# Preservation and enhancement of quantum correlations under Stark effect

Nitish Kumar Chandra1 E-mail<EMAIL_ADDRESS>Rajiuddin Sk 1
Corresponding E-mail<EMAIL_ADDRESS>Prasanta K. Panigrahi1
E-mail<EMAIL_ADDRESS>1Department of Physical Sciences,
Indian Institute of Science Education and Research Kolkata, India

###### Abstract

We analyze the dynamics of quantum correlations by obtaining the exact
expression of Bures distance entanglement, trace distance discord, and local
quantum uncertainty of two two-level atoms. Here, the atoms undergo two-photon
transitions mediated through an intermediate virtual state where each atom is
separately coupled to a dissipative reservoir at zero temperature in the
presence of the Stark shift effect. We have investigated the dynamics of this
atomic system for two different initial conditions of the environment. In the
first case, we have assumed the environment’s state to be in ground state and
in the other case, we have assumed the state to be in first excited state. The
second initial condition is significant as it shows the role played by both
the Stark shift parameters in contrast to only one of the Stark shift
parameters for the first initial condition. Our results demonstrate that
quantum correlations can be sustained for an extended period in the presence
of Stark shift effect in the case of both Markovian and non-Markovian
reservoirs. The effect in the non-Markovian reservoir is more prominent than
the Markovian reservoir, even for a very small value of the Stark shift
parameter. We observe that among the correlation measures considered, only
local quantum uncertainty is accompanied by a sudden change phenomenon, i.e.,
an abrupt change in the decay rate of a correlation measure. Our findings are
significant as preserving quantum correlations is one of the essential aspects
in attaining optimum performance in quantum information tasks.

###### keywords:

Stark Shift Effect, Bures Distance Entanglement, Trace Distance Discord, Local
Quantum Uncertainty

##  1 Introduction

Quantum correlations, a distinguishing feature of quantum systems, are a
crucial resource for quantum information science [1]. Their presence in any
bipartite or multipartite quantum system results in non-local interactions
between the subsystems. Quantum entanglement, one of the most widely
considered quantum correlations, is a useful resource for several quantum
tasks like teleportation [2], quantum key distribution [3], quantum metrology
[4], and several others [5]. The presence of nonclassical correlations, even
in certain separable states, has attracted considerable attention to
understand their origin and usefulness as quantum resources [6]. For mixed
states, entanglement fails to account for all non-local correlations,
resulting in a slew of works devoted to introducing quantum correlation
quantifiers beyond entanglement [7]. The entropic quantum discord is one such
fundamental quantifier that has gained significant interest in the literature
due to its role in quantum speedups using deterministic quantum computation
with one quantum bits [8, 9] and other quantum tasks [10]. It can be
experimentally estimated as a function of temperature for an antiferromagnetic
Heisenberg system [11]. However, computing entropic quantum discord for a two-
qubit system is NP-complete; hence very hard to calculate for a general case.
Only partial findings in the form of closed expressions exist for a few two-
qubit states [7]. As a result, geometric versions of quantum discord with
various norms have been developed. These geometric discord measures offer
easier computability and have found operational significance in quantum tasks
such as state distinguishability [12]. For special cases like two-qubit X
state, the exact form of trace distance discord is available in closed form
[13].

Recently, Girolami et al. [14] proposed a computable discord type measure
based on local quantum uncertainty (LQU). It uses skew information and
fulfills all the conditions to be a physical quantum correlation measure [15].
The calculation of this measure avoids the difficult step of optimization and
fulfills contractivity, which is not adequately proven by geometric discord
based on the Hilbert-Schmidt norm [16]. In particular, LQU plays a significant
role in determining the critical points of quantum phase transitions in
multipartite spin systems better than quantum discord and exhibits
correlations that are not captured by quantum entanglement, or quantum discord
[17]. LQU is also associated with quantum Fisher information, which is known
to play an essential role in quantum parameter estimation [18].

Understanding the impact of decoherence on the quantum system is an essential
component of quantum information research. In general, the dynamics of systems
interacting with their environment are immensely complicated to solve. The
quantum system is classified as either an open quantum system or a closed
quantum system based on the type of system-environment interaction. Memory
effects are a crucial aspect of non-Markovian quantum behavior and are implied
by information flow from the surrounding environment to the open system. The
process is Markovian if the information flow is unidirectional from the open
system to the environment. The Markovian process is based on the premise that
the system’s characteristic times are significantly longer than the
environment’s and presupposes that there is always a weak coupling between the
system and environment. The dynamics of quantum correlations based on Bures
norm, Trace norm, entropy, Fisher information, and quantum uncertainty have
been investigated under various noise in both Markovian and non-Markovian
regimes [19, 20, 21, 22, 23, 24].

Quantum correlations are very sensitive to environmental disturbances and
decay quickly in the presence of noise, resulting in the phenomenon of
entanglement sudden death [25, 26, 27, 28]. However, for a non-Markovian
environment, the revival and oscillation of entanglement are observed owing to
the memory effect of the environment [29]. Due to the relevance of
entanglement in various applications, numerous techniques like quantum Zeno
effect [30], weak measurement [31, 32], decoherence-free subspace method [33]
and others [34, 35] have been proposed to prevent the system to decohere.

The atomic two-photon excitation mechanism, governed by time-dependent quantum
dynamics, has been explored in the context of spectroscopic precision in Ref.
[36]. Further, the dynamic Stark shift for two-photon transitions in bound
two-body Coulomb systems is computed showing high quantum efficiency in
resonant ionisation spectroscopy. In Ref. [37], the authors show the
application of a dc field based on Stark shifts to regulate spontaneous
emission in a cavity. The impact of the Stark effect on the entanglement
dynamics in atomic systems with a quantized cavity field was investigated in
some works without considering the system’s dissipation [38, 39]. Recently, in
a dissipative environment, the entanglement dynamics of two two-level atoms
were investigated and shown that entanglement captured by concurrence [40] can
be protected in the presence of Stark shift [41]. We show that correlations
beyond entanglement can be effectively protected in the presence of Stark
shift for two-level atoms having two-photon transitions, where each atom is
separately attached to a dissipative reservoir at zero temperature. Our
investigations indicate that by altering the Stark shift in the two-qubit
system, the quantum correlations captured by Bures distance entanglement,
trace distance discord, and local quantum uncertainty can be enhanced for both
Markovian and non-Markovian reservoirs. The enhancement of quantum
correlations is more effective in the case of a non-Markovian reservoir in
comparison to a Markovian reservoir for a specific magnitude of the Stark
shift parameter.

