Unitary-gate synthesis for continuous-variable systems

We investigate the synthesis of continuous-variable two-mode unitary gates in the setting where two modes A and B are coupled by a fixed quadratic Hamiltonian H. The gate synthesis consists of a sequence of evolutions governed by Hamiltonian H, which are interspaced by local phase shifts applied to A and B. We concentrate on protocols that require the minimum number of necessary steps and we show how to implement the beam splitter and the two-mode squeezer in just three steps. Particular attention is paid to Hamiltonian xApB that describes the effective off-resonant interaction of light with the collective atomic spin