Simulation of supersymmetric quantum mechanics in a Cooper-pair box shunted by a Josephson rhombus

Supersymmetries in quantum mechanics offer a way to obtain degeneracies in the excitation spectrum which do not originate from selection rules. The mechanism behind the degeneracies is the same as the one that leads to the miraculous cancellations of divergences in supersymmetric field theories found in the high energy physics context. Even though of importance, there is up to now no realistic proposal of non-integrable systems that show level degeneracies due to a supersymmetric structure. Here, we propose an implementation of a quantum-mechanical supersymmetry in a Cooper-pair box shunted by a Josephson junction rhombus which is effectively $π$-periodic in the superconducting phase difference. For a characteristic ratio between the strength of the $2π$- and the $π$-periodic junction, we find a two-fold degeneracy of all the energy levels all the way from the weak junction/charge qubit limit to the strong junction/transmon regime. We provide explicit values for the parameters of the rhombus and show that tuning in and out of the supersymmetric point is easily achieved by varying an external gate voltage. We furthermore discuss a microwave experiment to detect the supersymmetry and conclude that it could indeed be simulated with currently existing Josephson junction technology.

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