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

TL;DR

The work addresses the missing phase information in Gaussian unitaries generated by quadratic bosonic Hamiltonians by showing how to extract the phase from the vacuum-to-vacuum amplitude $c = \langle 0|U|0\rangle$ with $e^{i\varphi} = c/|c|$. It develops a scalable framework that leverages symplectic diagonalization (including Williamson’s theorem), the Bargmann representation, and, for time-dependent cases, a Riccati-equation-based approach to compute $c$ without Fock-space truncation. The contributions include closed-form amplitudes for positive-definite or symplectically diagonalizable Hamiltonians, a polynomial-cost method for general multimode Hamiltonians, and a time-dependent generalization that remains efficient via linear differential equations and trotterization. This enables a complete unitary characterization of Gaussian operations, with practical implications for Gaussian-state processing, quantum optics, and hybrid quantum information protocols.

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.