In this work, the anisotropic quantum Rabi model with different coupling strengths of the rotating-wave and counter-rotating wave terms is studied by using two kinds of extended coherent states (ECS). By the first kind of ECS, we can derive a so-called
G-function, by which both the regular and exceptional solutions can be given. The exceptional solution are just corresponding to the crossing points of two energy levels with different parities, so is doubly degenerate. By the second kind of ECS, a general scheme for the eigensolutions is derived analytically in a unified way. The zero-order approximation is just the adiabatic approximation, and the first-order approximation is actually a generalized rotating-wave approximation. The algebraic formulae for the eigensolutions are given explicitly in two approximations. The generalized rotating-wave approximations work well in a wide range of two different coupling strengths and the qubit detunings.
In the frame of the basic quantum theory,from the one-dimensional infinite-height half-wall quantum potential well with time-dependent boundary conditions and from the three-dimensional finite-depth sphere-square potential well with time-dependent boundary conditions we constructed two kinds of new Hamiltonian operators separately which describe electric particle interacting with eletromagnetic field.In the adiabatic approximation are calculated the complex Berry phases of two kinds of new systems.