Showing papers on "Master equation published in 1992"

Atom-photon interactions

Abstract: This book of the physical phenomena resulting from the interactions between atoms and photons brings together and discusses different theoretical approaches which have up to now been dispersed in more specialized works. Each of these approaches is presented in a self-contained way and can be studied independently. Covers perturbative methods, resolvent method, master equations, Langevin equation, optical Block equations, and dressed atom method.

Wave-function approach to dissipative processes in quantum optics.

Abstract: A novel treatment of dissipation of energy from a ``small'' quantum system to a reservoir is presented. We replace the usual master equation for the small-system density matrix by a wave-function evolution including a stochastic element. This wave-function approach provides new insight and it allows calculations on problems which would otherwise be exceedingly complicated. The approach is applied here to a two- or three-level atom coupled to a laser field and to the vacuum modes of the quantized electromagnetic field.

A rigorous derivation of the chemical master equation

Abstract: It is widely believed that the chemical master equation has no rigorous microphysical basis, and hence no a priori claim to validity. This view is challenged here through arguments purporting to show that the chemical master equation is exact for any gas-phase chemical system that is kept well stirred and thermally equilibrated.

Atom-Photon Interactions: Basic Processes and Applications

Abstract: Transition Amplitudes in Electrodynamics. A Survey of Some Interaction Processes Between Photons and Atoms. Nonperturbative Calculation of Transition Amplitudes. Radiation Considered as a Reservoir: Master Equation for the Particles. Optical Bloch Equations. The Dressed Atom Approach. Exercises. Appendix. References. Index.

Quantum Brownian motion in a general environment: Exact master equation with nonlocal dissipation and colored noise.

Abstract: We use the influence functional path-integral method to derive an exact master equation for the quantum Brownian motion of a particle linearly coupled to a general environment (ohmic, subohmic, or supraohmic) at arbitrary temperature and apply it to study certain aspects of the loss of quantum coherence.

The quantum-state diffusion model applied to open systems

Abstract: A model of a quantum system interacting with its environment is proposed in which the system is represented by a state vector that satisfies a stochastic differential equation, derived from a density operator equation such as the Bloch equation, and consistent with it. The advantages of the numerical solution of these equations over the direct numerical solution of the density operator equations are described. The method is applied to the nonlinear absorber, cascades of quantum transitions, second-harmonic generation and a measurement reduction process. The model provides graphic illustrations of these processes, with statistical fluctuations that mimic those of experiments. The stochastic differential equations originated from studies of the measurement problem in the foundations of quantum mechanics. The model is compared with the quantum-jump model of Dalibard (1992), Carmichael and others, which originated among experimenters looking for intuitive pictures and rules of computation.

Monte Carlo simulation of the atomic master equation for spontaneous emission.

Abstract: A Monte Carlo simulation of the atomic master equation for spontaneous emission in terms of atomic wave functions is developed. Realizations of the time evolution of atomic wave functions are constructed that correspond to an ensemble of atoms driven by laser light undergoing a sequence of spontaneous emissions. The atomic decay times are drawn according to the photon count distribution of the driven atom. Each quantum jump of the atomic electron projects the atomic wave function to the ground state of the atom. Our theory is based on a stochastic interpretation and generalization of Mollow's pure-state analysis of resonant light scattering, and the Srinivas-Davies theory of continuous measurements in photodetection. An extension of the theory to include mechanical light effects and a generalization to atomic systems with Zeeman substructure are given. We illustrate the method by simulating the solutions of the optical Bloch equations for two-level systems, and laser cooling of a two-level atom in an ion trap where the center-of-mass motion of the atom is described quantum mechanically.

Wave-function quantum stochastic differential equations and quantum-jump simulation methods

Abstract: The quantum-stochastic-differential-equation formulation of driven quantum-optical systems is carried out in the interaction picture, and quantum stochastic differential equations for wave functions are derived on the basis of physical principles. The Ito form is shown to be the most practical, since it already contains all the radiation reaction terms. The connection between this formulation and the master equation is shown to be very straightforward. In particular, a direct connection is made to the theory of continuous measurements, which leads directly to the method of quantum-jump simulations of solutions of the master equation. It is also shown that all conceivable spectral and correlation-function information in output fields is accessible by means of an augmentation of the simulation process. Finally, the question of the reality of the jumps used in the simulations is posed.

