Self-adjoint realizations of higher-order squeezing operators

• URL: http://arxiv.org/abs/2508.09044v2
• Date: Wed, 13 Aug 2025 11:53:55 GMT
• Title: Self-adjoint realizations of higher-order squeezing operators
• Authors: Felix Fischer, Daniel Burgarth, Davide Lonigro,
• Abstract summary: Higher-order squeezing captures non-Gaussian features of quantum light by probing moments of the field beyond the variance.<n>We study a class of operators of the form $xi (adag)kal+xiast (adag)lak+f(adag a)$, which arise naturally in the analysis of higher-order quantum fluctuations.
• Score: 1.4260624980098286
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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 $\xi (a^\dag)^ka^l+\xi^\ast (a^\dag)^la^k+f(a^\dag a)$, 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 $f(a^\dag a)$, 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.

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