key: cord-1014396-4vwwyu2l
authors: Haze, Shinsuke; D'Incao, Jos'e P.; Dorer, Dominik; Deiss, Markus; Tiemann, Eberhard; Julienne, Paul S.; Denschlag, Johannes Hecker
title: Spin-conservation propensity rule for three-body recombination of ultracold Rb atoms
date: 2021-12-27
journal: Phys Rev Lett
DOI: 10.1103/physrevlett.128.133401
sha: 4f20cdc9f8d5d7975e16576371d512b8e255df91
doc_id: 1014396
cord_uid: 4vwwyu2l

We explore the physical origin and the general validity of a propensity rule for the conservation of the hyperfine spin state in three-body recombination. This rule was recently discovered for the special case of $^{87}$Rb with its nearly equal singlet and triplet scattering lengths. Here, we test the propensity rule for $^{85}$Rb for which the scattering properties are very different from $^{87}$Rb. The Rb$_2$ molecular product distribution is mapped out in a state-to-state fashion using REMPI detection schemes which fully cover all possible molecular spin states. Interestingly, for the experimentally investigated range of binding energies from zero to $sim13:textrm{GHz}times h$ we observe that the spin-conservation propensity rule also holds for $^{85}$Rb. From these observations and a theoretical analysis we derive an understanding for the conservation of the hyperfine spin state. We identify several criteria to judge whether the propensity rule will also hold for other elements and collision channels.

The particular mechanisms of chemical reactions often give rise to selection and propensity rules. While selection rules express strict exclusion principles for product channels, propensity rules specify which product channels are more likely to be populated than others [1, 2] . Since the early days of quantum mechanics a central question in reaction dynamics is whether composite spins are conserved. Wigner's spinconservation rule, e.g., states that the total electronic spin has a propensity to be conserved [3] [4] [5] . The recent progress in the quantum state-resolved preparation and detection of ultracold atoms and molecules has now made it possible to experimentally explore spin conservation rules that also involve nuclear spins. In a recent study of bimolecular reactions of ultracold KRb molecules the conservation of the total nuclear spin was found [6, 7] . In a study on the final state distribution of three-body recombination of ultracold 87 Rb atoms our group found a propensity for the conservation of the hyperfine state of the atom pair forming the molecule [8, 9] . More precisely, this spin propensity rule states that the angular momentum quantum numbers F, f a , f b and m F = m f a + m f b are conserved in the reaction. Here, f a , f b correspond to the total angular momenta of the two atoms (a, b) forming the molecule, and F = f a + f b .

Formally, there is no selection rule that forbids spin exchange between all three atoms in the recombination process, and a corresponding change in the quantum numbers. In fact, recent calculations predict spin exchange to occur in threebody recombination of 7 Li [10] and 39 K [11, 12] , although being suppressed for 87 Rb [10] . In order to explain the observed spin propensity rule for 87 Rb one can justifiably argue that 87 Rb is special since here the singlet (a s ) and triplet (a t ) scattering lengths are nearly identical (a s ≈ 90a 0 , a t ≈ 99a 0 , where a 0 is the Bohr radius). This leads to a strong suppression of two-atom spin exchange reactions [13] [14] [15] .

In order to explore the validity of the spin propensity rule further, we investigate here, both experimentally and theoretically, three-body recombination of ultracold 85 Rb atoms which have very different two-body scattering properties from 87 Rb atoms. The singlet and triplet scattering lengths for 85 Rb atoms are a s = 2720a 0 and a t = −386.9a 0 [16] , respectively, and the three-body recombination rate constant L 3 for 85 Rb is about four orders of magnitude larger than for 87 Rb. Nevertheless, as a central result of our work, we find the spin propensity rule to also hold for 85 Rb, within the investigated range of binding energies from zero to 13 GHz × h and the resolution of our experiment [17] . This result is corroborated by the fact that our measured product distributions are well reproduced by numerical three-body calculations based on a single-spin channel. The spin propensity rule that we find for 87 Rb and 85 Rb will also hold for other elements if certain conditions are met, which are formulated in the present work.

