 + ∆2c(6.10)

The final cooling temperature T for the atom is
kB T =

hp2 i
D∆2 + κ2 /4
=
=~ c
,
m
2mα
4∆c

(6.11)

since from Eq. (6.10) and Eq. (6.9), the diffusion and friction coefficient, D and α, are given byκS 2 /4
D = ~2 k 2 sin2 (kx) 2 d,
κ /4 + ∆2c
κS 2 /4
2∆c k/m
α = ~k sin2 (kx) 2 d.
2
κ /4 + ∆c ∆2c + (κ/2)2(6.12)

63
According to Eq. (6.11), the minimum temperature that can be achieved by cavity cooling is
kB T = ~κ/4 at ∆c = κ/2.
The many-atom effects of cavity-assisted cooling were theoretically discussed by Ritsch andcollaborators [31] and experimentally reported [5, 17]. It is shown that the many-atom case exhibitsa threshold, which is mainly determined by the intensity of the transverse laser. Below the threshold,the atoms distribute homogeneously along the cavity axis. The physics is the same as the singleatom case discussed above. Above threshold, self-organized atomic structure (Bragg gratings)
form, which superradiantly scatters photons into the cavity. Recently, it has been shown thatthe long-range nature of the cavity-mediated interaction between atoms gives rise to interesting
prethermalization behavior [118]. In spite of the intrinsic many-body nature, the underlying coolingmechanism shares much with the single-atom case, and indeed the final temperature observed inthese systems is limited by the cavity linewidth.

6.2

Supercoolingȁ‫ۄ‬

Repump
laser

ȁ‫ۄ‬

Rapid decay݃ ሺ݇‫ ݔ‬ሻ

ȁ‫ۄ‬
ߛ

ߠ

ȁ‫ ۄ‬՞ ȁ‫ۄ‬

݇Ԣ

ߢ
‫ݔ‬ො

Figure 6.2: Atoms with ultranarrow transition g ↔ e are confined to the axis of a standing-wave
mode of an optical cavity. Different implementations of pumping may be considered [12, 93]. In thesimplest scenario shown, a transition is driven from the ground state |gi to an auxiliary state |aithat rapidly decays to the excited state |ei. In this way |ai can be adiabatically eliminated and a
two-state pseudospin description in the {|gi, |ei} subspace used, with repumping corresponding toan effective rate w from |gi to |ei. If the repumping laser is directed normal to the cavity axis, theabsorption does not modify the momentum. Momentum recoil is induced by the on-axis componentof the wavevector k~0 of the dipole radiation pattern for the |ai ↔ |ei transition.

In this section, we demonstrate that the mechanical action of the atom-cavity coupling takes

64
on a dramatically new character for atoms in the regime of steady-state superradiance [11, 12,
91, 93, 140, 142]. Specifically, the frictional force on a single atom is significantly enhanced, andthe final temperature is much lower than the temperature that can be achieved in cavity-assisted
cooling [51, 125]. Furthermore, as the atom number increases, the cooling may become faster due tothe increasing rate of superradiant collective emission. We show that ability to achieve much lower
temperatures than for single-atom cavity-assisted cooling derives from the emergence of atom-atom
dipole correlations in the many-body atomic ensemble.
We now consider a specific situation of an ensemble of N point-like two-level atoms with transition frequency ωa and natural linewidth γ, interacting with a single-mode cavity with resonancefrequency ωc and linewidth κ, as shown in Fig. 6.2. The atoms are restricted to move freely along
the direction of the cavity axis (x-axis), a situation that can be realized by tightly confining theatoms in the other two directions. The atom-cavity coupling strength is given by g cos(kx), where
g is the vacuum Rabi frequency at the field maximum and cos(kx) describes the one-dimensional
cavity mode function. The atoms are incoherently repumped at rate w, thus providing the sourceof photons.
The Hamiltonian describing the atom-cavity system in the rotating frame of the atomic
transition frequency is given by,Ĥ = ~∆â† â +

N
X
p̂2j
j=1

2m

N

+~

gX † −(â σ̂j + σ̂j+ â) cos(kx̂j ) ,2(6.13)

j=1

where ∆ = ωc − ωa . We have introduced the bosonic annihilation and creation operators, â and â† ,for cavity photons. The j-th atom is represented by Pauli pseudospin operators, σ̂jz and σ̂j− = (σ̂j+ )† ,and position and momentum x̂j and p̂j , respectively.
In the presence of dissipation, the evolution of the system is described by the Born-Markov
quantum master equation for the density matrix ρ̂ for the cavity and atoms,
N Z 1
i
X
1 h
d
0
ρ̂ =
Ĥ, ρ̂ + κL[â]ρ + w
duN (u)L[σ̂j+ eiuk x̂j ]ρ ,
dt
i~
−1

(6.14)

j=1

The term proportional to κ describes the cavity decay. The repumping is the term proportional
to w and is modeled by spontaneous absorption with recoil [44]. The recoil is parametrized by the65normalized emission pattern N (u) and wavevector k 0 . It will generally be a good approximation toneglect the effect of free-space spontaneous emission of the atoms outside of the cavity, since thenatural linewidth γ is assumed to be extremely small for atoms with an ultraweak-dipole transition.
The typical relaxation time of the cavity mode is of the order of TC ∼ |κ + i∆|−1 , while the

n√
o−1
p
one of the atoms is given by TA ∼ max
N n̄g, w, k hp2 i/m, where n̄ is the mean photon
number in the cavity. In steady state superradiance, the relaxation time of the cavity is assumed to
be much shorter than the timescale on which the atoms are evolving, namely TA  TC . Therefore,
the cavity field can be quasiadiabatically eliminated as in Section 6.1. To this end, we start with
the quantum Langevin equation for the cavity field according to the quantum master equation
Eq. (6.14),√dκgâ = − â − i∆â − i Jˆ− + κξˆκ (t) ,dt
22where Jˆ− =(6.15)−
ˆ
j=1 σ̂j cos(kx̂j ) is redefined in this case to include the atomic position and ξκ isPN

again the quantum white noise originating from the cavity output with hξˆκ (t)ξˆκ† (t0 )i = δ(t − t0 ).
The formal solution to Eq. (6.15) isâ(t) = ewhere F̂(t) =−(κ/2+i∆)∆t

