Fig. 2. Experimental setup schematic.

(A) State initialization: Single photons with Gaussian transverse mode emerge from a single-mode (SM) fiber 1. A half-wave plate (HWP1) and PBS (PBS1) prepare horizontal polarization $\mid 0\rangle$. A beam splitter (BS) reflects 10% for photon number normalization. (B) Spatial-pointer coupling: PBS2 splits light into $\mid 0\rangle$ and $\mid 1\rangle$ polarization branches. The $\mid 0\rangle$ branch passes through CYL and slit, projecting onto position state $\mid x_{k-1}\rangle$; the $\mid 1\rangle$ branch remains unchanged. Recombination occurs at another PBS3. Piezo-driven prism stabilizes optical path difference. An 852-nm reference laser stabilizes Mach-Zehnder interferometer (MZI), with optical power detector (PD) providing feedback. (C) State evolution: 4f-system1 (equivalents to two convex lenses) maps slit exit wavefront to evolution region (see Materials and Methods). Initial time $t_{k-1}$ adjusted via slit-to-4f-system1 distance; position $x_{k-1}$ controlled by slit and CYL translation. (D) Position postselection and readout: Second 4f-system2 relays wavefront at measurement positions ($z={\mathit{ct}}_k$) to ICMOS camera (CISS, 2DSPC). The camera gated by SPDC idler triggers (see Materials and Methods). Longitudinal camera adjustment accesses different $t_k$ planes. Q/HWP and PBS4 enable pointer expectation measurements.