Achieving the ultimate optical resolution

We determine the ultimate potential of quantum imaging for boosting the resolution of a far-field, diffractionlimited, linear imaging device within the paraxial approximation. First we show that the problem of estimating the separation between two point-like sources is equivalent to the estimation of the loss parameters of two lossy bosonic channels, i.e., the transmissivities of two beam splitters. Using this representation, we establish the ultimate precision bound for resolving two point-like sources in an arbitrary quantum state, with a simple formula for the specific case of two thermal sources. We find that the precision bound scales with the number of collected photons according to the standard quantum limit. Then we determine the sources whose separation can be estimated optimally, finding that quantum-correlated sources (entangled or discordant) can be super-resolved at the sub-Rayleigh scale. Our results set the upper bounds on any present or future imaging technology, from astronomical observation to microscopy, which is based on quantum detection as well as source engineering. Introduction. Quantum imaging aims at harnessing quantum features of light to obtain optical images of high resolution beyond the boundary of classical optics. Its range of potential applications is very broad, from telescopy to microscopy and medical diagnosis, and has motivated a substantial research activity [1][2][3][4][5][6][7][8][9][10]. Typically, quantum imaging is scrutinized to outperform classical imaging in two ways. First, to resolve details below the Rayleigh length (sub-Rayleigh imaging). Second, to improve the way the precision scales with the number of photons, by exploiting non-classical states of light. It is well known that a collective state of N quantum particles has an effective wavelength that is N times smaller than individual particles [11][12][13][14][15][16][17][18]. If N independent photons are measured one expects that the blurring of the image scales as 1/ √ N (known as standard quantum limit or shot-noise limit), while for N entangled photons one can sometimes achieve a 1/N scaling (known as the Heisenberg limit).

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“…Our findings generalize the seminal Ref. [10] which has led to several experimental advances in quantum imaging [20][21][22][23].…”

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confidence: 86%

Ultimate Precision Bound of Quantum and Subwavelength Imaging

2016

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“…[16] that approaches the QCR bound for sub-Rayleigh separations. Recent experimental work [17][18][19][20] inspired by the above proposals has substantiated the surprising findings of the above papers.…”

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confidence: 53%

Quantum limit for two-dimensional resolution of two incoherent optical point sources

2017

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We obtain the multiple-parameter quantum Cramér-Rao bound for estimating the transverse Cartesian components of the centroid and separation of two incoherent optical point sources using an imaging system with finite spatial bandwidth. Under quite general and realistic assumptions on the point-spread function of the imaging system, and for weak source strengths, we show that the Cramér-Rao bounds for the x and y components of the separation are independent of the values of those components, which may be well below the conventional Rayleigh resolution limit. We also propose two linear optics-based measurement methods that approach the quantum bound for the estimation of the Cartesian components of the separation once the centroid has been located. One of the methods is an interferometric scheme that approaches the quantum bound for sub-Rayleigh separations. The other method using fiber coupling can in principle attain the bound regardless of the distance between the two sources.

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“…mechanism emerges for a lack of fidelity that might otherwise have been more loosely attributed to noise. This is an issue of considerable significance in the context of factors more widely limiting optical resolution [28]. And thirdly, the form of dependence on linear susceptibility does indicate that optically dense materials exhibit a greater propensity for nonlocal down-conversion; as noted earlier, exploiting preresonance enhancement of SPDC in the wings of an optical absorption band disproportionately favors more delocalized emission.…”

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confidence: 88%

Nonlocalized Generation of Correlated Photon Pairs in Degenerate Down-Conversion

2017

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The achievement of optimum conversion efficiency in conventional spontaneous parametric downconversion requires consideration of quantum processes that entail multisite electrodynamic coupling, actively taking place within the conversion material. The physical mechanism, which operates through virtual photon propagation, provides for photon pairs to be emitted from spatially separated sites of photon interaction; occasionally pairs are produced in which each photon emerges from a different point in space. The extent of such nonlocalized generation is influenced by individual variations in both distance and phase correlation. Mathematical analysis of the global contributions from this mechanism provides a quantitative measure for a degree of positional uncertainty in the origin of down-converted emission. DOI: 10.1103/PhysRevLett.118.133602 Spontaneous parametric down-conversion (SPDC) is a process in which light passing through an optically nonlinear medium can generate double-wavelength output. At a fundamental level, the process entails the conversion of pump photons into conascent, phase-matched pairs, executed by material interactions that entail the secondorder nonlinear optical susceptibility: a third-order electric dipole response. Each generated pair of photons has a combined energy and momentum equal to that of the corresponding annihilated photon, and they also exhibit correlated polarization. When the emergent photons equally share the energy of the input, the process is known as degenerate down-conversion (DDC); an alternative perspective is to regard the conversion as an exact time reversal of second-harmonic generation [1,2]. Much work has been carried out on correlated photon pairs, including their generation [3][4][5], manipulation [6][7][8][9], and application [10][11][12]. The entanglement of the quantum states in each pair has important applications in quantum computing and communication [13], and potential utilization clinically [14].In this Letter we develop an expanded theory of SPDC, highlighting and quantifying an important contribution from nonlocalized couplings. While SPDC is one of the main sources of entangled photon pairs [15], an exact location for the creation of each output photon cannot of course be inferred by direct observation-although pump photon annihilation and down-converted photon emission are generally assumed to be colocated. Of course, the diffuse nature of atomic and molecular orbitals precludes exact identification of the location for any photon creation event; equally, even the emission of two correlated photons cannot be considered as precisely colocated in origin. However, the spatial extent of the region from within which a pair of down-converted photons may emerge is considerably larger than may usually be supposed.In fact, there is a possibility that for each input photon the process may entail correlated photon interactions at two separate locations, creating one down-converted photon at each as indicated in Fig. 1. Accounting for such delocalized interact...

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Achieving the ultimate optical resolution

Cited by 197 publications

References 15 publications

Ultimate Precision Bound of Quantum and Subwavelength Imaging

Ultimate Precision Bound of Quantum and Subwavelength Imaging

Quantum limit for two-dimensional resolution of two incoherent optical point sources

Nonlocalized Generation of Correlated Photon Pairs in Degenerate Down-Conversion

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