Quantum cooling and squeezing of a levitating nanosphere via time-continuous measurements

With the purpose of controlling the steady state of a dielectric nanosphere levitated within an optical cavity, we study its conditional dynamics under simultaneous sideband cooling and additional timecontinuous measurement of either the output cavity mode or the nanosphere's position. Wefind that the average phonon number, purity and quantum squeezing of the steady-states can all be made more non-classical through the addition of time-continuous measurement.Wepredict that the continuous monitoring of the system, together with Markovian feedback, allows one to stabilize the dynamics for any value of the laser frequency driving the cavity. By considering state of the art values of the experimental parameters, we prove that one can in principle obtain a non-classical (squeezed) steadystatewith an average phonon number nph ≈ 0.5.