Quantum acoustics with tunable nonlinearity in the superstrong coupling regime

Precise control of mechanical modes in the quantum regime is a key resource for quantum technologies, offering promising pathways for quantum sensing with macroscopic systems and scalable architectures for quantum simulation. In this work, we realise a multimode mechanical cavity coupled to a superconducting Kerr resonator, which induces nonlinearity in the mechanical modes. The Kerr mode is realised by a flux-tunable SQUID array resonator, while the mechanical modes are implemented by a surface acoustic wave (SAW) cavity. Both mechanical and electromagnetic modes are individually addressable via dedicated measurement lines, enabling full spectroscopic characterisation. We introduce a straightforward protocol to measure the SQUID array resonator's participation ratio in the hybrid acoustic modes, quantifying the degree of hybridisation. The participation ratio reveals that our device operates at the onset of the multimode coupling regime, where multiple acoustic modes simultaneously interact with the nonlinear superconducting element. Furthermore, this platform allows controllable Kerr-type nonlinearities in multiple acoustic modes, with the participation ratio serving as the key parameter determining both the dissipation rates and nonlinear strengths of these hybridised modes. Close to the resonant regime, we measure a cross-Kerr interaction between seven pairs of mechanical modes, which is controllable via the SQUID array resonator detuning. These results establish a platform for engineering nonlinear multimode mechanical interactions, offering potential for future integration with superconducting qubits and implementation of multiple mechanical qubits.