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Physical Review A

SU(2) coherent states in parametric down-conversion

Alfredo Luis, Jan Peřina

Published Mar 1, 1996• 38 citations
Publisher

Abstract

The states generated from the vacuum in two parametric down-conversion crystals with aligned idler beams are studied. It is shown that after a measurement of the photon number in some of the modes, the emerging states can be related to SU(2) and SU(1,1) coherent and minimum-uncertainty states. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physreva.53.1886

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The states generated from the vacuum in two parametric down-conversion crystals with aligned idler beams are studied. It is shown that after a measurement of the photon number in some of the modes, the emerging states can be related to SU(2) and SU(1,1) coherent and minimum-uncertainty states. \textcopyright{} 1996 The American Physical Society.
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An interference experiment with signal and idler photons produced by parametric down-conversion in two nonlinear crystals is described and analyzed theoretically. It is found that when the idlers for the two crystals are superposed and aligned, the idler photon from the first crystal can induce coherence between the two signals, but without inducing any additional emission. Blocking the first idler wipes out the interference. The implications for the interpretation of the quantum state vector are discussed, and this leads to the conclusion that the state reflects not only what is known, but to some extent also what is knowable in principle.
Using a multidimensional Gaussian approximation of the wave function for the signal and idler light generated by spontaneous parametric down-conversion, we derive analytical expressions for the second-order coherence function and the fourth-order coherence function (which is proportional to the signal-idler photon coincidence rate). The magnitudes of these functions are expressed as products of Gaussian functions of the azimuthal angles, the polar angles, and the time delay. Their widths determine six parameters: the coherence angles and coherence time, and the entanglement angles and entanglement time. We show how these parameters are governed by the pump-beam waist, the pump spectral width, and the crystal length. We thereby derive relations analogous to the van Cittert--Zernike theorem and the Siegert relation for thermal light. We show that the normalized photon coincidence rate decreases sharply as the signal and idler apertures become mismatched or misaligned. We experimentally confirm this latter prediction by using parametrically down-converted light obtained from a LiIO3{\mathrm{LiIO}}_{3}LiIO3​ crystal pumped by Kr+{\mathrm{Kr}}^{+}Kr+-ion laser radiation at 413 nm. \textcopyright{}1996 The American Physical Society.