Nuclear coherent population transfer to the ${}^{229m}\mathrm{Th}$ isomer using x-ray pulses

Population of the 8 eV ${}^{229m}\mathrm{Th}$ isomer via the second nuclear excited state at 29.19 keV by means of coherent x-ray pulses is investigated theoretically. We focus on two nuclear coherent population transfer schemes using partially overlapping x-ray pulses known from quantum optics: stimulated Raman adiabatic passage, and successive $\pi$ pulses. Numerical results are presented for three possible experimental setups. Our results identify the Gamma Factory as the most promising scenario, where two ultraviolet pulses combined with relativistically accelerated ions deliver the required intensities for efficient isomer population. Our simulations require knowledge of the in-band and cross-band nuclear transition probabilities. We give theoretically predicted values for the latter and discuss them in the context of recent experiments.

Figure 1

Three-level $\mathrm{\Lambda }$ system, comprised of the ground and two first-excited states of ${}^{229}\mathrm{Th}$. Each state is labeled with its angular momentum, parity, Nilsson quantum numbers [26] and energy. The letters P, S correspond to the pulse type (pump, Stokes) coupling to the respective levels. Energies are not to scale.

Figure 2

(a) Generic pulse sequence for STIRAP. $\mathrm{\Delta }\tau ={\tau }_p-{\tau }_s$ indicates the delay between the pulses. (b) Generic pulse sequence for NCPT via $\pi$ pulses.

Figure 3

(a) STIRAP transfer rate as a function of time delay (in the laboratory frame) for different scaling factors of the laser intensities. (b) STIRAP population transfer as a function of time for $0.3I_{0,P/S}$ and $\mathrm{\Delta }\tau =2.2\mathrm{ps}$ (lab frame). (c) $\pi$-pulse transfer rate as a function of time delay between pulses $\left|\mathrm{\Delta }\tau \right|$ (in the laboratory frame) for different ion energy spreads. (d) Single-$\pi$-pulse population transfer sequence for the parameters of the two black dots in panel (c): (i) $\mathrm{\Delta }\tau =-0.7\mathrm{ps}, 6I_{0,P/S}$ and $\mathrm{\Delta }\gamma /\gamma ={10}^{-4}$, and (ii) $\mathrm{\Delta }\tau =-5\mathrm{ps}, I_{0,P/S}$ and $\mathrm{\Delta }\gamma /\gamma ={10}^{-5}$ (in the laboratory frame).

Figure 4

(a) STIRAP transfer rate as a function of delay (in the laboratory frame) for different XFEL intensity values for $\mathrm{\Delta }\gamma /\gamma ={10}^{-4}$. (b) Single STIRAP sequence (lab frame) for $0.8I_{0,P/S}$ and $\mathrm{\Delta }\tau =7.5\mathrm{fs}$. (c) $\pi$-pulse transfer rate as a function of delay $\left|\mathrm{\Delta }\tau \right|$ (lab frame) for different ion energy spreads. (d) Single-$\pi$-pulse sequence (lab frame) for $\mathrm{\Delta }\gamma /\gamma ={10}^{-6}$ and $\mathrm{\Delta }\tau =-90\mathrm{fs}$.

Figure 5

(a) STIRAP transfer rate as a function of pulse delay with scaled intensities (XFELO parameters from Ref. [37]). (b) Related transfer rate as a function of intensity for different pulse delays. The vertical lines on the right- and left-hand side correspond to $3I_{0,P/S}$ and $0.01I_{0,P/S}$. (c) $\pi$-pulse transfer rate as a function of pulse delay $\left|\mathrm{\Delta }\tau \right|$ for XFELO parameters [37] (solid line), and [73] (dashed line). (d) Single $\pi$-pulse sequence for set [73] with delay $\mathrm{\Delta }\tau =-16\mathrm{ps}$.