Showing papers on "Master equation published in 1993"
TL;DR: The complete quantum theory of covariant closed strings was constructed in detail in this article, where the nonpolynomial action is defined by elementary vertices satisfying recursion relations that give rise to Jacobi-like identities for an infinite chain of string field products.
748 citations
TL;DR: A quantum theory of feedback in which the homodyne photocurrent alters the dynamics of the source cavity is presented, and it is shown that under ideal conditions the noise spectra of the output light exhibit perfect squeezing on resonance.
Abstract: We present a quantum theory of feedback in which the homodyne photocurrent alters the dynamics of the source cavity. To the nonlinear stochastic (Ito) evolution of the conditioned system state we add a feedback term linear in the instantaneous stochastic (Stratonovich) photocurrent. Averaging over the photocurrent gives a feedbackk master equation which has the desired driftlike term, plus a diffusionlike term. We apply the model to phase locking a regularly pumped laser, and show that under ideal conditions the noise spectra of the output light exhibit perfect squeezing on resonance.
536 citations
TL;DR: Quantum Langevin equations and a master equation are derived for a two-atom system in which the first atom is driven by coherent field, and the fluorescent light is used to drive a second atom.
Abstract: Quantum Langevin equations and a master equation are derived for a two-atom system in which the first atom is driven by coherent field, and the fluorescent light used to drive a second atom. We show that the light beams from both atoms are antibunched, and that they are mutually anticorrelated.
374 citations
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01 Jan 1993TL;DR: Theoretical applications of non-equilibrium thermodynamics to solids have been investigated in this paper, where the authors evaluate nuclear magnetic relaxation rates and random walk theories of atomic diffusion.
Abstract: 1. Atomic movements in solids - phenomenological equations 2. Imperfections in solids 3. Statistical thermodynamics of crystals containing point defects 4. Non-equilibrium thermodynamics of atomic transport processes in solids 5. Some applications of non-equilibrium thermodynamics to solids 6. Microscopic theories - the master equation 7. Kinetic theory of relaxation processes 8. Kinetic theory of isothermal diffusion processes 9. The theory of random walks 10. Random walk theories of atomic diffusion 11. Transport coefficients of dilute solid solutions - results and applications 12. The evaluation of nuclear magnetic relaxation rates 13. Theories of concentrated and highly defective systems Epilogue References.
371 citations
TL;DR: It is shown that the evolution of an open quantum system whose density operator obeys a Markovian master equation can in some cases be meaningfully described in terms of stochastic Schrodinger equations (SSE’s) for its state vector.
Abstract: It is shown that the evolution of an open quantum system whose density operator obeys a Markovian master equation can in some cases be meaningfully described in terms of stochastic Schrodinger equations (SSE’s) for its state vector. A necessary condition for this is that the information carried away from the system by the bath (source of the irreversibility) be recoverable. The primary field of application is quantum optics, where the bath consists of the continuum of electromagnetic modes. The information lost from the system can be recovered using a perfect photodetector. The state of the system conditioned on the photodetections undergoes stochastic quantum jumps. Alternative measurement schemes on the outgoing light (homodyne and heterodyne detection) are shown to give rise to SSE’s with diffusive terms. These three detection schemes are illustrated on a simple quantum system, the two-level atom, giving new perspectives on the interpretation of measurement results. The reality of these and other stochastic processes for state vectors is discussed.
300 citations
TL;DR: The complete analytical solution of the cavity QED of a two-level atom and a field mode at zero temperature is presented, which includes both dissipation of the field due to a finite Q value ofThe cavity and incoherent decay mechanisms for the atom.
