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\title{Quantum dynamics of a wave packet for a Cantor spectrum}
\author{Ariane SORET$^{1,2}$\\
Evgeni GUREVICH$^1$, Eric AKKERMANS$^1$}
\institute{1 - Technion Institute of Technology ; 2 - \'Ecole Normale Supérieure Cachan }
\date{IPS meeting 2014 - Be'er Sheva}

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%\section{Introduction}

%\begin{frame}

%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.

%\end{frame}

\section{Introduction}

\begin{frame}

%Dynamics of systems with a fractal feature (spatial and/or spectral) ?

\medskip

Consider systems with a (spatial and/or spectral) \textcolor{blue}{fractal feature}.

\medskip

Motivation :
\begin{itemize}
\item \textcolor{blue}{singular continuous} spectra are ubiquitous in physical systems, but not well known
\item Explain experimental and numerical observations
\end{itemize}

Objectives :
\begin{itemize}
\item Identify typical behaviour of relevant physical quantities in the presence of a fractal feature (spatial and/or spectral) $\rightarrow$ power-law modulated by log-periodic oscillations 
\item Derive analytical expressions, and relate them to experimental results
\end{itemize} 

%\begin{figure}
%\includegraphics[scale=0.4]{cantor.png}
%\caption{Triadic Cantor set}
%\end{figure}
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\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\$
%\end{block}
\begin{columns}
\column{.5\textwidth}
%\begin{block}{Fractal - definition}
Fractal :
\begin{itemize}
\item a set with a \textcolor{red}{discrete scaling symmetry}
\item fractal dimension $d_f$ \\
(length scale $l$ ; volume scales as $l^{d_f}$)
\end{itemize}
%\end{block}
\column{.5\textwidth}
\begin{figure}
\includegraphics[scale=0.1]{sierpinski.png}
\caption{Sierpinski gasket ; $d_f = \frac{\ln 3}{\ln 2}$}
\end{figure}
\end{columns}
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\section{Analytical study in the case of a Cantor set}

\begin{frame}
Quantum dynamics for a triadic Cantor set spectrum.

\medskip

$H = -\frac{d^{2}}{dx^{2}} + V(x)$ on $L^2(\mathbb{R})$, with associated triadic Cantor set spectrum.
%\begin{columns}
%\column{.5\textwidth}
%\begin{itemize} 
%\item Eigenfunctions $\{|\phi(\epsilon)\rangle\}$ of $H$
%\item Spectral decomposition $ |\psi(t)\rangle = \int_{K} g(\epsilon) e^{-i\epsilon t} |\phi(\epsilon)\rangle d\mu(\epsilon)$
%\item Analytical expression
%\item Inversement : quelles quantit\'{e}s donnent informations sur le spectre ?  
%\end{itemize}
%\column{.5\textwidth}
%\begin{block}{}
\begin{figure}
\includegraphics[scale=0.8]{cantor.png}
\caption{Triadic Cantor set. Fractal dimension $d_f = \frac{\ln 2}{\ln 3}$}
\end{figure}
%\end{block}
%\end{columns}
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\subsection{Appropriate measure}


\begin{frame} 
Triadic Cantor set $K$: zero (Lebesgue) measure. Degenerate density of states $\rho(\epsilon)$. Define measure :$d\mu(\epsilon) \equiv \rho(\epsilon)d\epsilon$ :

\begin{figure}
\includegraphics[scale=0.5]{mesure4.JPG}
\end{figure}


 %$\int_K f(x)d\mu(x) = \lim_{n \to + \infty} \left(\frac{3}{2}\right)^{n} \sum_{P_{n}} \int_{a_{j,n}}^{a_{j,n}+3^{-n}} f(x)dx$
%with $P_{n} = \{a_{j,n}\}$ the left edges of the remaining segments after $n$ steps 



Verifies : $ \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)$
(\textcolor{red}{scaling property})





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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}
%\end{block}
%\end{columns}
%\end{frame}


\subsection{Time-averaged return probability}
\begin{frame}
How does the fractal feature affect the dynamics of the wave packet $ |\psi(t)\rangle$ ?

\begin{enumerate}
\item RMS displacement $\Delta x (t)= \sqrt{\int(x(t)-x_0)^2 |\psi(x,t)|^2 dx}$
\item return probability $p(t) = |\langle\psi(0)|\psi(t)\rangle|^{2}$
\item ...
\end{enumerate}


\end{frame}

\begin{frame}
A relevant quantity : the \textcolor{blue}{time-averaged return probability} :
$$C(t) = \frac{1}{t}\int_0^t |\langle\psi(0)|\psi(t')\rangle|^{2} dt' $$

