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As a unique feature, the $^{229}$Th nucleus has an isomeric transition in thevacuum ultraviolet that can be accessed by optical lasers. The interferenceeffects occurring in the interaction between coherent optical light and anensemble of $^{229}$Th nuclei are investigated theoretically. We consider thescenario of nuclei doped in vacuum ultraviolet-vacuum ultraviolet transparentcrystals and take into account the effect of different doping sites andtherefore different lattice fields that broaden the nuclear transition width.This effect is shown to come in interplay with interference effects due to thehyperfine splitting of the ground and isomeric nuclear states. We investigatepossible experimentally available situations involving two-, three- andfour-level schemes of quadrupole sublevels of the ground and isomeric nuclearstates coupling to one or two coherent fields. Specific configurations whichoffer clear signatures of the isomer excitation advantageous for the moreprecise experimental determination of the transition energy are identified.Furthermore, it is shown that population trapping into the isomeric state canbe achieved. This paves the way for further nuclear quantum optics applicationswith $^{229}$Th such as nuclear coherent control.

Quantum interference effects in an ensemble of $^{229}$Th nuclei interacting with coherent light