Abstract: We present the complete analytical solution of the cavity QED of a two-level atom and a field mode at zero temperature It includes both dissipation of the field due to a finite Q value of the cavity and incoherent decay mechanisms for the atom This analytical solution is provided by a powerful method for treating general master equations that appear in quantum optics As distinct from the usual approaches we first deal with that part of the master equation which describes the dissipative coupling of the field and the atom to their reservoirs Rather than using number-state or dressed-state bases we expand the density operator into the eigenstates of the nonunitary parts of the master equation which model the dissipative part of the dynamics The set of these eigenstates is the damping basis With the aid of this expansion we find the eigenvalues and eigenstates of the total Liouville operator The evolution of an arbitrary initial state is then known We employ these results to give an exact solution of the dynamics of the photon field in realistic experiments with one-atom masers at very low temperatures It includes detuning, cavity leakage effects, spontaneous decay mechanisms for the atoms, a Fizeau-type velocity distribution for the atomic beam, and a statistical parameter for the probability of the excitation of incoming atoms, covering the limits of Poissonian pumping and of regular pumping On the same grounds one can treat the one-atom laser, consisting of a single atom which stays in permanent interaction with the field mode and which is continuously pumped by external heat baths
213 citations
TL;DR: The master equation describing non-equilibrium one-dimensional problems like diffusion limited reactions or critical dynamics of classical spin systems can be written as a Schr\"odinger equation in which the wave function is the probability distribution and the Hamiltonian is that of a quantum chain with nearest neighbor interactions as discussed by the authors.
Abstract: The master equation describing non-equilibrium one-dimensional problems like diffusion limited reactions or critical dynamics of classical spin systems can be written as a Schr\"odinger equation in which the wave function is the probability distribution and the Hamiltonian is that of a quantum chain with nearest neighbor interactions. Since many one-dimensional quantum chains are integrable, this opens a new field of applications. At the same time physical intuition and probabilistic methods bring new insight into the understanding of the properties of quantum chains. A simple example is the asymmetric diffusion of several species of particles which leads naturally to Hecke algebras and $q$-deformed quantum groups. Many other examples are given. Several relevant technical aspects like critical exponents, correlation functions and finite-size scaling are also discussed in detail.
210 citations
IBM1
TL;DR: In this paper, it was shown that the lattice Boltzman equation for hydrodynamics can be extended in such a way as to describe non-relativistic quantum mechanics.
193 citations
TL;DR: In this paper, it was shown that a systematic Markovian approximation yields a given new term to the known master equation, ensuring conservation of positivity for arbitrary initial conditions and for all times.
Abstract: It is shown that a systematic Markovian approximation yields a given new term to the known master equation, ensuring conservation of positivity for arbitrary initial conditions and for all times.
128 citations
TL;DR: In this article, two additional damping terms in the master equation have been derived and shown to assure the Lindblad form of the equation, and a transient term in the density matrix has been found and expressed in a simple form.
Abstract: The Markovian approximation of quantum dissipation has been reconsidered in the model of Caldeira and Leggett [Physica A 121 (1983) 587]. Their high temperature master equation has been generalized to medium temperatures. Two additional damping terms in the master equation have been derived and shown to assure the Lindblad form of the equation. A transient term in the density matrix has been found and expressed in a simple form.
121 citations
TL;DR: In this article, a method for studying thermal relaxation in multiminima energy landscapes is introduced, where all the configurations connected to a given energy minimum by paths never exceeding a chosen energy lid are found, each equipped with a set of pointers to its neighbours.
Abstract: A method is introduced for studying thermal relaxation in multiminima energy landscapes. All the configurations connected to a given energy minimum by paths never exceeding a chosen energy lid are found, each equipped with a set of pointers to its neighbours. This information defines a phase space pocket around the minimum, in which the master equation for the relaxation process is directly solved. As an example we analyse some instances of the Travelling-Salesman Problem. We find that i) the number of configurations accessible from a given suboptimal tour grows exponentially with the energy lid, ii) the density of states within the pocket also shows exponential growth, iii) the low-temperature dynamical behaviour is characterized by a sequence of local equilibrations in increasingly larger regions of phase space and finally iv) the propagator decays algebraically with a temperature-dependent exponent. These observations are related to both theoretical models and experimental findings on relaxation in complex systems.
