# Would quantum coherence be increased by curvature effect in de Sitter space?

Shu-Min<EMAIL_ADDRESS>Chun-Xu Wang1, Dan-Dan Liu1, Xiao-Li Huang1222
<EMAIL_ADDRESS>Hao-Sheng<EMAIL_ADDRESS>(corresponding author) 1 Department of Physics, Liaoning Normal University,
Dalian 116029, China
2 Department of Physics, Hunan Normal University, Changsha 410081, China

###### Abstract

We study the quantum coherence in de Sitter space for the bipartite system of
Alice and Bob who initially share an entangled state between the two modes of
a free massive scalar field. It is shown that the space-curvature effect can
produce both local coherence and correlated coherence, leading to the increase
of the total coherence of the bipartite system. These results are sharp
different from the Unruh effect or Hawking effect, which, in the single mode
approximation, cannot produce local coherence and at the same time destroy
correlated coherence, leading to the decrease of the total coherence of the
bipartite systems. Interestingly, we find that quantum coherence has the
opposite behavior compared with the quantum correlation in de Sitter space. We
also find that quantum coherence is most severely affected by the curvature
effect of de Sitter space for the cases of conformal invariance and
masslessness. Our result reveals the difference between the curvature effect
in the de Sitter space and the Unruh effect in Rindler spacetime or the
Hawking effect in black hole spacetime on quantum coherence.

###### pacs:

04.70.Dy, 03.65.Ud,04.62.+v

##  I Introduction

The coherent superposition of states is one of the characteristic features,
which mark the departure of quantum mechanics from the classical physics, if
not the most essential one L1 . Quantum entanglement and quantum coherence are
arguably the two cardinal attributes of quantum theory, originating from the
tensor product structure and the quantum superposition principle, respectively
L2 ; L3 ; L4 . Even the phenomena of quantum entanglement and discord can also
be assigned to the nonlocal superpositions of quantum states L5 ; LL5 ; LLL5 .
Analogous to quantum entanglement and discord, quantum coherence is also a
powerful resource, which is widely used in such as quantum information,
quantum metrology, thermodynamics, solid-state physics and biological systems
L6 ; L7 ; L9 ; L10 ; L11 ; L12 ; L13 ; L14 ; L15 ; L16 . Despite the
fundamental importance of quantum coherence, it received a lot of attention
until Baumgratz $et$ $al.$ put forward quantification of coherence, including
the $l_{1}$ norm coherence and the relative entropy of coherence L17 .

Quantum mechanics and relativity theory represent two fundamentals of modern
physics, but unification of the two theories remains an open problem. Much
effort have been put forward to bridge the gap between them, which gives birth
to quantum field theory (QFT). The important predictions in QFT are the Unruh
effect, the Hawking radiation, and the generation of entanglement entropy in
an expanding universe and de Sitter space, which tells us that the vacuum of
the field depends on the observer L18 ; L19 ; L20 ; L20L ; L20LL . It has been
found that (i) quantum entanglement, discord, and coherence decrease
monotonically with the increase of acceleration in Rindler spacetime L21 ; L22
; L23 ; L24 ; L25 ; L26 ; L27 ; L28 ; L29 ; L30 ; L31 ; L32 ; L33 ; L34 ; (ii)
quantum entanglement, discord, and coherence suffer from a degradation due to
the Hawking radiation in the background of black holes QWE81 ; QWE82 ; L35 ;
L36 ; L37 ; L38 ; L39 ; L40 ; (iii) quantum entanglement and discord decrease
monotonically with the growth of the curvature in de Sitter space LL42 ; L42 ;
L43 ; Lqw43 . From these researches, we find that quantum coherence in both
Rindler and black hole spacetimes decreases monotonically. A question
naturally arises: How does curvature effect affect quantum coherence in de
Sitter space?

Our universe, undergoing an accelerated expansion, can be approximated by an
exponential de Sitter space with a positive cosmological constant. It is also
widely believed that de Sitter space is a better description of the early
stage of universe inflation. The importance comes from the fact that the de
Sitter space is the unique maximally symmetric curved space. Moreover, it is
one of the cornerstones of the accelerating expansion of the cosmology that
the large scale structure of our universe and temperature fluctuations of the
cosmic microwave background (CMB) originate from quantum fluctuations during
the initial inflationary era. Therefore, the investigation in curved spacetime
may itself lead to a more precise prediction for cosmological observations and
offer a better understanding of the initial stages of our universe. To this
aim, quantum entanglement in de Sitter space has been widely studied LL42 ;
L42 ; L43 ; Lqw43 ; L44 ; LL44 ; LLL44 . As quantum coherence can reflect
nonclassical world better than quantum entanglement, the study of quantum
coherence in the maximally symmetric curved space has a deeper understanding
of the quantum cosmology.

In this paper, we study the quantum coherence of an entangled state between
two free field modes of a massive scalar field in de Sitter space. Assume that
two observers: Alice in a global chart and Bob in an open chart of de Sitter
space, our aim is to study how the space background of de Sitter space
influences the quantum coherence of the two-mode system observed by Alice and
Bob. Interestingly, we find that the quantum coherence increases monotonically
under the curvature effect of de Sitter space. In other words, quantum
coherence has the opposite behavior compared with the quantum correlation
(entanglement and discord) for the two field modes in de Sitter space L42 ;
L43 . Our result also reveals the difference of the de Sitter space with the
Rindler or black hole spacetimes; in the latter two spacetimes quantum
coherence decreases monotonically with the strengthening of Unruh effect L28 ;
L29  or Hawking effect L40 .

The paper is organized as follows. In Sec. II, we review briefly the
quantization of the free massive scalar field in de Sitter space. In Sec. III,
we study the behaviors of quantum coherence in de Sitter space. The last
section is devoted to a brief conclusion.

##  II Quantization of scalar fields in de Sitter space

Figure 1: De Sitter space and its Penrose diagram. The $R$ and $L$ regions are
the causally disconnected open charts of de Sitter space.

The quantization of scalar fields in de Sitter space has been described many
times before. Here we only give a brief review following the references L44 ;
L45 . We consider a free scalar field $\phi$ with mass $m$ in de Sitter space
with metric $g_{\mu\nu}$ . The action of the scalar field is given by

$\displaystyle S=\int
d^{4}x\sqrt{-g}\left[\,-\frac{1}{2}\,g^{\mu\nu}\partial_{\mu}\phi\,\partial_{\nu}\phi-\frac{m^{2}}{2}\phi^{2}\,\right]\,.$
(1)

The metric for the coordinate system that covers the whole Euclidean de Sitter
space is given by

$\displaystyle ds^{2}_{E}=H^{-2}[d\tau^{2}+\cos^{2}\tau(d\rho^{2}+\sin^{2}\rho
d\Omega^{2})].$ (2)

The Lorentzian region can be divided into three open charts of de Sitter space
that consist of three parts: $C$, $R$ and $L$ as shown in Fig1 L44 ; L45 ,
whose coordinates are related to the fundamental ones via the relationships

$\displaystyle t_{R}$ $\displaystyle=$ $\displaystyle
i(\tau-\frac{\pi}{2})\qquad\qquad(t_{R}\geq 0),$ $\displaystyle r_{R}$
$\displaystyle=$ $\displaystyle i_{\rho}\qquad\qquad\qquad\qquad(r_{R}\geq
0),$ $\displaystyle t_{C}$ $\displaystyle=$
$\displaystyle\tau\qquad\qquad\qquad\qquad(\frac{\pi}{2}\geq
t_{C}\geq-\frac{\pi}{2}),$ $\displaystyle r_{C}$ $\displaystyle=$
$\displaystyle i(\rho-\frac{\pi}{2})\qquad\qquad(\infty>r_{C}>-\infty),$
$\displaystyle t_{L}$ $\displaystyle=$ $\displaystyle
i(-\tau-\frac{\pi}{2})\qquad\qquad(t_{L}\geq 0),$ $\displaystyle r_{L}$
$\displaystyle=$ $\displaystyle i_{\rho}\qquad\qquad\qquad\qquad(r_{L}\geq
0),$ (3)

and their metrics are given, respectively, by

$\displaystyle ds^{2}_{C}$ $\displaystyle=$ $\displaystyle
H^{-2}\left[dt^{2}_{C}+\cos^{2}t_{C}\left(-dr^{2}_{C}+\cosh^{2}r_{C}\,d\Omega^{2}\right)\right]\,,$
$\displaystyle ds^{2}_{R}$ $\displaystyle=$ $\displaystyle
H^{-2}\left[-dt^{2}_{R}+\sinh^{2}t_{R}\left(dr^{2}_{R}+\sinh^{2}r_{R}\,d\Omega^{2}\right)\right]\,,$
$\displaystyle ds^{2}_{L}$ $\displaystyle=$ $\displaystyle
H^{-2}\left[-dt^{2}_{L}+\sinh^{2}t_{L}\left(dr^{2}_{L}+\sinh^{2}r_{L}\,d\Omega^{2}\right)\right]\,,$
(4)

where $H^{-2}$ is the square of Hubble radius and $d\Omega^{2}$ is the metric
on the two-sphere. The $L$ region is causally disconnected from the $R$ region
in de Sitter space. The $C$ region is covered by the coordinate
$(t_{C},r_{C})$. Since the $L$ and $R$ regions are completely symmetric, we
write their coordinates by $(t,r)=(t_{R},r_{R})$ or $(t_{L},r_{L})$,
respectively.

