37.1 Introduction

For more than two decades now, techniques have been designed and experimentally implemented enabling physicists to get rid of, or at least to reduce, quantum fluctuations in optical measurements. The same techniques lead also to the production of strong quantum correlations and entanglement in light. This domain has recently been successfully developed in the direction of quantum information processing and is presented extensively in the present book.

So far the quantum noise reduction, or the correlations, was effective when the total intensity of light beams was recorded. But there is another part of optics, which presents a great interest from the point of view of information: The domain of optical images, which are a privileged medium to convey a great quantity of information in a parallel way. “Pixellized” detectors (such as CCD cameras or detectors arrays) are used to record such information, either in the photon counting regime or with macroscopic intensities. Due to the quantum nature of light, this information is inevitably affected by uncontrolled fluctuations, the “quantum noise” or shot noise, which limits the reliability of the information extraction from the image, or the ultimate resolution for the detection of small details in the image. In these optical measurements, the fluctuations that come into play are the local spatial quantum fluctuations.

Researches made in the last decade at the theoretical level showed that it was possible to tailor these local spatial quantum fluctuations of light (of course within the constraint imposed by Heisenberg inequalities), and also to produce spatial quantum entanglement, that is, to create strong quantum correlations in the measurements performed at different points of the optical image. Quantum techniques have the potentiality to improve the sensitivity of measurements performed in images and to increase the optical resolution beyond the wavelength limit, not only at the single photon counting level, but also with macroscopic beams of light. These new techniques could then be of interest in many domains where light is used as a tool to convey information in very delicate physical measurements, such as ultraweak absorption spectroscopy or atomic force microscopy. Detecting details in images smaller than the wavelength has obvious applications in the fields of microscopy and pattern recognition, and also in optical data storage, where it is now envisioned to store bits on areas much smaller than the square of the wavelength. Furthermore, spatial entanglement leads to completely novel and fascinating effects, such as two‐photon imaging, in which the camera is illuminated by light, which did not interact with the object to image, or “quantum microlithography,” where the quantum entanglement is able to act upon matter at a scale smaller than the wavelength. Finally, there is a natural extension of quantum information protocols to multimode quantum information and computing using images that is still in its very early days.

This kind of study forms a newly emerging subject of quantum optics, and few pioneer experimental demonstrations have been already performed. The investigations made so far concern mainly the ways of producing and characterizing spatially entangled nonclassical light and also first simple implementations of applications, which showed that it is possible using such concepts to improve information extraction from images. To illustrate these somewhat abstract considerations, we will give in the following a short description of several achievements obtained in the domain, and conclude by mentioning some perspectives and open problems, which seem promising and deserve therefore more investigations in the future. Readers interested in more details can find them in some review articles (1).

37.2 The Quantum Laser Pointer

Experiments have demonstrated that the sensitivity of optical measurements performed on the global intensity of a light beam, or on its global phase, can be improved by using single mode nonclassical states of light, such as sub‐Poissonian or squeezed states. This is no longer true for measurements performed in optical images, in which one monitors a variation of the transverse distribution of the light and not of the total intensity. One needs in this case more complex nonclassical beams, which are superpositions of different transverse modes, that is, multimode nonclassical states of light.

The simplest of these measurements is that of the position of the center of a beam, which is obtained using a quadrant detector (Figure 37.1): If the four partial intensities are equal, the beam is exactly centered on the detector, and any imbalance between the four signals gives information about the transverse displacement of the beam. This is actually a highly sensitive measurement, at the nanometer scale. But as all optical measurements, it is limited by the standard quantum noise, or shot noise, present on the four parts of the quadrant detector. It was first shown theoretically (2) that the pointing sensitivity can be improved beyond such a standard quantum limit by using, instead of a usual laser beam, the superposition of a single‐mode squeezed beam with a coherent beam having its two halves in the transverse plane shifted with each other by π. This curious mixing actually creates a perfect quantum correlation between the intensities measured on the halves of the total beam, and therefore on the photocurrents detected on the corresponding pixels of the detector.

