Test for descriptions of relativistic spin dynamics by using ultraintense lasers

The relativistic spin operator cannot be uniquely defined within relativistic quantum mechanics. Searching for relativistic equations that describe both the evolution of the spin and its influence on the motion of particles with spin represents a problem with almost centenary history. We develop a self-consistent module for modeling relativistic particles with spin motion into three-dimensional particle-in-cell simulations, where the expression of the spin-induced force can be chosen by various semiclassical models, such as the Frenkel and noncovariant Derbenev-Kondratenko models. Through simulations, we propose a potential experimental scheme for observing and testing theoretical descriptions on relativistic spin dynamics, where ultraintense lasers are used to collide with relativistic electron beams. The basis for the scheme is that different models predict different transverse polarization distributions of the scattered electrons after interactions, which can be measured in experiments.

Figure 1

(a) Schematic of the proposed scheme for testing theoretical predictions of different semiclassical models. Red and blue spheres represent electrons polarized parallel and antiparallel to the $z$ direction, respectively. (b) and (c) Transverse spatial distributions of the polarizations of scattered electrons in simulations with the Frenkel and noncovariant DK models, respectively. (d) Initial polarization distribution of the electron beam.

Figure 2

(a) According to the first term of Eq. (14) in the Frenkel model, time evolution of the SG force in the $y$ direction on electrons with opposite spin (red curves for $s_z=+1$ and black curves for $s_z=-1$) and different $\gamma$, where the laser amplitude is fixed as $a_0=80$. (b) The corresponding time evolutions for different $a_0$, where the Lorentz factor of the electron is fixed as $\gamma =800$. (c) The momentum accumulated in the $y$ direction derived by integrating Eq. (14) from $t=0$ to $t=T_0$ under various laser intensities $a_0$ and various electron energies $\gamma$ for an electron with spin $s_z=+1$. (d) The corresponding momenta for an electron with spin $s_z=-1$. (e)–(h) Same as (a)–(d), respectively, but according to the second and third terms of Eq. (15) in the noncovariant DK model [(e) and (f) have the same curve colors as in (a) and (b)].

Figure 3

Three-dimensional PIC simulation results: (a) The averaged spin ${\overline{s}}_z$ (red solid curve) and density (blue dashed curve) of electrons that distributed various intervals of ${\theta }_y$ under the Frenkel model, where ${\theta }_y=arctan(p_y/p_x)$. (c) Time evolution of averaged SG force in the $y$ direction for electrons with opposite spins dominated by the Frenkel model. (e) Forty typical electron trajectories in real space $(x,y,z)$, where the colors of the trajectories denote the spin state of electrons, red for spin up ($s_z>0$) and blue for spin down ($s_z<0$); isosurface distributions for polarization of the electron beam at $t=30,32,34,36,38T_0$, where the color bar displays the polarization of the isosurface distributions. The projection diagram shows the time evolution of the accumulated momentum in the $y$ direction by the SG force for electrons with opposite spins dominated by the Frenkel model. (b), (d), and (f) Same as (a), (c), and (e), respectively, but for the noncovariant DK model.

Figure 4

Three-dimensional PIC simulation results with consideration of radiation under the Frenkel model: (a) Isosurface distribution for polarization $s_z$ of the electron beam after interactions. (b) The averaged spin ${\overline{s}}_z$ (red solid curve) and density (blue dashed curve) of electrons that distributed various intervals of ${\theta }_y$. (c) and (d) Time evolution of $|{\overline{P}}_x|/m_e/c$ and ${\overline{P}}_y/m_e/c$, respectively, for electrons with opposite spins. (e) and (f) Time evolution of the averaged SG force in the $y$ direction and accumulated momentum by it, respectively, in the $y$ direction for electrons with opposite spins. (g) Time evolution of $\gamma$ of typical electrons, where the color of curves represents the evolutions of $s_z$ of the electrons. (h) Same as (g), but for the case without taking radiation into account.

Figure 5

Impacts of the (a) laser amplitude $a_0$, (b) initial energy of the electron beam $\gamma$, and (c) laser duration $\tau$ on the transverse angular distribution for density (black curves) and polarization (red curves) of the electron beam after interacting with the probe laser. In (a), the solid, dashed, and dash-dotted curves represent the cases $a_0=60, a_0=80$, and $a_0=100$, respectively. In (b), the solid, dashed, and dash-dotted curves represent the cases $\gamma =600, \gamma =800$, and $\gamma =1000$, respectively. In (c), the solid, dashed, and dash-dotted curves represent the cases $\tau =10T_0, \tau =20T_0$, and $\tau =30T_0$, respectively. Other parameters of the probe laser and electron beam are the same as those in Fig. 3.

Figure 6

Three-dimensional simulation results as a benchmark against which the results of previous publications (Refs. [46, 56]) can be compared. (a) and (b) Difference of the SG force components in the direction of laser propagation ($x$) subjected by two electrons with opposite spin ($s_z=\pm 1$) and initial momentum of (a) $p_0=(-m_{e}c,0,0)$ and (b) $p_0=(-2m_{e}c,0,0)$ during the interaction with a LP laser pulse. (c) Trajectories of electrons with opposite spins in a LP focused laser. (d) Trajectories of electrons with initial momentum of $3.56m_{e}c$ in a homogeneous magnetic field with strength of $4.7\times {10}^8\mathrm{T}$, with (blue curve for a spin-up electron and orange curve for a spin-down electron) and without (green curve) consideration of the SG force.