Dynamics of interacting bosons using the Herman-Kluk semiclassical initial value representation

TL;DR

This paper evaluates the Herman-Kluk semiclassical propagator for interacting bosons, demonstrating its accuracy in reproducing quantum spectra and dynamics, especially in the large particle number limit, and compares it with the frozen Gaussian approximation.

Contribution

It shows that the Herman-Kluk propagator accurately describes interacting bosons in the semiclassical limit and highlights the limitations of the frozen Gaussian approximation.

Findings

HK propagator conserves unitarity at large n

FGA violates unitarity for non-zero interactions

HK reproduces the exact spectrum in the large n limit

Abstract

Recent experimental progress using ultracold gases in optical lattices necessitates a quantitative theoretical description for a significant number of bosons. In the present paper, we investigate if time-dependent semiclassical initial value methods, with propagators expressed as integrals over phase space using classical trajectories, is suitable to describe interacting bosons, concentrating on a single mode. Despite the nonlinear contribution from the self-interaction, the corresponding classical dynamics allows for a largely analytical treatment of the semiclassical propagator. We find that the Herman-Kluk (HK) propagator conserves unitarity in the semiclassical limit ($n\to\infty$), but a decay of the norm is seen for low $n$. The frozen Gaussian approximation (FGA, i.e. HK with unit prefactor) is explicitly shown to violate unitarity in the present system for non-vanishing interaction strength, even in the semiclassical limit. Furthermore, we show by evaluating the phase space integral in steepest descent approximation, that the HK propagator reproduces the exact spectrum correctly in the limit $n\to \infty$. An error is, however, incurred in next-to-next-to-leading order (small parameter $1/n$), as seen upon numerical evaluation of the integral or by considering finite $n$ corrections to steepest descent. The FGA, in contrast, is only accurate to lowest order, and an erroneous next-to-leading order term in the energy spectrum was found analytically. Finally, as an example application, we study the dynamics of wave packets by computing the time evolution of the Wigner function. While the often-used truncated Wigner approximation cannot capture any interferences present in the exact quantum mechanical solution (known analytically), we find that the HK approach, despite also using classical information only, reproduces the salient features of the exact solution correctly.