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

As a unique feature, the $^{229}$Th nucleus has an isomeric transition in the vacuum ultraviolet that can be accessed by optical lasers. The interference effects occurring in the interaction between coherent optical light and an ensemble of $^{229}$Th nuclei are investigated theoretically. We consider the scenario of nuclei doped in vacuum ultraviolet-vacuum ultraviolet transparent crystals and take into account the effect of different doping sites and therefore different lattice fields that broaden the nuclear transition width. This effect is shown to come in interplay with interference effects due to the hyperfine splitting of the ground and isomeric nuclear states. We investigate possible experimentally available situations involving two-, three- and four-level schemes of quadrupole sublevels of the ground and isomeric nuclear states coupling to one or two coherent fields. Specific configurations which offer clear signatures of the isomer excitation advantageous for the more precise experimental determination of the transition energy are identified. Furthermore, it is shown that population trapping into the isomeric state can be achieved. This paves the way for further nuclear quantum optics applications with $^{229}$Th such as nuclear coherent control.