Space-time-resolved quantum field approach to Klein-tunneling dynamics across a finite barrier

Letter

Space-time-resolved quantum field approach to Klein-tunneling dynamics across a finite barrier

M. Alkhateeb and A. Matzkin

Phys. Rev. A 106, L060202 – Published 19 December 2022

Abstract

We investigate Klein tunneling through finite potential barriers with space-time-resolved solutions to relativistic quantum field equations. We find that no particle actually tunnels through a finite supercritical barrier, even in the case of resonant tunneling. The transmission is instead mediated by modulations in pair production rates, at each edge of the barrier, caused by the incoming electron. We further examine the effect of the barrier's width on the numbers of produced pairs in the fermionic case (characterized by saturation) and in the bosonic case (characterized by exponential superradiance). This work paves the way to precise studies of the radiating dynamics of supercritical barriers, and could be applied to certain analogs of Klein tunneling observed in systems modeled by relativistic wave equations.

Received 14 June 2022
Revised 21 September 2022
Accepted 2 December 2022

DOI:https://doi.org/10.1103/PhysRevA.106.L060202

Physics Subject Headings (PhySH)

Light-matter interaction Quantum field theory Relativistic & quantum electrodynamic effects in atoms, molecules,& ions Relativistic wave equations

General PhysicsParticles & FieldsGravitation, Cosmology & AstrophysicsAtomic, Molecular & Optical

Authors & Affiliations

M. Alkhateeb and A. Matzkin

Laboratoire de Physique Théorique et Modélisation, CNRS Unité 8089, CY Cergy Paris Université, 95302 Cergy-Pontoise Cedex, France

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Issue

Vol. 106, Iss. 6 — December 2022

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Images

Figure 1
(a) The spatial density $ρ (x_{1}, x_{2})$ of positron couples at positions $x_{1}$ and $x_{2}$ created inside the barrier at time $2 \times 10^{- 3}$ expressed in atomic units (a.u., defined by $ℏ = m = 1, c = 137.036$ , where $m$ is the electron mass); the barrier has width $L = 0.2$ a.u., height $V_{0} = 3 m c^{2}$ , and the smoothness has been set to $ε = 0.3 / c$ . (b) Same as (a) but without the exchange interaction terms in the computations [see Eq. (6)].
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Figure 2
(a) Time dependence (in a.u.) of the total number of created electron-positron pairs $N (t)$ for a fermionic barrier with the parameters given in Fig. 1 except for the width $L$ , indicated in the legend. The rate vanishes except in the limit of a potential step (note that in order to compare the step and finite barrier cases, the curve shown for a step has been multiplied by two). (b) The total number of created boson-antiboson pairs $N_{bo} (t)$ for a massive bosonic field obeying the Klein-Gordon equation interacting with a background potential barrier with $V_{0} = 3 m_{bo} c^{2}, ε = 0.3 / c$ , and varying widths; $m_{bo}$ is the boson mass which renormalizes the atomic units, so that $L$ and $t$ are given in units of $λ = m_{bo} ℏ / c$ and $λ / c$ , respectively.
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Figure 3
Dynamics of Klein tunneling: An electron wave packet initially ( $t = 0$ ) centered at $x = - 3$ a.u. to the left of the supercritical barrier is launched with mean momentum $p_{0} = 100$ a.u. The barrier ( $V_{0} = 3 m c^{2}, ε = 0.3 / c$ ) lies in the region $- 0.0474 < x < 0.0474$ a.u., chosen so that the width $L$ obeys the resonance condition [Eq. (8)]. (a) Snapshot of particle densities at $t = 10^{- 2}$ a.u., as the electron wave packet (dotted orange line) approaches the barrier. The solid black line gives the total electron density (due to pair creation as well as the wave packet). The positron density lies outside the scale of the main plot and is shown in the inset (thick gray line); the thin blue line is the contribution of the modulations due to the interaction between the wave packet and the barrier. (b) Snapshot at $t = 3 \times 10^{- 2}$ a.u. as the wave packet reaches the barrier. (c) Zoom around the barrier region at $t = 4 \times 10^{- 2}$ a.u., as the electron wave packet is reformed at the right edge of the barrier; note there is no electron density inside the barrier, but the positron modulations (thin line in the inset) are of greater amplitude. (d) At $t = 5 \times 10^{- 2}$ a.u. most of the wave packet (dotted orange line) is “transmitted” to the right, while a smaller fraction is reflected.
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Figure 4
The transmission mechanism detailed in the text is illustrated for an electron wave packet colliding on a smooth rectangular barrier (region between dashed lines, $x$ in a.u.). The ratio $ρ_{e l} (x, t) / ρ_{B el} (x, t)$ is shown in black at different times ( $ρ_{el}$ is the total electron density and $ρ_{B el}$ the electron density in the absence of an incoming wave packet). The corresponding positron number density ratio $ρ_{pos} (x, t) / ρ_{B pos} (x, t)$ is shown in gray (online blue). At $t = 2 \times 10^{- 2}$ a.u., the front tail of the electron wave packet reaches the left edge of the filled barrier. This creates a dip in the positron density, visible at $t = 3 \times 10^{- 2}$ a.u. The dip propagates inside the barrier ( $t = 4 \times 10^{- 2}$ a.u.), and reaches the right edge, stimulating pair production. The additional created electron excitations (relative to vacuum polarization) to the right of the barrier are seen at $t = 5 \times 10^{- 2}$ a.u. to correspond to the front tail of the wave packet that appears as having been transmitted.
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