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 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 into 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.

Figure 1

Quadrupole splitting of ${}^{229}$Th doped in VUV-transparent crystals such as LiCaAlF${}_6$ or CaF${}_2$. Here $m$ is the magnetic quantum number. The splittings are on the order of MHz [23].

Figure 2

(a) Level scheme for VUV excitation of the doped ${}^{229}$Th in the crystal lattice. All doping thorium nuclei share the same environment and lattice-generated fields. (b) Level schemes for two different doping sites. The darker (blue) thorium nuclei are influenced by a different environment than the lighter (yellow) ones. See text for further explanations.

Figure 3

NFS time spectra scattered off an ensemble of two-level nuclei. The dotted (black) line shows the results for single-site doping while the solid (red) one with yellow shading is for two-site doping. Here $\gamma =0.356{\Gamma }_0$, ${\eta }^{\left(1\right)}={\eta }^{\left(2\right)}=\eta /2\sim 91$ Hz/cm for $\xi ={10}^6$ with the pulse and detuning parameters ${\Omega }_0={10}^6{\Gamma }_0$, $t_0=0.1$ ms, $\tau =0.01$ ms, and $\delta \left(B\right)={10}^{\left(8\right)}{\Gamma }_0$, respectively.

Figure 4

Level schemes of the doped ${}^{229}$Th in the crystal lattice exposed to two VUV fields. (a) The doping occurs in one site only and thus all the thorium nuclei experience the same environment. (b) Two-site doping with different hyperfine splittings due to different magnetic environments in the crystal lattice. See text for further explanations.

Figure 5

The time domain spectrum of NFS for a $\Lambda$ three-level configuration with two-fields setup. The black dotted (solid red) curve with yellow shaded region represents the scattering signal for single (two-site) doping. Here we have considered equal doping density for the two nuclear sites such that ${\eta }_{13}^{\left(1\right)}={\eta }_{13}^{\left(2\right)}={\eta }_{13}/2\simeq 227$ Hz/cm for $\xi \simeq {10}^6$. The additional detuning introduced by the two different doping sites is ${\delta }_i\left(B\right)={10}^8{\Gamma }_0$. The spontaneous decay rates of the transitions are respectively ${\gamma }_1=0.088{\Gamma }_0$ and ${\gamma }_2=0.889{\Gamma }_0$. The control and probe fields are equally detuned from the level $|3/2,3/2\rangle$ by $\Delta ={10}^5{\Gamma }_0$. The control field is assumed to have a Rabi frequency of ${10}^8{\Gamma }_0$.

Figure 6

Time domain pulse shape of the control and probe fields shown by the dashed (red) and solid (blue) line. The control pulse has 10 times the width of the probe pulse with ${\Omega }_{c0}={10}^8{\Gamma }_0$ and $t_{c0}=5$ ms, ${\tau }_c=0.5$ ms, while the probe parameters are ${\Omega }_{p0}={10}^6{\Gamma }_0$ and $t_{p0}=5$ ms, ${\tau }_p=0.05$ ms, respectively.

Figure 7

The time-domain spectrum of NFS for three-level configuration with a two-pulsed-field setup. The black dotted with yellow shading (red solid) curve is for a single-doping (two-doping) site case. Due to doping at a different site an additional shift of ${\delta }_2\left(B\right)={10}^8{\Gamma }_0$ of the energy level is induced in the second group of nuclei. Here we have considered equal doping density for the two groups such that ${\eta }_{13}^{\left(1\right)}={\eta }_{13}^{\left(2\right)}={\eta }_{13}/2\simeq 227$ Hz/cm and ${\eta }_{12}^{\left(1\right)}={\eta }_{12}^{\left(2\right)}={\eta }_{12}/2\simeq 22.7$ Hz/cm for $\xi \simeq {10}^6$. The spontaneous emission rates are the same as those of Fig. 5. The control and probe fields are equally detuned from the level $|3/2,3/2\rangle$ by $\Delta ={10}^5{\Gamma }_0$.