Solving the Klein-Gordon equation in different open regions, one can obtain

$\displaystyle u_{\sigma p\ell m}(t_{R(L)},r_{R(L)},\Omega)$
$\displaystyle\sim$ $\displaystyle\frac{H}{\sinh
t_{R(L)}}\,\chi_{p,\sigma}(t_{R(L)})\,Y_{p\ell m}(r_{R(L)},\Omega)\,,\qquad$
$\displaystyle-{\rm\bf L^{2}}Y_{p\ell m}$ $\displaystyle=$
$\displaystyle\left(1+p^{2}\right)Y_{p\ell m}\,,$ (5)

with $Y_{p\ell m}$ being harmonic functions on the three-dimensional
hyperbolic space [see appendix A]. The positive frequency mode functions
$\chi_{p,\sigma}(t_{R(L)})$ that support on the $R$ and $L$ regions are
defined by

$\displaystyle\chi_{p,\sigma}(t_{R(L)})=\left\\{\begin{array}[]{l}\frac{e^{\pi
p}-i\sigma
e^{-i\pi\nu}}{\Gamma(\nu+ip+\frac{1}{2})}P_{\nu-\frac{1}{2}}^{ip}(\cosh
t_{R})-\frac{e^{-\pi p}-i\sigma e^{-i\pi\nu}}{\Gamma(\nu-
ip+\frac{1}{2})}P_{\nu-\frac{1}{2}}^{-ip}(\cosh t_{R})\,,\\\ \\\ \frac{\sigma
e^{\pi
p}-i\,e^{-i\pi\nu}}{\Gamma(\nu+ip+\frac{1}{2})}P_{\nu-\frac{1}{2}}^{ip}(\cosh
t_{L})-\frac{\sigma e^{-\pi p}-i\,e^{-i\pi\nu}}{\Gamma(\nu-
ip+\frac{1}{2})}P_{\nu-\frac{1}{2}}^{-ip}(\cosh t_{L})\,,\end{array}\right.$
(9)

where $P^{\pm ip}_{\nu-\frac{1}{2}}$ are the associated Legendre functions,
and $\sigma=\pm 1$ distinguishes two independent solutions for each region.
These solutions can be normalized by the factor

$\displaystyle N_{p}=\frac{4\sinh\pi p\,\sqrt{\cosh\pi
p-\sigma\sin\pi\nu}}{\sqrt{\pi}\,|\Gamma(\nu+ip+\frac{1}{2})|}\,,$ (10)

where the eigenvalue $p$ normalized by $H$ is a positive real parameter. As
the effect of the curvature of the three-dimensional hyperbolic space starts
to appear when $p$ approaches to $1$, we can regard $p$ as the description of
the curvature degree of the de Sitter space. The curvature effect gets
stronger as $p$ becomes smaller than $1$ L42 ; L43 ; Lqw43 . Therefore, $p$
can be considered as the curvature parameter of the de Sitter space. The mass
parameter $\nu$ is defined by $\nu=\sqrt{\frac{9}{4}-\frac{m^{2}}{H^{2}}}$. We
will later comment on the parameter region:
$0\leq\frac{m^{2}}{H^{2}}\leq\frac{9}{4}$ to make the discussion clear. When
$\nu=\frac{1}{2}$ (or $\frac{m^{2}}{H^{2}}=2$), we have a conformally coupled
scalar field and the system is conformally invariant. The minimally coupled
massless scalar field corresponds to $\nu=\frac{3}{2}$.

One can use the eigenfunctions of Eq.(II) to expand the scalar field as

$\displaystyle\hat{\phi}(t,r,\Omega)=\frac{H}{\sinh t}\int
dp\sum_{\sigma,\ell,m}\left[\,a_{\sigma p\ell m}\,\chi_{p,\sigma}(t)+a_{\sigma
p\ell-m}^{\dagger}\,\chi^{*}_{p,\sigma}(t)\right]Y_{p\ell m}(r,\Omega)\,,$
(11)

where $a_{\sigma p\ell m}$ ($a_{\sigma p\ell m}^{\dagger}$) is the
annihilation (creation) operator. The Bunch-Davies vacuum $|0\rangle_{\rm BD}$
can be defined as $a_{\sigma p\ell m}|0\rangle_{\rm BD}=0$. For simplicity,
hereafter, we omit the indices $p$, $\ell$, $m$ of the mode functions and
operators. For example, the mode functions and the associated Legendre
functions can be rewritten in the simple forms
$\chi_{p,\sigma}(t)\rightarrow\chi^{\sigma}$, $P_{\nu-1/2}^{ip}(\cosh
t_{R,L})\rightarrow P^{R,L}$, and $P_{\nu-1/2}^{-ip}(\cosh t_{R,L})\rightarrow
P^{R*,L*}$.

Next, we consider the positive frequency mode functions corresponding to the
the R-vacuum (L-vacuum), which has supported only on the $R$ ($L$) region. We
define the mode functions in the each $R$ and $L$ regions of open charts of de
Sitter space by

$\displaystyle\varphi^{q}=\left\\{\begin{array}[]{ll}\frac{|\Gamma(1+ip)|}{\sqrt{2p}}P^{q}{}&\mbox{in
region}~{}q\,,\\\ 0{}&\mbox{in the opposite region}\,,\end{array}\right.$ (14)

where $q=(R,L)$. Since the field should be the same under the different mode
functions, we can obtain

$\displaystyle\phi(t)=a_{\sigma}\,\chi^{\sigma}+a_{\sigma}^{\dagger}\,\chi^{\sigma}{}^{*}=b_{q}\,\varphi^{q}+b_{q}^{\dagger}\,\varphi^{q}{}^{*}\,.$
(15)

Where we have introduced the new annihilation and creation operators
($b_{q},b_{q}^{\dagger}$), and the corresponding vacuum is defined as
$b_{R}|0\rangle_{R}=0$ (R-vacuum) and $b_{L}|0\rangle_{L}=0$ (L-vacuum). In
terms of the Bogoliubov transformations between operators
($a_{q},a_{q}^{\dagger}$) and ($b_{q},b_{q}^{\dagger}$), one can find that the
Bunch-Davies vacuum state and the single particle excitation state can be
written as L42

$\displaystyle|0\rangle_{\rm
BD}=\sqrt{1-|\gamma_{p}|^{2}}\,\sum_{n=0}^{\infty}\gamma_{p}^{n}|n\rangle_{\rm
R}|n\rangle_{\rm L}\,,$ (16) $\displaystyle|1\rangle_{\rm
BD}=\frac{1-|\gamma_{p}|^{2}}{\sqrt{2}}\sum_{n=0}^{\infty}\gamma^{n}_{p}\sqrt{n+1}[|(n+1)\rangle_{\rm
L}|n\rangle_{\rm R}+|n\rangle_{\rm L}|(n+1)\rangle_{\rm R}],$ (17)

with the parameter $\gamma_{p}$ given by

$\displaystyle\gamma_{p}=i\frac{\sqrt{2}}{\sqrt{\cosh 2\pi p+\cos
2\pi\nu}+\sqrt{\cosh 2\pi p+\cos 2\pi\nu+2}}\,.$ (18)

Now we expound the claim that $p$ can be considered as the curvature parameter
of the de Sitter space. By using Eq.(16), the reduced density matrix from
Bunch-Davies basis to the basis of the open chart $L$ region is found to be

$\displaystyle\rho_{\rm L}=\rm{Tr_{R}(|0\rangle_{BD}\langle
0|)}=(1-|\gamma_{p}|^{2})\sum_{n=0}^{\infty}|\gamma_{p}|^{2n}|n\rangle_{\rm
L}\langle n|.$ (19)

For the conformal invariance ($\nu=1/2$) and masslessness ($\nu=3/2$), we
obtain $|\gamma_{p}|=e^{-\pi p}$. Then the reduced density matrix is given by

$\displaystyle\rho_{\rm L}=(1-e^{-2\pi p})\sum_{n=0}^{\infty}e^{-2\pi
pn}|n\rangle_{\rm L}\langle n|.$ (20)

Similarly, we can also get the reduced density matrix of Minkowski vacuum in
region I of Rindler spacetime

$\displaystyle\rho_{\rm
I}=(1-e^{-2\pi\frac{\omega}{a}})\sum_{n=0}^{\infty}e^{-2\pi
n\frac{\omega}{a}}|n\rangle_{\rm I}\langle n|,$ (21)

where $a$ is the acceleration L21 ; L28 . Comparing Eq.(20) with Eq.(21), we
obtain the relation $p=\frac{\omega}{a}$, which means that the increasing in
acceleration is related to the decreasing of $p$ with $a\rightarrow\infty$
corresponding to $p\rightarrow 0$. For the more general $\nu$, the comparing
between Eq.(19) and Eq.(21) gives $a=-\pi\omega/\ln|\gamma_{p}|$. For the
given $\nu$ and $\omega$, one can find that $a$ is a monotonic decreasing
function of $p$. From the view of this comparison, we say that $p$ is the
description of the curvature of de Sitter space. The smaller the $p$, the
larger the curvature. In fact, this claim has been used already L42 ; L43 ;
Lqw43 .

##  III  Quantum coherence in de Sitter space

Quantum coherence, being at the heart of interference phenomena, originates
from the quantum state superposition principle and plays an important role in
physics as it enables applications which are impossible within ray optics or
classical mechanics L3 . The rise of quantum mechanics as a unified picture of
particles and waves strengthened the prominent role of quantum coherence in
physics. Indeed, quantum coherence underlies phenomena such as multipartite
interference and is the precondition of quantum correlation. In this section,
we study the effect of space curvature of de Sitter space on quantum
coherence. For comparison, we quantify quantum coherence in terms of two kinds
of measures: the $l_{1}$ norm coherence and the relative entropy of coherence
L17 . In a given reference basis $\\{|i\rangle\\}_{i=1,\ldots,n}$ of a
$n$-dimensional system, the $l_{1}$-norm of quantum coherence is defined as
the sum of the absolute values of all the off-diagonal elements of the system
density matrix $\rho$ L17

$C_{l_{1}}(\rho)=\sum_{i\neq j}|\rho_{i,j}|,$ (22)

and the relative entropy of quantum coherence is given by

$C_{\rm
RE}\left(\rho\right)=S\left({\rho_{\rm{diag}}}\right)-S\left(\rho\right),$
(23)

where $S(\rho)$ is the von Neumann entropy of quantum state $\rho$, and
$\rho_{\rm diag}$ is the state obtained from $\rho$ by deleting all off-
diagonal elements.

It should be emphasized that quantum coherence depends on the choice of
reference basis. Using different reference basis to represent the same quantum
state may lead to different values of coherence. In practice, the reference
basis may be dictated by the physics of the problem under consideration. For
example, one may focus on the energy eigenbasis when addressing coherence in
the transport phenomena and thermodynamics. In the quantum description of
Young’s two-slit interference, the path basis is favorable. In this paper, we
are in the particle number representation to study the dynamics of coherence
for the scalar fields in de Sitter space. In quantum optics, the coherent
superposition of number states with different number of photons is very
important, which plays important roles in various optical interference
settings Scully . The well-known coherent states and squeezed states of
optical fields are the typical examples of this coherent superposition. For
the two-mode optical fields, the coherent superposition in the photon-number
bases can give rise to entanglement between the photons of the two modes (such
as the two-mode squeezed states), which is also a kind of important resource.
Quantum coherence is closely related to the quantum interference. The
coherence of superposition between photon-number states gives rise to usually
the nonuniform distribution of optical intensity with respect to position or
time, i.e., produce interference fringes, which can be detected through
suitable settings. All this mature technologies in quantum optics provide the
warrant for us to explore the counterpart for the scalar fields in de Sitter
space.

As for the detection of quantum coherence in a given reference basis, it may
be done through the so-called tomography of quantum states. For a quantum
state $\rho_{2}$ of the qubit system, it contain three independent parameters
and may be expressed as
$\rho_{2}=\frac{1}{2}(\hat{I}+\sum_{i=1}^{3}b_{i}\hat{\sigma}_{i})$ with
$b_{i}={\rm Tr}(\rho_{2}\hat{\sigma}_{i})$. It suffices to measure the average
of the three Pauli operators $\hat{\sigma}_{i}$ ($i=1,2,3$). Generically, for
the $n$ level systems, the density matrix may be written as Gunter
$\rho_{n}=\frac{1}{n}\hat{I}+\frac{1}{2}\sum_{i=1}^{n-1}(\lambda_{i}\hat{\lambda}_{i})$,
where $\hat{\lambda}_{i}$ ($i=1,2,\ldots,n-1$) are the generating operators of
the group $SU(n)$ and $\lambda_{i}={\rm Tr}(\rho_{n}\hat{\lambda}_{i})$. This
is to say, we need only measuring the average of the $n-1$ generating
operators, then we can reconstruct the quantum state $\rho_{n}$. Note that the
quantum coherence of qubit systems via tomography has been measured in recent
experiments Xu2020 ; Wang2017 .

