We measure the dispersive energy-level shift of an LC resonator magnetically coupled to a superconducting qubit, which clearly shows that our system operates in the ultrastrong coupling regime. The large mutual kinetic inductance provides a coupling energy of ≈ 0.82 GHz, requiring the addition of counter-rotating-wave terms in the description of the Jaynes-Cummings model. We find a 50 MHz Bloch-Siegert shift when the qubit is in its symmetry point, fully consistent with our analytical model.

superconducting circuits, semiconductor quantum wells, and other hybrid quantum systems. Finally, anticipated applications are highlighted utilizing USC and DSC regimes, including novel quantum optical phenomena, quantum simulation, and quantum computation. CONTENTSCaltech Caltech Caltech LKB Paris LKB Paris LKB Paris LKB Paris LKB Paris Harvard Harvard Würzburg U Tokyo ETH U Tokyo Stanford Princeton (MW) ETH (MW) Yale Delft, NTT IMS WMI, Delft WMI NICT ETH ETH UPD ISSP UPD UPD U Reg IMS U Tokyo CNRS CNRS NCU CNR ICL Caltech Caltech Caltech LKB Paris LKB Paris LKB Paris Harvard Harvard Würzburg LKB Paris LKB Paris NICT Yale Delft, NTT WMI, Delft WMI UPD IMS UPD UPD IMS ETH ETH

The study of light-matter interaction has led to many fundamental discoveries as well as numerous important technologies. Over the last decades, great strides have been made in increasing the strength of this interaction at the single-photon level, leading to a continual exploration of new physics and applications. Recently, a major achievement has been the demonstration of the so-called strong coupling regime [1, 2], a key advancement enabling great progress in quantum information science. Here, we demonstrate light-matter interaction over an order of magnitude stronger than previously reported, reaching the nonperturbative regime of ultrastrong coupling (USC). We achieve this using a superconducting artificial atom tunably coupled to the electromagnetic continuum of a one-dimensional waveguide. For the largest coupling, the spontaneous emission rate of the atom exceeds its transition frequency. In this USC regime, the description of atom and light as distinct entities breaks down, and a new description in terms of hybrid states is required [4, 8]. Our results open the door to a wealth of new physics and applications. Beyond light-matter interaction itself, the tunability of our system makes it a promising tool to study a number of important physical systems such as the well-known spin-boson [9] and Kondo models [12]. * These authors contributed equally to this work.

We propose different designs of switchable coupling between a superconducting flux qubit and a microwave transmission line. They are based on two or more loops of Josephson junctions which are directly connected to a closed (cavity) or open transmission line. In both cases the circuit induces a coupling that can be modulated in strength, reaching the so-called ultrastrong coupling regime in which the coupling is comparable to the qubit and photon frequencies. Furthermore, we suggest a wide set of applications for the introduced architectures. DOI: 10.1103/PhysRevLett.105.023601 PACS numbers: 42.50.Àp, 03.67.Lx, 85.25.Àj Superconducting quantum circuits [1] possess ingredients for quantum information processing and for developing on-chip microwave quantum optics [2]. After the first manipulations of few-level superconducting systems (qubits) [3][4][5], the real boost came with the achievement of the strong coupling regime between qubits and confined microwave photons [6][7][8]. The initial qubit-cavity couplings of 10-100 MHz exceeded by orders of magnitude the rate at which photons leak out of the resonator, but the use of the transmon qubit [9] improved those numbers by a factor of 2-3 reaching a strength that is comparable only to the state of the art in microwave quantum optics [10,11]. More recently, proof-of-principle theoretical and experimental studies have paved the way to the ultrastrong coupling regime [12][13][14], where the coupling approaches the qubit transition frequency and the Jaynes-Cummings model of cavity QED [10,14] breaks down [15,16], and a door opens to the rather unexplored physics beyond the rotating-wave approximation [17,18].The strong coupling regime in circuit QED has made possible an incredible variety of experiments, such as dispersive readouts of qubits [19], resolving the photon numbers in cavity [20], multiphoton excitations of the Jaynes-Cummings model [21], preparing nonclassical states of a resonator [22], full quantum tomography of the microwave radiation field [23], or the TavisCummings model [24], etc. However, all those experiments have something in common: The microwave field is confined inside a resonator. In other words, the transmission line spectrum is discrete and the coupling between qubits and photons could be switched on and off by tuning the qubit [25] or cavity frequency [26]. While the switchability of the coupling has been proposed for open lines [27,28], this has not been achieved in the ultrastrong coupling regimes.In this work, we will introduce a novel circuit QED design where the qubit is ultrastrongly coupled to a transmission line, open or not, with a coupling that can be tuned in strength and kind by applying an external flux bias. Our proposal uses the type of designs shown in Fig. 1, where the qubit is built in direct contact with the transmission line. It has been shown theoretically [14], and demonstrated experimentally [13], that the system admits an effective description based on a two-level system-the current in the loop-ultrastrongly coup...

