Effective Hamiltonian approach to the exact dynamics of open system by complex discretization approximation for environment

Abstract: The discretization approximation method commonly used to simulate the dynamics of quantum system coupled to the environment in continuum often suffers from the periodically partial recovery of initial state because of the effect of finite dimension, dubbed the recurrence. To address this issue, we proposes a generalization of the discretization approximation method into the complex frequency space basing on complex Gauss quadratures. An effective Hamiltonian can be established by this way, which is non-Hermitian and demonstrates the complex energy modes with negative imaginary part, describing the dissipation of the system. This method is applied to examine the dynamics in two exactly solvable models, the dephasing model and the single-excitation dissipative dynamics in the Aubry-Andr\'{e}-Harper model. By comparison with the exact numerics and analytical results, it is found that our approach not only significantly reduces the effect of recurrence and improve the effectiveness of calculation, but also provide a unique perspective into the dynamics of open system from the point of complex energy levels. Furthermore, we establish a simple relationship between the parameters in computation and the effectiveness of simulation by analyzing the computational error.