Observation of topological transitions in interacting quantum circuits


Topology, with its abstract mathematical constructs, often manifests itself in physics and has a pivotal role in our understanding of natural phenomena. Notably, the discovery of topological phases in condensed-matter systems has changed the modern conception of phases of matter1,2,3,4,5. The global nature of topological ordering, however, makes direct experimental probing an outstanding challenge. Present experimental tools are mainly indirect and, as a result, are inadequate for studying the topology of physical systems at a fundamental level. Here we employ the exquisite control afforded by state-of-the-art superconducting quantum circuits to investigate topological properties of various quantum systems. The essence of our approach is to infer geometric curvature by measuring the deflection of quantum trajectories in the curved space of the Hamiltonian6. Topological properties are then revealed by integrating the curvature over closed surfaces, a quantum analogue of the Gauss–Bonnet theorem. We benchmark our technique by investigating basic topological concepts of the historically important Haldane model7 after mapping the momentum space of this condensed-matter model to the parameter space of a single-qubit Hamiltonian. In addition to constructing the topological phase diagram, we are able to visualize the microscopic spin texture of the associated states and their evolution across a topological phase transition. Going beyond non-interacting systems, we demonstrate the power of our method by studying topology in an interacting quantum system. This required a new qubit architecture8,9 that allows for simultaneous control over every term in a two-qubit Hamiltonian. By exploring the parameter space of this Hamiltonian, we discover the emergence of an interaction-induced topological phase. Our work establishes a powerful, generalizable experimental platform to study topological phenomena in quantum systems.

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Figure 1: Dynamical measurement of Berry curvature and .
Figure 2: Dynamical measurement of .
Figure 3: Dynamic measurement of the topological phase diagram and adiabatic visualization of phases.
Figure 4: Topological phase diagram of an interacting system.


  1. 1

    Klitzing, K. v., Dorda, G. & Pepper, M. New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance. Phys. Rev. Lett. 45, 494–497 (1980)

    ADS  Article  Google Scholar 

  2. 2

    Tsui, D. C., Stormer, H. L. & Gossard, A. C. Two-dimensional magnetotransport in the extreme quantum limit. Phys. Rev. Lett. 48, 1559–1562 (1982)

    CAS  ADS  Article  Google Scholar 

  3. 3

    Bernevig, B. A., Hughes, T. L. & Zhang, S.-C. Quantum spin Hall effect and topological phase transition in HgTe quantum wells. Science 314, 1757–1761 (2006)

    CAS  ADS  Article  Google Scholar 

  4. 4

    Hasan, M. Z. & Kane, C. L. Colloquium: topological insulators. Rev. Mod. Phys. 82, 3045–3067 (2010)

    CAS  ADS  Article  Google Scholar 

  5. 5

    Moore, J. E. The birth of topological insulators. Nature 464, 194–198 (2010)

    CAS  ADS  Article  Google Scholar 

  6. 6

    Gritsev, V. & Polkovnikov, A. Dynamical quantum Hall effect in the parameter space. Proc. Natl Acad. Sci. USA 109, 6457–6462 (2012)

    CAS  ADS  Article  Google Scholar 

  7. 7

    Haldane, F. D. M. Model for a quantum Hall effect without Landau levels: condensed-matter realization of the “parity anomaly”. Phys. Rev. Lett. 61, 2015–2018 (1988)

    CAS  ADS  MathSciNet  Article  Google Scholar 

  8. 8

    Geller, M. et al. Tunable coupler for superconducting Xmon qubits: perturbative nonlinear model. Preprint at http://arXiv.org/abs/1405.1915 (2014)

  9. 9

    Chen, Y. et al. Qubit architecture with high coherence and fast tunable coupling. Preprint at http://arXiv.org/abs/1402.7367 (2014)

  10. 10

    Thouless, D. J., Kohmoto, M., Nightingale, M. P. & den Nijs, M. Quantized Hall conductance in a two-dimensional periodic potential. Phys. Rev. Lett. 49, 405–408 (1982)

