Ermakov–Milne equation and pair creation by time-dependent electric fields

In this work, we have studied the creation of charged particles by time-dependent electric fields using the Ermakov–Lewis invariant method. We have first shown that the complex scalar field, in the presence of a time-dependent electric field, possesses an Ermakov–Lewis like invariant which has a well-defined ground state (invariant vacuum). Then, we have established the Bogoliubov transformation between the creation and annihilation operators associated with the adiabatic vacuum and those associated with the invariant vacuum and obtained the time-dependent number of created particles. We have, also, proven that the general solution to the nonlinear Ermakov–Milne equation proposed by Pinney for the harmonic oscillator can be generalized to the case of a complex field and is always the modulus of the solution of the linear Klein–Gordon equation. As application, we have considered the constant electric field, the exponentially decreasing field and the exponentially switch-off electric field. It is shown that the rapid switch-off of the electric field defines unequivocally the notion of particles and vacuum states and it leads to a huge amplification of the number of created particles.