Direct observation of coherent inter-orbital spin-exchange dynamics
Giacomo Cappellini, Marco Mancini, Guido Pagano, Pietro Lombardi, Lorenzo Livi, Mario Siciliani de Cumis, Pablo Cancio, Marco Pizzocaro, Davide Calonico, Filippo Levi, Carlo Sias, Jacopo Catani, Massimo Inguscio, Leonardo Fallani
DDirect observation of coherent inter-orbital spin-exchange dynamics
G. Cappellini , , M. Mancini , , G. Pagano , , P. Lombardi , L. Livi , M. Siciliani de Cumis ,P. Cancio , , , M. Pizzocaro , D. Calonico , F. Levi , C. Sias , , J. Catani , , M. Inguscio , , , L. Fallani , , LENS European Laboratory for Nonlinear Spectroscopy, Sesto Fiorentino, Italy Department of Physics and Astronomy, University of Florence, Italy INO-CNR Istituto Nazionale di Ottica del CNR, Sezione di Sesto Fiorentino, Italy Scuola Normale Superiore di Pisa, Italy INRIM Istituto Nazionale di Ricerca Metrologica, Torino, Italy
We report on the first direct observation of fast spin-exchange coherent oscillations between differentlong-lived electronic orbitals of ultracold
Yb fermions. We measure, in a model-independent way,the strength of the exchange interaction driving this coherent process. This observation allows us toretrieve important information on the inter-orbital collisional properties of
Yb atoms and pavesthe way to novel quantum simulations of paradigmatic models of two-orbital quantum magnetism.
PACS numbers: 03.75.Ss, 34.50.Fa, 34.50.Cx, 37.10.Jk, 67.85.Lm
Alkaline-earth-like (AEL) atoms are providing a newvaluable experimental platform for advancing the possi-bilities of quantum simulation with ultracold gases [1].For instance, the purely nuclear spin of ground stateAEL fermionic isotopes results in the independence ofthe atom-atom scattering properties from the nuclearspin projection. This feature has enabled the investi-gation of multi-component
Yb fermions with SU(N)interaction symmetry both in optical lattices [2] and inone-dimensional quantum wires [3]. In addition to theirnuclear spin, AEL atoms offer experimental access tosupplementary degrees of freedom, in particular to a long-lived electronically-excited state | e (cid:105) = | P (cid:105) which can becoherently populated from the ground state | g (cid:105) = | S (cid:105) by optical excitation on an ultranarrow clock transition.The possibility of coherently manipulating both the or-bital and the spin degree of freedom has recently beenenvisioned to grant the realization of paradigmatic modelsof two-orbital magnetism, like the Kondo model [4]. Inthis context, the two electronic states | g (cid:105) and | e (cid:105) play therole of two different orbitals.Recent experiments have investigated the SU(N) sym-metry in | g (cid:105) - | e (cid:105) ultracold collisions of two-electron atoms[5] and reported on first signatures of spin-exchange in-teractions between atoms in the two electronic states [6].Spin-exchange interactions arise from the difference inthe spin-singlet and spin-triplet potential curves in thescattering of one | g (cid:105) and one | e (cid:105) atom. Let us assume thatthe two interacting atoms are in different nuclear spinstates | ↑(cid:105) and | ↓(cid:105) (where the arrows are placeholdersfor two arbitrary nuclear spin states) and that they sharethe same spatial wavefunction. At zero magnetic field thedegeneracy of the configurations | g ↑ , e ↓(cid:105) and | g ↓ , e ↑(cid:105) ,which are associated to a well-defined spin in each orbital[7], is lifted by the atom-atom interaction and the eigen-states are the orbital-symmetric (spin-singlet) | eg + (cid:105) andthe orbital-antisymmetric (spin-triplet) | eg − (cid:105) states [4] | eg ± (cid:105) = 1 √ | g ↑ , e ↓(cid:105) ∓ | g ↓ , e ↑(cid:105) ) . (1) FIG. 1. Two-orbital spin-exchange interaction in AEL atoms.a) One atom in the ground state | g (cid:105) and one atom in thelong-lived electronic state | e (cid:105) periodically “exchange” theirnuclear spins because of the different interaction energy in thespin-singlet | eg + (cid:105) and spin-triplet | eg − (cid:105) two-particle states(note that in the graphical notation the two-particle exchangesymmetry is implicit [7]). b) Dependence of the two-particleenergy on the magnetic field B . The spin dynamics is initiatedby exciting the two atoms to the | eg L (cid:105) state at finite B andthen quenching the magnetic field to zero in order to create asuperposition of the | eg + (cid:105) and | eg − (cid:105) states (dashed arrows). Owing to the different atom-atom scattering properties,these two states have different interaction energies U ± eg , assketched in Fig. 1. Preparing the two atoms in the initialstate | ψ (cid:105) = | g ↑ , e ↓(cid:105) = √ [ | eg + (cid:105) + | eg − (cid:105) ] would resultin a spin-exchange dynamics in which the spins of the | g (cid:105) and | e (cid:105) atoms are periodically flipped at a frequency a r X i v : . [ c ond - m a t . qu a n t - g a s ] M a y V ex /h = | U − eg − U + eg | /h , with probability of finding aground-state atom in the | g ↑(cid:105) state being given by P ( | g ↑(cid:105) )( t ) = 12 (cid:20) (cid:18) V ex (cid:126) t (cid:19)(cid:21) . (2)Recent measurements have suggested that in Yb thescattering lengths associated to the spin-triplet and spin-singlet scattering are quite different [6], resulting in a largeinter-orbital spin-exchange interaction energy V ex . How-ever, spin oscillations induced by such interaction havenot been observed, and no demonstration of the coherenceof this exchange process has been shown. Here we reporton the first, time-resolved observation of inter-orbitalspin oscillations. This measurement clearly demonstratesthe coherent nature of the exchange interaction, whichis fundamentally important for its applications in quan-tum simulation. By measuring the oscillation frequencywe determine the interaction strength V ex in a model-independent way, finding it to be much larger than boththe Fermi energy E F = k B T F and k B T (where k B , T F and T are the Boltzmann constant, the Fermi and the gastemperature, respectively). Moreover, our measurementsallow us to determine the scattering length associatedwith the orbital-symmetric scattering potential.The experiment is performed on quantum degenerateFermi gases of Yb in a balanced mixture of two dif-ferent states out of the I = 5 / | m I = +5 / (cid:105) ≡ | ↑(cid:105) and | m I = − / (cid:105) ≡ | ↓(cid:105) . Theatoms, at an initial temperature T (cid:39) . T F (cid:39)
25 nK,are trapped in a 3D optical lattice, with a variable depthup to s = 40, where s measures the lattice depth in unitsof the recoil energy E R = h / mλ L , λ L and m beingthe lattice wavelength and atomic mass, respectively. Inour experimental conditions (see Supplemental Material[8]), the site occupancy in the center of the trap is n (cid:39) | e (cid:105) state is populatedby exciting the S → P intercombination transitionwith linearly-polarized light coming from a λ = 578 nmultranarrow laser stabilized to an ULE (Ultra Low Expan-sion) glass optical resonator with a closed-loop linewidthbelow 10 Hz [9]. The lattice is operating at the magicwavelength λ L = 759 .
