Electric-field-dependent $g$ factor for the ground state of lead monofluoride, PbF

The electric-field-dependent $g$ factor and the electron electric dipole moment (eEDM)-induced Stark splittings for the lowest rotational levels of ${}^{207,208}\mathrm{PbF}$ are calculated. Observed and calculated Zeeman shifts for ${}^{207}\mathrm{PbF}$ are found to be in very good agreement. It is shown that the ${}^{207}\mathrm{PbF}$ hyperfine sublevels provide a promising system for the eEDM search and related experiments.

Figure 1

Rotational Zeeman spectra for the ${}^{207}\mathrm{PbF}$ $F_L=3/2\to F_U=5/2$ transition at 22 541.912 MHz, showing the comparable (left) eight $\mathrm{\Delta }{M}_F=\pm 1$ ($B_0=0.465$ G) and (right) four $\mathrm{\Delta }{M}_F=0$ ($B_0=0.458$ G) splittings, respectively. Note the doubled transitions due to the COBRA Doppler effect.

Figure 2

A depiction of rotational Zeeman spectra shown in Fig. 1. The $\mathrm{\Delta }{M}_F=\pm 1$ are marked by blue arrows. The $\mathrm{\Delta }{M}_F=0$ components are marked by red arrows.

Figure 3

The eEDM-induced Stark splitting ($\mathrm{\Delta }E$) between $\pm M_F$ pairs of hyperfine states. (a) ${}^{208}\mathrm{PbF}$. The solid (red) line corresponds to the $|M_F|=1$ lower-lying $\mathrm{\Omega }$-doublet states whereas the dashed (green) line corresponds to the higher-lying $|M_F|=1$ states. (b) ${}^{207}\mathrm{PbF}$. The solid (red) line corresponds to the $|M_F|=3/2$ lower-lying $\mathrm{\Omega }$-doublet states whereas the dashed (green) line corresponds to the higher-lying $|M_F|=3/2$ states. (c) ${}^{207}\mathrm{PbF}$. The solid (red) line corresponds to the lower-lying $F=3/2$, $|M_F|=1/2$ $\mathrm{\Omega }$-doublet states, the dashed (green) line corresponds to the higher-lying $F=3/2$, $|M_F|=1/2$ states, the dotted (blue) line corresponds to the lower-lying $F=1/2$, $|M_F|=1/2$ states, whereas the dashed-dotted (violet) line corresponds to the higher-lying $F=1/2$, $|M_F|=1/2$ states

Figure 4

Calculated (circles) and experimental (horizontal bands, bandwidths corresponding to two standard deviation uncertainty) shifts for the $F_L=3/2\to F_U=5/2$ transition at $\nu =22541.912$ MHz. $M{F}_L$ values are on the $x$ axis, and $M{F}_U$ values are marked in the figure. Note the excellent agreement and the eightfold/fourfold natures of the $\mathrm{\Delta }{M}_F=\pm 1$ and $\mathrm{\Delta }{M}_F=0$ transitions clearly apparent in Fig. 1.

Figure 5

Calculated $g$ factors. (a) ${}^{208}\mathrm{PbF}$. The solid (red) and dotted (blue) lines correspond to the $|M_F|=1$ lower-lying $\mathrm{\Omega }$-doublet states whereas the dashed (green) and the dashed-dotted (violet) lines correspond to the higher-lying $|M_F|=1$ states. The dotted (blue) and the dashed-dotted (violet) lines were calculated without interaction with other rotational levels taken into account. (b) ${}^{207}\mathrm{PbF}$. The solid (red) line corresponds to the $|M_F|=3/2$ lower-lying $\mathrm{\Omega }$-doublet states whereas the dashed (green) line corresponds to the higher-lying $|M_F|=3/2$ states. (c) ${}^{207}\mathrm{PbF}$. The solid (red) line corresponds to the lower-lying $F=3/2$, $|M_F|=1/2$ $\mathrm{\Omega }$-doublet states, the dashed (green) line corresponds to the higher-lying $F=3/2$, $|M_F|=1/2$ states, the dotted (blue) line corresponds to the lower-lying $F=1/2$, $|M_F|=1/2$ states, whereas the dashed-dotted (violet) line corresponds to the higher-lying $F=1/2$, $|M_F|=1/2$ states.