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Journal of Optics B Quantum and Semiclassical Optics

Comparative study of photon antibunching of non-stationary fields

Adam Miranowicz, J. Bajer, Hideaki Matsueda, Mohamed Ridza Wahiddin, R. Tanaś

Published Aug 13, 1999• 18 citations
Publisher

Abstract

Two standard criteria of the photon antibunching for non-stationary fields are compared. A new criterion, obtained by direct application of the Cauchy-Schwarz inequality, is proposed. All three definitions, based on the two-time correlation functions, are equivalent for stationary fields. However, the photon antibunching in the non-stationary regime is demonstrated to be not uniquely defined, since different criteria can lead to self-contradictory predictions. As an example, photon correlations of the signal mode in the parametric frequency converter are analysed analytically.

DOI
10.1088/1464-4266/1/5/302

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Two standard criteria of the photon antibunching for non-stationary fields are compared. A new criterion, obtained by direct application of the Cauchy-Schwarz inequality, is proposed. All three definitions, based on the two-time correlation functions, are equivalent for stationary fields. However, the photon antibunching in the non-stationary regime is demonstrated to be not uniquely defined, since different criteria can lead to self-contradictory predictions. As an example, photon correlations of the signal mode in the parametric frequency converter are analysed analytically.
Previous work on the two-time photon-number correlations of quantum non-stationary fields has focused on the demonstration of self-contradictory predictions of the photon antibunching according to different definitions. In this paper we analyse the usefulness of the conventional and generalized definitions of photon bunching in a description of classical non-stationary light. It is proved that the generalized definition applied to classical fields predicts photon bunching but never antibunching. In contrast, the two conventional definitions are shown to improperly classify certain classical non-stationary fields as being antibunched.
Recently, Miranowicz et al compared three different definitions of antibunching in non-stationary fields and found that they are by no means equivalent. In particular, they may lead to contradictory predictions of the photon antibunching. We consider another violation of the Cauchy-Schwarz inequality in non-classical and non-stationary fields which we term as hyperbunching. Similarly, this leads to three different definitions of hyperbunching in non-stationary fields which, again, are by no means equivalent and may lead to contradictory predictions of this phenomenon. We analyse these different concepts of anti- and hyperbunching in the non-stationary resonance fluorescence from a bichromatically driven two-level atom.
We analyze the most popular approaches to photon antibunching as exemplified in a sim-ple model of second-harmonic generation. We find the exact evolution of the fundamental and harmonic fields for initial Fock states with small photon numbers. We show explicitly that definitions of photon antibunching based on the unnormalized, G(2)(t, t′), and normal-ized, g(2)(t, t′), two-time correlation functions describe distinct phenomena for nonstationary fields. For completeness, we also compare antibunching effects defined by the two-time and single-time correlations. 1 Photon antibunching Photon antibunching and sub-Poisson photon statistics (also called antibunching) reveal the quan-tum nature of light and therefore have received a great deal of attention during the last decades. The first detection of antibunched light by Kimble, Dagenais and Mandel [1] and first detection of sub-Poisson light by Short and Mandel [2] are regarded as milestones in the progress in quantum optics. In spite of many experimental and theoretical achievements, a deeper analysis of these effects seems to be necessary, particularly in the case of nonstationary fields. We shall clarify some of these aspects by referring to an example of second-harmonic generation. There are several commonly used definitions of photon antibunching for a single-mode radiation field (e.g., Refs. [1]–[9]): Definition I: The photon antibunching (see, e.g., Ref. [3]) occurs if the two-time light intensity correlation function G(2)(t, t+ τ) increases from its initial value at τ = 0,
Some general properties of photon correlations are discussed in a simple way through an analysis of the two-detector measurement scheme. It is shown that the assumption of the discreteness of the random process leads directly to the conclusion that the zero-delay value of the correlation function is only bound to be non-negative. The adopted approach allows discussing in a more intuitive way the photon correlation properties of different optical fields, including non-classical fields presenting an apparent violation of the Cauchy-Schwarz inequality. The comparison between the two- and the single-detector experiment clarifies the role of the operator ordering in the definition of the correlation function.
Under the classical (strong) pump condition, the Hamiltonian involving the signal and the idler modes of a non-degenerate parametric oscillator is exhibited. Without using the usual rotating wave approximation (RWA), the analytical solutions of the field operators are used to investigate the antibunching of photons of the input radiation field coupled to the non-degenerate parametric oscillator. By using the symbolic calculation, the antibunching of photons for both the signal and idler modes are investigated. In particular, the effects of the inclusion of rotating wave approximated terms on the antibunching of photons are clearly indicated. To substantiate the analytical results, the temporal evolution of signal photon and idler photons, and the antibunching effects of the signal and idler modes are investigated numerically by using the QuTip 3.1.0. The exact numerical results obtained by QuTip 3.1.0 matches extremely well with those of the analytical results. The present article and hence the analytical method might be of use for investigating the situations of having ultra-strongly and deep-strongly coupled systems where the possibilities of using RWA is completely ruled out.