\Part[
    \eng{Quantum Computing}
    \ger{Quantencomputing}
]{qc}

\Section[
    \eng{Qubits}
    \ger{Qubits}
    ]{qubit}
    \begin{formula}{bloch_sphere}
        \desc{Bloch sphere}{}{}
        \desc[german]{Bloch-Sphäre}{}{}
        \eq{
            \ket{\psi} &= \alpha \ket{0} + \beta \ket{1} \\
                       &= \cos \frac{\theta}{2} \e^{i\phi_\alpha} \ket{0} + \sin{\frac{\theta}{2} \e^{i\phi_\beta}} \ket{1} \\
                       &= \e^{i\phi_\alpha} \cos\frac{\theta}{2} \ket{0} + \sin\frac{\theta}{2} \e^{i\phi} \ket{1}
        }
    \end{formula}

\Section[
    \eng{Gates}
    \ger{Gates}
]{gates}
    \begin{formula}{gates}
        \desc{}{}{}
        \desc[german]{}{}{}
        \begin{alignat}{2}
            & \text{\gt{bitflip}:}      & \hat{X} &= \sigma_x = \sigmaxmatrix \\
            & \text{\gt{bitphaseflip}:} & \hat{Y} &= \sigma_y = \sigmaymatrix \\
            & \text{\gt{phaseflip}:}    & \hat{Z} &= \sigma_z = \sigmazmatrix \\
            & \text{\gt{hadamard}:}     & \hat{H} &= \frac{1}{\sqrt{2}}(\hat{X}-\hat{Z}) = \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}
        \end{alignat}
    \end{formula}
    % \begin{itemize}
    %     \item \gt{bitflip}: $\hat{X} = \sigma_x = \sigmaxmatrix$
    %     \item \gt{bitphaseflip}: $\hat{Y} = \sigma_y = \sigmaymatrix$
    %     \item \gt{phaseflip}: $\hat{Z} = \sigma_z = \sigmazmatrix$ \item \gt{hadamard}: $\hat{H} = \frac{1}{\sqrt{2}}(\hat{X}-\hat{Z}) = \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}$
    % \end{itemize}
\Section[
    \eng{Superconducting qubits}
    \ger{Supraleitende qubits}
    ]{scq}

    \Subsection[
        \eng{Building blocks}
        \ger{Bauelemente}
        ]{elements}
        \Subsubsection[
            \eng{Josephson Junction}
            \ger{Josephson-Kontakt}
            ]{josephson_junction}
            \begin{ttext}[desc]
                \eng{When two superconductors are separated by a thin isolator, Cooper pairs can tunnel through the insulator. The Josephson junction is a non-linear inductor.}
                \ger{Wenn zwei Supraleiter durch einen dünnen Isolator getrennt sind, können Cooper-Paare durch den Isolator tunneln. Der Josephson-Kontakt ist ein nicht-linearer Induktor.} 
            \end{ttext}

            \begin{formula}{hamiltonian}
                \desc{Josephson-Hamiltonian}{}{}
                \desc[german]{Josephson-Hamiltonian}{}{}
                \eq{
                    \hat{H}_\text{J} &= - \frac{E_\text{J}}{2} \sum_n [\ket{n}\bra{n+1} + \ket{n+1}\bra{n}]
                }
            \end{formula}

            \begin{formula}{1st_josephson_relation}
                \desc{1. Josephson relation}{Dissipationless supercurrent accros junction at zero applied voltage}{$I_\text{C}=\frac{2e}{\hbar}E_\text{J}$ critical current, $\delta$ phase difference accross junction}
                \desc[german]{1. Josephson Gleichung}{Dissipationsloser Suprastrom durch die Kreuzung ohne angelegte Spannung}{$I_\text{C}=\frac{2e}{\hbar}E_\text{J}$ kritischer Strom, $\delta$ Phasendifferenz zwischen den Supraleitern}
                \eq{\hat{I}\ket{\delta} = I_\text{C}\sin\delta \ket{\delta}}
            \end{formula}

            \begin{formula}{2nd_josephson_relation}
                \desc{2. Josephson relation}{superconducting phase change is proportional to applied voltage}{$\varphi_0=\frac{\hbar}{2e}$ reduced flux quantum}
                \desc[german]{2. Josephson Gleichung}{Supraleitende Phasendifferenz is proportional zur angelegten Spannung}{$\varphi_0=\frac{\hbar}{2e}$ reduziertes Flussquantum}
                \eq{\odv{\hat{\delta}}{t}=\frac{1}{i\hbar}[\hat{H},\hat{\delta}] = -\frac{2eU}{i\hbar}[\hat{n},\hat{\delta}] = \frac{1}{\varphi_0} U}
            \end{formula}

