Driven electronic bridge processes via defect states in ${}^{229}\mathrm{Th}$-doped crystals

The electronic defect states resulting from doping ${}^{229}\mathrm{Th}$ in ${\mathrm{CaF}}_2$ offer a unique opportunity to excite the nuclear isomeric state ${}^{229\mathrm{m}}\mathrm{Th}$ 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 ${}^{229}\mathrm{Th}$ 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 ${}^{229\mathrm{m}}\mathrm{Th}$.

Figure 1

EB process for the excitation of ${}^{229\mathrm{m}}\mathrm{Th}$ from the ground state $|g\rangle$ to the isomeric state $|m\rangle$ (right graph) [26]. The initially populated electronic defect states $|d\rangle$ lie in the crystal band gap above or below the isomer energy. The EB process occurs either spontaneously (left graph) or assisted by an optical laser in the stimulated or absorption schemes (middle graphs). In all cases, EB proceeds via a virtual electronic state $|v\rangle$ and ends in the ground state $|o\rangle$. The conduction band states are given by the set $|c\rangle$.

Figure 2

Electron density illustrations for the defect states labeled $\left\{\right|d\rangle \}=\left\{\right|d_1\rangle ,...,|d_8\rangle \}$.

Figure 3

Average spontaneous EB rate ${\mathrm{\Gamma }}_{E1}^{sp}$ as a function of the maximum energy of the included states measured with respect to the electronic ground state $|o\rangle$.

Figure 4

Average spontaneous EB rate ${\mathrm{\Gamma }}_{E1}^{sp}$ as a function of number of conduction band states included in the intermediate summation of $|n\rangle$ and $|k\rangle$ seen in (5). The solid line is the total rate, while the dotted line shows the ${\mathcal{T}}_{E2}$ contribution alone.

Figure 5

Normalized average driven EB rate ${\rho }_d{\mathrm{\Gamma }}_{E1}^{\zeta }\left(\right|g,\left\{d\right\}\rangle \to |m,o\rangle )/\left(I_d{I}_p\right)$ [in units of ${\mathrm{m}}^4/\left({\mathrm{W}}^2{\mathrm{s}}^3\right)]$ with $\zeta =ab/st$ as a function of average defect state energy. The solid line is the total EB rate, whereas the dotted line is the contribution from ${\mathcal{M}}_{E2}$. The blue vertical line shows the isomer energy considered here, $E_m=8.28\mathrm{eV}$. Left (right) of this line, $\zeta =ab$ ($\zeta =st$).

Figure 6

Normalized average driven EB rates ${\rho }_d{\mathrm{\Gamma }}_{\mu L}^{\zeta }\left(\right|g,\left\{d\right\}\rangle \to |m,o\rangle )/\left(I_d{I}_p\right)$ [in units of ${\mathrm{m}}^4/\left({\mathrm{W}}^2{\mathrm{s}}^3\right)]$, with $\zeta =ab/st$, as a function of average defect state energy where (a) $\mu L=M1$ and (b) $\mu L=E2$. The blue vertical lines mark the isomer energy $E_m=8.28\mathrm{eV}$.

Figure 7

Normalized $E1$ bridge rates ${\rho }_d{\mathrm{\Gamma }}_{E1}^{\zeta }\left(\right|g,d_i\rangle \to |m,o\rangle )/\left(I_d{I}_p\right)$ [in units of ${\mathrm{m}}^4/\left({\mathrm{W}}^2{\mathrm{s}}^3\right)]$ as a function of initial defect state $|d_i\rangle$ energy considering (a) $i=3$, (b) $i=5$, (c) $i=7$. The blue vertical lines mark the isomer energy $E_m=8.28\mathrm{eV}$, with $\zeta =ab$ ($\zeta =st$) left (right) thereof.

Figure 8

Normalized (a) $\mu L=M1$, (b) $\mu L=E2$ bridge rates ${\rho }_d{\mathrm{\Gamma }}_{\mu L}^{\zeta }\left(\right|g,d_7\rangle \to |m,o\rangle )/\left(I_d{I}_p\right)$ [in units of ${\mathrm{m}}^4/\left({\mathrm{W}}^2{\mathrm{s}}^3\right)]$ as a function of initial defect state energy. The blue vertical lines mark the isomer energy $E_m=8.28\mathrm{eV}$, with $\zeta =ab$ ($\zeta =st$) left (right) thereof.

Figure 9

Quenching processes for the deexcitation of ${}^{229\mathrm{m}}\mathrm{Th}|m\rangle \to |g\rangle$. The electronic defect states $|d\rangle$ can be thereby excited via a spontaneous (or additionally stimulated) EB scheme provided they lie below the isomer energy. If the defect states lie higher in energy than the isomer, absorption of a laser photon is required. Wiggly arrows in red depict the EB photons related to the quenching of the isomeric state, either spontaneously emitted or externally pumped by a laser for the stimulated or absorption schemes. Photons in blue result from the subsequent spontaneous decay of the defect state $|d\rangle \to |o\rangle$.

Figure 10

Ultimate clock fractional instability ${\sigma }_y\sqrt{\tau }=\delta f\sqrt{t}/\omega$ as a function of excitation rate $R$ for an optimized interrogation cycle with (black solid curve) and without (red dotted curve) laser-enhanced quenching during measurement phases. See text for further explanations and used parameters.