# Measure the temperature of the spin via weak measurement

Yusuf Turek<EMAIL_ADDRESS>School of Physics and Electronic
Engineering, Xinjiang Normal University, Urumqi, Xinjiang 830054, China

###### Abstract

In this study, we give a new proposal to measure the temperature of the spin
in a sample in magnetic resonance force microscopy system by using
postselected weak measurement, and investigate the fisher information to
estimate the precision of our scheme. We showed that in high temperature
regime the temperature of the spin can be measured via weak measurement
technique with proper postselection and our scheme able to increase the
precision of temperature estimation.

###### pacs:

03.65.Ta, 42.50.-p,06.20.Dk,05.70.-a

##  I Introduction

Measurement is a basic concept in physics and any information of the system
can be obtained from the measurement. Unlike the classical measurement, in
most of the quantum measurement processes the information of the system only
can be obtained or estimated by using indirect measurements. Using quantum
measurement terminology, in a quantum measurement process, there have an
interaction between measuring device (probe) and measured system, and for
guarantee the measurement precision the interaction time must be very short so
that the system and the probe itself doesn’t effect the results of
measurements. In mathematics, these requirements of quantum measurement can be
expressed by von Newmann Hamiltonian $H=g\hat{A}\otimes\hat{P}$Neumann . Here,
$\hat{A}$ is the operator of measured system we want to measure, $\hat{P}$ the
canonical momentum of measuring device, and $g$ represent the measuring
strength. If the interaction strength between the system and the probe is
strong, i.e. $g\gg 1$, we can get the information of the system by single
trial with very small error, the Stern-Gerlah experiment is a typical example.
In contrast, if the interaction strength between probe and system is very
small, i.e. $g\ll 1$, there have still interference between different
eigenvalues of the system observable we want to measure, and after single
trial we can not obtain the information of the system precisely.

However, the weak measurement, as a generalized von Neumann quantum
measurement theory, was proposed by Aharonov, Albert, and Vaidman in 1988
opens a new route to measure the system information in weak coupling
measurement problemsAharonov(1988) . In weak measurement technique, the
coupling between the pointer and the measured systems is sufficiently weak,and
the obtained information by single trial is trivial, but using its inherent
feature one can get enough information as precisely as possibleTollaksen(2010)
. One of the distinguished properties of weak measurement compared with strong
measurement is that its induced weak value of the observable on the measured
system can be beyond the usual range of the eigenvalues of that
observableqparadox(2005) . The feature of weak value usually referred to as an
amplification effect for weak signals rather than a conventional quantum
measurement and used to amplify many weak but useful information in physical
systemsDressel(2014) ; Nori ; Shikano(2010) . So far, the weak measurement
technique has been applied in different fields to investigate very tiny
effects, such as beam deflectionPfeifer(2011) ; Hosten2008 ; Hogan(2011) ;
Zhou(2013) ; Starling(2009) ; Dixon(2009) , frequency shiftsStarling(2010)-1 ,
phase shiftsStarling(2010) , angular shiftsMagana(2013) ; Bertulio(2014) ,
velocity shiftsviza(2013) , and even temperature shiftEgan(2012) .
Furthermore, it has been applied to solve some fundamentals of quantum physics
such as quantum paradoxes(Hardy’s paradoxAharonov(2002) ; Lundeen(2009) ;
Yokota(2009)  and the three-box paradoxResch(2004) ), quantum correlation and
quantum dynamicsAharonov(2005) ; Aharonov(2008) ; Holger(2013) ; Shikano(2012)
; Aharonov(2011) ; Shikano(2011) , quantum state tomographyLundeen(2012) ;
Braveman(2013) ; Kocsis(2011) ; Malik(2014) ; Salvail(2013) ; Lundeen(2011) ,
violation of the generalized Leggett-Garg inequalitiesPalacios ; Suzuki(2012)
; Emary(2014) ; Goggin(2011) ; Groen(2013) ; Dressel(2011)  and the violation
of the initial Heisenberg measurement-disturbance relationshipLee(2012) ;
Eda(2014) ,etc.

