# Anomalous spontaneous emission dynamics at chiral exceptional points

Yu-Wei Lu  School of Physics and Optoelectronics, South China University of
Technology, Guangzhou 510641, China  School of Physics and Optoelectronic
Engineering, Foshan University, Foshan 528000, China  Yanhui Zhao  School of
Physics and Optoelectronic Engineering, Ludong University, Yantai 264000,
China  Runhua Li<EMAIL_ADDRESS>School of Physics and Optoelectronics,
South China University of Technology, Guangzhou 510641, China  Jingfeng Liu
College of Electronic Engineering, South China Agricultural University,
Guangzhou 510642, China

###### Abstract

An open quantum system operated at the spectral singularities where
dimensionality reduces, known as exceptional points (EPs), demonstrates
distinguishing behavior from the Hermitian counterpart. Here, we present an
analytical description of local density of states (LDOS) for microcavity
featuring chiral EPs, and unveil the anomalous spontaneous emission dynamics
from a quantum emitter (QE) due to the non-Lorentzian response of EPs.
Specifically, we reveal that a square Lorentzian term of LDOS contributed by
chiral EPs can destructively interfere with the linear Lorentzian profile,
resulting in the null Purcell enhancement to a QE with special transition
frequency, which we call EP induced transparency. While for the case of
constructive interference, the square Lorentzian term can narrow the linewidth
of Rabi splitting even below that of bare components, and thus significantly
suppresses the decay of Rabi oscillation. Interestingly, we further find that
an open microcavity with chiral EPs supports atom-photon bound states for
population trapping and decay suppression in long-time dynamics. As
applications, we demonstrate the advantages of microcavity operated at chiral
EPs in achieving high-fidelity entanglement generation and high-efficiency
single-photon generation. Our work unveils the exotic cavity quantum
electrodynamics unique to chiral EPs, which opens the door for controlling
light-matter interaction at the quantum level through non-Hermiticity, and
holds great potential in building high-performance quantum-optics devices.

††preprint: APS/123-QED

##  I Introduction

Exceptional points (EPs), the spectral degeneracies where two or more
eigenvalues and the associated eigenstates simultaneously coalesce, are the
central concept in non-Hermitian physics [1, 2, 3, 4, 5]. A plethora of
intriguing effects and exotic phenomena emerge around EPs due to their
nontrivial topological properties and the dimensionality reduction, including
ultrasensitive sensing [6, 7, 8, 9, 10, 11, 12], laser mode selection [13,
14], chiral mode conversion [15, 16, 17], and unidirectional invisibility [18,
19]. Harnessing these peculiar features of non-Hermitian degeneracies for
building novel devices with unprecedented performance has been experimentally
demonstrated in various classical dissipative platforms, ranging from
nanophotonics [2, 20, 21, 22, 23], acoustics [24], to macroscopic facilities,
such as fiber network [25], electric circuits [26] and heat diffusive system
[27]. In recent years, great efforts have been dedicated to accessing the
quantum EPs by implementing the non-Hermiticity in quantum systems [11, 17,
28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], and investigating the quantum
states control through EPs [17, 31, 32, 33, 38, 40, 41, 42]. In this respect,
pioneer works have shown the ability of EPs to tune the photon statistics [32,
33], enhance the sensitivity of quantum sensing [43, 44], and steer the
evolution of single quantum system [17, 31, 38, 40, 41, 42, 45, 46, 47].

Despite these promising results, quantum effects of EPs beyond wave mechanics
is still largely unexplored. Recently, the emission properties of a quantum
emitter (QE) in electromagnetic environment supporting EPs attracts growing
attention [38, 40, 42, 42, 45], since the modification of spontaneous emission
(SE) exhibits a non-Lorentzian feature around EPs, contrast to the Lorentzian
response in conventional single-mode cavity, which can lead to a greater
enhancement of SE rate [40, 42, 45]. It implies that the formation of EPs in
nanophotonic structures significantly alters the local density of states
(LDOS), which fully governs the interaction between a QE and arbitrary
electromagnetic environment [48, 49, 50]. Tailoring the LDOS of
electromagnetic environment is crucial for optimizing the performance of many
practical applications, ranging from traditional optoelectronic devices like
lasers [51, 52] and solar cells [53], to advanced quantum technologies, such
as quantum light sources [54, 55, 56] and logical gate [57, 58]. Therefore, a
LDOS theory that can capture the effects of EPs is of both fundamental and
applied significance. The formalism of LDOS based on the classical Green
function separates the EPs contribution from the usual Lorentzian term [40,
41, 59], while it leaves the origin of EPs unclear, i.e., information of the
underlying cavity resonances (frequency and decay rate) is ambiguous, the
coupling parameters of coalescent cavity modes that forms EPs are implicit,
and thus the EPs condition cannot be given explicitly. As a consequence, there
are at least two obvious limitations that hinder the exploration of quantum
effects of EPs using macroscopic quantum electrodynamics (QED) based on the
classical Green function: the quick and accurate design of nanophotonic
structures possessing EPs remains challenging, and the nonlinear regime of
cavity QED involving multiphoton interaction is exclusive.

Figure 1:  Schematic diagrams of cavity QED models. (a) A whispering-gallery-
mode (WGM) cavity represented by microring resonator supports the degenerate
clockwise (CW) and counterclockwise (CCW) modes. A linearly polarized quantum
emitter (QE) is simultaneously coupled to both CW and CCW modes with equal
coupling rate $g$. (b) A microring cavity coupled to a semi-infinite waveguide
with a mirror at the one side can create chiral EPs due to the chiral coupling
between the CW and CCW modes. (c) Conceptual illustration of the equivalent
cavity QED model for the system in (b) with a perfect mirror, where the
original CW mode is flipped to another CCW mode through mirror symmetry.
Accordingly, the linearly polarized QE is replaced by a circularly polarized
one. The distance between two waveguide-cavity junctions is $L$.

To overcome the aforementioned limitations of current LDOS theory for EPs and
explore the quantum state control through EPs, in this work we propose an
equivalent cavity QED model that can provide an intuitive and flexible quantum
master equation approach for a class of novel microcavity with chiral EPs [8,
11, 60, 61, 62, 63], also known as exceptional surface, which is a collection
of EPs in high-dimensional parameter space rather than a single EP. An
analytical description of LDOS for such EP cavity is derived in Sec. II, and
parameterized by the resonance frequencies, decay rates and dissipative
coupling rate of discrete cavity modes, as well as the coupling rates between
the cavity and the QE. Therefore, it can be used to determine the parameters
in quantum master equation from LDOS of a given photonic structure obtained by
electromagnetic simulations, and establishes an explicit connection with
cavity QED at EPs. Furthermore, the formulation of SE spectrum is also
obtained to understand the peculiar quantum dynamics at EPs. Our LDOS theory
not only explains the previously known properties of SE dynamics related to
EPs only with the special parameters choice [41, 42], but also allows the
entire formalization and exploring more extraordinary effects of EPs in open
quantum systems.

