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

Theorem: Short-time concurrence

Scaling and short-time parameter dependence · lines 680–725 · proof included or linked

Em04-A@2#thm:shorttime, Em04-A@2#eq:shortC

For the default localized preparation and $Γ_{2} > 0$ ,

C_{01} (τ) = 2 τ - \frac{2 ( 1 + r )}{χ} τ^{2} + O (τ^{3}), τ ↓ 0.

The formula holds for arbitrary fixed real detunings and includes $r = 0, 1$ . In physical time, for all nonnegative rates, it reads

C_{01} (t) = 2 J t - 2 J (γ_{r} + γ_{d}) t^{2} + O (t^{3}) .

Proof. Set $P = ∣0 ⟩ ⟨ 0∣$ , $d_{i} = - ϵ_{i} / J$ , and $κ = 2 (1 - r) / χ$ . For $A (M) = - i [h, M] - κ (M - Δ M)$ , analyticity gives

S = P + τ A (P) + \frac{τ ^{2}}{2} A^{2} (P) + O (τ^{3}) .

Direct matrix multiplication on the open chain yields

A (P)_{01} = i, - [h, [h, P]]_{01} = d_{0} - d_{1}, A^{2} (P)_{01} = d_{0} - d_{1} - i κ .

Thus

S_{01} = i τ + \frac{d _{0} - d _{1} - i κ}{2} τ^{2} + O (τ^{3}), ∣ S_{01} ∣ = τ - \frac{κ}{2} τ^{2} + O (τ^{3}) .

The detuning term affects the modulus only at third order. Multiplying by $2 q$ , with $q = 1 - (2 r / χ) τ + O (τ^{2})$ , proves (eq:shortC). The same calculation with physical coefficients proves the last formula, including zero dissipation. All remainders concern a fixed model, not unbounded detuning families.

Displayed formulas in this passage

Em04-A@2#eq:shortC · line 683
$C_{01} (τ) = 2 τ - \frac{2 ( 1 + r )}{χ} τ^{2} + O (τ^{3}), τ ↓ 0.$
unlabeled · line 691
$C_{01} (t) = 2 J t - 2 J (γ_{r} + γ_{d}) t^{2} + O (t^{3}) .$
unlabeled · line 702
$S = P + τ A (P) + \frac{τ ^{2}}{2} A^{2} (P) + O (τ^{3}) .$
unlabeled · line 707
$A (P)_{01} = i, - [h, [h, P]]_{01} = d_{0} - d_{1}, A^{2} (P)_{01} = d_{0} - d_{1} - i κ .$
unlabeled · line 714
$S_{01} = i τ + \frac{d _{0} - d _{1} - i κ}{2} τ^{2} + O (τ^{3}), ∣ S_{01} ∣ = τ - \frac{κ}{2} τ^{2} + O (τ^{3}) .$

Theorem: Short-time concurrence

Displayed formulas in this passage

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