Abstract
Simple models of interacting spins have an important role in physics. They capture the properties of many magnetic materials, but also extend to other systems, such as bosons and fermions in a lattice, gauge theories, hightemperature superconductors, quantum spin liquids, and systems with exotic particles such as anyons and Majorana fermions^{1,2}. To study and compare these models, a versatile platform is needed. Realizing such systems has been a longstanding goal in the field of ultracold atoms. So far, spin transport has only been studied in systems with isotropic spin–spin interactions^{3,4,5,6,7,8,9,10,11,12}. Here we realize the Heisenberg model describing spins on a lattice, with fully adjustable anisotropy of the nearestneighbour spin–spin couplings (called the XXZ model). In this model we study spin transport far from equilibrium after quantum quenches from imprinted spinhelix patterns. When spins are coupled only along two of three possible orientations (the XX model), we find ballistic behaviour of spin dynamics, whereas for isotropic interactions (the XXX model), we find diffusive behaviour. More generally, for positive anisotropies, the dynamics ranges from anomalous superdiffusion to subdiffusion, whereas for negative anisotropies, we observe a crossover in the time domain from ballistic to diffusive transport. This behaviour is in contrast with expectations from the linearresponse regime and raises new questions in understanding quantum manybody dynamics far away from equilibrium.
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The data that support the findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
We thank M. D. Lukin, N. Y. Yao and M. Knap for discussions, Y. K. Lee for experimental assistance, as well as C. M. E. Paus for sharing computing resources, and J. de Hond for comments on the manuscript. We acknowledge support from the NSF through the Center for Ultracold Atoms and grant number 1506369, NSF (grant number OAC193471), AROMURI NonEquilibrium ManyBody Dynamics (grant number W911NF1410003), AFOSRMURI Photonic Quantum Matter (grant number FA95501610323), AFOSRMURI Quantum Phases of Matter (grant number FA95501410035), ONR (grant number N000141712253), ARO (grant number W911NF2010163), the Vannevar Bush Faculty Fellowship, and the Gordon and Betty Moore Foundation EPiQS Initiative (grant number GBMF4306).
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P.N.J., J.A.G., I.D. and W.K. conceived the experiment. P.N.J., J.A.G. and I.D. developed the experimental setup and took the data. P.N.J. led the data analysis. W.W.H. and E.D. developed the theoretical analysis. W.W.H. performed the numerical simulations. All authors discussed the results and contributed to the writing of the manuscript.
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Extended data figures and tables
Extended Data Fig. 1 Determination of the Heisenberg parameters.
a, Corrections for tunnelling \((\tilde{t}\tilde{t}{}^{(0)})/\tilde{t}{}^{(0)}\) (green), onsite interactions (U −U^{(0)})/U^{(0)} (blue), superexchange (J − J^{(0)})/J^{(0)} (purple) and offsite interactions −2V/J^{(0)} (orange), where \(\tilde{t}\)^{(0)}, U^{(0)} and \({J}^{(0)}=4{(\tilde{t}{}^{(0)})}^{2}/{U}^{(0)}\) are the uncorrected values and \(\tilde{t}\), U and \(J\,=\,4{\tilde{t}}^{2}/U\) include corrections, at a lattice depth of 13E_{R} (solid line) and 9E_{R} (dotted line). b, As a function of magnetic field B we show the scattering lengths a_{⇈} (blue), a_{⇅} (purple) and a_{⇊} (orange) measured in our previous work (points)^{33}. Here a_{0} is the Bohr radius. c, d, We interpolate the data in b using hyperbolic fits (solid lines) and calculate values for the transverse coupling constant J_{xy} (purple), the longitudinal coupling constant J_{z} (yellow) and the anisotropy Δ = J_{z}/J_{xy} (black), without corrections (dashed line) and including corrections (solid line) for a lattice depth of 11E_{R}. The excluded region (a_{⇈} > 700a_{0}) is around a Feshbach resonance in the ↑⟩ state near 845.4 G.
Extended Data Fig. 2 Contrast measurement.
a–d, The distribution of atoms in the ↑⟩ state. Every pixel is a local measurement of the column density (number of atoms per unit area). The y and z axes are displayed in units of the lattice spacing a = 0.532 μm. The images are projected (integrated) along the y direction from y = −30a to +30a to obtain the linear density (number of atoms per unit length). The resulting 1D distributions are fitted with \(f(z)=g(z)[1+{\mathscr{C}}\,\cos (Qz+\theta )]/2\) (solid line), where g(z) is a Gaussian envelope (dashed line), between z = ±54a. The data in a–d were measured at different evolution times t = 0 (a), 2.3ħ/J_{xy} (b), 6.3ħ/J_{xy} (c) and 12.0ħ/J_{xy} (d), for anisotropy Δ ≈ 0 and wavelength λ = 10.4a. The obtained contrast \({\mathscr{C}}(t)\) is shown in Fig. 2a. In general, we also normalize by the initial contrast \({\mathscr{C}}(0)\) to correct for finite optical imaging resolution. This is important for shorter wavelengths λ close to the optical resolution of 3 μm, where the measured contrast \({\mathscr{C}}(t)\) is reduced compared to the real contrast \(c(t)={\mathscr{C}}(t)/{\mathscr{C}}(0)\).
