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\title{Quantum dynamics for a fractal spectrum}
\author{Ariane SORET}
\institute{Supervised by Eric AKKERMAN \\
ENS Cachan - Technion Institute of Technology}
\date{6 August 2014}

\begin{document}

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\section{Mathematical prerequisites}

\begin{frame}
\begin{block}{Mandelbrot's definition of a fractal(1986)}
\textit{A fractal is a shape made of parts similar to the whole in some way}\\
\end{block}
\begin{block}{More mathematical formulation (1982)}
\textit{A fractal is by definition a set for which the Hausdorff-Besicovitch dimension strictly exceeds the topological dimension}
\end{block}

In our case : objects with a scaling symmetry.
\begin{columns}
\column{.5\textwidth}
\begin{exampleblock}{Example}
Sierpinsky gasket
\end{exampleblock}
\column{.5\textwidth}
\begin{block}{}
\begin{figure}
\includegraphics[scale=0.08]{sierpinski.png}
\caption{}
\end{figure}
\end{block}
\end{columns}


\end{frame}

\begin{frame}

Scaling property and log-periodic oscillations :
$$f(x) = \frac{1}{b}f(ax)$$
$$f(x) = x^{ln(b)/ln(a)}g\left(\frac{ln(x)}{ln(a)}\right) $$
$g$ periodic of period 1.

\begin{columns}
\column{.5\textwidth}
\begin{exampleblock}{Another example : triadic Cantor set}
  "zoom in" by a factor 3 and get the same object, with total mass or number of states rescaled by factor $\frac{1}{2}$
\end{exampleblock}
\column{.5\textwidth}
\begin{block}{}
\begin{figure}
\includegraphics[scale=0.7]{cantor.png}
\caption{Triadic Cantor set}
\end{figure}
\end{block}
\end{columns}

\end{frame}

\section{Experimental example : Fibonacci cavity}

\begin{frame}

\begin{columns}
\column{.5\textwidth}
\begin{exampleblock}{Fibonacci cavity}
  Spacially varying potential yields modification of the density of states of the eigenmodes ;  
   quasiperiodic potential -> ???
  \medskip
  Longitudinal cavity of quasiperiodically varying width according to Fibonacci order -> Cantor like spectrum.
\end{exampleblock}
\column{.5\textwidth}
\begin{block}{}
\begin{figure}
\includegraphics[scale=0.2]{cavite-fibo.png}
\caption{Fibonacci cavity. Source : [1]}
\end{figure}
\end{block}
\end{columns}


Previous numerical and experimental results \footnote{D. Tanese, E. Gurevich, A. Lemaitre, E. Galopin, I. Sagnes, A. Amo, J. Bloch, E. Akkermans, arxiv:1311.3453v1} show log-periodic behaviour of some quantities...

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

\end{frame}


\section{Anaytical study in the case of a Cantor set}

\begin{frame}

Hamiltonian $H$, spectrum = triadic Cantor set.
\begin{itemize}
\item Dynamics of the wave paquet ?
\item Which physical quantities highlight the fractal feature ?
\item Analytical expression ?
\item Conversely : determine the spectrum from these quantities ?  
\end{itemize}

\end{frame}

\subsection{Measure}
\begin{frame} 
Define an appropriate measure :
$$\int_C f(x)d\mu(x) = \lim_{n \to + \infty} \left(\frac{3}{2}\right)^{n} \sum_{P_{n}} \int_{a_{j}}^{a_{j}+3^{-n}} f(x)dx$$
with $P_{n} = \{a_{j}\}$ the set of the left edges of the remaining segments after $n$ steps.
\medskip 
Satisfies :
\begin{itemize}
\item $\int_C d\mu(x) = 1$
\item $ \int_C f(x)d\mu(x) = \frac{1}{2}\int_C f(\frac{x}{3})d\mu(x) + \frac{1}{2}\int_C f(\frac{x+2}{3})d\mu(x)$
\end{itemize} 
\textit{Note : informally, one can think of $d\mu$ as $d\mu(\epsilon) \equiv \rho(\epsilon)d\epsilon$}

\end{frame}

\begin{frame}
Integrated density of states (IDOS) :
\begin{eqnarray*}
\lefteqn{N(\epsilon)=\int_{spectrum} \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') \\
&=& \frac{1}{2}N(3\epsilon) + g(\epsilon)
\end{eqnarray*}
\end{frame}





