From 04b9fde5ddf394980f0e3353cbfa81dcde8e3f78 Mon Sep 17 00:00:00 2001 From: Valentin Boettcher Date: Mon, 10 Jul 2023 16:27:48 -0400 Subject: [PATCH] explain frames better --- .../index.tex | 48 +++++++++++++------ 1 file changed, 33 insertions(+), 15 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 54a6639..ae35004 100644 --- a/roam/data/d6/e44fbc-975b-463d-87f9-878d50758b55/index.tex +++ b/roam/data/d6/e44fbc-975b-463d-87f9-878d50758b55/index.tex @@ -70,6 +70,9 @@ frequencies \(ω_{k_β} = c \abs{k_{β}}/n_{A}\), where \(n_{A}\) is the refractive index inside the cavity. For simplicity we set \(\hbar = 1\) such that time is measured in units of inverse energy. +\subsection{Transformations and Rotating Frames} +\label{sec:rotating-frames} + The bare \(a_{β}\) modes are linear combinations of the \(c_{m}\) and can be related through \begin{equation} @@ -85,24 +88,27 @@ terms from the interaction \label{eq:12} \tilde{c}(t)_{m} = c_{m}(t)\eu^{\iu ω^{0}_{m}t} \implies H_{A} \to \tilde{H}_{A}= ∑_{mn}V_{mn}(t) \eu^{-\iu t (ω^{0}_{n}-ω^{0}_{m})} - \tilde{c}_{m}^†\tilde{c}_{n} \approx ∑_{mn}V^{0}_{mn} \eu^{\iu (ε_{m}-ε_{n})t}\tilde{c}_{m}^†\tilde{c}_{n}. + \tilde{c}_{m}^†\tilde{c}_{n} \approx ∑_{mn}V^{0}_{mn} \eu^{\iu (ε_{m}-ε_{n})t}\tilde{c}_{m}^†\tilde{c}_{n}, \end{equation} +where \(\abs{ε_{m}-ε_{n}}\ll\abs{ω^{0}_{m} - ω^{0}_{n}}\). Upon changing into another rotating frame we can remove this residual time dependence \begin{equation} \label{eq:33} h_{n}(t) = \tilde{c}_{n}\eu^{-\iu ε_{n}t} \implies \tilde{H}'_{A} = - ∑_{mn}\bqty{V^{0}_{mn} + δ_{mn}ε_{m}} h_{m}^†h_{n} + ∑_{mn}\bqty{V^{0}_{mn} + δ_{mn}{ε_{m}-\iu η_{m}}} h_{m}^†h_{n}, \end{equation} +where we've added an ad-hock decay rate due to the coupling to the +transmission line that will be introduced more rigorously later on. We can subsequently find a unitary transformation that diagonalizes the RWA interaction \begin{equation} \label{eq:30} - ∑_{mn}\pqty{O^{†}}_{im}\bqty{V^{0}_{mn} + δ_{mn}ε_{m}}O_{nj} = ω_{j} δ_{ij}, + ∑_{mn}\pqty{O^{-1}}_{γm}\bqty{V^{0}_{mn} + δ_{mn}\pqty{ε_{m}-\iu η_{m}}}O_{nγ'} = ω_{γ} δ_{γ,γ'}. \end{equation} -where the columns of \(O\) are the normalized eigenvectors of -\(V_{mn}^{0}=\mel{m}{V^{0}}{n}\). So if \(\ket{ψ_{j}}\) is an +For \(η_{m}=0\) the columns of \(O\) are the normalized eigenvectors +of \(V_{mn}^{0}=\mel{m}{V^{0}}{n}\). So if \(\ket{ψ_{j}}\) is an eigenvector of \(V\), then \(\braket{i}{ψ_{j}} = O_{ij}\) \footnote{This is just a reminder for Valentin who can't seem to keep this in his head.}. @@ -110,14 +116,24 @@ eigenvector of \(V\), then \(\braket{i}{ψ_{j}} = O_{ij}\) Transforming the \(h_{m}\) according to \begin{equation} \label{eq:13} - d_{γ} = ∑_{n}O^{\ast}_{nγ}(0) h_{n} = ∑_{n}O^{\ast}_{nγ}(t) \tilde{c}_{n} \implies \tilde{H}''_{A} = ∑_{γ}ω_{γ} d_{γ}^†d_{γ} + d_{γ} = ∑_{n}O^{-1}_{γn}(0) h_{n} = ∑_{n}O^{-1}_{γn}(t) \eu^{-\iu + ε_{n}t}\tilde{c}_{n} \implies \iu \dot{d}_{γ} = ω_{γ}d_{γ} \end{equation} -where -\begin{equation} - \label{eq:35} - O^\ast_{nγ}(t)\equiv O^\ast_{nγ}\eu^{-\iu ε_{n}t} -\end{equation} -leaves us with a very simple Hamiltonian. +leaves us with a very simple equation of motion. + + +In summary, the bare modes of the resonators are denoted by +\(a_{j,α}\) where \(j\) refers to the resonator and \(α\) labels the +mode within that resonator. The eigenmodes \(c_{m}\) of the coupled +oscillators obeying \(H_{0}\) are related to the bare modes +\(α_{j,α}\) by \cref{eq:43}. The eigenmodes in the rotating frame of +\(H_{0}\) are called \(\tilde{c}_{m}\). The frame in which the +equations of motion for the modes reflects the target Hamiltonian +\(\tilde{H}_{A}'\) is given by rotating away the slow-oscillating +terms in \cref{eq:12} leading to the \(h_{n}\) modes. The resulting +equations of motion for the \(h_{n}\) can be decoupled by the +transformation \(O_{mγ}\) giving the eigenmodes of the target +Hamiltionaion \(d_{γ}\) including damping. Let us list the relation between the \(a\), \(c\), \(h\) and \(d\) operators for later reference @@ -126,12 +142,14 @@ for later reference c_{n} &= \eu^{-\iu ω^{0}_{n} t}\tilde{{c}}_{n} = \eu^{-\iu (ω^{0}_{n}-ε_{n}) t} h_{n}= \eu^{-\iu - (ω^{0}_{n}) t} ∑_{γ} O_{nγ}(t)d_{γ} \\ + (ω^{0}_{n}-ε_{n}) t} ∑_{γ} O_{nγ}d_{γ} \\ a_{β} &= ∑_{m} U_{β,m} c_{m} = ∑_{m} U_{β,m} \tilde{c}_{m}\eu^{-\iu - ω^{0}_{m} t} = ∑_{mγ} U_{β,m} \eu^{-\iu - ω^{0}_{m} t} O_{mγ}(t)d_{γ} + ω^{0}_{m} t} = ∑_{mγ} U_{β,m}O_{mγ} \eu^{-\iu + (ω^{0}_{m}-ε_{m}) t} d_{γ} \end{align} +\subsection{Coupling to the Transmission Line} +\label{sec:coupl-transm-line} The transmission line is considered to only have one polarization direction and one dimension of