Showing papers on "Master equation published in 1999"

Mixed quantum-classical dynamics

Abstract: Mixed quantum-classical equations of motion are derived for a quantum subsystem of light (mass m) particles coupled to a classical bath of massive (mass M) particles. The equation of motion follows from a partial Wigner transform over the bath degrees of freedom of the Liouville equation for the full quantum system, followed by an expansion in the small parameter μ=(m/M)1/2 in analogy with the theory of Brownian motion. The resulting mixed quantum-classical Liouville equation accounts for the coupled evolution of the subsystem and bath. The quantum subsystem is represented in an adiabatic (or other) basis and the series solution of the Liouville equation leads to a representation of the dynamics in an ensemble of surface-hopping trajectories. A generalized Pauli master equation for the evolution of the diagonal elements of the density matrix is derived by projection operator methods and its structure is analyzed in terms of surface-hopping trajectories.

Non-Markovian evolution of the density operator in the presence of strong laser fields

Abstract: We present an accurate, efficient, and flexible method for propagating spatially distributed density matrices in anharmonic potentials interacting with solvent and strong fields. The method is based on the Nakajima–Zwanzig projection operator formalism with a correlated reference state of the bath that takes memory effects and initial/final correlations to second order in the system–bath interaction into account. A key feature of the method proposed is a special parametrization of the bath spectral density leading to a set of coupled equations for primary and N auxiliary density matrices. These coupled master equations can be solved numerically by representing the density operator in eigenrepresentation or on a coordinate space grid, using the Fourier method to calculate the action of the kinetic and potential energy operators, and a combination of split operator and Cayley implicit method to compute the time evolution. The key advantages of the method are: (1) The system potential may consist of any numb...

Deriving fractional Fokker-Planck equations from a generalised master equation

Abstract: A generalised master equation is constructed from a non-homogeneous random walk scheme. It is shown how fractional Fokker-Planck equations for the description of anomalous diffusion in external fields, recently proposed in the literature, can be derived from this framework. Long-tailed waiting time distributions which cause slowly decaying memory effects, are demonstrated to give rise to a time-fractional Fokker-Planck equation that describes systems close to thermal equilibrium. An extension to include also Levy flights leads to a generalised Laplacian in the corresponding fractional Fokker-Planck equation.

Stochastic wave function method for non-markovian quantum master equations

Abstract: A generalization of the stochastic wave-function method to quantum master equations which are not in Lindblad form is developed. The proposed stochastic unraveling is based on a description of the reduced system in a doubled Hilbert space and it is shown that this method is capable of simulating quantum master equations with negative transition rates. Non-Markovian effects in the reduced systems dynamics can be treated within this approach by employing the time-convolutionless projection operator technique. This ansatz yields a systematic perturbative expansion of the reduced systems dynamics in the coupling strength. Several examples such as the damped Jaynes-Cummings model and the spontaneous decay of a two-level system into a photonic band gap are discussed. The power as well as the limitations of the method are demonstrated.

Approaches to the approximate treatment of complex molecular systems by the multiconfiguration time-dependent Hartree method

Abstract: A consistent treatment of environmental effects is proposed in the framework of the multiconfiguration time-dependent Hartree (MCTDH) method. The method is extended in view of treating complex molecular systems which require an exact quantum dynamics for a certain number of “primary” modes while an approximate dynamics is adequate for a class of “secondary” modes. The latter may correspond to the weakly coupled modes in a polyatomic molecule, or the first solvent shell in a solute-solvent complex. For these modes, a description in terms of parameterized functions is introduced. The MCTDH working equations are generalized to allow for the nonorthogonality of these functions, which may take, e.g., a multidimensional Gaussian form. The formalism is developed on the level of both the wave function description and the density matrix description. Dissipative effects are accounted for in terms of a stochastic Hamiltonian approach versus master equation approach in the respective descriptions.