This effect has been recently experimentally demonstrated (3): The two transverse displacements of the beam center were measured with a sensitivity better than the standard quantum limit. In order to measure simultaneously the two transverse coordinates below shot noise, one needs a three‐mode nonclassical state of light, consisting of the superposition of two squeezed states and a coherent state, each transverse mode having appropriate π phase shifts in the four quadrants of the transverse plane corresponding to the four detection regions (Figure 37.1).

Transverse displacement is the simplest measurement that can be performed with a multipixel detector, but there are many other parameters that can be extracted from an image: The motion of a very small scattering object, a very weak spatial modulation, the presence or absence of small holes carrying digitized information, for example, in CDs used in optical storage of information. The extraction of such an information is made through “image processing,” which consists in most cases in computing linear combinations of the local intensities measured by the different pixels. This problem has been analyzed in detail at the theoretical level (4). The transverse mode responsible for the noise in this kind of image processing has been identified for any linear processing. By reducing the noise in this specific mode, one improves the determination of the corresponding information. This kind of technique may improve many image processing and analysis functions, such as pattern recognition, image segmentation, or wavefront analysis.

On the quantum information side, techniques have recently been theoretically proposed for producing spatially entangled beams (5), which constitute an extension to the problem of transverse measurements of the entangled state proposed by Einstein, Podolsky, and Rosen for the measurement of X and P. In these “EPR‐entangled beams,” the measurements of the transverse position x of two light beams are perfectly correlated, whereas measurements of the angular tilt θ with respect to the optical axis are perfectly anticorrelated. In the transverse plane, x and θ are indeed the quantum‐conjugate quantities, which are analogous of X and P for a particle.

37.3 Manipulation of Spatial Quantum Noise

The experiment described in the previous section shows that it is possible to manipulate the transverse distribution of temporal quantum fluctuations in light. But in an image, there is also a “pure” spatial quantum noise, that is, the pixel‐to‐pixel fluctuations of the light intensity when it is integrated over the total duration of a single light pulse. It concerns only spatial averages, and no longer time averages.

Measuring pixel‐to‐pixel fluctuations at the quantum level is a new experimental challenge, and novel and delicate experimental techniques had to be developed in order to reach the shot noise level for spatial fluctuations. In particular, it is necessary to make a very precise calibration of each pixel of the detector, so that the intensity measured on each one can be properly normalized (6). It is only after all these technical problems have been solved that it was possible to observe the two specific spatial quantum effects that are described in the two following subsections.

37.3.1 Observation of Pure Spatial Quantum Correlations in Parametric Down Conversion

It is well known that parametric down conversion produces “twin photons,” which are perfectly correlated at the quantum level, not only temporally (they are produced at the same time) but also spatially (they are produced in symmetric directions). This effect has been extensively used in beautiful landmark experiments at the photon counting level. When the pump intensity is raised by a large factor, many twin photons are produced, and they can no longer be counted individually. One now obtains patterns, or “images” on the signal and idler beams, which are still temporally and spatially correlated at the quantum level. As can be seen in Figure 37.2, each image has large pixel‐to‐pixel fluctuations, but almost identical intensity values on pixels symmetrical with respect to the center of the figure.

The experiment (7) has been performed with an intense pulse laser as the pump of the spontaneous down‐conversion process. The parametric gain is high in such a regime (10 to 1000), and roughly 10 to 100 photons were recorded on average on each pixel. A pixel to pixel quantum correlation was found between the intensity distributions of the signal and idler transverse patterns recorded after a single pump laser shot. More precisely, the variance of the difference between the intensities recorded on the signal and idler modes on symmetrical pixels, averaged over the different points of the transverse plane, was measured to be well below the standard quantum limit, which is in this case the spatial shot noise corresponding to the total intensities measured on the photodetectors. The best spatial noise reduction observed was about 50% below the standard quantum limit. For very high gain, the quantum correlation turns out to disappear. This quantum‐to‐classical transition from the quantum to the classical regime is due to the spatial narrowing of the signal or idler beams generated by the nonlinear crystal with increased gain, which leads, through diffraction, to an extension of the zone in which the twin photons are distributed. The quantum spatial correlation that has been observed can now be used to improve information processing in images, for example, to improve the sensitivity in the detection of faint images below the standard quantum limit.

