Ultracold Rare-Earth Magnetic Atoms with an Electric Dipole Moment

arXiv:1803.04159v1 [physics.atom-ph] 12 Mar 2018 oainllvl o itmcmlcl (see molecule diatomic spectrum: a energy for the levels of rotational levels odd-parity and even mixes [2]. state Rydberg considered etly[02] atclratnini ado gases on paid is experi- attention reach particular A within [20–22]. nowadays mentally and are simulation Prospects chemistry quantum 19]. physics, ultracold [18, cu- many-body separation inverse to anisotropic spatial the related their highly as of a power varying through bic energy interact potential par- gas the long-range the the Indeed, in of [7–17]. active gases ticles very dilute ultracold currently of is context atoms, lanthanide cially 6]. spatial [5, the atom inside Rydberg bound a atom of it extension when state frame, ground laboratory a the combines per- in a observed moment exhibit dipole can been manent molecule has diatomic lab- it homonuclear the surprisingly, a in that More moment dipole frame. permanent revealing oratory a levels, of energy existence the on the shift Stark linear nounced (see Rydberg for atoms momenta angular orbital different with levels br tm,weeteidcddpl oetscales moment Ry- dipole in induced is maximized the as space where spectacularly of is atoms, effect symmetry dberg This spherical small the a broken. acquires as field neutral moment, electric a external Moreover, dipole an field. in orienta- the placed preferred of atom in direction a the placed acquiring along is in field, tion molecule manifest electric a will external such It an when frame axis. laboratory dipole interatomic the permanent its a along heteronuclear possesses moment an charges which by be- negative provided molecule, is separation and diatomic example a obvious positive An with of [1]. appears barycenter moment the tween dipole a tion, ttesnl-atcesae h xenleeti field electric external the scale, single-particle the At h erhfrsc ioa ytm,ivligespe- involving systems, dipolar such for search The Introduction. n 2 where , aec Lepers Maxence e.g. lrcl aeerhmgei tm iha lcrcdipol electric an with atoms magnetic rare-earth Ultracold tm.Tepi fnal eeeaeeeg eeso opposi of levels energy degenerate nearly momentum of angular pair electronic The atoms. ioemmn ssrnl eedn nteagebtenthe allowin between thus range, angle millisecond the the i in manipulation. on field found dependent Deby is magnetic levels 0.22 strongly a mixed to is When up moment moment diatomics. dipole dipole alkali-metal electric for predic an obtained we mo and experiments, dipole magnetons, current-day electric Bohr an to relevant inducing thus amplitudes field, electric external n epooeanwmto opouea lcrcadmgei di magnetic and electric an produce to method new a propose We stepicplqatmnme fthe of number quantum principal the is 4) nbt ae,ti ed oapro- a to leads this cases, both In [4]). obneUieste,UM nv ai ,CR M81,92 UMR8112, CNRS 6, Paris Univ. Universités, UPMC Sorbonne nacasclnurlcag distribu- charge neutral classical a In 3 nvri´ eBuggeFaceCm´,208Djn Fra Dijon, Franche-Comté, 21078 Bourgogne Université de 1 EM,Osraor ePrsMuo,PLRsac Univers Research PSL Paris-Meudon, de Observatoire LERMA, N ai-aly nvri´ ai-aly 10 Orsay, 