aainstitutetext: School of Physics, Xi’an Jiaotong University, Xi’an, Shaanxi
710049, Chinabbinstitutetext: Institute of Theoretical Physics, Xi’an Jiaotong
University, Xi’an, Shaanxi 710049, Chinaccinstitutetext: School of Mathematics
and Physics, The University of Queensland, Brisbane, QLD 4072, Australia

# Coherence revival and metrological advantage for moving Unruh-DeWitt
detector

Jun Feng,111Corresponding author. a Jiafan Wang a Shuaijie Li
<EMAIL_ADDRESS>
###### Abstract

In this paper, we investigate the quantum coherence extraction for two
accelerating Unruh-DeWitt detectors coupling to a scalar background in $3+1$
Minkowski spacetime. We find that quantum coherence as a sort of nonclassical
correlation can be generated through the Markovian evolution of the detectors
system just like quantum entanglement. However, with growing Unruh
temperature, in contrast to monotonous degrading entanglement, we find that
quantum coherence exhibits a striking revival phenomenon. That is, for certain
choice of detectors’ initial state, coherence measure will reduce to zero
firstly then grow up to an asymptotic value. We verify such coherence revival
by inspecting its metrological advantage on enhancing the quantum Fisher
information (QFI). Since the maximal QFI bounding the accuracy of a quantum
measurement, we conclude that the extracted coherence can be utilized as a
physical resource in quantum metrology.

###### Keywords:

Unruh Effect; Open Quantum System; Quantum Coherence

##  1 Introduction

Nonclassical correlation has long been regarded as a key physical resource. A
notably example is quantum entanglement which plays a central role in many
areas of fundamental physics Entg1 ; Entg2 , such as quantum information
theory, black hole physics and cosmology. Other types of quantum correlation
beyond entanglement, such as nonlocality Entg3  and quantum discord Entg4 ,
were also exploited. However, there are more general notions on quantum
resource charactering the nonclassicality that makes a system inherently
quantum. In recent years, a significant amount of work has been done on
quantum coherence, which encapsulates the defining features of quantum theory
in arbitrary dimensions, from superposition principle of single qubit to
quantum correlations among multipartite. Quantum coherence constitutes a
powerful resource Coh1 ; Coh2 ; Coh3 , which was wildly applied in quantum
metrology Coh4 , biological systems Coh5  and thermodynamics Coh6 . Moreover,
on a fundamental level, it has been shown Coh7 ; Coh8  that coherence and
entanglement are actually intimately related as they are interchangeable and
can be unified into a single resource (see also Coh9 ; Coh10  for a review).

On the other hand, it was realized that the intrinsic nonclassical correlation
of quantum vacuum can be extracted by designed physical system. For instance,
two initially uncorrelated Unruh-DeWitt (UDW) detectors in Minkowski spacetime
can generate nonlocal entanglement under relativistic acceleration motion
Entg5 ; Entg6 ; Entg7 , and be witnessed by various measures, such as
uncertainty relation F1 ; F2  and Bell-type inequalities Entg7+ ; F3 . In
above context, the residual entanglement at final equilibrium is a result of
competition between the Unruh decoherence and entanglement generation from
Markovian evolution of the detectors that interacts to background field. In
other word, even with the entanglement generation, Unruh effect plays solely
the role as an environment decoherence U1 .

Beyond entanglement existing among multipartite quantum system, it is
intriguing to exploit the possibility of extraction of more general
quantumness from quantum vacuum. Earlier study Coh-U1  has observed that the
extraction of quantum coherence is possible even for _single_ UDW detector.
For certain initial energy of the background field and the duration of the
interaction, a detector moving with uniform acceleration can extract a larger
amount of coherence from field state, comparing to a detector at rest. In
other word, the rate of coherence loss may become slower for a moving detector
than for a detector at rest sometimes. Similarly, for a spatially extended UDW
detector, coherence harvesting and swelling was also observed Coh-U2 . Such
extracted coherence was shown to be catalytic which means that the same amount
coherence can be repeatedly extracted.

In this paper, we concern about the quantum coherence extraction for a system
of _multi_ -UDW detectors, from a coupling to a massless scalar field in $3+1$
Minkowski spacetime. In particular, we exploit the quantum coherence of a
combined system with two UDW detectors, both under a uniform acceleration.
Motivated by the entanglement generation for multi-UDW detectors Entg6 , we
anticipate that quantum coherence can also be generated after the system
approaching its equilibrium, even with two detectors initially are in an
incoherent state. Starting from an arbitrary initial state, however, we are
interested in the necessary conditions that detectors should satisfy to ensure
the coherence extraction, i.e., the residual coherence at equilibrium is
larger than the initial quantum coherence of detectors.

