Showing papers on "Phase space published in 1990"
TL;DR: In this article, a general algorithm for constructing coherent states of dynamical groups for a given quantum physical system is presented, and the result is that the coherent states are isomorphic to a coset space of group geometrical space.
Abstract: In this review, a general algorithm for constructing coherent states of dynamical groups for a given quantum physical system is presented. The result is that, for a given dynamical group, the coherent states are isomorphic to a coset space of group geometrical space. Thus the topological and algebraic structure of the coherent states as well as the associated dynamical system can be extensively discussed. In addition, a quantum-mechanical phase-space representation is constructed via the coherent-state theory. Several useful methods for employing the coherent states to study the physical phenomena of quantum-dynamic systems, such as the path integral, variational principle, classical limit, and thermodynamic limit of quantum mechanics, are described.
1,354 citations
TL;DR: In this paper, a general definition of local symmetries on the manifold of field configurations is given that encompasses, as special cases, the usual gauge transformations of Yang-Mills theory and general relativity.
Abstract: The general relationship between local symmetries occurring in a Lagrangian formulation of a field theory and the corresponding constraints present in a phase space formulation are studied. First, a prescription—applicable to an arbitrary Lagrangian field theory—for the construction of phase space from the manifold of field configurations on space‐time is given. Next, a general definition of the notion of local symmetries on the manifold of field configurations is given that encompasses, as special cases, the usual gauge transformations of Yang–Mills theory and general relativity. Local symmetries on phase space are then defined via projection from field configuration space. It is proved that associated to each local symmetry which suitably projects to phase space is a corresponding equivalence class of constraint functions on phase space. Moreover, the constraints thereby obtained are always first class, and the Poisson bracket algebra of the constraint functions is isomorphic to the Lie bracket algebra of the local symmetries on the constraint submanifold of phase space. The differences that occur in the structure of constraints in Yang–Mills theory and general relativity are fully accounted for by the manner in which the local symmetries project to phase space: In Yang–Mills theory all the ‘‘field‐independent’’ local symmetries project to all of phase space, whereas in general relativity the nonspatial diffeomorphisms do not project to all of phase space and the ones that suitably project to the constraint submanifold are ‘‘field dependent.’’ As by‐products of the present work, definitions are given of the symplectic potential current density and the symplectic current density in the context of an arbitrary Lagrangian field theory, and the Noether current density associated with an arbitrary local symmetry. A number of properties of these currents are established and some relationships between them are obtained.
833 citations
TL;DR: In this paper, it was shown that the topological properties of the symplectic diffeomorphism groups are related to the topology of the phase space geometry of the manifold.
Abstract: In this paper we show that symplectic maps have surprising topological properties. In particular, we construct an interesting metric for the symplectic diffeomorphism groups, which is related, but not obviously, to the topological properties of symplectic maps and phase space geometry. We also prove a certain number of generalised symplectic fixed point theorems and give an application to a Hamiltonian system.
412 citations
TL;DR: In this article, it was shown that any hamiltonian system admits a Lax representation, at least locally at generic points in phase space, and introduced the most general Poisson bracket ensuring the involution property of the integrals of motion and existence of a lax pair.
302 citations
TL;DR: In this paper, the photon number distribution is shown to display unusual oscillations which are interpreted as interference in phase space, analogous to Franck-Condon oscillations in molecular spectra, and the possibility of detecting these oscillations is discussed, through the photodetection counting statistics of the displaced number states.
Abstract: Recent developments in quantum optics have led to new proposals to generate number states of the electromagnetic field using conditioned measurement techniques or the properties of atom-field interactions in microwave cavities in the micromaser. The number-state field prepared in such a way may be transformed by the action of a displacement operator; for the microwave micromaser state this could be implemented by the action of a classical current that drives the cavity field. We evaluate some properties of such displaced number states, especially their description in phase space. The photon number distribution is shown to display unusual oscillations, which are interpreted as interference in phase space, analogous to Franck-Condon oscillations in molecular spectra. The possibility of detecting these oscillations is discussed, through the photodetection counting statistics of the displaced number states. We show that the displaced-number-state quantum features are relatively robust when dissipation of the field energy is included.
286 citations
01 Jan 1990
TL;DR: In the present context we understand chaos to refer to irregular fluctuation that is, however, described by deterministic equations, as distinct from indeterminate fluctuations that obeys the definitions of randomness.
