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.
The Stark effect in atomic and molecular Rydberg states plays a central role in the state-selective detection of Rydberg atoms and molecules by field ionization [1,2], in the accurate determination of ionization energies [3][4][5][6][7][8][9][10], in quantum control and manipulation of interatomic interactions [11][12][13][14][15][16][17][18], and in electric-field-sensing applications [19][20][21][22][23][24][25][26]. For this reason, it has been extensively studied both experimentally and theoretically.
Rydberg states with a closed-shell ion core such as those of H and the alkali-metal atoms exhibit simple, characteristic spectral structures in the presence of electric fields [2,27]. At zero field, Rydberg states of high principal quantum number are characterized by high spectral densities of near degenerate nonpenetrating states with orbital angular momentum quantum number ℓ ≥ 4 and spectrally isolated penetrating states with ℓ < 4. Whereas the high-ℓ Rydberg states exhibit a linear Stark effect at low electric fields already and fan out into characteristic linear Stark manifolds, low-ℓ Rydberg states are subject to a quadratic Stark effect at low fields and gradually merge into the linear high-ℓ Stark manifolds as the electric field strength increases. If the * frederic.merkt@phys.chem.ethz.ch (F. Merkt) fine and hyperfine structures are neglected, the magnetic orbital quantum number m ℓ is a good quantum number when the electric field is homogeneous. There are n -|m ℓ | Stark states in each n, m ℓ -manifold and the states with |m ℓ | > 0 form pairs of degenerate levels with m ℓ = ±|m ℓ |. The Stark states are labeled by the electric quantum number k [28], which takes values ranging from -(n -|m ℓ | -1) to (n -|m ℓ | -1) in steps of 2, see Ref. [27] for the alkali-metal atoms and Fig. 8a below for the case of ortho-D 2 Rydberg states with zero nuclear-spin and rotational angular momenta. Hyperfine and spinorbit interactions of the Rydberg electron lead to weak splittings of the levels which very rapidly decrease with increasing n values and are only observable at ultrahigh resolution [8].
Rydberg atoms with open-shell ion cores have more complex structures because of additional exchange and spin-orbit interactions between the core and Rydberg electrons. The good quantum number is the magnetic quantum number M J associated with the total electronic angular momentum ⃗ J rather than m ℓ , where ⃗ J = ⃗ J + + ⃗ j is the sum of the ion-core ( ⃗ J + ) and Rydberg-electron ( ⃗ j) total electronic angular momenta. The Stark manifolds typically retain the general structure discussed above, but the Stark states split into multiple magnetic sublevels, see, e.g., Refs. [29][30][31][32][33][34] for the rare-gas atoms.
Further complexity is expected to arise when the ion core has a nonzero nuclear spin ⃗ I and its level struc-ture is split by the hyperfine interaction. In this case, which has not yet been studied experimentally, the good quantum number in a homogeneous field is the magnetic quantum number M F associated with the total angular momentum ⃗ F = ⃗ J + ⃗ I. The energy-level pattern in the presence of an electric field then depends on the degree of angular-momentum uncoupling induced by the electric field, in analogy to the uncoupling caused by magnetic fields (Paschen-Back effect). In the limit of full uncoupling, which arises at large field strengths and high n values, the Stark states should form independent manifolds for each hyperfine level of the ion core. The effect of the hyperfine interaction in atomic Rydberg states in electric fields has only been treated theoretically for low-n states in an early study based on perturbation theory and angular-momentum algebra [35]. The tensor formalism introduced in Ref. [35] is still used to calculate static and dynamical polarizabilities and Stark shifts in the ground and low-lying electronic states of atoms [36][37][38][39] and was used to analyze the Stark effect in n = 40 and 60 Rydberg states of Cs [40].
The Stark effect in Rydberg systems has also been extensively studied theoretically in the past decades. Collision-theory approaches based on multichannel quantum-defect theory [31,[41][42][43][44][45][46] have been applied to treat the Stark effect in the Rydberg states of the rare gas atoms [31,47,48], molecular hydrogen [49,50] and NO [51], however, without consideration of the nuclear spins. More commonly, the Hamiltonian Ĥ = Ĥ0 + eF ẑ (1) is expressed in matrix form, where Ĥ0 describes the atom or molecule in the absence of the electric field and the Stark effect is included through the additional potentialenergy term eF ẑ perceived by the Rydberg electron in the external electric field ⃗ F = (0, 0, F) † [1,2,27,52]. The Stark spectrum is obtained by determining the eigenvalues of the matrix under conditions where a sufficiently large range of n values are included to reach convergence. In both approaches, the effect of the external electric field
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