Universality of the Three-Body Parameter for Efimov States in Ultracold Cesium
M. Berninger, A. Zenesini, B. Huang, W. Harm, H.-C. Nägerl, F. Ferlaino, R. Grimm, P. S. Julienne, J. M. Hutson
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] S e p Universality of the Three-Body Parameter for Efimov States in Ultracold Cesium
M. Berninger, A. Zenesini, B. Huang, W. Harm, H.-C. N¨agerl, F. Ferlaino, R. Grimm,
1, 2
P. S. Julienne, and J. M. Hutson Institut f¨ur Experimentalphysik and Zentrum f¨ur Quantenphysik, Universit¨at Innsbruck, 6020 Innsbruck, Austria Institut f¨ur Quantenoptik und Quanteninformation, ¨Osterreichische Akademie der Wissenschaften, 6020 Innsbruck, Austria Joint Quantum Institute, NIST and the University of Maryland, Gaithersburg, Maryland 20899-8423, USA Department of Chemistry, Durham University, South Road, Durham, DH1 3LE, United Kingdom (Dated: November 1, 2018)We report on the observation of triatomic Efimov resonances in an ultracold gas of cesium atoms. Exploitingthe wide tunability of interactions resulting from three broad Feshbach resonances in the same spin channel, wemeasure magnetic-field dependent three-body recombination loss. The positions of the loss resonances yieldcorresponding values for the three-body parameter, which in universal few-body physics is required to describethree-body phenomena and in particular to fix the spectrum of Efimov states. Our observations show a robustuniversal behavior with a three-body parameter that stays essentially constant.
PACS numbers: 03.75.-b, 21.45.-v, 34.50.Cx, 67.85.-d
The concept of universality manifests itself in the fact thatdifferent physical systems can exhibit basically the same be-havior, even if the relevant energy and length scales differ bymany orders of magnitude [1]. Universality thus allows us tounderstand in the same theoretical framework physical situa-tions that at first glance seem completely different. In ultra-cold atomic collisions, the universal regime is realized whenthe s -wave scattering length a , characterizing the two-body in-teraction in the zero-energy limit, is much larger than the char-acteristic range of the interaction potential. Then the essentialproperties of the two-body system such as the binding energyof the most weakly bound dimer state and the dominating partof the two-body wave function can simply be described interms of a , independent of any other system-dependent param-eters. In the three-body sector, the description of a universalsystem requires an additional parameter, which incorporatesall relevant short-range interactions not already included in a .In few-body physics, this important quantity is commonly re-ferred to as the three-body parameter (3BP).In Efimov’s famous scenario [1, 2], the infinite ladder ofthree-body bound states follows a discrete scaling invariance,which determines the relative energy spectrum of the states.The 3BP fixes the starting point of the ladder and thus theabsolute energies of all states. The parameter enters the theo-retical description as a short-range boundary condition for thethree-body wave function in real space or as a high-frequencycut-off in momentum space. To determine the 3BP from the-ory would require precise knowledge of both the two-bodyinteractions and the genuine three-body interactions at shortrange. In real systems, this is extremely difficult and the 3BPneeds to be determined experimentally through the observa-tion of few-body features such as Efimov resonances.In the last few years, ultracold atomic systems have openedup the possibility to explore Efimov’s scenario experimentallyand to test further predictions of universal theory [3–14]. Thekey ingredient of such experiments is the possibility to control a by an external magnetic field B via the Feshbach resonancephenomenon [15]. This naturally leads to the important ques-tion whether the 3BP remains constant or whether it is affected by the magnetic tuning, in particular when different Feshbachresonances are involved.The current status of theoretical and experimental researchdoes not provide a conclusive picture on possible variationsof the 3BP. A theoretical study [16] points to strong possiblevariations when different two-body resonances are exploitedin the same system, and even suggests a change of the 3BPon the two sides