Generalized Squeezing Polynomials

• Generalized squeezing polynomials are algebraic frameworks that extend standard quadratic squeezing to fractional orders and non-linear deformations.
• They are constructed using gamma-function interpolation, matrix multi-orthogonal polynomials, and Hermite polynomial bases to capture diverse quantum dynamics.
• These frameworks elucidate spectral transitions, state recurrences, and metrological enhancements, with applications in quantum optics and conformal field theories.

Generalized squeezing polynomials denote polynomial structures that arise when squeezing is extended beyond the standard quadratic bosonic setting. In the literature, they appear in several distinct but related forms: as gamma-function interpolants of generalized squeezing Hamiltonians with fractional order $n$ (Ashhab, 22 Jan 2026), as $3\times 3$ matrix multi-orthogonal polynomials governing finite $\mathfrak{u}(2)$ squeezed-coherent states (Genest et al., 2012), as Hermite-polynomial realizations of squeezed bases in Bargmann-type spaces (Ali et al., 2013), as operator-valued polynomials generated by higher Virasoro/Witt transformations (Katagiri et al., 2019), and as polynomial dependencies of signal and noise on the squeezing parameter in generalized echo squeezing protocols (Li et al., 2022). Taken together, these works suggest that the subject is best understood as a family of algebraic frameworks attached to generalized squeezing transformations, rather than as a single canonical polynomial sequence.

1. Generalized squeezing beyond the quadratic oscillator

A standard generalized squeezing operator of order $n$ is written as

$$\hat{U}_n(r)=\exp\{-ir\hat{H}_n\}, \qquad \hat{H}_n=i\left[(\hat{a}^\dagger)^n-\hat{a}^n\right],$$

with $\hat{a}$ and $\hat{a}^\dagger$ the bosonic lowering and raising operators, and $r$ a real squeezing parameter (Ashhab, 22 Jan 2026). In the corresponding basis, the dynamics couples states in steps of $n$, so that the relevant sector is $\{|0\rangle, |n\rangle, |2n\rangle,\dots\}$.

Several generalizations depart from this integer-order, infinite-dimensional bosonic picture in different directions. One route replaces factorials by gamma functions and allows real $3\times 3$0, producing a fractional interpolation of generalized squeezing Hamiltonians (Ashhab, 22 Jan 2026). Another route replaces the Heisenberg algebra by $3\times 3$1, where generalized squeezed-coherent states are built from exponentials of $3\times 3$2 generators and their amplitudes are controlled by matrix multi-orthogonal polynomials (Genest et al., 2012). A third route realizes squeezing through SU(1,1) in Bargmann space, where Hermite polynomials in a complex variable provide a squeezed analytic basis (Ali et al., 2013). A fourth route replaces linear symplectic squeezing by higher Virasoro/Witt generators, yielding nonlinear local scale deformations whose conjugation laws are governed by explicit squeezing polynomials (Katagiri et al., 2019).

This multiplicity of constructions is not merely terminological. It reflects different algebraic choices for the dynamical generator, the state space, and the meaning of “squeezing,” ranging from spectral interpolation and finite-oscillator recurrences to conformal deformations and metrological phase magnification.

2. Fractional orders and the interpolation of generalized squeezing Hamiltonians

The fractional generalization of the generalized-squeezing problem is obtained by replacing factorials with gamma functions,

$3\times 3$3

which extends the squeezing order from integers to arbitrary real $3\times 3$4 (Ashhab, 22 Jan 2026). In matrix form, the off-diagonal elements of the fractional Hamiltonian are

$3\times 3$5

For fractional $3\times 3$6, the basis $3\times 3$7 is not physically interpretable as a genuine Fock basis; it is explicitly introduced as a mathematical device for smooth interpolation (Ashhab, 22 Jan 2026).

This interpolation reveals two critical points. For spectral behavior, the smallest positive eigenvalue $3\times 3$8 tends to zero for $3\times 3$9, indicating a continuous spectrum, while for $\mathfrak{u}(2)$0 it converges to a finite nonzero value, indicating a discrete spectrum. At $\mathfrak{u}(2)$1, numerics place the system at the boundary between these regimes. For dynamics, the expectation value of the renormalized number operator

$\mathfrak{u}(2)$2

diverges for $\mathfrak{u}(2)$3, diverges logarithmically at $\mathfrak{u}(2)$4, and converges to a finite constant for $\mathfrak{u}(2)$5; asymptotically it approaches $\mathfrak{u}(2)$6 in the large-$\mathfrak{u}(2)$7 regime (Ashhab, 22 Jan 2026).

$\mathfrak{u}(2)$8 range	Spectrum	$\mathfrak{u}(2)$9 behavior
$n$0	Continuous	Diverges
$n$1	Critical boundary	Diverges
$n$2	Discrete	Diverges more slowly
$n$3	Discrete	Logarithmic divergence
$n$4	Discrete	Finite saturation

The large-$n$5 regime is described by a strong hierarchy of matrix elements, so that low-energy dynamics becomes effectively confined to the first two basis states. This provides the intuitive explanation for the saturation of $n$6, while the smallest positive eigenvalue scales as

$n$7

with Stirling asymptotics

$n$8

for large $n$9 (Ashhab, 22 Jan 2026). Within this framework, generalized squeezing polynomials are implicit in the gamma-interpolated matrix structure that replaces the integer combinatorics of ordinary (n\