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| A '''NOON state''' is a quantum-mechanical many-body [[Quantum entanglement|entangled state]]:
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| : <math>|\psi_\text{NOON} \rangle = \frac{|N \rangle_a |0\rangle_b + e^{iN \theta} |{0}\rangle_a |{N}\rangle_b}{\sqrt{2}}, \, </math>
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| which represents a superposition of ''N'' particles in mode ''a'' with zero particles in mode ''b'', and vice versa. Usually, the particles are [[photon]]s, but in principle any [[bosonic field]] can support NOON states.
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| ==Applications==
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| NOON states are an important concept in [[quantum metrology]] and [[quantum sensing]] for their ability to make precision phase measurements when used in an optical [[Interferometry|interferometer]]. For example, consider the observable
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| : <math> A = |N,0\rangle\langle 0,N| + |0,N\rangle\langle N,0|. \, </math>
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| The expectation value of <math>A</math> for a system in a NOON state switches between +1 and −1 when the phase changes from 0 to <math>\pi/N</math>. Moreover, the error in the phase measurement becomes
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| : <math> \Delta \theta = \frac{\Delta A}{|d\langle A\rangle / d\theta|} = \frac{1}{N}. </math>
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| This is the so-called [[Heisenberg limit]], and gives a quadratic improvement over the [[quantum limit|standard quantum limit]]. NOON states are closely related to [[Schrödinger cat]] states and [[Greenberger-Horne-Zeilinger state|GHZ states]], and are extremely fragile.
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| ==Towards experimental realization==
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| There have been several theoretical proposals for creating photonic NOON states. [[Pieter Kok|Kok]], Lee, and [[Jonathan Dowling|Dowling]] proposed the first general method based on post-selection via photodetection.<ref>[http://ojps.aip.org/getpdf/servlet/GetPDFServlet?filetype=pdf&id=PLRAAN000065000005052104000001&idtype=cvips P. Kok et al., Phys. Rev. A 65, 052104 (2002)]</ref> The down-side of this method was its exponential scaling of the success probability of the protocol. Pryde and White<ref>[http://dx.doi.org/10.1103/PhysRevA.68.052315 G. J. Pryde and A. G. White, Creation of [[Maximally entangled state|maximally entangled]] photon number states using optical fibre multiports, Physical Review A, 68, 052315 (2003)]</ref> subsequently introduced a simplified method using intensity-symmetric multiport beam splitters, single photon inputs, and either heralded or conditional measurement. Their method, for example, allows heralded production of the ''N'' = 4 NOON state without the need for postselection or zero photon detections, and has the same success probability of 3/64 as the more complicated circuit of Kok et al. Cable and Dowling proposed a method that has polynomial scaling in the success probability, which can therefore be called efficient.<ref>[http://link.aps.org/doi/10.1103/PhysRevLett.99.163604 H. Cable and J. P. Dowling, Phys. Rev. Lett. 99, 163604 (2007)]</ref>
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| Two-photon NOON states, where ''N'' = 2, can be created deterministically from two identical photons and a 50:50 beam splitter. This is called the [[Hong–Ou–Mandel effect]] in [[quantum optics]]. Three- and four-photon NOON states cannot be created deterministically from single-photon states, but they have been created probabilistically via post-selection using [[spontaneous parametric down-conversion]].<ref>[http://www.nature.com/nature/journal/v429/n6988/abs/nature02552.html P. Walther et al., Nature 429, 158 (2004).]</ref><ref>[http://www.nature.com/nature/journal/v429/n6988/abs/nature02493.html M. W. Mitchell et al., Nature 429, 161 (2004).]</ref> A different approach, involving the interference of non-classical light created by [[spontaneous parametric down-conversion]] and a classical laser beam on a 50:50 beam splitter, was used by I. Afek, O. Ambar, and Y. Silberberg to experimentally demonstrate the production of NOON states up to ''N'' = 5.<ref>[http://www.sciencemag.org/content/328/5980/879.short I. Afek, O. Ambar, and Y. Silberberg, Science 328, 879-881 (2010)]</ref><ref>[http://pra.aps.org/abstract/PRA/v85/i2/e022115 Y. Israel, I. Afek, S. Rosen, O. Ambar, and Y. Silberberg, Phys. Rev. A 85, 022115 (2012)]</ref>
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| Super-resolution has previously been used as indicator of NOON state production, in 2005 Resch et al.<ref>[http://dx.doi.org/10.1103/PhysRevLett.98.223601 K. J. Resch, K. L. Pregnell, R. Prevedel, A. Gilchrist, G. J. Pryde, J. L. O'Brien, and A. G. White, Time-reversal and super-resolving phase measurements, Physical Review Letters 98, 223601 (2007), and arXiv http://arxiv.org/abs/quant-ph/0511214]</ref> showed that it could equally well be prepared by classical interferometry. They showed that only phase super-sensitivity is an unambiguous indicator of a NOON state; furthermore they introduced criteria for determining if it has been achieved based on the observed visibility and efficiency. Super resolution, but not phase super sensitivity as the efficiency was too low, of NOON states up to ''N'' = 4 photons was also demonstrated experimentally.<ref>[http://www.sciencemag.org/cgi/content/abstract/sci;316/5825/726 T. Nagata, et al., Science 316, 726 (2007)].</ref>
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| ==History and terminology== | |
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| NOON states were first introduced by Barry Sanders in the context of studying [[quantum decoherence]] in [[Schrödinger cat]] states.<ref>[http://link.aps.org/abstract/PRA/v40/p2417 B.C.Sanders, Phys. Rev. A 40, 2417 (1989)].</ref> They were independently rediscovered in 2000 by [[Jonathan P. Dowling]]'s group at [[JPL]], who introduced them as the basis for the concept of [[quantum lithography]].<ref>[http://link.aps.org/abstract/PRL/v85/p2733 A.N. Boto, et al., Phys. Rev. Lett. 85, 2733 (2000)].</ref> The term "NOON state" first appeared in print as a footnote in a paper published by Lee, [[Pieter Kok|Kok]], and [[Jonathan Dowling|Dowling]] on [[Quantum metrology]],<ref>[http://www.informaworld.com/smpp/content~content=a713825937~db=all~order=page H. Lee et al., J. Mod. Opt. 49, 2325-2338 (2002)].</ref> where it was spelled N00N, with zero's instead of Os.
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| ==References==
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| {{Reflist}}
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| [[Category:Quantum information science]]
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| [[Category:Quantum mechanics]]
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Nestor is the title my parents gave me but I don't like when people use my complete title. To play croquet is the hobby I will by no means quit performing. Delaware is our birth location. Bookkeeping is what he does.
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