# Optimal temperature estimation in polariton Bose-Einstein Condensate

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

###### Abstract

Improving the measurement precision of temperature is very important and
challenging, especially in the low temperature range. Based on the existence
of invariant subspaces during the polariton thermalization, we propose a new
way to enhance the measurement precision of the low temperature and obtain
Landau bound to avoid that the measurement uncertainty of the temperature
diverges as the temperature approaches zero. The measurement precision of the
low temperature increases significantly with the number of polariton states.
In order to resist the dissipation, the incoherent pumping is necessary for
obtaining the information of the temperature encoded in the steady state. It
should be noted that too strong incoherent pumping is wasteful due to that the
quantum Fisher information of the temperature becomes less and less dependent
on the total number of the polaritons.

Introduction.-Precise estimation of temperature is significant and crucial for
the fundamental natural science and the changing quantum technologylab1a ;
lab2a ; lab3a ; lab1 ; lab2 ; lab3 ; lab4 ; lab5 ; lab6 ; lab7 ; lab8 . Since
quantum devices generally work at low temperaturelab9 ; lab10 ; lab11  and the
development of the field of quantum thermodynamicslab12 ; lab13 ; lab14 ;
lab15 ; lab16  also need the low temperature to reserve the quantum
properties, precisely control and measurement of the low temperature is
becoming an important subject in quantum metrologylab8  and quantum sensing.

Enhancing the estimation precision of temperature with quantum resources and
investigating the fundamental limitations on temperature estimation have
attracted a lot of attentionlab17 ; lab18 ; lab19 ; lab20 ; lab21 ; lab22 ;
lab23 ; lab24 ; lab25 ; lab26 ; lab27 ; lab28 . Temperature measurements can
generally be divided into two categories: one is to measure the temperature
encoded in the thermal equilibrium state of the probe system, the other is in
the non-equilibrium state. Either way, the measurement of the low temperature
has always been a challenging and arduous task. The uncertainty of the
temperature diverges as the temperature approaches zerolab29 . Recently, some
works attempted to alleviate the divergence. Correa et al.lab30  showed that
the thermometric precision at low temperature could be significantly enhanced
by the strong probe-sampling coupling. Mukherjee et al.lab31  utilized the
periodic modulation to obtain the low-temperature thermometry with
temperature-independent relative uncertainty. More importantly, Zhang et
al.lab32  obtained the Landau boundlab33  by using a continuous-variable
system to detect the temperature of a non-Markovian reservoir.

In this work, we propose a new way to obtain the optimal low temperature
estimation, and the Landau bound can also be achieved. The exciton-polariton
Bose-Einstein Condensates (BECs) are used as the thermometer to measure the
temperature of the phonons in the semiconductors or the temperature of
intermolecular oscillations of the organic dyes. Different with the Bose
polaron modellab9  in BEC, there are invariant subspaces due to the polariton
thermalization, in which the final thermal state is not unique. For the
infinite states of the polaritons, the Landau bound can also be achieved at
the low temperature. In addition, we find that incoherent pumping can resist
the dissipation. But too strong incoherent pumping is useless in enhancing the
estimation precision of the temperature due to that the temperature estimation
precision will not increase with the increase of the total number of the
polaritons for enough polaritons.

The physical model of polaritons.- When cavity photons strongly interact with
an optical transition of active material, new eigenstates can be generated,
i.e., lower and upper exciton-polariton brancheslab34 ; lab35 . We only
consider the lower branch in which the BEC occurs. Due to pair particle
scattering, the exciton polaritons of the lower polariton branch can be
treated as harmonic oscillators by neglecting the nonlinearitylab36 . The
Hamiltonian of the polaritons can be described as

$\displaystyle H_{LP}=\sum_{j=0}^{M}\hbar\omega_{j}a^{\dagger}_{j}a_{j}.$ (1)

where the ground state (mode) is the state with $j=0$ and $a_{j}$
($a_{j}^{\dagger}$) denotes the bosonic annihilation (creation) operator. The
master equation for the the density matrix of the polaritons $\rho$, subject
only to the polariton thermalization, can be expressed as

