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Chaos An Interdisciplinary Journal of Nonlinear Science

The metamorphoses of the light cones of a relativistic particle on a ring

M. Ćosić, M. Hadžijojić

Published Dec 1, 2025• 0 citations
Publisher

Abstract

In this paper, we investigate the dynamics of a relativistic particle confined to a ring. The focus is on the metamorphoses of the functional form of the particle propagator induced by a change in the asymptotic parameter, proportional to the ratio of the ring length to the particle Compton wavelength. Tackling the divergent nature of the propagator enabled us to describe and classify all patterns produced by particle self-interference, regardless of the shape of the initial wave packet. We shall show under which conditions a quasiperiodic structure, known as quantum carpets, arises and demonstrate that its quartic part of the phase function, which structurally stabilizes the canonical carpet, is not just a correction but explains all features of the exact solution.

DOI
10.1063/5.0289963

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•Jan 1, 2017•0 citations
In this paper, we investigate the dynamics of a relativistic particle confined to a ring. The focus is on the metamorphoses of the functional form of the particle propagator induced by a change in the asymptotic parameter, proportional to the ratio of the ring length to the particle Compton wavelength. Tackling the divergent nature of the propagator enabled us to describe and classify all patterns produced by particle self-interference, regardless of the shape of the initial wave packet. We shall show under which conditions a quasiperiodic structure, known as quantum carpets, arises and demonstrate that its quartic part of the phase function, which structurally stabilizes the canonical carpet, is not just a correction but explains all features of the exact solution.
We have investigated the dynamics of a quantum particle in the optical lattice potential. Initially, the quantum particle was represented by a Gaussian wave packet, located in the center of the well. The corresponding Schrödinger equation was solved explicitly by the method of the Chebyshev global propagation. Obtained solutions were also used for the construction of the Wigner functions. We found a great number of local abrupt changes of the solution shape. To explain this behavior, we used the fact that structurally stable systems, which form the largest class of the low dimensional dynamical systems, can be modeled and classified according to the catastrophe theory. All important features of the exact solution were explained on the basis of the mathematical properties of the catastrophic model. Such an approach enabled us to extract relevant information out of numerical solutions without employing any kind of approximations. We have investigated the influence of the Wigner catastrophes on the details of the quantum-classical correspondence breakdown. The wave packet was found to expand rapidly, filling the whole classically available area of the phase space. It was found that its self-interference pattern saturates quickly. A region of the phase space emerges in which the Wigner function oscillations transform into the singularity driven fluctuations. Once this region covers the whole area of the phase space, a wave packet dynamics enters into the new regime where its Wigner function fluctuates around the ergodic average. It will be shown that all mentioned processes are caused by the proliferation of the catastrophes and their mutual interactions.
The freedom to define branch cuts of the complex function is used to derive an integral representation of the quantum carpet, thus producing a generalization of the Poincar\'e recurrence theorem in the case of the continuous spectrum. This approach provides a different way to renormalize resonant states to be both space and time convergent. The coherence of quantum carpets was related to the properties of the Wigner function in the canonical time-frequency phase space. It has been shown that the distortion of the Wigner function shape is directly responsible for the lack of the ability of the dynamics to produce revivals equally as sharp as the initial wave packet.
Generic channel and ridge structures are known to appear in the time-dependent position probability distribution of a one-dimensional quantum particle confined to a box. These structures are shown to have a detailed quantitative explanation in terms of a travelling-wave decomposition of the probability density, wherein each contributing term corresponds simultaneously to (i) a real wave propagating at a quantized velocity and (ii) to the time-averaged structure of the position distribution along a quantized direction in spacetime. The approach leads to new predictions of channel locations, widths and depths, and is able to provide more structural details than earlier approaches based on partial interference and Wigner functions. Results are also applicable to light diffracted by a periodic grating, and to the quantum rigid rotator.
Abstract We analyze the structures emerging in the spacetime representation of the probability density woven by a slightly relativistic particle caught in a one‐dimensional box. In particular, we evaluate the relativistic effects on the revival time and the specific changes produced in the intermode traces, which quantum carpets consist of. Moreover, we present a detailed mathematical analysis of such quantum carpets pursuing the approach of a kernel. Here we represent the probability distribution as a superposition of interfering Airy function‐type structures along straight world lines. We also show that this phenomenon can be enhanced by many orders of magnitude in semiconductors with narrow band‐gap (e.g. as in InSb) and small effective mass of the electron, whereby due to the strong nonparabolicity of the semiconductor conduction band, the electron energy vs momentum dispersion relation behaves in a pseudo‐relativistic way.
One of the most intriguing quantum phenomena of light is the wave–particle duality which is related to the inability of the classical wave and particle concepts to fully describe the behavior of quantum objects. Theoretically, the wave features of the electromagnetic field, such as interference, follow from the solution of Maxwell's equations while particle features are a consequence of the field quantization, in which the electromagnetic field is described to consist of discrete energy packets, photons. Due to the wave–particle duality, for example, simultaneous description of optical absorption, emission, and interference of propagating fields has conventionally not been considered feasible using local optical models but has required using Maxwell's equations with stochastic source terms accounting for the wave–particle duality. Also the description of the momentum of light in a medium has been problematic. In the research described in this thesis, the quantized fluctuational electrodynamics (QFED) model is derived based on combining the electromagnetic Green's functions to the field quantization to describe the position dependence of the photon number and local thermal balance in general non-equilibrium conditions, and to separate the electromagnetic field into parts propagating in different directions in resonant structures, which has conventionally been problematic due to interference effects. The QFED method is shown to resolve the previously found anomalies in the canonical commutation relations of photon ladder operators. The QFED method is also used to derive quantum optical field–matter interaction parameters fully capturing the interference, emission, damping, and scattering of propagating photons in stratified media leading to an interference-exact radiative transfer equation (RTE) model. Therefore, the derived interaction parameters solve the problem of simultaneously describing interference and losses in the widely used RTE framework. The particle aspects of light are also studied by directly applying the conservation laws of nature and the special theory of relativity to show that light propagating in a medium must be described as mass-polariton (MP) quasiparticles, covariant coupled states of the field and matter. These quasiparticles are shown to have momentum of the Minkowski form and a nonzero mass that is carried by a mass density wave associated with light. The field–matter interaction related to the MP quasiparticles also leads to the dissipation of photon energy. These particle model results are also derived using a wave-based optoelastic continuum dynamics (OCD) approach following from the electrodynamics of continuous media and the continuum mechanics. The obtained results verify the full agreement between the wave and particle models when they are correctly applied. The key finding that a light pulse propagating in a medium is inevitably associated with an experimentally measurable mass implies a fundamental change in the prevailing perceptions of light.; Eras kiehtovimmista valon…