Crafting the dynamical structure of synchronization by harnessing bosonic multilevel cavity QED Riccardo J. Valencia-Tortora1Shane P. Kelly1

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Crafting the dynamical structure of synchronization

by harnessing bosonic multilevel cavity QED

Riccardo J. Valencia-Tortora,1, ∗Shane P. Kelly,1

Tobias Donner,2Giovanna Morigi,3Rosario Fazio,4, 5 and Jamir Marino1

1Institut f¨ur Physik, Johannes Gutenberg-Universit¨at Mainz, D-55099 Mainz, Germany

2Institute for Quantum Electronics, Eidgen¨ossische Technische

Hochschule Z¨urich, Otto-Stern-Weg 1, CH-8093 Zurich, Switzerland

3Theoretical Physics, Department of Physics, Saarland University, 66123 Saarbr¨ucken, Germany

4The Abdus Salam International Center for Theoretical Physics (ICTP), I-34151 Trieste, Italy

5Dipartimento di Fisica, Universit`a di Napoli Federico II, Monte S. Angelo, I-80126 Napoli, Italy

(Dated: May 23, 2023)

Many-body cavity QED experiments are established platforms to tailor and control the collective

responses of ensembles of atoms, interacting through one or more common photonic modes. The

rich diversity of dynamical phases they can host, calls for a uniﬁed framework. Here we commence

this program by showing that a cavity QED simulator assembled from N-levels bosonic atoms,

can reproduce and extend the possible dynamical responses of collective observables occurring after

a quench. Speciﬁcally, by initializing the atoms in classical or quantum states, or by leveraging

intra-levels quantum correlations, we craft on demand the entire synchronization/desynchronization

dynamical crossover of an exchange model for SU(N) spins. We quantitatively predict the onset of

diﬀerent dynamical responses by combining the Liouville-Arnold theorem on classical integrability

with an ansatz for reducing the collective evolution to an eﬀective few-body dynamics. Among them,

we discover a synchronized chaotic phase induced by quantum correlations and associated to a ﬁrst

order non-equilibrium transition in the Lyapunov exponent of collective atomic dynamics. Our

outreach includes extensions to other spin-exchange quantum simulators and a universal conjecture

for the dynamical reduction of non-integrable all-to-all interacting systems.

I. INTRODUCTION

Tailoring light-matter interactions is at the root of

numerous technological or experimental applications

in quantum optics, and it has generated a persistent

drive for better control of atoms and photons since

the advent of modern molecular and atomic physics.

For instance, the pursuit to create precision clocks and

sensors has lead to the development of cavity QED

systems in which a cold gas couples to few or several

electromagnetic modes in an optical cavity [1–6]. Such

systems can be brought out of equilibrium to generate

reproducible many-body dynamics which show com-

plex behavior including self-organization [4, 7–15] and

dynamical phase transitions [1, 6, 12, 16–20], quantum

squeezed and non-Gaussian entangled states [21–27],

time crystals [28–31], and glassy dynamics [9, 13, 32, 33].

This rich phenomenology comes from a high degree

of tunability in such systems, allowing control over

local external ﬁelds, detunings between cavity mode

and applied drive ﬁelds, the ability to couple multiple

atomic levels to the cavity ﬁeld [5, 9, 34–39], and more

recently the realization of programmable geometries for

light-matter interactions [40–42].

Recently, the theoretical and experimental investiga-

tion of multilevel cavity systems has gathered increas-

ing attention. Current progress includes dissipative state

∗Corresponding author: rvalenci@uni-mainz.de

preparation of entangled dark states [43–45], multicriti-

cality in generalized Dicke-type models [46, 47], incom-

mensurate time crystalline phases [35, 48, 49], correlated

pair creations and phase-coherence protection via spin-

exchange interactions [37, 50, 51], spin squeezing and

atomic clock precision enhancement [52–54]. Yet, the

quenched dynamics in multilevel cavity systems is widely

unexplored and the few individual results lack an orga-

nizing principle.

