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_((DECOHERENCE AND ASYMPTOTIC ENTANGLEMENT
IN OPEN QUANTUM DYNAMICS
Author One1* and Author Two2
1Department of Physics, National Univercity 1, Moscow, Russia 119991
2Department of Physics, National Univercity 2, SaintPetersburg, Russia 119991
*Corresponding author email: author1 @moscow.rus
Abstract
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 longtime regime.
Keywords: open systems, quantum decoherence, quantum entanglement, nonseparable states.
1 Introduction
By quantum decoherence (QD) [1, 2] we understand the irreversible, uncontrollable, and persistent formation of quantum correlations (entanglement) of a system with its environment [1], expressed by the damping of the coherences present in the quantum state of the system, when offdiagonal 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.
2 Markovian Master Equation for a Harmonic Oscillator
The normalized evolution equation of the soliton propagating along the longitudinal direction z in onedimensional waveguides with periodic refractiveindex modulation in the transverse direction x and a Kerrtype selffocusing nonlinear media reads [3]
i "z = ""2u+2 "x2 " u2u " pV(x)u, (1)
where p is the modulationdepth parameter, V(x) = cos(Xx) is the periodicallymodulated refractive index determining the waveguide array, and T = 2/X describes the modulation period. In our analysis, the initial light beam has the form [3]
u(z = 0, x) = A sech (x " x0) sin(x " x0) + A sech (x + x0) sin(x + x0). (2)
3 Quantum Decoherence
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 offdiagonal 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 offdiagonal 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, 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 offdiagonal elements get smaller and smaller.
Figure 1: Density matrix in the coordinate representation for = 0.2, = 0.1, = 4, and r = 0:  at the initial time t = 0 (a) and " for C = 10 (b).
4 Summary
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.
Acknowledgments
The authors acknowledge the financial support provided within the Project RFBR 70129574.
References
[1] E. Joos, H. D. Zeh, C. Kiefer, et al., Decoherence and the Appearance of a Classical World in Quantum Theory, Springer, Berlin (2003).
[2] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge Univ. Press, Cambridge, MS (2000).
[3] S. Bandyopadhyay, P. O. Boykin, V. Roychowdhury, and F. Vatan, Algorithmica, 34, 512 (2002).
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