The non-Markovian stochastic Schrödinger equation for open systems

We present the non-Markovian generalization of the widely used stochastic Schrödinger equation. Our result allows to describe open quantum systems in terms of stochastic state vectors rather than density operators, without Markov approximation. Moreover, it unifies two recent independent attempts towards a stochastic description of non-Markovian open systems, based on path integrals on the one hand and coherent states on the other. The latter approach utilizes the analytical properties of coherent states and enables a microscopic interpretation of the stochastic states. The alternative first approach is based on the general description of open systems using path integrals as originated by Feynman and Vernon. 03.65.-w, 03.65.Bz, 42.50.Lc Typeset using REVTEX ∗e-mail: [email protected] †e-mail: [email protected] 1 In the last few years the description of open quantum systems in terms of stochastic Schrödinger equations has received remarkable attention. They are now widely used in different fields (measurement theory, quantum optics, quantum chaos, solid states [1–7]), wherever quantum irreversibility matters. They do not only serve as a fruitful theoretical concept but also as a practical method for computations in the form of quantum trajectories. Up to now, however, the Markov approximation was believed to be essential for a stochastic description [8]. For systems where non-Markovian effects are inevitable, as for non-equilibrium relativistic fields, especially in quantum cosmology [9] or solid state physics [10,11], an advantageous stochastic pure state description was missing. This Letter presents an exact non-Markovian stochastic Schrödinger equation. Traditionally, open systems are described by the reduced density operator ρ̂sys(t) = Trenv (ρ̂tot(t)) , obtained from the total density operator by tracing over the environmental degrees of freedom. In Markov approximation, it is well known that the system dynamics can either be described by a master equation for the reduced density operator ρ̂sys(t), or alternatively, by a stochastic Schrödinger equation for state vectors |ψZ(t)〉 [1–7]. In this latter approach, the reduced density operator is recovered as the ensemble average over these stochastic pure states: ρ̂sys(t) =MZ [|ψZ(t)〉〈ψZ(t)|] . (1) Here, |ψZ(t)〉 indicates the solution of the stochastic Schrödinger equation with a particular realization of the in this case Wiener stochastic process Z(t), the mean MZ [. . .] refers to the ensemble average over these processes. The states |ψZ(t)〉 may or may not be normalized depending on whether one utilizes the nonlinear [2] or linear [4] version of the stochastic Schrödinger equation. Both the linear and the non-linear equation lead to the correct reduced density operator according to (1) and they are mathematically equivalent by virtue of a redefinition of the stochastic processes Z(t) [12]. It is the aim of this paper to demonstrate that a stochastic decomposition just like (1) also holds in the general case, without any approximation, in particular without Markov approximation. We derive the linear version of the relevant non-Markovian stochastic Schrödinger equation, the corresponding non-linear, norm preserving theory can be found similar to the 2 Markovian case [13]. Our result can be based on two recent independent approaches to a stochastic description of non-Markovian open systems [14,15]. One approach [14] uses coherent states and has the advantage of offering an interpretation of the solutions of the stochastic Schrödinger equation from first principles. The other [15] is based on the Feynman-Vernon approach to open systems using path integrals [16] and is valid for arbitrary temperatures. To be specific, we use a standard model of open system quantum mechanics, a system coupled linearly via position coupling to an environment of harmonic oscillators [16] Ĥtot = Hsys(q̂, p̂)− q̂ ∑