35 nm, which is not shifting the S → P transition frequency [10].A typical spectrum for a long excitation time ( (cid:39) B (which was set to 28 G for the data shownin the figure). Due to the Zeeman shift, at a finite B the eigenstates of the system become an admixture ofspin-singlet and spin-triplet states | eg L (cid:105) = α | eg − (cid:105) + β | eg + (cid:105) ( | eg H (cid:105) = β ∗ | eg − (cid:105)− α ∗ | eg + (cid:105) ), with | α | = | β | =1 / B = 0 the ground state | gg (cid:105) is coupled only to the | eg L (cid:105) = | eg − (cid:105) state, because of the Clebsch-Gordan coef-ficients determining the strength of the Rabi couplings. FIG. 2. a) Spectrum of the λ = 578 nm clock transition forthe excitation of a two-spin mixture of Yb atoms trappedin a 3D lattice. The vertical axis shows the number of residual | g (cid:105) atoms after the excitation, while the horizontal axis showsthe offset with respect to the clock transition frequency. Thelabels below the plot indicate the most prominent featuresof the spectrum. The dependence of the peak centers on themagnetic field B allows us to attribute them to the excitationof one atom in either singly-occupied sites ( | e ↓(cid:105) and | e ↑(cid:105) ) or in doubly-occupied sites ( | eg L (cid:105) ) (see Ref. [6] for theassignment of the other peaks). b) Time-resolved detection ofspin-exchange oscillations. The points show the difference infractional population between | g ↑(cid:105) and | g ↓(cid:105) atoms. The datashown in figure have been taken at a lattice depth s = 30 . . (cid:39) | e ↓(cid:105) , | e ↑(cid:105) peaks in panel a). The points are averages over 5 repeatedmeasurements and the line is the result of a fit with a dampedsinusoidal function (a global error bar based on the fit residualshas been assigned to the points). The inset shows a differentdataset taken at s = 35 with oscillations extending to longertimes. FIG. 3. a) The points show the measured spin-exchangefrequency as a function of the lattice depth s . The data havebeen corrected for the small bias magnetic field B = 3 . | eg + (cid:105) state, calculated asthe sum of the experimentally measured 2 V ex and the U − eg calculated by using a − eg = 219 . a [6]. The shaded area showsthe energy difference between ground and first excited latticeband. In order to initiate the spin dynamics we first excitethe atoms with a π -pulse resonant with the | eg L (cid:105) exci-tation frequency. The excitation is performed at a largelattice depth s in ≥
30, in order to avoid tunneling ofatoms during the excitation time, and at large magneticfield (60 G), in order to have a sizeable admixture of thespin-singlet state | eg + (cid:105) into the | eg L (cid:105) state ( | α | (cid:39) . | β | (cid:39) . t ramp = 25 µ s,which is fast enough to have a significant population ofthe | eg H (cid:105) (cid:39) | eg + (cid:105) state by nonadiabatic Landau-Zenerexcitation (see dashed arrows in Fig. 1) [11]. The cre-ation of a superposition of | eg − (cid:105) and | eg + (cid:105) states allowsus to start the spin dynamics, which is observed by de-tecting the fraction of ground-state atoms in the differentspin states by performing optical Stern-Gerlach (OSG)detection after different evolution times [12]. Figure 2bshows clear oscillations of the ground-state magnetization[ N ( g ↑ ) − N ( g ↓ )] / [ N ( g ↑ ) + N ( g ↓ )], which are drivenby the spin-exchange process. These oscillations, clearlyvisible for tens of periods (as shown in the inset), providea clear demonstration of the coherent nature of this spin-exchange interaction. The measurement of their frequencyprovides a direct, model-independent determination ofthe interaction strength, which is 2 V ex = h × (13 . ± . s = 30 . | eg + (cid:105) statein the | eg L (cid:105) state (due to excitation at a finite B ); 2)the finite switching time of the magnetic field, whichmakes the projection onto the new eigenstates at low B only partially diabatic; 3) the presence of singly-occupiedlattice sites not participating to the spin oscillation, yetcontributing to the background signal. We also havechecked that these spin oscillations disappear if no laserexcitation pulse is performed: collisions among | g (cid:105) atomscan only take place in the spin-singlet channel, and thestrong SU(N) interaction symmetry grants the absenceof spin-changing collisions [3]. We have also checked thatno other nuclear spin states, different from | ↑(cid:105) and | ↓(cid:105) ,are populated during the spin-exchange dynamics.In order to quantify the strength and the properties ofthe inter-orbital exchange interaction, we have measuredthe frequency of these spin oscillations as a function ofthe lattice depth s and of the magnetic field B .The points in Fig. 3a show the dependence of thespin oscillation frequency 2 V ex /h on the lattice depth,clearly exhibiting a monothonic increase with s . In thesemeasurements the optical excitation is performed at alattice depth s in ≥
