Stochastic Master Equation Research Articles

Dynamical Ising-like model for the two-step spin-crossover systems

In order to reproduce the two-step relaxation observed experimentally in spin-crossover systems, we investigate analytically the static and the dynamic properties of a two-sublattice Ising-like Hamiltonian. The formalism is based on a stochastic master equation approach. It is solved in the mean-field approximation, and yields two coupled differential equations that correspond to the HS fractions of the sublattices A and B.

An analysis of reading out the state of a charge quantum bit

We provide a unified picture for the master equation approach and the quantum trajectory approach to a measurement problem of a two-state quantum system (a qubit), an electron coherently tunneling between two coupled quantum dots (CQD's) measured by a low transparency point contact (PC) detector. We show that the master equation of ``partially'' reduced density matrix can be derived from the quantum trajectory equation (stochastic master equation) by simply taking a ``partial'' average over the all possible outcomes of the measurement. If a full ensemble average is taken, the traditional (unconditional) master equation of reduced density matrix is then obtained. This unified picture, in terms of averaging over (tracing out) different amount of detection records (detector states), for these seemingly different approaches reported in the literature is particularly easy to understand using our formalism. To further demonstrate this connection, we analyze an important ensemble quantity for an initial qubit state readout experiment, P(N,t), the probability distribution of finding N electron that have tunneled through the PC barrier(s) in time t. The simulation results of P(N,t) using 10000 quantum trajectories and corresponding measurement records are, as expected, in very good agreement with those obtained from the Fourier analysis of the ``partially'' reduced density matrix. However, the quantum trajectory approach provides more information and more physical insights into the ensemble and time averaged quantity P(N,t). Each quantum trajectory resembles a single history of the qubit state in a single run of the continuous measurement experiment. We finally discuss, in this approach, the possibility of reading out the state of the qubit system in a single-shot experiment.

Influence of geometry on light harvesting in dendrimeric systems

The exact analytic expression for the mean time to trapping (or mean walklength) for a particle (electron/exciton) performing a random walk on a finite dendrimer lattice with a trap at the center of the dendrimer was obtained. Exact analytic expressions have also been obtained for articulated/extended dendrimeric systems. The full dynamical behavior was determined for each case studied via numerical solution of the stochastic master equation, and the results obtained were shown to be a direct consequence of the structural properties of the dendrimeric system. These studies are linked to the behavior observed in experiments on light harvesting in dendrimeric supermolecules.

Nucleation in physical and nonphysical systems

The aggregation of particles out of an initially homogeneous situation is well known in physics. Depending on the system under consideration and its control parameters, the cluster formation in a supersaturated (metastable or unstable) situation has been observed in nucleation physics as well as in other branches. We investigate the well-known example of condensation (formation of liquid droplets) in an undercooled vapour to conclude that the formation of bound states as a phase transition is related to transportation science. We present a comparison of nucleation in an isothermal–isochoric container with traffic congestion on a circular one-lane freeway. The analysis is based, in both cases, on the probabilistic description by stochastic master equations.

The Chemical Langevin and Fokker−Planck Equations for the Reversible Isomerization Reaction

This paper uses the simple reversible isomerization reaction to illustrate and clarify the roles played in chemical kinetics by recently proposed forms for the chemical Langevin equation and chemical Fokker−Planck equation. It is shown that the stationary solution of the chemical Fokker−Planck equation for this model reaction provides, for most purposes, an excellent approximation to the stationary solution of the chemical master equation. It is also shown that, when allowance is made for the stipulated macroscopic nature of the time increment dt in the chemical Langevin equation, the changes in molecular population during dt predicted by that equation for this model reaction closely approximate the changes prescribed by the chemical master equation. The discussion highlights the role of the chemical Langevin equation as not only a potential computational aid but also a conceptual bridge between the stochastic chemical master equation and the traditional deterministic reaction rate equation.

Feedback-stabilization of an arbitrary pure state of a two-level atom

Unit-efficiency homodyne detection of the resonance fluorescence of a two-level atom collapses the quantum state of the atom to a stochastically moving point on the Bloch sphere. Recently,Hofmann, Mahler, and Hess [Phys. Rev. A {\bf 57}, 4877 (1998)] showed that by making part of the coherent driving proportional to the homodyne photocurrent can stabilize the state to any point on the bottom half of the sphere. Here we reanalyze their proposal using the technique of stochastic master equations, allowing their results to be generalized in two ways. First, we show that any point on the upper or lower half, but not the equator, of the sphere may be stabilized. Second, we consider non-unit-efficiency detection, and quantify the effectiveness of the feedback by calculating the maximal purity obtainable in any particular direction in Bloch space.