Based on the proposed LDOS theory, in Sec. III we perform comprehensive study
to show how the chiral EPs tailors the LDOS and affects the SE dynamics of a
QE and compare to the conventional Lorentzian cavity. With the equivalent
cavity QED model, we further find out the formation of a special kind of
quantum state without decay at chiral EPs, and demonstrate its applications in
quantum state control. Finally, we conclude in Sec. IV.

##  II Model and theory

In this section, we build the quantum model to describe the interaction
between a QE and the microcavity with chiral EP (we call EP cavity hereafter).
In traveling wave optical resonators, such as the whispering-gallery-mode
(WGM) microcavity, the chiral EPs emerges via two unidirectionally coupled
cavity modes. Different strategies have been proposed to implement the
unidirectional coupling between the clockwise (CW) and the counterclockwise
(CCW) modes of WGM microcavity, by introducing two or more Rayleigh-type nano-
scatters [8, 62] or coupling to a waveguide terminated by a mirror at the one
side [11, 42, 63]. Despite different realizations, the underlying physical
mechanism is essentially the same, where the unidirectional coupling is
induced by auxiliary reservoir modes offered by either nano-scatters or
waveguide. Here we employ the latter scheme for the sake of theoretical
simplicity, but the results and conclusions presented in this work are
suitable for other types of EP cavity.

Fig. 1(a) depicts a basic cavity QED system consisting of a QE and a WGM
microcavity with infinite waveguide. The WGM microcavity has a decay rate
$\kappa$ due to the coupling to waveguide. Since there is no coupling between
the CW and CCW modes, the cavity has two degenerate modes, i.e., is operated
at diabolic points (DPs) [60, 64], and thus called DP cavity hereafter. The QE
is linearly polarized, and coupled to a pair of degenerate CW and CCW modes
with equal coupling rate $g$. The system is described by the two-mode Jaynes-
Cummings (JC) model. While by coupling the WGM microcavity to a semi-infinite
waveguide with a mirror at the right end, a unidirectional coupling from CCW
mode to CW mode is created, as Fig. 1(b) illustrates. This photonic structure
supports chiral EPs, and offers great tunability to SE rate of a QE.

In order to develop an intuitive cavity QED model for this novel EP cavity, we
transform the system to an equivalent one through mirror symmetry, as shown in
Fig. 1(c). The original CW mode is flipped to a mirrored CCW mode, and the
corresponding left-handed emission of QE is converted to the right-handed
emission into the mirrored CCW mode as well. As a result, the linearly
polarized QE becomes circularly polarized in the equivalent cavity QED model.
Two CCW modes in the equivalent model constitute a cascaded system [65, 66,
67], the quantum dynamics of system is described by the extended cascaded
quantum master equation $\left(\hbar=1\right)$

$\begin{gathered}\dot{\rho}=-i\left[H_{0}+H_{I},\rho\right]+\gamma\mathcal{L}\left[\sigma_{-}\right]\rho+\kappa\mathcal{L}\left[c_{L}\right]\rho+\kappa\mathcal{L}\left[c_{R}\right]\rho\\\
+\kappa|r|\left(e^{i\varphi}\left[c_{L}\rho,c_{R}^{\dagger}\right]+e^{-i\varphi}\left[c_{R},\rho
c_{L}^{\dagger}\right]\right)\end{gathered}$ (1)

with the free Hamiltonian

$H_{0}=\omega_{0}\sigma_{+}\sigma_{-}+\omega_{c}c_{L}^{\dagger}c_{L}+\omega_{c}c_{R}^{\dagger}c_{R}$
(2)

and the interaction Hamiltonian

$H_{I}=g\left(e^{-i\phi}c_{L}^{\dagger}\sigma_{-}+h.c.\right)+g\left(e^{i\phi}c_{R}^{\dagger}\sigma_{-}+h.c.\right)$
(3)

where $\sigma_{-}$ is the lowering operator of QE with transition frequency
$\omega_{0}$ and SE rate $\gamma$ in homogeneous medium.
$\gamma=\mu^{2}\omega_{0}^{3}n_{b}/\left(3\pi\hbar\varepsilon_{0}c^{3}\right)$,
where $\varepsilon_{0}$ is the permittivity of vacuum, $\mu$ is the dipole
moment of QE, and $n_{b}$ is the refractive index of background medium.
$c_{L}/c_{R}$ is the bosonic annihilation operator for CCW mode in the
left/right cavity with resonance frequency $\omega_{c}$. $\kappa$ is the
dissipative coupling rate between the cavity modes and the waveguide. The
intrinsic decay of two CCW modes is omitted, considering that the evanescent
coupling to waveguide dominates the cavity dissipation.
$\mathcal{L}[O]\rho=O\rho O^{\dagger}-\left\\{O^{\dagger}O,\rho\right\\}/2$ is
the Liouvillian superoperator for operator $O$. The second line of Eq. (1)
describes the unidirectional coupling from the left CCW mode to the right CCW
mode, where $\left|r\right|$ is the reflectivity of mirror, and $\varphi=\beta
L$ is the phase factor, with $\beta$ and $L$ being the propagation constant of
waveguide and the distance between two cavities, respectively. In the
interaction Hamiltonian, the phase factor $\phi$ originates from the traveling
wave nature of WGM modes and depends on the azimuthal orientation of QE
location [68, 64].

From Eq. (1), we take the operator expectation values to obtain the equations
of motion

$\frac{d}{dt}\vec{p}=-i\mathbf{M}\vec{p}$ (4)

where $\vec{p}=\left(\left\langle c_{L}\right\rangle,\left\langle
c_{R}\right\rangle,\left\langle\sigma_{-}\right\rangle\right)^{T}$ and the
$3\times 3$ matrix takes the form

$\mathbf{M}=\left(\begin{array}[]{ccc}\omega_{c}-i\frac{\kappa}{2}&0&ge^{-i\phi}\\\
-i\kappa|r|e^{i\varphi}&\omega_{c}-i\frac{\kappa}{2}&ge^{i\phi}\\\
ge^{i\phi}&ge^{-i\phi}&\omega_{0}-i\frac{\gamma}{2}\end{array}\right)$ (5)

The above equations do not include the time delay of itinerant photon. This is
justified if the propagation time $L/v_{g}$, where $v_{g}$ is the group
velocity of waveguide, is much smaller than the timescale of system evolution
set by $\min\left[g^{-1},\kappa^{-1},\gamma^{-1}\right]$. In the following
study, we consider the optical frequency and focus on the strong coupling
regime, and thus the timescale of system evolution is determined by $g^{-1}$.
For WGM cavity, $g$ is typically smaller than 1meV, and we evaluate that
$g^{-1}\gg L/v_{g}$ for $L<20\mu\mathrm{m}$; in this case, we can assume that
the photon propagates without time delay.