TL;DR: In this article, the authors considered a discrete, one-dimensional, heterogeneous and anisotropic phase space and proposed a master equation for the kinetics of classical particles that obey the exclusion principle.
Abstract: In this paper we analyze the kinetics of classical particles which obey an exclusion principle (EP) in the only-individual-transitions (OIT) approximation, and separately in the more rigorous contemporary-transitions (CT) description. In order to be able to include the EP into the kinetics equations we consider a discrete, one-dimensional, heterogeneous and anisotropic phase space and, after defining the reduced transition probabilities, we write a master equation. As a limit to the continuum of this master equation we obtain a generalized Fokker-Planck (FP) equation. This last is a nonlinear partial differential equation and reduces to the standard FP equation if the nonlinear term, which takes into account the EP, is neglected. The steady states of this equation, both in the OIT approximation and CT description, are considered. In the particularly interesting case of Brownian particles as a steady state in the OIT approximation we obtain the Fermi-Dirac (FD) distribution, while in the CT description we obtain another distribution which differs slightly from that of the FD. Moreover, our approach permits us to treat in an alternative and efficient way the problem of the determination of an effective potential to simulate the exclusion principle in classical many-body equations of motion.
Book•
01 Dec 1993
TL;DR: In this article, the origin of the Yang-Baxter equation was investigated and the quantum enveloping algebras quantum Clebsch-Gordan co-efficients simple YB equation trigonometric and rational solutions non-generic q values.
Abstract: Mathematical preliminaries origin of the Yang-Baxter equation classical Yang-Baxter equation quantum enveloping algebras quantum Clebsch-Gordan co-efficients simple Yang-Baxter equation trigonometric and rational solutions non-generic q values.
TL;DR: It is shown that truly continuous position measurements lead to a quantum Zeno effect in certain parameter regimes, manifest by the changing of the particle dynamics from coherent tunneling between the well minima to incoherent flipping, as in a random telegraph.
Abstract: We present a model of continuous (in time) position measurements on a quantum system using a single pseudoclassical meter. The nonselective evolution of the system is described by a master equation which is identical to that obtained from previous models. The selective evolution is described by a stochastic nonlinear Schrodinger equation. The significance of this equation is that the stochastic term has a physical interpretaion. By carefully choosing the parameters which define the meter and the system-meter coupling, we obtain a meter pointer with well-defined position which undergoes fluctuations. This ‘‘jitter’’ in the pointer position gives rise to the stochastic dynamical collapse of the system wave function. By the inclusion of feedback on the meter, the pointer is made to relax towards an appropriate readout. We apply this model to the selective measurement of the position of a particle in a double-well potential. In contrast to a recent claim [H. Fearn and W. E. Lamb, Jr., Phys. Rev. A 46, 1199 (1992)] we show that truly continuous position measurements lead to a quantum Zeno effect in certain parameter regimes. This is manifest by the changing of the particle dynamics from coherent tunneling between the well minima to incoherent flipping, as in a random telegraph. As the measurement strength increases, the average length of time the particle remains stuck in one well increases proportionally.
TL;DR: It is found that the one-atom laser exhibits most of the typical features of a normal laser, but in the region far below threshold some aspects, among them the linewidth, are changed due to eigenvalues of the master equation with imaginary parts.
Abstract: We present a detailed numerical study of the one-atom laser, that is, a single two-level atom interacting with one lasing mode, whereby both the atom and the photon field are coupled to reservoirs. The stationary as well as the dynamical properties of the model are calculated directly from the quantum master equation with the help of two numerical methods. These numerical methods do not need any quasi-probability representation and they do not require approximations. We find that the one-atom laser exhibits most of the typical features of a normal laser. In the region far below threshold some aspects, among them the linewidth, are changed due to eigenvalues of the master equation with imaginary parts
TL;DR: It is demonstrated that substantial coherence is retained when such light interacts with a two-photon-absorbing reservoir and is responsible for the preservation of squeezing in the steady state despite the effect of dissipation.
