Nonadiabatic effects on the structure of Rydberg states of alkaline-earth atoms are examined and found to be large in many cases. Calculations for the quantum defects of the singlet $F$ and $G$ series in calcium, including dynamical contributions, are shown to be in good agreement with recent experimental data. This is in contrast to predictions based on the adiabatic core-polarization model, which significantly overestimates the quantum defects. Values for quantum defects of Ca for $l\ge 4$ are presented.

We report measurements of the degree of linear polarization of scattered light as a function of excitation frequency near the Doppler-broadened $\mathrm{Na} D2$ resonance line. For resonant excitation of the $D2$ line, the hyperfine interaction strongly depolarizes the emitted light due to precession of the electronic angular momentum $\overrightarrow{\mathrm{J}}$ prior to spontaneous emission. However, as the laser frequency is tuned away from resonance, the polarization degree sharply rises to the level predicted using a simple $J{M}_J$ basis, i.e., neglecting any coupling of the electronic angular momentum $\overrightarrow{\mathrm{J}}$ with the nuclear spin $\overrightarrow{\mathrm{I}}$. Roughly speaking, there is a transition from $F$ selection rules ($\overrightarrow{\mathrm{F}}=\overrightarrow{\mathrm{I}}+\overrightarrow{\mathrm{J}}$) for resonant excitation to $J$ selection rules for off-resonant excitation. We present a rigorous calculation of the polarization degree with the use of multipole moments of the density matrix which agrees quantitatively with our data and also predicts complicated interference structure when Doppler broadening is negligible. We show that this structure may shift peaks observed in precision laser-spectroscopy experiments. For Doppler-broadened transitions we discuss a simplified approach (based on an approximation to the Voigt profile) which treats separately the contributions of resonant and off-resonant scattering.

We consider the sideband instabilities that can occur in free-electron lasers which operate in the regime where the bulk of the energetic electron beam is trapped in the ponderomotive potential generated by the beating of the electromagnetic pulse with the static wiggler field. The general dynamical equations which govern the evolution of the system in the presence of sideband modes are derived for modes propagating along the axis of symmetry. All space-charge effects are included in a self-consistent manner, and the gains for both the primary signal and the upper and lower sidebands are computed. We find the lower sideband to be excited, and discuss the consequences for operation of both free-electron-laser amplifiers and oscillators. The effect of the space-charge waves on the gain depends upon the number of bounces undergone by the trapped electrons in the length of the system, and can act either to enhance or reduce the gain of the sideband.

The Einstein-coefficient method is used to calculate the growth rate of a low-gain free-electron laser for a tenuous relativistic electron beam propagating in the combined axial and transverse helical wiggler fields $B_0{\hat{e}}_z-(\delta B cos{k}_{0}z{\hat{e}}_x+\delta B sin{k}_{0}z{\hat{e}}_y)$. The analysis assumes that the system is close to resonance between the electron-cyclotron frequency in the guide field (${\omega }_c=\frac{e{B}_0}{\gamma \mathrm{mc}}$) and the wiggler frequency in the beam frame (${\omega }_0=k_0{V}_b$). It is shown that the gain near resonance can be substantially enhanced relative to the gain obtained in the region far from resonance.

An S-conserving theory of collective resonance fluorescence by $N$ two-level atoms driven by a uniform strong field is presented. The detuning between field and atoms can be varied. A canonical transformation to dress the atoms by the driving field and Langevin equations for dressed operators are used to investigate the behavior of the system. For sufficiently large detuning the dressed atoms are shown to reach a steady state after a transient super-radiant-like decay in times short, compared with the long decay time which is attained on resonance and which is the same as the single-atom spontaneous-decay time. In the transient regime the parameters of the quasi-super-radiant decay are shown to vary with detuning and with the driving-field amplitude. Both the amplitude spectrum and the integrated intensity of the fluorescence field, emitted off resonance by the dressed atoms during the transient, are shown to bear pulselike traces of the quasi-super-radiant decay. The elastic components of the off-resonance amplitude spectrum at the end of the super-radiant-like transient are broadened with respect to resonance, and the integrated intensity exhibits a sudden transient decrease in an appropriate range of detuning, which has been given the name "superdarkening" and which disappears on resonance. After the steady-state regime is attained, the resonance intensity spectrum is shown to coincide with that obtained by other authors, while off resonance the inelastic lines are broadened in a super-radiant-like fashion.

