Main

A chemical reaction in a low-density gas phase is typically described well by a fully coherent quantum mechanical evolution. Therefore, such a gas is a promising test bed for quantum control of chemical processes. In fact, recent platforms based on ensembles of ultracold atoms or molecules have paved the way for extended quantum mechanical steering of reactions. Demonstrated control schemes include the use of photoassociation1,2,3, Feshbach resonances4,5,6,7,8,9, microwave-engineered collisions10,11,12,13,14, electric-field-controlled reactions15, relative positioning of traps16,17,18,19, confinement-induced effects20,21,22,23,24,25 and quantum interference26,27 and can rely on propensity rules and conservation laws28,29. This progress has been further promoted by emerging technologies that enable state-to-state measurements (for example, refs. 30,31,32,33,34).

A prominent tool for controlling chemical reactions is a tunable Feshbach resonance. A Feshbach resonance in atomic gases occurs when the energy of the scattering state of two colliding atoms is tuned into degeneracy with that of a molecular state, leading to the mixing of two such states8. As they offer unique control over the interparticle interaction, tunable Feshbach resonances have been essential for the development of the ultracold-quantum-gas platform. An established application of Feshbach resonances for chemical reactions is the controlled production of ultracold molecules. By magnetically ramping over a Feshbach resonance, ultracold pairs of atoms can be converted into an extremely weakly bound molecule, the Feshbach molecule4,5,8,35,36,37. In three-body recombination where three free atoms collide to form a diatomic molecule, Feshbach resonances have been used to tune the total molecular production rate and, specifically, to suppress atom loss38,39,40,41, and to demonstrate the Efimov effect40,42,43. Feshbach resonances and resonant scattering have also been proposed for controlling complex few-body reactions. See, for example, refs. 44,45.

Here we demonstrate the use of a Feshbach resonance in a three-body recombination process to steer the reaction flux between families of molecular product channels with different spin states. More specifically, by tuning the magnetic field towards a Feshbach resonance, we can gradually increase the initially negligible reaction rate into a specific spin channel, so that it becomes close to the total rate into all channels. The process is coherent, and by using the applied magnetic field as a precisely tunable control knob, it enables the state-selective control of molecular production rates.

Our experiments were carried out with an 860-nK-cold cloud of about 2.5 × 105 85Rb atoms. Each atom i was in the hyperfine state \((\,{f}_{i},{m}_{{f}_{i}})=(2,-2)\) of the electronic ground state. The atoms were confined in a far-detuned crossed optical dipole trap. For more details, see Methods and ref. 46. In the atom cloud, three-body recombination spontaneously occurred, predominantly producing weakly bound molecules in states of the coupled molecular complex \({\rm{X}}^{1}{\Sigma }_\mathrm{g}^{+}\!-{\rm{a}}^{3}{\Sigma }_\mathrm{u}^{+}\). By tuning the magnetic field B in the vicinity of the s-wave Feshbach resonance at B = 155 G (refs. 47,48), we could control the product distribution of the molecules. For details of the Feshbach resonance, see Supplementary Section 1. The molecules were state-selectively detected with resonance-enhanced multiphoton ionization (REMPI). See Methods for details.

Our scheme for controlling the reaction flux into different spin channels is illustrated in Fig. 1. In the three-body recombination process, the Rb atoms a, b and c collide and the pair (a, b) forms a molecule (Fig. 1a). In the particular reactions we study here, the third atom c is far enough away, so that it interacts with the atoms a and b merely mechanically and no spin flip between c and the pair (a, b) occurs. For this regime, the following conditions need to be met29. First, the molecular state must be bound weakly enough so that the molecular size is not much smaller than the van der Waals length. Second, spin admixtures due to a spin-exchange interaction in the two-atom collision wavefunction need to be negligible for atomic distances larger than about one half of the van der Waals length. Note that these conditions are fulfilled well for the weakly bound states of Rb considered here, but not, for example, for Li (ref. 49).

