We present a detailed study to analyze the Dicke quantum phase transition within the thermodynamic limit for an optomechanically driven Bose—Einstein condensate in a cavity. The photodetection-based quantum optical measurements have been performed to study the dynamics and excitations of this optomechanical Dicke system. For this, we discuss the eigenvalue analysis, fluorescence spectrum and the homodyne spectrum of the system. It has been shown that the normal phase is negligibly affected by the mechanical mode of the mirror while it has a significant effect in the superradiant phase. We have observed that the eigenvalues and the spectra both exhibit distinct features that can be identified with the photonic, atomic and phononic branches. In the fluorescence spectra, we further observe an asymmetric coherent energy exchange between the three degrees of freedom of the system in the superradiant phase arising as a result of optomechanical interaction and Bloch—Siegert shift.
We present a detailed study to analyze the Dicke quantum phase transition within the thermodynamic limit for an optomechanically driven Bose—Einstein condensate in a cavity. The photodetection-based quantum optical measurements have been performed to study the dynamics and excitations of this optomechanical Dicke system. For this, we discuss the eigenvalue analysis, fluorescence spectrum and the homodyne spectrum of the system. It has been shown that the normal phase is negligibly affected by the mechanical mode of the mirror while it has a significant effect in the superradiant phase. We have observed that the eigenvalues and the spectra both exhibit distinct features that can be identified with the photonic, atomic and phononic branches. In the fluorescence spectra, we further observe an asymmetric coherent energy exchange between the three degrees of freedom of the system in the superradiant phase arising as a result of optomechanical interaction and Bloch—Siegert shift.
Recent experiments have demonstrated an open system realization of the Dicke quantum phase transition in the motional degrees of freedom of an optically driven Bose–Einstein condensate in a cavity. Relevant collective excitations of this light–matter system are polaritonic in nature, allowing access to the quantum critical behavior of the Dicke model through light leaking out of the cavity. This opens the path to using photodetection-based quantum optical techniques to study the dynamics and excitations of this elementary quantum critical system. We first discuss the photon flux observed at the cavity face and find that it displays a different scaling law near criticality than that obtained from the mean-field theory for the equivalent closed system. Next, we study the second-order correlation measurements of photons leaking out of the cavity. Finally, we discuss a modulation technique that directly captures the softening of polaritonic excitations. Our analysis takes into account the effect of the finite size of the system, which may result in an effective symmetry-breaking term.
We make a semi-classical steady state analysis of the influence of mirror motion on the quantum phase transition for an optomechanical Dicke model in the thermodynamic limit. An additional external mechanical pump is shown to modify the critical value of atom–photon coupling needed to observe the quantum phase transition. We further show how to choose the mechanical pump frequency and cavity–laser detuning to produce extremely cold condensates. The present system can be used as a quantum device to measure weak forces.
