Stochastic Equations in Hilbert Space with Application to Navier-Stokes Equations in Any Dimension

AbstractWe give an existence theorem for an abstract nonlinear stochastic evolution equation in a Hilbert space. The result is applicable to the stochastic Navier-Stokes equation in any dimension with a nonlinear noise term

Measurement-induced Squeezing of a Bose-Einstein Condensate

We discuss the dynamics of a Bose-Einstein condensate during its nondestructive imaging. A generalized Lindblad superoperator in the condensate master equation is used to include the effect of the measurement. A continuous imaging with a sufficiently high laser intensity progressively drives the quantum state of the condensate into number squeezed states. Observable consequences of such a measurement-induced squeezing are discussed.Comment: 4 pages, 2 figures, submitted to PR

Robustness and diffusion of pointer states

Classical properties of an open quantum system emerge through its interaction with other degrees of freedom (decoherence). We treat the case where this interaction produces a Markovian master equation for the system. We derive the corresponding distinguished local basis (pointer basis) by three methods. The first demands that the pointer states mimic as close as possible the local non-unitary evolution. The second demands that the local entropy production be minimal. The third imposes robustness on the inherent quantum and emerging classical uncertainties. All three methods lead to localized Gaussian pointer states, their formation and diffusion being governed by well-defined quantum Langevin equations.Comment: 5 pages, final versio

Measurements continuous in time and a posteriori states in quantum

Measurements continuous in time were consistently introduced in quantum mechanics and applications worked out, mainly in quantum optics. In this context a quantum filtering theory has been developed giving the reduced state after the measurement when a certain trajectory of the measured observables is registered (the a posteriori states). In this paper a new derivation of filtering equations is presented, in the cases of counting processes and of measurement processes of diffusive type. It is also shown that the equation for the a posteriori dynamics in the diffusive case can be obtained, by a suitable limit, from that one in the counting case. Moreover, the paper is intended to clarify the meaning of the various concepts involved and to discuss the connections among them. As an illustration of the theory, simple models are worked out.Comment: 31 page. See also related papers at http://www.maths.nott.ac.uk/personal/vpb/research/mes_fou.html and http://www.maths.nott.ac.uk/personal/vpb/research/fil_con.htm

Non-Markovian stochastic Schr\"odinger equations: Generalization to real-valued noise using quantum measurement theory

Do stochastic Schr\"odinger equations, also known as unravelings, have a physical interpretation? In the Markovian limit, where the system {\em on average} obeys a master equation, the answer is yes. Markovian stochastic Schr\"odinger equations generate quantum trajectories for the system state conditioned on continuously monitoring the bath. For a given master equation, there are many different unravelings, corresponding to different sorts of measurement on the bath. In this paper we address the non-Markovian case, and in particular the sort of stochastic \sch equation introduced by Strunz, Di\' osi, and Gisin [Phys. Rev. Lett. 82, 1801 (1999)]. Using a quantum measurement theory approach, we rederive their unraveling which involves complex-valued Gaussian noise. We also derive an unraveling involving real-valued Gaussian noise. We show that in the Markovian limit, these two unravelings correspond to heterodyne and homodyne detection respectively. Although we use quantum measurement theory to define these unravelings, we conclude that the stochastic evolution of the system state is not a true quantum trajectory, as the identity of the state through time is a fiction.Comment: 17 pages, 3 figure

Open system dynamics with non-Markovian quantum trajectories

A non-Markovian stochastic Schroedinger 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 operator without any approximation. We illustrate the power of this approach with several examples, including exponentially decaying bath correlations and extreme non-Markovian cases, where the `environment' consists of only a single oscillator. The latter case shows the decay and revival of a `Schroedinger 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.Comment: 4 pages RevTeX, 3 figure

Covariant Quantum Dynamical Semigroups: Unbounded generators

A survey of the probabilistic approaches to quantum dynamical semigroups with unbounded generators is given. An emphasis is made upon recent advances in the structural theory of covariant Markovian master equations. The relations with the classical Levy-Khinchin formula are elucidated. As an example, a complete characterizations of the Galilean covariant irreversible quantum Markovian evolutions is given in terms of the corresponding quantum master and Langevin equations. Important topics for future investigation are outlined.Comment: 14 pages,Latex, no figures, submitted to the Semigroup Volume, Group 21, Goslar 199

The Post-Decoherence Density Matrix Propagator for Quantum Brownian Motion

Using the path integral representation of the density matrix propagator of quantum Brownian motion, we derive its asymptotic form for times greater than the localization time, (\hbar / \gamma k T )^{\half}, where $\gamma$ is the dissipation and $T$ the temperature of the thermal environment. The localization time is typically greater than the decoherence time, but much shorter than the relaxation time, $\gamma^{-1}$. We use this result to show that the reduced density operator rapidly evolves into a state which is approximately diagonal in a set of generalized coherent states. We thus reproduce, using a completely different method, a result we previously obtained using the quantum state diffusion picture (Phys.Rev. D52, 7294 (1995)). We also go beyond this earlier result, in that we derive an explicit expression for the weighting of each phase space localized state in the approximately diagonal density matrix, as a function of the initial state. For sufficiently long times it is equal to the Wigner function, and we confirm that the Wigner function is positive for times greater than the localization time (multiplied by a number of order 1).Comment: 17 pages, plain Tex, submitted to Physical Review

Quantum State Diffusion, Density Matrix Diagonalization and Decoherent Histories: A Model

We analyse the quantum evolution of a particle moving in a potential in interaction with an environment of harmonic oscillators in a thermal state, using the quantum state diffusion (QSD) picture of Gisin and Percival, in which one associates the usual Markovian master equation for the density operator with a class of stochastic non-linear Schr\"odinger equations. We find stationary solutions to the Ito equation which are Gaussians, localized around a point in phase space undergoing classical Brownian motion. We show that every initial state approaches these stationary solutions in the long time limit. We recover the density operator corresponding to these solutions, and thus show, for this particular model, that the QSD picture effectively supplies a prescription for approximately diagonalizing the density operator in a basis of phase space localized states. The rate of localization is related to the decoherence time, and also to the timescale on which thermal and quantum fluctuations become comparable. We use these results to exemplify the general connection between the QSD picture and the decoherent histories approach.Comment: 32 pages, plain Tex