Showing papers in "Physical Review A in 1978"

Self-phase-modulation in silica optical fibers

Abstract: We report measurements of frequency broadening due to self-phase-modulation (SPM) in optical fibers. The use of single-mode silica-core fibers and mode-locked argon-laser pulses leads to the simplest and cleanest measurements yet made of SPM. The qualitative features of the frequency spectrum are in good agreement with theoretical expectations. The experiment provides an independent measurement of ${n}_{2}$, the self-focusing coefficient. The results also point to some simple and useful techniques based on fibers for the measurement and analysis of mode-locked laser pulses.

Perturbation analysis of fluxon dynamics

Abstract: A two-stage scheme is presented to study structural perturbations of the sine-Gordon equation. Although the method is based upon the inverse scattering transform, detailed knowledge of this technical apparatus is not necessary in order to effect the calculations. In the first stage, slow modulations of speeds and positions for the soliton components are computed. The radiation resulting from the acceleration of the solitons is then calculated as a first-order correction through an easily constructed radiative Green's function. The method is exemplified by using it to study several outstanding problems that arise in applications of the Josephson transmission line. In particular we consider: (i) the pinning of flux quanta by microshorts, (ii) the quantum flux shuttle, (iii) annihilation conditions for fluxon-antifluxon collisions, (iv) breather decay, and (v) radiation from a moving fluxon.

Optical bistability and cooperative effects in resonance fluorescence

Abstract: We discuss a mean-field quantum-mechanical model which describes the dynamics of a homogeneously broadened system of two-level atoms contained in a pencil-shaped resonant cavity and driven by a coherent resonant field ${E}_{I}$. The model is treated in the semiclassical approximation. The model is justified on the basis of Maxwell-Bloch equations with two coherently coupled directions of propagation, with boundary conditions for the fields taken into account. Above a suitable critical density of atoms the system exhibits a bistable behavior including both the stationary situation and the transient, both the light transmitted in the forward direction and the fluorescent light. Bistability is shown to be a consequence of atomic cooperation. We give a simple description of optical bistability leading to new predictions for the transient behavior of the transmitted light and for the spectrum of the fluorescent light. The damping constant which characterizes the rate of appraoch to the stationary situation exhibits a hysteresis cycle. In the low-transmission regime the approach is monotonic, whereas in the highly transmitting situation the approach is oscillatory. One finds a critical slowing down in correspondence with the values ${{E}_{I}}^{(+)}$, ${{E}_{I}}^{(\ensuremath{-})}$ of the incident field where the transmitted field (as well as the total fluorescence intensity, the rate of approach to the stationary state, etc.) changes discontinuously. On the basis of the regression hypothesis we give a qualitative description of the spectrum of the fluorescent light. This is shown to undergo hysteresis and discontinuous changes at the same values ${E}_{I}^{(+)}$, ${E}_{I}^{(\ensuremath{-})}$ of the incident field. Below the critical density of atoms one recovers the usual picture of resonance fluorescence, with a continuous transition from a single-line spectrum to a three-peaked structure when the Rabi frequency becomes equal to the natural linewidth. Above the critical density the triplet appears only when the Rabi frequency becomes equal to the cooperative linewidth of pure super-fluorescence (i.e., for ${E}_{I}={E}_{I}^{(+)}$). Moreover, it appears discontinuously: when ${E}_{I}$ crosses the value ${E}_{I}^{(+)}$ the spectrum changes from a single narrow line to a triplet with well-separated sidebands.

Bifurcation in Onsager's model of the isotropic-nematic transition

Abstract: In this paper Onsager's theory of the orientational order in a three-dimensional system of hard rods is reanalyzed as a nonlinear eigenvalue problem. Bifurcation is found and the equation of state is calculated from the orientational distribution function for a nematic phase. We also investigate the corresponding twodimensional system of hard lines. The existence and order of a phase transition are shown to depend on both the direction of bifurcation and on properties of the global solutions. The analysis can be adapted to other nonlinear equations obtained in theories of liquid crystals.

