Fractional squeezing: spectra and dynamics from generalized squeezing Hamiltonian with fractional orders

Sahel Ashhab

REVIEW 4 major objections 5 minor 24 references

Fractional squeezing: spectra and dynamics from generalized squeezing Hamiltonian with fractional orders

T0 review · 4 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read By treating the generalized-squeezing order n as a continuous variable, this paper argues that the squeezing Hamiltonian's spectrum switches from continuous to discrete exactly at n=2, and that the oscillation amplitude switches from diverg

desk verdict A clever fractional-order interpolation of the squeezing Hamiltonian with finite-size scaling identifies n=2 and n=4 as critical points, but the conclusions may depend on the interpolation, which is not tested. read the letter →

arxiv 2601.15693 v2 pith:YHOYXEV5 submitted 2026-01-22 quant-ph

Sahel Ashhab This is my paper · ORCID

classification quant-ph

keywords GeneralizedsqueezingFractionalorderGamma-functioninterpolationSpectrumcontinuityCriticalpointsPhoton-numberdivergenceFinite-sizescalingQuantumoptics

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

This paper tries to settle where qualitative changes occur in generalized squeezing—the process that creates photons in groups of n—by allowing n to take fractional values. The author's central claim is that the spectrum of the squeezing Hamiltonian is continuous for n up to 2 and discrete for n above 2, and that the photon-number oscillation amplitude diverges for n below 4 but becomes finite for n at or above 4, with n=4 exhibiting a slow logarithmic divergence. A sympathetic reader would care because the integer cases n=2 and n=4 are exactly the ones that resist direct numerical study, and the paper's interpolation-and-extrapolation strategy turns them from unreachable points into predictable boundaries. The payoff is a clearer map of how multiphoton squeezing behaves as the order changes, with practical guidance for which truncations and approximations work.

What carries the argument

The central object is the fractional-order squeezing Hamiltonian matrix H(n) whose off-diagonal elements are sqrt(Gamma((k+1)n+1)/Gamma(kn+1)), obtained by replacing factorials with Gamma functions so that n becomes a continuous parameter. The extrapolation machinery consists of fitting the truncation-size dependence of the smallest positive eigenvalue to E_min = E_min,∞ + δE_min N^{-α} and the renormalized photon number to ⟨m⟩ = A N^{-α} + B, which lets the paper read off the infinite-N limits and the scaling exponents. A secondary explanatory device is a toy 'hierarchical' Hamiltonian with rapidly increasing off-diagonal elements, which shows why the middle-spectrum eigenstates become loca

What would settle it

Compute the smallest positive eigenvalue E_min for n=2 using much larger truncation sizes than those reported; if the extrapolated E_min tends to a positive finite value rather than zero, the claimed continuous spectrum at n=2 is wrong. Likewise, at n=4, compute ⟨m⟩ for truncation sizes beyond the range used here: if the growth is faster than logarithmic (e.g., a small power law), the claimed logarithmic scaling and the n=4 critical boundary would be refuted.

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Extended reading notes

Core claim

The author generalizes the generalized-squeezing Hamiltonian to fractional order by replacing the factorial ratios in the matrix elements with Gamma-function ratios, so that the squeezing order n can be any positive real number. Using finite-N truncations and power-law extrapolations of the smallest positive eigenvalue and the renormalized photon number, the paper identifies two critical points: n=2, where the spectrum changes from continuous to discrete, and n=4, where the photon-number oscillation amplitude changes from asymptotically infinite to finite. At n=4 the data follows a logarithmic scaling law, which is why the power-law fit fails there; for n>4 the extrapolated photon number app

Load-bearing premise

The argument rests on trusting finite-N truncations and power-law fits to extrapolate to the infinite-N limit near n=2 and n=4, even though fractional n lacks a physical multiphoton interpretation and the fit at n=4 fails, as the paper itself acknowledges.

Editorial extensions

If this is right

The spectrum of two-photon squeezing, previously debated in finite-N simulations, is continuous, consistent with the standard continuous-spectrum treatment of two-photon squeezing dynamics.
At n=4 the photon-number oscillation amplitude diverges only logarithmically, so it is infinite in the infinite-size limit but weaker than any power-law divergence.
For n>4 the photon-number oscillation amplitude is finite, and for n≥5 it stabilizes close to the value 0.5 predicted by a two-level truncation.
The fractional-order extrapolation method can predict behavior at computationally hard integer points, offering a template for studying other parameter-dependent quantum-optical Hamiltonians such as the multiphoton quantum Rabi model.
In the large-n limit, the low-energy physics of generalized squeezing is governed almost entirely by the two lowest Fock states, giving a simple asymptotic description.

