A pump photon striking the nonlinear crystal only occasionally splits into a correlated pair: a lower-energy "signal" photon and a lower-energy "idler" photon, created at the exact same instant but sent off in two different directions. Because the pair is always born together, a click on the idler-arm detector heralds — announces — that a signal photon must exist in the signal arm right now, without ever measuring the signal photon itself. That's what makes this a genuinely on-demand, verified single-photon source, unlike an attenuated laser which fires at unpredictable moments with no advance warning.
P(n pairs) = μⁿ / (1+μ)ⁿ⁺¹ (single-mode thermal statistics)
P(click) = μ / (1+μ)
purity = P(n=1 | click) = 1 / (1+μ)
- Pump power — sets the mean pair number μ per pulse. Higher power = more pairs generated per second, but also a fatter tail of multi-pair pulses.
- Herald click — the idler detector firing. It cannot tell whether one pair or several were created that pulse — it only fires or doesn't.
- Multi-pair leak — a pulse where 2+ pairs were created simultaneously. The herald still reports "one photon ready", but the signal arm actually held more than one — corrupting the single-photon guarantee.
- Heralding purity — the fraction of herald clicks that truly correspond to exactly one signal photon. It's highest at low pump power (rare, clean pairs) and degrades as power rises (more multi-pair contamination) — the core rate-vs-purity trade-off of every heralded photon source.
Real-world relevance: every quantum-optics experiment that needs "exactly one photon, and I know precisely when" — quantum key distribution, loophole-free Bell tests, linear-optical quantum computing — runs its SPDC crystal at a deliberately low pump power for this reason, trading generation rate for purity.