Collisions Between Ultracold Molecules and Atoms in a Magnetic Trap
S. Jurgilas, A. Chakraborty, C. J. H. Rich, L. Caldwell, H. J. Williams, N. J. Fitch, B. E. Sauer, Matthew D. Frye, Jeremy M. Hutson, M. R. Tarbutt
CCollisions Between Ultracold Molecules and Atoms in a Magnetic Trap
S. Jurgilas, A. Chakraborty, C. J. H. Rich, L. Caldwell, H. J. Williams, N. J.Fitch, B. E. Sauer, Matthew D. Frye, Jeremy M. Hutson, and M. R. Tarbutt ∗ Centre for Cold Matter, Blackett Laboratory, Imperial College London, Prince Consort Road, London SW7 2AZ UK Joint Quantum Centre (JQC) Durham-Newcastle, Department ofChemistry, Durham University, South Road, Durham DH1 3LE, UK
We prepare mixtures of ultracold CaF molecules and Rb atoms in a magnetic trap and study theirinelastic collisions. When the atoms are prepared in the spin-stretched state and the molecules inthe spin-stretched component of the first rotationally excited state, they collide inelastically with arate coefficient of k = (6 . ± . × − cm / s at temperatures near 100 µ K. We attribute thisto rotation-changing collisions. When the molecules are in the ground rotational state we see noinelastic loss and set an upper bound on the spin relaxation rate coefficient of k < . × − cm / swith 95% confidence. We compare these measurements to the results of a single-channel loss modelbased on quantum defect theory. The comparison suggests a short-range loss parameter close tounity for rotationally excited molecules, but below 0.04 for molecules in the rotational ground state. The formation and control of ultracold molecules is ad-vancing rapidly, motivated by a broad range of excitingapplications [1] including tests of fundamental physics,the exploration of many-body quantum physics, quan-tum information processing, and the study and controlof chemical reactions at the quantum level. Collisions arecrucial to this field, just as they have been for the fieldof ultracold atoms. They are a rich source of informa-tion about the interactions and reactions of atoms andmolecules in a fully quantum-mechanical regime wherethe internal states of the reactants and the partial wavesdescribing their relative motion are all resolved. Theircontrol is important for evading losses and controllingreactivity [2], and they can be harnessed for sympatheticor evaporative cooling [3, 4]. The former is especially im-portant for laser-cooled molecules. Having already pro-duced molecules at a few µ K [5–7], direct laser coolingis unlikely to lower the temperature much further, andat present the densities are insufficient for evaporativecooling. Instead, sympathetic cooling with evaporativelycooled atoms [8] is a promising way to increase the phase-space density and bridge the gap to quantum degeneracy.Several methods have been used to study collisions inthe temperature range between 10 mK and 1 K. In thiscold regime, crossed and merged beams have been usedto study quantum state resolved collisions [9–13] and col-lisions have been investigated in electric and magnetictraps loaded by buffer gas cooling, Stark decelerationand Zeeman deceleration [14–19]. However, these meth-ods are not suitable for studying collisions at µ K tem-peratures. Molecules in this ultracold regime have beenproduced by atom association [20], optoelectrical cool-ing [21], and direct laser cooling [22, 23]. For moleculesproduced by atom association, molecule-molecule colli-sions in optical traps have been an important topic ofstudy. These collisions lead to rapid trap loss eitherdue to chemical reactions [2] or, when reactions are en- ∗ [email protected] ergetically forbidden, due to the formation of long-livedcomplexes that are subsequently excited by the trappinglaser [24, 25]. The loss rate coefficients are found to beclose to those predicted by a single-channel model withuniversal loss, in which molecules are lost with unit prob-ability once they reach short range [26, 27]. Recently, re-active losses of KRb molecules have been suppressed byusing an electric field and confining the