Complexes formed in collisions between ultracold alkali-metal diatomic molecules and atoms

We explore the properties of 3-atom complexes of alkali-metal diatomic molecules with alkali-metal atoms, which may be formed in ultracold collisions. We estimate the densities of vibrational states at the energy of atom-diatom collisions, and find values ranging from 2.2 to 350~K$^{-1}$. However, this density does not account for electronic near-degeneracy or electron and nuclear spins. We consider the fine and hyperfine structure expected for such complexes. The Fermi contact interaction between electron and nuclear spins can cause spin exchange between atomic and molecular spins. It can drive inelastic collisions, with resonances of three distinct types, each with a characteristic width and peak height in the inelastic rate coefficient. Some of these resonances are broad enough to overlap and produce a background loss rate that is approximately proportional to the number of outgoing inelastic channels. Spin exchange can increase the density of states from which laser-induced loss may occur.

All the alkali-metal diatomic molecules produced so far have been found to undergo collisional loss in optical traps [27,42,46,48,51,54], even in cases where there is no energetically allowed 2-body reaction. In most systems the loss rate coefficients approach the predictions of a "universal loss" model [55,56] in which every molecular pair that reaches short range is lost from the trap. For RbCs, however, detailed loss measurements that include temperature dependence [54] have been used to determine the parameters of a non-universal model [57] in which there is partial reflection at short range.
Mayle et al. [58,59] proposed that the observed trap loss is due to "sticky collisions", in which an initial bimolecular collision forms a long-lived complex that survives long enough to collide with a third molecule. They estimated the densities of states ρ for the 4-atom complexes at the energy of the colliding molecules, and used arguments based on random-matrix theory to estimate the resulting mean lifetime τ of the complex. For (KRb) 2 they obtained ρ > 3000 µK −1 and τ > 150 ms. In subsequent work, Christianen et al. [60] obtained improved estimates of ρ, taking fuller account of angular momentum constraints and using a more accurate representa-tion of the potential energy surface. The corresponding lifetimes, τ = 6 µs for (NaK) 2 , are too short for most complexes to collide with a third molecule at the experimental densities. Christianen et al. [61] proposed that the complexes are instead excited by the trapping laser, and showed that this can occur fast enough to account for the observed trap loss. This proposal is supported by experiments on collisions of RbCs [62,63] and 40 KRb [64], though recent experiments on Na 40 K [65] and on Na 39 K and Na 87 Rb [66] suggest that the complexes have longer lifetimes than predicted in the absence of the trapping laser.
In parallel with the work on molecule-molecule collisions, experiments have been carried out on atommolecule collisions. The systems studied experimentally include 40 K 87 Rb with 40 K and Rb [27,67], 87 RbCs with 87 Rb and Cs [63], Na 39 K with Na and 39 K [42,68] and Na 40 K with 40 K [50,69]. For each molecule, reaction is energetically allowed in a collision with the lighter atom but forbidden in a collision with the heavier one. Fast collisional loss has been observed in all cases where the reaction is energetically allowed.
For non-reactive atom-molecule systems, the picture is more complicated. Experiments have been carried out on a number of systems including 40 K 87 Rb with Rb [27,67], 87 RbCs with Cs [63], Na 39 K with 39 K [42,68] and Na 40 K with 40 K [50]. For the last of these, Yang et al. [50] observed narrow Feshbach resonances as a function of magnetic field; these resonances have been assigned as due to long-range states of triatomic complexes [69]. A wide variety of behavior has been observed, ranging from nearuniversal loss to very slow loss, and no consistent picture is yet available. Recently, Nichols et al. [67] directly probed complexes in collisions of 40 K 87 Rb with Rb; they measured lifetimes of 0.4 ms in the absence of the trapping laser, 5 orders of magnitude larger than theoretical predictions [60].
The purpose of the present paper is to explore the properties of the collision complexes that can be formed in collisions between alkali-metal diatomic molecules and atoms. The structure of the paper is as follows. In section II, we consider the angular momenta that are present in alkali-metal triatomic systems, and basic aspects of the coupling between them. In section III, we consider the relationship between densities of states, lifetimes of complexes, and loss rates, including threshold effects. In section IV, we estimate the densities of vibrational states for short-range 3-atom complexes near the atom-diatom collision threshold. In section V we consider the electronic structure of the complexes and the effect of orbital near-degeneracy. In section VI, we present a model for the Fermi contact interaction between electron and nuclear spins, and show that it can drive spin exchange between atomic and molecular spins. In section VII, we consider the effect of spin exchange in atom-molecule collisions. We show that it can cause Feshbach resonances of three distinct types, each of which produces peaks in the loss rate coefficient with characteristic widths and peak heights. Some of these resonances are broad enough to overlap and produce a background loss rate that is approximately proportional to the number of outgoing inelastic channels. Finally, section VIII presents perspectives and conclusions of the work.

