Conventional Master Equation Research Articles

Analytical and Numerical Treatment of Continuous Ageing in the Voter Model

The conventional voter model is modified so that an agent’s switching rate depends on the ‘age’ of the agent—that is, the time since the agent last switched opinion. In contrast to previous work, age is continuous in the present model. We show how the resulting individual-based system with non-Markovian dynamics and concentration-dependent rates can be handled both computationally and analytically. The thinning algorithm of Lewis and Shedler can be modified in order to provide an efficient simulation method. Analytically, we demonstrate how the asymptotic approach to an absorbing state (consensus) can be deduced. We discuss three special cases of the age-dependent switching rate: one in which the concentration of voters can be approximated by a fractional differential equation, another for which the approach to consensus is exponential in time, and a third case in which the system reaches a frozen state instead of consensus. Finally, we include the effects of a spontaneous change of opinion, i.e., we study a noisy voter model with continuous ageing. We demonstrate that this can give rise to a continuous transition between coexistence and consensus phases. We also show how the stationary probability distribution can be approximated, despite the fact that the system cannot be described by a conventional master equation.

Mode Interference Effect in Optical Emission of Quantum Dots in Photonic Crystal Cavities

Radiation properties of a pointlike source of light, such as a molecule or a semiconductor quantum dot, can be tailored by modifying its photonic environment. This phenomenon lies at the core of cavity quantum electrodynamics (CQED). Quantum dots in photonic crystal microcavities have served as a model system for exploring the CQED effects and for the realization of efficient single-photon quantum emitters. Recently, it has been suggested that quantum interference of the exciton recombination paths through the cavity and free-space modes can significantly modify the radiation. In this work, we report an unambiguous experimental observation of this fundamental effect in the emission spectra of site-controlled quantum dots positioned at prescribed locations within a photonic crystal cavity. The observed asymmetry in the polarization-resolved emission spectra strongly depends on the quantum dot position, which is confirmed by both analytical and numerical calculations. We perform quantum interferometry in the near-field zone of the radiation, retrieving the overlap and the position-dependent relative phase between the interfering free-space and cavity-mode-mediated radiative decays. The observed phenomenon is of importance for realization of photonic-crystal light emitters with near unity quantum efficiency. Our results suggest that the full description of light-matter interaction in the framework of CQED requires a modification of the conventional quantum master equation by also considering the radiation mode interference.12 MoreReceived 20 October 2021Revised 12 February 2022Accepted 13 April 2022Corrected 22 June 2022DOI:https://doi.org/10.1103/PhysRevX.12.021042Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasCavity quantum electrodynamicsLight-matter interactionNanophotonicsOpen quantum systemsPhotonic crystalsPhotonicsQuantum description of light-matter interactionQuantum interference effectsSingle photon sourcesPhysical SystemsQuantum cavitiesQuantum dotsTechniquesFinite-difference time-domain methodFinite-element methodInterferometryOptical spectroscopyPhotoluminescenceQuantum master equationTime-resolved photoluminescenceCondensed Matter, Materials & Applied PhysicsAtomic, Molecular & OpticalGeneral Physics

Analysis of Length Biases in Single-Cell RNA Sequencing of Unspliced mRNA by Markov Modeling

Recent experimental advances in single-cell RNA sequencing (scRNA-seq) have enabled the quantification of transcriptomes with single-molecule resolution. However, thus far, the stochastic modeling of transcription has been separate from the discussion of the statistics of the sequencing process, leading to simplifications that may obfuscate transcriptional dynamics, and technical artifacts in the assays. For example, imputation, normalization, and smoothing, used to correct for stochastic sequencing phenomena, make experimental molecule count data incompatible with a discrete representation, thus rendering the data uninterpretable in the context of conventional Chemical Master Equation (CME) models.

A master equation for spin systems far from equilibrium.

The quantum dynamics of spin systems is often treated by a differential equation known as the master equation, which describes the trajectories of spin observables such as magnetization components, spin state populations, and coherences between spin states. The master equation describes how a perturbed spin system returns to a state of thermal equilibrium with a finite-temperature environment. The conventional master equation, which has the form of an inhomogeneous differential equation, applies to cases where the spin system remains close to thermal equilibrium, which is well satisfied for a wide variety of magnetic resonance experiments conducted on thermally polarized spin systems at ordinary temperatures. However, the conventional inhomogeneous master equation may fail in the case of hyperpolarized spin systems, when the spin state populations deviate strongly from thermal equilibrium, and in general where there is a high degree of nuclear spin order. We highlight a simple case in which the inhomogeneous master equation clearly fails, and propose an alternative master equation based on Lindblad superoperators which avoids most of the deficiencies of previous proposals. We discuss the strengths and limitations of the various formulations of the master equation, in the context of spin systems which are far from thermal equilibrium. The method is applied to several problems in nuclear magnetic resonance and to spin-isomer conversion.

