  The electronic defect states resulting from doping $^{229}$Th in CaF$_2$
offer a unique opportunity to excite the nuclear isomeric state $^{229m}$Th at
approximately 8 eV via electronic bridge mechanisms. We consider bridge schemes
involving stimulated emission and absorption using an optical laser. The role
of different multipole contributions, both for the emitted or absorbed photon
and nuclear transition, to the total bridge rates are investigated
theoretically. We show that the electric dipole component is dominant for the
electronic bridge photon. In contradistinction, the electric quadrupole channel
of the $^{229}$Th isomeric transition plays the dominant role for the bridge
processes presented. The driven bridge rates are discussed in the context of
background signals in the crystal environment and of implementation methods. We
show that inverse electronic bridge processes quenching the isomeric state
population can improve the performance of a solid-state nuclear clock based on
$^{229m}$Th.
