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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?)

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There are many many many many more. You can have a look at some two-mode examples if you can have access to

  • Generalized two-mode harmonic oscillator: dynamical group and squeezed states, José M Cerveró and Juan D Lejarreta, Journal of Physics A: Mathematical and General, Volume 29, Number 23.

Other examples in higher dimensions include

  • Bartlett, S. D., et al. Unitary transformations for testing Bell inequalities Physical Review A 63.4 (2001): 042310 available from ArXiv
  • Bartlett, Stephen D., Hubert de Guise, and Barry C. Sanders. "Quantum encodings in spin systems and harmonic oscillators." Physical Review A 65.5 (2002): 052316 also available from ArXiv.

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 $sp(2,\mathbb{R})\sim su(1,1)$.

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