Wigner function approach to the quantum Brownian motion of a particle in a potential

Abstract

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.

Nonlinear noninertial response of a quantum Brownian particle in a tilted periodic potential to a strong ac force as applied to a point Josephson junction

“…[19][20][21] If one wishes to treat accurately the GHz and THz regions, the complete phase-space distribution W͑ , p , t͒ ͑p = m˙͒ must be used, giving rise, [15][16][17][18]30 on expansion of the momentum part of the distribution in orthogonal Hermite polynomials, to a hierarchy of partial differential recurrence relations in configuration space. The actual configuration-space distribution P͑ , t͒ must then be extracted from the hierarchy, usually by continued fractions, with the much simpler QSE naturally emerging 15,17 from the hierarchy in the high-damping limit …”

“…The sole exception, however, is the overdamped ͑or noninertial͒ limit, where the governing equation is the quantum Smoluchowski equation 15,17 ‫ץ‬ P͑x,t͒…”

“…15 The QSE ͑6͒ is formally equivalent to the diffusion equation in configuration space describing classical noninertial Brownian motion in a potential with coordinatedependent diffusion coefficient D. Thus, the dynamics of the system described by the QSE ͑6͒ may be equivalently described using a quantum analog of the noninertial Langevin equation with multiplicative noise ͑see Appendix A͒. In the classical limit ⌳ = 0, Eq.…”

“…8-16͒ has recently been used 15,17 to derive a quantum Smoluchowski equation ͑QSE͒ governing the time evolution of the configuration-space distribution function P͑x , t͒ for particles with separable and additive Hamiltonians in the overdamped ͑or noninertial͒ limit. In the present context, pertaining to a quantum Brownian particle of mass m moving along the x axis under the influence of a potential V͑x͒, the canonical variables are the position x and the momentum p of the particle.…”

“…͑1͒ in phase space then describes the relaxation of W͑x , p , t͒ to the stationary state given by W 0 ͑x , p͒ in the long time limit. 15,17 The ansatz that the Wigner stationary distribution W 0 ͑x , p͒ renders the right-hand side of the master Eq. ͑1͒ zero ͑whence the diffusion coefficients must depend on the derivatives of the potential͒ may be used if the interactions between the Brownian particle and the heat bath are small enough to allow one to use the weak-coupling limit, and if the correlation time characterizing the bath is so short that one can regard the stochastic process originating in the bath as Markovian.…”