Phases of cold atoms interacting via photon-mediated long-range forces

Abstract

Abstract. Atoms in high-finesse optical resonators interact via the photons they multiply scatter into the cavity modes. The dynamics is characterized by dispersive and dissipative optomechanical long-range forces, which are mediated by the cavity photons, and exhibits a steady state for certain parameter regimes. In standingwave cavities the atoms can form stable spatial gratings. Moreover, their asymptotic distribution is a Maxwell-Boltzmann whose effective temperature is controlled by the laser parameters. In this work we show that in a two-mode standing-wave cavity the stationary state possesses the same properties and phases of the Generalized Hamiltonian Mean Field model in the canonical ensemble. This model has three equilibrium phases: a paramagnetic, a nematic, and a ferromagnetic one, which here correspond to different spatial orders of the atomic gas and can be detected by means of the light emitted by the cavities. We further discuss perspectives for investigating in this setup the ensemble inequivalence predicted for the Generalized Hamiltonian Mean Field model. arXiv:1702.07653v1 [quant-ph]

Quenches across the self-organization transition in multimode cavities

“…This gives rise to an effective model, where the atoms experience a long-range interaction mediated by the cavity photons, while retardation effects and fluctuations of the cavity field are responsible for friction forces and diffusion. In the semi-classical limit one can derive a Fokker-Planck equation for the atoms' position and momentum distribution, assuming that the single-atom momentum distribution has a width Dp which, at all instants of time, is orders of magnitude greater than the photon recoil k:  D  p k [13,14,20]. The corresponding stochastic differential equations read as .…”

“…The corresponding stochastic differential equations read as . Another possible realisation, where w w » c,1 c,2 , has been discussed in [13]; it uses two optical single-mode cavities crossing at an angle of 60°. For a similar experimental setup see also [19]…”

“…In particular, Q = | | 1 n when the atoms are localized either at the maxima or at the minima of ( ) nkx cos , which is the configuration which maximizes the intracavity field intensity. We identify Q n with the order parameter for self-organization in the corresponding cavity mode [13]. Below, we denote by 'long-wavelength order' any configuration with a nonvanishing value of Q 1 , corresponding to a Bragg grating with period l p = k 2…”

“…In a multimode cavity and for several illumination frequencies, competing ordering processes are present and lead to richer phase dynamics. In a two-mode cavity, like the one depicted in figure 1(a), the transition to self-organization can be a phase transition of the first or second order depending on the laser intensities and on their relative strength [13]. The corresponding self-ordered phases can exhibit superradiant scattering either in one or in both cavity modes, as illustrated in figure 1(b), while the asymptotic distribution of the atoms can be thermal provided that the lasers' frequencies are suitably chosen [13].…”

“…In particular, for quenches starting with ensembles at low temperatures, the buildup of long-range order requires longer times than that for higher initial temperatures does.Our work is organized as follows: in section 2 we introduce the system and the semi-classical equations describing the dynamics. The atoms' stationary properties are then summarized in a phase diagram, which is derived from [13]. In section 3 we numerically study the real-time dynamics when the parameters are varied within the phase diagram according to different quench protocols.…”