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\title{Quantum dynamics for a fractal spectrum}
\author{Ariane SORET, Evgeni GUREVICH, Eric AKKERMAN}
\institute{
ENS Cachan - Technion Institute of Technology}
\date{21 December 2014}

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\section{Introduction}

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Dynamics of system with a fractal feature (scaling symmetry) : turbulence, random walk on fractals, singular continuous energy spectrum...

Non euclidean geometry, singular continuous spectrum.

Ubiquity of singular continuous spectra.

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\section{Definitions - Signature of fractals}

\begin{frame}
%\begin{block}{}
%"Fractal" : introduit par Mandelbrot pour d\'{e}finir les objets tellement irr\'{e}guliers qu'ils sortent du cadre de la g\'{e}om\'{e}trie classique
%\end{block}
\begin{block}{Mandelbrot (1986)}
\textit{"Fractal" : so irregular that it cannot be described with classical geometry ; composed of parts similar to the whole in some way (scaling symmetry)}\\
\end{block}

\begin{columns}
\column{.5\textwidth}
\begin{exampleblock}{Example}
Sierpinski gasket 
\end{exampleblock}
\column{.5\textwidth}
\begin{block}{}
\begin{figure}
\includegraphics[scale=0.08]{sierpinski.png}
\caption{}
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Signature of a fractal spectrum in relevant physical quantities 

Integrated density of states (IDOS) :
\tiny{$$
N(\epsilon)=\int_{spectre} \theta(\epsilon - \epsilon')d\mu(\epsilon')= \frac{1}{2}\int_{spectre} \theta\left(\epsilon - \frac{\epsilon'}{3}\right)d\mu(\epsilon') + \frac{1}{2}\int_{spectre} \theta\left(\epsilon - \frac{\epsilon'+2}{3}\right)d\mu(\epsilon')$$\\
$$ N(\epsilon)= \frac{1}{2}N(3\epsilon) + g(\epsilon)$$}

AJOUTER IMAGE CANTOR

\begin{figure}
\includegraphics[scale=0.3]{idos_cantor.png}
\caption{IDOS for a triadic Cantor set spectrum}
\end{figure}

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\section{Experiment : cavity polaritons in a Fibonacci potential}

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\begin{columns}
\column{.5\textwidth}
\begin{exampleblock}{\small{Fibonacci cavity}}
\includegraphics[scale=0.3]{fibo_seq.JPG}
\caption{Fibonacci cavity. Source : [1]}
\end{exampleblock}
\column{.5\textwidth}
\begin{block}{}
\begin{figure}
\includegraphics[scale=0.3]{cavite-fibo.png}
\caption{Fibonacci cavity. Source : [1]}
\end{figure}
\end{block}
\end{columns}

[1]
%R\'{e}sultats num\'{e}riques et exp\'{e}rimentaux \footnote{D. Tanese, E. Gurevich, A. Lemaitre, E. Galopin, I. Sagnes, A. Amo, J. Bloch, E. Akkermans, arxiv:1311.3453v1}: log-p\'{e}riodicit\'{e} de certaines quantit\'{e}s.

%\begin{figure}
%\includegraphics[scale=0.2]{idos.png}
%\caption{Source : [1]}
%\end{figure}
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\begin{frame}

\begin{columns}
\column{.5\textwidth}
Photon described by $\psi(x,y,z)$, confined in $z$ direction :
$$E\psi(x,y) = -\frac{\hbar^{2}}{2m_{ph}}\Delta_{\perp}\psi(x,y)$$
 with $\Delta_{\perp} =  \partial_{x} + \partial_{y} $, $E \sim \hbar\omega - \frac{ck_{z}}{n}$, $m_{ph}=\frac{\hbar n k_{z}}{c}$ effective mass of the photon.
 
