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

We examine a model for a massive one-dimensional particle in a singular periodic potential receiving kicks from a gas. Our model is described by a Lindblad equation where the Hamiltonian is a Shrödinger operator with a periodic $\delta$-potential and the noise has a frictionless form arising in a Brownian limit where time is not rescaled. We prove that there is an emergent Markov process governing the quasimomentum distribution in a semi-classical limit. The main result is a proof of a central limit theorem for an integral of this quasimomentum process, which is closely related to the position of the particle. When normalized by $t^{\frac{5}{4}}$, the position process converges to a time-changed Brownian motion whose diffusion rate is determined by the absolute value of the quasimomentum process. The scaling $t^{\frac{5}{4}}$ contrasts with that of $t^{\frac{3}{2}}$ which would be expected for the case of a smooth periodic potential or for a comparable classical processes. The difference is a wave effect driven by Bragg reflections occurring when the particle's momentum is kicked near the half-spaced reciprocal lattice.