Towards Superresolution Surface Metrology: Quantum Estimation of Angular and Axial Separations.

We investigate the localization of two incoherent point sources with arbitrary angular and axial separations in the paraxial approximation. By using quantum metrology techniques, we show that a simultaneous estimation of the two separations is achievable by a single quantum measurement, with a precision saturating the ultimate limit stemming from the quantum Cramér-Rao bound. Such a precision is not degraded in the subwavelength regime, thus overcoming the traditional limitations of classical direct imaging derived from Rayleigh's criterion. Our results are qualitatively independent of the point spread function of the imaging system, and quantitatively illustrated in detail for the Gaussian instance. This analysis may have relevant applications in three-dimensional surface measurements.

[1]  Mankei Tsang,et al.  Interferometric superlocalization of two incoherent optical point sources. , 2015, Optics express.

[2]  Richard Leach,et al.  Open questions in surface topography measurement: a roadmap , 2015 .

[3]  Christoph Simon,et al.  Far-field linear optical superresolution via heterodyne detection in a higher-order local oscillator mode , 2016, 1606.02662.

[4]  Richard K. Leach,et al.  Applications of super-resolution imaging in the field of surface topography measurement , 2013 .

[5]  Animesh Datta,et al.  Quantum Enhanced Estimation of a Multidimensional Field. , 2015, Physical review letters.

[6]  Hugo Ferretti,et al.  Beating Rayleigh's Curse by Imaging Using Phase Information. , 2016, Physical review letters.

[7]  Sudhakar Prasad,et al.  Quantum Limited Superresolution of an Incoherent Source Pair in Three Dimensions. , 2018, Physical review letters.

[8]  M. Tsang.  Quantum nonlocality in weak-thermal-light interferometry. , 2011, Physical review letters.

[9]  Liang Jiang,et al.  Modern description of Rayleigh's criterion , 2018, Physical Review A.

[10]  J. Wolf.  Coherent Quantum Control in Biological Systems , 2011 .

[11]  Peter J. de Groot,et al.  Lateral resolution and instrument transfer function as criteria for selecting surface metrology instruments , 2012 .

[12]  M. G. A. Paris,et al.  Optimal estimation of joint parameters in phase space , 2012, 1206.4867.

[13]  S. Pirandola,et al.  Ultimate Precision of Adaptive Noise Estimation. , 2016, Physical review letters.

[14]  Marco G. Genoni,et al.  Entangling measurements for multiparameter estimation with two qubits , 2017, 1704.03327.

[15]  Marco Genovese,et al.  Real applications of quantum imaging , 2016, 1601.06066.

[16]  Sammy Ragy,et al.  Compatibility in multiparameter quantum metrology , 2016, 1608.02634.

[17]  Leonhard Möckl,et al.  Super-resolved fluorescence microscopy: Nobel Prize in Chemistry 2014 for Eric Betzig, Stefan Hell, and William E. Moerner. , 2014, Angewandte Chemie.

[18]  K. Banaszek,et al.  Superresolution imaging via two-photon interference , 2018 .

[19]  Mankei Tsang,et al.  Quantum theory of superresolution for two incoherent optical point sources , 2015, 1511.00552.

[20]  P. Humphreys,et al.  Quantum enhanced multiple phase estimation. , 2013, Physical review letters.

[21]  Jonas Zmuidzinas,et al.  Cramér-Rao sensitivity limits for astronomical instruments: implications for interferometer design. , 2003, Journal of the Optical Society of America. A, Optics, image science, and vision.

[22]  N. Treps,et al.  Quantum parameter estimation using general single-mode Gaussian states , 2013, 1307.5318.

[23]  Saikat Guha,et al.  Fundamental limit of resolving two point sources limited by an arbitrary point spread function , 2017, 2017 IEEE International Symposium on Information Theory (ISIT).

[24]  S. Braunstein,et al.  Statistical distance and the geometry of quantum states. , 1994, Physical review letters.

[25]  M. Paris.  Quantum estimation for quantum technology , 2008, 0804.2981.

[26]  W E Moerner,et al.  New directions in single-molecule imaging and analysis , 2007, Proceedings of the National Academy of Sciences.

[27]  Mankei Tsang,et al.  Quantum limit for two-dimensional resolution of two incoherent optical point sources , 2016, 2016 Conference on Lasers and Electro-Optics (CLEO).

