  High dimensional quantum states are of fundamental interest for quantum
information processing. They give access to large Hilbert spaces and, in turn,
enable the encoding of quantum information on multiple modes. One method to
create such quantum states is parametric down-conversion (PDC) in waveguide
arrays (WGAs) which allows for the creation of highly entangled photon-pairs in
controlled, easily accessible spatial modes, with unique spectral properties.
In this paper we examine both theoretically and experimentally the PDC process
in a lithium niobate WGA. We measure the spatial and spectral properties of the
emitted photon-pairs, revealing strong correlations between spectral and
spatial degrees of freedom of the created photons. Our measurements show that,
in contrast to prior theoretical approaches, spectrally dependent coupling
effects have to be taken into account in the theory of PDC in WGAs. To
interpret the results, we developed a theoretical model specifically taking
into account spectrally dependent coupling effects, which further enables us to
explore the capabilities and limitations for engineering the spatial
correlations of the generated quantum states.
