Gas of sub-recoiled laser cooled atoms described by infinite ergodic theory

Abstract

The velocity distribution of a classical gas of atoms in thermal equilibrium is the normal Maxwell distribution. It is well known that for sub-recoiled laser cooled atoms Lévy statistics and deviations from usual ergodic behaviour come into play. Here we show how tools from infinite ergodic theory describe the cool gas. Specifically, we derive the scaling function and the infinite invariant density of a stochastic model for the momentum of laser cooled atoms using two approaches. The first is a direct analysis of the master equation and the second following the analysis of Bertin and Bardou using the lifetime dynamics. The two methods are shown to be identical, but yield different insights into the problem. In the main part of the paper we focus on the case where the laser trapping is strong, namely the rate of escape from the velocity trap is $R(v)\propto |v{|}^{\alpha }$ for $v\to 0$ and $\alpha >1$. We construct a machinery to investigate the time averages of physical observables and their relation to ensemble averages. The time averages are given in terms of functionals of the individual stochastic paths, and here we use a generalisation of Lévy walks to investigate the ergodic properties of the system. Exploring the energy of the system, we show that when $\alpha =3$ it exhibits a transition between phases where it is either an integrable or non integrable observable, with respect to the infinite invariant measure. This transition corresponds to very different properties of the mean energy, and to a discontinuous behaviour of the fluctuations. Since previous experimental work showed that both $\alpha =2$ and $\alpha =4$ are attainable we believe that both phases could be explored also experimentally.

arXiv comment32 pages, 11 figures

References in corpus (34)

• Laser cooling of a diatomic molecule
• Lévy walks
• Random Time-Scale Invariant Diffusion and Transport Coefficients
• Universal emission intermittency in quantum dots, nanorods, and nanowires
• Stochastic Ergodicity Breaking: a Random Walk Approach
• Observation of anomalous diffusion and fractional self-similarity in one dimension
• Exact and explicit probability densities for one-sided Levy stable distributions
• The Continuous Time Random Walk, still trendy: Fifty-year history, state of art, and outlook
• From the area under the Bessel excursion to anomalous diffusion of cold atoms
• Theory of fractional-Lévy kinetics for cold atoms diffusing in optical lattices
• The Single Big Jump Principle in Physical Modelling
• Pesin-Type Identity for Weak Chaos
• Single-atom quantum probes for ultracold gases using nonequilibrium spin dynamics
• A Single Atom Thermometer for Ultracold Gases
• From Non-normalizable Boltzmann-Gibbs statistics to infinite-ergodic theory
• Distributional Response to Biases in Deterministic Superdiffusion
• Aging generates regular motions in weakly chaotic systems
• Anomalous diffusion in random dynamical systems
• Distributional Ergodicity in Stored-Energy-Driven Lévy Flights
• Anomalous spatial diffusion and multifractality in optical lattices
• Renewal theory with fat tailed distributed sojourn times: typical versus rare
• Phase Diagram in Stored-Energy-Driven Lévy Flight
• Statistics of the longest interval in renewal processes
• Infinite ergodic theory meets Boltzmann statistics
• Transitions in the ergodicity of subrecoil-laser-cooled gases
• Infinite invariant density in a semi-Markov process with continuous state variables
• From laser cooling to aging: a unified Levy flight description
• Distributional behavior of time averages of non-$L^1$ observables in one-dimensional intermittent maps with infinite invariant measures
• Infinite density for cold atoms in shallow optical lattices
• Statistics of occupation time and connection to local properties of non-homogeneous random walks
• Statistics of the occupation time of renewal processes
• Extreme value theory for constrained physical systems
• Infinite ergodic theory for three heterogeneous stochastic models with application to subrecoil laser cooling
• Renewal processes and fluctuation analysis of molecular motor stepping

Cited by in corpus (3)

• Infinite ergodic theory for three heterogeneous stochastic models with application to subrecoil laser cooling
• Anomalous statistics in the Langevin equation with fluctuating diffusivity: from Brownian yet non-Gaussian diffusion to anomalous diffusion and ergodicity breaking
• Non-self-averaging of current in a totally asymmetric simple exclusion process with quenched disorder