├── README.md ├── course1 ├── 2.png ├── 3.png ├── 4.png ├── 5.png ├── 6.png ├── 7.png ├── 8.png ├── 9.png ├── download.png ├── logo.png ├── presentation_1.aux ├── presentation_1.log ├── presentation_1.nav ├── presentation_1.out ├── presentation_1.pdf ├── presentation_1.snm ├── presentation_1.synctex.gz ├── presentation_1.tex ├── presentation_1.toc ├── presentation_1.vrb ├── screenshot001.png ├── screenshot002.png ├── screenshot007.png └── texput.log ├── course2 ├── DressedState.png ├── PRA_philip.png └── Quantum_Optics.ipynb └── course3 ├── Quantum_Optics3.ipynb ├── fig1.png ├── fig2.png └── fig3.jpeg /README.md: -------------------------------------------------------------------------------- 1 | # Quantum-Optics-with-Python 2 | This is a repository for notes on Quantum Optics. The examples are implemented by QuTip using Python. 3 | -------------------------------------------------------------------------------- /course1/2.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/caidish/Quantum-Optics-with-Python/198974a1598d5b39de0abfe0c070ee70176ffbce/course1/2.png -------------------------------------------------------------------------------- /course1/3.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/caidish/Quantum-Optics-with-Python/198974a1598d5b39de0abfe0c070ee70176ffbce/course1/3.png -------------------------------------------------------------------------------- /course1/4.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/caidish/Quantum-Optics-with-Python/198974a1598d5b39de0abfe0c070ee70176ffbce/course1/4.png -------------------------------------------------------------------------------- /course1/5.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/caidish/Quantum-Optics-with-Python/198974a1598d5b39de0abfe0c070ee70176ffbce/course1/5.png -------------------------------------------------------------------------------- /course1/6.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/caidish/Quantum-Optics-with-Python/198974a1598d5b39de0abfe0c070ee70176ffbce/course1/6.png -------------------------------------------------------------------------------- /course1/7.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/caidish/Quantum-Optics-with-Python/198974a1598d5b39de0abfe0c070ee70176ffbce/course1/7.png -------------------------------------------------------------------------------- /course1/8.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/caidish/Quantum-Optics-with-Python/198974a1598d5b39de0abfe0c070ee70176ffbce/course1/8.png -------------------------------------------------------------------------------- /course1/9.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/caidish/Quantum-Optics-with-Python/198974a1598d5b39de0abfe0c070ee70176ffbce/course1/9.png -------------------------------------------------------------------------------- /course1/download.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/caidish/Quantum-Optics-with-Python/198974a1598d5b39de0abfe0c070ee70176ffbce/course1/download.png -------------------------------------------------------------------------------- /course1/logo.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/caidish/Quantum-Optics-with-Python/198974a1598d5b39de0abfe0c070ee70176ffbce/course1/logo.png -------------------------------------------------------------------------------- /course1/presentation_1.aux: -------------------------------------------------------------------------------- 1 | \relax 2 | \providecommand\hyper@newdestlabel[2]{} 3 | \providecommand\HyperFirstAtBeginDocument{\AtBeginDocument} 4 | \HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined 5 | \global\let\oldcontentsline\contentsline 6 | \gdef\contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}} 7 | \global\let\oldnewlabel\newlabel 8 | \gdef\newlabel#1#2{\newlabelxx{#1}#2} 9 | \gdef\newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}} 10 | \AtEndDocument{\ifx\hyper@anchor\@undefined 11 | \let\contentsline\oldcontentsline 12 | \let\newlabel\oldnewlabel 13 | \fi} 14 | \fi} 15 | \global\let\hyper@last\relax 16 | \gdef\HyperFirstAtBeginDocument#1{#1} 17 | \providecommand\HyField@AuxAddToFields[1]{} 18 | \providecommand\HyField@AuxAddToCoFields[2]{} 19 | \@writefile{nav}{\headcommand {\slideentry {0}{0}{1}{1/1}{}{0}}} 20 | \@writefile{nav}{\headcommand {\beamer@framepages {1}{1}}} 21 | \@writefile{nav}{\headcommand {\slideentry {0}{0}{2}{2/2}{}{0}}} 22 | \@writefile{nav}{\headcommand {\beamer@framepages {2}{2}}} 23 | \@writefile{toc}{\beamer@sectionintoc {1}{An Introduction to Qutip}{3}{0}{1}} 24 | \@writefile{nav}{\headcommand {\sectionentry {1}{An Introduction to Qutip}{3}{An Introduction to Qutip}{0}}} 25 | \@writefile{nav}{\headcommand {\beamer@sectionpages {1}{2}}} 26 | \@writefile{nav}{\headcommand {\beamer@subsectionpages {1}{2}}} 27 | \@writefile{nav}{\headcommand {\slideentry {1}{0}{1}{3/3}{}{0}}} 28 | \@writefile{nav}{\headcommand {\beamer@framepages {3}{3}}} 29 | \@writefile{nav}{\headcommand {\slideentry {1}{0}{2}{4/4}{}{0}}} 30 | \@writefile{nav}{\headcommand {\beamer@framepages {4}{4}}} 31 | \@writefile{nav}{\headcommand {\slideentry {1}{0}{3}{5/5}{}{0}}} 32 | \@writefile{nav}{\headcommand {\beamer@framepages {5}{5}}} 33 | \@writefile{toc}{\beamer@subsectionintoc {1}{1}{Installation and Some Dos and Don'ts}{6}{0}{1}} 34 | \@writefile{nav}{\headcommand {\beamer@subsectionentry {0}{1}{1}{6}{Installation and Some Dos and Don'ts}}\headcommand {\beamer@subsectionpages {3}{5}}} 35 | \@writefile{nav}{\headcommand {\slideentry {1}{1}{1}{6/6}{Installation and Some Dos and Don'ts}{0}}} 36 | \@writefile{nav}{\headcommand {\beamer@framepages {6}{6}}} 37 | \@writefile{nav}{\headcommand {\slideentry {1}{1}{2}{7/7}{Installation and Some Dos and Don'ts}{0}}} 38 | \@writefile{nav}{\headcommand {\beamer@framepages {7}{7}}} 39 | \@writefile{toc}{\beamer@subsectionintoc {1}{2}{Finite Hilbert Space!}{8}{0}{1}} 40 | \@writefile{nav}{\headcommand {\beamer@subsectionentry {0}{1}{2}{8}{Finite Hilbert Space!}}\headcommand {\beamer@subsectionpages {6}{7}}} 41 | \@writefile{nav}{\headcommand {\slideentry {1}{2}{1}{8/8}{Finite Hilbert Space!