Past exam of the laser science and quantum information processing course of the University of Oxford / 2022 / 6 / a

The two output ports are orthogonal single-photon states. Conservation of total probability at a lossless beam splitter gives $2c^2=1$. With $c$ real and positive,In the ordered input basis $(0,1^{\prime })$ and output basis $(0^{\prime },1)$, the beam splitter scattering matrix isFor the usual reciprocal lossless beam splitter, reversing propagation uses $S^{-1}=S^{†}=S$. Consequently,The relative phase in the first rule is essential to the repeated-path calculation. Overall port phases may be changed consistently; they cannot be changed on one pass alone while retaining the original scattering rules. In the following, take unit mirror reflection amplitudes by incorporating fixed reflection phases into the propagation-mode convention, and keep any unremoved propagation phase explicitly if needed.