#  $\mathcal{PT}$-symmetric quantum Rabi model

Xilin Lu1,2,3 Hui Li1 Jia-Kai Shi1 Li-Bao Fan1 Vladimir Mangazeev3 Zi-Min Li1
<EMAIL_ADDRESS>Murray T. Batchelor2 1Institute 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
2Mathematical Science Institute, Australian National University, Canberra ACT
2601, Australia
3Research School of Physics, Australian National University, Canberra ACT
2601, Australia

###### Abstract

In this work, we explore the $\mathcal{PT}$-symmetric quantum Rabi model,
which describes a $\mathcal{PT}$-symmetric qubit coupled to a quantized light
field. By employing the adiabatic approximation (AA), we are able to solve
this model analytically in the parameter regime of interest and analyze
various physical aspects. We investigate the static and dynamic properties of
the model, using both the AA and numerical diagonalization. Our analysis
reveals a multitude of exceptional points (EPs) that are closely connected
with the exactly solvable points in the Hermitian counterpart of the model.
Intriguingly, these EPs vanish and revive depending on the light-matter
coupling strength. Furthermore, we discuss the time evolution of physical
observables under the non-Hermitian Hamiltonian. Rich and exotic behaviors are
observed in both strong and ultra-strong coupling regimes. Our work extends
the theory of $\mathcal{PT}$ symmetry into the full quantum light-matter
interaction regime and provides insights that can be readily enlarged to a
broad class of quantum optical systems.

##  I Introduction

Quantum mechanics assumes Hamiltonians that describe physical systems to be
Hermitian to ensure real energy eigenvalues and unitary time evolution.
However, there has been a recent surge of interest in non-Hermitian systems
[1, 2], especially those with parity-time ($\mathcal{PT}$) symmetry that
manifest a transition from purely real to complex conjugate spectra [3].
$\mathcal{PT}$ symmetry not only is of fundamental importance but also has
been applied across myriad fields [4, 5, 6].

The $\mathcal{PT}$-symmetric qubit, a two-level system with balanced gain and
loss, is a paradigmatic model to demonstrate the $\mathcal{PT}$ symmetry [7].
This simple solvable model effectively reveals the essential consequences of
$\mathcal{PT}$ symmetry and has been realized across multiple platforms [8,
9]. Further research has explored the interaction between the
$\mathcal{PT}$-symmetric qubit and classical light fields [10, 11, 12], with
detailed analyses of energy spectra and phase transitions in these semi-
classical models. However, $\mathcal{PT}$ symmetry in pure quantum systems,
particularly in the context of light-matter interactions, has rarely been
visited.

The quantum interaction between light and matter is typically characterized by
the quantum Rabi model (QRM), which describes a two-level system coupled to a
single-mode quantized light field [13, 14, 15]. Despite its simplicity, the
QRM exhibits rich physics and has found applications ranging from quantum
optics [16] and condensed matter physics [17], to molecular physics and state-
of-the-art superconducting circuit quantum electrodynamics [18, 19, 20].

Motivated by the QRM, here we generalize the theory of $\mathcal{PT}$ symmetry
into full quantum light-matter interaction systems. We begin by constructing a
$\mathcal{PT}$-symmetric Hamiltonian from a realistic physical system and
verifying its symmetry. Next, we interpret the model in the displaced
oscillator picture and solve it analytically using the adiabatic approximation
[21, 22, 23]. We then investigate the static properties, including energy
eigenvalues and eigenstates, both analytically and numerically. Interestingly,
we observe an infinite number of exceptional points (EPs), which vanish and
revive depending on the coupling strength. Additionally, we study the dynamics
governed by the non-Hermitian Hamiltonian, with a particular focus on the time
evolution of the mean photon number in the cavity and the qubit population.

##  II Model Hamiltonian

The system we propose to study the $\mathcal{PT}$ symmetry is depicted in Fig.
1(a), in which a non-Hermitian qubit is coupled to a single-mode cavity. This
hybrid system can be described by the Hamiltonian

$H=\dfrac{\Delta}{2}\sigma_{z}+\dfrac{i\epsilon}{2}\sigma_{x}+\omega
a^{\dagger}a+g\sigma_{x}\left(a^{\dagger}+a\right),$ (1)

where $\sigma_{x,z}$ are the spin 1/2 Pauli matrices, with $\Delta$ being the
qubit level splitting and $\epsilon$ term being the driving or coupling
between the two qubit states. The cavity mode with frequency $\omega$ is
governed by the bosonic creation and annihilation operators $a^{\dagger}$ and
$a$. The coupling strength between the qubit and the cavity is denoted by $g$.
Without loss of generality, all system parameters in this work are assumed to
be non-negative.

We begin our analysis with the $\mathcal{PT}$-symmetric qubit

$H_{q}^{z}=\dfrac{\Delta}{2}\sigma_{z}+\dfrac{i\epsilon}{2}\sigma_{x},$ (2)

for which the parity operator is $\sigma_{z}$ and the time reversal operator
is the complex conjugate operator. The eigenvalues are simply
$\pm\sqrt{\Delta^{2}-\epsilon^{2}}$, indicating an exceptional point (EP) at
$\epsilon/\Delta=1$, where the two eigenvectors coalesce and the eigenvalues
become degenerate. When $\epsilon/\Delta$ crosses the EP, the eigenvalues
change from real to complex, signaling the phase transition from
$\mathcal{PT}$-symmetric to $\mathcal{PT}$-broken. More details on the
$\mathcal{PT}$-symmetric qubit are given in the Appendix A.

Figure 1: (a) Schematic of the system consisting of a $\mathcal{PT}$-symmetric
qubit coupled to a cavity. (b) Two displaced oscillators, one with gain and
the other with balanced loss, are coupled by the qubit term
$\frac{\Delta}{2}\sigma_{z}$.

Regarding the light field, the effects of the $\mathcal{PT}$ operators are
equivalent to those of the bosonic parity transformation, leading to
$a\rightarrow-a$ and $a^{\dagger}\rightarrow-a^{\dagger}$.

Therefore, the combined parity operator for the hybrid quantum optical system
(1) is $\mathcal{P}=\sigma_{z}e^{i\pi a^{\dagger}a}$, which may be interpreted
as the parity of the total excitation number in the system [24]. Meanwhile,
the time-reversal operation is still to take the complex conjugation. The
symmetry of the full Hamiltonian can therefore be readily verified as

$(\mathcal{PT})^{\dagger}H\mathcal{PT}=H.$ (3)

We thus refer to this system as the $\mathcal{PT}$-symmetric quantum Rabi
model (PTQRM). Unlike previous approaches that change the coupling parameter
$g$ of the semi-classical Rabi model into an imaginary value [10, 11, 12],
here we have taken a more physically intuitive route to obtain the pure
quantum model by coupling a $\mathcal{PT}$-symmetric qubit to a quantized
field.

The PTQRM reduces to the well-known asymmetric quantum Rabi model (AQRM) if
the imaginary term is replaced by the real version
$\frac{\epsilon}{2}\sigma_{x}$ [14, 25, 26, 27, 28]. It reduces further to the
standard QRM if $\epsilon=0$.

##  III Adiabatic approximation

The non-Hermitian nature of the system, as described by Eq. (1), arises from
the nonzero value of $\epsilon$. Furthermore, the interplay between $\epsilon$
and $\Delta$ underpins our understanding of non-Hermitian physics [4, 7].
Consequently, neither $\Delta/\omega$ nor $\epsilon/\omega$ need to have large
absolute values. Under these conditions, the criteria for the adiabatic
approximation (AA), given by $\Delta/\omega<1$, remains applicable [22, 23].
Thus, the physical phenomena of PTQRM can be explored through the analytical
framework provided by the AA. It is worth noting that when the system deviates
from the AA conditions, specifically when $\Delta/\omega\gtrsim 1$, the
observed physics remains largely similar. However, in this regime, analytical
treatments become challenging.

