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 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.


I. INTRODUCTION
Since their prediction almost twenty years ago [1] and boosted by their first observation [2], ultralong-range Rydberg molecules have become a research area of major interest (for a review, see, e.g. [3]). In such exotic molecules the scattering of the Rydberg electron off the ground state atom leads to molecular bound states. So far, a number of experimental studies have been carried out on different atomic species (Rb, Cs, Sr), investigating molecules correlated with S, P, or D Rydberg atomic states (e.g. [2,[4][5][6][7][8][9][10][11][12][13][14]). These studies comprise also trilobite dimers [1,6,14], which can feature huge permanent electric dipole moments [6], and butterfly dimers [15,16] that were observed recently [7]. While the binding mechanism for trilobite molecules relies on s-wave scattering, p-wave scattering has to be included in order to theoretically describe butterfly molecules. Typically, the formation of molecules takes place in the closed-wells of the potential energy curves connected to the Rydberg electron wave function. However, bound states can also be obtained by internal quantum reflection at a shape resonance [17], i.e. even in the absence of a regular inner potential well.
In the present work we investigate molecular states associated with the 16P 3/2 + 5S 1/2 atomic asymptote of 87 Rb. Here, for the given low principal quantum number n = 16 already the second outermost molecular potential well is heavily distorted by the p-wave shape resonance. This leads to intersecting potential energy curves, which entail avoided crossings characterized by energy gaps and ledges in the potential energy curves. Overall, this gives rise to a complicated molecular level structure. As our analysis will show the observed energy level positions and lifetimes of Rb 2 bound states can only be explained by the combined closed-and open-well character of the potential curves, the latter being associated with a quantum reflection process.
For our measurements we use ultracold samples of 87 Rb atoms, which are confined in an optical dipole trap. Two different types of samples can be prepared such that the atoms are spin-polarized either in the hyperfine state F = 1 or F = 2. We perform photoassociation spectroscopy, where the molecules are detected via their ionic decay products, however, using a novel scheme. Once an ion is formed it is captured in an ion trap which is overlaid with the optical dipole trap. The ion elastically collides with atoms inflicting atom loss, i.e. a small number of formed molecules can be converted into a significant signal.
Applying the given scheme, we study vibrational manifolds of potential energy curves within a coupled complex. Interestingly, in our data patterns of singlet, doublet, and triplet resonance features are resolved. For the interpretation of the spectra, we make use of a comparison of data for the different initial atomic hyperfine states F = 1 and F = 2 exploiting the selective addressing of Rydberg molecular levels according to the dipole transition selection rules (see also [8]). On the basis of calculations presented here we can explain the given multiplet substructures as arising from a combination of different spin-spin and spin-orbit coupling mechanisms.
In our theoretical description of ultralong-range Rydberg molecules, the spin-orbit interaction of the Rydberg electron and the hyperfine interaction for the ground state atom are included. In the presence of spin-dependent scattering channels, this hyperfine coupling effectively mixes the collective electronic singlet and triplet state of the valence electrons [9]. In a series of studies, the described mixing was investigated and related scattering lengths were extracted [8,12,[18][19][20]. Furthermore, we encounter a spin-orbit coupling connected to the relative orbital angular momentum between the Rydberg electron and the ground state atom. Including this effect in arXiv:1901.08792v1 [cond-mat.quant-gas] 25 Jan 2019 theoretical models [21][22][23] leads to a modification of the butterfly curves and dipole moments of the corresponding molecular states, especially when the kinetic energy of the Rydberg electron is close to the p-wave shape resonance. Although some preliminary indication of this spin-orbit interaction has been found recently [21,24] no clear experimental evidence has been demonstrated so far.
In our model, we include all of the above mentioned mechanisms employing the pseudopotential Hamiltonian developed in [21]. As a result of our analysis, the observed series of line patterns can unambiguously be assigned to potential energy curves that are energetically separated from each other due to the combined effects of p-wave scattering and internal molecular spin-couplings. Within an individual series each pattern represents the substructure of a vibrational state. The calculated line splittings within these multiplet line structures and the separation of vibrational states agree reasonably well with the measured data in view of the complexity of the potential structure to be modeled here.
This article is structured as follows. In section II we provide a simplistic description of the investigated Rydberg molecules and the corresponding potential energy curves in order to establish a basis for the following discussion. Subsequently, in section III we introduce our experimental scheme and the obtained spectroscopic data are described in detail in section IV. Section V is devoted to a discussion of our theoretical model including different types of spin-spin and spin-orbit coupling mechanisms. This model will be used to interpret our experimental data in section VI by comparing the results of theoretical simulations assuming adiabaticity or diabaticity.

II. RYDBERG MOLECULAR INTERACTIONS
In the present work, we explore energetically low-lying Rydberg molecular states associated with n = 16. Here, the pwave shape resonance plays an important role as it cuts already through the second outermost potential well, thereby distorting the associated vibrational energy levels. In general, due to the given resonance, for the regime of small n, there is a large variation of the shape of the potential energy curves with varying n. This renders energetically low-lying Rydberg states interesting, however, entailing also complications in the theoretical description.
