Quantum Chaos in Ultracold Collisions of Erbium
Albert Frisch, Michael Mark, Kiyotaka Aikawa, Francesca Ferlaino, John L. Bohn, Constantinos Makrides, Alexander Petrov, Svetlana Kotochigova
QQuantum Chaos in Ultracold Collisions of Erbium
Albert Frisch, Michael Mark, Kiyotaka Aikawa, and Francesca Ferlaino ∗ Institut für Experimentalphysik, Universität Innsbruck, Technikerstraße 25, 6020 Innsbruck, Austria
John L. Bohn
JILA, University of Colorado and National Institute of Standards and Technology, Boulder, Colorado 80309-0440, USA
Constantinos Makrides, Alexander Petrov, † and Svetlana Kotochigova Department of Physics, Temple University, Philadelphia, Pennsylvania 19122, USA (Dated: December 9, 2013)
Atomic and molecular samples reduced to tem-peratures below 1 microkelvin, yet still in the gasphase, afford unprecedented energy resolution inprobing and manipulating how their constituentparticles interact with one another. As a resultof this resolution, atoms can be made to scatterresonantly at the experimenter’s whim, by pre-cisely controlling the value of a magnetic field[1]. For simple atoms, such as alkalis, scatter-ing resonances are extremely well-characterized[2]. However, ultracold physics is now poisedto enter a new regime, where far more complexspecies can be cooled and studied, including mag-netic lanthanide atoms and even molecules. Formolecules, it has been speculated [3, 4] that adense forest of resonances in ultracold collisioncross sections will likely express essentially ran-dom fluctuations, much as the observed energyspectra of nuclear scattering do [5]. Accordingto the Bohigas-Giannoni-Schmit conjecture, thesefluctuations would imply chaotic dynamics of theunderlying classical motion driving the collision[6, 7]. This would provide a paradigm shift in ul-tracold atomic and molecular physics, necessitat-ing new ways of looking at the fundamental inter-actions of atoms in this regime, as well as perhapsnew chaos-driven states of ultracold matter.In this report we provide the first experimen-tal demonstration that random spectra are indeedfound at ultralow temperatures. In the experi-ment, an ultracold gas of erbium atoms is shownto exhibit many Fano-Feshbach resonances, forbosons on the order of 3 per gauss. Analysis oftheir statistics verifies that their distribution ofnearest-neighbor spacings is what one would ex-pect from random matrix theory [8]. The densityand statistics of these resonances are explainedby fully-quantum mechanical scattering calcula-tions that locate their origin in the anisotropy ofthe atoms’ potential energy surface. Our resultstherefore reveal for the first time chaotic behaviorin the native interaction between ultracold atoms.
In the common perception, atoms are regarded as sim-ple systems in sharp contrast to complex molecules, whosebehavior is dictated by many (rotational and vibrational) degrees of freedom. The recent realization of dipolarBose-Einstein condensates and Fermi gases of magneticlanthanides [9–12] made available a novel class of atomsin the ultracold regime. These exotic species, such as er-bium (Er), allow to bridge the enormous conceptual gapbetween simple atoms and molecules, potentially provid-ing a natural testbed to explore complex scattering dy-namics in a controlled environment. The rich scatteringbehavior of lanthanides has been pointed out in pioneer-ing experiments at millikelvin temperatures [13, 14] andtheoretical work on cold collisions of atoms with non-zeroangular momenta [15, 16].A wealth of intriguing properties in Er, which is thefocus of this paper, originates from its exotic electronicconfiguration. Er is a submerged-shell atom with elec-tron vacancies in the inner anisotropic f shell, whichlies beneath a filled s shell. As a consequence, it notonly has a large magnetic moment of Bohr magnetons( µ B ) but also has a large electronic orbital (total) angularmomentum quantum number of L = 5 ( J = 6 ); note thatfor bosonic (fermionic) isotope the nuclear angular quan-tum number is I = 0 ( I = 7 / ). Large values for L and J are sources of anisotropy in the interatomic interac-tion. Moreover, the two-body scattering is controlled byas many as 91 electronic Born-Oppenheimer (BO) inter-action potentials, each potential accounting for a specificorientation of J with respect to the internuclear axis.All BO potentials are anisotropic and include at large in-ternuclear separations a strong dipole-dipole interaction(DDI) and anisotropic van der Waals dispersion poten-tials. This situation is in contrast to that of conven-tional ultracold atoms, such as alkali-metal atoms, wherethe scattering is determined mainly by the isotropic sin-glet and triplet BO potentials [2]. Recent theoreticalwork predicted the existence of anisotropy-induced Fano-Feshbach resonances in magnetic lanthanides [17]. Thisgreater complexity brings significant new challenges inunderstanding and exploiting scattering processes.Our experimental study is based on high-resolutiontrap-loss spectroscopy of Fano-Feshbach resonances in anoptically-trapped ultracold sample of Er atoms in theenergetically lowest magnetic Zeeman sublevel. We pre-pare the ultracold sample by following a similar coolingand trapping approach to that described in Ref. [11] forbosons and Ref. [12] for fermions (Method Summary). a r X i v : . [ c ond - m a t . qu a n t - g a s ] D ec
