††thanks: These authors contributed equally to this work.††thanks: These
authors contributed equally to this work.

# Anisotropic Rabi model with two-photon relaxation

Hui Li  Jia-Kai Shi  Li-Bao Fan  Zi-Min Li<EMAIL_ADDRESS>Chuan-Cun Shu
<EMAIL_ADDRESS>Institute of Quantum Physics, Hunan Key Laboratory of
Nanophotonics and Devices, Hunan Key Laboratory of Super-Microstructure and
Ultrafast Process, School of Physics, Central South University, Changsha
410083, China

###### Abstract

The interplay of three light-matter interaction processes - rotating and
counter-rotating interactions and two-photon relaxation of the light field -
is a topic of interest in quantum optics and quantum information processing.
In this work, we theoretically investigate the three light-matter interaction
processes using the anisotropic Rabi model, which accounts for different
strengths of rotating and counter-rotating interactions and the unique
occurrence of photon escape exclusively in pairs. By numerically solving the
Lindblad master equation, we analyze the excitation-relaxation dynamics and
derive a non-Hermitian effective Hamiltonian to gain further physical
insights. To explore the individual effects of these interactions, we examine
three analytically tractable limits of the effective Hamiltonian. Our analysis
reveals that the three competitive light-matter interaction processes exhibit
sensitivity to parity, leading to intriguing phenomena in both transient and
steady states. Particularly interesting dynamical patterns resembling quantum
phase transitions emerge when these three interaction terms compete. This work
deepens the understanding of ultrastrong light-matter interaction in open
quantum systems and offers valuable insights into cavity-based quantum
computations.

##  I Introduction

The quantum Rabi model (QRM), which characterizes the dynamic interplay
between a two-level quantum system (referred to as a qubit) and a single-mode
quantized light field confined within a cavity, stands as one of the most
elementary interaction physical models [1, 2, 3, 4, 5, 6]. Despite its
fundamental nature, the QRM manifests a diverse array of physical phenomena
and has discovered practical utility across an extensive spectrum of
scientific disciplines. These include, but are not limited to, the realms of
quantum optics [7], condensed matter physics [8, 9], and the forefront of
cutting-edge quantum technologies [10, 11].

The QRM features a competitive interplay between two distinct modes of light-
matter interaction: the “rotating terms” (RTs) and the “counter-rotating
terms” (CRTs). The RTs characterize processes involving the exchange of
excitations between the qubit and the cavity, preserving the total excitation
count. In contrast, the CRTs give rise to events where both a qubit excitation
and a photon are simultaneously created or annihilated, maintaining only the
parity of the excitation number. The conservation of parity within the QRM
leads to a multitude of significant physical consequences, including its
solvability [2, 12, 5, 13, 14], the emergence of level crossing points [15,
16], the presence of hidden symmetries [17, 18, 19], and the occurrence of a
super-radiant phase transition [20, 21, 22].

While most research on the QRM has focused on idealized closed systems without
relaxation, real-world situations involve environmental influences. Accounting
for the environment requires the use of the Lindblad master equation. In
certain cases, a non-Hermitian effective Hamiltonian can be derived to
describe the system in the Schrödinger formalism, offering deeper physical
insights. For example, an intriguing two-photon relaxation process has been
postulated and investigated [23, 24, 25]. In this context, the escape of
photons from a cavity occurs exclusively in pairs, preserving the conservation
of parity within the QRM. As a consequence, the excitation-relaxation dynamics
of the system, including both transient and steady states, exhibit a
sensitivity to parity, giving rise to various remarkable phenomena that are
absent in the closed QRM. The two-photon relaxation mechanism has been
proposed to facilitate universal quantum computing. Specifically, it has been
theoretically and experimentally demonstrated that any initial state of the
cavity, subjected to two-photon relaxation, evolves into two Schrödinger cat
states, resulting in a qubit that relies on the characteristics of a cavity
[25, 26, 27, 28, 29, 30, 31]. The intricate interplay between the light-matter
interaction and the two-photon relaxation has been explored in Ref. [24].
However, there is still a lack of comprehensive understanding regarding the
competitive dynamics involving RTs, CRTs, and relaxation terms.

In this work, we adopt the theoretical framework established in Ref. [24] and
investigate the competition among different interaction processes.
Specifically, we consider the anisotropic Rabi model, which has been
extensively studied in previous works [32, 33, 34, 35, 36, 37, 38, 39, 40, 41,
42]. In this model, the coupling strengths of the RTs and CRTs are different.
By adjusting the coupling strengths and the two-photon relaxation rate, we
find that the excitation-relaxation dynamics exhibit interesting parity-
sensitive behavior in both transient and steady states. We explain these
phenomena by analyzing the competition among different interaction processes.
To achieve this, we numerically solve the quantum master equation and utilize
the effective Hamiltonian to gain further physical insights.

The structure of this paper is as follows. In Section II, we present the
system Hamiltonian and the Lindblad master equation that incorporates two-
photon relaxation. We subsequently explore three solvable limits in Section
III, which are indicative of the effects arising from distinct transition
processes. Section IV showcases exceptional phenomena observed in both
transient dynamics and steady states. These are elucidated by examining the
underlying principles of competitive interactions. Finally, a summary is given
in Section V.

##  II Model Hamiltonian

We employ the anisotropic Rabi model (ARM) [33, 32, 36, 43] to examine the
consequences of distinct light-matter interaction components. Different from
dealing with the standard QRM, we encounter two types of interactions
characterized by different strengths in the ARM. The ARM Hamiltonian reads

$\mathcal{H}=\mathcal{H}_{0}+\mathcal{H}_{1}+\mathcal{H}_{2},$ (1)

where the free Hamiltonian $\mathcal{H}_{0}$, the Jaynes-Cummings interaction
Hamiltonian $\mathcal{H}_{1}$ and the counter-rotating interaction Hamiltonian
$\mathcal{H}_{2}$ are given by ($\hbar=1$)

$\begin{split}&\mathcal{H}_{0}=\omega
a^{\dagger}a+\frac{\Delta}{2}\sigma_{z},\\\
&\mathcal{H}_{1}=g_{1}\left(a\sigma_{+}+a^{\dagger}\sigma_{-}\right),\\\
&\mathcal{H}_{2}=g_{2}\left(a^{\dagger}\sigma_{+}+a\sigma_{-}\right).\\\
\end{split}$ (2)

Here, $a^{\dagger}$ and $a$ are the creation and annihilation operators for
the quantized light field in the cavity, with a resonance frequency $\omega$.
Meanwhile, $\sigma_{z}=\ket{e}\bra{e}-\ket{g}\bra{g}$,
$\sigma_{+}=\ket{e}\bra{g}$, and $\sigma_{-}=\ket{g}\bra{e}$ are the atomic
raising and lowering operators linked to the qubit with a transition frequency
$\Delta$. The two types of interactions between the cavity and qubit are
characterized by the coupling strengths $g_{1}$ and $g_{2}$. The ARM
simplifies to the standard QRM with $g_{1}=g_{2}$. When $g_{2}=0$, it
transforms into the Jaynes-Cummings model (JCM). When $g_{1}=0$, it
corresponds to the anti-JCM (AJCM). Physical implementations or simulations of
the QRM and ARM are achievable, e.g., in circuit quantum electrodynamics
setups [10, 36] and in synthetic antiferromagnets with intrinsic asymmetry of
magnetic anisotropy [44].

