  Activating transitions between internal states of physical systems has
emerged as an appealing approach to create lattices and complex networks. In
such a scheme, the internal states or modes of a physical system are regarded
as lattice sites or network nodes in an abstract space whose dimensionality may
exceed the systems' apparent (geometric) dimensionality. This introduces the
notion of synthetic dimensions, thus providing entirely novel pathways for
fundamental research and applications. Here, we analytically show that the
propagation of multi-photon states through multi-port waveguide arrays gives
rise to synthetic dimensions where a single waveguide system generates a
multitude of synthetic lattices. Since these synthetic lattices exist in
photon-number space, we introduce the concept of pseudo-energy and demonstrate
its utility for studying multi-photon interference processes. Specifically, the
spectrum of the associated pseudo-energy operator generates a unique ordering
of the relevant states. Together with generalized pseudo-energy ladder
operators, this allows for representing the dynamics of multi-photon states by
way of pseudo-energy term diagrams that are associated with a synthetic atom.
As a result, the pseudo-energy representation leads to concise analytical
expressions for the eigensystem of $N$ photons propagating through $M$
nearest-neighbor coupled waveguides. In the regime where $N>2$ and $M>2$,
non-local coupling in Fock space gives rise to hitherto unknown all-optical
dark states which display intriguing non-trivial dynamics.
