= Spontaneous parametric down-conversion
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In <spontaneous parametric down-conversion>, a strong pump drives a second-order <nonlinear optical susceptibility>, producing paired output <photons> in initially unseeded modes. The two outputs are conventionally called the <signal photon> and <idler photon>. They need not have equal <frequencies>. For a monochromatic pump, <conservation of energy> gives
$$
\omega_p=\omega_s+\omega_i.
$$
Efficient bulk generation also requires <phase matching>: the nonlinear source and emitted waves must remain in phase. In the ideal plane-wave geometry this is $\mathbf k_p=\mathbf k_s+\mathbf k_i$. Different <refractive indices> and propagation directions determine which pairs satisfy it. A finite interaction <length> supplies a mismatch bandwidth rather than an infinitely sharp <wavevector> constraint. <Calculating Characteristics of Non-collinear Phase-matching in Uniaxial and Biaxial Crystals by N. Boeuf and others>[Sections I–II.1, printed pp. 2–3, equations 1–2]{ref}
At weak gain, the <SPDC pair state> contains a large <vacuum> component and a small pair amplitude. Detecting one output conditions the other, producing a <heralded single-photon source>. This is probabilistic: stronger pumping also increases unwanted multiple-pair events. Experiments collect two output modes with <optical filters> and <single-photon detectors>, comparing coincident detections with background counts. <Sources of Nonclassical Light for Quantum Networks by NIST>[Spontaneous Parametric Down-Conversion and Figure 2, pair generation and heralding]{ref}
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