Abstract: The dynamics of two weakly interacting spins coupled to separate bosonic baths is studied. An analytical solution in the Born approximation for arbitrary spectral density functions of the bosonic environments is found. It is shown that in the non-Markovian cases concurrence 'lives' longer or reaches greater values.
Abstract: The classical product state capacity of a noisy quantum channel with memory is investigated. A forgetful noise-memory channel is constructed by Markov switching between two depolarizing channels which introduces non-Markovian noise correlations between successive channel uses. The computation of the capacity is reduced to an entropy computation for a function of a Markov process. A reformulation in terms of algebraic measures then enables its calculation. The effects of the hidden-Markovian memory on the capacity are explored. An increase in noise correlations is found to increase the capacity.
Abstract: Recently a generalized master equation was derived that extends the Lindblad theory to highly non-Markovian quantum processes [H.-P. Breuer, Phys. Rev. A 75, 022103 (2007)]. We perform a stochastic unraveling of this master equation by considering n random state vectors that satisfy the corresponding stochastic differential equation for a piecewise deterministic process. As an application we consider a two-state system randomly coupled to an environment consisting of two energy bands with finite number of levels. Our numerical results are compared to results obtained from the time-convolutionless projection operator method using correlated projection superoperators and the exact solution of the Schrodinger equation for this system.
Abstract: A non-Markovian model of quantum repeated interactions between a small quantum system and an infinite chain of quantum systems is presented. By adapting and applying usual projection operator techniques in this context, discrete versions of the integro-differential and time-convolutionless master equations for the reduced system are derived. Next, an intuitive and rigorous description of the indirect quantum measurement principle is developed and a discrete non-Markovian stochastic master equation for the open system is obtained. Finally, the question of unravelling in a particular model of non-Markovian quantum interactions is discussed.
Abstract: The quantum dynamics of a spin chain interacting with multiple bosonic baths is described in a mixed Wigner-Heisenberg representation. The formalism is illustrated by simulating the time evolution of the reduced density matrix of two coupled spins, where each spin is also coupled to its own bath of harmonic oscillators. In order to prove the validity of the approach, an analytical solution in the Born-Markov approximation is found. The agreement between the two methods is shown.
Abstract: We stress the relevance of the two features of translational invariance and atomic nature of the gas in the quantum description of the motion of a massive test particle in a gas, corresponding to the original picture of Einstein used in the characterization of Brownian motion. The master equation describing the reduced dynamics of the test particle is of Lindblad form and complies with the requirement of covariance under translations.
Abstract: Thermal fluctuations in time-dependent quantum processes are treated by a constant-temperature generalization of Wigner's formulation of quantum mechanics in phase space. To this end, quantum Nose-Hoover dynamics is defined by generalizing the Moyal bracket. Computational applications of the formalism, together with further theoretical developments, are discussed.
Abstract: The dynamics of a simple spin chain (two spins) coupled to bosonic baths at different temperatures is studied. The analytical solution for the reduced density matrix of the system is found. The dynamics and temperature dependence of spin-spin entanglement is analyzed. It is shown that the system converges to a steady state. If the energy levels of the two spins are different, the steady-state concurrence assumes its maximum at unequal bath temperatures. It is found that a difference in local energy levels can make the steady-state entanglement more stable against high temperatures.
Abstract: A parametrization of density operators for bipartite quantum systems is prop-osed. It is based on the particular parametrization of the unitary group found recently by Jarlskog. It is expected that this parametrization will find interesting applications in the study of quantum properties of multipartite systems.
Abstract: Already in the case of finite dimensional Hilbert spaces H the general form of density matrices rho is not known. The main reason for this lack of knowledge is the nonlinear constraint for these matrices. We propose a representation of density matrices on finite dimensional Hilbert spaces interms of finitely many independent parameters. For dimensions 2, 3, and 4 we write down this representation explicitly. As a further application of this representation we study the time dependence of density matrices rho(t) which in our case is implemented through time dependence of the independent parameters. Under obvious differentiability assumptions the explicit form of d/dt rho(t) is determined. As a special case we recover, for instance, the Lindblad form.
Abstract: Decoherence phenomena in a small quantum system coupled to a complex environment can be modelled with random matrices. We propose a simple deterministic model in the limit of a high dimensional environment. The model is investigated numerically and some analytically addressable questions are singled out.
Abstract: The time evolution of a spin-1/2 particle under the influence of a locally applied external magnetic field, and interacting with anisotropic spin environment in thermal equilibrium at temperature T is studied. The exact analytical form of the reduced density matrix of the central spin is calculated explicitly for a finite number of bath spins. The case of an infinite number of environmental spins is investigated using the convergence of the rescaled bath operators to normal Gaussian random variables. In this limit, we derive the analytical form of the components of the Bloch vector for antiferromagnetic interactions within the bath, and we investigate the short-time and long-time behavior of reduced dynamics. The effect of the external magnetic field, the anisotropy, and the temperature of the bath on the decoherence of the central spin are discussed.
Abstract: We discuss two methods of an exact stochastic representation of the non-Markovian quantum dynamics of open systems. The first method employs a pair of stochastic product vectors in the total system's state space, while the second method uses a pair of state vectors in the open system's state space and a random operator acting on the state space of the environment. Both techniques lead to an exact solution of the von Neumann equation for the density matrix of the total system. Employing a spin star model describing a central spin coupled to the bath of surrounding spins, we perform Monte Carlo simulations for both variants of the stochastic dynamics. In addition, we derive an analytical expression for the expectation values of the stochastic dynamics to obtain the exact solution for the density matrix of the central spin.
Abstract: The exact solution of a Cauchy problem related to a linear second-order difference equation with constant noncommutative coefficients is reported. Copyright (c) 2007 John Wiley & Sons, Ltd.
