Showing papers on "Master equation published in 2010"
TL;DR: In this article, an information-theoretic approach for quantitatively characterizing the non-Markovianity of open quantum processes is presented. But the approach is restricted to a class of time-local master equations, where the Fisher Information (QFI) flow is decomposed into additive subflows according to different dissipative channels.
Abstract: We establish an information-theoretic approach for quantitatively characterizing the non-Markovianity of open quantum processes. Here, the quantum Fisher information (QFI) flow provides a measure to statistically distinguish Markovian and non-Markovian processes. A basic relation between the QFI flow and non-Markovianity is unveiled for quantum dynamics of open systems. For a class of time-local master equations, the exactly analytic solution shows that for each fixed time the QFI flow is decomposed into additive subflows according to different dissipative channels.
456 citations
TL;DR: A formulation of stochastic thermodynamics for systems subjected to nonequilibrium constraints and furthermore driven by external time-dependent forces is proposed, leading to a splitting of the second law leading to three second-law-like relations.
Abstract: We propose a formulation of stochastic thermodynamics for systems subjected to nonequilibrium constraints (i.e. broken detailed balance at steady state) and furthermore driven by external time-dependent forces. A splitting of the second law occurs in this description leading to three second-law-like relations. The general results are illustrated on specific solvable models. The present paper uses a master equation based approach.
340 citations
TL;DR: In this article, the authors derived an analytical expression for the dipole-field coupling strength and the Purcell factor and showed that simple semiclassical theory fails to predict the correct scattered field spectrum even in the weak-field limit.
Abstract: We derive a full quantum optical model of interactions between a dipole and a metal nanoparticle. The electromagnetic field of the nanoparticle is quantized from the time-harmonic solution to the wave equation. We derive an analytical expression for the dipole-field coupling strength and the Purcell factor. The semiclassical theory, derived from the Maxwell-Bloch equations, is compared to the full quantum calculations based on numerical solution of the master equation. The metal nanoparticle-dipole system is found to be in an interesting regime of cavity quantum electrodynamics where dipole decay is dominated by dephasing, but the dipole-field coupling strength is still strong enough to achieve large cooperativity. In the presence of large dephasing, we show that simple semiclassical theory fails to predict the correct scattered field spectrum even in the weak-field limit. We reconcile this discrepancy by applying the random-phase-jump approach to the cavity photon number instead of the dipole operator. We also investigate the quantum fluctuations of the scattered field and show that they are significantly dependent on the dephasing rate.
239 citations
TL;DR: It is shown that any dynamical map representing the evolution of such a system may be described either by a nonlocal master equation with a memory kernel or equivalently by an equation which is local in time.
Abstract: We analyze non-Markovian evolution of open quantum systems. It is shown that any dynamical map representing the evolution of such a system may be described either by a nonlocal master equation with a memory kernel or equivalently by an equation which is local in time. These two descriptions are complementary: if one is simple, the other is quite involved, or even singular, and vice versa. The price one pays for the local approach is that the corresponding generator keeps the memory about the starting point ``${t}_{0}$.'' This is the very essence of non-Markovianity. Interestingly, this generator might be highly singular; nevertheless, the corresponding dynamics is perfectly regular. Remarkably, the singularities of the generator may lead to interesting physical phenomena such as the revival of coherence or sudden death and revival of entanglement.
231 citations
TL;DR: In this paper, the authors derived an effective master equation describing two-phon cooling of the mechanical oscillator and demonstrated how to achieve mechanical squeezing by driving the cavity with two beams.
Abstract: We explore the physics of optomechanical systems in which an optical cavity mode is coupled parametrically to the square of the position of a mechanical oscillator. We derive an effective master equation describing two-phonon cooling of the mechanical oscillator. We show that for high temperatures and weak coupling, the steady-state phonon number distribution is nonthermal (Gaussian) and that even for strong cooling the mean phonon number remains finite. Moreover, we demonstrate how to achieve mechanical squeezing by driving the cavity with two beams. Finally, we calculate the optical output and squeezing spectra. Implications for optomechanics experiments with the membrane-in-the-middle geometry or ultracold atoms in optical resonators are discussed.
224 citations
TL;DR: The crux of the theory is a dissipative variant of WKB (Wentzel-Kramers-Brillouin) approximation which assumes that the typical population size in the metastable state is large, and yields both entropic barriers to extinction and pre-exponential factors.