        \Subsubsection[
            \eng{SQUID}
            \ger{SQUID}
            ]{squid}
            \ctikzsubcircuitdef{squidloop}{n, s, nw, ne, se, sw}{
                % start at top
                coordinate(#1-n)
                (#1-n)
                to ++(-1, 0) coordinate(#1-nw)
                to[josephsoncap=$\phi_1$] ++(0,-2) coordinate(#1-sw)
                to ++(1,0) coordinate(#1-s) to ++(1,0) coordinate(#1-se)
                to[josephsoncap=$\phi_2$] ++(0,2) coordinate(#1-ne)
                to ++(-1,0)
                (#1-s)  % leave at bottom
            }
            \begin{formula}{circuit}
                \desc{SQUID}{Superconducting quantum interference device, consists of parallel \hyperref{sec:qc:scq:josephson_junction}{josephson junctions}, can be used to measure extremely weak magnetic fields}{}
                \desc[german]{SQUID}{Superconducting quantum interference device, besteht aus parralelen \hyperref{sec:qc:scq:josephson_junction}{Josephson Junctions} und kann zur Messung extrem schwacher Magnetfelder genutzt werden}{}
                \centering
                \begin{tikzpicture}
                    \draw (0, 0) \squidloop{loop}{};
                \end{tikzpicture}
            \end{formula}
            \begin{formula}{hamiltonian}
                \desc{Hamiltonian}{}{$\hat{\phi}$ phase difference across the junction}
                \desc[german]{Hamiltonian}{}{$\hat{\phi}$ Phasendifferenz an einer Junction}
                \eq{\hat{H} &= -E_{\text{J}1} \cos\hat{\phi}_{1} - E_{\text{J}2} \cos\hat{\phi}_{2}}
            \end{formula}

        \Subsection[
            \eng{Josephson Qubit??}
            \ger{TODO}
            ]{josephson_qubit}
            \begin{tikzpicture}
              \draw (0,0) to[capacitor] (0,2);
              \draw (0,0) to (2,0);
              \draw (0,2) to (2,2);
              \draw (2,0) to[josephson] (2,2);

              \draw[->] (3,1) -- (4,1);
              \draw (5,0) to[josephsoncap=$C_\text{J}$] (5,2);
            \end{tikzpicture}
            \TODO{Include schaltplan}

            \begin{tikzpicture}
              \draw (0,0) to[sV=$V_\text{g}$] (0,2);
              \draw (0,2) to[capacitor=$C_\text{g}$] (2,2);
              \draw (2,2) to (4,2);
              \draw (2,0) to[josephsoncap=$C_\text{J}$] (2,2);
              \draw (4,0) to[capacitor=$C_C$] (4,2);
              \draw (0,0) to (2,0);
              \draw (2,0) to (4,0);
            \end{tikzpicture}

            \begin{formula}{charging_energy}
                \desc{Charging energy / electrostatic energy}{}{}
                \desc[german]{Ladeenergie?}{}{}
                \eq{E_\text{C} = \frac{(2e)^2}{C}}
            \end{formula}

            \begin{formula}{josephson_energy}
                \desc{Josephson energy}{}{}
                \desc[german]{Josephson-Energie?}{}{}
                \eq{E_\text{J} = \frac{I_0 \phi_0}{2\pi}}
            \end{formula}
            \TODO{Was ist I0}

            \begin{formula}{inductive_energy}
                \desc{Inductive energy}{}{}
                \desc[german]{Induktive Energie}{}{}
                \eq{E_\text{L} = \frac{\varphi_0^2}{L}}
            \end{formula}

            \begin{formula}{gate_charge}
                \desc{Gate charge}{or offset charge}{}
                \desc[german]{Gate Ladung}{auch Offset charge}{}
                \eq{n_\text{g}=\frac{C_g V_\text{g}}{2e}}
            \end{formula}

            \begin{formula}{anharmonicity}
                \desc{Anharmonicity}{}{}
                \desc[german]{Anharmonizität}{}{}
                \eq{\alpha \coloneq \omega_{1\leftrightarrow 2} - \omega_{0\leftrightarrow 1}}
            \end{formula}