While quantum fluctuation of background field plays a role of an environment,
it is properly in our context to treat two-detectors as an open quantum system
Op1 , whose reduced density matrix evolves via a quantum master equation,
derived by tracing out the environmental degrees of freedom. For the detectors
with uniform acceleration $a$, this formalism may lead us to a naively
expectation that just as illustrated for entanglement in various scenarios
Entg8 ; Entg9 , Unruh effect or environmental decoherence should also result a
decay of quantum interference in two-detectors system. However, we will show
that quantum coherence indeed exhibits a striking _revival_ phenomenon that
for certain choice of detectors’ initial state, coherence measure Coh1  will
reduce to zero firstly then grow up and finally approach to an asymptotic
value, while Unruh temperature $T_{U}=a/2\pi$ keep growing during the process.

As an important physical resource, we expect that above coherence revival can
be verified in some practical quantum process by providing an enhanced
performance. We note that in metrological tasks, nonclassicality rather than
entanglement is a necessary resource to achieve quantum advantages metro1 ;
metro2 . As a particular nonclassicality which is more general than
entanglement, quantum coherence is intimately related to the quantum Fisher
information (QFI) Coh11 ; Coh12 , a key quantity bounding the ultimate
precision of the estimation of a parameter encoded in a quantum state222For
instance, in the context of quantum thermodynamics, coherence cost of any
state is determined by its QFI Coh13 .. Therefore, motivated by the close
relation between the QFI and quantum coherence, we will evaluate the QFI for
the two-UDW detectors, with Unruh temperature as a parameter chosen to be
optimally estimated. We will show that the revival coherence corresponds to
the enhanced QFI, thus may provide a significant quantum advantage in
metrological task.

The paper is organized as follows. In Sec. 2, we review the open quantum
system approach to resolve the reduced state of two accelerating UDW
detectors. In Sec. 3, we evaluate the quantum coherence of final equilibrium
state of two-detectors system, and demonstrate the related coherence revival
phenomenon. We give the condition of initial state preparation for detectors
to ensure the occurring of coherence revival. In Sec. 4, we evaluate the QFI
for the system and show a significant enhancement of QFI can be obtained due
to the coherence resource. We conclude in Sec. 5 with some remarks on possible
root of coherence revival and its distinction to entanglement. Throughout the
paper, we use units where $k_{B}=c=\hbar=1$.

##  2 Accelerating UDW detectors as open quantum system

To proceed, we first recall the dynamics of two accelerating UDW detector in
$3+1$-dimensional Minkowski spacetime Entg6 . With a coupling to the bath of
fluctuating background, we treat two-detectors as an open quantum system,
whose density matrix is governed by a Lindblad master equation and evolves
non-unitarily, due to environment decoherence or dissipation on the system.

Modeling each detector as a two-level atom, the total Hamiltonian of the
combined system of detectors and background field is

$H=\frac{\omega}{2}\Sigma_{3}+H_{\Phi}+\mu H_{I}$ (1)

where $\omega$ is the energy level spacing of the atom and $\Sigma_{3}$ is one
of symmetrized bipartite operators
$\Sigma_{i}\equiv\sigma_{i}^{(A)}\otimes\mathbf{1}^{(B)}+\mathbf{1}^{(A)}\otimes\sigma_{i}^{(B)}$,
defined by Pauli matrices $\\{\sigma^{(\alpha)}_{i}|i=1,2,3\\}$ with
superscript $\\{\alpha=A,B\\}$ labeling distinct atoms. $H_{\Phi}$ is the
Hamiltonian of free massless scalar fields $\Phi(t,\mathbf{x})$ satisfying
standard Klein-Gordon equation. The interaction Hamiltonian can be written in
a dipole form Op2
$,H_{I}=(\sigma_{2}^{(A)}\otimes\mathbf{1}^{(B)})\Phi(t,\mathbf{x}_{1})+(\mathbf{1}^{(A)}\otimes\sigma_{2}^{(B)})\Phi(t,\mathbf{x}_{2})$,
between the atoms and the fluctuating field bath.

