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COLLECTIVE AND INTERATOMIC EFFECTS IN STRONG-FIELD RESONANCE FLUORESCENCE

Daniel Griffin

Published Mar 1, 1982• 0 citations
OpenAlex

Abstract

We have calculated the spectrum of resonance fluorescence and the second-order correlation function for a system of two two-level atoms interacting via the dipole-dipole interaction and via radiative coupling. The method employed is the "master equation" for the reduced atomic density operator. The scattered spectrum is symmetric about the resonant frequency and is altered significantly by the presence of the dipole-dipole term. For large incident field strengths pairs of sidebands are displaced from resonance by (+OR-)(omega)(,R) and (+OR-)2(omega)(,R), where (omega)(,R) is the Rabi frequency. Additional peaks arise when the dipole-dipole interaction is of the same order as the atom-field interaction. For weak incident fields the spectrum consists of a central peak and two peaks displaced symmetrically by (omega)(,d), the strength of the dipole-dipole interaction. The spectrum when averaged over the atomic separation and orientation consists of the central peak and two pairs of sidebands displaced by (+OR-)(omega)(,R) and (+OR-)2(omega)(,R). The central and first pair of sidebands become weaker and broader as the number density increases, whereas the second pair of sidebands is enhanced and broadened. The first effect is probably observable for suitable atomic number densities. Perhaps the second pair of sidebands could also be observed in a careful experiment. The second-order correlation function of this system predicts incomplete antibunching for strong incident field strengths. At weak field strengths the antibunching depends strongly on the dipole-dipole energy.

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We have calculated the spectrum of resonance fluorescence and the second-order correlation function for a system of two two-level atoms interacting via the dipole-dipole interaction and via radiative coupling. The method employed is the "master equation" for the reduced atomic density operator. The scattered spectrum is symmetric about the resonant frequency and is altered significantly by the presence of the dipole-dipole term. For large incident field strengths pairs of sidebands are displaced from resonance by (+OR-)(omega)(,R) and (+OR-)2(omega)(,R), where (omega)(,R) is the Rabi frequency. Additional peaks arise when the dipole-dipole interaction is of the same order as the atom-field interaction. For weak incident fields the spectrum consists of a central peak and two peaks displaced symmetrically by (omega)(,d), the strength of the dipole-dipole interaction. The spectrum when averaged over the atomic separation and orientation consists of the central peak and two pairs of sidebands displaced by (+OR-)(omega)(,R) and (+OR-)2(omega)(,R). The central and first pair of sidebands become weaker and broader as the number density increases, whereas the second pair of sidebands is enhanced and broadened. The first effect is probably observable for suitable atomic number densities. Perhaps the second pair of sidebands could also be observed in a careful experiment. The second-order correlation function of this system predicts incomplete antibunching for strong incident field strengths. At weak field strengths the antibunching depends strongly on the dipole-dipole energy.
The spectrum of resonance fluorescence and the second-order correlation function are calculated for a system of two identical two-level cooperative systems interacting via the dipole-dipole interaction. The results are valid for arbitrary incident-field strengths and for atomic separations much less than the resonant wavelength. The intensity of the sidebands at the Rabi frequency is diminished by the dipole interaction, whereas that of the sidebands at twice the Rabi frequency is enhanced. Both the cooperative behavior and the dipole-dipole interaction act to diminish the photon antibunching, as shown by the second-order correlation function.
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.
The excitation spectrum arising from the interaction between two identical atoms (molecules), one of which is excited in the presence of a resonant strong electromagnetic field (pump field), is investigated. The excitation spectrum is found to consist of those describing the symmetric and antisymmetric modes, respectively. The form of both spectra depends on the relation between the distance RRR separating the atoms and the wavelength $\ensuremath{\lambda}$ of the transition from the ground state to the excited state. For $R<\ensuremath{\lambda}$, and when certain conditions prevail, the spectral function for the symmetric modes consists of three Lorentzian lines describing the central peak and the two sidebands whose radiative widths are equal to ${\ensuremath{\gamma}}_{0}$, $\frac{3{\ensuremath{\gamma}}_{0}}{2}$, and $\frac{3{\ensuremath{\gamma}}_{0}}{2}$, which are twice the corresponding ones arising from an isolated single atom interacting with the pump field. For the antisymmetric modes the central peak has a $\ensuremath{\delta}$-function distribution indicating the stability of the mode in question, while, when certain conditions are satisfied, the two sidebands are described by Lorentzian lines each having a radiative width of the order of $\frac{{\ensuremath{\gamma}}_{0}}{2}$, which is equal to the natural linewidth for a photon spontaneously emitted from an isolated atom. For both the symmetric and antisymmetric modes, the dipole-dipole interaction between the atoms brings about small energy shifts. For $R>\ensuremath{\lambda}$, apart from the small energy shifts caused by the dipole-dipole interaction, the spectral functions for symmetric and antisymmetric modes are similar to that for the single atom interacting with the pump field.
The excitation spectrum arising from photon density-density correlations has been considered in a system of two interacting identical atoms. The two-level atoms are pumped near resonance by a strong electromagnetic field while they interact through their dipole-dipole interaction and radiate to each other as well. In such a system the photon density-density correlations arise from the interference spectrum due to the beating of the frequency of the pump field and the atomic transition frequency, as well as from the spectrum induced by the dipole-dipole and radiative interactions between the atoms. The spectral function of this system has been calculated in the limit of high photon densities. It consists of four terms describing four bands which are similar to those appearing in the two-atom resonance fluorescence spectrum for the symmetric modes when the central peak is missing. The collective behavior as well as the dynamics of the system are discussed in detail. By using the spectral function, an expression for the photon density-density correlation function at different times has been derived where thermal effects have been also included.
We have studied the fluorescence spectra arising from the interaction of a three-level atom with two strong electromagnetic fields whose initially populated modes are equal to the two atomic transition frequencies, respectively. The Green's function formalism has been used to calculate the excitation spectrum of the system. An expression for the spectral function describing the excitation spectrum of the system has been derived in a closed form in the limit of high photon densities. Numerical computation of the expression for the spectral function indicates that at each transition frequency there may exist either one pair or two pairs or three pairs of sidebands, in addition to the central peak, depending upon the relative strength of the Rabi frequencies involved.
We suggest that the statistical properties of the scattered radiation in resonance-fluorescence experiments may be affected significantly by the existence of atomic correlations. The scattered light spectrum from two- and three-atom collective systems has been calculated and compared with the one-atom spectrum. The differences are quite significant for weak fields, but become less pronounced as the intensity of the driving field is increased. In addition, we have calculated the scattered intensity correlation function for collectively interacting systems, and found that its behavior is very different from that of the single-atom intensity correlation function, both for weak and strong incident fields. The implications of our findings for the observation of photon antibunching are also discussed.