# Entanglement dynamics in $\kappa$-deformed spacetime

Xiaobao Liua, Zehua Tianb,c111Corresponding author. Email<EMAIL_ADDRESS>Jiliang Jingd222Corresponding author. Email<EMAIL_ADDRESS>
a Department of physics and electrical engineering, Liupanshui Normal
University, Liupanshui 553004, Guizhou, China
b CAS Key Laboratory of Microscale Magnetic Resonance and School of Physical
Sciences, University of Science and Technology of China, Hefei 230026, China
c CAS Center for Excellence in Quantum Information and Quantum Physics,
University of Science and Technology of China, Hefei 230026, China
d Department of Physics, Key Laboratory of Low Dimensional Quantum Structures
and Quantum Control of Ministry of Education, and Synergetic Innovation Center
for Quantum Effects and Applications, Hunan Normal University, Changsha, Hunan
410081, China

###### Abstract:

We treat two identical and mutually independent two-level atoms that are
coupled to quantum field as an open quantum system. The master equation that
governs their evolution is derived by tracing over the degree of freedom of
field. With this, we comparatively study the entanglement dynamics of the two
atoms moving with different trajectories in $\kappa$-deformed spacetime and
Minkowski spacetime. It is found that when there is no the environment-induced
interatomic interaction, the entanglement dynamics of two static atoms in
$\kappa$-deformed spacetime are reduced to the case in Minkowski spacetime in
the case that the spacetime deformation parameter $\kappa$ is huge enough as
theoretically predicted. However, if the atoms undergo relativistic motion, no
matter inertial or non-inertial, their entanglement dynamics in
$\kappa$-deformed spacetime behave quite differently with that in Minkowski
spacetime even $\kappa$ is huge. We investigate various entanglement
behaviors, such as decay and generation, and discuss how different
relativistic motion, such as uniform motion in a straight line and circular
motion, amplifies the difference of entanglement dynamics between the
$\kappa$-deformed spacetime case and the Minkowski spacetime case. Besides,
when the environment-induced interatomic interaction is considered, we find
that it may also enhance the difference of entanglement dynamics between in
these two spacetimes. So, in principle, one can tell whether he is in
$\kappa$-deformed spacetime or in Minkowski spacetime by checking the
entanglement behaviors between two atoms in certain circumstances.

††preprint:

##  1 Introduction

Quantum entanglement, as a crucial physical resource in technologies based on
quantum effects, is the essential feature underlying quantum information,
cryptography, quantum computation [1, 2, 3], and so on. Recently, quantum
entanglement has been investigated together with the theories of relativity
and quantum fields. For example, entanglement dynamics [4, 5, 6, 7, 8, 9, 10,
11, 12, 13, 14, 15, 16, 17, 18, 19, 20], and entanglement harvesting [21, 22,
23, 24, 25, 26, 27, 28, 29, 30, 31] have been studied in a variety of
different relativistic scenes and settings. The aim of these researches is to
explore the effects of relativistic motion and gravity on quantum
entanglement, and in turn, entanglement could also be used to probe the
structure of spacetime and analyze gravitational-matter interaction. On the
one hand, the unavoidable coupling between a quantum system and the external
environment usually leads to decoherence and dissipation to quantum system. It
may also cause disentanglement or even entanglement sudden death [32, 33, 34],
and thus has been thought as one of the major obstacles to the realization of
the quantum information technologies. However, on the other hand, a common
bath may induce the indirect interactions between otherwise independent atoms
immersed in which as a consequence of environment correlations. Thus in this
case entanglement could be created for atoms although initially in separable
states [35, 36, 37, 38], and even the destroyed entanglement may also revive
[39]. What is more, entanglement for two atoms with vanishing separation still
persists even in the asymptotic equilibrium regime [4, 5]. In Ref. [40] it has
been shown that the environment-induced interatomic interaction would assist
entanglement generation. Particularly, recently the entanglement character of
two particle detectors in two different while quite similar spacetimes, e.g.,
in the de Sitter spacetime and in the thermal Minkowski spacetime [41, 42, 43,
44], has been discussed. It is found that the relevant different entangling
power of spacetime could be expected to distinguish these different universes
in principle [13, 41, 42, 43, 44, 45, 46, 47]. Such investigations
aforementioned remind us that the study of entanglement dynamics within a
framework of relativity can tell us about the nature of spacetime, which is
what makes sense for us to explore related fields.

The Minkowski spacetime has a continuous structure, whose spacetime
coordinates are commutative, and the scalar field theory is well constructed
in this commutative spacetime. However, on the microscopic level side, one of
the significant topics of quantum gravity theories concerns the modification
of the notion of spacetime and the quantization of spacetime coordinates. This
requirement may result in that the notions of symmetry of the spacetime are
modified. The symmetry algebra of a certain quantum gravity model is known to
be the $\kappa$-Poincaré algebra. The correponding “Lie algebra”-type of
noncommutative spacetime is named the $\kappa$-Minkowski spacetime [48, 49].
In this respect, the exploration of this noncommutative spacetime can help us
deepen our understanding of the structure and properties of spacetime in the
microscopic areas. There exists substantially related works on the $\kappa$
spacetime, as well as the construction and relevant investigations of field
theory models on this spacetime [50, 51, 52, 53, 54, 55, 56, 57, 58] (and
references cited therein). Usually, the quantum field theory in
$\kappa$-deformed spacetime is quite complicated as a result of the non-
commutation of spacetime coordinates. However, recently in Ref. [59] an
interesting model starts with $\kappa$-deformed Klein-Gordon theory which is
invariant under a $\kappa$-Poincaré algebra and written in commutative
spacetime. Inspired by this approach, we in Ref. [60] investigated the quantum
Fisher information in $\kappa$-deformed spacetime and found that the
relativistic motion can effectively improve the quantum precision in the
estimation of spacetime deformation. In this regard, we note that the possible
and expected $\kappa$ deformation of spacetime is quite weak so that usually
all the physical theory consists with that in Minkowski spacetime. Hence, it
is worthy to ask whether it is possible to distinguish the $\kappa$-deformed
spacetime from the Minkowski spacetime?

In this paper we investigate the entanglement dynamics for two-atom system
coupled to a scalar field in $\kappa$-deformed spacetime, and compare the
results with that in Minkowski spacetime. Firstly, the master equation that
governs the system evolution is derived, and the standard field theory
techniques developed for the commutative spacetime is used to explore this
$\kappa$-deformed scalar theory [59, 60]. Then we consider the evolution
behaviors of entanglement between the two atoms with different trajectories in
$\kappa$-deformed spacetime and Minkowski background. Our results demonstrate
that the relativistic motion of the atoms can help us to distinguish the
entanglement dynamics between in $\kappa$-deformed spacetime and in Minkowski
spacetime. Besides, when choosing special initial states which may introduce
the environment-induced interatomic interaction, the atomic entanglement
dynamics could also behave quite differently in these two universes. Thus this
entanglement dynamics difference of two atoms can be used to distinguish the
$\kappa$-deformed spacetime and the Minkowski spacetime in principle.

The paper is organized as follows. In section 2, basic formulaes of the master
equation for two-atom system interacting with scalar field are reviewed. We
also take a review of $\kappa$-deformed scalar theory written in commutative
spacetime and its propagator. In section 3, we consider the dynamics of
entanglement for two-atom system in $\kappa$-deformed spacetime and Minkowski
spacetime without the enviornment-induced interatomic interaction. In section
4, the influence of the enviornment-induced interatomic interaction on the
behaviors of the entanglement dynamics is explored in detail. Finally, we end
up with conclusions in section 5.

Throughout the whole paper we employ natural units $c=\hbar=1$. Relevant
constants are restored when needed for the sake of clarity.

##  2 Master equation and Scalar field propagator

We simply review the master equation of two atoms which interact with a scalar
field in the vacuum fluctuation, and introduce the corresponding scalar field
propagator in $\kappa$-deformed spacetime with the commutative theory.

###  2.1 Master equation of two-atom system

Let us consider two identical and mutually independent atoms weakly coupled to
a bath of fluctuating scalar field in its vacuum state. We also applied this
model in the relativistic scenario recently [61, 62, 63, 64, 65, 66, 67]. The
total Hamiltonian $H$ for the complete system, i.e., the two atoms together
with the external scalar field, reads

$\displaystyle H=H_{S}+H_{E}+H_{I}.$ (1)

Here the free two-atom Hamiltonian $H_{S}$ is given by

$\displaystyle
H_{S}=\frac{1}{2}\omega_{0}\sigma_{3}^{(1)}+\frac{1}{2}\omega_{0}\sigma_{3}^{(2)},$
(2)

where $\sigma_{i}^{(1)}=\sigma_{i}\otimes\mathbf{I}$,
$\sigma_{i}^{(2)}=\mathbf{I}\otimes\sigma_{i}$, $\sigma_{i}$ with
$i\in\\{1,2,3\\}$ are Pauli matrices and $\mathbf{I}$ is the $2\times 2$ unit
matrix, and $\omega_{0}$ denotes the energy level spacing of an individual
atom. Note that $H_{E}$ is the free Hamiltonian of the scalar field, whose
explicit expression is not required here, and $H_{I}$ represents the
interaction Hamiltonian between atoms and field. We assume that the coupling
between the two atoms and the scalar field takes the form, specifically in
analogy to the electric dipole interaction [68],

$\displaystyle
H_{I}=\mu[\sigma_{2}^{(1)}\phi(\mathbf{x}_{1}(\tau))+\sigma_{2}^{(2)}\phi(\mathbf{x}_{2}(\tau))].$
(3)

Here $\mu$ is the coupling constant that we assume to be small, and
$\phi(\mathbf{x}_{\alpha}(\tau))$ with $\alpha\in\\{1,2\\}$ corresponds to the
scalar field operator with $\tau$ being the proper time of the atoms.

In the frame of atoms, the time evolution of the total system is governed by
the von Neumann equation

$\displaystyle\frac{\partial\rho_{\mathrm{tot}}(\tau)}{\partial\tau}=-i[H,\rho_{\mathrm{tot}}(\tau)].$
(4)

We assume the initial density matrix of the atoms-field system as
$\rho_{\mathrm{tot}}=\rho(0)\otimes|0\rangle\langle 0|$, in which $\rho(0)$ is
the initial reduced density matrix of the two-atom system and $|0\rangle$ is
the vacuum state of the scalar field. It is worth mentioning that we are
interested in the time evolution of the two-atom system, thus we trace over
the field degrees of freedom, i.e.,
$\rho(\tau)=\mathrm{Tr}_{E}[\rho_{t\mathrm{ot}}(\tau)]$. Under the Born-Markov
approximation [69], the reduced dynamics of the two-atom system can be
described by the Kossakowski-Lindblad form [70, 71, 72] in the limit of weak
coupling

$\displaystyle\frac{\partial\rho(\tau)}{\partial\tau}$ $\displaystyle=$
$\displaystyle-i[H_{\mathrm{eff}},\rho_{A}(\tau)]+\mathcal{D}[\rho(\tau)],$
(5)

where the effective Hamiltonian $H_{\mathrm{eff}}$ is

$\displaystyle
H_{\mathrm{eff}}=H_{S}-\frac{i}{2}\sum_{\alpha,\beta=1}^{2}\sum_{i,j=1}^{3}H_{ij}^{(\alpha\beta)}\sigma_{i}^{(\alpha)}\sigma_{j}^{(\beta)},$
(6)

and the dissipator $\mathcal{D}[\rho(\tau)]$ is

$\displaystyle\mathcal{D}[\rho(\tau)]=\frac{1}{2}\sum_{\alpha,\beta=1}^{2}\sum_{i,j=1}^{3}C_{ij}^{(\alpha\beta)}[2\sigma_{j}^{(\beta)}\rho\sigma_{i}^{(\alpha)}-\sigma_{i}^{(\alpha)}\sigma_{j}^{(\beta)}\rho-\rho\sigma_{i}^{(\alpha)}\sigma_{j}^{(\beta)}].$
(7)

