Theory of photon echoes from interacting impurities in crystals with inhomogeneously broadened absorption spectra

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Abstract
We present a theory of photon echoes from impurity molecules in mixed crystals that takes into account electronic interactions among impurity molecules and inhomogeneous broadening of the electronic absorption spectrum. The system is modeled as a collection of randomly placed interacting two level systems with a distribution of transition energies. The observable of the echo experiment is expanded in powers of the density of two level systems, and the first two terms of this expansion are calculated. A scaling argument suggests that the density expansion is rapidly convergent for concentrations of experimental interest, and the density expansion is used to construct a Padé approximant. At times comparable to or less than the inverse of the width of the inhomogeneous distribution of resonance frequencies, the time dependence of the signal is determined primarily by the interactions between the impurities and is independent of the inhomogeneous width. At longer times the inhomogeneity becomes significant and the echo signal displays a much weaker time dependence. We examine the effect of different microscopic models of inhomogeneous broadening on the echo observable. We demonstrate that the photon echo experiment can provide a sensitive probe of interactions between impurities in mixed crystals.