An Ultracold Gas of Bosonic 23 Na 39 K Ground-State Molecules
Kai K. Voges, Philipp Gersema, Mara Meyer zum Alten Borgloh, Torben A. Schulze, Torsten Hartmann, Alessandro Zenesini, Silke Ospelkaus
AAn Ultracold Gas of Bosonic Na K Ground-State Molecules
Kai K. Voges, ∗ Philipp Gersema, Mara Meyer zum Alten Borgloh, Torben A. Schulze, Torsten Hartmann, Alessandro Zenesini,
1, 2 and Silke Ospelkaus † Institut f¨ur Quantenoptik, Leibniz Universit¨at Hannover, 30167 Hannover, Germany INO-CNR BEC Center and Dipartimento di Fisica, Universit`a di Trento, 38123 Povo, Italy (Dated: August 13, 2020)We report the creation of ultracold bosonic dipolar Na K molecules in their absolute rovi-brational ground state. Starting from weakly bound molecules immersed in an ultracold atomicmixture, we coherently transfer the dimers to the rovibrational ground state using an adiabaticRaman passage. We analyze the two-body decay in a pure molecular sample and in molecule-atommixtures and find an unexpectedly low two-body decay coefficient for collisions between moleculesand K atoms in a selected hyperfine state. The preparation of bosonic Na K molecules opensthe way for future comparisons between fermionic and bosonic ultracold ground-state molecules ofthe same chemical species.
Heteronuclear polar ground-state molecules haveattracted considerable attention in recent years. Theyserve as a new platform for controlled quantum chem-istry [1, 2], novel many-body physics [3, 4] and quantumsimulations [5, 6]. Their permanent electric dipolemoment gives rise to anisotropic and tunable long-rangeinteractions which can be induced in the lab frame viaelectric fields or resonant microwave radiation [7, 8].This gives exquisite control over additional quantumdegrees of freedom. In recent years there has beencontinuous progress in the production of ultracoldbialkali molecules. Fermionic K Rb [9], Na K [10]and Li Na [11] as well as bosonic Rb Cs [12] and Na Rb [13] molecules have been prepared.Up to now, not a single molecule has been availableboth as a bosonic and a fermionic molecular quantumgas, which makes findings among different species andquantum statistics challenging to interpret and tocompare. For bialkali molecules only combinations withLi or K offer the possibility to prepare the bosonic andfermionic molecule, as Li and K are the only alkalimetals which possess long-lived fermionic and bosonicisotopes. Among these molecules (LiK, LiNa, LiRb,LiCs, NaK, KRb, KCs) all possible combinations witha Li atom as well as the KRb molecule are knownto undergo exothermic atom exchange reactions inmolecule-molecule collisions [14]. This leaves only NaKand KCs [15] as chemically stable molecules for a com-parison of scattering properties of the same molecularspecies but different quantum statistics.Both chemically reactive and non-reactive spin-polarizedfermionic molecular ensembles have been reported tobe long-lived due to the centrifugal p -wave collisionalbarrier limiting the two-body collisional rate to thetunneling rate [1, 10]. The lifetime of bosonic molecularensembles, however, has been observed to be significantlyshorter and limited by the two-body universal scatteringrate [13, 16]. Two-body collisions involving moleculescan lead to the formation of collisional complexes dueto a large density of states. The complexes can either decay to new chemical species for chemically reactivemolecules [17] or within the lifetime of the complexesare removed from the trap by light excitation [18–20] orcollisions with a third scattering partner [16, 21].In this letter, we report on the production of ultracoldbosonic Na K rovibrational ground-state molecules.The preparation follows the pioneering experiments forthe creation of K Rb molecules [9] with Feshbachmolecule creation and subsequent STImulated RamanAdiabatic Passage (STIRAP) transfer [22] to a se-lected hyperfine state in the rovibrational ground-statemanifold. We model our STIRAP transfer through aneffective 5-level master equation model and work out anefficient pathway to create spin-polarized ground-statemolecular ensembles. We prepare pure molecular ensem-bles as well as molecule-atom mixtures and extract theresulting collisional loss rate coefficients. We find theloss rate for the Na K+ K mixture to be drasticallysuppressed which opens interesting perspectives forsympathetic cooling.The experiments start from ultracold weakly boundmolecules. As previously described in [23], we asso-ciate Na K Feshbach dimers by applying a radiofrequency pulse to an ultracold mixture of bosonic Na and K held in a 1064 nm crossed-beam op-tical dipole trap (cODT) with temperatures below350 nK. We create 6 × dimers in the least boundvibrational state | f i with a total angular momen-tum projection M F = − h ×
100 kHz at a magnetic field of 199 . | f i is mainly com-posed of α | m i, Na = − / , m i, K = − / , M S = − i + α | m i, Na = − / , m i, K = − / , M S = 0 i . M S is thetotal electron spin projection, m i, Na/K are the nuclearspin projections and α / denote the state admixtures.For detection, we use a standard absorption techniqueof K atoms directly from the weakly-bound molecularstate.For the STIRAP transfer, we make use of external- a r X i v : . [ c ond - m a t . qu a n t - g a s ] A ug cavity diode laser systems as already described in[24]. Both lasers are referenced simultaneously to a10 cm-long high-finesse ULE cavity using a sidebandPound-Drever-Hall locking scheme [25]. The cavity’sfinesses for the Pump and Stokes laser are 24900 and37400, respectively, the free spectral range is 1 .