of a zero crossing of the scattering length.Other theoretical papers point to the importance of the partic-ular character of the Feshbach resonance [15]. While closed-channel dominated (“narrow”) resonances involve an addi-tional length scale that may fix the 3BP [17–19], the case ofentrance-channel dominated (“broad”) resonances leaves the3BP in principle open. However, predictions based on two-body scattering properties exist that apparently fix the 3BP forbroad resonances as well [20–22]. The available experimen-tal observations provide only fragmentary information. Thefirst observation of Efimov physics in an ultracold Cs gas [3]is consistent with the assumption of a constant 3BP on bothsides of a zero crossing. A later observation on K [5] indi-cated different values of the 3BP on both sides of a Feshbachresonance. A similar conclusion was drawn from experimentson Li [6], but other experiments on Li showed universal be-havior with a constant 3BP for the whole tuning range of asingle resonance [7] and for another spin channel [12]. Be-sides these observations on bosonic systems, experiments onfermionic gases of Li [8–10] can be interpreted based on aconstant 3BP [23]. A recent experiment on Li, however, in-dicates small variations of the 3BP [24].In the present work, we investigate universality in an ultra-cold gas of Cs atoms, which offers several broad Feshbach res-onances in the same spin channel and thus offers unique pos-sibilities to test for variations of the 3BP. In the lowest hyper-fine and Zeeman sublevel | F = , m F = i , Cs features a vari-ety of broad and narrow Feshbach resonances in combinationwith a large background scattering length [25]. Of particu-lar interest are three broad s -wave Feshbach resonances in therange up to 1000 G [26], with poles near −
10 G, 550 G, and800 G [15, 20, 25]. The character of these three resonances
FIG. 1: Illustration of the three broad s -wave Feshbach resonancesfor Cs in the absolute atomic ground state | F = , m F = i . The opencircle corresponds to the previous observation of a triatomic Efimovresonance at 7 . d - and g -wave molecular states [25] are not shown forthe sake of clarity. The region with B < | F = , m F = − i , which is not stable against two-body decay. Scat-tering lengths are given in units of Bohr’s radius a . is strongly entrance-channel dominated, as highlighted by thelarge values of their resonance strength parameter s res [15] of560, 170, and 1470, respectively. The resulting magnetic-fielddependence a ( B ) is illustrated in Fig. 1. In our previous work[3, 4] we have focussed on the low-field region up to 150 G.After a major technical upgrade of our coil set-up, we are nowin the position to apply magnetic fields B of up to 1.4 kG withprecise control down to the 20 mG uncertainty level and thusto explore the resonance regions at 550 G and 800 G [27].Our ultracold sample consists of about 2 × opticallytrapped Cs atoms, close to quantum degeneracy. Thepreparation is based on an all-optical cooling approach as pre-sented in Refs. [28, 29]. The final stage of evaporative coolingis performed in a crossed-beam dipole trap (laser wavelength1064 nm) and stopped shortly before Bose-Einstein condensa-tion is reached. Finally, the trap is adiabatically recompressedto twice the initial potential depth to suppress further evapo-ration loss. At this point, the mean trap frequency is about10 Hz and the temperature is typically 15 nK.Our experimental observable is the three-body loss coeffi-cient L , which in the framework of universal theory is con-veniently expressed as L = C ( a ) ¯ ha m [30], m denotes theatomic mass. The expression separates a log-periodic func-tion C ( a ) from the general a -scaling of three-body loss. For a <
