Thermodynamic measurements of magnetic fluxes and I-V characteristics in SQUIDs offer promising paths to the characterization of topological superconducting phases. We consider the problem of macroscopic quantum tunneling in an rf-SQUID in a topologi
cal superconducting phase. We show that the topological order shifts the tunneling rates and quantum levels, both in the parity conserving and fluctuating cases. The latter case is argued to actually enhance the signatures in the slowly fluctuating limit, which is expected to take place in the quantum regime of the circuit. In view of recent advances, we also discuss how our results affect a $pi$-junction loop.
We study the bilayer quantum Hall system at total filling factor u_T = 1 within a bosonization formalism which allows us to approximately treat the magnetic exciton as a boson. We show that in the region where the distance between the two layers is
comparable to the magnetic length, the ground state of the system can be seen as a finite-momentum condensate of magnetic excitons provided that the excitation spectrum is gapped. We analyze the stability of such a phase within the Bogoliubov approximation firstly assuming that only one momentum Q0 is macroscopically occupied and later we consider the same situation for two modes pm Q0. We find strong evidences that a first-order quantum phase transition at small interlayer separation takes place from a zero-momentum condensate phase, which corresponds to Halperin 111 state, to a finite-momentum condensate of magnetic excitons.
