Hanbury-Brown and Twiss effect

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The Hanbury Brown and Twiss effect (HBT) is any of a variety of correlation and anti-correlation effects in the intensities received by two detectors from a beam of particles. HBT effects can generally be attributed to the dual wave-particle nature of the beam, and the results of a given experiment depend on whether the beam is composed of fermions or bosons. Devices which utilize the effect are commonly called intensity interferometers and were originally used in astronomy, although they are also heavily used in the field of quantum optics.

1 History
2 Wave mechanics
3 References
4 See also
5 External links

[edit] History

In 1956, Robert Hanbury Brown and Richard Q. Twiss published A test of a new type of stellar interferometer on Sirius, in which two photomultiplier tubes (PMTs), separated by about 6 meters, were aimed at the star Sirius. Light was collected into the PMTs using mirrors from searchlights. An interference effect was observed between the two intensities, revealing a positive correlation between the two signals, despite the fact that no phase information was collected. Hanbury Brown and Twiss used the interference signal to determine the apparent angular size of Sirius, claiming excellent resolution.

An example of an intensity interferometer that would observe anti-correlation.

This result met with much skepticism in the physics community. Although intensity interferometry had been widely used in radio astronomy where Maxwell's equations are valid, at optical wavelengths the light would be quantised into a relatively small number of photons. 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 papers (see References below) which 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 secondly 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.

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 1986. Finally, bosons have a tendency to clump together, but due to the Pauli exclusion principle, fermions tend to spread apart, and so when the Morgan and Mandel experiment is performed on electrons, an anti-correlation in arrival times was observed for the first time in 1999. All of these are considered HBT like effects.

[edit] Wave mechanics

The HBT effect can in fact be predicted solely by treating the incident electromagnetic radiation as a classical wave. Suppose we have a single incident wave with frequency $ω$ on two detectors. Since the detectors are separated, say the second detector gets the signal delayed by a phase of $φ$ . Since the intensity at a single detector is just the square of the wave amplitude, we have for the intensities at the two detectors

$i_1=E^2\sin^2(\omega t)\,$

$i_2=E^2\sin^2(\omega t + \phi)=E^2(\sin(\omega t)\cos(\phi)+\sin(\phi)\cos(\omega t))^2\,$

which makes the correlation

$\langle i_1i_2\rangle = \lim_{T\rightarrow\infty}\frac{E^4}{T}\int^T_0 \sin^2(\omega t)(\sin(\omega t)\cos(\phi)+\sin(\phi)\cos(\omega t))^2\,dt$

$= \frac{E^4}{4}+\frac{E^4}{8}\cos(2\phi).$

A constant plus a phase dependent component. 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 $\Delta i = i-\langle i\rangle$ , where $\langle i\rangle$ is the average intensity, ought to be correlated. In general

$\langle\Delta i_1\Delta i_2\rangle = \langle(i_1-\langle i_1\rangle)(i_2-\langle i_2\rangle)\rangle =\langle i_1i_2\rangle-\langle i_1\langle i_2\rangle\rangle -\langle i_2\langle i_1\rangle\rangle +\langle i_1\rangle \langle i_2\rangle$

$=\langle i_1i_2\rangle -\langle i_1\rangle \langle i_2\rangle,$

and since the average intensity at both detectors in this example is $E 2 / 2$ ,

$\langle \Delta i_1\Delta i_2\rangle=\frac{E^4}{8}\cos(2\phi),$

so our constant vanishes. The average intensity is $E 2 / 2$ because the time average of $sin 2 (ω t)$ is 1/2.

[edit] References

Note that Hanbury Brown is not hyphenated.

E. Brannen, H. Ferguson (1956). "The question of correlation between photons in coherent light beams". Nature 178: 481. - paper which (incorrectly) disputed the existence of the Hanbury Brown and Twiss effect

R. Hanbury Brown and R. Q. Twiss (1956). "A Test of a New Type of Stellar Interferometer on Sirius". Nature 178: 1046-1048. - experimental demonstration of the effect

E. Purcell (1956). "The Question of Correlation Between Photons in Coherent Light Rays". Nature 178: 1449-1450.

R. Hanbury Brown and R. Q. Twiss (1957). "Interferometry of the intensity fluctuations in light. I. Basic theory: the correlation between photons in coherent beams of radiation". Proc of the Royal Society of London A 242: 300-324. download as PDF

R. Hanbury Brown and R. Q. Twiss (1958). "Interferometry of the intensity fluctuations in light. II. An experimental test of the theory for partially coherent light". Proc of the Royal Society of London A 243: 291-319. download as PDF

B. L. Morgan and L. Mandel (1966). "Measurement of Photon Bunching in a Thermal Light Beam". Phys. Rev. Lett. 16: 1012-1014.

P. Grangier, G. Roger, and A. Aspect (1986). "Experimental Evidence for a Photon Anticorrelation Effect on a Beam Splitter: A New Light on Single-Photon Interferences". Europhysics Letters 1: 173-179.