Reading between the lines

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

If the Gamma-function interpolation is more than a numerical device, it may be possible to define fractional powers of the creation and annihilation operators algebraically, and to check whether the resulting spectra match physical implementations of nonlinear interactions.
The linear scaling of the exponent α on both sides of n=2 hints at a closed-form scaling law for the small-n discrete-to-continuous transition that the paper does not derive; extracting that law from the fit data could be a direct follow-up.
The same finite-N fitting strategy could be applied to the multiphoton quantum Rabi model to locate its critical coupling thresholds, which the paper mentions only as a suggestion.
The parity-sensitivity of dynamics for n>4, explained by the hierarchical toy model, suggests that experiments probing high-order squeezing may need to control the truncation parity of the effective Hilbert space, an implication the paper leaves implicit.

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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

Dear colleague,

Quick take: this paper is a clear, honest numerical study that introduces a Gamma-function interpolation of the generalized-squeezing Hamiltonian to sweep the squeezing order n continuously, and uses finite-size scaling to pin down n=2 (continuous-to-discrete spectrum) and n=4 (divergence-to-finite oscillation amplitude). The idea is genuinely useful and the paper is well-written. But the central claims rest on a single interpolation choice, so the exact critical orders and exponents are not as secure as the abstract suggests.

What's new: the interpolation itself, and the large-n explanation via a hierarchical toy model that reproduces the numerical behavior. The paper is also fair about its own limits: it flags the conceptual oddness of fractional n, acknowledges the fit fails at n=4, and attributes the n=2 deviation to numerical error. That's refreshing.

Soft spots: first, the Gamma function is only one of infinitely many smooth extensions of the factorial. The paper never argues it's the natural choice, and no robustness check is done with an alternative interpolation. The asymptotic growth of the matrix elements is indeed interpolation-independent, which makes a threshold near n=2 and n=4 plausible, but the exact locations and scaling exponents could shift. Second, the finite-size fits (Eqs. 7 and 9) are three-parameter fits without error bars, and the critical cases are exactly where the fit is least reliable. The n=4 point is inferred from the failure of a power-law fit rather than from a successful fit. Third, no code or data is provided (only "available on request"), which makes it hard to reproduce or test the interpolation dependence.

That said, the conclusions are probably correct. They align with prior expectations for integer n, and the paper is careful not to overclaim—the word 'likely' appears where it should. I don't think the interpolation ambiguity is fatal, but it should be addressed in revision.

Who this is for: people working on generalized squeezing, multi-photon Rabi-like models, and nonlinear bosonic systems. They'll find it a useful computational tool and a plausible resolution to the conflicting claims in Refs. [17-21].

Recommendation: send to peer review. A serious referee can push for a robustness check across interpolations and for error bars or a more principled extrapolation scheme. With those additions, it could be a solid contribution.

Referee Report

4 major / 5 minor

Summary. The manuscript generalizes the generalized-squeezing Hamiltonian H_n = i[(a^†)^n - a^n] to real (fractional) orders n by replacing the factorial matrix elements in Eq. (4) with Gamma functions, giving Eq. (5). The author performs finite-N truncation diagonalizations for n ∈ [0,6] and fits the lowest positive eigenvalue E_min and the renormalized photon-number expectation value ⟨m⟩ to power-law-plus-offset forms (Eqs. (7) and (9)). From these fits, the paper infers that the spectrum is continuous for n ≤ 2 and discrete for n > 2, that the generalized-squeezing oscillation amplitude diverges for n < 4 and becomes finite for n ≥ 4, with n = 4 exhibiting logarithmic scaling, and that large-n behavior is captured by a hierarchical toy Hamiltonian. The central claim is that fractional-order calculations provide a reliable way to locate and characterize the critical points n = 2 and n = 4, which are hard to treat by direct integer-order simulation.

