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The Journal of Chemical Physics

Non-Markovian theory of collisional dephasing in near resonance light scattering

Shaul Mukamel

Published Oct 1, 1979• 46 citations
Publisher

Abstract

We present a general microscopic theory for near resonance light scattering (RLS) from collisionally perturbed atoms and molecules. The theory is based on the tetradic scattering formalism of Fano and Ben-Reuven. We perform a systematic density expansion and show that the four-time, many-body, dipole correlation functions necessary for the evaluation of the RLS cross section may be rigorously expressed (to lowest order in pressure) in terms of three single-particle, two-time, correlation functions, two of which are associated with the absorption and emission line broadening and the third is a cross correlation function. Our most general expressions are valid for an arbitrary value of the bath correlation time (relative to the broadening), and interpolate smoothly all the way from the static limit of inhomogenous broadening up to the reverse, the Markovian (homogenous) limit. The stochastic Gaussian model of Takagahara, Hanamura, and Kubo is obtained also as a special case of the present formulation.

DOI
10.1063/1.438689

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In this thesis, two new types of solvable stochastic model are proposed and applied to light scattering phenomena. The theoretical framework for each model Is made based on a time-convolutlon-type equation In the damping theory. Each model Is used to develop a theory of line shape, in which the Influence of an environment for a relevant system is idealized so as to be represented with a stochastic process. Next, each model Is used to formulate a light scattering theory, especially focusing on the problem of "Raman versus Luminescence", or on a correlation among incident and scattered photons. For qualitative understanding of a problem like the light scattering, a stochastic approach such as employed here is really powerful. Recently, according to the development in experimental techniques of generating an ultra-short pulse excitation, various kinds of information upon relaxation phenomena have become obtainable in the real time domain. The models considered here may be of good use for phenomena in the ultra-short time domain. Two typical stochastic models are frequently used; one is the two-state-jump Markoff process and the other Is the Gaussian process. They are the opposite extremes with each other, whereas many real processes assume intermediate character between the two. Thus In part I a model (the first model) which bridges smoothly between these two extremes is introduced as a superposition of N two-state-jump Markoff processes. For a single two-state-jump Markoff process, the "partial cumulants" of higher than the second order vanish Identically, so that, its power spectrum can be expressed by the first order continued fraction. Because of this simple property, the general structure of the partial cumulants in case of an. arbitrary number N can be found. Thus, summing up contributions with the aid of a diagrammatic technique, the power spectrum of the proposed model is determined in a form of the N-step continued fraction. For a clear illustration of this model, a simple example of the random frequency modulation is considered, and then numerical calculations for its power spectra and its relaxation functions are given. This model is further adopted to analyze the stationary light scattering; a rigorous expression of the scattering Intensity is obtained. This result generalizes that of the stochastic model proposed by Takagahara, Hanamura and Kubo. According to an approach based on a generalized master (time-convolution-type) equation, It is found that the inhomogeneous term plays an essential role, whereas this term plays no important role In existing treatments of the relaxation. This Is due to the relaxation mechanism of the light scattering phenomena. Behavior of the light scattering spectrum is systematically investigated by the present theory with the aid of numerical calculations. It is shown how the spectrum changes from the "Raman" to the "Luminescence" under the Influence of environment. The multi-peak structure of the luminescence-like spectrum gradually disappears to become a smooth curve when the process is changed from the two-state-jump to the Gaussian. The transient response for a pulse excitation Is also considered with the use of this new model. From the above-mentioned result of the stationary light scattering, the expression for the time-resolved spectrum is readily obtained. This expression can be calculated numerically and a problem of the coherence is discussed. In part II, a further generalized model (the second model) is proposed. While the usual two-state-jump Markoff process assumes equal weight population for the two states, the newly introduced two-state-jump Markoff process Includes an asymmetric population. As a superposition of such constituent processes with asymmetric property, the new model becomes more flexible than the previous (first) model. In other words, the model Is composed of hopping processes among (N+1)-states with the binomial distribution in equilibrium. Namely, the distribution of the n-th state in equilibrium is given by p(n)=_NC_nx^ (1-x)^n, (O≤X≤1). This model is applied to various problems; the random frequency modulation, the stationary light scattering and the transient light scattering. The spectrum of this model is quite analogous to the one due to the phonon side bands, though the direct comparison must be done carefully. Our second model may be powerful in treating various systems except the ordinary light scattering, for Instance, γ-ray or X-ray scattering in solids. Moreover the formulation developed here may have wide range of applicability to various other phenomena. These remain to be solved in future.

