Feshbach resonances in 23 Na+ 39 K mixtures and refined molecular potentials for the NaK molecule
Torsten Hartmann, Torben A. Schulze, Kai K. Voges, Philipp Gersema, Matthias W. Gempel, Eberhard Tiemann, Alessandro Zenesini, Silke Ospelkaus
FFeshbach resonances in Na+ K mixtures and refined molecular potentials for theNaK molecule
Torsten Hartmann, Torben A. Schulze, Kai K. Voges, Philipp Gersema,Matthias W. Gempel, Eberhard Tiemann, Alessandro Zenesini, and Silke Ospelkaus ∗ Institut f¨ur Quantenoptik, Leibniz Universit¨at Hannover, 30167 Hannover, Germany (Dated: April 3, 2019)We present a detailed study of interspecies Feshbach resonances of the bosonic Na+ K mixturefor magnetic fields up to 750 G in various collision channels. A total of fourteen Feshbach resonancesare reported, as well as four zero crossings of the scattering length and three inelastic two-body lossfeatures. We use the observed magnetic field locations of the resonant features together with theknown data on Na+ K to refine the singlet and triplet ground state potentials of NaK and achievea consistent description of Feshbach resonances for both, the Bose-Bose mixture of Na+ K aswell as the Bose-Fermi mixture of Na+ K. We also discuss the influence of the interplay betweeninelastic two-body and three-body processes on the observation of a Feshbach resonance.
I. INTRODUCTION
Mixtures of ultracold atoms have recently attractedgreat interest as they enable the study of exciting quan-tum many-body effects [1]. Furthermore, molecules intheir ro-vibrational ground state are promising candi-dates for the study of dipolar many-body physics [2–4]. Ensembles of Na+K feature several predicted in-terspecies Feshbach resonances [5] and their heteronu-clear molecules possess a large electric dipole momentof 2 .
72 Debye [6], the freedom to prepare both bosonicand fermionic NaK molecules and chemical stability inmolecular two-body collisions [7].Up to now, Feshbach resonances for the mixture ofbosonic Na and fermionic K have been reported [8, 9]and fermionic ground state Na K molecules have beenprepared by association of Feshbach molecules [10] andsubsequently by following two different STIRAP paths[7, 11]. Thanks to these results and to previously ob-tained spectroscopic data [12, 13], the interatomic molec-ular potentials of the singlet and triplet ground stateshave been refined [9], leading to predictions for Fesh-bach resonances in the bosonic mixtures by isotope massrescaling. However, the very different hyperfine couplingbetween Na and K compared to Na and K lead todifferent singlet-triplet mixing within the Feshbach mani-fold. The accuracy of the predictions strongly depend onresidual correlations in the determination of the singletand triplet potential from measured Na+ K Feshbachresonances. A direct measurement of the Feshbach res-onance positions in the bosonic mixture is necessary tofurther refine the potentials for NaK and in particularto minimize correlations between the singlet and tripletmolecular potentials.Experimental investigations of the Feshbach resonancespectrum of the bosonic pair Na+ K have startedrecently. The identification of Feshbach resonances inthe | f = 1 , m f = − (cid:105) Na + | f = 1 , m f = − (cid:105) K channel has ∗ [email protected] been the basis for the preparation of a dual-species Bose-Einstein condensate of Na and K atoms in the vicinityof a Feshbach resonance at about 247 G [14]. Here, f de-notes the total angular momentum of the respective atomand m f denotes the projection onto the quantizationaxis. By comparing the measured resonance positionswith predictions by Viel et al. [5] significant deviationsbecame apparent, whereas predictions making use of therecent evaluation in [9], including measured d-wave reso-nances of Na+ K, reduce these deviations. Since theFeshbach resonances observed in the | , − (cid:105) Na + | , − (cid:105) K mixture are only a small subset of the many possible res-onances, the observed deviations have motivated a thor-ough search for the remaining structures in order to fur-ther improve the potential energy curves (PECs).In this paper we present a detailed study of Feshbachresonances in a variety of hyperfine combinations in theground state manifold of Na+ K for a magnetic fieldrange from 0 to 750 G. Our approach follows the itera-tive procedure of prediction, measurement and model re-finement that is typical for molecular spectroscopy. Thepaper is structured accordingly:In Sec. II we first give a brief summary of Feshbach res-onance predictions for the bosonic Na+ K mixture,which have been derived from molecular potentials ob-tained by conventional spectroscopy of the NaK moleculeand measurements of Feshbach resonances in the Bose-Fermi mixture of Na+ K by isotope mass rescaling. Inthe second part of the section the experimental sequencefor the measurements is described.In Sec. III we present our measurements of loss featuresarising from elastic and inelastic scattering resonances.In Sec. IV we describe and discuss the updated molec-ular potentials and how they improve the current knowl-edge of the scattering properties of both the Bose-Boseand Bose-Fermi mixtures.In Sec. V a brief discussion of an inelastic loss featureobserved in the | , (cid:105) Na + | , − (cid:105) K channel and its influ-ence on the possible observation of a close lying Feshbachresonance is given. a r X i v : . [ phy s i c s . a t o m - ph ] A p r II. THEORETICAL AND EXPERIMENTALOVERVIEWA. Theoretical predictions from known molecularpotentials and a brief discussion of atom-lossspectroscopy
Molecular potentials for the X Σ + state and the a Σ + state, correlating with the atomic ground state asymp-tote, have been derived from extensive classical spec-troscopy [12, 13] and refined by measurements of Fesh-bach resonances of the atom pair Na+ K [8, 9]. Thesecan be used to predict Feshbach resonances for the pair Na+ K, if the Born-Oppenheimer approximation isassumed to remain valid and by incorporating the properhyperfine and Zeeman interaction of K . Details aboutthe potential representations and the interpretation ofthe coupled channel calculations are given in Sec. IV.From the potentials we predict 32 resonances in themagnetic field region from 0 to 750 G by coupled-channelcalculations for atom pairs with magnetic quantum num-bers f Na = 1, m f Na = {− , , +1 } and f K = 1, m f K = {− , , +1 } . Additionally, resonances for f = 2, m f Na , K = − µ K and therefore onlys-wave resonances are considered. Comparing theoreticaland experimental results in [14], the accuracy of our pre-dictions is expected to be on the order of a few Gauss,significantly simplifying the search in the experiment.We locate Feshbach resonance positions by atom-lossspectroscopy, making use of the strongly enhanced scat-tering length and associated large on-resonance three-body loss in the atomic ensemble [15]. The detectedlosses are referred to as an elastic loss signal through-out this paper. Atom loss is also observed in the case ofinelastic two-body collisions, where the energy releasedby the spin exchange is sufficiently high to lead