Particle Trajectories for Quantum Maps

TL;DR

This paper investigates how quantum particle trajectories under repeated measurements on a quantized torus converge to classical trajectories, with convergence rates influenced by the system's chaos level, supported by numerical simulations.

Contribution

It demonstrates the convergence of quantum trajectories to classical ones in a semiclassical setting and introduces the concept of uniform defect measures.

Findings

Quantum trajectories converge to classical trajectories up to Ehrenfest time.

Less chaotic systems exhibit longer convergence times.

Numerical simulations support theoretical results.

Abstract

We study the trajectories of a semiclassical quantum particle under repeated indirect measurement by Kraus operators, in the setting of the quantized torus. In between measurements, the system evolves via either Hamiltonian propagators or metaplectic operators. We show in both cases the convergence in total variation of the quantum trajectory to its corresponding classical trajectory, as defined by propagation of a semiclassical defect measure. This convergence holds up to the Ehrenfest time of the classical system, which is larger when the system is less chaotic. In addition, we present numerical simulations of these effects. In proving this result, we provide a characterization of a type of semi-classical defect measure we call uniform defect measures. We also prove derivative estimates of a function composed with a flow on the torus.