Capacitively Coupled Charge Qubits

This project started in the Fall of 2014, focusing on the operation of two or more charge qubits which are capacitively coupled.

General Formulation

Energy levels of the 2 qubit system as a function of both detunings.

For a single charge qubit, the Hamiltonian is

$H_{i}=\left({\begin{matrix}\epsilon _{i}/2&\Delta _{i}\\\Delta _{i}&-\epsilon _{i}/2\end{matrix}}\right)$

We will refer to $\epsilon _{i}$ and $\Delta _{i}$ as the detuning and tunnel coupling of qubit $i$, respectively.

We can further write down the full Hamiltonian explicitly:

$H=\left({\begin{matrix}{\frac {1}{2}}(\epsilon _{1}+\epsilon _{2})+g&\Delta _{2}&\Delta _{1}&0\\\Delta _{2}&{\frac {1}{2}}(\epsilon _{1}-\epsilon _{2})-g&0&\Delta _{1}\\\Delta _{1}&0&{\frac {1}{2}}(-\epsilon _{1}+\epsilon _{2})-g&\Delta _{2}\\0&\Delta _{1}&\Delta _{2}&{\frac {1}{2}}(-\epsilon _{1}-\epsilon _{2})+g\end{matrix}}\right)$

where $g$ is the capacitive coupling between the qubits.

Sweet Spots

A major issue with charge qubits is that they are very susceptible to charge noise, which occurs when charge fluctuations outside the system induce undesired shifts in the parameters of the Hamiltonian. The goal of a sweet spot is to find a point in the parameter space where the energy levels are as invariant to the shifts as possible.

First Order Detuning Noise

The most dominant noise source is due to the shifts in the detuning. It is believed that the only point at which the first order dependence on the detunings disappears is at $\epsilon _{1}=\epsilon _{2}=0$. This has been confirmed analytically in the limit of small $g$, and no exceptions have been observed numerically.

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  Gap between 01 and 10 states

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  Gap between 00 and 11 states

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  Gap between 00 and 01 states

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  Gap between 00 and 10 states

Second Order Detuning Noise

For $\epsilon _{2}=0$, we have the following energy levels:

$\lambda _{1}=-{\sqrt {\Delta _{1}^{2}+\Delta _{2}^{2}+{\frac {\epsilon _{1}^{2}}{4}}+g^{2}+{\sqrt {4\Delta _{1}^{2}\Delta _{2}^{2}+\epsilon _{1}^{2}(\Delta _{2}+g^{2})}}}}$

$\lambda _{2}=-{\sqrt {\Delta _{1}^{2}+\Delta _{2}^{2}+{\frac {\epsilon _{1}^{2}}{4}}+g^{2}-{\sqrt {4\Delta _{1}^{2}\Delta _{2}^{2}+\epsilon _{1}^{2}(\Delta _{2}+g^{2})}}}}$

$\lambda _{3}={\sqrt {\Delta _{1}^{2}+\Delta _{2}^{2}+{\frac {\epsilon _{1}^{2}}{4}}+g^{2}-{\sqrt {4\Delta _{1}^{2}\Delta _{2}^{2}+\epsilon _{1}^{2}(\Delta _{2}+g^{2})}}}}$

$\lambda _{4}={\sqrt {\Delta _{1}^{2}+\Delta _{2}^{2}+{\frac {\epsilon _{1}^{2}}{4}}+g^{2}+{\sqrt {4\Delta _{1}^{2}\Delta _{2}^{2}+\epsilon _{1}^{2}(\Delta _{2}+g^{2})}}}}$

which to second order in $\epsilon _{1}$ are:

$\lambda _{1}\approx \lambda _{1}^{0}+{\frac {\Delta _{2}(\Delta _{1}+\Delta _{2})+g^{2}}{8\Delta _{1}\Delta _{2}\lambda _{1}^{0}}}\epsilon _{1}^{2}$

$\lambda _{2}\approx \lambda _{2}^{0}+{\frac {\Delta _{2}(\Delta _{1}+\Delta _{2})-g^{2}}{8\Delta _{1}\Delta _{2}\lambda _{2}^{0}}}\epsilon _{1}^{2}$

$\lambda _{3}\approx \lambda _{3}^{0}+{\frac {\Delta _{2}(\Delta _{1}+\Delta _{2})-g^{2}}{8\Delta _{1}\Delta _{2}\lambda _{3}^{0}}}\epsilon _{1}^{2}$

$\lambda _{4}\approx \lambda _{4}^{0}+{\frac {\Delta _{2}(\Delta _{1}+\Delta _{2})+g^{2}}{8\Delta _{1}\Delta _{2}\lambda _{4}^{0}}}\epsilon _{1}^{2}$

The second order terms cannot be tuned such that all gaps are invariant to second order noise. However, $g$ can be tuned such that some of the transitions become invariant to some of the second order detuning shifts. To make the transition between Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle |01\rangle} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle |10\rangle} flat with respect to second order fluctuations in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \epsilon_1} , we can set

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle g^2 = \Delta_2(\Delta_1+\Delta_2) }

Similarly, to make the transitions between Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle |00\rangle} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle |01\rangle} and between Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle |10\rangle} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle |11\rangle} flat to second order in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \epsilon_1} , we can set

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle g^2 = (\Delta_1+\Delta_2)\sqrt{(\Delta_1+\Delta_2)(\Delta_1^3-\Delta_1^2\Delta_2+3\Delta_1\Delta_2^2+\Delta_2^3)}-\Delta_1^3-\Delta_1^2\Delta_2-\Delta_1\Delta_2^2-\Delta_2^3 }

Tunnel Coupling and Capacitive Coupling Noise

Assuming that we sit at the sweet spot Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \epsilon_1 = \epsilon_2 = 0} , the energies are relatively simple, so we can easily see the effect of noise on the other parameters.