to a two-body loss process in the trapped sample. In both caseswe identify the local maximum of the atom loss with aresonance position. Typically, three-body processes re-quire higher particle densities than two-body processesto obtain comparable loss rates. Thus in our case inelas-tic loss processes are expected to elapse on a shorter timescale than the three-body loss and will therefore lead tostrong losses already for short hold times.Care has to be taken when investigating a zero cross-ing of the scattering length as function of magnetic field.The minimum of the detected losses is in general notidentical to the zero of the scattering length [16], and ameasurement similar to Sec. II B can give misleading re-sults. We instead localize this magnetic field position byexploiting the two-body losses that appear during opticalevaporation, similar to the work in [14]. In this proce-dure the magnetic field strength is set to the target valuebefore the start of the optical evaporation. Because theoptical trapping potential depth ratio is U K ≈ . U Na ,predominantly Na is ejected from our crossed opticaldipole trap (cODT), while K is sympathetically cooled. On a zero crossing two body collisions are suppressed andtherefore also losses resulting from the evaporation pro-cess [17].
B. Experimental procedure
The experimental setup is described in detail in [18]and [19]. The experimental sequence is based on theexperiences of [14]. Since a precise knowledge of thegenerated magnetic field strength is necessary, a calibra-tion has been performed repetitively in the course of themeasurement campaign. The calibration method is thesame as already described in [14]. We use microwavespectroscopy on a sample of Na with a temperatureof ∼
800 nK confined in the cODT. For a defined elec-tric current we measure the microwave frequency of the | f = 1 , m f = − (cid:105) → | f = 2 , m f = 0 (cid:105) transition using theatom loss in the | f = 1 , m f = − (cid:105) state as signal and cal-culate the corresponding magnetic field using the Breit-Rabi formula. The transition frequency is determinedwith an uncertainty of about 10 kHz leading to an un-certainty on the value for the magnetic field strength onthe order of 30 mG. This is typically small compared tothe statistical uncertainty originating from the resonanceloss measurements. In the following we give a brief sum-mary of the experiment sequence and explain the appliedmodifications in comparison to [14].First, an optically plugged magnetic quadrupole trap isloaded from a dual-species magneto-optical trap (MOT).The atoms are then transferred to a cODT where weprepare an ultracold mixture of Na and K, both in | f = 1 , m f = − (cid:105) , by optical evaporation. The tempera-ture is ∼ µ K for both species, as measured by time-of-flight (TOF) expansions.After the evaporation in the cODT has been com-pleted, we transfer Na and K to the spin-state combi-nation of interest, making use of rapid-adiabatic-passage[20] sequences. Their efficiency is close to unity and nei-ther heating of the sample nor atom loss due to the trans-fers is observed in our experiment.For the atom loss spectroscopy, we vary the atom num-bers of the two species, preparing one species as the ma-jority component and the other one as the minority com-ponent. The peak densities in the cODT are between1 . · cm − and 7 . · cm − for Na and between3 . · cm − and 1 . · cm − for K. Detected losswithin the minority component provides the primary sig-nal. We use different tuning knobs in the experimentalsequence to adjust the atom numbers. The first one isgiven by the loading times of the dual-species MOT. Thesecond tuning knob is the depth of the forced microwaveevaporation we perform in our optically plugged mag-netic quadrupole trap. Due to the smaller repulsion of K by the blue-detuned plug laser light, a deeper evap-oration and thereby colder atomic sample leads to anincreased K density close to the magnetic trap cen-ter compared to Na. This increases losses in the Kcloud. Hence, the deeper the evaporation is performedthe more the atom ratio inside the cODT is shifted to-wards a prevalence of Na.We ramp the magnetic field strength in a few millisec-onds to the target value. The loss measurement is re-peated multiple times for every magnetic field value. Forevery resonance under investigation we experimentallydetermine the appropriate holding time, ensuring thatthe minority cloud is not depleted completely at the mini-mum but the loss feature is well visible. The holding timevaries between 10 ms and 1000 ms and its magnitude canbe an indication whether inelastic two-body or inelasticthree-body processes dominate the losses.The number of remaining atoms is recorded by absorp-tion imaging of the majority component in the cODT andof the minority component after a short TOF. Where pos-sible, we ramp down the magnetic field to zero in 5 to40 ms (depending on the initial magnetic field value) andimage both species at zero magnetic field.For some spin state combinations we find that low fieldFeshbach resonances and/or high background scatteringlengths lead to sizable losses, rendering imaging bothspecies at zero magnetic field unfavorable. To circum-vent these additional losses, we perform high-field imag-ing on the K D -line. For magnetic field strengths be-yond 200 G the Paschen-Back regime is reached for K,where the electron angular momentum ( j, m j ) and thenuclear spin ( i, m i ) decouple from each other in bothelectronic states and instead align directly relative tothe external magnetic field. With this, f is no longer agood quantum number. Together with the selection rule∆ m i = 0, it is always possible to find a transition with∆ m j = ±
1, which then serves as closed imaging tran-sition. We choose a magnetic field for which no otherresonance has to be crossed and for which the scatteringrate of the state combination under investigation is low.We then image K as the majority component in thecODT and follow the scheme above for the Na detec-tion.To improve the signal-to-noise ratio of the atomic cloudpictures, especially for low atom numbers, the absorptionimages of Na and K are post-processed. The back-ground of every picture is reconstructed using an algo-rithm based on principal component analysis [21] andthis background is subtracted from the picture. Thepictures taken at equal magnetic field values are thenaveraged and the resulting image is fitted with a two-dimensional Gaussian. From the fit, the atom number isderived. The errors on the atom number result from thestandard deviations of the fit. They vary between the dif-ferent resonance measurements because they incorporateshot-to-shot atom number fluctuations which can origi-nate from the different required spin preparation stepsand the number of resonances which need to be crossedto reach the magnetic field value under investigation. Forlucidity they are only shown for two exemplary measure-ments in Fig. 1. The errors are propagated to the profilefit of the loss feature and therefore contribute to the un- certainties of the resonance positions (see Tab. I).
