Self-adjoint realizations of higher-order squeezing operators

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 ξ ( a † ) k a l + ξ ∗ ( a † ) l a 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 ⩾ 3 , 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.