Abstract: In this paper we investigate the non-Markovian dynamics of a qubit by comparing two generalized master equations with memory. In the case of a thermal bath, we derive the solution of the recently proposed post-Markovian master equation [A. Shabani and D. A. Lidar, Phys. Rev. A 71, 020101(R) (2005)] and we study the dynamics for an exponentially decaying memory kernel. We compare the solution of the post-Markovian master equation with the solution of the typical memory kernel master equation. Our results lead to a new physical interpretation of the reservoir correlation function and bring to light the limits of usability of master equations with memory for the system under consideration.
Abstract: We derive the exact reduced dynamics of a central two-qubit system in a spin star configuration. The exact evolution of the reduced system density matrix is obtained and we compute the limit of an infinite number of environment spins. Initially pure states of the central system evolve into mixed ones and we determine the decoherence-free states of the model. The long-time behavior is studied, partial decoherence is shown to be a result of the coupling of the qubits to the environment, and entanglement evolution of the central system is investigated.
Abstract: The dynamics of a free charged particle, initially described by a coherent wave packet, interacting with an environment, i.e., the electromagnetic field characterized by a temperature T, is studied. Using the dipole approximation, the exact expressions for the evolution of the reduced density matrix both in momentum and configuration space and the vacuum and the thermal contribution to decoherence are obtained. The time behavior of the coherence lengths in the two representations are given. Through the analysis of the dynamic of the field structure associated with the particle the vacuum contribution is shown to be linked to the birth of correlations between the single momentum components of the particle wave packet and the virtual photons of the dressing cloud.
Abstract: We consider the dynamics of a charged free particle, initially described by a coherent wave packet, interacting with an electromagnetic field characterized by the temperature T, considered as the environment. We have used dipole approximation neglecting the potential vector quadratic term in the minimal coupling Hamiltonian. This leads to the loss of coherence in the momentum representation, described by the decay of the off diagonal elements of the particle reduced density matrix, while the populations remain constant. Here we extend the analysis to the coordinate representation. We compute the particle reduced density matrix in this basis, analyzing in particular the mixing of various effects, such as free spreading, vacuum dressing and vacuum and thermal decoherence, in the wave packet width and coherence length evolution.
Abstract: We demonstrate a scaling method for non-Markovian Monte Carlo wave-function simulations used to study open quantum systems weakly coupled to their environments. We derive a scaling equation, from which the result for the expectation values of arbitrary operators of interest can be calculated, all the quantities in the equation being easily obtainable from the scaled Monte Carlo wave-function simulations. In the optimal case, the scaling method can be used, within the weak coupling approximation, to reduce the size of the generated Monte Carlo ensemble by several orders of magnitude. Thus, the developed method allows faster simulations and makes it possible to solve the dynamics of the certain class of non-Markovian systems whose simulation would be otherwise too tedious because of the requirement for large computational resources.
Abstract: A fully quantum treatment of Einstein's Brownian motion is given, stressing in particular the role played by the two original requirements of translational invariance and connection between dynamics of the Brownian particle and atomic nature of the medium. The former leads to a clearcut relationship with a generator of translation-covariant quantum-dynamical semigroups recently characterized by Holevo, the latter to a formulation of the fluctuation-dissipation theorem in terms of the dynamic structure factor, a two-point correlation function introduced in seminal work by Van Hove, directly related to density fluctuations in the medium and therefore to its atomistic, discrete nature. A microphysical expression for the generally temperature dependent friction coefficient is given in terms of the dynamic structure factor and of the interaction potential describing the single collisions. A comparison with the Caldeira-Leggett model is drawn, especially in view of the requirement of translational invariance, further characterizing general structures of reduced dynamics arising in the presence of symmetry under translations.
Abstract: Quite recently quantum features exhibited by a mesoscopic field interacting with a single Rydberg atom in a microwave cavity has been observed [A. Auffeves et al., Phys. Rev. Lett. 91, 230405 (2003)]. In this paper we theoretically analyze all the phases of this articulated experiment considering from the very beginning cavity losses. Fully applying the theory of quantum open systems, our modelization succeeds in predicting fine aspects of the measured quantity, reaching qualitative and quantitative good agreement with the experimental results. This fact validates our theoretical approach based on the fundamental atom-cavity interaction model and simple mathematical structure of dissipative superoperators.
Abstract: Computational physics as a university undergraduate programme, or as a choice of specialized modules or laboratories within the mainstream physics programme, has grown considerably over the past several years, but this trend is yet to develop fully in South Africa. This paper provides a motivation for developing computational physics at our universities and suggests how such a programme may be designed. We argue that students develop a more intuitive feel for physics, that they learn useful, transferable skills that make our graduates sought-after by industry and commerce, and that such graduates are better prepared to tackle research problems at the master's and doctoral levels both in theoretical and experimental physics. We also discuss current research topics in computational physics that are of relevance to South Africa.
Abstract: We consider a free charged particle interacting with an electromagnetic bath at zero temperature. The dipole approximation is used to treat the bath wavelengths larger than the width of the particle wave packet. The effect of these wavelengths is then described by a linear Hamiltonian whose form is analogous to the phenomenological Hamiltonians previously adopted to describe the free particle-bath interaction. We study how the time dependence of decoherence evolution is related with initial particle-bath correlations. We show that decoherence is related to the time dependent dressing of the particle. Moreover, because decoherence induced by the T = 0 bath is very rapid, we make some considerations on the conditions under which interference may be experimentally observed.
Abstract: The reduced dynamics of a central spin coupled to a bath of N spin-1/2 particles arranged in a spin star configuration is investigated. The exact time evolution of the reduced density operator is derived, and an analytical solution is obtained in the limit N-->infinity of an infinite number of bath spins, where the model shows complete relaxation and partial decoherence. It is demonstrated that the dynamics of the central spin cannot be treated within the Born-Markov approximation. The Nakajima-Zwanzig and the time-convolutionless projection operator technique are applied to the spin star system. The performance of the corresponding perturbation expansions of the non-Markovian equations of motion is examined through a comparison with the exact solution.