Significance. If the inferred critical points are robust, this would provide a clean global picture of generalized squeezing and resolve ambiguities left by integer-order studies, e.g., the continuity of the spectrum at n = 2 and the divergence behavior at n = 4. The paper is honest in emphasizing that fractional n is a mathematical tool rather than a physical parameter (Sec. II) and in acknowledging the fit failure at n = 4 (Sec. III.B). The Gamma-function interpolation is explicit and reproducible from the equations, and the hierarchical toy model gives a simple, testable explanation of the large-n trend. However, the central inference rests on a non-unique interpolation and on three-parameter fits without error bars at exactly the points where the fits are least reliable. The manuscript does not yet establish that the critical values n = 2 and n = 4 are independent of the interpolation choice or that the finite-size extrapolations are converged.

major comments (4)

[Sec. II, Eq. (5)] The continuous family Ĥ(n) is defined by replacing the factorial matrix elements of Eq. (4) with Gamma functions. This is one of infinitely many smooth extensions that agree at integer n, and the paper itself states that fractional n has no physical interpretation. The inferred critical points n = 2 and n = 4 and the associated scaling exponents could, in principle, depend on the specific interpolation: the leading large-k growth h_k ~ (kn)^{n/2} is interpolation-independent in a broad class of extensions, but sub-leading terms can shift the thresholds. Because the paper's conclusions about the physical integer cases are drawn from the fractional-n data, please provide a robustness check under alternative factorial interpolations (e.g., Γ(n+1) times a slowly varying or subexponential factor) or an analytic derivation of the thresholds from an interpolation-independent quantity. Without
[Sec. III.A, Eq. (7)] The conclusion that E_min,∞ = 0 for n ≤ 2 is obtained by fitting E_min(N) to E_min,∞ + δE_min N^{-α}. In the critical region near n = 2, the fitted α deviates from the extrapolated linear behavior, and the paper attributes this to 'likely' numerical errors without supporting statistics. Since E_min,∞ is itself the fitted intercept, the zero/nonzero status of the asymptotic eigenvalue at n = 2 is read off from the same fitting function used to define it. Please provide convergence evidence that is not based solely on this fit: for example, include odd-N data, Richardson extrapolation, or a rigorous upper/lower bound on E_min(N), and report confidence intervals for the fitted parameters. The current graphical evidence is suggestive but not conclusive at the critical point.
[Sec. III.B, Eq. (9)] The fit ⟨m⟩ = A N^{-α} + B is explicitly stated to fail at n = 4 ('the fitting fails and generates an error in our numerical fitting calculations'). This is precisely the value of n at which the paper locates the transition from divergent to finite ⟨m⟩. The logarithmic scaling at n = 4 is taken from Ref. [17] rather than demonstrated by the present fitting procedure. Consequently, the boundary n = 4 and the claim that α crosses zero at n = 4 are inferred indirectly from fits that are unreliable exactly at the point of interest. Please add a dedicated analysis at n = 4 (e.g., a direct logarithmic fit with residuals), and quantify the uncertainty in the inferred transition point, for instance by bootstrapping over truncation-size ranges or by comparing fits with and without the n = 4 data.
[Sec. III, Figs. 2 and 5] All load-bearing conclusions rely on three-parameter nonlinear fits (Eqs. (7) and (9)) to truncation-size data. No error bars, residual analyses, or goodness-of-fit metrics are reported, and only even values of N are used. This is especially concerning because the fit models are used to locate critical points at which the models themselves break down. Even if the qualitative picture is correct, the paper would be substantially strengthened by reporting uncertainties on E_min,∞, α, A, and B and by showing that the results are stable under removing the smallest few N data points.

minor comments (5)

[Sec. III.B] Typo: 'intuitively undestood' should be 'intuitively understood'; 'logarithim scaling law' should be 'logarithmic scaling law'.
[Figure 2 caption] The inset and axis labels contain apparent rendering issues: '10□5' and '10□3' should presumably be 10^{-5} and 10^{-3} or similar. Please check the figure output.
[Sec. IV] The sentence 'We can therefore restrict ourselves to the state space extending from 1 to N−2' after removing the two highest states is slightly confusing about indexing; clarify whether states are numbered 1,...,N and which indices remain.
[Sec. II] The paragraph on fractional calculus and fractional spatial dimensions is useful context, but it is not connected to the rest of the paper. A sentence explaining why these analogies are not pursued further would help set expectations.
[Sec. III.A, Eq. (7)] In Eq. (7), the notation δE_min is used as a fitting parameter, which is nonstandard and could be confused with a small error. Consider renaming it c or E_1 to avoid ambiguity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the fractional-n construction is an independent interpolation, and the extrapolated critical points are inferred from numerical fits rather than inserted by definition.