Specifically, we investigate Rydberg molecules correlated with the atomic asymptotes 16P 3/2 + 5S 1/2 and 16P 1/2 + 5S 1/2 of 87 Rb. A pictorial representation of the setup and the basic binding mechanism are presented in Fig. 1(a). In the molecular state the Rydberg P-orbital is aligned along the internuclear axis R and exhibits radial oscillations of the probability density. Without loss of generality we assume that R lies on the z-axis, i.e. R = Rê z . When the ground state atom enters the electronic orbital it will modify the electronic energy. We model this interaction in terms of quantum scattering by a short-range, Fermi-type pseudopotental including sand p-wave channels (for more details see Sec. V). Modifications due to the s-wave interaction can be seen from the Born-Oppenheimer potential energy curves (PEC) presented Energy (GHz × h) F  1. (a) Sketch of a Rb atom in a Rydberg P state, with its ionic core located in the center and its electronic orbital. A ground state Rb atom is located at position R = Rê z relative to the ionic core. (b) The molecular Born-Oppenheimer potentials of the above setup when pwave interactions are neglected. Two oscillatory potentials featuring the nodes of the electronic wave function emerge from each atomic asymptote: One deeper PEC associated with pure triplet scattering and one shallower PEC associated with mixed singlet/triplet scattering.
in Fig. 1(b): For separations larger than the extension of the Rydberg cloud (here ∼ 400 a 0 , where a 0 is the Bohr radius) all curves approach the atomic asymptotes that are characterized by the Rydberg levels (16P 1/2 , 16P 3/2 ) and the internal hyperfine state (F = 1, F = 2) of the ground state atom. For smaller separations, however, the ground state atom splits and perturbs the electronic energies. Additionally, the hyperfine states can become mixed (color coding of the curves). The strength of this interaction is proportional to the electronic density and, consequently, the shape of the PEC reflects the oscillations of the Rydberg orbital. Whether this interaction leads to a blue-or to a red-shift of the Rydberg energies depends, however, on the complex interplay of the different degrees of freedom in the system.
A pictorial overview of these degrees of freedom is presented in Fig. 2. In the frame centered around the ionic core of the Rydberg atom (coordinate origin), the Rydberg electron has an angular momentum l and a spin s 1 . Both are coupled via spin-orbit interaction to form the total angular momentum zoom FIG. 2. Setup of the molecular system: A ground state atom is located at position R relative to the ionic core of a Rydberg atom. The Rydberg electron with spin s 1 is located at position r with orbital angular momentum l relative to the ionic core and at position X with orbital angular momentum L relative to the ground state atom. Furthermore, the ground state atom possesses an electronic spin s 2 and a nuclear spin I. j = l + s 1 (fine structure of the Rydberg atom). The ground state atom is located at a position R and has an electronic spin s 2 as well as a nuclear spin I with I = 3/2. These spins are coupled to F = I + s 2 (hyperfine structure of the ground state atom). However, neither F nor j are conserved quantities, since the electron atom interaction couples the spins s 1 and s 2 . Additionally, there is a spin-orbit interaction between the total electronic spin S = s 1 + s 2 and the angular momentum of the Rydberg electron L with respect to the scattering center located at the ground state atom's position R.
In the model potential for the electron-atom interaction we include sand p-wave scattering, where s and p refer to L = 0 and L = 1, respectively. The corresponding scattering amplitudes depend therefore on S and the combined angular momentum J = S + L. In general, the obtained PEC can be labeled by the good quantum number Ω = m l + m 1 + m 2 + m I , which represents the projection of the total angular momentum of the molecular system on the internuclear axis in terms of the projections of the orbital angular momentum m l , the Rydberg electronic spin m 1 , the ground state atom's valence electron's spin m 2 , and the ground state atom's nuclear spin m I . Due to the axial symmetry of the system, electronic states appear always as energetically degenerate pairs with quantum numbers ±Ω.
The PEC in Fig.1(b) correspond to a simplified scenario where p-wave interaction is neglected. This is typically a good approximation when the ground state atom lies close to the semiclassical turning points of the Rydberg electron (most outer wells of the PEC), such that the kinetic energy of the electron is low. By contrast, at smaller separations, the impact of p-wave interaction becomes crucial. This is illustrated in Fig. 3(a). At distances R < 300 a 0 additional p-wave dominated PEC appear that cross through the earlier discussed s-wave dominated PEC near the location of the second to outmost potential well, therefore having a major impact on the PEC even at relatively large internuclear separations. This crossing is a consequence of p-wave shape resonances of the electron-87 Rb interaction at energies of approximately E r = 23 meV [25], which induce a strong coupling of the Rydberg levels. Related to the 'butterfly'-shape of the associated electronic densities, the p-wave dominated PEC are also called butterfly curves [15,23]. Due to level repulsion with the lower n = 13 hydrogenic manifold, these butterfly curves form deep potential wells at energies around −400 GHz × h that support several bound vibrational states [7].
In our work we focus on the region where the crossing between the butterfly curve and the 16P states happens. This region is shown in Fig. 3(b). Compared to Fig. 1(b) we observe three major features of the PEC that can be related to this crossing and the increased p-wave interactions: An opening of the potential wells, a strong mixing of hyperfine states (Fmixing) and a splitting of the PEC near the avoided crossings. Furthermore, these features are correlated. Here, the combined spin operator N = s 1 + s 2 + I is useful for structuring the PEC. States with mixed F-character correspond to N = 3/2 and split into two curves of different |Ω| = {1/2, 3/2}. Similarly, states with dominant F = 2 are attributed to N = 5/2 and split accordingly into three PEC, while states with F = 1 correspond to N = 1/2 and do not split (see Sec. V for more details).

