  Coherent states and their generalisations, displaced Fock states, are of
fundamental importance to quantum optics. Here we present a direct observation
of a classical analogue for the emergence of these states from the eigenstates
of the harmonic oscillator. To this end, the light propagation in a
Glauber-Fock waveguide lattice serves as equivalent for the displacement of
Fock states in phase space. Theoretical calculations and analogue classical
experiments show that the square-root distribution of the coupling parameter in
such lattices supports a new family of intriguing quantum correlations not
encountered in uniform arrays. Due to the broken shift-invariance of the
lattice, these correlations strongly depend on the transverse position.
Consequently, quantum random walks with this extra degree of freedom may be
realised in Glauber-Fock lattices.