\subsection{Correlation function}

\begin{frame}
\begin{block}{Correlation function or return probability}
$C(t) = \frac{1}{t}\int_0^t |<\psi(0)|\psi(t')>|^{2} dt' $
\end{block}
 
Results for the Fibonacci cavity : log-periodic oscillations.\\
Likely to be the same for Cantor.\\
\medskip
$|\psi(t)>$ wave function. \\


%Definition of the time-averaged correlation function


\end{frame}

\begin{frame}
Decomposition over the eigenfunction (orthonormal basis) oh the Hamiltonian $H$
$$ |\psi(t)> = \int_{C} g(\epsilon) e^{-i\epsilon t} |\phi(\epsilon)>d\mu(\epsilon)$$

Normalization :

\begin{eqnarray*}
\lefteqn{<\psi(t)|\psi(t)> = \int_{C} |g(\epsilon)|^{2} d\mu(\epsilon)} \\
&=& 1
\end{eqnarray*}

Define : $d\tilde\mu(\epsilon) = |g(\epsilon)|^{2} d\mu(\epsilon)$

\end{frame}

\begin{frame}

\begin{eqnarray*}
\lefteqn{C(t) = \frac{1}{t}\int_0^t |<\psi(0)|\psi(t')>|^{2} dt'}\\
&=& \frac{1}{t} \int_0^t |\int_{C} |g(\epsilon|^{2}e^{-i\epsilon t'}d\mu(\epsilon)|^{2}dt' \\
&=& \frac{1}{t} \int_0^t \iint_{C\times C} e^{i(\epsilon' - \epsilon)t'}d\tilde\mu(\epsilon)d\tilde\mu(\epsilon')\\
&=& \frac{1}{t} \iint_{C\times C} \frac{e^{i(\epsilon' - \epsilon)t} -1}{i(\epsilon' - \epsilon)} d\tilde\mu(\epsilon)d\tilde\mu(\epsilon') \\
&=& \iint_{C\times C} sinc((\epsilon' - \epsilon)t)d\tilde\mu(\epsilon)d\tilde\mu(\epsilon') + \underbrace{i\iint_{C\times C} \frac{sin^{2}((\epsilon' - \epsilon)t/2)}{(\epsilon' - \epsilon)t}}_{\mbox{nul by symmetry}} 
\end{eqnarray*}

\end{frame}

\begin{frame}
$$ C(t) = \iint_{C\times C} sinc((\epsilon' - \epsilon)t)d\tilde\mu(\epsilon)d\tilde\mu(\epsilon')$$
\begin{columns}
\column{.5\textwidth}
\begin{block}{}
Change of variable : $l = |\epsilon' - \epsilon|$ \\
$dS(l,\mu) = $area for measure $\mu$ of the Cantor carpet in the strips\\
\end{block}
\begin{alertblock}{Correlation function becomes }
$C(t) = \int_0^1 sinc(lt) dS(l,\mu)$
\end{alertblock}
\column{.5\textwidth}
\begin{block}{}
\includegraphics[scale=0.4]{cantor_carpet.PNG}
\end{block}
\end{columns}
\end{frame}

\begin{frame}
\begin{figure}
\includegraphics[scale=0.4]{c_mu_l.png}
\caption{$S(l,\mu) $for a triadic Cantor set}
\end{figure}
\end{frame}

\begin{frame}
Scaling property + Mellin to calculate $dS(l,\mu)$. \\
\begin{block}{Mellin transform}
\begin{itemize}
\item $M_{f}(s) = \int_0^{\infty}f(x)x^{s-1}dx$
\item $M^{-1}_{\phi}(x) = \frac{1}{2i\pi}\int_{\gamma -i\infty}^{\gamma +i\infty} \phi(s)x^{-s}ds$
\end{itemize}
\end{block}

\begin{eqnarray*}
\lefteqn{\zeta(\alpha) = \int_0^1 l^{-\alpha}dS(l,\mu)}\\
&=& \iint_{C\times C} \frac{d\mu(x)d\mu(y)}{|x-y|^{\alpha}} \\
&=& \frac{1}{2}\iint_{C\times C} \frac{d\mu(x)d\mu(y)}{3^{-\alpha}|x-y|^{\alpha}} + \iint_{C\times C} \frac{d\mu(x)d\mu(y)}{3^{-\alpha}|x-y+2|^{\alpha}} \\
&=& \frac{3^{\alpha}}{2}\zeta(\alpha) + \epsilon(\alpha)
\end{eqnarray*}
\end{frame}