We carry out the measurements with an ultracold cloud of 2.5 × 10 5 85 Rb ground state atoms at a temperature of 860 nK and at near-zero magnetic field B. The atoms have spin f = 2, m f = −2 and are trapped in a far-detuned optical dipole trap [18] . Three-body recombination produces weaklybound molecules in the mixed singlet X 1 Σ + g and triplet a 3 Σ + u states [see Fig. 1 (a)], which are coupled by hyperfine interaction [16] . We measure the yields of molecular products, observing a range of rovibrational states with vibrational and rotational quantum numbers v, L R , respectively, from v = −1 to −4 [22] and from L R = 0 to 6. A main result of our experiments is that we only find population in molecular states which are in the same spin state F = 4, f a = 2, f b = 2 (in short F f a f b = 422) as the reacting atom pair, although the investigated range of binding energies covers many bound states with different spin states. Because the two 85 Rb atoms (a, b) are identical bosons, the state F f a f b = 422 goes along with only having even angular momenta L R and a positive total parity. We have extended our previous state-selective detection scheme [8] so that it now covers all symmetries of the dimer product state space, including spin triplet and singlet states with their respective u/g symmetry. We apply two-step resonance-enhanced multiphoton ionization (REMPI), similar 15 [16, 20, 21] . (b,c) REMPI spectra of product molecules using path (S) for (b) and path (T) for (c). Shown is the normalized remaining atom number N/N 0 as a function of the REMPI laser frequency ν, where ν 0 = 497603.591 GHz (b), ν 0 = 497831.928 GHz (c), are the resonance positions of the photoassociation signals (PA) of the intermediate levels with J = 1. J is the total angular momentum excluding nuclear spin. The vertical lines represent calculated resonance positions, where black (gray) color indicates experimentally observed (unobserved) states. Each line is marked with the vibrational quantum number v. L R and J are given by the linestyle and the plot symbols, respectively (see legends).

to [21, [23] [24] [25] [26] , but with a cw-laser. By two different REMPI pathways, (S) or (T), we probe product molecules via singlet or triplet character [see Fig. 1(a) ]. Both schemes use identical photons for the two REMPI steps at wavelengths around 602.2 nm (see [18] ). The intermediate states are deeply-bound levels of (2) 1 Σ + u and (2) 3 Π g for REMPI (S) and (T), respectively. When ions are produced via REMPI, they are directly trapped and detected in an eV-deep Paul trap which is centered on the atom cloud. Subsequently, elastic atom-ion collisions inflict tell-tale atom loss while the ions remain trapped. From the atom loss which is measured via absorption imaging of the atom cloud [8, 27, 28] the ion number can be inferred [18] .

Figures 1(b) and (c) show REMPI spectra of Rb 2 product molecules following three-body recombination, using path (S) and (T), respectively. Apart from three signals stemming from the photoassociation of two atoms (indicated by PA) [29] , each resonance line of loss corresponds to a molecular product state. The photoassociation lines serve as references for the | f = 2, m f = −2 + | f = 2, m f = −2 asymptote corresponding to zero binding energy at about zero magnetic field. The vertical lines in Figs. 1(b) and (c) are predicted frequency positions for product states for the spin state F f a f b = 422. These predictions are obtained from coupled-channel calculations for the X 1 Σ + g − a 3 Σ + u complex [16] . Coincidences of predicted and observed lines allow for an assignment. As a consistency check for the assignment of signals in Figs. 1(b) and (c) we make use of product states with L R > 0 since these give rise to two resonance lines corresponding to the transitions towards J = L R ± 1. Indeed, the data in Figs. 1(b) and (c) confirm this consistency. Inspection clearly shows, that all experimentally observed spectral lines in Figs.1(b) and (c) can be explained as signals from product molecules with the spin F f a f b = 422. As an additional check for the line assignment, we show in [18] that our experimental spectra do not match up with molecular spin states other than F f a f b = 422. This clearly indicates that the same spin propensity rule previously observed for 87 Rb also holds for 85 Rb.

For each product signal in the singlet REMPI path (S) we obtain a corresponding signal in the triplet REMPI path (T). This is because the spin state F f a f b = 422, m F = −4 has sizeable singlet (≈ 15%) and triplet (≈ 85%) admixtures. The spectra of Figs. 1(b) and (c) generally look different since the rotational constants differ for (2) 1 Σ + u (B v = 289(1) MHz) and for (2) 3 Π g , 0 + g (B v = 389(2) MHz), see, e.g., [30] . The linewidths in Fig. 1 (c) are typically on the order of 100 MHz (FWHM) which is larger than the typical linewidths in Fig. 1 (b) of about 30 MHz. This is a consequence of the larger hyperfine splitting of the (2) 3 Π g state, which is not resolved in our measurements. In Fig. 1 (c) at ν − ν 0 = 15.90 GHz there is a photoassociation signal which belongs to the 0 − g component of (2) 3 Π g . We show the corresponding REMPI spectrum in [18] . It exhibits the same molecular states as in Figs. 1(b) and (c).