To study the quantum dynamics of a QE located in a realistic photonic
structure, especially for the interactions involving multiphoton process, we
need to determine the system parameters in quantum master equation from LDOS
obtained by electromagnetic simulation. Therefore, an analytical LDOS theory
is crucial to extract the system parameters. The normalized LDOS of
electromagnetic environment, i.e., Purcell factor, is expressed as
$F_{P}(\omega)=[J(\omega)+J_{0}(\omega)]/J_{0}(\omega)$, where $J(\omega)$ and
$J_{0}(\omega)=\gamma/2\pi$ are the spectral density of photonic structures
and background medium (free space), respectively. For EP cavity shown in Fig.
1(b), the spectral density of cavity is given by
$J(\omega)=\int_{-\infty}^{+\infty}d\tau
e^{i\omega\tau}g^{2}\left\langle\left[c_{L}^{\dagger}(0)+e^{-i2\phi}c_{R}^{\dagger}(0)\right]\left[c_{L}(\tau)+e^{i2\phi}c_{R}(\tau)\right]\right\rangle$
[69, 70, 71], where the environmental correlation functions $\left\langle
c_{L}^{\dagger}(0)c_{L}(\tau)\right\rangle$, $\left\langle
c_{L}^{\dagger}(0)c_{R}(\tau)\right\rangle$ and $\left\langle
c_{R}^{\dagger}(0)c_{R}(\tau)\right\rangle$ can be calculated by applying the
quantum regression theorem $\left\langle
O_{i}(t)O_{j}(t+\tau)\right\rangle=\operatorname{Tr}\left\\{O_{j}(0)e^{\mathcal{L}^{\prime}\tau}\left[\rho(t)O_{i}(0)\right]\right\\}$
[72], with $\mathcal{L}^{\prime}\rho^{\prime}=\dot{\rho}^{\prime}$, and
$\dot{\rho}^{\prime}$ is given by

$\begin{gathered}\dot{\rho}^{\prime}=-i\omega_{c}\left[c_{L}^{\dagger}c_{L}+c_{R}^{\dagger}c_{R},\rho\right]+\kappa\mathcal{L}\left[c_{L}\right]\rho+\kappa\mathcal{L}\left[c_{R}\right]\rho\\\
+\kappa|r|\left(e^{i\varphi}\left[c_{L}\rho,c_{R}^{\dagger}\right]+e^{-i\varphi}\left[c_{R},\rho
c_{L}^{\dagger}\right]\right)\end{gathered}$ (6)

From the above quantum master equation, we have

$\frac{d}{dt}\vec{v_{LL}}=\left[\begin{array}[]{cc}\omega_{c}-i\frac{\kappa}{2}&0\\\
-i\kappa|r|e^{i\varphi}&\omega_{c}-i\frac{\kappa}{2}\end{array}\right]\vec{v_{LL}}$
(7)

where $\vec{v_{LL}}=\left(\left\langle
c_{L}^{\dagger}(0)c_{L}(\tau)\right\rangle,\left\langle
c_{L}^{\dagger}(0)c_{R}(\tau)\right\rangle\right)^{T}$. Since the initial
state of electromagnetic environment is the vacuum state, we can obtain the
analytical expression of $\left\langle
c_{L}^{\dagger}(0)c_{L}(\tau)\right\rangle$ with $\left\langle
c_{L}^{\dagger}(0)c_{L}(0)\right\rangle=1$; other correlation functions can be
solved in similar fashion. We finally arrive at

$J(\omega)=J_{\mathrm{DP}}(\omega)+J_{\mathrm{EP}}(\omega)$ (8)

with the DP and EP contributions

$\displaystyle J_{\mathrm{DP}}(\omega)$
$\displaystyle=-g^{2}\operatorname{Im}\left[\chi_{\mathrm{DP}}(\omega)\right]$
(9) $\displaystyle J_{\mathrm{EP}}(\omega)$
$\displaystyle=-g^{2}\operatorname{Im}\left[\chi_{\mathrm{EP}}(\omega)\right]$

where $\chi_{\mathrm{DP}}(\omega)$ is the usual linear Lorentzian response
given by $\left\langle c_{L}^{\dagger}(0)c_{L}(\tau)\right\rangle$ and
$\left\langle c_{R}^{\dagger}(0)c_{R}(\tau)\right\rangle$

$\chi_{\mathrm{DP}}(\omega)=\frac{1}{\pi}\frac{2}{\left(\omega-\omega_{c}\right)+i\kappa/2}$
(10)

and $\chi_{\mathrm{EP}}(\omega)$ is the EP term contributed by $\left\langle
c_{L}^{\dagger}(0)c_{R}(\tau)\right\rangle$

$\chi_{\mathrm{EP}}(\omega)=\frac{1}{\pi}\frac{-i\kappa|r|e^{i\Delta\phi}}{\left[\left(\omega-\omega_{c}\right)+i\kappa/2\right]^{2}}$
(11)

where $\Delta\phi=\varphi-2\phi$ is the phase difference. The factor 2 in the
numerator of $\chi_{\mathrm{DP}}(\omega)$ represents that the QE is coupled to
two cavity modes. Eq. (11) clearly indicates the square Lorentzian profile of
$\chi_{\mathrm{EP}}(\omega)$, a signature of second-order EPs. However, it
also indicates that any loss of reflection amplitude, i.e., an imperfect
mirror with $\left|r\right|<1$, will degrade the quantum effects of EPs. In
the following study, we take $\left|r\right|=1$ unless specially noted. We
highlight that our approach can also be applied to a more general case where
the coupling between two cavities are bidirectional but asymmetrical, for
example, introducing a Rayleigh-type nano-scatter in close proximity of
microring [8, 19, 73]. As a result, the system is drawn out of chiral EPs, and
such configuration can be utilized to study the emergence of quantum effects
of EPs. Furthermore, one can obtain the LDOS of target nanophotonic structure
that owns desirable quantum effects of EPs for electromagnetic design. This is
important for structure optimization, especially for these consisting of
absorbing and dispersing medium like the plasmonic-photonic cavity [74, 75,
76, 77], where the coupling rate between cavity modes is hard to evaluate due
to their distinct features, contract to the EP cavity studied in this work. On
the other hand, the quantum-optics properties of a given nanophotonic
structure are predictable, by parametrizing the extended cascaded quantum
master equation (Eq. (1)) using the simple curve fitting of LDOS obtained from
electromagnetic simulations. Therefore, Eqs. (8)-(11) build the bridge between
the electromagnetic design of nanophotonic structures and the quantum state
control by EPs.

Figure 2:  Comparison of analytical and numerical spectral density of EP
cavity. (a) and (b) show the results of $\Delta\phi=0$ and $\pi$,
respectively. The design parameters of EP cavity are: outer radius
$R=5\mu\mathrm{m}$, width $w=0.25\mu\mathrm{m}$, refractive index
$n_{c}=3.47$, edge-to-edge separation to the waveguide $d=0.2\mu\mathrm{m}$.
The width of waveguide is $d$, and the mirror is made of 100-nm thick silver.
The whole structure is embedded in the background medium with refractive index
of $n_{b}=1.44$. The insets show the corresponding electric field distribution
at the resonance frequency of DP cavity $\omega_{c}=0.78122\mathrm{eV}$.

