# Non-Markovian quantum thermometry

Ning Zhang  Lanzhou Center for Theoretical Physics and Key Laboratory of
Theoretical Physics of Gansu Province, Lanzhou University, Lanzhou 730000,
China  Chong Chen<EMAIL_ADDRESS>Department of Physics and The Hong
Kong Institute of Quantum Information of Science and Technology, The Chinese
University of Hong Kong, Shatin, New Territories, Hong Kong, China  Si-Yuan
Bai  Lanzhou Center for Theoretical Physics and Key Laboratory of Theoretical
Physics of Gansu Province, Lanzhou University, Lanzhou 730000, China  Wei Wu
Lanzhou Center for Theoretical Physics and Key Laboratory of Theoretical
Physics of Gansu Province, Lanzhou University, Lanzhou 730000, China  Jun-Hong
An<EMAIL_ADDRESS>Lanzhou Center for Theoretical Physics and Key
Laboratory of Theoretical Physics of Gansu Province, Lanzhou University,
Lanzhou 730000, China

###### Abstract

The rapidly developing quantum technologies and thermodynamics have put
forward a requirement to precisely control and measure the temperature of
microscopic matter at the quantum level. Many quantum thermometry schemes have
been proposed. However, precisely measuring low temperature is still
challenging because the obtained sensing errors generally tend to diverge with
decreasing temperature. Using a continuous-variable system as a thermometer,
we propose non-Markovian quantum thermometry to measure the temperature of a
quantum reservoir. A mechanism to make the sensing error $\delta T$ scale with
the temperature $T$ as the Landau bound $\delta T\simeq T$ in the full-
temperature regime is discovered. Our analysis reveals that it is the quantum
criticality of the total thermometer-reservoir system that causes this
enhanced sensitivity. Efficiently avoiding the error-divergence problem, our
result gives an efficient way to precisely measure the low temperature of
quantum systems.

##  I Introduction

Precise sensing of temperature is of significance in fields ranging from the
fundamental natural sciences to the rapidly developing quantum technologies
[1, 2, 3, 4]. People’s increased capabilities of controlling and using quantum
characters of microscopic matter have led to the development of the field of
quantum thermodynamics [5, 6, 7, 8], where the precise measuring of
thermodynamic quantities at the quantum level invalidates classical sensing
schemes and calls for advanced ones. On the other hand, temperature is one of
the main reasons causing decoherence, which is a main bottleneck in the
practical realization of quantum technology protocols. Thus, from an
application viewpoint, quantum devices generally work at ultralow temperature,
e.g., cold-atom and ion-trap systems [9, 10, 11, 12, 13], which also calls for
the ability to precisely control and sense temperature.

Quantum thermometry aims to realize the precise measurement of temperature
using quantum features [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26,
27, 28, 29, 30, 31, 32, 13]. A quantum system is chosen as a thermometer and
is brought into thermal contact with the measured system in thermal
equilibrium. The temperature is measured in either the equilibrium thermal
state or nonequilibrium dynamical state of the thermometer through certain
observables. It has been explored in various platforms, including nitrogen-
vacancy centers in diamond [33, 34], optical nanofibres [35], cavity
optomechanical systems [36], and quantum dots [37]. The advantage of using
quantum features in thermometry is that an enhanced precision can be achieved
in certain temperature regimes due to quantum coherence [14, 15, 16], strong
coupling [19, 11], quantum correlation [36, 27, 28], periodic driving [23], or
nonequilibrium dynamics [29, 30, 31, 32, 13]. A challenge of almost all of the
existing schemes is that their sensing errors tend to diverge with decreasing
temperature [38, 39], although some ways to slow down the divergence via
strong coupling [11], periodic driving [23], and finite measurement resolution
[24, 25] have been explored. Therefore, quantum thermometry performing well in
the low-temperature regime is still absent.

In this paper, we propose a quantum thermometry scheme efficiently avoiding
the error-divergence problem in the low-temperature regime. Using a
continuous-variable system as a thermometer to measure the temperature of a
quantum reservoir, our scheme permits us to achieve a scaling of sensing error
as $\delta T\simeq T$, which is called the Landau bound [40], in the full-
temperature regime. We find that it is the combined action of the non-
Markovian dynamical encoding of the thermometer and quantum criticality of the
total thermometer-reservoir system that causes this notable performance. The
quantum criticality occurs as a consequence of quantum phase transition of the
total thermometer-reservoir system with the abrupt formation of a bound state
out of its continuous energy band. Supplying a useful idea to design quantum
thermometry, our study may potentially prompt advances of low-temperature
sensing in quantum thermodynamics and technologies.

##  II Quantum thermometry

Quantum sensing to a quantity $\theta$ of a certain system generally involves
three steps [41, 42]. One first prepares a quantum sensor in certain state
$\rho_{\text{in}}$. Then the interaction of the sensor with the measured
system is switched on to encode $\theta$ into the sensor state
$\rho_{\theta}=\check{\mathcal{E}}\rho_{\text{in}}$. Acting on the Liouvillian
space of the density matrix, the superoperator $\check{\mathcal{E}}$ may be
either unitary or nonunitary. Lastly, one measures the sensor and infers the
value of $\theta$ from the results. The inevitable errors mean that one cannot
estimate $\theta$ exactly. The ultimate estimation error of $\theta$ is
constrained by the quantum Cramér-Rao bound
$\delta\theta=(\mathcal{N}\mathcal{F}_{\theta})^{-1/2}$ [43, 44]. Here
$\mathcal{N}$ is the measurement times and
$\mathcal{F}_{\theta}=\text{Tr}(\hat{L}_{\theta}^{2}\rho_{\theta})$, with
$\hat{L}_{\theta}$ determined by
$\partial_{\theta}\rho_{\theta}=(\hat{L}_{\theta}\rho_{\theta}+\rho_{\theta}\hat{L}_{\theta})/2$,
is the quantum Fisher information (QFI) describing the most information for
estimating $\theta$ from $\rho_{\theta}$. Due to the independence of
$\mathcal{F}_{\theta}$ on $\mathcal{N}$, we choose $\mathcal{N}=1$. It has
been found that entanglement [45, 46], squeezing [47, 48, 49, 50], chaos [51],
and criticality [52, 53, 54] can act as quantum resources to beat the
sensitivity limit of classical sensors.