The paper is organized as follows: In Section $2$, we briefly review the
definition of quantum correlation measures. In Section $3$, we show the
dynamics of the two-qubit system’s correlation measures in Markovian and non-
Markovian settings under the influence of the Stark shift effect when the
environments are initially in vacuum states. In Section $4$, we discuss the
dynamics of correlation measures when the environments are in the first
excited states. Finally, in Section $5$, we conclude by summarizing the
results of the paper.

##  2 Quantum Correlation Measures

In the following, we review the definitions of Bures distance entanglement,
Trace distance discord, and local quantum uncertainty, which are robust and
reliable measures of quantum correlations.

###  2.1 Bures Distance Entanglement

The Bures distance entanglement measure is defined using Bures distance norm,
[42] is given by

$B(t)=\sqrt{2-\sqrt{2+2\sqrt{1-\mathcal{C}(t)^{2}}}},$ (1)

where $\mathcal{C}(t)$ is concurrence, defined as [40]

$\mathcal{C}(t)=\max\left\\{0,\sqrt{\lambda_{1}}-\sqrt{\lambda_{2}}-\sqrt{\lambda_{3}}-\sqrt{\lambda_{4}}\right\\}.$
(2)

Here $\lambda_{1}\geq\lambda_{2}\geq\lambda_{3}\geq\lambda_{4}$ are the
eigenvalues of the matrix,
$T=\rho_{AB}\left(\sigma_{y}\otimes\sigma_{y}\right)\rho^{*}_{AB}\left(\sigma_{y}\otimes\sigma_{y}\right)$,
for the bi-partite density matrix $\rho_{AB}$. When $B(t)=\sqrt{2-\sqrt{2}}$,
the two-qubit state corresponds to a maximally entangled state, and for a
separable state, $B(t)=0$.

###  2.2 Trace Distance Discord

Trace distance discord is another measure of quantum correlation which will be
used in our work. It quantifies quantum correlations based on Schatten 1-norm
of a given state $\rho$, from the zero discord classical quantum state
$\rho_{\mathrm{CQ}}$ :

$\mathscr{D}(\rho)=\min_{\rho_{\mathrm{CQ}}\in\Delta_{0}}\left\|\rho-\rho_{\mathrm{CQ}}\right\|_{1},$
(3)

where $\Delta_{0}$ is the set of classical-quantum states having zero discord
and $\mathcal{|}{X}\|_{1}=$
$\operatorname{Tr}\left(\sqrt{\mathcal{X}^{\dagger}\mathcal{X}}\right)$ is the
trace norm of an operator $\mathcal{X}$. The zero-discord classical-quantum
states $\rho_{{CQ}}$ can be written as

$\rho_{\mathrm{CQ}}=\sum_{i=1}^{2}p_{i}\Pi_{i}^{P}\otimes\rho_{i}^{Q},$ (4)

where $\left\\{p_{i}\right\\}$ is the probability distributions with $0\leq
p_{i}\leq 1;\left\\{\Pi_{i}^{A}\right\\}$ represents a set of orthogonal
projectors corresponding to the subsystem $P$ and $\rho_{i}^{Q}$ is the
reduced density matrix of the subsystem $Q$. The closed expression of trace
distance discord for a two-qubit $X$ state is given by [13]

$\mathscr{D}\left(\rho_{PQ}\right)=\sqrt{\frac{\mathcal{R}_{11}^{2}\mathcal{R}_{\max}^{2}-\mathcal{R}_{22}^{2}\mathcal{R}_{\mathrm{min}}^{2}}{\mathcal{R}_{\max}^{2}-\mathcal{R}_{\min}^{2}+\mathcal{R}_{11}^{2}-\mathcal{R}_{22}^{2}}}.$
(5)

We have written the above expression in the Fano-Bloch representation [43],
using which one can write the density matrix $\rho_{PQ}^{\prime}$ as

$\rho_{PQ}^{\prime}=\frac{1}{4}\sum_{\alpha,\beta=0}^{3}\mathcal{R}_{\alpha\beta}\sigma_{\alpha}\otimes\sigma_{\beta},$
(6)

where,
$\mathcal{R}_{\alpha\beta}=\operatorname{Tr}\left(\rho_{PQ}^{\prime}\sigma_{\alpha}\otimes\sigma_{\beta}\right)$.
We know that a bonafide correlation measure is invariant under local
transformations, so through suitable local unitary transformations [13], the
non zero components $\mathcal{R}_{\alpha\beta}$ are given by

$\begin{array}[]{ll}\mathcal{R}_{00}=\operatorname{Tr}\rho=1,&\mathcal{R}_{03}=1-2\left(\rho_{22}+\rho_{44}\right),\quad\mathcal{R}_{30}=1-2\left(\rho_{33}+\rho_{44}\right)\\\
\mathcal{R}_{11}=2\left(\left|\rho_{23}\right|+\left|\rho_{14}\right|\right),&\mathcal{R}_{22}=2\left(\left|\rho_{23}\right|-\left|\rho_{14}\right|\right),\quad\mathcal{R}_{33}=1-2\left(\rho_{22}+\rho_{33}\right)\end{array}$
(7)

with
$\mathcal{R}_{\max}^{2}=\max\left\\{\mathcal{R}_{33}^{2},\mathcal{R}_{22}^{2}+\mathcal{R}_{30}^{2}\right\\}$
and
$\mathcal{R}_{\min}^{2}=\min\left\\{\mathcal{R}_{11}^{2},\mathcal{R}_{33}^{2}\right\\}$.