91405 Université Paris-Saclay, Paris-Saclay, ENS , 2 u Li Hui , 2 aoaor nedsilnieCro eBugge CNRS Bourgogne, de Carnot Interdisciplinaire Laboratoire 1 aoaor i´ otn NS Université Paris-Sud, CNRS, Aimé Cotton, Laboratoire 1 Jean-Fran¸cois Wyart , J 0 n t1545 cm 17514.50 at and 10, = e.g. Dtd ac 3 2018) 13, March (Dated: 3) or [3]), ev ioa ffcs ntecnrr,teodd-parity ob- the to contrary, sufficient the not On is level [36], effects. to dipolar equal (a.u.) serve moment, units dipole atomic transition 0.015 reduced How- their [33–35]. ever, measurements fundamental for employed hc rsn eue rniindpl oeto 3 of moment dipole transition reduced a present which ataie.Hsoial,tepi flvl t19797.96 at levels of of spectra pair rich cm the the Historically, electric- in the accidentally lanthanides. energy on appear opposite-parity which based dys- quasi-degenerate levels, is ultracold of method of mixing Our gas field dipolar atoms. magnetic prosium and electric an [23–32]. diatomics polar paramagnetic up of which moment, consist dipole now magnetic to a and electric an with vnprt level even-parity ei ioemmns(Ds fadsrsu tmin atom levels dysprosium of a superposition of a (MDMs) moments dipole netic [37]. gases dipolar for promising very are a.u., xeiet,w rdc D of MDM a current-day predict to we relevant experiments, respective amplitudes arbitrary field an For with orientation. field magnetic a and electric eos oorkoldetelretvleosre in of observed EDM value an largest and the experiments, knowledge ultracold our to netons, ioemmn,wihrne rm0to 0 from electric the ranges of which control molecules moment, strong diatomic a dipole of demonstrate values also We the to [38]. similar is which bye, l ooi n emoi isotopes. fermionic and bosonic all d els ino h nl ewe h ed.Because fields. the between angle the of tion e ilscn o h ee hrceie by characterized level obtain the and for parameters, millisecond fields radiative few the a atomic of the functions calculate as also lifetime we levels, excited are 1 max , 3 nti etr epooeanwmto oproduce to method new a propose we Letter, this In ecluaeteeege,eeti EM)admag- and (EDMs) electric energies, the calculate We Model. − ole Quéméner Goulven , | a 1 ial,w hwta u ehdi plcbefor applicable is method our that show we Finally, . i | o utbeeprmna eeto and detection experimental suitable for g a iheetoi nua momenta angular electronic with eti h aoaoyfae o field For frame. laboratory the in ment and i antcdpl oetu o13 to up moment dipole magnetic a t − at 1 rsn,w hwta h electric the that show we present, s with ecnie nao yn ntoeeg lev- energy two in lying atom an consider We | eprt,a 71.3cm 17513.33 at parity, te E b i ed.Telftm ftefield- the of lifetime The fields. oa a futaoddysprosium ultracold of gas polar a ,wihi iia otevalues the to similar is which e, fenergies of , 17513 = J | b ,cnb ie ihan with mixed be can 9, = i 9 edn France Meudon, 195 at nce France E 1 . | 3cm 33 b ∗ a n lve Dulieu Olivier and , 17514 = i E and ity, and i n oa nua momen- angular total and − moment e 1 | b i with . n umte oan to submitted and , µ 0cm 50 − max 1 J with d J 3Bh mag- Bohr 13 = a d − max max 0hsbeen has 10 = 1 0adthe and 10 = with 1 0 = | safunc- a as a µ i max . J and 2De- 22 b 