Quantumlike Dynamics of Classical Particles in Ponderomotive Potentials

The average dynamics of a classical particle under the action of a high-frequency radiation resembles quantum particle motion in a conservative field with an effective de Broglie wavelength $\lambda$ equal to the particle average displacement on the oscillation period. In a quasiclassical field, with a spatial scale large compared to $\lambda$, the guiding-center motion is adiabatic. Otherwise, a particle exhibits quantized eigenstates in ponderomotive potential wells, tunnels through “classically forbidden” regions, and experiences stochastic reflection from attractive potentials.

Figure 1

Poincaré mapping (fragment) for a particle exhibiting 1D oscillations in the field $E(z,t)={\overline{E}}_0{sinh}^2(z/L)\sin \omega t$ with $\widehat{ϵ}\equiv e{\overline{E}}_0/m{\omega }^{2}L=2$: $\mathcal{E}$ (a.u.) and $\theta =\omega t$ are taken at particle crossing $z=0$ with $v>0$. Stable oscillations (continuous curves) are observed near resonant orbits (one in the center and five on the sides); chaos is developed at the periphery. At larger $\widehat{ϵ}$, only particles near the central orbit are confined.

Figure 2

First five stationary eigenstates of a guiding center trapped within a ponderomotive potential formed by an ac field with the amplitude $E(z)=q|z{|}^{\alpha }$, $\alpha =0.6$, $q={10}^{-3}$: (a) phase plane $(z,v_z)$, (b) energy diagram [dashed, numerical; solid gray, quasiclassical; solid black, Miller potential $\Phi (z)$] ($E$ is measured in units $mc\Omega /e$; $z$ is measured in units $c/\Omega$; $v_z$ is measured in units $\widehat{v}$; ${\mathcal{E}}_{\perp }$ is measured in arbitrary units).

Figure 3

Longitudinal velocity $v_z$ vs $z$ for a particle being trapped and released by a Gaussian attractive ponderomotive potential in a magnetic field with $\widehat{ϵ}=6\pi$ ($E_{\max }=0.001$, $\Lambda =100$, $L=0.33$; ${\mathcal{E}}_{\perp ,0}=0$; same notation as in Fig. 2): $v_z=0.30$ (black) and $v_z=0.31$ (gray).

Figure 4

First three stationary eigenstates of a free particle traveling through a crystal formed of multiple ponderomotive barriers (same notation and parameters as in Fig. 3): shaded, $\Phi (z)$; solid, longitudinal energy ${\mathcal{E}}_{\parallel }$ vs $z$.