Quadrature operator eigenstates and wavefunctions of $f$-deformed oscillators

Adipta Pal; Aditi Pradeep; C. Sudheesh; S. Anupama

REVIEW 4 major objections 3 minor 33 references

Quadrature operator eigenstates and wavefunctions of $f$-deformed oscillators

T0 review · 4 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read For any f-deformed oscillator, all excited-state quadrature wavefunctions are a new orthogonal polynomial J_n times the ground-state wavefunction.

desk verdict Useful generalization of q-deformed quadrature wavefunctions, but the 'explicit' claim is hollow: every excited state is given in terms of an undetermined ground state, and the derivation of the central recurrence has a sign inconsistency. read the letter →

arxiv 1908.01480 v2 pith:IQUB32D4 submitted 2019-08-05 quant-ph

S. Anupama , Aditi Pradeep , Adipta Pal , C. Sudheesh This is my paper · ORCID

classification quant-ph

keywords f-deformedoscillatorsquadratureoperatorwavefunctionsorthogonalpolynomialshomodynedetectionq-deformation(pq)-deformationdeformedFockspace

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper sets out to give explicit quadrature wavefunctions for the most general f-deformed oscillator, covering the math-type q-deformed, physics-type q-deformed, and (p,q)-deformed oscillators as special cases. Its central result is a factorization: every excited-state wavefunction in the quadrature basis can be written as $\Psi_n(X_\theta)=e^{-in\theta} J_n(X_\theta) \Psi_0(X_\theta)$, where $J_n$ is a new family of orthogonal polynomials fixed by a three-term recurrence that depends only on the deformation parameter $Q$ and the deformed occupation numbers $[n]$. This reduces the whole excited-state manifold to one unknown ground-state wavefunction plus deterministic polynomials. Because quadrature wavefunctions are what homodyne detection measures, the result would give deformed-state experimentalists a direct route from measured quadrature distributions to the deformed quantum state. In the limit $Q\to 1$, $f(n)=1$, the construction and the polynomials reduce to the standard harmonic oscillator and its Hermite polynomials.

What carries the argument

The central object is the deformed quadrature operator $\hat X_\theta$, the f-deformed analogue of the homodyne quadrature operator, together with the orthogonal polynomials $J_n(X_\theta)$ it generates through the recurrence in Eq. (28). The polynomials are the deformed counterpart of the Hermite polynomials: they are defined purely by the deformation data $Q$ and $[n]$, they are proven orthogonal by Favard's theorem, and they carry the $n$-dependence of every excited state. The entire argument moves by converting the eigenvalue equation $\hat X_\theta |X_\theta\rangle = X_\theta |X_\theta\rangle$ into a recurrence for the Fock-basis components, solving that recurrence in terms of $J_n$, and then specializing $Q$ and $[n]$ for each deformation model.

What would settle it

For a fixed deformation and fixed $\theta$, take a quadrature eigenvalue $X_\theta$, numerically diagonalize $\hat X_\theta$ in a truncated deformed Fock basis, and read off the eigenvector components $c_n(X_\theta)$. The central claim predicts $c_n(X_\theta)/c_0(X_\theta)=e^{-in\theta}J_n(X_\theta)$ for the $J_n$ of Eq. (28); a mismatch for any $n$ would falsify the factorization.

Watch

Extended reading notes

Core claim

Starting from the deformed ladder operators $\hat A = \hat a f(\hat n)$ and the commutation relation $[\hat A,\hat A^\dagger] = \varphi(\hat n)$, the authors define a deformed homodyne quadrature operator $\hat X_\theta = \sqrt{(1+Q)/2}(\hat A e^{-i\theta} + \hat A^\dagger e^{i\theta})$ whose eigenstates $|X_\theta\rangle$ are expanded in the deformed Fock basis. The eigenvalue equation produces a two-term recurrence for the components $\Psi_n(X_\theta)=\langle X_\theta|n\rangle_f$, and its solution has the form $\Psi_n(X_\theta)=e^{-in\theta} J_n(X_\theta) \Psi_0(X_\theta)$, with $J_0=1$ and $J_{n+1} = (1/\sqrt{[n+1]})[(2X_\theta/\sqrt{1+Q}) J_n - \sqrt{[n]} J_{n-1}]$. Favard's theorem is invoked to show these $J_n$ form a genuine orthogonal-polynomial family. Written this way, the excited-state wavefunction problem for an arbitrary deformation function $f(n)$ is reduced to computing the single ground-state wavefunction $\Psi_0(X_\theta)$, and the three concrete deformations are worked out as illustrations.

