# Exceptional point in self-consistent Markovian master equations

Dong Xie<EMAIL_ADDRESS>College of Science, Guilin University of
Aerospace Technology, Guilin, Guangxi 541004, People’s Republic of China
Chunling Xu  College of Science, Guilin University of Aerospace Technology,
Guilin, Guangxi 541004, People’s Republic of China

###### Abstract

Exceptional point (EP) denotes the non-Hermitian degeneracy, in which both
eigenvalues and eigenstates become identical. By the conventional local
Markovian master equation, EP can be constructed by parity-time (PT) or anti-
PT symmetry in a system composed of coupled subsystems. However, the coupling
between two systems makes the conventional local Markovian master equation
become inconsistent. By using self-consistent Markovian master equation, we
show that there is no EP in the system composed of two bosonic subsystems. The
conventional local master equation can be valid only when the coupling
strength is much smaller than the difference in resonance frequency between
the two subsystems. In a system composed of three bosonic subsystems, EP can
be obtained by adiabatically eliminating one of the three subsystems.

##  I Introduction

The Hamiltonian governing the evolution of the closed system is Hermitian, and
thus only degeneracy of the energy levels is possible. The inevitable coupling
to the surrounding environment makes the effective Hamiltonian of the open
system become non-Hermitian. The non-Hermitian degeneracy, known as EP lab1 ;
lab2 , denotes that both eigenvalues and eigenstates coalesce. EPs have
recently attracted more and more research, mainly by finding a large number of
meaningful applications and exotic phenomena such as loss-induced lasing lab3
, stopped lightlab3a , quantum state control lab4 ; lab5 , asymmetric
backscattering lab6 , asymmetric mode switching lab7 ; lab8 , energy transfer
lab9 , phase accumulation lab10 ; lab11 , enhancement of Quantum Heat Engine
lab12 . More importantly, EPs have been found to play an important role in
improving the measurement sensitivity lab13 ; lab14 ; lab15 ; lab16 ; lab17 ;
lab18 ; lab19 ; lab20 .

EPs can appear in PT and anti-PT symmetrical systems. In the presence of PT
symmetry lab21 ; lab22 , non-Hermitian Hamiltonians can have entirely real
eigenvalues. EPs are the separate points between purely real eigenvalues and
the normally complex eigenvalues. Similarly, EPs are the separate points
between purely imaginary eigenvalues and the normally complex eigenvalues of
anti-PT symmetrical non-Hermitian Hamiltonians lab23 ; lab24 .

PT and anti-PT symmetrical non-Hermitian effective Hamiltonian can be
constructed by the coupled bosonic systems suffering from the local Markovian
dissipation or driving lab25 . By transforming the conventional Lindblad
master equation into the quantum Heisenberg-Langevin equation, the effective
PT and anti-PT symmetrical non-Hermitian effective Hamiltonian for the
evolution of bosonic modes can be obtained. However, the local master equation
may fail when there is coupling between the systems. It has been shown that
local master equation may violate the second law of thermodynamics lab26  and
give rise to non-physical results lab27 ; lab28 ; lab29 ; lab30 , even in the
limit of small bath couplings. Recently, it is shown that local master
equation may fail to describe dissipative critical behavior lab31 . It is
therefore necessary to be careful about the coupling between systems when
constructing EPs.

Taking into account light-matter interaction, a self-consistent nonlocal
Markovian master equation in the dressed picture has been proposed lab32 ;
lab33 . Recently, self-consistent nonlocal Markovian dissipation master
equation for open quadratic quantum system has been derived lab34 . In this
article, we further derive the self-consistent nonlocal Markovian driving
master equation for constructing the PT symmetrical system. We show that a
fermionic bath with a strong enough chemical potential is required to
obtaining the incoherent driving. By transforming the self-consistent equation
into the corresponding quantum Heisenberg Langevin equation, we prove that EP
can not appear in the system composed of two subsystems. The conventional
local Markovian master equation is reasonable, requiring that not only the
coupling strength is much smaller than the bare resonance frequency difference
but also that the baths are symmetric. Finally, we show that adiabatically
eliminating one of the three coupled subsystems can construct EPs.

This article is organized as follows. In Section II, we introduce the EPs in
PT symmetrical non-Hermitian Hamiltonian by the conventional local Markovian
master equation. In Section III, the self-consistent Markovian master equation
for general system is reviewed. In Section IV, we obtain the dressed Markovian
master equation for the general quadratic system and the condition of
incoherent driving is proposed. In Section V, we show that there is no EP in
the system composed of two bosonic subsystems. In Section IV, EP can be
obtained by adiabatically eliminating one of the three coupled subsystems. A
simple summary and the possibility of experimentation are proposed in Section
VII.

##  II EPs by the conventional local Markovian master equation

A typical non-Hermitian system is composed of coupled cavities with two
resonant modes $a_{1}$ and $a_{2}$, as shown in Fig. 1, with the non-Hermitian
Hamiltonian (setting $\hbar=1$)

$\displaystyle
H=(\omega_{1}-i\gamma_{1})a_{1}^{\dagger}a_{1}+(\omega_{2}+i\gamma_{2})a_{2}^{\dagger}a_{2}+g(a_{1}^{\dagger}a_{2}+a_{2}^{\dagger}a_{1}),$
(1)

where $\omega_{1}$ and $\omega_{2}$ are the resonance frequencies of the modes
1 and 2, respectively; $\gamma_{1}$ and $\gamma_{2}$ are the total loss/gain
rates of the modes 1 and 2, respectively.

Figure 1: Schematic diagram of non-Hermitian Hamiltonian. Two cavity modes are
coupled with the strength $g$. Cavity mode 1 suffers from the dissipation bath
1 with the loss rate $\gamma_{1}$. Cavity mode 2 is incoherent driven by the
bath 2 with the gain rate $\gamma_{2}$.

When resonance frequencies are tuned to be equal (i.e.
$\omega_{1}=\omega_{2}=\omega$) and the gain rate is equal to the loss rate
(i.e. $\gamma_{1}=\gamma_{2}=\gamma$), the non-Hermitian Hamiltonian is PT
symmetrical. The eigenvalues of the non-Hermitian Hamiltonian $H$ are given by

$\displaystyle E_{\pm}=\omega\pm\sqrt{g^{2}-\gamma^{2}}.$ (2)

When $g=\gamma$, the eigenvalues and the eigenstates are degenerate. Hence,
$g=\gamma$ represents the EP.