            \begin{minipage}{0.8\textwidth}
                \begingroup
                    \setlength{\tabcolsep}{0.9em}  % horizontal
                    \renewcommand{\arraystretch}{2}  % vertical
                \begin{tabular}{  p{0.5cm} |p{0.8cm}||p{2.2cm}|p{1.9cm}|p{1.9cm}|p{1.8cm}|}
                    \multicolumn{1}{c}{}& \multicolumn{1}{c}{} &\multicolumn{4}{c}{$E_L/(E_J-E_L)$} \\
                    \cline{3-6}
                    \multicolumn{1}{c}{} & & $0$ & $\ll$ 1 & $\sim 1$ & $\gg 1$\\
                    \hhline{~|=====|}
                    \multirow{4}{*}{$\frac{E_J}{E_C}$} & $\ll 1$ & cooper-pair box  &     &  & \\
                    \cline{2-6}
                    & $\sim 1$ & quantronium &   fluxonium  &    &\\
                    \cline{2-6}
                    & $\gg 1$ &transmon & & &  flux qubit\\
                    \cline{2-6}
                    & $\ggg 1$ & & & phase qubit & \\
                    \cline{2-6}
                \end{tabular}
                \endgroup
            \end{minipage}
            \begin{minipage}{0.2\textwidth}
                \begin{tikzpicture}[scale=2]
                    \draw[-latex,line width=2pt] (0,1)--++(0,1) node[midway,above,sloped] () {charge noise};
                    \draw[-latex,line width=2pt] (0,1)--++(0,1) node[midway,below,sloped] () {sensitivity};
                    \draw[-latex,line width=2pt] (0,0)--++(1,1) node[midway,above,sloped] () {flux noise};
                    \draw[-latex,line width=2pt] (0,0)--++(1,1) node[midway,below,sloped] () {sensitivity};
                    \draw[-latex,line width=2pt] (0,0)--++(1,-1) node[midway,above,sloped] () {critical current};
                    \draw[-latex,line width=2pt] (0,0)--++(1,-1) node[midway,below,sloped] () {noise sensitivity};
                \end{tikzpicture}
            \end{minipage}


        \Subsection[
            \eng{Cooper Pair Box (CPB) qubit}
            \ger{Cooper Paar Box (QPB) Qubit}
            ]{cpb}
            \begin{ttext}
                \eng{
                    = voltage bias junction\\= charge qubit?
                }
                \ger{}
            \end{ttext}
            \begin{formula}{circuit}
                \desc{Cooper Pair Box / Charge qubit}{
                    \begin{itemize}
                        \gooditem large anharmonicity
                        \baditem sensitive to charge noise
                    \end{itemize}
                }{}
                \desc[german]{Cooper Pair Box / Charge Qubit}{
                    \begin{itemize}
                        \gooditem Große Anharmonizität
                        \baditem Sensibel für charge noise
                    \end{itemize}
                }{}
                \centering
                \begin{tikzpicture}
                  \draw (0,0) to[sV=$V_\text{g}$] (0,2);
                  % \draw (0,0) to (2,0);
                  \draw (0,2) to[capacitor=$C_\text{g}$] (2,2);
                  \draw (2,0) to[josephsoncap=$C_\text{J}$] (2,2);
                  \draw (0,0) to (2,0);
                \end{tikzpicture}
            \end{formula}


            \begin{formula}{hamiltonian}
                \desc{Hamiltonian}{}{}
                \desc[german]{Hamiltonian}{}{}
                \eq{\hat{H} &= 4 E_C(\hat{n} - n_\text{g})^2 - E_\text{J} \cos\hat{\phi} \\
                            &=\sum_n \left[4 E_C  (n-n_\text{g})^2 \ket{n}\bra{n} - \frac{E_\text{J}}{2}\ket{n}\bra{n+1}+\ket{n+1}\bra{n}\right] }
            \end{formula}

        \Subsection[
            \eng{Transmon qubit}
            \ger{Transmon Qubit}
            ]{transmon}
            \begin{formula}{circuit}
                \desc{Transmon qubit}{
                    Josephson junction with a shunt \textbf{capacitance}.
                    \begin{itemize}
                        \gooditem charge noise insensitive
                        \baditem small anharmonicity
                    \end{itemize}
                }{}
                \desc[german]{Transmon Qubit}{
                    Josephson-Kontakt mit einem parallelen \textbf{kapzitiven Element}.
                    \begin{itemize}
                        \gooditem Charge noise resilient
                        \baditem Geringe Anharmonizität $\alpha$
                    \end{itemize}
                }{}
                \centering
                \begin{tikzpicture}
                    % \draw (0,0) to[sV=$V_\text{g}$] ++(0,3)
                    % to[capacitor=$C_\text{g}$] ++(2,0)
                    \draw (0,0) to ++(2,0) to ++(0,-0.5) to[josephsoncap=$C_\text{J}$] ++(-0,-2) to ++(0,-0.5) to ++(-2,0)
                     to[capacitor=$C_C$] ++(0,3);
                \end{tikzpicture}
            \end{formula}

