# Dynamical Crossover from Markovian to Non-Markovian dynamics in the strong
coupling regime

Md. Manirul Ali<EMAIL_ADDRESS>Centre for Quantum Science and
Technology, Chennai Institute of Technology, Chennai 600069, India
Chandrashekar Radhakrishnan<EMAIL_ADDRESS>Centre for Quantum
Information, Communication and Computing, Indian Institute of Technology
Madras, Chennai 600036, India  Centre for Quantum Science and Technology,
Chennai Institute of Technology, Chennai 600069, India

(August 29, 2024)

###### Abstract

The transient dynamics of quantum coherence of Gaussian states are
investigated. The state is coupled to an external environment which can be
described by a Fano-Anderson type Hamiltonian. Solving the quantum Langevin
equation, we obtain the Greens functions which are used to compute the time
evolved first and second moments of the quadrature operators. From the
quadrature operator moments, we construct the covariance matrix which is used
to measure the coherence in the system. The coherence is measured using the
relative entropy of coherence measure. We consider three different classes of
spectral densities in our analysis viz, the Ohmic, the sub-Ohmic, and the
super-Ohmic densities. In our work, we study the dynamics of the coherent
state, squeezed state, and displaced squeezed state. For all these states we
observe that when the coupling with the system and the environment is weak,
the coherence monotonically decreases and eventually vanishes in a long time.
Thus all the states exhibit Markovian evolution in the weak coupling limit. In
the strong coupling limit, the dynamics for the initial period is Markovian
and after a certain period, it becomes non-Markovian where we observe an
environmental backaction on the system. Thus in the strong coupling limit, we
observe a dynamical crossover from Markovian nature to non-Markovian behavior.
This crossover is very abrupt under some environmental conditions and for some
parameters of the quantum state. Using a quantum master equation approach we
verify the crossover from the dynamics of the dissipation and fluctuation
parameters and the results endorse those obtained from coherence dynamics.

##  I Introduction

Quantum coherence arises due to the superposition principle and is an
essential property of quantum systems. Basic features of coherence were well
studied through [1], where they were investigated in the context of quantum
optics. But these studies have focused only on the detection of quantum
coherence while a method to estimate coherence was not available. A procedure
to measure coherence using a quantum information theoretic framework was
introduced by Baumgratz et. al. in Ref. [2]. This method is used to estimate
the coherence of finite dimensional systems. This led to some fundamental
results in the field of resource theory of quantum coherence [3, 4, 5, 6, 7].
Subsequently quantum coherence has been studied in the contexts, where the
system is either in a non-inertial frame of reference [8, 9] or where the
system is in contact with an external environment [10, 11, 12, 13]. But most
of these investigations have been carried out on quantum systems with finite
number of degrees of freedom like qubits.

Quantum communication requires a quantum description of the interaction and
propagation of electromagnetic waves. An infinite number of degrees of freedom
with continuous spectrum is needed to describe the electromagnetic waves.
Hence from both theoretical and experimental perspective, continuous variable
(CV) states [14, 15, 16, 17] of infinite dimensional systems are very
important. Continuous variable systems constitute an extremely powerful
resource to quantum information processing. One particular class of continuous
variable states is the Gaussian states [16, 18, 17] for which the quantum
state has a representation in terms of Gaussian functions. To a first
approximation, the ground states of the thermal states of a quantum system is
a Gaussian state. Similarly there are dynamical operations in which quantum
states are transformed to another Gaussian states and these are referred to as
Gaussian operations. The Gaussian operations are linear in nature and any
nonlinear operation can be approximated to a Gaussian operation to a
satisfactory level of accuracy. In light of these reasons, the Gaussian states
occupy a special place in the study of continuous variable systems. The first
quantum resource to be studied extensively in a Gaussian state is its
entanglement. Since a covariance matrix along with the displacement vectors
completely characterizes a Gaussian state, the entanglement can be quantified
using a covariance matrix based approach. To measure the quantum coherence of
Gaussian states a measure based on relative entropy, using covariance matrix
and displacement vectors was introduced in Ref. [19]. The fundamental
postulates of (i) positivity (ii) monotonicity and (iii) convexity were
verified for this relative entropy measure.

In most of the investigations on discrete and continuous variable systems, we
do not consider the effect of the external environment on the system. But in
reality the quantum systems are exposed to an environment, which acts as a
bath and affects the characteristics of the system by constantly interacting
with it. To understand the effects of the environment and incorporate them in
the characterization of the quantum resources, we take an open quantum system
approach. Here we model the environment as a quantum many-body object which is
coupled to the system. Based on the strength of the coupling and the other
characteristics of the system and the environment, there is a dynamical change
in the quantum resource. Depending on the bath spectrum and whether the system
is weakly or strongly coupled to the environment, the system can exhibit a
Markovian or non-Markovian dynamics [20, 21]. A quantum resource can
completely disappear and then reappear a phenomenon known as revival. These
features are dependent not only on the initial properties of the bath and the
system but also on how they are coupled to each other. Understanding this
dynamical change is essential to the fabrication of quantum devices [22, 23,
24, 25, 26, 27, 28, 29]. It provides us a clear idea of the operational time
of the quantum devices, i.e., the time within which the quantum operations
have to be completed before the quantum resources vanish. The time evolution
of the entanglement of finite dimensional systems has been investigated in a
very detailed manner and some unique features like sudden death of
entanglement [30, 31, 32] and entanglement revival [33, 34, 35, 36] due to
environmental back action have been observed. For continuous variable systems,
investigations on the dynamics of entanglement have been done [37, 38, 39]. An
open quantum system study of coherence has been carried out using an atom-
field interaction model [11] and a central spin model [12]. The dynamics of
coherence and its distribution was extensively investigated for the different
bipartite and tripartite states. But both these investigations were carried
out on finite dimensional system. For the continuous variable systems, the
quantum coherence dynamics has not been investigated so far. In this work we
investigate the dynamics of coherence of a single mode Gaussian states viz the
coherent state, the squeezed state and the displaced squeezed state.

The plan of the manuscript is as follows: In Sec. II we give a brief
introduction to the continuous variable quantum state and explain the Gaussian
states in detail. Also we describe the covariance matrix based coherence
measure introduced in Ref. [19]. The description of the Gaussian state in
contact with a non-Markovian environment is described in Sec. III. For the
single mode coherent states the dynamics of quantum coherence is discussed in
Sec. IV. Next we study the effect of non-Markovian environment on squeezed
states in Sec. V. An analysis of coherence in the single mode displaced
squeezed state is shown in Sec. VI. Finally in Sec. VII, we investigate the
crossover from Markovian to non-Markovian dynamics. We present our conclusions
in Sec. VIII.

##  II Gaussian States and Quantification of coherence

###  II.1 Gaussian states

A continuous variable quantum system [14, 15] has an infinite dimensional
Hilbert space in which observables have a continuous eigenspectrum. An example
of a continuous-variable system is the electromagnetic field in which the
quantized radiation modes are represented by the bosonic modes. The tensor
product Hilbert space corresponding to these modes is
$\otimes_{k=1}^{n}H_{k}$, where ‘$n$’ is the number of bosonic modes. For a
given mode ‘$k$’ of the bosonic field, the operators $a_{k}^{\dagger}$ and
$a_{k}$ are the corresponding creation and annihilation operators. Apart from
the bosonic field operators, the continuous-variable system can also be
described using the quadrature operators $\\{x_{k},p_{k}\\}$. The quadrature
field operators can be arranged in the form of a $2n$-dimensional vector as
${\bm{\xi}}=\left(x_{1},p_{1},...,x_{n},p_{n}\right)$. Here the
$2n$-dimensional vectors ${\bm{\xi}}$ contains the quadrature pairs of all the
$n$ modes. In terms of the bosonic field operators, the quadrature operators
$\\{x_{k},p_{k}\\}$ are expressed as

$\displaystyle x_{k}=(a_{k}+a_{k}^{\dagger}),\qquad
p_{k}=-i(a_{k}-a_{k}^{\dagger}),$

and they are canonically conjugate to each other. These quadrature field
operators act similar to the canonically conjugate position and momentum
operators of a harmonic oscillator.

