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Annalen der Physik

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

H. E. Ponath, Max Schubert

Published Jan 1, 1976• 8 citations
Publisher

Abstract

Abstract 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.

DOI
10.1002/andp.19764880603

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Fluctuation and Relaxation Properties of a Two‐Level‐System Influenced by a Nonstationary External Field

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Abstract The structural dependence of the longitudinal and transversal relaxation times of a two‐level‐system on the transition frequency ω0 and the temperature of the coupled dissipative system has been derived. The fluctuation operators are explicitely given in dependence on the operators and the constants of the dissipative and the two‐level‐system; the corresponding fluctuation correlators have been calculated. If additionally a time‐dependent external force F(t) acts on the system, then the fluctuation correlators depend on the temporal behavior of this force. Due to this effect the run of nonstationary processes which are initiated by quantum noise (e.g., scattering processes) is affected. These considerations have been applied to the case of spontaneous anti‐Stokes Raman scattering with biharmonic vibrational excitation by ultra‐short light pulses. The strength of the external force, which gives rise to an appreciable time dependence of the correlators, has been estimated.

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Polaron formation in a medium of damped excitations

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On the Nonstationary Processes in Exciton Systems

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Abstract In consequence of the nonconservation of excitons, the excitation number operator as well as the exciton pair creation and annihilation operators are expressed as functions of time and of the corresponding stationary operators. The nonstationary pair correlation processes and the nonstationary excitation density fluctuation processes are analysed using the nonstationary operators. It is found that the nonstationary pair correlations are characterized by a specific kind of dissipation that is linearly proportional with time. In contradistinction to stationary processes, the corresponding nonstationary ones, lead to elementary excitations having a double energy: E (nonstationary) = 2E (stationary). It is shown that in contrast to exciton density fluctuations, pair correlations have a tendency of localization related to a lattice point. It is also shown that the exciton non‐statinoary pair correlations are in many respect similar to the density fluctuations that take place in the crystal due to a near slow neutron passage. Finally, a general method for the evaluation of the correlation function, applicable even in the case when the Green's function has a n‐th order pole, is developed in the Appendix.

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A theoretical study of resonance Raman scattering from molecules. II. Temperature effect

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The temperature effect on the resonance Raman scattering from molecules is investigated. Considering a total system consisting of a molecular system, a heat bath, and the weak radiation field, a general expression for the relaxation processes is derived utilizing the Liouville operator technique for the time evolution of the density matrix of the total system. By using the perturbation method with respect to the radiation and matter up to second order, rate constants for zero photon (radiationless processes), one-photon and two-photon processes are derived after taking the long time limit in the Markoff approximation. Assuming a weak interaction between the molecular system and heat bath, a temperature-dependent resonance Raman scattering cross section is formulated in the adiabatic basis set for the molecular systems. In the displaced harmonic oscillator model, we obtain an analytical expression for the temperature-dependent Raman scattering cross section for the multimode case, which is suitable for both weak coupling and strong coupling cases. Analytical expressions for the Stokes and anti-Stokes bands of the jth order vibrational transition in the one oscillator model can easily be obtained from the general expression for the resonance Raman scattering cross section derived. An approximate Stokes to anti-Stokes ratio is given up to the second order of the dimensionless displacement Δ. The resonance Raman scattering cross section and the excitation energy profiles are numerically calculated as a function of physical constant such as temperature, molecular vibrational frequency and Δ. It is shown that the excitation profiles of the anti-Stokes band of the jth order vibrational transition have the similar envelope of the corresponding Stokes band shifted by jh/ω. The inverse Raman effect which is closely related to the ordinary Raman effect is also discussed.

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In the previous chapter, we developed the equation of motion for a system as it evolved under the influence of an unobserved (reservoir) system. We used the density matrix approach and worked in the interaction picture. In this chapter, we consider the same problem of the system-reservoir interaction using a quantum operator approach. We again eliminate the reservoir variables. The resulting equations for the system operators include, in addition to the damping terms, the noise operators which produce fluctuations. These equations have the form of classical Langevin equations, which describe, for example, the Brownian motion of a particle suspended in a liquid. The Heisenberg–Langevin approach discussed in this chapter is particularly suitable for the calculation of two-time correlation functions of the system operator as is, for example, required for the determination of the natural linewidth of a laser. We first consider the damping of the harmonic oscillator by an interaction with a reservoir consisting of many other simple harmonic oscillators. This system describes, for example, the damping of a single-mode field inside a cavity with lossy mirrors. The reservoir, in this case, consists of a large number of phonon-like modes in the mirrors. We also consider the decay of the field due to its interaction with an atomic reservoir. An interesting application of the theory of the system–reservoir interaction is the evolution of an atom inside a damped cavity. It is shown that the spontaneous transition rate of the atom can be substantially enhanced if it is placed in a resonant cavity.

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A model of nonstationary backward stimulated Raman scattering in an extended system: Interaction of forward and backward Stokes waves

N. I. Shamrov

A semiclassical model of nonstationary stimulated Raman scattering with account of forward and backward waves of different frequencies is proposed. Equations for the elements of a collective density matrix and electric-field amplitudes are obtained applying the averaging method. It is shown that forward waves are generated by a united polarization component, whereas each of the backward waves is formed by its own component. The description is based on the approximation of small scattering angles and assumes that the quantities considered do not depend on transverse coordinates. The emission dynamics of forward-backward Stokes pulses is investigated both for the linear case and under the strong depletion of pumping and level populations. The domain of applicability of the model of nonstationary forward stimulated Raman scattering is determined.

