The occurrence of multistability and normal mode splitting in an optomechanical system

Highlights: • Introduce the position-dependent mass approach in an optomechanical system. • The position-dependent mass parameter shifts the system from a bistable to a multistable behavior. • The separation between the peaks of the normal mode increases with increasing the effective mass parameter. • The results will be helpful to think all-optical transistors and to study the feedback cooling and mechanical squeezing. -- Abstract: We provide a theoretical technique to study the occurrence of multistability and normal mode splitting in an optomechanical system driven by a single mode cavity field. For this purpose, we consider the position-dependent mass mechanical resonator (PDMR) which produces the nonlinearity in the system. This shifts the system from monostable to bistable regime and bistable to a multistable regime at different conditions of the nonlinearity. Further, we use the Fourier transform technique to find out the fluctuation in the position of the nano-mechanical resonator and their energy spectrum as well. Moreover, we used the input-output theory to analyze the transmitted field spectrum as well as x and y quadratures spectra of the output field. We explain the effect of the non-linearity on the normal mode splitting of the former and later spectra. In addition, we also explain the normal mode splitting as a function of cavity decay rate and laser power, respectively.