Systematic effects in the ${\mathrm{HfF}}^{+}$-ion experiment to search for the electron electric dipole moment

The energy splittings for $J=1$, $F=3/2$, $|m_F|=3/2$ hyperfine levels of the ${}^3{\mathrm{\Delta }}_1$ electronic state of ${}^{180}\mathrm{Hf}{}^{19}\mathrm{F}^{+}$ ion are calculated as functions of the external variable electric and magnetic fields within two approaches. In the first one, the transition to the rotating frame is performed, whereas in the second approach, the quantization of rotating electromagnetic field is performed. Calculations are required for understanding possible systematic errors in the experiment to search for the electron electric dipole moment ($e\mathrm{EDM}$) with the ${}^{180}\mathrm{Hf}{}^{19}\mathrm{F}^{+}$ ion.

Figure 1

Calculated energy splittings for the $H^3{\mathrm{\Delta }}_1$ ($J=1,F=3/2$, $|m_F|=3/2$) Stark pairs as functions of ${\mathcal{B}}_{\mathrm{rot}}$. ${\mathcal{E}}_{\mathrm{rot}}=24\mathrm{V}$/cm, ${\omega }_{\mathrm{rot}}/2\pi =250$ kHz, $\tilde{\mathcal{R}}=+1$ in the calculations. Lines are calculated within approach I. The solid (red) line corresponds to the lower ($\tilde{\mathcal{D}}=+1$) Stark pair, ${\mathcal{E}}_z=0$; the dashed (green) line corresponds to the upper ($\tilde{\mathcal{D}}=-1$) Stark pair, ${\mathcal{E}}_z=0$; the dotted (blue) line corresponds to the lower Stark pair, ${\mathcal{E}}_z=0.3$ mV/cm; and the dot-dashed (purple) line corresponds to the upper Stark pair, ${\mathcal{E}}_z=0.3$ mV/cm. Figures are calculated within approach (II) with $N=3$. Circles (red) correspond to the lower Stark pair, ${\mathcal{E}}_z=0$; squares (green) correspond to the upper Stark pair, ${\mathcal{E}}_z=0$; up-triangles (blue) correspond to the lower Stark pair, ${\mathcal{E}}_z=0.3$ mV/cm; and down-triangles (purple) correspond to the upper Stark pair, ${\mathcal{E}}_z=0.3$ mV/cm.

Figure 2

Calculated $e\mathrm{EDM}$-induced $f^{\mathcal{B}\mathcal{D}}$ splitting. The solid (red) line corresponds to ${\omega }_{\mathrm{rot}}/2\pi =250$ kHz, ${\mathcal{E}}_{\mathrm{rot}}=24$ V/cm; the dashed (green) line corresponds to ${\omega }_{\mathrm{rot}}/2\pi =250$ kHz, ${\mathcal{E}}_{\mathrm{rot}}=20$ V/cm; the dotted (blue) line corresponds to ${\omega }_{\mathrm{rot}}/2\pi =150$ kHz, ${\mathcal{E}}_{\mathrm{rot}}=24$ V/cm; and the dot-dashed (purple) line corresponds to ${\omega }_{\mathrm{rot}}/2\pi =150$ kHz, ${\mathcal{E}}_{\mathrm{rot}}=20$ V/cm.

Figure 3

Calculated $f^{\mathcal{D}}$ as a function of ${\mathcal{B}}_{\mathrm{rot}}$. Solid (red) curve: Interactions with both ${}^3{\mathrm{\Delta }}_2$ and ${}^3{\mathrm{\Pi }}_{0^{\pm }}$ states are taken into account. Dashed (green) curve: Only interactions with the ${}^3{\mathrm{\Pi }}_{0^{\pm }}$ states are taken into account. Dotted (blue) curves: Interactions with both ${}^3{\mathrm{\Delta }}_2$ and ${}^3{\mathrm{\Pi }}_{0^{\pm }}$ states are omitted. ${\mathcal{E}}_{\mathrm{rot}}=24\mathrm{V}$/cm, ${\omega }_{\mathrm{rot}}/2\pi =250$ kHz in the calculations.

Figure 4

Calculated $f^{\mathcal{D}}$ as a function of $f^0$. Solid (red) curve: Interactions with both ${}^3{\mathrm{\Delta }}_2$ and ${}^3{\mathrm{\Pi }}_{0^{\pm }}$ states are taken into account. Dashed (green) curve: Only interactions with the ${}^3{\mathrm{\Pi }}_{0^{\pm }}$ states are taken into account. Dotted (blue) curves: Interactions with both ${}^3{\mathrm{\Delta }}_2$ and ${}^3{\mathrm{\Pi }}_{0^{\pm }}$ states are omitted. ${\mathcal{E}}_{\mathrm{rot}}=24\mathrm{V}$/cm, ${\omega }_{\mathrm{rot}}/2\pi =150$ kHz. The circle is the experimental value.