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Canadian Journal of Physics

Many body theory of resonant dielectric susceptibility in a gas

Haider R. Zaidi

Published Jul 15, 1977• 9 citations
Publisher

Abstract

A theory is developed for the resonant dielectric susceptibility of a high density gas which consists of two level atoms interacting through the dipole–dipole potential. A many body collision model, which includes the binary as well as the shielding effects, is investigated. The linear response of the dielectric is given by a generalized transport equation. Self-consistent coupled equations are derived for the various collision operators, using the propagator techniques. In the two limiting cases, the results of the binary collision model and those of the shielded interaction model, respectively, are recovered. The equations are simplified by neglecting the nonresonant contributions. Finally, a solution is obtained in the resonance broadening approximation. The resulting expression for the dielectric susceptibility includes the two body collision as well as the many body shielding effects. In particular, it includes the important collective excitation transfer effects. The proposed many body collision model should be appropriate for the resonant, linear response of high density gases.

DOI
10.1139/p77-159

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•Jan 1, 2016•0 citations
A theory is developed for the resonant dielectric susceptibility of a high density gas which consists of two level atoms interacting through the dipole–dipole potential. A many body collision model, which includes the binary as well as the shielding effects, is investigated. The linear response of the dielectric is given by a generalized transport equation. Self-consistent coupled equations are derived for the various collision operators, using the propagator techniques. In the two limiting cases, the results of the binary collision model and those of the shielded interaction model, respectively, are recovered. The equations are simplified by neglecting the nonresonant contributions. Finally, a solution is obtained in the resonance broadening approximation. The resulting expression for the dielectric susceptibility includes the two body collision as well as the many body shielding effects. In particular, it includes the important collective excitation transfer effects. The proposed many body collision model should be appropriate for the resonant, linear response of high density gases.
The combined effect of two collective processes in the resonant dielectric susceptibility is investigated. The first is the coherent excitation transfer process, which gives rise to the Lorentz–Lorenz shift. The second is the many body shielding of the resonant interaction between the atoms. The combined treatment of these effects leads to a qualitatively different behaviour of the susceptibility of a high density gas. The frequency dependence of the susceptibility is obtained from the detailed numerical solution of the self-consistent equations. In the quasi-static limit, an analytical solution is also obtained. In this limit, the result depends upon two parameters: one proportional to the gas density, and the other proportional to the square root of it.
The resonance broadening of a transition between two excited states of a radiator in a high density gas is investigated. The important many body shielding contributions are included to all orders, and the two body correlations for the close collisions are approximately taken into account. Corrections to the line width and shift at moderately high densities are calculated. The results agree with the experimental data for krypton. The density dependence of the width parameter is examined at very high densities also. It is found that this parameter, after reaching a peak value, decreases with further increase in the density. Moreover, the square root dependence of the line widths on the density is not valid over any extended range of gas densities.
The many-body linear response expressions obtained in previous papers [J. Math. Phys. 19, 1345 (1978); 20, 2573 (1979)] are applied to systems of weakly interacting particles. General expressions for the susceptibility and conductivity in such systems are obtained. The diagonal parts depend on the scattering processes, for which we consider interactions with bosons with mass and electron-phonon interaction. For elastic collisions simple closed forms result. For general two-body collisions, the closed expressions are cumbersome, except when the current is due to collisional current through localized states, such as Landau orbits; in that event a generalized Adams-Holstein result is obtained. The nondiagonal electrical conductivity is shown to be of paramount importance for the quantum mechanical Hall effect. We also derive quantum mechanical Boltzmann equations, both for the diagonal occupancy operator 〈nζ〉t and for the nondiagonal operator 〈c+ζ′ cζ\〉t. The total Boltzmann equation is shown to be fully equivalent with the linear response results. Finally, in the last part we derive the Boltzmann equation for the Wigner function of inhomogeneous systems. In the classical limit this yields the usual Boltzmann transport equation. This equation has therefore been obtained by first principles from the von Neumann equation.
The response of a collision-broadened gas sample to driving coherent radiation is studied theoretically, taking into account effects of coherent excitations of two or more atoms (or molecules). In analogy to the Bloch-type master equation for one-atom coherences, describing the motion of a single atom "dressed" by the relevant (incident and detected) field modes, a master equation is derived for two-atom coherences, including an effective interaction of the Bethe-Salpeter---type, accounting for the mutual interaction of the coherently driven pair. The master equation includes also all symmetrization effects owing to resonance exchange between identical atoms, and is limited to nonreactive gas atoms (or molecules) undergoing binary collisions, including otherwise all (internal and translational) relaxation effects. The self-energy kernels are expressed in a nonperturbative fashion, in terms of binary-collision scattering amplitudes, and include renormalization effects due to coincidence of radiative couplings with the collisions (optical and radiative collisions). The concept of two-atom coherences is generalized to higher coherence ranks by constructing a hierarchy of master equations, including vertex operators that upgrade or downgrade the coherence rank, as a prelude to a diagrammatic method for calculating continuous-wave spectra. This hierarchy is compared with prevalent Agarwal-type master equations, based on the Dicke pseudospin method, and used in the study of two-level atoms.
General expressions are derived for the spectral line shapes of resonance absorption and scattering of coherent radiation in collision-broadened gases, taking into account effects of coherent excitation of two or more atoms (or molecules), as steady-state solutions of a hierarchy of master equations described in a previous publication (paper I). Coupling between the coherent motions of the atoms, provided by a Bethe-Salpeter---type effective interaction, in the binary-collision approximation, forms the essential mechanism for introducing cooperative coherent effects into the steady-state spectra. Explicit expressions are given for the effects of two-atom coherence in the binary-collision approximation, in which the Bloch-type dressed-atom self-energy superoperator is modified by the presence of collisions in which both atoms retain memory of their coherent propagation before the collision. The self-energies include the effects of resonance exchange symmetrization in self-broadening, and are renormalized by the coincidence of radiative transitions during the collisions. The impact (near-resonance) and the quasistatic (line-wing) limits of the applied-frequency detunings are discussed. In the quasistatic limit, coherent many-atom excitations become irrelevant; however, interactions of both collision partners with the radiation during the collision accounts for such phenomena as collision-induced absorption or radiative collisions. In the impact limit, the inclusion of the Bethe-Salpeter interactions allows for the appearance of two-atom resonances. Magnitude estimates of these effects are discussed. Effects of higher-rank (many-body) coherences are formally discussed with the help of a diagrammatic method, leading into implicit bootstrap equations that can be solved by iterative or other procedures.
At high density, dipole-dipole interactions between the atoms may have a major impact on light propagation in a dense gas. We have developed a classical-electrodynamics simulation to study the cooperative response of a near-resonant gas to light. We take into account the motion of the radiators using classical trajectories, including collisions with the walls of the container and atom-atom collisions, and describe the transition from homogeneously broadened to inhomogeneously broadened phenomenology.