  Quantum states of light, particularly at optical frequencies, are considered
necessary to realize a host of important quantum technologies and applications,
spanning Heisenberg-limited metrology, continuous-variable quantum computing,
and quantum communications. Nevertheless, a wide variety of important quantum
light states are currently challenging to deterministically generate at optical
frequencies. In part, this is due to a relatively small number of schemes that
prepare target quantum states given nonlinear interactions. Here, we present an
especially simple concept for deterministically generating and stabilizing
important quantum states of light, using only simple third-order optical
nonlinearities and engineered dissipation. We show how by considering either a
nonlinear cavity with frequency-dependent outcoupling, or a chain of nonlinear
waveguides, one can "filter" out all but a periodic ladder of photon number
components of a density matrix. As examples of this phenomenon, we show
cavities which can stabilize squeezed states, as well as produce
"photon-number-comb" states. Moreover, in these types of filter cavities,
Glauber coherent states will deterministically evolve into Schrodinger cat
states of a desired order. We discuss potential realizations in quantum
nonlinear optics. More broadly, we expect that combining the techniques
introduced here with additional "phase-sensitive" nonlinearities (such as
second-order nonlinearity) should enable passive stabilization and generation
of a wider variety of states than shown here.
