Universality of the three-body Efimov parameter at narrow Feshbach resonances
Sanjukta Roy, Manuele Landini, Andreas Trenkwalder, Giulia Semeghini, Giacomo Spagnolli, Andrea Simoni, Marco Fattori, Massimo Inguscio, Giovanni Modugno
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] M a r Universality of the three-body Efimov parameter at narrow Feshbach resonances
Sanjukta Roy , Manuele Landini , Andreas Trenkwalder , Giulia Semeghini , GiacomoSpagnolli , Andrea Simoni , Marco Fattori , , Massimo Inguscio , , and Giovanni Modugno , LENS and Dipartimento di Fisica e Astronomia, Universit´a di Firenze,and Istituto Nazionale di Ottica, CNR, 50019 Sesto Fiorentino, Italy INFN, Sezione di Firenze, 50019 Sesto Fiorentino, Italy and Institut de Physique de Rennes, UMR 6251, CNRS and Universit´e de Rennes 1, 35042 Rennes Cedex, France (Dated: October 29, 2018)We measure the critical scattering length for the appearance of the first three-body bound state,or Efimov three-body parameter, at seven different Feshbach resonances in ultracold K atoms.We study both intermediate and narrow resonances, where the three-body spectrum is expectedto be determined by the non-universal coupling of two scattering channels. We observe insteadapproximately the same universal relation of the three-body parameter with the two-body van derWaals radius already found for broader resonances, which can be modeled with a single channel.This unexpected observation suggests the presence of a new regime for three-body scattering atnarrow resonances.
PACS numbers: 34.50.-s; 34.50.Cx
Recent experiments on ultracold atoms with Feshbachresonances [1–13] have opened up a new path to studythe Efimov spectrum of the three-body bound states thatarise in the presence of resonant two-body interactions[14]. Under these conditions, the details of the underlyingforces become irrelevant, leading to a universal behaviorof completely different systems. This phenomenon wasfirst described in the context of nuclear physics, but isnow explored also in atomic, molecular and condensed-matter systems [15–18]. The resonant interaction is ex-pected to give rise to a three-body potential scaling as1 /R , where R is the hyperradius that parametrizes themoment of inertia of the system. This leads to an infiniteseries of trimer states with a universal geometrical scal-ing for the binding energies. For a finite, negative two-body scattering length a , the three-body potential has along-range cutoff at R ≃ | a | , and only a finite numberof bound states exist. The critical scattering length a − for the appearance of the first Efimov state at the three-body threshold, often called the three-body parameter,was expected to be the only parameter to be influencedby non-universal physics, i.e. by the microscopic detailsof two or even three-body forces [14, 16]. While a clearevidence of the universal scaling of the Efimov spectrumis still missing, recent experiments on identical bosonssuggested that also a − might be universal [19]. This sur-prising result has been interpreted in a recent series oftheoretical studies [20–22]. The underlying idea is thatthe sharp drop in the two-body interaction potential ata distance of the order of the van der Waals radius R vdW results in an effective barrier in the three-body poten-tial at a comparable distance [22]. This prevents thethree particles to come sufficiently close to explore non-universal features of the interactions at short distances,and leads to a three-body parameter set by R vdW alone, a − ≃ − . R vdW [19, 21, 22].However, this scenario is realized only for the broad Feshbach resonances studied so far in most experiments,which can be described in terms of a single scatteringchannel, the so-called open channel. For narrow reso-nances one must instead take into account the couplingof the open and a second closed channel [23]. It hasbeen shown that in this case a new length scale that de-pends on the details of the specific Feshbach resonance,the so-called intrinsic length R ∗ > R vdW , must be in-troduced to parameterize the two-body scattering. Thethree-body potentials are also modified, with an expecteddeviation from the Efimovian dependence into 1 / ( R ∗ R )for distances R < R ∗ [24]. This tends to reduce thedepth of the three-body potential, and leads to the non-universal result a − = − . R ∗ [24, 25], which is muchlarger than that obtained for broad resonances. This pre-diction is valid only close to resonance, where | a | ≫ R ∗ .It is still unclear how a − scales in the intermediate regimeof | a | ≃ R ∗ or generally for resonances of intermediatewidths. Various general models have been proposed [26–31], but they are either not fully predictive, or give con-tradicting results.In this Letter we address this problem by performingan experimental study of three-body collisions in ultra-cold bosonic K atoms, where we determine the three-body parameter a − at several Feshbach resonances of in-termediate or narrow width. In particular, our measure-ments probe for the first time the regime of very smallresonance strengths, s res = 0 . R vdW /R ∗ ≃ .
