Majorana fermions revealing the true nature of quantum phase slip junctions

Quantum phase slips (QPS) are famously considered to be the exact dual to the Josephson effect. However, if taken at face value, this duality stipulates that charge transported across a QPS junction is continuous, as opposed to regular Josephson junctions, which perserve charge quantization. We here analyse a circuit consisting of a QPS junction shunted by a topological Josephson junction hosting Majoranas, the latter being sensitive to charge transport in units of half a Cooper pair charge. Contrary to common phenomenological approaches, we consider a microscopic derivation of the phase slip processes, using a large Josephson junction array to model the QPS junction. We find that there is a cross-over when increasing the number of junctions in the array, from a regime where the standard, dual description of phase slip junctions is valid, to a regime where charge has to be quantized in integer units of the Cooper pair charge. This cross-over isstraightforwardly visible in the eigenspectrum of the circuit as a function of an externally applied flux. Importantly, we expect that the latter (quantized) regime renders QPS junctions useless for circuit realizations of the Gottesman-Kitaev-Preskill code.