Figure 1: (a) A schematic of the anisotropic Rabi model with two-photon
relaxation. (b) and (c) illustrate the possible transitions induced by the
rotating interaction terms (green), counter-rotating terms (blue), and the
two-photon relaxation term (red), in even and odd parity subspaces of photon-
qubit bare basis, respectively.

The Hamiltonian has the $Z_{2}$ symmetry in the sense that it is invariant
under the parity transformation generated by the combined parity operator

$\mathcal{P}=-\sigma_{z}e^{i\pi a^{\dagger}a}.$ (3)

As a physical consequence, the ARM conserves the parity of the total
excitation number in the system [17]. To be specific, we consider the photon-
qubit bare basis, defined by $\ket{n,e(g)}\equiv\ket{n}\otimes\ket{e(g)}$,
with $\ket{n}$ being the Fock states, $\ket{e}$ and $\ket{g}$ being the
excited and ground states of $\sigma_{z}$. Owing to the conserved parity, from
an initial state with definite parity, the evolution under the ARM Hamiltonian
goes along the corresponding parity chain [45, 24], given by

$\begin{split}p=+1,\quad\left\\{\ket{0,g},\ket{1,e},\ket{2,g},\ket{3,e},\dots\right\\},\\\
p=-1,\quad\left\\{\ket{0,e},\ket{1,g},\ket{2,e},\ket{3,g},\dots\right\\}.\\\
\end{split}$ (4)

Regarding the effects of the environment, a relaxation scheme where photons
can only leak in pairs has been investigated [24]. In this case, the
dissipation does not break the parity conservation, and the dynamical
behaviors are expected to be parity-sensitive, i.e., different for initial
states from two parity subspaces.

When two-photon relaxation is considered, the time-dependent density matrix
$\rho(t)$ of the system can be described by the Lindblad master equation [46,
47, 48]

${\dot{\rho}(t)}=-i\left[\mathcal{H},\rho(t)\right]+2\kappa
a^{2}\rho(t)(a^{\dagger})^{2}-\kappa\\{(a^{\dagger})^{2}a^{2},\rho(t)\\},$ (5)

with $\kappa$ being the relaxation rate.

By rearranging Eq. (5), we obtain

${\dot{\rho}(t)}=-i\left(\mathcal{H}_{\text{eff}}\rho(t)-\rho(t)\mathcal{H}_{\text{eff}}^{\dagger}\right)+\mathcal{L}_{\text{jump}}(\rho),$
(6)

with the effective Hamiltonian

$\mathcal{H}_{\text{eff}}=\mathcal{H}-i\kappa(a^{\dagger})^{2}a^{2},$ (7)

and the quantum jump operator

$\mathcal{L}_{\text{jump}}(\rho)=2\kappa
a^{2}\rho(t)\left(a^{\dagger}\right)^{2}.$ (8)

When the quantum jump effects can be neglected, the dynamics of the system can
be approximately described in Schrödinger formalism with the effective
Hamiltonian $\mathcal{H}_{\text{eff}}$. The imaginary parts of eigenvalues of
$\mathcal{H}_{\text{eff}}$ represent the decays of the eigenstates, and thus
the leaks of probabilities from the system into the environment. Therefore,
when calculating the expectation value of a physical observable in non-
Hermitian systems, one needs to renormalize the wavefunction. Specifically,
the imaginary term $i\kappa(a^{\dagger})^{2}a^{2}$ in Eq. (7) indicates that
higher photon levels have larger decay rates and thus decrease faster in
probabilities. As a result, the states with slower photon decay become
dominant. This is effectively a relaxation process with probabilities
transiting from high photon levels to low photon levels, coinciding with the
effects of the quantum jump operator $\mathcal{L}_{\text{jump}}(\rho)$. In
this way, the TDSE of the effective Hamiltonian captures the main features of
the dynamics apart from the quantum jump effects.

Figure 1 shows the transition diagram induced by the rotating interaction
$\mathcal{H}_{1}$, the counter-rotating term $\mathcal{H}_{2}$ and the quantum
jump operator $\mathcal{L}_{\text{jump}}(\rho)$. The rotating interactions
(denoted with green arrows between $\ket{n,e}$ and $\ket{n+1,g}$) exchange
excitations between the qubit and the cavity, and thus conserve the total
number of excitations. The CRTs (indicated in blue arrows between $\ket{n,g}$
and $\ket{n+1,e}$) create (or annihilate) excitations in both the qubit and
cavity simultaneously, and thus only conserve the parity of excitation
numbers. The quantum jump operator $\mathcal{L}_{\text{jump}}(\rho)$ (shown in
red arrows) only happens from high levels to low levels. The decay rates are
larger in the states with more photons. The interplay of these processes
induces the exotic dynamical behaviors of the system.

##  III Complex energy spectrum

###  III.1 Spectrum of the effective Hamiltonian

Figure 2: (a) Real and (b) imaginary parts of the eigenvalues of the ARM with
two-photon relaxation, with respect to the coupling strength $g_{1}/\omega$.
(c) Eigenvalues of the system Hamiltonian on the complex plane with various
values of the coupling strength $g_{1}/\omega$. The fixed parameters are
$\Delta/\omega=0.8$, $\kappa/\omega=0.02$, $\lambda=g_{2}/g_{1}=0.5$.

As mentioned, the open system dynamics can be described by the Schödinger
equation of the effective non-Hermitian Hamiltonian. In the following, we
analyze the spectrum of this Hamiltonian.

There is no simple closed-form solution to the ARM and we thus rely on
numerical diagonalization, as well as approximations. We first calculate the
spectrum by numerical diagonalization and give physical interpretations to the
complex eigenvalues. The spectrum is complex as shown in Fig. 2. The real
parts of eigenvalues denote the frequencies and the imaginary parts
characterize the effective decay rates. In Fig. 2, the eigenvalues are labeled
with $\psi_{n}^{p}$, where $n$ is the level index and the value of $p$ takes
$e$ for even parity subspace and $o$ for odd parity subspace.

The non-Hermitian term $i\kappa a^{\dagger}a$ in the effective Hamiltonian
does not directly induce any transition between bare states. However, it
contributes imaginary eigenvalues associated with different photon numbers,
and thus effectively results in decay processes. Therefore, the effective
decay term in the Schrödinger formalism approximates the effects of the
relaxation processes in the Lindblad formalism. In this way, we see
intuitively how the non-Hermitian effective Hamiltonian can describe the full
dynamics that are determined by the quantum master equation.
We turn to solvable limits of the effective Hamiltonian for further physical
intuitions. The JCM and AJCM are both exactly solvable and thus the effects of
the RTs and CRTs can be analyzed separately. From the analytic results, we can
deduce the competition between different types of interactions. Even the
trivial decoupled limit, with $g_{1}=g_{2}=0$, provides additional intuition
to the decay process.