A flux qubit biased at its symmetry point shows a minimum in the energy splitting (the gap), providing protection against flux noise. We have fabricated a qubit of which the gap can be tuned fast and have coupled this qubit strongly to an LC oscillator. We show full spectroscopy of the qubit-oscillator system and generate vacuum Rabi oscillations. When the gap is made equal to the oscillator frequency νosc we find the largest vacuum Rabi splitting of ∼ 0.1νosc. Here being at resonance coincides with the optimal coherence of the symmetry point.Superconducting qubits coupled to quantum oscillators have demonstrated a remarkable richness of physical phenomena in the last few years. After the first reports of coherent state transfer and strong coupling [1, 2], we have witnessed a rapid development of the field called circuit quantum electrodynamics (CQED) using high quality superconducting oscillators in realizing quantum gates [3], algorithms [4] as well as non-classical states of light and matter in artificially fabricated structures [5,6]. Among the different implementations the transmon [1,[3][4][5] and the phase qubit [6] dominated this development. With flux qubits the avoided crossing between qubit and oscillator level was observed [7,8] and the coherent single-photon exchange between qubit and oscillator was demonstrated [8]. However the, coherence of the flux qubit is optimally preserved only in the symmetry point for flux bias, where the energy splitting is minimal. This minimal splitting (h∆) is called the gap and depends (exponentially) on the properties of the Josephson junctions. Therefore, the gap is hard to control in fabrication and it is impossible to make it coincide with a fixed oscillator frequency. We now have developed a flux qubit of which the gap ∆ can be tuned over a broad range on sub-ns time scales [9]. With the use of this control we demonstrate strong coupling of a flux qubit with good coherence to a lumped-element LC oscillator, showing fast and longlived vacuum Rabi oscillations.Parameters of the superconducting qubits can be to a large extent chosen in the design phase. For strong coupling, where the interaction strength g exceeds the cavity and qubit loss rates, the rotating-wave approximation (RWA) can be applied and the system can be described by a Jaynes-Cummings type Hamiltonian. If g approaches the qubit or oscillator frequencies the RWA no longer holds, leading into the ultra-strong coupling regime [10,11]. For a flux qubit the ratio g/ν osc can be an order of magnitude larger than for charge and phase qubits [12], while these latter devices have a coupling that can be several orders of magnitude larger than the atomlight interaction energy [1]. For good coherence, operating the qubit at its spectral symmetry point is required. Therefore, experimentally combining galvanic coupling of oscillator and flux qubit with this symmetry point operation provides a major step forward in the development of CQED systems. For the flux qubit at the symmetry point the anharmonicity (distance ...

We propose a superconducting circuit to implement a two-photon quantum Rabi model in a solid-state device, where a qubit and a resonator are coupled by a two-photon interaction. We analyze the input-output relations for this circuit in the strong coupling regime and find that fundamental quantum optical phenomena are qualitatively modified. For instance, two-photon interactions are shown to yield single-or two-photon blockade when a pumping field is either applied to the cavity mode or to the qubit, respectively. In addition, we derive an effective Hamiltonian for perturbative ultrastrong two-photon couplings in the dispersive regime, where twophoton interactions introduce a qubit-state-dependent Kerr term. Finally, we analyze the spectral collapse of the multi-qubit two-photon quantum Rabi model and find a scaling of the critical coupling with the number of qubits. Using realistic parameters with the circuit proposed, three qubits are sufficient to reach the collapse point.

In recent years, the notion of ‘Quantum Materials’ has emerged as a powerful unifying concept across diverse fields of science and engineering, from condensed-matter and coldatom physics to materials science and quantum computing. Beyond traditional quantum materials such as unconventional superconductors, heavy fermions, and multiferroics, the field has significantly expanded to encompass topological quantum matter, two-dimensional materials and their van der Waals heterostructures, Moiré materials, Floquet time crystals, as well as materials and devices for quantum computation with Majorana fermions. In this Roadmap collection we aim to capture a snapshot of the most recent developments in the field, and to identify outstanding challenges and emerging opportunities. The format of the Roadmap, whereby experts in each discipline share their viewpoint and articulate their vision for quantum materials, reflects the dynamic and multifaceted nature of this research area, and is meant to encourage exchanges and discussions across traditional disciplinary boundaries. It is our hope that this collective vision will contribute to sparking new fascinating questions and activities at the intersection of materials science, condensed matter physics, device engineering, and quantum information, and to shaping a clearer landscape of quantum materials science as a new frontier of interdisciplinary scientific inquiry. We stress that this article is not meant to be a fully comprehensive review but rather an up-to-date snapshot of different areas of research on quantum materials with a minimal number of references focusing on the latest developments.

Spontaneous parametric downconversion (SPDC) has been a key enabling technology in exploring quantum phenomena and their applications for decades. For instance, traditional SPDC, which splits a high energy pump photon into two lower energy photons, is a common way to produce entangled photon pairs. Since the early realizations of SPDC, researchers have thought to generalize it to higher order, e.g., to produce entangled photon triplets. However, directly generating photon triplets through a single SPDC process has remained elusive. Here, using a flux-pumped superconducting parametric cavity, we demonstrate direct three-photon SPDC, with photon triplets generated in a single cavity mode or split between multiple modes. With strong pumping, the states can be quite bright, with flux densities exceeding 60 photon/s/Hz. The observed states are strongly non-Gaussian, which has important implications for potential applications. In the single-mode case, we observe a triangular star-shaped distribution of quadrature voltages, indicative of the longpredicted "star state". The observed star state shows strong third-order correlations, as expected for a state generated by a cubic Hamiltonian. By pumping at the sum frequency of multiple modes, we observe strong three-body correlations between multiple modes, strikingly, in the absence of secondorder correlations. We further analyze the third-order correlations under mode transformations by the symplectic symmetry group, showing that the observed transformation properties serve to "fingerprint" the specific cubic Hamiltonian that generates them. The observed non-Gaussian, thirdorder correlations represent an important step forward in quantum optics and may have a strong impact on quantum communication with microwave fields as well as continuous-variable quantum computation.