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arXiv:2607.25627

The Natural Formulation of Cubic-Phase Dynamics in the Linear Potential

Maximilian L. D. D. Pellner, Georgi Gary Rozenman

quant-phphysics.opticsSubmitted Jul 28, 2026• Announced Jul 29, 2026• 0 citations• Revised Oct 27, 2026 (v2)
v1 · Jul 28, 2026v2 · Oct 27, 2026
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Abstract

A quantum wave packet in a linear potential, i.e. a constant force such as gravity, accumulates a cubic-in-time phase, universal across Schrödinger-type platforms. The natural formulation of the cubic phase dynamics follows from their canonical realization, the Airy eigenstates. The eigenforce, the potential gradient for which the Airy profile is stationary and hence the intrinsic force behind its known self-acceleration absent any applied force, derives from the eigenvalue problem and characterizes the state uniquely up to a rigid translation. Alongside the applied force, it underlies the analytically closed form of these dynamics, comprising three known contributions, intrinsic, force-induced, and a cross term. The force-induced one alone is shape-independent, whereas the Airy eigenstate renders the other two non-dispersing. The cubic coefficient factorizes into two Newtonian force balances, vanishing at the stationary eigenstate and at a non-trivial zero where the phase cancels while the packet keeps accelerating. This exposes the eigenforce as an effective antagonist to the applied force, not only in the caustic's self-acceleration but here also within the phase, while leaving the centroid untouched by Ehrenfest's theorem. Spatially uniform within each packet, it escapes direct measurement, accessible only as the relative phase of two colliding packets, each in its own potential. The relative cubic coefficient is symmetry-forbidden for identical packets and activated by preparation asymmetry. Simulated interference of two truncated Airy packets demonstrates the extraction. Spanning ultracold atoms, paraxial optics, and surface-gravity water waves, it provides a platform-independent probe of the cubic phase and, through its force dependence, of the constant force itself.

Comments
9 pages, 4 figures. Revised version. Corrected numerical results, expanded derivation and a new appendix with the simulation parameters. Title changed

