# Thermometry with a Dissipative Heavy Impurity

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

Improving the measurement precision of low temperature is significant in
fundamental science and advanced quantum technology application. However, the
measurement precision of temperature $T$ usually diverges as $T$ tends to 0.
Here, by utilizing a heavy impurity to measure the temperature of a Bose gas,
we obtain the Landau bound to precision $\delta^{2}T\propto T^{2}$ to avoid
the divergence. Moreover, when the initial momentum of the heavy impurity is
fixed and non-zero, the measurement precision can be $\delta^{2}T\propto
T^{3}$ to break the Landau bound. We derive the momentum distribution of the
heavy impurity at any moment and obtain the optimal measurement precision of
the temperature by calculating the Fisher information. As a result, we find
that enhancing the expectation value of the initial momentum can help to
improve the measurement precision. In addition, the momentum measurement is
the optimal measurement of the temperature in the case of that the initial
momentum is fixed and not equal to 0. The kinetic energy measurement is the
optimal measurement in the case of that the expectation value of the initial
momentum is 0. Finally, we obtain that the temperatures of two Bose gases can
be measured simultaneously. The simultaneous measurement precision is
proportional to $T^{2}$ when two temperatures are close to $T$.

##  I Introduction

With the rapid development of quantum technology, precise measurement of low
temperature is becoming an important and significant subject in the field of
quantum metrologylab1 ; lab2 ; lab3 ; lab4 ; lab5 ; lab6 ; lab7  and quantum
thermodynamicslab8 ; lab9 ; lab10 . The measurement of low temperature has
always been a challenging task due to that the uncertainty of temperature
diverges as the temperature tends to zerolab11 ; lab12 . A strong
couplinglab13 , periodic drivinglab14  and correlations among multiple probes
induced by the common bathlab15  have been used to slow down the divergence.
In order to truly avoid the divergence, quantum system with a non-vanishing
gaplab16 , a non-Markovian reservoirlab17 , and invariant subspaces due to the
polariton thermalizationlab18  have been utilized to obtain the Landau bound
to precision $\delta^{2}T\propto T^{2}$. In these cases, the measurement
uncertainty is not only not divergent but also getting smaller and smaller as
the temperature decreases to 0.

With the progress of experimental techniques in quantum gases, the high-
resolution imaging of distinguishable impurities has been realizedlab19 ;
lab20 ; lab21 . The strength of the impurity-environment coupling can be tuned
to obtain many significant physical results. The motion of mobile impurities
through one-dimensional quantum liquid or gas has been extensively
studiedlab22 ; lab23 ; lab24 ; lab25 ; lab26 . A heavy impurity moving through
a Luttinger liquidlab27  was explored to find that the friction force
experienced by the impurity behaves as the fourth power of temperature
($T^{4}$). By controlling the system parameter, the friction force can
dramatically change its temperature dependence from $T^{4}$ to $T^{8}$ in Bose
liquidlab28 ; lab29 . In recent worklab30 , the low-temperature $T^{2}$
dependence of the friction force has been obtained for a strong coupling
between a heavy impurity and a Bose gas, which is contrasted with the expected
$T^{4}$ scaling for a weak coupling. In order to obtain the friction force
from the Bose gas, the temperature of the Bose gas should be measured
accurately in advance.

In this article, we use a heavy impurity to measure the temperature of a Bose
gas. Based on that the relation between the friction force and the momentum is
linear, we analytically derive the momentum distribution of the heavy impurity
at any time given by the initial Gaussian distribution. The optimal
temperature measurement precision can be analytically derived by achieving the
Fisher information from the momentum distribution of the heavy impurity. As a
result, we show that the Landau bound to precision $\delta^{2}T\propto T^{2}$
can be obtained in the general case. More importantly, when the initial
momentum of the heavy impurity is fixed and non-zero, the measurement
precision can be $\delta^{2}T\propto T^{3}$ to break the Landau bound. And we
find that enhancing the expectation value of the initial momentum can help to
improve the measurement precision. In the case of that the initial momentum is
fixed and not equal to 0, we show that the momentum measurement is the optimal
measurement of the low temperature. When the initial momentum is not fixed and
the expectation value is not equal to 0, the measurement precision
proportional to $T^{2}$ can still be obtained by using the momentum
measurement. The the kinetic energy measurement just happens to be the optimal
measurement in the case of that the expectation value of the initial momentum
is 0. Finally, we show that the simultaneous measurement precision is
proportional to $T^{2}$ when the temperatures of two gases are close to $T$.

