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.Comment: 18 pages, 6 figure