Focus on Cavity and Circuit Quantum Electrodynamics in Solids

We introduce the works collected in the focus issue on Cavity and circuit quantum quantum electrodynamics in solids.

Solid state-based light emitters such as semiconductor quantum dots (QDs) have been demonstrated to be versatile candidates to study the fundamentals of light–matter interaction. In contrast to optics with isolated atomic systems, in the solid-state dissipative processes are induced by the inherent coupling to the environment and are typically perceived as a major obstacle toward stable performances in experiments and applications. In this theoretical model study we show that this is not necessarily the case. In fact, in certain parameter regimes, the memory of the solid-state environment can enhance coherent quantum optical effects. In particular, we demonstrate that the non-Markovian coupling to an incoherent phonon bath can exhibit a stabilizing effect on the coherent QD cavity-quantum electrodynamics by inhibiting irregular oscillations and allowing for regular collapse and revival patterns. For self-assembled GaAs/InAs QDs at low photon numbers we predict dynamics that deviate dramatically from the well-known atomic Jaynes–Cummings model. Even if the required sample parameters are not yet available in recent experimental achievements, we believe our proposal opens the way to a systematic and deliberate design of photon quantum effects via specifically engineered solid-state environments.

We present a concept for performing direct parity measurements on three or more qubits in microwave structures with superconducting resonators coupled to Josephson-junction qubits. We write the quantum-eraser conditions that must be fulfilled for the parity measurements as requirements for the scattering phase shift of our microwave structure. We show that these conditions can be fulfilled with present-day devices. We present one particular scheme, implemented with two-dimensional cavity techniques, in which each qubit should be coupled equally to two different microwave cavities. The magnitudes of the couplings that are needed are in the range that has been achieved in current experiments. A quantum calculation indicates that the measurement is optimal if the scattering signal can be measured with near single-photon sensitivity. A comparison with an extension of a related proposal from cavity optics is presented. We present a second scheme, for which a scalable implementation of the four-qubit parities of the surface quantum error correction code can be envisioned. It uses three-dimensional cavity structures, using cavity symmetries to achieve the necessary multiple resonant modes within a single resonant structure.

We present a scheme for simulating relativistic quantum physics in circuit quantum electrodynamics. By using three classical microwave drives, we show that a superconducting qubit strongly coupled to a resonator field mode can be used to simulate the dynamics of the Dirac equation and Klein paradox in all regimes. Using the same setup we also propose the implementation of the Foldy–Wouthuysen canonical transformation, after which the time derivative of the position operator becomes a constant of the motion.

Interaction between light and matter generates optical nonlinearities, which are particularly pronounced in the quantum strong coupling regime. When a single bosonic mode couples to a single fermionic mode, a Jaynes–Cummings (JC) ladder is formed, which we realize here using cavity photons and quantum dot excitons. We measure and model the coherent anharmonic response of this strongly coupled exciton–cavity system at resonance. Injecting two photons into the cavity, we demonstrate a $\sqrt {2}$ larger polariton splitting with respect to the vacuum Rabi splitting. This is achieved using coherent nonlinear spectroscopy, specifically four-wave mixing, where the coherence between the ground state and the first (second) rung of the JC ladder can be interrogated for positive (negative) delays. With increasing excitation intensity and thus rising average number of injected photons, we observe spectral signatures of the quantum-to-classical crossover of the strong coupling regime.

We demonstrate non-perturbative coupling between a single self-assembled InGaAs quantum dot and an external fiber-mirror-based microcavity. Our results extend the previous realizations of tunable microcavities while ensuring spatial and spectral overlap between the cavity mode and the emitter by simultaneously allowing for deterministic charge control of the quantum dots. Using resonant spectroscopy, we show that the coupled quantum dot cavity system is at the onset of strong coupling, with a cooperativity parameter of C ≈ 2.0 ± 1.3. Our results constitute a milestone in the progress toward the realization of a high-efficiency solid-state spin–photon interface.

