Conditional Phase Sensitivity and Atom–RestCorrelations in a Diamond Four-LevelOptomechanical Cavity

Abstract We investigate how phase information initially stored in a diamond four-level atom is redistributed when the atom interacts with a quantized cavity field and a mechanical mode. The phase $\phi$ is encoded in the atomic superposition $(|1\rangle+e^{i\phi}|2\rangle)/\sqrt{2}$, while the optical and mechanical subsystems start in Fock states. The dynamics are generated by a coarse-grained optomechanical Hamiltonian together with normalized non-Hermitian no-jump evolution. A central point of the analysis is that the induced $|2\rangle\leftrightarrow|3\rangle$ channel contains the distinct photon operators $a^{\dagger k}a^k$ and $a^ka^{\dagger k}$; they cannot be merged without changing the dynamics. The numerical study is deliberately restricted to the stationary, single-photon case $k=1$, for which the calculation is reproducible and the Hilbert-space truncation can be checked directly. We track the reduced-atomic quantum Fisher information (QFI), von Neumann entropy, purity, and total $l_1$ coherence. In the parameter range considered, increasing the optomechanical coupling $G$ reduces the maximum conditioned atom--rest entropy from about $1.676$ to $0.374$ bits and raises the minimum atomic purity from about $0.356$ to $0.876$. At the same time, the time-averaged local QFI decreases from about $0.450$ to $0.336$, while the mean total coherence changes only weakly and non-monotonically. Thus stronger optomechanical coupling does not improve all quantum resources simultaneously: it keeps the conditioned atomic state purer and less entangled with the rest, but it also reduces the amount of phase information locally accessible from the atom. To expose this contrast more clearly, the time-dependent observables are plotted for two complementary combinations of $G$ values, and the full nine-point coupling scan is summarized separately.