The interaction between the electromagnetic field inside a cavity and natural or artificial atoms has played a crucial role in developing our understanding of light-matter interaction,and is central to various quantum technologies. Recently, new regimes beyond the weak and strong light-matter coupling have been explored in several settings. These regimes, where the interaction strength is comparable (ultrastrong) or even higher (deep-strong) than the transition frequencies in the system, can give rise to new physical effects and applications. At the same time, they challenge our understanding of cavity QED. When the interaction strength is so high, fundamental issues like the proper definition of subsystems and of their quantum measurements, the structure of light-matter ground states, or the analysis of time-dependent interactions are subject to ambiguities leading to even qualitatively distinct predictions. The resolution of these ambiguities is also important for understanding and designing next-generation quantum devices that will exploit the ultrastrong coupling regime. Here we discuss and provide solutions to these issues.

Non-equilibrium quantum many-body systems, which are difficult to study via classical computation, have attracted wide interest. Quantum simulation can provide insights into these problems.Here, using a programmable quantum simulator with 16 all-to-all connected superconducting qubits, we investigate the dynamical phase transition in the Lipkin-Meshkov-Glick model with a quenched transverse field. Clear signatures of the dynamical phase transition, merging different concepts of dynamical criticality, are observed by measuring the non-equilibrium order parameter, nonlocal correlations, and the Loschmidt echo. Moreover, near the dynamical critical point, we obtain the optimal spin squeezing of −7.0±0.8 decibels, showing multipartite entanglement useful for measurements with precision five-fold beyond the standard quantum limit. Based on the capability of entangling qubits simultaneously and the accurate single-shot readout of multi-qubit states, this superconducting quantum simulator can be used to study other problems in non-equilibrium quantum many-body systems.

Already in the first edition of this book (Barone and Paterno, „Fundamentals and Physics and Applications of the Josephson Effect“, Wiley 1982), a great number of interestingand important applications for Josephson junctions were discussed. In the decades that have passed since then, several new applications have emerged. This chapter treats one such new class of applications: quantum optics and quantum information processing (QIP) based on superconducting circuits with Josephson junctions. In this chapter, we aim to explain the basics of superconducting quantum circuits with Josephson junctions and demonstrate how these systems open up new prospects, both for QIP and for the study of quantum optics and atomic physics.

Dispersive coupling based on the Rabi model with large detuning is widely used for quantum nondemolition (QND) qubit readout in quantum computation. However, the measurement speed andfidelity are usually significantly limited by the Purcell effects, i.e.: Purcell decay, critical photon numbers, and qubit-dependent Kerr nonlinearity. To avoid these effects, we propose how to realize an ideal QND readout of a gradiometric flux qubit with a tunable gap via its direct dispersive coupling to a boundary-tunable measurement resonator. We show that this novel readout mechanism is free of dipole-field interactions, and that the qubit-QND measurement is not deteriorated by intracavity photons. Both qubit-readout speed and fidelity can avoid the Purcell limitations. Moreover, this direct dispersive coupling can be conveniently turned on and off via an external control flux. We show how to extend this proposal to a multi-qubit architecture for a joint qubit readout.

The ability to entangle quantum systems is crucial for many applications in quantum technology, including quantum communication and quantum computing. Here, we propose a new, simple,and versatile setup for deterministically creating Bell and Greenberger-Horne-Zeilinger (GHZ) states between photons of different frequencies in a two-step protocol. The setup consists of a quantum bit (qubit) coupled ultrastrongly to three photonic resonator modes. The only operations needed in our protocol are to put the qubit in a superposition state, and then tune its frequency in and out of resonance with sums of the resonator-mode frequencies. By choosing which frequency we tune the qubit to, we select which entangled state we create. We show that our protocol can be implemented with high fidelity using feasible experimental parameters in state-of-the-art circuit quantum electrodynamics. One possible application of our setup is as a node distributing entanglement in a quantum network.

We discuss level splitting and sideband transitions induced by a modulated coupling between a superconducting quantum circuit and a nanomechanical resonator. First, we show how to achievean unconventional time-dependent longitudinal coupling between a flux (transmon) qubit and the resonator. Considering a sinusoidal modulation of the coupling strength, we find that a first-order sideband transition can be split into two. Moreover, under the driving of a red-detuned field, we discuss the optical response of the qubit for a resonant probe field. We show that level splitting induced by modulating this longitudinal coupling can enable two-color electromagnetically induced transparency (EIT), in addition to single-color EIT. In contrast to standard predictions of two-color EIT in atomic systems, we apply here only a single drive (control) field. The monochromatic modulation of the coupling strength is equivalent to employing two eigenfrequency-tunable mechanical resonators. Both drive-probe detuning for single-color EIT and the distance between transparent windows for two-color EIT, can be adjusted by tuning the modulation frequency of the coupling.

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 the past 20 years, impressive progress has been made both experimentally and theoretically in superconducting quantum circuits, which provide a platform for manipulating microwavephotons. 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.

Spontaneous parametric down-conversion is a well-known process in quantum nonlinear optics in which a photon incident on a nonlinear crystal spontaneously splits into two photons. Herewe propose an analogous physical process where one excited atom directly transfers its excitation to a pair of spatially-separated atoms with probability approaching one. The interaction is mediated by the exchange of virtual rather than real photons. This nonlinear atomic process is coherent and reversible, so the pair of excited atoms can transfer the excitation back to the first one: the atomic analogue of sum-frequency generation of light. The parameters used to investigate this process correspond to experimentally-demonstrated values in ultrastrong circuit quantum electrodynamics. This approach can be extended to realize other nonlinear inter-atomic processes, such as four-atom mixing, and is an attractive architecture for the realization of quantum devices on a chip. We show that four-qubit mixing can efficiently implement quantum repetition codes and, thus, can be used for error-correction codes.

We describe a hybrid quantum system composed of a micrometer-size carbon nanotube (CNT) longitudinally coupled to a flux qubit. We demonstrate the usefulness of this device for generatinghigh-fidelity nonclassical states of the CNT via dissipative quantum engineering. Sideband cooling of the CNT to its ground state and generating a squeezed ground state, as a mechanical analogue of the optical squeezed vacuum, are two additional examples of the dissipative quantum engineering studied here. Moreover, we show how to generate a long-lived macroscopically-distinct superposition (i.e., a Schr\“odinger cat-like) state. This cat state can be trapped via dark-state methods assuming that the CNT dissipation is negligible compared to the qubit dissipation, and can be verified by detecting the optical response of control fields.