To validate our theory and as an example, in Fig. 2 we compare the spectral
density given by Eqs. (8)-(11) with the electromagnetic simulations using a
realistic EP cavity. The QE is $Z$-oriented (perpendicular to the cavity
plane), and located at one-quarter of the perimeter from the waveguide-cavity
junction ($X=0$) in the clockwise direction. For QE with a diploe moment of
$\mu=60$ Debye, we can evaluate
$(\omega_{c},\kappa,g)=(0.78122\mathrm{eV},152.8\mu\mathrm{eV},24.9\mu\mathrm{eV})$
by curve fitting using Eq. (10) from the corresponding DP cavity, i.e., EP
cavity without the mirror in waveguide; subsequently, the spectral density of
EP cavity can be analytically obtained from Eqs. (8)-(11). Fig. 2 shows the
spectral density of EP cavity, where we can see the good accordance between
the analytical expressions and the numerical simulation for $\Delta\phi=0$ and
$\pi$. The results confirm the validness of our LDOS theory for EP cavity.
With the obtained parameters, the quantum dynamics of system can be
investigated by the extended cascaded quantum master equation.

The exotic quantum dynamics at EPs can be well understood from the spectral
properties of QE. The emission spectrum of QE is defined as
$S(\omega)=(2\pi)^{-1}\int_{0}^{\infty}dt_{1}\int_{0}^{\infty}dt_{2}e^{i\omega\left(t_{2}-t_{1}\right)}\left\langle\bm{E}^{-}\left(\bm{r},t_{1}\right)\cdot\bm{E}^{+}\left(\bm{r},t_{2}\right)\right\rangle$
[72], with
$\bm{E}^{-}\left(\bm{r},t_{1}\right)=\left[\bm{E}^{+}\left(\bm{r},t_{1}\right)\right]^{\dagger}$
and
$\bm{E}^{+}(\bm{r},t)=e^{-i\omega_{0}t}\int_{0}^{t}dt^{\prime}\bm{G}\left(t,t^{\prime}\right)\sigma_{-}\left(t^{\prime}\right)e^{i\omega_{0}t^{\prime}}$,
where $\bm{G}\left(t,t^{\prime}\right)$ is a kind of propagator dependent on
the detection method. For assuming an initially excited QE and
$\bm{r}=\bm{r}_{0}$, where $\bm{r}_{0}$ is the QE position, $S(\omega)$
becomes the SE spectrum, also called the polarization spectrum [78, 79], which
reflects the local dynamics of a QE, and is given by
$S(\omega)=\int_{0}^{\infty}dt_{2}\int_{0}^{\infty}dt_{1}e^{i\omega\left(t_{2}-t_{1}\right)}\left\langle\sigma_{+}\left(t_{1}\right)\sigma_{-}\left(t_{2}\right)\right\rangle$.
Denoting $\tau=t_{2}-t_{1}$ and taking the limit of
$t_{1}=t\rightarrow\infty$, the SE spectrum can be expressed as
$S(\omega)=\lim_{t\rightarrow\infty}\operatorname{Re}\left[\int_{0}^{\infty}d\tau\left(\sigma_{+}(t+\tau)\sigma_{-}(t)\right\rangle
e^{i\omega\tau}\right]$, where
$\left\langle\sigma_{+}(t+\tau)\sigma_{-}(t)\right\rangle$ can be calculated
using the quantum regression theorem. Then the SE spectrum can be analytically
obtained

$S(\omega)=\frac{1}{\pi}\frac{\gamma+\Gamma(\omega)}{\left[\omega-\omega_{0}-\Delta(\omega)\right]^{2}+\left[\frac{\gamma+\Gamma(\omega)}{2}\right]^{2}}$
(12)

with the photonic Lamb shift

$\Delta(\omega)=g^{2}\operatorname{Re}\left[\chi_{\mathrm{DP}}(\omega)+\chi_{\mathrm{EP}}(\omega)\right]$
(13)

and the local coupling strength

$\Gamma(\omega)=-2g^{2}\operatorname{Im}\left[\chi_{\mathrm{DP}}(\omega)+\chi_{\mathrm{EP}}(\omega)\right]$
(14)

Note that the SE dynamics of QE can be retrieved from
$\mathcal{F}[S(\omega)]$, the Fourier transform of SE spectrum. Eqs. (12)-(14)
indicate that LDOS is crucial for understanding the exotic behavior of quantum
dynamics at EPs. The contributions of DP and EP are separated in Eqs. (13) and
(14), which is beneficial to unravel how the emergence of EPs alters the
quantum dynamics.

##  III Results and discussion

###  III.1 Exceptional point induced transparency

We first consider the weak coupling regime, where the Purcell effect is
expected to modify the SE rate of a QE. Note that the quality factor of modes
in EP cavity is typically $10^{4}$ due to the dissipative coupling to
waveguide, therefore, we will focus on the case of $\kappa\gg\gamma$ in the
following discussion. Fig. 3(a) shows the SE dynamics of QEs with different
$\gamma$, while the coupling rate $g$ and the cooperativity
$C=8g^{2}/\kappa\gamma=0.2$ are set to be fixed. For resonant QE-cavity
coupling, the maximum Purcell factor of DP cavity is equal to $(C+1)$,
obtained by substituting Eq. (8) into
$F_{P}(\omega_{c})=J(\omega_{c})/J_{0}(\omega_{c})+1$. Therefore, QEs
experience the Purcell enhancement and the corresponding SE dynamics manifests
a faster decay, as shown by the dashed lines in Fig. 3(a). On the contrary,
the SE dynamics of QEs in EP cavity is counterintuitive, which shows good
accordance with QEs in the free space as if the cavity is absent. Therefore,
we call this intriguing phenomenon as EP induced transparency. The same effect
has been reported in Ref. [42] and explained as a consequence of decoupling
between the QE and the cavity modes due to the formation of stand wave pattern
of electric field at EPs. Different from the previous work, here we unravel
from the perspective of LDOS that the EP induced transparency results from the
precise cancellation of EP and DP contributions of $J\left(\omega\right)$ at
special frequency point, giving arise to null Purcell enhancement for a QE
with the same transition frequency.

Figure 3:  EP induced transparency and spectral density of EP cavity. (a)
Comparison of the spontaneous emission dynamics of a QE resonantly and weakly
coupled to the EP (solid lines) and DP (dashed lines) cavities with different
SE rates $\gamma$. The QE dynamics in the free space is also shown for
comparison (circles). The parameters are $g=\gamma_{0}$ and
$g^{2}/\kappa\gamma=1/40$, where $\gamma_{0}$ represents the unit rate of QE
decay. (b) Normalized spectral density of EP cavity $J(\omega)/J_{0}(\omega)$
versus frequency and phase difference $\Delta\phi$ with $g=\gamma$ and
$\kappa=20\gamma$. The gray dashed line indicates the parameter region for
Purcell factor enhancement $\eta>1$. (c) Decomposition of $J(\omega)$ of EP
cavity versus frequency for $\Delta\phi=0$ and $\pi/2$. The DP and EP
contributions are plotted by the solid blue lines and the dashed color lines,
respectively. (d) $\eta$ of EP cavity as the function of $\Delta\phi$ for
various $\Delta\omega$. $\left|r\right|$ declines from 1.0 to 0.2 in a linear
fashion for curves with the same color from dark to light. The parameters of
(c) and (d) are the same as (b).

