  Engineering multiphoton states is an outstanding challenge with applications
in multiple fields, such as quan- tum metrology, quantum lithography or even
biological systems. State-of-the-art methods to obtain them rely on
post-selection, multi-level systems or Rydberg atomic ensembles. Recently, it
was shown that a strongly driven two-level system interacting with a detuned
cavity mode can be engineered to continuously emit n-photon states. In the
present work, we show that spectral filtering of its emission relaxes
considerably the requirements on the system parameters even to the more
accessible bad-cavity situation, opening up the possibility of implementing
this protocol in a much wider landscape of different platforms. This
improvement is based on a key observation: in the imperfect case where only a
certain fraction of emission is composed of n-photon states, these have a well
defined energy separated from the rest of the signal, which allows to reveal
and purify multiphoton emission just by frequency filtering. We demonstrate
these results by obtaining analytical expressions for relevant figures of merit
of multiphoton emission, such as the n-photon coupling rate between cavity and
emitter, the fraction of light emitted as n-photon states, and n-photon
emission rates. This allows us to make a systematic study of such figures of
merit as a function of the system parameters and demonstrate the viability of
the protocol in several relevant types of cavity QED setups, where we take into
account the impact of their respective experimental limitations.