\end{frame}

\begin{frame}
A relevant quantity : the \textcolor{blue}{time-averaged return probability} :
$$C(t) = \frac{1}{t}\int_0^t |\langle\psi(0)|\psi(t')\rangle|^{2} dt'=\iint_{K\times K} sinc((\epsilon' - \epsilon)t)d\mu(\epsilon)d\mu(\epsilon')$$
Note : $\epsilon$ and $t$ are coupled variables.\\
Method
\begin{enumerate}
\item use the scaling symmetry of the measure to obtain a functional equation
\item apply the Mellin transform   
\end{enumerate}
\end{frame}

\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 2D Cantor set ("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}
\tiny{(cf. energy integrals :  D. Bessis et al., Phys. Rev. A,
Vol. 36, number 2, July 15, 1987)}
\end{frame}

\begin{frame}
The main steps :
\begin{enumerate}
\item Mellin transform : $\zeta(\alpha) = \int_0^1 l^{-\alpha}dS(l,\mu) = \iint_{K\times K} \frac{d\mu(x)d\mu(y)}{|x-y|^{\alpha}}$
\item scaling property : \\
\small{$\iint_{K\times K} \frac{d\mu(x)d\mu(y)}{|x-y|^{\alpha}} = \frac{1}{2}\iint_{K\times K} \frac{d\mu(x)d\mu(y)}{3^{-\alpha}|x-y|^{\alpha}} + \underbrace{\frac{1}{2}\iint_{K\times K} \frac{d\mu(x)d\mu(y)}{3^{-\alpha}|x-y+2|^{\alpha}}}_{g(\alpha)} $}

\medskip

\small{$= \frac{3^{\alpha}}{2}\zeta(\alpha) + g(\alpha)$}
\item $\zeta(\alpha) = \frac{g(\alpha)}{1-\frac{3^{\alpha}}{2}}$
\item inverse Mellin transform $\rightarrow dS(l,\mu) \rightarrow C(t)$ 
\end{enumerate}
\textit{Remind} : 
\begin{itemize}
\item Mellin transform of $f$ : $M_{f}(s) = \int_0^{\infty}f(x)x^{s-1}dx$
\item Inverse Mellin transform : $M^{-1}_{\phi}(x) = \frac{1}{2i\pi}\int_{c-i\infty}^{c+i\infty}\phi(s)x^{-s}ds$
\end{itemize}
\end{frame}
   
\begin{frame}
\begin{block}{Time-averaged return probability for triadic Cantor set}
\small{$$C(t) =  \frac{t^{-\ln 2/\ln 3}}{\ln(3)}\sum_{n}\epsilon_{n}e^{-\frac{2i\pi n}{ln(3)}\ln t}\sin\left(\frac{\pi}{2}(d_{n}-1)\right)
\Gamma(d_{n}-1)$$}

\end{block}
\small{$d_{n}=\frac{\ln(2)}{\ln(3)}+\frac{2i\pi n}{\ln(3)}$, $\epsilon_{n} = \int_{K\times K} \frac{d\mu(x) d\mu(y)}{|x-y+2|^{d+\frac{2i\pi n}{\ln(3)}}}$, $\Gamma$ the Euler gamma function}
$\Rightarrow C(t) = t^{-\ln 2/\ln 3}G_p(\ln t/\ln 3)$, $G_p(y+1) = G_p(y)$\\
\begin{enumerate}
\item \textcolor{blue}{power-law} modulated by \textcolor{blue}{log-periodic oscillations}. 
\item $C(t) \propto t^{-d_f}$, with $d_f = \frac{\ln 2}{\ln 3}$ (fractal dimension) for a triadic Cantor set.
\end{enumerate}
\end{frame}


\begin{frame}

\begin{figure}
\includegraphics[scale=0.3]{ct_loglog2.eps}
%\includegraphics[scale=0.3]{ct.eps}
\caption{Return probability $C(t)$ for a Cantor spectrum}
%$C(t) = t^{-\ln 2/\ln 3}G_p(\ln t/\ln 3)$, $G_p(y+1) = G_p(y)$}
\end{figure}

\end{frame}

\begin{frame}
\begin{figure}
\includegraphics[scale=0.3]{ct_per.eps}
\caption{Return probability divided by $t^{-\ln 2/\ln 3}$, $C(t)t^{\ln 2/\ln 3}$ for a Cantor spectrum}
\end{figure}
\end{frame}

\begin{frame}
Partial summary :
\begin{itemize}
\item General method :\\
scaling property \\
$\rightarrow$ functional equation $f(x) = \frac{1}{b}f(ax)+g(x)$ \\
$\rightarrow$ Mellin transform \\ 
$\rightarrow f(x) = x^{\ln(b)/\ln(a)}G_p\left(\frac{\ln{x}}{\ln{a}}\right)$, $G_p(y+1) = G_p(y)$
\item Typical behaviour due to a fractal spectrum : power-law modulated by log-periodic oscillations 
\item The method gives an intuition of which are the relevant physical quantities to observe typical behaviour
\end{itemize}
\end{frame}


