Spin-one-half particles in strong electromagnetic fields: Spin effects and radiation reaction

Various classical models of electrons including their spin degrees of freedom are commonly applied to describe the electron dynamics in strong electromagnetic fields. We demonstrate that different models can lead to different or even contradicting predictions regarding how the spin degree of freedom modifies the electron's orbital motion when the electron moves in strong electromagnetic fields. This discrepancy is rooted in the model-specific energy dependency of the spin-induced Stern-Gerlach force acting on the electron. The Frenkel model and the classical Foldy-Wouthuysen model are compared exemplarily in the nonrelativistic and the relativistic limits in order to identify parameter regimes where these classical models make different predictions. This allows for experimental tests of these models. In ultrastrong laser setups in parameter regimes where effects of the Stern-Gerlach force become relevant, radiation-reaction effects are also expected to set in. We incorporate the radiation reaction classically via the Landau-Lifshitz equation and demonstrate that although radiation-reaction effects can have a significant effect on the electron trajectory, the Frenkel model and the classical Foldy-Wouthuysen model remain distinguishable also if radiation-reaction effects are taken into account. Our calculations are also suitable to verify the Landau-Lifshitz equation for the radiation reaction of electrons and other spin-1/2 particles.

Figure 1

Trajectories of an electron with initial momentum $3.65mc$ in an ultrastrong homogeneous magnetic field (perpendicular to the plane of projection) of strength $2000\mathrm{a}.\mathrm{u}.=4.7\times {10}^8$ T as predicted by the Dirac equation (solid gray line), the classical Foldy-Wouthuysen model (dotted red line), and the Frenkel model (solid blue and dashed blue lines). The electron's initial spin orientation is parallel or antiparallel to the magnetic field. Trajectories of electrons with parallel spin (solid blue line) differ from trajectories of electrons with antiparallel spin (dashed blue line) within the Frenkel model, whereas the Dirac theory and the classical Foldy-Wouthuysen model do not lead to spin-dependent trajectories for homogeneous fields. In order to illustrate radiation-reaction effects, the dot-dashed green line shows the trajectory for the classical Foldy-Wouthuysen model with radiation damping included via the Landau-Lifshitz equation. As an illustration the density of the electron's quantum mechanical wave packet is also indicated at five points in time.

Figure 2

(a) Inhomogeneous magnetic field $B_x$ with gradient ${\mathbf{\nabla }}_x{B}_x=\pm \widehat{B}/\ell$ in the range $x\in [-\ell ,\ell ]$ (solid black curve) and the field $B_x^{\prime }=\widehat{B}/{\left(1+{\pi }^2{x}^2/{\ell }^2\right)}^{3/2}$ (solid gray curve) generated by a single-loop solenoid with radius $\ell /\pi$. Their gradients are shown by dashed curves of the respective colors. (b) Trajectories of electrons with spin up and spin down predicted by the nonrelativistic spin model (indicated by the index NR). Electrons propagate in magnetic field $B_x$ along the $x$ axis with an initial velocity $v_0={10}^5$ m/s. The right $(x-v_{0}t)$ coordinate in SI units corresponds to a maximum magnetic field $\widehat{B}=10$ T and the characteristic length $\ell /\pi =1$ cm, similar to those presented in Ref. [6]. (c) The same trajectories of electrons predicted by the nonrelativistic spin model, classical the Foldy-Wouthuysen model (indicated by the index FW), and Frenkel model (indicated by the index F), respectively. The initial velocity of electrons is $v_0=2\times {10}^8$ m/s.

Figure 3

Trajectories of electrons in a plane-wave laser pulse (39) with the initial spin orientation parallel to the $z$ direction and initial momentum ${\mathit{p}}_{\mathrm{kin}}\left(0\right)={(-mc,0,0)}^{\mathsf{T}}$ as predicted by the various considered models. Laser parameters are $\widehat{E}=5000\mathrm{a}.\mathrm{u}.=2.57\times {10}^{15}$ V/m, $\lambda =20\mathrm{a}.\mathrm{u}.=1.06$ nm, and $n=6$. In the case of the Dirac equation the wave packet had a width of $0.5\mathrm{a}.\mathrm{u}.=0.026$ nm initially.