molecules to twodimensions, and this has led to the formation of a stablequantum degenerate gas of these molecules [4, 28, 29].Collisions have also been studied between ultracold CaFmolecules in tweezer traps [30, 31]. Rapid inelastic losseswere observed in these experiments too, both for ground-state molecules and those in excited hyperfine and rota-tional states. Again, the loss rate coefficent was not farfrom the one predicted by the universal loss model.Ultracold atom-molecule collisions have been investi-gated extensively by theory [32–37], but there are veryfew experimental studies. Recently, elastic collisions be-tween optically trapped Na atoms and NaLi moleculesproduced by atom association were observed and usedfor sympathetic cooling of the molecules [3]. Similarly,elastic collisions of KRb molecules with K atoms havebeen shown to maintain thermal equilibrium in the for-mation of a quantum-degenerate Fermi gas [38]. Mix-tures of laser-cooled molecules and atoms present ex-citing new opportunities to study and exploit ultracoldcollisions. Here, we produce the first such mixture anduse it to study inelastic atom-molecule processes in the µ K regime. We produce laser-cooled CaF molecules andRb atoms, load them into a magnetic trap, and mea-sure the collision-induced loss rate from the trap. Whenthe molecules are in a rotationally excited state, the pres-ence of Rb increases their loss rate, which we attribute tofast rotation-changing collisions. The loss rate coefficientis close to the value predicted by a single-channel uni-versal loss model and is not suppressed when the atomsand molecules are in spin-stretched states. By contrast,when the molecules are in the ground rotational state, nocollision-induced losses are observed. We use this obser- a r X i v : . [ phy s i c s . a t o m - ph ] J a n vation to set an upper limit to the spin-relaxation ratecoefficient. Our development of atom-molecule mixtures,and study of the inelastic processes within these mix-tures, are important steps towards sympathetic cooling.The starting point of the experiments is a dual-speciesmagneto-optical trap (MOT) of CaF molecules and Rbatoms. Each experiment begins by accumulating Rbatoms from a 2D MOT into the dual-species 3D MOTat a rate of about 2 × atoms/s. Once the desirednumber of atoms has been loaded, the 2D MOT is turnedoff and the CaF MOT is loaded using the methods de-scribed previously [39–42]. To lower the temperature ofthe molecules, we ramp down the intensity of the mainCaF cooling laser to 20% of its initial value over 4 ms,then hold it at this value for 10 ms. An image of theCaF MOT is acquired during this 10 ms period, which weuse to determine the initial number of molecules, N .Next, the quadrupole magnetic field is turned off andboth species are cooled simultaneously in two indepen-dent optical molasses for 10 ms. For CaF, we follow themolasses procedure described in Ref. [42]. For Rb we lin-early ramp the detuning and intensity of the cooling lightto −
58 MHz and 0.36 mW/cm over the 10 ms period.The molecules cool to 100 µ K, and the atoms to 40 µ K.Next, we prepare the molecules in a single, selectedquantum state | N, F, M F (cid:105) by optical pumping and mi-crowave transfer [43] in an applied magnetic field of230 mG along z . Here, N is the rotational quantumnumber and F and M F are the quantum numbers for thetotal angular momentum and its projection onto z . Atthe same time, the atoms are optically pumped into thestate | F = 2 , M F = 2 (cid:105) . Then, all the laser light is blockedusing mechanical shutters and the magnetic quadrupoletrap is turned on for a time t at an axial field gradientof 30 G/cm. We measure the final number of molecules, N CaF ( t ), by recapturing them in the MOT and imagingtheir fluorescence for 10 ms. Molecules prepared in N = 0are transferred back to N = 1 using a microwave pulseprior to recapture in the MOT. We measure the finalnumber of atoms, N Rb ( t ), by taking an absorption imageof the cloud shortly after releasing it