Probabilistic Description of Traffic Flow

The study of traffic flow is investigated by different means. Well established theories are (i) kinematic models based on partial differential equations to describe traveling density waves, and (ii) deterministic models using nonlinear car‐following equations to determine trajectories of moving cars, as well as (iii) large-scale simulation hopping models like cellular automata. An important intermediate approach is (iv) the stochastic or probabilistic attempt to understand phenomena like “Stau aus dem Nichts” (phantom jam) on long crowded roads. Initiated by the old argument that road traffic is a stochastic process, we develop a new probabilistic theory based on Markov processes to improve our understanding of traffic flow and its three phases (free flow, synchronized motion, wide moving jams) discovered by Kerner. As an introductory example, first we consider a dissolution of a car queue described by the stochastic master equation as a one-step decay process. Furtheron more realistic models are developed to investigate the nucleation, growth and condensation as well as dissolution of car clusters on a circular one-lane freeway. In analogy to usual aggregation phenomena such as the formation of liquid droplets in supersaturated vapour the clustering behavior in traffic flow is described by the master equation. At overcritical densities the transition from the initial free particle situation (free flow of vehicles) to the final congested state, where one or several big aggregates of cars have been formed, is shown. In dependence on the concentration of cars on the road the stationary solution of the master equation is derived analytically. The obtained fundamental diagram as flow-density-relation indicates clearly the different regimes of traffic flow (free jet of cars, coexisting phase of jams and isolated cars, highly viscous heavy traffic). In the (thermodynamic) limit of infinite number of vehicles on an infinite long road the analytical solution for the fundamental diagram is in agreement with experimental traffic flow data. As a particular example we take into account measurements from German highways presented by Kerner and Rehborn.

An analytical solution of the stochastic master equation for reversible bimolecular reaction kinetics

The kinetics of the binding reaction A+B⇄C are solved exactly via the stochastic master equation, in which molecular populations are considered as time-dependent, integer-valued random variables. This transient solution extends previous work on the irreversible bimolecular reaction and the equilibrium state of the reversible bimolecular reaction. For small ensembles of reactants, comparisons are made between the results of the deterministic and stochastic approaches to the problem, and methods are presented to numerically evaluate the solution.

Evaluation of heating effects on atoms trapped in an optical trap

We solve a stochastic master equation based on the theory of Savard et al. [T. A. Savard, K. M. O'Hara, and J. E. Thomas, Phys. Rev. A 56, R1095 (1997)] for heating arising from fluctuations in the trapping laser intensity. We compare with recent experiments of Ye et al. [J. Ye, D. W. Vernooy, and H. J. Kimble, Phys. Rev. Lett. 83, 4987 (1999)], and find good agreement with the experimental measurements of the distribution of trap occupancy times. The major cause of trap loss arises from the broadening of the energy distribution of the trapped atom, rather than the mean heating rate, which is a very much smaller effect.

State determination in continuous measurement

The possibility of determining the state of a quantum system after a continuous measurement of position is discussed in the framework of quantum trajectory theory. Initial lack of knowledge of the system and external noises are accounted for by considering the evolution of conditioned density matrices under a stochastic master equation. It is shown that after a finite time the state of the system is a pure state and can be inferred from the measurement record alone. The relation to emerging possibilities for the continuous experimental observation of single quanta, as for example in cavity quantum electrodynamics, is discussed.

Approaches to the approximate treatment of complex molecular systems by the multiconfiguration time-dependent Hartree method

A consistent treatment of environmental effects is proposed in the framework of the multiconfiguration time-dependent Hartree (MCTDH) method. The method is extended in view of treating complex molecular systems which require an exact quantum dynamics for a certain number of “primary” modes while an approximate dynamics is adequate for a class of “secondary” modes. The latter may correspond to the weakly coupled modes in a polyatomic molecule, or the first solvent shell in a solute-solvent complex. For these modes, a description in terms of parameterized functions is introduced. The MCTDH working equations are generalized to allow for the nonorthogonality of these functions, which may take, e.g., a multidimensional Gaussian form. The formalism is developed on the level of both the wave function description and the density matrix description. Dissipative effects are accounted for in terms of a stochastic Hamiltonian approach versus master equation approach in the respective descriptions.

