Transport of a quantum particle in a time-dependent white-noise potential

Abstract

We show that a quantum particle in ${\mathbb{R}}^d$, for $d\ge 1$, subject to a white-noise potential, moves super-ballistically in the sense that the mean square displacement $\int \Vert x{\Vert }^2⟨\rho (x,x,t)⟩dx$ grows like $t^3$ in any dimension. The white noise potential is Gaussian distributed with an arbitrary spatial correlation function and a delta correlation function in time. This is a known result in one dimension (see refs. Fischer, Leschke, Müller and Javannar, Kumar}. The energy of the system is also shown to increase linearly in time. We also prove that for the same white-noise potential model on the lattice ${\mathbb{Z}}^d$, for $d\ge 1$, the mean square displacement is diffusive growing like $t^1$. This behavior on the lattice is consistent with the diffusive behavior observed for similar models in the lattice ${\mathbb{Z}}^d$ with a time-dependent Markovian potential (see ref. Kang, Schenker).

18 pages