Theory of a lossless nonlinear Fabry-Perot interferometer

Abstract: A Fabry-Perot interferometer filled with a medium whose refractive index depends upon intensity has a multiple-valued transmission-vs-intensity characteristic. When the incident light is circularly polarized, and the nonlinearity is only cubic in the fields, Maxwell's equations may be solved exactly for the plane-parallel resonator field in terms of elliptic functions. An accurate approximate analysis is given for more general cases, including resonators with spherical mirrors and finite beams, where self-focusing is important. The theory is developed to yield design information for operation as an optical switch, "transistor," and power limiter.

Stimulated emission from relativistic electrons passing through a spatially periodic transverse magnetic field

Abstract: Stimulated emission from a relativistic beam of electrons passing through a spatially periodic right-hand circularly polarized magnetic field is considered. The amplification is found to be due to a ponderomotive bunching of the electrons. The effect is completely classical and for an infinite interaction distance a dispersion relation, which takes into account space-charge effects, describing the scattered field is derived. Conditions on the pump field amplitude, beam density, and momentum spread of the beam for emission from individual electrons to occur or for emission from plasma oscillations to occur are examined. Also, emission from individual electrons over a finite interaction distance is considered and gain is determined for distances less than an $e$-folding length.

Optical levitation and partial-wave resonances

Abstract: Sharp peaks observed recently in a levitation-power experiment are caused by partial-wave resonances. The theoretical results computed from the Mie-Debye theory are compared with experimental observations. The first resonances in each partial wave are extremely narrow and they were not detected in the experiment. We conjecture that the peaks in levitation power do not exist if levitated particles are irregularly shaped.

Connection between the maximal entropy and the scattering theoretic analyses of collision processes

Abstract: The information-theoretic maximal-entropy procedure for the analysis of collision processes is derived as a consequence of the dynamics, be they quantal or classical. The method centers attention on the minimal number of operators (the "dynamic constraints") whose expectation values are both necessary and sufficient to completely characterize the collision dynamics. For a given Hamiltonian and initial state, the constraints required to obtain an exact solution of the equations of motion are determined by a purely algebraic procedure. It is furthermore found possible to derive equations of motion for the conjugate Lagrange parameters. Immediate applications are noted, e.g., a family of similar reactions is shown to have a common set of dynamic constraints and simple illustrative applications are provided. The determination of the scattering matrix is discussed, with examples. The general formalism consists in solving the scattering problem in two stages. The first is purely algebraic. At the end of this stage one obtains the functional form of, say, the scattering matrix or of the density matrix after the collision expressed in terms of parameters whose number equals the number of dynamic constraints. The end result of this algebraic stage suffices to analyze the scattering pattern for any initial state. The second stage is the predictive procedure. Explicit coupled first-order nonlinear differential equations are obtained for the parameters.

Investigation of two-time correlations in photon emissions from a single atom

Abstract: Two-time correlations in the emissions of photons by a single atom of sodium in the presence of a coherent exciting field near resonance have been investigated. In the experiment sodium atoms in an atomic beam are excited by a perpendicular light beam from a tunable dye laser, and they are prepared by optical pumping to behave as two-level quantum systems. The fluorescent light is collected by a microscope objective in a direction that is approximately orthogonal to both beams and imaged on two photomultipliers. Photoelectric pulse correlations are measured in the presence of exciting fields of various strengths and for various detunings from the atomic resonance, and are found to exhibit significant nonclassical features. The results show clearly that the emitted photons exhibit antibunching, in good quantitative agreement with the predictions of quantum electrodynamics.

Extremely compact formulas for molecular two-center one-electron integrals and Coulomb integrals over Slater-type atomic orbitals

Abstract: Extremely compact analytical formulas are derived for molecular two-center one-electron integrals and Coulomb integrals, which occur in linear-combination-of-atomic-orbitals calculations with Slater-type orbitals (STO's). The derivation is based on the connection of the STO's with the reduced Bessel functions (RBF's) and makes use of the convolution theorems of RBF's. The final results are of a surprisingly simple structure and are, therefore, especially useful for practical applications.

