Schwinger pair production in spacetime fields: Moir\'e patterns, Aharonov-Bohm phases and Sturm-Liouville eigenvalues

We use a worldline-instanton formalism to study the momentum spectrum of Schwinger pair production in spacetime fields with multiple stationary points. We show that the interference structure changes fundamentally when going from purely time-dependent to space-time-dependent fields. For example, it was known that two time-dependent pulses give interference if they are anti-parallel, i.e. $E_z(t)-E_z(t-\Delta t)$, but here we show that two spacetime pulses will typically give interference if they instead are parallel, i.e. $E_z(t,z)+E_z(t-\Delta t,z-\Delta z)$. We take into account the fact that the momenta of the electron, $p_z$, and of the positron, $p'_z$, are independent for $E_z(t,z)$ (it would be $p_z+p'_z=0$ for $E(t)$), and find a type of fields which give moir\'e patterns in the $p_z-p'_z$ plane. Depending on the separation of two pulses, we also find an Aharonov-Bohm phase. We also study complex momentum saddle points in order to obtain the integrated probability from the spectrum. Finally, we calculate an asymptotic expansion for the eigenvalues of the Sturm-Liouville equation that corresponds to the saddle-point approximation of the worldline path integral, use that expansion to compute the product of eigenvalues, and compare with the result obtained with the Gelfand-Yaglom method.