The quantum theory of the three-energy-level two-mode laser in which all the levels are being pumped is investigated and the master equation of the laser operation is obtained. It contains two variables, $n_1$, the photon number of the first mode, and $n_2$, the photon number of the second mode. There appear four terms in the master equation which represent two-photon processes between the two modes, the absorption of a photon from one mode, and emission of a photon to the other. They are absent in the single-mode case and in the two-mode case with pumping only to the upper level. The master equation can be represented by a diagram of probability in two dimensions which can be extended to infinity, and each arrow of it represents a term on the right-hand side of the master equation. By summing each variable ($n_1$ or $n_2$) of the master equation, two equations can be obtained. Each equation contains only one variable, and can be expressed by a diagram of probability flow in one dimension. By considering the correspondence between macroscopic equilibrium and microscopic detailed balance the equations of motion in the steady state are obtained. By introducing a parameter $H$, the equation of motion is simplified and a formal solution can be deduced which is dependent on $H$. It appears that the parameter $H$ cannot be fully determined, but some properties of it can be deduced which prove that the introduction of $H$ is reasonable. The operation characteristics of the laser under different conditions are discussed on the basis of the formal solution: among them are the threshold condition, the condition for one-mode operation, and the change of the photon-statistical distribution. We especially discuss the laser output power curves under two-mode operation above threshold. The curve of one mode, which oscillates first, shows a bending-down phenomenon as the excitation increases when the other mode goes beyond its threshold as demonstrated experimentally by Otsuka. This phenomenon is explained qualitatively and is attributed to the two-photon processes between the two modes.

Time-dependent optical Autler-Townes spectra are calculated for a four-level system in which the ground state is a doublet with frequency separation much larger than any of the Rabi frequencies. The finite linewidths of the lasers are taken into account in the phase-diffusion model. Optical-pumping effects cause a transfer of population with a resulting significant change in the spectra with time. Numerically evaluated spectra are presented, and it is found that the nature of the asymmetry can change drastically with time.

The theory of coherent anti-Stokes Raman scattering (CARS) is extended to include the effect of pump fluctuations. The intensities and spectra of lines in resonant CARS are calculated to all orders in fields assuming a phase-diffusion model for waves at the two pump frequencies. The bandwidth of the two lasers enters in a much more complicated way than following a simple scaling of $T_1$ or $T_2$. Various resonances in CARS spectra due to dynamic splitting of the energy levels are discussed for a range of detunings, field intensities, and bandwidths. In contrast to the usual spectra in strong fields, the Rabi sidebands appear as dispersion-shaped structures. The laser linewidth is shown to change dramatically the CARS line shape. The case of no saturation is also treated, thus allowing for the inclusion of more general line shapes and fluctuations of the pump waves and the nonlinear susceptibility tensor ${\chi }^{\mathrm{}\left(3\right)\mathrm{}}$. Gaussian statistics of the pump field are shown to lead to enhancement factors in CARS intensity, similar to those appearing in the context of multiphoton absorption processes.

Photon statistics in some fully quantized models of the interaction of a two-level atom with a single-mode radiation field have been studied using the operator equations of motion. Expressions for the photon number distribution and the mean photon number are presented for various initial conditions. It is found that the mean photon number may show decays and revivals of coherence similar to those of the atomic inversion in the coherent-state Jaynes-Cummings model. Application of these models to the study of multiphoton laser, absorption, and emission processes is also discussed.