Abstract: We study the open system dynamics of a harmonic oscillator coupled with an artificially engineered reservoir. We single out the reservoir and system variables governing the passage between Lindblad-type and non-Lindblad-type dynamics of the reduced system's oscillator. We demonstrate the existence of conditions under which virtual exchanges of energy between system and reservoir take place. We propose to use a single trapped ion coupled to engineered reservoirs in order to simulate quantum Brownian motion.
Abstract: The dynamics of a typical open quantum system, namely a quantum Brownian particle in a harmonic potential, is studied focusing on its non-Markovian regime. Both an analytic approach and a stochastic wave-function approach are used to describe the exact time evolution of the system. The border between two very different dynamical regimes, the Lindblad and non-Lindblad regimes, is identified and the relevant physical variables governing the passage from one regime to the other are singled out. The non-Markovian short-time dynamics is studied in detail by looking at the mean energy, the squeezing, the Mandel parameter, and the Wigner function of the system.
Abstract: The dynamics of two two-level dipole-dipole interacting atoms coupled to a common electro-magnetic bath and closely located inside a lossy cavity, is reported. Initially injecting only one excitation in the two-atom cavity system, loss mechanisms asymptotically drive the matter sample toward a stationary maximally entangled state. The role played by the closeness of the two atoms, with respect to such a cooperative behavior, is carefully discussed. Stationary radiation trapping effects are found and transparently interpreted.
Abstract: A master equation approach to the numerical solution of option pricing models is developed. The basic idea of the approach is to consider the Black-Scholes equation as the macroscopic equation of an underlying mesoscopic stochastic option price variable. The dynamics of the latter is constructed and formulated in terms of a master equation. The numerical efficiency of the approach is demonstrated by means of stochastic simulation of the mesoscopic process for both European and American options. (C) 2002 Elsevier Science B.V. All rights reserved.
Abstract: The emergence of decoherence in quantum electrodynamics is investigated. On combining superoperator methods with functional techniques from field theory, the degrees of freedom of a thermal radiation field are eliminated and the influence phase functional is derived which governs the reduced dynamics of the matter variables. Employing a prototypical interference device, a decoherence functional is developed which provides a gauge invariant relativistic measure for the degree of decoherence. It is demonstrated that the decoherence functional describes the destruction of quantum coherence through the emission of bremsstrahlung which is caused by the relative motion of the interfering components of a superposition. Explicit analytical expressions for the vacuum and the thermal contribution to the decoherence functional and for the corresponding coherence lengths are determined. These expressions reveal that bremsstrahlung leads to a fundamental decoherence mechanism which dominates for short times and which is present even in the electromagnetic field vacuum at zero temperature. The influence of bremsstrahlung on the center of mass coordinate of a system of many identical charged particles is also studied and is shown to lead to a strong suppression of quantum coherence.
Abstract: The non-Markovian dynamics of a continuous-wave atom laser model is studied which includes gravitational effects and interactions inside the Bose-Einstein condensate. With the help of the time-convolutionless projection operator technique both the occupation number of the condensate and the spectrum of the atoms coupled out of the condensate are determined. A correct modeling of memory effects in the atom laser requires a generalization of the projection operator technique which takes into account correlations in the initial state of the combined atom-reservoir system. The non-Markovian evolution is shown to yield substantial deviations from results obtained in the Born-Markov approximation.
Abstract: The time-convolutionless projection operator method is used to investigate the non-Markovian dynamics of open quantum systems. On the basis of this method a systematic perturbation expansion for the reduced density matrix equation is obtained involving a time-dependent generator which is local in time. This formalism is generalized to enable the treatment of system-environment correlations in the initial state, which arise in the computation of equilibrium correlation functions or from the preparation of the system by a quantum measurement. The general method is illustrated by means of the damped harmonic oscillator and of the spin-boson model. The perturbation expansion of the equation of motion is applied to a study of relaxation and dephasing processes and to the determination of the stationary state and of equilibrium correlation functions. Special emphasis is laid on the construction of general, computable error estimates which allow the explicit validation of the obtained results. In particular, the parameter regime for which an expansion of the equation of motion to fourth order yields reliable results is determined. The results clearly reveal that a large range of physically relevant parameters, in particular those that might be interesting for experiments on macroscopic quantum coherence phenomena, can already be treated using the expansion to fourth order, It is thus demonstrated that the time-convolutionless projection operator technique provides a transparent and technically feasible method to go beyond the Markovian approximation in the study of open quantum systems. (C) 2001 Academic Press.
Abstract: In a recent experiment the progressive decoherence of a mesoscopic superposition of two coherent field states in a high-Q cavity, known as Schrodinger cat state; has been measured for the first time [Brune et al., Phys. Rev. Lett. 77, 4887 (1966)]. Here, the full master equation governing the coupled dissipative dynamics of the atom-field system studied in the experiment is formulated and solved numerically for the experimental parameters. The model simulated avoids the approximations underlying an analytically solvable model which is based on a harmonic expansion of the energies of the dressed atomic states and on a treatment of their dynamics within the adiabatic approximation. In particular, the numerical simulations reveal that the coupling of the cavity field mode to its environment causes important decoherence effects already during the initial preparation phase of the Schrodinger cat state. This phenomenon is investigated in detail with the help of a measure for the purity of states. Moreover, the Hilbert-Schmidt distance of the intended target state, the Schrodinger cat, to the state that is actually prepared in the experiment is determined.
Abstract: The dynamics of periodically driven quantum systems coupled to a thermal environment is investigated. The interaction of the system with the external coherent driving field is taken into account exactly by making use of the Floquet picture. Treating the coupling to the environment within the Born-Markov approximation one finds a Pauli-type master equation for the diagonal elements of the reduced density matrix in the Floquet representation. The stationary solution of the latter yields a quasistationary, time-periodic density matrix which describes the long-time behavior of the system. Taking the example of a periodically driven particle in a box, the stationary solution is determined numerically for a wide range of driving amplitudes and temperatures. It is found that the quasistationary distribution differs substantially from a Boltzmann-type distribution at the temperature of the environment. For large driving fields it exhibits a plateau region describing a nearly constant population of a certain number of Floquet states. This number of Floquet states turns out to be nearly independent of the temperature. The plateau region is sharply separated from an exponential tail of the stationary distribution which expresses a canonical Boltzmann-type distribution over the mean energies of the Floquet states. These results are explained in terms of the structure of the matrix of transition rates for the dissipative quantum system. Investigating the corresponding classical, nonlinear Hamiltonian system, one finds that in the semiclassical range essential features of the quasistationary distribution can be understood from the structure of the underlying classical phase space.