III. EXPERIMENTAL SCHEME
Once produced, a Rydberg molecule can decay into an ion. In general, different ionization processes are possible such as photoionization which leads to a Rb + atomic ion or associative ionization entailing a Rb + 2 molecular ion [26,27]. Typically, Rydberg molecules are detected via field ionization of the Rydberg state or via their ionic reaction products using microchannel plates and time-of-flight sequences. We demonstrate here a different method that relies on elastic atom-ion collisions. It is a further development of a technique that we have introduced recently to detect molecules in the electronic ground state of Rb 2 [28,29]. Using this technique even a single produced ion leads to sizeable atom loss.
The general scheme is illustrated in Fig. 4. Our measurements are carried out in a hybrid atom-ion setup as described in detail, e.g. in [30]. The experimental sequence starts by preparing an ultracold sample of spin-polarized 87 Rb atoms. We can initialize the atomic sample either in the hyperfine state (F = 1, m F = −1) or (F = 2, m F = +2) within the 5S 1/2 electronic ground state. Here, m F is the projection quantum number corresponding to F. The atom cloud is confined in a crossed optical dipole trap at a wavelength of about 1065 nm featuring a potential depth of about 20 µK × k B . We work with thermal samples at temperatures of around 1 µK, typically consisting of about 3 − 4 × 10 6 atoms. The cloud is Gaussian-shaped with a size of σ x,y,z ≈ (70, 10, 10) µm along the three directions of space.
In a first step [(i) in Fig. 4 ], Rb 2 Rydberg molecules are created by means of photoassociation employing a single laser at a wavelength of about 302 nm (for technical details on the photoassociation laser setup, see Appendix A). These The molecular PEC associated with the 5S 1/2 + 16P j atomic asymptotes for j ∈ {1/2, 3/2} and different hyperfine states F ∈ {1, 2} of the 5S 1/2 atom. The color code represents the expectation value of the quantum number F. In contrast to Fig. 1(b), p-wave interactions are included here and lead (due to a p-wave shape resonance at internuclear separations R ≈ 250 a 0 ) to steep curves that cut through the PEC creating a multitude of avoided crossings with the oscillating P-state curves. So called butterfly molecules can form in the potential wells at energies in the range of about −500 to −100 GHz × h. (b) This study focuses on P-state molecules strongly influenced by the butterfly PEC in the zoomed in region. At energies between about 10 to 20 GHz × h three plateaus of different hyperfine character F = 2, mixed F, and F = 1 emerge that form a triplet, doublet, and singlet structure, respectively, in the vicinity of the butterfly crossing at R ≈ 260 a 0 . We label the corresponding interaction channels by N = 5/2, N = 3/2, and N = 1/2, respectively. molecules can decay into ions (ii). In our setup, we have an ion trap available. It is a linear Paul trap with a potential depth of about 1 eV. Thus, any ions are captured right after their formation. The centers of the Paul trap and the optical dipole trap are overlaid such that the accumulated ions are immersed into the atom cloud and atom-ion collisions take place. Since the ions are micromotion driven in the Paul trap their kinetic energies are orders of magnitude larger than those of the ultracold atoms. Therefore, due to the collisions, the ions continuously expel atoms out of the shallow dipole trap [see (iii)] [31]. By measuring the remaining number of atoms as a function of the frequency of the photoassociation laser light we obtain a spectrum of atomic and molecular states. In our studies we find that the strengths of spectroscopic signals can be tuned via optimizing the ionic kinetic energy using the driven micromotion. Furthermore, signals can even be enhanced after the photoassociation laser is already off since the accumulated ions still collide with neutral atoms. In order to discriminate between Rb + and Rb + 2 ions we have carried out mass spectrometry with our Paul trap. For this, the rf-drive is modulated with the trapping frequency for Rb + and Rb + 2 , respectively, which expels the corresponding ionic species out of the trap and therefore leads to a decrease in atom loss. Using the given spectrometry we have only observed Rb + ions. However, it is still not possible to exclude an initial production of Rb + 2 since Rb + may be formed out of Rb + 2 by collisions with neutral atoms in the cloud. Indeed, experimentally, we find indications that associative ionization leading to Rb + 2 might be the dominant ionization process. However, further investigation is necessary for a conclusive statement. We emphasize that for our detection method of Rydberg molecules it is not essential whether Rb + or Rb + 2 is formed in a secondary process, since the scheme works for any of those charged particles.

