Non-markovian continuous quantum measurement of retarded observables

We reconsider the non-Markovian time-continuous measurement of a Heisenberg observable x[over ] and show for the first time that it can be realized by an infinite set of entangled von Neumann detectors. The concept of continuous readout is introduced and used to rederive the non-Markovian stochastic Schr?dinger equation. We can prove that, contrary to recent doubts, the resulting non-Markovian quantum trajectories are true single system trajectories and correspond to the continuous measurement of a retarded functional of x[over ].     

Weak noise expansions through functional integrals for colored noise

We use path integral methods to obtain expansions for the correlation functions of the non-Markovian stochastic processes generated by stochastic differential equations with colored noise.     

Non-Markovian relaxation of a three-level system: quantum trajectory approach

The non-Markovian dynamics of a three-level quantum system coupled to a bosonic environment is a difficult problem due to the lack of an exact dynamic equation such as a master equation. We present for the first time an exact quantum trajectory approach to a dissipative three-level model. We have established a convolutionless stochastic Schr?dinger equation called the time-local quantum state diffusion (QSD) equation without any approximations, in particular, without Markov approximation. Our exact time-local QSD equation opens a new avenue for exploring quantum dynamics for a higher dimensional quantum system coupled to a non-Markovian environment.     

Non-Markovian diffusion equations and processes: Analysis and simulations

In this paper we introduce and analyze a class of diffusion type equations related to certain non-Markovian stochastic processes. We start from the forward drift equation which is made non-local in time by the introduction of a suitable chosen memory kernel K(t). The resulting non-Markovian equation can be interpreted in a natural way as the evolution equation of the marginal density function of a random time process l(t). We then consider the subordinated process Y(t)=X(l(t)) where X(t) is a Markovian diffusion. The corresponding time evolution of the marginal density function of Y(t) is governed by a non-Markovian Fokker-Planck equation which involves the memory kernel K(t). We develop several applications and derive the exact solutions. We consider different stochastic models for the given equations providing path simulations.     

Memory effects on stochastic resonance

We study the phenomenon of stochastic resonance (SR) in a bistable system with internal colored noise. In this situation the system possesses time-dependent memory friction connected with noise via the fluctuation-dissipation theorem, so that in the absence of periodic driving the system approaches the thermodynamic equilibrium state. For this non-Markovian case we find that memory usually suppresses stochastic resonance. However, for a large memory time SR can be enhanced by the memory.     

Relationship between a non-Markovian process and Fokker–Planck equation

We demonstrate the equivalence of a non-Markovian evolution equation with a linear memory-coupling and a Fokker–Planck equation (FPE). In case the feedback term offers a direct and permanent coupling of the current probability density to an initial distribution, the corresponding FPE offers a non-trivial drift term depending itself on the diffusion parameter. As the consequence the deterministic part of the underlying Langevin equation is likewise determined by the noise strength of the stochastic part. This memory induced stochastic behavior is discussed for different, but representative initial distributions. The analytical calculations are supported by numerical results.     

Non-Markovian stochastic resonance

The phenomenological linear response theory of non-Markovian stochastic resonance (SR) is put forward for stationary two-state renewal processes. In terms of a derivation of a non-Markov regression theorem we evaluate the characteristic SR-quantifiers; i.e., the spectral power amplification (SPA) and the signal-to-noise ratio (SNR), respectively. In clear contrast to Markovian-SR, a characteristic benchmark of genuine non-Markovian SR is its distinctive dependence of the SPA and SNR on small (adiabatic) driving frequencies; particularly, the adiabatic SNR becomes strongly suppressed over its Markovian counterpart. This non-Markovian SR-theory is elucidated for a fractal gating dynamics of a potassium ion channel possessing an infinite variance of closed sojourn times.     

A class of collision processes with memory and application to simple chemical reactions in a solvent

We apply the general formalism of a class of non-Markovian processes which we have studied elsewhere to three simple models of chemical reactions: dissociation, isomerization, and diffusion in a double-well potential. Our method leads to explicitly solvable models and numerical computations. The results are in agreement with numerical simulation and stochastic dynamics studies of other authors.     

Microscopic preparation and macroscopic motion of a Brownian particle

We study the effect of a non-equilibrium state of the bath on the macroscopic motion of a Brownian particle (B.p.) in a linear chain. The macroscopic motion is described by a stochastic process which is non-Markovian and nonstationary due to the initial non-equilibrium of the bath. We derive generalized Langevin equations for this proces. We solve them explicitely for the case of a free B.p. and discuss the resulting mean values. A Markov approximation is valid only in the long time region, non-Markovian transients cannot be neglected. In the long time region the non-equilibrium state of the bath has the same effect as a modified macroscopic initial condition for the B.p.     

