Coherent and Squeezed States

The number states diagonalize the Hamiltonian but look nothing like a classical oscillator. Each has $⟨\hat{x}⟩=⟨\hat{p}⟩=0$ for all time, a probability cloud frozen in place. A pendulum, by contrast, has a definite position that swings sinusoidally. The quantum states that reproduce that behavior as closely as the uncertainty principle allows are the coherent states, and they are singled out by one clean condition: they are eigenstates of the annihilation operator.¹

Eigenstates of the annihilation operator

Ask for a state $|\alpha ⟩$ obeying

The eigenvalue $\alpha$ is complex because $\hat{a}$ is not Hermitian. The raising operator has no such eigenstates — ${\hat{a}}^{†}$ pushes probability up the ladder without bound, and no normalizable superposition survives — but the lowering operator does, because it terminates on the vacuum. Expand $|\alpha ⟩={\sum }_n{c}_n|n⟩$ and impose the eigenvalue condition. Using $\hat{a}|n⟩=\sqrt{n}|n-1⟩$,

Matching the coefficient of $|n⟩$ gives the recursion $c_{n+1}\sqrt{n+1}=\alpha c_n$, so

Normalization fixes $c_0$. Since ${\sum }_n|c_n{|}^2=|c_0{|}^2{\sum }_n|\alpha {|}^{2n}/n!=|c_0{|}^2{e}^{|\alpha {|}^2}=1$,

Poissonian photon statistics

The probability of finding $n$ quanta in a coherent state is $P(n)=|⟨n|\alpha ⟩{|}^2$, which reads off the expansion:

This is a Poisson distribution with mean $\bar{n}=|\alpha {|}^2$. The mean and variance of a Poisson law are equal, so

The fractional spread in photon number shrinks as the field grows, so a bright coherent beam has a nearly definite intensity even though the number is not sharp. The photon-counting fluctuations $\Delta N=\sqrt{\bar{n}}$ are the shot noise of an ideal laser, whose output is very close to a coherent state.²

The displacement operator

The coherent state has a compact operator description. Define the displacement operator

which is unitary because its exponent is anti-Hermitian. Its exponent splits by the Baker–Campbell–Hausdorff identity: with $\hat{A}=\alpha {\hat{a}}^{†}$ and $\hat{B}=-{\alpha }^{\ast }\hat{a}$, the commutator $[\hat{A},\hat{B}]=|\alpha {|}^2$ is a number, so

Acting on the vacuum, $e^{-{\alpha }^{\ast }\hat{a}}|0⟩=|0⟩$ because $\hat{a}|0⟩=0$, while $e^{\alpha {\hat{a}}^{†}}|0⟩={\sum }_n({\alpha }^n/n!)({\hat{a}}^{†}{)}^n|0⟩={\sum }_n({\alpha }^n/\sqrt{n!})|n⟩$. The two combine into exactly the coherent-state expansion:

A coherent state is a displaced vacuum. The name is literal: conjugating the ladder operators shows the displacement shifts phase space by $\alpha$,

Minimum uncertainty and classical motion

Because a coherent state is the vacuum shifted rigidly, its widths equal the vacuum's. From $\hat{a}|\alpha ⟩=\alpha |\alpha ⟩$ and its adjoint,

and a short calculation with $⟨{\hat{a}}^2⟩={\alpha }^2$, $⟨{\hat{a}}^{†}\hat{a}⟩=|\alpha {|}^2$ gives the widths

Every coherent state saturates the uncertainty bound, and the widths are independent of $\alpha$: the wave packet keeps the shape of the ground-state Gaussian while its center sits anywhere in phase space. The vacuum is the special case centered at the origin.

The center then moves classically. In the Heisenberg picture$\hat{a}(t)=\hat{a}(0)e^{-i\omega t}$, so the eigenvalue rotates, $⟨\hat{a}{⟩}_t={\alpha }_0{e}^{-i\omega t}$, and

The expectation traces a sinusoid at the classical frequency with amplitude set by $|{\alpha }_0|$. Equivalently, under time evolution a coherent state stays coherent, $|{\alpha }_0⟩\to e^{-i\omega t/2}|{\alpha }_0{e}^{-i\omega t}⟩$: the Gaussian blob keeps its shape and orbits the phase-space origin on a circle of radius $|{\alpha }_0|$, the closest a quantum state comes to a point particle on a classical trajectory.

Squeezed states

A coherent state divides its uncertainty equally between two directions in phase space. Nothing forces that split. Define the dimensionless quadrature operators

the amplitudes of the $\cos \omega t$ and $\sin \omega t$ parts of the motion. Their commutator $[{\hat{X}}_1,{\hat{X}}_2]=i/2$ enforces $\Delta X_1\Delta X_2\ge \frac{1}{4}$. The vacuum and every coherent state are isotropic: $\Delta X_1=\Delta X_2=\frac{1}{2}$, a circle of uncertainty. A squeezed state keeps the product at the floor $\frac{1}{4}$ but makes the ellipse eccentric,

with $r$ the squeeze parameter. It is generated by the squeeze operator

whose quadratic exponent conjugates $\hat{a}$ into a mixture of $\hat{a}$ and ${\hat{a}}^{†}$, stretching one quadrature and compressing the orthogonal one.

Squeezing has a direct experimental payoff. An interferometer limited by the shot noise of coherent light can beat that limit by injecting squeezed light into its dark port, aligning the reduced quadrature with the measured phase. Gravitational-wave detectors run this way, reading out a strain smaller than the vacuum fluctuations of ordinary laser light would permit.³ The three regimes of photon statistics label where a state's number fluctuations sit relative to the coherent benchmark.

The coherent state is the boundary case: the most classical pure state of the field, and the reference against which quantum-enhanced measurement is defined.