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Annalen der Physik

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

H. E. Ponath, M. Schubert

Published Jan 1, 1977• 2 citations
Publisher

Abstract

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.

DOI
10.1002/andp.19774890607

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

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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.

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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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It is shown that the Stokes and anti-Stokes components, related by the conditions ω s + ω α = 2 ω L and k s + k α =2 k L , are correlated. When the temperature of the scattering material is high [ kT ≫ h ( ω L – ω s )] and/or stimulated scattering exists, this correlation assumes a classical form and is analogous to the Brown-Twiss intensity correlation. This intensity correlation is readily explained by modulation of the pump wave due to molecular vibrations so that the thermal fluctuations of energy of these vibrations are synchronously transferred into the s and α sidebands. At low temperatures and pump intensities (as well as outside Raman resonances) a photon correlation is found which is explained by the simultaneous production of pairs of s and α photons as a result of the spontaneous decay of two pump photons. The statistics of the scattering field in this case assumes a nonclassical form characterized by a relatively high ( m ≫1) probability of pair production compared with the product of the probabilities for the production of separate s and α photons (with the intensity correlation m≤1). An estimate of the rate of coincidences for the scattering of 1-W focused TEM 00 pump radiation in liquid nitrogen yielded ~6 sec –1 and m~10 3 . Analogs of the Kirchhoff equation were obtained for the inelastic scattering of light. These formulas, derived from the conditions of unitarity of the scattering process and equilibrium of matter, define the statistics of the scattered light in terms of the Raman gain and the parametric conversion coefficient.

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Quantum decoherence dynamics in stochastically fluctuating environments

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We theoretically study the decoherence of a two-level quantum system coupled to noisy environments exhibiting linear and quadratic fluctuations within the framework of a stochastic Liouville equation. It is shown that the intrinsic energy levels of the quantum system renormalize under either the linear or quadratic influence of the environmental noise. In the case of quadratic dependence, the renormalization of the energy levels of the system emerges even if the environmental noise exhibits stationary statistical properties. This is in contrast to the case under linear influence, where the intrinsic energy levels of the system renormalize only if the environmental noise displays nonstationary statistics. We derive the analytical expressions of the decoherence function in the cases where the fluctuation of the frequency difference depends linearly and quadratically on the nonstationary Ornstein-Uhlenbeck noise (OUN) and random telegraph noise (RTN) processes, respectively. In the case of the linear dependence of the OUN, the environmental nonstationary statistical property can enhance the dynamical decoherence. However, the nonstationary statistics of the environmental noise can suppress the quantum decoherence in this case under the quadratic influence of the OUN. In the presence of the RTN, the quadratic influence of the environmental noise does not give rise to decoherence but only causes a determinate frequency renormalization in dynamical evolution. The environmental nonstationary statistical property can suppress the quantum decoherence of the case under the linear influence of the RTN.

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Sub-shot noise photon-number correlation in stimulated and spontaneous Raman processes

Biswajit Sen, Vlasta Peřinová, A. Lukš +2 more

The light initially prepared in a coherent state runs the stimulated and spontaneous Raman processes in a second-order nonlinear medium. Using analytical expressions of field operators for the light modes, we determine the quantum statistical properties of the modes in the stimulated and spontaneous Raman processes. By a choice of phase, we obtain a boson-pair property correlating the pump and vibration modes and such a property correlating the pump and anti-Stokes modes. The Stokes and vibration modes exhibit the boson-pair property for all initial phase differences between the pump mode and the other ones. In the spontaneous Raman process only the Stokes and vibration modes show such a property.

Journal of Physics B Atomic Molecular and Optical Physics•Apr 27, 2011•13 citations
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.
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.
The time-dependent (physical) spectrum of stimulated Raman scattering (SRS) from a two-level system is calculated, proving the influence of the spectral width Γ of the detector on the obtained spectral line shape. In addition, the time evolution of the process leading to the Raman line is investigated. For a partially excited system, the scattered field intensity exhibits oscillations in time, and Γ is found to affect the amplitude of the oscillations in the Raman line intensity. A numerical analysis of the result is performed for the case of monochromatic external fields.
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.
It is shown that the Stokes and anti-Stokes components, related by the conditions ω s + ω α = 2 ω L and k s + k α =2 k L , are correlated. When the temperature of the scattering material is high [ kT ≫ h ( ω L – ω s )] and/or stimulated scattering exists, this correlation assumes a classical form and is analogous to the Brown-Twiss intensity correlation. This intensity correlation is readily explained by modulation of the pump wave due to molecular vibrations so that the thermal fluctuations of energy of these vibrations are synchronously transferred into the s and α sidebands. At low temperatures and pump intensities (as well as outside Raman resonances) a photon correlation is found which is explained by the simultaneous production of pairs of s and α photons as a result of the spontaneous decay of two pump photons. The statistics of the scattering field in this case assumes a nonclassical form characterized by a relatively high ( m ≫1) probability of pair production compared with the product of the probabilities for the production of separate s and α photons (with the intensity correlation m≤1). An estimate of the rate of coincidences for the scattering of 1-W focused TEM 00 pump radiation in liquid nitrogen yielded ~6 sec –1 and m~10 3 . Analogs of the Kirchhoff equation were obtained for the inelastic scattering of light. These formulas, derived from the conditions of unitarity of the scattering process and equilibrium of matter, define the statistics of the scattered light in terms of the Raman gain and the parametric conversion coefficient.
We theoretically study the decoherence of a two-level quantum system coupled to noisy environments exhibiting linear and quadratic fluctuations within the framework of a stochastic Liouville equation. It is shown that the intrinsic energy levels of the quantum system renormalize under either the linear or quadratic influence of the environmental noise. In the case of quadratic dependence, the renormalization of the energy levels of the system emerges even if the environmental noise exhibits stationary statistical properties. This is in contrast to the case under linear influence, where the intrinsic energy levels of the system renormalize only if the environmental noise displays nonstationary statistics. We derive the analytical expressions of the decoherence function in the cases where the fluctuation of the frequency difference depends linearly and quadratically on the nonstationary Ornstein-Uhlenbeck noise (OUN) and random telegraph noise (RTN) processes, respectively. In the case of the linear dependence of the OUN, the environmental nonstationary statistical property can enhance the dynamical decoherence. However, the nonstationary statistics of the environmental noise can suppress the quantum decoherence in this case under the quadratic influence of the OUN. In the presence of the RTN, the quadratic influence of the environmental noise does not give rise to decoherence but only causes a determinate frequency renormalization in dynamical evolution. The environmental nonstationary statistical property can suppress the quantum decoherence of the case under the linear influence of the RTN.
The light initially prepared in a coherent state runs the stimulated and spontaneous Raman processes in a second-order nonlinear medium. Using analytical expressions of field operators for the light modes, we determine the quantum statistical properties of the modes in the stimulated and spontaneous Raman processes. By a choice of phase, we obtain a boson-pair property correlating the pump and vibration modes and such a property correlating the pump and anti-Stokes modes. The Stokes and vibration modes exhibit the boson-pair property for all initial phase differences between the pump mode and the other ones. In the spontaneous Raman process only the Stokes and vibration modes show such a property.