Microscopic theory of optical line narrowing of a coherently driven solid

The optical Bloch equations which incorporate the phenomenological population $\left(T_1\right)$ and dipole dephasing $\left(T_2\right)$ times have been tested recently by optical free-induction-decay (FID) measurements on an impurity-ion crystal ${\mathrm{Pr}}^{3+}$:${\mathrm{La}}_3$ at 1.6 K. At low optical fields, the observed ${\mathrm{Pr}}^{3+}$ optical linewidth is dominated by magnetic fluctuations arising from pairs of fluorine nuclear flip-flops where the condition $T_1\gg T_2$ prevails. At elevated fields, this nuclear broadening mechanism is quenched and the Bloch equations are violated with $T_2\to T_1$. In this paper, a microscopic theory appropriate for a low-temperature impurity solid is presented which reveals the above features both for optical and radio frequencies, and a simple physical interpretation of this line narrowing phenomenon is given. Modified Bloch equations of a novel form are derived to second order and yield analytic FID solutions over the entire range of optical-field strength. A discussion of the earlier NMR theories is given, pointing out similarities and differences.