Precision spectroscopy of the fine and hyperfine structures of high molecular Rydberg-Stark states: Metrology of molecular hydrogen ions
The Stark effect in autoionizing high-$n$ Rydberg states decouples the Rydberg electron from the ion core through $\ell$ mixing with core-nonpenetrating high-$\ell$ states. The Rydberg states become long-lived, which is ideal for precision spectroscopy, and their structures reflect the fine and hyperfine structures of the ion-core levels. We report on precision measurements, in weak electric fields, of the fine and hyperfine structures of two distinct categories of high autoionizing molecular Rydberg-Stark states differing by the nature of the ion-core angular momentum: Rydberg states of para-H$_2$ (total nuclear spin $I=0$) with a rotationally excited ($N^+=2$) H$_2^+$ ion core and Rydberg states of ortho-D$_2$ ($I=2$) with a rotationless ($N^+=0$) ion core. The spectra reveal striking differences which are interpreted as arising from the dominance of anisotropic charge-quadrupole interactions between the rotating quadrupolar ion core and the Rydberg electron in para-H$_2$ and the absence of such interactions in rotationless ortho-D$_2$ Rydberg states. In ortho-D$_2$, the dominant interaction, the magnetic Fermi-contact hyperfine interaction in the ion core, does not significantly affect the motion of the Rydberg electron. By analyzing these spectra based on a treatment combining multichannel quantum-defect theory and matrix diagonalization, we derive new experimental values of the hyperfine coupling constant $b_F$ = 139.84(5) MHz of D$_2^+ (v^+=1, N^+=0)$, the spin-rotation coupling constant $c_e$ = 39.62(11) MHz of H$_2^+ (v^+=1, N^+=2)$ and the fundamental vibrational interval of ortho-D$_2^+$ (47279980.8(1.9) MHz). The approach followed here in the study of molecular Rydberg-Stark states is general and broadly applicable to measurements of the fine and hyperfine structures of molecular cations.