 \medskip
 
 Mapping to a 1D problem with effective quasi-periodic potential $V(x)$ :
 $$E\psi(x) = \frac{\hbar^{2}}{2m_{ph}}\left[-\frac{d^{2}}{dx^{2}} + V(x)\right]\psi(x)$$
\column{.5\textwidth}
\begin{block}{}
\begin{figure}
\includegraphics[scale=0.3]{cavite-fibo.png}
\caption{Fibonacci cavity. Source : [1]}
\end{figure}
\end{block}
\end{columns}
 \end{frame} 
 
 \begin{frame}
 \begin{columns}
 \column{.5\textwidth}
 \begin{figure}
 \includegraphics[scale=0.4]{idos_im2.png}
 %\caption{\tiny{}
 \end{figure}
 \column{.5\textwidth}
 \begin{figure}
 \includegraphics[scale=0.4]{idos_im3.png}
 %\caption{\tiny{Cavit\'{e}de 144 lettres. (a)Emission en champ lointain. (b) R\'{e}sultats num\'{e}rique avec potentiel effectif ; gaps en $[p,q]$ indiqu\'{e}s par fl\`{e}ches rouges. (c) Mesure de l'intensit'{e} totale \'{e}mise (int'{e}grale angulaire)$I(\epsilon')$. (d) Int\'{e}grale spectrale de l'intensit'{e} $\int_{E_{0}}^{\epsilon} I(\epsilon')d\epsilon'$ ($E_{0}$ \'{e}nergie minimale). (e) DOS liss\'{e} calcul\'{e} pour comparaison avec (c). (f) Calcul de IDOS normalis\'{e}. Source : [1]}}
 \end{figure}
 %\column{.5\textwidth}
 %\begin{block}{}
 %\begin{figure}
 %\includegraphics[scale=0.6]{c_t_fibo.png}
 %\caption{Fonction de corr\'{e}lation du paquet d'onde pour la cavit\'{e} de Fibonacci. Source : [1]}
 %\end{figure}
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 Time-averaged return probability :
 $$C(t) = \frac{1}{t}\int_0^t |<\psi(0)|\psi(t')>|^{2} dt' $$
 \begin{figure}
 \includegraphics[scale=0.6]{c_t_fibo.png}
 \caption{Fonction de corr\'{e}lation du paquet d'onde pour la cavit\'{e} de Fibonacci. Source : [1]}
 \end{figure}
 
 Log-periodic oscillation, signature of the fractal feature of the spectrum.
 
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\section{Anaytical study in the case of a Cantor set}

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Consider $H = -\frac{d^{2}}{dx^{2}} + V(x)$ on $L^2(\mathbb{R}$, with associated triadic Cantor set spectrum.
\begin{columns}
\column{.5\textwidth}
\begin{block}{Objectives}
\begin{itemize}
\item Dynamics of the wave packet $i\hbar\partial^{\psi(x,t)}_{x} = \left[-\frac{d^{2}}{dx^{2}} + V(x)\right]\psi(x,t)$ 
\item What is the signature of the fractal spectrum ? Which are the relevant physical quantities ?
\item Analytical expression
%\item Inversement : quelles quantit\'{e}s donnent informations sur le spectre ?  
\end{itemize}
\end{block}
\column{.5\textwidth}
\begin{block}{}
\begin{figure}
\includegraphics[scale=0.5]{cantor.png}
\caption{Triadic Cantor set}
\end{figure}
\end{block}
\end{columns}
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\subsection{Appropriate measure}


