A simple approximate expression is obtained for the critical coupling strength at which an array of N evanescent coupled semiconductor lasers will lose stability. The critical coupling decreases with increasing N but reaches a limiting value for large N which is one-half of the value for N=2.

A simple and direct approach to studying squeezed states is proposed. It is based on treating the corresponding differential equations and their solutions directly on the Fock-Bargmann space. The method is exemplified by giving quite general solutions to the case of amplitude-squared squeezed states. We show the importance of this approach by deriving the overlap between these solutions and connecting them with the overlap integral associated with the Franck-Condon principle.

The practical aspect of the moment problem for a density matrix for single-mode radiation is solved; namely, two simple explicit expressions for the density matrix in terms of its moments are presented. A biorthogonal set of operators is established, so that two bases are available for expressing an arbitrary operator.

A simple difference-differential equation derived by Scully and Lamb [Phys. Rev. 179, 368 (1969)] is used to describe the photoelectric detection of the radiation in a cavity totally isolated except for its interaction with the detector. As results of the solution, the expressions for the distributions of unconditional and conditional waiting times are obtained. Based on the definition that we have antibunching whenever the conditional waiting time is greater than the unconditional waiting time, the criterion for the occurrence of antibunching is established.

Using a Debye-potential approach we present a theoretical formulation of surface-emitting second-harmonic generation (SESHG) from an optically trapped microsphere with Mie size. A useful analytical expression for evaluating the conversion efficiency is derived. With the surface-emitting scheme where nonlinear interactions among the whispering-gallery cavity modes (fundamental waves) and the scattering mode (second-harmonic wave) are considered, problems that would arise in the phase matching among discrete cavity modes are avoidable. Through numerical calculations for the SESHG from a ${\mathrm{LiNbO}}_3$ microsphere in water, we predict drastic enhancement of the process for certain sizes of the sphere.

The Infeld-Hull recurrence relation for the radial part of the dipole matrix elements in atomic hydrogen is considered. Conditions are investigated under which reduction of this relation to the differential equation of Price and Harmin [Phys. Rev. A 42, 2555 (1990)] is justified.

We thank P Braun [preceding Comment, Phys. Rev. A 46, 6108 (1992)] for clarifying how errors can arise in attempting to use the differential equation beyond its region of applicability. The principal results of our paper are correct for their stated domains.

We present a continuation method for experimentally tracking unstable periodic orbits by slowly varying an available system parameter in a dynamical system. The method does not depend on an explicit model, but on the signal analysis of a measured time series. Unstable periodic orbits can be tracked through various bifurcations. We apply this to a Duffing-like circuit and compare the results to an approximate model of the circuit.

We consider wave systems in which rays split on reflection from sharp boundaries. Examples include the Schrödinger equation with the potential changing discontinuously across a surface, electromagnetic waves in a region with a discontinuous dielectric constant, elastic media with a clamped or free boundary, etc. By introducing a Monte Carlo treatment of the rays, it is possible to define chaotic rays via the standard Lyapunov number criterion. Numerical implementation of the Monte Carlo ray technique is carried out for the example of elastic media, and is utilized to investigate the extent to which these systems are globally ergodic. It is suggested that results from previous extensive work on quantum chaos without ray splitting can be extended to these ray splitting problems. In particular, we indicate a generalization of the Gutzwiller trace formula to cover ray splitting.

Extensions of the one-dimensional two-state trough model introduced by Carlson, Chayes, Grannan, and Swindle (CCGS) [Phys. Rev. A 42, 2467 (1990)] are considered. In particular, I investigate what kinds of physical processes are relevant to its scaling behavior. Short-range rearrangements of trough positions (slide events), which were neglected by CCGS, are shown to be irrelevant. By a simple modification of the dynamics, however, I obtain universality classes characterized by a single parameter. For trough models in general, including the two-state and the ‘‘limited local’’ sandpile models, asymptotically exact relations between the distribution of trough-trough distances and that of the mass of avalanches are derived. They yield moment relations in agreement with Krug’s [J. Stat. Phys. 66, 1635 (1992)]. All results are verified by simulations.

Transient properties of the Lotka-Volterra model perturbed by external white noise have been investigated. Stochastic metastability was observed for a degenerate case in which both order parameters shared the same rates and initiated from the common deterministic fixed point. Near this unique state, fluctuations have been shown analytically to be diverging with time, while mean values and the Lyapunov function remained unchanged. Fluctuations in order parameters due to additive noises have been found to have the same time dependence as Brownian particles. Those due to multiplicative noises have shown deviations at larger times and for larger noise intensities. The time duration over which this metastable state could persist has been estimated for three types of noises. Noise effects associated with different nonlinearities were compared systematically and explicitly.