We now carry out a more quantitative analysis where we compare the experimental signal strengths of the various REMPI paths and also compare them to theoretical calculations. For this, we measure the ion production rate for each assigned resonance line in Fig. 1 and Fig. S1 [18] . For a given REMPI path, the ion rate signal is expected to be proportional to the product molecule population rate, if we assume equal REMPI ionization efficiencies for the states. Our data show that this assumption is indeed fulfilled for most data points within the uncertainty limits of the recordings. Figure 2 shows the extracted ion production rates γ i of each molecular product state for the three REMPI paths (via (2) 1 Σ + u , (2) 3 Π g 0 + g , and (2) 3 Π g 0 − g ). If a state is observed via (scaled) ion production rates γ i for the product states (v, L R ) with binding energies E b are given for the three REMPI paths together with calculations for the three-body recombination channel rate constants L 3 (v, L R ) (see the legend for the color coding). The ion signals for the (2) 3 Π g 0 − g and (2) 1 Σ + u paths have been scaled (see text).

two or three different J levels for a given path, we plot the average of the rates and mark the standard deviation from the mean with an error bar. In order to ease the comparison between the three data sets we have globally scaled the ion signals for the (2) 3 Π g 0 − g and (2) 1 Σ + u paths by a factor of 7.3 and 3.2, respectively, so that the signal bars for all REMPI paths in Fig. 2 are the same height for the state (v = −2, L R = 4), see black arrow. These scaling factors compensate the differences in the ionization efficiencies of the different REMPI paths, which are due to the different singlet and triplet components of the product molecule as well as differences in the electric dipole transition moments, which are generally not very well known yet [31] . After the scaling the signals of the three paths for a given bound state are consistent over the set of detected product states.

In addition to the experimental data, we plot in Fig. 2 calculated channel rate constants L 3 (v, L R ) for a temperature of 0.8 µK. The calculations use a single-spin model [18] which is similar to the one used in Ref. [8] in order to solve the three-body Schrödinger equation in an adiabatic hyperspherical representation [32, 33] . For this, we use pairwise additive long range van der Waals potentials with a scattering length of −443a 0 [34] for 85 Rb and with a truncated number Z of L R = 0 bound states (Z = 9 for Fig. 2 ). The calculated total recombination rate constant at 0.8 µK (including thermal averaging) is L 3 = 3.07 × 10 −25 cm 6 /s and is consistent with the values found in Ref. [35] .

All data sets in Fig. 2 display the trend that the population of a molecular state due to three-body recombination typically decreases with increasing binding energy E b of the product state, which is consistent with the work for 87 Rb [8] . The overall agreement between theory and experiment in Fig. 2 is good, as the experimentally observed relative strengths of the signals for low L R are in general well reproduced by the calculated recombination rates. This suggests that our singlespin model fully captures the characteristics of the three-body chemical reaction in the given parameter regime, which can be viewed as additional evidence for the spin propensity rule.

Based on our theoretical analysis, we conclude that the spin propensity rule in our experiment is a consequence of the following scenario. a) The reaction takes place at interparticle distances where the interactions of particles a and b (forming the molecule) with particle c are nearly spin-independent. b) In the investigated range all possible product molecules have quantum states with good quantum numbers F, f a , f b . c) In the reaction region, the spin composition of the reacting pair a, b is essentially given by F f a f b = 422. As a consequence of conditions a), b) and c) the molecule formation is driven by mechanical forces from atom c while the spin state of the reacting pair is not affected.

To show that this scenario holds for our experiments, we first analyze the typical interparticle distances where the reaction occurs. Our numerical calculations [18] show that the formation of 85 Rb 2 molecules mainly takes place near a hyperradius R ≈ 1.5r vdW , extending from R ≈ 1.1r vdW to 2r vdW . Here, r vdW = (2µC 6 /h 2 ) 1/4 = 82a 0 denotes the van der Waals length for Rb, and µ and C 6 are the reduced mass and the van der Waals coefficient of the two-particle system, respectively. The hyperradius R describes the characteristic size of the three-body system and is given by

where r i is the location of particle i and d 2 = 2/ √ 3 [32] . The fact that the reactions occur at these large R can be understood within the framework of the adiabatic hyperspherical representation, where an effective repulsive barrier for the three-body entrance channel forms at a hyperradius of about R = 1.7r vdW [18, 36] .