[28]  Marco G. Genoni,et al.  Joint estimation of phase and phase diffusion for quantum metrology , 2014, Nature Communications.

[29]  M. Mitchell,et al.  Quantum-enhanced measurements without entanglement , 2017, Reviews of Modern Physics.

[30]  L. Pezzè,et al.  Quantum metrology with nonclassical states of atomic ensembles , 2016, Reviews of Modern Physics.

[31]  Jing Yang,et al.  Optimal measurements for quantum multiparameter estimation with general states , 2018, Physical Review A.

[32]  Stefano Pirandola,et al.  Quantum Fidelity for Arbitrary Gaussian States. , 2015, Physical review letters.

[33]  Marcin Jarzyna,et al.  On superresolution imaging as a multiparameter estimation problem , 2017, 1709.08392.

[34]  C. Helstrom.  Quantum detection and estimation theory , 1969 .

[35]  Z Hradil,et al.  Optimal measurements for resolution beyond the Rayleigh limit. , 2017, Optics letters.

[36]  Stefano Pirandola,et al.  Ultimate Precision Bound of Quantum and Subwavelength Imaging. , 2016, Physical review letters.

[37]  S. Lloyd,et al.  Advances in quantum metrology , 2011, 1102.2318.

[38]  Mankei Tsang,et al.  Quantum limit to subdiffraction incoherent optical imaging , 2018, Physical Review A.

[39]  J. Rehacek,et al.  Optimal measurements for quantum spatial superresolution , 2017, Physical Review A.

[40]  Mankei Tsang,et al.  Far-Field Superresolution of Thermal Electromagnetic Sources at the Quantum Limit. , 2016, Physical review letters.

[41]  M. Barbieri,et al.  Metrology with Unknown Detectors. , 2015, Physical review letters.

[42]  Tillmann Baumgratz,et al.  Multi-parameter quantum metrology , 2016, 1604.02615.

[43]  F. Illuminati,et al.  Measurement of damping and temperature: Precision bounds in Gaussian dissipative channels , 2010, 1010.0442.

[44]  P. Humphreys,et al.  Optimal Measurements for Simultaneous Quantum Estimation of Multiple Phases. , 2017, Physical review letters.

[45]  Karsten Danzmann,et al.  Quantum-dense metrology , 2012, Nature Photonics.

[46]  L. Mandel.  Fluctuations of Photon Beams: The Distribution of the Photo-Electrons , 1959 .

[47]  G. Tóth,et al.  Quantum metrology from a quantum information science perspective , 2014, 1405.4878.

[48]  J. Goodman.  Introduction to Fourier optics , 1969 .

[49]  Mankei Tsang,et al.  Quantum limits to optical point-source localization , 2014, 1411.2954.

[50]  Gerardo Adesso,et al.  Multiparameter Gaussian quantum metrology , 2017, Physical Review A.

[51]  Keiji matsumoto.  A new approach to the Cramér-Rao-type bound of the pure-state model , 2002 .

[52]  L. Mandel,et al.  Optical Coherence and Quantum Optics , 1995 .

[53]  Lord Rayleigh F.R.S..  LVI. Investigations in optics, with special reference to the spectroscope , 1879 .

[54]  R. Nair.  Quantum-Limited Loss Sensing: Multiparameter Estimation and Bures Distance between Loss Channels. , 2018, Physical review letters.

[55]  Mankei Tsang.  Subdiffraction incoherent optical imaging via spatial-mode demultiplexing , 2017 .

[56]  Gerardo Adesso,et al.  Estimating phase with a random generator: strategies and resources in multiparameter quantum metrology , 2017 .

[57]  Alexander Ling,et al.  Fault-tolerant and finite-error localization for point emitters within the diffraction limit. , 2016, Optics express.

[58]  Paul A Knott,et al.  Multiparameter Estimation in Networked Quantum Sensors. , 2017, Physical review letters.

[59]  J. Rehacek,et al.  Multiparameter quantum metrology of incoherent point sources: Towards realistic superresolution , 2017, 1709.07705.

[60]  D. Gottesman,et al.  Longer-baseline telescopes using quantum repeaters. , 2011, Physical review letters.

[61]  Jaroslav Rehacek,et al.  Achieving the ultimate optical resolution , 2016, EPJ Web of Conferences.

[62]  Marco Barbieri,et al.  Tradeoff in simultaneous quantum-limited phase and loss estimation in interferometry , 2012, 1206.0043.