Abstract: Different methods for the numerical solution of stochastic differential equations arising in the quantum mechanics of open systems are discussed. A comparison of the stochastic Euler and Heun schemes, a stochastic variant of the fourth order Runge-Kutta scheme, and a second order scheme proposed by Platen is performed. By employing a natural error measure the convergence behaviour of these schemes for stochastic differential equations of the continuous spontaneous localization type is investigated. The general theory is tested by two examples from quantum optics. The numerical tests confirm the expected convergence behaviour in the case of the Euler, the Heun and the second order scheme. On the contrary, the heuristic Runge-Kutta scheme turns out to be a first order scheme such that no advantage over the simple Euler scheme is obtained. The results also clearly reveal that the second order scheme is superior to the other methods with regard to convergence behaviour and numerical performance. (C) 2000 Elsevier Science B.V. All rights reserved.
Abstract: A generalization of the stochastic wave-function method to quantum master equations which are not in Lindblad form is developed. The proposed stochastic unraveling is based on a description of the reduced system in a doubled Hilbert space and it is shown that this method is capable of simulating quantum master equations with negative transition rates. Non-Markovian effects in the reduced systems dynamics can be treated within this approach by employing the time-convolutionless projection operator technique. This ansatz yields a systematic perturbative expansion of the reduced systems dynamics in the coupling strength. Several examples such as the damped Jaynes-Cummings model and the spontaneous decay of a two-level system into a photonic band gap are discussed. The power as well as the limitations of the method are demonstrated. [S1050-2947(99)08102-0].
Abstract: Recently a master equation for the three-dimensional Navier-Stokes equation in k-space has been proposed. It has been shown, that the Hopf-equation can be derived from the time evolution of the stochastic process given by the master equation. Therefore it reproduces exactly the correct turbulence moment hierarchy. Here we present the results of a Monte Carlo simulation of turbulence in three space dimensions using the proposed master equation. We simulate the underlying stochastic process defined by the master equation by producing realizations and calculating averages. The results of the simulation at a Taylor-Reynolds number R-lambda of about 112 show a -5/3 scaling range for the energy spectrum and the Kolmogorov-constant is about 1.6. (C) 1999 Published by Elsevier Science B.V. All rights reserved.
Abstract: A generalization of the stochastic wave function method is presented that allows the unraveling of arbitrary linear quantum master equations that are not necessarily in Lindblad form and, moreover, the explicit treatment of memory effects by employing the time-convolutionless projection operator technique. The crucial point of this construction is the description of the open system in a doubled Hilbert space, which has already been successfully used for the computation of multitime correlation functions.
Abstract: The dynamics of atom lasers with a continuous output coupler based on two-photon Raman transitions is investigated. With the help of the time-convolutionless projection operator technique the quantum master equations for pulsed- and continuous-wave (cw) atom lasers are derived. In the case of the pulsed atom laser the power of the time-convolutionless projection operator technique is demonstrated through comparison with the exact solution. It is shown that in an intermediate coupling regime where the Born-Markov approximation fails, the results of this algorithm agree with the exact solution. To study the dynamics of a continuous-wave atom laser a pump mechanism is included in the model. Whereas the pump mechanism is treated within the Born-Markov approximation, the output coupling leads to non-Markovian effects. The solution of the master equation resulting from the time-convolutionless projection operator technique exhibits strong oscillations in the occupation number of the Bose-Einstein condensate. These oscillations are traced back to a quantum interference which is due to the non-Markovian dynamics and which decays slowly in time as a result of the dispersion relation for massive particles. [S1050-2947(99)01610-8].
Abstract: A relativistic generalization of the quantum-state diffusion model is developed. The model describes a Dirac electron which is coupled to an external electromagnetic field and a dissipative environment. A relativistically covariant stochastic Dirac equation is obtained by regarding the state vector as a functional on a certain set of spacelike hypersurfaces in Minkowski space and by the definition;bf an appropriate Hilbert bundle on this set. The integrability condition of the stochastic process and the corresponding covariant density matrix equation are derived. Further, the relativistic equations governing the dynamical state-vector localization are deduced.
Abstract: A stochastic simulation algorithm for the computation of multitime correlation functions which is based on the quantum-state diffusion model of open systems is developed. The crucial point of the proposed scheme is a suitable extension of the quantum master equation to a doubled Hilbert space which is then unravelled by a stochastic differential equation.
Abstract: It has been demonstrated recently by the authors that the non-relativistic quantum state diffusion model can be generalized to yield a consistent, relativistically covariant theory. It is shown here that the 'counterexample' constructed by Diosi can easily be disproven.
Abstract: A fast simulation algorithm for the calculation of multitime correlation functions of open quantum systems is presented. It is demonstrated that any stochastic process which "unravels" the quantum Master equation can be used for the calculation of matrix elements of reduced Heisenberg picture opera tors, and thus for the calculation of multitime correlation functions, by extending the stochastic process to a doubled Hilbert space. The numerical performance of the stochastic simulation algorithm is investigated by means of a standard example.
Abstract: A Lorentz covariant generalization of quantum state diffusion is constructed by associating a Dirac wave function and a Lindblad operator to each flat, spacelike hypersurface in Minkowski space. The proof of relativistic covariance is given and the necessary integrability condition for the stochastic process is derived. (C) 1998 Elsevier Science B.V.
