Leonard Mandel

Sub-Poissonian photon statistics in resonance fluorescence

Expressions are derived for the probability p(n) that n photons are emitted in a given time in the steady state by a two-level atom, when it is placed in a resonant, coherent, exciting field. The distribution p(n) is shown to be narrower than Poissonian. The ratio [<(Deltan)(2)> - ] is negative and has an absolute maximum value of 3/4. The possibility of observing the sub-Poissonian statistics is discussed briefly.

Spectral coherence and the concept of cross-spectral purity*

A new measure of correlations in optical fields, introduced in recent investigations on radiometry with partially coherent sources, is studied and applied to the analysis of interference experiments. This measure, which we call the complex degree of spectral coherence, or the spectral correlation coefficient, characterizes the correlations that exist between the spectral components at a given frequency in the light oscillations at two points in a stationary optical field. A relation between this degree of correlation and the usual degree of coherence is obtained and the role that the complex degree of spectral coherence plays in the spectral structure of a two-beam interference pattern is examined. It is also shown that the complex degree of spectral coherence provides a clear insight into the physical significance of cross-spectral purity. When the optical field at two points is cross-spectrally pure, the absolute value of the complex degree of spectral coherence at these points is found to be the same for every frequency component of the light. This fact is reflected in the visibility of the spectral components of the interference fringes formed by light from these points.

Fluctuations of Photon Beams: The Distribution of the Photo-Electrons

The probability distribution p(n, T) of the number of counts n from a photoelectric detector illuminated by coherent light for a time T is studied, by associating photons stochastically with Gaussian random waves. The cumulants of the distribution are derived and it is shown to be of the expected form for a boson assembly in a limited volume of phase space. The distribution depends strongly on the degeneracy of the light beam. It approaches the Poisson form for classical particles at low degeneracies and the distribution characteristic of classical waves at high degeneracies. The analysis leads, incidentally, to an expression for the extent of the unit cell of phase space in the direction of the beam. It is argued that this should be adopted as the measure of coherence length.

Fluctuations of Photon Beams and their Correlations

The distribution of counts from a photoelectric detector illuminated by light of bandwidth Δν0 is analysed by associating the photons with Gaussian random waves. This is shown to lead to a full statistical description of the counts. It is shown that the number nT in a time interval T > 1/Δν0 are simply the density fluctuations of a boson assembly in a phase space of ~ Δν0T cells. The correlation coefficient ρ of the fluctuations of counts from two detectors illuminated by partially coherent beams is found to be proportional to the local time average of the square of the coherence function γ122. The correlation is shown to depend on the degeneracy of the beams in such a way that ρ → 2γ122 for highly degenerate beams. The results are all consistent with those obtained by Hanbury Brown and Twiss in 1957.

Theory of photoelectric detection of light fluctuations

The basic formulae governing the fluctuations of counts registered by photoelectric detectors in an optical field are derived. The treatment, which has its origin in Purcells explanation of the Hanbury Brown-Twiss effect, is shown to apply to any quasi-monochromatic light, whether stationary or not, and whether of thermal origin or not. The representation of the classical wave amplitude, of the light by Gabors complex analytic signal appears naturally in this treatment. It is shown that the correlation of counts registered by N separate photodetectors at N points in space is determined by a 2Nth order correlation function of the complex classical field. The variance of the individual counts is shown to be expressible as the sum of terms representing the effects of classical particles and classical waves, in analogy to a well-known result of Einstein relating to black-body radiation. Since the theory applies to correlation effects obtained with any type of light it applies, in particular, to the output of an optical maser, although, for a maser operating on one mode, correlation effects are likely to be very small.

Quantum Effects in One-Photon and Two-Photon Interference

Although interference is intrinsically a classical wave phenomenon, the superposition principle which underlies all interference is also at the heart of quantum mechanics. Feynman has referred to interference as really ‘‘the only mystery’’ of quantum mechanics. Furthermore, in some interference experiments we encounter the idea of quantum entanglement, which has also been described as really the only quantum mystery. Clearly interference confronts us with some quite basic questions of interpretation. Despite its long history, going back to Thomas Young at the beginning of the 19th century, optical interference still challenges our understanding, and the last word on the subject probably has not yet been written. With the development of experimental techniques for fast and sensitive measurements of light, it has become possible to carry out many of the Gedanken experiments whose interpretation was widely debated in the 1920s and 1930s in the course of the development of quantum mechanics. Although this article focuses entirely on experiments with light, interference has also been observed with many kinds of material particles like electrons, neutrons, and atoms. We particularly draw the reader’s attention to the beautiful experiments with neutron beams by Rauch and co-workers and others (see, for example, Badurek et al., 1988). Quantum optical interference effects are key topics of a recent book (Greenstein and Zajonc, 1997), an extended rather thorough review (Buzek and Knight, 1995) and an article in Physics Today (Greenberger et al., 1993). The essential feature of any optical interference experiment is that the light from several (not necessarily primary) sources like SA and SB (see Fig. 1) is allowed to come together and mix, and the resulting light intensity is measured at various positions. We characterize interference by the dependence of the resulting light intensities on the optical path length or phase shift, but we need to make a distinction between the measurement of a single realization of the optical field and the average over an ensemble of realizations or over a long time. A single realization may exhibit interference, whereas an ensemble average may not. We shall refer to the former as transient interference, because a single realization usually exists only for a short time. Transient interference effects have been observed in several optical experiments in the 1950s and 1960s. (Forrester et al., 1955; Magyar and Mandel, 1963; Pfleegor and Mandel, 1967, 1968).

Coherence and indistinguishability

It has long been known that the interference produced by two light beams is related both to their mutual coherence and also to the intrinsic indistinguishability of the photon paths. With the help of a decomposition of the density operator it is shown that the degree of indistinguishability equals the degree of coherence. This provides the fundamental link between the wave and the particle descriptions.

Experimental demonstration of the violation of local realism without Bell inequalities

Abstract We report on a two-photon coincidence experiment that demonstrates the violation of local realism, as defined by Einstein, Podolsky and Rosen, by about 45 standard deviations without explicit use of Bell inequalities. The experiment is an implementation of ideas put forward by Hardy and Jordan; it depends on showing that certain coincidence rates are zero while another rate is non-zero.

Interference and the Alford and Gold Effect

A simple optical superposition experiment is analyzed in detail and a comparison is made between the photoelectric signal modulation which is characteristic of interference and the Alford and Gold effect. Some significant differences are pointed out. It is shown that the Alford and Gold effect is of the first order in the degeneracy parameter and of the second order in the degree of coherence, so that it resembles the intensity correlation effect discovered by Hanbury Brown and Twiss.

Optimum conditions for heterodyne detection of light

The problem of detecting a coherent light beam in the presence of unwanted background radiation by the heterodyne method is examined. For a sufficiently strong local-oscillator field, the detectability of the signal is unaffected by the presence of the background radiation. It is shown that, in general, there exists an optimum receiver size that maximizes the signal-to-noise ratio. This result is illustrated by several examples. A procedure for the detection of a light signal of unknown direction is suggested.