We investigate the relaxation of multilevel systems under the influence of stochastic fields with finite correlation time ${\mathrm{\tau }}_{\mathit{c}}$. First, we study a system with spin S=1/2 in a static magnetic field in the presence of dichotomic perturbations. The spectra of the components of the averaged spin vector S¯ are evaluated exactly by resorting to the memory-function approach. The model is then extended to include the influence on the relaxation of S¯ of a different species with spin I=1/2. The relaxation rates ${\mathit{T}}_1$ and ${\mathit{T}}_2$ are given in terms of cumulant expansions and compared with the exact results. Fourth-order calculations recover the inequality ${\mathit{T}}_2$/2${\mathit{T}}_1$≤1 for correlated noise fields. Furthermore, it is found that ${\mathit{T}}_1$ can be defined also for long ${\mathrm{\tau }}_{\mathit{c}}$. ${\mathit{T}}_1$ is longer than ${\mathrm{\tau }}_{\mathit{c}}$ provided that ${\mathit{r}}^{\mathrm{}\left(\mathrm{est}\right)\mathrm{}}$<1, where ${\mathit{r}}^{\mathrm{}\left(\mathrm{est}\right)\mathrm{}}$ is the squared ratio between the total amplitude of the fluctuating fields and the largest one between the Larmor frequency and 1/${\mathrm{\tau }}_{\mathit{c}}$. It is also found that 2${\mathit{r}}^{\mathrm{}\left(\mathrm{est}\right)\mathrm{}}$ is an upper bound of the relative error between the exact and the second-order cumulant expansion of ${\mathit{T}}_1$. Generalizations on systems with S=1/2, I>1/2, and arbitrary kinds of noise are discussed.

We study coupled systems whose elements are chaotic maps, with the coupling ranging from ‘‘local’’ (with the interaction spreading over K neighbors) to ‘‘global’’ (mean-field-type coupling). We find that well-defined peaks emerge in the power spectrum of the mean field, indicating a subtle coherence among the elements, as the extent of coupling, i.e., K, is increased. We observe that the significant quantity here is not the ratio of the number of elements coupled K, to lattice size N, but the magnitude of K. After a critical value of K equal to ${\mathit{K}}_{\mathit{c}}$, the coupling takes on a ‘‘global’’ character, and is practically indistinguishable from mean-field interaction. Interestingly, the value of ${\mathit{K}}_{\mathit{c}}$≊${\mathit{N}}_{\mathit{c}}$, where ${\mathit{N}}_{\mathit{c}}$ is the critical lattice size after which the power spectra in globally coupled systems saturate. We also find that the mean-square deviation of the mean field grows linearly with coupling strength, up to ${\mathit{K}}_{\mathit{c}}$.

Topological entropy is defined and its behavior investigated near bifurcations of chaotic scattering. The result shows that for different types of bifurcations the topological entropy exhibits distinct behavior.

The fragmentation properties of percolation clusters yield information about their structure. Monte Carlo simulations and exact cluster enumeration for a square bond lattice and exact calculations for the Bethe lattice are used to study the fragmentation probability ${\mathrm{a}}_{\mathrm{s}}$(p) of clusters of mass s at an occupation probability p and the likelihood ${\mathrm{b}}_{\mathrm{s}\prime \mathrm{s}}$(p) that fragmentation of an s cluster will result in a daughter cluster of mass s′. Evidence is presented to support the scaling laws ${\mathrm{a}}_{\mathrm{s}}$(${\mathrm{p}}_{\mathrm{c}}$)∼s and ${\mathrm{b}}_{\mathrm{s}\prime \mathrm{s}}$(${\mathrm{p}}_{\mathrm{c}}$)=${\mathrm{s}}^{\mathrm{-}\mathrm{\phi }}$g(s′/s), with φ=2-σ given by the standard cluster-number scaling exponent σ. Simulations for d=2 verify the finite-size-scaling form ${\mathrm{c}}_{\mathrm{s}\prime \mathrm{sL}}$(${\mathrm{p}}_{\mathrm{c}}$)=${\mathrm{s}}^{1\mathrm{-}\mathrm{\phi }}$g̃(s′/s,s/${\mathrm{L}}^{{\mathrm{d}}_{\mathrm{f}}}$) of the product ${\mathrm{c}}_{\mathrm{s}\prime \mathrm{s}}$(${\mathrm{p}}_{\mathrm{c}}$)=${\mathrm{a}}_{\mathrm{s}}$(${\mathrm{p}}_{\mathrm{c}}$)${\mathrm{b}}_{\mathrm{s}\prime \mathrm{s}}$(${\mathrm{p}}_{\mathrm{c}}$), where L is the lattice size and ${\mathrm{d}}_{\mathrm{f}}$ is the fractal dimension. Exact calculations of the fragmentation probability ${\mathrm{f}}_{\mathrm{st}}$ of a cluster of mass s and perimeter t indicate that branches are important even on the maximum perimeter clusters. These calculations also show that the minimum of ${\mathrm{b}}_{\mathrm{s}\prime \mathrm{s}}$(p) near s′=s/2, where the two daughter masses are comparable, deepens with increasing p.

