Symmetries and Collective Excitations in Large Superconducting Circuits

Abstract

In this work, we present theoretical tools suitable for quantitative modeling of large superconducting circuits that include one-dimensional Josephson-junction arrays. The large number of low-energy degrees of freedom and the peculiar interactions between them induced by flux quantization present a considerable challenge to the detailed modeling of such circuits. For the concrete example of the fluxonium device, we show how to address this challenge. Starting from the complete degrees of freedom of the circuit, we employ the relevant collective modes and circuit symmetries to obtain a systematic approximation scheme. Important circuit symmetries include approximate invariance under the symmetric group and lead to considerable simplifications of the theory. Selection rules restrict the possible coupling among different collective modes and help explain the remarkable accuracy of previous simplified models. Using this strategy, we obtain new predictions for the energy spectrum of the fluxonium device that can be tested with current experimental technology.

Collective modes in the fluxonium qubit

“…It can be shown [24] that L S is symmetric under the action of the symmetric group S N . Due to the large ratio E a J /E a C , fluctuations of ξ μ are small; for fluctuations in φ small compared to πN, we can then expand the last term in U S to quadratic order to find (up to a constant term)…”

“…Nonetheless, in the ideal case a permutation symmetry is still present; the effect of this symmetry and its breaking on the fluxonium spectrum has been studied in Ref. [24]. Aspects of the quasiparticle-induced decoherence in the fluxonium have been studied theoretically in Refs.…”

“…The Lagrangian description of this circuit without the coupling capacitors is discussed in detail in Ref. [24]. In Appendix A, we briefly summarize (for the paper to be self-contained) and generalize (to include the coupling capacitors) the relevant parts of that work.…”

“…We note that the symmetry under permutations ensures [24] that the anharmonic terms that we neglect cannot couple any state given by the direct product between a qubit eigenstate and a collective modes singly excited state to a state which is the direct product between any qubit state and the collective modes ground state. One of the main objectives of this work is to understand the interactions induced by the presence of ground and coupling capacitors; we will show, for example, that the coupling between the states just discussed is in general present, albeit weak.…”

“…While there are N + 1 junctions in the circuit, due to charge conservation and flux quantization there are only N -independent degrees of freedom [24]: the qubit mode φ and the collective modes ξ μ , μ = 1, . .…”