Robert Gilmore

Geometry of symmetrized states

Abstract A projection operator is defined in terms of coset representatives and “coset harmonics.” This operator is then used to construct fully symmetrized states describing N identical two-level atoms. The properties and quantum numbers of the symmetrized states are fully discussed. Fully symmetrized and labeled states for N identical 3 (or r) level atoms are constructed by a natural extension of this procedure. The interaction Hamiltonian, which describes transitions in an r-level atom, is constructed from the shift operators in the algebra of SU(r). When each of the N identical atoms “sees” the same field dependence, transitions are only allowed between energy eigenstates occurring within the same SU(r) unitary irreducible representation. Physical dispersion of the particles out of the coherence region of the field leads to “leakage” out of an SU(r) representation. If each two-level atom evolves from the ground state under the same field dependence, the total system state is a coherent superposition of bases belonging to the fully symmetric representation {N = 2J}2. These states exist in 1-1 correspondence with points in the surface of a unit sphere. They are therefore called Bloch states. The properties of Bloch states are derived and are seen to resemble the properties of coherent photon states. The connection is made manifest by a group contraction process. All properties of Glauber states can be constructed by contraction from the corresponding properties of the Bloch states. The states corresponding to N-identical r-level particles evolving under a coherent field are constructed in the same way. These states exist in 1-1 correspondence with points in the coset space SU(r) U(r−1) , which is a Riemannian-symmetric space. The eigenstate expansions, inner products, eigenvalue equations, and uncertainty relations for these states are derived. More generally, systems of N identical (p + q) level atoms, evolving from the lowest q levels under a coherent field, are described by states existing in 1 - 1 correspondence with Riemannian symmetric space SU(p + q) S[U(p)×U(q)] . The properties of these spaces are used to give a geometric analytic interpretation to these coherent atomic states.

The Topology of Chaos

Intro.-01 Intro.-02 Intro.-03 Exp’tal-01 Exp’tal-02 Exp’tal-03 Exp’tal-04 Exp’tal-05 Exp’tal-06 Exp’tal-07 Exp’tal-08 Embed-01 Embed-02 Embed-03 Embed-04a Embed-04b Topology of Orbits-01 Topology of Orbits-02 Topology of Orbits-03a Topology of Orbits-03b Topology of Orbits-04 Topology of Orbits-04a Topology of Orbits-04b Topology of Orbits-05a Topology of Orbits-05b Topology of Orbits-06 Topology of Orbits-07 Topology of Orbits-08 Topology of Orbits-09 Topology of Orbits-10 Topology of Orbits-11 Topology of Orbits-12 Topology of Orbits-13 Topology of Orbits-14 Topology of Orbits-15 Topology of Orbits-16 Topology of Orbits-17 Program-01 Program-02 Program-03 Program-04 Program-05 Program-06 Program-07 Program-08 Program-09a Program-09b Program-10a Program-10b Program-11 Program-12 Reps-01 Reps-02 Reps-03 Reps-04 Reps-05 Reps-06 Reps-07 Reps-08 Reps-09 Class-01 Class-02 Class-03 Class-04 Class-05 Bases-01 Bases-02 Bases-03 Bases-04 Tori-01 Tori-02 Tori-03 Tori-04 Tori-05 Tori-06 Tori-07 Tori-08 Tori-09 Tori-10 Tori-11 Tori-12 Tori-13 Tori-14 Tori-15 Tori-16a Tori-16b Tori-16c Tori-17 Tori-18 Tori-19 Tori-20 Summary-01 Summary-02 Summary-03 Summary-04 Summary-05 Summary-06 Summary-07 Summary-08 Summary-09 Summary-10 The Topology of Chaos

Topological analysis and synthesis of chaotic time series

Abstract We have developed a topological procedure for analyzing chaotic time series which identifies the stretching and squeezing mechanisms responsible for chaotic behavior in low-dimensional dynamical systems. These mechanisms, quantitatively described by a “template” or “knot-holder”, can then be used to model the processes which generate the original chaotic data set.

