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International Journal of Modern Physics B

Phase-space description of a particle in a quartic double-well potential

Adina Ceausu–Velcescu, Paul Blaise, Yuri P. Kalmykov

Published Jul 3, 2014• 5 citations
Publisher

Abstract

The stationary Wigner functions (WFs) have been calculated for particles evolving in a quartic double-well potential V(x) = ax2/2+bx4/4(a < 0 and b > 0), at temperature T. In the high temperature limit, the results totally agree with those obtained using Wigner's perturbative method of deriving quantum corrections to the classical distribution function. Comparison with the perturbative approach allows one to establish the range of applicability of the latter. For illustration, the second moments of the position and momentum have been calculated for the double-well potential. Furthermore, the time-evolution of the WFs for a state initially located at one of the wells has been also investigated to show the tunneling effect.

DOI
10.1142/s0217979214501641

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The stationary Wigner functions (WFs) have been calculated for particles evolving in a quartic double-well potential V(x) = ax2/2+bx4/4(a < 0 and b > 0), at temperature T. In the high temperature limit, the results totally agree with those obtained using Wigner's perturbative method of deriving quantum corrections to the classical distribution function. Comparison with the perturbative approach allows one to establish the range of applicability of the latter. For illustration, the second moments of the position and momentum have been calculated for the double-well potential. Furthermore, the time-evolution of the WFs for a state initially located at one of the wells has been also investigated to show the tunneling effect.
In the second part of this paper on micro-canonical ensemble a new numerical approach for consideration of quantum dynamics and calculations of the average values of quantum operators and time correlation functions in the Wigner representation of quantum statistical mechanics has been developed. The time correlation functions have been presented in the form of the integral of the Weyl's symbol of the considered operators and the Fourier transform of the product of matrix elements of the dynamic propagators. For the latter function the integral Wigner–Liouville's type equation has been derived. The initial condition for this equation has been obtained in the form of the Fourier transform of the Wiener path integral representation of the matrix elements of the propagators at initial time. A numerical procedure for solving this equation combining both molecular dynamics and Monte Carlo methods has been developed. Numerical results have been obtained for series of average values of the quantum operators as well as for the time correlation function characterizing the energy level structure, the momentum flow of tunnelling particles at barrier crossing and the absorption spectra of electrons in a potential well. The developed quantum dynamics method was tested by comparison of numerical results with analytical estimations. Tunnelling transitions and the effect of the quasi stationary state have been considered as the reasons for the peculiarities in behaviour of the time correlation functions and position and momentum dispersions. The possibility of applying the developed approach to the theory of classical wave propagation in random media has also been considered. For classical waves some results have been obtained for Gaussian beam propagation in 2D and 3D waveguides.
Recent progress in our understanding of quantum effects on the Brownian motion in an external potential is reviewed. This problem is ubiquitous in physics and chemistry, particularly in the context of decay of metastable states, for example, the reversal of the magnetization of a single domain ferromagnetic particle, kinetics of a superconducting tunnelling junction, etc. Emphasis is laid on the establishment of master equations describing the diffusion process in phase space analogous to the classical Fokker-Planck equation. In particular, it is shown how Wigner's [E. P. Wigner, Phys. Rev., 1932, 40, 749] method of obtaining quantum corrections to the classical equilibrium Maxwell-Boltzmann distribution may be extended to the dissipative non-equilibrium dynamics governing the quantum Brownian motion in an external potential V(x), yielding a master equation for the Wigner distribution function W(x,p,t) in phase space (x,p). The explicit form of the master equation so obtained contains quantum correction terms up to o(h(4)) and in the classical limit, h --> 0, reduces to the classical Klein-Kramers equation. For a quantum oscillator, the method yields an evolution equation coinciding in all respects with that of Agarwal [G. S. Agarwal, Phys. Rev. A, 1971, 4, 739]. In the high dissipation limit, the master equation reduces to a semi-classical Smoluchowski equation describing non-inertial quantum diffusion in configuration space. The Wigner function formulation of quantum Brownian motion is further illustrated by finding quantum corrections to the Kramers escape rate, which, in appropriate limits, reduce to those yielded via quantum generalizations of reaction rate theory.
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Quantum tunneling effects of Gaussian wave packet in one- and two-dimensional double well potentials are investigated using quantum Liouville equation for time evolution of Wigner distribution in phase space. It is shown that a trajectory-based solution of this problem can be constructed by the entangled trajectory ensemble simulating the evolving quantum state. Quantum effects arise in this approach as a breakdown of the statistical independence of the trajectory ensemble. © 2014 Wiley Periodicals, Inc.
We present a comprehensive study of semiclassical phase-space propagation in the Wigner representation, emphasizing numerical applications, in particular as an initial-value representation. Two semiclassical approximation schemes are discussed. The propagator of the Wigner function based on van Vleck's approximation replaces the Liouville propagator by a quantum spot with an oscillatory pattern reflecting the interference between pairs of classical trajectories. Employing phase-space path integration instead, caustics in the quantum spot are resolved in terms of Airy functions. We apply both to two benchmark models of nonlinear molecular potentials, the Morse oscillator and the quartic double well, to test them in standard tasks such as computing autocorrelation functions and propagating coherent states. The performance of semiclassical Wigner propagation is very good even in the presence of marked quantum effects, e.g., in coherent tunneling and in propagating Schrodinger cat states, and of classical chaos in four-dimensional phase space. We suggest options for an effective numerical implementation of our method and for integrating it in Monte-Carlo-Metropolis algorithms suitable for high-dimensional systems.
A study is provided of dynamics and dissipative tunneling in a symmetric quartic double well potential. The numerical solution for the position autocorrelation function obtained through the Wigner–Fokker–Planck equation is compared with numerically exact results of Stockburger and Mak [J. Chem. Phys. 110, 4983 (1999)]. We find that the Wigner–Fokker–Planck dynamics agree well with the numerically exact computations, they account for both quantum coherences as well as quantum tunneling phenomena. This, in contrast to the mixed quantum classical approximation, which does not perform as well.