Multimode Fock states with large photon number: effective descriptions and applications in quantum metrology

We develop general tools to characterise and efficiently compute relevant observables of multimode N-photon states generated in nonlinear decays in one-dimensional waveguides. We then consider optical interferometry in a Mach-Zender interferometer where a d-mode photonic state enters in each arm of the interferometer. We derive a simple expression for the quantum Fisher information in terms of the average photon number in each mode, and show that it can be saturated by number-resolved photon measurements that do not distinguish between the different d modes.