Exploring non-classicality and phase estimation of Schrödinger cat states generated in a Mach-Zehnder interferometer...

We explore the non-classicality and precision limits of coherent superposition mechanical states generated in a Mach-Zehnder interferometer (MZI). In this setup, a photonic NOON state interacts with a mechanical oscillator, producing such coherent superposition states that give rise to Schrödinger cat (SC) states at specific times and phase differences. The non-classicality is quantified via the negativity volume (NV) of the Wigner function, and the precision limit of the mechanical state is characterized by the quantum Fisher information (QFI). We observe that NV and QFI exhibit correlated trends at selected times and phase differences, indicating that higher non-classicality may enhance the precision limits. Notably, for certain instances, QFI can remain high even when fidelity between the coherent superposition state and the target SC state is low, indicating that conditional states can encode substantial phase information. Using parity detection, we demonstrate that the quantum Cramér-Rao bound (CRB) can be saturated, reaching the Heisenberg limit (HL) for certain evolution times and confirming it as the optimal detection strategy. These results provide insights into the quantum dynamics of optomechanical systems for high-precision measurements and enable the creation of high-fidelity SC states.