  We simulate the dynamics of systems with $N$ = 1-20 qubits coupled to a
cavity in order to assess their potential for quantum metrology of a parameter
in the open systems limit. The qubits and the cavity are both allowed to have
losses and the system is studied under various coupling strength regimes. The
focus is primarily on the coupling between the qubits using the quantum Fisher
information as the measured parameter. Some results on estimating the
qubit-cavity detuning parameter are also presented. We investigate the scaling
of the uncertainty in the estimate of the qubit-cavity coupling with the number
of qubits and for different initial states of the qubits that act as the
quantum probe. As initial probe states, we consider Dicke states with varying
excitation numbers, the GHZ state, and separable X-polarized states. It is
shown that in the strong coupling regime, i.e., when the coupling between the
qubits and the cavity is greater than the decay parameters of both the qubits
and the cavity, Dicke states with a large excitation number can achieve the
Heisenberg limit, with the precision scaling improving as the excitation number
increases. A particularly intriguing finding of our study is that in the weak
coupling regime, as well as in situations where either the qubit or cavity
decay parameters exceed the coupling, the separable $X$-polarized state is the
best in terms of scaling and is even able to achieve the Heisenberg limit in
these lossy regimes for the range of $N$ considered.
