Pair Production in time-dependent Electric field at Finite times

We investigate the finite time behavior of pair production from the vacuum by time-dependent Sauter pulsed electric fields in spinor quantum electrodynamics (QED). Using the exact analytic solution of the mode function, we find the one-particle distribution function in momentum space. The longitudinal momentum spectrum of particles shows oscillatory behavior at a finite time in a small window of longitudinal momentum where the electric field diminishes to around one-hundredth of its maximum magnitude and its oscillation time is close to Compton time. This oscillation is asymmetric, i.e., the amplitude of oscillation is maximum for negative longitudinal momentum compared to positive rate. The change in the longitudinal momentum spectrum can occur due to the quantum interference effect, and this quantum interference effect comes from the result of dynamical tunneling. The transverse momentum spectrum shows the Gaussian structure with a peak at zero transverse momentum when \(t = 0\). After \(t \approx \tau/2\), the smooth Gaussian design becomes distorted, and we see inconstancy in spectrum structure, either a dip at the origin with an off-axis maximum or a peak at zero transverse momentum with small mountains up to \(t \approx 2{\tau}\) observed. After that, the spectrum shows a maximum height at zero transverse momentum with weakly pronounced peaks.