Polytropic equilibrium and normal modes in cold atomic traps

Abstract

The compressibility limit of a cold gas confined in a magneto-optical trap due to multiple scattering of light is a long-standing problem. This scattering mechanism induces long-range interactions in the system, which is responsible for the occurrence of plasma-like phenomena. In the present paper, we investigate the importance of the long-range character of the mediated atom-atom interaction in the equilibrium and dynamical features of a magneto-optical trap. Making use of a hydrodynamical formulation, we derive a generalized Lane-Emden equation modeling the polytropic equilibirum of a magneto-optical trap, allowing us to describe the cross-over between the two limiting cases: temperature dominated and multiple-scattering dominated traps. The normal collective modes of the system are also computed.Introduction. Since the first realizations of cold atomic gases [1], both theoretical and experimental investigations reveal that magneto-optical traps (MOT) pave a stage for very exciting and complex physical phenomena [2]. The interest in studying the basic properties of MOTs have, however, considerably decreased after the production of Bose-Einstein condensates [3,4], as they started being used mainly as a riding horse to achieve quantum degeneracy. However, the study of the dynamical properties of MOTs have received much attention recently, which revives the investigation of the basic properties of laser cooled gases. Examples of such a growing interest can be found in the work realized by Kim et al. [5], where a parametric instability is excited by an intensity modulated laser beam, and in the works of di Steffano and co-workers [6][7][8], where the feedback of retroreflected laser beams can induce stochastic or deterministic chaos for a large optical thickness of the MOT.A route for the most intriguing complex behavior in magneto-optical traps relies exactly on the multiple scattering of light, a mechanism which have been described since the early stages of MOTs as the principal limitation for the compressibility of the cloud [9,10]. Under these circumstances, the atoms experience a mediated long-range interaction potential similar to a Coulomb potential (∼ 1/r) [11] and the system can therefore be regarded as a one-component trapped plasma. In a series of previous works, we have put in evidence the important consequences of such plasma description of a cold atomic traps [12], whereas the formal analogy and the application of plasma physics techniques reveal to be important in the description of driven mechanical instabilities [14] or even more exciting instability phenomena, like photon bubbles [15], phonon lasing [16] or even the appearance of a roton minimum in the classical regime [17].In this work, we investigate the hydrodynamic equilibrium and normal modes of cold atomic traps. For that

Wave-kinetic approach to the Schrödinger–Newton equation

“…Neglecting small kinetic effects, and assuming harmonic confinement, this would lead to the well-known Thomas-Fermi equilibrium, in the case of a BEC. Similarly, using a polytropic equation of state, different equilibrium profiles could be established for a lasercooled gas [28]. The validity of this approach was confirmed by recent detailed experiments in a large MOT [29].…”

“…The validity of this approach was confirmed by recent detailed experiments in a large MOT [29]. It should be noticed, in this context, that the equilibrium profiles of the laser-cooled gas are described by a generalised Lane-Emden equation [28], very similar to the equation describing matter equilibrium in a gravitational field [30,31]. Application of this approach to BEC equilibrium and its implications to dark matter research has also been considered [32,33].…”

“…For the purpose of our present analysis, it is useful to rewrite the dispersion relation (28) as 1−χ (ω, k)=0, where χ ≡ χ (ω, k) is the susceptibility of the medium. We can further use χ=χ r +i χ i , where the real part is determined by the principal part of the integral in (28) and is determined by as seen before, whereas for the imaginary part we have to retain the pole contributions to the integral, and use This then leads to w G -…”

Towards a measurement of the Debye length in very large magneto-optical traps

“…The analogy is not perfect: for instance the non potential part of the shadow effect is neglected, the friction and diffusion in a MOT are much stronger than in a non neutral plasma, and the typical optical depth in an experiment is not very small. Nevertheless, it is a basic model to analyze MOT physics, and has been used recently to predict new plasma related phenomena in MOTs (see for instance [16,40]).…”

Stability analysis of orbital modes for a generalized Lane–Emden equation

“…1 Ω 1/n , 0 will be shown to be unstable (see Theorem II.10), for n ∈ N. Before we state the stability/instability results for the nonlinear system (16) we first need to introduce some terminology that is used in [15].…”