In the master equation (5) that the dissipator $\mathcal{D}[\rho(\tau)]$
describes the environment-induced decoherence and dissipation, which means
that the evolution of the quantum system is nonunitary. There is also
modification of the free Hamiltonian of the two-atom system which incarnates
in the effective Hamiltonian $H_{\mathrm{eff}}$. The coefficients of the
matrix $C_{ij}^{(\alpha\beta)}$ in Eq. (7) reads

$\displaystyle
C^{(\alpha\beta)}_{ij}=A^{(\alpha\beta)}\delta_{ij}-iB^{(\alpha\beta)}\epsilon_{ijk}\delta_{3k}-A^{(\alpha\beta)}\delta_{3i}\delta_{3j},$
(8)

where

$\displaystyle
A^{(\alpha\beta)}=\frac{\mu^{2}}{4}[\mathcal{G}^{(\alpha\beta)}(\omega)+\mathcal{G}^{(\alpha\beta)}(-\omega)],$
$\displaystyle
B^{(\alpha\beta)}=\frac{\mu^{2}}{4}[\mathcal{G}^{(\alpha\beta)}(\omega)-\mathcal{G}^{(\alpha\beta)}(-\omega)].$
(9)

In the above, we have defined

$\displaystyle\mathcal{G}^{(\alpha\beta)}(\lambda)=\int_{-\infty}^{+\infty}d\Delta\tau
e^{i\lambda\Delta\tau}\langle\phi(\mathbf{x}_{\alpha}(\tau))G^{(\alpha\beta)}(\Delta\tau)\rangle,$
(10)

which is the Fourier transform of the field correlation functions

$\displaystyle
G^{(\alpha\beta)}(\Delta\tau)=\langle\phi(\mathbf{x}_{\alpha}(\tau))\phi(\mathbf{x}_{\beta}(\tau^{\prime}))\rangle.$
(11)

Similarly, $H_{ij}^{(\alpha\beta)}$ in the above expressions can be derived by
replacing the Fourier transform $\mathcal{G}^{(\alpha\beta)}(\lambda)$ with
the Hilbert transform $\mathcal{K}^{(\alpha\beta)}(\lambda)$, which is

$\displaystyle\mathcal{K}^{(\alpha\beta)}(\lambda)=\frac{P}{\pi
i}\int_{-\infty}^{+\infty}d\omega\frac{\mathcal{G}^{(\alpha\beta)}(\lambda)}{\omega-\lambda},$
(12)

with $P$ denoting the principal value. It was shown in Refs. [4, 5] that the
effective Hamiltonian $H_{\mathrm{eff}}=\tilde{H}_{S}+H_{\mathrm{eff}}^{(12)}$
includes two pieces. The first term is the renormalization of the transition
frequencies, i.e., the Lamb shift of each individual atom, and is derived by
replacing $\omega$ in the atom’s Hamiltonian $H_{S}$ (2) with a renormalized
energy level spacing

$\displaystyle\tilde{\omega}=\omega+\frac{i\mu^{2}}{2}[\mathcal{K}^{(11)}(-\omega)-\mathcal{K}^{(11)}(\omega)].$
(13)

Note that this term can be regarded as a rescaling of the gap of the energy
level, we shall not consider it any further. The second term is an
environment-induced coupling between the atoms, which is

$\displaystyle
H_{\mathrm{eff}}^{(12)}=-\sum_{i,j=1}^{3}\Omega_{ij}^{(12)}\sigma_{i}\otimes\sigma_{j},$
(14)

where

$\displaystyle\Omega_{ij}^{(12)}=D\delta_{ij}-D\delta_{3i}\delta_{3j},$ (15)

with

$\displaystyle
D=\frac{i\mu^{2}}{4}[\mathcal{K}^{(12)}(-\omega)+\mathcal{K}^{(12)}(\omega)].$
(16)

As a result, we can rewrite the master equation (5) as

$\displaystyle\frac{\partial\rho(\tau)}{\partial\tau}$ $\displaystyle=$
$\displaystyle-i\tilde{\omega}\sum_{\alpha=1}^{2}[\sigma_{3}^{(\alpha)},\rho_{(}\tau)]+i\sum_{i,j=1}^{3}\Omega_{ij}^{(12)}[\sigma_{i}\otimes\sigma_{j},\rho(\tau)]$
(17)
$\displaystyle+\frac{1}{2}\sum_{\alpha,\beta=1}^{2}\sum_{i,j=1}^{3}C^{(\alpha\beta)}_{ij}[2\sigma_{j}^{(\beta)}\rho\sigma_{i}^{(\alpha)}-\sigma_{i}^{(\alpha)}\sigma_{j}^{(\beta)}\rho-\rho\sigma_{i}^{(\alpha)}\sigma_{j}^{(\beta)}].$

To study the dynamics of the two-atom system, it is convenient for us to solve
the master equation (5) in the coupled basis, i.e., in the set:
$\\{|G\rangle=|00\rangle,|A\rangle=\frac{1}{\sqrt{2}}(|10\rangle-|01\rangle),|S\rangle=\frac{1}{\sqrt{2}}(|10\rangle+|01\rangle),|E\rangle=|11\rangle\\}$.
Moreover, with the help of Eqs. (8)-(15), we can write the coefficient of the
dissipator in the master equation (17) as

$\displaystyle
C^{(11)}_{ij}=A_{1}\delta_{ij}-iB_{1}\epsilon_{ijk}\delta_{3k}-A_{1}\delta_{3i}\delta_{3j},$
$\displaystyle
C^{(22)}_{ij}=A_{2}\delta_{ij}-iB_{2}\epsilon_{ijk}\delta_{3k}-A_{2}\delta_{3i}\delta_{3j},$
$\displaystyle
C^{(12)}_{ij}=A_{3}\delta_{ij}-iB_{3}\epsilon_{ijk}\delta_{3k}-A_{3}\delta_{3i}\delta_{3j},$
$\displaystyle
C^{(21)}_{ij}=A_{4}\delta_{ij}-iB_{4}\epsilon_{ijk}\delta_{3k}-A_{4}\delta_{3i}\delta_{3j},$
$\displaystyle\Omega^{(12)}_{ij}=D\delta_{ij}-D\delta_{3i}\delta_{3j}.$ (18)

Then, the master equation (17) in terms of coupled basis can be rewritten as
[73]

$\displaystyle\dot{\rho}_{GG}$ $\displaystyle=$
$\displaystyle-2(A_{1}+A_{2}-B_{1}-B_{2})\rho_{GG}+(A_{1}+A_{2}-A_{3}-A_{4}+B_{1}+B_{2}-B_{3}-B_{4})\rho_{AA}$
$\displaystyle+(A_{1}+A_{2}+A_{3}+A_{4}+B_{1}+B_{2}+B_{3}+B_{4})\rho_{SS}+(A_{1}-A_{2}-A_{3}+A_{4}+B_{1}$
$\displaystyle-
B_{2}-B_{3}+B_{4})\rho_{AS}+(A_{1}-A_{2}+A_{3}-A_{4}+B_{1}-B_{2}+B_{3}-B_{4})\rho_{SA},$
$\displaystyle\dot{\rho}_{EE}$ $\displaystyle=$
$\displaystyle-2(A_{1}+A_{2}+B_{1}+B_{2})\rho_{EE}+(A_{1}+A_{2}-A_{3}-A_{4}-B_{1}-B_{2}+B_{3}+B_{4})\rho_{AA}$
$\displaystyle+(A_{1}+A_{2}+A_{3}+A_{4}-B_{1}-B_{2}-B_{3}-B_{4})\rho_{SS}+(-A_{1}+A_{2}+A_{3}-A_{4}+B_{1}$
$\displaystyle-
B_{2}-B_{3}+B_{4})\rho_{AS}+(-A_{1}+A_{2}-A_{3}+A_{4}+B_{1}-B_{2}+B_{3}-B_{4})\rho_{SA},$
$\displaystyle\dot{\rho}_{AA}$ $\displaystyle=$
$\displaystyle-2(A_{1}+A_{2}-A_{3}-A_{4})\rho_{AA}+(A_{1}+A_{2}-A_{3}-A_{4}-B_{1}-B_{2}+B_{3}+B_{4})\rho_{GG},$
$\displaystyle+(A_{1}+A_{2}-A_{3}-A_{4}+B_{1}+B_{2}-B_{3}-B_{4})\rho_{EE}+(-B_{1}+B_{2}+B_{3}-B_{4})\rho_{AS}$
$\displaystyle+(-B_{1}+B_{2}+B_{3}-B_{4})\rho_{SA},$
$\displaystyle\dot{\rho}_{SS}$ $\displaystyle=$
$\displaystyle-2(A_{1}+A_{2}+A_{3}+A_{4})\rho_{SS}+(A_{1}+A_{2}+A_{3}+A_{4}-B_{1}-B_{2}-B_{3}-B_{4})\rho_{GG},$
$\displaystyle+(A_{1}+A_{2}+A_{3}+A_{4}+B_{1}+B_{2}+B_{3}+B_{4})\rho_{EE}+(-B_{1}+B_{2}+B_{3}-B_{4})\rho_{AS}$
$\displaystyle+(-B_{1}+B_{2}+B_{3}-B_{4})\rho_{SA},$
$\displaystyle\dot{\rho}_{AS}$ $\displaystyle=$
$\displaystyle(A_{1}-A_{2}-A_{3}+A_{4}-B_{1}+B_{2}+B_{3}-B_{4})\rho_{GG}+(-A_{1}+A_{2}+A_{3}-A_{4}-B_{1}$
$\displaystyle+B_{2}+B_{3}-B_{4})\rho_{EE}-2(A_{1}+A_{2}+2iD)\rho_{AS},$
$\displaystyle\dot{\rho}_{SA}$ $\displaystyle=$
$\displaystyle(A_{1}-A_{2}+A_{3}-A_{4}-B_{1}+B_{2}-B_{3}+B_{4})\rho_{GG}+(-A_{1}+A_{2}-A_{3}+A_{4}-B_{1}$
$\displaystyle+B_{2}-B_{3}+B_{4})\rho_{EE}-2(A_{1}+A_{2}-2iD)\rho_{SA},$
$\displaystyle\dot{\rho}_{GE}$ $\displaystyle=$
$\displaystyle-2(A_{1}+A_{2})\rho_{GE},\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\dot{\rho}_{EG}=-2(A_{1}+A_{2})\rho_{EG},$
(19)

where $\rho_{IJ}=\langle I|\rho|J\rangle$, $I,J\in\\{G,A,S,E\\}$, and
$\dot{\rho}_{IJ}$ is the derivative with respect to the atomic proper time
$\tau$. Note that the parameter $D$ contains the environment-induced
interatomic interaction, so if $D=0$, Eq. (2.1) recovers to scenario where the
environment-induced interatomic interaction does not exist for the two atoms
coupled to the scalar field as shown in Ref. [73]. It is worthy mentioning
that if the initial matrix takes the X form, namely: states with nonzero
elements only along the diagonal and antidiagonal of the density matrix, in
the decoupled basis $\\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\\}$, the X
structure will be maintained during the evolution. To learn the entanglement
dynamics of the two-atom system, we use concurrence to characterize quantum
entanglement, which is introduced by Wootters [74]. For the X states, the
concurrence is analytically given by [75]

$\displaystyle C[\rho(\tau)]=\textrm{max}\\{0,K_{1}(\tau),K_{2}(\tau)\\},$
(20)

where

$\displaystyle K_{1}(\tau)$ $\displaystyle=$
$\displaystyle\sqrt{[\rho_{AA}(\tau)-\rho_{SS}(\tau)]^{2}-[\rho_{AS}(\tau)-\rho_{SA}(\tau)]^{2}}-2\sqrt{\rho_{GG}(\tau)\rho_{EE}(\tau)},$
(21) $\displaystyle K_{2}(\tau)$ $\displaystyle=$
$\displaystyle-\sqrt{[\rho_{AA}(\tau)+\rho_{SS}(\tau)]^{2}-[\rho_{AS}(\tau)+\rho_{SA}(\tau)]^{2}}+2|\rho_{GE}(\tau)|.\
$ (22)

###  2.2 Scalar field propagator in $\kappa$-deformed spacetime and Minkowski
spacetime

We are interested in entanglement dynamics in $\kappa$-deformed spacetime and
Minkowski spacetime. Before that we here simply review the $\kappa$-deformed
Klein-Gordon theory, especially the field correlation function in
$\kappa$-deformed spacetime, which plays a very important role in the
following calculation.

