# Time evolution of quantum correlations in presence of state dependent bath

Mehboob Rashid1 Muzaffar Qadir Lone2 Prince A Ganai1 1Department of Physics,
National Institute of Technology, Srinagar-190006 India.
2 Department of Physics, University of Kashmir, Srinagar-190006 India

###### Abstract

The emerging quantum technologies heavily rely on the understanding of
dynamics in open quantum systems. In the Born approximation, the initial
system-bath correlations are often neglected which can be violated in the
strong coupling regimes and quantum state preparation. In order to understand
the influence of initial system-bath correlations, we study the extent to
which these initial correlations and the distance of separation between qubits
influence the dynamics of quantum entanglement and coherence. It is shown that
at low temperatures, the initial correlations have no role to play while at
high temperatures, these correlations strongly influence the dynamics.
Furthermore, we have shown that distance of separation between the qubits in
presence of collective bath helps to maintain entanglement and coherence at
long times.

##  I Introduction

Quantum correlations (entanglement) 1  form a key concept towards the
foundational understanding of quantum mechanics, and has been realized as a
precious resource for various tasks that are impossible in classical domian
like quantum teleportation, cryptograpgy, and various other processings2 ; 3 ;
4 ; 5 .The deeper insights into the dynamics of quantum correlations is of
great importance, since the real quantum system always interacts with its bath
that leads to decoherence of quantum superpositions and entanglement
degradation 6 ; 7 ; 8 ; 9 ; 10 . This system-bath dynamics is categorized
either as Markovian or non-Markovian 11 ; 12 ; 13 . In a Markov process, the
system loses information to environment irreversibly. However, many quantum
systems posses a pronounced non-Markovian behavior in which there is flow of
information back to system, signifying the presence of quantum memory effects
14 ; 15 ; 16 ; 17 ; 18 ; 19 ; 20 ; 21 ; 22 . The physical realization and
control of dynamical processes in open quantum systems plays a decisive role,
for example, in recent proposals for the generation of entangled states 23 ;
24 , for schemes of dissipative quantum computation 25 ; 26 , for the design
of quantum memories 27  and for the enhancement of the efficiency in quantum
metrology 28 . Moreover, in the regime where non-Markovian effects are
important, the presence of system-bath correlations invalidates the initial
state in which the system and the bath are independent. Especially in the
experimentally relevant case where the qubit system is excited out of
equilibrium and the subsequent dynamics is probed, a proper treatment of the
initial state is crucial.

A complete understanding of the dynamics of entanglement 8 ; 9 ; 10  relies on
available measure that can reflect the time variation of the system of
interest. Here, we utilize Concurrence 29 ; 30  as the entanglement measure to
understand the underlying dynamics in presence of initial system-bath
correlations. In addition to the quantum entanglement, coherence has been
proposed as an alternative resouce for quantum information processings 31 ; 32
; 33 ; 34 ; 35 .It has attracted much attention over the past decade both in
theory and experiment 36 ; 37 ; 38 ; 39 . It has been shown that the coherence
exists in photosynthetic complexes 40 ; 41 ; 42 , therefore can play an
important role in explaining the high efficiency of the these complexes, which
in turn have technological benifits. There exist several quantifiers for
coherence 32  in a given quantum system like $l_{1}$-norm, relative entropy
etc. In this work, we utilize $l_{1}-$norm which is in simple terms represents
the sum of off-diagonal terms of a density matrix. Due to its important role
in quantum mechanics43 ; 44 , quantum information 45 ; 46 , and quantum
biology 47 ; 48 ; 49 ; 50 , the behavior of coherence during the evolution of
a system is necessary to be investigated.

There has been tremendous amount of work towards understanding the non-
Markovian behaviour in single and many qubits coupled to a bosonic bath51 ; 52
; 53 ; 54 ; 55 ; 56 ; 57 . In most of these cases, Born approximation is used
which means that the joint initial state of the system and bath are assumed to
be uncorrelated. However this assumption is often violated when there exist
strong interaction between system and the bath 21 ; 58 ; 59 ; 60 ; 61 ; 62 ;
63 ; 64 ; 65 ; 66 . In particular, quantum state preparation can lead to
strong system-bath correlations thus effecting the subsequent dynamics. In
this regard, various works have critcally examined the initial system-bath
correlations in spin-Boson model67 , superconducting qubits 68 , quantum dots
69  etc. For example, Mozorov et. al. 61 ; 67 , has considered an exactly
solvable dephasing model of a single qubit interacting a bosonic in presence
of initial system-bath correlations, while in 70  has considered Jaynes-
Cunnings model and 71  has studied in detail dynamics in presence of spin
bath. These models however, consider either the single qubit or many coupled
with individual baths. Furthermore, the distance of separation between the
qubits in presence of initial correlations is not taken into account in these
works. Our main objective in this work is to study the effect of distance
dependent interaction between qubits and the collective bath in presence of
initial system-bath correlations. Such kind of settings can be obtained using
cold atom impurities immersed in a Bose-Einestien condensate (BEC) 72 . The
positioning of immersing of cold atoms in the BEC would yeild a distance
dependent interaction with non-trivial bath spectral density. Thus the present
work would be mainly important for quantum information processings using cold
atoms.

This paper is organised in the following way. In section II, we introduce the
model of qubits interacting with a collective bath in presence of initial
system-bath correlations and calculate the time evolution of two qubits
density matrix. In section III, we discuss the dynamics of entanglement given
by concurrence and coherence. Finally we conclude in section IV.

##  II Model Calculations

In this section, we introduce our model of qubits interacting with a bosonic
bath. In spin representation, the total Hamiltonian of the model is written as
( $\hbar=1$).

$\displaystyle H$ $\displaystyle=$ $\displaystyle H_{S}+H_{B}+H_{int}$
$\displaystyle=$
$\displaystyle\frac{\omega_{0}}{2}\sum_{i}\sigma_{i}^{z}+\sum_{k}\omega_{k}b_{k}^{\dagger}b_{k}+\sum_{ik}\sigma_{i}^{z}(g_{k}e^{-i\vec{k}.\vec{r_{i}}}b_{k}+h.c.)$

where $\omega_{0}$ is the energy splitting of the qubit, $\sigma_{z}$ is the
$z$-Pauli matrix, $\vec{r_{i}}$ is the position vector of $ith$-qubit, and
h.c. means Hermitian conjugate. We assume distance of separation between two
qubits to $\vec{L}=\vec{r}_{1}-\vec{r}_{2}$. $g_{k}$ is the system-bath
coupling. This kind of model can be realized in ultracold setting by immersing
an ultracold gas trapped in an optical lattice in Bose Einestien condensate 72
; 73 . The low lying exciations of BEC i.e. Bogoliubov phonons will act as the
bosonic bath. We assume linear dispersion for bath modes (phonon or photon
type). In BEC setting it will correspond to phonon like excitations in long
wavelength domain instead of particle like excitations at large momentum.

In this work, we will assume a particular type of initial state of the bath
and system as a case study to dynamics of quantum correlations. Before $t=0$,
we initially start in the state for the combined system and bath that are
thermal states of the whole system

$\displaystyle\rho=\frac{e^{-\beta H}}{Z}$ (2)

where $Z$ is the partion function and $\beta=\frac{1}{T}$. This should be
compared to usual situtaion where the thermal state is considered only with
respect to the bath states i.e. $\rho_{B}=\frac{e^{-\beta H_{B}}}{Z_{B}}$.
This type of state could arise in a situation where the measurement apparatus
is prepared in the vicinity of the system prior to the state preparation.
Since the system is rather small, the time taken to reach thermal equilibrium
can be rather short, and the state (2) is attained quickly before the state
preparation is performed.