We now consider the formation of a molecule state with a size < ∼ r hf ≈ 0.6r vdW . Only for such a state can spin components other than F f a f b = 422 be substantial (for more details see [18] ). Here, r hf = (C 6 /E hf ) 1/6 is the hyperfine radius and E hf = 3.04 GHz × h is the atomic hyperfine splitting. Given the hyperradius R > 1.1r vdW and the interparticle distance r ab ≡ | r a − r b | < r hf , the distances of particle c to the others must be r ca , r cb > 0.6r vdW . Since the interaction between two Rb atoms is essentially spin-independent for distances > ∼ 0.25r vdW [18] , this validates point a) of the scenario described above. Concerning point b), our coupled-channel calculations show that the weakly-bound molecular states up to a binding energy of about 50 GHz × h have almost pure spin states F f a f b . This can be explained by the fact that the 85 Rb triplet and singlet scattering lengths are large in magnitude. For 85 Rb, where these scattering lengths are of opposite sign, this leads to a near energetical degeneracy of the triplet vibrational levels v T with the singlet vibrational levels v S = v T − 1 [37, 38] . Since, in addition, the singlet and triplet vibrational wavefunctions are very similar at long range, the interaction between the two atoms is effectively spin-independent. As a consequence the atomic hyperfine interaction of each atom is essentially unperturbed, which leads to the atomic hyperfine structure with the quantum numbers f a , f b for the molecule.

Subsequent coupling of f a , f b in the molecule forms a total F. To show point c), we first note that due to the magnitude of R the three-body system can be effectively decomposed (with respect to spin) into a two-body collision of two f = 2, m f = −2 atoms (a and b) and a third atom (c) which is spin-wise only a spectator, see also [18] . During the two-body collision of atoms a and b, spin admixtures to the original 422 state can occur at close distance. However, these admixtures are only on the % level for 85 Rb even for short atomic distances r i j ∼ 0.25r vdW , and therefore negligible. That this admixture is small can be derived from the fact, that the twobody scattering wavefunction is very similar to the one of the corresponding last molecular bound state at short distance. The scattering state therefore shares the relatively pure F f a f b spin state character of the weakly-bound molecular states, as discussed before.

Our coupled-channel calculations using the potentials of [16] show that for molecular bound states with a binding energy larger than 50 GHz × h the spin decomposition changes, because the splitting of adjacent singlet and triplet vibrational levels becomes larger than the hyperfine splitting. This is shown in Fig.3(a) . In our experiment we only observe product states down to the v = −4 level in the 422 family, for which the norm of the 422 entrance channel remains between 0.99 and unity. All other unobserved spin families have a norm of the 422 channel much less than unity. In Fig. 3(b) we plot the ratio ρ of the triplet and singlet level splitting and the hyperfine splitting, ρ = |E T (v) − E S (v − 1)|/E hf . For ρ > 1 hyperfine mixing is increasingly suppressed and the propensity rule is expected to break down.

Similar arguments can be used to also explain the spin propensity rule for 87 Rb as observed in [8, 9] . Here, the singlet and triplet scattering lengths are almost equal, leading to a near degeneracy of the singlet and triplet vibrational states for v T = v S . Furthermore, the atomic hyperfine splitting in 87 Rb is larger by a factor of ≈ 2.4 than in 85 Rb. As a consequence the spin propensity rule holds for even larger binding energies of up to 100 GHz × h (see Fig S5 in [18] ).

In the future, it will be interesting to investigate the breakdown of the spin propensity rule for 85 Rb by studying product molecules with binding energies larger than ∼ 50 GHz × h. For these measurements the ability to state-selectively detect singlet and triplet product molecules as demonstrated here, will be essential. The spin propensity rule will also break down when F is not a good quantum number anymore, e.g., by applying strong magnetic fields. 85 Rb features a broad Feshbach resonance at 153.3 G [39] where spin-mixing in the incoming channel naturally becomes important [11, 12, 40] . Besides for Rb the spin-conservation propensity rule may hold for other elements. Cs, e.g., might be a good candidate when working in a regime where dipolar relaxation processes are negligible.

This work was financed by the Baden-Württemberg Stiftung through the Internationale Spitzenforschung program (contract BWST ISF2017-061) and by the German Research Foundation (DFG, Deutsche Forschungsgemeinschaft) within contract 399903135. J. P. D. also acknowledges partial support from the U.S. National Science Foundation 

The ultracold 85 Rb atoms are initially prepared in the spin state f = 2, m f = −2 of the electronic ground state. This state is stable in two-body collisions due to energetic closure of other atomic exit channels and due to small dipolar relaxation rates. Indeed, our coupled-channel calculations have verified that the two-body spin-relaxation rate constants for 85 Rb remain below 10 −14 cm 3 /s when the magnetic field is not near a Feshbach resonance. The scattering length for the collision of two f = 2, m f = −2 atoms is a = −460a 0 .

