Quantum Langevin approach for non-Markovian quantum dynamics of the spin-boson model

One longstanding difficult problem in quantum dissipative dynamics is to solve the spin-boson model in a non-Markovian regime where a tractable systematic master equation does not exist. The spin-boson model is particularly important due to its crucial applications in quantum noise control and manipulation as well as its central role in developing quantum theories of open systems. Here we solve this important model by developing a non-Markovian quantum Langevin approach. By projecting the quantum Langevin equation onto the coherent states of the bath, we can derive a set of non-Markovian quantum Bloch equations containing no explicit noise variables. This special feature offers a tremendous advantage over the existing stochastic Schrödinger equations in numerical simulations. The physical significance and generality of our approach are briefly discussed.

Figure 1

Time evolution of $\langle {\sigma }_z\rangle$ for a bath with Ornstein-Uhlenbeck correlation function $\alpha (t,s)=\frac{\mathrm{\Gamma }\gamma }{2}{e}^{-\gamma |t-s|}$. (a) $\mathcal{N}=100$, and the inverse of the correlation time is chosen to be $\gamma =0.2,0.4$, and $0.8$, respectively. (b) $\gamma =0.2$, and the hierarchical order is chosen to be $\mathcal{N}=0$, 3, 10, and 100, respectively. Also, the coupling strength is chosen to be $\gamma \mathrm{\Gamma }=0.2$ in both (a) and (b).