Fig. 5 . Observing the generalized thermalization.

a Fidelity (dots) between the measured time-averaged coin states ${\overline{\rho }}_t$ and GME prediction ${\rho }_{\mathrm{GME}}$ against the time steps, and b their values (opaque bars) after a ten-step walk. The initial Gaussian states have five different widths Δk but the same coin state $∣{\downarrow }_y⟩$, and then evolve governed by ${\mathcal{H}}_{\mathrm{eff}}\left(7\pi /9,\pi /9\right)$. The colored solid lines in (a) and transparent bars in (b) represent the theoretical simulations. The inset in (a) shows the effective band structures (solid green lines with left-hand scale), the occupations of the two bands for one initial state with ${\mathrm{\Delta }}_k=0.3$ as an example (gray curves with right-hand scale). The expected energy and momentum of the initial state is E₀ and k₀, respectively. After expanding the initial state, the shift of the mean momentum of each band occupation relative to the k₀ takes the following form: ${\sum }_{\delta k}\mathcal{P}\left(k\right)\mathcal{O}\left(k-k_0\right)$, where $\mathcal{O}$ denotes the order of the function. Moreover, the vertical gray and horizontal green regions depict the chosen momentum window with $\delta k_{\mathrm{GME}}=0.4$ and associated energy window, respectively. The dashed line in (b) indicates the lower bound (99.42%) of the measured $\mathcal{F}\left({\overline{\rho }}_{10}\mid {\rho }_{\mathrm{DE}}\right)$