In our set-up the atoms are trapped in a far-detuned optical dipole trap with trapping frequencies ω x,y,z = 2π × (156, 148, 18) Hz. The optical dipole trap operates at a wavelength of about 1064 nm.

For REMPI we use a laser wavelength around 602.2 nm. The laser light is either provided by a cw dye laser (Matisse, Sirah Lasertechnik GmbH) or an optical parametric oscillator (C-Wave, Hübner GmbH). Each laser is stabilized to a cavity and has a short-term linewidth of less than 1 MHz. Longer term drifts are compensated by a lock to a wavelength meter and we have a shot-to-shot frequency stability on the order of ±5 MHz. The laser beam has a power of 100 mW and a beam waist (1/e 2 radius) of 1 mm at the location of the molecules. It has a mixture of σ -and π-polarization and can be essentially considered as unpolarized.

REMPI produces Rb + 2 molecules with binding energies larger than 479 cm −1 in the X 2 Σ + g state. We have no indication of a resonance structure for the ionization transition, in agreement with [24, 43] where the ionization range is referred to as a diffuse band.

Coincidences of predicted and observed lines in REMPI spectra allow for an assignment of the observed lines. The precision of coincidences was typically around 10 MHz, as mainly determined by slight drifts of the wavemeter.

We note that besides the triplet and singlet paths discussed in detail in this publication we also have tested a REMPI path via the A 1 Σ + u state (associated with the 5s + 5p asymptote) similar to the one described in [8] , however, the overall detection efficiencies are much lower in that case. Furthermore, the spectra of Figs. 1(b) and (c) and Fig. S1 are very clean in a sense that essentially no unidentified resonance signals are visible which represents a significant improvement as compared to [8] .

The (2) 3 Π g , 0 − g intermediate state Figure S1 shows the REMPI spectrum using the intermediate state (2) 3 Π g , 0 − g . For 0 − g the total parity is given by (−1) J +1 . Therefore, since the initial product molecules have positive total parity, only transitions towards intermediate states with even quantum number J are possible. Concerning photoassociation we now observe two lines, one for J = 0 and the other one for J = 2. The vertical lines are calculations for the expected REMPI signals, where we use the same rotational constant of B v = 389(2) MHz for the intermediate state as for (2) 3 Π g , 0 + g . The calculated resonance positions agree well with the measurements. As a consistency check for the assignment of signals in Fig. S1 we use that product states can give rise to up to three resonance lines corresponding to the transitions towards J = L R and J = |L R ± 2|. The data in Fig. S1 confirm this consistency.

For each molecular resonance line in the spectra of Fig. 1 and Fig. S1 we have measured the ion production rate in our experiment. For this, we tune the ionization laser onto a given resonance and turn it on for a short enough pulse time so that only a few ions (< 20) are produced. We are then in a regime where the ion number grows linearly with time (see inset of Fig. S2 for a typical measurement). Afterwards we count the number of ions by inserting them into a fresh atom cloud with 2.5 × 10 5 atoms. After an interaction time of 500 ms, during which the ions inflict atom loss due to (mostly) elastic atom-ion collisions [28] , we measure the remaining number of atoms with absorption imaging. Using the calibration curve in Fig. S2 we can convert the measured atom fraction into an ion number. Dividing the ion number by the pulse time we obtain the ion production rate.

