Tunable Single-Ion Anisotropy in Spin-1 Models Realized with Ultracold Atoms
Woo Chang Chung, Julius de Hond, Jinggang Xiang, Enid Cruz-Colón, Wolfgang Ketterle
TTunable Single-Ion Anisotropy in Spin-1 Models Realized with Ultracold Atoms
Woo Chang Chung, ∗ Julius de Hond, ∗ Jinggang Xiang ( 项 晶 罡 ), Enid Cruz-Col´on, and Wolfgang Ketterle Research Laboratory of Electronics, MIT-Harvard Center for Ultracold Atoms, Department of Physics,Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA (Dated: January 6, 2021)Mott insulator plateaus in optical lattices are a versatile platform to study spin physics. Usingsites occupied by two bosons with an internal degree of freedom, we realize a uniaxial single-ionanisotropy term proportional to ( S z ) , which plays an important role in stabilizing magnetism forlow-dimensional magnetic materials. Here we explore non-equilibrium spin dynamics and observea resonant effect in the spin anisotropy as a function of lattice depth when exchange coupling andon-site anisotropy are similar. Our results are supported by many-body numerical simulations andare captured by the analytical solution of a two-site model. Mott insulators of ultracold atoms in optical latticescomprise a widely used platform for quantum simula-tions of many-body physics [1]. Since the motion ofatoms is frozen out, the focus is on magnetic ordering andspin dynamics in a system with different (pseudo-)spinstates. As suggested in 2003, Mott insulators with two-state atoms realize quantum spin models with tunableexchange interactions and magnetic anisotropies [2, 3].Experimental achievements for spin-1/2 systems includethe observation of antiferromagnetic ordering of fermions[4] and the study of spin transport in a Heisenberg spinmodel with tunable anisotropy of the spin-exchange cou-plings [5]. Spin dynamics for
S > i, j proportional to (cid:80) (cid:104) ij (cid:105) S ki S kj (where k ∈ { x, y, z } ) and to Zeeman couplings to effec-tive magnetic fields, proportional to (cid:80) i S zi . For Mottinsulators with two or more atoms per site, the Hubbardmodel has direct on-site interactions which can give riseto a nonlinear term D (cid:80) i ( S zi ) , where D is the so-calledsingle-ion anisotropy constant. ( S z ) terms, which arepresent for S ≥ FIG. 1. Experimental sequence for the measurement of spinalignment and doublon fractions. (i) The lattices are rampedup to initialize a single-component Mott insulator with a max-imal site occupancy of two. (ii) Microwave pulses prepare asuperposition of two hyperfine states ( | a (cid:105) − i | b (cid:105) ) / √
2. Ramp-ing down the longitudinal lattice initiates spin exchange dy-namics. (iii) Ramping up the lattices stops the exchange dy-namics. Microwave pulses transfer the two components to apair of states with a Feshbach resonance. (iv) Either | ab (cid:105) dou-blons or all doublons are removed with the help of Feshbach-enhanced inelastic losses. Remaining atoms are transferredback to the F = 1 hyperfine states and are counted via ab-sorption imaging to measure N p or N d . systems is demonstrated by various studies on differentplatforms [17–19].In this Letter, we use cold atoms in optical latticesto implement a spin-1 Heisenberg Hamiltonian using aMott insulator of doubly occupied sites and demonstrateunique dynamical features of the single-ion anisotropy.For spin-exchange interactions studied thus far in opticallattices, the only time scale for dynamics is second-ordertunneling (i.e. superexchange) which monotonically slowsdown for deeper lattices. In contrast, as we show here, a r X i v : . [ c ond - m a t . qu a n t - g a s ] J a n the single-ion anisotropy introduces a new time scale, andwe find a dynamical behavior which is