The Schwinger Effect in Time Dependent Electric Fields

Date of Completion

January 2012


Physics, Electricity and Magnetism




We study the Schwinger effect which is the non-perturbative production of e e+ pairs from the vacuum by an external electric field. Basic quantitative analysis consists of extracting pair production probabilities which could be done exactly for only few cases. Thus, as far as realistic electric fields are concerned, application of numerical methods becomes essential for the analysis. We present two well known numerical methods which are developed for the electric fields varying only in one dimension and we give the treatment for the electric fields varying in time. The first methods employs a Riccati type equation and relates the reflection probability of the associated scattering problem to the produced particle density. The second approach makes use of a time-dependent number operator whose evolution given by the quantum Vlasov equation, and its asymptotic value gives the density of the produced pairs. We show that these two methods are equivalent and the evolution equations can be reduced to a second order nonlinear differential equation which is free of phase integrals. Pair production probabilities are expressed for fixed values of longitudinal momentum which is conserved during the evolution, and in this way momentum spectrum for the created e e + pairs are computed. In the following, we present a detailed semiclassical analysis both within the framework of Jeffreys-Wentzel-Kramers-Brillouin (JWKB) approximation and worldline instanton formalism. In the former method analytical continuation rules for the approximate JWKB solutions are introduced and the reflection amplitude is obtained for a generic time dependent potential with an arbitrary number of critical points located in the complex plane. The latter method deals with classical tunneling trajectories such that evaluation of path integral on the semiclassical trajectories gives the same results with JWKB solutions. The semiclassical treatment shows how the qualitative features of the momentum spectrum such as interference effects and shift along the momentum axis can effectively be linked to the critical point distribution of the effective potential. In conjunction with the experimental studies, we compute the momentum spectra for realistic laser pulses with sub-cycle structure which might be relevant for the planned laser facilities. ^