Abstract: A generalization of the stochastic wave-function method to open quantum systems under the influence of strong laser fields is derived. To this end, quantum-statistical ensembles of reduced state vectors are represented in terms of probability distributions on the projective Hilbert space of the open system. By employing the Floquet expansion of the quantum time evolution, this representation is shown to lead in the Markov approximation to a piecewise deterministic stochastic process for the reduced wave function. The realizations of the stochastic process consist of smooth deterministic parts that are interrupted by sudden jumps. The jump operators are the eigenoperators of the Floquet Hamiltonian and describe instantaneous transitions between Floquet states. Two examples are given that serve to illustrate the method and the stochastic simulation technique of the process.
Abstract: The stochastic dynamics of open quantum systems interacting with a zero temperature environment is investigated by employing a formulation of quantum statistical ensembles in terms of probability distributions on projective Hilbert space. It is demonstrated that the open system dynamics can consistently be described by a stochastic process on the reduced state space. The physical meaning of reduced probability distributions on projective Hilbert space is derived from a complete, orthogonal measurement of the environment. The elimination of the variables of the environment is shown to lead to a piecewise deterministic process in Hilbert space defined by a differential Chapman-Kolmogorov equation. A Hilbert space path integral representation of the stochastic process is constructed. The general theory is illustrated by means of three examples from quantum optics. For these examples the microscopic derivation of the stochastic process is given and the general solution of the differential Chapman-Kolmogorov equation is constructed by means of the path integral representation.
Abstract: Numerical investigations of open quantum systems, which are widely performed in such fields as photochemistry, quantum optics and nuclear magnetic resonance, can, in the Markovian regime, be based either on the master equation for the reduced density operator or on a stochastic process in the Hilbert space of the reduced system. It is shown that the CPU time consumptions of the two methods depend on the system size N as Nalpha+1 and as R(N)N-alpha, respectively. The exponent alpha is characteristic of the specific system. R(N) is the number of process realizations generated in the simulation and is defined by prescribing the tolerable statistical error of the result. Since R(N) is a non-increasing function of N, the stochastic method is found to be always faster for large systems. This is demonstrated for the example of the dissipative Morse oscillator excited by an intense short laser pulse. (C) 1997 Elsevier Science B.V.
Abstract: Within the framework of probability distributions on projective Hilbert space, a scheme for the calculation of multitime correlation functions is developed. The starting point is the Markovian stochastic wave-function description of an open quantum system coupled to an environment consisting of an ensemble of harmonic oscillators in arbitrary pure of mixed states. it is shown that matrix elements of reduced Heisenberg picture operators and general time-ordered correlation functions can be expressed by time-symmetric expectation values of extended operators in a doubled Hilbert space. This representation allows the construction of a stochastic process in the doubled Hilbert space which enables a determination of arbitrary matrix elements and correlation functions. The numerical efficiency of the resulting stochastic simulation algorithm is investigated and compared with an alternative Monte Carlo wave function method proposed first by Dalibard, Castin, and Molmer [Phys. Rev. Lett. 68, 580 (1992)]. By means of a standard example the suggested algorithm is shown to be more efficient numerically and to converge faster. Finally, some specific examples from quantum optics are presented in order to illustrate the proposed method, such as the coupling of a system to a vacuum, a squeezed vacuum within a finite solid angle, and a thermal mixture of coherent states.
Abstract: Three-dimensional homogeneous isotropic turbulence is formulated in terms of a discrete stochastic process in Fourier space. The time-dependent joint probability distribution of the stochastic Fourier modes is governed by a multivariate master equation. It is demonstrated that the characteristic functional of the stochastic process obeys the Hopf functional equation. As a first application of the method stochastic simulations of the Burgers's turbulence model are performed and shown to yield the expected energy spectrum.
Abstract: A fast method for the Monte Carlo solution of the balance equations that arise in nonequilibrium thermodynamics is suggested. It is applicable to chemical kinetics, reaction-diffusion processes, fluid dynamics, and heat conduction. The method is based on a multivariate master equation that is constructed in such a way that the simulation algorithm avoids time-consuming transition selection procedures, and thus becomes particularly efficient. For the example of a large chemical reaction scheme, the proposed method is shown to perform significantly faster than conventional methods, which rely on a birth-and-death master equation in discrete occupation number space.
Abstract: A specific model of quantum measurement [D. F. Walls, M. J. Collet, and G. J. Milburn, Phys. Rev. D 32, 3208 (1985)] based on a generic system-meter-environment measurement scheme is investigated in terms of probability distributions on projective Hilbert space. Starting from the unitary evolution of the total system a stochastic process for the wave function of the reduced system-meter system is derived by elimination of the degrees of freedom of the environment. The stochastic process is interpreted in terms of the direct detection of the photoelectrons created by the meter quanta in the environment. Two general variances are introduced which represent, respectively, the average intrinsic quantum fluctuations and the statistical fluctuations of the quantum expectation of the measured observable. Involving fourth moments of the wave function, these variances allow to monitor the transformation from quantum to classical probabilities during the process of measurement.
Abstract: A stochastic formulation of open quantum systems is applied to a microscopic system-meter-reservoir model of quantum measurement. The quantum statistical variance is decomposed into two variances which allow a quantitative description of the transformation from quantum to classical probabilities during the process of measurement.
Abstract: On rod disc membranes, single photoactivated rhodopsin (R*) molecules catalytically activate many copies of the G-protein (G(t)), which in turn binds and activates the effector (phosphodiesterase). We have performed master equation simulations of the underlying diffusional protein interactions on a rectangular 1-mu m(2) model membrane, divided into 15 x 15 cells. Mono- and bimolecular reactions occur within cells, and diffusional transitions occur between (neighboring) cells. Reaction and diffusion constants yield the related probabilities for the stochastic transitions. The calculated kinetics of active effector form a response that is essentially determined by the stochastic lifetime distribution of R* (with characteristic time tau(R)*) and the reaction constants of G(t) activation. Only a short tau(R)* (similar to 0.3 s) and a high catalytic rate (3000-4000 G(t) s(-1)R*(-1)) are consistent with electrophysiological data. Although R* shut-off limits the rise of the response, the lifetime distribution of free R* is not translated into a corresponding variability of the response peaks, because 1) the lifetime distribution of catalytically engaged R* is distorted, 2) small responses are enlarged by an overshoot of active effector, and 3) larger responses tend to undergo saturation. Comparison of these results to published photocurrent waveforms may open ways to understand the relative uniformity of the rod response.