In the present work we study a single-mode continuous variable system. The two
quadrature operators corresponding to the single mode continuous variable
system are $\xi_{1}=x=(a+a^{\dagger})$ and $\xi_{2}=p=-i(a-a^{\dagger})$ and
the 2D-vector ${\bm{\xi}}=\\{\xi_{1},\xi_{2}\\}$. The quadrature operators
satisfy the canonical commutation relations
$\left[\xi_{i},\xi_{j}\right]=2i\Omega_{ij}$, where $\Omega_{ij}$ are the
elements of the matrix

$\displaystyle\bm{\Omega}=\left(\begin{array}[]{cc}0&1\\\ -1&0\\\
\end{array}\right).$ (3)

In particular, we study the coherence dynamics of the Gaussian states. The
Gaussian states [16, 18] are defined as the states with Gaussian Wigner
function. For a Gaussian state, the Wigner quasiprobability distribution is

$\displaystyle W({\bm{\xi}})=\frac{1}{2\pi\sqrt{\rm{det}\leavevmode\nobreak\
\bm{V}}}\exp\left\\{-\frac{1}{2}({\bm{\xi}-\bm{\overline{\xi}}}).\bm{V}^{-1}({\bm{\xi}-\bm{\overline{\xi}}})^{T}\right\\}$

A Gaussian state is completely characterized by the first and the second
moments of the quadrature field operators. Here the vector of the first
moments ${\overline{\bm{\xi}}}=(\langle\xi_{1}\rangle,\langle\xi_{2}\rangle)$
and the covariance matrix ${\bm{V}}$

$\displaystyle V_{ij}=\langle\\{\Delta\xi_{i},\Delta\xi_{j}\\}\rangle={\rm
Tr}\left(\\{\Delta\xi_{i},\Delta\xi_{j}\\}\rho\right),$ (4)

where $\Delta\xi_{i}=\xi_{i}-\langle\xi_{i}\rangle$ is the fluctuation
operator and
$\\{\Delta\xi_{i},\Delta\xi_{j}\\}=(\Delta\xi_{i}\Delta\xi_{j}+\Delta\xi_{j}\Delta\xi_{i})/2$.
For the single mode quantum system under consideration, the matrix elements of
the covariance matrix are

$\displaystyle V_{ii}$ $\displaystyle=$
$\displaystyle\langle\xi_{i}^{2}\rangle-\langle\xi_{i}\rangle^{2},$ (5)
$\displaystyle V_{ij}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\langle\xi_{i}\xi_{j}+\xi_{j}\xi_{i}\rangle-\langle\xi_{i}\rangle\langle\xi_{j}\rangle.$
(6)

To reflect the positivity of the density matrix, the covariance matrix has to
satisfy the uncertainty relation [40] $\bm{V}+i\bm{\Omega}\geq 0$.

###  II.2 Measurement of Coherence

The quantum coherence of a qubit system is measured as follows: First we
define the set of incoherent states $\mathcal{I}$, i.e., the states with zero
coherence. Next we use a suitable distance measure to find the distance
between the arbitrary density matrix and the closest incoherent states. In
Ref. [2], the authors used the relative entropy measure to quantify the amount
of coherence in the system. In general the relative entropy distance between
two density matrices is

$\displaystyle\mathcal{D}(\rho)=\min_{\sigma}S(\rho\|\sigma)=\min_{\sigma}{\rm
Tr}(\rho\log_{2}\rho-\rho\log_{2}\sigma),$ (7)

where $\rho$ is the given quantum state and $\sigma$ is the reference state.
Using this formulation we can find the coherence in the system by assuming the
reference state to be the incoherent state. It has been proved that for the
relative entropy of coherence, the closest incoherent state is
$\rho_{d}=\sum_{n}\rho_{nn}|n\rangle\langle n|$, which is the diagonal state
of the density matrix. Hence it is not necessary to perform the minimization
to determine the quantum coherence. The relative entropy measure reads:

$\displaystyle C(\rho)=S(\rho\|\rho_{d})=S(\rho_{d})-S(\rho),$ (8)

where $S(\rho)=-{\rm tr}(\rho\log_{2}\rho)$ is the vonNeumann entropy of the
system. So far, the relative entropy of coherence is the most widely used
measure. The quantum coherence of continuous variable system was first
investigated in [41]. Here the authors used a relative entropy based
quantifier of coherence by considering the density matrix $\rho$ in the Fock
basis. While this method is a valid process to quantify coherence, it does not
give a closed form expression for general Gaussian states. Alternatively in
Ref. [19], the coherence measure for a one-mode Gaussian state $\rho$ was
defined as

$\displaystyle C(\rho)=\inf_{\delta}S(\rho\|\delta).$ (9)

Here $S(\rho\|\delta)$ is the relative entropy measure with $\delta$ being an
incoherent Gaussian state and the infimum runs over all incoherent Gaussian
states. Also, it was proved that a one mode Gaussian state is incoherent if
and only if it is a thermal state. Hence for a Gaussian state, the
minimization is achieved for thermal states of the form

$\displaystyle\rho_{d}={\overline{\rho}}=\sum_{n=0}^{\infty}\frac{\overline{n}^{n}}{(1+\overline{n})^{n+1}}|n\rangle\langle
n|,$ (10)

where $\overline{n}={\rm Tr}[a^{\dagger}a{\overline{\rho}}]$ is the mean
number. For the Gaussian states, the entropy of a given state $\rho$ can be
written as

$\displaystyle
S(\rho)=\frac{\nu+1}{2}\log_{2}\frac{\nu+1}{2}-\frac{\nu-1}{2}\log_{2}\frac{\nu-1}{2},$
(11)

where $\nu=\sqrt{{\rm det}{\bm{V}}}$. Based on these expressions, the relative
entropy based coherence measure for the Gaussian states is

$\displaystyle C(\rho)$ $\displaystyle=$ $\displaystyle
S(\rho\|{\overline{\rho}})={\rm
Tr}(\rho\log_{2}\rho-\rho\log_{2}{\overline{\rho}})$ $\displaystyle=$
$\displaystyle\frac{\nu-1}{2}\log_{2}\frac{\nu-1}{2}-\frac{\nu+1}{2}\log_{2}\frac{\nu+1}{2}$
$\displaystyle+$
$\displaystyle(\overline{n}+1)\log_{2}(\overline{n}+1)-\overline{n}\log_{2}\overline{n}.$
(13)

Hence the quantum coherence of a Gaussian state is completely determined by
the determinant of the covariance matrix and the mean number $\overline{n}$
associated with the Gaussian state. But this measure can be used to compute
coherence of only Gaussian states. It is important to note that when the
system-environment interaction Hamiltonian is bilinear in the creation and
annihilation operators, the Gaussian nature of the quantum states is preserved
during the time evolution [42, 18].