We are interested in measuring the temperature of a quantum reservoir with
infinite degrees of freedom. We choose a continuous-variable system as the
quantum thermometer. It may be an $LC$ oscillator [18], harmonic potential
trapped BEC [4], or mechanical oscillator [55]. Its interaction with the
reservoir for encoding the temperature into the state of the thermometer reads
($\hbar=1$)

$\hat{H}=\omega_{0}\hat{a}^{\dagger}\hat{a}+\sum_{k}[\omega_{k}\hat{b}^{\dagger}_{k}\hat{b}_{k}+g_{k}(\hat{a}^{\dagger}\hat{b}_{k}+\hat{b}^{\dagger}_{k}\hat{a})],$
(1)

where $\hat{a}$ and $\hat{b}_{k}$ are the annihilation operators of the
thermometer with frequency $\omega_{0}$ and the $k$th reservoir mode with
frequency $\omega_{k}$, and $g_{k}$ is their coupling strength. Their coupling
is further characterized by the spectral density
$J(\omega)=\sum_{k}g^{2}_{k}\delta(\omega-\omega_{k})$. We consider the Ohmic-
family spectral density
$J(\omega)=\eta\omega^{s}\omega_{c}^{1-s}e^{-\omega/\omega_{c}}$, where $\eta$
is a dimensionless coupling constant, $\omega_{c}$ is a cutoff frequency, and
$s$ is an Ohmicity index [56]. It is widely used to describe the noises in
circuit QED [57, 58, 59], ion trap [60], and waveguide [61] systems. Depending
on the dispersion relation and density of states of the reservoir,
$\omega_{c}$ characterizes the dominate modes coupled to the thermometer and
relates to the typical time scale of the correlation function of the
reservoir. The index $s$ generally is determined by the reservoir dimension
[62]. The temperature $T$ is carried by the initial state
$\rho_{\text{R}}(0)=\prod_{k}e^{-\beta\omega_{k}\hat{b}^{\dagger}_{k}\hat{b}_{k}}/\text{Tr}[e^{-\beta\omega_{k}\hat{b}^{\dagger}_{k}\hat{b}_{k}}]$,
where $\beta=(K_{B}T)^{-1}$ and $K_{B}$ is the Boltzmann constant.

Setting the initial state of the total system as
$\rho_{\text{tot}}(0)=\rho(0)\otimes\rho_{\text{R}}(0)$, we can derive the
exact non-Markovian master equation of the quantum thermometer by the path-
integral influence-functional method [63, 64, 65] as

$\displaystyle\dot{\rho}(t)$ $\displaystyle=$
$\displaystyle-i\Omega(t)[\hat{a}^{{\dagger}}\hat{a},\rho(t)]+[\Gamma(t)+{\Gamma^{\beta}(t)}/{2}]\check{\mathcal{L}}_{\hat{a}}\rho(t)$
(2)
$\displaystyle+{\Gamma^{\beta}(t)}/{2}\check{\mathcal{L}}_{\hat{a}^{\dagger}}\rho(t),$

where
$\check{\mathcal{L}}_{\hat{o}}\cdot=2\hat{o}\cdot\hat{o}^{\dagger}-\cdot\hat{o}^{\dagger}\hat{o}-\hat{o}^{\dagger}\hat{o}\cdot$
is the Lindblad superoperator. Here $\Omega(t)=-\textrm{Im}[\dot{u}(t)/u(t)]$
is the renormalized frequency, $\Gamma(t)=-\textrm{Re}[\dot{u}(t)/u(t)]$ and
$\Gamma^{\beta}(t)=\dot{v}(t)+2v(t)\Gamma(t)$ are the dissipation and noise
coefficients. The functions $u(t)$ and $v(t)$ are determined by

$\displaystyle\dot{u}(t)+i\omega_{0}u(t)+\int_{0}^{t}dt_{1}\mu(t-t_{1})u(t_{1})=0,$
(3) $\displaystyle
v(t)=\int_{0}^{t}dt_{1}\int_{0}^{t}dt_{2}u^{*}(t_{1})\nu(t_{1}-t_{2})u(t_{2}),$
(4)

under $u(0)=1$. The kernel functions $\mu(x)=\int_{0}^{\infty}d\omega
J(\omega)e^{-i\omega x}$ and $\nu(x)=\int_{0}^{\infty}d\omega
J(\omega)\bar{n}(\omega)e^{-i\omega x}$ with
$\bar{n}(\omega)=(e^{\beta\omega}-1)^{-1}$. Keeping the same Lindblad form as
the Born-Markovian master equation [66], Eq. (2) incorporates all the non-
Markovian effects induced by the backactions of the reservoir into these time-
dependent coefficients self-consistently. It can be seen that the temperature
of the reservoir is successfully encoded into the thermometer state $\rho(t)$
via $\Gamma^{\beta}(t)$. It is worth noting that the encoding in our scheme is
different from that of either quantum metrology schemes based on Ramsey [67]
and Mech-Zehnder [68] interferometers, where the encoding dynamics is unitary,
or quantum sensing to the spectral density of a reservoir [69, 70], where,
although the encoding is also nonunitary, the sensed quantities are carried by
the sensor-reservoir interactions.

We consider that the initial state of the thermometer is a coherent state,
i.e., $\rho(0)=|\alpha_{0}\rangle\langle\alpha_{0}|$. Governed by Eq. (2), it
evolves to (see Appendix A)

$\rho(t)=\sum_{n=0}^{\infty}{M(t)^{n+1}v(t)^{n}}\hat{\mathcal{D}}_{t}|n\rangle\langle
n|\hat{\mathcal{D}}_{t}^{\dagger},$ (5)