Assuming a weak coupling between two-atom system and environment ($\mu\ll 1$),
such that the initial state of combined system is approximated as
$\rho_{tot}(0)=\rho_{AB}(0)\otimes|0\rangle\langle 0|$, where $\rho_{AB}(0)$
is the initial state of the detectors and $|0\rangle$ is the field vacuum.
Obviously, the dynamics of $\rho_{tot}$ should be governed by a unitary
evolution via von Neumann equation
$\dot{\rho}_{tot}(\tau)=-i[H,\rho_{tot}(\tau)]$, where $\tau$ is the proper
time of the atom. Whenever the typical time scale of the environment is much
smaller than that of the detectors, we can further assume them under a
Markovian evolution. The reduced dynamics of the detectors is then obtained by
integrating over the background field degrees from the $\rho_{tot}(\tau)$,
driven by a quantum dynamical semigroup of completely positive map. This can
further be represented by a Kossakowski-Lindblad master equation Op3 ; Op4

$\frac{\partial\rho_{AB}(\tau)}{\partial\tau}=-i[H_{\tiny\mbox{eff}},\rho_{AB}(\tau)]+\mathcal{L}\left[\rho_{AB}(\tau)\right]$
(2)

where

$\mathcal{L}\left[\rho\right]=\sum_{\begin{subarray}{c}i,j=1,2,3\\\
\alpha,\;\beta=A,B\end{subarray}}\frac{C_{ij}}{2}\big{[}2\sigma_{j}^{(\beta)}\rho_{AB}\sigma_{i}^{(\alpha)}-\\{\sigma_{i}^{(\alpha)}\sigma_{j}^{(\beta)},\rho_{AB}\\}\big{]}$
(3)

representing a dissipative evolution attributed to the interaction between the
detectors and external field. The Kossakowski matrix $C_{ij}$ can be
explicitly resolved. After introducing the Wightman function of scalar field
$G^{+}\left(x,x^{\prime}\right)=\left\langle
0\left|\Phi(x)\Phi\left({x}^{\prime}\right)\right|0\right\rangle$, its Fourier
transform

$\mathcal{G}(\lambda)=\int_{-\infty}^{\infty}d\tau~{}e^{i\lambda\tau}G^{+}(\tau)=\int_{-\infty}^{\infty}d\tau~{}e^{i\lambda\tau}\left\langle\Phi(\tau)\Phi(0)\right\rangle$
(4)

determines the coefficients $C_{ij}$ by a decomposition

$C_{ij}=\frac{\gamma_{+}}{2}\delta_{ij}-i\frac{\gamma_{-}}{2}\epsilon_{ijk}\delta_{3,k}+\gamma_{0}\delta_{3,i}\delta_{3,j}$
(5)

where

$\gamma_{\pm}=\mathcal{G}(\omega)\pm\mathcal{G}(-\omega),~{}~{}~{}\gamma_{0}=\mathcal{G}(0)-\gamma_{+}/2$
(6)

Moreover, the interaction with external scalar field would also induce a Lamb
shift contribution for the detector effective Hamiltonian $H_{\mbox{\tiny
eff}}=\frac{1}{2}\tilde{\omega}\sigma_{3}$, in terms of a renormalized
frequency $\tilde{\omega}=\omega+i[\mathcal{K}(-\omega)-\mathcal{K}(\omega)]$,
where
$\mathcal{K}(\lambda)=\frac{1}{i\pi}\mbox{P}\int_{-\infty}^{\infty}d\omega\frac{\mathcal{G}(\omega)}{\omega-\lambda}$
is Hilbert transform of Wightman functions.

Following a trajectory of the accelerating detectors, one can find that the
field Wightman function fulfills the Kubo-Martin-Schwinger (KMS) condition,
i.e., $G^{+}(\tau)=G^{+}(\tau+i\beta)$, where $\beta\equiv 1/T_{U}=2\pi/a$ is
recognized. Translating it into frequency space, one has

$\mathcal{G}(\lambda)=e^{\beta\omega}\mathcal{G}(-\lambda)$ (7)

Using translation invariance $\langle
0|\Phi(x(0))\Phi(x(\tau))|0\rangle=\langle
0|\Phi(x(-\tau))\Phi(x(0))|0\rangle$, after some algebras, we find (6) can be
resolved as

$\displaystyle\gamma_{+}$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}d\tau~{}e^{i\lambda\tau}\langle
0|\left\\{\Phi(\tau),\Phi(0)\right\\}|0\rangle=\left(1+e^{-\beta\omega}\right)\mathcal{G}(\omega),$
$\displaystyle\gamma_{-}$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}d\tau~{}e^{i\lambda\tau}\langle
0|\left[\Phi(\tau),\Phi(0)\right]|0\rangle=\left(1-e^{-\beta\omega}\right)\mathcal{G}(\omega)$
(8)

holding true for generic interacting fields. For later use, we also introduce
the ratio

$\gamma\equiv\gamma_{-}/\gamma_{+}=\frac{1-e^{-\beta\omega}}{1+e^{-\beta\omega}}=\tanh(\beta\omega/2)$
(9)

which depends solely on the Unruh temperature $T_{U}$ due to the frequency KMS
condition (7), but has nothing to do with local correlator of the background.

