490 lines
23 KiB
TeX
490 lines
23 KiB
TeX
\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}
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|