III. LOSS RESONANCES AND ZEROCROSSINGS
We have located 21 features, including the ones alreadypresented in [14]. Fourteen features are assigned to pre-dicted Feshbach resonances, resulting in three-body loss,three result from inelastic loss channels and four are as-signed to zero crossings of the two-body scattering length.Figure 1 shows the measured features. To determine theresonance positions we do a weighted fit to our data witha phenomenological Gaussian function. For every mea-surement the atom number is normalized on the baselineof the fit. In Fig. 1, the normalized atom numbers of thedifferent measurements for every channel are set equalto one. This leads to an artificial increase in the atomnumber above one in case of a measured zero crossing.The resulting positions of all measured features are sum-marized in Tab. I. In cases of loss features being clearlyvisible for both species, the position of the resonance isthe weighted average of the center positions from the twofits. The error estimate of the experimentally determinedresonance positions, given in Tab. I, includes the uncer-tainty in the profile fit (which includes the errors fromthe atom number determination as explained above) aswell as the uncertainty in the calibration of the magneticfield strength.As summarized in Tab. I, some calculated resonancesremain undetected. The main reasons are: • Some of the state combinations experience a veryhigh background scattering rate over the com-plete investigated magnetic field range. For thesespin mixtures, the resonances are hidden since theatomic samples experience large losses already dur-ing the state preparation and/or the ramp to thetarget magnetic field. • While pure Na does not show significant loss fea-tures in the investigated range of the magnetic fieldstrength, K exhibits several Feshbach resonancesin different spin channels. We remeasured the Kresonances relevant for our investigations and foundall resonance positions to be within the experimen-tal uncertainties of previous publications [22, 23].Some of them are critical for our heteronuclearmeasurements since they are located near or evenoverlap with the resonance positions predicted for Na+ K. These cases are mentioned in Tab. I.Additionally, the K Feshbach resonances are in-dicated in Fig. 1 as vertical blue dashed lines.
IV. THEORY AND CALCULATIONS
The theoretical modeling of two-body collisions of twoalkali atoms in their electronic ground state is well es-
326 327 328 329 3300.00.40.81.21.62.0 550 560 570 580 590 6000.00.40.81.21.60 25 50 750.00.51.01.5
490 500 510 520 530 540 5500.00.51.01.52.02.53.0
500 505 510 5150.00.40.81.21.62.0 * *
384 387 390 393 3960.00.40.81.21.62.0
FIG. 1. Collection of resonant features in different spin mixtures of Na+ K. M is the total magnetic quantum numberof the pair | f, m f (cid:105) Na + | f, m f (cid:105) K , f is for increasing magnetic field strength B only an approximate quantum number. Opencircles and solid squares correspond to resonances observed by loss of Na and of K, respectively. Insets show zooms tothe detected narrow resonance features. For two measurements, in | , (cid:105) Na + | , (cid:105) K and | , (cid:105) Na + | , − (cid:105) K marked with (*),error bars are given, representing the variation in the errors for different loss measurements (for details, see text). For eachrecording, the holding time and the initial atom numbers are independently optimized. The data are normalized accordingto the respective phenomenological Gaussian fit (red solid line) of the feature with a baseline set to one. Zero crossingsappear therefore artificially as enhancement of the atom number. Vertical gray dotted lines indicate the calculated positions of Na+ K Feshbach resonances as listed in Tab. I. Vertical dashed blue lines mark the positions of K resonances, taken from[22, 23]. The trace for the | , − (cid:105) Na + | , − (cid:105) K mixture corresponds to data from Ref. [14]. tablished and described in many publications (see for ex-ample [24]). The Hamiltonian contains the conventionalkinetic and potential energy for the relative motion of thetwo particles and needs for the coupling of the molecularstates X Σ + and a Σ + the hyperfine and Zeeman terms.For finer details of the partial waves with l ≥
