  We demonstrate a direct single-source measurement of absolute nonlinearity of
single-photon detectors with unprecedented accuracy. We discover a surprising
supralinear behavior of single-photon avalanche diodes and show that it cannot
be explained using known theoretical models. We also fully characterize sub-
and supra-linear operation regimes of superconducting nanowire single-photon
detectors and uncover the supralinearity under faint continuous illumination.
The results identify new detector anomalies that supersede existing knowledge
of nonlinear effects at the single-photon level.
