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Symmetries and Collective Excitations in Large Superconducting Circuits

David G. Ferguson, A. A. Houck, and Jens Koch
Phys. Rev. X 3, 011003 – Published 17 January 2013
Physics logo See Synopsis: Setting Ground Rules for Quantum 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.

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  • Received 4 October 2012

DOI:https://doi.org/10.1103/PhysRevX.3.011003

This article is available under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Synopsis

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Setting Ground Rules for Quantum Circuits

Published 17 January 2013

Theoretical calculations describe the behavior of a large quantum circuit by taking advantage of certain symmetrical relationships between superconducting elements.

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Authors & Affiliations

David G. Ferguson1, A. A. Houck2, and Jens Koch1

  • 1Department of Physics and Astronomy, Northwestern University, Evanston, Illinois 60208, USA
  • 2Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA

Popular Summary

The huge success of modern electronics is an indisputable testimony to the power of the concept of electronic circuits. At the heart of the concept are modularity based on elementary building blocks and limitless possibilities to combine those blocks into large networks for new functionality. A thorough scientific understanding of how the elements work individually and in combination together with the ease of fabrication has made the concept a technologically powerful one. Can the concept of superconducting circuits, in which electric voltages and currents are governed by the laws of quantum physics, similarly find its power in quantum-information processing? Quantum behavior in such circuits is fragile, and experiments are only now starting to probe larger circuit networks. While the energy levels of such large quantum circuits can be sensitively probed by experiments, their theoretical prediction poses a significant challenge. In our work, we address this challenge by leveraging the symmetry properties of certain superconducting circuits and achieving dramatic simplifications in the computation of energy levels.

The basic building block of a superconducting circuit is a Josephson junction—a superconducting wire interrupted by an insulating link. A circuit containing an array of more than 40 Josephson junctions, the so-called fluxonium device, is already experimentally realized. In pace with this experimental development and in anticipation of further experimental advances to come, we have developed a theory that enables, for the first time, quantitative modeling of energy spectra of large circuits. What gives the theory its capability is our realization and employment of the power of the symmetry properties of the circuits. Symmetries such as the approximate invariance of the energy spectra under permutations of individual junction variables divide the energy spectrum of a fluxonium device into subspectra corresponding to nearly degenerate states. This conceptual simplification also allows us to obtain new, otherwise-difficult-to-obtain predictions of the energies of the collective excitations in the circuit under perturbation.

Experiments have shown that coherence times of fluxonium devices are on par with those of much simpler circuits. Our findings should encourage further research that explores quantum coherence in superconducting circuit networks of increasing complexity, and the approach of harnessing the symmetries of circuit networks will be an important ingredient for quantitative theory in the future.

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Vol. 3, Iss. 1 — January - March 2013

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