Production of large 41 K Bose-Einstein condensates using D1 gray molasses
Hao-Ze Chen, Xing-Can Yao, Yu-Ping Wu, Xiang-Pei Liu, Xiao-Qiong Wang, Yu-Xuan Wang, Yu-Ao Chen, Jian-Wei Pan
PProduction of large K Bose-Einstein condensates using D1 gray molasses
Hao-Ze Chen,
1, 2, 3
Xing-Can Yao,
1, 2, 3, 4
Yu-Ping Wu,
1, 2, 3
Xiang-Pei Liu,
1, 2, 3
Xiao-Qiong Wang,
1, 2, 3
Yu-Xuan Wang,
1, 2, 3
Yu-Ao Chen,
1, 2, 3 and Jian-Wei Pan
1, 2, 3, 4 Shanghai Branch, National Laboratory for Physical Sciences at Microscale and Department of Modern Physics,University of Science and Technology of China, Hefei, Anhui 230026, China CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics,University of Science and Technology of China, Shanghai, 201315, China CAS-Alibaba Quantum Computing Laboratory, Shanghai, 201315, China Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg,Im Neuenheimer Feld 226, 69120 Heidelberg, Germany
We use D1 gray molasses to achieve Bose-Einstein condensation of a large number of K atomsin an optical dipole trap. By combining a new configuration of compressed-MOT with D1 graymolasses, we obtain a cold sample of . × atoms with a temperature as low as 42 µ K. Aftermagnetically transferring the atoms into the final glass cell, we perform a two-stage evaporativecooling. A condensate with up to . × atoms in the lowest Zeeman state | F = 1 , m F = 1 (cid:105) isachieved in the optical dipole trap. Furthermore, we observe two narrow Feshbach resonances in thelowest hyperfine channel, which are in good agreement with theoretical predictions. I. INTRODUCTION
Quantum degenerate gases of neutral atoms not onlyprovide platforms for studying few- or many-body quan-tum systems [1, 2], but are also promising candidates forhigh-precision quantum metrology [3] and scalable quan-tum computation [4]. In the past few decades, tremendousexperimental efforts have been devoted to achieving quan-tum degenerate gases with alkali [5], alkaline-earth [6],and rare-earth metals [7]. Due to the existence of bothfermionic and bosonic isotopes, a dilute gas of potassiumis of particular interest for studying Bose-Fermi and Fermi-Fermi mixtures with tunable interactions. For instance, amixture of K and K represents an ideal test bed forexploring impurity problems [8] and strongly interactingBose-Fermi mixtures [9].A Bose-Einstein condensate (BEC) of K was firstachieved by sympathetic cooling with Rb [10] in 2001.With a positive background scattering length, K couldbe directly evaporatively cooled into a Bose-Einstein con-densate [11, 12]. Moreover, K has proven to be anefficient coolant for Li and K, due to their favorableinterspecies background scattering lengths [9, 13]. Formost alkali-metal atoms, the temperature can be eas-ily reduced below the Doppler limit through Sisyphuscooling in optical molasses. However, the D2 excitedhyperfine splitting of K is only 17 MHz, which is unre-solved in comparison to its natural linewidth ( Γ =6 MHz)(see Fig. 1). Therefore, conventional sub-Doppler cool-ing techniques are ineffective, making direct evaporativecooling of K an arduous task. In order to overcomethis obstacle, different laser-cooling techniques have beendeveloped in recent years [14, 15]. For example, usingthe narrower 4s-5p transition of potassium, temperaturesaround 60 µ K have been observed, equivalent to half ofthe D2 transition Doppler-cooling limit [15]. However,additional laser sources at 405 nm and optical setup arerequired for the implementation of this scheme, increas- ing its experimental complexity. Recently, a so-calledgray molasses sub-Doppler cooling technique has beenintensively exploited; this method takes advantage ofSisyphus cooling [16] and velocity selective coherent pop-ulation trapping (VSCPT) [17] to cool atoms well belowthe Doppler limit. To date, gray molasses has become awell-established method for achieving sub-Doppler cool-ing of Li [18, 19], Li [20], Na [21], K [22, 23], and K [18, 24]; however, to the best of our knowledge, theapplication of the gray molasses technique to K has yetto be reported.
