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<div>In [https://en.wikipedia.org/wiki/Physics physics], the '''Hanbury Brown and Twiss''' ('''HBT''') '''effect''' is any of a variety of correlation and anti-correlation effects in the [https://en.wikipedia.org/wiki/Intensity_(physics) intensity (physics)] received by two detectors from a beam of particles. HBT effects can generally be attributed to the [https://en.wikipedia.org/wiki/Wave–particle_duality wave-particle duality] of the beam, and the results of a given experiment depend on whether the beam is composed of [[fermion]]s or [[boson]]s. Devices which use the effect are commonly called [[intensity interferometer]]s and were originally used in [[astronomy]], although they are also heavily used in the field of [[quantum optics]].<br />
<br />
== History ==<br />
In 1954, [[Robert Hanbury Brown]] and [[Richard Q. Twiss]] introduced the [[intensity interferometer]] concept to [[radio astronomy]] for measuring the tiny angular size of stars, suggesting that it might work with visible light as well.<ref name="BrownTwiss2010">{{cite journal|last1=Brown|first1=R. Hanbury|last2=Twiss|first2=R.Q.|title=A new type of interferometer for use in radio astronomy|journal=[[Philosophical Magazine]]|volume=45|issue=366|year=1954|pages=663–682|issn=1941-5982|doi=10.1080/14786440708520475}}</ref> Soon after they successfully tested that suggestion: in 1956 they published an in-lab experimental mockup using blue light from a [[mercury-vapor lamp]],<ref name="BrownTwiss1956">{{cite journal|last1=Brown|first1=R. Hanbury|last2=Twiss|first2=R. Q.|title=Correlation between Photons in two Coherent Beams of Light|journal=Nature|volume=177|issue=4497|year=1956|pages=27–29|issn=0028-0836|doi=10.1038/177027a0}}</ref> and later in the same year, they applied this technique to measuring the size of [[Sirius]].<ref>{{Cite journal |doi = 10.1038/1781046a0|title = A Test Of A New Type Of Stellar Interferometer On Sirius |journal = Nature|volume = 178|pages = 1046-1048|year = 1956|last1 = Hanbury Brown|first1 = R.|last2 = Twiss|first2 = Dr R.Q.|url = http://www.cmp.caltech.edu/refael/league/hanbury.pdf|bibcode = 1956Natur.178.1046H}}</ref> In the latter experiment, two [[photomultiplier tube]]s, separated by a few meters, were aimed at the star using crude telescopes, and a correlation was observed between the two fluctuating intensities. Just as in the radio studies, the correlation dropped away as they increased the separation (though over meters, instead of kilometers), and they used this information to determine the apparent [[angular size]] of Sirius.<br />
<br />
[[File:Correlation-interferometer.svg|frame|150px|right|An example of an intensity interferometer that would observe no correlation if the light source is a coherent laser beam, and positive correlation if the light source is a filtered one-mode thermal radiation. The theoretical explanation of the difference between the correlations of photon pairs in thermal and in laser beams was first given by [[Roy J. Glauber]], who was awarded the 2005 [[Nobel Prize in Physics]] "for his contribution to the quantum theory of [[Coherence (physics)|optical coherence]]".]]<br />
<br />
This result was met with much skepticism in the physics community. The radio astronomy result was justified by [[Maxwell's equations]], but there were concerns that the effect should break down at optical wavelengths, since the light would be quantised into a relatively small number of [[photon]]s that induce discrete [[photoelectron]]s in the detectors. Many [[physicists]] worried that the correlation was inconsistent with the laws of thermodynamics. Some even claimed that the effect violated the [[uncertainty principle]]. Hanbury Brown and Twiss resolved the dispute in a neat series of articles (see [[#References|References]] below) that demonstrated, first, that wave transmission in quantum optics had exactly the same mathematical form as Maxwell's equations, albeit with an additional noise term due to quantisation at the detector, and second, that according to Maxwell's equations, intensity interferometry should work. Others, such as [[Edward Mills Purcell]] immediately supported the technique, pointing out that the clumping of bosons was simply a manifestation of an effect already known in [[statistical mechanics]]. After a number of experiments, the whole physics community agreed that the observed effect was real.<br />
<br />
The original experiment used the fact that two bosons tend to arrive at two separate detectors at the same time. Morgan and Mandel used a thermal photon source to create a dim beam of photons and observed the tendency of the photons to arrive at the same time on a single detector. Both of these effects used the wave nature of light to create a correlation in arrival time – if a single photon beam is split into two beams, then the particle nature of light requires that each photon is only observed at a single detector, and so an anti-correlation was observed in 1977 by [[H. Jeff Kimble]].<ref>{{Cite journal |doi = 10.1103/PhysRevLett.39.691|title = Photon Antibunching in Resonance Fluorescence|journal = Physical Review Letters|volume = 39|issue = 11|pages = 691–695|year = 1977|last1 = Kimble|first1 = H. J.|last2 = Dagenais|first2 = M.|last3 = Mandel|first3 = L.|url = https://authors.library.caltech.edu/6051/1/KIMprl77.pdf|bibcode = 1977PhRvL..39..691K}}</ref> Finally, bosons have a tendency to clump together, giving rise to [[Bose–Einstein correlations]], while fermions due to the [[Pauli exclusion principle]], tend to spread apart, leading to Fermi–Dirac (anti)correlations. Bose–Einstein correlations have been observed between pions, kaons and photons, and Fermi–Dirac (anti)correlations between protons, neutrons and electrons. For a general introduction in this field, see the textbook on Bose–Einstein correlations by [[Richard M. Weiner]]<ref>Richard M. Weiner, Introduction to Bose–Einstein Correlations and Subatomic Interferometry, John Wiley, 2000.</ref> A difference in repulsion of [[Bose–Einstein condensate]] in the "trap-and-free fall" analogy of the HBT effect<ref>[https://arxiv.org/abs/cond-mat/0612278 Comparison of the Hanbury Brown-Twiss effect for bosons and fermions].</ref> affects comparison.<br />
<br />
Also, in the field of [[particle physics]], [[Gerson Goldhaber|Goldhaber]] et al. performed an experiment in 1959 in [[University of California, Berkeley|Berkeley]] and found an unexpected angular correlation among identical [[pion]]s, discovering the [[rho meson|ρ<sup>0</sup> resonance]], by means of <math>\rho^0 \to \pi^-\pi^+</math> decay.<ref><br />
{{cite journal<br />
|author1=G. Goldhaber<br />
|author2=W. B. Fowler<br />
|author3=S. Goldhaber<br />
|author4=T. F. Hoang<br />
|author5=T. E. Kalogeropoulos<br />
|author6=W. M. Powell<br />
|year=1959<br />
|title=Pion-pion correlations in antiproton annihilation events<br />
|journal=Phys. Rev. Lett.<br />
|volume=3 |issue=4 |page=181<br />
|doi=10.1103/PhysRevLett.3.181<br />
|bibcode=1959PhRvL...3..181G|url=http://www.escholarship.org/uc/item/7nw6p1br<br />
}}</ref> From then on, the HBT technique started to be used by the [[High-energy nuclear physics|heavy-ion community]] to determine the space–time dimensions of the particle emission source for heavy-ion collisions. For recent developments in this field, see for example the review article by Lisa.<ref>M. Lisa, et al., ''Annu. Rev. Nucl. Part. Sci.'' '''55''', p. 357 (2005), [https://arxiv.org/abs/nucl-ex/0505014 ArXiv 0505014].</ref><br />
<br />
== Wave mechanics ==<br />
The HBT effect can, in fact, be predicted solely by treating the incident [[electromagnetic radiation]] as a classical [[wave]]. Suppose we have a monochromatic wave with frequency <math>\omega</math> on two detectors, with an amplitude <math>E(t)</math> that varies on timescales slower than the wave period <math>2\pi/\omega</math>. (Such a wave might be produced from a very distant [[point source]] with a fluctuating intensity.)<br />
<br />
Since the detectors are separated, say the second detector gets the signal delayed by a time <math>\tau</math>, or equivalently, a [[Phase (waves)|phase]] <math>\phi = \omega\tau</math>; that is,<br />
<br />
:<math> E_1(t) = E(t) \sin(\omega t),</math><br />
<br />
:<math> E_2(t) = E(t - \tau) \sin(\omega t - \phi).</math><br />
<br />
The intensity recorded by each detector is the square of the wave amplitude, averaged over a timescale that is long compared to the wave period <math>2\pi/\omega</math> but short compared to the fluctuations in <math>E(t)</math>:<br />
<br />
:<math><br />
\begin{align}<br />
i_1(t) &= \overline{E_1(t)^2} = \overline{E(t)^2 \sin^2(\omega t)} = \tfrac{1}{2} E(t)^2, \\<br />
i_2(t) &= \overline{E_2(t)^2} = \overline{E(t - \tau)^2 \sin^2(\omega t - \phi)} = \tfrac{1}{2} E(t - \tau)^2,<br />
\end{align}<br />
</math><br />
where the overline indicates this time averaging. For wave frequencies above a few [[Terahertz radiation|terahertz]] (wave periods less than a [[picosecond]]), such a time averaging is unavoidable, since detectors such as [[photodiode]]s and [[photomultiplier tube]]s cannot produce photocurrents that vary on such short timescales.<br />
<br />
The correlation function <math>\langle i_1 i_2 \rangle(\tau)</math> of these time-averaged intensities can then be computed:<br />
:<math><br />
\begin{align}<br />
\langle i_1 i_2 \rangle(\tau) &= \lim_{T \to \infty} \frac{1}{T} \int\limits_0^T i_1(t) i_2(t)\, \mathrm{d}t \\<br />
&= \lim_{T \to \infty} \frac{1}{T} \int\limits_0^T \tfrac{1}{4} E(t)^2 E(t-\tau)^2 \, \mathrm{d}t.<br />
\end{align}<br />
</math><br />
<br />
Most modern schemes actually measure the correlation in intensity fluctuations at the two detectors, but it is not too difficult to see that if the intensities are correlated, then the fluctuations <math>\Delta i = i - \langle i\rangle</math>, where <math>\langle i\rangle</math> is the average intensity, ought to be correlated, since<br />
:<math>\begin{align}<br />
\langle\Delta i_1\Delta i_2\rangle &= \big\langle(i_1 - \langle i_1\rangle)(i_2 - \langle i_2\rangle)\big\rangle = \langle i_1 i_2\rangle - \big\langle i_1\langle i_2\rangle\big\rangle - \big\langle i_2\langle i_1\rangle\big\rangle + \langle i_1\rangle \langle i_2\rangle \\<br />
&=\langle i_1 i_2\rangle -\langle i_1\rangle \langle i_2\rangle.<br />
\end{align}</math><br />
<br />
In the particular case that <math>E(t)</math> consists mainly of a steady field <math>E_0</math> with a small sinusoidally varying component <math>\delta E \sin(\Omega t)</math>, the time-averaged intensities are<br />
:<math><br />
\begin{align}<br />
i_1(t) &= \tfrac{1}{2} E_0^2 + E_0\,\delta E \sin(\Omega t) + \mathcal{O}(\delta E^2), \\<br />
i_2(t) &= \tfrac{1}{2} E_0^2 + E_0\,\delta E \sin(\Omega t-\Phi) + \mathcal{O}(\delta E^2),<br />
\end{align}<br />
</math><br />
with <math>\Phi = \Omega \tau</math>, and <math>\mathcal{O}(\delta E^2)</math> indicates terms proportional to <math>(\delta E)^2</math>, which are small and may be ignored.<br />
<br />
The correlation function of these two intensities is then<br />
:<math><br />
\begin{align}<br />
\langle \Delta i_1 \Delta i_2 \rangle(\tau) &= \lim_{T \to \infty} \frac{(E_0\delta E)^2}{T} \int\limits_0^T \sin(\Omega t) \sin(\Omega t - \Phi) \, \mathrm{d}t \\<br />
&= \tfrac{1}{2} (E_0 \delta E)^2 \cos(\Omega\tau),<br />
\end{align}<br />
</math><br />
showing a sinusoidal dependence on the delay <math>\tau</math> between the two detectors.<br />
<br />
== Quantum interpretation ==<br />
[[File:Photon bunching.svg|thumb|400px|Photon detections as a function of time for a) antibunching (e.g. light emitted from a single atom), b) random (e.g. a coherent state, laser beam), and c) bunching (chaotic light). τ<sub>c</sub> is the coherence time (the time scale of photon or intensity fluctuations).]]<br />
The above discussion makes it clear that the Hanbury Brown and Twiss (or photon bunching) effect can be entirely described by classical optics. The quantum description of the effect is less intuitive: if one supposes that a thermal or chaotic light source such as a star randomly emits photons, then it is not obvious how the photons "know" that they should arrive at a detector in a correlated (bunched) way. A simple argument suggested by [[Ugo Fano]] [Fano, 1961] captures the essence of the quantum explanation. Consider two points <math>a</math> and <math>b</math> in a source that emit photons detected by two detectors <math>A</math> and <math>B</math> as in the diagram. A joint detection takes place when the photon emitted by <math>a</math> is detected by <math>A</math> and the photon emitted by <math>b</math> is detected by <math>B</math> (red arrows) ''or'' when <math>a</math>'s photon is detected by <math>B</math> and <math>b</math>'s by <math>A</math> (green arrows). The quantum mechanical probability amplitudes for these two possibilities are denoted by <br />
<math>\langle A|a \rangle \langle B|b \rangle</math> and <br />
<math>\langle B|a \rangle \langle A|b \rangle</math> respectively. If the photons are indistinguishable, the two amplitudes interfere constructively to give a joint detection probability greater than that for two independent events. The sum over all possible pairs <math>a, b</math> in the source washes out the interference unless the distance <math>AB</math> is sufficiently small. <br />
<br />
[[File:Two-photon Amplitude.svg|thumb|right|Two source points ''a'' and ''b'' emit photons detected by detectors ''A'' and ''B''. The two colors represent two different ways to detect two photons.]]<br />
<br />
Fano's explanation nicely illustrates the necessity of considering two-particle amplitudes, which are not as intuitive as the more familiar single-particle amplitudes used to interpret most interference effects. This may help to explain why some physicists in the 1950s had difficulty accepting the Hanbury Brown and Twiss result. But the quantum approach is more than just a fancy way to reproduce the classical result: if the photons are replaced by identical fermions such as electrons, the antisymmetry of wave functions under exchange of particles renders the interference destructive, leading to zero joint detection probability for small detector separations. This effect is referred to as antibunching of fermions [Henny, 1999]. The above treatment also explains [[photon antibunching]] [Kimble, 1977]: if the source consists of a single atom, which can only emit one photon at a time, simultaneous detection in two closely spaced detectors is clearly impossible. Antibunching, whether of bosons or of fermions, has no classical wave analog.<br />
<br />
From the point of view of the field of quantum optics, the HBT effect was important to lead physicists (among them [[Roy J. Glauber]] and [[Leonard Mandel]]) to apply quantum electrodynamics to new situations, many of which had never been experimentally studied, and in which classical and quantum predictions differ.<br />
<br />
== See also ==<br />