Memory effects in the relaxation of quantum open systems

Abstract: A close examination of the validity of the Markovian approximation in the context of relaxation theory is presented. In particular, we examine the question of positivity of various approximations to the reduced dynamics of an open system in interaction with a heat reservoir. It is shown that the Markovian equations of motion obtained in the weak coupling limit (Redfield equations) are a consistent approximation to the actual reduced dynamics only if supplemented by a slippage in the initial conditions. This slippage captures the effects of the non‐Markovian evolution that takes place in a short transient time, of the order of the relaxation time of the isolated bath.

Monte Carlo simulation of master equations in quantum optics for vacuum, thermal, and squeezed reservoirs.

Abstract: Wave-function simulation of the master equation in terms of quantum jumps is illustrated for vacuum, thermal, and squeezed reservoirs. We discuss simulation techniques for (i) atomic density matrices, and resonance fluoresence and weak-field absorption spectra of atoms, (ii) decay of a two-level system in a squeezed vacuum, and (iii) a strongly coupled atom-cavity system driven by thermal light.

One-atom lasers.

Abstract: One-atom lasers are important because their governing equations can be solved exactly, even with a quantized field. We present a fully quantum-mechanical treatment of one-atom lasers modeled by quantum-optical master equations. These are solved numerically without any significant approximations. We show that laser action is possible with one atom, and that it might be achievable experimentally. Laser action is characterized by the dominance of stimulated emission over spontaneous emission. We use the one-atom laser model to investigate, without approximation, some interesting generic laser phenomena. Under certain conditions lasers produce intensity squeezed light, and then the laser linewidth increases with the pumping rate, in contrast with standard lasers. We also report ``self-quenching'' behavior: lasers with incoherent pumping out of the lower laser level turn off when the pumping is sufficiently fast because the coherence between the laser levels is destroyed.

Closed String Field Theory: Quantum Action and the BV Master Equation

Abstract: The complete quantum theory of covariant closed strings is constructed in detail. The action is defined by elementary vertices satisfying recursion relations that give rise to Jacobi-like identities for an infinite chain of string field products. The genus zero string field algebra is the homotopy Lie algebra $L_\infty$, and the higher genus algebraic structure implies the Batalin-Vilkovisky (BV) master equation. From these structures on the off-shell state space, we show how to derive the $L_\infty$ algebra, and the BV equation on physical states, recently constructed in d=2 string theory. The string diagrams are surfaces with minimal area metrics, foliated by closed geodesics of length $2\pi$. These metrics generalize quadratic differentials in that foliation bands can cross. The string vertices are succinctly characterized; they include the surfaces whose foliation bands are all of height smaller than $2\pi$. --While this is not a review paper, an effort was made to give a fairly complete and accessible account of the quantum closed string field theory.--

Collective response in globally coupled bistable systems.

Abstract: We consider a system of globally coupled bistable systems under the influence of noise and periodic modulations. The hopping process between the stable states is described by a nonlinear master equation. We observe an unusually large amplification of the periodic modulations for certain values of the noise strength due to collective dynamics of the coupled bistable elements.

Classical theory for shot noise in resonant tunneling.

Abstract: We show that shot noise for electrons can be suppressed in resonant tunneling through a double barrier, using a classical description based on the rate equation for ``sequential'' tunneling. The suppression is greatest when the escape rates through the two barriers are equal, in agreement with experiment and with the quantum-mechanical ``coherent'' model of resonant tunneling. A master equation is needed to calculate the noise, but cannot be uniquely derived from the rate equation; choices differ in the way that they describe transport between the emitter and the resonant state. Our choice for the rates, which are consistent with the exclusion principle, gives a suppression of the shot noise. We briefly discuss the results of choosing rates that are consistent with classical or Bose statistics instead of Fermi statistics. Finally, we apply our results to the two-state regime of the classical Coulomb blockade.

