Experimentally simulating the dynamics of quantum light and matter at ultrastrong coupling using circuit QED (1) - implementation and matter dynamics -*

The field dipole coupling of quantum light and matter, described by the quantum Rabi model, leads to exotic phenomena when the coupling strength $g$ becomes comparable or larger than the atom and photon frequencies $\omega_{q, r}$. In this ultra-strong coupling regime, excitations are not conserved, leading to collapse-revival dynamics in atom and photon parity and Schr{\"o}dinger-cat-like atom-photon entanglement. We realize a quantum simulation of the Rabi model using a transmon qubit coupled to a resonator. In this first part, we describe our analog-digital approach to implement up to 90 symmetric Trotter steps, combining single-qubit gates with the Jaynes-Cummings interaction naturally present in our circuit QED system. Controlling the phase of microwave pulses defines a rotating frame and enables simulation of arbitrary parameter regimes of the Rabi model. We demonstrate measurements of qubit parity dynamics showing revivals at $g/\omega_r>0.8$ for $\omega_q=0$ and characteristic dynamics for nondegenerate $\omega_q$ from $g/4$ to $g$.