Load-bearing premise

The load-bearing premise is that the ground-state wavefunction $\Psi_0(X_\theta)$ exists and can actually be computed for the deformed oscillator at hand; the paper expresses every excited state in terms of it but does not itself supply $\Psi_0$.

Editorial extensions

If this is right

Only the ground-state wavefunction has to be found for each deformation; every excited state follows automatically from the $J_n$ recurrence.
The $J_n$ polynomials provide ready-made position-space wavefunctions ($\theta=0$) for math-type $q$-, physics-type $q$-, and $(p,q)$-deformed oscillators.
Homodyne-detection setups for deformed states can compare measured quadrature distributions against these predictions to reconstruct the deformed density matrix.
The exact recovery of Hermite polynomials and oscillator wavefunctions at $Q\to 1$ gives a built-in consistency check.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The orthonormality of the wavefunctions imposes $\int J_m(X_\theta)J_n(X_\theta)|\Psi_0(X_\theta)|^2\,dX_\theta = \delta_{mn}$, so the missing ground state is itself the solution of the moment problem for the $J_n$ family; solving it would make the construction fully explicit.
The same factorization should carry over to deformed coherent states, whose quadrature overlap would be a generating function of the $J_n$ polynomials and could yield closed expressions for Q functions.
A natural test for moderate $q$ values is to compare the predicted $c_n(X_\theta)$ ratios against truncated-Fock diagonalization; where the ratio deviates will show how quickly the deformed algebra's truncation matters.
Because $J_n$ depends only on $Q$ and $[n]$, any newly proposed deformation function $f(n)$ can be plugged in directly without reworking the derivation.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: this is a legitimate but incomplete paper. It generalizes the q-deformed quadrature construction from ref [19] to arbitrary f-deformed oscillators, derives a recurrence for the Fock-state wavefunctions in the quadrature basis, and defines a new set of orthogonal polynomials J_n. That part is real and, as far as I can tell, correct in substance. The problem is that the advertised result—'explicitly the wavefunctions'—is not delivered. Every excited state is written as e^{-inθ} J_n(Xθ) Ψ0(Xθ), and Ψ0 is never computed, not even for the specific deformations they plot. The recurrence (28) determines only the ratios J_n = Ψn/Ψ0; it says nothing about Ψ0. Figures 1–3 plot |Ψ0|^2 and |Ψ1|^2 without specifying Ψ0's functional form or normalization, so the plots are unreproducible. The paper itself acknowledges this reduction ('reducing the unknowns to just the ground state wavefunction'), but the abstract overclaims.

There's also a technical inconsistency I'd want fixed. Solving Eq. (21) as printed gives a factor e^{+iθ} in the recurrence for Ψ_{n+1}, while Eq. (23) uses e^{-iθ}. The recurrence (23) passes the Q→1, |f(n)|^2=1 limit and reproduces the standard Hermite recursion, so I suspect (23) is correct and the printed derivation has a sign/notation slip, likely related to the ambiguous definition of Ψ_n as either ⟨Xθ|n⟩ or ⟨n|Xθ⟩. That needs cleanup.

What's actually new: the extension from q-deformation to general f-deformation is a genuine parameter generalization, and the polynomial family J_n is new as far as I can tell. The paper cites [19] properly and the homodyne-detection motivation is reasonable.

Who this is for: groups working on deformed oscillators and quantum state reconstruction. It's a tool improvement, not a new phenomenon. The completeness gap reduces its value, but the recurrence and polynomial structure are useful on their own.

My view: this deserves a serious referee, but not acceptance as is. The authors should either solve for Ψ0 for at least the three deformations they plot, or honestly reframe the contribution as a reduction of the wavefunction problem to the ground state, and fix the derivation of (23). If they do that, it could be a solid paper. Right now, the central claim is unsupported.