Fig. 1: Scheme for controlling the reaction flux into different spin channels using a two-body Feshbach resonance.
figure 1

a, Atoms a, b and c undergo three-body recombination, where (a, b) form a molecule. During this process, atom c is outside the ranges (cyan areas) for spin-exchange interactions with the other atoms. b, Schematic representation of the Born–Oppenheimer potential energy curves for the atom pair (a, b). At close distances, the incoming scattering state (1) with spin \(\left\vert \uparrow \right\rangle\) experiences admixing with the bound state (2), which has spin \(\left\vert \downarrow \right\rangle\). Upon collision with the third atom c (not shown here), the scattering state can then relax into molecular bound states (3) or (4), with their respective spin states \(\left\vert \downarrow \right\rangle\) and \(\left\vert \uparrow \right\rangle\).

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Therefore, for the present work, the spin physics aspects of the reaction can be understood to a large extent in a two-body picture, in which atom a collides with atom b (Fig. 1b). At large internuclear distances, the (a, b) scattering state has the hyperfine spin quantum numbers (F, fa, fb, mF) = (4, 2, 2, −4), where F denotes the total angular momentum of the molecule excluding rotation, and \({m}_{F}={m}_{{f}_{\mathrm{a}}}+{m}_{{f}_{\mathrm{b}}}\) represents its projection. We denote this spin state by \(\left\vert \uparrow \right\rangle\). At short internuclear distances, the scattering state couples to an energetically nearby molecular bound level, giving rise to the Feshbach resonance. This level has the spin state (F, fa, fb, mF) = (4, 3, 3, −4), which we denote by \(\left\vert \downarrow \right\rangle\). Furthermore, the level has rotational angular momentum LR = 0, and its vibrational quantum number is v = −3, counting down from the fa = fb = 3 atomic threshold, starting with v = −1 for the most weakly bound state. (The vibrational levels for \(\left\vert \uparrow \right\rangle\) states are counted analogously, however, starting from the fa = fb = 2 atomic threshold.)

The coupling leads to an admixture of the \(\left\vert \downarrow \right\rangle\) state with the initial scattering state with spin \(\left\vert \uparrow \right\rangle\). The strength of this admixture can be magnetically controlled. Next, in the mechanical collision with atom c, the scattering state of (a, b) can transition into a molecular bound state. Due to conservation of angular momentum, the spin state of the newly formed molecule must, however, overlap either with the spin state \(\left\vert \uparrow \right\rangle\) or with \(\left\vert \downarrow \right\rangle\). This is closely linked to a spin-conservation propensity rule that was observed in our recent work29, which stated that the three-body recombination process for Rb conserves the hyperfine spin state of the atom pair forming the molecule. In fact, by tuning the \(\left\vert \downarrow \right\rangle\) admixture of the (a, b) scattering state, we can control the reaction flux into molecular product channels with spin \(\left\vert \downarrow \right\rangle\).

We now demonstrate this control scheme experimentally. Figure 2 presents REMPI spectra in a selected frequency range. It shows signals from three different molecular product states. The spectra have been taken within a range of magnetic fields B between 4.6 and 159.3 G. (Note the non-uniform step size of B.) ν represents the REMPI laser frequency, and ν0 is the reference frequency (Methods). Each dip in a trace for a chosen B field corresponds to a state-specific product molecule signal. Coloured diamonds mark the known resonance positions predicted from coupled-channel calculations.

Fig. 2: Observation of \(\left\vert \uparrow \right\rangle\) and \(\left\vert \downarrow \right\rangle\) molecules.
figure 2

REMPI spectra as a function of the REMPI laser frequency ν for various magnetic fields B. Here, ν0 = 497,603.591 GHz. Each dip in a trace corresponds to a signal from a distinct molecular level. The REMPI signals are normalized, ranging from 0 to 1, as indicated by the vertical bar. The bar is valid for all data traces. Diamonds mark the theoretical positions of possible molecular signals, and their colours indicate the spin state as well as the vibrational and rotational levels (v, LR). The faint colour bands connecting the diamonds are guides for the eye. Note that the binding energy of the \(\left\vert \uparrow \right\rangle\) level is smaller than that of the two \(\left\vert \downarrow \right\rangle\) levels. In the spectra shown, however, the signal for \(\left\vert \uparrow \right\rangle\) is at higher frequency ν, as the intermediate rotational level for the REMPI is different. See also Methods.