Kinetic transitions in diffusion-reaction space. II. Geometrical effects

We extend the stochastic master equation approach described earlier [J. J. Kozak and R. Davidson, J. Chem. Phys. 101, 6101 (1994)] to examine the influence on reaction efficiency of multipolar correlations between a fixed target molecule and a diffusing coreactant, the latter constrained to move on the surface of a host medium (e.g., a colloidal catalyst or molecular organizate) modeled as a Cartesian shell [Euler characteristic, χ=2]. Our most comprehensive results are for processes involving ion pairs, and we find that there exists a transition between two qualitatively different types of behavior in diffusion-reaction space, viz., a regime where the coreactant’s motion is totally correlated with respect to the target ion, and a regime where the coreactant’s motion is effectively uncorrelated. This behavior emerges both in the situation where correlations between the ion pair are strictly confined to the surface of the host medium or where correlations can be propagated through the host medium. The effects of system size are also examined and comparisons with diffusion-reaction processes taking place on surfaces characterized by Euler characteristic χ=0 are made. In all cases studied, the most dramatic effects on the reaction efficiency are uncovered in the regime where the Onsager (thermalization) length is comparable to the mean displacement of the coreactant, a conclusion consistent with results reported in earlier work.

Retroactive Quantum Jumps in a Strongly Coupled Atom-Field System

We investigate a novel type of conditional dynamic that occurs in the strongly-driven Jaynes-Cummings model with dissipation. Extending the work of Alsing and Carmichael [Quantum Opt. {\bf 3}, 13 (1991)], we present a combined numerical and analytic study of the Stochastic Master Equation that describes the system's conditional evolution when the cavity output is continuously observed via homodyne detection, but atomic spontaneous emission is not monitored at all. We find that quantum jumps of the atomic state are induced by its dynamical coupling to the optical field, in order retroactively to justify atypical fluctuations in ocurring in the homodyne photocurrent.

Realistic master equation modeling of relaxation on complete potential energy surfaces: Kinetic results

Using the potential surface information for (KCl)5 and Ar9 and partition function models introduced in the preceding paper [Ball and Berry, J. Chem. Phys. 109, 8541 (1998)] we construct a stochastic master equation for each system using Rice–Ramsperger–Kassel–Marcus (RRKM) theory for transition rates between adjacent minima. We test several model approximations to reactant and transition-state partition functions by comparing their master equation predictions of isothermal relaxation for (KCl)5 and Ar9 with the results of molecular dynamics simulations of relaxations performed in the canonical ensemble. Accurate modeling of the transition-state partition functions is more important for (KCl)5 than for Ar9 in reproducing the relaxation observed in simulation. For both systems, several models yield qualitative agreement with simulation over a large temperature range. This full treatment of small systems using realistic partition function models is a necessary first step in the application of the master equation method to larger systems, for which one can only expect to have statistical samples of the potential energy surfaces.

Realistic master equation modeling of relaxation on complete potential energy surfaces: Partition function models and equilibrium results

To elucidate the role that potential surface topography plays in shaping the evolution of a cluster toward equilibrium, entire sets of kinetically accessible bound-state configurations and transition states on the model potential energy surfaces of (KCl)5 and Ar9 are mapped and compared. To describe the stochastic dynamics on these surfaces in terms of transition-state theory, we require adequate approximations of the partition functions of the minima and transition states. In this paper we introduce several partition function models derived from harmonic and anharmonic approximations and compare their predicted equilibrium population distributions with those determined from canonical-ensemble molecular dynamics. We perform this comparison for both (KCl)5 and Ar9 in order to evaluate the relative performance of the models for two different types of potential surfaces. For each system, particular models are found to give results that agree better with simulation than do the results using the simple harmonic approximation. However, no one unparameterized model gives acceptable results for all minima, and the best parameter-free strategies differ for (KCl)5 and Ar9. Nevertheless, a one-parameter version of one of the models is shown to give the best agreement with simulation for both systems. In an accompanying paper, the best partition function models are used to construct a stochastic master equation which makes predictions of relaxation behavior. These predictions are compared with results from molecular dynamics.

Relative Stability of Multiple Stationary States Related to Fluctuations

We apply the eikonal approximation to the stochastic master equation for reaction−diffusion systems. Its stationary solution is expressed as an excess work and is shown to be a Lyapunov function for the deterministic evolution of inhomogeneous systems. From this result we establish a new stochastic criterion of relative stability and equistability of systems with multiple homogeneous stationary states in terms of inhomogeneous fluctuations.

