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Capacitively Coupled Charge Qubits: Difference between revisions
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Capacitively Coupled Charge Qubits (view source)
Revision as of 22:47, 25 October 2014
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===Second Order Detuning Noise===
For <math>\epsilon_2 = 0</math>, we have the following energy levels:
<math>
\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)}}
</math>
<math>
\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)}}
</math>
<math>
\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)}}
</math>
<math>
\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)}}
</math>
which to second order in <math>\epsilon_1</math> are:
<math>
\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
</math>
<math>
\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
</math>
<math>
\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
</math>
<math>
\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
</math>
The second order terms cannot be tuned such that all gaps are invariant to second order noise.
===Tunnel Coupling and Capacitive Coupling Noise===
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