Time scales and optical dephasing measurements: Investigation of dynamics in complex systems

A detailed theoretical analysis of optical dephasing measurements performed on complex systems, e.g., glasses, proteins, or complex crystals, is presented. Unlike simple crystals, such systems undergo dynamical processes which have a very broad rate distribution. Dynamics can occur on a variety of time scales ranging from subpicoseconds to seconds, or longer. The formalism is based on a four-point correlation-function description of line-narrowing experiments. The results of optical dephasing measurements (time domain) or linewidth measurements (frequency domain) depend on a time $T_W$ which defines the time scale associated with the particular experimental technique. Detailed information about the rate distribution of the system’s dynamics is obtained from the change in the optical dephasing rate as the experimental time scale is changed, not from the dephasing rate measured in any individual experiment. A fundamental result, which is independent of the nature of the rate distribution or the coupling to the optical center, is proven; i.e., the derivative of the optical dephasing rate with respect to $T_W$, the experimental time scale, is directly proportional to the Laplace transform of the fluctuation rate distribution. As examples, the formal results are applied to two types of systems, optical centers in a glass and in a complex crystal. For the crystalline system ${\mathrm{Pr}}^{3+}$:${\mathrm{CaF}}_2$, $T_W$-dependent optical dephasing data from the literature are analyzed quantitatively, where previously only a qualitative description was possible.