*[[Bose–Einstein correlations]]<br />
*[[Degree of coherence]]<br />
*[[Timeline of electromagnetism and classical optics]]<br />
<br />
== References ==<br />
<references/><br />
Note that Hanbury Brown is not hyphenated.<br />
<br />
* {{cite journal |author1=E. Brannen |author2=H. Ferguson | title=The question of correlation between photons in coherent light beams | journal=Nature | year=1956 | volume=178 | pages=481–482 | doi=10.1038/178481a0 | issue=4531|bibcode = 1956Natur.178..481B }} – paper which (incorrectly) disputed the existence of the Hanbury Brown and Twiss effect<br />
* {{cite journal |author1=R. Hanbury Brown |author2=R. Q. Twiss | title=A Test of a New Type of Stellar Interferometer on Sirius | journal=Nature | year=1956 | volume=178 | pages=1046–1048 | doi=10.1038/1781046a0 | issue=4541|bibcode = 1956Natur.178.1046H }} – experimental demonstration of the effect<br />
* {{cite journal | author=E. Purcell | title=The Question of Correlation Between Photons in Coherent Light Rays | journal=Nature | year=1956 | volume=178 | pages=1449–1450 | doi=10.1038/1781449a0 | issue=4548|bibcode = 1956Natur.178.1449P }}<br />
* {{cite journal |author1=R. Hanbury Brown |author2=R. Q. Twiss | title=Interferometry of the intensity fluctuations in light. I. Basic theory: the correlation between photons in coherent beams of radiation | journal=Proceedings of the Royal Society A | year=1957 | volume=242 | pages=300–324 | doi=10.1098/rspa.1957.0177 | issue=1230|bibcode = 1957RSPSA.242..300B }} [https://web.archive.org/web/20050124033547/http://www.strw.leidenuniv.nl/~tubbs/classic_papers/hanbury_brown_et_twiss_1957.pdf download as PDF]<br />
* {{cite journal |author1=R. Hanbury Brown |author2=R. Q. Twiss | title=Interferometry of the intensity fluctuations in light. II. An experimental test of the theory for partially coherent light | journal=Proceedings of the Royal Society A | year=1958 | volume=243 | pages=291–319 | doi=10.1098/rspa.1958.0001 | issue=1234|bibcode = 1958RSPSA.243..291B }} [https://web.archive.org/web/20050124033547/http://www.strw.leidenuniv.nl/~tubbs/classic_papers/hanbury_brown_et_twiss_1958a.pdf download as PDF]<br />
*{{cite journal |last=Fano |first=U. |title=Quantum theory of interference effects in the mixing of light from phase independent sources |journal=American Journal of Physics |volume=29 |year=1961 |pages=539–545 |doi=10.1119/1.1937827 |issue=8 |bibcode = 1961AmJPh..29..539F }}<br />
* {{cite journal |author1=B. L. Morgan |author2=L. Mandel | title=Measurement of Photon Bunching in a Thermal Light Beam | journal=Phys. Rev. Lett. | year=1966 | volume=16 | pages=1012–1014 | doi=10.1103/PhysRevLett.16.1012 | issue=22 | bibcode=1966PhRvL..16.1012M|citeseerx=10.1.1.713.7239 }}<br />
*{{Cite journal |last1=Kimble |first1=H. J. |last2=Dagenais|first2=M. |last3=Mandel|first3=L.|title=Photon antibunching in resonance fluorescence |journal=Physical Review Letters |volume=39 |year=1977 |pages=691–695|doi=10.1103/PhysRevLett.39.691 |issue=11 |bibcode=1977PhRvL..39..691K|url=https://authors.library.caltech.edu/6051/1/KIMprl77.pdf }}<br />
*{{Cite journal |last1=Dayan |first1=B. |last2=Parkins |first2=A. S.|last3=Aoki |first3=T.|last4=Ostby |first4=E. P.|last5=Vahala |first5=K. J. |last6=Kimble |first6=H. J.|title=A Photon Turnstile Dynamically Regulated by One Atom |journal=Science |volume=319 |issue=5866 |year=2008 |pages=1062–1065|doi=10.1126/Science.1152261|bibcode = 2008Sci...319.1062D |pmid=18292335|url=https://authors.library.caltech.edu/35067/2/Dayan.SOM.pdf }} – the cavity-QED equivalent for Kimble & Mandel's free-space demonstration of photon antibunching in resonance fluorescence<br />
* {{cite journal |author1=P. Grangier |author2=G. Roger |author3=A. Aspect | title=Experimental Evidence for a Photon Anticorrelation Effect on a Beam Splitter: A New Light on Single-Photon Interferences | journal=Europhysics Letters | year=1986 | volume=1 | pages=173–179 | doi=10.1209/0295-5075/1/4/004 | issue=4|bibcode = 1986EL......1..173G |citeseerx=10.1.1.178.4356 }}<br />
* {{cite journal | author=M. Henny| title=The Fermionic Hanbury Brown and Twiss Experiment | journal=Science | year=1999 | volume=284 | pages=296–298 | doi=10.1126/science.284.5412.296 | pmid=10195890 | issue=5412|bibcode = 1999Sci...284..296H |display-authors=etal| url=https://epub.uni-regensburg.de/3370/1/ScienceHBT.pdf }}<br />
* {{cite book | author=R Hanbury Brown| title=BOFFIN : A Personal Story of the Early Days of Radar, Radio Astronomy and Quantum Optics | publisher=Adam Hilger | year=1991 | isbn=978-0-7503-0130-5}}<br />
* {{cite book | author=Mark P. Silverman | title=More Than One Mystery: Explorations in Quantum Interference | url=https://archive.org/details/morethanonemyste0000silv | url-access=registration | publisher=Springer | year=1995 | isbn=978-0-387-94376-3}}<br />
* {{cite book | author=R Hanbury Brown | title=The intensity interferometer; its application to astronomy | publisher=Wiley | year=1974 |id=ASIN B000LZQD3C | isbn=978-0-470-10797-3}}<br />
* {{cite journal |author1=Y. Bromberg |author2=Y. Lahini |author3=E. Small |author4=Y. Silberberg | title=Hanbury Brown and Twiss Interferometry with Interacting Photons | journal=Nature Photonics| year=2010 | volume=4 | pages=721–726 | doi=10.1038/nphoton.2010.195 | issue=10|bibcode = 2010NaPho...4..721B }}<br />
<br />
== External links ==<br />
* http://adsabs.harvard.edu//full/seri/JApA./0015//0000015.000.html<br />
* http://physicsweb.org/articles/world/15/10/6/1<br />
* https://web.archive.org/web/20070609114114/http://www.du.edu/~jcalvert/astro/starsiz.htm<br />
* http://www.2physics.com/2010/11/hanbury-brown-and-twiss-interferometry.html<br />
*[https://www.becker-hickl.com/applications/antibunching-experiments/ Hanbury-Brown-Twiss Experiment] (Becker & Hickl GmbH, web page)<br />
<br />
[[Category:Quantum optics]]</div>Tomasikhttps://wiki.thor-cost.eu/index.php?title=Femtoscopy&diff=68Femtoscopy2021-05-10T13:14:41Z<p>Tomasik: </p>
<hr />
<div>In [https://en.wikipedia.org/wiki/Physics physics], the '''Hanbury Brown and Twiss''' ('''HBT''') '''effect''' is any of a variety of [[correlation]] and anti-correlation effects in the [[intensity (physics)|intensities]] received by two detectors from a beam of particles. HBT effects can generally be attributed to the [https://en.wikipedia.org/wiki/Wave–particle_duality wave-particle duality] of the beam, and the results of a given experiment depend on whether the beam is composed of [[fermion]]s or [[boson]]s. Devices which use the effect are commonly called [[intensity interferometer]]s and were originally used in [[astronomy]], although they are also heavily used in the field of [[quantum optics]].<br />
<br />
== History ==<br />
In 1954, [[Robert Hanbury Brown]] and [[Richard Q. Twiss]] introduced the [[intensity interferometer]] concept to [[radio astronomy]] for measuring the tiny angular size of stars, suggesting that it might work with visible light as well.<ref name="BrownTwiss2010">{{cite journal|last1=Brown|first1=R. Hanbury|last2=Twiss|first2=R.Q.|title=A new type of interferometer for use in radio astronomy|journal=[[Philosophical Magazine]]|volume=45|issue=366|year=1954|pages=663–682|issn=1941-5982|doi=10.1080/14786440708520475}}</ref> Soon after they successfully tested that suggestion: in 1956 they published an in-lab experimental mockup using blue light from a [[mercury-vapor lamp]],<ref name="BrownTwiss1956">{{cite journal|last1=Brown|first1=R. Hanbury|last2=Twiss|first2=R. Q.|title=Correlation between Photons in two Coherent Beams of Light|journal=Nature|volume=177|issue=4497|year=1956|pages=27–29|issn=0028-0836|doi=10.1038/177027a0}}</ref> and later in the same year, they applied this technique to measuring the size of [[Sirius]].<ref>{{Cite journal |doi = 10.1038/1781046a0|title = A Test Of A New Type Of Stellar Interferometer On Sirius |journal = Nature|volume = 178|pages = 1046-1048|year = 1956|last1 = Hanbury Brown|first1 = R.|last2 = Twiss|first2 = Dr R.Q.|url = http://www.cmp.caltech.edu/refael/league/hanbury.pdf|bibcode = 1956Natur.178.1046H}}</ref> In the latter experiment, two [[photomultiplier tube]]s, separated by a few meters, were aimed at the star using crude telescopes, and a correlation was observed between the two fluctuating intensities. Just as in the radio studies, the correlation dropped away as they increased the separation (though over meters, instead of kilometers), and they used this information to determine the apparent [[angular size]] of Sirius.<br />
<br />
[[File:Correlation-interferometer.svg|frame|150px|right|An example of an intensity interferometer that would observe no correlation if the light source is a coherent laser beam, and positive correlation if the light source is a filtered one-mode thermal radiation. The theoretical explanation of the difference between the correlations of photon pairs in thermal and in laser beams was first given by [[Roy J. Glauber]], who was awarded the 2005 [[Nobel Prize in Physics]] "for his contribution to the quantum theory of [[Coherence (physics)|optical coherence]]".]]<br />
<br />
This result was met with much skepticism in the physics community. The radio astronomy result was justified by [[Maxwell's equations]], but there were concerns that the effect should break down at optical wavelengths, since the light would be quantised into a relatively small number of [[photon]]s that induce discrete [[photoelectron]]s in the detectors. Many [[physicists]] worried that the correlation was inconsistent with the laws of thermodynamics. Some even claimed that the effect violated the [[uncertainty principle]]. Hanbury Brown and Twiss resolved the dispute in a neat series of articles (see [[#References|References]] below) that demonstrated, first, that wave transmission in quantum optics had exactly the same mathematical form as Maxwell's equations, albeit with an additional noise term due to quantisation at the detector, and second, that according to Maxwell's equations, intensity interferometry should work. Others, such as [[Edward Mills Purcell]] immediately supported the technique, pointing out that the clumping of bosons was simply a manifestation of an effect already known in [[statistical mechanics]]. After a number of experiments, the whole physics community agreed that the observed effect was real.<br />
<br />
The original experiment used the fact that two bosons tend to arrive at two separate detectors at the same time. Morgan and Mandel used a thermal photon source to create a dim beam of photons and observed the tendency of the photons to arrive at the same time on a single detector. Both of these effects used the wave nature of light to create a correlation in arrival time – if a single photon beam is split into two beams, then the particle nature of light requires that each photon is only observed at a single detector, and so an anti-correlation was observed in 1977 by [[H. Jeff Kimble]].<ref>{{Cite journal |doi = 10.1103/PhysRevLett.39.691|title = Photon Antibunching in Resonance Fluorescence|journal = Physical Review Letters|volume = 39|issue = 11|pages = 691–695|year = 1977|last1 = Kimble|first1 = H. J.|last2 = Dagenais|first2 = M.|last3 = Mandel|first3 = L.|url = https://authors.library.caltech.edu/6051/1/KIMprl77.pdf|bibcode = 1977PhRvL..39..691K}}</ref> Finally, bosons have a tendency to clump together, giving rise to [[Bose–Einstein correlations]], while fermions due to the [[Pauli exclusion principle]], tend to spread apart, leading to Fermi–Dirac (anti)correlations. Bose–Einstein correlations have been observed between pions, kaons and photons, and Fermi–Dirac (anti)correlations between protons, neutrons and electrons. For a general introduction in this field, see the textbook on Bose–Einstein correlations by [[Richard M. Weiner]]<ref>Richard M. Weiner, Introduction to Bose–Einstein Correlations and Subatomic Interferometry, John Wiley, 2000.</ref> A difference in repulsion of [[Bose–Einstein condensate]] in the "trap-and-free fall" analogy of the HBT effect<ref>[https://arxiv.org/abs/cond-mat/0612278 Comparison of the Hanbury Brown-Twiss effect for bosons and fermions].</ref> affects comparison.<br />
<br />
Also, in the field of [[particle physics]], [[Gerson Goldhaber|Goldhaber]] et al. performed an experiment in 1959 in [[University of California, Berkeley|Berkeley]] and found an unexpected angular correlation among identical [[pion]]s, discovering the [[rho meson|ρ<sup>0</sup> resonance]], by means of <math>\rho^0 \to \pi^-\pi^+</math> decay.<ref><br />
{{cite journal<br />
|author1=G. Goldhaber<br />
|author2=W. B. Fowler<br />
|author3=S. Goldhaber<br />
|author4=T. F. Hoang<br />
|author5=T. E. Kalogeropoulos<br />
|author6=W. M. Powell<br />
|year=1959<br />
|title=Pion-pion correlations in antiproton annihilation events<br />
|journal=Phys. Rev. Lett.<br />
|volume=3 |issue=4 |page=181<br />
|doi=10.1103/PhysRevLett.3.181<br />
|bibcode=1959PhRvL...3..181G|url=http://www.escholarship.org/uc/item/7nw6p1br<br />
}}</ref> From then on, the HBT technique started to be used by the [[High-energy nuclear physics|heavy-ion community]] to determine the space–time dimensions of the particle emission source for heavy-ion collisions. For recent developments in this field, see for example the review article by Lisa.<ref>M. Lisa, et al., ''Annu. Rev. Nucl. Part. Sci.'' '''55''', p. 357 (2005), [https://arxiv.org/abs/nucl-ex/0505014 ArXiv 0505014].</ref><br />
<br />
== Wave mechanics ==<br />
The HBT effect can, in fact, be predicted solely by treating the incident [[electromagnetic radiation]] as a classical [[wave]]. Suppose we have a monochromatic wave with frequency <math>\omega</math> on two detectors, with an amplitude <math>E(t)</math> that varies on timescales slower than the wave period <math>2\pi/\omega</math>. (Such a wave might be produced from a very distant [[point source]] with a fluctuating intensity.)<br />
<br />
Since the detectors are separated, say the second detector gets the signal delayed by a time <math>\tau</math>, or equivalently, a [[Phase (waves)|phase]] <math>\phi = \omega\tau</math>; that is,<br />
<br />
:<math> E_1(t) = E(t) \sin(\omega t),</math><br />
<br />
:<math> E_2(t) = E(t - \tau) \sin(\omega t - \phi).</math><br />
<br />
The intensity recorded by each detector is the square of the wave amplitude, averaged over a timescale that is long compared to the wave period <math>2\pi/\omega</math> but short compared to the fluctuations in <math>E(t)</math>:<br />
<br />
:<math><br />
\begin{align}<br />
i_1(t) &= \overline{E_1(t)^2} = \overline{E(t)^2 \sin^2(\omega t)} = \tfrac{1}{2} E(t)^2, \\<br />
i_2(t) &= \overline{E_2(t)^2} = \overline{E(t - \tau)^2 \sin^2(\omega t - \phi)} = \tfrac{1}{2} E(t - \tau)^2,<br />
\end{align}<br />
</math><br />
where the overline indicates this time averaging. For wave frequencies above a few [[Terahertz radiation|terahertz]] (wave periods less than a [[picosecond]]), such a time averaging is unavoidable, since detectors such as [[photodiode]]s and [[photomultiplier tube]]s cannot produce photocurrents that vary on such short timescales.<br />