1, one alsoneeds the spin-spin interaction. The molecular PECs arerepresented in a power expansion of an appropriate func-tion ξ ( R ) of the internuclear separation Rξ ( R ) = R − R m R + b R m , (1)to describe the anharmonic form of the potential functionfor R → ∞ or R → R m is an internuclear separationclose to the minimum of the respective PEC and b is aparameter to optimize the potential slopes left and rightof R m with few terms in the power expansion. The fullPECs are extended by long-range terms and short-rangerepulsive ones, see [24] and the supplement of this paper.We calculate the two-body collision rate at the kineticenergy that corresponds to the temperature of the pre-pared ensemble. Thermal averaging is not performed,which would need significant computing time, but, moreimportantly, for a complete description we would have toconsider the two cases of two- and three-body effects inthe modeling. Here, we take the calculated maximum inthe two-body collision rate constant to be equal to the ob-served Feshbach resonance and the calculated minimumto the observed loss minimum in optical evaporation.The most recent fit of Feshbach resonances was re-ported for Na+ K in [9] and the present evaluationstarts from that result. Calculating the resonances forthe observed cases with those derived PECs, we find sig-nificant deviations between observation and theory, thusdemanding for new fits. They include all former observa-tions and additionally the measured Feshbach resonancesand zero crossings presented in this paper, in total 82 in-dependent data points from Feshbach spectroscopy. Thisallows to further reduce correlations in the determinationof the triplet and singlet potentials. The improved molec-ular potentials lead to a higher consistency between themeasured and theoretically predicted resonance featuresfor the bosonic Na+ K mixture as well as the Bose-Fermi mixture of Na+ K. The sum of squared resid-uals of calculated and experimentally determined reso-nance positions, weighted by the experimental uncertain-ties, improved from 337.0 to 255.31. We give a full listingof the data points and the evaluation with the differentpotential approaches in the supplement. Furthermore,we find that no inclusion of Born-Oppenheimer correctionis needed to achieve this improvement. The parametersof the refined PECs can be found in the supplement.Table I lists the calculated resonance positions whichare derived using the updated potential energy curves.In several channels, the calculations show maxima forthe elastic and inelastic scattering rates to appear closeto each other. Such a constellation can lead to a shiftedminimum in the atom-loss measurement. For one res- onance a remarkable shift was observed in our experi-ment and will be discussed in the following section. Notethat closely located maxima of elastic and inelastic scat-tering rates can also lead to asymmetric broadening ofloss signals. This can be an additional reason why forseveral measurements of Feshbach resonances shifts andasymmetric broadening of the loss signals were reported[25–30]. R e m a i n i n g a t o m f r a c t i o n inelasticelastic 10 − − − S c a tt e r i n g r a t e c o n s t a n t ( c m / s ) Magnetic fi eld strength (G) FIG. 2. Remaining atom fraction of Na (open circles) andGaussian fit (red curve) as well as elastic (green dotted line)and inelastic (black dashed line) collision rate constants forthe | , (cid:105) Na + | , − (cid:105) K channel. The rate constants are calcu-lated for a kinetic energy of 1 µ K. The peak densities in thecODT for this measurement were 2 . · cm − for Na and4 . · cm − for K. V. INELASTIC LOSS FEATURE IN | , (cid:105) Na + | , − (cid:105) K Our theoretical model predicts a Feshbach resonanceat 15 G for the | , (cid:105) Na + | , − (cid:105) K channel. However, ourmeasurements show a broad loss signal at 26 .
