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Pierre-Francois Loos 2020-07-23 16:20:23 +02:00
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RapportStage/Rapport.tex
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@ -116,7 +116,7 @@ Laboratoire de Chimie et Physique Quantiques
31062 Toulouse - France}
\url{https://www.irsamc.ups-tlse.fr/}
\url{https://www.irsamc.ups-tlse.fr/loos}
} %fin de la commande \parbox encadrant / laboratoire d'accueil
\vspace{0.5cm}
@ -227,7 +227,7 @@ In the real $\lambda$ axis, the point for which the states are the closest ($\la
The ``shape'' of the avoided crossing is linked to the magnitude of $\Im(\lambda_\text{EP})$: the smaller $\Im(\lambda_\text{EP})$, the sharper the avoided crossing is.
Around $\lambda = \lambda_\text{EP}$, Eq.~\eqref{eq:E_2x2} behaves as \cite{MoiseyevBook}
\begin{equation}
\begin{equation} \label{eq:E_EP}
E_{\pm} = E_\text{EP} \pm \sqrt{2\lambda_\text{EP}} \sqrt{\lambda - \lambda_\text{EP}},
\end{equation}
and following a complex contour around the EP, i.e.~$\lambda = \lambda_\text{EP} + R \exp(i\theta)$, yields
@ -242,26 +242,33 @@ and we have
\end{align}
This evidences that encircling non-Hermitian degeneracies at EPs leads to an interconversion of states, and two loops around the EP are necessary to recover the initial energy.
The eigenvectors associated to the energies \eqref{eq:E_2x2} are
\begin{equation}
\phi_{\pm}=\begin{pmatrix}
\frac{1}{2\lambda}(\epsilon_1 - \epsilon_2 \pm \sqrt{(\epsilon_1 - \epsilon_2)^2 + 4\lambda^2}) \\ 1
\end{pmatrix},
The eigenvectors associated to the eigenenergies \eqref{eq:E_2x2} are
\begin{equation} \label{eq:phi_2x2}
\phi_{\pm}=
\begin{pmatrix}
E_{\pm}/\lambda
\\
1
\end{pmatrix},
\end{equation}
and for $\lambda=\lambda_\text{EP}$ they become
and, for $\lambda=\lambda_\text{EP}$, they become
\begin{equation}
\phi_{\pm}=\begin{pmatrix}
\mp i \\ 1
\end{pmatrix},
\end{equation}
which are clearly self-orthogonal. The equation (7) can be rewrite as
which are clearly self-orthogonal.
\titou{what do you mean by self-orthogonal?}
Using Eq.~\eqref{eq:E_EP}, Eq.~\eqref{eq:phi_2x2} can be recast as
\begin{equation}
\phi_{\pm}=\begin{pmatrix}
\frac{1}{\lambda}(\frac{\epsilon_1 - \epsilon_2}{2} \pm \sqrt{2\lambda_\text{EP}} \sqrt{\lambda - \lambda_\text{EP}}) \\ 1
\end{pmatrix},
\phi_{\pm}=
\begin{pmatrix}
E_\text{EP}/\lambda \pm \sqrt{2\lambda_\text{EP}/\lambda} \sqrt{1 - \lambda_\text{EP}/\lambda}
\\
1
\end{pmatrix},
\end{equation}
we can see that if we normalise them they will behave as
$(\lambda - \lambda_\text{EP})^{-1/4}$ resulting in the following pattern when looping around one EP:
where one clearly see that, if normalised, they behave as $(\lambda - \lambda_\text{EP})^{-1/4}$ resulting in the following pattern when looping around one EP:
\begin{align}
\phi_{\pm}(2\pi) & = \phi_{\mp}(0),
&
@ -270,10 +277,9 @@ $(\lambda - \lambda_\text{EP})^{-1/4}$ resulting in the following pattern when l
&
\phi_{\pm}(8\pi) & = \phi_{\pm}(0),
\end{align}
showing that 4 loops around the EP are necessary to recover the initial state. We can also that looping around the other way round leads to a different pattern.
\titou{Maybe you should add a few equations here to highlight the self-orthogonality process.
What do you think?
You could also show that the behaviour of the wave function when one follows the complex contour around the EP.}
\titou{I don't think it is too obvious that the eigenvectors behave as $(\lambda - \lambda_\text{EP})^{-1/4}$.}
In plain words, four loops around the EP are necessary to recover the initial state.
We can also see that looping the other way around leads to a different pattern.
%============================================================%
\section{Perturbation theory}