from the magnetictrap. The collisional loss rate is determined by measuringthe fraction of molecules remaining, N CaF ( t ) /N , as afunction of t , both with and without atoms in the trap.Our simulations of the trap loading show that the sizesand positions of the clouds reach a steady state within300 ms, which is the minimum value of t we use.The density of the atomic sample exceeds that of themolecules by 6 orders of magnitude, so we can ignoremolecule-molecule collisions and the collision-inducedloss of atoms. In this case, the loss of molecules fromthe trap is described by˙ N CaF = (cid:0) − Γ − Γ Rb-CaF e − Γ Rb t (cid:1) N CaF (1)whose solution is N CaF ( t ) = N CaF (0) exp (cid:18) − Γ t − Γ Rb-CaF Γ Rb (1 − e − Γ Rb t ) (cid:19) . (2) N C a F ( t ) / N C a F ( ) Rb CaF ( s ) F r e q u e n c y Single speciesDual species
FIG. 1. Fraction of molecules, N CaF ( t ) /N CaF (0), remaining inthe magnetic trap after a hold time t , with Rb (open points)and without Rb (filled points). The molecules are preparedin the | , , (cid:105) state. Each point is the average and standarddeviation of 6 measurements. For the single species data theline is a fit to an exponential decay model. For the two-speciesdata the line is a fit to Eq.(2). Inset: frequency of Γ Rb-CaF estimates using the re-sampling method described in the text,together with a fit to a normal distribution.
Here Γ is the loss rate of CaF in the absence of Rb,Γ Rb is the loss rate of Rb, and Γ Rb-CaF is the collision-induced loss rate of CaF at t = 0. This isΓ Rb-CaF = k N Rb (0) (cid:90) f Rb ( (cid:126)r ) f CaF ( (cid:126)r ) d (cid:126)r = k ζ (3)where k is the inelastic rate coefficient and f s ( (cid:126)r ) is thedensity distribution of species s normalised such that (cid:82) f s ( (cid:126)r ) d (cid:126)r = 1. The equation defines ζ , an effective Rbdensity that accounts for the overlap between the twodistributions.Figure 1 is an example of one measurement showing thefraction of molecules remaining in the trap as a functionof t , for molecules prepared in the state | , , (cid:105) . In theabsence of Rb (filled points), the data fit well to a single-exponential decay with a loss rate of Γ = 0 . − .This is the natural loss rate due to collisions with resid-ual background gas and vibrational excitation by black-body radiation [43]. The Rb data, N Rb ( t ), also fit wellto a single exponential with Γ Rb = 0 . − . Thesesingle-species loss rates vary negligibly throughout theexperiments. The open points show the loss of moleculesin the presence of N Rb (0) = 1 . × atoms with apeak number density of 5(1) × cm − . We fit thesedata to Eq. (2) with N CaF (0) and Γ
Rb-CaF as free param-eters. Fixing Γ
CaF0 and Γ Rb to the values above yieldsΓ Rb-CaF = 0 . − for the data shown. To accountproperly for the uncertainties in Γ CaF0 and Γ Rb , we re-sample each measurement of N CaF ( t ) /N CaF (0) with re-placement, draw random values of Γ Rb and Γ CaF0 fromtheir measured distributions, then fit to Eq. (2). Repeat-ing this several thousand times yields the distribution of (10 cm ) R b C a F ( s ) Position (cm) c C a F ( c m ) Position (cm) c C a F ( c m ) c R b ( c m ) c R b ( c m ) Rb CaF
FIG. 2. Points: collisional loss rate, Γ
Rb-CaF , as a func-tion of effective density, ζ , for molecules in the | , , (cid:105) state.Line: straight line fit to the data. Insets: radial density dis-tributions, c s ( x ) = (cid:82) N s f s ( x, y, z ) dy dz , of the atoms andmolecules for small and large ζ . Γ Rb-CaF values shown in the inset of the figure. The meanand standard deviation of this distribution give our bestestimate for this dataset, Γ