Topological analysis of chaotic time series data from the Belousov-Zhabotinskii reaction

SummaryWe have applied topological methods to analyze chaotic time series data from the Belousov-Zhabotinskii reaction. First, the periodic orbits shadowed by the data set were identified. Next, a three-dimensional embedding without self-intersections was constructed from the data set. The topological structure of that flow was visualized by constructing a branched manifold such that every periodic orbit in the flow could be held by the branched manifold. The branched manifold, or induced template, was computed using the three lowest-period orbits. The organization of the higher-period orbits predicted by this induced template was compared with the organization of the orbits reconstructed from the data set with excellent results. The consequences of the presence of certain knots found in the data are discussed.

Vector coherent state representation theory

A vector coherent state theory is formulated as a natural extension of standard coherent state theory. It is shown that the Godement representations and the coherent state representations of the Sp(N,R) groups of Rowe and of Deenen and Quesne are special cases of this more general theory.

Phase transitions in nuclear matter described by pseudospin Hamiltonians

Abstract Upper and lower bounds on the ground-state energy per nucleon E g N and the free energy per nucleon F(β) N are constructed for nuclear systems described by pseudospin Hamiltonians. In the limit of large numbers of nucleons these bounds become equal. A simple algorithm is developed for computing E g N , F(β) N and S N (entropy per nucleon) exactly in the N → ∞ limit. The values of E g N and F(β) N are obtained by computing the minimum value of associated potential functions hc and Φ. These potentials are constructed very simply from the pseudospin Hamiltonian. Groundstate energy phase transitions are determined by investigating how the minima of the potential hc change as a function of changing nuclear interaction parameters. Thermodynamic phase transitions are determined by investigating how the minima of the potential Φ change as a function of changing nuclear temperature. Concise conditions are given for the occurrence of a second-order phase transition of either type. These conditions define critical values of the nuclear interaction parameters at which a ground-state energy second-order phase transition occurs and the critical temperature at which a thermodynamic second-order phase transition occurs. A “crossover theorem” relates the occurrence of a ground-state energy phase transition to a thermodynamic phase transition.

An efficient algorithm for fast O(N∗ln(N)) box counting

Abstract A new topological ordering is defined which significantly reduces the time requirements for the fast box counting method proposed in a recent paper by Liebovitch and Toth. Only one sorting is necessary in this algorithm.

Influence of coexisting attractors on the dynamics of a laser system

Abstract We study the interaction among coexisting attractors in a laser model with modulated parameters, and show that these interactions lead to a deterministic truncation of the period doubling bifuraction sequence and to crises of chaotic attractors.

Group theoretical approach to semiclassical dynamics: Single mode case

A recently proposed procedure for computing the S matrix for collinear molecular collisions is extended from the single‐mode case to the multimode case. In this procedure the semiclassical Hamiltonian describing the molecular collision is integrated in a finite‐dimensional, faithful nonunitary representation of the dynamical group. When the integration is completed, the resulting group operation is mapped into the infinite‐dimensional unitary representation to describe the S matrix which acts on the molecular Hilbert space and describes collisional excitation of the internal vibrational modes. This procedure is implemented to study the two‐mode collision process N2+O2 and the two three‐mode collision processes N2+CO2 and NO+CO2. The results compare favorably with other treatments of these collinear collision processes.

Relative Rotation Rates for Driven Dynamical Systems

Relative rotation rates for two-dimensional driven dynamical systems are defined with respect to arbitrary pairs of periodic orbits. These indices describe the average rate, per period, at which one orbit rotates around another. These quantities are topological invariants of the dynamical system, but contain more physical information than the standard topological invariants for knots, the linking and self-linking numbers„ to which they are closely related. This definition can also be extended to include noisy periodic orbits and strange attractors. A table of the relative rotation rates for a dynamical system, its intertwining matrix, can be used to determine whether orbit pairs can undergo bifurcation and, if so, the order in which the bifurcations can occur. The relative rotation rates are easily computed and measured. They have been computed for a simple model, the laser with modulated parameter. By comparing these indices with those of a zero-torsion lift of a horseshoe return map, we have been able to determine that the dynamics of the laser are governed by the formation of a horseshoe. Additional stable periodic orbits, besides the principal subharmonics previously reported, are predicted by the dynamics. The two additional period-five attractors have been located with the aid of their logical sequence names, and their identification has been confirmed by computing their relative rotation rates.