The main purpose of our work concerns the long-time behavior of initial distributions with diverging second moments and to show that in the mean the behavior is still exponential. We consider solutions $F(x,t)\mathrm{}$ of the Boltzmann equation for a Maxwell-like model for which $\int 0^{\infty }\mathrm{dx}{x}^{2}F(x,0\mathrm{})\mathrm{}=\infty$ and $\int 0^{\infty }\mathrm{dx}F(x,0\mathrm{})\mathrm{}=\int 0^{\infty }\mathrm{dx}\mathrm{xF}(x,0\mathrm{})\mathrm{}=1\mathrm{}$. It is shown that $F(x,t)\mathrm{}$ can be approximated arbitrarily well in the mean by $\overline{F}\mathrm{}(x,t)\mathrm{}$ for $0\le t<\mathrm{}\infty$ and $\overline{F}\mathrm{}(x,t)\mathrm{}\simeq e^{-x}[1+K{e}^{-\frac{t}{3}}{L}_2\mathrm{}(x\left)\right]$ as $t\to \infty$. Consequently, the Hilbert space of square-integrable functions, viz., $\int 0^{\infty }\mathrm{dx}{F}^2\mathrm{}(x,0\mathrm{})\mathrm{}<\infty$, is complete in the sense of convergence in the mean for $0\le t<\mathrm{}\infty$, and thus the time evolution of initial singularities—for instance, an $F(x,0\mathrm{})\mathrm{}$ with $\delta$-function peaks, jump discontinuities, etc.—is not physically meaningful.

The structure of a region of stable period three for a nonlinear dissipative system described by the Rössler equations and a two-parameter cubic map is studied. The intricate configuration of this region, which is bordered by intermittent-type chaos on one side, subharmonic cascades on the other, and possesses several sharp features, is shown to be associated with bistability and hysteresis of the orbits of the flow or map. Locally, the two-parameter cubic map successfully models many features of the differential flow. The mechanism which gives rise to the hysteresis is quite general and corresponds to a cusp catastrophe. This process is described in detail for the map and related to the same phenomenon in the flow.

Systems are considered that undergo a continuous forward Hopf bifurcation into an oscillatory steady state, breaking a continuous $U\mathrm{}\left(1\right)\mathrm{}$ symmetry. The control parameter is assumed to consist of a systematic constant part and a superimposed Gaussian white-noise part. The Fokker-Planck equation describing the dynamics of fluctuations near instability is solved exactly, generalizing earlier results for nonoscillatory bifurcation. A check of the completeness of the eigenfunctions by a sum rule is presented. Correlation functions and transient moments are evaluated from the Fokker-Planck solution on both sides of the bifurcation point and their asymptotic long-time behavior is discussed. Below the bifurcation point transient moments are found to exist whose decay constants are smaller than the smallest eigenvalue in the Fokker-Planck spectrum. Two methods for calculating such moments from the Fokker-Planck spectrum are presented and exemplified.

The dependence of the universal bifurcation ratio on the order of the local maximum is studied for a family of maps $f\mathrm{}\left(x\right)=1-a{x}^z$($z=2,4,6,8\mathrm{}$). Although these maps all exhibit period doubling, the precise bifurcation ratio depends on the order $z$, and seems to be an increasing function of $z$. We have employed a renormalization-group method to calculate $\delta \mathrm{}\left(z\right)\mathrm{}$ to second order. They agree very well with the exact values.

Experimental results are presented for atomic-capture probabilities of negative pions in organic molecules. The data are analyzed in terms of atomic and molecular models. This analysis shows that the Fermi-Teller law ($Z$ law) and its modifications do not give an adequate description of the data, but that a mesomolecular model together with hydrogen transfer contains the features essential to fit the data. Clear evidence is given for chemical effects in the pion-capture process.

Starting from a macroscopic nonequilibrium entropy, we obtain an expression for the nonequilibrium fluctuations of the electric current in a metallic resistor. Our method goes further than previous theories of irreversible thermodynamics and, as well as microscopic entropies, it leads to results of the same order of magnitude but not completely coincident with the full nonequilibrium corrections obtained from kinetic methods by Tremblay et al.