Figure 1. (Color Online) The energy-level diagram of Katoms. The natural linewidth of the D2-transition of K is 6MHz, which is comparable to its excited hyperfine splitting.The laser detuning for gray molasses is defined as δ for bothcooling and repumping transitions. In this letter, we report sub-Doppler cooling of K byusing D1 gray molasses. This technique allows us to lower a r X i v : . [ c ond - m a t . qu a n t - g a s ] A ug the initial temperature of . × atoms from 418 µ K to42 µ K. With the help of two-stage evaporative cooling, weare able to produce a K BEC of up to . × atoms inoptical dipole trap without a discernible thermal fraction.Furthermore, we investigate the collisional properties of K theoretically and experimentally.
II. EXPERIMENTAL SET-UP
The experimental setup for potassium consists of threemain parts: a 2D MOT chamber, a 3D MOT chamber,and a science chamber (Fig. 2). The shadow region isto be used for double-species experiments in the future.The 2D MOT chamber is a 4.5 cm × × × − mbar, which greatly increases the lifetimeof the atoms. Figure 2. (Color Online) Schematics of the vacuum assembly.The shadowing represents the vacuum manifold for futuredouble-species experiments. In the inset, we zoom-in on the2D MOT chamber to illustrate the optics for our longitudinalmolasses. The red and blue optics represent the absorptivepolarizer and mirror, respectively.
III. 2D + MOT & 3D MOT
Our experimental procedure starts with pre-coolingand confinement of a K atomic vapor using an en-hanced 2D + MOT. In a standard 2D + MOT, a pairof counter-propagating beams are applied to the longitu-dinal molasses. The intensity ratio between incident and retro-reflected beams strongly affect the performance ofthe 2D + MOT. Thanks to the polarizer and mirror insidethe chamber (see Fig. 2), we can easily tune the intensityratio by changing the polarization of an incident laserbeam. In our case, the intensity ratio is about 4:1, whichis optimized for atomic fluorescence signals. Moreover,an additional blue-detuned laser beam with a 1/e radiusof 0.6 mm is applied in the axial direction to push theatoms into the 3D MOT chamber. We find this pushbeam to be particularly efficient for achieving a further50% enhancement of the atomic flux beyond that allowedby the standard 2D + MOT. Special care is required toslightly misalign the push beam with the captured atomiccloud in the 3D MOT. We also install two UV LED arraysaround the 2D MOT chamber, which serve to increasethe loading efficiency due to light-induced atom desorp-tion [25]. In a 3D MOT chamber, the atomic clouds arecaptured by six independent laser beams with 1/e radiiof 9 mm. The total intensity of each beam is I (cid:39) I sat ,where I sat =1.75 mW/cm is the saturation intensity ofthe K D2 transition. We can load more than × atoms at a temperature of 5.6 mK in 2 s. IV. GRAY MOLASSES
Gray molasses makes use of both Sisyphus cooling andVSCPT, which is capable of cooling atoms near the single-photon recoil energy. It is composed of two frequencies,which are each blue-detuned by either the K D1 coolingtransition | F = 2 (cid:105) → | F (cid:48) = 2 (cid:105) or the repumping transition | F = 1 (cid:105) → | F (cid:48) = 2 (cid:105) . When the detunings of the two lasersfulfill the Raman condition, a set of long-lived dresseddark states are generated. Dark states and bright statesvary throughout real space because of the polarizationgradient created by the radiation field. The atoms thatare initially in the bright states consume their kineticenergy by climbing the potential hill before being pumpedback into dark states, whereas the atoms in dark statesthat are not sufficiently cold turn into bright states andreenter the cooling circles. This procedure terminateswhen the atoms in the dark states have a sufficientlylong lifetime, leading to a narrow peak in the momentumdistribution.A simplified layout of the D1 laser setup employed forgray molasses is presented in Fig. 3(a). A commercialdiode laser (Toptica: DL pro) provides the laser at 770nm, which is locked to the D1 transition of K. Thelaser passes through a resonant electro-optical modula-tor (EOM: Qubig EO-K41L3) operating at 254 MHz forgeneration of the required repumper frequency. The laserbeams of the D1 and D2 transitions are superimposedusing a customized dichroic mirror before being injectedinto a tapered amplifier (New Focus: TA 7613-H). Twoindividual acoustic-optical modulators (AOMs) serve asfast switches, allowing us to rapidly change the output fre-quency for different cooling phases. With this scheme, thesame optical setup is used for both D1 