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

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