We report on the study of the phase dynamics of high critical temperature superconductor Josephson junctions. We realized YBa$_2$Cu$_3$O$_{7-x}$ (YBCO) grain boundary (GB) biepitaxial junctions in the submicron scale, using low loss substrates, and a
nalyzed their dissipation by comparing the transport measurements with Monte Carlo simulations. The behavior of the junctions can be fitted using a model based on two quality factors, which results in a frequency dependent damping. Moreover, our devices can be designed to have Josephson energy of the order of the Coulomb energy. In this unusual energy range, phase delocalization strongly influences the devices dynamics, promoting the transition to a quantum phase diffusion regime. We study the signatures of such a transition by combining the outcomes of Monte Carlo simulations with the analysis of the devices parameters, the critical current and the temperature behavior of the low voltage resistance $R_0$.
We describe the transport properties of mesoscopic devices based on the two dimensional electron gas (2DEG) present at the LaAlO$_3$/SrTiO$_3$ interface. Bridges with lateral dimensions down to 500~nm were realized using electron beam lithography. Th
eir detailed characterization shows that processing and confinement do not alter the transport parameters of the 2DEG. The devices exhibit superconducting behavior tunable by electric field effect. In the normal state, we measured universal conductance fluctuations, signature of phase-coherent transport in small structures. The achievement of reliable lateral confinement of the 2DEG opens the way to the realization of quantum electronic devices at the LaAlO$_3$/SrTiO$_3$ interface.
