What are the reasons/considerations for setting the transmon qubit frequency between around 3-6 GHz? What undesirable consequences will result if the frequency is out of the range?
1 Answer
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The transmon frequency $f_t$ is linked both to the charging energy $E_C$ and the inductive energy $E_L$ according to the formula:
$$ f_t = \sqrt{8E_JE_C}. $$
where $E_C = e^2 / 2C$ and $E_L = (\phi_0 / 2 \pi)^2 / L$. $E_C$ is typically fixed to $200MHz$ so fixing the frequency fixes $E_J$.
For small transmon frequency, you need a big inductance meaning a small junction which can be challenging for the fab. Besides, even lower frequency would mean more thermal population.
For big transmon frequency, you have small inductance, no fab problem here. However, you start to reach the high frequency range of microwave pulse.