Numerically exact full counting statistics of the energy current in the Kondo regime

We use the inchworm Quantum Monte Carlo method to investigate the full counting statistics of particle and energy currents in a strongly correlated quantum dot. Our method is used to extract the heat fluctuations and entropy production of a quantum thermoelectric device, as well as cumulants of the particle and energy currents. The energy--particle current cross correlations reveal information on the preparation of the system and the interplay of thermal and electric currents. We furthermore demonstrate the signature of a crossover from Coulomb blockade to Kondo physics in the energy current fluctuations, and show how the conventional master equation approach to full counting statistics systematically fails to capture this crossover.

Dynamics of Multi-Level Open Quantum Systems with Initial System-Environment Correlations

Exact and approximate master equations were derived by the projection operator method for the reduced statistical operator of a multi-level open quantum system with finite number $$N$$ of quantum eigenstates interacting with arbitrary external deterministic fields and dissipative environment simultaneously. Unlike conventional master equations derived by the projection operator method for the quantum systems of this kind so far, these equations are homogeneous in the reduced statistical operator while, at the same time, taking into account initial correlations between the multi-level system and the environment. Consistent representation of the multi-level system evolution in terms of the $$SU(N)$$ operator algebra facilitates efficient numerical analysis of these equations derived for various physically realistic quantum models.

Capturing Brownian dynamics with an on-lattice model of hard-sphere diffusion

Conventional master equation approaches approximate the diffusion of molecules in continuum space by the process of particles hopping on a spatial lattice. The hopping probability from one voxel (spatial lattice point) to its neighbor is usually considered to be constant throughout space. Such an assumption is only consistent with pointlike molecules and thus neglects volume-exclusion effects due to finite particle size. A few studies have attempted to introduce volume-exclusion effects by choosing the hopping probability from one voxel to a neighboring one to be a linear function of the number density. Here, we formulate an alternative master equation in which the hopping probability is equal to the fraction of available space in the neighboring voxel as estimated using scaled particle theory. This leads to the hopping probability being a nonlinear function of the number density. A mean-field approximation (mfa) leads to a partial differential equation of the advection-diffusion type. We show that the time evolution of the particle number density sampled using the stochastic simulation algorithm associated with the new master equation and the number density obtained by numerical integration of the mfa are in good agreement with each other. They are also distinctly different than the time evolution predicted by the conventional master equation and those with hopping probabilities which are linear functions of the number density. The results from the new lattice description are also shown to be in very good agreement with the lattice-free method of Brownian dynamics, even for highly crowded scenarios.

Model reduction for stochastic chemical systems with abundant species.

Biochemical processes typically involve many chemical species, some in abundance and some in low molecule numbers. We first identify the rate constant limits under which the concentrations of a given set of species will tend to infinity (the abundant species) while the concentrations of all other species remains constant (the non-abundant species). Subsequently, we prove that, in this limit, the fluctuations in the molecule numbers of non-abundant species are accurately described by a hybrid stochastic description consisting of a chemical master equation coupled to deterministic rate equations. This is a reduced description when compared to the conventional chemical master equation which describes the fluctuations in both abundant and non-abundant species. We show that the reduced master equation can be solved exactly for a number of biochemical networks involving gene expression and enzyme catalysis, whose conventional chemical master equation description is analytically impenetrable. We use the linear noise approximation to obtain approximate expressions for the difference between the variance of fluctuations in the non-abundant species as predicted by the hybrid approach and by the conventional chemical master equation. Furthermore, we show that surprisingly, irrespective of any separation in the mean molecule numbers of various species, the conventional and hybrid master equations exactly agree for a class of chemical systems.