$\displaystyle\frac{d}{dt}\rho(t)=\mathcal{L}\rho=-i[H,\rho]+L_{\textmd{thermal}}(\rho),$
(2)

where the polariton thermalization is described by the Lindblad superoperator

$\displaystyle L_{\textmd{thermal}}(\rho)=$
$\displaystyle\sum_{j=0}^{M}\sum_{k=0}^{M}\Gamma_{jk}(a_{j}a_{k}^{\dagger}\rho
a_{k}a_{j}^{\dagger}-\frac{1}{2}\rho
a_{k}a_{j}^{\dagger}a_{k}a_{j}^{\dagger}-\frac{1}{2}a_{k}a_{j}^{\dagger}a_{k}a_{j}^{\dagger}\rho),$
(3)

where $\Gamma_{jk}$ is the transition rate from the $j$th polariton state to
the $k$th state. The thermalization rates obey the Kubo-Martin-Schwinger
relationlab36a ; lab37a
$\Gamma_{jk}=\exp[\frac{\hbar(\omega_{j}-\omega_{k})}{\kappa_{B}T}]\Gamma_{kj}$,
where $\kappa_{B}$ is the Boltzmann constant and $T$ is the temperature of
intermolecular oscillations of the organic dyes or the temperature of the
phonons in the semiconductors that we want to estimate. There are different
underlying mechanisms of the polariton thermalization, which is dependent on
the detail system. For example, the polariton thermalization comes from the
nonlinear interaction with low frequency vibrations in organic polariton
systemslab37 ; lab38 .

The operator of the total polariton number
$\sum_{j=0}^{M}a_{j}^{\dagger}a_{j}$ is the constant of motion during the
thermalization process. The constant of motion implies that there are
invariant subspaces $|n_{0},n_{1},...,n_{M}\rangle\langle
n_{0},n_{1},...,n_{M}|$ with the total number of polaritons equal to
$\sum_{j=0}^{M}n_{j}=N$. When there are invariant subspaces, the stationary
solution is not uniquelab39 . The Gibbs distribution over the states of a
given invariant subspace is also a stationary solution, which is given bylab36

$\displaystyle\rho_{s}=\sum_{N=0}^{\infty}P_{N}(0)\frac{1}{Z_{N}}\sum_{n_{0}+...+n_{M}=N}\nu_{0}^{n_{0}}...\nu_{M}^{n_{M}}$
$\displaystyle\times|n_{0},n_{1},...,n_{M}\rangle\langle
n_{0},n_{1},...,n_{M}|,$ (4)

where $P_{N}(0)$ denotes the probability that there are $N$ polaritons in
total in the low polariton branch at the initial time, which is given by
$P_{N}(0)=\sum_{n_{0}+...+n_{M}=N}\textmd{Tr}[\rho(0)|n_{0},n_{1},...,n_{M}\rangle\langle
n_{0},n_{1},...,n_{M}|]$ with $\sum_{j=0}^{M}n_{j}=N$,
$\nu_{j}=\exp[\frac{\hbar(\omega_{0}-\omega_{j})}{k_{B}T}]$, and $Z_{N}$ is
the partition function discribed by

$\displaystyle
Z_{N}=\sum_{n_{0}+...+n_{M}=N}\nu_{0}^{n_{0}}...\nu_{M}^{n_{M}}.$ (5)

The estimation uncertainty of the unbiased estimator $\delta^{2}T$ is bounded
by the quantum Cramér-Rao lower bound as $\delta^{2}T\geq 1/F_{T}$lab40 ;
lab41 , where $F_{T}$ is the quantum Fisher information (QFI)lab42  of
temperature $T$ in the steady state $\rho_{s}$, which is given by

$\displaystyle
F_{T}=\sum_{N=0}^{\infty}\sum_{n_{0}+...+n_{M}=N}\frac{(\partial_{T}P_{N,n_{0},..,n_{M}})^{2}}{P_{N,n_{0},..,n_{M}}},$
(6)

where
$P_{N,n_{0},..,n_{M}}=\frac{P_{N}(0)}{Z_{N}}\nu_{0}^{n_{0}}...\nu_{M}^{n_{M}}$
denotes the probability of projection into the state
$|n_{0},n_{1},...,n_{M}\rangle$ with $\sum_{j=0}^{M}n_{j}=N$, and the
shorthand notation $\partial_{T}=\frac{\partial}{\partial T}$. Hereafter, we
set $\hbar=\kappa_{B}=1$ for convenience.

Without loss of generality, we first consider that the system is consisted of
$M+1$ states equidistant in frequency, i.e.,
$\omega_{j}=\omega_{0}+\omega\times j/M$ with $\omega=\omega_{M}-\omega_{0}$.

