Amplitude damping, phase damping and depolarisation

Units are SI and nothing here is relativistic, so the course metric (+, −, −, −) never enters. Write ρ_eg = ⟨e|ρ|g⟩ and take the Bloch vector with z = ρₑₑ − ρ_gg and |ρ_eg| = ½√(x² + y²), so the ground state is the south pole. Let the environment start empty, in |0_E⟩. The ground state has nothing to give, U|g⟩|0_E⟩ = |g⟩|0_E⟩, while U|e⟩|0_E⟩ = √(1 − p)|e⟩|0_E⟩ + √p|g⟩|1_E⟩. Topic 14.01's rule Kₖ = ⟨k_E|U|0_E⟩ gives K₀ = |g⟩⟨g| + √(1 − p)|e⟩⟨e| and K₁ = √p|g⟩⟨e|, with K₀†K₀ + K₁†K₁ = I, derived. So ρₑₑ → (1 − p)ρₑₑ, ρ_gg → ρ_gg + pρₑₑ and ρ_eg → √(1 − p)ρ_eg. The square root is the point: coherence survives only in the K₀ branch, where the excited amplitude shrank by √(1 − p) and the ground amplitude did not, while in the K₁ branch the environment holds a photon, an orthogonal record. At p = 0.36 the excited population keeps 0.64 and the coherence 0.80. The ball becomes an ellipsoid with semi-axes √(1 − p), √(1 − p) and 1 − p, centred at z = −p and touching the ground pole; p = 0 is the identity, p = 1 sends every state to |g⟩.

A coupling that commutes with σ_z cannot move population; it can only let the environment learn which state the qubit is in. Then U = |g⟩⟨g| ⊗ U_g + |e⟩⟨e| ⊗ Uₑ, and from |E⟩ the environment ends in |E_g⟩ = U_g|E⟩ or |Eₑ⟩ = Uₑ|E⟩. Tracing it out leaves ρₑₑ and ρ_gg alone and multiplies ρ_eg by ⟨E_g|Eₑ⟩, derived: the better the two records can be told apart, the less coherence survives. Make the environment a qubit tilted by ±θ, |E_g⟩ = cos θ|0_E⟩ + sin θ|1_E⟩ and |Eₑ⟩ = cos θ|0_E⟩ − sin θ|1_E⟩. Then K₀ = cos θ I and K₁ = −sin θ σ_z, a phase flip with probability q = sin²θ, and the coherence factor 1 − 2q = cos 2θ is the overlap: θ = 30° gives q = 0.25 and keeps half the coherence. An environment that flips only when the qubit is excited, with probability λ, gives the different-looking pair |g⟩⟨g| + √(1 − λ)|e⟩⟨e| and √λ|e⟩⟨e|, the same channel whenever √(1 − λ) = 1 − 2q: topic 14.01's unitary freedom. On the ball x and y shrink by 1 − 2q and z stays, so states slide towards the z axis; q = ½ is a σ_z measurement nobody read.

Replace the qubit by I/2 with probability p: ρ → (1 − p)ρ + p I/2. Kraus operators need I/2 written as an average. With σ₀ = I, the rules σᵢσⱼσᵢ = −σⱼ for i ≠ j and σⱼσⱼσⱼ = σⱼ give Σ_μ σ_μσⱼσ_μ = (1 + 1 − 1 − 1)σⱼ = 0, while Σ_μ σ_μIσ_μ = 4I. So for any ρ = ½(I + r⋅σ), Σ_μ σ_μρσμ = 2I, derived: applying I, σₓ, σ_y or σ_z at random, each with probability ¼, erases every Bloch vector. Hence ρ → (1 − 3p/4)ρ + (p/4)(σₓρσₓ + σ_yρσ_y + σ_zρσ_z), with K₀ = √(1 − 3p/4) I and Kᵢ = √(p/4) σᵢ, and r → (1 − p)r: the only one of the three channels with no preferred axis, and the only one that drives every state to the centre. Every pure state keeps fidelity 1 − p/2, so p = 0.06 leaves 0.97. The weight 1 − 3p/4 stays non-negative up to p = 4/3, where r → −r/3, so the map stays physical past the point where p reads as a probability. At p = 2 it would give r → −r, which maps the ball onto itself, yet on half of a singlet it leaves an eigenvalue of −½: positive, not completely positive, as topic 14.01 warned.