Let us firstly give the basic ingredients for the field correlation function
of the scalar field in $\kappa$-deformed spacetime. For more details, one can
refer to Refs. [59, 60], where the $\kappa$-deformed Klein-Gordon theory has
been investigated in the commutative spacetime itself. This processing allows
us to explicitly define the trajectory of the motional atoms in the
commutative spacetime. Specifically, in $\kappa$-deformed spacetime, the time
and space coordinates are not commutative but obey the Lie algebra type
commutation relations

$\displaystyle[\hat{x}_{i},\hat{x}_{j}]=0,\;\;\;[\hat{x}_{0},\hat{x}_{i}]=\frac{i}{\kappa}\hat{x}_{i},$
(23)

with $i,j\in\\{1,2,3\\}$ and the positive parameter $\kappa$ representing the
deformation of the spacetime. In Refs. [48, 49] the authors indicated that the
symmetry of $\kappa$-deformed spacetime is well known to be the
$\kappa$-Poincaré algebra, in which the defined relations of this algebra
involve the deformation parameter $\kappa$, and when
$\kappa\rightarrow\infty$, it is reduced to the Poincaré algebra. In order to
construct the $\kappa$-Poincaré algebra, we can seek realizations of the
noncommutative coordinates $\hat{x}_{\mu}$ in terms of ordinary commutative
coordinates $x_{\mu}$ and corresponding derivatives $\partial_{\mu}$:
$\partial_{\mu}=\frac{\partial}{\partial x_{\mu}}$. These realizations define
a unique mapping between the functions on noncommutative space to functions on
commutative space. In such references, a general ansatz for noncommutative
coordinates satisfying the algebra (23) is given by

$\displaystyle\hat{x}_{i}=x_{i}\varphi(A),\;\;\;\;\hat{x}_{0}=x_{0}\psi(A)+\frac{i}{\kappa}x_{i}\partial_{i}\gamma(A),$
(24)

where $\varphi$, $\psi$ and $\gamma$ are functions of
$A=-\frac{i}{\kappa}\partial_{0}$. Inserting this ansatz (24) into (23), one
has

$\displaystyle\gamma=1+\frac{\varphi^{\prime}}{\varphi}\psi,$ (25)

where $\varphi^{\prime}$ is the derivative of $\varphi$ with respect to $A$.
Note that here $\varphi$, $\psi$, and $\gamma$ are positive functions with the
boundary conditions

$\displaystyle\varphi(0)=1,\;\;\;\;\psi(0)=1,$ (26)

and $\gamma(0)=1+\varphi^{\prime}(0)$ has to be finite. It is worth mentioning
that, in the above equations, $\varphi$ characterizes arbitrary realizations
of the noncommutative coordinates in terms of the commutative coordinates and
their derivatives.

Furthermore, let $M_{\mu\nu}$ denote the generators obeying the ordinary
undeformed ${\textit{s}o}(n-1,n)$ algebra:

$\displaystyle[M_{\mu\nu},M_{\lambda\rho}]=\eta_{\nu\lambda}M_{\mu\rho}-\eta_{\mu\lambda}M_{\nu\rho}-\eta_{\nu\rho}M_{\mu\lambda}+\eta_{\mu\rho}M_{\nu\lambda},$
$\displaystyle
M_{\mu\nu}=-M_{\nu\mu},\;\;\;\;\eta_{\mu\nu}=\textrm{diag}(-1,1,1,1).$ (27)

It is required that the commutator $[M_{\mu\nu},\hat{x}_{\lambda}]$ between
the generators $M_{\mu\nu}$ and the noncommutative coordinates
$\hat{x}_{\lambda}$, is antisymmetric with respect to the indices $\mu$ and
$\nu$, and is linear functions of $\hat{x}_{\lambda}$ and $M_{\mu\nu}$. Note
that as $\kappa\rightarrow\infty$, we have a smooth commutative limit.
Therefore, in this regard, there emerges two classes of possible realizations,
given by $\psi=1$ and $\psi=1+2A$. We will focus on $\psi=1$ case, and the
explicit corresponding form of $M_{\mu\nu}$ are

$\displaystyle M_{i0}$ $\displaystyle=$ $\displaystyle
x_{i}\partial_{0}\varphi\frac{e^{2A}-1}{2A}-x_{0}\partial_{i}\frac{1}{\varphi}+\frac{i}{\kappa}x_{i}\Delta\frac{1}{2\varphi}-\frac{i}{\kappa}x_{k}\partial_{k}\partial_{i}\frac{\gamma}{\varphi},$
$\displaystyle M_{ij}$ $\displaystyle=$ $\displaystyle
x_{i}\partial_{j}-x_{j}\partial_{i},$ (28)

where $\Delta=\partial_{k}\partial_{k}$. In Refs. [76, 77, 78, 79, 80, 81, 82]
the Dirac derivatives $D_{\mu}$ and the invariant Laplace operator $\Box$ have
been introduced, to obtain the generalized Klein-Gordon equation invariant
under the $\kappa$-Poincaré algebra, through the following relations

$\displaystyle[M_{\mu\nu},D_{\lambda}]=\eta_{\nu\lambda}D_{\mu}-\eta_{\mu\lambda}D_{\nu},\;\;\;[D_{\mu},D_{\nu}]=0,$
$\displaystyle\;\;\;\;\;\;[M_{\mu\nu},\Box]=0,\;\;\;\;\;[\Box,\hat{x}_{\mu}]=2D_{\mu},$
(29)

with

$\displaystyle
D_{i}=\partial_{i}\frac{e^{-A}}{\varphi},\;\;\;D_{0}=\partial_{0}\frac{\sinh
A}{A}+\frac{i}{\kappa}\Delta(\frac{e^{-A}}{2\varphi^{2}}),$
$\displaystyle\Box=\Delta(\frac{e^{-A}}{\varphi^{2}})+2\partial_{0}^{2}\frac{1-\cosh
A}{A^{2}}.$ (30)

Note that $D_{\mu}$ and $M_{\mu\nu}$ given above generate the
$\kappa$-Poincaré algebra, whose relations are the same as that of the usual
Poincaré algebra. However, the explicit form of these generators are modified
and those modifications are dependent on the deformation parameter.

With Eq. (2.2), one can find that the Casimir of this algebra,
$D_{\mu}D_{\mu}$, can be expressed in terms of the $\Box$ operator as

$\displaystyle D_{\mu}D_{\mu}=\Box(1-\frac{1}{4\kappa^{2}}\Box).$ (31)

When $\kappa\rightarrow\infty$, we have
$D_{\mu}D_{\mu}\rightarrow\partial_{\mu}\partial_{\mu}$, which reduces to the
usual relativistic dispersion relation. Generalizing the notions from
commutative space, it is nature to write the generalized Klein-Gordon equation
using the Casimir which is invariant under the $\kappa$-Poincaré algebra as

$\displaystyle\bigg{(}\Box\big{(}1-\frac{1}{4\kappa^{2}}\Box\big{)}-m^{2}\bigg{)}\phi(\mathbf{x})=0\;.$
(32)

As a result of the realizations of the noncommutative coordinates in term of
the commuting ones and corresponding derivatives, both the generators and
Casimir of the $\kappa$-Poincaré algebra can be expressed in terms of the
commutative coordinates and their derivatives, then the scalar field and
operators appearing in the $\kappa$-deformed Klein-Gordon equation (32) are
well defined in the commutative spacetime. Therefore, we can use the standard
tools of field theory defined in the commutative spacetime to analyze the
$\kappa$-deformed Klein-Gordon theory. The deformed dispersion relation
rooting in Eq. (32) reads

$\displaystyle
4\kappa^{2}\sinh^{2}(\frac{p_{0}}{2\kappa})-p_{i}^{2}\frac{e^{-\frac{p_{0}}{\kappa}}}{\varphi^{2}(\frac{p_{0}}{\kappa})}-\frac{1}{4\kappa^{2}}\bigg{[}4\kappa^{2}\sinh^{2}(\frac{p_{0}}{2\kappa})-p_{i}^{2}\frac{e^{-\frac{p_{0}}{\kappa}}}{\varphi^{2}(\frac{p_{0}}{\kappa})}\bigg{]}^{2}=m^{2},$
(33)

where $p_{0}=i\partial_{0}$ and $p_{i}=-i\partial_{i}$. We can see from (33)
that the characters of field, i.e., nonlocal and noncausal, appear. The
Hamiltonian for this field is too complicated and difficult to be expressed in
a compact form. Therefore, in order to obtain the two-point correlation
function in $\kappa$-deformed spacetime for simplicity, we chose
$\varphi(\frac{p_{0}}{\kappa})=e^{-\frac{p_{0}}{2\kappa}}$ as done in Refs.
[59, 60]. Moreover, from here onwards we keep only term up to second order in
$1/\kappa$ since $\kappa$ is expected to be quite big in the theory. Through
the specific calculation, we find the two-point correlation function reads

$\displaystyle
G^{+}(\mathbf{x},\mathbf{x}^{\prime})=\frac{1}{4\pi^{2}}\frac{1}{(\mathbf{x}-\mathbf{x}^{\prime})^{2}-(t-t^{\prime})^{2}}$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;-\frac{1}{16\pi^{2}\kappa^{2}}\frac{(\mathbf{x}-\mathbf{x}^{\prime})^{2}+3(t-t^{\prime})^{2}}{[(\mathbf{x}-\mathbf{x}^{\prime})^{2}-(t-t^{\prime})^{2}]^{3}}$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;-\frac{1}{4\pi^{2}\kappa^{2}}\frac{[(\mathbf{x}-\mathbf{x}^{\prime})^{2}+(t-t^{\prime})^{2}](t-t^{\prime})^{2}}{[(\mathbf{x}-\mathbf{x}^{\prime})^{2}-(t-t^{\prime})^{2}]^{4}}.$
(34)

Note that for $\kappa\rightarrow\infty$, the two-point correlation function in
Eq. (2.2) recovers to the Minkowski spacetime case [83], as expected, which is

$\displaystyle
G^{+}(\mathbf{x},\mathbf{x}^{\prime})=\frac{1}{4\pi^{2}}\frac{1}{(\mathbf{x}-\mathbf{x}^{\prime})^{2}-(t-t^{\prime})^{2}}.$
(35)

In what follows, these two correlation functions will be used to explore the
two-atom system entanglement dynamics.

##  3 Entanglement dynamics for two atoms without the environment-induced
interatomic interaction

Now let us consider the entanglement dynamics of two-atom system interacting
with an external environment. Three different initial state cases will be
considered: 1) the separable state $|E\rangle$, 2) the symmetric entangled
state $|S\rangle$, 3) the antisymmetric entangled state $|A\rangle$. Note that
all of these chosen initial states won’t result in the environment-induced
interatomic interaction. We focus on how the relativistic motion affects the
entanglement dynamics. Specifically, we will analyse the entanglement dynamics
for static atoms, inertial atoms moving with a constant velocity and
circularly accelerated atoms, respectively, which are coupled with a massless
scalar field in $\kappa$-deformed spacetime and Minkowski spacetime. In
particular, we will comparatively investigate the phenomena of entanglement
generation and degradation in these two universes.