Abstract: We investigate the dynamical evolution of nonclassical states of light undergoing a two-photon absorption process. We consider two distinct cases of initial states, a squeezed coherent state and an eigenstate of the two-photon annihilation operator (a superposition of macroscopically distinct coherent states). We analyze the fluctuations in the photon-number operator and in the quadrature components of the field. Whereas one-photon linear damping rapidly destroys quantum features such as squeezing, we demonstrate that substantial coherence is retained when such light interacts with a two-photon-absorbing reservoir. This surviving coherence is responsible for the preservation of squeezing in the steady state despite the effect of dissipation. We relate the origin of squeezing of initially unsqueezed light interacting with two-photon absorbers with the squeezing generated by simple superposition states of light.
TL;DR: In this paper, a configurational master equation for the description of behavioral changes by pair interactions of individuals is developed, and three kinds of social pair interactions are distinguished: avoidance processes, compromising processes, and imitative processes.
Abstract: In the last decade, stochastic models have shown to be very useful for quantitative modelling of social processes. Here, a configurational master equation for the description of behavioral changes by pair interactions of individuals is developed. Three kinds of social pair interactions are distinguished: Avoidance processes, compromising processes, and imitative processes. Computational results are presented for a special case of imitative processes: the competition of two equivalent strategies. They show a phase transition that describes the self-organization of a behavioral convention. This phase transition is further analyzed by examining the equations for the most probable behavioral distribution, which are Boltzmann-like equations. Special cases of Boltzmann-like equations do not obey the H-theorem and have oscillatory or even chaotic solutions. A suitable Taylor approximation leads to the so-called game dynamical equations (also known as selection-mutation equations in the theory of evolution).
TL;DR: In this article, the antifields of the Batalin-Vilkovisky lagrangian quantization are standard antighosts of certain collective fields, which ensure that Schwinger-Dyson equations are satisfied as a consequence of the gauge symmetry algebra.
TL;DR: The weak interaction of such a field with a heat bath of arbitrary temperature is shown to preserve the Gaussian form of the characteristic function, and simple analytic formulas for the counting distribution and its factorial moments are derived.
Abstract: We consider a single-mode radiation field initially in a displaced squeezed thermal state. The weak interaction of such a field with a heat bath of arbitrary temperature is shown to preserve the Gaussian form of the characteristic function. Accordingly, the study of the time development of the density operator reduces to our previous description [P. Marian and T. A. Marian, preceding paper, Phys. Rev. A 47, 4474 (1993)] of the initial quantum state. As examples, photon statistics and squeezing properties of the damped field are analyzed. Based on the close relation between field dissipation and photon detection, we derive simple analytic formulas for the counting distribution and its factorial moments. Nonclassical features of a displaced squeezed thermal state, such as oscillations of the photon-number distribution, survive in the counting process, provided that the quantum efficiency of the detector is high enough.
TL;DR: In this paper, the authors studied the quantal dynamics of a harmonically driven quartic double well in the presence of dissipation and derived a master equation for the reduced density operator in the Floquet representation.
Abstract: We study the quantal dynamics of a harmonically driven quartic double well in the presence of dissipation. A master equation for the reduced density operator is derived in the Floquet representation. In the classical limit, this system corresponds to a Duffing oscillator with Ohmic damping. We present numerical results for the transient time evolution and for the stationary sate. The influence of the weak dissipation on interference effects in the context of driven tunnelling is discussed on the basis of these results. We find that the coherent suppression of tunnelling can be stabilized by reservoir-induced noise for a suitably chosen temperature.
TL;DR: In this paper, a Fock-space formalism is proposed which allows to get dynamical equations for averaged quantities from a master equation on a lattice with occupation numbers 0 and 1.