By expressing the reduced density matrix of the two-atom system as

$\rho_{AB}(\tau)=\frac{1}{4}[\mathbf{1}^{A}\otimes\mathbf{1}^{B}+\sum_{i=1}^{3}\rho_{i}\Sigma_{i}+\sum_{i,j=1}^{3}\rho_{ij}\sigma_{i}^{A}\otimes\sigma_{j}^{B}]$
(10)

and inserting it back into (2), one can derive F1  the reduced density matrix
of two-UDW detector at equilibrium with an X-type structure

$\rho_{AB}(\omega,\beta,\Delta_{0})=\left(\begin{array}[]{cccc}A&0&0&0\\\
0&C&D&0\\\ 0&D&C&0\\\ 0&0&0&B\end{array}\right)$ (11)

where

$\displaystyle\displaystyle A$ $\displaystyle=$
$\displaystyle\frac{(3+\Delta_{0})(\gamma-1)^{2}}{4\left(3+\gamma^{2}\right)}~{}~{}~{}~{},~{}~{}~{}~{}\displaystyle
B=\frac{(3+\Delta_{0})(\gamma+1)^{2}}{4\left(3+\gamma^{2}\right)}$
$\displaystyle\displaystyle C$ $\displaystyle=$
$\displaystyle\frac{3-\Delta_{0}-(\Delta_{0}+1)\gamma^{2}}{4\left(3+\gamma^{2}\right)}~{}~{},~{}~{}\displaystyle
D=\frac{\Delta_{0}-\gamma^{2}}{2\left(3+\gamma^{2}\right)}$ (12)

One can observe that the final equilibrium state of two-detectors system now
depends on the ratio $\gamma$ characterizing the thermal nature of Unruh
effect, as well as the choice of the initial state encoded in
$\Delta_{0}=\sum_{i}\text{Tr}[\rho_{AB}(0)\sigma^{(A)}_{i}\otimes\sigma^{(B)}_{i}]$,
a dimensionless constant of motion satisfying $-3\leqslant\Delta_{0}\leqslant
1$ to keep $\rho_{AB}(0)$ positive.

##  3 Quantum coherence and its revival

###  3.1 Quantum coherence monotone

We first review some elementary concepts concerning coherence measures. The
notion of coherence admitted in this paper is identified to the one in Coh1 ,
where in order to specify a reasonable measure of quantum coherence, the
following set of axioms should be satisfied. For a given fixed basis
$\\{|i\rangle\\}$, the set of _incoherent states_ $\mathcal{I}$ is the set of
quantum states with diagonal density matrices with respect to this basis.
Incoherent completely positive and trace preserving maps (ICPTP) are maps that
map every incoherent state to another incoherent state. Given this, we say
that $\mathcal{C}$ is a proper measure of quantum coherence if it satisfies
following properties:

(C1). $\mathcal{C}(\rho)\geqslant 0$ for any quantum state $\rho$ and equality
holds iff $\rho\in\mathcal{I}$.

(C2a). The measure is non-increasing under a ICPTP map $\Phi$, i.e.,
$\mathcal{C}(\rho)\geqslant\mathcal{C}(\Phi(\rho))$.

(C2b). Monotonicity for average coherence under selective outcomes of ICPTP:
$\mathcal{C}(\rho)\geqslant$ $\sum_{n}p_{n}\mathcal{C}\left(\rho_{n}\right)$,
where $\rho_{n}=K_{n}\rho K_{n}^{\dagger}/p_{n}$ and
$p_{n}=\operatorname{Tr}\left[K_{n}\rho K_{n}^{\dagger}\right]$ for all
$K_{n}$ with $\sum_{n}K_{n}^{\dagger}K_{n}=\mathbf{1}$ and
$K_{n}\mathcal{I}K_{n}^{\dagger}\subseteq\mathcal{I}$.

(C3). Convexity, i.e.
$\lambda\mathcal{C}(\rho)+(1-\lambda)\mathcal{C}(\sigma)\geqslant\mathcal{C}(\lambda\rho+(1-\lambda)\sigma)$,
for any density matrix $\rho$ and $\sigma$ with $0\leqslant\lambda\leqslant
1$.

A general distance-based coherence quantifier can be found Coh9  which
satisfies all the conditions mentioned above, in terms of the minimal distance
between target state and a given incoherent state. Using quantum relative
entropy

$S(\rho\|\sigma)=\operatorname{Tr}\left[\rho\log\rho\right]-\operatorname{Tr}\left[\rho\log\sigma\right]$
(13)

One coherence monotone is $\min_{\sigma\in\mathcal{I}}S(\rho\|\sigma)$ which
can be further written as Coh1

$\mathcal{C}_{\text{R.E. }}({\rho})=S\left({\rho}_{\text{diag
}}\right)-S({\rho})$ (14)

where $S(\hat{\rho})=-\operatorname{Tr}(\hat{\rho}\log\hat{\rho})$ is the von
Neumann entropy for the state
$\hat{\rho}=\sum_{ij}\rho_{ij}\left|i\right\rangle\left\langle j\right|$, and
$\hat{\rho}_{\text{diag }}=\sum_{i}\rho_{ii}\left|i\right\rangle\left\langle
i\right|$ is derived from a dephasing operation on a density matrix.