##  III Single mode Gaussian states in a Non-Markovian environment

A study of the dynamics of quantum coherence of qubit systems have been
investigated both theoretically [10, 11, 12] and experimentally [13, 43]. On
the theoretical side, the dynamics was characterized in atomic systems and
spin systems for both Markovian and non-Markovian environments. Experimentally
in [13, 43], the dynamics of coherence, entanglement and mutual information
were compared in qubit system. In the present work we consider a continuous
variable system consisting of a single bosonic mode of frequency $\omega_{0}$
coupled to a general non-Markovian environment [44, 45, 46, 47, 48] at finite
temperature. The single bosonic mode can be either a quantum optical field, a
superconducting resonator or a nano-mechanical oscillator. A structured
bosonic reservoir with a collection of infinite modes of varying frequencies
can describe a general non-Markovian environment. The entire system comprising
of the single bosonic mode and the reservoir can be described by the Fano-
Anderson Hamiltonian [49, 50]. This Hamiltonian was introduced in the context
of atomic [50] and condensed mater physics [49] and has been used to study
several different models. The Fano-Anderson Hamiltonian of the system and bath
combination is:

$\displaystyle
H=\hbar\omega_{0}a^{\dagger}a+\hbar\sum_{k}\omega_{k}b_{k}^{\dagger}b_{k}+\hbar\sum_{k}\left(\mathcal{V}_{k}a^{\dagger}b_{k}+\mathcal{V}_{k}^{\ast}b_{k}^{\dagger}a\right),$
(14)

where $\omega_{0}$ is the system frequency and $a^{{\dagger}}$ ($a$) is the
creation (annihilation) operator corresponding to the bosonic mode (system)
and $b^{{\dagger}}_{k}$ ($b_{k}$) is the creation (annihilation) operator of
the $k^{th}$ mode of the bosonic reservoir with frequency $\omega_{k}$. The
factor $\mathcal{V}_{k}$ represents the coupling strength between the system
and the environment.

The dynamics of the single mode system and the environment is solved using the
Heisenberg equation of motion approach. The time evolution of the bosonic
field operator and the environment operators reads:

$\displaystyle a(t)=e^{\frac{iHt}{\hbar}}ae^{-\frac{iHt}{\hbar}},\quad\quad
b_{k}(t)=e^{\frac{iHt}{\hbar}}b_{k}e^{-\frac{iHt}{\hbar}}.$ (15)

In the Heisenberg picture, these operators satisfy the following equations of
motion,

$\displaystyle\frac{d}{dt}a(t)=-\frac{i}{\hbar}\left[a(t),H\right]=-i\omega_{0}a(t)-i\sum_{k}\mathcal{V}_{k}b_{k}(t),\quad$
(16)
$\displaystyle\frac{d}{dt}b_{k}(t)=-\frac{i}{\hbar}\left[b_{k}(t),H\right]=-i\omega_{k}b_{k}(t)-i\mathcal{V}_{k}^{\ast}a(t).\quad$
(17)

Solving for Eqn. (17) for $b_{k}$ we get the solution

$\displaystyle
b_{k}(t)=b_{k}(0)e^{-i\omega_{k}t}-i\mathcal{V}_{k}^{\ast}\int_{0}^{t}d\tau\leavevmode\nobreak\
a(\tau)\leavevmode\nobreak\ e^{-i\omega_{k}(t-\tau)}.$ (18)

Substituting Eq. (18) in (16) we arrive at the following quantum Langevin
equation [51]

$\displaystyle{\dot{a}}(t)+i\omega_{0}a(t)+\int_{0}^{t}d\tau
g(t,\tau)a(\tau)=-i\sum_{k}\mathcal{V}_{k}b_{k}(0)e^{-i\omega_{k}t}.$ (19)

Here the integral kernel
$g(t,\tau)=\sum_{k}|\mathcal{V}_{k}|^{2}e^{-i\omega_{k}(t-\tau)}$
characterizes the non-Markovian memory effects between the system and the
environment. For the continuous environment spectrum
$g(t,\tau)=\int_{0}^{\infty}d\omega J(\omega)e^{-i\omega(t-\tau)}$, where
$J(\omega)=\varrho(\omega)|\mathcal{V}(\omega)|^{2}$ is the spectral density
which characterizes all the non-Markovian memory of the environment on the
system. Here $\varrho(\omega)$ being the density of states of the environment
and $\omega$ is the continuously varying frequency of the bath. Due to the
linearity of Eq. (19) the time evolved bosonic operator $a(t)$ can be
expressed [52] in terms of the initial field operators $a(0)$ and $b_{k}(0)$
of the system and the environment as

$\displaystyle a(t)=u(t)a(0)+f(t).$ (20)

The time-dependent coefficient $u(t)$ and noise operator $f(t)$ are determined
by the quantum Langevin equation, and they satisfy the two integrodifferential
equations given below:

$\displaystyle\frac{d}{dt}u(t)$ $\displaystyle=$
$\displaystyle-i\omega_{0}u(t)-\int_{0}^{t}d\tau g(t,\tau)u(\tau),$ (21)
$\displaystyle\frac{d}{dt}f(t)$ $\displaystyle=$
$\displaystyle-i\omega_{0}f(t)-\int_{0}^{t}d\tau g(t,\tau)f(\tau)$ (22)
$\displaystyle-i\sum_{k}\mathcal{V}_{k}b_{k}(0)e^{-i\omega_{k}t}.$

To determine $u(t)$ we solve Eqn. (21) numerically with the initial condition
$u(0)=1$. Here the solution for the noise operator obtained using the initial
condition $f(0)=0$ is

$\displaystyle f(t)=-i\sum_{k}\mathcal{V}_{k}b_{k}(0)\int_{0}^{t}d\tau
e^{-i\omega_{k}\tau}u(t,\tau).$ (23)

The nonequilibrium thermal fluctuation can be evaluated using the quantity

$\displaystyle\langle
f^{\dagger}(t)f(t)\rangle=v(t)=\\!\\!\int_{0}^{t}\\!\\!\\!d\tau_{1}\int_{0}^{t^{\prime}}\\!\\!\\!d\tau_{2}u(t,\tau_{1}){\widetilde{g}}(\tau_{1},\tau_{2})u^{\ast}(t^{\prime},\tau_{2}),$
(24)

where we consider the initial state of the total system to be
$\rho_{tot}(0)=\rho_{S}(0)\otimes\rho_{E}(0)$. Here the initial environment
state $\rho_{E}(0)=\exp(-\beta H_{E})/{\rm Tr}[\exp(-\beta H_{E})]$ is the
thermal state for the Hamiltonian
$H_{E}=\sum_{k}\hbar\omega_{k}b_{k}^{\dagger}b_{k}$, where $\beta=1/k_{B}T$ is
the inverse temperature and $k_{B}$ is the Boltzmann constant.

When the environment has a continuous spectrum, the time correlation function
reads:

$\displaystyle{\widetilde{g}}(\tau_{1},\tau_{2})=\int_{0}^{\infty}d\omega
J(\omega){\bar{n}}(\omega)e^{-i\omega(\tau_{1}-\tau_{2})},$ (25)

where ${\bar{n}}(\omega)=1/(e^{\hbar\omega/k_{B}T}-1)$ is the initial particle
number distribution of the bosonic environment. From the Eqs. (20) to (24),
one can obtain the time-dependent average values of the system operators as
follows:

$\displaystyle\langle a(t)\rangle=u(t)\langle a(0)\rangle,\leavevmode\nobreak\
\leavevmode\nobreak\ \langle a^{\dagger}(t)\rangle=u^{\ast}(t)\langle
a^{\dagger}(0)\rangle,$ (26) $\displaystyle\langle
a(t)a(t)\rangle=(u(t))^{2}\langle a(0)a(0)\rangle,$ (27) $\displaystyle\langle
a^{\dagger}(t)a^{\dagger}(t)\rangle=(u^{\ast}(t))^{2}\langle
a^{\dagger}(0)a^{\dagger}(0)\rangle,$ (28) $\displaystyle\langle
a^{\dagger}(t)a(t)\rangle=|u(t)|^{2}\langle a^{\dagger}(0)a(0)\rangle+v(t).$
(29)

