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arXiv:physics/0309001 · Physical Review A

Influence of radiative damping on the optical-frequency susceptibility

Peter W. Milonni, Robert W. Boyd

Published Feb 27, 2004• 33 citations
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Abstract

Motivated by recent discussions concerning the manner in which damping appears in the electric polarizability, we show that (a) there is a dependence of the nonresonant contribution on the damping and that (b) the damping enters according to the ``opposite sign prescription.'' We also discuss the related question of how the damping rates in the polarizability are related to energy-level decay rates.

DOI
10.1103/physreva.69.023814

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Journal of Physics B Atomic Molecular and Optical Physics•Jul 24, 2006•35 citations
Motivated by recent discussions concerning the manner in which damping appears in the electric polarizability, we show that (a) there is a dependence of the nonresonant contribution on the damping and that (b) the damping enters according to the ``opposite sign prescription.'' We also discuss the related question of how the damping rates in the polarizability are related to energy-level decay rates.
The effect of spontaneous decay on the linear polarizability of an atom is typically included by adding imaginary parts to the frequency denominators that appear in the Kramers-Heisenberg formula. It has been shown for a two-level atom with radiative broadening that these (frequency-dependent) imaginary parts must be included in both the resonant and antiresonant frequency denominators [P. W. Milonni and R. W. Boyd, Phys. Rev. A 69, 023814 (2004)]; however, the expression obtained by Milonni and Boyd for the polarizability does not satisfy the optical theorem, if contributions from non-rotating-wave terms are included. In this paper, we derive a more accurate expression for the polarizability. The calculations are rather complicated and require that we go beyond the standard Weisskopf-Wigner approximation. We present calculations carried out in both the Heisenberg and Schr\"odinger pictures, since they offer complementary methods for understanding the dynamics of the Rayleigh scattering associated with the atomic polarizability. Moreover, it is shown that the shifts associated with the excited state are not the Lamb shifts of an isolated atom, but depend on the dynamics of the atom-field interaction. Our results for the polarizability are consistent with those obtained recently by Loudon and Barnett using a Green's-function approach.
Different expressions for the linear polarizability of a two-level atom with radiative corrections have been derived recently. We show that an expression said to differ from that obtained by the present authors is in fact consistent with it. The same-sign and opposite-sign prescriptions for linewidths are revisited with respect to the polarizability, the scattering amplitude, and the optical theorem. Both prescriptions represent approximations to more general expressions in the two-level case, and neither is correct for transitions between excited atomic states, as we demonstrate by calculating the linear polarizability of a three-level atom.
If the distance between two or more energy levels is comparable to their natural breadth, the levels cease to act independently. The coupling between the levels is then describable by means of a "damping constant matrix." The usual discussions concern themselves with a diagonal damping constant. Below a systematic treatment of emission, absorption and fluorescence is given without this restriction and it is shown how in general the frequencies of emission lines are affected by this radiation coupling. The theory is applied to the calculation of the polarization of fluorescence radiation. It is shown that the coupling state of the lower group of levels of the fluorescence line does not influence the polarization. Results of detailed calculations are given for the polarization of $\mathrm{H}\ensuremath{\alpha}$ excited by absorption of the second line of the Lyman series.
We have investigated the optical response of superradiant atoms, which undergoes three different damping mechanisms: radiative dissipation (γrγ_rγr​), dephasing (γdγ_dγd​), and nonradiative dissipation (γnγ_nγn​). Whereas the roles of γdγ_dγd​ and γnγ_nγn​ are equivalent in the linear susceptibility, the third-order nonlinear susceptibility drastically depends on the ratio of γdγ_dγd​ and γnγ_nγn​: When γd≪γnγ_d \ll γ_nγd​≪γn​, the third-order susceptibility is essentially that of a single atom. Contrarily, in the opposite case of γd≫γnγ_d \gg γ_nγd​≫γn​, the third-order susceptibility suffers the size-enhancement effect and becomes proportional to the system size.
The polarizability of a two-level atom in the presence of radiative damping alone is derived by a Green function calculation. The rotating-wave approximation is not made, either in the interaction of the atom with the incident light or in its interaction with the scattered optical field. In contrast to previous work, the derivation takes account of the modification of the system ground state brought about by the non-rotating-wave part of the latter interaction. The final expression satisfies the optical theorem in addition to the other general requirements on the polarizability.