where $M(t)=[1+v(t)]^{-1}$ and
$\hat{\mathcal{D}}_{t}=\exp[\alpha_{0}u(t)\hat{a}^{\dagger}-\alpha_{0}^{*}u^{*}(t)\hat{a}]$.
As a Gaussian state, the characteristic function of Eq. (5) is of Gaussian
form [71]:
$\chi({\boldsymbol{\gamma}})\equiv\text{Tr}[\rho\hat{\mathcal{D}}({\gamma})]=\exp(-{\frac{1}{4}}{\boldsymbol{\gamma}}^{{\dagger}}{\boldsymbol{\sigma}}{\boldsymbol{\gamma}}-i{\mathbf{d}}^{{\dagger}}K{\boldsymbol{\gamma}})$,
where ${\boldsymbol{\gamma}}=(\gamma,\gamma^{*})^{T}$, $K=\text{diag}(1,-1)$,
and the elements of the displacement vector ${\mathbf{d}}$ and the covariant
matrix ${\boldsymbol{\sigma}}$ are $d_{i}=\text{Tr}(\rho\hat{A}_{i})$ and
$\sigma_{ij}=\text{Tr}[\rho\\{\Delta\hat{A}_{i},\Delta\hat{A}_{j}^{{\dagger}}\\}]$
with $\hat{\mathbf{A}}=(\hat{a},\hat{a}^{\dagger})^{T}$ and
$\Delta\hat{A}_{i}=\hat{A}_{i}-d_{i}$. Its QFI for $\theta$ reads
$\mathcal{F}_{\theta}=\frac{1}{2}[\text{vec}(\partial_{\theta}{\boldsymbol{\sigma}})]^{\dagger}\mathcal{M}^{-1}\text{vec}(\partial_{\theta}{\boldsymbol{\sigma}})+2(\partial_{\theta}\mathbf{d})^{\dagger}\boldsymbol{\sigma}^{-1}\partial_{\theta}\mathbf{d}$,
where
$\mathcal{M}={\boldsymbol{\sigma}}^{*}\otimes{\boldsymbol{\sigma}}-K\otimes
K$, with $\boldsymbol{\sigma}^{*}$ being the complex conjugate of
$\boldsymbol{\sigma}$ [72]. The QFI of the temperature for Eq. (5) reads

$\mathcal{F}_{T}(t)={M(t)[\partial_{T}v(t)]^{2}/v(t)}.$ (6)

It can be proven that the QFI (6) is reached by measuring the number operator
$\hat{a}^{\dagger}\hat{a}$ of the thermometer.

As a special case, we first revisit the precision under the Born-Markovian
approximation. When the coupling is weak and the time scale of the reservoir
correlation function is much smaller than that of the thermometer, we can make
this approximation and obtain
$u_{\text{MA}}(t)=e^{-[\kappa+i(\omega_{0}+\Delta(\omega_{0}))]t}$ and
$v_{\text{MA}}(t)=\bar{n}(\omega_{0})(1-e^{-2\kappa t})$ (Appendix B). Here
$\kappa=\pi J(\omega_{0})$ and
$\Delta(\omega_{0})=\mathcal{P}\int{J(\omega)\over\omega_{0}-\omega}d\omega$,
with $\mathcal{P}$ denoting the Cauchy principal value. Then the coefficients
in Eq. (2) reduce to $\Gamma_{\text{MA}}(t)=\kappa$,
$\Omega_{\text{MA}}(t)=\omega_{0}+\Delta(\omega_{0})$, and
$\Gamma_{\text{MA}}^{\beta}(t)=2\kappa\bar{n}(\omega_{0})$. The unique steady
state is a canonical state
$\rho_{\text{MA}}(\infty)=e^{-\beta\omega_{0}\hat{a}^{\dagger}\hat{a}}/\text{Tr}[e^{-\beta\omega_{0}\hat{a}^{\dagger}\hat{a}}]$,
which is independent of the initial state. Thus the thermometer equilibrates
to a thermal state with the same temperature as the reservoir in Born-
Markovian dynamics. According to Eq. (6), we obtain the QFI

$\mathcal{F}_{T}^{\text{MA}}(t)=\bar{F}_{T}(\omega_{0}){\bar{n}(\omega_{0})+1\over\bar{n}(\omega_{0})+(1-e^{-2\kappa
t})^{-1}},$ (7)

where
$\bar{F}_{T}(\omega_{0})\equiv(\beta\omega_{0})^{2}\bar{n}(\omega_{0})[1+\bar{n}(\omega_{0})]/T^{2}$.
We have neglected the constant $\Delta(\omega_{0})$, which is generally
renormalized into $\omega_{0}$ [56]. Equation (7) reveals that the QFI
increases with time and saturates to $\bar{F}_{T}$ corresponding to the QFI of
$\rho_{\text{MA}}(\infty)$. The equilibrium-state performance of the
thermometer reads $\lim_{T\rightarrow\infty}\bar{F}_{T}(\omega_{0})=T^{-2}$ in
the high-temperature regime, which is called the Landau bound [40]. However,
it is unfortunate to find that the QFI tends to zero in the low-temperature
limit. Being consistent with previous works [14, 15, 16, 17, 18, 19, 20, 21,
22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 13], this means that the
thermometer becomes insufficient for measuring low temperatures under the
Born-Markovian approximation.

The non-Markovian solution of Eq. (3) is [69]

$u(t)=Ze^{-iE_{b}t}+\int_{0}^{\infty}dE\Theta(E)e^{-iEt},$ (8)

with $Z=[1+\int_{0}^{\infty}\frac{J(\omega)}{(E_{b}-\omega)^{2}}d\omega]^{-1}$
and $\Theta(E)=\frac{J(E)}{[E-\omega_{0}-\Delta(E)]^{2}+[\pi J(E)]^{2}}$. Here
$E_{b}$ is a possibly formed isolated root of the transcendental equation

$~{}y(E)\equiv\omega_{0}-\int_{0}^{\infty}\frac{J(\omega)}{\omega-E}d\omega=E.$
(9)

The solutions of Eq. (9) are the eigenenergies of Eq. (1) in the single-
excitation subspace. Since $y(E)$ is a decreasing function in the regime
$E<0$, Eq. (9) has one isolated root $E\equiv E_{b}$ provided $y(0)<0$. We
call the eigenstate corresponding to this isolated eigenenergy the bound
state. On the contrary, it has infinite roots in the regime $E>0$, which form
a continuous energy band. Since an extra band gap is induced by the formation
of the bound state, we claim that a quantum phase transition occurs in the
total system. Such a quantum phase transition has a profound impact on the
dynamics of the thermometer. Contributed by the continuous energy band, the
integral in Eq. (8) gradually vanishes with time due to out-of-phase
interference. Thus, if the bound state is formed, then
$\lim_{t\rightarrow\infty}u(t)=Ze^{-iE_{b}t}$, leading to a dissipationless
dynamics; while if it is absent, then $\lim_{t\rightarrow\infty}u(t)=0$,
characterizing a complete decoherence. It has been found that, when the bound
state is absent, the thermometer equilibrates to a thermal state at an
effective temperature; whenever the bound state is present, it tends to a
steady state no longer capable of being described by thermal state [65]. For
the Ohmic-family spectral density, the bound state is formed when
$\omega_{0}/\omega_{c}<\eta\gamma(s)$, where $\gamma(s)$ is Euler’s $\gamma$
function. Different from the previous schemes based on the thermal state [14,
20, 24, 29, 38, 73, 39, 40], our dynamical scheme enables us to reveal its
full performance in both the transient process and steady state regardless of
whether the thermometer equilibrates to a thermal state or not.

