In cavity quantum electrodynamics (QED) 1-3 , light-matter interaction is probed at its most fundamental level, where individual atoms are coupled to single photons stored in three-dimensional cavities. This unique possibility to experimentally explore the foundations of quantum physics has greatly evolved with the advent of circuit QED 4-13 , where on-chip superconducting qubits and oscillators play the roles of two-level atoms and cavities, respectively. In the strong coupling limit, atom and cavity can exchange a photon frequently before coherence is lost. This important regime has been reached both in cavity and circuit QED, but the design flexibility and engineering potential of the latter allowed for increasing the ratio between the atom-cavity coupling rate g and the cavity transition frequency ωr above the percent level 8,14,15 . While these experiments are well described by the renowned Jaynes-Cummings model 16 , novel physics is expected when g reaches a considerable fraction of ωr. Promising steps towards this so-called ultrastrong coupling regime 17,18 have recently been taken in semiconductor structures 19,20 . Here, we report on the first experimental realization of a superconducting circuit QED system in the ultrastrong coupling limit and present direct evidence for the breakdown of the Jaynes-Cummings model. We reach remarkable normalized coupling rates g/ωr of up to 12 % by enhancing the inductive coupling of a flux qubit 21 to a transmission line resonator using the nonlinear inductance of a Josephson junction 22 . Our circuit extends the toolbox of quantum optics on a chip towards exciting explorations of the ultrastrong interaction between light and matter.In the strong coupling regime, the atom-cavity coupling rate g exceeds the dissipation rates κ and γ of both, cavity and atom, giving rise to coherent light-matter oscillations and superposition states. This regime was reached in various types of systems operating at different energy scales [1][2][3][23][24][25] . At microwave frequencies, strong coupling is feasible due to the enormous engineerability of superconducting circuit QED systems 4,5 . Here, small cavity mode volumes and large dipole moments of artificial atoms 26 enable coupling rates g of about 15 1 % of the cavity mode frequency ω r . Nevertheless, as in cavity QED, the quantum dynamics of these systems follows the Jaynes-Cummings model, which describes the coherent exchange of a single excitation between the atom and the cavity mode. Although the Hamiltonian of a realistic atom-cavity system contains so-called counterrotating terms allowing the simultaneous creation ior annihilation of an excitation in both atom and cavity mode, these terms can be safely neglected for small normalized coupling rates g/ω r . However, when g becomes a significant fraction of ω r , the counterrotating terms are expected to manifest, giving rise to exciting effects in QED.The ultrastrong coupling regime is difficult to reach in traditional quantum optics, but was recently realized in a solid-stat...

† authors with equal contribution to this work Superconducting qubits 1,2 behave as artificial two-level atoms and are used to investigate fundamental quantum phenomena. In this context, the study of multi-photon excitations 3,4,5,6,7 occupies a central role. Moreover, coupling superconducting qubits to on-chip microwave resonators has given rise to the field of circuit QED 8,9,10,11,12,13,14,15 . In contrast to quantum-optical cavity QED 16,17,18,19 , circuit QED offers the tunability inherent to solid-state circuits. In this work, we report on the observation of key signatures of a two-photon driven Jaynes-Cummings model, which unveils the upconversion dynamics of a superconducting flux qubit 20 coupled to an on-chip resonator. Our experiment and theoretical analysis show clear evidence for the coexistence of one-and two-photon driven level anticrossings of the qubit-resonator system. This results from the symmetry breaking of the system Hamiltonian, when parity becomes a not well-defined property 21 . Our study provides deep insight into the interplay of multiphoton processes and symmetries in a qubit-resonator system.In cavity QED, a two-level atom interacts with the quantized modes of an optical or microwave cavity. The information on the coupled system is encoded both in the atom and in the cavity states. The latter can be accessed spectroscopically by measuring the transmission properties of the cavity 16 , whereas the former can be read out by suitable detectors 18,19 . In circuit QED, the solid-state counterpart of cavity QED, the first category of experiments was implemented by measuring the microwave radiation emitted by a resonator (acting as a cavity) strongly coupled to a charge qubit 8 . In a dual experiment, the state of a flux qubit was detected with a DC superconducting quantum interference device (SQUID) and vacuum Rabi oscillations were observed 10 . More recently, both approaches have been exploited to extend the toolbox of quantum optics on a chip 11,12,13,14,15,22 . Whereas all these experiments employ one-photon driving of the coupled qubit-resonator system, multi-photon studies are available only for sideband transitions 15 or bare qubits 3,4,5,6,7 . The experiments discussed in this work explore, to our knowledge for the first time, the physics of the two-photon driven Jaynes-Cummings dynamics in circuit QED. In this context, we show that the dispersive interaction between the qubit and the two-photon driving enables real level transitions. The nature of our experiment can be understood as an upconversion mechanism, which transforms the two-photon coherent driving into single photons of the Jaynes-Cummings dynamics. This process requires energy conservation and a not well-defined parity 21 of the interaction Hamiltonian due to the symmetry breaking of the qubit potential. Our experimental findings reveal that such symmetry breaking can be obtained either by choosing a suitable qubit operation point or by the presence of additional spurious fluctuators 23 .The main elements of our setup, ...

In order to gain a better understanding of the origin of decoherence in superconducting flux qubits, we have measured the magnetic field dependence of the characteristic energy relaxation time (T(1)) and echo phase relaxation time (T(2)(echo)) near the optimal operating point of a flux qubit. We have measured T(2)(echo) by means of the phase cycling method. At the optimal point, we found the relation T(2)(echo) approximately 2T(1). This means that the echo decay time is limited by the energy relaxation (T(1) process). Moving away from the optimal point, we observe a linear increase of the phase relaxation rate (1/T(2)(echo)) with the applied external magnetic flux. This behavior can be well explained by the influence of magnetic flux noise with a 1/f spectrum on the qubit.