A beam splitter with real amplitude transmission and reflection (so ) acts on its two…

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Problem

A beam splitter with real amplitude transmission

t=0.8

and reflection

r=0.6

(so

t^2+r^2=1

) acts on its two input modes as the unitary matrix

U=\begin{pmatrix}t & r\\ r & -t\end{pmatrix}.

One photon enters each input port, and the two photons are indistinguishable in every other respect (same polarisation, arrival time and colour). Using the rule that the amplitude for a given output click pattern is the permanent of the corresponding submatrix of

U

, find the probability of a coincidence detection (exactly one photon in each output port).

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