36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 7010
36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 7010 A t o m nu m be r A t o m nu m be r Magnetic field (G)
Magnetic field (G) ba Figure 1 | Fano-Feshbach spectrum of
Er and
Er from to
70 G . The trap-loss spectroscopy is performedin an optically trapped sample of Er atoms in the energetically lowest Zeeman sublevel m J = − at a temperature of
330 nK . The atom number is measured after a holding time of
400 ms . a , We observe Fano-Feshbach resonancesfor
Er and b , resonances for Er. Resonance positions are extracted by fitting a Gaussian shape to individualloss features; a full list is given in the Supplementary Information.After the preparation procedure, the ultracold sampletypically contains about atoms at a temperaturearound
400 nK . High-resolution trap-loss spectroscopyconsists of many experimental cycles. In each cycle, weramp the magnetic field to a target value B and hold theatoms for
400 ms in the optical dipole trap, during whichthey undergo elastic and inelastic collisions. To probe theloss of atoms from the trap, we record the atom numberby applying standard time-of-flight absorption imagingat zero magnetic field. In the next experimental cycle,we vary the magnetic field value from to
70 G with stepsizes of a few mG and repeat the measurement. Fig-ure 1 shows the loss spectra for Er and
Er. Forboth isotopes, we observe an enormous number of res-onant loss features, which we interpret as being causedby Fano-Feshbach resonances [2]. We identify res-onances for
Er and resonances for
Er, mean-ing that we observe about 3 resonances per gauss. Weperformed similar spectroscopic measurements with the fermionic isotope
Er, revealing a much higher densityof resonances that exceeds 20 resonances per gauss (Ex-tended Data Fig. 1). The fermionic case is complicatedby its additional hyperfine structure and detailed studieswill be subject of future work.The immense density of resonances in Er is withoutprecedent in ultracold quantum gases. For comparison,the density of resonances observed in experiments withultracold alkali-metal atoms or even mixtures is abouttwo orders of magnitude lower than Er (c. f. Ref. [18, 19]).In Er, it is unclear whether a quantitative mapping ofthe observed resonances is possible at all. In principlethere are at least 91 unknown parameters, correspond-ing to the phase shifts introduced by the BO potentials[17]. Instead, we focus our theoretical analysis on fun-damental questions, such as: Can the observed densityof resonances be reproduced by microscopic calculations?Do our results imply the presence of highly anisotropicinteractions, which call into play resonant states of highorbital momentum? We answer these questions in theaffirmative using full coupled-channel (CC) calculations,supported by an analytical model.We construct a first-principle CC model for Er+Erscattering to calculate the spectrum of Fano-Feshbachresonances for the experimental conditions. FollowingRef. [17], our model uses the atomic basis set and Hamil-tonian (Methods) that includes the radial kinetic and ro-tational energy operators, the Zeeman interaction, andthe 91 anisotropic BO potentials. For small interatomicseparations R , the BO potentials are calculated usingthe ab initio relativistic multi-reference configuration-interaction method [20]. At intermediate to large R , theBO potentials are expressed as a sum of multipolar in-teraction terms. The van der Waals dispersion interac-tion potentials ( ∝ /R ) are determined from experimen-tal data on atomic transition frequencies and oscillatorstrengths [21, 22]. An important point is that the dis-persion potentials have both isotropic and anisotropiccontributions. The latter comes from the non- S statecharacter of the Er electronic ground state. The BO po-tentials induce thus either isotropic ( (cid:96) and m (cid:96) conserv-ing) or anisotropic ( (cid:96) or m (cid:96) changing) couplings. Here, (cid:96) and m (cid:96) are the partial wave quantum number and itsprojection on the magnetic-field quantization axis.We perform CC calculations for bosonic Er, consid-ering s -wave ( (cid:96) = 0 ) collisions and couplings to molecularstates with even (cid:96) up to L max = 20 . We calculate the elas-tic collisional rate coefficient as a function of magneticfield to obtain the Fano-Feshbach resonance spectrum.For L max = 20 , we observe a very dense resonance spec-trum with about 1.5 resonances per gauss, which qual-itatively reproduces our experimental observation (Ex-tended Data Fig. 2). To get deeper insight into the roleof the anisotropy of the potentials, we calculate the meandensity of resonances ρ from our CC calculations for dif-ferent values of the maximum partial wave L max (Fig. 2).For L max up to 20, we observe that ρ increases with L max in a quadratic manner. This dependence stands in starkcontrast to alkali-metal atoms, where high-partial-waveresonances tend to be too narrow to be observed.Since our limited computational resources do not al-low us to perform calculations for L max > , it is worthestimating the density of resonances in a simpler way,based on the separated atom quantum numbers [3]. Thekey ideas of our model are the following. For each chan-nel | j m J, , j m J, , (cid:96)m (cid:96) (cid:105) we define the long-range poten-tial − C /R + (cid:126) (cid:96) ( (cid:96) + 1) / (2 µR ) + gµ B ( m J, + m J, ) B ,with the isotropic van der Waals C coefficient of theBO potentials. Here µ is the reduced mass, g is theatomic g-factor, and for ground state Er C = 1723 a.u.. Fano-Feshbach resonances in our open ( m J, = −