<br />
The correlation function <math>\langle i_1 i_2 \rangle(\tau)</math> of these time-averaged intensities can then be computed:<br />
:<math><br />
\begin{align}<br />
\langle i_1 i_2 \rangle(\tau) &= \lim_{T \to \infty} \frac{1}{T} \int\limits_0^T i_1(t) i_2(t)\, \mathrm{d}t \\<br />
&= \lim_{T \to \infty} \frac{1}{T} \int\limits_0^T \tfrac{1}{4} E(t)^2 E(t-\tau)^2 \, \mathrm{d}t.<br />
\end{align}<br />
</math><br />
<br />
Most modern schemes actually measure the correlation in intensity fluctuations at the two detectors, but it is not too difficult to see that if the intensities are correlated, then the fluctuations <math>\Delta i = i - \langle i\rangle</math>, where <math>\langle i\rangle</math> is the average intensity, ought to be correlated, since<br />
:<math>\begin{align}<br />
\langle\Delta i_1\Delta i_2\rangle &= \big\langle(i_1 - \langle i_1\rangle)(i_2 - \langle i_2\rangle)\big\rangle = \langle i_1 i_2\rangle - \big\langle i_1\langle i_2\rangle\big\rangle - \big\langle i_2\langle i_1\rangle\big\rangle + \langle i_1\rangle \langle i_2\rangle \\<br />
&=\langle i_1 i_2\rangle -\langle i_1\rangle \langle i_2\rangle.<br />
\end{align}</math><br />
<br />
In the particular case that <math>E(t)</math> consists mainly of a steady field <math>E_0</math> with a small sinusoidally varying component <math>\delta E \sin(\Omega t)</math>, the time-averaged intensities are<br />
:<math><br />
\begin{align}<br />
i_1(t) &= \tfrac{1}{2} E_0^2 + E_0\,\delta E \sin(\Omega t) + \mathcal{O}(\delta E^2), \\<br />
i_2(t) &= \tfrac{1}{2} E_0^2 + E_0\,\delta E \sin(\Omega t-\Phi) + \mathcal{O}(\delta E^2),<br />
\end{align}<br />
</math><br />
with <math>\Phi = \Omega \tau</math>, and <math>\mathcal{O}(\delta E^2)</math> indicates terms proportional to <math>(\delta E)^2</math>, which are small and may be ignored.<br />
<br />
The correlation function of these two intensities is then<br />
:<math><br />
\begin{align}<br />
\langle \Delta i_1 \Delta i_2 \rangle(\tau) &= \lim_{T \to \infty} \frac{(E_0\delta E)^2}{T} \int\limits_0^T \sin(\Omega t) \sin(\Omega t - \Phi) \, \mathrm{d}t \\<br />
&= \tfrac{1}{2} (E_0 \delta E)^2 \cos(\Omega\tau),<br />
\end{align}<br />
</math><br />
showing a sinusoidal dependence on the delay <math>\tau</math> between the two detectors.<br />
<br />
== Quantum interpretation ==<br />
[[File:Photon bunching.svg|thumb|400px|Photon detections as a function of time for a) antibunching (e.g. light emitted from a single atom), b) random (e.g. a coherent state, laser beam), and c) bunching (chaotic light). τ<sub>c</sub> is the coherence time (the time scale of photon or intensity fluctuations).]]<br />
The above discussion makes it clear that the Hanbury Brown and Twiss (or photon bunching) effect can be entirely described by classical optics. The quantum description of the effect is less intuitive: if one supposes that a thermal or chaotic light source such as a star randomly emits photons, then it is not obvious how the photons "know" that they should arrive at a detector in a correlated (bunched) way. A simple argument suggested by [[Ugo Fano]] [Fano, 1961] captures the essence of the quantum explanation. Consider two points <math>a</math> and <math>b</math> in a source that emit photons detected by two detectors <math>A</math> and <math>B</math> as in the diagram. A joint detection takes place when the photon emitted by <math>a</math> is detected by <math>A</math> and the photon emitted by <math>b</math> is detected by <math>B</math> (red arrows) ''or'' when <math>a</math>'s photon is detected by <math>B</math> and <math>b</math>'s by <math>A</math> (green arrows). The quantum mechanical probability amplitudes for these two possibilities are denoted by <br />
<math>\langle A|a \rangle \langle B|b \rangle</math> and <br />
<math>\langle B|a \rangle \langle A|b \rangle</math> respectively. If the photons are indistinguishable, the two amplitudes interfere constructively to give a joint detection probability greater than that for two independent events. The sum over all possible pairs <math>a, b</math> in the source washes out the interference unless the distance <math>AB</math> is sufficiently small. <br />
<br />
[[File:Two-photon Amplitude.svg|thumb|right|Two source points ''a'' and ''b'' emit photons detected by detectors ''A'' and ''B''. The two colors represent two different ways to detect two photons.]]<br />
<br />
Fano's explanation nicely illustrates the necessity of considering two-particle amplitudes, which are not as intuitive as the more familiar single-particle amplitudes used to interpret most interference effects. This may help to explain why some physicists in the 1950s had difficulty accepting the Hanbury Brown and Twiss result. But the quantum approach is more than just a fancy way to reproduce the classical result: if the photons are replaced by identical fermions such as electrons, the antisymmetry of wave functions under exchange of particles renders the interference destructive, leading to zero joint detection probability for small detector separations. This effect is referred to as antibunching of fermions [Henny, 1999]. The above treatment also explains [[photon antibunching]] [Kimble, 1977]: if the source consists of a single atom, which can only emit one photon at a time, simultaneous detection in two closely spaced detectors is clearly impossible. Antibunching, whether of bosons or of fermions, has no classical wave analog.<br />
<br />
From the point of view of the field of quantum optics, the HBT effect was important to lead physicists (among them [[Roy J. Glauber]] and [[Leonard Mandel]]) to apply quantum electrodynamics to new situations, many of which had never been experimentally studied, and in which classical and quantum predictions differ.<br />
<br />
== See also ==<br />
*[[Bose–Einstein correlations]]<br />
*[[Degree of coherence]]<br />
*[[Timeline of electromagnetism and classical optics]]<br />
<br />
== References ==<br />
<references/><br />
Note that Hanbury Brown is not hyphenated.<br />
<br />
* {{cite journal |author1=E. Brannen |author2=H. Ferguson | title=The question of correlation between photons in coherent light beams | journal=Nature | year=1956 | volume=178 | pages=481–482 | doi=10.1038/178481a0 | issue=4531|bibcode = 1956Natur.178..481B }} – paper which (incorrectly) disputed the existence of the Hanbury Brown and Twiss effect<br />
* {{cite journal |author1=R. Hanbury Brown |author2=R. Q. Twiss | title=A Test of a New Type of Stellar Interferometer on Sirius | journal=Nature | year=1956 | volume=178 | pages=1046–1048 | doi=10.1038/1781046a0 | issue=4541|bibcode = 1956Natur.178.1046H }} – experimental demonstration of the effect<br />
* {{cite journal | author=E. Purcell | title=The Question of Correlation Between Photons in Coherent Light Rays | journal=Nature | year=1956 | volume=178 | pages=1449–1450 | doi=10.1038/1781449a0 | issue=4548|bibcode = 1956Natur.178.1449P }}<br />
* {{cite journal |author1=R. Hanbury Brown |author2=R. Q. Twiss | title=Interferometry of the intensity fluctuations in light. I. Basic theory: the correlation between photons in coherent beams of radiation | journal=Proceedings of the Royal Society A | year=1957 | volume=242 | pages=300–324 | doi=10.1098/rspa.1957.0177 | issue=1230|bibcode = 1957RSPSA.242..300B }} [https://web.archive.org/web/20050124033547/http://www.strw.leidenuniv.nl/~tubbs/classic_papers/hanbury_brown_et_twiss_1957.pdf download as PDF]<br />
* {{cite journal |author1=R. Hanbury Brown |author2=R. Q. Twiss | title=Interferometry of the intensity fluctuations in light. II. An experimental test of the theory for partially coherent light | journal=Proceedings of the Royal Society A | year=1958 | volume=243 | pages=291–319 | doi=10.1098/rspa.1958.0001 | issue=1234|bibcode = 1958RSPSA.243..291B }} [https://web.archive.org/web/20050124033547/http://www.strw.leidenuniv.nl/~tubbs/classic_papers/hanbury_brown_et_twiss_1958a.pdf download as PDF]<br />
*{{cite journal |last=Fano |first=U. |title=Quantum theory of interference effects in the mixing of light from phase independent sources |journal=American Journal of Physics |volume=29 |year=1961 |pages=539–545 |doi=10.1119/1.1937827 |issue=8 |bibcode = 1961AmJPh..29..539F }}<br />
* {{cite journal |author1=B. L. Morgan |author2=L. Mandel | title=Measurement of Photon Bunching in a Thermal Light Beam | journal=Phys. Rev. Lett. | year=1966 | volume=16 | pages=1012–1014 | doi=10.1103/PhysRevLett.16.1012 | issue=22 | bibcode=1966PhRvL..16.1012M|citeseerx=10.1.1.713.7239 }}<br />
*{{Cite journal |last1=Kimble |first1=H. J. |last2=Dagenais|first2=M. |last3=Mandel|first3=L.|title=Photon antibunching in resonance fluorescence |journal=Physical Review Letters |volume=39 |year=1977 |pages=691–695|doi=10.1103/PhysRevLett.39.691 |issue=11 |bibcode=1977PhRvL..39..691K|url=https://authors.library.caltech.edu/6051/1/KIMprl77.pdf }}<br />
*{{Cite journal |last1=Dayan |first1=B. |last2=Parkins |first2=A. S.|last3=Aoki |first3=T.|last4=Ostby |first4=E. P.|last5=Vahala |first5=K. J. |last6=Kimble |first6=H. J.|title=A Photon Turnstile Dynamically Regulated by One Atom |journal=Science |volume=319 |issue=5866 |year=2008 |pages=1062–1065|doi=10.1126/Science.1152261|bibcode = 2008Sci...319.1062D |pmid=18292335|url=https://authors.library.caltech.edu/35067/2/Dayan.SOM.pdf }} – the cavity-QED equivalent for Kimble & Mandel's free-space demonstration of photon antibunching in resonance fluorescence<br />
* {{cite journal |author1=P. Grangier |author2=G. Roger |author3=A. Aspect | title=Experimental Evidence for a Photon Anticorrelation Effect on a Beam Splitter: A New Light on Single-Photon Interferences | journal=Europhysics Letters | year=1986 | volume=1 | pages=173–179 | doi=10.1209/0295-5075/1/4/004 | issue=4|bibcode = 1986EL......1..173G |citeseerx=10.1.1.178.4356 }}<br />
* {{cite journal | author=M. Henny| title=The Fermionic Hanbury Brown and Twiss Experiment | journal=Science | year=1999 | volume=284 | pages=296–298 | doi=10.1126/science.284.5412.296 | pmid=10195890 | issue=5412|bibcode = 1999Sci...284..296H |display-authors=etal| url=https://epub.uni-regensburg.de/3370/1/ScienceHBT.pdf }}<br />
* {{cite book | author=R Hanbury Brown| title=BOFFIN : A Personal Story of the Early Days of Radar, Radio Astronomy and Quantum Optics | publisher=Adam Hilger | year=1991 | isbn=978-0-7503-0130-5}}<br />
* {{cite book | author=Mark P. Silverman | title=More Than One Mystery: Explorations in Quantum Interference | url=https://archive.org/details/morethanonemyste0000silv | url-access=registration | publisher=Springer | year=1995 | isbn=978-0-387-94376-3}}<br />
* {{cite book | author=R Hanbury Brown | title=The intensity interferometer; its application to astronomy | publisher=Wiley | year=1974 |id=ASIN B000LZQD3C | isbn=978-0-470-10797-3}}<br />
* {{cite journal |author1=Y. Bromberg |author2=Y. Lahini |author3=E. Small |author4=Y. Silberberg | title=Hanbury Brown and Twiss Interferometry with Interacting Photons | journal=Nature Photonics| year=2010 | volume=4 | pages=721–726 | doi=10.1038/nphoton.2010.195 | issue=10|bibcode = 2010NaPho...4..721B }}<br />
<br />
== External links ==<br />
* http://adsabs.harvard.edu//full/seri/JApA./0015//0000015.000.html<br />
* http://physicsweb.org/articles/world/15/10/6/1<br />
* https://web.archive.org/web/20070609114114/http://www.du.edu/~jcalvert/astro/starsiz.htm<br />
* http://www.2physics.com/2010/11/hanbury-brown-and-twiss-interferometry.html<br />
*[https://www.becker-hickl.com/applications/antibunching-experiments/ Hanbury-Brown-Twiss Experiment] (Becker & Hickl GmbH, web page)<br />
<br />
[[Category:Quantum optics]]</div>Tomasikhttps://wiki.thor-cost.eu/index.php?title=Femtoscopy&diff=67Femtoscopy2021-05-10T13:13:05Z<p>Tomasik: </p>
<hr />
<div>{{Use dmy dates|date=June 2016}}<br />
In [[physics]], the '''Hanbury Brown and Twiss''' ('''HBT''') '''effect''' is any of a variety of [[correlation]] and anti-correlation effects in the [[intensity (physics)|intensities]] received by two detectors from a beam of particles. HBT effects can generally be attributed to the [https://en.wikipedia.org/wiki/Wave–particle_duality wave-particle duality] of the beam, and the results of a given experiment depend on whether the beam is composed of [[fermion]]s or [[boson]]s. Devices which use the effect are commonly called [[intensity interferometer]]s and were originally used in [[astronomy]], although they are also heavily used in the field of [[quantum optics]].<br />
<br />
== History ==<br />
In 1954, [[Robert Hanbury Brown]] and [[Richard Q. Twiss]] introduced the [[intensity interferometer]] concept to [[radio astronomy]] for measuring the tiny angular size of stars, suggesting that it might work with visible light as well.<ref name="BrownTwiss2010">{{cite journal|last1=Brown|first1=R. Hanbury|last2=Twiss|first2=R.Q.|title=A new type of interferometer for use in radio astronomy|journal=[[Philosophical Magazine]]|volume=45|issue=366|year=1954|pages=663–682|issn=1941-5982|doi=10.1080/14786440708520475}}</ref> Soon after they successfully tested that suggestion: in 1956 they published an in-lab experimental mockup using blue light from a [[mercury-vapor lamp]],<ref name="BrownTwiss1956">{{cite journal|last1=Brown|first1=R. Hanbury|last2=Twiss|first2=R. Q.|title=Correlation between Photons in two Coherent Beams of Light|journal=Nature|volume=177|issue=4497|year=1956|pages=27–29|issn=0028-0836|doi=10.1038/177027a0}}</ref> and later in the same year, they applied this technique to measuring the size of [[Sirius]].<ref>{{Cite journal |doi = 10.1038/1781046a0|title = A Test Of A New Type Of Stellar Interferometer On Sirius |journal = Nature|volume = 178|pages = 1046-1048|year = 1956|last1 = Hanbury Brown|first1 = R.|last2 = Twiss|first2 = Dr R.Q.|url = http://www.cmp.caltech.edu/refael/league/hanbury.pdf|bibcode = 1956Natur.178.1046H}}</ref> In the latter experiment, two [[photomultiplier tube]]s, separated by a few meters, were aimed at the star using crude telescopes, and a correlation was observed between the two fluctuating intensities. Just as in the radio studies, the correlation dropped away as they increased the separation (though over meters, instead of kilometers), and they used this information to determine the apparent [[angular size]] of Sirius.<br />
<br />
[[File:Correlation-interferometer.svg|frame|150px|right|An example of an intensity interferometer that would observe no correlation if the light source is a coherent laser beam, and positive correlation if the light source is a filtered one-mode thermal radiation. The theoretical explanation of the difference between the correlations of photon pairs in thermal and in laser beams was first given by [[Roy J. Glauber]], who was awarded the 2005 [[Nobel Prize in Physics]] "for his contribution to the quantum theory of [[Coherence (physics)|optical coherence]]".]]<br />
<br />
