Generalized Master Equation Research Articles

Nonadiabatic transitions between adiabatic surfaces: Phase diffusion in superconducting atomic point contacts

Motivated by experiments with current-biased superconducting atomic point contacts the general problem of nonadiabatic transitions between adiabatic surfaces in presence of strong dissipation is studied. For a single-channel device the supercurrent is determined by the diffusive motion of the superconducting phase difference on two Andreev levels. These surfaces are uncoupled only in the adiabatic limit of low to moderate transmissions while for high transmissions curve crossings are important. Starting from a general master equation of the full density matrix an approximate time evolution equation for the populations on the adiabatic surfaces in the overdamped limit is derived from which the relevant observables can be obtained. Specific results for the case of atomic point contacts are in agreement with experimental observations that cannot be explained by conventional theory.

Nonlinear generalized master equations and accounting for initial correlations

We develop a new method based on using a time-dependent operator (generally not a projection operator) converting a distribution function (statistical operator) of a total system into the relevant form that allows deriving new exact nonlinear generalized master equations (GMEs). The derived inhomogeneous nonlinear GME is a generalization of the linear Nakajima-Zwanzig GME and can be viewed as an alternative to the BBGKY chain. It is suitable for obtaining both nonlinear and linear evolution equations. As in the conventional linear GME, there is an inhomogeneous term comprising all multiparticle initial correlations. To include the initial correlations into consideration, we convert the obtained inhomogeneous nonlinear GME into the homogenous form by the previously suggested method. We use no conventional approximation like the random phase approximation (RPA) or the Bogoliubov principle of weakening of initial correlations. The obtained exact homogeneous nonlinear GME describes all evolution stages of the (sub)system of interest and treats initial correlations on an equal footing with collisions via the modified memory kernel. As an application, we obtain a new homogeneous nonlinear equation retaining initial correlations for a one-particle distribution function of the spatially inhomogeneous nonideal gas of classical particles. In contrast to existing approaches, this equation holds for all time scales and takes the influence of pair collisions and initial correlations on the dissipative and nondissipative characteristics of the system into account consistently with the adopted approximation (linear in the gas density). We show that on the kinetic time scale, the time-reversible terms resulting from the initial correlations vanish (if the particle dynamics are endowed with the mixing property) and this equation can be converted into the Vlasov-Landau and Boltzmann equations without any additional commonly used approximations. The entire process of transition can thus be followed from the initial reversible stage of the evolution to the irreversible kinetic stage.

Geometrical effects and signal delay in time-dependent transport at the nanoscale

Nonstationary and steady-state transport through a mesoscopic sample connected to particle reservoirs via time-dependent barriers is investigated by the reduced density operator method. The generalized master equation is solved via the Crank–Nicolson algorithm by taking into account the memory kernel which embodies the non-Markovian effects that are commonly disregarded. The lead–sample coupling takes into account the match between the energy of the incident electrons and the levels of the isolated sample, as well as their overlap at the contacts. Using a tight-binding description of the system, we investigate the effects induced in the transient current by the spectral structure of the sample and by the localization properties of its eigenfunctions. In strong magnetic fields, the transient currents propagate along edge states. The behavior of populations and coherences is discussed, as well as their connection to the tunneling processes that are relevant for transport.

Charge conservation breaking within a generalized master equation description of electronic transport through dissipative double quantum dots

We report an observation of charge conservation breaking in a model study of electronic current noise of transport through a dissipative double quantum dot within a generalized master equation formalism. We study the current noise through a double quantum dot coupled to two electronic leads in the high bias limit and a dissipative heat bath in the weak coupling limit. Our calculations are based on the solution of a Markovian generalized master equation. The zero-frequency component of the current noise calculated within the system, i.e., between the two dots, via the quantum regression theorem exhibits unphysical negative values. On the other hand, the current noise calculated for currents between the dots and the leads by the counting variable approach shows no anomalies and seems physically plausible. We inquire into the origin of this discrepancy between two nominally equivalent approaches and show that it stems from the simultaneous presence of the two types of baths, i.e., the electronic leads and the dissipative bosonic bath. This finding raises interesting questions concerning conceptual foundations of the theory describing multiple-baths open quantum systems widely encountered in nanoscience.