###  III.2 Decoupled limit

With $g_{1}=g_{2}=0$, the qubit is decoupled from the field, and the effective
Hamiltonian becomes

$\mathcal{H}_{\text{eff}}^{g=0}=\mathcal{H}_{0}-i\kappa(a^{\dagger})^{2}a^{2}.$
(9)

This case is trivially solvable with the eigenvalues given as

$E^{g=0}_{n,\pm}=n\omega\pm\frac{\Delta}{2}-i\kappa n(n-1),$ (10)

and the eigenstates are simply bare states

$\psi^{g=0}_{n,\pm}=\ket{n,\pm}\equiv\ket{n}\otimes\ket{\pm}.$ (11)

We see from the eigenvalues that the decay rates of eigenstates increase with
the photon number.

###  III.3 Jaynes-Cummings model

When $g_{2}$ is set to 0 and only rotating interactions are taken into
account, we reach the JCM [49, 50, 51, 52]. The effective Hamiltonian now
reads

$\mathcal{H}_{\text{eff}}^{\text{JC}}=\mathcal{H}_{0}+g_{1}\left(a\sigma_{+}+a^{\dagger}\sigma_{-}\right)-i\kappa(a^{\dagger})^{2}a^{2},$
(12)

which is exactly solvable.

The eigenvalues are

$E_{n,\pm}^{\text{JC}}=\left(n+\frac{1}{2}\right)\omega-i\kappa
n^{2}\pm\frac{1}{2}\sqrt{(A_{n}^{-})^{2}+B_{n}^{2}}$ (13)

where

$\begin{split}&A_{n}^{\pm}=(\omega\pm\Delta)-2i\kappa n,\\\
&B_{n}=2g\sqrt{n+1}.\end{split}$ (14)

The definition of $A_{n}^{+}$ will be used for the anti-Jaynes-Cummings model
later.

A special case that can be analytically dealt with is when the qubit and
cavity are in resonance, with $\delta=\omega-\Delta=0$. The corresponding
eigenvalues are

$E_{n,\pm}^{\text{JC}}=\left(n+\frac{1}{2}\right)\omega-i\kappa
n^{2}\pm\sqrt{(n+1)g_{1}^{2}-n^{2}\kappa^{2}}.$ (15)

The two eigenstates with the same index $n$ are degenerate when the square
root in Eq. (15) is 0, leading to an exceptional point (EP) due to the passive
$\mathcal{PT}$-symmetry, see further discussion in the Appendix D.

###  III.4 Anti-Jaynes-Cummings model

We now consider the anti-JCM to explore the effects brought by CRTs. The
effective Hamiltonian now reads

$\mathcal{H}_{\text{eff}}^{\text{AJC}}=\mathcal{H}_{0}+g_{2}\left(a^{\dagger}\sigma_{+}+a\sigma_{-}\right)-i\kappa(a^{\dagger})^{2}a^{2},$
(16)

which is also exactly solvable.

The corresponding eigenvalues are

$E_{n,\pm}^{\text{AJC}}=\left(n+\frac{1}{2}\right)\omega-i\kappa
n^{2}\pm\frac{1}{2}\sqrt{(A_{n}^{+})^{2}+B_{n}^{2}},$ (17)

with $A_{n}^{+}$ and $B_{n}$ given in Eq. (14).

##  IV Excitation and relaxation dynamics

In this section, we analyze the dynamics of the ARM with two-photon
relaxation. To this end, we tune the coupling strengths $g_{1}$ and $g_{2}$,
as well as the two-photon relaxation rate $\kappa$, to explore the interplay
and competition of the different interaction terms.

We focus on both transient states and steady states by calculating the
corresponding physical observables, namely the cavity photon number and the
qubit population. In particular, we discuss the excitation-relaxation dynamics
with three approaches. Firstly, we calculate the dynamics by numerically
solving the Lindblad master equation with a specific initial state $\rho(0)$.
We illustrate the overall dynamics in two parity subspaces and discuss the
transient and steady states in detail. Secondly, we map the bare states onto
the eigenbasis of the effective Hamiltonian and understand the relaxation
dynamics by interpreting the corresponding complex eigenvalues as decay rates.
Thirdly, we understand some interesting steady-state behaviors from the
perspective of the transition processes of the bare states.

###  IV.1 Dynamical effects of the interaction terms

To begin with, we intuitively analyze the impact of the three types of
interactions on system dynamics. The relaxation processes involved in Eq. (5)
is the simplest case as it does not include the interaction with the qubit. In
addition, there is no process to regenerate photons into the system after the
dissipation. In the decoupled limit, if we start from a bare state
$\ket{n,e(g)}$, photons will decay at an exponential rate given by $\kappa
n(n-1)$ and the cavity reaches the vacuum state $\ket{0,e(g)}$ if $n$ is even.
For initial states with odd photons, however, there will be a single photon
left in the system since there is no mechanism to dissipate it. Since the
qubit and cavity are decoupled in this limit, the qubit state is immune to
relaxation and remains unchanged.

As shown in Fig. 1, the JC interaction terms generate transitions within
subspaces that conserve excitation numbers. This leads to Rabi oscillations
between two basis states that share the same number of excitations.
Particularly, if we start with the initial state $\ket{n,e}$, the probability
of state $\ket{n+1,g}$ is oscillatory, given by

$P^{\text{JC}}_{n}=\frac{g_{1}^{2}\sin^{2}(\Omega_{n}^{\text{JC}}t)}{(\omega-\Delta)^{2}+g_{1}^{2}},$
(18)

with the Rabi frequencies

$\Omega_{n}^{\text{JC}}=\sqrt{(\omega-\Delta)^{2}+4g_{1}^{2}(n+1)}.$ (19)

From Eq. (18), we see that the complete population transfer only occurs in the
resonant case. The oscillation frequencies increase with the coupling strength
$g_{1}$ and the photon number $n$.

Figure 3:  Excitation-relaxation dynamics of the ARM in even parity subspace,
with the initial state $\ket{2,g}$, and parameters $\lambda=g_{2}/g_{1}=0.5$,
$\Delta/\omega=0.8$, $\kappa/\omega=0.02$. Time evolution of (a)(c)(e) mean
photon number and (b)(d)(f) qubit population with various values of coupling
strength $g_{1}/\omega$ in the time range of $t/T_{c}$, with
$T_{c}\equiv\pi/\omega$. (g) The probability of eigenstates when mapping the
initial state onto the eigenbasis. (h) The transient states of mean photon
number and qubit population with various values of coupling strength
$g_{1}/\omega$ at time $t/T_{c}=60$.

