Manuscript Psy02-A@1 · Psy02-A.tex · theorem · d92ce402

Theorem: Finite regularized reduction semigroup

A separately declared reduction model › Unconditional dynamics · lines 2364–2410 · proof included or linked

Psy02-A@1#thm:collapse-channel, Psy02-A@1#eq:collapse-solution

The unique evolution $C_{s} = e^{s L_{c}}$ is CPTP and satisfies

[C_{s} (ρ)]_{ab} = e^{- i (E_{a} - E_{b}) s /ℏ} e^{- E_{G, ab} s /ℏ} ρ_{ab} .

All populations are preserved. For $E_{G, ab} > 0$ , the $ab$ coherence has e-folding time

τ_{ab} = \frac{ℏ}{E _{G, ab}} .

Proof. Each matrix unit is an eigenvector of the generator, giving (eq:collapse-solution). For complete positivity, let $W_{α} (s)$ be independent real Brownian variables of variance $s$ , and set

V_{s} = exp (- \frac{i H s}{ℏ} - i α \sum L_{α} W_{α} (s)) .

Then

C_{s} (ρ) = E (V_{s} ρ V_{s}^{†}),

because Gaussian averaging multiplies the $ab$ matrix unit by

exp [- \frac{s}{2} α \sum (ℓ_{α a} - ℓ_{α b})^{2}] .

An average of unitary channels is CPTP. Finite linear-ODE uniqueness gives the specified evolution.

Displayed formulas in this passage

Psy02-A@1#eq:collapse-solution · line 2369
$[C_{s} (ρ)]_{ab} = e^{- i (E_{a} - E_{b}) s /ℏ} e^{- E_{G, ab} s /ℏ} ρ_{ab} .$
unlabeled · line 2379
$τ_{ab} = \frac{ℏ}{E _{G, ab}} .$
unlabeled · line 2389
$V_{s} = exp (- \frac{i H s}{ℏ} - i α \sum L_{α} W_{α} (s)) .$
unlabeled · line 2397
$C_{s} (ρ) = E (V_{s} ρ V_{s}^{†}),$
unlabeled · line 2402
$exp [- \frac{s}{2} α \sum (ℓ_{α a} - ℓ_{α b})^{2}] .$

Theorem: Finite regularized reduction semigroup

Displayed formulas in this passage

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