Dynamically assisted Klein tunneling in the Furry picture

TL;DR

The paper investigates 1D relativistic fermion scattering from a stationary step potential in the presence of a temporally oscillating electric field, using Furry-picture perturbation theory to treat the field perturbatively while the step remains non-perturbative. A wave-packet formalism is developed because the time-dependent background precludes standard stationary scattering, and it is shown that a positive-frequency incident wave can access the negative-frequency region below the step by exchanging energy $\omega$, yielding a nonzero transmission at second order. The central finding is dynamically assisted Klein tunneling, which can occur even when neither the stationary potential nor the oscillating field alone is supercritical, provided $V_0+\omega>2m$; explicit expressions for $P_{\mathrm{refl}}^{(0,1,2)}$ and $P_{\mathrm{trans}}^{(0,1,2)}$ are derived and validated against numerical simulations. The results illuminate cooperative nonperturbative-perturbative effects in relativistic scattering and point to extensions to more general field configurations and to graphene-like systems.

Abstract

One-dimensional scattering of a wave packet of a relativistic fermion under a temporally oscillating electric field superimposed on a potential step is discussed by using the Furry-picture perturbation theory, where the oscillating electric field is treated as a perturbation. Reflection and transmission probabilities of the wave packet, which in its single-mode limit are consistent with those in the stationary scattering off the potential step alone, are investigated up to the second order. We show that even in the absence of the so-called Klein region, a positive-frequency incoming wave can penetrate the negative-frequency region below the potential step by emitting its energy to the oscillating electric field with a finite tunneling probability.

Figures (5)

• Figure 1: Scattering behavior of the left-incident and positive-frequency solution $\psi_s^{(E_p)}$ off the overcritical potential step. Gray-shaded regions represent the forbidden region or the mass gap. Blue arrows for (i), (ii), or (iii) show the directions of the incident, reflected, and transmitted waves for each.
• Figure 2: Schematic picture of the dynamically assisted Klein tunneling under the subcritical potential step and the oscillating field. The thin blue arrow in $z < 0$ represents the positive-frequency left-incident mode $\psi_s^{(E_p)}$ in the initial wave packet, which shows the total reflection. With the aid of the oscillating electric field, the transition to the negative-frequency right-incident mode $\phi_{s'}^{(V_0 - E_{q'})}$ (the thin blue arrow in $z > 0$, which also shows the total reflection) by emitting its energy $\omega$ (the orange wavy line) can occur. Then, the wave packet can reach the right infinity according to the reflected wave of $\phi_{s'}^{(V_0 - E_{q'})}$. Thus, the oscillating electric field enables the tunneling process of the wave packet from the positive-frequency region to the negative-frequency region, as indicated by the sky-blue thick arrow.
• Figure 3: Transmission probability of the wave packet $P_\mathrm{trans}$ for the central momentum $\bar{p}/m = 1.0$ and the potential height $V_0/m = 1.5$. Other parameters are $\sigma/m = 0.1$, $e\mathcal{E}_z/m^2 = 0.04$, and $l/{\mkern0.75mu\mathchar '26\mkern -9.75mu\lambda} = 1.0$, where ${\mkern0.75mu\mathchar '26\mkern -9.75mu\lambda}$ is the Compton wavelength. The analytical results up to the first and second orders are shown in the black dotted line and the thick red line, respectively. The yellow dash-dotted line (the light green solid line) represents the analytical result up to the second order except for the first (second) term on the right-hand side of $P_\mathrm{trans}^{(2)}$ in eq. \ref{['eq:(sec5)P_refl(2)']}. The numerical result is plotted in blue-green.
• Figure 4: Transmission probability of the wave packet $P_\mathrm{trans}$ for the central momentum $\bar{p}/m = 2.0$ and the assistance energy $\omega = \bar{E}-(V_0 - m)$. Other parameters are $\sigma/m = 0.1$, $e\mathcal{E}_z/m^2 = 0.04$, and $l/{\mkern0.75mu\mathchar '26\mkern -9.75mu\lambda} = 1.0$. The numerical result is shown by the blue-green plots, while the analytical results up to the first and second orders are represented by the black dotted line and the thick red line, respectively.
• Figure 5: The probability densities of the wave packets under the potential step with height $V_0 = 5.0$ at times $t = -10.0$ (left panel) and $t = 10.0$ (right panel), in units of $m = 1$. The red dashed line and the green solid line correspond to Klein tunneling and Klein paradox, respectively. The sky-blue region is the numerical simulation.