Coherent States

Definition

A coherent state is defined by three equivalent statements — each is a valid way to say the same thing.

1. Eigenstate of the annihilation operator. A coherent state |α⟩ is the right-eigenstate of the (non-Hermitian) annihilation operator â:

â|α⟩ = α|α⟩

Here â is the lowering operator of the quantum harmonic oscillator, and the eigenvalue α = |α|e^iφ is any complex number: |α| sets the amplitude and φ sets the phase of the oscillation. Because â is not Hermitian, α need not be real.

2. Displaced vacuum. Apply the displacement operator D(α) to the ground state |0⟩:

|α⟩ = D(α)|0⟩,   D(α) = exp(α·â† − α*·â)

D(α) is unitary; it rigidly slides the vacuum's Gaussian noise blob out to the point α in phase space without changing its shape.

3. A specific superposition of number states. Expanded in the Fock (number) basis {|n⟩}:

|α⟩ = e^(−|α|²/2) · Σₙ (αⁿ / √(n!)) |n⟩

All three definitions describe the same object. The coefficients above give the photon-number probabilities |⟨n|α⟩|² = e^−|α|²|α|²ⁿ/n! — a Poisson distribution.

Symbols and units

Symbol	Meaning	Units / value
|α⟩	Coherent state with complex amplitude α	dimensionless ket
α = |α|eiφ	Complex eigenvalue: amplitude |α|, phase φ	dimensionless
â, â†	Annihilation and creation operators, [â, â†] = 1	dimensionless
n̂ = â†â	Number operator; ⟨n̂⟩ = |α|²	photons
D(α)	Displacement operator, exp(α↠− α*â)	unitary
x̂, p̂	Position and momentum quadratures	m, kg·m/s
ω	Oscillator angular frequency	rad/s
ħ	Reduced Planck constant	1.055 × 10⁻³⁴ J·s

Minimum uncertainty and non-spreading

Write â in terms of the dimensionless quadratures X̂ = (â + â†)/2 and P̂ = (â − â†)/2i, which obey [X̂, P̂] = i/2 so the Heisenberg bound is ΔX·ΔP ≥ 1/4. In a coherent state both variances are equal to the vacuum value and are time-independent:

ΔX = ΔP = 1/2   ⇒   ΔX·ΔP = 1/4   (equality — minimum uncertainty)

In physical units this is the saturated Heisenberg relation Δx·Δp = ħ/2. The key feature is that the wave packet, which is a rigidly displaced ground-state Gaussian, keeps a constant width. As the phase evolves the packet's center follows the classical trajectory

⟨x̂(t)⟩ = x₀ cos(ωt + φ),   ⟨p̂(t)⟩ = −mωx₀ sin(ωt + φ)

while its shape never changes. This is exactly what Schrödinger built in 1926: a quantum state whose center of mass sweeps the potential well like a classical ball, without the dispersion that plagues a generic free-particle wave packet. In phase space the state is a circular Gaussian blob of fixed radius (the vacuum noise) whose center orbits the origin at angular frequency ω.

Photon statistics — Poissonian by construction

Measuring the photon number of a coherent state gives a Poisson distribution:

P(n) = e^(−|α|²) · |α|^(2n) / n!

with

⟨n⟩ = |α|²,   Var(n) = |α|²,   Δn = |α| = √⟨n⟩

The variance equals the mean, so the Fano factor F = Var(n)/⟨n⟩ = 1 exactly. The fractional fluctuation Δn/⟨n⟩ = 1/|α| shrinks as the field gets stronger — a bright laser beam of ⟨n⟩ = 10¹⁰ photons has a relative amplitude noise of only 10⁻⁵. The normalized second-order coherence is g⁽²⁾(0) = 1: no bunching, no antibunching. Light with g⁽²⁾(0) < 1 (sub-Poissonian, e.g. from a single-photon emitter) is strictly nonclassical and cannot be a coherent state.

Light source	Photon statistics	g(2)(0)	Classical?
Ideal laser (well above threshold)	Poissonian (coherent state)	1	Yes (most classical)
Thermal / chaotic (bulb, star)	Super-Poissonian (bunched)	2	Classical field, noisy
Single-photon source	Sub-Poissonian (antibunched)	≈ 0	No — nonclassical
Fock state |n⟩	Number-squeezed, Δn = 0	1 − 1/n	No — nonclassical

The displacement operator, step by step

The whole family of coherent states is generated from the single point α = 0 (the vacuum) by translations in phase space.

1. Definition. D(α) = exp(α↠− α*â). Because the exponent is anti-Hermitian, D(α) is unitary and D†(α) = D(−α).
2. It shifts operators by a c-number. D†(α) â D(α) = â + α. So in the displaced frame the annihilation operator is just â + α, and acting on the vacuum where â|0⟩ = 0 immediately gives â|α⟩ = α|α⟩.
3. It translates phase space. The mean quadratures become ⟨x̂⟩ ∝ Re α and ⟨p̂⟩ ∝ Im α, moving the noise blob from the origin to (√2 Re α, √2 Im α) without deforming it.
4. Composition carries a phase. Two displacements do not simply add: D(α)D(β) = exp(i·Im(αβ*)) D(α + β). The extra phase is the (signed) area swept in phase space — the same geometric phase that appears in the Aharonov–Bohm effect and Berry phase.