            \begin{formula}{hamiltonian}
                \desc{Hamiltonian}{}{}
                \desc[german]{Hamiltonian}{}{}
                \eq{\hat{H} &= 4 E_C\hat{n}^2 - E_\text{J} \cos\hat{\phi}}
            \end{formula}

            \Subsubsection[
                \eng{Tunable Transmon qubit}
                \ger{Tunable Transmon Qubit}
                ]{tunable}
                \begin{formula}{circuit}
                    \desc{Frequency tunable transmon}{By using a \fqSecRef{qc:scq:elements:squid} instead of a \fqSecRef{qc:scq:elements:josephson_junction}, the qubit is frequency tunable through an external field}{}
                    \desc[german]{}{Durch Nutzung eines \fqSecRef{qc:scq:elements:squid} anstatt eines \fqSecRef{qc:scq:elements:josephson_junction}s, ist die Frequenz des Qubits durch ein externes Magnetfeld einstellbar}{}
                    \centering
                    \begin{tikzpicture}
                        % \draw (0,0) to[sV=$V_\text{g}$] ++(0,3)
                        % to[capacitor=$C_\text{g}$] ++(2,0)
                        \draw (0,0) to ++(-2,0)
                        to ++(3,0) to ++(0,-0.5) \squidloop{loop}{SQUID} to ++(0,-0.5) to ++(-3,0)
                        to[capacitor=$C_C$] ++(0,3);
                    \end{tikzpicture}
                \end{formula}

                \begin{formula}{energy}
                    \desc{Josephson energy}{}{$d=(E_\text{J1}-E_\text{J2})/(E_\text{J1}+E_\text{J2})$ asymmetry}
                    \desc[german]{Josephson Energie}{}{$d=(E_\text{J1}-E_\text{J2})/(E_\text{J1}+E_\text{J2})$ Asymmetrie}
                    \eq{E_\text{J,eff}(\Phi_\text{ext}) = (E_\text{J1}+E_\text{J2}) \sqrt{\cos^2\left(\pi\frac{\Phi_\text{ext}}{\Phi_0}\right) + d^2 \sin \left(\pi\frac{\Phi_\text{ext}}{\Phi_0}\right)}}
                \end{formula}
                \begin{formula}{hamiltonian}
                    \desc{Hamiltonian}{}{}
                    \desc[german]{Hamiltonian}{}{}
                    \eq{\hat{H} = 4E_C \hat{n}^2 - \frac{1}{2} E_\text{J,eff}(\Phi_\text{ext}) \sum_{n}\left[\ket{n}\bra{n+1} + \ket{n+1}\bra{n}\right]}
                \end{formula}

                \begin{figure}[h]
                    \centering
                    \includegraphics[width=0.8\textwidth]{img/qubit_transmon.pdf}
                    \caption{Transmon and so TODO}
                    \label{fig:img-qubit_transmon-pdf}
                \end{figure}


        \Subsection[
            \eng{Phase qubit}
            \ger{Phase Qubit}
            ]{phase}
            \begin{formula}{circuit}
                \desc{Phase qubit}{}{}
                \desc[german]{Phase Qubit}{}{}
                \centering
                \begin{tikzpicture}
                    % \draw (0,0) to[sV=$V_\text{g}$] ++(0,3)
                    % to ++(2,0) coordinate(top1)
                    % to ++(2,0) coordinate(top2)
                    % to ++(2,0) coordinate(top3);
                    % \draw (0,0)
                    % to ++(2,0) coordinate(bot1)
                    % to ++(2,0) coordinate(bot2)
                    % to ++(2,0) coordinate(bot3);
                    \draw[color=gray] (0,0) to[capacitor=$C_C$] (0,-2);
                    % \draw (top1) to ++(0,-0.5) to[josephsoncap=$C_\text{J}$] ++(-0,-2) to (bot2);
                    \draw(0,0) to ++(2,0) to[josephsoncap=$C_\text{J}$] ++(0,-2) to ++(-2,0);
                    \draw (2,0) to ++(2,0) to[cute inductor=$E_L$] ++(0,-2) to ++(-2,0);
                    \node at (3,-1.5) {$\Phi_\text{ext}$};
                \end{tikzpicture}
                \\\TODO{Ist beim Fluxonium noch die Voltage source dran?}
            \end{formula}
            \begin{formula}{hamiltonian}
                \desc{Hamiltonian}{}{$\delta = \frac{\phi}{\phi_0}$}
                \desc[german]{Hamiltonian}{}{}
                \eq{\hat{H} = E_C \hat{n}^2 - E_J \cos \hat{\delta} + E_L(\hat{\delta} - \delta_s)^2}
            \end{formula}

