Generalized Master Equation Research Articles

Ward identity for nonequilibrium Fermi systems

A nonequilibrium Ward identity (NE WI) connecting the scalar transport vertex correction with one-particle self-energy is derived using the global $\mathrm{U}(1)$ symmetry of the fermion nonequilibrium Green's functions (NGF). The nonperturbative derivation does not depend on the details of the many-body system. A renormalized multiplicative composition rule for the NGF, reflecting time coherence, is obtained and related to the NE WI. Applications involve (i) testing the consistency of approximations shown in the example of a self-consistent Born approximation for disorder scattering, and (ii) in the general quantum transport theory, the formalism permits one to assess routes to generalized master equations, in particular those based on the generalized Kadanoff-Baym ansatz.

Vibrational Coherences in Single Electron Tunneling through Nanoscale Oscillators

Charging a nanoscale oscillator by single electron tunneling leads to an effective double-well potential due to image charges. We combine exact numerical diagonalizations with generalized master equations and show that the resulting quantum tunneling of the mechanical degree of freedom can be visualized in the electronic current noise spectrum.

Density fluctuation of the3He reaction—diffusion model in nonextensive statistical mechanics

In this paper the time-space correlation of density fluctuation of the 3He reaction–diffusion model is investigated where the equilibrium distribution is described by the generalized Maxwell–Boltzmann distribution in the framework of Tsallis statistics. By using the density operator technique, the nonextensive pressure effect is introduced into the master equation and thus the generalized master equation is derived for the nonextensive system. This paper uses the 3He reaction–diffusion model to analyse the effect of nonextensive pressure on the fluctuation and finds that the nonextensive parameter q plays a very important role in determining the characteristics of the fluctuation waves.

Advanced Theory of Excitation Energy Transfer in Dimers

Previously, we developed a unified theory of the excitation energy transfer (EET) in dimers, which is applicable to all of the cases of excitonic coupling strength (Kimura, A.; Kakitani, T.; Yamato, T. J. Phys. Chem. B 2000, 104, 9276). This theory was formulated only for the forward reaction of the EET. In the present paper, we advanced this theory so that it might include the backward reaction of the EET as well as the forward reaction. This new theory is formulated on the basis of the generalized master equation (GME), without using physically unclear assumptions. Comparing the present result with the previous one, we find that the excitonic coupling strengths of criteria between exciton and partial exciton and between hot transfer and hopping (Förster) mechanisms are reduced by a factor of 2. The critical coherency eta c is also reduced significantly.

Spin-boson dynamics beyond conventional perturbation theories

A generalized approximation scheme is proposed to describe the dynamics of the spin-boson problem. Being nonperturbative in the coupling strength nor in the tunneling frequency, it gives reliable results over a wide regime of temperatures and coupling strength to the thermal environment for a large class of bath spectral densities. We use a path-integral approach and start from the exact solution for the two-level system population difference in the form of a generalized master equation (GME). Then, we approximate interblip and blip-sojourn interactions up to linear order, while retaining all intrablip correlations to find the kernels entering the GME in analytical form. Our approximation scheme, which we call weakly interacting blip approximation, fully agrees with conventional perturbative approximations in the tunneling matrix element (noninteracting-blip approximation) or in the system-bath coupling strength in the proper parameter regime.

Continuous time random walks in periodic systems: fluid limit and fractional differential equations on the circle

In this paper, the continuous time random walk on the circle is studied. We derive the corresponding generalized master equation and discuss the effects of topology, especially important when Lévy flights are allowed. Then, we work out the fluid limit equation, formulated in terms of the periodic version of the fractional Riemann–Liouville operators, for which we provide explicit expressions. Finally, we compute the propagator in some simple cases. The analysis presented herein should be relevant when investigating anomalous transport phenomena in systems with periodic dimensions.

Dynamics of decoherence without dissipation in a squeezed thermal bath

We study a generic open quantum system where the coupling between the system and its environment is of an energy-preserving quantum nondemolition (QND) type. We obtain the general master equation for the evolution of such a system under the influence of a squeezed thermal bath of harmonic oscillators. From the master equation it can be seen explicitly that the process involves decoherence or dephasing without any dissipation of energy. We work out the decoherence-causing term in the high- and zero-temperature limits and check that they match with known results for the case of a thermal bath. The decay of the coherence is quantified as well by the dynamics of the linear entropy of the system under various environmental conditions. We make a comparison of the quantum statistical properties between QND and dissipative types of evolution using a two-level atomic system and a harmonic oscillator.