Now we would like to prepare the state of the system in a manner such that a
projective measurement is made on the system. For the system the projection
operators are

$\displaystyle
P_{S}=\\{|\psi\rangle\langle\psi|,|\psi_{\perp}\rangle\langle\psi_{\perp}|\\}$
(3)

where $|\psi\rangle$ is the initial state of the system and
$|\psi_{\perp}\rangle$ is the state that is orthogonal to this state. Now we
postselect on the state $|\psi\rangle$ such that the system is ensured to have
the state

$\displaystyle\rho_{S}(0)=|\psi\rangle\langle\psi|.$ (4)

and the bath states then take a form

$\displaystyle\rho_{B}^{\psi}=\frac{\bra{\psi}\exp(-\beta
H)\ket{\psi}}{Tr_{B}\bra{\psi}\exp(-\beta H)\ket{\psi}}$ (5)

where there is a dependence on the state of the system because the original
state (2) were thermal states in the space of the system and bath. The initial
state of the whole system is then the product state

$\displaystyle\rho(0)=\rho_{S}(0)\otimes\rho_{B}^{\psi}(0).$ (6)

The primary difference to uncorrelated case is that the bath state according
to this preparation depends in a non-trivial way on the system state. We
assume the initial state of the system to be a general two qubit state
$|\psi\rangle=a|00\rangle+b|01\rangle+c|10\rangle+d|11\rangle$ with
$|a|^{2}+|b|^{2}+|c|^{2}+|d|^{2}=1$. For this state, we have (for detailed
calculations see appendix):

$\displaystyle\rho_{B}^{\psi}(0)=\frac{1}{Z}\big{[}|a|^{2}e^{-\beta\omega_{0}}e^{-\beta
H_{B1}^{+}}+|b|^{2}e^{-\beta H_{B1}^{-}}+|c|^{2}e^{-\beta
H_{B2}^{+}}+|d|^{2}e^{\beta\omega_{0}}e^{-\beta H_{B2}^{-}}\big{]}$ (7)

Next, our main interest is to calculate the reduced density matrix of the
system by tracing out degrees of freedom of the bath:

$\displaystyle\rho_{S}(t)={\rm Tr}_{B}[U(t)\rho(0)U(t)^{\dagger}]$ (8)

where $U(t)=T~{}e^{-i\int_{0}^{t}dt^{\prime}H_{I}(t^{\prime})}$ is the time
evolution operator and $H_{I}(t)$ is the interaction Hamiltonian in interactin
picture. We write
$U(t)=e^{i\phi(t)}e^{\Sigma_{i}\sigma_{i}^{z}\hat{\Lambda}_{i}(t)}$ where
$\phi(t)$ is a function of time only and
$\hat{\Lambda}_{i}(t)=\Sigma_{k}(\alpha_{ik}(t)b_{k}-\alpha_{ik}^{*}(t)b_{k}^{\dagger})$
with $\alpha_{ik}(t)=g_{ik}\frac{(1-e^{-i\omega_{k}t})}{\omega_{k}}$ and
$\phi(t)$ is a function of time only. Therefore, we can write

$\displaystyle\rho_{S}(t)$ $\displaystyle=$ $\displaystyle{\rm
Tr}_{B}[U(t)\rho(0)U(t)^{\dagger}]$ (9) $\displaystyle=$ $\displaystyle{\rm
Tr}_{B}[e^{\Sigma_{i}\sigma_{i}^{z}\hat{\Lambda}_{i}(t)}|\psi\rangle\langle\psi|\otimes\rho_{B}^{\psi}e^{-\Sigma_{i}\sigma_{i}^{z}\hat{\Lambda}_{i}(t)}]$
(10)

after a cumbersome calculations (see appendix ), we get

$\displaystyle\rho_{s}(t)=\begin{pmatrix}|a|^{2}&ab^{*}\varphi(t)&ac^{*}\zeta(t)&ad^{*}\kappa(t)\\\
ba^{*}\varphi^{*}(t)&|b|^{2}&bc^{*}\bar{\kappa}(t)&bd^{*}\bar{\zeta}(t)\\\
ca^{*}\zeta^{*}(t)&cb^{*}\bar{\kappa}^{*}(t)&|c|^{2}&cd^{*}\bar{\varphi}(t)\\\
da^{*}\kappa^{*}(t)&db^{*}\bar{\zeta}^{*}(t)&dc^{*}\bar{\varphi}^{*}(t)&|d|^{2}\end{pmatrix}$
(11)

where the different functions are written explicitly in appendix. In the next
section we examine the entanglement and coherence in various approximations.

##  III Dynamics of Entanglement and Coherence

###  III.1 Concurrence

In this section, we analyze quantum entanglement measured by concurrence in a
two qubit system considered in this work. For a density matrix $\rho$, the
concurrence is defined as

$\displaystyle\mathcal{C}={\rm
max}\\{0,\sqrt{\lambda_{1}}-\sqrt{\lambda_{2}}-\sqrt{\lambda_{3}}-\sqrt{\lambda_{4}}\\},$
(12)

where $\lambda_{i},i=1,2,3,4$ are the eigen values of the matrix
$\rho\tilde{\rho}$ taken in descending order and
$\tilde{\rho}=(\sigma^{y}_{1}\otimes\sigma^{y}_{2})\rho^{\star}(\sigma^{y}_{1}\otimes\sigma^{y}_{2})$
is the time reversal density matrix. $\rho^{\star}$ is the conjugation
obtained in standard basis. For an unentangled state $\mathcal{C}=0$ while for
maximally entangled state $\mathcal{C}=1$. In order to simplify our
calculations, we assume $a=\sqrt{p}$, $d=\sqrt{1-p}$ and $b=0=c$ so that our
initial state is the class of states
$\psi=\sqrt{p}|00\rangle+\sqrt{1-p}|11\rangle$, with $0\leq p\leq 1$. The same
analysis holds for other types of states as well. For this state, time evolved
concurrence can be written as

$\displaystyle\mathcal{C}(t)=2{\rm max}\\{0,C(t)\\},$ (13)

with

$\displaystyle C(t)$ $\displaystyle=$
$\displaystyle\sqrt{\cos^{2}\Phi+\sin^{2}\Phi\tanh^{2}(\beta\omega_{0})}e^{-\gamma(t)}C(0)$
(14)

where

$\displaystyle\Phi(t)$ $\displaystyle=$ $\displaystyle
8\sum_{k}\frac{|g_{k}|^{2}}{\omega_{k}^{2}}\sin\omega_{k}t\Big{[}1+\cos(\vec{k}.\vec{L})\Big{]}$
$\displaystyle\gamma(t)$ $\displaystyle=$ $\displaystyle
4\sum_{k}|g_{k}|^{2}\Big{[}1+\cos(\vec{k}.\vec{L})\Big{]}\frac{1-\cos\omega_{k}t}{\omega_{k}^{2}}\coth\frac{\beta\omega_{k}}{2}$

We rewrite the concurrence $C(t)=e^{-\Gamma(t)}C(0)$, with
$C(0)=2\sqrt{p(1-p)}$ as the initial entanglement of the system. Thus
unentangled states $p=0$ and $p=1$ remain always unetangled. This is due to
the dephasing nature of the interaction. Thus all values of $0<p<1$ show the
same type of behavour. Therefore, without loss any generality we take $p=1/2$.
Also, $\Gamma(t)=\gamma_{0}(t)+\gamma(t)$ with

$\displaystyle\gamma_{0}(t)=-\frac{1}{2}\log[\cos^{2}\Phi+\sin^{2}\Phi\tanh^{2}(\beta\omega_{0})]$
(16)

as the decoherence function due to initial correlations and $\gamma(t)$ is the
standard decoherence function which is present even in absence of initial
system-bath correlations.

###  III.2 Coherence

Quantum coherence directly stems from the superposition principle that enables
it to show quantum interference phenomena. There has been several proposals to
quantify coherence in a given quantum system. However, we take $l_{1}$-norm
quantification of coherence in the present work. It is an intuitive measure
related to off-diagonal terms of the density matrix. It is defined as

$\displaystyle\mathcal{N}=\sum_{i\neq j}||\rho_{ij}-\mathcal{I}_{ij}||$ (17)

where the optimization is to be carried out over all possible incoherent
states $\mathcal{I}$. After the optimization, we get the following expression
for the coherence in the standard basis

$\displaystyle\mathcal{N}=\sum_{i\neq j}|\langle i|\rho|j\rangle|.$ (18)

This simply represents the sum of off-diagonal elements of the density matrix
under consideration and therefore captures the notion of interference in a
quantum state. For the states under consideration we have

$\displaystyle\mathcal{N}=2\sqrt{\cos^{2}\Phi+\sin^{2}\Phi\tanh^{2}(\beta\omega_{0})}e^{-\gamma(t)}=2C(t).$
(19)

This is true for other types of
states($\sqrt{p}|01\rangle+\sqrt{1-p}|10\rangle$) as well. Thus, pairwise
entanglemet given by concurrence is half to that of the coherence given by
$l_{1}$ norm. Thus it is sufficient to analyze the concurrence $C(t)$.
Furthermore, this provides an explicit example where entanglement can be
measured by coherence34 ; 38 .

