Three-Dimensional and Selective Displacement Sensing of a Levitated Nanoparticle via Spatial Mode Decomposition

We propose and experimentally demonstrate a novel detection method that significantly improves the precision of real-time measurement of the three-dimensional displacement of a levitated dipolar scatterer. Our technique relies on spatial mode sorting of the light scattered by the levitated object, allowing us to selectively extract the position information of all translational degrees of freedom with minimal losses. To this end, we collect all the light back-scattered from a levitated nanoparticle using a parabolic mirror and couple it into a spatial mode sorter. We measure displacement sensitivities ($\sqrt{S_{\mathrm{imp}, x}}, \sqrt{S_{\mathrm{imp}, y}}, \sqrt{S_{\mathrm{imp}, z}}$) $=$ (1.7, 2.4, 1.0) $\times$ $10^{-14}$ m/$\sqrt{\mathrm{Hz}}$ below the zero-point motion ($x_{\mathrm{zpm}}, y_{\mathrm{zpm}}, z_{\mathrm{zpm}}$) $=$ (2.2, 2.4, 1.6) $\times$ $10^{-12}$ m of the levitated particle considered here. In the regime where environmental decoherence is not limited by gas collision we estimate that our method can reach measurement efficiencies of $(\eta_{^{\mathrm{tot}}}^{_{x}}, \eta_{^{\mathrm{tot}}}^{_{y}}, \eta_{^{\mathrm{tot}}}^{_{z}}) = (0.13, 0.18, 0.33)>1/9$, which would enable the 3D motional quantum ground state of a levitated optomechanical system.