In this work, we propose a unifying framework for the

dynamics after a quench of all-to-all connected multilevel

systems. We show that the ﬂexible control endowed by

bosonic multilevel atoms is suﬃcient to reproduce estab-

lished dynamical phases and beyond. We explain how the

dynamical response can be crafted into these new and ex-

isting dynamical phases by introducing a reduction of dy-

namics to a few-body eﬀective classical evolution, valid

regardless of the underlying integrability of the model.

Of particular note, we demonstrate how quantum corre-

lations in the initial state can drive a transition between

a regular and chaotic synchronized phases.

Our analysis extends the established phenomenology

of the two-level Tavis-Cummings model with local inho-

mogeneous ﬁelds. This two-level model is integrable [55],

and allows for the emergent collective many-body dy-

namics to be exactly described through an eﬀective few-

body Hamiltonian [56–66]. In particular, the few body

model yields predictions for the dynamical responses of

collective observables S(t), such as the collective spin

raising operator, given by the macroscopic sum of sev-

eral individual constituents [56, 59, 60, 63–67]. The re-

sulting dynamical phases are best presented in terms of

arXiv:2210.14224v4 [cond-mat.quant-gas] 20 May 2023

the possible synchronization between the local atomic

degree of freedoms (spins-1/2) which evolve with a fre-

quency set by the competition of their local ﬁeld and

collective photon-mediated interactions. In the desyn-

chronized phase, which we call Phase-I as shorthand,

all the spins evolve independently as a result of domi-

nant classical dephasing processes imprinted by the lo-

cal inhomogeneous ﬁelds, thus S(t) relaxes to zero. In

the synchronized phase, collective interactions lock the

phase precession and we can distinguish three diﬀerent

scenarios in which S(t) either relaxes to a stationary

value (Phase-II), up to a phase of a Goldstone mode [56]

associated to a global U(1) symmetry, or its magnitude

enters self-generated oscillatory dynamics, corresponding

to a Higgs mode [56], either periodic (Phase-III), or ape-

riodic (Phase-IV). While Phase-I and Phase-II describe

relaxation to a steady state up to an irrelevant global

phase, Phase-III and Phase-IV are instead examples of

a self-generated oscillating synchronization phenomenon

without an external driving force [68–71].

A. Summary of results

In this work we investigate dynamics beyond two-level

approximations by considering Nabosonic atoms, each

hosting Nlevels which realize SU(N) spins. The addi-

tional structure due to the bosonic statistics allows us

to naturally consider both classical and quantum initial

states (c.f. Sec. II C). Using this ﬂexibility in the initial

state, and also the tunability of Hamiltonian parameters,

we show how to craft not only the dynamical responses

present in the two-level integrable setup (from Phase-I

up to Phase-IV), but also how to access a novel chaotic

dynamical response. This chaotic response, which we re-

fer to as Phase-IV?, again has all atoms synchronized but

with the dynamics of the average atomic coherences char-

acterized by exponential sensitivity to initial conditions.

The self-generated chaotic Phase-IV?emerges from the

interplay of initial quantum correlations, and the collec-

tive interactions mediated by the cavity ﬁeld. It is there-

fore qualitatively diﬀerent from chaos induced by other

mechanisms as due to additional local interactions [72]

or external pump [30, 73–76].

In order to show how to craft and control these dy-

namical responses, we introduce a generalization of the

reduction hypothesis used for two level systems. Specif-

ically, we propose that the diﬀerent dynamical phases

(Phase-I up to Phase-IV∗) all correspond to a diﬀerent ef-

fective few body Hamiltonian that depends on the global

symmetries of the many body system, degree of inho-

mogeneity, W, number of atomic levels N, and degree

of quantum correlations in the initial state, quantiﬁed

by a parameter p(cf. Sec. IV B). Then, by consider-

ing an appropriate classical limit arising in the limit of

large system size (cf. Sec. II B), we apply the Liouville-

Arnold theorem to the eﬀective Hamiltonian to identify

a correspondence between the dynamical phases and the

eﬀective Hamiltonians. Using physical arguments for the

FIG. 1. Cartoon of the possible dynamical responses of intra-

level phase coherence in a photon-mediated spin-exchange

model between SU(3) spins, as a function of the degree of

inhomogeneity of the local ﬁelds W, and of quantum corre-

lations in the initial state parameterized by p. At p= 0

each site is initialized in the same bosonic coherent state. For

p > 0, there are ﬁnite quantum correlations in the system.

