# Probing cosmic string spacetime through parameter estimation

Ying Yang111Corresponding author, Email<EMAIL_ADDRESS>Department of
Physics, Key Laboratory of Intelligent Sensors and Advanced Sensor Materials,
Hunan University of Science and Technology, Xiangtan, Hunan 411201, P. R.
China.  Jiliang Jing  Department of Physics, Key Laboratory of Low Dimensional
Quantum Structures and Quantum Control of Ministry of Education, and
Synergetic Innovation Center for Quantum Effects and Applications, Hunan
Normal University, Changsha, Hunan 410081, P. R. China.  Zehua
Tian222Corresponding author, Email<EMAIL_ADDRESS>CAS Key Laboratory of
Microscale Magnetic Resonance and Department of Modern Physics, University of
Science and Technology of China, Hefei 230026, China

###### Abstract

Quantum metrology studies the ultimate precision limit of physical quantities
by using quantum strategy. In this paper we apply the quantum metrology
technologies to the relativistic framework for estimating the deficit angle
parameter of cosmic string spacetime. We use a two-level atom coupled to
electromagnetic fields as the probe and derive its dynamical evolution by
treating it as an open quantum system. We estimate the deficit angle parameter
by calculating its quantum Fisher information(QFI). It is found that the
quantum Fisher information depends on the deficit angle, evolution time,
detector initial state, polarization direction, and its position. We then
identify the optimal estimation strategies, i.e., maximize the quantum Fisher
information via all the associated parameters, and therefore optimize the
precision of estimation. Our results show that for different polarization
cases the QFIs have different behaviors and different orders of magnitude,
which may shed light on the exploration of cosmic string spacetime.

metrology; quantum Fisher information; cosmic string spacetime.

##  I Introduction

In the context of quantum field theory, cosmic strings have attracted
considerable attention due to their significance on the research of
astrophysical, gravitational and cosmological Kaiser ; Bennett ; Vilenkin ;
Hindmarsh . Although the cosmic microwave background radiation shows that the
cosmic strings are not abundant, the evolution of string brings distinct
astrophysical effects, such as gammaray bursts Brandenberger1 ; MacGibbon ,
producing detectable gravitational waves Damour ; Brandenberger2 ; Jackson ,
and high-energy cosmic rays Cheng . Among these interesting astrophysical
consequences caused by cosmic string, topology defect plays an important role
Kibble . The interest to the research of topology defects in cosmic string
spacetime is amplified by various theoretical studies, which have created a
promising perspective for the development of this field Vilenkin ; Hindmarsh ;
Davis1 ; Siemens .

Based on the unique signatures of cosmic string Linet ; Frolov ; Davies1 ;
Gott ; Charnock ; Foreman , various investigations have been conducted to
detect the string Dvorkin ; Eiichiro ; Aghanim ; Urrestilla ; Helliwell . The
simplest cosmic string spacetime is characterized by a flat metric with a
deficit angle, which is described by an infinite, straight and static
cylindrically symmetric cosmic string, and many quantum effects exhibit
significant characteristics in such spacetime Sousa ; Bezerra ; Iliadakis ;
Berezinsky ; Davies ; Bilge .

Quantum metrology which aims to achieve higher precision than the classical
ways by using quantum techniques, has been paid a lot of attention recently
Giovannetti ; Giovanetti2 ; Paris ; advance . There are considerable interests
in applying quantum metrological techniques in relativistic settings,
specifically, to relativistic quantum fields Ahmadi ; Huang ; Kish ; Mann R B
; xiaobao . One of the purpose is to explore, in the relativistic quantum
regimes, how the relativistic effects on quantum system affect the precision
of certain measurements Parameter estimation ; relativistic motion ; Liu1 ;
M-Ahmadi ; Liu2 . The other is how to measure the physical quantity of
relativity, e.g., Unruh-Hawking temperature, with quantum techniques to obtain
higher precision beyond the classical case Tian ; Adesso ; TianZH . Previous
research has investigated the metrology of a wide range of relativistic
phenomena, including measuring the distance to achieve adequate sensitivity
for the detection of gravitational wave gravitational wave , high precision
estimation of the Earth’s Schwarzschild spacetime parameters by quantum
experiments Schwarzschild spacetime parameters , estimation of gravitational
redshift from matter wave interference gravitational redshift , and so on.

Since in practice a lot of quantities of interest to us do not correspond to
quantum observable, direct observation of them is not accessible. In these
situations, one has to infer the interested values of the quantities by
examining a set of data from the measurement of a different observable, or a
set of observable. Therefore, any conceivable strategy aimed at evaluating the
quantity of interest eventually reduces to a parameter-estimation problem
Helstrom ; Holevo . In this regard, let us note that the quantum Fisher
information (QFI) is the key figure of merit in the framework of quantum
estimation theory. The QFI links to the Cramér-Rao bound and determines the
ultimate bound on the precision of the estimator Helstrom . Larger QFI means
more optimal precision for estimating parameter can be possibly achieved in a
metrological task. Therefore, how to maximize the QFI in a certain system via
different system parameters and different approaches is particularly
significant.

We now study quantum metrology in the relativistic framework and focus on
estimating the deficit angle in cosmic string spacetime. We use a two-level
atom as probe which is coupled to electromagnetic fields in the background of
cosmic string spacetime. We study the dependence of the QFI on various
parameters, and identify strategies for maximizing it. Note that although
relativistic metrology has been considered in the cosmic string spacetime, all
of them jin ; huang  paid their attention to how the spacetime property
affects on estimating the quantum state parameter of probe, and overlooked the
estimation of the spacetime parameter, e.g., the deficit angle. However, this
overlooked issue plays a very important role in the exploration of cosmic
string physics conicity , since it might tell us how to obtain the ultimate
limit of precision for estimating cosmic string and what the ultimate
precision bounded by quantum mechanics is. In our previous research ying , we
estimated the deficit angle of a cosmic string spacetime using a moving
detector that is coupled to a massless scalar field at vacuum state. To
further elaborate the parameter estimation of cosmic string spacetime, we here
extent the relevant investigation to a more practical scenario by replacing
the massless scalar field with a vector field, i.e., an electromagnetic field.
We would like to emphasize that compared with the massless scalar field case,
the unique emerging electromagnetic polarization will cause significant
effects on the QFI for estimating the deficit angle. Specifically, the QFIs
not only have different behaviors via dependent parameters, but also there is
more than two orders of QFI magnitude difference for different polarization
cases, as shown in the following.