Z ∆tg
â(t0 ) − i2dse−(κ/2+i∆)s Jˆ− (t − s) + F̂(t) ,(6.16)

0√ R ∆tκ 0 dse−(κ/2+i∆)s ξˆκ (t − s) is the noise term and ∆t = t − t0 . Under the ap-

proximation of coarse graining (TA  ∆t  TC ), the first term on the right-hand side (RHS) of
Eq. (6.16) vanishes, and it is shown in Appendix A thathF̂(t)Fˆ† (t0 )i ≈√κ
κ2 /4 + ∆2

δ(t − t0 ) .

(6.17)

γˆIt would be convenient to choose F̂(t) = −i g/2C ξ(t),
with
γC =

g 2 κ/4.
κ2 /4 + ∆2

(6.18)

Furthermore, the integral in Eq. (6.16) can be expanded in powers of 1/(κ/2 + i∆) (see Appendix
A). As a result we obtainâ(t) ≈

d−i g2 Jˆ−(−i g2 Jˆ− )− dt+ F̂(t) + O[(κ/2 + i∆)−3 ] .
κ/2 + i∆ (κ/2 + i∆)2

(6.19)

66
As can be seen from Eq. (6.19), the retardation effects between the cavity field and atomic variables
are included.
The force on the j-th atom F̂j is given by
F̂j =

d
g
p̂j = ~k sin(kx̂j ) (σ̂j+ â + â† σ̂j− ) + N̂jpump ,dt
2(6.20)

where N̂jpump represents the random force due to recoil of the incoherent pumping process. Substituting Eq. (6.19) into the above equation, we haveγC dp̂j ≈ ~k sin(kx̂j )−iσ̂j+ Jˆ− + iJˆ+ σ̂j−
dt
2


N
γ∆ X
+ −
+ −
+ d −
∗ d + −
cos(kxl ) σ̂j σ̂l + σ̂l σ̂j − β1 σ̂j σ̂l − β1 σ̂l σ̂j− ~k sin(kx̂j )2
dt
dt(6.21)

l=1N

γC X η
− sin(kx̂j )[sin(kx̂l ), p̂l ]+ σ̂j+ σ̂l− + σ̂l+ σ̂j− + β2 σ̂j+ σ̂l− + β2∗ σ̂l+ σ̂j− + N̂j ,
2
2
l=1

h
i
where Â, B̂ = ÂB̂ + B̂ Â is the anticommutator and the coefficients are+γ∆ =g 2 ∆/2
κκ2 /4 − ∆2
κ2 /4 − ∆2
4ωr ∆
,
β
=
+
i
,
β
=
i
,η = 2. (6.22)
1
2
2
2
2
2
2
2
κ /4 + ∆κ /4 + ∆∆(κ /4 + ∆ )κ∆κ /4 + ∆2

Here N̂j = N̂jcav + N̂jpump is the sum of the noise processes originating from the cavity output N̂jcav
and repumping N̂jpump . In the first line of equation (6.21) we neglect β1 because in the steadyd −state superradiance regime it holds that |β1 |hσ̂j+ dt
σ̂l i ∼ wκ hσ̂j+ σ̂l− i  hσ̂j+ σ̂l− i. This has also beenchecked numerically. Therefore we getddp̂j = p̂0j + N̂j ,dt
dt(6.23)

where we define the force without noise as
N




d 0
γC 
γ∆ X
p̂j ≈ ~k sin(kx̂j )−iσ̂j+ Jˆ− + iJˆ+ σ̂j− − ~k sin(kx̂j )cos(kxl ) σ̂j+ σ̂l− + σ̂l+ σ̂j−
dt
2
2
l=1

γC
− sin(kx̂j )
2

N
X
l=1

(6.24)

η
[sin(kx̂l ), p̂l ]+ σ̂j+ σ̂l− + σ̂l+ σ̂j− + β2 σ̂j+ σ̂l− + β2∗ σ̂l+ σ̂j− .
2We work at the detuning ∆ = κ/2 so that η is maximized and β2 vanishes. Note that at ∆ = κ/2,

67
γC = g 2 /2κ = ΓC /2. As a result we obtain

d 0
γC 
p̂j ≈ ~k sin(kx̂j )−iσ̂j+ Jˆ− + iJˆ+ σ̂j− − σ̂j+ Jˆ− − Jˆ+ σ̂j−
dt
2
N


γC X η
[sin(kx̂l ), p̂l ]+ σ̂j+ σ̂l− + σ̂l+ σ̂j− .
− sin(kx̂j )2
2(6.25)

l=1

The first term on the RHS of Eq. (6.25) represents forces originating from the adiabatic componentof the cavity field, while the second term represents the frictional force arising from retardation
effects. The noise term N̂j in equation (6.23) gives rise to momentum diffusion due to quantum
noises associated with incoherent processes. So we derive the equations of motion for the second
moments of momenta,
*
+