gray molasses and A t o m N u m be r T e m pe r a t u r e ( K ) μ Duration (ms) A t o m N u m be r Final D1 Laser Intensity (I/I ) sat T e m pe r a t u r e ( K ) μ A t o m N u m be r T e m pe r a t u r e ( K ) μ Detuning ( )δ Γ (a) (b)(c) (d) Figure 3. (Color Online) (a) Simplified layout of the optical setup for gray molasses. (b) Atom number (blue squares) andtemperature (red circles) varying with the final cooling intensity of a D1 laser. The power ratio between the cooler and repumperis fixed at 5:1. Atom temperature scales linearly with the final cooling intensity from . I sat (42 µ K) to I sat (160 µ K). (c)Atom number (blue squares) and temperature (red circles) as a function of the duration, τ m . The lowest temperature of 47 µ Kis observed with a 1/e cooling time constant of τ =3.84 ms. (d) Atom number (blue squares) and temperature (red circles) as afunction of the detuning, δ . Gray Molasses is found to work best for δ ∈ [4 Γ , . D2 MOT, greatly reducing the experimental complexity.Before the gray molasses phase, a novel hybrid D1-D2 compressed-MOT (CMOT) technique is implementedto increase the density of the atomic cloud, in a similarmanner to that described in G. Salomon’s work in Ref [23].Compared with a conventional CMOT, the D1 coolingtransition | F = 2 (cid:105) → | F (cid:48) = 2 (cid:105) is used instead of theD2 cooling transition. During this stage, we shut off theinput signal of the EOM immediately and ramp up themagnetic gradient from 13 G/cm to 21 G/cm over 10ms. In order to suppress light-assisted collision, cooling(repumping) intensity is linearly reduced from . I sat ( I sat ) to . I sat ( . I sat ) and detuning is increasedfrom . (- . ) to . (- . ). During this process,the RMS radius of the cloud is greatly reduced, yieldingan eightfold increase in the peak density. Meanwhile,atom temperature is lowered from 5.6 mK to 418 µ K.In the gray molasses stage, the D2 repumping laserand magnetic field are switched off while the input signalof the EOM is switched on. In the following analysis,we use the same global detuning, δ , for both D1 cooling and repumping transitions to fulfill the Raman conditionshown in Fig. 1. As the capture efficiency is stronglyaffected by the laser intensity [18], we set the laser powerto its maximum at the beginning of gray molasses phase,yielding a temperature of 160 µ K. With an initial atomnumber of . × in the CMOT, a maximum of . × atoms can be captured by the gray molasses, correspond-ing to a capture efficiency of 92%. We observe a furthercooling of atoms when the laser intensity is ramped downto a lower value. Moreover, the lowered intensity is stillcapable of capturing almost all of the atoms in the graymolasses. Thus, we first measure the atom number andtemperature of gray molasses by varying the final the D1laser intensity. The loading time is fixed at 2 ms as graymolasses reaches its maximum capture rate, and then thelaser intensity is linearly ramped down over the following15 ms. The cooling/repumping intensity ratio is fixedat a constant ( ∼ ) and the global detuning, δ , is . As shown in Fig. 3(b), the atom temperature scaleslinearly with the final cooling intensity from . I sat to I sat . We observe a minimum temperature (42 µ K) that
Sequence δ c I c δ r I r δ c I c δ r I r B(G/cm) T( µ K) N ρ Loading (2 s) -8.9 11.5 -5 17.5 - - - - 13 5600 × × − D1-D2 CMOT (10 ms) - - -5.6 → -8.4 1 → → → →
21 418 . × . × − Molasses Holding (2 ms) - - - - 5.3 12.5 5.3 2.5 - - - -Molasses Ramping (15 ms) - - - - 5.3 12.5 → → . × . × − Table I. Optimal experimental parameters before the optical pumping stage. The unit of δ is the natural linewidth of the D2transition of K ( Γ =6 MHz) and the unit of I is its saturation intensity ( I sat =1.75 mW/cm ). The subscripts 1 and 2 refer toD1 and D2 transitions, while c and r refer to cooling and repumping, respectively. We also present the time sequence, magneticfield gradient B, temperature T, atom number N, and phase-space density ρ at each stage. occurs at I cool = 1 . I sat . Both atom loss and inefficientcooling occur when the final intensity is below this value.Thus, we make use of this piecewise time sequence in ourfollowing measurements.Then, we study the efficiency of gray molasses as a func-tion of its duration, τ m , see Fig. 3(c). Though the initialatom temperature is 418 µ K, it rapidly decreases over thefirst 10 ms and reaches an asymptotic temperature of 47 µ K with a 1/e cooling time constant of τ =3.84 ms.Finally, we study the influence of global detuning, δ ,which is presented in Fig. 3(d). From the data, we observethat the atom number shows a weak dependence uponthe global detuning from to . For δ ∈ [2 Γ , ,the temperature decreases from 60 µ K to 49 µ K. Then,the temperature remains minimal for δ ∈ [4 Γ , andincreases rapidly when δ ≥ .A typical temperature after the gray molasses phase is42 µ K, and the obtained phase-space density is . × − .The optimized parameters before the optical pumpingstage and the time sequence are presented in Table. I. V. MAGNETIC TRANSPORT
