# Persisting quantum effects in the anisotropic Rabi model at thermal
equilibrium

He-Guang Xu  School of Physics, Dalian University of Technology, 116024
Dalian, China  V. Montenegro<EMAIL_ADDRESS>Institute of
Fundamental and Frontier Sciences, University of Electronic Science and
Technology of China, Chengdu 610051, China  Gao Xianlong<EMAIL_ADDRESS>Department of Physics, Zhejiang Normal University, Jinhua 321004, Zhejiang, P.
R. China  Jiasen Jin<EMAIL_ADDRESS>School of Physics, Dalian University
of Technology, 116024 Dalian, China  G. D. de Moraes Neto<EMAIL_ADDRESS>Department of Physics, Zhejiang Normal University, Jinhua 321004, Zhejiang, P.
R. China

###### Abstract

Quantum correlations and nonclassical states are at the heart of emerging
quantum technologies. Efforts to produce long-lived states of such quantum
resources are a subject of tireless pursuit. Among several platforms useful
for quantum technology, the mature quantum system of light-matter interactions
offers unprecedented advantages due to current on-chip nanofabrication,
efficient quantum control of its constituents, and its wide range of
operational regimes. Recently, a continuous transition between the Jaynes-
Cummings model and the Rabi model has been proposed by exploiting anisotropies
in their light-matter interactions, known as the anisotropic quantum Rabi
model. In this work, we study the long-lived quantum correlations and
nonclassical states generated in the anisotropic Rabi model and how these
indeed persist even at thermal equilibrium. To achieve this, we thoroughly
analyze several quantumness quantifiers, where the long-lived quantum state is
obtained from a dressed master equation that is valid for all coupling regimes
and with the steady state ensured to be the canonical Gibbs state.
Furthermore, we demonstrate a stark distinction between virtual excitations
produced beyond the strong coupling regime and the quantumness quantifiers
once the light-matter interaction has been switched off. This raises the key
question about the nature of the equilibrium quantum features generated in the
anisotropic quantum Rabi model and paves the way for future experimental
investigations, without the need for challenging ground-state cooling.

##  I Introduction

The resource theory of quantum information [1, 2] demonstrates that quantum
correlations found in nonclassical states are of utmost importance for
performing quantum processing tasks [3, 4]. From a theoretical perspective,
nonclassical states have proven highly valuable in investigating decoherence
[5], witnessing the potential quantum nature of gravity [6, 7], studying the
quantum-to-classical transition [8], enabling remote quantum control of the
weak value amplification [9], and serving as a powerful tool for witnessing
quantum phase transitions in critical systems [10]. From a practical
standpoint, quantum correlations have demonstrated their pivotal role as a
fundamental element in emerging quantum technologies [11, 12], including
secure quantum communication [13, 14], quantum sensing [15, 16, 17, 18, 19,
20, 21], and quantum simulation [22]. Nonetheless, the intrinsic sources of
noise and decoherence inherent in quantum dynamics render the production of
long-lived quantum states necessary for quantum technologies [23, 24]. Efforts
to advance efficient techniques for generating and protecting nonclassical
states against quantum noise [25] encompass approaches such as decoherence-
free subspaces [26, 27, 28], dynamical decoupling [29, 30], and reservoir
engineering schemes [31, 32, 33]. Therefore, proposing schemes that facilitate
the generation of long-lived highly correlated nonclassical states is one of
the primary tasks within the quantum information field.

Quantum systems undergoing light-matter interactions stand as diverse
platforms for generating nonclassical states and executing quantum information
tasks [34, 35]. The recently achieved ultrastrong-coupling regime (USC) [36,
37, 38] and deep strong coupling regime (DSC) [39, 40] of light-matter
interactions have advanced quantum correlation generation beyond the strong
coupling regime (SC) [41, 42, 43, 44, 45], i.e. when the light-matter coupling
strength exceeds both the decay rates and the natural frequencies of the
systems. Notably, a clear distinction between USC/DSC and the SC regime is the
emergence of nonclassicality in the ground state of the system, manifested
through squeezing and entanglement [46, 47]. Moreover, within the USC regime,
thermal photons can exhibit anti-bunching behavior [48], and emissions from
the matter subsystem (e.g., a two-level atom) can display photon bunching
[49], whereas the behavior is reversed in the SC regime.

The cornerstone model describing light-matter interaction is composed of a
two-level atom (qubit) coupled to a quantized single-mode cavity field through
dipole interaction, known as the quantum Rabi model (QRM) [50, 51, 52]. The
QRM has been the subject of extensive theoretical work in quantum optics [53],
for the generation of quantum entanglement [54], as an Otto quantum engine in
quantum thermodynamics [55], to exhibit the emergence of quantum phase
transitions [56, 57, 58], and motivated the development of a novel operational
criterion for integrability [59, 60]. Interestingly, the QRM transforms into
the Jaynes-Cummings model (JCM) [61] in situations where the natural
frequencies of the system significantly surpass the bare light-matter coupling
strength. In this scenario, one can invoke the rotating-wave approximation
(RWA), permitting the retention of the rotating-wave (RW) terms while omitting
the counter-rotating-wave (CRW) terms. However, this approximation becomes
invalid once the system transit towards the USC regime [60, 62, 63, 64, 65].

This motivates the introduction of a continuous transition between the JCM and
the QRM by varying the RW and CRW coupling strengths, known as the anisotropic
quantum Rabi model (AQRM). The AQRM has been explored in both closed systems
[66, 67] and open systems [68, 69], with novel quantum technological schemes
such as criticality-enhanced quantum sensing [70] and squeezed vacuum state
laser [71]. Notably, an exact solution for the AQRM with a biased term has
been recently derived [72], and certain analytical solutions have been found
via transcendental function extension [72, 73] and transformation techniques
[74, 75]. Experimental realizations of the AQRM have been achieved in two-
dimensional quantum wells [76], cavity quantum electrodynamics [77], and
superconducting circuits [72]. The flexibility to adjust the RW and CRW
coupling strengths renders the AQRM to be highly versatile. Indeed, this
grants the ability to explore a plethora of coupling regimes, thereby enabling
the creation of highly-correlated nonclassical states. A pivotal question
arises: can we generate long-lived highly-correlated nonclassical states that
persist even in the presence of a reservoir at finite temperature? This
question is of paramount importance for current experimental platforms.

