Coherent scattering 2D cooling in levitated cavity optomechanics

The strong light-matter optomechanical coupling offered by Coherent Scattering (CS) set-ups have allowed the experimental realisation 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 2D cooling of the full in-plane motion, in any direction in the transverse plane. By a simple adjustment of the trap polarisation, one obtains two nearly equivalent modes, with similar frequencies ω_xsimω_y and optomechanical couplings g_xsimeq 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_ylesssim1 at moderate vacuum $10^-6$ mbar is possible in a "Goldilocks" zone bounded by sqrt{κΓ/4}lesssim g_x,g_ylesssim|ω_x-ω_y|lesssimκ, where one balances the need to suppress dark modes whilst 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_ygtrsimκ in view one must consider the genuine three-way hybridisation 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.