Here we consider $\langle f(t)\rangle=\langle f^{\dagger}(t)\rangle=0$ and
also $\langle f(t)f(t)\rangle=\langle f^{\dagger}(t)f^{\dagger}(t)\rangle=0$,
since the reservoir is initially in a thermal state uncorrelated to the
system. Using the time-dependent average values in Eqs (26) -(29), we can
evaluate the time evolved first and second moments of the quadrature operators
namely, $\langle\xi_{1}(t)\rangle$, $\langle\xi_{2}(t)\rangle$,
$\langle\xi_{1}^{2}(t)\rangle$, $\langle\xi_{2}^{2}(t)\rangle$,
$\langle\xi_{1}(t)\xi_{2}(t)\rangle$, and
$\langle\xi_{2}(t)\xi_{1}(t)\rangle$. The elements of the time evolved
covariance matrix are

$\displaystyle V_{11}$ $\displaystyle=$ $\displaystyle
1+2v(t)+2|u(t)|^{2}\,{\rm Cov}(a^{\dagger}(0),a(0))$ (30)
$\displaystyle+(u(t))^{2}\,{\rm Var}(a(0))+(u^{\ast}(t))^{2}\,{\rm
Var}(a^{\dagger}(0)),\qquad$ $\displaystyle V_{22}$ $\displaystyle=$
$\displaystyle 1+2v(t)+2|u(t)|^{2}\,{\rm Cov}(a^{\dagger}(0),a(0))$ (31)
$\displaystyle-(u(t))^{2}\,{\rm Var}(a(0))-(u^{\ast}(t))^{2}\,{\rm
Var}(a^{\dagger}(0)),\qquad$ $V_{12}=i(u^{\ast}(t))^{2}\;{\rm
Var}(a^{{\dagger}}(0))-i((u(t))^{2}{\rm Var}(a(0)).$ (32)

where ${\rm Cov}(a,b)=\langle ab\rangle-\langle a\rangle\langle b\rangle$ and
${\rm Var}(a)={\rm Cov}(a,a)$ and $V_{12}=V_{21}$ due to the symmetry of the
covariance matrix.

Once the initial state and the bath parameters are known, the time evolved
covariance matrix elements are completely determined using the nonequilibrium
Green’s function $u(t)$ and $v(t)$. To calculate the Green’s function we need
to specify the spectral density $J(\omega)$ of the environment. In our work we
consider a Ohmic-type spectral density which can simulate a large class of
thermal baths [53]

$\displaystyle J(\omega)=\eta\leavevmode\nobreak\
\omega\left(\frac{\omega}{\omega_{c}}\right)^{s-1}\leavevmode\nobreak\
e^{-\omega/\omega_{c}},$ (33)

where $\eta$ is the coupling strength between the system and the environment
and $\omega_{c}$ is the frequency cut-off of the environmental spectra. A
localized mode is generated when the system-environment coupling approaches a
critical value $\eta_{c}=\omega_{0}/(\omega_{c}\Gamma(s))$ where $\Gamma(s)$
is the Gamma function. Depending on the value of $s$, the environment is
classified as Ohmic for $s=1$, sub-Ohmic for $s<1$ and super-Ohmic for $s>1$.
Throughout our work we use a scaled temperature
$T_{s}=k_{B}T/\hbar\omega_{0}$, where $\omega_{0}$ is the system frequency.
The coherence dynamics for different environmental conditions, and various
system-environment couplings is studied for pure Gaussian states through our
work. Since the Hamiltonian under consideration (14) is bilinear in the
creation and annihilation operators, during evolution, the Gaussian states
preserve their form and remain Gaussian.

Figure 1: The time evolution of quantum coherence of a coherent state with
parameter $\alpha$ in contact with a Ohmic bath is shown for weakly coupled
systems ($\eta=0.01\;\eta_{c}$) in (a) at low temperature $T_{s}=1$, (b) at
high temperature $T_{s}=20$ and for the strongly coupled systems
($\eta=2.0\;\eta_{c}$) in (c) at low temperature $T_{s}=1$, (d) at high
temperature $T_{s}=20$. We use the cut-off frequency
$\omega_{c}=5.0\;\omega_{0}$.

##  IV Quantum Coherence dynamics of coherent states in a Non-Markovian
environment

Glauber [1] defined a coherent state $|\alpha\rangle$ as the eigenstate of the
annihilation operator $a$ with eigenvalue $\alpha\in\mathbb{C}$. We can
generate the coherent state from the ground state $|0\rangle$ of an oscillator
through the action of the displacement operator $D(\alpha)=\exp(\alpha
a^{{\dagger}}-\alpha^{\ast}a)$, where $a^{{\dagger}}$ and $a$ are the
annihilation and creation operators of the standard harmonic oscillator. In
the number basis, the coherent states can be expressed as

$\displaystyle|\alpha\rangle=\exp\left(-\frac{1}{2}|\alpha|^{2}\right)\sum_{n=0}^{\infty}\frac{\alpha^{n}}{\sqrt{n!}}|n\rangle.$
(34)

where $|n\rangle$ is the oscillator number basis. The coherent states are
Gaussian wavepackets which does not spread and has minimal uncertainty. Thus
for a free classical mode, the closest quantum analogue is the coherent state.
Hence the coherent state is very useful in quantum optics especially to study
the state of the quantized electromagnetic field. The time evolution of the
quantum coherence of a coherent state subjected to a spectral density of the
form given in Eq. (33) is reported in this section. Depending on the value of
the parameter $s$, the environments are classified as Ohmic $s=1$, sub-Ohmic
$s<1$ and super-Ohmic $s>1$.

Ohmic bath (s=1):
The time dynamics of quantum coherence of a coherent state for a Ohmic bath
with spectral density $J(\omega)=\eta\omega\exp(-\omega/\omega_{c})$ is shown
in Fig. 1. From the plots Figs 1 (a) - (d) we find that higher the value of
the coherent state parameter ‘$\alpha$’ higher the initial amount of coherence
and also the value at a given time. In Fig 1(a) and 1(b) the time variation of
coherence, when the system is weakly coupled to the environment i.e.,
$\eta=0.01\,\eta_{c}$ is shown. Particularly in Fig. 1(a) we look at the low
temperature limit and in Fig. 1(b), the high temperature regime of the
coherence dynamics. From both these plots we find that quantum coherence
decreases monotonically with time. But it falls faster at higher temperature
because apart from the environmental effects, the thermal effects also
contribute to the system decoherence. The dynamics of coherence when the
system is strongly coupled to the environment i.e., for $\eta=2.0\,\eta_{c}$
is shown in Fig. 1(c) and 1(d). Here we can see that the coherence initially
decreases monotonically and then there is an oscillatory phase. This
oscillatory phase is because of the environmental back action due to the non-
Markovian nature of the bath. For lower tempertures as shown in Fig. 1(c), the
coherence saturates at a higher value compared to the high temperature limit
given in Fig. 1(d).

Figure 2: The coherence dynamics of a coherent mode with parameter $r$ in
contact with sub-Ohmic bath is studied for weakly coupled systems
($\eta=0.01\;\eta_{c}$) in (a) low temperature $T_{s}=1$, (b) high temperature
$T_{s}=20$. For the strongly coupled systems ($\eta=2.0\;\eta_{c}$) the plots
are (c) at low temperature $T_{s}=1$ and (d) at high temperature $T_{s}=20$.
We use the cut-off frequency $\omega_{c}=5.0\;\omega_{0}$.

