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Optica Acta International Journal of Optics

Higher Harmonics in Cooperative Resonance Fluorescence

Vladislav Cheltsov

Published Oct 1, 1984• 4 citations
Publisher

Abstract

In the Stark representation, in third-order perturbation theory with respect to fluorescence, it is shown that probability of the ωa + 4Ω harmonic emission in the cooperative fluorescence of two two-level atoms is small and proportional to γ2 / Ω2, Ω2 / ω2 a, contributions to this probability from non-resonant processes being of the order of γ2 / ω2 a.

DOI
10.1080/713821408

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In the Stark representation, in third-order perturbation theory with respect to fluorescence, it is shown that probability of the ωa + 4Ω harmonic emission in the cooperative fluorescence of two two-level atoms is small and proportional to γ2 / Ω2, Ω2 / ω2 a, contributions to this probability from non-resonant processes being of the order of γ2 / ω2 a.
The resonance fluorescence spectrum of one and two two-level atoms in an ideal cavity with one resonant mode is investigated as a function of the initial excitation of the system. The analysis is carried out in the Stark representation with respect to the cavity's field and in first-order perturbation theory with respect to the fluorescence field. In the absence of the field of the cavity (n c = 0) and under an initial excitation of one atom only, the spectrum consists of two lines which are symmetrical with respect to ωa. In all other cases the spectrum has a central line at ωa and other lines at ωa ± 2 Ω. The intensity of the higher (ωa ± 4 Ω) harmonics is small and decreases as 1/ 2 c c with increasing n>c. The nonlinearity of the interaction hamiltonian results in the parametric (with respect to the field of the cavity) amplification of the ωa + 2 Ω harmonic and the attenuation of the ωa − 2 Ω harmonic. The stimulated emission of the ωa ± 2 Ω harmonics, vanishing in a strong (n c » 1) field, is shown to exist for two atoms.
The Stark representation of the resonator field and the first order of perturbation theory of the fluorescence field are used to study the fluorescence spectrum of one and two two-level atoms in an ideal resonator as a function of the initial excitation of the system. It is shown that in the absence of resonator photons (nc = 0) the spectrum obtained for initial excitation of one atom consists of two lines located symmetrically relative to the central frequency ωα. In all other cases the spectrum consists of the central harmonic ωα and of ωα ± 2Ω satellites (Ω is the Rabi frequency). The intensity of collective higher harmonics ωα ± 4Ω is low and it falls on increase in the resonator field as I/nc2. The nonlinearity of the interaction Hamiltonian results in parametric amplification of the ωα + 2Ω harmonics and in attenuation of the ωα–2Ω harmonics. In the case of two atoms it is possible to achieve stimulated fluorescence at the harmonics ωα ± 2Ω.
We have considered the fluorescent spectra arising from the interaction of a four-level atom with two strong electromagnetic fields whose initially populated modes ω a and ω b are in resonance with three of the atomic transition frequencies. Green's function formalism has been used in the limit of high photon densities to calculate the excitation spectra near the frequencies ω = ±(2ω a - ω b ). Expressions for the total relative intensity of the system have been derived as a function of the parameter ζ = Ω2 b /Ω2 a , where Ω a and Ω b are the Rabi frequencies of the two laser fields, respectively. For ζ = 0, the spectra consist of a central peak at ω = 2ω a - ω b and a pair of sidebands at ω = 2ω a - ω b ± Ω a √2, both having spectral widths of the order of γ0, where γ0 is the spontaneous emission probability. The relative intensity of the central peak for Ω a ≠ 0 is reduced by 25 per cent of its original value obtained for Ω a = 0. For ζ ≠ 0, the relative intensity of the central peak increases slowly as ζ increases while that of the pair of sidebands at ω = 2ω b - ω b ± Ω a √2 is reduced by a factor of two of its given value for ζ = 0. This destructive interference is due entirely to the presence of the second laser field with Ω b ≠ 0. A new pair of sidebands, which arise from the cooperative effect of both laser fields, is peaked at ω = 2ω a - ω b ± Ω a √(2 + 0·5ζ) and has a spectral width equal to γ0. A set of four pairs of sidebands also appear at ω = 2ω a - ω b ± Cj (ζ)Ω a with j = 1, 2, 3, 4, respectively, which have spectral widths of the order of 3γ0/2. The existence of these pairs of sidebands depends strongly on the value of the parameter ζ. Numerical estimates for the coefficients Cj (ζ) and the relative intensities of these sidebands are tabulated for a wide range of values of ζ and discussed in detail.
We have considered the excitation spectra of three-photon processes arising from the interaction of a three-level atom, in the cascade configuration, with two strong electromagnetic fields whose initially populated modes ω a and ω b are near resonance with the atomic transition frequencies ω21 and ω32, respectively. In the limit of high photon densities, expressions for the Green's functions are derived in a closed form describing third order nonlinear processes in the region of frequencies ω = ω a + 2ω b and ω = ω b + 2ω a , respectively. Numerical calculations are then performed and the computed spectra are presented graphically for different values of the Rabi frequencies Ω a and Ω b , and detunings ζ a and ζ b , respectively. The excitation spectrum near ω = ω a + 2ω b is calculated as a function of the parameters η a = Ω a /γ0, η b = Ω b /γ0, ζ a = (ω21 - ω a )/γ0 and ζ b = (ω32 - ω b )/γ0 where γ0 is the spontaneous transition probability. With no detuning the spectral intensities vary slowly with respect to variations of the Rabi frequencies and are shifted by amounts governed by the field strengths η a and η b . In the presence of detunings, asymmetries, characteristics of such off-resonance situations, arise enhancing certain spectral peaks while diminishing the intensity of others.
A QED calculation of the resonance fluorescence spectra of a two-level atom in a strong, nearly resonant field, is reported, taking into account multiple spontaneous emissions (in any order) and interference effects. The results of this calculation are compared with the results of the quantum statistical theories, and the role of the elementary processes analysed. At resonance, in the high-field limit, the exact agreement of the QED and statistical theories is demonstrated. It is pointed out that the interference effects are of the greatest importance.