The displaced oscillator picture offers a particularly intuitive understanding
of the AA. In this representation, light-matter interactions manifest as
qubit-dependent displacements in the light field [21]. Such a representation
becomes feasible by considering the degenerate qubit limit, where $\Delta=0$.
In this case, the Hamiltonian becomes

$H_{\text{do}}=\omega
a^{\dagger}a+g\sigma_{x}(a^{\dagger}+a)+\frac{i\epsilon}{2}\sigma_{x}.$ (4)

With respect to the basis of $\ket{\pm x}$, where $\sigma_{x}\ket{\pm
x}=\pm\ket{\pm x}$, the Hamiltonian can be rewritten in the form of displaced
creation and annihilation operators, namely

$H_{\text{do}}^{\pm}=\omega\left[\left(a^{\dagger}\pm\frac{g}{\omega}\right)\left(a\pm\frac{g}{\omega}\right)\right]\pm\frac{i\epsilon}{2}-\frac{g^{2}}{\omega}.$
(5)

It follows that $H_{\text{do}}$ is diagonalized by a unitary transformation
generated by the position displacement operator
$\mathcal{D}(\alpha)=\exp[-\alpha(a^{\dagger}-a)]$, with the displacement
amplitude $\alpha=\pm g/\omega$ associated with the qubit states. This leads
to the solutions

$\displaystyle\psi^{\text{do}}_{N,\pm}$ $\displaystyle=\ket{n_{\pm},\pm
x}=\mathcal{D}(\pm g)\ket{n}\otimes\ket{\pm x},$ (6) $\displaystyle
E^{\text{do}}_{N,\pm}$
$\displaystyle=n\omega-\frac{g^{2}}{\omega}\pm\frac{i\epsilon}{2},$ (7)

where $\ket{n_{\pm}}=\mathcal{D}(\pm g/\omega)\ket{n}$ are the displaced Fock
states, also known as the generalized or extended coherent states.

At an intuitive level, Eq.(5) describes two displaced harmonic oscillators.
These oscillators have a displacement amplitude of $\pm g/\omega$ and a gain
or loss rate of $i\epsilon/2$, corresponding to the qubit states $\ket{\pm
x}$, as depicted in Fig.1(b). This framework, termed the “displaced oscillator
picture”, is a well-established tool in light-matter interaction studies.

Next, we express the QRM Hamiltonian in terms of the displaced oscillator
eigenstates, denoted by Eq. (6), and introduce the parameter $\Delta$. The
qubit term, $\frac{\Delta}{2}\sigma_{z}$, facilitates the coupling of these
eigenstates, a process that can be analogized as tunneling between the two
displaced oscillators. Central to the AA is the assumption that only the
tunneling processes between eigenstates sharing the same quantum number $n$
are taken into account, effectively disregarding higher-order terms. By doing
so, the PTQRM Hamiltonian is decomposed into block-diagonal form, consisting
of infinitely many $2\times 2$ blocks

$H_{n}=n\omega-\dfrac{g^{2}}{\omega}+\dfrac{1}{2}\begin{pmatrix}i\epsilon&\Omega_{n}\\\
\Omega_{n}&-i\epsilon\end{pmatrix},$ (8)

where

$\Omega_{n}=\Delta
e^{-2g^{2}/\omega^{2}}L_{n}\left(\dfrac{4g^{2}}{\omega^{2}}\right),$ (9)

with $L_{n}(x)$ being the $n$th-order Laguerre polynomial. The $2\times 2$
matrix presented in Eq. (8) resembles a $\mathcal{PT}$-symmetric qubit. Yet,
it also encompasses a parameter-dependent driving, $\Omega_{n}$, that
integrates the effects of the light-matter interaction. It is important to
highlight that the AA essentially effectuates a change of basis. Specifically,
it shifts from the (bare) photon-qubit basis

$\ket{n,\pm}=\ket{n}\otimes\ket{\pm z}$ (10)

as detailed in Eq.(1), transitioning to the displaced oscillator basis
$\ket{n_{\pm},\pm x}$ as described in Eq.(6). Considering the qubit’s degree
of freedom, this basis transformation results in a $\pi/2$ rotation on the
Bloch sphere, while maintaining the $\mathcal{PT}$ symmetry. A more
comprehensive discussion on the $\mathcal{PT}$ symmetry postrotation is given
in Sect. VI.

To ascertain the eigenvalues of the PTQRM, we diagonalize $H_{n}$, resulting
in the eigenvalues

$E_{n}^{\pm}=n\omega-\dfrac{g^{2}}{\omega}\pm\dfrac{\sqrt{\Omega_{n}^{2}-\epsilon^{2}}}{2}.$
(11)

Owing to the transformation from the infinite-dimensional matrix $H$ to the
two-dimensional matrix $H_{n}$, the energy levels can now be characterized by
a multitude of level pairs, denoted by the eigenvalues $E_{n}^{\pm}$. In our
analysis, we initiate from the zeroth level $(n=0)$. The term
$\sqrt{\Omega_{n}^{2}-\epsilon^{2}}$ reveals the non-Hermiticity of the $n$th
level pair. Notably, exceptional points (EPs) emerge when $\epsilon$ equates
to $|\Omega_{n}|$.

The eigenstates corresponding to these eigenvalues are

$\ket{\psi_{n}^{\pm}}=\mathcal{N}_{n}^{\pm}\begin{pmatrix}\dfrac{i\epsilon\pm\sqrt{\Omega_{n}^{2}-\epsilon^{2}}}{\Omega_{n}}\\\
\vspace{-.5em}\\\ 1\end{pmatrix},$ (12)

which are expressed with respect to the displaced oscillator basis, as
outlined in Eq. (6). The normalization factors for these eigenstates can be
deduced as

$\mathcal{N}_{n}^{\pm}=\begin{cases}\dfrac{1}{\sqrt{2}},&\absolutevalue{\Omega_{n}}\geq\epsilon,\\\
\dfrac{\Omega_{n}}{\sqrt{2\epsilon^{2}\pm
2\epsilon\sqrt{\epsilon^{2}-\Omega_{n}^{2}}}},&\absolutevalue{\Omega_{n}}<\epsilon.\end{cases}$
(13)

When $\epsilon=0$, the derived eigensystem aligns with the AA of the standard
QRM, as detailed in Ref. [22].

Utilizing the AA results, we can derive analytical expressions for various
physical observables. For instance, consider the fidelity between the two
eigenstates of the $n$th level pair:

$F_{n}^{\text{AA}}=\absolutevalue{\innerproduct{\psi_{n}^{+}}{\psi_{n}^{-}}}^{2}=\begin{cases}{\epsilon^{2}}/{\Omega_{n}^{2}},&\absolutevalue{\Omega_{n}}\geq\epsilon,\\\
{\Omega_{n}^{2}}/{\epsilon^{2}},&\absolutevalue{\Omega_{n}}<\epsilon.\end{cases}$
(14)

Additionally, the mean cavity photon number is described as

$\expectationvalue{a^{\dagger}a}_{n}^{\pm}=n+g^{2}/\omega^{2}.$ (15)

The qubit expectation value is given analytically by

$\expectationvalue{\sigma_{z}}_{n}^{\pm}=\begin{cases}\pm\dfrac{\sqrt{\Omega_{n}^{2}-\epsilon^{2}}}{\Omega_{n}},&\absolutevalue{\Omega_{n}}\geq\epsilon,\\\
0,&\absolutevalue{\Omega_{n}}<\epsilon.\end{cases}$ (16)

Using this expectation value, the qubit population can be determined as

$W_{n}^{\pm}=\frac{1}{2}\left(1+\expectationvalue{\sigma_{z}}_{n}^{\pm}\right).$
(17)

As depicted in Fig. 3, there is a clear agreement between the AA expressions
and exact results. This consistency provides deeper insights into the
intricate dynamics of $\mathcal{PT}$-symmetric physics.

##  IV Spectrum, exceptional points, and phase transitions

Figure 2: Vanishing and revival of the EPs. (a)-(d) Real (blue thick) and
imaginary (red thin) energy spectra of the PTQRM with respect to the qubit
parameter $\epsilon$, with the coupling strength $g/\omega=0,0.2,0.5$, and
$1.8$, respectively. For each figure, $\Delta/\omega$ is set to 0.5. For
clarity, imaginary parts of the eigenvalues have been rescaled by adding the
corresponding level index $n$. The arrows in the figures denote the moving
directions, blue for vanishing and orange for reviving, of the EPs if $g$ is
to be increased.