At the end of the gray molasses phase, atoms are depo-larized and randomly distributed into hyperfine groundstates. Hence, optical pumping is required to prepareatoms in the low-field seeking state | F = 2 , m F = 2 (cid:105) formagnetic trapping. In order to minimize the absorption-emission cycles during optical pumping, we use D1 lineoptical pumping, which causes the target state behavelike a dark state in the presence of a bias magnetic field.Furthermore, we apply a pair of balanced retro-reflectingoptical pumping beams to decrease the displacement ofthe atoms. With an optimized optical pumping phase,we achieve a loading efficiency of 70% for the magnetictrap. Then, we adiabatically transfer the atoms a dis-tance of 54 cm over 3 s from the MOT chamber to thescience chamber. The magnetic transport consists of fif-teen pairs of overlapping quadrupole coils which generatea moving trapping potential by applying time-varyingcurrents. The transfer path has an angle of 45 ◦ , such thatthe science cell is out of the line-of-sight of the potassiumoven (see Fig. 2). We finally obtain × atoms with atemperature of 200 µ K at a 115 G/cm magnetic gradientin the glass cell.
VI. EVAPORATIVE COOLING
Although a K BEC has already been achieved in themagnetic trap, the fact that only the low-field seekingstate can be magnetically confined limits its further ap-plications. In contrast to magnetic traps, the trappingpotential of optical traps is independent of magnetic sub-states, making the latter well-suited to investigating ofspin-mixture [26, 27] and Feshbach resonance [28]. There-fore, it is desirable to achieve a K BEC in the opticaldipole trap. In our experiment, due to heating and atomloss in the magnetic transport phase, directly loadingatoms into an optical dipole trap is inefficient. To solvethis problem, we adopt a two-stage evaporative coolingstrategy. Atoms are first evaporatively cooled in the op-tically plugged magnetic trap, and then transferred intothe optical dipole trap for subsequent evaporation.In order to avoid Majorana spin-flip, a tightly focused532 nm laser is used to plug the center of the quadrupoletrap. The laser power is 11 W with a 1/e radius of34 µ m, which provides a potential barrier of 360 µ K.Evaporation is performed by driving the | F = 2 , m F =2 (cid:105) → | F = 1 , m F = 1 (cid:105) transition, which is 254 MHzunder zero magnetic field. In the first trail, we ramp upthe magnetic gradient from 115 G/cm to 198 G/cm over200 ms to increase the collision rate and then scan theRF-knife from 340 MHz to 265 MHz linearly over 11 s.In the second trail, in order to suppress three-body loss,we decompress the trap back to 110 G/cm over 160 ms.Afterwards, the RF-knife is further reduced from 261.5MHz to 257 MHz over 2 s. Finally, we obtain a cloud of . × atoms at 6.5 µ K with an achieved phase-spacedensity of 0.26. The evolution of phase-space density asa function of atom number is plotted in Fig. 4. The totalefficiency, Γ , of evaporative cooling is . ± . , where Γ = − d ( lnP SD ) /d ( lnN ) .After evaporative cooling is performed in the opticallyplugged magnetic trap, we transfer the cold sample intoa single beam dipole trap. Our trapping potential isgenerated by a 170 mW, 1064 nm laser with a single spatialand longitudinal mode. The beam is focused into a 1/e radius of 24 µ m, which corresponds to a 23 µ K trap depthfor K. Experimentally, we adiabatically ramp downthe magnetic gradient and increase the laser intensitysimultaneously over 100 ms. With optimized parameters,the total atom number transferred into the optical trapis . × at 12.2 µ K. Then, we immediately transferatoms into the hyperfine ground state | F = 1 , m F =1 (cid:105) by performing a Landau-Zener sweep. The forcedevaporation is accomplished by exponentially decreasingthe laser power by a factor of 24 in 3 s. Additional axialconfinement is provided by the magnetic curvature undera 300 G bias field, which becomes dominant near the endof evaporation. Finally, we observe a K BEC of up to . × atoms without a discernible thermal fraction(see the inset of Fig. 4). The calculated axial and radialtrapping frequencies are about π × . Hz and π × Hz, respectively.