Non-Markovian quantum state diffusion: Perturbation approach

Abstract: We present a perturbation theory for non-Markovian quantum state diffusion (QSD), the theory of diffusive quantum trajectories for open systems in a bosonic environment [Physical Review A 58, 1699, (1998)]. We establish a systematic expansion in the ratio between the environmental correlation time and the typical system time scale. The leading order recovers the Markov theory, so here we concentrate on the next-order correction corresponding to first-order non-Markovian master equations. These perturbative equations greatly simplify the general nonMarkovian QSD approach, and allow for efficient numerical simulations beyond the Markov approximation. Furthermore, we show that each perturbative scheme for QSD naturally gives rise to a perturbative scheme for the master equation which we study in some detail. Analytical and numerical examples are presented, including the quantum Brownian motion model.

Algebraic Structure of Discrete Zero Curvature Equations and Master Symmetries of Discrete Evolution Equations

Abstract: An algebraic structure related to discrete zero curvature equations is established. It is used to give an approach for generating master symmetries of the first degree for systems of discrete evolution equations and an answer to why there exist such master symmetries. The key of the theory is to generate nonisospectral flows (λt=λl, l⩾0) from the discrete spectral problem associated with a given system of discrete evolution equations. Three examples are given.

Slippage of initial conditions for the Redfield master equation

Abstract: For a slow open quantum subsystem weakly coupled to a fast thermal bath, we derive the general form of the slippage to be applied to the initial conditions of the Redfield master equation. This slippage is given by a superoperator which describes the non-Markovian dynamics of the subsystem during the short-time relaxation of the thermal bath. We verify in an example that the Redfield equation preserves positivity after the slippage superoperator has been applied to the initial density matrix of the subsystem. For δ-correlated baths, the Redfield master equation reduces to the Lindblad master equation and the slippage of initial conditions vanishes consistently.

Master-equation approach to the study of electronic transport in small semiconductor devices

Abstract: Previously [J. Appl. Phys. 83, 270 (1998)] the Pauli master equation has been argued to constitute an equation suitable for the simulation of steady-state electron transport in semiconductor devices of length L smaller than the dephasing length ${\ensuremath{\lambda}}_{\ensuremath{\varphi}}$ in the contacts. Here, the master equation is derived emphasizing the role played by the dissipative interactions of the Van Hove-type, by the Markov approximation, and by the Van Hove limit in establishing irreversibility. An extension of the method to realistic band structures is also presented. Finally, the approach is applied to simulate electron transport in a simple one-dimensional Si $\mathrm{nin}$ diode at 77 K.

Lectures on the dynamical Yang-Baxter equations

Abstract: This paper contains a systematic and elementary introduction to a new area of the theory of quantum groups -- the theory of the classical and quantum dynamical Yang-Baxter equations. It arose from a minicourse given by the first author at MIT in the Spring of 1999, when the second author extended and improved his lecture notes of this minicourse. The quantum dynamical Yang-Baxter equation is a generalization of the ordinary quantum Yang-Baxter equation, considered in a physical context by Gervais and Neveu, and later from a mathematical viewpoint by Felder. Felder attached to every solution of this equation a quantum group, and also considered the classical analogue of the quantum dynamical Yang-Baxter equation -- the classical dynamical Yang-Baxter equation. Since then, the theory of dynamical Yang-Baxter equations and the corresponding quantum groups was systematically developed in many papers. By now, this theory has many applications, in particular to integrable systems and representation theory. The goal of this paper is to discuss this theory and some of its applications.

Non-Markovian stochastic Schrödinger equation

Abstract: We report a study of a stochastic Schrodinger equation corresponding to the Redfield master equation with slipped initial conditions, which describes the dynamics of a slow subsystem weakly coupled to a fast thermal bath. Using the projection-operator method of Feshbach, we derive a non-Markovian stochastic Schrodinger equation of the generalized Langevin type, which simulates the time evolution of the quantum wave functions of the subsystem driven by the fluctuating bath. For δ-correlated baths, the non-Markovian stochastic Schrodinger equation reduces to the previously derived Markovian one. Numerical methods are proposed to simulate the time evolution under our non-Markovian stochastic Schrodinger equation. These methods are illustrated with the spin-boson model.