Different from the JC interaction terms, the CRTs induce transitions where two
excitations emerge or disappear simultaneously, leading to anti-Rabi
oscillations. With the initial state being $\ket{n,g}$, the population of the
state $\ket{n+1,e}$ is given by

$P^{\text{AJC}}_{n}=\frac{g_{2}^{2}\sin^{2}(\Omega_{n}^{\text{AJC}}t)}{(\omega+\Delta)^{2}+g_{2}^{2}},$
(20)

with the anti-Rabi frequencies

$\Omega_{n}^{\text{AJC}}=\sqrt{(\omega+\Delta)^{2}+4g_{2}^{2}(n+1)}.$ (21)

We can observe from the above Equations that for small values of $g_{2}$, the
frequencies are large while the amplitudes are small. Therefore, the effects
of CRTs are negligible for small $g_{2}$ while prominent for large $g_{2}$.

The intricate interplay of the above dynamical processes leads to the
interesting time evolution of the full system.

###  IV.2 Transient states

We now start our discussions on the dynamics with the transient states in the
even parity subspace. The full dynamics with even parity are presented in Fig.
3. The system is initially prepared in the bare state $\ket{2,g}$, indicating
that there are two photons in the cavity and the qubit is in its ground state
at $t=0$. We calculate the mean photon number and qubit population by solving
the Lindblad master equation numerically. In the calculations, we set
$g_{2}/g_{1}=\lambda$ and vary the value of $g_{1}$ to obtain the dynamics
concerning coupling strengths. To better see how RTs and CRTs interact with
the relaxation separately, we have chosen the coupling ratio $\lambda=0.5$ to
further suppress the effects of CRTs for lower coupling strength regimes. The
overall results are displayed in Fig. 3(a) and 3(b). It is clear to observe
that the interplay of the three types of interactions depends on the coupling
strengths. The dynamical behaviors can be physically interpreted from the
perspective of transition processes induced by the three types of
interactions, as illustrated in Fig. 1(a).

We take the case $g_{1}/\omega=0.1$ as the first example, with the
corresponding cavity photon number and qubit population displayed in Fig. 3(c)
and (d), respectively. The “ripples” for short-range time mainly result from
the interplay between the JC interactions and the two-photon relaxation, since
the effects of CRTs are weak with the marginal value of $g_{2}/\omega=0.05$.
Through the relaxation channel $\ket{2,g}\Rightarrow\ket{0,g}$, the two
photons in the initial state quickly escape from the cavity with a certain
probability. Meanwhile, the Rabi oscillations transfer some populations
through the process $\ket{2,g}\leftrightarrow\ket{1,e}$ and thus prevent the
relaxation. The effect of Rabi oscillations can be more obviously observed
from the qubit dynamics in Fig. 3(d), where the qubit is not directly affected
by the relaxation process and undergoes several periods of oscillations at the
constant Rabi frequency. Later on, since the Rabi oscillations are reversible,
the re-excited populations undergo the relaxation again, which gives rise to
an overall decaying behavior for both cavity and qubit. Consequently, after
some periods of Rabi oscillations, the populations concentrate in the state
$\ket{0,g}$, with both cavity and qubit sitting around their ground states.

When $g_{1}$ increases, the effects of CRTs become non-negligible, and the
interference between these effects leads to non-trivial dynamical patterns. To
demonstrate this, we take the case $g_{1}/\omega=0.9$, and thus
$g_{2}/\omega=0.45$, as an example. The results are displayed in Fig. 3(e) and
(f). The current case has two main differences from the previous one: the
photons decay slower at the initial stage, and more photons are left in the
cavity in the final state. Since CRTs induce fast small-amplitude
oscillations, the dynamics at beginning times are largely different from the
previous case. For this time range, the main transition channel owing to the
competition between CRTs and relaxation is

$\ket{2,g}\Leftrightarrow\ket{3,e}\Rightarrow\ket{1,e},$ (22)

where the relaxation is weakened to a much greater extent. As a consequence,
the mean photon number and qubit population in this case are both more than in
the previous case at the beginning. After this initial competition, the
reversible nature of Rabi and anti-Rabi oscillations still offers chances to
the relaxation channel and thus leads to the overall decaying behavior. The
relaxation processes smooth these oscillatory dynamics for a long time range,
where all the transitions reach a dynamical equilibrium, with a higher number
of excitations in the final state.

We note that any other initial states with even parity give rise to almost the
same physics since they decay quickly to the bare state $\ket{2,g}$ that we
consider in the above example. However, higher initial states will lead to
more complicated dynamics at the beginning times, since the Rabi and anti-Rabi
frequencies increase with photon number $n$. The final dynamical equilibrium
only depends on the system parameters and the parity of the initial state,
ignoring the specific form of the wavefunctions.

If we focus on the medium-range time that is long (e.g., $t=60\pi$ in our
case), but not long enough for the system to reach the final steady state, we
can see how the equilibrium is affected by the coupling strengths. For the
even parity subspace, we observe some dynamical behaviors that are rather
steady, as illustrated in Fig. 3(d). These medium-range states are already
quite similar to the final steady states except for the local maximum at
around $g_{1}=0.05$. This peak was noticed in Ref. [24] and was believed to
exist in the steady state for the coupling strength $g_{1}=g_{2}=\kappa$.

Here, we show that this local maximum is absent in the final steady states and
its position is irrelevant to the relaxation rate $\kappa$. It turns out to be
a consequence of the competition between JC interactions and two-photon
relaxation, which is most clearly seen in the dissipative JCM, where CRTs are
absent. In a more direct interpretation, the local maximum originates from the
slow Rabi oscillations for small $g_{1}$, whose position can be analytically
determined from the JCM. We see from Eq. (19) that the Rabi frequency
increases with the coupling strength $g_{1}$. Therefore, for a given time $t$,
oscillations associated with different coupling strengths are going through
different stages. There might be one that happens to be at its maximum,
whereas the neighboring ones are not. At this specific time, we then observe
the local maximum with respect to the coupling strength, as displayed in Fig.
4(h). With this reasoning, it is clear that the corresponding $g_{1}$ value
decreases with time $t$, and thus the peak disappears for a long enough time.

Figure 4:  Excitation-relaxation dynamics of the ARM in odd parity subspace,
with the initial state $\ket{3,g}$, and parameters $\lambda=g_{2}/g_{1}=0.5$,
$\Delta/\omega=0.8$, $\kappa/\omega=0.02$. Time evolution of (a)(c)(e) mean
photon number and (b)(d)(f) qubit population with various values of coupling
strength $g_{1}/\omega$ in the time range of $t/T_{c}$, with
$T_{c}\equiv\pi/\omega$. (g) The probability of eigenstates when mapping the
initial state onto the eigenbasis. (h) The transient states of mean photon
number and qubit population with various values of coupling strength
$g_{1}/\omega$ at time $t/T_{c}=60$.