1, where R ∗ might be expected to be the relevant length-scale thatdetermines a − . Surprisingly, we find values of a − thatare around the same − . R vdW measured for broad res-onances, suggesting the existence of a novel intermediateregime of three-body scattering.The investigation of closed-channel dominated Fesh-bach resonances is particularly favoured in K, whichhas several resonances with moderate magnetic width∆ and relatively small background scattering length − a bg ≃ a [32]. These parameters, together withthe difference of the magnetic moments of the closedand open channels, δµ , determine the intrinsic length R ∗ = ¯ h / ( ma bg ∆ δµ ) [23]. In particular, we investigatedseven different resonances with s res in the range 0.1-2.8in the three magnetic sub-levels of the hyperfine groundstate F =1 [32]. One of the broadest of those resonanceswas already studied before [2], and our new data clarifiesan apparent deviation from the universal behaviour.A detailed description of the experimental set-up andmethods for preparing Bose-Einstein condensates of Katoms by direct evaporation is given elsewhere [33]. Thethree-body parameter was determined by finding themaximum of the three-body loss coefficient K in theregion of negative a at each Feshbach resonance, as inprevious experiments [1–13]. In the presence of three-body losses, both the atom number N and tempera-ture T evolve according to dN/dt = − K h n i N and dT /dt = ( K / h n i T , where h n i = (1 /N ) R n ( ~x ) d x is the mean square density [34]. The temperature in-crease is due to the preferential removal of atoms in thehigh-density region around the trap center. The typi-cal starting condition was a non-condensed sample with3-80 × atoms in a temperature range of 20-400 nK,depending on the spin channel and Feshbach resonance[35]. The atoms were held in a purely optical trap (or inan optical trap with an additional magnetic confinement,depending on the specific resonance) at sufficiently lowdensity to have a negligible mean-field interaction en-ergy. Care was taken to have a trap depth sufficientlylarge to avoid an evaporation associated to the heating.The samples were initially prepared at small negative a inproximity of the Feshbach resonances; the measurementsstarted 10 ms after the scattering length was ramped tothe final value in about 2 ms. A t o m nu m b e r ( ) T e m p e r a t u r e ( K ) t (s) FIG. 1: Example of the time-evolution of the atom num-ber (circles) and temperature (triangles), fitted to Eq.1 (solidline) and Eq.2 (dashed line) to determine the three-body loss-coefficient K . Fig. 1 shows a typical evolution of N and T , as mea-sured by absorption imaging after a free expansion. They were simultaneously fitted with N ( t ) = N / (1 + 3 β √ N T K t ) / , (1) T ( t ) = T (1 + 3 β √ N T K t ) / . (2)Here N and T are the initial atom number and temper-ature, respectively, and β = ( m ¯ ω / πk B ) / , with ¯ ω themean trap frequency. In such a fit, one-body losses wereneglected, since they occur on a much longer timescale.Crucial ingredients for a reliable measurement of the K dependence on the scattering length were an accuratecalibration of the magnetic field B and the use of a high-quality coupled-channel (CC) model for a ( B ), based ona large number of experimental observations for the posi-tions and widths of the Feshbach resonances [32, 35]. Thecenters and widths of the Feshbach resonances were rede-termined in the present work, finding a good agreementwith the theoretical ones. An additional confirmation ofthe CC model was derived from a direct measurement ofthe dimer binding energy at the two narrowest resonancesby radio-frequency spectroscopy. The magnetic field hada stability of better than 0.1 G, and was calibrated bymeans of radio-frequency or microwave spectroscopy withan accuracy of 0.1 G. The inhomogeneity of B across theatomic samples was estimated to be less than 0.01 G inall cases.We observed for all Feshbach resonances a clear peak in K in the region of | a | =600-1000 a , as shown in Figs.2-3. We compared the