The non-Hermitian Hamiltonian comes from the conventional local Markovian
master equation lab25

$\displaystyle\frac{d\rho}{dt}=-i[H_{S},\rho]+\gamma\mathcal{L}(a_{1})\rho+\gamma\mathcal{L}(a_{2}^{\dagger})\rho,$
(3)

where the superoperator $\mathcal{L}(a)\rho=2a\rho
a^{\dagger}-a^{\dagger}a\rho-\rho a^{\dagger}a$ with $a=\\{a_{1},\ a_{2}\\}$,
and the Hamiltonian
$H_{S}=\omega_{1}a_{1}^{\dagger}a_{1}+\omega_{2}a_{2}^{\dagger}a_{2}+g(a_{1}^{\dagger}a_{2}+a_{2}^{\dagger}a_{1}).$
The quantum Langenvin equation can be achieved by the formula lab35 ; lab36 ;
lab37

$\displaystyle\frac{da}{dt}=i[H_{S},a]-[a,a_{1}^{\dagger}](\gamma
a_{1}-\sqrt{2\gamma}a_{1\textmd{in}})+(\gamma
a_{1}^{\dagger}-\sqrt{2\gamma}a^{\dagger}_{1\textmd{in}})$
$\displaystyle[a,a_{1}]-[a,a_{2}](\gamma
a_{2}^{\dagger}-\sqrt{2\gamma}a^{\dagger}_{2\textmd{in}})+(\gamma
a_{2}-\sqrt{2\gamma}a_{2\textmd{in}})[a,a_{2}^{\dagger}],$ (4)

where the noise operators satisfy that

$\displaystyle\langle a_{j\textmd{in}}\rangle=0,\ \ \ \ \ \langle
a_{j\textmd{in}}a_{k\textmd{in}}\rangle=0,$ (5) $\displaystyle\langle
a^{\dagger}_{j\textmd{in}}a_{k\textmd{in}}\rangle=0,\langle
a_{j\textmd{in}}a^{\dagger}_{k\textmd{in}}\rangle=\delta_{jk}.$ (6)

##  III self-consistent Markovian master equation

However, the master equation in Eq. (3), is not self-consistent due to the
coupling between the two subsystems. A self-consistent master equation can be
derived by the dressed master equation lab32 ; lab33 ; lab34 ; lab38

$\displaystyle\frac{d\rho}{dt}=-i[H_{S}+H_{LS},\rho]+\mathcal{D}[\rho],$ (7)

where
$H_{LS}=\sum_{\alpha,\beta,\omega}L_{\alpha\beta}(\omega)O^{\dagger}_{\alpha}(\omega)O_{\beta}(\omega)$
is a Lamb-shift correction, and the superoperator $\mathcal{D}[\rho]$ is
described by

$\displaystyle\mathcal{D}[\rho]=\sum_{\alpha,\beta,\omega}\lambda_{\alpha\beta}(\omega)[2O_{\beta}(\omega)\rho
O_{\alpha}^{\dagger}(\omega)-O_{\beta}(\omega)O_{\alpha}^{\dagger}(\omega)\rho-$
$\displaystyle\rho O_{\beta}(\omega)O_{\alpha}^{\dagger}(\omega)].$ (8)

Here, the dressed operator is given by

$\displaystyle
O_{\alpha}(\omega)=\sum_{k,q}\delta_{\omega_{q}-\omega_{k}}|k\rangle\langle
k|Q_{\alpha}|q\rangle\langle q|,$ (9)

and the factors are

$\displaystyle\lambda_{\alpha\beta}(\omega)$
$\displaystyle=\frac{1}{2}\int_{-\infty}^{\infty}d\tau
e^{i\omega\tau}\langle{\tilde{R}}^{\dagger}_{\alpha}(\tau)R_{\beta}\rangle,$
(10) $\displaystyle L_{\alpha\beta}(\omega)$
$\displaystyle=\frac{1}{2i}\int_{0}^{\infty}d\tau[e^{i\omega\tau}\langle{\tilde{R}}^{\dagger}_{\alpha}(\tau)R_{\beta}\rangle-e^{-i\omega\tau}\langle{\tilde{R}}^{\dagger}_{\alpha}(\tau)R_{\beta}\rangle],$
(11)

where $\omega_{k}$ and $|k\rangle$ are the $k$th eigenvalue and eigenstate of
the system Hamiltonian $H_{S}$, the undressed operator $O_{\alpha}$ acts on
the system in the interaction Hamiltonian
$H_{\textmd{int}}=\sum_{\alpha}O_{\alpha}\bigotimes R_{\alpha}$ ($R_{\alpha}$
acts on the environment), $\langle.\rangle$ denotes the expectation value
calculated with the environment density operator $\rho_{E}$, and
$\tilde{R}_{\alpha}(\tau)=e^{iH_{E}\tau}{R}_{\alpha}e^{-iH_{E}\tau}$ with the
environment Hamiltonian $H_{E}$.

##  IV dressed master equation for general quadratic system

For a general quadratic bosonic system, the Hamiltonian is described by lab39

$\displaystyle H^{\prime}=\sum_{n=1}^{N}H_{n}+\sum_{i=1,i<j}^{N}H_{ij},$ (12)

in which,

$\displaystyle
H_{n}=\omega_{n}a^{\dagger}_{n}a_{n}+(\frac{\chi_{n}}{2}a_{n}^{2}+h.c.),$ (13)
$\displaystyle
H_{ij}=(g_{ij}a_{i}a_{j}+\lambda_{ij}a_{i}a_{j}^{\dagger})+h.c.,$ (14)

where $H_{n}$ denotes the Hamiltonian for the $n$th subsystem with
$n=\\{1,...,N\\}$, $\omega_{n}$ is the resonance frequency of the bosonic
subsystem with the annihilation operator $a$ and the creation operator
$a^{\dagger}$, $|\chi_{n}|$ denotes the strength of two-photon driving, and
$\lambda_{ij}$ ($g_{ij}$) denotes the coupling strength of the rotating
(counterrotating)-wave interaction between the two subsystems.

