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Journal of Modern Optics

Quantum signatures in the interference of macroscopic signal–idler beams

Luigi A. Lugiato, A. Gatti, E. Brambilla, Lucia Caspani

Published Feb 17, 2010• 1 citation
Publisher

Abstract

The interference of signal and idler waves produced by optical parametric down-conversion was the object of a classic paper by Ghosh and Mandel [Phys. Rev. Lett. 1987, 59, 1903–1905] many years ago. In this paper we investigate the problem in the continuous variable regime, with the aim of identifying quantum signatures that arise from the entanglement of signal and idler waves. By using a model which allows for analytical calculations we show first of all that no quantum features appear for the interference in the fourth-order correlation function in the high-gain regime of parametric down-conversion. The quantum signatures emerge instead from the spatio-temporal correlation function of a generic quadrature component, and we demonstrate that they are clearly linked to the signal/idler entanglement. Relevant and useful links with the theory of quantum images complete the picture.

DOI
10.1080/09500340903576441

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The interference of signal and idler waves produced by optical parametric down-conversion was the object of a classic paper by Ghosh and Mandel [Phys. Rev. Lett. 1987, 59, 1903–1905] many years ago. In this paper we investigate the problem in the continuous variable regime, with the aim of identifying quantum signatures that arise from the entanglement of signal and idler waves. By using a model which allows for analytical calculations we show first of all that no quantum features appear for the interference in the fourth-order correlation function in the high-gain regime of parametric down-conversion. The quantum signatures emerge instead from the spatio-temporal correlation function of a generic quadrature component, and we demonstrate that they are clearly linked to the signal/idler entanglement. Relevant and useful links with the theory of quantum images complete the picture.
In the Wigner formalism, after giving a general description of a light beam, the theory of parametric down-conversion is developed to second-order in the coupling constant. We then describe the detection process by calculating the auto-correlation and cross-correlations of the signal and idler beams. Four recent experiments are analyzed in detail: interference on a screen, Franson's experiments [Phys. Rev. Lett. 62, 2205 (1989)], Rarity and Tapster's experiments [Phys. Rev. Lett. 64, 2495 (1990)], and induced coherence without induced emission [X. Y. Zou, L. J. Wang, and L. Mandel, Phys. Rev. Lett. 67, 318 (1991)].
A theoretical construct is presented for fourth-order interference between the signal and the idler beams of a parametric downconverter. Previous quantum treatments of fourth-order interference have employed correlated single-photon wave packets. The introduced approach, however, relies on Gaussian-state field correlations, which were previously used to characterize quadrature-noise squeezing produced by an optical parametric amplifier and nonclassical twin-beam generation in an optical parametric oscillator. Three principal benefits accrue from the correlation-function formalism. First, the quantum theory of fourth-order interference is unified with that for the other nonclassical effects of χ(2) interactions, i.e., squeezing and twin-beam production. Second, the semiclassical photodetection limit on Gaussian-state fourth-order interference is established; a purely quantum effect can be claimed at fringe visibilities substantially below the 50% level. Finally, both photon-coincidence counting (within the low-photon-flux regime) and intensity interferometry (in the high-photon-flux limit) are easily analyzed within a common framework.
We explore semiclassical and quantum descriptions of fourth-order interference that are built upon classical and quantum Gaussian-state models for the signal and idler outputs from a parametric down converter. Our approach unifies the analysis of fourth-order interference with those for squeezing and photon twins. It also demonstrates that a purely-quantum fourth-order interference effect can be claimed at fringe visibilities substantially below the commonly accepted threshold of 50%. We start with a pair of models - one quantum and one classical - for the signal and idler fields produced by parametric down conversion.>
We herein study the rate of coincidences in a Hong–Ou–Mandel interferometer exit whose arms have been supplemented by two optical cavities which reflect the photons generated by parametric down conversion. The fourth-order correlation function at the beam-splitter exit is calculated. In the regime where the coherence length of the photons is larger than the cavities lengths, photon coalescence and anti-coalescence interferences are observed. In particular, complete anti-coalescence between photons generated by parametric down conversion Type I is evidenced. The photons reshaping as they are reflected by the cavities are also studied and path diagrams for the indistinguishable processes that lead to quantum interference are presented.
This chapter discusses two reported experiments in which the interference between signal and idler photons generated in the process of parametric down-conversion were investigated. Although two-photon interference effects have been treated theoretically before, it analyzes the two experiments in some detail to show how differences between the predictions of classical and quantum theories appear. The chapter compares the predictions of classical and quantum theories with the experimental results obtained. As practically all detectors of light rely on the photoelectric effect, in which a free electron is produced at the expense of a photon, the photon annihilation operator plays a key role in describing the measurement of light in the quantum theory. The time correlation between signal and idler photons produced in the parametric down-conversion process was first measured with a resolving time of about 10 nanoseconds. The interference effect can be made the basis of a measurement of the length of a photon wave packet with femtosecond time resolution.