beam splitters photon unitary
Both beam splitters are modelled as the unitary matrix 



B
=


1

2





(



1


i




i


1



)




{\displaystyle B={\frac {1}{\sqrt {2}}}{\begin{pmatrix}1&i\\i&1\end{pmatrix}}}

, which means that when a photon meets the beam splitter it will either stay on the same path with a probability amplitude of 



1

/



2




{\displaystyle 1/{\sqrt {2}}}

, or be reflected to the other path with a probability amplitude of 



i

/



2




{\displaystyle i/{\sqrt {2}}}

. The phase shifter on the upper arm is modelled as the unitary matrix 



P
=


(



1


0




0



e

i
Δ
Φ





)




{\displaystyle P={\begin{pmatrix}1&0\\0&e^{i\Delta \Phi }\end{pmatrix}}}

, which means that if the photon is on the "upper" path it will gain a relative phase of 



Δ
Φ


{\displaystyle \Delta \Phi }

, and it will stay unchanged if it is in the lower path.