faster in deeperlattices, due to a resonance effect when the energies ofsuperexchange and single-ion anisotropy are comparable.We present a protocol to directly measure theanisotropy in the spin distribution and find pronouncedtransient behaviour of this quantity when the resonancecondition is met. Transients change sign along with thethe single-ion anisotropy. We find good agreement withtheoretical simulations, and explain the most salient fea-tures using a two-site model with an exact solution.In the Mott insulator regime the optical lattices aresufficiently deep such that first-order tunneling is sup-pressed, and exchange processes are only possible viasecond-order tunneling. For two atoms per site, theBose–Hubbard Hamiltonian is approximated by an ef-fective spin Hamiltonian H = − J (cid:88) (cid:104) ij (cid:105) S i · S j + D (cid:88) i ( S zi ) − B (cid:88) i S zi , (1)where S i are spin-1 operators, (cid:104) ij (cid:105) are pairs of nearest-neighboring sites, J is the exchange constant, D is theuniaxial single-ion anisotropy constant, and B is a ficti-tious magnetic bias field. The spin-1 operators are re-lated to the boson creation/annihilation operators via S zi = ( a † i a i − b † i b i ) / S + i = a † i b i , S − i = b † i a i underthe constraint a † i a i + b † i b i = 2, where a i and b i areboson annihilation operators at site i for state a andstate b respectively. In terms of the tunneling ampli-tude t and interaction energies U σσ (cid:48) : J = 4 t /U ab and D = ( U aa + U bb ) / − U ab , where U σσ (cid:48) represents theon-site interaction energy between atoms in two states σ, σ (cid:48) ∈ { a, b } . The term proportional to B can bedropped if the total longitudinal magnetization (cid:80) i S zi is constant, as it is in the experiment.For the species studied here, Rb, all U σσ (cid:48) differ byless than 1%, and therefore all spin exchange couplingsare almost equal resulting in isotropic spin Hamiltoniansfor site occupancy ν = 1. However, for ν = 2, we cantune the relevant anisotropy parameter D/J over a largerange of values, because J decreases exponentially withlattice depth, while D —a differential on-site energy—slowly increases.The experimental sequence begins by preparing aBose–Einstein condensate (BEC) of Rb atoms in the | F = 1 , m F = − (cid:105) hyperfine state inside a crossed op-tical dipole trap. It proceeds by loading the BEC intoa deep three-dimensional optical lattice formed by retro-reflected lasers with wavelengths of λ = 1064 nm. Thelattices are ramped to final depths of 30 E R in 250 ms,where E R = h / (cid:0) mλ (cid:1) is the recoil energy for atomicmass m . Experimental parameters are chosen to maxi-mize the size of the ν = 2 Mott-insulator plateau withoutsignificant population of sites with ν = 3 [see Fig. 1(i)and the Supplemental Material]. To allow for spin dynamics, all atoms are rotated intoan equal superposition of two hyperfine states ( | a (cid:105) − i | b (cid:105) ) / √ D arerealized with the pairs | a (cid:105) = | , − (cid:105) , | b (cid:105) = | , (cid:105) , and | a (cid:105) = | , − (cid:105) , | b (cid:105) = | , (cid:105) , respectively (see the Supple-mental Material). The spin exchange dynamics in one-dimensional chains is initiated by a 3-ms quench, duringwhich we ramp down the longitudinal lattice to a vari-able depth, while the transverse lattices are ramped upto 35 E R [Fig. 1(ii)]. After a variable evolution time,the final spin configuration is “frozen in” by ramping thelongitudinal lattice to 35 E R as well [Fig. 1(iii)].Our observable for the anisotropy in the spin distribu-tion is the longitudinal spin alignment A = S ( S + 1) − (cid:104) ( S z ) (cid:105) , measured in the ν = 2 plateau. (cid:104) ( S z ) (cid:105) = (cid:80) Ni =1 (cid:104) ( S zi ) (cid:105) /N is the average on-site longitudinal spincorrelation. A is defined to be zero for a random dis-tribution of spins. Since S z = 1 , , − | aa (cid:105) , | ab (cid:105) and | bb (cid:105) doublons, respectively, A can be obtained bymeasuring the relative abundance of the different dou-blons. Specifically, we refer to the fraction of | ab (cid:105) dou-blons as the “spin-paired doublon fraction” f . Since (cid:104) ( S z ) (cid:105) = 1 − f , we obtain A = 3 f −
