Green's function treatment of Rydberg molecules with spins

TL;DR

This work develops a fully spin-dependent Coulomb Green's function framework to compute the adiabatic potential energy curves of long-range Rydberg molecules, incorporating fine-structure, hyperfine, and spin–orbit couplings. By reframing the problem as a determinantal root search for a spin-aware Green's function, it avoids the nonuniqueness and convergence problems of zero-range diagonalization and provides a direct link between electron–atom scattering phaseshifts, quantum defects, and molecular potentials. The method yields accurate PECs for high-lying Rydberg states and reveals how hyperfine and P-/D-wave contributions shape butterfly and trilobite states, with implications for spectroscopic interpretation and phase-shift extraction. Overall, this spin-inclusive Green's function approach offers a robust, self-consistent path to quantitatively connect Rydberg-molecule spectra with underlying scattering physics, and it enables high-n calculations without exponential basis growth.

Abstract

The determination of ultra-long-range molecular potential curves has been reformulated using the Coulomb Greens function to give a solution in terms of the roots of an analytical determinantal equation. For a system consisting of one Rydberg atom with fine structure and a neutral perturbing ground state atom with hyperfine structure, the solution yields potential energy curves and wavefunctions in terms of the quantum defects of the Rydberg atom and the electron-perturber scattering phaseshifts and hyperfine splittings. This method provides a promising alternative to the standard currently utilized method of diagonalization, which suffers from problematic convergence issues and nonuniqueness, and can potentially yield a more quantitative relationship between Rydberg molecule spectroscopy and electron-atom scattering phaseshifts.

Figures (12)

• Figure 1: Electron-Rb scattering phase shifts used in this paper. The black (dark) curves are calculated ab initio using a relativistic model potential khuskivadzeAdiabatic2002. The blue (light) curves show phase shifts which were modified (in the triplet case only) by Ref. engelPrecision2019 to produce electronic potential curves whose vibrational energies match those observed in experiment .
• Figure 2: Potential energy curves with threshold values at the $25p_j,f=1,2$ levels, presented relative to the $25p_{1/2},f=1$ threshold. The deep wells at smaller $R$ values host butterfly molecular states. The calculation uses the fitted phase shifts of Ref. engelPrecision2019.
• Figure 3: The deepest butterfly potential energy curve near the $25p_{1/2},f=1$ state representing the zero of the energy scale, calculated using our Green's function method (in black) and compared with diagonalization in 6 different basis sizes (colored and dashed). The basis sets include states from the degenerate Rydberg manifolds $22\le n \le 23$ (purple), $21\le n \le 23$ (blue), $21\le n \le 24$ (cyan), $21\le n \le 25$ (green), $20\le n \le 24$ (orange), and $20\le n \le 25$ (red), as well as all quantum defect-shifted states within these energy ranges. These curves show the potential curves due to electron-atom scattering only, i.e. without the additional polarization interaction from Eq. \ref{['eq:potencurves']}.
• Figure 4: Potential energy curves for Rb$^*$Rb supporting butterfly molecules with $M_\text{tot}=\frac{1}{2}$. At large $R$ these potential curves approach threshold values at the non-interacting $25p_j,f=2$ energy levels. The black curves were calculated with the fitted phase shifts of Ref. engelPrecision2019, and those in blue using the calculated phase shifts of Ref. khuskivadzeAdiabatic2002. Vibrational levels and their dipole moments were reported in Ref. niederprumObservation2016 in the energy range from $-50$ GHz to $-40$ GHz.
• Figure 5: Potential energy curves for Rb$^*$Rb with threshold values at the $16p_j,f=1,2$ levels, presented relative to the $16p_{1/2},f=1$ threshold. The upper left (black) Green's function calculation uses the fitted phase shifts of Ref. engelPrecision2019, while the upper right (blue) Green's function calculation uses the calculated phase shifts of Ref. khuskivadzeAdiabatic2002. The bottom two panels show results using the fitted phase shifts and obtained via diagonalization with two different basis sizes. In the bottom left (orange) the basis includes states with $12\le n \le 14$; in the bottom right (magenta) the basis include $12 \le n \le 15$