We study a family of Jacobi operators in which the diagonal entries are multiplied by a coupling parameter $λ\geq0$ . Under suitable conditions, the operator is self-adjoint for every $λ>0$ , while the formal limit at $λ=0$ is a symmetric Jacobi operator admitting a one-parameter family of self-adjoint extensions. A central ingredient of our analysis is the derivation of uniform bounds for square-summable generalized eigenvectors in the small- $λ$ regime, which combines discrete WKB methods with Airy-function asymptotics. Using these estimates, we analyze the limiting behavior $λ\to0$ in the strong resolvent sense, proving that for every sequence $λ_j\to0$ one can extract a subsequence along which the corresponding Jacobi operators converge to some self-adjoint extension of the limiting operator; conversely, every such extension can be obtained in this way. We call this behavior an essentially singular limit, by analogy with essential singularities in complex analysis. As an application, we study higher-order squeezing operators arising in quantum optics. Using the connection with Jacobi operators, we show that when the relative strength of the free-field term tends to zero, different self-adjoint extensions of the squeezing operator are selected along different sequences. In particular, this limit does not single out a physically distinguished self-adjoint extension, but instead identifies a distinguished subclass of extensions compatible with the underlying symmetry.We study a family of Jacobi operators in which the diagonal entries are multiplied by a coupling parameter $λ\geq0$. Under suitable conditions, the operator is self-adjoint for every $λ>0$, while the formal limit at $λ=0$ is a symmetric Jacobi operator admitting a one-parameter family of self-adjoint extensions. A central ingredient of our analysis is the derivation of uniform bounds for square-summable generalized eigenvectors in the small-$λ$ regime, which combines discrete WKB methods with Airy-function asymptotics. Using these estimates, we analyze the limiting behavior $λ\to0$ in the strong resolvent sense, proving that for every sequence $λ_j\to0$ one can extract a subsequence along which the corresponding Jacobi operators converge to some self-adjoint extension of the limiting operator; conversely, every such extension can be obtained in this way. We call this behavior an essentially singular limit, by analogy with essential singularities in complex analysis. As an application, we study higher-order squeezing operators arising in quantum optics. Using the connection with Jacobi operators, we show that when the relative strength of the free-field term tends to zero, different self-adjoint extensions of the squeezing operator are selected along different sequences. In particular, this limit does not single out a physically distinguished self-adjoint extension, but instead identifies a distinguished subclass of extensions compatible with the underlying symmetry.

math-phCross-listedarXiv:2605.21355May 21, 2026

Bound, unbound operators and the squeezing effect

B. Baseia, H. Dias, Vanderlei Salvador Bagnato

Il Nuovo cimento della Società italiana di fisica. B/Il Nuovo cimento BAug 1, 19950 citations

SQUEEZING EFFECTS OF THE ATOMIC OPERATORS

WORLD SCIENTIFIC eBooksJul 1, 19980 citations

Tunable passive squeezing of squeezed light through unbalanced double homodyne detection

Niels Tripier-Mondancin, David Barral, Ganaël Roeland +3 more

The full characterization of quantum states of light is a central task in quantum optics and information science. Double homodyne detection provides a powerful method for the direct measurement of the Husimi Q quasi-probability distribution, offering a complete state representation in a simple experimental setting and a limited time frame. Here, we demonstrate that double homodyne detection can serve as more than a passive characterization tool. By intentionally unbalancing the input beamsplitter that splits the quantum signal, we show that the detection scheme itself performs an effective squeezing or anti-squeezing transformation on the state being measured. The resulting measurement directly samples the Q function of the input state as if it were acted upon by a squeezing operator whose strength is a tunable experimental parameter: the beamsplitter's reflectivity. We experimentally realize this technique using a robust polarization-encoded double homodyne detection to characterize a squeezed vacuum state. Our results demonstrate the controlled deformation of the measured Q function's phase-space distribution, confirming that unbalanced double homodyne detection is a versatile tool for simultaneous quantum state manipulation and characterization.The full characterization of quantum states of light is a central task in quantum optics and information science. Double homodyne detection provides a powerful method for the direct measurement of the Husimi Q quasi-probability distribution, offering a complete state representation in a simple experimental setting and a limited time frame. Here, we demonstrate that double homodyne detection can serve as more than a passive characterization tool. By intentionally unbalancing the input beamsplitter that splits the quantum signal, we show that the detection scheme itself performs an effective squeezing or anti-squeezing transformation on the state being measured. The resulting measurement directly samples the Q function of the input state as if it were acted upon by a squeezing operator whose strength is a tunable experimental parameter: the beamsplitter's reflectivity. We experimentally realize this technique using a robust polarization-encoded double homodyne detection to characterize a squeezed vacuum state. Our results demonstrate the controlled deformation of the measured Q function's phase-space distribution, confirming that unbalanced double homodyne detection is a versatile tool for simultaneous quantum state manipulation and characterization.

quant-pharXiv:2512.08540Dec 10, 20251 citation

We study a family of Jacobi operators in which the diagonal entries are multiplied by a coupling parameter

λ\geq0

. Under suitable conditions, the operator is self-adjoint for every

λ>0

, while the formal limit at

λ=0

is a symmetric Jacobi operator admitting a one-parameter family of self-adjoint extensions. A central ingredient of our analysis is the derivation of uniform bounds for square-summable generalized eigenvectors in the small-

λ

regime, which combines discrete WKB methods with Airy-function asymptotics. Using these estimates, we analyze the limiting behavior

λ\to0

in the strong resolvent sense, proving that for every sequence

λ_j\to0

one can extract a subsequence along which the corresponding Jacobi operators converge to some self-adjoint extension of the limiting operator; conversely, every such extension can be obtained in this way. We call this behavior an essentially singular limit, by analogy with essential singularities in complex analysis. As an application, we study higher-order squeezing operators arising in quantum optics. Using the connection with Jacobi operators, we show that when the relative strength of the free-field term tends to zero, different self-adjoint extensions of the squeezing operator are selected along different sequences. In particular, this limit does not single out a physically distinguished self-adjoint extension, but instead identifies a distinguished subclass of extensions compatible with the underlying symmetry.