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The molecular levels are labelled by their spin states (\(\left\vert \uparrow \right\rangle\) or \(\left\vert \downarrow \right\rangle\)) and by their vibrational (v) and rotational (LR) quantum numbers48. Figure 2 shows one molecular state with spin \(\left\vert \uparrow \right\rangle\) and two molecular states with spin \(\left\vert \downarrow \right\rangle\). The resonance positions changed in a characteristic way with the B field due to the Zeeman effect. We used this as fingerprint information for identifying the molecular levels. The strength of each signal roughly reflected the recombination rate towards each respective state. In the magnetic field range up to about 120 G, each REMPI spectrum exhibited only a single resonance dip, which can be unambiguously assigned to the state \(\left\vert \uparrow \right\rangle\). At about 120 G, two more molecular signals started to appear. These stemmed from molecular states in the spin state \(\left\vert \downarrow \right\rangle\). The strengths of the signals of these states became close to the \(\left\vert \uparrow \right\rangle\) signals on approaching the Feshbach resonance at 155 G. For magnetic fields above the Feshbach resonance, all signals decreased very quickly within a few gauss.

We extracted from our REMPI spectra molecule detection rates for each observed molecular state. These rates were roughly proportional to the partial three-body recombination rates for the flux into individual product channels46 (Methods). The rates obtained for the states in Fig. 2 are shown in Fig. 3a for the magnetic field region in the vicinity of the Efimov and Feshbach resonances, which are at 140 and 155 G, respectively.

Fig. 3: Opening a product spin channel.
figure 3

a, Molecule detection rates for three product states for which the quantum numbers \(\left\vert \uparrow /\downarrow \right\rangle ({v},{L}_\mathrm{R})\) are given in the legend. The grey dashed line marks the experimental detection limit and the light green and purple shaded areas indicate the positions of the Efimov and Feshbach resonances, respectively. Each data point represents the mean value of ten repetitions of the experiment, and the error bars in the plot indicate one standard deviation (1σ). b, Calculated three-body recombination rate coefficients. The black dashed line is the total rate coefficient L3,tot. Solid coloured lines correspond to partial rates L3 for the states under discussion. Grey lines correspond to other molecular states. No calculations are shown for 152 G ≤ B ≤ 157 G (see text). We expect theoretical errors up to a few tens of percent for the partial rate for vibrational levels down to v = −4, judging from when more vibrational states are included in our effective potentials46. c, The normalized reaction rate coefficients L3/L3,tot do not exhibit a maximum at the Efimov resonance.

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The data show that the rates for the \(\left\vert \downarrow \right\rangle\) states indeed strongly increased from below the detection limit (grey dashed line) to about a factor of 50 above the detection limit as the magnetic field was increased from B < 115 G towards the Efimov resonance. The detection limit was mainly determined by the background noise of our REMPI scheme. By contrast, the rate for the \(\left\vert \uparrow \right\rangle\) state was rather constant for all B fields below the Efimov resonance. At the position of the Efimov resonance at 140 G, we observed a clear enhancement of the rates for all molecular states. In fact, the signals for all three states attained similar strengths, which demonstrates the large relative tuning range of our scheme. At this point, the spin product channel \(\left\vert \downarrow \right\rangle\) has been fully opened up for the reaction flux.

The relatively constant production rate for the \(\left\vert \uparrow \right\rangle\) state below the Efimov resonance might be unexpected at first in view of the known a4 scaling of the recombination rate in the limit of zero temperature6,50, where a is the scattering length. It can, however, be explained to a large extent as an effect of our finite temperature of 860 nK (refs. 51,52), as further discussed below.

