Observation of spin-orbit-dependent electron scattering using long-range Rydberg molecules

We present experimental evidence for spin-orbit interaction of an electron as it scatters from a neutral atom. The scattering process takes place within a ${\mathrm{Rb}}_2$ ultralong-range Rydberg molecule, consisting of a Rydberg atomic core, a Rydberg electron, and a ground state atom. The spin-orbit interaction leads to characteristic level splittings of vibrational molecular lines which we directly observe via photoassociation spectroscopy. We benefit from the fact that molecular states dominated by resonant $p$-wave interaction are particularly sensitive to the spin-orbit interaction. Our work paves the way for studying novel spin dynamics in ultralong-range Rydberg molecules. Furthermore, it shows that the molecular setup can serve as a microlaboratory to perform precise scattering experiments in the low-energy regime of a few meV.

Figure 1

Composition of the molecular system (see text).

Figure 2

(a) The molecular PECs correlated to the $5S_{1/2}+16P_j$ atomic asymptotes for $j\in \{1/2,3/2\}$ and different hyperfine states $F\in \{1,2\}$ of the $5S_{1/2}$ atom. The color code represents the expectation value of the quantum number $F$. Calculations of PECs are described in Sec. 4a. (b) Zoom into the dashed rectangle in (a) indicating the region of interest for the present work. $N$ and $\left|\mathrm{\Omega }\right|$ are quantum numbers which label the PECs. The $N=5/2$, $3/2$, $1/2$ branches are composed of a triplet, doublet, and singlet substructure of $\left|\mathrm{\Omega }\right|$ states, respectively.

Figure 3

Illustration of the experimental setup and scheme. The orange solid line indicates the dipole trap potential for the ultracold neutral atoms while the dark-green solid line represents the Paul trap potential for ions. (i) Inside the atom cloud (indicated by the gray shaded area) ${\mathrm{Rb}}_2$ Rydberg molecules are produced by means of the UV photoassociation laser (blue dashed lines and blue arrow). (ii) The molecules can decay into ions, where processes leading to ${\mathrm{Rb}}^{+}$ and ${\mathrm{Rb}}_2^{+}$ are possible (see, e.g., [33, 34]). A resulting ion is captured by the Paul trap. (iii) The micromotion-driven ion elastically collides with Rb atoms leading to atom loss from the dipole trap.

Figure 4

Spectra measured for atomic samples initially prepared in the hyperfine state $F=1$ (a) and $F=2$ (b), respectively. Shown is the atom loss $L$ as a function of the frequency $\nu$ of the UV spectroscopy laser light. The frequency $\nu$ is given in terms of $\mathrm{\Delta }\nu =\nu -{\nu }_0$ (a) and $\mathrm{\Delta }\tilde{\nu }=\nu -{\nu }_0+2\times {\nu }_{\text{hfs}}$ (b), where ${\nu }_0=991.55264\text{THz}$ is the resonance frequency for the $16P_{1/2}$ atomic Rydberg line when starting with $F=1$ atoms. The data of (a) are obtained for a pulse duration of $125\text{ms}$ of the spectroscopy light while for (b) $200\text{ms}$ are used (the light intensities, micromotion energies, and ion-atom cloud interaction times are about the same for both scans). Horizontal and vertical black arrows mark resonances assigned to atomic transitions. The black solid lines with denotations (i), (ii), and (iii) point to line multiplets which are investigated in Fig. 6 with higher resolution. Vertical red solid (dashed) lines in (a) illustrate the frequency positions of observed strong (weak) three-line multiplets for $F=2$, while vertical blue solid (dashed) lines in (b) mark strong (weak) single-line peaks for $F=1$.

Figure 5

Comparison of measured molecular term energies (purple horizontal lines) and calculated molecular term energies for $N=1/2$ (a), $N=3/2$ (b), and $N=5/2$ (c). The theoretical results for the different $\left|\mathrm{\Omega }\right|$ states are indicated by the line color and line style as given in the legend. Here, the predicted term energies and the PECs are shifted by $1.94\text{GHz}\times h$ to higher energies as compared to Table 2 of the Appendix and Fig. 2, respectively, for better comparison to the experimental data. Then, the calculated position for the vibrational ground state of $N=1/2$ (which has even symmetry) coincides with the lowest observed singlet line at $16.30\text{GHz}\times h$ (which is a strong signal). Such a shift is well within the uncertainty of a few GHz × $h$ of absolute energy determinations in the perturbative electronic structure calculations [7, 37].