On the markoffian approximation in the description of relaxation in open systems

Most traditional discussions of the Markoffian Born approximation of relaxation phenomena (e.g., Bloch equations) are based on the assumption of widely separated time scales in the non-Markoffian weak coupling generalized master equation. By exploiting the physically reasonable Gaussian factorization property of multitime correlation functions which enter through the system reservoir interaction, recent work has shed significant new light on the relationship between the Markoffian-Born approximation and the exact generalized master equation (GME) describing the evolution of the relaxing subsystem. Two limits for which the GME becomes the Born approximation relaxation equation arc discussed: (1) the Van Hove (λ2t) limit rigorously discussed for open systems by Davies, (2) the zero memory (white noise) suggested by Middleton—Schieve and recently discussed for open systems by Gorini—Kossakawsky and Palmer. We have outlined the proof of this limit. We then discuss the physical meaning of these two exact limits in the context of the general theory of kinetic equations of Resibois. The dominant pole contributions are isolated to all orders by means of an expansion in which the terms are in norm proportional to powers of the ratio of the time scales τc/τB where τc is the two point correlation memory time and τB the relaxation time in the Born approximation. Each of the above exact limits corresponds to the vanishing of this ratio. Comments are made on nonexponential decay.

Event-driven power-law relaxation in weak turbulence: A liquid crystals mesoscopic experiment bridging quantum dots and the integration theory for the brain

We illustrate a theoretical approach to derive generalized master equations, in both quantum and classical physics, compatible with the occurrence of crucial events. We show how to derive out of it a new Fluctuation-Dissipation Theorem (FDT), called phenomenological FDT. We fail deriving from the same approach another FDT form, called dynamic FDT, which is proved by us to drive the response of liquid crystals to external perturbations. We argue that this experimental effect is incompatible with the adoption of a density approach, in the same way as the decoherence theory fails explaining the wave function collapse, if this corresponds to crucial events. We also argue that the same perspective applies to the brain, according to the tenets of the integration theory of consciousness.

Exact decoherence dynamics of a single-mode optical field

We apply the influence-functional method of Feynman and Vernon to the study of a single-mode optical field that interacts with an environment at zero temperature. Using the coherent-state formalism of the path integral, we derive a generalized master equation for the single-mode optical field. Our analysis explicitly shows how non-Markovian effects manifest in the exact decoherence dynamics for different environmental correlation time scales. Remarkably, when these are equal to or greater than the time scale for significant change in the system, the interplay between the backaction-induced coherent oscillation and the dissipative effect of the environment causes the non-Markovian effect to have a significant impact not only on the short-time behavior but also on the long-time steady-state behavior of the system.

Stochastic wave-function unraveling of the generalized Lindblad master equation

Recently a generalized master equation was derived that extends the Lindblad theory to highly non-Markovian quantum processes (H.-P. Breuer, Phys. Rev. A \textbf{75}, 022103 (2007)). We perform a stochastic unravelling 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 (TCL) projection operator method using correlated projection superoperators and the exact solution of the Schrodinger equation for this system.

General theory of excitation energy transfer in donor-mediator-acceptor systems

General theory of the excitation energy transfer (EET) in the case of donor-mediator-acceptor system was constructed by using generalized master equation (GME). In this theory, we consider the direct and indirect transitions in the EET consistently. Hence, our theory includes the quantum mechanical interference between the direct and indirect transitions automatically. Memory functions in the GME were expressed by the overlap integrals among the time-dependent emission spectrum of the donor, the absorption spectrum of the mediator, the time-dependent emission spectrum of the mediator, and the absorption spectrum of the acceptor. In the Markov limit of the memory functions, we obtained the rate of EET which consists of three terms due to the direct transition, the indirect transition, and the interference between them. We found that the interference works effectively in the limit of slow thermalization at the intermediate state. The formula of EET rate in this limit was expressed by the convolution of the EET interaction and optical spectra. The interference effect strongly depends on the width of the absorption spectrum of mediator molecule and the energy gap between the donor and the mediator molecules.

