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 ${\omega }_x\sim {\omega }_y$ and optomechanical couplings $g_x\simeq g_y$—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 $n_x+n_y\lesssim 1$ at moderate vacuum (${10}^{-6}$ mbar) is possible in a “Goldilocks” zone bounded by $\sqrt{\kappa \mathrm{\Gamma }/4}\lesssim g_x,g_y\lesssim |{\omega }_x-{\omega }_y|\lesssim \kappa$, where one balances the need to suppress dark modes while avoiding far-detuning of either mode or low cooperativities, and $\kappa$ ($\mathrm{\Gamma }$) is the cavity decay rate (motional heating rate). With strong-coupling regimes $g_x,g_y\gtrsim \kappa$ 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.

中文翻译：

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悬浮腔光机中的相干散射二维冷却

相干散射装置提供的强光-物质光-机械耦合使得悬浮基纳米粒子轴向运动的量子基态腔冷却实验得以实现。Delić等。，科学 367，892（2020）]。现在，一个有吸引力的里程碑是在平面内任何方向上的完整平面内运动的量子二维（2D）冷却。通过简单地调整陷波极化，可以得到两个近似等效的模式，具有相似的频率${\omega }_X〜{\omega }_{ÿ}$ 和光机械耦合 $G_X\simeq G_{ÿ}$-在此实验配置中，我们确定了最佳陷阱陷阱椭圆率，纳米球尺寸和腔体线宽，可以实现高效的2D冷却。而且，我们发现2D制冷占位${ñ}_X+{ñ}_{ÿ}\lesssim \mathrm{1个}$ 在中等真空度下（${10}^{-6}$ mbar）可以在以 $\sqrt{\kappa \mathrm{\Gamma }/4}\lesssim G_X，G_{ÿ}\lesssim |{\omega }_X-{\omega }_{ÿ}|\lesssim \kappa$，其中平衡了抑制暗模式的需求，同时又避免了模式或低协作性的严重失谐，以及 $\kappa$ （$\mathrm{\Gamma }$）是腔衰变率（运动加热率）。具有强耦合机制$G_X，G_{ÿ}\gtrsim \kappa$ 鉴于必须考虑真正的三元杂交 $X，ÿ$腔体光模式导致亮/暗模式混合。最后，我们显示出悬浮设置中的亮/暗模式具有简单的几何解释，与横向平面中的旋转有关，对方向感测有影响。