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Stochastic Theory of the Resonant Scattering of Photons

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The resonant scattering of photons from a perturbed two-level system is studied in detail. Adiabatic perturbations are approximated by a randomly fluctuating term in the level splitting. Off-diagonal perturbations are accounted for in the linewidth and the time dependence of the correlation functions. Explicit expressions for the cross section are obtained in the statistical and motional-narrowing limits. The detailed-balancing symmetry of the scattering is established. Minor modifications of the approximate expressions for the cross sections are proposed in order to preserve this symmetry. The effects of off-diagonal perturbations, which couple the atom to a thermal bath, are investigated at length for a model interaction.

Physical Review•Jun 10, 1968•75 citations
We present a general microscopic theory for near resonance light scattering (RLS) from collisionally perturbed atoms and molecules. The theory is based on the tetradic scattering formalism of Fano and Ben-Reuven. We perform a systematic density expansion and show that the four-time, many-body, dipole correlation functions necessary for the evaluation of the RLS cross section may be rigorously expressed (to lowest order in pressure) in terms of three single-particle, two-time, correlation functions, two of which are associated with the absorption and emission line broadening and the third is a cross correlation function. Our most general expressions are valid for an arbitrary value of the bath correlation time (relative to the broadening), and interpolate smoothly all the way from the static limit of inhomogenous broadening up to the reverse, the Markovian (homogenous) limit. The stochastic Gaussian model of Takagahara, Hanamura, and Kubo is obtained also as a special case of the present formulation.
A theoretical study is presented of the effects of collisional perturbations on the time-resolved resonant and near-resonant photon scattering from a molecular system. The photon counting rate is formulated in terms of tetradic Green's functions in Liouville space. The collisional and radiative damping effects are handled by an effective-Liouvillian formalism. The formal expressions are disentangled by invoking a series of successive approximations, assuming short correlation times of the thermal bath, neglecting relaxation processes in the ground electronic state, considering rotational relaxation as providing the dominant damping mechanism in the excited electronic manifold, specializing to weak electromagnetic fields, neglecting some off-resonance processes and disregarding collision-induced radiative processes. The photon counting rate is expressed as a triple convolution of the photon counting rate for the free molecule with the Doppler Gaussian profile and with a Lorentz profile. The latter incorporates purely collisional effects due to intrastate cross relaxations and proper T2{T}_{2}T2​ processes. The theory accounts well for the recent experimental data of Williams, Rousseau, and Dworetsky on collisionally perturbed I2{\mathrm{I}}_{2}I2​ molecules.
A non-Markovian microscopic theory for the spontaneous emission spectrum of a two-level atom driven by a strong laser field and undergoing dephasing via the interaction with a thermal bath is developed. The present theory is valid for an arbitrary correlation time scale of the bath relative to the linewidth and interpolates from the Markovian to the static limits. Model calculations using the stochastic line-shape function of Kubo show how the characteristic triplet spectrum is modified by the interaction with the bath. The present results generalize our previous studies of resonance Raman scattering in a three-level system in a strong radiation field.
We develop an expansion for the resonance Raman line shape of a three-level system ∣a〉|a〉∣a〉, ∣b〉|b〉∣b〉, and ∣c〉|c〉∣c〉 in an arbitrary strong radiation field, being perturbed by dephasing collisions with foreign gas particles. The microscopic information relevant to the line broadening is expresssed in terms of a hierarchy of nnn-time correlation functions of the transition dipole operator (n=2,4,6,…n=2,4,6,\dots{}n=2,4,6,…). To lowest order (n=2n=2n=2) the spectrum is expressed in terms of the three ordinary line-shape functions corresponding to the ab\mathrm{ab}ab, bc\mathrm{bc}bc, and ac\mathrm{ac}ac transitions. The present theory is not restricted to the impact limit and, depending on the collision time scale, may yield very different results from the well-known impact spectra. Explicit calculations of the predicted spectra are presented for a model system.
In this thesis, two new types of solvable stochastic model are proposed and applied to light scattering phenomena. The theoretical framework for each model Is made based on a time-convolutlon-type equation In the damping theory. Each model Is used to develop a theory of line shape, in which the Influence of an environment for a relevant system is idealized so as to be represented with a stochastic process. Next, each model Is used to formulate a light scattering theory, especially focusing on the problem of "Raman versus Luminescence", or on a correlation among incident and scattered photons. For qualitative understanding of a problem like the light scattering, a stochastic approach such as employed here is really powerful. Recently, according to the development in experimental techniques of generating an ultra-short pulse excitation, various kinds of information upon relaxation phenomena have become obtainable in the real time domain. The models considered here may be of good use for phenomena in the ultra-short time domain. Two typical stochastic models are frequently used; one is the two-state-jump Markoff process and the other Is the Gaussian process. They are the opposite extremes with each other, whereas many real processes assume intermediate character between the two. Thus In part I a model (the first model) which bridges smoothly between these