Figure 3:  The energy spectra of (a) the QRM and (b) the PTQRM with respect to
the coupling strength $g/\omega$. The real and imaginary parts of the PTQRM
eigenvalues are denoted with blue and red lines, respectively. For the third
pair levels, the plots are (c) the energy spectrum, (d) the fidelity, (e)
qubit populations, and (f) mean photon numbers for the corresponding two
eigenstates, against coupling strength $g/\omega$. Relevant parameter values
are $\Delta/\omega=0.5$ and $\epsilon/\omega=0.1$. Analytic results of AA are
depicted in dashed lines. The system is in the $\mathcal{PT}$-symmetric phase
in the blue (dark gray) regions and the $\mathcal{PT}$-broken phase in the red
(light gray) regions.

In our subsequent analysis, we delve into the spectral structure and the novel
non-Hermitian dynamics inherent in the PTQRM. We employ both numerical
diagonalization and analytical insights from the AA for a more comprehensive
understanding. Analyzing the spectrum, we discern numerous EPs, akin to those
observed in the $\mathcal{PT}$-symmetric qubits. Notably, these EPs exhibit
additional modulation attributable to the cavity. Intriguingly, the cavity
introduces an exotic phenomenon: the vanishing and revival of EPs.

To observe the interesting EP phenomena under light-matter interaction, we
tune the ratio of the qubit parameters $\Delta$ and $\epsilon$, as shown in
Fig. 2. In the decoupled case, all level pairs behave the same as the spectrum
of the $\mathcal{PT}$-symmetric qubit, only with the energy raised by the
cavity eigenvalues. When the coupling strength $g$ is increased, EPs are
shifted due to the interaction, and their overall behavior goes through three
stages: vanishing, oscillating revival, and vanishing again. It can be seen
from Fig. 2(b) that EPs at higher levels move faster. Therefore, EPs from high
to low vanish in sequence with the increase of $g$. Owing to the light-matter
interaction, the EPs may reappear in an oscillatory manner. Finally, when the
interaction approaches a certain strength, all EPs tend to disappear in
sequence, as illustrated in Fig. 2(d) with $g/\omega=1.8$.

The phenomenon of EPs’ vanishing and revival can be inferred from the PTQRM
spectrum with respect to the coupling strength $g$, as shown in Fig. 3(b). It
is noteworthy that the majority of EPs stem from the exactly solvable Juddian
points, the level crossings, in the QRM [27], whose spectrum is depicted in
Fig. 3(a). The locations and numbers of Juddian points on each pair of energy
levels are predicted by the defining polynomials [29, 30, 27, 31, 32]. In the
AQRM, where the $\epsilon$ term is real, the Juddian points manifest as
diabolic points (DPs) in the three-dimensional parameter space [33, 34]. Each
DP bifurcates into a pair of EPs when $\epsilon$ assumes an imaginary value.

In addition to the spectrum, the eigenstates also exhibit novel
$\mathcal{PT}$-symmetric physics due to the light-matter interaction. To
explore the eigenstates, we take the second-level pair, determined by $H_{2}$,
as an example. The eigenspectrum together with the corresponding fidelity,
qubit population, and mean photon number is shown in Fig. 3, with the AA
results denoted by the dashed lines.

Owing to the intricate modulation induced by the light-matter interaction,
there is no standard overall real-to-complex phase transition when considering
all energy levels [3]. For a given set of parameter values, there will always
be imaginary eigenvalues when photon numbers reach sufficiently high values.
Nevertheless, it proves beneficial to delineate “local” phases for individual
level pairs of the system’s eigenstates. To be precise, for a specified level
index $n$, the system is considered to be in the $\mathcal{PT}$-symmetric
(PTS) phase if the eigenvalues $E_{n}^{\pm}$ are real, and in the
$\mathcal{PT}$-broken (PTB) phase if they are complex. Such a definition
becomes particularly pertinent when the cavity contains only a limited number
of photons and the Hilbert space remains finite-dimensional. These phases can
be visualized in Fig. 3, distinguished by blue (PTS) and red (PTB)
backgrounds.

In Hermitian systems, the eigenstates associated with different eigenvalues
are orthogonal. Contrastingly, in non-Hermitian scenarios, eigenstates are
typically non-orthogonal and even parallel at EPs. For two pure states, their
fidelity is defined as
$F_{12}=\left|\innerproduct{\psi_{1}}{\psi_{2}}\right|^{2}$. Two states are
orthogonal if $F_{\text{12}}=0$ and parallel if $F_{12}=1$. As illustrated in
Fig. 3(d) that the two eigenstates are generally not orthogonal, with non-zero
fidelity. Remarkably, eigenstates align perfectly at multiple EPs, with the
fidelity being unity, a characteristic of EPs that does not exist in Hermitian
systems. Another intriguing characteristic observed is that the eigenstates
are orthogonal at some specific points between the EPs, which is absent in
normal non-Hermitian systems. Interestingly, these orthogonal points coincide
with the original Juddian points in the standard QRM [27], and their positions
are independent of the value of $\epsilon$. The nature of these exotic
orthogonal points can be understood from the perspective of the AA. In the AA
for the standard QRM, Juddian points appear when the Laguerre polynomials in
$\Omega_{n}$ vanish. Consequently, the eigenvectors in Eq. (12) become
$\psi_{n}^{+}=(1,0)^{T}$ and $\psi_{n}^{-}=(0,1)^{T}$, which are trivially
orthogonal.

Figure. 3(e) depicts the qubit population using dashed lines, while the
analytical results from Eq. (16) are represented by solid lines. This behavior
is similar to the $\mathcal{PT}$-symmetric qubit dynamics, as discussed in
Ref. [7]. Specifically, for a zero coupling strength, $g/\omega=0$, the qubit
and cavity are decoupled, with the PTQRM eigenstates manifesting as the bare
states $\ket{n,\pm}$ from Eq. (11). Here, the corresponding qubit populations
are trivially 1 for $\ket{+}$ states and 0 for $\ket{-}$ states. Upon
introducing non-zero coupling, entanglement between the qubit and cavity
states ensues, instigating a blend of the two qubit states and a resultant
shift in qubit populations. The eigenstates coalesce at the first EP, aligning
proximally to the Bloch sphere’s equator, leading to equal qubit populations
for both eigenstates. In the subsequent PTB phase post this EP, the qubit
states consistently traverse the Bloch sphere’s equator, maintaining this
uniform population distribution. As $g$ increases, the system passes through
several PTS and PTB zones, all of which exhibit analogous behavior.

In Fig. 3(f), we present the variation of the cavity photon number as a
function of the coupling strength. The associated AA analytical model is
captured by Eq. (15). Consistent with expectations, the mean photon number
within the cavity grows with increasing coupling strength, aligning with
findings from Ref. [35]. It is worth noting that discrepancies arise between
the exact results and the AA photon number, given by
$\expectationvalue{a^{\dagger}a}_{n}^{\pm}=n+g^{2}/\omega^{2}$, particularly
within the PTS regions. This divergence can be attributed to the inherent
assumptions within the AA framework. In the reduced subspace, the photon
number is definite for a specific $n$ and displacement $g$. Therefore, the
deviations in Fig. 3(d) cannot be captured solely by the AA and require the
inclusion of higher-order tunneling effects [23].

##  V Dynamics of physical observables

The dynamics of the PTQRM manifests rich physics absent in its Hermitian or
semi-classical counterparts. To reveal this, we focus on the time evolution of
the mean photon number and qubit population by numerically solving the time-
dependent Schrödinger equation (TDSE) across varied parameter domains.

###  V.1 Dynamics in the strong coupling regime

Figure 4: Time evolution of the (a) norm of the wave function (which is
rescaled with logarithms and capped at 10), (b) cavity photon number, and (c)
qubit population of the PTQRM in the strong coupling regime, with
$g/\omega=0.05,\Delta/\omega=0.5$ and the value of $\epsilon/\omega$ ranging
from 0 to 0.7. The initial state is set as $\ket{0,+}$.

