  Cavity-magnon systems are emerging as a fruitful architecture for the
integration of quantum technologies and spintronic technologies, where magnons
are coupled to microwave photons via the magnetic-dipole interaction.
Controllable the photon-magnon (P-M) couplings provide a powerful means of
accessing and manipulating quantum states in such hybrid systems. Thus
determining the relevant P-M couplings is a fundamental task. Here we address
the quantum estimation problem for the P-M coupling strength in a
double-cavity-magnon system with drive and dissipation. The effects of various
physical factors on the estimation precision are investigated and the
underlying physical mechanisms are discussed in detail. Considering that in
practical experiments it is almost infeasible to perform measurements on the
global quantum state of this composite system, we identify the optimal
subsystem for performing measurements and estimations. Further, we evaluate the
performance of different Gaussian measurements, indicating that optimal
Gaussian measurement almost saturates the ultimate theoretical bound on the
estimation precision given by the quantum Fisher information.