Abstract: We investigate the phenomenon of extinction of a long-lived self-regulating stochastic population, caused by intrinsic (demographic) noise. Extinction typically occurs via one of two scenarios depending on whether the absorbing state $n=0$ is a repelling (scenario A) or attracting (scenario B) point of the deterministic rate equation. In scenario A the metastable stochastic population resides in the vicinity of an attracting fixed point next to the repelling point $n=0$. In scenario B there is an intermediate repelling point $n={n}_{1}$ between the attracting point $n=0$ and another attracting point $n={n}_{2}$ in the vicinity of which the metastable population resides. The crux of the theory is a dissipative variant of WKB (Wentzel-Kramers-Brillouin) approximation which assumes that the typical population size in the metastable state is large. Starting from the master equation, we calculate the quasistationary probability distribution of the population sizes and the (exponentially long) mean time to extinction for each of the two scenarios. When necessary, the WKB approximation is complemented (i) by a recursive solution of the quasistationary master equation at small $n$ and (ii) by the van Kampen system-size expansion, valid near the fixed points of the deterministic rate equation. The theory yields both entropic barriers to extinction and pre-exponential factors, and holds for a general set of multistep processes when detailed balance is broken. The results simplify considerably for single-step processes and near the characteristic bifurcations of scenarios A and B.
189 citations
Book•
02 Jun 2010TL;DR: In this paper, the authors describe the drift-diffusion model and its application in particle-based simulation of nanoelectronic devices, as well as the applicability of the Drift-Diffusion model in commercial Semiconductor device modeling tools.
Abstract: Introduction to Computational Electronics Si-Based Nanoelectronics Heterostructure Devices in III-V or II-VI Technology Modeling of Nanoscale Devices The Content of This Book Introductory Concepts Crystal Structure Semiconductors Band Structure Preparation of Semiconductor Materials Effective Mass Density of States Electron Mobility Semiconductor Statistics Semiconductor Devices Semiclassical Transport Theory Approximations for the Distribution Function Boltzmann Transport Equation Relaxation-Time Approximation Rode's Iterative Method Scattering Mechanisms: Brief Description Implementation of the Rode Method for 6H-SiC Mobility Calculation The Drift-Diffusion Equations and Their Numerical Solution Drift-Diffusion Model Derivation Drift-Diffusion Application Example Hydrodynamic Modeling Introduction Extensions of the Drift-Diffusion Model Stratton's Approach Hydrodynamic (Balance, Blotekjaer) Equations Model The Need for Commercial Semiconductor Device Modeling Tools State-of-the-Art Commercial Packages The Advantages and Disadvantages of Hydrodynamic Models: Simulations of Different Generation FD SOI Devices Particle-Based Device Simulation Methods Direct Solution of Boltzmann Transport Equation: Monte Carlo Method Multi-Carrier Effects Device Simulations Coulomb Force Treatment within a Particle-Based Device Simulation Scheme Representative Simulation Results of Multiparticle and Discrete Impurity Effects Modeling Thermal Effects in Nano-Devices Some General Aspects of Heat Conduction Classical Heat Conduction in Solids Form of the Heat Source Term Modeling Heating Effects with Commercial Simulation Packages The ASU Particle-Based Approach to Lattice Heating in Nanoscale Devices Open Problems Quantum Corrections to Semiclassical Approaches One-Dimensional Quantum-Mechanical Space Quantization Quantum Corrections to Drift-Diffusion and Hydrodynamic Simulators The Effective Potential Approach in Conjunction with Particle-Based Simulations Description of Gate Current Models Used in Device Simulations Monte Carlo-k _ p-1D Schrodinger Solver for Modeling Transport in p-Channel Strained SiGe Devices Quantum Transport in Semiconductor Systems Tunneling General Notation Transfer Matrix Approach Landauer Formula and Usuki Method Far-From-Equilibrium Quantum Transport Mixed States and Distribution Function Irreversible Processes and MASTER Equations The Wigner Distribution Function Green's Functions Nonequilibrium Keldysh Green's Functions Low Field Transport in Strained-Si Inversion Layers NEGF in a Quasi-1D Formulation Quantum Transport in 1D-Resonant Tunneling Diodes Coherent High-Field Transport in 2D and 3D Conclusions Appendix A: Electronic Band Structure Calculation Appendix B: Poisson Equation Solvers Appendix C: Computational Electromagnetics Appendix D: Stationary and Time-Dependent Perturbation Theory Each chapter concludes with "Problems" and "References"
182 citations
TL;DR: This work analyzes a master equation formulation of stochastic neurodynamics for a network of synaptically coupled homogeneous neuronal populations each consisting of N identical neurons to derive the lowest order corrections to these rate equations for large but finite N.