For gradiometric three-Josephson-junction flux qubits, we perform a systematic study on the tuning of the minimal transition frequency, the so-called qubit gap. By replacing one of the qubit's Josephson junctions by a dc superconducting quantum interference device (SQUID), the critical current of this SQUID and, in turn, the qubit gap can be tuned in situ by a control flux threading the SQUID loop. We present spectroscopic measurements demonstrating a well-defined controllability of the qubit gap between zero and more than 10 GHz. This is important for tuning the qubit into and out of resonance with other superconducting quantum circuits in scalable architectures, while still operating it at its symmetry point with optimal dephasing properties. The experimental data agree very well with model calculations based on the full qubit Hamiltonian. From a numerical fit, we determine the Josephson coupling and the charging energies of the qubit junctions. The derived values agree well with those measured for other junctions fabricated on the same chip. We also demonstrate the biasing of gradiometric flux qubits near the symmetry point by trapping an odd number of flux quanta in the gradiometer loop. In this way, we study the effect of the significant kinetic inductance, thereby obtaining valuable information for the qubit design.

Dirac particles, massless relativistic entities, obey linear energy dispersions and hold important implications in particle physics. The recent discovery of Dirac fermions in condensed matter systems including graphene and topological insulators has generated a great deal of interest in exploring the relativistic properties associated with Dirac physics in solid-state materials. In addition, there are stimulating research activities to engineer Dirac particles, elucidating their exotic physical properties in a controllable setting. One of the successful platforms is the ultracold atom–optical lattice system, whose dynamics can be manipulated and probed in a clean environment. A microcavity exciton–polariton–lattice system offers the advantage of forming high-orbital condensation in non-equilibrium conditions, which enables one to explore novel quantum orbital order in two dimensions. In this paper, we experimentally construct the band structures near Dirac points, the vertices of the first hexagonal Brillouin zone with exciton–polariton condensates trapped in a triangular lattice. Due to the finite spectral linewidth, the direct map of band structures at Dirac points is elusive; however, we identify the linear part above Dirac points and its associated velocity value is ∼0.9–2 × 10⁸ cm s⁻¹, consistent with the theoretical estimate 1 × 10⁸ cm s⁻¹ with a 2 μm lattice constant. We envision that the exciton–polariton condensates in lattices would be a promising solid-state platform, where the system order parameter can be accessed in both real and momentum spaces.

A solid-state single-photon source emitting indistinguishable photons on-demand is an essential component of linear optics quantum computing schemes. However, the emitter will inevitably interact with the solid-state environment causing decoherence and loss of indistinguishability. In this paper, we present a comprehensive theoretical treatment of the influence of phonon scattering on the coherence properties of single photons emitted from semiconductor quantum dots. We model decoherence using a full microscopic theory and compare with standard Markovian approximations employing Lindblad-type relaxation terms. Significant differences between the two approaches are found.

The impact of non-resonant background emitters in semiconductor quantum-dot microcavity lasers is addressed within theoretical investigations based on the solution of the von Neumann equation. Off-resonant coupling between emitter resonances and the cavity mode is enabled via phonons, which are included in the von Neumann dynamics by an effective Lindblad contribution. The results show enhanced coherent emission from non-resonantly coupled quantum dots, while the frequently used phenomenological cavity feeding mechanism only enhances the thermal component of the emission.

We study several dynamical properties of a recently proposed implementation of the quantum transverse-field Ising chain in the framework of circuit quantum electrodynamics (QED). Particular emphasis is placed on the effects of disorder on the nonequilibrium behavior of the system. We show that small amounts of fabrication-induced disorder in the system parameters do not jeopardize the observation of previously predicted phenomena. Based on a numerical extraction of the mean free path of a wave packet in the system, we also provide a simple quantitative estimate for certain disorder effects on the nonequilibrium dynamics of the circuit QED quantum simulator. We discuss the transition from weak to strong disorder, characterized by the onset of Anderson localization of the system's wave functions, and the qualitatively different dynamics it leads to.

In this work, we theoretically analyze a circuit quantum electrodynamics design where propagating quantum microwaves interact with a single artificial atom, a single Cooper-pair box. In particular, we derive a master equation in the so-called transmon regime, including coherent drives. Inspired by recent experiments, we then apply the master equation to describe the dynamics in both a two-level and a three-level approximation of the atom. In the two-level case, we also discuss how to measure photon antibunching in the reflected field and how it is affected by finite temperature and finite detection bandwidth.