37.3.2 Noiseless Image Parametric Amplification

Optical amplification is one of the key techniques in the handling of optical information. Quantum theory shows that the amplification process induces inevitably a degradation of the signal to noise ratio by at least a factor 2 when oscillating signals are amplified in a way independent of the phase of the oscillation. In contrast, the amplification can be noiseless in the phase‐sensitive configuration. It is known that parametric amplification, in the frequency degenerate configuration, can operate in such a phase sensitive way. It can thus amplify an optical signal without degrading it.

This important property of degenerate parametric amplifiers had been demonstrated for the total intensities of the amplified signal beam in a pulsed parametric amplifier. It also holds for image amplification. The experimental demonstration of noiseless image amplification has been the first experimental demonstration of a quantum imaging effect. It concerned the temporal fluctuations measured at the different points of an image (8). The effect was also recently demonstrated for the pure pixel to pixel spatial fluctuations of an image amplified by a pulsed optical parametric amplifier and recorded on a pump laser single shot (9). In a very delicate experiment, the spatial noise figures were determined in the phase‐sensitive and phase insensitive schemes, and it was shown that in the low‐gain regime the phase sensitive amplifier does not add noise, while the phase insensitive amplifier leads to the degradation of the signal to noise ratio by a factor 2 (Figure 37.3). Amplification of faint images without degradation of their quality is obviously a domain, which may have important applications.

37.4 Two‐Photon Imaging

Two‐photon imaging, sometimes labeled as “ghost imaging,” is a striking effect based on the spatial correlations of light. It was demonstrated for the first time by using the spatial quantum correlations existing between the signal and idler twin photons produced by spontaneous parametric down‐conversion (10). Its principle is the following (see Figure 37.4): One inserts in the signal arm an object that one intends to observe. The image of this object is obtained in a rather paradoxical way, without using a pixellized detector but instead a nonimaging “bucket” detector on the signal beam, which measures only the total intensity transmitted through the object. On the other hand, one inserts a pixellized detector (i.e., a CCD camera) in the idler arm where there is no object. The image of the object is obtained by retaining the information on the CCD camera only when it is coincident with a photon measured by the “bucket” detector. This technique was implemented at the photon counting level in a number of beautiful experiments in the mid‐nineties, and was generally considered as the perfect example of a specific use of spatial quantum correlations in the photon‐counting regime.

It was then showed that the effect could be also observed using the same kind of imaging setup, but in the intense light produced in the high‐gain regime of parametric amplification. More precisely, the image appears on the correlation existing between the total intensity of the signal beam and the spatially resolved intensity distribution of the idler beam. It was also predicted that both the image itself (often called “near‐field image”) and its Fourier transform (obtained, for example, through diffraction in the far‐field regime, and called “far‐field image”) could be determined in the same experimental setup. As these two quantities play a role having some similarity with the conjugate variables position and momentum, this property was considered as being related to the EPR character of the spatial correlation between the signal and idler beams.

In a recent experiment (11), it was shown that a near‐field image could be obtained by the same technique using classically correlated beams produced by a beam‐splitter, and not twin photons. A lively worldwide discussion started on the precise assignment of classical and quantum features in “two‐photon imaging”: It was in particular discovered, and experimentally demonstrated (12), that the same imaging technique could provide both the near‐field and the far‐field images using a thermal beam divided into two parts on a beam‐splitter, instead of quantum correlated beams. Only some quantitative features, such as the contrast of the image are improved when one uses quantum correlated beams instead of classical correlations. From the point of view of applications, the fact that the mysterious “ghost imaging” can be realized using a simple beam‐splitter and a thermal lamp instead of twin beams produced by a complex setup is positive in terms of cost and simplicity: This shows that there is some practical interest in precisely assigning what is classical and what is quantum in a given phenomenon, a discussion that is generally considered as purely academic.

