Quantum Phase Transition In A Double-Cavity Optomechanical System

Quantum phase transition is an important research topic in condensed matter physics,and Dicke quantum phase transition was successfully observed in the experiment in 2010,which greatly promoted the study of quantum phase transition of the ultracold atoms in the optomechanical cavity.The main contents of this dissertation is included in:In Chapter 1,firstly,the optomechanical cavity system is briefly introduced,and then quantum phase transition is introduced,especially Dicke model's quantum phase transition.Secondly,we introduce the development of the geometric phase.Moreover,we give people's understanding of the relationship between geometric phase and quantum phase transition.In Chapter 2,this chapter is our main research content.Firstly,we study the multiple stable macroscopic quantum states of the Bose-Einstein condensate in the standard optomechanical cavity by adjusting mechanical oscillator photon-phonon coupling strength and nonlinear atom-photon interaction.Secondly,we study the interaction between coupled optomechanical double-cavity and Bose-Einstein condensate by means of the spin-coherent-state variational method.Based on given ground-state energy functional of the system in theory,we present the extreme value based on variation method and give the stable and unstable rich ground-state phase diagram in the resonance,which is determined by the Hessian matrix.We also plot the line figures with the change of the parameters according to the expressions of mean photon number,atomic population difference,and average energy,which is consistent with the phase diagrams.In Chapter 3,firstly we calculate the non-adiabatic geometric phase by using time-dependent gauge transformation.Then we present the non-adiabatic geometric phase in the double-cavity optomechanical system,which interacts with Bose-Einstein condensate.After one week of evolution,the non-adiabatic geometric phase is proportional to the expected value of thesum of the number of double-cavity photons and the phononIn Chapter 4,it's the summary and work outlook of the article.