Fig. 1: Sketch of the experimental setup.

a Implementation of the time-multiplexing split-step quantum walk. A heralded single photon with a central wavelength of 780 nm generated from beam-like spontaneous parametric down conversion is adopted as the walker. The anticorrelation parameter⁶³ measured in experiment is 0.031 ± 0.001. The polarization of the walker is prepared by a spin initialization module consisting of a PBS, HWP and QWP in sequence at the beginning and is measured by a spin analyzer consisting of a QWP, HWP and PBS in sequence at the end of the quantum walk. A full step of the split-step quantum walk is realized with two HWPs with their optical axes oriented at θ₁ and θ₂, respectively, for implementing the coin tossing, and two calcite crystals with their optical axes cut colinearly and orientated horizontally, for implementing the spin-orbit coupling. The final arrival time of the photon is measured by a homemade upconversion single-photon detector. In the inset of a, we show a diagram of the split-step quantum walk in the conventional time frame. To access the two shifted nonequivalent time frames, we change the rotation angle of the first HWP and add an extra HWP at the end for each complete step. The walker’s position space consists of the time bins (defining the arriving time of a single photon) with a pulse interval of 5 ps (determined by the length of the calcite crystal). The maximum repetition rate is 76 MHz, corresponding to a time of 13 ns. Here, we use ten quantum walk modules in total for a split-step quantum walk and one extra quantum walk module for adiabatically preparing some special initial states. b The complete topological phase diagram hidden in the split-step discrete-time quantum walk. In c, we present the probability distributions at times t = 0, t = 1 and t = 6 in the first configuration, i.e., the quench from a ground state of the Hamiltonian in the trivial phase (θ₂ = π) and ending in a nontrivial phase (θ₁ = 8π/9 and θ₂ = −π/3). The total coincidence counts between the idle photon and the upconversion signal are above 200 Hz, and for each basis, we set the integration time to 200 s. PBS polarized beam splitter, HWP half-wave plate, QWP quarter-wave plate.