We report the implementation of a near-quantum-limited, traveling-wave parametric amplifier that uses three-wave mixing (3WM). To favor amplification by 3WM, we use the superconductingnonlinear asymmetric inductive element (SNAIL) loops, biased with a dc magnetic flux. In addition, we equip the device with dispersion engineering features to create a stop-band at the second harmonic of the pump and suppress the propagation of the higher harmonics that otherwise degrade the amplification. With a chain of 440 SNAILs, the amplifier provides up to 20 dB gain and a 3-dB bandwidth of 1 GHz. The added noise by the amplifier is found to be less than one photon.

We review recent advances in the research on quantum parametric phenomena in superconducting circuits with Josephson junctions. We discuss physical processes in parametrically driventunable cavity and outline theoretical foundations for their description. Amplification and frequency conversion are discussed in detail for degenerate and non-degenerate parametric resonance, including quantum noise squeezing and photon entanglement. Experimental advances in this area played decisive role in successful development of quantum limited parametric amplifiers for superconducting quantum information technology. We also discuss nonlinear down-conversion processes and experiments on self-sustained parametric and subharmonic oscillations.

We report on the experimental observation of period multiplication in parametrically driven tunable superconducting resonators. We modulate the magnetic flux through a superconductingquantum interference device, attached to a quarter-wavelength resonator, with frequencies nω close to multiples, n=2,3,4,5, of the resonator fundamental mode and observe intense output radiation at ω. The output field manifests n-fold degeneracy with respect to the phase, the n states are phase shifted by 2π/n with respect to each other. Our demonstration verifies the theoretical prediction by Guo et al. in PRL 111, 205303 (2013), and paves the way for engineering complex macroscopic quantum cat states with microwave photons.

We investigate nondegenerate parametric oscillations in a multimode superconducting microwave resonator that is terminated by a SQUID. The parametric effect is achieved by modulatingmagnetic flux through the SQUID at a frequency close to the sum of two resonator-mode frequencies. For modulation amplitudes exceeding an instability threshold, self-sustained oscillations are observed in both modes. The amplitudes of these oscillations show good quantitative agreement with a theoretical model. The oscillation phases are found to be correlated and exhibit strong fluctuations which broaden the oscillation spectral linewidths. These linewidths are significantly reduced by applying a weak on-resonance tone, which also suppresses the phase fluctuations. When the weak tone is detuned, we observe synchronization of the oscillation frequency with the frequency of the input. For the detuned input, we also observe an emergence of three idlers in the output. This observation is in agreement with theory indicating four-mode amplification and squeezing of a coherent input.

We have observed period-tripling subharmonic oscillations, in a superconducting coplanar waveguide resonator operated in the quantum regime, kBT≪ℏω. The resonator is terminatedby a tunable inductance that provides a Kerr-type nonlinearity. We detected the output field quadratures at frequencies near the fundamental mode, ω/2π∼5GHz, when the resonator was driven by a current at 3ω with an amplitude exceeding an instability threshold. The output radiation was red-detuned from the fundamental mode. We observed three stable radiative states with equal amplitudes and phase-shifted by 120∘. The downconversion from 3ω to ω is strongly enhanced by resonant excitation of the second mode of the resonator, and the cross-Kerr effect. Our experimental results are in quantitative agreement with a model for the driven dynamics of two coupled modes.

We develop a theory for non-degenerate parametric resonance in a tunable superconducting cavity. We focus on nonlinear effects that are caused by nonlinear Josephson elements connectedto the cavity. We analyze parametric amplification in a strong nonlinear regime at the parametric instability threshold, and calculate maximum gain values. Above the threshold, in the parametric oscillator regime the linear cavity response diverges at the oscillator frequency at all pump strengths. We show that this divergence is related to the continuous degeneracy of the free oscillator state with respect to the phase. Applying on-resonance input lifts the degeneracy and removes the divergence. We also investigate the quantum noise squeezing. It is shown that in the strong amplification regime the noise undergoes four-mode squeezing, and that in this regime the output signal to noise ratio can significantly exceed the input value. We also analyze the intermode frequency conversion and identify parameters at which full conversion is achieved.

We present a new read-out technique for a superconducting qubit dispersively coupled to a Josephson parametric oscillator. We perform degenerate parametric flux pumping of the Josephsoninductance with a pump amplitude surpassing the threshold for parametric instability. We map the qubit states onto two distinct states of classical parametric oscillations: one oscillating state, with on average 180 photons in the resonator, and one with zero oscillation amplitude. We demonstrate single-shot readout performance, with a total state discrimination of 81.5%. When accounting for qubit errors, this gives a corrected fidelity of 98.7%, obviating the need for a following quantum-limited amplifier. An error budget indicates that the readout fidelity is currently limited by spurious switching events between two bistable states of the resonator.

We experimentally study the behavior of a parametrically pumped nonlinear oscillator, which is based on a superconducting lambda /4 resonator, and is terminated by a flux-tunable SQUID.We extract parameters for two devices. In particular, we study the effect of the nonlinearities in the system and compare to theory. The Duffing nonlinearity, \alpha, is determined from the probe-power dependent frequency shift of the oscillator, and the nonlinearity, \beta, related to the parametric flux pumping, is determined from the pump amplitude for the onset of parametric oscillations. Both nonlinearities depend on the parameters of the device and can be tuned in-situ by the applied dc flux. We also suggest how to cancel the effect of \beta by adding a small dc flux and a pump tone at twice the pump frequency.