Nonclassical effects such as squeezing, antibunching, and sub-Poissonian statistics of photons have been attracting attention in quantum optics over the last decade. Up to now most theoretical and experimental investigations have been carried out exclusively in the time domain while neglecting the spatial aspects by considering only one spatial mode of the electromagnetic field. In many situations such an approximation is well justified. There are, however, problems that do not allow in principle a single-mode consideration. This is the case when one wants to investigate the quantum fluctuations of light at different spatial points in the plane perpendicular to the direction of propagation of the light beam. Such an investigation requires a complete description of quantum fluctuations of light in both time and space and cannot be done within a single-mode theory. This space-time description brings about a natural generalization into the spatial domain of such notions as the standard quantum limit, squeezing, antibunching, etc. It predicts, for example, the possibility of generating a light beam with sub-Poissonian statistics of photons not only in time but also in the beam's transverse plane. Of particular relevance to the applications is a situation in which the cross section of the light beam contains several nonoverlapping areas with sub-Poissonian statistics of photons in each. Photodetection of such a beam produces several sub-shot-noise photocurrents depending on the number of independent areas with sub-Poissonian statistics. This is in marked contrast to the case of a single-mode sub-Poissonian light beam in which any attempt to collect light from only a part of the beam deteriorates the degree of shot-noise reduction. This property of multimode squeezed light opens a range of interesting new applications in optical imaging, optical parallel processing of information, parallel computing, and many other areas in which it is desirable to have a light beam with regular photon statistics across its transverse area. The aim of this review is to describe the recent development in this branch of quantum optics.
Nonclassical effects such as squeezing, antibunching, and sub-Poissonian statistics of photons have been attracting attention in quantum optics over the last decade. Up to now most theoretical and experimental investigations have been carried out exclusively in the time domain while neglecting the spatial aspects by considering only one spatial mode of the electromagnetic field. In many situations such an approximation is well justified. There are, however, problems that do not allow in principle a single-mode consideration. This is the case when one wants to investigate the quantum fluctuations of light at different spatial points in the plane perpendicular to the direction of propagation of the light beam. Such an investigation requires a complete description of quantum fluctuations of light in both time and space and cannot be done within a single-mode theory. This space-time description brings about a natural generalization into the spatial domain of such notions as the standard quantum limit, squeezing, antibunching, etc. It predicts, for example, the possibility of generating a light beam with sub-Poissonian statistics of photons not only in time but also in the beam's transverse plane. Of particular relevance to the applications is a situation in which the cross section of the light beam contains several nonoverlapping areas with sub-Poissonian statistics of photons in each. Photodetection of such a beam produces several sub-shot-noise photocurrents depending on the number of independent areas with sub-Poissonian statistics. This is in marked contrast to the case of a single-mode sub-Poissonian light beam in which any attempt to collect light from only a part of the beam deteriorates the degree of shot-noise reduction. This property of multimode squeezed light opens a range of interesting new applications in optical imaging, optical parallel processing of information, parallel computing, and many other areas in which it is desirable to have a light beam with regular photon statistics across its transverse area. The aim of this review is to describe the recent development in this branch of quantum optics.
Summary form only given.It has previously been shown that a traveling-wave optical parametric amplifier can generate multimode squeezed states of light. Homodyne detection of multimode squeezed light, under certain physical conditions, results in sub-Poissonian statistics of photons not only in time but also in the transverse plane of the light beam. Of particular relevance to applications is a situation when the cross section of the light beam contains several nonoverlapping areas with sub-Poissonian statistics of photons within each of them. Under photodetection of such a beam of light, several sub-shot-noise photocurrents can be produced according to the number of independent areas with sub-Poissonian statistics. This is in drastic contrast to the case of a single-mode sub-Poissonian light beam, where any attempt to collect light only from a part of the beam deteriorates the degree of shot-noise reduction. This property of multimode squeezed light opens a range of new interesting applications in optical imaging, optical parallel processing of information, parallel computing, and many other phenomena where it is desirable to have the regular statistics of photons across the transverse plane of the electromagnetic wave. We describe the application of multimode squeezed light in sub-shot-noise microscopy, noiseless amplifications of optical images, and optical imaging. We also consider the possibility of improving the resolution in optical imaging and microscopy beyond the Rayleigh limit via multimode squeezed light.
