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

We show that a quantum particle in Rd, for d ⩾ 1, subject to a white-noise potential, moves superballistically in the sense that the mean square displacement ∫∥x∥2⟨ρ(x, x, t)⟩dx grows like t3 in any dimension. The white-noise potential is Gaussian distributed with an arbitrary spatial correlation function and a delta correlation function in time. Similar results were established in one dimension by Jayannavar and Kumar [Phys. Rev. Lett. 48(8), 553–556 (1982)], and for any dimension using different methods by Fischer et al. [Phys. Rev. Lett. 73(12), 1578–1581 (1994)]. We also prove that for the same white-noise potential model on the lattice Zd, for d ⩾ 1, the mean square displacement is diffusive growing like t1. This behavior on the lattice is consistent with the diffusive behavior observed for similar models on the lattice Zd with a time-dependent Markovian potential by Kang and Schenker [J. Stat. Phys. 134, 1005–1022 (2009)].