The outline of this paper is as follows. In Sec. II, we recall the physics of
QFI. Then we introduce our detector, a two-level atom which is coupled to
electromagnetic fields in the cosmic string spacetime in Sec. III. In Sec. IV
we investigate the parameter estimation of the deficit angle parameter with
different polarizations by using a static atomic detector, and find the
effective strategies to maximize the QFI for estimating the deficit angle. In
Sec. V, we give a simple discussion about our model. Final remarks and
conclusions are given in Sec. VI.

##  II Quantum Fisher information

The quantum Cramér-Rao bound provides a fundamental limit on the precision
that quantum measurements can achieve Cramer ; Helstrom ; Holevo , and it is
expressed as

$\displaystyle{\rm Var}(\lambda)\geq\frac{1}{NF_{\lambda}}.$ (1)

Here $\rm{Var}(\lambda)=E_{\lambda}[(\hat{\lambda}-\lambda)^{2}]$ is the
variance of the estimator, $N$ is the number of measurements and
$F_{\lambda}=F(\lambda)$ denotes the QFI of parameter $\lambda$. The
definition of $F(\lambda)$ can be written as Cramer

$F(\lambda)\equiv Tr[\rho(\lambda)L(\lambda)^{2}]\;,$ (2)

where $\rho$ is the density matrix of the detector and $L(\lambda)$ is the
symmetric logarithmic derivative which satisfies
$\partial\rho(\lambda)=\frac{1}{2}(L(\lambda)\rho(\lambda)+\rho(\lambda)L(\lambda))$.
The quantum Cramér-Rao bound provides the ultimate bound of parameter
estimation accuracy for a state of the family $\rho(\lambda)$. Consider a two-
level quantum system, whose quantum state in the Bloch sphere is of the form

$\rho=\frac{1}{2}(I+\vec{\omega}\cdot\vec{\sigma})\;,$ (3)

where $\vec{\omega}=(\omega_{1},\omega_{2},\omega_{3})$ denotes the Bloch
vector, and $\vec{\sigma}=(\sigma_{1},\sigma_{2},\sigma_{3})$ are the Pauli
matrices. For such an elemental quantum system, one can find its QFI
expression analytically Zhong

$F_{\lambda}=\left\\{\begin{array}[]{l}|\partial_{\lambda}\vec{\omega}|^{2}+\frac{(\vec{\omega}\cdot\partial_{\lambda}\vec{\omega})^{2}}{1-|\vec{\omega}|^{2}},\;\;\;|\vec{\omega}|<1\;,\\\
|\partial_{\lambda}\vec{\omega}|^{2},\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;|\vec{\omega}|=1.\\\
\end{array}\right.$ (4)

##  III Open quantum system in cosmic string spacetime

In practice, any quantum systems inevitably interact with external
environment, which may lead to dissipation and decoherence. Correspondingly
the ambient noise will have effect on the QFI encoded in the system’s quantum
state and thus have a certain influence on the accuracy of parameter
estimation. Then, it is very necessary to explore the parameter estimation
issues in the area of open quantum system. In this section, we will introduce
a two-level detector which is subjected to the quantum vacuum fluctuations of
an electromagnetic field in the cosmic string spacetime. The detector is
treated as an open quantum system and its dynamics is derived by tracing over
the degree of freedom of the electromagnetic field.

Let us begin with the cosmic string spacetime, whose metric is of the form
Linet ; Skarzhinsky ; jin

$ds^{2}=dt^{2}-dr^{2}-r^{2}d{\alpha}^{2}-dz^{2}.$ (5)

As shown in Fig. 1, in the cylindrical coordinates the cosmic string spacetime
is supposed to have an string lies along the $z$ direction.

Figure 1:  Schematic representation of the cosmic string spacetime with an
infinite and straight cosmic string along $z$ direction.

The metric (5) is quite similar to the metric of Minkowski spacetime, the
difference is that for the Minkowski spacetime $0\leq\alpha<2\pi$, while for
the cosmic string spacetime $0\leq\alpha<\frac{2\pi}{\nu}$, where
$\nu=(1-4G\mu)^{-1}$ with $G$, the Newton constant, and $\mu$, the mass per
unit length of the string. Therefore, compared to the Minkowski spacetime, the
cosmic string spacetime is locally flat but with a deficit angle which is used
to characterize the nontrivial global topology. One can quantize the
electromagnetic field in the cosmic string spacetime, the corresponding vector
potential of electromagnetic field is expressed as Pavel

$A_{\xi}(t,\vec{x})=\int d\mu_{j}\;[\;c_{\xi j}(t)f_{\xi j}(\vec{x})+c_{-\xi
j}^{{\dagger}}(t)f_{-\xi j}^{\ast}(\vec{x})]$ (6)

with

$\int
d\mu_{j}=\sum_{m=-\infty}^{\infty}\int_{-\infty}^{\infty}dk_{3}\int_{0}^{\infty}dk_{\perp}k_{\perp}\;,$
(7)

in which $c_{\xi j}(t)=c_{\xi j}(0)e^{-i\omega t}$ is the annihilation
operator and $c^{{\dagger}}_{-\xi j}(t)=c^{{\dagger}}_{-\xi j}(0)e^{i\omega
t}$ is the creation operator with $\xi\in\\{0,\pm 1,3\\}$, $m\in Z$,
$k_{3}\in[-\infty,\infty]$, $k_{\perp}\in[0,\infty]$. The exact form of the
$f_{\xi j}(\vec{x})$ is $f_{\xi
j}(\vec{x})=\frac{1}{2\pi}\sqrt{\frac{\nu}{2\omega}}J_{|\nu
m+\xi|}(\kappa_{\bot}r)e^{i(\nu m\alpha+k_{3}z)}$. It is shown that

