A nonlinear stochastic Schrödinger equation for pure states describing non-Markovian diffusion of quantum trajectories and compatible with non-Markovian master equations is presented. This provides an unravelling of the evolution of any quantum system coupled to a finite or infinite number of harmonic oscillators without any approximation. Its power is illustrated by several examples, including measurement-like situations, dissipation, and quantum Brownian motion. Some examples treat this environment phenomenologically as an infinite reservoir with fluctuations of arbitrary correlation. In other examples the environment consists of a finite number of oscillators. In such a quasi-periodic case we show the reversible decay of a 'Schrödinger cat' state. Finally, our description of open systems is compatible with different positions of the 'Heisenberg cut' between system and environment.
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
A non-Markovian stochastic Schrödinger equation for a quantum system coupled to an environment of harmonic oscillators is presented. Its solutions, when averaged over the noise, reproduce the standard reduced density matrix without any approximation. We illustrate the power of this approach with several examples, including exponentially decaying memory correlations and extreme non-Markovian periodic cases, where the 'environment' consists of only a single oscillator. The latter case shows the decay and revival of a 'Schrödinger cat' state. For strong coupling to a dissipative environment with memory, the asymptotic state can be reached in a finite time. Our description of open systems is compatible with different positions of the 'Heisenberg cut' between system and environment.The dynamics of open quantum systems is a very timely problem, both to address fundamental questions (quantum decoherence, measurement problem) as well as to tackle the more practical problems of engineering the quantum devices necessary for the emerging fields of nanotechnology and quantum computing. So far, the true dynamics of open systems has almost always been simplified by the Markov approximation: environmental correlation times are assumed negligibly short compared to the system's characteristic time scale.For the numerical solution of Markovian open systems, described by a master equation of Lindblad form(where ρt denotes the density matrix, H the system's Hamiltonian and the operators Lm describe the effect of the environment in the Markov approximation), a breakthrough was achieved through the discovery of stochastic unravellings [1,2]. These are stochastic Schrödinger equations for states ψt(z), driven by a certain classical noise zt with distribution functional P (z). Crucially, the ensemble mean M [. . .] over the noise recovers the density operator,Hence, the solution of eq. (1) is reduced from a problem in the matrix space of ρ to a much simpler Monte Carlo simulation of quantum trajectories ψt(z) in the state space. For the Markov master eq.(1), several such unravellings are known. Some involve jumps at random times [1], others have continuous, diffusive solutions [2]. They have been used extensively over recent years, as they provide useful insight into the dynamics of continuously monitored (individual) quantum processes [3], or into the mechanism of decoherence [4]. In addition, they provide an efficient tool for the numerical solution of the master eq.(1). It is thus desirable to extend the powerful concept of stochastic unravellings to the more general case of non-Markovian evolution.The simplest unravellings are linear stochastic Schrödinger equations. In the Markov case (1), for a single L, the linear equationprovides such an unravelling, where, zt is a complex-valued Wiener process of zero mean and correlations M [z * t zs] = δ(t − s), M [ztzs] = 0, and where • denotes the Stratonovich product [5].However, eq. (3) is of limited value, since the norm ψt(z) of its solutions tends to 0 with probability 1 and to infini...
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