Universal cross sections for K-shell ionization by heavy charged particles. II. Intermediate particle velocities

Abstract: Experimental K-shell ionization cross sections of /sub 13/Al and /sub 28/Ni are reported for ions of /sup 1//sub 1/H, /sup 2//sub 1/H, /sup 4//sub 2/He, /sup 6//sub 3/Li, and /sup 7//sub 3/Li with kinetic energies in the range from 2 to 36 MeV, and of /sub 28/Ni for ions of /sup 12//sub 6/C, /sup 16//sub 8/O, and /sup 19//sub 9/F in the range from 4 to 90 MeV. The theory of direct Coulomb K-shell ionization, as developed in an earlier paper (Phys. Rev. A 7, 983 (1973)) for projectiles of atomic number Z/sub 1/, small compared to the target atomic number Z/sub 2/, and of velocities v/sub 1/ small compared to the target K-shell electron velocity v/sub 2K/, i.e., v/sub 1/ very-much-less-than v/sub 2K/, is extended to intermediate velocities v/sub 1/ approx. = v/sub 2K/. New effects appear. They add to the Z/sup 2//sub 1/-proportional cross sections one derives from linear-response theories for direct ionizations. They are attributed to the polarization of the target K shell in the field of the projectile, and to electron capture by the projectile. Guided by the perturbed stationary-state theory of atomic collisions, the polarization effects are incorporated so that the theory retains the unifyingmore » aspects of the cross sections derived in the plane-wave Born approximation, but the variables now contain the nonlinear effects as scaling factors. Electron-capture cross sections are added. When v/sub 1/ >> v/sub 2K/, such contributions subside, and one retrieves the cross sections of the linear-response approximation. The theory predicts K-shell ionization cross sections for projectiles with Z/sub 1//Z/sub 2/ < 0.5 at all velocities in a comprehensive manner. It agrees with experimental data covering six orders of magnitude for collisions partners with Z/sub 1//Z/sub 2/ ranging from 0.03 to 0.3 and v/sub 1//v/sub 2K/ from 0.07 to 2.« less

Diagrammatic analysis of the density operator for nonlinear optical calculations Pulsed and CW responses

Abstract: In the present paper a diagrammatic analysis of the density operator for the evaluation of nonlinear optical quantities is considered. The present approach extends earlier diagrammatic analysis by treating the time evolution of both the wave function and its complex conjugate. Time-ordered graphs result, each of which corresponds to a term in the density matrix. Examples involving the third-order susceptibility are discussed for both monochromatic and pulse excitation. In particular coherent rotational transient birefringence is discussed. The diagrams provide a convenient means by which nonlinear optical processes can be precisely defined and the susceptibility readily evaluated.

Multiatom and transit-time effects on photon-correlation measurements in resonance fluorescence

Abstract: An expression is derived for the expected number of photon pairs separated by a time interval τ that are detected in photoelectric correlation measurements of an atomic beam, when due account is taken of the fluctuations of the number of radiating atoms and of the effect of their finite transit time through the field of view. The theoretical expression is checked against some recent measurements and good agreement is obtained.

K- and L-shell ionization of atoms by relativistic electrons

Abstract: Results of a relativistic calculation of the cross section for the ionization from the K and L shells by high-energy electrons are presented. The calculation use the first-order Born approximation to treat the interaction between the scattered and atomic electrons. Plane waves are used to treat the high-energy scattered electron, while solutions of the Dirac equation in a Hartree-Slater central potential are used to describe the atomic electrons. Results are presented for a set of elements from Z = 18 to 92 and incident energies from 50 keV to 1 GeV. Calculated results are given for the correction due to the density effect arising from the polarizability of the medium. In the calculation of this correction, the medium is treated as composed of free electons.

