Exact quantum master equation via the calculus on path integrals

An exact quantum master equation formalism is constructed for the efficient evaluation of quantum non-Markovian dissipation beyond the weak system-bath interaction regime in the presence of time-dependent external field. A novel truncation scheme is further proposed and compared with other approaches to close the resulting hierarchically coupled equations of motion. The interplay between system-bath interaction strength, non-Markovian property, and required level of hierarchy is also demonstrated with the aid of simple spin-boson systems.     

Markovian approximation in the relaxation of open quantum systems

In this paper, we examine the validity of the Markovian approximation and the slippage scheme used to incorporate short time transient memory effects in the Markovian master equations (Redfield equations). We argue that for a bath described by a spectral function, J(omega), that is dense and smoothly spread out over the range omega(d), a time scale of tau(b) approximately 1/omega(d) exists; for times of t > tau(b), the Markovian approximation is applicable. In addition, if J(omega) decays to zero reasonably fast in both the omega --> 0 and omega --> infinity limits, then the bath relaxation time, tau(b), is determined by the width of the spectral function and is weakly dependent on the temperature of the bath. On the basis of this criterion of tau(b), a scheme to incorporate transient memory effects in the Markovian master equation is suggested. Instead of using slipped initial conditions, we propose a concatenation scheme that uses the second-order perturbation theory for short time dynamics and the Markovian master equation at long times. Application of this concatenation scheme to the spin-boson model shows that it reproduces the reduced dynamics obtained from the non-Markovian master equation for all parameters studied, while the simple slippage scheme breaks down at high temperatures.     

Padé spectrum decompositions of quantum distribution functions and optimal hierarchical equations of motion construction for quantum open systems

Pade? spectrum decomposition is an optimal sum-over-poles expansion scheme of Fermi function and Bose function [J. Hu, R. X. Xu, and Y. J. Yan, J. Chem. Phys. 133, 101106 (2010)]. In this work, we report two additional members to this family, from which the best among all sum-over-poles methods could be chosen for different cases of application. Methods are developed for determining these three Pade? spectrum decomposition expansions at machine precision via simple algorithms. We exemplify the applications of present development with optimal construction of hierarchical equations-of-motion formulations for nonperturbative quantum dissipation and quantum transport dynamics. Numerical demonstrations are given for two systems. One is the transient transport current to an interacting quantum-dots system, together with the involved high-order co-tunneling dynamics. Another is the non-Markovian dynamics of a spin-boson system.     

Quantum two-state dynamics driven by stationary non-Markovian discrete noise: Exact results

We consider the problem of stochastic averaging of a quantum two-state dynamics driven by non-Markovian, discrete noises of the continuous time random walk type (multistate renewal processes). The emphasis is put on the proper averaging over the stationary noise realizations corresponding, e.g., to a stationary environment. A two-state non-Markovian process with an arbitrary non-exponential distribution of residence times (RTDs) in its states with a finite mean residence time provides a paradigm. For the case of a two-state quantum relaxation caused by such a classical stochastic field we obtain the explicit exact, analytical expression for the averaged Laplace-transformed relaxation dynamics. In the limit of Markovian noise (implying an exponential RTD), all previously known results are recovered. We exemplify new more general results for the case of non-Markovian noise with a biexponential RTD. The averaged, real-time relaxation dynamics is obtained in this case by numerically exact solving of a resulting algebraic polynomial problem. Moreover, the case of manifest non-Markovian noise with an infinite range of temporal autocorrelation (which in principle is not accessible to any kind of perturbative treatment) is studied, both analytically (asymptotic long-time dynamics) and numerically (by a precise numerical inversion of the Laplace-transformed averaged quantum relaxation).     