            \Eng[TESTT]{This is only a test}
            \Ger[TESTT]{}
            \GT{TESTT}



        \Subsection[
            \eng{Flux qubit}
            \ger{Flux Qubit}
            ]{flux}
            \TODO{TODO}
            \begin{formula}{circuit}
                \desc{Flux qubit / Persistent current qubit}{}{}
                \desc[german]{Flux Qubit / Persistent current qubit}{}{}
                \centering
                \begin{tikzpicture}
                    \draw (0,0) to[josephsoncap=$\alpha E_\text{J}$, scale=0.8, transform shape] (0,-3);
                    \draw (0,0) to ++(3,0) 
                    to[josephsoncap=$E_\text{J}$] ++(0,-1.5)
                    to[josephsoncap=$E_\text{J}$] ++(0,-1.5)
                    to ++(-3,0);
                    \node at (1.5,-1.5) {$\Phi_\text{ext}$};
                \end{tikzpicture}
                    % \begin{tikzpicture}
                    %     \draw (0,0) to[sV=$V_\text{g}$] ++(0,3)
                    %     to ++(2,0) coordinate(top1)
                    %     to ++(2,0) coordinate(top2)
                    %     to ++(2,0) coordinate(top3);
                    %     \draw (0,0)
                    %     to ++(2,0) coordinate(bot1)
                    %     to ++(2,0) coordinate(bot2)
                    %     to ++(2,0) coordinate(bot3);
                    %     \draw[color=gray] (top1) to[capacitor=$C_C$] (bot1);
                    %     % \draw (top1) to ++(0,-0.5) to[josephsoncap=$C_\text{J}$] ++(-0,-2) to (bot2);
                    %     \draw[scale=0.8, transform shape] (top2) to[josephsoncap=$\alpha E_\text{J}$] (bot2);
                    %     \draw (top3) 
                    %     to[josephsoncap=$E_\text{J}$] ++(0,-1.5)
                    %     to[josephsoncap=$E_\text{J}$] (bot3);
                    %     \node at (5,0.5) {$\Phi_\text{ext}$};
                    % \end{tikzpicture}
            \end{formula}


        \Subsection[
            \eng{Fluxonium qubit}
            \ger{Fluxonium Qubit}
            ]{fluxonium}
            \begin{formula}{circuit}
                \desc{Fluxonium qubit}{
                    Josephson junction with a shunt \textbf{inductance}. Instead of having to tunnel, cooper pairs can move to the island via the inductance.
                    The inductance consists of many parallel Josephson Junctions to avoid parasitic capacitances.
                }{}
                \desc[german]{Fluxonium Qubit}{
                    Josephson-Kontakt mit einem parallelen \textbf{induktiven Element}.
                    Anstatt zu tunneln, können die Cooper-Paare über das induktive Element auf die Insel gelangen.
                    Das induktive Element besteht aus sehr vielen parallelen Josephson-Kontakten um parisitische Kapazitäten zu vermeiden.
                }{}
                \centering
                \begin{tikzpicture}
                    % \draw (0,0) to[sV=$V_\text{g}$] ++(0,3)
                    % to ++(2,0) coordinate(top1);
                    \draw[color=gray] (0,0) to ++(-2,0) to[capacitor=$C_C$] ++(0,-3) to ++(2,0);
                    \draw (0,0) to[josephsoncap=$C_\text{J}$] ++(-0,-3);
                    \draw (0,0) to ++(2,0) to[cute inductor=$E_L$] ++(0,-3) to ++(-2,0);
                    \node at (1,-0.5) {$\Phi_\text{ext}$};
                \end{tikzpicture}
                \\\TODO{Ist beim Fluxonium noch die Voltage source dran?}
            \end{formula}