Quantum statistical theory of optical-resonance phenomena in fluctuating laser fields

Abstract: A quantum statistical theory of resonant optical phenomena is developed for the case when the fluctuations of the laser, used for exciting transitions, are important. The general equations describing the dynamics of a relaxing two-level atom (TLA) are solved exactly for different types of mean values and the two-time amplitude and intensity correlations. The phase-diffusion model is adopted for laser fluctuations. The exact results are used to analyze the effect of laser fluctuations on a number of optical effects-optical free-induction decay, adiabatic following, the spectrum of the scattered light from a relaxing TLA, the energy-absorption spectrum from a weak field, Hanle resonances, etc. In each case, the laser fluctuations are found to affect in an important way the characteristics of the above optical-resonance phenomena. For example, the spectrum of scattered radiation from a TLA has the usual three-peak structure for fields at resonance and with strengths $\ensuremath{\alpha}$ above the threshold and for $\ensuremath{\alpha}\ensuremath{\gg}{\ensuremath{\gamma}}_{c}$; however, now the peak heights (widths) are in the ratio $3x$ ($\frac{2}{3x}$), $x=\frac{[{\ensuremath{\gamma}}_{c}+\frac{({T}_{1}^{\ensuremath{-}1}+{T}_{2}^{\ensuremath{-}1})}{3}]}{[2{\ensuremath{\gamma}}_{c}+{T}_{2}^{\ensuremath{-}1}]}$, where ${\ensuremath{\gamma}}_{c}^{\ensuremath{-}1}$ is the correlation time for laser phase fluctuations. For ${\ensuremath{\gamma}}_{c}\ensuremath{\gg}\ensuremath{\alpha}$, one gets a single-peak spectrum. The laser field is treated as a second-quantized field with excitation in either a coherent state or a Fock state. The results, obtained by straightforward perturbation theory, but valid for arbitrary values of the relaxation parameters, detuning, and the laser correlation time, are also presented.

Measurements of total scattering cross sections for low-energy positrons and electrons colliding with helium and neon atoms

Abstract: Total scattering cross sections have been measured for 0.3--31 eV positrons colliding with helium atoms and for 0.25--24 eV positrons colliding with neon atoms using a beam-transmission technique. These measurements have resulted in the first direct observations of Ramsauer-Townsend effects for positrons colliding wih helium and neon, and also provide clear indications of the onset of positronium formation in each of these gases. As an over all check of the reliability of the experimental approach for measuring total scattering cross sections for positrons, electron-atom scattering cross sections have been measured in the identical apparatus, using the same target gases, and the same technique as was used for the positron measurements.

Coupled-state calculations of proton—hydrogen-atom scattering using a scaled hydrogenic basis set

Abstract: Coupled-state calculations of proton---hydrogen-atom scattering using a scaled hydrogenic basis set have been performed, and the results are reported on in this paper. Thirty-five basis functions, centered about each proton, were included in the expansion of the electron wave function. Cross sections for direct excitation and charge transfer to the $n=1, 2, \mathrm{and} 3$ levels, and for ionization, have been calculated. The results for ionization indicate that charge transfer to the continuum dominates over direct ionization at proton energies below about 60 keV. The charge distribution has been plotted as a function of time at an energy of 40 keV and impact parameter of 1.5 a.u.; the plot illustrates the considerable distortion of the electron cloud caused by the passing proton.