We numerically calculated the partial rate coefficients L3 for each molecular quantum state using the adiabatic hyperspherical representation39,53,54 (Methods). This determined the partial recombination rate, L3(f) × ∫n3 d3r/3, into a molecular state f, where n is the atomic density distribution. The calculations took into account thermal averaging of the partial rate constants. The results are shown in Fig. 3b. The region from 152 to 157 G (the direct vicinity of the Feshbach resonance) was excluded as the numerical calculations quickly become computationally highly demanding in the resonant regime where the scattering length diverges. This was because for the scattering calculations we needed to calculate the three-body potentials and diabatic couplings up to a hyperradius at least an order of magnitude larger than the scattering length, all at a high resolution in length.

Among all the possible molecular states produced by recombination (grey dotted lines), we highlight in colour the molecular states under discussion. In addition, we present the total three-body recombination rate coefficient (dashed black lines).

Our calculations show that due to thermal averaging, the calculated recombination rate coefficient for the probed \(\left\vert \uparrow \right\rangle\) state increases only moderately towards the Efimov resonance. For more details on how finite temperature affects the recombination rate coefficient, see Supplementary Section 2. However, the theoretical curves still increase more quickly than the experimental data. This may mainly be explained by imperfections in the experiment. Already, during the B-field ramp, atoms were lost due to three-body recombination and the sample slightly heated up. As a consequence, the density of the atom cloud sank. In addition, the B-field ramp was not perfect but tended to lag behind and to overshoot, which could have averaged out the signals. Furthermore, there could have been a small variation in the REMPI efficiency as a function of magnetic field. These variations hamper a direct comparison between ion rates and L3 coefficients.

Nevertheless, the main characteristics of the experimental data are qualitatively well described. For example, the observed sharp drop of the recombination rate above the Feshbach resonance was clearly reproduced by the theory. The reason for this drop was the rapid decrease of the scattering length towards its zero crossing near B = 166 G and the close-by minimum in L3 due to Efimov physics39,55. Note that the Feshbach molecular state appears above the Feshbach resonance (see the dark green solid line in Fig. 3b) and accounts for the main fraction of the total reaction flux.

Our calculations show that the effect of the Efimov resonance was to increase the partial three-body recombination rate coefficients with the same overall factor but not favouring any particular product channels. This is evident from Fig. 3b, which shows that all partial rate coefficients exhibited a similar maximum at the location of the Efimov resonance. It was also manifest when normalizing the partial rate coefficients to the total rate coefficient (Fig. 3c), as each maximum at the Efimov resonance disappeared. The global enhancement occurs because the Efimov resonance is a shape resonance in a single three-body adiabatic channel. As such, approaching the resonance increases the overall amplitude of the three-body scattering wavefunction at short distances where the reaction takes place, therefore, enhancing all the partial rates by the same factor43.

Notably, near the Feshbach resonance, we also found, both experimentally and theoretically, molecular products in spin states other than \(\left\vert \uparrow \right\rangle\) and \(\left\vert \downarrow \right\rangle\), as shown in Fig. 4. This points towards physics beyond the \(\left\vert \uparrow \right\rangle\) and \(\left\vert \downarrow \right\rangle\) Feshbach mixing. In the experiments, these were molecular products with spins (F, fa, fb) = (4, 3, 2) and (5, 3, 2). In this notation, we omit mF, as it is always mF = −4. Thus, we observed product states in which only one of the fi has flipped and in which even the total angular momentum F could change. Following a similar analysis as used in refs. 29,46, the observation of such states can be understood as follows in terms of two-body physics. The spin state (4, 3, 2) can be produced by a two-body spin-exchange interaction at short distances, starting either from state \(\left\vert \uparrow \right\rangle \equiv (4,2,2)\) or from \(\left\vert \downarrow \right\rangle \equiv (4,3,3)\). Close to the Feshbach resonance, the amplitude of the scattering wavefunction is strongly enhanced at short range and with it also the rate for spin-exchange. Producing the spin state (5, 3, 2) is possible due to the presence of a finite B field, which breaks the global rotational symmetry and couples different F quantum numbers. A (5, 3, 2) state is typically energetically close to a corresponding (4, 3, 2) state, such that coupling between them is resonantly enhanced.