###  2.3 Local Quantum Uncertainty

The uncertainty principle limits our capacity to estimate with perfect
certainty the measurement outcomes of two incompatible observables at the same
time. The uncertainty relations are related to the unique characteristics of
quantum mechanics like non-locality, and quantum correlations [15, 44]. Local
quantum uncertainty is a quantum correlation measure that captures the minimal
quantum uncertainty in a quantum state as a result of local measurements on
one of the subsystems of a composite quantum system. It is a discord type
measure that offers easier computability, with exact expressions available for
qubit-qudit systems compared to the discord measures, which involves a
difficult step of optimization over all possible projective measurements, and
closed-form is available for just a few forms of two-qubit states [14]. The
total uncertainty caused by the effect of measuring a single observable $K$,
on a quantum state $\rho$ is given by variance,

$\operatorname{Var}(\rho,H)=\operatorname{Tr}(\rho
K^{2})-(\operatorname{Tr}(\rho K))^{2}.$ (8)

Both classical and quantum counterparts have their contribution in this
uncertainty. Wigner and Yanase developed the idea of skew information to
calculate the quantum counterpart of variance [45]. The skew information of a
bipartite density matrix $\rho$ is given by

$I(\rho,K)=-\frac{1}{2}\operatorname{Tr}([\sqrt{\rho},K]^{2}),$ (9)

where [.] is the commutator. It is worth noting that, unlike variance, Wigner-
Yanase skew information (WYSI) is unaffected by classical mixing. Girolami et
al. defined a measure to quantify quantum correlation based on the definition
of WYSI [14]. It possesses all of the necessary features for quantifying
quantum correlations for a bipartite system. This measure is termed as ‘Local
Quantum Uncertainty’ and is evaluated by minimizing the WYSI over the local
observables of subsystem $A$,

$\mathcal{Q}(\rho)=\min_{K_{A}}\mathcal{I}\left(\rho,K_{A}\otimes
I_{B}\right),$ (10)

where $K_{A}$ is an observable operating on subsystem $A$. The expression of
$\mathrm{LQU}$ is given by

$\mathcal{Q}(\rho)=1-\max\left[\lambda_{1},\lambda_{2},\lambda_{3}\right],$
(11)

where $\lambda_{i}$ ’s are eigenvalues of the $3\times 3$ matrix $M$ having
the elements

$[M]_{ij}\equiv\operatorname{tr}\left\\{\sqrt{\rho}\left(\sigma_{i}\otimes
I_{B}\right)\sqrt{\rho}\left(\sigma_{j}\otimes I_{B}\right)\right\\},$ (12)

where $i,j$ varies from $1,2,3$ and $\sigma_{i}$ ’s are Pauli matrices.

In case of pure bipartite states, this correlation measure reduces to linear
entropy of the reduced density matrix of the composite system. It becomes zero
in the case of classically correlated states and is invariant under local
unitary operations.

##  3 The Physical Model

We consider a model in which two effective two-level atoms (atom $P$ and atom
$Q$) are interacting locally with two independent bosonic reservoirs
$\Omega_{1}$ and $\Omega_{2}$, respectively in the presence of the Stark
effect at zero temperature [41]. With the presence of the Stark effect, we can
consider this as a qubit system interacting with their environment’s field
through degenerate two-photon transitions of frequency $2\omega_{j}$ between
the ground state $\ket{0}$ and excited state $\ket{1}$. This two photon
transition is mediated by an intermediate level $\ket{i}$, having energy in
between the ground state and excited state. The parameters $\eta$ and $\xi$
characterize the Stark shifts of two levels of each atom owing to virtual
transitions to the intermediate level. The two-mode environmental field
oscillates with frequencies $\omega_{j_{1}}$ and $\omega_{j_{2}}$ while
interacting with two two-level atoms. In rotating wave approximation (RWA),
the effective Hamiltonian of the system is given by

$\displaystyle\hat{H}_{\mathrm{eff}}=$
$\displaystyle\omega_{0}\left(\hat{S}_{+}^{P}\hat{S}_{-}^{P}+\hat{S}_{+}^{Q}\hat{S}_{-}^{Q}\right)+\sum_{j_{1}}\omega_{j_{1}}\hat{a}_{j_{1}}^{\dagger}\hat{a}_{j_{1}}+\sum_{j_{2}}\omega_{j_{2}}\hat{a}_{j_{2}}^{\dagger}\hat{a}_{j_{2}}+\sum_{j_{1}}\gamma_{j_{1}}\left(\hat{a}_{j_{1}}^{\dagger^{2}}\hat{S}_{-}^{P}+\hat{a}_{j_{1}}^{2}\hat{S}_{+}^{P}\right)$
$\displaystyle+\sum_{j_{1}}\hat{a}_{j_{1}}^{\dagger}\hat{a}_{j_{1}}\left(\eta_{j_{1}}\hat{S}_{-}^{P}\hat{S}_{+}^{P}+\xi_{j_{1}}\hat{S}_{+}^{P}\hat{S}_{-}^{P}\right)+\sum_{j_{2}}\gamma_{j_{2}}\left(\hat{a}_{j_{2}}^{\dagger^{2}}\hat{S}_{-}^{Q}+\hat{a}_{j_{2}}^{2}\hat{S}_{+}^{Q}\right)$
$\displaystyle+\sum_{j_{2}}\hat{a}_{j_{2}}^{\dagger}\hat{a}_{j_{2}}\left(\eta_{j_{2}}\hat{S}_{-}^{Q}\hat{S}_{+}^{Q}+\xi_{j_{2}}\hat{S}_{+}^{Q}\hat{S}_{-}^{Q}\right).$
(13)