This result was met with much skepticism in the physics community. The radio astronomy result was justified by [[Maxwell's equations]], but there were concerns that the effect should break down at optical wavelengths, since the light would be quantised into a relatively small number of [[photon]]s that induce discrete [[photoelectron]]s in the detectors. Many [[physicists]] worried that the correlation was inconsistent with the laws of thermodynamics. Some even claimed that the effect violated the [[uncertainty principle]]. Hanbury Brown and Twiss resolved the dispute in a neat series of articles (see [[#References|References]] below) that demonstrated, first, that wave transmission in quantum optics had exactly the same mathematical form as Maxwell's equations, albeit with an additional noise term due to quantisation at the detector, and second, that according to Maxwell's equations, intensity interferometry should work. Others, such as [[Edward Mills Purcell]] immediately supported the technique, pointing out that the clumping of bosons was simply a manifestation of an effect already known in [[statistical mechanics]]. After a number of experiments, the whole physics community agreed that the observed effect was real.<br />
<br />
The original experiment used the fact that two bosons tend to arrive at two separate detectors at the same time. Morgan and Mandel used a thermal photon source to create a dim beam of photons and observed the tendency of the photons to arrive at the same time on a single detector. Both of these effects used the wave nature of light to create a correlation in arrival time – if a single photon beam is split into two beams, then the particle nature of light requires that each photon is only observed at a single detector, and so an anti-correlation was observed in 1977 by [[H. Jeff Kimble]].<ref>{{Cite journal |doi = 10.1103/PhysRevLett.39.691|title = Photon Antibunching in Resonance Fluorescence|journal = Physical Review Letters|volume = 39|issue = 11|pages = 691–695|year = 1977|last1 = Kimble|first1 = H. J.|last2 = Dagenais|first2 = M.|last3 = Mandel|first3 = L.|url = https://authors.library.caltech.edu/6051/1/KIMprl77.pdf|bibcode = 1977PhRvL..39..691K}}</ref> Finally, bosons have a tendency to clump together, giving rise to [[Bose–Einstein correlations]], while fermions due to the [[Pauli exclusion principle]], tend to spread apart, leading to Fermi–Dirac (anti)correlations. Bose–Einstein correlations have been observed between pions, kaons and photons, and Fermi–Dirac (anti)correlations between protons, neutrons and electrons. For a general introduction in this field, see the textbook on Bose–Einstein correlations by [[Richard M. Weiner]]<ref>Richard M. Weiner, Introduction to Bose–Einstein Correlations and Subatomic Interferometry, John Wiley, 2000.</ref> A difference in repulsion of [[Bose–Einstein condensate]] in the "trap-and-free fall" analogy of the HBT effect<ref>[https://arxiv.org/abs/cond-mat/0612278 Comparison of the Hanbury Brown-Twiss effect for bosons and fermions].</ref> affects comparison.<br />
<br />
Also, in the field of [[particle physics]], [[Gerson Goldhaber|Goldhaber]] et al. performed an experiment in 1959 in [[University of California, Berkeley|Berkeley]] and found an unexpected angular correlation among identical [[pion]]s, discovering the [[rho meson|ρ<sup>0</sup> resonance]], by means of <math>\rho^0 \to \pi^-\pi^+</math> decay.<ref><br />
{{cite journal<br />
|author1=G. Goldhaber<br />
|author2=W. B. Fowler<br />
|author3=S. Goldhaber<br />
|author4=T. F. Hoang<br />
|author5=T. E. Kalogeropoulos<br />
|author6=W. M. Powell<br />
|year=1959<br />
|title=Pion-pion correlations in antiproton annihilation events<br />
|journal=Phys. Rev. Lett.<br />
|volume=3 |issue=4 |page=181<br />
|doi=10.1103/PhysRevLett.3.181<br />
|bibcode=1959PhRvL...3..181G|url=http://www.escholarship.org/uc/item/7nw6p1br<br />
}}</ref> From then on, the HBT technique started to be used by the [[High-energy nuclear physics|heavy-ion community]] to determine the space–time dimensions of the particle emission source for heavy-ion collisions. For recent developments in this field, see for example the review article by Lisa.<ref>M. Lisa, et al., ''Annu. Rev. Nucl. Part. Sci.'' '''55''', p. 357 (2005), [https://arxiv.org/abs/nucl-ex/0505014 ArXiv 0505014].</ref><br />
<br />
== Wave mechanics ==<br />
The HBT effect can, in fact, be predicted solely by treating the incident [[electromagnetic radiation]] as a classical [[wave]]. Suppose we have a monochromatic wave with frequency <math>\omega</math> on two detectors, with an amplitude <math>E(t)</math> that varies on timescales slower than the wave period <math>2\pi/\omega</math>. (Such a wave might be produced from a very distant [[point source]] with a fluctuating intensity.)<br />
<br />
Since the detectors are separated, say the second detector gets the signal delayed by a time <math>\tau</math>, or equivalently, a [[Phase (waves)|phase]] <math>\phi = \omega\tau</math>; that is,<br />
<br />
:<math> E_1(t) = E(t) \sin(\omega t),</math><br />
<br />
:<math> E_2(t) = E(t - \tau) \sin(\omega t - \phi).</math><br />
<br />
The intensity recorded by each detector is the square of the wave amplitude, averaged over a timescale that is long compared to the wave period <math>2\pi/\omega</math> but short compared to the fluctuations in <math>E(t)</math>:<br />
<br />
:<math><br />
\begin{align}<br />
i_1(t) &= \overline{E_1(t)^2} = \overline{E(t)^2 \sin^2(\omega t)} = \tfrac{1}{2} E(t)^2, \\<br />
i_2(t) &= \overline{E_2(t)^2} = \overline{E(t - \tau)^2 \sin^2(\omega t - \phi)} = \tfrac{1}{2} E(t - \tau)^2,<br />
\end{align}<br />
</math><br />
where the overline indicates this time averaging. For wave frequencies above a few [[Terahertz radiation|terahertz]] (wave periods less than a [[picosecond]]), such a time averaging is unavoidable, since detectors such as [[photodiode]]s and [[photomultiplier tube]]s cannot produce photocurrents that vary on such short timescales.<br />
<br />
The correlation function <math>\langle i_1 i_2 \rangle(\tau)</math> of these time-averaged intensities can then be computed:<br />
:<math><br />
\begin{align}<br />
\langle i_1 i_2 \rangle(\tau) &= \lim_{T \to \infty} \frac{1}{T} \int\limits_0^T i_1(t) i_2(t)\, \mathrm{d}t \\<br />
&= \lim_{T \to \infty} \frac{1}{T} \int\limits_0^T \tfrac{1}{4} E(t)^2 E(t-\tau)^2 \, \mathrm{d}t.<br />
\end{align}<br />
</math><br />
<br />
Most modern schemes actually measure the correlation in intensity fluctuations at the two detectors, but it is not too difficult to see that if the intensities are correlated, then the fluctuations <math>\Delta i = i - \langle i\rangle</math>, where <math>\langle i\rangle</math> is the average intensity, ought to be correlated, since<br />
:<math>\begin{align}<br />
\langle\Delta i_1\Delta i_2\rangle &= \big\langle(i_1 - \langle i_1\rangle)(i_2 - \langle i_2\rangle)\big\rangle = \langle i_1 i_2\rangle - \big\langle i_1\langle i_2\rangle\big\rangle - \big\langle i_2\langle i_1\rangle\big\rangle + \langle i_1\rangle \langle i_2\rangle \\<br />
&=\langle i_1 i_2\rangle -\langle i_1\rangle \langle i_2\rangle.<br />
\end{align}</math><br />
<br />
In the particular case that <math>E(t)</math> consists mainly of a steady field <math>E_0</math> with a small sinusoidally varying component <math>\delta E \sin(\Omega t)</math>, the time-averaged intensities are<br />
:<math><br />
\begin{align}<br />
i_1(t) &= \tfrac{1}{2} E_0^2 + E_0\,\delta E \sin(\Omega t) + \mathcal{O}(\delta E^2), \\<br />
i_2(t) &= \tfrac{1}{2} E_0^2 + E_0\,\delta E \sin(\Omega t-\Phi) + \mathcal{O}(\delta E^2),<br />
\end{align}<br />
</math><br />
with <math>\Phi = \Omega \tau</math>, and <math>\mathcal{O}(\delta E^2)</math> indicates terms proportional to <math>(\delta E)^2</math>, which are small and may be ignored.<br />
<br />
The correlation function of these two intensities is then<br />
:<math><br />
\begin{align}<br />
\langle \Delta i_1 \Delta i_2 \rangle(\tau) &= \lim_{T \to \infty} \frac{(E_0\delta E)^2}{T} \int\limits_0^T \sin(\Omega t) \sin(\Omega t - \Phi) \, \mathrm{d}t \\<br />
&= \tfrac{1}{2} (E_0 \delta E)^2 \cos(\Omega\tau),<br />
\end{align}<br />
</math><br />
showing a sinusoidal dependence on the delay <math>\tau</math> between the two detectors.<br />
<br />
== Quantum interpretation ==<br />
[[File:Photon bunching.svg|thumb|400px|Photon detections as a function of time for a) antibunching (e.g. light emitted from a single atom), b) random (e.g. a coherent state, laser beam), and c) bunching (chaotic light). τ<sub>c</sub> is the coherence time (the time scale of photon or intensity fluctuations).]]<br />
The above discussion makes it clear that the Hanbury Brown and Twiss (or photon bunching) effect can be entirely described by classical optics. The quantum description of the effect is less intuitive: if one supposes that a thermal or chaotic light source such as a star randomly emits photons, then it is not obvious how the photons "know" that they should arrive at a detector in a correlated (bunched) way. A simple argument suggested by [[Ugo Fano]] [Fano, 1961] captures the essence of the quantum explanation. Consider two points <math>a</math> and <math>b</math> in a source that emit photons detected by two detectors <math>A</math> and <math>B</math> as in the diagram. A joint detection takes place when the photon emitted by <math>a</math> is detected by <math>A</math> and the photon emitted by <math>b</math> is detected by <math>B</math> (red arrows) ''or'' when <math>a</math>'s photon is detected by <math>B</math> and <math>b</math>'s by <math>A</math> (green arrows). The quantum mechanical probability amplitudes for these two possibilities are denoted by <br />
<math>\langle A|a \rangle \langle B|b \rangle</math> and <br />
<math>\langle B|a \rangle \langle A|b \rangle</math> respectively. If the photons are indistinguishable, the two amplitudes interfere constructively to give a joint detection probability greater than that for two independent events. The sum over all possible pairs <math>a, b</math> in the source washes out the interference unless the distance <math>AB</math> is sufficiently small. <br />
<br />
[[File:Two-photon Amplitude.svg|thumb|right|Two source points ''a'' and ''b'' emit photons detected by detectors ''A'' and ''B''. The two colors represent two different ways to detect two photons.]]<br />
<br />
Fano's explanation nicely illustrates the necessity of considering two-particle amplitudes, which are not as intuitive as the more familiar single-particle amplitudes used to interpret most interference effects. This may help to explain why some physicists in the 1950s had difficulty accepting the Hanbury Brown and Twiss result. But the quantum approach is more than just a fancy way to reproduce the classical result: if the photons are replaced by identical fermions such as electrons, the antisymmetry of wave functions under exchange of particles renders the interference destructive, leading to zero joint detection probability for small detector separations. This effect is referred to as antibunching of fermions [Henny, 1999]. The above treatment also explains [[photon antibunching]] [Kimble, 1977]: if the source consists of a single atom, which can only emit one photon at a time, simultaneous detection in two closely spaced detectors is clearly impossible. Antibunching, whether of bosons or of fermions, has no classical wave analog.<br />
<br />
From the point of view of the field of quantum optics, the HBT effect was important to lead physicists (among them [[Roy J. Glauber]] and [[Leonard Mandel]]) to apply quantum electrodynamics to new situations, many of which had never been experimentally studied, and in which classical and quantum predictions differ.<br />
<br />
== See also ==<br />
*[[Bose–Einstein correlations]]<br />
*[[Degree of coherence]]<br />
*[[Timeline of electromagnetism and classical optics]]<br />
<br />
== References ==<br />
<references/><br />
Note that Hanbury Brown is not hyphenated.<br />
<br />
* {{cite journal |author1=E. Brannen |author2=H. Ferguson | title=The question of correlation between photons in coherent light beams | journal=Nature | year=1956 | volume=178 | pages=481–482 | doi=10.1038/178481a0 | issue=4531|bibcode = 1956Natur.178..481B }} – paper which (incorrectly) disputed the existence of the Hanbury Brown and Twiss effect<br />
* {{cite journal |author1=R. Hanbury Brown |author2=R. Q. Twiss | title=A Test of a New Type of Stellar Interferometer on Sirius | journal=Nature | year=1956 | volume=178 | pages=1046–1048 | doi=10.1038/1781046a0 | issue=4541|bibcode = 1956Natur.178.1046H }} – experimental demonstration of the effect<br />
* {{cite journal | author=E. Purcell | title=The Question of Correlation Between Photons in Coherent Light Rays | journal=Nature | year=1956 | volume=178 | pages=1449–1450 | doi=10.1038/1781449a0 | issue=4548|bibcode = 1956Natur.178.1449P }}<br />
* {{cite journal |author1=R. Hanbury Brown |author2=R. Q. Twiss | title=Interferometry of the intensity fluctuations in light. I. Basic theory: the correlation between photons in coherent beams of radiation | journal=Proceedings of the Royal Society A | year=1957 | volume=242 | pages=300–324 | doi=10.1098/rspa.1957.0177 | issue=1230|bibcode = 1957RSPSA.242..300B }} [https://web.archive.org/web/20050124033547/http://www.strw.leidenuniv.nl/~tubbs/classic_papers/hanbury_brown_et_twiss_1957.pdf download as PDF]<br />
* {{cite journal |author1=R. Hanbury Brown |author2=R. Q. Twiss | title=Interferometry of the intensity fluctuations in light. II. An experimental test of the theory for partially coherent light | journal=Proceedings of the Royal Society A | year=1958 | volume=243 | pages=291–319 | doi=10.1098/rspa.1958.0001 | issue=1234|bibcode = 1958RSPSA.243..291B }} [https://web.archive.org/web/20050124033547/http://www.strw.leidenuniv.nl/~tubbs/classic_papers/hanbury_brown_et_twiss_1958a.pdf download as PDF]<br />
*{{cite journal |last=Fano |first=U. |title=Quantum theory of interference effects in the mixing of light from phase independent sources |journal=American Journal of Physics |volume=29 |year=1961 |pages=539–545 |doi=10.1119/1.1937827 |issue=8 |bibcode = 1961AmJPh..29..539F }}<br />
* {{cite journal |author1=B. L. Morgan |author2=L. Mandel | title=Measurement of Photon Bunching in a Thermal Light Beam | journal=Phys. Rev. Lett. | year=1966 | volume=16 | pages=1012–1014 | doi=10.1103/PhysRevLett.16.1012 | issue=22 | bibcode=1966PhRvL..16.1012M|citeseerx=10.1.1.713.7239 }}<br />
*{{Cite journal |last1=Kimble |first1=H. J. |last2=Dagenais|first2=M. |last3=Mandel|first3=L.|title=Photon antibunching in resonance fluorescence |journal=Physical Review Letters |volume=39 |year=1977 |pages=691–695|doi=10.1103/PhysRevLett.39.691 |issue=11 |bibcode=1977PhRvL..39..691K|url=https://authors.library.caltech.edu/6051/1/KIMprl77.pdf }}<br />
