Stochastic Schrödinger equations with coloured noise

A natural non-Markovian extension of the theory of white noise quantum trajectories is presented. In order to introduce memory effects in the formalism an Ornstein-Uhlenbeck coloured noise is considered as the output driving process. Under certain conditions a random Hamiltonian evolution is recovered. Moreover, non-Markovian stochastic Schrödinger equations which unravel non-Markovian master equations are derived. As a prototypical example, an harmonic oscillator is considered, able to emit light and with memory terms in the dynamics. The signature of the non-Markovian dynamics is seen in the spectrum of the emitted light. Introduction. – Recently, stochastic wave function methods for the description of open quantum systems have received considerable attention [1–4]. These approaches have been mainly motivated by the continuous measurement description of detection schemes in quantum optics, namely direct photo-detection, homodyne and heterodyne detection. In general, selective, indirect and continuous quantum measurements allow for a description of the open quantum system in terms of the stochastic evolution of its own wave function. Typically, the random trajectories of the state vector involve quantum jumps or diffusion processes and are described by stochastic differential equations for the wave function ψt. The relationship to the more traditional approach to open quantum systems in terms of master equations for the density matrix ρt is easily established by realising that the latter can be expressed as ρt = E[|ψt〉〈ψt|], where E denotes the ensemble average over realisations of the stochastic process ψt. Thus, the master equation evolution can be reproduced by generating a large number of trajectories of the state vector. This unravelling procedure, called Monte Carlo wave function method, has gained considerable importance for the numerical simulation of complex open systems [3,5]. The procedure described above has been a major breakthrough in the description of the Markovian dynamics of open quantum systems [1–3,5, 6], where it has found many applications. At present an active line of research is to find generalisations of the stochastic Schrödinger equation (SSE) in the non-Markovian