full rationale

The paper's central move is to replace factorial matrix elements in Eq. (4) with Gamma functions in Eq. (5), producing a continuous family of truncated Hamiltonians. The integer-n results are not used to define this family: Eq. (5) is an independent extension that agrees with the integer cases but is not constrained by the conclusions (continuous spectrum for n<=2, finite oscillation amplitude for n>=4). The subsequent fits, Eqs. (7) and (9), estimate asymptotic intercepts and scaling exponents from finite-N data; the inferred critical values n=2 and n=4 are extrapolations of those fitted curves, not quantities already contained in the fit inputs. The paper explicitly flags the conceptual limitation that fractional n has no physical interpretation (Sec. II after Eq. (5)); this is a robustness concern about interpolation nonuniqueness, not circularity. The citations to Refs. [17] and [19] are to prior published work by the same group, but they are used as context and benchmarks (e.g., the n=4 logarithmic scaling in Ref. [17]) rather than as unverified uniqueness theorems that force the conclusions; the present paper's own finite-N data independently generates the fitted trends. No equation in the paper reduces to another by construction, and no fitted parameter is renamed as a prediction. The derivation chain is self-contained given the Gamma-interpolation ansatz; whether that ansatz is physically privileged is a validity question, not a circularity.

Assumptions & free parameters 4 free parameters · 5 assumptions · 1 invented entities

The central claims rest on a newly invented fractional-n Hamiltonian (Gamma interpolation), an assumed finite-size scaling form, and several fitted parameters. None of these is independently validated by external results, and the paper's own statements acknowledge the lack of physical meaning for fractional n and the fitting failure at n=4.

free parameters (4)

E_min,∞ (per-n asymptotic smallest eigenvalue) = ≈0 for n < 2; rises rapidly for n > 2
Fitted constant term in Eq. (7); the conclusion that the spectrum is continuous below n=2 and discrete above n=2 rests on this fitted intercept.
α (per-n convergence exponent for E_min) = Decreases linearly to 0 at n=2 for n<2; increases linearly for n>2
Fitted exponent in Eq. (7); extrapolation of the α lines to n=2 is the main evidence for locating the continuous/discrete boundary.
A, B, α in the ⟨m⟩ scaling fit = B→0.5 as n→∞; α changes sign at n=4
Parameters in Eq. (9); used to infer n=4 as the boundary between divergent and finite oscillation amplitude. The n=4 fit fails and is not reported.
Toy-model hierarchy ratio h_{j+1}/h_j = 10 or 100
Chosen by hand for the illustrative simulations in Sec. IV; not used for quantitative conclusions about the actual squeezing Hamiltonian.

assumptions (5)

ad hoc to paper Gamma-function interpolation of the squeezing-order Hamiltonian (replacing factorials by Γ) defines a meaningful continuous family interpolating integer squeezing orders.
Eq. (5) defines Ĥ(n) for fractional n; the paper acknowledges there is no physical interpretation for fractional n and treats it as a mathematical tool (Sec. II).
domain assumption Finite-size extrapolation using Eq. (7) correctly gives the infinite-N asymptotic E_min, with a power-law ansatz.
Used throughout Sec. III.A to infer E_min,∞=0 versus >0; no proof that the asymptotic form holds, especially at n=2.
domain assumption The two eigenstates closest to zero characterize the squeezed-vacuum dynamics and spectrum for n>2.
Sec. II states 'the vacuum state is typically well approximated by a superposition of only the two eigenstates...'; Sec. III.B notes this fails for n<2.
domain assumption Even truncation sizes N are sufficient; parity effects do not change the inferred boundaries.
The simulations always choose even N; parity sensitivity for n>4 is discussed qualitatively but not systematically accounted for in the fits.
standard math Standard finite-dimensional linear algebra and numerical diagonalization are valid for the truncated matrix.
Numerical diagonalization is used throughout; the paper explicitly sets aside questions of self-adjointness of the infinite operator in Sec. II.

invented entities (1)

Fractional-order squeezing Hamiltonian Ĥ(n) with Gamma-function matrix elements
purpose: Continuous interpolation to locate qualitative transitions in integer-order generalized squeezing
No physical process generates fractional-n photons; the paper explicitly says there is no simple physical interpretation (Sec. II). No predicted observable independently validates the construction.