*{{Cite journal |last1=Dayan |first1=B. |last2=Parkins |first2=A. S.|last3=Aoki |first3=T.|last4=Ostby |first4=E. P.|last5=Vahala |first5=K. J. |last6=Kimble |first6=H. J.|title=A Photon Turnstile Dynamically Regulated by One Atom |journal=Science |volume=319 |issue=5866 |year=2008 |pages=1062–1065|doi=10.1126/Science.1152261|bibcode = 2008Sci...319.1062D |pmid=18292335|url=https://authors.library.caltech.edu/35067/2/Dayan.SOM.pdf }} – the cavity-QED equivalent for Kimble & Mandel's free-space demonstration of photon antibunching in resonance fluorescence<br />
* {{cite journal |author1=P. Grangier |author2=G. Roger |author3=A. Aspect | title=Experimental Evidence for a Photon Anticorrelation Effect on a Beam Splitter: A New Light on Single-Photon Interferences | journal=Europhysics Letters | year=1986 | volume=1 | pages=173–179 | doi=10.1209/0295-5075/1/4/004 | issue=4|bibcode = 1986EL......1..173G |citeseerx=10.1.1.178.4356 }}<br />
* {{cite journal | author=M. Henny| title=The Fermionic Hanbury Brown and Twiss Experiment | journal=Science | year=1999 | volume=284 | pages=296–298 | doi=10.1126/science.284.5412.296 | pmid=10195890 | issue=5412|bibcode = 1999Sci...284..296H |display-authors=etal| url=https://epub.uni-regensburg.de/3370/1/ScienceHBT.pdf }}<br />
* {{cite book | author=R Hanbury Brown| title=BOFFIN : A Personal Story of the Early Days of Radar, Radio Astronomy and Quantum Optics | publisher=Adam Hilger | year=1991 | isbn=978-0-7503-0130-5}}<br />
* {{cite book | author=Mark P. Silverman | title=More Than One Mystery: Explorations in Quantum Interference | url=https://archive.org/details/morethanonemyste0000silv | url-access=registration | publisher=Springer | year=1995 | isbn=978-0-387-94376-3}}<br />
* {{cite book | author=R Hanbury Brown | title=The intensity interferometer; its application to astronomy | publisher=Wiley | year=1974 |id=ASIN B000LZQD3C | isbn=978-0-470-10797-3}}<br />
* {{cite journal |author1=Y. Bromberg |author2=Y. Lahini |author3=E. Small |author4=Y. Silberberg | title=Hanbury Brown and Twiss Interferometry with Interacting Photons | journal=Nature Photonics| year=2010 | volume=4 | pages=721–726 | doi=10.1038/nphoton.2010.195 | issue=10|bibcode = 2010NaPho...4..721B }}<br />
<br />
== External links ==<br />
* http://adsabs.harvard.edu//full/seri/JApA./0015//0000015.000.html<br />
* http://physicsweb.org/articles/world/15/10/6/1<br />
* https://web.archive.org/web/20070609114114/http://www.du.edu/~jcalvert/astro/starsiz.htm<br />
* http://www.2physics.com/2010/11/hanbury-brown-and-twiss-interferometry.html<br />
*[https://www.becker-hickl.com/applications/antibunching-experiments/ Hanbury-Brown-Twiss Experiment] (Becker & Hickl GmbH, web page)<br />
<br />
[[Category:Quantum optics]]</div>Tomasikhttps://wiki.thor-cost.eu/index.php?title=Femtoscopy&diff=66Femtoscopy2021-05-10T13:09:52Z<p>Tomasik: </p>
<hr />
<div>{{Use dmy dates|date=June 2016}}<br />
In [[physics]], the '''Hanbury Brown and Twiss''' ('''HBT''') '''effect''' is any of a variety of [[correlation]] and anti-correlation effects in the [[intensity (physics)|intensities]] received by two detectors from a beam of particles. HBT effects can generally be attributed to the <ref>https://en.wikipedia.org/wiki/Wave–particle_duality</ref> of the beam, and the results of a given experiment depend on whether the beam is composed of [[fermion]]s or [[boson]]s. Devices which use the effect are commonly called [[intensity interferometer]]s and were originally used in [[astronomy]], although they are also heavily used in the field of [[quantum optics]].<br />
<br />
== History ==<br />
In 1954, [[Robert Hanbury Brown]] and [[Richard Q. Twiss]] introduced the [[intensity interferometer]] concept to [[radio astronomy]] for measuring the tiny angular size of stars, suggesting that it might work with visible light as well.<ref name="BrownTwiss2010">{{cite journal|last1=Brown|first1=R. Hanbury|last2=Twiss|first2=R.Q.|title=A new type of interferometer for use in radio astronomy|journal=[[Philosophical Magazine]]|volume=45|issue=366|year=1954|pages=663–682|issn=1941-5982|doi=10.1080/14786440708520475}}</ref> Soon after they successfully tested that suggestion: in 1956 they published an in-lab experimental mockup using blue light from a [[mercury-vapor lamp]],<ref name="BrownTwiss1956">{{cite journal|last1=Brown|first1=R. Hanbury|last2=Twiss|first2=R. Q.|title=Correlation between Photons in two Coherent Beams of Light|journal=Nature|volume=177|issue=4497|year=1956|pages=27–29|issn=0028-0836|doi=10.1038/177027a0}}</ref> and later in the same year, they applied this technique to measuring the size of [[Sirius]].<ref>{{Cite journal |doi = 10.1038/1781046a0|title = A Test Of A New Type Of Stellar Interferometer On Sirius |journal = Nature|volume = 178|pages = 1046-1048|year = 1956|last1 = Hanbury Brown|first1 = R.|last2 = Twiss|first2 = Dr R.Q.|url = http://www.cmp.caltech.edu/refael/league/hanbury.pdf|bibcode = 1956Natur.178.1046H}}</ref> In the latter experiment, two [[photomultiplier tube]]s, separated by a few meters, were aimed at the star using crude telescopes, and a correlation was observed between the two fluctuating intensities. Just as in the radio studies, the correlation dropped away as they increased the separation (though over meters, instead of kilometers), and they used this information to determine the apparent [[angular size]] of Sirius.<br />
<br />
[[File:Correlation-interferometer.svg|frame|150px|right|An example of an intensity interferometer that would observe no correlation if the light source is a coherent laser beam, and positive correlation if the light source is a filtered one-mode thermal radiation. The theoretical explanation of the difference between the correlations of photon pairs in thermal and in laser beams was first given by [[Roy J. Glauber]], who was awarded the 2005 [[Nobel Prize in Physics]] "for his contribution to the quantum theory of [[Coherence (physics)|optical coherence]]".]]<br />
<br />
This result was met with much skepticism in the physics community. The radio astronomy result was justified by [[Maxwell's equations]], but there were concerns that the effect should break down at optical wavelengths, since the light would be quantised into a relatively small number of [[photon]]s that induce discrete [[photoelectron]]s in the detectors. Many [[physicists]] worried that the correlation was inconsistent with the laws of thermodynamics. Some even claimed that the effect violated the [[uncertainty principle]]. Hanbury Brown and Twiss resolved the dispute in a neat series of articles (see [[#References|References]] below) that demonstrated, first, that wave transmission in quantum optics had exactly the same mathematical form as Maxwell's equations, albeit with an additional noise term due to quantisation at the detector, and second, that according to Maxwell's equations, intensity interferometry should work. Others, such as [[Edward Mills Purcell]] immediately supported the technique, pointing out that the clumping of bosons was simply a manifestation of an effect already known in [[statistical mechanics]]. After a number of experiments, the whole physics community agreed that the observed effect was real.<br />
<br />
The original experiment used the fact that two bosons tend to arrive at two separate detectors at the same time. Morgan and Mandel used a thermal photon source to create a dim beam of photons and observed the tendency of the photons to arrive at the same time on a single detector. Both of these effects used the wave nature of light to create a correlation in arrival time – if a single photon beam is split into two beams, then the particle nature of light requires that each photon is only observed at a single detector, and so an anti-correlation was observed in 1977 by [[H. Jeff Kimble]].<ref>{{Cite journal |doi = 10.1103/PhysRevLett.39.691|title = Photon Antibunching in Resonance Fluorescence|journal = Physical Review Letters|volume = 39|issue = 11|pages = 691–695|year = 1977|last1 = Kimble|first1 = H. J.|last2 = Dagenais|first2 = M.|last3 = Mandel|first3 = L.|url = https://authors.library.caltech.edu/6051/1/KIMprl77.pdf|bibcode = 1977PhRvL..39..691K}}</ref> Finally, bosons have a tendency to clump together, giving rise to [[Bose–Einstein correlations]], while fermions due to the [[Pauli exclusion principle]], tend to spread apart, leading to Fermi–Dirac (anti)correlations. Bose–Einstein correlations have been observed between pions, kaons and photons, and Fermi–Dirac (anti)correlations between protons, neutrons and electrons. For a general introduction in this field, see the textbook on Bose–Einstein correlations by [[Richard M. Weiner]]<ref>Richard M. Weiner, Introduction to Bose–Einstein Correlations and Subatomic Interferometry, John Wiley, 2000.</ref> A difference in repulsion of [[Bose–Einstein condensate]] in the "trap-and-free fall" analogy of the HBT effect<ref>[https://arxiv.org/abs/cond-mat/0612278 Comparison of the Hanbury Brown-Twiss effect for bosons and fermions].</ref> affects comparison.<br />
<br />
Also, in the field of [[particle physics]], [[Gerson Goldhaber|Goldhaber]] et al. performed an experiment in 1959 in [[University of California, Berkeley|Berkeley]] and found an unexpected angular correlation among identical [[pion]]s, discovering the [[rho meson|ρ<sup>0</sup> resonance]], by means of <math>\rho^0 \to \pi^-\pi^+</math> decay.<ref><br />
{{cite journal<br />
|author1=G. Goldhaber<br />
|author2=W. B. Fowler<br />
|author3=S. Goldhaber<br />
|author4=T. F. Hoang<br />
|author5=T. E. Kalogeropoulos<br />
|author6=W. M. Powell<br />
|year=1959<br />
|title=Pion-pion correlations in antiproton annihilation events<br />
|journal=Phys. Rev. Lett.<br />
|volume=3 |issue=4 |page=181<br />
|doi=10.1103/PhysRevLett.3.181<br />
|bibcode=1959PhRvL...3..181G|url=http://www.escholarship.org/uc/item/7nw6p1br<br />
}}</ref> From then on, the HBT technique started to be used by the [[High-energy nuclear physics|heavy-ion community]] to determine the space–time dimensions of the particle emission source for heavy-ion collisions. For recent developments in this field, see for example the review article by Lisa.<ref>M. Lisa, et al., ''Annu. Rev. Nucl. Part. Sci.'' '''55''', p. 357 (2005), [https://arxiv.org/abs/nucl-ex/0505014 ArXiv 0505014].</ref><br />
<br />
== Wave mechanics ==<br />
The HBT effect can, in fact, be predicted solely by treating the incident [[electromagnetic radiation]] as a classical [[wave]]. Suppose we have a monochromatic wave with frequency <math>\omega</math> on two detectors, with an amplitude <math>E(t)</math> that varies on timescales slower than the wave period <math>2\pi/\omega</math>. (Such a wave might be produced from a very distant [[point source]] with a fluctuating intensity.)<br />
<br />
Since the detectors are separated, say the second detector gets the signal delayed by a time <math>\tau</math>, or equivalently, a [[Phase (waves)|phase]] <math>\phi = \omega\tau</math>; that is,<br />
<br />
:<math> E_1(t) = E(t) \sin(\omega t),</math><br />
<br />
:<math> E_2(t) = E(t - \tau) \sin(\omega t - \phi).</math><br />
<br />
The intensity recorded by each detector is the square of the wave amplitude, averaged over a timescale that is long compared to the wave period <math>2\pi/\omega</math> but short compared to the fluctuations in <math>E(t)</math>:<br />
<br />
:<math><br />
\begin{align}<br />
i_1(t) &= \overline{E_1(t)^2} = \overline{E(t)^2 \sin^2(\omega t)} = \tfrac{1}{2} E(t)^2, \\<br />
i_2(t) &= \overline{E_2(t)^2} = \overline{E(t - \tau)^2 \sin^2(\omega t - \phi)} = \tfrac{1}{2} E(t - \tau)^2,<br />
\end{align}<br />
</math><br />
where the overline indicates this time averaging. For wave frequencies above a few [[Terahertz radiation|terahertz]] (wave periods less than a [[picosecond]]), such a time averaging is unavoidable, since detectors such as [[photodiode]]s and [[photomultiplier tube]]s cannot produce photocurrents that vary on such short timescales.<br />
<br />
The correlation function <math>\langle i_1 i_2 \rangle(\tau)</math> of these time-averaged intensities can then be computed:<br />
:<math><br />
\begin{align}<br />
\langle i_1 i_2 \rangle(\tau) &= \lim_{T \to \infty} \frac{1}{T} \int\limits_0^T i_1(t) i_2(t)\, \mathrm{d}t \\<br />
&= \lim_{T \to \infty} \frac{1}{T} \int\limits_0^T \tfrac{1}{4} E(t)^2 E(t-\tau)^2 \, \mathrm{d}t.<br />
\end{align}<br />
</math><br />
<br />
Most modern schemes actually measure the correlation in intensity fluctuations at the two detectors, but it is not too difficult to see that if the intensities are correlated, then the fluctuations <math>\Delta i = i - \langle i\rangle</math>, where <math>\langle i\rangle</math> is the average intensity, ought to be correlated, since<br />
:<math>\begin{align}<br />
\langle\Delta i_1\Delta i_2\rangle &= \big\langle(i_1 - \langle i_1\rangle)(i_2 - \langle i_2\rangle)\big\rangle = \langle i_1 i_2\rangle - \big\langle i_1\langle i_2\rangle\big\rangle - \big\langle i_2\langle i_1\rangle\big\rangle + \langle i_1\rangle \langle i_2\rangle \\<br />
&=\langle i_1 i_2\rangle -\langle i_1\rangle \langle i_2\rangle.<br />
\end{align}</math><br />
<br />
In the particular case that <math>E(t)</math> consists mainly of a steady field <math>E_0</math> with a small sinusoidally varying component <math>\delta E \sin(\Omega t)</math>, the time-averaged intensities are<br />
:<math><br />
\begin{align}<br />
i_1(t) &= \tfrac{1}{2} E_0^2 + E_0\,\delta E \sin(\Omega t) + \mathcal{O}(\delta E^2), \\<br />
i_2(t) &= \tfrac{1}{2} E_0^2 + E_0\,\delta E \sin(\Omega t-\Phi) + \mathcal{O}(\delta E^2),<br />
\end{align}<br />
</math><br />
with <math>\Phi = \Omega \tau</math>, and <math>\mathcal{O}(\delta E^2)</math> indicates terms proportional to <math>(\delta E)^2</math>, which are small and may be ignored.<br />
<br />
The correlation function of these two intensities is then<br />
:<math><br />
\begin{align}<br />
\langle \Delta i_1 \Delta i_2 \rangle(\tau) &= \lim_{T \to \infty} \frac{(E_0\delta E)^2}{T} \int\limits_0^T \sin(\Omega t) \sin(\Omega t - \Phi) \, \mathrm{d}t \\<br />
&= \tfrac{1}{2} (E_0 \delta E)^2 \cos(\Omega\tau),<br />
\end{align}<br />
</math><br />
showing a sinusoidal dependence on the delay <math>\tau</math> between the two detectors.<br />
<br />
== Quantum interpretation ==<br />
[[File:Photon bunching.svg|thumb|400px|Photon detections as a function of time for a) antibunching (e.g. light emitted from a single atom), b) random (e.g. a coherent state, laser beam), and c) bunching (chaotic light). τ<sub>c</sub> is the coherence time (the time scale of photon or intensity fluctuations).]]<br />
The above discussion makes it clear that the Hanbury Brown and Twiss (or photon bunching) effect can be entirely described by classical optics. The quantum description of the effect is less intuitive: if one supposes that a thermal or chaotic light source such as a star randomly emits photons, then it is not obvious how the photons "know" that they should arrive at a detector in a correlated (bunched) way. A simple argument suggested by [[Ugo Fano]] [Fano, 1961] captures the essence of the quantum explanation. Consider two points <math>a</math> and <math>b</math> in a source that emit photons detected by two detectors <math>A</math> and <math>B</math> as in the diagram. A joint detection takes place when the photon emitted by <math>a</math> is detected by <math>A</math> and the photon emitted by <math>b</math> is detected by <math>B</math> (red arrows) ''or'' when <math>a</math>'s photon is detected by <math>B</math> and <math>b</math>'s by <math>A</math> (green arrows). The quantum mechanical probability amplitudes for these two possibilities are denoted by <br />