The parameter ptunes from bosonic coherent states (p= 0)

to a multimode Schr¨odinger cat state (p > 0) initialized on

each site. The susceptibility of the dynamical response to

quantum correlations is strictly linked to having SU(N) spins

with N > 2, thus cannot be achieved considering two-level

systems. Up to inhomogeneity W/(χNa)≈1, the system

is in the synchronized phase. At larger inhomogeneities, the

system enters in the desynchronized phase and all phase co-

herence is washed (Phase-I). In the synchronized phase, phase

coherence relaxes asymptotically to a nonzero value up to a

phase associated to a global U(1) symmetry (Phase-II), or its

magnitude enters a self-generated oscillatory dynamics, either

periodic (Phase-III), or aperiodic (Phase-IV), as well as po-

tentially chaotic (Phase-IV?). In this last case dynamics are

exponentially sensitive to changes in initial conditions.

nature of the eﬀective Hamiltonian, we then predict how

to tune between diﬀerent dynamical responses. The re-

sult is an intuitive control over the rich dynamical re-

sponse possible in multilevel cavity QED. See Fig. 1 for

a cartoon of the diﬀerent dynamical responses for N= 3

level atoms, using as a proxy the synchronized (or de-

synchronized) evolution of the magnitude of the average

intra-level coherences in the ensemble.

We conclude by discussing the potential universality of

the reduction hypothesis. In particular, we conjecture it

applies not only for state-of-the-art cavity QED experi-

ments (cf. Sec. VI), but could ﬁnd potential applications

in other ﬁelds. Following Refs. [77, 78], where cavity

QED platforms are proposed to model the dynamics of

s-wave and (p+ip)-wave BCS superconductors, our re-

sults could ﬁnd potential applications to lattice systems

with local SU(N) interactions, such as SU(N) Hubbard

models [79–82]. Another possible outreach of our results

could consist in noticing that the Nlevels of the atoms

could be used as a synthetic dimension, with the geom-

etry ﬁxed by the photon-mediated processes, as for in-

stance in a synthetic ladder system [83, 84]. Furthermore,

since we consider bosonic systems, our results could po-

tentially ﬁnd applications in spinor Bose-Einstein con-

densates [85, 86] or in molecules embedded in a cav-

ity, where bosons could be identiﬁed as their vibrational

modes [87, 88].

B. Organization of the manuscript

The paper is organized as follows. In Sec. II, we in-

troduce the model and initial states we investigate, and

we discuss the cumulant expansion we use to capture

quench dynamics. In Sec. III, we present the dynamical

reduction hypothesis, and discuss the diﬀerent classes of

eﬀective dynamics that can result from it. In Sec. IV

we show that in the homogeneous limit our hypothesis is

exact and demonstrate how local quantum correlations

in SU (3) atoms can induce a chaotic dynamical phase

with ﬁnite Lyapunov exponent. In Sec. V, we show that

the dynamical responses observed in the homogeneous

limit are robust against moderate inhomogeneity in the

local ﬁelds, and we provide numerical evidences that an

eﬀective few-body Hamiltonian is able to capture the dy-

namical periodic response of collective observables in the

three-level case. We conclude this section with a discus-

sion on the impact of inhomogeneity in the dynamical

responses of the system. In Sec. VI we propose an exper-

imental implementation potentially accessible in state-of-

the-art cavity QED systems.

