The Stark effect in molecular Rydberg states: Calculation of Rydberg-Stark manifolds of H$_2$ and D$_2$ including fine and hyperfine structures

Abstract

We present a general theoretical treatment and calculations of the fine and hyperfine structures in the spectra of high-$n$ molecular Rydberg states in static uniform electric fields. The treatment combines (i) multichannel quantum-defect theory and long-range polarization models to determine the field-free energies of $n\ell$ Rydberg states of the molecules ($\ell$ is the orbital-angular-momentum quantum number of the Rydberg electron), (ii) a matrix-diagonalization approach to calculate the Stark shifts including their hyperfine structure, and (iii) sequences of angular-momentum frame transformations to predict the line positions and intensities in Stark spectra as they would be observed in single or multiphoton excitation sequences. To clarify how the molecular rotation and the nuclear spins influence the fine and hyperfine structure of molecular Rydberg-Stark spectra, we compare calculated spectra of ortho-D$_2$ with a D$_2^+$ ion core in the rotational ground state ($N^+=0$) for total nuclear spins $I$ of 0 (i.e., without hyperfine structure) and 2 (i.e., with hyperfine structure) with the corresponding spectra of para-H$_2$ with an H$_2^+$ ion core in the first excited rotational state ($N^+=2$) but zero nuclear spin ($I=0$). The calculations show that the hyperfine interaction alone does not significantly modify the Stark effect, but splits each Stark state by almost exactly the hyperfine Fermi-contact splitting of the ion core. In contrast, the effect of the molecular rotation, which is coupled both to the ion-core electron spin by the magnetic spin-rotation interaction and to the Rydberg-electron orbital motion by the core-polarization and charge-quadrupole interactions, induces Stark-state specific splittings that significantly differ from the spin-rotation splitting of the ($N^+=2$) ion core.

Figures (16)

• Figure 1: Angular-momentum coupling diagrams showing the basis sets of matrices used in the calculations of the: MQDT energies ($|1 \rangle$, Section \ref{['subsec:mqdt']}), long-range interaction model ($|2 \rangle$, Section \ref{['subsec:longrange']}), Stark effect ($|3 \rangle$, Section \ref{['sec:stark']}), intensities ($|4 \rangle$, Section \ref{['sec:intensity_model']}).
• Figure 2: $\ell=0-3, n = 34, v^+=1$ zero-field matrices for (a) ortho-D$_2$, $I=2, N^+=0$ and (b) para-H$_2$, $I=0, N^+=2$ obtained from MQDT calculations including spins. The different entries within each diagonal block correspond to the possible values of $F$ resulting from the addition of $\vec{j}$ and $\vec{F}^+$, e.g., in the case of $\ell=0$, $F=1$ for $j=1/2$ and $F^+=3/2$, $F=2$ for $j=1/2$ and $F^+=3/2$ and $F^+=5/2$, and $F=3$ for $j=1/2$ and $F^+=5/2$. The diagonal elements are given with respect to $- R_{\textrm{D}_2 (\textrm{H}_2)} / {n^2}$. The blocks are labeled by $\ket{\ell jF^+F}$.
• Figure 3: Angular-momentum-coupling diagram in the $\ket{(\ell N^+)N(I)K(S^+)F_s(s)F}$ basis set (left) and subset of four interacting states with the total angular momentum $F$ (right).
• Figure 4: 4 $\times$4 matrices showing the electrostatic long-range ($\hat{H}_\textrm{lr}$), hyperfine ($\hat{H}_\textrm{hfs}$), spin-orbit ($\hat{H}_\textrm{so}$) and other-spin$-$orbit ($\hat{H}_\textrm{oso}$) interactions between $\ket{n v^+} \ket{(\ell N^+)N(I)K(S^+)F_s(s)F}$ states with $n=40, v^+=1, \ell=5, F=3$, for a) ortho-D$_2$ ($I=2, N^+=0$) and b) para-H$_2$ ($I =0, N^+=2$). The center of the color scale (corresponding to zero coupling) is set to white. The basis states are labeled as $\ket{K,F_s}$, corresponding to Fig. \ref{['fig:angmom_polmol']}.
• Figure 5: Evolution of electrostatic long-range, hyperfine, spin-orbit and other-spin$-$orbit interactions in the range $n=40-100$, for $\ell = 5, 15, 30$ and $v^+=1$, plotted as the diagonal matrix element with $K=\ell - 2$, $F_s=K-1/2 = \ell-5/2$, $F=K=\ell-2$. The results for ortho-D$_2$ ($I=2, N^+=0$) are drawn as dashed lines and those for para-H$_2$ ($I=0, N^+=2$) as solid lines.