0, effective field theory [1] provides the analytic expres-sion C ( a ) = ( h − ) sin [ s ln ( a / a − )] + sinh h − , (1)with s ≈ . h − is a non-universal quantity that depends on the deeply-bound molecular spectrum [23]. The scattering length a − marks the situation where an Efimov state intersects the three-atom threshold and the resulting triatomic Efimov resonanceleads to a giant three-body loss feature. In the following, thequantity a − will serve us as the representation of the 3BP.To measure L we record the time evolution of the atomnumber after quickly (within 10 ms) ramping B from the evap- oration to the target field strength. We determine the atomnumber N by absorption imaging. One-body decay, as causedby background collisions, is negligible under our experimen-tal conditions. Furthermore, two-body decay is energeticallysuppressed in the atomic state used. We can therefore modelthe decay by ˙ N / N = − L h n i , where the brackets denote thespatial average weighted with the atomic density distribution n . Additional, weaker loss contributions caused by four-bodyrecombination [31] can be described in terms of an effective L [32]. For fitting the decay curves and extracting L we usean analytic expression that takes into account the density de-crease resulting from anti-evaporation heating [30].The experimental results are a function of B whereas theoryexpresses L as a function of a . It is thus crucial to have a re-liable conversion function a ( B ) . We have obtained a ( B ) fromfull coupled-channel calculations on a Cs-Cs potential ob-tained by least-squares fitting to extensive new measurementsof binding energies, obtained by magnetic-field modulationspectroscopy [33], together with additional measurements ofloss maxima and minima that occur at resonance poles andzero crossings. The new potential provides a much improvedrepresentation of the bound states and scattering across thewhole range from low field to 1000 G. The experimental re-sults and the procedures used to fit them will be described ina separate publication.Figure 2 shows our experimental results on the magnetic-field dependent recombination loss near the two broad high-field Feshbach resonances (550 G and 800 G regions). Forconvenience we plot our data in terms of the recombinationlength r = ( mL / ( √ h )) / [34]. The three filled arrowsindicate three observed loss resonances that do not coincidewith the poles of two-body resonances. We interpret thesethree features as triatomic Efimov resonances.In the 800 G region, a single loss resonance shows up at853 G, which lies in the region of large negative values of a .We fit the L data based on Eq. (1) [35] and using the con-version function a ( B ) described above. The fit generally re-produces the experimental data well, apart from a small back-ground loss that apparently does not result from three-bodyrecombination [36]. For the 3BP the fit yields the resonanceposition of a − = − ( ) a , where the given error includesall statistical errors. For the decay parameter the fit gives h − = . ( ) .For the 550 G region, Fig. 1 suggests a qualitatively simi-lar behavior as found in the 800 G region. The experimentaldata, however, reveal a more complicated structure with threeloss maxima and a pronounced minimum. This behavior is ex-plained by a g -wave resonance (not shown in Fig. 1) that over-laps with the broader s -wave resonance. We have thoroughlyinvestigated this region by Feshbach spectroscopy. Thesestudies clearly identify the central maximum (554.06 G) andthe deep minimum (553.73 G) as the pole and zero crossingof the g -wave resonance (see inset). With s res = .
9, this res-onance is an intermediate case between closed-channel andentrance-channel dominated.The g -wave resonance causes a splitting that produces two FIG. 2: (color online) Recombination loss in the vicinity of the high-field Feshbach resonances. The measured recombination length r isshown for three different regions ( N , 552G < B < • , 554G < B < (cid:7) , 830G < B < a . The dashed lines show the predictions of effective field theory for a > a <
0. The insets show a ( B ) (solid line, full calculation; dash-dot line, s -wave states only). The arrows in the main figure andthe corresponding dots in the insets refer to the triatomic Efimov resonances. The small arrow indicates a recombination minimum. -2 -1 00510152025 -2 -1 0 -0.04-0.020.000.020.040.06 (c) (b) r e c . l eng t h ( a ) scattering length a (10 a )(a) l og . ( a - / a m ) FIG. 3: (color online) Efimov resonances and the 3BP. In (a) and(b), we compare the resonance previously observed [3] at 7.6 G tothe one found at 853 G. In (c), we plot the 3BPs obtained for all fourresonances measured in Cs. The dashed line corresponds to a meanvalue of a m = − a , calculated as a weighted average of the fourdifferent values. The logarithmic scale (to the basis of 22 .
