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Acta Physica Sinica

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

Wu Guang-Zhi, Wang Qiang, Zhou Cang-Tao, Fu Li-Bin

Published Jan 1, 2017• 0 citations
Publisher

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.

DOI
10.7498/aps.66.070301

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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.
The early papers by Klein, Sauter and Hund which investigate scattering off a high step potential in the context of the Dirac equation are discussed to derive the 'paradox' first obtained by Klein. The explanation of this effect in terms of electron-positron production is re-assessed. It is shown that a potential well or barrier in the Dirac equation can become supercritical and emit positrons or electrons spontaneously if the potential is strong enough. If the well or barrier is wide enough, a seemingly constant current is emitted. This phenomenon is transient whereas the tunnelling first calculated by Klein is time-independent. It is shown that tunnelling without exponential suppression occurs when an electron is incident on a high barrier, even when the barrier is not high enough to radiate. Klein tunnelling is therefore a property of relativistic wave equations and is not necessarily connected to particle emission. The Coulomb potential is investigated and it is shown that a heavy nucleus of sufficiently large Z will bind positrons. Correspondingly, as Z increases the Coulomb barrier should become increasingly transparent to positrons. This is an example of Klein tunnelling. Phenomena akin to supercritical positron emission may be studied experimentally in superfluid 3 He.
The time evolution of quantum mechanical states in a square well of infinite depth with a Dirac δ function at its center has been examined for cases where the initial state was localized in one of the wells. Let well A denote the well in which the initial state is localized, and let P(t) denote the integrated probability density Ψ*Ψ in well A. For opaque barriers the time-dependent system is adequately described by a two-state model in which only the pair of stationary states of even and odd parity are considered whose wave functions in well A, apart from an arbitrary phase factor, are largely identical with the wave function of the initial state. For P(t) a harmonic oscillation is observed whose frequency νt is well approximated by the well-known formula for the tunneling frequency νt≂ΔE/h, where ΔE represents the energy separation among the pair of states in the model. For the present model of a symmetric double well it has been shown that for highly transparent barriers a three-state model can describe the time evolution adequately. The three stationary states involved in this model are a state of odd parity whose wave function in well A is largely the same as the wave function of the initial state and a pair of stationary states of even parity which on the energy scale are immediately above and below the first state. In this three-state model the function P(t) is a superposition of two sinusoidal functions with nearly identical amplitudes and frequencies plus a constant. As a consequence, the amplitude of P(t) changes harmonically. In the present model a δ function has been used as a barrier in order to minimize the mathematical detail involved in the time-dependent treatment. It is to be expected that the beating in P(t) can be observed also in the time evolution of a state localized in one of the wells of a symmetric double-minimum potential with a more realistic ‘‘low’’ barrier if the density of energy levels near the energies of the levels to be considered in the corresponding three-state model varies slowly with energy.
Previous 1+1-dimensional Dirac wavepacket calculations showed that the tunneling component of a relativistic electron wavepacket can generate an arrival-time distribution whose peak occurs earlier than the corresponding free-photon peak. However, adapting superluminal tunneling to signaling leads to subluminal signaling due to the low tunneling probability. In the present work we note that the barriers used in those calculations are supercritical with respect to the Sauter-Schwinger effect. Consequently, the single-electron evolution must be accompanied by spontaneous electron-positron production from the vacuum. We derive compact formulas for the electron and positron densities when one additional electron is present, showing that the evolved wavepacket contribution adds to the vacuum-produced electron density, while Pauli blocking reduces the positron density by the negative-energy component of the propagated electron. We then apply these formulas to a fourth-order super-Gaussian barrier which produces superluminal tunneling of an electron. The resulting densities are shown explicitly at several times, and are compared with a semiclassical resonance model for the pair number. The semiclassical description reproduces the numerical growth of the pair yield and clarifies the role of Klein-zone resonance energies and widths. Finally, we outline the extension from 1+1 to 1+3 dimensions by integrating over transverse momenta, using scaling properties of the 1+1-dimensional pair number.
We performed time-evolved numerical simulation of a two-dimensional electron wave packet through a double slit by solving the time-dependent Schrödinger equation using the finite difference method to simulate the electron interference phenomenon of a single electron. The propagating directions of interference peaks show that the starting point of electron interference is the center between the two slit exits. The number of interference peaks and their angles are found to be independent of potential barrier thickness. These results are compared with the results of our previous calculation of electron diffraction through a single slit.
We present the mathematical model and numerical calculation results for the tunneling of the wave function in a time-periodic double-well potential. The biquadratic potential of a double-well form is used. Based on a mathematical model of the time evolution of the wave function, a numerical algorithm has been developed and a program has been created for solving the Schrödinger equation, which describes the time evolution of the particle's wave function. As a result of numerical modeling, modulation regimes have been obtained at which tunneling took place. In a stationary double well potential, no tunneling was observed. On the contrary, in the time periodic potential, tunneling was observed, the rate of which depended on the modulation frequency. Adjusting the modulation parameters, it is possible to control the particle wave function tunneling rate.