Using the Baker–Campbell–Hausdorff identity, D(α) = e^−|α|²/2 e^α↠e^−α*â, which acting on |0⟩ reproduces the number-state expansion above.

Non-orthogonal and overcomplete

Coherent states are not a set of orthogonal basis vectors. Their overlap is

⟨β|α⟩ = exp(−½|α|² − ½|β|² + β*α),   |⟨β|α⟩|² = e^(−|α−β|²)

which is nonzero for every finite separation — two coherent states become nearly orthogonal only when |α − β| ≫ 1. Yet they still resolve the identity,

(1/π) ∫ |α⟩⟨α| d²α = Î

so any state can be expanded over them, but the expansion is not unique. This overcompleteness is exactly what makes the phase-space representations of quantum optics work: the Sudarshan–Glauber P-function, the Husimi Q-function, and the Wigner function all trade the Hilbert-space ket for a quasi-probability distribution over the α-plane.

Worked example — a single-mode field with ⟨n⟩ = 10⁶

Take a single-mode laser field with mean photon number ⟨n⟩ = |α|² = 10⁶. Then:

• Amplitude: |α| = √(10⁶) = 1000, so the state is |α⟩ with |α| = 1000.
• Photon-number spread: Δn = |α| = 1000, i.e. a fractional fluctuation Δn/⟨n⟩ = 1/1000 = 0.1%. The shot noise is exactly the square root of the count.
• Probability of the vacuum: P(0) = e^−|α|² = e^−10⁶ ≈ 0 — you essentially never measure zero photons.
• Overlap with a slightly detuned field β with |α − β| = 3: |⟨β|α⟩|² = e⁻⁹ ≈ 1.2 × 10⁻⁴, so these two states are almost distinguishable.

The shot-noise limit Δn = √⟨n⟩ is not a technical imperfection — it is the irreducible quantum noise of the most classical possible light. Beating it requires squeezed states, which trade reduced noise in one quadrature for increased noise in the other and are used to enhance gravitational-wave detectors.

Why coherent states matter

• Laser light. The output of an ideal single-mode laser far above threshold is, to excellent approximation, a coherent state. This is the workhorse model of quantum optics.
• Classical limit of the field. Coherent states bridge quantum electrodynamics and classical electromagnetism — the field expectation value ⟨Ê⟩ is a genuine sinusoid, recovering Maxwell's waves.
• Quantum communication and sensing. Coherent-state pulses carry information in optical fiber; homodyne detection reads out their quadratures; their shot noise sets the standard quantum limit.
• Bose–Einstein condensates and superfluids. A macroscopically occupied mode is well described by a coherent-state order parameter, giving BECs a definite phase.
• Foundations. Superposing two well-separated coherent states, (|α⟩ + |−α⟩)/√2, builds a Schrödinger-cat state whose fragility under decoherence makes coherent states a natural pointer basis.

Common misconceptions

• "A coherent state has a definite photon number." No — its photon number is Poisson-distributed with spread Δn = |α|. What is well-defined is the amplitude and phase, not the count.
• "Coherent states are orthogonal like Fock states." They are never orthogonal; ⟨β|α⟩ is exponentially small but nonzero. They form an overcomplete, non-orthogonal set.
• "The uncertainty grows over time as the packet moves." It does not. The defining feature is that the Gaussian keeps constant width; only its center oscillates. Generic (squeezed or excited) states do breathe or spread.
• "Any laser is a perfect coherent state." Real lasers have phase diffusion, amplitude noise, and multimode structure. The coherent state is the idealized single-mode limit.
• "Squeezed light is a coherent state with less noise." No — squeezing pushes the state below the symmetric minimum-uncertainty circle in one quadrature, which a coherent state cannot do. Coherent states are the boundary, not the interior.
• "The eigenvalue α must be real because it's a measurement outcome." α is complex because â is non-Hermitian; its real and imaginary parts are the two quadratures, and its phase is the field's optical phase.

History

1926 — Schrödinger. Erwin Schrödinger constructed non-spreading Gaussian wave packets for the harmonic oscillator to show how quantum mechanics recovers classical motion. These were the first coherent states, though not yet named.

1963 — Glauber and Sudarshan. Roy J. Glauber's paper "Coherent and Incoherent States of the Radiation Field" (Physical Review 131, 2766) founded the quantum theory of optical coherence, defining |α⟩ as the annihilation-operator eigenstate and building the correlation-function hierarchy. George Sudarshan independently introduced the diagonal P-representation over coherent states the same year. John Klauder developed the general mathematical framework of overcomplete continuous representations.

2005 — Nobel Prize. Glauber received one half of the 2005 Nobel Prize in Physics "for his contribution to the quantum theory of optical coherence." Coherent states are now called Glauber states, and the machinery he built underlies modern quantum optics, laser physics, and continuous-variable quantum information.