Manuscript Psy02-B@2 · Psy02-B-v2.tex · theorem · cfd06e2e

Theorem: Exact colored dephasing

Track A1: carrier, matched branches, and exact dynamics › Finite thermal bath · lines 334–370 · proof included or linked

Psy02-B@2#thm:colored, Psy02-B@2#eq:chi

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For code inputs,

ρ_{01} (t) = e^{- i ω t - χ (t)} ρ_{01} (0),

where

χ (t) = 4 j = 1 \sum^{M} \frac{g _{j} ^{2}}{ν _{j} ^{2}} (1 - cos ν_{j} t) coth (\frac{β ℏ ν _{j}}{2}) .

Populations are preserved. The reduced maps are CPTP. At zero coupling they recover the carrier's own free dynamics.

Proof. For branch $z = \pm 1$ ,

ν_{j} b_{j}^{†} b_{j} + z g_{j} (b_{j} + b_{j}^{†}) = ν_{j} (b_{j}^{†} + z g_{j} / ν_{j}) (b_{j} + z g_{j} / ν_{j}) - g_{j}^{2} / ν_{j} .

The branch-independent scalar phases cancel. The relative displacement has squared modulus $8 g_{j}^{2} (1 - cos ν_{j} t) / ν_{j}^{2}$ . Thermal displacement expectation is

Tr (τ_{j} D (α)) = exp [- \abs α^{2} coth (β ℏ ν_{j} /2) /2] .

Multiplying the mode factors proves the formula. Unitary evolution and partial trace establish physicality.

Displayed formulas in this passage

Psy02-B@2#eq:chi · line 341
$χ (t) = 4 j = 1 \sum^{M} \frac{g _{j} ^{2}}{ν _{j} ^{2}} (1 - cos ν_{j} t) coth (\frac{β ℏ ν _{j}}{2}) .$
unlabeled · line 337
$ρ_{01} (t) = e^{- i ω t - χ (t)} ρ_{01} (0),$
unlabeled · line 354
$ν_{j} b_{j}^{†} b_{j} + z g_{j} (b_{j} + b_{j}^{†}) = ν_{j} (b_{j}^{†} + z g_{j} / ν_{j}) (b_{j} + z g_{j} / ν_{j}) - g_{j}^{2} / ν_{j} .$
unlabeled · line 364
$Tr (τ_{j} D (α)) = exp [- \abs α^{2} coth (β ℏ ν_{j} /2) /2] .$

Theorem: Exact colored dephasing

Displayed formulas in this passage

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