Within an oscillating field with high frequency, electron-positron pairs can be generated from vacuum as the result of multi-photon transition process. In this paper, through the computational quantum field theory and the split operator technique, we use a numerical method to solve the spatiotemporally dependent Dirac equation, the result of which enables us to discuss the process of creating electron-positron pair under a time-dependent and spatially localized external field. By monitoring the total number and the energy distribution of created pairs, the effect of the field width on the creating electron-positron pair is discussed.

For a wide width, the symmetric transition of single photon transition is dominant, because the momentum of the transition particle is approximately conserved due to a gradually varying space. For an oscillating field with frequency that exceeds the threshold 2mc^2, the energy of a single-photon is sufficient to cross the energy gap between the positive energy continuum and the negative energy continuum. As a result, the electron-positron pairs will be generated continuously, where a transition with symmetric energy has the maximum probability. Meanwhile, higher-order photon transition also arises, especially for three-photon transition with one photon transition completely inside the negative energy continuum. To observe the effect of this photon, we artificially cut the negative energy at a specific value. Accordingly, in the energy distribution of the created pairs, the peak corresponding to three-photon transition disappears, which indicates that the photon inside the negative energy continuum is indispensable in a three-photon transition process. For a narrow field width where the conservation of the momentum breaks down, the production corresponding to the asymmetric transition becomes obvious. In the energy distribution, the peaks representing two-photon transition and three-photon transition become wide and are split into two small peaks. For the three-photon transition, if we cut the negative energy at a specific value, it affects only the peak with lower energy, which indicates a different transition mode of the case corresponding to a wide field. Furthermore, in a narrow field the transition probability of double-photon transition greatly increases, even to a similar order of magnitude of the single photon transition. Apart from transitions with energy equal to integer multiple of the frequency of the photon appearing with asymmetric patterns, there also exists transitions with other energy. The multi-photon transition process of the particles for a narrow field width is more complicated than for a wide field width.