Capacitively Coupled Charge Qubits: Difference between revisions
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==Sweet Spots== |
==Sweet Spots== |
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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. |
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. |
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===First Order Detuning Noise=== |
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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 <math>\epsilon_1 = \epsilon_2 = 0</math>. |
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 <math>\epsilon_1 = \epsilon_2 = 0</math>. This has been confirmed analytically in the limit of small <math>g</math>, and no exceptions have been observed numerically. |
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<gallery widths=400px mode="nolines"> |
<gallery widths=400px mode="nolines"> |
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Image:00-10 gap.png |Gap between 00 and 10 states |
Image:00-10 gap.png |Gap between 00 and 10 states |
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</gallery> |
</gallery> |
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===Second Order Detuning Noise=== |
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===Tunnel Coupling and Capacitive Coupling Noise=== |
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Assuming that we sit at the sweet spot <math>\epsilon_1 = \epsilon_2 = 0</math>, the energies are relatively simple, so we can easily see the effect of noise on the other parameters. |
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{| border="1" cellpadding="2" |
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!width="50"|State |
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!width="225"|Energy |
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!width="225"|Effect of <math>\delta\Delta_1</math> |
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!width="225"|Effect of <math>\delta g</math> |
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|<math>|00\rangle</math> || <math>-\sqrt{(\Delta_1+\Delta_2)^2 + g^2}</math> || <math>-\frac{(\Delta_1+\Delta_2)}{\sqrt{(\Delta_1+\Delta_2)^2 + g^2}}\delta\Delta_1</math> || <math>-\frac{g}{\sqrt{(\Delta_1+\Delta_2)^2 + g^2}}\delta g</math> |
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|- |
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|<math>|01\rangle</math> || <math>-\sqrt{(\Delta_1-\Delta_2)^2 + g^2}</math> || <math>-\frac{(\Delta_1-\Delta_2)}{\sqrt{(\Delta_1-\Delta_2)^2 + g^2}}\delta\Delta_1</math> || <math>-\frac{g}{\sqrt{(\Delta_1-\Delta_2)^2 + g^2}}\delta g</math> |
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|- |
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|<math>|10\rangle</math> || <math>\sqrt{(\Delta_1-\Delta_2)^2 + g^2}</math> || <math>\frac{(\Delta_1-\Delta_2)}{\sqrt{(\Delta_1-\Delta_2)^2 + g^2}}\delta\Delta_1</math> || <math>\frac{g}{\sqrt{(\Delta_1-\Delta_2)^2 + g^2}}\delta g</math> |
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|- |
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|<math>|11\rangle</math> || <math>\sqrt{(\Delta_1+\Delta_2)^2 + g^2}</math> || <math>\frac{(\Delta_1+\Delta_2)}{\sqrt{(\Delta_1+\Delta_2)^2 + g^2}}\delta\Delta_1</math> || <math>\frac{g}{\sqrt{(\Delta_1+\Delta_2)^2 + g^2}}\delta g</math> |
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|} |
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==Rotations== |
==Rotations== |
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Revision as of 19:26, 25 October 2014
This project started in the Fall of 2014, focusing on the operation of two or more charge qubits which are capacitively coupled.
General Formulation
For a single charge qubit, the Hamiltonian is
- 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 H_i = \left( \begin{matrix} \epsilon_i/2 & \Delta_i \\ \Delta_i & -\epsilon_i/2 \end{matrix}\right) }
We will refer to 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_i} 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 \Delta_i} as the detuning and tunnel coupling of qubit 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 i} , respectively.
We can further write down the full Hamiltonian explicitly:
- 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 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 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} 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 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} . This has been confirmed analytically in the limit of small 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} , and no exceptions have been observed numerically.
Gap between 01 and 10 states
Gap between 00 and 11 states
Gap between 00 and 01 states
Gap between 00 and 10 states
Second Order Detuning Noise
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} |