A Self-Adjoint Angular Flux Equation

Abstract: The traditional second-order self-adjoint forms of the transport equation are the even- and odd-parity equations. A useful alternative to these equations exists in the form of a second-order self-a...

Wigner functions, squeezing properties, and slow decoherence of a mesoscopic superposition of two-level atoms

Abstract: Department of Quantum Physics, University of Ulm, D-89069 Ulm, Germany(Submitted to Physical Review A: March 26, 1999.)We consider a class of states in an ensemble of two-level atoms: a superposition of two distinctatomic coherent states, which can be regarded as atomic analogues of the states usually calledSchr¨odinger cat states in quantum optics. According to the relation of the constituents we definepolar and nonpolar cat states. The properties of these are investigated by the aid of the sphericalWigner function. We show that nonpolar cat states generally exhibit squeezing, the measure of whichdepends on the separation of the components of the cat, and also on the number of the constituentatoms. By solving the master equation for the polar cat state embedded in an external environment,we determine the characteristic times of decoherence, dissipation and also the characteristic time ofa new parameter, the non-classicality of the state. This latter one is introduced by the help of theWigner function, which is used also to visualize the process. The dependence of the characteristictimes on the number of atoms of the cat and on the temperature of the environment shows that thedecoherence of polar cat states is surprisingly slow.PACS number(s): 42.50.-p, 42.50.Fx, 03.65.BzI. INTRODUCTION

A Direct Transition State Theory Based Study of Methyl Radical Recombination Kinetics

Abstract: Multireference configuration interaction based quantum chemical estimates are directly implemented in a variational transition state theory based analysis of the kinetics of methyl radical recombination. Separations ranging from 5.5 to 1.9 A are considered for two separate forms for the reaction coordinate. The a priori prediction for the high-pressure limit rate constant gradually decreases with increasing temperature, with a net decrease of a factor of 1.7 from 300 to 1700 K. Near room temperature, this theoretical estimate is in quantitative agreement with the experimental data. At higher temperatures, comparison between theory and experiment requires a model for the pressure dependence. Master equation calculations employing the exponential down energy transfer model suggest that the theoretical and experimental high-pressure limits gradually diverge with increasing temperature, with the former being about 3 times greater than the latter at 1700 K. The comparison with experiment also suggests that the...

Structural relaxation in atomic clusters: master equation dynamics.

Abstract: The role of the potential energy landscape in determining the relaxation dynamics of model clusters is studied using a master equation. Two types of energy landscape are examined: a single funnel, as exemplified by 13-atom Morse clusters, and the double funnel landscape of the 38-atom Lennard-Jones cluster. Interwell rate constants are calculated using Rice-Ramsperger-Kassel-Marcus theory within the harmonic approximation, but anharmonic model partition functions are also considered. Decreasing the range of the potential in the Morse clusters is shown to hinder relaxation toward the global minimum, and this effect is related to the concomitant changes in the energy landscape. The relaxation modes that emerge from the master equation are interpreted and analyzed to extract interfunnel rate constants for the Lennard-Jones cluster. Since this system is too large for a complete characterization of the energy landscape, the conditions under which the master equation can be applied to a limited database are explored. Connections are made to relaxation processes in proteins and structural glasses.

Cavity-induced coherence effects in spontaneous emissions from preselection of polarization

Abstract: Spontaneous emission can create coherences in a multilevel atom having close-lying levels, subject to the condition that the atomic dipole matrix elements are nonorthogonal. This condition is rarely met in atomic systems. We report the possibility of bypassing this condition and thereby creating coherences by letting the atom with orthogonal dipoles interact with the vacuum of a preselected polarized cavity mode rather than the free-space vacuum. We derive a master equation for the reduced density operator of a model four-level atomic system, and obtain its analytical solution to describe the interference effects. We report the quantum beat structure in the populations.

