Theoretical Studies On The Qubit-oscillator Coupled System With Two-photon And Two-mode Interactions

Quantum Rabi model is one of the simplest systems to describe the qubit-oscillator in-teraction.Its original form includes an one-photon interaction between the two-level system and light field.In this paper,we consider generalizations of the single-photon quantum Rabi model,including the two-photon Rabi model,the two-mode Rabi model,and the mixed Rabi model which combines the one-and two-photon interactions.Then,the influences of different interactions are analyzed.The extended squeezed states are introduced based on the Bogoli-ubov operator approach,with which we achieve the analytical exact solutions.Furthermore,we analyze their integrability.All the models can be realized in the circuit QED experiments Therefore,our theoretical studies point out the direction for further experimental researches.In Chapter 1,we review the analytical methods used in the one-photon quantum Rabi model.Based on the Bargmann space and the Bogoliubov operator,the analytical exact solu-tions to the symmetric and asymmetric quantum Rabi models are achieved.We confirm that the methods based on the Bargmann space and the Bogoliubov operators are equivalent.Then,we briefly introduce Braak's criterion of quantum integrability.In Chapter 2,we study the two-photon Rabi model.The two-photon Rabi model exhibits a Z4 symmetry.By the Bogoliubov operator,we introduce the extended squeezed states and use them to construct the eigenstates.The su(1,1)Lie algebra and the Schrodinger equation lead to a three-term recurrence relation.Based on the symmetry of the Hamiltonian,G functions are achieved whose zeros correspond to the eigenvalues.The two-photon Rabi model is super-integrable according to Braak's criterion.Based on the pole structures,we analyze the spectral collapse at g??/2.With the help of finite-order expansions and variational methods,we achieve a simpler approximate results.In Chapter 3,we study the two-mode Rabi model,which has both U(1)symmetry and Z2 symmetry.We introduce a two-mode extended squeezed state based on the Bogoliubov operator.The symmetry of the Hamiltonian leads to its G function.We conclude that the two-mode Rabi model is integrable.Then,we analyze the pole structure and the spectral collapse at g??.In Chapter 4,we study the mixed Rabi model,which is much harder to be solved analyti-cally due to the lack of symmetry.We construct two forms of Bogoliubov operators,with which we get its G functions.The mixed Rabi model has two degrees of freedom while its eigenen-ergies only need one index,which indicates that it is non-integrable.When g2??/2,all the eigenenergies tends to negative infinite which lead to the spectral collapse.Furthermore,we construct an effective Hamiltonian with one-photon interaction which is equivalent to the orig-inal one.The effective Hamiltonian introduces a positive bias,suppressed photon frequency and enhanced one-photon coupling,which may be used to access the deep-strong one-photon coupling regime in the experiments.