Photon statistics

Light is detected in discrete photons, and the number counted in a fixed time fluctuates even when the average power is perfectly steady. How it fluctuates depends on the source. A beam of power $P$ at wavelength $\lambda$ delivers on average $P\lambda/(hc)$ photons per second, 7.8 × 10⁹ per second for 1 nW at 1550 nm and 7.8 million per second for 1 pW, and the spread of counts around that mean is the photon statistics.

Three kinds of light

Coherent light, the ideal output of a laser well above threshold, has a Poisson distribution: the variance of the count equals its mean $\bar n$, so the relative fluctuation is $1/\sqrt{\bar n}$. A measurement that collects 100 photons has a signal-to-noise ratio of 10; one that collects 10,000 photons, 100. This is the origin of shot noise, the standard quantum limit of direct detection.

Thermal or chaotic light, from lamps, the Sun, LEDs and amplified spontaneous emission, has a Bose-Einstein distribution within one mode, with variance $\bar n + \bar n^2$: photons tend to arrive in bunches. Because real thermal sources span many modes and detectors average over them, the excess noise is usually small in practice, but it is measurable, and in optical amplifiers it appears as signal-spontaneous and spontaneous-spontaneous beat noise.

Some nonclassical light has variance below the mean, sub-Poissonian statistics. A true single-photon source emits exactly one photon per trigger; squeezed light reduces the fluctuations of one quadrature below the shot-noise level. These states are the resources of quantum communication and sensing, such as quantum key distribution.

The second-order correlation

The distinction is measured with the normalized intensity correlation at zero delay,

which is 1 for coherent light, 2 for single-mode thermal light (photon bunching) and 0 for an ideal single-photon source (antibunching). A measured $g^{(2)}(0)$ below 0.5 is the usual evidence that a source emits single photons.

Measurement

$g^{(2)}$ is measured with a Hanbury Brown-Twiss setup: the light is split on a 50:50 beamsplitter onto two single-photon detectors, and a time tagger histograms the delays between detections on the two arms. Photon-number distributions are measured with photon-number-resolving detectors such as transition-edge sensors, or with superconducting nanowire arrays. For bright beams, the same physics is observed as the noise spectrum of a photocurrent, compared with the shot-noise level set by a balanced detector.

References: R. Loudon, The Quantum Theory of Light, 3rd ed. (Oxford University Press, 2000); R. Hanbury Brown, R. Q. Twiss, Nature 177, 27 (1956); M. Fox, Quantum Optics: An Introduction (Oxford University Press, 2006).