  We introduce a driven-dissipative two-mode bosonic system whose reservoir
causes simultaneous loss of two photons in each mode and whose steady states
are superpositions of pair-coherent/Barut-Girardello coherent states. We show
how quantum information encoded in a steady-state subspace of this system is
exponentially immune to phase drifts (cavity dephasing) in both modes.
Additionally, it is possible to protect information from arbitrary photon loss
in either (but not simultaneously both) of the modes by continuously monitoring
the difference between the expected photon numbers of the logical states.
Despite employing more resources, the two-mode scheme enjoys two advantages
over its one-mode cat-qubit counterpart with regards to implementation using
current circuit QED technology. First, monitoring the photon number difference
can be done without turning off the currently implementable dissipative
stabilizing process. Second, a lower average photon number per mode is required
to enjoy a level of protection at least as good as that of the cat-codes. We
discuss circuit QED proposals to stabilize the code states, perform gates, and
protect against photon loss via either active syndrome measurement or an
autonomous procedure. We introduce quasiprobability distributions allowing us
to represent two-mode states of fixed photon number difference in a
two-dimensional complex plane, instead of the full four-dimensional two-mode
phase space. The two-mode codes are generalized to multiple modes in an
extension of the stabilizer formalism to non-diagonalizable stabilizers. The
$M$-mode codes can protect against either arbitrary photon losses in up to
$M-1$ modes or arbitrary losses and gains in any one mode.
