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\begin{document}
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\newcommand{\rigmark}{\em Journal of Russian Laser Research}
\newcommand{\lemark}{\em Volume 30, Number 5, 2009}
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\begin{center} {\Large \bf
\begin{tabular}{c}
DECOHERENCE AND ASYMPTOTIC ENTANGLEMENT
\\[-1mm]
IN OPEN QUANTUM DYNAMICS
\end{tabular}
} \end{center}
\bigskip
\bigskip
\begin{center} {\bf
Author One$^{1*}$ and Author Two$^2$
}\end{center}
\medskip
\begin{center}
{\it
$^1$Department of Physics, National Univercity 1\\
Moscow, Russia 119991
\smallskip
$^2$Department of Physics, National Univercity 2\\
Saint-Petersburg, Russia 119991
}
\smallskip
$^*$Corresponding author e-mail:~~~author1~@~moscow.rus\\
\end{center}
\begin{abstract}\noindent
Within the framework of the theory of open systems based on completely
positive quantum dynamical semigroups, we
determine the degree of quantum decoherence of a harmonic oscillator
interacting with a thermal bath. It is found that the system
manifests a quantum decoherence which is more and more significant
in time. We also calculate the decoherence time and show that it has
the same scale as the time after which thermal fluctuations become
comparable with quantum fluctuations. We solve the master equation
for two independent harmonic oscillators interacting with an environment
in the asymptotic long-time regime. We give a description of
the continuous-variable asymptotic entanglement in terms of the covariance
matrix of quantum states of the considered system for an
arbitrary Gaussian input state. Using the Peres--Simon
necessary and sufficient condition for separability of two-mode
Gaussian states, we show that the two noninteracting systems
immersed in a common environment become asymptotically
entangled for certain environments, so that in the long-time regime
they manifest nonlocal quantum correlations.
\end{abstract}
\medskip
\noindent{\bf Keywords:}
open systems, quantum decoherence, quantum entanglement, nonseparable states.
\section{Introduction}
\pst
By quantum decoherence (QD) \cite{1,2} we understand the
irreversible, uncontrollable, and persistent formation of quantum
correlations (entanglement) of a system with its environment
\cite{1}, expressed by the damping of the coherences present in the
quantum state of the system, when off-diagonal elements of the
density matrix decay below a certain level, so that this density
matrix becomes approximately diagonal.
The paper is organized as follows.
In Sec. 2 we review the Markovian master equation for the damped harmonic
oscillator and solve it in the coordinate representation.
In Sec.~3 we investigate QD and calculate the
decoherence time of the system.
In Sec.~4 we write the equations of motion in the Heisenberg
picture for two independent harmonic oscillators interacting with
a general environment. With these equations, we
derive in Sec.~5 the asymptotic values of the variances and
covariances of the coordinates and momenta which enter the
asymptotic covariance matrix. Then, by using the Peres--Simon necessary
and sufficient condition for separability of two-mode Gaussian states~\cite{1,2,3},
we investigate the behavior of the environment-induced entanglement in the limit
of long times. We show that for certain classes of environments
the initial state evolves asymptotically to an equilibrium
state which is entangled, while for other values of the
parameters describing the environment, the entanglement is
suppressed and the asymptotic state is separable. The existence
of the quantum correlations between the two systems in the
asymptotic long-time regime is the result of the competition
between the entanglement and QD. A summary is given in
Sec. 6.
\section{Markovian Master Equation for a Harmonic Oscillator}
\pst
The normalized evolution equation of the soliton propagating along the
longitudinal direction $z$ in one-dimensional waveguides with periodic
refractive-index modulation in the transverse direction $x$ and a Kerr-type
self-focusing nonlinear media reads~\cite{3}
\be
i\frac{\partial u}{\partial z}=-\frac{1}{2}\,\frac{\partial^2u}{\partial x^2}-
|u|^2u-pV(x)u,
\ee
%(1)
where $p$ is the modulation-depth parameter, $V(x) = \cos(Xx)$ is the
periodically-modulated refractive index determining the waveguide array, and
$T =2\pi/X$ describes the modulation period.
In our analysis, the initial light beam has the form~\cite{3}
\be
u(z=0,x)=\delta A\,\mbox{sech}\,(x-x_0)\sin(x-x_0)+A\,\mbox{sech}\,(x+x_0)\sin(x+x_0).
\ee
%(2)
\section{Quantum Decoherence}
\pst
An isolated system has a unitary evolution and the coherence of the
state is not lost --- pure states evolve in time only to pure states.
The QD phenomenon, that is, the loss of coherence or the destruction
of off-diagonal elements representing coherences between quantum
states in the density matrix, can be achieved by introducing an
interaction between the system and environment --- an initial pure
state with a density matrix which contains nonzero off-diagonal
terms can nonunitarily evolve into a final mixed state with a
diagonal density matrix.
Figure 1 illustrates the dependence of the QD degree on temperature and time.
We see that, in general, $\delta_{QD}$ decreases and, therefore, QD
becomes stronger with increasing time and temperature, i.e., the density matrix
becomes more and more diagonal at higher $T$ and the contributions
of the off-diagonal elements get smaller and smaller.
\begin{figure}[ht]
\bc \includegraphics[width=8.6cm]{nopic.png}
\includegraphics[width=8.6cm]{nopic.png} \ec
\vspace{-4mm}
\caption{
Density matrix $\rho$ in the coordinate representation for
$\lambda=0.2$, $\mu=0.1$, $\delta=4$, and $r=0$: $|\rho|$ at the initial time
$t=0$~(a) and $\rho_{\infty}$ for $C=10$~(b).}
\end{figure}
\section{Summary}
\pst
We have studied QD with the Markovian equation of Lindblad for
a system consisting of an one-dimensional harmonic oscillator
in interaction with a thermal bath within the framework of the
theory of open quantum systems based on completely positive quantum dynamical
semigroups. Within the same framework, we investigated
the existence of the asymptotic quantum entanglement for a subsystem
composed of two uncoupled identical harmonic oscillators
interacting with an environment.
\section*{Acknowledgments}
\pst
The author acknowledges the financial support provided within
the Project RFBR~70-1295-74.
\begin{thebibliography}{99}
\bibitem{1}
E. Joos, H. D. Zeh, C. Kiefer, et al.,
{\it Decoherence and the Appearance of a Classical World
in Quantum Theory}, Springer, Berlin (2003).
\bibitem{2}
M. A. Nielsen and I. L. Chuang, {\it Quantum Computation and Quantum
Information}, Cambridge Univ. Press, Cambridge, MS (2000).
\bibitem{3} S. Bandyopadhyay, P. O. Boykin, V. Roychowdhury, and F. Vatan,
{\sl Algorithmica}, \textbf{34}, 512 (2002).
\end{thebibliography}
\end{document}