6) + ( m J, = − channel are due to couplings to themost-weakly bound rovibrational level of closed chan-nels. For a van der Waals potential [2, 23] this boundstate has a binding energy that must fall within the (cid:96) -dependent energy window [ − ∆ (cid:96) , with ∆ (cid:96) > . Theshort range potentials are not accurately known and, ( / G ) L max Figure 2 | Mean resonance density for bosonicEr as a function of largest included partial wave L max . CC calculations for L max up to 20 (circles) andRQDT calculation (solid line) for a magnetic-field regionfrom 0 G to 70 G. For calculations a collision energy of E/k B = 360 nK is assumed. The mean densities of reso-nances measured in the experiment are shown for Er(dashed line) and for
Er (dash-dotted line) with one-sigma confidence bands (shaded areas).for each closed channel, there is a probability dE b / ∆ (cid:96) of finding a bound state with a binding energy between E b and E b + dE b . From Ref. [23] and numerical simula-tions we find ∆ (cid:96) /E vdW ≈ . . (cid:96) + 3 . (cid:96) , where E vdW = (cid:126) / (2 µx ) and x vdW = (cid:112) µC / (cid:126) / . Eachclosed channel contributes gµ B δm/ ∆ (cid:96) to the mean reso-nance density, where gµ B δm > is the magnetic-momentdifference of the closed and open channels and δm istheir difference in molecular projection quantum num-bers. Adding the contributions for the closed channelsgives ρ . This counting technique, which we here namerandom quantum defect theory (RQDT), yields the meandensity of states shown in Fig. 2. For L max (cid:54) , the re-sults of our analytic RQDT agrees very well with theexact CC calculations. For larger L max , the density ofresonances keeps growing and eventually saturates to avalue comparable to the one observed in the experiment.RQDT shows that at least partial waves need to beconsidered to reproduce the experimental observations.Our microscopic models reproduce well the qualitativebehavior of the system. However, given the complexityof the scattering, the analysis of ultracold collision datacan not and should not aim anymore at the assignmentof individual resonances and the fundamental question ofhow to tackle complex scattering naturally arises. Histor-ically, spectra of great complexity have been understoodwithin the framework of random matrix theory (RMT),as originally developed by Wigner to describe heavy nu-clei containing a very large number of degrees of freedom[24]. This is an alternative view of the quantum mechan-ics of complex systems, where individual energy levelsand resonances are not theoretically reproduced one-by-one, yet their statistics can be described [25]. RMT char-acterizes spectra by fluctuations of their energy levels andclassifies their statistical behavior in terms of symmetryclasses, e. g. the Gaussian-orthogonal ensemble (GOE) inthe case of a system with time-reversal symmetry, suchas neutral atoms.Following RMT, the distribution of spacings betweenneighboring levels (or resonances) characterizes the spec-tral fluctuations of the system and reflects the absenceor the presence of level correlations in terms of a di-mensionless parameter, s , i. e. the actual spacing be-tween neighboring levels in units of the mean spacing, d = 1 /ρ . Whereas the nearest-neighbor spacing (NNS)distribution P ( s ) of non-interacting levels is Poissonian, P P = exp( − s ) , strongly interacting levels obey a totallydifferent distribution which, in the case of GOE statis-tics, is known as the Wigner-Dyson (WD) distribution or Wigner surmise [25] P WD = π s exp( − πs / , (1)which shows a strong level repulsion for small s , P WD (0) = 0 . The field of application of the WD distri-bution is so vast as to make it a universal feature of verycomplex systems, such as heavy nuclei, disordered con-ductors, zeros of the Riemann function in number theory,and even risk management models in finance [5]. Re-markably, the Bohigas-Giannoni-Schmit conjecture fur-ther enriched the field of applications of GOE statistics[6], showing that it applies generally to chaotic systems,such as Rydberg atoms in strong magnetic fields or Sinaibilliards, where only few degrees of freedom are relevant,but where motion in these degrees of freedom occurs on ahighly anisotropic potential energy surface [7]. Recently,it has been speculated that even cold and ultracold atom-molecule collisions will show essential features of GOEstatistics [3, 4].Inspired by these works, we statistically analyze boththe experimental and calculated Fano-Feshbach spectrumaccording to RMT. To obtain the NNS distribution ofresonances, we first derive ρ and the mean spacing be-tween resonances, d , by constructing the so-called stair-case function [7]. This step-like function counts the num-ber of resonances below a magnetic field value B and isdefined as N ( B ) = B (cid:82) dB (cid:48) (cid:80) i δ ( B (cid:48) − B i ) , with δ being thedelta function and B i the position of the i -th resonance.For our experimental data (Fig. 3a) the staircase functionshows an increase of the number of resonances with B ,which proceeds linearly at large B and flattens out to-wards lower magnetic-field values (Fig. 3b). The densityof resonances is given by the derivative of the staircasefunction. We evaluate ρ in the region above