Plugged Magnetic TrapOptical Dipole Trap
Atom Number P ha s e S pa c e D en s i t y
500 mμ
Figure 4. (Color Online) Phase-space density as a functionof atom number during two-stage evaporative cooling. Theblue squares (red circles) represent evaporation in the opticallyplugged magnetic trap (optical dipole trap). The inset showsthe false-color absorption image and the 1D-integrated densityprofile of a BEC after a 25 ms time-of-flight.
VII. DETECTION OF FESHBACHRESONANCE
Feshbach resonance is an essential ingredient for con-trolling interatomic interactions. Different theoreticalapproaches for predicting two-body collisional behaviorshave been developed in recent years and shown good agree-ment with experiment results [29–32]. Coupled-Channelcalculation is a numerical solution method of the cou-pled Schrödinger equations which can directly obtain thescattering length as a function of the bias field [29].Here, s-wave Feshbach resonances of K in the state | F = 1 , m F = 1 (cid:105) are theoretically simulated using aCoupled-Channel calculation, based on well-establishedsinglet X Σ and triplet a Σ potentials [11]. The theoreti-cal predictions of resonance locations (409.3 G & 660.6G) serve as a guide for the experiment. We perform in-elastic loss spectroscopy to detect Feshbach resonances where enhanced atom losses occur due to three-body de-cay. In the following measurements, the initial status of K is N = 3 . × and T = 300 nK. We first rampthe magnetic field to the desired value and then hold itthere for a period of time before measuring the residualatom number in the trap. In order to avoid measurementfluctuations during fast deactivation of the magnetic field,the atom number is directly measured by high-field ab-sorption imaging. Reproducible loss features are observedaround 409.2 G and 660.6 G, as shown in Fig. 5; the bluedashed lines are theoretical predictions of the scatteringlength based on the Coupled-Channel calculation. Theinelastic loss spectra are fitted with Lorentz functions,which satisfactorily agree with simulation results. Theaccuracy of the magnetic field is calibrated by RF spec-troscopy between the two lowest hyperfine states of Kfor several points between 400 G and 700 G, and foundto be better than 50 mG. S c a tt e r i ng Leng t hun i t s o f a )( S c a tt e r i ng Leng t hun i t s o f a )( Figure 5. (Color Online) Inelastic loss spectra in the lowesthyperfine channel of K. The measurement data (red circles)are fitted by Lorentz functions. The resonance points arelocated at 409.2 G (a) and 660.6 G (b), respectively. The bluedashed lines represent the theoretical predictions of scatteringlength in the units of Bohr radius according to a Coupled-Channel calculation; these predictions are in good agreementwith the experimental results.
VIII. CONCLUSION
In summary, we have achieved a large production ofa K BEC in an optical dipole trap. Our method isbased on the a combination of enhanced 2D + MOT,D1-D2 CMOT, D1 gray molasses, and a two-stage evap-oration strategy. D1 gray molasses provides an effectivesub-Doppler cooling mechanism for K. We make useof two successive cooling phases, yielding a high capturerate (92%) as well as a low temperature (42 µ K). Af-ter magnetically transferring the cloud into the sciencechamber, evaporative cooling is first performed in an op-tically plugged magnetic trap with an overall efficiencyof . ± . . This procedure allows us to enhance thephase-space density of atoms by more than four orders of magnitude. The subsequent evaporative cooling is per-formed in a single beam optical dipole trap, producing apure BEC of more than . × atoms, which is four timeslarger than that in any previous experiment [12]. Thecollisional properties of K in the state | F = 1 , m F = 1 (cid:105) are studied both theoretically and experimentally. Tworesonance positions are observed at 409.2 G and 660.6 G,as has been predicted theoretically. These results set anew benchmark for generation of a BEC with K atoms.We thank useful discussions with Jue Nan for Coupled-Channel calculations. This work has been supported bythe NSFC of China, the CAS, and the National Fundamen-tal Research Program (under Grant No. 2013CB922001).X.