\documentclass[12pt]{article} \setlength{\oddsidemargin}{0pt} \setlength{\evensidemargin}{0pt} \setlength{\textwidth}{6.5in} \setlength{\topmargin}{0pt} \setlength{\textheight}{9in} \setlength{\headheight}{0pt} \setlength{\headsep}{0pt} \begin{document} \pagestyle{empty} \begin{center} \Huge Classical and Quantum Dynamics in the Driven Triangle Well \vspace{.2in} \Large Todd Timberlake \end{center} \vspace{3.0in} \huge \begin{itemize} \item Classical Motion \vspace{.5in} \item Quantum Dynamics \vspace{.5in} \item Radiation Spectra \vspace{.5in} \item Quasienergies \vspace{.5in} \item Periodic Orbit Theory \end{itemize} \pagebreak \begin{center} \Huge Classical Motion of the Driven Triangle Well \end {center} \vspace{.5in} \begin{itemize} \LARGE \item Hamiltonian \Large \[H = \frac{p^{2}}{2m} - \epsilon_{0} x - \epsilon x \cos ( \omega t + \theta ) + V_{L}(x)\] \[V_{L}(x) = \left\{ \begin{array}{ll} 0 & \mbox{if $xL$} \end{array} \right. \] \vspace{.5in} \LARGE \item Equations of Motion \Large \[\dot{x} = \frac{p}{m}\] \[\dot{p} = \epsilon_{0} + \epsilon \cos ( \omega t + \theta ) \] \vspace{.5in} \LARGE \item Infinite Wall \Large \begin{itemize} \item To numerically integrate the equations of motion one must stop the integration when $x = L$ and reverse the momentum. \vspace{.1in} \item The NAG library has an ode integrator 'd02cje' which stops when $x = L$. \end{itemize} \pagebreak \LARGE \item Collision Maps \Large \begin{itemize} \item One way to study the dynamics of this system is to make a plot of the canonical action (J) and phase of the driving field ($\phi$) each time the particle hits the infinite wall. \vspace{.1in} \item These plots give a good indication of what the classical motion is like for this sytem. \vspace{.1in} \item As we increase $\epsilon$, these plots show the transition to chaotic motion. \end{itemize} \end{itemize} \pagebreak \begin{center} \Huge Quantum Dynamics of the Driven Triangle Well \end{center} \vspace{.5in} \begin{itemize} \LARGE \item Time-Dependent Schr\"{o}dinger's Equation \Large \[i \hbar \frac{\partial}{\partial t} |\psi (t) \rangle = [H_{0} - \xi (t) x \cos (\omega t + \theta)] |\psi (t) \rangle \] \[H_{0} = \frac{p^{2}}{2m} - \epsilon_{0} x \] \vspace{1mm} \[\xi (t) = \left \{ \begin{array}{ll} \epsilon \sin^{2} (\frac{\omega t}{4n}) & t < \frac{2 \pi n}{\omega}\\ \epsilon & t > \frac{2 \pi n}{\omega} \end{array} \right. \] \vspace{.5in} \LARGE \item Solution for Unperturbed Triangle Well \Large \[\psi_{i} (x) = \langle x|E_{i} \rangle = N_{i} \mbox{Ai} \left[ \mbox{\large $-(\frac{2m}{\hbar^{2}})^{\frac{1}{3}} \epsilon_{0}^{-\frac{2}{3}} E_{i} - (\frac{2m\epsilon_{0}}{\hbar^{2}})^{\frac{1}{3}} x$} \right] \] \vspace{.5in} \LARGE \item Decomposition of Time-Dependent Wavefunction \Large \[|\psi (t) \rangle = \sum_{i} c_{i} (t) |E_{i} \rangle \] \begin{itemize} \item Why not solve the PDE? \vspace{.1in} \item This method makes calculating spectra easy and allows study of the diffusion of energy. \end{itemize} \pagebreak \LARGE \item ODEs for C(t)'s \Large \[ \frac{\partial c_{i}(t)}{\partial t} = - \frac{iE_{i}}{\hbar} c_{i}(t) + \frac{i}{\hbar} \xi(t) \cos(\omega t + \theta) \sum_{j} x_{ij} c_{j}(t) \] \vspace{1mm} \[ x_{ij} = \left \{ \begin{array}{ll} - \frac{2E_{i}}{3\epsilon_{0}} & i = j\\ - \frac{\epsilon_{0}}{m(E_{i} - E_{j})^{2}} & i \neq j \end{array} \right. \] \vspace{.5in} \LARGE \item Acceleration Time Series \Large \[ \langle \psi (t)| \ddot{x} | \psi (t) \rangle = \sum_{ij} c_{j}^{*}(t) \ddot{x}_{ij} c_{i}(t) \] \[ \ddot{x}_{ij} = - \frac{1}{\hbar^{2}}(E_{i}-E_{j})^{2} x_{ij} + \frac{1}{\hbar^{2}} \xi(t) \cos(\omega t + \theta) \sum_{k} (2E_{k} - E_{j} - E_{i}) x_{ik} x_{kj} \] \vspace{.5in} \LARGE \item Radiation Spectra - High Harmonic Generation \end{itemize} \pagebreak \begin{center} \Huge Quasienergies \end{center} \begin{itemize} \LARGE \item Floquet Operator \[ \hat{U}_{T} |\psi (t) \rangle = |\psi (t+T) \rangle \] \vspace{.5in} \item If the system is periodic in time then $\hat{U}_{T}$ can be diagonalized. The eigenvalues of this operator take the form $e^{-\frac{i q_{n} T}{\hbar}}$. The $q_{n}$ are called the quasienergies of the system. They are only defined modulo $\omega \hbar$. \vspace{.5in} \item Quasienergies in time-periodic systems play a role similar to that of energies in conservative systems. \vspace{.5in} \item All of the analysis that might usually be done on the energy spectrum of a conservative system can be done on the quasienergy spectrum of a time-periodic system. \end{itemize} \pagebreak \begin{center} \Huge Periodic Orbits and Quasienergies \end{center} \vspace{.5in} \begin{itemize} \LARGE \item Periodic Orbits and Energy Spectra \Large \begin{itemize} \item Chaotic Systems - Gutzwiller \vspace{.1in} \item Regular Systems - Balian and Bloch, Berry and Tabor \end{itemize} \vspace{.5in} \LARGE \item Periodic Orbits and Quasienergy Spectra \Large \begin{itemize} \item A theory similar to theperiodic-orbit theory for conservative systems was developed by Tabor to study area-preserving maps. \vspace{.1in} \item Quasienergy density of states \[ \rho (Q) = \frac{T}{2 \pi i \hbar} \sum_{N=-\infty}^{\infty} e^{i(NQT/\hbar)} \mbox{Tr}(\hat{U}^{N}) = \sum_{M=-\infty}^{\infty} \sum_{n} \delta (M \omega \hbar - (Q-q_{n})) \] \vspace{.1in} \item Tr($\hat{U}^{N}$) is a sum of terms from each orbit that is periodic with period $NT$. \end{itemize} \pagebreak \LARGE \item To construct the quasienergy density of states: \Large \vspace{.1in} \begin{enumerate} \item locate all periodic orbits of period NT \vspace{.2in} \item calculate the stability matrix \textbf{$M_{j}$} for each orbit \[ \mathbf{M_{j}} = \prod_{i=0}^{N} \left[ \begin{array}{cc} \frac{dp_{i+1}}{dp_{i}} & \frac{dp_{i+1}}{dx_{i}} \\ \frac{dx_{i+1}}{dp_{i}} & \frac{dx_{i+1}}{dx_{i}} \end{array} \right] \] \vspace{.2in} \item calculate residue $R_{j}$ for each orbit \[ R_{j} = \frac{1}{4} (2-\mbox{Tr} \mathbf{M_{j}}) \] \vspace{.2in} \item calculate the classical action along each orbit \[ \bar{W}_{j} = \int_{0}^{NT} L(t) dt \] \vspace{.2in} \item repeat above procedure for all N \vspace{.2in} \item quasienergy density of states \[ \rho (Q) = \frac{T}{2 \pi \hbar} \sum_{j} \frac{1}{2} \left[ \frac{1}{R_{j}} \right]^{1/2} e^{(i/ \hbar)(N_{j}QT+ \bar{W}_{j})} \] \end{enumerate} \pagebreak \LARGE \vspace{.5in} \item Results for driven harmonic oscillator \begin{itemize} \Large \item Use shooting technique with Newton's method to find the sole periodic orbit with period T. \vspace{.1in} \item Calculate residue and action for this orbit and its repetitions. \vspace{.1in} \item Compare the density of states obtained from this numerical oribt-finding to the analytical result. \end{itemize} \pagebreak \LARGE \item Application to the Driven Triangle Well \begin{itemize} \Large \item Infinite wall adds difficulty to ODE integration. \vspace{.1in} \item The map $t \rightarrow t+T$ is non-linear in the initial coordinates, so several Newton iterations will be required to locate the periodic orbit. \vspace{.1in} \item There may be more than one periodic orbit with a given period. \item The basin of attraction of a given periodic orbit is finite. \item These last three probelems can be handled by using a mesh of starting points from which one searches for periodic orbits. \end{itemize} \end{itemize} \end{document}