Single-photon stimulated emission in waveguide quantum electrodynamics

We study the scattering of a single-photon pulse from quantum two-level system (qubit) coupled to a continuum spectrum of modes in a one-dimensional (1D) nanophotonic waveguide. We consider two different cases-single-excitation subspace, when photon is incident on a qubit in the ground state, and two-excitation subspace, when photon is incident on excited qubit. For both cases, we find the spatial distribution of photon fields inside a waveguide and derive a time-dependent analytical solution for qubit and photon amplitudes, which can be used to obtain probabilities to find excited qubit and forward or backward moving photons in different configurations. We find that incident wave has much higher probability to be transmitted through the excited qubit which is a signature of single-photon stimulated emission. This property together with a fact that a qubit in the ground state almost completely reflects resonant photons, which can be used to develop method to detect qubit state in an open waveguide. By analyzing two-photon amplitudes for photon incident on excited qubit, we show clear evidence of stimulated emission. Calculations are performed for two initial states of incident photon: plane wave and Gaussian pulse.