###  3.1 Entanglement dynamics of two static atoms

We first consider the entanglement dynamics of two static atoms which are
separated by a distance $L$ along the trajectories

$\displaystyle
t_{1}(\tau)=\tau,\;\;\;x_{1}(\tau)=0,\;\;\;y_{1}(\tau)=0,\;\;\;z_{1}(\tau)=0,$
$\displaystyle
t_{2}(\tau)=\tau,\;\;\;x_{2}(\tau)=0,\;\;\;y_{2}(\tau)=0,\;\;\;z_{2}(\tau)=L.$
(36)

Substituting the above trajectories into the two-point correlation function in
$\kappa$-deformed spacetime (2.2), we obtain

$\displaystyle G^{11}(x,x^{\prime})$ $\displaystyle=$ $\displaystyle
G^{22}(x,x^{\prime})=-\frac{1}{4\pi^{2}}\frac{1}{\triangle\tau^{2}}-\frac{1}{16\pi^{2}\kappa^{2}}\frac{1}{\triangle\tau^{4}},$
$\displaystyle G^{12}(x,x^{\prime})$ $\displaystyle=$ $\displaystyle
G^{21}(x,x^{\prime})=-\frac{1}{4\pi^{2}}\bigg{[}\frac{1}{\triangle\tau^{2}-L^{2}}+\frac{1}{4\kappa^{2}}\frac{3\triangle\tau^{2}+L^{2}}{(\triangle\tau^{2}-L^{2})^{3}}-\frac{1}{\kappa^{2}}\frac{\triangle\tau^{4}+L^{2}\triangle\tau^{2}}{(\triangle\tau^{2}-L^{2})^{4}}\bigg{]}.$

By invoking the method of residue theorem, the corresponding Fourier
transforms shown in (10) to the field correlation functions (3.1) are found to
be

$\displaystyle\mathcal{G}^{11}(\lambda)$ $\displaystyle=$
$\displaystyle\mathcal{G}^{22}(\lambda)=\frac{\lambda}{2\pi}\bigg{[}1-\frac{7\lambda^{2}}{96\kappa^{2}}\bigg{]}\theta(\lambda),$
$\displaystyle\mathcal{G}^{12}(\lambda)$ $\displaystyle=$
$\displaystyle\mathcal{G}^{21}(\lambda)=\frac{\lambda}{2\pi}\bigg{[}\frac{\sin\lambda
L}{\lambda L}-\frac{\lambda^{2}\cos\lambda
L}{24\kappa^{2}}\bigg{]}\theta(\lambda),$ (38)

where $\theta(x)$ is the step function. Substituting the Fourier transforms
(3.1) into Eq. (2.1), one can find

$\displaystyle
A_{1}=A_{2}=B_{1}=B_{2}=\frac{\Gamma_{0}}{4}\bigg{[}1-\frac{\omega^{2}}{24\kappa^{2}}\bigg{]},$
$\displaystyle
A_{3}=A_{4}=B_{3}=B_{4}=\frac{\Gamma_{0}}{4}\bigg{[}\frac{\sin\omega L}{\omega
L}-\frac{\omega^{2}\cos\omega L}{24\kappa^{2}}\bigg{]},$ (39)

with $\Gamma_{0}=\frac{\mu^{2}\omega}{2\pi}$ being the spontaneous emission
rate of each individual atom.

Notice that for $\kappa\rightarrow\infty$, the functions (3.1) in
$\kappa$-deformed spacetime recover to that of two atoms case at rest in
Minkowski spacetime as expected. Thus we have the relevant coefficients of Eq.
(2.1) for this case as

$\displaystyle A_{1}=A_{2}=B_{1}=B_{2}=\frac{\Gamma_{0}}{4},$ $\displaystyle
A_{3}=A_{4}=B_{3}=B_{4}=\frac{\Gamma_{0}}{4}\frac{\sin\omega L}{\omega L}.$
(40)

Preparing the initial state, e.g., $|E\rangle$, $|S\rangle$ or $|A\rangle$,
and inserting these relevant coefficients above into the Eqs. (2.1), we can
solve the master equation correspondingly. Then we can find the corresponding
entanglement in Eq. (20) is given by

$\displaystyle C[\rho(\tau)]=\textrm{max}\\{0,K_{1}(\tau)\\},$ (41)

where

$\displaystyle
K_{1}(\tau)=\sqrt{[\rho_{AA}(\tau)-\rho_{SS}(\tau)]^{2}}-\sqrt{\rho_{GG}(\tau)\rho_{EE}(\tau)},$
(42)

from which we can see that the concurrence is independent of the environment-
induced interatomic interaction. Explicit entanglement dynamics for various
situations is investigated in the following.

####  3.1.1 Two static atoms initially prepared in a separable state
$|E\rangle$

We start with the entanglement dynamics for static atoms initially prepared in
a separable state $|E\rangle$. From Eqs. (41)-(42), we can see that
entanglement can be generated only when the factor
$\sqrt{[\rho_{AA}(\tau)-\rho_{SS}(\tau)]^{2}}$ outweighs the threshold factor
$2\sqrt{\rho_{GG}(\tau)\rho_{EE}(\tau)}$. We note that such entanglement
generation phenomenon inevitably occurs when the system undergoes only
spontaneous emission evolution for a finite time. This phenomenon is called as
the delayed sudden birth of entanglement [37].

Let us consider the case where the interatomic separation vanishes
$(L\rightarrow 0)$. Then
$A_{i}=B_{i}=\frac{\Gamma_{0}}{4}[1-\frac{\omega^{2}}{24\kappa^{2}}]$ in Eq.
(3.1) and $A_{i}=B_{i}=\frac{\Gamma_{0}}{4}$ in Eq. (3.1) with
$i\in\\{1,2,3,4\\}$. Therefore $\rho_{AA}(\tau)$ remains zero during evolution
both in $\kappa$-deformed spacetime and Minkowski spacetime. In such case the
threshold always outweighs the population $\rho_{SS}(\tau)$ and no quantum
entanglement can be generated both in these two universes. For the interatomic
separation comparable with the transition wavelength $(L\sim\omega^{-1})$, we
solve Eq. (2.1) numerically. We show the corresponding results in Fig. 1. We
find that unlike the vanishing interatomic separation case, the delayed sudden
birth of entanglement occurs both for the $\kappa$-deformed spacetime case and
Minkowski spacetime case. Specifically, one can note that the time when the
entanglement begins to be generated depends on the interatomic separation and
the spacetime deformation. The larger the distance between the two atoms is,
the earlier the entanglement generation occurs. The deformation of spacetime
may delay the entanglement creation. Furthermore, the amplitude of the created
entanglement is also influenced by the interatomic separation and the
spacetime deformation, which we will discuss detailedly in the following. When
the value of spacetime deformation parameter is large, no matter how long the
atomic separation is, the points list substantially coincides with the solid
line, i.e., the entanglement generation for two static atoms during evolution
in $\kappa$-deformed spacetime is almost identical to the Minkowski spacetime
case in this situation.

(a) $|E\rangle$ Figure 1: Time evolution of concurrence for two static atoms
initially prepared in $|E\rangle$. _M_ denotes the Minkowski spacetime.

(a) $|E\rangle$ (b) $|E\rangle$

Figure 2: The maximum of concurrence during evolution for two static atoms via
the interatomic separation initially prepared in $|E\rangle$.

In Fig. 2, we study the effects of atomic separation on the maximum of
entanglement generated during the evolution. It is shown that the maximum of
entanglement is a periodic function of interatomic separation, and the
amplitude decays with the increasing interatomic separation both in
$\kappa$-deformed spacetime and Minkowski spacetime. Remarkably, when the
modification of spacetime is relatively strong, e.g., $\kappa/\omega=1$, we
find that the entanglement behaviors of two atoms via the interatomic distance
in $\kappa$-deformed spacetime are different from that in Minkowski spacetime
[see Fig. 2 (a)]. However, when the modification of spacetime is relatively
weak, e.g., $\kappa/\omega=1000$, we find the entanglement for the
$\kappa$-deformed spacetime case and the Minkowski spacetime case behaves
almost the same [see Fig. 2 (b)]. In this regard, we infer that for the larger
spacetime deformation parameter (LDP) case, all the laws of physics in
$\kappa$-deformed spacetime substantially recover to that in flat spacetime.
Thus in this case it seems to be difficult to distinguish these two
spacetimes.

####  3.1.2 Two static atoms initially prepared in entangled states
$|A\rangle$ and $|S\rangle$

Here we are going to investigate the entanglement degradation for two static
atoms initially prepared in the antisymmetric entangled state $|A\rangle$ and
the symmetric entangled state $|S\rangle$, both of which are maximally
entangled.

(a) $|A\rangle$ (b) $|S\rangle$ (c) $|A\rangle$ (d) $|S\rangle$

Figure 3:  Concurrence as a function of evolution time $\Gamma_{0}\tau$ (a, b)
and deformation parameter $\kappa/\omega$ (c, d) for static atoms initially
prepared in $|A\rangle$ (left) and $|S\rangle$ (right).

From Fig. 3, we find that the concurrence decreases monotonically via time and
goes to zero in the infinite time limit for both of these two universe cases.
Moreover, as the interatomic separation increases, the entanglement magnitude
difference between the $\kappa$-deformed spacetime case and Minkowski
spacetime case becomes smaller. This means that it becomes more difficult for
us to distinguish theses two universes through two-atom entanglement dynamics.
We also find that the response of entanglement magnitude to the interatomic
separation behaves quite differently for different initial entangled states
(antisymmetric entangled state and symmetric entangled state). Specifically,
the entanglement magnitude for the initial antisymmetric entangled state case
(initial symmetric entangled state) decreases (increases) as the increase of
the interatomic separation. In particular, for a fixed evolution time, the
concurrence drops to an asymptotic value as the spacetime deformation
parameter $\kappa$ increases [see Figs. 3 (c) and (d)]. In other words, when
the deformation parameter $\kappa$ is large, the entanglement dynamics curves
for two atoms in $\kappa$-deformed spacetime almost coincide with that for the
Minkowski spacetime case [also see Figs. 3 (a) and (b)].

We give a brief summary here. For the case of two static atoms initially
prepared in above three specific states, when the spacetime deformation
parameter $\kappa$ is large, the entanglement dynamics in $\kappa$-deformed
spacetime are almost indistinguishable with that in Minkowski spacetime
regardless of the choice of interatomic separation. This implies that we
cannot distinguish these two spacetimes using the entanglement dynamics of
static atoms with these initial states when the deformation parameter is
large. However, it is known from the theoretical hypothesis that the value of
deformation parameter should be large involved in $\kappa$-deformed spacetime
[84, 85, 86], thus the $\kappa$-deformed spacetime exhibits almost the same
properties as the Minkowski spacetime generally and obeys the Poincaré
algebra. Hence, an issue arises: can we find some external auxiliary
conditions, such as relativistic motion, to distinguish these two spacetimes
by means of entanglement dynamics when the spacetime deformation parameter is
large? This is what we will study in the following.

###  3.2 Entanglement dynamics of two uniformly moving atoms

In this section we will investigate the entanglement dynamics of two atoms
moving with a constant velocity in $\kappa$-deformed spacetime and Minkowski
spacetime. We mainly focus on how the velocity affects the entanglement
behaviors in these two different spacetimes.