Sub-Ohmic bath (s=1/2):
To investigate the coherence dynamics in the sub-Ohmic region we consider
$s=1/2$ in Eq. (33). The corresponding spectral density reads
$J(\omega)=\eta\sqrt{\omega\,\omega_{c}}\exp(-\omega/\omega_{c})$. The results
corresponding to this analysis is presented in Fig. 2. The time variation of
coherence for the weak coupling limit with $\eta=0.01\,\eta_{c}$ is given in
Fig. 2 (a) and (b). In Fig. 2 (a) the low temperature limit is analyzed and we
find that the coherence decreases monotonically with time. At higher
temperatures also, coherence decreases monotonically but the fall is much
faster as seen in Fig. 2 (b). This faster fall is because at high temperatures
apart from the dissipation, the thermal fluctuations also cause a decoherence
in the system. The coherence dynamics when the system is strongly coupled to
the environment $\eta=2.0\,\eta_{c}$ is presented in Fig. (2) (c) and (d)
corresponding to the low and high temperature cases respectively. In the low
temperature case illustrated in Fig. 2 (c), we find a revival of coherence due
to the non-Markovian effects of the bath. Such a coherence revival is not
observed in the high temperature limit even in the strong coupling case. This
is because the temperature affects the environmental back action on the
system, a feature which has also been observed in Ref. [].

Figure 3: A coherent mode in contact with a super-Ohmic bath is studied for
weakly interacting systems ($\eta=0.01\;\eta_{c}$) for (a) $T_{s}=1$ and (b)
$T_{s}=20$ and also for strongly interacting systems ($\eta=2.0\;\eta_{c}$)
with (c) $T_{s}=1$ and (d) $T_{s}=20$. The cut-off frequency used is
$\omega_{c}=5.0\;\omega_{0}$.

Super-Ohmic bath (s=3):
For the super-Ohmic bath we consider $s=3$ to investigate the dynamics of
coherence. The spectral density in this case reads
$J(\omega)=\eta\,\omega_{c}(\omega/\omega_{c})^{3}\exp(-\omega/\omega_{c})$.
The coherence evolution in this case is described through the plots in Fig. 3
(a) to (d) considering different coupling strengths and different
temperatures. In Fig. 3 (a) and (b) we analyze the coherence variation in the
weak coupling limit when $\eta=0.01\,\eta_{c}$. Here we can see that the
coherence decreases monotonically both in the low and high temperature limits.
Due to temperature dependent decoherence effects, the rate of decrease is
higher in Fig 3 (b). The nature of coherence variation in the strong coupling
limit ($\eta=2.0\,\eta_{c}$) is examined in Fig. 3 (c) and (d) respectively.
In both these cases we see that the coherence decreases initially and then
saturates at a finite value a phenomenon known as coherence freezing. But the
value at which coherence freezes is different in Fig. 3 (c) and Fig. 3 (d) and
is also dependent on the parameter $\alpha$. For a higher $\alpha$, the
coherence freezes at a higher value. The general decrease in coherence is due
to the environmental effects. For the same environment at different
temperatures, the system experiences temperature dependent decoherence
effects.

Figure 4: A squeezed mode in contact with a Ohmic bath is studied for weakly
interacting systems ($\eta=0.01\;\eta_{c}$) for (a) $T_{s}=1$ and $T_{s}=20$
and also for strongly coupled systems ($\eta=2.0\;\eta_{c}$) for (a) $T_{s}=1$
and $T_{s}=20$. The cut-off frequency used is $\omega_{c}=5.0\,\omega_{0}$.

##  V Non-Markovian dynamics of Squeezed states

A squeezed state has a minimum value for the product of the dispersion of the
position and momentum operators. For a squeezed state [54], the Hamiltonian
consists of terms quadratic in the creation and the annihilation operator. The
Gaussian unitary corresponding to the one mode squeezing operator is

$\displaystyle S(r)=\exp\left[r\left(a^{2}-(a^{{\dagger}})^{2}\right)\right].$

Here we assume $r\in\mathbb{R}$ for the sake of convenience. The Bogoliubov
transformation of the annihilation and creation operator based on the
squeezing operator reads:

$\displaystyle S^{{\dagger}}(r)aS(r)=a\cosh r-a^{{\dagger}}\sinh r,,$
$\displaystyle S^{{\dagger}}(r)a^{{\dagger}}S(r)=a^{{\dagger}}\cosh r-a\sinh
r,$

When this squeezing operator is applied to a vaccum state we can generate a
squeezed vaccum state

$\displaystyle|0,r\rangle=S(r)|0\rangle=\frac{1}{\sqrt{\cosh
r}}\sum_{n=0}^{\infty}\frac{\sqrt{(2n)!}}{2^{n}n!}\tanh^{n}r|2n\rangle.$

In this state, the variance of one of the quadrature operator is below the
quantum shot noise and hence it is called a squeezed state. To compensate the
squeezing in one quadrature, there is an antisqueezing in the other
quadrature. The dynamics of quantum coherence of the single mode squeezed
state is presented in this section, for the spectral density Eq. (33),
considering the Ohmic, sub-Ohmic and the super-Ohmic conditions.

Ohmic bath (s=1) :
The Ohmic bath has a spectral density
$J(\omega)=\eta\omega\exp(-\omega/\omega_{c})$ and the dynamics of quantum
coherence of a single mode squeezed state is given through the plots in Fig.
(4). When the squeezed single mode system is weakly coupled
($\eta=0.01\,\eta_{c}$) to the environment, the coherence evolution is shown
through Fig. 4(a) and Fig. 4(b) for the high and the low temperature limits
respectively. We do not observe non-Markovian effects in both the limits. But
the coherence falls faster in the high temperature limit due to temperature
dependent decoherence effects. Under the conditions when the system is
strongly coupled to the environment ($\eta=2.0\,\eta_{c}$), the coherence
evolution is shown in Fig. 4 (c) and Fig. 4 (d) respectively. We observe a
very small non-Markovian effects in the low temperature limit. It is absent in
the high temperature limit due to the thermal decoherence effects.

Figure 5: The transient dynamics of coherence for a single squeezed mode in
contact with a sub-Ohmic bath is studied for weakly interacting systems
($\eta=0.01\;\eta_{c}$) for (a) $T_{s}=1$ and (b) $T_{s}=20$ and strongly
interacting systems ($\eta=2.0\;\eta_{c}$) (c) $T_{s}=1$ and (d) $T_{s}=20$
and using a cut-off frequency of $\omega_{c}=5.0\;\omega_{0}$.  Figure 6: For
a squeezed mode in contact with a super-Ohmic bath, the coherence dynamics is
studied for weakly interacting systems ($\eta=0.01\;\eta_{c}$) for (a)
$T_{s}=1$ and (b) $T_{s}=20$ and strongly coupled systems
($\eta=2.0\;\eta_{c}$) (c) $T_{s}=1$ and (d) $T_{s}=20$. The cut-off frequency
used is $\omega_{c}=5.0\;\omega_{0}$.  Figure 7: The temporal evolution of the
quantum coherence $(C)$ as a function of time ($\omega_{0}t$) and coupling
strength ($\eta/\eta_{c}$) is given for the displaced squeezed state in
contact with an Ohmic bath. The plot is divided into four columns as (a)
$\alpha=0.1,r=0.1$, (b) $\alpha=0.1,r=2.0$ (c) $\alpha=4.0,r=0.1$ and (d)
$\alpha=4.0,r=2.0$. In each column there are two plots one for $T_{s}=1$ and
$T_{s}=20$. The cut-off frequency used is $\omega_{c}=5.0\;\omega_{0}$.

