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arXiv:1110.2231 · American Journal of Physics

Energy and momentum entanglement in parametric downconversion

Pablo L. Saldanha, Carlos Henrique Monken

Published Dec 20, 2012• 12 citations
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Abstract

We present a simple treatment of the phenomenon of spontaneous parametric downconversion consisting of the coherent scattering of a single pump photon into an entangled photon pair inside a nonlinear crystal. The energy and momentum entanglement of the quantum state of the generated twin photons are seen as a consequence of the fundamental indistinguishability of the time and the position in which the photon pair is created inside the crystal. We also discuss some consequences of photon entanglement.

DOI
10.1119/1.4757623

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We present a simple treatment of the phenomenon of spontaneous parametric downconversion consisting of the coherent scattering of a single pump photon into an entangled photon pair inside a nonlinear crystal. The energy and momentum entanglement of the quantum state of the generated twin photons are seen as a consequence of the fundamental indistinguishability of the time and the position in which the photon pair is created inside the crystal. We also discuss some consequences of photon entanglement.
In the process of spontaneous parametric downconversion, photons from a pump field are converted to signal and idler photon pairs in a nonlinear crystal. The reversed process, or upconversion of these pairs back to single photons in a second crystal, is also possible. Here, we present experimental measurements of the upconversion rate with a controlled time delay introduced between the signal and idler photons. As a function of delay, this rate presents a full width at half maximum of 27.9 fs under our experimental conditions[1].
A theory of spontaneous parametric downconversion in a nonlinear crystal that is placed inside a microcavity in the degenerate, collinear, type I arrangement is given. Both atom and field variables are fully quantized. The photon-correlation function for the entangled photon pairs produced has a much longer correlation time (~0.4 ns) than that without the cavity (of the order of subpicoseconds) and can be measured directly with fast photodetectors. We also calculate the resonant suppression and enhancement in the photon-pair count rate as a function of cavity length. The results are in qualitative agreement with recent downconversion experiments [Phys. Rev. Lett. 85, 2556 (1999)] in a 4.5-mm cavity.
The process of upconversion in a nonlinear crystal involves the annihilation of two photons and the generation of a single photon with frequency given by the sum of the incident photon frequencies. In this paper we discuss recent work where we exploit the upconversion of two photonic modes, each of them generated by the process of spontaneous parametric downconversion, either from the same or different crystals. On the one hand, we describe an experiment where we exploit upconversion for the temporal characterization of photon pairs. On the other hand, we describe an experimental technique, which relies on the detection of an upconverted photon from two distinct photon-pair sources, which permits two photons that never interacted to become entangled in frequency.
In spontaneous parametric down-conversion, pairs of photons are known to be created coherently and with equal probability over the entire length of the crystal. Then, there is no particular position in the crystal where a photon pair is created. We make the seemingly contradictory observation that the time delay between the photons in the pair depends on the distance from the crystal to the collection lens, as if the photons were actually collected preferentially from a particular position in the crystal. We resolve this contradiction by showing that the spatio–temporal correlations critically affect the temporal properties of the pair of photons. The theoretical model we present matches all our experimental results. We expect this to have important implications for experiments that require indistinguishable photons.
We study the process of spontaneous parametric downconversion in a one-dimensional defective quadratic nonlinear photonic crystal. It is shown that the strong confinements of both the pump and signal waves around the defective layer result in a giant enhancement of the entangled photon-pair generation. An enhancement factor as high as 3.4×106 is obtained in our defective structure based on the dual-localized modes. Furthermore, the linewidth of the downconverted fields is only 0.1 nm. Such a photonic crystal structure can be applied as a highly efficient source of entangled photon pairs for highly integrated all-optical circuits.