Figure 1: Evolution of the non-Markovian QFI for different $\omega_{c}$ by
numerically solving Eqs. (3) and (4). The red solid line is analytically
obtained from Eq. (10) and the black dashed line is the long-time QFI
$\bar{{F}}_{T}(\omega_{0})$ under the Born-Markovian approximation. We use
$\eta=0.1$, $s=1$, and $T=0.1\omega_{0}/K_{B}$.

Equation (4) is recast into $v(t)=\int^{\infty}_{0}d\omega
A_{\omega}(t)\bar{n}(\omega)$, where
$A_{\omega}(t)=J(\omega)|\tilde{u}_{\omega}(t)|^{2}$, with
$\tilde{u}_{\omega}(t)=\int_{0}^{t}d\tau u(\tau)e^{i\omega\tau}$, is called
the heat-exchange spectrum, which tends to
$A_{\omega}(\infty)=\Theta(\omega)+{Z^{2}J(\omega)/(\omega-E_{b})^{2}}$. Then
we obtain from Eq. (6) an upper bound of the QFI (see Appendix C)

$\mathcal{F}_{T}(\infty)\leq
M(\infty){\int_{0}^{\infty}d\omega\bar{{F}}_{T}(\omega)A_{\omega}(\infty)[1+\bar{n}(\omega)]}.$
(10)

This reveals that the non-Markovian QFI is fully determined by the overlap
integral between the QFI of an equilibration mode with given frequency
$\omega$ and the thermal-excitation dressed heat-exchange spectrum
$M(\infty)A_{\omega}(\infty)[1+\bar{n}(\omega)]$. First,
$\lim_{T\rightarrow\infty}\bar{{F}}_{T}(\omega)=T^{-2}$ reduces Eq. (10) to
$\lim_{T\rightarrow\infty}\mathcal{F}_{T}(\infty)\leq[1-Z^{2}M(\infty)]T^{-2}$
in the high-temperature limit, where $\int_{0}^{\infty}d\omega
A_{\omega}(\infty)=1-Z^{2}$ has been used. This scaling relation with
temperature matches with the Landau bound (7) under the Born-Markovian
approximation except for the prefactor when the bound state is formed. Second,
it is remarkable to find that $v(\infty)$ and $A_{\omega}(\infty)$ show an
infrared divergence at the critical point of forming the bound state.
Therefore, the integral in Eq. (10) is dominated by the infrared-frequency
regime. Using $\lim_{\omega\rightarrow 0}\bar{{F}}_{T}(\omega)=T^{-2}$ at the
critical point, we readily convert Eq. (10) into

$\mathcal{F}_{T}(\infty)|_{\text{CP}}\leq T^{-2}$ (11)

at the critical point of forming the bound state (see Appendix C). This
scaling relation works well in the full-temperature regime. Such quantum-
criticality-enhanced QFI succeeds in avoiding the problem that the QFI tends
to zero in the low-temperature regime in conventional quantum thermometry
schemes [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30,
31, 32, 13]. Being independent of $s$, our low-temperature scaling relation
surpasses the result $\mathcal{F}_{T}\propto T^{s-1}$ obtained in the spin-
boson model [25]. Note that our quantum-criticality-enhanced thermometry is
substantially different from that in Ref. [73], where a similar scaling is
obtained for the Caldeira-Leggett model without quantum criticality under the
condition of $\omega_{0}$ equal to zero.

Figure 2: Energy spectrum in the single-excitation space of the total system
(a), $|u(t)|^{2}$ (b), and $v(t)$ (c) as a function of $\omega_{c}$. (d) Heat-
exchange spectrum $A_{\omega}(\infty)$ for different $\omega_{c}$, which shows
an infrared divergence at the critical point $\omega_{0}=\eta\omega_{c}$. (e)
Steady-state QFI for different $T$ and $\omega_{c}$. The red solid line is
obtained via the analytical function $[1-Z^{2}M(\infty)]T^{-2}$. Other
parameters are the same as Fig. 1.

##  III Numerical calculations

Taking the Ohmic spectral density as an example, we plot in Fig. 1 the non-
Markovian evolution of $\mathcal{F}_{T}(t)$ for different cutoff frequencies
$\omega_{c}$. It can be seen that $\mathcal{F}_{T}(t)$ gradually increases
with time from zero to $\omega_{c}$-dependent stable values, which are larger
than the Markovian approximate one $\bar{F}_{T}(\omega_{0})$ in the full-
parameter regime. It indicates one of the advantages of our non-Markovian
quantum thermometer over the Markovian approximate ones. The stable
$\mathcal{F}_{T}(\infty)$ shows good matching with the analytical form in Eq.
(10). It verifies the validity of Eq. (10) in characterizing the steady-state
performance of our quantum thermometer. Another interesting feature is that an
obvious maximum of the QFI is present at $\omega_{0}=\eta\omega_{c}$. To
uncover the physical reason, we plot in Fig. 2(a) the energy spectrum in the
single-excitation space of the total system of Eq. (1). We really see that an
isolated eigenenergy with the associated eigenstate called the bound state is
present in the band-gap regime when $\omega_{0}<\eta\omega_{c}$. Its presence
opens an extra band gap in the energy spectrum, which signifies a quantum
phase transition of the total system. Accompanying its formation, the long-
time $|u(t)|^{2}$ abruptly increases from zero to a finite value [see Fig.
2(b)]. The functions $v(t)$ and $A_{\omega}(\infty)$, respectively, show a
long-time and low-frequency divergence at the critical point of forming the
bound state [see Figs. 2(c) and 2(d)]. Because
$\bar{{F}}_{T}(\omega)(\bar{n}(\omega)+1)$ is a decreasing function of
$\omega$, the overlap integral in Eq. (10) is dominated by the low frequencies
and leads to the maximum at the critical point. All the results confirm that
the maximum of $\mathcal{F}_{T}(\infty)$ at $\omega_{0}=\eta\omega_{c}$ is
intrinsically rooted in the quantum criticality induced by the bound state.

Figure 3: Energy spectra for different $\eta$ when $s=1$ (a) and for different
$s$ when $\eta=0.1125$ (c). (b), (d) The corresponding steady-state QFI for
different $T$. $\omega_{c}=10\omega_{0}$ is used.

