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Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences

Non–ergodic laser cooling

Claude N. Cohen-Tannoudji

Published Dec 15, 1997• 0 citations
Publisher

Abstract

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.

DOI
10.1098/rsta.1997.0120

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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.
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.
Laser cooling allows now the velocity spread of an ensemble of atoms to be reduced below the value corresponding to the recoi l \nvelocity of an atom emitting or absorbing a single photon (subrecoil cooling) . This is achieved by using a random walk in velocit y \nspace with a jump rate which vanisches at zero velocity . We will review a few possible theoretical descriptions of such a coolin g \nmechanism . The anomalous character of the corresponding random walk will be analysed using Lvy statistical distributions an d \ncertain nonergodic features of the atomic evolution will be emphasized . Recent experimental developments will be also described . \nThey have allowed temperatures of a few nanokelvins to be reached and measured .
Anomalous diffusion processes, dominated by rare events, are shown to exist and to play a central role in certain subrecoil laser cooling schemes. We present a new statistical analysis of these processes, in terms of L\'evy flights, which provides a precise analytical description of the cooled atoms in the long time limit, where the standard methods of quantum optics are inappropriate. These analytical predictions are quantitatively checked by comparison with the results of quantum Monte Carlo simulations of the cooling process at intermediate times.
Laser cooling of atoms provides an ideal case study for the application of Lévy statistics in a privileged situation where the statistical model can be derived from first principles. This book demonstrates how the most efficient laser cooling techniques can be simply and quantitatively understood in terms of non-ergodic random processes dominated by a few rare events. Lévy statistics are now recognised as the proper tool for analysing many different problems for which standard Gaussian statistics are inadequate. Laser cooling provides a simple example of how Lévy statistics can yield analytic predictions that can be compared to other theoretical approaches and experimental results. The authors of this book are world leaders in the fields of laser cooling and light-atom interactions, and are renowned for their clear presentation. This book will therefore hold much interest for graduate students and researchers in the fields of atomic physics, quantum optics, and statistical physics.
This book deals with the important developments that have recently occurred in two different research fields, laser manipulation of atoms on the one hand, non-Gaussian statistics and anomalous diffusion processes on the other hand. It turns out that fruitful exchanges of ideas and concepts have taken place between these two apparently disconnected fields. This has led to cross-fertilization of each of them, providing new physical insights into the most efficient laser cooling mechanisms as well as simple and mathematically soluble examples of anomalous random walks. We thought that it would be useful to present in this book a detailed report of these developments. Our ambition is to try to improve the dialogue between different communities of scientists and, hopefully, to stimulate new, interesting developments. This book is therefore written as a case study accessible to the non-specialist. Our aim is also to promote, within the atomic physics and quantum optics community, a way to approach and solve problems that is less based on exact solutions, but relies more on the identification of the physically relevant features, thus allowing one to construct simplified, idealized models and qualitative (and sometimes quantitative) solutions. This approach is of course common in statistical physics, where, often, details do not matter, and only robust global features determine the relevant physical properties. Laser cooling is an ideal case study, where the power of this methodology is clearly illustrated.