Propagators and related descriptors for non-Markovian asymmetric random walks with and without boundaries

There are many current applications of the continuous-time random walk (CTRW), particularly in describing kinetic and transport processes in different chemical and biophysical phenomena. We derive exact solutions for the Laplace transforms of the propagators for non-Markovian asymmetric one-dimensional CTRW's in an infinite space and in the presence of an absorbing boundary. The former is used to produce exact results for the Laplace transforms of the first two moments of the displacement of the random walker, the asymptotic behavior of the moments as t-->infinity, and the effective diffusion constant. We show that in the infinite space, the propagator satisfies a relation that can be interpreted as a generalized fluctuation theorem since it reduces to the conventional fluctuation theorem at large times. Based on the Laplace transform of the propagator in the presence of an absorbing boundary, we derive the Laplace transform of the survival probability of the random walker, which is then used to find the mean lifetime for terminated trajectories of the random walk.     

Fluctuation theorem for channel-facilitated membrane transport of interacting and noninteracting solutes

In this paper, we discuss the fluctuation theorem for channel-facilitated transport of solutes through a membrane separating two reservoirs. The transport is characterized by the probability, P(n)(t), that n solute particles have been transported from one reservoir to the other in time t. The fluctuation theorem establishes a relation between P(n)(t) and P-(n)(t): The ratio P(n)(t)/P-(n)(t) is independent of time and equal to exp(nbetaA), where betaA is the affinity measured in the thermal energy units. We show that the same fluctuation theorem is true for both single- and multichannel transport of noninteracting particles and particles which strongly repel each other.     

Generalized Langevin dynamics of a nanoparticle using a finite element approach: thermostating with correlated noise

A direct numerical simulation (DNS) procedure is employed to study the thermal motion of a nanoparticle in an incompressible Newtonian stationary fluid medium with the generalized Langevin approach. We consider both the Markovian (white noise) and non-Markovian (Ornstein-Uhlenbeck noise and Mittag-Leffler noise) processes. Initial locations of the particle are at various distances from the bounding wall to delineate wall effects. At thermal equilibrium, the numerical results are validated by comparing the calculated translational and rotational temperatures of the particle with those obtained from the equipartition theorem. The nature of the hydrodynamic interactions is verified by comparing the velocity autocorrelation functions and mean square displacements with analytical results. Numerical predictions of wall interactions with the particle in terms of mean square displacements are compared with analytical results. In the non-Markovian Langevin approach, an appropriate choice of colored noise is required to satisfy the power-law decay in the velocity autocorrelation function at long times. The results obtained by using non-Markovian Mittag-Leffler noise simultaneously satisfy the equipartition theorem and the long-time behavior of the hydrodynamic correlations for a range of memory correlation times. The Ornstein-Uhlenbeck process does not provide the appropriate hydrodynamic correlations. Comparing our DNS results to the solution of an one-dimensional generalized Langevin equation, it is observed that where the thermostat adheres to the equipartition theorem, the characteristic memory time in the noise is consistent with the inherent time scale of the memory kernel. The performance of the thermostat with respect to equilibrium and dynamic properties for various noise schemes is discussed.     

On stochastic models of dynamic disorder

In this paper we investigate some general aspects of stochastic models of dynamic disorder. First, we reexamine the Zwanzig model for the kinetics of escape through a fluctuating hole. We show that this model is trivially connected to the canonical model of the broadening of the zero-phonon line (ZPL) in crystals. This provides a new perspective of the Wang-Wolynes expression for the rate of escape from a geometric bottleneck with non-Markovian Gaussian fluctuations. Motivated by recent single-molecule experiments, we examine more general examples of fluctuation processes from the perspective of cumulant expansions. Finally, we discuss recent single-molecule experiments probing enzyme turnover performed by Xie and co-workers.     

The quantum solvation, adiabatic versus nonadiabatic, and Markovian versus non-Markovian nature of electron-transfer rate processes

In this work, we revisit the electron-transfer rate theory, with particular interests in the distinct quantum solvation effect and the characterizations of adiabatic/nonadiabatic and Markovian/non-Markovian rate processes. We first present a full account for the quantum solvation effect on the electron transfer in Debye solvents, addressed previously in J. Theor. Comput. Chem. 2006, 5, 685. Distinct reaction mechanisms, including the quantum solvation-induced transitions from barrier crossing to tunneling and from barrierless to quantum barrier crossing rate processes, are shown in the fast modulation or low viscosity regime. This regime is also found in favor of nonadiabatic rate processes. We further propose to use Kubo's motional narrowing line shape function to describe the Markovian character of the reaction. It is found that a non-Markovian rate process is most likely to occur in a symmetric system in the fast modulation regime, where the electron transfer is dominant by tunneling due to the Fermi resonance.     