In following section, we will investigate above coherence monotone for two-UDW
detector system. We should remark that we do not use another well-known
$l_{1}$-norm monotone as authors of Coh-U1 ; Coh-U2  did, because monotone
(14) is a more universal measure even for infinite-dimensional system to avoid
potential divergent Coh-U3 . Although in present case, these two monotones are
equivalent, we remain using (14) for possible extension in more general
context in future work.

###  3.2 Coherence revival

In our context, by diagonalizing the density matrix (11), we obtain the
related eigenvalues

$\displaystyle\lambda_{1}=\frac{(1-\gamma)^{2}(3+\Delta_{0})}{4\left(3+\gamma^{2}\right)}~{}$
,
$\displaystyle~{}\lambda_{2}=\frac{(1+\gamma)^{2}(3+\Delta_{0})}{4\left(3+\gamma^{2}\right)}$
$\displaystyle\lambda_{3}=\frac{(1-\gamma^{2})(3+\Delta_{0})}{3+\gamma^{2}}~{}$
, $\displaystyle~{}\lambda_{4}=\frac{1-\Delta_{0}}{4}$ (15)

and eigenvectors

$\displaystyle|\lambda_{1}\rangle=|00\rangle~{}~{}~{}$ ,
$\displaystyle~{}~{}~{}|\lambda_{2}\rangle=|11\rangle$
$\displaystyle|\lambda_{3}\rangle=|10\rangle-|01\rangle~{}~{}~{}$ ,
$\displaystyle~{}~{}~{}|\lambda_{4}\rangle=|01\rangle-|10\rangle$ (16)

We straightforwardly calculate the relative entropy as

$\mathcal{C}_{R.E.}(\omega,T_{U},\Delta_{0})=-2C\log
C+\lambda_{3}\log\lambda_{3}+\lambda_{4}\log\lambda_{4}$ (17)

which depends on the energy spacing of atom $\omega$, the initial state choice
of detectors encoded in $\Delta_{0}$, as well as Unruh temperature $T_{U}$
proportional to the detectors’ acceleration.

Since $\omega$ and $T_{U}$ are combined in $\gamma$ (9), for simplicity, we
specify $\omega=1$ and depict the related quantum coherence in Fig.1, as a
function of initial state choice $\Delta_{0}$ and Unruh temperature $T_{U}$.
Since Unruh effect in general is recognized as an environment decoherence
TAKAGI1 , we may naively expect that for arbitrary fixed initial state,
quantum coherence $\mathcal{C}_{R.E.}$ would be a monotone decreasing function
of Unruh temperature $T_{U}$. However, we can observe directly from Fig.1 that
corresponding to a class of initial states, the quantum coherence is a non-
monotone function of $T_{U}$. In particular, there is a revival of coherence,
i.e., as Unruh temperature $T_{U}$ growing, the quantum coherence degrades to
zero at first and then increases to a nonvanishing asymptotic value
$\mathcal{C}_{\text{asymp}}$, obtained from (17) by taking an infinite
acceleration limit $\gamma\rightarrow 0$.

Figure 1: When two-detector system evolving to an equilibrium, its quantum
coherence monotone $\mathcal{C}_{R.E.}$ is plotted as a function of initial
state choice $\Delta_{0}$ and Unruh temperature $T_{U}$, for fixed $\omega=1$.
For particular chosen initial states with $\Delta_{0}\in(0,1]$, a coherence
revival phenomenon occurs. That is as Unruh temperature $T_{U}$ growing,
$\mathcal{C}_{R.E.}$ degrades at first to zero and then revives to a
nonvanishing asymptotic value. In the inset, the dependence of asymptotic
quantum coherence $\mathcal{C}_{\text{asymp}}$ on the choice of initial state
is demonstrated.

In the upper line of Fig.2, we plot the quantum coherence as a function of
initial state choice $\Delta_{0}$ and Unruh temperature $T_{U}$, for different
value of detector’s energy spacing. By numerically analysis, we find that
above _coherence revival_ occurs when the initial state satisfying
$\Delta_{0}\in(0,1]$, bounded by the white solid curves in Fig.2(a)-2(c). We
also note that for larger energy spacing of a UDW detector, the revival of
coherence is more suppressed.

