\documentclass[11pt,fleqn]{article}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% template that does not use Revtex4
%%% but allows special fonts
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%
%%% Please use this template.
%%% Edit it using e.g. Notepad
%%% Ignore the header (do not change it)
%%% Process the file in the Latex site 
%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


% Page setup
\topmargin -1.5cm      
\oddsidemargin -0.04cm   
\evensidemargin -0.04cm  
\textwidth 16.59cm
\textheight 24cm 
\setlength{\parindent}{0cm} 
\setlength{\parskip}{0cm} 


% Fonts
\usepackage{latexsym}
\usepackage{amsmath} 
\usepackage{amssymb} 
\usepackage{bm}
\usepackage{enumerate}


% Math symbols I
\newcommand{\sinc}{\mbox{sinc}}
\newcommand{\const}{\mbox{const}}
\newcommand{\trc}{\mbox{trace}}
\newcommand{\intt}{\int\!\!\!\!\int }
\newcommand{\ointt}{\int\!\!\!\!\int\!\!\!\!\!\circ\ }
\newcommand{\ar}{\mathsf r}
\newcommand{\im}{\mbox{Im}}
\newcommand{\re}{\mbox{Re}}

% Math symbols II
\newcommand{\eexp}{\mbox{e}^}
\newcommand{\bra}{\left\langle}
\newcommand{\ket}{\right\rangle}

% Mass symbol
\newcommand{\mass}{\mathsf{m}} 
\newcommand{\Mass}{\mathsf{M}} 

% More math commands
\newcommand{\tbox}[1]{\mbox{\tiny #1}}
\newcommand{\bmsf}[1]{\bm{\mathsf{#1}}} 
\newcommand{\amatrix}[1]{\begin{matrix} #1 \end{matrix}} 
\newcommand{\pd}[2]{\frac{\partial #1}{\partial #2}}

% Other commands
\newcommand{\hide}[1]{}
\newcommand{\drawline}{\begin{picture}(500,1)\line(1,0){500}\end{picture}}
\newcommand{\bitem}{$\bullet$ \ \ \ }
\newcommand{\Cn}[1]{\begin{center} #1 \end{center}}
\newcommand{\mpg}[2][1.0\hsize]{\begin{minipage}[b]{#1}{#2}\end{minipage}}
\newcommand{\Dn}{\vspace*{3mm}}

% Figures
\newcommand{\putgraph}[2][0.30\hsize]{\includegraphics[width=#1]{#2}}

% heading
\newcommand{\heading}[1]{\begin{center} \Large {#1} \end{center}}
\newcommand{\auname}[1]{\begin{center} \bf Submitted by: #1 \end{center}}


\begin{document} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\heading{E1344: Oscillations between a site and a ring} 
\auname{Asaf Barak, Jeremy Gartner}


{\bf The problem:}

\Dn


A given ring with length $L=1$ has $N$ sites, which have equal potential ($V=0$). The hopping amplitude of a particle per time unit between neighbouring sites is $c$. Another site is added at the center of the ring. The relation energy of the particle in the central site is $\varepsilon_0$. The hopping amplitude per time unit from the central site to any of the other sites along the ring is $c_0$. The system is then placed in a magnetic field, so that the total flux through the ring is $\Phi$. The charge of the particle is $\mathrm{e}$. From the following analysis, it is derived that the particle will perform oscillations between the site and the ring. This is a generalization of oscillations in a system of two sites.

\Dn

\begin{enumerate}[(1)]
\item Write the Hamiltonian $H_{ring}(p)$ of a particle in the ring described above (without the central site).
\item Write the eigenenergies $E_n$ of the particle in the ring (without the central site).
\item Calculate the coupling $\langle E_n | H | \varepsilon_0 \rangle$ of the central site to the states of the ring.
\item What is the oscillation frequency $\Omega$ of the particle?
\item For a given magnetic flux - What should be the coupled energy $\varepsilon_0$ in order to get full oscillations?
\item For oscillation $\Omega(\Phi)$ there is periodical dependency in the magnetic flux. What is the period, based on Aharonov-Bohm?
\end{enumerate}

\Dn
Answer the questions, using given data only.