###  III.3 Decay of Concurrence

Next, we analyze the decoherenc functions $\Phi(t)$ and $\gamma(t)$ using
various approximations. Before, evaluating the sum in these equations, we
realize that $\gamma(t)$ becomes zero for certian modes of the bath. For
$\vec{k}.\vec{L}=(2n+1)\pi,n=0,1,..$, we have $\cos\vec{k}.\vec{L}=-1$, which
makes $\gamma(t)=0$ and $\Phi(t)=0$. It implies there exist certian bath modes
for $\theta=\cos^{-1}\Big{(}\frac{(2n+1)\pi}{kL}\Big{)}$ that do not lead to
the decay of entanglement (or coherence), $\theta$ is the angle between
$\vec{k}$ and $\vec{L}$. In other words, all bath modes do not couple to the
qubits which can lead to decoherence. This can be intuitively understood from
the fact that these bath modes do not resolve the separation of qubits, hence
suppresses the decay of correlations. Next, we define the bath spectral
density as

$\displaystyle J(\omega)=\sum_{k}|g_{1k}+g_{2k}|^{2}\delta(\omega-\omega_{k})$
(20)

where $g_{ik}=g_{k}e^{-i\vec{k}.\vec{r}_{i}}$. This spectral density has a
very complicated form and in general depends on the dimensionality $d$ of the
bath. This structure of the spectral density requires the bath to resolve the
distance between the qubits, hence the factor of $(1+\cos\vec{k}.\vec{L})$.
Next, assume that the form of
$|g(\omega)|^{2}=g_{0}\frac{\omega}{\Omega^{2}}e^{-\omega^{2}/\Omega^{2}}$, 74
where $g_{0}$ is the intrinsinc coupling between system and the bath, $\Omega$
is the cut-off frequency of the bath. Therefore, we write

$\displaystyle\gamma(t)$ $\displaystyle=$ $\displaystyle
16\int\frac{d^{3}k}{(2\pi)^{3}}|g_{k}|^{2}\cos^{2}(\vec{k}.\vec{L})\frac{\sin^{2}\frac{\omega_{k}t}{2}}{\omega_{k}^{2}}\coth\frac{\beta\omega_{k}}{2}$
(21) $\displaystyle=$
$\displaystyle\frac{4g_{0}}{\Omega^{2}\pi^{2}v^{3}}\int_{0}^{\infty}d\omega\omega
e^{-\frac{\omega^{2}}{\Omega^{2}}}\Big{(}1+\frac{\sin\omega s}{\omega
s}\Big{)}\sin^{2}\frac{\omega_{k}t}{2}\coth\frac{\beta\omega}{2}$

and

$\displaystyle\Phi(t)$ $\displaystyle=$ $\displaystyle
8\int\frac{d^{3}k}{(2\pi)^{3}}|g_{k}|^{2}\cos^{2}(\vec{k}.\vec{L})\frac{\sin\omega_{k}t}{\omega_{k}^{2}}$
(22) $\displaystyle=$
$\displaystyle\frac{4g_{0}}{\pi^{2}v^{3}\Omega^{2}}\int_{0}^{\infty}d\omega\omega
e^{-\frac{\omega^{2}}{\Omega^{2}}}\Big{(}1+\frac{\sin\omega s}{\omega
s}\Big{)}\sin\omega t$

where $s=\frac{L}{v}$ is the time scale provided by the interactions mediated
by the bath modes between qubits separated by the distance $L$ and $v$ is the
velocity of bath modes. In a typical experimental setup 75  for cold atoms, we
can vary distance between two qubits from $L=100$nm to $10\mu$m with speed of
sound $v=350ms^{-1}$, which yeilds $s=0.1$ns to $10$ns. Also, we have
different energy scales arising in our model. The highest energy scale is
given by the cutt off frequency $\Omega$ which provides the relaxation time
scale for the bath $\tau_{B}\sim\frac{1}{\Omega}$; the energy scale
$\omega_{0}$ provides a natural time scale for the relaxation of qubits
$\tau_{s}\sim\frac{1}{\omega_{0}}$. Now we can parametrize the above equations
21 and 22 in the following way: $\omega\to\frac{\omega}{\Omega},~{}t\to\Omega
t,s\to\Omega s$ and the temperature is measured with respect to $\Omega$:
$\beta\to\beta\Omega$. For notational convienence, we take $\Omega=1$ without
loss of generality.

Figure 1: Variation of decoherence functions $\gamma(t)$, $\gamma_{0}(t)$ and
depahsing function $\Phi(t)$ for different values of $s$: In (a) and (b) we
have $\gamma(t)$ vs $t$ in units of $\Omega$ for low temperature
$\beta\Omega>>1$ and high temperature $\beta\Omega<<1$ cases repectively. In
(c) we have $\gamma_{0}(t)$ vs $t$ for $\beta\Omega<<1$, (d) $\Phi(t)$ does
not have any temperature dependence and varies non-monotonically with qubit
separation. In the long time limit, in all cases, $\gamma(t)$ saturates at
some particular value, which suggest suprresion of decoherence due to long
wavelength modes of the bath.

Figure 2: (a) Time variation of entanglement $C(t)$ We see that for different
separation lengths between qubits, entanglement saturates over long time
scales with different values for (a) $\beta\Omega>>1$ and (b)
$\beta\Omega<<1$.

In order to understand the time evolution of concurrence, we first observe the
behaviour of $\gamma(t)$ , $\gamma_{0}(t)$ and $\Phi(t)$ for different
temperature regimes and time scales $s$. For this, we plot $\gamma(t)$ in
figure 1(a) for low temperature $\beta\Omega>>1$, while in figure 1(b) we have
$\gamma(t)$ is plotted for high temperature $\beta\Omega<<1$ case. From these
plots, we observe that in the long time limit $\gamma(t)$ saturates at a
certian value. We see that for short distances $L$ or small $s$, $\gamma(t)$
varies appreciably at high and low temperatures in comparison to large $s$
values i.e. large L, Thus $L$ plays an important role to protect system from
decoherence. Furthermore, from the non-monotonic variation of $\gamma(t)$
leads to the identification of three regions of dynamics. First, for very
short times $\gamma(t)\propto t^{2}$ showing strong non-Markovian behaviour
while for the intermediate decay $\gamma(t)\propto t$ and thus Markovian
behviour sets in the dynamics. Thirdly, the long time limit, there is a
saturation region that implies existence of finite value of quantum
correlations. In figure 1(c), we have decay function due to initial
correlations $\gamma_{0}(t)$, plotted for different values of $s$ and
$\beta\Omega<<1$. We see that for small $s$, the $\gamma_{0}(t)$ has
prounounced non-Markovian decay in comparison to large $s$. Since concurrence
has an $e^{-\gamma_{0}(t)}$ factor, which therefore implies initial system-
bath correlations contribute to concurrence $C(t)$ for small $L$, while for
large $L$, thermal fluctuations influence the decay. Also, in fig 1(d), we
plot $\Phi(t)$, for different values of $s$. we see that with the overall
behvaiour of $\Phi(t)$ is same for all values of $s$, in other words, distance
of separation of qubits doesnot influence the dephasing. In the long time
limit, $\Phi(t)$ goes to zero which implies that influnce of initial
correlations on the entanglement remains over long periods of time.

