Damped cavity mode: coherent states survive, cats do not

SI units throughout; nothing here moves relativistically, so the metric (+, −, −, −) never enters. The mode is an oscillator, H = ħωa†a with the zero-point term dropped, coupled through a partly transmitting mirror to a continuum of outside modes bₖ by ħΣ(gₖa†bₖ + gₖ*bₖ†a), with the outside at zero temperature. Topic 14.03's steps go through with σ replaced by a: the Born step keeps the outside in its vacuum, the Markov step needs its spectrum flat over a band much wider than κ around ω, and the secular step was taken in writing the coupling without the terms abₖ and a†bₖ†, which oscillate near 2ω. What survives is the master equation of the first card, with κ = 2π|g(ω)|² times the density of outside modes at ω, the golden-rule rate through the mirror, and a small frequency shift absorbed into ω. Every term of dρ/dt is in s⁻¹. The approximations are very good: a superconducting cavity at 51.1 GHz with photon lifetime T_c = 160 μs has κ = 6.25 × 10³ s⁻¹ and Q = ω/κ = 5.1 × 10⁷, so κ/ω = 2 × 10⁻⁸.

For any operator A the master equation gives d⟨A⟩/dt = iω⟨[a†a, A]⟩ + κ⟨a†Aa − ½a†aA − ½Aa†a⟩. For A = a the commutator is −a, and with aa† = a†a + 1 the dissipator is a†aa − ½a†aa − ½(a†a + 1)a = −½a, so d⟨a⟩/dt = −(iω + κ/2)⟨a⟩. For A = a†a it is a†a†aa − (a†a)² = −a†a, so d⟨n⟩/dt = −κ⟨n⟩. In general d⟨a†ᵐaⁿ⟩/dt = [i(m − n)ω − (m + n)κ/2]⟨a†ᵐaⁿ⟩: each normally ordered moment obeys its own closed equation. The amplitude decays at κ/2 and the energy at κ, as for a classical cavity of quality factor Q = ω/κ, and κ → 0 returns the free mode. For a single photon, ⟨n⟩ = e^(−κt) is the probability that it is still there. In 2007 the ENS group, in a 51.1 GHz cavity whose ring-down time was measured as T_c = 0.130 s, watched single photons appear and vanish in sudden jumps, the single-shot face of that probability, quoted.

Try ρ(t) = |β⟩⟨β| with β(t) = α₀e^{(−iωt − κt/2)}. Writing |β⟩ = e^{(−|β|²/2)}e^(βa†)|0⟩ and differentiating gives d|β⟩/dt = [κ|β|²/2 − (iω + κ/2)βa†]|β⟩, and βa†|β⟩ = a†a|β⟩. So d(|β⟩⟨β|)/dt = κ|β|²ρ − iω[a†a, ρ] − (κ/2)(a†aρ + ρa†a), which is the master equation's right side term for term, because its jump term is aρa† = |β|²ρ. The ansatz is exact. The state stays pure, its photon statistics stay Poissonian with mean |α₀|²e^(−κt), and no noise is added. The reason is that |β⟩ is an eigenstate of the jump operator, so losing a photon leaves it unchanged. Seen from outside, a reduced state that stays pure means a product: |α₀⟩|vac⟩ becomes |β⟩|E⟩, and since the coupling is linear in the mode operators, |E⟩ is a multimode coherent state carrying |α₀|²(1 − e^(−κt)) photons, exactly those the mode lost. That is why laser light sent through any passive loss is still laser light.

Start from N(|α₀⟩ + |−α₀⟩), with N² = 1/[2(1 + e^{(−2|α₀|²)})]. Of its four dyads the two diagonal ones are solved already. The equation is linear, so try c(t)X with X = |β⟩⟨−β|. The derivative of X again supplies −iω[a†a, X] − (κ/2)(a†aX + Xa†a) + κ|β|²X, but the jump term is now aXa† = β(−β)*X = −|β|²X, with the opposite sign. Matching leaves dc/dt = −2κ|β|²c = −2κ|α₀|²e^(−κt)c, so c = exp[−2|α₀|²(1 − e^(−κt))]; the exponent is a photon number times a pure number. With |α₀|² = 4, by κt = 0.125 each component has lost 12% of its photons and is still pure while c has fallen to 0.39, and by κt = 0.25 the loss is 22% and c = 0.17. For general components α₁ and α₂, |c| = exp[−½|α₁ − α₂|²(1 − e^(−κt))], with a phase Im(α₁α₂*)(1 − e^(−κt)). In the Wigner function, quoted, each component is a Gaussian hill and c sets the height of the fringes between them. Late on c → e^{(−2|α₀|²)}: both hills have shrunk into one vacuum.

The rate 2κ|α₀|² is twice the rate κ|α₀|² at which photons leave, and two pictures say why. First, the records: each component writes its own coherent state into the outside world, |±γ⟩ with |γ|² = |α₀|²(1 − e^(−κt)), and tracing out the outside multiplies the interference by their overlap, ⟨−γ|γ⟩ = e^(−2|γ|²), the factor the master equation gave. Decoherence is over once the two records can be told apart, long before the energy has gone. Second, the photons themselves: a(|α⟩ + |−α⟩) = α(|α⟩ − |−α⟩), so every escaping photon flips the relative sign. The number k that has escaped is Poisson-distributed with mean μ = |α₀|²(1 − e^(−κt)), and averaging the sign over it gives ⟨(−1)ᵏ⟩ = e^(−2μ), the same factor again. The one-photon limit is the opposite extreme: in |0⟩ + |1⟩ the coherence is ⟨a⟩ itself and decays at κ/2, half the energy rate, the T₂ = 2T₁ of amplitude damping in 14.02. Scaled up, an optical superposition with 10⁶ photons per component and T_c = 1 μs keeps its coherence for 0.5 ps, which is why superpositions of classical fields are never seen.

In 1996 Brune, Haroche and colleagues at the École Normale Supérieure watched this happen, quoted here. A coherent field of mean photon number 3.3 sat in a superconducting cavity near 51 GHz with a photon lifetime of about 160 μs. A rubidium atom in a superposition of two circular Rydberg states crossed it off resonance and shifted the field's phase one way or the other, leaving components |α₀e^(±iδ)⟩ at distance D = 2|α₀| sin δ, entangled with the atom. A second atom, sent a delay τ later, tested whether they could still interfere. The signal decayed well within the photon lifetime, faster for larger D, in agreement with the D²/2 rate derived here. The model has limits. A reservoir at temperature T raises the loss term by a factor nₜₕ + 1 and adds 14.04's gain term κnₜₕ(a†ρa − ½aa†ρ − ½ρaa†), so coherent states stop being exact solutions and the early rate becomes (2nₜₕ + 1)κD²/2, quoted; at 51.1 GHz and 0.8 K, nₜₕ = 0.05, a 10% increase. And once an atom couples to the mode faster than κ, the mode no longer decays on its own: that is Jaynes-Cummings dynamics, beyond this equation.