Physics
Coherent States
Quick fact
Coherent states are eigenstates of the annihilation operator, meaning they are unchanged (except for a factor) when the number of quanta is reduced by one.
Why this is interesting
You know laser light is intensely bright and smooth, but why does it behave so classically when it's actually a quantum phenomenon? The secret lies in a special quantum state called a coherent state.
Read the full explanation
Understanding Coherent States
Imagine a pendulum swinging back and forth. Classically, you can describe its position and momentum perfectly. In the quantum world, a harmonic oscillator (like a vibrating atom or a light wave) can't have both position and momentum known exactly—there's fundamental uncertainty. But some quantum states come close to the classical ideal: coherent states. They are wavepackets that oscillate without spreading, and their uncertainties in position and momentum are equal and minimal, just as the Heisenberg uncertainty principle allows. Light from a laser is a wonderful example: the photons are not in a definite number state but in a coherent superposition that produces a well-defined oscillating electric field.
A deeper explanation
Mathematically, a coherent state |α⟩ is defined as the eigenstate of the annihilation operator: â|α⟩ = α|α⟩, where α is a complex number representing the amplitude and phase of the oscillation. It is generated by applying the displacement operator D̂(α) = exp(α↠- αâ) to the vacuum state |0⟩: |α⟩ = D̂(α)|0⟩. This operator shifts the vacuum in phase space, creating a state that evolves in time like a classical harmonic oscillator while preserving its shape. Coherent states form an overcomplete basis, meaning any quantum state can be expanded in terms of them, making them a powerful tool for calculations. Their importance extends beyond light: they describe the quantum behavior of superconducting circuits, trapped ions, and other bosonic systems. In quantum information, coherent states are used for quantum key distribution and continuous-variable quantum computing.