Theoretical Study On Qubit-oscillator Coupled System With Nonliner Interaction

The interaction between light and matter is an important research field in condensed matter physics,quantum optics and quantum information science.In recent years,ultrastrong coupling,even deep strong coupling of light and matter can be achieved with the development of experimen-tal technology in artificial solid-state devices,such as superconducting circuit system,ion traps,cold-atom system.The simplest model describing the light-matter interaction is the quantum Rabi model and its generalized models.The enhancement of coupling between light and matter as well as the adjustability of artificial devices would modify the conventional physical pictures and lead to the emergence of novel physical phenomena,such as rich quantum phase,quantum phase transi-tions and quantum dynamics.According to the progress of the recent experiments,we study three generalized models which exhibit nonlinear coupling between the quantum qubits and oscillators theoretically.1)The Rabi model with both one-photon and two-photon coupling terms.This mixed inter-action is ubiquitous in the superconducting circuit electrodynamics system.It has been reported in the literature that it can be realized in quantum simulation based on ion traps easily.The un-mixed one-photon or two-photon Rabi model has parity symmetry,but the mixed model with both one-photon and two-photon interactions will break the parity symmetry naturally.Therefore,this generalized model is difficult to solve analytically.Even in its rotating-wave approximation,there is no analytical closed-form solution.The numerically exact solution and approximately analytical solution are obtained by the Bogoliubov transformation and the adiabatic approximation.Although the adiabatic solution is accurate if the atomic frequency is small,many physical observables ob-tained from the adiabatic approximation agree well with the numerical solutions in a wide coupling range.In the rotating-wave approximation,an approximate analytical solution is proposed based on the dominant non-perturbative states of the pure one-photon or two-photon model.For this generalized model,we find two Rabi frequencies.We apply the present analytical theory to the vacuum Rabi splitting.It is found that some different phenomena emerge because of the presence of the additional two-photon coupling term.2)Quantum Rabi-Stark model.In the Raman transition of an optical cavity electrodynamics system,the nonlinear Stark coupling emerges in the system,which can be described as by the so-called quantum Rabi-Stark model.The Stark-type nonlinear interaction between two level system and light field is added in the quantum Rabi model.By the Bogoliubov operator method,analyt-ical exact solution based on a transcendental function is derived,which is much simpler than that based on a few coupled transcendental equation in the Bargmann space.The zeros of the transcen-dental function correspond to the regular spectra,we also get two kinds of exceptional solutions by baseline of the transcendental functions.At the same time,when the strength of the nonlinear term is positive,the ground state will undergo first-order phase transition where the critical cou-pling strength is given analytically.The first-order phase transition does not exist in the isotropic quantum Rabi model.In addition,we also obtain the exact solution of the model by using tunable coherent states method.In the framework of tunable coherent states,we can obtain the analytical solution in the first-order approximation.The ground state energy and the average photon num-ber in the first-order approximation are in good agreement with the exact ones in a wide coupling range.When the coupling strength of the nonlinear term is twice the frequency of the cavity,the spectrum would collapse.We also derive the exact solution in this case and find the collapse point.Since there is no exact solution beyond the collapse point,we calculate the spectra by numerical diagonalization method.Finally,we find two kinds of quantum phase transitions.The first kind is the second-order phase transition like Rabi model with the infinite parameters,while another quantum phase transition occurs with finite parameters.By studying the scaling behavior of ener-gy gap and fidelity susceptibility of the latter phase transition,we observed that its universality is essentially different from the Rabi model with extreme parameters.3)The anisotropic two-photon Dicke model.This model is studied by the Schrieffer-Wolff method.The ground state energy and energy gap are obtained analytically.The second-order quan-tum phase transition occurs in the ground state when the frequency of the atom far less than the frequency of cavity.Spectral collapse is observed after the critical point of the phase transition.With the increase of coupling strength,the quantum phase transition from normal phase to super-radiant phase occurs in the ground state,and the symmetry breaks and energy gap disappears at the critical point.The second derivative of the ground state energy suddenly changes at the critical point,demonstrating the existence of the second-order phase transition.In addition,the critical exponent of the energy gap and the scaling exponent by the collapse of the fidelity susceptibility in the ground state show that the anisotropic two-photon Dicke model and the isotropic one-photon Dicke model share the same universality.