II. PRELIMINARIES

A. The model

We consider a system of Nabosonic atoms interact-

ing via a single photonic mode of a cavity. The atoms

are cooled to the motional ground state and evenly dis-

tributed among Ldiﬀerent atomic ensembles labeled by

a site index j. Within each site (ensemble), the atoms are

indistinguishable and can occupy Ndiﬀerent atomic lev-

els with energies that are site- and level-dependent. We

consider the atoms suﬃciently far apart for interatomic

interactions to be negligible. The photon-matter interac-

tion mediates atom number conserving processes where

the absorption and/or the emission of a cavity photon

results in an atom transitioning from level nto levels

n±1 within the same site, with a rate generally depen-

dent on the speciﬁc level n. The associated many-body

light-matter Hamiltonian reads

H=ω0ˆa†ˆa+

j=1

n=1

h(j)

nˆ

b†

n,jˆ

bn,j +

j=1

N−1

n=1 hgnˆ

b†

n+1,jˆ

bn,j ˆa+h.c.+

+λnˆ

b†

n+1,jˆ

bn,j ˆa†+h.c.i,

(1)

where ˆa(†)is the bosonic annihilation (creation) opera-

tor of the cavity photon; ˆ

b(†)

n,j is the bosonic annihilation

(creation) operator on site j∈[1, L] and level n∈[1, N],

with energy splitting h(j)

n;gnand λnare the single-

particle photon-matter couplings which controls rotat-

ing and co-rotating processes, respectively. Tuning gn

and λnenables us to pass from a generalized multilevel

Dicke model, when gn, λn6= 0, to the multilevel Tavis-

Cummings model, when λn= 0. In our work, we con-

sider dynamics on time scales where dissipative processes

are sub-dominant compared to coherent evolution (cf.

Sec. VII A).

When the cavity is far detuned from the atomic tran-

sitions, the photon does not actively participate in dy-

namics of Eq. (1) but instead mediates virtual atom-

atom interactions [89]. This occurs in the limit ω0

max{h(j)

n, gn√Na, λn√Na}, where the factor √Nacomes

from the cooperative enhancement given by the Na

atoms [90, 91]. The mediated interaction results in an

eﬀective atoms-only Hamiltonian of the form

j=1

n=1

h(j)

nˆ

Σ(j)

n,n+

−

N−1

m,n=1 hχn,m ˆ

Σn+1,n ˆ

Σm,m+1 +ζn,m ˆ

Σn,n+1 ˆ

Σm+1,m+

+νn,m ˆ

Σn+1,n ˆ

Σm+1,m +νm,n ˆ

Σn,n+1 ˆ

Σm,m+1i,

(2)

where χn,m ≡gngm/ω0;ζn,m ≡λnλm/ω0;νn,m ≡

λngm/ω0. For convenience, we have written the Hamil-

tonian in Eq. (2) as a function of the operators

Σ(j)

n,m =ˆ

b†

n,jˆ

bm,j ,(3)

Σn,m =

j=1

Σ(j)

n,m.(4)

The operators {ˆ

Σ(j)

n,m}are generators of the SU(N)

group [92, 93] and they obey the commutation relations

[ˆ

Σ(i)

n,m,ˆ

Σ(j)

k,l ] = δi,j (ˆ

Σ(i)

n,lδm,k −ˆ

Σ(j)

k,mδn,l), and (ˆ

Σ(j)

n,m)†=

Σ(j)

m,n.

The regime we are mostly interested in is νn,m =

ζn,m = 0, which translates to λn= 0. In this limit,

the Hamiltonian in Eq. (2) turns into a spin-exchange in-

teraction Hamiltonian between SU(N) spins with rates

{χn,m}and inhomogeneous ﬁelds, h(j)

n. In the follow-

ing, we set the collective spin-exchange rate χNa=

NaPN−1

n=1 χn,n as our energy scale, such that the time-

scales of our results are independent of the number of

atoms Nain the system. An implementation of the spin

exchange model in Eq. (2) is oﬀered in Sec. VI.