II. ANGULAR MOMENTUM COUPLING
There are 6 sources of angular momentum in a triatomic system AB+C formed from a singlet molecule and an alkali-metal atom: the electron spin S = 1/2, three nuclear spins i A , i B , i C , the diatomic rotation n and the partial-wave quantum number L for rotation of AB and C about one another.
An atom-diatom system in a single electronic state is governed by a 3-dimensional potential energy surface V (R, r, θ). This is written here in Jacobi coordinates where r is the diatom bond length and R and θ are the atom-diatom distance and angle. For the alkalimetal systems of interest here, V (R, r, θ) is deep (of order 50 THz) and provides strong coupling between the vibrational and rotational states of the diatomic molecule. Nevertheless, it is diagonal in the total spin-free angular momentum N , which is the resultant of n and L.
An alkali-metal atom C in a 2 S state, with electron spin S = 1/2 and nuclear spin i C , is characterized in zero field by its total spin f C = i C ± 1 2 . The two hyperfine states are separated by the hyperfine splitting (i C + 1 2 )ζ C , of order 1 GHz, where ζ C is the scalar hyperfine coupling constant that arises from the Fermi contact interaction. In a magnetic field, each hyperfine state is split into 2f C + 1 Zeeman states labeled by m f,C . As will be seen below, the Fermi contact interaction in a triatomic complex can depend strongly on geometry and provides a coupling that can be off-diagonal in f C and/or m f,C , while conserving the total spin projection m f,tot = m A + m B + m C + M S .
Our overall picture of the states of the triatomic complex is that the electronic interaction potential creates a strongly coupled and potentially chaotic manifold of states for each spin combination (m A , m B , f C , m f,C ), which correlates at long range with an atom in state (f C , m f,C ) and a molecule in state (m A , m B ). There is a weaker coupling between manifolds of the same m f,tot , due to the Fermi contact interaction, that may cause inelastic loss when there are suitable open channels.
Both the electronic interaction potential and the Fermi contact interaction conserve the total spin-free angular momentum N and its projection M N . There are weaker interactions arising from Zeeman, spin-rotation, and additional hyperfine interactions, some of which are off-diagonal in N and M N . These may play a role in sharp Feshbach resonances, but are unlikely to be strong enough to influence background loss.
Atom-molecule pairs that reach short range may be excited by the trapping laser, producing laser-induced loss analogous to that observed in molecule-molecule collisions. This is possible both for direct collisions and for collisions that form long-lived complex; laser-induced loss is not in itself evidence of complex formation.

III. RESONANCE WIDTHS AND LIFETIMES OF COLLISION COMPLEXES
A quasibound state is often thought of as characterised by a width Γ and a corresponding lifetime τ =h/Γ. However, these quantities need careful definition. When a bound state is embedded in a scattering continuum, it produces a resonance whose width Γ is governed by the matrix elements between the bound state and the continuum. If the state is well above the threshold of a single open channel, the scattering phase shift δ(E) follows the Breit-Wigner form, where δ bg is the background phase shift and E 0 is the resonance energy. If there are several open channels a, the S-matrix eigenphase sum [70] follows the form (1) and Γ is a sum of partial widths Γ a to the individual open channels. When a quasibound state is probed by absorption spectroscopy from a single initial state, and the continuum itself is dark, the spectrum has a Lorentzian lineshape with width Γ [71][72][73]. Conversely, excitation of the entire Lorentzian, by a laser that is broad compared to Γ, produces a non-stationary state (wavepacket) that decays into the continuum with lifetime τ =h/Γ [74]. Nevertheless, it is important to realize that, even when a bound state is spread out over a continuum, solutions of the time-independent Schrödinger equation still exist at each energy. Such solutions represent stationary states whose densities do not evolve in time. The rate at which a wavepacket can evolve is ultimately limited by its spread in energy, which can be much smaller than Γ. This is particularly important in ultracold systems, which may possess a very small energy spread characterized by their temperature. Complexes formed in ultracold systems may thus exhibit lifetimes limited by the temperature, which may be much longer than implied by the width of the underlying state. The width of a Feshbach resonance is energydependent when the state lies just above threshold, even if the underlying couplings are independent of energy. In quantum defect theory (QDT), the partial width for decay of a resonance to an open channel a is where Γ s a is the short-range width and C −2 (E a ) is a QDT function that depends on the kinetic energy E a for channel a [75][76][77][78]. For an interaction potential −C 6 R −6 , C −2 (E a ) depends on the background scattering length a bg and is a universal function when written in terms of the mean scattering lengthā = (2µC 6 /h 2 ) 1/4 × 0.4779888 . . . [79] and the corresponding energyĒ = h 2 /(2µā 2 ). Examples of C −2 (E a ) are shown for a variety of values of