0
0
dp̂
dp̂
d
j 0
hp̂j p̂l i = p̂0j l +p̂ + γC ~2 k 2 hsin(kx̂j ) sin(kx̂l )σ̂j+ σ̂l− i + wδjl ~2 k 02 u2 hσ̂j− σ̂l+ i ,
dt
dt
dt l
(6.26)
where δjl is the Kronecker delta, and u2 is the second moment of the dipole radiation pattern, i.e.,
Z 1
2
2
u =
duN (u)u2 = ,(6.27)
5−1√
where we have taken the dipole pattern N (u) = 23 |u| 1 − u2 .
We treat the external atomic motion classically under the assumption that the momentum width ofp
the particles hp2 i is larger than the single photon recoil ~k [Eq. (6.3)]. So we make the mapping
hp̂j i → pj and hx̂j i → xj . As a result this leads todd
pj = p0j + ξjp ,dt
dt(6.28)

with
N



X
d 0
pj = ~k sin(kxj )γC Im[hσ̂j+ Jˆ− i] − Re[hσ̂j+ Jˆ− i] − sin(kxj )γCηRe[hσ̂j+ σ̂l− i] sin(kxl )pl ,dt
l=1(6.29)
where ξjp is the classical noise acting on the momentum of j-th atom and hξjp (t)ξlp (t0 )i = Djl δ(t−t0 ).
The diffusion matrix Djl can be computed by making quantum-classical correspondence for thesecond moments. According to Eq. (6.28),*+
0
dp0j 0d0 dpl
hpj pl i = pj
p + Djl .
+
dt
dt
dt l

(6.30)

68
We use symmetric ordering of quantum operators for the quantum-classical correspondence, i.e.,
h
D
i 
E
dpl
dp̂l
1
→
p
p̂
,
j
j
2
dt
dt . Matching Eq. (6.26) and Eq. (6.30), we get+

Djl = γC ~2 k 2 sin(kxj ) sin(kxl )Re[hσ̂l+ σ̂j− i] + wδjl ~2 k 02 u2 hσ̂j− σ̂l+ i .

(6.31)

Therefore, we could simulate the external motion of atoms with Eq. (6.28) and the equation of
motion for xj
pj
d
xj =
.
dt
m

(6.32)

The classical noises ξjp with diffusion matrix Djl make sure that we have the right second ordermoments for momenta.
To begin with, we first consider the case in which the effect of recoil associated with therepumping is neglected, i.e. we set k 0 = 0. This will determine the ultimate temperature limit
imposed by the vacuum noise due to the cavity output. For the simple one-atom (labeled by 1)
case, we can then directly find the friction (α1 ) and diffusion (D1 ) coefficient from Eq. (6.28) andEq. (6.31). The steady-state temperature T for the single atom is
kB T =

hp21 i
D1
~κ
=
=
,
m
2mα1
4

(6.33)since
D1 = ~2 k 2 γC sin2 (kx1 )hσ̂1+ σ̂1− i ,α1 = ηγC sin2 (kx1 )hσ̂1+ σ̂1− i .

(6.34)

Note that this is precisely the same temperature limit previously found in the cavity-assisted coolingcase in Section 6.1. Here the rate of the decay into the cavity mode is proportional to γC hσ̂1+ σ̂1− i,which is applicable to the weak coupling regime of cavity QED [95]. In Fig. 6.3(a), we show anumerical simulation of the cooling trajectory of a single atom as a function of time. As expected,
the final temperature kB T asymptotes to ~κ/4 and the cooling rate is well approximated by RS =
ηγC hσ̂1+ σ̂1− i.
The cooling in the many-atom case exhibits a distinctly different character. A feature of thismodel is the pseudospin-to-motion coupling of the atoms. In order to close the evolution equations

69

Figure 6.3: Time evolution of the average momentum square (red dots) evaluated from 4000 trajectories simulated by integrating Eqs. (6.28) and (6.32) for 1 atom (a), 20 atoms (b), and 60
atoms (c). The blue solid line is a fit to an exponential decay. The parameters are ∆ = κ/2 = 100,γC = 0.1, and ωr = 0.25. The repumping rates are chosen such that the average atomic population
inversion in all cases is the same [w = 0.15 (a), 0.28 (b), 1.3 (c)]. Insets show the momentum
statistics. The blue solid line is a fit to a Gaussian distribution.

of the atomic motion as described by Eq. (6.28) and Eq. (6.32), it is also necessary to solve for the
dynamics of the pseudospins. For this purpose, we derive the effective quantum master equation
for the pseudospins.
For the internal dynamics of atoms in a superradiant laser, it is sufficient to keep the first70
order term in Eq. (6.19),â(t) ≈ −i

γC ˆ− γ∆ ˆ−J −J + F̂(t) .
gg(6.35)

Here, retardation effects are not included because they give rise to corrections that are of higher
order and their contribution is negligible. The adiabatic elimination of the cavity field leads to aneffective quantum master equation for the atomic spins onlyNX
1d
ρ = [Ĥeff , ρ] + γC L[Jˆ− ]ρ + w
dt
i~
j=1