To further reveal the superiority of such quantum-criticality-enhanced
thermometry, we plot in Fig. 2(e) the steady-state QFI
$\mathcal{F}_{T}(\infty)$ for different $T$ and $\omega_{c}$. It clearly
demonstrates that, in the high-temperature regime, $\mathcal{F}_{T}(\infty)$
matches with our analytical result $[1-Z^{2}M(\infty)]T^{-2}$. At the critical
point of forming the bound state, the QFI scales with the temperature as Eq.
(10) in the full-temperature regime, which means that the performance of our
non-Markovian quantum thermometry becomes better and better with a decrease of
the temperature. A similar performance can be obtained by tuning the coupling
constant $\eta$ [see Figs. 3(a) and 3(b)]. This successfully solves the
problem of the conventional schemes where the QFI tends to zero in the low-
temperature regime. It is noted that the sensitivity (10) is achievable not
only exactly at the critical point, but also in a relatively wide parameter
regime near the critical point. Via the criticality-scaling analysis, we find
that the condition to achieve (10) can be relaxed to
$|\omega_{0}-\eta\omega_{c}|\leq 1.52K_{B}T$ (see Appendix D). Thus, the
parameter regime supporting (10) becomes wider with increasing temperature, as
confirmed by Figs. 2(e) and 3(b). Given the fact that we never experiment
exactly at zero temperature, it endows our scheme with a fault tolerance to
the imprecise parameter tuning to reach the critical point. This parameter
regime also reveals that what really matters is the relative value
$\omega_{c}/\omega_{0}$. The critical regime is also achievable by tuning
$\omega_{0}$ for given $\omega_{c}$ and $\eta$.

##  IV Discussion and conclusions

Our result can be generalized to other forms of spectral density, where the
specific condition on the quantum criticality may be different, but the
conclusion remains unchanged [see Figs. 3(c) and 3(d)]. Many ways of
controlling the spectral density in circuit QED [57, 58, 59] and trapped ion
[60] platforms have been proposed. As the essential feature of our scheme, the
bound state and its dynamical effect have been observed in circuit QED [74]
and ultracold atom [75] systems. These progresses indicate that our scheme via
engineering the quantum criticality caused by the bound state is realizable in
the state-of-the-art technique of quantum optics experiments.

In summary, we propose a non-Markovian quantum thermometry scheme to measure
the equilibrium temperature of a quantum reservoir by use of a continuous-
variable system as a thermometer. A quantum criticality induced by the
formation of a thermometer-reservoir bound state and its crucial role in
enhancing the performance of the thermometer are revealed. It is revealed that
the scaling relation of the QFI in the long-time limit reaches the Landau
bound $T^{-2}$ in the full-temperature regime near the critical point. This
efficiently avoids the problem that the QFI of conventional quantum
thermometry schemes tends to zero in the low-temperature regime. Supplying a
mechanism for designing high-precision sensors to low temperature, our result
may be used in developing temperature-monitoring components in quantum
thermodynamics and quantum device fabrication.

## Acknowledgments

The work is supported by the National Natural Science Foundation (Grants No.
11875150, No. 11834005, and No. 12047501).

##  Appendix A Time-dependent state of the thermometer

Here, we give the derivation details of Eq. (5). According to the path-
integral influence-function theory, the evolution of the thermometer state in
the coherent-state representation [65] reads

$\displaystyle\rho(\bar{\alpha}_{f},\alpha_{f}^{\prime};t)$ $\displaystyle=$
$\displaystyle\int
d\mu(\alpha_{i})d\mu(\alpha_{i}^{\prime})\mathcal{J}(\bar{\alpha}_{f},\alpha_{f}^{\prime};t|\bar{\alpha}_{i}^{\prime},\alpha_{i};0)$
(12) $\displaystyle\times\rho(\bar{\alpha}_{i},\alpha_{i}^{\prime};0),$

where
$\rho(\bar{\alpha}_{f},\alpha_{f}^{\prime};t)\equiv\langle\bar{\alpha}_{f}|\rho\left(t\right)|\alpha_{f}^{\prime}\rangle$
and
$\rho(\bar{\alpha}_{i},\alpha_{i}^{\prime};0)=\langle\bar{\alpha}_{i}|\rho(0)|\alpha_{i}^{\prime}\rangle$,
with $|\alpha\rangle=e^{\alpha\hat{a}^{\dagger}}|0\rangle$ and
$d\mu(\alpha)=e^{-|\alpha|^{2}}d^{2}\alpha/\pi$. After eliminating the degrees
of freedom of the reservoir, the propagation functional
$\mathcal{J}(\bar{\alpha}_{f},\alpha_{f}^{\prime};t|\bar{\alpha}_{i}^{\prime},\alpha_{i};0)$
reads

$\displaystyle\mathcal{J}(\bar{\alpha}_{f},\alpha_{f}^{\prime};t|\bar{\alpha}_{i}^{\prime},\alpha_{i};0)=M(t)\exp\\{J_{1}(t)\bar{\alpha}_{f}\alpha_{i}$
$\displaystyle~{}~{}~{}+J_{2}\left(t\right)\bar{\alpha}_{f}\alpha_{f}^{\prime}-[J_{3}\left(t\right)-1]\bar{\alpha}_{i}^{\prime}\alpha_{i}+J_{1}^{\ast}(t)\bar{\alpha}_{i}^{\prime}\alpha_{f}^{\prime}\\},~{}~{}$
(13)

where

$\displaystyle M(t)$ $\displaystyle=$
$\displaystyle[1+v(t)]^{-1},~{}J_{1}(t)=M(t)u(t),$ (14) $\displaystyle
J_{2}(t)$ $\displaystyle=$ $\displaystyle
M(t)v(t),~{}J_{3}(t)=M(t)|u(t)|^{2}.$ (15)

The equations of motion of $u(t)$ and $v(t)$ take the forms of Eqs. (3) and
(4), respectively.