In our analysis, we begin by focusing on the strong coupling (SC) regime,
wherein the light-matter interaction strength is substantially lower than both
the qubit and photon frequencies, that is, $g\ll\omega,\Delta$. Traditionally,
the SC regime requires a coupling strength surpassing the system’s decay rates
[16, 36]. However, within the context of the non-Hermitian PTQRM, we still
identify this regime as SC even if the loss rate $\epsilon$ could exceed the
coupling strength $g$. This classification arises from the fact that the gain
rate $\epsilon$ ensures sustained dynamics prior the exponential decay [4].

Within the Hermitian SC regime, the rotating wave approximation (RWA) is
typically employed to the QRM, leading to the well-known Jaynes-Cummings model
(JCM) [37, 38, 16]. This approximation is valid, given that the effects of
counter-rotating terms (CRTs) (those terms which do not conserve excitation
numbers) are negligible in this regime. A key consequence of the RWA is that
the otherwise infinite-dimensional Hilbert space is decomposed into infinitely
many two-dimensional subspaces, thereby facilitating the analytic solvability
[23]. The SC regime’s dynamics are primarily characterized by Rabi
oscillations between various bare photon-qubit states, signaling the exchange
of excitations between the cavity and the qubit.

In this part, we explore the non-Hermitian dynamics in the SC regime.
Contrasting with the Hermitian case, the non-Hermitian terms (gain and loss)
couple all the basis states, rendering the Hilbert space undecomposable via
the RWA. To carry out the calculations, we start from the initial bare photon-
qubit state $\ket{0,+}=\ket{0}\otimes\ket{+z}$, indicating an empty cavity and
a qubit in its excited state. Subsequently, the non-Hermitian dynamics are
examined across a range of $\epsilon$ values, while holding the other system
parameters constant at $\Delta/\omega=0.5$ and $g/\omega=0.01$. Based on Fig.
2(a), the EPs for the lowest energy levels should lie proximately at
$\epsilon/\omega=0.5$.

We begin by investigating the wave function’s norm within the PTQRM framework.
As mentioned before, the norms of wave functions, defined by

$\norm{\psi(t)}=\sqrt{\innerproduct{\psi(t)}{\psi(t)}},$ (18)

deviate from conservation due to the Hamiltonian’s non-Hermitian nature.
However, it is worth noting that the expectation values of physical
observables in non-Hermitian systems necessitate wave function norm
renormalization, featuring the norm dynamics as a pivotal indicator of non-
Hermitian effects. The dynamics, after taking logarithms and capped at 10, are
illustrated in Fig. 4(a). The figure distinctly shows that, in contrast to the
Hermitian case where the norms are inherently conserved, the norm of the wave
function oscillates in the PTS regime ($\epsilon/\omega<0.5$) and diverges in
the PTB regime ($\epsilon/\omega>0.5$). These behaviors are similar to the
dynamics previously observed in the $\mathcal{PT}$-symmetric qubit [7].

Proceeding further, we calculate the renormalized cavity photon number,
represented as

$\expectationvalue{n}(t)=\expectationvalue{a^{\dagger}a}(t)=\frac{\bra{\psi(t)}a^{\dagger}a\ket{\psi(t)}}{\innerproduct{\psi(t)}{\psi(t)}},$
(19)

which holds significance in the realm of light-matter interaction analysis.
Referring to Eq. (19), the associated dynamics, depicted in Fig. 4(b), deviate
noticeably from those in the Hermitian context. The figure reveals that photon
numbers oscillate across both the PTS and PTB phases. Intriguingly, while the
PTS regime sees the oscillation characteristics – frequency and amplitude –
being influenced by the non-Hermitian term $\epsilon$, they stabilize and
remain invariant in the PTB regime.

The qubit population stands as a pivotal quantity in the domain of quantum
information sciences. We compute this quantity using Eq. (17), and thus its
time evolution

$W(t)=\frac{1}{2}\left(1+\bra{\psi(t)}{\sigma_{z}}\ket{\psi(t)}\right).$ (20)

Referring to the outcomes presented in Fig. 4(c), it is evident that within
the PTS regime, the quantity oscillates around the originating qubit state,
$\ket{+z}$. This particular state corresponds to the north pole on the Bloch
sphere. Transitioning to the PTB phase, the population gravitates towards the
equatorial state on the Bloch sphere, registering a population value of 0.5.
Notably, subtle oscillations remain manifest in this PTB phase.

Through the dynamics of the three quantities demonstrated in Fig. 4, the
transition from the PTS phase to the PTB phase becomes evident, specifically
around the point $\epsilon/\omega=0.5$. In detail: the norm shifts from
oscillatory behavior to a divergent trajectory; both the photon number and
qubit population evolve from oscillations dependent on $\epsilon$ to
oscillations that remain indifferent to variations in $\epsilon$. This
transition can be further elucidated when considering a low-dimensional
subsystem accommodating a maximum of 1 photon. Such a system can be described
by the $4\times 4$ Hamiltonian matrix

$H_{\mathrm{low}}=\begin{pmatrix}-\frac{\Delta}{2}&\frac{i\epsilon}{2}&0&g\\\
\frac{i\epsilon}{2}&\frac{\Delta}{2}&g&0\\\
0&g&\omega-\frac{\Delta}{2}&\frac{i\epsilon}{2}\\\
g&0&\frac{i\epsilon}{2}&\omega+\frac{\Delta}{2}\end{pmatrix},$ (21)

which is expressed with respect to the basis $\\{\ket{0,\pm},\ket{1,\pm}\\}$.
Within this matrix, coupling terms induce transitions across the basis states.
It is noteworthy to mention that this system encapsulates both Hermitian and
non-Hermitian transition mechanisms. For the Hermitian processes driven by
$g$, the RWA can be applied, allowing analysis through the JCM. This typically
results in the manifestation of Rabi oscillations. Conversely, the non-
Hermitian mechanisms stemming from $\epsilon$ terms bear similarities to the
dynamics of the $\mathcal{PT}$-symmetric qubit: they oscillate based on the
gain and loss rates in the PTS phase and stabilize in the PTB phase [7]. In
the PTS regime, the dynamics introduce an intriguing interplay, with non-
Hermitian oscillations challenging the established JCM behavior. In the PTB
phase, however, the non-Hermitian mechanisms quickly reach a steady state,
leaving only the JCM dynamics active and culminating in oscillations that are
indifferent to variations in $\epsilon$. Drawing a parallel, it is evident
that the PTQRM dynamics in the SC regime share considerable similarities with
those of the $\mathcal{PT}$-symmetric qubit [7]. Nonetheless, PTQRM introduces
a unique layer of complexity by fostering competition in non-Hermitian
dynamics, primarily by enabling energy exchanges between light and matter.

In the SC regime, while the immediate impact of CRTs – which couple distinct
subspaces – may seem insubstantial, their cumulative effects can be
significant over time. Consequently, with prolonged duration, the probability
may escape from the predominant subspace. For sufficiently high coupling
strengths, even over short timescales, this effect becomes non-negligible,
leading to markedly distinct dynamic behaviors.

###  V.2 Dynamics in the ultrastrong coupling regime

Figure 5:  (a) The real spectrum of the PTQRM, with background colors
indicating the mapping of the initial bare state $\ket{4,+}$ onto the
eigenstates. The vertical dashed green line indicates the coupling strength
$g/\omega=0.7$. (b) Initial probabilities with respect to the basis of
eigenstates with $g/\omega=0.7$, mapped from the initial bare state
$\ket{4,+}$. (c) Imaginary parts of eigenvalues for the lowest 101 eigenstates
with $g/\omega=0.7$. The dashed grid lines denote the max and min values
$\pm\epsilon/2$. Time evolution of (d) the cavity photon number and (e)
eigenstate probabilities with $\ket{\psi(0)}=\ket{4,+}$. The inset in (d) is
the long-term dynamics where all three stages are present. Relevant parameter
values are $\Delta/\omega=0.5$ and $\epsilon/\omega=0.1$.

