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.

Superconducting flux qubits are a promising candidate for realizing quantum information processing and quantum simulations. Such devices behave like artificial atoms, with the advantagethat one can easily tune the „atoms“ internal properties. Here, by harnessing this flexibility, we propose a technique to minimize the inhomogeneous broadening of a large ensemble of flux qubits by tuning only the external flux. In addition, as an example of many-body physics in such an ensemble, we show how to observe superradiance, and its quadratic scaling with ensemble size, using a tailored microwave control pulse that takes advantage of the inhomogeneous broadening itself to excite only a sub-ensemble of the qubits. Our scheme opens up an approach to using superconducting circuits to explore the properties of quantum many-body systems.

Single-photon devices at microwave frequencies are important for applications in quantum information processing and communication in the microwave regime. In this work, we describea proposal of a multi-output single-photon device. We consider two superconducting resonators coupled to a gap-tunable qubit via both its longitudinal and transverse degrees of freedom. Thus, this qubit-resonator coupling differs from the coupling in standard circuit quantum-electrodynamic systems described by the Jaynes-Cummings model. We demonstrate that an effective quadratic coupling between one of the normal modes and the qubit can be induced, and this induced second-order nonlinearity is much larger than that for conventional Kerr-type systems exhibiting photon blockade. Assuming that a coupled normal mode is resonantly driven, we observe that the output fields from the resonators exhibit strong sub-Poissonian photon-number statistics and photon antibunching. Contrary to previous studies on resonant photon blockade, the first-excited state of our device is a pure single-photon Fock state rather than a polariton state, i.e., a highly hybridized qubit-photon state. In addition, it is found that the optical state truncation caused by the strong qubit-induced nonlinearity can lead to an entanglement between the two resonators, even in their steady state under the Markov approximation.

Electromagnetically induced transparency (EIT) has been realized in atomic systems, but fulfilling the EIT conditions for artificial atoms made from superconducting circuits is a moredifficult task. Here we report an experimental observation of the EIT in a tunable three-dimensional transmon by probing the cavity transmission. To fulfill the EIT conditions, we tune the transmon to adjust its damping rates by utilizing the effect of the cavity on the transmon states. From the experimental observations, we clearly identify the EIT and Autler-Townes splitting (ATS) regimes as well as the transition regime in between. Also, the experimental data demonstrate that the threshold ΩAIC determined by the Akaike information criterion can describe the EIT-ATS transition better than the threshold ΩEIT given by the EIT theory.

Phonon blockade is a purely quantum phenomenon, analogous to Coulomb and photon blockades, in which a single phonon in an anharmonic mechanical resonator can impede the excitation ofa second phonon. We propose an experimental method to realize phonon blockade in a driven harmonic nanomechanical resonator coupled to a qubit, where the coupling is proportional to the second-order nonlinear susceptibility χ(2). This is in contrast to the standard realizations of phonon and photon blockade effects in Kerr-type χ(3) nonlinear systems. The nonlinear coupling strength can be adjusted conveniently by changing the coherent drive field.As an example, we apply this model to predict and describe phonon blockade in a nanomechanical resonator coupled to a Cooper-pair box (i.e., a charge qubit) with a linear longitudinal coupling. By obtaining the solutions of the steady state for this composite system, we give the conditions forobserving strong antibunching and sub-Poissonian phonon-number statistics in this induced second-order nonlinear system. Besides using the qubit to produce phonon blockade states, the qubit itself can also be employed to detect blockade effects by measuring its states. Numerical simulations indicate that the robustness of the phonon blockade, and the sensitivity of detecting it, will benefit from this strong induced nonlinear coupling.

Ultrastrong coupling in circuit quantum electrodynamics systems not only provides a platform to study the quantum Rabi model, but it can also facilitate the implementation of quantumlogic operations via high-lying resonator states. In this regime, quantum manifolds with different excitation numbers are intrinsically connected via the counter-rotating interactions, which can result in multi-photon processes. Recent experiments have demonstrated ultrastrong coupling in superconducting qubits electromagnetically coupled to superconducting resonators. Here we report the experimental observation of multiphoton sideband transitions of a superconducting flux qubit coupled to a coplanar waveguide resonator in the ultrastrong coupling regime. With a coupling strength reaching about 10% of the fundamental frequency of the resonator, we obtain clear signatures of higher-order red-sideband transitions and the first-order blue-sideband transition in a transmission spectroscopic measurement. This study advances the understanding of driven ultrastrongly-coupled systems.

Electromagnetically induced transparency (EIT) has been extensively studied in various systems. However, it is not easy to observe in superconducting quantum circuits (SQCs), becausethe Rabi frequency of the strong controlling field corresponding to EIT is limited by the decay rates of the SQCs. Here, we show that EIT can be achieved by engineering decay rates in a superconducting circuit QED system through a classical driving field on the qubit. Without such a driving field, the superconducting qubit and the cavity field are approximately decoupled in the large detuning regime, and thus the eigenstates of the system are approximately product states of the cavity field and qubit states. However, the driving field can strongly mix these product states and so-called polariton states can be formed. The weights of the states for the qubit and cavity field in the polariton states can be tuned by the driving field, and thus the decay rates of the polariton states can be changed. We choose a three-level system with Λ-type transitions in such a driven circuit QED system, and demonstrate how EIT and ATS can be realized in this compound system. We believe that this study will be helpful for EIT experiments using SQCs.