30, then the optical lattice is rampedto s in ∼ µ s, immediately before the quench whichinitiates the spin dynamics. The measured values of2 V ex are significantly large, ≈ U = (4 π (cid:126) a/m ) (cid:82) | w ( r ) | d r (where w ( r ) are the single-particle Wannier functions), is expected to fail. At largeinteraction strength the two-particle wavefunction cannotbe expressed in terms of lowest-band Wannier functionssince, in the limit of infinite repulsion, the two atomstend to spatially separate in each lattice site [14] and theprobability of finding them at the same position dropsto zero. For a system of two particles in a harmonicpotential it has been shown that, for a scattering length a significantly larger than the harmonic oscillator length a ho , the interaction energy saturates at the energy of thefirst excited harmonic oscillator state [15, 16].In order to relate our measurements to the values ofthe scattering lengths a ± eg we follow a similar treatmentto that adopted in Refs. [17, 18], where the interactionenergy for two particles in a true optical lattice potentialwas derived by evaluating the anharmonic corrections tothe lowest-order parabolic approximation of the potential.In our analysis we express the total Hamiltonian on abasis formed by wavefunctions for the relative motionand for the center-of-mass motion of the two particles.For the former, we use the wavefunctions for interactingparticles in a harmonic trap analytically derived in Ref.[15]; for the latter, harmonic oscillator wavefunctions areconsidered (see Supplemental Material [8] for more de- FIG. 4. Circles: measured spin-exchange frequency ( U Heg − U Leg ) /h at s = 30 as a function of the magnetic field. Squares:measured energy of the | eg L (cid:105) state derived from the spectro-scopic measurements exemplified in Fig. 2a. The solid linesshow the predictions of the model in Eq. (3) by using the a + eg value derived in Fig. 3. The dashed lines show a fit of thepoints to the same model leaving a + eg as free parameter (seemain text for more details). tails). We then evaluate the anharmonic terms (up to10 th order) on this basis and by numerical diagonaliza-tion of the total Hamiltonian we derive the dependenceof the interaction energy in the motional ground state U ( a, s ) as a function of the scattering length a and ofthe lattice depth s . In Fig. 3a we fit the experimen-tal data of the spin oscillation frequency vs. s with thefunction (cid:2) U ( a + eg , s ) − U ( a − eg , s ) (cid:3) /h (solid line), assumingthe value a − eg = 219 . a for the spin-triplet scatteringlength measured in Ref. [6] (where a is the Bohr radius).The result of the fit is a spin-singlet scattering length a + eg = (3300 ± a . This scattering length is remark-ably large and, as shown in Fig. 3b, causes the energyof the | eg + (cid:105) state to almost saturate to the energy gapbetween the first two lattice bands (grey curve).At a finite B the spin-exchange oscillation shows a fasterfrequency, as the Zeeman energy increasingly contributesto the energy difference between | eg L (cid:105) and | eg H (cid:105) (seeFig. 1). The circles in Fig. 4 show the measured spin-oscillation frequency ( U Heg − U Leg ) /h at s = 30 as a functionof B , while the squares indicate the energy of the | eg L (cid:105) state determined by fitting the position of the peaks in thespectroscopic measurements shown in Fig. 2a. These dataare compared to a simple single-band model in which theHamiltonian of the two-atom system including interactionenergy and Zeeman shift is written on the {| eg − (cid:105) , | eg + (cid:105)} basis as H = (cid:18) U + eg F ∆ B F ∆ B U − eg (cid:19) , (3) where ∆ B = ∆ µB is the Zeeman splitting (arising froma difference ∆ µ in the magnetic moments of the | g (cid:105) and | e (cid:105) states [19]) coupling the zero-field eigenstates | eg + (cid:105) and | eg − (cid:105) . Differently from Ref. [6], we have included aFranck-Condon factor F , defined as the overlap integral F = (cid:90) (cid:90) d r d r ψ + eg ( r , r ) ψ − eg ( r , r ) , (4)between the wavefunctions ψ ± eg of the two atoms interact-ing in the two different channels. The strong repulsion inthe spin-singlet channel causes indeed a strong modifica-tion of the wavefunction, resulting in an overlap integralthat is significantly smaller than unity ( F (cid:39) .
