Dynamical approximations for composite quantum systems: Assessment of error estimates for a separable ansatz

TL;DR

This paper evaluates the accuracy of a separable Hartree approximation in modeling the dynamics of composite quantum systems with different scales, comparing it to fully correlated solutions and confirming error estimates.

Contribution

It provides a numerical assessment of error estimates for a separable approximation in a specific quantum tunneling system, validating theoretical predictions.

Findings

The Hartree approximation's error is accurately predicted on moderate time scales.

Coupling and correlations significantly influence the approximation's accuracy.

The study demonstrates the effectiveness of theoretical error estimates in practical simulations.

Abstract

Numerical studies are presented to assess error estimates for a separable (Hartree) approximation for dynamically evolving composite quantum systems which exhibit distinct scales defined by their mass and frequency ratios. The relevant error estimates were formally described in our previous work [I. Burghardt, R. Carles, C. Fermanian Kammerer, B. Lasorne, C. Lasser, J. Phys. A. 54, 414002 (2021)]. Specifically, we consider a representative two-dimensional tunneling system where a double well and a harmonic coordinate are cubically coupled. The timedependent Hartree approximation is compared with a fully correlated solution, for different parameter regimes. The impact of the coupling and the resulting correlations are quantitatively assessed in terms of a time-dependent reaction probability along the tunneling coordinate. We show that the numerical error is correctly predicted on moderate time scales by a theoretically derived error estimate.