\begin{frame}
Therefore : $\zeta(\alpha) = \frac{\epsilon(\alpha)}{1-\frac{3^{\alpha}}{2}}$ 
and
\begin{eqnarray*}
\lefteqn{dS(l,\mu) = \frac{1}{2i\pi}\int_{\gamma -i\infty}^{\gamma +i\infty} \zeta(1-\alpha)l^{-\alpha}d\alpha}\\
&=& \sum_{n} \left[\frac{\alpha - (1-\alpha_{n})}{1-\frac{3^{1-\alpha}}{2}}l^{-\alpha}\epsilon(\alpha)\right]_{\alpha=\alpha_{n}}
\end{eqnarray*}
$1-\alpha_{n} = \frac{ln(2)}{ln(3)}$ , $d=\frac{ln(2)}{ln(3)}$
\begin{block}{}
$dS(l,\mu) = \frac{l^{d-1}}{ln(3)}\sum_{n}l^{\frac{2i\pi n}{ln(3)}}\epsilon\left(d+\frac{2i\pi n}{ln(3)}\right)=\frac{l^{d-1}}{ln(3)}\sum_{n}l^{\frac{2i\pi n}{ln(3)}}\epsilon_{n}$
\end{block}
\end{frame} 

\begin{frame}

%begin{eqnarray*}
%lefteqn{C(t) = \int_0^1 sinc(lt) dS(l,\mu)}\\
%=& \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{eqnarray*}
\begin{block}{Correlation function when the spectrum is a Cantor set}
\small{$$C(t) =  \frac{1}{ln(3)}\sum_{n}\epsilon_{n}t^{-\frac{2i\pi n}{ln(3)}}sin\left(\frac{\pi}{2}(d_{n}-1)\right)\Gamma(d_{n}-1)$$}
\small{$$C(t) =\left(\frac{E}{e.u.}\right)^{2}\left(\frac{Et}{\hbar}\right)^{-d} \sum_{n}\frac{\epsilon_{n}}{ln(3)}\left(\frac{Et}{\hbar}\right)^{-\frac{2i\pi n}{ln(3)}}sin\left(\frac{\pi}{2}(d_{n}-1)\right)\Gamma(d_{n}-1)$$}
\end{block}
%d_{n} = d+\frac{2i\pi n}{ln(3)}$\\

$d_{n} = d+\frac{2i\pi n}{ln(3)}$
\end{frame}

\begin{frame}


\begin{figure}
\includegraphics[scale=0.3]{correlation_function.png}
\caption{\tiny{Correlation function $C(t)$ for a triadic Cantor set embedded in [0,E]}}
\end{figure}

\end{frame}

\begin{frame}

\begin{figure}
\includegraphics[scale=0.3]{correlation_function_log.png}
\caption{Correlation function $C(t)$ for a triadic Cantor set embedded in [0,E], log scale}
\end{figure}


\end{frame}

\begin{frame}
Asymptotic behaviour : 
$$|\epsilon_{n}sin(\frac{\pi}{2}(d+\frac{2i\pi n}{ln(3)}-1))\Gamma(d+\frac{2i\pi n}{ln(3)}-1)| = O\left(\frac{1}{n^{5/2-d}}\right)$$
\end{frame}

\subsection{Participation ratio}


\begin{frame}
\begin{block}{Participation ratio}
$P_{r}(t) = \frac{1}{t}\int_0^t \frac{dt,}{|<x|\psi(t')>|^{4}} $
\end{block}