30 G , wherethe staircase function shows a linear progression (Sup-plementary Information) and we obtain ρ = 3 . − and d = 0 . . We perform a similar analysis with Er and find ρ = 3 . − and d = 0 . (Ex-tended Data Fig. 3). For CC-calculation data, we find ρ = 3 . − for L max = 20 (Fig. 2). We finally de-rive the NNS distribution for the experimental and CC-
36 38 40 42 4490100110 b Er B (G) a B (G)
Figure 3 | Loss-maxima position and staircasefunction for
Er. a , Positions of the measured lossmaxima of Fig. 1 are shown as vertical lines. b , The stair-case function shows a linear dependence on the magneticfield at large values. A linear fit to the data above
30 G is plotted in light colors. The inset shows a magnifica-tion of the data to emphasize the step-like nature of thestaircase function.calculated data by constructing a histogram of resonancespacings. We choose a number of bins on the order of √ N , with N being the number of Fano-Feshbach reso-nances used for analysis [26]. We then rescale the his-togram in terms of the dimensionless quantity s = B/d and normalize the distribution in order to obtain P ( s ) .Figure 4 is the main result of our statistical analy-sis for Er. The plot shows the NNS distribution ofthe experimental and the CC-calculated Fano-Feshbachresonances together with the parameter-free Poisson andWigner-Dyson distributions (Eq. 1). We see an impres-sive agreement between the experimental result and theCC calculations. Remarkably, both follow a distributionmuch closer to the WD than to the Poissonian one. Toquantify the agreement with the GOE statistics, we eval-uate the reduced chi squared, ˜ χ , between our data andthe Poisson and WD distribution. We find ˜ χ = 0 . and ˜ χ = 2 . for our experimental data and ˜ χ = 0 . and ˜ χ = 3 . for the data of the CC calculations. Thefact that ˜ χ (cid:54) confirms that our data are well de-scribed by a WD distribution. Similar results are foundfor Er (Extended Data Fig. 4).To further investigate the spectral correlations, we an-alyze our data in terms of other statistical quantities,such as the number variance and the two-gap distributionfunction (Supplementary Information) [27]. The numbervariance Σ (∆ B ) measures the fluctuations of the num-ber of resonances in a magnetic-field interval ∆ B (Meth-ods) [7] . For non-correlated (Poissonian-distributed)levels, Σ = ∆ B , indicating large fluctuations arounda mean value. For quantum chaotic systems, the correla-tions are strong and the fluctuations are thus less spreadout. In this case, Σ ∝ ln(∆ B ) . This slower increase ab P s ∆ B (G)
Figure 4 | NNS distribution and number vari-ance. a , Er NNS distribution above
30 G with a binsize of
160 mG . The plot shows the experimental data(circles) with the corresponding Brody distribution (solidline), the Brody distribution for the CC calculation with L max = 20 (dotted line), and the parameter free distri-butions P P (dashed line) and P WD (short-dashed line).The Brody distribution is given in the Methods section.For the error bars in the experimental data, we assumea Poisson counting error. b , Number variance for theexperimental data (solid line) with a two-sigma confi-dence band (shaded area), the CC-calculation data (dot-ted line), P P (dashed line), and P WD (short-dashed line).of the number variance is regarded as a strong spectralrigidity of the system [7]. Our observations clearly devi-ate from the Poissonian behavior showing that Σ tendsto the WD case (Fig. 4b) and confirm the presence ofcorrelations in our system.To conclude, our observations reproduce the salient features predicted by GOE statistics for chaotic systems,the level repulsion and the spectral rigidity. This impliesa degree of complexity in Er+Er cold collisions unprece-dented in any previous ultracold scattering system. Ourresults bring the powerful analytical tools of quantumchaos to bear [7]. In particular, these approaches connectthe large-scale structure of the spectra to simple featuressuch as the shortest closed classical orbits in the poten-tial energy surface, where these connections are made bythe Gutzwiller trace formula [28]. Identifying the mostimportant closed orbits will then shed light on the poten-tial energy surface itself, providing a route to describingultracold collisions that is complementary to the elabo-rate close-coupling calculations that will be difficult toconnect in detail with the data.Erbium represents the first occasion where statisticalanalyses and chaotic behavior are important to ultra-cold collisions, but they will not be the last. Specifi-cally, much experimental effort is being exerted towardproducing ultracold molecular samples, which also enjoyhighly anisotropic potential energy surfaces. Learning toread complex spectra, by acknowledging their essentiallychaotic nature, represents a turning point in how the fieldwill consider ultracold collisions in the future and providenew inroads into ultracold chemistry. METHODS SUMMARYSample preparation.
