Spin effects on the semiclassical trajectories of Dirac electrons

The relativistic semiclassical evolution of the position of an electron in the presence of an external electromagnetic field is studied in terms of a Newton equation that incorporates spin effects directly. This equation emerges from the Dirac equation and allows the identification of scenarios where spin effects are necessary to understand the main characteristics of the electron trajectories. It involves the eigenvalues of the non-Hermitian operator ${\mathrm{\Sigma }}_{\mu \nu }{F}^{\mu \nu }$, with ${\mathrm{\Sigma }}_{\mu \nu }$ and $F^{\mu \nu }$ as the spin and electromagnetic tensors. The formalism allows a deeper understanding on the physics behind known analytical solutions of the Dirac equation when translational dynamics seem to decouple from spin evolution. As an illustrative example, it is applied to an electron immersed in an electromagnetic field which exhibits chiral symmetry and optical vortices. It is shown that the polarization of intense structured light beams can be used to suppress or enhance spin effects on the electron semiclassical trajectory; the latter configuration yields a realization of a Stern-Gerlach apparatus for an electron.

Figure 1

${\mathrm{\Delta }}_{BE}^2$ for a TE Bessel mode with $k_{\perp }=0.04\times (\omega /c)$ and with (a) $m_z=0$ and (b) $m_z=1$; (c) radial derivative ${\partial }_{k_{\perp }\rho }{\mathrm{\Delta }}_{EB}^2$ and (d) phase derivative ${\partial }_{\mathrm{\Theta }}{\mathrm{\Delta }}_{EB}^2$ for the $m_z=1$. Results are shown at the plane $k_{z}z-\omega t=0$ and they are given in units of the squared amplitude $|{\mathcal{E}}^{TE}{|}^2$.

Figure 2

Trajectories of an electron in a $m_z=1$ TE Bessel beam of wavelength ${\lambda }_{\text{beam}}=0.1$ nm, $k_{\perp }=0.04\omega /c$, and ${\mathcal{E}}^{\text{TE}}=0.005mc\omega /q$: (a) $\ell =-{\mu }_B\sqrt{|{\mathrm{\Delta }}_{BE}^2|}$; (b) $\ell ={\mu }_B\sqrt{|{\mathrm{\Delta }}_{BE}^2|}$. (c) $z\left(t\right)$ for $\ell =-{\mu }_B\sqrt{|{\mathrm{\Delta }}_{BE}^2|}$ (dot line), $\ell ={\mu }_B\sqrt{|{\mathrm{\Delta }}_{BE}^2|}$ (dashed line), and neglecting spin effects (continuous line). The initial conditions are ${\rho }_0=0.05\lambda , z_0=0, d{\rho }_0/dct=0, d{\phi }_0/dct=-0.01, d{z}_0/dct=3\times {10}^{-5}$.

Figure 3

(a) Real and (b) imaginary part of ${\mathcal{F}}_{z\phi }^{(j:j)}$ for a TE Bessel beam. Results are shown at the plane $k_{z}z-\omega t=0$, and the divergence at the null planes were cut at the value ${10}^{-5}$ in order to illustrate the high localization of ${\mathcal{F}}_{z\phi }^{(j:j)}$.