Abstract: We analyze a master equation formulation of stochastic neurodynamics for a network of synaptically coupled homogeneous neuronal populations each consisting of N identical neurons. The state of the network is specified by the fraction of active or spiking neurons in each population, and transition rates are chosen so that in the thermodynamic or deterministic limit ($N\rightarrow\infty$) we recover standard activity-based or voltage-based rate models. We derive the lowest order corrections to these rate equations for large but finite N using two different approximation schemes, one based on the Van Kampen system-size expansion and the other based on path integral methods. Both methods yield the same series expansion of the moment equations, which at $\mathcal{O}(1/N)$ can be truncated to form a closed system of equations for the first- and second-order moments. Taking a continuum limit of the moment equations while keeping the system size N fixed generates a system of integrodifferential equations for the ...
179 citations
TL;DR: In this article, the authors derive Markovian master equations for single and interacting harmonic systems in different scenarios, including strong internal coupling, and compare the dynamics resulting from the corresponding master equations with numerical simulations of the global system's evolution.
Abstract: We derive Markovian master equations for single and interacting harmonic systems in different scenarios, including strong internal coupling. By comparing the dynamics resulting from the corresponding master equations with numerical simulations of the global system's evolution, we delimit their validity regimes and assess the robustness of the assumptions usually made in the process of deriving the reduced Markovian dynamics. The results of these illustrative examples serve to clarify the general properties of other open quantum system scenarios subject to treatment within a Markovian approximation.
179 citations
TL;DR: The results of these illustrative examples serve to clarify the general properties of other open quantum system scenarios subject to treatment within a Markovian approximation, and assess the robustness of the assumptions usually made in the process of deriving the reduced Markovians.
Abstract: We derive Markovian master equations of single and interacting harmonic systems in different scenarios, including strong internal coupling. By comparing the dynamics resulting from the corresponding Markovian master equations with exact numerical simulations of the evolution of the global system, we precisely delimit their validity regimes and assess the robustness of the assumptions usually made in the process of deriving the reduced dynamics. The proposed method is sufficiently general to suggest that the conclusions made here are widely applicable to a large class of settings involving interacting chains subject to a weak interaction with an environment.
161 citations
TL;DR: The main idea of the existence analysis is to reformulate the quantum Navier–Stokes equations by means of a so-called effective velocity involving a density gradient, leading to a viscous quantum Euler system.
Abstract: The global-in-time existence of weak solutions to the barotropic compressible quantum Navier–Stokes equations in a three-dimensional torus for large data is proved. The model consists of the mass conservation equation and a momentum balance equation, including a nonlinear third-order differential operator, with the quantum Bohm potential, and a density-dependent viscosity. The system has been derived by Brull and Mehats [Derivation of viscous correction terms for the isothermal quantum Euler model, 2009, submitted] from a Wigner equation using a moment method and a Chapman–Enskog expansion around the quantum equilibrium. The main idea of the existence analysis is to reformulate the quantum Navier–Stokes equations by means of a so-called effective velocity involving a density gradient, leading to a viscous quantum Euler system. The advantage of the new formulation is that there exists a new energy estimate which implies bounds on the second derivative of the particle density. The global existence of weak s...
TL;DR: In this article, the authors studied the excitonic dynamics of a driven quantum dot under the influence of a phonon environment and developed a master equation that can be valid over a much larger range of excitonphonon coupling strengths and temperatures than in the case of the standard weak-coupling approach.
