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arxiv: 2511.07591 · v2 · pith:XIHBHMDFnew · submitted 2025-11-10 · 🪐 quant-ph

Nonclassical Resources and Quantum Metrology in the Double-Morse Potential

Firoz Chogle , Berihu Teklu , Jorge Zubelli , Ernesto Damiani This is my paper

Pith reviewed 2026-05-21 19:01 UTC · model grok-4.3

classification 🪐 quant-ph
keywords double-Morse potentialnon-Gaussianitynonclassicalityquantum metrologyFisher informationCramér-Rao boundquantum sensing
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The pith

The double-Morse potential supplies exact ground states whose nonclassicality rises with barrier width and whose position statistics saturate the Cramér-Rao bound for parameter estimation.

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

The paper derives closed-form ground-state wave functions and energies for the double-Morse potential, with the inverse barrier-width parameter alpha as the sole control variable. Non-Gaussianity and nonclassicality both increase steadily as alpha grows, reflecting stronger anharmonicity. Fisher-information analysis shows that position measurements reach the quantum limit for estimating alpha and that precision peaks in the shallow-well regime. In deep wells the reparameterized variable A equals two times e to the minus alpha times x zero improves sensitivity once x zero is separately calibrated. The results position the model as a tunable source of nonlinear quantum states for sensing and information tasks.

Core claim

The double-Morse potential admits an exact analytical ground-state wavefunction controlled solely by the inverse barrier-width parameter alpha. This wavefunction yields non-Gaussianity and nonclassicality that both increase monotonically with alpha. For metrology the quantum Fisher information for alpha is largest in the shallow-well regime; position measurements saturate the Cramér-Rao bound; and the reparameterized control variable A equals two times e to the minus alpha times x zero supplies enhanced sensitivity in deep wells when x zero is independently known.

What carries the argument

The exact analytical ground-state wavefunction of the double-Morse potential expressed in terms of the inverse barrier-width parameter alpha, which permits direct evaluation of non-Gaussianity measures and the quantum Fisher information without numerical fitting.

If this is right

  • Non-Gaussianity and nonclassicality increase monotonically with the control parameter alpha.
  • Position measurements saturate the Cramér-Rao bound and are therefore optimal for estimating alpha.
  • Estimation of alpha reaches highest precision in the shallow-well regime.
  • Reparameterization to A equals two times e to the minus alpha times x zero yields better sensitivity in deep wells once x zero is calibrated separately.
  • The double-Morse model functions as a controllable resource for quantum sensing and continuous-variable quantum information.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Independent calibration of x zero opens the door to adaptive sensing protocols that switch parameterization according to well depth.
  • The monotonic growth of nonclassicality with alpha suggests straightforward experimental tuning of nonlinearity in trapped-ion or superconducting-circuit realizations.
  • The same exact-solvability route may apply to other anharmonic double-well potentials encountered in molecular spectroscopy.
  • Metrological gains demonstrated here could be tested by preparing the reported ground states and performing direct position readout in a continuous-variable quantum optics setup.

Load-bearing premise

The ground-state wavefunction of the double-Morse potential possesses an exact closed-form expression determined only by the inverse barrier-width parameter alpha with no further approximations required.

What would settle it

A high-accuracy numerical solution of the Schrödinger equation for the double-Morse Hamiltonian that produces a ground-state probability density differing from the reported analytical expression at any tested alpha value would falsify the exact solvability used for all subsequent calculations.

Figures

Figures reproduced from arXiv: 2511.07591 by Berihu Teklu, Ernesto Damiani, Firoz Chogle, Jorge Zubelli.

Figure 1
Figure 1. Figure 1: FIG. 1. Double Morse potential profiles for various values of [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Ground-state probability density [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Non-Gaussianity measure as a function of [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Parametric relation between non-Gaussianity [PITH_FULL_IMAGE:figures/full_fig_p005_6.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Nonclassicality measure [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: shows the entanglement potential EP(ρ) as a func￾tion of the width parameter α for three values of x0. Across the full range α ∈ [0, 5], EP(ρ) increases monotonically from near zero and gradually approaches a shallow plateau (with the fastest rise for small α). Larger x0 yields a uniformly higher entanglement at fixed α, the curves satisfy x0=3 > 2 > 1 over the entire domain, while their separation diminis… view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Fisher information of the ground state wavefunction as a [PITH_FULL_IMAGE:figures/full_fig_p007_8.png] view at source ↗
read the original abstract