77, seeSupplemental Material [8]). By diagonalizing Eq. (3) wefind the eigenstates (cid:8) | eg L (cid:105) , | eg H (cid:105) (cid:9) and the dependenceof the energies U Leg , U Heg on the magnetic field B (see alsoFig. 1). The solid lines in Fig. 4 show the predictionsof this model by using a − eg = 219 . a , a + eg = 3300 a (from the fit in Fig. 3) and the F factor calculated byusing the interacting wavefunctions obtained previously.The agreement with the experimental data is quite good,showing the substantial validity of the model in Eq. (3)as long as the overlap factor F between the interactingwavefunctions is considered. Alternatively, we have per-formed a simultaneous fit of the two datasets in Fig. 4with the eigenenergies of Eq. (3) by expressing U + eg and F as functions of the free parameter a + eg (obtained fromthe model discussed previously): the result (dashed lines)is a + eg = (4700 ± a , which is ∼ σ away from themore precise determination coming from the fit of thedata shown in Fig. 3. We note that a precise determina-tion of a + eg is complicated by the fact that, in this regimeof strong interactions, the dependence of U + eg on a + eg isextremely weak and small effects coming e.g. from cali-bration uncertainties or from higher-order contributionsin the theory could yield significant changes. We alsonote that in the presence of a tight trapping the inter-pretation of the results in terms of an effective scatteringlength should be considered [20]. However, we stress that,differently from a + eg , our determination of V ex is free fromany assumption or modeling and represents an accuratemeasurement of the spin-exchange coherent coupling inan actual experimental configuration.The 3D lattice setting that we have used in our ex-periments has allowed us to study the dynamics of anisolated two-atom system in which only one atom is in theexcited state, therefore significantly reducing the effectsof inelastic | e (cid:105) − | e (cid:105) collisions. Nevertheless, we measurea finite lifetime of the spin-exchange oscillations, on theorder of ∼ t wait betweenthe laser excitation to the | eg L (cid:105) state and the magneticfield quench. For t wait as large as 30 ms (more than oneorder of magnitude larger than the observed dampingtime) we still detect high-contrast spin-exchange oscil-lations. This rules out the explanation of the dampingin the inset of Fig. 2b in terms of either a detrimentaleffect of inelastic | g (cid:105) − | e (cid:105) collisions in doubly-occupiedsites, or a possibile collisional dephasing introduced by thetunneling of highly mobile atoms in excited lattice bands.After the exclusion of these fundamental mechanisms ofdecoherence, it seems highly plausible that the decay ofthe spin-exchange oscillations arises from technical im-perfections (associated e.g. to the fast switching of themagnetic field).In conclusion, we have observed for the first time fast,long-lived inter-orbital spin-exchange oscillations by ex-ploiting a system of ultracold AEL fermions trapped ina 3D optical lattice. The direct observation of severalperiods of these oscillations has allowed us to demonstratethe coherence of the process and to measure the exchangeinteraction strength in an accurate, model-independentway. We note that, if compared with the spin dynam-ics observed in other atomic systems, arising from eithersmall differences in the scattering lengths [21–23] or fromsecond-order tunnelling between adjacent sites of an opti- cal lattice [24], the oscillation that we have measured issignificantly fast. In particular, the exchange energy V ex ,on the order of ∼ h ×
