Master equation

About: Master equation is a research topic. Over the lifetime, 10541 publications have been published within this topic receiving 276095 citations.

Master equations and the theory of stochastic path integrals

Abstract: This review provides a pedagogic and self-contained introduction to master equations and to their representation by path integrals. We discuss analytical and numerical methods for the solution of master equations, keeping our focus on methods that are applicable even when stochastic fluctuations are strong. The reviewed methods include the generating function technique and the Poisson representation, as well as novel ways of mapping the forward and backward master equations onto linear partial differential equations (PDEs). Spectral methods, WKB approximations, and a variational approach have been proposed for the analysis of the PDE obeyed by the generating function. After outlining these methods, we solve the derived PDEs in terms of two path integrals. The path integrals provide distinct exact representations of the conditional probability distribution solving the master equations. We exemplify both path integrals in analysing elementary chemical reactions. Furthermore, we review a method for the approximation of rare event probabilities and derive path integral representations of Fokker-Planck equations. To make our review accessible to a broad community, we have used the language of probability theory rather than quantum (field) theory. The probabilistic structures underpinning various technical concepts, such as coherent states, the Doi-shift, and normal-ordered observables, are thereby made explicit.

Fluorescence into Flat and Structured Radiation Continua: An Atomic Density Matrix without a Master Equation

Abstract: We investigate an atomic $\ensuremath{\Lambda}$ system with one transition coupled to a laser field and a flat continuum of vacuum modes and the other transition coupled to field modes near the edge of a photonic band gap. The system requires simultaneous treatment of Markovian and non-Markovian dissipation processes, but the photonic-band-gap continuum cannot be eliminated within a density matrix treatment. Instead we propose a formalism based on Monte Carlo wave functions, and we present results relevant to the experimental characterization of a structured continuum.

Fewest Switches Surface Hopping in Liouville Space.

Abstract: The novel approach to nonadiabatic quantum dynamics greatly increases the accuracy of the most popular semiclassical technique while maintaining its simplicity and efficiency. Unlike the standard Tully surface hopping in Hilbert space, which deals with population flow, the new strategy in Liouville space puts population and coherence on equal footing. Dual avoided crossing and energy transfer models show that the accuracy is improved in both diabatic and adiabatic representations and that Liouville space simulation converges faster with the number of trajectories than Hilbert space simulation. The constructed master equation accurately captures superexchange, tunneling, and quantum interference. These effects are essential for charge, phonon and energy transport and scattering, exciton fission and fusion, quantum optics and computing, and many other areas of physics and chemistry.

Transport through an Anderson impurity: Current ringing, nonlinear magnetization, and a direct comparison of continuous-time quantum Monte Carlo and hierarchical quantum master equations

Abstract: We give a detailed comparison of the hierarchical quantum master equation (HQME) method to a continuous-time quantum Monte Carlo (CT-QMC) approach, assessing the usability of these numerically exact schemes as impurity solvers in practical nonequilibrium calculations. We review the main characteristics of the methods and discuss the scaling of the associated numerical effort. We substantiate our discussion with explicit numerical results for the nonequilibrium transport properties of a single-site Anderson impurity. The numerical effort of the HQME scheme scales linearly with the simulation time but increases (at worst exponentially) with decreasing temperature. In contrast, CT-QMC is less restricted by temperature at short times, but in general the cost of going to longer times is also exponential. After establishing the numerical exactness of the HQME scheme, we use it to elucidate the influence of different ways to induce transport through the impurity on the initial dynamics, discuss the phenomenon of coherent current oscillations, known as current ringing, and explain the nonmonotonic temperature dependence of the steady-state magnetization as a result of competing broadening effects. We also elucidate the pronounced nonlinear magnetization dynamics, which appears on intermediate time scales in the presence of an asymmetric coupling to the electrodes.

Impurity Quenching of Molecular Excitons. I. Kinetic Comparison of Förster—Dexter and Slowly Quenched Frenkel Excitons in Linear Chains

Abstract: The quenching kinetics of tightly bound excitons for two different one‐dimensional models are compared. The quenching of fully incoherent, or Forster—Dexter, excitons is described by a standard master equation, and that of fully coherent, or Frenkel, excitons by an ad hoc linear differential equation whose eigenvalues are complex. Moments of the chain excitation function (probability that excitation remains at time t) are calculated on each model for finite chains with either localized or uniform initial conditions, free‐end or periodic boundary conditions, one disruptive or one nondisruptive quencher, and various quencher locations, but with only nearest‐neighbor interactions included. The ad hoc equation is treated only in the limit that the quenching is slow enough not to affect the exciton wave functions in first order. In that limit of Frenkel exciton quenching, an analytic expression is given for the mean de‐excitation time in the presence of uniform decay processes such as fluorescence. The master ...