The calibration curve was obtained as follows. Initially, the ion trap and the optical dipole trap are spatially separated from each other so that the atoms cannot collide with the ions. A known number of laser-cooled 138 Ba + ions in a range between 1 to about 40 is prepared in the Paul trap. The cold ions form an ion crystal in the trap and are counted with single-particle resolution after fluorescence imaging. In parallel, we prepare a 85 Rb atom cloud in the crossed optical dipole trap. Next, the centers of the ion and atom traps are overlaid, immersing the Ba + ions into the atom cloud. Subsequently, the Ba + ions can undergo reactions with the Rb atoms, such as, e.g., charge exchange [41] . After a long interaction time of 2s we are predominantly left with Rb + ions, as confirmed by mass spectrometry. Although these ions can have high kinetic energy they are still confined in the eV-deep Paul trap, i.e., the initial number of trapped ions is conserved. Afterwards, the ion trap and the optical dipole trap are separated again from each other, the old atom cloud is discarded and a new atom cloud (with 2.5×10 5 atoms) is prepared. The ions are immersed into the new atom cloud for 500 ms and atom-ion collisions inflict again atom loss. Figure S2 shows the remaining atom fraction N/N 0 as a function of initial ion number N i . The dashed line . Total three-body recombination rate L 3 (black solid line) and corresponding partial rates for the six most weakly-bound molecular states currently observed, (v, L R ) = (-1,2), (-1,0), (-2,6), (-2,4), (-2,2) and (-2,0), as well as (-3,0) and (-4,0), calculated with a twobody interaction model supporting 5 s-wave bound states. These results indicate that recombination is likely to occur at R ≈ 1.5r vdW , while it is increasingly suppressed for R < 1.5r vdW likely due to the presence of a repulsive barrier on the income collision channel (see text) and for R > 1.5r vdW due to the decrease of the hyperradial inelastic couplings with R [42].

is a fit with the empirically-found function N/N 0 = ( N i α + 1) β . Here, α and β are fit coefficients.

Our three-body calculations for 85 Rb atoms were performed using the adiabatic hyperspherical representation [32, 33] where the hyperradius R determines the overall size of the system, while all other degrees of freedom are represented by a set of hyperangles Ω. Within this framework, the three-body adiabatic potentials U α and channel functions Φ α are determined from the solutions of the hyperangular adiabatic equation:

which contains the hyperangular part of the kinetic energy, expressed through the grand-angular momentum operator Λ 2 and the three-body reduced mass µ = m/ √ 3. To calculate the three-body recombination rate we solve the hyperradial Schrödinger equation [33] ,

where α is an index that labels all necessary quantum numbers to characterize each channel, and E is the total energy. From Eq.

(2) we determine the S-matrix and the recombination rate L 3 [33] .

In this present study, the interaction between 85 Rb atoms is modeled by a potential similar to that used in Ref. [8] , and given by a modified Lenard-Jones potential,

where C 6 = 4710.431E h a 6 0 , C 8 = 576722.7E h a 8 0 , and C 10 = 75916271E h a 10 0 , are the van der Waals dispersion coefficients from Ref. [16] . Here, f λ (r) = tanh(2r/λ ) is a cut-off function that suppresses the divergence of the C 8 and C 10 interaction terms for vanishing interatomic distance r. We can adjust the value of λ to have different numbers of diatomic bound states supported by the interaction, while still reproducing the value of the background scattering length a bg = −443a 0 [34] . Our calculations were performed using λ = 20.72577414766491a 0 , producing 9 s-wave (L R = 0) bound states, and a total of 101 bound states including higher partial-wave states, L R > 0. Our numerical calculations for three-body recombination through the solutions of Eq. (2) have included up to 200 hyperspherical channels leading to a total rate converged within a few percent. The calculated total recombination rate constant at 0.8 µK (including thermal averaging) is L 3 = 3.07 × 10 −25 cm 6 /s and is consistent with the values found in Ref. [35] .

In order to gain more insight on how three-body recombination occurs within the hyperspherical representation we introduced an artificial hyperradial hard-wall at R = R hw and performed recombination calculations for various values of R hw . The hard-wall prevents any flux into the range R < R hw and therefore reactions in this range are not possible. By analyzing how the total and partial reaction rates change as a function of R hw one can infer at what hyperradii R reactions dominantly FIG. S4. Hyperangular probability density, |Φ α (R; θ , ϕ)| 2 sin(2θ ) [36] , at R = 1.593r vdW for the initial collision channel (a) and the molecular product state (v, L R ) = (−4, 0) (b). A point on the θ -ϕ hyperangular plane specifies the geometry of the three-atom system. The regions within the circles marked by dashed lines, correspond to geometries for which two of the atoms are found at distances r i j ≤ r vdW and r i j ≤ r hf , respectively, while the third atom is further away.

