1

I am so sorry that I ask something not particularly helpful for future users. I am very new to Latex. I use a template for my report. When I typeset (Ctrl + T), I got an error. I looked up previous discussion on this site, but I could not find a particular answer. Here is an entire latex file that I am working on,

%----------------------------------------------------------------------------------------
%   PACKAGES AND OTHER DOCUMENT CONFIGURATIONS
%----------------------------------------------------------------------------------------

\documentclass[twoside]{article}

\usepackage{amsmath}
\usepackage{amsmath,amssymb}
\usepackage{braket}
\usepackage[english]{babel}
\usepackage[utf8x]{inputenc}
\usepackage{graphicx}
\usepackage[colorinlistoftodos]{todonotes}

\usepackage{lipsum} % Package to generate dummy text throughout this template

\usepackage[sc]{mathpazo} % Use the Palatino font
\usepackage[T1]{fontenc} % Use 8-bit encoding that has 256 glyphs
\linespread{1.05} % Line spacing - Palatino needs more space between lines
\usepackage{microtype} % Slightly tweak font spacing for aesthetics

\usepackage[hmarginratio=1:1,top=32mm,columnsep=20pt]{geometry} % Document margins
\usepackage{multicol} % Used for the two-column layout of the document
\usepackage[hang, small,labelfont=bf,up,textfont=it,up]{caption} % Custom captions under/above floats in tables or figures
\usepackage{booktabs} % Horizontal rules in tables
\usepackage{float} % Required for tables and figures in the multi-column environment - they need to be placed in specific locations with the [H] (e.g. \begin{table}[H])
\usepackage{hyperref} % For hyperlinks in the PDF

\usepackage{lettrine} % The lettrine is the first enlarged letter at the beginning of the text
\usepackage{paralist} % Used for the compactitem environment which makes bullet points with less space between them

\usepackage{abstract} % Allows abstract customization
\renewcommand{\abstractnamefont}{\normalfont\bfseries} % Set the "Abstract" text to bold
\renewcommand{\abstracttextfont}{\normalfont\small\itshape} % Set the abstract itself to small italic text

\usepackage{titlesec} % Allows customization of titles
\renewcommand\thesection{\Roman{section}} % Roman numerals for the sections
\renewcommand\thesubsection{\Roman{subsection}} % Roman numerals for subsections
\titleformat{\section}[block]{\large\scshape\centering}{\thesection.}{1em}{} % Change the look of the section titles
\titleformat{\subsection}[block]{\large}{\thesubsection.}{1em}{} % Change the look of the section titles

\usepackage{fancyhdr} % Headers and footers
\pagestyle{fancy} % All pages have headers and footers
\fancyhead{} % Blank out the default header
\fancyfoot{} % Blank out the default footer
\fancyhead[C]{Running title $\bullet$ November 2012 $\bullet$ Vol. XXI, No. 1} % Custom header text
\fancyfoot[RO,LE]{\thepage} % Custom footer text

\usepackage{mathtools}
\DeclarePairedDelimiter{\evdel}{\langle}{\rangle}
\newcommand{\ev}{\operatorname{E}\evdel} % Declare new command for expectation value

%----------------------------------------------------------------------------------------
%   TITLE SECTION
%----------------------------------------------------------------------------------------

\title{\vspace{-15mm}\fontsize{24pt}{10pt}\selectfont\textbf{Article Title}} % Article title

\author{
\large
\textsc{Sopheak Seng}\thanks{A thank you or further information}\\[2mm] % Your name
\normalsize QOQI Lab \\ % Your institution
\normalsize \href{mailto:sengsopheak@live.com}{sensopheak@live.com} % Your email address
\vspace{-5mm}
}
\date{}