Z 1
−1

~0

duN (u)L[σ̂j+ eik ·~xj ]ρ ,(6.36)

where the Hamiltonian Ĥeff = − ~γ2C Jˆ+ Jˆ− describes the coherent coupling between each pair ofatoms, and the collective decay [term γC L[Jˆ− ] in Eq. (6.36)] leads to dissipative coupling. We wantto emphasize that this atomic master equation is not sufficient for the external degrees of freedom,
for which retardation effects are not negligible.
The spin degrees of freedom of atoms scale exponentially with the number of atoms. To solve
Eq. (6.36), we thus use a semiclassical approximation that is applicable to large atom numbers inthe steady-state superradiance [91, 92]. Cumulants for the expectation values of spin operators
are expanded to second order. Because of the U (1) symmetry, hσ̂j± i = 0. Therefore, all nonzero
observables are expanded in terms of hσ̂j+ σ̂j− i and hσ̂j+ σ̂l− i (j 6= l). Their equations of motion canthen be found from the effective master equation,
1
1
d + −
hσ̂ σ̂ i = w(1 − hσ̂j+ σ̂j− i) − (γC + iγ∆ ) cos(kx̂j )hJˆ+ σ̂j− i − (γC − iγ∆ ) cos(kx̂j )hσ̂j+ Jˆ− i,
dt j j
2
2
d + −
1
1
hσ̂ σ̂ i = −whσ̂j+ σ̂l− i + (γC + iγ∆ ) cos(kx̂j )hJˆ+ σ̂l− σ̂jz i + (γC − iγ∆ ) cos(kx̂l )hσ̂lz σ̂j+ Jˆ− idt j l22≈ − w + (γC + iγ∆ ) cos2 (kx̂j )hσ̂j+ σ̂j− i + (γC − iγ∆ ) cos2 (kx̂l )hσ̂l+ σ̂l− i hσ̂j+ σ̂l− i
1
1+ (γC + iγ∆ ) cos(kx̂j )(2hσ̂j+ σ̂j− i − 1)hJˆ+ σ̂l− i + (γC − iγ∆ ) cos(kx̂l )(2hσ̂l+ σ̂l− i − 1)hσ̂j+ Jˆ− i,22(6.37)
describing the population inversion and spin-spin correlation respectively. In deriving Eq. (6.37),
we have dropped the third-order cumulants. In the simulations we integrate (6.28), (6.32) and(6.37) simultaneously.

71

Figure 6.4: (a) Cooling rate (in units of the psingle atom cooling rate RS ) as a function of atom
number. (b) Final momentum width (∆p = hp2 i, blue squares) and spin-spin correlation (reddots) as a function of atom number. The parameters are the same as those in Fig. 6.3.

Simulations of the cooling dynamics for many atoms are shown in Figs. 6.3(b) and (c).
Remarkably, we find the collective atomic effects to lead to a more rapid cooling rate, and simultaneously to generate a lower final temperature. Figure 6.4 shows the cooling rate (a) and the final
momentum width (b) as a function of the atom number. We note that the cooling rate exhibits two
kinds of behavior, hinting towards the existence of a N -dependent threshold, see Fig. 6.4(a). For
N . 20, the cooling rate is independent of N , while for N & 20, it increases monotonously. Correspondingly, in this regime, the momentum width has reached a minimum which is independentof N , see Fig. 6.4(b). Note that when the final temperature gets closer to the recoil temperature,
the momentum distribution is not Gaussian anymore, rendering the notion of temperature invalid.
The semiclassical treatment predicts a uniform distribution in the momentum interval [−~k,~k]
corresponding to the recoil limit, as shown in the inset of Fig. 6.3(c). We note that sub-Doppler
temperatures for a similar setup have been reported in Refs. [112], where spontaneous decay was

72
assumed to be the fastest incoherent process. Differing from that regime, the recoil limit is herereached thanks to the small spontaneous decay rate. When the temperature approaches the recoil
temperature, however, the validity of the semiclassical treatment of atomic motion is questionable
and a full quantum model is necessary in order to determine the asymptotic energy. These results
demonstrate that for atoms in the steady-state superradiant regime, not only is the cooling more
efficient due to the rapid rate of superradiant light emission, but also the final temperature isdetermined by the relaxation rate of the atomic dipole, and not the cavity linewidth.
The principal new feature here is that spin-spin correlations between atoms develop due tothe cavity-mediated coupling. In order to measure the extent of this effect, we introduce hσ̂ + σ̂ − iE
defined as averaged spin-spin correlations,hσ̂ + σ̂ − iE = hJˆ+ Jˆ− i −NXhσ̂j+ σ̂j− i cos2 (kxj ) /[N (N − 1)] .

(6.38)

j=1

Fig. 6.4(b) shows hσ̂ + σ̂ − iE as a function of the number of atoms. The equilibrium temperature is seen to decrease as the collective spin-spin correlation emerges. This is reminiscent of thelinewidth of the superradiant laser, where the synchronization of spins leads to a significant reduction of the linewidth to the order of γC [93, 140]. The establishment of spin-spin correlations is acompetition between dephasing due to both cavity output noise and repumping, and the dissipative
coupling between atoms which tends to synchronize the dipoles [140]. Since the coupling strengthscales with N , a sufficient atom number is required to establish strong spin-spin correlations [140].
Further characterizing the ultimate temperature limits, Fig. 6.5(a) shows the final momentum
width as a function of γC . We see that as γC is decreased, the final temperature reduces inproportion to γC until it hits the recoil limit. This effect is consistent with a significantly increased
friction coefficient providing a reduction of the order of the final temperature from the one to many
atom case from κ to γC .
In the large atom number limit, we are able to derive analytically the average frictional force
on a single atom moving in a large ensemble of atoms that are in steady-state superradiance.
We start by finding the steady state of the large ensemble of atoms in equilibrium. In the

73

Figure 6.5: (a) Final momentum width as a function of γC for 40 atoms. The parameters are∆ = κ/2 = 200, w = N γC /4, and ωr = 0.25. (b) Final momentum width as a function of
repumping strength for 40 atoms without (k 0 = 0, blue squares) and with recoil associated with
repumping (k 0 = k, red dots). The parameters are ∆ = κ/2 = 200, γC = 0.5, and ωr = 0.25.

