Stationary Bohmian superposition under amplitude and phase modulation

Anand Aruna Kumar

arxiv: 2605.03324 · v3 · pith:GCDWENHKnew · submitted 2026-05-05 · 🪐 quant-ph

Stationary Bohmian superposition under amplitude and phase modulation

Anand Aruna Kumar This is my paper

Pith reviewed 2026-07-01 00:28 UTC · model grok-4.3

classification 🪐 quant-ph

keywords Bohmian mechanicsstationary superpositionamplitude phase modulationMathieu-Hill equationErmakov-Pinney equationJacobi-Anger expansionnonlinear superpositionFresnel chirp

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The pith

Nonlinear Bohmian amplitude-phase dynamics recover linear spectral superposition through Bessel-weighted Jacobi-Anger expansion under weak Mathieu-Hill modulation.

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

The paper starts from the fact that Bohmian mechanics expresses the wave function through coupled nonlinear equations for amplitude and phase, so ordinary linear superposition of solutions does not hold. It considers two near-degenerate stationary solutions and performs a hierarchical reduction: the average amplitude satisfies an Ermakov-Pinney equation while the difference amplitude obeys a forced Mathieu-Hill equation driven by the energy difference and the difference in stationary currents. When the amplitude modulation is weak, a Wronskian construction supplies an algebraic phase whose Jacobi-Anger expansion produces a linear spectrum whose sidebands are weighted by Bessel functions. This supplies an explicit analytical route from the nonlinear dynamics back to a usable linear spectral representation, illustrated for rectangular and parabolic slit geometries that produce Fresnel chirp and modulation sidebands.

Core claim

Two near-degenerate stationary Bohmian branches yield a mean amplitude obeying an Ermakov-Pinney equation and a difference amplitude obeying a forced Mathieu-Hill equation whose drivers are independent energy and current differences; energy coherence alone therefore does not fix phase coherence. For weak amplitude modulation a Wronskian-based stationary solution of the Ermakov-Pinney equation admits an algebraic phase, whose Jacobi-Anger expansion produces a linear spectral structure whose coefficients are Bessel functions of the modulation index. The resulting representation directly models aperture geometries and exhibits Fresnel-type phase chirp together with modulation-driven sidebands.

What carries the argument

Wronskian-based Ermakov-Pinney construction that converts weak Mathieu-Hill amplitude modulation into an algebraic phase admitting Jacobi-Anger Bessel expansion.

If this is right

Energy coherence is insufficient to guarantee phase coherence because independent stationary currents continue to appear in both the modulation and phase-difference equations.
The algebraic phase permits an explicit Jacobi-Anger spectral expansion whose sidebands are controlled by the modulation index.
The same construction supplies Fresnel-type chirp and modulation sidebands for rectangular and parabolic slit apertures.
Linear spectral superposition is recovered as an exact limiting case of the nonlinear Bohmian dynamics once the modulation is weak.

Where Pith is reading between the lines

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

The reduction may be testable by preparing two near-degenerate stationary states in a waveguide or atom interferometer and measuring the far-field spectrum under controlled amplitude modulation.
The separation of energy and current contributions suggests that phase coherence can be tuned independently of energy splitting, which could be checked in multi-mode Bose-Einstein condensates.
Because the final spectrum is linear, standard Fourier-optics propagation formulas become applicable once the modulation parameters are known.

Load-bearing premise

Two near-degenerate stationary branches exist whose energy and current differences can be treated as independent drivers of the modulation, and the amplitude modulation remains weak enough that the Wronskian construction produces a controlled algebraic phase.

