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Topological lattices realized in superconducting circuit optomechanics

Nature volume 612, pages 666–672 (2022)Cite this article

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Abstract

Cavity optomechanics enables the control of mechanical motion through the radiation-pressure interaction1, and has contributed to the quantum control of engineered mechanical systems ranging from kilogramme-scale Laser Interferometer Gravitational-wave Observatory (LIGO) mirrors to nanomechanical systems, enabling ground-state preparation2,3, entanglement4,5, squeezing of mechanical objects6, position measurements at the standard quantum limit7 and quantum transduction8. Yet nearly all previous schemes have used single- or few-mode optomechanical systems. By contrast, new dynamics and applications are expected when using optomechanical lattices9, which enable the synthesis of non-trivial band structures, and these lattices have been actively studied in the field of circuit quantum electrodynamics10. Superconducting microwave optomechanical circuits2 are a promising platform to implement such lattices, but have been compounded by strict scaling limitations. Here we overcome this challenge and demonstrate topological microwave modes in one-dimensional circuit optomechanical chains realizing the Su–Schrieffer–Heeger model11,12. Furthermore, we realize the strained graphene model13,14 in a two-dimensional optomechanical honeycomb lattice. Exploiting the embedded optomechanical interaction, we show that it is possible to directly measure the mode functions of the hybridized modes without using any local probe15,16. This enables us to reconstruct the full underlying lattice Hamiltonian and directly measure the existing residual disorder. Such optomechanical lattices, accompanied by the measurement techniques introduced, offer an avenue to explore collective17,18, quantum many-body19 and quench20 dynamics, topological properties9,21 and, more broadly, emergent nonlinear dynamics in complex optomechanical systems with a large number of degrees of freedom22,23,24.

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Fig. 1: Optomechanical lattices composed of superconducting circuit optomechanical systems.
Fig. 2: Superconducting circuit optomechanical chain realizing the 1D SSH model.
Fig. 3: Optomechanical mode-shape measurement and Hamiltonian reconstruction of a topological SSH chain.
Fig. 4: 2D superconducting circuit optomechanical honeycomb lattice realizing the strained graphene model.

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Data availability

The data used to produce the plots within this paper are available on Zenodo (https://doi.org/10.5281/zenodo.6987358). All other data used in this study are available from the corresponding author on reasonable request.

Code availability

The code used to produce the plots within this paper is available on Zenodo (https://doi.org/10.5281/zenodo.6987358).

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Acknowledgements

We thank O. Yazyev and P. Delplace for critical discussions with respect to the topological properties of strained graphene. We thank T. Sugiyama for discussions on the Hamiltonian reconstruction. This work was supported by the EU H2020 research and innovation programme under grant no. 101033361 (QuPhon), and by the European Research Council (ERC) grant no. 835329 (ExCOM-cCEO). This work was also supported by the Swiss National Science Foundation (SNSF) under grant Nos. NCCR-QSIT: 51NF40_185902 and 204927. All devices were fabricated in the Center of MicroNanoTechnology (CMi) at EPFL.

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Author notes

  1. These authors contributed equally: Amir Youssefi, Shingo Kono, Andrea Bancora

Authors and Affiliations

  1. Laboratory of Photonics and Quantum Measurement (LPQM), Swiss Federal Institute of Technology Lausanne (EPFL), Lausanne, Switzerland

    Amir Youssefi, Shingo Kono, Andrea Bancora, Mahdi Chegnizadeh, Jiahe Pan, Tatiana Vovk & Tobias J. Kippenberg

  2. Center for Quantum Science and Engineering, EPFL, Lausanne, Switzerland

    Amir Youssefi, Shingo Kono, Andrea Bancora, Mahdi Chegnizadeh, Jiahe Pan, Tatiana Vovk & Tobias J. Kippenberg

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Contributions

A.Y. conceived the experiment. A.Y. and A.B. designed and simulated devices. S.K. provided the theoretical support with the assistance of A.Y. and J.P. A.Y. and A.B. performed the numerical analysis. A.Y. developed the fabrication process with the assistance of M.C. and T.V. M.C. and A.Y. fabricated the samples. The measurement technique was implemented by A.Y., A.B. and S.K. The data were collected by A.B. and S.K., with the assistance of A.Y. The data analysis was performed by A.B., A.Y. and S.K. The manuscript was written by A.Y., S.K. and A.B. with the assistance of T.J.K. and all the other authors. T.J.K. supervised the study.

Corresponding author

Correspondence to Tobias J. Kippenberg.

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Nature thanks Vittorio Peano and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.

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Extended data figures and tables

Extended Data Fig. 1 Fabrication process.

a, b, Etching a trench in a silicon wafer (325 nm depth). c, Aluminum deposition of the bottom plate (100 nm). d, Patterning of Al. e, SiO2 sacrificial layer deposition (3 μm). f, CMP planarization. g, Landing on the substrate using IBE etching. h, Top Al layer deposition and patterning (200 nm). i, Releasing the structure using HF vapour. Owing to compressive stresses, the top plate will buckle up. j, At cryogenic temperatures, the drumhead shrinks and flattens owing to the temperature-induced tensile stress.

Extended Data Fig. 2 Charecterization of 24-site 2D hanycomb lattice.

a, Optomechanically induced transparency (OMIT) responce of the 2D device measured on the highest microwave bulk mode. Increasing the trench radius results in a slight shift of the mechanical frequencies. b, Microwave resonance frequencies of the device, design targets (orange), and measured values (blue).

Extended Data Fig. 3 Modeshapes of 24-site 2D honeycomb lattice.

The amplitude of the modeshape \(| {\psi }_{i}^{k}| \) is encoded in the area of the circles. Only for modes that share the same colour bar can the size and colour of the circles be compared. Highlighted in purple are the four edge modes.

Extended Data Fig. 4 Hamiltonian reconstruction of 24-site 2D honeycomb lattice.

The reconstructed Hamiltonian of 24 site 2D honeycomb device (left) and the designed Hamiltonian including second nearest-neighbour couplings (right). The diagonal elements represent an individual site’s resonance frequency deviation from the average bare cavity frequency.

Supplementary information

Supplementary Information

Supplementary Sections 1–8 and references.

Peer Review File

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Youssefi, A., Kono, S., Bancora, A. et al. Topological lattices realized in superconducting circuit optomechanics. Nature 612, 666–672 (2022). https://doi.org/10.1038/s41586-022-05367-9

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