<math>\langle A|a \rangle \langle B|b \rangle</math> and <br />
<math>\langle B|a \rangle \langle A|b \rangle</math> respectively. If the photons are indistinguishable, the two amplitudes interfere constructively to give a joint detection probability greater than that for two independent events. The sum over all possible pairs <math>a, b</math> in the source washes out the interference unless the distance <math>AB</math> is sufficiently small. <br />
<br />
[[File:Two-photon Amplitude.svg|thumb|right|Two source points ''a'' and ''b'' emit photons detected by detectors ''A'' and ''B''. The two colors represent two different ways to detect two photons.]]<br />
<br />
Fano's explanation nicely illustrates the necessity of considering two-particle amplitudes, which are not as intuitive as the more familiar single-particle amplitudes used to interpret most interference effects. This may help to explain why some physicists in the 1950s had difficulty accepting the Hanbury Brown and Twiss result. But the quantum approach is more than just a fancy way to reproduce the classical result: if the photons are replaced by identical fermions such as electrons, the antisymmetry of wave functions under exchange of particles renders the interference destructive, leading to zero joint detection probability for small detector separations. This effect is referred to as antibunching of fermions [Henny, 1999]. The above treatment also explains [[photon antibunching]] [Kimble, 1977]: if the source consists of a single atom, which can only emit one photon at a time, simultaneous detection in two closely spaced detectors is clearly impossible. Antibunching, whether of bosons or of fermions, has no classical wave analog.<br />
<br />
From the point of view of the field of quantum optics, the HBT effect was important to lead physicists (among them [[Roy J. Glauber]] and [[Leonard Mandel]]) to apply quantum electrodynamics to new situations, many of which had never been experimentally studied, and in which classical and quantum predictions differ.<br />
<br />
== See also ==<br />
*[[Bose–Einstein correlations]]<br />
*[[Degree of coherence]]<br />
*[[Timeline of electromagnetism and classical optics]]<br />
<br />
== References ==<br />
<references/><br />
Note that Hanbury Brown is not hyphenated.<br />
<br />
* {{cite journal |author1=E. Brannen |author2=H. Ferguson | title=The question of correlation between photons in coherent light beams | journal=Nature | year=1956 | volume=178 | pages=481–482 | doi=10.1038/178481a0 | issue=4531|bibcode = 1956Natur.178..481B }} – paper which (incorrectly) disputed the existence of the Hanbury Brown and Twiss effect<br />
* {{cite journal |author1=R. Hanbury Brown |author2=R. Q. Twiss | title=A Test of a New Type of Stellar Interferometer on Sirius | journal=Nature | year=1956 | volume=178 | pages=1046–1048 | doi=10.1038/1781046a0 | issue=4541|bibcode = 1956Natur.178.1046H }} – experimental demonstration of the effect<br />
* {{cite journal | author=E. Purcell | title=The Question of Correlation Between Photons in Coherent Light Rays | journal=Nature | year=1956 | volume=178 | pages=1449–1450 | doi=10.1038/1781449a0 | issue=4548|bibcode = 1956Natur.178.1449P }}<br />
* {{cite journal |author1=R. Hanbury Brown |author2=R. Q. Twiss | title=Interferometry of the intensity fluctuations in light. I. Basic theory: the correlation between photons in coherent beams of radiation | journal=Proceedings of the Royal Society A | year=1957 | volume=242 | pages=300–324 | doi=10.1098/rspa.1957.0177 | issue=1230|bibcode = 1957RSPSA.242..300B }} [https://web.archive.org/web/20050124033547/http://www.strw.leidenuniv.nl/~tubbs/classic_papers/hanbury_brown_et_twiss_1957.pdf download as PDF]<br />
* {{cite journal |author1=R. Hanbury Brown |author2=R. Q. Twiss | title=Interferometry of the intensity fluctuations in light. II. An experimental test of the theory for partially coherent light | journal=Proceedings of the Royal Society A | year=1958 | volume=243 | pages=291–319 | doi=10.1098/rspa.1958.0001 | issue=1234|bibcode = 1958RSPSA.243..291B }} [https://web.archive.org/web/20050124033547/http://www.strw.leidenuniv.nl/~tubbs/classic_papers/hanbury_brown_et_twiss_1958a.pdf download as PDF]<br />
*{{cite journal |last=Fano |first=U. |title=Quantum theory of interference effects in the mixing of light from phase independent sources |journal=American Journal of Physics |volume=29 |year=1961 |pages=539–545 |doi=10.1119/1.1937827 |issue=8 |bibcode = 1961AmJPh..29..539F }}<br />
* {{cite journal |author1=B. L. Morgan |author2=L. Mandel | title=Measurement of Photon Bunching in a Thermal Light Beam | journal=Phys. Rev. Lett. | year=1966 | volume=16 | pages=1012–1014 | doi=10.1103/PhysRevLett.16.1012 | issue=22 | bibcode=1966PhRvL..16.1012M|citeseerx=10.1.1.713.7239 }}<br />
*{{Cite journal |last1=Kimble |first1=H. J. |last2=Dagenais|first2=M. |last3=Mandel|first3=L.|title=Photon antibunching in resonance fluorescence |journal=Physical Review Letters |volume=39 |year=1977 |pages=691–695|doi=10.1103/PhysRevLett.39.691 |issue=11 |bibcode=1977PhRvL..39..691K|url=https://authors.library.caltech.edu/6051/1/KIMprl77.pdf }}<br />
*{{Cite journal |last1=Dayan |first1=B. |last2=Parkins |first2=A. S.|last3=Aoki |first3=T.|last4=Ostby |first4=E. P.|last5=Vahala |first5=K. J. |last6=Kimble |first6=H. J.|title=A Photon Turnstile Dynamically Regulated by One Atom |journal=Science |volume=319 |issue=5866 |year=2008 |pages=1062–1065|doi=10.1126/Science.1152261|bibcode = 2008Sci...319.1062D |pmid=18292335|url=https://authors.library.caltech.edu/35067/2/Dayan.SOM.pdf }} – the cavity-QED equivalent for Kimble & Mandel's free-space demonstration of photon antibunching in resonance fluorescence<br />
* {{cite journal |author1=P. Grangier |author2=G. Roger |author3=A. Aspect | title=Experimental Evidence for a Photon Anticorrelation Effect on a Beam Splitter: A New Light on Single-Photon Interferences | journal=Europhysics Letters | year=1986 | volume=1 | pages=173–179 | doi=10.1209/0295-5075/1/4/004 | issue=4|bibcode = 1986EL......1..173G |citeseerx=10.1.1.178.4356 }}<br />
* {{cite journal | author=M. Henny| title=The Fermionic Hanbury Brown and Twiss Experiment | journal=Science | year=1999 | volume=284 | pages=296–298 | doi=10.1126/science.284.5412.296 | pmid=10195890 | issue=5412|bibcode = 1999Sci...284..296H |display-authors=etal| url=https://epub.uni-regensburg.de/3370/1/ScienceHBT.pdf }}<br />
* {{cite book | author=R Hanbury Brown| title=BOFFIN : A Personal Story of the Early Days of Radar, Radio Astronomy and Quantum Optics | publisher=Adam Hilger | year=1991 | isbn=978-0-7503-0130-5}}<br />
* {{cite book | author=Mark P. Silverman | title=More Than One Mystery: Explorations in Quantum Interference | url=https://archive.org/details/morethanonemyste0000silv | url-access=registration | publisher=Springer | year=1995 | isbn=978-0-387-94376-3}}<br />
* {{cite book | author=R Hanbury Brown | title=The intensity interferometer; its application to astronomy | publisher=Wiley | year=1974 |id=ASIN B000LZQD3C | isbn=978-0-470-10797-3}}<br />
* {{cite journal |author1=Y. Bromberg |author2=Y. Lahini |author3=E. Small |author4=Y. Silberberg | title=Hanbury Brown and Twiss Interferometry with Interacting Photons | journal=Nature Photonics| year=2010 | volume=4 | pages=721–726 | doi=10.1038/nphoton.2010.195 | issue=10|bibcode = 2010NaPho...4..721B }}<br />
<br />
== External links ==<br />
* http://adsabs.harvard.edu//full/seri/JApA./0015//0000015.000.html<br />
* http://physicsweb.org/articles/world/15/10/6/1<br />
* https://web.archive.org/web/20070609114114/http://www.du.edu/~jcalvert/astro/starsiz.htm<br />
* http://www.2physics.com/2010/11/hanbury-brown-and-twiss-interferometry.html<br />
*[https://www.becker-hickl.com/applications/antibunching-experiments/ Hanbury-Brown-Twiss Experiment] (Becker & Hickl GmbH, web page)<br />
<br />
[[Category:Quantum optics]]</div>Tomasikhttps://wiki.thor-cost.eu/index.php?title=Wave%E2%80%93particle_duality&diff=65Wave–particle duality2021-05-10T13:07:23Z<p>Tomasik: Created page with "[https://en.wikipedia.org/wiki/Wave–particle_duality]"</p>
<hr />
<div>[https://en.wikipedia.org/wiki/Wave–particle_duality]</div>Tomasikhttps://wiki.thor-cost.eu/index.php?title=Parton_energy_loss&diff=51Parton energy loss2018-10-29T22:05:21Z<p>Tomasik: Created page with "Hard [https://en.wikipedia.org/wiki/Parton_(particle_physics) partons] which traverse the deconfined medium loose energy towards the bulk matter."</p>
<hr />
<div>Hard [https://en.wikipedia.org/wiki/Parton_(particle_physics) partons] which traverse the deconfined medium loose energy towards the bulk matter.</div>Tomasikhttps://wiki.thor-cost.eu/index.php?title=Jets&diff=50Jets2018-10-29T22:04:05Z<p>Tomasik: Created page with "Jets which are created in nuclear collisions serve as probes of the hot and dense matter. Their production is quenched because of the parton energy loss."</p>
<hr />
<div>Jets which are created in nuclear collisions serve as probes of the hot and dense matter. Their production is quenched because of the [[parton energy loss]].</div>Tomasikhttps://wiki.thor-cost.eu/index.php?title=Heavy_quarks&diff=49Heavy quarks2018-10-29T22:02:55Z<p>Tomasik: Created page with "Heavy quarks are specific."</p>
<hr />
<div>Heavy quarks are specific.</div>Tomasikhttps://wiki.thor-cost.eu/index.php?title=Strangeness_production&diff=48Strangeness production2018-10-29T22:02:36Z<p>Tomasik: Created page with "Strangeness production is enhanced in heavy-ion collisions in comparison to proton-proton collisions."</p>
<hr />
<div>Strangeness production is enhanced in heavy-ion collisions in comparison to proton-proton collisions.</div>Tomasikhttps://wiki.thor-cost.eu/index.php?title=Baryon_number_fluctuations&diff=47Baryon number fluctuations2018-10-29T22:01:41Z<p>Tomasik: </p>
<hr />
<div>The baryon number fluctuations are related to [[fluctuations of conserved charges]].</div>Tomasikhttps://wiki.thor-cost.eu/index.php?title=Baryon_number_fluctuations&diff=46Baryon number fluctuations2018-10-29T22:01:25Z<p>Tomasik: </p>
<hr />
<div>The baryon number fluctuations are related to [fluctuations of conserved charges].</div>Tomasikhttps://wiki.thor-cost.eu/index.php?title=Baryon_number_fluctuations&diff=45Baryon number fluctuations2018-10-29T22:01:02Z<p>Tomasik: Created page with "The baryon number fluctuations are related to fluctuations of conserved charges."</p>
<hr />
<div>The baryon number fluctuations are related to fluctuations of conserved charges.</div>Tomasikhttps://wiki.thor-cost.eu/index.php?title=Femtoscopy&diff=44Femtoscopy2018-10-29T22:00:18Z<p>Tomasik: Created page with "Femtoscopy allows to observe the space-time structures and their sizes. It uses momentum correlations."</p>
<hr />
<div>Femtoscopy allows to observe the space-time structures and their sizes. It uses momentum correlations.</div>Tomasikhttps://wiki.thor-cost.eu/index.php?title=Collective_expansion&diff=43Collective expansion2018-10-29T21:59:31Z<p>Tomasik: Created page with "Effects of collective expansion can be seen in spectra of hadrons."</p>
<hr />
<div>Effects of collective expansion can be seen in spectra of hadrons.</div>Tomasikhttps://wiki.thor-cost.eu/index.php?title=Transport_models&diff=42Transport models2018-10-29T21:58:55Z<p>Tomasik: Created page with "Transport models effectively solve the [https://en.wikipedia.org/wiki/Boltzmann_equation Boltzmann equation]."</p>
<hr />
<div>Transport models effectively solve the [https://en.wikipedia.org/wiki/Boltzmann_equation Boltzmann equation].</div>Tomasikhttps://wiki.thor-cost.eu/index.php?title=Hydrodynamics&diff=41Hydrodynamics2018-10-29T21:58:01Z<p>Tomasik: Created page with "[https://en.wikipedia.org/wiki/Fluid_dynamics Hydrodynamics] is an effective theory which can very successfully be used for the description of heavy-ion dynamics."</p>
<hr />
<div>[https://en.wikipedia.org/wiki/Fluid_dynamics Hydrodynamics] is an effective theory which can very successfully be used for the description of heavy-ion dynamics.</div>Tomasikhttps://wiki.thor-cost.eu/index.php?title=Parton_distribution_functions&diff=40Parton distribution functions2018-10-29T21:48:58Z<p>Tomasik: Created page with "[https://en.wikipedia.org/wiki/Parton_(particle_physics)#Parton_distribution_functions Parton distribution functions] can be changed if nuclei collide instead of protons."</p>
<hr />
<div>[https://en.wikipedia.org/wiki/Parton_(particle_physics)#Parton_distribution_functions Parton distribution functions] can be changed if nuclei collide instead of protons.</div>Tomasikhttps://wiki.thor-cost.eu/index.php?title=Glauber_model&diff=39Glauber model2018-10-29T21:47:24Z<p>Tomasik: Created page with "Glauber model is a simple model for the description of the initial stage of a heavy-ion collisions. For the average quantities it can be formulated as optical Glauber model. I..."</p>
<hr />
<div>Glauber model is a simple model for the description of the initial stage of a heavy-ion collisions. For the average quantities it can be formulated as optical Glauber model. In simulations that require event-by-event fluctuations one must use a Glauber Monte Carlo.</div>Tomasikhttps://wiki.thor-cost.eu/index.php?title=Statistical_model&diff=38Statistical model2018-10-29T21:44:34Z<p>Tomasik: Created page with "Statistical model is very successful in describing the production of particles."</p>
<hr />
<div>Statistical model is very successful in describing the production of particles.</div>Tomasikhttps://wiki.thor-cost.eu/index.php?title=Transport_coefficients&diff=37Transport coefficients2018-10-29T21:43:30Z<p>Tomasik: Created page with "Transport coefficients, like [https://en.wikipedia.org/wiki/Viscosity shear viscosity] and [https://en.wikipedia.org/wiki/Volume_viscosity bulk viscosity], are important for t..."</p>
<hr />
<div>Transport coefficients, like [https://en.wikipedia.org/wiki/Viscosity shear viscosity] and [https://en.wikipedia.org/wiki/Volume_viscosity bulk viscosity], are important for the description of fireball expansion in heavy-ion collisions.</div>Tomasikhttps://wiki.thor-cost.eu/index.php?title=Fluctuations_of_conserved_charges&diff=36Fluctuations of conserved charges2018-10-29T21:41:06Z<p>Tomasik: Created page with "The fluctuations of different orders can be expressed with the help of susceptibilities which are derivatives of the QCD partition function."</p>
<hr />
<div>The fluctuations of different orders can be expressed with the help of susceptibilities which are derivatives of the QCD partition function.</div>Tomasikhttps://wiki.thor-cost.eu/index.php?title=Lattice_qcd&diff=35Lattice qcd2018-10-29T21:39:27Z<p>Tomasik: Replaced content with "obsolete"</p>
