Mesoscopic RLC Circuit and its Associated Occupation Number and Berry Phase

We consider the quantization of the time-dependent harmonic oscillator and its associated
Berry phase using the invariant operator method, as well as the occupation number of the induced
quasi-particle production. Furthermore, we point out that in the literature there exist different methods
for determining the solution to the Milne-Pinney equation, which leads to different results. By
measuring the time-dependent occupation number and associated Berry phase, one can, in principle,
determine which of these methods leads to physically realized results. As a concrete example, we
consider the mesoscopic RLC circuit and derive the occupation number and associated Berry phase
for each of these different methods. We find that, the solution to the Ermakov equations leads to a
time-dependent occupation number and associated Berry phase, while the particular solution to the
Milne-Pinney equation does not.