Magnetic Field Effects on Quantum-Dot Spin Valves

We study the magnetic field effects on the spin-polarized transport of the quantum dot (QD) spin valve in the sequential tunneling regime. A set of generalized master equation is derived. Based on that, we discuss the collinear and noncollinear magnetic field effects, respectively. In the collinear magnetic field case, we find that the Zeeman splitting can induce a negative differential conductance (NDC), which is quite different from the one found in previous studies. It has a critical polarization in the parallel arrangement and will disappear in the antiparallel configuration. In the noncollinear magnetic field case, the current shows two plateaus and their angular dependence is analyzed. Although sometimes the two current plateaus have similar angular dependence, their mechanisms are different. Our formalism is also suitable for calculating the transport in magnetic molecules, in which the spin splitting is induced not by a magnetic field but by the intrinsic magnetization.

Continuous-time random walks with internal dynamics and subdiffusive reaction-diffusion equations

We formulate the generalized master equation for a class of continuous-time random walks in the presence of a prescribed deterministic evolution between successive transitions. This formulation is exemplified by means of an advection-diffusion and a jump-diffusion scheme. Based on this master equation, we also derive reaction-diffusion equations for subdiffusive chemical species, using a mean-field approximation.

Anomalous transport in a one-dimensional Lorentz gas model

Employing the generalized master equation proposed in [R. Friedrich et al., Phys. Rev. Lett. 96, 230601 (2006)], we derive a kinetic equation for a random kick model. For a particular choice of the time evolution kernel, a fractional master equation is obtained, which can be related to a Levy walk. In one dimension, we use this model to describe a stochastic Lorentz gas with an annealed disorder. Exact moment relations are obtained in Laplace space, and the long-time behavior of the moments is discussed. The results are compared to those of related models.

Higher-order processes of excitation energy transfer in supramolecular complexes: Liouville space analysis of bridge molecule mediated transfer and direct photon exchange

Long-range electronic excitation energy transfer is studied in the framework of a generalized master equation approach, which offers a systematic account for higher-order processes. Bridge molecule mediated transfer is confronted with the direct excitation energy exchange via photon emission and absorption. It is the central aim of this paper to present a systematic study of fourth-order rates, which describe both types of transfer processes characterized by the presence of intermediate states. While such a Liouville space formulation of rates is known from bridge mediated transfer, it is new for the case of photon mediated processes. In the former case, however, a novel approach to account for intermediate state relaxation is introduced. Finally and for illustration, some estimates are offered for the length dependence of the various discussed transfer rates.

Angular anisotropy of characteristic x-radiation and auger electrons during the resonance coherent excitation of relativistic ions under planar channeling conditions

The results of new experiments on the resonance coherent excitation of relativistic highly charged ions under planar channeling conditions are analyzed by a method of the generalized master equation for the density matrix of ions. The results of calculations pertaining to the angular anisotropy of characteristic electromagnetic radiation, to the survival fraction of ions as a function of the trajectory of their motion in the channel, and to the excitation of autoionizing states of helium-like ions under the double resonance coherent excitation are in close agreement with measurement data. New experiments are proposed.

Waiting times and noise in single particle transport

AbstractThe waiting time distribution w(τ), i.e. the probability for a delay τ between two subsequent transition (‘jumps’) of particles, is a statistical tool in (quantum) transport. Using generalized Master equations for systems coupled to external particle reservoirs, one can establish relations between w(τ) and other statistical transport quantities such as the noise spectrum and the Full Counting Statistics. It turns out that w(τ) usually contains additional information on system parameters and properties such as quantum coherence, the number of internal states, or the entropy of the current channels that participate in transport.