Again, numerics for the Fibonacci cavity gave log-periodic oscillations ; so it is
likely to be the same for Cantor.\\
\end{frame}

\begin{frame}
Hamiltonian $H$, wave function $|\psi(t)>$, orthonormal eigenfunctions basis $\{|\phi(\epsilon)>\}$
\begin{block}{Green function}
$G(x,x',t) = <x|e^{-iHt}|x'> = \int_{spectrum} \phi(x,\epsilon)^{*}\phi(x',\epsilon)e^{-i\epsilon t}d\mu(\epsilon)$
\end{block}

\begin{block}{}
$G^{A,R}(x,x',t) = \pm i\theta(\mp t)\int_{spectrum} \phi(x,\epsilon)^{*}\phi(x',\epsilon)e^{-i\epsilon t}d\mu(\epsilon)$\\
$Im(G^{R})(x,x',t) = \frac{G^{A}(x,x',t) - G^{R}(x,x',t)}{i}$
\end{block}

\begin{block}{Fourier transform}
\begin{eqnarray*}
\lefteqn{G^{A,R}(x,x',\epsilon) = \int_{-\infty}^{+\infty} G^{A,R}(x,x',t) e^{i\epsilon t}dt}\\
&=& \lim_{\eta \to 0^{+}}\int_{spectrum}\frac{\phi(x,\epsilon)^{*}\phi(x',\epsilon)}{\epsilon - \epsilon' \mp i\eta}d\mu(\epsilon')
\end{eqnarray*}
\end{block}
\end{frame}

\begin{frame}
Using : $\frac{1}{x+i0} = PP\left(\frac{1}{x}\right) - i\pi\delta(x)$ one has 
\begin{block}{Relation between the Green function and the local density of states}
$Im(G^{R})(x,x',\epsilon) = 2\pi \phi(x,\epsilon)^{*}\phi(x',\epsilon)d\mu(\epsilon)$
\end{block}

\begin{block}{Density of states, local density of states} 
$"\rho(\epsilon)d\epsilon" = d\mu(\epsilon)$\\
$"\rho(\epsilon,x)d\epsilon" = |\phi(x,\epsilon)|^{2}d\mu(\epsilon)$\\
$"\rho(\epsilon,x,x')d\epsilon" = \phi(x,\epsilon)^{*}\phi(x',\epsilon)d\mu(\epsilon)$
\end{block}

\textit{These definition are generic ; in some cases, $\rho$ is not defined (like for fractals)-> quotation marks}.

\end{frame}

\begin{frame}

\begin{eqnarray*}
\lefteqn{|\psi(x,t)| = |<x|\psi(t)>|}\\
&=& \left|<x|e^{-iHt}|\psi(0)>\right|\\
&=& \left|\int_{space}<x|e^{-iHt}|x'>\psi(x',0)dx'>\right|\\
&=& \left|\int\limits_{space}{\int\limits_{spectrum} e^{-i\epsilon t} Im G(x,x',\epsilon)\psi(0,t)dx'd\mu(\epsilon)}\right|\\
&=& \left|\int\limits_{space}{\int\limits_{spectrum} e^{-i\epsilon t} \phi(x,\epsilon)^{*}\phi(x',\epsilon)\psi(x',0)dx'd\mu(\epsilon)}\right|
\end{eqnarray*}

\begin{alertblock}{} 
What can we say about $\phi(x,\epsilon)^{*}\phi(x',\epsilon)d\mu(\epsilon)$ ?
\end{alertblock}


\end{frame}

\section{Outlooks}
\subsection{Ideas for the participation ratio}

\begin{frame}
Ideas for continuing the study of the participation ratio :
\begin{enumerate}
\item Assume $\overline{\phi(0,\epsilon)^{*}\phi(x,\epsilon)} = f\left(\frac{x}{l_{n}}\right)$ ?
\item Find/assume dispersion relation and use Fourier theory
\item Prove that $\int_{band} \phi(x,\epsilon)^{*}\phi(x',\epsilon)d\mu(\epsilon) < a||x-x'||^{-b}$
\end{enumerate}

\begin{block}{}
\textit{Note}
(1) is reasonable for the eigenfunctions of the Laplacian ; try numerics on a system with (fractal ?) shape in space in order to have a fractal spectrum for the Laplacian.\\
(2) Related to definition and convergence of Fourier series on fractals.\\
(3) Such a result holds for diagonal terms of the resolvent\footnote{E. Akkermans,O. Benichou,G. V. Dunne,A. Teplyaev,and Raphael Voituriez,Physical Review E86,061125(2012)} : $G(x,x,t)$ 
\end{block}

\end{frame} 

\subsection{Spontaneous emission - superradiance - black body radiation}

\begin{frame}
One could apply the same kind of calculation to study spontaneous emission and superradiance, and thermodynamics on fractal structures. 
\end{frame}

%\section{Conclusion}

%\begin{frame}

%\begin{itemize}
%\item 



\end{document}
     