Figure 8

VUV excitation of the doping thorium nuclei in a three level V-configuration. The pulse with frequency ${\omega }_p$ is equally detuned from the upper doublet by $\Delta$. The time-dependent driving pulse ${\Omega }_d\left(t\right)$ creates a time-dependent coherent superposition of the upper levels. Population trapping in the excited doublet occurs as long as drive and probe pulse overlap. See text for further explanations.

Figure 9

The time domain spectrum of NFS for three-level $V$configuration in a two-field setup. The black dotted (red solid) curve is in the absence (presence) of the external field coupling between the upper doublet. The parameters used for computation are ${\gamma }_1=0.267{\Gamma }_0$, ${\gamma }_2=0.0889{\Gamma }_0$, ${\eta }_{13}\simeq 22.7$ Hz/cm, ${\eta }_{23}\simeq 68$ Hz/cm for $\xi \simeq {10}^6$, ${\Omega }_{d0}={10}^8{\Gamma }_0$, $t_{d0}=1$ ms, ${\tau }_d=0.1$ ms, ${\Omega }_{p0}={10}^9{\Gamma }_0$, $t_{p0}=1.1$ ms, and ${\tau }_p=0.1$ ms.

Figure 10

Population dynamics of the upper state doublets in $V$-configuration in (a) the absence and (b),(d) the presence of coherent coupling among the upper states. The solid (blue) and dashed (red) curves illustrate ${\rho }_{11}$ and ${\rho }_{22}$. The amplitude of the driving field is different in (b) and (d), being respectively ${\Omega }_d={10}^8{\Gamma }_0$ and ${\Omega }_d={10}^9{\Gamma }_0$. All other parameters are the same as in Fig. 9. In (a) the dashed (red) line is shifted along the time axis by 0.5 ms for better visibility. (c) shows the NFS spectra corresponding to the population dynamics in (d).

Figure 11

(a) NFS time spectra in the absence (dashed black line with shading) and presence (solid red line) of an external DC field coupling of the upper level doublet. (b) The solid (blue) and dashed (red) curves illustrate ${\rho }_{11}$ and ${\rho }_{22}$ when the upper states are coupled by a DC field with Rabi frequency ${\Omega }_d={10}^8{\Gamma }_0$. All other parameters are considered to be same as for Fig. 9.

Figure 12

(a) VUV excitation of the doped ${}^{229}$Th in the crystal lattice in the case of a single doping site. The laser with frequency ${\omega }_c$ is tuned near the $|5/2,5/2\rangle \to |3/2,3/2\rangle$ and $|5/2,3/2\rangle \to |3/2,1/2\rangle$ transitions. The relaxation and decoherence rates ${\gamma }_1$, ${\gamma }_{d1}$ ${\gamma }_2$ and ${\gamma }_{d2}$ are assumed for the addressed transitions. (b) The case of two doping sites. The blue group of thorium nuclei is influenced by a different environment than the yellow one which experiences the additional detunings ${\delta }_1$ and ${\delta }_2$ as a function of the magnetic field. See text for further explanations.

Figure 13

NFS time domain spectra for a multilevel sample with near degenerate transitions. The thin (black) and thick (red) curves with yellow shading are for samples with one and two doping sites, respectively, with laser detuning of $\Delta ={10}^9{\Gamma }_0$. A magnetic-field-dependent detuning of ${\delta }_i\left(B\right)={10}^8{\Gamma }_0$ in the case of two doping sites is considered. Here, we have considered ${\eta }_{13}^{\left(1\right)}={\eta }_{13}^{\left(2\right)}={\eta }_{13}/2\simeq 227$ Hz/cm and ${\eta }_{24}^{\left(1\right)}={\eta }_{24}^{\left(2\right)}={\eta }_{24}/2\simeq 137$ Hz/cm for $\xi \simeq {10}^6$. The spontaneous decay rates of the relevant transitions are, respectively, ${\gamma }_1=0.899{\Gamma }_0$ and ${\gamma }_2=0.533{\Gamma }_0$. The probe pulse has a duration of $\tau =0.01$ ms with a Gaussian shape centered on $t_0=0.1$ ms.