Quantum Chaos in the Extended Dicke Model

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arXiv:2210.05869v1 [quant-ph] 12 Oct 2022

Article

Quantum Chaos in the Extended Dicke Model

Qian Wang 2,1

Citation: Wang, Q.

.Preprints 2022,1, 0.

https://doi.org/

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Accepted:

Published:

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1CAMTP-Center for Applied Mathematics and Theoretical Physics, University of Maribor, SI-2000 Maribor,

Slovenia

2Department of Physics, Zhejiang Normal University, Jinhua 321004, China

Abstract: We systematically study the chaotic signatures in a quantum many-body system con-

sisting of an ensemble of interacting two-level atoms coupled to a single-mode bosonic ﬁeld, the

so-called extended Dicke model. The presence of the atom-atom interaction also leads us to explore

how the atomic interaction affects the chaotic characters of the model. By analyzing the energy spec-

tral statistics and the structure of eigenstates, we reveal the quantum signatures of chaos in the model

and discuss the effect of the atomic interaction. We also investigate the dependence of the boundary

of chaos extracted from both eigenvalue-based and eigenstate-based indicators on the atomic inter-

action. We show that the impact of the atomic interaction on the spectral statistics is stronger than

on the structure of eigenstates. Qualitatively, the integrablity-to-chaos transition found in the Dicke

model is ampliﬁed when the interatomic interaction in the extended Dicke model is switched on.

Keywords: quantum chaos; extended Dicke model; spectral statistics; eigenstate structure

1. Introduction

In recent years, the study of quantum chaos in many-body systems has attracted

much attention, both theoretically and experimentally, in different ﬁelds of physics, such

as statistical physics [1–5], condensed matter physics [6–13], high energy physics [14–19],

as well as quantum information science [20–26]. To some extent, this great interest in quan-

tum many-body chaos is due to the close connections of chaos to several fundamental

questions that arise in current theoretical and experimental studies. Although a full under-

standing of quantum many-body chaos is still lacking, much progress has been achieved.

It is known that chaos in interacting quantum many-body systems can lead to thermaliza-

tion [1–3], the fast scrambling of quantum information [14,27–29], an exponential growth

of quantum complexities [18,30–35], and diffusive transport [36].

The notion of chaos in the classical regime is usually deﬁned by the so-called butter-

ﬂy effect, namely the exponential separation of iniﬁtesimally nearby trajectories for initial

perturbations [37,38]. However, as the concept of trajectory is ill-deﬁned in quantum me-

chanics, the deﬁnition of quantum chaos remains an open question. Therefore, to probe the

signatures of chaos in quantum many-body systems becomes a central task in the studies

of quantum many-body chaos. To date, many complementary detectors of quantum chaos

and the limits of their usefulness have been widely investigated in literature [28,31–33,39–

54]. Important model systems in this context are billiards [40,55]. Another task, which has

recently also drawn great interest, is to unveil different factors that affect the chaotic prop-

erties of quantum many-body systems. While the impacts of the strength of disorder and

the choice of initial states on the development of quantum chaos in various many-body

systems have been extensively explored [4,56–61], more works are required in order to get

deeper insights into the universal aspects of quantum many-body chaos.

In the present work, we analyze the emergence of chaos in the extended Dicke model.

There are several different versions of the extended Dicke model [62–65]. Here, we focus

on the one that has been discussed in Ref. [63]. Different from the original Dicke model

2 of 14

[66], which consists of an ensemble of noninteracting two-level atoms interacts with a

single bosonic mode, the atoms in our considered extended Dicke model are permitted

to interact. This allows us to analyze the effects of the atomic interaction on the degree

of chaos of the model. Previously, the role of the atom-ﬁeld coupling in the Dicke model

for the emergence of quantum chaos has been investigated [67–70], while here we explore

how this transition is affected by additional atomic interaction. By performing a detailed

analysis of the energy spectral statistics and the structure of eigenstates, we systematically

study both the chaotic signatures of the extended Dicke model and examine the effect

of the atomic interaction on the chaotic features in the model. We show how the atomic

interaction affects the spectral statistics and the structure of eigenstates, respectively.

The article is structured as follows. The model is introduced in Sec. 2. The inﬂuences

of the atomic interaction on the energy spectral statistics are discussed in Sec. 3. The de-

tailed investigation of the consequence of the atomic interaction on the structure of eigen-

states is presented in Sec. 4. Finally, we conclude in Sec. 5with a brief summarize of our

results and outlook.

