Multiple photon effective Hamiltonians in linear quantum optical networks

We give an alternative derivation for the explicit formula of the effective Hamiltonian describing the evolution of the quantum state of any number of photons entering a linear optics multiport. The description is based on the effective Hamiltonian of the optical system for a single photon and comes from relating the evolution in the Lie group that describes the unitary evolution matrices in the Hilbert space of the photon states to the evolution in the Lie algebra of the Hamiltonians for one and multiple photons. Our derivation complements previous results with a point of view which reminds of the Schrödinger picture. These group theory results allow us to prove that quantum optical linear networks obey different proportionality rules which relate the expected photon number at each input and output ports. In particular, we show that for a uniform photon input the average photon number at the output is preserved for any linear multiport.