Showing papers on "Master equation published in 2006"
TL;DR: In this paper, the authors give an overview and extension of the stochastic Liouville equation focusing on its theoretical background and applications to help further the development of their works, and cast the equation into the hierarchy of equations which can be solved analytically or computationally in a nonperturbative manner including the effect of a colored noise.
Abstract: Half century has past since the pioneering works of Anderson and Kubo on the stochastic theory of spectral line shape were published in J. Phys. Soc. Jpn. 9 (1954) 316 and 935, respectively. In this review, we give an overview and extension of the stochastic Liouville equation focusing on its theoretical background and applications to help further the development of their works. With the aid of path integral formalism, we derive the stochastic Liouville equation for density matrices of a system. We then cast the equation into the hierarchy of equations which can be solved analytically or computationally in a nonperturbative manner including the effect of a colored noise. We elucidate the applications of the stochastic theory from the unified theoretical basis to analyze the dynamics of a system as probed by experiments. We illustrate this as a review of several experimental examples including NMR, dielectric relaxation, Mossbauer spectroscopy, neutron scattering, and linear and nonlinear laser spectroscop...
806 citations
TL;DR: A pedagogical treatment of the formalism of continuous quantum measurement is presented, using the simple and direct approach of generalized measurements to derive the stochastic master equation describing the continuous measurements of observables.
Abstract: We present a pedagogical treatment of the formalism of continuous quantum measurement. Our aim is to show the reader how the equations describing such measurements are derived and manipulated in a direct manner. We also give elementary background material for those new to measurement theory, and describe further various aspects of continuous measurements that should be helpful to those wanting to use such measurements in applications. Specifically, we use the simple and direct approach of generalized measurements to derive the stochastic master equation describing the continuous measurements of observables, give a tutorial on stochastic calculus, treat multiple observers and inefficient detection, examine a general form of the measurement master equation, and show how the master equation leads to information gain and disturbance. To conclude, we give a detailed treatment of imaging the resonance fluorescence from a single atom as a concrete example of how a continuous position measurement arises in a physical system.
456 citations
TL;DR: The application of master equation methods to problems in gas phase chemical kinetics, focusing on reactions that take place over multiple, interconnected potential wells and on the dissociation of weakly bound free radicals, is discussed.
Abstract: In this article, we discuss the application of master equation methods to problems in gas phase chemical kinetics. The focus is on reactions that take place over multiple, interconnected potential wells and on the dissociation of weakly bound free radicals. These problems are of paramount importance in combustion chemistry. To illustrate specific points, we draw on our experience with reactions we have studied previously.
381 citations
TL;DR: This article presents an exact solution formula of the CME for arbitrary initial conditions in the case where the underlying system is governed by monomolecular reactions, expressed in terms of the convolution of multinomial and product Poisson distributions with time-dependent parameters evolving according to the traditional reaction-rate equations.
Abstract: The stochastic dynamics of a well-stirred mixture of molecular species interacting through different biochemical reactions can be accurately modelled by the chemical master equation (CME). Research in the biology and scientific computing community has concentrated mostly on the development of numerical techniques to approximate the solution of the CME via many realizations of the associated Markov jump process. The domain of exact and/or efficient methods for directly solving the CME is still widely open, which is due to its large dimension that grows exponentially with the number of molecular species involved. In this article, we present an exact solution formula of the CME for arbitrary initial conditions in the case where the underlying system is governed by monomolecular reactions. The solution can be expressed in terms of the convolution of multinomial and product Poisson distributions with time-dependent parameters evolving according to the traditional reaction-rate equations. This very structured representation allows to deduce easily many properties of the solution. The model class includes many interesting examples. For more complex reaction systems, our results can be seen as a first step towards the construction of new numerical integrators, because solutions to the monomolecular case provide promising ansatz functions for Galerkin-type methods.
329 citations
TL;DR: In this article, a QME for the many-body density matrix of an open current-carrying system weakly-coupled to two metal leads is derived for arbitrary bias.
