The technique of describing scattering solutions of the Schrödinger equation by means of $L^2$ functions is illustrated for a variable coupling constant in a simple separable potential model which admits an exact solution. The orthogonal polynomials generated by the $L^2$ solution are shown to be positive definite only for a limited range of the coupling constant. Further, it is shown that the underlying Gaussian quadrature rule for the positive-definite region can be extended to the general case albeit with the consequence that the weights are no longer all positive. A simple expression for the weights in terms of the associated abscissas is derived for the model. To complete the analysis it is demonstrated that a renormalization procedure of the $L^2$ functions to recover the scattering solutions is justified for arbitrary values of the coupling constant. Finally, some numerical examples are presented to test the Heller equivalent weight prescription and it is concluded that it works for all ranges of coupling strengths in the model.

We transform our quantum theory of multiwave mixing, which is based on a reduced density operator, to the corresponding nondegenerate quantum-noise Langevin equations. We derive the diffusion coefficients from the drift coefficients using the generalized Einstein relation and illustrate the resulting quantum Langevin theory of multiwave mixing by specializing to the Reid and Walls degenerate theory of squeezed states of light.

Forward nondegenerate four-wave mixing in an optical-fiber ring resonator is proposed as a method to generate squeezed states of light. The nonlinear interactions are analyzed both with a self-consistent propagation-equation technique and with Fokker-Planck equations in the Glauber-Sudarshan P representation. Excellent squeezing is predicted at modest input power levels, with perfect quantum-noise squeezing at the critical points for optical bistability. A method to suppress the stimulated Brillouin effect is proposed and demonstrated experimentally, and the effects of forward spontaneous guided acoustic wave Brillouin scattering inside the resonator are analyzed. Methods are suggested for minimizing this noise under conditions where squeezing can be detected. Experimental apparatus and procedures are outlined for verifying the predictions of our theory and demonstrating squeezing of classical and quantum noise.

The phenomenon of parametric sideband amplification observed in modulated semiconductor lasers can be understood in terms of the general amplification mechanism occurring in bifurcating dynamical systems. An analysis of the semiconductor laser rate equations, viewed in this light, yields new insight into the behavior of these systems as small-signal amplifiers.

A Lie-group-theoretical approach to the analysis of interferometers is presented. Conventional interferometers such as the Mach-Zehnder and Fabry-Perot can be characterized by SU(2). We introduce a class of interferometers characterized by SU(1,1). These interferometers employ active elements such as four-wave mixers or degenerate-parametric amplifiers in their construction. Both the SU(2) and SU(1,1) interferometers can in principle achieve a phase sensitivity Δφ approaching 1/N, where N is the total number of quanta entering the interferometer, provided that the light entering the input ports is prepared in a suitable quantum state. SU(1,1) interferometers can achieve this sensitivity with fewer optical elements.

Externally injected class-B lasers are shown to be approximately described by a reversible three-dimensional flow. For critical values of the external field such a model displays symmetry-breaking bifurcations, where the structure of space changes from conservative to dissipative either in a continuous or discontinuous manner. As a consequence, the coexistence of the two behaviors is observed in suitable parameter ranges. Symmetry breaking is shown to explain the onset of the very stable periodic solutions exhibited by the physical system. Such bifurcations are then studied in the simpler context of two-dimensional flows and recognized as degenerate codimension-2 phenomena. Generic normal forms are finally introduced which describe the local bifurcation unfolding.

The output of an externally pumped passive nonlinear-optical ring resonator is known to exhibit chaotic dynamics under variation of one or more external control parameters. The dynamics of the internal transverse laser beam profile is described, in the good-cavity limit, by an infinite-dimensional discrete-time map in function space. We find that the asymptotic dynamical behavior of the internal resonator field is in marked contrast to earlier plane-wave predictions. Instead, self-focusing, self-defocusing nonlinearities, and linear diffraction, combined with the pump and feedback of the ring resonator, give rise to strong spatial modulation of the transverse profile. For a self-focusing nonlinearity, spatially coherent robust transverse solitonlike structures persist even though the beam is undergoing temporally chaotic motion. Using attractor embedding techniques on our numerical solutions, we isolate distinct routes to optical turbulence under variation of a number of different control parameters. In particular, we provide examples of Ruelle-Takens-Newhouse sequences, intermittency, and a nongeneric bifurcation involving period doubling of invariant circles. A main result of the paper is that many few-dimensional attractors can coexist in different regions of the infinite-dimensional phase space at fixed values of the external control parameters. These attractors are accessed by varying the systems initial conditions and can, under variation of an appropriate control parameter, independently undergo transition to chaos. Interaction between neighboring attractors appears to be responsible for numerically observed departures from generic behavior. We note striking similarities between some of our numerically generated instability sequences and recent experimental observations in low-aspect-ratio fluid systems.