IV. SPECTROSCOPIC DATA
We consider the spectra presented in Fig. 5 covering a large frequency interval. The plots show the measured normalized atom loss L = 1 −Ñ/Ñ 0 as a function of the frequency ν of the photoassociation laser light when the ion trap is turned on during the spectroscopy. Here,Ñ andÑ 0 represent the remaining numbers of atoms for the UV light being on and off, respectively. For scan (a) (blue data points) the atoms were initially spin-polarized in the hyperfine state F = 1, while for scan (b) (red data points) the atoms were prepared in F = 2. The laser frequency ν is expressed in terms of the detunings ∆ν = ν − ν 0 (for F = 1) and ∆ν = ν − ν 0 + 2 × ν hfs (for F = 2), respectively. We use ν 0 = 991.55264 THz as reference, which is determined by the resonance position of a strong loss feature observed for F = 1. This loss feature and all other loss features marked by an arrow in Fig. 5 correspond to atomic lines. A discussion of atomic lines is given in the Appendix B. By choosing the different detunings ∆ν and ∆ν, the data for F = 2 are shifted by two times the hyperfine splitting ν hfs = 6.8346 GHz [32] for the 5S 1/2 electronic ground state of 87 Rb, with respect to the data corresponding to F = 1. This is done to directly compare molecular resonance lines in the F = 1 spectra to those in the F = 2 spectra. We note that for n = 16 the hyperfine splitting for the excited Rydberg P state is expected to be negligible as compared to our measurement uncertainty, due to the ∝ n −3 eff scaling of the hyperfine constant, where n eff is the effective principal quantum number) [33].
Besides the loss features assigned to excited atomic states the spectra of Fig. 5 exhibit many more resonance lines which represent transitions towards Rydberg molecular energy levels below the 5S 1/2 + 16P 3/2 and 5S 1/2 + 16P 1/2 dissociation thresholds, respectively. In order not to complicate the discussion we focus in the following on the spectral frequency region of about 10 GHz < ∆ν < 40 GHz, i.e. the open potential wells shown in Fig.3(b) associated with the 5S 1/2 +16P 3/2 atom pair asymptote are addressed.
The molecular signals are characterized by rather symmetric lineshapes with typical linewidths (FWHM) on the order of several tens of MHz. For example, the blue data points in the left panel of Fig. 6 represent a detailed investigation of the molecular resonance indicated by (i) in Fig. 5(a). This resonance can be described quite well by a fit using a Gaussian curve. Remarkably, if the Paul trap is off during the spectroscopy laser pulse then the signal vanishes as can be seen from the magenta data points, which were obtained for the same UV light intensity and pulse duration as the blue ones. From this, we conclude that a low number of ions is produced which does not lead to significant loss of signal in the absorption imaging.
In the spectra for F = 1 atoms we do not only observe single lines but also double-line structures. One example is presented in the center panel of Fig. 6 (blue data points). This scan was obtained at the spectral region indicated by (ii) in Fig. 5(a). Interestingly, as can be seen from the red data points in the center panel of Fig.6, when using F = 2 atoms we find a very similar loss feature at the same corresponding frequency ∆ν. Such coincidences between F = 1 and F = 2 data are observed for almost all double-line resonances. The strengths of the double-line signals are generally lower for F = 2 as compared to F = 1 (except for the structure at ∆ν ∼ 18.7 GHz), such that some of them are only detected for F = 1. Typically, the separation of the individual resonances within such double-line structures is a few hundred MHz.
In contrast, single isolated resonance lines as in the left panel of Fig. 6 are exclusively observed for F = 1 atoms. In turn, only for F = 2 atoms we observe patterns of three lines. One example is presented in the right panel of Fig. 6 (detailed scan of (iii) in Fig. 5(b)). The three-line structures are generally characterized by a systematic behavior. The separation between adjacent resonances is typically on the order of about Overall, in the considered spectral range of 10 GHz < ∆ν < 40 GHz we find a systematic progression for each kind of multiplet line structures. These progressions will be identified as vibrational ladders corresponding to the respective PEC. For example, in the measurement for F = 2 atoms as in Fig. 5(b) adjacent three-line pattern are separated by a vibrational spacing between 1.5 to 1.8 GHz (positions of measured triplet structures are marked by vertical red lines). We note that in the spectrum of Fig. 5(b) only every second resonance pattern is visible (indicated by the solid vertical red lines). However, in experiments with higher signal-to-noise resolution also much weaker three-line structures located between the strong ones are observed (indicated by the dashed vertical red lines). This is discussed in Appendix C (see also corresponding Fig.9). The same alternating behavior and similar vibrational splittings are also found for single-line features and double-line resonances (see also Fig. 9). An overview of all measured resonance lines and splittings is given in Tab. II of the Appendix.
H ryd describes the Rydberg electron in the potential of the ionic core and has eigenstates φ nl jm j ( r) with energies E nl j . We take these energies from spectroscopic measurements [34][35][36] and use them as an input to determine analytically the longrange behavior (larger than a few a 0 ) of φ nl jm j ( r) in terms of appropriately phase shifted Coulomb wave functions. Knowledge on the wave functions for smaller radii is not necessary for our purpose. The internal hyperfine structure of the ground state atom is described by H g = A I · s 2 with A = 3.417 GHz taken from [33]. This Hamiltonian has eigenstates |Fm F . The atomic eigenstates of H ryd and H g become coupled by the electron-atom interaction V . We employ a generalized Fermi pseudopotential [21,37] where X = | r − R| is the absolute distance between the Rydberg electron and the ground state atom (see Fig. 2) and β is a multiindex that defines projectors onto the different scattering channels |β = |LSJM J . The quantum numbers L, S, J, and M J specify the total orbital angular momentum L in the reference frame centered on the ground state atom (L = 0 for s-wave and L = 1 for p-wave scattering), the total spin S (S = 0 for singlet and S = 1 for triplet scattering), and the total angular momentum J = L + S (J ∈ {1, 2, 3} with M J ∈ {−J, . . . , J}). The interaction strength in each channel depends on the scattering lengths a(L, S, J, k) for a free electron with wave number k that scatters off a 87 Rb ground state atom. We use scattering phase shifts as presented in [21]. The wave number is computed via the semiclassical relation 1/2k 2 − 1/R = −1/(2n 2 eff ). To compute the PEC in the vicinity of the 16P 1/2 state (which represents the closest asymptotic state) we use the respective quantum defect principle quantum number n eff = 13.3447. The cusps in the outer wells of the PEC, e.g. around R = 360 a 0 in Fig 1(b), occur due to the non-analytic behavior of k close to the classical turning point, where k becomes zero.