a bg /ā in Fig. 1; it is proportional to E 1/2 a at limitingly low energy and in all cases it approaches 1 when E a Ē . Because of this, Γ a Γ s a near threshold. For molecules with regular patterns of energy levels, such as most low-lying vibrational states, the widths are unrelated to the spacings between levels and are often much smaller than them. However, Mayle et al. [58,59] suggested that the vibrational states of an atom-molecule or molecule-molecule collision complex are chaotic in nature, and may be approximated by random-matrix theory (RMT). Under these circumstances the mean short-range width of the states may be written in terms of their mean spacing d [80],Γ where T s a is a transmission coefficient between 0 and 1 that governs the likelihood of complex formation or decay for collisions that reach short range in channel a. Neglecting threshold effects, it is related to the unitarity deficit of the mean S matrixS by T s a = 1 − |S aa | 2 [80]. Mayle et al. [58,59] gave estimates of lifetimes for collision complexes in atom-molecule and molecule-molecule collisions based onτ =h/Γ and further estimatedΓ as d/(2π) to obtain the mean lifetime asτ = 2πhρ. Christianen et al. [60] obtained better estimates for ρ but used the same procedure to estimateτ . However, this neglects the reduction of Γ both by threshold effects and by the short-range transmission coefficient T s a . The actual widths of the resonances are thus likely to be considerably smaller than the estimates of Mayle et al., and the lifetimes of the complexes, once formed, are likely to be considerably larger than τ = 2πhρ. In general, if the temperature T is high enough to average over many resonances, k B T d, thenτ = 2πhρ is still the correct mean collisional time delay [81], but only a fraction of collisions form complexes and those that do have extended lifetimes.
The effects of thresholds and of T s a can be combined into a single transmission coefficient T a , which is generally not a simple product of the two contributions. In the presence of partial reflection at short range, T a can be calculated from a non-universal QDT theory [55,57]. The formation of complexes, averaged over many resonances, can be described in the same way as inelastic loss [82] and characterized by a short-range parameter analogous to the loss parameter y of Idziaszek and Julienne [55], where T s a = 4y/(1 + y 2 ) [80]. The collision complex may subsequently decay back to the incoming channel, or be lost by a secondary process such as collision with a 3rd body, laser excitation, or inelastic decay. If the system behaves chaotically, the mean width for decay back to the incoming channel isΓ inc = T inc d/(2π). In the absence of other loss processes, this implies a mean lifetimē τ = 2πhρ/T inc , but only if the thermal energy spread is large compared toΓ inc as described above. Christianen et al. [83] recently considered a lossy QDT model of such resonances. They concluded that if the complexes are lost rapidly once formed andΓ s inc = d/2π, such that T s inc = 1 in Eq. (3), the loss rate is represented by y = 0.25, as opposed to the y = 1 implicit in the model of Croft et al. [82]. The reasons for this apparent disagreement are not clear at present, but both methods rely on averaging over a large number of resonances.
The densities of states for atom-molecule collision complexes are much lower than those for molecule-molecule complexes [60]. In particular, as seen below, the mean spacings between levels are far larger than both typical thermal energies and laser broadening. It is therefore not appropriate to average across the resonance widths. Instead, we need to consider how the possible presence of a single resonant state near threshold can enhance collisional loss. In the following, we consider this in terms of the resonant profile as the state crosses the incoming threshold, even if its energy relative to threshold is almost fixed. As shown in the Appendix, the rate coefficient for resonant inelastic loss due to a single resonance with no background loss, at limitingly low collision energy, shows a Lorentzian peak of the form (4) Here k univ 2 = 4πhā/µ is the universal rate coefficient at zero energy [55]; values ofā and k univ 2 are given in Table  I. Γ s inc is the short-range partial width for decay to the incoming channel, which may be well represented by Eq.
(3). Γ inel is the sum of the partial widths for secondary loss processes, including both laser excitation and inelastic decay. The width of the peak is determined by Γ inel , which is not subject to threshold effects in the incoming channel. It is entirely possible for the resonantly enhanced loss rate to exceed the universal rate, particularly when Γ s inc > Γ inel . Such supra-universal rates are observed, for example, near Feshbach resonances for Na 40 K with 40 K [50].
If Γ inel > ∼ d, which often occurs when there are many open channels, multiple resonant features may overlap. When only a few resonances overlap, the separate contributions may be approximately additive. However, when many resonances overlap, they reach the regime of Ericson fluctuations [80] and the total loss cannot be represented in this way. This situation has been considered by Christianen et al. [83]. Here we will consider only the case Γ inel < d, which is applicable to typical alkali atom+diatom collisions.

