Abstract

In the usual, predictive approach of quantum mechanics, one deals with the preparation of a quantum system followed by its time evolution and ultimately its measurement. In the retrodictive approach of quantum mechanics, one postselects the instances where a particular measurement outcome was observed and considers the probability of the preparation variable conditionally on this outcome. This can be interpreted as if the actually measured state had propagated backwards in time to the preparer. I will present an intermediate picture, called partial time reversal, where a composite system propagates partly forwards and partly backwards in time. As a striking application, I will focus on the simplest 2-mode linear-optical component, namely a beam splitter, and show that it transforms into a two-mode squeezer under partial time reversal. More generally, by building on the generating function of the matrix elements of Gaussian unitaries in Fock basis, I will build simple recurrence equations that are obeyed by the multiphoton transition probabilities (both for the passive and active linear coupling between two bosonic modes). These recurrence equations exhibit an interferometric suppression term which generalizes the Hong-Ou-Mandel effect for more than two indistinguishable photons impinging on a beam splitter of transmittance 1/2. It also exhibits an unexpected 2-photon suppression effect in an optical parametric amplifier of gain 2 originating from the timelike indistinguishability between the input and output photons (it is the partial time-reversed version of the usual spacelike indistinguishability that is at work in the Hong-Ou-Mandel effect).