Exact Solutions And Enery Spectrum Characteristics Of Anisotropic Qubit And Optical Cavity Nonlinear Coupling System

The coupling system of the qubits and optical cavities is an important disciplinary field of condensed matter physics and quantum optics.Its simplest model is given by the quantum Rabi model(QRM),which describes the basic interaction between a qubit and a single-mode cavity field.In the weak coupling regime,the QRM can be reduced to the Jaynes-Cummings model in rotating-wave approximation(RWA).Recently,both theories and experiments in this field have achieved great success.In many advanced solid devices,such as circuit quantum electrodynamics(QED)system,and trapped ions,the ultra-strong coupling,even deep strong coupling between the artificial atom and resonators have been accessed,and the RWA is demonstrated to be invalid.The full quantum Rabi model and its variants including the nonlinear coupling have attracted extensive attentions.With the increase of the coupling strength increases,many physical phenomena have emerged.In this thesis,we first review the derivation of the model describing the two-level atom and single-mode cavity coupling system.The two-level system can be represented by the Pauli matrix,the cavity is described by the quantization of the electromagnetic field with the creation and annihilation operators of photons.These two parts together with the interaction of the qubit and photon compose the whole Hamiltonian of the system.The anisotropy and the nonlinear coupling of the qubit and the cavity can be further introduced in this model.Then,we introduce the theoretical methodologies for the QRM.To obtain analytical exact solution and energy spectrum,we describe two methods in detail,namely the transcendental function(G function)method based on the Bogoliubov operators method and the extended coherent state method In the former method,after introducing a new Boson operator,the diagonal matrix elements of the model Hamiltonian can be transformed into that of free particles,facilitating the further study.The eigenwave function of the model can be expanded according to the complete basis of the new boson operator.Finally,we can derive the transcendental function(G function),its zeros are just the eigenenergies of the model,therefore the energy spectra are obtained.In the extended coherent state method,we expand the eigenfunction of the system in the activated coherent states.Selecting a truncation number,locating the zeros of a polynomial equation,we can also obtain the spectra.Both methods can yield the exact solutions to the model.In the main part of this thesis,we apply the above two methods to study the anisotropic Rabi-Stark model.Using the Bogoliubov operators method,the G function of this model is strictly derived,and the energy eigenvalue of the system is determined from the zeros of the G function.By the extended coherent state method,the zeros of the polynomial equation is solved,and we also get the energy spectra.We can obtain analytical solution of this model in the first-order approximation,Its ground-state energy is in good agreement with the exact solution obtained by the exact diagonalizing in a wide coupling regime.We find that the approximate coherent state approach can give very good eigenenergies especially at the ultra-strong coupling regime.In the energy spectra obtained within these two methods,we observed that the ground-state and the first excited states energies cross at the critical point,indicating the first-order quantum phase transitions in this model.In summary,in the presence of the nonlinear Stark coupling term,we find some novel physical phenomena,such as spectra collapse,quantum phase transitions.Moreover,the extended coherent state approach improved in this thesis can be used in more general Rabi model.