499 GHz.The linewidths of both locked lasers are estimated to bebelow 5 kHz. Furthermore, the power of the Pump laseris amplified by a tapered amplifier. Both lasers, Pumpand Stokes, are overlapped and focused to the positionof the molecules with 1 /e Gaussian beam waists of 35and 40 µ m, respectively. The direction of propagationis perpendicular to the direction of the magnetic field,thus π ( σ + / − )-transitions can be addressed by choosingthe polarization parallel(perpendicular) to the magneticfield.Possible transfer pathways to the ground state have beenpreviously investigated theoretically and experimentally[26],[24]. Figure 1(a) summarizes the relevant states in-volved in the transfer scheme. Starting from the weaklybound dimer state | f i with mainly triplet character,we make use of the triplet-singlet mixed excited state | e i to transfer the molecules into a selected hyperfinestate in the rovibrational ground state | g i with puresinglet character. For the excited state | e i we choose thestrongly spin-orbit coupled B Π | v = 8 i /c Σ + | v = 30 i state manifolds (see Fig.1(a)), which have a largestate admixture of 26 % /
74 % [24]. The hyperfinestructure of the | X Σ + , v = 0 , N = 0 i ground stateconsists of 16 states with a total angular momentumprojection M F = m i, Na + m i, K , which group into fourbranches with different m i, Na at high magnetic fields(see Fig.1(b)) [27]. At 199 . Knuclear momenta are also decoupled from the othernuclear and electronic angular momenta [28]. Therefore,dipole transitions only change the latter ones. Thislimits the number of accessible ground states to three,which are highlighted in Fig. 1(b). Accounting onlyfor π -transitions for the Pump transition to maximizethe coupling strength, only a single state is accessi-ble in the c Σ + hyperfine manifold, namely the | e i = | c Σ + , m i, Na = − / , m i, K = − / , M J = − , M F = − i .The transition yields an energy of 12242 . − (which corresponds to a wavelength of 816 . . − (572 . σ − -transition to the | g i = | X Σ + , m i, Na = − / , m i, K = − / , M J = 0 , M i = − i state. Nevertheless, our experimental setup alwayssupports σ − - and σ + -transitions at the same time. Con-sequently, the ground state is coupled to two additionalstates | e , i through σ + -transitions (see inset Fig. 1(a)).For the experiments and for the modeling we thus have Magnetic (cid:28)eld (G) E n e r g y / h ( k H z ) X Σ , v = 0 , N = 0 (b) m i, Na = − m i, Na = − m i, Na = m i, Na = X Σ + a Σ + c Σ + B Π Internuclear distance (¯) W a v e nu m b e r ( c m − ) Na (3 s ) + K (4 p ) Na (3 s ) + K (4 s ) (a) | e i | f i| g i ∆ δ P u m pS t o k e s M F = − M F = − × E n e r g y [ M H z ] E n e r g y / h ( M H z ) | e i| e i| e i MF = − MF = − MF = − FIG. 1. (a) Potential energy curves of the Na K molecule.The energy is shown in cm − as function of the internucleardistance. The solid green curve corresponds to the electronic X Σ + , the dotted light blue to the a Σ + and the dashed linesto the c Σ + and B Π potentials. Wavefunctions are shownas black lines with the corresponding shading. Amplitudes ofthe wavefunctions are not to scale. The black arrows indicatethe STIRAP transitions and the one(∆)- and two( δ )-photondetunings. The inset shows the magnetic field dependenceof the Pump transition to the excited states from the modelin [24]. (b) Magnetic field dependence of the ground-statehyperfine energy structure. The green lines are the states with M F = − M F = − m i, Na become visible. The magnetic field, wherethe molecule creation process is performed, is marked with across on the axis. to consider an effective 5-level system. The details ofthe model are described in the supplemental material.For STIRAP a high degree of phase coherence betweenthe two independent laser sources is imperative. To provethe coherence and to determine the explicit frequenciesfor the two-photon Raman transition we perform electro-magnetically induced transparency (EIT) experimentson the selected states. For the measurement shown inFig. 2(a) Rabi frequencies of Ω Pump = 2 π × . Stokes = 2 π × . µ s. The observed asymmetryof the molecule revival arises from a one-photon detuning∆ = 2 π × | e i . EIT reliesonly on coherent dark state effects and never populatesthe ground state. A coupling between the ground stateand the perturbing excited states | e , i does not alterthe coupling scheme as the two-photon condition is notfulfilled for these states. Consequently, a 3-level scheme p o pu l a t i o n Evolution time t ( µ s) / ( π ) ( M H z ) M o l ec u l e s t a t e R a b i f r e q . (b)(a) Pump laser detuning/ (2 π ) (MHz) R e m a i n i n g m o l ec u l e f r a c t i o n s p o pu l a t i o n STIRAPTransfer
