Pattern formation of atoms in high-finesse optical resonators results from the mechanical forces of light associated with superradiant scattering into the cavity mode. It occurs when the laser intensity exceeds a threshold value, such that the pumping processes counteract the losses. We consider atoms driven by a laser and coupling with a mode of a standing-wave cavity and describe their dynamics with a Fokker-Planck equation, in which the atomic motion is semiclassical but the cavity field is a full quantum variable. The asymptotic state of the atoms is a thermal state, whose temperature is solely controlled by the detuning between the laser and the cavity frequency and by the cavity loss rate. From this result we derive the free energy and show that in the thermodynamic limit selforganization is a second-order phase transition. The order parameter is the field inside the resonator, to which one can associate a magnetization in analogy to ferromagnetism, the control field is the laser intensity, however the steady state is intrinsically out-of-equilibrium. In the symmetrybroken phase quantum noise induces jumps of the spatial density between two ordered patterns: We characterize the statistical properties of this temporal behaviour at steady state and show that the thermodynamic properties of the system can be extracted by detecting the light at the cavity output. The results of our analysis are in full agreement with previous studies, extend them by deriving a self-consistent theory which is valid also when the cavity field is in the shot-noise limit, and elucidate the nature of the selforganization transition.

We investigate laser cooling of an ensemble of atoms in an optical cavity. We demonstrate that when atomic dipoles are sychronized in the regime of steady-state superradiance, the motion of the atoms may be subject to a giant frictional force leading to potentially very low temperatures. The ultimate temperature limits are determined by a modified atomic linewidth, which can be orders of magnitude smaller than the cavity linewidth. The cooling rate is enhanced by the superradiant emission into the cavity mode allowing reasonable cooling rates even for dipolar transitions with ultranarrow linewidth.PACS numbers: 37.10. Vz, 42.50.Nn, 37.30.+i, 03.65.Sq The discovery of laser cooling [1] has enabled a new world of quantum gas physics and quantum state engineering in dilute atomic systems [2]. Laser cooling is an essential technology in many fields, including precision measurements, quantum optics, and quantum information processing [3][4][5]. Doppler cooling [6,7] is perhaps the most elementary kind of laser cooling and relies on repeated cycles of electronic excitation by lasers followed by spontaneous relaxation. The temperatures that can be achieved in this way are limited by the atomic linewidth. Only specific ionic and atomic species can be Doppler cooled because they typically should possess an internal level structure that allows for closed cycling transitions.Cavity-assisted laser cooling [8,9] utilizes the decay of an optical resonator instead of atomic spontaneous emission as the energy dissipation mechanism. It is based on the preferential coherent scattering of laser photons into an optical cavity [10,11], rather than absorption of free-space laser photons as in conventional Doppler cooling. Temperatures that can be achieved in cavity-assisted cooling are limited by the cavity linewidth. Since the particle properties enter only through the coherent scattering amplitude, cavityassisted cooling promises to be applicable to any polarizable object [12][13][14][15][16][17][18][19][20], including molecules [17,18] and even mesoscopic systems such as nanoparticles [19,20].The many-atom effects of cavity-assisted cooling were theoretically discussed by Ritsch and collaborators [21] and experimentally reported in Refs. [22,23]. The cavity-mediated atom-atom coupling typically leads to a cooling rate that is faster for an atomic ensemble than for a single atom. Above a threshold of the pump laser, self-organization may occur and is observed as patterns in the atomic distribution that maximize the cooperative scattering. Recently, it has been shown that the long-range nature of the cavity-mediated interaction between atoms gives rise to interesting prethermalization behavior in the self-organization dynamics [24]. In spite of the intrinsic many-body nature, the underlying cooling mechanism shares much with the single-atom case, and indeed the final temperature observed in these systems is limited by the cavity linewidth.In this paper, we demonstrate that the mechanical action of the atom-cavity coupling takes o...

We theoretically characterize the semiclassical dynamics of an ensemble of atoms after a sudden quench across a driven-dissipative second-order phase transition. The atoms are driven by a laser and interact via conservative and dissipative long-range forces mediated by the photons of a singlemode cavity. These forces can cool the motion and, above a threshold value of the laser intensity, induce spatial ordering. We show that the relaxation dynamics following the quench exhibits a long prethermalizing behaviour which is first dominated by coherent long-range forces, and then by their interplay with dissipation. Remarkably, dissipation-assisted prethermalization is orders of magnitude longer than prethermalization due to the coherent dynamics. We show that it is associated with the creation of momentum-position correlations, which remain nonzero for even longer times than mean-field predicts. This implies that cavity cooling of an atomic ensemble into the selforganized phase can require longer time scales than the typical experimental duration. In general, these results demonstrate that noise and dissipation can substantially slow down the onset of thermalization in long-range interacting many-body systems. The quest for a systematic understanding of nonequilibrium phenomena is an open problem in theoretical physics for its importance in the description of dynamics from the microscopic up to astrophysical scales [1-3]. Aspects of these dynamics are studied in the relaxation of systems undergoing temporal changes (quenches) of the control field across a critical point [4][5][6]. Quenches across a non-equilibrium phase transition provide further insight into the interplay between noise and external drives on criticality and thermalization [7,8]. In this context photonic systems play a prominent role, thanks to their versatility [9][10][11][12][13][14][15].Polarizable particles in a high-finesse cavity, like in the setup illustrated in Fig. 1(a), offer a unique system to study relaxation in long-range interacting systems. Here, multiple photon scattering mediates particleparticle interactions whose range scales with the system size in a single-mode cavity [15][16][17][18]. In this limit, atomic ensembles in cavities are expected to share several features with other long-range interacting systems such as gravitational clusters and non-neutral plasmas in two or more dimensions [3,16,19]. The equilibrium thermodynamics of these systems can exhibit ensemble inequivalence [3,20], while quasi-stationary states (QSS) typically characterise the out-of-equilibrium dynamics [3,[21][22][23]. QSS are metastable states in which the system is expected to remain trapped in the thermodynamic limit, they are Vlasov-stable solutions and thus depend on the initial state. So far, however, evidence of QSS has been elusive. It has been conjectured that noise and dissipation can set a time scale that limits the QSS lifetime [24][25][26][27], and possibly gives rise to dynamical phase transitions [25]. In Ref. [28] it was shown that, in pr...

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