Introduction: States That Look Classical
Coherent states are eigenstates of the lowering operator \(\hat{a}\). They maintain minimum uncertainty, follow classical trajectories, and describe laser light.
Definition
\[\hat{a}|\alpha\rangle = \alpha|\alpha\rangle\]where \(\alpha\) is any complex number.
Explicit form:
\[|\alpha\rangle = e^{-|\alpha|^2/2}\sum_{n=0}^{\infty}\frac{\alpha^n}{\sqrt{n!}}|n\rangle\]Properties
- Minimum uncertainty: \(\Delta x \cdot \Delta p = \hbar/2\)
- Expectation values: \(\langle x\rangle = \sqrt{2\hbar/m\omega}\text{Re}(\alpha)\), oscillates like classical position
- Poisson distribution of photon number: \(P(n) = e^{-|\alpha|^2}|\alpha|^{2n}/n!\)
- Not orthogonal: \(\langle\beta|\alpha\rangle = e^{-|\alpha-\beta|^2/2}e^{(\bar{\beta}\alpha - \bar{\alpha}\beta)/2}\)
Time Evolution
\[|\alpha(t)\rangle = |e^{-i\omega t}\alpha\rangle\]The coherent state remains coherent! It just rotates in phase space.
The Quantum Connection
Coherent states describe laser light, where photon number fluctuates but the phase is well-defined. They're the closest quantum mechanics gets to classical oscillation. Squeezed states, used in gravitational wave detection, are generalizations where \(\Delta x\) or \(\Delta p\) is reduced below the coherent state value.