FIG. 2. EIT and time evolution of STIRAP. (a) EIT mea-surement together with a theory curve. The points are theremaining Feshbach molecule fraction normalized to the ini-tial Feshbach molecule number. The solid black line is thetheory curve from a 3-level master equation and the dashedlines with the enclosed shaded gray area correspond to theuncertainty of the Rabi frequencies. (b) Time evolution ofthe Feshbach and ground-state population during a round-trip STIRAP. Data points in the upper panel are the observedFeshbach molecule number normalized to the initial moleculenumber. The solid green(dashed black) line is a theory curvefor the ground-state(Feshbach molecule-state) population us-ing the model described in the text and the pulses from thelower panel. The pulse duration for both lasers is 10 µ s. Theramping up of the Pump pulse starts 1 µ s before the rampdown of the Stokes pulse begins. The lower panel showsthe pulse sequence of the Pump and Stokes laser during theSTIRAP. Rabi frequencies are obtained from one-photon lossmeasurements (not shown here). Error bars are the standarddeviation coming from different experimental runs. is sufficient for its description. Fig. 2(a) shows theexperimental data and the theoretical prediction (solidblack line) using experimentally determined parametersfor Rabi frequencies and laser detunings. The errors onthe parameters are displayed as dashed lines and grayshaded area. We find very good agreement of our datawith the model and consequently good conditions forthe STIRAP.For the creation of ground-state molecules, we performSTIRAP starting from Feshbach molecules. As theFeshbach molecule lifetime is very short, on the order of0.3 ms [23], STIRAP is completed 25 µ s after Feshbachmolecules creation. The STIRAP process itself takes11 µ s so that no significant loss from a decay of theweakly bound dimers is expected. Figure 2(b) shows atypical signal for ground-state molecule creation. Thefigure includes the STIRAP light pulse sequence (lowerpanel) and the populations of the Feshbach moleculesas well as the ground-state molecules during the pulsesequence calculated by a 5-level master equation. Start-ing with Feshbach molecules at t = 0, the molecules aretransferred to the ground state at t = 14 µ s where themolecules become dark for the imaging light. To imagethe molecules, we reverse the STIRAP sequence andtransfer ground-state molecules back to the Feshbachstate. Due to the additional coupling of the groundstate to the excited states | e , i , the STIRAP is highlydependent on the one-photon detuning (see Fig. 3).The states | e , i act as loss channels, into which theground-state molecules are pumped and consequentlyget lost. On resonance with one of the | e , i states,nearly no ground-state molecules revive (see Fig. 3).Clearly, in the vicinity of the | e , i states, the STIRAPbenefits from fast transfers, which is restricted by theadiabaticity in the limit of small pulse-overlap areas[22]. On the other hand, the pulse-overlap area canbe increased by raising the Rabi frequencies of thepulses, which accordingly also increases the undesiredcoupling to the states | e , i . We find the best results inour system for a pulse duration of 12 µ s with a Pumppulse delay of − µ s and resonant Rabi frequencies ofΩ Pump = 2 π × . Stokes = 2 π × . π × | e i . Under these conditions single-tripSTIRAP efficiency can get as high as 70 % whichcorresponds to a ground-state molecule number ofabout 4200 in a single hyperfine spin state (see insetof Fig. 3). Moreover, we do not observe heating effectsof the molecules due to the STIRAP (see supplementalmaterial), leading to a phase-space density of up to0 .
