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Properties and dynamics of generalized squeezed states

T0 review · 3 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read For $n\ge 3$, the generalized squeezing operator produces oscillatory dynamics: the vacuum develops $n$-fold symmetric peaks and returns almost fully to the vacuum at a finite $r$, with period set by the generator's eigenvalue gap.

desk verdict A solid numerical demonstration of oscillatory generalized squeezing for n≥3, but the infinite-N claims need a self-adjointness caveat and the 'squeezing diminishes' conclusion is under-supported. read the letter →

arxiv 2411.17022 v2 pith:UAL6BWKH submitted 2024-11-26 quant-ph cond-mat.other

Sahel Ashhab , Mohammad Ayyash This is my paper · ORCID
classification quant-phcond-mat.other
keywords generalizedsqueezinghigher-ordertrisqueezedvacuumoscillatorydynamicsFock-spacetruncationeigenvaluegapnon-Gaussianstatesquantumoptics
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 analyzes the family of generalized squeezing operators $U_n(r)=\exp(r(\hat a^dagger)^n-r^*\hat a^n)$ acting on the vacuum, and claims that for $n\ge 3$ the resulting dynamics are oscillatory rather than runaway. Starting from the vacuum, the state builds an $n$-fold symmetric multi-peak structure and then returns almost completely to the vacuum at a finite value of $r$, with vacuum probability $0.991$ at $r=1.75$ for $n=3$. The oscillation period is set by the gap between the two eigenstates of the generator $H_n=i[(\hat a^dagger)^n-\hat a^n]$ that dominate the vacuum, and the maximum achievable squeezing shrinks as $n$ increases. If correct, this draws a sharp distinction between two-photon squeezing and its higher-order generalizations, and guides the design of future non-Gaussian state generation.

What carries the argument

The load-bearing object is the Hermitian generator $H_n=i[(\hat a^dagger)^n-\hat a^n]$, which turns the squeezing parameter $r$ into a dimensionless time via $U_n(r)=\exp(-i H_n r)$. The key property is that for $n\ge 3$ the vacuum state is, to good approximation, a superposition of only two eigenstates of $H_n$; the resulting two-level oscillation has frequency $\Delta E$, the eigenvalue gap, and the low-lying probability distribution is strongly localized at small photon numbers. This two-state dominance is what replaces the divergent power-series behavior identified in earlier work with bounded, periodic dynamics, and it also explains why the period survives truncation while the oscillation amplitude does not.

What would settle it

Simulate the same squeezing dynamics with a different high-photon regularization, such as a soft cutoff of a different functional form or an added photon-number-dependent diagonal term, and check whether the revival at $r\approx1.75$ for $n=3$ and the eigenvalue gap $\Delta E\approx3.528$ persist exactly as $N$ grows; if the vacuum probability at the revival shifts systematically with the cutoff, or if the period fails to match $2\pi/\Delta E$, the central claim would collapse.

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Extended reading notes

Core claim

The central discovery is that third- and higher-order squeezing is qualitatively different from displacement and two-photon squeezing: the state $U_n(r)|0\rangle$ evolves in a near-periodic cycle instead of moving indefinitely toward larger photon numbers. For $n=3$ the vacuum probability drops below $5\times 10^{-4}$ near $r=0.9$, then revives to $0.991$ at $r=1.75$; for $n=4$ it revives to $0.9989$ at $r=0.75$. The revival time is governed by the eigenvalue gap $\Delta E$ of the generator $H_n$ between the two eigenstates that carry most of the vacuum overlap: for $n=3$, $\Delta E\approx 3.528$, predicting a period $2\pi/\Delta E\approx 1.781$. The paper argues that the oscillations are physical---not a truncation artifact---because hard and soft cutoffs over three orders of magnitude in basis size give nearly identical low-photon probabilities and a truncation-independent period, even though the average photon number in the first oscillation diverges as $N$ grows.

