Photonnumber

Photon number refers to the count of photons in a given mode of the quantized electromagnetic field or in a light state. In the quantum description, each mode is described by creation and annihilation operators a† and a, which satisfy the commutation relation [a, a†] = 1. The photon number operator for a single mode is n̂ = a† a. Its eigenstates |n⟩, called Fock states, have definite photon number n, with n̂|n⟩ = n|n⟩. The energy of a single mode is E = n ħ ω, where ω is the mode frequency, so states with a definite photon number have discrete energies. For multiple modes, the total photon number operator is N̂ = ∑k a_k† a_k, and the corresponding eigenstates are |n1, n2, …⟩.

Measurement and detectors: Photon counting detectors aim to measure the photon number in a mode or a set of modes. Photon-number-resolving detectors (PNRDs) can discriminate different photon counts, though real devices are imperfect, with finite quantum efficiency and dark counts. In experiments, the measured statistics reflect the underlying quantum state; for example, coherent states exhibit a Poisson distribution of photon numbers with mean ⟨n⟩.

States and statistics: Fock states have a definite photon number, while coherent states have a distribution of photon numbers. Some quantum states exhibit nonclassical statistics, such as sub-Poissonian distributions or squeezing, where photon-number correlations reveal quantum features beyond classical light.

Applications: Photon number is central in quantum optics, quantum information, and metrology, where control over light at the level of individual photons enables protocols for communication, computation, and precision measurement.