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European Journal of Physics

Quantum carpets in leaky boxes

Michael Berry

Published Jul 5, 2022• 3 citations
Publisher

Abstract

Abstract Quantum waves evolving or propagating inside one-dimensional boxes generate ‘quantum carpets’: intricate patterns in spacetime. The formally equivalent patterns in classical paraxial optics are observed as the Talbot effect. Boxes that are leaky, i.e. with penetrable walls, can be modelled by representing the walls as nonhermitian boundaries. The waves are superpositions of discrete eigenmodes with complex eigenvalues. Several different boundary conditions are explored, in which the Robin parameter—ratio of the mode derivatives and their values at the walls—may or may not depend on the eigenvalue. Nonhermitian models, which ignore the waves leaking outside the box, or represent them in simplified form, are approximations. For one physical model, the evolving wave in the full space can be calculated exactly, indicating that the nonhermitian model for the wave inside is highly accurate. Leaky boxes can be useful pedagogically: as introductory examples of nonhermiticity and biorthogonality (left and right eigenvectors), and as a rich source of postgraduate or advanced undergraduate student projects, exploring many variants.

DOI
10.1088/1361-6404/ac7e8d

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Abstract Quantum waves evolving or propagating inside one-dimensional boxes generate ‘quantum carpets’: intricate patterns in spacetime. The formally equivalent patterns in classical paraxial optics are observed as the Talbot effect. Boxes that are leaky, i.e. with penetrable walls, can be modelled by representing the walls as nonhermitian boundaries. The waves are superpositions of discrete eigenmodes with complex eigenvalues. Several different boundary conditions are explored, in which the Robin parameter—ratio of the mode derivatives and their values at the walls—may or may not depend on the eigenvalue. Nonhermitian models, which ignore the waves leaking outside the box, or represent them in simplified form, are approximations. For one physical model, the evolving wave in the full space can be calculated exactly, indicating that the nonhermitian model for the wave inside is highly accurate. Leaky boxes can be useful pedagogically: as introductory examples of nonhermiticity and biorthogonality (left and right eigenvectors), and as a rich source of postgraduate or advanced undergraduate student projects, exploring many variants.
The Talbot effect in quantum physics is known to produce intricate patterns in the probability distribution of a particle, known as ``quantum carpets,'' corresponding to the revival and replication of the initial wave function. Recently, it was shown that one can encode a DDD-level qudit in such a way that the Talbot effect can be used to process the DDD-dimensional quantum information [Far\'{\i}as et al., Phys. Rev. A 91, 062328 (2015)]. Here we introduce a scheme to produce free-propagating ``entangled quantum carpets'' with pairs of photons produced by spontaneous parametric down-conversion. First we introduce an optical device that can be used to synthesize arbitrary superposition states of Talbot qudits. Sending spatially entangled photon pairs through a pair of these devices produces an entangled pair of qudits. As an application, we show how the Talbot effect can be used to test a DDD-dimensional Bell inequality. Numerical simulations show that violation of the Bell inequality depends strongly on the amount of spatial correlation in the initial two-photon state. We briefly discuss how our optical scheme might be adapted to matter wave experiments.
Abstract The diffraction-like process displayed by a spatially localized matter wave is here analyzed in a case where the free evolution is frustrated by the presence of hard-wall-type boundaries (beyond the initial localization region). The phenomenon is investigated in the context of a nonrelativistic, spinless particle with mass m confined in a one-dimensional box, combining the spectral decomposition of the initially localized wave function (treated as a coherent superposition of energy eigenfunctions) with a dynamical analysis based on the hydrodynamic or Bohmian formulation of quantum mechanics. Actually, such a decomposition has been used to devise a simple and efficient analytical algorithm that simplifies the computation of velocity fields (flows) and trajectories. As it is shown, the development of space-time patters inside the cavity depends on three key elements: the shape of the initial wave function, the mass of the particle considered, and the relative extension of the initial state with respect to the total length spanned by the cavity. From the spectral decomposition it is possible to identify how each one of these elements contribute to the localized matter wave and its evolution; the Bohmian analysis, on the other hand, reveals aspects connected to the diffraction dynamics and the subsequent appearance of interference traits, particularly recurrences and full revivals of the initial state, which constitute the source of the characteristic symmetries displayed by these patterns. It is also found that, because of the presence of confining boundaries, even in cases of increasingly large box lengths, no Fraunhofer-like diffraction features can be observed, as happens when the same wave evolves in free space. Although the analysis here is applied to matter waves, its methodology and conclusions are also applicable to confined modes of electromagnetic radiation (e.g. light propagating through optical fibers).
Summary form only given. Wave packets (WPs) has recently emerged as a very active research field in quantum optics. We show here that the evolution of WPs may result in new class of large-scale interference effects: the formation of highly regular spatio-temporal patterns in the QM probability, |/spl Psi/|/sup 2/, with the key mechanism for this being pair interference between the eigen-modes of the system and the degeneracy of this interference. The new patterns have been discovered in the probability density |/spl Psi/|/sup 2/(x,t) of a particle in an infinitely deep square well, by numerical simulations of the evolution of a Gaussian WP; later work by a few groups used other initial conditions and media (e.g. Bose-Einstein condensate and angular WPs). In all these cases, the QM distribution shows a net of canals (minima), and ridges (maxima), running as straight lines in space-time (x,t). Similar patterns in the near field of a diffraction grating, and in matter waves in atom optics reveals a new aspects of the Talbot effect. So far the numerical exploration of these patterns has relied on the specific choice (i) of the initial conditions and (ii) of the square well; no physical explanation of the patterns has been suggested. Recently, we observed a quantum carpet under more general conditions, offered its universal explanation, and predicted distinct curved patterns in general case of an anharmonic potential.
Over the last decades, quantum optics has evolved from high-quality-factor cavities in the early experiments toward new cavity designs involving leaky modes. Despite very reliable models, in the concepts of cavity quantum electrodynamics, photon leakage is most of the time treated phenomenologically. Here, we take a different approach, and starting from first principles, we define an inside-outside representation which is derived from the original true-mode representation, in which one can determine the effective Hamiltonian and Poynting vector. Unlike the phenomenological model, they allow a full description of a leaking single photon produced in the cavity and propagating in free space. This is applied for a laser-driven atom-cavity system. In addition, we propose an atom-cavity nonresonant scheme for single-photon generation, and we rigorously analyze the outgoing single photon in time and frequency domains for different coupling regimes. Finally, we introduce a particular coupling regime ensuring adiabatic elimination for which the pulse shape of the outgoing single photon is tailored using a specifically designed driving field envelope.
Talbot and Talbot-Lau effects are frequently used in lensless imaging applications with light, ultrasound, x-rays, atoms and molecules--generally in situations where refractive optical elements are non-existent or not suitable. We here show an experimental visualization of the intriguing wave patterns that are associated with near-field interferometry behind a single periodic diffraction grating under plane wave illumination and which are often referred to as Talbot carpets or quantum carpets. We also show the patterns behind two separated diffraction gratings under nearly-monochromatic but spatially incoherent illumination that illustrate the nature of Talbot-Lau carpets.
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.