9, = and . | 21 b i 2 tum Ji (i = a, b). Firstly, we consider bosonic isotopes 17518 ε θ which have no nuclear spin, I = 0. In absence of field, = = 0 (a) each level |ii is (2Ji +1)-time degenerate, and the corre- ) 17516 -1 |b〉 sponding Zeeman subslevels are labeled with their mag- 17514 netic quantum number Mi. The atom is submitted both B e e to a magnetic field = B z, with z the unit vector 〉 Energy (cm 17512 |a in the z direction, taken as quantization axis, and to E u u electric field = E , with a unit vector in the di- 17510 rection given by the polar angles θ and φ = 0. In the 0 1000 2000 3000 4000 5000 basis {|Ma = −Jai, ..., | + Jai, |Mb = −Jbi, ..., | + Jbi} Magnetic field (G) spanned by the Zeeman sublevels of |ai and |bi, the ) -0.5

-1 o Hamiltonian can be written B = 100 G ; θ = 0 (b)

Ji -0.55 Hˆ = E |M ihM | + Wˆ + Wˆ . (1) i i i Z S |a〉 i=a,b Mi=−Ji –17513.92 (cm

X X n -0.6 E The Zeeman Hamiltonian Wˆ Z only contains diagonal terms equal to MigiµBB, with gi the Land´eg-factor of -0.65 Energy level |ii. The last term of Eq. (1) is the Stark Hamilto- 0 2 4 6 8 10 nian, which couples sublevels |Mai with sublevels |Mbi Electric field (kV/cm) as ) -0.5 -1 B = 100 G ; ε = 5 kV/cm (c) ˆ 4π dˆ hMa|WS|Mbi = − hak kbiE -0.55 s3(2Ja + 1) ∗ JaMa |a〉 × Y1 − (θ, 0) C − , (2) ,Ma Mb JbMb,1,Ma Mb –17513.92 (cm

n -0.6 E where hakdˆkbi is the reduced transition dipole moment, cγ -0.65

Ykq (θ, φ) a spherical harmonics and Caαbβ a Clebsch- Energy Gordan coeﬃcient [39]. For given values of E, B and 0 30 60 90 120 150 180 Angle (degrees) θ, we calculate the eigenvalues En and eigenvectors

Ji FIG. 1. Eigenvalues of the atom-field Hamiltonian (1) as func- |Ψni = cn,Mi |Mii (3) tions of: (a) the magnetic field B for vanishing electric field i=a,b Mi=−Ji and angle E = θ = 0; (b) the electric field E for B = 100 X X ◦ G and θ = 0 ; (c) the angle θ for B = 100 G and E = 5 of the Hamiltonian in Eq. (1). kV/cm. In panels (b) and (c), the origin of energies is taken at −1 The energy levels that we consider here are Ea = (Ea + Eb)/2 = 17513.92 cm . The blue curve with crosses −1 −1 17513.33 cm , Ja = 10 and Eb = 17514.50 cm , corresponds to the eigenstate converging towards |Ma = −10i Jb = 9. Their Landég-factors ga = 1.30 and gb = 1.32 when θ → 0. are experimental values taken from Ref. [40]. The reduced transition dipole moment hakdˆkbi is calculated using the method developed in our previous works [41–44]. Energies in electric and magnetic fields. Figure 1(a) Firstly, odd-level energies are taken from Ref. [43] which shows the eigenvalues of the Hamiltonian (1) as func- include the electronic configurations [Xe]4f 106s6p and tions of the magnetic field for E = θ = 0. The field splits [Xe]4f 95d6s2, [Xe] being the xenon core. Even-level en- levels |ai and |bi into 21 and 19 sublevels respectively, 10 2 ergies are calculated with the configurations [Xe]4f 6s , each one associated with a given Ma or Mb. On fig. 1(a), 10 9 2 [Xe]4f 5d6s and [Xe]4f 6s 6p [45]. Secondly, following we emphasize the lowest sublevel |Ma = −10i, in which Ref. [43], we adjust the mono-electronic transition dipole ultracold atoms are usually prepared. Due to the close moments by multiplying their ab initio values by