We determine the quantum Cramr-Rao bound for the precision with which the oscillator frequency and damping constant of a damped quantum harmonic oscillator in an arbitrary Gaussian state can be estimated. This goes beyond standard quantum parameter estimation of a single mode Gaussian state for which typically a mode of fixed frequency is assumed. We present a scheme through which the frequency estimation can nevertheless be based on the known results for single-mode quantum parameter estimation with Gaussian states. Based on these results, we investigate {the optimal measurement time}. For measuring the oscillator frequency, our results unify previously known partial results and constitute an explicit solution for a general single-mode Gaussian state. Furthermore, we show that with existing carbon nanotube resonators (see J. Chaste et al.~Nature Nanotechnology 7, 301 (2012)) it should be possible to achieve a {mass sensitivity of the order of an electron mass $\si{\hertz^{-1/2}}$.

中文翻译：

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谐波振荡器频率和阻尼的量子参数估计

我们确定量子Cramr-Rao的精度，以其精度可以估算任意高斯状态下的阻尼量子谐波振荡器的振荡器频率和阻尼常数。这超出了通常假设为固定频率模式的单模高斯状态的标准量子参数估计。我们提出了一种方案，通过该方案，频率估计仍可以基于已知的高斯态单模量子参数估计结果。基于这些结果，我们研究了{最佳测量时间}。为了测量振荡器频率，我们的结果统一了先前已知的部分结果，并为一般的单模高斯态构成了一个明确的解决方案。此外，我们证明了现有的碳纳米管谐振器（参见J. Chaste等。$ \ si {\ hertz ^ {-1/2}} $。