moving frame of atoms, the master equation for the system [Eq. (6.14)] becomes


N
N Z 1
X
d
1  gX † −
~0
+
ρ=
~
(â σ̂j + σ̂j â) cos(δj t), ρ + κD[â]ρ + w
duN (u)D[σ̂j+ eik ·~xj ]ρ ,
dt
i~
2
−1
j=1

(6.39)

j=1

where δj = kpj /m is the Doppler frequency for j-th atom. Following the cumulant expansion
method developed in Ref. [93], we get a set of closed equations from the above master equation,
1 + hσ̂lz igd † −hâ σ̂l i ≈ −(κ0 − i∆)hâ† σ̂l− i + i cos(δl t) hâ† âihσ̂lz i +
dt
2
2
gX
+i
cos(δj t)hσ̂j+ σ̂l− i ,2(6.40a)

j6=l


d + −
g
hσ̂j σ̂l i = −whσ̂j+ σ̂l− i + i cos(δl t)hσ̂lz ihσ̂j+ âi − cos(δj t)hσ̂jz ihâ† σ̂l− i ,dt2d zhσ̂ i = w(1 − hσ̂lz i) + ig (Xl − Xl∗ ) ,
dt l

d †
gX
hâ âi = −κhâ† âi − i
Xj − Xj∗ ,dt
2
j(6.40b)(6.40c)(6.40d)

74
where Xj = hâ† σ̂j− i cos(δj t) and κ0 = (κ + w)/2. In the adiabatic region where w  δj , we obtainfrom Eq. (6.40b) the spin-spin correlationhσ̂j+ σ̂l− i ≈ igcos(δl t)hσ̂lz iXj∗ / cos(δj t) − cos(δj t)hσ̂jz iXl / cos(δl t) .
2w

(6.41)

Plugging it into Eq. (6.40a) in the steady state,
Xl ≈

X
j6=l


g2
cos2 (δj t)hσ̂jz iXl − cos2 (δl t)hσ̂lz iXj∗ ,0
4w(κ − i∆)(6.42)

where we have dropped the single-atom terms in Eq. (6.40a), because the collective terms dominate
in the steady-state superradiance for many atoms [93].
For further simplification, we define the average of a quantity q in terms of ensemble averagePhqiE = N
j=1 qj /N and time average in steady state q. By assuming the ergodicity of the system,
hqiE = q. Performing time average on both sides of Eq. (6.42), we end up with a simple equation
hX iE ≈

N g 2 hσ̂ z iE
(hX iE − hX ∗ iE ) .
8w(κ0 − i∆)(6.43)

The approximation employed here is thathcos2 (δj t)hσ̂jz iiE =

N
X
j=1

1
cos2 (δj t)hσ̂jz i/N ≈ hσ̂ z iE ,2(6.44)

1
cos2 (δl t)hσ̂lz i ≈ hσ̂ z iE ,2assuming homogeneous density distribution of atoms. Furthermore, from Eq. (6.40c), we havew(hσ̂ z iE − 1) = ig(hX iE − hX ∗ iE ) .

(6.45)For convenience, we introduce a real quantity hσ̂ + σ̂ − iE ,hX iE =iN g
1
hσ̂ + σ̂ − iE .
0
2 κ − i∆(6.46)

Solving Eq. (6.43) and Eq. (6.45), we finally obtain in the steady statehσ̂ z iE =2wκ κ02 + ∆2
2w
≈
,
N g2
κ02
N γC

1
hσ̂ + σ̂ − iE ≈ hσ̂ z iE (1 − hσ̂ z iE ) ,4hâ† âi =N 2 γC hσ̂ + σ̂ − iE.
κ
(6.47)

75hσ̂ + σ̂ − iE can be understood as the averaged spin-spin correlation, becausehσ̂ + σ̂ − iE = hJˆ+ Jˆ− i/N 2 .

(6.48)

We now proceed to calculate the average frictional force on a single atom (labeled by 1)
moving in a large ensemble of atoms in steady state. From Eq. (6.39), this force is
f1 = ~kgRe[(S − R)/2i] .

(6.49)

where S = hâ† σ̂1− ieiδ1 t and R = hâ† σ̂1− ie−iδ1 t . The equation of motion for S isddgX
S = iδ1 S + eiδ1 t hâ† σ̂j− i ≈ iδ1 S − (κ0 − i∆)S + icos(δj t)eiδ1 t hσ̂j+ σ̂1− i ,dtdt
2(6.50)

j6=l

where we once again omit the single-atom terms. Since we only care about the average frictional
force in the steady state, this would require that eiδ1 t hσ̂j+ σ̂1− i is in steady state, i.e.
0=d + − iδ1 t= −(w − iδ1 )hσ̂j+ σ̂1− ieiδ1 thσ̂ σ̂ ie
dt j 1g+eiδ1 t i cos(δ1 t)hσ̂1z iXj∗ / cos(δj t) − cos(δj t)hσ̂jz ix1 / cos(δ1 t) .(6.51)
2

Solving the above equation for hσ̂j+ σ̂1− ieiδ1 t and plugging the result into Eq. (6.50), we can getXdg2
S = −(κ0 − i∆ − iδ1 )S +cos2 (δj t)hσ̂jz iS − eiδ1 t cos(δ1 t)hσ̂1z iXj∗ .
dt
4(w − iδ1 )(6.52)

j6=l

Invoking the approximation in Eq. (6.44), we end up with a simple equation for S(κ0 − i∆ − iδ1 )S ≈N g 2 hσ̂ z iE
(S − hX ∗ iE ) .
8(w − iδ1 )(6.53)(κ0 − i∆ + iδ1 )R ≈N g 2 hσ̂ z iE
(R − hX ∗ iE ) .
8(w + iδ1 )(6.54)

Similarly, for R

Plugging S and R into Eq. (6.49),
f1 ≈ −~kg 2

δ1 ∆κ0 /2 4κ0
N hσ̂ + σ̂ − iE .
(κ02 + ∆2 )2 w

(6.55)In deriving the above equation, we only keep terms to the first order in δ1 /w. Recall the one-atom
friction coefficient α1 in Eq. (6.34)
α1 = γC sin2 (kx1 )ηhσ̂1+ σ̂1− i .

