Pointer states: what decoherence explains and leaves open

SI units with ħ kept explicit; no four-vector appears, and where the course needs one its metric is (+, −, −, −). Quoted from Quantum Mechanics II: a measurement entangles, c↑|↑⟩|P↑⟩ + c↓|↓⟩|P↓⟩, and leaves the system's coherence multiplied by ⟨P↓|P↑⟩. Now make the pointer a bath: H = ½ħω₀σ_z + Σₖħωₖbₖ†bₖ + ħσ_zΣₖ(gₖbₖ† + gₖ*bₖ), every mode in its vacuum at t = 0, so each mode is topic 11.07's driven oscillator with a constant drive whose sign is set by σ_z. In the interaction picture the coupling is ħσ_zΣₖ(gₖbₖ†e^(iωₖt) + gₖ*bₖe^(−iωₖt)). Its commutator at two times is a c-number times σ_z² = 1, so the Magnus series ends at second order and U(t) = e^(iφ) exp[σ_zΣₖ(αₖbₖ† − αₖ*bₖ)], with φ(t) a c-number phase and αₖ = gₖ(1 − e^(iωₖt))/ωₖ: derived. That is a displacement by +αₖ when σ_z = +1 and by −αₖ when σ_z = −1, and the phase φ is the same for both. So |↑⟩|0⟩ becomes |↑⟩|+α⟩, still a product state, and |↓⟩|0⟩ becomes |↓⟩|−α⟩. Only superpositions become entangled.

Start from (c↑|↑⟩ + c↓|↓⟩)|0⟩. At time t the state is c↑|↑⟩|+α⟩ + c↓|↓⟩|−α⟩, and tracing out the bath gives ρ↑↓ = c↑c↓* ∏ₖ⟨−αₖ|αₖ⟩. Quoted from topic 11.07: ⟨β|α⟩ = exp(−|α|²/2 − |β|²/2 + β*α), so ⟨−α|α⟩ = e^(−2|α|²), real and positive. With |1 − e^(iωt)|² = 2(1 − cos ωt), each mode contributes 2|αₖ|² = 4|gₖ|²(1 − cos ωₖt)/ωₖ², and restoring the qubit's own precession, ρ↑↓(t) = ρ↑↓(0) e^(−iω₀t) exp[−Γ(t)] with Γ(t) = Σₖ4|gₖ|²(1 − cos ωₖt)/ωₖ²: derived. Checks. Dimensions: gₖ/ωₖ is a pure number. No coupling: Γ = 0. Short times: Γ ≈ 2t²Σₖ|gₖ|², quadratic as second-order perturbation theory requires, so there is no rate until the bath has had time to respond. One mode: Γ swings between 0 and 8|g|²/ω² and is zero again at t = 2π/ω, when the displaced oscillator has come back to its vacuum and the record is wiped. Monroe and co-workers (1996) drove a ⁹Be⁺ ion's motion 83 nm apart in its two spin branches, against a 7 nm wavepacket: an overlap of e^(−17.6) ≈ 2 × 10⁻⁸.

For many modes define J(ω) = 4Σₖ|gₖ|²δ(ω − ωₖ), in s⁻¹, so that Γ(t) = ∫J(ω)(1 − cos ωt)/ω² dω. Two questions decide the outcome. Do the cosines dephase? For a continuum, ∫J cos(ωt)/ω² dω dies away once t exceeds the inverse bandwidth, by the Riemann-Lebesgue lemma, and Γ settles at S = ∫J/ω² dω. Is S finite? For an ohmic bath, J = λω exp(−ω/ω_c), it is not, and the exact integral is Γ = (λ/2)ln(1 + ω_c²t²): derived. The coherence falls to zero, but only as (ω_c t)^(−λ), carried there by ever slower modes. For acoustic phonons coupled through a deformation potential, J ∝ ω³ at low ω, S is finite and the coherence stops at e^(−S): the record is incomplete. At zero temperature e^(−S) is also the weight of the zero-phonon line, quoted, and the NV centre in diamond, which puts about 3% of its light there, has S ≈ 3.5. A thermal bath multiplies each term by coth(ħωₖ/2k_B T), quoted, which turns the ohmic power law into exponential decay. A discrete bath with spacing δ returns after 2π/δ; for any macroscopic body that is longer than anyone will wait.

|↑⟩ and |↓⟩ are special because the coupling contains σ_z and nothing else, so the bath copies them undisturbed. The general test is how fast a pure state loses purity: for a dissipator with jump operator L and rate γ, −d Tr ρ²/dt = 2γ(⟨L†L⟩ − |⟨L⟩|²) at ρ = |ψ⟩⟨ψ|, derived because Tr(ρ[H, ρ]) = 0. It vanishes only when ψ is an eigenstate of L. When the coupling outweighs the system's own Hamiltonian H_S, those eigenstates are the pointer states and their superpositions are what the environment destroys; Zurek called the test the predictability sieve. An H_S that does not commute with L rotates them into superpositions, so selection is a competition. A weakly damped oscillator settles it. By topic 14.05, L = a, whose eigenstates are coherent states, and ħωa†a turns |α⟩ into |αe^(−iωt)⟩, still one. So |α⟩ stays pure, while the cat |α⟩ + |−α⟩, with ⟨a⟩ = 0, loses purity at 2κ|α|²tanh(|α|²), topic 14.05's rate 2κ|α|² once |α|² ≳ 2. If H_S dominates and the bath cannot exchange energy at the system's frequencies, energy eigenstates are selected instead (Paz and Zurek, 1999, quoted).

It explains two things. First, lost interference: once Γ ≫ 1, every measurement on the qubit alone returns the statistics of ρ_S = |c↑|²|↑⟩⟨↑| + |c↓|²|↓⟩⟨↓|, so no Ramsey fringe and no ⟨σₓ⟩ reveals the superposition. The loss is exactly the information now in the bath: for equal weights the fringe visibility is V = |⟨E↓|E↑⟩|, two pure records differ by the trace distance D = √(1 − V²), quoted, and so V² + D² = 1: what the fringes lose, the records gain. Second, a preferred basis. Quantum Mechanics II's pointer could not say why the record is of |↑⟩ and |↓⟩ rather than of (|↑⟩ ± |↓⟩)/√2, since an entangled state with equal weights can be expanded either way. Here the Hamiltonian decides: only eigenstates of the coupled operator stay unentangled, so the basis is fixed by physics, not bookkeeping. Both are seen. Brune and co-workers (1996) watched the interference between two coherent field states in a cavity fade faster the further apart they were, and Hornberger and co-workers (2003) saw fullerene fringes fade exponentially with background-gas pressure, each collision adding to the record.

Decoherence never removes a branch. The qubit-plus-bath state is c↑|↑⟩|E↑⟩ + c↓|↓⟩|E↓⟩, as pure as it began, and ρ_S is diagonal only because the trace discards the correlations. Read ρ_S as the qubit being up or down, unknown which, and an improper mixture, obtained by ignoring a partner, has been mistaken for a proper one, an ensemble of definite states. The two agree on every measurement of the qubit and disagree on joint ones, where an observable sensitive to both branches still shows the interference. One mode does hand the coherence back at t = 2π/ω; with many, only the impracticality of reversing the bath prevents it. So decoherence explains why interference becomes unobservable and which basis is preferred. It does not explain why one outcome occurs, and the weights |c↑|² and |c↓|² enter through the trace rule, which it assumes. That is the measurement problem, and it stands. Copenhagen, Everett's relative states, de Broglie-Bohm pilot waves and objective-collapse models such as GRW answer it differently, the last with testable predictions; this course names them only.