Time-dependent quantum transport: An efficient method based on Liouville-von-Neumann equation for single-electron density matrix

Basing on our hierarchical equations of motion for time-dependent quantum transport [X. Zheng, G. H. Chen, Y. Mo, S. K. Koo, H. Tian, C. Y. Yam, and Y. J. Yan, J. Chem. Phys. 133, 114101 (2010)], we develop an efficient and accurate numerical algorithm to solve the Liouville-von-Neumann equation. We solve the real-time evolution of the reduced single-electron density matrix at the tight-binding level. Calculations are carried out to simulate the transient current through a linear chain of atoms, with each represented by a single orbital. The self-energy matrix is expanded in terms of multiple Lorentzian functions, and the Fermi distribution function is evaluated via the Padè spectrum decomposition. This Lorentzian-Padè decomposition scheme is employed to simulate the transient current. With sufficient Lorentzian functions used to fit the self-energy matrices, we show that the lead spectral function and the dynamics response can be treated accurately. Compared to the conventional master equation approaches, our method is much more efficient as the computational time scales cubically with the system size and linearly with the simulation time. As a result, the simulations of the transient currents through systems containing up to one hundred of atoms have been carried out. As density functional theory is also an effective one-particle theory, the Lorentzian-Padè decomposition scheme developed here can be generalized for first-principles simulation of realistic systems.

Mixed Bose-Fermi statistics: Kinetic equation and navigation through a network

The conventional master equation is generalized to include Bose and Fermi moves. The obtained equation describes the time evolution of a system with mixed Bose-Fermi statistics which are controlled by the fraction of Fermi moves (α) ; the latter plays a role of an exploratory tendency (ET) factor. The theoretical consideration is illustrated with numerical results for navigation through a model scale-free network. Depending on the ET factor and the number of participants involved in the navigation (N(tot)) , a broad variety of the behavior scenarios has been observed. The cases of N(tot)=1 and N(tot)>>1 have been found drastically different. Of particular interest is the case of N(tot)>>1 , when the participants interfere with each other's motion, which is characteristic of many real systems (urban traffic, Internet, epidemic spreading, etc.). It has been found that at α<1 all participants reach a preselected (target) state but an excessive intention to try detour routes (α→1) leads to a critical slowdown in their passage to this state. At the same time, if α=1 (solely Fermi moves, no possibility to return to a previously visited state), only one of the participants attains the target state but it does it much faster than when a single copy navigates through the network (N(tot)=1) , i.e., this one optimizes the way to the target state at the expense of other participants. The procedure used to include Bose and Fermi moves in the master equation offers a framework for derivation of kinetic equations for more complex and general statistics.

Quantum measurement of single electron state by a quantum point contact

Closely related to the quantum information processing in solid states, we study the quantum measurement of single electron state by a mesoscopic charge-sensitive detector, namely the quantum point contact (QPC). We find that the conventional Lindblad-type master equation is not appropriate for describing the underlying measurement dynamics. The treatment developed in this work properly accounts for the energy-exchange between the detector and the measured system, and its role on the detailed-balance relation. A valid description for the QPC measurement dynamics is provided which may have impact on the study of quantum measurement and quantum feedback control in solid states.

Monte Carlo wavefunction approach to the exciton dynamics of molecular aggregates with exciton–phonon coupling

We develop the second-order Monte Carlo wavefunction (MCWF) approach to the exciton dynamics of molecular aggregate systems composed of dipole-coupled two-state monomers. The explicit form of Lindblad operator, which is indispensable for applying the MCWF approach, for population relaxation among exciton states is derived based on the quantum master equation involving weak exciton–phonon coupling. The exciton migration behaviors obtained by the MCWF approach are turned out to coincide with those by the conventional master equation approach, indicating the high potential of the MCWF approach to the dissipative exciton dynamics of extended molecular aggregates or supermolecules.

Master-equation for cascade damage modeling

This paper is a continuation of our effort to understand and model irradiation effects under cascade damage. The conventional master equation for the evolution of a void ensemble is reformulated to include the stochastic effects due to the random cascade initiation. Numerical solution of the modified master equations to model the evolution of an ensemble of voids explicitly shows that stochastic fluctuations due to random cascade initiation produce important effects that cannot be accounted for by the conventional mean-field rate theory approach. The results are in good agreement with those derived previously using the Fokker–Planck equation.

Output spectrum of a detector measuring quantum oscillations

We consider a two-level quantum system (qubit) which is continuously measured by a detector and calculate the spectral density of the detector output. In the weakly coupled case the spectrum exhibits a moderate peak at the frequency of quantum oscillations and a Lorentzian-shape increase of the detector noise at low frequency. With increasing coupling the spectrum transforms into a single Lorentzian corresponding to random jumps between two states. We prove that the Bayesian formalism for the selective evolution of the density matrix gives the same spectrum as the conventional master equation approach, despite the significant difference in interpretation. The effects of the detector nonideality and the finite-temperature environment are also discussed.

