Deep strong coupling in a circuit QED system (3) - data and analysis -

We have experimentally achieved deep-strong coupling between a superconducting flux qubit and a superconducting LC circuit, where the coupling energy, $\hbar g$, exceeds both the transition energy of the flux qubit, $\hbar\omega_{\rm q}$, and the resonant energy of the LC circuit, $\hbar\omega_{\rm r}$. At the optimal flux bias of the flux qubit, the qubit-resonator system is described by the Rabi model, which is one of the simplest quantum models of atom-cavity systems. The Hamiltonian of the Rabi model can be written as $\mathcal{H}_{\rm Rabi}$ = $-\frac{\hbar}{2}\omega_{\rm q}\sigma_z$ + $\hbar\omega_{\rm r}(a^{\dagger}a + \frac{1}{2})$ + $\hbar g \sigma_x (a + a^{\dagger})$, where $\sigma_{x(z)}$ is a Pauli matrix and $a(a^{\dagger})$ is an annihilation (creation) operator. In this presentation, we will show the spectroscopy data of qubit-resonator systems in the deep-strong-coupling regime. Transition frequencies calculated from $\mathcal{H}_{\rm Rabi}$ fit the measured data well. We have also observed that $\hbar\omega_{\rm q}$ is largely suppressed due to the Lamb shift caused by the deep-strong coupling to the resonator. In this regime, the ground state is predicted to be an entangled state of the qubit's persistent-current states and the resonator's coherent states.