appro- Landég-factors, the two Zeeman manifolds look very sim- priate scaling factors [42], equal to 0.794 for h6s|rˆ|6pi, ilar, i.e. the branches characterized by the same values 0.97 for h4f|rˆ|5di and 0.80 for h5d|rˆ|6pi. From the result- Ma = Mb are almost parallel. For B ≥ 1000 Gauss, the ing Einstein coefficients, we can extract hakdˆkbi = 3.21 two Zeeman manifolds overlap; but because the magnetic 4 −1 a.u., as well as the natural linewidth γb =2.98 × 10 s . field conserves parity, the sublevels of |ai and |bi are not At the electric-dipole approximation, γa vanishes; con- mixed. Provoking that mixing is the role of the electric sidering electric-quadrupole and magnetic-dipole transi- field. tions, it can be estimated with the Cowan codes [46] as On figure 1(b), we plot the 21 lowest eigenvalues of −2 −1 γa =3.56 × 10 s . Eq. (1) as functions of the electric field for B = 100 3

◦ Gauss and θ =0 . We focus on the eigenstates converg- 0.3 B = 100 G ; θ = 0o (a) ing to the sublevels of |ai when E → 0. In the range 0.2 of field amplitudes chosen in Figs. 1(a) and (b), which 0.1 corresponds to current experimental possibilities, the influence of E is much weaker than the influence of B. On 0 Fig. 1(b), the energies decrease quadratically with the -0.1 electric field, because the sublevels of |ai are repelled by Electric dipole (D) -0.2 ◦ the sublevels of |bi. Since θ = 0 , the z component of -0.3 the total angular momentum is conserved, and so, the 0 1 2 3 4 5 6 sublevels for which Ma = Mb are coupled in pairs. In Electric field (kV/cm) consequence, the sublevels |Ma = ±10i are insensitive to the electric field, as they have no counterparts among the 0.2 (b) sublevels of |bi (recalling that Jb = 9). 0.1 The only way to couple the |Ma = ±10i sublevels to 0 the other ones is to rotate, say, the electric field, and B = 100 G ; ε = 5 kV/cm thus break the cylindrical symmetry around the z axis. -0.1

On figure 1(c), the 21 lowest eigenvalues of Eq. (1) are Electric dipole (D) now shown as function of the angle θ, for fixed field am- -0.2 plitudes, E = 5 kV/cm and B = 100 Gauss. Even if 0 30 60 90 120 150 180 the corresponding eigenvectors are not associated with a Angle (degrees) single sublevel |Mii (unlike Figs. 1(a) and (b)), they can conveniently be labeled |M ii after their field-free or θ =0 FIG. 2. Electric dipole moments, see Eq. (5), associated with the eigenstates of the atom-field Hamiltonian (1) as functions counterparts. For a given eigenstate, the θ-dependence ◦ of: (a) the electric field E for B = 100 G and θ = 0 ; (b) the of energy is weak. However for |M = ±10i, the energy a angle θ for B = 100 G and E = 5 kV/cm. The blue curve decrease reveals the repulsion with sublevels of |bi, which with crosses corresponds to the eigenstate |M a = −10i. is maximum for θ = 90◦. Magnetic and electric dipole moments. The z component of the MDM associated with the eigenvector |Ψni is As shows figure 2(b), the EDMs change dramatically equal to as function of the angle θ. In particular, the EDM of the eigenstate |M a = −10i ranges continuously from 0 Ji ◦ 2 to a maximum dmax = 0.224 Debye for θ = 90 . The µn = −µB gi |cn,Mi | Mi. (4) eigenstate |M a = 10i follows a similar evolution, except