A striking difference between quantum coherence and entanglement generated
under Unruh effect Entg5  need to be highlighted. It has been shown that the
survive of entanglement for the equilibrium state of two accelerating
detectors is a result of competition between the Unruh decoherence and
entanglement enhancement from Markovian dynamics. In other word, during the
process of entanglement generation, Unruh effect solely plays a role of
decoherence, therefore, as Unruh temperature growing, the survived
entanglement degrades. To see this, we directly calculate the negativity for
the final equilibrium state of two accelerating UDW detectors, which is a
measure of distillable entanglement. It is defined by
$\mathcal{E}_{\text{neg}}(\rho)=\frac{1}{2}\sum_{i}\left(\left|\lambda_{i}\right|-\lambda_{i}\right)=-\sum_{\lambda_{i}<0}\lambda_{i}$,
where $\lambda_{i}$ are the negative eigenvalues of partial transposed density
matrix. The value of negativity ranges from 0 , for a separable state, to
$0.5$, for a maximally entangled state. Form (11), we can straightforwardly
obtain F1

$\mathcal{E}_{\text{neg}}=\max\left\\{\frac{\sqrt{\left(\Delta_{0}-\gamma^{2}\right)^{2}+\gamma^{2}\left(3+\Delta_{0}\right)^{2}}}{2\left(3+\gamma^{2}\right)}-\frac{\lambda_{1}+\lambda_{2}}{2},0\right\\}$
(18)

which reaches maximum $0.5$ at $\Delta_{0}=-3$. In the lower line of Fig.2, we
plot the negativity for the final equilibrium state of two-detector system for
$\omega=1,3,5$. For varying initial state preparation, the negativity (18) is
vanishing at $\Delta_{0}=\frac{5\gamma^{2}-3}{3-\gamma^{2}}$, presented by the
red solid curves in Fig.2(d)-2(f). It is obvious that entanglement always
degrades monotonously with growing Unruh temperature, i.e., no revival of
entanglement can happen for any initial state preparation. This clearly
demonstrate the distinction between the entanglement and coherence.

(a) $\mathcal{C}_{\text{R.E. }}(\omega=1)$ (b) $\mathcal{C}_{\text{R.E.
}}(\omega=3)$

(c) $\mathcal{C}_{\text{R.E. }}(\omega=5)$ (d)
$\mathcal{E}_{\text{neg}}(\omega=1)$

(e) $\mathcal{E}_{\text{neg}}(\omega=3)$ (f)
$\mathcal{E}_{\text{neg}}(\omega=5)$

Figure 2: (Upper) The quantum coherence of the final equilibrium state of two-
detectors system as a function of initial state preparation encoded by
$\Delta_{0}$ and Unruh temperature $T_{U}$. We demonstrate the coherence
monotone $\mathcal{C}_{R.E.}$ for fixed energy spacing of detector with
$\omega=1,3,5$. In each case, coherence revival can exist for the initial
states with $\Delta_{0}\in(0,1]$. (Lower) Entanglement measured by negativity
is plotted as a function of initial state preparation encoded by $\Delta_{0}$
and Unruh temperature $T_{U}$. For any fixed initial state, no revival of
entanglement with respect to Unruh temperature can occur.

###  3.3 A concrete example

We illustrate our result in a concrete example. Choosing the initial state of
two-detector in a product form

$\rho_{\text{initial}}=\rho_{A}(0)\otimes\rho_{B}(0)$ (19)

The state of each detector can be written in Bloch form

$\rho_{A}(0)=\frac{1}{2}\left(\mathbf{1}+\mathbf{n}\cdot\bm{\sigma}\right),~{}~{}~{}\rho_{B}(0)=\frac{1}{2}\left(\mathbf{1}+\mathbf{m}\cdot\bm{\sigma}\right)$
(20)

where $\mathbf{n}$ and $\mathbf{m}$ are two unit Bloch vectors. Without loss
of generality, taking $\mathbf{n}=(0,0,1)$ and
$\mathbf{m}=(0,\sin\theta,\cos\theta)$, we have
$\Delta_{0}=\mathbf{n}\cdot\mathbf{m}=\cos\theta$ where $\theta\in[0,\pi]$ is
angle between two vectors giving $\Delta_{0}\in[-1,1]$.

The quantum coherence of the initial state (19) is

$\mathcal{C}_{R.E.}(0)=H_{\text{binary}}\left(\frac{\cos\theta+1}{2}\right)$
(21)

where $H_{\text{binary}}(x)\equiv-x\log x-(1-x)\log(1-x)$ is binary entropy of
variable $x$.