Dissipation, decoherence and preparation effects in the spin-boson system

Abstract: The dynamics of the reduced density matrix of the driven dissipative two-state system is studied for a general diagonal/off-diagonal initial state. We derive exact formal series expressions for the populations and coherences and show that they can be cast into the form of coupled nonconvolutive exact master equations and integral relations. We show that neither the asymptotic distributions, nor the transition temperature between coherent and incoherent motion, nor the dephasing rate and relaxation rate towards the equilibrium state depend on the particular initial state chosen. However, in the underdamped regime, effects of the particular initial preparation, e.g. in an off-diagonal state of the density matrix, strongly affect the transient dynamics. We find that an appropriately tuned external ac-field can slow down decoherence and thus allow preparation effects to persist for longer times than in the absence of driving.

Quantum kinetic theory for a condensed bosonic gas

Abstract: We present a kinetic theory for Bose-Einstein condensation of a weakly interacting atomic gas in a trap. Starting from first principles, we establish a Markovian kinetic description for the evolution towards equilibrium. In particular, we obtain a set of self-consistent master equations for mean fields, normal densities, and anomalous fluctuations. These kinetic equations generalize the Gross-Pitaevskii mean-field equations, and merge them consistently with a quantum-Boltzmann-equation approach.

Quantum Trajectories for Brownian Motion

Abstract: We present the stochastic Schr\"odinger equation for the dynamics of a quantum particle coupled to a high temperature environment and apply it to the dynamics of a driven, damped, nonlinear quantum oscillator. Apart from an initial slip on the environmental memory time scale, in the mean, our result recovers the solution of the known non-Lindblad quantum Brownian motion master equation. A remarkable feature of our powerful stochastic approach is its localization property: individual quantum trajectories remain localized wave packets for all times, even for the classically chaotic system considered here, the localization being stronger as $\ensuremath{\Elzxh}\ensuremath{\rightarrow}0$.

Master-equation model of a single-quantum-dot microsphere laser

Abstract: We present a theoretical model for the electrically pumped single-quantum-dot microsphere laser. We solve the master equation of the system and analyze the steady state and dynamical properties of the optical field, such as output power, photon number fluctuation, and linewidth, for realistic experimental parameters. The laser threshold power is several orders of magnitude lower than is currently possible with semiconductor microlasers. A semiclassical approximation for the output power and laser linewidth is derived and compared to the exact solution. Electrical pumping together with Coulomb blockade effect allows for the realization of regular pumping in the system. We discuss the possibility for the generation of heralded single photons and of sub-Poissonian laser light. @S1050-2947~99!05506-7#

Spin relaxation in Mn12-acetate

Abstract: We present a comprehensive theory of the magnetization relaxation in a Mn12-acetate crystal based on thermally assisted spin tunneling induced by quartic anisotropy and weak transverse magnetic fields. The overall relaxation rate as a function of the magnetic field is calculated and shown to agree well with data including all resonance peaks. The Lorentzian shape of the resonances is also in good agreement with recent data. A generalized master equation including resonances is derived and solved exactly. It is shown that many transition paths with comparable weight exist that contribute to the relaxation process. Previously unknown spin-phonon coupling constants are calculated explicitly.

Quantum open-systems approach to current noise in resonant tunneling junctions

Abstract: A quantum Markovian master equation is derived to describe the current noise in resonant tunnelling devices. This equation includes both incoherent and coherent quantum tunnelling processes. We show how to obtain the population master equation by adiabatic elimination of quantum coherences in the presence of elastic scattering. We calculate the noise spectrum for a double well device and predict sub-shot noise statistics for strong tunnelling between the wells. The method is an alternative to Green's functions methods, and population master equations for very small coherently coupled quantum dots.

Quantum noise in the position measurement of a cavity mirror undergoing Brownian motion

Abstract: We perform a quantum theoretical calculation of the noise power spectrum for a phase measurement of the light output from a coherently driven optical cavity with a freely moving rear mirror. We examine how the noise resulting from the quantum back action appears among the various contributions from other noise sources. We do not assume an ideal (homodyne) phase measurement, but rather consider phase-modulation detection, which we show has a different shot noise level. We also take into account the effects of thermal damping of the mirror, losses within the cavity, and classical laser noise, We relate our theoretical results to experimental parameters, so as to make direct comparisons with current experiments simple. We also show that in this situation, the standard Brownian motion master equation is inadequate for describing the thermal damping of the mirror, as it produces a spurious term in the steady-state phase-fluctuation spectrum. The corrected Brownian motion master equation CL. Diosi, Europhys. Lett. 22, 1 (1993)] rectifies this inadequacy. [S1050-2917(99)02107-1].