Ultrastrong coupling (USC) between light and matter is reached when the
coupling strength $g$ is comparable to the cavity or qubit frequencies [36,
39]. In the USC regime, CRTs significantly influence the dynamics, making the
application of the RWA infeasible for decomposing the Hamiltonian into
$2\times 2$ matrices [22, 23]. Consequently, any wave function becomes a
superposition of all eigenstates. Decomposing an arbitrary wave function with
respect to the eigenstate basis reveals components both within the PTS and PTB
regimes. This leads to a system evolution that intricately blends PTS and PTB
dynamics in the USC regime, which is sharply different from the SC regime. Our
subsequent analysis will delve into the time evolution of the cavity mean
photon number, determined numerically via the TDSE, highlighting the profound
effects of the ultrastrong light-matter interaction.

The long-term dynamics of the mean photon number in the USC regime can be
classified into three distinct stages: firstly, it exhibits oscillations
around the initial state; secondly, it converges towards eigenstates
characterized by the locally largest imaginary eigenvalues; and finally, in
the long term, it diverges towards infinite-photon states. Notably, the
duration of the first two stages is modulated by both system parameters and
the chosen initial states.

In the following, we discuss in detail the case $g/\omega=0.7$ as an example.
We start from the initial bare state $\ket{4,+}=\ket{4}\otimes\ket{+z}$, which
represents a system configuration with four photons in the cavity and the
qubit in its excited state. We can express this initial state in the context
of the eigenstate basis. To determine the component probability amplitudes
$p_{n}$ , we project the initial state onto these eigenstates, yielding

$p_{n}=\absolutevalue{\innerproduct{4,+}{\psi_{n}}}^{2}.$ (22)

Here, eigenstates are denoted $\ket{\psi_{n}}$, with $n$ indexed in order of
increasing eigenvalues. For computational considerations, our analysis
includes only the first 101 eigenstates. It is worth noting that expanding the
analysis to a broader set of eigenstates would still result in the observed
three-stage behavior.

Figure 5(a) illustrates the real eigenvalues of the lowest 22 eigenstates of
the PTQRM. The background colors represent the respective probabilities of
these eigenstates in the initial configuration $\ket{4,+}$. Tracing the line
$g/\omega=0.7$ in Fig. 5(a), we observe several pairs of eigenstates in the
PTB phase. Notably, the ninth pair exhibits the largest imaginary energies.

To further elucidate, Fig. 5(b) presents the probability distributions across
these eigenstates. While the majority of probabilities cluster around the
lower eigenstates, we emphasize that none of the higher eigenstates have a
strict probability of 0. This phenomenon follows from above that every wave
function manifests as a superposition of all the eigenstates in the USC
regime. Although these probabilities in higher eigenstates may initially
appear insignificant, their amplification by imaginary eigenvalues can propel
them to prominence. Corroborating this, we plot the imaginary eigenvalues
Im($E_{n}$) of the lowest 101 eigenstates in Fig. 5(c). Notably, imaginary
energies are more prominent in higher-level eigenstates, and their locally
maximal imaginary eigenvalues are closer to the theoretical maximum of
$\epsilon/2$. The interplay between these probability distributions and
imaginary eigenvalues gives rise to the distinct three-stage dynamics, which
will be further explored.

In Fig. 5(d), we present the evolution of the mean photon number, starting
from the bare state $\ket{4,+}$. Initially, the system exhibits oscillations
around its initial state, stemming from the interplay among dynamics of the
high-probability eigenstate components. As the evolution progresses, the
system stabilizes, with the mean photon number settling around 9. This plateau
arises as the norms of the PTB states amplify and oscillatory behaviors
diminish, notably post $100\pi$. The stable behavior is sustained until the
system eventually diverges towards infinite-photon states, which is evident
from the long-term dynamics demonstrated in the inset. In numerical
calculations, however, the infinity is unattainable due to our Hilbert space
truncation, and the maximal photon number caps at approximately 100.

The transition between the first two stages stands out as particularly
interesting. In this process, the imaginary energies $\imaginary(E_{n})$
dictate the growth rates of the corresponding eigenstates $\ket{\psi_{n}}$.
Over time, an eigenstate with the largest imaginary energy prevails through
exponential growth, overshadowing their initial probabilities. This phenomenon
is exemplified when projecting the bare state $\ket{4,+}$ onto the
eigenstates. Remarkably, the dominant eigenstate $\ket{\psi_{20}}$, despite
its initial probability being less than 0.01, quickly rises to prominence, as
illustrated in Fig. 5(e).

##  VI Further discussions

In the preceding sections, we have proposed the PTQRM Hamiltonian and analyzed
its exotic properties in both statics and dynamics. Moving forward, we aim to
provide a deeper understanding of the PTQRM Hamiltonian. To this end, we
demonstrate an alternative representation of the PTQRM to frame the non-
Hermitian Hamiltonian within tangible physical contexts. We also juxtapose the
non-Hermitian dynamics against the outcomes derived from the quantum master
equation, which is the standard approach to deal with open systems. Lastly, we
explore the potential experimental realization of the PTQRM based on
superconducting circuits. By including these aspects, we hope to offer a
comprehensive perspective on the PTQRM Hamiltonian and its potential
implications in the realm of quantum physics.

###  VI.1 Alternative representation of the PTQRM

In this part, we further justify the $\mathcal{PT}$ symmetry of the PTQRM by
drawing parallels with the conventional form of $\mathcal{PT}$-symmetric qubit
in the existing literature [7, 8, 5, 4]. Although the $\mathcal{PT}$ symmetry
in the non-Hermitian qubit, as described by Eq. (2), might not seem obvious at
first glance, a deeper examination reveals its underlying symmetry. Here in
Eq. (2), the superscript $z$ signifies the chosen basis where $\sigma_{z}$ is
diagonalized. Therefore, the two basis states of the qubit are
$\ket{\pm}=\ket{\pm z}$. In this framework, $\Delta$ represents the level
splitting between the two states and $i\epsilon$ delineates the coupling
between them.

The direct physical interpretation of the imaginary coupling in $H_{q}^{z}$
might pose challenges. To render it more intuitive, we can implement a unitary
transformation generated by

$\quad U=\frac{1}{\sqrt{2}}\begin{pmatrix}1&1\\\ 1&-1\end{pmatrix},$ (23)

with the effects

$\quad U^{\dagger}\sigma_{x}U=\sigma_{z},\quad
U^{\dagger}\sigma_{z}U=\sigma_{x}.$ (24)

We then arrive at the new qubit Hamiltonian

$H_{q}^{x}=U^{\dagger}H_{q}^{z}U=\frac{\Delta}{2}\sigma_{x}+\frac{i\epsilon}{2}\sigma_{z}=\frac{1}{2}\begin{pmatrix}i\epsilon&\Delta\\\
\Delta&-i\epsilon\end{pmatrix},$ (25)

where the superscript $x$ indicates our transition to the eigenbasis of
$\sigma_{x}$. With $H_{q}^{x}$, the two states of the qubit are now $\ket{\pm
x}$. The term $\Delta$ describes the mutual coupling of these states, while
the imaginary diagonal terms $i\epsilon$ intuitively represent the gain for
the excited state and the corresponding loss for the ground state.

Conventionally, a $\mathcal{PT}$-symmetric qubit is expressed in the form of
$H_{q}^{x}$ due to the explicit physical interpretation of the imaginary terms
[7]. It is readily seen that $H_{q}^{x}$ remains invariant under the
transformation

$(\mathcal{P}_{x}\mathcal{T})^{\dagger}H_{q}^{x}(\mathcal{P}_{x}\mathcal{T})=H_{q}^{x},$
(26)

where $\mathcal{P}_{x}=\sigma_{x}$ and $\mathcal{T}$ is complex conjugation.