Exponential Convergence for the Stochastically Forced Navier-Stokes Equations and Other Partially Dissipative Dynamics

 We prove that the two dimensional Navier-Stokes equations possess an exponentially attracting invariant measure. This result is in fact the consequence of a more general ``Harris-like' ergodic theorem applicable to many dissipative stochastic PDEs and stochastic processes with memory. A simple iterated map example is also presented to help build intuition and showcase the central ideas in a less encumbered setting. To analyze the iterated map, a general ``Doeblin-like' theorem is proven. One of the main features of this paper is the novel coupling construction used to examine the ergodic theory of the non-Markovian processes. Received: 23 March 2001 / Accepted: 2 April 2002 Published online: 14 October 2002     

Exact c-number representation of non-Markovian quantum dissipation

The reduced dynamics of a quantum system interacting with a linear heat bath finds an exact representation in terms of a stochastic Schr?dinger equation. All memory effects of the reservoir are transformed into noise correlations and mean-field friction. The classical limit of the resulting stochastic dynamics is shown to be a generalized Langevin equation, and conventional quantum state diffusion is recovered in the Born-Markov approximation. The non-Markovian exact dynamics, valid at arbitrary temperature and damping strength, is exemplified by an application to the dissipative two-state system.     

Noise-induced coupling and phase transition in initially homogeneous bistable system

We show that a colored spatial noise induces a heterogeneous behavior and coupling of initially uncoupled single bistable units. A formal approximation reduces a non-Markovian stochastic process described by the initial set of equations into Markovian process in terms of Langevin equation, for which a simple piecewise linear emulation was used to represent the nonlinear deterministic force. It turned out that the coupling leads to a phase transition due to the noise-induced diffusive term. As an example, a typical bistable noisy system with symmetric double-well potential was studied.     

Microdynamics and time-evolution of macroscopic non-Markovian systems. II

We study the time-evolution of the joint and the conditional probability of macroscopic variables of a closed system from a microscopic point of view. We derive an exact generalized master equation for their time rate of change which consists of two parts, one instantaneous and local in state space, the other retarded and nonlocal in state space. It is represented by stochastic operators depending both on the initial preparation and on the initial macrodistribution, which reflects the non-Markovian character of the process. The connection with the time-evolution of the single-event probability is discussed.Work Supported by the Swiss National Science Foundation     

Pure-state quantum trajectories for general non-Markovian systems do not exist

Since the first derivation of non-Markovian stochastic Schr?dinger equations, their interpretation has been contentious. In a recent Letter [Phys. Rev. Lett. 100, 080401 (2008)10.1103/Phys. Rev. Lett.100.080401], Diósi claimed to prove that they generate "true single system trajectories [conditioned on] continuous measurement." In this Letter, we show that his proof is fundamentally flawed: the solution to his non-Markovian stochastic Schr?dinger equation at any particular time can be interpreted as a conditioned state, but joining up these solutions as a trajectory creates a fiction.     

Non-Markovian dynamics of open quantum systems: Stochastic equations and their perturbative solutions

We treat several key stochastic equations for non-Markovian open quantum system dynamics and present a formalism for finding solutions to them via canonical perturbation theory, without making the Born–Markov or rotating wave approximations (RWA). This includes master equations of the (asymptotically) stationary, periodic, and time-nonlocal type. We provide proofs on the validity and meaningfulness of the late-time perturbative master equation and on the preservation of complete positivity despite a general lack of Lindblad form. More specifically, we show how the algebraic generators satisfy the theorem of Lindblad and Gorini, Kossakowski and Sudarshan, even though the dynamical generators do not. These proofs ensure the mathematical viability and physical soundness of solutions to non-Markovian processes. Within the same formalism we also expand upon known results for non-Markovian corrections to the quantum regression theorem. Several directions where these results can be usefully applied to are also described, including the analysis of near-resonant systems where the RWA is inapplicable and the calculation of the reduced equilibrium state of open systems.     

Stochastic Resonance in Nonequilibrium Single-Stable System

The harmonic noise is used as an external stochastic source in non-Markovian equilibrium system. A bi-peak resonance phenomenon is carried out due to the external nohe frequency close to the frequencies of damped linear-oscillatpr and internal noise, respectively.     

Entanglement evolution in a non-Markovian environment

We extend recent theoretical studies of entanglement dynamics in the presence of environmental noise, following the long-time interest of Krzysztof Wodkiewicz in the effects of stochastic models of noise on quantum optical coherences. We investigate the quantum entanglement dynamics of two spins in the presence of classical Ornstein-Uhlenbeck noise, obtaining exact solutions for evolution dynamics. We consider how entanglement can be affected by non-Markovian noise, and discuss several limiting cases.