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The Journal of Atomic and Molecular Sciences

Time Evolution of Wigner Function in Dissipative Quantum Systems Using Entangled Trajectory Molecular Dynamics

Lifei Wang Lifei Wang, Yan Wang Yan Wang, Qin Zhang Qin Zhang, Haibo Sun, Juan Zhao Juan Zhao

Published Jul 1, 2016• 0 citations

Abstract

The dissipative quantum systems are treated using Klein-Kramers equation, combined with the Gaussian kernel trajectory ensemble, for time evolution of Wigner function $ρ_ω(q, p, t)$ in phase space. The entangled trajectory molecular dynamics approach is used to obtain trajectory solutions for the Klein-Kramers equation with three models: free particle, damped harmonic oscillator and metastable potential. It is found that the performance of semiclassical Wigner propagation is effectively for the relaxation of damped harmonic oscillator and dissipative decay of a metastable state. In addition, the energy of trajectory ensemble decays faster with smaller friction value and changes slightly with variable temperature parameters.

DOI: 10.4208/jams.053016.072116a

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The dissipative quantum systems are treated using Klein-Kramers equation, combined with the Gaussian kernel trajectory ensemble, for time evolution of Wigner function

ρ_ω(q, p, t)

in phase space. The entangled trajectory molecular dynamics approach is used to obtain trajectory solutions for the Klein-Kramers equation with three models: free particle, damped harmonic oscillator and metastable potential. It is found that the performance of semiclassical Wigner propagation is effectively for the relaxation of damped harmonic oscillator and dissipative decay of a metastable state. In addition, the energy of trajectory ensemble decays faster with smaller friction value and changes slightly with variable temperature parameters.

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.

We obtain analytic solutions to various models of dissipation of the quantum harmonic oscillator, employing a simple method in the Wigner function Fourier transform description of the system; and study as an exemplification, the driven open quantum harmonic oscillator. The environmental models we use are based on optical master equations for the zero and finite temperature bath and whose open dynamics are described by a Lindblad master equation, and also we use the Caldeira-Leggett model for the high temperature limit, in the the under damped an the over damped case. Under the Wigner Fourier transform or chord function as it has been called, it becomes particularly simple to solve the dynamics of the open oscillator in the sense that the dynamics of the system are reduced to the application of an evolution matrix related to the damped motion of the oscillator.

In this paper we present a trajectory-based formulation of entanglement dynamics by employing the Wigner function. The linear entropy of a single trajectory is derived based on the trajectory evolution of the Wigner function. The entanglement dynamics with a separable Gaussian initial state is investigated using different values of Planck's constant $\ensuremath{\hbar}$ in the evolution of the trajectories, while quantum-classical correspondence in entanglement dynamics is investigated. In addition, we show why chaos can increase entanglement rapidly through analytical results and physical pictures of the underlying dynamics in phase space.

In this paper, we extend the entangled trajectory molecular dynamics (ETMD) method to multidimensional systems. The integrodifferential form of the evolution equation for the Wigner function is employed, allowing general potentials not represented as a polynomial to be treated. As the example, the method is applied to a two-dimensional model of scattering from an Eckart barrier. The results of ETMD are in good agreement with quantum hydrodynamics and exact quantum simulations. By comparing the quantum and classical trajectory in phase space, the quantum tunneling phenomenon is interpreted vividly.