The author reviews methods for generating sub-Poissonian light and related concepts. This light has energy fluctuations reduced below the level which corresponds to a classical Poissonian process (shot-noise level). After an introduction to the concept of nonclassical light, an overview is given of the main methods of quantum-noise reduction. Sub-Poissonian processes are exemplified in different areas of optics, ranging from single-atom resonance fluorescence to nonlinear optics, laser physics, and cavity quantum electrodynamics. Emphasis is placed on the conceptual foundations, and on developments in laser theory that lead to the possibility, already demonstrated experimentally, of linewidth narrowing and sub-Poissonian light generation in lasers and masers. The sources of quantum noise in these devices are analyzed, and four noise-suppression methods are discussed in detail: regularization of the pumping, suppression of spontaneous-emission noise, nonadiabatic evolution of the atomic variables, and twin-beam generation.
The statistical fluctuations of light can, under certain circumstances, exhibit a number of interesting nonclassical features. These include photon antibunching, sub-Poissonian statistics, and squeezing, all of which exist only in a quantum field. The three effects are distinct, and do not necessarily go together, although the first two are sometimes mistakenly treated as interchangeable. Both photon antibunching and sub-Poissonian statistics have been observed in experiments on resonance fluorescence from single atoms, which is described. Squeezing has not yet been seen experimentally, although many nonlinear processes are expected to give rise to it. The squeezed state can be recognized in heterodyne or homodyne experiments, where the squeezing becomes manifest as sub-Poissonian photon fluctuations.
Although light quanta were known since the works by Planck and Einstein, for several decades after these works, there was no need to treat light quantum mechanically, and all experiments could be well explained using the quantum description of matter and classical description of light (the semiclassical approach). The situation changed with the development of sources of so-called nonclassical light, i.e., light whose properties cannot be understood in the framework of the semiclassical approach. Two simple examples can be given here. The first one is the suppression of shot noise in optical detectors. Shot noise, i.e., fluctuations of photocurrent scaling as the square root of the photocurrent mean value, can be, in principle, attributed to the discrete nature of the charge generated by the detector. At the same time, the effect of shot-noise suppression, which is observed with certain light sources (squeezed light) is clearly related to the statistics of light, and quantum mechanics is completely necessary for its description. Another example, which brings us closer to the subject of the present paper, is two-photon light. This type of light contains only pairs of photons (often called biphotons), and hence, a pair of properly aligned photon-counting detectors registering two-photon light, in an ideal case, will produce only simultaneous photocounts. This behavior, again, can only be explained in terms of a quantum-mechanical description for the light.
Two-beam states obtained by partial photon-number-resolving detection in one beam of a multi-mode twin beam are experimentally investigated using an intensified CCD camera. In these states, sub-Poissonian photon-number distributions in one beam are accompanied by sub-shot-noise fluctuations in the photon-number difference of both beams. Multi-mode character of the twin beam implying the beam nearly Poissonian statistics is critical for reaching sub-Poissonian photon-number distributions, which contrasts with the use of a two-mode squeezed vacuum state. Relative intensities of both nonclassical effects as they depend on the generation conditions are investigated both theoretically and experimentally using photon-number distributions of these fields. Fano factor, noise-reduction parameter, local and global nonclassicality depths, degree of photon-number coherence, mutual entropy as a non-Gaussianity quantifier, and negative quasi-distributions of integrated intensities are used to characterize these fields. Spatial photon-pair correlations as means for improving the field properties are employed. These states are appealing for quantum metrology and imaging including the virtual-state entangled-photon spectroscopy.
This paper provides a simple physical consideration of a new type of squeezed state of light: space-time squeezed state. As it was shown recently, spatial-multimode broadband squeezed light can generate (by homodyne detection) photon counting statistics that are regular not only in time, but also in the plane of observation. A possible advantage of application of space-time squeezed light in a number of areas of quantum optics (i.e., low-noise generation, detection of optical images, Fourier optics, and holography) is discussed from an elementary point of view.