We consider the one-mode quantum-statistical model of optical bistability, and show that atom-atom correlations are negligible in the good-cavity limit, in all effects that can be described by a linearized treatment. In the light of this result, we discuss the concept of ‘‘atomic cooperation’’ in quantum optical systems. In the purely absorptive case, the atomic polarization exhibits a squeezing effect which arises both in the good- and in the bad-cavity limit. This phenomenon has the same origin of the squeezing effect in resonance fluorescence.

Correlations between the emission of photons from an excited three-level atom with one transition more strongly driven than the other are studied. Antibunching, anticorrelation, and superbunching are exhibited in the four possible configurations: ‘‘vee’’ (V), ‘‘lambda’’ (Λ), and two cascades with the upper or lower transition strongly driven. We pay particular attention to the projection of the atom into particular states by the measurement of fluorescence, and its significance for detecting quantum jumps in the weak transition by observing the fluorescence from the strong transition. For incoherent excitation we find that cross antibunching (the anticorrelation between emissions of different types of photons) manifests itself as prolonged periods of darkness. For the V and the cascade configuration with a strongly driven lower transition, these periods herald the emission of a weak-transition photon; for the other cascade and the Λ configuration the periods are heralded by the emission of a weak-transition photon. For on-resonance coherent excitation, dark periods are extremely infrequent for the former two configurations, and are extremely long or infinite for the latter two.

We study the scaling behavior of points on the Feigenbaum attractor away from the peak. We label these points by ‘‘CS sequences’’ which are related to the binary representations on a Cantor set. [A ‘‘CS sequence’’ (‘‘change’’-‘‘same’’) is based on whether a term in an initial ‘‘binary sequence’’ changes from or remains the same as the previous term.] A periodic point on the attractor has an infinite CS sequence with cyclic ending. In the neighborhood of a period-N point, $2^n$ iterations lead asymptotically to a family of N universal functions and a new scaling parameter α. We obtain the first few families of universal functions and their scale parameters both by direct iterations and by renormalization-group calculations.

We test the validity of Hamilton-equation methods for determining the time evolution of trial state vectors in quantum mechanics. Given a trial state vector, we are able to construct a differential operator under which a scalar Hamilton function must be invariant. State vectors composed of single-particle states, coherent-state products, and mixed single-particle states and coherent-state products are considered explicitly. In the latter category, we consider state vectors of the form proposed by Davydov in his treatment of the quantum soliton problem. We find that Davydov’s wave vector, as determined by the Hamilton-equation method, is not a solution of the Schrödinger equation for the Fröhlich Hamiltonian except under very restrictive circumstances. The theoretical justification for a number of conclusions about soliton transport in Fröhlich-type systems is thus called into question.

We consider two applications of a method introduced by Davydov for the purpose of modeling energy transport in deformable molecular chains. By specializing to the immobile-exciton limit in which the quantum-mechanical problem is exactly solvable, each application of the method is shown to suffer disagreement with the exact dynamics. Expectation values providing measures of the chain deformation and of the energy content of the chain vibrations are calculated explicitly, as are inner products of model state vectors with corresponding exact state vectors. We show that predictions based on Davydov’s methods may differ substantially from the corresponding exact results under experimentally relevant conditions.

The Minkowski stress tensor is statistically derived from the Maxwell-Lorentz field equations and the Liouville equation for a system of composite particles subject to an electromagnetic field. The derivation is subject to the condition that the fluctuations of field vectors are not statistically correlated. It is also shown by deriving the electromagnetic-field-momentum balance equation and the mechanical-momentum balance equation that the electromagnetic field momentum may be taken with $c^{\mathrm{-}1}$D×B instead of $c^{\mathrm{-}1}$E×H. The pressure in ponderable media is shown to contain an electromagnetic contribution: p=$p_0$+(D⋅E+B⋅H)/2, where $p_0$ is the pressure in the case of no field. The electromagnetic contribution does not contain an adjustable, or undetermined, parameter unlike that in the existing theories.

In this paper we report an extensive study of the relevant static and dynamic parameters connected to microscopic order and reorientational motion of single molecules in liquid-crystalline mesophases. In particular, we have studied the temperature dependence of the second- and fourth-order parameters 〈$P_2$〉 and 〈$P_4$〉, and of the spinning and tumbling diffusion coefficients $D_{?}^r$ and $D_{\perp }^r$ in the nematic phase of several mesogens; some smectic phases were studied also. These results were obtained using a combined theoretical-experimental method newly developed by us, which allows the simultaneous determination of both static and dynamic parameters from the study of Raman and infrared band shapes by Fourier-transform analysis. The results are discussed in the framework of reorientational diffusion processes in anisotropic potentials. Implications on the nature of the nematic-isotropic phase transition are also given.