This article is organized as follows. In Section II, we introduce the model of
the heavy impurity in the Bose gas. In Section III, the momentum distribution
of the heavy impurity is derived given by the initial Gaussian distribution.
In Section IV, the optimal estimation precision is obtained by the Fisher
information. In Section V, two practical measurements are used to obtain the
temperature measurement precision. The simultaneous estimation of two
temperatures is discussed in Section VI. We make a conclusion and a discussion
on the feasibility of the experiment in Section VII.

##  II Heavy Impurity in a Bose Gas

We consider the system of a mobile heavy impurity in a Bose gas composed of
one-dimensional interacting bosons. The total Hamiltonian is described bylab31

$\displaystyle H_{T}=H_{B}-\int
dx\hat{\Psi}^{\dagger}\frac{\hbar^{2}}{2M}\partial_{x}^{2}\hat{\Psi}+G\int
dx\hat{\Psi}^{\dagger}\hat{\Psi}\hat{\psi}^{\dagger}\hat{\psi},$ (1)

where $H_{B}$ denotes the Lieb-Liniger Hamiltonian of the Bose gas

$\displaystyle H_{B}=-\int
dx\hat{\psi}^{\dagger}\frac{\hbar^{2}}{2m}\partial_{x}^{2}\hat{\psi}+g\int
dx\hat{\psi}^{\dagger}\hat{\psi}^{\dagger}\hat{\psi}\hat{\psi},$ (2)

$\hat{\Phi}(x)=\Psi(x)\ [\psi(x)]$ denotes the bosonic field operator for the
heavy impurity with the mass $M$ [ the single boson with the mass $m$], which
satisfies the commutation relations
$[\hat{\Phi}^{\dagger}(x),\hat{\Phi}^{\dagger}(x^{\prime})]=\delta(x-x^{\prime})$
and $[\hat{\Phi}(x),\hat{\Phi}(x^{\prime})]=0$ with $\delta(x-x^{\prime})$
being the Dirac delta function. In the Bose gas, the contact interaction
between bosons are repulsive and weak, i.e., $\hbar^{2}n_{0}/m\gg g>0$ with
$n_{0}$ being the mean density of the bosons. In the case of the weak
interaction between bosons, the quasi-particles of the Hamiltonian $H_{B}$
have the Bogoliubov dispersion relationlab32
$\epsilon_{P}=\sqrt{v^{2}p^{2}+p^{4}/4m^{2}}$ with the sound velocity
$v=\sqrt{gn_{0}/m}$. And $G$ denotes the density-density interaction strength
between the heavy impurity and the Bose gas.

##  III Distribution function of the impurity

Due to that the heavy impurity collides with thermally excited bosons in the
Bose gas, the heavy impurity motion is stochastic. The momentum distribution
function $f(t,P)$ of the heavy impurity ($M\gg m$) can be characterized by the
Fokker-Planck formlab33 ; lab34

$\displaystyle\frac{\partial f(t,P)}{\partial t}=\frac{\partial}{\partial
P}[-Ff(t,P)+\frac{1}{2}\frac{\partial Df(t,P)}{\partial P}],$ (3)

where $P$ represents the momentum of the heavy impurity. $F$ denotes the
friction force due to scattering off thermally excited quasiparticles in the
Bose gas, which is given by

$\displaystyle F=-\frac{m^{2}v^{2}P}{2\pi\hbar
M\tilde{T}}\int_{0}^{\infty}dk\frac{k^{2}|r(k,\tilde{G})|^{2}(2+k^{2})}{\sinh^{2}(k\frac{\sqrt{4+k^{2}}}{4\tilde{T}})\sqrt{4+k^{2}}},$
(4)

where $\tilde{T}=\frac{T}{mv^{2}}$, and $\tilde{G}=\frac{G}{\hbar v}$.
$r(k,\tilde{G})$ denotes the reflection amplitude of the Bogoliubov
quasiparticles scattering off a heavy impurity, which has been studied by the
Bogoliubov-de Gennes theorylab34a . At low temperatures, $\tilde{T}\ll 1$, in
the cases of weak coupling ($\tilde{G}\ll 1$) and strong coupling
$\tilde{G}\gg 1$ the friction force $F$ can be simplified aslab30