Applicability of the conventional master equation for the description of microstructure evolution under cascade-producing irradiation

The physical basis of the conventional master equation used for the description of microstructure evolution under continuous irradiation is examined. It is rigorously shown that in this equation only fluctuations of point-defect fluxes caused by the randomness of migratory jumps alone are considered. Taking into account also the stochastic effects caused by the random nature of the cascade occurrence shows that this contribution is very important and its omission may introduce significant errors in the description of microstructure evolution by the conventional master equation.

One-atom maser: Statistics of detector clicks.

We present a general theory for the calculation of various characteristic properties of the beam of atoms emerging from a resonator in one-atom-maser experiments. The beam is described in terms of the statistics of the detector clicks. The evolution of the state of the maser photons between clicks is governed by a nonlinear master equation. The nonlinearity originates in the necessity to account for the atoms that escape detection. Despite the permanent reductions of the photon state, resulting from the detections, the steady state of the conventional linear master equation determines the statistics of the detector clicks. The whole process is ergodic, in the sense that a single run of the experiment contains all reproducible statistical data, provided the duration of the run is much longer than all relevant correlation times. The nonlinear master equation is used to calculate the distribution of waiting times between detector clicks. Other statistical properties of the clicks that are derived include correlation functions and variances of the counting statistics. The formalism is applied to standard one-atom-maser experiments and to parity measurements on both unpumped and pumped cavities. We find that, for the standard one-atom-maser operation, none of the said statistical properties is a simple immediate indicator for a sub-Poissonian variance of the photon number. For example, the detector clicks may be antibunched although the photon distribution has a super-Poissonian variance.

Directed random walk with spatially correlated random transfer rates.

We have investigated the random walk of particles in the frame of a conventional master equation for directed random walks. The transfer rates are supposed to be random variables and we incorporate the possibility of correlations. We assume that the chain consists of successive segments of random lengths. Within a given segment, the transfer rates are equal to a single random variable. The transfer rates belonging to two different segments are supposed to be independent and distributed according to the same probability law. We have calculated the time-asymptotic behavior of the mean coordinate of the particle. The resulting character of the motion emerges from the interplay between two basic features: the probability of having a small value of the transfer rate and the probability of having long segments. If the first moment of the segment-length distribution diverges, the asymptotic regime undergoes radical changes as compared to the noncorrelated model.

Kinetic method for waves in random media

New tools of statistical analysis for systems obeying differential equations in partial (and ordinary) derivatives with randomly alternating parameters of Markovian type and edge boundary conditions are presented. While the conventional master equation methods are valid only for the stochastic equations with given initial conditions, the method presented is more general and incorporates no embedding to an initial value problem. We develop a simple formalism by introduction of special double-sided conditional averages of dynamic variables. Its main advantage is that the averaging procedure for linear systems with multiplicative random influence is linear, yielding the possibility to obtain exact analytical solutions. The method is applied to a typical model of the theory of wave propagation in random media and exact results of its analysis are presented.

Dynamics of excitation in systems with a randomly modulated decay channel

We have investigated the kinetics of excitation in the frame of the conventional Pauli master equation for a molecular aggregate with a decay channel attached to one distinguished molecule (or site). As a principally new feature we assume the rate of probability flow through the decay channel to be a randomly modulated function of time. The modulation is described by a Markoff stochastic process and we do not invoke the white-noise assumption. An exact calculation is given for a broad family of stochastic processes which bridges (and includes) the asymmetric random telegraph signal and the Gaussian process. The general method is then applied in three physically relevant situations: one-molecule aggregate, cyclic-antenna system and infinite linear chain. In these models, we discuss the site-occupation probabilities, the aggregate-excitation function and the mean de-excitation time.

Intermolecular forces, spontaneous emission, and superradiance in a dielectric medium: Polariton-mediated interactions.

A reduced equation of motion that describes the excited-state dynamics of interacting two-level impurity molecules in a dielectric host crystal is derived starting from a microscopic model for the total system. Our theory generalizes the derivation of the conventional superradiance master equation for molecules in vacuum; the role of photons in the conventional theory is played by polaritons (mixed crystal-radiation excitations) in our approach. Our final equation thus contains dispersive and superradiant polariton-mediated intermolecular interactions. The effect of the dielectric host is completely contained within a rescaling of these interactions with the transverse dielectric function \ensuremath{\epsilon}(\ensuremath{\omega}) of the crystal taken at the impurity's transition frequency. Our theory yields all local field and screening factors for both the dispersive and the dissipative couplings from a single, unified starting point. Known scaling laws for the spontaneous-emission rate and the instantaneous dipole-dipole interaction are extended to the frequency region where the dispersion of \ensuremath{\epsilon}(\ensuremath{\omega}) is important.