Numerical Approach to the Evolution of the Spin-boson Systems and its Application on the Buck-Sukumar Model

We present a systematic numerical iteration approach to study the evolution properties of the spin-boson systems, which works well in whole coupling regime. This approach involves the evaluation of a set of coefficients for the formal expansion of the time-dependent Schr"{o}dinger equation $\vert t\rangle=e^{-i\hat{H}t}\vert t=0\rangle$byexpandingtheinitialstate$\vert t=0\rangle$ in Fock space. The main advantage of this method is that this set of coefficients is unique for the Hamiltonian studied, which allows one to calculate the time evolution based on the different initial states. To complement our numerical calculations, the method is applied to the Buck-Sukumar model. Furthermore, we pointed out that, when the ground state energy of the model is unbounded and no ground state exists in a certain parameter space, the unstable time evolution of the physical quantities is the natural results. Furthermore, we test the performance of the numerical method to the Hamiltonian that use anti-Hermitian terms for modeling open quantum systems.