Optics and Spectroscopy•Oct 1, 2008•2 citations
Abstract 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.
Abstract The structural dependence of the longitudinal and transversal relaxation times of a two‐level‐system on the transition frequency ω0 and the temperature of the coupled dissipative system has been derived. The fluctuation operators are explicitely given in dependence on the operators and the constants of the dissipative and the two‐level‐system; the corresponding fluctuation correlators have been calculated. If additionally a time‐dependent external force F(t) acts on the system, then the fluctuation correlators depend on the temporal behavior of this force. Due to this effect the run of nonstationary processes which are initiated by quantum noise (e.g., scattering processes) is affected. These considerations have been applied to the case of spontaneous anti‐Stokes Raman scattering with biharmonic vibrational excitation by ultra‐short light pulses. The strength of the external force, which gives rise to an appreciable time dependence of the correlators, has been estimated.
Two schemes of the quantization of dynamical variables of dissipative systems are studied with an example of a two-level atom interacting with phonons scattered from an environment. Transformations of relevant Fock-space operators or rigged-Fock-space operators are proposed in order to diagonalize their equations of motion. It is shown that the damping of dispersionless phonons leads to the formation of a quasi-stationary state of the system—a polaron, which is found in both approaches. A time dependence of the polaron energy and the spectral lineshape of the two-level subsystem are evaluated. A transformation diagonalizing the equations of motion of the relevant classical problem (of a harmonic oscillator coupled to a field of damped waves) is found to be similar to the quantum case.
Abstract In consequence of the nonconservation of excitons, the excitation number operator as well as the exciton pair creation and annihilation operators are expressed as functions of time and of the corresponding stationary operators. The nonstationary pair correlation processes and the nonstationary excitation density fluctuation processes are analysed using the nonstationary operators. It is found that the nonstationary pair correlations are characterized by a specific kind of dissipation that is linearly proportional with time. In contradistinction to stationary processes, the corresponding nonstationary ones, lead to elementary excitations having a double energy: E (nonstationary) = 2E (stationary). It is shown that in contrast to exciton density fluctuations, pair correlations have a tendency of localization related to a lattice point. It is also shown that the exciton non‐statinoary pair correlations are in many respect similar to the density fluctuations that take place in the crystal due to a near slow neutron passage. Finally, a general method for the evaluation of the correlation function, applicable even in the case when the Green's function has a n‐th order pole, is developed in the Appendix.
The temperature effect on the resonance Raman scattering from molecules is investigated. Considering a total system consisting of a molecular system, a heat bath, and the weak radiation field, a general expression for the relaxation processes is derived utilizing the Liouville operator technique for the time evolution of the density matrix of the total system. By using the perturbation method with respect to the radiation and matter up to second order, rate constants for zero photon (radiationless processes), one-photon and two-photon processes are derived after taking the long time limit in the Markoff approximation. Assuming a weak interaction between the molecular system and heat bath, a temperature-dependent resonance Raman scattering cross section is formulated in the adiabatic basis set for the molecular systems. In the displaced harmonic oscillator model, we obtain an analytical expression for the temperature-dependent Raman scattering cross section for the multimode case, which is suitable for both weak coupling and strong coupling cases. Analytical expressions for the Stokes and anti-Stokes bands of the jth order vibrational transition in the one oscillator model can easily be obtained from the general expression for the resonance Raman scattering cross section derived. An approximate Stokes to anti-Stokes ratio is given up to the second order of the dimensionless displacement Δ. The resonance Raman scattering cross section and the excitation energy profiles are numerically calculated as a function of physical constant such as temperature, molecular vibrational frequency and Δ. It is shown that the excitation profiles of the anti-Stokes band of the jth order vibrational transition have the similar envelope of the corresponding Stokes band shifted by jh/ω. The inverse Raman effect which is closely related to the ordinary Raman effect is also discussed.
In the previous chapter, we developed the equation of motion for a system as it evolved under the influence of an unobserved (reservoir) system. We used the density matrix approach and worked in the interaction picture. In this chapter, we consider the same problem of the system-reservoir interaction using a quantum operator approach. We again eliminate the reservoir variables. The resulting equations for the system operators include, in addition to the damping terms, the noise operators which produce fluctuations. These equations have the form of classical Langevin equations, which describe, for example, the Brownian motion of a particle suspended in a liquid. The Heisenberg–Langevin approach discussed in this chapter is particularly suitable for the calculation of two-time correlation functions of the system operator as is, for example, required for the determination of the natural linewidth of a laser. We first consider the damping of the harmonic oscillator by an interaction with a reservoir consisting of many other simple harmonic oscillators. This system describes, for example, the damping of a single-mode field inside a cavity with lossy mirrors. The reservoir, in this case, consists of a large number of phonon-like modes in the mirrors. We also consider the decay of the field due to its interaction with an atomic reservoir. An interesting application of the theory of the system–reservoir interaction is the evolution of an atom inside a damped cavity. It is shown that the spontaneous transition rate of the atom can be substantially enhanced if it is placed in a resonant cavity.
A semiclassical model of nonstationary stimulated Raman scattering with account of forward and backward waves of different frequencies is proposed. Equations for the elements of a collective density matrix and electric-field amplitudes are obtained applying the averaging method. It is shown that forward waves are generated by a united polarization component, whereas each of the backward waves is formed by its own component. The description is based on the approximation of small scattering angles and assumes that the quantities considered do not depend on transverse coordinates. The emission dynamics of forward-backward Stokes pulses is investigated both for the linear case and under the strong depletion of pumping and level populations. The domain of applicability of the model of nonstationary forward stimulated Raman scattering is determined.