Sensitivity of the YbOH molecule to $\mathcal{P}\mathcal{T}$-odd effects in an external electric field

An electron electric dipole moment (eEDM) search using laser-coolable triatomics such as YbOH is one of the most sensitive probes for physics beyond the standard model. The eEDM-induced energy shift is proportional to the polarization ($P$) of the molecule. Similarly to diatomics with an $\mathrm{\Omega }$-doubling structure, it was assumed that for triatomics with an $l$-doubling structure, related to the vibrational angular momentum, $P$ can be easily saturated to almost 100% with a moderate external electric field. We developed a method for the calculation of the properties of triatomic molecules and applied it to a calculation of $P$ of ${}^{174}\mathrm{YbOH}$ in the first excited $v=1$ bending mode. Calculations showed that most of the levels reach less than 50% efficiency. We showed that this fact is related to the Hund's case $b$ coupling scheme of YbOH. As the coupling scheme (for $\mathrm{\Omega }=1/2$ molecules) approaches the $a$ (or $c$) case, polarization increases up to 100%. The results of our calculations should be used for the correct extracting of the eEDM value from YbOH experiments and similar calculations are required for other molecules.

Figure 1

Orientation of the YbOH relative to the molecular frame. Orientation of the molecular frame relative to the laboratory one is given by Euler's angles (not presented in the figure) $\alpha , \beta , \gamma =0$. $R,r,\theta$ are Jacobi coordinates.

Figure 2

Polarization $P$ [see Eq. (1)] for the lowest $N=1$ rotational level of the first excited $v=1$ bending vibrational mode of ${}^{174}\mathrm{YbOH}$ as a function of the external electric field. (a) Solid lines correspond to the $M_F=M_J+M_I=1$ levels. Dashed lines correspond to the $M_J=1/2$ levels. Colors (numbering) of lines correspond to colors (numbering) of lines in Fig. 3. (b) $M_F=M+M_I=2$ levels. Solid lines correspond to the real matrix element (17), and dashed lines correspond to the reduced matrix element (19).

Figure 3

Energies of the lowest $N=1$ rotational level of the first excited $v=1$ bending vibrational mode of ${}^{174}\mathrm{YbOH}$ as functions of the external electric field. Solid lines correspond to the $M_F=M_J+M_I=1$ levels. Dashed lines correspond to the $M_J=1/2$ levels. Colors (numbering) of lines correspond to colors (numbering) of lines in Fig. 2.