\subsection{Other relevant quantities}

\begin{frame}
Other relevant quantities :
\begin{itemize}
\item Participation ratio : $P_{r}(t) = \frac{1}{t}\int_0^t \frac{dt,}{|\langle x|\psi(t')\rangle|^{4}} $\\
\item RMS displacement : $\Delta x (t)= \sqrt{\int(x(t)-x_0)^2 |\psi(x,t)|^2 dx}$  
\end{itemize}

\end{frame}


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

%\begin{frame}

%\begin{columns}
%\column{.5\textwidth}
%\begin{exampleblock}{\small{Fibonacci cavity}}
%\begin{figure}
%\includegraphics[scale=0.6]{fibo_seq.JPG}
%\caption{$S_1 = B$, $S_2 = A$, $S_{j+1}=S_{j-1}S_{j}$}
%\end{figure}
%\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] Tanese, E. Gurevich, A. Lematre, E. Galopin, I. Sagnes, A. Amo, J. Bloch, E.
%Akkermans, Physical Review Letters 112, 146404 (2014)
%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}
%\end{frame}

\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^2_{x} + \partial^2_{y} $,  $m_{ph}$ effective mass of the photon.
Cavity polaritons (quasi-particles arising from the strong coupling between photons and excitons) placed in a quasi-periodic potential.

 \medskip
 
 Mapping to a 1D problem with effective \textcolor{blue}{quasi-periodic potential} $V(x)$ with a \textcolor{blue}{Cantor like spectrum} :
 $$E\psi(x) = \frac{\hbar^{2}}{2m_{ph}}\left[-\frac{d^{2}}{dx^{2}} + V(x)\right]\psi(x)$$
\column{.5\textwidth}
%\begin{block}{}
\begin{figure}
\includegraphics[scale=0.4]{cav-seq-fibo.JPG}
\caption{Fibonacci cavity}
\end{figure}
%\end{block}
\end{columns}
 \end{frame} 
 
 \begin{frame}
Numerical and experimental results 

 \begin{figure}
 \includegraphics[scale=0.5]{idos-fibo2.JPG}
 %\caption{\tiny{}
 \end{figure}


%\tiny{[1] Akkermans, Gurevich et al., Physical Review Letters 112, 146404 (2014)} 
 \end{frame}
 
 \begin{frame}
\includegraphics[scale=0.5]{idos-synth-3.JPG}
\end{frame}
 
 %Integrated density of states (IDOS) :
 %$$
 %N(\epsilon)=\int_{spectre} \theta(\epsilon - \epsilon')d\mu(\epsilon')$$
 %$$N(\epsilon/3) = \frac{1}{2}N(\epsilon) \Rightarrow N(\epsilon) = \epsilon^{\ln 2/\ln 3}g_p(\ln \epsilon/\ln 3), g(y+1) = g(y)$$
 
 

 
 \begin{frame}
 Time-averaged return probability :$C(t) = \frac{1}{t}\int_0^t |\langle\psi(0)|\psi(t')\rangle|^{2} dt' $
 \begin{columns}
 \column{.5\textwidth}
 \begin{figure}
 \includegraphics[scale=0.7]{c_t_fibo.PNG}
 \caption{\tiny{Time-averaged return probability of a wave packet in a Fibonacci potential.}}
 \end{figure}
  \column{.5\textwidth}
 \begin{figure}
  \includegraphics[scale=0.2]{ct_loglog2.eps}
  \caption{\tiny{Time-averaged return probability for a triadic Cantor spectrum.}}
  \end{figure}
  \end{columns} 
 %Log-periodic oscillation, signature of the fractal feature of the spectrum.
 
 \end{frame}


\section{Summary}

\begin{frame}
In summary :
\begin{itemize}
 
\item We identified the signature of a fractal spectrum through measurable physical quantities : power-law modulated by log-periodic oscillation, for dynamical and spectral quantities ($C(t)$, integrated density of states)
\item General method to study theoretically systems with a fractal feature
\item Oscillations difficult to observe ; easier with dynamical quantities 
\item Verified experimentally with polaritons in a Fibonacci potential (Cantor like spectrum) 
 
%\item applied to the return probability for a Cantor spectrum
\end{itemize}
\end{frame}

\begin{frame}
Next : other relevant quantities in quantum mechanics 
\begin{itemize}
%\item Pursue the study of the participation ratio and RMS displacement
\item Signature of the fractal spectral feature in the Thouless coefficient $\gamma(\epsilon)$
\item Spatial correlation of the eigenfunction ; generalization of Thouless formula : $\overline{\phi(0,\epsilon)^{*}
\phi(x,\epsilon)} \propto e^{-|x|\gamma(\epsilon)}$   
\item Spontaneous emission of a single atom coupled to a fractal spectrum 
\item Superradiance of a two atoms system coupled to a fractal spectrum
\end{itemize}
\end{frame}

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Questions ?
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