Monte Carlo Method for Evaluating the Quantum Real Time Propagator

Abstract

A new exact representation of the quantum propagator is derived in terms of semiclassical initial value representations. The resulting expression may be expanded in a series, of which the leading order term is the semiclassical one. Motion of a Gaussian wave packet on a symmetric double well potential is used to demonstrate numerical convergence of the series and the ability to compute each element in the series using Monte Carlo methods.

Tunneling of Ultracold Atoms in Time-Independent Potentials

“…The crucial point is that even for nondissipative systems a semiclassical expression for the time-evolution operator (Equation 15.4) in the long time limit where deep tunneling occurs, is not known. Some progress has been made by analyzing the phase space dynamics of minimal action paths in the complex plane [7,11,12] or by using initial value representations for the quantum propagator [13].…”

Tunneling in two-dimensional systems using a higher-order Herman–Kluk approximation

“…7,8 Although it is sometimes possible to overcome this difficulty by an analysis which provides optimal values for the parameters, 7 this approach has been implemented only for the one-dimensional ͑1D͒ Eckart potential and its application to more general systems seems difficult. A number of additional ways to describe tunneling within the framework of HK-like approximations have been proposed [9][10][11][12][13][14][15][16][17][18] but, in most cases, it is not clear whether they are sufficiently general and numerically efficient for the practical treatment of tunneling in multidimensional systems in the absence of thermal averaging.…”

Non-Markovian Dissipative Semiclassical Dynamics

“…The HK propagator was recently shown to be the leading term of consistent series expansions of the exact quantum propagator of a Hamiltonian system [17,18]. Currently, efforts are under way to directly extend this approach to dynamics with quantum memory effects [19].…”