In this work, we demonstrate that long-lived highly-correlated nonclassical
states of the AQRM can indeed persist even in the presence of a reservoir at
finite temperature. To support our theoretical findings, we use a quantum
Markovian master equation to describe the evolution of the AQRM. This approach
remains applicable across all coupling regimes, including degenerate points in
the energy spectrum, i.e. without resorting to secular approximation.
Consequently, such a quantum open system genuinely approaches a thermal Gibbs
state as its steady state. Furthermore, through a series of complementary
analyses, we characterize the quantumness of the steady state in the finite-
temperature-coupling strength phase diagram, showing that highly-correlated
nonclassical states can be achieved across a wide parameter region.

The rest of the paper is structured as follows: In Sec. II we present the open
system description of our model. In Sec. III we briefly defined the measures
and witnesses of nonclassicality thorughout our work. In Sec. IV, we outline
our results, including the effects of temperature in the system. Finally, we
present our concluding remarks in Sec. VI.

##  II The model and methods

The Hamiltonian of the AQRM [72] is ($\hbar=1$)

$\hat{H}=\omega\hat{a}^{\dagger}\hat{a}+\frac{\Delta}{2}\hat{\sigma}_{z}+\lambda_{1}(\hat{a}\hat{\sigma}^{+}+\hat{a}^{\dagger}\hat{\sigma}^{-})+\lambda_{2}(\hat{a}^{\dagger}\hat{\sigma}^{+}+\hat{a}\hat{\sigma}^{-}).$
(1)

This quantum system couples a two-level system (qubit) with a single-mode
bosonic field undergoing both the RW and the CRW interactions terms unevenly.
Here, the qubit is described by Pauli matrices
$\hat{\sigma}_{x,y,z},\hat{\sigma}^{\pm}=(\hat{\sigma}_{x}\pm
i\hat{\sigma}_{y})/2$ and transition frequency $\Delta$, whereas the boson
field is described with frequency $\omega$ and annihilation (creation)
operator $\hat{a}$ ($\hat{a}^{\dagger}$). Finally, $\lambda_{1}$
($\lambda_{2}$) accounts for the light-matter coupling strength for the RW
(CRW) interaction, where $\lambda_{2}=0$ is the JCM limit, $\lambda_{1}=0$ is
the anti-Jaynes-Cummings model (AJCM), and $\lambda_{1}=\lambda_{2}$ is the
isotropic QRM.

One key aspect of the AQRM in Eq. (1) is that preserves the parity symmetry
$\mathbb{Z}_{2}$ as in the isotropic model [38]. This fact can be evidenced by
considering the total number of excitations operator
$\hat{n}=\hat{a}^{\dagger}\hat{a}+\hat{\sigma}^{+}\hat{\sigma}^{-}$ and the
parity operator $\hat{\pi}=\mathrm{exp}\left(i\pi\hat{n}\right)$. One can
readily notice that, as opposed to the JCM, for the AQRM case
$[\hat{H},\hat{n}]\neq 0$ and $[\hat{H},\hat{\pi}]=0$. The above has two
immediate consequences: (i) even in the ground state, the expected mean number
of qubit and boson excitations is non-zero, and (ii) it implies that
$\hat{\pi}$ possesses two eigenvalues, $\left\langle\hat{\pi}\right\rangle=\pm
1$, depending on whether the total number of excitations are even or odd. This
in turns enable to solve the AQRM analytically by similar techniques used for
the isotropic case, for instance, the Bargmann-Fock space of analytic
functions and the Bogoliubov operator approach.

To gain a clearer understanding of the energy levels of the AQRM, in Fig. 1,
we plot the energy spectrum in relation to the ground state energy as a
function of the light-matter coupling $\lambda_{1}$ ($\lambda_{2}$) for
several coupling ratios $\lambda_{2}/\lambda_{1}$ ($\lambda_{1}/\lambda_{2}$).
As seen from the figure, the spectrum exhibits level crossings where the
ground state parity changes sign (see green circles), leading the system to
undergo infinite first-order quantum phase transitions as the coupling
strength increases. In addition to the above degenerate crossing, several
other non-degenerate and quasi-degenerate points can be observed. The analysis
of the AQRM spectrum, as shown in Fig. 1, is important for understanding the
physics underlying the emergence of thermal correlations, a study that we will
address in later sections.

Figure 1: Energy level differences are plotted as functions of the light-
matter coupling strengths $\lambda_{1}$ ($\lambda_{2}$) for different coupling
ratios $\lambda_{2}/\lambda_{1}$ ($\lambda_{1}/\lambda_{2}$). In (c) and (h),
the green circles represent the (degenerate) crossing of the energy levels and
the plus and minus signs denote the change of parity of the energy levels.
Other system parameters are specified as $\Delta=1$ and $\omega=1$.

As stated before, the central objective of our work is to study the quantum
correlations and quantum nonclassicality, which remain present even at thermal
equilibrium —the so-called thermal correlations and thermal nonclassicality of
the AQRM, respectively. See quantum thermalization and thermal entanglement in
the open JCM [78] and QRM [79] as particular cases of the AQRM. To do so, we
obtain the AQRM steady-state when such a system interacts unavoidably with a
reservoir at temperature $T$. It is essential to point out that, even though
the standard Born-Markov master equation accurately describes the quantum
state within the weak coupling regime $\lambda_{i}\ll\\{\omega,\Delta\\}$, it
breaks down for stronger coupling regimes such as the USC and DSC regimes [80,
37, 38]. Consequently, the steady-state at thermal equilibrium cannot be fully
captured using the standard master equation in those regimes. To ensure that
the steady-state is the actual quantum state at thermal equilibrium for an
arbitrary set of coupling strengths $\lambda_{i}$, one needs to switch to a
Born-Markov master equation in a dressed picture [81], see Refs. [82, 83] for
a similar approach in the JC case. In this more suitable representation, the
quantum jumps occur between the dressed eigenstates ladder of the full
Hamiltonian as opposed to the standard master equation where the upward and
downward jumps only consider the free energy of the Hamiltonian.