Sub-Ohmic bath (s=1/2):
For a sub-Ohmic bath with spectral density,
$J(\omega)=\eta\sqrt{\omega\,\omega_{c}}\exp(-\omega/\omega_{c})$, the
coherence dynamics of the single mode squeezed state is shown via the plots in
Fig. 5 (a) - (d). In Fig. 5 (a) and 5(b), the system is analyzed when it is
weakly coupled to the bath ($\eta=0.01\,\eta_{c}$). Here we see that at high
temperature the coherence falls faster due to thermal decoherence. The strong
coupling effects ($\eta=2.0\,\eta_{c}$) are analyzed in the plots 5(c) and
5(d) respectively. In the low temperature limit, there is a small non-
Markovian effect, but in the high temperature limit, there is no environmental
back action. Overall, we find that coherence falls faster with increase in
temperature.

Super-Ohmic bath (s=3):
To study the effects of a super-Ohmic environment, we consider a bath with a
spectral density
$J(\omega)=\eta\,\omega_{c}(\omega/\omega_{c})^{3}\exp(-\omega/\omega_{c})$.
The plots in Fig. 6 (a) - (d) show the dynamics of the quantum coherence of
the squeezed single mode state. The change of coherence with time, when the
system is weakly coupled to the bath ($\eta=0.01\,\eta_{c}$) is shown in Fig.
6 (a) and (b). The coherence falls monotonically in the weak coupling limit
and the rate of fall of coherence increases with increase in the temperature.
In the strong coupling limit i.e., when ($\eta=2.0\,\eta_{c}$), the coherence
initially decreases but saturates at a finite value, a phenomenon known as
coherence freezing. The rate of fall of coherence and the saturation value of
coherence is dependent on the temperature and for higher temperature the
coherence saturates at a lower value. In general we find that the amount of
coherence is directly proportional to the squeezing parameter $r$, under all
the environmental conditions.

##  VI Quantum coherence evolution of displaced squeezed states

A displaced squeezed [55] state has a general form as given below:

$\displaystyle|\alpha,r\rangle=D(\alpha)S(r)|0\rangle,$

where $|0\rangle$ is the vaccum state and $D(\alpha)$ and $S(r)$ are the
displacement and squeezing operators given below:

$\displaystyle D(\alpha)=\exp(\alpha a^{{\dagger}}-\alpha^{\ast}a);\quad\quad
S(r)=\exp(ra^{2}-r(a^{{\dagger}})^{2}).$

Here $\alpha\in\mathbb{C}$ and $r\in\mathbb{R}$ are the relevant parameters.
In this state, we investigate the transient dynamics of quantum coherence for
different values of the displacement parameter $\alpha$ and the squeezing
parameter $r$. The single mode displaced squeezed state is examined for the
spectral density Eq. (33), considering the Ohmic, sub-Ohmic and the super-
Ohmic conditions.

Ohmic bath (s=1) :
Ohmic bath has a spectral density
$J(\omega)=\eta\omega\exp(-\omega/\omega_{c})$ and the coherence dynamics of
the single mode displaced squeezed state for this spectrum is given through
the $3D$ plots in Fig. 7. The coherence $C(\rho)$ being the vertical axis and
the parameters $\omega_{0}t$ and $\eta/\eta_{c}$ along the orthogonal
horizontal axes. The figure is split in to four columns labelled from (a) to
(d) corresponding to the four different states of the displaced squeezed
states. The first row gives the plots in the low temperature limit ($T_{s}=1$)
and the second row the high temperature regime ($T_{s}=20$).

In the first column i.e., Fig. 7 (a), we present the dynamics for the
parameter values of ($\alpha=0.1,r=0.1$). Both in the high and low temperature
limit, the coherence falls monotonically to zero in a very fast manner which
can be referred to as coherence sudden death in analogy with the sudden death
of entanglement. In the low temperature limit the coherence revives and
saturates at a finite value, which does not happen when the temperature is
high. After revival in the low temperature limit, the coherence attains
saturation and here it shows a non-Markovian feature that survives for a long
time. When the squeezing is increased to high values ($\alpha=0.1,r=2.0$), the
coherence behavior is shown in Fig. 7 (b). Here we again find a coherence
sudden death in both the low and high temperature limits. When the
displacement values are higher, the coherence is shown in Fig. 7 (c) and 7 (d)
for the values ($\alpha=4.0,r=0.1$) and ($\alpha=4.0,r=2.0$) respectively. The
system showns coherence sudden death after which there is a coherence revival
and saturation in this limit. The system exhibits non-Markovian behavior for
the parameters after the revival. From all the plots Fig. 7 (a) to (d) we can
see that the saturation value is higher when ($\alpha=4.0,r=0.1$) and on
increasing the squeezing parameter it decreases as can be seen from Fig 7 (d)
corresponding to the parameter values ($\alpha=4.0,r=2.0$).

Figure 8: The coherence dynamics of a displaced squeezed mode is shown in the
figure above as a function of time $\omega_{0}t$, and $\eta/\eta_{c}$ when it
is in contact with a sub-Ohmic bath. The plot is divided into four columns (a)
$\alpha=0.1,r=0.1$, (b) $\alpha=0.1,r=2.0$, (c) $\alpha=4.0,r=0.1$, and (d)
$\alpha=4.0,r=2.0$. In each column there are two plots for $T_{s}=1$ and
$T_{s}=20$. The cut-off frequency used is $\omega_{c}=5.0\;\omega_{0}$. Figure
9: The time evolution of coherence of a displaced squeezed mode in contact
with a super-Ohmic bath is shown in the figure above as a funciton of
$\omega_{0}t$, and $\eta/\eta_{c}$. There are four columns in the plot (a)
$\alpha=0.1,r=0.1$, (b) $\alpha=0.1,r=2.0$, (c) $\alpha=4.0,r=0.1$, and (d)
$\alpha=4.0,r=2.0$. In each column there are two plots for $T_{s}=1$ and
$T_{s}=20$. The cut-off frequency used is $\omega_{c}=5.0\;\omega_{0}$.

Sub-Ohmic bath (s=1/2):
The single displaced squeezed mode on being affected by a sub-Ohmic bath is
given through the plots in Fig. 8. The corresponding spectral density is
$J(\omega)=\eta\sqrt{\omega\,\omega_{c}}\exp(-\omega/\omega_{c})$. The $3D$
plots describing the variation of coherence given as a set of plots, where the
first row represents the low temperature regime ($kT=1$) and the second row
representing the high temperature regime ($kT=20$). We label the columns from
(a) to (d) in the plots.

For the parameter values of ($\alpha=0.1$, $r=0.1$) and ($\alpha=0.1$,
$r=2.0$), the coherence dynamics is given in Fig. 8 (a) and (b) respectively.
Here in the low temperature limit we observe both coherence sudden death and
coherence revival and also a non-Markovian effect in the coherence obtained
after the revival. Meanwhile in the high temperature limit, we observe only
coherence sudden death and there is no revival. In the regimes ($\alpha=4.0$,
$r=0.1$) and ($\alpha=4.0$, $r=2.0$) we see the coherence again decaying
within a short interval of time as shown through the plots in Fig. 8 (c) and
(d) respectively. The coherence sudden death and the coherence revival is
observed both in the low and high temperature limit. Also the non-Markovian
effects are clearly seen in these plots. In the sub-Ohmic limit, we see
coherence revivals only in the high $\alpha$ region.

Figure 10: The $3$-D plots $\frac{{\rm d}C}{{\rm d}\eta_{s}}$ Vs $\eta_{s}$ Vs
$\omega_{0}t$ for the squeezed coherent state with parameters
$\alpha=4.0,r=0.1$ is given above. In Fig. (a) and (c) we study the low
temperature ($T_{s}=1$) and the high temperature limit ($T_{s}=20$) when the
state is in contact with an Ohmic bath. The low temperature ($T_{s}=1$) and
high temperature limit ($T_{s}=20$) is shown in Figs. (b) and (d)
respectively, when the state is exposed to a sub-Ohmic bath. The cut-off
frequency used is $\omega_{c}=5.0\;\omega_{0}$.

