Efficient excitation of a two level atom by a single photon in a propagating mode

State mapping between atoms and photons, and photon-photon interactions play an important role in scalable quantum information processing. We consider the interaction of a two-level atom with a quantized \textit{propagating} pulse in free space and study the probability \(P_e(t)\) of finding the atom in the excited state at any time \(t\). This probability is expected to depend on (i) the quantum state of the pulse field and (ii) the overlap between the pulse and the dipole pattern of the atomic spontaneous emission. We show that the second effect is captured by a single parameter \(\Lambda\in[0,8\pi/3]\), obtained by weighting the dipole pattern with the numerical aperture. Then \(P_e(t)\) can be obtained by solving time-dependent Heisenberg-Langevin equations. We provide detailed solutions for both single photon Fock state and coherent states and for various temporal shapes of the pulses.