Understanding the non-deterministic behavior of deterministic nonlinear systems has been an implicit dream since Lorenz named it the „butterfly effect“. A prominent exampleis the hysteresis and bistability of the Duffing oscillator, which in the classical description is attributed to the coexistence of two steady states in a double-well potential. However, this interpretation fails in the quantum-mechanical perspective, where a single unique steady state is allowed in the whole parameter space. Here, we measure the non-equilibrium dynamics of a superconducting Duffing oscillator and reconcile the classical and quantum descriptions in a unified picture of quantum metastability. We demonstrate that the two classically regarded steady states are in fact metastable states. They have a remarkably long lifetime in the classical hysteresis regime but must eventually relax into a single unique steady state allowed by quantum mechanics. By engineering the lifetime of the metastable states sufficiently large, we observe a first-order dissipative phase transition, which mimics a sudden change of the mean field in a 11-site Bose-Hubbard lattice. We also reveal the two distinct phases of the transition by quantum state tomography, namely a coherent-state phase and a squeezed-state phase separated by a critical point. Our results reveal a smooth quantum state evolution behind a sudden dissipative phase transition, and they form an essential step towards understanding hysteresis and instability in non-equilibrium systems.

The field of propagating quantum microwaves has started to receive considerable attention in the past few years. Motivated at first by the lack of an efficient microwave-to-opticalplatform that could solve the issue of secure communication between remote superconducting chips, current efforts are starting to reach other areas, from quantum communications to sensing. Here, we attempt at giving a state-of-the-art view of the two, pointing at some of the technical and theoretical challenges we need to address, and while providing some novel ideas and directions for future research. Hence, the goal of this paper is to provide a bigger picture, and — we hope — to inspire new ideas in quantum communications and sensing: from open-air microwave quantum key distribution to direct detection of dark matter, we expect that the recent efforts and results in quantum microwaves will soon attract a wider audience, not only in the academic community, but also in an industrial environment.

We describe a unified quantum approach for analyzing the scattering coefficients of superconducting microwave resonators with a variety of geometries. We also generalize the methodto a chain of resonators in either hanger- or necklace-type, and reveal interesting transport properties similar to a photonic crystal. It is shown that both the quantum and classical analyses provide consistent results, and they together form a solid basis for analyzing the decoherence effect in a general microwave resonator. These results pave the way for designing and applying superconducting microwave resonators in complex circuits, and should stimulate the interest of distinguishing different decoherence mechanisms of a resonator mode beyond free energy relaxation.

We describe a unified classical approach for analyzing the scattering coefficients of superconducting microwave resonators with a variety of geometries. To fill the gap between experimentand theory, we also consider the influences of small circuit asymmetry and the finite length of the feedlines, and describe a procedure to correct them in typical measurement results. We show that, similar to the transmission coefficient of a hanger-type resonator, the reflection coefficient of a necklace- or bridge-type resonator does also contain a reference point which can be used to characterize the electrical properties of a microwave resonator in a single measurement. Our results provide a comprehensive understanding of superconducting microwave resonators from the design concepts to the characterization details.

We propose a tunable coupler consisting of N off-resonant and fixed-frequency qubits that can tune and even amplify the effective interaction between two general circuit components.The tuning range of the interaction is proportional to N, with a minimum value of zero and a maximum that can exceed the physical coupling rates in the system. The effective coupling rate is determined by the collective magnetic quantum number of the qubit ensemble, which takes only discrete values and is free from collective decay and decoherence. Using single-photon pi-pulses, the coupling rate can be switched between arbitrary initial and final values within the dynamic range in a single step without going through intermediate values. A cascade of the couplers for amplifying small interactions or weak signals is also discussed. These results should not only stimulate interest in exploring the collective effects in quantum information processing, but also enable development of applications in tuning and amplifying the interactions in a general cavity-QED system.