We know that temperature is a basic concept in thermodynamics and most of the
properties of a matter is directly related with its temperature. According to
the thermodynamics the temperature is an intensive quantity and independent to
sample’s volume and mass. Thus, it is possible to measure the temperature of
nanoscale objects which inserted in large sample. The motion of nanoscale
objects obey the quantum mechanics and with the recent progress to
manipulation of individual quantum systems, it has been possible to
temperature readings with nanometric spatial resolutionNeu ; Kubo ; Toyli .
Quantum thermometery also applied to precisely estimate the temperature of
fermonicHauck ; Haupt  and bosonicWhite ; Braun  hot reservoirs,
micromechanical resonatorsBruelli2 ; Brunelli ; Higgins  and nuclear
spinsRaitz . The nanoscale system is too fragile and measuring the temperature
of such systems requires that the measurement process should not disturb them
to much while keeping the high measurement precision. Furthermore, as
investigated in previous studiesDressel(2014) , the weak measurement almost
doesn’t destroy the measured system and can get the desired system information
via statistical techniques with high precision. Thus,here raising an
intriguing question as to whether one can measure the temperature of nanoscale
objects by using postselected weak measurement technique. In recent
studyPati(2019) , they investigated the method to measure the temperature of a
bath using the postselected weak measurement scheme with a finite dimensional
probe, and anticipated more applications of post weak measurement method in
thermometry.

In this paper, motivated by the previous studyPati(2019) , we investigate to
measure the temperature of nanoscale system(spin) which inserted in large
thermal reservoir by using postselected weak measurement. We assume that there
have a weak interaction between spin and cantilever in magnetic resonance
force microscopy(MRFM) systemBerman . We show that the temperature of the spin
which stayed in hot reservoir with termpreature $T$, can be measured by
postselected weak measurement technique with proper postselected states of the
spin, and the weak value can be read out from optical experimental processes.
In temperature estimation process the fluctuation always exists due to the
indirect measurement, but the quantum estimation theory provides the tool to
evaluate lower bounds to the amount of fluctuations for a given measurement.
To investigate the precision of our method, we evaluate the fisher
information(FI) for the estimation of temperature via postselected weak
measurement. We found that in high temperature regime the FI is more larger
than unity with proper postselected states of the spin, and this enable us to
show that postselected meak measurement method is indeed cane be used in
temperature estimation process in MRFM system with high presicion.

The rest of the paper is organized as follows. In Sec. II, we give the setup
for our system. In Sec. III, we give the details how to measure the
temperature of a spin via postselected weak measurement. In Sec. IV, we
investigate the Fisher information to investigate the precision of our
scheme.We give a conclusion to our paper in Sec. V.

##  II Model setup

As illustrated in Fig.1, two ends of a cantilever used in MRFM is fixed and a
small ferromagnetic particle is attached at the middle of the cantileverBerman
. The force produced by a single spin on the ferromagnetic particle effects
the parameters of the mechanical vibrations of the cantilever. There have a
external permanent magnetic field $B_{ext}$ directed in the $z$ direction
exerts on the spin and a ferromagnetic particle attached on the middle of the
cantilever also exerts a gradient dipole magnetic field produced by the
ferromagnetic particle on the spin $\partial B_{d}$/$\partial z$ with
$B_{d}=\frac{2}{3}\mu_{0}M_{0}(\frac{R_{0}}{R_{0}+d+z})^{3},$ where $\mu_{0}$
is the permeability of the free space, $d$ is the distance between the bottom
of the ferromagnetic particle and the spin is initially at equilibrium
position($z=0$), $R_{0}$ and $M_{0}$ are the radius and magnetization of the
ferromagnetic particle,respectively. The spin also exposed to the transversal
frequency modulated rf magnetic field $\vec{B}_{1}(t)=B_{1}\left(\cos(\omega
t+\phi(t)),\sin(\omega t+\phi(t)),0\right)$ in $x-y$ plane. The motion of the
cantilever with ferromagnetic particle can be modeled as a simple harmonic
oscillator with effective mass $m$ and frequency $\omega_{c}$. we can express
the oscillation position $z$ of the cantilever with creation and annihilation
operators $\hat{a}$ and $\hat{a}^{\dagger}$ by
$\hat{z}=\sigma(\hat{a}+\hat{a}^{\dagger})$, with
$\sigma=\sqrt{\text{$\hbar$}/2m\omega_{c}}$. In the rotating system
coordinates(RSC) the Hamiltonian of this system can be written asBerman