IV. DENSITIES OF STATES FOR COLLISION COMPLEXES
The densities of states are much lower for 3-atom than for 4-atom complexes. Christianen et al. [60] have developed a procedure for evaluating the density of states, based on a semiclassical phase-space integral incorporating angular momentum constraints. They obtain Here g N Jp is a parity factor that accounts for the absence of a conserved parity in classical phase space, which is 1 for the systems considered here; m X is the mass of atom is the atom-diatom reduced mass. Christianen et al. [60] included a degeneracy factor g ABC in this expression to account for equivalent nuclei, but here we take that into account in considering hyperfine states below. The resulting density of vibrational states ρ(E) is for a single electronic state and a single value of the spinfree total angular momentum N and its projection M N . 1 It also neglects fine and hyperfine structure. Equation (5) can be integrated numerically if a full interaction potential V (R, r, θ) is available. However, if we make some approximations we can obtain analytic expressions. We wish to estimate the density of states that are strongly enough coupled to form a chaotic bath. Such states exist principally at short range. We therefore fix the first term in the integrand to its value at the equilibrium geometry, R = R e and r = r e . We also approximate the potential to be isotropic and harmonic around the minimum, The integral can now be evaluated analytically, giving Here the energy is relative to the potential minimum, and for the present purpose we evaluate ρ at E = D e , the energy of the triatomic minimum with respect to the energy of the separated atom and diatomic molecule. We therefore conclude that the density of short-range states around the atom-diatom threshold is likely to scale approximately as The equilibrium geometries and binding energies D e for all the alkali-metal 3-atom systems containing 2 or 3 identical atoms have been obtained from electronic structure calculations [84]. For each system, we estimate the force constant k R for the atom-diatom vibrations assuming a Lennard-Jones potential in R. This gives where D e is the binding energy of the trimer with respect to the atom+diatom threshold and R e = (C 6 /D e ) 1/6 ; for XY + X systems we use C 6 coefficients from [85] and for X + X 2 systems we use twice the C 6 coefficient for X + X [86]. The force constant k r is taken to be the same as that determined for the free diatomic molecule XY from electronic spectroscopy . The resulting densities of short-range states ρ and mean level spacings d for the systems considered here are given in Table I. We have selected one representative isotope of each element, and the small variations due to isotopic substitution can be calculated from Eq. 8 if needed. The values range from 60.5 MHz for Cs 2 +Cs to 9.48 GHz for Li 2 +Li. Most of the differences come from the atomic and reduced masses, which vary by up to factor of 20 between systems. The value obtained by this method for K 2 +Rb is within 25% of that obtained by Christianen et al. [60] by evaluating Eq. 5 using a different model potential.

V. ORBITAL NEAR-DEGENERACY
To understand the chemical bonding in the collision complexes, it is useful first to consider the homonuclear alkali-metal triatomic molecules [115]. At a geometry corresponding to an equilateral triangle (point group D 3h ), these systems are orbitally degenerate, with a single electron in an orbital of symmetry e . The resulting state has symmetry 2 E, so is subject to a Jahn-Teller distortion; the actual equilibrium geometry is an obtuse isosceles triangle (point group C 2v ). At this geometry the ground state has 2 B 2 symmetry, but there is a low-lying excited state of 2 A 1 symmetry. At lower-symmetry geometries the two states are mixed. The two resulting surfaces intersect along seams of conical intersections that include one at equilateral geometries. The three equivalent potential minima on the lower surface are connected by low-energy pathways through scalene and acute isosceles geometries; motion along these pathways gives rise to the phenomenon known as pseudorotation, with characteristic energy-level patterns.
For heteronuclear 3-atom systems X 2 Y , the situation is more complicated [84]. Some systems have ground states of 2 B 2 symmetry with minima at isosceles geometries, while others have ground states of 2 A symmetry at scalene geometries (point group C s ). Nevertheless, the principle remains that there are two electronic states of similar well depth that cross and avoided-cross as a function of nuclear coordinates. The resulting short-range states are strongly coupled to one another, so the neardegeneracy of the partially filled orbitals produces almost a doubling in the densities of states from the values of Section IV.