After the Markovian evolution, the final equilibrium state of two detectors
possess quantum coherence (17). Therefore, the quantum coherence extracted
through the interaction between the detectors’ system and the background field
is

$\delta\mathcal{C}_{R.E.}=\mathcal{C}_{R.E.}(\omega,T_{U},\theta)-\mathcal{C}_{R.E.}(0)$
(22)

which is depicted for $\omega=1$ in Fig.3.

Figure 3: The quantum coherence extraction for two UDW detectors (fixed
$\omega=1$). For initial states preparation with $\theta\in[0,\frac{\pi}{2}]$,
coherence revival occurs for sufficiently large Unruh temperature. Further,
for initial states with $\theta$ chosen inside the regions bounded by the red
solid lines, a coherence extraction can happen.

Starting from an incoherent initial state with $\theta=0,\pi$, one has
$\mathcal{C}_{R.E.}(0)=0$ but a nonvanishing $\delta\mathcal{C}_{R.E.}>0$ for
two-detector at final equilibrium, which indicates a generation of quantum
coherence through its evolution. For general initial states, we know that a
coherence revival can occur with choice $\theta\in[0,\pi/2)$ (i.e.,
$\Delta_{0}\in(0,1]$) for sufficiently large Unruh temperature. However, by
numerically evaluating, we recognize only regions bounded by the red solid
lines in Fig.3 can have $\delta\mathcal{C}_{R.E.}>0$, referring to a coherence
extraction for sufficiently large $T_{U}$, i.e., the coherence of two-
detector’s final state is enhanced compared to its initial state.
Nevertheless, the region in Fig.3 without coherence extraction just indicates
that the generated coherence via Markovian evolution of the detectors cannot
counteract the effect of Unruh decoherence.

##  4 Metrological advantage

In general, quantum coherence is recognized as a physical resource that can be
utilized to improve the performance of various quantum information tasks. One
particular example is the close relation between non-classical coherence and
quantum metrological tasks, where the quantumness encoded in coherence can be
used to enhance the accuracy of estimation of extremely sensitive parameter,
e.g., Unruh temperature, spacetime curvature in relativistic context Wang ;
Tian ; F4 . Our purpose in this section is to examine the metrological
advantage fo coherence revival found before. In particular, we will evaluate
the QFI for the two-UDW detector, with Unruh temperature as a parameter chosen
to be optimally estimated. We will show that the revival coherence corresponds
to the enhanced QFI, therefore may provide a significant quantum advantage in
metrological task.

In any quantum metrological task, QFI gives a lower bound to the mean-square
error in the parameter estimation via the Cramér-Rao inequality
$\operatorname{Var}(X)\geqslant\left[N\mathcal{F}_{X}\right]^{-1}$, where $N$
is the number of repeated measurements. In terms of the symmetric logarithmic
derivative (SLD) operator $L_{X}$, which satisfies
$\partial_{X}\rho=\frac{1}{2}\\{\rho,L_{X}\\}$, the QFI is defined as
$\mathcal{F}_{X}=\mbox{Tr}[\rho(X)L^{2}_{X}]$. For a diagonalized density
matrix like $\rho=\sum_{i=\pm}\lambda_{i}|\psi_{i}\rangle\langle\psi_{i}|$,
the related QFI can be further written as QFI1 ; QFI2

$\mathcal{F}_{X}=\sum_{i=\pm}\frac{(\partial_{X}\lambda_{i})^{2}}{\lambda_{i}}+\sum_{i\neq
j=\pm}\frac{2(\lambda_{i}-\lambda_{j})^{2}}{\lambda_{i}+\lambda_{j}}|\langle\psi_{i}|\partial_{X}\psi_{j}\rangle|^{2}$
(23)

where the summations run over all eigenvalues satisfying $\lambda_{i}\neq 0$
and $\lambda_{i}+\lambda_{j}\neq 0$.

We choose to estimate Unruh temperature $T_{U}$. The related QFI can be
straightforwardly calculated from (23), by substituting the eigenvalues (15)
of diagonalized density matrix of detectors system. According to different
preparations of initial state encoded in $\Delta_{0}\in[-3,1]$, we come to two
classes of QFI:

(i) $\Delta_{0}=-3$, only one nonvanishing eigenvalue $\lambda_{4}=1$, which
gives $\mathcal{F}_{T}=0$;

(ii) $\Delta_{0}\in(-3,1]$, all eigenvalues (15) are nonvanishing. Since
$\partial_{T}\lambda_{4}=0$, we have