Optomechanical cavity is a powerful connection between a nanomechanical oscillator and a quantized electromagnetic field. In this system, a novel photon-phonon nonlinear interaction arising from the nanomechanical oscillation is produced through the radiation pressure. Now this nonlinear photon-phonon interaction has become an important resource for implementing high-precision measurements and processing quantum information. Motivated by T. Esslinger group’s experiment, it is very meaningful to explore the exotic quantum phenomena when a ultra-cold BEC is trapped in an optomechanical cavity. In this paper, we mainly investigate phase transition and the finite-temperature thermodynamic properties of a Bose-Einstein condensate in an optomechanical cavity. It’s worth mentioning that at zero temperature many different mean-field approximate methods have been used to analyze the ground state properties of a Bose-Einstein condensate in an optomechanical cavity. Two common methods are Holstein-Primakoff transformation and spin coherent state variation. In this paper, an interesting imaginary-time path integral approach has been introduced to study finite temperature thermodynamic properties and phase transition of a Bose-Einstein condensate in an optomechanical cavity. First, we obtained system's partition function by taking imaginary-time path integration. Meanwhile, an effective action has been obtained by means of this method, which is the basic of the variation to get the numerical solution of photon number and the expression of the atomic number. At zero temperature, these results are consistent with what we have obtained by Holstein-Primakoff transformation or spin coherent state variational method. By adjusting the atom-field coupling strength and other parameters the second-order phase transition from the normal phase to the superradiant phase has been revealed. Meanwhile, a new unstable superradiant state was also found. And we found that in addition to the normal phase and superradiation phase, there exists an un-solution region of the mean photon number. Meanwhile, we find that the nonlinear photon-phonon interaction does not affect the normal phase. However, in the superradiant phase, the nonlinear photon-phonon interaction can enhance the macroscopic collective excitations. At the same time, the thermodynamic properties of the system are also discussed. According to the obtained distribution function, we can derive the analytical expression of the average energy and the free energy. Furthermore, the expression of entropy at finite temperature can also be obtained. we find the nonlinear photon-phonon interaction does not affect the average energy in the normal phase, but the average energy in the superradiant phase can deeply deviate in the large nonlinear photon-phonon interaction. It’s worth mentioning that the mean photon number and average energy in the finite-temperature tend to be consistent with the case in absolute zero temperature in the strong coupling region, while the entropy in the superradiant phase is rapidly reduced to zero as the atom-field coupling strength increases. In other words, strongly coupled collective excited states are highly ordered and are not affected by thermal fluctuations in the temperature range we are considering. The thermodynamic properties, such as the entropy and corresponding specific heat, characterize the Dicke phase transition.
In this paper we investigate the finite-temperature properties of a Bose-Einstein condensate (BEC)-cavity system with a strong nonlinear atom-photon interaction by means of a functional path-integral approach. It is shown that the experimentally observed phase diagram [Baumann, Guerlin, Brennecke, and Esslinger, Nature (London) 464, 1301 (2010)] can be better explained in our finite-temperature theory. More importantly, we identify a new dynamical unstable phase in this experiment. By tuning various experimental parameters, we reveal some rich temperature-driven phase diagrams and, in particular, predict a four-phase coexistence point. Finally, we find analytically that the specific heat in the superradiant phase increases exponentially at lower temperatures. Moreover, it has a large jump at the temperature-driven critical point where the superradiant-normal phase transition occurs. As a result, we argue that the specific heat can serve as a powerful tool to probe the thermodynamic properties of the BEC-cavity system.
Abstract Motivated by the experimental realization of Dicke model in optical cavities, we model an optomechanical system consisting of two-level BEC atoms with transverse pumping. We investigate the transition from normal and inverted state to the superradiant phase through a detailed study of the phase portraits of the system. The rich phase portraits generated by analytical arguments display two types of superradiant phases, regions of coexistence and some portion determining the persistent oscillations. We study the time evolution of the system from any phase and discuss the role of mirror frequency in reaching their attractors. Further, we add an external mechanical pump to the mirror which is capable of changing the mirror frequency through radiation pressure and study the impact of the pump on the phase portraits and the dynamics of the system. We find the external mirror frequency changing the phase portraits and even shifting the critical transition point, thereby predicting a system with controllable phase transition.
We theoretically examine collective excitations of an optically driven atomic Bose-Einstein condensate, coupled to a high-finesse optical cavity. This open system has been recently used for the experimental demonstration of the Dicke superradiance of cavity photons, which is simultaneously and mutually triggered by spontaneous breaking of translational symmetry of the condensate into a crystalline order. We first develop a Hartree-Fock mean field dynamical model of the physical system. Using this model, we compute the dynamics of the cavity photons, the condensate density profile and the Dicke phase transition diagram. Both the imaginary-time and real-time evolution methods are used in the calculations. Collective excitations are determined by the solving Bogoliubov-de Gennes equations. The spectrum, softening of the modes and energetic hierarchy of excitations are determined.