Abstract: We study the excitonic dynamics of a driven quantum dot under the influence of a phonon environment, going beyond the weak exciton-phonon coupling approximation. By combining the polaron transform and time-local projection operator techniques, we develop a master equation that can be valid over a much larger range of exciton-phonon coupling strengths and temperatures than in the case of the standard weak-coupling approach. For the experimentally relevant parameters considered here, we find that the weak-coupling and polaron theories give very similar predictions for low temperatures (below 30 K), while at higher temperatures we begin to see discrepancies between the two. This is because, unlike the polaron approach, the weak-coupling theory is incapable of capturing multiphonon effects, while it also does not properly account for phonon-induced renormalization of the driving frequency. In particular, we find that the weak-coupling theory often overestimates the damping rate when compared to the polaron theory. Finally, we extend our theory to include non-Markovian effects and find that, for the parameters considered here, they have little bearing on the excitonic Rabi rotations when plotted as a function of pulse area.
TL;DR: The theory of effective mesoscopic rate equations generalizes the conventional rate equations of physical chemistry to describe kinetics in systems of mesoscopic size such as biological cells and shows that these differences depend sensitively on the Fano factors and on the inherent structure and topology of the chemical network.
Abstract: Chemical master equations provide a mathematical description of stochastic reaction kinetics in well-mixed conditions. They are a valid description over length scales that are larger than the reactive mean free path and thus describe kinetics in compartments of mesoscopic and macroscopic dimensions. The trajectories of the stochastic chemical processes described by the master equation can be ensemble-averaged to obtain the average number density of chemical species, i.e., the true concentration, at any spatial scale of interest. For macroscopic volumes, the true concentration is very well approximated by the solution of the corresponding deterministic and macroscopic rate equations, i.e., the macroscopic concentration. However, this equivalence breaks down for mesoscopic volumes. These deviations are particularly significant for open systems and cannot be calculated via the Fokker-Planck or linear-noise approximations of the master equation. We utilize the system-size expansion including terms of the order of Omega(-1/2) to derive a set of differential equations whose solution approximates the true concentration as given by the master equation. These equations are valid in any open or closed chemical reaction network and at both the mesoscopic and macroscopic scales. In the limit of large volumes, the effective mesoscopic rate equations become precisely equal to the conventional macroscopic rate equations. We compare the three formalisms of effective mesoscopic rate equations, conventional rate equations, and chemical master equations by applying them to several biochemical reaction systems (homodimeric and heterodimeric protein-protein interactions, series of sequential enzyme reactions, and positive feedback loops) in nonequilibrium steady-state conditions. In all cases, we find that the effective mesoscopic rate equations can predict very well the true concentration of a chemical species. This provides a useful method by which one can quickly determine the regions of parameter space in which there are maximum differences between the solutions of the master equation and the corresponding rate equations. We show that these differences depend sensitively on the Fano factors and on the inherent structure and topology of the chemical network. The theory of effective mesoscopic rate equations generalizes the conventional rate equations of physical chemistry to describe kinetics in systems of mesoscopic size such as biological cells.
TL;DR: A stochastic model of neuronal population dynamics with intrinsic noise is analyzed, reducing the dynamics to a neural Langevin equation, and showing how the intrinsic noise amplifies subthreshold oscillations (quasicycles).
Abstract: We analyze a stochastic model of neuronal population dynamics with intrinsic noise. In the thermodynamic limit N -> infinity, where N determines the size of each population, the dynamics is described by deterministic Wilson–Cowan equations. On the other hand, for finite N the dynamics is described by a master equation that determines the probability of spiking activity within each population. We first consider a single excitatory population that exhibits bistability in the deterministic limit. The steady–state probability distribution of the stochastic network has maxima at points corresponding to the stable fixed points of the deterministic network; the relative weighting of the two maxima depends on the system size. For large but finite N, we calculate the exponentially small rate of noise–induced transitions between the resulting metastable states using a Wentzel–Kramers–Brillouin (WKB) approximation and matched asymptotic expansions. We then consider a two-population excitatory/inhibitory network that supports limit cycle oscillations. Using a diffusion approximation, we reduce the dynamics to a neural Langevin equation, and show how the intrinsic noise amplifies subthreshold oscillations (quasicycles).
TL;DR: In this paper, the authors generalize the method of third quantization to a unified exact treatment of Redfield and Lindblad master equations for open quadratic systems of n fermions in terms of diagonalization of a 4n×4n matrix.