Abstract: A quantum master equation QME is derived for the many-body density matrix of an open current-carrying system weakly-coupled to two metal leads. The dynamics and the steady-state properties of the system for arbitrary bias are studied using projection operator techniques, which keep track of the number of electrons in the system. We show that coherences between system states with different number of electrons, n Fock space coherences, do not contribute to the transport to second order in system-lead coupling. However, coherences between states with the same n may effect transport properties when the damping rate is of the order of or faster than the system Bohr frequencies. For large bias, when all the system many-body states lie between the chemical potentials of the two leads, we recover previous results. In the rotating wave approximation when the damping is slow compared to the Bohr frequencies, the dynamics of populations and coherences in the system eigenbasis are decoupled. The QME then reduces to a birth and death master equation for populations.
233 citations
TL;DR: In this article, the authors considered the general physical situation of a quantum system interacting with a chain of exterior systems, one after the other, during a small interval of time h and following some Hamiltonian H on H, and showed the necessity for three different time scales in H. They also showed that the typical Hamiltonian allowing this passage to the limit shows up three different parts which correspond to the free evolution, to a weak coupling limit part and to a scattering interaction part.
Abstract: We consider the general physical situation of a quantum system \(\mathcal{H}_{0} \) interacting with a chain of exterior systems \( \otimes _{{\mathbb{N}^{ * } }} \mathcal{H},\) one after the other, during a small interval of time h and following some Hamiltonian H on \(\mathcal{H}_{0} \otimes \mathcal{H}.\) We discuss the passage to the limit to continuous interactions (h → 0) in a setup which allows to compute the limit of this Hamiltonian evolution in a single state space: a continuous field of exterior systems \( \otimes _{{\mathbb{R}^{ + } }} \mathcal{H}.\) Surprisingly, the passage to the limit shows the necessity for three different time scales in H. The limit evolution equation is shown to spontaneously produce quantum noises terms: we obtain a quantum Langevin equation as limit of the Hamiltonian evolution. For the very first time, these quantum Langevin equations are obtained as the effective limit from repeated to continuous interactions and not only as a model. These results justify the usual quantum Langevin equations considered in continual quantum measurement or in quantum optics. Physically, the typical Hamiltonian allowing this passage to the limit shows up three different parts which correspond to the free evolution, to an analogue of a weak coupling limit part and to a scattering interaction part. We apply these results to give an Hamiltonian description of the von Neumann measurements. We also consider the approximation of continuous time quantum master equations by discrete time ones; in particular we show how any Lindblad generator is naturally obtained as the limit of completely positive maps.
210 citations
TL;DR: In this article, a general method and geometrical picture is developed and shown to agree with effective action results, commonly derived through path integration, for perturbations around a harmonic oscillator ground state.
Abstract: In many situations, one can approximate the behavior of a quantum system, i.e. a wave function subject to a partial differential equation, by effective classical equations which are ordinary differential equations. A general method and geometrical picture is developed and shown to agree with effective action results, commonly derived through path integration, for perturbations around a harmonic oscillator ground state. The same methods are used to describe dynamical coherent states, which in turn provide means to compute quantum corrections to the symplectic structure of an effective system.
181 citations
TL;DR: Based on a real-time measurement of the motion of a single ion in a Paul trap, electromechanical cooling below the Doppler limit by homodyne feedback control (cold damping) is demonstrated by a model based on a quantum mechanical master equation.
Abstract: Based on a real-time measurement of the motion of a single ion in a Paul trap, we demonstrate its electromechanical cooling below the Doppler limit by homodyne feedback control (cold damping). The feedback cooling results are well described by a model based on a quantum mechanical master equation.
180 citations
TL;DR: Shabani et al. as mentioned in this paper investigated the non-Markovian dynamics of a qubit by comparing two generalized master equations with memory and derived the solution of the recently proposed post Markovian master equation.
Abstract: In this paper we investigate the non-Markovian dynamics of a qubit by comparing two generalized master equations with memory. In the case of a thermal bath, we derive the solution of the recently proposed post-Markovian master equation [A. Shabani and D. A. Lidar, Phys. Rev. A 71, 020101(R) (2005)] and we study the dynamics for an exponentially decaying memory kernel. We compare the solution of the post-Markovian master equation with the solution of the typical memory kernel master equation. Our results lead to a new physical interpretation of the reservoir correlation function and bring to light the limits of usability of master equations with memory for the system under consideration.
173 citations
TL;DR: A two-dimensional model of transcription, translation, and nuclear membrane transport in eukaryotic cells is presented to demonstrate the feasibility of the Gillespie method in studying cell-wide biological processes.
