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arXiv:quant-ph/0306116 · Physical Review A

Simultaneous near-field and far-field spatial quantum correlations in the high-gain regime of parametric down-conversion

E. Brambilla, A. Gatti, Morten Bache, Luigi A. Lugiato

Published Feb 9, 2004• 197 citations
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Abstract

We study the spatial correlations of quantum fluctuations that can be observed in multimode parametric down-conversion in the regime of high gain. We investigate both a type-I and a type-II phase-matching configuration: in the latter case spatial correlations at the quantum level are shown to exist both in the near-field and in the far-field zones of the down-converted light. In the stationary and plane-wave approximation we treat the problem analytically. A stochastic model is solved numerically to obtain quantitative results beyond this approximation. The finite transverse size and pulse duration of the pump beam and other features of the system, such as spatial walk-off and diffraction are taken into account, and we show that correlations beyond the standard quantum limit exist for values of parameters consistent with realistic experiments.

DOI
10.1103/physreva.69.023802

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We study the spatial correlations of quantum fluctuations that can be observed in multimode parametric down-conversion in the regime of high gain. We investigate both a type-I and a type-II phase-matching configuration: in the latter case spatial correlations at the quantum level are shown to exist both in the near-field and in the far-field zones of the down-converted light. In the stationary and plane-wave approximation we treat the problem analytically. A stochastic model is solved numerically to obtain quantitative results beyond this approximation. The finite transverse size and pulse duration of the pump beam and other features of the system, such as spatial walk-off and diffraction are taken into account, and we show that correlations beyond the standard quantum limit exist for values of parameters consistent with realistic experiments.
Using a 1 GW, 1 ps pump laser pulse in high-gain parametric down conversion allows us to detect sub-shot-noise spatial quantum correlation with up to 100 photoelectrons per mode by means of a high efficiency charge coupled device. The statistics is performed in single shot over independent spatial replica of the system. Evident quantum correlations were observed between symmetrical signal and idler spatial areas in the far field. In accordance with the predictions of numerical calculations, the observed transition from the quantum to the classical regime is interpreted as a consequence of the narrowing of the down-converted beams in the very high-gain regime.
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We consider travelling-wave parametric down-conversion in the high-gain regime and present the experimental demonstration of the quantum character of the spatial fluctuations in the system. In addition to showing the presence of sub-shot noise fluctuations in the intensity difference, we demonstrate that the peak value of the normalized spatial correlations between signal and idler lies well above the line marking the boundary between the classical and the quantum domain. This effect is equivalent to the apparent violation of the Cauchy-Schwartz inequality, predicted by some of us years ago, which represents a spatial analogue of photon antibunching in time. Finally, we analyse numerically the transition from the quantum to the classical regime when the gain is increased and we emphasize the role of the inaccuracy in the determination of the symmetry center of the signal/idler pattern in the far-field plane.
We consider travelling-wave parametric down-conversion in the high-gain regime and present the experimental demonstration of the quantum character of the spatial fluctuations in the system. In addition to showing the presence of sub-shot noise fluctuations in the intensity difference, we demonstrate that the peak value of the normalized spatial correlations between signal and idler lies well above the line marking the boundary between the classical and the quantum domain. This effect is equivalent to the apparent violation of the Cauchy–Schwartz inequality, predicted by some of us years ago, which represents a spatial analogue of photon antibunching in time. Finally, we analyse numerically the transition from the quantum to the classical regime when the gain is increased and we emphasize the role of the inaccuracy in the determination of the symmetry centre of the signal–idler pattern in the far-field plane.
We analyze the spatial aspects of the down-converted field emitted by a cavityless optical parametric amplifier, both from theoretical/numerical and experimental viewpoints. Our model neglects pump depletion and includes a classical description of the noise which initiates the process. The ringshaped angular distribution in the far field is determined basically by the phase-matching conditions. In the near field, the configuration arises from the superposition of contributions from a broad frequency band; it is shown that the finite duration of the pump pulse introduces a correlation among neighboring frequencies that allows for the appearance of a spotlike spatial modulation. When the spatial configuration of the pump field is changed from a plane wave to a narrow Gaussian, the near-field pattern assumes the form of a roll structure. The experimental observations obtained with a lithium triborate crystal agree qualitatively well with the theoretical picture.
The state of spatially correlated down-converted photons is usually treated as a two-mode Gaussian entangled state. While intuitively this seems to be reasonable, it is known that new structures in the spatial distributions of these photons can be observed when the phase-matching conditions are properly taken into account. Here, we study how the variances of the near- and far-field conditional probabilities are affected by the phase-matching functions, and we analyze the role of the EPR-criterion regarding the non-Gaussianity and entanglement detection of the spatial two-photon state of spontaneous parametric down-conversion (SPDC). Then we introduce a statistical measure, based on the negentropy of the joint distributions at the near- and far-field planes, which allows for the quantification of the non-Gaussianity of this state. This measure of non-Gaussianity requires only the measurement of the diagonal covariance sub-matrices, and will be relevant for new applications of the spatial correlation of SPDC in CV quantum information processing.