Coherent States—From Single Photons to Beams of Light

Abstract

A beam of light is composed of single photon states. The corresponding electric field can be expressed as a combination of coherent states that are designed to be the eigenstates of the electric field. We show that the coherent states can be expressed in terms of the number states, and develop a displacement operator that will create a coherent state from the vacuum. This development shows that the coherent states are a combination of number states distributed according to a Poisson distribution. An example of a beam of coherent light is a laser operating well above threshold. Photon emission from a laser is well described by Poisson statistics, and the second-order correlation coefficient is 1 for all times. That is, photons emitted from a laser are uncorrelated. Photons emitted from a thermal source obey Bose-Einstein statistics. Thus, they are not coherent. Correlations between photons emitted from a thermal source were first measured by Hanbury Brown and Twiss. Their experiment raised the curtain on the field of quantum optics. We show that the second-order correlation coefficient \( g^{2} \left( {\tau = 0} \right) \) for photons obeying Bose-Einstein statistics is 2. The Hanbury Brown and Twiss experiment confirms this result. If one photon is emitted by a thermal source, another photon will be close at hand. Photons are bunched together. Single photons represent a third type of photon state. A single photon that encounters a beam-splitter will be either reflected or transmitted. Simultaneous detection of transmission and detection never occurs, and the photons are anti-correlated. A photon beam having this characteristic is called non-classical light.