For bosonic sample prepa-ration we follow the approach of Ref. [11]. We obtainabout × optically-trapped atoms at a density of × cm − . The trap-loss spectroscopy is performed ina trap with frequencies of ( ν x , ν y , ν z ) = (65 , , .The temperature of the cloud is measured by time-of-flight imaging at . and gives T = 326(4) nK and T = 415(4) nK , respectively. We ramp the magneticfield within
10 ms to a probe value between and
70 G ,and hold the atomic cloud for
400 ms in the optical dipoletrap. We observe an increase of the temperature up to
560 nK at a magnetic field of about
50 G due to the ramp-ing over many Fano-Feshbach resonances. For fermionicsample preparation we follow the approach of Ref. [12].We obtain about . × fermionic atoms at a densityof × cm − and at a temperature of . T F , where T F = 1 . µ K is the Fermi temperature. The trap fre-quencies are (427 , , . ∗ [email protected]; corresponding author † Alternative address: St. Petersburg Nuclear Physics In-stitute, Gatchina, 188300; Division of Quantum Mechan-ics, St. Petersburg State University, 198904, Russia.[1] S. Inouye, M. R. Andrews, J. Stenger, H.-J. Miesner,D. M. Stamper-Kurn, and W. Ketterle, “Observationof Feshbach resonances in a Bose-Einstein condensate.”Nature , 151 (1998).[2] C. Chin, R. Grimm, P. S. Julienne, and E. Tiesinga,“Feshbach resonances in ultracold gases.” Rev. Mod.Phys. , 1225 (2010). [3] M. Mayle, B. P. Ruzic, and J. L. Bohn, “Statistical as-pects of ultracold resonant scattering.” Phys. Rev. A ,062712 (2012).[4] M. Mayle, G. Quéméner, B. P. Ruzic, and J. L. Bohn,“Scattering of ultracold molecules in the highly resonantregime.” Phys. Rev. A , 012709 (2013).[5] T. Guhr, A. Müller-Groeling, and H. A. Weidenmüller,“Random-matrix theories in quantum physics: commonconcepts.” Physics Reports , 189 (1998).[6] O. Bohigas, M. J. Giannoni, and C. Schmit, “Charac-terization of chaotic quantum spectra and universality of level fluctuation laws.” Phys. Rev. Lett. , 1 (1984).[7] H. A. Weidenmüller and G. E. Mitchell, “Random ma-trices and chaos in nuclear physics: Nuclear structure.”Rev. Mod. Phys. , 539 (2009).[8] T. A. Brody, “A statistical measure for the repulsion ofenergy levels.” Lettere Al Nuovo Cimento Series 2 , 482(1973).[9] M. Lu, N. Q. Burdick, S. H. Youn, and B. L. Lev,“Strongly dipolar Bose-Einstein condensate of dyspro-sium.” Phys. Rev. Lett. , 190401 (2011).[10] M. Lu, N. Q. Burdick, and B. L. Lev, “Quantum degen-erate dipolar Fermi gas.” Phys. Rev. Lett. (2012).[11] K. Aikawa, A. Frisch, M. Mark, S. Baier, A. Rietzler,R. Grimm, and F. Ferlaino, “Bose-Einstein condensationof erbium.” Phys. Rev. Lett. , 210401 (2012).[12] K. Aikawa, A. Frisch, M. Mark, S. Baier, R. Grimm,and F. Ferlaino, “Reaching Fermi degeneracy via uni-versal dipolar scattering.” ArXiv e-prints (2013),arXiv:1310.5676.[13] C. I. Hancox, S. C. Doret, M. T. Hummon, L. Luo, andJ. M. Doyle, “Magnetic trapping of rare-earth atoms atmillikelvin temperatures.” Nature , 281 (2004).[14] C. B. Connolly, Y. S. Au, S. C. Doret, W. Ketterle,and J. M. Doyle, “Large spin relaxation rates in trappedsubmerged-shell atoms.” Phys. Rev. A , 010702 (2010).[15] V. Kokoouline, R. Santra, and C. H. Greene, “Multichan-nel cold collisions between metastable Sr atoms.” Phys.Rev. Lett. , 253201 (2003).[16] R. V. Krems, G. C. Groenenboom, and A. Dalgarno,“Electronic interaction anisotropy between atoms in ar-bitrary angular momentum states.” J. Phys. Chem. A , 8941 (2004).[17] A. Petrov, E. Tiesinga, and S. Kotochigova, “Anisotropy-induced Feshbach resonances in a quantum dipolar gasof highly magnetic atoms.” Phys. Rev. Lett. , 103002(2012).[18] M. Berninger, A. Zenesini, B. Huang, W. Harm, H.-C.Nägerl, F. Ferlaino, R. Grimm, P. S. Julienne, and J. M.Hutson, “Feshbach resonances, weakly bound molecularstates, and coupled-channel potentials for cesium at highmagnetic fields.” Phys. Rev. A , 032517 (2013).[19] T. Takekoshi, M. Debatin, R. Rameshan, F. Ferlaino,R. Grimm, H.-C. Nägerl, C. R. Le Sueur, J. M. Hut-son, P. S. Julienne, S. Kotochigova, and E. Tiemann,“Towards the production of ultracold ground-state RbCsmolecules: Feshbach resonances, weakly bound states,and the coupled-channel model.” Phys. Rev. A ,032506 (2012).