Pair Production in Time-Dependent Electric Field at Finite Times

We examine the problem of pair production in a time-varying spatially uniform electric field \(E(t)=(0,0,E_0 sech^2(t/\tau ))\) at the finite time in an adiabatic basis. To understand the dynamical behavior of particle production, the one-particle momentum distribution function \(f(p_\parallel ,p_\perp , t)\) is an important quantity. The time evolution of the particle distribution function, \(f(p_\parallel ,p_\perp , t)\) in momentum space is studied for a non-perturbative regime. The longitudinal momentum spectrum of the created particle shows a complex behavior of splitting and manifests oscillation arising at the finite time where the electric field nearly vanished, and this oscillating structure can be understood as the result of quantum inference. We further investigate the impact of the transverse momentum spectrum in which created particle’s spectrum shows only the smooth structure’s splitting and the absence of quantum interference, which is an obvious interference effect that occurs only in the direction of vacuum polarization.