The trajectories of the two inertial atoms, which are moving with a constant
velocity and are separated from each other by a distance $L$, can be described
as

$\displaystyle
t_{1}(\tau)=\gamma\tau,\;\;\;x_{1}(\tau)=\upsilon\gamma\tau,\;\;\;y_{1}(\tau)=0,\;\;\;z_{1}(\tau)=0,$
$\displaystyle
t_{2}(\tau)=\gamma\tau,\;\;\;x_{2}(\tau)=\upsilon\gamma\tau,\;\;\;y_{2}(\tau)=0,\;\;\;z_{2}(\tau)=L,$
(43)

where $\upsilon$ denotes the velocity and $\gamma=1/\sqrt{1-\upsilon^{2}}$ is
the usual Lorentz factor.

Substituting the trajectories (3.2) into Eq. (2.2), the two-point correlation
functions in $\kappa$-deformed spacetime can be rewritten as

$\displaystyle
G^{11}(x,x^{\prime})=G^{22}(x,x^{\prime})=-\frac{1}{4\pi^{2}}\frac{1}{\triangle\tau^{2}}+\frac{1}{16\pi^{2}\kappa^{2}}\frac{\gamma^{2}(3+\upsilon^{2})}{\triangle\tau^{4}}-\frac{1}{4\pi^{2}\kappa^{2}}\frac{\gamma^{4}(\upsilon^{2}+1)}{\triangle\tau^{4}},$
$\displaystyle
G^{12}(x,x^{\prime})=G^{21}(x,x^{\prime})=-\frac{1}{4\pi^{2}}\frac{1}{\triangle\tau^{2}-L^{2}}+\frac{1}{16\pi^{2}\kappa^{2}}\frac{\gamma^{2}(3+\upsilon^{2})\triangle\tau^{2}+L^{2}}{(\triangle\tau^{2}-L^{2})^{4}}$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;-\frac{1}{4\pi^{2}\kappa^{2}}\frac{\gamma^{4}(\upsilon^{2}+1)\triangle\tau^{4}+L^{2}\gamma^{2}\triangle\tau^{2}}{(\triangle\tau^{2}-L^{2})^{4}}.$
(44)

Subsequently the Fourier transforms of the above correlation functions can be
calculated with residue theorem

$\displaystyle\mathcal{G}^{11}(\lambda)$ $\displaystyle=$
$\displaystyle\mathcal{G}^{22}(\lambda)=\frac{\lambda}{2\pi}\bigg{[}1-\frac{\lambda^{2}}{24\kappa^{2}}\frac{1+\upsilon^{2}}{(1-\upsilon^{2})^{2}}-\frac{\lambda^{2}}{96\kappa^{2}}\frac{3+v^{2}}{1-v^{2}}\bigg{]}\theta(\lambda),$
$\displaystyle\mathcal{G}^{12}(\lambda)$ $\displaystyle=$
$\displaystyle\mathcal{G}^{21}(\lambda)=\frac{\lambda}{2\pi}\bigg{[}\frac{\sin\lambda
L}{\lambda
L}+\frac{f(\lambda,L,\upsilon)}{24\lambda\kappa^{2}L^{3}}\bigg{]}\theta(\lambda),$
(45)

where $f(\lambda,L,\upsilon)=\frac{(3\upsilon^{4}\lambda
L-\lambda^{3}L^{3})\cos\lambda
L-3\upsilon^{2}(\upsilon^{2}+2\lambda^{2}L^{2})\sin\lambda
L}{(1-\upsilon^{2})^{2}}$. Submitting the Fourier transforms (3.2) into Eq.
(2.1), we can obtain the coefficients in Eq. (2.1) for the $\kappa$-deformed
spacetime case

$\displaystyle
A_{1}=A_{2}=B_{1}=B_{2}=\frac{\Gamma_{0}}{4}\bigg{[}1-\frac{\omega^{2}}{6\kappa^{2}}\frac{1+\upsilon^{2}}{(1-\upsilon^{2})^{2}}+\frac{\omega^{2}}{24\kappa^{2}}\frac{3+\upsilon^{2}}{1-\upsilon^{2}}\bigg{]},$
$\displaystyle
A_{3}=A_{4}=B_{3}=B_{4}=\frac{\Gamma_{0}}{4}\bigg{[}\frac{\sin\omega L}{\omega
L}+\frac{1}{24\omega\kappa^{2}L^{3}}f(\omega,L,\upsilon)\bigg{]}.$ (46)

Similarly, the coefficients in Eq. (2.1) for two uniformly moving atoms in
Minkowski spacetime can be derived as

$\displaystyle A_{1}=A_{2}=B_{1}=B_{2}=\frac{\Gamma_{0}}{4},$ $\displaystyle
A_{3}=A_{4}=B_{3}=B_{4}=\frac{\Gamma_{0}}{4}\frac{\sin\omega L}{\omega L}.$
(47)

Note that the coefficients in Eq. (3.2) are similar to that of two static
atoms for Minkowski spacetime case (3.1). It means that, in Minkowski
spacetime, all the laws of physics for two uniformly moving atoms keep the
same for two static atoms.

According to the above coefficients, we can infer that the dynamics for two
uniformly moving atoms in $\kappa$-deformed spacetime are related to the
velocity of atoms, while that in Minkowski spacetime are not. Therefore, it is
worth to investigate how the velocity affects on the entanglement dynamics, so
that to seek whether it is possible to distinguish these two spacetime with
the help of relativistic motion.

####  3.2.1 Two uniformly moving atoms initially prepared in a separable state
$|E\rangle$

Let us start with the effects of velocity on the entanglement generation for
two uniformly moving atoms initially prepared in a separable state
$|E\rangle$. When the interatomic separation is vanishingly small
$(L\rightarrow 0)$, it can be directly obtained from Eqs. (3.2) and (3.2) that
in this limit, $A_{i}=B_{i}$, and thus the evolution of the population
$\rho_{AA}(\tau)$ remains zero. Therefore, the threshold factor
$2\sqrt{\rho_{GG}(\tau)\rho_{EE}(\tau)}$ always overweights the population
$\rho_{SS}(\tau)$, meaning that with the vanishing interatomic distance there
is no generated entanglement for the two atoms in both of these two universes.
For the interatomic separation comparable with the transition wavelength
($L\sim\omega^{-1}$), we make a comparison between the entanglement dynamics
for the $\kappa$-deformed spacetime case and Minkowski spacetime case in Fig.
4. We find that the waiting time to generate entanglement depends on the
velocity and interatomic separation. The farther the interatomic separation
is, the earlier the entanglement is generated. Besides, with the increase of
the velocity of atoms, the waiting time for generating entanglement becomes
longer, i.e., the sudden creation of entanglement become later. However, here
we want to note that with the increase of velocity, the difference of the
entanglement dynamics between that for the $\kappa$-deformed spacetime case
and that for the Minkowski spacetime case become more distinguishable. That
is, even when the spacetime deformation parameter $\kappa$ is large, it is
still in principle possible to discriminate these two spacetimes with the help
of relativistic motion. This result is quite different from that of two static
atoms case discussed above. We also note that the maximum entanglement
amplitude created quite depends on the interatomic separation. This result
could also be seen from Fig. 5 (a) detailedly.

(a) $|E\rangle$ Figure 4: Time evolution of concurrence for two uniformly
moving atoms initially prepared in $|E\rangle$.

(a) $|E\rangle$ (b) $|E\rangle$

Figure 5: The maximum of concurrence during evolution for two uniformly moving
atoms initially prepared in $|E\rangle$.

In Fig. 5 (b), we show the behavior of the maximum of entanglement generated
during evolution with the assistance of the atomic motion. We find that when
the velocity of atoms is large enough, the maximum of concurrence for the
$\kappa$-deformed spacetime case and that for the Minkowski spacetime case
behaves differently even the spacetime deformation parameter is relative
large. This result is completely different from that shown in Fig. 2 (b).
Moreover, in the Minkowski spacetime the maximum of concurrence is
irrespective of velocity and remains a constant, see Fig. 5 (b), which is also
given by Eq. (3.2). In addition, as the velocity increases, the maximum of
concurrence presents an increasing difference between the $\kappa$-deformed
spacetime case and the Minkowski spacetime case. It tells us that when the
velocity is large enough, the entanglement behaviors of two atoms in principle
can be used to distinctly discriminate these two universes even the spacetime
deformation parameter, $\kappa$, is relative large.

####  3.2.2 Two uniformly moving atoms initially prepared in entangled states
$|A\rangle$ and $|S\rangle$

(a) $|A\rangle$ (b) $|S\rangle$

Figure 6:  Time evolution of concurrence for two uniformly moving atoms
initially prepared in $|A\rangle$ (a) and $|S\rangle$ (b).

Now we investigate the effects of velocity on the entanglement dynamics for
two uniformly moving atoms initially prepared in two kinds of maximally
entangled states, i.e., $|A\rangle$ and $|S\rangle$, as shown in Fig. 6. We
find that as the two-atom system evolves, their concurrence decreases
monotonously and finally decays to zero in the infinite time limit both for
the $\kappa$-deformed spacetime case and Minkowski spacetime case. However, we
want to note that although the spacetime deformation parameter $\kappa$ is
relatively large, the atomic concurrence curve in $\kappa$-deformed spacetime
with a LDP still does not overlap with that for the Minkowski spacetime case,
which is totally different compared with those shown in Fig. 3 for two static
atoms. This result originates from the influence of atomic velocity. Moreover,
we also find that as the velocity of the atoms increases the difference
between the entanglement dynamics in $\kappa$-deformed spacetime and that in
Minkowski spacetime becomes more distinct. For the initial sates, $|A\rangle$
and $|S\rangle$, their evolution responds to the interatomic distance behave
differently. In the former case, the entanglement decreases with increase of
the interatomic distance, while in the latter case the entanglement increases
as the interatomic distance increases. It means that the symmetry of atomic
entangled state may play a very important role in distinguishing the two
universes with the entanglement dynamics.

###  3.3 Entanglement dynamics of two circularly accelerated atoms

In the following, we will explore the entanglement dynamics of two circularly
accelerated atoms (with a LDP) in $\kappa$-deformed spacetime and Minkowski
spacetime. We are interested in whether the uniform circular motion of the
atoms is more readily able to tell us the difference of $\kappa$-deformed
spacetime from the Minkowski spacetime.

We assume that these two atoms rotate synchronically with a separation $L$
perpendicular to the rotating plane, whose trajectories are described as

$\displaystyle t_{1}(\tau)$ $\displaystyle=$
$\displaystyle\gamma\tau,\;\;x_{1}(\tau)=R\cos\frac{\gamma\upsilon\tau}{R},y_{1}(\tau)=R\sin\frac{\gamma\upsilon\tau}{R},\;\;z_{1}(\tau)=0,$
$\displaystyle t_{2}(\tau)$ $\displaystyle=$
$\displaystyle\gamma\tau,\;\;x_{2}(\tau)=R\cos\frac{\gamma\upsilon\tau}{R},y_{2}(\tau)=R\sin\frac{\gamma\upsilon\tau}{R},\;\;z_{2}(\tau)=L,$
(48)

where $R$ denotes the radius of the circular orbit. In the rest frame of the
atoms, the centripetal acceleration is $a=\frac{\gamma^{2}\upsilon^{2}}{R}$.

