## Time-reversal-symmetry breaking in circuit-QED-based photon lattices

Breaking time-reversal symmetry is a prerequisite for accessing certain interesting many-body states such as fractional quantum Hall states. For polaritons, charge neutrality prevents magnetic fields from providing a direct symmetry-breaking mechanism and, similar to the situation in ultracold atomic gases, an effective magnetic field has to be synthesized. We show that in the circuit-QED architecture, this can be achieved by inserting simple superconducting circuits into the resonator junctions.

## Nonequilibrium delocalization-localization transition of photons in circuit quantum electrodynamics

We show that photons in two tunnel-coupled microwave resonators each containing a single superconducting qubit undergo a sharp nonequilibrium delocalization-localization (self-trapping) transition due to strong photon-qubit coupling. We find that self-trapping of photons in one of the resonators (spatial localization) forces the qubit in the opposite resonator to remain in its initial state (energetic localization). This allows for an easy experimental observation of the transition by local readout of the qubit state.

## Nonlinear response of the vacuum Rabi resonance

On the level of single atoms and photons, the coupling between atoms and the electromagnetic field is typically very weak. By using a cavity to confine the field, the strength of this interaction can be increased by many orders of magnitude, to a point where it dominates over any dissipative process. This strong-coupling regime of cavity quantum electrodynamics has been reached for real atoms in optical cavities, and for artificial atoms in circuit quantum electrodynamics and quantum dot systems.

## Proposal for generating and detecting multi-qubit GHZ states in circuit QED

We propose methods for the preparation and entanglement detection of multi-qubit Greenberger–Horne–Zeilinger (GHZ) states in circuit quantum electrodynamics. Using quantum trajectory simulations appropriate for the situation of a weak continuous measurement, we show that the joint dispersive readout of several qubits can be utilized for the probabilistic production of high-fidelity GHZ states. When employing a nonlinear filter on the recorded homodyne signal, the selected states are found to exhibit values of the Bell–Mermin operator exceeding 2 under realistic conditions.

## Randomized benchmarking and process tomography for gate errors in a solid-state qubit

We present measurements of single-qubit gate errors for a superconducting qubit. Results from quantum process tomography and randomized benchmarking are compared with gate errors obtained from a double π pulse experiment. Randomized benchmarking reveals a minimum average gate error of 1.1±0.3% and a simple exponential dependence of fidelity on the number of gates. It shows that the limits on gate fidelity are primarily imposed by qubit decoherence, in agreement with theory.