14. To model the influence of the states | e , i on theSTIRAP we apply a 5-level master equation model fit(solid curve in Fig. 3) and compare it to an ideal 3-levelone (dashed curve in Fig.3). The model is described indetail in the supplemental material. In the comparisonbetween the 5- and 3-level model the influence of the S T I R A P r o und - t r i p e (cid:30) c i e n c y One-photon detuning ∆ / (2 π ) (MHz) δ (kHz) | e i| e i| e i FIG. 3. One- and two-photon detuning for STIRAP. Theround-trip efficiency for STIRAP is shown as a function ofthe one-photon detuning ∆. The pulse sequence and laserintensities for these measurements were kept constant corre-sponding to the optimal values given in the text. The verti-cal solid blue(dashed red)[dotted red] line is the position ofthe | e i state deduced from measurements and the modeldeveloped in [24]. The solid black curve is a fit using the 5-level master equation model and the individual couplings ofthe Stokes laser to the | e , i states as free parameters. Thedashed gray curve is a theory curve from a 3-level model us-ing the same set of parameters. The inset shows the STIRAPround-trip efficiency dependent on the two-photon detuning δ with a phenomenological Gaussian fit. The error bars forboth plots are the standard error coming from different ex-perimental cycles. states | e , i gets clear. It indicates, that the STIRAPefficiency can be easily increased by choosing a differentexcited state, experimental geometric condition, such aslaser polarization relative to the magnetic field axis, andlarger STIRAP pulse overlap areas, which is discussedin the supplemental material.After the transfer to the ground state the moleculesare still immersed in a gas of Na and K atomsremaining from the creation process of the weakly bounddimers. Na atoms can be removed by applying a500 µ s resonant light pulse. K atoms can be removedby transferring them to the | f = 2 , m f = − i state bya rapid adiabatic passage and a subsequent resonantlight pulse for 500 µ s. By introducing a variable holdtime between the atom removals and the reversedSTIRAP pulse, we perform loss measurements, which weanalyze assuming a two-body decay model to extract thetwo-body decay rate coefficient. The model is describedin the supplemental material.First, we investigate the mixture of molecules and atoms.We observe fast losses from Na K+ Na collisions(see Fig. 4). The extracted loss rate coefficient is1 . × − cm s − , which is close to the theoreticalprediction of 1 . × − cm s − taken from [29]. Weassign the saturation of the losses to chemical reactions, S T I R A P r o und - t r i p e (cid:30) c i e n c y One-photon detuning ∆ / (2 π ) [MHz] δ [kHz] | e i| e i| e i R e m a i n i n g m o l ec u l e f r a c t i o n Hold time (ms) Na K Na K+ K Na K+ Na+ K FIG. 4. Loss measurements of pure ground-state moleculesand with remaining atoms. The open triangles are measure-ments without atom removal. The fast loss originates fromthe chemical reaction with Na atoms. The gray circles aremeasurements with only Na removed while still K atomsremain in the trap. The solid circles are measurements per-formed with a pure molecular ensemble. The data is normal-ized to the molecule number without holding time obtainedfrom the individual fits. The curves are fits using a coupleddifferential equation system for modeling the losses. For thecorresponding loss rate coefficients see text. All error bars arethe standard deviation resulting from different experimentalruns. in which Na dimers are formed. Thus, Na atomsare generally removed as fast as possible from the trapas the ground-state molecule number suffers from thestrong losses.In a next step, we measure losses in a pure molecularensemble (see Fig.4). The two-body loss rate coefficientis measured to be 4 . . × − cm s − . This lossrate coefficient is comparable to the universal limit[30] and is possibly resulting from sticky collisions[21] and subsequent removal of the tetramers from thetrap. Comparable observations have been made inexperiments with other bosonic ground-state molecules,such as Rb Cs and Na Rb [12, 13]. However,the loss rate coefficient for the fermionic counterpart Na K is 6 × − cm s − [10]. The difference canbe assigned to the absence of the centrifugal barrier inbosonic s -wave collisions.Next, we investigate collisions between the moleculesand K atoms. Surprisingly, even a high density of K atoms in the non-stretched | f = 1 , m f = − i K statecolliding with Na K in the non-stretched hyperfineground state does not increase the molecular loss(compare Fig. 4), although sticky collisions with trimerformation are also expected in mixtures of