    CAS  ADS  Article  Google Scholar 

  11. 11

    Wen, X.-G. Quantum Field Theory of Many-body Systems (Oxford Univ. Press, 2004)

  12. 12

    Bernevig, B. A. & Hughes, T. L. Topological Insulators and Topological Superconductors (Princeton Univ. Press, 2013)

    Google Scholar 

  13. 13

    Berry, M. V. Quantal phase factors accompanying adiabatic changes. Proc. R. Soc. Lond. A 392, 45–57 (1984)

    ADS  MathSciNet  Article  Google Scholar 

  14. 14

    Wilczek, F. & Shapere, A. Geometric Phases in Physics (World Scientific, 1989)

    Google Scholar 

  15. 15

    Neeley, M. et al. Emulation of a quantum spin with a superconducting phase qudit. Science 325, 722–725 (2009)

    CAS  ADS  Article  Google Scholar 

  16. 16

    Leek, P. J. et al. Observation of Berry’s phase in a solid-state qubit. Science 318, 1889–1892 (2007)

    CAS  ADS  MathSciNet  Article  Google Scholar 

  17. 17

    Atala, M. et al. Direct measurement of the Zak phase in topological Bloch bands. Nature Phys. 9, 795–800 (2013)

    CAS  ADS  Article  Google Scholar 

  18. 18

    Mariantoni, M. et al. Implementing the quantum von Neumann architecture with superconducting circuits. Science 334, 61–65 (2011)

    CAS  ADS  Article  Google Scholar 

  19. 19

    Houck, A. A., Türeci, H. E. & Koch, J. On-chip quantum simulation with superconducting circuits. Nature Phys. 8, 292–299 (2012)

    CAS  ADS  Article  Google Scholar 

  20. 20

    Buluta, I. & Nori, F. Quantum simulators. Science 326, 108–111 (2009)

    CAS  ADS  Article  Google Scholar 

  21. 21

    Barends, R. et al. Coherent Josephson qubit suitable for scalable quantum integrated circuits. Phys. Rev. Lett. 111, 080502 (2013)

    CAS  ADS  Article  Google Scholar 

  22. 22

    Schroer, M. D. et al. Measuring a topological transition in an artificial spin-1/2 system. Phys. Rev. Lett. 113, 050402 (2014)

    CAS  ADS  Article  Google Scholar 

  23. 23

    Xu, C., Poudel, A. & Vavilov, M. G. Nonadiabatic dynamics of a slowly driven dissipative two-level system. Phys. Rev. A 89, 052102 (2014)

    ADS  Article  Google Scholar 

  24. 24

    Feynman, R. P. Simulating physics with computers. Int. J. Theor. Phys. 21, 467–488 (1982)

    MathSciNet  Article  Google Scholar 

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We acknowledge discussions with J. Moore, C. Nayak, M. Niu, A. Rahmani, T. Souza, M. Vavilov, D. Weld and A. Yazdani. This work was supported by the NSF (grants DMR-0907039 and DMR-1029764), the AFOSR (FA9550-10-1-0110), and the ODNI, IARPA, through ARO grant W911NF-10-1-0334. Devices were made at the UCSB Nanofab Facility, part of the NSF-funded NNIN, and the NanoStructures Cleanroom Facility.

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P.R., C.N. and Y.C. designed and fabricated the sample, performed the experiment, analysed the data, and with M.K. and C.Q. co-wrote the manuscript and Supplementary Information. M.K. and A.P. provided theoretical assistance. All members of the UCSB team contributed to the experimental set-up. All authors contributed to the manuscript preparation.

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Correspondence to J. M. Martinis.

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The authors declare no competing financial interests.

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Roushan, P., Neill, C., Chen, Y. et al. Observation of topological transitions in interacting quantum circuits. Nature 515, 241–244 (2014). https://doi.org/10.1038/nature13891

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