Applying the trajectories (3.3) to the $\kappa$-deformed two-point correlation
function (2.2), we have

$\displaystyle G^{11}(x,x^{\prime})$ $\displaystyle=$ $\displaystyle
G^{22}(x,x^{\prime})=-\frac{1}{4\pi^{2}}\frac{1}{\triangle\tau^{2}[1+\frac{1}{12}(a^{2}\triangle\tau^{2})]}-\frac{1}{16\pi^{2}\kappa^{2}}\frac{(4\gamma^{2}-1)-\frac{1}{12}a^{2}\triangle\tau^{2}}{\triangle\tau^{4}(1+\frac{1}{12}a^{2}\triangle\tau^{2})^{3}}$
(49)
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;-\frac{1}{4\pi^{2}\kappa^{2}}\frac{[(2\gamma^{2}-1)-\frac{1}{12}a^{2}\triangle\tau^{2}]\gamma^{2}}{\triangle\tau^{4}(1+\frac{1}{12}a^{2}\triangle\tau^{2})^{4}},$

and

$\displaystyle
G^{12}(x,x^{\prime})=G^{21}(x,x^{\prime})=-\frac{1}{4\pi^{2}}\frac{1}{\triangle\tau^{2}[1+\frac{1}{12}(a^{2}\triangle\tau^{2})]-L^{2}}$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;-\frac{1}{16\pi^{2}\kappa^{2}}\frac{(4\gamma^{2}-1)\triangle\tau^{2}-\frac{1}{12}a^{2}\triangle\tau^{4}+L^{2}}{[\triangle\tau^{2}(1+\frac{1}{12}a^{2}\triangle\tau^{2})-L^{2}]^{3}}$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;-\frac{1}{4\pi^{2}\kappa^{2}}\frac{[(2\gamma^{2}-1)\triangle\tau^{2}-\frac{1}{12}a^{2}\triangle\tau^{4}+L^{2}]\gamma^{2}\triangle\tau^{2}}{[\triangle\tau^{2}(1+\frac{1}{12}a^{2}\triangle\tau^{2})-L^{2}]^{4}}.$
(50)

Using the residual theorem, we can straightly derive the corresponding
expression of the coefficients $A_{i}$ and $B_{i}$ in Eqs. (2.1) for this
circularly acceleration case in $\kappa$-deformed spacetime. However, the
expression is too complex to exhibit here. Note that when
$\kappa\rightarrow\infty$ for this case, we can recover the result obtained in
Ref. [87] for a circularly accelerated two-atom system in Minkowski spacetime
as expected.

####  3.3.1 Two circularly accelerated atoms initially prepared in a separable
state $|E\rangle$

To study the entanglement generation for two circularly accelerated atoms in
$\kappa$-deformed spacetime with a LDP, we assume that these two atoms
initially to be prepared in a separable state $|E\rangle$.

(a) $|E\rangle$, $\omega L=1$ (b) $|E\rangle$, $a/\omega=1$

Figure 7:  Time evolution of concurrence for circularly accelerated atoms
initially prepared in $|E\rangle$, varying the centripetal acceleration
$a/\omega$ (a) and the interatomic distance $\omega L$ (b).

We focus on the case where the intermediate separations comparable with the
transition wavelength of the atoms ($L\sim\omega^{-1}$). There exists a
delayed feature of entanglement generation for two circularly accelerated
atoms in $\kappa$-deformed spacetime and Minkowski spacetime, as depicted in
Fig. 7. We note that, the waiting time to generate entanglement is not only
related to the centripetal acceleration, but also associated to the
interatomic separation. Even the spacetime deformation parameter is relatively
large, the maximum generated entanglement and the waiting time are different
for these two spacetime cases concerned. Especially, for the case of
circularly accelerated atoms with a fixed separation distance in Minkowski
spacetime, there exists a critical value, i.e., $a/\omega\approx 1.35$ for the
centripetal acceleration, beyond which entanglement generation does not
happen. However, in $\kappa$-deformed spacetime, this critical value will be
modified and increases with decreasing the value of spacetime deformation
parameter. This tells us that, in some cases, entanglement can be generated in
$\kappa$-deformed spacetime while in Minkowski spacetime this cannot, so this
presence/absence of entanglement in principle gives us a good criterion to
check which universe we are living in (see the detailed discussion of this
criterion in the following). Furthermore, we also find that the existing time
of entanglement generated depends on the centripetal acceleration $a/\omega$
and interatomic distance $\omega L$. Meanwhile, although the two-atom system
undergoes the same conditions, i.e., the same $a/\omega$ and $\omega L$, as a
result of the spacetime deformation, the existing time of entanglement
generated is quite different for these two universes concerned. This property
may also help us to distinguish the $\kappa$-deformed spacetime and the
Minkowski spacetime in principle.

In Fig. 8, we plot how the maximal concurrence during evolution is affected by
the interatomic separation and centripetal acceleration of atoms. As shown in
Figs. 8 (a) and (b), when the two atoms are static ($a/\omega=0$) the maximal
entanglement for the $\kappa$-deformed spacetime case and that for the
Minkowski spacetime case can not be distinguished. However, one can find they
become more distinguishable with the increase of the centripetal acceleration.
The maximal entanglement behavior via the centripetal acceleration depends on
the interatomic separation: it would first increase to a maximum and then
decay to zero with the increase of the centripetal acceleration when the
interatomic separation is relatively small; or decay monotonically to zero
when the interatomic separation is relatively far. Furthermore, Figs. 8 (c)
and (d) show how the maximum entanglement generated depends on the interatomic
separation with fixed centripetal acceleration. Remarkably, the interatomic
distance regime where entanglement can be created is the acceleration-
dependent. Therefore, we can find that even the spacetime deformation
parameter $\kappa$ is relatively large, and one can find the spatial region
where entanglement can be created in the $\kappa$-deformed spacetime while
cannot in the Minkowski spacetime with the assistance of the centripetal
acceleration (see Fig. 8 (d)).

(a) $|E\rangle$, $\omega L=1/2$ (b) $|E\rangle$, $\omega L=2$ (c) $|E\rangle$,
$a/\omega=1$ (d) $|E\rangle$, $a/\omega=1.35$

Figure 8:  Comparison between the maximum of concurrence during evolution for
circularly accelerated atoms initially prepared in $|E\rangle$ via the
centripetal acceleration $a/\omega$ (a, b) and the interatomic distance
$\omega L$ (c, d).  Figure 9:  Entanglement profile for two-atom system
initially prepared in $|E\rangle$. Region A: two atoms in $\kappa$-deformed
spacetime with a LDP cannot get entangled while two atoms in Minkowski
spacetime can. Region B: two atoms in both of these universes can get
entangled. Region C: two atoms in $\kappa$-deformed spacetime with a LDP can
get entangled while two atoms in Minkowski spacetime cannot. Region D: two
atoms in both of these two universes can not get entangled. Here, we fixed
$\kappa/\omega=1000$.

From the above analysis, we can find that, in the presence of the centripetal
acceleration, the entanglement generation of two-atom system behaves quite
differently in $\kappa$-deformed spacetime and Minkowski spacetime. Therefore,
an interesting issue arises: in which parameter regions we can discriminate
these two universes concerned. In Fig. 9, we show detailedly the parameter
regions of the centripetal acceleration and interatomic separation where the
entanglement can/cannot be generated in $\kappa$-deformed spacetime and
Minkowski spacetime. It is found that there exists different regions which
indicate different properties of entanglement in these two universes. We can
see from this diagram that the two atoms could get entangled only in some
special regime of the centripetal acceleration and interatomic separation.
There exists upper bounds of centripetal acceleration and interatomic
separation larger than which entanglement cannot be generated. Another fact
shown in Fig. 9 is that the possible region of entanglement generation for two
atoms in $\kappa$-deformed spacetime with a LDP does not completely overlap
with that for two atoms in Minkowski spacetime. Thus, using the difference
properties of entanglement generation as a criterion, one can in principle
distinguish these two universes.

####  3.3.2 Two circularly accelerated atoms initially prepared in entangled
states $|A\rangle$ and $|S\rangle$

We study the entanglement degradation when the two-atom system is initially
prepared in two kinds of maximally entangled states, $|A\rangle$ and
$|S\rangle$. In Fig. 10, we show the concurrence as a function of the
evolution time by fixing the centripetal acceleration and interatomic
separation in $\kappa$-deformed spacetime and in Minkowski spacetime. For the
initial antisymmetry entangled state case, we can see from Figs. 10 (a) and
(b) that with a fixed interatomic separation the entanglement decays rapidly
with the evolution time for the two circularly accelerated atoms in these two
spacetimes. The larger the centripetal acceleration is, the faster the
entanglement decays, and so does the variation of entanglement via the
interatomic separation with a fixed centripetal acceleration. For the initial
symmetry entangled state case in Figs. 10 (c) and (d), we can also find that
with a fixed interatomic separation the two-atom entanglement decays rapidly
with the evolution time in these two spacetimes. The larger the centripetal
acceleration is, the faster the entanglement decays. However, the variation of
entanglement via the interatomic separation with a fixed centripetal
acceleration is just the opposite. This is quite different from the initial
antisymmetry entangled state case. Furthermore, for both of the initial
entangled state cases, the entanglement for two circularly accelerated atoms
in Minkowski spacetime decays more quickly than that in $\kappa$-deformed
spacetime. In this sense, with the help of the centripetal acceleration one
can also exploit the entanglement behaviors of two atoms to discriminate the
$\kappa$-deformed spacetime and the Minkowski spacetime in principle.

(a) $|A\rangle$, $\omega L=1$ (b) $|A\rangle$, $a/\omega=2$ (c) $|S\rangle$,
$\omega L=1$ (d) $|S\rangle$, $a/\omega=2$

Figure 10:  Time evolution of concurrence for two circularly accelerated
atoms, varying the centripetal acceleration $a/\omega$ (left) and the
interatomic distance $\omega L$ (right), initially prepared in $|A\rangle$ (a,
b)and $|S\rangle$ (c, d).

##  4 Entanglement dynamics for two atoms with the environment-induced
interatomic interaction

In this section we consider the two atoms initially prepared in a separable
state $|10\rangle$ and the superposition state of $|A\rangle$ and $|S\rangle$,
i.e., $|\psi\rangle=\sqrt{p}|A\rangle+\sqrt{1-p}|S\rangle$ $(0<p<1,p\neq
1/2)$, to address how the environment-induced interatomic interaction affects
the entanglement dynamics. More precisely, we try to find out whether the
environment-induced interatomic interaction can help us to distinguish the
$\kappa$-deformed spacetime with a LDP and the Minkowski spacetime through the
atomic entanglement dynamics.

To address the above issue, we here study the evolution process of
entanglement for two atoms with $\rho_{AS}(0)=\rho_{SA}(0)\neq 0$. According
to Eq. (2.1), it is easy to see that only the density elements, $\rho_{AS}$
and $\rho_{SA}$, are affected by the environment-induced interatomic
interaction in the coupled basis. According to the calculation aforementioned,
the time evolution of the density matrix elements $\rho_{AS}(\tau)$ and
$\rho_{SA}(\tau)$ can be calculated as

$\displaystyle\rho_{AS}(\tau)=\rho_{AS}(0)e^{-4(A_{1}+iD)\tau},\;\;\;\;\;\;\rho_{SA}(\tau)=\rho_{SA}(0)e^{-4(A_{1}-iD)\tau}.$
(51)

Inserting Eq. (51) into the definition of concurrence (20), we get

$\displaystyle C[\rho(\tau)]=\textrm{max}\\{0,K_{1}(\tau)\\},$ (52)

where

$\displaystyle K_{1}(\tau)$ $\displaystyle=$
$\displaystyle\sqrt{[\rho_{AA}(\tau)-\rho_{SS}(\tau)]^{2}+[\rho_{AS}(0)e^{-4(A_{1}+iD)\tau}-\rho_{SA}(0)e^{-4(A_{1}-iD)\tau}]^{2}}$
(53) $\displaystyle-\sqrt{\rho_{GG}(\tau)\rho_{EE}(\tau)}.$

We can see that there exists an extra term
$[\rho_{AS}(0)e^{-4(A_{1}+iD)\tau}-\rho_{SA}(0)e^{-4(A_{1}-iD)\tau}]^{2}$ in
Eq. (53) due to the effects of the environment-induced interatomic
interaction. It is worth emphasizing that the presence of this extra term may
result in a number of intriguing physical properties, shown detailedly in the
following.