Consider two observers, Alice and Bob, who initially share an entangled state
between the two modes of a free massive scalar field in de Sitter space

$\displaystyle|\Phi\rangle_{\rm{AB}}=\cos\theta|0_{\rm
A}\rangle_{\rm{BD}}|0_{\rm
B}\rangle_{\rm{BD}}+\sin\theta|1_{\rm{A}}\rangle_{\rm{BD}}|1_{\rm
B}\rangle_{\rm{BD}},$ (24)

where $\theta\in(0,\frac{\pi}{2})$. Now let Alice stay in the global chart,
while Bob is restricted to the $L$ region of the de Sitter open chart. From
Eqs.(16) and (17), we find that the vacuum and one-particle excitation states
seen by the observer in the global chart become the entangled states in the
open charts. Thus Eq.(24) can be written as

$\displaystyle|\Psi\rangle_{\rm{AB_{L}B_{R}}}$ $\displaystyle=$
$\displaystyle\cos\theta\sqrt{1-|\gamma_{p}|^{2}}\sum_{n=0}^{\infty}\gamma_{p}^{n}|0\rangle_{\rm{BD}}|n\rangle_{\rm
L}|n\rangle_{\rm R}$ (25) $\displaystyle+$
$\displaystyle\frac{\sin\theta(1-|\gamma_{p}|^{2})}{\sqrt{2}}\sum_{n=0}^{\infty}\gamma_{p}^{n}\sqrt{n+1}|1\rangle_{\rm{BD}}[|n+1\rangle_{\rm
L}|n\rangle_{\rm R}+|n\rangle_{\rm L}|(n+1)\rangle_{\rm R}].$

For simplicity, hereafter, we omit the indices $\rm A$ and $\rm B$ in $|n_{\rm
A}\rangle$ and $|n_{\rm B}\rangle$, unless there may be any confusion.

Both Alice and Bob have detectors which can detect respectively the mode $\rm
A$ and mode $\rm B$ of the entangled state. Since Bob is causally disconnected
from the state in the $R$ region of the de Sitter open chart, we should trace
over the degrees of freedom in the open chart $R$ and obtain

$\displaystyle\rho_{\rm{AB_{L}}}=(1-|\gamma_{p}|^{2})\sum_{n=0}^{\infty}|\gamma_{p}|^{2n}\rho_{\rm{AB_{L}}}^{n},$
(26)

where

$\displaystyle\rho_{\rm{AB_{L}}}^{n}$ $\displaystyle=$
$\displaystyle\cos^{2}\theta|0n\rangle\langle
0n|+\sin^{2}\theta\frac{1-|\gamma_{p}|^{2}}{2}(n+1)(|1n\rangle\langle
1n|+|1n+1\rangle\langle 1n+1|)$ (27) $\displaystyle+$
$\displaystyle\cos\theta\sin\theta\sqrt{\frac{1-|\gamma_{p}|^{2}}{2}}\sqrt{n+1}(\gamma_{p}|0n+1\rangle\langle
1n|+\gamma_{p}^{*}|1n\rangle\langle 0n+1|)$ $\displaystyle+$
$\displaystyle\cos\theta\sin\theta\sqrt{\frac{1-|\gamma_{p}|^{2}}{2}}\sqrt{n+1}(|0n\rangle\langle
1n+1|+|1n+1\rangle\langle 0n|)$ $\displaystyle+$
$\displaystyle\sin^{2}\theta\frac{1-|\gamma_{p}|^{2}}{2}\sqrt{(n+1)(n+2)}(\gamma_{p}|1n+2\rangle\langle
1n|+\gamma_{p}^{*}|1n\rangle\langle 1n+2|),$

where $|nm\rangle=|n\rangle_{\rm{BD}}|m\rangle_{\rm{L}}$. Therefore, the
$l_{1}$-norm coherence of state $\rho_{\rm{AB_{L}}}$ is

$\displaystyle C_{l_{1}}(\rho_{\rm{AB_{L}}})$ $\displaystyle=$
$\displaystyle(1-|\gamma_{p}|^{2})\sum_{n=0}^{\infty}|\gamma_{p}|^{2n}[\sin
2\theta\sqrt{\frac{1-|\gamma_{p}|^{2}}{2}}(|\gamma_{p}|+1)\sqrt{n+1}$
$\displaystyle+$
$\displaystyle\sin^{2}\theta|\gamma_{p}|(1-|\gamma_{p}|^{2})\sqrt{(n+1)(n+2)}].$

In the bases
$\\{|0,n\rangle,|0,n+1\rangle,|0,n+2\rangle,|1,n\rangle,|1,n+1\rangle,|1,n+2\rangle\\}$,
we can rewrite Eq.(27) in the matrix,

$\displaystyle\rho_{\rm{AB_{L}}}^{n}=\left(\\!\\!\begin{array}[]{cccccccc}\rho_{11}^{n}&0&0&0&\rho_{15}^{n}&0\\\
0&0&0&\rho_{24}^{n}&0&0\\\ 0&0&0&0&0&0\\\
0&\rho_{42}^{n}&0&\rho_{44}^{n}&0&\rho_{46}^{n}\\\
\rho_{51}^{n}&0&0&0&\rho_{55}^{n}&0\\\ 0&0&0&\rho_{64}^{n}&0&0\\\
\end{array}\\!\\!\right),$ (35)

where the matrix elements are written by

$\displaystyle\rho_{11}^{n}$ $\displaystyle=$ $\displaystyle\cos^{2}\theta,$
$\displaystyle\rho_{44}^{n}$ $\displaystyle=$
$\displaystyle\rho_{55}^{n}=\sin^{2}\theta\frac{(1-|\gamma_{p}|^{2})(1+n)}{2},$
$\displaystyle\rho_{15}^{n}$ $\displaystyle=$ $\displaystyle\rho_{51}^{n}=\sin
2\theta\frac{\sqrt{(1-|\gamma_{p}|^{2})(n+1)}}{2\sqrt{2}},$ (36)
$\displaystyle\rho_{24}^{n}$ $\displaystyle=$
$\displaystyle(\rho_{42}^{n})^{*}=\sin
2\theta\frac{\sqrt{(1-|\gamma_{p}|^{2})(n+1)}\gamma_{p}}{2\sqrt{2}},$
$\displaystyle\rho_{46}^{n}$ $\displaystyle=$
$\displaystyle(\rho_{64}^{n})^{*}=\sin^{2}\theta\frac{(1-|\gamma_{p}|^{2})\sqrt{(n+1)(n+2)}\gamma_{p}^{*}}{2}.$

The matrix of Eq.(35) has three nonzero eigenvalues

$\displaystyle\lambda_{1,n}=\frac{1}{4}\big{\\{}3+n-|\gamma_{p}|^{2}-n|\gamma_{p}|^{2}+[1+|\gamma_{p}|^{2}+n(|\gamma_{p}|^{2}-1)]\cos
2\theta\big{\\}},$ (37) $\displaystyle\lambda_{2/3,n}$ $\displaystyle=$
$\displaystyle\frac{1}{8}\bigg{\\{}1+n-|\gamma_{p}|^{2}-n|\gamma_{p}|^{2}+\cos
2\theta(1+n)(|\gamma_{p}|^{2}-1)\bigg{\\}}$ $\displaystyle\pm$
$\displaystyle\frac{\sqrt{2}}{8}\bigg{\\{}(1+n)(|\gamma_{p}|^{2}-1)\bigg{[}1+n+15|\gamma_{p}|^{2}+3n|\gamma_{p}|^{2}-8|\gamma_{p}|^{4}-4n|\gamma_{p}|^{4}$
$\displaystyle+$ $\displaystyle\cos
2\theta[-1+|\gamma_{p}|^{2}+8|\gamma_{p}|^{4}+n(-1-3|\gamma_{p}|^{2}+4|\gamma_{p}|^{4})]\bigg{]}\sin^{2}\theta\bigg{\\}}^{\frac{1}{2}}.$

Thus, the relative entropy of coherence of state $\rho_{\rm{AB_{L}}}$ is

$\displaystyle C_{RE}(\rho_{\rm{AB_{L}}})$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}(1-|\gamma_{p}|^{2})|\gamma_{p}|^{2n}\big{\\{}\lambda_{1,n}\log_{2}[(1-|\gamma_{p}|^{2})|\gamma_{p}|^{2n}\lambda_{1,n}]$
$\displaystyle+$
$\displaystyle\lambda_{2,n}\log_{2}[(1-|\gamma_{p}|^{2})|\gamma_{p}|^{2n}\lambda_{2,n}]+\lambda_{3,n}\log_{2}[(1-|\gamma_{p}|^{2})|\gamma_{p}|^{2n}\lambda_{3,n}]$
$\displaystyle-$
$\displaystyle\sin^{2}\theta{(1-|\gamma_{p}|^{2})(1+n)}\log_{2}[\sin^{2}\theta|\gamma_{p}|^{2n}\frac{(1-|\gamma_{p}|^{2})^{2}(1+n)}{2}]$
$\displaystyle-$
$\displaystyle\cos^{2}\theta\log_{2}[\cos^{2}\theta(1-|\gamma_{p}|^{2})|\gamma_{p}|^{2n}]\big{\\}}.$

Obviously, the quantum coherence $C_{l_{1}}(\rho_{\rm{AB_{L}}})$ and
$C_{RE}(\rho_{\rm{AB_{L}}})$ depends on not only the initial parameter
$\theta$, but also the curvature parameter $p$ and mass parameter $\nu$ of the
de Sitter space.

Figure 2: Quantum coherence $C_{l_{1}}(\rho_{\rm{AB_{L}}})$ and
$C_{RE}(\rho_{\rm{AB_{L}}})$ as functions of the curvature parameter $p$ for
different $\theta$. The mass parameter $\nu$ is fixed as $\nu=1/2$.

In Fig2, we plot quantum coherence $C_{l_{1}}(\rho_{\rm{AB_{L}}})$ and
$C_{RE}(\rho_{\rm{AB_{L}}})$ as functions of the curvature parameter $p$ of de
Sitter space for different $\theta$. It shows that both
$C_{l_{1}}(\rho_{\rm{AB_{L}}})$ and $C_{RE}(\rho_{\rm{AB_{L}}})$ increases
monotonically with the decrease of curvature parameter $p$ in de Sitter space,
meaning that space curvature enhances quantum coherence of the system of Alice
and Bob. This result is opposite to the quantum correlation (entanglement and
discord) which decreases monotonically under the curvature effect of de Sitter
space LL42 ; L42 ; L43 . The different behaviors reflect different
characteristics between quantum coherence and quantum correlation.

