Markovian embedding procedures for non-Markovian stochastic Schrödinger equations

We present embedding procedures for the non-Markovian stochastic Schrödinger equations, arising from studies of quantum systems coupled with bath environments. By introducing auxiliary wave functions, it is demonstrated that the non-Markovian dynamics can be embedded in extended, but Markovian, stochastic models. Two embedding procedures are presented. The first method leads to nonlinear stochastic equations, the implementation of which is much more efficient than the non-Markovian stochastic Schrödinger equations. The stochastic Schrödinger equations obtained from the second procedure involve more auxiliary wave functions, but the equations are linear, and a closed-form generalized quantum master equation for the density-matrix can be obtained. The accuracy of the embedded models is ensured by fitting to the power spectrum. The stochastic force is represented using a linear superposition of Ornstein-Uhlenbeck processes, which are incorporated as multiplicative noise in the auxiliary Schrödinger equations. It is shown that the asymptotic behavior of the spectral density in the low frequency regime, which is responsible for the long-time behavior of the quantum dynamics, can be preserved by using correlated stochastic processes. The approximations are verified by using a spin-boson system as a test example.