Fig. 3: Experimental apparatus for the 2D-QW implementation.

Photon preparation. Two photons are generated by a spontaneous parametric down-conversion source and independently injected into single-mode fibers. Polarization controllers are employed to change their polarization state, while delay lines enable controlling their degree of indistinguishability. For two-photon inputs, both photons are injected in the QW implementation and they propagate along two parallel paths. A lens system enlarges their waist radius and introduces a relative angle between the optical modes. The relative inclinations of the optical modes represent different lattice positions. In this way, the photons start the walk at positions (−1, 0) and (1, 0) of the lattice. In the single-particle case, one of the two photons is directly measured to act as a trigger. QW implementations. The quantum walk is performed by using waveplates and g-plates arranged in a cascade configuration. Each g-plate is controlled independently by tuning its phase retardation via a voltage controller. Measurement stage. A lens system at the output stage converts the different momentum values into a spatial grid. Then, for single-photon acquisition, an array of micro-lenses is used to efficiently inject the output modes in a 2D square-lattice fiber array. Finally, each output fiber is plugged into an avalanche photodiode detector connected to a coincidence electronic system. For the coherent light data acquisition, we inserted a beamsplitter between the last lens and the micro-lense array, and we positioned the CCD on the reflected path to perform image acquisition.