observations to the known relationfor identical bosons at zero collision energy and in thezero-range approximation, for a < K ( a ) = 4590 3¯ ha m sinh(2 η − )sin [ s ln( a/a − )] + sinh η − . (3)Here s ≃ . η − isthe decay parameter which sets the width of the Efimovresonance and incorporates short-range inelastic transi-tions to deeply bound molecular states [16]. At the fi-nite temperature of the experiment, there is a limita-tion in the maximum observable K set by unitarity at K max = 36 √ π ¯ h / ( k B T ) m [37, 38]. Therefore, weused an effective rate of the form (1 /K ( a )+1 /K max ) − .As shown in Fig.2, the experimental K ( a ) for the fivebroadest resonances is in good agreement with Eq.3, be-sides a multiplicative factor of order 3 that can be jus-tified with the experimental uncertainty in the determi-nation of the density [35]. We extracted the relevantparameters a − and η − with a fit.Also the two narrowest Feshbach resonances featurea maximum in K around -1000 a , as shown in Fig.3.There is however a slower background variation of K with a . It was shown that for narrow resonances oneshould expect a slower evolution in the regime | a | < R ∗ ,with K ∝ | a | / [24], but also this behavior does notseem to reproduce the data. Since these two resonancesare in an excited spin state, there is in principle also acontribution of two-body processes in the losses, whichare expected to have a slower dependence on a [23].While it was not possible to distinguish in a reliable waytwo- from three-body losses in the experiment, we haveverified that only the observed K ( a ) far from the lossmaxima might be partially attributed to two-body losses[35]. The low- | a | tail at the Feshbach resonance centeredat B = 58 .
92 G might also be affected by a nearby nar-row d -wave resonance [35]. While the deviations of themeasured K from theory for these narrow resonanceswill deserve further investigation, in the present work weidentified the position of the Efimov resonance with thepeak of a Gaussian fit of the measured maximum in K as shown in Fig. 3.
300 400 500 600 800 1000 2000 3000 s res = 2.8 (m F =0) s res = 1.05s res = 2.5s res = 2.6s res = 2.8 (m F =+1) K ( - c m / s ) |a| (a ) FIG. 2: Three-body loss rate measured in the proximity offive Feshbach resonances of intermediate strength (see Table1 for the assignment of the spin state). The experimental data(squares) is fitted to Eq.3 incorporating the effect of unitarityat finite temperature (solid line).
A summary of our analysis is reported in Table I. For
200 1000 10000 res = 0.11 |a| (a ) s res = 0.14 K ( - c m s - ) FIG. 3: Three-body loss rate measured in the proximity of twonarrow Feshbach resonances in the m F = 0 state. The exper-imental data (squares) is compared with Eq.3, using η − =0.1(solid line), and fitted with a Gaussian (dashed line) to de-termine a − from the position of the loss maximum. the calculation of a ( B ), we used the experimentally de-termined Feshbach resonance centers B exp and the reso-nance widths and the background scattering lengths fromthe CC model. The uncertainties in B exp include those inthe calibration of B and in the determination of B fromthe loss resonances. Particular care was put in the deter-mination of B for the two narrowest resonances, wherewe found a rather good agreement between independentmeasurements of the atom losses and of the binding en-ergy [35]. The uncertainties in a − include the statisti-cal uncertainties from the fit of the K data and fromthe determination of a ( B ). For the two narrowest reso-nances, the dominant source of uncertainty comes fromthe determination of B . The reported values of R ∗ aredetermined from the on-resonance predictions of our CCmodel [32, 35]. We observe a whole range of values of η − for the different Efimov resonances; this is probably aconsequence of the different measurement temperatures,but possibly also of the non-universal nature of η − .A comparison of the results in Table 1 leads to thestriking conclusion that the three-body parameter a − stays around values