The thermometer is initially in a coherent state

$\rho(\bar{\alpha}_{i},\alpha_{i}^{\prime};0)=\exp(-\left|\alpha_{0}\right|^{2}+\bar{\alpha}_{i}\alpha_{0}+\bar{\alpha}_{0}\alpha_{i}^{\prime}).$
(16)

Substituting (16) into Eq. (12) and performing the Gaussian integration, we
get

$\displaystyle\rho(\bar{\alpha}_{f},\alpha_{f}^{\prime};t)$ $\displaystyle=$
$\displaystyle
M(t)\exp[-J_{3}(t)\left|\alpha_{0}\right|^{2}+J_{1}(t)\bar{\alpha}_{f}\alpha_{0}$
(17)
$\displaystyle+J_{2}(t)\bar{\alpha}_{f}\alpha_{f}^{\prime}+J_{1}^{\ast}(t)\bar{\alpha}_{0}\alpha_{f}^{\prime}].$

Remembering that the density matrix is expressed as $\rho(t)=\int
d\mu(\alpha_{f})d\mu(\alpha_{f}^{\prime})\rho(\bar{\alpha}_{f},\alpha_{f}^{\prime};t)|\alpha_{f}\rangle\langle\bar{\alpha}_{f}^{\prime}|$,
we obtain

$\displaystyle\rho(t)$ $\displaystyle=$ $\displaystyle
M(t)\exp[-J_{3}(t)\left|\alpha_{0}\right|^{2}]\exp\left[J_{1}(t)\alpha_{0}a^{{\dagger}}\right]$
(18) $\displaystyle\times\exp\left[\ln
J_{2}(t)a^{{\dagger}}a\right]\exp\left[J_{1}^{\ast}(t)\alpha_{0}^{\ast}a\right].$
$\displaystyle=$ $\displaystyle
M(t)\hat{\mathcal{D}}_{t}e^{\hat{a}^{{\dagger}}\hat{a}\ln
J_{2}(t)}\hat{\mathcal{D}}^{{\dagger}}_{t},$

where
$\hat{\mathcal{D}}_{t}=\exp[\alpha_{0}u(t)\hat{a}^{\dagger}-\alpha_{0}^{*}u^{*}(t)\hat{a}]$.
The last equality of Eq. (18) can be proven as follows.

Proof. Temporally neglecting the arguments “$(t)$” of each time-dependent
functions for brevity, we have

$\displaystyle\hat{\mathcal{D}}_{t}e^{\hat{a}^{{\dagger}}\hat{a}\ln
J_{2}}\hat{\mathcal{D}}^{{\dagger}}_{t}$ $\displaystyle=$
$\displaystyle\exp[(a^{{\dagger}}-u^{\ast}\alpha_{0}^{\ast})(a-u\alpha_{0})\ln
J_{2}]$ (19) $\displaystyle=$ $\displaystyle
J_{2}^{\left|u\alpha_{0}\right|^{2}}\exp[-\ln
J_{2}u\alpha_{0}\hat{a}^{{\dagger}}+\ln J_{2}\hat{a}^{{\dagger}}\hat{a}$
$\displaystyle-\ln J_{2}u^{\ast}\alpha_{0}^{\ast}\hat{a}].$

Then according to the rule of disentangling an exponential of operators of a
harmonic oscillator algebra, i.e.,
$e^{(\beta_{+}\hat{a}^{{\dagger}}+\beta_{0}\hat{a}^{{\dagger}}\hat{a}+\beta_{-}\hat{a})}=e^{f_{+}\hat{a}^{{\dagger}}}e^{f_{0}\hat{a}^{{\dagger}}\hat{a}}e^{f_{-}\hat{a}}e^{g}$,
with $f_{\pm}={\beta_{\pm}(e^{\beta_{0}}-1)}/{\beta_{0}}$, $f_{0}=\beta_{0}$,
and $g={\beta_{+}\beta_{-}(e^{\beta_{0}}-1-\beta_{0})}/{\beta_{0}^{2}}$, we
readily obtain

$\displaystyle\hat{\mathcal{D}}_{t}e^{\hat{a}^{{\dagger}}\hat{a}\ln
J_{2}}\hat{\mathcal{D}}^{{\dagger}}_{t}$ $\displaystyle=$ $\displaystyle
e^{-\left|u\alpha_{0}\right|^{2}(1-J_{2})}e^{(1-J_{2})u\alpha_{0}\hat{a}^{{\dagger}}}e^{\ln
J_{2}\hat{a}^{{\dagger}}\hat{a}}$ (20) $\displaystyle\times
e^{(1-J_{2})u^{\ast}\alpha_{0}^{\ast}\hat{a}}$ $\displaystyle=$ $\displaystyle
e^{-J_{3}\left|\alpha_{0}\right|^{2}}e^{J_{1}\alpha_{0}\hat{a}^{{\dagger}}}e^{\ln
J_{2}\hat{a}^{{\dagger}}\hat{a}}e^{J_{1}^{*}\alpha_{0}^{\ast}\hat{a}},$

which just the first equality of Eq. (19).

After inserting the complete basis $\sum_{n}|n\rangle\langle n|=1$ into Eq.
(19), we have

$\displaystyle\rho(t)$ $\displaystyle=$ $\displaystyle
M(t)\hat{\mathcal{D}}_{t}\sum_{n}e^{n\ln J_{2}(t)}|n\rangle\langle
n|\hat{\mathcal{D}}^{{\dagger}}_{t}$ (21) $\displaystyle=$
$\displaystyle\hat{\mathcal{D}}_{t}\sum_{n}{v(t)^{n}\over[1+v(t)]^{n+1}}|n\rangle\langle
n|\hat{\mathcal{D}}^{{\dagger}}_{t}.$

##  Appendix B Born-Markovian limit of $u(t)$ and $v(t)$

Defining $u(t)=e^{-i\omega_{0}t}u^{\prime}(t)$, we rewrite Eq. (3) as

$\dot{u}^{\prime}(t)+\int_{0}^{t}d\tau\int_{0}^{\infty}d\omega
J(\omega)e^{-i(\omega-\omega_{0})(t-\tau)}u^{\prime}(\tau)=0.$ (22)

When the thermometer-reservoir coupling is weak and the time scale of the
reservoir correlation function is much smaller than that of the thermometer,
we can calculate the Born-Markovian approximate solution of Eq. (22) via
neglecting the memory effect [66], i.e., $u^{\prime}(\tau)\simeq
u^{\prime}(t)$, and extending the upper limit of the integral to infinity,
i.e. $\int_{0}^{t}d\tau\simeq\int_{0}^{\infty}d\tau$. Utilization of the
identity $\lim_{t\rightarrow\infty}\int_{0}^{t}d\tau
e^{-i(\omega-\omega_{0})(t-\tau)}=\pi\delta(\omega-\omega_{0})+i\mathcal{P}{1\over\omega_{0}-\omega}$,
with $\mathcal{P}$ being the Cauchy principal value, results in
$u^{\prime}_{\text{MA}}(t)=e^{-[\kappa+i\Delta(\omega_{0})]t}$, where
$\kappa=\pi J(\omega_{0})$ and
$\Delta(\omega_{0})=\mathcal{P}\int_{0}^{\infty}{J(\omega)\over\omega_{0}-\omega}d\omega$.
We thus have the Born-Markovian approximate solution of $u(t)$ as
$u_{\text{MA}}(t)=e^{-[\kappa+i(\omega_{0}+\Delta(\omega_{0}))]t}$.