We have fabricated and studied a system of two tunable and coupled nonlinear superconducting resonators. The nonlinearity is introduced by galvanically coupled dc-SQUIDs. We simulatethe system response by means of a circuit model, which includes an additional signal path introduced by the electromagnetic environment. Furthermore, we present two methods allowing us to experimentally determine the nonlinearity. First, we fit the measured frequency and flux dependence of the transmission data to simulations based on the equivalent circuit model. Second, we fit the power dependence of the transmission data to a model that is predicted by the nonlinear equation of motion describing the system. Our results show that we are able to tune the nonlinearity of the resonators by almost two orders of magnitude via an external coil and two on-chip antennas. The studied system represents the basic building block for larger systems, allowing for quantum simulations of bosonic many-body systems with a larger number of lattice sites.

Quantum Fourier transform (QFT) is a key ingredient of many quantum algorithms. In typical applications such as phase estimation, a considerable number of ancilla qubits and gates areused to form a Hilbert space large enough for high-precision results. Qubit recycling reduces the number of ancilla qubits to one, but it is only applicable to semi-classical QFT and requires repeated measurements and feedforward within the coherence time of the qubits. In this work, we explore a novel approach based on resonators that forms a high-dimensional Hilbert space for the realization of QFT. By employing the perfect state-transfer method, we map an unknown multi-qubit state to a single resonator, and obtain the QFT state in the second oscillator through cross-Kerr interaction and projective measurement. A quantitive analysis shows that our method allows for high-dimensional and fully-quantum QFT employing the state-of-the-art superconducting quantum circuits. This paves the way for implementing various QFT related quantum algorithms.

Quantum communication is a holy grail to achieve secure communication among a set of partners, since it is provably unbreakable by physical laws. Quantum sensing employs quantum entanglementas an extra resource to determine parameters by either using less resources or attaining a precision unachievable in classical protocols. A paradigmatic example is the quantum radar, which allows one to detect an object without being detected oneself, by making use of the additional asset provided by quantum entanglement to reduce the intensity of the signal. In the optical regime, impressive technological advances have been reached in the last years, such as the first quantum communication between ground and satellites, as well as the first proof-of-principle experiments in quantum sensing. The development of microwave quantum technologies turned out, nonetheless, to be more challenging. Here, we will discuss the challenges regarding the use of microwaves for quantum communication and sensing. Based on this analysis, we propose a roadmap to achieve real-life applications in these fields.

Superconducting 3D microwave cavities offer state-of-the-art coherence times and a well controlled environment for superconducting qubits. In order to realize at the same time fastreadout and long-lived quantum information storage, one can couple the qubit both to a low-quality readout and a high-quality storage cavity. However, such systems are bulky compared to their less coherent 2D counterparts. A more compact and scalable approach is achieved by making use of the multimode structure of a 3D cavity. In our work, we investigate such a device where a transmon qubit is capacitively coupled to two modes of a single 3D cavity. The external coupling is engineered so that the memory mode has an about 100 times larger quality factor than the readout mode. Using an all-microwave second-order protocol, we realize a lifetime enhancement of the stored state over the qubit lifetime by a factor of 6 with a Z-fidelity of 82%. We also find that this enhancement is not limited by fundamental constraints.

The concept of parity describes the inversion symmetry of a system and is of fundamental relevance in the standard model, quantum information processing, and field theory. In quantumelectrodynamics, parity is conserved and selection rules (SRs) appear when matter is probed with electromagnetic radiation. However, typically large field gradients are required to engineer the parity of the light-matter interaction operator for natural atoms. In this work, we instead irradiate a specifically designed superconducting artificial atom with spatially shaped microwave fields to select the interaction parity in situ. In this way, we observe dipole and quadrupole SRs for single state transitions and induce transparency via longitudinal coupling. Furthermore, we engineer an artificial potassium-like atom with adjustable wave function parity originating from an artificial orbital momentum provided by a resonator. Our work advances light-matter interaction to a new level with promising application perspectives in simulations of chemical compounds, quantum state engineering, and relativistic physics.