By using an Hopfield-Bogoliubov (HB) transformation lab40 ; lab41 , for the
stable normal phase, the total Hamiltonian can be rewritten as a diagonal form

$\displaystyle
H^{\prime}=\sum_{n=1}^{N}\Omega_{n}b_{n}^{\dagger}b_{n}+\frac{1}{2}(\Omega_{n}-\omega_{n}),$
(15)

where the collective bosonic mode operators $b_{n}$ satisfy the commutation
relation: $[b_{i},b_{j}^{\dagger}]=\delta_{ij}$,

$\displaystyle
b_{n}=\sum_{i=1}^{N}(\mu_{ni}a_{i}+\nu_{ni}a_{i}^{\dagger})/\xi_{n},$ (16)

in which, the normalization factor is described by
$\xi_{n}=\sqrt{\sum_{i=1}^{N}(|\mu_{ni}|^{2}-|\nu_{ni}|^{2})}$. In the Nambu
space, it can be rewritten as

$\displaystyle\vec{\mathbf{b}}=\mathbb{T}\vec{\mathbf{a}},$ (17)

where the Nambu field vector is defined as

$\displaystyle\vec{\mathbf{x}}=(x_{1},...,x_{N},x^{\dagger}_{1},...,x^{\dagger}_{N}),$
(18)

and the canonical transformation matrix

$\mathbb{T}=\left(\begin{array}[]{ll}\mathbb{\tilde{\mu}}\ \ \
\mathbf{\tilde{\nu}}\\\ \mathbf{\tilde{\nu}}^{*}\ \
\mathbf{\tilde{\mu}}^{*}\\\ \end{array}\right),$

where the elements of the matrix are $\tilde{\mu}_{ni}=\mu_{ni}/\xi_{n}$ and
$\tilde{\nu}_{ni}=\nu_{ni}/\xi_{n}$. The coefficient vectors
$(\mu_{n1},...,\mu_{nN},\nu_{n1},...,\nu_{nN})^{T}$ are eigenvectors of the HB
matrix M, which is derived by the commutation relation
$[b_{n},H^{\prime}]=\Omega_{n}b_{n}$ lab39

$\mathbf{M}=\left(\begin{array}[]{ll}\mathbf{A}\ \ \ -\mathbf{B}\\\
\mathbf{B}^{*}\ \ -\mathbf{A}^{*}\\\ \end{array}\right),$

with submatrixs

$\mathbf{A}=\left(\begin{array}[]{ll}\omega_{1}\ \ \lambda_{12}\
\ldots\lambda_{1N}\\\ \lambda_{12}^{*}\ \omega_{2}\ \ldots\ \lambda_{2N}\\\ \
\vdots\ \ \ \ \ \vdots\ \ \ \ddots\ \ \ \vdots\\\ \lambda_{1N}^{*}\
\lambda_{2N}^{*}\ \ldots\omega_{N}\end{array}\right),$
$\mathbf{B}=\left(\begin{array}[]{ll}\chi_{1}\ \ g_{12}\ \ldots\ g_{1N}\\\
g_{12}\ \ \chi_{2}\ \ldots\ g_{2N}\\\ \ \vdots\ \ \ \ \ \ \ \vdots\ \ \ddots\
\ \ \vdots\\\ g_{1N}\ g_{2N}\ \ldots\ \chi_{N}\end{array}\right).$

###  IV.1 Incoherent dissipation and driving

For the dissipation environment, there are $N$ independent thermal baths with
the Hamiltonian described by

$\displaystyle H_{E,n}=\int
dk(\epsilon_{n}(k)-\eta_{n})c_{n}^{\dagger}(k)c_{n}(k),$ (23)

where the spectrum $\epsilon_{n}(k)\geq 0$ is non-negative and $\eta_{n}$
denotes the chemical potential of the $n$th thermal bath at temperature
$T_{n}$.

The bath operators $c_{n}(k)$ satisfy the rules

$\displaystyle\\{c_{n}(k),c^{\dagger}_{n}(q)\\}_{\zeta_{n}}=\delta(k-q),$ (24)
$\displaystyle\\{c_{n}(k),c_{n}(q)\\}_{\zeta_{n}}=0,\ [c_{n}(k),c_{m}(q)]=0,$
(25)

where $\\{X,Y\\}_{\zeta_{n}}=XY+\zeta_{n}YX$, and $\zeta_{n}=+1\ (-1)$ belongs
to fermionic (bosonic) systems. The density matrix of thermal bath can be
described by

$\displaystyle\rho_{E}=\bigotimes_{n=1}^{N}\frac{e^{-H_{E,n}/T_{n}}}{\textmd{Tr}(e^{-H_{E,n}/T_{n}})},$
(26)

Based on the above equations, the two-point expectation values can be obtained

$\displaystyle\langle c_{n}(k)c_{m}(q)\rangle=0,$ (27) $\displaystyle\langle
c^{\dagger}_{n}(k)c_{m}(q)\rangle=\delta_{mn}\delta(k-q)f_{n}(\epsilon_{n}(k)),$
(28)

in which,

$\displaystyle
f_{n}(\epsilon)=[\zeta_{n}+e^{(\epsilon-\eta_{n})/T_{n}}]^{-1}.$ (29)

We consider that $N$ subsystems interact linearly with the corresponding bath
separately, which are described by

$\displaystyle H_{\textmd{int}}$
$\displaystyle=\sum_{n=1}^{N}(a_{n}+a_{n}^{\dagger})\otimes\int
dkg_{n}(k)[c_{n}(k)+c^{\dagger}_{n}(k)]$
$\displaystyle=\sum_{n=1}^{N}O_{n}\otimes R_{n}.$ (30)

In the basis of the Hamiltonian $H_{S}$, the eigenoperator associated with
$O_{n}$ can be obtained

$\displaystyle
O_{n}(\omega)=\sum_{k=1}^{N}[\phi_{n,k}\delta_{\omega,\Omega_{k}}b_{k}+\phi_{n,k}^{*}\delta_{\omega,-\Omega_{k}}b^{\dagger}_{k}],$
(31)

where the element
$\phi_{n,k}=(\mathbb{T}^{-1})_{n,k}+(\mathbb{T}^{-1})^{*}_{n,k+N}$.

The factor $\lambda_{nm}$ in Eq. (10) can be derived by the correlation
functions lab34

$\displaystyle\lambda_{nm}(\omega)=\delta_{nm}\lambda_{nn}(\omega),$ (32)
$\lambda_{nn}(\omega)=\left\\{\begin{array}[]{ll}\mathcal{J}_{n}(\omega)[1-\zeta_{n}f_{n}(\omega)]&\textmd{if}\
\ \omega>0\\\ \mathcal{J}_{n}(-\omega)f_{n}(-\omega)&\textmd{if}\ \
\omega<0\\\ \mathcal{J}_{n}(0)[1+(1-\zeta_{n})f_{n}(0)]&\textmd{if}\ \
\omega=0\end{array}\right\\},$

where the spectral density for the $n$th bath is given by

$\displaystyle\mathcal{J}_{n}(\omega)=\pi\int
dk|g_{n}(k)|^{2}\delta(\omega-\epsilon_{n}(k)).$ (34)

In a general case, the system is not degenerate (i.e.
$\Omega_{n}\neq\Omega_{m}$ when $n\neq m$) and the eigenspectrum is not zero
(i.e. $\Omega_{n}\neq 0$), the superoperator $\mathcal{D}(\rho)$ can be
described by

$\displaystyle\mathcal{D}(\rho)=\sum_{n,k}\gamma_{n,k}[[1-\zeta_{n}f_{n}(\Omega_{k})]\mathcal{L}(b_{k})\rho+f_{n}(\Omega_{k})\mathcal{L}(b^{\dagger}_{k})\rho],$
(35)

where the coupling constants
$\gamma_{n,k}=\mathcal{J}_{n}(\Omega_{k})|\phi_{n,k}|^{2}$.