Rb-CaF = 0 . − .To determine ζ , we measure the density distributionsof both species in the magnetic trap by turning off themagnetic field gradient and then imaging their distribu-tions. For Rb we use absorption imaging as described inmore detail in the Supplemental Material [44], whereasfor CaF we turn on the MOT light for 1 ms and image thefluorescence. We fit the radial distributions to Gaussiansand the axial distributions to the equilibrium distributionfor a magnetic quadrupole trap plus gravity [44]. We usethese fits to calculate the overlap integral in Eq. (3). Ourchoice of axial distribution automatically accounts for thedifferential gravitational sag between the two clouds. Thetemperatures in the magnetic trap are measured usingthe standard ballistic expansion technique. We find that,when the CaF is loaded into the trap, its 1 /e radius ex-pands from 1 . . µ K. This is because the ini-tial position of the CaF cloud is displaced from the trapcenter. The Rb cloud is well centered because of thegreater flexibility in controlling the intensity balance ofthe Rb MOT light. Its geometric-mean temperature inthe magnetic trap is 71(1) µ K.We repeat the measurements of Γ
Rb-CaF and ζ for var-ious values of N Rb (0). The Rb density distribution, f Rb ,depends on the number of atoms, so we measure it inevery case. Figure 2 shows how Γ Rb-CaF varies with ζ ,together with the measured radial cloud distributions forthe smallest and largest atom clouds used. The gradientof the straight-line fit gives the value of k . We account N C a F ( t ) / N C a F ( ) Rb CaF ( s ) F r e q u e n c y N=0N=1
FIG. 3. Loss of molecules from the magnetic trap in thepresence of 1 . × atoms, for molecules in the | , , (cid:105) state(open points) and the | , , (cid:105) state (filled points). Lines: fitsto Eq.(2). Inset: frequency of Γ Rb-CaF estimates using there-sampling method described in the text, together with fitsto normal distributions. for the uncertainties in the measured number of atomsand cloud sizes using a Monte-Carlo re-sampling of therelevant parameters from their measured distributions.The result is k = (6 . ± . × − cm / s for the state | , , (cid:105) . As discussed later, we attribute this loss to col-lisions with Rb that change the rotational state of themolecule from N = 1 to N = 0. This results in a loss ofmolecules for two reasons: (i) we detect molecules onlyin N = 1; (ii) the collision releases 1 K of energy, whichis far greater than the trap depth.For the measurement described above, both the atomsand molecules are in spin-stretched states, an arrange-ment that often suppresses inelastic collisions [3, 14]. Totest whether that is the case here, we repeat the measure-ment using molecules in the | , + , (cid:105) state [45], whichis not spin stretched. We obtain k = (5 . ± . × − cm / s, showing that these rotation-changing colli-sions have no strong dependence on the choice of hyper-fine or Zeeman level.Next, we make the same measurement for moleculesprepared in the ground rotational state | , , (cid:105) . Here,the only open loss channel is spin relaxation. Figure3 compares the loss rates for molecules in N = 0 and N = 1, in the presence of N Rb (0) = 1 . × atoms, dra-matically illustrating how the choice of rotational stateinfluences these ultracold inelastic collisions. For thesedata, the collisional loss rate is 0 . − for moleculesin | , , (cid:105) and 0 . − for molecules in | , , (cid:105) .Averaging several similar measurements we obtain k = − . ± . × − cm / s. This result is consistent withzero and yields the upper bound k < . × − cm / swith 95% confidence. This is an order of magnitudesmaller than the rotation-changing rate coefficient formolecules in the first excited rotational state.To interpret the experimental loss rates, we compare FIG. 4. Thermally averaged loss rate coefficient k from thesingle-channel model as a function of loss parameter y andshort-range phase shift δ s . The mean and standard errorof the experimentally measured k for molecules in | , , (cid:105) are shown as black lines. The 95% upper bound for k formolecules in | , , (cid:105) is shown as a dashed red line. to calculations with a single-channel non-universal modelbased on quantum defect theory [27, 44]. This models thelong-range interaction potential by its asymptotic form − C R − and takes full account of temperature depen-dence, which is important under the