Figure 6

Line multiplets observed for atomic samples initially prepared in the hyperfine state $F=1$ [blue data points for (i) and (ii)] and $F=2$ [red data points for (ii) and (iii)], respectively. The individual lines are labeled with the $\left|\mathrm{\Omega }\right|$ quantum number of the corresponding assigned molecular state (see also Fig. 5). In the left panel for the magenta data points the Paul trap was off during the spectroscopy pulse, and no loss signal is visible. Here, the pulse duration of the spectroscopy light was $200\text{ms}$ for the $F=1$ data and $300\text{ms}$ for the $F=2$ data. For better visibility the magenta data points and also the red data points in the center panel are shifted in the vertical direction by 0.3. The error bars represent the statistical uncertainty. Dashed blue and red lines are the results of Gaussian fits.

Figure 7

Values of the molecular PECs at an internuclear distance $R=265a_0$ as a function of the interaction control parameters ${\lambda }_1$ and ${\lambda }_2$ (see also Table 1). On the left, where ${\lambda }_1={\lambda }_2=0$, interaction is identical for singlet and triplet states. $s$-wave and $p$-wave interactions are, however, not identical. When going to the right ${\lambda }_1$ and ${\lambda }_2$ are subsequently turned on. Parameter ${\lambda }_1$ modifies the singlet $s$-wave and $p$-wave scattering and introduces a splitting of the lines in two respects. First, branches of mixed $F$ character separate from branches of pure $F$ character. Second, the $\left|\mathrm{\Omega }\right|$ components within these branches slightly split off from each other. Parameter ${\lambda }_2$ introduces a ${\vec{L}}_{\mathrm{p}}·\vec{S}$ type of interaction. This enhances the $\left|\mathrm{\Omega }\right|$ splittings by about two orders of magnitude.

Figure 8

The molecular PECs correlated to the $5S_{1/2}+16P_j$ atomic asymptotes for $j\in \{1/2,3/2\}$ and different hyperfine states $F\in \{1,2\}$ of the $5S_{1/2}$ atom. Here, the color code represents the expectation value of the quantum number $S$ of the total electronic spin.

Figure 9

Measured series of line multiplets for atomic samples initially prepared in the hyperfine state $F=1$ (a) and $F=2$ (b). The blue and red data curves in (a) and (b), respectively, are the same as in Fig. 4. All other spectra are obtained for individually optimized experimental parameters to locally increase the signal-to-noise ratio. For better visibility, light blue (cyan) data in (a) are shifted in the vertical direction by 0.5 (1.0), as well as the magenta (purple) data in (b). Blue and orange (red and orange) vertical lines at the bottom of plot (a) [at the top of plot (b)] indicate the frequency positions of measured resonances belonging to line singlets and line doublets (line triplets and line doublets) in (a) and (b), respectively. Orange color corresponds to light gray in gray-scale versions. The alternating signal strength behavior for each multiplet series is illustrated by the line style, where solid (dashed) lines represent strong (weak) signals.

Figure 10

The molecular Born-Oppenheimer potentials when $s$-wave interactions between the electron and the ground state atom are taken into account but $p$-wave interactions are neglected. The colors of the curves indicate the expectation value of the $F$ quantum number of the ground state atom. From each atomic asymptote two oscillatory potentials emerge. They have $|m_j|=1/2$ and feature the nodes of the electronic wave function. The deeper PECs (marked with A) are associated with pure triplet scattering and the shallower PECs (marked with B) are associated with mixed singlet-triplet scattering. The PECs labeled with C have $|m_j|>1/2$ and do not show an oscillatory behavior. In the inset a sketch of an ultralong-range Rydberg molecule is shown. It consists of a ${\mathrm{Rb}}^{+}$ ionic core, an electronic Rydberg $16P$ state orbital, and a ground state Rb atom which is located at position $\vec{R}=R{\widehat{e}}_z$ relative to the ionic core. The $16P$ electronic orbital is given in a contour plot representation.