Renormalization-group (RNG) analysis, as it has been applied to infrared similarity ranges in hydrodynamic turbulence, corroborates some scaling exponents given by qualitative physical reasoning, but fails to make them more convincing. This is because local-in-wave-number interactions mediate most of the energy transfer, except in the case of asymptotic freedom in the infrared, and these interactions do not yield to low-order perturbation theory. Ultraviolet RNG analysis, which attempts to eliminate bands of modes at the low-wave-number end of a similarity range, meets the additional severe problem that renormalized mode amplitudes exhibit spurious decay due to phase mixing. Perturbative RNG analysis and pre-RNG renormalized perturbation theory both attempt to handle strong nonlinearity by chopping the dynamics into small pieces, but in different ways. Both procedures are limited by divergence, inaccessible analyticity properties, and lack of computability except in low orders.

A shock is assumed to be a steady plane wave, and irreversible thermodynamics is assumed valid. The fluid is characterized by heat conduction and by viscous or viscoelastic response, according to the strain rate. It is shown that setting the viscosity zero produces a solution which constitutes a lower bound through the shock process for the shear stress, and upper bounds for the temperature, entropy, pressure, and heat current. It is shown that there exists an upper bound to the dynamic stresses which can be achieved during shock compression, that this bound corresponds to a purely elastic response of the fluid, and that solution for the shock process along this bound constitutes lower bounds for the temperature and entropy. It is shown that a continuous steady shock is possible only if the heat current is positive and the temperature is an increasing function of compression almost everywhere. In his theory of shocks in gases, Rayleigh showed that there is a maximum shock strength for which a continuous steady solution can exist with heat conduction but without viscosity. Two more limits are shown to exist for dense fluids, based on the fluid response in the leading edge of the shock: for shocks at the overdriven threshold and above, no solution is possible without heat transport; for shocks near the viscous fluid limit and above, viscous fluid theory is not valid, and the fluid response in the leading edge of the shock is approximately that of a nonplastic solid. The viscous fluid limit is estimated to be 13 kbar for water and 690 kbar for mercury.

By an extension to the dense-fluid regime of a method based upon maximization of entropy subject to constraints, first exploited by Lewis to obtain the Boltzmann equation, kinetic equations for one-particle and two-particle classical distribution functions are obtained. For the hard-sphere potential and a one-particle constraint, the kinetic equation (for the one-particle distribution function) of the revised Enskog theory is obtained; for a two-particle constraint a more general kinetic equation (for the two-particle distribution function) than that studied by Livingston and Curtiss is obtained. For a pair potential with hard-sphere core plus smooth attractive tail, a new mean-field kinetic equation is obtained on the one-particle level. In the Kac-tail limit the equation takes the form of an Enskog-Vlasov equation. The method yields an explicit entropy functional in each case. Explicit demonstration of an $H$ theorem is made for the one-particle theories in a novel way that illustrates the roles of the reversible and irreversible parts of the hard-sphere piece of the collision integral. The latter part leads to the classical form of entropy-production density as described by linear irreversible thermodynamics and so possesses many of the features of the Boltzmann collision integral. The former part introduces new elements into the entropy-production term. It is noted that the kinetic coefficients of the revised Enskog theory exhibit Onsager reciprocity in the linear regime. Upon consideration of the standard Enskog theory in the linear regime, we construct an entropy-production density and identify conjugate fluxes and forces and also kinetic coefficients which are shown to exhibit Onsager reciprocity. The standard theory is in disagreement, however, with the results of phenomenological irreversible thermodynamics for the conventional forms of fluxes and forces.

A kinetic-variational theory recently derived for a pure fluid on the basis of maximization of entropy is given for mixtures. Explicit results are found for particles interacting via hard cores and Kac tails (i.e., weak long-range attraction).

Neutron inelastic-scattering data from liquid ${\mathrm{N}}_2$ at wave-vector transfer $\kappa$ between 0.18 and 2.1 ${\mathrm{\AA }}^{-1\mathrm{}}$ and temperatures ranging from $T=65-77\mathrm{}$ K are presented. The data are corrected for the contribution from multiple scattering and incoherent scattering. The resulting dynamic structure factor $S(\kappa ,\omega )$ is compared with $S(\kappa ,\omega )$ determined by a molecular-dynamics calculation and with that calculated from a linearized Vlasov equation. For $\kappa \le 0.3$ ${\mathrm{\AA }}^{-1\mathrm{}}$ the experimental $S(\kappa ,\omega )$ is compared with the values predicted by hydrodynamic theory.