conditions of theexperiment. It has been used successfully to model lossesin collisions of ultracold RbCs molecules [46]. The modelcharacterizes the complicated short-range part of the in-teraction in terms of two parameters, y and δ s . The lossparameter y runs from 0 (no loss) to 1 (complete lossat short range, corresponding to a universal model). Itaccounts for all sources of collisional loss, including ro-tational inelasticity and spin-changing collisions. δ s isthe phase shift between the incoming wave and the re-flected wave at short range, and is related to the scatter-ing length.Figure 4 shows a contour plot of the rate coefficient ob-tained from the single-channel model as a function of y and δ s . The experimental value and range of uncertaintyfor | , , (cid:105) and the upper limit for | , , (cid:105) are indicatedas lines. The calculated loss rates show significant peaksaround δ s = 3 π/ π/ π/
8; these are associated with p-, d-, and f-wave shaperesonances, respectively. The corresponding s-wave peakaround δ s = π/
8, which is very prominent at lower tem-peratures [27, 46], is weakened at these temperatures dueto increased transmission past the long-range potentialand by thermal averaging.The measured loss rate for molecules in N = 1 is closeto the universal rate, which is 8 . × − cm s − atthis temperature. The most likely ranges of parameters have y > .
4, but the measurements do not rule outsignificantly lower y if there is resonant enhancement.Since the spin state is found to have little influence onthe loss rate and complex-mediated loss is unlikely tobe important for this system [44], we conclude that thisloss is dominated by rotation-changing collisions. Forthe rotational ground state, the measured loss rate isconsistent with loss parameters y < .
04. Overall, theresults are consistent with the expectation that rotationalrelaxation is very fast, but that spin relaxation is slow.Full coupled-channel scattering calculations includingnuclear spin [36] are not feasible for Rb+CaF, becausethe very deep potential well would require an impracti-cally large basis set for convergence. Even without nu-clear spin, they are very challenging. Morita et al. [47]have carried out such calculations on a single interac-tion potential for the related system Rb+SrF. However,the particular interaction potential they used produces ap-wave resonance very close to threshold; it would corre-spond to δ s ≈ π/ − cm − and a magneticfield of 1 G, their calculation gave a spin-relaxation crosssection of ∼
200 ˚A ; without thermal averaging, this cor-responds to k ≈ × − cm s − , which is well withinour experimental upper bound.In summary, we have produced the first mixtures oflaser-cooled atoms and molecules and have studied theirinelastic collisions in the ultracold regime. We have com-pared the results to a short-range loss model and find thatrotational relaxation proceeds rapidly, close to the uni-versal rate and independent of hyperfine and magneticsub-level, whereas spin relaxation is at least 10 timesslower. The latter observation is encouraging for theprospects of sympathetic cooling, where spin relaxationprocesses are expected to be a limiting factor.Our mixture can be used to investigate collisions inother settings. For example, studies in the dual-speciesMOT can shed light on inelastic collisions with excited-state atoms and light-assisted processes [48]. By captur-ing atoms and molecules from the mixture into tweezertraps, the collisions can be studied at the single-particlelevel [31]. It will be interesting to investigate the in-fluence of applied electric and magnetic fields on thesecollisional processes [15, 36]. The mixture could also beused to form ultracold triatomic molecules by photoasso-ciation or magnetoassociation. We are currently workingto trap more molecules, trap atoms at a higher density,improve the overlap of the clouds, and incorporate anoptical dipole trap. With these improvements, we aim tostudy the elastic collisions required for sympathetic cool-ing of molecules by evaporatively cooled atoms [8], whichis a promising way to increase the phase-space density to-wards quantum degeneracy.We acknowledge helpful discussions with Micha(cid:32)lTomza. This work was supported by EPSRC grantEP/P01058X/1. [1] L. D. Carr, D. DeMille, R. V. Krems, and J. 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I. DETERMINING ATOM DENSITY
