Static synthetic gauge field control of double optomechanically induced transparency in a closed-contour interaction scheme

We study theoretically an optical cavity and a parity-time ($\mathcal{PT}$)-symmetric pair of mechanical resonators, where all oscillators are pairwise coupled, forming an optomechanical system with a closed-contour interaction. Due to the presence of both gain and feedback, we explore its stability and the root loci over a wide coupling range. Under the red-sideband pumping and for the so-called $\mathcal{PT}$-unbroken phase it displays a double optomechanically induced transparency (OMIT) for an experimentally realizable parameter set. We show that both the transmission amplitude and the group delay can be continuously steered from the lower transmission window to the upper one by the loop coupling phase which breaks the time-reversal symmetry and introduces a static synthetic gauge field. In the $\mathcal{PT}$-unbroken phase both the gain-bandwidth and delay-bandwidth products remain constant over the full range of the controlling phase. Tunability in transmission and bandwidth still prevails in the $\mathcal{PT}$-broken phase, albeit over a reduced range. In essence, we suggest a simple scheme that grants coupling phase-dependent control of the single and double OMIT phenomena within an effective $\mathcal{PT}$-symmetric optomechanical system.