What’s my phase again? Computing the vacuum-to-vacuum amplitude of quadratic bosonic evolution

Abstract Quadratic bosonic Hamiltonians and their associated unitary transformations form a fundamental class of operations in quantum optics, modelling key processes such as squeezing, displacement and beam-splitting. Their Heisenberg-picture dynamics simplifies to linear (or possibly affine) transformations on quadrature operators, enabling efficient analysis and decomposition into optical gate sets using matrix operations. However, this formalism discards a phase, which, while often neglected, is essential for a complete unitary characterization. We present efficient methods to recover this phase directly from the vacuum-to-vacuum amplitude of the unitary, using calculations that scale polynomially with the number of modes and avoid Fock space manipulations. We reduce the general problem for time-dependent Hamiltonians to integration, and provide analytical results for key cases including time-independent Hamiltonians which are positive definite, passive, active or single-mode. Finally, we show that our results can be easily used to obtain the phase associated with any Gaussian state, be it mixed or pure.