The Hamiltonian H is constructed in a finite basis set that includes the 16S, 17S, 15P, 16P, 14D, 15D states, and the hydrogenic states with higher orbital angular momenta l ≥ 3 with principle quantum numbers n = 13 and n = 14. All these states are considered with all possible total angular momenta j, while the projections m j are truncated to include |m j | ≤ 3/2. According to the choice of the molecular axis lying on the z-axis, states with |m j | > 3/2 do not interact with the ground state atom. Additionally, the ground state atom nuclear and electronic spin are taken into account completely (m I = {±1/2, ±3/2} and m 2 = ±1/2). Note, that placing the perturber onto the z-axis significantly reduces the basis set. Since the scattering interaction V vastly exceeds the Zeeman energy for any magnetic fields occurring due to the experimental setup, the atomic orbitals align along the internuclear axis. This is different, however, when the interaction with an external field is comparable or larger than the scattering interaction [10]. An alternative approach to derive the PEC that circumvents a finite basis set are Green's function methods employed for example in [16,23]. However, these approaches do not incorporate hyperfine interactions that are crucial for the interpretation of our results.
Diagonalizing Eq. (1) yields the PEC presented in Fig. 3. The splitting of the PEC, which in Fig. 3 is especially visible near R = 260 a 0 , in the vicinity of the crossing of butterfly curves with the 16P curves, is essential for the interpretation of the experimental results. We discuss the underlying mechanism in more detail in Fig. 7 with an emphasis on the participating scattering channels. To this end, we introduce three control parameters λ 1 , λ 2 , and λ 3 that govern the strength of the different scattering channels. The mapping is summarized in Table I. When λ 1 = λ 2 = λ 3 = 0, the electron-atom interaction V is insensitive to the total electron spin S and the interaction can be simplified to [1,15] with a s (k) = a(0, 1, 1, k), a p (k) = a(1, 1, 1, k) and R = Rê z . As can be seen in Fig. 7, in that case there is no splitting of the PEC. This changes in the regime of finite values of λ 1 > 0, while λ 2 = λ 3 = 0. Here, the parameter λ 1 introduces a singlet channel in the s-wave scattering and leads to a splitting into a deep and a shallow curve, which is also visible in Fig. 1(b) [9,12,19]. In Fig. 7 deep curves are flat as a function of λ 1 because they interact exclusively via triplet s-wave scattering and are, hence, insensitive to the singlet s-wave channel. However, they are not pure triplet states. This becomes apparent by means of an example. Consider the electronic state of a Rydberg atom in a P state with total orbital angular momentum j = 1/2 and projection m j = 1/2 and a ground state atom in a polarized nuclear spin state F = 2 and m F = 2  Table I  eigenvalues of N 2 given by N(N + 1). One can easily show that states with the minimal value N = 1/2 are pure triplet states (S = 1) with F = 1, while states with maximal N = 5/2 are pure triplet states (S = 1) with F = 2. In contrast, states with N = 3/2 can have contributions from S = 0 and S = 1 as well as from F = 1 and F = 2 states. The degree of F-mixing depends on the relative strength of the scattering V and the hyperfine interaction H g . N reproduces the multiplicity for the PEC visible in Fig. 1 (b), Fig. 3 (b) and Fig. 7: For the F = 2 asymptote, the deep curve has six degenerate states of pure F = 2 character that interact via s-wave scattering exclusively in the N = 5/2 channel. For the F = 1 asymptote, the deep curve corresponds to N = 1/2 and has two degenerate states of pure F = 1 character. The shallow curves of both, the F = 1 and the F = 2 asymptote correspond to N = 3/2 and have four degenerate states of mixed F = 1 and F = 2 character each. The F-mixing due to the presence of a singlet s-wave scattering channel is essential for the spin flip effect observed in [8].
While this regime of interactions is sufficient to describe the PEC at the outer potential well (in this case for R > 300 a 0 ), additional p-wave related interactions become important for smaller internuclear separations and will be relevant in the next parameter regime of control parameters. The next regime is λ 1 = 1, λ 2 > 0 and λ 3 = 0, see Fig. 7. The parameter λ 2 introduces a singlet channel in the p-wave scattering. This makes also spinor components with m l = ±1, which only probe the p-wave interaction but not the s-wave interaction, susceptible to a singlet scattering channel and introduces a small splitting due to singlet/triplet mixing in pwave channels. Although the exemplary state with Ω = 5/2 in Eq. (4) is of pure triplet character in its m l = 0 component, it is of mixed singlet/triplet character in its m l = 1 component. Such a mixing will also occur for different |Ω| but with a different mixing ratio. A spin-selective p-wave interaction scattering channel model mapping 1, 1, k) TABLE I. Overview of the scattering lengths/volumes a(L, S, J, k) that are modified via control parameters λ 1 , λ 2 , and λ 3 in order to study the splitting mechanism in Fig. 7. splits these states therefore into different |Ω| states. This splitting, however, arises only due to the Rydberg fine structure and is, hence, not visible in s-state ultralong-range Rydberg molecules recently studied [38] due to the missing fine structure.
