  Non-Gaussian quantum states have been deterministically prepared and
autonomously stabilized in single- and two-mode circuit quantum electrodynamics
architectures via engineered dissipation. However, it is currently unknown how
to scale up this technique to multi-mode non-Gaussian systems. Here, we upgrade
dissipation engineering to collective (normal) modes of nonlinear resonator
arrays and show how to stabilize multi-mode Schrodinger cat states. These
states are multi-photon and multi-mode quantum superpositions of coherent
states in a single normal mode delocalized over an arbitrary number of
cavities. We consider tailored dissipative coupling between resonators that are
parametrically driven and feature an on-site nonlinearity, which is either a
Kerr-type nonlinearity or an engineered two-photon loss. For both types of
nonlinearity, we find the same exact closed-form solutions for the
two-dimensional steady-state manifold spanned by superpositions of multi-mode
Schrodinger cat states. We further show that, in the Zeno limit of strong
dissipative coupling, the even parity multi-mode cat state can be
deterministically prepared from the vacuum. Remarkably, engineered two-photon
loss gives rise to a fast relaxation towards the steady state, protecting the
state preparation against decoherence due to intrinsic single-photon losses,
which sets in at longer times. The relaxation time is independent of system
size making the state preparation scalable. Multi-mode cat states are naturally
endowed with a noise bias that increases exponentially with system size and can
thus be exploited for enhanced robust encoding of quantum information.
