Phase Shifter

Linear Optics

A phase shifter is a linear optical operator. The interaction Hamiltonian is given by

$$\hat{H}=\varphi {\hat{a}}^{†}\hat{a}$$

The action of a phase shifter on the Fock basis is given by

$${\hat{a}}^{†}\to {\hat{U}}^{†}{\hat{a}}^{†}\hat{U}=e^{i\varphi }{\hat{a}}^{†}$$

Here, we use the Heisenberg picture, with $\hat{U}=e^{-it\hat{H}}$.

Gaussian Formalism