Below, we consider both situations when the energies

of the atomic levels are homogenous and when they are

inhomogenous. In the latter situation, we expect our re-

sults to hold for various forms of inhomogenities, but we

will in particular focus on the situations when the atomic

levels on each site are in an evenly spaced ladder conﬁg-

uration with spacing ∆hj≡(h(j)

n+1 −h(j)

n) sampled from

a box distribution with zero average and width W. In

this case, the Hamiltonian is spatially homogeneous for

W= 0, and spatially inhomogeneous for W > 0. At

W= 0 we can make precise predictions of the dynami-

cal responses as a function of the features of the initial

state and multilevel structure. Then, we show numeri-

cally their robustness against many-body dynamics due

to inhomogenities (W > 0), in a fashion reminiscent of a

synchronization phenomenon.

Given an evenly spaced ladder conﬁguration within

each site, the Hamiltonians in Eq. (1) and Eq. (2) can,

for certain values of the couplings gnand λn, be written

in terms of the generators of a subgroup of SU (N). For

instance, in the N= 3 level case, if gn=gand λn=λ,

the Hamiltonian can be written as a function of the gen-

erators of a SU (2) subgroup of SU(3). Speciﬁcally, only

the SU (2) operators ˆ

S−

j=√2(ˆ

Σ(j)

1,2+ˆ

Σ(j)

2,3), ˆ

S+

j= ( ˆ

S−

j)†,

and ˆ

j= (ˆ

Σ(j)

3,3−ˆ

Σ(j)

1,1) are required to represent the

Hamiltonian, and as a consequence, the dynamics can be

more simply described by the dynamics of these SU (2)

spins. For instance, we recover the spin-1 Dicke model for

λ=gand the spin-1 Tavis-Cummings model for λ= 0

in Eq. (1). Since we aim to explore the impact of genuine

interactions between SU(N) spins, we ﬁx gnand λnsuch

that the dynamics cannot be restricted to a subgroup of

SU (N), if not otherwise speciﬁed. An important excep-

tion is the three-level case, where the system can enter in

a chaotic phase upon passing from interactions between

SU (2) to SU(3) spins (see Sec. IV B). We highlight that

while the interactions considered lead to nontrivial eﬀects

in the SU (N) degrees of freedom, they are not SU(N)-

symmetric.

B. Mean ﬁeld limit

Given a generic interacting Hamiltonian, the dynamics

of any n-point correlation function depends on higher

order correlation functions – a structure known as the

BBGKY hierarchy [94]. In fully connected systems, as

in our case, the hierarchy can be eﬃciently truncated

starting from separable states, or in other words, from a

Gutzwiller-type ansatz [95]

|Ψi=⊗L

j=1|ψji⊗|αi,(5)

where |ψjiis a generic state on the j-th atom, and |αi

is a bosonic coherent state describing the cavity ﬁeld.

Given |Ψiin Eq. (5), the hierarchy can be truncated as

hˆ

Σ(j)

n,mˆai=hˆ

Σ(j)

n,mihˆaiand hˆ

Σ(j)

n,m ˆ

Σr,si=hˆ

Σ(j)

n,mihˆ

Σr,siup

to 1/L corrections [55, 90, 96–98]. Here and from now on,

we assume all expectation values are taken with respect

to the state |Ψi, i.e. hˆo(t)i≡hΨ|ˆo(t)|Ψi. In the limit

L→ ∞ no additional quantum correlations build up in

time, hence the equation of motions of one-point and

two-points correlation functions are exactly closed at all

times and the state |Ψiremains an exact ansatz of the

many-body state.

Combining the large Llimit and the nature of the in-

teraction in the Hamiltonian, the dynamics of hˆ

Σ(j)iand

hˆaican be accordingly obtained in the mean ﬁeld limit of

the Hamiltonians in Eq. (1) and Eq. (2). This is achieved

replacing the operators ˆ

Σ(j)

n,m and ˆa(†)by classical SU(N)

spins and photon amplitude given by

Σ(j)

n,m =hˆ

Σ(j)

n,mi/(Na/L),

a=hˆai/pNa,(6)

with Na/L the average number of bosonic excitations per

site and by substituting the commutators with Poisson

brackets. The same dynamics can be obtained starting

from the Heisenberg equation of motions and then taking

the expectation value on the state |Ψiin Eq. (5) [93]

truncating the hierarchy as discussed above.