In the case of two polariton states.-For $M=1$, we can obtain the probability
of the state $|n_{0},n_{1}\rangle$ with $n_{0}+n_{1}=N$

$\displaystyle P_{N,n_{0},n_{1}}=\langle
n_{0},n_{1}|\rho_{s}(M=1)|n_{0},n_{1}\rangle=\frac{\lambda^{-n_{1}}(\lambda-1)}{\lambda-\lambda^{-N}},$
(7)

where the factor $\lambda$ is defined as $\lambda=\exp(\omega/T)$, and only
one invariant subspace is considered, i.e., $P_{N}(0)=1$. By utilizing the
formula in Eq. (6), the QFI with $M=1$ is analytically expressed as

$\displaystyle
F_{T}=\frac{\omega^{2}}{T^{4}}[\frac{1}{\lambda-1}+\frac{1}{(\lambda-1)^{2}}-\frac{(1+N)^{2}}{\lambda^{1+N}-1}-\frac{(1+N)^{2}}{(\lambda^{1+N}-1)^{2}}].$
(8)

For $N\gg 1$, we can obtain the simplified form
$F_{T}\simeq\frac{\lambda\omega^{2}}{(\lambda-1)^{2}T^{4}}$. This result is
equivalent to measuring the temperature $T$ in the thermal equilibrium state
of harmonic oscillator with the Hamiltonian $H_{e}=\omega a^{\dagger}a$lab25 .
This does not reflect the advantage of having invariant subspaces. This is
mainly due to the low number of states. Next, we investigate the QFI of the
temperature with a large number of states, i.e., $M>1$.

In the case of single polarization: $N=1$.- We then consider another simple
case, which can be analytically calculated for $N=1$. In this case, the
probability of projection into the state $|0,0,...,n_{j}=1,...0\rangle$ is
$P_{N=1,n_{j}=1}=\frac{\lambda^{-j/M}}{\sum_{j=0}^{M}\lambda^{-j/M}}=\frac{\lambda^{1-j/M}(\lambda^{1/M}-1)}{\lambda^{(}1+1/M)-1}$.
The corresponding QFI is given by

$\displaystyle F_{T}=2\omega^{2}\lambda^{1+2/M}\times$
$\displaystyle\frac{M(2+M)-(1+M)^{2}\cosh(\frac{\omega}{MT})+\cosh[(1+M)\frac{\omega}{MT}]}{(\lambda^{1/M}-1)^{2}[\lambda^{(1+M)/M}-1]^{2}M^{2}T^{4}}.$
(9)

For the infinite $M$, the QFI of the temperature $T$ tends to be

$\displaystyle
F_{T}|_{M\rightarrow\infty}=\frac{1}{T^{2}}-\frac{\lambda\omega^{2}}{(\lambda-1)^{2}T^{4}}.$
(10)

For $\omega\gg T$, we can obtain $F_{T}|_{M\rightarrow\infty}\simeq 1/T^{2}$,
which is the maximal QFI of the temperature. According to the quantum Cramér-
Rao lower bound, we obtain the Landau boundlab33 , i.e., $\delta T\geq T$. It
means that we can obtain the infinite QFI for the temperature $T\rightarrow
0$, leading to that the optimal estimation precision $\delta T\approx 0$ is
achieved. The result shows that we can perform very accurate low temperature
measurement in the case of the invariant subspace with infinite states. As
shown in Fig. 1, the QFI increases with the value of $M$ for $\omega\gg T$. On
the contrary, the QFI decreases with the value of $M$ for high temperature.
For $M$ is larger than a certain characteristic value $M_{c}$, the
relationship between QFI and $M$ becomes increasingly independent. From Fig.
1, we can see that the characteristic value $M_{c}$ decreases as the
temperature $T$ increases. In the low temperature region, the increasing
number of polariton states $M$ promotes the estimation precision of
temperature more obviously.

Figure 1: The change relation diagram of the ratio of the QFI corresponding to
finite $M$ value and the maximal QFI $1/T^{2}$ corresponding to infinite $M$,
i.e., $T^{2}F_{T}$, versus $M$ for three different temperatures in arbitrary
units: $T=0.5,\ 3.0,\ 10$. Here the dimensionless parameters are given by
$\omega=10$ and $N=1$.