$\displaystyle H$
$\displaystyle=\hbar\omega_{c}\hat{a}^{\dagger}\hat{a}+\hbar\omega_{z}\hat{S}_{z}+\hbar\omega_{R}\hat{S}_{x}-\hbar
g\hat{S}_{z}\otimes\hat{z}.$ (1)

Here, $\omega_{R}=\gamma B_{1}$, $\omega_{z}=\gamma
B_{0}-\omega-\frac{d\phi}{dt}$ , $g=\gamma\frac{\partial B_{d}}{\partial z}$,
$\gamma$ is the gyromagnetic ratio and $B_{0}=B_{ext}+B_{d}^{(0)}$ is the
total magnetic field on the spin when $z=0$ in $z$ direction, and the gradient
of the dipole field $\partial B_{d}$/$\partial z$ is taken at the spin
location when $z=0$. $\hat{S}_{z,x}$are defined by Pauli operators
$\hat{S}_{x}\rightarrow\frac{1}{2}\hat{\sigma}_{x}$ and
$\hat{S}_{z}\rightarrow\frac{1}{2}\hat{\sigma}_{z}$, respectively. As we can
see, our coupling system Hamiltonian consists of three parts(see Eq.1); first
term represents the motion of cantilever, middle two terms describes the spin
system and we denote with $H_{s}$, and the final term describes the
interaction of spin and cantilever, and we denote it with $H_{int}$. According
to the technique of MRFM, if the frequency of the spin oscillations matches
the resonance frequency of the cantilever vibrations, the spin force will
amplify the vibrations of the cantilever and its motion can be detected by
optical methodsBerman . In this study, we will focus on to measure the
temperature of the single spin which stayed in reservoir with temperature $T$
in MRFM system. Hereafter, in this paper we use the unit $\hbar=1$.

Figure 1: (Color online) The setup of our scheme. A spin and the cantilever
are coupled with each other by a tiny ferromagnetic particle attached to the
middle of the cantilever. $d$ is the distance between the bottom of the
ferromagnetic particle and the spin at equilibrium position, $z=0$.

##  III Measure the temperature of the spin

As given in Eq.(1), the interaction Hamiltonian between the system(spin) and
pointer(cantilever) is given by the standard von Neumman
Hamiltonianqparadox(2005)

$H_{ini}=-g\hat{S}_{z}\otimes\hat{z}$ (2)

Here, $g$ is a coupling coefficient between spin and cantilever, and its value
depends on gyromagnetic ratio $\gamma$ and the gradient magnetic field at the
spin position produced by ferromagnetic particle. $\hat{z}=\int
z|z\rangle\langle z|dz$ is the position operator,while the conjugate momentum
operator is $\hat{p}=\int p|p\rangle\langle p|dp$ where
$[\hat{x},\hat{p}]=i\text{$\hat{I}$}$. We consider the spin as system and
cantilever as measuring device(pointer) to measure the temperature of the
spin. We assume that initially the system (spin) is in a heat bath with
temperature $T$, and reached the thermal equilibrium state
$\text{$\rho$}_{s}=e^{-\beta H_{s}}/Tr(e^{-\beta H_{s}})$. Here
$\beta=1/k_{B}T$ , $k_{B}$ is Boltzmann constant and it taken as unity
hereafter. We assume that the initial state of measuring device and system are
prepared as $\phi(z)=\langle z|\phi\rangle$ and $\rho(0)=\rho_{s}$,
respectively. Then, under the action of unitary evolution operator
$U(t)=\exp(-i\int_{0}^{t}H_{ini}d\tau)$, the initial state of total system
$\rho(0)=\rho_{s}\otimes|\phi\rangle\langle\phi|$ will be evolved to