To ascertain the $\mathcal{PT}$ symmetry of our original qubit Hamiltonian
$H_{q}^{z}$, we only need to revert to the $\sigma_{z}$ basis by doing the $U$
transformation again:

$H_{q}^{z}=U^{\dagger}H_{q}^{x}U.$ (27)

This implies the parity operator adopts the form $\mathcal{P}_{z}=\sigma_{z}$,
whereas the $\mathcal{T}$ operator remains the same. With this, the
$\mathcal{PT}$ symmetry is evident from

$(\mathcal{P}_{z}\mathcal{T})^{\dagger}H_{q}^{z}(\mathcal{P}_{z}\mathcal{T})=H_{q}^{z}.$
(28)

The PTQRM Hamiltonian, when expressed in this rotated representation, becomes

$\displaystyle H_{\text{PTQRM}}^{x}=$
$\displaystyle\frac{\Delta}{2}\sigma_{x}+\frac{i\epsilon}{2}\sigma_{z}+\omega
a^{\dagger}a+g\sigma_{z}\left(a^{\dagger}+a\right)$ (29) $\displaystyle=$
$\displaystyle H_{\text{QRM}}^{x}+\frac{i\epsilon}{2}\sigma_{z}.$

Notably, the Hermitian $H_{\text{QRM}}^{x}$ aligns with the convention of the
QRM within the context of superconducting circuits [22]. Verification confirms
that the Hamiltonian (29) has exactly the same eigenspectrum as the PTQRM. The
corresponding eigenstates, along with other physical properties, remain
congruent upon a rotation transformation.

###  VI.2 The passive $\mathcal{PT}$ symmetry

In experimental realizations, the passive form of $\mathcal{PT}$ symmetry is
often favored due to its practicality [8, 9]. The passive
$\mathcal{PT}$-symmetric qubit is described by

$\tilde{H}_{q}^{x}=\frac{\Delta}{2}\sigma_{x}+i\epsilon\sigma_{+}\sigma_{-}=H^{x}_{q}+\frac{i\epsilon}{2}\mathbb{I}.$
(30)

Upon examination, it becomes evident that the passive $\mathcal{PT}$-symmetric
qubit $\tilde{H}_{q}^{x}$ possesses the same topology as its
$\mathcal{PT}$-symmetric counterpart, with the sole difference being a
constant shift ${i\epsilon}/{2}$ in eigenvalues. Within the passive
$\mathcal{PT}$ symmetry framework, only loss exists. Nonetheless, it retains
all the salient features inherent to standard $\mathcal{PT}$ symmetry,
including EPs [4, 5].

Incorporating a cavity into the qubit as expressed in Eq. (30) naturally
extends to the QRM under passive $\mathcal{PT}$ symmetry, represented by

$\displaystyle\tilde{H}^{x}_{\text{PTQRM}}$
$\displaystyle=\frac{\Delta}{2}\sigma_{x}+i\epsilon\sigma_{+}\sigma_{-}+\omega
a^{\dagger}a+g\sigma_{z}\left(a^{\dagger}+a\right)$ (31)
$\displaystyle=H^{x}_{\text{PTQRM}}+\frac{i\epsilon}{2}\mathbb{I}.$

The spectrum now has a constant shift $i\epsilon/2$ in energy, compared to
both the PTQRM (1) and its alternative representation $H^{x}_{\text{PTQRM}}$.
Moreover, the non-Hermitian dynamics generated by the Hamiltonian
$\tilde{H}^{x}_{\text{PTQRM}}$ are also identical to those of the PTQRM, upon
undergoing the renormalization outlined in Eq. (19).

Up to this point, we have delineated several forms of the PTQRM Hamiltonians
as given in Eqs. (1), (29) and (31). These distinct representations are
interrelated as illustrated by the sequence

$H\xrightarrow{\text{rotation}}H^{x}_{\text{PTQRM}}\xrightarrow{\text{energy
shift}}\tilde{H}^{x}_{\text{PTQRM}}.$ (32)

Crucially, their respective eigenspectra and dynamics coincide when
considering energy offsets or appropriate wave function renormalizations.

###  VI.3 Lindblad master equation approach

While closed quantum systems evolve according to the Schrödinger equation,
open systems, those that interact with an external environment, often require
a more intricate treatment. In such scenarios, the Lindblad master equation
(LME), a widely accepted formulation of the quantum master equation, becomes
instrumental [40]. This equation offers a framework to account for effects
such as decoherence and dissipation that arise due to environmental
interactions. Here, we illustrate that the passive PTQRM (31) can be perceived
as the non-Hermitian effective Hamiltonian that emerges from the underlying
dynamics captured by the LME.

To better elucidate the link between the passive PTQRM and the dynamics driven
by environmental effects, we consider a dissipative qubit coupled to a cavity.
The dissipative qubit may be described by a three-level system, and the
corresponding basis states are $\\{\ket{2},\ket{1},\ket{0}\\}$ with decreasing
energies [8]. We specify the decay rate, denoted by $\gamma$, from $\ket{1}$
to $\ket{0}$. Such dynamics can be accurately captured through the LME, given
by

$\frac{d\rho}{dt}=-i\left[H_{\mathrm{QRM}}^{x},\rho\right]+\gamma\mathcal{D}\left[\ket{0}\bra{1}\right]\rho,$
(33)

where $H_{\mathrm{QRM}}^{x}$ is the standard QRM Hamiltonian in the rotated
frame given in Eq. (29). Here $\rho$ is the density matrix of the system,
which encapsulates both the state information and coherence properties.
$\mathcal{D}[A]\rho\equiv A\rho A^{\dagger}-\frac{1}{2}(A^{\dagger}A\rho+\rho
A^{\dagger}A)$ defines the Lindblad superoperator.

We then expand the superoperator and rearrange the terms to obtain

$\displaystyle\frac{d\rho}{dt}=$
$\displaystyle-i\left(H_{\mathrm{QRM}}^{x}-i\frac{\gamma}{2}\ket{1}\bra{1}\right)\rho$
(34)
$\displaystyle+i\rho\left(H_{\mathrm{QRM}}^{x}+i\frac{\gamma}{2}\ket{1}\bra{1}\right)+\gamma\ket{0}\rho_{11}\bra{0}.$

Upon confining our attention to the subspace
$\mathrm{span}\\{\ket{1},\ket{2}\\}$ and with the identification
$\gamma=2\epsilon$, the governing dynamics of the system, in terms of the LME,
morphs into

$\frac{d\rho}{dt}=-i\left(\tilde{H}\rho-\rho\tilde{H}^{\dagger}\right)+2\epsilon\ket{0}\rho_{11}\bra{0}.$
(35)

Here, the effective Hamiltonian $\tilde{H}=\tilde{H}^{x}_{\text{PTQRM}}$ is
synonymous with our passive PTQRM (31) in the rotated frame, which is
equivalent to our standard PTQRM Hamiltonian (1). The last term in Eq. (34),
known as the quantum jump operator, incorporates the transitions from the
state space of the effective Hamiltonian to the larger Hilbert space of the
environment.

Upon neglecting the quantum jump term, the LME reduces to the von Neumann
equation, which is formally equivalent to the TDSE of the effective
Hamiltonian [40]. Consequently, the system’s dynamics are again governed by
the effective non-Hermitian Hamiltonian $\tilde{H}^{x}_{\text{PTQRM}}$, which
parallels the PTQRM Hamiltonian (1). The distinction between LME and TDSE
arises from the influence of the quantum jump term.

Experimentally, when one solely focuses on the effective Hamiltonian, a post-
selection scheme can be applied. This technique eliminates the effects of
quantum jump terms [8], rendering the system dynamics to be wholly governed by
the non-Hermitian Hamiltonian.

###  VI.4 Potential physical realizations

Figure 6: Possible circuit QED realization, the transmon qubit circuit is
embedded in a cavity (blue box in this diagram) with controls, for driving and
read-out, explained in Ref. [8] in detail. The lowest three energy levels are
used for constructing a $\mathcal{PT}$-symmetric qubit, which is consistent
with the master equation approach. Decay rates between these states are
controlled by the magnetic flux $\Phi$.

From the perspective of circuit quantum electrodynamics (QED), we could
envision a synthesis by merging the realization of QRM in Ref. [19] and the
$\mathcal{PT}$-symmetric qubit in Ref. [8]. The resulting superconducting
circuit is depicted in Fig. 6, where a transmon qubit contained in a cavity is
coupled to an LC resonator. Drawing from the transmon circuit’s lowest three
energy levels, we proceed with the derivation of the master equation for the
passive PTQRM: the ground state functions as a sink for the excited (qubit)
states. It was shown in [8] that, by modulating the magnetic flux $\Phi$
through the transmon circuit, we can control the two decay rates such that
$\ket{1}$ to $\ket{0}$ is much faster than $\ket{2}$ to $\ket{1}$. Given this
differential in decay rates, we can judiciously overlook the latter
transition, leading to a characterization of this system via our master
equation (35) in the absence of the quantum jump term.

