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.
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.
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 S R ( 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.
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 (τ 0 , p 0 , p D , Δ p , p max , τ b 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 τ b 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. We only consider here the limit of small laser intensities (and a null detuning for VSCPT) but it is clear that the calculations can easily be generalized if needed. Velocity Selective Coherent Population Trapping We first present the quantum optics treatment of one-dimensional σ + /σ − VSCPT (Section A.1.1).
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.
Long ago, Paul Lévy invented a strange family of random walks – where each segment has a very broad probability distribution. These flights, when they are observed on a macroscopic scale, do not follow the standard Gaussian statistics. When I was a student, Lévy's idea appeared to me as (a) amusing, (b) simple – all the statistics can be handled via Fourier transforms – and (c) somewhat baroque: where would it apply? As often happens with new mathematical ideas, the fruits came later. For example, é. Bouchaud proved that adsorbed polymer chains often behave like Lévy flights. In a very different sector, J.P. Bouchaud showed the role of Lévy distributions in risk evaluation. Now we meet a third major example, which is described in this book: cold atoms. The starting point is a jewel of quantum physics: we think of an atom in a state of 0 translational momentum p = 0 (zero Doppler effect), inside a suitably prescribed laser field. For instance, with an angular momentum J = 1 we can have two ground states │+〉 and │−〉, and one excited state │0〉. The particular state │+〉+│−〉 has an admirable property: it is entirely decoupled from the radiation and can live for an indefinitely long time. It is thus possible to create a trap (around p = 0 in momentum space) in which the atoms will live for very long times: this so-called ‘ subrecoil laser cooling’ has been a major advance of recent years.
Motivation As explained in Chapter 2, the Lévy statistics treatment of subrecoil laser cooling has been introduced after a series of simplifications, where we have dropped details of the quantum microscopic description to only keep the main features of the physical process. Such a way of reasoning is standard in statistical physics. It is difficult, however, to be sure a priori that one has not dropped important features, and the validity of the statistical approach needs to be checked. An important step of this verification, although not a rigorous proof, is to compare a posteriori the predictions of the statistical approach with experimental results as well as with the predictions of microscopic theoretical approaches. This chapter presents such comparisons. We present in Section 8.2 the approaches (theoretical and experimental) to which our statistical approach can be compared. We then proceed to compare the results obtained by the different approaches. First, in Section 8.3, we treat in detail the predictions for a global quantity, the proportion of trapped atoms. This is done for the three recycling models introduced in this work, in the one-dimensional case. Then, in Section 8.4, we study another physical quantity with a richer content, the momentum distribution of cooled atoms. In Section 8.5, we investigate the influence of the dimensionality of the problem, and the role of friction during the recycling periods – which are crucial predictions of the Lévy statistics analysis.
Atoms can now be laser cooled, in one, two and three dimensions, below the limit corresponding to the recoil energy of an atom absorbing or emitting a single photon. Recent works have shown that all subrecoil cooling schemes are based on an anomalous random walk of the atom in momentum space which does not obey the central limit theorem and which introduces a fundamental non–ergodicity in the cooling process. Lévy statistics turn out to be very useful for analysing such a situation since they provide quantitative results for the long time limit of the momentum distribution. In this paper, we briefly review these theoretical works and we give a few references where more details can be found. We also mention a few recent experiments which have been stimulated by these theoretical developments.