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arXiv:1310.1995 · Physical Review A

Unified dynamics of electrons and photons viaZitterbewegungand spin-orbit interaction

Cody C. Leary, Karl H. Smith

Published Feb 19, 2014• 10 citations
arXivHTMLPDF

Abstract

We show that when an electron or photon propagates in a cylindrically symmetric waveguide, it experiences both a Zitterbewegung effect and a spin-orbit interaction leading to identical propagation dynamics for both particles. Applying a unified perturbative approach to both particles simultaneously, we find that to first order in perturbation theory, their Hamiltonians each contain identical Darwin (Zitterbewegung) and spin-orbit terms, resulting in the unification of their dynamics. The presence of the Zitterbewegung effect may be interpreted physically as the delocalization of the electron on the scale of its Compton wavelength, or the delocalization of the photon on the scale of its wavelength in the waveguide. The presence of the spin-orbit interaction leads to the prediction of several rotational effects: the spatial or time evolution of either particle's spin or polarization vector is controlled by the sign of its orbital angular momentum quantum number or, conversely, its spatial wave function is controlled by its spin angular momentum.

DOI
10.1103/physreva.89.023831

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We show that when an electron or photon propagates in a cylindrically symmetric waveguide, it experiences both a Zitterbewegung effect and a spin-orbit interaction leading to identical propagation dynamics for both particles. Applying a unified perturbative approach to both particles simultaneously, we find that to first order in perturbation theory, their Hamiltonians each contain identical Darwin (Zitterbewegung) and spin-orbit terms, resulting in the unification of their dynamics. The presence of the Zitterbewegung effect may be interpreted physically as the delocalization of the electron on the scale of its Compton wavelength, or the delocalization of the photon on the scale of its wavelength in the waveguide. The presence of the spin-orbit interaction leads to the prediction of several rotational effects: the spatial or time evolution of either particle's spin or polarization vector is controlled by the sign of its orbital angular momentum quantum number or, conversely, its spatial wave function is controlled by its spin angular momentum.
We show that when an electron or photon propagates in a cylindrically symmetric waveguide, its spin angular momentum (SAM) and its orbital angular momentum (OAM) interact. Remarkably, we find that the dynamics resulting from this spin-orbit interaction are quantitatively described by a single expression applying to both electrons and photons. This leads to the prediction of several rotational effects: the spatial or time evolution of either particle's spin-polarization vector is controlled by its OAM quantum number or, conversely, its spatial wave function is controlled by its SAM. We show that the common origin of these effects in electrons and photons is a universal geometric phase. We demonstrate how these phenomena can be used to reversibly transfer entanglement between the SAM and OAM degrees of freedom of two-particle states.
In the Dirac theory for the motion of free relativistic electrons, highly oscillatory components appear in the time evolution of physical observables such as position, velocity, and spin angular momentum. This effect is known as zitterbewegung. We present a theoretical analysis of rather different Hamiltonians with gapped and/or spin-split energy spectrum (including the Rashba, Luttinger, and Kane Hamiltonians) that exhibit analogs of zitterbewegung as a common feature. We find that the amplitude of oscillations of the Heisenberg velocity operator v(t)\mathbf{v}(t)v(t) generally equals the uncertainty for a simultaneous measurement of two linearly independent components of v\mathbf{v}v. It is also shown that many features of zitterbewegung are shared by the simple and well-known Landau Hamiltonian, describing the dynamics of two-dimensional (2D) electron systems in the presence of a magnetic field perpendicular to the plane. Finally, we also discuss the oscillatory dynamics of 2D electrons arising from the interplay of Rashba spin splitting and a perpendicular magnetic field.
The detailed motion of a free Dirac electron is investigated by examining the expectation values of the position r and of r×ṙ in a wave packet. It is shown that the well-known zitterbewegung may be looked upon as a circular motion about the direction of the electron spin, with a radius equal to the Compton wavelength (divided by 2π) of the electron. It is further shown that the intrinsic spin of the electron may be looked upon as the “orbital angular momentum” of this motion. The current produced by the zitterbewegung is seen to give rise to the intrinsic magnetic moment of the electron.
A classical circularly polarized electromagnetic wave carries angular momentum, and represents the classical limit of a photon, which carries quantized spin. It is shown that a very similar picture of a circularly polarized coherent wave can account for both the spin of an electron and its quantum wave function, in a Lorentz-invariant fashion. The photon-electron interaction occurs through the usual electromagnetic potentials, modulating the frequency and wavevector (energy and momentum) of this rotating spin field. Other quantum particles can also be represented either as rotating spin fields, or as composites of such fields. Taken together, this picture suggests an alternative interpretation of quantum mechanics based solely on coherent wave packets, with no point particles present.
Zitterbewegung (ZBW), the trembling motion predicted by the Dirac equation, has long remained unobservable in free electrons due to its sub-Compton scale. We elaborately construct a relativistic vortex electron wave packet as a coherent superposition of both positive- and negative-energy Dirac states and derive their space-time dynamics. Our analysis demonstrates that introducing orbital angular momentum provides a mechanism for amplifying the ZBW amplitude far beyond that of conventional Gaussian packets, while maintaining coherence. The resulting relativistic vortex states unify Gaussian and Bessel-Gaussian models within a single framework and opens new possibilities for observing relativistic quantum dynamics in structured electron wave packets.