Quantum Fisher information of a ◊-type four-level atom interacting with a single-mode quantized field in an optomechanical cavity

In science and technology, precision measurement of physical quantities is crucial, and the quantum Fisher information (QFI) plays a significant role in the study of quantum systems. In this work, we explore the dynamics of QFI in a hybrid optomechanical system, which consists of a lozenge-type four-level atom interacting with a single-mode quantized field via a multi-photon process. We account for various sources of dissipation, including the decay rates of the atom, the cavity and the mechanical modes. Using an effective Hamiltonian, we analytically derive the explicit form of the state vector of the entire system via the time-dependent Schr & ouml;dinger equation. We then investigate the atomic QFI for the estimation precision of the decay rate of the mechanical oscillator. Furthermore, we examine how optomechanical and atom-field coupling strengths, dissipation parameters and multi-photon transition influence the dynamics of atomic QFI. Our numerical results suggest that the estimation precision of the decay rate of the mechanical oscillator can be controlled by these parameters.