We consider successive measurements on quantum-mechnical systems and investigate the way in which sequences of measurements produce information. The eigenvalues of suitable projection operators form symbolic sequences that characterize the quantum system under consideration. For several model systems with finite-dimensional state space, we explicitly calculate the probabilities to observe certain symbol sequences and determine the corresponding Rényi entropies K(β) with the help of the transfer-matrix method. A nontrivial dependence on β is observed. We show that the Rényi entropies as well as the symbol-sequence probabilities of the quantum-mechanical measurement process coincide with those of appropriate classes of one-dimensional chaotic maps.

Gravitational scattering of a single star by a stellar pair, in an initially circular orbit, is investigated numerically through the generation of initial-value space images showing final characteristics of the system versus initial values. This space is composed of regions for which simple deflection or exchange occurs after a single close approach between the star and the pair, and, separating these regions, a network of resonance ‘‘rivers,’’ for which the three stars become temporarily mutually bound, exhibiting complicated, chaotic behavior before one particle is eventually ejected. We discuss the general features of the initial-value space, with emphasis on the resonance regions, which are found to contain small-scale, bandlike structure, and explain the origin of this structure with a straightforward dynamical argument. An expression for the scaling of the bands is derived. High-resolution one-dimensional slices are taken through the initial-value space images, which indicate that the small-scale structure can be considered to be continuous in one direction and discontinuous in the transverse direction. Similarities with standard chaotic scattering systems are discussed. Finally, we relate the locations of the resonance regions to geometrical degeneracies in the positions of the three stars at close approach.

We investigate a mechanical model consisting of a one-dimensional chain of blocks connected by springs. Each block is in contact with a lower rough surface, and the chain is pulled at one end with a small constant velocity. This system was introduced by Burridge and Knopoff [Bull. Seismol. Soc. Am. 57, 341 (1967)] to model the dynamics of earthquakes. In a wide range of the parameter space, we observe the existence of self-organized criticality, that is, robust power-law distributions limited only by the size of the chain. The model is completely deterministic.

The properties of the anisotropic random sequential adsorption of dimers on a square lattice are determined by Monte Carlo simulation and time-series expansion. The fractions of vertical and horizontal bonds occupied by dimers are calculated as a function of v, the probability of choosing vertical bonds to place dimers. In the limit v→0, the asymptotic fraction of vertical bonds occupied by dimers does not vanish but has the exact value ${\mathit{e}}^{\mathrm{-}2}$[1-exp(-2${\mathit{e}}^{\mathrm{-}2}$)]/2=0.016 046. . . .

The mean-square radius of gyration 〈${\mathit{R}}_{\mathit{G}}^2$〉 and a shape parameter Σ=〈${\mathit{R}}_{\mathit{G}\min }^2$〉/〈${\mathit{R}}_{\mathit{G}\max }^2$〉 are studied as a function of the number of bonds, bends, and branches of self-avoiding lattice trees on the square, triangular, and honeycomb lattices. We identify the universality classes, and exhibit the crossover scaling functions that connect them. We find (despite doubts recently raised) that there is a universal crossover from rods to self-avoiding walks, embodied in 〈${\mathit{R}}_{\mathit{G}}^2$〉∼${\mathit{N}}^2$U(Nw), where w(z) is an appropriately chosen nonlinear scaling field reducing to the stiffness fugacity z as z→0 ; that ‘‘rigid trees’’ (which are bond clusters that branch but do not bend) are in the same universality class as branched polymers or free trees; that the crossover from rods to rigid trees has the universal form 〈${\mathit{R}}_{\mathit{G}}^2$〉∼${\mathit{N}}^2$W(${\mathit{Ny}}^2$), where y is the branching fugacity; and that the crossover from self-avoiding walks to branched polymers has the universal form 〈${\mathit{R}}_{\mathit{G}}^2$〉∼${\mathit{N}}_{\mathit{s}}^{2\nu }$Y(${\mathit{Ny}}^{\mathrm{\phi }}$), with ${\nu }_{\mathit{s}}$=3/4 and φ=55/32.