Extended Data Fig. 4: Effect of finite hole concentration.
From: Spin transport in a tunable Heisenberg model realized with ultracold atoms
By varying the thermal fraction Nth/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 11ER. a, Decay curves c(t) for varying hole concentrations ranging from low (blue) to high (orange) thermal fraction. Solid lines are fits c(t) = [a0 + b0cos(ωt)]e−t/τ + c0. b, The background contrast c0 increases monotonously with thermal fraction Nth/N. A linear fit (solid line) extrapolates to c0 = 0.01(2), consistent with zero, for Nth/N = 0. This suggests that all of the background contrast is due to hole excitations. c, Higher hole concentrations suppress the oscillating fraction b0/(a0 + b0). 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 Nth/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 work7 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 on-site 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 first-order tunnelling is suppressed by the gradient of the (harmonic) trapping potential50. 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/Jxy (middle) and 21.7ħ/Jxy (bottom). The two in situ images in both f and g are for t = 0 (top) and 21.7ħ/Jx (bottom). The dashed lines indicate contours of constant radius, r = 30a (f) and r = 20a (g).
