Can we have a coherent state between a fermion and a photon! I mean can there ever be a fermionic polariton?
You may have to clarify your question a bit. Imagine a simple atomic system with one electron in a magnetic field. The photon can interact with the system by flipping the electrons spin from down to up, total change of spin by 1, but the electron remains a fermion. What situation were you thinking of? What part of the system should be fermionic?
Thank you Zarqon for replying, I did some search and got my answer. The Wiki page says :"The polariton is a bosonic quasiparticle, and should not be confused with the polaron, a fermionic one, e.g. an electron plus attached phonon cloud. ". Precisely something I was looking for, if you are still interested we can discuss
Note also that you cannot have a superposition of states with an even and odd number of fermions. This is calles univalence superselection rule. It is easiest to see from the fact that such a superposition is not invariant (not even up to a phase) under a rotation by 2π.