  We present a new mechanism that harnesses extremely weak Kerr-type
nonlinearities in a single driven cavity to deterministically generate single
photon Fock states, and more general photon-blockaded states. Our method is
effective even for nonlinearities that are orders-of-magnitude smaller than
photonic loss. It is also completely distinct from so-called unconventional
photon blockade mechanisms, as the generated states are non-Gaussian, exhibit a
sharp cut-off in their photon number distribution, and can be arbitrary close
to a single-photon Fock state. Our ideas require only standard linear and
parametric drives, and is hence compatible with a variety of different photonic
platforms.
