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

TL;DR

This paper introduces efficient methods to compute the phase of quadratic bosonic evolutions from vacuum-to-vacuum amplitudes, enabling complete characterization of Gaussian unitaries in quantum optics.

Contribution

It provides polynomial-scaling techniques to recover the phase directly from vacuum amplitudes, including analytical solutions for key Hamiltonian classes and applications to Gaussian states.

Findings

Methods scale polynomially with the number of modes

Analytical solutions for time-independent Hamiltonians

Applicable to mixed and pure Gaussian states

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.