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 standing-wave 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.
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 standing-wave 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.
Photons mediate long-range optomechanical forces between atoms in high-finesse resonators, which can induce the formation of ordered spatial patterns. When a transverse laser drives the atoms, the system undergoes a second-order phase transition that separates a uniform spatial density from a Bragg grating maximizing scattering into the cavity and is controlled by the laser intensity. Starting from a Fokker-Planck equation describing the semiclassical dynamics of the N-atom distribution function, we systematically develop a mean-field model and analyze its predictions for the equilibrium and out-of-equilibrium dynamics. The validity of the mean-field model is tested by comparison with the numerical simulations of the N-body Fokker-Planck equation and by means of a Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy. The mean-field theory predictions well reproduce several results of the N-body Fokker-Planck equation for sufficiently short times and are in good agreement with existing theoretical approaches based on field-theoretical models. The mean field, on the other hand, predicts thermalization time scales which are at least one order of magnitude shorter than the ones predicted by the N-body dynamics. We attribute this discrepancy to the fact that the mean-field ansatz discards the effects of the long-range incoherent forces due to cavity losses.
This is a review of state-of-the-art theory and experiment of the motion of cold and ultracold atoms coupled to the radiation field within a high-finesse optical resonator in the dispersive regime of the atom-field interaction with small internal excitation. The optical dipole force on the atoms together with the backaction of atomic motion onto the light field gives rise to a complex nonlinear coupled dynamics. As the resonator constitutes an open driven and damped system, the dynamics is nonconservative and in general enables cooling and confining the motion of polarizable particles. In addition the emitted cavity field allows for real-time monitoring of the particle's position with minimal perturbation up to subwavelength accuracy. For many-body systems, the resonator field mediates controllable long-range atom-atom interactions, which set the stage for collective phenomena. Besides the correlated motion of distant particles, one finds critical behavior and nonequilibrium phase transitions between states of different atomic order in conjunction with superradiant light scattering. Quantum-degenerate gases inside optical resonators can be used to emulate optomechanics as well as novel quantum phases such as supersolids and spin glasses. Nonequilibrium quantum phase transitions as predicted by, e.g., the Dicke Hamiltonian can be controlled and explored in real time via monitoring the cavity field. In combination with optical lattices, the cavity field can be utilized for nondestructive probing Hubbard physics and tailoring long-range interactions for ultracold quantum systems.
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 self-organization 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, but the steady state is intrinsically out of equilibrium. In the symmetry-broken phase, quantum noise induces jumps of the spatial density between two ordered patterns: We characterize the statistical properties of this temporal behavior 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; we 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 self-organization transition.
Atoms can spontaneously form spatially ordered structures in optical resonators when they are transversally driven by lasers. This occurs when the laser intensity exceeds a threshold value and results from the mechanical forces on the atoms associated with superradiant scattering into the cavity mode. We treat the atomic motion semiclassically and show that, while the onset of spatial ordering depends on the intracavity-photon number, the stationary momentum distribution is a Gaussian function whose width is determined by the rate of photon losses. Above threshold, the dynamics is characterized by two time scales: after a violent relaxation, the system slowly reaches the stationary state over time scales exceeding the cavity lifetime by several orders of magnitude. In this transient regime the atomic momenta form non-Gaussian metastable distributions, which emerge from the interplay between the long-range dispersive and dissipative mechanical forces of light. We argue that the dynamics of self-organization of atoms in cavities offers a test bed for studying the statistical mechanics of long-range interacting systems.
