Zur Quantentheorie der stimulierten RAMAN-Streuung in Molekülkristallen. II. Spezielle Fälle eines stationären RAMAN-Oszillators

Two kinds of stationary RAMAN oscillators are investigated theoretically for molecular crystals. The calculations are done firstly for the generation of one first order anti-STOKES mode and secondly for the generation of one second order STOKES mode. By using a quantum theoretical model described in an earlier paper for treatment of molecular crystals RAMAN scattering is assumed to be polariton scattering. Within this framework coupled nonlinear equations for the polariton operators of the excited modes are derived, stationary occupation numbers for the different modes and threshold conditions are calculated. The influence of phase fluctuations of the pump wave on the line widths of the RAMAN modes are investigated.     

Hyper-Raman scattering by vibrational excitations in crystals,glasses and liquids

Results of experimental study of hyper-Raman scattering by phonons and polaritons in centrosymmetric and noncentrosymmetric crystals, inorganic oxide glasses and molecular liquids are discussed. Experimental techniques developed for investigation of this weak scattered light are described. We discuss a number of interesting results such as (a) detection of silent modes forbidden in Raman scattering and infrared absorption, (b) determination of the soft mode frequency damping at various temperatures, (c) the study of vibrational polaritons and their resonances with vibrational excitations in centrosymmetrical media, and (d) existence of the vibrational spectra dependence on both value and orientation of momentum transferred to vibrational system in the scattering process in glasses and liquids.     

General N-Body Theory of Nonrelativistic Quantum Scattering

 The two-Hilbert-space theory of scattering is reviewed with particular reference to its application to nonrelativistic multichannel quantum- mechanical scattering theory. In Part I the abstract assumptions of the theory are collected, transition operators (both on- and off-energy-shell) are defined, the dynamical equations that determine the off-shell transition operators are presented and their real-energy limits examined, and the convergence of sequences of approximate transition operators is established. A section on how to incorporate group symmetries into the formalism reports new work. The material of Part I is relevant to a variety of both classical and quantum scattering systems. In Part II attention is directed specifically to N-body nonrelativistic quantum scattering systems in which the particles interact via short-range pair potentials. A method of constructing approximate transition operators is presented and shown to satisfy all the abstract assumptions of Part I. The dynamical equations that determine the half-on-shell approximate transition operators are shown to be coupled one-dimensional integral equations that have compact kernels and unique solutions when considered as operators on a Hilbert space of H?lder continuous functions. Moreover, the on-shell parts of those approximate transition amplitudes are shown to converge to the exact on-shell amplitudes as the order of the approximation increases. Detailed formulas for the kernels of the integral equations are written down for systems of particles that are distinguishable and for systems containing identical particles. Finally, some important open problems are described. Received July 2, 1999; accepted in final form October 27, 1999     

Quantum Gelfand-Levitan equations for the multicomponent nonlinear Schrödinger model with supermatrices

The quantum Gelfand-Levitan equations for the multicomponent nonlinear Schrödinger model with supermatrices are derived, and the commutation relations for scattering data operators which are needed for calculating Green functions and correlation functions are given.     

Quantum mechanical theory of fluctuations and relaxation in semiconductor lasers

Pumping, spontaneous emission and electron-electron scattering lead to relaxation terms in the mean equations of motion for the electron system. These terms are derived from a quantum mechanical basis by second order perturbation theory and by appropriate reservoir averaging. If the electron distribution is not too strongly degenerate, then a relaxation time approximation can be derived. The quantum mechanical Langevin method is used to include noise. The correlation functions of all fluctuation operators are calculated. From the knowledge of the equations of motion and of the correlation functions it is possible to calculate the noise properties of the laser light and of the junction current, as will be shown in other publications.     

Quantum Maps with Memory from Generalized Lindblad Equation

In this paper, we proposed the exactly solvable model of non-Markovian dynamics of open quantum systems. This model describes open quantum systems with memory and periodic sequence of kicks by environment. To describe these systems, the Lindblad equation for quantum observable is generalized by taking into account power-law fading memory. Dynamics of open quantum systems with power-law memory are considered. The proposed generalized Lindblad equations describe non-Markovian quantum dynamics. The quantum dynamics with power-law memory are described by using integrations and differentiation of non-integer orders, as well as fractional calculus. An example of a quantum oscillator with linear friction and power-law memory is considered. In this paper, discrete-time quantum maps with memory, which are derived from generalized Lindblad equations without any approximations, are suggested. These maps exactly correspond to the generalized Lindblad equations, which are fractional differential equations with the Caputo derivatives of non-integer orders and periodic sequence of kicks that are represented by the Dirac delta-functions. The solution of these equations for coordinates and momenta are derived. The solutions of the generalized Lindblad equations for coordinate and momentum operators are obtained for open quantum systems with memory and kicks. Using these solutions, linear and nonlinear quantum discrete-time maps are derived.     

