Justification of the canonical quantization of the Josephson effect and its modifications due to high capacitance energy

Quantum devices based on the Josephson effect in superconductors are usually described by a Hamiltonian obtained by commonly used canonical quantization. However, this recipe has not yet been rigorously and systematically justified. We show that this approach is indeed correct in a certain range of parameters. We find the condition of validity of such quantization and the systematic corrections to the Josephson energy E-J: namely, the capacitance energy E-C = e(2)/2C must be much smaller than the superconducting gap Delta. Moreover, we find an experimentally testable modification of Josephson energy at large capacitance energy also with nonlinear capacitance.