Abstract: In the present context we understand chaos to refer to irregular fluctuation that is, however, described by deterministic equations, as distinct from indeterminate fluctuation that obeys the definitions of randomness (see Fig. 10.1a). The problem of nonlinear dynamics originates in planetary motion. Henri Poincare was the first to investigate the complex behavior of simple mathematical systems. He analyzed topological structures in phase space and discovered that the equation for the motion of planets could display an irregular or chaotic motion (Poincare 1892). A mathematical basis for this behavior was later given by Birkhoff (1932).
266 citations
TL;DR: This work considers the problem of prediction and system identification for time series having broadband power spectra that arise from the intrinsic nonlinear dynamics of the system and finds that the parameter values that minimize the least-squares criterion do not, in general, reproduce the invariants of the dynamical system.
Abstract: We consider the problem of prediction and system identification for time series having broadband power spectra that arise from the intrinsic nonlinear dynamics of the system. We view the motion of the system in a reconstructed phase space that captures the attractor (usually strange) on which the system evolves and give a procedure for constructing parametrized maps that evolve points in the phase space into the future. The predictor of future points in the phase space is a combination of operation on past points by the map and its iterates. Thus the map is regarded as a dynamical system and not just a fit to the data. The invariants of the dynamical system, the Lyapunov exponents and optimum moments of the invariant density on the attractor, are used as constraints on the choice of mapping parameters. The parameter values are chosen through a constrained least-squares optimization procedure, constrained by the values of these invariants. We give a detailed discussion of methods to extract the Lyapunov exponents and optimum moments from data and show how to equate them to the values for the parametric map in the constrained optimization. We also discuss the motivation and methods we utilize for choosing the form of our parametric maps. Their form has a strong similarity to the work in statistics on kernel density estimation, but the goals and techniques differ in detail. Our methodology is applied to "data" from the H\'enon map and the Lorenz system of differential equations and shown to be feasible. We find that the parameter values that minimize the least-squares criterion do not, in general, reproduce the invariants of the dynamical system. The maps that do reproduce the values of the invariants are not optimum in the least-squares sense, yet still are excellent predictors. We discuss several technical and general problems associated with prediction and system identification on strange attractors. In particular, we consider the matter of the evolution of points that are off the attractor (where few or no data are available), onto the attractor where long-term motion takes place. We find that we are able to realize maps that give a least-squares approximation to the data with rms variation over the attractor of 0.5% or less and still reproduce the dynamical invariants to 5% or better. The dynamical invariants are the classifiers of the dynamical system producing the broadband time series in the first place, so this quality of the maps is essential in representing the correct dynamics.
213 citations
TL;DR: In this article, a multimode Brownian oscillator model is used to account for high-frequency molecular vibrations and local intermolecular modes as well as collective solvent motions.
Abstract: A theory for ultrafast pump-probe spectroscopy of large polyatomic molecules in condensed phases is developed. A multimode Brownian oscillator model is used to account for high-frequency molecular vibrations and local intermolecular modes as well as collective solvent motions. A semiclassical picture is provided using the density matrix in Liouville space. The pump field creates a doorway state that propagates for a specified time interval, and the spectrum is calculated by finding its overlap with a window state, prepared by the probe pulse. The doorway and the window states are wave packets in phase space. For high-frequency modes and with long pulses they are expanded in the vibronic eigenstates, whereas for low-frequency modes and with impulsive pulses the Wigner (phase-space) representation is more adequate. Conditions for the observation of quantum beats, spectral diffusion, and solvation dynamics (dynamical Stokes shift) are specified.
211 citations
TL;DR: In this article, a new type of molecular dynamics is proposed to solve approximately the many-body problem of interacting identical fermions with spin, where the interacting system is represented by an antisymmetrized manybody wave function consisting of single-particle states which are localized in phase space.
205 citations
TL;DR: In this article, the Husimi distribution is computed for a particle in a double-well potential and an oscillatory driving force, and coherent tunneling between two disjoint stable tubes of regular orbits is found at a rate many orders of magnitude greater than the rate of ordinary undriven tunneling.
Abstract: The Husimi distribution is computed for a particle in a double-well potential and an oscillatory driving force. The extended phase space of the classical system contains two disjoint stable tubes of regular orbits, embedded in a chaotic sea. For the quantum system we find coherent oscillatory tunneling between these stability tubes, at a rate many orders of magnitude greater than the rate of ordinary undriven tunneling.
198 citations
TL;DR: In this paper, the authors explore the potential of the homogeneous relaxation model (HRM) as a basis for the description of adiabatic, one-dimensional, two-phase flows.
