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

Finite thermal bath

Track A1: carrier, matched branches, and exact dynamics › Finite thermal bath · lines 313–333

Psy02-B@2#eq:Htot

During sensing, define

H_{tot} = \frac{ℏ ω}{2} Z \otimes 1 + 1 \otimes ℏ j = 1 \sum^{M} ν_{j} b_{j}^{†} b_{j} + ℏ Z \otimes j = 1 \sum^{M} g_{j} (b_{j} + b_{j}^{†}),

with finite $M$ , $ν_{j} > 0$ , $g_{j} \in R$ , and an independent thermal bath of inverse temperature $β > 0$ :

τ_{B} = j ⨂ (1 - e^{- β ℏ ν_{j}}) n = 0 \sum^{\infty} e^{- β ℏ ν_{j} n} ∣ n ⟩ ⟨ n ∣ .

Conditional oscillator Hamiltonians are defined by displacement of the free oscillator Hamiltonians.

Displayed formulas in this passage

Psy02-B@2#eq:Htot · line 316
$H_{tot} = \frac{ℏ ω}{2} Z \otimes 1 + 1 \otimes ℏ j = 1 \sum^{M} ν_{j} b_{j}^{†} b_{j} + ℏ Z \otimes j = 1 \sum^{M} g_{j} (b_{j} + b_{j}^{†}),$
unlabeled · line 326
$τ_{B} = j ⨂ (1 - e^{- β ℏ ν_{j}}) n = 0 \sum^{\infty} e^{- β ℏ ν_{j} n} ∣ n ⟩ ⟨ n ∣ .$