The frequency corresponding to null Purcell enhancement can be found from Eqs.
(8)-(11) by letting $J(\omega)=0$. The solution takes a simple form

$\Delta\omega=\Delta\omega_{m}\equiv-\frac{\kappa}{2}\tan\left(\frac{\Delta\phi}{2}\right)$
(15)

where $\Delta\omega=\omega-\omega_{c}$ is the frequency detuning. Fig. 3(b)
displays the normalized LDOS of cavity $J(\omega)/J_{0}(\omega)$ versus
$\Delta\omega$ and $\Delta\phi$, which can be easily tuned by varying the
mirror location, i.e., the waveguide length $L$. We can see that the zero
point of $J\left(\omega\right)$ goes away from cavity resonance as
$\Delta\phi$ increase from 0 to $\pi$, and the opposite tendency is observed
for $\Delta\phi\in[\pi,2\pi]$, where the zero point of frequency is larger
than cavity resonance in this case. Especially, the null Purcell enhancement
is achieved at the cavity resonance with $\Delta\phi=0$, which is just the
circumstance studied in Ref. [42]. Fig. 3(c) plots the decomposition of
$J\left(\omega\right)$ for $\Delta\phi=0$ and $\pi/2$, where we plot
$-J_{\mathrm{EP}}(\omega)$, i.e., we reverse the curve of EP term by taking a
negative sign, to clearly display the cancellation of EP and DP contributions,
see the shaded circles for indicating $\Delta\omega_{m}$. It shows that the EP
term can be negative and weakens the Purcell effect. For $\Delta\phi=0$, the
EP response features an even symmetry and a narrow linewidth, compared to a DP
(Lorentzian) cavity. The upper panel of Fig. 3(c) shows $J(\omega)>0$ for
$\left|\omega-\omega_{c}\right|>\kappa/2$, while $J(\omega)<0$ for
$\left|\omega-\omega_{c}\right|<\kappa/2$. The resultant
$J\left(\omega\right)$ is slightly enhanced at a wide frequency range far
detuned from the cavity, but strongly suppressed around the cavity resonance.
The EP response cancels the DP contribution at $\Delta\omega$, resulting in
the vanishing Purcell effect for a QE resonantly coupled to EP cavity. The
lower panel of Fig. 3(c) shows that the EP response exhibits a totally
different profile for $\Delta\phi=\pi/2$, which becomes odd symmetry and
changes its sign at cavity resonance. As a result, the disappeared Purcell
effect occurs at the left side of cavity resonance
$(\Delta\omega=-\kappa/2<0)$, while the enhanced Purcell effect is observed at
the whole frequency range of $\omega>\omega_{c}$, leading to a strongly
asymmetrical lineshape of $J\left(\omega\right)$.

Fig. 3(d) plots the Purcell effect enhancement of EPs as the function of
$\Delta\phi$ for various $\Delta\omega$ and $\left|r\right|$, which is defined
as $\eta\equiv
J(\omega_{c})/J_{\mathrm{DP}}(\omega_{c})=J_{\mathrm{EP}}(\omega_{c})/J_{\mathrm{DP}}(\omega_{c})+1$.
It shows that EP cavity attains greatest tunabiliy to Purcell effect from
chiral EPs. Particularly, with resonant QE-cavity coupling, a double increase
of Purcell enhancement compared to a DP cavity can be realized with a prefect
mirror and $\Delta\phi=\pi$, permitting the stronger light-matter interaction.
For a QE detuned from cavity, the null Purcell enhancement still exists with
$\left|r\right|=1$, but the maximum $\eta$ decreases. $\eta>1$ can be achieved
inside the parameter region indicated by the gray dashed line in Fig. 3(b).
Fig. 3(d) also shows that the maximum $\eta$ drops as the loss of reflection
amplitude increases, but a mirror with $\left|r\right|>0.8$ still holds great
tunability of Purcell effect, where, for example, the maximum $\eta$ is
greater than $1.75$ at $\Delta\omega=0$. It is worth noting that
$\left|r\right|\sim 0.98$ is achievable [42, 55], and thus a practical mirror
will not significantly weaken the ability of chiral EPs to tune the Purcell
effect in experiment.

###  III.2 Decay suppression of Rabi oscillation at chiral EPs

The above analysis reveals that the chiral EPs have the ability to
significantly modify the SE process of a QE, from completely suppressed to
enhanced Purcell effect, and thus a EP cavity can provide greater degrees of
freedom to control the light-matter interaction than a DP cavity. We now go
beyond the weak coupling regime and investigate the effects of chiral EPs on
the coherent energy exchange between the QE and the cavity, i.e., the Rabi
oscillation.

Figure 4:  Decay suppression of Rabi oscillation in EP cavity. (a) Population
dynamics for QE (blue line) and cavity modes of EP (yellow and pink lines) and
DP (thin green line with shading) cavities. SE dynamics of a bare QE (in free
space) with the same decay rate is also shown for comparison (dashed red
line). The parameters are $g=10\gamma$, $\kappa=20\gamma$,
$\omega_{0}=\omega_{c}$ and $\Delta\phi=\pi/2$. The inset plots the
corresponding normalized LDOS of cavity. (b) is the same as (a) but for
$g=100\gamma$. The upper panels of (c) and (d) plot the normalized SE spectra
corresponding to (a) and (b) for EP (blue lines) and DP (red lines) cavities,
respectively. While the lower panels show the SE spectrum versus $\Delta\phi$.
The inset in the upper panel shows the peak of Rabi doublet with lower energy,
where the SE spectrum of a bare QE is also presented for comparison (black
dashed line). The white dashed lines in the lower panel track the eigenenergy
levels given by Eqs. (16) and (17).  Figure 5:  Enhanced entanglement
generation in EP cavity. (a) Schematic of spontaneous entanglement generation
between two identical qubits in EP cavity. (b) and (c) Dynamical concurrence
between two qubits in DP (red lines) and EP cavities (blue lines) for
$\Delta_{0c}=2.32g$ and $\Delta_{0c}=0$, respectively. Other parameters are
$g=100\gamma$, $\kappa=20\gamma$, and $\Delta\phi=\pi$. The results are
numerically calculated using QuTip [80] with one qubit initially in the
excited state while another in the ground state.

Fig. 4(a) shows the time evolution of populations with an initially excited QE
resonantly coupled to EP cavity, where the Rabi oscillation is evident. The
parameters are $g=10\gamma$, $\kappa=20\gamma$, and $\Delta\phi=\pi$. It also
shows that the maximum population of $c_{L}$ cavity is slightly lower than the
DP cavity, due to the lack of direct energy blackflow from $c_{R}$ cavity. On
the other hand, the Rabi oscillation of $c_{L}$ cavity sustains for a longer
time and four cycles can be observed, indicating less energy dissipation in EP
cavity. While the Rabi oscillation of DP cavity manifests a faster decay, and
the population of second-cycle oscillation is much smaller than that of EP
cavity and hard to recognize. Meanwhile, the maximum population of $c_{R}$
cavity is more than threefold compared to the DP cavity, and reaches $\sim
0.66$. However, though the higher population, the period of Rabi oscillation
is not obviously changed. The SE spectra shown in the upper panel of Fig. 4(c)
reveals that, it is because the splitting width is slightly enlarged, but the
linewidth of Rabi peaks are greatly reduced. Eq. (12) indicates that the
reduction of linewidth and the resultant decay suppression of Rabi oscillation
are attributed to the modification of LDOS by the EP term, see the inset of
Fig. 4(a). It shows that for $\Delta\phi=\pi$, EP cavity exhibits the
simultaneously enhanced LDOS and narrower linewidth compared to the DP cavity.

