A coherent state is a specific type of quantum state of a harmonic oscillator that exhibits properties reminiscent of classical oscillators. In quantum mechanics, a coherent state is often associated with the eigenstates of the annihilation operator in the context of quantum optics and quantum mechanics of harmonic oscillators. Key characteristics of coherent states include: 1. **Minimum Uncertainty**: Coherent states minimize the uncertainty relation between position and momentum; they achieve the Heisenberg uncertainty principle's lower bound.
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A coherent state of one bosonic mode is a normalized eigenstate of its annihilation operator: . Equivalently it is a displaced vacuum, , with complex amplitude specifying its mean quadratures.
Its occupation probabilities are Poisson distributed with mean and variance . In a quantum harmonic oscillator, the state evolves to a phase times , so its mean position follows the classical sinusoidal motion while its width stays fixed. This combination makes it a useful quantum description of a classical-like oscillating field. It is not a state of definite photon number, and a global phase multiplying its vector differs from the physical phase encoded in the relative number-state amplitudes.