Picosecond spectroscopy of polaritons in anthracene crystals II. Luminescence

The low-temperature luminescence spectrum of anthracene crystals is investigated by applying simultaneous time and frequency resolution. The complicated kinetics of the emission in the polariton bottleneck region reflects directly the evolution and relaxation of the polariton distribution in the crystal. Three distinct relaxation stages are distinguished: (1) the ultrafast decay of initial vibronic excitations, mediated by optical phonons and resulting in a broad distribution of polaritons near the band bottom; (2) the formation of a narrow distribution of polaritons with a characteristic time of 30 ps, which is caused by scattering on acoustic phonons; (3) relaxation through the bottleneck region on a subnanosecond time scale. It is suggested that the polaritons immediately below the resonance frequency are responsible for the observed excitonic energy transfer in anthracene crystals.     

Dispersionless polaritons at symmetrically oriented surfaces of biaxial crystals

A set of dispersion relations is derived for surface polaritons in optically biaxial crystals at the surfaces parallel to the symmetry planes of the permittivity tensor ?. The domains of existence, as well as the sectors of the propagation directions of dispersionless surface polaritons which arise at positive components of the tensor ?, are analyzed. Three nonoverlapping domains of the dielectric-anisotropy parameters where dispersionless polaritons can exist are found for weakly anisotropic crystals. In each of these domains, polaritons exist at two different mutually orthogonal surfaces of the crystal. In optically biaxial crystals, in contrast to optically uniaxial media, polaritons arise not only in positive but also in negative crystals. The evolution of the optical-axis configuration is traced as the anisotropy parameters vary in the domains of existence of polaritons.     

The virtual Compton amplitude in the generalized Bjorken region: Twist-2 contributions

A systematic derivation is presented of the twist-2 anomalous dimensions of the general quark and gluon light-ray operators in the generalized Bjorken region in leading order both for unpolarized and polarized scattering. Various representations of the anomalous dimensions are derived in the non-local and local light cone expansion and their properties are discussed in detail. Evolution equations for these operators are derived using different representations. General two- and single-variable evolution equations are presented for the expectation values of these operators for non-forward scattering. The Compton amplitude is calculated in terms of these distribution amplitudes. In the limit of forward scattering a new derivation of the integral relations between the twist-2 contributions to the structure functions is given. Special limiting cases which are derived from the general relations are discussed, as the forward case, near-forward scattering, and vacuum-meson transition. Solutions of the two-variable evolution equations for non-forward scattering are presented.     

Theoretical Treatment of Nonstationary Scattering by Phonons and Polaritons Part I. Derivation of the Basic Equations

The equations of motion of a two-level system and a harmonic oscillator coupled to a dissipative system are discussed; the dissipative system is assumed to consist of a large number of radiation oscillators. Special equations for the determination of the correlation functions of the fluctuation forces are derived under the condition of large time values, for which the atomic system has “forgotten” its initial state. The expectation values of the correlation functions are connected with the damping constant and the population operator of the excited state of the atomic system is in thermal equilibrium. Taking into account the influence of the coherent radiation field on the atomic system, the basic equations for the treatment of the nonstationary Raman scattering by polaritons are derived; the temporal range of validity is discussed. Using a time-dependent “variable” Fourier transformation, the nonstationary time- and spacedependent spectral densities are related to the correlation functions of the fields; here the Wiener-Khintchine theorem is applied in a nonstationary form. The limiting cases of the stationary scattering process as well as the usually introduced correlators of the slowly varying amplitudes are discussed.     

Semiclassical expansion of quantum characteristics for many‐body potential scattering problem

In quantum mechanics, systems can be described in phase space in terms of the Wigner function and the star‐product operation. Quantum characteristics, which appear in the Heisenberg picture as the Weyl's symbols of operators of canonical coordinates and momenta, can be used to solve the evolution equations for symbols of other operators acting in the Hilbert space. To any fixed order in the Planck's constant, many‐body potential scattering problem simplifies to a statistical‐mechanical problem of computing an ensemble of quantum characteristics and their derivatives with respect to the initial canonical coordinates and momenta. The reduction to a system of ordinary differential equations pertains rigorously at any fixed order in ?. We present semiclassical expansion of quantum characteristics for many‐body scattering problem and provide tools for calculation of average values of time‐dependent physical observables and cross sections. The method of quantum characteristics admits the consistent incorporation of specific quantum effects, such as non‐locality and coherence in propagation of particles, into the semiclassical transport models. We formulate the principle of stationary action for quantum Hamilton's equations and give quantum‐mechanical extensions of the Liouville theorem on conservation of the phase‐space volume and the Poincaré theorem on conservation of 2p‐forms. The lowest order quantum corrections to the Kepler periodic orbits are constructed. These corrections show the resonance behavior.     