VI. THE FERMI CONTACT INTERACTION
The strongest hyperfine term for both the free atom and the triatomic collision complex is the Fermi contact interaction. At long range this couples S to i C for the free atom to form f C . At short range, however, all three nuclei experience significant spin densities and S couples to all of i A , i B and i C to give resultant f tot . The coupling is of the form whereŜ andî X are the operators for the electron and nuclear spin angular momenta. The coupling coefficients ζ X are proportional to the product of the corresponding nuclear magnetic moment and the electron spin density at nucleus X. The spin densities are strongly dependent on geometry; as any one nucleus is pulled away from the other two, the electron spin localizes on the separating atom, until the full hyperfine coupling of the free atom is achieved as R → ∞. However, in the strongly interacting region the spin density is distributed between the three atoms and shifts substantially from one atom to another as the complex vibrates. The spin density at the nucleus is sensitive only to the spin population in atomic s orbitals, and is reduced if population is transferred to p orbitals. There is some controversy over the Fermi contact interactions in alkali-metal triatomics near their equilibrium geometries. For Na 3 [116] and K 3 [117], electron-spin resonance (ESR) studies of matrix-isolated species show that the s-orbital spin densities on the 3 atoms sum to around 0.9 at the equilibrium geometry of the 2 B 2 state, with most of the density on the two equivalent atoms. For Li 3 [118] and Na 2 Li [119] the distribution is similar, but the densities sum to only 0.69 and 0.78, respectively. The Li 3 result has been confirmed by a molecular-beam study [120]. These results accord with a physical picture in which relatively little spin density is located in p orbitals. However, a molecular-beam microwave study on Na 3 [121] suggests that the Fermi contact interactions are much smaller than the ESR spin densities imply, and this is supported by electronic structure calculations [122]. These details do not affect the basic physics discussed in the present paper: the important feature is that the spin densities shift substantially between atoms as a function of wide-amplitude motions.
We have modeled the spin densities using a valencebond method. This is related to the London-Eyring-Polanyi-Sato (LEPS) [123][124][125][126] approach, which has been widely used for interaction potentials, including those of the alkali-metal triatomic molecules [127,128]. In the LEPS approach, the two doublet surfaces are [123] where R XY is the separation of atoms X and Y , and J XY and K XY are 2-atom Coulomb and exchange integrals. Values of these integrals can be estimated from the potential curves for the corresponding 2-atom systems by writing their singlet and triplet curves as with only the two-body singlet and triplet potentials  as input. The London equation (10) [123] gives only the energies of the surfaces. However, Slater's derivation [129] TABLE I. Properties of atom+diatom systems: mean scattering lengthā and zero-temperature universal rate constant k univ 2 , together with densities of short-range states ρ and mean spacings d calculated from Eqs. (5) and (7). and one has projection − 1 2 . Remarkably, there are some atomic arrangements near isosceles geometries where one of the three configurations dominates. For such arrangements, the spin density on the isolated atom is actually dominated by m s = − 1 2 , with the opposite sign to the overall spin. This is a result that is quite impossible for a spin-restricted Hartree-Fock (RHF) wavefunction with an unpaired electron in a single molecular orbital.
In a magnetic field, the Fermi contact interaction can mediate spin exchange between the electron and nuclear spins (and indirectly among the nuclear spins) while conserving the total spin projection m f,tot . Each rovibrational state of the triatomic complex will split into N hf spin sublevels. If the atomic and molecular states are spin-stretched, m f,tot = ±f max , where f max = S + i A + i B +i C , only a single sublevel exists for each rovibrational state of the complex. However, the number of sublevels increases as |m f,tot | decreases. Neglecting exchange symmetry, there are N hf = 4 sublevels for f max −|m f,tot | = 1; three of these have even exchange parity and one is odd. For larger deviations the numbers increase, and are easily evaluated, but depend on the specific values of the spins.
An example of the effect of shifting spin density is shown in Fig. 2. This shows the Fermi contact contribution to the hyperfine energy for the reaction 87 Rb + 40 K 87 Rb → 87 Rb 2 + 40 K along a selected reaction path that includes a rearrangement by pseudorotation. It is obtained by diagonalizing the Fermi-contact Hamiltonian (9) at each geometry, with the coupling coefficients ζ X calculated from the valence-bond model. The resulting hyperfine adiabats are shown for m f,tot = −3/2, which corresponds to the lowest hyperfine state of the reactants. In this case, f max − |m f,tot | = 6 and there are N hf = 32 hyperfine sublevels. At the center of the diagram, where there is substantial exchange of spin density, the levels are spread out across an energy range comparable to the atomic hyperfine splitting and their energies depend strongly on geometry. There are many avoided crossings with a variety of widths, particularly along the pseudorotation section of the path, that are likely to result in spin valence-bond model. Along the coordinate ξ, the system starts from the separated Rb atom and KRb molecule (ξ = 0); these approach each other with fixed Rb-K-Rb angle, to the absolute minimum (ξ = 1), which is an obtuse isosceles triangle. The system then pseudorotates along the minimum-energy path to a local minimum at a scalene geometry (ξ = 2), and finally dissociates by moving the K atom away from Rb2, with fixed Rb-Rb-K bond angle (ξ = 3). The inset shows an expansion of the adiabats around the midpoint of the pseudorotation, illustrating the presence of close avoided crossings that are likely to result in spin exchange.

