From e1ddd40e512f7cfad58eaf7e63ae36a27f9fb971 Mon Sep 17 00:00:00 2001
From: Ian Jauslin <jauslin@ias.edu>
Date: Mon, 1 Oct 2018 22:59:41 +0000
Subject: As presented at Rutgers on 2018-09-27

---
 Jauslin_Rutgers_2018.tex | 259 +++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 259 insertions(+)
 create mode 100644 Jauslin_Rutgers_2018.tex

(limited to 'Jauslin_Rutgers_2018.tex')

diff --git a/Jauslin_Rutgers_2018.tex b/Jauslin_Rutgers_2018.tex
new file mode 100644
index 0000000..50b9bea
--- /dev/null
+++ b/Jauslin_Rutgers_2018.tex
@@ -0,0 +1,259 @@
+\documentclass{ian-presentation}
+
+\usepackage[hidelinks]{hyperref}
+\usepackage{graphicx}
+\usepackage{array}
+
+\begin{document}
+\pagestyle{empty}
+\hbox{}\vfil
+\bf\Large
+\hfil Solution of time-dependent Schr\"odinger equation\par
+\smallskip
+\hfil  for field emission\par
+\vfil
+\large
+\hfil Ian Jauslin
+\normalsize
+\vfil
+\hfil\rm joint with {\bf Ovidiu Costin}, {\bf Rodica Costin}, and {\bf Joel L. Lebowitz}\par
+\vfil
+arXiv:{\tt \href{http://arxiv.org/abs/1808.00936}{1808.00936}}\hfill{\tt \href{http://ian.jauslin.org}{http://ian.jauslin.org}}
+\eject
+
+\setcounter{page}1
+\pagestyle{plain}
+
+\title{Field emission}
+\vfill
+\hfil\includegraphics[height=5cm]{emitter.jpg}
+\vfill
+\eject
+
+\title{Emitter}
+$$
+  V(x)=U\Theta(x)
+$$
+\hfil\includegraphics[height=5cm]{potential_square.pdf}
+\vfill
+\eject
+
+\title{Semi-classical model}
+$$
+  V_t(x)=\Theta(x)(U-E_tx)
+  ,\quad
+  E_t=e_0+e_1\cos(\omega t)
+$$
+\hfil\includegraphics[height=5cm]{potential.pdf}
+\vfill
+\eject
+
+\title{Triangular barrier}
+$$
+  V(x)=\Theta(x)(U-Ex)
+$$
+\hfil\includegraphics[height=5cm]{potential.pdf}
+\vfill
+\eject
+
+\title{Field emission through a triangular barrier}
+\begin{itemize}
+  \item \href{https://doi.org/10.1073\%2Fpnas.14.1.45}{[Millikan, Lauritsen, 1928]}: experimental plot of the logarithm of the current against $1/E$
+\end{itemize}
+\hfil\includegraphics[height=4.5cm]{Millikan-Lauritsen_current.png}
+\vfill
+\eject
+
+\title{Field emission through a triangular barrier}
+\vfill
+\begin{itemize}
+  \item \href{https://doi.org/10.1098/rspa.1928.0091}{[Fowler, Nordheim, 1928]}: predicted that the current is, for small $E$,
+  $$
+    J\approx Ce^{-\frac aE}
+  $$
+  \item (\href{https://doi.org/10.1088/1751-8113/44/5/05530@}{[Rokhlenko, 2011]}: studied the range of applicability of the approximation, and found more accurate approximations for larger fields.)
+\end{itemize}
+\vfill
+\eject
+
+\title{Fowler-Nordheim equation}
+\begin{itemize}
+  \item Schr\"odinger equation
+  $$
+    i\partial_t\psi=-\Delta\psi+\Theta(x)(U-Ex)\psi
+  $$
+  \item Fowler-Nordheim: quasi-stationary solution: $\psi_{\mathrm{FN}}(x,t)=e^{-ik^2t}\varphi_{\mathrm{FN}}(x)$
+  $$
+    \varphi_{\mathrm{FN}}(x)=
+    \left\{ \begin{array}{l@{\ }l}
+      e^{ikx}+R_Ee^{-ikx} & x<0\\
+      T_E\mathrm{Ai}(e^{-\frac{i\pi}3}(E^{\frac13}x-E^{-\frac23}(U-k^2)) & x>0
+    \end{array}\right.  
+  $$
+  $R_E$ and $T_E$ are chosen so that $\varphi_{\mathrm{FN}}$ and $\partial\varphi_{\mathrm{FN}}$ are continuous at $x=0$.
+\end{itemize}
+\vfill
+\eject
+
+\title{Fowler-Nordheim equation}
+\vfill
+\hfil\includegraphics[height=5.5cm]{asymptotic.pdf}
+\vfill
+\eject
+
+\title{Initial value problem}
+\begin{itemize}
+  \item Initial condition:
+  $$
+    \psi(x,0)=
+    \left\{ \begin{array}{l@{\ }l}
+      e^{ikx}+R_0e^{-ikx} & x<0\\
+      T_0 e^{-\sqrt{U-k^2}x} & x>0
+    \end{array}\right.  
+  $$
