Self-adjoint realizations of higher-order squeezing operators
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| Format: | Preprint |
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2025
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| author | Fischer, Felix Burgarth, Daniel Lonigro, Davide |
| author_facet | Fischer, Felix Burgarth, Daniel Lonigro, Davide |
| contents | Higher-order squeezing captures non-Gaussian features of quantum light by probing moments of the field beyond the variance, and is associated with operators involving nonlinear combinations of creation and annihilation operators. Here we study a class of operators of the form $ξ(a^†)^ka^l+ξ^\ast (a^†)^la^k+f(a^†a)$, which arise naturally in the analysis of higher-order quantum fluctuations. The operators are defined on the linear span of Fock states. We show that the essential self-adjointness of these operators depends on the asymptotics of the real-valued function $f(n)$ at infinity. In particular, pure higher-order squeezing operators ($k\geq3$, $l=0$, and $f(n)=0$) are not essentially self-adjoint, but adding a properly chosen term $f(a^†a)$, like a Kerr term, can have a regularizing effect and restore essential self-adjointness. In the non-self-adjoint regime, we compute the deficiency indices and classify all self-adjoint extensions. Our results provide a rigorous operator-theoretic foundation for modeling and interpreting higher-order squeezing in quantum optics, and reveal interesting connections with the Birkhoff-Trjitzinsky theory of asymptotic expansions for recurrence relations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_09044 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Self-adjoint realizations of higher-order squeezing operators Fischer, Felix Burgarth, Daniel Lonigro, Davide Mathematical Physics Functional Analysis Quantum Physics 81Q10, 81Q12, 47B25, 46N50 Higher-order squeezing captures non-Gaussian features of quantum light by probing moments of the field beyond the variance, and is associated with operators involving nonlinear combinations of creation and annihilation operators. Here we study a class of operators of the form $ξ(a^†)^ka^l+ξ^\ast (a^†)^la^k+f(a^†a)$, which arise naturally in the analysis of higher-order quantum fluctuations. The operators are defined on the linear span of Fock states. We show that the essential self-adjointness of these operators depends on the asymptotics of the real-valued function $f(n)$ at infinity. In particular, pure higher-order squeezing operators ($k\geq3$, $l=0$, and $f(n)=0$) are not essentially self-adjoint, but adding a properly chosen term $f(a^†a)$, like a Kerr term, can have a regularizing effect and restore essential self-adjointness. In the non-self-adjoint regime, we compute the deficiency indices and classify all self-adjoint extensions. Our results provide a rigorous operator-theoretic foundation for modeling and interpreting higher-order squeezing in quantum optics, and reveal interesting connections with the Birkhoff-Trjitzinsky theory of asymptotic expansions for recurrence relations. |
| title | Self-adjoint realizations of higher-order squeezing operators |
| topic | Mathematical Physics Functional Analysis Quantum Physics 81Q10, 81Q12, 47B25, 46N50 |
| url | https://arxiv.org/abs/2508.09044 |