Combining Eq. (4) and Eq. (35), we can see that the total change rate of the
$n$th mode $b_{k}$ is

$\displaystyle\Gamma_{k}=\sum_{n}\Gamma_{k,n}=\sum_{n}\gamma_{n,k}[1-\zeta_{n}f_{n}(\Omega_{k})-f_{n}(\Omega_{k})].$
(36)

When all the baths are composed of bosons, the total change rate
$\Gamma_{k}=\sum_{n}\gamma_{n,k}>0$ denotes that the mode is suffering from
the dissipation process.

In order to obtain the incoherent gain, several baths must be composed of
fermions, i.e., $\zeta_{n}=1$. In this case, the change rate
$\Gamma_{k,n}=\gamma_{n,k}[1-2f_{n}(\Omega_{k})]$ should be negative. As a
result, it leads to that $\Omega_{k}<\eta_{n}$, which means that a strong
enough chemical potential is needed to drive the system incoherently. It can
also be considered that the bath is composed of spins, which are in the
excited states. In other words, bosonic bath is not suitable for implementing
incoherent driving.

##  V EP does not exist in two-boson systems

In this section, we investigate whether EP can exist in the system composed of
two bosonic subsystems with the self-consistent Markovian master equation.

The bosonic system composed of two subsystems is dominated by the Hamiltonian

$\displaystyle H_{S1}=\omega a_{1}^{\dagger}a_{1}+\omega
a_{2}^{\dagger}a_{2}+g(a_{1}^{\dagger}a_{2}+a_{2}^{\dagger}a_{1}).$ (37)

The subsystem 1 interacts with the bath 1 composed of bosons at zero
temperature, and the subsystem 2 interacts with the bath 2 composed of
fermions at zero temperature. The corresponding interaction Hamiltonian is
described by

$\displaystyle
H_{\textmd{int1}}=\sum_{n=1}^{2}(a_{n}+a_{n}^{\dagger})\otimes\int
dkg_{n}(k)[c_{n}(k)+c^{\dagger}_{n}(k)].$ (38)

The Hamiltonian can be diagonalized as

$\displaystyle
H_{{S}1}=(\omega+g)b_{1}^{\dagger}b_{1}+(\omega-g)b_{2}^{\dagger}b_{2},$ (39)

where the dressed operators are $b_{1}=(a_{1}+a_{2})/\sqrt{2}$ and
$b_{2}=(a_{1}-a_{2})/\sqrt{2}$. Then, using the consistent master equation in
Eq. (35) for $g\neq 0$, we can achieve

$\displaystyle\frac{d\rho}{dt}=$
$\displaystyle-i[H_{S1},\rho]+[{J}_{1}(\omega+g)\mathcal{L}(b_{1})\rho+{J}_{2}(\omega+g)\mathcal{L}(b_{1}^{\dagger})\rho$
$\displaystyle+{J}_{1}(\omega-g)\mathcal{L}(b_{2})\rho+{J}_{2}(\omega-g)\mathcal{L}(b_{2}^{\dagger})\rho].$
(40)

We assume that baths have a very large bandwidth, leading to that
$\gamma={J}_{j}(\omega+g)={J}_{j}(\omega-g)$ with $j=1,2$ and the Lambshift
correction $H_{{LS}1}=0$. The quantum-Heisenberg Langevin equation are given
by using Eq. (4)

$i\left(\begin{array}[]{ll}\dot{b}_{1}\\\ \dot{b}_{2}\\\
\end{array}\right)=\left(\begin{array}[]{ll}\omega+g\ \ \ \ 0\\\ \ \ \ 0\ \ \
\ \omega-g\\\ \end{array}\right)\left(\begin{array}[]{ll}{b}_{1}\\\ {b}_{2}\\\
\end{array}\right)+\sqrt{2\gamma}\left(\begin{array}[]{ll}b_{1\textmd{in}}+b^{\dagger}_{1\textmd{in}}\\\
b_{2\textmd{in}}+b^{\dagger}_{2\textmd{in}}\\\ \end{array}\right),$

where $b_{\textmd{in}}$ denotes the noise operator. The effective Hamiltonian
is described by

$H_{\textmd{eff}1}=\left(\begin{array}[]{ll}\omega+g\ \ \ \ 0\\\ \ \ \ 0\ \ \
\ \omega-g\\\ \end{array}\right).$

Due to that $H_{\textmd{eff}1}$ is Hermitian, EP does not exist. This result
shows that EP can not be constructed in a resonance-coupled driven-dissipative
system. Because of the resonant coupling, both the driving and the dissipation
act synchronously on each subsystem. The asymmetric effects of driving and
dissipation can not be obtained.

###  V.1 The self-consistent master equations with degenerate eigenvalues

In the non-degenerate case, the dressed modes $b_{k}$ are independent, leading
to that EP does not exist. Then, we consider the degenerate eigenenergies of
the system Hamiltonian.

The system possesses $M$ different energy eigenspaces, labeled by an index
$\iota=1,...,M$. There are $N_{\iota}$ eigenvectors for eigenvalue
$\omega_{\iota}$. The consistent Markovian master equation for the baths with
large bandwidth can be given by lab34

$\displaystyle\frac{d\rho}{dt}=-i[H_{S},\rho]+\sum_{n,\iota}\sum_{\alpha=1,\beta=1}^{N_{\iota}}[\Phi_{\mu\nu}^{(n,\iota)}\lambda_{nn}(\omega_{\iota})(2b_{\mu}\rho
b^{\dagger}_{\nu}-$
$\displaystyle\\{b^{\dagger}_{\nu}b_{\mu},\rho\\}_{+})+\Phi_{\nu\mu}^{(n,\iota)}\lambda_{nn}(-\omega_{\iota})(2b^{\dagger}_{\mu}\rho
b_{\nu}-\\{b_{\nu}b^{\dagger}_{\mu},\rho\\}_{+})],$ (43)

where the factors $\Phi_{\mu\nu}^{(n,\iota)}=\phi_{n,\mu}\phi^{*}_{n,\nu}$.