Extended Data Fig. 3 Dispersion relations.
For all positive anisotropies Δ ≥ 0, the time evolution of the contrast c(t) shows a damped oscillatory component, in addition to the overall exponential decay. For larger Δ, the oscillations become smaller. a–c, Decay and weak oscillation at the isotropic point Δ ≈ 1 measured for different wavelengths λ, at three different lattice depths 9E_{R} (orange), 11E_{R} (blue) and 13E_{R} (yellow). Solid lines are fits c(t) = [a_{0} + b_{0}cos(ωt)]e^{−t/τ} + c_{0} and dashed lines show the overall decay a_{0}e^{−t/τ} + c_{0}, around which the oscillations take place. The oscillations become more pronounced for short wavelengths λ, because the decay time (τ ∝ λ^{2}) decreases with smaller wavelength more strongly than the oscillation period (T ∝ λ). d, The oscillation frequencies follow linear dispersion relations ω(Q) = vQ shown for Δ = −0.12 (red), 0.35 (orange), 0.78 (yellow), 1.01 (blue) and 1.27 (light blue). e, The obtained velocities v decrease with increasing anisotropy Δ. For Δ = 1.58 (open symbol), oscillations are small and the measurement was limited to large values of Q, which precluded recording a full dispersion relation. We note that although the oscillations are difficult to discern by eye (for example, in a), especially for large anisotropies Δ and small wavevectors Q, the fitted oscillation frequencies ω all fall very well on linear dispersion relations, which demonstrates that those barely visible oscillations are real. The linear scaling ω(Q) = vQ persists even in the superdiffusive, diffusive and subdiffusive regimes, where the powerlaw scaling of the decay time constant τ ∝ Q^{−α} is strongly nonlinear. This small ballistic (oscillatory) component may be related to our initial condition of a spin helix, which in the mapping to lattice fermions is a 100% density modulation, which reduces scattering at early times.
Extended Data Fig. 4 Effect of finite hole concentration.
By varying the thermal fraction N_{th}/N of the Bose–Einstein condensate before it is loaded into the optical lattice, we vary the energy and entropy of the atoms in the spin chain, and therefore the concentration of holes. (For our conditions, doubly occupied sites have higher energies than holes). Measurements are shown here for Δ ≈ 0 and λ = 10.4a, at a lattice depth of 11E_{R}. a, Decay curves c(t) for varying hole concentrations ranging from low (blue) to high (orange) thermal fraction. Solid lines are fits c(t) = [a_{0} + b_{0}cos(ωt)]e^{−t/τ} + c_{0}. b, The background contrast c_{0} increases monotonously with thermal fraction N_{th}/N. A linear fit (solid line) extrapolates to c_{0} = 0.01(2), consistent with zero, for N_{th}/N = 0. This suggests that all of the background contrast is due to hole excitations. c, Higher hole concentrations suppress the oscillating fraction b_{0}/(a_{0} + b_{0}). d, Holes do not affect the oscillation period T = 2π/ω. e, Holes decrease the decay time τ, albeit slightly. b–e show that almost all of our measurements are not sensitive to a small thermal fraction, which is usually N_{th}/N ≤ 0.05 throughout this work. The behaviour shown in c and e is most probably caused by mobile holes in the central part of the Mott insulator. Indeed, numerical simulations of the \(\tilde{t}\)–J model reproduce such effects (Fig. 2a). Note though that for the isotropic case Δ ≈ 1, a previous work^{7} found a ~50% change in decay time when the hole concentration changed from 0 to 5%. Our numerical simulations (Extended Data Fig. 8b) do not show such strong sensitivity (for any anisotropy, even at Δ = 1), possibly owing to asymmetry in the onsite interactions (U_{⇈} ≠ U_{⇅} ≠ U_{⇊}) in our system. On the other hand, a finite background contrast (b) is probably caused by immobile holes located in the outer parts of the atom distribution where firstorder tunnelling is suppressed by the gradient of the (harmonic) trapping potential^{50}. Immobile holes disrupt spin transport, and so we expect that the imprinted spin modulation in these regions will not (or only very slowly) decay. f, g, The region with immobile holes is visible as a shell of low atomic density surrounding the Mott insulator in the in situ images for large hole concentration (f) and is absent for low hole concentration (g). The three curves in both f and g show the local contrast as a function of distance r from the centre of the atom cloud for the evolution times t = 0 (top), 2.7ħ/J_{xy} (middle) and 21.7ħ/J_{xy} (bottom). The two in situ images in both f and g are for t = 0 (top) and 21.7ħ/J_{x} (bottom). The dashed lines indicate contours of constant radius, r = 30a (f) and r = 20a (g).