Referee Report

4 major / 3 minor

Summary. The paper aims to determine the quadrature-operator eigenstates and wavefunctions of f-deformed oscillators. It defines a deformed quadrature operator Xθ = sqrt(1+Q)/2 (A e^{-iθ} + A† e^{iθ}), expands its eigenstates in the deformed Fock basis, derives a three-term recurrence for the quadrature wavefunctions, and introduces a family of polynomials J_n(Xθ) through which the wavefunctions are written as Ψ_n(Xθ) = e^{-inθ} J_n(Xθ) Ψ_0(Xθ). The formalism is then applied to math-type q-deformed, physics-type q-deformed, and (p,q)-deformed oscillators, and probability densities for the ground and first excited states are plotted. The main claim is that these are explicit wavefunctions for general f-deformed oscillators.

Significance. The reduction of all excited-state wavefunctions to the ground state via a recurrence is a useful structural observation, and the recurrence (28) is a legitimate generalization of the Hermite recurrence; the Favard argument for the orthogonality of the J_n is also sound. The paper correctly identifies the parameter replacement needed to extend the q-deformed framework of reference [19] to the general f-deformed case. However, because Ψ_0 is never obtained, the central claim of explicit wavefunctions is not delivered; the contribution is at present an incomplete framework rather than a solution.

major comments (4)

[Section 3, Eq. (22) and Eqs. (27), (36)] The ground-state wavefunction Ψ_0 is introduced but never derived, specified, or numerically determined. The text after Eq. (22) states that the problem reduces to the ground-state wavefunction, but no equation for Ψ_0 is given. Eq. (27) and Eq. (36) therefore express every wavefunction in terms of an undetermined function Ψ_0, and the recurrence (28) determines only the ratios J_n = Ψ_n/Ψ_0. Figures 1–3 plot |Ψ_0|² and |Ψ_1|² without providing the functional form of Ψ_0 or its normalization. This directly contradicts the abstract's claim that the wavefunctions are obtained explicitly.
[Section 3, Eq. (21) vs. Eq. (23)] The displayed Eq. (21) is internally inconsistent with the recurrence (23) that is derived from it. Solving Eq. (21) for Ψ_{n+1} gives factors e^{+iθ} (and, for the second term, e^{+2iθ}), not the e^{-iθ} factors used in Eq. (23). If one maintains the convention Ψ_n = ⟨Xθ|n⟩_f from Eq. (20), then f⟨n|Xθ⟩ = Ψ_n^* and the correct eigenvalue equation has the phase factors reversed relative to Eq. (21). Although Eq. (23) may be correct after this phase correction and passes the q→1 Hermite limit, the derivation as printed is not self-consistent and must be rewritten.
[Section 3, Eqs. (31) and (32)] The explicit formulas for J_2 and J_3 do not follow from the recurrence (28). For Q = 1 and [n] = n, the recurrence gives J_2 = (2x² − 1)/√2 and J_3 = (2x³ − 3x)/√3, whereas Eqs. (31) and (32) give J_2 = (4x² − 2)/(4√2) and J_3 = (8x³ − 12x)/(8√6). The discrepancy is a spurious factor (1+Q) in the denominator of Eq. (31) and an incorrect combination of [1] and [2] in Eq. (32). Consequently, the claimed second and third polynomials are not correct, and the Hermite limit (34) is not recovered from these expressions.
[Section 3, Eq. (28) and Figures 1–3] The orthogonality and normalization of the quadrature basis are not established. The quantum-mechanical normalization requires a square-integrable Ψ_0 such that ∫ dXθ |Ψ_0|² e^{i(m−n)θ} J_m J_n = δ_{mn}; in other words, |Ψ_0|² must be the weight function for the J_n. Favard's theorem guarantees the existence of some measure for the polynomials, but it does not show that this measure is the physical ground-state density, and the paper gives no normalization constants. Without this, the probability densities plotted in Figures 1–3 are not well defined.

minor comments (3)

[Section 3, sentence before Eq. (21)] The reference to '(2.)' should be Eq. (2), not '(2.)'.
[Section 2, after Eq. (11)] The line 're f| f (n)|2n = [n]' contains a typographical error; it should read '|f(n)|² n = [n]'.
[Section 3, conventions] The notation for Ψ_n is ambiguous: Eq. (20) defines Ψ_n = ⟨Xθ|n⟩_f, while Eq. (21) uses f⟨n|Xθ⟩ as though it were equal to Ψ_n. The conjugation convention should be stated explicitly.

