  An arbitrary initial state of an optical or microwave field in a lossy driven
nonlinear cavity can be changed, in the steady-state limit, into a partially
incoherent superposition of only the vacuum and the single-photon states. This
effect is known as single-photon blockade, which is usually analyzed for a
Kerr-type nonlinear cavity parametrically driven by a single-photon process
assuming single-photon loss mechanisms. We study photon blockade engineering
via a squeezed reservoir, i.e., a quantum reservoir, where only two-photon
absorption is allowed. Namely, we analyze a lossy nonlinear cavity
parametrically driven by a two-photon process and allowing two-photon loss
mechanisms, as described by the master equation derived for a two-photon
absorbing reservoir. The nonlinear cavity engineering can be realized by a
linear cavity with a tunable two-level system via the Jaynes-Cummings
interaction in the dispersive limit. We show that by tuning properly the
frequencies of the driving field and the two-level system, the steady state of
the cavity field can be the single-photon Fock state or a partially incoherent
superposition of several Fock states with photon numbers, e.g., (0,2), (1,3),
(0,1,2), or (0,2,4). We observe that an arbitrary initial coherent or
incoherent superposition of Fock states with an even (odd) number of photons
can be changed into a partially incoherent superposition of a few Fock states
of the same photon-number parity. A general solution for an arbitrary initial
state is a weighted mixture of the above two solutions with even and odd photon
numbers, where the weights are given by the probabilities of measuring the even
and odd numbers of photons of the initial cavity field, respectively. Thus, in
contrast to the standard photon blockade, we prove that the steady state in the
engineered photon blockade, can depend on its initial state.
