Dynamical Phase Transition And Selective Energy Exchange In Optomechanical Cavity

In recent years,quantum mechanical behaviors have been deeply researched in optomechanical cavity,in which quantum nonlinear properties have developed an important aspect.The evolution equation of phase transition is about that the phase ratio's change as the function of the time when the system occurs phase transition.At present,the new phase ratio's change is characterized by the driving force of phase transition,the temperature and the relaxation time of the phase transition,which is based on phenomenology.However,this phenomenology has the obvious disadvantage because the physical mechanism has been ignored when it occurs the phase transition.Moreover,the mode compression,which is based on the quantum uncertainty relation,becomes one of the important research topic in the optomechanical cavity.The dynamic phase transition and selective energy exchange has been respectively investigated in the single-mode and two-mode optomechanical cavity.We find that the two systems will both undergo a dynamic phase transition,which is like to Dicke-Hepp-Lieb superradiant type phase transition.Meanwhile,a new dynamic critical point appears by modulating the coupling strength between the optical mode and the mechanical mode or the two-cavity coupling strength.We found that the external field driving in the optomechanical cavity without the two-cavity coupling strength is equivalent to the two-cavity coupling strength in the optomechanical dual-cavity,where there didn't exists the external field driving.By modulating the coupling strength between the optical mode and the mechanical mode or the two-cavity coupling strength,the selective energy exchange has been realized between any two modes,and the critical coupling point corresponds to the selective energy exchange.So a conclusion that mode squeezing is a signature of energy transfer is obtained,and the mode squeezing of any two modes is determined by the energy exchange between specific modes.The main research contents are as follows:(1)We have introduced the research background of the optomechanical cavity,mainly about the following aspects: mechanical vibrator,optomechanical cavity,phase transition and mode squeezing.(2)We have presented the dynamical phase transition and selective energy exchange in a single-mode optomechanical cavity.Starting from the rotating transition for the Hamiltonian of the single-mode optomechanical cavity,an effective Hamiltonian of the system is got.Then,we derive the Heisenberg equation of motion for the related operators based on Heisenberg-Langevin equation and yield the excitation energy of optical mode and mechanical mode in the normal phase.From the graph of the excitation energy as the function of the coupling parameter,we find that the dynamical phase transition in the critical point corresponds to quantum phase transition from the normal phase to the superradiant phase in Dicke model.The stable phase of the dynamical phase transition corresponds to the normal phase,while the unstable phase corresponds to the superradiant phase.For just two modes the system is not strictly quantum phase transition.By means of quantum uncertainty relations and Bogoliubov transform,the position variances and the momentum variances are given.Moreover,mode squeezing is a signature of energy transfer for the position variances are unsqueezed and the momentum variances are completely sequeezed at the dynamical critical point.(3)We have studied the dynamical phase transition and selective energy exchange in a dual-mode optomechanical cavity,in which there exists the two-cavity coupling strength or none.The research method here is the same as the one in Chapter 2.However,the increasing of the optical mode results some new properties of the dynamical phase transition and new dynamical critical points occur.In the critical points,there exists maximum squeezing,which explains that the signature of energy transfer is mode squeezing.By Fourier transform,we study mode splitting and show that it is possible to select different modes to energy exchange by modulating different coupling parameters in a dual-mode optomechanical cavity.