Quantum chaos and critical behavior on a chip

The Dicke model describes N qubits (or two-level atoms) homogeneously coupled to a bosonic mode. Here we examine an open-system realization of the Dicke model, which contains critical and chaotic behaviors. In particular, we extend this model to include an additional open transport qubit (coupled to the bosonic mode) for passive and active measurements. We illustrate how the scaling (in the number of qubits N) of the super-radiant phase transition can be observed in both current and current-noise measurements through the transport qubit. Using a master equation, we also investigate how the phase transition is affected by the back action from the transport qubit and losses in the cavity. In addition, we show that the nonintegrable quantum chaotic character of the Dicke model is retained in an open-system environment. We propose how all of these effects could been seen in a circuit-QED system formed from an array of superconducting qubits, or an atom chip, coupled to a quantized resonant cavity (e.g., a microwave transmission line).