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arXiv:1606.02188 · Physical Review A

Heisenberg scaling of imaging resolution by coherent enhancement

Robert McConnell, Guang Hao Low, Theodore J. Yoder, Colin Bruzewicz, Isaac L. Chuang, John Chiaverini, Jeremy Sage

Published Nov 9, 2017• 4 citations
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Abstract

Classical imaging works by scattering photons from an object to be imaged, and achieves resolution scaling as $1/\sqrt{t}$, with $t$ the imaging time. By contrast, the laws of quantum mechanics allow one to utilize quantum coherence to obtain imaging resolution that can scale as quickly as $1/t$ -- the so-called ``Heisenberg limit.'' However, ambiguities in the obtained signal often preclude taking full advantage of this quantum enhancement, while imaging techniques designed to be unambiguous often lose this optimal Heisenberg scaling. Here we demonstrate an imaging technique which combines unambiguous detection of the target with Heisenberg scaling of the resolution. We also demonstrate a binary search algorithm which can efficiently locate a coherent target using the technique, resolving a target trapped ion to within 0.3% of the $1/{e}^{2}$ diameter of the excitation beam.

DOI
10.1103/physreva.96.051801

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Classical imaging works by scattering photons from an object to be imaged, and achieves resolution scaling as 1/t1/\sqrt{t}1/t​, with ttt the imaging time. By contrast, the laws of quantum mechanics allow one to utilize quantum coherence to obtain imaging resolution that can scale as quickly as 1/t1/t1/t -- the so-called ``Heisenberg limit.'' However, ambiguities in the obtained signal often preclude taking full advantage of this quantum enhancement, while imaging techniques designed to be unambiguous often lose this optimal Heisenberg scaling. Here we demonstrate an imaging technique which combines unambiguous detection of the target with Heisenberg scaling of the resolution. We also demonstrate a binary search algorithm which can efficiently locate a coherent target using the technique, resolving a target trapped ion to within 0.3% of the 1/e21/{e}^{2}1/e2 diameter of the excitation beam.
Conventional wisdom dictates that to image the position of fluorescent atoms or molecules, one should stimulate as much emission and collect as many photons as possible. That is, in this classical case, it has always been assumed that the coherence time of the system should be made short, and that the statistical scaling ∼1/√t defines the resolution limit for imaging time t. However, here we show in contrast that given the same resources, a long coherence time permits a higher resolution image. In this quantum regime, we give a procedure for determining the position of a single two-level system and demonstrate that the standard errors of our position estimates scale at the Heisenberg limit as ∼1/t, a quadratic, and notably optimal, improvement over the classical case.
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Quantum imaging exploits the spatial correlations between photons to image object features with a higher resolution than a corresponding classical light source could achieve. Using a quantum correlated N-photon state, the method of optical centroid measurement (OCM) was shown to exhibit a resolution enhancement by improving the classical Rayleigh limit by a factor of 1/N. In this work, the theory of OCM is formulated within the framework of an imaging formalism and is implemented in an exemplary experiment by means of a conventional entangled photon pair source. The expected resolution enhancement of a factor of two is demonstrated. The here presented experiment allows for single-shot operation without scanning or iteration to reproduce the object in the image plane. Thereby, photon detection is performed with a newly developed integrated time-resolving detector array. Multi-photon interference effects responsible for the observed resolution enhancement are discussed and possible alternative implementation possibilities for higher photon number are proposed.
The quantum mechanical phenomenon of entanglement can be utilized to beat the the Rayleigh limit, the classical bound on image resolution. This is done by entangling the photons that are used as the signal states. Using entanglement, the best possible image resolution is instead given by the Heisenberg limit, an improvement proportional to the number of entangled photons in the signal. Here, we present a novel application of entanglement by inverting the above procedure. We show that the resolution obtained by an imaging system utilizing separable photons can be achieved by an imaging system making use of entangled photons, but with the advantage of a smaller aperture. This results in a smaller, lighter imaging system. Smaller imaging systems can be especially valuable in satellite imaging where weight and size play a vital role.
We introduce quantum microscopy by coincidence (QMC) featuring balanced pathlengths, which facilitates super-resolution imaging at the Heisenberg limit, drastically boosting speed and contrast-to-noise ratio (CNR) compared to existing wide-field quantum imaging methods. QMC uses correlated photons traversing symmetric paths, behaving like a photon with half the wavelength for twice the resolution. It withstands 155 times stronger stray light than classical signals, promising non-destructive bioimaging. Our approach propels quantum imaging to microscopic scale by imaging cancer cells. Experimental and theoretical results endorse this balanced pathlength configuration as a path to quantum-enhanced coincidence imaging at the Heisenberg limit.
The spatial resolution of an imaging apparatus is limited by the Rayleigh diffraction bound, a consequence of the imager's finite spatial extent. We show some NNN-photon strategies that permit resolution of details that are smaller than this bound, attaining either a 1∕N1∕\sqrt{N}1∕N​ enhancement (standard quantum limit) or a 1∕N1∕N1∕N enhancement (Heisenberg-like scaling) over standard techniques. In the incoherent imaging regime, the methods presented are loss resistant, since classical light sources suffice. Our results may be of importance in many applications: microscopy, telescopy, lithography, metrology, etc.