In the past 20 years, impressive progress has been made both experimentally and theoretically in superconducting quantum circuits, which provide a platform for manipulating microwave photons. This emerging field of superconducting quantum microwave circuits has been driven by many new interesting phenomena in microwave photonics and quantum information processing. For instance, the interaction between superconducting quantum circuits and single microwave photons can reach the regimes of strong, ultra-strong, and even deep-strong coupling. Many higher-order effects, unusual and less familiar in traditional cavity quantum electrodynamics with natural atoms, have been experimentally observed, e.g., giant Kerr effects, multi-photon processes, and single-atom induced bistability of microwave photons. These developments may lead to improved understanding of the counterintuitive properties of quantum mechanics, and speed up applications ranging from microwave photonics to superconducting quantum information processing. In this article, we review experimental and theoretical progress in microwave photonics with superconducting quantum circuits. We hope that this global review can provide a useful roadmap for this rapidly developing field.Comment: Review article, 170 pages (main text 101 pages), 35 figures, 5 tables, 1362 references; v2: a few more references added, typos corrected; Physics Reports (in press

Ultrastrong coupling between light and matter has, in the past decade, transitioned from theoretical idea to experimental reality. It is a new regime of quantum light-matter interaction, going beyond weak and strong coupling to make the coupling strength comparable to the transition frequencies in the system. The achievement of weak and strong coupling has led to increased control of quantum systems and applications like lasers, quantum sensing, and quantum information processing. Here we review the theory of quantum systems with ultrastrong coupling, which includes entangled ground states with virtual excitations, new avenues for nonlinear optics, and connections to several important physical models. We also review the multitude of experimental setups, including superconducting circuits, organic molecules, semiconductor polaritons, and optomechanics, that now have achieved ultrastrong coupling. We then discuss the many potential applications that these achievements enable in physics and chemistry.

Quantum information can be stored in micromechanical resonators, encoded as quanta of vibration known as phonons. The vibrational motion is then restricted to the stationary eigenmodes of the resonator, which thus serves as local storage for phonons. In contrast, we couple propagating phonons to an artificial atom in the quantum regime, and reproduce findings from quantum optics with sound taking over the role of light. Our results highlight the similarities between phonons and photons, but also point to new opportunities arising from the unique features of quantum mechanical sound. The low propagation speed of phonons should enable new dynamic schemes for processing quantum information, and the short wavelength allows regimes of atomic physics to be explored which cannot be reached in photonic systems.The quantum nature of light is revealed and explored in its interaction with atoms, which can be either elemental or artificial. Artificial atoms typically have transition frequencies in the microwave range and can be designed on a microchip with parameters tailored to fit specific requirements. This makes them well suited as tools to investigate fundamental phenomena of atomic physics and quantum optics. In the form of superconducting qubits, they have seen extensive use in closed spaces (electromagnetic cavities), where they have ample time to interact with confined microwave radiation (1-3). These experiments have recently been extended to quantum optics in open one-dimensional (1D) transmission lines, where the atom interacts with itinerant microwave photons (4-7). We present an acoustic equivalent of such a system, where the quantum properties of sound are explored, rather than those of light.At the intersection between quantum informatics and micromechanics, recent milestones include the coupling between a superconducting qubit and a vibrational mode (8,9), hybrids of mechanical resonators and electrical microwave cavities (10), and the use of mechanics to interface between microwaves and optical photons (11,12). The system we present here is another manifestation of mechanics in the quantum regime, but one that differs fundamentally from the suspended resonators mentioned above. In our case, the phonons are not bound to the eigenmodes of any structure, but consist of Surface Acoustic Waves (SAWs) which propagate freely over long distances, before and after interacting with an atom in their path.In the domain of quantum information, SAWs with high power have been used to transport electrons and holes in semiconductors (13)(14)(15). This stands in contrast with our use of SAWs, where the power is much too low to transport charge carriers, and we instead focus on the quantum nature of the phonons themselves.We do this by coupling an artificial atom directly to the SAWs via piezoelectricity, so that this mode of interaction becomes the dominant one for the atom. This means that we can communicate with the atom bidirectionally through the SAW channel, exciting it acoustically as well as listening to its emission...

We investigate the effective interaction between two microwave fields, mediated by a transmon-type superconducting artificial atom which is strongly coupled to a coplanar transmission line. The interaction between the fields and atom produces an effective cross-Kerr coupling. We demonstrate average cross-Kerr phase shifts of up to 20 degrees per photon with both coherent microwave fields at the single-photon level. Our results provide an important step toward quantum applications with propagating microwave photons.