Laser-cooled atoms in optical resonators can self-organize in spatially-ordered structures. Self organization sets in when the strength of the cooling laser, which directly pumps the atoms, exceeds a threshold value. Then, the ordered structures are stable and support coherent scattering of the atoms into the resonator, so that the cavity optical potential is maximized . In turn, the cavity field mediates infinitely long-range interactions between the atoms, giving rise to novel dynamics. The dynamics of the onset of self organization in this system is characterized by several peculiar features, which emerge because of the long-range interparticle potential. In this contribution we present an analytical and numerical study of the dynamics of self organization, which is based on a Fokker-Planck equation we derived assuming that the atomic motion is in the semiclassical regime . We show that at steady state the sample is in a thermal distribution, whose temperature does not depend on the pump strength but is only limited by the cavity line width. On the other hand, above threshold the steady state exhibits density-density correlations, to which one can associate a behaviour analogous to magnetization. Using numerical simulations we show that the dynamics leading to the stationary state is characterized by two time
scales: after a violent relaxation, the system slowly reaches the stationary states over time scales which exceed the cavity lifetime by several orders of magnitude . We analyze in details the corresponding field at the cavity output as a tool to monitor this behaviour. Finally, we draw analogies with the Hamiltonian-Mean-Field model  and argue that this system can be used as a testbed for studying the predictions of the statistical mechanics of long-range interacting systems.
 H. Ritsch, P. Domokos, F. Brennecke, and T. Esslinger, Rev. Mod.
Phys. 85, 553 (2013)
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