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, modeling 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 to integration the general problem, including the case of time-dependent Hamiltonians, and provide analytical results for key cases including when the Hamiltonian is positive definite, passive, active, or single-mode.