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Cite this review

Pith. "Pith review of Fractional squeezing: spectra and dynamics from generalized squeezing Hamiltonian with fractional orders." pith.science (2026). https://pith.science/paper/YHOYXEV5

@misc{pith2026260115693,
  author       = {Pith},
  title        = {Pith review of: Fractional squeezing: spectra and dynamics from generalized squeezing Hamiltonian with fractional orders},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YHOYXEV5}},
  note         = {Machine review of arXiv:2601.15693}
}

abstract

We generalize the generalized-squeezing problem to include fractional values of the squeezing order $n$. This approach allows us to determine the locations of critical points at which qualitative changes in behaviour occur and accurately predict the behaviour at these critical points, which are challenging for conventional computational methods. Based on our numerical calculations, we identify with a high degree of confidence the point at which the spectrum turns from continuous to discrete and the point at which oscillations turn from having asymptotically infinite amplitudes to having finite amplitudes. Furthermore, we numerically investigate the behaviour in the large $n$ regime and provide an intuitive explanation for the numerical results.

Figures

Figures reproduced from arXiv: 2601.15693 by the authors.

**Figure 2.** FIG. 2: (a) The asymptotic value of [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** FIG. 3: (a) Expectation value of the renormalized photon number operator [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]

**Figure 5.** Figure 5: FIG. 5: Fitting parameters obtained by fitting the ( [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

24 extracted references · 4 linked inside Pith

[17]

Ashhab and M

S. Ashhab and M. Ayyash, Properties and dynamics of generalized squeezed states, New J. Phys.27, 054104 (2025)

2025
[1]

D. F. Walls and G. J. Milburn,Quantum Optics(Springer, Berlin, 1994)

1994
[2]

M. O. Scully and M. S. Zubairy,Quantum Optics(Cambridge University Press, 1997). 13

1997
[3]

P. D. Drummond and M. Hillery,The Quantum Theory of Nonlinear Optics(Cambridge University Press, 2014)

2014
[4]

Felicetti, D

S. Felicetti, D. Z. Rossatto, E. Rico, E. Solano, and P. Forn-D ´ ıaz, Two-photon quantum Rabi model with superconducting circuits, Phys. Rev. A97, 013851 (2018)

2018
[5]

C. W. S. Chang, C. Sab ´ ın, P. Forn-D ´ ıaz, F. Quijandr ´ ıa, A. M. Vadiraj, I. Nsanzineza, G. Johansson, and C. M. Wilson, Observation of three-photon spontaneous parametric down- conversion in a superconducting parametric cavity, Phys. Rev. X10, 011011 (2020)

2020
[6]

A. M. Eriksson, T. S´ epulcre, M. Kervinen, T. Hillmann, M. Kudra, S. Dupouy, Y. Lu, M. Khanahmadi, J. Yang, C. Castillo-Moreno, P. Delsing, and S. Gasparinetti, Universal control of a bosonic mode via drive-activated native cubic interactions, Nature Commun.15, 2512 (2024)

2024
[7]

Ayyash, X

M. Ayyash, X. Xu, S. Ashhab, and M. Mariantoni, Driven multiphoton qubit-resonator inter- actions, Phys. Rev. A110, 053711 (2024)

2024

Show all 24 references

[8]

B˘ az˘ avan, S

O. B˘ az˘ avan, S. Saner, D. J. Webb, E. M. Ainley, P. Drmota, D. P. Nadlinger, G. Araneda, D. M. Lucas, C. J. Ballance, and R. Srinivas, Squeezing, trisqueezing, and quadsqueezing in a spin-oscillator system, arXiv:2403.05471

arXiv
[9]

Saner, O

S. Saner, O. B˘ az˘ avan, D. J. Webb, G. Araneda, D. M. Lucas, C. J. Ballance, and R. Srini- vas, Generating arbitrary superpositions of nonclassical quantum harmonic oscillator states, arXiv:2409.03482

arXiv
[10]

R. A. Fisher, M. M. Nieto and V. D. Sandberg, Impossibility of naively generalizing squeezed coherent states, Phys. Rev. D29, 1107 (1984)

1984
[11]

Hillery, M

M. Hillery, M. S. Zubairy, and K. W´ odkiewicz, Squeezing in higher order nonlinear optical processes, Phys. Lett. A103, 259 (1984)

1984
[12]