}{0}}} 42 | \@writefile{nav}{\headcommand {\beamer@framepages {8}{8}}} 43 | \@writefile{toc}{\beamer@subsectionintoc {1}{3}{States and Operators}{9}{0}{1}} 44 | \@writefile{nav}{\headcommand {\beamer@subsectionentry {0}{1}{3}{9}{States and Operators}}\headcommand {\beamer@subsectionpages {8}{8}}} 45 | \@writefile{nav}{\headcommand {\slideentry {1}{3}{1}{9/9}{States and Operators}{0}}} 46 | \@writefile{nav}{\headcommand {\beamer@framepages {9}{9}}} 47 | \@writefile{toc}{\beamer@subsectionintoc {1}{4}{Solving Equation of Quantum System by Qutip}{10}{0}{1}} 48 | \@writefile{nav}{\headcommand {\beamer@subsectionentry {0}{1}{4}{10}{Solving Equation of Quantum System by Qutip}}\headcommand {\beamer@subsectionpages {9}{9}}} 49 | \@writefile{nav}{\headcommand {\slideentry {1}{4}{1}{10/10}{Solving Equation of Quantum System by Qutip}{0}}} 50 | \@writefile{nav}{\headcommand {\beamer@framepages {10}{10}}} 51 | \@writefile{nav}{\headcommand {\slideentry {1}{4}{2}{12/12}{Solving Equation of Quantum System by Qutip}{0}}} 52 | \@writefile{nav}{\headcommand {\beamer@framepages {12}{12}}} 53 | \@writefile{nav}{\headcommand {\slideentry {1}{4}{3}{14/14}{Solving Equation of Quantum System by Qutip}{0}}} 54 | \@writefile{nav}{\headcommand {\beamer@framepages {14}{14}}} 55 | \@writefile{nav}{\headcommand {\slideentry {1}{4}{4}{16/16}{Solving Equation of Quantum System by Qutip}{0}}} 56 | \@writefile{nav}{\headcommand {\beamer@framepages {16}{16}}} 57 | \newlabel{fig:download}{{14}{17}{Solving Equation of Quantum System by Qutip}{Doc-Start}{}} 58 | \@writefile{snm}{\beamer@slide {fig:download}{17}} 59 | \@writefile{nav}{\headcommand {\slideentry {1}{4}{5}{17/17}{Solving Equation of Quantum System by Qutip}{0}}} 60 | \@writefile{nav}{\headcommand {\beamer@framepages {17}{17}}} 61 | \@writefile{nav}{\headcommand {\slideentry {1}{4}{6}{18/18}{Solving Equation of Quantum System by Qutip}{0}}} 62 | \@writefile{nav}{\headcommand {\beamer@framepages {18}{18}}} 63 | \@writefile{nav}{\headcommand {\slideentry {1}{4}{7}{19/19}{Solving Equation of Quantum System by Qutip}{0}}} 64 | \@writefile{nav}{\headcommand {\beamer@framepages {19}{19}}} 65 | \@writefile{toc}{\beamer@sectionintoc {2}{An Introduction to Quasi-probability Theory}{22}{0}{2}} 66 | \@writefile{nav}{\headcommand {\sectionentry {2}{An Introduction to Quasi-probability Theory}{22}{An Introduction to Quasi-probability Theory}{0}}} 67 | \@writefile{nav}{\headcommand {\beamer@sectionpages {3}{21}}} 68 | \@writefile{nav}{\headcommand {\beamer@subsectionpages {10}{21}}} 69 | \@writefile{toc}{\beamer@subsectionintoc {2}{1}{Motivation}{22}{0}{2}} 70 | \@writefile{nav}{\headcommand {\beamer@subsectionentry {0}{2}{1}{22}{Motivation}}\headcommand {\beamer@subsectionpages {22}{21}}} 71 | \@writefile{nav}{\headcommand {\slideentry {2}{1}{1}{22/22}{Motivation}{0}}} 72 | \@writefile{nav}{\headcommand {\beamer@framepages {22}{22}}} 73 | \@writefile{toc}{\beamer@subsectionintoc {2}{2}{P-representation}{24}{0}{2}} 74 | \@writefile{nav}{\headcommand {\beamer@subsectionentry {0}{2}{2}{24}{P-representation}}\headcommand {\beamer@subsectionpages {22}{23}}} 75 | \@writefile{nav}{\headcommand {\slideentry {2}{2}{1}{24/24}{P-representation}{0}}} 76 | \@writefile{nav}{\headcommand {\beamer@framepages {24}{24}}} 77 | \@writefile{nav}{\headcommand {\slideentry {2}{2}{2}{25/25}{P-representation}{0}}} 78 | \@writefile{nav}{\headcommand {\beamer@framepages {25}{25}}} 79 | \@writefile{nav}{\headcommand {\slideentry {2}{2}{3}{26/26}{P-representation}{0}}} 80 | \@writefile{nav}{\headcommand {\beamer@framepages {26}{26}}} 81 | \@writefile{toc}{\beamer@subsectionintoc {2}{3}{Q-representation}{27}{0}{2}} 82 | \@writefile{nav}{\headcommand {\beamer@subsectionentry {0}{2}{3}{27}{Q-representation}}\headcommand {\beamer@subsectionpages {24}{26}}} 83 | \@writefile{nav}{\headcommand {\slideentry {2}{3}{1}{27/27}{Q-representation}{0}}} 84 | \@writefile{nav}{\headcommand {\beamer@framepages {27}{27}}} 85 | \@writefile{toc}{\beamer@subsectionintoc {2}{4}{Wigner-function}{28}{0}{2}} 86 | \@writefile{nav}{\headcommand {\beamer@subsectionentry {0}{2}{4}{28}{Wigner-function}}\headcommand {\beamer@subsectionpages {27}{27}}} 87 | \@writefile{nav}{\headcommand {\slideentry {2}{4}{1}{28/28}{Wigner-function}{0}}} 88 | \@writefile{nav}{\headcommand {\beamer@framepages {28}{28}}} 89 | \@writefile{nav}{\headcommand {\slideentry {2}{4}{2}{29/29}{Wigner-function}{0}}} 90 | \@writefile{nav}{\headcommand {\beamer@framepages {29}{29}}} 91 | \@writefile{toc}{\beamer@sectionintoc {3}{Some Examples and Exercise}{30}{0}{3}} 92 | \@writefile{nav}{\headcommand {\sectionentry {3}{Some Examples and Exercise}{30}{Some Examples and Exercise}{0}}} 93 | \@writefile{nav}{\headcommand {\beamer@sectionpages {22}{29}}} 94 | \@writefile{nav}{\headcommand {\beamer@subsectionpages {28}{29}}} 95 | \@writefile{toc}{\beamer@subsectionintoc {3}{1}{Cat State(Demo)}{30}{0}{3}} 96 | \@writefile{nav}{\headcommand {\beamer@subsectionentry {0}{3}{1}{30}{Cat State(Demo)}}\headcommand {\beamer@subsectionpages {30}{29}}} 97 | \@writefile{nav}{\headcommand {\slideentry {3}{1}{1}{30/30}{Cat State(Demo)}{0}}} 98 | \@writefile{nav}{\headcommand {\beamer@framepages {30}{30}}} 99 | \@writefile{nav}{\headcommand {\slideentry {3}{1}{2}{32/32}{Cat State(Demo)}{0}}} 100 | \@writefile{nav}{\headcommand {\beamer@framepages {32}{32}}} 101 | \@writefile{toc}{\beamer@subsectionintoc {3}{2}{Kerr Effect to Create Cat State(Exercise)}{33}{0}{3}} 102 | \@writefile{nav}{\headcommand {\beamer@subsectionentry {0}{3}{2}{33}{Kerr Effect to Create Cat State(Exercise)}}\headcommand {\beamer@subsectionpages {30}{32}}} 103 | \@writefile{}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {Vacuum state displacing to a cohrent state}}}} 104 | \@writefile{}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {Cohrent state squezing to a squezeed state}}}} 105 | \newlabel{fig:screenshot002}{{1}{33}{Kerr Effect to Create Cat State(Exercise)}{Doc-Start}{}} 106 | \@writefile{snm}{\beamer@slide {fig:screenshot002}{33}} 107 | \@writefile{nav}{\headcommand {\slideentry {3}{2}{1}{33/33}{Kerr Effect to Create Cat State(Exercise)}{0}}} 108 | \@writefile{nav}{\headcommand {\beamer@framepages {33}{33}}} 109 | \@writefile{nav}{\headcommand {\beamer@partpages {1}{33}}} 110 | \@writefile{nav}{\headcommand {\beamer@subsectionpages {33}{33}}} 111 | \@writefile{nav}{\headcommand {\beamer@sectionpages {30}{33}}} 112 | \@writefile{nav}{\headcommand {\beamer@documentpages {33}}} 113 | \@writefile{nav}{\headcommand {\def \inserttotalframenumber {26}}} 114 | -------------------------------------------------------------------------------- /course1/presentation_1.log: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/caidish/Quantum-Optics-with-Python/198974a1598d5b39de0abfe0c070ee70176ffbce/course1/presentation_1.log -------------------------------------------------------------------------------- /course1/presentation_1.nav: -------------------------------------------------------------------------------- 1 | \headcommand {\slideentry {0}{0}{1}{1/1}{}{0}} 2 | \headcommand {\beamer@framepages {1}{1}} 3 | \headcommand {\slideentry {0}{0}{2}{2/2}{}{0}} 4 | \headcommand {\beamer@framepages {2}{2}} 5 | \headcommand {\sectionentry {1}{An Introduction to Qutip}{3}{An Introduction to Qutip}{0}} 6 | \headcommand {\beamer@sectionpages {1}{2}} 7 | \headcommand {\beamer@subsectionpages {1}{2}} 8 | \headcommand {\slideentry {1}{0}{1}{3/3}{}{0}} 9 | \headcommand {\beamer@framepages {3}{3}} 10 | \headcommand {\slideentry {1}{0}{2}{4/4}{}{0}} 11 | \headcommand {\beamer@framepages {4}{4}} 12 | \headcommand {\slideentry {1}{0}{3}{5/5}{}{0}} 13 | \headcommand {\beamer@framepages {5}{5}} 14 | \headcommand {\beamer@subsectionentry {0}{1}{1}{6}{Installation and Some Dos and Don'ts}}\headcommand {\beamer@subsectionpages {3}{5}} 15 | \headcommand {\slideentry {1}{1}{1}{6/6}{Installation and Some Dos and Don'ts}{0}} 16 | \headcommand {\beamer@framepages {6}{6}} 17 | \headcommand {\slideentry {1}{1}{2}{7/7}{Installation and Some Dos and Don'ts}{0}} 18 | \headcommand {\beamer@framepages {7}{7}} 19 | \headcommand {\beamer@subsectionentry {0}{1}{2}{8}{Finite Hilbert Space!}}\headcommand {\beamer@subsectionpages {6}{7}} 20 | \headcommand {\slideentry {1}{2}{1}{8/8}{Finite Hilbert Space!}{0}} 21 | \headcommand {\beamer@framepages {8}{8}} 22 | \headcommand {\beamer@subsectionentry {0}{1}{3}{9}{States and Operators}}\headcommand {\beamer@subsectionpages {8}{8}} 23 | \headcommand {\slideentry {1}{3}{1}{9/9}{States and Operators}{0}} 24 | \headcommand {\beamer@framepages {9}{9}} 25 | \headcommand {\beamer@subsectionentry {0}{1}{4}{10}{Solving Equation of Quantum System by Qutip}}\headcommand {\beamer@subsectionpages {9}{9}} 26 | \headcommand {\slideentry {1}{4}{1}{10/10}{Solving Equation of Quantum System by Qutip}{0}} 27 | \headcommand {\beamer@framepages {10}{10}} 28 | \headcommand {\slideentry {1}{4}{2}{12/12}{Solving Equation of Quantum System by Qutip}{0}} 29 | \headcommand {\beamer@framepages {12}{12}} 30 | \headcommand {\slideentry {1}{4}{3}{14/14}{Solving Equation of Quantum System by Qutip}{0}} 31 | \headcommand {\beamer@framepages {14}{14}} 32 | \headcommand {\slideentry {1}{4}{4}{16/16}{Solving Equation of Quantum System by Qutip}{0}} 33 | \headcommand {\beamer@framepages {16}{16}} 34 | \headcommand {\slideentry {1}{4}{5}{17/17}{Solving Equation of Quantum System by Qutip}{0}} 35 | \headcommand {\beamer@framepages {17}{17}} 36 | \headcommand {\slideentry {1}{4}{6}{18/18}{Solving Equation of Quantum System by Qutip}{0}} 37 | \headcommand {\beamer@framepages {18}{18}} 38 | \headcommand {\slideentry {1}{4}{7}{19/19}{Solving Equation of Quantum System by Qutip}{0}} 39 | \headcommand {\beamer@framepages {19}{19}} 40 | \headcommand {\sectionentry {2}{An Introduction to Quasi-probability Theory}{22}{An Introduction to Quasi-probability Theory}{0}} 41 | \headcommand {\beamer@sectionpages {3}{21}} 42 | \headcommand {\beamer@subsectionpages {10}{21}} 43 | \headcommand {\beamer@subsectionentry {0}{2}{1}{22}{Motivation}}\headcommand {\beamer@subsectionpages {22}{21}} 44 | \headcommand {\slideentry {2}{1}{1}{22/22}{Motivation}{0}} 45 | \headcommand {\beamer@framepages {22}{22}} 46 | \headcommand {\beamer@subsectionentry {0}{2}{2}{24}{P-representation}}\headcommand {\beamer@subsectionpages {22}{23}} 47 | \headcommand {\slideentry {2}{2}{1}{24/24}{P-representation}{0}} 48 | \headcommand {\beamer@framepages {24}{24}} 49 | \headcommand {\slideentry {2}{2}{2}{25/25}{P-representation}{0}} 50 | \headcommand {\beamer@framepages {25}{25}} 51 | \headcommand {\slideentry {2}{2}{3}{26/26}{P-representation}{0}} 52 | \headcommand {\beamer@framepages {26}{26}} 53 | \headcommand {\beamer@subsectionentry {0}{2}{3}{27}{Q-representation}}\headcommand {\beamer@subsectionpages {24}{26}} 54 | \headcommand {\slideentry {2}{3}{1}{27/27}{Q-representation}{0}} 55 | \headcommand {\beamer@framepages {27}{27}} 56 | \headcommand {\beamer@subsectionentry {0}{2}{4}{28}{Wigner-function}}\headcommand {\beamer@subsectionpages {27}{27}} 57 | \headcommand {\slideentry {2}{4}{1}{28/28}{Wigner-function}{0}} 58 | \headcommand {\beamer@framepages {28}{28}} 59 | \headcommand {\slideentry {2}{4}{2}{29/29}{Wigner-function}{0}} 60 | \headcommand {\beamer@framepages {29}{29}} 61 | \headcommand {\sectionentry {3}{Some Examples and Exercise}{30}{Some Examples and Exercise}{0}} 62 | \headcommand {\beamer@sectionpages {22}{29}} 63 | \headcommand {\beamer@subsectionpages {28}{29}} 64 | \headcommand {\beamer@subsectionentry {0}{3}{1}{30}{Cat State(Demo)}}\headcommand {\beamer@subsectionpages {30}{29}} 65 | \headcommand {\slideentry {3}{1}{1}{30/30}{Cat State(Demo)}{0}} 66 | \headcommand {\beamer@framepages {30}{30}} 67 | \headcommand {\slideentry {3}{1}{2}{32/32}{Cat State(Demo)}{0}} 68 | \headcommand {\beamer@framepages {32}{32}} 69 | \headcommand {\beamer@subsectionentry {0}{3}{2}{33}{Kerr Effect to Create Cat State(Exercise)}}\headcommand {\beamer@subsectionpages {30}{32}} 70 | \headcommand {\slideentry {3}{2}{1}{33/33}{Kerr Effect to Create Cat State(Exercise)}{0}} 71 | \headcommand {\beamer@framepages {33}{33}} 72 | \headcommand {\beamer@partpages {1}{33}} 73 | \headcommand {\beamer@subsectionpages {33}{33}} 74 | \headcommand {\beamer@sectionpages {30}{33}} 75 | \headcommand {\beamer@documentpages {33}} 76 | \headcommand {\def \inserttotalframenumber {26}} 77 | -------------------------------------------------------------------------------- /course1/presentation_1.out: -------------------------------------------------------------------------------- 1 | \BOOKMARK [2][]{Outline0.1}{An Introduction to Qutip}{}% 1 2 | \BOOKMARK [3][]{Outline0.1.1.6}{Installation and Some Dos and Don'ts}{Outline0.1}% 2 3 | \BOOKMARK [3][]{Outline0.1.2.8}{Finite Hilbert Space!