Heavy-tailed phase-space distributions beyond Boltzmann-Gibbs and equipartition: Statistics of confined cold atoms

TL;DR

This paper explores the stationary phase-space distributions of cold atoms under Sisyphus cooling, revealing heavy tails and deviations from Boltzmann-Gibbs statistics, especially under strong confinement and high energies.

Contribution

It provides analytical expressions and numerical analysis of non-thermal phase-space distributions, highlighting violations of energy equipartition and detailed balance in cold atom systems.

Findings

Phase-space density exhibits heavy power-law tails.

Boltzmann-Gibbs approximation holds for deep lattices and certain conditions.

Explicit violations of detailed balance in non-equilibrium steady states.

Abstract

The Boltzmann-Gibbs density, a central result of equilibrium statistical mechanics, relates the energy of a system in contact with a thermal bath to its equilibrium statistics. This relation is lost for non-thermal systems such as cold atoms in optical lattices, where the heat bath is replaced by the laser beams of the lattice. We investigate in detail the stationary phase-space probability for Sisyphus cooling under harmonic confinement. In particular, we elucidate whether the total energy of the system still describes its stationary state statistics. We find that this is true for the center part of the phase-space density for deep lattices, where the Boltzmann-Gibbs density provides an approximate description. The relation between energy and statistics also persists for strong confinement and in the limit of high energies, where the system becomes underdamped. However, the phase-space density now exhibits heavy power-law tails. In all three cases we find expressions for the leading order phase-space density and corrections which break the equivalence of probability and energy and violate energy equipartition. The non-equilibrium nature of the steady state is confounded by explicit violations of detailed balance. We complement these analytical results with numerical simulations to map out the intricate structure of the phase-space density.