Abstract: We generalize the method of third quantization to a unified exact treatment of Redfield and Lindblad master equations for open quadratic systems of n fermions in terms of diagonalization of a 4n×4n matrix. Non-equilibrium thermal driving in terms of the Redfield equation is analyzed in detail. We explain how one can compute all the physically relevant quantities, such as non-equilibrium expectation values of local observables, various entropies or information measures, or time evolution and properties of relaxation. We also discuss how to exactly treat explicitly time-dependent problems. The general formalism is then applied to study a thermally driven open XY spin 1/2 chain. We find that the recently proposed non-equilibrium quantum phase transition in the open XY chain survives the thermal driving within the Redfield model. In particular, the phase of long-range magnetic correlations can be characterized by hypersensitivity of the non-equilibrium steady state to external (bath or bulk) parameters. Studying the heat transport, we find negative differential thermal conductance for sufficiently strong thermal driving as well as non-monotonic dependence of the heat current on the strength of the bath coupling.
TL;DR: The curse of dimensionality is revisits and an efficient strategy for circumventing such challenging issue is proposed, based on the use of a Proper Generalized Decomposition, is specially well suited to treat the multidimensional parametric equations.
TL;DR: In this paper, a tensor decomposition of the Stokes vector of the light field with moments of the atomic spin state is proposed to describe the spin dynamics in the presence of both nonlinear dynamics and photon scattering.
TL;DR: A fractional Fokker-Planck equation for subdiffusion in a general space- and time-dependent force field from power law waiting time continuous time random walks biased by Boltzmann weights is derived.
Abstract: We derive a fractional Fokker-Planck equation for subdiffusion in a general space- and time-dependent force field from power law waiting time continuous time random walks biased by Boltzmann weights. The governing equation is derived from a generalized master equation and is shown to be equivalent to a subordinated stochastic Langevin equation.
TL;DR: In this article, the quantum entanglement dynamics of two spins in the presence of classical Ornstein-Uhlenbeck noise were investigated, and exact solutions for evolution dynamics were obtained.
TL;DR: An adaptive method which compresses the problem very efficiently by representing the solution in a sparse wavelet basis that is updated in each step that is chosen adaptively according to estimates of the temporal and spatial approximation errors.
TL;DR: The irreversibility and the entropy production in nonequilibrium interacting particle systems described by a Fokker-Planck equation by the use of a suitable master equation representation and the connection between the definition of entropy production rate and the Jarzynski equality is shown.
Abstract: We analyze the irreversibility and the entropy production in nonequilibrium interacting particle systems described by a Fokker-Planck equation by the use of a suitable master equation representation. The irreversible character is provided either by nonconservative forces or by the contact with heat baths at distinct temperatures. The expression for the entropy production is deduced from a general definition, which is related to the probability of a trajectory in phase space and its time reversal, that makes no reference a priori to the dissipated power. Our formalism is applied to calculate the heat conductance in a simple system consisting of two Brownian particles each one in contact to a heat reservoir. We show also the connection between the definition of entropy production rate and the Jarzynski equality.
TL;DR: An explicit fitting formula is derived to extract the sequence of decay rate ratios from the measurements of averaged current in a periodically driven device, which provides a device-specific fingerprint which allows us to compare different architectures, and predict the upper limits of initialization accuracy from low precision measurements.
Abstract: Dynamic quantum dots can be formed by time-dependent electrostatic potentials, such as in gate- or surface-acoustic-wave-driven electron pumps. In this work we propose and quantify a scheme to initialize quantum dots with a controllable number of electrons. It is based on a rapid increase of the electron potential energy and simultaneous decoupling from the source lead. The full probability distribution for the final number of captured electrons is obtained by solving a master equation for stochastic cascade of single electron escape events. We derive an explicit fitting formula to extract the sequence of decay rate ratios from the measurements of averaged current in a periodically driven device. This provides a device-specific fingerprint which allows us to compare different architectures, and predict the upper limits of initialization accuracy from low precision measurements.
TL;DR: In this paper, the Exner equation and population exchange model were used to track the number of moving particles in a river bed and showed that particle entrainment and deposition can be modeled as population exchanges between the stream and the bed.