Since the total system conserves the parity of the total excitation number, it
is expected that the dynamics are parity-sensitive. Indeed, the results with
the initial odd-parity state $\ket{3,g}$, shown in Fig. 4, demonstrate that
the dynamics of both cavity photon number and qubit population are
significantly different from the even parity case. The major difference is
that the oscillations are much more persistent in odd subspace. The dynamics
can also be understood in the same intuitive way and can be largely deduced
from the solvable limits. In the decoupled case where $g_{1}=g_{2}=0$, the
initial state $\ket{3,g}$ quickly loses two photons and reaches the final
state $\ket{1,g}$. In this case, there is no mechanism to excite the qubit or
dissipate the photon. Things become more interesting when light-matter
interactions are introduced. For small coupling strength, the system
oscillates between states $\ket{1,g}$ and $\ket{0,e}$ owing to the JC
interactions. If $g_{2}=0$, this oscillation persists forever and the steady
state is never reached. This is verified by the constant oscillations depicted
in Figs. 4(c) and 4(d). For non-zero $g_{2}$, however, the weak effects of
CRTs induce a tiny amount of transitions to higher photon states which are
then dissipated by the relaxation processes. This is the reason why the system
can still reach steady states after long-time oscillations. When $g_{2}$ grows
to the regimes where the effects of CRTs are prominent, the Rabi oscillations
are destructed by the CRTs and then quickly smoothed by the relaxation terms,
as shown in Figs. 4(e) and 4(f). It is clear from the quasi-steady states in
Fig. 4(h) that the photon and qubit excitation left in the system is more than
those of the even parity case. This is because there are always some
populations of $\ket{1,g}$ and $\ket{0,e}$ that can not be dissipated.

Since the connections between the Lindblad master equation and the non-
Hermitian effective Hamiltonians have been established in the preceding
sections, the overall dynamics can also be understood with physical intuitions
by mapping the initial state on the eigenbasis of the Hamiltonian (1). This
mapping shows which modes are more active at a given value of $g$ in each
parity subspace. By doing so, we can check the corresponding frequencies and
decay rates from Fig. 2 and thus understand the dynamical behavior. Here, we
map the initial bare state $\ket{2,g}$ onto the eigenstates of the effective
Hamiltonian, and the probabilities of the eigenstates are displayed in Fig.
3(g).

###  IV.3 Steady states

Figure 5:  (a) The cavity photon number and (b) the qubit population in the
steady state of even parity subspace with respect to the coupling strengths
$g_{1}$ and $g_{2}$. (c) The cavity photon number and the qubit population
with fixed coupling $g_{2}/\omega=0.15$ and varying coupling strength
$g_{1}/omega$. (d) The eigenvalue spectrum with the same parameter values as
in (c). The peaks in (c) coincide with the avoided crossings in (d).

After a long enough time, the system arrives at a dynamical equilibrium of the
three types of transitions, i.e. the steady state. With steady states, it is
easy to identify the interplay of the interaction processes. The steady states
for the cavity and the qubit in even and odd parity subspaces are calculated
with respect to the coupling strengths $g_{1}$ and $g_{2}$ and displayed in
Figs. 5 and 6, respectively. We can observe different behaviors by following
different paths on the $g_{1}$-$g_{2}$ plane. For example, following the path
$g_{2}=0$ we observe the steady states of the JCM with respect to $g_{1}$. On
the other hand, we have the steady states of the AJCM with the constraint
$g_{1}=0$. Setting $g_{1}=g_{2}$ we have the standard QRM, which exhibits
parabolic behaviors with respect to the coupling strengths. We note that the
steady states are determined solely by the parity, rather than any specific
forms, of the initial states. Nevertheless, for the sake of clarity and
simplicity, we base our discussions on the particular initial states
$\ket{2,g}$ for even parity and $\ket{3,g}$ for odd parity in the following.

To gain more physical intuitions, we again turn to the three solvable limits
and study the interplay of coupling and relaxation terms. For the even parity
case, the decoupled limit leads to the trivial consequence that both photons
escape from the system through the relaxation $\ket{2,g}\Rightarrow\ket{0,g}$.
Therefore, there is no photon in the cavity after a long enough time, and the
qubit stays in the ground state. In the JCM limit where $g_{2}=0$, the
different situation gives rise to the same result. As can be seen from the
transient states of the JCM, the Rabi oscillation between
$\ket{2,g}\leftrightarrow\ket{1,e}$ diminishes with time because the total
populations decay exponentially to the vacuum bare state through the
irreversible relaxation process $\ket{2,g}\Rightarrow\ket{0,g}$. With this, it
is again verified that the small peak in Fig. 3(h) is not a steady state, but
rather a “quasi-steady” one.

Following another path $g_{1}=0$, the dynamics can be described by the AJCM,
in which case the transition processes are more complicated. The anti-Rabi
oscillation $\ket{2,g}\Leftrightarrow\ket{3,e}$ competes with the relaxation,
leading to the lower level anti-Rabi oscillation
$\ket{0,g}\Leftrightarrow\ket{1,e}$ that is immune to the further relaxation
process. The amount of populations that are transferred by the anti-Rabi
oscillation is determined by the $P_{1}^{\text{AJC}}$ in Eq. (20).
Consequently, the excitation numbers in the AJCM case increase with the value
of $g_{2}$. This anti-Rabi oscillation persists forever unless JC interactions
associated with $g_{1}$ are introduced.

For the general case, with all three types of interactions present, the steady
state of the ARM is a non-trivial mix of JCM and AJCM with two-photon
relaxation. Generally, the steady states of cavity and qubit increase with
respect to both $g_{1}$ and $g_{2}$, as can be observed from Figs. 5(a) and
5(b) for cavity and qubit, respectively. The results along the path
$g_{1}=g_{2}$ are consistent with the results presented in Ref. [24]. A
particularly interesting phenomenon, absent in the standard QRM, of the steady
states in even parity subspace is the peaks along $g_{1}$ with a small value
of $g_{2}$, as shown in Fig. 5(c). These peaks are found sitting at the
avoided level crossings of the eigenspectrum, as displayed in Fig. 5(d). These
avoided crossings can be analytically determined by calculating genuine level
crossings of the JCM. The position for the $m$-th peak, with respect to
$g_{1}$, is then given by

$\frac{g_{1}}{\omega}=\sqrt{(2m-1)+\frac{\Delta}{\omega}},\quad
m\in\mathbb{N}_{+}.$ (23)

To understand the physical origins of these peaks, we go back to the Hermitian
case. We start with JCM with $U(1)$ symmetry. The energy levels in even parity
subspace cross with each other, since they now all belong to different
symmetric sectors. If we add perturbative $g_{2}$ into the system. The $U(1)$
symmetry is broken and reduced to the $\mathbb{Z}_{2}$ symmetry. The spectrum
with a specific parity subspace will repulse with each other, and thus the
previous level crossings become avoided. At avoided crossings induced by small
$g_{2}$, eigenstates strongly mixed and thus the corresponding physical
observables drastically change. In JCM, the initial state $\ket{0,g}$ is the
ground state living in a 1-dimensional subspace and it does not interact with
any other eigenstates. When we switch on the CRTs with small $g_{2}$, the
initial state $\ket{0,g}$ strongly interacts with other states at the avoided
crossings. Since the initial state $\ket{0,g}$ is no longer the eigenstate of
the system, the dynamics exhibit Rabi oscillation between the two original
eigenstates at the avoided crossings. In the vicinity of the first peak at
$g_{1}/\omega=\sqrt{1+\Delta/\omega}$, the following oscillation occur,

