Positron wave interference and Klein tunnel during the production of pairs in the double-well potential

Abstract

In this paper, the electron-positron creation process in a double well scheme is investigated. A series of simulations is conducted by solving the quantized Dirac equation numerically. Here the split operator scheme is used to solve the Dirac equation, and the Fourier analysis is adopted to study the evolution of the wave function. The evolution starts from the state that all the negative energy eigenstates are occupied. By projecting the time dependent wave function to the positive energy eigenstates, the distributions of electrons and positrons in coordinate space and momentum space would be calculated. The total number of the electrons and positrons can be obtained by integrating the momentum distributions, and the number of the positrons in different parts of coordinate space can be achieved by integrating the space distributions. At first the electron-positron is created at the double-well edge, and positrons are emitted from the edges of double-well potential and propagate out while the electrons are bounded by the barriers. It is found that when the positron waves from different double-well edges encounter in the double-well for the first time, there occurs no positron wave interference phenomenon. The wave interference emerges after the positron no indent wave is reflected by the barriers. At the same time, because of Klein tunneling the number of positrons outside the double well begin to surpass the positrons inside the double well. After a piece of time, the amplitude of interference wave would reach its peak, and then collapses since Klein tunneling. If the double-well potential meets the standing-wave conditions, a stationary wave would be found before the interference wave reaches its peak if the distance between the double wells is short, and a stationary wave would be found after the interference wave has reached its peak if the distance between the double wells is long. And the stationary wave occurs when the positron wave is reflected by the barriers for the second time. The occurring of the stationary wave would affect the pairs producing process by making the number of pairs fluctuate. Because of Klein tunneling, the wave packages close to the double-well would disappear first, and the others can last for a longer time when the standing-wave condition is fulfilled, but all of the stationary wave packages disappear in the double well finally. And there is barely no positrons left inside the double well to the end since Klein tunneling.