For the general case, i.e., $N>1$ and $M>1$, there are no analytical
solutions. It can be calculated numerically to investigate the effects of the
total number of polaritons $N$ and the number of polariton states $M+1$ on the
QFI of the temperature $T$. As shown in Fig. 2, with the increase of the
polaritons $N$, the QFI of the temperature $T$ increases when the temperature
$T$ above a certain value. However, the enhancement effect becomes less and
less obvious as $N$ increases. This result is consistent with the previous
analytical result of $M=1$, which is independent of the total number of
polaritons $N$ for large number $N\gg 1$. By contrast, increasing $M$ can
significantly improve the QFI with $N=4$, especially in the low-temperature
areas, as shown in Fig. 3. This conclusion also supports the previous
analytical results.

Figure 2: The QFI of the temperature changes with the temperature $T$ for the
four numbers of polaritons: $N=1,\ 2,\ 3,\ 4$. Here the dimensionless
parameters are given by $\omega=10$ and $M=10$. Figure 3: The QFI of the
temperature changes with the temperature $T$ for the three numbers of
polariton states: $M=2,\ 4,\ 6$. Here the dimensionless parameters are given
by $\omega=10$, and $N=4$.

As a summary, one of our main results is that increasing the number of
polariton states can greatly enhance the measurement precision of the low
temperature. However, the total number of polaritons $N$ plays a smaller and
smaller role in enhancing the measurement precision of the temperature as $N$
increases.

The polariton dissipation and incoherent pumping.- We consider that there is a
polariton dissipation in the polariton system, which is generally unavoidable
in real quantum systems. We assume that the coupling between the polariton
system and the environment is weak, leading to that the Born-Markov
approximationlab42a ; lab43a  can be utilized. Therefore, the polariton
dissipation can be described by the Lindblad superoperator

$\displaystyle L_{\textmd{diss}}(\rho)=\sum_{j=0}^{M}\gamma_{j}(a_{j}\rho
a_{j}^{\dagger}-\frac{1}{2}\rho
a_{j}^{\dagger}a_{j}-\frac{1}{2}a_{j}^{\dagger}a_{j}\rho),$ (11)

where $\gamma_{j}$ is the dissipation rate of the $j$th polariton state.
Without extra pumping, the total number of the polaritons in the steady state
will be 0 due to the dissipation, i.e., the steady system is in the vacuum
state $|0,0,...,0\rangle$. The information of temperature can not be obtained
by the steady state due to that the dissipation erases all information. In
order to get the information of the temperature, it is necessary to take the
measurements in advance before the system reaches the steady state. In this
work, we assume that the thermalization process is much faster than the
dissipation process, i.e., $\Gamma_{0j}(1+\langle
a_{0}^{\dagger}a_{0}\rangle)\gg\ \gamma_{j}$. The density matrix in Eq. (4)
can be obtained approximately at time $\gamma_{j}^{-1}\gg
t\gg\Gamma_{0j}^{-1}(1+\langle a_{0}^{\dagger}a_{0}\rangle)^{-1}$. In thus
case, the optimal interrogation time should be much smaller than the
characteristic time of the dissipation process and larger than that of the
thermalization process.

In order not to control the interrogation time, extra pumping is required to
obtain a non-vacuum steady state. We consider that there is a incoherent
pumping. The energy transfers from the upper branch and uncoupled excitons
towards the lower branch can be treated as an effective incoherent pumping,
which can be described by the Lindblad master equation

$\displaystyle L_{\textmd{pump}}(\rho)=\sum_{j=0}^{M}\kappa_{j}(a_{j}\rho
a_{j}^{\dagger}-\frac{1}{2}\rho
a_{j}^{\dagger}a_{j}-\frac{1}{2}a_{j}^{\dagger}a_{j}\rho)+$
$\displaystyle\sum_{j=0}^{M}\kappa_{j}(a_{j}^{\dagger}\rho
a_{j}-\frac{1}{2}\rho
a_{j}a_{j}^{\dagger}-\frac{1}{2}a_{j}a_{j}^{\dagger}\rho),$ (12)

where $\kappa_{j}$ is the pumping rate of the $j$th polariton state. Including
the dissipation and the incoherent pumping, the density matrix of the
polaritons $\rho$ is dominated by the master equation

$\displaystyle\frac{d}{dt}\rho=-i[H,\rho]+L_{\textmd{thermal}}(\rho)+L_{\textmd{diss}}(\rho)+L_{\textmd{pump}}(\rho).$
(13)

