Atom-assisted quadrature squeezing of a mechanical oscillator inside a dispersive cavity

We present a hybrid optomechanical scheme to achieve dynamical squeezing of position quadrature of a mesoscopic mechanical oscillator, that can be externally controlled by classical fields. Amembrane-in-the-middle setup is employed, in which an atom in Lambda configuration is considered to be trapped on either side of the membrane inside the cavity. We show that a considerable amount of squeezing (beyond the 3-dB limit) can be achieved and maintained at a transient time scale that is not affected by the spontaneous emission of the atom. Squeezing depends upon the initial preparation of atomic states. Further, a strong effective coupling (larger than the relevant decay rates) between the atom and the oscillator can be attained by using large control fields that pump the atom and the cavity. The effects of cavity decay and the phononic bath on squeezing are studied. The results are supported by the detailed analytical calculations.