1. The doublonstatistics can be measured by selectively introducing afast loss process that targets a specific type of doublon,and by comparing the remaining total numbers of atoms,which are measured via absorption imaging. Specifically,if N a is the average total atom number in the wholecloud, N p the average number of remaining atoms af-ter removing | ab (cid:105) doublons, and N d the average num-ber of remaining atoms after removing all doublons, then f = ( N a − N p ) / ( N a − N d ) [Fig. 1(iv)]. Fast losses of dou-blons are induced by transferring the atoms to hyperfinestates for which inelastic two-body loss is enhanced neartwo narrow Feshbach resonances around a magnetic fieldof 9 G [20] (also see the Supplemental Material). Since f and A are obtained from the ratio of differences in atomnumbers, good atom number stability in the experiment(the deviation from mean being typically < A with sufficiently small uncertainties.For the initial state, f = 1 / A = 1 /
2. Overtimes that are long compared to spin exchange time scale¯ h/J , heating processes drive the system towards thermalequilibrium with A = 0. At short times, coherent spindynamics is observed: If D is negative, the | aa (cid:105) and | bb (cid:105) doublons are energetically favorable, and we expect f and A to decrease. If D is positive, the | ab (cid:105) doublons are fa-vorable and we expect f and A to increase. If D is zero,the system is described by an isotropic spin-1 HeisenbergHamiltonian of which the initial state is an eigenstate.By fixing the hold time and scanning the value of thelattice depth for the spin chains, we can monitor the im-pact of D/J on the dynamical change in A . For positive(negative) D , we chose a hold time of 70 ms (25 ms). 6 S L Q D O L J Q P H Q W A / D W W L F H G H S W K ( E R ) 6 S L Q D O L J Q P H Q W A 6 L Q J O H L R Q D Q L V R W U R S \ D (1/ J ) FIG. 2. Transient enhancement and reduction of the spinalignment A by coherent spin dynamics. The change in A isstrongest when | D/J | ∼
2. Measurements were done for bothpositive (top) and negative (bottom) values of
D/J . Theatoms were held for 70 ms and 25 ms, respectively (also seeFig. 4). The top axis in both figures indicates the
D/J ra-tio. Solid lines are the results of MPS-TEBD calculations.The error bars represent the standard error of the mean for A , obtained by error propagation after averaging three mea-surements for each of N a , N p , and N d . For the lowest latticedepths, the spin model may not fully represent the Bose–Hubbard model. These hold times are chosen to be comparable to ¯ h/J when | D/J | ∼ | D/J | (cid:28) | D/J | (cid:29) A stays near its initial value of 1 /
2. However, when
D/J ∼
