Coherent States (Intro)

Number states $|n\rangle$ are the energy eigenstates, but they are deeply non-classical: a number state has $\langle\hat{x}\rangle = 0$ at all times, so it never oscillates the way a classical mass on a spring does. The states that do behave classically are the coherent states, and they have an elegant definition in terms of the annihilation operator.

Definition: eigenstates of the annihilation operator

A coherent state $|\alpha\rangle$ is an eigenstate of the (non-Hermitian) annihilation operator,

a|\alpha\rangle = \alpha\,|\alpha\rangle,

where the eigenvalue $\alpha$ is any complex number. Because $a$ is not Hermitian, $\alpha$ need not be real — and the two real degrees of freedom in $\alpha$ will turn out to encode the classical position and momentum.

Expansion in number states

Writing $|\alpha\rangle = \sum_n c_n |n\rangle$ and imposing $a|\alpha\rangle = \alpha|\alpha\rangle$ with $a|n\rangle = \sqrt{n}|n-1\rangle$ gives the recursion $c_n = \alpha\,c_{n-1}/\sqrt{n}$ , hence $c_n = \alpha^n c_0/\sqrt{n!}$ . Normalizing fixes $c_0 = e^{-|\alpha|^2/2}$ , so

|\alpha\rangle = e^{-|\alpha|^2/2}\sum_{n=0}^{\infty}\frac{\alpha^n}{\sqrt{n!}}\,|n\rangle.

A coherent state is therefore a specific superposition of all number states, not an energy eigenstate. The ground state is the special case $\alpha = 0$ : $|0\rangle$ is itself coherent.

Photon-number statistics are Poissonian

The probability of finding $n$ quanta in $|\alpha\rangle$ is

P(n) = |c_n|^2 = e^{-|\alpha|^2}\,\frac{(|\alpha|^2)^n}{n!},

a Poisson distribution with mean $\bar{n} = |\alpha|^2$ . The number of quanta is therefore not sharp; it fluctuates with variance also equal to $|\alpha|^2$ (the hallmark of Poisson statistics). This is exactly the photon-counting distribution of an ideal laser, which is why laser light is modelled by coherent states.

Why they are "quasi-classical"

Two properties make coherent states the closest quantum analogue of a classical oscillator:

Minimum uncertainty. Like the ground state, $|\alpha\rangle$ saturates $\Delta x\,\Delta p = \hbar/2$ , with equal spread in $x$ and $p$ . It is just the ground-state Gaussian displaced in phase space to $(\langle\hat{x}\rangle, \langle\hat{p}\rangle)$ determined by $\alpha$ .
Classical motion. Under time evolution the eigenvalue rotates, $\alpha \to \alpha e^{-i\omega t}$ , so the centre of the wave packet traces the classical ellipse $\langle\hat{x}\rangle(t) \propto \cos(\omega t)$ in phase space — without spreading. A coherent state oscillates like a classical particle while remaining a minimum-uncertainty blob.

Looking ahead

Coherent states are generated from the ground state by the unitary displacement operator $D(\alpha) = e^{\alpha a^\dagger - \alpha^* a}$ , satisfying $|\alpha\rangle = D(\alpha)|0\rangle$ . That construction, and the squeezed states that go beyond minimum uncertainty, are central to quantum optics and to continuous-variable quantum information — natural next steps once the ladder algebra of this module is fluent.