High precision spectroscopy of trilobite Rydberg molecules

We perform three-photon photoassociation to obtain high resolution spectra of $^{87}$Rb trilobite dimers for the principal quantum numbers $n = 22,24,25,26$, and $27$. The large binding energy of the molecules in combination with a relative spectroscopic resolution of $10^{-4}$ provides a rigorous benchmark for existing theoretical models. A recently developed Green's function framework, which circumvents the convergence issues that afflicted previous studies,, is employed to theoretically reproduce the vibrational spectrum of the molecule with high accuracy. The relatively large molecular binding energy are primarily determined by the low energy $S$-wave electron-atom scattering length, thereby allowing us to extract the $^3S_1$ scattering phase shift with unprecedented accuracy, at low energy regimes inaccessible to free electrons.