of the order of -10 R vdW for all theFeshbach resonances explored in K, including the oneswith R ∗ as large as ∼ a , hence much larger than R vdW . We note that in the earlier measurement at theresonance in the m F =1 state [2] a similar value of a − wasfound, but we cannot confirm the additionally observedfeature at 1500 a . We suspect this was an artefact ofthe analysis of the limited time-dependent data.Fig.4 shows the measured | a − | /R vdW as a function of s res . One notes just a moderate deviation of our datafrom the mean value 9.73(3) measured for open-channel TABLE I: Theoretical and experimental parameters for thethree-body resonances at Feshbach resonances in the m F spinchannels: measured resonance center B exp ; intrinsic length R ∗ and strength s res of the Feshbach resonances from the CCmodel; measured three-body parameter a − and decay param-eter η − ; initial temperature T of the atomic sample. For K, R vdW =64.49 a .m F B exp (G) R ∗ ( a ) s res − a − ( a ) η − T (nK)0 471.0 (4) 22 2.8 640 (100) 0.065 (11) 50 (5)+1 402.6 (2) 22 2.8 690 (40) 0.145 (12) 90 (6)-1 33.64 (15) 23 2.6 830 (140) 0.204 (10) 120 (10)-1 560.72 (20) 24 2.5 640 (90) 0.22 (2) 20 (7)-1 162.35 (18) 59 1.1 730 (120 0.26 (5) 40 (5)0 65.67 (5) 456 0.14 950 (250) 330 (30)0 58.92 (3) 556 0.11 950(150) 400 (80) -1.0 -0.5 0.0 0.51015202530 | a - | / R v d W log(s res ) FIG. 4: (color online) Ratio of the measured three-body pa-rameter to the van der Waals radius for K atoms as a func-tion of the strength of the Feshbach resonance (blue circles)and comparison to the predictions of theoretical models inRef.[21] (gray shaded region) and Ref.[30] (solid line). Thedashed lines show the maximum and minimum of the scatterin the experimental data for the three-body parameters mea-sured at broad Feshbach resonances in other atomic species. dominated resonances [12, 19, 21, 36], and also for otherintermediate resonances [4–6, 13, 19, 36]. This observa-tion is far from the already mentioned prediction for nar-row resonances [24, 25], which indicates that the Efimovresonances should appear at scattering lengths that aremultiples of a − = − . R ∗ by a factor exp( π/s ) ≃ . | a | ≫ R ∗ ≫ | a bg | , where the three-body potential at large hyperradii R > R ∗ has an Efi-movian character [24]. The present experiment does notaccess this extreme limit but is in an intermediate regimealso for the two narrowest resonances, which show indeed R ∗ ≃ | a − | .Other models for the three-body physics at Feshbachresonances of intermediate strength have been proposed[26–31]. The specific problem of connecting the resultsfor the three-body parameter in the open-channel dom-inated regime, where a − is determined by R vdW , andthe closed channel limit, where it is R ∗ which sets thescale for a − , has been addressed recently [30, 31], findinghowever considerably different results. In particular, themodel of Ref.[30] predicts that a crossover between thetwo regimes of broad and narrow resonances would takeplace around s res ≃
1, as shown in Fig.4. Additionally,the regime of a − = − . R ∗ should be reached only forexcited Efimov states, while the first one has a slightlysmaller a − = − . R ∗ . Although an increase of | a − | with decreasing s res might be present in the experimentaldata, there is a clear disagreement with such predictions.Experiments on Li and
Cs have also measured sim-ilar values for a − at three intermediate resonances with s res = 0 . − K.We note that for the two narrowest resonances | a − | isonly a factor of two larger than R ∗ . This observationseems to indicate that the three-body potential can sup-port a bound state that resides only in the region withhyperradius R ≤ R ∗ . This is a regime that was notaccessible in previous one-channel models, and a multi-channel approach will be presumably necessary to modelthe experimental observations.In conclusion, our study showed an apparent universalbehaviour of the three-body parameter on several dif-ferent Feshbach resonances of the same atomic species,down to a resonance strength s res ≃ .
