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 Shr\"odinger 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.