Fig. 4: Spin families of molecular products.
figure 4

a,b, Signals of product molecules in various spin families detected by REMPI spectroscopy at a magnetic field of B = 4.6 G (a) and B = 155 G (b). Each data point in a and b represents the mean value of 4 and 3 repetitions of the experiment, respectively. The error bars indicate one standard deviation (1σ). Vertical lines correspond to the calculated resonance positions for molecular states assigned to spin families (F, fa, fb) according to the legend on top of the figure. The marked individual states have vibrational quantum numbers in the range from v = −1 to −7 and rotational quantum numbers LR = 0, 2, 4 or 6. The REMPI path was through a 3Πg intermediate state (Methods). We have the frequency reference νB = ν0 = 497,831.928 GHz for a and νB = ν0 − 228 MHz = 497,831.700 GHz for b. The 228 MHz shift compensated for the Zeeman shift to give a better comparison of the two spectra. c, Calculated summation of the molecular product fraction for each spin family (F, fa, fb), as indicated next to the curves.

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Figure 4a,b compares REMPI spectra for magnetic fields B = 4.6 and 155 G, respectively. The spectrum at high magnetic field exhibits many more resonance lines than the spectrum at low field. In a thorough analysis of the spectra, similarly as in ref. 30, we identified a total of four spin families for B = 155 G and only a single one for B = 4.6 G. The individual spin states are marked with coloured bars in Fig. 4a,b.

Figure 4c shows our numerical calculations for the product fractions of the molecules in the different spin families (F, fa, fb) as a function of the external magnetic field B. For each individual family, we summed the populations for all corresponding molecular states having the same spin characteristics. In agreement with our previous discussion, spin-exchange was strongly enhanced on approaching the Feshbach resonance. Furthermore, Fig. 4c reveals a hierarchy in the propensity for the production of spin states. Changing the total angular momentum F was more strongly suppressed than changing an atomic f quantum number. See also ref. 56.

In summary, we have demonstrated a powerful scheme for controlling the reaction pathway in a three-body recombination process of ultracold atoms. Using a magnetically tunable Feshbach resonance, we admixed a well-defined spin state with the reaction complex of three atoms and, thus, steered the reaction flux between the corresponding spin channels. We found that a large fraction of the total reaction flux can be redirected in this way. Furthermore, we showed that in contrast to the Feshbach resonance, an Efimov resonance enhances globally the reaction rate and maintains the relative flux between reaction channels. We investigated our control scheme both experimentally and theoretically using high-resolution state-to-state measurements and state-of-the-art numerical three-body scattering calculations, respectively.

The demonstrated reaction control holds large promise for general few-body reactions, as it is simple and can easily be extended. Feshbach resonances are ubiquitous in cold atomic and molecular gases. The scheme is fully coherent and can thus be used as a central building block in interferometric control, where the Feshbach resonance functions as a beam splitter for the incoming wavefunction. The split parts can then potentially follow different pathways towards the same final product state where they interfere. For example, the final product state could have a tunable spin-mixed character, which can be set by further control methods, such as state dressing with optical or microwave fields. In this way, further tuning of the interference can be achieved.

Methods

Preparation of an ultracold atomic sample

The experimental sequence started with capturing 85Rb atoms in a magneto-optical trap. After a magnetic transport over 40 cm, the atoms were subsequently loaded into an optical dipole trap, where they were evaporatively cooled. They were then transported to the centre of a Paul trap by a moving one-dimensional optical lattice. In the final stage of sample preparation, the atoms were confined in a far-detuned crossed-dipole trap formed by 1,064 nm lasers. The trapping frequencies were ωx,y,z = 2π × (156, 148, 18) Hz. The resulting atom cloud consisted of a pure sample in the \(({f}_{i},{m}_{{f}_{i}})=(2,-2)\) hyperfine spin state with typically 2.5 × 105 particles. The temperature of the atoms was 860 nK. This temperature was chosen as it provided the strongest recombination signals at a reasonably cold temperature.

