Dual approach to circuit quantization using loop charges

Abstract

The conventional approach to circuit quantization is based on node fluxes and traces the motion of node charges on the islands of the circuit. However, for some devices, the relevant physics can be best described by the motion of polarization charges over the branches of the circuit that are in general related to the node charges in a highly nonlocal way. Here, we present a method, dual to the conventional approach, for quantizing planar circuits in terms of loop charges. In this way, the polarization charges are directly obtained as the differences of the two loop charges on the neighboring loops. The loop charges trace the motion of fluxes through the circuit loops. We show that loop charges yield a simple description of the flux transport across phase-slip junctions. We outline a concrete construction of circuits based on phase-slip junctions that are electromagnetically dual to arbitrary planar Josephson junction circuits. We argue that loop charges also yield a simple description of the flux transport in conventional Josephson junctions shunted by large impedances. We show that a mixed circuit description in terms of node fluxes and loop charges yields an insight into the flux decompactification of a Josephson junction shunted by an inductor. As an application, we show that the fluxonium qubit is well approximated as a phase-slip junction for the experimentally relevant parameters. Moreover, we argue that the $0$-$π$ qubit is effectively the dual of a Majorana Josephson junction.