In this chapter, we first recall (in Section 2.1) a few properties of the most usual laser cooling schemes, which involve a friction force. In such standard situations, the motion of the atom in momentum space is a Brownian motion which reaches a steady-state, and the recoil momentum of an atom absorbing or emitting a single photon appears as a natural limit for laser cooling. We then describe in Section 2.2 some completely different laser cooling schemes, based on inhomogeneous random walks in momentum space. These schemes, which are investigated in the present study, allow the ‘recoil limit’ to be overcome. They are associated with non-ergodic statistical processes which never reach a steady-state. Section 2.3 is devoted to a brief survey of various quantum descriptions of subrecoil laser cooling, which become necessary when the ‘recoil limit’ is reached or overcome. The most fruitful one, in the context of this work, is the ‘quantum jump description’ which will allow us in Section 2.4 to replace the microscopic quantum description of subrecoil cooling by a statistical study of a related classical random walk in momentum space. It is this simpler approach that will be used in the subsequent chapters to derive some quantitative analytical predictions, in cases where the quantum microscopic approach is unable to yield precise results, in particular in the limit of very long interaction times, and/or for a momentum space of dimension D higher than 1.

We establish here the correspondence between the statistical models introduced in Chapter 3 and the quantum evolution of atoms undergoing subrecoil laser cooling. This enables us to establish analytical expressions connecting the parameters of the statistical models (τ0, p0, pD, Δp, pmax, τb and) to atomic and laser parameters relevant to subrecoil laser cooling.

Such a ‘dictionary’ is useful for the numerical estimation of the results derived in this book (see Chapter 8). It also leads to analytical relations between τb and, which are used for cooling optimization (see Chapter 9).

We first treat in detail Velocity Selective Coherent Population Trapping in Section A.1. Analytical expressions are given for the statistical parameters. Special attention is given to the p-dependences of the jump rates both for small p and for large p, because they control the asymptotic behaviours of the trapping and recycling times. It is thus important to include these p-dependences correctly in the simplified jump rates in order to ensure the validity of the statistical model. Raman cooling is then briefly treated in Section A.2.

Velocity Selective Coherent Population Trapping

We first present the quantum optics treatment of one-dimensional σ+/σ− VSCPT (Section A.1.1).

We now have all the mathematical tools in hand to address the important questions for the cooling process, namely: what is the proportion ftrap(θ) of ‘trapped’ atoms (i.e. those which have a very small momentum p < ptrap); 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 ftrap(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 ftrap(θ) as the probability of finding the atom in the trap at time t = θ. Therefore, ftrap(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 ftrap(θ) and the time average only coincide when 〈τ〉 and are finite, whereas they differ when either µ or is smaller than one.

Introduction

The statistical approach presented in this book provides not only a deeper physical understanding of subrecoil cooling, but also analytical expressions for the various characteristics of the momentum distribution of the cooled atoms. A great confidence in the validity of these predictions has been obtained in the previous chapter, by comparing them with experimental and numerical results. Therefore, we are now entitled to apply the approach developed in this work to specific problems, such as the optimization of one particular feature of the cooling process, namely the height of the peak of cooled atoms. This is the subject of this chapter.

Finding empirically the optimum conditions for a subrecoil cooling experiment is a difficult task. There are a priori many parameters to be explored and each experiment with a given set of parameters is in itself lengthy. The same can also be said of numerical simulations. One needs guidelines such as those provided by the present statistical approach to reduce the size of the parameter space to be explored.

There is a variety of optimization problems that can be considered. Following usual motivations of laser cooling, like the increase of atomic beam brightness or the search for quantum degeneracy, we will concentrate here on optimizing the height h(θ) of the peak of the momentum distribution of the cooled atoms, which corresponds also to the gain in phase space provided by the cooling (see Section 6.2.3).

In the previous chapters, several important quantities characterizing the cooled atoms have been introduced and calculated. We now discuss the physical content of these results. We first show (Section 7.1) that the momentum distribution (p, θ) can be interpreted as the solution of a rate equation describing competition between rate of entry and rate of departure. This provides a new insight into the sprinkling distribution SR(t) which appears as a ‘source term’ for the trapped atoms. We then consider the tails of the momentum distribution (Section 7.2) and we show that they appear as a steady-state or ‘quasi-steady’-state solution of the rate equation describing the evolution of the momentum distribution. On the contrary, in the central part of this distribution, atoms do not have the time to reach a steady-state or a quasi-steady-state because their characteristic evolution times are longer than the observation time θ. One can understand in this way the θ-dependence of the height of the peak of the cooled atoms (Section 7.3). We also investigate (Section 7.4) the important case where the jump rate R(p) does not exactly vanish when p = 0 and we show that, when θ is increased, there is a cross-over between a regime where Lévy statistics is relevant, as in the previous case, and a regime where a true steady-state can be reached for the whole momentum distribution.

In this chapter, we introduce the main concepts and tools of Lévy statistics that will be used in subsequent chapters in the context of laser cooling. In Section 4.1, we show how statistical distributions with slowly decaying power-law tails can appear in a physical problem. Then, in Section 4.2, we introduce the generalized Central Limit Theorem enabling one to handle statistically ‘Lévy sums’, i.e. sums of independent random variables, the distributions of which have power-law tails. We also sketch, in a part that can be skipped at first reading, the proof of the theorem and present a few mathematical properties concerning distributions with power-law tails and Lévy distributions. In Section 4.3, we present some properties of Lévy sums which will turn out to be crucial for the physical discussion presented in subsequent chapters: the scaling behaviour, the hierarchy and fluctuation problems. These properties are illustrated using numerical simulations. Finally, in Section 4.4, we present the distribution S(t), called the ‘sprinkling distribution’. This distribution presents unexpected features and will play an essential role in the following chapters.

Power-law distributions. When do they occur?

Situations where broad distributions appear and where rare events play a dominant role are more and more frequently encountered in physics, as well as in many other fields, such as geology, economy and finance. The term ‘broad distributions’ usually refers to distributions decaying very slowly for large deviations, typically as a power law, implying that some moments of the distribution are formally infinite.