Decoherence effects in Bose–Einstein condensate interferometry I. General theory

The present paper outlines a basic theoretical treatment of decoherence and dephasing effects in interferometry based on single component Bose–Einstein condensates in double potential wells, where two condensate modes may be involved. Results for both two mode condensates and the simpler single mode condensate case are presented. The approach involves a hybrid phase space distribution functional method where the condensate modes are described via a truncated Wigner representation, whilst the basically unoccupied non-condensate modes are described via a positive P representation. The Hamiltonian for the system is described in terms of quantum field operators for the condensate and non-condensate modes. The functional Fokker–Planck equation for the double phase space distribution functional is derived. Equivalent Ito stochastic equations for the condensate and non-condensate fields that replace the field operators are obtained, and stochastic averages of products of these fields give the quantum correlation functions that can be used to interpret interferometry experiments. The stochastic field equations are the sum of a deterministic term obtained from the drift vector in the functional Fokker–Planck equation, and a noise field whose stochastic properties are determined from the diffusion matrix in the functional Fokker–Planck equation. The stochastic properties of the noise field terms are similar to those for Gaussian–Markov processes in that the stochastic averages of odd numbers of noise fields are zero and those for even numbers of noise field terms are the sums of products of stochastic averages associated with pairs of noise fields. However each pair is represented by an element of the diffusion matrix rather than products of the noise fields themselves, as in the case of Gaussian–Markov processes. The treatment starts from a generalised mean field theory for two condensate modes, where generalised coupled Gross–Pitaevskii equations are obtained for the modes and matrix mechanics equations are derived for the amplitudes describing possible fragmentations of the condensate between the two modes. These self-consistent sets of equations are derived via the Dirac–Frenkel variational principle. Numerical studies for interferometry experiments would involve using the solutions from the generalised mean field theory in calculations for the stochastic fields from the Ito stochastic field equations.     

Generation of terahertz radiation in cubic non-centrosymmetric crystals

The conditions of parametric radiation generation on polaritons in cubic noncentrosymmetric crystals are studied. The possibility of such generation is theoretically justified. The polariton radiation frequencies are calculated for GaP, ZnSe, ZnTe, and GaAs crystals. The obtained generation frequencies are compared to the experimental results on Raman scattering on polaritons. The block diagram of the terahertz radiation generator operation using a GaP crystal and a pulsed laser with high peak power at low energy of laser pump pulses is presented. The lasing frequency shift is analyzed depending on the scattering geometry. The coefficient of exciting radiation conversion to the terahertz range is determined.     

Light scattering spectra of guided wave polaritons in thin crystals: Experiment

Guided wave polaritons (GWP) of thin crystals have been studied by near- forward Raman scattering experiments on GaP crystals of ≈ 5–35 μm thickness. The dispersion of these modes agrees well with theoretical predictions, but their Raman scattering intensities cannot be described completely by bulk Raman tensors. We discuss the temperature dependence of the observed frequencies of the modes.     

Bose condensation of exciton polaritons in an optical microcavity

Two methods are considered for producing traps for exciton polaritons in an optical microcavity with an embedded quantum well. The first method for controlling polaritons consists in producing a polariton trap governed by the longitudinal confinement of photons. Traps of this type can be created using an optical microcavity with a variable width. In traps of the second type, the exciton confinement is ensured by a weak potential that is applied to a quantum well with excitons or when this well is subjected to an inhomogeneous deformation. The behavior of a two-component Bose condensate of photons and excitons is analyzed theoretically. The Bose condensate is described by the coupled system of equations of the Gross-Pitaevskii type. The approximate wave functions and the spatial profiles of coupled photon and exciton condensates are obtained.     

Conservation Laws for Linear Equations on Quantum Minkowski Spaces

The general, linear equations with constant coefficients on quantum Minkowski spaces are considered and the explicit formulae for their conserved currents are given. The proposed procedure can be simplified for *-invariant equations. The derived method is then applied to Klein–Gordon, Dirac and wave equations on different classes of Minkowski spaces. In the examples also symmetry operators for these equations are obtained. They include quantum deformations of classical symmetry operators as well as an additional operator connected with deformation of the Leibnitz rule in non-commutative differential calculus. Received: 4 April 1997 / Accepted: 10 June 1997     

Resonance Raman scattering by excitonic polaritons in CuGaS2

Resonance Raman scattering by exciton polaritons in crystals of CuGaS₂ under excitation with the 4880 and 4765 Å lines of an Ar⁺ laser at 9 K is studied. Lines of one-and two-phonon scattering of excitonic polaritons are found and studied. It is shown that the 1LO and 2LO phonons are arranged in accordance with their energies as the Stokes shifts move farther away from the excitation energy.