Collective effects in ${}^{229}\mathrm{Th}$-doped crystals

Vacuum-ultraviolet-transparent crystals have been proposed as host lattice for the coherent driving of the unusually low-lying isomer excitation in ${}^{229}\mathrm{Th}$ for metrology and quantum optics applications. Here the possible collective effects occurring for the coherent pulse propagation in the crystal system are investigated theoretically. We consider the effect of possible doping sites, quantization axis orientation, and pulse configurations on the scattered light intensity and signatures of nuclear excitation. Our results show that for narrow-pulse driving, the rather complicated quadrupole splitting of the level scheme is significantly simplified. Furthermore, we investigate complex driving schemes with a combination of pulsed fields and investigate the occurring interference process. Our theoretical results support experimental attempts for first direct driving of the nuclear transition with coherent light.

Figure 1

Th:${\mathrm{CaF}}_2$ structure, dopant orientation with (a) ${90}^{\circ }$ and (b) ${180}^{\circ }$ fluoride interstitials.

Figure 2

Nuclear quadrupole splitting level scheme ${}^{229}\mathrm{Th}$:${\mathrm{CaF}}_2$ (not to scale) for the ${180}^{\circ }$ fluoride interstitials doping configuration. Both ground state and isomeric state have positive parity. The possible projections of the nuclear spin angular momentum on the quantization axis are denoted by $m$. Assuming that only the two lower ground-state hyperfine levels are populated, probe (${\mathrm{\Omega }}_p$) and couple (${\mathrm{\Omega }}_c$) VUV laser pulses can couple to the two- and three-level schemes depicted in the panels on the right-hand side. ${\mathrm{\Delta }}_{p/c}$ stands for the corresponding detunings of the probe and couple laser pulses.

Figure 3

Scattered intensity for the case of the two-level system for $\mathrm{\Delta }=0\text{–}{10}^8\mathrm{\Gamma }$ (black solid line) and $\mathrm{\Delta }={10}^9\mathrm{\Gamma }$ (red dashed line). The excitation occurs for $\mathrm{\Delta }<{10}^{10}\mathrm{\Gamma }\approx \hslash /E_p$.

Figure 4

Scattered intensity for the case of the three-level system for $\mathrm{\Delta }=0\text{–}{10}^6\mathrm{\Gamma }$ (black solid line) and $\mathrm{\Delta }={10}^8\mathrm{\Gamma }$ (red dashed line). The excitation occurs for $\mathrm{\Delta }<{10}^{10}\mathrm{\Gamma }\approx \hslash /E_p$.

Figure 5

NFS intensity of the three-level thorium system resulting from abruptly switching the couple laser on and off (black solid line, left axis) together with the relative intensity of the couple pulse (red dashed line, right axis).

Figure 6

NFS scattered intensity for the three-level thorium system with a Gaussian pulse couple laser (black solid line, left axis) together with the couple pulse relative intensity (red dashed line, right axis) for $t_c=t_p=50\mu \mathrm{s}$ and $T_c=4$ ms.

Figure 7

Scattered intensity for the two-level thorium system with $\mathrm{\Delta }=0$, $t_0=50\mu \mathrm{s}$, and $T=0.1\mu \mathrm{s}$. (Blue dashed line) NFS intensity after the first pulse impacting at $t=0$. (Black solid line) With the additional excitation of a second pulse in phase with the first $\varphi =0$. (Black dotted line) With the additional excitation of a second pulse with $\varphi =\pi$.

Figure 8

Scattered intensity for the two-level thorium system with $\mathrm{\Delta }={10}^9\mathrm{\Gamma }$ and $T=0.1\mu \mathrm{s}$ for (a) a single pulse at $t=0$, (b) the first pulse at $t=0$ followed by the second one at $t_0=\pi /\mathrm{\Delta }$, (c) the same but with delay between pulses $t_0=3\pi /2\mathrm{\Delta }$, and (d) for delay between pulses $t_0=2\pi /\mathrm{\Delta }$. (Black solid line) NFS intensity $|\mathrm{\Omega }/{\mathrm{\Omega }}_0{|}^2$ when using two pulses of the same phase. (Black dotted line) NFS intensity $|\mathrm{\Omega }/{\mathrm{\Omega }}_0{|}^2$ when using two pulses with phase shift $\pi$. (Red dashed line) ${(\text{Re}\left\{\mathrm{\Omega }\right\}/{\mathrm{\Omega }}_0)}^2$. (Green dash-dotted line) ${(\text{Im}\left\{\mathrm{\Omega }\right\}/{\mathrm{\Omega }}_0)}^2$.

