Representation theory of inhomogeneous Gaussian unitaries

Abstract Gaussian unitaries, generated by quadratic Hamiltonians, are fundamental in quantum optics and continuous-variable computing. They form projective representations of the symplectic (bosons) and orthogonal (fermions) group, respectively, which leaves their global phases undetermined. In physical scenarios where this phase matters, one is naturally required to move to the respective double covers, namely the metaplectic and spin group. The homogeneous case (no displacements, only group elements connected to the identity) has been resolved through a parameterization constructed in a recent work [arXiv:2409.11628]. We extend the previous framework to inhomogeneous Gaussian unitaries, which includes displacements for bosons and group elements disconnected from the identity for fermions. We provide simple closed formulas for the multiplication law in our parametrization and how our parameters determine the global complex phase of the respective Gaussian unitary.