In general, the above equation is difficult to be solved numerically and
analytically. It can be approximately solved by assuming that the
thermalization is much faster than the dissipation and the incoherent pumping,
i.e., $\Gamma_{0j}(1+\langle
a_{0}^{\dagger}a_{0}\rangle)\gg\gamma_{j},\kappa_{j}$lab43 ; lab44 . The
general expression for the density matrix in the steady state $\rho_{s}$ is
also described by

$\displaystyle\rho_{s}=\sum_{N=0}^{\infty}P_{N}\frac{1}{Z_{N}}\sum_{n_{0}+...+n_{M}=N}\nu_{0}^{n_{0}}...\nu_{M}^{n_{M}}$
$\displaystyle\times|n_{0},n_{1},...,n_{M}\rangle\langle
n_{0},n_{1},...,n_{M}|,$ (14)

where $P_{N}$ denotes the probability that there are $N$ polaritons in total
in the steady state. In this case, $P_{N}$ is independent of the initial value
$P_{N}(0)$. Without loss of generality, we consider $\kappa_{j}=\kappa$ and
$\gamma_{j}=\gamma$. For only two polariton states ($M=1$), we can obtain the
general form of the probabilities (see Appendix A for details)

$\displaystyle P_{N}=(\frac{\kappa}{\kappa+\gamma})^{N}(N+1)P_{0},$ (15)
$\displaystyle P_{0}=\frac{1}{1+\frac{\kappa(\kappa+2\gamma)}{\gamma^{2}}}.$
(16)

In the steady state, the probability of the state $|n_{0},n_{1}\rangle$ with
$n_{0}+n_{1}=N$ is given by

$\displaystyle
P_{N,n_{0},n_{1}}=P_{N}\lambda^{-n_{1}}(\lambda-1)/(\lambda-\lambda^{-N}),$
(17)

where $n_{1}$ ranges from $0$ to $N$. The corresponding QFI can be calculated
by combing Eq. (6) and Eq. (17), and the results are directly shown in Fig. 4.
The QFI can increase with the pumping rate $\kappa$ for different temperature,
but not all the time. When the pumping rate $\kappa$ is larger than a certain
value, the QFI is independent of the pumping rate. It is because that the
number of polaritons $N$ is significantly increased by the strong enough
pumping, i.e., $\sum_{N\gg 1}P_{N}\sim 1$. As shown in Eq. (8), the QFI is
independent of the number of polaritons $N$ in the case of $N\gg 1$.
Therefore, too strong pumping is wasteful in improving the measurement
precision of the temperature $T$. Only the appropriate strength of the
incoherent pumping is required to resist the effects of dissipation.

Figure 4: Diagram of the QFI $F_{T}$ as a function of pumping rate $\kappa$
with four different temperature: $T=1,3,5,7$. Here, the dimensionless
parameters are given by $\omega=10$, and $\gamma=10$.

M degenerate states.- In order to deal with the case of $N\gg 1$ and $M\gg 1$,
we consider that there are $M$ degenerate states (modes), i.e.,
$\omega_{j}=\omega_{0}+\omega$ for $j\geq 1$. The general form of the
partition function $Z^{d}_{N}$ in the case of $M$ degenerate states are
obtained (see Appendix B for details)

$\displaystyle
Z^{d}_{N}=\frac{\lambda^{M}}{(\lambda-1)^{M}}-\frac{(M+N)!_{2}F_{1}[1,1+M+N,2+N;\frac{1}{\lambda}]}{(M-1)!(N+1)!\lambda^{(1+N)}},$
(18)

where
${}_{2}F_{1}(a,b,c;d)=\sum_{k=0}^{\infty}\frac{(a)_{k}(b)_{k}d^{k}}{(c)_{k}k!}$
denotes the generalized Hypergeometric function with
$(r)_{k}=\Gamma(r+k)/\Gamma(r)$. The general form of the probability of the
state $|n_{0},...,n_{M}\rangle$ with $n_{0}+...+n_{M}=N$ is given by

$\displaystyle
P^{d}_{N,n_{0},..,n_{M}}=\frac{\lambda^{n_{0}-N}P^{d}_{N}}{Z_{N}},$ (19)

where $P^{d}_{N}$ denotes the probability that there are $N$ polaritons in
total in the steady state: $P^{d}_{N}=P_{N}(0)$ in the absence of the
dissipation and the incoherent pumping; $P^{d}_{N}=P_{N}$ with the dissipation
and the incoherent pumping. We note that there are
$(N-n_{0}+M-1)!/[(N-n_{0})!(M-1)!]$ different states $|n_{0},...,n_{M}\rangle$
with the same probability $P^{d}_{N,n_{0},..,n_{M}}$ due to the $M$ degenerate
modes. This simplifies the calculation somewhat. We numerically calculate the
corresponding QFI $F_{T}^{d}$ with the large $N$ and $M$.