In addition to the circuit QED platform, trapped ion experiments, employing
the ${}^{40}\mathrm{Ca}^{+}$ ion [9], also offer a prospective avenue for
realizing our system, specifically by incorporating the lasing mode. A
noteworthy investigation has recently delved into this avenue, adding an
auxiliary 1-photon state, thus revealing a more intricate landscape of EP
structures [41].

Another possible direction is considering the substitution of the traditional
qubit in QRM experiments [42] with $\mathcal{PT}$-symmetric ones. This
potential pivot points to an interesting direction in the field.

##  VII Summary

In this work, we explored the intricacies of the $\mathcal{PT}$-symmetric
quantum Rabi model (PTQRM) and formulated an analytical solution via the
adiabatic approximation (AA).

Our exploration of static properties revealed multiple exceptional points
(EPs) within the spectrum. Intriguingly, these EPs vanish and revive depending
on the coupling strength. A significant observation was the identification of
unique points where distinct eigenstates stand orthogonal – a characteristic
absent in typical non-Hermitian systems. In particular, these orthogonal
points coincide with the level crossing points of the Hermitian QRM.

We conducted a comprehensive examination of non-Hermitian dynamics in the
strong coupling (SC) and ultrastrong coupling (USC) regimes. In the SC regime,
the intricate interplay between Hermitian and non-Hermitian couplings
manifests strikingly distinct dynamical phenomena on either side of the EPs.
In contrast, within the USC regime, our investigation revealed a three-stage
evolution pattern in the cavity photon number: oscillation around the initial
state, convergence to nearby eigenstates with large imaginary energies, and
divergence to infinite-photon states.

This work broadens the theoretical horizon of $\mathcal{PT}$ symmetry,
encompassing the comprehensive quantum light-matter interaction domain.
Furthermore, it offers insights that have the potential to be extrapolated
across a wide spectrum of quantum optical systems [24, 43, 44]. We are
optimistic about the practical implementation of the PTQRM across various
experimental platforms, such as circuit QED [8] and trapped ions [9].

###### Acknowledgements.

This work is supported by the National Natural Science Foundation of China
Grant No. 12205383, and the Natural Science Foundation of Changsha Grant No.
kq2202082, and Australian Research Council Grant No. DP210102243. X. Lu, H.
Li, and J.-K. Shi contributed equally to this work.

##  Appendix A $\mathcal{PT}$ symmetry

Figure 7: The energy eigenvalues of the $\mathcal{PT}$-symmetric qubit
$H^{z}_{q}$. The real part is in blue lines and the imaginary part is in red
lines.

It is straightforward to calculate the eigenstates and eigenvalues of the non-
Hermitian qubit defined in the main text, i.e.,

$H_{q}^{z}=\frac{\Delta}{2}\sigma_{z}+\frac{i\epsilon}{2}\sigma_{x}.$ (36)

The eigensystem can be divided into two cases, depending on the values of
$\Delta$ and $\epsilon$, as explained below.

Case I: $\Delta\geq\epsilon$. The eigenvalues and corresponding eigenstates
are

$\displaystyle E_{\pm}^{z}$
$\displaystyle=\pm\frac{1}{2}\sqrt{\Delta^{2}-\epsilon^{2}},$ (37)
$\displaystyle\ket{\psi^{z}_{\pm}}$
$\displaystyle=\begin{pmatrix}\dfrac{-i\Delta\mp
i\sqrt{\Delta^{2}-\epsilon^{2}}}{\epsilon}\\\ \vspace{-.5em}\\\
1\end{pmatrix}.$

It is easy to verify that $\ket{\psi^{z}_{\pm}}$ are also the eigenstates of
the combined $\mathcal{PT}$ operator, namely

$\mathcal{PT}\ket{\psi^{z}_{\pm}}=-\ket{\psi^{z}_{\pm}}.$ (38)

Therefore, the system is in the $\mathcal{PT}$-symmetric phase.

Case II: $\Delta<\epsilon$. The eigenvalues and corresponding eigenstates are

$\displaystyle E_{\pm}^{z}$
$\displaystyle=\pm\frac{i}{2}\sqrt{\epsilon^{2}-\Delta^{2}},$ (39)
$\displaystyle\ket{\psi^{z}_{\pm}}$
$\displaystyle=\begin{pmatrix}\dfrac{-i\Delta\pm\sqrt{\epsilon^{2}-\Delta^{2}}}{\epsilon}\\\
\vspace{-.5em}\\\ 1\end{pmatrix},$

which now yields

$\mathcal{PT}\ket{\psi^{z}_{\pm}}=\begin{pmatrix}\dfrac{i\Delta\pm\sqrt{\epsilon^{2}-\Delta^{2}}}{\epsilon}\\\
\vspace{-.5em}\\\ -1\end{pmatrix}=-\ket{\psi_{\mp}^{z}}\neq
c\ket{\psi^{z}_{\pm}},$ (40)

where $c\in\mathbb{C}$. This case is termed as $\mathcal{PT}$-broken because
the wave functions do not have $\mathcal{PT}$ symmetry even though the
Hamiltonian still commutes with the combined $\mathcal{PT}$ operator. An
interesting fact is that the combined $\mathcal{PT}$ operator flips the two
eigenstates in the $\mathcal{PT}$-broken case, as shown in Eq. (40). The
energy spectrum discussed above is depicted in Fig. 7, with the real part and
imaginary part presented in blue and red lines, respectively. The transition
from real to complex eigenvalues emerges at the exceptional point
$\epsilon/\Delta=1$.

Regarding the $\mathcal{PT}$ symmetry of the light field, we recall the
general effects of parity and time operators $\mathcal{P}$ and $\mathcal{T}$
described as

$\begin{split}\mathcal{P}:i\rightarrow i,\quad x\rightarrow-x,\quad
p\rightarrow-p\\\ \mathcal{T}:i\rightarrow-i,\quad x\rightarrow x,\quad
p\rightarrow-p\end{split}$ (41)

It follows that by applying the $\mathcal{PT}$ transformation to the bosonic
creation and annihilation operators, we have

$a\rightarrow-a,\quad a^{\dagger}\rightarrow-a^{\dagger},$ (42)

which is equivalent to the bosonic parity transformation [24].