We determine the number densities of the trappedatom clouds by absorption imaging. A collimated laserbeam traverses the atom cloud whose shadow is imagedonto a CCD camera using a pair of achromatic lenses.The first lens has a focal length of f = 300 mm and is adistance f + δl away from the atoms, while the secondlens has f = 125 mm and is a distance f + δl fromthe camera. The lenses are separated by f + f . Here δl and δl account for positioning errors in the imagingsystem. Due to its refractive index and Gaussian densitydistribution, the atom cloud acts like a lens whose focallength is approximately f A = − π Γ σ ∆ λτ (S1)where ∆ is the detuning of the light, λ is the wavelength,Γ is the decay rate of the excited state, σ is the stan-dard deviation of the Gaussian density distribution and τ is the optical depth at the center of the cloud. When∆ < f A is positive, meaning that the cloud acts as aconverging lens. The magnification of the entire imagingsystem is M = δl f f f A + f ( δl − f A ) f f A . (S2)For a resonant probe, f A → ∞ and the magnificationis simply M = − f /f and is robust against δl and δl .On resonance however, the degree of absorption by theatom cloud exceeds the dynamic range of the camera, sowe take absorption images with ∆ = − . δl = δl = 0)the lensing effect of the atom cloud is canceled and againthe magnification is M = − f /f . To investigate theresidual systematic shift due to lensing when there arepositioning errors ( δl , (cid:54) = 0), we acquire absorption im-ages for a range of detunings. Figure S1 shows how theapparent cloud size varies with ∆. As expected fromthe qualitative behaviour described above, there is a dif-ference of about 10% in apparent size between positiveand negative ∆ over the range explored. We concludethat in our standard absorption images the cloud size isunderestimated by 3–5% (depending on the atom den-sity) and we apply a systematic correction to all the Rbdata to account for this. This leads to a 4% systematiccorrection to the values of k , which we have applied toour results. The uncertainty in making this correctioncontributes negligibly to the overall uncertainty of themeasurement.In the absorption imaging, the effective absorptioncross section depends on the polarization of the probe
30 20 10 0 10 20 30 40
Detuning (MHz) S i z e ( mm ) FIG. S1. Apparent RMS radial size of the atom cloud forvarious probe detunings. The sizes are determined by fitting2D Gaussian distributions to the acquired images. beam relative to the applied magnetic field, the inten-sity and detuning of the probe beam, the initial statedistribution of the atomic sample and the duration ofthe imaging pulse. We optically pump atoms to thestate | F = 2 , M F = 2 (cid:105) prior to magnetic trapping, andthe magnetic field gradient of the trap is slightly belowthe value needed to levitate atoms in | , (cid:105) or | , − (cid:105) against gravity. This removes any uncertainty aboutthe state distribution. We use a set of three orthogonalHelmholtz coil pairs to cancel the background magneticfield and apply a 200 mG bias field along the direction ofthe probe beam. The coils are calibrated by microwavespectroscopy of the CaF molecules using a magnetically-sensitive hyperfine component of the rotational transi-tion [43]. The probe laser beam is circularly polarizedusing a quarter wave plate. The variation of the opticaldepth with the angle of the quarter wave plate matcheswell with the results of a model of the absorption imag-ing where we solve the set of rate equations describingthe atom-light interaction for various polarizations. Thisgives us high confidence that the imaging is well under-stood. In the experiments, we set the polarization todrive σ + transitions from the | , (cid:105) state, so that the res-onant absorption cross section is 3 λ / (2 π ). II. EQUILIBRIUM DISTRIBUTION INMAGNETIC QUADRUPOLE TRAP