            \def\temp{$E_\text{C} = \frac{(2e)^2}{2C}, E_\text{L} = \frac{\varphi_0^2}{2L}, \delta_\text{s} = \frac{\varphi_\text{s}}{\varphi_0}$}
            \begin{formula}{hamiltonian}
                \desc{Hamiltonian}{}{\temp}
                \desc[german]{Hamiltonian}{}{\temp}
                \eq{\hat{H} = 4E_\text{C} \hat{n}^2 - E_\text{J} \cos \hat{\delta} + E_\text{L}(\hat{\delta} - \delta_\text{s})^2}
            \end{formula}

            \begin{figure}[h]
                \centering
                \includegraphics[width=\textwidth]{img/qubit_flux_onium.pdf}
                \caption{img/}
                \label{fig:img-}
            \end{figure}


    \Section[
        \eng{Two-level system}
        \ger{Zwei-Niveau System}
        ]{stuff}

        \begin{formula}{resonance_frequency}
            \desc{Resonance frequency}{}{}
            \desc[german]{Ressonanzfrequenz}{}{}
            \eq{\omega_{21} = \frac{E_2 - E_1}{\hbar}}
        \end{formula}
        \TODO{sollte das nicht 10 sein?}

        \begin{formula}{rabi_oscillation}
            \desc{Rabi oscillations}{}{$\omega_{21}$ resonance frequency of the energy transition, $\Omega$ Rabi frequency}
            \desc[german]{Rabi-Oszillationen}{}{$\omega_{21}$ Resonanzfrequenz des Energieübergangs, $\Omega$ Rabi-Frequenz}
            \eq{\Omega_ \text{\TODO{TODO}}}
        \end{formula}

        \Subsection[
            \eng{Ramsey interferometry}
            \ger{Ramsey Interferometrie}
            ]{ramsey}

            \begin{ttext}
                \eng{$\ket{0} \xrightarrow{\frac{\pi}{2}\,\text{pulse}}$ precession in $xy$ plane for time $\tau$ $\xrightarrow{\frac{\pi}{2}\,\text{pulse}}$ measurement}
                \ger{q}
            \end{ttext}

    \Section[
        \eng{Noise and decoherence}
        \ger{Noise und Dekohärenz}
        ]{noise}
        \begin{formula}{long}
            \desc{Longitudinal relaxation rate}{$\Gamma_{1\downarrow}$: $\ket{1}\rightarrow \ket{0}$ \\ $\Gamma_{1\uparrow}$: $\ket{0}\rightarrow \ket{1}$}{}
            \desc[german]{Longitudinale Relaxationsrate}{$\Gamma_{1\downarrow}$: $\ket{1}\rightarrow \ket{0}$ \\ $\Gamma_{1\uparrow}$: $\ket{0}\rightarrow \ket{1}$}{}
            \eq{\Gamma_1 = \frac{1}{T_1} = \Gamma_{1\uparrow} + \Gamma_{1\downarrow}}
        \end{formula}

        \begin{ttext}[long]
            \eng{$\Gamma_{1\uparrow}$ is supressed at low temperatures because of detailed balance}
            \ger{$\Gamma_{1\uparrow}$ ist bei niedrigen Temperaturen unterdrückt wegen detailed balance}
        \end{ttext}

        \begin{formula}{dephasing}
            \desc{Pure dephasing rate}{}{}
            \desc[german]{Reine Phasenverschiebung}{}{}
            \eq{\Gamma_\phi}
        \end{formula}

        \begin{formula}{trans}
            \desc{Transversal relaxation rate}{}{}
            \desc[german]{Transversale Relaxationsrate}{}{}
            \eq{\Gamma_2 = \frac{1}{T_2} = \frac{\Gamma_1}{2} + \Gamma_\phi}
        \end{formula}

        \begin{formula}{bloch_redfield}
            \desc{Bloch-Redfield density matrix}{2-level System weakly coupled to noise sources with short correlation time}{}
            \desc[german]{Bloch-Redfield Dichtematrix}{2-Niveau System schwach an Noise Quellen mit kurzer Korrelationszeit gekoppelt}{}
            \eq{\rho_\text{BR} = \begin{pmatrix} 1+(\abs{\alpha}^2-1)\e^{-\Gamma_1 t} & \alpha \beta^* \e^{-\Gamma_2 t} \\ 
            \alpha^*\beta \e^{-\Gamma_2 t} & \abs{\beta}^2 \e^{-\Gamma_1 t} \end{pmatrix} }
        \end{formula}
        


        
    



