# Multi-photon states in photonic quantum computing?

Within photonic quantum computing, one of the ways to represent information is the dual-rail representation of single-photon states ($$c_0|01\rangle \ + \ c_1|10\rangle$$). Is it possible to utilize multi-photon states (for example, two optical cavities with total energy $$2\hbar\omega$$, using a state like $$c_0|02\rangle \ + \ c_1|20\rangle \ + \ c_2|11\rangle$$, where $$|02\rangle$$ represents two photons in one cavity, $$|20\rangle$$ represents two photons in the other cavity, and $$|11\rangle$$ represents one photon in each cavity?

This is very much possible. And is a very general technique of how product systems in composite states are coupled. Here of the form $$|n_1\rangle |n_2\rangle$$. This kind of general ket is a solution of the Hamiltonian interaction/coupling terms like $$V\sim (a_1^\dagger a_2 +h.c)$$ which describe the exchange of one quanta (between the two optical cavities here). $$a_i (a_i^\dagger)$$ are the annihilation and creator operators for $$i^{th}$$ cavity in second quantisation. It follows similarly for more systems coupled in the form of product states. A very general and frequently used is a spin (with an excited and ground state as $$|e\rangle,|g\rangle$$) in an optical cavity with solutions like $$|\psi\rangle=c_1|e,n\rangle+c_2|g,n+1\rangle$$ where $$n$$ is the number of photons in the cavity, and so on for more complicated systems.