VII. INELASTIC LOSS MEDIATED BY SPIN EXCHANGE
There is a manifold of short-range vibrational states associated with each hyperfine sublevel. Each such manifold is likely to be chaotic in nature. Spin exchange allows a collision between atoms and molecules in one pair of states to access bound states in any manifold with the same m f,tot . The density of states is enhanced to N hf ρ, and all the states can cause Feshbach resonances. However, these states are not necessarily equivalent, because the Fermi contact interaction that drives spin exchange is weaker than the anisotropic potential coupling. A given incoming channel couples strongly to one of these manifolds, but it may couple more weakly to the remaining N hf − 1 manifolds. This may be quantified with a spinexchange parameter z, analogous to the isospin-mixing parameter used for reactions of compound nuclei [130], where H ab is a matrix element of the coupling between states in manifolds a and b. When z 1, the mean partial width connecting the incoming channel to states in these other manifolds is zΓ a . Conversely, when z 1, states in the N hf different manifolds are fully mixed and all contribute to the effective density of states that enters into RMT.
A full calculation of the spin-exchange parameter is beyond the scope of this work. It requires not only the mean level spacing d but also the wavefunctions of the states in the two chaotic manifolds. Such a calculation is near the limits of current theoretical methods, but may be achievable for lighter atom-diatom systems within a few years. Here we follow the practice in studies of compound nuclei [130], and treat z as a phenomenological parameter to be inferred from experimental results.
Spin exchange can also produce inelastic loss if there are channels lower in energy than the incoming state but with the same m f,tot . We again take 40 K 87 Rb+ 87 Rb as our example [27,67]. The lowest state is (m K , m Rb , f Rb , m f,Rb ) = (−4, 3/2, 1, 1). Excitation of either nuclear spin of the molecule individually does not produce any inelastic channels that are accessible by spin exchange, but the combined excitation to (m K , m Rb ) = (−3, 1/2) has the same m f,tot as the ground state, so spin exchange is allowed. However, the states are split by only the nuclear Zeeman effect, so the kinetic energy release is small, equivalent to 15 µK at a representative magnetic field of 200 G. The corresponding QDT function C −2 (E) is typically around 0.1, so decay by this path is partially suppressed by threshold effects. If the atom is in an excited state, spin exchange is usually energetically allowed. In such cases the kinetic energy releases are much larger, equivalent to 7 mK at 200 G, such that C −2 (E) is close to 1, and loss is not suppressed by threshold effects.
Even when there are open channels that are energetically accessible, non-resonant spin exchange is likely to be slow. The avoided crossings in Fig. 2 occur deep in the potential well. Even for the widest such crossings, a Landau-Zener treatment estimates inelastic transition probabilities of the order of 10 −3 for a double crossing in Rb+KRb. Significant loss is therefore expected only when there is resonant enhancement. If z < 1, there are three general cases of resonant inelastic decay, which correspond to different values of Γ s inc and Γ inel in Eq. (4). In the following, we relate the mean heights of resonant loss features to the universal rate, modified by the factor [1 + (1 − a bg /ā) 2 ] in Eq. 4, and their mean widths toΓ s a of Eq. 3.
The first case is where the resonant state is part of the manifold associated with the incoming channel and coupled to N out inelastic channels by spin exchange. We refer to the resulting resonances as incoming-manifold resonances. The coupling to the incoming channel is characterized byΓ s inc =Γ s a and that to the inelastic channels byΓ inel = zΓ s a . The height of the peak in the loss rate is thus multiplied by a factor 1/(zN out ), but its width is divided by the same factor. When it is appropriate to average over many such resonances, their total contribution is thus independent of z, but in the atom+molecule case, where the density of states is too low for averaging to be appropriate, the narrowing decreases the probability of hitting such a resonance.
The second case is where the resonant state is part of a manifold associated with an open channel other than the incoming channel. We refer to these as outgoing-manifold resonances. In this caseΓ s inc = zΓ s a andΓ inel =Γ s a . The height of the peak in loss rate is thus multiplied by a factor z. However the widths of the resonances are unaffected by z, so may be comparable to their spacings, according to Eq. 3. If there are N out open channels, each of them will support resonances of this type.