d -dimensional turbulence

Abstract: $d$-dimensional homogeneous isotropic incompressible turbulence is defined, for arbitrary nonintegral $d$, by analytically continuing the Taylor expansion in time of the energy spectrum ${E}_{k}(t)$, assuming Gaussian initial conditions. If $dl2$, the positivity of the energy spectrum is not necessarily preserved in time. For $d\ensuremath{\ge}2$ all steady-state and initial-value calculations have been made with a realizable second-order closure, the eddy-damped quasi normal Markovian approximation. Near two dimensions the enstrophy (mean square vorticity) conservation law is weakly broken, enough to allow ultraviolet singularities to develop in a finite time but not enough to prevent energy from cascading in the infrared direction. A systematic investigation is made of zero-transfer (inertial) steady-state scaling solutions ${E}_{k}\ensuremath{\propto}{k}^{\ensuremath{-}m}$ and of their stability. Energy-inertial solutions with $m=\frac{5}{3}$ exist for arbitrary $d$; the direction of the energy cascade reverses at $d={d}_{c}\ensuremath{\simeq}2.05$. For $dl{{d}^{\ensuremath{'}}}_{c}\ensuremath{\simeq}2.06$ there are in addition, as in the cascade model studied by Bell and Nelkin, inertial solutions with zero energy flux; their exponents $m(d)$ are given by a roughly parabolic curve in the ($m, d$) plane, linking enstrophy cascade ($m=3$, $d=2$) to enstrophy equipartition ($m=1$, $d=2$) For any point in the ($m, d$) plane such that the transfer integral is finite and negative, a steady-state scaling solution ${E}_{k}\ensuremath{\propto}{k}^{\ensuremath{-}\mathrm{m}}$ is obtained when the fluid is subject to random forces with spectrum ${F}_{k}\ensuremath{\propto}{k}^{\frac{3(m\ensuremath{-}1)}{2}}$. A special case is the "model B" [$m=\ensuremath{-}1=\frac{2}{3}\ensuremath{\epsilon}+O({\ensuremath{\epsilon}}^{2})$, $d=4\ensuremath{-}\ensuremath{\epsilon}$] obtained by Forster, Nelson, and Stephen using a dynamical renormalization-group procedure. Forced steady-state solutions are actually not resticted to the neighborhood of $m=\ensuremath{-}1$, $d=4$; they are amenable to renormalization-group calculations on the primitive equations for arbitrary $dg2$ when $m$ is close to the crossover -1 and, perhaps, also near the crossover +3.

Statistical reduction for strongly driven simple quantum systems

Abstract: We derive reduced equations of motion for simple quantum systems which are strongly driven by an external field and are modulated stochastically by a coupling to a bath. In the derivation we make use of the cumulant-expansion method of Kubo using two different time-ordering prescriptions. We demonstrate how the choice of the ordering prescription is related to the statistical properties of the bath, once the cumulant expansion is truncated. Our equations of motion are valid for arbitrary time scale for the motions of the bath relative to those of the system, and they change smoothly from the static to the Markov (motional narrowing) limit. As examples, we consider the problems of a randomly modulated and driven harmonic oscillator and a modulated and damped two-level system. In the Markovian limit both ordering prescriptions yield Bloch-type equations of motion; in general, however, the driving and modulation interfere and the different statistical properties of the bath, as determined by the two truncated ordering prescriptions, lead to different results.

Single-electron capture by multiply charged ions of carbon, nitrogen, and oxygen in atomic and molecular hydrogen

Abstract: Cross sections for electron capture by ${\mathrm{N}}^{+q}$, ${\mathrm{O}}^{+q}(1\ensuremath{\le}q\ensuremath{\le}5)$, and ${\mathrm{C}}^{+q}(1\ensuremath{\le}q\ensuremath{\le}4)$ incident on atomic and molecular hydrogen have been measured in the velocity range (0.3-5.2) \ifmmode\times\else\texttimes\fi{} ${10}^{8}$ cm/s. The capture cross sections for the higher incident charge states are large, in many cases exceeding 1 \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}15}$ ${\mathrm{cm}}^{2}$. For relative velocities above the Bohr velocity, all the measured cross sections were found to decrease monotonically with increasing velocity and to scale in magnitude roughly as ${q}^{2}$. Furthermore, for these higher velocities the cross sections for electron capture from ${\mathrm{H}}_{2}$ are systematically larger than the corresponding cross sections for electron capture from H. For relative velocities below the Bohr velocity, no simple behavior of the measured cross sections was observed. The behavior of the cross sections at low velocities is discussed qualitatively using the Landau-Zener formalism. For higher velocities reasonable agreement was found with the classical trajectory Monte Carlo calculations of Olson and Salop.

