Coherent-scattering two-dimensional cooling in levitated cavity optomechanics

The strong light-matter optomechanical coupling offered by coherent scattering set-ups have allowed the experimental realization of quantum ground-state cavity cooling of the axial motion of a levitated nanoparticle [U. Delić et al., Science 367, 892 (2020)]. An appealing milestone is now quantum two-dimensional (2D) cooling of the full in-plane motion, in any direction in the transverse plane. By a simple adjustment of the trap polarization, one obtains two nearly equivalent modes, with similar frequencies ω x ∼ ω y and optomechanical couplings g x ≃ g y —in this experimental configuration we identify an optimal trap ellipticity, nanosphere size, and cavity linewidth which allows for efficient 2D cooling. Moreover, we find that 2D cooling to occupancies n x + n y ≲ 1 at moderate vacuum ( 10 − 6 mbar) is possible in a “Goldilocks” zone bounded by √ κ Γ / 4 ≲ g x , g y ≲ ∣ ∣ ω x − ω y ∣ ∣ ≲ κ , where one balances the need to suppress dark modes while avoiding far-detuning of either mode or low cooperativities, and κ ( Γ ) is the cavity decay rate (motional heating rate). With strong-coupling regimes g x , g y ≳ κ in view one must consider the genuine three-way hybridization between x , y and the cavity light mode resulting in hybridized bright/dark modes. Finally, we show that bright/dark modes in the levitated set-up have a simple geometrical interpretation, related by rotations in the transverse plane, with implications for directional sensing