Pair production in inhomogeneous electric fields with phase modulation

Electron-positron pair production in spatial inhomogeneous electric fields with sinusoidal phase modulation is studied within the Dirac-Heisenberg-Wigner formalism.The focus is on discussing the effects of the modulation parameters on the momentum spectrum and the reduced particle number at various spatial scales.For the momentum spectrum,the interference effect becomes more and more remarkable with the increase of modulated amplitude or frequency,while the symmetry is severely destroyed with modulated amplitude.For the reduced particle number,it is greatly enhanced by about a few times and evenly one order of magnitude when modulation parameters are applied.Moreover,the effect of spatial scales on the reduced particle number is carefully examined,and it is found that it increases rapidly at small spatial scales,while it tends to be a constant at large spatial scales.We also obtain the optimal pair production that can be achieved through different modulations.These results can provide a possibility for realizing the optimal pair production by combining the advantages of field spatial inhomogeneity with different choices of phase modulation.