Multiphoton interference effects in passive and active Gaussian transformations

We develop a method which, building on the generating function of the matrix elements of Gaussian unitaries in Fock basis, gives access to the multiphoton transition probabilities via unexpectedly simple recurrence equations. The method is developed for Gaussian unitaries effecting both passive and active linear coupling between two bosonic modes, but could also be extended to Bogoliubov transformations acting on many modes. The recurrence includes an interferometric suppression term which generalizes the Hong-Ou-Mandel effect for more than two indistinguishable photons impinging on a balanced beam splitter. It also exhibits an unsuspected 2-photon suppression effect in optical parametric amplification of gain 2, which originates from the indistinguishability between the input and output photon pairs. The resulting framework should be quite valuable for characterizing bosonic linear coupling in various physical situations, as well as non-Gaussian bosonic channels.