Semiclassical analysis of a voltage-biased series of two Josephson junctions: attractors and chaos.

The purpose of this thesis is the characterization of dynamics of the superconducting phase emerging from a semiclassical description of a superconducting circuit composed by two Josephson junctions in series, in the presence of an external voltage source. The topology of the circuit identifies two superconducting leads and a central superconducting island. Starting from a semiclassical Lagrangian derived by a lumped element description of the superconducting circuit, the Hamiltonian is then derived and, consequently the Hamilton's equations. Further manipulations of the latter leads to a non-linear second order differential equation controlling the evolution of the island superconducting phase. The Resistive and Capacitatively Shunted Junction (RSCJ) model has been used to take account of the dissipation in the system. Very interestingly, this equation can be easily mapped into the dynamical equation of a vertically and periodically driven pendulum on a horizontal plane, i.e. a zero-gravity Kapitza pendulum: this draws a correspondence between the superconducting circuit and the Kapitza pendulum mechanical problem. Numerical simulations have been performed in order to characterize the dynamics of the superconducting phase (related to the island super-current) in terms of the dimensionless driving amplitude (related to the external voltage). The dynamical parameters space is thus composed by the initial superconducting phase value and by the dimensionless driving amplitude. The characteristic attractors have been detected and their distribution in the parameters space has been found. The latter has a clear fractal structure, showing sensitivity to initial conditions emblematic of chaos. Moreover, the limit-cycles solutions present a diversification in terms of the nods' number, showing a very intricate internal chaotic structure in the parameters space: the Hausdorff's dimensions of all the attractors' regions have been computed, revealing that their fractal dimension is non-integer. In the end, the regime of validity of such a semiclassical regime has been discussed, comparing the superconducting circuit with the superconducting Single Electron Transistors (SSET), two devices having the same lumped element description. What has been found is that, in order to observe the semiclassical phase dynamics the regime has to be opposite to the one in which SSETs operate.