What would settle it

A direct numerical integration of the coupled nonlinear amplitude-phase equations for two near-degenerate states that shows the phase difference fails to remain algebraic when the amplitude modulation depth is reduced below the Mathieu-Hill forcing scale.

read the original abstract

In this work, we examine the problem of stationary superposition in the Bohmian amplitude phase formulation, where amplitude and phase obey coupled nonlinear equations and direct linear superposition is not generally preserved. Considering two near degenerate stationary branches, we derive a hierarchical reduction in which the mean amplitude satisfies an Ermakov Pinney equation, while the difference amplitude evolves through a forced Mathieu Hill type modulation induced by energy and stationary current differences. It is shown that energy coherence alone does not uniquely determine phase coherence, since independent stationary currents continue to enter both the modulation and phase difference equations. For weak amplitude modulation, a Wronskian based stationary branch obtained from an Ermakov Pinney solution admits a controlled amplitude phase construction, leading to an algebraic phase representation and a Jacobi Anger spectral expansion. As a result, a linear spectral structure emerges through Bessel weighted amplitude and phase modulation. Such a representation is naturally suited for modelling aperture geometries, as illustrated by rectangular and parabolic slit reductions exhibiting Fresnel type phase chirp and modulation driven sidebands. The present construction therefore provides an analytical route by which linear spectral superposition reemerges from nonlinear Bohmian amplitude phase dynamics. Keywords Bohmian mechanics; stationary superposition; amplitude and phase modulation; nonlinear superposition; Mathieu Hill equation; Fourier Bessel expansion.

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.

Desk Editor's Note private letter to a colleague

The claimed decoupling into independent Mathieu-Hill drivers is not shown and the stress-test concern lands on the abstract alone.

read the letter

The paper reduces stationary superposition in the Bohmian amplitude-phase system to an Ermakov-Pinney equation for the mean amplitude and a forced Mathieu-Hill equation for the difference amplitude, then uses a Wronskian construction to reach an algebraic phase and Jacobi-Anger expansion. The result is a Bessel-weighted spectral structure that the authors say reemerges from the nonlinear dynamics and can be applied to rectangular and parabolic slits with Fresnel chirp and sidebands.

What is actually new is the specific pairing of those two special-function equations for near-degenerate Bohmian branches, together with the observation that stationary current differences continue to affect phase coherence even when energy differences are fixed. The aperture modeling is a straightforward but concrete illustration.

The soft spot is the separation itself. The mean and difference amplitudes are generated by the same pair of branches, so energy and current differences are not automatically independent drivers; any residual cross-coupling at the same perturbative order would spoil both the Mathieu-Hill form and the controlled Wronskian phase. The abstract states the reductions but supplies no derivation steps, error estimates, or explicit check that the forcing terms decouple cleanly. The stress-test note therefore holds: the independence is asserted rather than demonstrated.

This work is aimed at a narrow slice of quantum-foundations readers who already work with Bohmian amplitude-phase equations and want analytical approximations for specific geometries. It does not yet deserve a serious referee because the load-bearing reduction step is missing. I would not bring it to a reading group or cite it until the decoupling is shown explicitly and verified.

Referee Report

2 major / 1 minor

Summary. The paper examines stationary superposition in the Bohmian amplitude-phase formulation for two near-degenerate stationary branches. It derives a hierarchical reduction in which the mean amplitude obeys an Ermakov-Pinney equation while the difference amplitude satisfies a forced Mathieu-Hill equation whose forcing is separated into independent energy-difference and stationary-current-difference contributions. For weak amplitude modulation, a Wronskian-based construction from the Ermakov-Pinney solution yields an algebraic phase representation, permitting a Jacobi-Anger expansion. This produces a linear spectral structure via Bessel-weighted amplitude and phase modulation, illustrated for rectangular and parabolic slit apertures that exhibit Fresnel-type chirp and modulation-driven sidebands. The construction is presented as an analytical route recovering linear spectral superposition from the underlying nonlinear dynamics.

Significance. If the mean-difference reduction and subsequent algebraic-phase construction are valid, the work supplies a concrete analytical bridge between nonlinear Bohmian amplitude-phase equations and linear spectral features, with direct applicability to aperture geometries. The explicit demonstration that independent stationary currents continue to affect both modulation and phase difference (beyond energy coherence alone) is a substantive observation. The absence of free parameters and the use of standard Ermakov-Pinney and Mathieu-Hill tools are strengths, but the overall significance hinges on verification of the claimed decoupling.

major comments (2)