Note that, in Rindler spacetime, both quantum correlation and coherence
decrease monotonically with the increase of acceleration L21 ; L22 ; L23 ; L24
; L25 ; L26 ; L27 ; L28 ; L29 ; L30 ; L31 ; L32 ; L33 ; L34 . This means that
the Unruh effect and the curvature effect of de Sitter space play similar
roles to quantum correlation (destroy quantum correlation), but play different
roles to quantum coherence (Unruh effect destroys, while the curvature effect
of de Sitter space improves quantum coherence). Also note that the behaviors
of quantum resources (entanglement, discord and coherence) in the background
of black holes are similar to that in Rindler spacetime QWE81 ; QWE82 ; L35 ;
L36 ; L37 ; L38 ; L39 ; L40 .

Quantum coherence in a bipartite system may be distinguished as the local and
correlated LL5 . The local coherence belongs to each single subsystem, which
comes from the coherent superposition between the levels of the particular
subsystem; While the correlated coherence reflects the quantum correlation
between the two subsystems, which cannot be attributed to particular
subsystems. The correlated coherence for a bipartite system $A$ and $B$ is
defined as

$\displaystyle C^{cc}(\rho_{AB})$ $\displaystyle=$ $\displaystyle
C(\rho_{AB})-C(\rho_{A})-C(\rho_{B}),$ (40)

where $C(\rho_{AB})$ denotes the total coherence of the bipartite system, and
$C(\rho_{A})$ and $C(\rho_{B})$ are the local coherence for subsystems $A$ and
$B$, respectively. What we have studied in the above is the total quantum
coherence of the system $\rm{AB_{L}}$. Note that the initial state given by
Eq.(24) has only correlated coherence, and the local coherence of each
subsystem is zero. We wonder if this kind of distribution of coherence remains
unchanged under the curvature effect of de Sitter space. So we continue to
study the local coherence of each systems and their correlated coherence. By
tracing over the modes $\rm{B_{L}}$ or $\rm A$ in the state
$\rho_{\rm{AB_{L}}}$, we obtain the reduced state of each subsystem

$\displaystyle\rho_{\rm A}=\cos^{2}\theta|0\rangle\langle
0|+\sin^{2}\theta|1\rangle\langle 1|,$ (41)
$\displaystyle\rho_{\rm{B_{L}}}=(1-|\gamma_{p}|^{2})\sum_{n=0}^{\infty}|\gamma_{p}|^{2n}\rho_{\rm{B_{L}}}^{n},$
(42)

where

$\displaystyle\rho_{\rm{B_{L}}}^{n}$ $\displaystyle=$
$\displaystyle\cos^{2}\theta+\sin^{2}\theta\frac{1-|\gamma_{p}|^{2}}{2}(n+1)]|n\rangle\langle
n|$ (43) $\displaystyle+$
$\displaystyle\sin^{2}\theta\frac{1-|\gamma_{p}|^{2}}{2}(n+1)|n+1\rangle\langle
n+1|$ $\displaystyle+$
$\displaystyle\sin^{2}\theta\frac{1-|\gamma_{p}|^{2}}{2}\sqrt{(n+1)(n+2)}(\gamma_{p}|n+2\rangle\langle
n|+\gamma_{p}^{*}|n\rangle\langle n+2|).$

The matrix representation of $\rho_{\rm{B_{L}}}^{n}$ in the subspace $\\{|\rm
n\rangle,|\rm n+1\rangle,|\rm n+2\rangle\\}$ is given by

$\displaystyle\rho_{\rm{B_{L}}}^{n}=\left(\\!\\!\begin{array}[]{cccccccc}\cos^{2}\theta+\sin^{2}\theta\frac{(1-|\gamma_{p}|^{2})(1+n)}{2}&0&\sin^{2}\theta\frac{(1-|\gamma_{p}|^{2})\sqrt{(n+1)(n+2)}\gamma_{p}^{*}}{2}\\\
0&\sin^{2}\theta\frac{(1-|\gamma_{p}|^{2})(1+n)}{2}&0\\\
\sin^{2}\theta\frac{(1-|\gamma_{p}|^{2})\sqrt{(n+1)(n+2)}\gamma_{p}}{2}&0&0\end{array}\\!\\!\right),$
(47)

which has three nonzero eigenvalues:

$\displaystyle\lambda_{4/5,n}$ $\displaystyle=$
$\displaystyle\frac{1}{16}\bigg{\\{}6+2n-2|\gamma_{p}|^{2}-2n|\gamma_{p}|^{2}+2\cos
2\theta-2n\cos 2\theta+2|\gamma_{p}|^{2}\cos 2\theta$ $\displaystyle+$
$\displaystyle 2n|\gamma_{p}|^{2}\cos
2\theta\pm\sqrt{2}\big{\\{}19+10n+3n^{2}+(14+20n+6n^{2})|\gamma_{p}|^{2}-45|\gamma_{p}|^{4}$
$\displaystyle-$ $\displaystyle
66n|\gamma_{p}|^{4}-21n^{2}|\gamma_{p}|^{4}+24|\gamma_{p}|^{6}+36n|\gamma_{p}|^{6}+12n^{2}|\gamma_{p}|^{6}-4[-3+6|\gamma_{p}|^{2}$
$\displaystyle-$ $\displaystyle
15|\gamma_{p}|^{4}+8|\gamma_{p}|^{6}+n^{2}(-1+|\gamma_{p}|^{2})^{2}(1+4|\gamma_{p}|^{2})+2n(-1+|\gamma_{p}|^{2})^{2}$
$\displaystyle\times$ $\displaystyle(1+6|\gamma_{p}|^{2})]\cos
2\theta+[1+10|\gamma_{p}|^{2}-15|\gamma_{p}|^{4}+8|\gamma_{p}|^{6}+n^{2}(-1+|\gamma_{p}|^{2})^{2}$
$\displaystyle\times$
$\displaystyle(1+4|\gamma_{p}|^{2})+2n(-1+6|\gamma_{p}|^{2}-11|\gamma_{p}|^{4}+6|\gamma_{p}|^{6})]\cos
4\theta\big{\\}}^{\frac{1}{2}}\bigg{\\}},$
$\displaystyle\lambda_{6,n}=-\frac{\sin^{2}\theta}{2}(1+n)(-1+|\gamma_{p}|^{2}).$
(49)

From these, we get the local and the correlated coherences as

$\displaystyle C_{l_{1}}(\rho_{\rm A})=C_{RE}(\rho_{\rm A})=0,$ (50)
$\displaystyle
C_{l_{1}}(\rho_{\rm{B_{L}}})=\sin^{2}\theta|(1-|\gamma_{p}|^{2})^{2}\sum_{n=0}^{\infty}|\gamma_{p}|^{2n+1}\sqrt{(n+1)(n+2)},$
(51) $\displaystyle C_{RE}(\rho_{\rm{B_{L}}})$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}(1-|\gamma_{p}|^{2})|\gamma_{p}|^{2n}\bigg{\\{}\lambda_{4,n}\log_{2}[(1-|\gamma_{p}|^{2})|\gamma_{p}|^{2n}\lambda_{4,n}]$
$\displaystyle+$
$\displaystyle\lambda_{5,n}\log_{2}[(1-|\gamma_{p}|^{2})|\gamma_{p}|^{2n}\lambda_{5,n}]+\lambda_{6,n}\log_{2}[(1-|\gamma_{p}|^{2})|\gamma_{p}|^{2n}\lambda_{6,n}]$
$\displaystyle-$
$\displaystyle[\cos^{2}\theta+\sin^{2}\theta\frac{(1-|\gamma_{p}|^{2})(1+n)}{2}]\log_{2}\big{\\{}(1-|\gamma_{p}|^{2})|\gamma_{p}|^{2n}$
$\displaystyle\times$
$\displaystyle[\cos^{2}\theta+\sin^{2}\theta\frac{(1-|\gamma_{p}|^{2})(1+n)}{2}]\big{\\}}$
$\displaystyle-$
$\displaystyle\sin^{2}\theta\frac{(1-|\gamma_{p}|^{2})(1+n)}{2}\log_{2}[\sin^{2}\theta|\gamma_{p}|^{2n}\frac{(1-|\gamma_{p}|^{2})^{2}(1+n)}{2}]\bigg{\\}},$
$\displaystyle C_{l_{1}}^{cc}(\rho_{\rm A\rm{B_{L}}})$ $\displaystyle=$
$\displaystyle\sin
2\theta\sqrt{\frac{(1-|\gamma_{p}|^{2})^{3}}{2}}(1+|\gamma_{p}|)\sum_{n=0}^{\infty}|\gamma_{p}|^{2n}\sqrt{n+1},$
(53)

and

$\displaystyle C_{RE}^{cc}(\rho_{\rm A\rm{B_{L}}})=C_{RE}(\rho_{\rm
A\rm{B_{L}}})-C_{RE}(\rho_{\rm{B_{L}}}).$ (54)

Obviously, the observer Bob in the $L$ region of the open chart has local
quantum coherence, meaning that the curvature effect of de Sitter space can
generate local coherence. This is in sharp contrast with the Unruh effect
which cannot produce local coherence [see appendix B].

Figure 3: Local quantum coherence and correlated coherence of state
$\rho_{\rm{AB_{L}}}$ as functions of the curvature parameter $p$ for different
$\theta$. The mass parameter $\nu$ is fixed as $\nu=1/2$.

Figure 4:  The total coherence, local coherence and correlated coherence of
state $\rho_{\rm{AB_{L}}}$ as functions of the mass parameter $\nu$ and the
initial parameter $\theta$. The curvature parameter $p$ is fixed as $p=0$.

In Fig3, we plot the local quantum coherence and correlated coherence as
functions of the curvature parameter $p$ for different $\theta$. It is shown
that both local quantum coherence and correlated coherence of quantum state
$\rho_{\rm{AB_{L}}}$ increase monotonically with the decrease of parameter
$p$, meaning that curvature effect of de Sitter space can not only produce the
local coherence but also enhance the correlated coherence of the underlying
system. Note that Unruh effect and Hawking effect neither produce local
coherence, nor enhance the correlated coherence in Rindler and black hole
spacetimes [see appendix B].

In Fig4, we plot the total coherence, local coherence and correlated coherence
of state $\rho_{\rm{AB_{L}}}$ as functions of the mass parameter $\nu$ and the
initial parameter $\theta$. It is shown that quantum coherence is most
severely affected by the curvature of de Sitter space for the cases $\nu=1/2$
(conformal invariance) and $\nu=3/2$ ( masslessness). With the growth of the
$\theta$, the total coherence $C_{l_{1}}(\rho_{\rm{AB_{L}}})$ and
$C_{RE}(\rho_{\rm{AB_{L}}})$, and the correlated coherence
$C_{l_{1}}^{cc}(\rho_{\rm A\rm{B_{L}}})$ and $C_{RE}^{cc}(\rho_{\rm
A\rm{B_{L}}})$ increase from zero to the maximum and then decrease, while the
local coherence $C_{l_{1}}(\rho_{\rm{B_{L}}})$ and $C_{RE}(\rho_{\rm{B_{L}}})$
increases monotonically. Note that these results are consistent with the
results shown in Fig2 and Fig3.