Present-day experiments have started to couple traditional simulations of ultracold atom experiments with quantum light-fields in cavities. This has provided a wealth of op- portunities to enlarge the number of interaction potentials: cavity mediated long-range interactions compete with kinetic energies, longitudinal fields, short-ranged collisional or magnetic spin-spin interactions. The intracavity many-body lattice models often have to be maintained far from equilibrium through the presence of external driving lasers that help to boost and engineer the various interaction potentials. The steady influx of energy is compensated by a steady stream of energy out of the atom-cavity system for example by photon losses or atomic spontaneous emission. Several experiments have demonstrated that such environments can give rise to new competing quantum phases. \nBut this long-standing ambition to push for models with tailorable interaction potentials can bring with it also considerable challenges in their theoretical description, since sponta- neous symmetry breaking transitions in many body lattice systems coupled to dynamical light-fields with single-photon character occur in the presence of drive and dissipation for the photonic force carriers. This clearly calls for model systems where the above men- tioned interplay of interactions, drive, dissipation and cooperative many-body behaviour can be theoretically studied to provide simple, experimentally verifiable predictions. \nThe Dicke model is, through its simplicity (an exactly solved ferromagnet with infinite range atom-atom interactions mediated by a single cavity mode), an exceptionally well- suited candidate. As the generic model for atom-light interactions, it has been experimen- tally realized in a variety of modern quantum optical systems, highlighting its relevance for present-day research. The Dicke model is also highly versatile itself. It has been extended into the dissipative realm, was promoted to account for multiple optical light modes and was used to describe multiple, coupled single-mode cavity structures. It was adapted to treat spin-selective coupling to a cavity to describe superradiance phase tran- sitions in multi-level atomic systems. Moreover, it was realised in electronic circuits where the dipole coupling of real atoms to single mode fields is replaced by a capacitive cou- pling of artificial atoms to a resonator mode. This illustrates that the Dicke model and its extended variants are ’future-proof’ and continue to be of relevance for fundamental light-matter interactions and for driven-dissipative phase transitions. \nIn this thesis, we investigate magnetic phase transitions in driven-dissipative atomic en- sembles interacting with quantum light. We present three research projects on variants of cooperative radiation of an ensemble of laser driven two-level atoms in a single mode optical cavity, as described by the Dicke model. \nThroughout the chapters 2,3 and 4 that contain the main body of research of this thesis, we investigate phase transitions between non-equilibrium stationary states in engineered quantum-optical systems each of which extends the conventional Dicke model physics. As a starting point, we map the quantum equations of motion onto a set of semiclassical nonlinear stochastic equations and analyse their stationary states and instabilities with master equations for atomic spin and photon mean-field amplitudes. These are used to obtain experimentally relevant parameters such as critical atom-light couplings for phase transitions, phase diagrams and properties of stationary non-equilibrium states in addition to cavity output spectra that identify the imprint of magnetic correlations in the light-field. \nIn chapter 2, we help resolve a discrepancy between earlier experimental investigations of the critical atom-light coupling strength for the superradiance transition in the Dicke model: higher external pumping strengths than theoretically predicted were needed to observe a coherent, superradiant state of the light field in an optical cavity. By including incoherent spontaneous emission of atomic excitations, we extend the dissipative Dicke model to a two loss channel variant containing both photon leakage and atomic decay that reproduces the experimentally observed critical atom-light coupling. \nRecent experiments have started to interface quantum many body lattice models with coherent cavity fields, thereby allowing to realize new quantum phases through competing atom-cavity and atom-atom interactions. In chapter 3, we consider a simplified model for such a set-up where a single quantized mode of the light-field interacts with an ensemble of Rydberg-dressed atoms inside a high finesse optical cavity. This model provides a base case for further studies of quantum magnets in optical cavities. At the heart of this model is a competition of short- (dipolar atom-atom) and long-range (atom-light) interactions at the Hamiltonian level in the presence of both spontaneous emission and photon leak- age through the cavity mirrors. We show that different magnetic phases can coexist with coherent atomic radiation and provide clear experimental signatures to identify the mag- netic structure and intra-cavity dynamics. We suggest an experimental level-scheme for a quantum optical implementation of our model. \nIn chapter 4 we consider a generic, collective decay for many-body excitations in the paradigmatic Dicke model. This extension drastically enriches the dynamics as it induces a bicritical point and a bistable regime dominated by true non-equilibrium fluctuations that induce a dissipative first-order phase transition that can only be resolved by including finite fluctuation corrections with the help of stochastic Langevin equations. We investigate the hysteretic response to time-dependent ramps of the atom-light coupling. Discontinuous first-order phase transitions where metastable states coexist in a hysteresis domain have been investigated in recent dissipative quantum-optical experiments. We review noise- activation far from thermal equilibrium in chapter 5.
Summary form only given. We report on the observation of the many-atom dipole-dipole interaction in an ensemble of ultracold rubidium atoms strongly coupled to the standing-wave field of a weakly driven high-finesse optical cavity. The atoms split the cavity resonance into two normal modes. The many-atom light force changes the spatial distribution of the atoms, creating an asymmetry in the usually symmetric normal-mode spectrum, for intracavity photon numbers much less than one. This asymmetry is investigated as a function of the number of atoms and as a function of the intracavity photon number. Moreover, analytical expressions have been derived for the dipole force and the diffusion coefficient for an atom in presence of other atoms. These expressions allow us to simulate the trajectories of several atoms simultaneously. The calculated and measured spectra agree only if the full motional dynamics of all atoms is taken into account.