The concept of critical ambient pressure is introduced in this paper. The particle escapes from its trap when the ambient pressure becomes comparable with or smaller than a critical value, even if the particle motion is cooled by one of the feedback cooling (or cavity cooling) schemes realized so far. The critical ambient pressure may be so small that it is not a limiting factor in ground-state cooling, but critical feedback cooling rates, which are also introduced in this paper, are limiting factors. The particle escapes from its trap if any of the feedback cooling rates (corresponding to the components of the particle motion) becomes comparable with or larger than its critical value. The critical feedback cooling rate is different from the well-known manifestation of the measurement noise. The critical feedback cooling rate corresponding to a certain component of the particle motion is usually smaller than the optimum feedback cooling rate at which the standard quantum limit happens, unless that component is cooled by the Coulomb force (instead of the optical gradient force). In addition, given that the measurement noise for the $z$ component of the particle motion is smaller than the measurement noises for the other two components (assuming that the beam illuminating the particle for photodetection propagates parallel to the $z$ axis), the feedback scheme in which the $z$ component of the particle motion is cooled by the Coulomb force has the best performance. This conclusion is in agreement with the experimental results published after writing the first version of this paper. The dependence of the critical ambient pressure, the critical feedback cooling rates, and the minimum achievable mean phonon numbers on the parameters of the system is derived in this paper, and can be verified experimentally. The insights into and the subtle points about the electromagnetic (EM) force (including the gradient force, radiation pressure, and recoil force), the EM force fluctuations, and the measurement noise presented in this paper are all of theoretical and practical importance, and might be useful in many systems besides those examined in this paper.

中文翻译：

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电磁悬浮纳米粒子光力学中的临界环境压力和临界冷却速率

本文介绍了临界环境压力的概念。当环境压力变得与临界值相当或小于临界值时，即使粒子运动被迄今为止实现的反馈冷却（或腔体冷却）方案之一冷却，粒子也会从其陷阱中逃脱。临界环境压力可能很小，以至于它不是基态冷却的限制因素，但临界反馈冷却速率（也在本文中介绍）是限制因素。如果任何反馈冷却速率（对应于粒子运动的分量）变得与其临界值相当或大于其临界值，则粒子从其陷阱中逃脱。临界反馈冷却速率不同于众所周知的测量噪声表现形式。与粒子运动的某个分量对应的临界反馈冷却速率通常小于标准量子极限发生时的最佳反馈冷却速率，除非该分量由库仑力（而不是光学梯度力）冷却。此外，考虑到测量噪声粒子运动的$z$分量小于其他两个分量的测量噪声（假设照射粒子进行光电探测的光束平行于$z$轴传播），其中$z$的反馈方案粒子运动的分量被库仑力冷却具有最佳性能。该结论与撰写本文第一版后发表的实验结果一致。本文导出了临界环境压力、临界反馈冷却速率和最小可实现平均声子数对系统参数的依赖性，并可通过实验进行验证。本文提出的对电磁（EM）力（包括梯度力、辐射压力和反冲力）、电磁力波动和测量噪声的见解和细微之处都具有理论和实践意义，并且除了本文中讨论的那些系统之外，它可能在许多系统中都有用。