Equation (4) is rewritten as

$v(t)=\int_{0}^{\infty}d\omega\bar{n}(\omega)A_{\omega}(t),$ (23)

where $A_{\omega}(t)=J(\omega)|\tilde{u}_{\omega}(t)|^{2}$ with
$\tilde{u}_{\omega}(t)=\int_{0}^{t}d\tau u(\tau)e^{i\omega\tau}$. The function
$A_{\omega}(t)$ is further recast into

$\int_{0}^{\infty}d\omega
A_{\omega}(t)=\int_{0}^{t}d\tau_{1}\int_{0}^{t}d\tau_{2}u^{*}(\tau_{2})\mu(\tau_{2}-\tau_{1})u(\tau_{1}).$
(24)

Its derivative with respect to time reads

$\displaystyle{d\over dt}\int_{0}^{\infty}d\omega A_{\omega}(t)$
$\displaystyle=$ $\displaystyle
u(t)\Big{[}\int_{0}^{t}d\tau_{2}u(\tau_{2})\mu(t-\tau_{2})\Big{]}^{*}$ (25)
$\displaystyle+u^{*}(t)\int_{0}^{t}d\tau_{1}\mu(t-\tau_{1})u(\tau_{1})$
$\displaystyle=$ $\displaystyle-{d|u(t)|^{2}\over dt},$

where Eq. (3) has been used in the last equality. Then integrating Eq. (25)
with time and using $\int_{0}^{\infty}d\omega A_{\omega}(0)=0$, we readily
obtain

$\int_{0}^{\infty}d\omega A_{\omega}(t)=1-|u(t)|^{2}.$ (26)

Under the Born-Markovian approximation, we safely replace $\bar{n}(\omega)$ in
Eq. (23) by $\bar{n}(\omega_{0})$. Then we have

$\displaystyle v_{\text{MA}}(t)$ $\displaystyle=$
$\displaystyle\bar{n}(\omega_{0})\int_{0}^{\infty}d\omega
A_{\omega}(t)=\bar{n}(\omega_{0})[1-|u_{\text{MA}}(t)|^{2}]$ (27)
$\displaystyle=$ $\displaystyle\bar{n}(\omega_{0})(1-e^{-2\kappa t}).$

##  Appendix C Derivation of the QFI

To calculate the long-time QFI, $v(t)$ and $A_{\omega}(t)$ in Eq. (23) should
be known. It can be proven that

$A_{\omega}(\infty)=J(\omega)|\tilde{u}_{\omega}(\infty)|^{2}=\Theta(\omega)+{Z^{2}J(\omega)\over(\omega-
E_{b})^{2}}.$ (28)

Proof. The exact solution of $u(t)$ is

$~{}u(t)=Ze^{-iE_{b}t}+\int_{0}^{\infty}dE\Theta(E)e^{-iEt},$ (29)

with $Z=[1+\int_{0}^{\infty}\frac{J(\omega)}{(E_{b}-\omega)^{2}}d\omega]^{-1}$
and $\Theta(E)=\frac{J(E)}{[E-\omega_{0}-\Delta(E)]^{2}+[\pi J(E)]^{2}}$. It
induces

$\displaystyle\tilde{u}_{\omega}(\infty)$ $\displaystyle=$ $\displaystyle\pi
Z\delta(\omega-E_{b})+i\mathcal{P}{Z\over\omega-E_{b}}$ (30)
$\displaystyle+\pi\Theta(\omega)+i\mathcal{P}\int_{0}^{\infty}dE{\Theta(E)\over\omega-E},$

where $\lim_{t\rightarrow\infty}\int_{0}^{t}d\tau
e^{-i(E-\omega)\tau}=\pi\delta(\omega-E)+i\mathcal{P}{1\over\omega-E}$ has
been used. Rewriting $\Theta(x)={i\over\pi}\text{Im}[{1\over
E-\omega_{0}-\Delta(x)+i\pi J(x)}]$ with $x=\omega$ and $E$ and using the
Hilbert transform [76]
$\text{Re}[h(x)]={1\over\pi}\mathcal{P}\int_{-\infty}^{+\infty}d\omega{\text{Im}[h(x)]\over\omega-x}$
for an analytical function $h(x)$, we recast the last two terms of Eq. (30) as

$\pi\Theta(\omega)+i\mathcal{P}\int_{0}^{\infty}dE{\Theta(E)\over\omega-E}={1\over
E-\omega_{0}-\Delta(\omega)+i\pi J(\omega)}.$ (31)

It is noted that, in deriving Eq. (31), we have extended the lower limit of
the integral of $E$ from zero to $-\infty$ based on the fact that $J(E)=0$ in
the regime $E\in(-\infty,0)$. Because of $E_{b}<0$, the first term of Eq. (30)
has no contribution to the integral in Eq. (23) and the Cauchy principal value
in the second term of Eq. (30) is equal to the function itself. Therefore,
using Eq. (31), we can safely convert Eq. (30) into

$\tilde{u}_{\omega}(\infty)={iZ\over\omega-E_{b}}+{1\over
E-\omega_{0}-\Delta(\omega)+i\pi J(\omega)},$ (32)

which results in

$\displaystyle|\tilde{u}_{\omega}(\infty)|^{2}$ $\displaystyle=$
$\displaystyle{1\over[\omega-\omega_{0}-\Delta(\omega)]^{2}+[\pi
J(\omega)]^{2}}+{Z^{2}\over(\omega-E_{b})^{2}}$ (33) $\displaystyle+{\pi
Z\Theta(\omega)\over\omega-E_{b}}.~{}~{}~{}~{}~{}~{}~{}~{}$

The last term also has no contribution to the integral in Eq. (23) because,
according to the residue theorem, the residue of its only pole $\omega=E_{b}$
is equal to zero due to $J(E_{b})=0$. We thus finally arrive at Eq. (28).

We are now ready to prove Eq. (11).