2, which corresponds to a longitudinal latticedepth of 14 E R (11 E R ) for positive D (negative D ), wesee that A reaches a maximum (minimum). This non-monotonic change of A with lattice depth is indicativeof the interplay between spin exchange and single-ionanisotropy. In addition, we observe that the change in A is smaller for positive D than for negative D .Several aspects of the observed spin dynamics can becaptured by a two-site model. Although states on twospin-1 sites span a 9-dimensional Hilbert space, we can re-duce the spin dynamics to a beat note between two states.Since exchange interactions do not change the total mag-netization (cid:80) Ni =1 S zi , the Hilbert space factorizes to sub-spaces with the same total magnetization (although S zi can differ within a subspace). Furthermore, the initialsuperposition state is symmetric between the left and +( ) / p ˆ x ˆ y +( ) / p ˆ x ˆ y 7 L P H /J 6 S L Q D O L J Q P H Q W A D/J = 0 . 7 L P H /J 6 S L Q D O L J Q P H Q W A D/J = − . FIG. 3. Coherent spin oscillations in a two-site model. Whilethe full basis contains nine states, the oscillations in the spinalignment A involve only a 2 × J = 0 the effective magnetic field points along ˆ z , andthe purely azimuthal precession will not change A . If J > A (right). The frequency of the oscillation, in units of J/ ¯ h ,is given by Ω = (cid:113) D/J + 4 (
D/J ) , and its amplitude is2 ( D/J ) / (cid:2) D/J + 4 (
D/J ) (cid:3) . This shows that the direc-tion of oscillation depends on the sign of D/J (compare topand bottom panels). Note that while the initial value of A for this subspace is 1, the contribution of other states sets theinitial A of the whole system to 1/2. right wells, and any change in A comes from the two cou-pled states: | ab (cid:105) L | ab (cid:105) R and ( | aa (cid:105) L | bb (cid:105) R + | bb (cid:105) L | aa (cid:105) R ) / √ A are 2 and − J and D suddenly changes the strength and theorientation of this external field and induces a precessionof the state vector around the new external field (see theSupplemental Material). This results in an oscillationof A with amplitude 2 ( D/J ) / (cid:104) D/J + 4 (
D/J ) (cid:105) .This function has local extrema for D/J = ± /
2, but isnot symmetric around
D/J = 0. This explains the non-monotonic behaviour as a function of lattice depth, andshows why the contrast is smaller for positive
D/J thanfor negative
D/J .One would expect that for a larger number of sites,additional precession frequencies appear, turning the pe-riodic oscillation for two sites into a relaxation towardan asymptotic value. Comparison between the two-sitemodel and a many-site model numerically simulated us-ing the time-evolution block-decimation algorithm formatrix-product states (MPS-TEBD) shows that the ini-tial change in A is indeed well captured by the two-site model (see the Supplemental Material). Due to thespin dynamics, the system evolves from a product stateinto a highly correlated state with entanglement betweensites; this has been the focus of recent theoretical works[21, 22]. In the two-site model, the von Neumann en-tanglement entropy can reach up to ∼ . × ln (3) dueto the interplay between single-ion anisotropy and ex-change terms. This corresponds to an almost maximallyentangled state since ln (3) is the maximum entropy fora spin-1 site.To illustrate that changes in the spin alignment A re-sult from competition between the exchange interactionand the single-ion anisotropy, we study the time evolutionof A at two different lattice depths (Fig. 4). For positive D , MPS-TEBD simulations predict very little change in A at a lower lattice depth, where the exchange constant isrelatively large, but the anisotropy is small, while it pre-dicts a noticeable change in A at a higher lattice depth,where the exchange constant and the anisotropy term be-comes comparable. While the simulation predicts equi-libration of A to an asymptotic value (thin lines), mea-surements show that it decays toward a lower value forpositive D and does not decrease as much as the simu-lation prediction for negative D . The measurements areconsistent with the fact that at high spin temperature,the spin distribution becomes isotropic and A vanishes.Indeed, when we ramp