Finally, for the regime λ 1 = λ 2 = 1, λ 3 > 0 the full interaction introduced in Eq. (2) is realized by adding different pwave triplet channels for each J (J-channels) that result from the L · S-coupling. Due to the varying resonance energy in the three J-channels, the triplet states experience different interaction strengths at a specific internuclear separation, which leads to increased splitting. This is visible for the lower mixed F curve and the F = 2 curve in Fig.7 and also exists for the upper mixed F curve, however, barely visible for that at the given scale. The various J-channels are all mixed for each |Ω|, except for |Ω| = 5 2 , where the J = 0 channel is not present. This splitting persists even in the case of s-state ultralong-range Rydberg molecules with negligible fine-structure [38].
The fact that the splittings introduced via the control parameters λ 1 , λ 2 , and λ 3 are not constant over R causes a non-trivial vibrational level structure of molecular states that can form on the PEC. To determine the vibrational states, we employ two different approaches. Using an adiabatic approach specific adiabatic PEC are considered individually (blue curves in Fig. 8). In the regime investigated here, molecular states form due to destructive interference at a potential drop and are referred to as quantum reflection states [17,18]. The positions and lifetimes of the respective resonances are obtained in terms of the phase Θ of the complex reflection coefficient of an outgoing plane wave reflected at the potential curve. A molecular state is found at energies E, at which the phase Θ undergoes a sudden change. These energies are shown as blue horizontal lines in Fig. 8. According to Wigner's time delay interpretation, the corresponding lifetime equals τ =¯h 2 dΘ dE [39]. In the diabatic approach, we connect two adiabatic PEC at the avoided crossing due to the butterfly curves, to form a new diabatic potential (black curves in Fig. 8). The bound states in the resulting potential are obtained via standard finite differences techniques employing hard wall boundary conditions. Within this approach, decay of such states is neglected and their energies are shown as black horizontal lines in Fig.8.

VI. INTERPRETATION OF EXPERIMENTAL RESULTS USING MODEL CALCULATIONS
From the considerations presented in Secs. II and V we can explain the systematic behavior of the observed molecular resonances. We start with a rather qualitative interpretation. The multiplet structures are due to the energy splitting of the Ω-states as a consequence of spin-orbit and spin-spin coupling mechanisms (see also Fig. 7). Measured single-line, double-line, and triple-line pattern reflect the substructures of the states characterized by the interactions channels N = 1/2, N = 3/2, and N = 5/2 (see Fig. 3) given by the corresponding |Ω| = 1/2, |Ω| = {1/2 , 3/2}, and |Ω| = {1/2 , 3/2 , 5/2} components, respectively. Note, that Ω is not the projection of N as it omits the projection of the orbital angular momentum of the Rydberg electron. The states associated to N = 1/2 and N = 5/2 feature pure triplet character (S = 1) and the state associated to N = 3/2 has mixed singlet/triplet character. Accordingly, the state associated to N = 3/2 is of mixed F character, i.e. 1 < F < 2 (dependent on the internuclear distance). Therefore, it can be addressed via the spectroscopy laser when using atomic samples either spin-polarized in F = 1 or F = 2. This is exactly what we find in our data [see double-line resonances e.g. in Fig. 6(b), and Fig. 9(a) and (b)]. Similarly, the single-line and three-line patterns are attributed to molecular states associated with the N = 1/2 and N = 5/2 potential energy curves, respectively. Again, this is in agreement with the experimental finding that single-line (three-line) structures can only be detected when starting with F = 1 (F = 2) atoms.
The frequency distances between adjacent multiplet resonance features of the same kind are identified as the vibrational splittings of the molecular levels. We note that for the N = 3/2 and N = 5/2 curves each Ω component has its individual vibrational ladder. However, the experimental data reveal that for a given quantum number N the vibrational spacings for the different Ω states are similar, at least for the investigated range of term energies. In addition, in the measurements an alternation in the signal strengths of adjacent multiplet structures is found (see also Tab. II). This alternation can be explained as a consequence of the change of g/u symmetry of the vibrational molecular wave function from one vibrational state to another. The excitation rate to a molecular state is proportional to the Franck-Condon factor (see, e.g. [11]). Indeed, for molecular wave functions with g symmetry the Franck-Condon factors are in general significantly higher as compared to molecular wave functions featuring u symmetry. We now compare our experimental data to calculations using either the adiabatic approach or the diabatic approach as described in Sec. V. For this, we start by discussing the results when assuming full adiabaticity in the dynamics, i.e. the potential energy curves indicated by blue color in Fig. 8 are considered. Here, the subplots (a), (b), and (c) correspond to the states characterized by N = 1/2, N = 3/2, and N = 5/2 (see also Fig. 3), with their respective Ω-substructure. The obtained molecular energy level positions for the adiabatic approach are given by the blue horizontal lines in Fig. 8 and the term frequencies and level splittings are provided in Tab. II of the Appendix. The experimental observations are shown by the red horizontal lines in Fig. 8.