(6.56)

76
Therefore, the average hα1 i = γC ηhσ̂1+ σ̂1− i/2, andf1 ≈ −hα1 ip1 ·κ N hσ̂ + σ̂ − iE,w hσ̂1+ σ̂1− i/2(6.57)+ −hσ̂ σ̂ iE. Because
where we can identify a significant enhancement of the friction by a factor wκ Nhσ̂
+ −
σ̂ i
1

N hσ̂ + σ̂ − iE
γC hJˆ+ Jˆ− i/N=,hσ̂1+ σ̂1− i/2
γC hσ̂1+ σ̂1− i/2

1(6.58)

the enhancement is due to increased single-atom photon emission rate by superradiance and thefactor κ/w. Plugging in w, we immediately see a great increase of the average frictional force ona single atom by a factor of order κ/γC . Note also that hσ̂ + σ̂ − iE should have finite value, or else
there would be no superradiant emission.
From Eq. (6.57), we find a significantly increased friction coefficient for the many-atom laser
cooling case due to both the increased photon emission rate due to superradiance (i.e. N hσ̂ + σ̂ − iE hσ̂1+ σ̂1− i/2) and the large ratio κ/w. The friction coefficient is enhanced by a factor of κ/γC
compared to the one-atom case. This is consistent with the reduction of the order of the final
temperature from the one to many atom case from κ to γC .
So far our discussion has neglected the recoil associated with repumping. We have done that
because its effect on the final temperature will depend crucially on specifics of its implementation,
including factors such as the polarizations and directions of repump lasers, the atomic system, andthe transitions used. However, in the specific repumping model shown in Fig. 6.2, the magnitudeof k 0 controls the recoil effect of the repumping on the momentum diffusion. Fig. 6.5(b) shows the
final momentum width as a function of repumping for k 0 = 0 and k 0 = k. Again, in the regionof small and large repumping, where spin-spin correlations are very small, the final temperature
is high. When the recoil due to repumping is included, the final temperature becomes higher andis eventually determined by wu2 . However for weak repumping, with w not significantly larger
then γC it is still possible to achieve temperatures not much higher than that predicted whenpump recoil was neglected. This is especially promising for the implementation of supercooling inrealistic experimental systems. Note that k = k 0 is more or less a worst case scenario, since by77
using a dipole allowed transition for the relaxation from the auxiliary state to the excited state,
one could in principle use a much reduced frequency with correspondingly small recoil.
To summarize, we have proposed supercooling of the atomic motion along the axis of an
optical cavity. The superradiant emission was observed to lead to an enhanced cooling rate andextremely low final temperature. The ultimate temperatures were constrained by the relaxationof the atomic dipole, and may be orders of magnitude lower than for single atom cooling where
temperatures are limited by the cavity linewidth. From a broader viewpoint, we have demonstrated
an example of many-body laser cooling in which all motional degrees of freedom of a collective
system are simultaneously cooled and in which macroscopic spin-spin correlations are essential andmust develop for the cooling mechanism to work.

Chapter 7

Summary and outlook

Time and frequency are the most accurately measured physical quantities so far. State-of-theart optical lattice clocks achieve accuracy in the realm of 10−18 [9, 49]. Up to now, all these highprecision optical clocks are passive frequency standards, where the frequency of a narrow linewidthlaser (local oscillator), stabilized to an ultrastable optical resonator, is used for the interrogation oftrapped ions or neutral atoms [60]. In these systems, the optical resonator undergoes unavoidable
thermal vibrations of the dielectric coatings on the cavity mirrors [65], giving rise to frequency
fluctuations of the local oscillator. Instability of the local oscillator is one of the main limiting
factors for the short-term stability of modern optical clocks [83]. Improvement may be attainedby minimizing the thermal vibration of the reference cavity [22] or by locking the local oscillator
to another reference that is more stable than the reference cavity. A promising candidate for thisreference is an active optical frequency standard. For example, in the microwave domain, theHydrogen maser is used as a flywheel for the Caesium timekeeping system. In the optical domain,
there is still no working system whose stability could compete with the stability provided by thereference cavity [60]. The steady-state superradiant laser offers such potential, which is supported
by what we have done in this thesis. The synchronization and supercooling mechanism revealedin this thesis lay the theoretical foundation for an active frequency standard. Combined withthe recent demonstration of Dicke superradiance on the clock transition of strontium atoms in an
optical cavity [99], an eventual experimental realization of the superradiant laser may be on thehorizon.

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Appendix A

Adiabatic elimination

There is a systematic way to perform adiabatic elimination of fast varaibles. Suppose wehave an differential equation such asdy(t) = −(κ/2 + i∆)y(t) + x(t).
dt

(A.1)

The formal solution to it isy(t) = e−(κ/2+i∆)tZ ty(0) +

dse−(κ/2+i∆)s x(t − s).