$\displaystyle F=-\Gamma({G})PT^{n}.$ (5)

For $1/\tilde{G}\ll\tilde{T}\ll 1$, $n=2$ and $\Gamma({G})=\frac{2\pi}{3\hbar
Mv^{2}}$. For $\tilde{T}\ll 1/\tilde{G}\ll 1$, $n=4$ and
$\Gamma({G})=\frac{8\pi^{3}G^{2}}{15\hbar^{3}m^{2}Mv^{8}}$. For $\tilde{G}\ll
1\ \textmd{and}\ \tilde{T}\ll 1$, $n=4$ and
$\Gamma({G})=\frac{2\pi^{3}G^{2}}{15\hbar^{3}m^{2}Mv^{8}}$. And $D$ is the
impurity diffusion coefficient, which is given by

$\displaystyle D=-2FMT/P.$ (6)

When the friction force $F$ is proportional to the momentum $P$ as shown in
Eq. (5), the Fokker-Planck form in Eq. (3) can be solved analytically. In
general, the initial momentum distribution $f(0,P)$ is Gaussian, which is
described by

$\displaystyle
f(0,P)=\frac{1}{\sqrt{\pi\Delta}}\exp[-\frac{(P-P_{0})^{2}}{\Delta}],$ (7)

where $\Delta/2$, $P_{0}$ represent the variance and the expectation value of
the initial momentum, respectively.

By analytically solving the Fokker-Planck equation in Eq. (3), we can obtain
the momentum distribution at time $t$ with the initial Gaussian momentum
distribution as shown in Eq. (7)

$\displaystyle f(t,P)$
$\displaystyle=\frac{1}{\sqrt{2MT(1-e^{-2t/\tau}(1-\Delta^{\prime}))}}$
$\displaystyle\times\exp[-\frac{(P-P_{0}e^{-t/\tau})^{2}}{2MT(1-e^{-2t/\tau}(1-\Delta^{\prime}))}],$
(8)

where $\Delta^{\prime}=\Delta/(2MT)$ and $\tau=1/(\Gamma T^{n})$ with the
abbreviation $\Gamma\equiv\Gamma(G)$ throughout the rest of this article.

##  IV Fisher Information from Gaussian Momentum Distribution

According to the Cramér-Rao boundlab35 , the measurement precision of the
temperature can be given by

$\displaystyle(\delta T)^{2}\geq\frac{1}{N\mathcal{F}[T]},$ (9)

where $N$ represents the total number of repeated experiments. Since the
content of our next study has nothing to do with the number of measurements,
we simply set $N=1$. $\mathcal{F}[T]$ denotes the Fisher information of the
temperature $T$, which is described by

$\displaystyle\mathcal{F}(T)=\int_{-\infty}^{\infty}dP\frac{[\partial_{T}f(t,P)]^{2}}{f(t,P)},$
(10)

where the partial derivative $\partial_{T}f(t,P)=\frac{\partial
f(t,P)}{\partial T}$.

When $\Delta$ is not an infinitesimal quantity, i.e., $\Delta>0$, and $t$ is
not infinite, we can obtain the Fisher information in the low temperature
limit $T^{n}t\Gamma\ll 1$

$\displaystyle\mathcal{F}(T,\Delta>0)=\frac{2(nt\Gamma)^{2}T^{2n-2}(P_{0}^{2}+\Delta)}{\Delta}.$
(11)

From the above equation, we can see that the Fisher information can be
increased by increasing the initial expectation values of momentum. When
$\tilde{T}\ll 1/\tilde{G}$, the coefficient $\Gamma$ increases by 4 times with
the coupling strength,i.e., $\Gamma(\tilde{G}\gg 1)=4\Gamma(\tilde{G}\ll 1)$.
Namely, the Fisher information increase 4 times with the coupling strength.
When the coupling strength continues to increase until $\tilde{G}\gg
1/\tilde{T}$, the power of temperature goes from $n=4$ to $n=2$. Hence, in the
case of $\tilde{G}\gg 1/\tilde{T}$, the Fisher information is proportional to
the square of the temperature ($\mathcal{F}(T)\propto T^{2}$). And it is
independent of the coupling strength, which shows that the measurement
precision of temperature will not be improved when the coupling strength
increases to a certain extent. As the temperature $T$ approaches to 0, the
Fisher information $\mathcal{F}(T)$ will approach to 0. It means that the
uncertainty of the temperature $T$ is divergent at $T=0$.