$\displaystyle c_{\xi j}(0)$ $\displaystyle=$ $\displaystyle i\int
d^{3}\vec{x}f^{\ast}_{\xi
j}(t,x)\overleftrightarrow{\partial_{t}}A_{\xi}(t,x)\;,$ (8) $\displaystyle
c^{{\dagger}}_{\xi j}(0)$ $\displaystyle=$ $\displaystyle-i\int
d^{3}\vec{x}f_{\xi j}(t,x)\overleftrightarrow{\partial_{t}}A_{\xi}(t,x)\;,$
(9)

where the commutation relations are as follows

$\displaystyle[c_{\xi j}(t),c_{\xi
j^{\prime}}^{{\dagger}}(t)]=\delta_{j,j^{\prime}}\quad\;for\quad\xi=\pm 1,3,$
(10)
$\displaystyle[c_{0j}(t),c_{0j^{\prime}}^{{\dagger}}(t)]=-\delta_{j,j^{\prime}}\;\;for\quad\xi=0.$
(11)

For more details of the quantization of the electromagnetic field in the
cosmic string spacetime, one can refer to Ref. Pavel .

We model a two-level atom as the detector which is coupled to an
electromagnetic field in the cosmic string spacetime. The total Hamiltonian of
the whole system is

$\displaystyle H=H_{A}+H_{F}+H_{I}\;,$ (12)

where $H_{A}$, $H_{F}$ and $H_{I}$ are the atomic Hamiltonian, the field
Hamiltonian and the interaction Hamiltonian, respectively. The specific
expression of the atomic Hamiltonian is
$H_{A}=\frac{1}{2}\hbar\omega_{0}\sigma_{3}$, where $\hbar\omega_{0}$ is the
energy spacing of the atom and $\sigma_{3}$ represents the Pauli matrix. The
interaction Hamiltonian is described as
$H_{I}=-e\texttt{r}\cdot\texttt{E}(x(\tau))$, where $e$ is electric charge,
$e\texttt{r}$ denotes the atomic electric dipole moment, and
$\texttt{E}(x(\tau))$ represents the strength of the electric field along the
atom’s trajectory $x(\tau)$.

We assume the total density matrix of the detector-field system at the
beginning can be written as $\hat{\rho}_{\rm
tot}(0)=\hat{\rho}(0)\otimes|0\rangle\langle 0|$, with $\hat{\rho}(0)$ being
the atomic initial density matrix, and $|0\rangle\langle 0|$ representing the
vacuum state of the field. In the atomic frame, the entire quantum system
follows the von Neumann equation Heinz

$\displaystyle\frac{\partial\hat{\rho}_{\rm tot}(\tau)}{\partial\tau}$
$\displaystyle=$ $\displaystyle-\frac{i}{\hbar}[H,\hat{\rho}_{\rm
tot}(\tau)]\;,$ (13)

where $\tau$ is the proper time. After tracing over the field degree of
freedom $\hat{\rho}(\tau)=Tr_{F}[\hat{\rho}_{\rm tot}(\tau)]$, the master
equation of the atom is given under the Born-Markov approximation as Lindblad

$\displaystyle\frac{\partial\hat{\rho}(\tau)}{\partial\tau}$ $\displaystyle=$
$\displaystyle-\frac{i}{\hbar}[H_{eff},\hat{\rho}(\tau)]+\mathcal{L}[\hat{\rho}(\tau)]\;,$
(14)

here ${\cal L}[\rho]={1\over
2}\sum_{i,j=1}^{3}a_{ij}\big{[}2\,\sigma_{j}\rho\,\sigma_{i}-\sigma_{i}\sigma_{j}\,\rho-\rho\,\sigma_{i}\sigma_{j}\big{]}$
with
$a_{ij}=A\delta_{ij}-iB\epsilon_{ijk}\delta_{k3}-A\delta_{i3}\delta_{j3}$, and
we have used the definition
$A=\frac{1}{4}[{\cal{G}}(\omega_{0})+{\cal{G}}(-\omega_{0})]$ and
$B=\frac{1}{4}[{\cal{G}}(\omega_{0})-{\cal{G}}(-\omega_{0})]$. By defining the
two-point functions of the electromagnetic field

$G^{+}(x-x^{\prime})={e^{2}\over\hbar^{2}}\sum_{i,j=1}^{3}\langle+|r_{i}|-\rangle\langle-|r_{j}|+\rangle\,\langle
0|E_{i}(x)E_{j}(x^{\prime})|0\rangle\;,$ (15)

with $\langle
0|E_{i}(x)E_{j}(x^{\prime})|0\rangle=\partial_{i}\partial^{\prime}_{j}\langle
0|A_{0}(x)A_{0}(x^{\prime})|0\rangle+\partial_{0}\partial^{\prime}_{0}\langle
0|A_{i}(x)A_{j}(x^{\prime})|0\rangle$, we achieve its Fourier and Hilbert
transforms as follows

${\cal
G}(\lambda)=\int_{-\infty}^{\infty}d\Delta\tau\,e^{i{\lambda}\Delta\tau}\,G^{+}\big{(}\Delta\tau\big{)}\;,$
(16)

and

${\cal K}(\lambda)=\frac{P}{\pi i}\int_{-\infty}^{\infty}d\omega\ \frac{{\cal
G}(\omega)}{\omega-\lambda}\;.$ (17)

