From 92154d89c8abb22cad8cea76cd711640aff3c416 Mon Sep 17 00:00:00 2001 From: Valentin Boettcher Date: Tue, 20 Jun 2023 13:52:51 -0400 Subject: [PATCH] write up changes when concidering apriori damping --- .../index.tex | 142 +++++++++++++++++- 1 file changed, 141 insertions(+), 1 deletion(-) 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 d794da4..3e69ac1 100644 --- a/roam/data/d6/e44fbc-975b-463d-87f9-878d50758b55/index.tex +++ b/roam/data/d6/e44fbc-975b-463d-87f9-878d50758b55/index.tex @@ -576,13 +576,153 @@ In the above we have assumed that \(H_{0}\) is hermitian. This, however, ceases to be the case when we assume some a-priori phenomenological decay in the bare components of the system and we cannot write \(H_{0}=H^{H}_{0} - \iu η \id\) with \(H^{H}_{0}\) -hermitian. +hermitian. To retain consistency, the decay rates have to be +introduced on the level of the equations of motion of the mode +operators \(a_{i,α}\) after deriving them from the hermitian +Hamiltonian. The equations of motion can then still be decoupled by +diagonalizing the non-hermitian that includes the phenomenological +decay. + +We find\footnote{Assuming that the non-hermiticity is small enough for +the matrix to remain diagonalizable.} +\begin{equation} + \label{eq:53} + ∑_{iα;jβ}\pqty{T^{-1}}_{m;i,α} \pqty{H_{0}}_{i,α;j,β}T_{j,β;n} = + \pqty{ω_{m}^{0}-\iu η_{m}^{0}}δ_{nm}, +\end{equation} +where \(T\) is the matrix whose rows are the eigenvectors of +\(H_{0}\). Note that \(T\) is not unitary anymore. For notational +convenience we will write \(T^{-1}_{m;i,α}\) instead of +\(\pqty{T^{-1}}_{m;i,α}\) and use explicit fractions if we want to +express the multiplicative inverse. The mode operators transform as +\begin{equation} + \label{eq:60} + c_{m} = ∑_{i,α} T^{-1}_{m;i,α}a_{i,α}, +\end{equation} +which are \emph{not} to be identified with bosons anymore, as the +non-unitarity of \(T\) breaks the bosonic commutation +relations. Again, we express the modes that are in contact with the +transmission line as \(a_{α}=a_{i_{0},α}\) and find +\begin{equation} + \label{eq:69} + α_{α} = ∑_{α} T_{i_{0},α;m}c_{m} \equiv ∑_{α}U_{αm} c_{m}. +\end{equation} +For convenience we define +\begin{equation} + \label{eq:70} + U^{-1}_{mα}\equiv T^{-1}_{m;iα}. +\end{equation} + +The modulation term \(V\) transforms as +\begin{equation} + \label{eq:74} + V_{mn}=∑_{iα;jβ}\pqty{T^{-1}}_{m;i,α} V_{i,α;j,β}T_{j,β;n}, +\end{equation} +and is no longer hermitian. + +We start by writing down the equations of motion for the original +modes, assuming \(H_{0}\) to be hermitian, introduce the non-hermitian +terms and express everything in terms of the \(c_{m}\) using +\(T\). Subsequently, we change into a rotating frame +\begin{equation} + \label{eq:66} + \tilde{c}_{m} = c_{m}\eu^{\iu ω^{0}_{m}t}, +\end{equation} +rotating away only the unitary evolution. Applying the rotating wave +and first Markov approximations works out precisely as in +\cref{sec:rotating-wave-first}. +To account for non-unitarity we have to make the following +replacements along the way +\begin{align} + \label{eq:68} + \tilde{G}_{m}(k) &\rightarrow \tilde{G}_{m}(k)= \frac{gΔx}{\sqrt{L_{A}}} ∑_{β} U_{βm} + G_{β}(k) δ_{\sgn(β),\sgn(k)}\\ + \tilde{G}^\ast_{m}(k) &\rightarrow\tilde{G}^{-1}_{m}(k) = \frac{g^\astΔx}{\sqrt{L_{A}}} ∑_{β} U^{-1}_{mβ} + G^\ast_{β}(k) δ_{\sgn(β),\sgn(k)}\\ + g_{m,σ}&\rightarrow g_{m,σ}=∑_{β}g^{0}_{β} U_{βm}δ_{\sgn(β),σ}\\ + g^\ast_{m,σ}&\rightarrow g^{-1}_{m,σ}=∑_{β}\pqty{g^{0}_{β}}^\ast U^{-1}_{mβ}δ_{\sgn(β),σ}\\ +\end{align} +which gives us +\begin{align} + \label{eq:72} + η_{m}=\frac{π n_{B}}{c} ∑_{σ} g_{mσ}g^{-1}_{mσ}, +\end{align} +which might have an imaginary part. +This leaves us with +\begin{equation} + \label{eq:73} + \dot{\tilde{c}}_{m}= -\iu\bqty{∑_{n}V^{0}_{mn}\eu^{\iu + \pqty{ε_{m}-ε_{n}}t} + \frac{\eu^{\iu ω^{0}_{m}t}}{\sqrt{ω^{0}_{m}}}∑_{σ=\pm}g_{mσ}^{-1}b_{\inputf,σ}^{m}} - \pqty{η_{m} + + η_{m}^{0}}\tilde{c}_{m}. +\end{equation} +To remove the residual explicit time dependence in \cref{eq:73} we +define +\begin{equation} + \label{eq:75} + h_{m}=\tilde{c}_{m}\eu^{-\iu ε_{m}t} +\end{equation} +and find +\begin{equation} + \label{eq:76} + \dot{h}_{m}= -\iu\bqty{∑_{n}\Bqty{V^{0}_{mn}+ \bqty{ε_{m}-\iu + \pqty{η_{m}^{0}+η_{m}}}δ_{nm}}h_{m}} + \frac{\eu^{\iu \pqty{ω^{0}_{m}-ε_{m}}t}}{\sqrt{ω^{0}_{m}}}∑_{σ=\pm}g_{mσ}^{-1}b_{\inputf,σ}^{m}. +\end{equation} +Diagonalizing +\begin{equation} + \label{eq:77} + ∑_{mn}O^{-1}_{γ'm}\bqty{∑_{n}\Bqty{V^{0}_{mn}+ \bqty{ε_{m}-\iu + \pqty{η_{m}^{0}+η_{m}}}δ_{nm}}h_{m}}O_{nγ} = \pqty{ω_{γ}-\iu λ_{γ}}δ_{γ,γ'} +\end{equation} +and defining +\begin{equation} + \label{eq:78} + d_{γ} = ∑_{n}O^{-1}_{γn}h_{n} = ∑_{n}O^{-1}_{γn}\eu^{-\iu + ε_{n}t}\tilde{c}_{n}\implies h_{n}=∑_{γ}\eu^{\iu ε_{n}t}O_{nγ}d_{γ} +\end{equation} +will give us the equivalent of \cref{eq:32} +\begin{equation} + \label{eq:80} + \dot{d}_{γ}=-\iu\pqty{ω_{γ}+\sqrt{κ^\ast}∑_{σ=\pm}U^{σ}_{γ}\frac{b_{\inputf,σ}}{\sqrt{ω_{0}}}}d_{γ} + - λ_{γ}d_{γ} +\end{equation} +where we have set \(g_{β}^{0}=\sqrt{κ}\) and defined +\begin{equation} + \label{eq:82} + U^{σ}_{γ} = + ∑_{mβ}\eu^{\iu\pqty{ω^{0}_{m}-ε_{m}}t}O^{-1}_{γm}U^{-1}_{mβ}δ_{\sgn(β),σ}. +\end{equation} +This also simplifies \cref{eq:64} to +\begin{equation} + \label{eq:88} + η_{m}=\abs{κ}\frac{πn_{B}}{c}∑_{σ=\pm,β,β'}U_{βm}U^{-1}_{βm'}δ_{\sgn(β),σ} δ_{\sgn(β'),σ}. +\end{equation} +Further defining +\begin{align} + \label{eq:83} + \bar{U}^{σ}_{γ}&=∑_{mβ}\eu^{-\iu\pqty{ω^{0}_{m}-ε_{m}}t}O_{mγ}U_{βm}δ_{\sgn(β),σ}\qq{and}& + χ_{γ}&=\abs{κ}\eu^{-\pqty{\iu ω_{γ}+λ_{γ}}t}, +\end{align} +we obtain +\begin{equation} + \label{eq:86} + \ev{{b_{\outputf}(x,t)}} = + \ev{b_{\inputf}(x,t)} - ∑_{σ=\pm}∫_{0}^{τ(x,t)}χ_{\sgn(x),σ}(τ(x,t),s) \ev{b_{\inputf,σ}(s)} \dd{s} +\end{equation} +with the time non-local susceptibility for the left and right moving +input fields +\begin{equation} + \label{eq:87} + χ_{δ,σ}(t,s) = \frac{π n_{B}}{c}Θ(t) ∑_{γ}\bar{U}^{δ}_{γ}(t)χ_{γ}(t-s)U^{σ}_{γ}(s). +\end{equation} +These equations are essentially the same as \cref{eq:39,eq:40}, +accounting for the non-unitary transformations and the apriori decay +rates when diagonalizing the equations of motion for the \(\tilde{c}_{m}\). \section{Application to the Non-Markovian Quantum Walk} \label{sec:appl-non-mark}