Stochastic jump processes for non-Markovian quantum dynamics

It is shown that non-Markovian master equations for an open system which are local in time can be unravelled through a piecewise deterministic quantum jump process in its Hilbert space. We derive a stochastic Schrödinger equation that reveals how non-Markovian effects are manifested in statistical correlations between different realizations of the process. Moreover, we demonstrate that possible violations of the positivity of approximate master equations are closely connected to singularities of the stochastic Schrödinger equation, which could lead to important insights into the structural characterization of positive non-Markovian equations of motion. Relaxation and decoherence phenomena in open quantum systems [1] can often be modelled with sufficient accuracy by a quantum Markov processes in which the open system’s density matrix is governed by a relatively simple quantum Markovian master equation with Lindblad structure [2, 3]. However, non-Markovian quantum systems featuring strong memory effects play an increasingly important role in many fields of physics such as quantum optics [4], solid state physics [5], and quantum information science [6]. Further applications include non-Markovian extensions of quantum process tomography, quantum control [7], and quantum transport [8]. The non-Markovian quantum dynamics of open systems is characterized by pronounced memory effects, finite revival times and non-exponential behavior of damping and decoherence, resulting from long-range correlation functions and from the dynamical relevance of large correlations and entanglement in the initial state. As a consequence the theoretical treatment of non-Markovian quantum dynamics is generally extremely demanding, both from the analytical and from the computational point of view [9]. Even if one is able to derive an appropriate nonMarkovian master equation or some other mathematical formulation of the dynamics, the numerical simulation of such processes turns out to be a very difficult and timeconsuming task, especially for high-dimensional Hilbert spaces. From classical physics it is known that Monte Carlo (a)E-mail: [email protected] (b)E-mail: [email protected] techniques provide efficient tools for the numerical simulation of complex systems. This fact was one of the motivations to introduce the Monte Carlo wave function technique [10–12] which provides efficient quantum simulation techniques in the regime of Markovian dynamics. Several generalizations of the Monte Carlo approach to nonMarkovian dynamics have been developed which are based on suitable extensions of the underlying reduced system’s Hilbert space [13–16]. Recently, an efficient alternative simulation algorithm for the treatment of non-Markovian open system dynamics has been proposed [17] that does not require any extension of the state space. The purpose of the present paper is to develop a mathematical formulation of this algorithm in terms of a stochastic Schrödinger equation (SSE) in the open system’s Hilbert space. We demonstrate that this formulation gives rise to a new type of piecewise deterministic quantum jumps process (PDP). Quantum master equations are often derived from an underlying microscopic theory by employing some approximation scheme. An appropriate scheme is the time-convolutionless (TCL) projection operator technique which leads to a time-local first-order differential equation for the density matrix [18–20]. It will be shown that TCL master equations allow a stochastic unravelling of the form developed here. Generally, the use of a certain approximation technique may lead to violations of the positivity of the master equation. We demonstrate that positivity violations are closely linked to singularities of the SSE at which the stochastic process breaks down. Hence, a great