two extremes is introduced as a superposition of N two-state-jump Markoff processes. For a single two-state-jump Markoff process, the "partial cumulants" of higher than the second order vanish Identically, so that, its power spectrum can be expressed by the first order continued fraction. Because of this simple property, the general structure of the partial cumulants in case of an. arbitrary number N can be found. Thus, summing up contributions with the aid of a diagrammatic technique, the power spectrum of the proposed model is determined in a form of the N-step continued fraction. For a clear illustration of this model, a simple example of the random frequency modulation is considered, and then numerical calculations for its power spectra and its relaxation functions are given. This model is further adopted to analyze the stationary light scattering; a rigorous expression of the scattering Intensity is obtained. This result generalizes that of the stochastic model proposed by Takagahara, Hanamura and Kubo. According to an approach based on a generalized master (time-convolution-type) equation, It is found that the inhomogeneous term plays an essential role, whereas this term plays no important role In existing treatments of the relaxation. This Is due to the relaxation mechanism of the light scattering phenomena. Behavior of the light scattering spectrum is systematically investigated by the present theory with the aid of numerical calculations. It is shown how the spectrum changes from the "Raman" to the "Luminescence" under the Influence of environment. The multi-peak structure of the luminescence-like spectrum gradually disappears to become a smooth curve when the process is changed from the two-state-jump to the Gaussian. The transient response for a pulse excitation Is also considered with the use of this new model. From the above-mentioned result of the stationary light scattering, the expression for the time-resolved spectrum is readily obtained. This expression can be calculated numerically and a problem of the coherence is discussed. In part II, a further generalized model (the second model) is proposed. While the usual two-state-jump Markoff process assumes equal weight population for the two states, the newly introduced two-state-jump Markoff process Includes an asymmetric population. As a superposition of such constituent processes with asymmetric property, the new model becomes more flexible than the previous (first) model. In other words, the model Is composed of hopping processes among (N+1)-states with the binomial distribution in equilibrium. Namely, the distribution of the n-th state in equilibrium is given by p(n)=_NC_nx^ (1-x)^n, (O≤X≤1). This model is applied to various problems; the random frequency modulation, the stationary light scattering and the transient light scattering. The spectrum of this model is quite analogous to the one due to the phonon side bands, though the direct comparison must be done carefully. Our second model may be powerful in treating various systems except the ordinary light scattering, for Instance, γ-ray or X-ray scattering in solids. Moreover the formulation developed here may have wide range of applicability to various other phenomena. These remain to be solved in future.
The nature of resonance emission of light, which appears as a mixture of (coherent) resonance scattering and (incoherent) resonance fluorescence, is considered. The basic conceptual elements which distinguish the scattering and fluorescence components for the linear excitation limit are developed in terms of a simple formulation whereby a molecule interacts with a coherent Lorentzian light pulse which increases exponentially with time for t<0 and is turned off at t = 0. These results together with the ideas, currently existing in the literature, that stochastic modulation of a molecule by its medium is essential for the appearance of fluorescence provide the basis for the further formulation of a theory which does not require the ad hoc introduction of a random potential. The theory developed considers the microscopic origins of manifest stochastic modulation effects in terms of a bath potential for the molecule–medium (bath) degrees of freedom. This potential, termed the sudden potential, is associated with the change in interaction with the medium that a molecule experiences upon intramolecular state transitions which are induced by external interactions (e.g., by light). Development of the reduced molecule–field density operator leads naturally to a separation of terms which describes scattering and (the propensity for) fluorescence. Critical to the theory is a four-time correlation function for the bath (sudden) potential. The difficulties inherent in defining absorption (spectrum) resonances and the corresponding distinctions between resonance and relaxed emission are discussed. Implications are considered for the further understanding of experimental results on molecular excitation dynamics in the gas phase and in supersonic jets. Sources of stochastic modulation other than material media are mentioned. In particular, the possibility is suggested of the pervasive existence of stochastic modulation through interaction of a molecule with the vacuum state of the radiation field. This raises questions concerning the fundamental relations between quantum mechanics and thermodynamics.
The resonant scattering of photons from a perturbed two-level system is studied in detail. Adiabatic perturbations are approximated by a randomly fluctuating term in the level splitting. Off-diagonal perturbations are accounted for in the linewidth and the time dependence of the correlation functions. Explicit expressions for the cross section are obtained in the statistical and motional-narrowing limits. The detailed-balancing symmetry of the scattering is established. Minor modifications of the approximate expressions for the cross sections are proposed in order to preserve this symmetry. The effects of off-diagonal perturbations, which couple the atom to a thermal bath, are investigated at length for a model interaction.