Load-bearing premise

The load-bearing premise is that the truncated Fock-space simulations, using up to $6\times 10^4$ basis states with hard and soft cutoffs, faithfully represent the infinite-dimensional dynamics for the low-photon-number probabilities and the oscillation period, even though the average photon number diverges with $N$ for $n=3$ and $n=4$.

Editorial extensions

If this is right

  • For $n=3$, the vacuum revival probability is $0.991$ at $r=1.75$, and the period set by $\Delta E\approx 3.528$ is about $1.781$.
  • The oscillation period decreases rapidly with $n$, following an exponential scaling of $\Delta E$ with $n$ for $n\ge 3$.
  • The maximum average photon number in the first oscillation diverges with the truncation size for $n=3$ ($\propto N^{0.562}$) and $n=4$ ($\propto \log N$), but the low-photon probabilities and the period converge.
  • The maximum achievable squeezing diminishes as $n$ increases, so the straightforward higher-order squeezing operator becomes increasingly ineffective for generating non-Gaussian states.
  • Alternative Hamiltonians, such as $\hat p^n$ with $\hat p=(\hat a^dagger-\hat a)/i$ or engineered couplings that mimic two-photon squeezing, can produce states that keep growing and do not undergo the periodic revival.

Reading between the lines

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

  • If the two-eigenstate description holds beyond the simulated orders, the revival could act as a built-in 'reset' for higher-order squeezing: the same nonlinear process that creates a non-Gaussian state also erases it at a predictable squeezing strength, which might be exploited in time-domain quantum control.
  • The truncation-independent period suggests that realistic systems with natural high-photon cutoffs, such as from higher-order nonlinearities, should still exhibit the revival at moderate photon numbers, making the effect testable in current circuit-QED setups rather than requiring the large photon numbers that would make the idealized model unphysical.
  • The divergence of the average photon number while low-photon probabilities converge points to a well-defined low-energy effective dynamics in the infinite-dimensional limit; an analytic proof of that limit would be a natural extension of the numerical evidence.
  • Because the period shrinks as $n$ grows, higher-order squeezing may be most useful in short bursts or in combination with other operations, rather than as a standalone route to highly nonclassical states.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. The paper studies the generalized squeezing operators U_n(r)=exp(r(a†)^n - r*a^n) applied to the vacuum for n=1,...,8. Using truncated Fock-space simulations with hard and soft cutoffs and N up to 6×10^4, it reports that for n≥3 the dynamics are oscillatory: the state initially develops n-fold symmetric multi-peak structure, then returns almost completely to the vacuum at a finite value of r (vacuum probability 0.991 at r=1.75 for n=3, 0.9989 at r=0.75 for n=4). The period is linked to the gap ΔE between the two eigenstates of the generator with largest vacuum overlap, and the maximum average photon number is found to decrease with n. The paper also discusses alternative Hamiltonians, the analogy with Bloch oscillations, and the relevance to experiments.

Significance. The paper's strongest asset is its extensive numerical cross-validation: hard and soft cutoffs, N from 300 to 6×10^4, agreement with exact solutions for n=1 and n=2, and independent Mathematica and Numpy implementations. If the infinite-N claims were made rigorous, the qualitative difference between n≤2 and n≥3 would be valuable for higher-order squeezing experiments. However, as written, the two headline claims—that the oscillations are physical (not artefacts) and that maximum squeezing decreases with n—are not fully established, because of the non-essential self-adjointness of the generator and the use of average photon number as a proxy for squeezing.

