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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Quantumlike corrections and semiclassical description of charged-particle beam transport

Renato Fedele, Vladimir I. Man’ko

Published Jul 1, 1998• 29 citations
Publisher

Abstract

It is shown that the standard classical picture of charged-particle beam transport in paraxial approximation may be conveniently replaced by a Wigner-like picture in a semiclassical approximation. In this effective description, the classical phase-space equation for electronic rays is replaced by a von Neumann--like equation, where the transverse emittance plays the role of $\ensuremath{\Elzxh}$. Relevant remarks concerning the quantumlike corrections for an arbitrary potential in comparison with the standard classical description of the beam transport are given.

DOI
10.1103/physreve.58.992

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It is shown that the standard classical picture of charged-particle beam transport in paraxial approximation may be conveniently replaced by a Wigner-like picture in a semiclassical approximation. In this effective description, the classical phase-space equation for electronic rays is replaced by a von Neumann--like equation, where the transverse emittance plays the role of \ensuremath\Elzxh\ensuremath{\Elzxh}\ensuremath\Elzxh. Relevant remarks concerning the quantumlike corrections for an arbitrary potential in comparison with the standard classical description of the beam transport are given.
An alternative procedure to the one by Gloge and Marcuse [J. Opt. Soc. Am. 59, 1629 (1969)] for performing the transition from geometrical optics to wave optics in the paraxial approximation is presented. This is done by employing a recent "deformation" method used to give a quantumlike phase-space description of charged-particle-beam transport in the semiclassical approximation. By taking into account the uncertainty relation (diffraction limit) that holds between the transverse-beam-spot size and the rms of the light-ray slopes, the classical phase-space equation for light rays is deformed into a von Neumann-like equation that governs the phase-space description of the beam transport in the semiclassical approximation. Here, Planck's constant and the time are replaced by the inverse of the wave number, not lambda, and the propagation coordinate, respectively. In this framework, the corresponding Wigner-like picture is given and the quantumlike corrections for an arbitrary refractive index are considered. In particular, it is shown that the paraxial-radiation-beam transport can also be described in terms of a fluid motion equation, where the pressure term is replaced by a quantumlike potential in the semiclassical approximation that accounts for the diffraction of the beam. Finally, a comparison of this fluid model with Madelung's fluid model is made, and the classical-like picture given by the tomographic approach to radiation beams is advanced as a future perspective.
An alternative procedure to the one by Gloge and Marcuse [J. Opt. Soc. Am. 59, 1629 (1969)] for performing the transition from geometrical optics to wave optics in the paraxial approximation is presented. This is done by employing a recent "deformation" method used to give a quantumlike phase-space description of charged-particle-beam transport in the semiclassical approximation. By taking into account the uncertainty relation (diffraction limit) that holds between the transverse-beam-spot size and the rms of the light-ray slopes, the classical phase-space equation for light rays is deformed into a von Neumann-like equation that governs the phase-space description of the beam transport in the semiclassical approximation. Here, Planck's constant and the time are replaced by the inverse of the wave number, not lambda, and the propagation coordinate, respectively. In this framework, the corresponding Wigner-like picture is given and the quantumlike corrections for an arbitrary refractive index are considered. In particular, it is shown that the paraxial-radiation-beam transport can also be described in terms of a fluid motion equation, where the pressure term is replaced by a quantumlike potential in the semiclassical approximation that accounts for the diffraction of the beam. Finally, a comparison of this fluid model with Madelung's fluid model is made, and the classical-like picture given by the tomographic approach to radiation beams is advanced as a future perspective.
A recent proposal (see quant-ph/9803068) to simulate semiclassical corrections to classical dynamics by suitable classical stochastic fluctuations is applied to the specific instance of charged beam dynamics in particle accelerators. The resulting picture is that the collective beam dynamics, at the leading semiclassical order in Planck constant can be described by a particular diffusion process, the Nelson process, which is time-reversal invariant. Its diffusion coefficient Nλc\sqrt{N}λ_{c}N​λc​ represents a semiclassical unit of emittance (here NNN is the number of particles in the beam, and λcλ_{c}λc​ is the Compton wavelength). The stochastic dynamics of the Nelson type can be easily recast in the form of a Schroedinger equation, with the semiclassical unit of emittance replacing Planck constant. Therefore we provide a physical foundation to the several quantum-like models of beam dynamics proposed in recent years. We also briefly touch upon applications of the Nelson and Schroedinger formalisms to incorporate the description of collective coherent effects.
In this paper we briefly review the main aspects of a recent proposal to simulate semiclassical corrections to classical dynamics by suitable classical stochastic fluctuations, and we apply it to the specific instance of charged beams in particle accelerators. The resulting picture is that the collective beam dynamics, at the leading semiclassical order in Planck constant can be described by a particular diffusion process, the Nelson process, that is time--reversal invariant. Its diffusion coefficient p N c represents a semiclassical unit of emittance (here N is the number of particles in the beam, and c is the Compton wavelength). The stochastic dynamics of the Nelson type can be easily recast in the form of a Schroedinger equation, with the semiclassical unit of emittance replacing Planck constant. Therefore we provide a physical foundation to the several quantum--like models of beam dynamics proposed in recent years. We also briefly touch upon applications of the Nelson and Schro...
A semiclassical approximation is derived by using a family of wavepackets to map arbitrary wavefunctions into phase space. If the Hamiltonian can be approximated as linear over each individual wavepacket, as often done when presenting Ehrenfest's theorem, the resulting approximation is a linear first-order partial differential equation on phase space, which will be referred to as the Schrödinger-Ehrenfest or SE equation. This advectively transports wavefunctions along classical trajectories, so that as a trajectory is followed in time the amplitude remains constant while the phase changes by the action divided by ℏ\hbarℏ. The wavefunction's squared-magnitude is a plausible phase space density and obeys Liouville's equation for the classical time evolution. This is a derivation of the Koopman-von~Neumann (KvN) formulation of classical mechanics, which previously was postulated but not derived. With the time-independent SE equation, continuity of the wavefunction requires the change of phase around any closed path in the torus covered by a classical trajectory to be an integer multiple of 2π2π2π, giving a standing wave picture of old quantum mechanics. While this applies to any system, for separable systems it gives Bohr-Sommerfeld quantization.