High-dimensional states of light with full control of transverse path and OAM degrees of freedom

We present here a compact scheme for the generation of high-dimensional states of light encoded in the transverse path variable of photons that carry orbital angular momentum. We use a programmable spatial light modulator in phase configuration to create correlations between these two spatial degrees of freedom. With our setup we are able to control, independently, the relative phases and amplitudes of the path superposition in addition to the topological charge of each path. Moreover, we engineer correlations that emulate bipartite quantum states of dimensions $d\times m$. Experimental results from the characterization of different generated states of dimensions up to $9\times 5$ are in excellent agreement with the numerical simulations. Fidelity with the target state is, for all cases, above $95\%$.
Comment: minor changes