Then the effective Hamiltonian $H_{\rm eff}$ is

$H_{\rm eff}=\frac{1}{2}\hbar\Omega\sigma_{3}={\hbar\over
2}\\{\omega_{0}+{i\over 2}[{\cal K}(-\omega_{0})-{\cal
K}(\omega_{0})]\\}\,\sigma_{3}\;,$ (18)

where $\Omega$ is the effective level spacing of the atom, containing the free
term $\omega_{0}$ and the Lamb shift expressed as the second term on the right
of Eq. (18). We assume that the detector is initially prepared in an arbitrary
state
$|\psi(0)\rangle=\cos\frac{\theta}{2}|+\rangle+e^{i\phi}\sin\frac{\theta}{2}|-\rangle$,
where $\theta$ and $\phi$ denote the weight parameter and phase parameter, and
$|-\rangle$, $|+\rangle$ represent the ground state and excited state of the
atom, respectively. With this initial condition, by solving Eq. (14) we find
the time-dependent Bloch vector of the detector as Kimura

$\displaystyle\omega_{1}(\tau)=\sin\theta\cos(\Omega\tau+\phi)e^{-\frac{1}{2}A\tau}\;,$
$\displaystyle\omega_{2}(\tau)=\sin\theta\sin(\Omega\tau+\phi)e^{-\frac{1}{2}A\tau}\;,$
$\displaystyle\omega_{3}(\tau)=\cos\theta
e^{-A\tau}-\frac{B}{A}(1-e^{-A\tau})\;.$ (19)

The interaction between the detector and field is encoded into the parameters,
$A$ and $B$, which determines the detector’s evolution and contains the
information of spacetime due to scattering off the quantum field. Therefore,
one can infer the related parameters of spacetime (what we will estimate in
the following) through the relevant outcomes by performing the measurements on
the detector.

##  IV Parameter estimation in cosmic string spacetime

In this section, we intend to exploit local quantum estimation theory to find
the ultimate limits of precision of the deficit angle parameter in the
detector-field model. Specifically, we study how the QFI is affected by the
detector’s state parameters, such as evolution time, detector initial state,
polarization direction, and so on.

###  IV.1 Static detector in cosmic string spacetime

We now probe the cosmic string spacetime by using a static detector which is
coupled to a vacuum fluctuating electromagnetic field. The trajectory of the
detector isjin

$\displaystyle t(\tau)=\tau\;,~{}~{}~{}~{}~{}r(\tau)=r\;,$ (20)
$\displaystyle\alpha(\tau)=\alpha\;,~{}~{}~{}~{}~{}z(\tau)=z\;,$

which is described in the polar coordinates. Substituting the above trajectory
(20) into the Wightman function of the electromagnetic field, we find its
final form along the detector’s trajectory can be written as

$\displaystyle\langle 0|E_{r}(x)E_{r}(x^{\prime})|0\rangle$ $\displaystyle=$
$\displaystyle\frac{\nu}{8\pi^{2}}\int
d\mu_{j}e^{i\omega(\tau-\tau^{\prime})}\biggl{[}\frac{\omega}{2}(J_{|\nu
m+1|}^{2}(k_{\perp}r)+J_{|\nu
m-1|}^{2}(k_{\perp}r))-\frac{1}{\omega}\biggl{(}\frac{dJ_{|\nu
m|}(k_{\perp}r)}{dr}\biggr{)}^{2}\biggr{]}\;,$ (21) $\displaystyle\langle
0|E_{\alpha}(x)E_{\alpha}(x^{\prime})|0\rangle$ $\displaystyle=$
$\displaystyle\frac{\nu r^{2}}{8\pi^{2}}\int
d\mu_{j}e^{i\omega(\tau-\tau^{\prime})}\biggl{[}\frac{\omega}{2}(J_{|\nu
m+1|}^{2}(k_{\perp}r)+J_{|\nu
m-1|}^{2}(k_{\perp}r))-\frac{\nu^{2}m^{2}}{\omega r^{2}}J_{|\nu
m|}^{2}(k_{\perp}r)\biggr{]}\;,$ (22) $\displaystyle\langle
0|E_{z}(x)E_{z}(x^{\prime})|0\rangle$ $\displaystyle=$
$\displaystyle\frac{\nu}{8\pi^{2}}\int
d\mu_{j}e^{i\omega(\tau-\tau^{\prime})}\frac{k_{\perp}^{2}}{\omega}J_{|\nu
m|}^{2}(k_{\perp}r)\;.$ (23)

Then the corresponding Fourier transform is given by

${\cal G}(\lambda)=\sum_{i}{e^{2}|\langle-|r_{i}|+\rangle|^{2}\lambda^{3}\over
3\pi}f_{i}(\lambda,r,\nu)\Theta(\lambda),$ (24)

where $\Theta(\lambda)$ is the step function and

$\displaystyle f_{r}(\omega,r,\nu)$ $\displaystyle=$
$\displaystyle\frac{3\nu}{4}\sum_{m}\int_{0}^{1}d\eta\frac{\eta}{\sqrt{1-\eta^{2}}}\bigg{[}(2-\eta^{2})J^{2}_{|\nu
m+1|}(\omega r\eta)+\eta^{2}J_{|\nu m|-1}(\omega r\eta)J_{|\nu m|+1}(\omega
r\eta)\bigg{]}\;,$ $\displaystyle f_{\alpha}(\omega,r,\nu)$ $\displaystyle=$
$\displaystyle\frac{3\nu}{4}\sum_{m}\int_{0}^{1}d\eta\frac{\eta}{\sqrt{1-\eta^{2}}}\bigg{[}(2-\eta^{2})J^{2}_{|\nu
m+1|}(\omega r\eta)-\eta^{2}J_{|\nu m|-1}(\omega r\eta)J_{|\nu m|+1}(\omega
r\eta)\bigg{]}\;,$ $\displaystyle f_{z}(\omega,r,\nu)$ $\displaystyle=$
$\displaystyle\frac{3\nu}{2}\sum_{m}\int_{0}^{1}d\eta\frac{\eta^{3}}{\sqrt{1-\eta^{2}}}J^{2}_{|\nu
m|}(\omega r\eta)\;.$ (25)

Figure 2: Polarization schematic: the left panel to the right panel represent
the radial polarization, the tangential polarization and the parallel
polarization, respectively.

