Disorder-induced broadening of quantum momentum distribution

TL;DR

The paper analyzes a non-interacting 2D quantum gas in a weak, long-range correlated random potential, showing that finite disorder broadens the initial momentum peak and drives isotropization. Using a cumulant expansion and continuum limits, it derives a closed equation for the momentum-space density, then obtains an explicit long-time energy distribution with exponential broadening governed by ε. It further develops a Boltzmann equation on the momentum ring, extracts angular relaxation times, and connects to diffusion in real space via a Green-Kubo framework, yielding a diffusion constant that scales as D ∝ ζ k⁵/ε². The results shed light on disorder-induced mixing and potential thermalization, with validation against numerical simulations and discussion of localization effects in 2D.

Abstract

We study the long-time behavior of a non-interacting two-dimensional quantum gas in a weak random potential with long-range correlations. Any peaked initial momentum distribution will eventually become isotropic and broaden due to scattering events with the random potential. We derive an expression for the long-time average of the momentum distribution and test it against computer simulations. We also discuss momentum isotropization and spatial diffusion.

Figures (3)

• Figure 1: Time evolution of disorder-averaged density field $n = \langle |\phi|^2\rangle$ generated by Eq. \ref{['eq:schrodinger_kspace']} showing momentum isotropization. In the first panel the brightest spot corresponds to the initial peak. Dynamics are averaged over 50 realizations and the parameters are $\epsilon = \frac{1}{32} k_\text{p}^2$, $\zeta=12 k_\text{p}^{-1}$. The densities are normalized s.t. the different panels are on the same scale. See App. \ref{['sec:numerical-methods']} for details on the simulations.
• Figure 2: The profile of the long-time number density $\bar{n}(k)$ is plotted against numerical data. The inset shows the same data on a logarithmic scale revealing the asymmetric behavior around the peak. The numerical data presented here is calculated for $\zeta = 12/k_\text{p}$ and $\epsilon = k_\text{p}^2/32$ and averaged over 100 realizations. For details see App. \ref{['sec:numerical-methods']}.
• Figure 3: The initial peak decays as $\propto e^{-t/t_\text{c}}$, where $t_\text{c}$ is given by Eq. \ref{['eq:decay_rate']}. The numerical results show the correct scaling with $\epsilon$ and $\zeta$. In the first plot $\zeta = 12/k_\text{p}$ and fitting $1/t_{\text{c}} = C_\epsilon \epsilon^2$ gives $C_\epsilon = 0.96$ times the theoretical prediction. In the second plot $\epsilon = 1/24 k_\text{p}^2$ and $1/t_\text{c} = C_\zeta \zeta$; fitting gives $C_\zeta=0.98$ times the theoretical value. In both cases each point is calculated by averaging over 1000 simulations.