Statistical Theory for the Stochastic Burgers Equation in the Inviscid Limit

Abstract: A statistical theory is developed for the stochastic Burgers equation in the inviscid limit. Master equations for the probability density functions of velocity, velocity difference and velocity gradient are derived. No closure assumptions are made. Instead closure is achieved through a dimension reduction process, namely the unclosed terms are expressed in terms of statistical quantities for the singular structures of the velocity field, here the shocks. Master equations for the environment of the shocks are further expressed in terms of the statistics of singular structures on the shocks, namely the points of shock generation and collisions. The scaling laws of the structure functions are derived through the analysis of the master equations. Rigorous bounds on the decay of the tail probabilities for the velocity gradient are obtained using realizability constraints. We also establish that the probability density function of the velocity gradient has a left power tail with exponent -7/2.

Born and Markov approximations for atom lasers

Abstract: We discuss the use of the Born and Markov approximations in describing the dynamics of an atom laser. In particular, we investigate the applicability of the quantum optical Born-Markov master equation for describing output coupling. We derive conditions based on the atomic reservoir and atom dispersion relations for when the Born-Markov approximations are valid and discuss parameter regimes where these approximations fail in our atom laser model. Differences between the standard optical laser model and the atom laser are due to a combination of factors, including the parameter regimes in which a typical atom laser would operate, the different reservoir state that is appropriate for atoms, and the different dispersion relations between atoms and photons. We present results based on an exact method in the regimes in which the Born-Markov approximation fails. The exact solutions in some experimentally relevant parameter regimes give a nonexponential loss of atoms from a cavity.

State determination in continuous measurement

Abstract: The possibility of determining the state of a quantum system after a continuous measurement of position is discussed in the framework of quantum trajectory theory. The initial lack of knowledge of the system and external noises are accounted for by considering the evolution of conditioned density matrices under a stochastic master equation. It is shown that after a finite time the state of the system is a pure state, and can be inferred from the measurement record alone. The relation to emerging possibilities for the continuous experimental observation of single quanta, as for example in cavity quantum electrodynamics, is discussed.

Bounds for Kac's master equation

Abstract: Mark Kac considered a Markov Chain on the n-sphere based on random rotations in randomly chosen coordinate planes. This same walk was used by Hastings on the orthogonal group. We show that the walk has spectral gap bounded below by c/n 3. This and curvature information are used to bound the rate of convergence to stationarity.

Master equation analysis of intermolecular energy transfer in multiple-well, multiple-channel unimolecular reactions. II. Numerical methods and application to the mechanism of the C2H5+O2 reaction

Abstract: Having elucidated a full theoretical analysis of the master equation for intermolecular and intramolecular energy transfer in multiple-well, multiple-channel chemically or thermally activated reactions [J. Chem. Phys. 107, 8904 (1997)], we now present efficient methods of numerical analysis for the computational examination of the dynamics of the master equation. We suggest the use of a Krylov-subspace method to determine the uppermost portions of the internal spectrum of the master equation kernel. Such a computation is pivotal in determining whether there exists a state of secular equilibrium for the population of the moieties and whether there exists within the possible state of secular equilibrium, a state wherein the dynamics are represented by an isolated dominating mode; for only in the state of secular equilibrium can one write rate equations for the dissociating processes that are local in time. And, if such a state is possible, we suggest the use of a Hermite–Laguerre orthogonal collocation method for obtaining highly accurate solutions to the population of the moieties. The theory and numerical analysis is then applied to study the dynamics of the chemically-activated reaction C2H5+O2. Comparison of the master equation treatment with modified strong-collision theory is also given for this system of multiple-well, multiple-channel reactions.