  Stimulated emission can be defined as the process when an incoming photon
stimulates an additional quantum of energy from an atom into the same
electromagnetic mode as the impinging photon. Hence, the two outgoing photons
are identical. In a waveguide or free space, this intuition is typically found
through Fermi's Golden rule, however, this does not properly account for the
wave-like nature of the photons. Here, I present an exact solution to
stimulated emission from a quantum two-level atom that properly accounts for
the incoming and outgoing electromagnetic modes. This result is valid over a
huge range of incident photon numbers. For a single incident photon, it shows
how the photon must properly mode match the two-level system to cause
stimulated emission. For a Fock state drive with large photon number, the exact
solution shows how a two-level system Rabi oscillates with the traveling Fock
mode as it passes by. I additionally use the same formalism to show that
stimulated emission by a coherent pulse cannot be understood as an additional
photon being emitted into the incident coherent state because the two-level
system's state factorizes with the electromagnetic field's coherent state.
Recent advances in superconducting circuits make them an ideal platform to test
these predictions.