i=a,b Mi=−Ji X X that its curve is sharper around its maximum. In con- Since the eigenvectors are mostly determined by their trast, the EDM of the eigenstate |M a =0i, which is the ◦ ◦ field-free counterparts, µn does not change significantly largest for θ = 0 , becomes the smallest for 90 . Com- in our range of field amplitudes; it is approximately equal pared to the eigenstates |M ai, the curves corresponding to µn ≈ −M agaµB for n ∈ [1; 21] and µn ≈ −M bgbµB to the eigenstates |M bi exhibit an approximate reflection for n ∈ [22; 40]. For instance, the state |M a = −10i has symmetry around the y axis. Finally, it is important to the maximal value µmax = 13.0 × µB. mention that the influence of the magnetic field on the The mean EDM dn = hΨn|d · u|Ψni associated with EDMs is weak in the amplitude range of Fig. 1(a). the eigenvector |Ψni in the direction u of the electric field Radiative lifetimes. The radiative lifetime τn =1/γn is associated with eigenvector |Ψni is such that γn is an arithmetic average of the natural line widths of |ai and 1 ∗ d = − c c hM | Wˆ |M i + c.c. , (5) |bi, n E n,Ma n,Mb a S b Ma,Mb X −1 Ji ˆ −1 2 where the matrix element of WS is given in Eq. (2). Fig- τn = γ = γi |cn,Mi | . (6) n   ure 2(a) presents the EDMs as functions of the electric i=a,b Mi=−Ji ◦ X X field E, for B = 100 G and θ = 0 . In this case, the   graph is symmetric about the y axis. All the curves Figure 3 displays the lifetimes of all eigenstates of Eq. (1) vary linearly with E; all, except the lowest and high- as functions of the angle θ for E = 5 kV/cm and B = 100 est ones, correspond to two eigenstates. In agreement G. Because the natural line widths γa and γb differ by with Fig. 1(b), the curve dn = 0 is associated with 6 orders of magnitude, the lifetimes τn are also spread |M a = ±10i (n = 1 and 21). The lowest and highest over a similar range. At the field amplitudes of Fig. 3, curves belong to M a,b = 0, for which by contrast, the the eigenvectors |M ai are composed at least of 90 % MDM vanishes. of sublevels of |ai, and similarly for eigenstates |M bi. 4

TABLE I. Maximal electric dipole moment dmax, dipolar 10 10 B = 100 G length ad [49] and minimal lifetime τmin obtained for different θ o 1 = 90 isotopes of dysprosium for an electric field E = 5 kV/cm, a ◦ 1 0.1 magnetic field B = 100 G and an angle θ = 90 . The results of 162Dy are also valid for the other bosonic isotopes 156Dy, 0.01 158Dy, 160Dy and 164Dy. 0.1 Lifetime (s) 0 1 2 3 4 5 6 dmax (D) ad (a.u.) τmin (ms) Electric field (kV/cm) 161 0.01 Dy 0.225 2299 4.18 162Dy 0.224 2293 4.22 0 30 60 90 120 150 180 163Dy 0.222 2266 4.23 Angle (degrees)

FIG. 3. Radiative lifetimes, see Eq. (6), associated with the eigenstates |M ai as functions of the angle θ for B = 100 G Wˆ Z and Stark Hamiltonians Wˆ S are calculated by assum- and E = 5 kV/cm. The blue curve with crosses corresponds ing that they do not act on the nuclear quantum number to the eigenstate |M a = −10i. The inset shows the lifetime of ◦ MI , and by using the formulas without hyperfine struc- this eigenstate as a function of E for B = 100 G and θ = 90 . ture (see Eq. (2) and text above). After diagonalizing Eq. (7), one obtains 240 eigenstates (compared to 40 in