We numerically study intermittency associated with the coexistence of multiple attractors in the damped, driven pendulum. For some ranges of the control parameters the basins of attraction for attractors with positive and negative average angular velocities are intricately interwoven, and the boundary between basins is a fractal set. We observe intrinsic intermittency due to a crisis, in which two chaotic attractors collide with the fractal boundary that divides their basins of attraction. For control parameters near the crisis, the fractal dimension of the basin boundaries approaches that of the phase space, and a random external forcing torque easily induces extrinsic intermittency. Both noise- and crisis-induced intermittency can produce power spectra S(ω)∝1/${\omega }^{\alpha }$, with αapeq21, over several decades in frequency ω.

It is pointed out that for temperatures T<0.5$T_c$, where $T_c$ is the critical temperature, the classical steady-state nucleation-rate formalism of Becker and Doring predicts an approximate critical supersaturation ratio $S_{\mathrm{cr}}$ (for the onset of nucleation) given by ln$S_{\mathrm{cr}/{\Omega }^{3/2}\sim 0.53(T_c/T\mathrm{-}{1)}^{3/2}\mathrm{.}\$\mathrm{OMEGA}—\mathrm{is}a\mathrm{material}-\mathrm{dependent}\mathrm{quantity}\mathrm{approximately}\mathrm{equal}\mathrm{to}\mathrm{the}\mathrm{excess}\mathrm{surface}\mathrm{entropy}\mathrm{per}\mathrm{molecule}.\mathrm{For}\mathrm{most}\mathrm{substances}\mathit{\Omega }\mathit{\sim }2.0\mathrm{and}\mathrm{for}\mathrm{associated}\mathrm{liquids}\mathit{\Omega }\mathit{\sim }1.5.Theexperimentaldata(\mathrm{for}\mathrm{nucleation}\mathrm{from}\mathrm{vapor}\mathrm{to}\mathrm{liquid})\mathrm{from}\mathrm{diffusion}\mathrm{chamber}\mathrm{and}\mathrm{nozzle}\mathrm{beam}\mathrm{studies}\mathrm{are}\mathrm{found}\mathrm{to}\mathrm{be}\mathrm{consistent}\mathrm{with}\mathrm{the}\mathrm{above}\mathrm{expression}.\mathrm{The}\mathrm{classical}\mathrm{theory}\mathrm{also}\mathrm{predicts}\mathrm{that}\mathrm{for}a\mathrm{supersaturation}\mathrm{ratio}\mathit{S}\mathrm{corresponding}\mathrm{to}\mathrm{constant}\mathit{J},ln\mathit{S}/\ln S_{\mathrm{cr}\mathrm{-}1\sim \ln \mathit{J}/2\ln J_c}}$, where ln$J_c$ is a quantity evaluated at the critical point and is ∼72 for most materials. Expansion cloud-chamber data for nonane, toluene, and water are also found to be consistent with these approximate scaling laws.

The path-probability method (PPM) is applied for the first time to the time development of the fluctuation from the most probable path, where the kinetic Ising ferromagnetism in the pair approximation is treated as an example. The evolution equations of the variance as well as the average motion of order parameters are derived and compared critically with the results obtained by the master-equation method (MEM). In the pair approximation, the PPM results agree completely with those of the MEM. This agreement is to be compared with the disagreement in the triangle approximation presented in Wada et al. (the following paper).

Comparison of the path-probability method (PPM) with the master-equation method (MEM) is extended to a higher degree of approximation (triangle approximation) for the kinetics of Ising ferromagnetism. The evolution equations of the averages and of the fluctuations around them are derived analytically in this high degree of approximation for the first time for both methods. Contrary to the case of pair approximation presented in Ishikawa et al. (the preceding paper), it was not possible to bring the evolution equations derived by the MEM into a complete agreement with those by the PPM even with the use of the most reasonable closure relations.

We have developed a time-dependent, collisional-radiative model to calculate radiative power and electron cooling rates for oxygen at intermediate densities (${10}^{16}$≤$n_e$≤${10}^{20}$ ${\mathrm{cm}}^{\mathrm{-}3}$) where the usual coronal approximation is not valid. Large differences from coronal values are predicted. The behavior of the steady-state radiative-power-loss coefficient, $L_z$, is investigated as the electron density is increased. Generalized power-loss coefficients applicable to transient plasmas are derived and applied to ionizing and recombining oxygen plasmas. Time-dependent effects are found to play a large role both in terms of the total radiated power and the net electron-energy-loss rate.

We show that the imaginary-time Green’s functions of a system of quantum charged particles at equilibrium satisfy a set of sum rules which express the screening of a classical external charge. We also provide a new derivation of the perfect screening property ${\lim }_{\mathrm{k}\to 0}$${\mathrm{\epsilon }}^{\mathrm{-}1}$(k)=0 where ε(k) is the static dielectric function. The proof relies on exact equations for the Green’s functions and certain assumptions of spatial clustering. It is nonperturbative, independent of the statistics, and applies to jellium as well as to multicomponent systems.