Interrelations between stochastic equations for systems with pair interactions

Abstract: Several types of stochastic equations are important in thermodynamics, chemistry, evolutionary biology, population dynamics and quantitative social science. For systems with pair interactions four different types of equations are derived, starting from a master equation for the state space: First, general mean value and (co)variance equations. Second, Boltzmann-like equations. Third, a master equation for the configuration space allowing transition rates which depend on the occupation numbers of the states. Fourth, a Fokker-Planck equation and a “Boltzmann-Fokker-Planck equation”. The interractions of these equations and the conditions for their validity are worked out clearly. A procedure for a self-consistent solution of the nonlinear equations is proposed. Generalizations to interactions between an arbitrary number of systems are discussed.

Rate parameters for coupled vibration-dissociation in a generalized SSH approximation

Abstract: A theoretical study of vibrational excitations and dissociations of nitrogen undergoing a nonequilibrium relaxation process upon heating and cooling is reported. The rate coefficients for collisional induced vibrational transitions and transitions from a bound vibrational state into a dissociative state have been calculated using an extension of the theory originally proposed by Schwarz (SSH) et al. (1952). High-lying vibrational states and dissociative states were explicitly included but rotational energy transfer was neglected. The transition probabilities calculated from the SSH theory were fed into the master equation, which was integrated numerically to determine the population distribution of the vibrational states as well as bulk thermodynamic properties. The results show that: (1) the transition rates have a minimum near the middle of the bound vibrational levels, causing a bottleneck in the vibrational relaxation and dissociation rates; (2) high vibrational states are always in equilibrium with the dissociative state; (3) for the heating case, only the low vibrational states relax according to the Landau-Teller theory; (4) for the cooling case, vibrational relaxation cannot be described by a rate equation; (5) Park's (1985, 1988) two-temperature model is approximately valid; and (6) the average vibrational energy removed in dissociation is about 30 percent of the dissociation energy.

The master equation approach to nonlinear economics

Abstract: A concept for modelling nonlinear economic dynamics is presented and exemplified by a concrete model. Generally, a configuration of macro-economic variables is considered whose probabilistic evolution is coupled to the decision making of agents and is described by a master equation. The transition rates in the master equation are modelled in terms of utility measures of the agents. Nonlinear dynamic meanvalue equations can be derived from the master equation.

Lattice-gas cellular-automaton method for semiclassical transport in semiconductors.

Abstract: A cellular-automaton method for solving the Boltzmann equation for semiclassical transport is presented and applied to nonlinear transport in semiconductors. It is shown that the Boltzmann equation for semiconductor transport can be transformed into a Boolean master equation, which represents a cellular automaton with nearest-neighbor interaction in position space. The resulting numerical algorithm is physically equivalent to the ensemble Monte Carlo method and tailored to modern vector or parallel processing. The algorithm is well suited for carrier systems with pronounced spatial inhomogeneities, large density variations, and scattering kernels involving single- and more-particle interactions. Several tests of the cellular-automaton technique for nonlinear transport in Si and GaAs are presented. The results agree very well with published Monte Carlo calculations.

Time reversal in stochastic processes and the Dirac equation.

Abstract: We consider the motion of a classical particle in (1+1)-dimensional space-time. Four probability distributions govern the trajectory of the particle; these give the probability of moving to the left or right in space while moving backwards or forwards in time. If these probabilities are randomly distributed and if the probability of moving backwards in time is related to the probability of moving forwards in time in a prescribed manner, then the master equations for these probabilities give rise to the Dirac equation without recourse to direct analytic continuation. In contrast, when a particle always moves forward in time, an analytic continuation is required to recover the Dirac equation.

Theory of thermal unimolecular reactions at low pressures. III. Superposition of weak and strong collisions

Abstract: Biexponential models for collisional energy transfer are formulated representing superimposed weak and strong collisions. Analytical solutions of the master equation for low pressure thermal unimolecular reactions at steady state are derived for such models. Nonequilibrium populations of excited states as well as collision efficiencies βc in the low pressure rate constant differ considerably from the corresponding quantities for simple exponential collision models. The results are analyzed in terms of general expressions of the collision efficiency βc as a function of the average energy transfer quantities 〈ΔE〉 and 〈ΔE2〉.

