  We explore the multiparticle transition probabilities in Gaussian unitaries
effected by a two-mode Bogoliubov bosonic transformation on the mode
annihilation and creation operators. We show that the transition probabilities
can be characterized by remarkably simple, yet unsuspected recurrence equations
involving a linear combination of probabilities. The recurrence exhibits an
interferometric suppression term - a negative probability - which generalizes
the seminal Hong-Ou-Mandel effect to more than two indistinguishable photons
impinging on a beam splitter of rational transmittance. Unexpectedly,
interferences thus originate in this description from the cancellation of
probabilities instead of amplitudes. Our framework, which builds on the
generating function of the non-Gaussian matrix elements of Gaussian unitaries
in Fock basis, is illustrated here for the most common passive and active
linear coupling between two optical modes driven by a beam splitter or a
parametric amplifier. Hence, it also allows us to predict unsuspected
multiphoton interference effects in an optical amplifier of rational gain. In
particular, we confirm the newly found two-photon interferometric suppression
effect in an amplifier of gain 2 originating from timelike indistinguishability
(arXiv:2012.15165 [quant-ph]). Overall, going beyond standard two-mode optical
components, we expect our method will prove valuable for describing general
quantum circuits involving Bogoliubov bosonic transformations.
