Nonlinear dynamics of open quantum systems



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Abstract

The evolution of a composite closed system using the integral wave equation with the kernel in the form of path integral is considered. It is supposed that a quantum particle is a subsystem of this system. The evolution of the reduced density matrix of the subsystem is described on the basis of the integral wave equation for a composite closed system. The equation for the density matrix for such a system is derived. This equation is nonlinear and depends on the history of the processes in the closed system. It is shown that, in general, the reduced density matrix trace does not conserve in the evolution processes progressing in open systems and the procedure of the trace normalization is necessary as the mathematical image of a real nonlocal physical process. The wave function collapse and EPR correlation are described using this approach.

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1. Introduction. The principle differences between the dynamics of an open system and the evolution of a closed one can not solely restricted by the irreversibility of the former. In general, there exist nonlinear transformations of the reduced density matrices of open systems. Depending on the specific properties of the system, these nonlinear processes can take the form of the wave function collapse in process of the measurement, the decoherence phenomenon, etc. Any open system can be considered as a subsystem of a large closed system obeying the linear evolution law. The impossibility to describe the nonlinear state transformation of an open system under the measurement using the Schrödinger equation 214 Nonlinear dynamics of open quantum systems led to the necessity to formulate a particular reduction postulate [1] (the quantum jump notion [2]). The peculiarity of the problem is that there is no cause, expressed in precise physical terms, determining the form of the transformation of the quantum state [3]. Except for nonlinearity the open system dynamics, in general, has one more specific property - the dependence on the evolution history. This property already emerges in the correlation of the uncertainty of the measured value of the stationary state energy with the duration of the measurement process and becomes apparent when considering the EPR paradox [4]1 . We assume that the Schrödinger equation is absolutely accurate when describing the evolution of closed quantum systems for infinitesimal time intervals2. The unique strict generalization of Schrödinger’s equation on finite time intervals is the integral wave equation with the kernel in the form of path integral [6, 7]. The action functionals entering into the integral evolution operator generates the dependence of the quantum system state on the evolution history. Besides, the mathematical form of this law supposes the existence of a subsystem nonlinear evolution [8, 9]. Open system quantum states can be described by reduced density matrices. A corresponding evolution equation is usually derived by considering a large closed system including this open system
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About the authors

Alexey Yu Samarin

Samara State Technical University

Email: samarinay@yahoo.com
Cand. Phys. & Math. Sci.; Associate Professor; Dept. of General Physics and Physics of Oil and Gas Production 244, Molodogvardeyskaya st., Samara, 443100, Russian Federation

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