A comparative study on the reliability of non-Hermitian effective Hamiltonian approach for modeling open quantum systems

In this paper, we conduct a comprehensive analysis on the reliability of non-Hermitian effective Hamiltonian approach for modeling quantum dissipative dynamics. It is found that one of the major drawbacks to adopting this theoretical approach is that it fails to reproduce correctly both the dynamics and the time evolution of observables of the system. With this in mind, we propose a method that allows correcting the non-Hermitian effective Hamiltonian approach when there exists a conserved quantity in the system. In order to illustrate the performance of our corrected non-Hermitian effective Hamiltonian approach, we consider a two-level system interacting with an electromagnetic cavity mode where the effects of the environment are included in a phenomenological manner. More precisely, the dissipation and pump in the system are considered through non-Hermitian Hamiltonian terms. Our numerical results have been focused on the time evolution of density matrix elements, observables as well as the emission spectrum of the system, and they demonstrate that our proposed methodology is in excellent agreement with the predictions of the Lindblad master equation approach.