  We propose a method to generate multi-mode N00N states with arrays of
ultracold atoms. Our protocol requires a strong relative offset among the wells
and a drive of the interparticle interaction at a frequency resonant with the
offsets. The proposal is demonstrated by a numerical and a Floquet analysis of
the quantum dynamics of a ring-shaped atomtronics circuit made of M weakly
coupled optical traps. We generate a hierarchy of energy scales down to very
few low-energy states where N00N dynamics takes place, making multi-mode N00N
states appear at nearly regular time intervals. The production of multi-mode
N00N states can be probed by time-of-flight imaging. Such states may be used to
build a multiple beam splitter.
