Fig. 1: QND gate between an atomic ensemble and a mechanical oscillator.

A squeezed quantum light pulse with a rectangular temporal profile first passes the atomic ensemble in a cavity and then the optomechanical cavity via circulators \({C}_{1,2}\) and then goes to the homodyne detector (LO—rectangularly shaped local oscillator). Within the cavities the optical pulse is coupled to atoms and mechanics, respectively, via QND interactions enabled by strong classical pumps. The homodyne detection data are used to control the optical feedforward procedure after the detection to shift the atomic quadratures. Canonical variables \(\{{\hat{X}}_{{\mathrm {a}}},{\hat{P}}_{{\mathrm {a}}}\}\), \(\{{\hat{X}}_{{\mathrm {m}}},{\hat{P}}_{{\mathrm {m}}}\}\), \(\{{\hat{x}}_{{\mathrm {c}}},{\hat{p}}_{{\mathrm {c}}}\}\), and \(\{{\hat{x}}_{{\mathrm {c}}}^{\prime},{\hat{p}}_{{\mathrm {c}}}^{\prime}\}\) are the quadratures of the collective atomic spin, mechanical oscillator, and intracavity modes, respectively; non-canonical variables \(\{{\hat{x}}^{{\mathrm {in}}},{\hat{p}}^{{\mathrm {in}}}\}\), \(\{{\hat{x}}^{{\mathrm {out}}},{\hat{p}}^{{\mathrm {out}}}\}\), \(\{{\hat{x}}^{\prime {\mathrm {in}}},{\hat{p}}^{\prime {\mathrm {in}}}\}\), and \(\{{\hat{x}}^{\prime {\mathrm {out}}},{\hat{p}}^{\prime {\mathrm {out}}}\}\) are the quadratures of the light field outside the cavities in free space at the corresponding parts of the scheme. The homodyne measurement and magnetic feedforward control via magnetic field phase shifter (MFPS) are optimized to perform the QND interaction and the squeezed light is used to achieve large entangling power.