Genuine quantum trajectories for non-Markovian processes

A large class of non-Markovian quantum processes in open systems can be formulated through time-local master equations which are not in Lindblad form. It is shown that such processes can be embedded in a Markovian dynamics which involves a time-dependent Lindblad generator on an extended state space. If the state space of the open system is given by some Hilbert space $\mathcal{H}$, the extended state space is the triple Hilbert space $\mathcal{H}\otimes {\mathbb{C}}^3$ which is obtained by combining the open system with a three-state system. This embedding is used to derive an unraveling for non-Markovian time evolution by means of a stochastic process in the extended state space. The process is defined through a stochastic Schrödinger equation which generates genuine quantum trajectories for the state vector conditioned on a continuous monitoring of an environment. The construction leads to a continuous measurement interpretation for non-Markovian dynamics within the framework of the theory of quantum measurement.

Figure 1

Illustration of the embedding theorem: The initial density matrix $\rho \left(0\right)$ evolves into $\rho \left(t\right)$ according to the given non-Markovian master equation (25). This evolution can be embedded in a Markovian dynamics on the extended state space $\tilde{\mathcal{H}}=\mathcal{H}\otimes {\mathbb{C}}^3$ in which the density matrix $W\left(0\right)$ evolves into $W\left(t\right)$ following the master equation (2).

Figure 2

The extended state space $\mathcal{H}\otimes {\mathbb{C}}^3={\mathcal{H}}_1\oplus {\mathcal{H}}_2\oplus {\mathcal{H}}_3$ and the action of the operators $J_i$ defined in Eqs. (36, 37, 38, 39).

Figure 3

Simulation of the SSE (6) for the non-Markovian decay of a two-state system. Top: ground-state probability $p_g$ obtained from a sample of ${10}^5$ quantum trajectories (dots) and analytical solution (solid line). Bottom: expectation value $E\left[2⟨{\psi }_2\mid {\psi }_1⟩\right]$. Parameters: ${\gamma }_0∕\lambda =25$, $\Delta ∕{\gamma }_0=0.2$.