down the lattices and retrieve aBose–Einstein condensate, we observe a significant re-duction of condensate fraction after 300 ms.In conclusion, we have implemented a spin-1 Heisen-berg model with a single-ion anisotropy using the ν = 2plateau of a Mott insulator, and have observed the subtleinterplay between spin exchange and on-site anisotropyin coherent spin dynamics. Much larger values of D canbe implemented with spin-dependent lattices, which willallow us to observe much faster anisotropy-driven dy-namics, and will also enable mapping out the phase dia-gram of the anisotropic spin Hamiltonian [11]. It shouldalso be noted that it is possible to change the sign of J with the gradient of an optical dipole potential [23, 24],which will permit exploration of the antiferromagneticsector with bosons. Interesting dynamical features ofanisotropic spin models have been predicted [25] includ-ing transient spin currents, implying counterflow super-fluidity.Regarding quantum simulations, single-ionanisotropies play a crucial role in magnetic materi-als (e.g. monolayers containing chromium [26, 27]). Insuch materials, crystal field effects lift the degeneracyof d -orbitals, and spin-orbit interaction transfers this E r D / J = 0.1 7 L P H P V E r D / J = 2.2 6 S L Q D O L J Q P H Q W A E r D / J = 0.7 7 L P H P V E r D / J = 3.4 6 S L Q D O L J Q P H Q W A FIG. 4. Coherent dynamics of the spin alignment A after aquench in D/J . Varying the hold time at characteristic latticedepths for both positive and negative values of
D/J (top andbottom pairs of panels, respectively) reveals that strong tran-sients in A only occur at intermediate lattice depth for which D and J are comparable. The vertical, dash-dotted lines indi-cate the hold times used for these pairs in Fig. 2. Dashed linesare the results of the MPS-TEBD simulation. The shadedregions denote the MPS-TEBD results with ± . E R uncer-tainty in the lattice depths, and include exponential decaytowards a thermal spin state with A = 0 with empirical 1 /e times of 400 ms ( D >
0) and 100 ms (
D < anisotropy to the electronic spins responsible for themagnetism [28]. Here we have simulated this anisotropyby selecting a pair of atomic hyperfine states wherethe interspecies scattering length is different from theaverage of the intraspecies values. This illustratesthe potential for ultracold atoms in optical lattices toimplement idealized Hamiltonians describing importantmaterials.We thank Colin Kennedy, William Cody Burton andWenlan Chen for contributions to the development ofexperimental techniques, and Ivana Dimitrova for crit-ical reading of the manuscript. We acknowledge supportfrom the NSF through the Center for Ultracold Atomsand Grant No. 1506369, ARO-MURI Non-equilibriumMany-Body Dynamics (Grant No. W911NF14-1-0003),AFOSR-MURI Quantum Phases of Matter (Grant No.FA9550-14-1-0035), ONR (Grant No. N00014-17-1-2253),and a Vannevar-Bush Faculty Fellowship. W.C.C. ac-knowledges additional support from the Samsung Schol-arship. ∗ These authors contributed equally to this work.[1] I. Bloch, J. Dalibard, and W. Zwerger, Reviews of mod-ern physics , 885 (2008).[2] L.-M. Duan, E. Demler, and M. D. Lukin, Physical Re-view Letters , 090402 (2003).[3] A. B. Kuklov and B. V. Svistunov, Phys. Rev. Lett. ,100401 (2003).[4] A. Mazurenko, C. S. Chiu, G. Ji, M. F. Parsons,M. Kan´asz-Nagy, R. Schmidt, F. Grusdt, E. Demler,D. Greiff, and M. Greiner, Nature , 462 (2017).[5] P. N. Jepsen, J. Amato-Grill, I. Dimitrova, W. W. Ho,E. Demler, and W. Ketterle, Nature , 403 (2020).[6] A. de Paz, A. Sharma, A. Chotia, E. Mar´echal, J. H.Huckans, P. Pedri, L. Santos, O. Gorceix, L. Vernac, andB. Laburthe-Tolra, Phys. Rev. Lett. , 185305 (2013).