34 G, seeFig. 1 and Tab. I. The large deviation can be explainedby looking at both elastic and inelastic loss contributions.Figure 2 shows the atom-loss measurement together withthe elastic loss rate which has a maximum at 15.4 G andan enhancement of the inelastic loss rate at 28 . | , (cid:105) Na + | , (cid:105) K channel and gains strength through a Feshbach resonanceat 29 . TABLE I. Measured magnetic field positions B exp and uncertainties ( ± ) together with calculated positions B th , applying theimproved potentials. M is the total magnetic quantum number of the pair | f, m f (cid:105) Na + | f, m f (cid:105) K , f is in most cases only anapproximate quantum number. Subscripts ” res ” and ” ZC ” stand for resonance and zero crossing, respectively. In some cases,maxima of the elastic (el.) and inelastic (in.) scattering rate are listed respectively. A.1 marks the inelastic loss featurediscussed in section V. The measurements with (*) have been previously presented in [14]. M Na f,m f K f,m f B exp, ZC (G) B th, ZC (G) B exp, res (G) B th, res (G)2 1,1 1,1 380.88 (3.83) 381.43 411.33 (1.28) 410.1- 507.0 508.73 (0.83) 508.811 1,1 1,0 6.72 (2.09) 6.6- 328.5 329.12 (0.77) 328.96- 442.5 close to KK res. 467- 577.5 579.94 (0.88) 580.491,0 1,1 - 7.5 (in.) 9.0 (el.)- 336.0 close to KK res. 419.0- 508.5 (in.) 512.0 (el.)0 1,1 1,-1 26.34 (3.31) A.1 15.4 (el.) 28.2 (in.)- 393.0 393.61 (0.76) 393.59515.85 (1.68) 516.4 536.07 (0.94) 536.671,0 1,0 - 4.25 5.47 (1.01) 5.631.86 (1.69) 29.8- 407 close to KK res. 475.5 (in.) 476.0 (el.)- 581.5 (in.) 585.5 (el.)1,-1 1,1 - 516.0 (el.) 522.5 (in.)-1 1,-1 1,0 - 13.0245.76 (1.45) 244.75- 588.5 (el.) 593.0 (in.)1,0 1,-1 - 107.5541.09 (1.50) 540.51,1 2,-2 - 88.5 (in.)- 134.0 (in.) 138.0 (el.)- 4712,-2 1,1 - 272.5 (in., weak)- 314.5 (in.)- 465.5 (in.) 473.5 (el.)-2 1,-1 1,-1 32.5 (0.8)(*) 33.13117.2 (0.2)(*) 117.08 247.1 (0.2)(*) 247.57646.6 (1.5)(*) 651.5 (el./in.)686.2 (1.5)(*) 686.7 (in.)1,0 2,-2 228.48 (1.49) 229.5570.29 (2.55) 574.3 - 619.02,-2 1,0 - 358.5 (in.)- 528.0 (in.) 533.0 (el.) the Feshbach resonance position with higher accuracy, abinding energy measurement should be performed [31]which is left for future investigation. Similar findings onthe interplay of inelastic two-body and three-body pro-cesses have been reported in [9]. VI. CONCLUSION AND OUTLOOK
In this paper we presented a detailed study of Fesh-bach resonances in many possible hyperfine combinationsof the ground state manifold of Na+ K in a magneticfield range from 0 to 750 G. We compared these mea-surements to theoretical predictions based on the cur-rently available data for NaK molecular potentials andused our data to refine those potentials. The improvedpotentials lead to a higher consistency between the exper-imental data and theoretical predicted resonance featuresfor both, the bosonic Na+ K mixture and the Bose-Fermi mixture of Na+ K. With the incorporation ofthe experimental data of the bosonic Feshbach resonancesand the finding that the Born-Oppenheimer approxima-tion remains valid for the investigated partial waves, reli-able predictions based on the new potentials will be also possible for the other, still unexplored, bosonic mixtureof Na+ K.Moreover, the observation of the inelastic loss fea-ture deviating significantly from the corresponding elas-tic peak indicates that careful theoretical investigationis recommended in case unexpected deviations appear inan analysis of a performed Feshbach resonance atom-lossmeasurement.The measurements and refined molecular potentialswill greatly aid the future investigation of interspin phe-nomena such as droplet formation [32] as well as in pro-ducing stable Na K molecules in their absolute groundstate via a STIRAP process [33].
ACKNOWLEDGEMENTS
We gratefully acknowledge financial support from theEuropean Research Council through ERC Starting GrantPOLAR and from the Deutsche Forschungsgemeinschaft(DFG) through CRC 1227 (DQ-mat), project A03 andFOR2247, project E5. K.K.V. thanks the DeutscheForschungsgemeinschaft for financial support throughResearch Training Group 1991. [1] C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Rev.Mod. Phys. , 1225 (2010).[2] M. Baranov, Physics Reports , 71 (2008).[3] T. Lahaye, C. Menotti, L. Santos, M. Lewenstein, andT. Pfau, Reports on Progress in Physics , 126401 (2009).[4] M. A. Baranov, M. Dalmonte, G. Pupillo, and P. Zoller,Chemical Reviews , 5012 (2012).[5] A. Viel and A. Simoni, Phys. Rev. A , 042701 (2016).[6] M. Aymar and O. Dulieu, The Journal of ChemicalPhysics , 204302 (2005).