In order to facilitate the physical analysis, we rewrite the expression of the
$l_{1}$-norm of coherence of state $\rho_{\rm{AB_{L}}}$

$\displaystyle C_{l_{1}}(\rho_{\rm{AB_{L}}})=\sin
2\theta\alpha(p,\nu)+\sin^{2}\theta\beta(p,\nu),$ (55)

where
$\alpha(p,\nu)=\sqrt{\frac{(1-|\gamma_{p}|^{2})^{3}}{2}}(1+|\gamma_{p}|)\sum_{n=0}^{\infty}|\gamma_{p}|^{2n}\sqrt{n+1}$
and
$\beta(p,\nu)=(1-|\gamma_{p}|^{2})^{2}\sum_{n=0}^{\infty}|\gamma_{p}|^{2n+1}\sqrt{(n+1)(n+2)}$.
Naturally, we obtain

$\displaystyle C_{l_{1}}^{cc}(\rho_{\rm A\rm{B_{L}}})=\sin
2\theta\alpha(p,\nu),\quad
C_{l_{1}}(\rho_{\rm{B_{L}}})=\sin^{2}\theta\beta(p,\nu)\quad\theta\in(0,\frac{\pi}{2}).$
(56)

From Eqs.(55) and (56), we see that the correlated coherence
$C_{l_{1}}^{cc}(\rho_{\rm A\rm{B_{L}}})$, local coherence
$C_{l_{1}}(\rho_{\rm{B_{L}}})$ and total coherence
$C_{l_{1}}(\rho_{\rm{AB_{L}}})$ include three variables, $\theta$, $p$ and
$\nu$. Now, we study the behavior of these coherences when only one of the
variables changes and the other two variables remain unchanged. Firstly, when
$\theta$ changes in the area $(0,\pi/2)$, the local coherence
$C_{l_{1}}(\rho_{\rm{B_{L}}})$ changes monotonically, and the correlated
coherence $C_{l_{1}}^{cc}(\rho_{\rm A\rm{B_{L}}})$ and total coherence
$C_{l_{1}}(\rho_{\rm{AB_{L}}})$ each have a peak. Secondly, both
$\alpha(p,\nu)$ and $\beta(p,\nu)$ are the increasing functions of
$|\gamma_{p}|$, and thus the decreasing functions of $p$. Therefore, the three
coherences are all the decreasing functions of $p$, meaning that the space
curvature leads to the increase of the coherences. Finally, both $\nu=1/2$ and
$\nu=3/2$ are the maximum points of $\alpha(p,\nu)$ and $\beta(p,\nu)$. Thus,
the three coherences appear peaks for the cases $\nu=1/2$ (conformal
invariance) and $\nu=3/2$ ( masslessness). This analysis is in agreement with
Fig4 and clearly shows the influences of curvature effect and mass parameter
on quantum coherence.

The mass parameter $\nu$ of the scalar field in de Sitter space depends on the
two scales: the mass $m$ and the Hubble radius $H^{-1}$. For $\nu=1/2$
($m^{2}/H^{2}=2$), the system is conformally invariant. We first consider the
meaning of the Bunch-Davies vacuum for the case $\nu=1/2$. The $\alpha$-vacuum
for the case $\nu=1/2$ is $\alpha$-independent and is equivalent to Bunch-
Davies vacuum. This is because that conformally flat space is
indistinguishable from Minkowski space for the case $\nu=1/2$ LL44 . Then, the
Legendre function for the case $\nu=1/2$ reduces to an elementary function as

$\displaystyle P^{ip}_{0}(\cosh t)=\frac{e^{-ip\eta}}{ip\Gamma(-ip)},$ (57)

where $\eta$ is related to $t$ by the relation, $\sinh\eta=\frac{-1}{\sinh
t}$. Since we focus on the $L$ region, a natural choice of the positive
frequency function is ${\chi}_{p}=\frac{e^{-ip\eta}}{\sqrt{2p}}$, which means
that the Bunch-Davies vacuum state is seen as a thermal state L45 . In other
words, for the case $\nu=1/2$, the reduced density matrix of Eq.(19) reduces
to a thermal state with temperature $T=H/(2\pi)$ and the generated
entanglement entropy is maximum LL42 ; LLL44 . Similarly, for the case
$\nu=1/2$, it is beneficial to the curvature effect to generate larger quantum
coherence. In addition, quantum coherence in de Sitter space depends on the
oscillatory behavior that comes from the factor $\cos 2\pi\nu$ in
$|\gamma_{p}|$. For fixed curvature parameter $p$, $|\gamma_{p}|$ is maximum
for the case $\nu=1/2$. Therefore, quantum coherence is most severely affected
by the curvature effect of de Sitter space for the cases of conformal
invariance ($\nu=1/2$).

From the above analysis, we see that quantum coherence in de Sitter space has
completely different behavior compared with that in the Rindler spacetime or
in the black hole spacetime. In the following, we want from the perspective of
Bogoliubov transformation to clarify how the different behaviors have been
produced. As the Bogoliubov transformations in the black hole spacetime and in
the Rindler spacetime are similar, we thus take the Rindler spacetime as the
example. Unfortunately, we can present the discussion only based on the
$l_{1}$-norm of coherence, because of its simplicity. The Minkowski vacuum and
excitation states observed by an inertial observer relate to the states of $I$
and $II$ regions in Rindler spacetime as follows L21 ; L28

$\displaystyle|0\rangle_{\rm
M}=\frac{1}{\cosh(r)}\sum_{n=0}^{\infty}\tanh^{n}(r)|n\rangle_{\rm
I}|n\rangle_{\rm{II}},$ (58) $\displaystyle|1\rangle_{\rm
M}=\frac{1}{\cosh^{2}(r)}\sum_{n=0}^{\infty}\tanh^{n}(r)\sqrt{n+1}|n+1\rangle_{\rm
I}|n\rangle_{\rm{II}},$ (59)

with $r$ the acceleration parameter. We find that this transformation is
asymmetric with respect to the $I$ and $II$ regions. However, the
transformation given by Eqs.(16) and (17) is symmetric with respect to the $R$
and $L$ regions. It is this symmetry and asymmetry of transformations that
lead to different behaviors of quantum coherence. Specifically, Eqs.(16) and
(17) lead to the evolution of the density matrices from Bunch-Davies basis to
the basis of the open chart $L$ region as

$\displaystyle\rm{Tr_{R}(|0\rangle_{BD}\langle
0|)}=(1-|\gamma_{p}|^{2})\sum_{n=0}^{\infty}|\gamma_{p}|^{2n}|n\rangle_{\rm
L}\langle n|,$ (60) $\displaystyle\rm{Tr_{R}(|0\rangle_{BD}\langle 1|)}$
$\displaystyle=$
$\displaystyle(1-|\gamma_{p}|^{2})^{\frac{3}{2}}\sum_{n=0}^{\infty}|\gamma_{p}|^{2n}\sqrt{\frac{n+1}{2}}$
(61) $\displaystyle(\gamma_{p}|n+1\rangle_{\rm L}\langle n|+|n\rangle_{\rm
L}\langle n+1|),$ $\displaystyle\rm{Tr_{R}(|1\rangle_{BD}\langle 1|)}$
$\displaystyle=$
$\displaystyle\frac{(1-|\gamma_{p}|^{2})^{2}}{2}\sum_{n=0}^{\infty}|\gamma_{p}|^{2n}[(n+1)(|n\rangle_{\rm
L}\langle n|+|n+1\rangle_{\rm L}\langle n+1|)$ (62) $\displaystyle+$
$\displaystyle\sqrt{(n+1)(n+2)}(\gamma_{p}|n+2\rangle_{\rm L}\langle
n|+\gamma_{p}^{*}|n\rangle_{\rm L}\langle n+2|)].$

Similarly, Eqs.(58) and (59) lead to the evolution of the density matrices
from Minkowski basis to the basis of $I$ region of Rindler spacetime given by

$\displaystyle\rm{Tr_{II}(|0\rangle_{M}\langle
0|)}=\frac{1}{\cosh^{2}r}\sum_{n=0}^{\infty}\tanh^{2n}r|n\rangle_{\rm
I}\langle n|,$ (63) $\displaystyle\rm{Tr_{II}(|0\rangle_{M}\langle
1|)}=\frac{1}{\cosh^{3}r}\sum_{n=0}^{\infty}\tanh^{2n}r\sqrt{n+1}|n\rangle_{\rm
I}\langle n+1|,$ (64) $\displaystyle\rm{Tr_{II}(|1\rangle_{M}\langle
1|)}=\frac{1}{\cosh^{4}r}\sum_{n=0}^{\infty}\tanh^{2n}r(n+1)|n+1\rangle_{\rm
I}\langle n+1|.$ (65)

By comparing Eqs.(63)-(65) with Eqs.(60)-(62), we find the difference between
the evolutions of density matrices: The curvature effect of de Sitter space
can make diagonal elements of the density matrix in the Bunch-Davies basis,
$(|1\rangle_{\rm{BD}}\langle 1|)$, to produce non-diagonal elements in the
basis of the open chart $L$ region [see Eq.(62)]. Further, the produced non-
diagonal elements have different positions with the non-diagonal elements
produced by $(|0\rangle_{\rm{BD}}\langle 1|)$ and its conjugation [see
Eq.(61)]. Obviously, the diagonal element $(|1\rangle_{\rm{BD}}\langle 1|)$
always has positive contribution to the $l_{1}$-norm of quantum coherence via
curvature effect of de Sitter space. This positive contribution can clearly
produce the local coherence $C_{l_{1}}(\rho_{\rm{B_{L}}})$. Actually it is
also the reason for enhancing the correlated coherence
$C_{l_{1}}^{cc}(\rho_{\rm A\rm{B_{L}}})$ as shown in Fig3(b). Therefore the
total coherence increases under the curvature effect of de Sitter space. In
contrast, the acceleration effect cannot make diagonal elements of the density
matrix in the Minkowski basis to produce non-diagonal elements in the
Rindler-$I$ spacetime [see Eqs.(63) and (65)], and the coherence in the
Rindler-$I$ spacetime comes completely from the evolution of the non-diagonal
elements $(|0\rangle_{\rm M}\langle 1|)$ and its conjugation. Thus Unruh
effect cannot produce the local $l_{1}$-norm of coherence. As the correlated
coherence under the Unruh effect degrades (see appendix B), the total
coherence also degrades.

##  IV  Conclusions

Based on the two measures of $l_{1}$-norm of coherence and relative entropy of
coherence, the effect of space curvature of de Sitter space on quantum
coherence of an entangled state between two free modes of a massive scalar
field has been investigated. We assume that the one mode (named by mode $\rm
A$) of the scalar field is observed by Alice who is in the global chart, and
another mode (named by mode $\rm{B_{L}}$) is observed by Bob who is in the $L$
region of the open chart of de Sitter space. We have shown that the curvature
effect of de Sitter space on the one hand can produce local coherence of Bob,
and on the other hand can enhance the correlated coherence between Alice and
Bob, so that the total coherence of the bipartite system of Alice and Bob
increases monotonically with the increase of curvature. This makes sharp
contrast with the quantum correlation (quantum entanglement and discord) which
decreases monotonically with the increase of curvature of de Sitter space L42
; L43 . For the first time, we have found that quantum coherence and quantum
correlation show completely opposite behavior in the framework of relativity.

