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

TL;DR

This paper demonstrates that a quantum particle in a white-noise potential exhibits super-ballistic transport in continuous space and diffusive behavior on a lattice, with energy increasing linearly over time.

Contribution

It extends known results by proving super-ballistic motion in continuous space and diffusive motion on the lattice for quantum particles under white-noise potentials.

Findings

Mean square displacement grows like t^3 in continuous space.

Energy increases linearly with time.

On the lattice, mean square displacement grows like t.

Abstract

We show that a quantum particle in $\mathbb{R}^d$, for $d \geq 1$, subject to a white-noise potential, moves super-ballistically in the sense that the mean square displacement $\int \|x\|^2 \langle \rho(x,x,t) \rangle ~dx$ 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, M\"uller 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 $d \geq 1$, 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).