Quantum Phase Transition Induced By Multi-photon Transitions And The Mechanical Oscillators In Rabi Model

In recent years,the interaction between the light and the matter has attracted the widespread attention.Superconducting circuits,cavity quantum electrodynamics(cavity QED),ion traps,and spin-orbit coupled quantum dots have been successfully simulated experimentally using Rabi model.By discussing the integrability of Rabi model,researchers has broken the research limitations of the coupling strength between the light and the atoms.Such research has not only provided a certain theoretical basis for the development of the experiment,but also provided the interesting in studying the physical properties of Rabi model.With the development of the nanomechanical oscillator,Rabi model has been widely researched from optical cavity to optomechanical cavity,which provides a new perspective for deep exploring the macroscopic mechanical motion.Based on this,this paper mainly discusses quantum phase transition in an optical cavity trapped in one or two two-level atom in the influence of multi-photon transitions and the mechanical oscillators.The main results are as follows:(I)Quantum phase transition by adjusting the atom-field coupling strength and the transition photon-number in the generalized Rabi model.The energy analytical solution is obtained by means of the spin-coherent-state variational method in Rabi model for multiphoton transitions.Meanwhile,the first-order and second-order derivatives of the energy function are obtained.Then the phase transition boundary is also solved.The average photon number,the atomic population,and the energy functional are also presented.Based on the phase diagram depicting the variation of the average photon number with the cavity frequency and the atom-field coupling strength during in single-photon and two-photon transitions,in the positive frequency region,the system in single-photon transition undergoes a second-order quantum phase transition from the normal phase to the superradiant phase,while in two-photon transition it occurs the coexistence of the normal phase and the inverted states in all regions,as well as an unstable non-zero photon number solutions still occurs just beyond the phase transition boundary.In the negative frequency region,only a stable inversion state exists beyond the phase transition boundary in singlephoton transition.And in two-photon transition only a stable non-zero photon number solution,which comes from the inverted energy functional,exists beyond the phase transition boundary.In multi-photon transition,the stable non-zero photon number solution exists in whole negative frequency region,and the zero frequency of the cavity is the boundary between the coexisting normal phases and the inverted superradiation phase.(II)Under resonance conditions,quantum phase transition and the multi-stable macroscopic quantum states in extended Rabi model were investigated by adjusting the atom-field coupling strength,the coupling strength of the mechanical oscillator,and the non-equilibrium parameters.The energy analytical solution,the first-order and secondorder derivatives of the energy function variational are given in Rabi model contained a mechanical oscillator by means of the spin-coherent-state variational method.From phase diagram about the average photon number as a function of the atom-field coupling strength and the coupling strength of the mechanical oscillator in a standard optomechanical cavity,we found that the normal state region is not affected with the increase of the coupling strength of the mechanical oscillator before the phase boundary,while the region of the superradiation state undergoes the collapse,especially an complete collapse when the coupling strength of the mechanical oscillator reaches some critical value.When a second atom is injected into the optomechanical cavity and there is no interaction between two atoms,rich phase diagrams and multi-stable macroscopic quantum states are displayed.The second atom made another energy functional appear.Meanwhile,every energy functional has stable zero or non-zero photon number solutions,which made phase diagram appear multi-stable macroscopic quantum states.The non-equilibrium parameter is introduced to better understand the effect of the interaction between the different atom and the cavity.Rich phase diagrams are also depicted about the non-equilibrium parameter.