A simple degenerate case is one in which there is only paring coupling between
two bosonic subsystems with Hamiltonian

$\displaystyle H_{S2}=\omega a_{1}^{\dagger}a_{1}+\omega
a_{2}^{\dagger}a_{2}+g(a_{1}a_{2}+a_{1}^{\dagger}a_{2}^{\dagger}),$ (44)

The subsystem 1 interacts with the bosonic bath 1 at zero temperature, and the
subsystem 2 interacts with the fermionic bath 2 at zero temperature.

The transformation matrix between Nambu field vector
$\vec{\mathbf{a}}=(a_{1},a_{2},a_{1}^{\dagger},a_{2}^{\dagger})$ and
$\vec{\mathbf{b}}=(b_{1},b_{2},b_{1}^{\dagger},b_{2}^{\dagger})$ can be
expressed as

$\mathbb{T}^{-1}=\left(\begin{array}[]{ll}W_{+}\ \ 0\ \ \ \ \ 0\ \ \ W_{-}\\\
\ 0\ \ \ W_{+}\ \ W_{-}\ \ 0\\\ \ 0\ \ \ W_{-}\ \ W_{+}\ \ 0\\\ W_{-}\ \ 0\ \
\ \ \ 0\ \ \ W_{+}\end{array}\right),$

where the values are defined as
$W_{\pm}=\pm\sqrt{{\frac{\omega}{2\sqrt{\omega^{2}-g^{2}}}\pm\frac{1}{2}}}$.

In the rotating reference frame, we can obtain the dynamic of the expectation
values of the dressed operators $b_{1},\ b_{2}$ according to Eq. (43)

$i\left(\begin{array}[]{ll}\dot{\langle b_{1}\rangle}\\\ \dot{\langle
b_{2}\rangle}\\\
\end{array}\right)=\left(\begin{array}[]{ll}\frac{-i(W\lambda_{-}+\lambda_{+})}{2}\
\ \ \frac{-i\sqrt{W^{2}-1}\lambda_{-}}{2}\\\
\frac{-i\sqrt{W^{2}-1}\lambda_{-}}{2}\ \ \ \
\frac{-i(W\lambda_{-}-\lambda_{+})}{2}\\\
\end{array}\right)\left(\begin{array}[]{ll}\langle b_{1}\rangle\\\ \langle
b_{2}\rangle\\\ \end{array}\right),$

where the factors are defined as $W=\frac{\omega}{2\sqrt{\omega^{2}-g^{2}}}$
and
$\lambda_{\pm}=\mathcal{J}_{1}(\sqrt{\omega^{2}-g^{2}})\pm\mathcal{J}_{2}(\sqrt{\omega^{2}-g^{2}})$.
Therefore, the effective Hamiltonian $H_{\textmd{eff}2}$ can be expressed as

$H_{\textmd{eff}2}=\left(\begin{array}[]{ll}\frac{-i(W\lambda_{-}+\lambda_{+})}{2}\
\ \ \frac{-i\sqrt{W^{2}-1}\lambda_{-}}{2}\\\
\frac{-i\sqrt{W^{2}-1}\lambda_{-}}{2}\ \ \ \
\frac{-i(W\lambda_{-}-\lambda_{+})}{2}\\\ \end{array}\right).$

The eigenvalues of $H_{\textmd{eff}2}$ can be derived, which are given by

$\displaystyle
E_{\pm}=\frac{i}{2}[\sqrt{\lambda_{+}^{2}+(W^{2}-1)\lambda_{-}^{2}}\pm
W\lambda_{-}].$ (48)

Due to that the eigenvalues are still imaginary, there are no EPs that
separate the real and imaginary values. Therefore, in the degenerate
eigenspace, EPs are not allowed in the self-consistent Markovian master
equation.

As a summary, EPs can not appear in the driven-dissipative bosonic system
irrespective of whether the eigenvalues of the eigensystem are degenerate or
not.

##  VI EPs by adiabatic elimination

In this section, we try to construct the EP by adiabatic elimination in
multiple boson systems.

Firstly, we find the condition that the conventional local Markovian master
equation can be close to the nonlocal self-consistent master equation. For two
non-resonant coupled subsystems, the Hamiltonian is described by

$\displaystyle
H_{S3}=\omega_{1}a_{1}^{\dagger}a_{1}+\omega_{2}a_{2}^{\dagger}a_{2}+g(a_{1}^{\dagger}a_{2}+a_{2}^{\dagger}a_{1}),$
(49)

In the diagonalized form, the Hamiltonian is rewritten as
$H_{S3}=\Omega_{+}b_{1}^{\dagger}b_{1}+\Omega_{-}b_{2}^{\dagger}b_{2}$ with
the eigenvalues
$\Omega_{\pm}=\frac{\omega_{1}+\omega_{2}\pm\sqrt{4g^{2}+\Delta^{2}}}{2}$. The
canonical transformation matrix is given by

$\mathbb{T}^{-1}=\left(\begin{array}[]{ll}\sqrt{\frac{1+Y}{2}}\ \
-\sqrt{\frac{1-Y}{2}}\ \ \ \ \ \ 0\ \ \ \ \ \ \ \ \ \ \ 0\\\
\sqrt{\frac{1-Y}{2}}\ \ \ \ \sqrt{\frac{1+Y}{2}}\ \ \ \ \ \ \ \ 0\ \ \ \ \ \ \
\ \ \ \ 0\\\ \ \ \ 0\ \ \ \ \ \ \ \ \ \ \ \ 0\ \ \ \ \ \ \ \
\sqrt{\frac{1+Y}{2}}\ \ \ -\sqrt{\frac{1-Y}{2}}\\\ \ \ \ 0\ \ \ \ \ \ \ \ \ \
\ \ 0\ \ \ \ \ \ \ \ \sqrt{\frac{1-Y}{2}}\ \ \ \
\sqrt{\frac{1+Y}{2}}\end{array}\right),$

where the value $Y$ is defined as $Y=\frac{\Delta}{\sqrt{\Delta^{2}+4g^{2}}}$,
and the resonance frequency difference $\Delta$ is given by
$\Delta=\omega_{1}-\omega_{2}$. Without loss of generality, next we consider
that both the resonance frequency difference and the coupling strength are
larger than 0, i.e., $\Delta>0$ and $g>0$.