Now, we plot time evolution of concurrence for different values of $s$ and
temperature. At low temperatures $\beta\Omega>>1$, we see from the equations
14, 21 and 22 that initial correlations $\gamma_{0}(t)$ do not contribute to
entanglement dynamics. This can be understood as follows: at low temperatures,
we have joint state of the system and bath $\rho=e^{-\beta
H}/Z\to|GS\rangle\langle GS|$, where $|GS\rangle$ is some ground state of the
Hamiltonian system (S) plus bath (B) $H_{S}+H_{B}$. For the model under
consideration, $|GS\rangle=|11\rangle_{S}\otimes|0\rangle_{B}$. Thus
$\rho_{B}^{\psi}=\frac{\bra{\psi}\exp(-\beta
H)\ket{\psi}}{Tr_{B}\bra{\psi}\exp(-\beta H)\ket{\psi}}\to|0\rangle\langle
0|_{B}$. Therefore, $\rho_{B}^{\psi}$ has no non-trivial dependence on the
system parameters and thus we have an initally uncorrelated state. However, at
high temperatures $\beta\Omega<<1$, we have $\gamma_{0}(t)$ influencing the
concurrence. This can be seen from the plot for $\gamma_{0}(t)$ in figure
1(c). We see from this plot for small qubit separations, the correlations are
least influenced by thermal fluctuations of the bath than the qubits with
larger separation. The overall contribution to concurrence from initial
system-bath correlations and damping factors $\gamma(t)$ is plotted in figures
2 (a) and (b) . In fig 2(a), we have concurrence plotted with respect $t$ at
low temperatur $\beta\Omega>>1$. From this plot, we see that concurence decays
in a non-Markovian way with saturation to a non-zero value of entaglement in
the long time limit. Furthermore, concurrence variation at the high
temperature case ($\beta\Omega<<1$) ( figure 2(b)) is mostly similiar in
behaviour to low temperature case with slightly lower vlaue for long time
entanglement. We see from these plots, that qubit separation plays an
important role in long time entangelemnt. The parameter can used to protect
the entaglement decay in an optimal way. We observe from figures 2(a) and (b)
that initial correlation superimposed with thermal fluctuations enhance the
long time entanglement for large $s$ values in comparison to small $s$ or
shorter qubits distances $L$. This can be intuitively understood in the
following way. Since we are assuming a common bath coupled to two qubits which
are separated by certian distance. Therefore, there will be less energetic
modes to scatter the qubits that would cause decoherence. In other words, at
large $L$, there will be only few modes that couple to the qubits and lead to
low decoherence rate while for small qubit separations, the number of modes
increase (as we are integrating over solid angle in equation 21) that cause
fast decoherence. However, the initial correlations suppress this rate so that
we have finite concurrence in the long time limit.

##  IV Conclusions

In conclusion, we have studied quantum entanglement between two qubits coupled
via distance dependent interaction with a common bath. The initial correlated
state is obtained via a projective measurement on the system while assuming a
joint thermal equillibrium state of system and bath. Such procedure is
important towards quantum state preparation where bath state can depend on the
system parameters. Assuming, such initial system-bath correlations, we studied
their influence, and distance of separation between the qubits on dynamics of
concurence and coherence in a large class of two qubit states. It is shown
that concurrence is half of the coherence for all such kind of states, thus
enabling to measure entanglement in terms of coherence.

Next, we have shown that at low temperatures, initial system-bath correlations
play no role in dynamics as the system and bath become uncorrelated. While at
high temperatures, these correlations substantially modify the entanglement
decay. Furthermore, we have shown distance between the qubits forms an
important parameter to control decoherece effects both at low and high
temperature, i,e. we can tune $L$ in such a way that there exist only few
modes which cause decoherence and some modes do not interact with qubits. In
order to characterize these modes and their influence in case of interacting
qubits in equillbrium as well as non-equillibrium scenarios will be treated
separately76 .

###### Acknowledgements.

The authors would like to thank Dr. Javid A Naikoo at University of Warsaw for
helpful discussions.

## Appendix

###  .1 Bath Density Matrix $\rho_{B}^{\psi}(0)$

In this appendix, we derive the state dependent bath density matrix
$\rho_{B}^{\psi}(0)$. Since we assume a two qubit state
$|\psi\rangle=a|00\rangle+b|01\rangle+c|10\rangle+d|11\rangle$ with
$|a|^{2}+|b|^{2}+|c|^{2}+|d|^{2}=1$ and bath density matrix is

$\displaystyle\rho_{B}^{\psi}=\frac{\bra{\psi}\exp(-\beta
H)\ket{\psi}}{Tr_{B}\bra{\psi}\exp(-\beta H)\ket{\psi}}.$ (23)

Using $\sigma^{z}\ket{o}=\ket{0},\sigma^{z}\ket{1}=-\ket{1}$, we have

$\displaystyle e^{-\beta H}\ket{00}=e^{-\beta\omega_{0}}e^{-\beta
H_{B1}^{+}}\ket{00},~{}~{}~{}e^{-\beta H}\ket{01}=e^{-\beta
H_{B1}^{-}}\ket{01}$ (24) $\displaystyle e^{-\beta H}\ket{10}=e^{-\beta
H_{B2}^{+}}\ket{10}~{}~{}~{}e^{-\beta H}\ket{11}=e^{\beta\omega_{0}}e^{-\beta
H_{B2}^{-}}\ket{11}$ (25)

where

$\displaystyle H_{B1}^{\pm}=H_{B}+(B_{1k}\pm
B_{2k})~{}~{}~{}H_{B2}^{\pm}=H_{B}-(B_{1k}\pm B_{2k})$ (26)

and $B_{ik}=g_{ik}b_{k}+g_{ik}^{\star}b_{k}^{\dagger}$. Therefore, we write

$\displaystyle\langle\psi|e^{-\beta
H}|\psi\rangle=|a|^{2}e^{-\beta\omega_{0}}e^{-\beta
H_{B1}^{+}}+|b|^{2}e^{-\beta H_{B1}^{-}}+|c|^{2}e^{-\beta
H_{B2}^{+}}+|d|^{2}e^{\beta\omega_{0}}e^{-\beta H_{B2}^{-}}.$ (27)

Next, the partition function $Z$, after a straight forward calculation, we
write

$\displaystyle Z=Tr_{B}\langle\psi|e^{-\beta
H}|\psi\rangle=\big{[}(|a|^{2}e^{-\beta\omega_{0}}+|d|^{2}e^{\beta\omega_{0}})e^{\beta\frac{\Sigma_{k}|g_{1k}+g_{2k}|^{2}}{\omega_{k}}}+(|b|^{2}+|c|^{2})e^{\beta\frac{\Sigma_{k}|g_{1k}-g_{2k}|^{2}}{\omega_{k}}}\big{]}Z_{B}\equiv
z^{\prime}Z_{B}.$ (28)

where $Z_{B}=Tr_{B}e^{-\beta H_{B}}$ and $z^{\prime}$ is the rest of the
expression. Therefore, using 27 and 28 in equation 23, we get the bath density
matrix depending on the system parameters.

###  .2 Time evolved density matrix $\rho_{s}(t)$

In this appendix, we give explicit derivation of the time evolved density
matrix given in equation 11. Suppose at time $t=0$ the state of the composite
system is described by the initial density matrix $\rho(0)$, then at time $t$
the density matrix in interaction picture is given by

$\displaystyle\rho(t)=U(t)\rho(0)U(t)^{\dagger}$ (29)

where $U(t)=T~{}e^{-i\int_{0}^{t}dt^{\prime}H_{I}(t^{\prime})}$ is the time
evolution operator and $H_{I}(t)$ is the interaction Hamiltonian in the
interaction picture. Our main interest is to calculate the reduced density
matrix of the system by tracing out degrees of freedom of the bath:

$\displaystyle\rho_{S}(t)={\rm Tr}_{B}[U(t)\rho(0)U(t)^{\dagger}].$ (30)

We write $U(t)=e^{i\phi(t)}e^{\Sigma_{i}\sigma_{i}^{z}\hat{\Lambda}_{i}(t)}$
where $\phi(t)$ is a function of time only and
$\hat{\Lambda}_{i}(t)=\Sigma_{k}(\alpha_{ik}(t)b_{k}-\alpha_{ik}^{*}(t)b_{k}^{\dagger})$
with $\alpha_{ik}(t)=g_{ik}\frac{(1-e^{-i\omega_{k}t})}{\omega_{k}}$ and
$\phi(t)$ is a function of time only. Therefore, we can write