1. This givesimportant information on the three-body physics in thisnarrow-resonance regime, where one expects a combinedrole of the open and closed molecular channels. Our re-sults will provide a benchmark for three-body multichan-nel models. By employing a narrower, low-field Fesh-bach resonance in K [32], further experiments probinga regime of even smaller s res ≃ .
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Given the importance of the quality of the coupled-channels (CC) model to predict a reliable a ( B ), we con-firmed experimentally several of its predictions for theseven Feshbach resonances investigated in this work. Oneexample of the typical calibration measurements is shownin Fig.5 for the narrow resonance at B ≃ m F =0 state. For example, the centers of the Feshbachresonances were experimentally determined, by measur-ing the atom losses versus the magnetic field after a con-stant waiting time. The resonance centers were identifiedwith the loss maxima. Also the positions of the zero-crossings of a were determined as minima in the elas-tic collisional rate, after a forced evaporation procedure.The zero crossings were identified with the maxima in thetemperature. In all cases we found a very good agreementwithin the typical experimental and theoretical uncer-tainties, as shown for example for the resonance centersin Table II. The uncertainties in the experiment are ofthe order of 0.1 G for the centers and 1 G for the widths.The corresponding uncertainties for the CC calculationsare of the order of 0.1 G.Note that the theoretical parameters for the Feshbachresonances are slightly different from those previously re-ported [32], which considered only s -wave scattering; thepresent calculations include instead also d - and g -wavescattering. In Table II we also report a narrow d -waveresonance, also predicted by the CC model, in the vicin-ity of the narrow resonance at B ≃ | a | ≃ a .We used particular care in determining the centers ofthe two narrow resonances, where the interesting regionof | a | ≃ R ∗ appears at detunings B − B of the order of0.1 G, which is also our typical accuracy in the calibra-tion of the magnetic field. In these cases, the resonancecenter was determined also from the measurement of the B(G)
B(G)
35 40 45 5080100120140 b) c)d) A t o m nu m b e r ( ) B(G) T e m p e r a t u r e ( a r b . un it s ) a) -23 -22 -21 K ( c m s - )
35 40 45 50 57.5 58.0 58.5 59.0 59.5 60.0-1000-50005001000 S ca tt e r i ng l e ng t h ( a ) B (G) x FIG. 5: Example of the calibration measurements for the narrow Feshbach resonance centered at B ≃ m F =0state. a) The magnetic-field position of the zero crossing was determined by evaporation measurements to be at B zc =43(2) G.b) The position of the resonance center was determined via loss measurements to be at B =58.9(2) G. c) The center wasindependently determined from the three-body loss data to be at B =58.92(3)(10) G, where the first uncertainty is statistical,and the second one is the accuracy in the absolute calibration of B . Lines are Gaussian fits to the data. d) The scatteringlength vs the magnetic field from the CC model of Ref.[32] predicts the zero crossing at B zc =44.3(2) G and the center at B =58.8(1) G. To make the zero-crossing more visible, the scattering length in that region was multiplied by a factor 50. three-body loss coefficient on both sides of the resonance,thus achieving a precision as low as 0.03 G. One measure-ment is shown in Fig.5c. In absence of a detailed modelof the loss rates very close to the resonance center, weperformed a Gaussian fit of the loss maximum, takingthe center as B and the halfwidth at 1/ e as the uncer-tainty. The values for B measured via the plain lossesand K are in good agreement between themselves andwith the theory, with a typical deviation below 0.1 G. Measurement of the dimer binding energy andestimation of R ∗ We obtained another confirmation of the quality ofthe CC model by directly measuring the binding energyof the weakly-bound dimers at the two Feshbach reso-nances. We employed a standard technique: a weakly-interacting Bose-Einstein condensate was prepared in the m F =-1 state, and then transferred to the m F =0 state bya radio-frequency