+  $R_0$ and $T_0$ ensure that $\psi$ and $\partial\psi$ are continuous.
+  \item Behaves asymptotically like $\psi_{\mathrm{FN}}$:
+  $$
+    \psi(x,t)e^{ik^2t}\mathop{\longrightarrow}_{t\to\infty}\varphi_{\mathrm{FN}}(x)
+  $$
+\end{itemize}
+\vfill
+\eject
+
+\title{Initial value problem}
+\begin{itemize}
+  \item Laplace transform:
+  $$
+    \hat\psi_p(x):=\int_0^\infty dt\ e^{-pt}\psi(x,t)
+  $$
+  \item Schr\"odinger equation:
+  $$
+    (-\Delta+\Theta(x)V(x)-ip)\psi_p(x)=-i\psi(x,0)
+    ,\quad
+    V(x):=U-Ex
+  $$
+\end{itemize}
+\vfill
+\eject
+
+\title{Solution in Laplace space}
+\begin{itemize}
+  \item For simplicity, $R_0\equiv T_0\equiv0$.
+  \item Solution:
+  $$
+    \hat\psi_p(x)=
+    \left\{\begin{array}{>\displaystyle l@{\ }l}
+      C_1(p)e^{\sqrt{-ip}x}-\frac{ie^{ikx}}{-ip+k^2}
+      &\mathrm{if\ }x<0\\[0.5cm]
+      C_2(p)\varphi_p(x)
+      &\mathrm{if\ }x> 0
+    \end{array}\right.
+  $$
+  with
+  $$
+    (-\Delta+V(x)-ip)\varphi_p(x)=0
+  $$
+  $$
+    \varphi_p(x)=\mathrm{Ai}\left(e^{-\frac{i\pi}3}\left(E^{\frac13}x-E^{-\frac23}(U-ip)\right)\right)
+  $$
+\end{itemize}
+\vfill
+\eject
+
+\title{Solution in Laplace space}
+\begin{itemize}
+  \item $C_1$ and $C_2$ ensure that $\hat\psi_p(x)$ and $\partial\hat\psi_p(x)$ are continuous at $x=0$:
+  $$
+    C_1(p)=\frac{i(ik\varphi_p(0)-\partial\varphi_p(0))}{(-ip+k^2)(\sqrt{-ip}\varphi_p(0)-\partial\varphi_p(0))}
+  $$
+  $$
+    C_2(p)=-\frac{i}{(\sqrt{-ip}+ik)(\sqrt{-ip}\varphi_p(0)-\partial\varphi_p(0))}.
+  $$
+\end{itemize}
+\vfill
+\eject
+
+\title{Poles in Laplace plane}
+\vfill
+\hfil\includegraphics[height=5.5cm]{contour.pdf}
+\vfill
+\eject
+
+\title{Asymptotic behavior}
+\begin{itemize}
+  \item As $t\to\infty$:
+  $$
+    \psi(x,t)
+    =\psi_{\mathrm{FN}}(x,t)+\left(\frac{t}{\tau_E(x)}\right)^{-\frac32}+O(t^{-\frac52})
+    .
+  $$
+
+  \item If $k<0$ (reflected wave), then there is no pole on the imaginary axis, so there is no contribution as $t\to\infty$.
+  \item Similarly, the transmitted wave in the initial condition does not contribute.
+\end{itemize}
+\vfill
+\eject
+
+\title{Laser field}
+\begin{itemize}
+  \item Time dependent potential:
+  $$
+    V_t(x)=\Theta(x)(U-\epsilon\omega\cos(\omega t)x)
+  $$
+  \item Magnetic gauge:
+  $$
+    \Psi(x,t):=
+    :=\psi(x,t)e^{-ix\Theta(x)A(t)}
+    ,\quad
+    A(t):=\int_0^t ds\ \epsilon\omega\cos(\omega s)
+  =
+  \epsilon\sin(\omega t)
+  $$
+  satisfies
+  $$
+    i\partial_t\Psi(x,t)=\left((-i\nabla+\Theta(x)A(t))^2+\Theta(x)U\right)\Psi(x,t)
+  $$
+\end{itemize}
+\vfill
+\eject
+
+\title{Periodic solution}
+\begin{itemize}
+  \item A solution:
+  $$
+    \Psi(x,t)=\left\{\begin{array}{ll}
+      \Psi_I(x,t)+\Psi_R(x,t)&\mathrm{\ if\ }x<0\\
+      \Psi_T(x,t)&\mathrm{\ if\ }x>0
+    \end{array}\right.
+  $$
+  $$
+    \Psi_I(x,t)=e^{ikx}\exp\left(-ik^2t\right)
+  $$
+  $$
+    \Psi_R(x,t)=\sum_{M\in\mathbb Z}R_Me^{iq_Mx}\exp\left(-iq_M^2t\right)
+  $$
+  $$
+    \Psi_T(x,t)=\sum_{M\in\mathbb Z}T_Me^{ip_Mx}\exp\left(-iUt-i\int_0^td\tau\ (p_M+A(\tau))^2\right)
+  $$
+\end{itemize}
+\vfill
+\eject
+
+\title{Periodic solution}
+\begin{itemize}
+  \item Choose $q_M$ and $p_M$ to make the solution periodic (up to the phase $e^{ik^2t}$):
+  $$
+    q_M=\pm\sqrt{k^2+M\omega}
+    ,\quad
+    p_M=\pm\sqrt{k^2-U+M\omega-U_V}
+  $$
+  and
+  $$
+    U_V:=\frac\omega{2\pi}\int_0^{\frac{2\pi}\omega} d\tau\ A^2(\tau)
+    =\frac{\epsilon^2}2
+    .
+  $$
+\end{itemize}
+
+
+\end{document}
-- 
cgit v1.2.3-70-g09d2