2. Extended Dicke model

As an extension of the original Dicke model [66–68], the extended Dicke model stud-

ied here consists of Nmutual interacting two-level atoms with energy gap ω0coupled

to a single cavity mode with frequency ω. By employing the collective spin operators

Jx,y,z=∑N

i=1ˆ

σ(i)

x,y,z(ˆ

σx,y,zare the Pauli matrices), the Hamiltonian of the extended Dicke

model can be written as (hereafter we set ¯h=1) [63,71]

H=ωa†a+ω0Jz+2λ

√NJx(a+a†) + κ

NJ2

z, (1)

where a(a†)denotes the bosonic annihilation (creation) operator, λis the coupling stength

between atom and ﬁeld, and κrepresent the strength of the atomic interaction.

The conservation of total spin operator J2=J2

x+J2

y+J2

zfor the Hamiltonian (1) leads

to the Hamiltonian matrix being block diagonal in J2representation. In this work, we will

focus on the maximum spin sector j=N/2, which involves the experimental realizations

and includes the ground state. Moreover, the commutation between Hamiltonian (1) and

the parity operator Π=eiπ(j+Jz+a†a)enables us to further separate the Hamiltonian matrix

into even- and odd-parity subspaces. Here, we will restrict our study to the even-parity

subspace.

To numerically diagonalize the Hamiltonian (1), we work in the usual Fock-Dicke

basis {|n,mi} ={|ni ⊗ |j,mi}. Here, |niare the Fock states of bosonic mode with n=

0, 1, 2, . . ., ∞and |j,mirepresent the so-called Dicke states with m=−j,−j+1, . . ., j. Then,

the elements of the Hamiltonian matrix are given by

hn′,m′|H|n,mi=(nω+mω0)δn′,nδm′,m+κ

Nm2δm′,m+λ

√Nh√nδn′,n−1+√n+1δn′,n+1i

×qj(j+1)−m(m+1)δm′,m+1+qj(j+1)−m(m−1)δm′,m−1. (2)

We remark that the value of nis unbounded from above, the actual dimension of the

Hilbert space is inﬁnite, regardless of the value of j. In practice, we need to cut off the

bosonic number states at a larger but ﬁnite value Nc. Moreover, the dependence of the

chaoticity in the Dicke model on the energy [63,72,73] further implies that it is also nec-

essary to cut off the energy in order to get the ﬁnite number of considered states. In our

numerical simulations, we set Nc=320 and restrict our analysis on the eigenstates with

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Figure 1. Level spacing distribution of the extended Dicke model for several combinations of λand κ.

The considered energy levels are the ones that have energies E/N∈[0.4, 4]. The total atom number

is N=2j=32 and the cut off in bosinic Hilbert space is Nc=320. The Poissonian (blue solid

lines) and Wigner-Dyson distributions (red dashed lines) are, respectively, plotted in each panel for

comparison. Other parameters are: ω=ω0=1. All quantities are dimensionless.

energies E/N∈[0.4, 4], the convergence of our results has been carefully examined. For

our selected energy interval, we have checked that our main results are still hold for other

choices of Nc, as long as N≥16 and the convergence of Fock-Dicke basis is fulﬁlled.

The extended Dicke model exhibits both ground state and excited state quantum

phase transitions and displays a transition from integrable to chaotic behavior with in-

creasing the system energy, like in the Dicke model. The features of these transitions have

been extensively investigated in the semiclassical regime [63]. It is worth to mention that

several possible experimental realizations of the extended Dicke model have been pointed

out in Refs. [63,71,74]

3. Energy spectrum statistics

In this section we will explore the transition from integrable regime to chaos in the

extended Dicke model by analyzing its energy level spacing distribution. In our study,

we focus on the energy levels with energies change from E/N=0.4 to E/N=4. We

will compare our results to the level distributions of fully integrable and chaotic cases,

respectively. We are aiming to characterize the quantum signatures of chaos in the model

and unveil the impact of the atomic interaction on its chaotic properties.

3.1. Level spacing statistics

As the most frequently used probe of quantum chaos, the distribution P(s)of the

spacings sof the consecutive unfolded energy levels quantiﬁes the degree of correlations

between levels. For integrable systems, where the energy levels are allowed to cross, the

distribution P(s)is given by the Poissonian distribution [75],

PP(s) = exp(−s). (3)

On the other hand, the energy levels in chaotic systems exhibit level replusion and the

distribution P(s)follows the Wigner-Dyson distribution [39–41]. For the systems with

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摘要：

arXiv:2210.05869v1[quant-ph]12Oct2022ArticleQuantumChaosintheExtendedDickeModelQianWang2,1Citation:Wang,Q..Preprints2022,1,0.https://doi.org/Received:Accepted:Published:Publisher’sNote:MDPIstaysneutralwithregardtojurisdictionalclaimsinpublishedmapsandinstitutionalafﬁl-iations.1CAMTP-CenterforApplied...

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