Phonon-induced two-axis spin squeezing with decoherence reduction in a hybrid spin-optomechanical system

We propose a scheme to implement Heisenberg-limited spin squeezing in a hybrid cavity optomechanical-spin system. In our system, $N$ two-level systems are coupled via Tavis-Cummings interactions to a mechanical resonator (MR) in a standard optomechanical setup. Within the dispersive coupling regime, adiabatic elimination of the optical mode induces a squeezing effect on the MR, which, in the squeezed representation, effectively transforms the collective spin operators into a Bogoliubov form. Under large detuning conditions, the phonon mode mediates interactions among the Bogoliubov collective spins, thereby enabling a two-axis twisting squeezing protocol through appropriate parameter tuning. Both theoretical analysis and numerical simulations show that in the presence of dephasing and phonon dissipation, the maximum squeezing degree asymptotically converges to a constant as $N$ increases, which implies the metrological precision asymptotically approaches the standard quantum limit without parameter optimization. Nevertheless, in parameter optimization we extract a scaling relation of the optimal squeezing which surpasses existing schemes in the literature. Moreover, the optimization also leads to a considerable reduction of the preparation time for the optimal squeezing. Our work may provide insights into dissipation effects in spin squeezing and offer a potential route for high-precision quantum metrology in many-body systems.