Numerical algorithm for the time fractional Fokker–Planck equation

Anomalous diffusion is one of the most ubiquitous phenomena in nature, and it is present in a wide variety of physical situations, for instance, transport of fluid in porous media, diffusion of plasma, diffusion at liquid surfaces, etc. The fractional approach proved to be highly effective in a rich variety of scenarios such as continuous time random walk models, generalized Langevin equations, or the generalized master equation. To investigate the subdiffusion of anomalous diffusion, it would be useful to study a time fractional Fokker–Planck equation. In this paper, firstly the time fractional, the sense of Riemann–Liouville derivative, Fokker–Planck equation is transformed into a time fractional ordinary differential equation (FODE) in the sense of Caputo derivative by discretizing the spatial derivatives and using the properties of Riemann–Liouville derivative and Caputo derivative. Then combining the predictor–corrector approach with the method of lines, the algorithm is designed for numerically solving FODE with the numerical error O ( k min { 1 + 2 α , 2 } ) + O ( h 2 ) , and the corresponding stability condition is got. The effectiveness of this numerical algorithm is evaluated by comparing its numerical results for α = 1.0 with the ones of directly discretizing classical Fokker–Planck equation, some numerical results for time fractional Fokker–Planck equation with several different fractional orders are demonstrated and compared with each other, moreover for α = 0.8 the convergent order in space is confirmed and the numerical results with different time step sizes are shown.

Properties of the generalized master equation: Green’s functions and probability density functions in the path representation

The Green's function for the master equation and the generalized master equation in path representation is an infinite sum over the length of path probability density functions (PDFs). In this paper, the properties of path PDFs are studied both qualitatively and quantitatively. The results are used in building efficient approximations for Green's function in 1D, and are relevant in modeling and in data analysis.

Continuous source of phase-controlled entangled two-photon laser

We show that an absolute coherent phase of a laser can be used to manipulate the entanglement of photon pairs in two-photon laser. We present simple physics behind a general master equation for two-photon laser. Our focus is on the generation of a continuous source of macroscopically entangled photon pairs in the double $\ensuremath{\Lambda}$ (or Raman) scheme. We show how the steady-state photon numbers and entanglement depend on the laser parameters, especially the phase. We obtain a relationship between entanglement and two-photon correlation. We derive conditions that give steady-state entanglement for the spontaneous Raman-electromagnetic-induced transparency scheme and use it to identify the region with macroscopic entanglement. No entanglement is found for the double resonant Raman scheme.

Phonon-mediated decay of an atom in a surface-induced potential

We study phonon-mediated transitions between translational levels of an atom in a surface-induced potential. We present a general master equation governing the dynamics of the translational states of the atom. In the framework of the Debye model, we derive compact expressions for the rates for both upward and downward transitions. Numerical calculations for the transition rates are performed for a deep silica-induced potential allowing for a large number of bound levels as well as free states of a cesium atom. The total absorption rate is shown to be determined mainly by the bound-to-bound transitions for deep bound levels and by bound-to-free transitions for shallow bound levels. Moreover, the phonon emission and absorption processes can be orders of magnitude larger for deep bound levels as compared to the shallow bound ones. We also study various types of transitions from free states. We show that, for thermal atomic cesium with a temperature in the range from $100\phantom{\rule{0.3em}{0ex}}\ensuremath{\mu}\mathrm{K}\phantom{\rule{0.3em}{0ex}}\text{to}\phantom{\rule{0.3em}{0ex}}400\phantom{\rule{0.3em}{0ex}}\ensuremath{\mu}\mathrm{K}$ in the vicinity of a silica surface with a temperature of $300\phantom{\rule{0.3em}{0ex}}\mathrm{K}$, the adsorption (free-to-bound decay) rate is about two times larger than the heating (free-to-free upward decay) rate, while the cooling (free-to-free downward decay) rate is negligible.

Fluctuation in nonextensive reaction–diffusion systems

The density fluctuation in a nonextensive reaction–diffusion system is investigated, where the nonequilibrium stationary-state distribution is described by the generalized Maxwell–Boltzmann distribution in the framework of Tsallis statistics (or nonextensive statistics). By using the density operator technique, the nonextensive pressure effect is introduced into the master equation and thus the generalized master equation is derived for the system. As an example, we take the3He reaction–diffusion model inside stars to analyse the nonextensive effect on the density fluctuation and we find that the nonextensive parameter q different from one plays a very important role in determining the characteristics of the fluctuation waves.