When $\Delta=0$ and $t$ is not infinite, the Fisher information in the low
temperature limit $T^{n}t\Gamma\ll 1$ is given by

$\displaystyle\mathcal{F}(T,\Delta=0)=\frac{(1+n)^{2}}{2T^{2}}+\frac{n^{2}P_{0}^{2}T^{n-3}t}{2M}.$
(12)

When $T$ is close to 0, the Fisher information,
$\mathcal{F}(T)\approx\frac{25}{2T^{2}}$, will be close to infinite. This
result is significant, which shows that the divergent question of the
uncertainty at $T=0$ is completely solved by using a defined initial momentum,
i.e., $\Delta=0$. In the case of a fixed initial momentum,
$f(0,P)=\delta(P-P_{0})$, the uncertainty of $T$ will be close to 0 as $T$ is
close to 0. A simple summary is that reducing the uncertainty of the initial
momentum distribution can effectively improve the measurement precision of the
temperature, especially when the uncertainty of the initial momentum
distribution is 0, the measurement precision of the temperature changes from
infinity to 0 at the temperature $T=0$.

Next, we consider that the measurement time $t$ can be arbitrarily large. By
optimizing the measurement time to obtain a better measurement precision, we
consider that the measurement time is the characteristic time of the
distribution function $f(t,P)$, i.e., $t=\tau=1/(\Gamma T^{m})$. When $T$ is
close to 0 and $\Delta>0$, the Fisher information is given by

$\displaystyle\mathcal{F}_{\tau}(\Delta>0)=\frac{2n^{2}(P_{0}^{2}+\Delta)}{T^{2}\Delta}.$
(13)

Contrary to the result in Eq. (5), we find that $\mathcal{F}_{\tau}(\Delta>0)$
becomes larger and larger as the temperature $T$ decreases. The essential
reason is that the measurement precision of temperature close to 0 is improved
by using the time resource tending to be infinite.

When $T$ is close to 0, $\Delta=0$ and $P_{0}\neq 0$, the Fisher information
is given by

$\displaystyle\mathcal{F}_{\tau}(\Delta=0)=\frac{n^{2}P_{0}^{2}}{MT^{3}(e^{2}-1)}.$
(14)

Figure 1: Evolution diagram of the ratio of the Fisher information at
different low temperatures. Here, $\gamma$ denotes the ratio of the Fisher
information $\mathcal{F}(T)$ at time $t$ and the Fisher information
$\mathcal{F}_{\tau}(\Delta=0)$ at the characteristic time $\tau$. The
dimensionless parameters chosen are given by: $P_{0}=1$, $M=1$, $n=4$, and
$\Gamma=1$.

As shown in Fig. 1, we can see that the maximum ratio
$\gamma=\mathcal{F}(T)/\mathcal{F}_{\tau}(\Delta=0)$ approaches to 1. It means
that $\mathcal{F}_{\tau}(\Delta=0)$ is close to the optimal Fisher
information. When $\Delta=0$, the characteristic time $\tau$ is close to the
optimal measurement time at low temperature. Comparing Eq. (12) and Eq. (14),
the Fisher information changes from being proportional to $1/T^{2}$ to
$1/T^{3}$ in the case of $P_{0}\neq 0$. This is also due to the use of time
resources that tend to be infinite.

When the initial momentum $P_{0}$ is 0, the Fisher information at the
characteristic time $\tau$ is given by

$\displaystyle\mathcal{F}_{\tau}(\Delta=0,P_{0}=0)=\frac{(2n+e^{2}-1)^{2}}{2T^{2}(e^{2}-1)^{2}}.$
(15)

This result shows that the initial non-zero momentum ($P_{0}$) can help to
effectively use the time resources, and thus improve the measurement precision
of the temperature.

##  V temperature measurement with momentum

In the previous section, we give the optimal measurement precision for a given
probability distribution by the Fisher information. Then we will use the error
propagation formula to show whether the specific measurement can approach the
optimal measurement result.