Abstract: The paper explores the potential of the homogeneous relaxation model (HRM) as a basis for the description of adiabatic, one-dimensional, two-phase flows. To this end, a rigorous mathematical analysis highlights the similarities and differences between this and the homogeneous equilibrium model (HEM) emphasizing the physical and qualitative aspects of the problem. Special attention is placed on a study of dispersion, characteristics, choking and shock waves. The most essential features are discovered with reference to the appropriate and convenient phase space Ω for HRM, which consists of pressure P , enthalpy h , dryness fraction x , velocity w , and length coordinate z . The geometric properties of the phase space Ω enable us to sketch the topological pattern of all solutions of the model. The study of choking is intimately connected with the occurrence of singular points of the set of simultaneous first-order differential equations of the model. The very powerful centre manifold theorem allows us to reduce the study of singular points to a two-dimensional plane Π , which is tangent to the solutions at a singular point, and so to demonstrate that only three singular-point patterns can appear (excepting degenerate cases), namely saddle points, nodal points and spiral points. The analysis reveals the existence of two limiting velocities of wave propagation, the frozen velocity a f and the equilibrium velocity a e . The critical velocity of choking is the frozen speed of sound. The analysis proves unequivocally that transition from ω a f to w > a f can take place only via a singular point. Such a condition can also be attained at the end of a channel. The paper concludes with a short discussion of normal, fully dispersed and partly dispersed shock waves.
Journal Article•
TL;DR: Emploi d'un modele d'oscillateur brownien multimode pour traiter les vibrations moleculaires de haute frequence, les modes locaux intermoleculaires et les mouvements collectifs du solvant dans l'espace de Lionville.
Abstract: A theory for ultrafast pump-probe spectroscopy of large polyatomic molecules in condensed phases is developed. A multimode Brownian oscillator model is used to account for high-frequency molecular vibrations and local intermolecular modes as well as collective solvent motions. A semiclassical picture is provided using the density matrix in Liouville space. The pump field creates a doorway state that propagates for a specified time interval, and the spectrum is calculated by finding its overlap with a window state, prepared by the probe pulse. The doorway and the window states are wave packets in phase space. For high-frequency modes and with long pulses they are expanded in the vibronic eigenstates, whereas for low-frequency modes and with impulsive pulses the Wigner (phase-space) representation is more adequate. Conditions for the observation of quantum beats, spectral diffusion, and solvation dynamics (dynamical Stokes shift) are specified.
TL;DR: In this paper, the role of periodic trajectories and other classical structures on single eigenfunctions of the quantized version of the baker's transformation is studied in phase space by means of a special positive definite representation adapted to discreteness of the map.
TL;DR: In this paper, the phase flow is attracted to a unique trajectory, the slow manifold M, before it reaches the point equilibrium of the system, and the line set M is found by solution of a functional equation derived from the flow differential equations.
Abstract: The time evolution of two model enzyme reactions is represented in phase space Γ. The phase flow is attracted to a unique trajectory, the slow manifold M, before it reaches the point equilibrium of the system. Locating M describes the slow time evolution precisely, and allows all rate constants to be obtained from steady‐state data. The line set M is found by solution of a functional equation derived from the flow differential equations. For planar systems, the steady‐state (SSA) and equilibrium (EA) approximations bound a trapping region containing M, and direct iteration and perturbation theory are formally equivalent solutions of the functional equation. The iteration’s convergence is examined by eigenvalue methods. In many dimensions, the nullcline surfaces of the flow in Γ form a prism‐shaped region containing M, but this prism is not a simple trap for the flow. Two of its edges are EA and SSA. Perturbation expansion and direct iteration are now no longer equivalent procedures; they are compared in a...
TL;DR: In this article, the authors present methods to compute thermodynamic properties of classical systems which involve extending the phase space by two degrees of freedom, which are used to replicate the coupling of the original system to the infinite degrees of free space of a heat bath.
TL;DR: In this article, the authors considered the quantisation of the two-dimensional toric and spherical phase spaces in analytic coherent state representations, and showed that the distribution of the zeros in the phase space becomes one-dimensional for integrable systems, and highly spread out (conceivably uniform) for chaotic systems.
Abstract: The quantisation of the two-dimensional toric and spherical phase spaces is considered in analytic coherent state representations. Every pure quantum state admits therein a finite multiplicative parametrisation by the zeros of its Husimi function. For eigenstates of quantised systems, this description explicitly reflects the nature of the underlying classical dynamics: in the semiclassical regime, the distribution of the zeros in the phase space becomes one-dimensional for integrable systems, and highly spread out (conceivably uniform) for chaotic systems. This multiplicative representation thereby acquires a special relevance for semiclassical analysis in chaotic systems.
TL;DR: In this article, the authors studied the field entropy in two-photon cases and linked the fluctuations in the field phase to the changes in field entropy, and also calculated the statistical Q function of the field and showed how the periodicity of the twophoton dynamics is linked to a periodic splitting of the Q function in phase space.
