Fig. 3: Time evolution of the local mean density and particle-number fluctuations.

a, Imbalance as a function of evolution time for J_⊥/J ≈ 0.0, 0.5 and 1.0. Each data point was obtained from averaging over the region of interest with 40 × 40 sites and about 35 fluorescence images. Solid curves are Bessel function fits to the experimental data (see text for details). The dotted curve is the theoretical expectation for the 1D chain, which was derived from free-fermion theory and takes into account the imperfect initial experimental state. Inset, Fitted 1/e decay constant τ as a function of J_⊥/J. Error bars denote the standard deviation. b, Normalized atom-number variances \({\overline{{{{\rm{Var}}}}}}_{L}(t)\) in ladder subsystems of size 2L. The data points indicate the time t_sat when the variance reached 80% of its fitted saturation value. The grey area in the right-hand panel marks the regime of very large subsystems for which we could not reliably determine the saturation value as the fluctuation growth was too slow. The error bars indicate the standard error of the fit used to determine t_sat. The solid lines are the same fits as in c. c, Threshold time t_sat as a function of subsystem size L and J_⊥/J in log–log scale. The solid lines are linear fits used to obtain the dynamical exponent \(L \propto{t}_{{{{\rm{sat}}}}}^{1/z}\). For reference, the dashed lines indicate ideal slopes corresponding to z = 1 or 2 for ballistic and diffusive dynamics. d, Atom-number variance for a subsystem of size L = 16 as a function of evolution time. The solid line is the FHD prediction fitted to the experimental data, which yields a diffusion constant D = 1.11(25) Ja²/ℏ, where a denotes the lattice spacing. Error bars denote the standard deviation.