Here, the operator $\hat{a}^{\dagger}_{j}$ and $\hat{a}_{j}$ are the creation
and annihilation operators of the $j^{th}$ mode of environment and
$\hat{S}^{P,Q}_{\pm}$ represent the raising and lowering operators for the
qubits describing atoms P and Q. The coefficients $\gamma_{j_{1}}$ and
$\gamma_{j_{2}}$ are coupling coefficients describing the two-photon strength
of the qubit for the environmental modes $j_{1}$ and $j_{2}$ respectively, and
the parameters $\eta$ and $\xi$ are Stark shift coefficients [46]. In the
interaction picture, the above Hamiltonian takes the following form:

$\displaystyle\hat{H}_{\mathrm{int}}=$
$\displaystyle\sum_{j_{1}}\gamma_{j_{1}}\left(\hat{a}_{j_{1}}^{\hat{2}}\hat{S}_{-}^{P}\mathrm{e}^{-\mathrm{i}\left(\omega_{0}-2\omega_{j_{1}}\right)t}+\hat{a}_{j_{1}}^{2}\hat{S}_{+}^{P}e^{i\left(\omega_{0}-2\omega_{j_{1}}\right)t}\right)+\sum_{j_{1}}\hat{a}_{j_{1}}^{\dagger}\hat{a}_{j_{1}}\left(\eta_{j_{1}}\hat{S}_{-}^{P}\hat{S}_{+}^{P}+\xi_{j_{1}}\hat{S}_{+}^{P}\hat{S}_{-}^{P}\right)$
$\displaystyle+\sum_{k_{2}}\gamma_{j_{2}}\left(\hat{a}_{j_{2}}^{\dagger^{2}}\hat{S}_{-}^{Q}\mathrm{e}^{-\mathrm{i}\left(\omega_{0}-2\omega_{j_{2}}\right)t}+\hat{a}_{j_{2}}^{2}\hat{S}_{+}^{Q}\right)e^{i\left(\omega_{0}-2\omega_{j_{2}}\right)t}+\sum_{j_{2}}\hat{a}_{j_{2}}^{\dagger}\hat{a}_{j_{2}}\left(\eta_{j_{2}}\hat{S}_{-}^{Q}\hat{S}_{+}^{Q}+\xi_{j_{2}}\hat{S}_{+}^{Q}\hat{S}_{-}^{Q}\right).$
(14)

We take the initial state of the system as an entangled state in the form:

$|\Phi(0)\rangle=\left(x|0\rangle_{P}|1\rangle_{Q}+\sqrt{1-x^{2}}|1\rangle_{P}|0\rangle_{Q}\right)\otimes\left|0_{j_{1}}\right\rangle_{\Omega_{1}}\mid
0_{j_{2}}\rangle_{\Omega_{2}},$ (15)

with $x\in[0,1]$, where $\left|0_{j_{1}}\right\rangle_{\Omega_{1}}$ and
$\left|0_{j_{2}}\right\rangle_{\Omega_{2}}$ indicate the vacuum state of the
environment of P and Q, and $\ket{0}$ and $\ket{1}$ are the ground state and
excited state of the two-level atoms. The time evolution of the total system
for time $t>0$ can be written as

$\begin{split}|\Phi(t)\rangle=(b_{1}(t)\ket{1}_{P}\ket{0}_{Q}+b_{2}(t)\ket{0}_{P}\ket{1}_{Q})\ket{0_{j_{1}}}_{\Omega_{1}}\ket{0_{j2}}_{\Omega_{2}}+\sum_{k_{1}}b_{j_{1}}(t)|0\rangle_{P}|0\rangle_{Q}\left|2_{j_{1}}\right\rangle_{\Omega_{1}}\left|0_{j_{2}}\right\rangle_{\Omega_{2}}\\\
+\sum_{j_{2}}b_{j}(t)|0\rangle_{P}|O\rangle_{Q}\left|0_{j_{1}}\right\rangle_{\Omega_{1}}\left|2_{j_{2}}\right\rangle_{\Omega_{2}},\end{split}$
(16)

where $\left|2_{j}\right\rangle_{\Omega_{j}}$ implies the presence of two
photons in the $j^{th}$ mode. So, each of the environment has two states
$\left|0_{j}\right\rangle$ and $\left|2_{j}\right\rangle$. We get the
following differential equations of probability amplitude from the Schrodinger
equation using Eqs. 3 and 16 as

$\dot{b}_{j}(t)=-\mathrm{i}\sqrt{2}\sum_{j_{m}}\gamma_{j_{m}}b_{j_{m}}(t)\mathrm{e}^{\mathrm{i}\left(\omega_{0}-\omega_{j}\right)t}$
(17)

and

$\dot{b}_{j_{m}}(t)=-\mathrm{i}\sqrt{2}\gamma_{j_{m}}^{*}b_{j}(t)\mathrm{e}^{-\mathrm{i}\left(\omega_{0}-\omega_{j_{m}}\right)t}-2\mathrm{i}\eta_{j_{m}}b_{j_{m}}(t),m=1,2.$
(18)

The parameters $\xi_{j_{1}}$ and $\xi_{j_{2}}$ do not affect the dynamics of
the quantum correlations because
$\hat{a}_{j_{1}}^{\dagger}\hat{a}_{j_{1}}\xi_{j_{1}}\hat{S}_{+}^{P}\hat{S}_{-}^{P}|\Phi(t)\rangle$
and
$\hat{a}_{j_{2}}^{\dagger}\hat{a}_{j_{2}}\xi_{j_{2}}\hat{S}_{+}^{Q}\hat{S}_{-}^{Q}|\Phi(t)\rangle$
becomes zero [41]. This happens due to the nature of $|\Phi(t)\rangle$ which
is a result of assuming the initial state of the environment to be in ground
state. We observe that the parameters $\xi_{j_{1}}$ and $\xi_{j_{2}}$ do not
play any role on the dynamics and further on the quantum correlations of the
model considered in our work. If we change the initial state of the
environments, we can observe the role played by both the Stark shift
parameters. After integrating Eq. 18, and using Eq. 17, we obtain the
following differential equation:

$\dot{b}_{j}(t)=-2\int_{0}^{t}\mathrm{~{}d}t^{\prime}g\left(t-t^{\prime}\right)b_{j}\left(t^{\prime}\right),$
(19)

where the correlation function can expressed as

$g\left(t-t^{\prime}\right)=\int\mathrm{d}\omega_{j}J\left(\omega_{j}\right)\exp\left[\mathrm{i}\left(\omega_{0}-2\omega_{k}-2\eta\right)\left(t-t^{\prime}\right)\right].$
(20)

Here we consider $\eta_{j_{1}}=\eta_{j_{2}}=\eta$ and
$\omega_{j_{1}}=\omega_{j_{2}}=\omega_{j}$. The discrete sum over the
reservoir modes can be approximated using the integral form when there are a
large number of reservoir modes,
$\sum_{k}\left|\gamma_{j}\right|^{2}\rightarrow\int\mathrm{d}\omega_{j}J\left(\omega_{j}\right)$,
where $J\left(\omega_{j}\right)$ represents the electromagnetic field’s
spectral density within a lossy cavity.