A channel has a strength, not yet a clock. Let amplitude damping over a time t have strength p(t), and suppose each interval meets a fresh environment, the photon emitted earlier having left: the Markov condition, which topic 14.03 earns. Running t and then s must equal running t + s, and the factors on the excited population multiply: 1 − p(t + s) = [1 − p(t)][1 − p(s)], with p(0) = 0. Logarithms make ln[1 − p(t)] additive, and an additive continuous function is linear, so 1 − p(t) = e^(−t/T₁), derived. Then ρₑₑ(t) = ρₑₑ(0)e^(−t/T₁) and the coherence factor is √(1 − p) = e^{(−t/(2T₁))}: amplitude damping alone gives T₂ = 2T₁. For t ≪ T₁, p ≈ t/T₁ is the golden-rule probability Γt, so T₁ = 1/Γ, Unit 12's Weisskopf-Wigner rate, quoted. Hydrogen 2p has Γ = 6.26 × 10⁸ s⁻¹, so T₁ = 1.60 ns, as measured, and T₂ = 3.19 ns. A coherence decaying as e^(−t/T₂) gives a Lorentzian line of full width 1/(πT₂); for rubidium-87's D2 line, measured lifetime 26.24 ns, that is 6.07 MHz, the natural width of the line. Phase damping composes the same way, 1 − 2q(t) = e^(−t/T_φ).

Phase damping leaves populations alone and both channels multiply ρ_eg by a number, so they commute and their coherence factors multiply: e^{(−t/(2T₁))}e^(−t/T_φ) = e^(−t/T₂), with 1/T₂ = 1/(2T₁) + 1/T_φ, derived. Rates add, times do not, and since 1/T_φ ≥ 0, T₂ ≤ 2T₁. The bound outlives these models. A 2 × 2 density matrix has det ρ ≥ 0, so |ρ_eg|² ≤ ρₑₑρ_gg. Take a pure state with a small excited population a, so |ρ_eg(0)|² = a(1 − a). If populations relax to the ground state in T₁ and coherences decay in T₂, positivity at time t needs a(1 − a)e^(−2t/T₂) ≤ ae^(−t/T₁)[1 − ae^(−t/T₁)], and as a → 0 that is e^(−2t/T₂) ≤ e^(−t/T₁): T₂ ≤ 2T₁ for every environment, derived. A coherence is bounded by the square root of the excited population, so it can fall at most half as fast. A transmon measured at T₁ = 62 μs and T₂ = 85 μs therefore has 1/T_φ = 1/85 − 1/124 μs⁻¹, T_φ = 270 μs. A report of T₂ = 120 μs beside T₁ = 50 μs cannot be right: for a = 0.01, after 100 μs ρ would have the eigenvalue −5.2 × 10⁻⁴.

Everything above rests on the fresh environment, which needs the environment's correlation time τ_c to be short against every time of interest. For an atom in vacuum τ_c is about 1/ω₀, 6.5 × 10⁻¹⁷ s for Lyman-α, seven orders of magnitude below T₁ = 1.60 ns. It fails in three ways. At times comparable with τ_c, survival falls quadratically in t, as the short-time expansion of e^(−iHt/ħ) requires, not linearly; the deviation is real but far too early to resolve for an atom. In a cavity whose coupling g exceeds both its loss and Γ, the excitation comes back, the population oscillates at Unit 11's vacuum Rabi frequency and no composition law holds, so topic 14.03's master equation fails there too. And slow noise remembers: if a qubit's frequency is fixed during each run but scatters between runs with r.m.s. spread σ, the coherence factor averages to e^{(−σ²t²/2)}, a Gaussian; for σ/2π = 10 kHz it reaches 1/e at 22.5 μs. That decay, written T₂*, reverses under a spin echo, which no composing channel allows. Finite temperature, which adds absorption in proportion to n̄, is topic 14.04's.