We have also compared the influences of the curvature effect of de Sitter
space with Unruh effect or Hawking effect on quantum coherence. We have found
that Unruh effect and Hawking effect cannot produce local coherence and at the
same time destroy the correlated coherence, so that the total coherence of the
bipartite system of Alice and Bob decreases monotonically with the increase of
acceleration in Rindler spacetime L28 ; L29  or with the strengthening of
Hawking radiation in the black hole spacetime L40 . Therefore, quantum
coherence exhibits the opposite trend in de Sitter space comparing to Rindler
spacetime or black hole spacetime. Because de Sitter space is the unique
maximally symmetric curved space, the transformation given by Eqs.(16) and
(17) is symmetric with respect to the $R$ and $L$ regions of de Sitter space.
However, the transformation given by Eq.(59) is asymmetric with respect to the
$I$ and $II$ regions in Rindler spacetime. Symmetry of de Sitter space leads
to curvature effect that can increase quantum coherence, while asymmetry of
Rindler spacetime leads to Unruh effect that can decrease quantum coherence.

The dynamics of the underlying quantum coherence in de Sitter space encodes
much information about the expansion of Universe, which may help us to
understand the behaviors of expansion of Universe. We expect our results can
play roles in the prediction of cosmological observations, and in the
simulating of the Universe expansion in superconducting circuit and in an
ultracold quantum fluid of light L45m ; L45m1 .

###### Acknowledgements.

This work is supported by the National Natural Science Foundation of China
(Grant Nos. 12205133, 1217050862, 11275064, 11975064 and 12075050 ),
LJKQZ20222315 and 2021BSL013.

##  Appendix A The mode functions of de Sitter space

We expand a free scalar field in de Sitter space

$\displaystyle\hat{\phi}(x)=\sum_{\Lambda}(\hat{a}_{\Lambda}u_{\Lambda}(x)+\hat{a}_{\Lambda}^{{\dagger}}{u_{\Lambda}(x)}^{*}),$
(66)

where {$u_{\Lambda}(x)$} and their complex conjugate form a complete set of
mode functions denoted by $\Lambda$ that satisfy the field equation

$\displaystyle[g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}-m^{2}]u_{\Lambda}(x)=0,$
(67)

and normalized with respect to the Klein-Gordon inner product L45 . We can
determines the vacuum state $\hat{a}_{\Lambda}|0\rangle=0$ for any $\Lambda$
associated with a specified set of mode functions.

One can write down the field equation via the coordinates in either of the $R$
or $L$ regions to find the Euclidean vacuum mode functions on the open chart
of de Sitter space. Since the $R$ and $L$ regions are completely symmetric, we
obatin

$\displaystyle[\frac{1}{a^{3}(t)}\frac{\partial}{\partial
t}a^{3}(t)\frac{\partial}{\partial t}-\frac{H^{-2}}{a^{2}(t)}{\rm\bf
L^{2}}+\frac{9}{4}-\nu^{2}]u_{\sigma p\ell m}(t,r,\Omega)=0,$ (68)

where $(t,r)=(t_{R},r_{R})$ or $(t_{L},r_{L})$, $a(t)=H^{-1}\sinh t$,
$\nu=\sqrt{\frac{9}{4}-\frac{m^{2}}{H^{2}}}$ and ${\rm\bf
L^{2}}=\frac{1}{\sinh^{2}r}\frac{\partial}{\partial
r}(\sinh^{2}r\frac{\partial}{\partial r})+\frac{1}{\sinh^{2}r}{\rm\bf
L^{2}}_{\Omega}$. Here, ${\rm\bf L^{2}}_{\Omega}$ denotes the usual Laplacian
operator on the unit two-sphere. Then the eigenvalue equation for the operator
$-{\rm\bf L^{2}}$ on a unit three-dimensional hyperboloid can be given by

$\displaystyle-{\rm\bf L^{2}}Y_{plm}(r,\Omega)=(1+p^{2})Y_{plm}(r,\Omega),$
(69)

where the eigenfunction $Y_{plm}$ that is regular at $r=0$ can be written as

$\displaystyle Y_{plm}(r,\Omega)$ $\displaystyle=$ $\displaystyle
f_{pl}(r)Y_{lm}(\Omega),$ $\displaystyle f_{pl}(r)$ $\displaystyle:=$
$\displaystyle\frac{\Gamma(ip+l+1)}{\Gamma(ip+1)}\frac{p}{\sqrt{\sinh
r}}P^{-l-\frac{1}{2}}_{ip-\frac{1}{2}}(\cosh r)$ (70) $\displaystyle=$
$\displaystyle(-1)^{l}\sqrt{\frac{2}{\pi}}\frac{\Gamma(-ip+1)}{\Gamma(-ip+l+1)}\sinh^{l}r\frac{d^{l}}{d(\cosh
r)^{l}}(\frac{\sin pr}{\sinh r}),$

where $P^{\nu}_{\mu}(z)$ is the associated Legendre function of the first
kind, $\Gamma(z)$ is the gamma function and $Y_{lm}(\Omega)$ is the normalized
spherical harmonic function on the unit two-sphere L45 . For real positive
values of $p$, the eigenfunctions $Y_{plm}$ form an orthonormal complete set
for squareintegrable functions on the unit three-hyperboloid, and are
normalized as

$\displaystyle\int^{\infty}_{0}dr\sinh^{2}r\int d\Omega
Y_{plm}(r,\Omega){Y^{*}_{p^{\prime}l^{\prime}m^{\prime}}(r,\Omega)}$
$\displaystyle=\delta(p-p^{\prime})\delta_{ll^{\prime}}\delta mm^{\prime}.$
(71)

Now the harmonic expansion of the mode functions can be given by

$\displaystyle u_{\sigma p\ell m}(t,r,\Omega)=\frac{H}{\sinh t}\chi_{\sigma
plm}(t)Y_{plm}(r,\Omega).$ (72)

##  Appendix B Quantum coherence in Rindler spacetime

In this appendix, we present the influence of Unruh effect on quantum
coherence. Assume that Alice and Bob initially share a state
$|\Phi\rangle_{\rm{AB}}$ [given by Eq.(24)] of bosonic fields in inertial
frame. Then Alice stays in the inertial frame, while Bob moves with a uniform
acceleration. According to the transformation of Eqs.(58)-(59), we get

$\displaystyle|\Psi\rangle_{\rm{AB_{I}B_{II}}}=\frac{\cos\theta}{\cosh
r}\sum_{n=0}^{\infty}\tanh^{n}r|0nn\rangle+\frac{\sin\theta}{\cosh^{2}r}\sum_{m=0}^{\infty}\tanh^{m}r\sqrt{m+1}|1(m+1)m\rangle,$
(73)

where we abbreviate $|i_{\rm A}j_{\rm{B_{I}}}k_{\rm{B_{II}}}\rangle$ as
$|ijk\rangle$. Tracing over the inaccessible mode ${\rm B_{II}}$ in region
$II$, we obtain the density matrix

$\displaystyle\rho_{\rm{AB_{I}}}=\frac{1}{\cosh^{2}r}\sum_{n=0}^{\infty}(\tanh^{2n}r)\rho_{\rm{AB_{I}}}^{n},$
(74)

where

$\displaystyle\rho_{\rm{AB_{I}}}^{n}$
$\displaystyle=\cos^{2}\theta|0n\rangle\langle
0n|+\frac{\sin^{2}\theta(n+1)}{\cosh^{2}r}|1n+1\rangle\langle 1n+1|$
$\displaystyle+\frac{\sin 2\theta\sqrt{n+1}}{2\cosh r}|0n\rangle\langle
1n+1|+\frac{\sin 2\theta\sqrt{n+1}}{2\cosh r}|1n+1\rangle\langle 0n|.$

In the subspace of $\\{|0,\rm n\rangle,|0,\rm n+1\rangle,|1,\rm
n\rangle,|1,\rm n+1\rangle\\}$, $\rho_{\rm{AB_{I}}}^{n}$ has its matrix

$\displaystyle\rho_{\rm{AB_{I}}}^{n}=\left(\\!\\!\begin{array}[]{cccccccc}\cos^{2}\theta&0&0&\frac{\sin
2\theta\sqrt{n+1}}{2\cosh r}\\\ 0&0&0&0\\\ 0&0&0&0\\\ \frac{\sin
2\theta\sqrt{n+1}}{2\cosh
r}&0&0&\frac{\sin^{2}\theta(n+1)}{\cosh^{2}r}\end{array}\\!\\!\right),$ (80)

with nonzero eigenvalues:

$\displaystyle\frac{\tanh^{2n}}{\cosh^{2}r}\bigg{[}\cos^{2}\theta+\frac{\sin^{2}\theta(1+n^{2})}{\cosh^{2}r}\bigg{]}.$
(81)

According to Eqs.(22) and (23), the coherence of this state in Rindler
spacetime can be expressed as

$\displaystyle C_{l_{1}}(\rho_{\rm{AB_{I}}})=\frac{\sin
2\theta}{\cosh^{3}r}\sum_{n=0}^{\infty}\sqrt{n+1}\tanh^{2n}r,$ (82)
$\displaystyle C_{RE}(\rho_{\rm{AB_{I}}})$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\bigg{\\{}\frac{\tanh^{2n}}{\cosh^{2}r}\bigg{[}\cos^{2}\theta+\frac{\sin^{2}\theta(1+n^{2})}{\cosh^{2}r}\bigg{]}\log_{2}\bigg{[}\frac{\tanh^{2n}}{\cosh^{2}r}\bigg{(}\cos^{2}\theta$
(83) $\displaystyle+$
$\displaystyle\frac{\sin^{2}\theta(1+n^{2})}{\cosh^{2}r}\bigg{)}\bigg{]}-\frac{\cos^{2}\theta\tanh^{2n}}{\cosh^{2}r}\log_{2}\bigg{[}\frac{\cos^{2}\theta\tanh^{2n}}{\cosh^{2}r}\bigg{]}$
$\displaystyle-$
$\displaystyle\frac{\sin^{2}\theta\tanh^{2n}(1+n^{2})}{\cosh^{4}r}\log_{2}\bigg{[}\frac{\sin^{2}\theta\tanh^{2n}(1+n^{2})}{\cosh^{4}r}\bigg{]}\bigg{\\}}.$

In Fig.5, we plot the coherence $C_{l_{1}}(\rho_{\rm{AB_{I}}})$ and
$C_{RE}(\rho_{\rm{AB_{I}}})$ as a function of the acceleration parameter $r$.
It is shown that the coherence reduces monotonously with the increase of the
acceleration parameter $r$. This means that the Unruh effect reduces the total
coherence of the bipartite system of Alice and Bob.

Figure 5: Quantum coherence $C_{l_{1}}(\rho_{\rm{AB_{I}}})$ and
$C_{RE}(\rho_{\rm{AB_{I}}})$ as a function of acceleration parameter $r$ for
different $\theta$.