Firstly, we obtain the information of the temperature $T$ by the measurement
of the momentum $P$. Given by the momentum distribution in Eq. (2), we achieve
the expectation value $\bar{P}$ and the variance
$\delta^{2}P={\bar{P^{2}}}-{\bar{P}}^{2}$, which are described by

$\displaystyle{\bar{P}}$ $\displaystyle=P_{0}e^{-t/\tau},$ (16)
$\displaystyle\delta^{2}P$
$\displaystyle=MT[1-e^{-2t/\tau}(1-\Delta^{\prime})].$ (17)

The uncertainty of the temperature $T$ can be derived by the error propagation
formulalab36

$\displaystyle\delta^{2}T=\frac{\delta^{2}P}{|\partial_{T}{\bar{P}}|^{2}}=\frac{MT(1-e^{-2t/\tau}(1-\Delta^{\prime}))}{(P_{0}nt\Gamma)^{2}T^{2n-2}e^{-2t/\tau}}.$
(18)

From the above equation, we can derive that the optimal measurement time can
be close to the characteristic time $\tau$. At time $\tau$, the corresponding
measurement precision of the temperature $T$ obtained by the measurement of
momentum can be given by

$\displaystyle\delta^{2}T|_{t=\tau}=\frac{MT(1-e^{-2}(1-\Delta^{\prime}))}{(P_{0}n)^{2}T^{-2}e^{-2}}$
(19) $\displaystyle\approx\frac{(e^{2}-1)MT^{3}+T^{2}\Delta}{(P_{0}n)^{2}}$
(20)

When $P_{0}\neq 0$ and $\Delta=0$, we can find that
$\delta^{2}T|_{t=\tau}=1/\mathcal{F}_{\tau}(\Delta=0)$. It shows that the
momentum measurement is the optimal measurement of the low temperature in the
case of that the initial momentum is fixed ($\Delta=0$) and not equal to 0
($P_{0}\neq 0$). When the initial momentum is not fixed ($\Delta>0$) and the
expectation value is not equal to 0 ($P_{0}\neq 0$), the measurement precision
proportional to $T^{2}$ can still be obtained by using the momentum
measurement. This means that the momentum measurement is close to the optimal
measurement as $T$ tends to 0 in the case of $P_{0}\neq 0$ and $\Delta>0$.

However, when $P_{0}=0$, one can obtain that
$\delta^{2}T|_{t=\tau}\rightarrow\infty$. It shows that the measurement of
momentum is the worst measurement, which can not obtain any information of the
temperature $T$ in the case of $P_{0}=0$.

###  V.1 Temperature measurement with kinetic energy

In order to deal with the case of $P_{0}=0$, we try to use the kinetic energy
measurement $P^{2}/(2M)$. The expectation value and the variance of $P^{2}$
are given by

$\displaystyle{\bar{P^{2}}}$
$\displaystyle=MT[1-e^{-2t/\tau}(1-\Delta^{\prime})];$ (21)
$\displaystyle\delta^{2}P^{2}$
$\displaystyle=2M^{2}T^{2}[1-e^{-2t/\tau}(1-\Delta^{\prime})]^{2}.$ (22)

By substituting the above equations into the error propagation formula, the
measurement precision of the temperature is derived

$\displaystyle\delta^{2}T=\frac{T^{2}[2(e^{2t/\tau}-1)MT+\Delta]^{2}}{2[MT(e^{2t/\tau}-1)+nt(2MT-\Delta)/\tau]^{2}}.$
(23)

For $t\rightarrow\infty$, the measurement precision of the low temperature
obtained by the kinetic energy measurement is $\delta^{2}T=2T^{2}$, which is
consistent with the results obtained by the Fisher information in the previous
section. For $t=\tau$ and $\Delta=0$, the measurement precision of the low
temperature is $\delta^{2}T=\frac{2(e^{2}-1)^{2}T^{2}}{(e^{2}-1+2n)^{2}}$,
which is consistent with the result as shown in Eq. (15). For $\Delta>0$, the
measurement precision of the low temperature is
$\delta^{2}T=\frac{T^{2}}{2n^{2}}$, which is also consistent with the result
by the Fisher information in Eq. (13). Therefore, in the case of $P_{0}=0$,
the temperature measurement with the kinetic energy is the optimal
measurement.

