Manuscript Em03-F@1 · Em03-F-v1.tex · section · 0d4d148b

Mode normalization

Exterior field, scattering coordinates, and infrared control › Mode normalization · lines 358–422

Em03-F@1#eq:coefficient, Em03-F@1#eq:Unorm

Let $f_{+} (x, ω) \sim e^{i ω x}$ at $+ \infty$ and $f_{-} (x, ω) \sim e^{- i ω x}$ at $- \infty$ . Define

W (ω) = f_{-} f_{+}^{'} - f_{-}^{'} f_{+}, τ (ω) = \frac{2 i ω}{W ( ω )},

q_{H} = τ f_{+}, q_{I} = τ f_{-} .

Let $ρ_{I}$ be the outgoing-infinity reflection amplitude of the infinity-incident mode. The spectral convention is

(U u)_{j} (ω) = \frac{1}{2 π} \int q_{j} (x, ω) u (x) d x .

The canonical field expansion is

Ψ (t, x) = j = H, I \sum \int_{0}^{\infty} \frac{q _{j} ( x , ω ) a _{j} ( ω ) e ^{- i ω t} + q _{j} ( x , ω ) a _{j} ( ω ) ^{†} e ^{i ω t}}{4 π ω} d ω .

Annihilation tests are conjugate-linear:

a (k) = j \sum \int k_{j} (ω) a_{j} (ω) d ω .

Write

X (k) = a (k) + a (k)^{†}, W (k) = e^{i X (k)},

σ (k, l) = 2 Im ⟨ k, l ⟩, W (k) W (l) = e^{- i σ (k, l) /2} W (k + l) .

For a compact reduced test,

k_{j} (f) (ω) = \frac{1}{4 π ω} \int f (t, x) q_{j} (x, ω) e^{i ω t} d t d x, ϕ (F [f]) = X (k (f)) .

The common covariance norm is

∥ k ∥_{U}^{2} = \int_{0}^{\infty} [(1 + 2 n_{H}) ∣ k_{H} ∣^{2} + ∣ k_{I} ∣^{2}] d ω .

The ordinary two-channel norm is denoted $∥ k ∥_{0}$ . It is dominated by $∥ k ∥_{U}$ . In the baseline,

ω_{U} (X (k)^{2}) = ∥ k ∥_{U}^{2}, ω_{U} (W (k)) = e^{- ∥ k ∥_{U}^{2} /2} .

Displayed formulas in this passage

Em03-F@1#eq:coefficient · line 398
$k_{j} (f) (ω) = \frac{1}{4 π ω} \int f (t, x) q_{j} (x, ω) e^{i ω t} d t d x, ϕ (F [f]) = X (k (f)) .$
Em03-F@1#eq:Unorm · line 409
$∥ k ∥_{U}^{2} = \int_{0}^{\infty} [(1 + 2 n_{H}) ∣ k_{H} ∣^{2} + ∣ k_{I} ∣^{2}] d ω .$
unlabeled · line 363
$W (ω) = f_{-} f_{+}^{'} - f_{-}^{'} f_{+}, τ (ω) = \frac{2 i ω}{W ( ω )},$
unlabeled · line 367
$q_{H} = τ f_{+}, q_{I} = τ f_{-} .$
unlabeled · line 372
$(U u)_{j} (ω) = \frac{1}{2 π} \int q_{j} (x, ω) u (x) d x .$
unlabeled · line 378
$Ψ (t, x) = j = H, I \sum \int_{0}^{\infty} \frac{q _{j} ( x , ω ) a _{j} ( ω ) e ^{- i ω t} + q _{j} ( x , ω ) a _{j} ( ω ) ^{†} e ^{i ω t}}{4 π ω} d ω .$
unlabeled · line 386
$a (k) = j \sum \int k_{j} (ω) a_{j} (ω) d ω .$
unlabeled · line 390
$X (k) = a (k) + a (k)^{†}, W (k) = e^{i X (k)},$
unlabeled · line 393
$σ (k, l) = 2 Im ⟨ k, l ⟩, W (k) W (l) = e^{- i σ (k, l) /2} W (k + l) .$
unlabeled · line 418
$ω_{U} (X (k)^{2}) = ∥ k ∥_{U}^{2}, ω_{U} (W (k)) = e^{- ∥ k ∥_{U}^{2} /2} .$