When the coupling strength is much less than the resonance frequency
difference ($g\ll\Delta$), we can obtain that

$\displaystyle a_{1}$ $\displaystyle\simeq
b_{1}-\frac{g}{\Delta}b_{2}+O(\frac{g^{2}}{\Delta^{2}})b_{1}+O(\frac{g^{3}}{\Delta^{3}})b_{2},$
(51) $\displaystyle a_{2}$ $\displaystyle\simeq
b_{2}+\frac{g}{\Delta}b_{1}+O(\frac{g^{2}}{\Delta^{2}})b_{2}+O(\frac{g^{3}}{\Delta^{3}})b_{1},$
(52) $\displaystyle\Omega_{1}$
$\displaystyle\simeq\omega_{1}+\frac{g^{2}}{\Delta}+O(\frac{g^{3}}{\Delta^{3}}),$
(53) $\displaystyle\Omega_{2}$
$\displaystyle\simeq\omega_{2}-\frac{g^{2}}{\Delta}-O(\frac{g^{3}}{\Delta^{3}}),$
(54) $\displaystyle\gamma_{21}$ $\displaystyle=\gamma_{12}\simeq
O(\frac{g^{2}}{\Delta^{2}}),$ (55)

where $O(\frac{g^{2}}{\Delta^{2}})$ denotes the second order small quantity,
and $O(\frac{g^{3}}{\Delta^{3}})$ denotes the third order small quantity.
Ignoring all the small quantities, the local Markovian master equation is
recovered

$\displaystyle\frac{d\rho}{dt}\approx\sum_{k=1}^{2}\\{-i[\omega_{k}a_{k}^{\dagger}a_{k},\rho]+\mathcal{L}_{\textmd{loc}}(\rho)\\},$
(56) $\displaystyle\textmd{in which},$
$\displaystyle\mathcal{L}_{\textmd{loc}}(\rho)=\gamma_{k,k}[[1-\zeta_{k}f_{k}(\omega_{k})]\mathcal{L}(a_{k})\rho+f_{k}(\omega_{k})\mathcal{L}(a^{\dagger}_{k})\rho].$
(57)

In the above local Markovian master equation, the two bosonic modes $a_{k}$
are independent. Therefore, EPs can not occur in the local Markovian master
equation.

Up to the first order small quantity (i.e. on the order of $g/\Delta$), the
self-consistent Markovian master equation can be given by

$\displaystyle\frac{d\rho}{dt}\approx-i[H_{S3},\rho]+\mathcal{L}_{loc}(\rho)+\frac{g}{\Delta}\sum_{k=1}^{2}(-1)^{k-1}\gamma_{k,k}\\{[1-$
$\displaystyle\zeta_{k}f_{k}(\omega_{k})](2a_{1}\rho
a_{2}^{\dagger}+2a_{2}\rho
a_{1}^{\dagger})+f_{k}(\omega_{k})(2a_{1}^{\dagger}\rho
a_{2}+2a_{2}^{\dagger}\rho a_{1})$
$\displaystyle-[1+(1-\zeta_{k})f_{k}(\omega_{k})](a_{1}^{\dagger}a_{2}+a_{2}^{\dagger}a_{1})\rho+\rho(a_{1}^{\dagger}a_{2}+a_{2}^{\dagger}a_{1})\\}.$
(58)

When the two baths are identical and their temperatures are close to zero
degrees ($\zeta_{1}f_{1}(\omega_{1})=\zeta_{2}f_{2}(\omega_{2})$), and the
couplings between the subsystems and the corresponding baths are the same
($\gamma_{11}=\gamma_{22}$), we can achieve the conventional Markovian master
equation from the above equation

$\displaystyle\frac{d\rho}{dt}\approx-i[H_{S3},\rho]+\mathcal{L}_{loc}(\rho).$
(59)

It shows that the conventional local Markovian master equation is reasonable,
requiring that not only the coupling strength $g$ is much smaller than the
resonance frequency difference $\Delta$ (rather than the resonance
frequencies) but also that the baths are symmetric.

Then, we consider a system composed of three bosonic subsystems, with the
Hamiltonian described by

$\displaystyle
H_{S4}=\sum_{i=1}^{3}\omega_{i}a_{i}^{\dagger}a_{i}+g(a_{1}^{\dagger}a_{3}+a_{3}^{\dagger}a_{1})+g^{\prime}(a_{2}^{\dagger}a_{3}+a_{3}^{\dagger}a_{2}),$
(60)

where $\omega_{i}$ denotes the frequencies of the $i$th subsystem, and $g$
($g^{\prime}$) denotes the coupling strength between modes 1 (2) and 3.

In the rotating reference frame, the Hamiltonian $H_{S4}$ can be rewritten as

$\displaystyle H_{S4}=$
$\displaystyle\Delta^{\prime}a_{1}^{\dagger}a_{1}+(\Delta^{\prime}-\varepsilon)a_{2}^{\dagger}a_{2}+g(a_{1}^{\dagger}a_{3}+a_{3}^{\dagger}a_{1})$
$\displaystyle+g^{\prime}(a_{2}^{\dagger}a_{3}+a_{3}^{\dagger}a_{2}),$ (61)

where $\Delta^{\prime}=\omega_{1}-\omega_{3}$ and
$\varepsilon=\omega_{1}-\omega_{2}$.

We consider that the condition satisfies $\Delta^{\prime}\gg g$, so that the
self-consistent Markovian master equation is close to the local Markovian
master equation according to Eq. (56). Then, we obtain the unitary
transformation matrix $U$ defined in
$(b_{1},b_{2},b_{3})^{T}=U(a_{1},a_{2},a_{3})^{T}$ for $\varepsilon\simeq
O(g)$, which is close to

$U\approx\left(\begin{array}[]{ll}\ 1\ \ \ \ \ \frac{g}{\Delta^{\prime}}\ \
\frac{g}{\Delta^{\prime}}\\\ \frac{-g}{\Delta^{\prime}}\ \ \ \ 1\ \ \ \
\frac{g}{\Delta^{\prime}}\\\ \frac{-g}{\Delta^{\prime}}\ \ \
\frac{-g}{\Delta^{\prime}}\ \ \ 1\end{array}\right).$

When the temperature in all three baths is zero, the self-consistent Markovian
master equation can give the Langevin equation with the diagonalized modes