Abstract: The excitation of Rydberg atom transitions by submillimeter-wavelength radiation in high-Q cavities forms the basis of the micromaser. The excitation dynamics of a micromaser is known to be dependent on the detailed photon statistics in the interaction cavity and can exhibit well-known collapses and revivals of the atomic inversion, dipole moment, and photon number. We study these effects in a two-photon model in which the time evolution is exactly periodic. We study the field entropy in two two-photon cases and link the fluctuations in the field phase to the changes in the field entropy. We also calculate the statistical Q function of the field and show how the periodicity of the two-photon dynamics is linked to a periodic splitting of the Q function in phase space. Finally this periodicity is linked to the nature of the atom-field dressed states involved in two-photon resonance.
TL;DR: In this article, the Weinhold-Ruppeiner metric is derived from the microscopic entropy of the thermodynamic phase space, and a non-degenerate bilinear form on the phase space is constructed, whose restriction to Gibbs space can serve as an alternative to the metric proposed by Gilmore.
Abstract: We show how both the contact structure and the metric structure of the thermodynamic phase space arise in a natural way from a generalized canonical probability distribution \ensuremath{\rho} In particular, the metric form and the contact form are found to be derived from the microscopic entropy s=-ln\ensuremath{\rho} Thus the first law and the second law of thermodynamics can be given the geometric interpretation that a thermodynamic system must possess both a contact and a compatible metric structure We proceed to construct explicitly a new nondegenerate bilinear form on the thermodynamic phase space, whose restriction to state space yields the Weinhold-Ruppeiner metric, and whose restriction to Gibbs space can serve as an alternative to the metric proposed by Gilmore
TL;DR: In this article, it was shown that the multiplicity distribution in a simple model (α-model) realized by a self-similar random cascade process in d=2 (or d=3) dimensions of momentum space has intermittency in the d-dimensional analysis but disappears in the one-dimensional projection.
TL;DR: A discrete, topological criterion for phase-space localization is presented, analogous to a quantized Hall conductivity, which when nonzero reflects phase- space delocalization.
Abstract: We study quantized classically chaotic maps on a toroidal two-diensional phase space. A discrete, topological criterion for phase-space localization is presented. To each eigenfunction an integer is associated, analogous to a quantized Hall conductivity, which when nonzero reflects phase-space delocalization. A model system is studied, and a correspondence between delocalization and chaotic classical dynamics is discussed.
TL;DR: In this paper, a generalized Husimi transform is used to obtain a phase space representation of the time-dependent Schrodinger equation directly from the coordinate representation, which governs the time evolution of densities such as the Husimi density.
Abstract: We present a time evolution equation that provides a novel basis for the treatment of quantum systems in phase space and for the investigation of the quantum‐classical correspondence. Through the use of a generalized Husimi transform, we obtain a phase space representation of the time‐dependent Schrodinger equation directly from the coordinate representation. Such an equation governs the time evolution of densities such as the Husimi density entirely in phase space, without recourse to a coordinate or momentum representation. As an application of the phase‐space Schrodinger equation, we compute the eigenfunctions of the harmonic oscillator in phase space, relate these to the Husimi transform of coordinate representation eigenstates, and investigate the coherent state, its time evolution, and classical limit (ℏ→0) for the probability density generated by this state. Finally, we discuss our results as they relate to the quantum‐classical correspondence, and quasiclassical trajectory simulations of quantum d...
TL;DR: A general expression in 1+1 dimensions for the phase and the associated gauge potential is derived, and the application of this formalism to the classical, continuous, antiferromagnetic Heisenberg spin chain is discussed.
Abstract: We show that the time evolution of a space curve is associated with a geometric phase. This phase arises from the path dependence of the rotation of the natural Frenet-Serret triad with respect to a nonrotating (Fermi-Walker) frame. We derive a general expression in 1+1 dimension for the phase and the associated gauge potential, and discuss the application of this formalism to the classical, continuous, antiferromagnetic Heisenberg spin chain.
TL;DR: In this paper, it was shown that in the (2 k − 1)-dimensional level sets of the Hamiltonian of a class of k-degree-of-freedom Hamiltonian systems, stable and unstable manifolds of normally hyperbolic invariant (2k − 3)-dimensional spheres form partial barriers to transport.
TL;DR: In this paper, the properties of almost Kahlerian manifolds are studied, making reference to the formalism developed in Part I to formulate Schrodinger quantum mechanics, and they are studied in terms of the properties and properties of Kahlerians in general.