Abstract: A Hilbert space path integral for the dissipative dynamics of matter in thermal radiation fields is derived from the Hamiltonian of quantum electrodynamics. This path integral represents the conditional transition probability of a stochastic Markov process as a sum over sample trajectories in Hilbert space. The realizations of the process are piecewise deterministic paths broken by instantaneous quantum jumps. It is shown that the operators which define the possible quantum jumps form a continuous family parametrized by the polarization vector of the emitted or absorbed photons. The stochastic process is shown to be representation-independent and to be invariant with respect to space rotations. The precise physical interpretation of the stochastic process is given. In particular, the expansion of the density matrix in terms of quantum jumps is derived for finite temperatures from the Hilbert space path integral.
Abstract: Extensive numerical simulation of a reaction-diffusion system reveals an unusual system size dependence of the fluctuation magnitude. If Ohm denotes the system size parameter, e.g. particle number, fluctuations are usually predicted to be of order Ohm(0.5) (stable case) or Ohm(1) (diffusion-type case). In contrast, a scaling like Ohm(0.84) is observed in a combined birth-death and random walk process, which is described by a multivariate chemical master equation and corresponds to the Fisher equation in the mascoscopic limit.
Abstract: The dynamics of open quantum systems is formulated in terms of a probability distribution on the underlying Hilbert space. Defining the time-evolution of this probability distribution by means of a Liouville-master equation the time-dependent wave function of the system becomes a stochastic Markov process in the sense of classical probability theory. It is shown that the equation of motion for the two-point correlation function of the random wave function yields the quantum master equation for the statistical operator. Stochastic simulations of the Liouville-master equation are performed for a simple example from quantum optics and are shown to be in perfect agreement with the analytical solution of the corresponding equation for the statistical operator.
Abstract: Typical complex systems, e. g., complex chemical reactions, reaction-diffusion systems, and turbulent fluids are described on a macroscopic level, that is, neglecting fluctuations, with the help of deterministic equations for corresponding variables. In this article it is shown on a phenomenological level, that these systems can be described in terms of integer- or real-valued Markov processes as well, which are governed by master equations. The latter are constructed such that the macroscopic law and the fluctuations around it are reproduced correctly. Stochastic processes defined through master equations can easily be simulated. The efficiency, the stability and the parallelization of the algorithms for stochastic simulations are discussed for some examples. In the last part of the paper it is shown that the same phenomenological approach can be successfully applied to open quantum systems. The wave function is assumed to be a complex valued stochastic process in Hilbert space and the quantum master equation for the statistical operator is regarded as the equation of motion for the two-point correlation function.
Abstract: The dynamics of an open quantum system coupled to an external reservoir is studied on the basis of a recently proposed formulation of quantum statistical ensembles in terms of probability distributions on projective Hilbert space. The previous result is generalized to include interaction Hamiltonians of the form Sigma(i) A(i) x B-i, where A(i) and B-i are operators acting on the Hilbert space of the reduced system and of the reservoir, respectively. The differential Chapman-Kolmogorov equation governing the dynamics of the conditional transition probability of the reduced system is derived from the underlying microscopic theory based on the Schrodinger equation for the total system. The stochastic process turns out to be a piecewise deterministic Markovian jump process in the projective Hilbert space of the reduced system. The sample paths are derived and shown to be similar to those of the Monte Carlo wave function simulation methods proposed in the literature. Finally, a diffusion-noise expansion of the Liouville master equation is performed and demonstrated to yield a stochastic differential equation for the state vector of the open system.
Abstract: The kinetics of irreversible coagulation phenomena in spatially homogeneous systems is formulated in terms of a multivariate stochastic process. The latter is governed by a master equation for the joint probability distribution of the numbers of reacting species. An efficient numerical algorithm is used to simulate the complete time evolution of the stochastic process. The method is illustrated by simulating the coagulation reaction with configuration-dependent reaction kernels, K(ij) = (ij)omega, for clusters of mass i and j with 1/2 < omega less-than-or-equal-to 1, which are designed to model gelation phenomena. It is demonstrated that the stochastic simulation allows the determination of critical exponents and the gel point directly from the master equation. The results are compared to predictions of the rate equation approach to the sol-gel transition.
Abstract: It is show that a recently proposed stochastic formulation of fluid dynamics makes possible a new approach to heat conduction and to temperature fluctuations. A master equation is presented which describes the time evolution of the probability distribution of a multivariate stochastic temperature variable. Splitting the stochastic variable in a macroscopic and a small fluctuating part and performing an expansion in powers of the inverse number of degrees of freedom per volume element of fluid the corresponding equations of fluctuating hydrodynamics are recovered. Stochastic simulations based on the generation of realizations of the stochastic process demonstrate the computational power of the approach.
Abstract: Stochastic differential equations are investigated which reduce in the deterministic limit to the canonical equations of motion of a Hamiltonian system with one degree of freedom. For example, stochastic differential equations of this type describe synchrotron oscillations of particles in storage rings under the influence of external fluctuating electromagnetic fields. In the first part of the article new numerical integration algorithms are proposed which take into account the symplectic structure of the deterministic Hamiltonian system. It is demonstrated that in the case of small white noise the algorithm is more efficient than conventional schemes for the integration of stochastic differential equations. In the second part the algorithms are applied to synchrotron oscillations. Analytical approximations for the expectation value of the squared longitudinal phase difference between the particle and the reference particle on the design orbit are derived. These approximations are tested by comparison with numerical results which are obtained by use of the symplectic integration algorithms.