###  4.1 Entanglement dynamics of two static atoms

To obtain the term of the environment-induced coupling between the two static
atoms, we plug Eqs. (3.1)-(3.1) into Eq. (12), and the Hilbert transforms of
the correlation functions for two static atoms in $\kappa$-deformed spacetime
read

$\displaystyle\mathcal{K}^{12}(\omega)=\frac{P}{\pi
i}\int^{\infty}_{-\infty}d\omega^{\prime}\frac{1}{\omega^{\prime}-\omega}\frac{\omega^{\prime}}{2\pi}\bigg{[}\frac{\sin\omega^{\prime}L}{\omega^{\prime}L}-\frac{\omega^{\prime
2}\cos\omega^{\prime}L}{24\kappa^{2}}\bigg{]},$
$\displaystyle\mathcal{K}^{12}(-\omega)=\frac{P}{\pi
i}\int^{\infty}_{-\infty}d\omega^{\prime}\frac{1}{\omega^{\prime}+\omega}\frac{\omega^{\prime}}{2\pi}\bigg{[}\frac{\sin\omega^{\prime}L}{\omega^{\prime}L}-\frac{\omega^{\prime
2}\cos\omega^{\prime}L}{24\kappa^{2}}\bigg{]}.$ (54)

Then, with the help of Eqs. (15)-(16), we can obtain

$\displaystyle
D=\Gamma_{0}\frac{1}{2\pi}P\int^{\infty}_{0}dx\bigg{[}\frac{x}{x-1}+\frac{x}{x+1}\bigg{]}\bigg{[}\frac{\sin
x\omega L}{x\omega L}-\frac{x^{2}\cos x\omega
L}{24(\frac{\kappa}{\omega})^{2}}\bigg{]}.$ (55)

Similarly, for the case of two static atoms in Minkowski spacetime, we have

$\displaystyle
D=\Gamma_{0}\frac{1}{2\pi}P\int^{\infty}_{0}dx\bigg{[}\frac{x}{x-1}+\frac{x}{x+1}\bigg{]}\frac{\sin
x\omega L}{x\omega L}.$ (56)

Note that for the case $\kappa\rightarrow\infty$, the result given by Eq. (55)
recovers to that in Eq. (56) for the Minkowski spacetime case.

####  4.1.1 Two static atoms initially prepared in a separable state
$|10\rangle$

We first study how the environment-induced interatomic interaction affects the
entanglement dynamics of two atoms initially prepared in state $|10\rangle$.
In this case, the extra term in Eq. (53) can be written as

$\displaystyle[\rho_{AS}(0)e^{-4(A_{1}+iD)\tau}-\rho_{SA}(0)e^{-4(A_{1}-iD)\tau}]^{2}=\sin^{2}(4D\tau)e^{-8A_{1}\tau}.$
(57)

(a) $|10\rangle$ (b)

Figure 11:  (a) Time evolution of concurrence for two static atoms initially
prepared in $|10\rangle$. (b) The factor $\sin^{2}(4D\tau)e^{-8A_{1}\tau}$ is
a function of time.

In Fig. 11, we plot the evolution process of concurrence for two static atoms
initially prepared in state $|10\rangle$ in $\kappa$-deformed spacetime with a
LDP and Minkowski spacetime, with fixed $\kappa/\omega=1000$. Seen from Fig.
11 (a), the environment-induced interatomic interaction has a significant
impact on the concurrence during initial period in $\kappa$-deformed
spacetime, but after long time the asymptotic concurrence for the case of
$\kappa$-deformed spacetime almost coincide with that of Minkowski spacetime
case. This interesting behaviors are caused by the parameter term
$\sin^{2}(4D\tau)e^{-8A_{1}\tau}$ shown in Eq. (53), which is dominated by the
trigonometric term $\sin^{2}(4D\tau)$ in a short initial time, and it is
determined by the exponential term $e^{-8A_{1}\tau}$ after a long enough time
[see Fig. 11 (b)]. Therefore, as a result of the environment-induced
interatomic interaction, we can find for the $\kappa$-deformed spacetime case
the entanglement generated evolves periodically in early time, and finally
decays to zero asymptotically. The interesting phenomenon is quite different
from that of the Minkowski spacetime case. In this sense, the environment-
induced interatomic interaction between two static atoms would help us to
distinguish these two universes in principle.

####  4.1.2 Two static atoms initially prepared in entangled state
$|\psi\rangle=\sqrt{p}|A\rangle+\sqrt{1-p}|S\rangle$

We will investigate the entanglement behaviors for two static atoms initially
prepared in the entangled states
$|\psi\rangle=\sqrt{p}|A\rangle+\sqrt{1-p}|S\rangle$ $(0<p<1,p\neq 1/2)$ under
the effects of the environment-induced interatomic interaction. We note here
that different $p$ denotes the different weights of the symmetric entangled
state and the antisymmetric entangled state to construct the initial entangled
state. In this case, we can calculate the extra term in Eq. (53) as

$\displaystyle[\rho_{AS}(0)e^{-4(A_{1}+iD)\tau}-\rho_{SA}(0)e^{-4(A_{1}-iD)\tau}]^{2}=4p(1-p)\sin^{2}(4D\tau)e^{-8A_{1}\tau}.$
(58)

(a) $\frac{1}{2}|A\rangle+\frac{\sqrt{3}}{2}|S\rangle$ (b)
$\frac{\sqrt{3}}{2}|A\rangle+\frac{1}{2}|S\rangle$ (c)

Figure 12:  Time evolution of concurrence for static atoms initially prepared
in $\frac{1}{2}|A\rangle+\frac{\sqrt{3}}{2}|S\rangle$ (a) and
$\frac{\sqrt{3}}{2}|A\rangle+\frac{1}{2}|S\rangle$ (b). (c) The factor
$\frac{3}{4}\sin^{2}(4D\tau)e^{-8A_{1}\tau}$ is a function of time.

For $p=1/4$ [see Fig. 12 (a)], meaning that the symmetric entangled state
contributes to the initial entangled state mainly, we find that the
concurrence for the Minkowski spacetime case behaves quite differently
compared with the initial entangled state $|A\rangle$ and $|S\rangle$ cases
(discussed above). Although this entanglement decays with the increase of
time, it is non-monotonous as a result of the environment-induced interatomic
interaction. Moreover, in early period this concurrence shows an oscillatory
behavior in $\kappa$-deformed spacetime with a LDP compared with the case in
Minkowski spacetime under the effects of environment-induced interatomic
interaction. However, in the long enough time limit, the entanglement
behaviors for atoms in $\kappa$-deformed spacetime almost coincide with that
for the Minkowski spacetime case, which implies that all the laws of physics
in $\kappa$-deformed spacetime with a LDP recover to that in flat spacetime.
For $p=3/4$ [see Fig. 12 (b)], it means that the antisymmetric entangled state
contributes to the initial entangled state mainly. We can also find that with
the environment-induced interatomic interaction, the entanglement dynamics for
the $\kappa$-deformed spacetime with a LDP case behave distinguishably
compared with that of the Minkowski spacetime case. We note that these
different behaviors are ultimately due to the trigonometric term
$\sin^{2}(4D\tau)$ in a short initial time, while after a long enough time the
entanglement behaviors are dominated by the exponential term $e^{-8A_{1}\tau}$
[see Fig. 12 (c)].

###  4.2 Entanglement dynamics of two uniformly moving atoms

Now we reconsider the above case while assuming that the two atoms move with a
uniformly speed. When the two-atom system moves uniformly in $\kappa$-deformed
spacetime, inserting Eqs. (3.2)-(3.2) into Eq. (12), we can calculate the
corresponding Hilbert transform of the correlation function as

$\displaystyle\mathcal{K}^{12}(\omega)=\frac{P}{\pi
i}\int^{\infty}_{-\infty}d\omega^{\prime}\frac{1}{\omega^{\prime}-\omega}\frac{\omega^{\prime}}{2\pi}\bigg{[}\frac{\sin\omega^{\prime}L}{\omega^{\prime}L}+\frac{f(\omega^{\prime},L,\upsilon)}{24\omega^{\prime}\kappa^{2}L^{3}}\bigg{]},$
$\displaystyle\mathcal{K}^{12}(-\omega)=\frac{P}{\pi
i}\int^{\infty}_{-\infty}d\omega^{\prime}\frac{1}{\omega^{\prime}+\omega}\frac{\omega^{\prime}}{2\pi}\bigg{[}\frac{\sin\omega^{\prime}L}{\omega^{\prime}L}+\frac{f(\omega^{\prime},L,\upsilon)}{24\omega^{\prime}\kappa^{2}L^{3}}\bigg{]}.$
(59)

Using Eq. (4.2), one has

$\displaystyle
D=\Gamma_{0}\frac{1}{2\pi}P\int^{\infty}_{0}dx\bigg{[}\frac{x}{x-1}+\frac{x}{x+1}\bigg{]}\bigg{[}\frac{\sin
x\omega L}{x\omega L}+\frac{f(x,\omega
L,\upsilon)}{24x(\frac{\kappa}{\omega})^{2}(\omega L)^{3}}\bigg{]}.$ (60)

Similarly, for two uniformly moving atoms in Minkowski spacetime, we have

$\displaystyle
D=\Gamma_{0}\frac{1}{2\pi}P\int^{\infty}_{0}dx\bigg{[}\frac{x}{x-1}+\frac{x}{x+1}\bigg{]}\frac{\sin
x\omega L}{x\omega L},$ (61)

which is the same as the case of two static atoms in Minkowski spacetime, and
completely unaffected by the velocity.

####  4.2.1 Two uniformly moving atoms initially prepared in a separable state
$|10\rangle$

(a) $|10\rangle$, $v=0.01$ (b) $v=0.01$ (c) $|10\rangle$, $v=0.5$ (d) $v=0.5$

Figure 13:  Time evolution of concurrence initially prepared in $|10\rangle$
with velocity $v=0.01$ (a) and $v=0.5$ (c). The factor
$\sin^{2}(4D\tau)e^{-8A_{1}\tau}$ is a function of time with velocity $v=0.01$
(b) and $v=0.5$ (d).

To analyze how the environment-induced interatomic interaction for two
uniformly moving atoms affects the entanglement generation in these two
different spacetime, we show the dynamics of concurrence with different
velocity in Fig. 13, for fixed $\kappa/\omega=1000$, and $\omega L=1$. When
the environment-induced interatomic interaction is introduced, there is an
oscillatory manner in the time evolution of concurrence during initial period
in $\kappa$-deformed spacetime, and this oscillation frequency increases as
the velocity of atoms increases. Besides, the oscillation is damped during
evolution, so the asymptotic concurrence will be consistent to the
entanglement behavior of Minkowski spacetime case. We note that this is
because the oscillatory behavior is dominated by the trigonometric term
$\sin^{2}(4D\tau)$ during initial stage, while it determined by the
exponential term $e^{-8A_{1}\tau}$ and decays to the case of Minkowski
spacetime after a long time. Hence, when the environment-induced interatomic
interaction is taken into account, the difference of entanglement dynamics
between the $\kappa$-deformed spacetime with a LDP and Minkowski spacetime is
more pronounced during initial period. Therefore, the environment-induced
interatomic interaction between two uniformly moving atoms can assist us in
distinguishing these two universes.

####  4.2.2 Two uniformly moving atoms initially prepared in entangled state
$|\psi\rangle=\sqrt{p}|A\rangle+\sqrt{1-p}|S\rangle$

(a) $\frac{1}{2}|A\rangle+\frac{\sqrt{3}}{2}|S\rangle$, $v=0.01$ (b)
$\frac{\sqrt{3}}{2}|A\rangle+\frac{1}{2}|S\rangle$, $v=0.01$ (c)
$\frac{1}{2}|A\rangle+\frac{\sqrt{3}}{2}|S\rangle$, $v=0.5$ (d)
$\frac{\sqrt{3}}{2}|A\rangle+\frac{1}{2}|S\rangle$, $v=0.5$ (e) $v=0.01$ (f)
$v=0.5$

Figure 14:  Time evolutin of concurrence for two uniformly moving atoms
initially prepared in $\frac{1}{2}|A\rangle+\frac{\sqrt{3}}{2}|S\rangle$ (a)
and $\frac{\sqrt{3}}{2}|A\rangle+\frac{1}{2}|S\rangle$ (b).