Figure 11: The dissipation coefficient $\gamma(t)$ is calculated from the
exact solution of $u(t)$. We present here the contour plot of the dissipation
coefficient $\gamma(t)$ and its time derivative ${\rm d}\gamma/{\rm d}t$ with
varying time and coupling strength. In Fig. (a) and (b) we show the transient
behavior of $\gamma(t)$ for Ohmic ($s=1$) and sub-Ohmic ($s=1/2$) bath
spectrum respectively. We next plot the time derivative ${\rm d}\gamma/{\rm
d}t$ with varying time and coupling strength when the system is in contact
with an Ohmic bath (c) and sub-Ohmic bath (d) respectively. The cut-off
frequency $\omega_{c}=5.0\omega_{0}$.

Super-Ohmic bath (s=3):
The quantum coherence dynamics of the single displaced squeezed mode, when it
is exposed to a super-Ohmic environment is given here. The spectral density
used for this computation is
$J(\omega)=\eta\,\omega_{c}(\omega/\omega_{c})^{3}\exp(-\omega/\omega_{c})$.
The results pertaining to this study is given as $3D$ plots in Fig. 9, with
the coherence $C(\rho)$ being along the vertical axis and the parameters
$\omega_{0}t$ and $\eta/\eta_{c}$ along the orthogonal horizontal axes. The
first row represents the low temperature regime ($T_{s}=1$) and the second row
the high temperature regime ($T_{s}=20$) and the columns in the figure are
labelled from (a) to (d).

Figure 12: The fluctuation coefficient $\widetilde{\gamma}(t)$ is calculated
from the exact solution of $u(t)$ and $v(t)$. We present here the contour plot
of the fluctuation coefficient $\widetilde{\gamma}(t)$ for Ohmic ($s=1$)
spectrum in the low temperature (a) $T_{s}=1$ and in the higher temperature
(c) $T_{s}=20$ respectively. We next show the contour plot of fluctuation
coefficient $\widetilde{\gamma}(t)$ for sub-Ohmic ($s=1/2$) spectrum in the
low temperature (b) $T_{s}=1$ and in the high temperature (d) $T_{s}=20$. The
cut-off frequency in all the cases $\omega_{c}=5.0\omega_{0}$.

In Fig. 9 (a) and (b) the coherence dynamics is given for parameter values of
($\alpha=0.1$, $r=0.1$) and ($\alpha=0.1$, $r=2.0$) respectively. For the
parameter values ($\alpha=4.0$, $r=0.1$) and ($\alpha=4.0$, $r=2.0$), the
dynamics is give through the plots in Fig. 9 (c) and (d). From all the plots
we observer that the coherence decays monotonically and then attains a
saturation value at which the coherence freezes for the rest of the evolution.
But the fall of coherence and the value at which the coherence freezes is
dependent on the temperature and higher the temperature, lower the saturation
value. This is because apart from the loss of coherence due to dissipation,
some of the coherence is also lost due to the thermal decoherence. From Fig. 9
(a) we can see that initial coherence is low when both the $\alpha$ and $r$
values are low. On increasing either the displacement parameter $\alpha$ or
the squeezing parameter $r$, we find that the system has a higher amount of
initial coherence i.e., coherence at $t=0$ as shown in Fig. 9 (b) and (c).
Further, from these two cases, we also observe that the saturation value is
higher when $\alpha$ is higher. Finally in 9 (d) we examine the situation
where both the displacement parameter ($\alpha$) and the squeezing parameter
($r$) are higher. In this case we find that the initial amount of coherence is
comparable to the cases in Fig. 9 (b) and (c), but the fall of coherence and
the saturation value more closely resembles the case in Fig. 9 (c), where
($\alpha=4.0$, $r=0.1$). An increase in either the displacement parameter or
the squeezing parameter increases the quantum coherence in the system.

##  VII Markovian to non-Markovian crossover

The present work considers a Gaussian state in contact with an external non-
Markovian environment. When we consider a Ohmic or sub-Ohmic bath coupled to
the system, we initially observe a Markovian evolution for the coherent state,
squeezed state and displaced squeezed state. Depending on whether the coupling
is weak or strong, the dynamics continues to remain either Markovian or
switches over to non-Markovian behavior. In this current section we examine
the crossover behavior in the dynamics of a displaced squeezed state.

Let us consider the evolution of a displaced squeezed state with the parameter
values of $\alpha=4.0$ and $r=0.1$. When the coupling between the bath and the
system is weak, we observe that the dynamics of the system is completely
Markovian where the system experiences either a death of coherence or
coherence freezing. In this case, the correlation time of the bath is much
shorter than the system evolution time. The irreversible loss of information
occurs, because the bath states are restored very quickly and so any
information received from the state is lost. For systems which have a stronger
coupling with the bath, the initial coherence decay is Markovian and then
there is a sudden switch where the dynamics becomes non-Markovian. In the non-
Markovian phase there is a back flow of information from the environment to
the system. To check this behaviour, we look at Fig. 10, the $3D$ plots of
$\frac{{\rm d}C}{{\rm d}\eta_{s}}$ Vs $\eta_{s}$ Vs $\omega_{0}t$ in both the
low temperature ($T_{s}=1$) and the high temperature ($T_{s}=20$) limits. In
Fig. 10 (a), the plot shows ${\rm d}C/{\rm d}\eta_{s}$ for the Ohmic spectrum
in the low temperature limit. Here we find that initially, the slope is $-ve$
and monotonic implying a Markovian nature. Increasing the coupling strength,
we observe that the slope changes from $-ve$ to $+ve$ indicating a change from
Markovian to non-Markovian behavior. For the Ohmic bath in the low temperature
regime this crossover in the behavior of dynamics is very abrupt and this
sudden change is reminiscent of a phase transition. In the initial phase we
observe $\tau_{b}\ll\tau_{s}$, where $\tau_{b}$ is the relaxation time of the
bath and $\tau_{s}$ is the evolution time of the system. After the crossover
(i.e., transition) the time scales are related as $\tau_{b}\approx\tau_{s}$
indicative of a new phase. In the low temperature limit $(T_{s}=1)$, the sub-
Ohmic regime is described in Fig. 10 (b). Here there is a region where the
coherence remains constant before the transition to the non-Markovian
evolution. The high temperature $(T_{s}=20)$ case of the $3$D plots are shown
in Fig. 10 (c) and (d) for the Ohmic and sub-Ohmic baths respectively. From
the plots we can observe that transition region between the Markovian and non-
Markovian regimes is more pronounced. Also the non-Markovian effect is
decreased indicating a lesser amount of environmental back flow of coherence.
This reduced amount of back flow is due to the decoherence caused by thermal
effects.

