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arXiv:1504.01846 · The Astrophysical Journal

QUANTUM OPTIMALITY OF PHOTON COUNTING FOR TEMPERATURE MEASUREMENT OF THERMAL ASTRONOMICAL SOURCES

Ranjith Nair, Mankei Tsang

Published Jul 27, 2015• 12 citations
arXivHTMLPDF

Abstract

Using the quantum Cramér–Rao bound from quantum estimation theory, we derive a fundamental quantum limit on the sensitivity of a temperature measurement of a thermal astronomical source. This limit is expressed in terms of the source temperature T s , input spectral bandwidth , and measurement duration T , subject to a long measurement time assumption . It is valid for any measurement procedure that yields an unbiased estimate of the source temperature. The limit agrees with the sensitivity of direct detection or photon counting, and also with that of the ideal radiometer in the regime for which the Rayleigh–Jeans approximation is valid, where is the center frequency at which the radiometer operates. While valid across the electromagnetic spectrum, the limit is especially relevant for radio astronomy in this regime, since it implies that no ingenious design or technological improvement can beat an ideal radiometer for temperature measurement. In this connection, our result refutes the recent claim of a radio astronomy technique with much-improved sensitivity over the radiometer.

DOI
10.1088/0004-637x/808/2/125

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Using the quantum Cramér–Rao bound from quantum estimation theory, we derive a fundamental quantum limit on the sensitivity of a temperature measurement of a thermal astronomical source. This limit is expressed in terms of the source temperature T s , input spectral bandwidth , and measurement duration T , subject to a long measurement time assumption . It is valid for any measurement procedure that yields an unbiased estimate of the source temperature. The limit agrees with the sensitivity of direct detection or photon counting, and also with that of the ideal radiometer in the regime for which the Rayleigh–Jeans approximation is valid, where is the center frequency at which the radiometer operates. While valid across the electromagnetic spectrum, the limit is especially relevant for radio astronomy in this regime, since it implies that no ingenious design or technological improvement can beat an ideal radiometer for temperature measurement. In this connection, our result refutes the recent claim of a radio astronomy technique with much-improved sensitivity over the radiometer.
We obtain the fundamental quantum limit for resolving two thermal point sources using an imaging system with limited spatial bandwidth. Using the quantum Cram\'er-Rao bound, we show that the standard Rayleigh limit is not fundamental and can be surpassed by concrete coherent measurement techniques. Our results are valid for all values of the source strength, all ranges of the electromagnetic spectrum, and for any imaging system with an inversion-symmetric point-spread function. Our findings have applications to many areas of metrology including microscopy, astronomy, and standoff target sensing.
We obtain the ultimate quantum limit for estimating the transverse separation of two thermal point sources using a given imaging system with limited spatial bandwidth. We show via the quantum Cramér-Rao bound that, contrary to the Rayleigh limit in conventional direct imaging, quantum mechanics does not mandate any loss of precision in estimating even deep sub-Rayleigh separations. We propose two coherent measurement techniques, easily implementable using current linear-optics technology, that approach the quantum limit over an arbitrarily large range of separations. Our bound is valid for arbitrary source strengths, all regions of the electromagnetic spectrum, and for any imaging system with an inversion-symmetric point-spread function. The measurement schemes can be applied to microscopy, optical sensing, and astrometry at all wavelengths.
Lieu et al. (2015) have recently claimed that it is possible to substantially improve the sensitivity of radio astronomical observations. In essence, their proposal is to make use of the intensity of the photon shot noise as a measure of the photon arrival rate. Lieu et al. (2015) provide a detailed quantum-mechanical calculation of a proposed measurement scheme that uses two detectors and conclude that this scheme avoids the sensitivity degradation that is associated with photon bunching. If correct, this result could have a profound impact on radio astronomy. Here I present a detailed analysis of the sensitivity attainable using shot-noise measurement schemes that use either one or two detectors, and demonstrate that neither scheme can avoid the photon bunching penalty. I perform both semiclassical and fully quantum calculations of the sensitivity, obtaining consistent results, and provide a formal proof of the equivalence of these two approaches. These direct calculations are furthermore shown to be consistent with an indirect argument based on a correlation method that establishes an independent limit to the sensitivity of shot-noise measurement schemes. Collectively, these results conclusively demonstrate that the photon bunching sensitivity penalty applies to shot noise measurement schemes just as it does to ordinary photon counting, in contradiction to the fundamental claim made by Lieu et al. (2015). The source of this contradiction is traced to a logical fallacy in their argument.
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I propose classical and quantum limits to the statistical resolution of two incoherent optical point sources from the perspective of minimax parameter estimation. Unlike earlier results based on the Cramér-Rao bound (CRB), the limits proposed here, based on the worst-case error criterion and a Bayesian version of the CRB, are valid for any biased or unbiased estimator and obey photon-number scalings that are consistent with the behaviours of actual estimators. These results prove that, from the minimax perspective, the spatial-mode demultiplexing measurement scheme recently proposed by Tsang, Nair, and Lu [Phys. Rev. X 2016, 6 031033.] remains superior to direct imaging for sufficiently high photon numbers.