Fig. 5 : Observing the generalized thermalization.

a Fidelity (dots) between the measured time-averaged coin states \(\bar \rho _t\) 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 \(\left| { \downarrow _y} \right\rangle\), and then evolve governed by \({{{\mathcal{H}}}}_{{{{\mathrm{eff}}}}}(7\pi /9,\pi /9)\). 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 \({\Delta}_k = 0.3\) 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: \(\mathop {\sum}\nolimits_{\delta k} {{{{\mathcal{P}}}}(k){{{\mathcal{O}}}}(k - k_0)}\), 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 \({{{\mathcal{F}}}}(\bar \rho _{10}|\rho _{{{{\mathrm{DE}}}}})\)