%----------------------------------------------------------------------------------------

\begin{document}

\maketitle % Insert title

\thispagestyle{fancy} % All pages have headers and footers

%----------------------------------------------------------------------------------------
%   ABSTRACT
%----------------------------------------------------------------------------------------

\begin{abstract}



\end{abstract}

%----------------------------------------------------------------------------------------
%   ARTICLE CONTENTS
%----------------------------------------------------------------------------------------

\begin{multicols}{2} % Two-column layout throughout the main article text

\section{Introduction}

Quantum Memory is vital for the development of devices to use in quantum information processing {1}, including a synchronization toll that matches various processes within a quantum computer, an identity quantum gate that leaves any state unchanged, and a mechanism to convert heralded photons to on demand photons. Quantum memories basic roles are writing, storing, and reading quantum bits without destroying the quantum states. Many protocols for quantum memories were proposed such as the DLCZ, Photon-echo quantum memories, and Optical delay lines and cavities.\\
The authors of {2} discussing the ability to record, store, and read out the quantum properties of qubits. They base their study on high-speed and adiabatic models of quantum memories in $\lambda$ configuration with strong coupling resonants external fields. Two quantum memory protocols based on the resonant interaction between an atomic ensemble and fields - signal and driving. The schemes  differ in duration of interaction that leads to differences in formation of the atomic coherence, on which the state of the signal field is mapped. One of their main issues here is to analyze the quantum efficiency. They desire to find out whether this characteristic of memory is as universal for non-beam-splitter-like protocols as for beam-splitter-like ones; i.e., could they predict how well a certain quantum state will survive in the memory, if they know only the efficiency of this memory?\\
To demonstrate their arguments, the authors of {2} consider the storage of squeezing in the schemes of adiabatic and high-speed quantum memory. There to they analyze how the squeezed light from a particular source with the desired properties is mapped on the atomic ensemble and then read out. They also solve the eigenfunction problem for two considered memory protocols. Based on the analysis of eigenfunctions, they compare the spectral bandwidth of the memories.\\

%------------------------------------------------

\section{Squeezed State}

A general class of minimum-uncertainty states are called squeezed states. Let us calculate the variances for the position and momentum operators for the harmonic oscillation,
\[
\begin{split}
q&=\sqrt{\frac{\hbar}{2\omega}}\left(a+a^{\dagger}\right)\\
p&=i\sqrt{\frac{\hbar\omega}{2}}\left(a-a^{\dagger}\right)
\end{split}
\]
The variance is given by,
\[
\left(\Delta A\right)^{2}=\ev{A^{2}}-\ev{A}^{2}
\]
In a coherent state, we can obtain,
\[
\begin{split}
\ev{Q}&=\sqrt{\frac{2\hbar}{\omega}}\alpha\\
\ev{P}&=\sqrt{2\hbar\omega}\alpha\\
\ev{Q^{2}}&=\frac{\hbar}{2\omega}\left[2\alpha+1\right]\\
\ev{P^{2}}&=\frac{\hbar\omega}{2}\left[2\alpha+1\right]\\
\end{split}
\]
Thus,
\[
\begin{split}
\left(\Delta Q\right)^{2}&=\frac{\hbar}{2\omega}\\
\left(\Delta P\right)^{2}&=\frac{\hbar\omega}{2}\\
\end{split}
\]
The uncertainty relation is given by,
\[
\Delta Q\Delta P=\frac{\hbar}{2}
\]
This indicates there exists a sense in which the description of the state of an oscillator by a coherent state represents as close an approach to classical localization as possible. For a single mode field, we write the annihilation operation $a$ as a linear combination of two Hermitian operators, $X_{1}$ and $X_{2}$,
\[
a\equiv\frac{X_{1}+iX_{2}}{2}
\]
where $X_{1}$ and $X_{2}$ are the real and imaginary parts of the complex amplitude.
\[
\left[X_{1},X_{2}\right]=2i
\]
The corresponding uncertainty principle is,
\[
\Delta X_{1}\Delta X_{2}\leq1
\]
This relation with the equals sign defines a family of minimum-uncertainty states/ The coherent states are a particular minimum-uncertainty state with,
\[
\Delta X_{1}=\Delta X_{2}=1
\]
The coherent state $\Ket{\alpha}$ has the mean complex amplitude $\alpha$ and it is a minimum-uncertainty state for $X_{1}$and $X_{2}$, with equal uncertainties in the two quadrature phases. A coherent state many be represented by an error circle in a complex amplitude plane whose axes are $X_{1}$ and $X_{2}$ (Fig. 1). The center of the error circle lies at $\frac{1}{2}\ev{X_{1}+iX_{2}}=\alpha$ and the radius $\Delta X_{1}=\Delta X_{2}=1$ accounts for the uncertainties in $X_{1}$ and $X_{2}$.
\begin{figure}[p]
\centering
\includegraphics[width=\textwidth]{Report2N1.jpg}
\caption{Phase space representation showing contours of constant uncertainty for \textbf{(a)} coherent state and \textbf{(b)} squeezed state $\Ket{\alpha,\varepsilon}$}
\label{fig:1}
\end{figure}