With the above Fourier transform, we can find the coefficients of the
Kossakowski matrix $a_{ij}$ is given by

$A=B={\gamma_{0}\over 4}\sum_{i}\zeta_{i}f_{i}(\lambda,r,\nu),$ (26)

where $\gamma_{0}=e^{2}|\langle-|{\bf r}|+\rangle|^{2}\,\omega_{0}^{3}/3\pi$,
and $\zeta_{i}=|\langle-|r_{i}|+\rangle|^{2}/|\langle-|{\bf r}|+\rangle|^{2}$
represents the relative polarizability which satisfies $\sum_{i}\zeta_{i}=1$,
as shown in Fig. 2. We will adopt the notation
$f(\lambda,r,\nu)=\sum_{i}\zeta_{i}f_{i}(\lambda,r,\nu)$ for simplification in
the following. From Eqs. (4), (III), and (26), we can find the corresponding
QFI for estimating the deficit angel parameter, $\nu$, is given by

$\displaystyle F_{\nu}(\omega,\nu,\tau,\theta,r)$ $\displaystyle=$
$\displaystyle\frac{e^{-f(\lambda,r,\nu)\gamma_{0}\tau}(\partial_{\nu}f(\lambda,r,\nu))^{2}(\gamma_{0}\tau)^{2}\cos^{2}\frac{\theta}{2}}{2(e^{f(\lambda,r,\nu)\gamma_{0}\tau}-1)}$
(27) $\displaystyle\times(2e^{f(\lambda,r,\nu)\gamma_{0}\tau}-1+\cos\theta).$

For convenience, we adopt the notations $f_{r}$, $f_{\alpha}$ and $f_{z}$ as
$f_{r}(\omega_{0},r,\nu)$, $f_{\alpha}(\omega_{0},r,\nu)$ and
$f_{z}(\omega_{0},r,\nu)$, and adopt the notation $F_{\nu}$ as
$F_{\nu}(\omega,\nu,\tau,\theta,r)$ in the following. Note that different
polarizations have different contributions to the parameters $A$ and $B$
discussed above, thus different effects on the QFI. Finding out what kind of
polarization is optimal for the QFI should be a significant issue for the
detection of cosmic string. We will investigate the QFI of deficit angel
parameter $\nu$ with the radial polarization, the tangential polarization and
the parallel polarization in the following.

###  IV.2 The estimation of deficit angel parameter for different
polarizations

For the radial polarization case, we have
$(\zeta_{r},\zeta_{\alpha},\zeta_{z})=(1,0,0)$ and the corresponding QFI is
given by

$F_{\nu}=\frac{e^{-f_{r}\gamma_{0}\tau}(\partial_{\nu}f_{r})^{2}(\gamma_{0}\tau)^{2}(2e^{f_{r}\gamma_{0}\tau}-1+\cos\theta)\cos^{2}\frac{\theta}{2}}{2(e^{f_{r}\gamma_{0}\tau}-1)}\;.$
(28)

The QFI is independent on the phase $\phi$ of the initial state, and it only
depends on the evolution time, $\tau$, the deficit angel, $\nu$, the initial
state parameter, $\theta$, and distance of the detector relative to the
string, $r$. Here $r$ is in the unit of $\frac{c}{\omega_{0}}$, and $\tau$ is
in the unit of $\frac{1}{\gamma_{0}}$. The units of $F_{\nu}$ depends on the
deficit angle parameter. Since the deficit angle parameter is dimensionless,
we obtain $F_{\nu}$ is dimensionless here. In order to ensure that the
coordinates quantity in the following figures are dimensionless, we have
worked with quantities by rescaling the time and the distance

$\displaystyle\tau\longmapsto\widetilde{\tau}\equiv\gamma_{0}\tau,~{}~{}~{}~{}r\longmapsto\widetilde{r}\equiv
r\omega_{0}.$ (29)

For convenience, we continue to term $\widetilde{\tau}$ and $\widetilde{r}$ as
$\tau$ and $r$, respectively in the following.

For the tangential polarization
$(\zeta_{r},\zeta_{\alpha},\zeta_{z})=(0,1,0)$, we find that the corresponding
QFI

$F_{\nu}=\frac{e^{-f_{\alpha}\tau}(\partial_{\nu}f_{\alpha})^{2}\tau^{2}(2e^{f_{\alpha}\tau}-1+\cos\theta)\cos^{2}\frac{\theta}{2}}{2(e^{f_{\alpha}\tau}-1)}\;.$
(30)

Besides, for the scenario where the polarization is parallel to the string we
have $(\zeta_{r},\zeta_{\alpha},\zeta_{z})=(0,0,1)$, then the corresponding
QFI is given by

$F_{\nu}=\frac{e^{-f_{z}\tau}(\partial_{\nu}f_{z})^{2}\tau^{2}(2e^{f_{z}\tau}-1+\cos\theta)\cos^{2}\frac{\theta}{2}}{2(e^{f_{z}\tau}-1)}\;.$
(31)

Note that for both of the tangential polarization and parallel polarization
cases, we find that their corresponding QFIs are also independent of quantum
phase $\phi$ of the initial sate. Besides, if the effective distance $r\ll 1$,
we can find $f_{r}(r,\nu)\approx
f_{\alpha}(r,\nu)\approx\frac{3\nu^{2}(\nu+1)}{\Gamma[2\nu+2]}r^{2(\nu-1)}$,
and $f_{z}(r,\nu)\approx\nu$. Then the QFIs discussed above can be reduced to
a simpler form, and it is interesting to note that in this case the QFI for
the radial and tangential polarization cases is the same.

