Atomic excitation probability for Fock and coherent-state pulses: asymptotic results

For a two-level atom in a cavity or waveguide, interacting with a single-photon pulse, the excitation probability $P_e$ can never equal one unless the pulse shape is the exact time-reverse of the spontaneous decay. For pulses with the ``wrong'' shape, we investigate (following the work of Wang et al.\footnote{Y. Wang, J. Min\'{a}\v{r}, and V. Scarani, Phys. Rev. A {\bf 86}, 023811 (2012)}) how many photons it takes to bring the excitation probability close to 1. For square pulses we find analytically that for large average photon numbers $\bar n$, $P_e-1$ scales as $1/\sqrt{\bar n}$, with a coefficient that is the same for Fock states as for coherent states. We also present analytical and numerical results for how the presence of additional losses affects $P_e$ and makes it necessary to increase the number of photons, even for the optimal-shape pulse.