# How do frequency-tunable microwave resonators work?

In superconducting quantum computing resonators are typically an LC circuit. If L and C are fixed the resonator will have a fixed resonance frequency. However, I heard that in the group of Prof. John Martinis they are using a different type of resonators that their frequency is tunable. The tunability is somehow possible by applying a DC flux bias. Can anyone elaborate on the subject? How does it work? Usages?

There's a superconducting circuit element called Josephson junction, which is roughly a nonlinear inductor. The inductance of a Josephson junction depends on current via the relation $$L(I) = \frac{L_0}{\sqrt{1 - (I/I_c)^2}}$$ where $$L_0$$ is the inductance of the junction with no bias current and $$I_c$$ is the so-called "critical current" which is the maximum current that can pass through the junction before it stops being superconducting. The resonance frequency of an LC resonator is $$\omega_0 = 1 / \sqrt{LC}$$, so if you change $$L$$ you change the resonance frequency. Therefore, if you can include a Josephson junction into the resonator, you have a current-dependent resonance frequency.
• But if I understand correctly adding a junction will result in a non-linear inductance $L_0$ , and because the energy of the inductance term of the circuit is $\Phi^2/2L$ then the circuit will not be an harmonic oscillator anymore. Is it in a regime where the non-linearity is negligible?