[20] S. Kotochigova, H. Levine, and I. Tupitsyn, “Corre-lated relativistic calculation of the giant resonance in theGd absorption spectrum.” Int. J. Quant. Chem. ,575 (1998).[21] A. Kramida, Y. Ralchenko, and J. Reader, “NIST atomicspectra database (ver. 5.0).” (2013).[22] J. E. Lawler, J. Wyart, and E. A. D. Hartog, “Atomictransition probabilities of Er I.” J. Phys. B: At. Mol. Opt.Phys. , 235001 (2010).[23] B. Gao, “Zero-energy bound or quasibound states andtheir implications for diatomic systems with an asymp-totic van der Waals interaction.” Phys. Rev. A ,050702(R) (2000).[24] E. P. Wigner, “On a class of analytic functions from thequantum theory of collisions.” Annals of MathematicsSecond Series, , 36 (1951). [25] F. J. Dyson and M. L. Mehta, “Statistical theory of theenergy levels of complex systems. IV.” Journal of Math-ematical Physics , 701 (1963).[26] J. Taylor, An Introduction to Error Analysis: The Studyof Uncertainties in Physical Measurements , A series ofbooks in physics (University Science Books, 1997).[27] T. A. Brody, J. Flores, J. B. French, P. A. Mello,A. Pandey, and S. S. M. Wong, “Random-matrix physics:Spectrum and strength fluctuations.” Rev. Mod. Phys. , 385 (1981).[28] M. Gutzwiller, Chaos in Classical and Quantum Mechan-ics (Springer, 1990).[29] A. Frisch, K. Aikawa, M. Mark, A. Rietzler, J. Schindler,E. Zupanič, R. Grimm, and F. Ferlaino, “Narrow-linemagneto-optical trap for erbium.” Phys. Rev. A ,051401 (2012).[30] J. D. Watts, J. Gauss, and R. J. Bartlett, “Coupled-cluster methods with noniterative triple excitations forrestricted open-shell Hartree-Fock and other general sin-gle determinant reference functions: Energies and ana-lytical gradients.” J. Chem. Phys. , 8718 (1993). Author contributions
A.F., M.M, K.A., and F.F. did the experimental workand statistical analysis of the data, C.M., A.P., and S.K.did the theoretical work on CC calculations and RQDT,J.L.B. did the theoretical work on RMT. The manuscriptwas written with substantial contributions from all au-thors.
Acknowledgements
The Innsbruck group thanks R. Grimm for inspiring dis-cussions and S. Baier, C. Ravensbergen, and M. Brownuttfor careful reading of the manuscript. S. K. andA. P. thank E. Tiesinga for useful discussions. J. L. B. issupported by an ARO MURI. The Innsbruck team issupported by the Austrian Science Fund (FWF) througha START grant under Project No. Y479-N20 and by theEuropean Research Council under Project No. 259435.K. A. is supported within the Lise-Meitner program ofthe FWF. Research at Temple University is supportedby AFOSR and NSF PHY-1308573.
METHODS
Experimental procedures.
For bosonic samplepreparation we follow the approach of Ref. [11]. In brief,after the magneto-optical trap [29], we load the atomsin an optical dipole trap composed of two laser beamsin horizontal ( ,
064 nm , . , single-mode) and ver-tical direction ( ,
064 nm , . , broadband Yb fiber-laser). In the trap, we force evaporation by rampingdown the power of both trapping laser beams within . , in the presence of a homogeneous magnetic fieldof . to prevent spin-flips to other Zeeman states.We stop evaporative cooling before the onset of Bose-Einstein condensation. Our final trap has frequencies of ( ν x , ν y , ν z ) = (65 , , and contains about × atoms at a density of × cm − . The temperature ofthe atomic cloud is measured by time-of-flight imagingfor both isotopes at . and gives T = 326(4) nK and T = 415(4) nK . We ramp the homogeneous mag-netic probe field up to
70 G within
10 ms and hold theatomic cloud for
400 ms in the optical dipole trap. Themagnetic field is suddenly ( < , limited by eddy cur-rents) switched off and the atom number and size of thecloud is measured via absorption imaging after a time offlight of
15 ms . We observe an increase of the tempera-ture up to
560 nK at a magnetic field of about
50 G dueto ramping over many Fano-Feshbach resonances. Forfermionic sample preparation we follow the approach ofRef. [12]. We obtain about . × fermionic atoms at adensity of × cm − and at a temperature of . T F ,where T F = 1 . µ K is the Fermi temperature. Thetrap frequencies are (427 , , . Magnetic-field control.