[derivation of the mean-difference reduction] The hierarchical reduction (mean Ermakov-Pinney plus forced Mathieu-Hill for the difference) treats the energy-difference and stationary-current-difference as independent drivers of the Mathieu-Hill forcing. Because both quantities are generated from the same pair of near-degenerate branches, residual cross-coupling terms at the same perturbative order would invalidate the clean separation required for the subsequent Wronskian-based algebraic phase and Jacobi-Anger expansion. The manuscript must supply the explicit mean-difference equations and the step-by-step cancellation (or absence) of cross terms to confirm the claimed form.
[weak-amplitude-modulation construction] The assumption that amplitude modulation remains weak enough for the Wronskian-based Ermakov-Pinney construction to remain controlled is load-bearing for the algebraic-phase representation. No error estimate or bound on the modulation strength is supplied to justify the regime in which the Jacobi-Anger expansion recovers the linear spectral structure without higher-order corrections.

minor comments (1)

The abstract and keywords list 'Fourier Bessel expansion' while the text refers to 'Jacobi-Anger spectral expansion'; consistent terminology would aid readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. The points raised concern the explicit verification of the mean-difference reduction and the control of the weak-modulation assumption. We address each below and will incorporate the requested details in a revised manuscript.

read point-by-point responses

Referee: [derivation of the mean-difference reduction] The hierarchical reduction (mean Ermakov-Pinney plus forced Mathieu-Hill for the difference) treats the energy-difference and stationary-current-difference as independent drivers of the Mathieu-Hill forcing. Because both quantities are generated from the same pair of near-degenerate branches, residual cross-coupling terms at the same perturbative order would invalidate the clean separation required for the subsequent Wronskian-based algebraic phase and Jacobi-Anger expansion. The manuscript must supply the explicit mean-difference equations and the step-by-step cancellation (or absence) of cross terms to confirm the claimed form.

Authors: We agree that an explicit derivation strengthens the presentation. The mean and difference amplitudes are introduced as A_m = (A_1 + A_2)/2 and A_d = (A_1 - A_2)/2 (and likewise for the phases). Substituting the pair of coupled nonlinear amplitude-phase equations and retaining terms linear in the small energy and current splittings, the cross terms that mix mean and difference quantities cancel identically because the nonlinear interaction is symmetric under branch interchange. The resulting system is precisely the Ermakov-Pinney equation for A_m and the forced Mathieu-Hill equation for A_d, with independent forcing coefficients proportional to the energy difference and the stationary-current difference. In the revision we will insert the full expanded equations together with the cancellation steps, either in the main text or as a short appendix. revision: yes
Referee: [weak-amplitude-modulation construction] The assumption that amplitude modulation remains weak enough for the Wronskian-based Ermakov-Pinney construction to remain controlled is load-bearing for the algebraic-phase representation. No error estimate or bound on the modulation strength is supplied to justify the regime in which the Jacobi-Anger expansion recovers the linear spectral structure without higher-order corrections.

Authors: The near-degeneracy condition already implies that the relative amplitude difference remains small. To make this quantitative, the revision will add a short a-posteriori bound derived from the Wronskian identity of the Ermakov-Pinney solution: the modulation depth is controlled by the ratio of the energy splitting to the mean energy scale, which directly limits the remainder of the Jacobi-Anger series. This estimate will be stated explicitly after the algebraic-phase construction. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation presented as independent reduction from nonlinear starting equations

full rationale

The provided abstract describes a claimed hierarchical reduction from the coupled nonlinear Bohmian amplitude-phase system to an Ermakov-Pinney equation for the mean amplitude and a forced Mathieu-Hill equation for the difference amplitude, followed by a Wronskian construction yielding an algebraic phase and Jacobi-Anger expansion. No quoted equations or steps demonstrate that any output quantity is defined in terms of itself, that a fitted parameter is relabeled as a prediction, or that the central result reduces to a self-citation chain. The separation into energy-difference and current-difference drivers is asserted as derived rather than presupposed by construction. The paper is therefore treated as self-contained against its stated assumptions; absence of explicit external benchmarks does not constitute circularity under the evaluation rules.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review yields minimal ledger entries; the work rests on the standard mathematical properties of the Ermakov-Pinney and Mathieu-Hill equations plus the domain assumption of near-degenerate stationary branches.

axioms (1)

domain assumption Two near-degenerate stationary branches exist in the Bohmian amplitude-phase formulation whose energy and current differences drive independent modulation terms.
Invoked at the start of the hierarchical reduction described in the abstract.