Na K+ K[31]. In these collisional trimer complexes nuclear spintransitions can occur leading to subsequent loss ofmolecules from the prepared hyperfine state. We analyzethe observed decay of the molecular cloud using themodel fit described in the supplemental material. We findthe loss rate coefficient for the two-body Na K+ Kcollisions to be consistent with zero with an upper limitof 1 . × − cm s − . The corresponding universal limitis calculated by using the prediction from [31, 32] andparameters from [33] and results in 1 . × − cm s − .Note that this corresponds to a suppression of thetwo-body decay in comparison to the universal limitby more than three orders of magnitude. This is incontrast to experiments reported for fermionic moleculesin collisions with bosonic atoms ( K Rb + Rb[1])and fermionic atoms ( Na K + K [34]), where suchsuppression of losses far below the universal limit hasnot been observed for sticky molecule-atom collisions.The only experiment describing such a suppression hasbeen performed in a mixture of the fermionic molecule Li Na with the bosonic atom Na with both particlesin their lowest stretched hyperfine states [35]. Here, wenow report collisions in non-stretched states with lossrates far below the universal limit, which might resultfrom a low density of resonant states [31]. Individualresonances might thus be resolvable in this system anddemand for further investigations of loss rates in otherspin channels and magnetic fields. Moreover, with thelow loss rate between Na K molecules and K atomsin the named hyperfine state it might be possible to use K atoms as a coolant for bosonic Na K moleculesto further increase the molecular phase-space density[35].In conclusion, we have reported the first creationof an ultracold high phase-space density gas of bosonic Na K ground-state molecules. We have investigatedthe creation process and find very good agreementwith our 5-level model. The spin-polarized molecularensemble yields up to 4200 molecules and is chemicallystable. We extract the two-body decay coefficient forthe bosonic Na K molecules. For molecule-atom col-lisions, we find a significant suppression of the two-bodydecay rate in collisions between Na K molecules and K atoms in non-stretched states. This unexpectedresult demands for further experiments including theanalysis of collisions between molecules and atoms indifferent hyperfine states and as a function of magneticfield to identify possible scattering resonances. Theseexperiments can be extended to a detailed comparisonof collision properties between same species molecules ofdifferent quantum statistics.We thank M. Siercke for enlightening commentsand discussions and the group of P. O. Schmidt, PTBBrunswick, for providing scientific material for the Ra-man laser system. We gratefully acknowledge financialsupport from the European Research Council throughERC Starting Grant POLAR and from the DeutscheForschungsgemeinschaft (DFG) through CRC 1227(DQ-mat), project A03 and FOR 2247, project E5. P.G. thanks the DFG for financial support through RTG1991. ∗ [email protected] † [email protected][1] S.Ospelkaus, K.-K. Ni, D. Wang, M. H. G. de Miranda,B. Neyenhuis, G. Qu´em´ener, P. S. Julienne, J. L. Bohn,D. S. Jin, and J. Ye, Science , 853 (2010).[2] M. Guo, X. Ye, J. He, M. L. Gonz´alez-Mart´ınez, R. Vex-iau, G. Qu´em´ener, and D. Wang, Phys. Rev. X , 041044(2018).[3] S. A. Moses, J. P. Covey, M. T. Miecnikowski, D. S. Jin,and J. Ye, Nature Physics , 13-20 (2016).[4] L. De Marco, G. Valtolina, K. Matsuda, W. G. Tobias,J. P. Covey, and J. Ye, Science , 853 (2019).[5] D. DeMille, Phys. Rev. Lett. , 067901 (2002).[6] B. Yan, S. Moses, B. Gadway, J. Covey, K. Hazzard,A. Rey, D. Jin, and J. Ye, Nature 501, 521-525 (2013).[7] M. H. G. de Miranda, A. Chotia, B. Neyenhuis, D. Wang,G. Qu´em´ener, S. Ospelkaus, J. L. Bohn, J. Ye, and D. S.Jin, Nature Physics , 502-507 (2011).[8] S. A. Will, J. W. Park, Z. Z. Yan, H. Loh, and M. W.Zwierlein, Phys. Rev. Lett. , 225306 (2016).[9] K.-K. Ni, S. Ospelkaus, M. H. G. de Miranda, A. Peer,B. Neyenhuis, J. J. Zirbel, S. Kotochigova, P. S. Julienne,D. S. Jin, and J. Ye, Science , 231 (2008).[10] J. W. Park, S. A. Will, and M. W. Zwierlein, Phys. Rev.Lett. , 205302 (2015).[11] T. M. Rvachov, H. Son, A. T. Sommer, S. Ebadi, J. J.Park, M. W. Zwierlein, W. Ketterle, and A. O. Jamison,Phys. Rev. Lett. , 143001 (2017).[12] T. Takekoshi, L. Reichs¨ollner, A. Schindewolf, J. M. Hut-son, C. R. Le Sueur, O. Dulieu, F. Ferlaino, R. Grimm,and H.-C. N¨agerl, Phys. Rev. Lett. , 205301 (2014).[13] M. Guo, B. Zhu, B. Lu, X. Ye, F. Wang, R. Vex-iau, N. Bouloufa-Maafa, G. Qu´em´ener, O. Dulieu, andD. Wang, Phys. Rev. Lett. , 205303 (2016).