Abstract: A method is developed for incorporating diffusion of chemicals in complex geometries into stochastic chemical kinetics simulations. Systems are modeled using the reaction-diffusion master equation, with jump rates for diffusive motion between mesh cells calculated from the discretization weights of an embedded boundary method. Since diffusive jumps between cells are treated as first order reactions, individual realizations of the stochastic process can be created by the Gillespie method. Numerical convergence results for the underlying embedded boundary method, and for the stochastic reaction-diffusion method, are presented in two dimensions. A two-dimensional model of transcription, translation, and nuclear membrane transport in eukaryotic cells is presented to demonstrate the feasibility of the method in studying cell-wide biological processes.
162 citations
TL;DR: It is shown by theory and example that stochastic effects are better captured using this technique while still maintaining the computational advantages of the reaction rate approach.
TL;DR: In this article, the authors investigate the heat conduction properties of molecular junctions comprising nonlinear interactions and find that these interactions can lead to phenomena such as negative differential thermal conductance and heat rectification.
Abstract: We investigate the heat conduction properties of molecular junctions comprising nonlinear interactions. We find that these interactions can lead to phenomena such as negative differential thermal conductance and heat rectification. Based on analytically solvable models we derive an expression for the heat current that clearly reflects the interplay between internal molecular anharmonic interactions, the strength of molecular coupling to the thermal reservoirs, and junction asymmetry. This expression indicates that negative differential thermal conductance shows up when the molecule is strongly coupled to the thermal baths, even in the absence of internal molecular nonlinearities. In contrast, diodelike behavior is expected for a highly anharmonic molecule with an inherent structural asymmetry.
TL;DR: In this article, the response of continuous-time random walks to an oscillating external field within the generalized master equation approach was studied, focusing on the time dependence of the two first moments of the walker's displacement.
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 two first moments of the walker's displacement. We show that for power-law waiting-time distributions with $0l\ensuremath{\alpha}l1$ 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.
TL;DR: This work presents a model reduction method for study of stochastic chemical kinetic systems that takes advantage of multiple time scales and is implemented in a novel numerical algorithm that exploits the time scale separation to achieve model order reductions while enabling error checking and control.
Abstract: The dynamics of chemical reaction networks often takes place on widely differing time scales--from the order of nanoseconds to the order of several days. This is particularly true for gene regulatory networks, which are modeled by chemical kinetics. Multiple time scales in mathematical models often lead to serious computational difficulties, such as numerical stiffness in the case of differential equations or excessively redundant Monte Carlo simulations in the case of stochastic processes. We present a model reduction method for study of stochastic chemical kinetic systems that takes advantage of multiple time scales. The method applies to finite projections of the chemical master equation and allows for effective time scale separation of the system dynamics. We implement this method in a novel numerical algorithm that exploits the time scale separation to achieve model order reductions while enabling error checking and control. We illustrate the efficiency of our method in several examples motivated by recent developments in gene regulatory networks.
TL;DR: In this article, the authors examine the internal consistency of a set of minimal assumptions entering the theory of fault-tolerant quantum error correction for Markovian noise, and point out that these assumptions may not be mutually consistent in light of rigorous formulations of the Markovians approximation.
Abstract: We critically examine the internal consistency of a set of minimal assumptions entering the theory of fault-tolerant quantum error correction for Markovian noise. These assumptions are fast gates, a constant supply of fresh and cold ancillas, and a Markovian bath. We point out that these assumptions may not be mutually consistent in light of rigorous formulations of the Markovian approximation. Namely, Markovian dynamics requires either the singular coupling limit (high temperature), or the weak coupling limit (weak system-bath interaction). The former is incompatible with the assumption of a constant and fresh supply of cold ancillas, while the latter is inconsistent with fast gates. We discuss ways to resolve these inconsistencies. As part of our discussion we derive, in the weak coupling limit, a new master equation for a system subject to periodic driving.
TL;DR: In this paper, the authors discuss the theoretical treatment of multidimensional quantum dynamics of hydrogen-bonded systems in the context of infrared spectroscopy, where the multidimensionality is reflected in the complex shape of linear infrared absorption spectra.
