In the last few years, several groups have proposed and developed their own platforms demonstrating quantum-limited linear parametric amplification, with evident applications in quantum information and computation, electrical and optical metrology, radio astronomy, and basic physics concerning axion detection. Here, we propose a short review on the physics behind parametric amplification via metamaterials composed by coplanar waveguides embedding several Josephson junctions. We present and compare different schemes that exploit the nonlinearity of the Josephson current-phase relation to mix the so-called signal, idler, and pump tones. The chapter then presents and compares three different theoretical models, developed in the last few years, to predict the dynamics of these nonlinear systems in the particular case of a 4-wave mixing process and under the degenerate undepleted pump assumption. We will demonstrate that, under the same assumption, all the results are comparable in terms of amplification of the output fields.
Part of the book: Advances in Condensed-Matter and Materials Physics