$\text{$\rho(t)=U^{\dagger}(t)\rho(0)U(t)$}.$ (3)

This is the state of the total system after interaction, but we only
interested on final state of measuring device(cantilever) which contains
system information. If we assume that the interaction strength $g$ is very
small , then it is enough to consider the expansion of unitary operator up to
its first order, and Eq.(3) becomes as

$\rho(t)\approx\rho_{s}\otimes|\phi\rangle\langle\phi|+igt\left[\hat{S}_{z}\otimes\hat{z},\rho_{s}\otimes|\phi\rangle\langle\phi|\right].$
(4)

According the standard terminology and procedure of weak measurement theory,
if we choose the state $|\psi_{f}\rangle$ as the final state of the spin, and
project it onto the total system state, Eq.(4), it gives the unnormalized
final state of measuring device as

$\phi(t)\approx\langle\psi_{f}|\rho_{s}|\psi_{f}\rangle
e^{igt\text{$\hat{S}$}_{w}\hat{z}}|\phi\rangle\langle\phi|e^{-igt\hat{S}_{w}\hat{z}}.$
(5)

Here,

$S_{w}$=$\frac{\langle\psi_{f}|\hat{S}_{z}\rho_{s}|\psi_{f}\rangle}{\langle\psi_{f}|\rho_{s}|\psi_{f}\rangle}$
(6)

is the weak value of spin $z$ component and it related to the initial state of
the spin which reached thermal equilibrium with a heat bath with temperature
$T$. As can see from Eq.(5) and Eq.(6), we can deduce the spin temperature by
properly choosing the final state$|\psi_{f}\rangle$. However, we have to note
that $|\psi_{f}\rangle$ cannot be eigenstate of operator $\hat{S}_{z}$,
otherwise the temperature will not related to the weak value and our scheme
will lose its validity to measure the temperature of the spin.

To read the temperature of the spin system, next we will study the property of
weak value of spin operator $\hat{S}_{z}$ by assuming that it in the hot bath
with high temperature, i.e. $\beta\ll 1$. Furthermore, we assume that the
eigenvalues of $\hat{S}_{x}$ and $\hat{S}_{y}$ are $\epsilon_{m}$and
$\varepsilon_{m}$, and the corresponding eigenstates are $|\psi_{m}\rangle$
and $|\varphi_{m}\rangle$,
i.e.,$\hat{S}_{x}|\psi_{m}\rangle=\epsilon_{m}|\psi_{m}\rangle$ and
$\hat{S}_{z}|\varphi_{m}\rangle=\varepsilon_{m}|\varphi_{m}\rangle$,
respectively. Then, the weak value of the spin $z$ component can be rewritten
as

$\displaystyle S_{w}$
$\displaystyle=\frac{\langle\psi_{f}|\hat{S}_{z}e^{-\beta
H_{s}}|\psi_{f}\rangle}{\langle\psi_{f}|e^{-i\beta H_{s}}|\psi_{f}\rangle}$
$\displaystyle=\frac{\sum_{n,m}e^{-\beta(\omega_{z}\varepsilon_{n}+\omega_{R}\epsilon_{m})}\langle\psi_{n}|\phi_{m}\rangle\langle\psi_{f}|\text{$\hat{S}_{z}$}|\psi_{n}\rangle\langle\phi_{m}|\psi_{f}\rangle}{\sum_{n,m}e^{-\beta(\omega_{z}\varepsilon_{n}+\omega_{R}\epsilon_{m})}\langle\psi_{n}|\phi_{m}\rangle\langle\psi_{f}|\psi_{n}\rangle\langle\phi_{m}|\psi_{f}\rangle}$
$\displaystyle=\frac{\langle\psi_{f}|\hat{S_{z}}|\psi_{f}\rangle-\beta\langle\psi_{f}|\hat{S_{z}}\hat{H}_{s}|\psi_{f}\rangle}{1-\beta\langle\psi_{f}|\hat{H}_{s}|\psi_{f}\rangle}.$
(7)