State	Energy	Effect of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \delta\Delta_1}	Effect of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \delta g}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle |00\rangle}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle -\sqrt{(\Delta_1+\Delta_2)^2 + g^2}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle -\frac{(\Delta_1+\Delta_2)}{\sqrt{(\Delta_1+\Delta_2)^2 + g^2}}\delta\Delta_1}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle -\frac{g}{\sqrt{(\Delta_1+\Delta_2)^2 + g^2}}\delta g}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle |01\rangle}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle -\sqrt{(\Delta_1-\Delta_2)^2 + g^2}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle -\frac{(\Delta_1-\Delta_2)}{\sqrt{(\Delta_1-\Delta_2)^2 + g^2}}\delta\Delta_1}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle -\frac{g}{\sqrt{(\Delta_1-\Delta_2)^2 + g^2}}\delta g}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle |10\rangle}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \sqrt{(\Delta_1-\Delta_2)^2 + g^2}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{(\Delta_1-\Delta_2)}{\sqrt{(\Delta_1-\Delta_2)^2 + g^2}}\delta\Delta_1}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{g}{\sqrt{(\Delta_1-\Delta_2)^2 + g^2}}\delta g}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle |11\rangle}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \sqrt{(\Delta_1+\Delta_2)^2 + g^2}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{(\Delta_1+\Delta_2)}{\sqrt{(\Delta_1+\Delta_2)^2 + g^2}}\delta\Delta_1}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{g}{\sqrt{(\Delta_1+\Delta_2)^2 + g^2}}\delta g}

Rotations

SECTION IN PROGRESS

After bringing the system adiabatically to the sweet spot Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \epsilon_1=\epsilon_2=0} , we can apply an AC pulse to some of our parameters to induce a rotation within the system.

AC Pulse	Resulting Matrix (Lab Basis)	Resulting Matrix (Energy Basis)
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \epsilon_1}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{1}{2}\left( \begin{matrix} B & 0 & 0 & 0 \\ 0 & B & 0 & 0 \\ 0 & 0 & -B & 0 \\ 0 & 0 & 0 & -B \end{matrix}\right)\cos{(\omega_{AC}t)}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{1}{2}\left( \begin{matrix} 0 & A_1 & A_2 & 0 \\ A_1 & 0 & 0 & A_3 \\ A_2 & 0 & 0 & A_4 \\ 0 & A_3 & A_4 & 0 \end{matrix}\right)\cos{(\omega_{AC}t)}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \epsilon_2}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{1}{2}\left( \begin{matrix} B & 0 & 0 & 0 \\ 0 & -B & 0 & 0 \\ 0 & 0 & B & 0 \\ 0 & 0 & 0 & -B \end{matrix}\right)\cos{(\omega_{AC}t)}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{1}{2}\left( \begin{matrix} 0 & 0 & B & 0 \\ 0 & 0 & 0 &B \\ B & 0 & 0 & 0 \\ 0 & B & 0 & 0 \end{matrix}\right)\cos{(\omega_{AC}t)}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Delta_1}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{1}{2}\left( \begin{matrix} 0 & 0 & B & 0 \\ 0 & 0 & 0 &B \\ B & 0 & 0 & 0 \\ 0 & B & 0 & 0 \end{matrix}\right)\cos{(\omega_{AC}t)}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{1}{2}\left( \begin{matrix} -B\frac{\Delta_1+\Delta_2}{\lambda_1} & 0 & 0 & -B\frac{g}{\lambda_1} \\ 0 & -B\frac{\Delta_1-\Delta_2}{\lambda_2} & -B\frac{g}{\lambda_2} & 0 \\ 0& -B\frac{g}{\lambda_2} & B\frac{\Delta_1-\Delta_2}{\lambda_2} & 0\\ -B\frac{g}{\lambda_1}& 0 & 0 &B\frac{\Delta_1+\Delta_2}{\lambda_1} \end{matrix}\right)\cos{(\omega_{AC}t)}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Delta_2}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{1}{2}\left( \begin{matrix} 0 & B & 0 & 0 \\ B & 0 & 0 & 0 \\ 0 & 0 & 0 & B \\ 0 & 0 & B & 0 \end{matrix}\right)\cos{(\omega_{AC}t)}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{1}{2}\left( \begin{matrix} 0 & 0 & B & 0 \\ 0 & 0 & 0 &B \\ B & 0 & 0 & 0 \\ 0 & B & 0 & 0 \end{matrix}\right)\cos{(\omega_{AC}t)}}