Rydberg states with a liquid core

TL;DR

The paper develops a self-consistent framework for Rydberg electrons in a finite, polarizable liquid core, such as a superfluid droplet, by formulating a droplet-dressed Rydberg (DDR) Hamiltonian and a spherically symmetric reference environment. It identifies two universal spectral classes, oDDR and iDDR, arising from the droplet potential and provides estimators for the droplet’s influence on angular momentum via $\ell_d$ and $\ell_M$, along with a hydrogenic basis expansion $u_{N\ell}(r)=\sum_n C^n_{N\ell}\phi_{n\ell}(r)$. The back-action of iDDR states on the droplet is quantified through a Skyrme functional, showing negligible effects for typical excitations ($\delta(n) \propto n^{-6}$, with $n \ge 10$ effectively safe to neglect); lifetimes of iDDR are obtained semiclassically, revealing rapid decreases with growing radial quantum number $\bar{n}$. The work also maps out selection rules and multi-step transitions between oDDR and iDDR, offering a route to probe droplet structure and anisotropy spectroscopically, with potential applications to dressed Rydberg molecules and droplets featuring crystalline fractions.

Abstract

We develop a self-consistent approach that provides an explicit potential for a Rydberg electron whose ionic core consists of a polarizable medium, typically realized with superfluid droplets. The electron's motion remains separable in spherical coordinates, but the radial force exerted by the droplet breaks degeneracy of the angular momentum states non-perturbatively. The ensuing electron spectrum reveals intriguing properties dependent on droplet size and electron excitation. Deviations of the polarizable medium from the continuous spherical distribution can be taken into account as a perturbation of this redefined Rydberg dynamics. We discuss specific but paradigmatic examples for superfluid helium and also propose a way to probe droplet properties including its possible crystallized fraction through stimulated transitions of the Rydberg electron.

Figures (6)

• Figure 1: Radial potential of \ref{['eq:H0']} for droplets of radius $70\,a_0$ (orange) and $250\,a_0$ (cyan). Horizontal dashed lines (orange and cyan) indicate the oDDR ground state energy for each droplet. Insets show cylindrical-coordinate surface plots of the Rydberg electron probability for the quasi-bound iDDR states (a,c) and the bound oDDR states (b,d), where the white circle marks the droplet boundary. The quantum numbers $\ket{\bar{n},\ell,m} = \ket{5,1,0}$ and $\ket{12,6,0}$ for (a) and (c), and $\ket{\nu,\ell,m} = \ket{5,1,0}$ and $\ket{15,6,0}$ for (b) and (d) reflect the nodal patterns of the respective states. The eigenenergies of the states are shown as dashed gray lines. Note that, the corresponding effective radial potentials have in addition to the ones shown for $\ell=0$ centrifugal barriers $\ell(\ell+1)/(2r^2)$.
• Figure 2: DDR spectrum for a droplet of radius $250\,a_0$ for the quasi-bound states (a) and the bound states (b). Ticks on the vertical axis are related by \ref{['eq:quasi-bound_energies']}. Colors indicate the dominant hydrogenic manifold, $\bar{n}$, contributing to each state. Black points denote states with a single hydrogenic component $C^n>0.99$. Their onset is estimated from \ref{['eq:estimator_lmin_for_hydrog_states']} (black line). $\ell=15$ marks the estimated kink position in the spectrum (see \ref{['eq:estimator_l_kink']}).
• Figure 3: Relative energy shift \ref{['eq:shift']} for a helium droplet of radius $R_d=250$ doped with different alkali ion impurities. (a) Quasi-bound states are labeled $\tilde{\alpha}$, and (b) bound states are labeled $\alpha$. For this illustration atomic quantum defects are omitted as well as DDR basis states with $\ell > 10$ (see text).
• Figure S1: Energy of the oDDR ground state as a function of droplet radius. Vertical ticks mark the unperturbed hydrogenic energy levels $E_n$. The red dashed curve shows the analytical estimate from \ref{['eq:quantum_prediction_electron_energy']}, while the blue points denote the computed oDDR ground state energies. The inset displays a log--log plot of the absolute oDDR ground state energy on the vertical axis.
• Figure S2: Estimated lifetimes of selected iDDR states $\ket{\bar{n}\ell m}$ for $\ell=0$ and $\ell=4$, for a droplet of radius $R_d=250$. As $\bar{n}$ increases, the states shift to higher energy and their electronic probability extends further inside the droplet towards the droplet boundary (see Fig. \ref{['fig:spectrum_rad250']}(a) and Fig. \ref{['fig:potential']} (c)). For $\bar{n}=11$ the lifetime is of the order of $0.1$ ps.