有限发射带宽下的量子有限源局域化和对超分辨
Quantum Limited Source Localization and Pair Superresolution under Finite Emission Bandwidth
摘要：
将单个准单色源光学地定位到亚衍射精度需要在光子计数极限中的最小光子成本，该最小光子成本与光学系统的点扩散函数（PSF）的宽度$w$和所寻求的定位精度$d$的平方比成比例，即，作为$α(w/d)^2$。对于具有有限发射频谱的源，虽然逆二次缩放预期保持不变，但是系数$α$必须增加，因为随着成像带宽增加和PSF经历频率依赖性加宽，定位的保真度降低。我们专门解决如何迅速$α$必须增加与增加的平顶光谱分布的发射的点源被定位在两个维度的成像仪与一个清晰的圆形孔径通过计算量子费舍尔信息（QFI），其逆产生最低可能的无偏估计方差的源定位误差。新颖的使用长球形波函数作为基础，获得的QFI计算所需的单光子密度算子的本征值问题的解决方案，帮助我们开发的概念的有效维度的连续状态问题的相关的空间带宽参数。随后，我们扩展我们的考虑QFI治疗有限带宽对超分辨问题在两个维度上，获得类似的结果。我们还考虑推广到任意配置文件的发射功率谱。
英文摘要：
Optically localizing a single quasi-monochromatic source to sub-diffractive precisions entails, in the photon-counting limit, a minimum photon cost that scales as the squared ratio of the width, $w$, of the optical system's point-spread function (PSF) and the sought localization precision, $d$, i.e., as $α(w/d)^2$. For sources with a finite emission-frequency spectrum, while the inverse quadratic scaling is expected to remain unchanged, the coefficient $α$ must increase due to a degrading fidelity of localization as the imaging bandwidth increases and PSF undergoes a frequency-dependent widening. We specifically address how rapidly $α$ must increase with increasing width of a flat-top spectral profile of emission of a point source being localized in two dimensions by an imager with a clear circular aperture by calculating quantum Fisher information (QFI), whose inverse yields the lowest possible unbiased-estimation variance of source-localization error. The novel use of prolate spheroidal wave functions as a basis for obtaining a solution of the eigenvalue problem of the single-photon density operator needed for the QFI calculation helps us develop the notion of an effective dimensionality of the continuous-state problem in terms of the associated space-bandwidth parameter. We subsequently extend our considerations of QFI to treat the finite-bandwidth pair superresolution problem in two dimensions, obtaining similar results. We also consider generalizations to emission power spectra of arbitrary profiles.
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