Qualitatively, the calculations are in agreement with the observed multiplicity of resonance lines and the approximate shapes of the vibrational ladders. However, quantitatively there are discrepancies, e.g. when considering the splittings within individual multiplet line structures. As can be seen from Fig. 8(b) and Tab. II, for the double-line pattern associated with the N = 3/2 curve the calculated spacings between the |Ω| = 1/2 and |Ω| = 3/2 components are generally too large. Even more, for N = 5/2 no pattern of isolated three-line structures is obtained from the adiabatic model as the order of the |Ω| = 1/2, 3/2, and 5/2 states is intermixed regarding the term energies. Regarding the vibrational splittings, there are also quantitative differences, however, e.g. for N = 1/2 and N = 3/2 the discrepancies between predicted and measured values are less than about a factor of two. Finally, from the adiabatic approach we expect resonance linewidths on the order of 100 to 400MHz for the molecular levels while in the experiment these are typically only about several tens of MHz. Therefore, the measurements can just partially be explained when assuming full adiabaticity.
In the second approach we assume full diabaticity in our model. This means that we ignore the gaps (at the left side of the potential wells in Fig. 3, see Sec. V for details), which arise from the avoided crossings due to the p-wave shape resonance. Instead, we connect the separated appropriate parts of the potential energy curves to form closed potential wells. The black solid lines in Fig. 8 are obtained from spline interpolations. We calculate the corresponding molecular energy level positions which are given by the horizontal black lines (see also Tab. II). Now, as in the experiment we find a clear sequence of isolated double-line as well as three-line pattern for the N = 3/2 and N = 5/2 curve, respectively. However, the calculated frequency splittings of molecular energy levels within individual multiplet structures are generally lower than the measured ones. For the double-line pattern corresponding to N = 3/2 the discrepancies are about one order of magnitude, while for the main part of three-line patterns associated with the N = 5/2 curve the differences are only about a factor of two. In terms of vibrational splittings the diabatic model calculations are in good agreement with the observations. We note that for the given comparisons of experimental data and calculations we have only considered relative energy level positions. This is because the uncertainty of absolute energy determinations in the perturbative electronic structure calculations is on the order of a few GHz [21,40] leading to a corresponding general offset with respect to the measured resonance positions.
From the given comparisons we conclude that both characteristics of adiabaticity and of diabaticity can be seen in our spectra. For example, the obtained frequency splittings for molecular energy levels within individual multiplet structures are too large when assuming adiabaticity and too small when assuming diabaticity. The diabatic model fits better than the adiabatic model as revealed by the good description of the vibrational level spacings. Probably, the reality lies in between the two limits. This interpretation is in line with our physical picture presented in Sec. II. Due to the p-wave shape resonance a steep potential barrier is created at which quantum reflection states can form (see Fig. 3). At the gaps resulting from the avoided crossings molecular probability flux can leak out. However, still the closed-well characters of the potentials are preserved to a substantial degree.
We expect that an improved agreement between theory and experiment can be achieved by performing full vibronic structure investigations that take into account non-adiabatic couplings between the involved electronic states associated to different (adiabatic) PEC. However, the complexity of the here considered system exceeds significantly those of standard vibronic coupling problems for energetically low-lying molecular states [41]. This is due to the importance of the various spin degrees of freedom, the resonant p-wave interaction, and the sheer number of strongly coupled electronic states. On top of that, the effective pseudopotential of Eq. (2) provides us only with an incomplete knowledge of the underlying electronic interactions and the non-adiabatic coupling elements. For these very reasons a full vibronic structure investigation requires a huge computational effort and is beyond the scope of the present work.

VII. CONCLUSIONS
In conclusion, we have examined Rydberg molecules in a previously unexplored parameter regime, for which the molecular energy level structure is dictated by the interplay of internal spin structure and p-wave electron-neutral scattering. This interplay leads to a fine structure for the vibrational energy levels and causes singlet, doublet and triplet line patterns. Via comparison with theory based on a Fermi model, we have demonstrated that the observed line multiplicity provides valuable means for assigning molecular states to their respective potential energy curves, specifically when the corresponding vibrational ladders overlap in energy. Moreover, our quantitative analysis of the experimental data hints at the need for a more elaborate multi-channel treatment beyond the Born-Oppenheimer approximation for the molecular states studied here.
We expect our work to spark increased theoretical efforts to explain subtle effects of interaction processes in Rydberg molecules which become apparent in complex energy level structures. In future studies, this complexity can be reduced by considering Rydberg molecules associated to Rydberg S 1/2 states, instead of P 1/2 and P 3/2 states. Due to the lower electronic angular momentum of the Rydberg electron, the l · s 1 coupling has only a minor impact, which will allow to attribute experimentally observed fine structures of vibrational energy levels more directly to the L · S coupling. Additionally, it has been proposed to employ homogeneous weak magnetic fields to verify the L · S coupling by aligning S 1/2 -states dominated Rydberg molecules [38]. The molecular states are addressed using a single photon photoassociation scheme starting from the 5S 1/2 + 5S 1/2 atom pair asymptote. The light at wavelengths of around 302 nm is generated by a frequency-doubled cw dye laser featuring a narrow short-time linewidth of a few hundred kilohertz. In addition it is stabilized to a wavelength meter (High Finesse WS7). The wavelength meter is repeatedly calibrated to an atomic 87 Rb reference signal at a wavelength of 780 nm in intervals of hours. We achieve a shot-to-shot frequency stability of below ±10 MHz in the UV.
