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arXiv:0907.1615 · Physical Review Letters

Non-Markovian Quantum Trajectories: An Exact Result

Angelo Bassi, Luca Ferialdi

Published Jul 28, 2009• 36 citations
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Abstract

We analyze the non-Markovian stochastic Schrödinger equation describing a particle subject to spontaneous collapses in space (in the language of collapse models), or subject to a continuous measurement of its position (in the language of continuous quantum measurement). For the first time, we give the explicit general solution for the free particle case (H=p(2)/2m) and discuss the main properties. We analyze the case of an exponential correlation function for the noise, giving a quantitative description of the dynamics and of its dependence on the correlation time.

DOI
10.1103/physrevlett.103.050403

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We analyze the non-Markovian stochastic Schrödinger equation describing a particle subject to spontaneous collapses in space (in the language of collapse models), or subject to a continuous measurement of its position (in the language of continuous quantum measurement). For the first time, we give the explicit general solution for the free particle case (H=p(2)/2m) and discuss the main properties. We analyze the case of an exponential correlation function for the noise, giving a quantitative description of the dynamics and of its dependence on the correlation time.
We analyze the non-Markovian dynamics of a quantum system subject to spontaneous collapse in space. After having proved, under suitable conditions, the separation of the center-of-mass and relative motions, we focus our analysis on the time evolution of the center of mass of an isolated system (free-particle case). We compute the explicit expression of the Green's function, for a generic Gaussian noise, and analyze in detail the case of an exponential correlation function. We study the time evolution of average quantities, such as the mean position, momentum, and energy. We eventually specialize to the case of Gaussian wave functions and prove that all basic facts about collapse models, which are known to be true in the white-noise case, hold also in the more general case of non-Markovian dynamics.
Starting from a generalization of the quantum trajectory theory [based on the stochastic Schr\"odinger equation (SSE)], non-Markovian models of quantum dynamics are derived. In order to describe non-Markovian effects, the approach used in this article is based on the introduction of random coefficients in the usual linear SSE. A major interest is that this allows a consistent theory of quantum measurement in continuous time to be developed for these non-Markovian quantum trajectory models. In this context, the notions of ``instrument,'' ``a priori,'' and ``a posteriori'' states can be introduced. The key point is that by starting from a stochastic equation on the Hilbert space of the system, we are able to respect the complete positivity of the mean dynamics for the statistical operator and the requirements of the axioms of quantum measurement theory. The flexibility of the theory is next illustrated by a concrete physical model of a noisy oscillator where non-Markovian effects come from the random environment, colored noises, randomness in the stimulating light, and delay effects. The statistics of the emitted photons and the heterodyne and homodyne spectra are studied, and we show how these quantities are sensitive to the non-Markovian features of the system dynamics, so that, in principle, the observation and analysis of the fluorescent light could reveal the presence of non-Markovian effects and allow for a measure of the spectra of the noises affecting the system dynamics.
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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 H\mathcal{H}H, the extended state space is the triple Hilbert space $\mathcal{H}\ensuremath{\bigotimes}{\mathbb{C}}^{3}$ 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 Schr\"odinger 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.
Spontaneous collapse models (SCM) employ an imaginary noise term in the modification of Schrödinger's equation in order to achieve a stochastic collapse process. Such noise term is typically assumed to be white, and therefore uncorrelated in time, which results in the dynamics being Markovian. In the first part of this paper I discuss the reasons why physicists are exploring the possibility to instead employ non-white stochastic noise terms in the dynamics of SCM, and I then briefly introduce these models. In the second part, my aim is to provide an evaluation of the major conceptual and philosophical consequences of the generalised collapse models with colored noise. In particular, I will focus on the broad class of implications having to do with the non-Markovian nature of the dynamics, with its resulting temporal non-locality, and on the possibility to consider the noise field as a real physical entity.
We discuss the time evolution of the wave function which is the solution of a stochastic Schrödinger equation describing the dynamics of a free quantum particle subject to spontaneous localizations in space. We prove global existence and uniqueness of solutions. We observe that there exist three time regimes: the collapse regime, the classical regime and the diffusive regime. Concerning the latter, we assert that the general solution converges almost surely to a diffusing Gaussian wave function having a finite spread both in position as well as in momentum. This paper corrects and completes earlier works on this issue.