\Dn\Dn

{\bf The solution:}

\Dn
\begin{enumerate}[(1)]

\item
From the lecture notes [9.1], we know that:
\begin{equation*}
\widehat{H}=ce^{-ia(\widehat{p}-A)}+ce^{ia(\widehat{p}-A)}+const.
\end{equation*}

Because of gauge consideration, we can determine that $const=0$.
\\ \\
In addition, we know that $a=\dfrac{L}{N}=\dfrac{1}{N}$, and $A=\dfrac{\mathrm{e} \Phi}{L}$. Therefore:

\begin{equation*}
\widehat{H}=ce^{-i\frac{1}{N}(\widehat{p}-\mathrm{e}\Phi)}+ce^{i\frac{1}{N}(\widehat{p}-\mathrm{e}\Phi)}=2c\cdot \mathrm{cos}\left(\frac{1}{N}\left(\widehat{p}-\mathrm{e}\Phi\right)\right)
\end{equation*}

\item
Considering the hamiltonian, the eigenstates are the momentum states and the eigenvalues are as follows:

\begin{equation*}
E_n=2c\cdot \mathrm{cos}\left(\frac{1}{N}\left(2\pi n-\mathrm{e}\Phi\right)\right)
\end{equation*}

\item
Due to the fact that $\widehat{H}=\widehat{H}(\hat{p})$, the eigenstates $|E_n\rangle = |k_n \rangle$. Hence, 
\begin{align*}
\langle E_n | H | \varepsilon_0 \rangle & = \langle k_n | H | \varepsilon_0 \rangle \\
& = \dfrac{1}{\sqrt{N}} \sum_{x=1}^{N}e^{ik_nx}\langle x | H | \varepsilon_0 \rangle \\
& = \dfrac{1}{\sqrt{N}} \sum_{x=1}^{N}e^{ik_nx}\cdot c_0 \\
& = \dfrac{1}{\sqrt{N}} N \delta_{n,0}c_0 \\
& = \sqrt{N}c_0\delta_{n,0}
\end{align*}

\item
Due to the coupling, there will be oscillations between the central site to the ring with 0 momentum: 
\[
\left(
\begin{array}{c c|c c c c }
  \varepsilon_0 & c_0\sqrt{N} &   & &  &\\ 
  c_0\sqrt{N} & E_0 &  & &  &\\ \hline
   &  & E_1 & & &  \\
   &  &     & E_2 & & \\
   &  &     &     & \ddots & \\
   &  &     &     &   & E_n \\
\end{array}
\right)
\]

Because of gauge consideration, we can determine for the $2\times 2$ subspace that:
 
\[
\left(
\begin{array}{c c}
  \dfrac{\varepsilon_0-E_0}{2} & c_0\sqrt{N} \\
  c_0\sqrt{N} & -\dfrac{\varepsilon_0-E_0}{2}  
\end{array}
\right)
\]

From the lecture notes [36.1], we learn that the evolution of any system whose states form a dim=2 Hilbert space can always be described by using a precession picture. Therefore, the subspace can be rewritten as:

\begin{equation*}
\dfrac{\varepsilon_0-E_0}{2}\cdot\sigma_z + c_0\sqrt{N}\cdot\sigma_x = \vec{\Omega} \cdot \vec{S}
\end{equation*}

Where $\vec{\Omega}= \left(2c_0\sqrt{N},0,\varepsilon_0-E_0\right)$, and $E_0=2c\cdot \mathrm{cos} \left(\frac{\mathrm{e}\Phi}{N}\right)$.\\
Therefore:


\begin{equation*}
\Omega = \sqrt{\left( \varepsilon_0 - 2c\cdot \mathrm{cos} \left(\frac{\mathrm{e}\Phi}{N}\right)\right)^2+4c_0^2N}
\end{equation*}

\item
Using the Rabi formula: 

\begin{equation*}
P(t)=1-\mathrm{sin}^2(\theta_0)\mathrm{sin}^2\left(\dfrac{\Omega t}{2}\right)
\end{equation*}

Where $\theta_0$ is the angle between the $\widehat{z}$ axis and the rotation axis ($\vec{\Omega}$).\\ 
The condition to get a full oscillation is $\theta_0=\dfrac{\pi}{2}$, and in our case $\theta_0$ is:

\begin{equation*}
\theta_0=\mathrm{arctan}\left(\dfrac{2c_0\sqrt{N}}{\varepsilon_0-2c\cdot \mathrm{cos} \left(\frac{\mathrm{e}\Phi}{N}\right)}\right)
\end{equation*}


Therefore:

\begin{equation*}
\varepsilon_0 = 2c\cdot \mathrm{cos} \left(\frac{\mathrm{e}\Phi}{N}\right)
\end{equation*}

\item
The oscillation $\Omega(\Phi)$ has a period depending on the the cosine function, as we can see in the equation from section (4).\\
Therefore, the necessary condition is:

\begin{equation*}
\frac{\mathrm{e}\Phi}{N}=2\pi k\,, \;\; k\in \mathbb{Z} \;\;\;\Longrightarrow \;\;\;\; \Phi=\frac{2\pi N}{\mathrm{e}}k
\end{equation*}

and the condition is:

\begin{equation*}
\Phi \longmapsto \Phi +\frac{2\pi N}{\mathrm{e}}k
\end{equation*}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{enumerate}
\end{document} 
