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 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 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.
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 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).
In 1979, Berry and Balazs [M V Berry and N L Balazs 1979 Am. J. Phys. 47 264] obtained a strict solution of the Schrödinger equation with Airy function used as the initial condition, and described the wave function represented by such solution as the Airy wave-packets. They discovered that infinite Airy wave-packet has unique properties such as non-spreading and free acceleration, proving that it is the only nontrivial non-spreading solution of the time-dependent Schrödinger equation in one dimension. However, the observing of the finite Airy beam seems to be more meaningful since wave-packets in reality is inevitably band limited. A certain form of finite Airy beam was investigated by Siviloglou et al. in 2007 [Siviloglou G A, Broky J, Dogariu A, Christodoulides D N 2007 Phys. Rev. Lett. 99 213901; Siviloglou G A, Christodoulides D N 2007 Opt. Lett. 32 979]. They noted that the Airy wave packet still exhibits its most exotic feature, i.e., its trend toward free acceleration. While in the present paper we discuss the properties of Airy beam in a few steps further and propose several conclusions. On the one hand, a theoretic explanation is given to solve the matter of the centre of mass of infinite Airy beam. On the other hand, deeper research is conducted on the unique properties of finite Airy beam. Another form of finite Airy beam is discussed by reduction to absurdity, and its field distribution is put forward by numerical simulation. We find that the trajectory of the centroid holds its position, which means that the beam cannot accelerate freely as a whole. Ultimately, we have the conclusion that finite Airy beam can neither freely accelerate nor be non-diffractive.
In a previous paper, we showed that the Hamilton equations of motion of a quantum particle are obtained as group velocities of the wave packet describing this particle only if the time dependent phase of a particle wave is proportional to the Lagrangian, not to the Hamiltonian as in a solution of the conventional Schrodinger equation. When a Lagrangian of relativistic form is considered, the wave packet of a quantum particle takes a physical form, with a finite spectrum of a cut off velocity (c). Based on a relativistic quantum principle, asserting the invariance of the time dependent phase for an arbitrary change of coordinates, we obtained the relativistic kinematics and dynamics, the electromagnetic field equations, the spin and the electromagnetic and gravitational interactions. When the Lagrangian is considered as a function of the Hamiltonian, we obtain a Schrodinger type equation with an additional term depending on the velocity and the momentum operator. Based on this equation, we investigate the dynamics of a relativistic quantum particle. In this framework, such a particle is described as a continuous distribution of conservative matter, according to the general theory of relativity. In an electromagnetic field, any time dependent phase variation is modified with a term proportional to a vector potential conjugated to the spatial coordinates and a scalar potential conjugated to time. However, this classical description tells us nothing about the structure of the physical world. Only Quantum Mechanics tells us something about the structure of this world. Namely, that this world is composed of species of identical quantum particles. On one hand, experimentally, it has been found that these particles are of a wavy nature. On the other hand, one could find that the simplest way to define a quantum particle is by a wave packet, with momenta conjugated to coordinates and energy conjugated to time Consider a quantum particle as a wave packet, and find that the group velocities in the coordinate and momentum spaces are in agreement with the Hamilton equations only when the Lagrangian is considered in the time dependent phases, instead of the Hamiltonian in the conventional forms of these waves as solutions of the Schrodinger equation. We define a relativistic quantum principle, and derive a wave equation for a relativistic quantum particle, the relativistic kinematics and dynamics of the particle waves, the Maxwell equations and the Lorentz force of a field interacting with the particle waves, the relativistic transform of such a field, and the spin as a characteristic of the particle waves. We consider a quantum particle as a distribution of conservative matter propagating according to the General Theory of Relativity. We obtain the dynamics of this matter in a gravitational field, the propagation in plane waves perpendicular to geodesic tracks, and equations of conservation. We describe the matter dynamics as a positively defined density and show that, according to the general theory of relativity, such a distribution can be conceive only as of a fragment of matter with a finite mass equal to a mass , as a characteristic of the matter dynamics, – the matter quantization. The group velocities of the Fourier conjugate representations in the coordinate and momentum spaces describe the dynamics of a quantum particle in agreement with the Hamiltonian equations. Under the action of an external (non-gravitational) field, the acceleration of the quantum matter has two components: 1) A component perpendicular to the velocity, given by the relativistic mechanical component of the time dependent phase, and 2) A component given by the additional field terms of this phase. A free quantum particle is described by a non-dispersing wave function , contrary to the solution of the Schrodinger equation. A coherent electromagnetic field, in resonance with a system of active quantum particles in a Fabry-Perot cavity, has a wave vector approximately proportional to the metric elements, as the resonance frequency is approximately constant – a gravitational wave can be detected by the transmission characteristics of an active Fabry-Perot cavity. In a constant gravitational field, a quantum particle undertakes a velocity and an acceleration, which, at the boundary of a black hole are null – absorption and evaporation processes at the boundary of a black hole arise only by gravitational perturbations. Generally, a quantum particle is described by a time-space volume, called graviton, with a spin 2, and a distribution of a specific matter in this volume, with a half-integer spin for Fermions and an integer spin for Bosons. A graviton Lagrangian is obtained as a curvature integral on a graviton volume, and a Hamiltonian tensor is obtained for the gravitational coordinates and velocities. In a gravitational field, the time space coordinates are deformed. In such a field, any plane wave remains perpendicular geodesic, while an additional acceleration is possible in the wave plane.