From 5e66bfafa0308b7ec65d8d6706f7bada2bd6f9cc Mon Sep 17 00:00:00 2001 From: Valentin Boettcher Date: Tue, 20 Jun 2023 11:23:01 -0400 Subject: [PATCH] fix absolute value of coupling constant --- roam/data/d6/e44fbc-975b-463d-87f9-878d50758b55/index.tex | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/roam/data/d6/e44fbc-975b-463d-87f9-878d50758b55/index.tex b/roam/data/d6/e44fbc-975b-463d-87f9-878d50758b55/index.tex index 33e5648..bc53a37 100644 --- a/roam/data/d6/e44fbc-975b-463d-87f9-878d50758b55/index.tex +++ b/roam/data/d6/e44fbc-975b-463d-87f9-878d50758b55/index.tex @@ -482,13 +482,13 @@ constant. With these considerations in mind we can simplify \cref{eq:32} to \begin{equation} \label{eq:64} - η_{m}=κ\frac{πn_{B}}{c}∑_{σ=\pm,β,β'}U_{βm}U^\ast_{β'm}δ_{\sgn(β),σ} δ_{\sgn(β'),σ} + η_{m}=\abs{κ}\frac{πn_{B}}{c}∑_{σ=\pm,β,β'}U_{βm}U^\ast_{β'm}δ_{\sgn(β),σ} δ_{\sgn(β'),σ} \end{equation} and \begin{gather} \label{eq:34} \dot{d}_{γ} = - -\iu\bqty{\pqty{ω_{γ}-\iu \tilde{η}_{γ}}d_{γ} + \sqrt{κ} ∑_{σ=\pm} + -\iu\bqty{\pqty{ω_{γ}-\iu \tilde{η}_{γ}}d_{γ} + \sqrt{κ^\ast} ∑_{σ=\pm} U^{σ}_{γ}(t) \frac{b_{\inputf}(t)}{\sqrt{ω_{0}}}}\\ U^{σ}_{γ}(t) = ∑_{m,β} δ_{\sgn({β}),σ}U^\ast_{βm}\pqty{O^{-1}(t)}_{γm} \eu^{\iu ω_{m}^{0}t}= ∑_{m,β} δ_{\sgn({β}),σ}U^\ast_{βm}\pqty{O^{-1}}_{γm}\eu^{\iu (ω_{m}^{0}-ε_{m})t}. \end{gather} @@ -506,7 +506,7 @@ We can now proceed to integrate \cref{eq:34} to obtain with \begin{equation} \label{eq:37} - χ_{γ}(t) = κ \eu^{-\pqty{\iu ω_{γ} + \tilde{η}_{γ}}t}. + χ_{γ}(t) = \abs{κ} \eu^{-\pqty{\iu ω_{γ} + \tilde{η}_{γ}}t}. \end{equation} When constructing the total output field, we have to remember how the