Abstract: A Fock-space formalism is proposed which allows to get dynamical equations for averaged quantities from a master equation on a lattice with occupation numbers 0 and 1. Kinetic equations for restricted diffusion, non-penetrating diffusion-limited aggregation (DLA) and reaction-limited interface growth (RLA) are derived. We find a critical concentration of immobile particles above which diffusion disappears. The new DLA equations lead to a stable steady-state solution. As a special case of RLA the Kardar-Parisi-Zhang equation can be obtained.
TL;DR: An approximate solution is given for the Jaynes-Cummings model with cavity losses, i.e., the problem of a two-level atom interacting with a single mode of the quantized radiation field, in the rotating-wave approximation, when the field is damped by a reservoir at zero temperature.
Abstract: An approximate solution is given for the Jaynes-Cummings model with cavity losses, i.e., the problem of a two-level atom interacting with a single mode of the quantized radiation field, in the rotating-wave approximation, when the field is damped by a reservoir at zero temperature. The approximate solution is derived for initial coherent field states with moderately large numbers of photons. It is simpler in form than earlier results derived by other authors and, over the appropriate parameter range, substantially more accurate than some of them, as shown by direct numerical integration of the master equation. In particular, it is found that an earlier treatment of this problem based on a secular approximation is seriously flawed, in that the conditions for its validity are much more restrictive than was previously believed. Among the results derived it is shown that, just as for the lossless case, when the atom is initially prepared in one of the semiclassical eigenstates the evolution is very simple, with the field and the atomic dipole drifting together in phase. For moderate losses this leads, as in the lossless case, to a ``state preparation''; i.e., to a good approximation, the state of the atom at a specific time can be made independent of its initial state. The effect of losses on the recently discovered ``Schr\"odinger cat'' state of the field is also analyzed. It is found that, although the dissipation destroys the coherence of the macroscopic superposition very rapidly, preparation and observation of the ``cat'' should be possible with the cavity quality factors reported in recent micromaser experiments.
TL;DR: Using the Wigner function formalism in phase space, it is shown that the decay rate of quantum coherences in phase-sensitive reservoirs can be significantly modified compared to the Decay rate in ordinary (phase-insensitive) thermal reservoirs.
Abstract: Using the Wigner function formalism in phase space, we analyze the decay of quantum coherences in phase-sensitive reservoirs. We show that the decay rate of quantum coherences in phase-sensitive reservoirs can be significantly modified compared to the decay rate in ordinary (phase-insensitive) thermal reservoirs. Depending on the phases of the quantum system (field mode) and the squeezed reservoir, the decay rate of the quantum coherence can be either enhanced or significantly suppressed, which is in agreement with the results obtained recently by other methods [T. A. B. Kennedy and D. F. Walls, Phys. Rev. A 37, 152 (1988)]. We show that in an ideally squeezed reservoir with a high degree of squeezing, the decay rate of the quantum coherence (i.e., the decay rate of off-diagonal terms of the density matrix in the coherent-state basis) can be equal to the decay rate of the energy of the system (i.e., the decay rate of diagonal terms of the density matrix). Suppression of the decay rate of the quantum coherence leads to preservation of nonclassical effects such as the oscillations in the photon number distribution. Moreover, we find that some initial superposition states of light exhibiting super-Poissonian photon statistics can be transformed into intermediate sub-Poissonian states under the influence of phase-sensitive reservoirs.
TL;DR: This paper examines theoretically the recent experiments of Lange and Walther on the dynamical interaction of Rydberg atoms in a microwave cavity in the presence of a strong driving field and derives the master equation for the atomic dynamics by adiabatically eliminating the cavity-field variables while treating the intense driving field nonperturbatively.
Abstract: We examine theoretically the recent experiments of Lange and Walther on the dynamical interaction of Rydberg atoms in a microwave cavity in the presence of a strong driving field. In particular, we study how the intense field renormalizes the cavity-induced spontaneous emission. For this purpose we derive the master equation for the atomic dynamics by adiabatically eliminating the cavity-field variables, while treating the intense driving field nonperturbatively. We present analytical and numerical solutions of the master equation, taking into account the turn on and turn off of the atom-field coupling in the rest frame of the atoms, as well as the velocity distribution of the atomic beam. We obtain good agreement between theoretical results and experiments.
