Signatures and characterization of dominating Kerr nonlinearity between two driven systems with application to a suspended magnetic beam

We consider a model of two harmonically driven damped harmonic oscillators that are coupled linearly and with a cross-Kerr coupling. We show how to distinguish this combination of coupling types from the case where a coupling of optomechanical type is present. This can be useful for the characterization of various nonlinear systems, such as mechanical oscillators, qubits, and hybrid systems. We then consider a hybrid system with linear and cross-Kerr interactions and a relatively high damping in one of the modes. We derive a quantum Hamiltonian of a doubly clamped magnetic beam, showing that the cross-Kerr coupling is prominent there. We discuss, in the classical limit, measurements of its linear response as well as the specific higher-harmonic responses. These frequency-domain measurements can allow estimating the magnitude of the cross-Kerr coupling or its magnon population.