Access to Klein Tunneling via Space-Time Modulation

TL;DR

The work shows that space-time modulation of electromagnetic potentials enables access to Klein tunneling at substatic thresholds by inducing oblique energy–momentum transitions. Using a subluminal modulation, the authors derive closed-form lab-frame kinematics via a comoving-frame energy conservation and Lorentz spinor boosts, yielding explicit expressions for reflected and transmitted channels. They demonstrate that the Klein gap is tunable through the modulation velocity $v_m$ and the vector-to-scalar offset ratio $r_{A/V}$, and reveal a velocity-dependent Klein paradox where transmission vanishes within a finite $v_m$ window and reappears as $v_m\to1^-$. The results suggest experimental pathways with flying-focus fronts and relativistic electron beams to realize Klein tunneling under practicable field strengths, offering a new knob for tunable electron-wave control.

Abstract

We show that space-time modulation of electromagnetic potentials enables Klein tunneling far below the static threshold. The derived kinematics reveal oblique transitions that can connect opposite-energy continua without requiring their overlap, yielding a velocity-tunable Klein gap where transmission vanishes within a finite velocity window and reemerges beyond. The associated reduction in energy thresholds -- by up to four orders of magnitude -- suggests the potential for experimental realization using flying-focus fronts and relativistic electron beams.

Figures (5)

• Figure 1: Conventional Klein paradox at a spatial scalar potential step. (a) Single interface at $z_0$ between $V_1$ and $V_2$ (we set $V_1=0$) with incident wave $\psi_\mathrm{i}$, reflected wave $\psi_\mathrm{r}$ and transmitted wave $\psi_\mathrm{t}$. (b) Current-ratio probabilities $R=j_\mathrm{r}/j_\mathrm{i}$ and $T=j_\mathrm{t}/j_\mathrm{i}$ versus step height $q\Delta V$ at fixed $E_\mathrm{i}$, with subcritical region $q\Delta V_\mathrm{A}<E_\mathrm{i}-m$, Klein gap $E_\mathrm{i}-m<q\Delta V_\mathrm{B}<E_\mathrm{i}+m$ (evanescent $\psi_\mathrm{t}$) and supercritical or Klein-region $q\Delta V_\mathrm{C}>E_\mathrm{i}+m$ (coupling to the negative-energy branch). (c) Corresponding dispersion diagrams labeling $\mathrm{i},\mathrm{r},\mathrm{t}$.
• Figure 1: Critical modulation velocities. Dispersions $E(p)$ of medium 1 (gray) and medium 2 (brown). The transition lines from the incident state (red) define the critical slopes, equal to the modulation velocities, $v_\mathrm{m,2}^{\mathrm{up,min}}$, $v_\mathrm{m,2}^{\mathrm{low,max}}$, $v_\mathrm{m,2}^{\mathrm{up,tan}}$ and $v_\mathrm{m,2}^{\mathrm{low,tan}}$.
• Figure 2: Scattering at a space-time modulation interface. (a) Direct space schematic, with an interface between the four-potentials $A_1^\mu$ and $A_2^\mu$ moving at velocity $v_\mathrm{m}$ (top: subluminal; bottom: superluminal) and channels: i (incident), r (reflected), t (transmitted), b (later-backward), f (later-forward). (b) Dispersion diagram with transitions: horizontal = pure-space, vertical = pure-time, oblique = space-time (sub/superluminal). Vertical/horizontal offsets correspond to scalar/vector steps $q\Delta V$ and $q\Delta A$.
• Figure 2: Medium-2 hyperbola and tangent lines from the point i in Fig. \ref{['fig:s_critical_vel']}, with tangent points $P^+$ and $P^-$.
• Figure 3: Klein-region response to a subluminal space-time modulation step with $r_{ \!\!{A}\!/\!{V}}\!=\!-1$. (a) Transmission probability $T$ for an electron with $E_{\mathrm i}-qV_1\!=\!4m$, versus $q\Delta V/m$ and $v_{\mathrm m}$. Brown point: static threshold $(E_{\mathrm i}\!-\!qV_1\!+\!m)/m$; gray point: minimum static threshold $q\Delta V^{\mathrm{th}}_{\mathrm{static,min}}/m\!=\!2$; red point: dynamic threshold. (b) Normalized Klein-region threshold $q\Delta V^{\mathrm{th}}/m$ for different values of $v_{\mathrm m}$ near the speed of light, corresponding to higher electron energies, where the Klein gap is too narrow to be visible on the scale of (a). Solid curves: $v_{\mathrm m}\!=\!1\!-\!\delta_n$, $\delta_n\!=\!10^{-n}$; dashed curve: velocity-matching limit $v_{\mathrm m}\!=\!v_{\mathrm g}$; horizontal gray line: minimum static threshold $q\Delta V^{\mathrm{th}}_{\mathrm{static,min}}/m\!=\!2$.