<hr />
<div>obsolete</div>Tomasikhttps://wiki.thor-cost.eu/index.php?title=Lattice_QCD&diff=34Lattice QCD2018-10-29T21:38:39Z<p>Tomasik: </p>
<hr />
<div>Lattice QCD is a formulation of [https://en.wikipedia.org/wiki/Statistical_physics Statistical physics] where the matter is described by QCD Lagrangian.</div>Tomasikhttps://wiki.thor-cost.eu/index.php?title=Main_Page&diff=33Main Page2018-10-29T21:38:24Z<p>Tomasik: </p>
<hr />
<div><br />
== Summary of CA15213 Theory of hot matter and relativistic heavy-ion collisions ==<br />
<br />
<br />
[[File:Logo01.jpg|200 px|left]]<br />
[[File:THOR poster print c.jpg|150 px|right]]<br />
<br />
Matter under extreme conditions in terms of temperature and density, as in the early Universe or in compact stellar objects (e.g. neutron stars) can be created and studied with the help of relativistic heavy ion collisions. Scientifically, the main aim is to explore and reconstruct the matter’s transport properties, phase structure, in-medium properties of hadrons and active degrees of freedom of Quantum Chromodynamics (QCD) from the experimental measurements of individual quantities. While the experimental activities are organised and optimised in large international collaborations, there is no such structure for theoretical activities.<br />
<br />
The proposed COST Action “Theory of hot matter and relativistic heavy-ion collisions” (THOR) creates a theoretical community platform as counterpart to the ongoing vigorous experimental activities. THOR will for the first time allow to fully exploit Europe’s exceptional potential in this field of theoretical research. THOR will pioneer novel approaches to the theoretical understanding of the properties of QCD from first principles and on the interpretations of these properties by effective models and numerical simulations of the system’s evolution. By this, THOR will provide new insights on the paramount questions of the field. Therefore THOR aims at bringing together excellent researchers in order to pinpoint and discuss the challenges that the field meets currently and in the near future for creating a vibrant, innovative and world- leading pan-European research environment.<br />
<br />
From this page one can continue to pages which discuss specific topics:<br />
<br />
<br />
= Strongly interacting matter =<br />
* [[Lattice QCD]]<br />
* [[Effective theories]]<br />
* [[Equation of State]]<br />
* [[Phase diagram]]<br />
* [[Fluctuations of conserved charges]]<br />
* [[Transport coefficients]]<br />
* [[Statistical model]]<br />
<br />
=Dynamics of heavy-ion collisions=<br />
* [[Glauber model]]<br />
* [[Parton distribution functions]]<br />
* [[Hydrodynamics]]<br />
* [[Transport models]]<br />
* [[Collective expansion]]<br />
* [[Femtoscopy]]<br />
* [[Baryon number fluctuations]]<br />
* [[Strangeness production]]<br />
* [[Heavy quarks]]<br />
* [[Jets]]<br />
* [[Parton energy loss]]<br />
<br />
<br />
<br />
Consult the [https://meta.wikimedia.org/wiki/Help:Contents User's Guide] for information on using the wiki software.<br />
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== Getting started ==<br />
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* [https://www.mediawiki.org/wiki/Special:MyLanguage/Localisation#Translation_resources Localise MediaWiki for your language]<br />
* [https://www.mediawiki.org/wiki/Special:MyLanguage/Manual:Combating_spam Learn how to combat spam on your wiki]</div>Tomasikhttps://wiki.thor-cost.eu/index.php?title=Main_Page&diff=32Main Page2018-10-29T21:35:52Z<p>Tomasik: </p>
<hr />
<div><br />
== Summary of CA15213 Theory of hot matter and relativistic heavy-ion collisions ==<br />
<br />
<br />
[[File:Logo01.jpg|200 px|left]]<br />
[[File:THOR poster print c.jpg|150 px|right]]<br />
<br />
Matter under extreme conditions in terms of temperature and density, as in the early Universe or in compact stellar objects (e.g. neutron stars) can be created and studied with the help of relativistic heavy ion collisions. Scientifically, the main aim is to explore and reconstruct the matter’s transport properties, phase structure, in-medium properties of hadrons and active degrees of freedom of Quantum Chromodynamics (QCD) from the experimental measurements of individual quantities. While the experimental activities are organised and optimised in large international collaborations, there is no such structure for theoretical activities.<br />
<br />
The proposed COST Action “Theory of hot matter and relativistic heavy-ion collisions” (THOR) creates a theoretical community platform as counterpart to the ongoing vigorous experimental activities. THOR will for the first time allow to fully exploit Europe’s exceptional potential in this field of theoretical research. THOR will pioneer novel approaches to the theoretical understanding of the properties of QCD from first principles and on the interpretations of these properties by effective models and numerical simulations of the system’s evolution. By this, THOR will provide new insights on the paramount questions of the field. Therefore THOR aims at bringing together excellent researchers in order to pinpoint and discuss the challenges that the field meets currently and in the near future for creating a vibrant, innovative and world- leading pan-European research environment.<br />
<br />
From this page one can continue to pages which discuss specific topics:<br />
<br />
<br />
= Strongly interacting matter =<br />
* [[Lattice qcd]]<br />
* [[Effective theories]]<br />
* [[Equation of State]]<br />
* [[Phase diagram]]<br />
* [[Fluctuations of conserved charges]]<br />
* [[Transport coefficients]]<br />
* [[Statistical model]]<br />
<br />
=Dynamics of heavy-ion collisions=<br />
* [[Glauber model]]<br />
* [[Parton distribution functions]]<br />
* [[Hydrodynamics]]<br />
* [[Transport models]]<br />
* [[Collective expansion]]<br />
* [[Femtoscopy]]<br />
* [[Baryon number fluctuations]]<br />
* [[Strangeness production]]<br />
* [[Heavy quarks]]<br />
* [[Jets]]<br />
* [[Parton energy loss]]<br />
<br />
<br />
<br />
Consult the [https://meta.wikimedia.org/wiki/Help:Contents User's Guide] for information on using the wiki software.<br />
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== Getting started ==<br />
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* [https://www.mediawiki.org/wiki/Special:MyLanguage/Localisation#Translation_resources Localise MediaWiki for your language]<br />
* [https://www.mediawiki.org/wiki/Special:MyLanguage/Manual:Combating_spam Learn how to combat spam on your wiki]</div>Tomasikhttps://wiki.thor-cost.eu/index.php?title=Lattice_QCD&diff=31Lattice QCD2018-10-29T21:35:35Z<p>Tomasik: Created page with "jhh"</p>
<hr />
<div>jhh</div>Tomasikhttps://wiki.thor-cost.eu/index.php?title=Main_Page&diff=30Main Page2018-10-29T21:35:11Z<p>Tomasik: </p>
<hr />
<div><br />
== Summary of CA15213 Theory of hot matter and relativistic heavy-ion collisions ==<br />
<br />
<br />
[[File:Logo01.jpg|200 px|left]]<br />
[[File:THOR poster print c.jpg|150 px|right]]<br />
<br />
Matter under extreme conditions in terms of temperature and density, as in the early Universe or in compact stellar objects (e.g. neutron stars) can be created and studied with the help of relativistic heavy ion collisions. Scientifically, the main aim is to explore and reconstruct the matter’s transport properties, phase structure, in-medium properties of hadrons and active degrees of freedom of Quantum Chromodynamics (QCD) from the experimental measurements of individual quantities. While the experimental activities are organised and optimised in large international collaborations, there is no such structure for theoretical activities.<br />
<br />
The proposed COST Action “Theory of hot matter and relativistic heavy-ion collisions” (THOR) creates a theoretical community platform as counterpart to the ongoing vigorous experimental activities. THOR will for the first time allow to fully exploit Europe’s exceptional potential in this field of theoretical research. THOR will pioneer novel approaches to the theoretical understanding of the properties of QCD from first principles and on the interpretations of these properties by effective models and numerical simulations of the system’s evolution. By this, THOR will provide new insights on the paramount questions of the field. Therefore THOR aims at bringing together excellent researchers in order to pinpoint and discuss the challenges that the field meets currently and in the near future for creating a vibrant, innovative and world- leading pan-European research environment.<br />
<br />
From this page one can continue to pages which discuss specific topics:<br />
<br />
<br />
= Strongly interacting matter =<br />
* [[Lattice QCD]]<br />
* [[Effective theories]]<br />
* [[Equation of State]]<br />
* [[Phase diagram]]<br />
* [[Fluctuations of conserved charges]]<br />
* [[Transport coefficients]]<br />
* [[Statistical model]]<br />
<br />
=Dynamics of heavy-ion collisions=<br />
* [[Glauber model]]<br />
* [[Parton distribution functions]]<br />
* [[Hydrodynamics]]<br />
* [[Transport models]]<br />
* [[Collective expansion]]<br />
* [[Femtoscopy]]<br />
* [[Baryon number fluctuations]]<br />
* [[Strangeness production]]<br />
* [[Heavy quarks]]<br />
* [[Jets]]<br />
* [[Parton energy loss]]<br />
<br />
<br />
<br />
Consult the [https://meta.wikimedia.org/wiki/Help:Contents User's Guide] for information on using the wiki software.<br />
<br />
== Getting started ==<br />
* [https://www.mediawiki.org/wiki/Special:MyLanguage/Manual:Configuration_settings Configuration settings list]<br />
* [https://www.mediawiki.org/wiki/Special:MyLanguage/Manual:FAQ MediaWiki FAQ]<br />
* [https://lists.wikimedia.org/mailman/listinfo/mediawiki-announce MediaWiki release mailing list]<br />
* [https://www.mediawiki.org/wiki/Special:MyLanguage/Localisation#Translation_resources Localise MediaWiki for your language]<br />
* [https://www.mediawiki.org/wiki/Special:MyLanguage/Manual:Combating_spam Learn how to combat spam on your wiki]</div>Tomasikhttps://wiki.thor-cost.eu/index.php?title=Phase_diagram&diff=29Phase diagram2018-10-29T21:34:32Z<p>Tomasik: Created page with "[https://en.wikipedia.org/wiki/Phase_diagram Phase diagram] of strongly interacting matter cannot currently be calculated from first principles. Calculation with the help of [..."</p>
<hr />
<div>[https://en.wikipedia.org/wiki/Phase_diagram Phase diagram] of strongly interacting matter cannot currently be calculated from first principles. Calculation with the help of [[Lattice QCD]] is impossible due to the [[sign problem]].</div>Tomasikhttps://wiki.thor-cost.eu/index.php?title=Equation_of_State&diff=28Equation of State2018-10-29T21:32:26Z<p>Tomasik: Created page with "This page will be specifically about the Equation of State of strongly interacting matter."</p>
<hr />
<div>This page will be specifically about the Equation of State of strongly interacting matter.</div>Tomasikhttps://wiki.thor-cost.eu/index.php?title=Effective_theories&diff=27Effective theories2018-10-29T21:31:13Z<p>Tomasik: Created page with "There are many effective theories which capture the symmetries of the quantum chromodynamics and are easier to handle. Examples are * [https://en.wikipedia.org/wiki/Nambu–Jo..."</p>
<hr />
<div>There are many effective theories which capture the symmetries of the quantum chromodynamics and are easier to handle. Examples are<br />
* [https://en.wikipedia.org/wiki/Nambu–Jona-Lasinio_model Nambu Jona Lasinio model]<br />
* [[Polyakov Nambu Jona-Lasinio model]]<br />
* [https://en.wikipedia.org/wiki/Chiral_perturbation_theory Chiral perturbation theory]<br />
and others...</div>Tomasikhttps://wiki.thor-cost.eu/index.php?title=Main_Page&diff=26Main Page2018-10-29T21:27:09Z<p>Tomasik: </p>
<hr />
<div><br />
== Summary of CA15213 Theory of hot matter and relativistic heavy-ion collisions ==<br />
<br />
<br />
[[File:Logo01.jpg|200 px|left]]<br />
[[File:THOR poster print c.jpg|150 px|right]]<br />
<br />
Matter under extreme conditions in terms of temperature and density, as in the early Universe or in compact stellar objects (e.g. neutron stars) can be created and studied with the help of relativistic heavy ion collisions. Scientifically, the main aim is to explore and reconstruct the matter’s transport properties, phase structure, in-medium properties of hadrons and active degrees of freedom of Quantum Chromodynamics (QCD) from the experimental measurements of individual quantities. While the experimental activities are organised and optimised in large international collaborations, there is no such structure for theoretical activities.<br />
<br />
The proposed COST Action “Theory of hot matter and relativistic heavy-ion collisions” (THOR) creates a theoretical community platform as counterpart to the ongoing vigorous experimental activities. THOR will for the first time allow to fully exploit Europe’s exceptional potential in this field of theoretical research. THOR will pioneer novel approaches to the theoretical understanding of the properties of QCD from first principles and on the interpretations of these properties by effective models and numerical simulations of the system’s evolution. By this, THOR will provide new insights on the paramount questions of the field. Therefore THOR aims at bringing together excellent researchers in order to pinpoint and discuss the challenges that the field meets currently and in the near future for creating a vibrant, innovative and world- leading pan-European research environment.<br />
<br />
From this page one can continue to pages which discuss specific topics:<br />
<br />
<br />
= Strongly interacting matter =<br />
* [[Lattice qcd]]<br />
* [[Effective theories]]<br />
* [[Equation of State]]<br />
* [[Phase diagram]]<br />
* [[Fluctuations of conserved charges]]<br />
* [[Transport coefficients]]<br />
* [[Statistical model]]<br />
<br />
=Dynamics of heavy-ion collisions=<br />
* [[Glauber model]]<br />
* [[Parton distribution functions]]<br />
* [[Hydrodynamics]]<br />
* [[Transport models]]<br />
* [[Collective expansion]]<br />
* [[Femtoscopy]]<br />
* [[Baryon number fluctuations]]<br />
* [[Strangeness production]]<br />
* [[Heavy quarks]]<br />
* [[Jets]]<br />
* [[Parton energy loss]]<br />
<br />
<br />
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* [https://www.mediawiki.org/wiki/Special:MyLanguage/Manual:Combating_spam Learn how to combat spam on your wiki]</div>Tomasikhttps://wiki.thor-cost.eu/index.php?title=Main_Page&diff=25Main Page2018-10-29T21:26:27Z<p>Tomasik: </p>
<hr />
<div><br />
= Summary of CA15213 Theory of hot matter and relativistic heavy-ion collisions =<br />
<br />
<br />
[[File:Logo01.jpg|200 px|left]]<br />
[[File:THOR poster print c.jpg|150 px|right]]<br />
<br />