$\ket{0,g}\leftrightarrow\ket{1,-}_{\text{JC}}.$ (24)

The JC eigenstate (or dressed state) $\ket{1,-}_{\text{JC}}$, explicitly given
in Eq. (34), is a superposition of the bare states $\ket{0,e}$ and
$\ket{1,g}$, and contains 1 excitation. Correspondingly, the photon number and
qubit population oscillate between 0 and 1. With the relaxation process
considered, the oscillations will be smoothed and arrive at steady states with
larger mean photon numbers. In the meantime, for other values of $g_{1}$ that
are far from the avoided crossings, the oscillation in Eq. (24) is weak and
the final excitation is marginal. If the initial state is not $\ket{0,g}$, the
JC interaction and relaxation will make the system jump to the lowest levels
and thus lead to the same results as the initial $\ket{0,g}$.

Figure 6:  (a) The cavity photon number and (b) the qubit population in the
steady state of odd parity subspace with respect to the coupling strengths
$g_{1}$ and $g_{2}$. (c) The cavity photon number and the qubit population
with fixed coupling $g_{1}/\omega=0.01$ and varying coupling strength
$g_{2}/omega$. (d) The eigenvalue spectrum with the same parameter values as
in (c). The peak and valley in (c) coincide with the avoided crossing in (d).

As expected, the steady states in the odd parity subspace are shapely
different, as shown in Fig. 6. In the trivial decoupled limit, the initial
state $\ket{3,g}$ irreversibly decays to the final state $\ket{1,g}$. In the
JCM limit, the Rabi oscillation $\ket{1,g}\leftrightarrow\ket{0,e}$ persists
forever, giving rise to the total excitation being 1. In the AJCM case, the
reversible oscillation $\ket{1,g}\Leftrightarrow\ket{2,e}$ competes with the
irreversible relaxation process $\ket{2,e}\Rightarrow\ket{0,e}$, and
eventually transfer all population to the state $\ket{0,e}$. Similar to the
even parity case, we also observe a peak (valley) in the mean photon number
displayed in Fig. 6(a) (the qubit population in Fig. 6(b)) at around
$g_{2}/\omega=0.42$ with very small $g_{1}/\omega$. These local extrema result
from the Rabi-type oscillation induced by avoided crossings, i.e.

$\ket{0,g}\leftrightarrow\ket{0,-}_{\text{AJC}},$ (25)

where $\ket{0,-}_{\text{AJC}}$ is a superposition of the bare states
$\ket{0,g}$ and $\ket{1,e}$, as given in Eq. (44). With simple calculations,
we know that the avoided crossings are present at

$\frac{g_{2}}{\omega}=\sqrt{(2m-1)-\frac{\Delta}{\omega}},\quad
m\in\mathbb{N}_{+},$ (26)

with small value of $g_{1}$.

##  V Summary

In summary, our investigation delves into the interplay among three types of
interactions - the Jaynes-Cummings interaction, the counter-rotating terms,
and the two-photon relaxation - within the framework of the anisotropic Rabi
model.

To comprehensively explore these competitive dynamics, we employ three
approaches: numerical solution of the master equation, interpretation of the
transition diagram induced by the interaction terms, and understanding of the
dynamics through the eigenbasis of the effective Hamiltonian.

We observe that the dynamics of cavity photon number and qubit population are
sensitive to parity, which can be elucidated through the analysis of three
solvable limits of the non-Hermitian effective Hamiltonian. Specifically, we
identify that the intriguing local maximum in the transient state for small
$g_{1}$ is absent in the steady state, and offer an intuitive explanation for
its origin. Furthermore, we discover peaks in the steady states of cavity
photon number and qubit population, attributed to avoided level crossings in
the eigenvalue spectrum.

Our work contributes to a deeper understanding of the competition between
Jaynes-Cummings interactions and counter-rotating terms, as well as the
effects of ultrastrong coupling in open quantum systems. We provide
clarification on the effects of the quantum jump operator and offer a
generalized framework for exploring the interplay among different types of
interactions. These findings may have implications for cavity-based quantum
computations.

###### Acknowledgements.

This work has been supported by the National Natural Science Foundation of
China Grant No. 12205383, 12274470, 62273361 and the Natural Science
Foundation of Changsha Grant No. kq2202082.

##  Appendix A Dynamics calculated from the non-Hermitian effective
Hamiltonian

We calculate the system dynamics by numerically solving the master equation in
the main text. Here, we study the changes in the cavity photon number and
qubit population of the system with the coupling strength after an extended
period of evolution, utilizing the time-dependent Schr$\ddot{\rm o}$dinger
equation(TDSE) derived from the effective Hamiltonian. In addition, by
comparing the results obtained by the two approaches, we find the crucial role
of the quantum jump term in reaching the ultimate steady state.

The general states of the system can be extended in the order of the basis
vectors of the Hilbert space. In the even parity subspace, when the total
photon number $N$ is even, the system’s state can be represented as

$\ket{\psi(t)}=\sum_{n=0}^{N/2}C_{2n}^{g}(t)\ket{2n,g}+\sum_{n=0}^{N/2-1}C_{2n+1}^{e}(t)\ket{2n+1,e},$
(27)

with the probability amplitudes $C_{2n}^{g}(t)$, $C_{2n+1}^{e}(t)$, which can
be obtained by solving the TDSE

$i\dot{\ket{\psi(t)}}=H_{\rm eff}\ket{\psi(t)}.$ (28)

with the initial state $\ket{\psi(0)}=\ket{2,g}$. As shown in 7, after a
sufficiently long evolution time $(10^{3}\pi)$, we observe variations in
$\langle a^{\dagger}a\rangle$ for different coupling strengths $g_{1},g_{2}$
under the evolution dominated by the effective Hamiltonian. The positions of
characteristic peaks align with those obtained by solving the master equation.
However, some oscillations are present in these steady state results,
indicating they are not fully stabilized but rather more stable states.

The master equation approach represents the behavior of the true final steady
state due to its consideration of all effects including the quantum jump term
$\gamma\sigma_{-}\rho\sigma_{+}$, whereas the effective Hamiltonian omits it.
Therefore, we can conclude that the inclusion of the quantum jump terms allows
the system to reach the ultimate steady state.

Figure 7: The cavity photon number of steady state in even parity subspace,
calculated through (a) the Lindblad master equation and (b) the TDSE of the
non-Hermitian effective Hamiltonian, with $\Delta/\omega=0.8$,
$\kappa/\omega=0.02$.