Single-electron tunneling processes through a double quantum dot can induce a lasing state in an electromagnetic resonator which is coupled coherently to the dot system. Here we study the noise properties of the transport current in the lasing regime, i.e. both the zero-frequency shot noise and the noise spectrum. The former shows a remarkable super-Poissonian behavior when the system approaches the lasing transition, but a sub-Poissonian behavior deep in the lasing state. The noise spectrum contains information about the coherent dynamics of the coupled dot–resonator system. It shows pronounced structures at frequencies matching that of the resonator due to the excitation of photons. For strong interdot Coulomb interaction, we observe asymmetries in the auto-correlation noise spectra of the left and right junctions, which we trace back to asymmetries in the incoherent tunneling channels.

We propose microwave-controlled rotations for qubits realized as Majorana bound states. To this end, we study an inhomogeneous Kitaev chain in a microwave cavity. The chain consists of two topologically nontrivial regions separated by a topologically trivial, gapped region. The Majorana bound states at the interfaces between the left (right) regions and the central region are coupled, and their energies are split by virtual cotunneling processes. The amplitude for these cotunneling processes decreases exponentially with the number of sites of the gapped region, and the decay length diverges as the gap of the topologically trivial region closes. We demonstrate that microwave radiation can exponentially enhance the coupling between the Majorana bound states, both for classical and quantized electric fields. By solving the appropriate Liouville equation numerically, we show that microwaves can drive Rabi oscillations in the Majorana sector. Our model emerges as an effective description of a topological semiconductor nanowire in a microwave cavity. Thus, our proposal provides an experimentally feasible way to obtain full single-qubit control necessary for universal quantum computation with Majorana qubits.

We report on electrically pumped quantum dot–microlasers in the presence of polarized self-feedback. The high-β microlasers show two orthogonal, linearly polarized emission modes which are coupled via the common gain medium. This coupling is explained in terms of gain competition between the two lasing modes and leads to distinct differences in their input–output characteristics. By applying polarized self-feedback via an external mirror, we are able to control the laser characteristics of the emission modes in terms of the output power, the coherence time and the photon statistics. We find that linearly polarized self-feedback stabilizes the lasing of a given mode, while cross-polarized feedback between the two modes reduces strongly the intensity of the other emission mode showing particular high-intensity fluctuations and even super-thermal values of the photon autocorrelation function g⁽²⁾(τ) at zero delay. Measurements of g⁽²⁾(τ) under external feedback also allow us to detect revival peaks associated with the round trip time of the external cavity. Analyzing the damping and shape of the g⁽²⁾(τ) revival peaks by a phenomenological model provides us insight into the underlying physics such as the effective exciton lifetime and gain characteristics of the quantum dots in the active region of these microlasers.

A quantum dot can be used as a source of one- and two-photon states and of polarization entangled photon pairs. The emission of such states is investigated here from the point of view of frequency-resolved two-photon correlations. These follow from a spectral filtering of the dot emission, which can be achieved either by using a cavity or by placing a number of interference filters before the detectors. A combination of these various options is used to iteratively refine the emission in a 'distillation' process and arrive at highly correlated states with a high purity. The so-called 'leapfrog processes', where the system undergoes a direct transition from the biexciton state to the ground state by direct emission of two photons, are shown to be central to the quantum features of such sources. Optimum configurations are singled out in a global theoretical picture that unifies the various regimes of operation.

A theory of nonlinear emission of localized excitons coupled to the optical mode of the microcavity is presented. Numerical results are compared with analytical ones. The effects of exciton–exciton interaction within the quantum dots and with the reservoir formed by non-resonant pumping are considered. It is demonstrated that the nonlinearity due to the interaction strongly affects the shape of the emission spectra. The collective superradiant mode of the excitons is shown to be stable against the nonlinear effects.

We study a driven-dissipative array of coupled nonlinear optical resonators by numerically solving the von Neumann equation for the density matrix. We demonstrate that quantum correlated states of many photons can also be generated in the limit where the nonlinearity is much smaller than the losses, contrary to common expectations. Quantum correlations in this case arise from the interference between different pathways that the system can follow in the Hilbert space to reach its steady state under the effect of coherent driving fields. We characterize, in particular, two systems: a linear chain of three coupled cavities and an array of eight coupled cavities. We demonstrate the existence of a parameter range where the system emits photons with continuous-variable bipartite and quadripartite entanglement, in the case of the first and the second system, respectively. This entanglement is shown to survive realistic rates of pure dephasing and opens up a new perspective for the realization of quantum simulators or entangled photon sources without the challenging requirement of strong optical nonlinearities.

