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Optics Express

Super-resolved phase measurements at the shot noise limit by parity measurement

Lior Cohen, D. Istrati, Liat Dovrat, H. S. Eisenberg

Published May 9, 2014• 55 citations
Publisher

Abstract

Classically, the resolution of optical measurements is limited by the Rayleigh limit and their sensitivity by the shot noise limit. However, non-classical measurements can surpass these limits. Measuring the photon number parity using a photon-number resolving detector, super resolved phase measurements up to 144 better than the Rayleigh limit are presented, with coherent states of up to 4,200 photons on average. An additional measurement that can be implemented with standard single-photon detectors is proposed and demonstrated. With this scheme, super resolution at the shot noise limit is demonstrated with coherent states of up to 200 photons on average.

DOI
10.1364/oe.22.011945

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Classically, the resolution of optical measurements is limited by the Rayleigh limit and their sensitivity by the shot noise limit. However, non-classical measurements can surpass these limits. Measuring the photon number parity using a photon-number resolving detector, super resolved phase measurements up to 144 better than the Rayleigh limit are presented, with coherent states of up to 4,200 photons on average. An additional measurement that can be implemented with standard single-photon detectors is proposed and demonstrated. With this scheme, super resolution at the shot noise limit is demonstrated with coherent states of up to 200 photons on average.
The ultimate sensitivity of optical measurements is a key element of many recent works. Classically, it is mainly limited by the shot noise limit. However, a measurement setup that incorporates quantum mechanical principles can surpass the shot noise limit and reach the Heisenberg limit. Nevertheless, many of those experiments fail to break even the classical shot-noise limit. Following a recent proposal, we present here the results of optical phase measurements with a photon-number resolving detector using coherent states of up to 4200 photons on average. An additional scheme that can be implemented using standard single-photon detectors is also presented, and the results of the two schemes are compared. These measurements present deterministic single-shot sub-wavelength super-resolution up to 288 better than the optical wavelength. The results follow the classically limited sensitivity, up to 86 times better than the wavelength.
There has been much recent interest in quantum optical interferometry for applications to metrology, subwavelength imaging, and remote sensing such as in quantum laser radar (LADAR). For quantum LADAR, atmospheric absorption rapidly degrades any quantum state of light, so that for high-photon loss the optimal strategy is to transmit coherent states of light, which suffer no worse loss than the Beer law for classical optical attenuation, and which provides sensitivity at the shot-noise limit. We show that coherent light coupled with photon-number-resolving detectors can provide a super-resolution much below the Rayleigh diffraction limit, with sensitivity no worse than shot noise in terms of the detected photon power.
We investigate the performance of the super-resolving quantum lidar with the entangled coherent states of light in the presence of loss and noise, especially in the noisy case. An exact analytical expression of the output signal has been derived with the binary-outcome photon counting measurements. Numerical results show that the resolution of our scheme with parity detection is √N (N) times enhanced relative to that of the coherent-state strategy with the same (intensity) detection in the lossless and noiseless cases. The influences of phase diffusion on resolution and sensitivity have been analyzed and discussed. It is found that the super-resolution emerges in the whole diffusion rate regions, whereas the super-sensitivity just exists in the high and low diffusion rate regimes. Comparisons are made with the well known N00N states, the results show that the entangled coherent states performs better resolution and sensitivity than those of the N00N scheme in the whole diffusion regimes. In addition, the effects of photon loss on resolution and sensitivity have also been studied. The phase sensitivity can beat the shot noise limit and the resolution is much better than the Rayleigh diffraction limit in the whole loss regions. Finally, the zero-nonzero photon counting measurement gives much worse sensitivity than that of the parity detection, which is just opposite from the case as demonstrated in a recent coherent-light Mach-Zehnder experiment.