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arXiv:1903.08551

Open quantum systems in Heisenberg picture

Fardin Kheirandish

Published Mar 20, 2019• 0 citations
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Abstract

In the framework of the Heisenberg picture, an alternative derivation of the reduced density matrix of a driven dissipative quantum harmonic oscillator as the prototype of an open quantum system is investigated. The reduced density matrix for different initial states of the combined system is obtained from a general formula, and different limiting cases are studied. Exact expressions for the corresponding characteristic function in quantum thermodynamics and Wigner quasi distribution function are found. A possible generalization based on the Magnus expansion of the evolution operator is presented.

DOI
10.48550/arxiv.1903.08551

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The harmonic oscillator with dissipation is studied within the framework of the Lindblad theory for open quantum systems. By using the Wang-Uhlenbeck method, the Fokker-Planck equation, obtained from the master equation for the density operator, is solved for the Wigner distribution function, subject to either the Gaussian type or the δδδ-function type of initial conditions. The obtained Wigner functions are two-dimensional Gaussians with different widths. Then a closed expression for the density operator is extracted. The entropy of the system is subsequently calculated and its temporal behaviour shows that this quantity relaxes to its equilibrium value.
In the framework of the Lindblad theory for open quantum systems the damping of the harmonic oscillator is studied. A generalization of the fundamental constraints on quantum mechanical diffusion coefficients which appear in the master equation for the damped quantum oscillator is presented; the Schrödinger and Heisenberg representations of the Lindblad equation are given explicitly. On the basis of these representations it is shown that various master equations for the damped quantum oscillator used in the literature are particular cases of the Lindblad equation and that the majority of these equations are not satisfying the constraints on quantum mechanical diffusion coefficients. Analytical expressions for the first two moments of coordinate and momentum are also obtained by using the characteristic function of the Lindblad master equation. The master equation is transformed into Fokker-Planck equations for quasiprobability distributions. A comparative study is made for the Glauber PPP representation, the antinormal ordering QQQ representation and the Wigner WWW representation. It is proven that the variances for the damped harmonic oscillator found with these representations are the same. By solving the Fokker-Planck equations in the steady state, it is shown that the quasiprobability distributions are two-dimensional Gaussians with widths determined by the diffusion coefficients. The density matrix is represented via a generating function, which is obtained by solving a time-dependent linear partial differential equation derived from the master equation. Illustrative examples for specific initial conditions of the density matrix are provided.