[7] M. Kitagawa and M. Ueda, Phys. Rev. A , 5138 (1993).[8] Y. Li, M. R. Bakhtiari, L. He, and W. Hofstetter, Phys.Rev. B , 144411 (2011).[9] Y. Li, L. He, and W. Hofstetter, Phys. Rev. A , 033622(2016).[10] E. Altman, W. Hofstetter, E. Demler, and M. D. Lukin,New Journal of Physics , 113 (2003).[11] J. Schachenmayer, D. M. Weld, H. Miyake, G. A.Siviloglou, W. Ketterle, and A. J. Daley, Phys. Rev.A , 041602(R) (2015).[12] F. D. M. Haldane, Physics Letters A , 464 (1983).[13] F. D. M. Haldane, Phys. Rev. Lett. , 1153 (1983).[14] F. D. M. Haldane, Rev. Mod. Phys. , 040502 (2017).[15] N. D. Mermin and H. Wagner, Phys. Rev. Lett. , 1133(1966).[16] J. Streˇcka, D. J´an, and L. ˇCanov´a, Chinese Journal ofPhysics , 329 (2008).[17] J. P. Renard, M. Verdaguer, L. P. Regnault, W. A. C.Erkelens, J. Rossat-Mignod, and W. G. Stirling, Euro-physics Letters (EPL) , 945 (1987).[18] P. Chauhan, F. Mahmood, H. J. Changlani, S. M. Kooh-payeh, and N. P. Armitage, Phys. Rev. Lett. , 037203(2020).[19] C. Senko, P. Richerme, J. Smith, A. Lee, I. Cohen,A. Retzker, and C. Monroe, Phys. Rev. X , 021026(2015).[20] A. M. Kaufman, R. P. Anderson, T. M. Hanna,E. Tiesinga, P. S. Julienne, and D. S. Hall, Phys. Rev.A , 050701(R) (2009).[21] I. Morera, A. Polls, and B. Juli´a-D´ıaz, Scientific Reports , 9424 (2019).[22] A. Venegas-Gomez, J. Schachenmayer, A. S. Buyskikh,W. Ketterle, M. L. Chiofalo, and A. J. Daley, QuantumScience and Technology , 045013 (2020). [23] I. Dimitrova, N. Jepsen, A. Buyskikh, A. Venegas-Gomez, J. Amato-Grill, A. Daley, and W. Ketterle,Phys. Rev. Lett. , 043204 (2020).[24] H. Sun, B. Yang, H.-Y. Wang, Z.-Y. Zhou, G.-X. Su, H.-N. Dai, Z.-S. Yuan, and J.-W. Pan, arXiv:2009.01426(2020).[25] A. Venegas-Gomez, A. S. Buyskikh, J. Schachenmayer,W. Ketterle, and A. J. Daley, Phys. Rev. A , 023321(2020).[26] C. Gong, L. Li, Z. Li, H. Ji, A. Stern, Y. Xia, T. Cao,W. Bao, C. Wang, Y. Wang, et al. , Nature , 265(2017).[27] C. Xu, J. Feng, H. Xiang, and L. Bellaiche, npj Compu-tational Materials (2018), 10.1038/s41524-018-0115-6.[28] D. Dai, H. Xiang, and M.-H. Whangbo, Journal of com-putational chemistry , 2187 (2008).[29] N. Marzari and D. Vanderbilt, Phys. Rev. B , 12847(1997).[30] W. Kohn, Physical Review , 809 (1959).[31] P. Cheinet, S. Trotzky, M. Feld, U. Schnorrberger,M. Moreno-Cardoner, S. F¨olling, and I. Bloch, Phys.Rev. Lett. , 090404 (2008).[32] G. K. Campbell, J. Mun, M. Boyd, P. Medley, A. E.Leanhardt, L. G. Marcassa, D. E. Pritchard, andW. Ketterle, Science , 649 (2006).[33] D. M. Stamper-Kurn and M. Ueda, Reviews of ModernPhysics , 1191 (2013).[34] J. Hauschild and F. Pollmann, SciPost Physics LectureNotes , 5 (2018).[35] G. Vidal, Phys. Rev. Lett. , 040502 (2004). SUPPLEMENTAL MATERIALCalculation of D and J The superexchange parameter J and single-ionanisotropy D were calculated using maximally localizedWannier functions for a simply cubic lattice [29, 30] andthe scattering lengths in Table I.The sign of D is important for the qualitative behav-ior. Of the F = 1 states, the only combination with D < | , − (cid:105) and | , (cid:105) states. Any pairinvolving the | , (cid:105) state has a positive value of D ; wechose the | , − (cid:105) and | , (cid:105) combination because it wasthe easiest to prepare from the initial | , − (cid:105) state. Asmentioned in the main text, the value of D is propor-tional to the various onsite interactions, which have alinear dependence on the scattering lengths. This meansthat D ∝ ( a aa + a bb ) / − a ab which equals − . a and0 . a for the two chosen pairs. Through the Wannierfunctions, D and J depend on the lattice depth, whichdependence is shown in Fig. 5. Confinement parameters