The reservoir’s spectral density is assumed to be Lorentzian [47],

$J\left(\omega_{j}\right)=\frac{1}{2\pi}\frac{\gamma_{0}\Lambda^{2}}{\left(\omega_{0}-2\omega_{j}\right)^{2}+\Lambda^{2}},$
(21)

where the parameter $\Lambda$ indicates the width of Lorentzian distribution
and the parameter $\omega_{0}$ is the transition frequency of the atom. The
relationship between reservoir correlation time $\tau_{\Omega}$ and $\Lambda$
is $\tau_{\Omega}\approx\Lambda^{-1}$. The decay rate of the excited atom is
represented by $\gamma_{0}$, and is directly related with the relaxation time
$\tau_{s}\approx\gamma_{0}^{-1}$ [48]. The correlation function
$g\left(t-t^{\prime}\right)$ obtained using Eqs. 20 and 21 is given by

$g\left(t-t^{\prime}\right)=\frac{\gamma_{0}\Lambda}{2}\exp\left[-(\Lambda+2\mathrm{i}\eta)\left(t-t^{\prime}\right)\right].$
(22)

We can find the exact solution $b_{1}(t)$ and $b_{2}(t)$ by making use of
Laplace’s method on Eq. 19, and using above correlation function:

$b_{1}(t)=x\phi(t),\quad b_{2}(t)=\sqrt{1-x^{2}}\phi(t).$ (23)

where,

$\phi(t)=e^{-\frac{(\Lambda+2i\eta)t}{2}}\left[\cosh\left(\frac{\sigma
t}{2}\right)+\frac{\Lambda+2\mathrm{i}\eta}{\sigma}\sinh\left(\frac{\sigma
t}{2}\right)\right]$ (24)

, and

$\sigma=\sqrt{-4\gamma_{0}\Lambda+(\Lambda+2\mathrm{i}\eta)^{2}}.$ (25)

If we write, $\Lambda=n_{1}\gamma_{0}$ and $\eta=n_{2}\gamma_{0}$ where
$n_{1}$ and $n_{2}$ are some positive numbers. By putting in the values of
Stark shift coefficients in terms of $\gamma_{0}$,

$\sigma=\sqrt{-4n_{1}\gamma_{0}+(n_{1}+2\mathrm{i}n_{2})^{2}}\gamma_{0}=\sigma_{1}\gamma_{0},$
(26)

where,

$\sigma_{1}=\sqrt{-4n_{1}\gamma_{0}+(n_{1}+2\mathrm{i}n_{2})^{2}}.$ (27)

Thus,

$\phi(\gamma_{0}t)=e^{-\frac{(n_{1}+2in_{2})\gamma_{0}t}{2}}\left[\cosh\left(\frac{\sigma_{1}\gamma_{0}t}{2}\right)+\frac{n_{1}+2\mathrm{i}n_{2}}{\sigma_{1}}\sinh\left(\frac{\sigma_{1}\gamma_{0}t}{2}\right)\right].$
(28)

We note that the scaled time is given by ’$\tau=\gamma_{0}t$’, The behavior of
$b_{1}(t)$ and $b_{2}(t)$ is differentiated by two regimes depending on the
weak or strong coupling between the open system and environment [49]. In the
weak coupling regime, we have, $\gamma_{0}<\Lambda/2$ and
$\tau_{s}<2\tau_{\Omega}$. The relaxation time in this region is greater than
the reservoir’s correlation time, and we observe a decaying process as time
progresses. This behavior of the qubit reservoir system is known as Markovian.
On the strong coupling regime, we have $\gamma_{0}>\Lambda/2$ and
$\tau_{s}>2\tau_{\Omega}$ implying the relaxation time is less than the qubit-
reservoir correlation time. The dynamics in this region are known as non-
Markovian dynamics, in which memory effect comes into play, and the revival of
quantum correlation is observed with damped oscillation. In this work, we
investigate both the Markovian and non-Markovian dynamics by considering the
Stark shift effect.

###  3.1 Dynamics of Bures Distance Entanglement

The reduced density matrix of the system (consisting of atoms P and Q) in the
atomic basis
$\left\\{|1\rangle_{P}|1\rangle_{Q},|1\rangle_{P}|0\rangle_{Q},|0\rangle_{P}|1\rangle_{Q},|0\rangle_{P}|0\rangle_{Q}\right\\}$
is given by

$\rho(t)=\left(\begin{array}[]{cccc}0&0&0&0\\\
0&\left|b_{1}(t)\right|^{2}&b_{1}(t)b_{2}^{*}(t)&0\\\
0&b_{1}^{*}(t)b_{2}(t)&\left|b_{2}(t)\right|^{2}&0\\\
0&0&0&1-\left|b_{1}(t)\right|^{2}-\left|b_{2}(t)\right|^{2}.\end{array}\right)$
(29)

Using the procedure of finding concurrence as discussed earlier, for the above
density matrix, one finds

$C(t)=2\left|b_{1}(t)b_{2}^{*}(t)\right|,$

where the Bures distance entanglement measure is given by

$B(t)=\sqrt{2-\sqrt{2+2\sqrt{1-C(t)^{2}}}}$

(a)

(b)

Figure 1: Dynamics of Bures distance entanglement in (a) Markovian reservoir
for $\Lambda=10\gamma_{0}$ with Stark shift values
$\eta=0,5\gamma_{0},10\gamma_{0},15\gamma_{0}$ (b) non-Markovian reservoir for
$\Lambda=0.1\gamma_{0}$, with
$\eta=0,0.5\gamma_{0},1.0\gamma_{0},1.5\gamma_{0}$.