$i\left(\begin{array}[]{ll}\dot{\langle b_{1}\rangle}\\\ \dot{\langle
b_{2}\rangle}\\\ \dot{\langle
b_{3}\rangle}\end{array}\right)\approx-\left(\begin{array}[]{ll}\Upsilon_{1}\
0\ \ 0\\\ 0\ \ \Upsilon_{2}\ 0\\\ 0\ \ \ 0\ \
\Gamma_{2}\end{array}\right)\left(\begin{array}[]{ll}\langle b_{1}\rangle\\\
\langle b_{2}\rangle\\\ \langle b_{3}\rangle\end{array}\right),$

where the values are defined as $\Upsilon_{1}=\Gamma_{1}+i\Delta^{\prime}$,
$\Upsilon_{2}=\Gamma_{2}+i(\Delta^{\prime}-\varepsilon)$,
$\Gamma_{1}=\mathcal{J}_{1}(\omega_{1})=\Gamma_{2}=\mathcal{J}_{2}(\omega_{2})$,
and $\Gamma_{3}=\mathcal{J}_{3}(\omega_{3})$. Based on the above eqaution, the
evolution of modes $a_{j}$ can be derived by

$i\left(\begin{array}[]{ll}\dot{\langle a_{1}\rangle}\\\ \dot{\langle
a_{2}\rangle}\\\ \dot{\langle
a_{3}\rangle}\end{array}\right)\approx-U^{-1}\left(\begin{array}[]{ll}\Upsilon_{1}\
0\ \ 0\\\ 0\ \ \Upsilon_{2}\ 0\\\ 0\ \ \ 0\ \
\Gamma_{2}\end{array}\right)U\left(\begin{array}[]{ll}\langle a_{1}\rangle\\\
\langle a_{2}\rangle\\\ \langle a_{3}\rangle\end{array}\right),$

After calculation from the above equation, we can achieve the detail evolution
dynamic

$i\left(\begin{array}[]{ll}\dot{\langle a_{1}\rangle}\\\ \dot{\langle
a_{2}\rangle}\\\ \dot{\langle
a_{3}\rangle}\end{array}\right)\approx-\Lambda\left(\begin{array}[]{ll}\langle
a_{1}\rangle\\\ \langle a_{2}\rangle\\\ \langle
a_{3}\rangle\end{array}\right),$

where the evolution matrix is given by

$\Lambda=\left(\begin{array}[]{ll}\ \ \ \ \Upsilon^{\prime}\ \ \ \ \
\frac{g^{2}(\Gamma_{3}-\Gamma_{1})}{\Delta^{\prime 2}}\
\frac{g(\Gamma_{1}-\Gamma_{3})}{\Delta^{\prime}}\\\
\frac{g^{2}(\Gamma_{3}-\Gamma_{1})}{\Delta^{\prime 2}}\ \ \ \
\Upsilon^{\prime}\ \ \ \ \ \
\frac{g(\Gamma_{1}-\Gamma_{3})}{\Delta^{\prime}}\\\
\frac{g(\Gamma_{1}-\Gamma_{2})}{\Delta^{\prime}}\
\frac{g(\Gamma_{1}-\Gamma_{3})}{\Delta^{\prime}}\ \ \ \ \ \
\Gamma_{3}\end{array}\right),$

in which,
$\Upsilon^{\prime}=\Gamma_{1}+\frac{g^{2}(2\Gamma_{1}+\Gamma_{3})}{\Delta^{\prime
2}}$. When $\Gamma_{2}\gg\Gamma_{1}\gg\Delta^{\prime}$, the mode $\langle
a_{3}\rangle$ can be eliminated due to that the mode $\langle a_{3}\rangle$
reaches the steady state much faster than the other two modes. By assuming
$\langle\dot{a}_{3}\rangle=0$, we can obtain

$\displaystyle\langle
a_{3}\rangle\approx\frac{g}{\Delta^{\prime}\Gamma_{2}}(\Gamma_{2}-\Gamma_{1})(\langle
a_{1}\rangle+\langle a_{2}\rangle),$ (67)

Using the above equation, we can obtain the evolution equation for the modes 1
and 2

$i\left(\begin{array}[]{ll}\dot{\langle a_{1}\rangle}\\\ \dot{\langle
a_{2}\rangle}\end{array}\right)=H_{\textmd{eff}3}\left(\begin{array}[]{ll}\langle
a_{1}\rangle\\\ \langle a_{2}\rangle\end{array}\right),$

where the effective Hamiltonian of the reduced two subsystems is depicted as

$H_{\textmd{eff}3}=i\left(\begin{array}[]{ll}-\Gamma_{1}-i\Delta^{\prime}\ \ \
\ \ -\frac{g^{2}\Gamma_{1}}{\Delta^{\prime 2}}\\\ \ \
-\frac{g^{2}\Gamma_{1}}{\Delta^{\prime 2}}\ \ \ \ \
-\Gamma_{1}-i(\Delta^{\prime}-\varepsilon)\end{array}\right).$

Moving to the reference frame rotating with frequency
$\Delta^{\prime}-\varepsilon/2$, the effective Hamiltonian can be reduced to

$H_{\textmd{eff}3}=i\left(\begin{array}[]{ll}-\Gamma_{1}-i\varepsilon/2\ \ \
-\frac{g^{2}\Gamma_{1}}{\Delta^{\prime 2}}\\\ \ \
-\frac{g^{2}\Gamma_{1}}{\Delta^{\prime 2}}\ \ \ \ \
-\Gamma_{1}+i\varepsilon/2\end{array}\right).$

The eigenvalues of $H_{\textmd{eff}3}$ are given by
$\Omega_{\pm}=i(-\Gamma_{1}\pm\sqrt{g^{4}\Gamma_{1}^{2}/\Delta^{\prime
4}-\varepsilon^{2}/4})$. As a consequence, the EP appears at
$\varepsilon=2g^{2}\Gamma_{1}/\Delta^{\prime 2}$, which separates the purely
imaginary eigenvalue and the normally complex eigenvalue. Therefore, EP
appears in the effective anti-PT symmetrical Hamiltonian.