##  Appendix B Physical interpretation of the Lindblad terms

In this section, we discuss the three dissipation terms in the Lindblad master
equation. Here, we consider two bare states in the basis of the main body
system, denoted as $\ket{1}=\ket{n-2,g}$ and $\ket{2}=\ket{n,g}$, which are
connected by the two-photon relaxation. Consequently, the density operator can
be expressed as $\rho=\sum_{i,j=1,2}\rho_{ij}\ket{i}\bra{j}$ or matrix form

$\rho=\begin{pmatrix}\rho_{22}&\rho_{21}\\\ \rho_{12}&\rho_{11}\end{pmatrix},$
(29)

where the diagonal elements represent population $(\sum_{i}\rho_{i,i}=1$ with
$\rho_{ii}\in\mathcal{R_{0}^{+}})$, while the off-diagonal elements denote
coherence $(\rho_{ij}=\rho_{ji}^{*}$ with $\rho_{ij}\in\mathcal{C})$.

The dynamics of the system can be fully described by employing the Lindblad
master equation, where the relaxation of photon pairs is represented by the
following dissipator

$\mathcal{D}[a^{2}]\rho=2\kappa
a^{2}\rho(a^{\dagger})^{2}-\kappa(a^{\dagger})^{2}a^{2}\rho-\kappa\rho(a^{\dagger})^{2}a^{2}.$
(30)

with relaxation rate $\kappa$. The effect of the dissipator on the density
operator can be divided into two parts: the continuous nonunitary dissipation
terms given by $\kappa\\{(a^{\dagger})^{2}a^{2},\rho\\}$, and the quantum jump
terms represented by $2\kappa a^{2}\rho(a^{\dagger})^{2}$. We can obtain a
matrix form for the dissipator as

$\mathcal{D}[a^{2}]\rho=n(n-1)\begin{pmatrix}-2\kappa\rho_{22}&-\kappa\rho_{21}\\\
-\kappa\rho_{12}&2\kappa\rho_{22}\end{pmatrix}.$ (31)

By examining each term individually within this matrix, we intuitively reveal
the effects of each element. Specifically speaking:

$\bullet$ The nonunitary dissipation terms impact elements such as
$\rho_{21}$, $\rho_{12}$ along with coherence loss between bare states, which
are represented by $-\kappa\rho_{21}$, $-\kappa\rho_{12}$ respectively.

$\bullet$ Additionally, the nonunitary dissipation terms also lead to energy
and information losses from the upper state into the environment, related to
$-2\kappa\rho_{22}$.

$\bullet$ On the contrary to the above-mentioned effects, the quantum jump
terms contribute towards an increase in lower states’ population, that is,
$2\kappa\rho_{22}$.

Overall, the excited number of the two bare states is conserved over time
i.e., $Tr([\mathcal{D}[a^{2}]\rho)=0$.

##  Appendix C Derivations of the JC and AJC results

In this section, we study the eigensystems of the JCM and AJCM with two-photon
relaxation, whose Hilbert space split into two unconnected parity chains due
to the $Z_{2}$-symmetry.

When $g_{2}=0$, the adjacent states in each parity subspace are solely coupled
by rotating terms, resulting in Hilbert space partitioned into a collection of
JC doublets $\\{\ket{n,e},\ket{n+1,g}\\}$. The effective Hamiltonian of the
JCM with two-photon relaxation can be expressed in matrix form as

$H_{\rm eff}^{\rm
JC}=\begin{pmatrix}(n+\frac{1}{2})\omega-\frac{\delta}{2}-i\kappa
n(n-1)&g_{1}\sqrt{n+1}\\\
g_{1}\sqrt{n+1}&(n+\frac{1}{2})\omega+\frac{\delta}{2}-i\kappa
n(n+1)\end{pmatrix},$ (32)

with $\delta=\omega-\Delta$ denoting the detuning. The complex eigenvalues of
the block Hamiltonian are given by

$E_{n,\pm}^{\rm JC}=(n+\frac{1}{2})\omega-i\kappa
n^{2}\pm\frac{1}{2}\Omega_{n}^{\rm JC},$ (33)

with Rabi frequency $\Omega_{n}^{\rm JC}=\sqrt{(\delta-2i\kappa
n)^{2}+4g_{1}^{2}(n+1)}$. Meanwhile, the corresponding eigenstates are found
as

$\displaystyle\ket{n,+}_{\rm
JC}=\cos\theta_{n}\ket{n,e}+\sin\theta_{n}\ket{n+1,g},$ (34)
$\displaystyle\ket{n,-}_{\rm
JC}=-\sin\theta_{n}\ket{n,e}+\cos\theta_{n}\ket{n+1,g},$ (35)

where the probability coefficients in the states are defined as

$\displaystyle\cos\theta_{n}=\sqrt{\frac{\Omega_{n}^{\rm JC}-\delta+2i\kappa
n}{2\Omega_{n}^{\rm JC}}},$ (36)
$\displaystyle\sin\theta_{n}=\sqrt{\frac{\Omega_{n}^{\rm JC}+\delta-2i\kappa
n}{2\Omega_{n}^{\rm JC}}}.$ (37)

Because of the excitation number conserving with this system, this leads the
Rabi oscillations between two bare states $\ket{n,e}$ and $\ket{n+1,g}$. In
this way, we can assume the form of the time evolution of the system states as

$\ket{\psi(t)}=[C_{n}^{e}(t)\ket{n,e}+C_{n+1}^{g}(t)\ket{n+1,g}]e^{-i(n+1/2)\omega
t}.$ (38)

where the $C_{n}^{e}(t)$ and $C_{n+1}^{g}(t)$ are time-dependent probability
amplitudes for each state. Applying the TDSE $i\dot{\ket{\psi(t)}}=H_{\rm
eff}^{\rm JC}\ket{\psi(t)}$, we then obtain the equations that the amplitudes
satisified,

$\begin{aligned} &\dot{C}_{n}^{e}(t)=[i\frac{\delta}{2}-\kappa
n(n-1)]C_{n}^{e}(t)-ig_{1}\sqrt{n+1}C_{n+1}^{g}(t),\\\
&\dot{C}_{n+1}^{g}(t)=[-i\frac{\delta}{2}-\kappa
n(n+1)]C_{n+1}^{g}(t)-ig_{1}\sqrt{n+1}C_{n}^{e}(t).\end{aligned}$ (39)

When the initial states meet $C_{n}^{e}(0)=1,C_{n+1}^{g}(0)=0$, and
$\delta=0$, we can calculate the time evolution equation of the system state
as follows

$\begin{aligned} \ket{\psi(t)}=&[(\cos\frac{\Omega_{n}^{\rm
JC}t}{2}+\frac{2\kappa n}{\Omega_{n}^{\rm JC}}\sin\frac{\Omega_{n}^{\rm
JC}t}{2})\ket{n,e}\\\ &-i\frac{2g_{1}\sqrt{n+1}}{\Omega_{n}^{\rm
JC}}\sin\frac{\Omega_{n}^{\rm JC}t}{2}\ket{n+1,g}]e^{-i(n+1/2)\omega t-\kappa
n^{2}t}.\end{aligned}$ (40)