TL;DR: In this article, the authors identify experimental signatures in the currentvoltage (I-V) characteristics of weakly contacted molecules directly arising from excitations in their many electron spectrum using a multielectron master equation in the Fock space of an exact diagonalized model manybody Hamiltonian for a prototypical molecule.
Abstract: We identify experimental signatures in the current-voltage (I-V) characteristics of weakly contacted molecules directly arising from excitations in their many electron spectrum. The current is calculated using a multielectron master equation in the Fock space of an exact diagonalized model many-body Hamiltonian for a prototypical molecule. Using this approach, we explain several nontrivial features in frequently observed I-Vs in terms of a rich spectrum of excitations that may be hard to describe adequately with standard one-electron self-consistent field theories.
TL;DR: It is shown that the master constraint programme has a wide range of applicability but that there are many, physically interesting subtleties that must be taken care of in doing so.
Abstract: Recently, the master constraint programme for loop quantum gravity (LQG) was proposed as a classically equivalent way to impose the infinite number of Wheeler?DeWitt constraint equations in terms of a single master equation. While the proposal has some promising abstract features, it was until now barely tested in known models. In this series of five papers we fill this gap, thereby adding confidence to the proposal. We consider a wide range of models with increasingly more complicated constraint algebras, beginning with a finite-dimensional, Abelian algebra of constraint operators which are linear in the momenta and ending with an infinite-dimensional, non-Abelian algebra of constraint operators which closes with structure functions only and which are not even polynomial in the momenta. In all these models, we apply the master constraint programme successfully; however, the full flexibility of the method must be exploited in order to complete our task. This shows that the master constraint programme has a wide range of applicability but that there are many, physically interesting subtleties that must be taken care of in doing so. In particular, as we will see, that we can possibly construct a master constraint operator for a nonlinear, that is, interacting quantum field theory underlines the strength of the background-independent formulation of LQG. In this first paper, we prepare the analysis of our test models by outlining the general framework of the master constraint programme. The models themselves will be studied in the remaining four papers. As a side result, we develop the direct integral decomposition (DID) programme for solving quantum constraints as an alternative to refined algebraic quantization (RAQ).
TL;DR: In this article, a density-matrix formalism is used for the determination of electron transport through molecular wires, where the full system is partitioned into a relevant part and a thermal fermionic bath, and the derivation of a quantum master equation, either based on a timeconvolutionless or time-convolution projection operator approach, incorporates the interaction with time-dependent laser fields nonperturbatively and is valid at low temperatures for weak system-bath coupling.
Abstract: New features of molecular wires can be observed when they are irradiated by laser fields. These effects can be achieved by periodically oscillating fields but also by short laser pulses. The theoretical foundation used for these investigations is a density-matrix formalism where the full system is partitioned into a relevant part and a thermal fermionic bath. The derivation of a quantum master equation, either based on a time-convolutionless or time-convolution projection-operator approach, incorporates the interaction with time-dependent laser fields nonperturbatively and is valid at low temperatures for weak system-bath coupling. From the population dynamics the electrical current through the molecular wire is determined. This theory including further extensions is used for the determination of electron transport through molecular wires. As examples, we show computations of coherent destruction of tunneling in asymmetric periodically driven quantum systems, alternating currents and the suppression of the directed current by using a short laser pulse.
01 Dec 2006
TL;DR: The generic nature and the power of the pertinent conceptual, analytical and computational frameworks helps eliminate some of the traditional language barriers, which often unnecessarily impede scientific progress and the interaction of researchers between disciplines such as physics, chemistry, biology, applied mathematics and engineering.