During the derivation of Eq.(7), we use $e^{\pm x}\approx 1\pm x$, $(x\ll 1)$,
and completeness of the basis
$|\psi_{m}\rangle,\text{$|\varphi_{m}\rangle,$i.e.
$\text{$\sum_{m}|\psi_{m}\rangle\langle\psi_{m}|=\sum_{m}|\varphi_{m}\rangle\langle\varphi_{m}|=1.$}$}$Since
$(1-x)^{-1}\approx 1-x,(x\ll 1)$, we can rewrite the Eq.(7) as below

$S_{w}=\left[\langle\psi_{f}|\hat{S_{z}}|\psi_{f}\rangle-\beta\langle\psi_{f}|\hat{S_{z}}\hat{H}_{s}|\psi_{f}\rangle\right]\left(1-\beta\langle\psi_{f}|\hat{H}_{s}|\psi_{f}\rangle\right).$
(8)

From this relation, we can express the temperature parameter $\beta$ by using
the weak value $S_{w}$ as

$\beta=\frac{S_{w}-\langle\hat{S}_{z}\rangle}{Conv(S_{z},H_{s})-\langle\psi_{f}|\hat{S}_{z}\hat{H_{s}}|\psi_{f}\rangle},$
(9)

where
$Conv(S_{z},H_{s})=\langle\hat{S}_{z}\rangle\langle\psi_{f}|\hat{H}_{s}|\psi_{f}\rangle$
and $\langle\hat{S}_{z}\rangle=\langle\psi_{f}|\hat{S}_{z}|\psi_{f}\rangle$,
respectively. From Eq.(9) we can see that since the value of
$\langle\hat{S}_{z}\rangle$ and $Conv(S_{z},H_{s})$ can be find by using
postselection $|\psi_{f}\rangle$, if we can find the value of $S_{w}$, then
the temperature of the spin can be estimated very easily. For example, if we
take $|\psi_{f}\rangle=\frac{1}{\sqrt{2}}(|\psi_{1}\rangle+|\psi_{2}\rangle)$,
then Eq.(9) reduced to

$\beta=\frac{2(1-2S_{w})}{\omega_{R}+3\omega_{z}}.$ (10)

From the above theoretical results we can deduce that the estimation of spin
in MRFM system depends on the postselected weak value. Thus, in the remaining
part of this section we will discuss the possibility to find the weak value of
our scheme. We can assume that the motion of the cantilever can be described
by simple harmonic oscillator, and its initial state can be written as

$\phi_{0}\left(z\right)=\langle
z|0\rangle=\left(\frac{1}{2\pi\sigma^{2}}\right)^{\frac{1}{4}}\exp\left(-\frac{z^{2}}{4\sigma^{2}}\right)$
(11)

where $\sigma$ is the width of the oscillation beam. The normalized final
state of the cantilever after postselection, i.g. Eq.(5), can be written as

$|\phi_{f}\rangle=\text{$\kappa$}[|0\rangle+ig_{0}S_{w}|1\rangle],$ (12)

where $g_{0}=gt\sigma$, $\kappa=(1+g_{0}^{2}|S_{w}|^{2})^{-\frac{1}{2}}$ and
we the $\phi_{0}(z)=\langle z|0\rangle$ and $\phi_{1}(z)=\langle z|1\rangle$
represent the ground and first excited states of catilever, respectively. The
expectation values the position and momentum operators under the final state
of the catilever $|\phi_{f}\rangle$ can be calculated as