The third case is where the resonant state is part of a manifold associated with a closed channel. We refer to these as closed-manifold resonances. In this caseΓ s inc = zΓ s a andΓ inel = zN outΓ s a . The height of the peak in loss rate is thus reduced by a factor N out from the universal rate, and the width of the peak is multiplied by a factor zN out . Figure 3 shows two schematic examples of the combined effect of these three types of resonance. For the purpose of illustration, we choose spin-exchange parameter z = 0.15 and short-range transmission coefficient T s a = 1.0. For each manifold, the bound states are randomly generated from a suitable chaotic distribution 2 [131,132] with mean spacing d. The horizontal axis represents scanning the energy of the incoming channel across the pattern of short-range bound states. Figure 3(a) shows an example with N hf = 9 and N out = 4, so that within each interval d there is 1 incomingmanifold resonance, 4 outgoing-manifold resonances, and 4 closed-manifold resonances. It may be seen that the outgoing-manifold and closed-manifold resonances make similar overall contributions to the inelastic rate, but the closed-manifold contribution is more structured because the underlying resonances are narrower. The incomingmanifold resonances give peaks that are high but relatively narrow (with width dependent on T s a ), and usually will not overlap. Figure 3(b) shows an example with N hf = 9 and N out = 2, with the resonances in the same locations as before to facilitate comparison. Within each interval d there are now 1, 2 and 7 incoming-, outgoingand closed-manifold resonances, respectively. The major differences are that the peaks for incoming-manifold resonances and closed-manifold are higher and sharper, while the overall contribution of outgoing-channel resonances is reduced by the smaller N out . An important feature that the two examples share is a weakly structured background loss from the outgoing-manifold resonances, whose mean height is 2 We randomly generate matrices of dimension 20 from the Gaussian orthogonal ensemble, and unfold the resulting eigenvalues analytically using Wigner's semicircle law. independent of d.
The loss rates in Fig. 3 are shown as a function of the mean spacing d. In ultracold scattering, however, such rates are usually measured as a function of magnetic field, or some other external variable such as electric field. As a function of magnetic field B, channel threshold shift with respect to one another by B∆µ, where ∆µ is the difference in magnetic moments. For free alkali-metal atoms at low field, ∆µ/h ≈ 1.4/(i C + 1 2 ) MHz/G for states with m f,C that differ by 1, and is typically between 0.7 MHz/G for atoms with i C = 3 2 ( 23 Na, 39 K, 41 K, 87 Rb) and 0.3 MHz/G for i C = 4 ( 40 K). The magnetic moments of triatomic complexes are more complicated, but will be of similar magnitude. The horizontal axes in Fig. 3 correspond to 3d and thus covers the equivalent of hundreds or even thousands of Gauss. The general conclusion is that atom-molecule resonances due to states that belong to the RMT bath are very broad indeed.
The largest body of experimental results for losses in alkali-metal atom-molecule systems is for Na 40 K with 40 K. Yang et al. [50] measured loss rates as a function of magnetic field for more than 20 combinations of atomic and molecular states. Their main focus was on narrow Feshbach resonances, but they also measured loss rate coefficients in several windows of magnetic field, including near 90 G and near 102 G, shown in Figs. S2 and S3 of their Supplemental Material. In cases where no resonant features are visible, the background loss rates generally increase with N out , reaching values between 0.5 and 1 times the universal loss rate when N out is 3 to 5. However, there are also a few state combinations that show unstructured loss at up to twice the universal rate. For Na 40 K with 40 K, d = 1033 MHz and a typical value of ∆µ/h is 0.3 MHz/G, so even the narrowest resonances in Fig. 3(b) have widths of order 200 G for z = 0.15. Both the general increase in background loss rates with N out and the occasional state combinations with suprauniversal rates are entirely consistent with the behaviour shown in Fig. 3.
In addition to inelastic loss to lower-lying open channels, there is the possibility of loss due to laser excitation of complexes by the trapping laser. This is likely to introduce loss from all channels accessible by spin exchange, with a contributionΓ laser toΓ inel that depends on the laser intensity but is independent ofΓ s a and hence of the density of states. The effect of this depends on the relative strength of this decay to any existing inelastic loss. Nichols et al. [67] have measured decay equivalent toΓ laser /h ≈ 5 MHz for 40 KRb+Rb at typical optical trap intensities, and it seems likely that other alkali-metal atom+molecule systems will be comparable. This value is substantially smaller than our estimates ofΓ s a for any of the systems considered here. Laser-induced loss is thus likely to have little effect on loss due to any outgoingmanifold resonances, but it may reduce the height and increase the width of incoming-manifold or closed-manifold resonances ifΓ laser > ∼ zΓ s a . More importantly, it may induce both incoming-manifold and closed-manifold resonances even in systems with no open inelastic channels, such as those involving atoms and molecules in their absolute ground states. However, the likelihood of encountering such a resonance for a particular incoming state at a single field is approximately (N hf − 1)Γ laser /d, which is usually fairly small.

