Optimization of spatially localized electric fields for electron-positron pair creation

Using numerical solutions to the quantum field theoretical Dirac equation, we study the electron-positron pair-creation process from the vacuum due to a spatially localized supercritical electric field. By varying the spatial profile of this external field, we search for optimal field configurations that maximize the pair-creation rate in the steady state. We find that for the class of pulse shapes with a single maximum and fixed total energy, the rate depends nonmonotonically on the field's spatial width and it is insensitive to other characteristics of the pulse shape. It turns out that the Schwinger rate can be corrected such that it can provide analytical estimates for the threshold behavior as well as finite pulse effects.