the bosonic case). Despite Therefore, the lifetimes of eigenstates |M bi (not shown on their large number of curves, the plots of energies, EDMs Fig. 3) are approximately 1/γb, and they weakly depend and lifetimes show similar features to figures 1–3. The on θ. As for the eigenstates |M ai, their small |bi compo- ′ 2 eignestates |Ψ i can be labeled |F iM Fi i after their field- nents, say ε, induces lifetimes roughly equal to ≈ τb/ε . n free counterparts |FiMFi i. Moreover, the “stretched” For |M a = ±10i, the lifetime ranges from τmin = 4.22 ◦ ◦ eigenstates |F aM Fa i = |25/2, ±25/2i are not sensitive ms for θ = 90 to τa = 28.1 s for θ = 0 . Again, this to the electric field for θ = 0◦, and maximally coupled illustrates that the coupling with the sublevels of |bi is ◦ maximum for perpendicular fields and absent for colinear for θ = 90 ; and so, their EDMs range from 0 up to dmax ones. and their lifetimes from τa down to τmin. As shows Table The inset of figure 3 shows the lifetime of the eigen- I, for the same field characteristics, the values of dmax and τmin are very similar from one isotope to another. state |M a = −10i as function of E. In this range of −2 field amplitude, τ1 scales as E . So, a large amplitude Table I also contains the so-called dipolar length ad = 2 2 E can strongly affect the lifetime of the atoms; but on mdmax/~ [49]. It characterizes the length at and beyond the other hand, E needs to be sufficient to induce a no- which the dipole-dipole interaction between two particles 161 ticeable EDM. So, there is a compromise to find between is dominant. For the Dy isotope, one can reach a dipo- EDM and lifetime, by tuning the electric-field amplitude lar length of ad = 2299 a.u.. To compare with, at E =5 and the angle between the fields. kV/cm and an induced dipole moment of 0.22 D [38], 40 87 Fermionic isotopes. There are two fermionic isotopes K Rb has a length of ad = 1734 a.u.. Similarly, a of dysprosium, 161Dy and 163Dy, both with a nuclear spin length of ad = 1150 a.u. was reached [11] for magnetic 168 I = 5/2. A given hyperfine sublevel is characterized by dipolar Feshbach molecules of Er2. With the particu- the total (electronic+nuclear) angular momentum Fi and lar set-up of electric and magnetic fields employed in this its z-projection MFi , where |Ji − I| ≤ Fi ≤ Ji + I, and study, we show that one can reach comparable and even

−Fi ≤ MFi ≤ Fi. Namely, Fa ranges from 15/2 to 25/2, stronger dipolar character with atoms in excited states and Fb ranges from 13/2 to 23/2. The hyperfine sub- than with certain diatomic molecules. levels are constructed by angular-momentum addition of Conclusion. We have demonstrated the possibility to J I FiMFi induce a strong electric dipole moment on atomic dys- i and , i.e. |FiMFi i = MiMI CJiMiIMI |JiMii|IMI i. Compared to Eq. (1), the Hamiltonian Hˆ ′ is modified as prosium, in addition to its large magnetic dipole mo- P ment. To do so, the atoms should be prepared in a ˆ ′ ˆ ˆ superposition of nearly degenerate excited levels using H = EFi |FiMFi ihFiMFi | + WZ + WS (7) an electric and a magnetic field of arbitrary orientations. i=a,b Fi MFi X X X We show a remarkable control of the electric dipole mo- where EFi is the hyperfine