By tracing over the mode $\rm A$ or $\rm{B_{I}}$ from $\rho_{\rm{AB_{I}}}$, we
further obtain

$\displaystyle\rho_{\rm A}=\cos^{2}\theta|0\rangle\langle
0|+\sin^{2}\theta|1\rangle\langle 1|,$ (84)
$\displaystyle\rho_{\rm{B_{I}}}=\sum_{n=0}^{\infty}\tanh^{2n}r\left[\frac{\cos^{2}\theta}{\cosh^{2}r}|n\rangle\langle
n|+\frac{\sin^{2}\theta}{\cosh^{4}r}(n+1)|n+1\rangle\langle n+1|\right].$ (85)

Obviously, the local quantum coherences for both Alice and Bob are equal to
zero. Therefore, the correlated coherence is equal to the total coherence,
$C^{cc}_{l_{1}}(\rho_{\rm{AB_{I}}})=C_{l_{1}}(\rho_{\rm{AB_{I}}})$ and
$C^{cc}_{RE}(\rho_{\rm{AB_{I}}})=C_{RE}(\rho_{\rm{AB_{I}}})$, which also
reduces monotonously with the increase of the acceleration parameter $r$.

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# The neutrino-antineutrino annihilation to electron-positron pair around a
Schwarzschild black hole in asymptotic safety

Yuxuan Shi, Hongbo Cheng111E-mail address<EMAIL_ADDRESS>
Department of Physics,
East China University of Science and Technology,
Shanghai 200237, China
The Shanghai Key Laboratory of Astrophysics,
Shanghai 200234, China

###### Abstract

We research on the neutrino pair annihilation
$\nu+\overline{\nu}\longrightarrow e^{-}+e^{+}$ around a massive source in
asymptotic safety. The ratio $\frac{\dot{Q}}{\dot{Q}_{Newt}}$ corresponding to
the energy deposition per unit time over that in the Newtonian case is derived
and calculated. We find that the quantum corrections to the black hole
spacetime affect the emitted energy rate ratio for the annihilation. It is
interesting that the more considerable quantum effect reduces the ratio value
slightly. The corrected annihilation process can become a source of gamma ray
burst. We also investigate the derivative $\frac{d\dot{Q}}{dr}$ relating to
the star’s radius $r$ to show that the quantum effect for the black hole will
drop the ratio. The more manifest quantum gravity influence leads the weaker
neutrino pair annihilation.

PACS number(s):
Keywords: quantum gravity; back hole; Gamma-ray burst;

I. Introduction

A lot of efforts from the astrophysical community have been contributed to
explain the gamma-ray burst phenomenon with enough energy source. There are
neutrinos and antineutrinos emitted by the hot accretion disk [1-14]. The
process of the neutrino-antineutrino annihilation into electrons and positrons
can become an energy source of gamma-ray bursts, so the gamma-ray bursts may
be thought as systems powered by newborn, stellar-mass black holes accreting
matter at hyper-critical rates [1-14]. The so-called gamma-ray bursts must be
based on more powerful energy sources, and we should understand the origins.
The annihilation $\nu+\overline{\nu}\longrightarrow e^{-}+e^{+}$ augment the
neutrino heating of the envelope generating a supernova explosion, further the
pair $e^{-}e^{+}$ near the surface of collapsing neutron star sets free the
gamma ray that may provide a possible explanation of the observed bursts [15].
In addition to the gravitational source in various stages respectively, the
structures of backgrounds and the generalization of general relativity can put
manifest influence on the energy deposition [15]. It was found that the
efficiency of neutrino-antineutrino annihilation into electron-positron pair
is enhanced over the Newtonian values up to a factor of more than 4 in the
case of Type II supernovae, or up to a factor of 30 near the surface of
collapsing neutron stars [15]. The relativistic effects on the energy
deposition rate according to the neutrino pair annihilation near the rotation
axis of a Kerr black hole with a thin accretion disk was studied [16, 17]. The
stronger off-axis contributions to the energy-momentum deposition rate from
the $\nu\overline{\nu}$ pair collisions above a Kerr black hole within thin
accretion disk were also discussed [18, 19]. It was shown that the
generalizations of general relativity such as charged Galileon, Einstein
dilation Gauss-Bonnet, Brans-Dicke, Eddington-inspired Born-Infeld, Born-
Infeld generalization of Reissner-Nordstrom solution and higher derivative
gravity raise the energy deposition significantly in the neutron star and
supernova envelope [20]. The authors of Ref. [21] consider the neutrino pair
annihilation efficiency around a gravitational object belonging to the black
hole in the presence of quintessence field to show that the quintessence
powers the emitted energy rate ratio, so the enhancement could be a source for
the gamma-ray burst. We also discuss the neutrino pair annihilation around a
massive source with an $f(R)$ global monopole to find that this kind of
stellar bodies can be well qualified as a source of gamma-ray bursts [22].
Within the frames of generalized gravities such as Hartle-Thorne spacetime,
Born-Infeld Reissner-Nordstrom metric and quintessence background, it was
found that the neutrino-antineutrino pair annihilation process is mediated by
the $Z^{\prime}$ gauge boson from the $U(1)_{B-L}$ extension of the standard
model and the maximum gamma ray bursts energy can be reached [23].

The black holes attract a lot of attentions from physical community for
several decades. As solutions to Einstein’s field equation in the general
relativity, the black holes are also the end-results of some stars during the
process of gravitational collapse [23-25]. In the case of black holes, the
classical description of spacetime can not be used to define the surrounding
of singularities inside the event horizon [23-25]. In order to overcome the
plague from the singular background, it is necessary to generalize the general
relativity up to the high energy scale [26-35]. The generalization can lead
the quantum gravity which could resolve the unphysical black hole
singularities [26-35]. Several proposals such as loop quantum gravity [26-30],
Horava-Lifshitz gravity [31-35], etc. have been put forward. Here we are going
to focus on the powerful issue named as asymptotic safety scenario (AS) which
is utilized to get rid of the singularities of the black holes [35]. The
techniques of the functional renormalization group (FRG) was hired to limit
the results of the theory at trans-Planckian energies, making the physical
quantities to avoid the divergences at all scales [35]. The quantum effects
are always shown as corrections to the classical metrics [36-44]. A series of
corrected metrics include Schwarzschild [36], Kerr [37], Schwarzschild-AdS
[38], Kerr-AdS [39] and Reissner-Nordstrom-AdS [40] without the classical
singularity. More attentions have been contributed to the RGI-Schwarzschild
black holes. The evaporation of the RGI-Schwarzschild black hole was
investigated [45, 46], so was the mini-black hole production in colliders
[47]. The radial accretion of matter onto a RGI-Schwarzschild black hole was
also explored [48-50]. The shadows of black holes have been studied in the AS
scenario for quantum gravity [51-54]. Recently, the quantum gravity effects on
the radiation properties of a thin accretion disk surrounding a RGI-
Schwarzschild black hole within the frame of IR-limit of the AS theory were
discussed [55].

Here we will explore the energy deposition rate by the neutrino pair
annihilation process around a Schwarzschild black hole in asymptotic safety.
The metric of Schwarzschild black hole modified by the quantum effects needs
to be studied in different directions. The motion of test particle around RGI-
Schwarzschild black hole was described [56]. The electromagnetic ray
reflection spectroscopy for the corrected black hole was also considered [25,
57-59]. The quantum gravity effects on radiation characteristics of thin
accretion disks around a renormalization group improved Schwarzschild black
hole were shown [55]. Besides the black hole features mentioned above, it is
important to probe the annihilation energy deposition rate in the background
of this kind of black holes to show the implications of asymptotic safety
introducing the quantum corrections to the spacetime structures of the
gravitational sources. However, to the best of our knowledge, little
contributions have been paid for this topic. In this paper, we derive the
integral form of the neutrino pair annihilation efficiency based on the
Schwarzschild black hole with running Newton coupling $G(r)$ corresponds to
the quantum effects. Secondly, we calculate the ratio of total energy
deposition to total Newtonian energy deposition for parameter $\xi$ encoding
the quantum effects on the spacetime geometry With the help of our numerical
estimation, we wil discuss the influence from a RGI-Schwarzschild black hole
in the IR-limit of the asymptotic safety scenario on the possibility that the
astrophysical bodies attract the annihilation process generating the gamma-ray
burst. The results and conclusions are listed in the end.

II. The energy deposition rate by the neutrino annihilation process in the
renormalization group improved Schwarzschild spacetime

We adopt the Schwarzschild metric modified by the quantum gravity in the
infrared limit as follow [60-64],

$ds^{2}=f(r)dt^{2}-\frac{dr^{2}}{f(r)}-r^{2}(d\theta^{2}+\sin^{2}\theta
d\varphi^{2})$ (1)

where

$f(r)=1-\frac{2MG(r)}{r}$ (2)

with mass $M$. For the quantum-gravity-corrected Schwarzschild metric in the
infrared limit, the running coupling $G(r)$ takes the form [61],

$G(r)=G_{0}(1-\frac{\xi}{r^{2}})$ (3)

where $\xi$ is a parameter with dimensions of length squared associated to the
scale identification between the momentum scale and the radial coordinate. We
can choose $c=G_{0}=1$ in natural units. The radius of the event horizon is a
root of $f(r)=0$ [55]. According to the calculation and discussion, the
dimensionless parameter $\overline{\xi}=\frac{\xi}{M^{2}}$ and the mass $M$
should be limited as [55],

$\displaystyle\\{\begin{array}[]{c}0\leq\overline{\xi}\leq\overline{\xi}_{c}\\\
M>M_{c}\\\ \end{array}$ (6)

where $\overline{\xi}_{c}=\frac{16}{27}$ and $M_{c}=\sqrt{\frac{27}{16}\xi}$,
to keep the metric with two horizons [55]. It is significant to investigate
the quantum gravity effects on the annihilation process within the region (4).

