Damping of Bloch oscillations: Variational solutions of the Boltzmann equation beyond linear response

TL;DR

This paper extends the variational approach to solve the Boltzmann equation beyond linear response, analyzing Bloch oscillation damping in lattice models with weak interactions relevant for ultracold atom experiments.

Contribution

It develops a simplified dynamic theory for high-entropy lattice systems, enabling analytic solutions for Bloch oscillation damping beyond linear response regimes.

Findings

Derived damping coefficients for Bloch oscillations in the Hubbard model.

Computed damping in a one-dimensional toy model.

Solved long-time dynamics showing subdiffusive scaling.

Abstract

Variational solutions of the Boltzmann equation usually rely on the concept of linear response. We extend the variational approach for tight-binding models at high entropies to a regime far beyond linear response. We analyze both weakly interacting fermions and incoherent bosons on a lattice. We consider a case where the particles are driven by a constant force, leading to the well-known Bloch oscillations, and we consider interactions that are weak enough not to overdamp these oscillations. This regime is computationally demanding and relevant for ultracold atoms in optical lattices. We derive a simple theory in terms of coupled dynamic equations for the particle density, energy density, current and heat current, allowing for analytic solutions. As an application, we identify damping coefficients for Bloch oscillations in the Hubbard model at weak interactions and compute them for a one-dimensional toy model. We also approximately solve the long-time dynamics of a weakly interacting, strongly Bloch-oscillating cloud of fermionic particles in a tilted lattice, leading to a subdiffusive scaling exponent.