Abstract: Making reference to the formalism developed in Part I to formulate Schrodinger quantum mechanics, the properties of Kahlerian functions in general, almost Kahlerian manifolds, are studied.
TL;DR: Its phase space is divided into transmitting and absorbing regions, which are responsible for the abrupt termination of a period-doubling sequence and the onset of a locking regime, in which an appropriately defined winding number has intriguing scaling properties.
Abstract: We consider a ball under the influence of gravity on a vibrating platform where the ball-platform collisions are completely inelastic. We present for the first time several remarkable features in the temporal behavior of the system; its phase space is divided into transmitting and absorbing regions, which are responsible for the abrupt termination of a period-doubling sequence and the onset of a locking regime, in which an appropriately defined winding number has intriguing scaling properties
TL;DR: In this article, a method is developed for estimating the transport rates of phase space areas for a class of two-dimensional diffeomorphisms and flows, defined by the topological structure of their stable and unstable manifolds, and hence are universal.
TL;DR: The phase space portrait of a cosmological model with a scalar field coupled to curvature is discussed in detail, analytically and numerically, for any value of the coupling constant ξ and any power law (ϕ2n) potential.
Abstract: The Phase Space portrait of a cosmological model with a scalar field coupled to curvature is discussed in detail, analytically and numerically, for any value of the coupling constant ξ and any power law (ϕ2n) potential. The results, particularly intuitive from the graphical point of view, generalize previous studies on the phase space with minimal coupling (ξ = 0) and quadratic or quartic potentials to the entire parameter space (ξ, n). We find global inflationary attractors, often in analytical form, with or without the correct Friedmannian limit. If the coupling constant is negative, escaping regions may occur, while, if it is positive, a forbidden region cuts out a large part of the phase space. Semiclassical instability of vacuum states and singularity-free trajectories are also discussed.
TL;DR: In this paper, a generalization of the local-to-normal transition seen in symmetric triatomics is considered for nonsymmetric molecules and 2:1 Fermi resonance systems.
Abstract: The generalization of the local‐to‐normal transition seen in symmetric triatomics is considered for nonsymmetric molecules and 2:1 Fermi resonance systems. A straightforward generalization based on a division of phase space into local and normal regions is not possible. Instead, classification of the phase space bifurcation structure is presented as the complete generalization of the local–normal concept for all spectroscopically relevant systems of two vibrations interacting via a single nonlinear resonance. The polyad phase sphere (PPS) is shown to be the natural arena to analyze the bifurcation structure for resonances of arbitrary order. For 1:1 and 2:1 resonances, the bifurcation problem is reduced to one or two great circles on the phase sphere. All bifurcations are shown to be examples of elementary bifurcations of vector fields in one dimension. The classification of the bifurcation structure is therefore governed and greatly simplified by the theory of the universal unfolding and codimension of e...
TL;DR: In this paper, a catastrophe manifold is used to classify the dynamics of spectra of resonantly coupled vibrations, based on earlier work on the bifurcation structure of the Darling-Dennison and 2:1 Fermi resonance fitting Hamiltonians.
Abstract: Catastrophe theory is used to classify the dynamics of spectra of resonantly coupled vibrations, based on earlier work on the bifurcation structure of the Darling–Dennison and 2:1 Fermi resonance fitting Hamiltonians. The goal is a generalization of the language of the ‘‘normal–local transition’’ to analyze experimental spectra of general resonant systems. The set of all fixed points of the Hamiltonian on the polyad phase sphere for all possible molecular parameters constitutes the catastrophe manifold. The projection of this manifold onto the subspace of molecular parameters is the catastrophe map. The map is divided into zones; each zone has its own characteristic phase sphere structure. The taxonomy of global phase sphere structures within all zones gives the classification of the semiclassical dynamics. The 1:1 system, with normal–local transition, is characterized by cusp catastrophes, with elementary pitchfork bifurcations. In contrast, the 2:1 system is characterized by fold catastrophes, with elem...
TL;DR: This work shows that the kicked rotator on a torus model exhibits localization effects which produce deviations from random-matrix-theory predictions which display a scaling behavior which is a counterpart of the scaling theory of one-dimensional Anderson localization in finite samples.
Abstract: The kicked rotator on a torus is a system with a bounded phase space in which a chaotic diffusion occurs for a large enough perturbation strength. The quantum version of this model exhibits localization effects which produce deviations from random-matrix-theory predictions. We show that these localization effects display a scaling behavior which is a counterpart of the scaling theory of one-dimensional Anderson localization in finite samples. We suggest that this behavior can be highly relevant to some general problems of quantum chaos.