Statistical mechanical theory for steady state systems. V. Nonequilibrium probability density

The phase space probability density for steady heat flow is given. This generalizes the Boltzmann distribution to a nonequilibrium system. The expression includes the nonequilibrium partition function, which is a generating function for statistical averages and which can be related to a nonequilibrium free energy. The probability density is shown to give the Green-Kubo formula in the linear regime. A Monte Carlo algorithm is developed based upon a Metropolis sampling of the probability distribution using an umbrella weight. The nonequilibrium simulation scheme is shown to be much more efficient for the thermal conductivity of a Lennard-Jones fluid than the Green-Kubo equilibrium fluctuation method. The theory for heat flow is generalized to give the generic nonequilibrium probability densities for hydrodynamic transport, for time-dependent mechanical work, and for nonequilibrium quantum statistical mechanics.     

Effects of Configurational Fluctuation on Electronic Coupling for Charge Transfer Dynamics

We advance a theory for the effects of bridge configurational fluctuations on the electronic coupling for electron transfer reactions in donor-bridge-acceptor systems. The theory of radiationless transitions was applied for activationless electron transfer, where the nuclear Franck–Condon constraints are minimized, with the initial vibronic state interacting directly with the final vibronic manifold, without the need for thermal activation. Invoking the assumption of energy-independent coupling, the time-dependent initial state population probability was analyzed in terms of a cumulant expansion. Two limiting situations were distinguished, i.e. the fast configurational fluctuation limit, where the electron transfer rate is given in terms of the configurational average of me squared electronic coupling, and the slow configurational fluctuation limit, where the dynamics is determined by a configurational averaging over a static distribution of electron transfer probability densities. The correlation times for configurational fluctuations of the electronic coupling will be obtained from the analysis of molecular dynamics, in conjunction with quantum mechanical calculations of the electronic coupling, to establish the appropriate limit for electron transfer dynamics.     

Nonperturbative vibrational energy relaxation effects on vibrational line shapes

A general formulation of nonperturbative quantum dynamics of solutes in a condensed phase is proposed to calculate linear and nonlinear vibrational line shapes. In the weak solute-solvent interaction limit, the temporal absorption profile can be approximately factorized into the population relaxation profile from the off-diagonal coupling and the pure-dephasing profile from the diagonal coupling. The strength of dissipation and the anharmonicity-induced dephasing rate are derived in Appendix A. The vibrational energy relaxation (VER) rate is negligible for slow solvent fluctuations, yet it does not justify the Markovian treatment of off-diagonal contributions to vibrational line shapes. Non-Markovian VER effects are manifested as asymmetric envelops in the temporal absorption profile, or equivalently as side bands in the frequency domain absorption spectrum. The side bands are solvent-induced multiple-photon effects which are absent in the Markovian VER treatment. Exact path integral calculations yield non-Lorentzian central peaks in absorption spectrum resulting from couplings between population relaxations of different vibrational states. These predictions cannot be reproduced by the perturbative or the Markovian approximations. For anharmonic potentials, the absorption spectrum shows asymmetric central peaks and the asymmetry increases with anharmonicity. At large anharmonicities, all the approximation schemes break down and a full nonperturbative path integral calculation that explicitly accounts for the exact VER effects is needed. A numerical analysis of the O-H stretch of HOD in D(2)O solvent reveals that the non-Markovian VER effects generate a small recurrence of the echo peak shift around 200 fs, which cannot be reproduced with a Markovian VER rate. In general, the nonperturbative and non-Markovian VER contributions have a stronger effect on nonlinear vibrational line shapes than on linear absorption.     

