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Physical Review A

Analytical solution for the spectrum of resonance fluorescence of a cooperative system of two atoms and the existence of additional sidebands

G. S. Agarwal, Ragini Saxena, Lorenzo M. Narducci, D.H. Feng, Robert Gilmore

Published Jan 1, 1980• 66 citations
Publisher

Abstract

The analytical solution for the spectrum of resonance fluorescence from a collective system of two atoms is presented. It is shown that the spectrum is a decreasing function of frequency in the neighborhood $\ensuremath{\omega}\ensuremath{-}{\ensuremath{\omega}}_{L}=\ifmmode\pm\else\textpm\fi{}2\ensuremath{\Omega}$ ($\ensuremath{\Omega}$ is the Rabi frequency) for moderately large $\ensuremath{\Omega}$ and exhibits resonant structure only for extremely large $\ensuremath{\Omega}\ensuremath{\approx}100\ensuremath{\gamma}$. For moderately large $\ensuremath{\Omega}$ the resonant contributions at $\ensuremath{\omega}\ensuremath{-}{\ensuremath{\omega}}_{L}=\ifmmode\pm\else\textpm\fi{}2\ensuremath{\Omega}$ are destroyed by the nonresonant contributions; hence the additional structures at $\ifmmode\pm\else\textpm\fi{}2\ensuremath{\Omega}$ can be resolved only by filtering the nonresonant contributions.

DOI
10.1103/physreva.21.257

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The analytical solution for the spectrum of resonance fluorescence from a collective system of two atoms is presented. It is shown that the spectrum is a decreasing function of frequency in the neighborhood \ensuremathω\ensuremath−\ensuremathωL=\ifmmode±\else\textpm\fi2\ensuremathΩ\ensuremath{\omega}\ensuremath{-}{\ensuremath{\omega}}_{L}=\ifmmode\pm\else\textpm\fi{}2\ensuremath{\Omega}\ensuremathω\ensuremath−\ensuremathωL​=\ifmmode±\else\textpm\fi2\ensuremathΩ (\ensuremathΩ\ensuremath{\Omega}\ensuremathΩ is the Rabi frequency) for moderately large \ensuremathΩ\ensuremath{\Omega}\ensuremathΩ and exhibits resonant structure only for extremely large \ensuremathΩ\ensuremath≈100\ensuremathγ\ensuremath{\Omega}\ensuremath{\approx}100\ensuremath{\gamma}\ensuremathΩ\ensuremath≈100\ensuremathγ. For moderately large \ensuremathΩ\ensuremath{\Omega}\ensuremathΩ the resonant contributions at \ensuremathω\ensuremath−\ensuremathωL=\ifmmode±\else\textpm\fi2\ensuremathΩ\ensuremath{\omega}\ensuremath{-}{\ensuremath{\omega}}_{L}=\ifmmode\pm\else\textpm\fi{}2\ensuremath{\Omega}\ensuremathω\ensuremath−\ensuremathωL​=\ifmmode±\else\textpm\fi2\ensuremathΩ are destroyed by the nonresonant contributions; hence the additional structures at \ifmmode±\else\textpm\fi2\ensuremathΩ\ifmmode\pm\else\textpm\fi{}2\ensuremath{\Omega}\ifmmode±\else\textpm\fi2\ensuremathΩ can be resolved only by filtering the nonresonant contributions.
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.
The fluorescence spectrum of a system of a large number of two-level atoms driven by a strong resonant field has been treated, so far, for atoms contained in a region that is small compared to a wavelength. For high laser intensities resonances displaced from the line center by twice the Rabi frequency were predicted1-3 in addition to the usual Stark sidebands.4 We show that the explicit spatial distribution of the two-level atoms affects the shape of the radiation spectra and leads to the appearance of additional sidebands.
We report on the semiclassical solution of the Bloch equations for cooperative resonance fluorescence with collective damping. These equations are shown to be equivalent to the relevant master equation, in the limit of large cooperation number and time scales faster than the one-atom lifetime (reciprocal Einstein AAA coefficient). The resulting spectrum is derived for a coherent driving field of arbitrary intensity. Additional spectral peaks, as well as the central triplet, are predicted above threshold. A physical explanation of the extra peaks is proposed.
We investigate the resonance fluorescence features of a small ensemble of closely packed and moderately laser-pumped two-level emitters at resonance. The mean distance between any two-level radiators is smaller than the corresponding emission wavelength, such that the dipole-dipole interactions are not negligible. We have found that under the secular approximation, the collective resonance fluorescence spectrum consists of 2 N + 1 spectral lines, where N is the number of emitters from the sample. The 2 N spectral sidebands, symmetrically located around the generalized Rabi frequency with respect to the central line at the laser frequency, are distinguishable if the dipole-dipole coupling strength is larger than the collective spontaneous decay rate. This way, one can estimate the radiators number within the ensemble via measuring of the spontaneously scattered collective resonance fluorescence spectrum. Contrarily, if the dipole-dipole coupling is of the order of or smaller than the cooperative spontaneous decay rate, but still non-negligible, the spectrum turns into a Mollow-like fluorescence spectrum, where the two sidebands' spectral lines broaden proportional to the dipole-dipole coupling strength. Published by the American Physical Society 2024