Controllable optical response by modifying the gain and loss of a mechanical resonator and cavity mode in an optomechanical system

We theoretically study a strongly-driven optomechanical system which consists of a passive optical cavity and an active mechanical resonator. When the optomechanical coupling strength is varied, phase transitions, which are similar those observed in $mathcal{PT}$-symmetric systems, are observed. We show that the optical transmission can be controlled by changing the gain of the mechanical resonator and loss of the optical cavity mode. Especially, we find that: (i) for balanced gain and loss, optical amplification and absorption can be tuned by changing the optomechanical coupling strength through a control field; (ii) for unbalanced gain and loss, even with a tiny mechanical gain, both optomechanically-induced transparency and anomalous dispersion can be observed around a critical point, which exhibits an ultra-long group delay. The time delay $tau$ can be optimized by regulating the optomechanical coupling strength through the control field and improved up to several orders of magnitude ($tausim2$ $mathrm{ms}$) compared to that of conventional optomechanical systems ($tausim1$ $mumathrm{s}$). The presence of mechanical gain makes the group delay more robust to environmental perturbations. Our proposal provides a powerful platform to control light transport using a $mathcal{PT}$-symmetric-like optomechanical system.