Figure 3: (color online). QFI of the deficit angle parameter $\nu$ as a
function of the effective time $\tau$ and the initial state parameter
$\theta$. The left to right panels correspond to the radial polarization
$(\zeta_{r},\zeta_{\alpha},\zeta_{z})=(1,0,0)$, the tangential polarization
$(\zeta_{r},\zeta_{\alpha},\zeta_{z})=(0,1,0)$ and the parallel polarization
$(\zeta_{r},\zeta_{\alpha},\zeta_{z})=(0,0,1)$, respectively. We take the
effective distance $r=0.1$ and the deficit angle parameter $\nu=1.5$. Here
$F_{\nu}$ and $\nu$ are dimensionless. $\theta$ is the state parameter, which
is expressed in radian. $\tau$ and $r$ are also dimensionless by rescaling,
actually $r$ is in the unit of $\frac{c}{\omega_{0}}$, and $\tau$ is in the
unit of $\frac{1}{\gamma_{0}}$. Figure 4: (color online). QFI of the deficit
angle parameter $\nu$ as a function of the effective time $\tau$ and the
effective distance $r$. The left to right panels correspond to the radial
polarization $(\zeta_{r},\zeta_{\alpha},\zeta_{z})=(1,0,0)$, the tangential
polarization $(\zeta_{r},\zeta_{\alpha},\zeta_{z})=(0,1,0)$ and the parallel
polarization $(\zeta_{r},\zeta_{\alpha},\zeta_{z})=(0,0,1)$, respectively. We
take the deficit angle parameter $\nu=1.5$ and the initial state parameter
$\theta=0$. Here $F_{\nu}$ and $\nu$ are dimensionless. $\theta$ is the state
parameter, which is expressed in radian. $\tau$ and $r$ are also dimensionless
by rescaling, actually $r$ is in the unit of $\frac{c}{\omega_{0}}$, and
$\tau$ is in the unit of $\frac{1}{\gamma_{0}}$.

Figure 5: (color online). QFI of the deficit angle parameter $\nu$ as a
function of the effective distance $r$ with fixed values of the parameter
$\nu=1.5,1.8,2.0$. The left to right panels correspond to the radial
polarization $(\zeta_{r},\zeta_{\alpha},\zeta_{z})=(1,0,0)$, the tangential
polarization $(\zeta_{r},\zeta_{\alpha},\zeta_{z})=(0,1,0)$ and the parallel
polarization $(\zeta_{r},\zeta_{\alpha},\zeta_{z})=(0,0,1)$, respectively. We
take the effective time $\tau=4$ and the initial state parameter $\theta=0$.
Here $F_{\nu}$ and $\nu$ are dimensionless. $\theta$ is the state parameter,
which is expressed in radian. $\tau$ and $r$ are also dimensionless by
rescaling, actually $r$ is in the unit of $\frac{c}{\omega_{0}}$, and $\tau$
is in the unit of $\frac{1}{\gamma_{0}}$.

###  IV.3 A comparison of the estimation for different polarizations

In this section, we would like to numerically compare the QFIs of different
polarization cases discussed above. To better illustrate our results, we
present several figures below.

The QFIs of the deficit angle parameter $\nu$ are plotted as a function of the
effective time $\tau$ and the initial states (denoted as different $\theta$),
which are shown in Figs. 3. It is found that the QFIs for different
polarization cases behave the same: With the increase of the evolution time,
they increase at the beginning to a maximum, then decrease monotonously, and
finally vanish. It means that there is an optimal detection time at which the
highest precision for estimating the deficit angle can be possibly achieved
during the exploration of cosmic string spacetime. Note that the QFIs for
$\theta=\pi$ cases (i.e., the initial detector state is ground state) keep
zero over the whole evolution time, which behave quite different from other
initial states cases. The reason is that the quantum vacuum fluctuating fields
in the cosmic string spacetime can not induce the spontaneous excitation of
the static detector, however, can induce its spontaneous emission. Therefore,
the detector has no response to the quantum vacuum fluctuating fields when it
is initially prepared at the ground state. However, if the detector is
initially prepared at the excited state or its quantum superposition with the
ground state, its evolution as a consequence of spontaneous emission would
depend on the spacetime characteristic through the detector-field interaction.
Note that for all the considered polarization cases, the QFIs always achieve
their peak values with $\theta=0$ at a certain time, i.e., the initial excited
state. Thus the excited state is the most sensitive state for the detection of
cosmic string spacetime. This is because that the interaction between detector
and fields, as discussed above, only causes the static atom to emit
spontaneously, and thus only the excited state of the detector will response
to the vacuum- fluctuation fields as a consequence. We conclude that the
maximum sensitivity in the estimation for the deficit angle parameter $\nu$
can be obtained by initial preparation of the detector in the excited state.
Note that this conclusion is universal, and it is actually valid for any
deficit angle parameter $\nu$ and the effective distance $r$. To verify its
universality, we can see from Eq. (27) that when the initial ground state is
prepared (i.e., $\theta=\pi$), the QFI keeps vanishing for any defect angle
parameter, evolution time, polarization direction, and effective distance,
then we take derivative of the QFI with respective to the initial state
parameter $\theta$, and find

$\partial_{\theta}F_{\nu}=-\frac{(\partial_{\nu}f(\lambda,r,\nu))^{2}\tau^{2}{(e^{f(\lambda,r,\nu)\tau}+\cos\theta)}\sin\theta}{2e^{f(\lambda,r,\nu)\tau}(e^{f(\lambda,r,\nu)\tau}-1)}.$
(32)

Clearly, the solutions to $\partial_{\theta}F_{\nu}=0$ are $\theta=0,\pi$,
which means these two values are the points at which the QFI has extreme
value. Since $\partial_{\theta}F_{\nu}\leq 0$ for all $\theta\in[0,\pi]$, we
obtain $\theta=0$ corresponds to the maximum point and $\theta=\pi$
corresponds to the minimum point. Besides, as shown in the Figure the
polarization direction affects the magnitude of the QFI significantly. The
maximum QFI magnitude of the radial polarization and the tangential
polarization case are almost two orders bigger than the parallel polarization
case. We can also find that the response width of the QFI to evolution time
behaves differently for different polarizations. The QFI for the parallel
polarization case has the narrowest response width compared with the other two
cases.