[7] J. W. Park, S. A. Will, and M. W. Zwierlein, Phys. Rev.Lett. , 205302 (2015).[8] J. W. Park, C.-H. Wu, I. Santiago, T. G. Tiecke, S. Will,P. Ahmadi, and M. W. Zwierlein, Phys. Rev. A , 051602(2012).[9] M.-J. Zhu, H. Yang, L. Liu, D.-C. Zhang, Y.-X. Liu,J. Nan, J. Rui, B. Zhao, J.-W. Pan, and E. Tiemann,Phys. Rev. A , 062705 (2017).[10] C.-H. Wu, J. W. Park, P. Ahmadi, S. Will, and M. W.Zwierlein, Phys. Rev. Lett. , 085301 (2012).[11] F. Seeßelberg, N. Buchheim, Z.-K. Lu, T. Schneider, X.-Y. Luo, E. Tiemann, I. Bloch, and C. Gohle, Phys. Rev.A , 013405 (2018).[12] A. Gerdes, M. Hobein, H. Kn¨ockel, and E. Tiemann,Eur. Phys. J. D , 67 (2008).[13] I. Temelkov, H. Kn¨ockel, A. Pashov, and E. Tiemann,Phys. Rev. A , 032512 (2015).[14] T. A. Schulze, T. Hartmann, K. K. Voges, M. W. Gem-pel, E. Tiemann, A. Zenesini, and S. Ospelkaus, Phys.Rev. A , 023623 (2018).[15] T. Weber, J. Herbig, M. Mark, H.-C. N¨agerl, andR. Grimm, Phys. Rev. Lett. , 123201 (2003). [16] Z. Shotan, O. Machtey, S. Kokkelmans, andL. Khaykovich, Phys. Rev. Lett. , 053202 (2014).[17] This procedure is most efficient for a low K intraspeciesscattering length because self-evaporation is suppressed inthis case.[18] M. W. Gempel, Ph.D. thesis (2016).[19] T. A. Schulze, Ph.D. thesis (2018).[20] J. C. Camparo and R. P. Frueholz, Journal of Physics B:Atomic and Molecular Physics , 4169 (1984).[21] X. Li, M. Ke, B. Yan, and Y. Wang, Chin. Opt. Lett. ,128 (2007).[22] C. D’Errico, M. Zaccanti, M. Fattori, G. Roati, M. Ingus-cio, G. Modugno, and A. Simoni, New Journal of Physics , 223 (2007).[23] S. Roy, M. Landini, A. Trenkwalder, G. Semeghini,G. Spagnolli, A. Simoni, M. Fattori, M. Inguscio, andG. Modugno, Phys. Rev. Lett. , 053202 (2013).[24] S. Knoop, T. Schuster, R. Scelle, A. Trautmann, J. App-meier, M. K. Oberthaler, E. Tiesinga, and E. Tiemann,Phys. Rev. A , 042704 (2011).[25] K. Dieckmann, C. A. Stan, S. Gupta, Z. Hadzibabic,C. H. Schunck, and W. Ketterle, Phys. Rev. Lett. ,203201 (2002).[26] T. Bourdel, J. Cubizolles, L. Khaykovich, K. M. F. Ma-galh˜aes, S. J. J. M. F. Kokkelmans, G. V. Shlyapnikov,and C. Salomon, Phys. Rev. Lett. , 020402 (2003).[27] C. Weber, G. Barontini, J. Catani, G. Thalhammer,M. Inguscio, and F. Minardi, Phys. Rev. A , 061601(2008).[28] O. Machtey, D. A. Kessler, and L. Khaykovich, Phys.Rev. Lett. , 130403 (2012).[29] S. Zhang and T.-L. Ho, New Journal of Physics ,055003 (2011).30] A. Y. Khramov, A. H. Hansen, A. O. Jamison, W. H.Dowd, and S. Gupta, Phys. Rev. A , 032705 (2012).[31] J. Ulmanis, S. H¨afner, R. Pires, E. D. Kuhnle, M. Wei-dem¨uller, and E. Tiemann, New Journal of Physics ,055009 (2015).[32] D. S. Petrov, Phys. Rev. Lett. , 155302 (2015).[33] U. Gaubatz, P. Rudecki, M. Becker, S. Schiemann,M. K¨ulz, and K. Bergmann, Chemical Physics Letters , 463 (1988). Appendix A: Molecular potential curves
The parametrization of the molecular potentials is de-scribed for example in [1]. The potentials are representedin three parts: the repulsive short-range part U SR ( R ),the intermediate range U IR ( R ) and the asymptotic longrange part U LR ( R ), which are given by the following ex-pressions: U SR ( R ) = A + BR q for R < R i , (A1) U IR ( R ) = n (cid:88) k =0 a k ξ ( R ) k for R i ≤ R ≤ R o , (A2)with ξ ( R ) = R − R m R + bR m , (A3) and U LR ( R ) = − C R − C R − C R − ... ± E ex for R < R o , (A4)where the exchange energy is given by E ex = A ex R γ exp ( − βR ) . (A5)It is negative for the singlet and positive for the tripletpotential.The parameters of the refined NaK singlet and tripletmolecular potential curves are listed in Tab. II. A com-puter code in FORTRAN for calculating the potentialfunctions can be found in the supplement of [2]. Appendix B: Evaluation of Feshbach spectroscopy