10 kHz, is much larger than eitherthe Fermi ( k B T F ) and the thermal ( k B T ) energies, whichmakes Yb remarkably interesting for the observation ofquantum magnetism in a two-orbital system with SU(N)interaction symmetry [4]. The direct measurement of V ex has also allowed us to provide a determination of theinter-orbital spin-triplet scattering length, which exceedsthe spin-singlet one by ∼
20 times. Besides, from a widerpoint of view, this strong spin-exchange interaction en-tangles two stable internal degrees of freedom of the atom[25], which can be independently and coherently manipu-lated, opening new realistic possibilities for both quantuminformation processing and quantum simulation.We would like to acknowledge N. Fabbri, M. Fattori,C. Fort and A. Simoni for useful discussions. This workhas been financially supported by EU FP7 Projects SIQS(Grant 600645) and SOC-2 (Grant 263500), MIUR ProjectPRIN2012 AQUASIM, ERC Advanced Grant DISQUA(Grant 247371). [1] M. Inguscio and L. Fallani,
Atomic Physics: Precise Mea-surements and Ultracold Matter (Oxford University Press,2013).[2] S. Taie et al., Nat. Phys. , 825 (2012).[3] G. Pagano et al., Nat. Phys. , 198 (2014).[4] V. Gorshkov et al., Nat. Phys. , 289 (2010).[5] X. Zhang et al., Science , 1467 (2014).[6] F. Scazza et al., Nat. Phys. , 779 (2014).[7] In the notation | g ↑ , e ↓(cid:105) the two-particle exchange sym-metry is implicit. The full state has to be intended as | g ↑ , e ↓(cid:105) = ( | g ↑(cid:105) | e ↓(cid:105) − | e ↓(cid:105) | g ↑(cid:105) ) / √
2, which is an-tisymmetric for the exchange of fermion 1 with fermion2.[8] See Supplemental Material for additional technical detailson the experimental procedure and on the theoreticalmodel.[9] M. Pizzocaro et al., IEEE T. Ultrason. Ferr. , 3 (2012).[10] Z. Barber et al., Phys. Rev. Lett , 103002 (2008).[11] In the limit t ramp → | eg + (cid:105) and | eg − (cid:105) states would be β /α .[12] T. Sleator, T. Pfau, V. Balykin, O. Carnal, and J. Mlynek,Phys. Rev. Lett. , 1996 (1992).[13] We note that the finite bias magnetic field B (cid:39) . V ex /h (by ∼
100 Hz). In order to showthe zero-field oscillation frequency the datapoints in Fig. 3have been corrected by using the finite- B model describedlater in the text.[14] G. Z¨urn et al., Phys. Rev. Lett. , 075303 (2012).[15] T. Busch, B.-G. Englert, K. Rza˙zewski, and M. Wilkens,Found. Phys. , 549 (1998).[16] M. K¨ohl, K. G¨unter, T. St¨oferle, H. Moritz, andT. Esslinger, J. Phys. B: At. Mol. Opt. Phys. , S47(2006).[17] F. Deuretzbacher et al., Phys. Rev. A , 032726 (2008).[18] J. Mentink and S. Kokkelmans, Phys. Rev. A , 032709(2009).[19] S. G. Porsev, A. Derevianko, and E. N. Fortson, Phys.Rev. A , 021403(R) (2004).[20] E. L. Bolda, E. Tiesinga, and P. S. Julienne, Phys. Rev.A , 013403 (2002).[21] A. Widera et al., Phys. Rev. Lett. , 190405 (2005).[22] J. S. Krauser et al., Nat. Phys. , 813 (2012).[23] J. S. Krauser et al., Science , 157 (2014).[24] S. Trotzky et al., Science , 295 (2008).[25] M. Anderlini et al., Nature , 452 (2007). Supplemental Material for“Direct observation of coherent inter-orbital spin-exchange dynamics”
G. Cappellini, M. Mancini, G. Pagano, P. Lombardi, L. Livi, M. Siciliani de Cumis,P. Cancio, M. Pizzocaro, D. Calonico, F. Levi, C. Sias, J. Catani, M. Inguscio, L. Fallani
S.I. EXPERIMENTAL SEQUENCE
Fig. S1 shows a diagram with the time sequence of ourexperiments. We start with a two-component
Yb Fermigas ( m I = ± / × atoms are left at a temperature T (cid:39) . T F (cid:39) λ L = 759 .