Simulation of First-Order Chemical Kinetics Using Cellular Automata

Cellular automata are dynamical systems composed of arrays of cells that change their states in a discrete manner following local, but globally applied, rules. It is shown that a two-dimensional asynchronous cellular automaton simulates both the deterministic and the stochastic features of first-order chemical kinetic processes in an especially simple manner, avoiding the chore of solving either the deterministic coupled differential rate equations or the stochastic master equation. Processes illustrated include first-order decay, opposing first-order reactions, consecutive reactions, the steady-state approximation, a rate-limiting step, pre-equilibrium, and parallel competing reactions. The deterministic solutions are seen to emerge as statistical averages in the limit of large cell numbers. Some additional stochastic and statistical features of the solutions are examined.

Renewal analysis for higher moments in stochastic transport

This paper considers steady-state particle transport and radiative transfer through a stochastic mixture consisting of two immiscible materials. The statistics of the mixing of the two components of the medium are allowed to be very general; in particular, they are not necessarily Markovian. In the absence of scattering, the theory of alternating renewal processes is used to obtain an exact set of integral equations describing the ensemble average of any integer power of the stochastic intensity field. These equations are solved in an infinite medium with homogeneous statistics, and this solution is used to develop a simpler, but approximate, formalism for treating particle transport in a non-Markovian mixture. This simplied description is in the form of differential, rather than integral, equations, and results from a modification of the differential Markovian equations which arise from the use of the stochastic Liouville master equation.

Energy migration and rotational motion within bichromophoric molecules. II. A derivation of the fluorescence anisotropy

A generalized Förster theory is presented which includes reorientation of the interacting molecules. The stochastic master equation is, for the first time, derived from the stochastic Liouville equation, so that it accounts for the molecular origin to the stochastic transitions rates. A formal solution to the stochastic master equation is given. This equation is compared with its truncated cumulant expansion. The second-order cumulant contains the correlation function 〈κ2(0)κ2(t)〉, where κ denotes the orientational dependence on dipole–dipole coupling. The solution of the master equation is used to formulate the time-dependent fluorescence anisotropy, which is the relevant observable of energy transfer within donor–donor (dd) pairs, or bichromophoric molecules. Depending on symmetry of the local orientational distributions of the donor molecules, and their rates of reorientation, the fluorescence anisotropy decay becomes more or less complicated. Different simplifying conditions are given. The orientational distribution of the bichromophoric molecules is assumed to be isotropic and their rotational motion is taken to be negligible. The effect of using the cumulant expansion on the excitation probability and the fluorescence anisotropy was numerically examined. Brownian dynamics simulations are used to describe isotropic rotation of the d molecules. In the fast case (or dynamic limit), where the rates of transfer are much slower than those of reorientation, the cumulant expansion is always valid. In the intermediate case the approximation becomes questionable, while for the static limit, a numerical evaluation of the formal solution must be performed. The theory presented here is easily modified to account for the influence of reorientation in studies of donor–acceptor transfer.

Electronic excitation transfer in chains modulated by conformational dynamic disorder

Electronic excitations along sites that undergo spatial and temporal fluctuations due to conformational chain motion have been studied in the picture of the stochastic master equation by means of the dynamic Monte Carlo (DMC) and the cumulant expansion (CE) approach. An incoherent site-to-site hopping which is adiabatic relative to the changes of conformational site coordinates has been assumed. The elementary act of conformational change has been considered to be fast, whereas the electronic transfer during the time period of the conformational event has been assumed to be negligibly small. The time evolution of electronic intersite coupling is thus controlled by chromophore sites that, in particular, correspond to the conformational minima of the potential energy landscape. The generalized equations of motion adapted for both the DMC and the CE analysis have been reduced to formulate donor site excitation probabilities 〈Piexc(t)〉 and donor excitation survival functions 〈PD(t)〉 for a simplified chain. In this polymer model, (i) specific nearest-neighbor electronic coupling occurs with two distinct transfer rates W1 and W2 corresponding to two different spatial arrangements of the pendant sites in the pair and (ii) transitions between two definite conformational states occur both in the correlated and in the uncorrelated regime. For short chains and a moderate number of sites in the rotational dyads the whole range from the dynamic to the static limit in the interplay between excitation transfer and correlated conformational motion has been calculated by the DMC method. By means of the cumulant technique well-behaved solutions could be obtained only in the fast conformational transition regime which allows a direct comparison with the DMC results. For longer chains up to 100 sites, in the limit case of uncorrelated conformational motion, preliminary cumulant approaches have been given which, for very rapid conformational rates, agree well with the dynamic effective medium approximation (DEMA) solutions.