$\mathcal{F}_{T}=\sum_{i=1,2,3}\frac{(\partial_{T}\lambda_{i})^{2}}{\lambda_{i}}=\frac{\gamma^{6}-9\gamma^{4}-\gamma^{2}+9}{\gamma^{6}+9\gamma^{4}+27\gamma^{2}+27}\cdot\frac{\omega^{2}(3+\Delta_{0})}{2T_{U}^{4}}$
(24)

With specific energy spacing of detector $\omega=1,3,5$, we depict the QFI in
Fig.4 as a function of Unruh temperature and initial state choice
$\Delta_{0}$. The first important thing steps out is that, like quantum
coherence rather than entanglement, with fixed initial state choice, the QFI
of two-UDW detector is not a monotone function of Unruh temperature. We note
that a local peak exists as Unruh temperature growing, which means the largest
precision of estimation on $T_{U}$ can be achieved for detectors at a relative
low acceleration. Such non-monotonicity of the QFI with respect to Unruh
temperature was also found in single-detector case recently nomono1 ; nomono2
.

(a) $\mathcal{F}_{T}(\omega=1)$ (b) $\mathcal{F}_{T}(\omega=3)$

(c) $\mathcal{F}_{T}(\omega=5)$

Figure 4: The quantum Fisher information (QFI) of the final equilibrium state
of two-detectors system as a function of initial state preparation encoded by
$\Delta_{0}$ and Unruh temperature $T_{U}$. We choose Unruh temperature as a
parameter to be estimated by quantum measurement. The related QFI
$\mathcal{F}_{T}$ is demonstrated for fixed energy spacing of detector as
$\omega=1,3,5$. In each case, local peak of QFI can exist, with its maximal
value is remarkable enhanced in a range of $\Delta_{0}$ where coherence
revival occurs. As revival of quantum coherence is suppressed for larger
$\omega$, the maximal value of QFI also degrades significantly.

Comparing Fig.4 and the upper line of Fig.2, we find further similarity
between quantum coherence and the QFI. It is obviously that the maximal value
of the QFI peak can be achieved for initial state preparation with
$\Delta_{0}=1$, i.e., as approaching $\Delta_{0}=1$ the increasing of QFI
corresponds to enhanced quantum coherence revival. For larger $\omega$, the
coherence revival is heavily suppressed and correspondingly we find that the
maximal value of QFI peak also decreases conspicuously.

In general, the quantumness of physical system can be utilized as a physical
resource in various quantum processes. While both quantum coherence and
entanglement can serve as a resource, with above analysis, we conclude that it
is the coherence rather than the later dominates the QFI’s local peak and the
maximal value. In other word, with coherence revival one can significantly
improve the precision in the estimation of Unruh temperature, thus endows
quantum coherence a metrological advantage.

##  5 Summary and discussions

In this paper, we investigate the quantum coherence extracted for two
accelerating Unruh-DeWitt detectors coupling to a massless scalar field in
$3+1$ Minkowski spacetime. We find coherence can be generated through the
Markovian evolution of the detectors similar as the known entanglement
generation. However, we find that with growing Unruh temperature, quantum
coherence exhibits a striking revival phenomenon. For certain choice of
detectors’ initial state, coherence measure will firstly reduce to zero then
grow up and eventually approach to an asymptotic value. We verify such
coherence revival by inspecting its advantage on enhancing the QFI of a
metrological task. As the maximal QFI bounding the accuracy of a quantum
measurement, we conclude that the extracted coherence can be utilized as a
physical resource in quantum metrology.

It seems curious at first glance that coherence monotone (14) and quantum
entanglement have distinctive behavior under Unruh decoherence, as many works
have indicated Coh9  that both of them are physical resource that be
interchangeable to each other. However, we note that unlike entanglement,
coherence is a basis-dependent concept. If we adopt the viewpoint that a
preferred basis of coherence only emerges by the environment Disc1 , with
varying Unruh temperature, we would better cautiously refer the related
background states as different environment. In this context, the coherence
revival may be identified as a demonstration of ”einselection” in relativistic
framework.

On the other hand, the intimate relation between entanglement dynamics and
coherence generation of multi-UDW detectors may be reexamined from a
thermodynamic perspective. In particular, for an open quantum system, its
evolution irreversibility can be characterized by the entropy production
arising from non-equilibrium quantum processes, which is ascribed to system
correlations. It has been recently shown Disc2  that the entropy production of
an open quantum system can further be interpreted as an interplay between
population dynamics and coherence dynamics. Extending this analysis to the
context of Unruh decoherence may shed new light on our understanding on the
quantum nature of Unruh effect. We would exploit these interesting topics in
future.

## Acknowledgement

This work is supported by the National Natural Science Foundation of China
(No.12075178). J.F. would like to thank Hen Fan for enlightening discussions.
He would also like to thank the kind hospitality at the University of
Queensland where most of this work was accomplished.

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