35 nm and, during the same time, the opticaldipole trap intensity is ramped to zero in order to letthe atoms be trapped only by the lattice potential. Theaverage filling is 0 . ≤ n ≤ s in ≥ E R = h / mλ L , where m isthe atomic mass).The atoms are excited by a 578 nm π -pulse, resonantwith the | gg (cid:105) → | eg L (cid:105) transition, at a high magnetic field B (cid:39)
60 G. After the excitation pulse the lattice depthis quickly ramped to s in ≈ µ s and, immediatelyafter, the magnetic field is quenched to the final B valuein 25 µ s, sufficiently fast to have a significant projectionof the atomic state onto | eg + (cid:105) . At this point the spin-exchange oscillation | g ↑ , e ↓(cid:105) ↔ | g ↓ , e ↑(cid:105) is started.After a variable oscillation time t osc , the optical latticeis switched off and the populations in the | g ↑(cid:105) , | g ↓(cid:105) states are measured after an optical Stern-Gerlach pulse,followed by a time of flight of 4.5 ms. S.II. THEORETICAL MODEL
Here we describe the model that we have developed inorder to relate the large interaction energies measured inthe experiment to the values of the scattering lengths a ± eg describing the s -wave collisions of two Yb atoms in the | g (cid:105) + | e (cid:105) channel. This model is valid also for strong interac-tions, when the relation between the Hubbard interactionenergy and the scattering length a is no longer linear, asit is in the usual expression U Hub = π (cid:126) m a (cid:82) | w ( r ) | d r ,where w ( r ) is the lowest-band Wannier function for a(noninteracting) atom localized at a lattice site [S1].The Hamiltonian describing two atoms interacting in alattice potential well is: H = p m + p m + V lat ( r )+ V lat ( r )+ V int ( r − r ) , (S.1)where V lat ( r ) = V (cid:80) i = x,y,z sin ( kr i ) is the latticepotential experienced by each atom and V int ( r ) = FIG. S1. Typical experimental sequence (see text for details). π (cid:126) m a δ ( r ) ∂∂r r is the interaction potential, expressed inthe form of a regularized pseudopotential [15].In order to take into account the anharmonicity ofthe lattice potential (which is essentially important for aquantitative comparison with the experimental data), weexpand V lat ( r ) around the origin up to the 10 th order: V lat ( r ) = V (cid:88) i = x,y,z ( k r i − k r i + 245 k r i + ... ) . (S.2)This order of expansion is high enough to describe properlythe shape of an individual lattice well (in order to considerthe effects of tunneling, which are important only at lowlattice depth, the potential should be expanded to ahigher order, at least to the 20 th , making the problemcomputationally much longer to solve). Introducing ω =2 √ sE rec / (cid:126) we can rewrite the Hamiltonian as H = p m + p m + 12 mω r + 12 mω r + V int ( r − r ) + V anh ( r , r ) , (S.3)where V anh ( r , r ) contains the anharmonic terms comingfrom the expansion of the lattice potential. By makingthe substitution R = r + r √ and r = r − r √ , we can writethe Hamiltonian in terms of center-of-mass { R , P } andrelative { r , p } coordinates: H = P m + p m + 12 mω R + 12 mω r + V int ( r )+ V anh ( R , r ) . (S.4)The harmonic+interaction part of the Hamiltonian (in-cluding all the terms in Eq. (S.4) except the last one)was solved analytically by Busch et al. [S2]. This workshowed that the interaction energy for two atoms in theground state of the trap saturates at the energy of thefirst excited vibrational state in the limit of a → + ∞ . Fora true lattice potential, the anharmonic terms V anh ( R , r )couple the relative and center-of-mass motion, makingthe problem impossibile to be solved analytically.In order to extend the results of Busch et al. to thecase of a lattice potential well, we diagonalize numericallythe full Hamiltonian in Eq. (S.4) written on a basisof wavefunctions which are solutions of the harmonicproblem: Ψ N,L,M ( R ) φ n,l,m ( r ) , (S.5)where N ( n ) is the principal quantum number and L, M ( l, m ) are the angular momentum quantum numbers forthe center-of-mass (relative) motion. For the relativewavefunctions φ n,l,m ( r ) we choose the solutions of the 3Disotropic harmonic oscillator for l (cid:54) = 0, while for l = 0 wetake the interacting wavefunctions derived in Ref. [S2] φ ( r ) = A exp (cid:18) − r a ho (cid:19) Γ (cid:18) − E (cid:126) ω + 34 (cid:19) U (cid:18) − E (cid:126) ω + 34 , , r a ho (cid:19) , (S.6)where U are the confluent hypergeometric functions, A is a normalization factor, a ho = (cid:112) (cid:126) /mω is the harmonicoscillator length and E is the total energy, given by thesolution of the equation √ (cid:0) − E (cid:126) ω + (cid:1) Γ (cid:0) − E (cid:126) ω + (cid:1) = a ho a . (S.7)For the center-of-mass wavefunctions Ψ N,L,M ( R ) we al-ways choose the solutions of the harmonic oscillator prob-lem. We found that taking N max = n max = 4 (corre-sponding to 196 states forming the basis) is sufficient toensure convergence in the calculation of the ground-stateenergy.In Fig. S2 we plot the results for the interaction energy(defined as the total energy minus the total energy inthe noninteracting case) as a function of the scatteringlength a for two values of the lattice depth s = 11 and s = 30. The curves are based on three different mod-els: 1) our model, containing anharmonic terms and thecoupling between relative and center-of-mass motion ( U , FIG. S2. a) Interaction energies for two particles in a latticesite, calculated for two lattice depths s = 11 and s = 30according to three different models (see text). The interactionenergy U calculated with our model is well approximated bythe usual Hubbard relation U Hub at small scattering length a . b) The same results are plotted up to larger values of a .For large a the interaction energy U saturates at the energydifference between the ground and the first-excited latticeband, here represented by the grey regions (the width of theseregions reflects the finite width of the energy bands caused bytunnelling). solid lines); 2) the model of Ref. [S2], containing onlythe harmonic part of the potential ( U Busch , dotted lines);3) the usual expression for the interaction energy in theHubbard model [S1], which takes into account the fulllattice potential and depends linearly on a ( U Hub , dashedlines). In addition, the first band gaps for s = 11 and for s = 30 are shown. The interaction energy derived fromour model saturates at the first excited band of the latticefor large values of the scattering length and, for low a ,it is well approximated by the usual Hubbard expression U Hub . Instead, the U Busch curves saturate at a higherenergy, coincident with (cid:126) ω = 2 √ sE rec . FIG. S3. Franck-Condon factor F ( a , a ) = (cid:104) ψ ( a ) | ψ ( a ) (cid:105) describing the overlap of the ground-state wavefunctions fortwo different scattering lengths a and a . By evaluating the eigenstates of the interacting system,we can compute the Franck-Condon factors F that mustbe put in the off-diagonal elements of the matrix in Eq. (3)of the main text. The Franck-Condon factor F is definedas the overlap (cid:104) ψ ( a ) | ψ ( a ) (cid:105) where ψ ( a ) is an eigenstate ofthe Hamiltonian in Eq. (S.4) with the scattering length a .In Fig. S3 we plot the Franck-Condon factors between twoground states of the system for different scattering lengths.We can see that along the diagonal (where a = a ) theFranck-Condon factor is unity, as expected since the twostates coincide, while it drops down to ∼ . S.III. ULTRANARROW 578 nm LASER
The laser radiation at 578 nm used to excite the atomsto the metastable | e (cid:105) = P state is produced by second-harmonic generation of the 1156 nm infrared light emittedby a quantum dot laser. Employing a bow-tie opticalcavity to enhance the efficiency of the frequency doublingprocess, we obtain up to 50 mW of 578 nm light. A smallpart of this radiation is coupled into a 10 cm long ULE(Ultra-Low Expansion) glass cavity, originally employedto realize the clock laser for the Yb optical lattice clockexperiment running at INRIM [S3].The laser frequency is locked to the ULE cavity witha 500 kHz bandwidth feedback system, and the in-looplinewidth of the laser can be estimated from the frequencynoise spectrum to be below 10 Hz [S4]. The ULE cav-ity, surrounded by a thermally-stabilized copper shield,is located in a 10 − mbar vacuum chamber to greatlyreduce its mechanical and thermal sensitivity. The wholesystem is placed on an antivibration platform to furtherreduce seismic noise, and is enclosed in an isolation boxto decouple the system from the lab environment.The long-term drift of the cavity has been characterizedand is corrected excluding a residual drift on the order of100 Hz/day. However, erratic fluctuations of some Hz/s,that we ascribe to an imperfect thermal stabilization of theULE cavity, limit our mid-term stability and represent oneof the limitations in the observation of long spin-exchangeoscillations. [S1] D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P.Zoller, Phys. Rev. Lett. , 3108 (1998).[S2] T. Busch, B.-G. Englert, K. Rza˙zewski, and M. Wilkens,Found. Phys. , 549 (1998). [S3] M. Pizzocaro et al., IEEE T. Ultrason. Ferr. , 3 (2012).[S4] J. L. Hall and M. Zhu, An Introduction to Phase-StableOptical Sources , in