TL;DR: In this article, a nonlinear stochastic differential equation for density matrices or for pure vectors is introduced, where the driving noise appearing in such an equation is not an external one, but its probability law is determined by the system itself (it is the probability measure on the trajectory space given by the theory of continuous measurements).
Abstract: In recent years a consistent theory describing measurements continuous in time in quantum mechanics has been developed. The result of such a measurement is a“trajectory”for one or more quantities observed with continuity in time. Applications are connected especially with detection theory in quantum optics. In such a theory of continuous measurements one can ask what is the state of the system given that a certain trajectory up to timet has been observed. The response to this question is the notion ofa posteriori states and a“filtering”equation governing the evolution of such states: this turns out to be a nonlinear stochastic differential equation for density matrices or for pure vectors. The driving noise appearing in such an equation is not an external one, but its probability law is determined by the system itself (it is the probability measure on the trajectory space given by the theory of continuous measurements).
TL;DR: In this article, a model neural network that is composed of two-state neurons obeying elementary stochastic transition rates is considered and a master-equation approach is proposed to obtain concise expressions for multipoint moments and an equation of motion.
Abstract: A master-equation approach to the stochastic neurodynamics proposed by Cowan [in Advances in Neural Information Processing Systems 3, edited by R. P. Lippman, J. E. Moody, and D. S. Touretzky (Morgan Kaufmann, San Mateo, 1991), p. 62] is investigated in this paper. We deal with a model neural network that is composed of two-state neurons obeying elementary stochastic transition rates. We show that such an approach yields concise expressions for multipoint moments and an equation of motion. We apply the formalism to a (1+1)-dimensional system. Exact and approximate expressions for various statistical parameters are obtained and compared with Monte Carlo simulations.
01 Dec 1993
TL;DR: It is shown that such an approach to the stochastic neurodynamics proposed by Cowan yields concise expressions for multipoint moments and an equation of motion and is applied to a (1+1)-dimensional system.
TL;DR: In this article, a technique for the solution of the master equation and of the diffusion equation via a random walk simulation is proposed and applied to a variety of problems of interest in the fields of physics, biology, and chemistry.
TL;DR: In this paper, a model of the stochastic behavior of the electrochemical interface is presented when electrochemical reactions limited by diffusion of the reacting species occur on the electrode surface, where the random fluctuations of the state variables (concentrations and voltage) are supposed to come from Poisson elementary noise sources which are directly acting on the elementary fluxes, either reactive or diffusive, of the particles.
Abstract: A model of the stochastic behavior of the electrochemical interface is presented when electrochemical reactions limited by diffusion of the reacting species occur on the electrode surface. The random fluctuations of the state variables (concentrations and voltage) are supposed to come from Poisson elementary noise sources which are directly acting on the elementary fluxes, either reactive or diffusive, of the particles. The evolution of the state variables is governed by Langevin equations obtained from the linearization of the nonlinear electrochemical equations deduced from the heterogeneous electrochemical kinetics. The Langevin noise sources are derived from the Poisson elementary noise sources. Their first and second order moments are calculated and appear to be coherent with the master equation approach. Finally, the control and observation variables are defined for the electrochemical interface and the power spectral density of the observable quantity (current or potential) are derived from the Poi...
TL;DR: In this article, a stochastic analysis of a chemical system obeying to mass-action kinetics and giving rise to deterministic chaos is carried out using the master equation approach.
Abstract: The stochastic analysis of a chemical system obeying to mass‐action kinetics and giving rise to deterministic chaos is carried out using the master equation approach. Quantitative comparison with the predictions of the deterministic description based on the rate equations shows that the system’s attractor and the probabilistic properties are robust toward fluctuations.