We consider the scenario where the initial state of two uniformly moving atoms
is prepared in the superposition state
$|\psi\rangle=\sqrt{p}|A\rangle+\sqrt{1-p}|S\rangle$ $(0<p<1,p\neq 1/2)$,
which may induce the environment-induced interatomic interaction. In Figs. 14
(a) and (c), we take $p=1/4$ and plot the relevant dynamics of entanglement
with different fixed velocity of atoms. One can see that the concurrence
decays with the increase of evolution time, while it decays differently for
these two spacetimes. Also, the oscillatory decay still dominates the
entanglement behaviors of the $\kappa$-deformed spacetime case at the early
stage. Meanwhile, the oscillatory frequency is velocity dependent, which
increases with the increase of the atomic velocity. At the late stage, the
entanglement in $\kappa$-deformed spacetime case share the same behaviors with
that in Minkowski spacetime case. Furthermore, we prepare $p=3/4$ initial
state case and plot its entanglement dynamics in Figs. 14 (b) and (d). It is
easy to see at the early evolution stage that the oscillatory manners of
entanglement are also dependent on the atomic velocity, but they are different
from the $p=1/4$ case. Similarly, at the late evolution stage one cannot
distinguish these two spacetimes through the atomic entanglement behaviors.
Remarkably, as shown in Figs. 14 (e) and (f), the oscillatory behaviors of
entanglement appears as a consequence of the environment-induced interatomic
interaction, which is dominated by the trigonometric term $\sin^{2}(4D\tau)$
during initial stage, while it determined by the exponential term
$e^{-8A_{1}\tau}$ for a long time.

Compared with Fig. 12, we can see from Fig. 14 that under the influence of
velocity, the behaviors of entanglement in $\kappa$-deformed spacetime are
different from that in Minkowski spacetime: the higher the velocity, the
greater the difference between entanglement behaviors in these two universes.
This tells us that even when the spacetime deformation parameter $\kappa$ is
relatively large, we can more easy to distinguish these two spacetimes with
the help of environment-induced interatomic interaction between two uniformly
moving atoms in principle.

###  4.3 Entanglement dynamics of two circularly accelerated atoms

In the following, we investigate how the entanglement dynamics between two
circularly accelerated atoms are dependent on the environment-induced
interatomic interaction in $\kappa$-deformed spacetime and Minkowski
spacetime, respectively. With the trajectories of the two circularly
accelerated atoms (3.3) and the Fourier transforms of correlation function, we
can straightforwardly derive an analytic expression for the corresponding
Hilbert transforms in Eq. (16). It is needed to note that the expression is
too long to exhibit here. Similarly, when the two circularly accelerated atoms
interact with a bath of fluctuating massless scalar field in Minkowski vacuum,
we can also calculate the corresponding Hilbert transforms for the uniform
circular motion case straightly.

####  4.3.1 Two circularly accelerated atoms initially prepared in a separable
state $|10\rangle$

We assume that the two atoms are initially prepared at the separable state
$|10\rangle$. In Fig. 15 we take $\kappa/\omega=1000$ and $\omega L=1$ and
plot the concurrence dynamics in $\kappa$-deformed spacetime and Minkowski
spacetime. This chosen initial sate may induce the environment-induced
interatomic interaction, which is embodied by the extra term
$\sin^{2}(4D\tau)e^{-8A_{1}\tau}$ and has significant contributes to the two-
atom entanglement dynamics.

(a) $|10\rangle$, $a/\omega=1$ (b) $a/\omega=1$ (c) $|10\rangle$, $a/\omega=2$
(d) $a/\omega=2$

Figure 15:  Time evolution of concurrence for two circularly accelerated atoms
initially prepared in $|10\rangle$ with centripetal acceleration $a/\omega=1$
(a) and $a/\omega=2$ (c). The factor $\sin^{2}(4D\tau)e^{-8A_{1}\tau}$ is a
function of time with centripetal acceleration $a/\omega=1$ (b) and
$a/\omega=2$ (d).

We first note that entanglement will be generated as a result of the vacuum
fluctuation of quantum field and the motion of atoms. We can also find that,
as above, there is an oscillatory manner for the entanglement dynamics during
the initial period, which is from the environment-induced interatomic
interaction dominated by the trigonometric term $\sin^{2}(4D\tau)$. The
oscillation frequency decreases as the growth of the centripetal acceleration.
In addition, this oscillation form decays to the case of Minkowski spacetime
after a long time, which is determined by the exponential term
$e^{-8A_{1}\tau}$. Therefore, the difference of entanglement dynamics between
the $\kappa$-deformed spacetime case with a LDP and Minkowski spacetime case
is more obvious during initial period when the environment-induced interatomic
interaction is considered. This tells us that the environment-induced
interatomic interaction between two circularly accelerated atoms is beneficial
to discriminate these two universes through entanglement generation dynamics.

####  4.3.2 Two circularly accelerated atoms initially prepared in entangled
state $|\psi\rangle=\sqrt{p}|A\rangle+\sqrt{1-p}|S\rangle$

Here we study the effects of the environment-induced interatomic interaction
on the entanglement dynamics for two circularly accelerated atoms initially
prepared in $|\psi\rangle=\sqrt{p}|A\rangle+\sqrt{1-p}|S\rangle$ $(0<p<1,p\neq
1/2)$. In Fig. 16, it is shown that the enhancement behaviors depend on the
atomic initial state prepared. When $p=1/4$ is taken, that is to say, the
symmetry entangled state contributes to the initial state mainly, and we find
under the effect of the environment-induced interatomic interaction, from
Figs. 16 (a) and (c), that the two-atom entanglement exhibits an intriguing
phenomenon: its decay and revival are quite different from the static atoms
case in Fig. 12 (a) and the uniformly moving atoms cases in Figs. 14 (a) and
(c). In particular, when the centripetal acceleration is large there could be
no entanglement revival and entanglement will suffer “sudden death”. Besides,
even at the late evolution stage, unlike the above cases, the entanglement
dynamics in $\kappa$-deformed spacetime does not consist with that in
Minkowski spacetime. In Figs. 16 (b) and (d), the entanglement dynamics of the
$p=3/4$ initial state case is shown. We can see that the initial entanglement
can be enhanced in the initial phase, while finally decays to zero
asymptotically. However, the entanglement dies off in an oscillatory manner in
$\kappa$-deformed spacetime and will never consist with the Minkowski
spacetime case at anytime. Remarkably, as the centripetal acceleration
increasing, the difference between the $\kappa$-deformed spacetime and
Minkowski spacetime becomes larger. Moreover, entanglement dynamics in both
the $\kappa$-deformed spacetime and the Minkowski spacetime could suffer
entanglement “sudden death”, which means that the entanglement decays to zero
at finite time. A quite interesting phenomenon for the entanglement in the
large acceleration situation is that at certain time interval the entanglement
for the $\kappa$-deformed spacetime exists, while that for the Minkowski
spacetime vanishes. This distinct character of entanglement dynamics may
assist us to distinguish these two universes.

(a) $\frac{1}{2}|A\rangle+\frac{\sqrt{3}}{2}|S\rangle$, $a/\omega=1$ (b)
$\frac{\sqrt{3}}{2}|A\rangle+\frac{1}{2}|S\rangle$, $a/\omega=1$ (c)
$\frac{1}{2}|A\rangle+\frac{\sqrt{3}}{2}|S\rangle$, $a/\omega=2$ (d)
$\frac{\sqrt{3}}{2}|A\rangle+\frac{1}{2}|S\rangle$, $a/\omega=2$ (e)
$a/\omega=1$ (f) $a/\omega=2$

Figure 16:  Time evolution of concurrence for two circularly accelerated
atoms, varying the centripetal acceleration $a/\omega$ (left) and the
interatomic distance $\omega L$ (right), initially prepared in
$\frac{1}{2}|A\rangle+\frac{\sqrt{3}}{2}|S\rangle$ (a, b) and
$\frac{\sqrt{3}}{2}|A\rangle+\frac{1}{2}|S\rangle$ (c, d).

We also note that the environment-induced interatomic interaction is embodied
in the extra term $\frac{3}{4}\sin^{2}(4D\tau)e^{-8A_{1}\tau}$. In Figs. 16
(e) and (f) we show how this term performs under the effects of atomic
acceleration in $\kappa$-deformed spacetime and Minkowski spacetime. Because
of the acceleration, the extra term seems to be always determined by the term
$\sin^{2}(4D\tau)$ for the $\kappa$-deformed spacetime case and shows
differently compared with the Minkowski spacetime case. This character is
quite different from the situations of static and uniformly moving atoms that
have been discussed above. Besides, when the acceleration increases, for the
$\kappa$-deformed spacetime case the oscillation frequency decreases, so does
the oscillation amplitude.

Therefore, here we conclude to that even the spacetime deformation parameter
$\kappa$ is large, by checking the atoms entanglement dynamics one may
distinguish these two universes with the help of the environment-induced
interatomic interaction between two circularly accelerated atoms in principle.

##  5 Conclusions

In this paper, we have investigated the dynamical behaviors of entanglement
for a pair of static, uniformly moving and circularly accelerated atoms with
different initial states coupled with the massless scalar field in
$\kappa$-deformed spacetime and Minkowski spacetime. Through numerical
evaluation, two different scenarios, i.e., with and without the environment-
induced interatomic interaction, are considered. We have shown that the
relativistic motion as well as the environment-induced interatomic interaction
may have a significant effect on the entanglement dynamics of a two-atom
system.

On the one hand, when the two static atoms are initially prepared in the
excited state, antisymmetric entangled state and symmetric entangled state
respectively, it is shown that, without the environment-induced interatomic
interaction, the differences of entanglement dynamics between in the
$\kappa$-deformed spacetime and in the Minkowski spacetime are not obvious
when the deformation parameter is large. However, when the inertial atoms
moves with a constant velocity, the entanglement dynamics is different between
in these two universe cases when the velocity is large. Furthermore, for two
circularly accelerated atoms with a nonvanishing separation, the entanglement
evolves quite differently with respective to the acceleration and the
interatomic distance or other parameters. More importantly, for certain
conditions, the circularly accelerated atoms initially prepared in the excited
state in $\kappa$-deformed spacetime (in Minkowski spacetime) can get
entangled, while they would not become entangled in the corresponding
Minkowski case (in the corresponding $\kappa$-deformed case). Thus, the
relativistic motion of the atoms significantly influences on the difference of
entanglement dynamics between these two universes.

On the other hand, when the atoms are initially prepared in a separable state
$|10\rangle$ and a superposition entangled states, we have demonstrated how
the environment-induced interatomic interaction affects the entanglement
dynamics of atoms coupled with the massless scalar fields. The numerical
results have told us that when the environment-induced interatomic interaction
is considered, it is more beneficial to distinguish the $\kappa$-deformed
spacetime from the Minkowski spacetime.

###### Acknowledgments.

Xiaobao Liu thanks Shifeng Huang and Jiaozhen She for advice and discussions.
This work was supported by the National Natural Science Foundation of China
under Grant Nos. 12065016 and 11905218; X. Liu thanks for the talent
recruitment program of Liupanshui normal university of China under Grant No.
LPSSYKYJJ201906 and the Discipline-Team of of Liupanshui Normal University of
China under Grant No. LPSSY2023XKTD11.

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