Figure 5: Diagram of the QFI $F^{d}_{T}$ as the function of the temperature
$T$ with three different values of $M$: $M=100,200,300$. The number of the
polaritons is fixed, i.e., $P_{N}^{d}=1$. Here, the dimensionless parameters
are given by $\omega=10$, and $N=100$.

As shown in Fig. 5, the QFI increases with the number of states $M$ with the
fixed number of the polaritons $N=100$, especially in the low temperature. It
means that the number of polariton states $M$ is an important resource for
enhancing the estimation precision of the temperature, which is consistent
with the previous results by using the small $N$ and $M$ in the case of states
equidistant in frequency.

As shown in Fig. 6, in the case of large $M$, the line $N=100$ is coincident
with the line $N=200$. It implies that the QFI of the temperature is
independent of the number of the polaritons for large $N$ in the case of large
$M$. The value of $M$ can not change the relation between the QFI and the
number of the polaritons $N$.

Figure 6: Diagram of the QFI $F_{T}$ as the function of the temperature $T$
with three different numbers of the polaritons: $N=10,200,300$. Here, the
dimensionless parameters are given by $\omega=10$, $P_{N}^{d}=1$, and $M=100$.

Discussion and Conclusion.- We have proposed a new mechanism to improve the
measurement precision of the low temperature and obtained Landau bound $\delta
T\simeq T$. It is based on the existence of invariant subspaces during the
polariton thermalization. The measurement precision of the low temperature
increases significantly with the number of polariton states. It is interesting
to apply our mechanism to different systems beyond polariton BECs by looking
for invariant subspaces with many modes during the thermalization.

When the polaritons suffer from the dissipation, the incoherent pumping can be
used to enhancing the estimation precision of temperature. Due to that the QFI
becomes less and less dependent on the total number of the polaritons, too
strong incoherent pumping is a waste of energy. Whether the periodic coherent
modulation and nonlinearity can be used to enhancing the estimation precision
of low tempearature deserves further study.

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 and National Key R&D Program of
China under Grant No. 2018YFB1601402-2.

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## appendix A

We consider that there are polariton dissipation and incoherent pumping of the
lower polariton states, which are described by the Lindblad superoperators

$\displaystyle L_{\textmd{diss}}(\rho)=\sum_{j=0}^{M}\gamma_{j}(a_{j}\rho
a_{j}^{\dagger}-\frac{1}{2}\rho
a_{j}^{\dagger}a_{j}-\frac{1}{2}a_{j}^{\dagger}a_{j}\rho)$ (S1) $\displaystyle
L_{\textmd{pump}}(\rho)=\sum_{j=0}^{M}\kappa_{j}(a_{j}\rho
a_{j}^{\dagger}-\frac{1}{2}\rho
a_{j}^{\dagger}a_{j}-\frac{1}{2}a_{j}^{\dagger}a_{j}\rho)+$
$\displaystyle\sum_{j=0}^{M}\kappa_{j}(a_{j}^{\dagger}\rho
a_{j}-\frac{1}{2}\rho
a_{j}a_{j}^{\dagger}-\frac{1}{2}a_{j}a_{j}^{\dagger}\rho),$ (S2)

where $\gamma_{j}$ ($\kappa_{j}$) is the dissipation (the incoherent pumping)
rate of the $j$th state. The general solution is difficult and we also
consider that the thermalization process is very rapid, i.e.,
$\Gamma_{0j}(1+\langle a_{0}^{\dagger}a_{0}\rangle)\gg\kappa_{j},\
\gamma_{j}$, which has been proved to be valid in ref.lab36 . In the first
stage, i.e., $\kappa_{j}^{-1},\ \gamma_{j}^{-1}\gg
t\gg\Gamma_{0j}^{-1}(1+\langle a_{0}^{\dagger}a_{0}\rangle)^{-1}$, the density
matrix $\rho(t)$ obeys the approximate differential equation by ignoring the
effect of the dissipation and the incoherent pumping,

