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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2507.06154 |
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| _version_ | 1866917120533594112 |
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| author | Quesada, Nicolás |
| author_facet | Quesada, Nicolás |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_06154 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | What's my phase again? Computing the vacuum-to-vacuum amplitude of quadratic bosonic evolution Quesada, Nicolás Quantum Physics 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. |
| title | What's my phase again? Computing the vacuum-to-vacuum amplitude of quadratic bosonic evolution |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2507.06154 |