Fig. 4(b) shows the time evolution of populations for $g=100\gamma$, where the
decay of Rabi oscillation is slower than not only the DP cavity but also a
bare QE. This is counterintuitive, since the coupling to a lossy cavity will
increase the dissipation rate due to the Purcell enhancement, and result in a
faster decay. To gain insight into the problem, we obtain the approximate
expressions of eigenenergies from the denominator of SE spectrum (Eq. (12)),
which are given by

$\omega_{1,2}\approx\pm\sqrt{2}g+\frac{\kappa}{4}\sin(\Delta\phi)-i\frac{[\cos(\Delta\phi)+1]\kappa+\gamma}{4}$
(16)
$\omega_{3}\approx\frac{\kappa}{2}\sin(\Delta\phi)\left[\frac{\cos(\Delta\phi)}{C}-1\right]-i\frac{\kappa}{2}\left[\cos(\Delta\phi)-1\right]$
(17)

where we expand the eigenenergies up to second order with respect to $\kappa$
and $\gamma$. The white dashed lines in the lower panel of Fig. 4(d) track the
energies of eigenenergy levels, where we find that two eigenenergies in Eq.
(16) correspond to the Rabi doublet in the SE spectrum shown in the upper
panel of Fig. 4(d). Same as the case of $g=10\gamma$, the SE spectrum of
$g=100\gamma$ exhibits the significant narrowing of Rabi doublet. However, Eq.
(16) indicates that the linewidth narrowing in this case is anomalous, since
the minimum decay achieved at $\Delta\phi=\pi$ is
$-2\operatorname{Im}\left[\omega_{1,2}\right]\approx\gamma/2$, a half of a
bare QE, see the inset in the upper panel of Fig. 4(d), where the SE spectrum
of a bare QE is shown for comparison. Eq. (16) also indicates that the
reduction of decay rate originates from the unidirectional coupling
($\kappa\cos(\Delta\phi)$ term) between two cavity modes for chiral EPs. The
condition $\Delta\phi=\pi$ implies that the decay suppression of Rabi
oscillation can be understood as a result of the backflow of partial leaky
energy through the reflection of mirror, and thus unique to EP cavity. The
lower panels of Fig. 4(c) and (d) display the SE spectra as the function of
$\Delta\phi$ for $g=10\gamma$ and $100\gamma$. It is interesting to see that
though both are under the strong coupling regime with the equal splitting seen
in SE spectrum at $\Delta\phi=\pi$, the underlying evolution of Rabi splitting
is distinguishing. The Rabi splitting in the case of $g=100\gamma$ are nearly
unchanged as $\Delta\phi$ varies, while for $g=10\gamma$, the peak of Rabi
splitting with lower energy evolves from cavity resonance, and the
corresponding linewidth is much narrower than the peak with higher energy
until $\Delta\phi$ approaches to $\pi$.

Figure 6:  Atom-photon bound states in EP cavity. Time evolution of QE (a),
$c_{L}$ cavity (b) and $c_{R}$ cavity (c) as the function of $\Delta\phi$ for
$\Delta_{\mathrm{0c}}=0$, $g=20\gamma$ and $\kappa=20\gamma$. The insets show
the time evolution for $\Delta\phi=0$ and $\Delta\phi_{\mathrm{BIC}}$. The
results of DP cavity are also shown for comparison. The time evolution is
obtained by numerically solving Eq.(4). (d) Imaginary part of eigenvalues of
Eq. (5) versus $\Delta\phi$. The Hopfield coefficients of QE and two cavities
contributed to eigenenergy levels $\omega_{2}$ and $\omega_{3}$ are shown in
(e) and (f), respectively.

The above results indicate that EP cavity together with sufficiently strong
QE-cavity interaction will have significant advantages in quantum-optics
applications, such as the spontaneous entanglement generation (SEG) between
qubits, benefiting from the decoherence suppression by chiral EPs. Here we
demonstarte the EP enhanced two-qubit SEG in EP cavity, see Fig. 5(a) for
schematic. Extending our model into the multi-QE case is straightforward

$\begin{gathered}\dot{\rho}_{M}=-i\left[H^{M},\rho\right]+\gamma\sum_{i}\mathcal{L}\left[\sigma_{-}^{(i)}\right]\rho+\kappa\mathcal{L}\left[c_{L}\right]\rho+\kappa\mathcal{L}\left[c_{R}\right]\rho\\\
+\kappa|r|\left(e^{i\varphi}\left[c_{L}\rho,c_{R}^{\dagger}\right]+e^{-i\varphi}\left[c_{R},\rho
c_{L}^{\dagger}\right]\right)\end{gathered}$ (18)

with full Hamiltonian $H^{M}=H_{0}^{M}+H_{I}^{M}$, where the free Hamiltonian
$H_{0}^{M}$ and the interaction Hamiltonian $H_{I}^{M}$ are given by

$H_{0}^{M}=\omega_{0}\sum_{i}\sigma_{+}^{(i)}\sigma_{-}^{(i)}+\omega_{c}c_{L}^{\dagger}c_{L}+\omega_{c}c_{R}^{\dagger}c_{R}$
(19)
$\begin{gathered}H_{I}^{M}=g\sum_{i}\left(e^{-i\phi_{i}}c_{L}^{\dagger}\sigma_{-}^{(i)}+e^{i\phi_{i}}\sigma_{+}^{(i)}c_{L}\right)\\\
+g\sum_{i}\left(e^{i\phi_{i}}c_{R}^{\dagger}\sigma_{-}^{(i)}+e^{-i\phi_{i}}\sigma_{+}^{(i)}c_{R}\right)\end{gathered}$
(20)

where we assume the identical qubits. With an initially excited qubit, the
generated entanglement is quantified by the concurrence $C(t)$ between qubits,
which in our system is given by the simple expression
$C(t)=2\left|C_{eg}(t)C_{ge}^{*}(t)\right|$ [81, 71], with $C_{eg}(t)$ and
$C_{ge}(t)$ being the probability amplitudes of two single-excitation states
that one qubit in the excited state while another in the ground state.

In the case of resonant qubit-cavity coupling, the upper bound of maximum
concurrence is 0.5. As shown in Fig. 5(b), the concurrence exponentially
decays with a rate of $\gamma$ for a DP cavity; while in the EP cavity, the
evolution of $C(t)$ manifests two distinguishing stages with different decay
rates: the decay of $C(t)$ is identical as DP cavity for $t<1.6\gamma^{-1}$;
after that, it is found to decay with a rate $\sim\kappa^{3}/32g^{2}$,
obtained from the eigenenergies of Eq.(18) in the single-excitation subspace
[71]. With a large coupling rate $g=100\gamma$,
$\kappa^{3}/32g^{2}=1/40\gamma\ll\gamma$, and thus the decay of $C(t)$ for
$t>1.6\gamma^{-1}$ is much slower than a DP cavity. It also indicates that for
a larger $g$, the decay suppression of quantum entanglement will be more
evident. Fig. 5(c) displays the $C(t)$ corresponding to the maximum
concurrence $C_{\mathrm{max}}={\mathrm{max}}\left[C(t)\right]=0.9866$,
achieved with the optimal qubit-cavity detuning
$\Delta_{0c}\equiv\omega_{0}-\omega_{c}=2.32g$. In this case, the concurrence
decays much faster in a DP cavity than the EP cavity, exhibiting the great
potential of EP cavity to preserve the generated entanglement.