-C. Yao acknowledges support from the Alexander vonHumboldt Foundation. [1] I. Bloch, J. Dalibard, and W. Zwerger, Reviews of ModernPhysics , 885 (2008).[2] I. Bloch, J. Dalibard, and S. Nascimbène, Nature Physics , 267 (2012).[3] A. Peters, K. Y. Chung, and S. Chu, Metrologia , 25(2001).[4] S. Trotzky, P. Cheinet, S. Fölling, M. Feld, U. Schnor-rberger, A. M. Rey, A. Polkovnikov, E. Demler, M. Lukin,and I. Bloch, Science , 295 (2008).[5] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E.Wieman, and E. A. Cornell, Science , 198 (1995).[6] S. Stellmer, M. K. Tey, B. Huang, R. Grimm, andF. Schreck, Physical Review Letters , 200401 (2009).[7] M. Lu, N. Q. Burdick, S. H. Youn, and B. L. Lev, PhysicalReview Letters , 190401 (2011).[8] A. Schirotzek, C.-H. Wu, A. Sommer, and M. W. Zwier-lein, Physical Review Letters , 230402 (2009).[9] C.-H. Wu, I. Santiago, J. W. Park, P. Ahmadi, and M. W.Zwierlein, Physical Review A , 011601 (2011).[10] G. Modugno, G. Ferrari, G. Roati, R. Brecha, A. Simoni,and M. Inguscio, Science , 1320 (2001).[11] S. Falke, H. Knöckel, J. Friebe, M. Riedmann, E. Tiemann,and C. Lisdat, Physical Review A , 012503 (2008).[12] T. Kishimoto, J. Kobayashi, K. Noda, K. Aikawa,M. Ueda, and S. Inouye, Physical Review A , 031602(2009).[13] E. Tiemann, H. Knöckel, P. Kowalczyk, W. Jastrzebski,A. Pashov, H. Salami, and A. Ross, Physical Review A , 042716 (2009).[14] M. Landini, S. Roy, L. Carcagní, D. Trypogeorgos, M. Fat-tori, M. Inguscio, and G. Modugno, Physical Review A , 043432 (2011).[15] D. McKay, D. Jervis, D. Fine, J. Simpson-Porco, G. Edge,and J. Thywissen, Physical Review A , 063420 (2011).[16] J. Dalibard and C. Cohen-Tannoudji, JOSA B , 2023(1989).[17] A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste,and C. Cohen-Tannoudji, Physical Review Letters ,826 (1988).[18] F. Sievers, N. Kretzschmar, D. R. Fernandes, D. Suchet, M. Rabinovic, S. Wu, C. V. Parker, L. Khaykovich, C. Sa-lomon, and F. Chevy, Physical Review A , 023426(2015).[19] A. Burchianti, G. Valtolina, J. Seman, E. Pace, M. De Pas,M. Inguscio, M. Zaccanti, and G. Roati, Physical ReviewA , 043408 (2014).[20] A. T. Grier, I. Ferrier-Barbut, B. S. Rem, M. Delehaye,L. Khaykovich, F. Chevy, and C. Salomon, PhysicalReview A , 063411 (2013).[21] G. Colzi, G. Durastante, E. Fava, S. Serafini, G. Lam-poresi, and G. Ferrari, Physical Review A , 023421(2016).[22] D. Nath, R. K. Easwaran, G. Rajalakshmi, and C. Un-nikrishnan, Physical Review A , 053407 (2013).[23] G. Salomon, L. Fouché, P. Wang, A. Aspect, P. Bouyer,and T. Bourdel, EPL (Europhysics Letters) , 63002(2014).[24] D. R. Fernandes, F. Sievers, N. Kretzschmar, S. Wu,C. Salomon, and F. Chevy, EPL (Europhysics Letters) , 63001 (2012).[25] C. Klempt, T. Van Zoest, T. Henninger, E. Rasel, W. Ert-mer, J. Arlt, et al. , Physical Review A , 013410 (2006).[26] K. O’hara, S. Hemmer, M. Gehm, S. Granade, andJ. Thomas, Science , 2179 (2002).[27] S. Jochim, M. Bartenstein, A. Altmeyer, G. Hendl,S. Riedl, C. Chin, J. H. Denschlag, and R. Grimm,Science , 2101 (2003).[28] C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Reviewsof Modern Physics , 1225 (2010).[29] H. Stoof, J. Koelman, and B. Verhaar, Physical ReviewB , 4688 (1988).[30] E. Wille, F. Spiegelhalder, G. Kerner, D. Naik,A. Trenkwalder, G. Hendl, F. Schreck, R. Grimm,T. Tiecke, J. Walraven, et al. , Physical Review Letters , 053201 (2008).[31] M. Houbiers, H. Stoof, W. McAlexander, and R. Hulet,Physical Review A , R1497 (1998).[32] T. M. Hanna, E. Tiesinga, and P. S. Julienne, PhysicalReview A79