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 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.
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.
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.
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.
We have experimentally realized a parabolic potential barrier for surface gravity water waves. The analogy between the resulting wave equation and the Schrodinger equation for the inverted harmonic oscillator (IHO) enables us to study the propagation of quantum-mechanical wave packets with different average energies in this iconic scattering model. We observe a clear boundary in the phase-space dynamics, namely the separatrix, which distinguishes wave packets with energies below the maximum of the IHO potential from those with energies above it. In the former case, the wave packet is blocked, whereas in the latter case, it is transmitted. We also measure the corresponding variation in momentum during this process.
Abstract In 1952 David Bohm proposed an interpretation of quantum mechanics, in which the evolution of states results from trajectories governed by classical equations of motion but with an additional potential determined by the wave function. There exist only a few experiments that test this concept and they employed weak measurement of non-classical light. In contrast, we reconstruct the Bohm trajectories in a classical hydrodynamic system of surface gravity water waves, by a direct measurement of the wave packet. Our system is governed by a wave equation that is analogous to the Schrödinger equation which enables us to transfer the Bohm formalism to classical waves. In contrast to a quantum system, we can measure simultaneously their amplitude and phase. In our experiments, we employ three characteristic types of surface gravity water wave packets: two and three Gaussian temporal slits and temporal Airy wave packets. The Bohm trajectories and their energy flows follow the valleys and bounce off the hills in the corresponding quantum potential landscapes.