Abstract: Computation of Invariant Manifolds.- A New Model Reduction Method for Nonlinear Dynamical Systems Using Singular PDE Theory.- A Versatile Algorithm for Computing Invariant Manifolds.- Covering an Invariant Manifold with Fat Trajectories.- "Ghost" ILDM-Manifolds and Their Identification.- Dynamic Decomposition of ODE Systems: Application to Modelling of Diesel Fuel Sprays.- Model Reduction of Multiple Time Scale Processes in Non-standard Singularly Perturbed Form.- Coarse-Graining and Ideas of Statistical Physics.- Basic Types of Coarse-Graining.- Renormalization Group Methods for Coarse-Graining of Evolution Equations.- A Stochastic Process Behind Boltzmann's Kinetic Equation and Issues of Coarse Graining.- Finite Difference Patch Dynamics for Advection Homogenization Problems.- Coarse-Graining the Cyclic Lotka-Volterra Model: SSA and Local Maximum Likelihood Estimation.- Relations Between Information Theory, Robustness and Statistical Mechanics of Stochastic Uncertain Systems via Large Deviation Theory.- Kinetics and Model Reduction.- Exactly Reduced Chemical Master Equations.- Model Reduction in Kinetic Theory.- Novel Trajectory Based Concepts for Model and Complexity Reduction in (Bio)Chemical Kinetics.- Dynamics of the Plasma Sheath.- Mesoscale and Multiscale Modeling.- Construction of Stochastic PDEs and Predictive Control of Surface Roughness in Thin Film Deposition.- Lattice Boltzmann Method and Kinetic Theory.- Numerical and Analytical Spatial Coupling of a Lattice Boltzmann Model and a Partial Differential Equation.- Modelling and Control Considerations for Particle Populations in Particulate Processes Within a Multi-Scale Framework.- Diagnostic Goal-Driven Reduction of Multiscale Process Models.- Understanding Macroscopic Heat/Mass Transfer Using Meso- and Macro-Scale Simulations.- An Efficient Optimization Approach for Computationally Expensive Timesteppers Using Tabulation.- A Reduced Input/Output Dynamic Optimisation Method for Macroscopic and Microscopic Systems.
TL;DR: In this article, the authors considered charge transport through two quantum dots coupled in series and investigated the effect of quantum coherence on the zero-frequency cumulants of the transport process, focusing on noise and skewness.
Abstract: We theoretically consider charge transport through two quantum dots coupled in series. The corresponding full counting statistics for noninteracting electrons is investigated in the limits of sequential and coherent tunneling by means of a master equation approach and a density matrix formalism, respectively. We clearly demonstrate the effect of quantum coherence on the zero-frequency cumulants of the transport process, focusing on noise and skewness. Moreover, we establish the continuous transition from the coherent to the incoherent tunneling limit in all cumulants of the transport process and compare this with decoherence described by a dephasing voltage probe model.
Book Chapter•
01 Jan 2006
TL;DR: A Krylov FSP algorithm based on a combination of state-space truncation and inexact matrix-vector product routines that allows larger-scale models to be studied and solutions for larger final times to be computed in a realistic execution time.
Abstract: Biochemical reactions underlying genetic regulation are often modelled as a continuous-time, discrete-state, Markov process, and the evolution of the associated probability density is described by the so-called chemical master equation (CME). However the CME is typically difficult to solve, since the state-space involved can be very large or even countably infinite. Recently a finite state projection method (FSP) that truncates the state-space was suggested and shown to be effective in an example of a model of the Pap-pili epigenetic switch. However in this example, both the model and the final time at which the solution was computed, were relatively small. Presented here is a
Krylov FSP algorithm based on a combination of state-space truncation and inexact matrix-vector product routines. This allows larger-scale models to be studied and solutions for larger final times to be computed in a realistic execution time. Additionally the new method computes the solution
at intermediate times at virtually no extra cost, since it is derived from Krylov-type methods for computing matrix exponentials. For the purpose of comparison the new algorithm is applied to the model of the Pap-pili epigenetic switch, where the original FSP was first demonstrated. Also
the method is applied to a more sophisticated model of regulated transcription. Numerical results indicate that the new approach is significantly faster and extendable to larger biological models.
TL;DR: The state-to-state chemical kinetic model, which considers a kinetic equation for each vibrational state of diatomic molecules, has been applied to some supersonic flow regimes and in particular in boundary layer, nozzle expansion, and shock wave as discussed by the authors.
Abstract: The state-to-state chemical kinetic model, which considers a kinetic equation for each vibrational state of diatomic molecules, has been applied to some supersonic flow regimes and in particular in boundary layer, nozzle expansion, and shock wave. Nonequilibrium vibrational distribution obtained in the calculations shows strong departure from equilibrium-inducing non-Arrhenius global chemical rates, which differ substantially from macroscopic rates commonly used in fluid-dynamic codes. The evolution properties of the distribution have been investigated by a zero-dimensional numerical code in controlled conditions. We are trying to obtain from zero-dimensional results the approach to find appropriate macroscopic rate models to be used in fluid-dynamic codes accounting for the vibrational nonequilibrium. A comparison of analytical fitting of the zero-dimensional data and fluid dynamic global rates has been performed. Nomenclature ci = coefficients for the solution of the master equation Ev = energy of the vth vibrational level k = Boltzmann constant k d = dissociation rate constant k p = rates of the process p
TL;DR: In this paper, the authors show that quantum mechanical entanglement can prevail in noisy open quantum systems at high temperature and far from thermodynamical equilibrium, despite the deteriorating effect of decoherence.