We emphasize that, the AQRM spectrum (see Fig. 1) shows non-degenerate and
degenerate eigenvalues which are appropriately modeled by the secular
approximation in the derivation of the Markovian master equation, but it fails
in the quasi-degenerate case. Following the steps of Ref. [84], we can derive
a valid master equation for the entire spectrum of the Hamiltonian Eq. (1), as
long as we consider weak damping, a high bath cut-off frequency, and a flat
spectral density, these are sufficient conditions to ensure positivity and
Markovianity. Indeed, the dissipative dynamics undergoes the following dressed
master equation

$\displaystyle\frac{d}{dt}\hat{\rho}$ $\displaystyle=$
$\displaystyle-i[\hat{H},\hat{\rho}]+\sum_{\begin{subarray}{c}u=a,\sigma^{-}\\\
k<j\end{subarray}}\\{\Gamma^{jk}_{u}n_{u}(\Delta_{jk})\mathfrak{D}[|\phi_{j}{\rangle}{\langle}\phi_{k}|,\hat{\rho}]$
(2)
$\displaystyle+\Gamma^{jk}_{u}[1+n_{u}(\Delta_{jk})]\mathfrak{D}[|\phi_{k}{\rangle}{\langle}\phi_{j}|,\hat{\rho}]\\},$

with Lindbladian superoperator term defined as

$\mathfrak{D}[\hat{O},\hat{\rho}]=\frac{1}{2}[2\hat{O}\hat{\rho}\hat{O}^{{\dagger}}-\hat{\rho}\hat{O}^{{\dagger}}\hat{O}-\hat{O}^{{\dagger}}\hat{O}\hat{\rho}].$
(3)

In Eq. (2), $|\phi_{k}{\rangle}$ is the eigenvector of the AQRM, namely

$\hat{H}|\phi_{k}{\rangle}=E_{k}|\phi_{k}{\rangle},$ (4)

where $E_{k}$ is the $k$th eigenenergy associated to the eigenstate
$|\phi_{k}{\rangle}$. We also define the dissipation rates

$\Gamma^{jk}_{u}=\gamma_{u}(\Delta_{jk})|S^{jk}_{u}|^{2},$ (5)

with two explicit contributions,

(i) The spectral function:

$\gamma_{u}(\Delta_{jk})=2\pi\sum_{k}|\lambda_{k,u}|^{2}\delta(\Delta_{jk}-\omega_{k}),$
(6)

where $\lambda_{k,u}$ accounts for the $u$th thermal bath coupled to a single-
mode boson field with the frequency $\omega_{k}$, and energy gap

$\Delta_{jk}=E_{j}-E_{k}.$ (7)

(ii) And the transition coefficients:

$\displaystyle{S}^{jk}_{q}$ $\displaystyle=$
$\displaystyle{\langle}\phi_{j}|(\hat{\sigma}_{+}+\hat{\sigma}_{-})|\phi_{k}{\rangle},$
(8) $\displaystyle{S}^{jk}_{c}$ $\displaystyle=$
$\displaystyle{\langle}\phi_{j}|(\hat{a}^{{\dagger}}+\hat{a})|\phi_{k}{\rangle}.$
(9)

To satisfy the necessary conditions for the validity of the above master
equation, we consider the Ohmic case:

$\gamma_{u}(\Delta_{jk})=\pi\alpha\Delta_{jk}e^{-|\Delta_{jk}|/\omega_{c}},$
(10)

where $\alpha$ is the coupling strength between the system and the environment
and $\omega_{c}$ is the cutoff frequency of thermal baths.

Finally, the temperature of the bath is encoded in the Bose-Einstein
distribution ($k_{B}=1$)

$n_{u}(\Delta_{jk},T_{u})=\frac{1}{e^{\Delta_{jk}/T_{u}}-1}.$ (11)

It is important to emphasize that, when both reservoirs have the same
temperature $T_{a}=T_{\sigma^{-}}$ (or only one of the subsystems is coupled
to the reservoir) the system reaches a thermal equilibrium. Consequently, the
dressed master equation steady-state solution of Eq. (2) results in the
density matrix of the canonical ensemble [a straightforward numerical
simulation proves that the Gibbs state is indeed the steady-state solution of
Eq. (2)]

$\displaystyle\hat{\rho}_{\mathrm{ss}}=\sum_{k}\frac{e^{-E_{k}/T}}{\mathcal{Z}}|\phi_{k}{\rangle}{\langle}\phi_{k}|,$
(12)

where $\mathcal{Z}=\sum_{k}e^{-E_{k}/T}$ is the partition function and the
steady-state population is

$\displaystyle P_{k}=\frac{e^{-E_{k}/T}}{\mathcal{Z}}.$ (13)

Notice that, for unequal reservoir temperatures $T_{a}\neq T_{\sigma^{-}}$,
the steady solution of the dressed master equation cannot be written in terms
of the canonical ensemble as in Eq. (12). In what follows, we assume equal
temperatures reservoir such that we can always extract the statistical
properties of the quantum state at thermal equilibrium from Eq. (12).

##  III Quantum Correlation and Nonclassicality measures

The concept of quantumness (nonclassicality) is related to the impossibility
of describing physical phenomena by a deterministic or probabilistic classical
theory. To understand the impact of the system’s light-matter anisotropy on
the thermal quantum correlations and thermal nonclassicality, i.e. quantum
correlations and nonclassicality that are present at thermal equilibrium, we
thoroughly study the following quantities: (i) the zero-delay second-order
correlation function $g^{(2)}(0)$, (ii) the bosonic field quadrature squeezing
$\zeta^{2}$, (iii) a phase-space interference macroscopicity measure
$\mathcal{I}(\rho)$ for the quantum state $\rho$, (iv) the negativity
$\mathcal{N}(\rho)$, and (v) the quantum discord $\mathcal{D}(\rho)$. In the
subsequent sections, we briefly explain the above quantities.

###  III.1 Zero-delay second-order correlation function

While the Poissonian and the super-Poissonian photon statistics of a light
beam can be entirely explained in terms of a classical theory of light, the
occurrence of sub-Poissonian photon statistics characterizes the quantumness
of photonic states without a classical counterpart. It is also relevant to
point out that, although this property does not consistently manifest in all
quantum states of a field mode, a state can be classified as nonclassical when
it is present. In this context, sub-Poissonian photon statistics serve as an
authentic signature of the quantum nature of light.

Alternatively, one can classify the quantumness of a light beam by the study
of the probabilities in measuring photons at a detector in a defined time
interval $t_{2}-t_{1}\equiv\tau$, the so-called second-order correlation
function $g^{(2)}(\tau)$. This definition classifies the light beam in a
threefold fashion, namely: (i) $g^{(2)}(0)>1$ bunched light where photons
populate themselves together (classically a chaotic description), (ii)
$g^{(2)}(0)=1$ random photon stream (classically a coherent description), and
(iii) $g^{(2)}(\tau)<1$ antibunched light where photons distribute separately
(with no classical analogy) [85, 86], where we have considered an
infinitesimal zero-delay time window $\tau\rightarrow 0$. Indeed, thermal
photons emitted from non-interacting quantum modes or in the strong coupling
regime is bunched ($g^{(2)}(0)=2$), being the standard example of an
incoherent source of light [85, 86, 87].

