  The interplay of $\pi$-flux and lattice geometry can yield full localization
of quantum dynamics in lattice systems, a striking interference phenomenon
known as Aharonov-Bohm caging. At the level of the single-particle energy
spectrum, this full-localization effect is attributed to the collapse of Bloch
bands into a set of perfectly flat (dispersionless) bands. In such lattice
models, the effects of inter-particle interactions generally lead to a breaking
of the cages, and hence, to the spreading of the wavefunction over the lattice.
Motivated by recent experimental realizations of analog Aharonov-Bohm cages for
light, using coupled-waveguide arrays, we hereby demonstrate that caging always
occurs in the presence of local nonlinearities. As a central result, we focus
on special caged solutions, which are accompanied by a breathing motion of the
field intensity, that we describe in terms of an effective two-mode model
reminiscent of a bosonic Josephson junction. Moreover, we explore the quantum
regime using small particle ensembles, and we observe quasi-caged
collapse-revival dynamics with negligible leakage. The results stemming from
this work open an interesting route towards the characterization of nonlinear
dynamics in interacting flat band systems.
