  Circuit quantum electrodynamics (QED) has emerged as a promising platform for
implementing quantum computation and simulation. Typically, junctions in these
systems are of a sufficiently small size, such that only the lowest plasma
oscillation is relevant. The interplay between the Josephson effect and
charging energy renders this mode nonlinear, forming the basis of a qubit. In
this work, we introduce a novel QED architecture based on extended Josephson
Junctions (JJs), which possess a non-negligible spatial extent. We present a
comprehensive microscopic analysis and demonstrate that each extended junction
can host multiple nonlinear plasmon modes, effectively functioning as a
multi-qubit interacting system, in contrast to conventional JJs. Furthermore,
the phase modes exhibit distinct spatial profiles, enabling individual
addressing through frequency-momentum selective coupling to photons. Our
platform has potential applications in quantum computation, specifically in
implementing single- and two-qubit gates within a single junction. We also
investigate a setup comprising several driven extended junctions interacting
via a multimode electromagnetic waveguide. This configuration serves as a
powerful platform for simulating the generalized Bose-Hubbard model, as the
photon-mediated coupling between junctions can create a lattice in both real
and synthetic dimensions. This allows for the exploration of novel quantum
phenomena, such as topological phases of interacting many-body systems.
