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Register a new userZeeman interaction in $^3Delta_1$ state of HfF$^+$ to search for the electron electric-dipole-moment
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We report the theoretical investigation of the suppression of magnetic systematic effects in HfF$^+$ cation for the experiment to search for the electron electric dipole moment. The g-factors for $J = 1$, $F=3/2$, $|M_F|=3/2$ hyperfine levels of the $^3Delta_1$ state are calculated as functions of the external electric field. The lowest value for the difference between the g-factors of $Omega$-doublet levels, $Delta g = 3 times 10^{-6}$, is attained at the electric field 7 V/cm. The body-fixed g-factor, $G_{parallel}$, was obtained both within the electronic structure calculations and with our fit of the experimental data from [H. Loh, K. C. Cossel, M. C. Grau, K.-K. Ni, E. R. Meyer, J. L. Bohn, J. Ye, and E. A. Cornell, Science {bf 342}, 1220 (2013)]. For the electronic structure calculations we used a combined scheme to perform correlation calculations of HfF$^+$ which includes both the direct 4-component all-electron and generalized relativistic effective core potential approaches. The electron correlation effects were treated using the coupled cluster methods. The calculated value $G_{parallel}=0.0115$ agrees very well with the $G_{parallel}=0.0118$ obtained in the our fitting procedure. The calculated value $D_{parallel}=-1.53$ a.u. of the molecule frame dipole moment (with the origin in the center of mass) is in agreement with the experimental value $D_{parallel}=-1.54(1)$ a.u. [H. Loh, Ph.D. thesis, Massachusetts Institute of Technology (2006)].
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The energy splitings for $J = 1$, $F=3/2$, $|M_F|=3/2$ hyperfine levels of the $^3Delta_1$ electronic state of $^{180}$Hf$^{19}$F$^+$ ion are calculated as functions of the external variable electric and magnetic fields within two approaches. In the first one 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 electron electric dipole moment (eEDM) on $^{180}$Hf$^{19}$F$^+$ ion.
The current limit on the electrons electric dipole moment, $|d_mathrm{e}|<8.7times 10^{-29} e {cdotp} {rm cm}$ (90% confidence), was set using the molecule thorium monoxide (ThO) in the $J=1$ rotational level of its $H ^3Delta_1$ electronic state [Science $bf 343$, 269 (2014)]. This state in ThO is very robust against systematic errors related to magnetic fields or geometric phases, due in part to its $Omega$-doublet structure. These systematics can be further suppressed by operating the experiment under conditions where the $g$-factor difference between the $Omega$-doublets is minimized. We consider the $g$-factors of the ThO $H^3Delta_1$ state both experimentally and theoretically, including dependence on $Omega$-doublets, rotational level, and external electric field. The calculated and measured values are in good agreement. We find that the $g$-factor difference between $Omega$-doublets is smaller in $J=2$ than in $J=1$, and reaches zero at an experimentally accessible electric field. This means that the $H,J=2$ state should be even more robust against a number of systematic errors compared to $H,J=1$.
We present an updated EDM effective electric field of $E_{text{eff}} = 75.2left[frac{rm GV}{rm cm}right]$ and the electron-nucleon scalar-pseudoscalar interaction constant $W_S=107.8$ [kHz] for the ${^3Delta}_1$ science state of ThO. The criticisms made in reference [J. Chem. Phys. 142, 024301 (2015)] are addressed and largely found to be unsubstantiated within the framework of our approach.
We investigate the merits of a measurement of the permanent electric dipole moment of the electron ($e$EDM) with barium monofluoride molecules, thereby searching for phenomena of CP violation beyond those incorporated in the Standard Model of particle physics. Although the BaF molecule has a smaller enhancement factor in terms of the effective electric field than other molecules used in current studies (YbF, ThO and ThF$^+$), we show that a competitive measurement is possible by combining Stark-deceleration, laser-cooling and an intense primary cold source of BaF molecules. With the long coherent interaction times obtainable in a cold beam of BaF, a sensitivity of $5times10^{-30}$ e$cdot$cm for an $e$EDM is feasible. We describe the rationale, the challenges and the experimental methods envisioned to achieve this target.
The outstanding progress has been made in reducing the upper bounds on EDM of several particles. Even if significant challenges must be overcome to further improve these limits, it is still one of the best chances to detect new type of interactions beyond the standard model. Analyzing several examples, we highlight a common thread that is visible in different set-ups used for the EDM detection. The electric dipole moment is one of the clear consequences of CP- or T-violating interactions, however it is not the only one. These symmetry-violating interactions enable extra phenomena that unavoidably accompany the EDM-induced spin precession, and they must be taken into account in planning and executing sensitive experiments. After reviewing three typical cases, we suggest conditions for improving the sensitivity of detecting the intrinsic EDM.
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