##  VI simultaneous measurement of two temperatures

We consider that the heavy impurity passes through the first Bose gas with
temperature $T_{1}$ and the second Bose gas with temperature $T_{2}$. Given
the initial Gaussian momentum distribution of the Heavy impurity as shown in
Eq. (7), the momentum distribution at time $t_{1}+t_{2}$ is described by

$\displaystyle f(t_{1}+t_{2},P)=$
$\displaystyle\frac{1}{\sqrt{2MT_{2}[1-e^{-2t_{2}/\tau_{2}}(1-\frac{\Delta_{1}}{2MT_{2}})]}}\times$
$\displaystyle\exp[-\frac{(P-P_{0}e^{-t_{1}/\tau_{1}-t_{2}/\tau_{2}})^{2}}{2MT_{2}[1-e^{-2t_{2}/\tau_{2}}(1-\frac{\Delta_{1}}{2MT_{2}})]}],$
(24)

where $t_{1}$ ($t_{2}$) is the interaction time between the heavy impurity and
the first (second) Bose gas,
$\Delta_{1}=2MT_{1}[1-e^{-2t_{1}/\tau_{1}}(1-\frac{\Delta}{2MT_{1}})]$ and the
characteristic time of the $i$th Bose gas is
$\tau_{i}=1/(\Gamma_{i}T_{i}^{n})$ for $i=\\{1,2\\}$.

At time $t=t_{1}+t_{2}$, the two temperatures can be simultaneously measured
by the momentum distribution. The estimation precision of $(T_{1},T_{2})$ is
governed by its covariance matrix $\textmd{Cov}(T_{1},T_{2})$, which is lower
bounded via the multi-parameter Cramér-Rao boundlab37

$\displaystyle\textmd{Cov}(T_{1},T_{2})\geq\frac{1}{\mathbf{\chi}},$ (25)

where $\mathcal{\chi}$ is the Fisher information matrix, which is derived by

$\displaystyle\mathcal{\chi}_{jk}=\int_{-\infty}^{\infty}dP\frac{\partial_{T_{j}}f(t,P)\partial_{T_{k}}f(t,P)}{f(t,P)},$
(26)

where $j,k=\\{1,2\\}$. The total simultaneous uncertainty of the two
temperatures is achieved by

$\displaystyle(\delta^{2}T_{1}+\delta^{2}T_{2})|_{\textmd{sim}}=\textmd{tr}[\textmd{Cov}(T_{1},T_{2})]$
(27)
$\displaystyle\geq\frac{\mathcal{\chi}_{11}+\mathcal{\chi}_{22}}{\mathcal{\chi}_{11}\mathcal{\chi}_{22}-|\mathcal{\chi}_{12}|^{2}}$
(28)

The simultaneous uncertainty of the two temperatures can be analytically
derived by the above equation. Especially, when $\Delta=0$, $P_{0}\neq 0$,
$T_{1}=T_{2}=T$ and $t_{j}=\tau_{j}$, the simultaneous estimation precision of
the two low temperatures is given by

$\displaystyle(\delta^{2}T_{1}+\delta^{2}T_{2})|_{\textmd{sim}}\geq
6.89625T^{2}$ (29)

This result shows that the simultaneous estimation of the two temperatures can
also obtain very high measurement precision, which is proportional to $T^{2}$
when both temperatures are close to $T$. However, the scaling $T^{3}$ can not
be obtained like the measurement precision of a single temperature. This is
mainly due to the fact that the variance of the momentum distribution,
$\Delta_{1}/2$, after passing through the first Bose gas is not 0. Therefore,
by controlling the initial momentum distribution, the measurement precision
obtained by two separate temperature measurements is higher than that obtained
by the simultaneous two-temperature measurement.