occur. Figure S3 shows the total rate L 3 (black solid line) and the partial rates for some of the currently observed molecular states, (v, L R ) = (-1,2), (-1,0), (-2,6), (-2,4), (-2,2) and (-2,0). This particular calculation was performed with a two-body interaction model (Eq. 3) supporting 5 s-wave bound states in order to reduce some numerical instabilities associated with the hard-wall interaction. The figure clearly indicates that recombination is most likely to occur at R ≈ 1.5r vdW . For R hw < 1.5r vdW the partial rates are affected by interference effects related to the different collision pathways three atoms can follow to form a molecule [32] . Interference effects can be seen as the fast oscillations on the partial rates in Fig. S3 . For R > 1.5r vdW partial rates are increasingly suppressed due to the decrease of the inelastic couplings with R [42]. The peaking of the reaction rates at around R ≈ 1.5 r vdW in Fig. S3 can be explained as a consequence of a repulsive barrier in the hyperspherical effective potential of the entrance channel at R ≈ 1.7r vdW . Such a barrier was first identified in Ref. [36] where it occurred at R ≈ 2r vdW for a = ±∞. We now analyze the geometric characteristics of the threeatom system near the region where recombination is most likely to occur in order to identify which type of coupling is dominating the inelastic transitions. Figure S4 shows the hyperangular probability density at R = 1.593r vdW in terms of the hyperspherical channel functions Φ α and hyperangles θ and ϕ as: |Φ α (R; θ , ϕ)| 2 sin(2θ ) (see Ref. [36] ). Each point in the θ -ϕ hyperangular plane corresponds to a specific geometry of the three-atom system. In particular, the regions within the dashed half-circles represent geometries where two of the atoms are found at a distance r ab ≤ r vdW and r ab ≤ r hf , respectively, and the third atom remains at distances r ca , r cb > r hf (i.e., r ci ≥ 0.71r vdW and r ci ≥ 1.2r vdW , respectively). In Fig. S4(a) we show the probability density for the initial collision channel, which indicates that the most likely configuration is where all three atoms are located at distances typically larger than r hf . In contrast, the probability density for the target molecular states is pronounced inside the small circle. As an example, Fig. S4(b) shows the probability density for (v, L R ) = (−4, 0). In order to produce this state, the two atoms forming the bound state obviously need to be closer than r hf and the third atom is at interparticle distances r ci > r hf . For a spin flip to happen during the reaction, the two atoms forming the molecule must be at a distance r ab < r hf and the wavefunctions of either the two-body scattering state or the two-body target bound state must have a sizeable spin admixture. In the range of binding energies discussed here for 85 Rb these admixtures are on the %-level and therefore quite small. The admixtures are not shown in Fig. S4 .

The interaction between two Rb atoms is effectively spin-independent for interparticle distances r i j ≡ | r i − r j | > ∼ 0.25r vdW . This is because at these distances the exchange splitting between the electronic singlet and triplet ground state potentials is smaller than the atomic hyperfine splitting E hf = 3.04 GHz × h. As a consequence the atomic hyperfine states are only weakly perturbed by the exchange interaction and therefore essentially remain eigenstates in this realm. In other words the exchange interaction is not strong enough to decouple the atomic hyperfine spins and flip them.

Next, we consider the low-energy collision of two f = 2, m f = −2 Rb atoms, corresponding to the channel F f a f b = 422. At vanishing magnetic field and neglecting dipolar relaxation processes, F and m F are conserved throughout the collision. At a close enough distance < 0.25r vdW spin exchange interaction, however, can admix the spin states F f a f b = 423, 433 (For an overview of all spin channels of two colliding ground state 85 Rb atoms see [44] .). Interestingly, this admixture is restricted to distances r ab < r hf ≈ 0.6r vdW due to energetic closure of the f a f b = 23, 33 scattering channels at larger distances. For m F = −4, the channel F f a f b = 422 is the lowest in energy. The hyperfine distance r hf = (C 6 /E hf ) 1/6 is the distance at which the potential curve of channel f a f b = 23 crosses the collision energy (which approximately corresponds to the asymptotic energy of the 422 channel). This crossing is also relevant in another regard. It effectively limits the size of molecular bound states of the spin channels f a f b = 23, 33 which are energetically located below the F f a f b = 422 entrance channel to be < ∼ r hf ≈ 0.6r vdW . We now consider a three-body collision of three f = 2, m f = −2 atoms and assume the hyperradius to be R > 1.1r vdW . Using the expression R 2 = ( r b − r a ) 2 /d 2 + d 2 ( r c − ( r a + r b )/2) 2 , where r i is the location of particle i and d 2 = 2/ √ 3 [32] , we calculate that for R > 1.1r vdW at least one of the three atoms, say c, has distances r ca , r cb > 0.6r vdW from both other atoms. This is outside the distance range r ca , r cb < r hf ≈ 0.6r vdW for atom c to exhibit a spin admixture. There-fore, atom c should have a pure spin state f = 2, m f = −2, even if it had possibly undergone spin exchange interaction in a previous collision with one of the other two atoms. Reciprocally, we can conclude that the two other colliding atoms a and b must be in the collision channel F f a f b = 422. Thus, at a hyperradius R > 1.1r vdW the three-body system can be effectively decomposed (with respect to spin) into a two-body collision of two f = 2, m f = −2 atoms and a third atom which is spin-wise only a spectator.

