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Annual Meeting Optical Society of America

Photon statistics of superimposed nonclassical fields

Malvin Carl Teich, Bahaa E. A. Saleh

Published Jan 1, 1988• 0 citations
Publisher

Abstract

Antibunching, photon-number squeezing, and quadrature squeezing represent characteristics of nonclassical electromagnetic fields that have been observed in the laboratory. Experiments involving nonclassical light often make use of beam splitters to superimpose light beams. We discuss the photon statistics that arise from the superimposition of two arbitrary (not necessarily independent) quantum fields at a device such as a beam splitter. The photomixing density operator of the superimposition field is presented, and the behavior of the balanced mixer is examined in the general case.1 The failure of classical probability theory as a descriptor of the relationship between the photon statistics at the output and at the input of the beam splitter is elucidated.

DOI
10.1364/oam.1988.ma2

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Antibunching, photon-number squeezing, and quadrature squeezing represent characteristics of nonclassical electromagnetic fields that have been observed in the laboratory. Experiments involving nonclassical light often make use of beam splitters to superimpose light beams. We discuss the photon statistics that arise from the superimposition of two arbitrary (not necessarily independent) quantum fields at a device such as a beam splitter. The photomixing density operator of the superimposition field is presented, and the behavior of the balanced mixer is examined in the general case.1 The failure of classical probability theory as a descriptor of the relationship between the photon statistics at the output and at the input of the beam splitter is elucidated.
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.
Using the quantum theory of a lossless beam splitter, we determine the joint and marginal photon-number distributions at its output ports, for two arbitrary (not necessarily independent) input fields. The joint probability at the output of the beam splitter turns out to be a Fourier series in the relative phase between the output beams. Photomixing with the number states exhibits no phase sensitivity. An elementary example of phase sensitive dependence in the photon-number domain is obtained with the Bernoulli states, which are a subclass of the binomial states.1 For a joint input Bernoulli state, the probability of detecting one photon (or zero photons) at either output port varies sinusoidally with the relative phase. For independent Bernoulli states incident on the beam splitter, the results differ and depend on whether the input is in a pure or mixed state. Photomixing of other nonclassical states exhibit higher orders of phase-induced modulation of the output photon-number probabilities.
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.
Properties of the nonclassical light (NCL) have been considered with an emphasis on experimentally-observed features, which are underlain by the well-known Mandel's formula connecting the statistics of photocounts with that of light falling on the detector. A systematic operational approach is presented to study the NCL using two parallel sets of numbers measured: probabilities of photocounts {pm} and normalised factorial moments of counts {gk}. Two particular examples are examined in detail: a 'heated' squeezed vacuum and a 'heated' one-photon state. An alternative method is proposed to discover the week nonclassicality using 'generalised' moments {ak(s)}. The effect of the linear absorption (amplification) and of the beam-splitting of the NCL, and the relation between the NCL and the absolute calibration of photodetectors are considered. The conditions are elucidated whereat the beam-splitter realises a mathematical operation of superposition of two one-mode fields useful in studying the NCL.