Quantum electromechanical L designs with discrete charge: Second harmonic (2 q_e / h) in the persistent current

A simple model of a purely inductive continuously deformed quantum circuit with charge discreteness is introduced. Without deformations, the persistent current is periodic in the pseudoflux variable (frequency frac(q_e, h)). Spatial dynamic deformations in the geometry are modeled as a mechanical harmonic oscillator (U (x) ∼ x²). As consequence of the electromechanical interaction there are important corrections to the electrical current crossing the inductance (persistent current). At first order, a generation of a second harmonic (ω = 2 frac(q_e, h)) appears. For non-harmonic mechanical systems (U (x) ∼ x^{2 n}, n > 1) we conjecture the generation of multiple modes.