REMPI detection

To state-selectively detect the product molecules, we applied REMPI with a continuous-wave laser with a linewidth of ~1 MHz. The laser beam was roughly an equal mixture of σ- and π-polarized light. It had a power of 100 mW and a beam waist (1/e2 radius) of 0.1 mm at the location of the atom cloud. We used identical photons for the two REMPI steps at a wavelength of around 602 nm. For Figs. 2 and 3, the intermediate REMPI states were levels of \({(2)}^{1}{\Sigma }_\mathrm{u}^{+}\) with \({J}^{{\prime} }=3\) for \(\left\vert \uparrow \right\rangle\) states and \({J}^{{\prime} }=1\) for \(\left\vert \downarrow \right\rangle\) states29, where \({J}^{{\prime} }\) represents the total angular momentum excluding nuclear spin. The \({J}^{{\prime} }=1\) and \({J}^{{\prime} }=3\) levels were split by 2.9 GHz. The photoassociation laser frequency towards the intermediate level \({J}^{{\prime} }=1\) was ν0 = 497,603.591 GHz at B = 4.6 G. The binding energies of the experimentally observed molecular states in Figs. 2 and 3 were 4.7 GHz × h for \(\left\vert \uparrow \right\rangle\) and spanned a range between 6.4 and 7.3 GHz × h for \(\left\vert \downarrow \right\rangle\), where h is the Planck constant. The binding energy was determined relative to the B-field-dependent (4,2,2) threshold. For Fig. 4, the intermediate REMPI states were deeply bound levels of \({(2)}^{3}\Pi \,{0}_\mathrm{g}^{+}\) with \({J}^{{\prime} }=1,3\) or 5 (ref. 29). Here, ν0 = 497,831.928 GHz is the photoassociation frequency towards \({J}^{{\prime} }=1\) at B = 4.6 G. The binding energies of the molecular states observed in Fig. 4a,b span a range between 0.6 and 12.6 GHz × h. Again, the binding energy was determined relative to the (4,2,2) threshold. In general, the Zeeman effects for our intermediate states were negligible compared to those for the ground state. We made an effort to ensure that the REMPI efficiencies were similar for the states that we probed, including at various magnetic fields, but we have not yet precisely calibrated the REMPI efficiency. Note that the first REMPI step was generally not saturated.

When ions were produced by REMPI, they were directly trapped and detected in an electronvolt-deep Paul trap that was centred on the atom cloud. Elastic atom–ion collisions inflicted a telltale atom loss while the ions remained trapped. From the atom loss, which was measured by absorption imaging of the atom cloud, we inferred the number of ions. For details, see ref. 29. From the number of ions and the interaction time, we obtained an ion production rate (the molecular detection rate) that was generally proportional to the state-selective molecular production rate and the three-body recombination loss rate constant.

Model calculations

Our numerical simulations used the adiabatic hyperspherical representation approach in which the coordinates of three particles are given in terms of the hyperradius R for the overall size of the system and a set of hyperangles Ω for the internal motion39,53,54,57. The three-body Schrödinger equation was solved by adiabatically separating the hyperradial motion:

$$\left[-\frac{{\hslash }^{2}}{2\mu }\frac{\mathrm{d}^{2}}{\mathrm{d}{R}^{2}}+{U}_{\nu }(R)\right]{F}_{\nu }(R)+\sum\limits_{{\nu }^{{\prime} }}{W}_{\nu {\nu }^{{\prime} }}(R){F}_{{\nu }^{{\prime} }}(R)=E{F}_{\nu }(R),$$
(1)

from the internal motion:

$${\hat{H}}_{{\rm{ad}}}{\varPhi }_{\nu }(R;\varOmega )={U}_{\nu }(R){\varPhi }_{\nu }(R;\varOmega ),$$
(2)

where the hyperradius R appears only as a parameter. Fν is the three-body hyperradial wavefunction, ν is the index of the three-body channel and E is the energy of the system. The diagonalization of the hyperangular adiabatic Hamiltonian \({\hat{H}}_{{\rm{ad}}}\) gives the three-body potentials Uν and the channel functions Φν, which were also used to compute the non-adiabatic couplings \({W}_{\nu {\nu }^{{\prime} }}\) for the hyperradial equation.

