Phase control of multi-photon electron-positron pair creation from vacuum

We investigate the creation of electron-positron pairs by two spatiotemporally inhomogeneous electric fields with a relative phase, employing computational quantum field theory. We find that, when the two fields are closely spaced, the pair yield exhibits a cosine-like dependence on the relative phase. This suggests that the relative phase provides an effective way to enhance multi-photon transition channels. Furthermore, our analysis reveals that the response of pair-creation channels to the relative phase changes substantially with the photon order of the transition. For one-photon transitions, the rate exhibits a $2\pi$ periodicity, whereas a reduced periodicity of $\pi$ is observed for two-photon transitions. To clarify the underlying mechanism, we map the quantum field-theoretical framework onto a time-dependent perturbation approach. By extending this approach to $n$-photon processes, we show that the transition probability is periodic in the relative phase with period $2\pi/n$. This observation suggests that the relative phase offers an effective means of identifying the order of multi-photon transitions.