Large dispersive shift in superconducting flux qubit

We study dispersive readout in superconducting flux qubits which are capacitively coupled to a superconducting cavity with $\sim$ 10 GHz resonant frequency $f_r$. To discriminate the state of the qubit precisely, large magnitude of the dispersive shift $\chi$ is desirable. For the two-level system, $\chi$ is given by $g^2/\Delta$ where $g$ is the coupling strength and $\Delta$ is the detuning between the qubit and the cavity. For the multilevel system such as superconducting qubits, however, this formula is modified due to the contributions from higher levels [1]. It has been pointed out that if $f_r$ lies between 01 and 12 transition frequencies of the qubit ($f_{01}$ and $f_{12}$, respectively), $|\chi|$ becomes large because of constructive contributions from different levels [1]. Our flux qubit has $f_{01}=$ 5 GHz and $f_{12}=$ 15 GHz at the optimal flux bias, thus satisfying this condition. Moreover, because of the large anharmonicity ($|f_{12} - f_{01}|$) of the flux qubit, we can easily make $g$ as large as $\sim$ 100 MHz, while staying in the deep dispersive limit. Both of these enhance $|\chi|$ and we have obtained $\chi$ of 80 MHz at the optimal flux bias, which agrees well with the prediction by the energy band calculation. [1] J. Koch et al., PRA 76, 042319 (2007)