The hierarchy can be further truncated at ﬁrst order

in the bosonic operators if the one-body reduced den-

sity matrix Σ(j), with matrix elements Σ(j)

n,m, is pure

(Tr[(Σ(j))2] = 1), namely there are no quantum corre-

lations on a given site j. For instance, if the state |ψjiin

Eq. (5) is a bosonic coherent state on each level of site j,

the matrix Σ(j)is pure and straightforwardly factorized

as Σ(j)

n,m =hˆ

b†

n,j ihˆ

bm,j i. The truncation at ﬁrst order in

the bosonic operators well approximates the full dynam-

ics up to corrections which are suppressed [99] in both

the number of sites Land the occupation on each site

Na/L. Therefore, in the limit Na→ ∞, the hierarchy is

exactly truncated at ﬁrst order in the bosonic amplitudes

hˆ

b(†)

n,j iand hˆai, at all times. In this limit, their dynam-

ics can be equivalently obtained in the classical limit of

the Hamiltonians in Eq. (1) and Eq. (2) by replacing the

bosonic operators ˆ

b(†)

n,j and ˆaby the classical ﬁelds

bn,j =hˆ

bn,j i/pNa/L,

a=hˆai/pNa,(7)

and replacing commutators with Poisson brackets.

In the following sections we will investigate the collec-

tive dynamical response of multilevel atoms in both mean

ﬁeld limits. We will show that the dynamical response

could be highly susceptible to quantum correlations in

the multilevel atom case, while it is insensitive in the

two level case.

C. Initial states

In this work we derive general results which can be

applied to any state of the form given in Eq. (5). As

discussed in Sec. II B we distinguish two diﬀerent clas-

sical limits, arising in the large Llimit, corresponding

to the one-body reduced density matrix Σ(j)on site j

being pure or mixed, respectively. For the sake of con-

creteness we now present a few states corresponding to

the two cases discussed above. The ﬁrst two states are a

bosonic coherent state and a SU (N) spin-coherent state,

both having no quantum correlations and a one-body re-

duced density matrix that is pure. While the other is

a multimode Schr¨odinger cat state, whose one-body re-

duced density matrix on a given site is mixed reﬂecting

the presence of quantum correlations.

1. Coherent states

The most general bosonic coherent state |ψjion a given

site jreads

|ψji= exp γj·ˆ

b†

j−h.c.|0i ≡ |eγji,

γj≡(γ1,j , γ2,j , . . . , γN,j ),

b†

j≡(ˆ

b†

1,j ,ˆ

b†

2,j ,...,ˆ

b†

N,j ),

(8)

with γn,j ∈Cthe amplitude of the bosonic coherent state

on the n-th level and site j, so that the average number of

particles per site is PN

n=1 |γn,j |2=Na/L. We highlight

that the state in Eq. (8) does not have an exact number

of particles. Nonetheless, since the ﬂuctuations of the

number of particles are subleading with respect to the

mean in the limit we consider (Na/L → ∞), the mean

ﬁeld treatment is unaﬀected. Such a state has a pure

single particle reduced density matrix, and will have an

evolution captured by a mean ﬁeld limit characterized by

the classical variables bn,j and a.

2. SU(N)spin-coherent states

The second example of state with pure one-body re-

duced density matrix is given by the superposition:

|ψji=PN

n=1 γn,jˆ

b†

n,j |0i, which has one excitation per

site. Once again, in this case, the mean ﬁeld limit ap-

plies. Furthermore, the choice to truncate to one parti-

cle per site is insensitive of particles’ statistics: either a

fermion or boson could be the single particle occupying

the site, as we further elaborate in the concluding sec-

tion, Sec. VII B. Such a state is the single particle limit

of the more general Na/L particle SU (N) spin-coherent

state [92] deﬁned by

|ψji=1

p(Na/L)! N

n=1

γn,jˆ

b†

n,j !Na/L

|0i,(9)

which again has a pure one-body reduced density ma-

trix reﬂecting a lack of quantum correlations. Thus, the

dynamics of the classical variables bn,j and aperfectly

describe the dynamics of both the bosonic and SU(N)

spin coherent states in the limit of a large number of

bosons Na. Below we will present numerical results sim-

ulating these classical dynamics; they can be interpreted

as describing the evolution of either of these two states.