C. K. Hong and L. Mandel, Higher-order squeezing of a quantum field, Phys. Rev. Lett.54, 323 (1985)

1985
[13]

S. L. Braunstein and R. I. McLachlan, Generalized squeezing, Phys. Rev. A35, 1659 (1987)

1987
[14]

S. L. Braunstein and C. M. Caves, Phase and homodyne statistics of generalized squeezed states, Phys. Rev. A42, 4115 (1990)

1990
[15]

Hillery, Photon number divergence in the quantum theory of n-photon down conversion, Phys

M. Hillery, Photon number divergence in the quantum theory of n-photon down conversion, Phys. Rev. A42, 498 (1990)

1990
[16]

Braak, Thek-photon quantum Rabi model, to appear in T

D. Braak, Thek-photon quantum Rabi model, to appear in T. Takagiet al.(Eds.), 14 Mathematical Foundations for Post-Quantum Cryptography(Springer, Singapore, 2026); arXiv:2401.02370

2026 arXiv
[18]

Properties and dynamics of generalized squeezed states

R. Gordillo and R. Puebla, Comment on “Properties and dynamics of generalized squeezed states”, arXiv:2507.12250
[19]

Ashhab, F

S. Ashhab, F. Fischer, D. Lonigro, D. Braak, and D. Burgarth, Finite-dimensional approxi- mations of generalized squeezing, Phys. Rev. A113, 013703 (2026)

2026
[20]

Fischer, D

F. Fischer, D. Burgarth, and D. Lonigro, Self-adjoint realizations of higher-order squeezing operators, arXiv:2508.09044

arXiv
[21]

Properties and dynamics of generalized squeezed states

S. Ashhab and M. Ayyash, Reply to Comment on “Properties and dynamics of generalized squeezed states”, arXiv:2601.07227
[22]

Laskin, Fractional Quantum Mechanics, Phys

N. Laskin, Fractional Quantum Mechanics, Phys. Rev. E62, 3135 (2000)

2000
[23]

Herrmann,Fractional Calculus: An Introduction for Physicists(World Scientific, Singa- pore, 2011)

R. Herrmann,Fractional Calculus: An Introduction for Physicists(World Scientific, Singa- pore, 2011)

2011
[24]

Goldenfeld,Lectures On Phase Transitions And The Renormalization Group(Taylor & Francis, Boca Raton, 1992)

N. Goldenfeld,Lectures On Phase Transitions And The Renormalization Group(Taylor & Francis, Boca Raton, 1992). 15

1992

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

[1] [17]

Ashhab and M

S. Ashhab and M. Ayyash, Properties and dynamics of generalized squeezed states, New J. Phys.27, 054104 (2025)

2025

[2] [1]

D. F. Walls and G. J. Milburn,Quantum Optics(Springer, Berlin, 1994)

1994

[3] [2]

M. O. Scully and M. S. Zubairy,Quantum Optics(Cambridge University Press, 1997). 13

1997

[4] [3]

P. D. Drummond and M. Hillery,The Quantum Theory of Nonlinear Optics(Cambridge University Press, 2014)

2014

[5] [4]

Felicetti, D

S. Felicetti, D. Z. Rossatto, E. Rico, E. Solano, and P. Forn-D ´ ıaz, Two-photon quantum Rabi model with superconducting circuits, Phys. Rev. A97, 013851 (2018)

2018

[6] [5]

C. W. S. Chang, C. Sab ´ ın, P. Forn-D ´ ıaz, F. Quijandr ´ ıa, A. M. Vadiraj, I. Nsanzineza, G. Johansson, and C. M. Wilson, Observation of three-photon spontaneous parametric down- conversion in a superconducting parametric cavity, Phys. Rev. X10, 011011 (2020)

2020

[7] [6]

A. M. Eriksson, T. S´ epulcre, M. Kervinen, T. Hillmann, M. Kudra, S. Dupouy, Y. Lu, M. Khanahmadi, J. Yang, C. Castillo-Moreno, P. Delsing, and S. Gasparinetti, Universal control of a bosonic mode via drive-activated native cubic interactions, Nature Commun.15, 2512 (2024)

2024

[8] [7]

Ayyash, X

M. Ayyash, X. Xu, S. Ashhab, and M. Mariantoni, Driven multiphoton qubit-resonator inter- actions, Phys. Rev. A110, 053711 (2024)

2024