}{Outline0.1}% 3 4 | \BOOKMARK [3][]{Outline0.1.3.9}{States and Operators}{Outline0.1}% 4 5 | \BOOKMARK [3][]{Outline0.1.4.10}{Solving Equation of Quantum System by Qutip}{Outline0.1}% 5 6 | \BOOKMARK [2][]{Outline0.2}{An Introduction to Quasi-probability Theory}{}% 6 7 | \BOOKMARK [3][]{Outline0.2.1.22}{Motivation}{Outline0.2}% 7 8 | \BOOKMARK [3][]{Outline0.2.2.24}{P-representation}{Outline0.2}% 8 9 | \BOOKMARK [3][]{Outline0.2.3.27}{Q-representation}{Outline0.2}% 9 10 | \BOOKMARK [3][]{Outline0.2.4.28}{Wigner-function}{Outline0.2}% 10 11 | \BOOKMARK [2][]{Outline0.3}{Some Examples and Exercise}{}% 11 12 | \BOOKMARK [3][]{Outline0.3.1.30}{Cat State\(Demo\)}{Outline0.3}% 12 13 | \BOOKMARK [3][]{Outline0.3.2.33}{Kerr Effect to Create Cat State\(Exercise\)}{Outline0.3}% 13 14 | -------------------------------------------------------------------------------- /course1/presentation_1.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/caidish/Quantum-Optics-with-Python/198974a1598d5b39de0abfe0c070ee70176ffbce/course1/presentation_1.pdf -------------------------------------------------------------------------------- /course1/presentation_1.snm: -------------------------------------------------------------------------------- 1 | \beamer@slide {fig:download}{17} 2 | \beamer@slide {fig:screenshot002}{33} 3 | -------------------------------------------------------------------------------- /course1/presentation_1.synctex.gz: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/caidish/Quantum-Optics-with-Python/198974a1598d5b39de0abfe0c070ee70176ffbce/course1/presentation_1.synctex.gz -------------------------------------------------------------------------------- /course1/presentation_1.tex: -------------------------------------------------------------------------------- 1 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 2 | % Beamer Presentation 3 | % LaTeX Template 4 | % Version 1.0 (10/11/12) 5 | % 6 | % This template has been downloaded from: 7 | % http://www.LaTeXTemplates.com 8 | % 9 | % License: 10 | % CC BY-NC-SA 3.0 (http://creativecommons.org/licenses/by-nc-sa/3.0/) 11 | % 12 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 13 | 14 | %---------------------------------------------------------------------------------------- 15 | % PACKAGES AND THEMES 16 | %---------------------------------------------------------------------------------------- 17 | 18 | \documentclass{beamer} 19 | 20 | \mode { 21 | 22 | % The Beamer class comes with a number of default slide themes 23 | % which change the colors and layouts of slides. Below this is a list 24 | % of all the themes, uncomment each in turn to see what they look like. 25 | 26 | %\usetheme{default} 27 | %\usetheme{AnnArbor} 28 | %\usetheme{Antibes} 29 | %\usetheme{Bergen} 30 | %\usetheme{Berkeley} 31 | %\usetheme{Berlin} 32 | %\usetheme{Boadilla} 33 | %\usetheme{CambridgeUS} 34 | %\usetheme{Copenhagen} 35 | %\usetheme{Darmstadt} 36 | %\usetheme{Dresden} 37 | %\usetheme{Frankfurt} 38 | %\usetheme{Goettingen} 39 | %\usetheme{Hannover} 40 | %\usetheme{Ilmenau} 41 | %\usetheme{JuanLesPins} 42 | %\usetheme{Luebeck} 43 | \usetheme{Madrid} 44 | %\usetheme{Malmoe} 45 | %\usetheme{Marburg} 46 | %\usetheme{Montpellier} 47 | %\usetheme{PaloAlto} 48 | %\usetheme{Pittsburgh} 49 | %\usetheme{Rochester} 50 | %\usetheme{Singapore} 51 | %\usetheme{Szeged} 52 | %\usetheme{Warsaw} 53 | 54 | % As well as themes, the Beamer class has a number of color themes 55 | % for any slide theme. Uncomment each of these in turn to see how it 56 | % changes the colors of your current slide theme. 57 | 58 | %\usecolortheme{albatross} 59 | %\usecolortheme{beaver} 60 | %\usecolortheme{beetle} 61 | %\usecolortheme{crane} 62 | %\usecolortheme{dolphin} 63 | %\usecolortheme{dove} 64 | %\usecolortheme{fly} 65 | %\usecolortheme{lily} 66 | %\usecolortheme{orchid} 67 | %\usecolortheme{rose} 68 | %\usecolortheme{seagull} 69 | %\usecolortheme{seahorse} 70 | %\usecolortheme{whale} 71 | %\usecolortheme{wolverine} 72 | 73 | %\setbeamertemplate{footline} % To remove the footer line in all slides uncomment this line 74 | %\setbeamertemplate{footline}[page number] % To replace the footer line in all slides with a simple slide count uncomment this line 75 | 76 | %\setbeamertemplate{navigation symbols}{} % To remove the navigation symbols from the bottom of all slides uncomment this line 77 | \def\mathfamilydefault{\rmdefault} 78 | %\setbeamertemplate{navigation symbols}{} % To remove the navigation symbols from the bottom of all slides uncomment this line 79 | \logo{\includegraphics[height=1.1cm]{logo.png}} 80 | \setbeamertemplate{navigation symbols}{} % To remove the navigation symbols from the bottom of all slides uncomment this line 81 | } 82 | 83 | \usepackage{graphicx} % Allows including images 84 | \usepackage{subfigure} 85 | \usepackage{booktabs} % Allows the use of \toprule, \midrule and \bottomrule in tables 86 | \usepackage{listings} 87 | \lstset{ 88 | numbers=left, 89 | numberstyle= \tiny, 90 | keywordstyle= \color{ blue!70}, 91 | commentstyle= \color{red!50!green!50!blue!50}, 92 | frame=shadowbox, % 阴影效果 93 | rulesepcolor= \color{ red!20!green!20!blue!20} , 94 | escapeinside=``, % 英文分号中可写入中文 95 | xleftmargin=2em,xrightmargin=2em, aboveskip=1em, 96 | framexleftmargin=2em 97 | } 98 | \lstset{language=python} 99 | \lstset{breaklines} 100 | \usepackage{xcolor} 101 | 102 | %---------------------------------------------------------------------------------------- 103 | % TITLE PAGE 104 | %---------------------------------------------------------------------------------------- 105 | 106 | \title[Quantum Optics with Python]{Quantum Optics with Python: \\ 107 | Lecture2:Qutip and Quasi-probability theory} % The short title appears at the bottom of every slide, the full title is only on the title page 108 | 109 | \author{Cai Jiaqi} % Your name 110 | \institute[HUST] % Your institution as it will appear on the bottom of every slide, may be shorthand to save space 111 | { 112 | Huazhong University of Science and Technology \\ % Your institution for the title page 113 | \medskip 114 | \textit{caidish@hust.edu.cn} % Your email address 115 | } 116 | \date{\today} % Date, can be changed to a custom date 117 | 118 | \begin{document} 119 | 120 | \begin{frame} 121 | \titlepage % Print the title page as the first slide 122 | \end{frame} 123 | 124 | \begin{frame} 125 | \frametitle{Overview} % Table of contents slide, comment this block out to remove it 126 | \tableofcontents % Throughout your presentation, if you choose to use \section{} and \subsection{} commands, these will automatically be printed on this slide as an overview of your presentation 127 | \end{frame} 128 | 129 | %---------------------------------------------------------------------------------------- 130 | % PRESENTATION SLIDES 131 | %---------------------------------------------------------------------------------------- 132 | 133 | %------------------------------------------------ 134 | \section{An Introduction to Qutip} 135 | \begin{frame}{What is Qutip?} 136 | \begin{center} 137 | \includegraphics[width=0.7\linewidth]{screenshot001} 138 | \end{center} 139 | \begin{block}{Qutip} 140 | Quantum Toolbox in Python.URL:http://qutip.org/ \\ 141 | Cite them with Comp. Phys. Comm. 184, 1234 (2013) or Comp. Phys. Comm. 183, 1760 (2012)! 142 | \end{block} 143 | \end{frame} 144 | 145 | \begin{frame}{Why I choose Qutip?} 146 | \begin{itemize} 147 | \item Free 148 | \item Easy 149 | \item Partially Fast 150 | \item Interactive 151 | \item Universal and Scalable 152 | \end{itemize} 153 | \end{frame} 154 | 155 | \begin{frame}{How Many People are Using Qutip?} 156 | Many people in the community of Quantum optics and related subjects are now using Qutip. Qutip helps to generate beautiful figure and simulation results of many high-quality papers. 157 | 158 | In 2016,the Unique Visitors of qutip.org is larger than 25,473. 159 | \end{frame} 160 | 161 | \subsection{Installation and Some Dos and Don'ts} 162 | \begin{frame}{Windows} 163 | Qutip is first developed on Unix and tested mostly on Linux. But thanks to the portability of Python language, Windows users can also use Qutip as you like. 164 | 165 | The recommended installation steps are: 166 | 167 | \begin{itemize} 168 | \item Install MSVC++ 2015 or VS2015.Notice: The default installation of VS2015 DON'T contain MSVC++ any more! You need to change the default to a manual one. 169 | \item Install Anaconda. For CERNET user, I highly recommend you to download the distribution from Tsinghua Opensource Mirror 170 | \item Pip install Qutip. 171 | \item enjoy it! 172 | \end{itemize} 173 | 174 | \end{frame} 175 | 176 | \begin{frame}{Linux and macOs} 177 | \begin{itemize} 178 | \item Install gcc.\\Note: Ubuntu: sudo wget build-essentials. Mac: brew install gcc. 179 | \item Install Anaconda. 180 | \item pip install Qutip. 181 | \item enjoy it! 182 | \end{itemize} 183 | 184 | \end{frame} 185 | 186 | \subsection{Finite Hilbert Space!} 187 | \begin{frame}{Hilbert Space} 188 | A computer can only hold a finite representation of Hilbert space. For analytic method, the Fock state can be represented as: 189 | \[\left\langle {m} 190 | \mathrel{\left | {\vphantom {m n}} 191 | \right. \kern-\nulldelimiterspace} 192 | {n} \right\rangle = {\delta _{mn}},\left| n \right\rangle = A{\left( {{a^\dag }} \right)^n}\left| 0 \right\rangle \] 193 | However, this Fock state cannot be stored conveniently by a computer. 194 | 195 | In qutip, we often set an upper bound of Hilbert space, e.g. $n$ . Then, states are $1\times n$ vectors and operators are $n \times n$ matrix. 196 | 197 | To generate many-body Hamiltonian, we should construct a total space from the tensor product(e.g,kronecker product) of two spaces: 198 | \[{H^{total}} = {H^{\left( 1 \right)}} \otimes {H^{\left( 2 \right)}}\] 199 | \end{frame} 200 | 201 | \subsection{States and Operators} 202 | \begin{frame}{States and Operators in Qutip} 203 | static function:basis(), create(), destroy(), tensor(),mesolve()..... 204 | 205 | Qobj class, properties: 206 | dims, 207 | shape, 208 | type, 209 | data 210 | 211 | some selected function:*,dag(),eigenenergies(),eigenstates, 212 | \end{frame} 213 | \begin{lstlisting} 214 | d = 5 215 | #the dimension of the Hilbert space 216 | g = basis(d,0) 217 | e1 = basis(d,1) 218 | e2 = basis(d,2) 219 | print(g) 220 | print(e1) 221 | print(e2) 222 | 223 | a=destroy(d) 224 | a_dag = create(d) 225 | e1 = a_dag*g 226 | e2 = (a_dag * a_dag * g).unit() 227 | print(g) 228 | print(e1) 229 | print(e2) 230 | \end{lstlisting}\frametitle{States and Operators in Qutip} 231 | \subsection{Solving Equation of Quantum System by Qutip} 232 | \begin{frame}{Master Equation Solver} 233 | \begin{block}{Unitary Evolution} 234 | \[\psi \left( t \right) = U\left( {t,0} \right)\psi \left( 0 \right)\] 235 | mesolve(H,psi0,times,[],[(exp)]) 236 | \end{block} 237 | \begin{block}{Non-unitary Evolution} 238 | \[\frac{d}{{dt}}\rho \left( t \right) = - \frac{i}{\hbar }\left[ {H\left( t \right),\rho \left( t \right)} \right] + \sum\limits_n {\frac{1}{2}\left[ {2{C_n}\rho \left( t \right)C_n^\dag - \rho \left( t \right)C_n^\dag {C_n} - C_n^\dag {C_n}\rho \left( t \right)} \right]} \] 239 | 240 | mesolve(H,psi0,times,[c\_ops\_list],[(exp)]) 241 | \end{block} 242 | 243 | \end{frame} 244 | \begin{frame}{Spin Model and Spin with Dissipation} 245 | The Hamiltonian of a spin in x-direction is read as:$H = {\omega _z}{\sigma _x}$. The dynamic evolution of this system can be obtained by qutip. 