Synchronization of Bloch oscillations by gate voltage modulation

We propose to synchronize Bloch oscillations in a double phase-slip junction by modulating the gate voltage rather than the bias voltage. We show this is advantageous and the relatively small a.c. modulation of the gate voltage gives rise to the pronounced plateaux of quantized current of the width of the order of Coulomb blockade threshold. We theoretically investigate the setup distinguishing three regimes of the strong, weak, and intermediate coupling defined by the ratio of the gate capacitance $C$ and the effective capacitance of the phase-slip junctions. An important feature of the intermediate coupling regime is the occurrence of the fractional plateaux of the quantized current. We investigate the finite temperature effects finding an empirical scaling for the smoothing of integer plateaux.