Abstract: We show that quantum mechanical entanglement can prevail in noisy open quantum systems at high temperature and far from thermodynamical equilibrium, despite the deteriorating effect of decoherence. The system consists of a number N of interacting quantum particles, and can interact and exchange particles with some environments. The effect of decoherence is counteracted by a simple mechanism, where system particles are randomly reset to some standard initial state, e.g., by replacing them with particles from the environment. We present a master equation that describes this process, which we can solve analytically for small N. If we vary the interaction strength and the reset against decoherence rate, we find a threshold below which the equilibrium state is classically correlated and above which there is a parameter region with genuine entanglement.
TL;DR: The fluctuation theorem for currents is applied to several mesoscopic systems on the basis of Schnakenberg's network theory, which allows one to verify its conditions of validity as mentioned in this paper.
Abstract: The fluctuation theorem for currents is applied to several mesoscopic systems on the basis of Schnakenberg's network theory, which allows one to verify its conditions of validity. A graph is associated with the master equation ruling the random process and its cycles can be used to obtain the thermodynamic forces or affinities corresponding to the nonequilibrium constraints. This provides a method of defining the independent currents crossing the system in nonequilibrium steady states and to formulate the fluctuation theorem for the currents. This result is applied to out-of-equilibrium diffusion in a chain, to a biophysical model of ion channels in a membrane, and to electronic transport in mesoscopic circuits made of several tunnel junctions. In this latter, we show that the generalizations of Onsager's reciprocity relations to the nonlinear response coefficients also hold.
TL;DR: It is shown that the fluctuation theorems provide a method to determine the affinity or thermodynamic force driving the motor, given in terms of the free enthalpy of the chemical reactions.
Abstract: The fluctuation theorems for the currents and the dissipated work are considered for molecular motors which are driven out of equilibrium by chemical reactions. Because of the molecular fluctuations, these nonequilibrium processes are described by stochastic models based on a master equation. Analytical expressions are derived for the fluctuation theorems, allowing us to obtain predictions on the work dissipated in the motor as well as on its rotation near and far from thermodynamic equilibrium. We show that the fluctuation theorems provide a method to determine the affinity or thermodynamic force driving the motor. This affinity is given in terms of the free enthalpy of the chemical reactions. The theorems are applied to the F1 rotary motor which turns out to be a stiff system typically functioning in the nonlinear regime of nonequilibrium thermodynamics. We show that this nonlinearity confers a robustness to the functioning of the molecular motor.
TL;DR: In this paper, the authors study BEC fluctuations in the ideal Bose gas in a trap and explain why the grand canonical description goes very wrong for all moments 〈 ( n 0 − n ¯ 0 ) m 〉, except of the mean value.