We then obtain the time-dependent population on state $\ket{n+1,g}$ is

$P_{n+1,g}^{\rm JC}=\frac{4g_{1}^{2}(n+1)}{{\Omega_{n}^{\rm
JC}}^{2}}\sin^{2}\frac{\Omega_{n}^{\rm JC}t}{2}.$ (41)

Similarly, when $g_{1}=0$, the ARM is reduced to AJCM where adjacent states
within each parity subspace are solely coupled through counter-rotating terms,
thus the Hilbert space is divided into blocks spanned by the basis
$\\{\ket{n+1,e},\ket{n,g}\\}$. Consequently, the effective Hamiltonian of the
AJCM with two-photon relaxation can be written in matrix form as

$H_{\rm eff}^{\rm
AJCM}=\begin{pmatrix}(n+\frac{1}{2})\omega+\frac{\xi}{2}-i\kappa
n(n+1)&g_{2}\sqrt{n+1}\\\
g_{2}\sqrt{n+1}&(n+\frac{1}{2})\omega-\frac{\xi}{2}-i\kappa
n(n-1)\end{pmatrix},$ (42)

with $\xi=\omega+\Delta$. Then the complex eigenvalues of the partitioned
matrix are given by

$E_{n,\pm}^{\rm AJC}=(n+\frac{1}{2})\omega-i\kappa
n^{2}\pm\frac{1}{2}\Omega_{n}^{\rm AJC},$ (43)

with $\Omega_{n}^{\rm AJC}=\sqrt{(\xi-2i\kappa n)^{2}+4g_{2}^{2}(n+1)}$
denoting the Rabi frequency. The eigenstates can also be obtained

$\displaystyle\ket{n,+}_{\rm
AJC}=\cos\varphi_{n}\ket{n+1,e}+\sin\varphi_{n}\ket{n,g},$ (44)
$\displaystyle\ket{n,-}_{\rm
AJC}=-\sin\varphi_{n}\ket{n+1,e}+\cos\varphi_{n}\ket{n,g},$ (45)

and

$\displaystyle\cos\varphi_{n}=\sqrt{\frac{\Omega_{n}^{\rm AJC}+\xi-2i\kappa
n}{2\Omega_{n}^{\rm AJC}}},$ (46)
$\displaystyle\sin\varphi_{n}=\sqrt{\frac{\Omega_{n}^{\rm AJC}-\xi+2i\kappa
n}{2\Omega_{n}^{\rm AJC}}}.$ (47)

Because of the counter-rotating terms, when the initial state of the system is
$\ket{n+1,e}$, only $\ket{g,n}$ state can be coupled with it. Then we can
assume the form of the time evolution of the system states as

$\ket{\phi(t)}=[C^{e}_{n+1}(t)\ket{n+1,e}+C^{g}_{n}(t)\ket{g,n}]e^{-i(n+1/2)\omega
t}.$ (48)

with the $C_{n+1}^{e}(t)$ and $C_{n}^{g}(t)$ denoting the time-dependent
probability amplitudes for each state. Using the TDSE
$i\dot{\ket{\phi(t)}}=H_{\rm eff}^{\rm AJC}\ket{\phi(t)}$, we then get the
equations that the amplitudes satisfied,

$\begin{aligned} &\dot{C_{n+1}^{e}}(t)=[-i\frac{\xi}{2}-\kappa
n(n+1)]C_{n+1}^{e}(t)-ig_{2}\sqrt{n+1}C_{n}^{g}(t),\\\
&\dot{C_{n}^{g}}(t)=[i\frac{\xi}{2}-\kappa
n(n-1)]C_{n}^{g}(t)-ig_{2}\sqrt{n+1}C^{e}_{n+1}(t).\end{aligned}$ (49)

When the initial states meet $C_{n+1}^{e}(0)=1,C_{n}^{g}(0)=0$, we then get
the time evolution equation of the system state as follows

$\begin{aligned} \ket{\phi(t)}=&[(\cos\frac{\Omega_{n}^{\rm
AJC}t}{2}-\frac{i\xi+2\kappa n}{\Omega_{n}^{AJC}}\sin\frac{\Omega_{n}^{\rm
AJC}t}{2})\ket{n+1,e}\\\
&-i\frac{2g_{2}\sqrt{n+1}}{\Omega_{n}^{AJC}}\sin\frac{\Omega_{n}^{\rm
AJC}t}{2}\ket{g,n}]e^{-i(n+1/2)\omega t-\kappa n^{2}t}\end{aligned}$ (50)

We then obtain the time-dependent population on state $\ket{n,g}$ is

$P_{n,g}^{\rm AJC}=\frac{4g_{2}^{2}(n+1)}{{\Omega_{n}^{\rm
AJC}}^{2}}\sin^{2}\frac{\Omega_{n}^{\rm AJC}t}{2}.$ (51)

##  Appendix D Further discussions on the parity-time-symmetry

The energy spectrum of the JCM with two-photon dissipation is characterized by
parity-time symmetry and the presence of exceptional points (EPs). Based on
the above calculation, the eigenvalues of this system at resonate are

$E_{n,\pm}^{\rm JC}=(n+\frac{1}{2})\omega-i\kappa
n^{2}\pm\sqrt{g^{2}(n+1)-\kappa^{2}n^{2}},$ (52)

where $-i\kappa n^{2}$ is interpreted as a frequency shift. For
$g^{2}(n+1)>\kappa^{2}n^{2}$, the radical term has a real value, whereas for
$g^{2}(n+1)<\kappa^{2}n^{2}$, the value becomes the following imaginary form

$E_{n,\pm}^{\rm JC}=(n+\frac{1}{2})\omega-i\kappa n^{2}\pm
i\sqrt{\kappa^{2}n^{2}-g^{2}(n+1)}.$ (53)

According to the energy spectrum of the JC model with two-photon dissipation
when $n=1$, the energy level splitting is closed at
$g_{1}/\omega=\kappa/\sqrt{2}$ and the imaginary part of the energy after this
position is a constant $-i\kappa$, which means the entire system has one
dissipation channel at the same rate. In this case, it can be regarded as a
passive $\mathcal{PT}$ system. The imaginary part of the energy level is zero
after the crossing point by shifting the imaginary part of the energy level.
The region of all-real eigenvalues corresponds to the $\mathcal{PT}$ symmetric
phase of the system, as shown in the darker part of Fig.(8).

Generally, the system reaches an exceptional point (EP) when $\delta=0$, and
the position is

$g_{1}^{\rm EP}=\sqrt{\frac{\kappa^{2}n^{2}}{n+1}}.$ (54)

which varies with the photon number $n$.

Figure 8: Energy spectrum of the JCM with two-photon dissipation when
$\Delta=\omega$ and $n=1$. (a) and (b) describe the real and imaginary parts
of the splitting energy levels $E_{1,+}^{\rm JC}$ (red line) and $E_{1,-}^{\rm
JC}$ (blue line), respectively. It can be observed that there is an energy
overlap at $g_{1}/\omega=\sqrt{2}\times 10^{-2}$.

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