  We explore amplification and cross-Kerr nonlinearity by a three-level emitter
(3LE) embedded in a waveguide and driven by two light beams. The coherent
amplification and cross-Kerr nonlinearity were demonstrated in recent
experiments, respectively, with a V and a ladder-type 3LE coupled to an open
superconducting transmission line carrying two microwave fields. Here, we
consider $\Lambda$, V, and ladder-type 3LE, and compare the efficiency of
coherent and incoherent amplification as well as the magnitude of the
cross-Kerr phase shift in all three emitters. We apply the Heisenberg-Langevin
equations approach to investigate the scattering of a probe and a drive beams
both initially in a coherent state. We particularly calculate the regime of the
probe and drive powers when the 3LE acts most efficiently as a coherent
amplifier, and derive the second-order coherence of amplified probe photons.
Finally, we apply the Kramers-Kronig relations to correlate the amplitude and
phase response of the probe beam, which are used in finding the coherent
amplification and the cross-Kerr phase shift in these systems.
