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Solutions to linear dissipative quantum systems. (English) Zbl 1465.81059

Summary: We use the characteristic function of the Wigner function (its double Fourier transform) to give solution to any generic open quantum linear systems (systems whose Hamiltonian is at most quadratic). The solution is carried out in terms of the application of the transition matrix of the dynamical evolution in the Fourier space. We address two cases: the time-independent coefficients for which we give the solutions for several dissipative models of the quantum harmonic oscillator and the one-dimensional free particle. In the latter, we also derive a heuristic model for a pure damped motion with suppression of diffusion. For the time-dependent coefficient problem, we give some particular cases that are integrable and derive a second order approximation to the generic case in which all parameters are time-dependent. We additionally explore the solutions of the system when the diffusion processes of the dissipative model lie beyond the weak coupling limit.
©2021 American Institute of Physics

MSC:

81S30 Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics
81S22 Open systems, reduced dynamics, master equations, decoherence
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
81Q15 Perturbation theories for operators and differential equations in quantum theory
37L05 General theory of infinite-dimensional dissipative dynamical systems, nonlinear semigroups, evolution equations
35Q41 Time-dependent Schrödinger equations and Dirac equations
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