The description of an impurity atom in a Bose-Einstein condensate can be cast in the form of Fröhlich's polaron Hamiltonian, where the Bogoliubov excitations play the role of the phonons. An expression for the corresponding polaronic coupling strength is derived, relating the coupling strength to the scattering lengths, the trap size and the number of Bose condensed atoms. This allows to identify several approaches to reach the strong-coupling limit for the quantum gas polarons, whereas this limit was hitherto experimentally inaccessible in solids. We apply Feynman's path-integral method to calculate for all coupling strengths the polaronic shift in the free energy and the increase in the effective mass. The effect of temperature on these quantities is included in the description. We find similarities to the acoustic polaron results and indications of a transition between free polarons and self-trapped polarons. The prospects, based on the current theory, of investigating the polaron physics with ultracold gases are discussed for lithium atoms in a sodium condensate.

We experimentally explore the dynamic optical hysteresis of a semiconductor microcavity as a function of the sweep time. The hysteresis area exhibits a double power law decay due to the shot noise of the driving laser, which triggers switching between metastable states. Upon increasing the average photon number and approaching the thermodynamic limit, the double power law evolves into a single power law. This algebraic behavior characterizes a dissipative phase transition. Our findings are in good agreement with theoretical predictions, and the present experimental approach is promising for the exploration of critical phenomena in photonic lattices.Optical bistability -the existence of two stable states with different photon numbers for the same driving conditions -is a general feature of driven nonlinear systems described within the mean-field approximation (MFA) [1]. Beyond the MFA, a quantum treatment predicts that the steady-state of a nonlinear cavity is unique at any driving condition [2]. The origin of this apparent contradiction was noted by Bonifacio and Lugiato [3], and by Drummond and Walls [4]: quantum fluctuations (the lost feature in the MFA) trigger switching between states and the exact solution corresponds to a weighted average over the two metastable states. Experiments in the 80's with two-mode lasers evidenced extremely long switching times [5], which were predicted to diverge for weak fluctuations and/or large photon numbers [6]. Already in these early works, this dramatic slowing down of the system dynamics was linked to a first order phase transition [5][6][7].The physics of nonlinear resonators is receiving renewed interest in connection to predictions of quantum many-body phases [8][9][10][11][12][13], critical phenomena [5,[12][13][14][16][17][18], and dissipative phase transitions [4]. Impressive progress is being made in building lattices of nonlinear resonators, such as photonic crystal cavities [20,21], waveguides [22], superconducting microwave resonators [23,24], or optomechanical resonators [25,26]. In this context, semiconductor microcavities operating in the exciton-photon strong coupling regime provide a versatile platform where photon hopping and the pumping geometry can be controlled [27]. Lattices of different dimensionalities can be engineered [28,29], and the hybrid light-matter nature of their elementary excitations, namely cavity polaritons, provide a strong and tunable Kerr nonlinearity via the exciton component [1,[30][31][32].Recently, it was predicted that even in a single resonator, critical exponents could be retrieved from dynamical hysteresis measurements [17]. More precisely, when the driving power is swept at a finite speed across a bistability, the area of the hysteresis cycle is expected to close following a double power-law as a function of the sweep time [5, 6]. The long-time decay arises from quantum fluctuations, and presents a universal −1 exponent [5]. In the thermodynamic limit wherein the photon number in the bistability tends to infinity and fluctua...