The conventional definition of the normalized zero-delay second-order
correlation function is [88]

$g^{(2)}(0)=\frac{\langle(\hat{a}^{\dagger})^{2}(\hat{a})^{2}\rangle}{\langle\hat{a}^{\dagger}\hat{a}\rangle^{2}}.$
(14)

This quantity describes the probability of detecting two photons
simultaneously ($\tau\rightarrow 0$), which is normalized by the probability
of detecting two photons at once within a random photon source. Nonetheless,
this definition holds for weak light-matter couplings, where the intracavity
photons, described by $\hat{a}$, suffice to explain the observed photons
correlation. On the other hand, in the USC regime, where the qubit system
strongly dresses the bosonic field, the second-order correlation function
$g^{(2)}(0)$ is derived from the input-output formalism as [89, 90]

$G^{(2)}(0)=\frac{{\langle}(\hat{X}^{-})^{2}(\hat{X}^{+})^{2}{\rangle}}{{\langle}\hat{X}^{-}\hat{X}^{+}{\rangle}^{2}},~{}$
(15)

where

$\displaystyle~{}\hat{X}^{+}=-i\sum_{k>j}\Delta_{kj}X_{jk}|\phi_{j}{\rangle}{\langle}\phi_{k}|,$
(16)

with $\hat{X}^{-}=(\hat{X}^{+})^{{\dagger}}$, $\Delta_{kj}=E_{k}-E_{j}$ being
the energy gap, and
$X_{jk}={\langle}\phi_{j}|(\hat{a}^{\dagger}+\hat{a})|\phi_{k}{\rangle}$.
Here, ${X}^{+}_{jk}$ describes the transition from the higher eigenstate
$|\phi_{k}{\rangle}$ to the lower one $|\phi_{j}{\rangle}$. Notice that, in
the weak light-matter interaction limit (i.e. $\lambda_{i}\ll 1$), the
operator $\hat{X}^{+}$ is reduced to $\hat{X}^{+}=-i\omega\hat{a}$. Thus, the
correlation function in Eq. (15) simplifies to the conventional case.

One fundamental observation regarding the second-order correlation function in
Eq. (15) is that, in the eigenbasis of the AQRM Hamiltonian shown in Eq. (1),
the dressed light-matter jump operator $\hat{X}$ provides the accurate
expression for the average number of excitations in the ground state:
$\langle\phi_{0}|\hat{X}^{-}\hat{X}^{+}|\phi_{0}\rangle=0$. This is in
contrast to the seemingly incorrect result
$\langle\phi_{0}|\hat{a}^{\dagger}\hat{a}|\phi_{0}\rangle\neq 0$. Note that as
the dressed ground state can be spanned in the bare light-matter basis
$\\{|g\rangle,|e\rangle,|n\rangle\\}$, the dressed ground state will be
composed of certain amount of virtual excitations [91]. To convert the virtual
excitations from the dressed picture into measurable photonic excitations, one
can switch the interaction coupling strength on and off [91].

Indeed, consider a quantum state in the dressed basis $|\phi_{k}\rangle$. Once
the interaction is switched off, the state will still contain the excitations
that can now be detected using the intracavity photon operator $\hat{a}$.
Hence, $\langle\phi_{k}|\hat{a}^{\dagger}\hat{a}|\phi_{k}\rangle\neq 0$ will
accurately describe the number of photonic excitations. It is worth noting
that the non-adiabatic switch on/off needs to be of the order of the inverse
of the relevant frequencies in the system, here ${\omega,\Delta}$ [91].

The conversion between virtual excitations and real photons via switching the
interaction on and off is a consistent way to reconcile the correlation
functions and other quantities that describe the thermal quantum correlations
and nonclassicality in our work. Let us point out that the photons in the
ground state of the USC models are virtual and cannot be detected [33], unless
the coupling is suddenly switched off.

Throughout the paper, whenever a quantifier is defined using standard bosonic
and spin operators, we refer to a scenario in which the coupling has been
turned off after reaching thermal equilibrium.

###  III.2 Quadrature squeezing

The second-order correlation function fails as a nonclassical quantifier in
the super-Poissonian regime. For instance, nonclassical Gaussian states, such
as squeezed states may show either photon bunching or antibunching depending
on whether the amplitude fluctuations are increased or reduced. To reveal the
nonclassicality present in the steady state of the AQRM field mode, we compute
the degree of squeezing. To do so, we introduce the generalized rotated field
quadrature
$\sqrt{2}\hat{x}_{\theta}=\hat{a}e^{-i\theta}+\hat{a}^{\dagger}e^{i\theta}$,
where any two operators that differ by $\theta=\pi/2$ form a conjugate pair
which satisfies the position ($\theta=0$) momentum ($\theta=\pi/2$) commutator
relation. We consider the following squeezing parameter [92]:

$\zeta^{2}=\underset{\theta\in(0,2\pi)}{\mathrm{min}}(\Delta\hat{x}_{\theta})^{2}.$
(17)

Only a value of $\zeta^{2}<1$ indicates bosonic squeezing, whereas
$\zeta^{2}=1$ corresponds to the photonic coherent state of radiation field.
The minimization in the above definition can be easily obtained

$\zeta^{2}=1+2\langle\hat{a}^{\dagger}\hat{a}\rangle-2|\langle\hat{a}^{2}\rangle|,$
(18)

where we have assumed that once the steady-state is reached, the light-matter
interaction is switched off. Note that, Refs. [93, 94] studied the squeezing
of output quadratures in the USC/DSC regime through the dressed light-matter
jump operators for a vacuum input field, that is the thermal equilibrium case
considered here. Interestingly, the authors found that any open system in its
ground state cannot produce output squeezing, even if its ground state is a
squeezed state. Since most of the squeezing in the AQRM is present in the
ground state, these generalized measures leads to zero degree of squeezing in
the thermal state if the coupling is on. For this reason we only focus in
evaluating the degree of squeezing once the light-matter interaction is
switched off, i.e., $\zeta^{2}$.

###  III.3 Phase space interference measure

To further study the nonclassicality of the AQRM steady-state, we consider a
phase-space interference measure that quantifies the degree of macroscopicity
of the state [95], defined as:

$\mathcal{I}=-\frac{\pi}{2}\int\mathrm{d}p\mathrm{d}qW(q,p)\left(\frac{\partial^{2}}{\partial
q^{2}}+\frac{\partial^{2}}{\partial p^{2}}+1\right),$ (19)

where $W(q,p)$ is the Wigner function [53]. The macroscopicity $\mathcal{I}$
takes the value of $0$ for classical states, while for pure quantum states
such as superpositions of coherent states, NOON states, and Fock states, it
corresponds to the average number of photons $\langle n\rangle$. In the case
of the mixed state obtained by Eq. (12), the quantifier is constrained by the
inequality
$\mathcal{I}(\hat{\rho}_{\mathrm{ss}})<\mathrm{Tr}(\hat{\rho}_{\mathrm{ss}}\hat{a}^{\dagger}\hat{a})$.