Continuous-time random walks in an oscillating field: Field-induced dispersion and the death of linear response

Abstract We discuss the response of continuous-time random walks to an oscillating external field within the generalized master equation approach. We concentrate on the time dependence of the first two moments of the walker’s displacement and show that for power-law waiting-time distributions with 0 α 1 (corresponding to a semi-Markovian situation displaying aging) the mean particle position tends to a constant, while the response to the external perturbation dies out. The oscillating field is shown to lead to an additional new contribution to the dispersion of the particle’s position which is proportional to the square of its amplitude and grows with time.

GENERALIZED MASTER EQUATION FOR NON-EXTENSIVE REACTION-DIFFUSION SYSTEMS

Based on the theory of non-equilibrium statistics and density operator equation, the generalized master equation satisfied by characteristic function for the non-extensive reaction-diffusion systems affected by pressure is derived by calculating the time variation of probability distribution function, where the pressure of the non-extensive systems is given in the framework of Tsallis statistics. This new equation not only depends on the non-extensive parameter but also has more nonlinear terms as compared with the master equation in the phenomenological theory and thus has more generality.

A two-dimensional inverse parabolic potential within the Lindblad theory for application in nuclear reactions

Using Lindblad's axiomatic approach for the description of dissipation in quantum mechanics, we describe the dynamics of an open quantum system with a two-dimensional inverse parabolic potential. The derivation of the equations of motion for the first two moments of the quantum system from the general Markovian master equation is shown and their solution is given. We analyse the scattering of a Gaussian wave package in the considered potential and present an analytical expression for the probability density in position space and numerical results for the probabilities. A possible application of the developed formalism is shown for the nuclear physics example of the collision of 48Ca on 248Cm.

Field-Induced Dispersion in Subdiffusion

We discuss the response of continuous-time random walks to an oscillating external field within the generalized master equation approach. We concentrate on the time dependence of the two first moments of the walker's displacement. We show that for power-law waiting-time distributions with 0<alpha<1 corresponding to a semi-Markovian situation showing nonstationarity, the mean particle position tends to a constant; namely, the response to the external perturbation dies out. On the other hand, the oscillating field leads to a new additional contribution to the dispersion of the particle position, proportional to the square of its amplitude and growing with time. These new effects, amenable to experimental observation, result directly from the nonstationary property of the system.

Time evolution of non-Hermitian quantum systems and generalized master equations

We inquire into the time evolution of quantum systems associated with pseudo-or quasi-Hermitian Hamiltonians. We obtain, in the pseudo-Hermitian case, a generalized Liouville-von Neumann equation for closed systems. We show that quantum systems with quasi-Hermitian Hamiltonians admit the proper interpretation in terms of open quantum system and derive a generalized Lindblad-Kossakowski equation. Finally, we extend such formalism to the study of decaying systems.

Decoherence in Holonomic Quantum Computation

Geometric phases are an interesting field of research in quantum mechanics. Recently both abelian and nonabelian geometric phases have been proposed as a useful resource for the experimental implementation of quantum computation. In this paper we focus on a particular physical model and study the effect of a bosonic bath on a class of holonomic transformations. We write a general master equation for time-dependent Hamiltonians and derive analytical and numerical solutions for the system considered. The fidelity is analyzed in the adiabatic and nonadiabatic regime. We also determine an optimal finite operation time for this class of gates.

Linear response to perturbation of nonexponential renewal process: A generalized master equation approach

The work by Barbi, Bologna, and Grigolini [Phys. Rev. Lett. 95, 220601 (2005)] discusses a response to alternating external field of a non-Markovian two-state system, where the waiting time between the two attempted changes of state follows a power law. It introduced a new instrument for description of such situations based on a stochastic master equation with reset. In the present Brief Report we provide an alternative description of the situation within the framework of a generalized master equation. The results of our analytical approach are corroborated by direct numerical simulations of the system.

Efficient computation of the first passage time distribution of the generalized master equation by steady-state relaxation

The generalized master equation or the equivalent continuous time random walk equations can be used to compute the macroscopic first passage time distribution (FPTD) of a complex stochastic system from short-term microscopic simulation data. The computation of the mean first passage time and additional low-order FPTD moments can be simplified by directly relating the FPTD moment generating function to the moments of the local FPTD matrix. This relationship can be physically interpreted in terms of steady-state relaxation, an extension of steady-state flow. Moreover, it is amenable to a statistical error analysis that can be used to significantly increase computational efficiency. The efficiency improvement can be extended to the FPTD itself by modelling it using a gamma distribution or rational function approximation to its Laplace transform.