Finite-temperature Properties And Phase Transition Of Many-body System In The Optical Micro-cavities

Phase transition is the result of competition between order and disorder. The interaction gives rise to order and thermal motion leads to disorder. When the temperature down to a certain degree,the order induced by a particular interaction isn’t broken and then a new phase could emerge.Classical phase is drived by thermal motion and the phase transition induced by quantum fluctuation refers to the sudden change of ground state with the coupling parameter. Heisenberg uncertainty causes quantum fluctuation.Quantum phase transition in many fields of modern physics plays an important role, such as condensed matter physics, nuclear physics, and so on.It can be used to the manipulation of the geometric phase and entanglement, so plays an important role in quantum information and computing.Thus,it becomes a hot research topic in recent years.The quantum phase transition has been widely studied for the Dicke model as a typical example.But at the moment the phase transition caused by the change of the coupling strength is limited to zero temperature. Studying the characteristics of finite temperature stable state and discussing the influence of thermal excited states to the phase transition undoubtedly have important theoretical and practical significance.Because superradiance quantum phase transition is observed at finite temperature in the experiment.In 2007 Brennencke and others studyed the Bose-Einstein condensation in the optical cavity in the experiment.By introducing the momentum spin state, superradiance quantum phase transition of the system was successfully observed.In this system the rich phase diagram and dynamic properties have been found.In recent years, the light-mechanical system consisting of an optical cavity and a mechanical oscillator becomes one of research hot spots.It has a broad application prospect in many fields such as quantum entanglement and quantum information research.Quantum phase transition can also be observed in this system. In this paper,based on what has been described,we have studied finite temperature properties of N two-level atoms in a single-mode optic cavity and phase transition,ground-state properties of a Bose-Einstein condensate in an optomechanical cavity and finite temperature phase transition of the optomechanical BEC-cavity system.(1) We have obtained partition function of the Dicke model by using of imaginary-time functional path-integral approach and the thermodynamic equilibrium equation has been calculated via partition function.The scaled atomic population and the scaled mean-photon number as a function of the atom-photon coupling strength have also been given.Futhermore,we investigate the thermodynamic properties of the Dicke model.The phase transition from the normal phase to the superradiant phase obey the Landau continuous phase transition theory through numerical calculation.(2) By the same method,we have studied the ground-state properties of a Bose-Einstein condensate in an optomechanical cavity.The thermodynamic equilibrium equation has been derived.The atomic population and average photon number as analytic functions of the atom-photon coupling strength are found at zero temperature.As the nonlinear photon-phonon interaction increases, the average photon number increases sharply.Besides,we have calculated the ground state energy and free energy of the system.The results show that the larger photon-phonon interaction affects the ground-state energy of the system.(3) We have investigated finite temperature phase transition of the optomechanical BEC-cavity system by means of the same approach.At first,the thermodynamic equilibrium equation of the system through the partition function is obtained.The atomic population as a function of the atom-photon coupling strength is given.It is great that a superradiant bistable phenomenon appears with the photon-phonon interaction increasing and the coexisting phase arises(the stable normal phase and the metastable superradiant phase).In addition,the thermodynamic properties including free energy,energy and entropy of the system have been studied.Imaginary-time functional path-integral approach is simple and can easily study the finite temperature properties of the system.