###  III.3 Population trapping at chiral EPs with atom-photon bound states

Figure 7:  Steady-state population with atom-photon bound states at chiral
EPs. (a) Time dynamics of cavity population ($c_{L}+c_{R}$) as the function of
QE-cavity detuning $\Delta_{0c}$, with parameters $\Delta\phi=\pi/2$ and
$g=10\gamma$. The inset shows the time dynamics at $\Delta_{0c}=0$. The right
panel plots $\Gamma=-\operatorname{Im}\left[\omega_{\mathrm{BIC}}\right]$ as
the function of $\Delta_{0c}$, where $\omega_{\mathrm{BIC}}$ is the
eigenenergy corresponding to BIC. (b) Steady-state cavity population versus
$g$ and $\Delta_{0c}$. (c) Steady-state populations of QE (solid lines),
cavity (thin solid lines with shading) and the system (QE + cavity, dashed
lines) as the function of $g$ for various $\Delta\phi$. In all figures,
$\kappa=20\gamma$.

Inspired by the above results that the spectral linewidth can be smaller than
a bare QE, in the following we investigate the SE dynamics of an ideal QE at
chiral EPs, i.e., $\gamma=0$. Fig. 6(a)-(c) plot the time evolution of QE and
two cavity modes versus $\Delta\phi=0$ for resonant QE-cavity coupling, where
we can observe the population trapping at $\Delta\phi=0$ and $0.77\pi$, while
it cannot be achieved in a DP cavity, see the green shaded lines. We can also
see that for $\Delta\phi=0$, the steady-state population of QE is almost zero
and smaller than that of two cavity modes; however, the situation is reversed
for $\Delta\phi=0.77\pi$, and the trapped population is also higher than the
case of $\Delta\phi=0$. These phenomena indicate the formation of atom-photon
bound states in the system, but the underlying physical mechanisms are
different. For the former case ($\Delta\phi=0$), the destructive interference
of two cavity modes through the reflection of mirror gives rise to stand wave
inside the cavity for population trapping. While the mechanism of bound state
for $\Delta\phi\neq 0$ is similar to the accidental bound states in the
continuum (BIC) [82, 83, 84], also known as Friedrich-Wintgen BIC, which
results from the destructive interference of two coupling pathways between QE
and $c_{R}$ cavity, the direct coupling and the indirect coupling mediated by
$c_{L}$ cavity. The explicit expression of non-zero $\Delta\phi$ for BIC can
be obtained by finding purely real eigenvalues of characteristic matrix Eq.
(5), which is given by

$\Delta\phi_{\mathrm{BIC}}=2\arccos\left(\frac{\kappa}{2\sqrt{2}g}\right)$
(21)

In Fig. 6(d), we plot the imaginary part (decay) of eigenenergies in the
single-excitation subspace as the function of $\Delta\phi$, where it shows
that bound states for $\Delta\phi=0$ and $\Delta\phi_{\mathrm{BIC}}$ are
achieved at different eigenenergy levels, $\omega_{3}$ and $\omega_{2}$,
respectively. Fig. 6(e) shows the Hopfield coefficients (eigenvector
components) of $\omega_{2}$ versus $\Delta\phi$, where we can see that the
coefficients for cavity modes and QE are both equal to 0.5 at
$\Delta\phi_{\mathrm{BIC}}$; by contrast, the Hopfield coefficient of QE for
$\omega_{2}$ is less than 0.1 at $\Delta\phi=0$ and much smaller than cavity
modes. It explains the different distributions of steady-state population for
two bound states shown in Fig. 6(a)-(c).

Since BIC exhibits better energy preservation at chiral EPs, we then consider
a general case of arbitrary $\Delta\phi$, and find the QE-cavity detuning
corresponding to BIC. From the denominator of SE spectrum (Eq. (12)), we
obtain

$\Delta\omega_{\mathrm{BIC}}=\frac{2g^{2}}{\kappa}\sin(\Delta\phi)+\Delta\omega_{m}$
(22)

We can see that Eq. (22) encompasses the case of EP induced transparency, with
$g\ll\kappa$ in the weak coupling regime. As $g$ increases, the contribution
of the first term in Eq. (22) becomes significant, the resultant
$\Delta\omega_{\mathrm{BIC}}$ is no longer overlapped with the location of
null Purcell enhancement for $\Delta\phi\neq 0$.

Fig. 7(a) shows the time evolution of cavity population versus QE-cavity
detuning $\Delta_{0c}=\omega_{0}-\omega_{c}$, with parameters
$\Delta\phi=\pi/2$, $g=10\gamma$, and $\kappa=20\gamma$. Eq. (22), as well as
the decay of eigenenergies, indicates that BIC is achieved at $\Delta_{0c}=0$,
see the blue line in the right panel of Fig. 7(a). As a result, the decay of
cavity population is obviously slower around $\Delta_{0c}=0$, and the
populations of QE and two cavities are partially trapped after a few cycles of
Rabi oscillation at $\Delta_{0c}=0$, as shown by the inset of Fig. 7(a). It
shows that though the steady-state populations of two cavities are the same,
$c_{R}$ cavity exhibits stronger Rabi oscillation due to the unidirectional
energy transfer from $c_{L}$ cavity. The maximum population of $c_{R}$ cavity
reaches 0.4, while that of $c_{L}$ cavity is about 0.2.

Figure 8:  Decay suppression in long-time dynamics with atom-photon bound
states at chiral EPs. (a) Logarithmic plot of QE dynamics with detuning
$\Delta\omega_{\mathrm{BIC}}$ (solid lines) from cavity for various
$\Delta\phi$. The dynamics of bare QEs are shown for comparison (dashed
lines). The parameters are $g=5\gamma_{0}$ and $g^{2}/\kappa\gamma=5/4$, where
$\gamma_{0}$ is the unit rate of QE decay. Note that the SE rate $\gamma$ is
taken to be $\gamma_{0}$, $\gamma_{0}/2$, and $\gamma_{0}/4$ for
$\Delta\phi=0$, $\pi/4$, and $\pi/2$, respectively, to distinguish different
curves, but the corresponding $\Gamma_{m}$ is still the curve of $g=5\gamma$
in (b). (b) $\Gamma_{m}$ as the function of $\Delta\phi$ for various $g$. (c)
$g^{(2)}(0)$ (upper panel) and population (lower panel) of $c_{L}$ cavity as
the function of frequency detuning in the case of $c_{R}$ cavity drive. The
results are obtained by implementing a driving Hamiltonian
$H_{d}=\Omega\left(c_{R}^{\dagger}+c_{R}\right)$ in the extended cascaded
quantum master equation (Eq. (1)) and numerically calculating using QuTip
[80], with parameters $g=5\gamma$, $\Delta\phi=0$,
$\Delta_{0c}=\Delta\omega_{\mathrm{BIC}}$ and driving strength
$\Omega=0.2\gamma$. $\kappa=20\gamma$ for all figures.