pulse. At small detunings from the res-onance, we observed a transfer to both the atomic and amolecular state, as shown in the example in the inset ofFig.6. The atomic signal is power broadened because theRabi frequencies of the atom-molecule and atom-atomtransitions are very different already for small detun-ings. We fitted the spectra of the atomic and molecu- lar signals with Gaussian distributions. The molecularbinding energy E b was determined as twice the energydetuning between the atomic and molecular peaks, tak-ing conservatively as uncertainty the quadrature of theindividual widths of the fitted Gaussian. For example,Fig.6 shows the measured E b for the resonance centeredat B ≃ B ≃ K cannot be rigorously interpreted in terms of one-or two-channel models over a wide range of magnetic fieldvalues, because of the presence of broad avoided crossings[32]. This makes the calculation of the intrinsic lengththrough the equation R ∗ = ¯ h /ma bg δµ ∆ difficult, sincethe relative molecular magnetic moment δµ changes withthe magnetic field. The values of δµ previously reportedin [32], represent indeed the average magnetic momentin a finite region of magnetic fields close to the resonancecenters.To determine R ∗ more accurately, we first extractedfrom our numerically calculated collision phase shift thevalue r res e of the effective range at resonance ( a → ∞ ).We then used the relation between r res e and the intrinsic atom molecule N ( ) (MHz) atom B i nd i ng e n e r gy ( k H z ) Magnetic field (G)
FIG. 6: Dimer binding energy versus the magnetic field. E b /h is determined as the difference between the molecular (redsquares) and the atomic (green squares) maxima in the ra-diofrequency transfer. For the atomic transition we plot thedeviations of the observed radio frequency transitions to theexpected ones. The data from coupled-channels model ofRef.[32] (blue circles) has been shifted to lower fields by 0.08 Gto match the experiment. A fit of the theoretical binding en-ergy with the two-channel model of Ref.[29], also given in Eq.5 (blue line), predicts B = 58 . m F =-1 to the m F =0 state, where the signal is the num-ber of atoms left in m F =-1 after the radiofrequency transfer.Both atomic and molecular signals can be seen. resonance length r res e = − R ∗ + 23 π Γ (cid:18) (cid:19) R vdW (4)obtained within a quantum-defect model in Ref.[39].To gain more analytical insight we also used an ex-actly solvable two-channel model developed in Ref.[27,29], which has already been adapted to narrow andintermediate-strength Feshbach resonances such as theones in K. In this model, the binding energy for | a | ≫ a bg is expressed as E b = ¯ h q dim / m , with q dim = ( a − p a − ar e ) /ar e ) , (5)where the effective range is r e ≃ − R ∗ (1 − a bg /a ) + 4 b/ √ π − b /a . (6)Here b is a fitting parameter of the order of the van derWaals radius.We fitted the calculated binding energies for all sevenFeshbach resonances investigated in the experiment, us-ing the calculated values for the background scatteringlength a bg and for the resonance width ∆ to evaluate a ( B ), while leaving both b and R ∗ as fit parameters. Inthe fit we employed CC data in a range of detunings | B − B | < δB max , with a maximum δB max of the order of 0 .
5∆ for each resonance; we derived a characteristicuncertainty on the fitted quantities by comparing the re-sults for the cases δB max = 0 .
75∆ and δB max = 0 . R ∗ arein very good agreement with the numerically exact CCcalculation. Analysis of the K measurements We now discuss in more detail the analysis of theloss measurements that resulted in the data of Figs.2-3 of the main paper. We recall that to determine K we fitted the coupled equations for N ( t ) and T ( t )at each magnetic field, obtaining the three quantitiesΓ = 3 β N K / √ T , N and T . The typical statis-tical errors are: 20% for Γ; 3% for N ; 10% for T . Theoverall relative statistical error on K , obtained by sum-ming the individual contributions, typically amounts to60%, and is quantified by the error bars in Figs.2-3 of themain paper.In addition, we have two systematic sources of erroron K . The first one is a 30% uncertainty on N , whichcomes from our imperfect knowledge of the