$\displaystyle\rho_{S}(t)$ $\displaystyle=$ $\displaystyle{\rm
Tr}_{B}[U(t)\rho(0)U(t)^{\dagger}]$ (31) $\displaystyle=$ $\displaystyle{\rm
Tr}_{B}[e^{\Sigma_{i}\sigma_{i}^{z}\hat{\Lambda}_{i}(t)}|\psi\rangle\langle\psi|\otimes\rho_{B}^{\psi}e^{-\Sigma_{i}\sigma_{i}^{z}\hat{\Lambda}_{i}(t)}].$
(32)

Let $\alpha_{1k}(t)+\alpha_{2k}(t)=A_{k}$ ,
$\alpha_{1k}(t)-\alpha_{2k}(t)=B_{k}$ and
$\alpha_{1k}^{*}(t)+\alpha_{2k}^{*}(t)=A_{k}^{*}$,
$\alpha_{1k}^{*}(t)-\alpha_{2k}^{*}(t)=B_{k}^{*}$ such that

$\displaystyle
U(t)\ket{00}=e^{i\phi(t)}\exp\Sigma_{k}(A_{k}^{*}b^{\dagger}-A_{k}b_{k})\ket{00},~{}~{}~{}U(t)\ket{01}=e^{i\phi(t)}\exp\Sigma_{k}(B_{k}^{*}b^{\dagger}-B_{k}b_{k})\ket{01},$
(33) $\displaystyle
U(t)\ket{10}=e^{i\phi(t)}\exp\Sigma_{k}(B_{k}b_{k}-B_{k}^{*}b_{k}^{\dagger})\ket{10},~{}~{}~{}U(t)\ket{11}=e^{i\phi(t)}\exp\Sigma_{k}(A_{k}b_{k}-A_{k}^{*}b_{k}^{\dagger})\ket{11}.$
(34)

We see that diagonal terms do not change, therefore we look for off-diagonal
terms.

  1. 1.

$\boxed{\ket{00}\bra{01}}$ matrix element ($ab^{*}$):

$\begin{split}\varphi=&\bra{00}\frac{1}{Z}\operatorname{Tr}_{B}\Big{\\{}e^{\Sigma_{i}\sigma_{i}^{z}\hat{\Lambda}_{i}(t)}\ket{00}\bra{01}\otimes\Big{[}|a|^{2}e^{-\beta\omega_{0}}e^{-\beta
H_{B1}^{+}}+|b|^{2}e^{-\beta H_{B1}^{-}}+|c|^{2}e^{-\beta
H_{B2}^{+}}+|d|^{2}e^{\beta\omega_{0}}e^{-\beta
H_{B2}^{-}}\Big{]}e^{-\Sigma_{i}\sigma_{i}^{z}\hat{\Lambda}_{i}(t)}\Big{\\}}\ket{01}\\\
&=\frac{1}{Z}\operatorname{Tr}_{B}\Big{\\{}e^{\Sigma_{k}(A_{k}^{*}b_{k}^{\dagger}-A_{k}b_{k})}\Big{[}|a|^{2}e^{-\beta\omega_{0}}e^{-\beta
H_{B1}^{+}}+|b|^{2}e^{-\beta H_{B1}^{-}}+|c|^{2}e^{-\beta
H_{B2}^{+}}+|d|^{2}e^{\beta\omega_{0}}e^{-\beta
H_{B2}^{-}}\big{]}\Big{\\}}.\end{split}$ (35)

Next we define unitary transformations

$\displaystyle O_{\pm}$ $\displaystyle=$
$\displaystyle\exp\mp[\Sigma_{k}(\frac{g_{1k}+g_{2k}}{\omega_{k}})b_{k}-(\frac{g^{*}_{1k}+g^{*}_{2k}}{\omega_{k}}b^{\dagger}_{k})]$
(36) $\displaystyle\Theta_{\pm}$ $\displaystyle=$
$\displaystyle\exp\mp[\Sigma_{k}(\frac{g_{1k}-g_{2k}}{\omega_{k}})b_{k}-(\frac{g^{*}_{1k}-g^{*}_{2k}}{\omega_{k}}b^{\dagger}_{k})]$
(37)

such that

$\displaystyle O_{+}e^{-\beta H_{B1}^{+}}O_{+}^{\dagger}=e^{-\beta
H_{B}}e^{\beta\frac{\Sigma_{k}|g_{1k}+g_{2k}|^{2}}{\omega_{k}}}~{}~{}~{}~{}O_{-}e^{-\beta
H_{B2}^{+}}O_{-}^{-1}=e^{-\beta
H_{B}}e^{\beta\frac{\Sigma_{k}|g_{1k}+g_{2k}|^{2}}{\omega_{k}}}$ (38)
$\displaystyle\Theta_{+}e^{-\beta H_{B1}^{-}}\Theta_{+}^{-1}=e^{-\beta
H_{B}}e^{\beta\frac{\Sigma_{k}|g_{1k}-g_{2k}|^{2}}{\omega_{k}}}~{}~{}~{}\Theta_{-}e^{-\beta
H_{B2}^{-}}\Theta_{-}^{-1}=e^{-\beta
H_{B}}e^{\beta\frac{\Sigma_{k}|g_{1k}-g_{2k}|^{2}}{\omega_{k}}}.$ (39)

Next, using these unitary transformations, we write

$\displaystyle{\rm
Tr}_{B}\big{[}e^{\Sigma_{k}(A_{k}^{*}b_{k}^{\dagger}-A_{k}b_{k})}e^{-\beta
H_{B1}^{+}}e^{\Sigma_{k}(B_{k}b_{k}-B_{k}^{*}b_{k}^{\dagger})}$
$\displaystyle~{}~{}~{}~{}~{}={\rm
Tr}_{B}\big{[}e^{\Sigma_{k}(A_{k}^{*}b_{k}^{\dagger}-A_{k}b_{k})}O_{+}^{-1}O_{+}e^{-\beta
H_{B1}^{+}}O_{+}^{-1}O_{+}e^{\Sigma_{k}(B_{k}b_{k}-B_{k}^{*}b_{k}^{\dagger})}\big{]}$
$\displaystyle~{}~{}~{}~{}~{}=e^{\beta\Sigma_{k}\frac{|g_{1k}+g_{2k}|^{2}}{\omega_{k}}}\operatorname{Tr}_{B}\Big{\\{}e^{A_{k}^{*}b_{k}^{\dagger}}e^{-A_{k}b_{k}}e^{-\frac{1}{2}|A_{k}|^{2}}O_{+}^{-1}e^{-\beta
H_{B}}O_{+}e^{B_{k}b_{k}}e^{-B_{k}^{*}b_{k}^{\dagger}}e^{\frac{1}{2}|B_{k}|^{2}}\Big{\\}}$
$\displaystyle~{}~{}~{}~{}~{}=e^{\beta\Sigma_{k}\frac{|g_{1k}+g_{2k}|^{2}}{\omega_{k}}}\operatorname{Tr}_{B}\Big{\\{}O_{+}^{-1}O_{+}e^{A_{k}^{*}b_{k}^{\dagger}}O_{+}^{-1}O_{+}e^{-A_{k}b_{k}}e^{-\frac{1}{2}|A_{k}|^{2}}O_{+}^{-1}e^{-\beta
H_{B}}O_{+}e^{B_{k}b_{k}}O_{+}^{-1}O_{+}e^{-B_{k}^{*}b_{k}^{\dagger}}e^{\frac{1}{2}|B_{k}|^{2}}\Big{\\}}$
$\displaystyle~{}~{}~{}~{}=e^{\beta\Sigma_{k}\frac{|g_{1k}+g_{2k}|^{2}}{\omega_{k}}}e^{\Sigma_{k}\frac{(g_{1k}+g_{2k})(B_{k}^{*}-A_{k}^{*})}{\omega_{k}}}e^{\Sigma_{k}\frac{-(g_{1k}^{*}+g_{2k}^{*})(B_{k}-A_{k})}{\omega_{k}}}e^{\Sigma_{k}\frac{(A_{k}^{*}B_{k}-B_{k}^{*}A_{k})}{2}}e^{-\gamma_{1}(t)}$
(40)

where
$e^{-\gamma_{1}(t)}=\operatorname{Tr}_{B}\big{\\{}e^{(B_{k}-A_{k})b_{k}-(B_{k}^{*}-A_{k}^{*})b_{k}^{\dagger}}\rho_{B}\big{\\}}$.
In the similiar fashion, we can evluate other terms. Thus putting together all
these terms, and using the form of $Z$ in equation 35, we arrive at the
following expression for the matrix element $|00\rangle\langle 01|$ with
coefficient $a^{*}b\varphi(t)$ where

