Suppressed dispersion for a randomly kicked quantum particle in a Dirac comb

TL;DR

This paper investigates a quantum particle in a periodic delta potential under random kicks, revealing a suppressed dispersion effect and a unique scaling limit due to wave phenomena like Bragg reflections.

Contribution

It introduces a semi-classical limit analysis showing a novel $t^{5/4}$ scaling for the momentum integral, highlighting wave effects in quantum dispersion.

Findings

The momentum distribution follows an emergent Markov process.

A central limit theorem describes the time integral of momentum.

The scaling differs from classical or smooth potential cases, indicating wave effects.

Abstract

I study a model for a massive one-dimensional particle in a singular periodic potential that is receiving kicks from a gas. The model is described by a Lindblad equation in which the Hamiltonian is a Schr\"odinger operator with a periodic $\delta$-potential and the noise has a frictionless form arising in a Brownian limit. I prove that an emergent Markov process in a semi-classical limit governs the momentum distribution in the extended-zone scheme. The main result is a central limit theorem for a time integral of the momentum process, which is closely related to the particle's position. When normalized by $t^{5/4}$, the integral process converges to a time-changed Brownian motion whose rate depends on the momentum process. The scaling $t^{5/4}$ contrasts with $t^{3/2}$, which would be expected for the case of a smooth periodic potential or for a comparable classical process. The difference is a wave effect driven by Bragg reflections that occur when the particle's momentum is kicked near the half-spaced reciprocal lattice.