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arXiv:2002.07710

Time evolution of charged particle wave functions in optical crystal: The coherent Kapitza-Dirac effect for plasma-based proton beams

Sushanta Barman, Sudeep Bhattacharjee

Published Feb 18, 2020• 0 citations
arXivHTMLPDF

Abstract

The stationary eigenstates and eigenvalues for the ponderomotive potential of an optical crystal confined in a one-dimensional infinite square well are numerically obtained. The initial states of the incoming particles taken as Gaussian, are expanded in the basis of the stationary eigenstates of the ponderomotive potential, to obtain the subsequent time evolution of the wave function of the particle during the interaction with the optical crystal. From the results of the time evolution of the probability density, it is observed that the particles get localized at equidistant positions in the transverse direction, which results in the diffraction pattern. The temporal evolution of the diffraction pattern is analyzed. As an application, the diffraction of proton beams is studied, where the experimental parameters are optimized to observe the diffraction pattern for a microwave plasma-based proton beam system. The observations are important for design of proton based matter-wave interferometers.

DOI
10.48550/arxiv.2002.07710

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The stationary eigenstates and eigenvalues for the ponderomotive potential of an optical crystal confined in a one-dimensional infinite square well are numerically obtained. The initial states of the incoming particles taken as Gaussian, are expanded in the basis of the stationary eigenstates of the ponderomotive potential, to obtain the subsequent time evolution of the wave function of the particle during the interaction with the optical crystal. From the results of the time evolution of the probability density, it is observed that the particles get localized at equidistant positions in the transverse direction, which results in the diffraction pattern. The temporal evolution of the diffraction pattern is analyzed. As an application, the diffraction of proton beams is studied, where the experimental parameters are optimized to observe the diffraction pattern for a microwave plasma-based proton beam system. The observations are important for design of proton based matter-wave interferometers.
The diffraction of electrons by a standing wave of light was first proposed by Kapitza and Dirac in 1933. We observed this effect for the first time in September 2001 by showing separate electron diffraction peaks. It was first observed for a focused laser beam, where the standing wave of light serves as a thin light crystal. Later, the effect was also observed with an unfocused laser beam, where the standing wave serves as a thick light crystal, where scattering can only occur at the Bragg angle. The theory behind the effect has been formulated within both atomic optics and electron optics communities. It involves the ponderomotive potential that an electron experiences within the electromagnetic field of two counter-propagating laser beams. The derivations of the differential equations given by both communities have identical results with their own benefits when making numerical calculations. These theoretical calculations fit well to our data, which falls between the limits of the diffraction and Bragg regimes. To analyze the data collected in these experiments, the statistical behavior of the arrival of electrons, governed by the Poisson distribution, was taken into account. The Gaussian nature of the laser beam and its consequences on focusing needed to be addressed, along with how far from diffraction limited the laser beam actually was. There may be a possibility of new effects arising from the interaction of electrons and light. The effect is like KDE, but modified to include two counter-propagating lasers of different frequency, the consequence of the longitudinal velocity of the electrons, and the role that the polarization of the laser light plays.
Similar to the optical diffraction of light passing through a material grating, the Kapitza-Dirac effect occurs when an electron is diffracted by a standing light wave. In its original description, the effect is time independent. Here, we extended the Kapitza-Dirac effect to the time domain. By tracking the spatiotemporal evolution of a pulsed electron wave packet diffracted by a 60-femtosecond (where one femtosecond = 10 −15 seconds) standing wave pulse in a pump-probe scheme, we observed time-dependent diffraction patterns. The fringe spacing in the observed pattern differs from that generated by the conventional Kapitza-Dirac effect. By exploiting this time-resolved diffraction scheme, we can access the time evolution of the phase properties of a free electron and potentially image ionic potentials and electronic decoherences.
The Kapitza-Dirac effect is the diffraction of a well-collimated particle beam by a standing wave of light. Why is this interesting? Comparing this situation to the introductory physics textbook example of diffraction of a laser beam by a grating, the particle beam plays the role of the incoming wave and the standing light wave the role of the material grating, highlighting particle-wave duality. Apart from representing such a beautiful example of particle-wave duality, the diffracted particle beams are coherent. This allows the construction of matter interferometers and explains why the Kapitza-Dirac effect is one of the workhorses in the field of atom optics. Atom optics concerns the manipulation of atomic waves in ways analogous to the manipulation of light waves with optical elements. The excitement and activity in this new field of physics stems in part from the realization that the shorter de Broglie wavelengths of matter waves allow ultimate sensitivities for diffractive and interferometric experiments that in principle would far exceed their optical analogues. Not only is the Kapitza-Dirac effect an important enabling tool for this field of physics, but diffraction peaks have never been observed for electrons, for which it was originally proposed in 1933. Why has this not been observed? What is the relation between the interaction of laser light with electrons and the interaction of laser light with atoms, or in other words what is the relation between the ponderomotive potential and the lightshift potential? Would it be possible to build interferometers using the Kapitza-Dirac effect for other particles? These questions will be addressed in this paper.
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).
Theoretical spin properties of the Kapitza-Dirac effect beyond the plane-wave description are not known in detail. We develop a method for computing electron diffraction of the two-photon Kapitza-Dirac effect in a two-dimensional Gaussian beam standing light wave within a relativistic formulation. The solutions are computed on the basis of time-dependent perturbation theory, where a momentum space formulation with the use of a Fourier transformation of the external potential allows for the solving the perturbative time-integrals. An iteration over each possible quantum state combination leads to a quadratic scaling of our method with respect to spacial grid resolution, where time-stepping does not occur in the numeric implementation. The position- and momentum space grids are adapted to the two-photon interaction geometry at low resolution, for which our study only finds partial convergence of the simulated diffraction pattern. Further, the method has the advantage of having an easy implementable parallelization layout.
Localization of relativistic particles have been of great research interests over many decades. We investigate the time evolution of the Gaussian wave packets governed by the one dimensional Dirac equation. For the free Dirac equation, we obtain the evolution profiles analytically in many approximation regimes, and numerical simulations consistent with other numerical schemes. Interesting behaviors such as Zitterbewegung and Klein paradox are exhibited. In particular, the dispersion rate as a function of mass is calculated, and it yields an interesting result that super-massive and massless particles both exhibit no dispersion in free space. For the Dirac equation with random potential or mass, we employ the Chebyshev polynomials expansion of the propagator operator to numerically investigate the probability profiles of the displacement distribution when the potential or mass is uniformly distributed. We observe that the widths of the Gaussian wave packets decrease approximately with the power law of order o(s−ν)o(s^{-ν})o(s−ν) with $\frac{1}{2}