Quantum-classical model for the formation of Rydberg molecules

A fascinating aspect of Rydberg atoms is their ability to form huge but very weakly bound molecules with a ground-state atom, only held together by a scattering process between the latter and the Rydberg electron. Beyond the usual way of creating such molecules by laser excitation from two ground-state atoms with a distance of less than the Rydberg radius, we demonstrate that Rydberg molecules can also be formed by capturing a ground-state atom which is initially located outside the range of the Rydberg atom when it comes in contact with it. To demonstrate this effect, we investigate the scattering process between the Rydberg electron and the ground-state atom within a quantum-classical framework. In this picture, the capture results from a dissipative finite-mass correction term in the classical equations of motion. We show that, and under which conditions the capturing takes place.

Figure 1

Molecular potential of a rubidium Rydberg molecule as a function of the internuclear distance for a negative scattering length $a_0=-16.05$ and a polarizability $\alpha =319$ in Eq. (2), calculated from Eq. (3) for quantum numbers $n=31$, $l=0$, $m=0$. The numbers count the minima starting from the outermost one and the assignment of colors to the different potential wells will be the same in Fig. 4.

Figure 2

Schematic drawing of the quantum-classical model for the interaction between the Rydberg electron, orbiting the Rb${}^{+}$ core (small gray sphere), and the Rb ground-state atom (big red sphere). For the magnetic quantum number $m=0$ considered here, there are only two Kepler orbits on which the Rydberg electron can hit the ground-state atom  and each is traversed in clockwise and counterclockwise directions.

Figure 3

Internuclear distance between the Rydberg and the ground-state atom in dependence of time for different angles $\phi$ between the initial direction of the motion of the ground-state atom and the connecting line to the nucleus of the Rydberg atom. The initial velocity is set to $v_{\mathrm{Rb}}={10}^{-9}$ a.u. For all angles shown the ground-state atom will be captured but the selection of a specific local minimum strongly depends on the initial value of $\phi$.

Figure 4

(a) Result of the dynamics calculations of the ground-state atom for initial values $R_0=2500$ a.u., $v_{\mathrm{Rb}}={10}^{-9}$ to $v_{\mathrm{Rb}}={10}^{-8}$, and angles $\phi$ from $0^{\circ }$ to ${90}^{\circ }$. Each single point represents a set of initial conditions ($v_{\mathrm{Rb}}$,$\phi$) and the numbers indicate in which minimum of the potential the ground-state atom will come to rest (numbers and colors refer to Fig. 1). White means that there is no capturing, i.e., the ground-state atom will again leave the Rydberg atom. (b) Fraction $N_i/N$ of the $N_i$ ground-state atoms that are captured in the $i$th potential minimum out of a total of $N$ atoms as a function of their velocity $v_{\mathrm{Rb}}$. The solid line shows the total fraction of captured ground-state atoms.

Figure 5

(a) Comparison of the molecular potential $V\left(\mathit{r}\right)$ with (solid line) and without (dashed-dotted line) $p$-wave scattering. (b) Comparison of two trajectories of the ground-state atom in the influence of the Rydberg atom for the same initial conditions with (dashed-dotted line) and without (solid line) $p$-wave scattering. The detailed dynamics changes significantly; the fact that the ground-state atom is captured, however, is not affected.