[14] P. S. ˙Zuchowski and J. M. Hutson, Phys. Rev. A ,060703 (2010).[15] M. Gr¨obner, P. Weinmann, E. Kirilov, H.-C. N¨agerl, P. S.Julienne, C. R. Le Sueur, and J. M. Hutson, Phys. Rev.A , 022715 (2017).[16] P. D. Gregory, M. D. Frye, J. A. Blackmore, E. M. Bridge,R. Sawant, J. M. Hutson, and S. L. Cornish, Nature Com-munications (2019).[17] M.-G. Hu, Y. Liu, D. D. Grimes, Y.-W. Lin, A. H.Gheorghe, R. Vexiau, N. Bouloufa-Maafa, O. Dulieu,T. Rosenband, and K.-K. Ni, Science , 1111-1115(2019).[18] A. Christianen, M. W. Zwierlein, G. C. Groenenboom,and T. Karman, Phys. Rev. Lett. , 123402 (2019).[19] P. D. Gregory, J. A. Blackmore, S. L. Bromley, and S. L.Cornish, Phys. Rev. Lett. , 163402 (2020).[20] Y. Liu, M.-G. Hu, M. A. Nichols, D. D. Grimes, T. Kar-man, H. Guo, and K.-K. Ni, (2020), arXiv:2002.05140[physics.atom-ph].[21] M. Mayle, G. Qu´em´ener, B. P. Ruzic, and J. L. Bohn,Phys. Rev. A , 012709 (2013).[22] N. V. Vitanov, A. A. Rangelov, B. W. Shore, andK. Bergmann, Rev. Mod. Phys. , 015006 (2017). [23] K. K. Voges, P. Gersema, T. Hartmann, T. A. Schulze,A. Zenesini, and S. Ospelkaus, Phys. Rev. A , 042704(2020).[24] K. K. Voges, P. Gersema, T. Hartmann, T. A. Schulze,A. Zenesini, and S. Ospelkaus, New Journal of Physics , 123034 (2019).[25] R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough,G. M. Ford, A. J. Munley, and H. Ward, Applied PhysicsB , 97 (1983).[26] T. A. Schulze, I. I. Temelkov, M. W. Gempel, T. Hart-mann, H. Kn¨ockel, S. Ospelkaus, and E. Tiemann, Phys.Rev. A , 023401 (2013).[27] J. Aldegunde and J. M. Hutson, Phys. Rev. A , 042506(2017).[28] I. Temelkov, H. Kn¨ockel, A. Pashov, and E. Tiemann,Phys. Rev. A , 032512 (2015). [29] H. Li, M. Li, C. Makrides, A. Petrov, and S. Kotochigova,Atoms , 36 (2019).[30] P. S. Julienne, T. M. Hanna, and Z. Idziaszek, Phys.Chem. Chem. Phys. , 19114 (2011).[31] M. Mayle, B. P. Ruzic, and J. L. Bohn, Phys. Rev. A ,062712 (2012).[32] G. Qu´em´ener, J. L. Bohn, A. Petrov, and S. Kotochigova,Phys. Rev. A , 062703 (2011).[33] B. E. Londo˜no, J. E. Mahecha, E. Luc-Koenig, andA. Crubellier, Phys. Rev. A , 012510 (2010).[34] H. Yang, D.-C. Zhang, L. Liu, Y.-X. Liu, J. Nan, B. Zhao,and J.-W. Pan, Science , 261 (2019).[35] H. Son, J. J. Park, W. Ketterle, and A. O. Jamison,Nature , 197200 (2020). upplemental Material: An Ultracold Gas of Bosonic Na K Ground-State Molecules
Kai K. Voges, ∗ Philipp Gersema, Mara Meyer zum Alten Borgloh, Torben A. Schulze, Torsten Hartmann, Alessandro Zenesini,
1, 2 and Silke Ospelkaus † Institut f¨ur Quantenoptik, Leibniz Universit¨at Hannover, 30167 Hannover, Germany INO-CNR BEC Center and Dipartimento di Fisica, Universit`a di Trento, 38123 Povo, Italy (Dated: August 13, 2020)In this supplement, we provide additional details on the 5-level STIRAP model, alternative STI-RAP pathways for the Na K molecules and the temperature measurements of the molecules.Furthermore, we detail on the loss model used for the determination of the two-body decay losscoefficients for molecule-molecule and molecule-atom collisions.
STImulated Raman Adiabatic Passage (STIRAP) forthe transfer of weakly bound Feshbach molecules to theground state, and vice versa, is typically performed ina pure 3-level Λ-system [1]. In our case, the Feshbachmolecule state is named | f i , the ground state | g i andthe excited states are named | e i i . The laser beams forthe Pump and the Stokes transitions are copropagatingand perpendicular to the magnetic field. For bothbeams, linear polarizations parallel ( k ) to the magneticfield access π -transitions in the molecules and linearpolarizations perpendicular ( ⊥ ) to the magnetic fieldaccess always both σ + - and σ − -transitions.The molecular starting state | f i canbe described as a composed state of α | m i, Na = − / , m i, K = − / , M S = − i + α | m i, Na = − / , m i, K = − / , M S = 0 i , where M S isthe total electron spin projection and α / representstate admixtures. With the goal of maximizing the Rabifrequency Ω P(ump) , we choose excited states from thetriplet hyperfine manifold of the coupled triplet-singletstates | c Σ + , v = 30 i and | B Π , v = 8 i [2, 3]. Moreover,we choose the polarization of the Pump beam to be k .The only possible accessible excited state is the | e i = | m i, Na = − / , m i, K = − / , M J = − , M F = − i .Using ⊥ polarization for the Stokes laser, we reach the | g i = | m i, Na = − / , m i, K = − / , M J = 0 , M i = − i ground state with a σ − -transition. Other states in theground state cannot be reached, because the groundstate manifold has pure singlet character and is deeply inthe Paschen-Back regime. Thus, nuclear and electronicspins are