37.5 Other Topics in Quantum Imaging

Among the numerous problems that are currently studied under the general name of quantum imaging, the investigations concerning the quantum limits on optical resolution have a special importance, as they may lead to new concepts in microscopy and optical data storage. “Super‐resolution techniques” have been studied for a long time at the classical level in the perspective of beating the Rayleigh limit of resolution, on the order of the wavelength. In principle, deconvolution techniques are likely to extract the shape of a very small object from its image, even if it is completely blurred by diffraction. But the noise present in the image, and ultimately the quantum noise, will prevent such a perfect reconstruction procedure. It will reduce the quantity of information that can be obtained about the small object shape. This procedure of object reconstruction was recently revisited at the quantum level (13). It was shown that it was in principle possible to improve the performance of super‐resolution techniques by injecting nonclassical light in very specific transverse modes, namely the eigenmodes of the propagation through the imaging system. The precise generation scheme of such a multi‐mode nonclassical light was also obtained. The simplicity of the proposed scheme brings confidence that quantum‐enhanced super‐resolution technique can effectively be implemented in an actual experiment.

Transverse solitons are potential candidates for the role of spatial q‐bits, and soliton arrays for the role of q‐registers. The theoretical study of their quantum features has been recently undertaken in different configurations: Free propagating solitons through planar Kerr media (14), and cavity solitons appearing in degenerate parametric oscillators (15). As a first step in the investigation in the direction of quantum information processing, the existence of local noise reduction and spatial correlations has been predicted in such devices.

Nonlinear media have been used for a long time to process images. For example, up‐conversion of optical images from the infrared to the visible has been proposed and realized in order to take advantage of the higher quality of CCD detectors in the visible. This domain was also recently investigated at the quantum level in second harmonic generation: It has been shown that there existed configurations where the up‐conversion of any image to the second harmonic field was possible without adding quantum noise to the initial image (16).

It is also interesting to investigate how the now “classical” protocols of quantum information on global variables, such as teleportation or cryptography, could be extended into the domain of images, and in which respect the intrinsic parallelism peculiar to imaging could be used in these protocols. The quantum teleportation of images was particularly studied in detail (17). The proposed scheme has indeed a lot of similarities with the usual holographic technique, with the advantage that, in the quantum teleportation scheme, no quantum noise is added by the image reproduction device.

37.6 Conclusion and Perspectives

Microscopy, wavefront correction, image processing, optical data storage, and optical measurements in general constitute a very important domain of our present‐day technologies. They can benefit in various ways from the researches on quantum imaging, which is currently studied by a growing number of teams throughout the world.

Optical technologies can directly benefit from the improvements brought by quantum effects and demonstrated by laboratory experiments, but their present complexity is an obstacle to such applications. At a less ambitious level, but perhaps more realistic, many optical technologies could be significantly improved by using the highly sophisticated methods developed in quantum optics laboratories to reach, and go beyond, the level of quantum noise in images. A great deal of research work remains indeed to be done, on the experimental side, to improve the light sources and the detectors in order to obtain high levels of quantum spatial entanglement and, but also on the theoretical side, to find more practical applications of spatial entanglement to information technologies. A promising direction of research is certainly the use of orbital angular momentum of light to convey and process quantum information. So far, the spatial quantum effects are somewhat on the edges of quantum computing, as they have been essentially used in the domain of metrology and information storage. No proposition has been made up to now to use the parallelism of optical imaging in quantum computing algorithms. This subject is obviously a very difficult one, but undoubtedly interesting. It requires collaborative work between the quantum computing and quantum imaging communities.

Acknowledgment

Laboratoire Kastler Brossel, of the Ecole Normale Supérieure, Sorbonne Université and Collège de France, is associated with the Centre National de la Recherche Scientifique. This work was supported by the European Commission in the frame of the QUANTIM project (IST‐2000‐26019).

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