We plot the QFIs of the deficit angle parameter $\nu$ as a function of the
effective distance $r$ and the effective time $\tau$ with the deficit angle
parameter $\nu=1.5$ shown in Fig. 4. We find that although different
polarizations will cause different behaviors of QFI, for all the polarization
cases the optimal position at which the QFIs would achieve their maximum
values is the same, i.e., $r\ll 1$. It means the optimal precision would be
achieved there. We can also find that the QFIs for the radial polarization and
tangential polarization cases are more than two orders larger than that for
the parallel polarization case. Remarkably, when $r\ll 1$, the response time
for the radial and the tangential polarization cases is much longer than that
of parallel polarization.

To further illustrate the dependence of QFI on the position of the detector,
by fixing the evolution time we plot the QFI as a function of $r$ with
different fixed deficit angle parameter $\nu$ shown in Fig. 5. For the radial
polarization and the tangential polarization cases, the deficit angle
parameter affects the magnitude of the QFI dramatically when $r<1$, and the
QFIs keep oscillating while changed relatively slightly when $r>2$. In
addition, we can numerically find the maximum QFIs and the corresponding
parameters for all the polarization cases. In our example where three
different deficit angle parameters are chosen ($\nu=1.5,1.8,2$), we find that
for a fixed evolution time and initial state ($\theta=0$) the maximum QFIs for
the radial and tangential polarization occur when $\nu=1.5$ and the detector
is closed to the string at $r=0.14$. Correspondingly, the maximum QFIs are
$F_{\nu max}=8.513$ for the radial polarization case and $F_{\nu max}=7.796$
for the tangential polarization, respectively. However, unlike the former two
cases, the maximum QFI of parallel polarization ($F_{\nu max}=0.2285$) occurs
when $\nu=2$ and $r=2.29$. This means that for a fixed time the sensitivity of
the detector depends on both the polarization and the deficit angle parameter.
Comparing the three maximum QFIs, the former two cases are almost two orders
bigger than the latter one obviously.

Figure 6: (color online). QFI of the deficit angle parameter $\nu$ as a
function of itself and the effective time $\tau$ with fixed value of the
initial state parameter $\theta=0$. The left to right panels correspond to the
radial polarization $(\zeta_{r},\zeta_{\alpha},\zeta_{z})=(1,0,0)$, the
tangential polarization $(\zeta_{r},\zeta_{\alpha},\zeta_{z})=(0,1,0)$ and the
parallel polarization $(\zeta_{r},\zeta_{\alpha},\zeta_{z})=(0,0,1)$,
respectively. We take the effective distance $r=0.1$. Here $F_{\nu}$ and $\nu$
are dimensionless. $\theta$ is the state parameter, which is expressed in
radian. $\tau$ and $r$ are also dimensionless by rescaling, actually $r$ is in
the unit of $\frac{c}{\omega_{0}}$, and $\tau$ is in the unit of
$\frac{1}{\gamma_{0}}$.

Fig. 6 shows how the estimation of the deficit angle $\nu$ is affected by
itself. For the fixed initial states, we find that the QFIs for the radial
polarization and tangential polarization cases behave similarly, while they
are quite different from the parallel polarization case. Fixing the time, for
the radial and tangential polarization cases, with the increase of the deficit
angle $\nu$ the QFIs increase at the beginning, reach a maximum value, then
decrease monotonously, and go to zero eventually. However, for the parallel
polarization case, the QFI decreases monotonously to zero with the increase
the deficit angle $\nu$. Again, the QFIs for different polarization cases
achieve their maximum values at different deficit angle, and the corresponding
maximum values can differ by orders of magnitude.

##  V Discussion

Recently, many desktop experiments have been designed with atoms and molecules
to explore the new physics, and similar model to our scenario has been used
thereof to detect the electromagnetic fields induced by new interaction
Safronova . By analyzing the QFI, we in the above have discussed how a two-
level detector detects the quantum vacuum fluctuations of electromagnetic
fields scattered by the cosmic string spacetime. However, for a more realistic
cosmic environment, the thermal noise is inevitable. Consider the relevant
analysis in a thermal environment with temperature $T$, the corresponding
Wightman function of the electromagnetic fields along the detector’s
trajectory is given by H Cai ; huang

$\displaystyle G_{rr}(\tau-\tau^{\prime})$ $\displaystyle=$
$\displaystyle\frac{\nu}{8\pi^{2}}\int
d\mu_{j}\operatorname{csch}\frac{\omega_{j}}{T}\cos(\omega_{j}\vartriangle\tau+\frac{i\omega_{j}}{T})\biggl{[}\frac{\omega_{j}}{2}V-\frac{1}{\omega_{j}}\biggl{(}\frac{\partial
J_{|\nu m|}(k_{\perp}r)}{\partial r}\biggr{)}^{2}\biggr{]}\;,$ (33)
$\displaystyle G_{\alpha\alpha}(\tau-\tau^{\prime})$ $\displaystyle=$
$\displaystyle\frac{\nu r^{2}}{8\pi^{2}}\int
d\mu_{j}\operatorname{csch}\frac{\omega_{j}}{T}\cos(\omega_{j}\vartriangle\tau+\frac{i\omega_{j}}{T})\biggl{[}\frac{\omega_{j}}{2}V-\frac{\nu^{2}m^{2}}{\omega_{j}r^{2}}J_{|\nu
m|}^{2}(k_{\perp}r)\biggr{]}\;,$ (34) $\displaystyle
G_{zz}(\tau-\tau^{\prime})$ $\displaystyle=$
$\displaystyle\frac{\nu}{8\pi^{2}}\int
d\mu_{j}\operatorname{csch}\frac{\omega_{j}}{T}\cos(\omega_{j}\vartriangle\tau+\frac{i\omega_{j}}{T})\frac{k_{\perp}^{2}}{\omega_{j}}J_{|\nu
m|}^{2}(k_{\perp}r)\;,$ (35)

with $V=J_{|\nu m+1|}^{2}(k_{\perp}r)+J_{|\nu m-1|}^{2}(k_{\perp}r)$. Then the
associated Fourier transform is