A multi mode optical fiber is used to transfer the UV light to the experimental table. At the location of the atoms the spectroscopy beam has a waist (1/e 2 radius) of about 1.5 mm and the power is typically in the range of 4 to 10 mW. The light pulse has a rectangular shape and the atoms are exposed to the laser radiation for a duration on the order of 0.1 to 1 s.

B. Atomic lines
In each of the spectra (a) and (b) of Fig. 5 two resonance lines (indicated by the horizontal black arrows) are characterized by very strong atom loss close to 100%. For both scans the two strong loss features are separated by the same frequency distance of about 36.4 GHz. When comparing these two-line structures for F = 1 and F = 2 one finds that they are shifted by about ν hfs with respect to each other. Therefore, we assign the peak for F = 1 atoms at ∆ν = 0 to the atomic transition (5S 1/2 , F = 1) ↔ (16P 1/2 , F = 1) and the peak at ∆ν ∼ 36.4 GHz corresponds to (5S 1/2 , F = 1) ↔ (16P 3/2 , F = 1). The measured position ν = ν 0 = 991.55264 THz of the line associated with (16P 1/2 , F = 1) serves as frequency reference in our work. Accordingly, for F = 2 the most significant loss features are linked to the excited atomic Rydberg states (16P 1/2 , F = 2) and (16P 3/2 , F = 2), respectively.
The significant long tail on the red side of the atomic resonances possibly arises (partially) due to the electric fields of the Paul trap but also could be an indication of interaction between Rydberg atoms and confined ions as reported very recently in [42] (see also [43] for a different experimental scheme). Indeed, when turning off the ion trap in our setup, we observe symmetric lineshapes with smaller widths for the same atomic resonances. Further details will be provided elsewhere.
Now we turn to the loss features indicated by the vertical black arrows in the spectrum for F = 1 atoms in Fig. 5(a). These features are located at about ν hfs below the ones indicated by the horizontal black arrows and again the lineshapes are also asymmetric. Therefore, we assign these resonances to the transitions (5S 1/2 , F = 2) ↔ (16P 1/2 , F = 2) and (5S 1/2 , F = 2) ↔ (16P 3/2 , F = 2). This means that we have a fraction of F = 2 atoms present in the atom cloud. In general, signal strengths in our spectra are characterized by a highly nonlinear dependence on the ion number accumulated in the Paul trap [28]. Thus, the number of ions leading to the parasitic F = 2 atomic lines can be orders of magnitude less as compared to the F = 1 atomic lines. Similarly, for the F = 2 spectra we observe resonances at about ν hfs above the strong atomic lines, which we assign to the (5S 1/2 , F = 1) ↔ (16P 1/2 , F = 1) and (5S 1/2 , F = 1) ↔ (16P 3/2 , F = 1) transitions.

C. Collection of data
In general, we have various parameters available to tune signal strengths in our detection scheme. These are the intensity and duration of the spectroscopy laser light, but also the ionic micromotion energy and the interaction time between trapped ions and neutral atoms. Using these parameters we can significantly enhance the signal-to-noise ratio as can be seen from Fig. 9. The blue data in (a) and the red data in (b) are zooms into Figs. 5(a) and (b), respectively. More and more multiplet resonance structures are revealed, when optimizing the experimental parameters individually for these structures. The blue, orange, and red vertical lines represent the cen- ter frequency positions of the measured resonances of singleline, double-line and triple-line pattern, respectively. In total, within the given frequency range we observe the complete series of expected resonances for each multiplet structure. This is supported by our finding of a stringent alternation between strong (solid vertical lines) and weak (dashed vertical lines) signals for all three cases. We note that here, the weak doubleline resonances are only detected for F = 1 atoms and not for F = 2 atoms [see, e.g. missing lines at around ∆ν = 27 GHz in the purple data of Fig. 9(b)]. Table II summarizes the results of all our measurements and the model calculations using an adiabatic ad diabatic approach, respectively. The subscripts e, ta, and td denote results from the experiments and the adiabatic and diabatic model, respectively. ∆ν e and ∆ν e are measured resonance frequencies. ∆ν ta and ∆ν td give computed term frequencies (referenced to the calculated 5S 1/2 + 16P 1/2 dissociation threshold). The subscript s indicates splittings between Ω-states within individual multiplet structures. The subscript v marks vibrational splittings, which are calculated using the energetically lowest states of two subsequent corresponding multiplet structures. δ ν s,ta and δ ν v,ta are not provided for the triple-line pattern (N = 5/2), since using the adiabatic approach no clear series of triple-line structures is obtained. The signal strengths of measured resonance lines are classified in weak (w) or strong (s). For better comparison the results of the diabatic model calculations are only given for the energy ranges from about the lowest to the highest given experimental resonance positions. Here, n.o. stands for not-observed lines. Values of ∆ν e indicated by ( * ) characterize experimental signals which might come from different molecular states than considered here. The resonance at ∆ν e = 31.86 GHz marked with ( * * ) is rather broad and expected to consist of a N = 5/2 and a 3/2 molecular line which cannot be resolved. We just give this frequency for the corresponding lines of the double-as well as the triple-line pattern.