major comments (3)
  1. [II (Eqs. 5–7), IV.B (Fig. 10)] The claim that H_n is Hermitian and therefore U_n(r)=exp(-irH_n) is unitary is insufficient for n≥3. On each Fock subspace the operator is a half-line Jacobi matrix with off-diagonal entries ~ k^{n/2}; for n≥3 these entries grow fast enough that the operator is in the limit-circle case at infinity and is not essentially self-adjoint on finite-excitation states. It therefore admits multiple self-adjoint extensions, and the finite-N hard/soft cutoffs select one boundary condition (or a family of approximating operators). The agreement between hard and soft cutoffs for P_0 and the period (Figs. 3 and 6) demonstrates convergence to one limit, but not extension independence. Since the infinite-N gap ΔE=3.528 in Fig. 10(b) and the near-return P_0=0.991 in Fig. 3 are used as evidence that the oscillations are physical, the paper must state explicitly which self-adjoint extension (or which limiting procedure) is being used and argue that the observable quantities are independent of that choice, or else restrict the claims to finite-N effective models. The paper's own observation that ⟨N⟩_max diverges with N (Fig. 8) is a concrete symptom of this sensitivity, and Sec. II already acknowledges the Hamiltonian is unbounded from below and suggests finite-N results as the physically relevant reference point, which creates an internal tension with the abstract's infinite-N claim.
  2. [III (Figs. 7–8), Conclusion] The claim that maximum squeezing diminishes with increasing squeezing order is not supported by the evidence presented. The paper computes the maximum average photon number in the first oscillation, but ⟨a†a⟩ is not a squeezing measure; a state can have a small average photon number and still be strongly squeezed in a quadrature, and vice versa. The paper does not compute quadrature variances, squeezing parameters, or any higher-order squeezing criterion. Moreover, in the infinite-N limit ⟨N⟩_max diverges for n=3 (as N^{0.56}) and n=4 (as log N), so the apparent ordering 'maximum decreases with n' depends on the chosen truncation size. The authors should either quantify squeezing directly (for example, the minimum quadrature variance as a function of r and n) or rephrase the claim to refer to the maximum deviation from the vacuum state that is actually computed.
  3. [IV.C] The argument that the wave reflection in photon-number space is a physical effect rather than a finite-size artefact is based mainly on the weak N-dependence of low-photon-number probabilities. This is suggestive but not conclusive: the reflection point could lie at a photon number that grows with N while the low-k probabilities still appear converged. A more quantitative test, such as tracking the position of the probability wavefront or the onset of the oscillatory tail as a function of N, would strengthen the central claim that the return to the vacuum is not caused by the artificial boundary.
minor comments (6)
  1. [II (after Eq. 3)] The sentence 'we restrict our analysis to real values ofr' contains a typo, and the reduction of complex r to real r via a rotation should be stated explicitly rather than only citing Ref. [27].
  2. [III (near Fig. 2)] The symmetry relation is written as R^†(2π/n)H_n R(2π/2n)=H_n; the second rotation angle appears to be a typo and should presumably be 2π/n.
  3. [Fig. 3 caption] The color description ('red/magenta, light/dark green, blue/cyan and yellow/orange') is hard to parse; please use explicit line styles or labels in addition to colors.
  4. [II (matrix exponentiation)] The statement that Mathematica and Numpy 'showed no evidence of divergences' is not meaningful for finite matrix exponentials; please rephrase to describe the actual check, such as agreement between packages and with the repeated-small-step method.
  5. [References] Reference [1] gives the title as 'Quantum Computation and Quantum Communication'; the standard title is 'Quantum Computation and Quantum Information'.
  6. [Introduction] The phrase 'Implementations of nonlinear phenomena at to the single photon level' contains a typo; delete 'at' or 'to'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the dynamics are direct numerical solutions of the stated Hamiltonian; the eigenvalue-gap comparison is an internal consistency check, not a fitted input.

full rationale

The paper's central quantities are the states U_n(r)|0> obtained by numerically exponentiating the truncated Hamiltonian built from Eqs. (6) and (7). No parameter appearing in the dynamics is fitted to the reported oscillation period, the vacuum-return probability, or the squeezing suppression. The eigenvalue gap Delta E = 3.528 is computed from the same truncated Hamiltonian and then compared with the numerically observed period through 2*pi/Delta E = 1.781; this is an internal consistency check of the two-eigenstate approximation (Fig. 9), not a construction that inserts the target result into the input. The hard- and soft-cutoff comparisons and the N-scaling studies are convergence checks; they do not redefine the quantities being predicted. The paper also explicitly acknowledges limitations: the average photon number diverges with N for n = 3 and n = 4 (Figs. 7 and 8), and realistic systems will have cutoffs far below the infinite-N limit (Sec. III). These are mathematical well-posedness and physical-relevance caveats, not circular reasoning. The cited prior work [24,27,28] supplies background on known divergences, and neither it nor the peripheral self-citation [23] is used to force the central claim. Accordingly, there are no circular steps of any of the enumerated kinds.