Proof. The QFI is calculated via
$\mathcal{F}_{T}(t)={M(t)[\partial_{T}v(t)]^{2}/v(t)}$ as

$\displaystyle\mathcal{F}_{T}(t)$ $\displaystyle=$ $\displaystyle{M(t)\over
T^{2}v(t)}\Big{[}\int_{0}^{\infty}d\omega\sqrt{A_{\omega}(t)}\beta\omega\sqrt{\bar{n}(\omega)}[1+\bar{n}(\omega)]$
(34) $\displaystyle\times\sqrt{\bar{n}(\omega)A_{\omega}(t)}\Big{]}^{2}.$

Using Cauchy-Schwarz inequality $[\int_{0}^{\infty}d\omega
x(\omega)y(\omega)]^{2}\leqslant\int_{0}^{\infty}d\omega
x(\omega)^{2}\int_{0}^{\infty}d\omega y(\omega)^{2}$ [76], we obtain

$\displaystyle\mathcal{F}_{T}(t)$ $\displaystyle\leqslant$
$\displaystyle{M(t)\over T^{2}}\int_{0}^{\infty}d\omega
A_{\omega}(t)(\beta\omega)^{2}\bar{n}(\omega)[1+\bar{n}(\omega)]^{2}$ (35)
$\displaystyle=$ $\displaystyle
M(t)\int_{0}^{\infty}d\omega\bar{{F}}_{T}(\omega)A_{\omega}(t)[1+\bar{n}(\omega)],$

where
$\bar{{F}}_{T}(\omega)=(\beta\omega)^{2}\bar{n}(\omega)[1+\bar{n}(\omega)]/T^{2}$.
The long-time QFI reads

$\mathcal{F}_{T}(\infty)\leqslant{\int_{0}^{\infty}d\omega\bar{{F}}_{T}(\omega)A_{\omega}(\infty)[1+\bar{n}(\omega)]\over
1+v(\infty)}.$ (36)

At the critical point of forming the bound state,
$\omega_{0}=\eta\omega_{c}\gamma(s)$, $E_{b}=0$, $Z=0$, and
$\Delta(0)=-\omega_{0}$ from Eq. (9). We thus have

$A_{\omega}(\infty)\big{|}_{\text{CP}}=\Theta(\omega)=\frac{J(\omega)}{[\omega-\omega_{0}-\Delta(\omega)]^{2}+[\pi
J(\omega)]^{2}},$ (37)

which has a sharp peak at $\omega=0$ due to $\lim_{\omega\rightarrow
0}A_{\omega}(\infty)\big{|}_{\text{CP}}=\lim_{\omega\rightarrow
0}1/[\pi^{2}J(\omega)]=\infty$. Therefore, the integral in Eq. (36) is
dominated by $\omega=0$. According to $\lim_{\omega\rightarrow
0}\bar{{F}}_{T}(\omega)=T^{-2}$, we have

$\displaystyle\mathcal{F}_{T}(\infty)\Big{|}_{\text{CP}}$
$\displaystyle\leqslant$ $\displaystyle T^{-2}{\int_{0}^{\infty}d\omega
A_{\omega}(\infty)[1+\bar{n}(\omega)]\over 1+v(\infty)}$ (38) $\displaystyle=$
$\displaystyle T^{-2}{1-|u(\infty)|^{2}+v(\infty)\over
1+v(\infty)}=T^{-2},~{}~{}$

where Eqs. (23) and (26), and $|u(\infty)|^{2}=Z^{2}=0$ at the critical point
have been used.

Figure 4: (a) $A_{\omega}(\infty)$ for different $|\omega_{0}-\eta\omega_{c}|$
for the Ohmic spectral density. (b) Numerical fitting to the frequency
$\omega_{\text{max}}$ of the peak of $A_{\omega}(\infty)$ with respect to
$|\omega_{0}-\eta\omega_{c}|$.

##  Appendix D QFI near the critical point

The result $\mathcal{F}_{T}(\infty)\propto T^{-2}$ is also achievable when the
parameters are near but not exactly at the critical point. Here, we give the
proof of this conclusion. The function $A_{\omega}(\infty)$ shows a sharp peak
at $\omega=0$ when the parameters are exactly at the critical point. When the
parameters slightly deviate from the critical point, the sharp peak exhibits a
blue shift to $\omega_{\text{max}}$ [see Fig. 2(d)]. This implies that we can
approximately equate $A_{\omega}(\infty)$ in Eq. (36) as
$A_{\omega}(\infty)\simeq\delta(\omega-\omega_{\text{max}})$. Then we
calculate from Eq. (36) that

$\mathcal{F}_{T}(\infty)\leqslant\bar{{F}}_{T}(\omega_{\text{max}})\equiv{f(\beta\omega_{\text{max}})\over
T^{2}},$ (39)

where $f(x)=x^{2}\bar{n}(x)[1+\bar{n}(x)]$ and Eq. (23) have been used.
According to the fact that $f(x)$ is a monotonically decreasing function with
an increase of $x$, a QFI larger than $f(1)/T^{2}=0.92/T^{2}$ is obtained as
long as

$\beta\omega_{\text{max}}\leqslant 1.$ (40)

We now numerically explore the scaling law of $\omega_{\text{max}}$ near the
critical point. Figure 4(a) shows $A_{\omega}(\infty)$ for different
$|\omega_{0}-\eta\omega_{c}|$ near the critical point
$\big{|}\omega_{0}-\eta\omega_{c}\big{|}_{\text{CP}}=0$ for the Ohmic spectral
density. The numerical fitting in Fig. 4(b) reveals that the position of the
peak of $A_{\omega}(\infty)$ scales with $|\omega_{0}-\eta\omega_{c}|$ as

$\omega_{\text{max}}=0.66\omega_{0}|(\omega_{0}-\eta\omega_{c})/\omega_{0}|^{1.04}\simeq
0.66|\omega_{0}-\eta\omega_{c}|.$ (41)

Substituting Eq. (41) into Eq. (40), we conclude that the QFI takes as
$\mathcal{F}_{T}(\infty)\leqslant f(1)/T^{2}=0.92/T^{2}$ as long as the
parameters fall in a relatively wide parameter regime near the critical point,
i.e.,

$|\omega_{0}-\eta\omega_{c}|\leqslant 1.52K_{B}T.$ (42)

Equation (42) indicates that $\mathcal{F}_{T}(\infty)\leqslant 0.92/T^{2}$ is
achievable only at single parameter point $\omega_{0}=\eta\omega_{c}$ when and
only when the temperature is absolute zero. The parameter regime supporting
$\mathcal{F}_{T}(\infty)\leqslant 0.92/T^{2}$ becomes wider and wider with
increasing temperature. Given the fact that we never experiment at absolute
zero temperature, this wide parameter regime endows our scheme with a fault
tolerance to the imprecise tuning of the system parameters to reach the
critical point.

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