In thermal equilibrium at temperature T , the densitydistribution of N atoms of mass m and magnetic moment µ in a quadrupole magnetic trap with axial field gradient b is n ( x, y, z ) = N (cid:0) γ − (cid:1) πz exp (cid:32) − (cid:115) z z + x + y z − γ zz (cid:33) (S3)where γ = mg/µb , z = k B T /µb , and g is the accelerationdue to gravity, which is in the − z direction. Integratingover the x and y dimensions gives the one-dimensionalprobability density in z , g ( z ) = (cid:0) γ − (cid:1) z (cid:18) | z | z (cid:19) exp (cid:18) − | z | + γzz (cid:19) , (S4)normalised such that (cid:82) g ( z ) dz = 1. We fit the axial dis-tributions of both the atoms and molecules to g ( z ). Thedata fit reasonably well to this model, and the fits givetemperatures close to those measured by the ballistic ex-pansion method, even though the molecules are unlikelyto come into thermal equilibrium in the magnetic trap.We find that the radial distributions fit well to Gaussiandistributions h ( x ) = 1 √ πσ exp (cid:18) − x σ (cid:19) . (S5)We model the density distributions of each species as f s ( x, y, z ) = h s ( x ) h s ( y ) g s ( z ) ( s ∈ { CaF , Rb } ) using theparameters obtained from the fits. The overlap integralof these distributions is (cid:90) f ( x, y, z ) f ( x, y, z ) dx dy dz = A A z (cid:0) C (cid:0) z − z , z , (cid:1) − z (cid:0) z + z , z , (cid:1)(cid:1) π ( σ + σ ) (2 z + C ) ( B z , + B z , ) , where A i = γ i − ,B i = γ i − ,C = γ z , + γ z , ,z = 12 ( z , + z , ) . We use this analytical form of the overlap integral to eval-uate the effective density ζ (see Eq. (3)). This methodaccounts for the differential gravitational sag between thetwo species in a natural way. III. CALCULATION OF LOSS RATES
Full quantum calculations for systems like Rb+CaFare highly challenging due to the combination of large masses, deep and strongly anisotropic interaction poten-tials, and electron and nuclear spins. Even if they canbe performed, uncertainties in the interaction potentialslead to large uncertainties in cross sections and rates atultracold temperatures. This happens because the scat-tering length goes through a full cycle from −∞ to + ∞ as an extra bound state is added to or removed from thewell. Therefore, we instead use an approximate single-channel quantum defect theory (QDT) model which ac-curately reproduces the range of possible loss rates [27]at much lower computational cost. The model is basedon analytic solutions on the asymptotic C R − potential[49–51], for which we use C = 3084 E h a as estimatedin Ref. [8]. This produces a p-wave barrier height equiv-alent to 135 µ K. The complicated short-range physics,including loss processes, is represented simply by an ab-sorbing boundary condition. This is characterized bya non-unitary short-range S-matrix, which can be pa-rameterised as S = (1 − y )(1 + y ) exp( δ s − π/ y is the loss parameter of Idziaszek and Julienne [26]and δ s is the short-range phase shift. In the absenceof loss ( y = 0), this phase shift is related to the scat-tering length by a/ ¯ a = 1 + cot( δ s − π/ δ s = π/ a = 0 . . . . (2 µC / (cid:126) ) / = 35 . T = µ ( T Rb /m Rb + T CaF /m CaF ) =133 µ K. Using this, the loss rates are thermally aver-aged as k ( T ) = (cid:82) ∞ (2 / √ π ) k ( E ) x / exp( − x ) dx , where x = E/k B T .To consider whether complex-mediated loss may be asignificant factor for this system, we estimate the densityof states near threshold. Christianen et al. [53] give asimple analytic approximate expression for the density ofstates of diatom+diatom complexes, and we adapt theirmethodology to apply to atom+diatom systems [54]. Weuse the estimated parameters for the interaction of Rband CaF from Ref. [8] and the CaF diatom parametersfrom [55]. The resulting density of states is 4 K −1