Matter under extreme conditions in terms of temperature and density, as in the early Universe or in compact stellar objects (e.g. neutron stars) can be created and studied with the help of relativistic heavy ion collisions. Scientifically, the main aim is to explore and reconstruct the matter’s transport properties, phase structure, in-medium properties of hadrons and active degrees of freedom of Quantum Chromodynamics (QCD) from the experimental measurements of individual quantities. While the experimental activities are organised and optimised in large international collaborations, there is no such structure for theoretical activities.<br />
<br />
The proposed COST Action “Theory of hot matter and relativistic heavy-ion collisions” (THOR) creates a theoretical community platform as counterpart to the ongoing vigorous experimental activities. THOR will for the first time allow to fully exploit Europe’s exceptional potential in this field of theoretical research. THOR will pioneer novel approaches to the theoretical understanding of the properties of QCD from first principles and on the interpretations of these properties by effective models and numerical simulations of the system’s evolution. By this, THOR will provide new insights on the paramount questions of the field. Therefore THOR aims at bringing together excellent researchers in order to pinpoint and discuss the challenges that the field meets currently and in the near future for creating a vibrant, innovative and world- leading pan-European research environment.<br />
<br />
From this page one can continue to pages which discuss specific topics:<br />
<br />
<br />
== Strongly interacting matter ==<br />
* [[Lattice qcd]]<br />
* [[Effective theories]]<br />
* [[Equation of State]]<br />
* [[Phase diagram]]<br />
* [[Fluctuations of conserved charges]]<br />
* [[Transport coefficients]]<br />
* [[Statistical model]]<br />
<br />
==Dynamics of heavy-ion collisions==<br />
* [[Glauber model]]<br />
* [[Parton distribution functions]]<br />
* [[Hydrodynamics]]<br />
* [[Transport models]]<br />
* [[Collective expansion]]<br />
* [[Femtoscopy]]<br />
* [[Baryon number fluctuations]]<br />
* [[Strangeness production]]<br />
* [[Heavy quarks]]<br />
* [[Jets]]<br />
* [[Parton energy loss]]<br />
<br />
<br />
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Consult the [https://meta.wikimedia.org/wiki/Help:Contents User's Guide] for information on using the wiki software.<br />
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* [https://www.mediawiki.org/wiki/Special:MyLanguage/Manual:Combating_spam Learn how to combat spam on your wiki]</div>Tomasikhttps://wiki.thor-cost.eu/index.php?title=Main_Page&diff=24Main Page2018-10-29T21:21:58Z<p>Tomasik: /* Strongly interacting matter */</p>
<hr />
<div><br />
= Summary of CA15213 Theory of hot matter and relativistic heavy-ion collisions =<br />
<br />
<br />
[[File:Logo01.jpg|200 px|left]]<br />
[[File:THOR poster print c.jpg|150 px|right]]<br />
<br />
Matter under extreme conditions in terms of temperature and density, as in the early Universe or in compact stellar objects (e.g. neutron stars) can be created and studied with the help of relativistic heavy ion collisions. Scientifically, the main aim is to explore and reconstruct the matter’s transport properties, phase structure, in-medium properties of hadrons and active degrees of freedom of Quantum Chromodynamics (QCD) from the experimental measurements of individual quantities. While the experimental activities are organised and optimised in large international collaborations, there is no such structure for theoretical activities.<br />
<br />
The proposed COST Action “Theory of hot matter and relativistic heavy-ion collisions” (THOR) creates a theoretical community platform as counterpart to the ongoing vigorous experimental activities. THOR will for the first time allow to fully exploit Europe’s exceptional potential in this field of theoretical research. THOR will pioneer novel approaches to the theoretical understanding of the properties of QCD from first principles and on the interpretations of these properties by effective models and numerical simulations of the system’s evolution. By this, THOR will provide new insights on the paramount questions of the field. Therefore THOR aims at bringing together excellent researchers in order to pinpoint and discuss the challenges that the field meets currently and in the near future for creating a vibrant, innovative and world- leading pan-European research environment.<br />
<br />
From this page one can continue to pages which discuss specific topics:<br />
<br />
<br />
== Strongly interacting matter ==<br />
- [[Lattice qcd]]<br />
- [[Effective theories]]<br />
<br />
<br />
Consult the [https://meta.wikimedia.org/wiki/Help:Contents User's Guide] for information on using the wiki software.<br />
<br />
== Getting started ==<br />
* [https://www.mediawiki.org/wiki/Special:MyLanguage/Manual:Configuration_settings Configuration settings list]<br />
* [https://www.mediawiki.org/wiki/Special:MyLanguage/Manual:FAQ MediaWiki FAQ]<br />
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* [https://www.mediawiki.org/wiki/Special:MyLanguage/Manual:Combating_spam Learn how to combat spam on your wiki]</div>Tomasikhttps://wiki.thor-cost.eu/index.php?title=Lattice_qcd&diff=23Lattice qcd2018-10-29T21:20:44Z<p>Tomasik: </p>
<hr />
<div>Lattice qcd is a formulation of [https://en.wikipedia.org/wiki/Statistical_physics Statistical physics] where the matter is described by QCD Lagrangian.</div>Tomasikhttps://wiki.thor-cost.eu/index.php?title=Lattice_qcd&diff=22Lattice qcd2018-10-29T21:19:33Z<p>Tomasik: Created page with "Lattice qcd is a formulation of Statistical physics where the matter is described by QCD Lagrangian."</p>
<hr />
<div>Lattice qcd is a formulation of Statistical physics where the matter is described by QCD Lagrangian.</div>Tomasikhttps://wiki.thor-cost.eu/index.php?title=Main_Page&diff=21Main Page2018-10-29T21:18:26Z<p>Tomasik: </p>
<hr />
<div><br />
= Summary of CA15213 Theory of hot matter and relativistic heavy-ion collisions =<br />
<br />
<br />
[[File:Logo01.jpg|200 px|left]]<br />
[[File:THOR poster print c.jpg|150 px|right]]<br />
<br />
Matter under extreme conditions in terms of temperature and density, as in the early Universe or in compact stellar objects (e.g. neutron stars) can be created and studied with the help of relativistic heavy ion collisions. Scientifically, the main aim is to explore and reconstruct the matter’s transport properties, phase structure, in-medium properties of hadrons and active degrees of freedom of Quantum Chromodynamics (QCD) from the experimental measurements of individual quantities. While the experimental activities are organised and optimised in large international collaborations, there is no such structure for theoretical activities.<br />
<br />
The proposed COST Action “Theory of hot matter and relativistic heavy-ion collisions” (THOR) creates a theoretical community platform as counterpart to the ongoing vigorous experimental activities. THOR will for the first time allow to fully exploit Europe’s exceptional potential in this field of theoretical research. THOR will pioneer novel approaches to the theoretical understanding of the properties of QCD from first principles and on the interpretations of these properties by effective models and numerical simulations of the system’s evolution. By this, THOR will provide new insights on the paramount questions of the field. Therefore THOR aims at bringing together excellent researchers in order to pinpoint and discuss the challenges that the field meets currently and in the near future for creating a vibrant, innovative and world- leading pan-European research environment.<br />
<br />
From this page one can continue to pages which discuss specific topics:<br />
<br />
<br />
== Strongly interacting matter ==<br />
[[Lattice qcd]]<br />
<br />
<br />
Consult the [https://meta.wikimedia.org/wiki/Help:Contents User's Guide] for information on using the wiki software.<br />
<br />
== Getting started ==<br />
* [https://www.mediawiki.org/wiki/Special:MyLanguage/Manual:Configuration_settings Configuration settings list]<br />
* [https://www.mediawiki.org/wiki/Special:MyLanguage/Manual:FAQ MediaWiki FAQ]<br />
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* [https://www.mediawiki.org/wiki/Special:MyLanguage/Manual:Combating_spam Learn how to combat spam on your wiki]</div>Tomasikhttps://wiki.thor-cost.eu/index.php?title=Main_Page&diff=20Main Page2018-10-29T21:16:03Z<p>Tomasik: </p>
<hr />
<div><br />
= Summary of CA15213 Theory of hot matter and relativistic heavy-ion collisions =<br />
<br />
<br />
[[File:Logo01.jpg|200 px|left]]<br />
[[File:THOR poster print c.jpg|150 px|right]]<br />
<br />
Matter under extreme conditions in terms of temperature and density, as in the early Universe or in compact stellar objects (e.g. neutron stars) can be created and studied with the help of relativistic heavy ion collisions. Scientifically, the main aim is to explore and reconstruct the matter’s transport properties, phase structure, in-medium properties of hadrons and active degrees of freedom of Quantum Chromodynamics (QCD) from the experimental measurements of individual quantities. While the experimental activities are organised and optimised in large international collaborations, there is no such structure for theoretical activities.<br />
<br />
The proposed COST Action “Theory of hot matter and relativistic heavy-ion collisions” (THOR) creates a theoretical community platform as counterpart to the ongoing vigorous experimental activities. THOR will for the first time allow to fully exploit Europe’s exceptional potential in this field of theoretical research. THOR will pioneer novel approaches to the theoretical understanding of the properties of QCD from first principles and on the interpretations of these properties by effective models and numerical simulations of the system’s evolution. By this, THOR will provide new insights on the paramount questions of the field. Therefore THOR aims at bringing together excellent researchers in order to pinpoint and discuss the challenges that the field meets currently and in the near future for creating a vibrant, innovative and world- leading pan-European research environment.<br />
<br />
From this page one can continue to pages which discuss specific topics:<br />
<br />
<br />
== Strongly interacting matter ==<br />
Lattice qcd[http://www.example.com link title]<br />
<br />
<br />
Consult the [https://meta.wikimedia.org/wiki/Help:Contents User's Guide] for information on using the wiki software.<br />
<br />
== Getting started ==<br />
* [https://www.mediawiki.org/wiki/Special:MyLanguage/Manual:Configuration_settings Configuration settings list]<br />
* [https://www.mediawiki.org/wiki/Special:MyLanguage/Manual:FAQ MediaWiki FAQ]<br />
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* [https://www.mediawiki.org/wiki/Special:MyLanguage/Manual:Combating_spam Learn how to combat spam on your wiki]</div>Tomasikhttps://wiki.thor-cost.eu/index.php?title=Main_Page&diff=19Main Page2018-10-29T13:48:47Z<p>Tomasik: </p>
<hr />
<div><br />
== Summary of CA15213 Theory of hot matter and relativistic heavy-ion collisions ==<br />
<br />
<br />
[[File:Logo01.jpg|200 px|left]]<br />
[[File:THOR poster print c.jpg|150 px|right]]<br />
<br />
Matter under extreme conditions in terms of temperature and density, as in the early Universe or in compact stellar objects (e.g. neutron stars) can be created and studied with the help of relativistic heavy ion collisions. Scientifically, the main aim is to explore and reconstruct the matter’s transport properties, phase structure, in-medium properties of hadrons and active degrees of freedom of Quantum Chromodynamics (QCD) from the experimental measurements of individual quantities. While the experimental activities are organised and optimised in large international collaborations, there is no such structure for theoretical activities.<br />
<br />
The proposed COST Action “Theory of hot matter and relativistic heavy-ion collisions” (THOR) creates a theoretical community platform as counterpart to the ongoing vigorous experimental activities. THOR will for the first time allow to fully exploit Europe’s exceptional potential in this field of theoretical research. THOR will pioneer novel approaches to the theoretical understanding of the properties of QCD from first principles and on the interpretations of these properties by effective models and numerical simulations of the system’s evolution. By this, THOR will provide new insights on the paramount questions of the field. Therefore THOR aims at bringing together excellent researchers in order to pinpoint and discuss the challenges that the field meets currently and in the near future for creating a vibrant, innovative and world- leading pan-European research environment.<br />
<br />
Consult the [https://meta.wikimedia.org/wiki/Help:Contents User's Guide] for information on using the wiki software.<br />
<br />
== Getting started ==<br />
* [https://www.mediawiki.org/wiki/Special:MyLanguage/Manual:Configuration_settings Configuration settings list]<br />
* [https://www.mediawiki.org/wiki/Special:MyLanguage/Manual:FAQ MediaWiki FAQ]<br />
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* [https://www.mediawiki.org/wiki/Special:MyLanguage/Manual:Combating_spam Learn how to combat spam on your wiki]</div>Tomasikhttps://wiki.thor-cost.eu/index.php?title=Main_Page&diff=18Main Page2017-08-03T19:50:22Z<p>Tomasik: </p>
<hr />
<div>== This is a wiki page set up by THOR COST project. ==<br />
<br />
[[File:Logo01.jpg|200 px|left]]<br />
[[File:THOR poster print c.jpg|150 px|right]]<br />
<br />
In order to be able to edit wiki, you have to be logged to [http://thor-cost.eu THOR COST pages]. You can do it before coming to wiki or you have to click on Log in button on the right top corner. You will be redirected to [http://thor-cost.eu THOR COST pages], where you have to log in and go back to wiki. <br />
<br />
Note, that you will not be redirected back automatically, this is a work in progress feature..<br />
<br />
<br />
<br />
If you get ...Fatal exception of type MWException... when saving a page, just reload the browser and the changes will be saved correctly. This bug will be hopefully soon addressed. <br />
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<br />
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Consult the [https://meta.wikimedia.org/wiki/Help:Contents User's Guide] for information on using the wiki software.<br />
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== Getting started ==<br />
* [https://www.mediawiki.org/wiki/Special:MyLanguage/Manual:Configuration_settings Configuration settings list]<br />
* [https://www.mediawiki.org/wiki/Special:MyLanguage/Manual:FAQ MediaWiki FAQ]<br />
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* [https://www.mediawiki.org/wiki/Special:MyLanguage/Manual:Combating_spam Learn how to combat spam on your wiki]</div>Tomasik