decoupled so that only the electronic spinprojection can be changed by an optical transition.At the same time σ + -transitions couple thestate | g i to the excited state | e , i which haveboth state contributions in the atomic base from | m i, Na = − / , m i, K = − / , M J = 1 , M F = − i . Notethat the Pump beam does not couple the state | f i tothe states | e , i due to ∆ M F = 2.In summary, the experimental situation requires toextend the typical 3-level Λ-system (for the state | f i , | e i and | g i ) to a 5-level system (for the states | f i , | e i , | g i and | e , i ). The model Hamilton operator H ( t ) forthe light-molecule interaction and the molecular energiesin the rotating-wave-approximation is ~ P ( t ) / P ( t ) / P Ω S ( t ) / S ( t ) / P − ∆ S Ω S,1 ( t ) / S,2 ( t ) /
20 0 Ω
S,1 ( t ) / P − ∆ S,1
00 0 Ω
S,2 ( t ) / P − ∆ S,2 . The time dependent state vector is represented by( c f ( t ) , c e ( t ) , c g ( t ) , c e ( t ) , c e ( t )) T , where c i is the prob-ability amplitude of the corresponding state | i i . Ω P ( t )is the Rabi frequency for the Pump transition andΩ S(tokes) ( t ), Ω S,1 ( t ) and Ω S,2 ( t ) are the Rabi frequenciesfor the Stokes transition to the excited states | e i , | e i and | e i , respectively. Note, that all Rabi frequencies aretime dependent and real. ∆ P and ∆ S are the detuningsof the Pump and Stokes laser frequency to the respectivemolecular transition. The relative positions of the ex-cited states | e , i to | e i are ∆ S,1 = 2 π × ( −
10) MHz and∆
S,2 = 2 π × ( −
21) MHz, respectively, at 199.3 G and aretaken from our excited state model presented in [3]. Toadditionally model losses of the molecules from the ex-cited states, a sixth state | l i is introduced, which is notdirectly coupled to any other state. This is importantfor the numerical calculation, as it keeps the populationnormalized during the evaluation. The dynamics of thesystem can be modeled by solving the master equationin Lindblad representation with the density matrix ρ ( t )˙ ρ ( t ) = − i ~ [ H ( t ) , ρ ( t )] + X k γ k D [ A k ] ρ ( t ) . (1)The second term denotes the losses from the system,where γ k are the decay rates of the excited states whichwe set for all three states to γ = γ = γ = 2 π ×
11 MHzand D [ A k ] are the corresponding Lindblad superopera-tors with the jump operator A k from the excited state | e k i to the loss state | l i [4].For the fit of the experimental data in Fig. 3 we use thismodel with the Rabi frequencies Ω S,1 and Ω
S,2 as freeparameters as well as the STIRAP Rabi frequencies Ω P and Ω S constrained to their experimentally determineduncertainties. We assign the optimum of the fit within a r X i v : . [ c ond - m a t . qu a n t - g a s ] A ug these constrains, confirming the consistency of our data.Furthermore, this model was used to also calculate theSTIRAP time dynamics of Fig. 2(b).The 5-level model can be reduced to a 3-level one by set-ting the coupling to the excited state | e , i to zero. Weuse this to calculate the theoretical electromagneticallyinduced transparency curve in Fig. 2(a) and the optimalcurve for the one-photon detuning (gray dashed line) inFig. 3. Alternative STIRAP pathways
Alternative STIRAP pathways using states from the c Σ + potential may be possible with either another STI-RAP beam alignment, for example parallel to the mag-netic field direction, and/or with other polarizations.In case of a perpendicular alignment, as it is describedabove, alternative STIRAP pathways to the ground state | g i are possible when switching the laser polarizations,using now ⊥ polarization for the Pump and k polariza-tion for the Stokes beam; see Fig.S1.We identify two additional states | e a,1 i and | e a,2 i suitingthese pathways, both yielding state contributions fromthe | m i, Na = − / , m i, K = − / , M J = 0 , M F = − i inthe atomic base. Their transitions are +189 and −
146 MHz detuned from the original one | e i and donot possess neighboring states close by which may bepopulated through σ − -transitions to the state | f i . Theadditional STIRAP pathways are identified based on themodel of the excited states [3]. Simulations, utilizing themodel described above suggest round-trip efficiencies ofmore than 80 %. These states will be object of futureinvestigation. Loss coefficients
