London superconductor and time-varying mesoscopic LC circuits

Abstract

In this work we study the classical and quantum dynamics of a London superconductor and of a time-dependent mesoscopic or nanoscale LC circuit by assuming that the inductance and capacitance vary exponentially with time at constant rate. Surprisingly, we find that the behavior of these two systems are equivalent, both classically and quantum mechanically, and can be mapped into a standard damped harmonic oscillator which is described by the Caldirola-Kanai Hamiltonian. With the aid of the dynamical invariant method and Fock states, we solve the time-dependent Schr\"odinger equation associated with this Hamiltonian and calculate some important physical properties of these systems such as expectation values of the charge and magnetic flux, their variances and the respective uncertainty principle.