actual absorp-tion cross-section of the atoms. The second one comesfrom the uncertainty in the mean trap frequency ¯ ω , whichis contained in the parameter β ( β ∝ ¯ ω ). This was mea-sured by exciting a sloshing motion of the atomic samplealong the three trap axes, with a typical uncertainty of20%. The overall systematic uncertainty is therefore aslarge as 200%. This implies that the K scale is deter-mined only up to a factor of about 3.The unitarity-limited three-body rate contains in-stead only one experimental parameter, the temperature: K max = 36 √ π ¯ h / ( k B T ) m . In our analysis we usedthe initial temperature T to estimate K max , with anoverall uncertainty that is negligible with respect to thescaling factor above.When fitting the experimental data for the five broadresonances, we allowed for a single fitting factor in frontof the overall rate (1 /K ( a ) + 1 /K max ) − . In all caseswe found factors smaller than 3, in agreement with theestimated uncertainty. Three-body and two-body decay
In principle, atoms in the excited m F =0,-1 states candecay also via two-body relaxation. The two-body rateis normally very small, since it involves processes thatchange the total m F , but it can become relevant closeto the Feshbach resonances, where the universal behav-ior for the two body constant K ∝ a sets in. Whilethe measured K at the broader Feshbach resonances fol-lows the expected a behavior, a possible reason for theslower variation of K with a at the two narrow Feshbach TABLE II: Theoretical and experimental parameters for the Feshbach resonances in the spin channels m F : resonance centersfrom the CC model ( B th ) and the experiment ( B exp ); resonance widths ∆ and background scattering lengths a bg [32]; intrinsiclength ( R ∗ ) calculated from the on-resonance effective range (see text) or ( R ∗ ch ) derived from a fit to a two-channel model[27]; resonance strengths s res ; measured three-body parameter a − and decay parameter η − ; initial temperature T ; initial atomnumber N and mean trap frequency ¯ ω/ π . m F B th (G) B exp (G) − ∆(G) a bg ( a ) R ∗ R ∗ ch s res a − ( a ) η − T (nK) N (10 ) ¯ ω/ π (Hz)0 471.9 471.0 (4) 72 -28 22 20(2) 2.8 640 (100) 0.065 (11) 50 (5) 5 (2) 28+1 402.4 402.6 (2) 52 -29 22 22(3) 2.8 690 (40) 0.145 (12) 90 (6) 40 (3) 14-1 33.6 33.64 (20) -55 -19 23 23(2) 2.6 830 (140) 0.204 (10) 120 (7) 80 (20) 14-1 560.7 560.72 (15) 56 -29 24 23(2) 2.5 640 (90) 0.22 (2) 20 (5) 3 (2) 14-1 162.3 162.35 (18) 37 -19 59 59(3) 1.1 730 (120) 0.26 (5) 40 (8) 16 (3) 120 65.6 65.67 (5) 7.9 -18 456 449(8) 0.14 950 (250) - 330 (30) 5(2) 1400 58.8 58.92 (3) 9.6 -18 556 559(1) 0.11 950(150) - 400 (80) 7 (1) 1360 a a d -wave -13 -12 -11 -10 -13 -12 -11 -10 b) a) K ( c m s - ) |a| (a ) FIG. 7: Comparison of the measured K (dots) with the CCcalculations (solid line), for the narrow Feshbach resonancesaround: a) 58.8 G; b) 65.6 G. resonances might be indeed a contribution of two-bodyprocesses, namely dipolar relaxation of a pair of m F =0atoms into m F =1 atoms. To investigate this possibility,we tried to model the observed evolution of N ( t ) and T ( t )also as a two-body decay, for which the rate equations are dNdt = − K h n i N , (7) dTdt = K h n i T , (8) where h n i = (1 /N ) R n ( ~x ) d x is the mean density. Acompact solution for the average density can be found as h n ( t ) i = h n i h n i K t , (9)which is nominally different from the analogous solutionfor h n ( t ) i in presence of three-body losses h n ( t ) i = h n i h n i K t . (10)However, the two solutions are not too different, and theexperimental data can in general be fitted almost equallywell using the two models.Since K can be predicted exactly by the CC model,we could however compare a K fitted from the exper-imental data assuming only two-body losses, with thecalculated one. As shown in Fig.7, this comparison indi-cates that the two-body decay is unlikely to play a role inthe observed decay for the resonance at 58.9 G, but mightindeed contribute to the observed background losses forthe resonance at 65.6 G. Note however that we estimatea typical uncertainty of about 2 in the overall scale ofthe measured K , so that further detailed experiments,possibly with different T and N0