$\displaystyle\langle z\rangle$ $\displaystyle=-2\sigma g_{0}\kappa^{2}\Im
S_{w}$ (13)

and

$\displaystyle\langle p\rangle$
$\displaystyle=\frac{\kappa^{2}g_{0}}{\sigma}\Re(S_{w})$ (14)

respectively. According to the results of recent studiesYutaka(2014) ;
Lundeen(2019) , we can measure the real and imaginary parts of the weak values
with optical experiments. Thus, in our scheme the temperature of spin can be
measured in the Lab with proper optical experimental setups.

##  IV Precision of our scheme

Fisher information is the maximum amount of information about the parameter
that we can estimate from the system. For a pure quantum state $|\phi\rangle$,
the quantum fisher information estimating the parameter $\beta$ is

$F(\beta)=4[\langle\partial_{\beta}\text{$\phi$}|\partial_{\beta}\phi\rangle-|\langle\phi|\partial_{\beta}\phi\rangle|^{2}],$
(15)

where the state $|\phi\rangle$is the normalized final state of the cantilever,
i.e. Eq.(12). The variance of unknown parameter $\triangle\beta$ is bounded by
the Cramér-Rao bound

$Var\left(\beta\right)\geq\frac{1}{NF(\beta)},$ (16)

where $N$ is the number of measurements. From this definition of variance of
parameter $\beta$, we can see that the Fisher information set the minimal
possible estimate for parameter $\beta$, while higher Fisher information means
a better estimation. As the variance of $\beta$ is inverse proportional to the
measurement time, we consider $N=1$ throughout this paper.

We investigate the variation of quantum Fisher information for different
postselected states
$|\psi_{f}\rangle=\cos\frac{\theta}{2}|\uparrow\rangle+e^{i\varphi}\sin\frac{\theta}{2}|\downarrow\rangle$
where $\theta\in[0,\pi],\varphi\in[0,2\pi)$, and $|\uparrow\rangle$ and
$|\downarrow\rangle$ are eigenvector of the spin operator $\hat{S}_{z}$ with
corresponding eigenvalues $+\frac{1}{2}$ and $-\frac{1}{2}$, respectively. Our
numerical results in Fig.2 and Fig.3 show that the quantum Fisher information
is higher in high temperature regime, and phase of postselected state also can
effect of the precision of our scheme. However, as we mentioned in this
section, when the postselected state is an eigenstate of the spin operator
$\hat{S}_{z}$, the Fisher information become zero and our scheme lose its
validity of measure the temperature of the spin.

Figure 2: (Color online) Quantum fisher information for parameter estimation
$\beta.$ Here, we take $\phi=0,g_{0}=10^{-8},\omega_{z}=4.8\times 10^{6}Hz$ ,
$\omega_{R}=3\times 10^{9}Hz$. Figure 3: (Color online) Quantum fisher
information for parameter estimation $\beta$ with different phase of
postselected states. Here $\theta=\frac{\pi}{4}$ and other parameters are the
same with Fig.2.

##  V Conclusion

In this study, we have proposed a new scheme to measure the temperature of
spin which put it in thermal bath with temperature $T$ by using postselected
weak measurement method. We find that in high temperature regime, the
precision of our proposal is high enough and can be controlled by adjusting
the parameters of postselected states of the spin. Furthermore, in Ref.Brun ,
the authors studied that quantum Fisher information is higher in postselected
rather than non-postselected weak measurement.Thus, even though we only
consider the high temperature regime, but in our scheme if someone use the
postselected measurement, it will show its good performance in low temperature
regime rather than non-postselected measurement schemes.

###### Acknowledgements.

This work was supported by the National Natural Science Foundation of China
(Grant No. 11865017, No.11864042).

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