Two-body loss coefficients for molecule-molecule andmolecule-atom collisions are extracted from the decay ofthe Na K ground-state molecule ensemble.In a pure molecular ensemble, losses can be assigned totwo-body losses with tetramer formation and subsequentremoval or loss of the tetramers, see [5]. We obtain ananalytic solution for the two-body loss of the ground-state molecule number N NaK ( t ) [6] N NaK ( t ) = N NaK , (1 + (cid:15) k NaK , t ) / , (2)where N NaK , is the initial ground-state molecule num-ber, k NaK , the molecular two-body loss coefficient and (cid:15) = ( m NaK ¯ ω/ (2 πk B )) / with ¯ ω the average trap fre-quency and k B the Boltzmann constant. N NaK , and k NaK , were used as free parameters for the fit.For the model of the loss from molecule-atom collisions π | e , i| e i | e a, i| e a, i| f i| g i σ + σ − σ + π PumpPumpStokes Stokes
FIG. S1. STIRAP pathways. This figure shows the current(left) and the alternative (right) STIRAP pathway. Pathwaysstart from the Feshbach molecule state | f i (black solid line)and end in the ground state | g i (green solid line). The excitedstates | e i i are the ones described in the text. For the currentSTIRAP the Pump beam drives π -transitions and the Stokesbeam σ − -transitions, displayed as solid arrows. On contrary,the Stokes beam couples also with σ + -transition to the ex-cited states | e , i (dashed arrow). The alternative STIRAPpathways use σ + -transitions for the Pump and π -transitionsfor the Stokes beam. The states | e a, i are shown as orangedashed-dotted lines. we use the coupled differential equation system:˙ N NaK ( t ) = − (cid:15) k NaK , N NaK ( t ) T NaK ( t ) / − η k a N a , N NaK ( t )˙ T NaK ( t ) = (cid:15) k NaK , N NaK ( t )4 p T NaK ( t ) . (3) η is the density overlap between molecules and atoms, k a the loss coefficient for the molecule-atom collision, N a , the initial atom number and T NaK the temperature of theground-state molecules.Note, that in the model anti-evaporation effects may beconsidered for molecules only, or for both, molecules andatoms. The difference of these two cases is smaller thanour experimental uncertainties. The presented data onlyconsider effects on molecules.
Temperature measurement for molecular clouds
Temperature measurements for atomic clouds are typ-ically done through time-of-flight (TOF) measurementsafter releasing them from the trap and fitting a temper-ature dependent expansion curve to the clouds width.For ground-state molecules, this technique is limited bythe free expansion time, as the molecules might leave theregion of the STIRAP laser beams which is needed totransfer the ground-state molecules back to the Feshbachstate for imaging. In our experiment, the STIRAP beamfoci have 1 /e radii of 35 and 40 µ m, respectively, allow-ing for almost no free expansion time of the moleculesbefore leaving the STIRAP beam area.To still measure the temperature of the ground-statemolecule ensemble we reverse the entire molecule creationprocess, by means of STIRAP and Feshbach moleculedissociation, before performing the TOF and imagingon the dissociated atoms. Note, that for our Fesh-bach molecules, imaging normally takes place from theFeshbach molecule state itself, as the linewidth of theimaging transition in K is larger than the binding en-ergy of the weakly bound dimers [7]. A temperaturemeasurement in TOF with Feshbach molecules is alsonot possible, because the Feshbach molecule lifetime isvery short, about 1 ms in a pure molecular ensemble,and an appropriate signal would be lost very fast. Acomplete dissociation within 600 µ s using a resonant ra-dio frequency of 210 . | f = 1 , m f = − i Na + | f = 1 , m f = − i K is performed immediately after the backwards STIRAP. Temperature TOF measurementsare then performed on the long living atomic ensem-ble. All atoms involved in the temperature measurementcome originally from deeply bound molecules.The extracted temperature from the atoms show thesame temperature as the initial atoms measured beforethe molecule creation happened. Consequently, all trans-fers in between (Feshbach molecule creation, STIRAPs,atom state preparations and removals, Feshbach moleculedissociation) do not heat the molecule ensemble. ∗ [email protected] † [email protected][1] N. V. Vitanov, A. A. Rangelov, B. W. Shore, andK. Bergmann, Rev. Mod. Phys. , 015006 (2017).[2] T. A. Schulze, I. I. Temelkov, M. W. Gempel, T. Hart-mann, H. Kn¨ockel, S. Ospelkaus, and E. Tiemann, Phys.Rev. A , 023401 (2013).[3] K. K. Voges, P. Gersema, T. Hartmann, T. A. Schulze,A. Zenesini, and S. Ospelkaus, New Journal of Physics , 123034 (2019).[4] H. J. Carmichael, Springer-Verlag Berlin Heidelberg10.1007/978-3-662-03875-8 (1999).[5] P. D. Gregory, M. D. Frye, J. A. Blackmore, E. M. Bridge,R. Sawant, J. M. Hutson, and S. L. Cornish, Nature Com-munications (2019).[6] The derivation of the analytical solution will be presentedin a future publication.[7] K. K. Voges, P. Gersema, T. Hartmann, T. A. Schulze,A. Zenesini, and S. Ospelkaus, Phys. Rev. A101