Modified optical Bloch equations for solids

Recently, DeVoe and Brewer [Phys. Rev. Lett. 50, 1269 (1983)] observed a striking deviation from a prediction of the optical Bloch equations by monitoring optical free-induction decay in the impurity-ion crystal ${\mathrm{Pr}}^{3+}$:${\mathrm{LaF}}_3$. At low optical fields, the ${\mathrm{Pr}}^{3+}$ optical dephasing time $T_2$ arises from magnetic fluctuations of the local environment, but at elevated optical fields, $T_2$ is no longer a constant, as assumed in the Bloch equations, because the magnetic line-broadening process is quenched. Several theories have been developed to explain the phenomenon. In this paper we present a simple theory of relaxation in solids which allows for comparison with earlier work. A ‘‘strong-redistribution’’ model is proposed where the optically excited impurity ions experience frequency shifts ε induced by a thermal bath. Frequency jumps occur at an average rate Γ and with an rms value of ${\epsilon }_0$/ √2 , where ${\epsilon }_0$ is the thermal width associated with the frequency shifts. Modified Bloch equations (MBE) follow that are solved explicitly for two limiting cases, ${\epsilon }_0$≪Γ and ${\epsilon }_0$≫Γ, and qualitatively for the more general case where the ratio Γ/${\epsilon }_0$ is arbitrary. Since many of the earlier theories are equivalent to the strong-redistribution model in the limit ${\epsilon }_0$≪Γ, we can assess the validity of the approximations made by these authors. Finally, we compare our MBE to the analogous transport equations that describe the effects of collisions in atomic vapors. We conclude that the problem addressed here is a general one, common to solids and gases alike, and is not restricted to impurity-ion crystals but will occur whenever frequency fluctuations are important.