(a)

(b)

Figure 2: The surface plot of BDE with scaled time $\gamma_{0}t$ and state
parameter $x$ in (a) in Markovian reservoir for $\Lambda=10\gamma_{0}$ with
the value of Stark shift ($\eta=15\gamma_{0}$) and (b) in non-Markovian
reservoir for $\Lambda=0.1\gamma_{0}$, with the value of Stark shift parameter
($\eta=0.2\gamma_{0}$).

In Fig.1, we show the temporal dynamics of the Bures distance entanglement
(BDE) in the presence and absence of Stark shift for both Markovian and non-
Markovian reservoirs. In the Markovian regime, we observe the exponential
decay of BDE. However, in the non-Markovian regime, we observe periodic death
and revival of BDE. For a particular value of $\Lambda$, BDE is protected more
efficiently with the increase of Stark shift parameter $\eta$. Specifically,
in non-Markovian regime, the effect of $\eta$ is more significant. The surface
plot of BDE with scaled time and state parameter x is depicted in Fig.2. We
observe that for maximally entangled input state ($x=\frac{1}{\sqrt{2}})$, BDE
has maximum value of $\sqrt{2-\sqrt{2}}$.

###  3.2 Dynamics of Trace Distance Discord

We can compute trace distance discord for the X state using the expression 5.

(a)

(b)

Figure 3: The time evolution of TDD with scaled time $\gamma_{0}t$ with state
parameter $x=\frac{1}{\sqrt{2}}$ in (a) Markovian reservoir for
$\Lambda=10\gamma_{0}$ with the values of Stark shift
($\eta=0,5\gamma_{0},10\gamma_{0},15\gamma_{0}$) and (b) non Markovian
reservoir for $\Lambda=0.1\gamma_{0}$ with
$\eta=0,0.5\gamma_{0},1.0\gamma_{0},1.5\gamma_{0}$ respectively.

(a)

(b)

Figure 4: The surface plot of TDD with scaled time $\gamma_{0}t$ and state
parameter $x$ in (a) in Markovian reservoir for $\Lambda=10\gamma_{0}$, with
the value of Stark shift ($\eta=15\gamma_{0}$) and (b) in non Markovian
reservoir for $\Lambda=0.1\gamma_{0}$, with the value of Stark shift
($\eta=0.2\gamma_{0}$).

The time variation of TDD in the presence of Stark shift is depicted in Fig.3.
TDD sustains for a longer period of time with an increase in the Stark shift
parameter $(\eta)$.

The surface plots of TDD with scaled time and state parameter are shown in
Fig. 4. It is evident from the plot that for maximally entangled input state
($x=\frac{1}{\sqrt{2}})$, TDD has maximum value $1$.

###  3.3 Dynamics of Local Quantum Uncertainty

The calculation of LQU requires one to find the eigenvalues of the matrix $M$:

$[M]_{ij}\equiv\operatorname{tr}\left\\{\sqrt{\rho}\left(\sigma_{i}\otimes
I_{B}\right)\sqrt{\rho}\left(\sigma_{j}\otimes I_{B}\right)\right\\},$

with elements,

$M_{11}=M_{22}=\frac{2|b_{1}(t)|^{2}(1-|b_{1}(t)|^{2}-|b_{2}(t)|^{2})^{1/2}}{(|b_{1}(t)|^{2}+|b_{2}(t)|^{2})^{1/2}},$
(30)
$M_{33}=\frac{(|b_{1}(t)|^{2}+|b_{2}(t)|^{2}-4|b_{1}(t)|^{2}|b_{2}(t)|^{2})}{(|b_{1}(t)|^{2}+|b_{2}(t)|^{2})}.$
(31)

The elements of the diagonal matrix are the eigenvalues themselves, the
expression of LQU is given by

$\mathcal{Q}(\rho)=1-\max(M_{11},M_{22},M_{33}).$ (32)

(a)

(b)

Figure 5: The time evolution of LQU with scaled time $\gamma_{0}t$ with
$x=\frac{1}{\sqrt{2}}$ in (a) Markovian reservoir for $\Lambda=10\gamma_{0}$
with the values of Stark shift
($\eta=0,5\gamma_{0},10\gamma_{0},15\gamma_{0}$) and (b) non-Markovian
reservoir for $\Lambda=0.1\gamma_{0}$, with
$\eta=0,0.5\gamma_{0},1.0\gamma_{0},1.5\gamma_{0}$, respectively.

(a)

(b)

Figure 6: The surface plot of LQU with scaled time $\gamma_{0}t$ and state
parameter $x$ in (a) in Markovian reservoir for $\Lambda=10\gamma_{0}$ with
the value of Stark shift ($\eta=15\gamma_{0}$) and (b) in non Markovian
reservoir for $\Lambda=0.1\gamma_{0}$ with the value of Stark shift
($\eta=0.2\gamma_{0}$).

In Fig. 5, we have shown the time variation of local quantum uncertainty in
the presence of Stark shift. Similar to previous cases, in the Markovian
region, we see the exponential decay of TDD. However, we can observe from the
dynamics of LQU that it is accompanied with a sudden change in its behavior,
which is in contrast to the dynamics of BDE and TDD. This behavior is a result
of the maximization process involved in the step of evaluating LQU and happens
at a time when $M_{11}=M_{33}$. It must be noted that the sudden change
phenomenon of quantum correlations is associated with various physical
phenomena like classical and quantum phase transitions [50].

