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 behavior come into play. In a recent letter, we showed how tools from infinite ergodic theory describe the cool gas. Here, using the master equation, we derive the scaling function and the infinite invariant density of a stochastic model for the momentum of laser cooled atoms, recapitulating results obtained by Bertin and Bardou [Am. J. Phys. 76, 630 (2008)] using life-time statistics. We focus on the case where the laser trapping is strong, namely, the rate of escape from the velocity trap is R(v) ∝ |v|α for v → 0 and α > 1. We construct a machinery to investigate 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 generalization of Lévy walks to investigate the ergodic properties of the system. Exploring the energy of the system, we show that when α = 3, it exhibits a transition between phases where it is either an integrable or a 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 behavior of fluctuations. While the integrable phase is described by universal statistics and the Darling-Kac law, the more challenging case is the exploration of statistical properties of non-integrable observables. Since previous experimental work showed that both α = 2 and α = 4 are attainable, we believe that both phases could also be explored experimentally.
Highlights
Laser cooled atoms and molecules are important for fundamental and practical applications [1–4]
The upper limit is taken to be infinity and we find in the small p limit φ2 (0, p) ∼ 1 −γ πγ/vmax sin πγ
Inserting Eqs. (116, 118) in Eq (112) we find the small p limit of Ĥ1 (p) and Ĥ2 (p)
Summary
Laser cooled atoms and molecules are important for fundamental and practical applications [1–4]. It is well known that Lévy statistics describes some of the unusual properties of cooling processes [5–9]. For sub-recoil laser cooling a special atomic trap in momentum space is engineered. The most efficient cooling is found when a mean trapping time, defined more precisely below, diverges [7]. In this sense the dynamics is time-scale-free. The fact that the characteristic time diverges, implies that the processes involved are non-stationary. In the physics literature they are sometimes called non-ergodic
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