Characterizing the Capacitance of Josephson Junctions for Topologically Protected Qubits

Recent measurements and modeling of quantum stabilizers implemented with superconducting hardware have shown that the concatenation of π-periodic Josephson elements can lead to protection against dephasing from flux noise that is exponential in the number of such elements. However, the Josephson junctions that are used in these elements require much higher Josephson and charging energies than traditional superconducting qubits. In this regime, the simultaneously large Josephson and charging energies result in a large junction plasma frequency, which can approach the superconducting gap, particularly for the Al electrodes that are commonly used in qubits. This results in an extra electronic capacitance contribution from quasiparticles on top of the conventional geometric capacitance. This excess capacitance causes a reduction in the junction charging energy below the level required for the π-periodic Josephson elements. We use measurements of the self-resonance steps on the IV curves of dc SQUIDs to characterize this electronic capacitance contribution as a function of critical current density and the superconducting gap. Junctions formed from electrodes with a larger superconducting gap should mitigate this effect for reaching the required regime for topological protection.