Non-Markovian stochastic Schrödinger equations in different temperature regimes: a study of the spin-boson model

Stochastic Schrodinger equations are used to describe the dynamics of a quantum open system in contact with a large environment, as an alternative to the commonly used master equations. We present a study of the two main types of non-Markovian stochastic Schrodinger equations, linear and nonlinear ones. We compare them both analytically and numerically, the latter for the case of a spin-boson model. We show in this paper that two linear stochastic Schrodinger equations, derived from different perspectives by Diosi, Gisin, and Strunz [Phys. Rev. A 58, 1699 (1998)], and Gaspard and Nagaoka [J. Chem. Phys. 13, 5676 (1999)], respectively, are equivalent in the relevant order of perturbation theory. Nonlinear stochastic Schrodinger equations are in principle more efficient than linear ones, as they determine solutions with a higher weight in the ensemble average which recovers the reduced density matrix of the quantum open system. However, it will be shown in this paper that for the case of a spin-boson system and weak coupling, this improvement does only occur in the case of a bath at high temperature. For low temperatures, the sampling of realizations of the nonlinear equation is practically equivalent to the sampling of the linear ones. We study further this result by analyzing, for both temperature regimes, the driving noise of the linear equations in comparison to that of the nonlinear equations.     

Theoretical analysis and computer simulation of fluorescence lifetime measurements. I. Kinetic regimes and experimental time scales

The configuration-controlled regime and the diffusion-controlled regime of conformation-modulated fluorescence emission are systematically studied for Markovian and non-Markovian dynamics of the reaction coordinate. A path integral simulation is used to model fluorescence quenching processes on a semiflexible chain. First-order inhomogeneous cumulant expansion in the configuration-controlled regime defines a lower bound for the survival probability, while the Wilemski-Fixman approximation in the diffusion-controlled regime defines an upper bound. Inclusion of the experimental time window of the fluorescence measurement adds another dimension to the two kinetic regimes and provides a unified perspective for theoretical analysis and experimental investigation. We derive a rigorous generalization of the Wilemski-Fixman approximation [G. Wilemski and M. Fixman, J. Chem. Phys. 60, 866 (1974)] and recover the 1/D expansion of the average lifetime derived by Weiss [G. H. Weiss, J. Chem. Phys. 80, 2880 (1984)].     

非平衡体系-热库纠缠定理与热输运

将最近建立的体系-热库纠缠定理(SBET)扩展到非平衡的情形. 其中, 任意体系与处于不同温度的多个高斯型热库环境相耦合. 现有的SBET将体系-热库的纠缠响应函数与体系的局域响应函数联系起来, 而扩展的理论则关注通过分子结的非平衡稳态量子输运流. 新理论是基于广义Langevin方程建立的, 它与量子情形下的非平衡热力学密切相关.     

Calculation of reactive flux correlation functions for systems in a condensed phase environment: a multilayer multiconfiguration time-dependent Hartree approach

A numerically exact quantum mechanical approach is proposed to evaluate thermal rate constants for systems in a model condensed phase environment. Employing the reactive flux correlation function formalism, the approach efficiently combines the multilayer multiconfiguration time-dependent Hartree theory with an importance sampling scheme for thermal distribution of the initial states. The performance of the method is illustrated by applications to two models of condensed phase dynamics: the donor-acceptor electron transfer model also known as the spin-boson model and a model for proton transfer reactions in the condensed phase.     

Dynamics of quantum dissipation systems interacting with fermion and boson grand canonical bath ensembles: hierarchical equations of motion approach

A hierarchical equations of motion formalism for a quantum dissipation system in a grand canonical bath ensemble surrounding is constructed on the basis of the calculus-on-path-integral algorithm, together with the parametrization of arbitrary non-Markovian bath that satisfies fluctuation-dissipation theorem. The influence functionals for both the fermion or boson bath interaction are found to be of the same path integral expression as the canonical bath, assuming they all satisfy the Gaussian statistics. However, the equation of motion formalism is different due to the fluctuation-dissipation theories that are distinct and used explicitly. The implications of the present work to quantum transport through molecular wires and electron transfer in complex molecular systems are discussed.