We study the photon blockade phenomenon in a nanocavity containing a single four-level quantum emitter. By numerically simulating the second-order autocorrelation function of the intra-cavity field with realistic parameters achievable in a state-of-the-art photonic-crystal nanocavity, we show that significant photon blockade effects appear even outside the strong coupling regime. We introduce an intuitive picture of the photon blockade that explains the performance difference between the two-level and the four-level emitter schemes reported in previous works, as well as why—in contrast to a cavity containing a two-level atom—signatures of photon blockade appear and should be experimentally observable outside the strong coupling regime when a four-level emitter is used. Finally, we show that thanks to the emitter–cavity coupling achievable in a nanocavity, photon blockade can be realized despite the large frequency difference between the relevant optical transitions in realistic four-level emitters, which has so far prevented the experimental realization of this photon blockade scheme.

We present detuning-dependent spectral and decay-rate measurements to study the difference between the spectral and dynamical properties of single quantum dots embedded in micropillar and photonic crystal cavities. For the micropillar cavity, the dynamics is well described by the dissipative Jaynes–Cummings model, whereas systematic deviations are observed for the emission spectra. The discrepancy for the spectra is attributed to the coupling of other exciton lines to the cavity and interference of different propagation paths toward the detector of the fields emitted by the quantum dot. In contrast, quantitative information about the system can readily be extracted from the dynamical measurements. In the case of photonic crystal cavities, we observe an anti-crossing in the spectra when detuning a single quantum dot through resonance, which is the spectral signature of a strong coupling. However, time-resolved measurements reveal that the actual coupling strength is significantly smaller than anticipated from the spectral measurements and that the quantum dot is rather weakly coupled to the cavity. We suggest that the observed Rabi splitting is due to cavity feeding by other quantum dots and/or multi-exciton complexes giving rise to collective emission effects.

We propose the use of weakly nonlinear passive materials for prospective applications in integrated quantum photonics. It is shown that strong enhancement of native optical nonlinearities by electromagnetic field confinement in photonic crystal resonators can lead to single-photon generation only exploiting the quantum interference of two coupled modes and the effect of photon blockade under resonant coherent driving. For realistic system parameters in state of the art microcavities, the efficiency of such a single-photon source is theoretically characterized by means of the second-order correlation function at zero-time delay as the main figure of merit, where major sources of loss and decoherence are taken into account within a standard master equation treatment. These results could stimulate the realization of integrated quantum photonic devices based on non-resonant material media, fully integrable with current semiconductor technology and matching the relevant telecom band operational wavelengths, as an alternative to single-photon nonlinear devices based on cavity quantum electrodynamics with artificial atoms or single atomic-like emitters.

We address recent advances in microwave quantum optics with artificial atoms in one-dimensional (1D) open space. This field relies on the fact that the coupling between a superconducting artificial atom and propagating microwave photons in a 1D open transmission line can be made strong enough to observe quantum coherent effects, without using any cavity to confine the microwave photons. We investigate the scattering properties in such a system with resonant coherent microwaves. We observe the strong nonlinearity of the artificial atom and under strong driving we observe the Mollow triplet. By applying two resonant tones, we also observe the Autler–Townes splitting. Exploiting these effects, we demonstrate two quantum devices at the single-photon level in the microwave regime: the single-photon router and the photon-number filter. These devices provide important steps toward the realization of an on-chip quantum network.

Nitrogen–vacancy centers in diamond have outstanding quantum optical properties that enable applications in information processing and sensing. As with most solid-state systems for quantum photonic applications, the great promise lies in the capability to embed them in an on-chip optical network. Here we present basic integrated devices composed of diamond micro-ring resonators coupled to waveguides that are terminated with grating out-couplers. Strong enhancement is observed for the zero-phonon line of nitrogen–vacancy centers coupled to the ring resonance. The zero-phonon line is efficiently coupled from the ring into the waveguide and then scattered out of plane by the grating out-couplers.