%------------------------------------------------

\section{Beam-Splitters}

Beam-splitters are devices used to split beams of lights into two. Consider a case in which a beam splitter deflecting part of the incident signal field onto a detector. We assume that both the signal field and the probe field are single mode with annihilation operators $a$ and $b$ respectively.\\
The amplitude and phase quadratures of the signal and probe fields are defined as,
\[
\begin{split}
X_{a}&=a+a^{\dagger}\\
X_{\phi}&=-i\left(a-a^{\dagger}\right)\\
Y_{a}&=b+b^{\dagger}\\
Y_{\phi}&=-i\left(b+b^{\dagger}\right)\\
\end{split}
\]
According to the uncertainty principle,
\[
\Delta X_{a}\Delta X_{\phi}\leq1
\]
As we saw in the section above, $\Delta X_{a}$ and $\Delta X_{\phi}$ are the square root of the variances.\\
In the beam splitter, the phase change on reflection gives a coupling between the amplitude quadrature of the signal and the phase quadrature of the probe. We consider making a measurement of the phase quadrature of the prove in order to determine the amplitude quadrature of the signal. The input quadrature fields can be related to the output quadrature fields using the transformation at the beam splitter,
\[
\left(
\begin{array}{c}
X_{a}^{\textrm{out}}\\Y_{\phi}^{\textrm{out}}
\end{array}
\right)=\left(
\begin{array}{c c}
\sqrt{1-\mathcal{R}^{2}}&-\mathcal{R}\\\mathcal{R}&\sqrt{1-\mathcal{R}^{2}}
\end{array}
\right)\left(
\begin{array}{c}
X_{a}^{\textrm{in}}\\Y_{\phi}^{\textrm{in}}
\end{array}
\right)
\]
where $\mathcal{R}$ represents the mirror amplitude reflectivity.\\
Note that, there is a $\frac{\pi}{2}$ phase change upon each reflection.
\begin{figure}[p]
\centering
\includegraphics[width=\textwidth]{Report2N2,jpg}
\caption{An optical measurement scheme for the quadrature phase based on a beam splitter.}
\label{fig:2}
\end{figure}

%------------------------------------------------

\section{Efficiency and Storage of Squeezing in Beam-Splitter-Like Quantum Memory}

Ideal quantum memories map the quantum state of a light pulse into a long-lived coherence in the atomic ground state or metastable states using a two-photon transition, with the help of a strong classical control field (probe field). The signal retrieval, obtained through the interaction of the atomic coherence with the control field should provide a light pulse with the same classical and quantum properties as the input pulse. Several criteria have been proposed to determine the quality of performance of the quantum memory. However, we will only focus only on three main criteria, efficiency, fidelity, and TV criterion.\\
Efficiency is the ratio between the energies of the input and output signals. Efficiency can be easy to determine in experiments, but it is not account for any effects such as contamination of the output signal due to noise.\\
Fidelity $F$ is a measure of the conservation of the quantum features of the signal, and allows characterizing the memory as compared to a classical memory, for which the fidelity is limited to some maximal value. Fidelity is given by, $F=\Braket{\psi|\rho|\phi}$, where $\psi$ is the initial wave vector (pure state) of the light, and $\rho$ is the density matrix describing the state of light after retrieval.\\
An intuitive understanding of the connection between the storage of squeezing and the efficiency can be gained from the beam-splitter model of a quantum memory, where the full writing and reading cycle can be model of a quantum memory, where the full writing and reading cycle can be modeled by the transmission through a beam splitter. Indeed, let us write the relation between input signal $\hat{a}_{\textrm{in}}$ and output signal $\hat{a}_{\textrm{out}}$ in the form of the beam-splitter transformation,
\[
\hat{a}_{\textrm{out}}\left(t\right)=\sqrt{\mathcal{T}}\hat{a}_{\textrm{in}}\left(t\right)-\sqrt{1-\mathcal{T}}\hat{a}_{\textrm{vac}}\left(t\right)
\]
where, $\mathcal{T}$is the transmission coefficient modeling the efficiency of the quantum memory process.\\
In frequency domain,
\[
\hat{a}_{\textrm{out,}\omega}=\sqrt{\mathcal{T}}\hat{a}_{\textrm{in,}\omega}-\sqrt{1-\mathcal{T}}\hat{a}_{\textrm{vac,}\omega}
\]