##  VII conclusion

In this article, we have proposed that the temperature of the Bose gas can be
measured by the mobile heavy impurity. Due to the scattering off thermally
excited quasiparticles in the Bose gas, the momentum distribution of the heavy
impurity will carry the information about the temperature of the Bose gas.
Based on the Fisher information from the momentum distribution of the heavy
impurity, the optimal measurement precision can be analytically derived. Due
to that the momentum is continuous, the Landau bound to precision
$\delta^{2}T\propto T^{2}$ can be obtained in the general case. And we find
that enhancing the expectation value of the initial momentum can help to
improve the measurement precision. Reducing the uncertainty of the initial
momentum distribution can further improve the measurement precision of the
temperature. More importantly, when the initial momentum of the heavy impurity
is fixed and non-zero, we obtain the measurement precision beyond the Landau
bound to be $\delta^{2}T\propto T^{3}$. In addition, we show that the momentum
measurement is the optimal measurement when the initial momentum is fixed and
not equal to 0. The kinetic energy measurement just happens to be the optimal
measurement when the expectation value of the initial momentum is equal to 0.
Finally, we find that the simultaneous measurement precision of the
temperatures of the two Bose gases can also reach the Landau bound. However,
the scaling $T^{3}$ can not be obtained like the measurement precision of a
single temperature.

Our scheme can be performed in the setup with cold atoms such as, a one-
dimensional quantum liquid of 4He atoms confined within a porous materiallab38
, and an ultracold Rb gas with single neutral Cs impurity atomslab39 . The
interaction strength between the impurity and host atoms is tunable by the
Feshbach resonances and powerful measurement techniques have been
developedlab40 ; lab41 .

## Appendix

A. The solution of the Fokker-Planck form

The the Fokker-Planck form can be rewritten as

$\displaystyle\frac{\partial f(t,P)}{\partial
t}=-\frac{1}{\tau}\mathcal{H}f(t,P),$ (A1)

where $\tau=|P/F|=\Gamma T^{n}$ and the operator $\mathcal{H}$ is described by

$\displaystyle\mathcal{H}=-MT\frac{\partial^{2}}{\partial
P^{2}}-\frac{\partial}{\partial P}P.$ (A2)

The general form of the solution can de described as

$\displaystyle f(t,P)=e^{-\mathcal{H}t/\tau}f(0,P).$ (A3)

Defining $\tilde{y}=P/\sqrt{2MT}$ and $\tilde{p}=-i\frac{\partial}{\partial
P}$, the operator $\mathcal{H}$ is rewritten as

$\displaystyle\mathcal{H}=\frac{1}{2}(\tilde{p}-i\tilde{y})^{2}+\frac{1}{2}\tilde{y}^{2}-\frac{1}{2}.$
(A4)

Then, let us make a transformation to get an operator $\mathcal{\tilde{H}}$,
which is similar with the Hamiltonian of a harmonic oscillator

$\displaystyle\mathcal{\tilde{H}}=\exp(\tilde{y}^{2}/2)\mathcal{H}\exp(-\tilde{y}^{2}/2)=\frac{1}{2}\tilde{p}^{2}+\frac{1}{2}\tilde{y}^{2}-\frac{1}{2}.$
(A5)

The eigenvalues ($E_{n}$) and eigenvectors ($\phi_{n}(\tilde{y})$) of the
operator $\mathcal{\tilde{H}}$ can be given by

$\displaystyle
E_{n}=n,\phi_{n}(\tilde{y})=(-1)^{n}(\frac{1}{\sqrt{\pi}2^{n}n!})^{1/2}e^{y^{2}/2}\frac{d^{n}}{dy^{n}}e^{-y^{2}}.$
(A6)

Setting $f(t,P)=e^{-\tilde{y}^{2}/2}\tilde{f}(t,P)$, we can obtain that

$\displaystyle\frac{\partial\tilde{f}(t,\tilde{y})}{\partial
t}=-\frac{1}{\tau}\mathcal{\tilde{H}}\tilde{f}(t,\tilde{y}).$ (A7)

The solution of above equation can be achieved by

$\displaystyle\tilde{f}(t,\tilde{y})=e^{-\mathcal{\tilde{H}}t/\tau}\tilde{f}(0,\tilde{y})$
(A8)
$\displaystyle=\sum_{n}e^{-\mathcal{\tilde{H}}t/\tau}A_{n}\phi(\tilde{y})$
(A9) $\displaystyle=\sum_{n}e^{-nt/\tau}A_{n}\phi(\tilde{y}),$ (A10)

where the coefficient $A_{n}$ is derived by

$\displaystyle
A_{n}=\int_{-\infty}^{\infty}\tilde{f}(0,P)\phi^{*}(\tilde{y})dy.$ (A11)

We consider that the initial momentum distribution is Gaussian, which is
described by

