Classical and quantum dynamics of a charged scalar particle in a background of two counterpropagating plane waves

We consider a scalar particle in a background formed by two counterpropagating plane waves. Two cases are studied: (i) dynamics at a magnetic node and (ii) zero initial transverse canonical momentum. The Lorentz and Klein-Gordon equations are solved for these cases and approximations analyzed. For the magnetic-node solution (homogeneous, time-dependent electric field), the modified Volkov wave function which arises from a high-energy approximation is found to be inaccurate for all energies and the solution itself unstable when the photon emission (nonlinear Compton scattering) is included. For the zero initial transverse canonical momentum case, in both quantum and classical cases, forbidden parameter regimes, absent in the plane-wave model, are identified.

Figure 1

Demonstration of particle orbits at the magnetic node of a standing plane wave of ${\xi }_1={\xi }_2=100$, ${\omega }_1={\omega }_2=0.01m$, ${\overline{k}}^2=(0.01m{)}^2$. The units of the transverse coordinates $x·{ϵ}_{1,2}$ are of $1/{\omega }_1$. In order of increasing parameter $s={\mu }_{\perp }^2/{\varpi }_{\perp }^2$: thin solid line: $p_{\mathrm{in}}=-a_{\mathrm{in}}$, i.e., ${\mathrm{\Pi }}_{\mathrm{in}}=0$ and $s=0$. Dashed line: $p_{\mathrm{in}}·{ϵ}_1=a_{\mathrm{in}}·{ϵ}_1$; $p_{\mathrm{in}}·{ϵ}_2=0.2a_{\mathrm{in}}·{ϵ}_2$, $s=0.72$. Thick solid line: $p_{\mathrm{in}}=a_{\mathrm{in}}$, $s=0.89$. (In the right-hand plot $k=\overline{k}$.) As $s\to 1$, the trajectory tends to a straight line and the phase tends to a horizontal asymptote.

Figure 2

Demonstration of different particle orbits in a plane wave for $p_{\mathrm{in}}=m(20.0255,0,0,20.0005)$, ${\omega }_1={\omega }_2=0.01m$, $-k_{\mathrm{\Delta }}^2=(0.02m{)}^2$, where transverse coordinates $x·{ϵ}_{1,2}$ are in units of $1/{\omega }_1$. Left: ${\xi }_1={\xi }_2=10$. Right: ${\xi }_1=0.01$, ${\xi }_2=10$.

Figure 3

Regions of instability (gaps) in $\lambda \text{−}Q$ space where $\mathrm{Im}\nu \ne 0$, are indicated by the linear hatched regions. In both the magnetic-node and zero initial transverse canonical momentum cases, allowed parameters are in the shaded region between the dashed lines, $0\le Q<\lambda /2$. There are no stable states for $\lambda <0$, which corresponds to the classically forbidden region.

Figure 4

A log-log plot of how the effective mass varies with increasing transverse momentum for ${\xi }_{\mathrm{\Sigma }}=1$, ${\overline{k}}^2=(0.01\mathrm{m}{)}^2$. The multiscale result agrees exactly with the analytical result from the Mathieu characteristic exponent and hence the lines lie atop one another and are indistinguishable in the plot, whereas the high-energy approximation disagrees at both high and low energy.

Figure 5

A log-log plot of how the effective mass varies with increasing intensity parameter ${\xi }_{\mathrm{\Sigma }}$ for ${\overline{k}}^2=(5\mathrm{m}{)}^2$ and $p^{\perp }=2m$. The large value of ${\overline{k}}^2$ has been chosen to emphasize the band structure. For ${\overline{k}}^2\ll 1$, the discrepancies between the high-energy and plane-wave approximations with the multiscale and exact results persist.