%------------------------------------------------

\section{Appendix}

\subsection{Quadrature}

Quadrature loosely refer to the position and momentum observables,
\[
\begin{split}
\hat{x}&=\frac{1}{\sqrt{2}}\left(a+a^{\dagger}\right)\\
\hat{p}&=\frac{i}{\sqrt{2}}\left(a-a^{\dagger}\right)\\
\end{split}
\]
where $a$ and $a^{\dagger}$ are the lowering and raising operators and $\left[\hat{x},\hat{p}\right]=i\hbar$. These two (or measurements coming from these observables) are loosely called "quadratures" because, for a coherent or squeezed state $\psi$ the mean measurements $\Braket{\psi|\hat{x}|\psi}$ and $\Braket{\psi|\hat{p}|\psi}$ are sinusoidally-with-time varying quantities which are in phase quadrature, i.e. a quarter cycle out of phase.\\
The squeezed states are the most general states that saturate (i.e. actually achieve equality in) the Heisenberg product. A coherent state is a special case for which the normalized momentum and position uncertainties are equal, and their product is the uncertainty bound $\frac{\hbar}{2}$. A squeezed state has the same uncertainty product, but the uncertainty on one of position or momentum measurements is smaller than the uncertainty on the other, so one achieves smaller uncertainty than for the corresponding coherent state at the expense of the other. One can generalize the above comments for any "generalized" position and momentum, defined, for any $\phi\in\mathbb{R}$, by,
\[
\begin{split}
\hat{x}&=\frac{1}{\sqrt{2}}\left(e^{i\phi}a+e^{-i\phi}a^{\dagger}\right)\\
\hat{p}&=\frac{i}{\sqrt{2}}\left(e^{i\phi}a-e^{-i\phi}a^{\dagger}\right)\\
\end{split}
\]
which are conjugate in the quantum mechanics sense of fulfilling the canonical commutation relation and the means of whose measurements vary sinusoidally with time, again, in phase quadrature.\\

%----------------------------------------------------------------------------------------
%   REFERENCE LIST
%----------------------------------------------------------------------------------------

\begin{thebibliography}{99} % Bibliography - this is intentionally simple in this template



\end{thebibliography}

%----------------------------------------------------------------------------------------

\end{multicols}

\end{document}

Here is the error message,

[1

pdfTeX warning: pdflatex.EXE (file pdftex.map): cannot open font map file
! pdfTeX error (font expansion): auto expansion is only possible with scalable 
fonts.
\@EveryShipout@Output ...@Org@Shipout \box \@cclv 

l.140 \]

!  ==> Fatal error occurred, no output PDF file produced!
Transcript written on untitled-2.log.
texify: pdflatex failed for some reason (see log file).

The error message indicates that row number 140 is error. Here I copy to you row 134-141,

Thus,
\[
\begin{split}
\left(\Delta Q\right)^{2}&=\frac{\hbar}{2\omega}\\
\left(\Delta P\right)^{2}&=\frac{\hbar\omega}{2}\\
\end{split}
\]
The uncertainty relation is given by,

Please help me. And again, I am so sorry for asking not appropriate question. Please forgive my English.

  • 1
    I don't get any error from compiling your code. Some warnings, but nothing fatal. – Alenanno Jul 11 '15 at 12:43
  • Guessing: You have a \lettrine somewhere but are trying to use a font which is a metafont source font rather than type1, say. However, there's no occurrence in the posted code.... – cfr Jul 11 '15 at 12:43
  • 1
    Your document is using bitmap fonts. This doesn't work with microtype. The reason is this cannot open font map file. Run on the command line updmap. – Ulrike Fischer Jul 11 '15 at 13:04
  • 2
    @TBBT -- I'm glad the file can be produced, but I'm suspect that was not a great choice (not on Windows myself, however). Font-installation issues are among the more intricate of TeX problems. Why did you not follow Ulrike's suggestion? – jon Jul 11 '15 at 14:40
  • 2
    Definitely not good as @jon says. I'm not on Windows either but this is not the way to do it. It may work for now in some sense but it will make things more difficult later. – cfr Jul 11 '15 at 16:06

Browse other questions tagged or ask your own question.