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.

The laws of geometrical optics were known from experiments long before the electromagnetic theory of light was established [1]. Today we recognize that they constitute an approximate solution for Maxwell's field equations. This solution describes the propagation of light and radio waves in media that change gradually with position [2]. The wavelength is taken to be zero in this approximation and diffraction effects are completely ignored. The field is represented by signals that travel along ray paths connecting the transmitter and receiver. In most applications these rays can be approximated by straight lines. These trajectories are uniquely determined by the dielectric constant of the medium and by the antenna pattern of the transmitter. In this approach energy flows along these ray paths and the signal acts locally like a plane wave. Geometrical optics provides a convenient description for a wide class of propagation problems when certain conditions are met.

The assumption that the medium changes gradually means that geometrical optics cannot describe the scattering by objects of dimensions comparable to a wavelength. Similarly, it cannot describe the boundary region of the shadows cast by sharp edges. A further condition is that rays launched by the transmitter must not converge too sharply – as they do for focused beams. These conditions must be refined when ray theory is used to describe propagation in random media.

Geometrical optics is widely used to describe electromagnetic propagation in the nominal atmosphere of the earth, other planets and the interstellar medium.

Degradation of stellar images is the most familiar example of propagation through random media and is visible to the naked eye. When a star is viewed through a telescope this degradation manifests itself in three ways: (a) as a variation of the image intensity, (b) as image broadening and (c) as wandering of the centroid of the image. This chapter is devoted to the third effect, which has also been called quivering, dancing and jitter. Image wandering is influenced primarily by large irregularities in the lower atmosphere for which ray theory is a good description. Image motion and angle-of-arrival fluctuations are different manifestations of the same random ray bending by atmospheric irregularities.

Image motion is readily observed in photographic plates placed at the focal plane of a stationary telescope. If there were no atmosphere, the stellar source would trace a smooth star trail on the plate as the earth and telescope turn together. Actual star trails exhibit random angular fluctuations about this nominal trajectory of 1 or 2 arc seconds as indicated in Figure 7.1. This random motion is observed in all astronomical measurements, although the magnitude varies with time, altitude and location. The error ranges from 0.5 to 2.0 arc seconds at sea level. It decreases with altitude and is usually 0.5 arc seconds on Mauna Kea (14 000 ft) but is sometimes as small as 0.25 arc seconds. Geometrical optics provides a valid description for astronomical quivering over a wide range of applications [1][2][3][4].