Polaron Picture For The Quantum Rabi Model And Its Related Models

Light-matter interaction is a continuously fascinating research area.The system which contains an artificial atom(superconducting qubit,N-V centre,quantum dot)and a cavity field provides a good platform to research this topic.With the advanced experimental technologies,the new phenomena drive the theoretical study of the quantum Rabi model(QRM)which is the simplest model for the light-matter interaction system.Study of this model and it's generations not only contribute to the understanding of the experimental results,but also help us to understand the physical insights of those systems.In this thesis,we provide a new theoretical method for these basic models which describe the light-matter interaction.Firstly,we review the existing research of the quantum Rabi model.Secondly,we introduce the polaron method for the QRM.Thirdly,we extend this method to the two-photon QRM.Finally,we show how polaron method helps us to understand the rich physical phenomena inside a new theoretical model which contains the linear and nonlinear light-matter interaction.And the polaron method also gives out an analytical solution to this model.Firstly,we review the existing main methods and results of the quantum Rabi model.In the eyes of a direct numerical method,the energy spectrum of the system can be calculated,thus the statistical and dynamic properties of the energy spectrum of the quantum Rabi model can be further studied.In addition,approximation methods,such as the generalized rotational wave approximation,the generalized variational method and the mean photon number dependent variational method,show good results in some parameter regions or in the entire parameter region.Most importantly,significant research progress has been made in the analytical solution of the quantum Rabi model in recent years.The newly developed theoretical analytical method has greatly promoted further research on the quantum Rabi model and its related models.Secondly,our team proposed the polaron method with solving the QRM.The polaron method contains two steps.The first one is to provide an analysis of the physical picture where each term of the system plays their own role.We found the linear lightmatter interaction causes the spin-state related potential wells separating in the position space.In the meanwhile,the tunneling term induces an additional potential well for each of those two potential wells.The second step is to give the analytical solution of the model based on the previous polaron picture,such as constructing a trial state with polaron and antipolaron to study the properties of the ground state.The two-polaron method can also be extended to be a multi-polaron method by taking the high order tunneling process into consideration.The polaron method has two advantages when applied to the QRM.Firstly,when compared to the existed approximate methods,the polaron method works for the whole coupling regime.Secondly,when compared to the coherent state expansion method,the polaron method can reach a better result more efficiently.Thirdly,we apply the polaron method to the two-photon QRM.It is found that this method also works for the nonlinear model.The difference and similarity between polaron picture here and the previous one is revealed.The difference is that the lightmatter interaction term here causes the potential wells separating in the frequency space instead of the position space.The similarity is that the tunneling term also plays a role that helps the two separated potential exchange their own components.Based on such understanding,we can construct the ground state of the system and calculate observables accurately.Moreover,polaron method provides special understanding of the spectral collapse problem based on the understanding of the tunneling induced potential well.The polaron method takes the tunneling inducing process into consideration and thus provides better qualitative understanding of the spectral collapse behavior compared to the existed understanding.The successful application of the polaron method in this model reflects the importance of the idea about tunneling inducing process.It also inspires us to find more application of the polaron method.Finally,we study a new generalized QRM which contains the single-photon and two-photon interaction at the same time.This model is important because it takes the linear and nonlinear light-matter interaction term into consideration at the same time.Linear and nonlinear light-matter interaction process actually appear together in real systems,however the latter is usually neglected because the nonlinear process is believed much weaker than the former.In this new mixed model,it is found that the nonlinear light-matter interaction plays an important role while the linear light-matter coupling enters the strong coupling regime.A weak nonlinear coupling causes great changes of the system properties and leads to the spontaneous symmetry breaking.Further research shows that a strong nonlinear can also lead to the symmetry breaking(phase transition)while the linear coupling is weak.A full research reveals three different phase transition boundary inside this model.They show obvious differences in a finite ? and merge when ? ? 0.The physical picture which contains displaced and frequency shifted potential wells contributes to the understanding of this model.Considering the competition between potential energy and tunneling energy,the polaron method can provide an analytical phase transition boundary when ? ? 0 which matches the numeric result very well.The polaorn method plays an important role for the understanding of the rich physical phenomena,and deepens the knowledge of the models which describe the light-matter interaction,especially with the nonlinear process inside.In this thesis,we show that using the polaron method to study models which describe light-matter interaction can not only contribute to the understanding of these models,but also indicate that the polaron method has general applicability.