Abstract: The reaction-diffusion process corresponding to the Fisher-Kolmogorov equation is studied by means of a discrete multivariate master equation. For travelling wave fronts the stability criterion necessary for the applicability of a system-size expansion is shown to be violated due to the existence of a zero mode of the first variational equation. This zero mode is connected to the translational invariance of the system. Performing stochastic simulations of the master equation in a wide range of parameters it is demonstrated that for finite size of the system (up to about 107 particles in the frontal region) a rather large fluctuation effect on the wave propagation speed results: in general, the asymptotic wave speed lies below the stable, minimal speed which is given by a theorem of Kolmogorov for the macroscopic equation. The wave front position exhibits a diffusion-type behaviour associated with translative fluctuations along the propagation direction.
Abstract: A stochastic approach to the energy balance equation of nonequilibrium thermodynamics is suggested. The stochastic formulation interprets the temperature field as a multivariate stochastic process which is governed by a master equation. By means of a systematic expansion in powers of the inverse number of degrees of freedom per volume element it is shown that the expectation value of the stochastic process obeys the macroscopic Fourier equation. The expansion reveals that in the linear noise approximation the fluctuations superimposed on the macroscopic dynamics are governed by the equations of fluctuating hydrodynamics. The master equation formulation gives rise to a new stochastic simulation method which is illustrated by applying it to heat conduction and temperature fluctuations in a fluid between two infinite parallel planes which are kept at different temperatures.
Abstract: The dynamics of two-dimensional (2D) turbulence is studied by a stochastic simulation method. The latter is based on a representation of the random vorticity field and stream function by a multivariate stochastic process defined by a discrete master equation. It is demonstrated that in the continuum Limit the complete hierarchy of coupled moment equations for the statistical formulation of the 2D Navier-Stokes equation is obtained. The probabilistic time evolution leads to random stresses, which can be traced to thermal fluctuations and allow one to disentangle hydrodynamic and thermodynamic degrees of freedom by some kind of renormalization procedure. The stochastic simulations at a large-scale Reynolds number of 2.5 x 10(5) clearly show the existence of a k(-3) power law, where k is the wave number, in the inertial range of the energy spectrum, as is predicted by the Kraichnan-Batchelor theory.
Abstract: The solution of stochastic partial differential equations generally relies on numerical tools. However, conventional numerical procedures are not appropriate to solve such problems. In this paper an algorithm is proposed which allows the numerical treatment of a large class of stochastic partial differential equations. To this end we reduce stochastic partial differential equations to a system of stochastic ordinary differential equations which can be solved numerically by a well-known stochastic Euler-procedure. We apply our algorithm to two stochastic partial differential equations which are special examples because their stationary two-point correlation functions can be determined analytically. Our algorithm proves to work out very well when numerical results are compared with the analytic correlation function.
Abstract: Recently a mesoscopic approach to computational fluid dynamics has been proposed which is based on a stochastic description defined by a discrete master equation. Applying van Kampen's system size expansion to one-dimensional flows, the deterministic macroscopic equations, i.e., the Navier-Stokes and continuity equation are obtained. The velocity and density fluctuations around the macroscopic dynamics are governed by a Fokker-Planck equation. From the assumption of local thermodynamic equilibrium, a relation between the velocity fluctuations and the temperature is obtained which allows to derive the linear Langevin equations of fluctuating hydrodynamics. Thus, the suggested stochastic approach makes possible the simultaneous numerical simulation of both the macroscopic balance equations and the hydrodynamic fluctuations.
Abstract: We investigate the estimation of the mean first passage time of a stochastic differential equation by numerical methods. In order to determine the mean first passage time correctly, one needs a numerical procedure to generate trajectories which converge in the mean square limit to exact solutions of the stochastic differential equation. First we briefly review a suitable algorithm for this purpose which uses Gaussian distributed random numbers. Then we show that the algorithm remains appropriate for the estimation of the mean first passage time even if one replaces the Gaussian random numbers by uniformly distributed ones. Exploiting this fact it is possible to reduce substantially the computation time for the estimation of the mean first passage time.
Abstract: By means of Burgers's equation a stochastic description of turbulent fluid flows is explained which is based on a discrete master equation. The latter governs the dynamics of a discrete multivariate stochastic process representing the random velocity field of the fluid. From the characteristic function corresponding to this stochastic process, the Hopf functional equation of turbulence is obtained. This implies that the infinite hierarchy of correlation functions can be derived from the master equation. The master-equation description naturally leads to a simple stochastic simulation algorithm which is well suited to numerical implementation. Stochastic simulations of the Burgers model of turbulence are performed and are shown to yield very accurate results.
Abstract: The interplay between localization and nonlinearity is investigated for a modified Swift-Hohenberg equation. We introduced a spatially stochastic contribution etaxi(x) in the control parameter that mimics, for instance, the essential effects of irregularities at the top and bottom plate in Rayleigh-Benard-convection experiments. Near the threshold where the trivial solution u = 0 becomes unstable, this randomness leads to localized solutions. Furthermore, the threshold value Of the spatially averaged control parameter is reduced by the disorder etaxi(x). The interaction between localization and nonlinearity leads to a characteristic change of the nonlinear bifurcation behavior. When ramping the control parameter in time, the disorder leads to an earlier onset and to a less steep temporal evolution of the pattern. This static and dynamic nonlinear behavior has similarities with recent measurements on Rayleigh-Benard convection.
Abstract: A stochastic representation of the turbulent dynamics of the two-dimensional Navier-Stokes equation is developed. This stochastic formulation interprets the fluctuating velocity held as a discrete, multivariate stochastic process which is governed by a master equation. By deriving the Hopf functional equation of statistical fluid mechanics it is shown that this approach yields a complete description of the stochastic properties of the turbulent velocity field. On the basis of the multivariate master equation a stochastic simulation method for the two-dimensional Navier-Stokes equation is constructed. This method is applied to the stochastic simulation of high-Reynolds-number turbulent Bows.