energy depending on the ment and radiative lifetime by tuning the angle between magnetic-dipole and electric-quadrupole constants Ai the fields. Since the two levels are metastable, they are 163 and Bi. For Dy, they have been calculated in Ref. [47]: not accessible by one-photon transition from the ground Aa = 225 MHz, Ba = 2434 MHz, Ab = 237 MHz level. Instead, one could perform a Raman transition 161 and Bb = 706 MHz. For Dy, we apply the relations between the ground level |gi (Jg = 8) and the level |bi 161 163 161 163 10 ( Ai)= −0.714×( Ai) and ( Bi)=0.947×( Bi) (Jb = 9) of leading configuration [Xe]4f 5d6s, through given in Ref. [48]. The matrix elements of the Zeeman the upper levels at 23736.61, 23832.06 or 23877.74 cm−1, 5

10 whose [Xe]4f 6s6p character insures significant transi- [18] A. Stone, The Theory of Intermolecular Forces (Oxford tion strengths with |gi and |bi. In the spectrum of other University Press, New York, 1996). lanthanides, there exist pairs of quasi-degenerate levels [19] M. Lepers and O. Dulieu, arXiv preprint accessible from the ground state, for instance the levels arXiv:1703.02833 (2017). −1 [20] M. Baranov, Phys. Rep. 464, 71 (2008). at 24357.90 and 24660.80 cm in holmium, but in turn [21] T. Lahaye, C. Menotti, L. Santos, M. Lewenstein, and their radiative lifetime is much shorter [50]. T. Pfau, Rep. Prog. Phys. 72, 126401 (2009). [22] M. Baranov, M. Dalmonte, G. Pupillo, and P. Zoller, Acknowledgments. We acknowledge support from Chem. Rev. 112, 5012 (2012). “DIM Nano-K” under the project “InterDy”, and from [23] P. Zuchowski,˙ J. Aldegunde, and J. Hutson, “Agence Nationale de la Recherche” (ANR) under the Phys. Rev. Lett. 105, 153201 (2010). project “COPOMOL” (contract ANR-13-IS04-0004-01). [24] B. Pasquiou, A. Bayerle, S. Tzanova, S. Stellmer, We also acknowledge the use of the computing center J. Szczepkowski, M. Parigger, R. Grimm, and F. Schreck, Phys. Rev. A 88, 023601 (2013). “MésoLUM” of the LUMAT research federation (FR [25] J. Barry, D. McCarron, E. Norrgard, M. Steinecker, and LUMAT 2764). D. DeMille, Nature 512, 286 (2014). [26] A. Khramov, A. Hansen, W. Dowd, R. Roy, C. Makrides, A. Petrov, S. Kotochigova, and S. Gupta, Phys. Rev. Lett. 112, 033201 (2014). [27] M. Tomza, R. González-Férez, C. Koch, and R. Moszyn- ∗ [email protected] ski, Phys. Rev. Lett. 112, 113201 (2014). [1] J. Jackson, Classical Electrodynamics (John Wiley & [28] P. Zuchowski,˙ R. Guérout, and O. Dulieu, Sons, Inc., New York, 1999). Phys. Rev. A 90, 012507 (2014). [2] T. Gallagher, Rydberg atoms, Vol. 3 (Cambridge Univer- [29] M. Karra, K. Sharma, B. Friedrich, S. Kais, and D. Her- sity Press, 2005). schbach, J. Chem. Phys. 144, 094301 (2016). [3] M. Brieger, Chem. Phys. 89, 275 (1984). [30] G. Quéméner and J. Bohn, [4] M. Zimmerman, M. Littman, M. Kash, and D. Kleppner, Phys. Rev. A 93, 012704 (2016). Phys. Rev. A 20, 2251 (1979). [31] D. Reens, H. Wu, T. Langen, and J. Ye, [5] C. Greene, A. Dickinson, and H. Sadeghpour, Phys. Rev. A 96, 063420 (2017). Phys. Rev. Lett. 85, 2458 (2000). [32] T. Rvachov, H. Son, A. Sommer, S. Ebadi, J. Park, [6] W. Li, T. Pohl, J. Rost, S. Rittenhouse, H. Sadeghpour, M. Zwierlein, W. Ketterle, and A. Jamison, J. Nipper, B. Butscher, J. B. Balewski, V. Bendkowsky, Phys. Rev. Lett. 119, 143001 (2017). R. Löw, and T. Pfau, Science 334, 1110 (2011). [33] D. Budker, D. DeMille, E. D. Commins, and M. S. Zolo- [7] M. Lu, S. Youn, and B. Lev, torev, Phys. Rev. Lett. 70, 3019 (1993). Phys. Rev. Lett. 104, 063001 (2010). [34] A. Cingöz, A. Lapierre, A.