Extended Data Fig. 5 Decay behaviour as a function of anisotropy.
a, b, Decay behaviour ranging from negative (a) to positive (b) anisotropy, for a fixed wavelength λ = 10.4a. Using Δ ≈ 0 as a reference point, we show how the temporal profile of the decay curve c(t) changes when we introduce positive or negative interactions. Every data point is an average of two measurements at lattice depths 11E_{R} and 13E_{R}. In a, from bottom to top: Δ = −0.12 (red), −0.59 (pink), −0.81 (yellow), −1.02 (blue), −1.43 (green) and −1.79 (purple). In b, from bottom to top: Δ = −0.13 (red), 0.08 (purple), 0.35 (pink), 0.55 (orange), 0.78 (yellow), 1.01 (blue), 1.27 (light blue) and 1.58 (green). Regardless of sign, for increasing Δ the decay always slows down and the revivals damp more quickly. However, there is a big difference in how this slowdown happens: for increasing positive interactions Δ > 0, the initial rate of decay decreases continuously (b); by contrast, for all negative interactions Δ < 0, the initial rate of decay stays constant (and is ballistic), coinciding with the Δ ≈ 0 case (a). It is only after a critical time t_{0} that the decay suddenly starts slowing down (and becomes diffusive) for times t > t_{0}. This critical time t_{0} decreases with increasing negative interaction strength Δ.
Extended Data Fig. 6 Collapse of decay curves for positive anisotropies.
All decay curves c(t) for wavelengths λ = 15.7a, 13.4a, 11.7a, 10.4a, 9.4a, 8.5a, 7.8a, 7.2a and 6.7a collapse very well into a single curve for all evolution times t, when time units are rescaled by λ^{α}, where the exponent α is a function of anisotropy Δ, both for experiment (points) and theory (solid lines). Experimental points were measured for lattice depths 9E_{R} (red), 11E_{R} (blue) and 13E_{R} (yellow). a, b, Ballistic regime (α = 1). c, Superdiffusion (α = 1.5). d, Diffusion (α = 2). e, f, Subdiffusion (α = 2.5, 3 for experiment and α = 3.5, 4.5 for numerical simulations; in f, experiments covered a reduced range λ ≤ 10.4a). For all anisotropies Δ ≥ 0 (a–f) the experimentally measured oscillation frequencies ω follow linear dispersion relations (Extended Data Fig. 3) and have a scaling behaviour different from the decay rates. However, such oscillations are small outside the ballistic regime α ≈ 1, and therefore only lead to a small deviation from the collapse behaviour. Note also the different timescales in experiments and simulations for Δ > 1.
Extended Data Fig. 7 Collapse at short times for negative anisotropies.
All decay curves c(t) for different wavelengths λ collapse into a single curve at early times, when time units are rescaled by λ (indicating ballistic behaviour). For later times the decay is diffusive with different scaling. a–c, Theory (from top to bottom: λ = 31.3a, 23.5a, 18.8a, 15.7a, 13.4a, 11.7a, 10.4a, 9.4a, 8.5a, 7.8a, 7.2a, 6.7a and 6.3a). The dotted lines are exponential fits \({{\rm{e}}}^{t/{\tau }_{{\rm{II}}}}\) to the diffusive regime and the time constants τ_{II} are shown in Extended Data Fig. 8a. d–f, Experiment (from top to bottom: λ = 18.8a, 13.4a, 10.4a, 8.5a, 7.2a and 6.3a). Every data point is an average of two measurements at lattice depths 11E_{R} and 13E_{R}. The black dashed line indicates the ballistic case Δ ≈ 0 (see Fig. 2c).
Extended Data Fig. 8 Powerlaw scalings (theory) and diffusion coefficients.