Electronic resonance enhancement of coherent anti-Stokes Raman scattering

Abstract: We analyze the enhancement of the coherent anti-Stokes Raman-scattering (CARS) susceptibility when the frequencies of the waves involved are tuned into resonance with discrete and continuum one-photon absorptions, and discuss the applications. We first derive the expression for the susceptibility by means of the usual iterative treatment of density-matrix perturbations. We then show that this derivation can be done in a straightforward manner by means of a time-ordered diagrammatic representation, which brings novel physical insight into CARS mechanisms. This representation can also be used to analyze the transient behavior of CARS as the pump fields are turned on and off. In addition, we discuss resonant CARS spectroscopy in the gas phase. The spectrum is composed of the expected enhanced Raman lines and also of double-electronic-resonance lines. All these lines occur as doublets. We derive their relative intensities based on detunings, collisional broadening, Franck-Condon overlap integrals, and rotational transition moments. The line contours are predicted by representing the susceptibility in the complex plane. The problems arising from saturation and the optical Stark effect are also considered; all should be small below pump densities of 100 MW/${\mathrm{cm}}^{2}$ in gas mixtures near STP. Fluorescence interference is negligible, except at power densities high enough for the Stark effect to be large. Beam absorption is also negligible at STP if the resonant species' concentration is less than 1000 ppm; phase matching is then satisfied. Finally, an experimental resonant CARS spectrum of ${\mathrm{I}}_{2}$ at 1 mb in air near STP is presented and interpreted; the susceptibility is about 400 times larger than that of ${\mathrm{O}}_{2}$ off resonance and under the same thermodynamic conditions.

Picosecond relaxation measurements by polarization spectroscopy in condensed phases

Abstract: A frequency domain technique is demonstrated which permits the determination of the homogeneous linewidth and relaxation rates of inhomogeneously broadened transitions in crystals, liquids, and glasses. The density-matrix formalism is employed to explain the origins of the intensity-dependent dichroism and birefringence observed in these experiments and to extract the relevant relaxation times. The samples initially studied were liquid solutions of the organic dyes malachite green and 1,3\ensuremath{'} -diethyl-2,2\ensuremath{'} quinolylthiacarbocyanine iodide. The ground-state recovery times were found to be 1.2\ifmmode\pm\else\textpm\fi{}0.1 and 3.4\ifmmode\pm\else\textpm\fi{}0.4 ps, respectively, for the two dyes in water solution. The transverse relaxation times were estimated to be less than 0.02 ps. The present technique is compared to resonant Rayleigh-type three-wave mixing and to time-resolved spectroscopic techniques.

Numerical experiments on the free motion of a point mass moving in a plane convex region: Stochastic transition and entropy

Abstract: We numerically investigate the behavior of a simple dynamical system, a plane billiard which by a continuous deformation of the border passes from a completely integrable system to a well-defined type of completely stochastic system, namely a $K$ system. A stochastic transition is observed, with the usual coexistence of ordered and stochastic regions. Moreover, the stochastic region at certain intermediate stages appears to be separated into several invariant components. An estimate of the Kolmogorov entropy is presented.

Exact wave-function normalization constants for the B/sub 0/ tanhz-U/sub 0/ cosh/sup -/2z and Poeschl-Teller potentials

Abstract: We obtain, in exact closed form, the wave-function normalization constants for the Schr\"odinger equation with potential $V={B}_{0}tanhz\ensuremath{-}{U}_{0}{cosh}^{\ensuremath{-}2}z$. These constants are derived in terms of a variety of formulations and solutions of the equation. We give discussions of both mathematical aspects and physical motivations of the problem. The main results are gathered in two appendixes. Wave-function normalization constants for the related P\"oschl-Teller potential are given in a third appendix.

Growth of fluctuations in quenched time-dependent Ginzburg-Landau model systems

Abstract: A dynamical theory is presented for the enhanced fluctuations that occur in a time-dependent Ginzburg-Landau model system with the order parameter not conserved which is quenched from a thermodynamically stable to an unstable state. In a certain weak-coupling, long-time, and long-distance limit, diffusion and saturation effects can be treated separately. As a result explicit expressions are found for the probability distribution functional, the two-point reduced distribution function, and the pair correlation function of the fluctuations, which evolve from an arbitrary initial probability distribution functional. The behavior of the latter two functions is also displayed graphically. A central role is played by the time-independent nonlinear transformation of the order parameter which takes care of the saturation effects. The nature of such a transformation is discussed in a general context. If the problem is viewed as a nonequilibrium critical phenomenon, the theory corresponds to the Landau mean-field theory. An expansion in $\ensuremath{\epsilon}=4\ensuremath{-}d$ is suggested to improve our treatment, where $d$ is the dimensionality of space.