$\displaystyle\frac{d}{dt}\rho\approx-i[H,\rho]+L_{\textmd{thermal}}(\rho),$
(S3)

which leads to the general expression for the density matrix $\rho$lab36

$\displaystyle\rho=\sum_{N=0}^{\infty}P_{N}(t)\frac{1}{Z_{N}}\sum_{n_{0}+...+n_{M}=N}\nu_{0}^{n_{0}}...\nu_{M}^{n_{M}}$
$\displaystyle\times|n_{0},n_{1},...,n_{M}\rangle\langle
n_{0},n_{1},...,n_{M}|,$ (S4)

where $P_{N}(t)\approx P_{N}(0)$ for $\kappa_{j}^{-1},\ \gamma_{j}^{-1}\gg
t\gg\Gamma_{0j}^{-1}(1+\langle a_{0}^{\dagger}a_{0}\rangle)^{-1}$.

In the second stage, one can substitute the above equation into the whole
evolution equation of the density matrix $\rho$, which is described as

$\displaystyle\frac{d}{dt}\rho=-i[H,\rho]+L_{\textmd{thermal}}(\rho)+L_{\textmd{diss}}(\rho)+L_{\textmd{pump}}(\rho).$
(S5)

Substituting Eq. (S4) into Eq. (S5) and using the steady conditions
$\frac{d}{dt}P_{N}(t\rightarrow\infty)=0$, the steady-state solutions of the
probabilities $P_{N}=P_{N}(t\rightarrow\infty)$ are achievedlab36

$\displaystyle
P_{N+1}=\frac{(d_{N-1}+\beta_{N})Z_{N+1}}{d_{N}Z_{N}}P_{N}-\frac{\beta_{N-1}Z_{N+1}}{d_{N}Z_{N-1}}P_{N-1},$
(S6) $\displaystyle P_{1}=\frac{\beta_{0}Z_{1}}{d_{0}Z_{0}}P_{0},$ (S7)

where
$d_{N}=\sum_{n=0}^{N}\sum_{j=0}^{M}(\gamma_{j}+\kappa_{j})\nu_{j}^{n+1}Z_{N-n}$
and $\beta_{N}=\sum_{n=0}^{N}\sum_{j=0}^{M}\kappa_{j}\nu_{j}^{n}Z_{N-n}$.
Utilizing the above equations, we further obtain that

$\displaystyle P_{2}=\frac{\beta_{1}Z_{2}}{d_{1}Z_{1}}P_{1},$ (S8)
$\displaystyle P_{3}=\frac{\beta_{2}Z_{3}}{d_{2}Z_{2}}P_{2},$ (S9)
$\displaystyle P_{4}=\frac{\beta_{3}Z_{4}}{d_{3}Z_{3}}P_{3},$ (S10)
$\displaystyle P_{5}=\frac{\beta_{4}Z_{5}}{d_{4}Z_{4}}P_{4},$ (S11)
$\displaystyle...$

By induction, we can obtain the general form of the probability $P_{N}$ with
$N\geq 1$

$\displaystyle P_{N}=\frac{\beta_{N-1}Z_{N}}{d_{N-1}Z_{N-1}}P_{N-1}$ (S12)
$\displaystyle=\frac{\prod_{n=0}^{N-1}\beta_{n}Z_{N}}{\prod_{n=0}^{N-1}d_{n}}P_{0}.$
(S13)

Without loss of generality, we consider $\kappa_{j}=\kappa$ and
$\gamma_{j}=\gamma$. In the case of two states ($M=1$), we can obtain the
analytical results about the distribution

$\displaystyle
d_{N}=\frac{\lambda^{-(1+N)}(\lambda^{2+N}-1)(1+N)(\kappa+\gamma)}{(\lambda-1)},$
(S14)
$\displaystyle\beta_{N}=\frac{\lambda^{-N}(\lambda^{1+N}-1)(2+N)\kappa}{(\lambda-1)},$
(S15) $\displaystyle
Z_{N}=\sum_{n=0}^{N}\exp[-n\omega/T]=\frac{\lambda-\lambda^{-N}}{\lambda-1}.$
(S16)

Substituting the above equations into Eq. (S13), we can derive a simplified
form of the probability $P_{N}$, which is described as

$\displaystyle P_{N}=(\frac{\kappa}{\kappa+\gamma})^{N}(N+1)P_{0}.$ (S17)

Due to the normalization condition $\sum_{N=0}^{\infty}P_{N}=1$, we can obtain
that