$\begin{split}\varphi(t)=&\frac{e^{-\gamma_{1}(t)}}{z^{\prime}}\Bigg{\\{}|a|^{2}e^{-\beta\omega_{0}}e^{\beta\Sigma_{k}\frac{|g_{1k}+g_{2k}|^{2}}{\omega_{k}}}\exp\Sigma_{k}{\frac{4i|g_{k}|^{2}}{\omega_{k}^{2}}[1+\cos
k.(r_{1}-r_{2})\sin\omega_{k}t]}\\\
&+|b|^{2}e^{\beta\Sigma_{k}\frac{|g_{1k}-g_{2k}|^{2}}{\omega_{k}}}\exp{-\Sigma_{k}\frac{4i|g_{k}|^{2}}{\omega_{k}^{2}}[1-\cos
k.(r_{1}-r_{2})\sin\omega_{k}t]}\\\
&+|c|^{2}e^{\beta\Sigma_{k}\frac{|g_{1k}-g_{2k}|^{2}}{\omega_{k}}}\exp{\Sigma_{k}\frac{4i|g_{k}|^{2}}{\omega_{k}^{2}}[2\sin
k.(r_{1}-r_{2})(1-\cos\omega_{k}t)+\sin\omega_{k}t(1-\cos
k.(r_{1}-r_{2}))]}\\\
&+|d|^{2}e^{\beta\omega_{0}}e^{\beta\Sigma_{k}\frac{|g_{1k}+g_{2k}|^{2}}{\omega_{k}}}\exp\Sigma_{k}{\frac{4i|g_{k}|^{2}}{\omega_{k}^{2}}[2\sin
k.(r_{1}-r_{2})(1-\cos\omega_{k}t)-\sin\omega_{k}t(1+\cos
k.(r_{1}-r_{2}))]}\Bigg{\\}},\end{split}$ (41)

and after further simplication, we arrive at the final expression

$\displaystyle\varphi(t)$ $\displaystyle=$
$\displaystyle\Bigg{[}\big{\\{}|a|^{2}e^{-\beta\omega_{0}}e^{i\Phi_{+}(r,t)}+|d|^{2}e^{\beta\omega_{0}}e^{i(\chi(r,t)-\Phi_{+}(r,t))}\big{\\}}e^{\beta\Sigma_{k}\frac{|g_{1k}+g_{2k}|^{2}}{\omega_{k}}}$
(42)
$\displaystyle+\big{\\{}|b|^{2}e^{-i\Phi_{-}(r,t)}+|c|^{2}e^{i(\chi(r,t)+\Phi_{-}(r,t))}\big{\\}}e^{\beta\Sigma_{k}\frac{|g_{1k}-g_{2k}|^{2}}{\omega_{k}}}\Bigg{]}\exp(-4\Sigma_{k}\frac{|g_{k}|^{2}}{\omega_{k}^{2}}(1-\cos{\omega_{k}t})\coth{\frac{\beta\omega_{k}}{2}})$

where

$\displaystyle\Phi_{\pm}(r,t)$ $\displaystyle=$ $\displaystyle
4\sum_{k}\frac{|g_{k}|^{2}}{\omega_{k}^{2}}\Big{[}\sin{\omega_{k}t}(1\pm\cos\vec{k}.(\vec{r}_{1}-\vec{r}_{2}))\Big{]}$
(43) $\displaystyle\chi(r,t)$ $\displaystyle=$ $\displaystyle
4\sum_{k}\frac{|g_{k}|^{2}}{\omega_{k}^{2}}\big{[}\sin\vec{k}.(\vec{r}_{1}-\vec{r}_{2})(1-\cos\omega_{k}t)\big{]}$
(44)

In the similiar way, we obtain all other matrix elements which are given by:

  2. 2.

$|00\rangle\langle 10|$ matrix element ($ac^{*}$):

$\displaystyle\zeta(t)$ $\displaystyle=$
$\displaystyle\bra{00}\frac{1}{Z}\operatorname{Tr}_{B}\Big{\\{}e^{\Sigma_{i}\sigma_{i}^{z}\hat{\Lambda}_{i}(t)}\ket{00}\bra{10}\otimes\Big{[}|a|^{2}e^{-\beta\omega_{0}}e^{-\beta
H_{B1}^{+}}+|b|^{2}e^{-\beta H_{B1}^{-}}+|c|^{2}e^{-\beta
H_{B2}^{+}}+|d|^{2}e^{\beta\omega_{0}}e^{-\beta
H_{B2}^{-}}\Big{]}e^{-\Sigma_{i}\sigma_{i}^{z}\hat{\Lambda}_{i}(t)}\Big{\\}}\ket{10}$
(45) $\displaystyle=$
$\displaystyle\Bigg{[}\big{\\{}|a|^{2}e^{-\beta\omega_{0}}e^{i\Phi_{+}(r,t)}+|d|^{2}e^{\beta\omega_{0}}e^{-i{(\chi(r,t)-\Phi_{+}(r,t))}}\big{\\}}e^{\beta\Sigma_{k}\frac{|g_{1k}+g_{2k}|^{2}}{\omega_{k}}}$
$\displaystyle~{}+\big{\\{}|b|^{2}e^{-{i(\chi(r,t)-\Phi_{-}(r,t))}}+|c|^{2}e^{-i\Phi_{-}(r,t)}\big{\\}}e^{\beta\Sigma_{k}\frac{|g_{1k}-g_{2k}|^{2}}{\omega_{k}}}\Bigg{]}\exp(-4\Sigma_{k}\frac{|g_{k}|^{2}}{\omega_{k}^{2}}(1-\cos{\omega_{k}t})\coth{\frac{\beta\omega_{k}}{2}})$

  3. 3.

$|00\rangle\langle 11|$ matrix element ($ad^{*}$):

$\displaystyle\kappa(t)$ $\displaystyle=$
$\displaystyle\bra{00}\frac{1}{Z}\operatorname{Tr}_{B}\Big{\\{}e^{\Sigma_{i}\sigma_{i}^{z}\hat{\Lambda}_{i}(t)}\ket{00}\bra{11}\otimes\Big{[}|a|^{2}e^{-\beta\omega_{0}}e^{-\beta
H_{B1}^{+}}+|b|^{2}e^{-\beta H_{B1}^{-}}+|c|^{2}e^{-\beta
H_{B2}^{+}}+|d|^{2}e^{\beta\omega_{0}}e^{-\beta
H_{B2}^{-}}\Big{]}e^{-\Sigma_{i}\sigma_{i}^{z}\hat{\Lambda}_{i}(t)}\Big{\\}}\ket{11}$
(46) $\displaystyle=$
$\displaystyle\Bigg{[}\big{\\{}|a|^{2}e^{-\beta\omega_{0}}e^{2i\Phi_{+}(r,t)}+|d|^{2}e^{\beta\omega_{0}}e^{-2i{(\Phi_{+}(r,t)}}\big{\\}}e^{\beta\Sigma_{k}\frac{|g_{1k}+g_{2k}|^{2}}{\omega_{k}}}$
$\displaystyle~{}+\big{\\{}|b|^{2}e^{-{i(\chi(r,t)}}+|c|^{2}e^{i\chi(r,t)}\big{\\}}e^{\beta\Sigma_{k}\frac{|g_{1k}-g_{2k}|^{2}}{\omega_{k}}}\Bigg{]}\exp(-4\Sigma_{k}\frac{|g_{k}|^{2}}{\omega_{k}^{2}}\big{(}1+\cos{k.(r_{1}-r_{2})}\big{)}(1-\cos{\omega_{k}t})\coth{\frac{\beta\omega_{k}}{2}})$

  4. 4.

