  Bosonic two-ring ladders constitute an important class of atomtronic
circuits, where coherent current flows not only can offer a new insight into
many-body physics, but also can play the role of actual degrees of freedom, and
hence allow for a viable implementation of cold-atom based devices and qubit
systems. In this work, we exhaustively investigate the ground state properties
and the low-lying energy spectrum of two linearly coupled Bose-Hubbard rings.
We show that the competition among interactions, intra- and inter-ring hopping
processes gives place to a rather rich physical scenario, where Mott-like
states and (different kinds of) superfluid-like states emerge. The latter ones
depend also on the (in)commensurate filling of the atoms. Our analysis, carried
out within a simple analytical framework and by means of the exact numerical
diagonalization of the system Hamiltonian, provides one with a rather complete
characterization of the static properties of the two-ring ladder, including,
but not limited to, coherence, fragmentation, correlations, and entanglement.
We complement our investigation by studying how these indicators depend on the
commensurability of the total number of bosons with respect to the total number
of sites and show that the two stacked rings are always entangled for an odd
number of atoms.
