  We address the estimation of the coupling constant of the Jaynes-Cummings
Hamiltonian for a coupled qubit-oscillator system. We evaluate the quantum
Fisher Information (QFI) for the system undergone the Jaynes-Cummings
evolution, considering that the probe initial state is prepared in a Fock state
for the oscillator and in a generic pure state for the qubit; we obtain that
the QFI is exactly equal to the number of excitations present in the probe
state. We then focus on the two subsystems, namely the qubit and the oscillator
alone, deriving the two QFIs of the two reduced states, and comparing them with
the previous result. Next we focus on feasible measurements on the system, and
we find out that if population measurement on the qubit and Fock number
measurement on the oscillator are performed together, the Cramer-Rao bound is
saturated, that is the corresponding Fisher Information (FI) is always equal to
the QFI. We compare also the performances of these measurements performed
alone, that is when one of the two subsystem is ignored. We show that, when the
qubit is prepared in either the ground or the excited state, the local
measurements are still optimal.