###  III.4 Negativity

To explore the quantum correlations between the light-matter constituents, we
commence by calculating the quantum entanglement within bipartite systems.
Among the various entanglement quantifiers available, we opt to use the
negativity $\mathcal{N}(\rho)$ [96, 97, 98, 99, 100]

$\mathcal{N}(\rho)=\frac{\|\rho^{T_{A}}\|_{1}-1}{2},$ (20)

where $\rho^{T_{A}}$ is the partial transpose of the quantum state $\rho$ with
respect to subsystem A, $\|Y\|_{1}={\rm Tr}|Y|={\rm Tr}\sqrt{Y^{\dagger}Y}$ is
the trace norm or the sum of the singular value the operator $Y$.
Equivalently, the negativity can be computed as
$\mathcal{N}(\rho)=1/2\sum_{i}(|\varepsilon_{i}|-\varepsilon_{i})$, where
$\varepsilon_{i}$ are the eigenvalues of the partially transposed light-matter
density matrix $\rho$. Note that $\mathcal{N}(\rho)=0$ corresponds to
separable (not entangled) quantum states.

###  III.5 Quantum discord

Quantum entanglement is not the sole manifestation of quantum correlations. In
fact, quantum discord (QD) represents another measure of potential quantum
correlations within a quantum system. This measure of correlation emerges from
the observation that two classically equivalent methods of defining mutual
information yield different outcomes within the quantum domain [101]. The QD
of two subsystems $\mathcal{A}$ and $\mathcal{B}$ can be expressed as [101]:

$\mathcal{D}(\hat{\rho}_{\mathcal{AB}})=S(\hat{\rho}_{A})-S(\hat{\rho}_{\mathcal{AB}})+{{\rm
min}}_{\\{\Pi_{j}^{\mathcal{A}}\\}}S(\hat{\rho}_{\mathcal{B}|\\{\Pi_{j}^{\mathcal{A}}\\}}),$
(21)

where $S(\hat{\rho}_{i})=-{\rm Tr}\hat{\rho}_{i}{\rm log}\hat{\rho}_{i}$ is
the von Neumann entropy for the reduced density matrix, and
$S(\hat{\rho}_{\mathcal{B}|\\{\Pi_{j}^{A}\\}}$ is the entropy conditioned
through performing measurements on the $\mathcal{A}$ system, defined as:

$S(\hat{\rho}_{\mathcal{B}|\\{\Pi_{j}^{A}\\}})=\sum_{j}p_{j}S((\\{\Pi_{j}^{\mathcal{A}}\\}\otimes
I^{\mathcal{B}})\hat{\rho}_{\mathcal{AB}}(\\{\Pi_{j}^{\dagger\mathcal{A}}\\}\otimes
I^{\mathcal{B}})/p_{j}).$ (22)

In the above, $\Pi_{j}^{\mathcal{A}}$ is a von Neumann projection operator on
subsystems $\mathcal{A}$ and $p_{j}={\rm
Tr}(\\{\Pi_{j}^{\mathcal{A}}\\}\otimes
I^{\mathcal{B}}\hat{\rho}_{\mathcal{AB}})$ is the probability with measurement
outcome $j$. Note that, one of the most distinct differences between QD from
entanglement is that QD can be non-zero for certain separable states [102, 9,
103].

##  IV Quantumness at thermal equilibrium

We commence by examining the impact of the light-matter coupling strengths,
$\lambda_{1}$ and $\lambda_{2}$, on the generation of long-lived quantum
correlations in the AQRM. In Figs. 2(a)-(c), we plot the second-order
correlation function using the dressed jump operator $G^{(2)}(0)$ as a
function of the light-matter coupling strength $\lambda_{i}$ ($i=1,2$) for
several coupling ratios $\lambda_{i}/\lambda_{j}$ ($i\neq j$) and a given
temperature $T=0.1\omega$. As the panels (a) and (c) show, $G^{(2)}(0)$
generally transitions from photon antibunching ($G^{(2)}(0)<1$) to bunching
($G^{(2)}(0)>1$) as the light-matter coupling strength increases. In the limit
of the DSC regime, i.e. $\lambda\gg\\{\omega,\Delta\\}$, the correlation
function takes the value $G^{(2)}(0)=2$ which corresponds to thermal photon
emission (regardless of the coupling ratios). In Figs. 2(b)-(d), we depict the
second-order correlation function using the bare jump operator $g^{(2)}(0)$ as
a function of $\lambda_{i}$ ($i=1,2$) for several coupling ratios
$\lambda_{i}/\lambda_{j}$ ($i\neq j$) and the same temperature $T=0.1\omega$.
As the panels (b) and (d) show, the second-order correlation function
$g^{(2)}(0)$ exhibits nonclassical features only for a strong degree of
anisotropy (when the coupling strength ratio is small), i.e
$\lambda_{2}/\lambda_{1}\leqslant 0.3$ and $\lambda_{1}/\lambda_{2}\leqslant
0.3$, respectively. Interestingly, as the coupling ratios
$\lambda_{i}/\lambda_{j}$ increase, the nonclassical features (photonic
antibunching effect) are completely suppressed ($g^{(2)}(0)>1$). Furthermore,
with an increase in the light-matter coupling, $g^{(2)}(0)$ approaches unity
regardless of the choice of coupling ratios. In other words, the thermal
emission now displays the same statistical behavior as a coherent state. Note
that in computing $g^{(2)}(0)$ using the bare bosonic field, we make the
assumption that the coupling strength has been switched off [91] once the
system has reached its thermal equilibrium. Notably, there is a significant
distinction between the scenario in which the correlation function is computed
from $G^{(2)}(0)$ and the one obtained when the light-matter coupling is
switched off, resulting in $g^{(2)}(0)$. This subtlety demonstrates the
intricate impact of virtual photonic excitations on the thermal state
properties of the AQRM.