$\displaystyle
f(0,P)=\frac{1}{\sqrt{\pi\Delta}}\exp[-\frac{(P-P_{0})^{2}}{\Delta}].$ (A12)

The corresponding momentum distribution $f(0,\tilde{y})$ is given by

$\displaystyle
f(0,\tilde{y})=\frac{1}{\sqrt{\pi\Delta^{\prime}}}\exp[-\frac{(\tilde{y}-{P^{\prime}_{0}})^{2}}{\Delta^{\prime}}],$
(A13)

where $\Delta^{\prime}=\frac{\Delta}{2MT}$ and
$P^{\prime}_{0}=P_{0}/\sqrt{2MT}$. Next, the initial momentum distribution
after the transformation is given by

$\displaystyle\tilde{f}(0,\tilde{y})=\frac{1}{\sqrt{\pi\Delta^{\prime}}}\exp[-\frac{(\tilde{y}-{P^{\prime}_{0}})^{2}}{\Delta^{\prime}}+\frac{\tilde{y}^{2}}{2}].$
(A14)

Expanding it with the eigenvectors of the harmonic oscillator operator
$\mathcal{\tilde{H}}$, the expansion coefficient can be obtained

$\displaystyle A_{n}$
$\displaystyle=\int_{-\infty}^{\infty}\frac{1}{\sqrt{\pi\Delta^{\prime}}}\exp[-\frac{(\tilde{y}-{P^{\prime}_{0}})^{2}}{\Delta^{\prime}}+\frac{\tilde{y}^{2}}{2}]\phi^{*}(\tilde{y})dy$
(A15)
$\displaystyle=\int_{-\infty}^{\infty}d\omega\frac{(i\omega)^{n}\exp{[-\frac{\omega^{2}}{4}-\frac{\Delta^{\prime}\omega^{2}+4iP^{\prime}_{0}\omega-4{P^{\prime}_{0}}^{2}}{4(1-\Delta^{\prime})}]}}{\sqrt{4\pi\sqrt{\pi}2^{n}n!(1-\Delta^{\prime})}},$
(A16)

where we have utilized the following formulas

$\displaystyle e^{-{\tilde{y}^{2}}}$
$\displaystyle=\frac{1}{2\sqrt{\pi}}\int_{-\infty}^{\infty}d\omega
e^{i\omega\tilde{y}-\omega^{2}/4}$ (A17)
$\displaystyle\frac{d^{n}}{d\tilde{y}^{n}}(e^{-{\tilde{y}^{2}}})$
$\displaystyle=\frac{1}{2\sqrt{\pi}}\int_{-\infty}^{\infty}d\omega(i\omega)^{n}e^{i\omega\tilde{y}-\omega^{2}/4}.$
(A18)

Substituting the above equations into Eq. (A10) and making an inverse
transformation, we can obtain the solution of the Fokker-Planck formula in Eq.
(A1)

$\displaystyle f(t,P)=$
$\displaystyle\frac{1}{\sqrt{2MT(1-e^{-2t/\tau}(1-\Delta^{\prime}))}}$
$\displaystyle\exp[-\frac{(P-P_{0}e^{-t/\tau})^{2}}{2MT(1-e^{-2t/\tau}(1-\Delta^{\prime}))}].$
(A19)

B. General form of Fisher information

For the general initial Gaussian momentum distribution, the general form of
the Fisher information can be analytically achieved by substituting Eq. (8)
into Eq. (10)

$\displaystyle\mathcal{F}(T)=$ $\displaystyle
2\Gamma^{2}[T(2RMT+\Delta)]^{-2}\\{n^{2}t^{2}T^{2n}[(P_{0}^{2}-4MT)\Delta$
$\displaystyle+2MT(RP_{0}^{2}+2MT)+\Delta^{2}]+(RMT/\Gamma)^{2}$
$\displaystyle+2nMRtT^{n+1}(2MT-\Delta)/\Gamma\\},$ (A20)

where $R=e^{2t\Gamma T^{n}}-1$.

## Acknowledgements

This research was supported by the National Natural Science Foundation of
China (Grant No. 12365001 and No. 62001134), Guangxi Natural Science
Foundation ( Grant No. 2020GXNSFAA159047), and the Project of Improving the
Basic Scientific Research Ability of Young and Middle-aged Teachers in
Universities of Guangxi (Grant No. 2023KY0815).

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