Abstract: It is shown that a recently proposed stochastic formulation of partial differential equations makes possible the use of stochastic simulation methods for their numerical solution. The approach is based on the identification of the continuous field appearing in the partial differential equation with the expectation value of a discrete stochastic process. The dynamics of this stochastic process is governed by an appropriately constructed multivariate master equation which can be simulated by standard methods. The new approach is introduced by applying it to the one-dimensional Burgers equation. The stochastic simulation of shock waves solutions of the Burgers equation demonstrates the high accuracy and efficiency of the method. Furthermore, the flexibility of the approach can be exploited to formulate algorithms which suppress numerical viscosity and the Gibbs phenomenon. It is demonstrated that the method is stable for arbitrary values of the Reynolds number.
Abstract: Several predictions for a recently proposed mesoscopic model for polymer melts and concentrated solutions is presented. It is a single Kramers chain model in which elementary motions of the Orwoll-Stockmayer type are allowed. However, for this model, the bead jumps are no longer given by a Markovian probability, but rather are described by a fractal "waiting-time" distribution function, with a single adjustable parameter-beta, which describes the long-time behavior of the distribution: approximately 1/t1+beta. We find that the model predicts D approximately 1/N2 and eta-0 approximately N3.4 for beta almost-equal-to 1.4, where n is the degree of polymerization. The generalized model predicts that the relaxation spectrum has a plateau regime whose height is independent of N, but whose width is strongly N dependent, in agreement with experiment. The model also predicts that rings will diffuse somewhat more slowly than linear chains of the same molecular weight (about 80% as fast), with the same scaling dependence on N as linear chains, also in agreement with preliminary data.
Abstract: A stochastic simulation method is proposed which allows the efficient treatment of chemical reactions involving a large number of reactants. The reaction is regarded as a stochastic process which is described by a master equation. Interesting quantities, e.g. the time evolution of concentrations, are evaluated not by solving the master equation as an ordinary differential equation but with the help of a stochastic simulation which generates realizations of the underlying stochastic process. The practicability and flexibility of this approach is demonstrated by means of a coagulation reaction.
Abstract: A recently proposed mesoscopic description of fluid dynamics leads to a new approach to turbulence. In contrast to the classical statistical theory of turbulence the new approach introduces a probabilistic time evolution of the random velocity governed by a master equation. The mesoscopic approach is explained by means of the (1 + 1)-dimensional Burgers' model of turbulence. By a continuous time stochastic simulation realizations of turbulent velocity fields are generated. Cor-relation functions and energy spectra are evaluated from appropriate ensemble averages.
Abstract: In this paper a new approach to computational fluid dynamics is introduced in which the fluid is regarded as a stochastic dynamical system. The velocity of the fluid is related to a stochastic process governed by an appropriate master equation acting in a discrete phase space. The method is explained by means of (1 + 1)-dimensional flow phenomena. It is shown that the stochastic approach naturally leads to transparent numerical algorithms for stochastic simulations of fluid motion. By simulating plane Poiseuille flow it is demonstrated that the probabilistic approach yields a correct description of laminar fluid motion. Furthermore, soliton-like and shock wave solutions of Burgers' equation are generated by stochastic simulations of the underlying stochastic process.
Abstract: A transient network model is presented which is based on the ideas of Lodge and Yamamoto. The network thus consists of different types of strands and the creation and loss rates are configuration dependent. As the model cannot be solved analytically one has to rely upon numerical tools. Using a continuous time algorithm this network can be simulated very efficiently. The material functions can be calculated as time-dependent ensemble averages. Concentration is put onto a quantitative comparison between the rheological predictions of the model and experimental data for polyisobutylene subjected to simple and multiaxial elongational flows. The predictions of the model are found to be in good agreement with the experimental data.
Abstract: By means of the example of Burgers' equation a stochastic approach to computational fluid dynamics is proposed. The velocity field is regarded as the expectation value of a discrete stochastic process which is governed by a master equation. Soliton-like and shock wave solutions of Burgers' equation are generated by stochastic simulations of the master equation.
Abstract: A new mesoscopic model is presented for polymer melts and concentrated solutions. It is a single Kramers chain model in which elementary motions of the Orwoll-Stockmayer type are allowed. However, for this model, the bead jumps are no longer given by a Markovian probability, but rather are described by "a waiting time distribution function." Such a distribution is supposed to occur when the chain is "frozen" in space until a "gap" in the solution or melt meets with the bead or chain segment. The time a bead must wait to jump is given by a distribution function with a single adjustable parameter beta, which describes the long-time behavior of the distribution: Approximately 1/t1+beta. We find that the model predicts non-Fickian diffusion in agreement with experimental data and Fickian diffusion for longer times which scales with chain length as 1/N2/alpha-1, where alpha is a function of beta. For beta = 1.3, D approximately 1/N2.28. The autocorrelation of the end-to-end vector of the chain is a stretched-exponential form with a time constant which scales as the length of the chain to the 3.3 power for beta = 1.3.
Abstract: In this work the rheological properties of a concentrated polymer solution or melt are investigated. The polymeric material is described as a transient network and the specific model is of the Yamamoto type with configuration dependent creation and loss rates. For the solution of the model a continuous time simulation approach is used which is based on the stochastic interpretation of the fundamental equations of the theory. The qualitative predictions for the rheological behaviour turn out to be very good. For a quantitative test the unknown configuration dependence of the loss rate function is extracted from experimental data as the solution of an inverse problem. The comparison of experimental data for planar elongational flow with the simulation results using the fitted loss rate function shows that the agreement is satisfactory, but not yet perfect. A possible reason for this finding is given.
Abstract: Concentrated polymer solutions and melts are usually simulated by molecular dynamics, Brownian dynamics or Monte Carlo methods. In this paper we describe another simulation approach. It is based on transient polymer network theories which can correctly describe the rheological behaviour. Due to a stochastic interpretation of the fundamental equations of these theories it is possible to formulate algorithms to simulate the dynamics of the network. Network models that cannot be treated analytically can then be studied. One of the algorithms presented here works with small discrete time steps. It can be applied to the most general models. For some simpler models it is possible to formulate a continuous-time algorithm which is generally more efficient.