Figure 2:  Second-order correlation functions: Panels (a) and (c) show
$G^{(2)}(0)$ as a function of the coupling strength $\lambda_{1}$
($\lambda_{2}$) for several choices of coupling ratios
$\lambda_{2}/\lambda_{1}$ ($\lambda_{1}/\lambda_{2}$). The black dotted
horizontal line indicates $G^{(2)}(0)=2$. Panels (b) and (d) depict
$g^{(2)}(0)$ as a function of $\lambda_{1}$ ($\lambda_{2}$) for different
coupling ratios $\lambda_{2}/\lambda_{1}$ ($\lambda_{1}/\lambda_{2}$). The
black dashed–dotted horizontal line corresponds to $g^{(2)}(0)=1$. The other
system parameters are the same as in Fig. 1.

We can gain a clearer understanding of the second-order correlation function
$G^{(2)}(0)$ by considering the energy level crossing and its associated
change in parity, as similarly discussed in Ref. [104].

Consider the particular case of $G^{(2)}(0)$ as a function of $\lambda_{1}$
for a fixed coupling ratio $\lambda_{2}/\lambda_{1}=0.5$, as redrawn in Fig.
3(c) for clarity. The non-analytical (sharp) peaks result from the energy
level crossing depicted in Fig. 3(a) (see Fig. 1 for other choices of
parameters). Indeed, the closing of the energy gap between higher excited
states, namely the parity change between the second ($E_{2}$) and the third
($E_{3}$) energy levels are responsible for the anti-bunching peaks shown in
Fig. 3(c). Furthermore, the closing of the energy gap between the ground state
($E_{0}$) and its first energy level ($E_{1}$) gives rise to the pronounced
photonic bunched peak in Fig. 3(c) (around the value of
$\lambda_{1}\sim\omega$). Remarkably, the closing of the energy gap between
$E_{1}$ and its ground state $E_{0}$ reopens as the coupling strength
increases. This cycle of opening, closing, and reopening dynamics is
responsible for the transition between antibunched-bunched-antibunched
behavior of $G^{(2)}(0)$ shown in Fig. 3(c). A similar analysis can be
conducted for Fig. 3(d), in which we present $G^{(2)}(0)$ as a function of
$\lambda_{2}$ for a fixed coupling ratio $\lambda_{1}/\lambda_{2}=0.5$. In
contrast to the previous scenario, no non-analytic behavior is observed. This
is because although a crossing of energy levels between the second and third
eigenenergies occurs, the energy gap between the ground state and the first
excited state monotonically vanishes as the coupling strength increases. As a
result, a smooth transition towards a photonic bunching effect takes place.

Figure 3: Relationship between energy level crossing and the behavior of the
second-order correlation function $G^{(2)}(0)$. In panels (a) and (b), we
redraw the energy gap $E_{n}-E_{0}$ of the AQRM as a function of $\lambda_{1}$
($\lambda_{2}$) for $\lambda_{2}/\lambda_{1}=0.5$
($\lambda_{1}/\lambda_{2}=0.5$), see more cases of coupling ratios in Fig. 1.
In panels (c) and (d), we show the particular case of $G^{(2)}(0)$ as a
function of $\lambda_{1}$ ($\lambda_{2}$) for $\lambda_{2}/\lambda_{1}=0.5$
($\lambda_{1}/\lambda_{2}=0.5$). The vertical gray lines evidence the
correspondence between sharp peaks of $G^{(2)}(0)$ and the closing of energy
gaps. Other system parameters are chosen as in Fig. 1.

To investigate the degree of photonic squeezing, in Fig. 4(a) [Fig. 4(c)], we
plot the squeezing parameter $\zeta^{2}$ as a function of $\lambda_{1}$
[$\lambda_{2}$] for several coupling ratios $\lambda_{2}/\lambda_{1}$
[$\lambda_{1}/\lambda_{2}$] for a given temperature $T=0.1\omega$. As the
figures show, there is a clear degree of squeezing $\zeta^{2}<1$ of the AQRM
steady-state (after the interaction has been switched off) for certain
coupling ratios and a specific window of light-matter coupling. Note that for
the JCM ($\lambda_{2}/\lambda_{1}=0$), no squeezing is achieved, i.e.
$\zeta^{2}\geq 1$, for all values of $\lambda_{1}$. Conversely, for the QRM
($\lambda_{2}=\lambda_{1}$), the degree of squeezing tends to a photonic
coherent field as the light-matter coupling strength goes towards the USC/DSC
regime. Notably, no photonic squeezing occurs for the high anisotropy regime
($\lambda_{i}/\lambda_{j}\ll 1$, $i\neq j$). For instance, consider
$\lambda_{2}/\lambda_{1}=0.1$ and $\lambda_{1}/\lambda_{2}=0.1$ in Figs. 4(a)
and 4(c), respectively. As seen from the figures, no relevant photonic
squeezing is achieved with the increasing of the light-matter coupling. This
nonclassicality measure is in stark contrast with the second-order correlation
function $g^{(2)}(0)$ (we recall that both cases are evaluated by turning the
coupling off) for which it presents high degree of nonclassicality in the
region of high anisotropy.

Figure 4: Panel (a) [(c)] shows the squeezing parameter $\zeta^{2}$ as a
function of $\lambda_{1}$ [$\lambda_{2}$] for different coupling ratios
$\lambda_{2}/\lambda_{1}$ [$\lambda_{1}/\lambda_{2}$]. The black dashed
horizontal line corresponds to $\zeta^{2}=1$. Panel (b) [(d)] depicts the
macroscopicity interference-based measure $\mathcal{I}(\rho)$ as a function of
$\lambda_{1}$ [$\lambda_{2}$] for several coupling ratios
$\lambda_{2}/\lambda_{1}$ [$\lambda_{1}/\lambda_{2}$]. We consider a given
temperature $T=0.1\omega$ and other system parameters are the same as in Fig.
1.

To further evidence quantum features of the steady-state AQRM field that may
have not been captured by previous quantifiers, in Fig. 4(b) [Fig. 4(d)], we
show the macroscopicity interference-based measure $\mathcal{I}(\rho)$ as a
function of $\lambda_{1}$ [$\lambda_{2}$] for different coupling ratios
$\lambda_{2}/\lambda_{1}$ [$\lambda_{1}/\lambda_{2}$] and $T=0.1\omega$. As
the figures show, the higher degree of interference-based measure generally
occurs in the coupling region $1<\lambda_{i}/\omega<2$ ($i=1,2$) for most of
the coupling ratios of anisotropies, where quantum squeezing did not capture
such a nonclassical behavior, yet it appears to be closer to capturing the
second-order correlation quantifier. Interestingly, as shown in Fig. 4(b)
$\mathcal{I}(\rho)$ increases with increasing anisotropy, thus exhibiting
clear quantum signatures in the DSC regime. In contrast, in Fig. 4(d)
$\mathcal{I}(\rho)$ is maximum for $\lambda_{1}/\lambda_{2}=0.5$, which is a
maximum of nonclassicality that is hidden in the previous measures.

