Hyperfine-resolved laser excitation and detection of nuclear isomer in trapped $^{229}$Th$^{3+}$ ions

Abstract

We present a comprehensive theoretical investigation of hyperfine-resolved excitation and detection of the low-energy isomeric state of $^{229}$Th in trapped $^{229}\mathrm{Th}^{3+}$ ions. Using a quantum master equation approach, we quantitatively analyze the dependence of the isomeric population on laser linewidth, detuning, and irradiation time, showing that their proper matching is essential for efficient excitation. Going beyond earlier conceptual discussions of electronic-fluorescence-based nuclear-state detection, we propose two concrete nuclear-state detection schemes based on three hyperfine-resolved electronic fluorescence channels at 690, 984, and 1088 nm. Our quantitative analysis shows that, for 50 ions, the 690- and 984-nm scheme yields detectable photon count rates on the order of $10^3~\mathrm{s}^{-1}$ at 690 nm and $10^4~\mathrm{s}^{-1}$ at 984 nm, whereas the 1088-nm scheme achieves a detectable photon rate on the order of $10^3~\mathrm{s}^{-1}$. By quantifying the trade-off between irradiation time and scan-step size, we show that the nuclear transition can be located within one month for a 100-MHz uncertainty using currently available vacuum-ultraviolet laser technology. These results provide practical guidance for trapped-ion $^{229}\mathrm{Th}$ spectroscopy and the development of nuclear clocks.