7) coversone tenth of the Efimov period.
Efimov resonances instead of one in this region. This ex-plains the upper and the lower loss maxima, which are foundat 553.30(4) G and 554.71(6) G (arrows in Fig. 2). To de-termine the parameters of these Efimov resonances, we in-dependently fit the two relevant regions of negative scatteringlength using Eq. (1) [35]. This yields a − = − ( ) a and − ( ) a for the lower and the upper resonances, respec-tively.We now compare all our observations on triatomic Efimov B res (G) a − / a d / a d / a d / a h − − − − − B res at theresonance centers and the corresponding 3BPs together with theirfull statistical uncertainties. The individual error contributions d , d , and d refer to the statistical uncertainties from the fit to the L data, from the determination of the magnetic field strength, and fromthe a ( B ) -conversion, respectively. resonances in Cs. We also include the previous data of Ref.[3] on the low-field resonance (7.6 G), which we have refittedusing our improved a ( B ) conversion. The relevant parametersfor the four observed Efimov resonances are given in Table I.Figure 3(a) and (b) show the recombination data for the low-field resonance and the 853 G resonance, using a convenient r ( a ) representation. This comparison illustrates the strikingsimilarity between both cases. For all four Efimov resonances,Fig. 3(c) shows the 3BP on a logarithmic scale, which relatesour results to the universal scaling factor 22.7. Note that thefull scale is only one tenth of the Efimov period, i.e. a fac-tor 22 . ≈ .
37. The error bars indicate the correspondinguncertainties (one standard deviation), resulting from all sta-tistical uncertainties [37]. The data points somewhat scatteraround an average value of about − a (dashed line) withsmall deviations that stay within a few percent of the Efimovperiod. Taking the uncertainties into account, our data areconsistent with a constant 3BP for all four resonances. How-ever, between the values determined for the two broad reso-nances at 7.6 and 853 G we find a possible small aberration ofabout 2 . a >
0, three-body recombination minima arewell known features related to Efimov physics [1, 5, 6, 28].In the 800 G region, we observe a minimum at B = ( ) G(small arrow in Fig. 2), corresponding to a = + ( ) a ,which is very similar to the minimum previously observed inthe low-field region [28] and consistent with a universal con-nection to the a < a > . ( ) G and 855 . ( ) G, corresponding to scat-tering lengths of − ( ) a and − ( ) a . This excellentlyfits to universal relations [32] and our previous observations atlow magnetic fields [31].Our observations show that universality persists in a widemagnetic-field range across a series of Feshbach resonancesin the same spin channel and that the 3BP shows only minorvariations, if any. This rules out a scenario of large varia-tions as suggested by the model calculations of Ref. [16]. Theapparent fact that the relevant short-range physics is not sub-stantially affected by the magnetic field may be connected tothe strongly entrance-channel dominated character [15] of thebroad resonances in Cs. However, even the case of overlap-ping s - and g -wave Feshbach resonances, where the latter onehas intermediate character, is found to exhibit universal be-havior consistent with an essentially constant 3BP. Our obser-vation that universality is robust against passing through manypoles and zero crossings of the scattering length also impliesa strong argument in favor of a universal connection of bothsides of a single Feshbach resonance. This supports