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Triadic Cantor set $K$: zero (Lebesgue) measure. Degenerate density of states $\rho(\epsilon)$. Define measure :$d\mu(\epsilon) \equiv \rho(\epsilon)d\epsilon$, such that :
%D\'{e}finition d'une mesure appropri\'{e} :
%$$\int_K f(x)d\mu(x) = \lim_{n \to + \infty} \left(\frac{3}{2}\right)^{n} \sum_{P_{n}} \int_{a_{j}}^{a_{j}+3^{-n}} f(x)dx$$
%avec $P_{n} = \{a_{j}\}$ ensemble des extr\'{e}mit\'{e}s gauches des segments restant \`{a} l'\'{e}tape $n$ de la construction du Cantor.
%\medskip 
%V\'{e}rifie :
\begin{itemize}
\item uniform 
\item $\int_K d\mu(x) = 1$ 
\end{itemize}
Property :
\begin{itemize}
\item $ \int_K f(x)d\mu(x) = \frac{1}{2}\int_K f(\frac{x}{3})d\mu(x) + \frac{1}{2}\int_K f(\frac{x+2}{3})d\mu(x)$
\end{itemize} 
%\textit{Remarque : on peut abusivement voir $d\mu$ comme $d\mu(\epsilon) \equiv \rho(\epsilon)d\epsilon$}

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\begin{frame}
Integrated density of states (IDOS) :
$$
N(\epsilon)=\int_{spectre} \theta(\epsilon - \epsilon')d\mu(\epsilon')$$

\begin{columns}
\column{.5\textwidth}
\begin{block}{}
\begin{figure}
\includegraphics[scale=0.2]{idos_cantor.png}
\caption{IDOS for a triadic Cantor set}
\end{figure}
\end{block}
\column{.5\textwidth}
\begin{block}{}
\begin{figure}
\includegraphics[scale=0.4]{idos_im3.png}
\caption{IDOS for the Fibonacci cavity}
\end{figure}
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\subsection{Time-averaged return probability}
\begin{frame}
Time-averaged return probability :
$C(t) = \frac{1}{t}\int_0^t |<\psi(0)|\psi(t')>|^{2} dt' $
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\begin{frame}
Decomposition over the eigenfunctions $\{|\phi(\epsilon)>\}$ of $H$ :
$$ |\psi(t)> = \int_{K} g(\epsilon) e^{-i\epsilon t} |\phi(\epsilon)>d\mu(\epsilon)$$
Then :
\begin{eqnarray*}
\lefteqn{C(t) = \frac{1}{t}\int_0^t |<\psi(0)|\psi(t')>|^{2} dt'}\\
&=& \iint_{C\times C} sinc((\epsilon' - \epsilon)t)d\mu(\epsilon)d\mu(\epsilon')
\end{eqnarray*}

\small{\textit{Note} : wavelet transform}.

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\begin{frame}
$$ C(t) = \iint_{K\times K} sinc((\epsilon' - \epsilon)t)d\mu(\epsilon)d\mu(\epsilon')$$
\begin{columns}
\column{.5\textwidth}
\begin{block}{}
Change of variable : $l = |\epsilon' - \epsilon|$ \\
$dS(l,\mu) = $area in the sense of $\mu$ of the Cantor dust within the red strips\\
\end{block}
\begin{alertblock}{}
$C(t) = \int_0^1 sinc(lt) dS(l,\mu)$
\end{alertblock}
\column{.5\textwidth}
\begin{block}{}
\includegraphics[scale=0.4]{cantor_carpet.PNG}
\end{block}
\end{columns}
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\begin{frame}
Using Mellin transform to derive $dS(l,\mu)$ :
\begin{block}{Time-averaged return probability for triadic Cantor set}
\small{$$C(t) =  \frac{1}{ln(3)}\sum_{n}\epsilon_{n}t^{-\frac{2i\pi n}{ln(3)}}sin\left(\frac{\pi}{2}(d_{n}-1)\right)\Gamma(d_{n}-1)$$}

\end{block}
$d_{n}=\frac{\ln(2)}{\ln(3)}+\frac{2i\pi n}{\ln(3)}$, $\epsilon_{n} = \int_{K\times K} \frac{d\mu(x) d\mu(y)}{|x-y+2|^{d+\frac{2i\pi n}{\ln(3)}}}$
\end{frame}

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\begin{figure}
\includegraphics[scale=0.3]{correlation_function_log.png}
\caption{Return probability $C(t)$ for a Cantor spectrum}
\end{figure}

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