Topological lattices realized in superconducting circuit optomechanics.

Cavity optomechanics enables the control of mechanical motion through the radiation-pressure interaction 1 , 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 preparation 2,3 , entanglement 4,5 , squeezing of mechanical objects 6 , position measurements at the standard quantum limit 7 and quantum transduction 8 . Yet nearly all previous schemes have used single- or few-mode optomechanical systems. By contrast, new dynamics and applications are expected when using optomechanical lattices 9 , which enable the synthesis of non-trivial band structures, and these lattices have been actively studied in the field of circuit quantum electrodynamics 10 . Superconducting microwave optomechanical circuits 2 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 model 11,12 . Furthermore, we realize the strained graphene model 13,14 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 probe 15,16 . 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 collective 17,18 , quantum many-body 19 and quench 20 dynamics, topological properties 9,21 and, more broadly, emergent nonlinear dynamics in complex optomechanical systems with a large number of degrees of freedom 22-24 .