coherent state
A coherent state is a family of quantum states of the harmonic oscillator (e.g., an electromagnetic field mode) that minimize the Heisenberg uncertainty relation. Mathematically, it is an eigenfunction of the annihilation operator a, a|α⟩ = α|α⟩, where the complex number α specifies the mean amplitude and phase. Expanding in number (Fock) states reveals a Poissonian photon statistics: the probability of detecting exactly n photons is e^{-|α|²} |α|^{2n}/n!. The average energy is ħω|α|². Such states do not spread over time and preserve the shape of the wave packet, which is why they often serve as a bridge between quantum and classical descriptions.
History
Erwin Schrödinger discovered this state in 1926 while trying to find a quantum description as close as possible to the familiar motion of a pendulum. Later, with the invention of the laser, it became a practical tool.
How it works
Imagine a swing: if you push it rhythmically, the amplitude grows and the oscillation becomes smooth. In the quantum world, the 'swing' is the electromagnetic field. A coherent state arises when the field is pumped by a laser—it oscillates with a well-defined amplitude (extent) and phase (shift). In this case, the uncertainty is evenly distributed: we are slightly wrong in both amplitude and phase, but the total error is minimal.
💡 When reflecting coherent light from an ordinary mirror, it remains coherent—losses only reduce α. That's why laser pointers work even after hundreds of kilometers in optical fiber.