Dimension-dependent stimulated radiative interaction of a single electron quantum wavepacket

Abstract

In the foundation of quantum mechanics, the spatial dimensions of electron wavepacket are understood only in terms of an expectation valuethe probability distribution of the particle location. One can still inquire how the quantum electron wavepacket size affects a physical process. Here we address the fundamental physics problem of particle-wave duality and the measurability of a free electron quantum wavepacket. Our analysis of stimulated radiative interaction of an electron wavepacket, accompanied by numerical computations, reveals two limits. In the quantum regime of long wavepacket size relative to radiation wavelength, one obtains only quantum-recoil multiphoton sidebands in the electron energy spectrum. In the opposite regime, the wavepacket interaction approaches the limit of classical point-particle acceleration. The wavepacket features can be revealed in experiments carried out in the intermediate regime of wavepacket size commensurate with the radiation wavelength.      , where   0ˆz Ee z iq z z E e  represents the dominant slow component of the radiation wave, and transverse field components, as well astransverse variation of the field are neglected. We examplify our modeling here for a case of Smith-Purcell radiation (see Fig. 1), for which the radiation wave is a Floquent mode: 4   zm iq z m m ze   EEwith 0 2 zm z G q q m   , G  is the grating period, 0 Θ, / z q qcos q c   and Θ is the incidence angle of the radiation wave relative to the axial interaction dimension. The radiation wave number z zm qq  represents one of the space harmonics m that satisfies synchronizm condition with the electron [6]: 0 zm vq   .We note that the analysis would be similar for the Cerenkov interaction scheme with   Θ z q n cos   , and () n  the index of refraction of the medium. Furthermore, the analysis can be extended to the case of FEL and other interaction schemes [13,14].The solution of Schrodinger equation to zero order (i.e. free-space propagation) is well known. Assuming that the initial wavepacket, which is emitted at some point

Spontaneous and stimulated radiative emission of modulated free-electron quantum wavepackets—semiclassical analysis

“…We assert, though, that this conclusion does not carry over to the case of stimulated interaction (emission/absorption or acceleration/deceleration). We have shown in an earlier publication [35] that the classical phase-dependent acceleration/deceleration of a single electron in the point-particle limit is partially valid and also in the quantum-wavepacket regime. The momentum transfer from the laser field to the wavepacket is smaller than in the point-particle limit, and it diminishes in the inherent quantum limit, where the evolving wavepacket size exceeds the optical radiation period T=2π/ω of the interacting radiation wave are the standard derivation sizes of the momentum and minimal duration of a coherent electron wavepacket)is often used to define the quantum regime [23].…”

“…A proposed scheme for measuring spontaneous and stimulated radiation emission and electron energy spectrum of an electron wavepacket is shown in figure 1. This interaction scheme, based on the Smith-Purcell radiation effect was used in [35] to calculate the wavepacket-dependent energy spectrum due to radiative interaction with an input radiation field, injected into the interaction region above the grating, in controlled phase correlation with the incoming electron wavepacket. The wavepacket size depends on the drift time from the cathode to the grating.…”

“…Our model for the current of the electron that excites the radiation through Maxwell equations is a single electron model where the charge density of the electron wavepacket is the expectation value of the probability density of the electron wavefunction solution of a relativistic Schrodinger equation [35]. The space-charge plasma effects and inter-particle Coulomb interactions are thus neglected.…”

“…The momentum transfer from the laser field to the wavepacket is smaller than in the point-particle limit, and it diminishes in the inherent quantum limit, where the evolving wavepacket size exceeds the optical radiation period T=2π/ω of the interacting radiation wave are the standard derivation sizes of the momentum and minimal duration of a coherent electron wavepacket)is often used to define the quantum regime [23]. This is a sufficient condition, but equation (1), with s s s = > ( ) L t t D t 0 is a necessary condition to enter the quantum regime of dismissing acceleration [35]. Thus, measurements of the electron energy spectrum after interaction with radiation waves at different frequencies would enable determination of the historydependentelectron wavepacket size s ( ) L t D .…”

“…Thus, measurements of the electron energy spectrum after interaction with radiation waves at different frequencies would enable determination of the historydependentelectron wavepacket size s ( ) L t D . In the semi-classical analysis of electron wavepacket interaction with radiation [35], the wavepacketdependent energy (momentum) acceleration/deceleration was calculated using the first-order perturbation analysis and numerical solution of Schrodinger equation. Evidently, such a change in the electron energy spectrum must involve also corresponding change in the spectrum of the interacting radiation.…”