$\displaystyle P_{N}=(\frac{\kappa}{\kappa+\gamma})^{N}(N+1)P_{0},$ (S18)
$\displaystyle P_{0}=\frac{1}{1+\frac{\kappa(\kappa+2\gamma)}{\gamma^{2}}}.$
(S19)

## appendix B

In the case of $M$ degenerate states, $\omega_{j}=\omega_{0}+\omega$ for
$j\geq 1$, the corresponding density matrix in the steady state
$\rho^{d}_{s}=\rho^{d}(t\rightarrow\infty)$ is also written as

$\displaystyle\rho^{d}_{s}=\sum_{N=0}^{\infty}P^{d}_{N}\frac{1}{Z^{d}_{N}}\sum_{n_{0}+...+n_{M}=N}\nu_{0}^{n_{0}}...\nu_{M}^{n_{M}}$
$\displaystyle\times|n_{0},n_{1},...,n_{M}\rangle\langle
n_{0},n_{1},...,n_{M}|,$ (S20)

The partition function can be derived

$\displaystyle
Z^{d}_{N}=\sum_{n_{0}+...+n_{M}=N}\lambda^{-(n_{1}+...+n_{M})}=\sum_{n_{0}=0}^{N}\mu(n_{0})\lambda^{n_{0}-N}$
(S21)
$\displaystyle=\frac{\lambda^{M}}{(\lambda-1)^{M}}-\frac{(M+N)!_{2}F_{1}[1,1+M+N,2+N;\frac{1}{\lambda}]}{(M-1)!(N+1)!\lambda^{(1+N)}},$
(S22)

where the degeneration coefficient $\mu(n_{0})=(N-n_{0}+M-1)!/[N!(M-1)!]$
comes from the $M$ degenerate modes, and
${}_{2}F_{1}(a,b,c;d)=\sum_{k=0}^{\infty}\frac{(a)_{k}(b)_{k}d^{k}}{(c)_{k}k!}$
denotes the generalized Hypergeometric function with
$(r)_{k}=\Gamma(r+k)/\Gamma(r)$. The general form of the probability of the
state $|n_{0},n_{1},...,n_{M}\rangle$ with the total number
$n_{0}+n_{1}...+n_{M}=N$ is given by

$\displaystyle
P^{d}_{N,n_{0},..,n_{M}}=\frac{\lambda^{n_{0}-N}P^{d}_{N}}{Z^{d}_{N}},$ (S23)

where $P^{d}_{N}$ denotes the probability that there are $N$ polaritons in
total in the steady state: $P^{d}_{N}=P_{N}(0)$ in the absence of the
dissipation and the incoherent pumping; $P^{d}_{N}=P_{N}$ with the dissipation
and the incoherent pumping. In the case of $M$ degenerate states, the general
formula of QFI $F^{d}_{T}$ can be described as

$\displaystyle
F^{d}_{T}=\sum_{N=0}^{\infty}\sum_{n_{0}+...+n_{M}=N}\frac{(\partial_{T}P^{d}_{N,n_{0},..,n_{M}})^{2}}{P^{d}_{N,n_{0},..,n_{M}}}$
(S24)
$\displaystyle=\sum_{N=0}^{\infty}\sum_{n_{0}=0}^{N}\mu(n_{0})\times\frac{(\partial_{T}\frac{\lambda^{n_{0}-N}P^{d}_{N}}{Z^{d}_{N}})^{2}}{\frac{\lambda^{n_{0}-N}P^{d}_{N}}{Z^{d}_{N}}}$
(S25)
$\displaystyle=\sum_{N=0}^{\infty}\sum_{n_{0}=0}^{N}\mu(N-n_{0})\times\frac{(\partial_{T}\frac{\lambda^{-n_{0}}P^{d}_{N}}{Z^{d}_{N}})^{2}}{\frac{\lambda^{-n_{0}}P^{d}_{N}}{Z^{d}_{N}}}.$
(S26)

For the fixed number of the polaritons, i.e., $P^{d}_{N}=1$, the QFI can be
calculated by

$\displaystyle
F^{d}_{T}=\sum_{n_{0}=0}^{N}\mu(N-n_{0})\times\frac{(\partial_{T}\frac{\lambda^{-n_{0}}}{Z^{d}_{N}})^{2}}{\frac{\lambda^{-n_{0}}}{Z^{d}_{N}}}.$
(S27)