The three-dimensional lattice is created by retro-reflecting three 1064-nm wavelength laser beams. Thetwo horizontal beams have Gaussian beam waists of150 µ m, while the vertical lattice beam has a waist of270 µ m. During the entire experiment the atoms arebeing held in a crossed-beam optical dipole trap. Thisconsists of a vertical beam (which has isotropic trap fre-quencies of 2 π ×
24 Hz) intersecting a highly elongatedhorizontal beam that is at a 45 ◦ angle with respect tothe horizontal lattices. The latter primarily serves tohold the atoms against gravity, and it has trap frequen-cies of 2 π ×
13 Hz and 2 π ×
130 Hz along its horizontaland vertical axes, respectively.Using these parameters, we were able to calculatethe occupation statistics of the Mott insulator, and ob-tained plateau fractions analogous to those presented inRef. [31]. We desire a large ν = 2 Mott insulator plateau,while avoiding any population in the ν = 3 shell as thatwould interfere with the doublon measurements. Occa-sionally, we have monitored the population in the dif-ferent shells using clock-shift spectroscopy [32]. On aday-to-day basis, however, we use the total atom num-ber or the doublon fraction as indicators (note that ourdoublon detection scheme detects all the atoms on siteswith ν ≥ .
5, and the atom number below 40 × ; for theseparameters the population in ν = 3 should be negligible. J / h + ] | a = |1, 1 | b = |1, 1| a = |1, 1 | b = |1, 0 / R Q J L W X G L Q D O O D W W L F H G H S W K ( E R ) D / h + ] FIG. 5. Values of D and J as a function of longitudinal latticedepth. The results are based on the scattering lengths givenin Table I, and assume transverse lattice depths of 35 E r . | , − (cid:105) | , (cid:105) | , (cid:105)| , − (cid:105) | , (cid:105) | , (cid:105) a calculated usingthe values tabulated in Ref. [33]. State preparation & doublon measurement
The initial state is prepared by a diabatic Landau–Zener sweep from the initial | , − (cid:105) state to the | , (cid:105) state. The sweep parameters are set in such a way thatwe robustly create an equal superposition of the twostates. Depending on whether we want to probe pos-itive or negative D/J we either transfer the populationfraction in | , (cid:105) to | , (cid:105) using a π pulse (which has smallsensitivity to magnetic-field fluctuations), or to | , (cid:105) us-ing an adiabatic Landau–Zener sweep.As described in the main text, the doublon statis-tics are derived from three separate measurements of theatom number, two of them after inducing selective lossesthat depend on the doublon type. To measure the totaldoublon fraction, all doublons are removed, regardless oftheir internal states. Dipolar relaxation is too slow, soa Feshbach resonance between the | , (cid:105) and | , (cid:105) statescan be used. For this, the | , − (cid:105) component of the pairis transferred to the | , (cid:105) state using a Landau–Zenersweep, while the other pair component is left in or putinto the | , (cid:105) state. The pairs are removed by modulat-ing the magnetic bias field around the narrow Feshbachresonance at 9 .
045 G [20]. Since the composition of thepairs we want to remove is arbitrary (they can be either | aa (cid:105) , | ab (cid:105) , or | bb (cid:105) ), we employ a diabatic Landau–Zenersweep between | , (cid:105) and | , (cid:105) states simultaneously withthe bias modulation, to make sure any doublon spendssome time in the Feshbach pair state in order to be re-moved. In practice, a removal time of 80 ms is sufficient.In order to specifically remove paired doublons (i.e.those of the | ab (cid:105) type), we transfer the | , − (cid:105) componentof the pair to the | , − (cid:105) state, and ensure that the othercomponent is in the | , (cid:105) state. To remove these pairs,the bias field is modulated around the 9 .