Fig. 7(b) displays the steady-state cavity population versus $\Delta\phi$ and
$g$. It reveals that for a given $\Delta\phi$, there is an optimal $g$ for
maximum population, denoted as $P_{c}^{opt}(\Delta\phi)$. The maximal
$P_{c}^{opt}(\Delta\phi)$ is found to achieve at $\Delta\phi=0$, i.e.,
$P_{c}^{opt}(0)$ is the upper bound of steady-state cavity population.
$P_{c}^{opt}(\Delta\phi)$ is evaluated to decrease by 0.03 as $\Delta\phi$
varies from 0 to $0.9\pi$, and thus is robust against the variation of
$\Delta\phi$. For $\Delta\phi=0$, the steady-state populations can be
analytically given from Eq. (4), which are
$P_{e}^{ss}=\kappa^{4}/\left(8g^{2}+\kappa^{2}\right)^{2}$ for QE and
$P_{c}^{ss}=8g^{2}\kappa^{2}/(8g^{2}+\kappa^{2})^{2}$ for cavity. We can see
that $P_{e}^{ss}$ reduces as $g$ increase, exhibiting distinguishing feature
from $P_{c}^{ss}$. Furthermore, we can obtain the analytical expression of
leaky energy $P_{\kappa}^{ss}=8g^{2}/(8g^{2}+{\kappa}^{2})$, which is
monotonically increasing with respect to $g$. It implies that the optimal $g$
for maximum cavity population is the result of balance between the population
transfer from QE and the system dissipation. Fig. 7(c) plots the steady-state
populations versus $g$ for various $\Delta\phi$. It shows that for
$\Delta\phi>0.9\pi$, trapping with high population can be achieved for cavity
at a wide range of $g$.

Finally, we investigate how the atom-photon bound states modify the SE
dynamics at chiral EPs by taking into account the SE rate of QE to the free
space, i.e., $\gamma\neq 0$. Fig. 8(a) compares the SE dynamics of initial
excited QEs in the free space and in EP cavity with optimal detuning
$\Delta\omega_{\mathrm{BIC}}$ from the cavity. Remarkably, we can see from the
slope of decay curves that BIC at chiral EPs can offer different degrees of
suppression on the decoherence process, according to the value of
$\Delta\phi$. As a result, the population of QEs in EP cavity can be larger
than the bare QEs for $t>50g^{-1}$.

The damping rate of long-time dynamics depends on the minimum decay of
eigenenergies, i.e.,
$\Gamma_{m}=\min\left[-\operatorname{Im}\left[\omega_{i}\right]\right]$. Fig.
8(b) plots $\Gamma_{m}$ for various $g$, where it shows that $\Gamma_{m}$
reaches the minimum at $\Delta\phi=0$, and increases as $\Delta\phi$ varies
from 0 to $\pi$. Consistent with the time dynamics shown in Fig. 8(a),
$\Gamma_{m}$ of EP cavity can be smaller than $\gamma/2$, which is not
possible for a DP cavity with $\kappa\geq\gamma$, since $\Gamma_{m}$ of a
standard JC system is $\gamma/2$ for weak coupling and greater than $\gamma/2$
for strong coupling. Fig. 8(b) also shows that enhancing the QE-cavity
interaction is beneficial to reduce the decay rate, and a tenfold reduction
$(\Gamma_{m}\approx\gamma/20)$ can be achieved with a moderate QE-cavity
coupling $g=20\gamma$ at $\Delta\phi=0$. While for arbitrary $g$,
$\Gamma_{m}\rightarrow\gamma/2$ as $\Delta\phi$ approaches to $\pi$, implying
that QE is decoupled from cavity at $\Delta\phi=\pi$.

As applications, the significantly reduced decay of eigenenergies makes the EP
cavity-QE system advantageous for single-photon generation exploiting the
photon blockade [85, 86], where the photon antibunching takes place at the
frequency of one of eigenenergy levels in the single-excitation subspace with
a coherent input. Therefore, single-photon blockade can attain efficiency
improvement at chiral EPs. In the case of $c_{R}$ cavity drive, Fig. 8(c)
compares the single-photon purity $g^{(2)}(0)=\left\langle
c_{L}^{\dagger}c_{L}^{\dagger}c_{L}c_{L}\right\rangle/n_{L}^{2}$ and
population $n_{L}=\left\langle c_{L}^{\dagger}c_{L}\right\rangle$ of DP and EP
cavities, with parameters $\Delta\phi=0$, $g=5\gamma$ and $\kappa=20\gamma$.
We evaluate $g\approx g_{c}=\sqrt{(\kappa^{2}+\gamma^{2})}/4$, where $g_{c}$
is the critical coupling rate of strong coupling for a DP cavity [87]. It
implies that the strong antibunching is absent in DP cavity, since the system
just reaches the strong coupling regime. As shown in Fig. 8(c), $g^{(2)}(0)$
curve of DP cavity is flat, and the minimum $g^{(2)}(0)$ is $\sim 0.1$. With
the resonant QE-cavity coupling, which is also the optimal configuration
detuning according to Eq. (22), the $c_{R}$ cavity demonstrates a great
enhancement of single-photon purity by over an order of magnitude at
$\Delta\phi=0$, with the minimal $g^{(2)}(0)\sim 0.005$, accompanied by a
hundredfold improvement of population. While for $\Delta\phi=\pi/4$, the
single-photon purity and population are both improved by about an order of
magnitude at chiral EPs. Therefore, EP cavity shows great potential in
building high-efficiency single-photon source.

##  IV Conclusion

In this work, we study the spontaneous emission of a quantum emitter located
in microcavity supporting chiral EPs, and find the intriguing phenomena of EP
induced transparency and the anomalously small decay rate of transient and
long-time dynamics. An analytical description of LDOS and SE spectrum is
presented to unveil that the non-Lorentzian response and atom-photon bound
states are responsible for these peculiar quantum dynamics at chiral EPs. The
results demonstrate the striking ability of chiral EPs to suppress the
decoherence process occurred in both weak- and strong-coupling regimes.
Therefore, we envision that EPs can offer the robustness against the dephasing
for various quantum-optics applications, including but not limited to high-
efficiency single-photon sources [88, 89], nonlinear interaction at the
single-photon level [90], and high-fidelity entanglement generation and
transport [91, 92]. Besides the nanophotonic structures, the quantum effects
of chiral EPs demonstrated in this work can also be implemented in other kind
of platforms, such as superconducting [93, 94], cavity optomechanics [95, 96],
and open cavity magnonic systems [28, 97]. We believe that our work can
provide insights on the effects of EPs in diverse quantum systems and is
instructive for harnessing the non-Hermiticity for building novel quantum
devices.

###### Acknowledgements.

We are grateful to Haishu Tan and Yongyao Li for helpful discussions and
supports. This work is supported by the Postdoctor Startup Project of Foshan
and the High-level Talent Scientific Research Startup Project of Foshan
University (CGZ07001).

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