(A.2)

0

Under the approximation of coarse graining (κ is large), i.e., t  κ−1 , we have
Z tdse−(κ/2+i∆)s x(t − s)

Z t 
1
≈
ds x(t) − x0 (t)s + x00 (t)s2 ... e−(κ/2+i∆)s
2!
0
1
1
1
x(t) −x0 (t) +x00 (t) + · · · .
=2(κ/2 + i∆)(κ/2 + i∆)(κ/2 + i∆)3y(t) ≈

0(A.3)

We can see that it is a systematic expansion with respect to 1/κ. As a result, we can approximatey(t) in terms of x(t) and its time derivatives. To the first order, y(t) is locked to x(t).
√When x(t) contains the noise term such as κξˆκ (t) with hξˆκ (t)ξˆκ† (t0 )i = δ(t − t0 ), we need todeal with it seperately. Following Eq. (A.3), the noise entering into y(t) isZ tF̂(t) =
0√dse−(κ/2+i∆)s κξˆκ (t − s) =

Z t
0

√
dτ e−(κ/2+i∆)(t−τ ) κξˆκ (τ ) .

(A.4)

90
It is obvious that hF̂(t)i = 0. The non-zero second order correlation ishF̂(t)Fˆ† (t0 )i =

Z t0

Z t
dτ1
0

0

Z t

Z t0
dτ1

=

0
dτ2 e−(κ/2+i∆)(t−τ1 ) e−(κ/2−i∆)(t −τ2 ) κhξˆκ (τ1 )ξˆκ† (τ2 )i0dτ2 e−(κ/2+i∆)(t−τ1 ) e−(κ/2−i∆)(t −τ2 ) κδ(τ1 − τ2 ),00−κ(t+t0 )/2−i∆(t−t0 )

Z t0

Z t
dτ1

=e

0
0

(A.5)
dτ2 κeκ/2(τ1 +τ2 )+i∆(τ1 −τ2 ) δ(τ1 − τ2 )00≈ e−κ|t −t|/2−i∆(t−t )Let us now examine the function f (τ ) = e−κ|τ |/2−i∆τ . We first notice thatZ ∞dτ f (τ ) =
−∞

1
1
κ
+
= 2
.
κ/2 + i∆ κ/2 − i∆
κ /4 + ∆2(A.6)

Therefore,
κ2 /4 + ∆2 −κ|τ |/2−i∆τe= δ(τ ),κκ  τ −1 .

(A.7)

As a result,hF̂(t)Fˆ† (t0 )i =We can then chooseκκ2 /4 + ∆2

δ(t − t0 ).

√eiθ κ ˆF̂(t) =
ξ(t),κ/2 + i∆

(A.8)

(A.9)

ˆ is the normal white noise satisfying hξ(t)ˆ ξˆ† (t0 )i = δ(t − t0 ).
where θ is an arbitrary phase and ξ(t)Appendix B

SU(4) generators and λ-matrices

The hermitian 4 × 4 matrix generators of SU(4) group, analogous to the Pauli matrices ofSU(2) group are [101]:
0 1 0
1 0 0λ1 = 

0 0 0
0 0 0

0 0 1
0 0 0λ4 = 

1 0 0
0 0 0

0 0 00 0 −iλ7 = 

0 i 0


0 0 0


0


0
 ,

0


0

0


0
 ,

0


0

0


0
 ,

0


0





−i 0 0
0 0 0
 ,
0 0 0
0 0 0
0 −i 0
0 0 0
 ,
0 0 0


0 0 0


1 0 0 0
0 1 0 01 
λ8 = √ 3
0 0 −2 0
0 0 0 00i
λ2 = 

0


0

0


0
λ5 = 

i


0


1


0
λ3 = 

0


0

0


0
λ6 = 

0


0

,


0 0 0

−1 0 0,
0 0 0
0 0 0

0 0 0

0 1 0
,
1 0 0


0 0 0


0 0 0 10 0 0 0,
λ9 = 
0 0 0 0
1 0 0 092
0 0 0 −i0 0 0 0  ,λ10 = 
0 0 0 0 
i 0 0 0
0 0 0 00 0 0 0 ,λ13 = 
0 0 0 1

0 0 1 0


0


0
λ11 = 

0


0

0


0
λ14 = 

0


0

0 0
0 0
0 0
1 0
0 0
0 0
0 0
0 i




0
0 0 0 0 






1
0 0 0 −i , λ12 = 
,






0
0
0
0
0





0
0 i 0 00
1 0 0 0 






0
0
1
0
0


1
 , λ15 = √ 
.



6



−i
0
0
1
0





0
0 0 0 −3The λ matrices are traceless, orthogonal and satisfy
Tr(λµ )2 = 2 ,µ = 1, ..., 15 .

Appendix C|S, M ihS, M 0 | representation

In order to project the density operator from the SU(4) basis onto the |S, M ihS, M 0 | represen(s)tation, let us first show that M and M 0 are related to the Pq,q3 ,σ3 by q3 + σ3 = M and q3 − σ3 = M 0 .
P(3)
To see this, defining Jˆ3 = N
j=1 σj /2, we could get
1
(s)
(s)
(s)
Jˆ3 Pq,q= (α + γ − β − δ)Pq,q= (q3 + σ3 )Pq,q
,
3 ,σ3
3 ,σ3
3 ,σ3
2
1
(s)
(s)
(s)
= (q3 − σ3 )Pq,q,
Pq,qJˆ = (α + δ − β − γ)Pq,q
3 ,σ3
3 ,σ3
3 ,σ3 32and by definition, we have

(C.1)

Jˆ3 |S, M ihS, M 0 | = M |S, M ihS, M 0 |,(C.2)
0|S, M ihS, M |Jˆ3 = M 0 |S, M ihS, M 0 |.
(s)