In the non-Markovian region, we observe a damped oscillatory behavior. For
each value of $\Lambda$, LQU is enhanced with the increment of Stark shift
parameter $\eta$. In the case non-Markovian regime, the significant effect of
$\eta$ is observed similar to BDE and TDD. The surface plot of LQU with scaled
time and state parameter is shown in Fig.6. From the surface plot, we observe
a region of sudden change as result of the maximization process.

##  4 Dynamics of quantum correlation when the mode of environment has one
photon initially

In the previous Section 3, we have considered the scenario in which the
environment of the atoms is in vacuum state, suggesting the absence of photons
in the initial state of the environment. We showed that this assumption
simplifies to the case where the dynamics of the quantum correlations show
dependence on only one of the Stark shift parameters, i,e. ’$\eta$’ and with
the increase of this parameter, we observe an enhancement of quantum
correlations. In this section, we deal with the case when both the environment
corresponding to the atoms are in the excited state having one photon each.
Now, we suppose that the initial state of the system environment is
$|\Phi^{\prime}(0)\rangle$, where $j^{th}$ mode of each atom’s environment has
one photon:

$|\Phi^{\prime}(0)\rangle=\left(x|0\rangle_{P}|1\rangle_{Q}+\sqrt{1-x^{2}}|1\rangle_{P}|0\rangle_{Q}\right)\otimes\left|1_{j_{1}}\right\rangle_{\Omega_{1}}\mid
1_{j_{2}}\rangle_{\Omega_{2}},$ (33)

where $\ket{1_{j_{m}}}_{\Omega_{m}}$ indicates that there is one photon in
$j^{th}$ mode of the $m^{th}$ environment. The joint state of the system and
environment at time t can be written as

$\begin{split}|\Phi^{\prime}(t)\rangle=(b^{\prime}_{1}(t)\ket{1}_{P}\ket{0}_{Q}+b^{\prime}_{2}(t)\ket{0}_{P}\ket{1}_{Q})\otimes\ket{1_{j_{1}}}_{\Omega_{1}}\ket{1_{j2}}_{\Omega_{2}}+\sum_{j_{1}}b^{\prime}_{j_{1}}(t)|0\rangle_{P}|0\rangle_{Q}\left|3_{j_{1}}\right\rangle_{\Omega_{1}}\left|1_{j_{2}}\right\rangle_{\Omega_{2}}\\\
+\sum_{j_{2}}b^{\prime}_{j}(t)|0\rangle_{P}|O\rangle_{Q}\left|1_{j_{1}}\right\rangle_{\Omega_{1}}\left|3_{j_{2}}\right\rangle_{\Omega_{2}},\end{split}$
(34)

where $\ket{3_{j_{m}}}_{\Omega_{m}}$ indicates that there are three photons in
$j^{\text{th}}$ mode of the $m^{\text{th}}$ environment. In the interaction
picture, the effective Hamiltonian of the system is given by Eq. 3. The time
dependent Schrödinger equation in the interaction picture,
$\hat{H}_{int}\ket{\Phi^{\prime}(t)}=\mathrm{i}\frac{\partial}{\partial
t}\ket{\Phi^{\prime}(t)}$ gives the following differential equations:

$\dot{b^{\prime}}_{j}(t)=-\mathrm{i}\sqrt{6}\sum_{j_{m}}\gamma_{j_{m}}b^{\prime}_{j_{m}}(t)\mathrm{e}^{\mathrm{i}\left(\omega_{0}-2\omega_{j_{m}}\right)t}-3\mathrm{i}\eta{b^{\prime}}_{j}(t)-\mathrm{i}\xi{b^{\prime}}_{j}(t)$
(35)

, and

$\dot{b^{\prime}}_{j_{m}}(t)=-\mathrm{i}\sqrt{6}\gamma_{j_{m}}^{*}b^{\prime}_{j}(t)\mathrm{e}^{-\mathrm{i}\left(\omega_{0}-2\omega_{j_{m}}\right)t}-4\mathrm{i}\eta
b^{\prime}_{j_{m}}(t),m=1,2.$ (36)

Where, for simplicity we have assumed that, $\eta_{j_{1}}=\eta_{j_{2}}=\eta$ ,
$\xi_{j_{1}}=\xi_{j_{2}}=\xi$ and $\omega_{j_{1}}=\omega_{j_{2}}=\omega_{j}$.
In the process of solving the Schrödinger equation, we note that
$\hat{a}_{j_{1}}^{\dagger}\hat{a}_{j_{1}}\xi_{j_{1}}\hat{S}_{+}^{P}\hat{S}_{-}^{P}|\Phi^{\prime}(t)\rangle$
and
$\hat{a}_{j_{2}}^{\dagger}\hat{a}_{j_{2}}\xi_{j_{2}}\hat{S}_{+}^{Q}\hat{S}_{-}^{Q}|\Phi^{\prime}(t)\rangle$
are nonzero terms. So, the dynamics of quantum correlations also depend on the
stark shift parameter $\xi_{1}$ and $\xi_{2}$, when the initial mode of the
environment is not at vacuum state. We got the following differential equation
of probability amplitude using the same approach as before:

$\dot{b^{\prime}}_{j}(t)=-3\gamma_{0}\Lambda\int_{0}^{t}\exp{[(\Lambda+4\mathrm{i}\eta)(t-t^{\prime})]}b^{\prime}_{j}\left(t^{\prime}\right)\mathrm{~{}d}t^{\prime}-\mathrm{i}(\xi+3\eta){b^{\prime}}_{j}(t).$
(37)

In deriving the above differential equation, the discrete sum over the
reservoir modes is replaced by the integral form,
$\sum_{k}\left|\gamma_{j}\right|^{2}\rightarrow\int\mathrm{d}\omega_{j}J\left(\omega_{j}\right)$,
where $J\left(\omega_{j}\right)$ represents the Lorentzian spectral density of
electric field given in Eq. 21. We can find the exact solution
$b^{\prime}_{1}(t)$ and $b^{\prime}_{2}(t)$ by using Laplace technique on Eq.
37. We obtain the following solutions:

$b^{\prime}_{1}(t)=x\Theta(t),\quad
b^{\prime}_{2}(t)=\sqrt{1-x^{2}}\Theta(t),$ (38)

where,

$\Theta(t)=e^{-\frac{(\Lambda+\mathrm{i}(\xi+7\eta)t}{2}}\left[\cosh\left(\frac{\sigma
t}{2}\right)+\frac{\Lambda+\mathrm{i}(\eta-\xi)}{\sigma}\sinh\left(\frac{\sigma
t}{2}\right)\right],$ (39)

with

$\sigma=\sqrt{-12\gamma_{0}\Lambda-4\mathrm{i}(\Lambda+4\mathrm{i}\eta)(\xi+3\eta)+(\Lambda+\mathrm{i}(\xi+7\eta))^{2}}.$
(40)

We can write the above state in terms of scaled time ’$\gamma_{0}t$’, if we
write, $\Lambda=l_{1}\gamma_{0}$, $\xi=l_{2}\gamma_{0}$ and
$\eta=l_{3}\gamma_{0}$ where $l_{1},l_{2}$ and $l_{3}$ are some numbers. The
state $\Theta(t)$ in terms of scaled time is given by

$\Theta(\gamma_{0}t)=e^{-\frac{(l_{1}+\mathrm{i}(l_{2}+7l_{3})\gamma_{0}t}{2}}\left[\cosh\left(\frac{\sigma_{1}t}{2}\right)+\frac{l_{1}+\mathrm{i}(l_{3}-l_{2})}{\sigma_{1}}\sinh\left(\frac{\sigma_{1}\gamma_{0}t}{2}\right)\right],$
(41)

where,

$\sigma_{1}=\sqrt{-12l_{1}-4\mathrm{i}(l_{1}+4\mathrm{i}l_{3})(l_{2}+3l_{3})+(l_{1}+\mathrm{i}(l_{2}+7l_{3}))^{2}}.$
(42)

The probability amplitude $b^{\prime}_{1}(t)$ and $b^{\prime}_{2}(t)$ exhibit
different behaviours based on how well a system is coupled to its
surroundings.

Now, in the presence of the Stark shift effect, we examine the Markovian and
non-Markovian dynamics of Bures distance entanglement, trace distance discord,
and local quantum uncertainty, which are shown in Figs. 7a, 7b, 7c and Figs.
7d, 7e, 7f. We have numerically computed the BDE, TDD, and LQU for an
initially maximally entangled state and plotted them against scaled time
$\gamma_{0}t$ in Fig. 7 for various Stark shift parameters, similar to the
vacuum environment field taken into consideration in section 3. We found the
dynamics of the quantum correlations depend on both the stark shift parameter
$\eta$ and $\xi$. The term $(\Lambda+\mathrm{i}(\eta-\xi))/\sigma$ in
$\Theta(t)$ plays a significant role in determining the positive effect of
Stark shift on the dynamics of quantum correlation measures. From the plots
below, we can see that by increasing the value of the difference of the Stark
shift parameters $(\eta-\xi)$, the quantum correlations present in the system
can be protected effectively and on decreasing the value of $(\eta-\xi)$, we
observe quick decay of BDE, TDD and LQU.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 7: The time evolution of BDE, TDD and LQU with scaled time
$\gamma_{0}t$ with state parameter $x=\frac{1}{\sqrt{2}}$. Subfigures 7a, 7b
and 7c represent Markovian dynamics for $\Lambda=10\gamma_{0}$ with the
difference of Stark shift parameter
$(\eta-\xi)=0,5\gamma_{0},10\gamma_{0},15\gamma_{0}$ and subfigures 7d, 7e and
7f represent non Markovian dynamics for $\Lambda=0.1\gamma_{0}$ with
$(\eta-\xi)=0,1.5\gamma_{0},3.0\gamma_{0},10\gamma_{0}$ respectively.

The quantum correlations between two atoms in this case decay similarly as in
the previous scenario when the environments were in their vacuum state,
however, the correlations diminish rapidly when the environments are in the
excited state initially. It is important to note that when spectral width
($\Lambda$) decreases, indicating strong non-Markovianity, quantum
correlations persist for a longer period of time. For this initial condition
also, we observe sudden change behaviour in LQU, which is absent in BDE and
TDD.

##  5 Conclusion

In conclusion, we have investigated the dynamics of quantum correlations based
on Bures norm, Schatten-1 norm, and local quantum uncertainty for two two-
level atoms coupled to dissipative reservoirs at zero temperature in the
presence of the Stark effect. It is shown that the quantum correlations can be
adequately protected by tuning the magnitude of the Stark shift in the
dissipative reservoirs. The quantum correlations in Markovian reservoirs
dissipate quickly in comparison to non-Markovian reservoirs. In non-Markovian
reservoirs, information flows from the environment back to the system. As a
result, the correlations steadily deteriorate. In the presence of the Stark
effect, quantum correlations is considerably enhanced by tweaking non-
Markovianity or lowering the spectral width. The quantum correlation measures
decay with small oscillations for low values of the Stark shift. For high
values of the Stark shift, the influence of dissipation is reduced by the
Stark effect in non-Markovian reservoirs. It is worth mentioning that quantum
correlations quantified by local quantum uncertainty are accompanied by a
sudden change phenomenon which is not the case with other correlations based
on Bures norm and trace norm. We have looked at the dynamics under two
distinct initial conditions of the environment. We have assumed that the
initial state of the environment is in the ground state in the first case and
the first excited state in the second case. Compared to the first initial
condition, which shows the role of only one of the Stark shift parameters, the
second initial condition demonstrates the role of both the Stark shift
parameters. Our findings have implications for sustaining and controlling
quantum correlations beyond entanglement in experiments and the possible use
of more general correlation quantifiers in quantum information processing
tasks.

##  6 Acknowledgements

NKC and PKP acknowledge the financial support from DST, India through Grant
No. DST/ICPS/QuST/Theme1/2019/2020-21/01. RS acknowledges institute fellowship
provided by IISER Kolkata.

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