conclu-sions from experiments on Li as reported in Refs. [7, 12], incontrast to Ref. [6] and related work on K [5].With the present experimental data there is growing ex-perimental evidence that theories based on low-energy two-body scattering and the near-threshold dimer states [20–22]can provide reasonable predictions for the 3BP without invok-ing genuine short-range three-body forces, which are knownto be substantial for all the alkali metal trimers [38]. We alsostress a remarkable similarity [7] between the Cs data and ex-perimental results on both Li isotopes. When the 3BP is nor-malized to the mean scattering length ¯ a of the van der Waalspotential [15], our actual Cs value a − / ¯ a = − . ( ) is remark-ably close to corresponding values for Li [6, 12] and Li [8–10, 23], which vary in the range between − − [1] E. Braaten and H.-W. Hammer, Phys. Rep. , 259 (2006).[2] V. Efimov, Phys. Lett. B , 563 (1970).[3] T. Kraemer et al. , Nature , 315 (2006).[4] S. Knoop et al. , Nature Phys. , 227 (2009).[5] M. Zaccanti et al. , Nature Phys. , 586 (2009).[6] S. E. Pollack, D. Dries, and R. G. Hulet, Science , 1683(2009).[7] N. Gross et al. , Phys. Rev. Lett. , 163202 (2009).[8] J. H. Huckans et al. , Phys. Rev. Lett. , 165302 (2009).[9] T. B. Ottenstein et al. , Phys. Rev. Lett. , 203202 (2008).[10] J. R. Williams et al. , Phys. Rev. Lett. , 130404 (2009).[11] G. Barontini et al. , Phys. Rev. Lett. , 043201 (2009).[12] N. Gross et al. , Phys. Rev. Lett , 103203 (2010).[13] T. Lompe et al. , Phys. Rev. Lett , 103201 (2010).[14] S. Nakajima et al. , Phys. Rev. Lett , 023201 (2010).[15] C. Chin et al. , Rev. Mod. Phys. , 1225 (2010).[16] J. P. D’Incao, C. H. Greene, and B. D. Esry, J. Phys. B: At. Mol.Opt. Phys. , 044016 (2009).[17] D. S. Petrov, Phys. Rev. Lett. , 143201 (2004).[18] P. Massignan and H. T. C. Stoof, Phys. Rev. A , 030701(2008).[19] Y. Wang, J. P. D’Incao, and B. D. Esry, Phys. Rev. A , 042710(2011).[20] M. D. Lee, T. K¨ohler, and P. S. Julienne, Phys. Rev. A ,012720 (2007).[21] M. Jona-Lasinio and L. Pricoupenko, Phys. Rev. Lett. ,023201 (2010).[22] P. Naidon and M. Ueda, C.R. Physique , 13 (2011).[23] A. N. Wenz et al. , Phys. Rev. A , 040702(R) (2009).[24] S. Nakajima et al. , Phys. Rev. Lett. , 143201 (2011).[25] C. Chin et al. , Phys. Rev. A , 032701 (2004).[26] Units of gauss instead of the SI unit tesla (1G = − T) areused to conform to conventional usage in this field.[27] M. Berninger, Ph.D. thesis, Innsbruck University (2011).[28] T. Kraemer et al. , Appl. Phys. B , 1013 (2004).[29] M. Mark et al. , Phys. Rev. A , 042514 (2007).[30] T. Weber et al. , Phys. Rev. Lett. , 123201 (2003).[31] F. Ferlaino et al. , Phys. Rev. Lett. , 140401 (2009).[32] J. von Stecher, J. P. D’Incao, and C. H. Greene, Nature Phys. ,417 (2009).[33] A. D. Lange et al. , Phys. Rev. A , 013622 (2009).[34] B. D. Esry, C. H. Greene, and J. P. Burke, Phys. Rev. Lett. ,1751 (1999).[35] The fits include an additional free scaling factor l to accountfor possible systematic errors in the number density. For the800 G region, we obtain l = .
89. For the lower and upper550 G region, we obtain l = .
46 and l = .
06, respectively.[36] The smallest values obtained for r correspond to one-bodylifetimes exceeding 100 s.[37] Systematic uncertainties are not included in our error budget.They may result from model-dependent errors in the determi-nation of the scattering length from binding energy and scat-tering data and from finite-temperature shifts. All these errors,however, stay well below the statistical uncertainties.[38] P. Sold´an, M. T. Cvitaˇs, and J. M. Hutson, Phys. Rev. A67