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

Theorem: Exact two-coordinate scaling

Scaling and short-time parameter dependence · lines 643–671 · proof included or linked

Em04-A@2#thm:scaling, Em04-A@2#eq:scaledS, Em04-A@2#eq:scaledq

With $h = h / J$ , Protocol-B dynamics satisfy

S^{'} (τ) q (τ) = - i [h, S] - \frac{2 ( 1 - r )}{χ} (S - Δ S), = e^{- 2 r τ / χ} .

After fixing $N, S_{0}, ϵ_{i} / J$ , the pair $(χ, r)$ determines these trajectories. A common detuning shift has no effect.

Multiplying $J, ϵ_{i}, γ_{r}, γ_{d}$ by a common $c > 0$ replaces the physical-time trajectory $ρ (t)$ by $ρ (c t)$ and leaves the corresponding Protocol-B trajectory unchanged. Nevertheless two models with the same finite $χ$ and different $r$ cannot have identical full Protocol-B trajectories.

Proof. Substitution proves the scaled equations. A common detuning shift changes $h$ by a scalar identity, which disappears from the commutator. Simultaneous scaling changes the full generator to $c L$ , because collapse amplitudes scale by $c$ . Finally, the excitation expectation is $e^{- 2 r τ / χ}$ , which differs for different $r$ at every $τ > 0$ .

Displayed formulas in this passage

Em04-A@2#eq:scaledS · line 646
$S^{'} (τ) q (τ) = - i [h, S] - \frac{2 ( 1 - r )}{χ} (S - Δ S), = e^{- 2 r τ / χ} .$
Em04-A@2#eq:scaledq · line 646
$S^{'} (τ) q (τ) = - i [h, S] - \frac{2 ( 1 - r )}{χ} (S - Δ S), = e^{- 2 r τ / χ} .$

Theorem: Exact two-coordinate scaling

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