Coherent states of a mode and the classical field

Work in SI units; nothing here is written covariantly, so the course metric diag(1, −1, −1, −1) never enters. Keep one box mode of frequency ω and polarisation ε, with A = √(ħ/(2ε₀ωV))(a ε e^(ik⋅r) + a† ε* e^(−ik⋅r)) quoted from topic 11.03. A classical current density J(r, t), prescribed and never itself a dynamical variable, couples through −∫J⋅A d³r, so H = ħω a†a + f(t) a† + f*(t) a with f(t) = −√(ħ/(2ε₀ωV)) ∫J⋅ε* e^(−ik⋅r) d³r. In the interaction picture H_I(t) = f(t) e^(iωt) a† + f*(t) e^(−iωt) a, and [H_I(t), H_I(t′)] = f*(t) f(t′) e^{(−iω(t−t′))} − c.c., a number. Slice the evolution into short steps exp(−iH_I δt/ħ); because every pair commutes to a number, the quoted identity e^X e^Y = e^(X+Y) e^{([X, Y]/2)} merges the product into one exponential and a phase: U_I = e^(iχ) exp(α a† − α* a), with α(t) = −(i/ħ)∫f(t′) e^(iωt′) dt′, derived. Dimensions: f is an energy and dt′/ħ an inverse energy, so α is a pure number. Limit: no current, α = 0, and the vacuum stays the vacuum.

Call D(α) = exp(α a† − α* a) the displacement operator; it is unitary because its exponent is anti-Hermitian. With X = α a† and Y = −α* a, [X, Y] = |α|², so the quoted identity gives D(α) = e^{(−|α|²/2)} e^{(α a†)} e^{(−α* a)}. The last factor leaves |0⟩ alone, and with a†ⁿ|0⟩ = √(n!)|n⟩ the state is |α⟩ = D(α)|0⟩ = e^{(−|α|²/2)} Σ αⁿ/√(n!) |n⟩, derived. Since Σ |α|⁽²ⁿ⁾/n! = e^(|α|²), it is normalised. Apply a: a|n⟩ = √n|n − 1⟩ shifts the sum by one and returns α|α⟩, so the coherent state is an eigenstate of the annihilation operator, derived; equivalently D†(α) a D(α) = a + α. Free evolution only turns the label, e^{(−iωt a†a)}|α⟩ = |α e^(−iωt)⟩, so a coherent state stays coherent and its complex amplitude rotates at ω like the classical mode amplitude. The Heisenberg equation da/dt = −iωa − (i/ħ)f(t) says it in one line: a(t) = e^(−iωt)[a(0) + α(t)], the free operator plus a c-number, derived.

Take the free field E = iE₁(a ε e^{(i(k⋅r − ωt))} − a† ε* e^{(−i(k⋅r − ωt))}) of topic 11.03, with E₁ = √(ħω/(2ε₀V)), and write α = |α|e^(iφ) with ε real. The eigenvalue property turns every a into α, so ⟨α|E|α⟩ = −2E₁|α| sin(k⋅r − ωt + φ) ε, derived: a classical plane wave of amplitude 2E₁|α| whose phase is the phase of α. Now square the component along ε. E² contains a a† + a†a, and a a† = a†a + 1, so ⟨E²⟩ = ⟨E⟩² + E₁², derived: the variance is ħω/(2ε₀V), the vacuum value of topic 11.05, at every amplitude and every instant. The Heisenberg form says why: the current adds a number to a and leaves its fluctuations alone. The noise-to-signal ratio is therefore 1/(2|α|), 5 × 10⁻⁴ for |α|² = 10⁶. Hold the amplitude fixed and |α| = amplitude/(2E₁) grows as 1/√ħ, which is what the classical limit means here. A number state is the opposite: ⟨n|a|n⟩ = 0 gives ⟨E⟩ = 0 everywhere, while ⟨E²⟩ = (2n + 1)E₁² is all fluctuation.

Square the amplitudes: P(n) = |⟨n|α⟩|² = e^(−n̄) n̄ⁿ/n! with n̄ = |α|², a Poisson distribution, derived. The moments come from the eigenvalue property without any summing: ⟨a†a⟩ = |α|², and (a†a)² = a†a†aa + a†a gives ⟨(a†a)²⟩ = |α|⁴ + |α|², so the variance equals the mean and Δn/n̄ = 1/|α|. A coherent state with n̄ = 100 holds 100 ± 10 photons, and even the likeliest counts, 99 and 100, each turn up only 4.0% of the time. Physically the current emits each photon independently of the others. Limit: |α| → 0 leaves P(0) = 1, the vacuum. Measured: in 1965 Arecchi counted photons from a He-Ne laser running well above threshold and found the Poisson distribution, while the same light scattered from a moving ground-glass disc, a pseudo-thermal source, gave the Bose–Einstein distribution P(n) = n̄ⁿ/(1 + n̄)⁽ⁿ⁺¹⁾, quoted from statistical mechanics. Run near threshold, a laser gives counts broader than Poisson, because its intensity itself still fluctuates: the classical excess the final section bounds.

For one mode, g⁽²⁾(0) = ⟨a†a†aa⟩/⟨a†a⟩² compares the rate of coincidences at zero delay with the rate for independent arrivals. Since a†a†aa = a†a(a†a − 1) = n(n − 1), it equals 1 + (⟨(Δn)²⟩ − n̄)/n̄², derived. Coherent: the eigenvalue property gives |α|⁴/|α|⁴ = 1 at once. Thermal: ρ = Σ (1 − x)xⁿ|n⟩⟨n| with x = exp(−ħω/k_B T) is quoted from statistical mechanics; then n̄ = x/(1 − x) and ⟨n(n − 1)⟩ = 2x²/(1 − x)² = 2n̄², so g⁽²⁾(0) = 2 at every temperature, derived. Number state: n(n − 1)/n² gives g⁽²⁾(0) = 1 − 1/n, derived, which is 0 for one photon, because one photon cannot be detected twice. Limits: as n → ∞ the number state tends to 1, and thermal light keeps 2 however dim, since bunching is a ratio. Measured: Hanbury Brown and Twiss saw the thermal excess in the correlated photocurrents of two detectors in 1956, diluted by detectors slower than the coherence time; pseudo-thermal sources with long coherence times show the full factor of 2.

Let the current be random: amplitude jitter, phase drift, added noise. Each realisation leaves a coherent state, so the field is a mixture ρ = ∫P(α)|α⟩⟨α| d²α with P a probability density, and every normal-ordered moment is an average of numbers: g⁽²⁾(0) = ⟨|α|⁴⟩/⟨|α|²⟩² = 1 + Var(|α|²)/⟨|α|²⟩² ≥ 1, derived, because a variance cannot be negative. The same step for the intensity I of any classical stochastic wave gives ⟨I²⟩ ≥ ⟨I⟩², so g⁽²⁾(0) ≥ 1 for every classical field, with equality only for a stable intensity. Glauber's theory, named here, writes every state in this P form; light is nonclassical when no non-negative P exists, and a measured g⁽²⁾(0) below 1 proves it. Poisson counts, discrete clicks and ⟨E⟩ = 0 prove nothing: a stable classical wave on a square-law detector gives Poisson clicks, quoted from the semiclassical theory of photodetection. Measured: Kimble, Dagenais and Mandel first saw antibunching in 1977, in sodium resonance fluorescence, as coincidences rising away from zero delay, which the quoted classical bound g⁽²⁾(τ) ≤ g⁽²⁾(0) forbids; with a fluctuating number of atoms in view their g⁽²⁾(0) stayed near 1, and single quantum dots now reach g⁽²⁾(0) below 0.01.