Electromagnetically induced transparency with quantized fields in a dissipative optomechanical system

We study electromagnetically induced transparency (EIT) using quantized fields in a dissipative optomechanical system with two mechanical oscillators coupled to each other via the Coulomb interaction. The weak probe field is a finite bandwidth squeezed vacuum field. In the absence of the Coulomb coupling, we show that an EIT dip is observable in the homodyne spectrum of the output field even for the squeezed vacuum field at the single-photon level. We find that the thermal environment has a negative impact on the EIT behavior. In the presence of the Coulomb coupling, we show that double EIT dips appear in the homodyne spectrum of the output field. The separation between the two EIT dips can be used to measure the Coulomb coupling strength. Compared to the case of the purely dissipative coupling, the combined dispersive and dissipative coupling can make the EIT dip broader and the minimum value of the EIT dip smaller.