We have designed a tunable nonlinear resonator terminated by a SNAIL (Superconducting Nonlinear Asymmetric Inductive eLement). Such a device possesses a sweet spot in which the externalmagnetic flux allows to suppress the Kerr interaction. We have excited photons near this Kerr-free point and characterized the device using a transmon qubit. The excitation spectrum of the qubit allows to observe photon-number-dependent frequency shifts about nine times larger than the qubit linewidth. Our study demonstrates a compact integrated platform for continuous-variable quantum processing that combines large couplings, considerable relaxation times and excellent control over the photon mode structure in the microwave domain.

Significant progress has been made with multipartite entanglement of discrete qubits, but continuous variable systems may provide a more scalable path toward entanglement of large ensembles.We demonstrate multipartite entanglement in a microwave frequency comb generated by a Josephson parametric amplifier subject to a bichromatic pump. We find 64 correlated modes in the transmission line using a multifrequency digital signal processing platform. Full inseparability is verified in a subset of seven modes. Our method can be expanded to generate even more entangled modes in the near future.

Hosting non-classical states of light in three-dimensional microwave cavities has emerged as a promising paradigm for continuous-variable quantum information processing. Here we experimentallydemonstrate high-fidelity generation of a range of Wigner-negative states useful for quantum computation, such as Schrödinger-cat states, binomial states, Gottesman-Kitaev-Preskill (GKP) states, as well as cubic phase states. The latter states have been long sought after in quantum optics and were never achieved experimentally before. To do so, we use a sequence of interleaved selective number-dependent arbitrary phase (SNAP) gates and displacements. We optimize the state preparation in two steps. First we use a gradient-descent algorithm to optimize the parameters of the SNAP and displacement gates. Then we optimize the envelope of the pulses implementing the SNAP gates. Our results show that this way of creating highly non-classical states in a harmonic oscillator is robust to fluctuations of the system parameters such as the qubit frequency and the dispersive shift.

We provide an explicit construction of a universal gate set for continuous-variable quantum computation with microwave circuits. Such a universal set has been first proposed in quantum-opticalsetups, but its experimental implementation has remained elusive in that domain due to the difficulties in engineering strong nonlinearities. Here, we show that a realistic microwave architecture allows to overcome this difficulty. As an application, we show that this architecture allows to generate a cubic phase state with an experimentally feasible procedure. This work highlights a practical advantage of microwave circuits with respect to optical systems for the purpose of engineering non-Gaussian states, and opens the quest for continuous-variable algorithms based on a few repetitions of elementary gates from the continuous-variable universal set.

Spontaneous parametric downconversion (SPDC) has been a key enabling technology in exploring quantum phenomena and their applications for decades. For instance, traditional SPDC, whichsplits a high energy pump photon into two lower energy photons, is a common way to produce entangled photon pairs. Since the early realizations of SPDC, researchers have thought to generalize it to higher order, e.g., to produce entangled photon triplets. However, directly generating photon triplets through a single SPDC process has remained elusive. Here, using a flux-pumped superconducting parametric cavity, we demonstrate direct three-photon SPDC, with photon triplets generated in a single cavity mode or split between multiple modes. With strong pumping, the states can be quite bright, with flux densities exceeding 60 photon/s/Hz. The observed states are strongly non-Gaussian, which has important implications for potential applications. In the single-mode case, we observe a triangular star-shaped distribution of quadrature voltages, indicative of the long-predicted „star state“. The observed star state shows strong third-order correlations, as expected for a state generated by a cubic Hamiltonian. By pumping at the sum frequency of multiple modes, we observe strong three-body correlations between multiple modes, strikingly, in the absence of second-order correlations. We further analyze the third-order correlations under mode transformations by the symplectic symmetry group, showing that the observed transformation properties serve to „fingerprint“ the specific cubic Hamiltonian that generates them. The observed non-Gaussian, third-order correlations represent an important step forward in quantum optics and may have a strong impact on quantum communication with microwave fields as well as continuous-variable quantum computation.

The Hermiticity axiom of quantum mechanics guarantees that the energy spectrum is real and the time evolution is unitary (probability-preserving). Nevertheless, non-Hermitian but -symmetricHamiltonians may also have real eigenvalues. Systems described by such effective -symmetric Hamiltonians have been realized in experiments using coupled systems with balanced loss (dissipation) and gain (amplification), and their corresponding classical dynamics has been studied. A -symmetric system emerging from a quantum dynamics is highly desirable, in order to understand what -symmetry and the powerful mathematical and physical concepts around it will bring to the next generation of quantum technologies. Here, we address this need by proposing and studying a circuit-QED architecture that consists of two coupled resonators and two qubits (each coupled to one resonator). By means of external driving fields on the qubits, we are able to tune gain and losses in the resonators. Starting with the quantum dynamics of this system, we show the emergence of the -symmetry via the selection of both driving amplitudes and frequencies. We engineer the system such that a non-number conserving dipole-dipole interaction emerges, introducing an instability at large coupling strengths. The -symmetry and its breaking, as well as the predicted instability in this circuit-QED system can be observed in a transmission experiment.

In this Letter, we demonstrate the generation of multimode entangled states of propagating microwaves. The entangled states are generated by parametrically pumping a multimode superconductingcavity. By combining different pump frequencies, applied simultaneously to the device, we can produce different entanglement structures in a programable fashion. The Gaussian output states are fully characterized by measuring the full covariance matrices of the modes. The covariance matrices are absolutely calibrated using an in situ microwave calibration source, a shot noise tunnel junction. Applying a variety of entanglement measures, we demonstrate both full inseparability and genuine tripartite entanglement of the states. Our method is easily extensible to more modes.

We demonstrate that stationary localized solutions (discrete solitons) exist in a one dimensional Bose-Hubbard lattices with gain and loss in the semiclassical regime. Stationary solutions,by defi- nition, are robust and do not demand for state preparation. Losses, unavoidable in experiments, are not a drawback, but a necessary ingredient for these modes to exist. The semiclassical calculations are complemented with their classical limit and dynamics based on a Gutzwiller Ansatz. We argue that circuit QED architectures are ideal platforms for realizing the physics developed here. Finally, within the input-output formalism, we explain how to experimentally access the different phases, including the solitons, of the chain.

Based on a circuit QED qubit-cavity array a source of two-mode entangled
microwave radiation is designed. Our scheme is rooted in the combination of
external driving, collective phenomenaand dissipation. On top of that the
reflexion symmetry is broken via external driving permitting the appearance of
chiral emission. Our findings go beyond the applications and are relevant for
fundamental physics, since we show how to implement quantum lattice models
exhibiting criticality driven by dissipation.