Semiclassical propagation of Wigner functions

Abstract

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 Schrödinger 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.

Quantum process tomography by 2D fluorescence spectroscopy

“…A variety of applied and foundational problems can be addressed once the process tomography tensor is reconstructed. From a foundational viewpoint, if the process tomography tensor is translated into the phase-space representation of quantum mechanics, it reduces to the propagator of the Wigner function [32][33][34]. Based on this object, it is possible to experimentally reconstruct signatures of quantum chaos such as scars with sub-Planckian resolution [32].…”

Nonclassical phase-space trajectories for the damped harmonic quantum oscillator

“…(32) As in the undamped case, the paths are subject to the boundary conditions q ± (0) = q ′ ± and q ± (t) = q ′′ ± . The two equations of motion (32) replace the equations of motion (11) obtained for the undamped case. As a consequence of the coupling to the heat bath, the equations of motion (32) now include damping terms depending on the friction kernel (25).…”

“…At the same time, this prevents making direct use of semiclassical phase-space dynamics based on the van-Vleck approximation as in Refs. [10,11], since this only applies to sufficiently anharmonic potentials.…”

“…Thus, they do not speak the same language as classical mechanics, which certainly would be desirable in order to describe and study the quantum-classical transition [6][7][8]. In view of these drawbacks, semiclassical methods in phase space offer a conceptually clearer and possibly a numerically more efficient approach [9][10][11].…”