Fig. 3. Observing the spin subsystem equilibration.

a Measured initial Gaussian distributions (orange bars in the left two panels) $P\left(x\right)$ with corresponding ${\mathrm{\Delta }}_x=0.9$ and ${\mathrm{\Delta }}_x=0.5$ in the lattice space, and the associated distributions (solid orange lines in the right two panels) $\mathcal{P}\left(k\right)$ with corresponding ${\mathrm{\Delta }}_k=0.56$ and ${\mathrm{\Delta }}_k=1.0$ in momentum space. The gray bars and solid gray lines represent the theoretical simulations, and the gray regions denote the width ($2\delta k=2\sqrt{2ln2}{\mathrm{\Delta }}_k$) of the distributions. b Fidelity $\mathcal{F}\left({\overline{\rho }}_t\mid {\rho }_{\mathrm{DE}}\right)$ (yellow and green symbols) between the measured time-averaged coin state and diagonal ensemble prediction at each time step starting from the prepared two-bath states coupling to the coin state $∣{\downarrow }_y⟩$. The lines (solid yellow and dashed green) give the theoretical simulations. c Bloch sphere representations of the initial coin states (gray points), ten-step-averaged coin states (black points), and diagonal ensemble predictions (pointed by the orange arrows). All the error bars represent the statistical errors estimated by applying the Monte Carlo simulations