Now, we study the thermal quantum correlations between the photons and the
qubit through the quantifiers of negativity $\mathcal{N}(\rho)$ and quantum
discord $\mathcal{D}(\rho)$. In Fig. 5(a), we plot the steady-state negativity
$\mathcal{N}(\rho)$ as a function of $\lambda_{1}$ for several coupling ratios
$\lambda_{2}/\lambda_{1}$ for $T=0.1\omega$. As the figure shows, at some
coupling ratios such as $\lambda_{2}/\lambda_{1}=0.1,0.3,0.5,0.7$, the
negativity shows a clear dip near to zero between two maxima. This minima near
zero corresponds to the energy level crossing of the ground-state and the
first excited state, see Figs. 1(b)-(d). In the case of the JCM
($\lambda_{2}/\lambda_{1}=0$), the steady-state negativity $\mathcal{N}(\rho)$
is nearly zero for coupling strengths $\lambda_{1}\leq 0.7\omega$. However, as
the coupling strength $\lambda_{1}$ increases, the negativity gradually rises
until it reaches the highest degree of entanglement compared to any other
coupling rate. In the case of the QRM ($\lambda_{2}/\lambda_{1}=1$),
$\mathcal{N}(\rho)$ exhibits a single maximum with the highest degree of
entanglement for coupling strength $\lambda_{1}<\omega$. In Fig. 5(c), we
depict the steady-state negativity $\mathcal{N}(\rho)$ as a function of
$\lambda_{2}$ for different coupling ratios $\lambda_{1}/\lambda_{2}$ at a
fixed temperature of $T=0.1\omega$. As observed in the figure, the
$\mathcal{N}(\rho)$ exhibits a smooth behavior without a sudden drop near
zero. This is because, in contrast to the previous scenario, there is no
energy level crossing between the ground state and the first excited state, as
shown in Figs. 1(g)-(i). While the steady-state negativity of Fig. 5(c) is
qualitatively similar to the steady-state quantum discord $\mathcal{D}(\rho)$
of Fig. 5(d), the situation differs for Figs. 5(a)-(b). Indeed, in Fig. 5(b),
we plot the quantum discord $\mathcal{D}(\rho)$ as a function of $\lambda_{1}$
for several coupling ratios $\lambda_{2}/\lambda_{1}$ for $T=0.1\omega$. As
the figure shows, its overall profile is similar to that of Fig. 5(a), this
means that almost all non-local quantum correlations are due to entanglement.
However, unlike negativity shown in Fig. 5(a), at the crossing of the ground
state and the first excited state the quantum discord $\mathcal{D}(\rho)$ does
not decrease near to zero for some coupling strengths (e.g.
$\lambda_{2}/\lambda_{1}=0.1,0.3,0.5,0.7$). Thus, we can infer: (i) all non-
local quantum correlations are in the form of entanglement; (ii) the quantum
correlations increase with anisotropy, and (iii) for high anisotropy they
increase with the coupling strength presenting its maximum in the DSC regime.

It is worth noting that the system’s behavior for increasing light-matter
coupling, e.g., $\lambda_{i}\gtrsim 3\omega$ ($i=1,2$), qualitatively
resembles the system’s behavior for the decoupled light-matter case, i.e.,
$\lambda_{i}=0$. Such a qualitative correspondence implies a trivial behavior
of the quantifiers for very large light-matter interaction strengths,
$\lambda_{i}\gtrsim 3\omega$ ($i=1,2$), and for anisotropies with values
exceeding $\lambda_{i}/\lambda_{j}\geq 0.3$, where $i\neq j$, see Fig. 1. This
behavior can be understood as, in the depths of the DSC regime, the spectrum
becomes harmonic again (see Fig. 1 for $\lambda_{i}>3\omega$), with the
dressed states forming product states between displaced Fock photonic states
and atomic states that are $x$-polarized, leading to $G^{(2)}(0)=2$. The
nearly equal energy gaps (quasi-harmonic spectrum) and the eigenstates being
product states in the DSC regime suggest that: i) no degree of squeezing and
macroscopicity, ii) the vanishing of quantum entanglement and quantum discord
at all temperatures should be expected (as shown in Figs. 2 to 5).

Figure 5: Panel (a) [(c)] shows the steady-state negativity
$\mathcal{N}(\rho)$ as a function of $\lambda_{1}$ [$\lambda_{2}$] for
different coupling ratios $\lambda_{2}/\lambda_{1}$
[$\lambda_{1}/\lambda_{2}$]. Panel (b) [(d)] depicts the steady-state quantum
discord $\mathcal{D}(\rho)$ as a function of $\lambda_{1}$ [$\lambda_{2}$] for
several coupling ratios $\lambda_{2}/\lambda_{1}$ [$\lambda_{1}/\lambda_{2}$].
We consider a given temperature $T=0.1\omega$ and other system parameters are
the same as in Fig. 1.

##  V robustness of thermal quantumness at higher temperatures

As temperatures rise, thermal fluctuations increase, thereby modifying the
AQRM steady-state as described in Eq. (12). On one hand, higher temperatures
result in an increased expansion of states in the polariton basis, which may
potentially lead to more frequent energy level crossings among higher AQRM
eigenstates. On the other hand, these thermal fluctuations may cause the
system to lose its coherence, potentially having a detrimental impact on the
AQRM steady-state. To explore this non-trivial scenario, which involves a
trade-off between increased energy level interplay and the influence of
thermal fluctuations, we investigate how higher finite thermal bath
temperatures affect the generation of nonclassical and quantum correlations in
the AQRM, while considering all the above quantifiers.

Figure 6: Second-order correlation function $G^{(2)}(0)$ as a function of the
light-matter coupling $\lambda_{i}$ ($i=1,2$) for several choices of coupling
ratios $\lambda_{i}/\lambda_{j}$ ($i\neq j$) and different temperatures $T$.
The black dashed horizontal line corresponds to $G^{(2)}(0)=2$. Other system
parameters are as in Fig. 1.

