Manuscript Em04-A@2 · Em04-A-v2.tex · lemma · f5b28abc

Lemma: Invariant blocks

Physicality and exact reduction · lines 291–343 · proof included or linked

Em04-A@2#lem:blocks, Em04-A@2#eq:R, Em04-A@2#eq:p, Em04-A@2#eq:h

The Hamiltonian commutes with

N_{exc} = k = 0 \sum^{N - 1} n_{k} .

The linear class

ρ = p ∣Ω ⟩ ⟨ Ω∣ + V R V^{†}

is invariant under the generator, with equations

R ˙ p ˙ h_{ij} = - i [h, R] - γ_{r} R - 2 γ_{d} (R - Δ R), = γ_{r} Tr R, = J (δ_{i, j + 1} + δ_{i, j - 1}) - ϵ_{i} δ_{ij} .

No vacuum–single-excitation coherences are generated.

Proof. The identity

\frac{1}{2} (X_{k} X_{k + 1} + Y_{k} Y_{k + 1}) = a_{k}^{†} a_{k + 1} + a_{k + 1}^{†} a_{k}

shows excitation conservation by hopping. The detuning terms also commute with $N_{exc}$ . The vacuum energy is $E_{Ω} = \frac{1}{2} \sum_{k} ϵ_{k}$ , and $V^{†} (H - E_{Ω} I) V = h$ , proving the Hamiltonian term.

Since $a_{k} ∣ e_{i} ⟩ = δ_{k i} ∣Ω ⟩$ , the relaxation jump term is

k \sum γ_{r} a_{k} (V R V^{†}) a_{k}^{†} = γ_{r} Tr R ∣Ω ⟩ ⟨ Ω∣.

The anticommutator term on the one-excitation space is $- γ_{r} R$ , because $\sum_{k} n_{k}$ restricts to its identity.

Finally,

D [L_{k}^{d}] (M) = \frac{γ _{d}}{2} (Z_{k} M Z_{k} - M) .

For $i \neq = j$ , conjugation of $∣ e_{i} ⟩ ⟨ e_{j} ∣$ changes sign at exactly sites $i, j$ . Each contributes $- γ_{d}$ . Diagonal excitation matrix units and the vacuum projector are unchanged. These calculations prove all stated block equations.

Displayed formulas in this passage

Em04-A@2#eq:R · line 302
$R ˙ p ˙ h_{ij} = - i [h, R] - γ_{r} R - 2 γ_{d} (R - Δ R), = γ_{r} Tr R, = J (δ_{i, j + 1} + δ_{i, j - 1}) - ϵ_{i} δ_{ij} .$
Em04-A@2#eq:p · line 302
$R ˙ p ˙ h_{ij} = - i [h, R] - γ_{r} R - 2 γ_{d} (R - Δ R), = γ_{r} Tr R, = J (δ_{i, j + 1} + δ_{i, j - 1}) - ϵ_{i} δ_{ij} .$
Em04-A@2#eq:h · line 302
$R ˙ p ˙ h_{ij} = - i [h, R] - γ_{r} R - 2 γ_{d} (R - Δ R), = γ_{r} Tr R, = J (δ_{i, j + 1} + δ_{i, j - 1}) - ϵ_{i} δ_{ij} .$
unlabeled · line 294
$N_{exc} = k = 0 \sum^{N - 1} n_{k} .$
unlabeled · line 298
$ρ = p ∣Ω ⟩ ⟨ Ω∣ + V R V^{†}$
unlabeled · line 316
$\frac{1}{2} (X_{k} X_{k + 1} + Y_{k} Y_{k + 1}) = a_{k}^{†} a_{k + 1} + a_{k + 1}^{†} a_{k}$
unlabeled · line 328
$k \sum γ_{r} a_{k} (V R V^{†}) a_{k}^{†} = γ_{r} Tr R ∣Ω ⟩ ⟨ Ω∣.$
unlabeled · line 336
$D [L_{k}^{d}] (M) = \frac{γ _{d}}{2} (Z_{k} M Z_{k} - M) .$

Lemma: Invariant blocks

Displayed formulas in this passage

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