VIII. CONCLUSIONS
We have developed the theory of the triatomic complexes that can be formed in ultracold collisions between alkali-metal diatomic molecules and atoms. We have estimated the densities of vibrational states of the complexes near the energy of the colliding particles, based on the properties of the diatomic molecules and the calculated binding energies of the trimers. The resulting densities range from 2.2 to 350 K −1 . We have considered the angular momentum couplings present in the complexes and the resulting fine and hyperfine structure. The largest such term is the Fermi contact interaction between the electron and nuclear spins. We have presented a model of this interaction, based on valence-bond theory, and shown that it varies substantially with the geometry of the complex because the unpaired spin moves between the atoms as the complex vibrates.
Our overall picture is that each pair of atomic and molecular spin states is associated with a manifold of vibrational states of the triatomic complex. Each such manifold is likely to be chaotic in nature. The Fermi contact interaction can couple these manifolds, and can thus drive spin exchange between spins on the atom and the molecule. The degree of coupling between manifolds is characterized by a spin-exchange parameter z, which depends on the ratio between off-diagonal matrix elements of the Fermi contact interaction and the mean spacing between vibrational levels, d. This parameter may be substantial in the alkali-metal triatomic complexes.
We have developed the theory of resonant low-energy scattering in the presence of several chaotic manifolds that are weakly coupled to one another. We find that there can be Feshbach resonances of three distinct types, which we term incoming-, outgoing-and closed-manifold resonances. Each type of resonance can cause peaks in the rate coefficient for inelastic scattering with characteristic peak heights and widths that depend on z and the number of outgoing inelastic channels. For atom-diatom systems, the resonances due to states in the chaotic bath are very broad compared to the spread of kinetic energies at ultracold temperatures, so it is not appropriate to average over their widths in ultracold atom-diatom collisions. Instead, the scattering properties at a specific collision threshold depend on where it is placed with respect to the essentially random pattern of individual bath states. Nevertheless, the resonances due to outgoingmanifold resonances are broad enough that they may overlap to produce a background loss for most incoming states, particularly when there are several outgoing inelastic channels that are accessible from the incoming state by spin exchange.
Resonances due to states in the chaotic bath are different from those arising from near-threshold states of triatomic complexes, as observed in Na 40 K+ 40 K [69]. Such states spend most of their time in the long-range tail of the interaction potential; they are relatively weakly coupled to one another and to the incoming and outgoing scattering states, so they can produce much narrower resonances than the states considered here. They do not form part of the chaotic bath of states, so there is no reason to expect their widths to be related to their spacings.
Atom-diatom complexes have much lower densities of states than diatom-diatom complexes. Nevertheless, some features of the theory developed here may apply in the diatom-diatom case. In particular, the densities of states in diatom-diatom complexes [60] can be 4 orders of magnitude larger than those obtained here for atomdiatom complexes. Since the mixing between manifolds is governed by the ratio of couplings to level spacings, much smaller couplings are sufficient to cause mixing in the diatom-diatom case. Even very small interactions such as the nuclear electric quadrupole coupling [133] might be large enough. Hutson [134] has described the behavior of a Feshbach resonance close to threshold in the presence of inelastic decay. The general expression for an S-matrix element in the vicinity of a resonance is Here, E res is the resonance energy, g a is in general complex and characterizes the partial width Γ a = |g 2 a |, and Γ = a Γ a is the total width. We are interested in the diagonal S-matrix element in the incoming channel, S inc ; we take the collision energy E−E inc to be small, such that we can approximate the background S-matrix element to be S bg,inc = 1; the more general case |S bg,inc | = 1 gives the same results for inelastic loss due to resonances, but we choose this specific value for simplicity. With this choice, the product g 2 inc must be real and non-negative and can be replaced by Γ inc , the partial width to the incoming channel. This gives The total width is Γ = Γ inc + Γ inel , where Γ inel characterizes the decay to all loss channels, whether inelastic, reactive, or light-induced. The partial width in the incoming channel is narrowed by threshold effects, Γ inc = Γ s inc C −2 (E − E inc ), as described by Eq. 2. The QDT function C −2 (E kin ) can be calculated explicitly from either numerical or analytic solutions [135] with an appropriate asymptotic potential. Its leading term for an asymptotic potential −C 6 /R 6 is [77,78,136] C −2 (E kin ) = kā 1 + (1 − a bg /ā) 2 , (A. 3) where k = √ 2µE kin /h is the wavevector and a bg is the background (non-resonant) scattering length.
We could now directly calculate inelastic cross sections and rate coefficients from Eq. (A.2), but it is convenient first to consider the complex scattering length. This is defined as [134] a(k) = α − iβ (A.7) This can be converted to a rate through Eq. A.5. We make the simplifying assumption that k|a| 1, such that the denominator in Eq. A.5 reduces to just µ, and write the result in terms of k univ 2 = 4πhā/µ. This gives (A.8) which is Eq. 4. This is a Lorentzian peak with width determined only by Γ inel , and a peak height that reaches at least the universal rate if Γ s inc ≥ Γ inel .