Superluminal Tunneling and the Sauter-Schwinger Effect

Previous 1+1-dimensional Dirac wavepacket calculations showed that the tunneling component of a relativistic electron wavepacket can generate an arrival-time distribution whose peak occurs earlier than the corresponding free-photon peak. However, adapting superluminal tunneling to signaling leads to subluminal signaling due to the low tunneling probability. In the present work we note that the barriers used in those calculations are supercritical with respect to the Sauter-Schwinger effect. Consequently, the single-electron evolution must be accompanied by spontaneous electron-positron production from the vacuum. We derive compact formulas for the electron and positron densities when one additional electron is present, showing that the evolved wavepacket contribution adds to the vacuum-produced electron density, while Pauli blocking reduces the positron density by the negative-energy component of the propagated electron. We then apply these formulas to a fourth-order super-Gaussian barrier which produces superluminal tunneling of an electron. The resulting densities are shown explicitly at several times, and are compared with a semiclassical resonance model for the pair number. The semiclassical description reproduces the numerical growth of the pair yield and clarifies the role of Klein-zone resonance energies and widths. Finally, we outline the extension from 1+1 to 1+3 dimensions by integrating over transverse momenta, using scaling properties of the 1+1-dimensional pair number.