Thermal states and field modes

A ket is maximal information about a system, and a mode in contact with a bath does not have it: the two are entangled, so tracing the bath out leaves a mixed ρ̂. Fix the mean energy, maximise S = −k_B Tr(ρ̂ ln ρ̂), and the stationary point is the Gibbs operator ρ̂ = e^(−βĤ)/Z. Now use the structure of Ĥ = ℏω(N̂ + ½): the exponential of an operator is diagonal wherever that operator is, so ⟨m|ρ̂|n⟩ = pₙ δₘₙ in the Fock basis with no calculation at all. The zero-point term then drops out of every population, because each Boltzmann factor splits into a common e^(−x/2) times e^{(−n x)}, and that common factor cancels against Z — the half-quantum shifts the energy, never the statistics.

Write r = e^(−x) with x = ℏω/k_B T. The spectrum Eₙ = ℏω(n + ½) makes every Boltzmann factor a power of r: Z = Σₙ e^(−βEₙ) = e^(−x/2) Σₙ rⁿ = e^(−x/2)/(1 − r), which converges for any T > 0 because the spectrum is bounded below and evenly spaced — the two facts the ladder argument delivered. Symmetrising gives Z = 1/(2 sinh(x/2)). Differentiate: ⟨Ĥ⟩ = −∂(ln Z)/∂β = (ℏω/2) coth(x/2) = ℏω(n̄ + ½), and the same derivative returns n̄ = 1/(e^x − 1). At x = 1 that is Z = 0.960, n̄ = 0.582, ⟨Ĥ⟩ = 1.08 ℏω. One derivative more gives the Einstein heat capacity C = k_B x² e^x/(e^x − 1)², equal to 0.921 k_B at x = 1 and dying like x² e^(−x) beyond it. That exponential freeze-out is what a classical solid cannot produce.

Two conversions do most of the work: k_B T = 25.9 meV = 208.5 cm⁻¹ at 300 K, and hf/k_B = 48.0 mK per GHz. Then read off the regime. Silicon's 520 cm⁻¹ optical phonon at 300 K has x = 2.49, n̄ = 0.090, and 91.7% of the ensemble on the ground rung — cold, in a warm room. A 6.0 GHz circuit mode, ℏω/k_B = 0.288 K, has n̄ = 13.4 at the 4 K stage but 5.6 × 10⁻⁷ at 20 mK, which is why superconducting qubits live in dilution refrigerators. A 100 MHz LC mode at 300 K has x = 1.6 × 10⁻⁵ and n̄ = 6.25 × 10⁴, where n̄ ≈ k_B T/ℏω − ½ to five figures: classical. Visible light is the far end — a 500 nm mode at 300 K has x = 95.9 and n̄ ≈ 2 × 10⁻⁴², which is why a warm room is dark.

Take n̄ = 3 and compare the thermal ρ̂ with the pure state |χ⟩ = Σₙ √pₙ |n⟩ built from the same populations. Measure N̂ and the two are indistinguishable: identical histograms, p₀ = 0.250 for both. Everything else separates them. Tr ρ̂² = 1/(2n̄ + 1) = 0.143 against 1. ⟨â⟩ = 0 against ⟨â⟩ = 1.60, so ⟨x̂⟩ stays at zero for the mixture while |χ⟩ swings at ω. And ρ̂ commutes with Ĥ, so it is stationary — a thermal state has no dynamics to watch. Number statistics cannot split those two, but they do split the thermal mode from a coherent state of the same mean: (Δn)² = n̄ + n̄² = 12 against 3, and g⁽²⁾(0) = 2 against 1. That bunching is the Hanbury Brown–Twiss signature of thermal light.

Expand a free field in normal modes and each wavevector-polarisation pair carries its own quadratic Hamiltonian ℏωₖ(â†ₖ âₖ + ½), with ωₖ = c|k| for light. The operator â†ₖ adds one quantum carrying energy ℏωₖ and momentum ℏk, so the excitation is a photon or a phonon rather than a rung on an abstract ladder. Because Ĥ is a sum over independent modes, e^(−βĤ) factorises into one thermal state per mode, and n̄ = 1/(e^x − 1) evaluated at x = ℏω/k_B T is the Planck function itself: multiply by the mode density 8πν²/c³ and by ℏω and the blackbody spectrum falls out with nothing else added. A 160 GHz mode of the cosmic microwave background at 2.725 K has x = 2.82 and n̄ = 0.064. The zero-point sum Σ ℏωₖ/2 diverges and is normal-ordered away; only differences in it, Casimir and Lamb, are measurable.

The thermal state is the fixed point of dynamics this Hamiltonian does not contain. Under the mode's own quadratic Ĥ every eigenstate keeps its population forever, a coherent state stays coherent, and nothing relaxes — so equilibrium has to be imported from a coupling the model dropped: anharmonic phonon–phonon scattering, a cavity mode leaking into lossy walls, electron–photon scattering in a filament. That same coupling sets the linewidth, so a mode narrow enough to call sharp is a mode slow to thermalise; and once the coupling is not weak, the mode's own temperature stops being well defined, because ρ̂ for the mode alone is then no longer the Gibbs state of its own Ĥ. Mode independence carries the same warning: modes decouple only while the field is free.