Abstract: Even under flow equilibrium conditions, river bed topography continuously evolves with time, producing trains of irregular bed forms. The idea has recently emerged that the variability in the bed form geometry results from some randomness in sediment flux. In this paper, we address this issue by using the Exner equation and a population exchange model derived in an earlier paper. In this model, particle entrainment and deposition are idealized as population exchanges between the stream and the bed, which makes it possible to use birth‐death Markov process theory to track the number of moving grains. The paper focuses on nascent bed forms on initially planar beds, a situation in which the coupling between the stream and bed is weak. In a steady state, the number of moving particles follows a negative binomial distribution. Although this probability distribution does not enter the family of heavy‐tailed distributions, it may give rise to large and frequent fluctuations because the standard deviation can be much larger than the mean, a feature that is not accounted for with classic probability laws (e.g., Hamamori’s law) used so far for describing bed load fluctuations. In the large‐system limit, the master equation of the birth‐death Markov process can be transformed into a Fokker‐Planck equation. This transformation is used here to show that the number of moving particles can be described as an Ornstein‐Uhlenbeck process. An important consequence is that in the long term, the number of moving particles follows a Gaussian distribution. Laboratory experiments show that this approximation is correct when the mean number per unit length of stream, $\vec{N}$/L, is sufficiently large (typically two particles per centimeter in our experiments). The particle number fluctuations give rise to bed elevation fluctuations, whose spectrum falls off like $\omega^{-2}$ in the high‐frequency regime (with $\omega$ the angular frequency) and variance grows linearly with time. These features are in agreement with recent observations on bed form development (in particular, ripple growth).
TL;DR: This work has established a convolutionless stochastic Schrödinger equation called the time-local quantum state diffusion (QSD) equation without any approximations, in particular, without Markov approximation.
Abstract: The non-Markovian dynamics of a three-level quantum system coupled to a bosonic environment is a difficult problem due to the lack of an exact dynamic equation such as a master equation. We present for the first time an exact quantum trajectory approach to a dissipative three-level model. We have established a convolutionless stochastic Schrodinger equation called the time-local quantum state diffusion (QSD) equation without any approximations, in particular, without Markov approximation. Our exact time-local QSD equation opens a new avenue for exploring quantum dynamics for a higher dimensional quantum system coupled to a non-Markovian environment.
TL;DR: In this paper, the authors proposed a sliding window method to compute an approximate solution of the CME by performing a sequence of local analysis steps, where only a manageable subset of states is considered, representing a "window" into the state space.
Abstract: The chemical master equation (CME) is a system of ordinary differential equations that describes the evolution of a network of chemical reactions as a stochastic process. Its solution yields the probability density vector of the system at each point in time. Solving the CME numerically is in many cases computationally expensive or even infeasible as the number of reachable states can be very large or infinite. We introduce the sliding window method, which computes an approximate solution of the CME by performing a sequence of local analysis steps. In each step, only a manageable subset of states is considered, representing a "window" into the state space. In subsequent steps, the window follows the direction in which the probability mass moves, until the time period of interest has elapsed. We construct the window based on a deterministic approximation of the future behavior of the system by estimating upper and lower bounds on the populations of the chemical species. In order to show the effectiveness of our approach, we apply it to several examples previously described in the literature. The experimental results show that the proposed method speeds up the analysis considerably, compared to a global analysis, while still providing high accuracy. The sliding window method is a novel approach to address the performance problems of numerical algorithms for the solution of the chemical master equation. The method efficiently approximates the probability distributions at the time points of interest for a variety of chemically reacting systems, including systems for which no upper bound on the population sizes of the chemical species is known a priori.
TL;DR: In this paper, a detailed comparison between three prominent approaches to quantum transport: the fourth-order Bloch-Redfield quantum master equation (BR), the real-time diagrammatic technique (RT), and the scattering rate approach based on the T-matrix (TM) is presented.
Abstract: Various theoretical methods address transport effects in quantum dots beyond single-electron tunneling while accounting for the strong interactions in such systems. In this paper we report a detailed comparison between three prominent approaches to quantum transport: the fourth-order Bloch-Redfield quantum master equation (BR), the real-time diagrammatic technique (RT), and the scattering rate approach based on the T-matrix (TM). Central to the BR and RT is the generalized master equation for the reduced density matrix. We demonstrate the exact equivalence of these two techniques. By accounting for coherences (nondiagonal elements of the density matrix) between nonsecular states, we show how contributions to the transport kernels can be grouped in a physically meaningful way. This not only significantly reduces the numerical cost of evaluating the kernels but also yields expressions similar to those obtained in the TM approach, allowing for a detailed comparison. However, in the TM approach an ad hoc regularization procedure is required to cure spurious divergences in the expressions for the transition rates in the stationary (zero-frequency) limit. We show that these problems derive from incomplete cancellation of reducible contributions and do not occur in the BR and RT techniques, resulting in well-behaved expressions in the latter two cases. Additionally, we show that a standard regularization procedure of the TM rates employed in the literature does not correctly reproduce the BR and RT expressions. All the results apply to general quantum dot models and we present explicit rules for the simplified calculation of the zero-frequency kernels. Although we focus on fourth-order perturbation theory only, the results and implications generalize to higher orders. We illustrate our findings for the single impurity Anderson model with finite Coulomb interaction in a magnetic field.
