Prediction of half harmonic generation in stacked Josephson junctions and ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_x$ single crystals

We demonstrate analytically that parametric excitation of certain plasma resonance modes in ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_x$ single crystals is possible. The model we use is that of a Josephson stack, and the fundamental mechanism is that of half harmonic generation in time and space when a threshold of the applied rf signal is exceeded. The phenomenon is important as a diagnostic tool for the investigation of plasma resonance in ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_x$-like materials, as well as being a basis for making high-$T_c$ high-frequency parametric amplifiers. It has some unique features of space and time nonlinear behavior.