$|01\rangle\langle 10|$ matrix element ($bc^{*}$):

$\displaystyle\bar{\kappa}(t)$ $\displaystyle=$
$\displaystyle\bra{01}\frac{1}{Z}\operatorname{Tr}_{B}\Big{\\{}e^{\Sigma_{i}\sigma_{i}^{z}\hat{\Lambda}_{i}(t)}\ket{01}\bra{10}\otimes\Big{[}|a|^{2}e^{-\beta\omega_{0}}e^{-\beta
H_{B1}^{+}}+|b|^{2}e^{-\beta H_{B1}^{-}}+|c|^{2}e^{-\beta
H_{B2}^{+}}+|d|^{2}e^{\beta\omega_{0}}e^{-\beta
H_{B2}^{-}}\Big{]}e^{-\Sigma_{i}\sigma_{i}^{z}\hat{\Lambda}_{i}(t)}\Big{\\}}\ket{10}$
(47) $\displaystyle=$
$\displaystyle\Bigg{[}\big{\\{}|a|^{2}e^{-\beta\omega_{0}}e^{i\chi(r,t)}+|d|^{2}e^{\beta\omega_{0}}e^{-i(\chi(r,t)}\big{\\}}e^{\beta\Sigma_{k}\frac{|g_{1k}+g_{2k}|^{2}}{\omega_{k}}}$
$\displaystyle~{}+\big{\\{}|b|^{2}e^{2{i\Phi_{-}(r,t)}}+|c|^{2}e^{-2i\Phi_{-}(r,t)}\big{\\}}e^{\beta\Sigma_{k}\frac{|g_{1k}-g_{2k}|^{2}}{\omega_{k}}}\Bigg{]}\exp(-4\Sigma_{k}\frac{|g_{k}|^{2}}{\omega_{k}^{2}}\big{(}1-\cos{k.(r_{1}-r_{2})}\big{)}(1-\cos{\omega_{k}t})\coth{\frac{\beta\omega_{k}}{2}})$

  5. 5.

$|01\rangle\langle 11|$ matrix element ($bd^{*}$):

$\displaystyle\bar{\zeta}(t)$ $\displaystyle=$
$\displaystyle\bra{01}\frac{1}{Z}\operatorname{Tr}_{B}\Big{\\{}e^{\Sigma_{i}\sigma_{i}^{z}\hat{\Lambda}_{i}(t)}\ket{01}\bra{11}\otimes\Big{[}|a|^{2}e^{-\beta\omega_{0}}e^{-\beta
H_{B1}^{+}}+|b|^{2}e^{-\beta H_{B1}^{-}}+|c|^{2}e^{-\beta
H_{B2}^{+}}+|d|^{2}e^{\beta\omega_{0}}e^{-\beta
H_{B2}^{-}}\Big{]}e^{-\Sigma_{i}\sigma_{i}^{z}\hat{\Lambda}_{i}(t)}\Big{\\}}\ket{11}$
(48) $\displaystyle=$
$\displaystyle\Bigg{[}\big{\\{}|a|^{2}e^{-\beta\omega_{0}}e^{i(\chi(r,t)+\Phi_{-}(r,t))}+|d|^{2}e^{\beta\omega_{0}}e^{-i{\Phi_{+}(r,t)}}\big{\\}}e^{\beta\Sigma_{k}\frac{|g_{1k}+g_{2k}|^{2}}{\omega_{k}}}$
$\displaystyle~{}+\big{\\{}|b|^{2}e^{{i\Phi_{-}(r,t)}}+|c|^{2}e^{i(\chi(r,t)-\Phi_{-}(r,t))}\big{\\}}e^{\beta\Sigma_{k}\frac{|g_{1k}-g_{2k}|^{2}}{\omega_{k}}}\Bigg{]}\exp(-4\Sigma_{k}\frac{|g_{k}|^{2}}{\omega_{k}^{2}}(1-\cos{\omega_{k}t})\coth{\frac{\beta\omega_{k}}{2}})$

  6. 6.

$|10\rangle\langle 11|$ matrix element ($bc^{*}$):

$\displaystyle\bar{\varphi}(t)$ $\displaystyle=$
$\displaystyle\bra{10}\frac{1}{Z}\operatorname{Tr}_{B}\Big{\\{}e^{\Sigma_{i}\sigma_{i}^{z}\hat{\Lambda}_{i}(t)}\ket{10}\bra{11}\otimes\Big{[}|a|^{2}e^{-\beta\omega_{0}}e^{-\beta
H_{B1}^{+}}+|b|^{2}e^{-\beta H_{B1}^{-}}+|c|^{2}e^{-\beta
H_{B2}^{+}}+|d|^{2}e^{\beta\omega_{0}}e^{-\beta
H_{B2}^{-}}\Big{]}e^{-\Sigma_{i}\sigma_{i}^{z}\hat{\Lambda}_{i}(t)}\Big{\\}}\ket{11}$
(49) $\displaystyle=$
$\displaystyle\Bigg{[}\big{\\{}|a|^{2}e^{-\beta\omega_{0}}e^{-i(\chi(r,t)-\Phi_{+}(r,t))}+|d|^{2}e^{\beta\omega_{0}}e^{-i{\Phi_{+}(r,t)}}\big{\\}}e^{\beta\Sigma_{k}\frac{|g_{1k}+g_{2k}|^{2}}{\omega_{k}}}$
$\displaystyle~{}+\big{\\{}|b|^{2}e^{-{i(\chi(r,t)+\Phi_{-}(r,t))}}+|c|^{2}e^{i\Phi_{-}(r,t)}\big{\\}}e^{\beta\Sigma_{k}\frac{|g_{1k}-g_{2k}|^{2}}{\omega_{k}}}\Bigg{]}\exp(-4\Sigma_{k}\frac{|g_{k}|^{2}}{\omega_{k}^{2}}(1-\cos{\omega_{k}t})\coth{\frac{\beta\omega_{k}}{2}})$

Unisng all these equations, we get the time evolved density matrix used in the
main text 11:

$\displaystyle\rho_{s}(t)=\begin{pmatrix}|a|^{2}&ab^{*}\varphi(t)&ac^{*}\zeta(t)&ad^{*}\kappa(t)\\\
ba^{*}\varphi^{*}(t)&|b|^{2}&bc^{*}\bar{\kappa}(t)&bd^{*}\bar{\zeta}(t)\\\
ca^{*}\zeta^{*}(t)&cb^{*}\bar{\kappa}^{*}(t)&|c|^{2}&cd^{*}\bar{\varphi}(t)\\\
da^{*}\kappa^{*}(t)&db^{*}\bar{\zeta}^{*}(t)&dc^{*}\bar{\varphi}^{*}(t)&|d|^{2}\end{pmatrix}$
(50)