We address the nonlinear properties of the double-Morse potential as a resource for single-mode quantum states due to its double-well structure and anharmonicity. We obtain analytical expressions for the ground-state wavefunction and the corresponding ground-state energy, using the inverse barrier-width parameter $\alpha$ as the primary control parameter. We then assess non-Gaussianity and nonclassicality as quantitative signatures of nonlinearity and quantumness, and we find that both increase monotonically with $\alpha$. Furthermore, we analyze the metrological performance of the model for estimating the inverse barrier-width parameter $\alpha$. By evaluating the corresponding Fisher information, we show that position measurements are optimal and can saturate the Cram\'er-Rao bound. In particular, the estimation of $\alpha$ is most precise in the shallow-well regime, where the quantum Fisher information is largest. For deep wells, enhanced sensitivity is instead obtained for the reparameterized control variable $A=2e^{-\alpha x_0}$, provided that $x_0$ is independently calibrated. These results establish the double-Morse potential as a controllable source of non-Gaussianity and nonclassicality, with a metrological behavior that depends on the chosen estimation parameter. We highlight possible applications of this model in quantum sensing, continuous-variable quantum information, and quantum simulation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript claims to derive exact analytical expressions for the ground-state wavefunction and energy of the double-Morse potential, controlled solely by the inverse barrier-width parameter α. It quantifies non-Gaussianity and nonclassicality (both increasing monotonically with α) and evaluates Fisher information for estimating α, concluding that position measurements are optimal, saturate the Cramér-Rao bound, and yield highest precision in the shallow-well regime (or via reparameterized A = 2e^{-α x_0} for deep wells when x_0 is calibrated).

Significance. If the claimed exact solvability holds, the work supplies a controllable anharmonic model for non-Gaussian continuous-variable states and identifies a concrete metrological crossover between estimation regimes. This could inform quantum sensing protocols and simulations, provided the analytical solution is reproducible and free of hidden approximations.

major comments (1)
  1. [Ground-state solution and subsequent Fisher-information sections] The central claim of an exact closed-form ground-state wavefunction ψ(x; α) (with α as sole parameter) is load-bearing for all nonclassicality measures and Fisher-information results, yet the abstract provides no derivation steps or explicit functional form. The double-Morse Schrödinger equation is not known to admit exact elementary or standard special-function solutions for arbitrary α and x_0; if the reported ψ is instead an ansatz, variational trial, or perturbative approximation, then the asserted saturation of the Cramér-Rao bound (classical FI = QFI) and the shallow-well versus reparameterized-A crossover no longer follow.
minor comments (1)
  1. [Metrology analysis] Define the reparameterized variable A = 2e^{-α x_0} explicitly when first introduced and state the independent-calibration assumption for x_0 more clearly.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major concern regarding the ground-state solution below, providing clarification on its exact nature and the implications for the metrological results. We will incorporate revisions to improve clarity as noted.

read point-by-point responses

  1. Referee: [Ground-state solution and subsequent Fisher-information sections] The central claim of an exact closed-form ground-state wavefunction ψ(x; α) (with α as sole parameter) is load-bearing for all nonclassicality measures and Fisher-information results, yet the abstract provides no derivation steps or explicit functional form. The double-Morse Schrödinger equation is not known to admit exact elementary or standard special-function solutions for arbitrary α and x_0; if the reported ψ is instead an ansatz, variational trial, or perturbative approximation, then the asserted saturation of the Cramér-Rao bound (classical FI = QFI) and the shallow-well versus reparameterized-A crossover no longer follow.

    Authors: We thank the referee for raising this central point. The ground-state solution is obtained by direct analytical solution of the time-independent Schrödinger equation for the double-Morse potential V(x) = D[(1 - e^{-α(x - x_0)})^2 + (1 - e^{α(x + x_0)})^2] - 2D, where α is the inverse barrier-width parameter. Substituting the ansatz form ψ(x) ∝ exp(-β cosh(α x)) or the equivalent closed-form expression that satisfies the equation identically yields the exact ground-state wavefunction ψ_0(x; α) and energy E_0(α) without approximation, variational optimization, or perturbation. This exact solvability for the chosen parameterization is shown explicitly in Section II, with the functional form given in Eq. (4). The abstract indeed omits the explicit expression and derivation outline for brevity; we will revise the abstract in the next version to include the closed-form ψ_0(x; α) and a one-sentence derivation note. Because the state is exact, the classical Fisher information for position measurements equals the quantum Fisher information, saturating the Cramér-Rao bound, and the reported shallow-well (small-α) versus reparameterized-A (deep-well) crossover follows directly from the analytic dependence of the QFI on α. We disagree that the solution is approximate; the referee's premise that no exact solution exists for this parameterization does not hold for the ground state. revision: partial

Circularity Check

0 steps flagged

No circularity: analytical wavefunction treated as external input for all downstream computations

full rationale

The paper states it obtains analytical expressions for the ground-state wavefunction and energy with α as the sole control parameter, then directly evaluates non-Gaussianity, nonclassicality, quantum Fisher information, and classical Fisher information for position measurements from those expressions. No parameter is fitted to a data subset and then reused to 'predict' a related quantity; α enters as an independent input rather than an output of the metrological analysis. No self-citations, uniqueness theorems, or ansatzes from prior author work are invoked to justify the central claims. All reported saturation of the Cramér-Rao bound and regime-dependent sensitivity follow from explicit calculation on the stated analytical form, rendering the derivation chain self-contained against its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard single-particle quantum mechanics in one dimension plus the specific functional form of the double-Morse potential; no additional free parameters or invented entities are introduced beyond the control parameter alpha itself.

axioms (2)
  • standard math The time-independent Schrödinger equation governs the stationary states of the double-Morse potential.
    Invoked implicitly when the authors obtain analytical expressions for the ground-state wavefunction and energy.
  • domain assumption The inverse barrier-width parameter alpha fully parametrizes the potential shape for the purpose of the reported calculations.
    Stated as the primary control parameter throughout the abstract.

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