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A quantum wave packet in a linear potential, i.e. a constant force such as gravity, accumulates a cubic-in-time phase, universal across Schrödinger-type platforms. The natural formulation of the cubic phase dynamics follows from their canonical realization, the Airy eigenstates. The eigenforce, the potential gradient for which the Airy profile is stationary and hence the intrinsic force behind its known self-acceleration absent any applied force, derives from the eigenvalue problem and characterizes the state uniquely up to a rigid translation. Alongside the applied force, it underlies the analytically closed form of these dynamics, comprising three known contributions, intrinsic, force-induced, and a cross term. The force-induced one alone is shape-independent, whereas the Airy eigenstate renders the other two non-dispersing. The cubic coefficient factorizes into two Newtonian force balances, vanishing at the stationary eigenstate and at a non-trivial zero where the phase cancels while the packet keeps accelerating. This exposes the eigenforce as an effective antagonist to the applied force, not only in the caustic's self-acceleration but here also within the phase, while leaving the centroid untouched by Ehrenfest's theorem. Spatially uniform within each packet, it escapes direct measurement, accessible only as the relative phase of two colliding packets, each in its own potential. The relative cubic coefficient is symmetry-forbidden for identical packets and activated by preparation asymmetry. Simulated interference of two truncated Airy packets demonstrates the extraction. Spanning ultracold atoms, paraxial optics, and surface-gravity water waves, it provides a platform-independent probe of the cubic phase and, through its force dependence, of the constant force itself.
A quantum wave packet evolving in a linear potential, i.e. a constant force such as gravity, accumulates a cubic-in-time phase. This cubic phase is universal across Schrödinger-type platforms, and the Airy functions, as the eigenstates of the linear potential, provide its natural realization. Because the classical action is quadratic in the force, the cubic-in- time phase comprises exactly three contributions – intrinsic, force-induced, and a cross term – of which the force-induced component alone is shape-independent, while the Airy eigenstate renders the remaining, shape-dependent terms in clean, non-dispersing form. An eigenstate-based non-dimensionalization fixes the eigenforce, the intrinsic force behind the packet’s acceleration even without an applied force, as a natural parameter. As a function of this eigenforce and the applied force, the cubic coefficient takes an analytically closed, physically interpretable form that factors along two zero lines, the static Airy eigenstate and a non-trivial zero where the phase cancels while the packet keeps accelerating. This exposes the eigenforce as an effective antagonist to the applied force not only in the caustic’s self-acceleration but here also within the phase, yet it leaves the centroid untouched, in accordance with Ehrenfest’s theorem. Being spatially uniform within each packet, the phase escapes direct measurement and becomes accessible only as the relative phase between two colliding packets, each subject to its own linear potential. The resulting general relative cubic coefficient, symmetry-forbidden for identical packets and activated by the preparation asymmetry, is hence a designable signal. Extracted by heterodyne demodulation from the simulated interference of two Airy packets, its central value matches the theoretical prediction to sub-percent accuracy, within the fit uncertainty. Spanning ultracold-atom condensates, paraxial optics, and surface-gravity water waves, this framework provides a platform-independent probe of the cubic phase and, through its force dependence, of the constant force itself.
We theoretically study and successfully observe the evolution of Gaussian and Airy surface gravity water wave packets propagating in an effective linear potential. This potential results from a homogeneous and time-dependent flow created by a computer-controlled water pump. For both wave packets we measure the amplitudes and the cubic phases appearing due to the linear potential. Furthermore, we demonstrate that the self-acceleration of the Airy surface gravity water wave packets can be completely canceled by a linear potential.
We show that for a wave ψ in the form of an Airy function the probability density ‖ψ‖2 propagates in free space without distortion and with constant acceleration. This ’’Airy packet’’ corresponds classically to a family of orbits represented by a parabola in phase space; under the classical motion this parabola translates rigidly, and the fact that no other curve has this property shows that the Airy packet is unique in propagating without change of form. The acceleration of the packet (which does not violate Ehrenfest’s theorem) is related to the curvature of the caustic (envelope) of the family of world lines in spacetime. When a spatially uniform force F (t) acts the Airy packet continues to preserve its integrity. We exhibit the solution of Schrödinger’s equation for general F (t) and discuss the motion for some special forms of F (t).
Self-accelerating Airy matter waves offer a clean setting to access the intrinsic cubic-in-time phase. Here we reconstruct the relative phase of simulated Airy-shaped Bose-Einstein condensates from interference fringes in free space, a regime approached in microgravity. The cubic phase dynamics are quantified via approximately debiased, windowed polynomial fits with systematics-aware uncertainty estimates that account for window-induced correlations. We compare two physically feasible phase-extraction methods, heterodyne-based and density-based, and show that an Airy-Gaussian geometry yields substantially improved robustness to fit-window selection relative to an Airy-Airy collision. In the weakly interacting regime, the extracted cubic coefficient responds linearly to leading order to the effective interaction strength, its shift from the noninteracting value providing a calibrated probe of weak mean-field nonlinearities in self-accelerating condensates.
We analyze the propagation dynamics of nonspreading wave packets by decomposing the Hamiltonian H into two parts: H = □(t) + H_c(t). The first part □(t) is such that Ψ(x, t) is its instantaneous eigenstate, and is therefore irrelevant to the propagation of the packet. The second part H_c(t) is shown to be the effective Hamiltonian governing the propagation dynamics of nonspreading wave packets both quantum mechanically and classically. The Ehrenfest theorem is thus based on the Hamiltonian H_c(t) connecting quantum mechanics and classical mechanics. This analysis also works for non-square-integrable packets, such as Airy packets, and explains why Airy packets can self-accelerate in free space.
We present the theoretical models and review the most recent results of a class of experiments in the field of surface gravity waves. These experiments serve as demonstration of an analogy to a broad variety of phenomena in optics and quantum mechanics. In particular, experiments involving Airy water-wave packets were carried out. The Airy wave packets have attracted tremendous attention in optics and quantum mechanics owing to their unique properties, spanning from an ability to propagate along parabolic trajectories without spreading, and to accumulating a phase that scales with the cubic power of time. Non-dispersive Cosine-Gauss wave packets and self-similar Hermite-Gauss wave packets, also well known in the field of optics and quantum mechanics, were recently studied using surface gravity waves as well. These wave packets demonstrated self-healing properties in water wave pulses as well, preserving their width despite being dispersive. Finally, this new approach also allows to observe diffractive focusing from a temporal slit with finite width.