Distinguishing quantum features in classical propagation

TL;DR

This paper demonstrates that a classical approximation of quantum dynamics, the truncated chord approximation, captures key quantum features within a small phase space region, validated through numerical comparisons.

Contribution

It reveals that the truncated chord approximation aligns with the full semiclassical quantum characteristic function near the origin, capturing quantum features.

Findings

Truncated chord approximation matches semiclassical quantum evolution near the origin.

Quantum features like blind spots are contained within a small phase space region.

Numerical results confirm the approximation's validity for Kerr Hamiltonian evolution.

Abstract

The strictly classical propagation of an initial Wigner function, referred to as TWA or LSC-IVR, is considered to provide approximate averages, despite not being a true Wigner function: it does not represent a positive operator. We here show that its symplectic Fourier transform, the truncated chord approximation (TCA), coincides with the full semiclassical approximation to the evolved quantum characteristic function (or chord function) in a narrow neighbourhood of the origin of the dual chord phase space. Surprisingly, this small region accounts for purely quantum features, such as blind spots and local wave function correlations, as well as the expectation of observables with a close classical correspondence. Direct numerical comparison of the TCA with exact quantum results verifies the semiclassical predictions for an initial coherent state evolving under the Kerr Hamiltonian. The resulting clear criterion for any further features, which may be estimated by classical propagation, is that, within the chord representation, they are concentrated near the origin.