We present a detailed evaluation of 82 independentdata points obtained in Feshbach spectroscopy and takenfrom different publications. The quality of four differentPEC-representations is compared in terms of the sum ofweighted squared deviations in Tab. III. Note that thereference [3] only takes a subset of the given scatter-ing features into account. Therefore, a sum of weightedsquared deviations is only calculated for the last threecolumns of Tab. III. [1] A. Gerdes, M. Hobein, H. Kn¨ockel, and E. Tiemann, Eur.Phys. J. D , 67 (2008).[2] R. Pires, M. Repp, J. Ulmanis, E. D. Kuhnle, M. Wei-dem¨uller, T. G. Tiecke, C. H. Greene, B. P. Ruzic, J. L.Bohn, and E. Tiemann, Phys. Rev. A , 012710 (2014).[3] A. Viel and A. Simoni, Phys. Rev. A , 042701 (2016).[4] I. Temelkov, H. Kn¨ockel, A. Pashov, and E. Tiemann,Phys. Rev. A , 032512 (2015).[5] M.-J. Zhu, H. Yang, L. Liu, D.-C. Zhang, Y.-X. Liu,J. Nan, J. Rui, B. Zhao, J.-W. Pan, and E. Tiemann, Phys. Rev. A , 062705 (2017).[6] This work.[7] T. A. Schulze, T. Hartmann, K. K. Voges, M. W. Gempel,E. Tiemann, A. Zenesini, and S. Ospelkaus, Phys. Rev.A , 023623 (2018).[8] J. W. Park, C.-H. Wu, I. Santiago, T. G. Tiecke, S. Will,P. Ahmadi, and M. W. Zwierlein, Phys. Rev. A , 051602(2012).[9] J. Rui, H. Yang, L. Liu, D.-C. Zhang, Y. Liu, J. Nan,B. Zhao, and J.-W. Pan, Nature Physics (2017). TABLE II. Potential parameters of the X Σ + and a Σ + states of NaK given with respect to the Na(3s)+K(4s) asymptote. X Σ + a Σ + For
R < R i = 2 . R < R i = 4 . × cm − A -0.13272988 × cm − B 0.696124608 × cm − ˚A q B 0.212875317 × cm − ˚A q q 4.92948 q 1.844150For R i ≤ R ≤ R o For R i ≤ R ≤ R o b -0.4 b -0.27 R m R m a -5273.62315 cm − a -207.81119 cm − a -0.239542348630413837 × cm − a -0.474837907736683607 cm − a × cm − a × cm − a × cm − a -0.159602468357546013 × cm − a -0.393070200439200050 × cm − a -0.948541718924311908 × cm − a -0.169145814548076414 × cm − a -0.135179373273532747 × cm − a -0.374171063602873910 × cm − a -0.183565449370861752 × cm − a × cm − a × cm − a × cm − a -0.163160543217713166 × cm − a -0.216398544375193026 × cm − a -0.199688039882199257 × cm − a -0.101610099703415297 × cm − a × cm − a × cm − a × cm − a × cm − a -0.391885588318469822 × cm − a -0.154974082312119037 × cm − a × cm − a -0.782460601529465795 × cm − a -0.839469806952623278 × cm − a × cm − a × cm − a × cm − a -0.270560975156805658 × cm − a -0.130777134652790947 × cm − a × cm − a × cm − a -0.127583274381506557 × cm − a -0.547443981078124619 × cm − a × cm − a × cm − a -0.139350456346844196 × cm − a -0.514892853898448334 × cm − a × cm − a × cm − a -0.157575108054349303 × cm − a -0.507254397888037300 × cm − For
R > R o = 11 . R > R o = 11 . C × cm − ˚A C × cm − ˚A C × cm − ˚A C × cm − ˚A C × cm − ˚A C × cm − ˚A A ex × cm − ˚A − γ A ex × cm − ˚A − γ γ γ β − β − TABLE III. Listed are 82 scattering features measured with Feshbach spectroscopy by different groups (see column ”ref”). The”note” column contains information about the scattering feature. ZC refers to a zero crossing of the scattering length, in to aloss feature which results from an inelastic two-body collision process, ov indicates the presence of two overlapping structuresand an empty entry refers to a Feshbach resonance. M is the total magnetic quantum number of the the entrance channellisted in column | f, m f (cid:105) . l and l max give the partial waves considered in the calculation of the feature position. B exp gives theexperimentally determined position of the scattering feature and σ exp the error. The difference ∆ of experimentally determinedand calculated positions of the features for four different representations for the PECs are compared in the last four columns ofthe table. Subscripts indicate the publications the PECs are taken from. The question mark in the ∆(G)[3] column indicatesthat the correct assignment is unknown for that specific data point.Isotope note M | f, m f (cid:105) l l max B exp ( G ) σ exp (G) ref ∆(G)[3] ∆(G)[4] ∆(G)[5] ∆(G)[6]Na K Na K23 39 2.0 (1,1) (1,1) 0 0 411.334 1.276 [6] -31.18 -1.334 1.206 1.24523 39 2.0 (1,1) (1,1) 0 0 508.730 0.831 [6] -27.27 -2.486 -0.330 -0.07923 39 ZC in ZC ZC -2.0 (1,0) (2,-2) 0 0 570.286 2.546 [6] -29.15 -5.947 -4.033 -4.02123 39 -2.0 (1,-1) (1,-1) 0 0 32.475 0.830 [7] 30.47 1.043 -1.643 -0.66323 39 -2.0 (1,-1) (1,-1) 0 0 247.108 0.230 [7] 5.71 -0.329 -0.969 -0.46023 39 ZC -2.0 (1,-1) (1,-1) 0 0 117.189 0.150 [7] 41.48 3.405 0.024 0.10923 39 ov -2.0 (1,-1) (1,-1) 0 0 646.600 1.500 [7] - - -3.800 -4.58523 40 -3.5 (1,1) (9/2,-9/2) 0 0 78.320 0.150 [8] 0.54 0.008 -0.066 -0.02423 40 -3.5 (1,1) (9/2,-9/2) 0 0 89.700 0.250 [8, 9] 1.02 0.692 0.596 -0.08423 40 -2.5 (1,1) (9/2,-7/2) 0 0 81.620 0.160 [8] 0.20 -0.108 -0.160 -0.03223 40 -2.5 (1,1) (9/2,-7/2) 0 0 89.780 0.460 [8] -0.04 -0.594 -0.672 -0.60923 40 -2.5 (1,1) (9/2,-7/2) 0 0 108.600 3.000 [8] -0.31 -0.606 -0.719 -1.69423 40 -1.5 (1,1) (9/2,-5/2) 0 0 96.540 0.090 [8] 0.15 -0.091 -0.141 0.00223 40 -1.5 (1,1) (9/2,-5/2) 0 0 106.920 0.270 [8] 0.38 -0.155 -0.234 -0.15223 40 -1.5 (1,1) (9/2,-5/2) 0 0 138.560 1.000 [8, 9] 1.74 1.518 1.384 0.04223 40 -0.5 (1,1) (9/2,-3/2) 0 0 116.910 0.150 [8] -0.28 -0.350 -0.388 -0.22623 40 -0.5 (1,1) (9/2,-3/2) 0 0 130.640 0.030 [8, 9] 0.28 -0.127 -0.199 -0.10023 40 -0.5 (1,1) (9/2,-3/2) 0 0 175.000 5.000 [8] -2.44 -2.520 -2.674 -4.53123 40 -4.5 (1,1) (9/2,-9/2) 1 1 6.350 0.030 [8] -0.15 -0.034 -0.042 -0.03023 40 -2.5 (1,1) (9/2,-9/2) 1 1 6.410 0.030 [8] -0.16 0.003 -0.008 -0.00223 40 -3.5 (1,1) (9/2,-9/2) 1 1 6.470 0.030 [8] -0.16 0.006 -0.004 -0.00223 40 -3.5 (1,1) (9/2,-9/2) 1 1 6.680 0.030 [8] -0.17 0.013 0.005 0.01023 40 -2.5 (1,1) (9/2,-9/2) 1 1 19.100 0.100 [8] -0.22 -0.056 -0.060 -0.05223 40 -4.5 (1,1) (9/2,-9/2) 1 1 19.200 0.100 [8] -0.19 0.016 0.012 0.00823 40 -3.5 (1,1) (9/2,-9/2) 1 1 19.300 0.100 [8] -0.17 0.048 0.044 0.03623 40 -3.5 (1,1) (9/2,-7/2) 1 1 7.320 0.140 [8] -0.30 -0.119 -0.133 -0.13423 40 -1.5 (1,1) (9/2,-7/2) 1 1 7.540 0.040 [8] -0.36 -0.148 -0.160 -0.15623 40 -2.5 (1,1) (9/2,-7/2) 1 1 7.540 0.040 [8] -0.36 -0.164 -0.176 -0.16823 40 -3.5 (1,1) (9/2,-7/2) 1 1 7.540 0.040 [8] -0.34 -0.138 -0.150 -0.14423 40 -2.5 (1,1) (9/2,-7/2) 1 1 23.190 0.040 [8] -0.30 -0.048 -0.048 -0.04823 40 -3.5 (1,1) (9/2,-7/2) 1 1 23.290 0.040 [8] -0.31 -0.012 -0.012 -0.02423 40 in -2.5 (1,1) (9/2,-5/2) 1 1 9.230 0.340 [8] -0.11 0.108 0.092 0.09623 40 -1.5 (1,1) (9/2,-5/2) 1 1 9.600 0.040 [8] -0.29 0.032 0.016 0.02823 40 -2.5 (1,1) (9/2,-5/2) 1 1 9.600 0.040 [8] -0.28 0.004 -0.012 -0.00423 40 -0.5 (1,1) (9/2,-5/2) 1 1 9.600 0.040 [8] -0.29 -0.004 -0.016 -0.01223 40 -1.5 (1,1) (9/2,-5/2) 1 1 29.200 0.090 [8] -0.41 -0.080 -0.072 -0.064 Isotope note M | f, m f (cid:105) l l max B exp (G) σ exp (G) ref ∆(G)[3] ∆(G)[4] ∆(G)[5] ∆(G)[6]Na K Na K23 40 -2.5 (1,1) (9/2,-5/2) 1 1 29.520 0.090 [8] -0.41 0.041 0.046 0.02423 40 -0.5 (1,1) (9/2,-5/2) 1 1 29.450 0.090 [8] -0.43 -0.002 0.004 -0.00823 40 -0.5 (1,1) (9/2,-3/2) 1 1 12.510 0.050 [8] -0.53 -0.171 -0.183 -0.15923 40 0.5 (1,1) (9/2,-3/2) 1 1 12.680 0.050 [8] -0.52 -0.096 -0.113 -0.10823 40 -0.5 (1,1) (9/2,-3/2) 1 1 39.390 0.040 [8] -0.48 0.059 0.080 0.10023 40 0.5 (1,1) (9/2,-3/2) 1 1 39.850 0.040 [8] -0.52 0.192 0.215 0.18823 40 0.5 (1,1) (9/2,-1/2) 0 0 146.700 0.200 [5] - -0.251 -0.245 -0.06523 40 0.5 (1,1) (9/2,-1/2) 0 0 165.340 0.300 [5] - -0.340 -0.383 -0.26923 40 0.5 (1,1) (9/2,-1/2) 0 0 233.000 1.800 [5] - -4.994 -5.126 -7.72223 40 1.5 (1,1) (9/2, 1/2) 0 0 190.500 0.200 [5] - -0.620 -0.513 -0.31923 40 1.5 (1,1) (9/2, 1/2) 0 0 218.400 0.200 [5] - -0.667 -0.622 -0.50523 40 1.5 (1,1) (9/2, 1/2) 0 0 308.100 3.220 [5] - -19.413 -19.588 -23.11423 40 2.5 (1,1) (9/2, 3/2) 0 0 256.600 0.200 [5] - -0.965 -0.619 -0.42323 40 2.5 (1,1) (9/2, 3/2) 0 0 299.900 0.400 [5] - -1.900 -1.630 -1.52223 40 0.5 (1,1) (9/2,-1/2) 1 1 18.810 0.100 [5] - -0.024 -0.031 0.01223 40 1.5 (1,1) (9/2,-1/2) 1 1 19.150 0.100 [5] - 0.112 0.107 0.11623 40 0.5 (1,1) (9/2,-1/2) 1 1 58.320 0.100 [5] - -0.103 -0.036 -0.01223 40 1.5 (1,1) (9/2,-1/2) 1 1 59.100 0.100 [5] - 0.184 0.249 0.20023 40 1.5 (1,1) (9/2, 1/2) 1 1 35.170 0.100 [5] - -0.156 -0.116 -0.02523 40 2.5 (1,1) (9/2, 1/2) 1 1 35.830 0.100 [5] - 0.047 0.084 0.11223 40 1.5 (1,1) (9/2, 1/2) 1 1 100.360 0.100 [5] - -0.104 0.111 0.11223 40 2.5 (1,1) (9/2, 1/2) 1 1 101.310 0.100 [5] - 0.171 0.381 0.28523 40 -3.5 (1,1) (9/2,-9/2) 0 2 204.520 0.200 [5] - -3.75 0.084 0.11623 40 -3.5 (1,1) (9/2,-9/2) 0 2 279.800 0.200 [5] - -5.99 -0.202 0.02423 40 in -2.5 (1,1) (9/2,-7/2) 0 2 202.680 0.200 [5] - -3.51 0.105 0.11223 40 in -2.5 (1,1) (9/2,-7/2) 0 2 276.300 0.200 [5] - -5.50 -0.040 0.11323 40 in -1.5 (1,1) (9/2,-5/2) 0 2 201.660 0.200 [5] - -3.29 0.117 0.09623 40 in -1.5 (1,1) (9/2,-5/2) 0 2 274.600 0.200 [5] - -5.14 0.026 0.11523 40 in -0.5 (1,1) (9/2,-3/2) 0 2 201.440 0.200 [5] - -3.06 0.124 0.07623 40 in -0.5 (1,1) (9/2,-3/2) 0 2 274.800 0.200 [5] - -4.67 0.223 0.25123 40 in in in in2.5 (1,1) (9/2, 3/2) 0 2 283.700 0.900 [5] - -3.98 0.003 -0.160