Hanbury Brown and Twiss Interferometry with Interacting Photons
[This is an invited article based on a recently published work by the authors
Authors: Yoav Lahini, Yaron Bromberg, Eran Small and Yaron Silberberg
Affiliation: Department of Physics of Complex Systems, the Weizmann Institute of Science, Rehovot, Israel.
The next time you go out on a sunny day, take a minute to consider the sunlight you see reflected from the ground near you. If you could have frozen time, you would see that actually, the light pattern on the ground is not homogenous, but rather it is speckled – it is made out of patches of light and darkness, similar to the speckle pattern you see when a laser light hits a rough surface like a wall. For sunlight, the typical speckle size is around 100 microns, but that is not the reason why the sunlight speckles are not observed in everyday life. The real reason is that this speckle pattern changes much faster than the human eye – and in fact, faster than any man made detector – can follow. As a result we see an averaged, smeared homogenous light reflected around us.
To understand this phenomenon, and its relation to the Hanbury-Brown and Twiss effect and the birth of quantum optics, let’s first consider the sun as observed by a spectator on earth. The sun is an incoherent light source – that is, there is no fixed phase relation between the rays of light coming from different parts of the sun’s surface. In fact, it is more accurate to say that there is a phase relation between the rays only that this phase difference continuously fluctuates. The rate of the phase fluctuations is very fast, typically on the scale of femtoseconds. Nevertheless, let’s assume for a moment that we could freeze time while looking at the sunlight on Earth. What would we see? Since everything is “frozen”, the phase between all the rays coming from the sun is fixed, and the rays will interfere. The result of such interference of many rays with a random phase leads to the speckle pattern – patches of light and darkness. Bright regions are formed where the rays interfere constructively, and dark regions where the rays interfere destructively. The typical size of the patches is determined by the distance over which constructive interference changes to destructive. This happens when the path lengths from the emitters on the sun (or any incoherent source) to the Earth change by about half a wavelength. In fact it can be shown that the typical size of such speckle, if one could ever be photographed, goes like the wavelength over the angular size of the source, as seen from the earth . This means that the size of a typical speckle is larger the further the distance between the source and the observer - the speckles diffract, their size increases as they propagate.
As noted earlier, the sun will create a speckle pattern on Earth with a typical speckle size of 100 microns, while a distant star (with a much smaller angular size) will create a speckle size of a few meters and even kilometers. So in theory, if the speckle size can be somehow measured, it will allow to determine the angular size of stars, or any other incoherent light source.
In 1956, two astronomers, Hanbury Brown and Twiss, did just that . They found a way to determine the typical speckle size in starlight with just two detectors instead of a camera. The trick was to use two fast detectors, and look into the noise measured by the two detectors, instead of the averaged signal as we usually do in the lab. So how does it work? The intensities measured by the two detectors are noisy, since the speckle pattern that impinges on the detectors continuously varies. But, as long as the two detectors are separated by a distance smaller than the typical speckle size, they will be illuminated most of the time by the same speckle. The signals measured by the two detectors will therefore be noisy but correlated, i.e. the two signals will fluctuate together. However, if the two detectors are separated by a distance larger than the typical speckle size, the signals' fluctuations will be totally uncorrelated, since each detector sees different speckles. Therefore the distance in which noise in the two detectors becomes uncorrelated is a measure of the typical speckle size, and therefore a measure of the angular size of the observed star.
Hanbury Brown and Twiss (HBT) proved their theory several times [3,4], by giving accurate measures of the angular size of several stars using radio and optical interferometry. These experiments gave rise to a vigorous debate about the nature of light: it is easy to prove the HBT effect if you think of light as classical waves, but what happens if you try to take the particle view of light? How can two photons, coming from two distant atoms on the surface of a star and measures by two distant detectors on the surface of the earth, be correlated? The answer to this question was given only after a few years by the Nobel Prize laureate Roy Glauber , an answer that marked the birth of the field of Quantum Optics.
Since those days, the HBT technique was adopted and used in many different fields in physics as a tool to remotely measure properties of different sources. For example, the HBT method was used to measure the properties subatomic particles created in nuclear collisions , of Bose-Einstein Condensates (BEC) in lattice potentials [7,8] and other systems [9-13]. In a work recently published in Nature Photonics , we note that these modern uses of HBT interferometry rely on an assumption that there are no interactions between the particles on their way from the source to the detectors. Such interactions (or nonlinear effects in the case of classical waves) would affect the correlations while the particles (or waves) propagate from the source to the detectors. The assumption of no interactions is probably valid in the astronomical case (although due to the very long distances involved that might also be questioned), but is not necessarily true for atom-matter waves released from their confining potential, or for charged sub-atomic particles propagating from the point of interest to the detection.
To see how one can cope with such complications we analyzed the effect of interactions on the resulting HBT correlation by considering light propagation in a nonlinear medium – a scenario physically similar to matter waves released from a confining potential (the equations describing the dynamics of matter waves are identical, in certain limits, to the equations used in our paper). Using a spatial light modulator and diffusers we mimicked a spatially incoherent light source in a controlled manner, and measured the HBT correlations after propagation of the speckle field in a nonlinear medium. We investigated both repulsive and attractive interactions, in two and three dimensional space. Using these measurements, we have shown how the interactions modify the measured HBT correlations. While the fact the interactions modify correlations is expected, our work provides an intuitive picture for the source of this modification. The key idea is to follow the propagation of the speckle patterns in the nonlinear medium. As discussed above, when there are no interactions the speckles diffract along the propagation. But in the presence of interactions, or nonlinearity, each speckle can turn into what is known as a soliton – a self trapped entity, with a size that does not change along the propagation. This means that the size of the speckles is no longer a measure for the angular size of the source. It is in fact a measure for the strength of the interactions.
Experimental observation of a speckle pattern propagating in a nonlinear medium. In the interaction free case, the width of a typical speckle is inversely proportional to the width of the source, W. In the presence of interactions, one needs to take into account the strength of the intensity fluctuations as well. Image credit: Adi Natan
But perhaps more importantly, we provide a new framework that can include interactions in HBT interferometry. We found that the information on the source can still be retrieved if the interactions are taken into account correctly. We show that in the presence of interactions the angular size of the source can be recovered, but one needs in addition to the spatial correlation also to measure the strength of the signals' fluctuations. Intuitively, this stems from the fact that speckles which have became “solitons” still propagate at different angles. Since these “speckolitons” keep their size along the propagation, the chance that a speckoliton will hit the detectors goes down as the distance from the source to the detector increases. But the intensity the speckolitons carries is much higher than the intensity of a linear speckle which diffracts along the propagation. Careful analysis of this phenomena leads to the conclusion that in the presence of interactions the intensity fluctuations carry the missing information on the angular size of the source.
One can measure the strength of the fluctuations by simply looking at the variance of the detectors' readouts, which is closely related to the contrast of the bright to dark patches in the speckle pattern. As a possible application, consider HBT interferometry with trapped BEC. A recent paper  identified the complication of using HBT interferometry arising due to interactions during the time-of-flight, after the condensate is released from the trap. That paper suggests an intricate manipulation of the condensate during the time-of-flight, to scale out the effects of interactions. Our paper provides a framework to include the interactions in the analysis, without the need for such complicated experiments.
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