An analog feedback loopstabilizes the current for the homogeneous magnetic-fieldcoils with a relative short-term stability of better than × − . Calibration of the magnetic field is done by driv-ing a radio-frequency transition between Zeeman states m J = − and m J = − . Trap-loss spectroscopy is car-ried out in steps of
20 mG (out of resonance) and (on resonance). The long-term offset stability of the mag-netic field was observed during the data recording periodto be better than within one week.
Coupled-channel calculations.
We perform ex-act CC calculations for Er+Er scattering in the basis | j m J, , j m J, , (cid:96)m (cid:96) (cid:105) ≡ Y (cid:96)m (cid:96) ( θ, φ ) | j m J, (cid:105)| j m J, (cid:105) , where (cid:126)j a =1 , are the atomic angular momenta with space-fixedprojection m J,a =1 , along the magnetic-field direction,the spherical harmonics Y (cid:96)m (cid:96) ( θ, φ ) describe molecular ro-tation with partial wave (cid:126)(cid:96) , and where the angles θ and φ orient the internuclear axis relative to the magnetic field.For a closed-coupling calculation of the rovibrationalmotion and of the scattering of the atoms we need all elec-tronic potentials dissociating to two ground-state atoms.There are 91 BO potentials for Er , of which 49 aregerade and 42 are ungerade potentials. For collisionsof bosons in the same Zeeman state only gerade statesmatter. These potential surfaces have been obtained us-ing an ab initio relativistic multi-reference configuration- interaction method (RMRCI) [20], and converted into atensor operator form with R -dependent coefficients. Ex-amples of tensor operators are the exchange interaction V ex ( R ) (cid:126)j · (cid:126)j and the anisotropic quadrupole-rotation op-erator V Q ( R ) Y ( ˆ R ) · [ (cid:126)j ⊗ (cid:126)j ] coupling the quadrupoleoperator [ (cid:126)j ⊗ (cid:126)j ] of one atom with angular momentum j to the rotation of the molecule. See [17] for otheroperators.Collisions of submerged 4f-shell atoms at low temper-atures also depend on the intermediate to long-rangeisotropic and anisotropic dispersion, magnetic dipole-dipole and quadrupole-quadrupole interatomic interac-tions. The van der Waals dispersion potentials fortwo ground-state atoms are obtained using the transi-tion frequencies and oscillator strengths [21, 22]. Thequadrupole moment of Er is calculated using an unre-stricted atomic coupled-cluster method with single, dou-ble, and perturbative triple excitations uccsd(t) [30] andshown to be small at Q = 0 .
029 a . u . .We use a first-principle coupled-channel model tocalculate anisotropy-induced magnetic Fano-Feshbach-resonance spectra of bosonic Erbium. The model treatsthe Zeeman, magnetic dipole-dipole, and isotropic andanisotropic dispersion interactions on equal footing. TheHamiltonian includes H = − (cid:126) µ d dR + (cid:126)(cid:96) µR + H Z + V ( (cid:126)R, τ ) , where (cid:126)R describes the orientation of and separation be-tween the two atoms. The first two terms are the ra-dial kinetic and rotational energy operators, respectively.The Zeeman interaction is H Z = gµ B ( j z + j z ) B , where g is an atomic g-factor and j iz is the z component of theangular momentum operator (cid:126) i of atom i = 1 , . The in-teraction, V ( (cid:126)R, τ ) , includes the Born-Oppenheimer andthe magnetic dipole-dipole interaction potentials, whichare anisotropic, and τ labels the electronic variables. Fi-nally, µ is the reduced mass and for R → ∞ the inter-action V ( (cid:126)R, τ ) → . Coupling between the basis statesis due to V ( (cid:126)R, τ ) , inducing either isotropic ( (cid:96) and m (cid:96) conserving) or anisotropic ( (cid:96) or m (cid:96) changing) couplings.The Hamiltonian conserves M tot = m J, + m J, + m (cid:96) andis invariant under the parity operation so that only even(odd) (cid:96) are coupled. In the atomic basis set, the Zeemanand rotational interaction are diagonal. NNS probability distribution.
