Bloch´s oscillations in quantum circuits with charge discreteness

We consider the effect of charge discreteness in a quantum circuit with inductance L. The inductance is pierced by a time-depending external magnetic field, which creates a time-dependent magnetic flux φ_ext(t). When the external magnetic flux varies linearly with time, the induced current in the inductance oscillates with a frequency proportional to the flux variation and charge discreteness. This phenomenon is equivalent to well-known Bloch´s oscillation in crystal or periodic superlattices heterostructures. In fact, formally, the charge discreteness plays the role of a lattice constant. The same phenomenon occurs when the flux variations are replaced by a (constant) electromotive force. We expect this phenomena to be realized in micrometer sized solid-state systems.