$\displaystyle{\cal
G}(\lambda)=\bigg{\\{}\begin{array}[]{c}\sum_{i}\frac{e^{2}|\langle-|r_{i}|+\rangle|^{2}\lambda^{3}}{3\pi\,f^{-1}_{i}(\lambda,r,\nu)}(N(\omega)+1),~{}~{}(\omega>0),\\\
\sum_{i}\frac{e^{2}|\langle-|r_{i}|+\rangle|^{2}\lambda^{3}}{3\pi\,f^{-1}_{i}(\lambda,r,\nu)}N(|\omega|),~{}~{}~{}~{}~{}~{}~{}~{}(\omega<0),\end{array}$
(38)

where $N=1/(e^{\omega/T}-1)$ and $f_{i}(\lambda,r,\nu)$ is what we have show
in the above. The corresponding coefficients of the Kossakowski matrix
$a_{ij}$ is

$\displaystyle A$ $\displaystyle=$
$\displaystyle\frac{\gamma_{0}}{4}\sum_{i}\zeta_{i}f_{i}(\lambda,r,\nu)(2N+1),$
$\displaystyle B$ $\displaystyle=$
$\displaystyle\frac{\gamma_{0}}{4}\sum_{i}\zeta_{i}f_{i}(\lambda,r,\nu).$ (39)

For $N=\frac{1}{e^{\omega/T}-1}$, we have let $\hbar=k_{B}=1$. By considering
the dimension, we obtain $N=\frac{1}{e^{\hbar\omega/K_{B}T}-1}$. For the
reduced Planck constant $\hbar=1.0546\times 10^{-34}J\cdot s$, the Boltzmann
constant $k_{B}=1.38\times 10^{-23}J/K$, the typical transition frequency of
the hydrogen atom $\omega\sim 10^{15}s^{-1}$ JHu , and the temperature for a
realistic cosmic environment is $T\cong 2.76K$ Turner , we obtain that
$N=\frac{1}{e^{\hbar\omega/k_{B}T}-1}\sim\frac{1}{e^{10^{4}}}\rightarrow 0$.
Then we arrive

$A=B=\frac{\gamma_{0}}{4}\sum_{i}\zeta_{i}f_{i}(\lambda,r,\nu)\;.$ (40)

It means that one can actually neglect the thermal noise. The corresponding
quantum Fisher information is thus

$\displaystyle F_{\nu}(\omega,\nu,\tau,\theta,r)$ $\displaystyle=$
$\displaystyle\frac{e^{-f(\lambda,r,\nu)\gamma_{0}\tau}(\partial_{\nu}f(\lambda,r,\nu))^{2}(\gamma_{0}\tau)^{2}\cos^{2}\frac{\theta}{2}}{2(e^{f(\lambda,r,\nu)\gamma_{0}\tau}-1)}$
(41) $\displaystyle\times(2e^{f(\lambda,r,\nu)\gamma_{0}\tau}-1+\cos\theta),$

which is the same result to the vacuum case shown above. Therefore, in a more
realistic cosmic environment, the results for the vacuum case discussed above
are still valid.

Besides, note that actually the cosmic strings have been explored in the
analogue gravity with various experimental platforms, e.g., superfluid Hendry
and anisotropic medium Moraes , and so on. Our results can also be examined in
such analogue gravity systems. When one consider the quantum simulation of
cosmic string in a condensed matter system, the density fluctuations thereof
can be used to simulate the quantum fields, and an impurity can be introduced
as the detector coupled to the density fluctuations. This model is proposed in
Refs. Recati ; Fedichev , and has been experimentally realized recently Schmid
; Zipkes . By controlling the properties of the condensed matter and checking
the dynamics of the detector, one can in principle test our results here in
the analogue gravity.

##  VI conclusion

Motivated by detecting the topological defects in cosmic string spacetime, we
use a static two-level atom which is coupled to the electromagnetic fields as
the detector to estimate the deficit angle parameter $\nu$. To obtain optimal
estimation conditions, measurements are performed on the detector to make QFI
values be maximized for different parameters. We focus on the QFI of $\nu$
with different polarizations, and find that the polarization direction plays a
very important role in the estimation of the deficit angle $\nu$, and it not
only affects the behavior of the QFI, but also determines its maximum
magnitude. For different polarization cases, the QFIs can differ by orders of
magnitude. It is found that one can not extract any information about the
cosmic string spacetime if the detector is initially prepared at the ground
state. The optimal initial state to estimate the deficit angle $\nu$ for all
the polarization cases is the pure excited state. The optimal point via the
evolution time $\tau$, the effective distance $r$, and the deficit angle $\nu$
itself, at which the QFI achieves the maximum value and thus the possible
estimation precision is optimal, depends on the polarization direction. Our
results are important extension of quantum metrology to the relativistic
framework, and could facilitate the exploration of cosmic string spacetime
characteristic in the future.

###### Acknowledgements.

Z. T. was supported by the National Natural Science Foundation of China under
Grant No. 11905218, and the CAS Key Laboratory for Research in Galaxies and
Cosmology, Chinese Academy of Sciences (No. 18010203). Y. Y. was supported by
the National Natural Science Foundation of China under Grant No. 12105097, and
Scientific Research Fund of Hunan Provincial Education Department(No.
20C0787). J. J. was supported by the National Natural Science Foundation of
China under Grant No. 11875025.

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