As the density ofresonances is not constant below
30 G we restrict ouranalysis to resonances appearing from to
70 G . Weplot a histogram of spacings between adjacent resonancesgiven by d i = B i +1 − B i . For this an appropriate num-ber of bins is chosen on the order of √ N , with N be-ing the total number of Fano-Feshbach resonances ob-served up to
70 G . This ensures a bin size at least anorder of magnitude larger than the mean resolution ofthe trap-loss spectroscopy scan. For every bin a sta-tistical counting error according to a Poisson distribu-tion is assigned. Next, the magnetic-field axis of the his-togram is divided by the mean spacing of resonances toget the dimensionless quantity s = B/d . To calculatethe NNS probability distribution P ( s ) the histogram hasto be normalized such that ∞ (cid:82) ds P ( s ) = 1 . As shownin Ref. [27], the probability distribution of uncorrelatedrandom numbers is simply given by the Poisson distribu-tion P P ( s ) = exp( − s ) . A theoretical spacing distributionof random matrices can not be written in a simple formbut, according to the Wigner surmise, an excellent ap-proximation is given by the Wigner-Dyson distribution P WD ( s ) = π s exp( − πs / . A way of discriminating be-tween these two distributions is to fit the so-called Brodydistribution to the NNS distribution [8]. It is an empiricalfunction with a single fitting parameter η , which interpo-lates between P WD and P P and quantifies the tendency(and not the degree of chaoticty) of the observed distri-bution to be more Poisson-like ( η = 0 ) or more Wigner-Dyson-like ( η = 1 ). It is defined by P B ( s ) = As η exp( − αs η +1 ) A = ( η + 1) αα = (cid:20) Γ (cid:18) η + 2 η + 1 (cid:19)(cid:21) η +1 , where Γ denotes the Gamma function. From a least-squares fit to the experimental data, we obtain η =0 . for Er and η = 0 . for Er, and afit to the CC-calculation data gives η CC = 0 . . Number variance.
The number variance Σ is aquantity that depends on long-range correlations betweenresonance spacings within an interval ∆ B . It is definedby Σ (∆ B ) = n ( B , ∆ B ) − ( n ( B , ∆ B )) , with n ( B , ∆ B ) = N ( B + ∆ B ) − N ( B ) giving thenumber of resonances in the interval [ B , B + ∆ B ] and the bar denotes the mean value over all B .For a Poisson distribution, Σ = ∆ B . By con-trast, for a spectrum according to RMT one expects Σ = 1 /π (cid:0) ln(2 π ∆ B ) + γ + 1 − π / (cid:1) , for large ∆ B andwhere γ = 0 . ... is Euler’s constant [25]. This behav-ior reflects that there are only very small fluctuationsaround an average number of resonances within a giveninterval of size ∆ B (spectral rigidity). Compared to theNNS distribution the number variance is more suitable toprobe long distances in the spectrum. A clear signatureof level repulsion on the one hand and a large spectralrigidity on the other are central properties of strong cor-relations between levels according to RMT [27]. A t o m nu m be r Magnetic field (G)
Extended Data Figure 1 | Fano-Feshbach spectrum of fermionic
Er from to . . The trap-lossspectroscopy is performed in an optically trapped sample of fermionic Er atoms at a temperature of . T F , where T F = 1 . µ K is the Fermi temperature. The atoms are spin-polarized in the lowest Zeeman sublevel, m F = − / .We keep the atomic sample at the magnetic probing field for a holding time of
100 ms . We observe resonancesup to . , which we attribute to be Fano-Feshbach resonances between identical fermions. The corresponding meandensity is about resonances per gauss. -9 -10 -11 -12 E l a s t i c r a t e ( c m / s ) Extended Data Figure 2 | Elastic rate coefficient of m J = − Er collisions.
The s -wave elastic ratecoefficient as a function of magnetic field assuming a collision energy of E/k B =
360 nK. Partial waves (cid:96) up to 20 areincluded. A divergence of the elastic rate coefficient, i.e. the position of a Fano-Feshbach resonance, is marked withsquares.
36 38 40 42 448090100110 Er ab B (G)
B (G)
Extended Data Figure 3 | Statistical analysis of high-density Fano-Feshbach resonances of isotope
Er. a , Position of the resonances are marked with vertical lines. b , The staircase function shows a similar behaviorto Er (Fig. 3). A linear fit to the data above
30 G is plotted in light colors. From the staircase function wecalculate a mean density of resonances of ρ = 3 . − , which corresponds to a mean distance between resonancesof d = 0 . .0 ab P ∆ B (G)
Extended Data Figure 4 | NNS distribution and number variance. a , Er NNS distribution above
30 G with a bin size of
140 mG . For the error bars we assume a Poisson counting error. The plot shows the experimentaldata (circles) with the corresponding Brody distribution (solid line). The parameter free distributions P P (short-dashed line) and P WD are shown and reduced chi-squared values are ˜ χ = 2 . for the Poisson and ˜ χ = 1 . forthe Wigner-Dyson distribution. b , Number variance Σ2