For the sake of simplicity, we will explicitly refer to these

states as coherent states.

3. Schr¨odinger cat states

To consider a state in which the full two point corre-

lations of the bosons, Σ(j)

n,m, must be considered, we add

quantum correlations on site j. This ensures that the

one body reduced density matrix is not pure and can-

not be written in the mean ﬁeld approximation, Σ(j)

n,m 6=

b∗

n,j bm,j . As an example, we consider a state where

each site is initialized in a ‘multimode Schr¨odinger cat

state’ [100, 101], which are the multimode generaliza-

tion of ‘entangled coherent states’ [102–104], given by

the superposition of two bosonic coherent states |eγ(m)i

with average occupation Na/L, deﬁned in Eq. (8), with

m={1,2}

|ψji=1

D|eγ(1)

ji+|eγ(2)

ji.(10)

Here Dis a normalization constant. If |heγ(1)

j|eγ(2)

ji| = 1

the state in Eq. (10) reduces to the one in Eq. (8). In-

stead, if |heγ(1)

j|eγ(2)

ji| <1, the one-body reduced density

matrix is mixed, reﬂecting the presence of quantum corre-

lations on site j(hˆ

b†

n,jˆ

bm,j ic≡ hˆ

b†

n,jˆ

bm,j i−hˆ

b†

n,j ihˆ

bm,j i 6=

0). We anticipate that the collective dynamical response

could be highly susceptible to quantum correlations in

the multilevel atom case, while they do not play a role in

the two-level case. As an instance, we discover the on-

set of chaos as |hˆ

b†

n,jˆ

bm,j ic|increases in the N= 3 levels

case (cf. Sec. IV B). We highlight that quantum features

of the state can only enter in initial conditions since dy-

namics are incapable of building quantum correlations in

the mean ﬁeld limit (cf. Sec. II B).

III. CLASSIFICATION OF DYNAMICAL

RESPONSES

The main purpose of this work is to investigate and

classify the dynamical response of collective observables

in multilevel cavity QED systems in the long-time limit.

Speciﬁcally, we investigate the dynamics of the magni-

tude of the intra-level average coherences, deﬁned as

|PL

j=1 Σ(j)

n,m|/Na(for n6=m). To this end, we formu-

late the dynamical reduction hypothesis, which general-

izes a similar procedure used for the integrable SU (2)

limits of Eq. (1) and Eq. (2). The hypothesis conjec-

tures that the dynamics of collective observables can be

captured by the Hamiltonian dynamics of a few eﬀective

collective degrees of freedom (DOFs). In the integrable

case, the eﬀective Hamiltonian has been used to quanti-

tatively predict the dynamical responses observed, which

include relaxation and persistent oscillations either peri-

odic or aperiodic [56–66, 105]. Despite lack of integra-

bility, we still obtain in our case not simply relaxation,

but also the persistent oscillatory responses present in

the integrable case, together with the possibility to de-

velop chaos (see Fig. 3 for example) [105–107]. Due to

the generic non-integrable nature of multilevel systems

an exact procedure for extracting the eﬀective model is

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CraftingthedynamicalstructureofsynchronizationbyharnessingbosonicmultilevelcavityQEDRiccardoJ.Valencia-Tortora,1,ShaneP.Kelly,1TobiasDonner,2GiovannaMorigi,3RosarioFazio,4,5andJamirMarino11InstitutfurPhysik,JohannesGutenberg-UniversitatMainz,D-55099Mainz,Germany2InstituteforQuantumElectronics,Eid...

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Crafting the dynamical structure of synchronization by harnessing bosonic multilevel cavity QED Riccardo J. Valencia-Tortora1Shane P. Kelly1.pdf

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Crafting the dynamical structure of synchronization by harnessing bosonic multilevel cavity QED Riccardo J. Valencia-Tortora1Shane P. Kelly1

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