Fig. 2. Quantum walk model and the experimental setup.

a A diagram of an isolated QW for investigating the generalized thermalization, which contains two parts: the discrete infinite lattice $\mathcal{B}$ and spin (coin) subsystem $\mathcal{S}$, coupled by the spin–orbit interaction. b The experimental realization of QW involves the following four modules. (1) Pairs of photons are generated via the spontaneous parametric down-conversion (SPDC) in the BBO2 crystal, where a photon from each pair is directly detected by the APD1 as a trigger and the other signal one as the walker is sent to the subsequent module. (2) First, a localized state is initialized via a PBS-HWP-QWP setting. Second, the signal photons go through at most six steps of walk with each step involving two SU(2) coin tossings achieved by the combination QWP-HWP-QWP and two shift operators achieved by the calcite crystals. An arbitrary Gaussian wavepacket over the lattice space can be obtained after projecting the coin on an unbiased basis $∣+⟩$ using an HWP and PBS. (3) After re-initialization of the coin state (here the time step is also re-initialized as t = 0) and sending the signal photons into a 10-step walk whose coin-tossing rotations along the y-axis and achieved by utilizing the HWP, the relaxation evolution of the coin state can be monitored through a coin analyzer. (4) The time-evolving probability distributions, i.e., the pulse train of signal photons with a time interval of approximately 5 ps, can be analyzed by utilizing the up-converted detector, which comprised a frequency upconversion in the BBO3 crystal, a 4-f system for filtering the spectrum, and a PMT detector. The 4-f system includes a prism for introducing dispersion, a knife-edge for blocking the noise photons whose wavelength is longer than 395 nm, and a mirror reflecting the up-converted signal photons to the PMT. List of the abbreviations: lens (L); β-BaB₂O₄ (BBO); avalanche photodiode detector (APD); dichroic mirror (DM); polarization-dependent beam splitter (PBS); half-wave plate (HWP); quarter-wave plate (QWP); photomultiplier tube (PMT). Detailed descriptions are provided in the Methods