Abstract: We review the phenomenon of equilibrium fluctuations in the number of condensed atoms n 0 in a trap containing N atoms total. We start with a history of the Bose–Einstein distribution, a similar grand canonical problem with an indefinite total number of particles, the Einstein–Uhlenbeck debate concerning the rounding of the mean number of condensed atoms n ¯ 0 near a critical temperature T c , and a discussion of the relations between statistics of BEC fluctuations in the grand canonical, canonical, and microcanonical ensembles. First, we study BEC fluctuations in the ideal Bose gas in a trap and explain why the grand canonical description goes very wrong for all moments 〈 ( n 0 − n ¯ 0 ) m 〉 , except of the mean value. We discuss different approaches capable of providing approximate analytical results and physical insight into this very complicated problem. In particular, we describe at length the master equation and canonical-ensemble quasiparticle approaches which give the most accurate and physically transparent picture of the BEC fluctuations. The master equation approach, that perfectly describes even the mesoscopic effects due to the finite number N of the atoms in the trap, is quite similar to the quantum theory of the laser. That is, we calculate a steady-state probability distribution of the number of condensed atoms p n 0 ( t = ∞ ) from a dynamical master equation and thus get the moments of fluctuations. We present analytical formulas for the moments of the ground-state occupation fluctuations in the ideal Bose gas in the harmonic trap and arbitrary power-law traps. In the last part of the review, we include particle interaction via a generalized Bogoliubov formalism and describe condensate fluctuations in the interacting Bose gas. In particular, we show that the canonical-ensemble quasiparticle approach works very well for the interacting gases and find analytical formulas for the characteristic function and all cumulants, i.e., all moments, of the condensate fluctuations. The surprising conclusion is that in most cases the ground-state occupation fluctuations are anomalously large and are not Gaussian even in the thermodynamic limit. We also resolve the Giorgini, Pitaevskii and Stringari (GPS) vs. Idziaszek et al. debate on the variance of the condensate fluctuations in the interacting gas in the thermodynamic limit in favor of GPS. Furthermore, we clarify a crossover between the ideal-gas and weakly-interacting-gas statistics which is governed by a pair-correlation, squeezing mechanism and show how, with an increase of the interaction strength, the fluctuations can now be understood as being essentially 1/2 that of an ideal Bose gas. We also explain the crucial fact that the condensate fluctuations are governed by a singular contribution of the lowest energy quasiparticles. This is a sort of infrared anomaly which is universal for constrained systems below the critical temperature of a second-order phase transition.
TL;DR: In this paper, the authors present a computational study of elementary bifurcations that occur in a driven and damped cavity quantum electrodynamics (cavity QED) model at low intracavity photon number.
Abstract: The interplay of quantum fluctuations with nonlinear dynamics is a central topic in the study of open quantum systems, connected to fundamental issues (such as decoherence and the quantum-classical transition) and practical applications (such as coherent information processing and the development of mesoscopic sensors and amplifiers) With this context in mind, we here present a computational study of some elementary bifurcations that occur in a driven and damped cavity quantum electrodynamics (cavity QED) model at low intracavity photon number In particular, we utilize the single-atom cavity QED master equation and associated stochastic Schrodinger equations to characterize the equilibrium distribution and dynamical behavior of the quantized intracavity optical field in parameter regimes near points in the semiclassical (mean-field, Maxwell-Bloch) bifurcation set Our numerical results show that the semiclassical limit sets are qualitatively preserved in the quantum stationary states, although quantum fluctuations apparently induce phase diffusion within periodic orbits and stochastic transitions between attractors We restrict our attention to an experimentally realistic parameter regime
TL;DR: A new approach for calculating the memory kernel and inhomogeneous term for arbitrary initial state and system-bath coupling and the utility and self-consistency of the formalism are demonstrated by an explicit calculation on a spin-boson model.
Abstract: The Nakajima-Zwanzig generalized quantum master equation provides a general, and formally exact, prescription for simulating the reduced dynamics of a quantum system coupled to a quantum bath. In this equation, the memory kernel accounts for the influence of the bath on the system’s dynamics, and the inhomogeneous term accounts for initial system-bath correlations. In this paper, we propose a new approach for calculating the memory kernel and inhomogeneous term for arbitrary initial state and system-bath coupling. The memory kernel and inhomogeneous term are obtained by numerically solving a single inhomogeneous Volterra equation of the second kind for each. The new approach can accommodate a very wide range of projection operators, and requires projection-free two-time correlation functions as input. An application to the case of a two-state system with diagonal coupling to an arbitrary bath is described in detail. Finally, the utility and self-consistency of the formalism are demonstrated by an explicit calculation on a spin-boson model.
TL;DR: The equation for the quantum motion of aBrownian particle in a gaseous environment is derived by means of S-matrix theory and incorporates both the known equation for an infinitely massive Brownian particle and the classical linear Boltzmann equation as limiting cases.
Abstract: The equation for the quantum motion of a Brownian particle in a gaseous environment is derived by means of S-matrix theory. This quantum version of the linear Boltzmann equation accounts nonperturbatively for the quantum effects of the scattering dynamics and describes decoherence and dissipation in a unified framework. As a completely positive master equation it incorporates both the known equation for an infinitely massive Brownian particle and the classical linear Boltzmann equation as limiting cases.