In Fig. 6, we illustrate the second-order correlation function $G^{(2)}(0)$ as
a function of the light-matter coupling $\lambda_{i}$ ($i=1,2$) for several
choices of coupling ratios $\lambda_{i}/\lambda_{j}$ ($i\neq j$) and different
temperatures $T$. As evident from the figures, the higher the anisotropy, the
more robust the quantum correlations are against temperature effects.
Moreover, it exhibits a larger quantum region as a function of the coupling
strengths. Notably, the correlation function $G^{(2)}(0)$ is prone to thermal
decoherence, as signatures of nonclassicality (antibunching) survives only at
temperatures as high as $T\sim 0.15\omega$, as depicted in all panels of Fig.
6. For higher temperatures, the antibunching effect is largely suppressed with
increasing temperature for the same values of the coupling ratio, suggesting
that thermodynamic fluctuations consistently destroy nonclassical features in
high-temperature regions.

Figure 7: The phase diagram of the squeezing parameter $\zeta^{2}$,
macroscopicity $\mathcal{I}(\rho)$, negativity $\mathcal{N}(\rho)$, and
quantum discord $\mathcal{D}(\rho)$, as functions of the light-matter coupling
$\lambda_{i}$ ($i=1,2$) and the temperature $T$ for different coupling ratios
$\lambda_{i}/\lambda_{j}$ ($i\neq j$). Other system parameters are as in Fig.
1.

In Fig. 7, we show the temperature effects on the four quantifiers, namely
$\zeta^{2}$, $\mathcal{I}(\rho)$, $\mathcal{N}(\rho)$, and
$\mathcal{D}(\rho)$, as functions of the light-matter coupling $\lambda_{i}$
($i=1,2$) and the temperature $T$ for different coupling ratios
$\lambda_{i}/\lambda_{j}$ ($i\neq j$).

In the first row of Fig. 7, we plot the squeezing parameter $\zeta^{2}$ as
functions of $\lambda_{i}$ ($i=1,2$) and $T$ for different
$\lambda_{i}/\lambda_{j}$ ($i\neq j$) values. Specifically, as observed in
panels (1) and (4) of Fig. 7, no squeezing effect is achieved in the specific
cases of the JCM ($\lambda_{2}=0$) and the AJCM ($\lambda_{1}=0$). However,
for other coupling ratios, significant squeezing effects are observed at
temperatures around $T\approx 0.3\omega$. Such a high degree of squeezing is
attained by decreasing the anisotropy, making the Rabi model the ideal
scenario for obtaining this resource.

In the second row of Fig. 7, we depict the macroscopicity interference-based
measure $\mathcal{I}(\rho)$ as functions of $\lambda_{i}$ ($i=1,2$) and $T$
for different $\lambda_{i}/\lambda_{j}$ ($i\neq j$) values. Unlike in the case
of the squeezing parameter, the macroscopicity interference-based measure for
the JCM and AJCM cases is non-zero, as shown in panels (8) and (11) of Fig. 7.
It’s worth noting that macroscopicity takes non-zero values over a broader
range of coupling strengths compared to other ratios at low temperatures
$T<0.1\omega$. However, as the figure shows, this quantifier is more
susceptible to thermal fluctuations originating from the thermal reservoir and
practically disappears for temperatures $T>0.1\omega$.

Lastly, in the third and fourth rows of Fig. 7, we depict steady-state quantum
entanglement $\mathcal{N}(\rho)$ and quantum discord $\mathcal{D}(\rho)$. Both
quantifiers exhibit qualitatively similar behaviors. Similar to the
macroscopicity interference-based measure, both quantifiers are more
pronounced over a broader range of coupling strengths and temperatures. As the
ratio $\lambda_{2}/\lambda_{1}$ ($\lambda_{1}/\lambda_{2}$) increases, the
range in which both quantifiers appear gradually narrows. In the case of
$0<\lambda_{2}/\lambda_{1}<1$, as $\lambda_{1}$ changes, quantum entanglement
$\mathcal{N}(\rho)$ appears in two regions, with a value of zero between these
two regions. This result is due to the energy level crossing between the
ground state and the first excited state. However, for quantum discord
$\mathcal{D}(\rho)$, only one region appears. Interestingly, in the case of
high anisotropy, entanglement persists up to temperatures $T\approx
0.3\omega$, while quantum discord demonstrates greater robustness against
thermal fluctuations up to temperatures $T\lesssim 0.4\omega$.

Our theoretical proposal to reach nonclassical equilibrium states is not just
of academic interests and also meets the present-day nontrivial experimental
challenges, mainly involving the coupling strength and low temperature
environments. On the one hand, the physical model considered is general and
has been implemented in a large number of systems, such as cavity quantum
electrodynamics [77], and superconducting circuits [72]. Among them, the USC
regime has been experimentally demonstrated for more than a decade [36]
($\lambda/\omega>0.1$). Since then, the USC regime has been achieved in
several other systems [38]. On the other hand, the DSC regime is within the
reach of experimental techniques, notably in Ref. [39] in circuit quantum
electrodynamics setup implementing the QRM, for the highest reported strength
of light-matter interaction of $\lambda/\omega\approx 1.34$ with temperature
of approximately 45 mK ($k_{B}T/\hbar\omega\approx 0.16$), where thermal
entanglement was also confirmed.

##  VI Conclusion

In this work, we consider the steady-state of the AQRM by solving a dressed
master equation at thermal equilibrium. Such a master equation in the dressed
basis is valid for any light-matter coupling strength, and where the resulting
steady-state is indeed a Gibbs state. Our findings are two-fold: (i) we
investigated the generation of quantum correlations and nonclassicality in the
AQRM steady-state at thermal equilibrium. Through a comprehensive analysis of
various measures of quantum correlation and nonclassicality, we demonstrate
that quantum effects persist across a broad range of anisotropies and coupling
strengths, even at moderate thermal equilibrium temperatures; and (ii) such
quantum features significantly emerge in the USC regime and at the onset of
the DSC regime, while practically vanishing in the deeper DSC regime. In the
dressed picture, this results in virtual field excitations within the system.
Notably, we show a significant difference in the second-order correlation
function when computed using dressed jump operators compared to using bare
field operators (i.e., when the light-matter coupling is assumed to be
switched off after the system reaches thermal equilibrium). Both the
difference between these quantifiers and the persisting quantum features of
the AQRM steady-state at high temperatures, could allow for experimental
approaches to study the impact of virtual excitations in the USC regime. In
particular, as the quantum features emerging from the system avoid the need
for demanding ground-state cooling.

##  VII Acknowledgements

H.-G. X. and J. J. are supported by National Natural Science Foundation of
China under Grant No. 11975064. V.M. thanks the Postdoctoral Science
Foundation of China (Grant No. 2022T150098)). G.X. acknowledges the support of
NSF of China under Grants No. 12174346.

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