Circularity Check

0 steps flagged · score 2.0 of 10

No load-bearing circularity; derivation self-contained, but 'explicit' wavefunctions depend on an uncomputed Psi_0 and the only self-citations are non-load-bearing.

full rationale

The central derivation (Eqs. 17-28) is self-contained: the quadrature operator is defined, the eigenvalue equation is projected onto the deformed Fock basis, and the three-term recurrence for the wavefunctions and the resulting J_n polynomials follows by algebra, with the Q->1 limit correctly reducing to the Hermite recurrence (Eq. 33) and the standard quadrature wavefunctions (Eqs. 34-35). No parameter is fitted to a target data set, and no uniqueness or existence claim is imported from the authors' prior work, so none of the seven circularity patterns is present. The citations to [12] and [19], both sharing authors, are contextual rather than load-bearing: [19] is cited for the q-deformed special case that this paper generalizes, but the ansatz (27), the recurrence (28), and the Hermite limit are re-derived here from the eigenvalue equation rather than assumed from [19]. The paper's genuine weakness is incompleteness, not circularity: after Eq. (22) it says 'This simplifies the problem at hand by reducing the unknowns to just the ground state wavefunction,' and Eq. (27) still contains the free function Psi_0(X_theta), which is never derived or specified for any deformed case (only the non-deformed limit (35) is given). Figures 1-3 plot normalized densities without stating Psi_0 or the normalization/numerical procedure, so the abstract's phrase 'used to find explicitly the wavefunctions' is overstated. There is also a minor internal sign inconsistency: the claim that Eq. (23) is the complex conjugate of Eq. (21) does not follow literally from the factors printed in Eq. (21), although Eq. (23) is the correct direct consequence of the eigenvalue equation in the bra-side convention. These are reproducibility and exposition issues, not circular reductions, so the circularity score is low (2).

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no fitted parameters; the deformation parameters q, p, Q are inputs from previously defined models. The main unstated assumptions are the existence of the ground state and the validity of the homodyne-based quadrature operator definition.

assumptions (4)

standard math Favard's theorem guarantees orthogonality of J_n if [n] > 0 for all n.
Invoked in Section 3 after Eq. (30) to assert J_n are orthogonal polynomials; the positivity of [n] for the considered deformations is not explicitly verified but holds for the stated parameter ranges.
domain assumption The deformed algebra commutation relations and [n] = |f(n)|^2 n from prior literature.
Section 2 reviews f-oscillator algebra from refs [20,21]; the paper relies on these as given.
domain assumption The quadrature operator definition Xθ = √((1+Q)/2)(A e^{-iθ} + A† e^{iθ}) (Eq. 17) is asserted based on homodyne detection without a full derivation.
Section 3 states it 'can be easily verified' from photon-number difference; this is a modeling choice.
domain assumption The ground state Ψ_0 exists and is normalizable.
All excited wavefunctions and the plotted densities depend on Ψ_0, which is never constructed.

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Cite this review

Pith. "Pith review of Quadrature operator eigenstates and wavefunctions of $f$-deformed oscillators." pith.science (2026). https://pith.science/paper/IQUB32D4

@misc{pith2026190801480,
  author       = {Pith},
  title        = {Pith review of: Quadrature operator eigenstates and wavefunctions of $f$-deformed oscillators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IQUB32D4}},
  note         = {Machine review of arXiv:1908.01480}
}

abstract

This paper is dedicated to finding the quadrature operator eigenstates and wavefunctions of the most general $f$-deformed oscillators. A definition for quadrature operator for deformed algebra is derived to obtain the quadrature operator eigenstates. A new set of polynomials are obtained using this quadrature operator and these polynomials are used to find explicitly the wavefunctions of the deformed oscillators. We have plotted wavefunctions for three different types of deformations and compared it with the wavefunctions of the non-deformed oscillator. Our result will immensely help the research groups working in the quantum state reconstruction and quantum information theory of deformed states.

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [19]

M. P. Jayakrishnan, S. Dey, M. Faizal, C. Sudheesh, Ann. Phys. 385, 584 (2017)

work page 2017

[2] [1]

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work page 1982

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work page 2017

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work page 2011

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