Then, we try to construct the EP without the condition $g\ll\Delta^{\prime}$,
i.e., there is no need to approach the local Markovian master equation. We
redefine that $\varepsilon=2\Delta^{\prime}$. In this case, the unitary
transformation matrix $U$ can be exactly derived, which is described by

$U=\left(\begin{array}[]{ll}\frac{\Delta^{\prime}+\Delta_{g}}{2\Delta_{g}}\ \
\frac{g^{2}}{\Delta_{g}(\Delta_{g}+\Delta^{\prime})}\ \
\frac{g}{\Delta_{g}}\\\ \frac{\Delta^{\prime}-\Delta_{g}}{2\Delta_{g}}\ \
\frac{g^{2}}{\Delta_{g}(\Delta^{\prime}-\Delta_{g})}\ \
\frac{g}{\Delta_{g}}\\\ \ \frac{-g\Delta_{g}}{\Delta^{\prime 2}}\ \ \ \ \ \
\frac{g\Delta_{g}}{\Delta^{\prime 2}}\ \ \ \ \ \ \
\frac{\Delta_{g}}{\Delta^{\prime}}\end{array}\right),$

where $\Delta_{g}=\sqrt{\Delta^{\prime 2}+2g^{2}}$. In this case, the
evolution matrix $\Lambda$ defined in Eq. (VI) can be reformulated as

$\Lambda=U^{-1}\left(\begin{array}[]{ll}\Gamma_{1}+i\Delta^{\prime}\ \ \ \ \ \
\ 0\ \ \ \ \ \ \ \ \ \ \ 0\\\ \ \ \ \ \ 0\ \ \ \ \ \ \ \
\Gamma_{1}-i\Delta^{\prime}\ \ \ \ \ 0\\\ \ \ \ \ \ 0\ \ \ \ \ \ \ \ \ \ \ \
0\ \ \ \ \ \ \ \ \ \ \ \Gamma_{3}\end{array}\right)U=$
$\left(\begin{array}[]{ll}\ \ \ \ \ \frac{Z_{-}}{\Delta_{g}^{2}}\ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \frac{ig^{2}(\Gamma_{1}-\Gamma_{3})}{\Delta_{g}^{2}}\ \ \ \ \
\frac{ig\Delta^{\prime}(\Gamma_{3}-\Gamma_{1})-g\Delta_{g}^{2}}{\Delta_{g}^{2}}\\\
\ \frac{ig^{2}(\Gamma_{1}-\Gamma_{3})}{\Delta_{g}^{2}}\ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \frac{Z_{+}}{\Delta_{g}^{2}}\ \ \ \ \ \ \ \ \
\frac{ig\Delta^{\prime}(\Gamma_{1}-\Gamma_{3})-g\Delta_{g}^{2}}{\Delta_{g}^{2}}\\\
\frac{ig\Delta^{\prime}(\Gamma_{3}-\Gamma_{1})-g\Delta_{g}^{2}}{\Delta_{g}^{2}}\
\frac{ig\Delta^{\prime}(\Gamma_{1}-\Gamma_{3})-g\Delta_{g}^{2}}{\Delta_{g}^{2}}\
\frac{-i(2g^{2}\Gamma_{1}+\Gamma_{3}\Delta^{\prime
2})}{\Delta_{g}^{2}}\end{array}\right),$

where $Z_{\pm}=i[g^{2}(\Gamma_{1}+\Gamma_{2})+\Gamma_{1}\Delta^{\prime
2}]\pm\Delta^{\prime}\Delta_{g}^{2}$. For
$|\Gamma_{3}|\gg|\Gamma_{1}|\textmd{and}\ |\Delta^{\prime}|$, we can
adiabatically eliminate the mode $a_{3}$. As a result, the effective
Hamiltonian for modes $a_{1}$ and $a_{2}$ can be described by

$H_{\textmd{eff4}}=\left(\begin{array}[]{ll}{\Gamma_{2}\Delta^{\prime}\Delta_{g}^{2}-i\kappa}\
\ \ \ \ \ \ -i\chi\\\ \ \ \ \ \ -i\chi\ \ \ \ \ \ \
{-\Gamma_{2}\Delta^{\prime}\Delta_{g}^{2}-i\kappa}\end{array}\right),$

where $\kappa=\Delta^{\prime
2}(g^{2}+\Gamma_{1}\Gamma_{3})+g^{2}(2g^{2}+\Gamma_{1}^{2}+\Gamma_{1}\Gamma_{3})$
and $\chi=g^{2}(\Delta_{g}^{2}+\Gamma_{1}^{2}-\Gamma_{1}\Gamma_{3})$. The
eigenvalues of $H_{\textmd{eff4}}$ are given by
$E_{\pm}=-i\kappa\pm\sqrt{(\Gamma_{3}\Delta^{\prime}\Delta_{g}^{2})^{2}-\chi^{2}}$.
When the modes 1, 2 and 3 are suffering from incoherent dissipation, i.e.,
$\Gamma_{1}>0$ and $\Gamma_{3}>0$,
$|\chi|=\Gamma_{3}\Delta^{\prime}\Delta_{g}^{2}$ denotes the anti-PT
symmetrical EP. Therefore, the relation
$\Gamma_{1}\approx\frac{\Delta^{\prime}(\Delta^{\prime 2}+2g^{2})}{g^{2}}$ is
required to find the anti-PT symmetrical EP.

In order to find the PT symmetrical EP, the condition $\kappa=0$ is necessary.
The modes 1 and 2 are incoherent driven, i.e., $\Gamma_{1}<0$. As a result, we
can achieve that $\Gamma_{2}=\frac{g^{2}(2g^{2}+\Delta^{\prime
2}+\Gamma_{1}^{2})}{(g^{2}+\Delta^{\prime 2})|\Gamma_{1}|}$. The PT
symmetrical EP can also appear at
$\Gamma_{1}\approx\frac{\Delta^{\prime}(\Delta^{\prime 2}+2g^{2})}{g^{2}}$.

##  VII conclusion

We investigate the construction of EPs in the effective non-Hermitian
Hamiltonian from the self-consistent Markovian master equation. Unlike the
result from the conventional local Markovian master equation, we prove that
the EPs can not exist in the system composed of two bosons. For constructing
the PT symmetrical system, we further derive the self-consistent nonlocal
Markovian driving master equation. We show that fermionic bath with a strong
enough chemical potential is required to obtaining incoherent driving. It can
also be considered that the bath is composed of spins, which are in the
excited states by extra controls. Bosonic bath is not suitable for
implementing incoherent driving. And we show that the conventional local
Markovian master equation is reasonable, which requires not only that the
coupling strength is much smaller than the resonance frequency difference
(rather than the resonance frequencies) but also that the baths are
symmetrical. By adiabatically eliminating one of the three coupled subsystems,
we can reconstruct the EPs with two different parameter choices: one is that
the coupling strength is much less than the resonance frequency difference,
and the other is that the coupling strength is not much less than the
resonance frequency difference. The former can only construct anti-PT
symmetrical EPs. The latter can construct anti-PT and PT symmetrical EPs.

The system composed of three coupled bosonic subsystems in this article can be
realized in magnon-cavity-magnon coupled system lab24  or in a variety of
different photonic and phononic systemslab42 . Our work lays the foundation
for constructing real EPs in non-Hermitian systems.

## Acknowledgements

This research was supported by the National Natural Science Foundation of
China under Grant No. 62001134, Guangxi Natural Science Foundation under Grant
No. 2020GXNSFAA159047.

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