In our model, the hyperangular adiabatic Hamiltonian reads

$${\hat{H}}_{{\rm{ad}}}=\frac{\hat{\Lambda}^{2}(\varOmega )+15/4}{2\mu {R}^{2}}{\hslash }^{2}+\mathop{\sum}\limits_{\begin{array}{c}i,j={\mathrm{a,b,c}}\\ i\ne j\end{array}}{\hat{V}}_{ij}(R,\varOmega )+\mathop{\sum}\limits_{ i=\mathrm{a,b,c}}{\hat{H}}_{i}^{{\rm{sp}}}(B),$$
(3)

where \(\hat{\Lambda }\) denotes the hyperangular momentum operator53,57 and \(\mu =m/\sqrt{3}\) is the reduced mass of three identical atoms of mass m. The atomic spin Hamiltonian \({\hat{H}}_{i}^{{\rm{sp}}}\) for atom i contains the hyperfine and Zeeman interactions, and to a very good approximation within the present work, its eigenstates are \(\left\vert\,{f}_{i},{m}_{{f}_{i}}\right\rangle\). For two Rb atoms (for example, i and j) of the 5S1/2 + 5S1/2 asymptote, the pairwise interaction \({\hat{V}}_{ij}\) can be expressed in terms of the electronic singlet and triplet Born–Oppenheimer potentials. We used the potentials from ref. 58 with another repulsive term \(C/{r}_{ij}^{12}\) to reduce the number of bound states in our simulation. Here, rij is the interatomic distance. Removing deeply bound states mitigated the computational hardship without affecting the results too much, as generally, more deeply bound states play a less important role in the three-body recombination process46. In brief, two C parameters (Cs and Ct) were adjusted individually for the truncated singlet and triplet potentials so that they contained six and five s-wave bound states, respectively, and so that the known singlet and triplet scattering lengths were reproduced. Further fine-tuning of the two C parameters together with the atomic hyperfine splitting aimed to reproduce the Feshbach resonance at about 155 G. As a result, the atomic hyperfine splitting was reduced by about 5% compared to the literature value. We used \({C}_{{\rm{s}}}={(0.3242030{r}_{{\rm{vdW}}})}^{6}\times{C}_{6}\) and \({C}_{{\rm{t}}}={(0.3258900{r}_{{\rm{vdW}}})}^{6}\times{C}_{6}\), where \({r}_{{\rm{vdW}}}=\frac{1}{2}{({m{C}_{6}}/{{\hslash}^{2}})}^{1/4}\) is the van der Waals length and C6 is the van der Waals coefficient.

Interactions between the particles and with the external magnetic field B coupled various angular momenta. Therefore, the incoming spin channel \(\left\vert 2,-2\right\rangle \left\vert 2,-2\right\rangle \left\vert 2,-2\right\rangle\) could, in principle, be coupled to a range of spin channels \(\left\vert\,{f}_\mathrm{a},{m}_{{f}_\mathrm{a}}\right\rangle \left\vert\,{f}_\mathrm{b},{m}_{{f}_\mathrm{b}}\right\rangle \left\vert\,{f}_\mathrm{c},{m}_{{f}_\mathrm{c}}\right\rangle\), where fi can be 2 or 3. Essentially, we had only the restriction that \({M}_{{\rm{tot}}}={m}_{{f}_\mathrm{a}}+{m}_{{f}_\mathrm{b}}+{m}_{{f}_\mathrm{c}}\) is conserved, if spin–spin interactions can be neglected. However, motivated by previous work29, where we found a spin-conservation propensity rule in the three-body recombination of Rb atoms, we restricted the spin of the third atom to \(\left\vert\,{f}_\mathrm{c},{m}_{{f}_\mathrm{c}}\right\rangle =\left\vert 2,-2\right\rangle\) in our calculations. One reason for this restriction could be that the third atom c interacted mainly mechanically with the other two (a, b) while they were forming a molecule. This approximation led to a model of five coupled three-body channels with the quantum numbers (F, fa, fb) = (4, 2, 2), (4, 3, 2), (4, 3, 3), (5, 3, 2) and (6, 3, 3).