Spin composition of 87 Rb 2 molecules Figure S5 shows the binding energy and the spin composition of weakly-bound 87 Rb 2 molecules. For 87 Rb 2 the ratio ρ is given by ρ = |E T (v) − E S (v)|/E hf . It becomes > 1 for binding energies E b > ∼ 100 GHz × h. Around ρ = 1 a major change in the spin composition of the spin families occurs.

In the spectra of Fig. 1 and Fig. S1 we showed that our observed lines match with the calculated positions of molecular states with F f a f b = 422. For the sake of completeness we show here with a few examples, that our data do not match with the calculated spectra of other spin families. Figure S6 (a) shows a section of the spectrum in Fig. 1(b) together with calculated positions for product molecules with f a f b = 23 and 33. The positions are indicated by vertical red and purple lines, respectively. Each of the bound states with different F (from 0 to 6) is plotted separately in vertical direction. Clearly, there is no convincing match between these and the experimental lines within the detection limit. The observed lines only match calculated resonance frequency positions for molecules with F f a f b = 422. In Fig. S6 (b) we zoom in further into the spectrum shown in Fig. S6(a) and check for molecules with F f a f b = 022 and F f a f b = 222. Again, the observed signals only match in a convincing way for F f a f b = 422.

reactions of triplet alkali-metal dimers with alkalimetal atoms

Nuclear spin conservation enables state-to-state control of ultracold molecular reactions

Precision test of statistical dynamics with stateto-state ultracold chemistry

Stateto-state chemistry for three-body recombination in an ultracold rubidium gas

Hyperfine magnetic substate resolved state-to-state chemistry

Kokkelmans, Strong spin-exchange recombination of three weakly in

Precision test of the limits to universality in few-body physics

Observation of Efimov universality across a nonuniversal Feshbach resonance in 39 K

Impact of the 87 Rb singlet scattering length on suppressing inelastic collisions

Collisional stability of double Bose condensates

Verhaar, Role of collisions in creation of overlapping Bose condensates

Hyperfine, rotational, and vibrational structure of the a 3 Σ + u state of 87 Rb

While we do not resolve the molecular m F quantum number in the present work, the conservation of m F should hold judging from our previous work on

See Supplementary Material at ..., which includes Refs. [41-44] for further information about the experimental setup and methodology, the theoretical model, additional REMPI data, as well as aspects and tests of the spin propensity rule

Spectroscopy of states with ion-pair character near Rb(5s)+Rb(4d), Doctoral Dissertation

Upper bound to the ionization energy of 85 Rb 2

Detection by two-photon ionization and magnetic trapping of cold Rb 2 triplet state molecules

Here, v is counted downwards, starting with −1 for the last s-wave bound state just below the atom pair asymptote | f = 2, m f = −2 + | f = 2

Formation, detection and spectroscopy of ultracold Rb 2 in the X 1 Σ + g state

Cold rubidium molecules formed in a magnetooptical trap

Photodissociation of trapped Rb + 2 : Implications for simultaneous trapping of atoms and molecular ions

Observation of ultracold ground-state heteronuclear molecules

Minimization of ion micromotion using ultracold atomic probes

Population distribution of product states following three-body recombination in an ultracold atomic gas

Ultracold photoassociation spectroscopy: Long-range molecules and atomic scattering

Updated potential energy function of the Rb 2 a 3 Σ + u state in the attractive and repulsive regions determined from its joint analysis with the 2 3 Π 0g state

We note that because of these uncertainties and uncertainties in the atom number calibration we refrain from converting the measured ion production rates into three-body recombination rate constants for each product channel

Few-body physics in resonantly interacting ultracold quantum gases

Numerical study of three-body recombination for systems with many bound states

Very-highprecision bound-state spectroscopy near a 85 Rb Feshbach resonance

Magnetic field dependence of ultracold inelastic collisions near a Feshbach resonance

Origin of the three-body parameter universality in Efimov physics

Ultracold molecules from ultracold atoms: a case study with the KRb molecule

Molecular states near a collision threshold

Cornish, Feshbach resonances in ultracold 85 Rb

Kokkelmans, Multichannel nature of three-body recombination for ultracold 39 K