a, b, Decay time constants τ for different anisotropies Δ ranging from negative (a) to positive (b). Numerical results are shown in a for Δ = −1 (blue), −1.5 (green) and −2 (purple) and in b for Δ = 0 (red), 0.5 (orange), 0.85 (yellow), 1 (blue) and 1.5 (green). Solid lines are powerlaw fits (to the filled symbols). Open symbols are excluded from the fit owing to finitesize effects. Crossed symbols are results from \(\tilde{t}\)–J model simulations including 5% hole fraction. Fitted powerlaw exponents are shown in Fig. 3c. For positive anisotropies Δ ≥ 0 the decay time τ is defined as τ = τ′/ln(1/0.60) with c(τ′) = 0.60. For negative anisotropies Δ < 0, the decay time τ_{I} for short times (I) is defined as \({\tau }_{{\rm{I}}}=10{\tau }_{{\rm{I}}}^{{\prime} }\) with \(c({\tau }_{{\rm{I}}}^{{\prime} })=0.90\). For longer times (II), the decay time τ_{II} is obtained from exponential fits \({{\rm{e}}}^{t/{\tau }_{{\rm{II}}}}\) to the diffusive longtime tail (see dotted curves in Extended Data Fig. 7a–c). c, Diffusion coefficients for the diffusive longtime regime (II) obtained from theory (open symbols) and experiment (filled symbols). For negative anisotropies Δ < 0, values were determined from quadratic powerlaw fits 1/τ = DQ^{2} to the data points in a (theory) and Fig. 3a (experiment) for the diffusive regime (II). Note that for Δ ≥ 0 the system is only diffusive for Δ = +1, as shown in b (theory) and Fig. 3b (experiment). From the experimental diffusion coefficients, we estimate mean free paths δx using the velocity v = 0.76(1)v_{F} from the ballistic shorttime regime (I), and obtain δx = 3.35(15)a, 1.07(7)a and 0.66(4)a for Δ = −1.02, −1.43 and −1.79, respectively. d, Shorttime (t ≪ ħ/J_{xy}) decay constant τ = Γ^{−1} obtained from Taylor expansion of the contrast c(t) = 1 + Γ^{2}t^{2} + … as a function of Q, for Δ = 0 (red), 0.55 (orange), 0.85 (yellow), 0.95 (purple) and 1 (blue). For Δ < 1 all curves in the log–log plot asymptote to the same slope as Q → 0 (continuum limit), whereas there are deviations for larger wavevectors Q. For Δ = 1 the slope is instead different. This indicates that the powerlaw exponent α in τ ∝ Q^{−α} depends on the range of wavevectors Q used to determine it. e, Powerlaw exponents α determined for the shortwavelength regime between λ = 6a and 20a (filled symbols) as in experiments and numerics, and for the longwavelength regime between λ = 150a and 200a (open symbols) approaching the continuum limit. In the former case, the exponents show a smooth crossover from superballistic to diffusive as Δ → 1 similar to that in the experiments and numerics, whereas in the latter case the exponents show a sharp jump from ballistic to diffusive occurring exactly at Δ = 1.
Extended Data Fig. 9 Finitesize effects from the initial phase of the spin helix.
a, The time evolution of the contrast c(t) depends strongly on the initial phase θ, illustrated here by simulations for Δ = 0 and λ = 10.4a. b, c, The dynamics of the local magnetization \(\langle {S}_{i}^{z}(t)\rangle \) for phases θ = 0 (b) and π/2 (c) reveals that this arises owing to the reflection of ballistically propagating magnetization off the ends of the chain. Depending on the initial phase of the spin helix, the reflected magnetization interferes constructively or destructively with the pattern of the bulk magnetization.
Extended Data Fig. 10 Finitesize effects from the chain length.
a, Contrast c(t) obtained after a weighted average over all different chain lengths between L = 0 and 44a (shown in b), for Δ = 0 and λ = 10.4a. The averaged dynamics (orange, yellow, blue) shows almost no dependence on the phase θ, in contrast to the dynamics determined from a single chain length L = 40a (Extended Data Fig. 9a). Also overlaid are the contrasts for a fixed chain length (L = 40a) averaged over all initial phases 0 ≤ θ < 2π (black solid line), and averaged over only the two phases θ = 0 and π/2 (black dashed line). The close agreement implies that averaging over either chain lengths or phases suppresses the dependence on initial or boundary conditions. b, A cut through the spherical Mott insulator with diameter L_{max} = 44a (as in the experiment) illustrates the distribution of different chain lengths (oriented along the z direction). Averaging the local magnetization S^{z} over the x and y directions provides a 1D magnetization profile (bottom panel), which is an average over all chains. c, The number of chains with length L is given by (π/2)(L/a). The total number of chains is πL_{max}/(2a)^{2} ≈ 1,500. d, The number of atoms in chains with length L is given by (π/2)(L/a)^{2}. The contribution of each chain to the imaging signal is proportional to the atom number in the chain, and so the relevant average over chain lengths is weighted by the atom number and is ⟨L⟩ = (3/4)L_{max} = 33a.
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Jepsen, P.N., AmatoGrill, J., Dimitrova, I. et al. Spin transport in a tunable Heisenberg model realized with ultracold atoms. Nature 588, 403–407 (2020). https://doi.org/10.1038/s415860203033y
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DOI: https://doi.org/10.1038/s415860203033y
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