# Symmetry group of quantum optical interactions

Some quantum optical interactions such as the beamsplitter and two-mode squeezing are unitaries that belong to certain continuous groups of transformations.

For example, the beamsplitter is an $SU(2)$ unitary because it is generated by $\frac{1}{2}(a^\dagger b + ab^\dagger)$ which together with $-\frac{i}{2}(a^\dagger b - ab^\dagger)$ and $\frac{1}{2}(a^\dagger a - b^\dagger b)$ satisfies the $su(2)$ algebra commutation relations. Another example is the two-mode squeezing operator which is an $SU(1,1)$ unitary for an analogous reason.

Besides these two, what other quantum optical unitaries belong to symmetry groups? (four-wave mixing? kerr effect? self-phase modulation?)

The list is quite extensive and those are only a very limited sample of the work done - mostly with $u(n)$ and $sp(n,\mathbb{R})$, generalizing your examples of $su(2)$ and $sp(2,\mathbb{R})\sim su(1,1)$.