5 - The proportion of atoms trapped in quasi-dark states

Summary

We now have all the mathematical tools in hand to address the important questions for the cooling process, namely: what is the proportion f_trap(θ) of ‘trapped’ atoms (i.e. those which have a very small momentum p < p_trap); what is the ‘line shape’, i.e. the momentum distribution, after an interaction time θ?

In Section 5.1, we define precisely the trapped proportion f_trap(θ) in terms of an ensemble average and compare it to a time average defined as the mean fraction of the time spent in the trap. The two averages do not always coincide, as shown by the explicit computation of Section 5.2. This reveals the non-ergodic character of the cooling process, as discussed in Section 5.3.

Ensemble averages versus time averages

We define the trapped proportion f_trap(θ) as the probability of finding the atom in the trap at time t = θ. Therefore, f_trap(θ) corresponds to an ensemble average, over many independent realizations of the stochastic process of Fig. 3.1. It is instructive to consider also a time average, by examining how a given atom shares its time between the ‘inside’ and the ‘outside’ of the trap. Because of the non-ergodic character of subrecoil laser cooling, ensemble averages and time averages do not in general coincide. In fact, we will see later on that the ensemble average f_trap(θ) and the time average only coincide when 〈τ〉 and are finite, whereas they differ when either µ or is smaller than one.