We start to consider the energy deposition in the spacetime governed by Eq.(2)
and (3). The energy deposition per unit time and per unit volume for the
neutrino pair annihilation process is given by [15],

$\frac{dE(\mathbf{r})}{dtdV}=2KG_{F}^{2}F(r)\int\int
n(\varepsilon_{\nu})n(\varepsilon_{\overline{\nu}})(\varepsilon_{\nu}+\varepsilon_{\overline{\nu}})\varepsilon_{\nu}^{3}\varepsilon_{\overline{\nu}}^{3}d\varepsilon_{\nu}d\varepsilon_{\overline{\nu}}$
(7)

where

$K=\frac{1}{6\pi}(1\pm 4\sin^{2}\theta_{W}+8\sin^{4}\theta_{W})$ (8)

with the Weinberg angle $\sin^{2}\theta_{W}=0.23$. For various neutrino pairs,
the forms of Eq.(6) can be chosen as [15],

$K(\nu_{\mu},\overline{\nu}_{\mu})=K(\nu_{\tau},\overline{\nu}_{\tau})=\frac{1}{6\pi}(1-4\sin^{2}\theta_{W}+8\sin^{4}\theta_{W})$
(9)

and

$K(\nu_{e},\overline{\nu}_{e})=\frac{1}{6\pi}(1+4\sin^{2}\theta_{W}+8\sin^{4}\theta_{W})$
(10)

respectively [15]. Here the Fermi constant $G_{F}=5.29\times
10^{-44}cm^{2}MeV^{-2}$. The angular integration factor is represented by
[15],

$\displaystyle
F(r)=\int\int(1-\Omega_{\nu}\Omega_{\overline{\nu}})^{2}d\Omega_{\nu}d\Omega_{\overline{\nu}}$
$\displaystyle=\frac{2\pi^{2}}{3}(1-x)^{4}(x^{2}+4x+5)\hskip 28.45274pt$ (11)

where

$x=\sin\theta_{r}$ (12)

The angle $\theta_{r}$ is between the particle trajectory and the tangent
vector to a circle orbit at radius $r$. For one kind of neutrino and
antineutrino, $\Omega_{\nu}(\Omega_{\overline{\nu}})$ is the unit direction
vector and $d\Omega_{\nu}(d\Omega_{\overline{\nu}})$ is a solid angle. At
temperature $T$, $n(\varepsilon_{\nu})$ and $n(\varepsilon_{\overline{\nu}})$
are number densities for neutrino and antineutrino respectively in the phase
space [15],

$n(\varepsilon_{\nu})=\frac{2}{h^{3}}\frac{1}{e^{\frac{\varepsilon_{\nu}}{kT}}+1}$
(13)

where $h$ is Planck constant and $k$ is Boltzmann constant. With integrating
the Eq.(5), the expression of rate per unit time and unit volume is given by
[15],

$\frac{dE}{dtdV}=\frac{21\zeta(5)\pi^{4}}{h^{6}}KG_{F}^{2}F(r)(kT)^{9}$ (14)

The local temperature $T(\mathbf{r})$ measured by a local observer is defined
as $T(\mathbf{r})\sqrt{g_{00}(\mathbf{r})}=constant$ with $g_{00}$, a
component of spacetime metric [15]. The neutrino temperature at the
neutrinosphere can be explained as [15],

$T(r)\sqrt{g_{00}(r)}=T()R_{0}\sqrt{g_{00}(R_{0})}$ (15)

where $R_{0}$ is the radius of a gravitational source. The luminosity relating
to the redshift can be selected as [15],

$L_{\infty}=g_{00}(R_{0})L(R_{0})$ (16)

where the luminosity for a single neutrino species at the neutrinosphere is
[15],

$L()R_{0}=4\pi R_{0}^{2}\frac{7}{4}\frac{ac}{4}T^{4}(R_{0})$ (17)

where $a$ is the radiation constant and $c$ is the speed of light in vacuum.
In order to replace the temperature associated with the observer’s position,
we combine Eq.(13), Eq.(14) and Eq.(15) and substitute them into Eq.(12) to
obtain [15],

$\frac{dE(\mathbf{r})}{dtdV}=\frac{21\zeta(5)\pi^{4}}{h^{6}}KG_{F}^{2}k^{9}(\frac{7}{4}\pi
ac)^{-\frac{9}{4}}L_{\infty}^{\frac{9}{4}}F(r)\frac{(g_{00}(R))^{\frac{9}{4}}}{(g_{00}(r))^{\frac{9}{4}}}R_{0}^{-\frac{9}{2}}$
(18)

In order to compute the angular integration $F(r)$ from Eq.(9), we should
firther the discussion on the variable $x$ in Eq.(10). We follow the procedure
of Ref. [15] and solve the null geodesic in the spacetime of a spherically
symmetric gravitational object [15] to show [20],

$\displaystyle x^{2}=\sin^{2}\theta_{r}|_{\theta_{R_{0}}=0}$
$\displaystyle=1-\frac{R_{0}^{2}}{r^{2}}\frac{f(r)}{f(R_{0})}\hskip
14.22636pt$ (19)

Here $f(r)=g_{00}(r)$, a component of metric (2). We can proceed the
integration of rate per unit time and unit volume from Eq.(16) on the
spherically symmetric volume around the gravitational source [21],

$\displaystyle\dot{Q}=\frac{dE}{\sqrt{g_{00}}dt}\hskip 284.52756pt$
$\displaystyle=\frac{84\zeta(5)\pi^{5}}{h^{6}}KG_{F}^{2}k^{9}(\frac{7}{4}\pi
ac)^{-\frac{9}{4}}L_{\infty}^{\frac{9}{4}}(g_{00}(R_{0}))^{\frac{9}{4}}R_{0}^{-\frac{9}{2}}\int_{R_{0}}^{\infty}\frac{r^{2}\sqrt{-g_{11}(r)}F(r)}{g_{00}(r)}^{\frac{9}{2}}dr$
(20)

Here $\dot{Q}$ can reflect the total amount of energy converted from neutrinos
to electron-positron pairs at any radius [15]. It is significant to compare
the energy deposition rate (18) with the Newtonian quantities as [15, 20, 21],

$\frac{\dot{Q}}{\dot{Q}_{Newt}}=3(g_{00}(R_{0}))^{\frac{9}{4}}\int_{1}^{\infty}(x-1)^{4}(x^{2}+4x+5)\frac{y^{2}\sqrt{-g_{11}(R_{0}y)}}{g_{00}(R_{0}y)^{\frac{9}{2}}}dy$
(21)

with dimensionless variable $y=\frac{r}{R_{0}}$ and $g_{00}(r)$, $g_{11}(r)$
components of metric (1).

According to the Ref.[15], we can present $\frac{d\dot{Q}}{dr}$ as the
function of radial coordinate $r$ to exhibit the enhancement,

$\displaystyle\frac{d\dot{Q}}{dr}=4\pi(\frac{dE}{dtdV})\sqrt{-g_{11}(r)}r^{2}\hskip
270.30118pt$
$\displaystyle=\frac{168\zeta(5)\pi^{7}}{3h^{6}}KG_{F}^{2}k^{9}(\frac{7}{4}\pi
ac)^{-\frac{9}{4}}L_{\infty}^{\frac{9}{4}}(x-1)^{4}(x^{2}+4x+5)(\frac{g_{00}(R_{0})}{g_{00}(r)})^{\frac{9}{4}}R_{0}^{-\frac{5}{2}}\sqrt{-g_{11}(r)}(\frac{r}{R_{0}})^{2}$
(22)

It is important to discuss the ratio (19) in the quantum-gravity-corrected
Schwarzschild spacetime in the infrared limit. We can relate that
$g_{00}(r)=f(r)$ and $g_{11}(r)=-\frac{1}{f(r)}$ for metric (1). The ratio
(19) can be reformed as [15, 60-64],

$\frac{\dot{Q}}{\dot{Q}_{Newt}}=3(f(R))^{\frac{9}{4}}\int_{1}^{\infty}(x-1)^{4}(x^{2}+4x+5)\frac{y^{2}}{(f(Ry))^{5}}dy$
(23)

where

$f(R)=1-\frac{2M}{R}+\frac{2M\xi}{R^{3}}$ (24)

$f(Ry)=1-\frac{2M}{R}\frac{1}{y}+\frac{2M\xi}{R^{3}}\frac{1}{y^{3}}$ (25)

According to the metric function (2) and the running coupling (3), we rewrite
the variable (17) as,

$x^{2}=1-\frac{1}{y^{2}}\frac{1-\frac{2M}{R}\frac{1}{y}+\frac{2M\xi}{R^{3}}\frac{1}{y^{3}}}{1-\frac{2M}{R}+\frac{2M\xi}{R^{3}}}$
(26)

In order to show the influence from quantum gravity, we should quantify the
integral expression of ratio $\frac{\dot{Q}}{\dot{Q}_{Newt}}$ in EQ.(21) and
depict the dependence on $\frac{R_{0}}{M}$ with the effect parameter $\xi$ in
the Figures. In Figure 1, the shape of ratio curves due to different values of
parameter $\xi$ are similar. As a gravitational source, the values of ratio
$\frac{\dot{Q}}{\dot{Q}_{Newt}}$ are much more than one, so this kind of
stellar objects attracting the neutrino pair annihilation process may provide
with a source of gamma-ray burst. In contrast to the case of accretion disk
around a stellar object in asymptotic safety that the quantum gravity effect
can enhance the disk’s thermal properties [55], the more considerable
influence will reduce the ratio $\frac{\dot{Q}}{\dot{Q}_{Newt}}$.

We also elaborate the derivative $\frac{d\dot{Q}}{dr}$ as a function of radius
for several stellar masses denoted as $\frac{M}{R_{0}}$ respectively to
exhibit the promotion of $e^{-}e^{+}$ pair energy from the neutrino
annihilation in Figure 2\. It is similar that a slight larger variable $\xi$
from quantum gravity decreases the derivative $\frac{d\dot{Q}}{dr}$. The
Figure 2 also indicate that the structure with smaller $\frac{M}{R_{0}}$ lead
to the greater $\frac{d\dot{Q}}{dr}$. It is interesting that the increase is
much larger near the surface of the neutron star similar to the case of
Ref.[15] although the black hole surrounded by the annihilation includes the
quantum gravity influence.

III. Conclusion

The neutrino pair annihilation $\nu\bar{\nu}\longrightarrow e^{-}e^{+}$ around
a Schwarzschild black hole in asymptotic safety is discussed. The quantum
gravity influence on the black holes can be explored in different directions.
We derivative and calculate the emitted energy rate ratio as
$\frac{\dot{Q}}{\dot{Q}_{Newt}}$ in the background of quantum-gravity
corrected massive source. It is found that the quantum-gravity effect does not
advance the ratio $\frac{\dot{Q}}{\dot{Q}_{Newt}}$, but damps $\dot{Q}$, the
energy released per unit time, in contrast to the case of accretion disk under
the same conditions. The ratio $\frac{\dot{Q}}{\dot{Q}_{Newt}}$ shows
numerically that the annihilation $\nu+\bar{\nu}\longrightarrow e^{-}+e^{+}$
around the quantum-gravity corrected black hole can proceed as a source for
the gamma-ray burst. The larger $\xi$, the parameter encoding the quantum
gravity effects on the spacetime structure, will reduce the energy deposition
rate. This kind of annihilation can occur around the Type II supernova or
collapsing neutron stars. Within the region of $\xi$, this kind of
annihilation process could be proposed as a well-qualified candidate of gamma-
ray bursts.

Acknowledge

This work is partly supported by the Shanghai Key Laboratory of Astrophysics
18DZ2271600.

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Figure 1: The solid, dotted and dashed curves of the ratio
$\frac{\dot{Q}}{\dot{Q_{Newt}}}$ as functions of the ratio $\frac{R}{M}$ for
quantum-gravity factors $\xi=0,0.26,0.56$ respectively.

Figure 2: The solid, dotted and dashed curves of the ratio
$\frac{\dot{Q}}{\dot{Q_{Newt}}}$ as functions of the ratio $\frac{R}{M}$ for
quantum-gravity factors $\xi=0,0.26,0.56$ respectively.