## References

  * (1) R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Rev. Mod. Phys. 81, 865 (2009).
  * (2) C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, Phys. Rev. Lett. 70, (1993).
  * (3) D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, Nat. Phys. 390, 575 (1997).
  * (4) X.-S. Ma, T. Herbst, T. Scheidl, D. Wang, S. Kropatschek, W. Naylor, B. Wittmann, A. Mech, J. Koer, E. Anisimova, et al., Nat. Phys. 489, 269 (2012).
  * (5) M. A. Nielsen and I. L. Chuang, _Quantum Computation and Quantum Information_ (Cambridge University Press, Cambridge, 2000).
  * (6) H.-P. Breuer and F. Petruccione, _The Theory of Open Quantum Systems_ (Oxford University Press, 2007).
  * (7) A. Rivas and S. F. Huelga, _Open Quantum Systems, SpringerBriefs in Physics_ (Springer Berlin Heidelberg, Berlin, Heidelberg, 2012).
  * (8) Fanchini, Felipe Fernandes, Diogo de Oliveira Soares Pinto, and Gerardo Adesso, eds. _Lectures on General Quantum Correlations and their Applications_. Berlin: Springer, 2017.
  * (9) G. Adesso, T. R. Bromley, and M. Cianciaruso, J. Phys. A Math. Theor. 49, 473001 (2016).
  * (10) L. Aolita, F. de Melo, and L. Davidovich, Rep. Prog. Phys. 78, 042001 (2015).
  * (11) H.-P. Breuer, J. Phys. B- At. Mol. Opt. 45, 154001 (2012).
  * (12) Á. Rivas, S. F. Huelga, and M. B. Plenio, Rep. Prog. Phys. 77, 094001 (2014).
  * (13) I. De Vega and D. Alonso, Rev. Mod. Phys. 89, 015001 (2017).
  * (14) M. M. Wolf, J. Eisert, T. S. Cubitt, and J. I. Cirac, Phys. Rev. Lett. 101, 150402 (2008).
  * (15) D. Chruściński, and A. Kossakowski, Eur. Phys. J. D 68, 1 (2014).
  * (16) M. Q. Lone, Pramana 87, 1 (2016).
  * (17) M. Q. Lone and S. Yarlagadda, Int. J. Mod. Phys. B 30, 1650063 (2016).
  * (18) A. Dey, M. Lone, and S. Yarlagadda, Phys. Rev. B 92, 094302 (2015).
  * (19) W.-M. Zhang, P.-Y. Lo, H.-N. Xiong, M. W.-Y. Tu, and F. Nori, Phys. Rev. Lett. 109, 170402 (2012).
  * (20) S. Das, and G. S. Agarwal, J. Phys. B- At. Mol. Opt. 42, 205502 (2009).
  * (21) A. G. Dijkstra and Y. Tanimura, Phys. Rev. Lett. 104, 250401 (2010).
  * (22) J. H. Reina, L. Quiroga, and N. F. Johnson, Phys. Rev. A 65, 032326 (2002).
  * (23) F. Reiter, D. Reeb, and A. S. Sørensen, Phys. Rev. Lett. 117, 040501 (2016).
  * (24) M. J. Kastoryano, F. Reiter, and A. S. Sørensen, Phys. Rev. Lett. 106, 090502 (2011).
  * (25) S. Diehl, A. Micheli, A. Kantian, B. Kraus, B., H. P. Büchler, and P. Zoller, Nat. Phys. 4, 878-883 (2008).
  * (26) F. Verstraete, M. M. Wolf, and J. Ignacio Cirac, Nat. Phys. 5, 633 (2009).
  * (27) F. Pastawski, L. Clemente, and J. I. Cirac, Phys. Rev. A 83, 012304 (2011).
  * (28) A. W. Chin, S. F. Huelga, and M. B. Plenio, Phys. Rev. Lett. 109, 233601 (2012).
  * (29) X. Gao, A. Sergio, K. Chen, S. Fei, and X. Li-Jost,Front. Comput. Sci. 2, 114 (2008).
  * (30) W. K. Wootters, Phys. Rev. Lett. 80, 2245 (1998).
  * (31) T. Baumgratz, M. Cramer, and M. B. Plenio, Phys. Rev. Lett. 113, 140401 (2014).
  * (32) A. Winter and D. Yang, Operational resource theory of coherence, Phys. Rev. Lett. 116, 120404 (2016).
  * (33) A. Streltsov, G. Adesso, and M. B. Plenio, Rev. Mod. Phys. 89, 041003 (2017).
  * (34) A. Streltsov, U. Singh, H. S. Dhar, M. N. Bera, and G. Adesso, Phys. Rev. Lett. 115, 020403 (2015).
  * (35) T. Chanda and S. Bhattacharya,Ann. Phys. (N. Y.) 336, (2016).
  * (36) C. L. Liu, X.-D. Yu, G. F. Xu, and D. M. Tong, Quantum Inf. Process 15, 4189 (2016).
  * (37) C. Napoli, T. R. Bromley, M. Cianciaruso, M. Piani, N. Johnston, and G. Adesso, Phys. Rev. Lett. 116, 150502 (2016).
  * (38) N. Pathania and T. Qureshi, Int. J. Theor. Phys. 61, 25 (2022).
  * (39) Ignatyuk and Morozov, J. Phys. Condens. Matter 16, 34001 (2013).
  * (40) S. Lloyd, J. Phys. Conf. Ser. 302, 012037 (2011).
  * (41) E. Romero, R. Augulis, V.I. Novoderezhkin, M. Ferretti, J. Thieme, D. Zigmantas, and R. Van Grondelle, Nat. Phys. 10, 676 (2014).
  * (42) M. Thorwart, J. Eckel, J. H. Reina, P. Nalbach, and S. Weiss, Chem. Phys. Lett. 478, 234 (2009).
  * (43) M. Afrin and T. Qureshi, Eur. Phys. J. D 73, 1 (2019).
  * (44) A. Venugopalan, S. Mishra, and T. Qureshi, Phys. A: Stat. Mech. Appl. 516, 308 (2019).
  * (45) J. Naikoo, S. Dutta, and S. Banerjee, Phys. Rev. A 99, 042128 (2019).
  * (46) J. Naikoo and S. Banerjee, Quantum Inf. Process 19, 1 (2020).
  * (47) P. Rebentrost, M. Mohseni, and A. Aspuru-Guzik, J. Phys. Chem. B 113, 9942 (2009).
  * (48) M. B. Plenio and S. F. Huelga, New J. Phys. 10, 113019 (2008).
  * (49) F. Caruso, A. W. Chin, A. Datta, S. F. Huelga, and M. B. Plenio. J. Chem. Phys. 131, 09B612 (2009).
  * (50) Y. Cheng and R. J. Silbey, Phys. Rev. Lett. 96, 028103 (2006).
  * (51) Z. X. Man, Y. J. Xia, and R. L. Franco, Phys. Rev. A 97, 062104 (2018).
  * (52) B. Bellomo, G. Compagno, R. Lo Franco, A. Ridolfo, and S. Savasta, Phys. Scr. 143, 014004 (2011).
  * (53) B. Bellomo, R. L. Franco, S. Maniscalco, and G. Compagno, Phys. Scr. 140, 014014 (2010).
  * (54) B. Bellomo, R. Lo Franco, and G. Compagno, Adv. Sci. Lett. 2, 459 (2009).
  * (55) B. Bellomo, R. Lo Franco, and G. Compagno, Phys. Rev. A 77, 032342 (2008).
  * (56) B. Bellomo, R. Lo Franco, and G. Compagno, Phys. Rev. Lett. 99, 160502 (2007).
  * (57) K. Berrada, F. F. Fanchini, and S. Abdel-Khalek, Phys. Rev. A 85, 052315 (2012).
  * (58) H. Grabert, P. Schramm, and G.-L. Ingold, Phys.Rep. 168, 115 (1988).
  * (59) S. Banerjee and R. Ghosh, Phys. Rev. E 67, 056120 (2003).
  * (60) J. C. Halimeh and I. de Vega, Phys. Rev. A 95, 052108 (2017).
  * (61) A. Z. Chaudhry, and J. Gong, Phys. Rev. A 87, 012129 (2013).
  * (62) A. Z. Chaudhry and J. Gong, Phys. Rev. A 88, 052107 (2013).
  * (63) C. C. Chen, and H. S. Goan, Phys. Rev. A 93, 032113 (2016).
  * (64) V. Semin, I. Sinayskiy, and F. Petruccione, Phys. Rev. A 86, 062114 (2012).
  * (65) Y. J. Zhang, X. B. Zou, Y. J. Xia, and G. C. Guo, Phys. Rev. A 82, 022108 (2010).
  * (66) M. Campisi, P. Talkner, and P. Hänggi, Phys. Rev. Lett. 102, 210401 (2009).
  * (67) V. G. Morozov, S. Mathey, and G. Röpke, Phys. Rev. A 85, 022101 (2012).
  * (68) H. Ying, D.-W. Luo, and J.-B. Xu, J. Appl. Phys. 114, 164902 (2013).
  * (69) C. Uchiyama and M. Aihara, Phys. Rev. A 82, 044104 (2010).
  * (70) A. Smirne, H.-P. Breuer, J. Piilo, and B. Vacchini, Phys. Rev. A 82, 062114 (2010).
  * (71) M. Majeed, and A. Z. Chaudhry, Eur. Phys. J. D 73, 1 (2019).
  * (72) M. Bruderer, A. Klein, S. R. Clark, and D. Jaksch, Phys. Rev. A 76, 011605 (2007).
  * (73) C. Gross and I. Bloch, Science 357, 995 (2017).
  * (74) D. P. S. McCutcheon, A. Nazir, S. Bose, and A. J. Fisher, Phys. Rev. B 81, 235321 (2010).
  * (75) R. Grimm, M. Weidemüller, and Y. B. Ovchinnikov, Adv. At. Mol. Opt. Phys. 42, 95 (2000).
  * (76) M. Rashid, M. Q. Lone, and P. A. Ganai, Unpublished, Unpublished.