TL;DR: In this paper, exact master equations describing the decay of a two-state system into a structured reservoir are constructed by employing the exact solution for the model, analytical expressions are determined for the memory kernel of the Nakajima-Zwanzig master equation and for the generator of the corresponding time-convolutionless master equation.
Abstract: Exact master equations describing the decay of a two-state system into a structured reservoir are constructed. By employing the exact solution for the model, analytical expressions are determined for the memory kernel of the Nakajima-Zwanzig master equation and for the generator of the corresponding time-convolutionless master equation. This approach allows an explicit comparison of the convergence behavior of the corresponding perturbation expansions. Moreover, the structure of widely used phenomenological master equations with a memory kernel may be incompatible with a nonperturbative treatment of the underlying microscopic model. Several physical implications of the results on the microscopic analysis and the phenomenological modeling of non-Markovian quantum dynamics of open systems are discussed.
TL;DR: In this article, a non-equilibrium quantum theory for transient electron dynamics in nanodevices based on the Feynman-Vernon influence functional was presented, which enables the study transient quantum transport in nanostructures with back-reaction effects from the contacts, with non-Markovian dissipation and decoherence being fully taken into account.
Abstract: In this paper, we present a non-equilibrium quantum theory for transient electron dynamics in nanodevices based on the Feynman-Vernon influence functional. Applying the exact master equation for nanodevices we recently developed to the more general case in which all the constituents of a device vary in time in response to time-dependent external voltages, we obtained non-perturbatively the transient quantum transport theory in terms of the reduced density matrix. The theory enables us to study transient quantum transport in nanostructures with back-reaction effects from the contacts, with non-Markovian dissipation and decoherence being fully taken into account. For a simple illustration, we apply the theory to a single-electron transistor subjected to ac bias voltages. The non-Markovian memory structure and the nonlinear response functions describing transient electron transport are obtained.
TL;DR: A recently developed measure for non-Markovianity is utilized to elucidate the exciton-phonon dynamics in terms of the information flow between electronic and vibrational degrees of freedom and it is found that for a model dimer system and for the Fenna-Matthews-Olson complex the non- MarkovianITY is significant under physiological conditions.
Abstract: Non-Markovian and non-equilibrium phonon effects are believed to be key ingredients in the energy transfer in photosynthetic complexes, especially in complexes which exhibit a regime of intermediate exciton-phonon coupling. In this work, we utilize a recently-developed measure for non-Markovianity to elucidate the exciton-phonon dynamics in terms of the information flow between electronic and vibrational degrees of freedom. We study the measure in the hierarchical equation of motion approach which captures strong system-bath coupling effects and non-equilibrium molecular reorganization. We propose an additional trace-distance measure for the information flow that could be extended to other master equations. We find that for a model dimer system and the Fenna-Matthews-Olson complex that non-Markovianity is significant under physiological conditions.
TL;DR: In this paper, a unified description of all classes of exact solutions to these equations in terms of confluent Heun functions is presented and large classes of new exact solutions are found and classified with respect to their characteristic properties.
Abstract: The Teukolsky master equation is the basic tool for the study of perturbations of the Kerr metric in linear approximation. It admits separation of variables, thus yielding the Teukolsky radial equation and the Teukolsky angular equation. We present here a unified description of all classes of exact solutions to these equations in terms of the confluent Heun functions. Large classes of new exact solutions are found and classified with respect to their characteristic properties. Special attention is paid to the polynomial solutions which are singular ones and introduce collimated one-way running waves. It is shown that a proper linear combination of such solutions can present bounded one-way running waves. This type of waves may be suitable as models of the observed astrophysical jets.