092 G Feshbachresonance between the | , (cid:105) and | , − (cid:105) states [20]. Two-site model
In the limit of two sites, the spin Hamiltonian (1) re-duces to H = − J S · S + D (cid:104) ( S z ) + ( S z ) (cid:105) . (2)The initial state is a product state between site 1 andsite 2: | Ψ (cid:105) = | ψ (cid:105) ⊗ | ψ (cid:105) , where the single-site state isgiven by: | ψ (cid:105) = (cid:18) | a (cid:105) − i | b (cid:105)√ (cid:19) atom 1 ⊗ (cid:18) | a (cid:105) − i | b (cid:105)√ (cid:19) atom 2 = 12 (cid:16) | (cid:105) − i √ | (cid:105) − |− (cid:105) (cid:17) (3)The full Hilbert space describing the two spin-1 sites isnine-dimensional. However, the Hamiltonian is block di-agonal in the total spin projection, S z + S z , and also withregard to odd and even symmetry between the two sites.For the state prepared initially, all the dynamics takesplace in the symmetric S z + S z = 0 subspace, which con-tains only two states: (cid:8) ( | , − (cid:105) + |− , (cid:105) ) / √ , | , (cid:105) (cid:9) .The Hamiltonian is given by H = (cid:18) J + 2 D −√ J −√ J (cid:19) . (4)The projection of the initial state into this subspace is | ψ (cid:105) = (cid:114)
16 ( | , − (cid:105) + |− , (cid:105) ) + (cid:114) | , (cid:105) , (5)also see Fig. 3. Since the | , (cid:105) state has ( S zi ) = 0,and the ( | , − (cid:105) + |− , (cid:105) ) / √ S zi ) = 1, a Rabioscillation between them leads to an oscillation of thespin alignment A . Note that the components of the initialstate in other subspaces contribute a constant value to A .Inspection of the Hamiltonian (4) identifies J + 2 D asa z field, which is added to an x field equal to √ J . In adeep lattice with J ∼
0, the field is parallel to the z axis, Time (ms) L a tt i c e d e p t h ( E r ) Time (ms) L a tt i c e d e p t h ( E r ) Sp i n a li g n m e n t A D / J -0.5-2-5-20-50 D / J -0.6-0.30.00.3 Sp i n a li g n m e n t A FIG. 6. Time evolution of the spin alignment A for vari-ous lattice depths, calculated using the TEBD algorithm formatrix-product states. The top and bottom figures are calcu-lated for pairs with positive ( | , − (cid:105) and | , (cid:105) ) and negative( | , − (cid:105) and | , (cid:105) ) values of D/J , respectively. The solid linesindicate the inverse (lattice depth dependent) Rabi frequencyof Eq. (6) times π √
2, showing that the initial behavior isdictated by nearest neighbors. but lowering the lattice adds an x field, which tilts thefield vector and initiates a precession of the state vectoraround it (see Fig. 3).The Rabi frequency of this oscillation is given byΩ = (cid:112) J + 4 JD + 4 D / ¯ h, (6)while the amplitude of the oscillation in A is 2 JD/ Ω ,which is maximized for | D/J | = 3 / Matrix-product state simulations
We implemented the time-evolving block decimationalgorithm for matrix-product states (MPS-TEBD) [34,35] on 100 sites, using a maximum bond dimension of20. This was found to give results consistent with pub-lished data [25]. The modest bond dimension is suffi-cient because the transient behavior in A occurs withina few exchange times (¯ h/J ), during which correlationsonly build up between clusters of sites. This has theadditional benefit that the calculation can be run on adesktop computer.The simulated evolution of the spin alignment A asa function of lattice depth is shown in Fig. 6. Theseresults form the basis of the simulations presented in themain text. Comparing to the two-site model, we observethat the early time behavior is dominated by nearest-neighbor physics. Specifically, the first minima seen in Fig. 6 occur at the period of the Rabi oscillation given inEq. (6) divided by √ π/ √
2Ω at the lattice depth where | D/J | ∼ /
2. This number equals 67 and 17 ms for thepositive and negative