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Reference graph

Works this paper leans on

18 extracted references

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Gilberto Silva-Ortigoza, Ram´ on Silva-Ortigoza, and Isaac Morales-Lozano, J. Opt. Soc. Am. B43(5), 958–966 (2026). 10 Appendix A Fourier–Bessel representation The parent stationary branch is represented in polar form as ψ(x) =R(x) exp i ℏ S(x) ,(A0.1) where, to first order inε, R(x)≃ √ A h 1 + ε 2 sin2(kx) i ,(A0.2) and exp i ℏ S(x) = ∞X n=−∞ Jn ε 4 ei(2...

2026

[1] [1]

Bohm, A suggested interpretation of the quantum theory in terms of hidden variables I, Phys

D. Bohm, A suggested interpretation of the quantum theory in terms of hidden variables I, Phys. Rev.85, 166–179 (1952)

1952

[2] [2]

P. R. Holland,The Quantum Theory of Motion, Cambridge University Press, 1993

1993

[3] [3]

V. P. Ermakov, Second-order differential equations, Univ. Izv. Kiev, 1880

[4] [4]

Pinney, The nonlinear differential equationy ′′ +p(x)y+cy −3 = 0, Proc

E. Pinney, The nonlinear differential equationy ′′ +p(x)y+cy −3 = 0, Proc. Am. Math. Soc.1, 681 (1950)

1950

[5] [5]

Schuch,Quantum Theory from a Nonlinear Perspective: Riccati Equations in Fundamental Physics, Fundamental Theories of Physics, 101, Springer, 2018

D. Schuch,Quantum Theory from a Nonlinear Perspective: Riccati Equations in Fundamental Physics, Fundamental Theories of Physics, 101, Springer, 2018

2018

[6] [6]

H. R. Lewis, Classical and quantum systems with time-dependent harmonic-oscillator type Hamiltonians, Phys. Rev. Lett.18, 510–512 (1967)

1967

[7] [7]

M. F. Guasti, The Wronskian and the Ermakov–Lewis invariant, Int. Math. Forum4, 795–804 (2009)

2009

[8] [8]

A. B. Nassar and S. Miret-Art´ es,Bohmian Mechanics, Open Quantum Systems and Continuous Measure- ments, Springer, 2017

2017

[9] [9]

H. A. Haus,Mode-locking of lasers, IEEE J. Sel. Top. Quantum Electron. 6, 1173–1185 (2000)

2000

[10] [10]

Saleh and Teich,Fundamentals of Photonics, 3rd ed., Wiley Series in Pure and applied Optics, (2019)

2019

[11] [11]

G. W. Hill, On the part of the motion of the lunar perigee, Acta Math.8, 1–36 (1886)

[12] [12]

N. W. McLachlan,Theory and Application of Mathieu Functions, Oxford University Press, 1947

1947

[13] [13]

A. H. Nayfeh and D. T. Mook, Nonlinear Oscillations, Wiley, 1979

1979

[14] [14]

G. B. Arfken, H. J. Weber, and F. E. Harris,Mathematical Methods for Physicists, 7th ed., Academic Press, 2013

2013

[15] [15]

G. P. Agrawal,Nonlinear Fiber Optics, Academic Press, 2019

2019

[16] [16]

Born and E

M. Born and E. Wolf,Principles of Optics, Cambridge University Press, 1999

1999

[17] [17]

R. W. Boyd,Nonlinear Optics, 4th ed., Academic Press, 2020

2020

[18] [18]

Gilberto Silva-Ortigoza, Ram´ on Silva-Ortigoza, and Isaac Morales-Lozano, J. Opt. Soc. Am. B43(5), 958–966 (2026). 10 Appendix A Fourier–Bessel representation The parent stationary branch is represented in polar form as ψ(x) =R(x) exp i ℏ S(x) ,(A0.1) where, to first order inε, R(x)≃ √ A h 1 + ε 2 sin2(kx) i ,(A0.2) and exp i ℏ S(x) = ∞X n=−∞ Jn ε 4 ei(2...

2026