246 | 247 | \includegraphics[width=0.5\linewidth]{2} 248 | \includegraphics[width=0.5\linewidth]{3} 249 | 250 | 251 | \end{frame} 252 | 253 | \begin{lstlisting} 254 | H = 2 * np.pi * 0.1 * sigmax() 255 | psi0 = basis(2, 0) 256 | times = np.linspace(0.0, 10.0, 20) 257 | result = mesolve(H, psi0, times, [], [sigmaz(),sigmay()]) 258 | fig, ax = plt.subplots() 259 | ax.plot(result.times, result.expect[0]); 260 | ax.plot(result.times, result.expect[1]); 261 | ax.set_xlabel('Time'); 262 | ax.set_ylabel('Expectation values'); 263 | ax.legend(("Sigma-Z", "Sigma-Y")); 264 | plt.show() 265 | \end{lstlisting} 266 | \newpage 267 | \begin{lstlisting} 268 | H = 2 * np.pi * 0.1 * sigmax() 269 | psi0 = basis(2, 0) 270 | times = np.linspace(0.0, 10.0, 20) 271 | result = mesolve(H, psi0, times, [np.sqrt(0.05)*sigmax()], [sigmaz(),sigmay()]) 272 | fig, ax = plt.subplots() 273 | ax.plot(result.times, result.expect[0]); 274 | ax.plot(result.times, result.expect[1]); 275 | ax.set_xlabel('Time'); 276 | ax.set_ylabel('Expectation values'); 277 | ax.legend(("Sigma-Z", "Sigma-Y")); 278 | plt.show() 279 | \end{lstlisting} 280 | \begin{frame}{Mento Carlo Solver} 281 | \[\begin{gathered} 282 | {H_{eff}} = {H_{sys}} - \frac{i}{2}\sum\limits_i {C_n^\dag {C_n}} \hfill \\ 283 | \psi \left( {t + dt} \right) = {C_n}\psi \left( t \right)/\sqrt {\left\langle {C_n^\dag {C_n}} \right\rangle } \hfill \\ 284 | \end{gathered} \] 285 | usage:mesolve(H,psi0,times,[c\_ops\_list],[(exp)]) 286 | \end{frame} 287 | \begin{frame}{Steady State Solver} 288 | \[\frac{{d{\rho _{ss}}}}{{dt}} = L{\rho _{ss}} = 0\] 289 | 290 | usage:steadystates(H,c\_ops) 291 | \end{frame} 292 | \begin{frame}{Example:Optomechanical System} 293 | \begin{figure} 294 | \centering 295 | \includegraphics[width=0.5\linewidth]{download} 296 | \label{fig:download} 297 | \end{figure} 298 | The optomechanical Hamiltonian arises from the radiation pressure interaction of light in an optical cavity where one of the cavity mirrors is mechanically compliant: 299 | $$\frac{\hat{H}}{\hbar}=-\Delta\hat{a}^{+}\hat{a}+\omega_{m}\hat{b}^{+}\hat{b}+g_{0}(\hat{b}+\hat{b}^{+})\hat{a}^{+}\hat{a}+E\left(\hat{a}+\hat{a}^{+}\right)$$ 300 | Where $\Delta$is the detuning between pump($\omega_{p}$) and cavity($\omega_c$), $\omega_{m}$ is frequency of the oscillator. $g_0$ is the single-photon-phonon coupling strength and $E$ is the amplitude of the pump mode. 301 | 302 | \end{frame} 303 | \begin{frame}{Example:Optomechanical System} 304 | To obtain the dynamic evolution of the system , we'll follow the steps below: 305 | \begin{itemize} 306 | \item Set system parameter 307 | \item Build Hamiltonian and collapse operators 308 | \item Solve master equation 309 | \item Run steady state Solver 310 | \item Visualize the result 311 | \end{itemize} 312 | \end{frame} 313 | 314 | \begin{frame}{Example:Optomechanical System} 315 | \includegraphics[width=0.5\linewidth]{4} 316 | \includegraphics[width=0.5\linewidth]{5} 317 | 318 | \end{frame} 319 | 320 | \begin{lstlisting} 321 | # System Parameters (in units of wm) 322 | #----------------------------------- 323 | Nc = 10 # Number of cavity states 324 | Nm = 80 # Number of mech states 325 | kappa = 0.3 # Cavity damping rate 326 | E = 0.1 # Driving Amplitude 327 | g0 = 2.4*kappa # Coupling strength 328 | Qm = 1e4 # Mech quality factor 329 | gamma = 1/Qm # Mech damping rate 330 | n_th = 1 # Mech bath temperature 331 | delta = -0.43 # Detuning 332 | 333 | # Operators 334 | #---------- 335 | a = tensor(destroy(Nc), qeye(Nm)) 336 | b = tensor(qeye(Nc), destroy(Nm)) 337 | num_b = b.dag()*b 338 | num_a = a.dag()*a 339 | psi0=tensor(basis(Nc,5),basis(Nm,10)) 340 | ro0 = psi0*psi0.dag() 341 | # Hamiltonian 342 | #------------ 343 | H = -delta*(num_a)+num_b+g0*(b.dag()+b)*num_a+E*(a.dag()+a) 344 | 345 | # Collapse operators 346 | #------------------- 347 | cc = np.sqrt(kappa)*a 348 | cm = np.sqrt(gamma*(1.0 + n_th))*b 349 | cp = np.sqrt(gamma*n_th)*b.dag() 350 | c_ops = [cc,cm,cp] 351 | 352 | t_list = np.linspace(0,20,1000) 353 | result = mesolve(H,ro0,times,c_ops,[num_b,num_a]) 354 | \end{lstlisting} 355 | \section{An Introduction to Quasi-probability Theory} 356 | \subsection{Motivation} 357 | \begin{frame}{Motivation} 358 | \begin{block}{Classical version of fluctuating of complex E(t)} 359 | \[\left\langle {{E^*}\left( {{r_1},t} \right)E\left( {{r_2},t} \right)} \right\rangle = \int {{E^*}\left( {{r_1},t} \right)E\left( {{r_2},t} \right)P\left( {E,{E^*},t} \right){d^2}E} \] 360 | \end{block} 361 | Q: Can we construct a similar description for quantum field fluctuations? 362 | 363 | A:The quasi-probability theory in coherent state representation! 364 | 365 | But: \centering{\textbf{???$a^{\dagger}$ and $a$ are not commute???}} 366 | 367 | \begin{itemize} 368 | \item Normal ordering ${a^\dag }a$-----\emph{P-representation} 369 | \item Anti-normal ordering $a{a^\dag }$-----\emph{Q-representation} 370 | \item Symmetric ordering $\left( {a{a^\dag } + {a^\dag }a} \right)/2$-----\emph{Wigner function} 371 | \end{itemize} 372 | 373 | \end{frame} 374 | 375 | 376 | \subsection{P-representation} 377 | \begin{frame}{P-representation: Theoretical Case} 378 | We want to expand any operator of the light field with probability distribution, Which means: 379 | \[\left\langle {O\left( {a,{a^\dag }} \right)} \right\rangle = \int {{d^2}\alpha P\left( {\alpha ,{\alpha ^*}} \right)O\left( {\alpha ,{\alpha ^ * }} \right)} \] 380 | 381 | Basically, any operator can be expanded in a normal ordering: 382 | \[\delta \left( {{\alpha ^*} - {a^\dag }} \right)\delta \left( {\alpha - a} \right) = \frac{1}{{{\pi ^2}}}\int {{d^2}\beta \left[ {{e^{ - \beta \left( {{\alpha ^*} - {a^\dag }} \right)}}{e^{\beta \left( {\alpha - a} \right)}}} \right]} \] 383 | Then, take the expectation value of the operator: 384 | \[\left\langle {O\left( {a,{a^\dag }} \right)} \right\rangle = Tr\left[ {\rho O} \right] = \sum\limits_n {\sum\limits_m {{c_{nm}}Tr\left[ {\rho {{\left( {{a^\dag }} \right)}^n}{a^m}} \right]} } \] 385 | Define an operator: 386 | \[\delta \left( {{\alpha ^*} - {a^\dag }} \right)\delta \left( {\alpha - a} \right) = \frac{1}{{{\pi ^2}}}\int {{d^2}\beta \left[ {{e^{ - \beta \left( {{\alpha ^*} - {a^\dag }} \right)}}{e^{\beta \left( {\alpha - a} \right)}}} \right]} \] 387 | 388 | 389 | \end{frame} 390 | 391 | \begin{frame}{P-representation: Theoretical Case} 392 | Then, the expectation can be written as: 393 | \[\left\langle O \right\rangle = \int {{d^2}\alpha \sum\limits_n {\sum\limits_m {{c_{nm}}Tr\left[ {\rho \delta \left( {{\alpha ^*} - {a^\dag }} \right)\delta \left( {\alpha - a} \right)} \right]{{\left( {{\alpha ^*}} \right)}^n}{\alpha ^m}} } } \] 394 | 395 | So: 396 | \[P\left( {\alpha ,{\alpha ^*}} \right) \equiv Tr\left[ {\rho \delta \left( {{\alpha ^*} - {a^\dag }} \right)\delta \left( {\alpha - a} \right)} \right]\] 397 | 398 | A convenient way to calculate P-function is take the Fourier inverse of anti-diagonal matrix element of $\rho$ (exercise): 399 | \[P\left( {\alpha ,{\alpha ^*}} \right) = {\mathfrak{F}_{{x_\beta },{y_\beta } \to {x_\alpha },{y_\alpha }}}\left[ {\left\langle { - \beta } \right|\rho \left| \beta \right\rangle {e^{{{\left| \beta \right|}^2}}}} \right]\] 400 | 401 | \end{frame} 402 | 403 | \begin{frame}{P-representation:Quantum Fock state} 404 | Last time, it is mentioned that Fock state is a quantum state of light. Here comes the reason: 405 | 406 | $\rho = \left| n \right\rangle \left\langle n \right|$ is the density matrix of a Fock state, and 407 | \[\left\langle { - \beta } \right|\rho \left| \beta \right\rangle = \exp \left( { - {{\left| \beta \right|}^2}} \right)\frac{{{{\left( { - 1} \right)}^n}{{\left| \beta \right|}^{2n}}}}{{n!}}\] 408 | 409 | It then follows that: 410 | \[\begin{gathered} 411 | P\left( {\alpha ,{\alpha ^*}} \right) = \frac{{{e^{{{\left| \alpha \right|}^2}}}}}{{{\pi ^2}}}\frac{{{{\left( { - 1} \right)}^n}}}{{n!}}\int {{d^2}\beta } \left[ {{{\left| \beta \right|}^{2n}}{e^{ - \beta {\alpha ^*} + {\beta ^*}\alpha }}} \right] \hfill \\ 412 | = \frac{{{e^{{{\left| \alpha \right|}^2}}}}}{{{\pi ^2}}}\frac{{{{\left( { - 1} \right)}^n}}}{{n!}}\frac{{{\partial ^{2n}}}}{{\partial {\alpha ^n}\partial {\alpha ^{*n}}}}\int {{d^2}\beta } \left[ {{e^{ - \beta {\alpha ^*} + {\beta ^*}\alpha }}} \right] \hfill \\ 413 | = \frac{{{e^{{{\left| \alpha \right|}^2}}}}}{{{\pi ^2}}}\frac{{{{\left( { - 1} \right)}^n}}}{{n!}}\frac{{{\partial ^{2n}}}}{{\partial {\alpha ^n}\partial {\alpha ^{*n}}}}{\delta ^2}\left( {\alpha ,{\alpha ^*}} \right) \hfill \\ 414 | \end{gathered} \] 415 | 416 | Which turns out to be \textbf{negative}. In a classical version of distribution theory, probability is required to be nonnegative! These whose $P(\alpha,\alpha^{\star})$ is negative for some value are called non-classical. 417 | \end{frame} 418 | 419 | \subsection{Q-representation} 420 | \begin{frame}{Q-representation: Theoretical Case} 421 | The definition of Q-representation is: 422 | \[\begin{gathered} 423 | Q\left( {\alpha ,{\alpha ^*}} \right) = Tr\left[ {\rho \delta \left( {\alpha - a} \right)\delta \left( {{\alpha ^*} - {a^\dag }} \right)} \right] \hfill \\ 424 | Q\left( {\alpha ,{\alpha ^*}} \right) = \frac{1}{\pi }\left\langle \alpha \right|\rho \left| \alpha \right\rangle \hfill \\ 425 | \end{gathered} \] 426 | It then follows that(exercise): 427 | \[\left\langle {O\left( {a,{a^\dag }} \right)} \right\rangle = \int {Q\left( {\alpha ,{\alpha ^*}} \right)O\left( {\alpha ,{\alpha ^*}} \right)} {d^2}\alpha \] 428 | And $Q\left( {\alpha ,{\alpha ^*}}\right)$ is nonnegative and bounded(exercise): 429 | \[0 \leqslant Q\left( {\alpha ,{\alpha ^*}} \right) \leqslant \frac{1}{\pi }\] 430 | \end{frame} 431 | \subsection{Wigner-function} 432 | \begin{frame}{Wigner Distribution: Theoretical Case} 433 | Note that $P\&Q$ can be written in terms of \textbf{characteristic functions}. So: 434 | 435 | \[\begin{gathered} 436 | P\left( {\alpha ,{\alpha ^*}} \right) = \frac{1}{{{\pi ^2}}}\int {{d^2}\beta {e^{ - i\beta {\alpha ^*} - i{\beta ^*}\alpha }}{C^p}\left( {\beta ,{\beta ^*}} \right)} \hfill \\ 437 | {C^p}\left( {\beta ,{\beta ^*}} \right) = Tr\left[ {{e^{i\beta {a^\dag }}}{e^{i{\beta ^*}a}}\rho } \right] \hfill \\ 438 | Q\left( {\alpha ,{\alpha ^*}} \right) = \frac{1}{{{\pi ^2}}}\int {{d^2}\beta {e^{ - i\beta {\alpha ^*} - i{\beta ^*}\alpha }}{C^Q}\left( {\beta ,{\beta ^*}} \right)} \hfill \\ 439 | {C^Q}\left( {\beta ,{\beta ^*}} \right) = Tr\left[ {{e^{i{\beta ^*}a}}{e^{i\beta {a^\dag }}}\rho } \right] \hfill \\ 440 | \end{gathered} \] 441 | 442 | Wigner and Weyl introduced Wigner function, which is defined as: 443 | \[\begin{gathered} 444 | W\left( {\alpha ,{\alpha ^*}} \right) = \frac{1}{{{\pi ^2}}}\int {{d^2}\beta {e^{ - \beta {\alpha ^*} - i{\beta ^*}\alpha }}{C^W}\left( {\beta ,{\beta ^*}} \right)} \hfill \\ 445 | {C^W}\left( {\beta ,{\beta ^*}} \right) = Tr\left[ {{e^{i{\beta ^*}a + i\beta {a^\dag }}}} \right] \hfill \\ 446 | \end{gathered} \] 447 | Among three types of quasi-distribution, Wigner function is used mostly because it serves as a phase diagram of a quantum state. 448 | \end{frame} 449 | 450 | \begin{frame}{Wigner Distribution:Vacuum state, Coherent state and squeezed state} 451 | 452 | \begin{figure} 453 | \centering 454 | \subfigure[Vacuum state displacing to a cohrent state]{ 455 | \begin{minipage}[b]{0.4\textwidth} 456 | \includegraphics[width=1\textwidth]{6} \\ 457 | \includegraphics[width=1\textwidth]{8} 458 | \end{minipage} 459 | } 460 | \subfigure[Cohrent state squezing to a squezeed state]{ 461 | \begin{minipage}[b]{0.4\textwidth} 462 | \includegraphics[width=1\textwidth]{7} \\ 463 | \includegraphics[width=1\textwidth]{9} 464 | \end{minipage} 465 | } 466 | \end{figure} 467 | \end{frame} 468 | \begin{lstlisting} 469 | N = 50 470 | vac = basis(N,0) 471 | rho_vac = vac*vac.dag() 472 | rho_coh1 = coherent_dm(N,1+3j) 473 | rho_coh2 = coherent_dm(N,-1-3j) 474 | plot_wigner(rho_vac,colorbar=True,cmap='spectral') 475 | plot_wigner(rho_coh1,colorbar=True,cmap='spectral') 476 | plot_wigner(rho_coh2,colorbar=True,cmap='spectral') 477 | vac = basis(N,0) 478 | d = displace(N,1+3j) 479 | s = squeeze(N,0.25+0.25j) 480 | psi = s*d*vac 481 | rho = psi*psi.dag() 482 | plot_wigner(rho,colorbar=True,cmap='spectral') 483 | \end{lstlisting} 484 | \section{Some Examples and Exercise} 485 | \subsection{Cat State(Demo)} 486 | \begin{frame}{Cat State(Demo)} 487 | Consider a so-called Schrodinger-cat state 488 | \[\left| \psi \right\rangle = \left| \alpha \right\rangle + \left| { - \alpha } \right\rangle \] 489 | where $\left| \alpha \right\rangle $ is a coherent state. (a) Find the normalization constant N. (b) 490 | What is the photon distribution function and wigner function? 