Fig. 1. QWs in the space of light transverse momentum. (a) Photonic modes implementing the position states on the lattice. For each mode carrying mx and my units of transverse momentum Δk⊥ in the x and y directions, respectively, we plot the linear phase profile in the transverse xy plane. (b) LC pattern of a g-plate. The local molecular director forms an angle θ with the x axis. In a g-plate, we have θ(x)=πx/Λ, with Λ being the spatial period. The birefringence is uniform and electrically tunable by applying a voltage to the cell.³⁴

Fig. 2. Large-scale mode mixing via LCMSs. (a) Three LCMSs (Q1,Q2,Q3) implement the optical transformation corresponding to the desired multimode mixing U. The inset illustrates an LCMS with its LC optic-axis pattern. Squares of different colors reflect different values of θ(x,y), while the birefringence δ, i.e., the orientation with respect to the propagation axis z, is homogeneous. Off-diagonal elements of the LCMS Jones matrix flip the polarization handedness and add a space-dependent conjugate phase modulation on orthogonal circular polarization components |L⟩ and |R⟩. (b) The mode sorting is realized in the focal plane of a lens (F), where modes appear as a 2D array of Gaussian beams separated by Δk⊥. Each spot is a superposition of the polarization (coin) states {|L⟩,|R⟩}.

Fig. 3. Numerical optimization to retrieve 2D continuous LCMS optic-axis modulations. (a) Values of θ(x,y) obtained from a single set of analytical solutions of Eq. (7) are typically discontinuous. A numerical routine is devised to match different solutions at each transverse position to enforce continuity. The resulting pattern displays isolated vortices. (b) Different scenarios (i)–(ii)–(iii) are illustrated for the optimization routine, depending on the current position on the plate rij (yellow square). The violet arrow path contains discrete positions where the optimization algorithm has already been executed. The green crosses mark neighboring elements rn where a continuous modulation has already been found and are, therefore, involved in the optimization of the metric dij (see text). Neighboring elements where the algorithm has not been executed yet are marked by red crosses. (c) Full pattern of one of the LCMSs designed to implement a 10-step QW (3Λ×3Λ square, with Λ=5mm), imaged between crossed polarizers to reveal the LC’s in-plane orientation.

Fig. 4. 2D QWs via spin-orbit photonics. (a) Optic-axis modulation of the first metasurface [θ1(x,y)] employed for the simulation of the 2D QW. (b) Experimental images obtained for an |R⟩-polarized input state, from which the walker probability distribution Pexp(mx,my) is extracted (c) and compared with the theoretical prediction Pth(mx,my) (d). For each realization, we report the value of the similarity, computed as the average of four independent measurements. Rows refer to 3, 5, 10, and 20 time steps (t), respectively.

Fig. 5. Resolving the totality of the modes. A g-plate with a smaller spatial period Λg≪Λ placed before the Fourier lens allows us to resolve separately light with orthogonal circular polarizations. (a) Experimental images, (b) experimental reconstructions Pexp(mx,my), and (c) theoretical predictions Pth(mx,my) of the output distribution and its projections on |L⟩ and on |R⟩. A localized |R⟩-polarized input after five steps is considered. (d) Variance of the output distribution along x and y. The experimental points (dots) correctly reproduce the expected ballistic behavior (solid lines) extracted numerically.

Fig. 6. Unitary maps obtained by reconfiguring a sequence of three plates. (a) LCMSs’ optical birefringence parameters δi∈[0,2π), represented as the tilt of the LC molecules with respect to the propagation axis z, can be electrically tuned. Moreover, their lateral relative position can also be adjusted, both in the x and y directions (red arrows). (b) When shifting the plates, the overall transformation is still a unitary circuit coupling transverse wave vector modes. The three panels show the output intensity distribution computed numerically for an |R⟩-polarized input state when the LCMSs designed to implement the five-step QW are not shifted, and when the second and the third are laterally shifted in opposite directions along both x and y of ±1 and ±2mm, respectively. (c) 500 unitary maps U˜ are numerically generated by randomly varying the birefringence parameters, with δi∈[0,2π) (red), and 500 more by randomly varying the relative position in a range ≤0.15mm (blue) of the three LCMSs implementing the 20-step QW. Through the histogram, the distribution of generated unitaries can be investigated, both in terms of the number of activated output modes and the fidelity F(U˜) with respect to the reference QW process.

Fig. 7. Experimental implementation. (a) Experimental setup to engineer QW dynamics. The entire evolution is compressed within only three LCMSs. (b) Reconstruction of the probability distribution Pexp(mx,my) from the experimental image. After the central |mx,my⟩=|0,0⟩ spot has been determined, the probability of each site is computed as the normalized integrated intensity within the corresponding light spot.

Fig. 8. Optimization routine. (a) Single analytical solution for the first LCMS implementing the QW evolution U010(π/2). Discontinuities across extended lines are visible. (b) The optimization routine is executed to output a continuous modulation for the sets of three LCMSs, only embedding isolated vortices.