Suppressed Dispersion for a Randomly Kicked Quantum Particle in a Dirac Comb

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 Schrödinger operator with a periodic δ-potential and the noise has a frictionless form arising in a Brownian limit. I prove that an emergent Markov process in an adiabatic 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 \(t^{\frac{5}{4}}\) the integral process converges to a time-changed Brownian motion whose diffusion rate depends on the momentum process. The scaling \(t^{\frac{5}{4}}\) contrasts with \(t^{\frac{3}{2}}\), 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 momentum reflections that occur when the particle’s momentum is kicked near the half-spaced reciprocal lattice of the potential.