Waiting times and noise in single particle transport

The waiting time distribution $w(\tau)$, i.e. the probability for a delay $\tau$ between two subsequent transition (`jumps') of particles, is a statistical tool in (quantum) transport. Using generalized Master equations for systems coupled to external particle reservoirs, one can establish relations between $w(\tau)$ and other statistical transport quantities such as the noise spectrum and the Full Counting Statistics. It turns out that $w(\tau)$ usually contains additional information on system parameters and properties such as quantum coherence, the number of internal states, or the entropy of the current channels that participate in transport.

Irreversible evolution of many-electron systems coupled to a statistical environment

We study the evolution of a many-electron system that is confined in a finite spatial region and coupled to a statistical environment. The latter may be composed of several independent bath subsystems, which are held at some statistical equilibrium. From the master equations describing the evolution of the coarse-grained $N$-particle density matrix, we obtain the equations, which describe the evolution of the $n$-particle density operators ${\overline{D}}^{(n)}$ for $n<N$. These equations are hierarchically coupled through the electron-electron interaction. We show that the hierarchy can be truncated under the assumption that the residual interaction of the electrons in the considered system with the environment introduces a memory loss, which hinders the electronic system to build up more than two-particle correlations. We first consider a weakly excited electronic system, which looses memory but where energy exchange with the statistical environment can be neglected. This is the quantum analog of the classical Boltzmann gas in a box. We derive the master equations, which describe the irreversible evolution of the coarse-grained one-particle density matrix. Based on this result, we show that, in accord with the second law of thermodynamics, the corresponding von Neumann entropy either increases with time or it remains constant. Finally, allowing also for energy exchange with one or more bath subsystems provided, e.g., by phonons or photons, we obtain the corresponding general master equations that describe the evolution of a spatially confined interacting electron gas of metallic density.

Optical field control of charge transmission through a molecular wire. I. Generalized master equation description

A generalized master equation description of charge transmission through a single molecule that goes beyond a perturbative treatment of electron-vibrational coupling and accounts for nonequilibrium vibrational distributions is presented. By utilizing a projection superoperator formalism, exact expressions for the generalized master equation kernels that include intramolecular vibrational energy redistribution, excited electronic levels, as well as their optical excitation and de-excitation are derived. This enables one to study optical current switching in the case of weak electron-lead coupling and large reorganization energy upon charging. Additionally, it is possible to quantify the influence of intramolecular vibrational energy redistribution and excited state de-excitation on the current switching.

Transport of strong-coupling polarons in optical lattices

We study the transport of ultracold impurity atoms immersed in a Bose–Einstein condensate (BEC) and trapped in a tight optical lattice. Within the strong-coupling regime, we derive an extended Hubbard model describing the dynamics of the impurities in terms of polarons, i.e. impurities dressed by a coherent state of Bogoliubov phonons. Using a generalized master equation based on this microscopic model, we show that inelastic and dissipative phonon scattering results in (i) a crossover from coherent to incoherent transport of impurities with increasing BEC temperature and (ii) the emergence of a net atomic current across a tilted optical lattice. The dependence of the atomic current on the lattice tilt changes from ohmic conductance to negative differential conductance within an experimentally accessible parameter regime. This transition is accurately described by an Esaki–Tsu-type relation with the effective relaxation time of the impurities as a temperature-dependent parameter.

Diffusion in a time-dependent external field

The problem of diffusion in a time-dependent (and generally inhomogeneous) external field is considered on the basis of a generalized master equation with two times, introduced by Trigger and co-authors [S. A. Trigger, G. J. F. van Heijst, and P. P. J. M. Schram, Physica A 347, 77 (2005); J. Phys.: Conf. Ser. 11, 37 (2005)]. We consider the case of the quasi-Fokker-Planck approximation, when the probability transition function for diffusion (PTD function) does not possess a long tail in coordinate space and can be expanded as a function of instantaneous displacements. The more complicated case of long tails in the PTD will be discussed separately. We also discuss diffusion on the basis of hydrodynamic and kinetic equations and show the validity of the phenomenological approach. A type of "collision" integral is introduced for the description of diffusion in a system of particles, which can transfer from a moving state to the rest state (with some waiting time distribution). The solution of the appropriate kinetic equation in the external field also confirms the phenomenological approach of the generalized master equation.