Figure 9

Three-level thorium system driven by (a) a single pulse at $t=0$ or (b) a single pulse at $t=0$ followed by a second identical pulse at $t=t_0$, which is chosen as the first minimum of ${(\text{Im}\left\{\mathrm{\Omega }\right\}/{\mathrm{\Omega }}_0)}^2$. (Black solid line) NFS intensity $|\mathrm{\Omega }/{\mathrm{\Omega }}_0{|}^2$ when using two pulses of the same phase. (Black dotted line) NFS intensity $|\mathrm{\Omega }/{\mathrm{\Omega }}_0{|}^2$ when using two pulses with phase shift $\pi$. (Red dashed line) ${(\left\{\mathrm{\Omega }\right\}/{\mathrm{\Omega }}_0)}^2$. (Green dash-dotted line) ${(\text{Im}\left\{\mathrm{\Omega }\right\}/{\mathrm{\Omega }}_0)}^2$. For the calculation we have used $\mathrm{\Delta }={10}^8\mathrm{\Gamma }$ and $T=0.5\mu \mathrm{s}$.

Figure 10

Two ${}^{229}\mathrm{Th}$:${\mathrm{CaF}}_2$ target setup. A left circularly polarized probe field drives the $|\frac{5}{2},\frac{5}{2}\rangle \leftrightarrow |\frac{3}{2},\frac{3}{2}\rangle$ isomeric transitions in both crystals. The detuning of the probe to the unperturbed resonance frequency is denoted by ${\mathrm{\Delta }}_p$. ${\mathrm{\Delta }}_B$ denotes the total Zeeman shift due to the external magnetic field $B$.

Figure 11

Scattered intensity for the two crystal system with $B={10}^{-4}$ T for $\mathrm{\Delta }=0\text{–}{10}^6\mathrm{\Gamma }$ (black solid line) and $\mathrm{\Delta }={10}^8\mathrm{\Gamma }$ (red dashed line). Excitation occurs for $\mathrm{\Delta }<{10}^{10}\mathrm{\Gamma }$.

Figure 12

Scattered intensity for the two crystal system considering $\mathrm{\Delta }=0$. The applied external magnetic field takes the values $B=5\times {10}^{-4}$ T, 0 T, ${10}^{-4}$ T, and again $5\times {10}^{-4}$ T.

Figure 13

Th:${\mathrm{CaF}}_2$ structure with ${180}^{\circ }$ fluoride interstitials showing the three possible rotations allowed in a bulk crystal.

Figure 14

Polarization vectors of fields used to drive the $(\pi ,{\sigma }^{+},{\sigma }^{-})$ transitions relative to the laboratory-frame $(x,y,z)$.

Figure 15

Relative difference in spectrum intensity $(I_a-I_b)/I_a$ considering $\left(a\right)$ two-state system, $1q$-axis, $N/6$ initial population of single ground state $|\frac{5}{2},\frac{5}{2}\rangle$ and $\left(b\right)$ 10-state system, $1q$-axis, initial population $N/3$ split equally among two lowest states $|\frac{5}{2},\pm \frac{5}{2}\rangle$. The calculations were performed considering $\mathrm{\Delta }=0$ for the transition $|\frac{5}{2},\pm \frac{5}{2}\rangle \to |\frac{3}{2},\pm \frac{3}{2}\rangle$.

Figure 16

Scattered intensity for (black solid line) two-state system, $1q$-axis, initial population $N/6$ in $|\frac{5}{2},\frac{5}{2}\rangle$ and (red dashed line) four-state systems, $3q$-axes, $N$ initial population split equally among two lowest states $|\frac{5}{2},\pm \frac{5}{2}\rangle$ with $1/3$ in each possible $q$-axis orientation. The calculations were performed considering $\mathrm{\Delta }=0$ for the transition $|\frac{5}{2},\pm \frac{5}{2}\rangle \to |\frac{3}{2},\pm \frac{3}{2}\rangle$.

Figure 17

Energy eigenvalues of ${\widehat{H}}_{E2}$ for $I=I_e=3/2$ as a function of the asymmetry parameter $\eta$. Labeled with the spin projection $m_e$ with respect to the largest field component.

Figure 18

Squared projection of state vector $|\frac{3}{2},{\psi }_m\left(\eta \right)\rangle$ on overlapping eigenstates of angular momentum as a function of the asymmetry parameter $\eta$.

Figure 19

Energy eigenvalues of ${\widehat{H}}_{E2}$ for $I=I_g=5/2$ as a function of the asymmetry parameter $\eta$. Labeled with the spin projection $m_g$ with respect to the largest field component.

Figure 20

Squared projection of state vector $|\frac{5}{2},{\psi }_m\left(\eta \right)\rangle$ on overlapping eigenstates of angular momentum as a function of the asymmetry parameter $\eta$.