The drift velocity in the Rubinstein—Duke model for electrophoresis

Abstract: The average drift velocity of a polymer chain due to an applied electric field is calculated in the Rubinstein-Duke model for reptation. Imposing periodic boundary conditions on the reptating chain the stationary state is solved exactly from the master equation, and the drift velocity is calculated to first and third order in the field. Finite-size effects are included and the scaling hypothesis of Widom et al. is confirmed to third order in the field. An approximate expression is presented for the scaling function in the whole scaling range.

Thermal Equilibrium in the Jaynes-Cummings Model

Abstract: The master equation currently used to describe a two level atom interacting with a single mode field damped by contact with a thermal reservoir (the damped Jaynes-Cummings model) is shown not to have, as its steady state solution, the expected canonical density operator prescribed by the general principles of statistical mechanics for a system in thermal equilibrium. A modified master equation is derived here which satisfies this requirement. Except for a reservoir at zero temperature, this master equation differs from that which is currently used in that the damping terms contain contributions due to the atom-field interaction.

Integrability of Chern-Simons-Higgs Vortex Equations and a Reduction of the Self-Dual Yang-Mills Equations to Three Dimensions

Abstract: I would like to make brief presentations on two topics, both of which focus on an issue of ‘integrability’ in equations of interest in high energy physics. In my first talk, I would like to introduce the Chern-Simons-Higgs vortex equations, which describe classical solutions of a certain (2 + 1)-dimensional field theory. In flat space-time these equations are non-integrable, but in curved spacetime the ODE describing cylindrically symmetric vortices can, by correct choice of the metric, be made to be a degenerate case of the third Painleve equation, possessing rational solutions which can be written down. Remarkably, the overall features of the solutions (found numerically) for the flat spacetime case, are very similar to those found in the integrable case, suggesting that maybe the non-integrable case should be looked on as a ‘perturbation’ of the integrable case. My second talk is on the topic of a reduction of the self-dual Yang-Mills equations from four to three dimensions: there has been considerable interest recently in the self-dual Yang-Mills equations as a ‘master equation’, from which many integrable systems can be obtained by suitable reductions. Here I focus on a method to reduce to three dimensions, but the systems that emerge are really trivial generalizations of two dimensional integrable systems.

Forward and reverse rate coefficients in equilibrating isomerization reactions

Abstract: Three models for the relaxation kinetics of a reversible unimolecular isomerization reaction are formulated and analyzed: a generalization of the simple Lindemann–Hinshelwood scheme, a detailed model with the strong collision approximation, and a master equation solution. For such systems the use of a classical relaxation analysis has been questioned. In each case it is found that the relaxation analysis does not give forward and reverse rate constants appropriate to the pure irreversible reactions, but that the rate constants so obtained can be interpreted in terms of irreversible schemes which allow for back reaction before collisional stabilization. The accuracy of this decomposition is linked with the applicability of the steady‐state approximation for the populations of the reactive states, as is demonstrated analytically under the strong collision approximation, and numerically with the full master equation. An alternative approach using perturbation theory is shown to be unacceptably inaccurate.

Intrinsic fluctuations in chaotic dynamics

Abstract: The multivariate master equation of a chemical system exhibiting spatio-temporal chaos is investigated. It is shown that the probability distribution in the presence of fluctuations remains centered on the underlying deterministic attractor.

Higher-order squeezing in three- and four-wave mixing processes with loss.

Abstract: An investigation is carried out of higher-order squeezing in a class of three- and four-wave mixing processes that can be modeled by a master equation. In (2N)th-order squeezing, as defined by Hong and Mandel, the Nth-power of the variance of a field quadrature falls below the value it would have for a coherent state. Higher-order squeezing is of interest because (i) it occurs in many processes in which ordinary (or second-order) squeezing occurs, and (ii) it allows a much larger fractional noise reduction to be achieved than ordinary squeezing. The master equation we use models a degenerate three- or four-wave process and takes into account the losses in the signal mode. We convert the master equation into a Fokker-Planck equation for the Wigner distribution function and solve it for a wide range of initial signal states (including thermal, coherent, Fock states, and mixtures of these). Results are presented for the enhancement of fourth- and sixth-order squeezing over second-order squeezing. Differences in the character of higher-order squeezing for different input states are pointed out. The role of losses in limiting the amount of achievable squeezing in all orders is especially investigated.