We introduce a numerically exact and yet computationally feasible nonlinear response theory developed for lossy superconducting quantum circuits based on a framework of quantum dissipationin a minimally extended state space. Starting from the Feynman–Vernon path integral formalism for open quantum systems with the system degrees of freedom being the nonlinear elements of the circuit, we eliminate the temporally non-local influence functional of all linear elements by introducing auxiliary harmonic modes with complex-valued frequencies coupled to the non-linear degrees of freedom of the circuit. In our work, we propose a concept of time-averaged observables, inspired by experiment, and provide an explicit formula for producing their quasiprobability distribution. Furthermore, we systematically derive a weak-coupling approximation in the presence of a drive, and demonstrate the applicability of our formalism through a study on the dispersive readout of a superconducting qubit. The developed framework enables a comprehensive fully quantum-mechanical treatment of nonlinear quantum circuits coupled to their environment, without the limitations of typical approaches to weak dissipation, high temperature, and weak drive. Furthermore, we discuss the implications of our findings to the quantum measurement theory.
We observe the emission of bunches of k⩾1 photons by a circuit made of a microwave resonator in series with a voltage-biased tunable Josephson junction. The bunches are emitted atspecific values Vk of the bias voltage, for which each Cooper pair tunneling across the junction creates exactly k photons in the resonator. The latter is a micro-fabricated spiral coil which resonates and leaks photons at 4.4~GHz in a measurement line. Its characteristic impedance of 1.97~kΩ is high enough to reach a strong junction-resonator coupling and a bright emission of the k-photon bunches. We show that a RWA treatment of the system accounts quantitatively for the observed radiation intensity, from k=1 to 6, and over three orders of magnitude when varying the Josephson energy EJ. We also measure the second order correlation function of the radiated microwave to determine its Fano factor Fk, which in the low EJ limit, confirms with Fk=k the emission of k photon bunches. At larger EJ, a more complex behavior is observed in quantitative agreement with numerical simulations.
We show experimentally that a dc-biased Josephson junction in series with two microwave resonators emits entangled beams of microwaves leaking out of the resonators. In the absenceof a stationary phase reference for characterizing the entanglement of the outgoing beams, we measure second-order coherence functions for proving entanglement up to an emission rate of 2.5 billion photon pairs per second. The experimental results are found in quantitative agreement with theory, proving that the low frequency noise of the dc bias is the main limitation for the coherence time of the entangled beams. This agreement allows us to evaluate the entropy of entanglement of the resonators, and to identify the improvements that could bring this device closer to a useful bright source of entangled microwaves for quantum-technological applications.
We show experimentally that a dc biased Josephson junction in series with a high-enough impedance microwave resonator emits antibunched photons. Our resonator is made of a simple micro-fabricatedspiral coil that resonates at 4.4 GHz and reaches a 1.97 kΩ characteristic impedance. The second order correlation function of the power leaking out of the resonator drops down to 0.3 at zero delay, which demonstrates the antibunching of the photons emitted by the circuit at a rate of 6 10^7 photons per second. Results are found in quantitative agreement with our theoretical predictions. This simple scheme could offer an efficient and bright single-photon source in the microwave domain.