Therefore, the complex coefficients from the Pq,q3 ,σ3 basis contributing to the matrix element for|S, M ihS, M 0 | all satisfy q3 + σ3 = M and q3 − σ3 = M 0 .
With this in mind, we now describe a systematic algorithm to obtain the density matrixm,nm,nelements DS,M,M
We illustrate our method by
0 from the SU(4) expansion coefficients Cq,q3 ,σ3 .

considering in detail the elementary case of three atoms. The density matrix in the |S, M ihS, M 0 |representation is block diagonal in S; the block matrices for all S can be arranged in the shape ofa pyramid as shown in Fig. 3.1(c). For instance, the base layer corresponds to S = N/2, with thematrix dimension being (N + 1)2 . The second layer has S = N/2 − 1 and dimension (N − 1)2 , andP
so on. Furthermore there are nS copies associated with each layer, so that S (2S + 1)nS = 2N .
Taking N = 3 for example, there are two layers, S = 3/2 and S = 1/2 with n3/2 = 1 and n1/2 = 2,
so that the Hilbert space dimension is (3 + 1) + 2 × (1 + 1) = 23 .

94
The density matrix needs to be built from the bottom layer upwards. In the bottom layer, we(s)find that the only element contributing to |N/2, N/2ihN/2, N/2| is PN/2,N/2,0 . So the top left corner
P
m,n
m,n
−
is DN/2,N/2,N/2= CN/2,N/2,0. We next apply the lowering operator Jˆ− = N
j=1 σj to iterativelym,ngenerate DN/2,N/2,M, with M = N/2 − 1, . . . , −N/2. To do this, we need the recursion relation

hS, M |ρm,n Jˆ− |S, M 0 im,nm,nDS,M,M|S, M 0 − 1i = p
0 −1 = hS, M |ρ(S + M 0 )(S − M 0 + 1)

(C.3)

hS, M |(U+ + V+ )ρm,n |S, M 0 i= p.
(S + M 0 )(S − M 0 + 1)

Therefore, with the actions of the raising and lowering operators [Eq. (3.12)], we can derivem,nall DN/2,N/2,M
Using the fact that the density matrix
0 , i.e. the first row of the bottom layer.
m,n
m,n
∗
is Hermitian and Cq,q
3 ,σ3 = (Cq,q3,−σ3 ) , we could get all the elements for the first column by
m,n
m,n
∗
DN/2,M
0 ,N/2 = (DN/2,N/2,M 0 ) . By repeatedly applying the recursion relation [Eq. (C.3)] in eachrow, we then construct the full base layer. As an explicit example, we have constructed the bottom
layer, i.e. S = 3/2 for the three atom case,h 23 , 32 || 32 , 32 im,n
 C3/2,3/2,0

 C m,n 1,1,−1/2√ m,n 3
 C1/2,1/2,−1√| 32 , − 12 i 

3

m,n
3
3
| 2 , − 2 i C0,0,−3/2| 32 , 12 i

h 32 , 12 |h 23 , − 12 |h 32 , − 32 |m,nC1,1,1/2
√
3
m,n
m,n
+C1/2,1/2,0C3/2,1/2,03m,nC1/2,1/2,1
√
3
m,n
m,n
C1,0,1/2+C0,0,1/2
3

m,n
m,n
+C0,0,−1/2
C1,0,−1/2
3

m,n
m,n
+C1/2,−1/2,0C3/2,−1/2,03m,nC1/2,−1/2,−1√3m,nC1,−1,−1/2√3m,nC0,0,3/2





m,n
C1/2,−1/2,1

√

3
.
m,n
C1,−1,1/2 

√

3

m,n
C3/2,−3/2,0

(C.4)



m,n
In order to illustrate the use of the recursion relation, we now show how to get D3/2,1/2,1/2
from
(s)

(s)

(s)

(s)

m,n
m,n
D3/2,1/2,3/2. Because V+ P3/2,1/2,0 = P1,1,−1/2 and U+ P1/2,1/2,0 = P1,1,−1/2 , we have D3/2,1/2,1/2
=
√
√
m,n
m,n
(C3/2,1/2,0+ C1/2,1/2,0)/ 3/ 3.

To construct the next layer, we thus find out the top left matrix element first, and then apply
the same procedure as before to determine the rest of the matrix elements. Let us first examine
the three atom case. The S = 1/2 layer has two copies, each of which is a 2 × 2 matrix. To find them,n
top left element D1/2,1/2,1/2, noticing the constraint imposed by the trace of the density matrix, we
m,n
m,n
m,n
m,n
m,n
m,n
derive 2D1/2,1/2,1/2+ D3/2,1/2,1/2= C3/2,1/2,0
so that D1/2,1/2,1/2= (2C3/2,1/2,0− C1/2,1/2,0)/6. By

95
applying the same method as in the bottom layer, we construct the block matrix for S = 1/2 layerh 21 , 12 || 12 , 21 i
| 21 , − 12 im,nm,n2C3/2,1/2,0−C1/2,1/2,06
 C m,nm,n−2C0,0,−1/2
1,0,−1/2
6

h 21 , − 21 |m,nm,nC1,0,1/2−2C0,0,1/2
6
m,n
m,n
2C3/2,−1/2,0−C1/2,−1/2,0
6



.

(C.5)

Therefore in general, if we suppose that we have constructed the block matrix for S 0 > S,m,nthe formula to find the top left matrix element DS,S,S
for layer S is

Xm,nnS 0 DSm,n0 ,S,S = CN/2,S,0 .

(C.6)

S≤S 0 ≤N/2

Having the top left matrix element for each layer S, we can easily construct the (2S + 1) × (2S + 1)
block matrix by applying the recursion relation based on the angular momentum lowering operator.
Repeated iteration of these steps systematically fills in all sites of the pyramid.