491 | \end{frame} 492 | \begin{frame}{Cat State(Demo)} 493 | \begin{center} 494 | \includegraphics[width=0.4\linewidth]{screenshot007} 495 | \end{center} 496 | 497 | 498 | \end{frame} 499 | \subsection{Kerr Effect to Create Cat State(Exercise)} 500 | \begin{frame}{Kerr Effect to create Cat state(exercise)} 501 | The effective Hamiltonian will result in Cat state from an initial coherent state: 502 | \[H = \frac{1}{2}\chi {a^\dag }{a^\dag }aa = \frac{1}{2}\chi n\left( {n - 1} \right)\] 503 | Test and verify this by numerical simulations. 504 | \begin{figure} 505 | \centering 506 | \includegraphics[width=0.3\linewidth]{screenshot002} 507 | \caption{See:\emph{Kirchmair, G., Vlastakis, B., Leghtas, Z., Nigg, S. E., Paik, H., Ginossar, E., … Schoelkopf, R. J. (2012). Observation of quantum state collapse and revival due to the single-photon Kerr effect. Nature, 495(7440), 205–209. https://doi.org/10.1038/nature11902} for details} 508 | \label{fig:screenshot002} 509 | \end{figure} 510 | see: 511 | \end{frame} 512 | 513 | \end{document} -------------------------------------------------------------------------------- /course1/presentation_1.toc: -------------------------------------------------------------------------------- 1 | \beamer@sectionintoc {1}{An Introduction to Qutip}{3}{0}{1} 2 | \beamer@subsectionintoc {1}{1}{Installation and Some Dos and Don'ts}{6}{0}{1} 3 | \beamer@subsectionintoc {1}{2}{Finite Hilbert Space!}{8}{0}{1} 4 | \beamer@subsectionintoc {1}{3}{States and Operators}{9}{0}{1} 5 | \beamer@subsectionintoc {1}{4}{Solving Equation of Quantum System by Qutip}{10}{0}{1} 6 | \beamer@sectionintoc {2}{An Introduction to Quasi-probability Theory}{22}{0}{2} 7 | \beamer@subsectionintoc {2}{1}{Motivation}{22}{0}{2} 8 | \beamer@subsectionintoc {2}{2}{P-representation}{24}{0}{2} 9 | \beamer@subsectionintoc {2}{3}{Q-representation}{27}{0}{2} 10 | \beamer@subsectionintoc {2}{4}{Wigner-function}{28}{0}{2} 11 | \beamer@sectionintoc {3}{Some Examples and Exercise}{30}{0}{3} 12 | \beamer@subsectionintoc {3}{1}{Cat State(Demo)}{30}{0}{3} 13 | \beamer@subsectionintoc {3}{2}{Kerr Effect to Create Cat State(Exercise)}{33}{0}{3} 14 | -------------------------------------------------------------------------------- /course1/presentation_1.vrb: -------------------------------------------------------------------------------- 1 | \frametitle{States and Operators in Qutip} 2 | \begin{listing} 3 | d = 5 4 | #the dimension of the Hilbert space 5 | g = basis(d,0) 6 | e1 = basis(d,1) 7 | e2 = basis(d,2) 8 | print(g) 9 | print(e1) 10 | print(e2) 11 | 12 | a=destroy(d) 13 | a_dag = create(d) 14 | e1 = a_dag*g 15 | e2 = (a_dag * a_dag * g).unit() 16 | print(g) 17 | print(e1) 18 | print(e2) 19 | \end{listing} 20 | -------------------------------------------------------------------------------- /course1/screenshot001.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/caidish/Quantum-Optics-with-Python/198974a1598d5b39de0abfe0c070ee70176ffbce/course1/screenshot001.png -------------------------------------------------------------------------------- /course1/screenshot002.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/caidish/Quantum-Optics-with-Python/198974a1598d5b39de0abfe0c070ee70176ffbce/course1/screenshot002.png -------------------------------------------------------------------------------- /course1/screenshot007.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/caidish/Quantum-Optics-with-Python/198974a1598d5b39de0abfe0c070ee70176ffbce/course1/screenshot007.png -------------------------------------------------------------------------------- /course1/texput.log: -------------------------------------------------------------------------------- 1 | This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/W32TeX) (preloaded format=pdflatex 2017.6.27) 15 AUG 2017 21:31 2 | entering extended mode 3 | \write18 enabled. 4 | %&-line parsing enabled. 5 | **presentation_1F 6 | 7 | ! Emergency stop. 8 | <*> presentation_1F 9 | 10 | *** (job aborted, file error in nonstop mode) 11 | 12 | 13 | Here is how much of TeX's memory you used: 14 | 3 strings out of 492995 15 | 117 string characters out of 6136402 16 | 54074 words of memory out of 5000000 17 | 3658 multiletter control sequences out of 15000+600000 18 | 3640 words of font info for 14 fonts, out of 8000000 for 9000 19 | 1141 hyphenation exceptions out of 8191 20 | 0i,0n,0p,1b,6s stack positions out of 5000i,500n,10000p,200000b,80000s 21 | ! ==> Fatal error occurred, no output PDF file produced! 22 | -------------------------------------------------------------------------------- /course2/DressedState.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/caidish/Quantum-Optics-with-Python/198974a1598d5b39de0abfe0c070ee70176ffbce/course2/DressedState.png -------------------------------------------------------------------------------- /course2/PRA_philip.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/caidish/Quantum-Optics-with-Python/198974a1598d5b39de0abfe0c070ee70176ffbce/course2/PRA_philip.png -------------------------------------------------------------------------------- /course3/fig1.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/caidish/Quantum-Optics-with-Python/198974a1598d5b39de0abfe0c070ee70176ffbce/course3/fig1.png -------------------------------------------------------------------------------- /course3/fig2.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/caidish/Quantum-Optics-with-Python/198974a1598d5b39de0abfe0c070ee70176ffbce/course3/fig2.png -------------------------------------------------------------------------------- /course3/fig3.jpeg: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/caidish/Quantum-Optics-with-Python/198974a1598d5b39de0abfe0c070ee70176ffbce/course3/fig3.jpeg --------------------------------------------------------------------------------