Amplification and cross-Kerr nonlinearity in waveguide quantum electrodynamics

TL;DR

This paper investigates the efficiency of amplification and cross-Kerr nonlinearity in various three-level emitter configurations embedded in waveguides, using theoretical models to compare their performance and optimize conditions for quantum applications.

Contribution

It provides a comparative analysis of different three-level emitter types and their effectiveness in coherent amplification and cross-Kerr phase shifts using the Heisenberg-Langevin approach.

Findings

Lambda, V, and ladder-type emitters show different efficiencies in amplification.

Optimal power regimes for maximum amplification are identified.

Cross-Kerr phase shifts are quantified for each emitter type.

Abstract

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.