Numerical computation of one-photon mazer resonances for arbitrary field modes

We present a novel approach for solving numerically one-dimensional scattering problems and apply it for computing the emission probability of an ultracold atom interacting with an arbitrary field mode of a high-$Q$ cavity. Our method is efficient, stable and succeeds when other numerical integration methods fail. It also replaces and improves advantageously the WKB approximation. The cases of sinusoidal, $\mathrm{sech}^2$ and Gaussian field modes are studied and compared. Divergences with previous works, where WKB was used, are found.