Figure 4

Trajectory of a highly energetic electron with the initial $\gamma =100$ in a focused laser pulse with wavelength $\lambda =1.51\times {10}^4\mathrm{a}.\mathrm{u}.=800\mathrm{nm}$, amplitude of strength $\widehat{E}=1056\mathrm{a}.\mathrm{u}.=8.03\times {10}^{14}\mathrm{V}/\mathrm{m}$, duration (number of cycles) $n=20$, and focus radius $w_0=2\lambda$. Black curves denote the trajectories of a spinless particle and dashed dark blue and dashed light red curves correspond to those described by the Frenkel model (34) for electrons with the spin parallel (spin up) and antiparallel (spin down) to the $z$ axis. Trajectories without radiation-reaction force (thin black and dashed dark blue curves) show a smaller reflection angle than those with radiation reaction (thick black and dashed light red curves). Inset: Definition of the extra deflection angles $\mathrm{\Delta }{\theta }_{\uparrow ,\mathrm{F}}$ and $\mathrm{\Delta }{\theta }_{\downarrow ,\mathrm{F}}$ for spin-up and spin-down electrons. All trajectories shown start from position $(0,0)$, where the electron hits the front of the laser pulse. Note that the field gradient is three orders of magnitude lower than for the setup in Fig. 3.

Figure 5

Ponderomotive deflection angle $\theta$ and extra angles $\mathrm{\Delta }\theta$ induced by the radiation-reaction force and spin forces of the Frenkel and the classical Foldy-Wouthuysen models as functions of the particle's initial energy, represented by the Lorentz factor $\gamma$, with the electric-field-strength scaling $\widehat{E}=4\pi m{c}^2\sqrt{{\gamma }^2-1}/\left(\right|q\left|\lambda \right)$ in (a) and $\widehat{E}=2\pi m{c}^2\sqrt{{\gamma }^2-1}/\left(\right|q\left|\lambda \right)$ in (b). Dotted black curves correspond to the deflection of spinless particles without accounting for the radiation reaction. Dashed green lines correspond to the extra deflection $\mathrm{\Delta }\theta$ induced by the radiation reaction only, while dashed red and blue curves correspond to the extra deflection $\mathrm{\Delta }\theta$ induced by the spin force within the classical Foldy-Wouthuysen model and the Frenkel model, respectively. Solid orange and cyan curves correspond to extra deflections from the classical Foldy-Wouthuysen and Frenkel models including the radiation reaction. In all cases, electrons in the spin-up state (parallel to the magnetic field) are considered.

Figure 6

Spin- contribution to the deflection angle as a function of the initial energy of the particle represented by $\gamma$ for spin-up and spin-down electrons. (a) The difference in the deflection angles of a spinless particle and a spin-1/2 electron for the Frenkel vs the classical Foldy-Wouthuysen models, i.e., $\mathrm{\Delta }{\theta }_{\uparrow /\downarrow ,\mathrm{FW}/\mathrm{F}}={\theta }_{\uparrow /\downarrow ,\mathrm{FW}/\mathrm{F}}-\theta$. (b) The difference in the deflection angles of a spinless particle and a spin-1/2 electron for the Frenkel and the classical Foldy-Wouthuysen models taking into account the radiation-reaction force (35) for the spinless particles as well as for the electron, i.e., $\mathrm{\Delta }{\theta }_{\uparrow /\downarrow ,\mathrm{FW}/\mathrm{F}}^{\mathrm{R}}={\theta }_{\uparrow /\downarrow ,\mathrm{FW}/\mathrm{F},\mathrm{R}}-{\theta }_{\mathrm{R}}$. In both panels, the laser parameters are the same as in Fig. 4 but the electric-field strength scales with the initial $\gamma$ as $\widehat{E}=4\pi m{c}^2\sqrt{{\gamma }^2-1}/\left(\right|q\left|\lambda \right)$. Note the different scales on the $y$ axes for the Frenkel and the classical Foldy-Wouthuysen models.