Assumptions & free parameters 1 free parameters · 2 assumptions · 0 invented entities

No new particles, forces, or dimensions are introduced. The only free modeling choices are the numerical truncation and cutoff shape, which the paper tests for robustness.

free parameters (1)
  • Soft-cutoff suppression factor = sin[5π(N-k)/(2N)] applied for k>0.8N
    Hand-chosen to smooth the high-energy truncation; the authors test other soft-cutoff forms and find qualitatively similar results, so it is not load-bearing for the central claim.
assumptions (2)
  • domain assumption The truncated Fock-space dynamics with hard or soft cutoffs approximately represents the infinite-dimensional dynamics for low-photon-number probabilities and the oscillation period.
    Section II and III rely on this to claim the oscillations are physical; checks with N up to 6×10^4 and different cutoffs show weak dependence, but no analytical proof is given, and the mean photon number is non-convergent.
  • domain assumption The effective Hamiltonian H_n = i[(a†)^n - a^n] (Eq. 6) is the correct generator for generalized squeezing in experimental settings.
    Section IV.E derives this from the RWA Hamiltonian in Eq. (13) and argues it arises naturally; the experimental relevance of the predictions depends on this modeling assumption, including the neglect of higher-order nonlinearities beyond a photon-number cutoff.

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Cite this review

Pith. "Pith review of Properties and dynamics of generalized squeezed states." pith.science (2026). https://pith.science/paper/UAL6BWKH

@misc{pith2026241117022,
  author       = {Pith},
  title        = {Pith review of: Properties and dynamics of generalized squeezed states},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UAL6BWKH}},
  note         = {Machine review of arXiv:2411.17022}
}
read the original abstract

We analyze the properties and dynamics of generalized squeezed states. We find that, in stark contrast to displacement and two-photon squeezing, higher-order squeezing leads to oscillatory dynamics. The state is squeezed in the initial stages of the dynamics but the squeezing reverses at later stages, and the state reverts almost completely back to the initial state. We analyze various quantities to verify that the oscillatory dynamics is physical and not a mathematical artefact. We also show that the maximum squeezing diminishes with increasing squeezing order, rendering the squeezing mechanism increasingly ineffective. Our results provide important rules that can help guide the development of more effective higher-order squeezing techniques.

Figures

Figures reproduced from arXiv: 2411.17022 by the authors.

Figure 1
Figure 1. FIG. 1: Matrix elements for the matrix [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Q and Wigner functions for the trisqueezed vacuum state, i.e. [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Occupation probabilities of the lowest four relevant Fock states (i.e. [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Occupation probabilities of the lowest 11 relevant states ( [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: Q and Wigner functions for the four-photon squeezed vacuum state, i.e. [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Occupation probabilities of the lowest four relevant Fock states (i.e. [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: Average photon number [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: Maximum average photon number [PITH_FULL_IMAGE:figures/full_fig_p014_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9: Highest ten probabilities of the Hamiltonian eigenstates based on the overlap between these [PITH_FULL_IMAGE:figures/full_fig_p017_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10: The difference ∆ [PITH_FULL_IMAGE:figures/full_fig_p018_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11: Probability distribution (as a function of photon number [PITH_FULL_IMAGE:figures/full_fig_p019_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12: Q and Wigner functions for the squeezed vacuum state obtained using the squeezing [PITH_FULL_IMAGE:figures/full_fig_p022_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13: Q and Wigner functions for quantum states that are constructed by taking the amplitudes [PITH_FULL_IMAGE:figures/full_fig_p024_13.png]

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