Topics in the theory of small Josephson junctions and layered superconductors

Small Josephson junctions are fascinating quantum systems with numerous potential applications, especially in the still infant field of quantum computation. In the first part of the thesis. I have studied some of the quantum aspects of these systems. Specifically, I have investigated their interactions with a resonant cavity, the Berry phase of a superconducting nanocircuit, and some properties of small Josephson junction arrays. I found that at specific values of the gate voltage of a Cooper pair box, there will be a strong coupling between the Josephson junction and the cavity mode; in this case the junction behaves as a two level atom. I also, verified that if there are N identical junctions in the cavity and are tuned on resonance, the cavity-junction interaction is enhanced by a factor $N$. This behavior is a clear sign of a cooperative phenomena. Also I found that if a Josephson junction is on resonance with one of the modes of a two-mode cavity, there will be a coupling between the two modes and ultimately a frequency-up or down conversion. I have also studied the spin-wave-like excitations of a disordered two and three dimensional array of small Josephson junctions. I consider a simple form of diagonal disorder and solve for the spin-wave spectral functions using the coherent potential approximation. Finally, using quantum Monte Carlo techniques, I have constructed the phase diagram of a disordered two dimensional array of small Josephson junctions.; In the second part of the thesis, I have studied the dynamics of vortices in a clean layered high-temperature superconductor. I found that the c-axis conductivity at nonzero frequencies shows a strong but not divergent increase as the vortex lattice freezing temperature is approached from above, followed by an apparently discontinuous drop. The discontinuity is consistent with the occurrence of a first-order freezing of the vortex lattice. The calculated equilibrium properties agree with previous Monte Carlo studies of the system.