-T. Nguyen, N. Leefer, [8] D. Sukachev, A. Sokolov, K. Chebakov, A. Aki- D. Budker, S. Lamoreaux, and J. Torgerson, mov, S. Kanorsky, N. Kolachevsky, and V. Sorokin, Phys. Rev. Lett. 98, 040801 (2007). Phys. Rev. A 82, 011405 (2010). [35] N. Leefer, C. Weber, A. Cingöz, J. Torgerson, and [9] K. Aikawa, A. Frisch, M. Mark, S. Baier, D. Budker, Phys. Rev. Lett. 111, 060801 (2013). A. Rietzler, R. Grimm, and F. Ferlaino, [36] D. Budker, D. DeMille, E. Commins, and M. Zolotorev, Phys. Rev. Lett. 108, 210401 (2012). Phys. Rev. A 50, 132 (1994). [10] J. Miao, J. Hostetter, G. Stratis, and M. Saffman, [37] J.-F. Wyart, Physica 75, 371 (1974). Phys. Rev. A 89, 041401 (2014). [38] K.-K. Ni, S. Ospelkaus, D. Wang, G. Quéméner, [11] A. Frisch, M. Mark, K. Aikawa, S. Baier, R. Grimm, B. Neyenhuis, M. De Miranda, J. Bohn, J. Ye, and A. Petrov, S. Kotochigova, G. Quéméner, M. Lepers, D. Jin, Nature 464, 1324 (2010). O. Dulieu, and F. Ferlaino, Phys. Rev. Lett. 115, 203201 [39] D. Varshalovich, A. Moskalev, and V. Khersonskii, (2015). Quantum theory of angular momentum (World Scientific, [12] H. Kadau, M. Schmitt, M. Wenzel, C. Wink, T. Maier, 1988). I. Ferrier-Barbut, and T. Pfau, Nature 530, 194 (2016). [40] A. Kramida, Yu. Ralchenko, J. Reader, and and NIST [13] D. Dreon, L. Sidorenkov, C. Bouazza, ASD Team, NIST Atomic Spectra Database (ver. 5.3), W. Maineult, J. Dalibard, and S. Nascimbène, [Online]. Available: http://physics.nist.gov/asd arXiv preprint arXiv:1610.02284 (2016). [2016, March 8]. National Institute of Standards and [14] E. Lucioni, G. Masella, A. Fregosi, C. Gab- Technology, Gaithersburg, MD. (2015). banini, S. Gozzini, A. Fioretti, L. Del Bino, [41] M. Lepers, J.-F. Wyart, and O. Dulieu, J. Catani, G. Modugno, and M. Inguscio, Phys. Rev. A 89, 022505 (2014). Eur. Phys. J. Special Topics 226, 2775 (2017). [42] M. Lepers, Y. Hong, J.-F. Wyart, and O. Dulieu, [15] J. Ulitzsch, D. Babik, R. Roell, and M. Weitz, Phys. Rev. A 93, 011401R (2016). Phys. Rev. A 95, 043614 (2017). [43] H. Li, J.-F. Wyart, O. Dulieu, S. Nascimbène, and [16] J. Becher, S. Baier, K. Aikawa, M. Lepers, M. Lepers, J. Phys. B 50, 014005 (2017). J.-F. Wyart, O. Dulieu, and F. Ferlaino, [44] H. Li, J.-F. Wyart, O. Dulieu, and M. Lepers, Phys. Rev. A 97, 012509 (2018). Phys. Rev. A 95, 062508 (2017). [17] C. Ravensbergen, V. Corre, E. Soave, M. Kreyer, [45] J.-F. Wyart, private communication (2018). S. Tzanova, E. Kirilov, and R. Grimm, arXiv preprint arXiv:1801.05658 (2018). 6

[46] R. Cowan, The theory of atomic structure and spectra, 3 [49] P. Julienne, T. Hanna, and Z. Idziaszek, (Univ of California Press, 1981). Phys. Chem. Chem. Phys. 13, 19114 (2011). [47] V. Dzuba, V. Flambaum, and M. Kozlov, [50] E. Den Hartog, L. Wiese, and J. Lawler, Phys. Rev. A 50, 3812 (1994). J. Opt. Soc. Am. B 16, 2278 (1999). [48] E. Eliel, W. Hogervorst, G. Zaal, K. van Leeuwen, and J. Blok, J. Phys. B 13, 2195 (1980).