Study of ultra-fast relaxation processes by resonant Rayleigh-type optical mixing. I. Theory

Abstract: A comprehensive theory of a new type of three-wave optical mixing (${\ensuremath{\omega}}_{3}=2{\ensuremath{\omega}}_{1}\ensuremath{-}{\ensuremath{\omega}}_{2}$) is given for the spectroscopic purpose of determining ultra-short relaxation times (picoseconds or less) associated with excited states of condensed matter. Two incident light frequencies ${\ensuremath{\omega}}_{1}$ and ${\ensuremath{\omega}}_{2}$ are chosen so that they are both resonant with an inhomogeneously broadened optical transition and $|{\ensuremath{\omega}}_{2}\ensuremath{-}{\ensuremath{\omega}}_{1}|$ is in the vicinity of inverse relaxation times. The theory is developed for its major application to broad-band electronic transitions. The nonlinear susceptibility ${\ensuremath{\chi}}_{R}^{(3)}$ for this process (named resonant Rayleigh-type mixing) is calculated by a density-matrix formalism with the model of a two-level atomic system incorporating the distribution of resonance frequencies and longitudinal (${T}_{1}$) and transverse (${T}_{2}$) relaxation times. The result shows a frequency characteristic depending only on ${T}_{1}$, ${T}_{2}$, and ${\ensuremath{\omega}}_{2}\ensuremath{-}{\ensuremath{\omega}}_{1}$ in a broad-band limit, which serves for the determination of ${T}_{1}$ and ${T}_{2}$ in the frequency domain. The analysis is further extended to include the various effects as follows. The calculation of the saturation effect in the lowest order reveals that it modifies the shape of the frequency characteristic of the nonlinear susceptibility but does not affect seriously the determination of relaxation times. The effect of spectral cross relaxation within the inhomogeneous broadening is incorporated by a generalized density-matrix formalism. The resultant ${\ensuremath{\chi}}_{R}^{(3)}$ consists of two terms. The dominant term is the same as before except that ${T}_{1}$ is replaced by a combined relaxation time ${T}_{1}^{\ensuremath{'}}={({T}_{1}^{\ensuremath{-}1}+{T}_{3}^{\ensuremath{-}1})}^{\ensuremath{-}1}$, where ${T}_{3}$ is the cross-relaxation time. The other term originates from the inverse spectral-diffusion process. The effect of other energy levels located between the two levels under study is analyzed with a simple three-level model. The resultant ${\ensuremath{\chi}}_{R}^{(3)}$ also consists of two terms, the dominant one being the same as that for the two-level model except that ${T}_{1}^{\ensuremath{-}1}$ should be interpreted as total population decay rate of the upper level. For accurate derivation of ${\ensuremath{\chi}}_{R}^{(3)}$ from experiment, we discuss the interference between the resonant and nonresonant susceptibility terms and the light-wave propagation effect. Finally, the close analogy between this type of optical mixing and the photon echo is discussed.

Mode-coupling theory of simple classical liquids

Abstract: A microscopic theory for the dynamics of simple classical liquids is formulated with the longitudinal and transverse current-fluctuation spectra expressed in terms of relaxation kernels. The kernels are approximated by two-mode decay integrals coupling longitudinal and transverse excitations that result in a closed set of nonlinear equations which can be solved numerically. The only input required is the pair potential and the static two- and three-particle correlation functions. The theory gives correctly the zeroth, second, and fourth moments for the dynamical structure factor, the zeroth and second moments of the longitudinal and transverse current correlation functions, the free-particle limit, a proper hydrodynamical limit, and reflects the long-time anomalies of the dynamical transport coefficients.