## References

  * Moiseyev [2014] N. Moiseyev,  _Non-Hermitian Quantum Mechanics_ (Cambridge University Press, 2014).
  * Ashida  _et al._ [2020] Y. Ashida, Z. Gong, and M. Ueda, Non-hermitian physics, Adv. Phys. 69, 249 (2020).
  * Bender [2005] C. M. Bender, Introduction to PT-symmetric quantum theory, Contemp. Phys. 46, 277 (2005).
  * El-Ganainy  _et al._ [2018] R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, Non-hermitian physics and PT symmetry, Nat. Phys. 14, 11 (2018).
  * Özdemir  _et al._ [2019] Ş. K. Özdemir, S. Rotter, F. Nori, and L. Yang, Parity–time symmetry and exceptional points in photonics, Nat. Mater. 18, 783 (2019).
  * Miri and Alù [2019] M.-A. Miri and A. Alù, Exceptional points in optics and photonics, Science 363, 10.1126/science.aar7709 (2019).
  * Dogra  _et al._ [2021] S. Dogra, A. A. Melnikov, and G. S. Paraoanu, Quantum simulation of parity–time symmetry breaking with a superconducting quantum processor, Commun. Phys. 4, 26 (2021).
  * Naghiloo  _et al._ [2019] M. Naghiloo, M. Abbasi, Y. N. Joglekar, and K. W. Murch, Quantum state tomography across the exceptional point in a single dissipative qubit, Nat. Phys. 15, 1232 (2019).
  * Wang  _et al._ [2021] W.-C. Wang, Y.-L. Zhou, H.-L. Zhang, J. Zhang, M.-C. Zhang, Y. Xie, C.-W. Wu, T. Chen, B.-Q. Ou, W. Wu, H. Jing, and P.-X. Chen, Observation of pt-symmetric quantum coherence in a single-ion system, Phys. Rev. A 103, l020201 (2021).
  * Joglekar  _et al._ [2014] Y. N. Joglekar, R. Marathe, P. Durganandini, and R. K. Pathak, $\mathcal{PT}$ spectroscopy of the Rabi problem, Phys. Rev. A 90, 040101 (2014).
  * Lee and Joglekar [2015] T. E. Lee and Y. N. Joglekar, $\mathcal{PT}$-symmetric Rabi model: Perturbation theory, Phys. Rev. A 92, 042103 (2015).
  * Xie  _et al._ [2018] Q. Xie, S. Rong, and X. Liu, Exceptional points in a time-periodic parity-time-symmetric Rabi model, Phys. Rev. A 98, 052122 (2018).
  * Rabi [1936] I. I. Rabi, On the process of space quantization, Phys. Rev. 49, 324 (1936).
  * Braak [2011] D. Braak, Integrability of the Rabi Model, Phys. Rev. Lett. 107, 100401 (2011).
  * Xie  _et al._ [2017] Q. Xie, H. Zhong, M. T. Batchelor, and C. Lee, The quantum Rabi model: Solution and dynamics, J. Phys. A: Math. Theor. 50, 113001 (2017).
  * Fox [2006] M. Fox,  _Quantum Optics_ (Oxford University Press, 2006).
  * Irish [2007] E. K. Irish, Generalized rotating-wave approximation for arbitrarily large coupling, Phys. Rev. Lett. 99, 173601 (2007).
  * Forn-Díaz  _et al._ [2010] P. Forn-Díaz, J. Lisenfeld, D. Marcos, J. J. García-Ripoll, E. Solano, C. J. P. M. Harmans, and J. E. Mooij, Observation of the Bloch-Siegert shift in a qubit-oscillator system in the ultrastrong coupling regime, Phys. Rev. Lett. 105, 237001 (2010).
  * Yoshihara  _et al._ [2016] F. Yoshihara, T. Fuse, S. Ashhab, K. Kakuyanagi, S. Saito, and K. Semba, Superconducting qubit–oscillator circuit beyond the ultrastrong-coupling regime, Nat. Phys. 13, 44 (2016).
  * Yoshihara  _et al._ [2018] F. Yoshihara, T. Fuse, Z. Ao, S. Ashhab, K. Kakuyanagi, S. Saito, T. Aoki, K. Koshino, and K. Semba, Inversion of qubit energy levels in qubit-oscillator circuits in the deep-strong-coupling regime, Phys. Rev. Lett. 120, 183601 (2018).
  * Crisp [1992] M. D. Crisp, Application of the displaced oscillator basis in quantum optics, Phys. Rev. A 46, 4138 (1992).
  * Irish  _et al._ [2005] E. K. Irish, J. Gea-Banacloche, I. Martin, and K. C. Schwab, Dynamics of a two-level system strongly coupled to a high-frequency quantum oscillator, Phys. Rev. B 72, 195410 (2005).
  * Li and Batchelor [2021a] Z.-M. Li and M. T. Batchelor, Generalized adiabatic approximation to the quantum Rabi model, Phys. Rev. A 104, 033712 (2021a).
  * Li and Batchelor [2021b] Z.-M. Li and M. T. Batchelor, Hidden symmetry and tunneling dynamics in asymmetric quantum Rabi models, Phys. Rev. A 103, 023719 (2021b).
  * Chen  _et al._ [2012] Q.-H. Chen, C. Wang, S. He, T. Liu, and K.-L. Wang, Exact solvability of the quantum Rabi model using Bogoliubov operators, Phys. Rev. A 86, 023822 (2012).
  * Zhong  _et al._ [2014] H. Zhong, Q. Xie, X. Guan, M. T. Batchelor, K. Gao, and C. Lee, Analytical energy spectrum for hybrid mechanical systems, J. Phys. A: Math. Theor. 47, 045301 (2014).
  * Li and Batchelor [2015] Z.-M. Li and M. T. Batchelor, Algebraic equations for the exceptional eigenspectrum of the generalized Rabi model, J. Phys. A: Math. Theor. 48, 454005 (2015).
  * Li and Batchelor [2016] Z.-M. Li and M. T. Batchelor, Addendum to ‘algebraic equations for the exceptional eigenspectrum of the generalized Rabi model’, J. Phys. A: Math. Theor. 49, 369401 (2016).
  * Judd [1979] B. R. Judd, Exact solutions to a class of Jahn-Teller systems, J. Phys. C: Solid State Phys. 12, 1685 (1979).
  * Kuś [1985] M. Kuś, On the spectrum of a two-level system, J. Math. Phys. 26, 2792 (1985).
  * Wakayama [2017] M. Wakayama, Symmetry of asymmetric quantum Rabi models, J. Phys. A: Math. Theor. 50, 174001 (2017).
  * Kimoto  _et al._ [2021] K. Kimoto, C. Reyes-Bustos, and M. Wakayama, Determinant expressions of constraint polynomials and the spectrum of the asymmetric quantum rabi model, Int. Math. Res. Not. 2021, 9458 (2021).
  * Batchelor  _et al._ [2016] M. T. Batchelor, Z.-M. Li, and H.-Q. Zhou, Energy landscape and conical intersection points of the driven Rabi model, J. Phys. A: Math. Theor. 49, 01LT01 (2016).
  * Li  _et al._ [2021] Z.-M. Li, D. Ferri, D. Tilbrook, and M. T. Batchelor, Generalized adiabatic approximation to the asymmetric quantum rabi model: conical intersections and geometric phases, J. Phys. A: Math. Theor. 54, 405201 (2021).
  * Rossatto  _et al._ [2017] D. Z. Rossatto, C. J. Villas-Bôas, M. Sanz, and E. Solano, Spectral classification of coupling regimes in the quantum Rabi model, Phys. Rev. A 96, 013849 (2017).
  * Kockum  _et al._ [2019] A. F. Kockum, A. Miranowicz, S. D. Liberato, S. Savasta, and F. Nori, Ultrastrong coupling between light and matter, Nat. Rev. Phys. 1, 19 (2019).
  * Jaynes and Cummings [1963] E. T. Jaynes and F. W. Cummings, Comparison of quantum and semiclassical radiation theories with application to the beam maser, Proc. IEEE 51, 89 (1963).
  * Shore and Knight [1993] B. W. Shore and P. L. Knight, The Jaynes-Cummings model, J. Mod. Opt. 40, 1195 (1993).
  * Forn-Díaz  _et al._ [2019] P. Forn-Díaz, L. Lamata, E. Rico, J. Kono, and E. Solano, Ultrastrong coupling regimes of light-matter interaction, Rev. Mod. Phys. 91, 025005 (2019).
  * Manzano [2020] D. Manzano, A short introduction to the lindblad master equation, AIP Advances 10, 025106 (2020).
  * Kim  _et al._ [2023] J. Kim, T. Ha, D. Kim, D. Lee, K.-S. Lee, J. Won, Y. Moon, and M. Lee, Third-order exceptional point in an ion-cavity system, Appl. Phys. Lett. 123, 161104 (2023).
  * Cai  _et al._ [2021] M.-L. Cai, Z.-D. Liu, W.-D. Zhao, Y.-K. Wu, Q.-X. Mei, Y. Jiang, L. He, X. Zhang, Z.-C. Zhou, and L.-M. Duan, Observation of a quantum phase transition in the quantum rabi model with a single trapped ion, Nat. Comm. 12, 10.1038/s41467-021-21425-8 (2021).
  * Xie  _et al._ [2014] Q.-T. Xie, S. Cui, J.-P. Cao, L. Amico, and H. Fan, Anisotropic Rabi model, Phys. Rev. X 4, 021046 (2014).
  * Xie  _et al._ [2019] Y.-F. Xie, L. Duan, and Q.-H. Chen, Quantum Rabi–Stark model: solutions and exotic energy spectra, J. Phys. A: Math. Theor. 52, 245304 (2019).