The dynamical behavior of the continuous variable system can also be studied
using a quantum master equation. The crossover between Markovian and non-
Markovian can be perceived from the changing behavior of the decay rates in
the master equation. The total density matrix $\rho_{tot}$ describing the
continuous variable system and environment has a dynamics governed by the
quantum evolution operator
$\rho_{tot}(t)=\exp\left(-\frac{i}{\hbar}Ht\right)\rho_{tot}(0)\exp\left(\frac{i}{\hbar}Ht\right)$.
The initial state of the system is considered to be
$\rho_{tot}(0)=\rho_{s}(0)\otimes\rho_{E}(0)$, where $\rho_{E}(0)=\exp(-\beta
H_{E})/({\rm Tr}\exp(-\beta H_{E}))$ as proposed in [56, 57]. Tracing over the
environmental degrees of freedom we can get the reduced density matrix of the
system. The master equation for the reduced density matrix reads:

$\displaystyle\frac{{\rm d}\rho(t)}{{\rm d}t}$ $\displaystyle=$
$\displaystyle-i\omega_{0}^{\prime}(t)[a^{{\dagger}}a,\rho(t)]$ (35)
$\displaystyle+\gamma(t)[2a\rho(t)a^{{\dagger}}-a^{{\dagger}}a\rho(t)-\rho(t)a^{{\dagger}}a]$
$\displaystyle+\widetilde{\gamma}(t)[a\rho(t)a^{{\dagger}}+a^{{\dagger}}\rho(t)a-a^{{\dagger}}a\rho(t)-\rho(t)aa^{{\dagger}}]\qquad$

where the coefficients are

$\displaystyle\omega_{0}^{\prime}(t)={\rm
Im}\left[\frac{\dot{u}(t)}{u(t)}\right],\qquad\gamma(t)=-{\rm
Re}\left[\frac{\dot{u}(t)}{u(t)}\right]$
$\displaystyle\widetilde{\gamma}(t)=\dot{v}(t)-2v(t){\rm
Re}\left[\frac{\dot{u}(t)}{u(t)}\right].$

Here $\omega_{0}^{\prime}(t)$ is the renormalized frequency of the single mode
system and $\gamma(t)$ and $\widetilde{\gamma}(t)$ are the dissipation and
fluctuation coefficients. In Fig. 11 (a) and (b), we present a contour plot of
the dissipation coefficient $\gamma(t)$ varying with time and coupling
strength. The Ohmic case is described through the plot in 11(a), where in the
weak coupling limit $\eta_{s}\leq 1$ we find that the dissipation is almost a
constant throughout the entire evolution. For the strong coupling regime
$\eta_{s}\geq 1$, the dissipation rate $\gamma(t)$ increases initially and
then decreases. This change in the direction of the decay rate with time,
indicates a backflow of information from the environment to the system. A
similar behavior of the dissipation constant $\gamma(t)$ is observed in the
sub-Ohmic case as shown in 11(b). The reverse flow of information is more
transparent from the contour plot of the slope of $\gamma(t)$. We show contour
plot of the time derivative of the decay rate (${\rm d}\gamma/{\rm d}t$) for
the Ohmic case in Fig. 11 (c) and sub-Ohmic case in Fig. 11 (d). From the plot
we observe that the slope of the dissipation coefficient $\gamma(t)$,
transitions from being positive to negative. This is indicative of a crossover
from the Markovian to a non-Markovian regime.

Next, we study the variation of the fluctuation coefficient namely
$\widetilde{\gamma}(t)$ in Fig. 12. We show the contour plot of fluctuation
coefficient for Ohmic spectrum in the low temperature limit ($T_{s}=1$) in
Fig. 12 (a) and for the high temperature limit ($T_{s}=20$) in Fig. 12 (c). In
the strong coupling regime ($\eta_{s}\geq 1$), the fluctuation coefficient
$\widetilde{\gamma}(t)$ rises initially and then starts decreasing with time.
This changing behavior of $\widetilde{\gamma}(t)$ indicates the environmental
back action in the strong coupling regime. The contour plot of fluctuation
coefficient for sub-Ohmic spectrum in the low temperature ($T_{s}=1$) is shown
in Fig. 12 (b). For the high temperature limit ($T_{s}=20$) is given in Fig.
12 (d). We observe a similar dynamical crossover of $\widetilde{\gamma}(t)$
for the sub-Ohmic spectral density as well. Hence the environmental back flow
of information [58, 59, 60] i.e., the non-Markovian behavior can be obtained
from the dynamical behavior of the dissipation and the fluctuation
coefficients. The dynamical crossover observed in the coherence dynamics of a
single mode state, stands verified by the analysis carried out from the
dynamics of the fluctuation and dissipation coefficients of the master
equation of the state.

##  VIII Conclusion

The transient dynamics of quantum coherence of a single mode Gaussian state is
analyzed in the open quantum system formalism. For this we consider a non-
Markovian environment, which is a collection of infinite bosonic modes of
varying frequencies. The entire analysis is carried out in the finite
temperature limit with arbitrary system-bath coupling strength. To measure the
quantum coherence we use the relative entropy measure which estimates the
distance to the thermal state. Hence the quantum coherence can be completely
characterized by the determinant of the covariance matrix and the average of
the number operator of the Gaussian states. We solve the quantum Langevin
equation for the bosonic mode operators to obtain the time evolved covariance
matrix elements. The time evolution of the field operators are determined by
the two basic nonequilibrium Green’s functions namely $u(t)$ and $v(t)$.

The time dynamics of quantum coherence is studied for three distinct
experimentally realizable Gaussian states. The investigations have been
carried out under three different environmental spectral densitites viz Ohmic,
sub-Ohmic and super-Ohmic spectral densitites. When the system interacts
weakly with the environment, the quantum coherence decreases monotonically
with time. The coherence decay rate is relatively lower for the super-Ohmic
bath when compared with the Ohmic and the sub-Ohmic baths. Also the rate of
decay increases with the temperature. In general we observe a coherence death
for the Ohmic and the sub-Ohmic baths in the short time limit. For the super-
Ohmic bath we observe coherence freezing which is a saturation of coherence at
a finite value. In the strong coupling regime, the coherence decreases
initially and then starts increasing resulting in a revival of coherence for
the Ohmic and the sub-Ohmic baths. This is followed by a stabilization of
coherence with an oscillatory phase. This oscillatory behavior is due to the
environmental backaction which is a feature of non-Markovian dynamics in the
system. Hence in our study we observe a transition or crossover from the
Markovian dynamics to the non-Markovian dynamics. The non-Markovian memory
effect helps us to restore quantum resources in the strong coupling limit. The
rate at which the crossover in the dynamics occurs depends on the parameters
of the Gaussian state as well as on the environmental parameters. To verify
the existence of the crossover we study the dynamical behavior of the quantum
system using a master equation approach. Here we compute the dissipation and
fluctuation coefficients in the master equation of the reduced density matrix.
Throughout the evolution, the dissipation rate is almost a constant in the
weak coupling limit. In the strong coupling limit, the dissipation rate
increases initially and then decreases, which implies a Markovian to non-
Markovian transition. A plot of the time derivative of the decay rate shows
that the slope of the dissipation coefficient changes from being positive to
negative. This is a clear indication of the crossover from the Markovian to
the non-Markovian dynamics. This crossover feature is also verified from the
dynamics of the fluctuation coefficient. Thus we find that the quantum system
has both Markovian and non-Markovian behavior at different times and there is
a dynamical crossover from the Markovian to the non-Markovian regime. This
change from Markovian to non-Markovian regime is not a gradual change. Instead
it is either a sudden change or there is a time period between the Markovian
and non-Markovian regimes, where there is no dynamical change. This raises an
important question as to whether this dynamical change can be considered as
some kind of phase transition between two different regimes with totally
different relaxation time scales. While the results seems to point towards
such a conclusion, a more detailed study from the point of view of quantum
phase transitions is needed to confirm and validate this, and such a study
will form the scope of our future works. A similar observation on an abrupt
change from Markovian to non-Markovian dynamics has also been made in Ref.
[61]. Here the authors study a single qubit system in contact with a single
qubit environment. In our work we consider a single mode Gaussian state in
contact with a structured bosonic reservoir with a collection of infinite
modes to describe a general non-Markovian environment. The investigations
carried out in the present work can be experimentally verified by measuring
the quantum coherence of Gaussian systems [62, 63, 64, 65].

## Acknowledgements

Md. Manirul Ali was supported in part by the Centre for Quantum Science and
Technology, Chennai Institute of Technology, India, vide funding number
CIT/CQST/2021/RD-007. Chandrashekar Radhakrishnan was supported in part by a
seed grant from IIT Madras to the Centre for Quantum Information,
Communication and Computing.

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